Polarized LightT H I R D E D I T I O N

Polarized LightT H I R D E D I T I O N

Dennis H. Goldstein

CRC Press is an imprint of theTaylor & Francis Group, an informa business

Boca Raton London New York

CRC PressTaylor & Francis Group6000 Broken Sound Parkway NW, Suite 300Boca Raton, FL 33487-2742

© 2011 by Taylor and Francis Group, LLCCRC Press is an imprint of Taylor & Francis Group, an Informa business

No claim to original U.S. Government works

Printed in the United States of America on acid-free paper10 9 8 7 6 5 4 3 2 1

International Standard Book Number-13: 978-1-4398-3041-3 (Ebook-PDF)

This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material repro-duced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint.

Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers.

For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copy-right.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identifica-tion and explanation without intent to infringe.

Visit the Taylor & Francis Web site athttp://www.taylorandfrancis.com

and the CRC Press Web site athttp://www.crcpress.com

v

ContentsPreface to the Third Edition ............................................................................................................xvPolarized Light: A History ............................................................................................................xvii

IPart Introduction to Polarized Light

1Chapter Introduction ..................................................................................................................3

Reference ......................................................................................................................7

2Chapter Polarization in the Natural Environment .....................................................................9

2.1 Sources of Polarized Light ................................................................................92.2 Polarized Light in the Atmosphere ....................................................................9

2.2.1 The Sky: Rayleigh Scattering and Polarization ...................................92.2.2 Rainbows ............................................................................................ 102.2.3 Clouds, Halos, and Glories ................................................................. 14

2.2.3.1 Clouds ................................................................................. 142.2.3.2 Haloes ................................................................................. 142.2.3.3 Glories ................................................................................. 15

2.2.4 The Sun .............................................................................................. 152.3 Production of Polarized Light by Animals ..................................................... 16

2.3.1 Scarabaeidae (Scarab Beetles) ............................................................ 162.3.2 Squid and Cuttlefish ...........................................................................222.3.3 Mantis Shrimp ....................................................................................23

2.4 Polarization Vision in the Animal Kingdom ...................................................24References ..................................................................................................................28

3Chapter Wave Equation in Classical Optics ............................................................................ 31

3.1 Introduction ..................................................................................................... 313.2 The Wave Equation ......................................................................................... 31

3.2.1 Plane-Wave Solution ........................................................................... 333.2.2 Spherical Waves .................................................................................343.2.3 Fourier Transform Method ................................................................. 353.2.4 Mathematical Representation of the Harmonic Oscillator

Equation .............................................................................................363.2.5 Note on the Equation of a Plane ......................................................... 38

3.3 Young’s Interference Experiment .................................................................... 393.4 Reflection and Transmission of a Wave at an Interface .................................. 43

4Chapter The Polarization Ellipse ............................................................................................. 49

4.1 Introduction ..................................................................................................... 494.2 The Instantaneous Optical Field and the Polarization Ellipse ........................50

vi Contents

4.3 Specialized (Degenerate) Forms of the Polarization Ellipse........................... 524.4 Elliptical Parameters of the Polarization Ellipse ............................................54References .................................................................................................................. 58

5Chapter Stokes Polarization Parameters .................................................................................. 59

5.1 Introduction ..................................................................................................... 595.2 Derivation of Stokes Polarization Parameters .................................................60

5.2.1 Linear Horizontally Polarized Light (LHP) ....................................... 635.2.2 Linear Vertically Polarized Light (LVP) ............................................645.2.3 Linear +45° Polarized Light (L +45) ..................................................645.2.4 Linear −45° Polarized Light (L −45) ..................................................645.2.5 Right Circularly Polarized Light (RCP) .............................................645.2.6 Left Circularly Polarized Light (LCP) ...............................................65

5.3 Stokes Vector ...................................................................................................655.3.1 Linear Horizontally Polarized Light (LHP) .......................................665.3.2 Linear Vertically Polarized Light (LVP) ............................................665.3.3 Linear +45° Polarized Light (L +45) ..................................................665.3.4 Linear −45° Polarized Light (L −45) ..................................................665.3.5 Right Circularly Polarized Light (RCP) .............................................665.3.6 Left Circularly Polarized Light (LCP) ............................................... 67

5.4 Classical Measurement of Stokes Polarization Parameters ............................. 715.5 Stokes Parameters for Unpolarized and Partially Polarized Light ................. 755.6 Additional Properties of Stokes Polarization Parameters ...............................775.7 Stokes Parameters and the Coherency Matrix ................................................875.8 Stokes Parameters and the Pauli Matrices ......................................................90References .................................................................................................................. 91

6Chapter Mueller Matrices for Polarizing Components ............................................................93

6.1 Introduction .....................................................................................................936.2 Mueller Matrix of a Linear Diattenuator (Polarizer).......................................956.3 Mueller Matrix of a Linear Retarder ............................................................. 1006.4 Mueller Matrix of a Rotator .......................................................................... 1036.5 Mueller Matrices for Rotated Polarizing Components .................................. 1056.6 Generation of Elliptically Polarized Light .................................................... 1116.7 Mueller Matrix of a Depolarizer ................................................................... 114References ................................................................................................................ 115

7Chapter Fresnel Equations: Derivation and Mueller Matrix Formulation ............................. 117

7.1 Introduction ................................................................................................... 1177.2 Fresnel Equations for Reflection and Transmission ...................................... 117

7.2.1 Definitions ........................................................................................ 1177.2.2 Boundary Conditions ....................................................................... 1187.2.3 Derivation of Fresnel Equations ....................................................... 119

7.3 Mueller Matrices for Reflection and Transmission at an Air–Dielectric Interface ......................................................................................................... 127

7.4 Special Forms for Mueller Matrices for Reflection and Transmission.......... 1357.4.1 Normal Incidence ............................................................................. 1367.4.2 Brewster Angle ................................................................................. 137

Contents vii

7.4.3 45° Incidence ................................................................................... 1387.4.4 Total Internal Reflection ................................................................... 141

7.5 Emission Polarization ................................................................................... 145References ................................................................................................................ 147

8Chapter Mathematics of the Mueller Matrix ......................................................................... 149

8.1 Introduction ................................................................................................... 1498.2 Constraints on the Mueller Matrix ................................................................ 1508.3 Eigenvector and Eigenvalue Analysis............................................................ 1518.4 Example Eigenvector Analysis ...................................................................... 155

8.4.1 Eigenvector Analysis ........................................................................ 1568.4.2 Noise ................................................................................................. 157

8.5 The Lu–Chipman Decomposition ................................................................. 1608.6 Decomposition Order .................................................................................... 1708.7 Decomposition of Depolarizing Matrices with Depolarization

Symmetry ...................................................................................................... 1718.8 Decomposition Using Matrix Roots .............................................................. 1748.9 Summary ....................................................................................................... 174References ................................................................................................................ 174

9Chapter Mueller Matrices for Dielectric Plates ..................................................................... 177

9.1 Introduction ................................................................................................... 1779.2 The Diagonal Mueller Matrix and the Abcd Polarization Matrix .............. 1779.3 Mueller Matrices for Single and Multiple Dielectric Plates .......................... 186References ................................................................................................................ 199

10Chapter The Jones Matrix Formalism ................................................................................... 201

10.1 Introduction ................................................................................................... 20110.2 The Jones Vector ...........................................................................................20210.3 Jones Matrices for the Polarizer, Retarder, and Rotator ................................20610.4 Applications of the Jones Vector and Jones Matrices ................................... 21110.5 Jones Matrices for Homogeneous Elliptical Polarizers and Retarders ......... 222References ................................................................................................................230

11Chapter The Poincaré Sphere ................................................................................................ 233

11.1 Introduction ................................................................................................... 23311.2 Theory of the Poincaré Sphere ......................................................................234

11.2.1 Note on the Derivation of Law of Cosines and Law of Sines in Spherical Trigonometry ....................................................................244

11.3 Projection of the Complex Plane onto a Sphere ............................................25011.4 Applications of the Poincaré Sphere ............................................................. 258References ................................................................................................................266

12Chapter Fresnel–Arago Interference Laws ............................................................................ 267

12.1 Introduction ................................................................................................... 26712.2 Stokes Vector and Unpolarized Light............................................................ 267

viii Contents

12.3 Young’s Double Slit Experiment ...................................................................26812.4 Double Slit with Parallel Polarizers: The First Law ...................................... 27112.5 Double Slit with Perpendicular Polarizers: The Second Law ....................... 27312.6 Double Slit and the Third Law ...................................................................... 27412.7 Double Slit and the Fourth Law .................................................................... 276References ................................................................................................................ 278

IPart I Polarimetry

13Chapter Introduction .............................................................................................................. 281

14Chapter Methods of Measuring Stokes Polarization Parameters .......................................... 283

14.1 Introduction ................................................................................................... 28314.2 Classical Measurement Method: Quarter-Wave Retarder and

Polarizer Method ........................................................................................... 28314.3 Measurement of Stokes Parameters Using a Circular Polarizer ...................28714.4 Null-Intensity Method ................................................................................... 29114.5 Fourier Analysis Using a Rotating Quarter-Wave Retarder ..........................29414.6 Method of Kent and Lawson .........................................................................29714.7 Simple Tests to Determine the State of Polarization of an Optical Beam ....304References ................................................................................................................ 310

15Chapter Measurement of the Characteristics of Polarizing Elements ................................... 311

15.1 Introduction ................................................................................................... 31115.2 Measurement of Attenuation Coefficients of a Polarizer (Diattenuator) ....... 311

15.2.1 First Measurement Method .............................................................. 31315.2.2 Second Measurement Method .......................................................... 31615.2.3 Third Measurement Method............................................................. 317

15.3 Measurement of the Phase Shift of a Retarder .............................................. 31815.3.1 First Method ..................................................................................... 31815.3.2 Second Method ................................................................................. 32015.3.3 Third Method ................................................................................... 323

15.4 Measurement of Rotation Angle of a Rotator................................................ 32415.4.1 First Method ..................................................................................... 32415.4.2 Second Method ................................................................................. 326

16Chapter Stokes Polarimetry ................................................................................................... 327

16.1 Introduction ................................................................................................... 32716.2 Rotating Element Polarimetry ....................................................................... 327

16.2.1 Rotating Analyzer Polarimeter ........................................................ 32716.2.2 Rotating Analyzer and Fixed Analyzer Polarimeter ........................ 32916.2.3 Rotating Retarder and Fixed Analyzer Polarimeter ......................... 32916.2.4 Rotating Retarder and Analyzer Polarimeter ................................... 32916.2.5 Rotating Retarder and Analyzer Plus Fixed Analyzer

Polarimeter ....................................................................................... 33116.3 Oscillating Element Polarimetry ................................................................... 331

16.3.1 Oscillating Analyzer Polarimeter ..................................................... 332

Contents ix

16.3.2 Oscillating Retarder with Fixed Analyzer Polarimeter ................... 33416.3.3 Oscillating Retarder and Analyzer Polarimeter ............................... 335

16.4 Phase Modulation Polarimetry ..................................................................... 33716.4.1 Phase Modulator and Fixed Analyzer Polarimeter .......................... 33716.4.2 Dual Phase Modulator and Fixed Analyzer Polarimeter ................. 338

16.5 Techniques in Simultaneous Measurement of Stokes Vector Elements ............................................................................................. 33916.5.1 Division of Wavefront Polarimetry .................................................. 33916.5.2 Division of Amplitude Polarimetry ..................................................340

16.5.2.1 Four-Channel Polarimeter Using Polarizing Beam Splitters ...................................................................340

16.5.2.2 Azzam’s Four-Detector Photopolarimeter ........................34016.5.2.3 Division of Amplitude Polarimeters

Using Gratings ..................................................................34616.5.2.4 Division of Amplitude Polarimeter Using a

Parallel Slab ...................................................................... 34716.6 Optimization of Polarimeters ........................................................................348References ................................................................................................................ 351

17Chapter Mueller Matrix Polarimetry ..................................................................................... 353

17.1 Introduction ................................................................................................... 35317.1.1 Polarimeter Types ............................................................................. 35317.1.2 Rotating Element Polarimeters ........................................................ 35517.1.3 Phase-Modulating Polarimeters ....................................................... 356

17.2 Dual Rotating Retarder Polarimetry ............................................................. 35717.2.1 Polarimeter Description ................................................................... 35717.2.2 Mathematical Development: Obtaining the Mueller Matrix............ 35717.2.3 Modulated Intensity Patterns ........................................................... 36117.2.4 Error Compensation ......................................................................... 36217.2.5 Optical Properties from the Mueller Matrix .................................... 36717.2.6 Measurements ................................................................................... 36917.2.7 Spectropolarimetry........................................................................... 36917.2.8 Measurement Matrix Method........................................................... 370

17.3 Other Mueller Matrix Polarimetry Methods ................................................. 37117.3.1 Modulator-Based Mueller Matrix Polarimeter ................................. 37217.3.2 Mueller Matrix Scatterometer .......................................................... 37317.3.3 Four-Detector Photopolarimeter ...................................................... 374

References ................................................................................................................ 375

18Chapter Techniques in Imaging Polarimetry ......................................................................... 377

18.1 Introduction ................................................................................................... 37718.2 Historical Perspective .................................................................................... 37818.3 Measurement Considerations ........................................................................ 379

18.3.1 Spectral Considerations .................................................................... 37918.3.2 One-Dimensional Polarimeters ........................................................38018.3.3 Two-Dimensional Polarimeters ........................................................38018.3.4 Three-Dimensional Polarimeters ..................................................... 38118.3.5 Full Stokes Polarimeters .................................................................. 381

x Contents

18.3.6 Active Imaging Polarimeters ............................................................ 38118.3.6.1 Mueller Matrix and Other Active

Imaging Systems ............................................................... 38218.3.6.2 Lidar Systems ................................................................... 382

18.3.7 Spectropolarimetric Imagers ............................................................ 38318.4 Measurement Strategies and Data Reduction Techniques ............................384

18.4.1 Data Reduction Matrix Techniques ..................................................38418.4.2 Fourier Modulation Techniques ....................................................... 38518.4.3 Channeled Spectropolarimeters ....................................................... 387

18.5 General Measurement Strategies: Imaging Architecture for Integrated Polarimeters ................................................................................. 38818.5.1 Division of Time (DoTP) Polarimeter .............................................. 38818.5.2 Division of Amplitude Polarimeters (DoAmP)................................ 38918.5.3 Division of Aperture Polarimeter (DoAP) .......................................39018.5.4 Division of Focal Plane (DoFP) Array Polarimeters ....................... 391

18.6 System Considerations ................................................................................... 39218.6.1 Alignment and Calibration of Imaging Polarimeters ...................... 39218.6.2 Experimental Determination of Data Reduction Matrix ................. 39218.6.3 Calibration of Fourier-Based Rotating Retarder Systems ................ 39318.6.4 Polarization Aberrations and Image Misalignment ......................... 39318.6.5 Optimization..................................................................................... 393

18.7 Summary ....................................................................................................... 395References ................................................................................................................ 396

19Chapter Channeled Polarimetry for Snapshot Measurements ............................................... 401

19.1 Introduction ................................................................................................... 40119.2 Channeled Polarimetry ..................................................................................402

19.2.1 Introduction to Channeled Spectropolarimetry ...............................40219.2.2 Introduction to Channeled Imaging Polarimetry .............................40619.2.3 Calibration Algorithms ....................................................................408

19.2.3.1 CS Calibration ..................................................................40819.2.3.2 CIP Calibration ................................................................. 411

19.3 Channeled Spectropolarimetry ..................................................................... 41319.3.1 CS with a Dispersive Spectrometer .................................................. 41319.3.2 Fourier Transform CS ...................................................................... 415

19.4 Channeled Imaging Polarimetry ................................................................... 41619.4.1 Prismatic CIP ................................................................................... 41619.4.2 Savart Plate CIP ............................................................................... 42019.4.3 Dispersion Compensation in CIP ..................................................... 423

19.4.3.1 DC in Prismatic CIP ......................................................... 42319.4.3.2 DC in Savart Plate CIP .....................................................424

19.5 Sources of Error in Channeled Polarimetry .................................................. 42619.5.1 Reconstruction Artifacts (CS and CIP) ............................................ 42619.5.2 Temperature Variations (CS and CIP) .............................................. 42719.5.3 Dichroism (CS and CIP) ................................................................... 42819.5.4 Dispersion (CS) ................................................................................ 429

19.6 Mueller Matrix Channeled Spectropolarimeters ........................................... 42919.7 Channeled Ellipsometers ............................................................................... 431References ................................................................................................................ 432

Contents xi

IIPart I applications

20Chapter Introduction .............................................................................................................. 437

21Chapter Crystal Optics ........................................................................................................... 439

21.1 Introduction ................................................................................................... 43921.2 Review of Concepts from Electromagnetism ................................................44021.3 Crystalline Materials and Their Properties ...................................................44221.4 Crystals .......................................................................................................... 443

21.4.1 Index Ellipsoid .................................................................................44821.4.2 Natural Birefringence ....................................................................... 45121.4.3 Wave Surface .................................................................................... 45121.4.4 Wavevector Surface .......................................................................... 454

21.5 Application of Electric Fields: Induced Birefringence and Polarization Modulation ................................................................................ 455

21.6 Magneto-Optics ............................................................................................. 46121.7 Liquid Crystals ..............................................................................................46321.8 Modulation of Light .......................................................................................46521.9 Photoelastic Modulators ................................................................................46621.10 Concluding Remarks ..................................................................................... 467References ................................................................................................................468

22Chapter Optics of Metals ....................................................................................................... 471

22.1 Introduction ................................................................................................... 47122.2 Maxwell’s Equations for Absorbing Media ................................................... 47222.3 Principal Angle of Incidence Measurement of Refractive Index and

Absorption Index of Optically Absorbing Materials ..................................... 48122.4 Measurement of Refractive Index and Absorption Index at an Incident

Angle of 45° ................................................................................................... 489References ................................................................................................................ 501

23Chapter Polarization Optical Elements .................................................................................. 503

23.1 Introduction ................................................................................................... 50323.2 Polarizers ....................................................................................................... 503

23.2.1 Absorption Polarizers: Polaroid ....................................................... 50323.2.2 Absorption Polarizers: Polarcor .......................................................50923.2.3 Wire Grid Polarizers ........................................................................ 51023.2.4 Plasmonic Lenses as Circular Polarizers ........................................ 51123.2.5 Polarization by Refraction (Prism Polarizers).................................. 51223.2.6 Polarization by Reflection ................................................................ 514

23.3 Retarders ........................................................................................................ 51423.3.1 Birefringent Retarders ...................................................................... 51523.3.2 Variable Retarders ............................................................................ 51823.3.3 Achromatic Retarders....................................................................... 519

23.3.3.1 Infrared Achromatic Retarder .......................................... 52023.3.3.2 Achromatic Waveplate Retarders ..................................... 523

xii Contents

23.4 Rotators .......................................................................................................... 52423.4.1 Optical Activity ................................................................................ 52423.4.2 Faraday Rotation .............................................................................. 52623.4.3 Liquid Crystals ................................................................................. 526

23.5 Depolarizers .................................................................................................. 526References ................................................................................................................ 527

24Chapter Ellipsometry ............................................................................................................. 529

24.1 Introduction ................................................................................................... 52924.2 Fundamental Equation of Classical Ellipsometry ......................................... 53024.3 Classical Measurement of the Ellipsometric Parameters Psi (ψ) and

Delta (Δ) ......................................................................................................... 53224.4 Solution of the Fundamental Equation of Ellipsometry ................................ 541

24.4.1 Stokes’s Treatment of Reflection and Refraction at an Interface ..... 55924.5 Further Developments in Ellipsometry: Mueller Matrix

Representation of ψ and ∆ .............................................................................560References ................................................................................................................ 567

25Chapter Form Birefringence and Meanderline Retarders ..................................................... 569

25.1 Introduction ................................................................................................... 56925.2 Form Birefringence ....................................................................................... 56925.3 Meanderline Elements ................................................................................... 570References ................................................................................................................ 572

IPart V Classical and Quantum theory of radiation by accelerating Charges

26Chapter Introduction to Classical and Quantum Theory of Radiation by Accelerating Charges ............................................................................................... 575

References ................................................................................................................ 576

27Chapter Maxwell’s Equations for Electromagnetic Fields ..................................................... 577

Reference .................................................................................................................. 582

28Chapter The Classical Radiation Field .................................................................................. 583

28.1 Field Components of the Radiation Field ...................................................... 58328.2 Relation between Unit Vector in Spherical Coordinates and

Cartesian Coordinates ................................................................................... 58528.3 Relation between Poynting Vector and Stokes Parameters ........................... 588References ................................................................................................................ 594

Contents xiii

29Chapter Radiation Emitted by Accelerating Charges ............................................................ 595

29.1 Stokes Vector for a Linearly Oscillating Charge ........................................... 59529.2 Stokes Vector for an Ensemble of Randomly Oriented Oscillating

Charges .......................................................................................................... 59829.2.1 Note on Use of Hooke’s Law for a Simple Atomic System .............. 601

29.3 Stokes Vector for a Charge Rotating in a Circle............................................ 60129.4 Stokes Vector for a Charge Moving in an Ellipse .........................................604

30Chapter Radiation of an Accelerating Charge in the Electromagnetic Field .........................607

30.1 Motion of a Charge in an Electromagnetic Field ..........................................60730.1.1 Motion of an Electron in a Constant Electric Field ..........................60830.1.2 Motion of a Charged Particle in a Constant Magnetic Field ............ 61030.1.3 Motion of an Electron in a Crossed Electric and Magnetic Field .... 614

30.2 Stokes Vectors for Radiation Emitted by Accelerating Charges ................... 61830.2.1 Stokes Vector for a Charge Moving in an Electric Field .................. 62130.2.2 Stokes Vector for a Charge Accelerating in a Constant Magnetic

Field .................................................................................................. 62330.2.3 Stokes Vector for a Charge Moving in a Crossed Electric and

Magnetic Field .................................................................................. 625References ................................................................................................................ 625

31Chapter The Classical Zeeman Effect ................................................................................... 627

31.1 Historical Introduction .................................................................................. 62731.2 Motion of a Bound Charge in a Constant Magnetic Field ............................. 62831.3 Stokes Vector for the Zeeman Effect............................................................. 637References ................................................................................................................642

32Chapter Further Applications of the Classical Radiation Theory .........................................645

32.1 Relativistic Radiation and the Stokes Vector for a Linear Oscillator ............64532.2 Relativistic Motion of a Charge Moving in a Circle: Synchrotron

Radiation........................................................................................................ 65232.3 Čerenkov Effect ............................................................................................. 65932.4 Thomson and Rayleigh Scattering................................................................. 670References ................................................................................................................ 678

33Chapter The Stokes Parameters and Mueller Matrices for Optical Activity and Faraday Rotation .................................................................................................................... 679

33.1 Introduction ................................................................................................... 67933.2 Optical Activity .............................................................................................68033.3 Faraday Rotation in a Transparent Medium ..................................................68733.4 Faraday Rotation in a Plasma ........................................................................ 691References ................................................................................................................ 693

xiv Contents

34Chapter Stokes Parameters for Quantum Systems ................................................................ 695

34.1 Introduction ................................................................................................... 69534.2 Relation between Stokes Polarization Parameters and Quantum

Mechanical Density Matrix ...........................................................................69634.3 Note on Perrin’s Introduction of Stokes Parameters, the Density Matrix,

and Linearity of Mueller Matrix Elements .................................................... 70534.4 Radiation Equations for Quantum Mechanical Systems ............................... 71034.5 Stokes Vectors for Quantum Mechanical Systems ........................................ 714

34.5.1 Particle in an Infinite Potential Well ................................................ 71434.5.2 One-Dimensional Harmonic Oscillator ........................................... 71634.5.3 Rigid Rotator .................................................................................... 717

References ................................................................................................................ 721

Appendix A: Conventions in Polarized Light............................................................................ 723

Appendix B: Jones and Stokes Vectors ...................................................................................... 725

Appendix C: Jones and Mueller Matrices ................................................................................. 727

Appendix D: Relationships between the Jones and Mueller Matrix Elements ...................... 731

Appendix E: Vector Representation of the Optical Field: Application to Optical Activity ....................................................................................................................... 733

Bibliography ................................................................................................................................. 745

Index .............................................................................................................................................. 747

xv

Preface to the Third EditionPolarized light is pervasive in our world, and we must understand it, measure it, and be able to use it to our advantage. This book is a comprehensive reference on polarized light for scientists and engi-neers working in a variety of fields. It also can be used as a textbook for advanced undergraduates or graduate students who have had calculus and linear algebra and perhaps a course in introductory physics.

Polarized Light, Third Edition is an updated version of Polarized Light, Second Edition, Revised and Expanded as published by Marcel Dekker, 2003. Polarized Light takes the reader from a gen-eral description of light through a complete description of polarized light, and includes practical applications. It incorporates such basic topics as polarization by refraction and reflection, polar-ization elements, anisotropic materials, polarization formalisms (Mueller–Stokes and Jones), and polarimetry, the science of polarization measurement.

This third edition includes substantive new material, and figures that were not redrawn in the second edition have been replaced here with new graphics, and black and white photos and color plates have been added. A completely revised historical review entitled “Polarized Light: A History” is included. The first two chapters are completely new, and are intended to inspire the reader to study polarized light, with a new “Introduction” to polarized light as the first chapter, and a new chapter on “Polarization in the Natural Environment” as Chapter 2. Chapter 7 “Fresnel Equations: Derivation and Mueller Matrix Formulation” has been revised. A chapter on the “Fresnel–Arago Interference Laws” has been completely rewritten and is included here as Chapter 12. The chapter “Polarization Optical Elements,” Chapter 23, has been updated with the addition of photos and improved diagrams. Additional new chapters “Form Birefringence and Meanderline Retarders,” “Techniques in Imaging Polarimetry,” and “Channeled Polarimetry for Snapshot Measurements” are included. A new appendix covers “Conventions in Polarized Light.”

The book is divided into four parts and has been rearranged from previous editions. Part I covers some of the fundamental concepts of polarized light and its theoretical framework. Aspects of the science of measuring polarization and polarimetry comprise Part II. Applications of polarized light make up Part III. Part IV consists of the application of our polarized light framework to topics from physics such as accelerating charges and quantum systems.

Polarized Light as the first edition began in 1993 as a book by Edward Collett. Much of the book was based on his extensive and valuable publications in scientific journals. The second edi-tion required extensive editing of the text and equations, and many of the original figures were redrawn, particularly those that were graphs. Four chapters were added to bring the book up to date: “Stokes Polarimetry” and “Mueller Matrix Polarimetry,” “Polarization Optical Elements,” and “The Mathematics of the Mueller Matrix.” The chapter on “Crystal Optics” was replaced with completely new material, and the chapter “Mueller Matrices for Reflection and Transmission” was heavily modified. This third edition builds upon that foundation.

Working with so many gracious and amazingly proficient people has been one of the greatest rewards of this endeavor. This book would not have been possible without the contributions of col-laborators Michael W. Kudenov and J. Scott Tyo of the University of Arizona, Joseph A. Shaw of Montana State University, and David B. Chenault of Polaris Sensor Technologies, Inc. I am grate-ful for the contributions of stunning visual imagery of animals and the natural environment from Roy L. Caldwell of the University of California, Berkeley; Tsyr-Huei Chiou of the University of Queensland, Australia; and Thomas W. Cronin of the University of Maryland, Baltimore Campus.

xvi Preface to the Third Edition

I also express my gratitude to reviewers Arthur Lompado of Polaris Sensor Technologies, Inc., Martin F. Wehling of the Air Force Research Laboratory, and Robert R. Kallman of the University of North Texas.

This edition is dedicated to the memory of my father, M. N. Goldstein.

Dennis H. Goldstein Polaris Sensor Technologies, Inc.

xvii

Polarized Light: A HistoryThe historical development of the science of polarized light is interwoven with the fabric of the his-tory of optics and our fundamental physical understanding of the natural world. We trace this devel-opment to give the reader some perspective on these discoveries and other events in physics and the world, to give some feel for the personalities involved, and to provide references for those interested in pursuing any of these topics. Many of the most important historical papers in the development of polarized light were collected in the book Polarized Light by William Swindell [1], now out of print but available through libraries. The source of biographical information not specifically referenced comes primarily from Asimov [2] and the Encyclopedia britannica [3].

As with many other basic discoveries, we will never know when or by whom polarized light was first observed or used. A Danish archaeologist suggested in 1967 that the Vikings used crystals as navigation aids [4], observing the polarized sky even when overcast to determine sun position. Arguments against this Viking theory have been made [5] as well as counterarguments experimen-tally demonstrating the utility of these kinds of navigational observations [6].

The earliest publication we have that concerns a history of polarized light is from 1669. In that year, Erasmus Bartholin (sometimes Latinized as Bartholinus), a Danish physician and scientist, reported double refraction in what is now called Iceland spar (calcite from Iceland). Bartholin (born August 13, 1625, died November 4, 1698), the second son of Gaspard Bartholinus, was from a fam-ily of physicians. His father, who died when Erasmus was only four, was a professor of medicine at the University of Copenhagen, and his elder brother Thomas is known for his work on the lymphatic system. Erasmus was first to publish the observation that the image of an object seen through the calcite is double, and one of the images rotates around the other as the calcite crystal was rotated. Bartholin is the source of the terminology we use for double refraction; that is, the light rays that form the fixed image are called ordinary rays, and the light rays that form the rotating image are called extraordinary rays.

Christiaan Huygens (born April 14, 1629, died June 8, 1695), a Dutch scientist, was a contempo-rary of Bartholin. Huygens contributed to the fields of mathematics, astronomy and telescope con-struction, dynamics and clock-making, and optics. He developed the wave theory of light introduced by Hooke and recorded many of his experiments in a major work, Traité de la Lumière, published in 1690. The corpuscular theory of light of Newton (1642–1727) and Huygens’s wave theory com-peted for acceptance as the correct explanation for optical phenomena at this early period before the development of electromagnetic theory and before the transverse wave nature of light was known. Huygens, like Bartholin, observed the double refraction taking place in calcite, and was able to explain this behavior in terms of his wave theory. The need to understand double refraction and the production of polarized light in anisotropic media was a phenomenon that drove much of this early experimental and theoretical work. Resolution of Newton’s theory and Huygens’s wave theory is the heart of one of the most important principles of physics—that of wave–particle duality.

Thomas Young (born June 13, 1773, died May 10, 1829) was a prodigy who trained and prac-ticed as a physician, became independently wealthy after the death of an uncle, and made contribu-tions to physics and physiology. He was also one of the first successful decipherers of hieroglyphics and laid the groundwork for Champollion’s translation of the Rosetta Stone. Young’s most profound contribution is perhaps what we know in optics as “Young’s experiment,” the demonstration that two coherent sources can produce interference. The results of his experiment of 1803, which clearly was evidence of the wave nature of light, was met with opposition in England. Young calculated the wavelength that was required for visible light from his experiments.

xviii Polarized Light: A History

Étienne-Louis Malus (born July 23, 1775, died February 24, 1812) was a French army engineer by profession. Well over 100 years had passed since Bartholin and Huygens published their work on double refraction. But Malus was yet another inquisitive scientist who held calcite in his hands and discovered an interesting phenomenon. He observed that light reflected from a glass window was polarized as seen through the calcite. This was an important observation because it established that light could be polarized by reflection and it confirmed that polarization was an intrinsic property carried by the light beam. As Malus observed the reflected light he found that rotating the crystal changed the observed flux. The change was proportional to the square of the cosine of the angle between the direction of the polarization and the transmission axis of the polarizer, the calcite in his experiment. Malus published this work in 1809, and we know this cosine-squared relation as Malus’s Law. Earlier, Newton had referred to light passing through calcite as having different “sides” and made an analogy between these and the poles of a magnet. Malus used the terminology “polarized light” to describe the phenomenon.

Dominique François Jean Arago (born February 26, 1786, died October 2, 1853) was a Renaissance man who was intimately involved with many scientific and social events during his lifetime. Early in his career he carried out surveys in Spain with Jean Baptiste Biot. When he returned to France after three months in Spanish prisons and six months in Africa, he was appointed to a chair of analytical geometry at the École Polytechnique and also became an astronomer at the Royal Observatory. He constructed a polariscope and discovered in 1809 that the sky was polarized. In 1811 he discovered optical rotation in quartz. (Arago’s former friend and fellow surveyor, Biot, showed optical rotation in organic substances in 1815.) The pile-of-plates polarizer was invented by Arago in 1812. Arago became a proponent of the wave theory of light and called Fresnel’s attention to Young’s experiments. Arago and Fresnel were friends and collaborators, but Arago, although he favored wave theory, could not quite accept the transverse wave theory because of the issue of the medium that was thought to be required to support transverse waves, and Fresnel published this alone. The question of light propagation and whether it was corpuscular or wave, and then whether this wave was longitudinal or transverse, and if a wave, what supported its propagation, was one of great controversy and debate during this period, and of course one of the most important keys to our understanding of the universe. Arago actually designed an experiment to test the velocity of light in air and in a dense medium such as water or glass. Wave theory predicts retardance while corpuscular theory predicts a velocity increase. Because of the revolution in France in 1848 and failing health, Arago did not carry out his proposed experiment, but experiments before his death based on Arago’s experimental design, carried out by Fizeau and Foucault, did show retardance. After the revolution of 1848 as a member of the Republican Government, Arago abolished flogging for sailors and slavery in the French colonies.

Malus had tried to establish a relationship between the properties of materials and what was then called the polarizing angles, but gave up before he found it. Sir David Brewster (born December 11, 1781, died February 10, 1868) was more persistent, and measured the polarizing angle for a variety of materials. In a paper from 1815, he revealed what we now know as Brewster’s Law in the following simple form: “The index of refraction is the tangent of the angle of polarisation.” Brewster is also known as the inventor in 1816 of the kaleidoscope.

Augustin-Jean Fresnel (born May 10, 1788, died July 14, 1827) was a giant in the science of optics despite a life terminated prematurely by tuberculosis. He was employed by the French government as an engineer during most of his career. About 1814, he became interested in light and independently duplicated some of Young’s experiments. Fresnel was made aware of Young’s work by Arago, adopted Young’s view that light was a transverse wave, and constructed a wave theory of light based on transverse oscillations. For the first time, double refraction was satisfac-torily explained. At the time of Fresnel, it was thought that a medium, the ether, was necessary to support transverse waves. Existence of an ether introduced new problems, and many scientists at the time could not accept this model. It was only after the experiments of Michelson that the ether was discounted. Fresnel went on to provide the theoretical derivation of the laws of refraction and

Polarized Light: A History xix

reflection (remember that this is long before Maxwell’s equations), developed with Arago the laws that govern interference of polarized light, developed lenses for lighthouses (Fresnel lenses), and designed a rhombohedron of glass (the Fresnel rhomb) that produces circularly polarized light.

The first polarizing prism that successfully separated orthogonal polarizations so “… that only one Image may be seen at a time” [7] was designed in 1828 by William Nicol (born in 1770, died September 2, 1851). Polarimetry was possible for the first time, and Nicol went on to develop meth-ods of preparing thin sections of fossils and minerals and studied them in polarized light.

Most animals do not have eyes that are sensitive to polarized light or behavioral activities that require polarized light. Wilhelm Karl von Haidinger (born February 5, 1795, died March 19, 1871) published a paper in 1844 announcing his discovery that the human eye does perceive linearly polarized light. This visual sensation is manifested as two opposing paddle-shaped yellow regions with blue areas orthogonal to the yellow. This pattern, known as Haidinger’s brushes, is best seen when looking at a highly polarized white background.

Michael Faraday (born September 22, 1791, died August 25, 1867) was a brilliant experimen-talist who lacked mathematical training. Through his observational skills, meticulous recording of results, and his experimental intuition, he made huge contributions to the sciences of chemistry, cryogenics, electricity, and optics. In the course of his experiments with a polarized light beam passing through a magnetic field in 1845, he discovered what we now call the Faraday effect. The Faraday effect occurs when a magnetic field is applied to a material and a linearly polarized light beam is passed through the medium parallel to the field lines. The plane of polarization is rotated. Faraday tried many materials and found a considerable number that exhibited this effect. In addi-tion to his inherent modesty, he belonged to a religious sect that discouraged any display of vanity. Accordingly, Faraday turned down the presidency of the Royal Society and a knighthood.

Sir George Gabriel Stokes (born August 13, 1819, died February 1, 1903), British mathemati-cian and physicist, held the three offices that only Isaac Newton had held before—that of Lucasian professor at Cambridge, and secretary and president of the Royal Society. Stokes contributed to a number of areas in physics and mathematics, but is perhaps best known for his work in fluid mechanics and optics. In a paper from 1852 [8], Stokes set out a method to mathematically describe unpolarized and partially polarized light in terms of observational quantities. We know the quanti-ties he defined as the Stokes parameters, although they were not in general use until a century later. Stokes went from polarized light to a larger work that also appeared in 1852 in a paper on fluores-cence, a term he introduced. He never returned to the subject of polarized light.

John Kerr (born December 17, 1824, died August 15, 1907), a Scottish physicist and friend of Lord Kelvin, discovered that birefringence could be induced by an electric field in 1875. Thus began the field of electro-optics. Induced birefringence proportional to the square of the imposed electric field is named the Kerr effect.

Jules Henri Poincaré (born April 29, 1854, died July 17, 1912) was a French mathematician. Upon obtaining his degree, he took a position at the University of Paris. Poincaré made contribu-tions in many areas of mathematics as well as astronomy and physics. In polarization optics his name is associated with the Poincaré sphere, the three-dimensional surface on which any polariza-tion state can be represented. This representation was described in 1892 in Poincaré’s book Théorie Mathématique de la Lumière. Poincaré apparently was not aware of the Stokes parameters because he did not present the sphere as being generated using the Stokes parameters as values along orthog-onal axes in a Cartesian coordinate system, a representation we often use today. A first cousin of Poincaré’s, Raymond Poincaré, was president of France during World War I.

Albert Abraham Michelson (born December 19, 1852, died May 9, 1931) was a Prussian-born German-American physicist. He obtained his undergraduate education at the United States Naval Academy and served as an instructor there before traveling to Germany for his graduate educa-tion. Michelson studied under the great German physicist Hermann Helmholtz at the University of Berlin and then returned to a physics professorship at Case School of Applied Science in Cleveland, Ohio. A skilled experimentalist, Michelson is known for his measurements of the speed of light and

xx Polarized Light: A History

the construction of the Michelson interferometer (financed by Alexander Graham Bell). His most notable achievement, recorded in 1887, is one of the most elegant and significant experiments in all of physics, the Michelson–Morley experiment. The negative results of this experiment established that an ether was not detectable, allowing theories to take over that didn’t require an ether. Light as a transverse wave was finally the accepted theory without requiring justification. Michelson received the Nobel Prize in 1907 “for his optical precision instruments and the spectroscopic and metrologi-cal investigations carried out with their aid” [9], the first American to receive a Nobel Prize in the sciences.

Edwin Herbert Land (born May 12, 1909, died March 1, 1991) invented the sheet polarizer in 1928. This discovery was announced in a talk given by Land to the Harvard Physics Colloquium in 1932. Land’s first patent is from 1933, and the sheet polarizer material commonly used today, H-sheet, was invented around 1938. Land’s account of his work with sheet polarizers was not docu-mented until 1951 [10]. It was on reading David Brewster’s book on kaleidoscopes [11] that Land became interested in herapathite (iodoquinine sulfate), a dichroic crystal. Land’s first experiments were with herapathite crystals in suspension and subjected to a magnetic field. The crystals were aligned with the field turned on, and Land had produced his first polarizer.

R. Clark Jones (born June 30, 1916, died April 26, 2004) developed the mathematical frame-work for the matrix formalism for polarization elements that bears his name in a series of eight papers published from 1941 to 1956 in the Journal of the Optical Society of America. These papers are reprinted in Swindell [1].

The Mueller matrix of polarization mathematics is named for Hans Mueller, a professor of physics at the Massachusetts Institute of Technology (MIT). Shurcliff [12] credits Mueller with the invention of the Mueller–Stokes formalism in 1943 in the form of MIT course notes and a previously classified government report. A student of Mueller’s, N. G. Parke, referred to Mueller matrices in a paper from 1949 [13], and this may be the first use of this nomenclature. Mueller (born October 27, 1900, died June 10, 1965) was born in Switzerland and obtained his degrees from the Eidgenossische Technische Hochschule in Zurich. He came to MIT in 1925 and remained there for the next 40 years.

Subrahmanyan Chandrasekhar (born October 19, 1910, died August 21, 1995) was an American astrophysicist of Tamil Indian descent. He is credited with the reintroduction and practi-cal use of the Stokes parameters in his book Radiative Transfer [14] of 1950 and in prior journal papers from 1946.

Our historical review has brought us almost 300 years (roughly 1669–1956) from the earliest publication up to the point where most of the tools and formalisms that we use today in polarization were in place. The past 50 years have seen the development of the technology of detectors, electron-ics, fiber and integrated optics, fabrication techniques, and computers so that our modern era in classical polarization might be considered an age of data gathering and applications.

The RumfoRd medal

The Rumford Medal was initiated in 1800 by The Royal Society to be awarded in even years, although some years had no awards [15]. It is notable that eight of the people important to the his-tory of polarization were recognized with this medal. The statement for the award of the Rumford Medal, given by The Royal Society, is as follows: “The Rumford Medal is awarded biennially (in even years) in recognition of an outstandingly important recent discovery in the field of thermal or optical properties of matter made by a scientist working in Europe, noting that Rumford was con-cerned to see recognized discoveries that tended to promote the good of mankind.”

The award citations for these eight scientists from Malus to Maxwell are listed below. It is inter-esting to note that the award for Stokes was for the work he did immediately after the monograph in which he defined the quantities we know as the Stokes parameters.

Polarized Light: A History xxi

1810 (4th award) Etienne-Louis Malus. For the discovery of certain new properties of Reflected Light, published in the second volume of the Memoires d’Arcueil.

1818 (7th award) David Brewster. For his discoveries relating to the Polarization of Light.1824 (8th award) Augustin-Jean Fresnel. For his development of the Undulatory Theory as

applied to the Phenomena of Polarized Light, and for his various important discoveries in Physical Optics.

1840 (12th award) Jean Baptiste Biot. For his researches in, and connected with, the circular polarization of light.

1846 (14th award) Michael Faraday. For his discovery of the optical phenomena developed by the action of magnets and electric currents in certain transparent media, the details of which are pub-lished in the 19th series of his experimental researches in electricity, inserted in the Philosophical Transactions for 1845 and in the Philosophical Magazine.

1850 (16th award) Francois Jean Dominique Arago. For his experimental investigations on polar-ized light, the concluding memoirs on which were communicated to the Academy of Sciences of Paris during the last two years.

1852 (17th award) George Gabriel Stokes. For his discovery of the change in the refrangibility of light.

1860 (21st award) James Clerk Maxwell. For his researches on the composition of colors and other optical papers.

RefeReNCeS

1. Swindell, W., Ed., Polarized Light, Stroudsberg, PA: Dowden, Hutchinson & Ross, 1975. 2. Asimov, I., Asimov’s biographical Encyclopedia of Science and Technology, 2nd revised ed., Garden

City, NY: Doubleday & Company, 1982. 3. Encyclopedia britannica, William Benton, Chicago, 1959. 4. Ramskou, T., Solstenen, Skalk 2 (1967): 16–7. 5. Roslund, C., and C. Beckman, Disputing Viking navigation by polarized skylight, Appl. Opt. 33 (1994):

4754–5. 6. Barta, A., G. Horváth, and V. Benno Meyer-Rochow, Psychophysical study of the visual sun location

in pictures of cloudy and twilight skies inspired by Viking navigation, J. Opt. Soc. Am. A 22 (2005): 1023–34.

7. Nicol, W., On a method of so far increasing the divergency of the two rays in calcareous-spar, that only one image may be seen at a time, Edinburgh J. Phil. 6 (1828): 83–4.

8. Stokes, G. G., On the composition and resolution of streams of polarized light from different sources, Trans. cambridge Phil. Soc. 9 (1852): 399–416.

9. Official website of the Nobel Foundation, http://nobelprize.org/nobel_prizes/physics/laureates/1907/ 10. Land, E. H., Some aspects of the development of sheet polarizers, J. Opt. Soc. Am. 441 (1951):

957–63. 11. Brewster, D., The Kaleidoscope, Its History, Theory, and construction, 2nd ed., London: John Murray,

1858. 12. Shurcliff, W. A., Polarized Light: Production and Use, Cambridge, MA: Harvard University Press,

1962. 13. Parke, N. G., Optical algebra, J. Math. Physics, 28 (1949): 131–9. 14. Chandrasekhar, S., Radiative Transfer, 24-35, Mineola, NY: Dover Publications, 1960. 15. The Royal Society. Available from http://royalsociety.org/Content.aspx?id=3366

IPart

Introduction to Polarized Light

3

1 Introduction

The story of polarized light is integral to the development of the science of optics, which itself plays a central role in the history of physics. The story surely begins when someone first saw the double images one sees when looking through a calcite crystal, a form of calcium carbonate (CaCO3), as in Figure 1.1. As we pointed out in the historical review, the Vikings may have used these crystals for navigation by observation of the polarized sky patterns. In any event, they certainly knew about these crystals, which are also called Iceland spar, and they must have seemed magical. The histori-cal record begins in 1669 with Erasmus Bartholinus, the first modern scientist to describe the phe-nomenon. He was the first in a long line of eminent scientists who held these crystals in their hands and wondered about them. A means of explaining this double image provided impetus for the devel-opment of ideas about the nature of light, and as theories about the character of light developed, they had to explain and be compatible with the observations made when looking through calcite.

The wave theory of light of Huygens and the corpuscular theory of the light of Newton were the competing theories of these seventeenth century scientists, whose lives overlapped. Newton’s theory was dominant during the eighteenth century, but at the beginning of the nineteenth century, the inter-ference experiments of Young, and somewhat later the work on diffraction by Fresnel and Arago, gave the wave theory a legitimacy and attention it did not have before.

This early work eventually resulted in the principle of particle–wave duality, now one of the basic principles of physics, but this did not happen until after the work of Maxwell, who succeeded in set-ting forth a unified theory of electromagnetic radiation in rigorous mathematical form (1873), and the work of Michelson and Morley (1887), who showed that the medium then thought to exist and support propagation of light waves, the luminiferous aether, apparently did not exist.

We now know that electromagnetic radiation is a transverse wave; that is, an oscillation of elec-tric and magnetic fields in a direction perpendicular to the direction of propagation. What we refer to as light, in the broader sense electromagnetic radiation from ultraviolet to infrared, and in human visual experience from violet to red, the wavelength region from 400 to 700 nm, is a subset of the entire electromagnetic spectrum.

When we refer to the polarization of light, we refer to one of the basic properties of a light wave; that is, the polarization is defined to be the description of the vibration of the electric field. Linear polarization is then a vibration along one direction in three-dimensional space with the propagation along a second direction, as in Figure 1.2, where the curve traces the location of the tip of the elec-tric field vector as the light propagates through space. Linear polarization is one extreme of a con-tinuum of possible polarizations, called states, where circular polarization, illustrated in Figure 1.3, is the other extreme. In this case, the plot of the tip of the electric field vector results in a helix. Elliptical polarization is a general term that can be used to describe any state in the continuum from linear to circular.

As Clarke and Grainger point out [1], the term polarization is perhaps unfortunate, but it is now one that we are obliged to use as there is no convenient substitute. The term appears to come from Newton, who discussed the “sides” that light exhibited in double refraction, as in passing through calcite. Newton compares this to poles of magnets. Having a piece of iron magnetically polarized, or a molecule or electron that is polarized, has little to do with polarized light, so the term can be confusing.

The very essence of light, a spatially asymmetric electromagnetic wave, means that light is natu-rally polarized. Polarization, along with frequency of vibration, is a fundamental property. Where there is light, there is polarized light, and truly randomly polarized light is an elusive phenomenon.

4 Polarized Light, Third Edition

And even if randomly polarized light is achieved, any interaction whatsoever, through the typically asymmetric processes of reflection, transmission, or scattering, will induce a polarization.

A few examples will serve to illustrate polarization by reflection and transmission (polarization by scattering is shown in Chapter 2). Figure 1.4a shows a black and white image of an automobile in a field. Figure 1.4b is an image of the automobile where the linear polarizations in the +45° direction and –45º direction (with respect to horizontal) are encoded in the colors blue and red, respectively, and light areas have little polarization in these directions. The final image, Figure 1.4c, has the amount of polarization at each point in the image encoded as a color. This is the degree of polariza-tion, and is encoded so that dark areas are not polarized and red areas are very highly polarized. Light has been polarized through reflection from the smooth surfaces of the vehicle.

Figure 1.5 shows a sheet of mica in between crossed linear polarizers. An ideal linear polarizer will absorb light of one linear polarization and transmit light of the orthogonal polarization. In this case, light of one linear polarization is transmitted through the first polarizer and is blocked by the

figuRe 1.1 (See color insert following page 394.) The double image seen through a calcite crystal. (Photo courtesy of D. H. Goldstein.)

x

y

z

figuRe 1.2 Linear polarization.

Introduction 5

y

x

z

figuRe 1.3 Circular polarization.

(a) (b) (c)

figuRe 1.4 (See color insert following page 394.) Images of an automobile in a field; (a) black and white photograph, (b) linear ±45° polarization encoded in pseudocolor, and (c) degree of polarization encoded in pseudocolor. (Photos courtesy of D. H. Goldstein.)

figuRe 1.5 (See color insert following page 394.) Mica between crossed polarizers. (Photo courtesy of D. H. Goldstein.)

6 Polarized Light, Third Edition

second polarizer in all black areas of the photo. Mica is a silicate mineral that has a different refrac-tive index in each of the three Cartesian directions. The phase of polarized light is retarded upon passing through the mica, the retardation being dependent upon the thickness and the frequency, and the polarization of light is thus changed. Mica naturally occurs in very thin sheets, and the dif-ferent colors observed in Figure 1.5 correspond to those colors that have been rotated into a polar-ization that will pass through the polarizer closest to the viewer because of the passage of the light through different thicknesses in the sheet.

In Figure 1.6 we have another type of crystalline material, camphor, as photographed under a polarized light microscope with crossed polarizers. Camphor is an organic molecule that has chiral-ity, or handedness, and it can rotate the direction of polarization. In this photograph, the different colors correspond once again to different thicknesses of the camphor and thus where different colors have been rotated by the amount necessary to pass through the polarizer closest to the viewer.

As a further example, Figure 1.7 shows a bottle of corn syrup as seen through (a) aligned polar-izers, (b) polarizers at 45° to one another, and (c) crossed polarizers. Corn syrup is also a chiral material, able to rotate the polarization. Again we see different colors corresponding to different degrees of rotation of the polarization direction.

figuRe 1.6 (See color insert following page 394.) Camphor between crossed polarizers. (Photo courtesy of D. H. Goldstein.)

(a) (b) (c)

figuRe 1.7 (See color insert following page 394.) Corn syrup (a) between parallel polarizers, (b) between polarizers at 45° to one another, and (c) between crossed polarizers. (Photos courtesy of D. H. Goldstein.)

Introduction 7

These are all entertaining and colorful examples of polarized light phenomena. Polarized light has to be considered in almost any optics application, and it has many important practical uses. In Part I of this book, we will explore the basic physics of polarized light and the mathematical meth-ods that have been developed to describe it. In Part II, we will describe techniques used to measure polarization, an activity called polarimetry. In Part III, we will describe applications, and in Part IV we will incorporate the framework of optical polarization into the physics of accelerating charges, optical activity, and quantum systems.

We can see that polarization is a fascinating topic of study, but even in the absence of scientific interest, polarization can be useful for anyone who spends time outside on sunny days, driving, or fishing. Our final example of polarization by reflection in Figure 1.8 shows what a driver sitting in his vehicle would see looking out his windshield (a) without polarized sunglasses and (b) with polarized sunglasses. Polarized sunglass lenses consist of polarizers designed to block horizontally polarized light that would typically be present when light reflects from the horizontal surfaces of other vehicles or water. In this example, light reflecting off an object on the dashboard reflects back to the inner surface of the windshield and then to the driver, and is polarized horizontally. The sun-glasses eliminate this image as if by magic.

RefeReNCe

1. Clarke, D., and J. F. Grainger, Polarized Light and Optical Measurement, New York: Pergamon Press, 1971.

(a) (b)

figuRe 1.8 (See color insert following page 394.) View from vehicle as seen (a) without polarized sun-glasses, and (b) with polarized sunglasses. (Photo courtesy of D. H. Goldstein.)

9

2 Polarization in the Natural Environment

2.1 SouRCeS of PolaRiZed lighT

As we have seen in Chapter 1, light is a transverse wave and is therefore inherently asymmetrical. Interaction of light with asymmetric materials, and at arbitrary angles, just adds to this asymmetry. In the natural environment, polarized light is primarily a result of reflection and scattering. The sub-ject of polarized light in our environment is vast, and books have been written on various aspects of polarization in nature and the sensing of it [1–6]. In this chapter, we will necessarily limit ourselves to a few topics out of the rich array available on this subject. We will briefly describe a few effects that produce polarized light in the atmosphere, we will describe some animals that produce polar-ized light, and we will discuss animals that can see polarized light.

2.2 PolaRiZed lighT iN The aTmoSPheRe

2.2.1 The Sky: Rayleigh ScaTTeRing and PolaRizaTion

Light from the sun interacts with the molecules of our atmosphere such that the light that we see coming from the dome of air over our heads is scattered sunlight (or moonlight). The scattering of light in the atmosphere is a process of absorption and re-emission of radiation through a coupling of the incident radiation and the electric dipole in the atmospheric molecules.

Atmospheric gas molecules have diameters on the order of 3 Å. At 4000–7000 Å, the wave-lengths of visible light are much larger than the molecules. When the wavelength is much larger than the scattering particle, we have what is known as Rayleigh scattering. Lord Rayleigh showed in 1871 that the scattering efficiency depends upon the fourth power of the frequency of light, thus the efficiency of molecular scattering for blue light is roughly an order of magnitude larger than for red. This is the reason for the blue color of the sky. Here we describe Rayleigh scattering qualitatively. A mathematical treatment is given in Chapter 32.

The blue sky (and the night sky when the moon is out) is also polarized because of Rayleigh scat-tering. The reason for this is notionally represented in Figure 2.1. When the transverse field that is the light from the sun interacts with the dipolar air molecule above our observer on the surface of the earth, the re-radiation that occurs can only be in a plane that is transverse to the page. The polarization that is in the vertical direction in this diagram is scattered in the forward direction, and our observer does not see it. Thus the observer sees only the light polarized perpendicular to the page.

The polarization of the sky is represented in Figure 2.2. Think of the circle as a hemispherical dome that you pick up and hold over your head. The sun is at the horizon and the polarization in direction and magnitude is represented by the direction and length of the arrows. The highest polar-ization is 90° from the sun, and at twilight this can reach about 75%. There is no polarization in the direction of the sun. There is also no polarization at the neutral point of Arago, about 25° above the horizon opposite the sun, as indicated in Figure 2.2. Beyond the neutral point, the vertical polariza-tion is caused by multiple scattering. The moonlit night sky is polarized in exactly the same way, but because of the much lower light level is not as easily observed without an integrating detector (such as a time exposure with film or digital camera).

10 Polarized Light, Third Edition

Polarization of the sky has been and continues to be a subject of research, and the details of our understanding are somewhat more complex than the simplified picture presented by Figure 2.2. For example, there are two more named neutral points, the Babinet neutral point discovered in 1840, and the Brewster neutral point, predicted in 1842 and experimentally confirmed in 1846. Only two neutral points can be observed from the ground at any one time. The interested reader will find more detailed information on sky polarization in Gehrels [1], Hovarth and Varju [2], and in the technical literature. Figure 2.3 shows photos of meas ured sky polarization.

2.2.2 RainbowS

The rainbow, a commonly enjoyed sky phenomenon, is one of many atmospheric displays that pro-duce polarized light. Refraction of light through drops of water creates the bows, and observation of the colors does not require a polarizer. However, the light from a rainbow is strongly polarized [3,4,8,9], and an arc of the bow is readily removed by observing it through a polarizer whose axis is tangential to that part of the bow.

The path of a light ray that produces the primary bow is shown in Figure 2.4. The angle between the ray entering the drop and the ray leaving the drop is about 42°, assuming a refractive index

ExEy Ex

Ey

Ex

figuRe 2.1 Notional geometry for sky polarization.

Sun N

figuRe 2.2 Polarization pattern of the sky when the sun is at the horizon. N indicates the neutral point of Arago.

Polarization in the Natural Environment 11

Original image(horizontal pol)

1800h0 = +2.7°

1840h0 = –7.1°

1800h0 = +2.7°

1850h0 = –9.6°

1900h0 = –12.0°

1910h0 = –14.4°

1840h0 = –7.1°

1850h0 = –9.6°

1900h0 = –12.0°

1910h0 = –14.4°

1840h0 = –7.1°

1850h0 = –9.6°

1900h0 = –12.0°

1910h0 = –14.4°

0%

50%

100%

1810h0 = +0.5°

1820h0 = –2.2°

1830h0 = –4.7°

1810h0 = +0.5°

1820h0 = –2.2°

1830h0 = –4.7°

1800h0 = +2.7°

1810h0 = +0.5°

1820h0 = –2.2°

1830h0 = –4.7°

045

90

% polarization

e-vector angle

figuRe 2.3 (See color insert following page 394.) Linear polarization of the sky at sunset and during evening twilight. (From Cronin, T. W., Warrant, E. J., and Greiner, B., Appl. Opt., 45, 5582–9, 2006. With permission from Optical Society of America.) Full-sky images were acquired using a Nikon Coolpix 5700 digital camera with a fisheye lens attachment having a linear polarizing filter mounted between the lens and the camera itself. Data were acquired on September 15, 2004 at Lizard Island, Australia. Sunset was at 1814 local time. Each image is labeled with the local time at which data were acquired (at 10 minute intervals) as well as the solar elevation, h0. In all images, the zenith is in the center, north is to the top, and east to the left. The top set of images are original digital photographs acquired when the polarizing filter was oriented east–west (indicated by the double-headed arrow), emphasizing the dark band of north–south electric vector orientation. The middle set shows linear polar-ization in percentages in pseudocolor. The bottom set indicates electric vector angle, also in pseudocolor as coded in the key to the right. Note that clouds appear in some of the images and are particularly noticeable at 1850.

12 Polarized Light, Third Edition

for water of 4/3, so that the angle between the light source and the observed bow is about 138°. The internal reflection at the back of the drop is near the Brewster angle, and this produces a very high degree of polarization of the reflected light. The rainbow is therefore tangentially polarized as shown in Figure 2.5. Of course, dispersion of the beam is occurring also, and we see the colors distributed radially.

There can be more than one internal reflection, and sometimes a secondary bow can be seen. Bows beyond the secondary produce so little light that they are not normally seen. The deviation of the beam for the kth bow is given by

d ki r r= − + −2 2( ) ( ),θ θ π θ (2.1)

where d is the net deviation, θi and θr are the angles of incidence and reflection, respectively, and n is the number of internal reflections. Differentiating this expression we obtain

dd d k di r= − +2 2 1θ θ( ) (2.2)

and setting dd/dθi = 0 to find a minimum (or maximum) we have

dd

ki

r

θθ

= +1. (2.3)

42°

θi

θr

figuRe 2.4 Geometry of a light ray through a spherical water drop with one internal reflection.

figuRe 2.5 (See color insert following page 394.) Representation of the rainbow with tangential polariza-tion indicated with the arrows.

Polarization in the Natural Environment 13

Snell’s Law, where n is the index of refraction for water and the index of refraction of air is 1.000, is

sin sin .θ θi rn= (2.4)

Differentiating Equation 2.4,

cos cos ,θ θ θ θi i r rd n d= (2.5)

and rearranging,

dd

ni

r

r

i

θθ

θθ

= coscos

(2.6)

and from Equations 2.3 and 2.6 we have that

n kr

i

coscos

,θθ

= +1 (2.7)

or, rearranging,

( )cos cos .k ni r+ =1 θ θ (2.8)

Squaring both sides of Equation 2.8 and Snell’s Law Equation 2.4 and adding we find that

( )cos ,k k ni2 2 22 1+ + =θ (2.9)

or

cos( )

.θin

k k= −

+

2 12

(2.10)

This equation gives us the minimum deviation for the kth bow. We can show that these correspond to minima by taking the second derivative of d. This is

d dd

nddi

r

i

2

2

2

22 1

θθθ

= − +( ) , (2.11)

and after some algebra we can show that

dd

nn

r

i

i

r

2

2

2

3 3

1θθ

θθ

= −( )sincos

. (2.12)

Since n is positive, the second derivative of d is positive and we indeed have minima.Photographs of a rainbow taken through a polarizer (a) aligned with the rainbow polarization and

(b) crossed with the rainbow polarization are shown in Figure 2.6. This rainbow had a fairly short arc, and the rainbow polarization variation is small from one end to the other.

14 Polarized Light, Third Edition

2.2.3 cloudS, haloS, and gloRieS

Clouds of water droplets, ice crystal, dust, sand, and smoke all can produce polarized light [4]. The degree of polarization that results is dependent on particle type and size, particle shape, cloud density, method of production (reflection, refraction, or scattering), and relative positions of light source, cloud, and observer. As can be imagined, there are a large number of different effects that can be observed. A few phenomena that have specific names are mentioned in this section. Many of these atmospheric effects are illustrated at the Web site www.atoptics.co.uk [10]. This site is an excellent source of additional references, and it includes downloadable simulations for both halos and glories.

2.2.3.1 CloudsThere are two types of high clouds seen after sunset that exhibit polarization: noctilucent clouds and nacreous clouds. The noctilucent clouds are typically seen at high latitudes in summer months and may be at altitudes up to 80 km. The noctilucent clouds are tangentially polarized and can have a very high degree of polarization (0.96) at 90° from the sun’s position below the horizon. Nacreous clouds are formed at altitudes of about 25 km and appear about 20° from the sun, again typically in northern latitudes. These clouds are colored, tangentially polarized, and the colors change as a polarizer is rotated.

2.2.3.2 haloesHaloes are circles of light around the sun or moon typically formed by refraction in clouds of ice crystals. Three of the most common halo effects are the 22° halo, the 46° halo, and parhelia, or sun dogs. Considering the ice crystals to be planar hexagonal plates, the 22° halo is formed when light is refracted through alternate faces such that the crystal is effectively a 60° prism as shown in Figure 2.7. The minimum deviation for a light ray along this path is 22°. Since the refraction angle for red light is less, the halo is colored with red closest to the sun (or moon). The 22° halo is radially polarized, although its polarization, at around 4%, is too weak to be seen [4].

The halo is formed when the ice crystals are randomly oriented. When they are horizontal, the parhelia, or sun dogs, appear. These are two bright spots of colored light 22° on either side of the sun. They can appear to be very short segments of a rainbow. The phenomenon seen at night is called a paraselene, or moon dog. It is usually seen when the moon is at its brightest (i.e. when it is full, or nearly full). Because ice crystals are birefringent, there are actually two overlapping halos and/or

(a) (b)

figuRe 2.6 (See color insert following page 394.) Photographs of a rainbow. In (a) the rainbow polariza-tion is aligned with the polarization axis of the polarizer. A secondary bow, as well as supernumerary bows, are evident. In (b) the rainbow polarization is perpendicular to the polarization axis of the polarizer and the rainbow almost disappears. (Photos courtesy of D. H. Goldstein.)

Polarization in the Natural Environment 15

parhelia that are orthogonally polarized and shifted from each other by 0.11° [4]. This shift can be observed by rotating a linear polarizer.

Haloes and parhelia can sometimes be formed at 46° from the sun when light refracts through ice crystal faces that are 90° to each other. These effects are less common, not as bright, but more highly polarized than the 22° variety [4]. As before, there are two orthogonal polarizations present because of the ice crystal birefringence.

The reader can find additional information on these and other effects in the classic books by Humphreys [3] and Wood [8] and also in references mentioned by Cowley [10].

2.2.3.3 gloriesGlories are colored rings around the shadow of an observer cast on clouds or fog. The observer has to be above the clouds. This is accomplished by being on a mountain with clouds at a lower altitude or being in an airplane. The glory is produced by water droplets typically smaller than those found in rain, and is predicted by Mie scattering theory (average drop diameter in rain is on the order of a millimeter, whereas average drop diameter in clouds is on the order of 10 μm). The path of a light ray is similar to that for a rainbow, except that the angles of deviation are slightly larger so that that light turns a total of 180°. This is explained through the use of sur-face waves at the air–water interfaces in the smaller drops [10]. The surface waves result in the colored rings of the glory being radially polarized while a white region inside the rings is tangentially polarized.

2.2.4 The Sun

The sun is generally considered to be a source of unpolarized light, and this may be a reasonable assumption for spectrally and spatially averaged light. However, the polarization of light from the sun is routinely used to determine the direction and magnitude of the solar magnetic field [11–13]. When polarization of solar spectral lines is examined, the solar disk is far from uniform. Polarization information is a critical tool in understanding the processes going on in our nearest star.

Figure 2.8a shows a full-disk image of the sun taken with a solar polarimeter, and Figure 2.8b shows a total flux image of the sun on the same day. The polarimetric image clearly shows that the light from the sun is not polarized uniformly.

22°

figuRe 2.7 Geometry of a light ray through an alternate faces of a hexagonal ice crystal producing the 22° halo. The crystal is effectively a 60° prism.

16 Polarized Light, Third Edition

As photons are generated in the presence of the solar magnetic fields, spectral lines are split according to the Zeeman effect (see Chapter 31). The polarization states that are created are depen-dent upon the relative position of the observer with respect to the orientation of the magnetic field and the direction of oscillation of charges. Spectral lines that have been used to make the measurements include the Lyman-alpha line at 1216 Å, the H-alpha line at 6563 Å, and the FeI 5250 Å line.

2.3 PRoduCTioN of PolaRiZed lighT by aNimalS

There are several examples of animals that produce polarized light from unpolarized light. It is not always known whether the animals put this capability to use. This is an area of active research, and some of what is known about animal vision is discussed in the next section.

Scarab beetles are all known to produce circularly polarized light. Firefly larvae produce biolu-minescent circular polarization [2]. Butterfly wings are known to produce linearly polarized light from unpolarized light [14,15]. Sea creatures such as squid, mantis shrimp, and cuttlefish are also able to induce polarized light [16].

In this section, we examine three type of animals, the scarab beetles, cephalopods (squid and cuttlefish), and stomatopods, a type of marine crustacean also known as mantis shrimp.

2.3.1 ScaRabaeidae (ScaRab beeTleS)

The creation of polarized light is common in nature, but the production of circularly polarized light from unpolarized light is quite rare. A. A. Michelson seems to have been the first to note that, in 1911 in a paper entitled “On Metallic Colouring in Birds and Insects,” [17] reflected light from the scarab beetle Plusiotis resplendens, a beetle that appears to be fashioned out of brass or gold, is circularly polarized. Michelson looked at Plusiotis resplendens and discovered that “On exami-nation…[of Plusiotis resplendens]…it was found that the reflected light was circularly polarized even at normal incidence, whether the incident light was polarized or natural. The proportion of circularly polarized light is greatest in the blue, diminishing gradually in the yellow portion of the spectrum and vanishing in the yellow–orange—for which colour the light appears to be completely depolarized. On progressing toward the red end of the spectrum traces of circular polarization in the opposite sense appear, the proportion increasing until the circular polarization is nearly complete in the extreme red.”

figuRe 2.8 (See color insert following page 394.) Solar disk with (a) polarimetric image and (b) white light image.

Polarization in the Natural Environment 17

It has been found that of the beetles only scarabs possess the ability to produce circularly polar-ized light. Figure 2.9a shows Plusiotis resplendens in the absence of polarizing optics, and Figure 2.9b shows the animal with a right-circular polarizer in front of the camera.

The effect for this creature is more subtle than it is for other scarabs, and it is difficult to discern the difference between these two images. One reason for this is the absence of a black backing layer in the cuticle of this scarab. The other reason, the unique polarization properties of this scarab, will be discussed later. A more impressive example is Plusiotis gloriosa, shown in Figure 2.10.

The coloration of this brilliant green and gold-striped beetle disappears when a right-circular polarizer is placed in front of the camera lens. In the photographs of Figures 2.9 and 2.10, and the photograph of a third scarab (Plusiotis clypealis) shown in Figure 2.11, the light that is polarized is light that penetrates the structure of the scarab cuticle and is reflected back out. Light that is reflected from the surface of the cuticle and is not polarized or is linearly polarized appears as a highlight or glint, and thus is still visible as photographed through the circular polarizer (the polar-izer is immediately in front of the camera lens).

Beetle measurements [18] show the following: (i) Scarab beetles generally reflect left-handed circularly polarized light and (ii) there are scarab beetles such as Plusiotis resplendens that gener-ate one circular state at one end of the visible and the orthogonal circular state at the other end of the visible.

Michelson [17] hypothesized that the source of the circular polarization resulted from a “screw structure” within the scarab cuticle, but he did not pursue a structural analysis. Caveney [19] provides perhaps the most complete investigation of the structure and chemical composi-tion of the polarizing material in the scarab cuticle. Caveney [19] used electron microscopy to examine the structure in several scarabs, and chemical analysis found that the “helicoidal struc-ture,” a term drawn from explanations of the operation of cholesteric liquid crystals, is made

No polarizer Right circular polarizer in front of camera

(a) (b)

figuRe 2.9 (See color insert following page 394.) Photographs of Plusiotis resplendens (b) with and (a) without a polarizer. A photograph of the scarab with a left circular polarizer in front of the camera would appear as in (a). (Photos courtesy of D. H. Goldstein.)

18 Polarized Light, Third Edition

up of parallel planes of the birefringent material uric acid. Whether the scarab beetles use the circularly polarized light as a recognition mechanism or to perform any other survival functions is not known.

The relative reflectance spectrum for Plusiotis clypealis is shown in Figure 2.12 and shows that the spectrum of light from this insect is without major features over the whole of the visible spec-trum. The Mueller matrix for this scarab is shown in Figure 2.13, and shows an object that is extremely unusual for an animal. The Mueller matrix for Plusiotis clypealis shows a spectacular resemblance to a textbook example of a matrix for a near wavelength-independent homogeneous left circular polarizer; that is,

1 0 0 1

0 0 0 0

0 0 0 0

1 0 0 1

−

−

(2.13)

The circular diattenuation can be read from the Mueller matrix element m03 in Figure 2.13 and appears to average approximately 0.75 over the measured spectral range.

The results of measurements for the two scarabs in Figures 2.9 and 2.10 are shown in Figures 2.14 through 2.17. Figure 2.14 shows the relative spectral reflectance of Plusiotis gloriosa while Figure 2.16 shows the relative spectral reflectance for Plusiotis resplendens. (The large spike in each of these plots at 0.6328 μm is due to the helium–neon laser in the spectrometer.) From Figure 2.14, it is evident that most of the reflected light from Plusiotis gloriosa is in the green to yellow spectrum

No polarizer

(a) (b)

Right circular polarizer in front of camera.

figuRe 2.10 (See color insert following page 394.) Photographs of Plusiotis gloriosa (b) with and (a) without a polarizer. A photograph of the scarab with a left circular polarizer in front of the camera would appear as in (a). (Photos courtesy of D. H. Goldstein.)

Polarization in the Natural Environment 19

with two peaks at approximately 0.53 and 0.58 μm. Plusiotis resplendens has a more uniform reflec-tance distribution with most of the energy at the yellow to red end of the spectrum. The Mueller matrix for Plusiotis gloriosa has slightly more spectral variation than that for Plusiotis clypealis, and it is noisy at the blue end of the spectrum due to low reflectivity, but it is still clearly a left cir-cular polarizer in the visible.

Plusiotis clypealis

0

10

20

30

40

50

60

0.4 0.45 0.5 0.55 0.6 0.65 0.7Wavelength (micrometers)

Rela

tive r

efle

ctan

ce

figuRe 2.12 Spectral reflectance of Plusiotis clypealis.

No polarizer Right circular polarizerin front of camera

(a) (b)

figuRe 2.11 (See color insert following page 394.) Photographs of Plusiotis clypealis (b) with and (a) without a polarizer. A photograph of the scarab with a left circular polarizer in front of the camera would appear as in (a). This scarab looks as though it were made of silver. With the polarizer, much of the light is lost, although, like Plusiotis resplendens, this scarab has no black backing layer in its cuticle so it does not have the dramatic loss of color as does Plusiotis gloriosa. (Photos courtesy of D. H. Goldstein.)

20 Polarized Light, Third Edition

The behavior of the Mueller matrix for Plusiotis resplendens has additional features. The matrix is of an object that generates circularly polarized light from unpolarized light but with considerable variation from one end of the visible spectrum to the other. At the same time, at the extreme short wave end of the spectrum it has mirror-like qualities. The hand of the matrix com-ponent generating circularly polarized light from unpolarized light, m30 in Figure 2.17, actually reverses twice from 0.4 to 0.7 μm, going from left to right circular at approximately 0.49 μm,

45

35

40

30

25

20

Refle

ctan

ce

15

5

10

00.4 0.45 0.550.5 0.6 0.65 0.7

Wavelength (micrometers)

figuRe 2.14 Spectral reflectance of Plusiotis gloriosa.

1

0M00

–10.4 0.47 0.55 0.62

λ0.7

1

0M01

–10.4 0.47 0.55 0.62

λ0.7

1

0M02

–10.4 0.47 0.55 0.62

λ0.7

1

0M03

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

M10 M11 M12 M13

M20 M21 M22 M23

M30 M31 M32 M33

figuRe 2.13 Mueller matrix of Plusiotis clypealis. Wavelength λ is in micrometers.

Polarization in the Natural Environment 21

returning to left circular at approximately 0.55 μm, and finally returning to right circular at approximately 0.62 μm. (The spike in the Mueller matrix spectra at approximately 0.44 μm as well as the noisiness of these spectra at the short wavelength end of the spectra are measurement artifacts.)

The spectral Mueller matrices for these three scarab beetles show that the scarabs are predomi-nantly reflecting left circularly polarized light when unpolarized light is incident. It is also evident

1

0M00

–10.4 0.47 0.55 0.62

λ0.7

1

0M01

–10.4 0.47 0.55 0.62

λ0.7

1

0M02

–10.4 0.47 0.55 0.62

λ0.7

1

0M03

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

M10 M11 M12 M13

M20 M21 M22 M23

M30 M31 M32 M33

figuRe 2.15 Mueller matrix for Plusiotis gloriosa. Wavelength λ is in micrometers.

Refle

ctan

ce

0

2

4

6

8

10

12

14

16

0.4 0.45 0.550.5 0.6 0.65 0.7Wavelength (micrometers)

figuRe 2.16 Spectral reflectance of Plusiotis resplendens.

22 Polarized Light, Third Edition

that there are scarabs for which the hand of the circular polarization reverses from the blue end of the spectrum to the red. This behavior is highly unusual within the animal kingdom, and is particu-larly intriguing since it serves no known function.

2.3.2 Squid and cuTTlefiSh

Squid and cuttlefish have been found to have the ability to reflect polarized light under voluntary control and it is thought that this is used as a form of communication [20]. Other animals, for example, predators that may not have any polarization sensitivity, would not see these signals, and thus polarization would be a form of secure communications.

These animals have chromatophores, or pigment-containing cells, distributed all over their bod-ies. The cuttlefish in particular are known for their ability to change color in the blink of an eye in order to camouflage themselves against a background [21]. The cells responsible for this ability are also known as iridophores when there is an iridescence of the reflection. These cells, sometimes also called guanophores, are pigment cells that reflect light using plates of crystalline guanine, an organic compound that is one of the major constituents of the nucleic acids. Guanine is added to shampoos, for example, to provide iridescence.

Chiou et al. [20] examined iridophores in the arm stripes of squid and cuttlefish with an electron microscope and found that there are stacks of parallel plates in these cells that form reflecting units. Arm stripes in a species of squid, and two species of cuttlefish are shown in Figures 2.18 and 2.19, respectively. Chiou et al. hypothesize that the polarization seen in the pseudocolor images of these figures is caused by the multilayer reflectors in the iridophores.

1

0M00

–10.4 0.47 0.55 0.62

λ0.7

1

0M01

–10.4 0.47 0.55 0.62

λ0.7

0.4 0.47 0.55 0.62 0.7

0.4 0.47 0.55 0.62 0.7

0.4 0.47 0.55 0.62 0.7

1

0M02

–10.4 0.47 0.55 0.62

λ0.7

1

0M03

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–1

λ

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

M10 M11 M12 M13

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–1

λ

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

M20 M21 M22 M23

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–1

λ

1

0

–10.4 0.47 0.55 0.62

λ0.7

1

0

–10.4 0.47 0.55 0.62

λ0.7

M30 M31 M32 M33

figuRe 2.17 Mueller matrix for Plusiotis resplendens. Wavelength λ is in micrometers.

Polarization in the Natural Environment 23

2.3.3 ManTiS ShRiMP

The stomatopod crustaceans, commonly known as mantis shrimp, are known to reflect highly polar-ized light [22]. The stomatopod Odontodactylus cultrifer is shown in Figure 2.20 in six views where a linear polarizer has been rotated by 30º from one photo to the next. There is clearly a substantial change for each 90º change in the polarizer angle. There are two types of polarized light reflectors in these crustaceans named “red” and “blue” according to their visual color; however both have maximum polarization values around 500 nm, approximately the wavelength that has the highest transmission through seawater [22]. The red polarization reflectors are in the animal’s cuticle, and are thought to result from layered structures. The blue reflectors, found in smaller areas such as antennae and mouth parts, are thought to result from scattering from ovoid vesicles underneath the cuticle. Figure 2.21 show the same stomatopod with the keel from the telson (tail) in transmitted light. The keel is at the right-hand side of the photo of the whole animal. It appears to preferentially transmit one linear polarization when seen from either side, but transmits circularly polarized light of opposite handedness from opposite sides.

figuRe 2.18 (See color insert following page 394.) Sepioteuthis lessoniana, or Bigfin Reef Squid, photo-graphs in natural light (top) and a pseudocolor polarization image (bottom) with color scale to right indicat-ing degree of linear polarization from 0 to 100%. (Photographs courtesy of Tsyr-Huei Chiou, University of Queensland, Australia.)

24 Polarized Light, Third Edition

2.4 PolaRiZaTioN ViSioN iN The aNimal kiNgdom

A large number of creatures have some polarization sensitivity in their visual systems. The list includes bees, ants, scarab beetles (dung beetles), flies, crickets, butterflies, moths, locusts, cock-roaches, water dwelling insects, dragonflies, spiders, scorpions, crabs, crayfish, stomatopods (mantis shrimp), cephalopods (octopi, squid, and cuttlefish), fish (anchovies), amphibians (tiger salaman-ders), lizards (desert lizard), possibly some birds, humans, and other mammals [2].

(a) (b)

figuRe 2.19 (See color insert following page 394.) (a) Sepia plangon, or Mourning Cuttlefish, photo from the side in natural light (top) and a pseudocolor polarization image (bottom) with color scale to right indicating degree of linear polarization from 0 to 100%. (Photos courtesy of Tsyr-Huei Chiou, University of Queensland, Australia.) (b) Sepia officinalis, or Common Cuttlefish, photo from the front in natural light (top) and a pseudocolor polarization image (bottom). (Adapted from the Journal of Experimental Biologists, cover photo, Vol. 210(20), 2007. With permission from The Company of Biologists.)

Odontodactylus cultrifer malelinear polarizer 210 degrees

Odontodactylus cultrifer malelinear polarizer 240 degrees

Odontodactylus cultrifer malelinear polarizer 330 degrees

Odontodactylus cultrifer malelinear polarizer 360 degrees

Odontodactylus cultrifer malelinear polarizer 300 degrees

Odontodactylus cultrifer malelinear polarizer 270 degrees

figuRe 2.20 (See color insert following page 394.) Odontodactylus cultrifer as seen through a linear polarizer rotated through 150º. (Photos courtesy of Roy Caldwell, University of California, Berkeley.)

Polarization in the Natural Environment 25

It is notable that most of the animals on this list are arthropods. If we look at the types of light sensors in compound eyes such as those found in arthropods and those found in vertebrates, we see that the compound eye structure might lend itself to polarization sensing. If we look at the surface of the compound eye, very often we see an array of hexagonal convexities, the facets of the eye. Each of these convexities, and the optical receiving structure behind it, is called an ommatidium. In Figure 2.22 we show a notional representation of the part of the ommatidium called the rhabdom. This is essentially a light guide below the lens that houses the microvilli, structures that increase the surface area available for the rhodopsin molecules that actually perform the light absorption. The microvilli are perpendicular to the axis of the rhabdom, and are long and thin (on the order of 50 nm in diameter and 1000 nm long) so that the rhodopsin molecules tend to be aligned in the direction of the microvillus. Polarized light that is aligned with the rhodopsin molecules is preferentially absorbed and perceived by the animal.

In contrast, the vertebrate eye contains rod and cone cells, represented notionally in Figure 2.23, which are organized in disks perpendicular to the cell axis. The visual pigment molecules then can be oriented randomly within the disks, and polarized light is not perceived, although there are some exceptions that we will describe briefly.

We will briefly summarize some of what is known about animal polarization vision. Much of this material is covered in much more detail in the excellent book by Horváth and Varjú [2] and

The stomatopod crustacean odontodactylus cultrifer

The sail of O.cultrifer in transmitted light The sail of O.cultrifer in transmitted light

L_CPL L_CPL

R_CPL R_CPL

figuRe 2.21 (See color insert following page 394.) A natural-color photograph of the stomatopod crustacean Odontodactylus cultrifer showing the prominent sail-like keel on the telson (the posterior segment). The photographs in the lower panels show the keel from both the right and left sides as seen in transmitted light and photographed through linear and circular polarizers, as indicated by the double-headed arrows (electric vector orientation of linear polarization) or R-CPL and L-CPL for right and left circular polarization, respectively. Note that the keel preferentially transmits horizontally polarized light when seen from either side, but that it transmits circularly polarized light of opposite handedness on each side. (Photos courtesy of Roy Caldwell, University of California, Berkeley.)

26 Polarized Light, Third Edition

the cited references therein. Note that many of the studies that show polarization sensitivity are behavioral, with some supported by anatomical and/or electrophysiological evidence. Anatomical evidence comes from examination of the microvilli and their organization in the eye, and electro-physiological evidence is obtained through insertion of probes and measuring electrical response to optical stimulus.

Honeybees have been long known to use sky polarization as a navigation aid. Once they discover a food source, they not only find their way back to this food source using sky polarization, they communicate the location to other bees at the hive using the “waggle dance.” Electrophysiological and anatomical studies have confirmed polarization sensitivity. Note that in bees and certain other insects, there is a region of ommatidia at the top edge of the eye that is called the dorsal rim area (DRA) [23]. Very often, polarization sensitive ommatidia are located there, and this is true for the honeybees. There are some desert ants that have been shown to use sky polarization for navigation. Electrophysiological studies have established that they have polarization-sensitive ultraviolet recep-tors in their DRA. Certain crickets have been found to have polarization-sensitive blue receptors in their DRA through electrophysiological studies, and it has been suggested that this is again a

figuRe 2.22 Notional diagram of arthropod light sensor. Pigment molecules tend to be aligned in the direction of the microvilli, perpendicular to the rhabdom axis.

Polarization in the Natural Environment 27

navigational aid using sky polarization. There is evidence that butterflies may use polarization from light reflected from their wings as a mating signal [24,25]. Locusts and scarab beetles have polar-ization sensitive DRAs. In particular, dung beetles, a type of scarab, use polarization of the night sky to roll a ball of dung in a straight line away from the source [26]. When there is no moon, the route is not a straight line. Another scarab, Plusiotis gloriosa (also known as chrysina gloriosa) that was described earlier in conjunction with its ability to reflect circularly polarized light, has been shown to exhibit behavior that suggests that it is can sense circular polarization [27]. Anatomical, physiological, and behavioral evidence suggest that dragonflies have polarization sensitivity not only in their DRA but also in the ventral region of their eyes; they use this capability to recognize water surfaces. Spiders have a pair of primary eyes, and three pairs of secondary eyes. Behavioral and electroretinography studies have established that spiders have polarization sensitivity that they use for navigation through observation of the sky. Some spiders have their polarization sensitivity in their primary eyes, and some in their secondary eyes. Scorpions also have two sets of eyes, a primary dorsal pair, and three to four pairs of smaller lateral eyes. Orientational behavioral experi-ments have shown polarization sensitivity in the dorsal eyes, although the animals used in the stud-ies are nocturnal and there is some doubt that the animals perceive the night sky polarization.

Crustaceans were perhaps the first animals in which polarization sensitivity was demonstrated as early as 1940. Horseshoe crabs, other crabs, crayfish, grass shrimp, and water fleas have been shown through some combination of anatomical, behavioral, or electrophysiological studies to be polarization sensitive.

It has been noted above that stomatopods, or mantis shrimp reflect polarized light from their cuti-cles. Anatomical and physiological evidence for linear polarization sensitivity has been available for some time, but recent work has shown that these animals can distinguish handedness of circularly polarized light [28], and this is the first observation of useful circular polarization sensitivity in an

figuRe 2.23 Notional diagram of vertebrate light sensor. Pigment molecules are oriented randomly in layers perpendicular to the cell axis.

28 Polarized Light, Third Edition

animal. There is a structure in the stomatopod eye that acts as an achromatic retarder [29], enabling circular polarization sensitivity over the whole of the visual range. Good achromatic retarder optical elements are highly prized for their properties, and it is amazing to see this achieved in a structure of biological origin. There is a sexual dimorphism in the reflected polarization from these animals, so that it is postulated that it is used as a covert communication method for mating.

Cephalopods (i.e., octopi, squids, and cuttlefish) are known to have polarization sensitivity through anatomical, behavioral, and electrophysiological studies. We have seen that squid and cut-tlefish can voluntarily control reflection of polarized light from their bodies, and so one use of this capability is intraspecific communication. Squid also use polarization to pick out transparent prey (krill) against the natural background. The transparency of the krill is a camouflage method from many predators, but the squid have overcome this with polarization-sensitive vision [30]. Many fish use their silvery bodies as a camouflage mechanism. By the use of polarization vision, cuttlefish can negate this advantage [31] and detect their prey.

As we have seen, because of the structure of the light sensing elements in vertebrate eyes, polar-ization sensitivity is not expected; however, there are mechanisms that might provide some sensitiv-ity. If the disks in the rod and cone cells of the vertebrate eye are tilted or even on edge with respect to the incoming light, polarization sensitivity could be present. In addition, a type of mechanism called form birefringence has been studied theoretically for its application to the structure of the vertebrate eye [32].

Polarization sensitivity has been behaviorally demonstrated in only a couple of adult amphib-ians, tiger salamanders, and red-spotted newts. In the case of the tiger salamanders, the polarization sensitivity is not even in the eye, but in intracranial photoreceptors.

Only two lizard species have polarization sensitivity, the fringe-toed lizard (Uma notata) and the Australian sleepy lizard (Tiliqua rugosa), and the polarization sensitivity is thought to reside in the parietal, or “third” eye, which exists in most lizards. This photoreceptor acts as a light meter for thermoregulation and in maintaining the circadian rhythms of the animals. Evidence for polariza-tion sensitivity is behavioral.

There have been behavioral studies that purport to show that some birds can use polarization for orientation. Research with the homing pigeon columba livia from 1952 to 1990, both behavioral and electrophysiological, first showed polarization insensitivity, then polarization sensitivity, with the last conclusive work showing no sensitivity.

Humans, and presumably other large mammals with color vision, are able to detect polarized light. The ability has no practical use and is an artifact of the structure of the eye. A normal eye will perceive two sets of brush or fan-shaped regions, one yellow and one blue, when looking at strong, linearly polarized light coming from a bright uniform surface. These are called Haidinger brushes. If the direction of linear polarization is horizontal, the yellow regions are oriented up and down and the blue regions left and right. The author’s method of observing the brushes is to look through a linear polarizer at a brightly illuminated white diffuse surface and tilt his head from side to side. Haidinger’s brushes will tilt with the tilt of the head and evoke recognition through this motion.

RefeReNCeS

1. Gehrels, T., Ed., Planets, Stars and Nebulae Studied with Photopolarimetry, Tucson, AZ: University of Arizona Press, 1974.

2. Horváth G., and D. Varjú, Polarized Light in Animal Vision: Polarization Patterns in Nature, New York: Springer, 2004.

3. Humphreys, W. J., Physics of the Air, Mineola, NY: Dover, 1964. 4. Können, G. P., Polarized Light in Nature, Cambridge: Cambridge University Press, 1985. 5. Egan, W. G., Optical Remote Sensing: Science and Technology, New York: Marcel Dekker, 2004. 6. Schott, J. R., Fundamentals of Remote Sensing, Bellingham, WA: SPIE Press, 2009. 7. Cronin, T. W., E. J. Warrant, and B. Greiner, Celestial polarization patterns during twilight, Appl. Opt.

45 (2006): 5582–9.

Polarization in the Natural Environment 29

8. Wood, R. W., Physical Optics, Washington, DC: Optical Society of America, 1988. 9. Graham, G. R., Polarization of rainbows, Phys. Educ. 10 (1975): 50–1. 10. Cowley, L., Atmospheric Optics, http://www.atoptics.co.uk 11. West, E. A., and K. S. Balasubramaniam, Crosstalk in solar polarization measurements, Proc. SPIE 1746

(1992): 281–94. 12. November, L. J., and L. M. Wilkins, The liquid crystal polarimeter for solid-state imaging of solar vector

magnetic fields, Proc. SPIE 2265 (1994): 210–21. 13. West, E. A., and M. H. Smith, Polarization characteristics of the MSFC experimental vector magneto-

graph, Proc. SPIE 2265 (1994): 272–83. 14. Sweeney, A., C. Jiggins, and S. Johnsen, Polarized light as a butterfly mating signal, Nature 423 (2003):

31–2. 15. Stavenga, D. G., M. A. Giraldo, and H. L. Leertouwer, Butterfly wing colors: Glass scales of Graphium

sarpedon cause polarized iridescence and enhance blue/green pigment coloration of the wing membrane, J. Exp biol. 213 (2010): 1731–9.

16. Cronin, T. W., N. Shashar, R. L. Caldwell, J. Marshall, A. G. Cheroske, and T.-H. Chiou, Polarization vision and its role in biological signaling, Integr. comp. biol. 43 (2003): 549–58.

17. Michelson, A. A., On metallic colouring in birds and insects, Phil. Mag. 21 (1911): 554–67. 18. Goldstein, D., Polarization properties of Scarabaeidae, Appl. Opt. 45 (2006): 7944–50. 19. Caveney, S., Cuticle reflectivity and optical activity in scarab beetles: The role of uric acid, Proc. R. Soc.

London, Ser. b 178 (1971): 205–25. 20. Chiou, T.-H., L. M. Mäthger, R. T. Hanlon, and T. W. Cronin, Spectral and spatial properties of polarized

light reflections from the arms of squid (Loligo pealeii) and cuttlefish (Sepia officinalis L.), J. Exp. biol. 210 (2007): 3624–35.

21. See the Nova program, Kings of Camouflage, http://www.pbs.org/wgbh/nova/camo/ 22. Chiou, T.-H., T. W. Cronin, R. L. Caldwell, and J. Marshall, Biological polarized light reflectors in stom-

atopod crustaceans, Proc. SPIE 5888 (2005): 58881B-1–9. 23. Labhart, T., and E. P Meyer, Detectors for polarized skylight in insects: A survey of ommatidial special-

izations in the dorsal rim area of the compound eye, Microscopy Res. and Tech. 47 (1999): 368–79. 24. Kinoshita, M., M. Sato, and K. Arikawa, Spectral receptors of Nymphalid butterflies, Naturwissenschaften

84 (1997): 199–201. 25. Sweeney, A., C. Jiggins, and S. Johnson, Polarized light as a butterfly mating signal, Nature 423 (2003):

31–32. 26. Dacke, M., P. Nordström, and C. H. Scholtz, Twilight orientation to polarised light in the crepuscular

dung beetle Scarabaeus zambesianus. J. Exp. biol. 206 (2003): 1535–43. 27. Brady, P., and M. Cummings, Differential response to circularly polarized light by the Jewel Scarab

beetle chrysina gloriosa, Am. Nat. 175 (2010): 614–20. 28. Chiou, T.-H., S. Kleinlogel, T. Cronin, R. Caldwell, B. Loeffler, A. Siddiqi, A. Goldizen, and J. Marshall,

Circular polarization vision in a stomatopod crustacean, curr. biol. 18 (2008): 429–34. 29. Roberts, N. W., T.-H. Chiou, N. J. Marshall, and T. W. Cronin, A biological quarter-wave retarder with

excellent achromaticity in the visible wavelength region, Nature Photonics 3 (2009): 641–4. 30. Shashar, N., R. T. Hanlon, and A. deM. Petz, Polarization vision helps detect transparent prey, Nature 393

(1998): 222–3. 31. Shashar, N., R. Hagan, J. G. Boal, and R. T. Hanlon, Cuttlefish use polarization sensitivity in predation

on silvery fish, Vision Res. 40 (2000): 71–5. 32. Roberts, N. W., The optics of vertebrate photoreceptors: Anisotropy and form birefringence, Vision Res.

46 (2006): 3259–66.

31

3 Wave Equation in Classical Optics

3.1 iNTRoduCTioN

The concept of light as a wave, in particular a transverse wave, is fundamental to the phenomena of polarization and propagation. In this chapter, we will introduce the wave equation and its solutions, briefly discuss interference, and apply the wave equation solutions to the interaction of a wave with a boundary between two media.

The concept of the interference of waves, developed in mechanics in the eighteenth century, was introduced into optics by Thomas Young at the beginning of the nineteenth century. In the eighteenth century, the mathematical physicists Euler, d’Alembert, and Lagrange had developed the wave equation from Newtonian mechanics and investigated its consequences (e.g., propagating and standing waves). It is not always appreciated that Young’s “leap of genius” was to take the ideas developed in one field (i.e., mechanics) and apply them to the completely different field of optics.

In addition to borrowing the idea of wave interference, Young found that it was also necessary to use another idea from mechanics. He discovered that the superposition of waves was insufficient to describe the phenomenon of optical interference; it, alone, did not lead to the observed interference pattern. To describe the interference pattern, he also borrowed the concept of energy from mechan-ics. This concept had been developed in the eighteenth century, and the relation between the ampli-tude of a wave and its energy was clearly understood. In short, the mechanical developments of the eighteenth century were crucial to the work of Young and to the development of optics in the first half of the nineteenth century. It is difficult to imagine the rapid progress that took place in optics without these previous developments. In order to have a better understanding of the wave equation and how it arose in mechanics and was then applied to optics, we now derive the wave equation from Newton’s Laws of Motion.

3.2 The WaVe eQuaTioN

Consider a homogeneous string of length l fixed at both ends and under tension T0, as shown in Figure 3.1. The lateral displacements are assumed to be small compared with l. The angle θ between any small segment of the string and the straight line (dashed) joining the points of support are sufficiently small so that sin θ is closely approximated by tan θ. Similarly, the tension T0 in the string is assumed to be unaltered by the small lateral displacements; the motion is restricted to the x, y plane.

The differential equation of motion is obtained by considering a small element ds of the string and is shown exaggerated as the segment Ab in Figure 3.1. The y component of the force acting on ds consists of F1 and F2. If θ1 and θ2 are small, then

F T T Tyx A

1 0 1 0 1 0= = ∂∂( )sin tan ,θ θ (3.1)

F T T Tyx b

2 0 2 0 2 0= = ∂∂( )sin tan ,θ θ (3.2)

32 Polarized Light, Third Edition

where the derivatives are partials because y depends on time t as well as on the distance x. The sub-scripts signify that the derivatives are to be evaluated at points A and b. Using Taylor’s expansion theorem, we obtain the equations

∂∂( ) = ∂

∂− ∂

∂∂∂

= ∂∂

− ∂∂

yx

yx x

yx

dx yx

yx

dx

A 2 2

2

2, (3.3)

∂∂( ) = ∂

∂+ ∂

∂∂∂

= ∂∂

− ∂∂

yx

yx x

yx

dx yx

yx

dx

b 2 2

2

2, (3.4)

in which the derivatives without subscripts are evaluated at the midpoint of ds. The resultant force in the y direction is

F F Ty

xdx2 1 0

2

2− = ∂

∂( ) . (3.5)

If ρ is the mass per unit length of the string, the inertial reaction (force) of the element ds is ρds(∂2y/∂t2). For small displacements, ds can be written as ds dx . The equation of motion is then obtained by equating the inertial reaction to the applied force Equation 3.5, so we have

∂∂

= ∂∂

2

2

02

2

yt

T yxρ

. (3.6)

Equation 3.6 is the wave equation in one dimension. In optics, y(x, t) is equated with the “optical disturbance” u(x, t). Also, the ratio of the tension to the density in the string T/ρ is found to be related to the velocity of propagation v by

vT2 0=ρ

. (3.7)

The form of Equation 3.7 is easily derived by a dimensional analysis of Equation 3.6. Equation 3.6 can then be written as

∂

∂= ∂

∂

2

2 2

2

2

1u x tx v

u x tt

( , ) ( , ), (3.8)

A

B

Cdx

dy

ds

T0

T0

F11

1

2

figuRe 3.1 Derivation of the wave equation. Motion of a string under tension.

Wave Equation in Classical Optics 33

and this is the form that it appears in optics. Equation 3.8 describes the propagation of an optical disturbance u(x, t) in a direction x at a time t. For a wave propagating in three dimensions it is easy to show that the wave equation is

∂

∂+ ∂

∂+ ∂

∂= ∂2

2

2

2

2

2 2

21u r tx

u r ty

u r tz v

u r t( , ) ( , ) ( , ) ( , )∂∂t2

, (3.9)

where r = (x2 + y2 + z2)1/2. Equation 3.9 can be written as

∇ = ∂∂

22

2

2

1u r t

vu r t

t( , )

( , ), (3.10)

where ∇2 is the Laplacian operator,

∇ ≡ ∂∂

+ ∂∂

+ ∂∂

22

2

2

2

2

2x y z. (3.11)

Because of the fundamental importance of the wave equation in both mechanics and optics, it has been thoroughly investigated. Equation 3.9 shall now be solved in several ways. Each method of solu-tion yields useful insights.

3.2.1 Plane-wave SoluTion

Let r(x, y, z) be a position vector of a point Ρ in space, and s(sx, sy, sz) a unit vector in a fixed direc-tion. Any solution of Equation 3.9 of the form

u u t= ⋅( )s r, (3.12)

is said to represent a plane-wave solution, since at each instant of time u is constant over each of the planes

s r⋅ = constant. (3.13)

Equation 3.13 is the vector equation of a plane; a further discussion of plane waves and Equation 3.13 will be given later.

Figure 3.2 shows a Cartesian coordinate system Ox, Oy, Oz. We now choose a new set of Cartesian axes, Οξ, Οζ, Οη, with Οζ in the direction s ⋅ r = ζ. Then ∂/∂x = (∂ζ/∂x) ⋅ ∂/∂ζ, for example, so

s x s y s zx y z+ + = ζ (3.14)

and we can write

∂∂

= ∂∂

∂∂

= ∂∂

∂∂

= ∂∂x

sy

sz

sx y zζ ζ ζ. (3.15)

Since s s sx y z2 2 2 1+ + = , we find that

∇ = ∂∂

22

2u

uζ

, (3.16)

34 Polarized Light, Third Edition

so that Equation 3.10 becomes

∂∂

− ∂∂

=2

2 2

2

2

10

uv

utζ

. (3.17)

Thus, the transformation Equation 3.14 and Equation 3.15 reduces the three-dimensional wave equation to a one-dimensional wave equation. Next, we set

ζ ζ− = + =vt p vt q, (3.18)

and substitute Equation 3.18 into Equation 3.17 to find

∂

∂ ∂=

2

0u

p q. (3.19)

The solution of Equation 3.19 is

u u p u q= ( ) + ( )1 2 , (3.20)

as a simple differentiation quickly shows. Thus, the general solution of Equation 3.17 is

u u vt u vt= ⋅ −( ) + ⋅ +( )1 2s r s r , (3.21)

where u1 and u2 are arbitrary functions. The argument of u is unchanged when (ζ, t) is replaced by (ζ + ντ, t + τ), where τ is an arbitrary time. Thus, u1(ζ + ντ) represents a disturbance that is propa-gated with a velocity ν in the negative ζ direction. Similarly, u2(ζ – ντ) represents a disturbance that is propagated with a velocity ν in the positive ζ direction.

3.2.2 SPheRical waveS

Next, we consider solutions representing spherical waves; that is,

u = (r, t) (3.22)

x

y

z

O

sr

P

ξ

ζ

η

figuRe 3.2 Propagation of plane waves.

Wave Equation in Classical Optics 35

where r = |r| = (x2 + y2 + z2)1/2. Using the relations

∂∂

= ∂∂

∂∂

= ∂∂x

rx r

xr

xr

, .,etc (3.23)

one finds after a straightforward calculation that

∇ = ∂∂

22

2

1u

rrur( )

. (3.24)

The wave Equation 3.10 then becomes

∂

∂− ∂

∂=

2

2 2

2

2

10

( ) ( ).

rur v

rut

(3.25)

Following Equation 3.17, the solution of Equation 3.25 is

u r tu r vt

ru r vt

r( , )

( ) ( ),= − + +1 2 (3.26)

where u1 and u2 are, again, arbitrary functions. The first term in Equation 3.26 represents a spheri-cal wave diverging from the origin, and the second term is a spherical wave converging toward the origin where the velocity of propagation is ν in both cases.

3.2.3 fouRieR TRanSfoRM MeThod

The method for solving the wave equation requires a considerable amount of insight and experience. It would be desirable to have a formal method for solving partial differential equations of this type. This can be done by the use of Fourier transforms.

Let us again consider the one-dimensional wave equation

∂

∂= ∂

∂

2

2 2

2

2

1u tv

u tt

( , ) ( , ).

ζζ

ζ (3.27)

The Fourier transform pair for u(ζ, t) is defined in the time domain to be

u t u e di t( , ) ( , )ζπ

ζ ωω=−∞

∞

∫12

ω (3.28)

and

u t e dti t( , ) ( , ) .ζ ω ζ ω=−∞

∞−∫ u (3.29)

We can then write

∂

∂= ∂

∂−∞

∞

∫2

2

2

2

12

u t u ed

i t( , ) ( , )ζζ π

ζ ωζ

ωω

(3.30)

36 Polarized Light, Third Edition

and

∂

∂= −

−∞

∞

∫2

221

2u t

tu e di t( , )

( , )( ) ,ζ

πζ ω ω ωω (3.31)

so Equation 3.27 is transformed to

∂

∂= −2

2

2

2

u uv

( , ) ( , ).

ζ ωζ

ω ζ ω (3.32)

Equation 3.32 is recognized immediately as the equation of a harmonic oscillator whose solu-tion is

u A e b eik ik( , ) ( ) ( )ζ ω ω ωζ ζ= + − (3.33)

where k = ω/ν. We note that the “constants” of integration, Α(ω) and Β(ω), must be written as func-tions of ω because the partial differentiation in Equation 3.27 is with respect to ζ. The reader can easily check that Equation 3.33 is the correct solution by differentiating it according to Equation 3.32. The solution of Equation 3.27 can then be found by substituting u(ζ,ω) in Equation 3.33 into the Fourier transform u(ζ, t) in Equation 3.28 to obtain

u t A e b e e dik ik i t( , ) [ ( ) ( ) ]ζπ

ω ω ωζ ζ ω= +−∞

∞−∫1

2 (3.34)

or

u t A e d b ei t v i t( , ) ( ) ( )( / ) (ζπ

ω ωπ

ωω ζ ω= +−∞

∞+

−∞

∞

∫ ∫12

12

−−ζ ω/ ) .v d (3.35)

From the definitions of the Fourier transform, Equations 3.28 and 3.29, we then see that

u t u tv

u tv

( , )ζ ζ ζ= +( ) + −( )1 2 (3.36)

which is equivalent to the Solution 3.21.Fourier transforms are used throughout physics and provide a powerful method for solving par-

tial differential equations. The Fourier transform pair shows that the simplest sinusoidal solution of the wave equation is

u t A t k b t k( , ) sin( ) sin( ),ζ ω ζ ω ζ= + + − (3.37)

where A and Β are constants. The reader can easily check that Equation 3.37 is the solution of the wave Equation 3.27.

3.2.4 MaTheMaTical RePReSenTaTion of The haRMonic oScillaToR equaTion

Before we end the discussion of the wave equation, it is also useful to further discuss the harmonic oscillator equation. From mechanics, the differential equation of the harmonic oscillator motion is

md xdt

kx2

2= − (3.38)

Wave Equation in Classical Optics 37

or

d xdt

km

x x2

2 02= − = −ω , (3.39)

where m is the mass of the oscillator, k is the force constant of the spring, and ω0 = 2πf is the angular frequency where f is the frequency in cycles per second. Equation 3.39 can be solved by multiplying both sides of the equation by dx/dt = ν (v = velocity) to obtain

vdvdt

xdxdt

= −ω02 (3.40)

or

vdv x dx= −ω02 . (3.41)

Integrating both sides of Equation 3.41 yields

v

x A2

02

2 2

2 2= − +ω

(3.42)

where A2 is the constant of integration. Solving for v, we have

vdxdt

A x= = −( ) ,/202 2 1 2ω (3.43)

which can be written as

dx

A xdt

( ).

/202 2 1 2−

=ω

(3.44)

The solution of Equation 3.44 is well known from integral calculus and is

x a t= +sin( ),ω δ0 (3.45)

where α and δ are constants of integration. Equation 3.45 can be rewritten in another form by using the trigonometric expansion

sin( ) sin( ) cos cos( ) sin ,ω δ ω δ ω δ0 0 0t t t+ = + (3.46)

so

x t A t b t( ) sin cos ,= +ω ω0 0 (3.47)

where

A a b a= =cos sin .δ δ (3.48)

38 Polarized Light, Third Edition

Another form for Equation 3.47 is to express cos ω0t and sin ω0t in terms of exponents; that is,

cos ,ωω ω

0

0 0

2t

e ei t i t

= + − (3.49)

sin .ωω ω

0

0 0

2t

e ei

i t i t

= − − (3.50)

Substituting Equations 3.49 and 3.50 into Equation 3.47 and grouping terms leads to

x t ce dei t i t( ) ,= + −ω ω0 0 (3.51)

where

cA ib

dA ib= − = +

2 2 (3.52)

and c and d are complex constants. We see that the solution of the harmonic oscillator can be written in terms of purely real quantities or complex quantities.

The form of Equation 3.42 is of particular interest. The differential Equation 3.38 clearly describes the amplitude motion of the harmonic oscillator. Let us retain the original form of Equation 3.38 and multiply through by dx/dt = ν , so we can write

mvdvdt

kxdvdt

= − . (3.53)

We now integrate both sides of Equation 3.53, and we are led to

mv kx

c2 2

2 2= − + , (3.54)

where c is a constant of integration. We see that by merely carrying out a formal integration, we are led to a new form for describing the motion of the harmonic oscillator. At the beginning of the eighteenth century the meaning of Equation 3.54 was not clear. Only slowly did physicists come to realize that Equation 3.54 describes the motion of the harmonic oscillator in a completely new way, i.e., the description of motion in terms of energy. The terms mv2/2 and −kx2/2 correspond to the kinetic energy and the potential energy for the harmonic oscillator, respectively. Thus, early on in the development of physics, a connection was made between the amplitude and energy for oscillatory motion. The energy of the wave could be obtained by merely squaring the amplitude. This point is introduced because of its bearing on Young’s interference experiment specifically and on optics generally. The fact that a relation exists between the amplitude of the harmonic oscillator and its energy was taken directly over from mechanics into optics and was critical for Young’s interference experiment. In optics, however, the energy would become known as the intensity.

3.2.5 noTe on The equaTion of a Plane

The equation of a plane was stated to be

s r⋅ = constant. (3.55)

Wave Equation in Classical Optics 39

We can show that Equation 3.55 does indeed describe a plane by referring to Figure 3.2. Inspecting the figure, we see that r is a vector with its origin at the origin of the coordinates, so

r i j k= + +x x z (3.56)

and i, j, and k are unit vectors. Similarly, from Figure 3.2 we see that

s i j k= + +s s sx x z . (3.57)

Suppose we now have a vector r0 along s, and the plane is perpendicular to s. Then OP is the vector r – r0 and is perpendicular to s. Hence, the equation of the plane is

s r r⋅ −( ) =0 0, (3.58)

or

s r⋅ = ζ, (3.59)

where ζ = s ⋅ r0 is a constant. Thus, the name plane-wave solutions arises from the fact that the wave front is characterized by a plane of infinite extent.

3.3 youNg’S iNTeRfeReNCe eXPeRimeNT

In the previous section, we saw that the developments in mechanics in the eighteenth century led to the mathematical formulation of the wave equation and the concept of energy. Around the year 1800, Thomas Young performed a simple, but remarkable, optical experiment known as the two-pinhole interference experiment. He showed that this experiment could be understood in terms of waves; the experiment gave the first clear-cut support for the wave theory of light. In order to understand the pattern that he observed, he adopted the ideas developed in mechanics and applied them to optics, an extremely novel and radical approach. Until the advent of Young’s work, very little progress had been made in optics since the researches of Newton (the corpuscular theory of light) and Huygens (the wave theory of light). The simple fact was that by the year 1800, aside from Snell’s Law of Refraction and the few things learned about polarization, there was no theoretical basis on which to proceed. Young’s work provided the first critical step in the development and acceptance of the wave theory of light.

The experiment carried out by Young is shown in Figure 3.3. A source of light, σ, is placed behind two pinholes s1 and s2, equidistant from σ. The pinholes then act as secondary monochro-matic sources that are in phase, and the light waves from them are superposed on the screen Σ, and observed at an arbitrary point P. Remarkably, one does not see a uniform distribution of light on the screen. Instead, a distinct pattern consisting of bright bands alternating with dark bands is observed. In order to explain this behavior, Young assumed that each of the pinholes, s1 and s2, emitted waves of the form

u u t kl1 01 1= −sin( ),ω (3.60)

u u t kl2 02 2= −sin( ),ω (3.61)

where pinholes s1 and s2 are in the source plane A, and are distances l1 and l2 from a point P(x,y) in the plane of observation Σ. The pattern is observed on the plane Oxy normal to the perpendicular

40 Polarized Light, Third Edition

bisector of s s1 2 where the x axis is parallel to s s1 2 . The separation of the pinholes is d, and a is the distance between the line joining the pinholes and the plane of observation Σ. For the point P(x, y) on the screen, Figure 3.3 shows that

l a y xd

12 2 2

2

2= + + −( ) (3.62)

l a y xd

22 2 2

2

2= + + +( ) . (3.63)

Subtracting Equation 3.62 from Equation 3.63, we find

l l xd22

12 2− = . (3.64)

Equation 3.64 can be written as

( )( ) .l l l l xd2 1 1 2 2− + = (3.65)

Now if x and y are small compared to a, then l l a1 2 2+ and

l l lxda

2 1− = =∆ . (3.66)

At this point we now return to the wave theory. The secondary sources s1 and s2 are assumed to be equal, so u01 = u02 = u0. In addition, the assumption is made that the optical disturbances u1 and u2 can be superposed at P(x, y) (the principle of coherent superposition), so

u t u u u t kl t kl( ) [sin( ) sin( )].= + = − + −1 2 0 1 2ω ω (3.67)

A serious problem now arises. While Equation 3.67 certainly describes an interference behavior, the parameter of time enters in the term ωt. In the experiment, the observed pattern does not vary over time, so the time factor cannot have any effect over the final result. This suggests that we average the amplitude u(t) over the time of observation T. The time average of u(t), written as ⟨u(t)⟩, is then defined to be

s1

s2d

A

σ

x

y

a

P(x, y)

l1

l2

Σ

figuRe 3.3 Young’s interference experiment.

Wave Equation in Classical Optics 41

u tu t dt

dt Tu t dt

T

T

TT

T

( ) lim( )

lim ( ) .= =→∞ →∞

∫∫ ∫0

0

0

1 (3.68)

Substituting Equation 3.67 into Equation 3.68 yields

u tuT

t kl t kl dtT

T

( ) lim sin( ) sin( ) .= − + −[ ]→∞ ∫0

01 2ω ω (3.69)

Using the trigonometric identity

sin( ) sin cos cos sinω ω ωt kl t kl t kl− = − (3.70)

and averaging over one cycle in Equation 3.69 yields

u t( ) .= 0 (3.71)

This is not observed. That is, the time average of the amplitude is calculated to be zero, but observa-tion shows that the pattern exhibits nonzero intensities. At this point we must abandon the idea that the interference phenomenon can be explained only in terms of amplitudes u(t). Borrowing another idea from mechanics, we will describe the optical disturbance in terms of squared quantities, analogous to energy, u2(t). But this, too, contains a time factor. A time average is introduced again, and a new quantity, I, in optics called the intensity, is defined as

I u tT

u t dtT

T

= =→∞ ∫2

0

21( ) lim ( ) . (3.72)

Substituting u2(t) = (u0sin(ωt – kl))2 into Equation 3.72 and averaging over one cycle yields

I u tT

u t kl dtu

IT

T

= ( ) = −( ) = =→∞ ∫2

02 2 0

2

00

12

lim sin .ω (3.73)

This is a statement that the intensity is constant over time, and this is the behavior that is observed.The time average of u2(t) is now applied to the superposed amplitudes Equation 3.67. Squaring

u(t) yields

u t u t kl t kl t kl202 2

12

2 12( ) = −( ) + −( ) + −( )sin sin sinω ω ω ssin .ωt kl−( )[ ]2 (3.74)

The last term is called the interference. Using the well-known trigonometric identity

2 21 2 2 1sin sin cos cosω ω ωt kl t kl k l l t k l−( ) −( ) = −[ ]( ) − − 22 1+[ ]( )l , (3.75)

Equation 3.74 can be written as

u t u t kl t kl

k l l

202 2

12

2

2 1

( ) = −( ) + −( )

+ −[ ]

[sin sin

cos

ω ω

(( ) − − +[ ]( )cos ].2 2 1ωt k l l (3.76)

42 Polarized Light, Third Edition

Substituting Equation 3.76 into Equation 3.72, we obtain the intensity on the screen

I u t I k l l Ik l l= = + − = −

20 2 1 0

2 2 12 1 42

( ) [ cos ( )] cos( )

(3.77)

or

I Ikxd

a= 4

20

2cos , (3.78)

where we used Equation 3.66 for l2 –l1.Equation 3.78 is Young’s famous interference formula. We note that from Equation 3.73 we

would expect the intensity from a single source to be u I02

02/ = , so the intensity from two indepen-dent optical sources would be 2I. Equation 3.78 shows that when the intensity is observed from interference between two sources originating from a single primary source, the observed intensity varies between 0 and 4I0; the intensity can be double or even zero from that expected from two independent optical sources! We see from Equation 3.78 that there will be maximum intensities (of 4I0) at

xa n

dn= = ± ± …λ

0 1 2, , , (3.79)

and minimum intensities (nulls) at

xad

nn= +( ) = ± ± …λ 2 1

20 1 2, , , . (3.80)

Thus, in the vicinity of O on the plane Σ, an interference pattern consisting of bright and dark bands is aligned parallel to the OY axis (at right angles to the line s s1 2 joining the two sources).

Young’s experiment is of great importance because it was the first step in establishing the wave theory of light and was the first theory to provide an explanation of the observed interference pat-tern. It also provides a method, albeit one of low precision, of measuring the wavelength of light by measuring d, a, and the fringe spacing according to Equation 3.79 or Equation 3.80. The separation Δx between the central bright line and the first bright line is, from Equation 3.79,

∆x x xad

= − =1 0λ

. (3.81)

The expected separation on the observing screen can be found by assuming the values a = 100 cm, d = 0.1 cm, λ = 5 × 10–5 cm, and Δx = 0.05 cm = 0.5 mm. The resolution of the human eye at a distance of 25 cm is of the same order of magnitude, so the fringes can be observed with the naked eye.

Young’s interference experiment gave the first real support for the wave theory; however, aside from the important optical concepts introduced here to explain the interference pattern, there is another reason for discussing Young’s interference experiment. Around 1818, Fresnel and Arago repeated his experiments with polarized light to determine the effects, if any, on the interference phenomenon. The results were surprising to understand in their entirety. To explain these experi-ments it was necessary to understand the nature and properties of polarized light. Before we turn to the subject of polarized light, however, we discuss another topic of importance; namely, the reflec-tion and transmission of a wave at an interface separating two different media.

Wave Equation in Classical Optics 43

3.4 RefleCTioN aNd TRaNSmiSSioN of a WaVe aT aN iNTeRfaCe

The wave theory and the wave equation allow us to treat the reflection and transmission of a wave at an interface between two different media. Light is found to be partially reflected and partially transmitted at the boundary of two media characterized by different refractive indices. The treat-ment of this problem was first carried out in mechanics, and shows how the science of mechanics paved the way for the introduction of the wave equation into optics.

Two media can be characterized by their ability to support two different velocities v1 and v2. In Figure 3.4 we show an incident wave coming from the left, which is partially transmitted and reflected at the interface. We saw earlier that the solution of the wave equation in complex form is

u x Ae beikx ikx( ) ,= +− + (3.82)

where k = ω/v. The time factor exp(iωt) has been suppressed. The term Ae−ikx describes propagation to the right, and the term be+ikx describes propagation to the left. The fields to the left and right of the interface can be described by a superposition of waves propagating to the right and left; that is,

u x Ae be xik x ik x1

1 1 0( ) ,= + <− + (3.83)

u x ce de xik x ik x2

2 2 0( ) ,= + >− + (3.84)

where k1 = ω/v1 and k2 = ω/v2.We must now evaluate A, b, c, and d. To do this, we assume that at the interface the fields are

continuous; that is,

u x u xx x1 0 2 0( ) ( ) ,= == (3.85)

and that the slopes of u1(x) and u2(x), that is, the derivatives of u1(x) and u2(x), are continuous at the interface so that

∂

∂= ∂

∂= =

u xx

u xxx x

1

0

2

0

( ) ( ). (3.86)

We also assume that there is no source of waves in the medium to the right of the interface (i.e., d = 0). This means that the wave that propagates to the left on the left side of the interface is due only to reflection of the incident wave.

x

x = 0

x < 0 x > 0

k1 , v1 k2 , v2

figuRe 3.4 Reflection and transmission of a wave at the interface between two media.

44 Polarized Light, Third Edition

With d = 0, and applying the boundary conditions in Equations 3.85 and 3.86 to Equations 3.83 and 3.84, we easily find that

A b c+ = (3.87)

k A k b k c1 1 2− = . (3.88)

We solve for b and c in terms of the amplitude of the incident wave, A, and find

bk kk k

A= −+( )1 2

1 2

(3.89)

ck

k kA=

+( )2 1

1 2

. (3.90)

The value of b is associated with the reflected wave in Equation 3.83. If k1 = k2 (i.e., the two media are the same) then Equations 3.89 and 3.90 show that b = 0 and c = A; that is, there is no reflected wave and we have complete transmission as expected.

We can write Equation 3.83 as the sum of an incident wave ui(x) and a reflected wave ur(x) so that

u x u x u xi r1( ) = ( ) + ( ), (3.91)

and we can write Equation 3.84 as a transmitted wave

u x u xt2 ( ) = ( ). (3.92)

The energies corresponding to ui(x), ur(x), and ut(x), are then the squares of these quantities. We can use complex quantities to bypass the formal time-averaging procedure and define the energies of these waves to be

εi i iu x u x= ( ) ( ),* (3.93)

εr r ru x u x= ( ) ( ),* (3.94)

εt t tu x u x= ( ) ( ).* (3.95)

The principle of conservation of energy requires that

ε ε εi r t= + . (3.96)

The fields ui(x), ur(x), and ut(x) from Equations 3.83 and 3.84 are

u x Aeiik x( ) ,= − 1 (3.97)

u x berik x( ) = + 1 (3.98)

u x cetik x( ) .= − 2 (3.99)

Wave Equation in Classical Optics 45

The energies corresponding to Equations 3.97 through 3.99 are then substituted in Equation 3.96, and we find

A b c2 2 2= + (3.100)

or

bA

cA( ) + ( ) =

2 2

1. (3.101)

The quantities (b/A)2 and (c/A)2 are the normalized reflection and transmission coefficients, which we write as R and T, respectively. Thus, Equation 3.101 becomes

R T+ = 1, (3.102)

where

Rk kk k

= −+( )1 2

1 2

2

(3.103)

Tk

k k=

+( )2 1

1 2

2

(3.104)

from Equations 3.89 and 3.90. Equations 3.103 and 3.104 can be seen to satisfy the conservation condition Equation 3.102.

The coefficients b and c show an interesting behavior. From Equations 3.89 and 3.90 we write

bA

k kk k

= −+

11

2 1

2 1

//

(3.105)

cA k k

=+

21 2 1/

, (3.106)

where

kk

vv

vv

2

1

2

1

1

2

= =ωω

//

. (3.107)

If v2 = 0 (i.e., there is no propagation in the second medium) Equation 3.107 becomes

lim .v

kk

vv2 0

2

1

1

2→= = ∞ (3.108)

With this limiting value in Equation 3.108, we see that Equations 3.105 and 3.106 become

bA

ei= − =1 π (3.109)

cA

= 0. (3.110)

46 Polarized Light, Third Edition

Equation 3.109 shows that there is a 180° phase reversal upon total reflection, thus the reflected wave is completely out of phase with the incident wave and we have total cancellation. This behavior is described by the term standing wave. We now derive the equation that specifically shows that the resultant wave does not propagate.

The field to the left of the interface is given by Equation 3.83 and is

u x t e Ae be xi t ik x ik x1

1 1 0( , ) ,= +( ) <−ω (3.111)

where we have reintroduced the suppressed time factor exp(iωt). From Equation 3.109 we can then write

u x t Ae e e

Ae Ae

i t ik x ik x

i t k x i t

11 1

1

( , ) ( )

( ) (

= −

= −

−

− +

ω

ω ω kk x

u x t u x t

1 )

( , ) ( , ),= −− +

(3.112)

where

u x t Aei t k x−

−=( , ) ( )ω 1 (3.113)

u x t Aei t k x+

+=( , ) .( )ω 1 (3.114)

The phase velocity vp of a wave can be defined in terms of amplitude as

vu tu x

p = − ∂ ∂( )∂ ∂( )

//

. (3.115)

Inserting Equations 3.113 and 3.114 into this definition, we find that

vk

p( )− = ω1

(3.116)

vkp( ) ,+ = −ω

1

(3.117)

so the total velocity of the wave is

v v vp p= − + + =( ) ( ) .0 (3.118)

Thus, the resultant velocity of the wave is zero according to Equation 3.118; that is, the wave does not propagate and it appears to be standing in place. The equation for the standing wave is given by Equation 3.112, which can be written as

u x t Ae k xi t1 12 sin, .( ) = ( )ω (3.119)

Taking the real part of Equation 3.119, we have

u x t A t kx, ,( ) = ( ) ( )2 cos sinω (3.120)

Wave Equation in Classical Optics 47

where we have dropped the subscript 1. We see that there is no propagator ωt – kx, so Equation 3.120 does not describe propagation.

We see that the wave equation and wave theory lead to a correct description of the transmission and reflection of a wave at a boundary. While this behavior was first studied in mechanics in the eighteenth century, it was applied with equal success to optics in the following century. It appears that this was first done by Fresnel, who derived the equations for reflection and transmission at an interface between two media characterized by refractive indices n1 and n2. Fresnel’s equations are derived in Chapter 7.

49

4 The Polarization Ellipse

4.1 iNTRoduCTioN

Christian Huygens was the first to suggest that light was not a scalar quantity, based on his work on the propagation of light through crystals; it appeared that light had “sides” in the words of Newton. This vectorial nature of light is called polarization. If we follow mechanics and equate an optical medium to an isotropic elastic medium, it should be capable of supporting three independent oscil-lations (optical disturbances): ux(r, t), uy(r, t), and uz(r, t). Correspondingly, three independent wave equations are then required to describe the propagation of the optical disturbance,

∇ =∂

∂=2

2

2

2

1u r t

v

u r t

ti x y zi

i( , )( , )

, , , (4.1)

where ν is the velocity of propagation of the oscillation and r = r(x, y, z). In a Cartesian system, the components ux(r, t) and uy(r, t) are said to be the transverse components, and the component uz(r, t) is said to be the longitudinal component when the propagation is in the z direction. According to Equation 4.1 the optical field components should be

u t u tx x xr k r, ( · ),( ) = − +0 cos ω δ (4.2)

u t u ty y yr k r, ( · ),( ) = − +0 cos ω δ (4.3)

u t u tz z zr k r, ( · ).( ) = − +0 cos ω δ (4.4)

In 1818, Fresnel and Arago carried out a series of fundamental investigations on Young’s interfer-ence experiment using polarized light. After a considerable amount of experimentation, they were forced to conclude that the longitudinal component Equation 4.4 did not exist. That is, light con-sisted only of the transverse components Equations 4.2 and 4.3. If we take the direction of propaga-tion to be in the z direction, then the optical field in free space must be described only by

u z t u t kzx x x, cos( ),( ) = − +0 ω δ (4.5)

u z t u t kzy y y, ( ),( ) = − +0 cos ω δ (4.6)

where u0x and u0y are the maximum amplitudes and δx and δy are arbitrary phases. There is no rea-son, a priori, for the existence of only transverse components on the basis of an elastic medium (the “luminiferous aether” in optics). It was considered to be a defect in Fresnel’s theory. Nevertheless, Equations 4.5 and 4.6 were found to describe satisfactorily the phenomenon of interference using polarized light.

The “defect” in Fresnel’s theory was overcome by the development of a new theory, Maxwell’s electrodynamic theory and his resulting equations. One of the immediate results of solving his equa-tions was that in free space only transverse components arose; there was no longitudinal component. This was one of the first triumphs of Maxwell’s theory. Even so, Maxwell’s theory took nearly 40 years to be accepted in optics due in large part to the fact that, up to the end of the nineteenth century, it led to practically nothing that could not be explained or understood by Fresnel’s theory.

50 Polarized Light, Third Edition

Equations 4.5 and 4.6 are spoken of as the polarized or polarization components of the optical field. In this chapter, we consider the consequences of these equations. The results are very interest-ing and lead to a surprising number of revelations about the nature of light.

4.2 The iNSTaNTaNeouS oPTiCal field aNd The PolaRiZaTioN elliPSe

We have pointed out that the experiments of Fresnel and Arago led to the discovery that light con-sisted of only two transverse components. The components are perpendicular to each other and could be chosen for convenience to propagate in the z direction. The waves are said to be “ instantaneous” in the sense that the time duration for the wave to go through one complete cycle is only 10−15 sec at optical frequencies. In this chapter, we find the equation that arises when the propagator is elimi-nated between the transverse components. In order to do this, we show in Figure 4.1 the transverse optical field propagating in the z direction. The transverse components are represented by

E z t Ex x x, ( ),( ) = +0 cos τ δ (4.7)

E z t Ey y y, ( ),( ) = +0 cos τ δ (4.8)

where τ = ωt – kz is called the propagator. The subscripts x and y refer to the components in the x and y directions, E0x and E0y are the maximum amplitudes, and δx and δy are the phases. As the field propagates, Ex(z, t) and Ey(z, t) give rise to a resultant vector. This vector describes a locus of points in space, and the curve generated by those points will now be derived. In order to do this, Equations 4.7 and 4.8 are written as

EE

x

xx x

0

= −cos cos sin sin ,τ δ τ δ (4.9)

E

Ey

yy y

0

= −cos cos sin sin .τ δ τ δ (4.10)

z

y

x

Ex

Ey

figuRe 4.1 Propagation of the transverse optical field.

The Polarization Ellipse 51

Hence,

EE

E

Ex

xy

y

yx y x

0 0

sin sin cos sin( ),δ δ τ δ δ− = − (4.11)

EE

E

Ex

xy

y

yx y x

0 0

cos cos sin sin( ).δ δ τ δ δ− = − (4.12)

Squaring Equations 4.11 and 4.12 and adding gives

EE

E

EEE

E

Ex

x

y

y

x

x

y

y

2

02

2

02

0 0

22+ − =cos sin ,δ δ (4.13)

where

δ δ δ= −y x . (4.14)

Equation 4.13 is recognized as the equation of an ellipse and shows that, at any instant of time, the locus of points described by the optical field as it propagates is an ellipse. This behavior is spoken of as optical polarization, and Equation 4.13 is called the polarization ellipse. The ellipse is shown inscribed within a rectangle in Figure 4.2. The sides of the rectangle are parallel to the coordinate axes and their lengths are 2E0x and 2E0y.

We now determine the points where the ellipse is tangent to the sides of the rectangle. We can write Equation 4.13 as

E E E E E E E E Ex y x y x y y x x02 2

0 0 02 2

02 22 0− ( ) + −( ) =cos sinδ δ .. (4.15)

The solution of this quadratic equation is

EE E

E

E

EE Ey

y x

x

y

xx x= ± −( )0

0

0

002 2 1 2cos sin

./δ δ (4.16)

O

y

x

2E0x

2E0y

x´y´ A

B

C

D

figuRe 4.2 An elliptically polarized wave and the polarization ellipse.

52 Polarized Light, Third Edition

The slope of the graph of the ellipse is zero where the ellipse is tangent to the rectangle. We now differentiate Equation 4.16, set E′y = dEy/dEx = 0, and find that

E Ex x= ± 0 cos .δ (4.17)

Substituting Equation 4.17 into Equation 4.16, the corresponding values of Ey are found to be

E Ey y= ± 0 . (4.18)

Similarly, by considering Equation 4.16 where the slope is ′ = ∞Ey on the sides of the rectangle, the tangent points are

E Ex x= ± 0 (4.19)

E Ey y= ± 0 cosδ. (4.20)

Equations 4.18 and 4.19 show that the sides of the ellipse extend to Ex = ±E0x and Ey = ±E0y. In Figure 4.2, the ellipse is shown touching the rectangle at point A, b, c, and d, the coordinates of which are A : (E0x cos δ, E0y), b : (E0x , E0y cos δ), c : (–E0x cos δ, –E0y), and d : (–E0x , –E0y cos δ).

The presence of the cross term in Equation 4.13 shows that the polarization ellipse is, in general, rotated, and this behavior is shown in Figure 4.2 where the ellipse is shown rotated through an angle ψ.

It is also of interest to determine the maximum and minimum areas of the polarization ellipse, which can be inscribed within the rectangle. We see that along the x axis, the ellipse is tangent at x = −E0x and x = +E0x. The area of the ellipse above the x axis is given by

A E dxyE

E

x

x

=−

+

∫0

0

. (4.21)

Substituting Equation 4.16 into Equation 4.21 and evaluating the integrals, we find that the area of the polarization ellipse is

A E Ex y= π δ0 0 sin . (4.22)

The area of the polarization ellipse depends on the lengths of the major and minor axes, E0x and E0y, and the phase shift δ between the orthogonal transverse components. We see that for δ = π/2 the area is πE0x E0y, whereas for δ = 0 the area is zero. The significance of these results will soon become apparent.

In general, completely polarized light is elliptically polarized. However, there are certain degen-erate forms of the polarization ellipse that are continually encountered in the study of polarized light. Because of the importance of these special degenerate forms, we now discuss them as special cases in the following section. These are the cases where either E0x or E0y is zero or E0x and E0y are equal and/or where δ = 0, π/2, or π radians.

4.3 SPeCialiZed (degeNeRaTe) foRmS of The PolaRiZaTioN elliPSe

The polarization ellipse Equation 4.13 degenerates to special forms for certain values of E0x, E0y, and δ. We now consider these special forms.

1. E0y = 0. In this case Ey(z, t) is zero and we have for the transverse components

Ex(z, t) = E0x cos(τ + δx), (4.23)

The Polarization Ellipse 53

Ey(z, t) = 0. (4.24)

In this case there is an oscillation only in the x direction. The light is then said to be linearly polarized in the x direction, and we call this linear horizontally polarized light. Similarly, if E0x = 0 and Ey (z, t) ≠ 0, then we have a linear oscillation along the y axis, and we call this linear vertically polarized light.

2. δ = 0 or π. Equation 4.13 reduces to

EE

E

EEE

E

Ex

x

y

y

x

x

y

y

2

02

2

02

0 0

2 0+ ± = . (4.25)

Equation 4.25 can be written as

EE

E

Ex

x

y

y0 0

2

0±

= (4.26)

and so

EE

EEy

y

xx= ±( )0

0

. (4.27)

Equation 4.27 is recognized as the equation of a straight line with slope ±(E0y/E0x) and zero intercept. Thus, we can say that we have linearly polarized light with slope ±(E0y/E0x). The value δ = 0 yields a negative slope, and the value δ = π yields a positive slope. If E0y = E0x, then we see that

E Ey x= ± . (4.28)

The positive value is said to represent linear + 45° polarized light, and the negative value is said to represent linear −45° polarized light.

3. δ = π/2 or 3π/2. The polarization ellipse reduces to

EE

E

Ex

x

y

y

2

02

2

02

1+ = . (4.29)

This is the standard equation of an ellipse. Note that we cannot tell from this resultant equation whether δ = π/2 or 3π/2.

4. E0x = E0y = E0 and δ = π/2 or δ = 3π/2. The polarization ellipse now reduces to

EE

E

Ex y2

02

2

02

1+ = (4.30)

Equation 4.30 describes the equation of a circle. In this case, the light is said to be right or left circularly polarized for δ = π/2 or 3π/2, respectively. Note that we cannot tell from Equation 4.30 whether the value of δ is π/2 or 3π/2.

Recall that in the previous section, we showed that the area of the polarization ellipse was

A E Ex y= π δ0 0 sin . (4.31)

54 Polarized Light, Third Edition

We see that for δ = 0 or π, the area of the polarization ellipse is zero, which is to be expected for linearly polarized light. For δ = π/2 or 3π/2, the area of the ellipse is a maximum (i.e., πE0xE0y). It is important to note that even if the phase shift between the orthogonal components is δ = π/2 or 3π/2, the light is, in general, elliptically polarized. Furthermore, the polarization ellipse shows that it is in the standard form as given by Equation 4.29.

For the more restrictive condition where the orthogonal amplitudes are equal so that E0x = E0y = E0 and δ = π/2 or 3π/2, Equation 4.31 becomes

A E= π 02, (4.32)

which is, of course, the area of a circle.The previous special forms of the polarization ellipse are referred to as degenerate states. We can

summarize these results by saying that the degenerate states of the polarization ellipse are (1) linear horizontally or vertically polarized light, (2) linear +45° or −45° polarized light, and (3) right or left circularly polarized light. Aside from the fact that these degenerate states appear quite naturally as special cases of the polarization ellipse, there is a fundamental reason for their importance; they are relatively easy to create in an optical laboratory and can be used to create “null-intensity” conditions. Polarization instruments, which may be based on null-intensity conditions, enable very accurate measurements to be made.

4.4 elliPTiCal PaRameTeRS of The PolaRiZaTioN elliPSe

As we have seen, the polarization ellipse has the form

EE

E

EEE

E

Ex

x

y

y

x

x

y

y

2

02

2

02

0 0

22+ − =cos sin ,δ δ (4.33)

where δ = δy – δx. In general, the axes of the ellipse are not coincident with the coordinate axes. In Equation 4.33, the presence of the cross term ExEy shows that it is actually a rotated ellipse. The standard form of an ellipse does not contain the cross term. In this section, we find the mathematical relations between the parameters of the polarization ellipse, E0x, E0y, and δ, and the angle of rotation ψ and another important parameter, χ, the ellipticity angle.

The rotated ellipse is illustrated in Figure 4.3. Let x and y be the initial, unrotated axes, and let x′ and y′ be a new set of axes along the rotated ellipse. Furthermore, let ψ(0 ≤ ψ ≤ π) be the angle between x and the direction x′ of the major axis. The components ′Ex and ′Ey are

′ = +E E Ex x ycos sin ,ψ ψ (4.34)

′ = − +E E Ey x ysin cos .ψ ψ (4.35)

If 2a and 2b (a ≥ b) are the lengths of the major and minor axes, respectively, then the equation of the ellipse in terms of x′ and y′ can be written as

′ = + ′E ax cos( ),τ δ (4.36)

′ = ± + ′E by sin( ),τ δ (4.37)

where τ is the propagator and δ′ is an arbitrary phase. The ± sign describes the two possible senses in which the endpoint of the field vector can describe the ellipse.

The Polarization Ellipse 55

The forms of Equations 4.36 and 4.37 are chosen because it is easy to see that they lead to the standard form of the ellipse

′ + ′ =E

a

E

bx y2

2

2

21. (4.38)

We can relate a and b in Equations 4.36 and 4.37 to the parameters E0x and E0y in Equation 4.33 by recalling that the original equations for the optical field are

EE

x

xx

0

= +( )cos τ δ (4.39)

E

Ey

yy

0

= +( )cos .τ δ (4.40)

We then substitute Equations 4.36, 4.37, 4.39, and 4.40 into Equations 4.34 and 4.35, expand the terms, and obtain

a E x x x(cos cos sin sin ) (cos cos sin sin )τ δ τ δ τ δ τ δ′ − ′ = −0 ccos

(cos cos sin sin )sin ,

ψ

τ δ τ δ ψ+ −E y y y0

(4.41)

± ′ + ′ = − −b E x x(sin cos cos sin ) (cos cos sin sinτ δ τ δ τ δ τ δ0 xx

y y yE

)sin

(cos cos sin sin )cos .

ψ

τ δ τ δ ψ+ −0

(4.42)

Equating the coefficients of cos τ and sin τ leads to the equations

a E Ex x y ycos cos cos cos sin ,δ δ ψ δ ψ′ = +0 0 (4.43)

a E Ex x y ysin sin cos sin sinδ δ ψ δ ψ′ = +0 0 , (4.44)

± ′ = −b E Ex x y ycos sin sin sin cosδ δ ψ δ ψ0 0 , (4.45)

O

Ψ

y

x

x´y´

a

b

figuRe 4.3 The rotated polarization ellipse.

56 Polarized Light, Third Edition

± ′ = −b E Ex x y ysin cos sin cos cosδ δ ψ δ ψ0 0 . (4.46)

Squaring and adding Equations 4.43 and 4.44 and using δ = δy – δx, we find that

a E E E Ex y x y2

02 2

02 2

0 02= + +cos sin cos sin cos .ψ ψ ψ ψ δ (4.47)

Similarly, from Equations 4.45 and 4.46 we find that

b E E E Ex y x y2

02 2

02 2

0 02= + −sin cos cos sin cos .ψ ψ ψ ψ δ (4.48)

Hence,

a b E Ex y2 2

02

02+ = + . (4.49)

Next, we multiply Equation 4.43 by Equation 4.45, and Equation 4.44 by Equation 4.46, and add. This gives

± =ab E Ex y0 0 sinδ. (4.50)

Further, dividing Equation 4.46 by Equation 4.43 and Equation 4.45 by Equation 4.44 leads to

( )sin cos cosE E E Ex y x y02

02

0 02 2 2− =ψ δ ψ (4.51)

or

tancos

,22 0 0

02

02

ψ δ=−

E E

E Ex y

x y

(4.52)

which relates the angle of rotation ψ to E0x, E0y, and δ. We note that, for nonzero values of E0x and E0y, ψ is equal to zero only for phases of δ = 90° or 270°. Similarly, for nonzero values of δ, ψ is equal to zero only if E0x or E0y is equal to zero.

An alternative method for determining ψ is to transform Equation 4.33 directly to Equation 4.38. To show this we write Equations 4.34 and 4.35 as

E E Ex x y= ′ − ′cos sin ,ψ ψ (4.53)

E E Ey x y= ′ + ′sin cos .ψ ψ (4.54)

Equations 4.53 and 4.54 can be obtained from Equations 4.34 and 4.35 by solving for Ex and Ey or, equivalently, replacing ψ by − ψ, Ex by ′Ex, and Ey by ′Ey. On substituting Equations 4.53 and 4.54 into Equation 4.33, the cross term is seen to vanish only for the condition given by Equation 4.52.

It is useful to introduce an auxiliary angle α(0 ≤ α ≤ π/2)for the polarization ellipse defined by

tan .α = E

Ey

x

0

0

(4.55)

Equation 4.52 reduces to, using Equations 4.53 and 4.54,

tan costantan

cos ,22 2

10 0

02

02 2

ψ δ αα

δ= =−

E E

E Ex y

x y

(4.56)

The Polarization Ellipse 57

which then yields

tan 2 tan 2 cosψ α δ= ( ) . (4.57)

We see that for δ = 0 or π, the angle of rotation is

ψ α= ± . (4.58)

For δ = π/2 or 3π/2, we have ψ = 0, so the angle of rotation is also zero.Another important parameter of interest is the angle of ellipticity, χ. This is defined by

tan .χ π χ π= ± − ≤ ≤ba 4 4

(4.59)

We see that for linearly polarized light, b = 0, so χ = 0. Similarly, for circularly polarized light, b = a, so χ = ±π/4. Thus, Equation 4.59 describes the extremes of the ellipticity of the polarization ellipse.

Using Equations 4.49, 4.50, and 4.55, we find that

±

+=

+=2 2

22 2

0 0

02

02

aba b

E E

E Ex y

x y

sin (sin )sin .δ α δ (4.60)

Next, using Equation 4.59 we see that the left-hand side of Equation 4.60 reduces to sin 2χ, so we can write Equation 4.60 as

sin (sin ) sin ,2 2χ α δ= (4.61)

which is the relation between the ellipticity of the polarization ellipse and the parameters E0x, E0y, and δ of the polarization ellipse. We note that only for δ = π/2 or 3π/2 does Equation 4.61 reduce to

χ α= ± , (4.62)

which is to be expected.The results that we have obtained here will be used again, so it is useful to summarize them. The

elliptical parameters E0x, E0y, and δ of the polarization ellipse are related to the orientation angle ψ and ellipticity angle χ by the equations

tan (tan )cos ,2 2 0ψ α δ ψ π= ≤ ≤ (4.63)

sin (sin )sin ,2 24 4

χ α δ π χ π= − ≤ ≤ (4.64)

where 0 ≤ a ≤ π/2 and

a b E Ex y2 2

02

02+ = + , (4.65)

tan ,α =E

Ey

x

0

0

(4.66)

tan .χ = ±ba

(4.67)

We emphasize that the polarization ellipse can be described either in terms of the orientation and ellipticity angles ψ and χ on the left-hand sides of Equations 4.63 and 4.64 or the major and minor axes E0x, E0y, and δ on the right-hand sides of Equations 4.63 and 4.64.

58 Polarized Light, Third Edition

Finally, a few words must be said on the terminology of polarization. Two cases of polarization are distinguished according to the sense in which the endpoint of the field vector describes the ellipse. The convention is explained in detail in Clarke [1,2] and Shurcliff [3]. Right-handedness is associated with a positive sign and a right-hand helix, where the helix represents the tip of the elec-tric vector in space. For an observer looking at an oncoming optical beam, right circular polarization is that polarization such that the tip of the electric vector describes a circle in the clockwise sense. This is Shurcliff’s sectional pattern, and is an end view of the snapshot picture. The snapshot picture requires consideration of the right-hand helix, which has a handedness independent of the observer’s viewpoint. Visualize this nonrotating helix passing through any plane perpendicular to the beam path. The point at which the helix pierces this plane, as perceived by the observer, advances in a clockwise fashion as the helix advances with time along the beam path without rotation. As Clarke states, it is erroneous to consider that the helix screws its way through space. In the case of left cir-cular polarization, the tip of the electric vector describes a circle in the counterclockwise sense, and is associated with a left-hand helix piercing a plane. The helix of Figure 1.3 is a right hand helix.

RefeReNCeS

1. Clarke, D., Nomenclature of polarized light: Linear polarization, Appl. Opt. 13 (1974): 3–5. 2. Clarke, D., Nomenclature of polarized light: Elliptical polarization, Appl. Opt. 13 (1974): 222–4. 3. Shurcliff, W. A., Polarized Light, Cambridge, MA: Harvard University Press, 1962.

59

5 Stokes Polarization Parameters

5.1 iNTRoduCTioN

In Chapter 4, we saw that the elimination of the propagator between the transverse components of the optical field led to the polarization ellipse. Analysis of the ellipse showed that, for special cases, it led to forms that can be interpreted as linearly polarized light and circularly polarized light. This description of light in terms of the polarization ellipse is very useful because it enables us to describe by means of a single equation various states of polarized light. However, this representa-tion is inadequate for several reasons. As the beam of light propagates through space, we find that in a plane transverse to the direction of propagation the light vector traces out an ellipse or some special form of an ellipse, such as a circle or a straight line in a time interval of the order 10−15 sec. This period of time is clearly too short to allow us to follow the tracing of the ellipse. This fact, therefore, immediately prevents us from ever observing the polarization ellipse. Another limitation is that the polarization ellipse is only applicable to describing light that is completely polarized. It cannot be used to describe either unpolarized light or partially polarized light. This is a particularly serious limitation because light is usually only partially polarized. Thus, the polarization ellipse is an idealization of the true behavior of light; it is only correct at any given instant of time. These limitations force us to consider an alternative description of polarized light in which only observed or measured quantities enter. We are, therefore, in the same situation as when we dealt with the wave equation and its solutions, neither of which can be observed. We must again turn to using average values of the optical field, which in the present case requires that we represent polarized light in terms of observables.

In 1852, Sir George Gabriel Stokes (1819–1903) discovered that the polarization behavior could be represented in terms of observables [1]. He found that any state of polarized light could be com-pletely described by four measurable quantities now known as the Stokes polarization parameters. The first parameter expresses the total intensity of the optical field. The remaining three parameters describe the polarization state. Stokes was led to his formulation in order to provide a suitable mathematical description of the Fresnel–Arago Interference Laws (1818). These laws were based on experiments carried out with an unpolarized light source, a quantity that Fresnel and his successors were never able to characterize mathematically. Stokes succeeded where others had failed because he abandoned the attempts to describe unpolarized light in terms of amplitude. He resorted to an experimental definition; that is, unpolarized light is light whose intensity is unaffected when a polar-izer is rotated or by the presence of a retarder of any retardance value. Stokes also showed that his parameters could be applied not only to unpolarized light but to partially polarized and completely polarized light as well. Unfortunately, Stokes’s paper [1] was forgotten for nearly a century. Its importance was finally brought to the attention of the scientific community by the Nobel Laureate Chandrasekhar in 1947, who used the Stokes parameters to formulate the radiative transfer equa-tions for the scattering of partially polarized light [2]. The Stokes parameters have been a prominent part of the optical literature on polarized light ever since.

We saw earlier that the amplitude of the optical field cannot be observed. However, the quantity that can be observed is the intensity, which is derived by taking a time average of the square of the amplitude. This suggests that if we take a time average of the unobservable polarization ellipse, we will be led to the observables of the polarization ellipse. When this is done, as we

60 Polarized Light, Third Edition

shall show shortly, we obtain four parameters, which are exactly the Stokes parameters. Thus, the Stokes parameters are a logical consequence of the wave theory. Furthermore, the Stokes parameters give a complete description of any polarization state of light. Most importantly, the Stokes parameters are exactly those quantities that are measured. Aside from this important formulation, however, when the Stokes parameters are used to describe physical phenomena (e.g., the Zeeman Effect) one is led to a very interesting representation. Originally, the Stokes parameters were used only to describe the measured intensity and polarization state of the opti-cal field. But by forming the Stokes parameters in terms of a column matrix, the so-called Stokes vector, we are led to a formulation in which we obtain not only measurables but also observables, which can be seen in a spectroscope. As a result, we shall see that the formalism of the Stokes parameters is far more versatile than originally envisioned and possesses a greater usefulness than is commonly known.

5.2 deRiVaTioN of STokeS PolaRiZaTioN PaRameTeRS

We consider a pair of plane waves that are orthogonal to each other at a point in space, conveniently taken to be z = 0, and not necessarily monochromatic, to be represented by the equations

E t E t t tx x x( ) = ( ) + ( )[ ]0 cos ω δ (5.1)

E t E t t ty y y( ) = ( ) + ( )[ ]0 cos ω δ (5.2)

where E0x (t) and E0y (t) are the instantaneous amplitudes, ω is the instantaneous angular frequency, and δx(t) and δy(t) are the instantaneous phase factors [3]. At all times, the amplitudes and phase factors fluctuate slowly compared to the rapid vibrations of the cosinusoids. The explicit removal of the term ωt between Equations 5.1 and 5.2 yields the familiar polarization ellipse, which is valid, in general, only at a given instant of time; that is,

E tE t

E t

E t

E t E t

E t Ex

x

y

y

x y

x

2

02

2

02

0 0

2( )( )

( )( )

( ) ( )( )

+ −yy t

t t( )

cos ( ) sin ( ),δ δ= 2 (5.3)

where δ(t) = δy(t) – δx(t).For monochromatic radiation, the amplitudes and phases are constant for all time, so Equation

5.3 reduces to

E tE

E t

E

E t E t

E Ex

x

y

y

x y

x y

2

02

2

02

0 0

2( ) ( ) ( ) ( )cos sin+ − =δ 22 δ. (5.4)

While E0x, E0y, and δ are constants, Ex and Ey continue to be implicitly dependent on time, as we see from Equations 5.1 and 5.2; therefore, we have written Ex(t) and Ey(t) in Equation 5.4. In order to represent Equation 5.4 in terms of the observables of the optical field, we must take an average over the time of observation. Because this is a long period of time relative to the time for a single oscillation, this can be taken to be infinite. However, in view of the periodicity of Ex(t) and Ey(t), we need average Equation 5.4 only over a single period of oscillation. The time average is represented by the symbol ⟨…⟩, and so we write Equation 5.4 as

E tE

E t

E

E t E t

E Ex

x

y

y

x y

x y

2

02

2

02

0 0

2( ) ( ) ( ) ( )cos sin+ − =δ 22 δ (5.5)

Stokes Polarization Parameters 61

where

E t E tT

E t E t dt i j x yi jT

i j

T

( ) ( ) lim ( ) ( ) , , .= =→∞ ∫1

0 (5.6)

Multiplying Equation 5.5 by 4 02

02E Ex y, we see that

4 4 802 2

02 2

0 0E E t E E t E E E t E ty x x y x y x y( ) ( ) ( ) ( ) cos (+ − =δ 22 0 02E Ex y sin ) .δ (5.7)

From Equations 5.1 and 5.2, we then find that the average values of Equation 5.7 using Equation 5.6 are

E t Ex x2

021

2( ) ,= (5.8)

E t Ey y2

021

2( ) ,= (5.9)

E t E t E Ex y x y( ) ( ) cos .= 12

0 0 δ (5.10)

Substituting Equations 5.8, 5.9, and 5.10 into Equation 5.7 yields

2 2 2 202

02

02

02

0 02

0 0E E E E E E E Ex y x y x y x y+ − =( cos ) ( sinδ δ)) .2 (5.11)

Since we wish to express the final result in terms of intensity, this suggests that we add and sub-tract the quantity E Ex y0

404+ to the left-hand side of Equation 5.11; doing this leads to perfect squares.

Upon doing this and grouping terms, we are led to the result

E E E E E E E Ex y x y x y x02

02 2

02

02 2

0 02

0 02 2+( ) − −( ) − ( ) =cosδ yy sin .δ( )2 (5.12)

We now write the quantities inside the parentheses as

S E Ex y0 02

02= + , (5.13)

S E Ex y1 02

02= − , (5.14)

S E Ex y2 0 02= cos ,δ (5.15)

S E Ex y3 0 02= sin ,δ (5.16)

and we can then express Equation 5.12 as

S S S S02

12

22

32= + + . (5.17)

The four Equations 5.13 through 5.16 are the Stokes polarization parameters for a plane wave. They were introduced into optics by Sir George Gabriel Stokes in 1852. We see that the Stokes

62 Polarized Light, Third Edition

parameters are real quantities, and they are simply the observables of the polarization ellipse and, hence, the optical field. The first Stokes parameter S0 is the total intensity of the light. The parameter S1 describes the amount of linear horizontal or vertical polarization, the parameter S2 describes the amount of linear +45° or −45° polarization, and the parameter S3 describes the amount of right or left circular polarization contained within the beam; this correspondence will be shown shortly. We note that the four Stokes parameters are expressed in terms of intensities, and we again emphasize that the Stokes parameters are real quantities.

If we now have partially polarized light, then we see that the relations given by Equations 5.13 through 5.16 continue to be valid for very short time intervals, since the amplitudes and phases fluctuate slowly. Using Schwarz’s inequality, one can show that for any state of polarized light the Stokes parameters always satisfy the relation

S S S S02

12

22

32≥ + + . (5.18)

This is an equality when we have completely polarized light, and an inequality when we have partially polarized light or unpolarized light.

In Chapter 4, we saw that the orientation angle ψ of the polarization ellipse was given by

tancos

.22 0 0

02

02

ψ δ=−

E E

E Ex y

x y

(5.19)

Inspecting Equations 5.13 through 5.16, we see that if we divide Equation 5.15 by Equations 5.14, ψ  can be expressed in terms of the Stokes parameters as

tan .2 2

1

ψ = SS

(5.20)

Similarly from Chapter 4, the ellipticity angle χ was given by

sinsin

.22 0 0

02

02

χ δ=+

E E

E Ex y

x y

(5.21)

Dividing Equation 5.16 by Equation 5.13, we can see that χ can be expressed in terms of the Stokes parameters as

sin .2 3

0

χ = SS

(5.22)

The Stokes parameters enable us to describe the degree of polarization P for any state of polariza-tion. By definition,

PI

IS S S

SP= = + + ≤ ≤pol

tot

( ),

/12

22

32 1 2

0

0 1 (5.23)

where Ipol is the intensity of the sum of the polarization components and Itot is the total intensity of the beam. The value of P = 1 corresponds to completely polarized light, P = 0 corresponds to unpo-larized light, and 0 < P < 1 corresponds to partially polarized light.

Stokes Polarization Parameters 63

To obtain the Stokes parameters of an optical beam, one must always take a time average of the polarization ellipse. However, the time-averaging process can be formally bypassed by representing the (real) optical amplitudes, Equations 5.1 and 5.2, in terms of the complex amplitudes

E t E i t E i tx x x x( ) = +[ ] =0 exp ( ) exp( ),ω δ ω (5.24)

E t E i t E i ty y y y( ) = +[ ] =0 exp ( ) exp( ),ω δ ω (5.25)

where

E E ix x x= 0 exp( )δ (5.26)

E E iy y y= 0 exp( )δ (5.27)

are complex amplitudes. The Stokes parameters for a plane wave are now obtained from the formulas

S E E E Ex x y y0 = +* *, (5.28)

S E E E Ex x y y1 = −* *, (5.29)

S E E E Ex y y x2 = +* *, (5.30)

S i E E E Ex y y x3 = −( ).* * (5.31)

We shall use Equations 5.28 through 5.31, the complex representation, henceforth, as the defining equations for the Stokes parameters. Substituting Equations 5.26 and 5.27 into these equations gives

S E Ex y0 02

02= + , (5.13)

S E Ex y1 02

02= − , (5.14)

S E Ex y2 0 02= cos ,δ (5.15)

S E Ex y3 0 02= sin ,δ (5.16)

which are the Stokes parameters obtained formally from the polarization ellipse.As examples of the representation of polarized light in terms of the Stokes parameters, we con-

sider (1) linear horizontal and linear vertical polarized light, (2) linear +45° and linear −45° polar-ized light, and (3) right and left circularly polarized light.

5.2.1 lineaR hoRizonTally PolaRized lighT (lhP)

For this case, there is no vertical field component so E0y = 0. From Equations 5.13 to 5.16, we have

S E x0 02= , (5.32)

S E x1 02= , (5.33)

64 Polarized Light, Third Edition

S2 0= , (5.34)

S3 = 0. (5.35)

5.2.2 lineaR veRTically PolaRized lighT (lvP)

For this case, there is no horizontal field component, so E0x = 0. From Equations 5.13 to 5.16, we have

S E y0 02= , (5.36)

S E y1 02= − , (5.37)

S2 = 0, (5.38)

S3 = 0. (5.39)

5.2.3 lineaR +45° PolaRized lighT (l +45)

The conditions to obtain L +45 polarized light are E0x = E0y = E0 and δ = 0°; that is, this is a super-position of in-phase, equal-amplitude horizontal and vertical fields. Using these conditions and the definition of the Stokes parameters in Equations 5.13 through 5.16, we find that

S E0 022= , (5.40)

S1 0= , (5.41)

S E2 022= , (5.42)

S3 0= . (5.43)

5.2.4 lineaR −45° PolaRized lighT (l −45)

The conditions on the amplitude are the same as for L +45 light, but the phase difference is δ = 180°. From Equations 5.13 to 5.16, we see that the Stokes parameters are

S E0 022= , (5.44)

S1 0= , (5.45)

S E2 022= − , (5.46)

S3 0= . (5.47)

5.2.5 RighT ciRculaRly PolaRized lighT (RcP)

The conditions to obtain RCP light are E0x = E0y = E0 and δ = 90°. From Equations 5.13 to 5.16 the Stokes parameters are then

S E0 022= , (5.48)

Stokes Polarization Parameters 65

S1 = 0, (5.49)

S2 0= , (5.50)

S E3 022= . (5.51)

5.2.6 lefT ciRculaRly PolaRized lighT (lcP)

For LCP light, the amplitudes are again equal, but the phase shift between the orthogonal, trans-verse components is δ = –90°. The Stokes parameters from Equations 5.13 to 5.16 are then

S E0 022= , (5.52)

S1 = 0, (5.53)

S2 0= , (5.54)

S E3 022= − . (5.55)

Finally, the Stokes parameters for elliptically polarized light are, of course, given by Equations 5.13 through 5.16.

Inspection of the four Stokes parameters suggests that they can be arranged in the form of a column matrix. This column matrix is called the Stokes vector. This step, while simple, provides a formal method for treating numerous complicated problems involving polarized light using well-established linear algebra techniques. We now discuss the Stokes vector.

5.3 STokeS VeCToR

The four Stokes parameters can be arranged in a column vector and written as

S =

S

S

S

S

0

1

2

3

. (5.56)

The column vector Equation 5.56 is called the Stokes vector. Mathematically, it is not a vector, but through custom it is called a vector. The Stokes vector for elliptically polarized light is then written from Equations 5.13 through 5.16 as

S =

+−

E E

E E

E E

E E

x y

x y

x y

x y

02

02

02

02

0 0

0 0

2

2

cos

sin

δδ

. (5.57)

Equation 5.57 is also called the Stokes vector for a plane wave. The Stokes vectors for linearly and circularly polarized light are readily found from this equation, and we now define these Stokes vectors.

66 Polarized Light, Third Edition

5.3.1 lineaR hoRizonTally PolaRized lighT (lhP)

For this case E0y = 0, and we find from Equation 5.57 that

S =

I0

1

1

0

0

, (5.58)

where I E x0 02= is the total intensity.

5.3.2 lineaR veRTically PolaRized lighT (lvP)

For this case E0x = 0, and we find that Equation 5.57 reduces to

S =−

I0

1

1

0

0

, (5.59)

where I E y0 02= is the total intensity.

5.3.3 lineaR +45° PolaRized lighT (l +45)

In this case, E0x = E0y = E0 and δ = 0, so Equation 5.57 becomes

S =

I0

1

0

1

0

, (5.60)

where I E0 022= .

5.3.4 lineaR −45° PolaRized lighT (l −45)

Again, E0x = E0y = E0 but now δ = 180°. Then Equation 5.57 becomes

S =−

I0

1

0

1

0

, (5.61)

and I E0 022= .

5.3.5 RighT ciRculaRly PolaRized lighT (RcP)

In this case, E0x = E0y = E0 and δ = 90°. Equation 5.57 becomes

S =

I0

1

0

0

1

, (5.62)

and I E0 022= .

Stokes Polarization Parameters 67

5.3.6 lefT ciRculaRly PolaRized lighT (lcP)

Again, we have E0x = E0y, but now the phase shift δ between the orthogonal amplitudes is δ = –90°. Equation 5.57 then reduces to

S =

−

I0

1

0

0

1

, (5.63)

and I E0 022= .

We also see from Equation 5.57 that if δ = 0° or 180º, then Equation 5.57 reduces to

S =

+−

±

E E

E E

E E

x y

x y

x y

02

02

02

02

0 02

0

. (5.64)

We recall that the ellipticity angle χ and the orientation angle ψ for the polarization ellipse are given by

sin ,24 4

3

0

χ π χ π= − ≤ ≤SS

(5.65)

tan .2 02

1

ψ ψ π= ≤ <SS

(5.66)

We see that S3 is zero, so the ellipticity angle χ is zero and, hence, Equation 5.64 is the Stokes vector for linearly polarized light. The orientation angle according to Equation 5.66 is

tan .22 0 0

02

02

ψ = ±−

E E

E Ex y

x y

(5.67)

The form of Equation 5.64 is a useful representation for linearly polarized light. Another useful representation can be made by expressing the amplitudes Ε0x and E0y in terms of an angle. To show this, we first rewrite the total intensity S0 as

S E E Ex y0 02

02

02= + = . (5.68)

Equation 5.68 suggests Figure 5.1. From Figure 5.1 we see that

E Ex0 0= cos α (5.69)

E Ey0 0 02

= ≤ ≤sin .α α π (5.70)

68 Polarized Light, Third Edition

The angle α is called the auxiliary angle; it is identical to the auxiliary angle used to represent the orientation angle and ellipticity equations summarized earlier. Substituting Equations 5.69 and 5.70 into Equation 5.64 leads to the Stokes vector for linearly polarized light

S =

I0

1

2

2

0

cos

sin,

αα (5.71)

where I E0 02= is the total intensity. Equations 5.69 and 5.70 can also be used in the representation

for the Stokes vector for elliptically polarized light, Equation 5.57. Substituting Equations 5.69 and 5.70 into Equation 5.57 gives

S =

I0

1

2

2

2

cos

sin cos

sin sin

.α

α δα δ

(5.72)

It is customary to write the Stokes vector in normalized form by setting I0 = 1, and Equation 5.72 is then written as

S =

1

2

2

2

cos

sin cos

sin sin

.α

α δα δ

(5.73)

The orientation angle ψ and the ellipticity angle χ of the polarization ellipse are given by Equations 5.65 and 5.66. Substituting S1, S2, and S3 from Equation 5.73 into Equations 5.65 and 5.66 gives

α

E0y

E0x

E0

figuRe 5.1 Resolution of the optical field components, where the optic axis is out of the page.

Stokes Polarization Parameters 69

tan 2 tan 2 cosψ α δ= (5.74)

sin 2 sin 2 sinχ α δ= , (5.75)

which are identical to the relations we found earlier.The use of the auxiliary angle α enables us to express the orientation and ellipticity in terms of α

and δ. Expressing Equation 5.73 in this manner shows that there are two unique polarization states. For α = 45°, Equation 5.73 reduces to

S =

1

0

cos

sin

.δδ

(5.76)

Thus, the polarization ellipse is expressed only in terms of the phase shift δ between the orthogonal amplitudes. The orientation angle ψ is seen to be always 45°. The ellipticity angle, Equation 5.75, however, is

sin 2 sinχ δ= , (5.77)

so χ = δ/2. The Stokes vector Equation 5.76 expresses that the polarization ellipse is rotated 45° from the horizontal axis and that the polarization state of the light can vary from linearly polarized (δ = 0, 180°) to circularly polarized (δ = 90°, 270°).

Another unique polarization state occurs when δ = 90° or 270°. For this condition, Equation 5.73 reduces to

S =

±

1

2

0

2

cos

sin

.α

α

(5.78)

We see that we now have a Stokes vector and a polarization ellipse that depends only on the auxil-iary angle α. From Equation 5.74, the orientation angle ψ is always zero. However, Equations 5.75 and 5.78 show that the ellipticity angle χ is now given by

sin 2χ = ± sin 2α, (5.79)

so χ = ±α. In general, Equation 5.79 shows that we will have elliptically polarized light. For α = + 45° and −45°, we obtain right and left circularly polarized light. Similarly, for α = 0° and 90° we obtain linear horizontally and vertically polarized light.

The Stokes vector can also be expressed in terms of S0, ψ, and χ. To show this we write Equations 5.69 and 5.70 as

S S3 0 2= sin χ (5.80)

S S2 1 tan 2= ψ. (5.81)

70 Polarized Light, Third Edition

In Section 5.2 we found that

S S S S02

12

22

32= + + . (5.82)

Substituting Equations 5.80 and 5.81 into Equation 5.82, we find that

S S1 cos 2 cos 2= 0 χ ψ, (5.83)

S S2 cos 2 sin 2= 0 χ ψ, (5.84)

S S3 sin 2= 0 χ. (5.85)

Arranging these equations in the form of a Stokes vector, we have

S =

S0

1

2 2

2 2

2

cos cos

cos sin

sin

.χ ψχ ψ

χ

(5.86)

The Stokes parameters in Equations 5.83 through 5.85 are almost identical in form to the well-known equations relating Cartesian coordinates to spherical coordinates. We recall that the spheri-cal coordinates r, θ, and ϕ are related to the Cartesian coordinates x, y, and z by

x r= sin cosθ φ, (5.87)

y r= sin sin ,θ φ (5.88)

z r= cosθ. (5.89)

Comparing Equations 5.87 through 5.89 with Equations 5.83 through 5.85, we see that the equations are identical if the angles are related by

θ χ= ° −90 2 , (5.90)

φ ψ= 2 . (5.91)

In Figure 5.2, we have drawn a sphere whose center is also at the center of the Cartesian coordinate system. We see that expressing the polarization state of an optical beam in terms of χ and ψ allows us to describe its ellipticity and orientation on a sphere; the radius of the sphere is taken to be unity. The representation of the polarization state on a sphere was first introduced by Henri Poincaré in 1892 and is appropriately called the Poincaré sphere. However, at that time, Poincaré introduced the sphere in an entirely different way, namely, by representing the polarization equations in a complex plane and then projecting the plane onto a sphere, a so-called stereographic projection. In this way, he was led to Equations 5.83 through 5.85. He does not appear to have known that the Equations 5.83 through 5.85 were directly related to the Stokes parameters. Because the Poincaré sphere is of historical interest and is still used to describe the polarization state of light, we shall discuss it in detail later. It is especially useful for describing the change in polarized light when it interacts with polarizing elements.

Stokes Polarization Parameters 71

The discussion in this chapter shows that the Stokes parameters and the Stokes vector can be used to describe an optical beam that is completely polarized. We have, at first sight, only provided an alternative description of completely polarized light. All of the equations derived here are based on the polarization ellipse given in Chapter 4; that is, the amplitude formulation. However, we have pointed out that the Stokes parameters can also be used to describe unpolarized and partially polar-ized light, quantities that cannot be described within an amplitude formulation of the optical field. In order to extend the Stokes parameters to unpolarized and partially polarized light, we must now consider the classical measurement of the Stokes polarization parameters.

5.4 ClaSSiCal meaSuRemeNT of STokeS PolaRiZaTioN PaRameTeRS

The Stokes polarization parameters are immediately useful because, as we shall now see, they are directly accessible to measurement. This is due to the fact that they are an intensity formulation of the polarization state of an optical beam. In this section, we shall describe the measurement of the Stokes polarization parameters. This is done by allowing an optical beam to pass through two opti-cal elements known as a retarder and a polarizer. Specifically, the incident field is described in terms of its components, and the field emerging from the polarization elements is then used to determine the intensity of the emerging beam. Later, we shall carry out this same problem by using a more formal but powerful approach known as the Mueller matrix formalism. In the following chapter, we shall also see how this measurement method enables us to determine the Stokes parameters for unpolarized and partially polarized light.

We begin by referring to Figure 5.3, which shows a monochromatic optical beam incident on a polarization element called a retarder. This polarization element is then followed by another polar-ization element called a polarizer. The components of the incident beam are

E t E e ex xi i tx( ) = 0δ ω (5.92)

P

O

x

y

z

S0

S1

S2

S3

2ψ

2

figuRe 5.2 The Poincaré representation of polarized light on a sphere.

72 Polarized Light, Third Edition

E t E e ey yi i ty( ) .= 0δ ω (5.93)

In Section 5.2 we saw that the Stokes parameters for a plane wave written in complex notation could be obtained from

S E E E Ex x y y0 = +* *, (5.94)

S E E E Ex x y y1 = −* *, (5.95)

S E E E Ex y y x2 = +* *, (5.96)

S i E E E Ex y y x3 = −( )* * , (5.97)

where i = −1 and the asterisk represents the complex conjugate.In order to measure the Stokes parameters, the incident field propagates through a phase-shifting

element, which has the property that the phase of the x component (Ex) is advanced by ϕ/2 and the phase of the y component Ey is retarded by ϕ/2, written as −ϕ/2. The components ′Ex and ′Ey emerg-ing from the phase-shifting element component are then

′ =E E ex xiφ/ ,2 (5.98)

′ = −E E ey yiφ/ .2 (5.99)

In optics, a polarization element that produces this phase shift is called a retarder; it will be dis-cussed in more detail later.

Next, the field described by Equations 5.98 and 5.99 is incident on a component called a polarizer. It has the property that the optical field is transmitted only along an axis known as the transmis-sion axis. Ideally, if the transmission axis of the polarizer is at an angle θ, only the components of

′Ex and ′Ey in this direction can be transmitted perfectly; there is complete attenuation at any other angle. A polarizing element that behaves in this manner is called an ideal polarizer. This behavior is described in Figure 5.4. The component of ′Ex along the transmission axis is ′Ex cosθ. Similarly, the component of ′Ey is ′Ey sinθ. The field transmitted along the transmission axis is the sum of these components so that the total field Ε emerging from the polarizer is

E E Ex y= ′ + ′cos sin .θ θ (5.100)

θ x

y

– /2

+ /2

figuRe 5.3 Measurement of the Stokes polarization parameters.

Stokes Polarization Parameters 73

Substituting Equations 5.98 and 5.99 into Equation 5.100, the field emerging from the polarizer is

E E e E exi

yi= + −φ φθ θ/ /cos sin .2 2 (5.101)

The intensity of the beam is defined by

I E E= · * . (5.102)

Taking the complex conjugate of Equation 5.101 and forming the product in accordance with Equation 5.102, the intensity of the emerging beam is

I E E E E E E ex x y y x yiθ φ θ θ θ θφ, cos sin sin cos* * *( ) = + + +−2 2 EE E ex y

i* sin cos .φ θ θ (5.103)

Equation 5.103 can be rewritten by using the well-known trigonometric half-angle formulas

coscos

,2 1 22

θ θ= + (5.104)

sincos

,2 1 22

θ θ= − (5.105)

sin cossin

.θ θ θ= 22

(5.106)

Using these equations in Equation 5.103 and grouping terms, we find that the intensity I(θ, ϕ) becomes

IE E E E E E E E

E E

x x y y x x y y

x y

θ φθ

,cos* * * *

*( ) =

+( ) + −( )

+12

2

++( ) + −( )

E E i E E E Ey x x y y x* * *cos sin sin sinφ θ φ θ2 2

. (5.107)

The terms within parentheses are exactly the Stokes parameters given in Equations 5.28 through 5.31. Equation 5.107 was first derived by Stokes and is the manner in which the Stokes parameters

θ

E’y

E’x

E

figuRe 5.4 Resolution of the optical field components by a polarizer.

74 Polarized Light, Third Edition

were first introduced in the optical literature. Replacing the terms in Equation 5.107 by the defini-tions of the Stokes parameters given in Equations 5.28 through Equation 5.31, we arrive at

I S S S Sθ φ θ φ θ φ θ, cos cos sin sin sin .( ) = + + +[ ]12

2 2 20 1 2 3 (5.108)

Equation 5.108 is Stokes’s famous intensity formula for measuring the four Stokes parameters. Thus we see that the Stokes parameters are directly accessible to measurement; that is, they are observ-able quantities.

The first three Stokes parameters are measured by removing the retarder (ϕ = 0°) and rotating the transmission axis of the polarizer to the angles θ = 0°, +45°, and +90°, respectively. The final parameter, S3, is measured by reinserting a quarter-wave retarder (ϕ = 90°) into the optical path and setting the transmission axis of the polarizer to θ = 45°. The intensities are then found from Equation 5.108 to be

I S S0 012

0 1° °( ) = +[ ], , (5.109)

I S S45 012

0 2° °( ) = +[ ], , (5.110)

I S S90 012

0 1° °( ) = −[ ], , (5.111)

I S S45 9012

0 3° °( ) = +[ ], . (5.112)

Solving these equations for the Stokes parameters, we have

S I I0 0 0 0 0= ° °( ) + ° °( ), , ,9 (5.113)

S I I1 0 0 90 0= ° °( ) − ° °( ), , , (5.114)

S I I I2 2 45 0 0 0 90 0= ° °( ) − ° °( ) − ° °( ), , , , (5.115)

S I I I3 2 45 90 0 0 90 0= ° °( ) − ° °( ) − ° °( ), , , . (5.116)

Equations 5.113 through 5.116 are really quite remarkable. In order to measure the Stokes parameters, it is necessary to measure the intensity at four pairs of angles. We must remember, however, that in 1852 there were no devices to measure the intensity quantitatively. The intensities can be measured quantitatively only with an optical detector. But when Stokes introduced the Stokes parameters, such detectors did not exist. The only optical detector was the human eye (retina), a detector capable of measuring only the null or greater-than-null states of light, and so the above method for measuring the Stokes parameters could not be used! Stokes did not introduce the Stokes parameters to describe the optical field in terms of observables as is sometimes stated. The reason for his derivation of Equation 5.108 was not to measure the Stokes polarization parameters but to provide the solution to an entirely different problem; namely, a mathematical statement for unpolarized light. We shall soon see that Equation 5.108 is perfect for doing this. It is possible to measure all four Stokes parameters using the human eye, however, by using a null-intensity technique. This method is described in a later chapter on polarimetry.

Stokes Polarization Parameters 75

Unfortunately, after Stokes solved this problem and published his great paper on the Stokes parameters and the nature of polarized light, he never returned to this subject again. By the end of his researches on this subject, he had turned his attention to the problem of the fluorescence of solutions. This problem would become the major focus of his attention for the rest of his life. Aside from Lord Rayleigh in England and Emil Verdet in France, the importance of Stokes’s paper and the Stokes parameters was not fully recognized, and the paper was practically forgotten for nearly a century by the optical community. Fortunately however, Emil Verdet did understand the significance of Stokes’s paper and wrote a number of subsequent papers on the Stokes polarization parameters. He thus began a tradition in France of studying the Stokes parameters. The Stokes polarization parameters did not really appear in the English-speaking world again until they were “rediscovered” by Chandrasekhar in the late 1940s when he was writing his monumental papers on radiative transfer. Previous to Chandrasekhar, no one had included optical polarization in the equations of radiative transfer. In order to introduce polarization into his equations, he eventually found Stokes’s original paper. He immediately recognized that because the Stokes parameters were an intensity formulation of optical polarization, they could be introduced into radiative equations. It was only after the publication of Chandrasekhar’s papers that the Stokes parameters reemerged. They have remained in the optical literature ever since.

We now describe Stokes’s formulation for unpolarized light.

5.5 STokeS PaRameTeRS foR uNPolaRiZed aNd PaRTially PolaRiZed lighT

The intensity I(θ,ϕ) of a beam of light emerging from the retarder/polarizer combination was seen in the previous section to be

I S S S S( , ) [ cos sin cos sin sin ],θ φ θ θ φ θ φ= + + +12

2 2 20 1 2 3 (5.117)

where S0, S1, S2, and S3 are the Stokes parameters of the incident beam, θ is the rotation angle of the transmission axis of the polarizer, and ϕ is the phase shift of the retarder. By setting θ–0°, 45°, or 90° and ϕ–0° or 90°, with the proper pairings of angles, all four Stokes parameters can then be measured. However, it was not Stokes’s intention to merely cast the polarization of the opti-cal field in terms of the intensity rather than the amplitude. Rather, he was interested in finding a suitable mathematical description for unpolarized light. Stokes, unlike his predecessors and his contemporaries, recognized that it was impossible to describe unpolarized light in terms of ampli-tudes. Consequently, he abandoned the amplitude approach and sought a description based on the observed intensity.

To describe unpolarized light using Equation 5.117, Stokes observed that unpolarized light had a very unique property; namely, its intensity was unaffected by (1) rotation of a linear polarizer (when a polarizer is used to analyze the state of polarization, it is called an analyzer) or (2) the presence of a retarder. For unpolarized light, the only way the observed intensity I(θ,ϕ) could be independent of θ,ϕ was for Equation 5.117 to satisfy

I S( , )θ φ = 12

0 (5.118)

and

S S S1 2 3= = = 0. (5.119)

Equations 5.118 and 5.119 are the mathematical statements for unpolarized light; Stokes had finally provided a correct mathematical statement. From a conceptual point of view S1, S2, and S3 describe

76 Polarized Light, Third Edition

the polarizing behavior of the optical field. Since there is no polarization, Equations 5.118 and 5.119 must be the correct mathematical statements for unpolarized light. Later, we shall show how these equations are used to formulate the interference laws of Fresnel and Arago.

In this way, Stokes discovered an entirely different way to describe the polarization state of light. His formulation could be used to describe both completely polarized and completely unpolarized light. Furthermore, Stokes had been led to a formulation of the optical field in terms of measurable quantities (observables), the Stokes parameters. This was a unique point of view for nineteenth-century optical physics. The representation of radiation phenomena in terms of observables would not reappear again in physics until 1925 with the discovery of the laws of quantum mechanics by Werner Heisenberg.

The Stokes parameters described in Equation 5.117 arise from an experimental configuration. Consequently, they were associated for a long time with the experimental measurement of the polar-ization of the optical field. A study of classical optics shows that polarization was conceptually understood with the nonobservable polarization ellipse, whereas the measurement was made in terms of intensities, the Stokes parameters. In other words, there were two distinct ways to describe the polarization of the optical field.

We have seen, however, that the Stokes parameters are actually a consequence of the wave the-ory, and arise naturally from the polarization ellipse. It is only necessary to transform the nonob-servable polarization ellipse to the observed intensity domain whereupon we are led directly to the Stokes parameters. The Stokes polarization parameters must be considered as part of the conceptual foundations of the wave theory.

For a completely polarized beam of light, we saw that

S S S S02

12

22

32= + + , (5.120)

and we have just seen that for unpolarized light

S S S S02

1 2 30 0> = = =, . (5.121)

Equations 5.120 and 5.121 represent extreme states of polarization. Clearly, there must be intermedi-ate polarization states. These intermediate states are called partially polarized light. Thus, Equation 5.120 can be used to describe all three polarization conditions by writing it as

S S S S02

12

22

32≥ + + . (5.122)

For perfectly polarized light, “≥” is replaced by “=”; for unpolarized light, “≥” is replaced by “>” with S1 = S2 = S3 = 0; and for partially polarized light “≥” is replaced by “>”.

An important quantity that describes these various polarization conditions is the degree of polar-ization, P. This quantity can be expressed in terms of the Stokes parameters. To derive P we decom-pose the optical field into unpolarized and polarized portions, which are mutually independent. Then, and this will be proved later, the Stokes parameters of a combination of independent waves are the sums of the respective Stokes parameters of the separate waves. The four Stokes parameters, S0, S1, S2, and S3 of the beam are represented by S. The total intensity of the beam is S0. We subtract the polarized intensity ( ) /S S S1

222

32 1 2+ + from the total intensity S0 and we obtain the unpolarized

intensity. Thus, we have

S( )u

S S S S

=

− + +

0 12

22

32

0

0

0

(5.123)

Stokes Polarization Parameters 77

and

S( ) ,p

S S S S

S

S

S

=

− + +

0 12

22

32

1

2

3

(5.124)

where S(u) represents the unpolarized part and S(p) represents the polarized part. The degree of polarization P is then defined to be

PI

IS S S

SP= = + + ≤ ≤pol

tot

12

22

32

0

0 1. (5.125)

Thus, P = 0 indicates that the light is unpolarized, P = 1 means that the light is completely polar-ized, and a value of P where 0 < P < 1 means that the light is partially polarized.

The use of the Stokes parameters to describe polarized light rather than the use of the amplitude formulation enables us to deal directly with the quantities measured in an optical experiment. We can carry out analyses in the amplitude domain and then transform the amplitude results to the Stokes parameters using the defining equations. When this is done, we can easily relate the experimental results to the theoretical results. Furthermore, when we obtain the Stokes parameters, or rather the Stokes vector, we shall see that we are led to a description of radiation in which the Stokes param-eters not only describe the measured quantities, but can also be used to truly describe the observed spectral lines in a spectroscope. In other words, we shall arrive at observables in the strictest sense of the word.

5.6 addiTioNal PRoPeRTieS of STokeS PolaRiZaTioN PaRameTeRS

Before we proceed to apply the Stokes parameters to a number of problems of interest, we wish to discuss a few of their additional properties. We saw earlier that the Stokes parameters could be used to describe any polarization state of a beam of light. In particular, we saw how unpolarized light and completely polarized light could both be written in terms of a Stokes vector. The question remains as to how we can represent partially polarized light in terms of the Stokes parameters and the Stokes vector. To answer this question, we must establish a fundamental property of the Stokes parameters, the property of additivity, whereby the Stokes parameters of two completely indepen-dent beams can be added to yield the Stokes parameters of the combined beam. This property is another way of describing the principle of incoherent superposition. We now prove this property of additivity.

We recall that the Stokes parameters for an optical beam can be represented in terms of complex amplitudes by

S E E E Ex x y y0 = +* *, (5.126)

S E E E Ex x y y1 = −* *, (5.127)

S E E E Ex y y x3 = +* *, (5.128)

S i E E E Ex y y x3 = −( ).* * (5.129)

78 Polarized Light, Third Edition

Consider now that we have two optical beams each of which is characterized by its own set of Stokes parameters represented as

S E E E Ex x y y01

1 1 1 1( ) * * ,= + (5.130)

S E E E Ex x y y11

1 1 1 1( ) * * ,= − (5.131)

S E E E Ex y y x21

1 1 1 1( ) * * ,= + (5.132)

S i E E E Ex y y x31

1 1 1 1( ) * *( ),= − (5.133)

and

S E E E Ex x y y02

2 2 2 2( ) * * ,= + (5.134)

S E E E Ex x y y12

2 2 2 2( ) * * ,= − (5.135)

S E E E Ex y y x22

2 2 2 2( ) * * ,= + (5.136)

S i E E E Ex y y x32

2 2 2 2( ) * *( ).= − (5.137)

The superscripts and subscripts 1 and 2 refer to the first and second beams, respectively. These two beams are now superposed. By the principle of superposition for amplitudes, the total field in the x and y direction is

E E Ex x x= +1 2 (5.138)

E E Ey y y= +1 2 . (5.139)

We now form products of Equations 5.138 and 5.139 according to the expressions for the Stokes parameters, Equations 5.126 through 5.129, so that

E E E E E E

E E E E E E

x x x x x x

x x x x x

* *

* *

( )( )= + +

= + +

1 2 1 2

1 1 1 2 2 1xx x xE E* * ,+ 2 2

(5.140)

E E E E E E

E E E E E E

y y y y y y

y y y y y

* *

* *

( )( )= + +

= + +

1 2 1 2

1 1 1 2 2 1yy y yE E* * ,+ 2 2

(5.141)

E E E E E E

E E E E E E

x y x x y y

x y x y x

* *

* *

( )( )= + +

= + +

1 2 1 2

1 1 1 2 2 1yy x yE E* * ,+ 2 2

(5.142)

E E E E E E

E E E E E E

y x y y x x

y x y x y

* *

* *

( )( )= + +

= + +

1 2 1 2

1 1 2 1 1 2xx y xE E* * .+ 2 2

(5.143)

Stokes Polarization Parameters 79

Let us now assume that the two beams are completely independent of each other with respect to their amplitudes and phases. We describe the degree of independence by writing an overbar that signifies a time average over the product of Ex and Ey; that is, E E E Ex x y y

* *, , and so on, and we can express these products using indices i and j as

E E i j x yi j* , , .= (5.144)

Since the two beams are completely independent, we express this behavior by

E E E E i ji j i j1 2 2 1 0* * ,= = ≠ (5.145)

E E i j x yi j1 1 0* , , ,≠ = (5.146)

E E i j x yi j2 2 0* , , .≠ = (5.147)

The value of zero in Equation 5.145 indicates complete independence. On the other hand, the non-zero value in Equations 5.146 and 5.147 means that there is some degree of dependence. Operating on Equations 5.140 through 5.143 with an overbar and using the conditions expressed by Equations 5.145 through 5.147, we find that

E E E E E Ex x x x x x* * * ,= +1 1 2 2 (5.148)

E E E E E Ey y y y y y* * * ,= +1 1 2 2 (5.149)

E E E E E Ex y x y x y* * * ,= +1 1 2 2 (5.150)

E E E E E Ey x y x y x* * * .= +1 1 2 2 (5.151)

We now form the Stokes parameters according to Equations 5.126 through 5.129, drop the overbar because the noncorrelated terms have been eliminated, and group terms. The result is

S E E E E E E E E E E E Ex x y y x x y y x x y0 1 1 1 1 2 2 2 2= + = + + +* * * * *( ) ( yy* ), (5.152)

S E E E E E E E E E E E Ex x y y x x y y x x y1 1 1 1 1 2 2 2 2= − = − + −* * * * *( ) ( yy* ), (5.153)

S E E E E E E E E E E E Ex y y x x y y x x y y2 1 1 1 1 2 2 2 2= + = + + +* * * * *( ) ( xx* ), (5.154)

S i E E E E i E E E E i E Ex y y x x y y x x y3 1 1 1 1 2 2= − = − + −( ) ( ) (* * * * * EE Ey x2 2* ). (5.155)

From Equations 5.130 to 5.133 and Equations 5.134 through 5.147 we see that we can then write these equations as

S S S0 01

02= +( ) ( ) , (5.156)

S S S1 11

12= +( ) ( ) , (5.157)

80 Polarized Light, Third Edition

S S S2 21

22= +( ) ( ) , (5.158)

S S S3 31

32= +( ) ( ). (5.159)

The meaning of these equations is that the Stokes parameters of two completely independent opti-cal beams can be added and represented by the Stokes parameters of the combined beams. We can represent Equations 5.156 through 5.159 in terms of Stokes vectors; that is,

S

S

S

S

S

S

S

S

0

1

2

3

01

11

21

31

=

( )

( )

( )

( )

+

S

S

S

S

02

12

22

32

( )

( )

( )

( )

, (5.160)

or simply

S S S= +( ) ( ) ,1 2 (5.161)

so the Stokes vectors S i i( ) =, ,1 2 are also additive.As a first application of this result, Equation 5.160, we recall that the Stokes vector for unpolar-

ized light is

S =

I0

1

0

0

0

. (5.162)

We also saw that the Stokes vector could be written in terms of the orientation angle ψ and the ellipticity χ as

S =

I0

1

2 2

2 2

2

cos cos

cos sin

sin

.χ ψχ ψ

χ

(5.163)

For a beam of light, which may be a result of combining two beams of equal intensity I0/2, we see from Equation 5.160 that we can write Equation 5.162, using Equation 5.163, as

II

00

1

0

0

0

2

1

2 2

2 2

=cos cos

cos sin

χ ψχ ψ

ssin

cos cos

cos sin

2

2

1

2 2

20

χ

χ ψχ

+−−

I

22

2

ψχ−

sin

. (5.164)

We can also express Equation 5.160 in terms of two beams of equal intensity I0/2 using the form in Equation 5.163 as

II

00 1 1

1

1

0

0

0

2

1

2 2

2

=cos cos

cos si

χ ψχ nn

sin

cos cos

cos2

2

2

1

2 2

1

1

0 2 2

ψχ

χ ψ

+ I

22 2

22 2

2

χ ψχ

sin

sin

.

(5.165)

Stokes Polarization Parameters 81

In this last equation, we must have that

cos cos cos cos ,2 2 2 22 2 1 1χ ψ χ ψ= − (5.166)

cos sin cos sin ,2 2 2 22 2 1 1χ ψ χ ψ= − (5.167)

sin 2 sin 22 1χ χ= − . (5.168)

Equation 5.168 is only true if

χ χ2 1= − ; (5.169)

and if this is true, then the ellipticity of beam 2 is the negative of that of beam 1. We now substitute Equation 5.169 into Equations 5.166 and 5.167 and we have

cos cos2 22 1ψ ψ= − (5.170)

sin 2 sin 22 1ψ ψ= − . (5.171)

Equations 5.170 and 5.171 can only be satisfied if

2 21 2ψ ψ π= ± (5.172)

or

ψ ψ π2 1

2= ± , (5.173)

and this means that the polarization ellipse for the second beam is oriented 90° (π/2) from the first beam. The conditions in Equation 5.169, χ2 = –χ1, and Equation 5.173, ψ ψ π2 1 2= ± / are said to describe two polarization ellipses of orthogonal polarization. Thus, unpolarized light is a superpo-sition of two beams of equal intensity and orthogonal polarization. As special cases of Equation 5.164, we see that unpolarized light can be decomposed into independent beams of linear and/or circular polarized light; that is,

II I

02 0

1

0

0

0

2

1

1

0

0

2

1

=

+−−

1

0

0

, (5.174)

II I

02 0

1

0

0

0

2

1

0

1

0

2

1

=

+00

1

0

−

, (5.175)

82 Polarized Light, Third Edition

II I

02 0

1

0

0

0

2

1

0

0

1

2

1

=

+00

0

1−

. (5.176)

Of course, the intensity of each beam is half the intensity of the unpolarized beam. A useful principle may also be noted at this point; any polarization may always be decomposed into (or is a superposi-tion of) two orthogonal polarizations.

We now return to our original problem of representing partially polarized light in terms of the Stokes vector. Recall that the degree of polarization P is defined by

PS S S

SP= + + ≤ ≤1

222

32

0

0 1. (5.177)

This equation suggests that partially polarized light can be represented by a superposition of unpo-larized light and completely polarized light by using Equation 5.160. A little thought shows that if we have a beam that we assume is partially polarized light expressed as the Stokes vector

S =

S

S

S

S

0

1

2

3

, (5.178)

then Equation 5.178 can be written as

S =

= −

S

S

S

S

P S

0

1

2

3

01

1

0

0

0

( )

+

≤ ≤PSS PS

S PS

S PS

P01 0

2 0

3 0

1

0/

/

/

11. (5.179)

The first Stokes vector on the right-hand side of Equation 5.179 represents unpolarized light, and the second Stokes vector represents completely polarized light. For P = 0, unpolarized light, Equation 5.179 reduces to

S =

S0

0

0

0

, (5.180)

and for P = 1, completely polarized light, Equation 5.179 reduces to

S =

S

S

S

S

0

1

2

3

. (5.181)

Stokes Polarization Parameters 83

When P is neither 0 nor 1 (i.e., 0 < P < 1) we note that S0 on the left-hand side of Equation 5.179 always satisfies

S S S S0 12

22

32> + + , (5.182)

whereas S0 in the Stokes vector associated with polarized light on the right-hand side of Equation 5.179 always satisfies

S S S S0 12

22

32= + + . (5.183)

For a general Stokes vector where we may not have any knowledge of the degree of polarization, it is always true that

S S S S0 12

22

32≥ + + . (5.184)

Another representation of partially polarized light in terms of P is the decomposition of a beam into two completely polarized beams of orthogonal polarizations; that is,

S

S

S

S

PP

PS

S

S

S

0

1

2

3

0

1

2

3

12

= +

+ − −−−

< ≤12

0 1

0

1

2

3

PP

PS

S

S

S

P , (5.185)

where

PS S S S0 12

22

32= + + . (5.186)

Thus, partially polarized light can also be decomposed into two orthogonally polarized beams. We note that for partially polarized light, we can show that, using Equations 5.186 and 5.185, the intensi-ties of the two beams are given by

S S S S S01

0 12

22

321

212

( ) ,= + + + (5.187)

S S S S S02

0 12

22

321

212

( ) .= − + + (5.188)

Only for unpolarized light are the intensities of the two beams equal.While we have restricted this discussion to two beams, it is easy to see that we could have

described the optical field in terms of n beams; that is, we could have extended Equation 5.161 to

S S S S S

S

= + + + +

= ==

∑

( ) ( ) ( ) ( )

( ) , , .

1 2 3

1

n

i

i t

n

i n (5.189)

We have not done this for the simple reason that, in practice, dealing with two beams is sufficient. Nevertheless, the reader should be aware that the additivity law can be extended to n beams.

84 Polarized Light, Third Edition

It is of interest to express the parameters of the polarization ellipse in terms of the Stokes param-eters. To do this, we recall that

S E E Ix y0 02

02

0= + = , (5.190)

S E E Ix y1 02

02

0 2= − = cos ,α (5.191)

S E E Ix y2 0 0 02 2= =cos sin cos ,δ α δ (5.192)

S E E Ix y3 0 0 02 2= =sin sin sin .δ α δ (5.193)

Rearranging, we can obtain the equations

ES S

x02 0 1

2= +

, (5.194)

ES S

y02 0 1

2= −

, (5.195)

cos ,δ = SE Ex y

2

0 02 (5.196)

sin .δ = SE Ex y

3

0 02 (5.197)

We recall that the instantaneous polarization ellipse is

EE

E

E

E E

E Ex

x

y

y

x y

x y

2

02

2

02

0 0

22+ − =cos sin .δ δ (5.198)

Substituting the expressions in Equations 5.194 through 5.197 into the appropriate terms in Equation 5.198 gives us

2 2 42

0 1

2

0 1

2

02

12

32

02

12

ES S

E

S S

S E E

S SS

S Sx y x y

++

−−

−=

−, (5.199)

where we have used E E S Sx y02

02

02

12 4= −( )/ obtained from Equations 5.194 and 5.195. Multiplying

through Equation 5.199 by ( )S S S02

12

32− / then yields

2 2 4

10 12

32

0 12

32

2

32

( ) ( ).

S S ES

S S E

S

S E E

Sx y x y− + + − = (5.200)

Equation 5.200 is of the form

Ax cxy by2 22 1,− + = (5.201)

where we let Ex = x, Ey = y, and

AS S

S= −2 0 1

32

( ), (5.202)

Stokes Polarization Parameters 85

bS S

S= +2 0 1

32

( ), (5.203)

cS

S= 2 2

32

. (5.204)

We can now find the orientation and ratio of the axes in terms of the Stokes parameters. We first express x and y in the polar coordinates ρ and ϕ such that

x = ρ φcos (5.205)

y = ρ φsin . (5.206)

Substituting Equations 5.205 and 5.206 into Equation 5.201, we have

A c bρ φ ρ φ φ ρ φ2 2 2 2 2cos 2 sin cos sin 1.− + = (5.207)

Using the half-angle formulas for cos2ϕ and sin2ϕ, Equation 5.207 then becomes

A

cbρ φ ρ φ ρ φ2

221 2

22

1 22

1( cos )

sin( cos )

.+ − + − = (5.208)

We now introduce the parameter L defined in terms of ρ as

L = 22ρ

. (5.209)

Let us substitute Equation 5.209 into Equation 5.208, and write

L A b c A b= +( ) − + −( )2 sin 2 cos 2φ φ. (5.210)

The major and minor axes of the ellipse correspond to maximum and minimum values of ρ, respec-tively, whereas L is a minimum and maximum. The angle ϕ where this maximum and minimum occur can be found in the usual way by setting dL/dϕ = 0 and solving for ϕ. We have

dLd

c A bφ

φ φ= − − − =4 2 2 2 0cos ( )sin (5.211)

and

sincos

tan .22

22φ

φφ= = −

−c

A b (5.212)

Solving for ϕ, we find that

φ = −−

−12

21tan .c

A b (5.213)

To find the corresponding maximum and minimum values of L in Equation 5.210, we must express sin 2ϕ and cos 2ϕ in terms of A, b, and c. We can find unique expressions for sin 2ϕ and cos 2ϕ

86 Polarized Light, Third Edition

from Equation 5.212 by constructing the right triangle in Figure 5.5. We see from the right triangle that Equation 5.212 is satisfied by

sin( ) ( )

22

21 2 2

φ = −− + −

cc A b

(5.214)

cos( ) ( )

,22

1 2 2φ = −

− + −A b

c A b (5.215)

or

sin( ) ( ( ))

22

22 2 2

φ =+ − −

cc A b

(5.216)

cos( )

( ) ( ( )),2

22 2 2

φ = − −+ − −A b

c A b (5.217)

where the subscripts 1 and 2 correspond to the two possible solutions for the triangle in Equations 5.214 through 5.217.

Substituting Equations 5.214 and 5.215 into Equation 5.210 yields

L A b c A bmax ( ) ( ) ( )= + + − + −2 2 2 (5.218)

and, similarly, substituting Equations 5.216 and 5.217 into Equation 5.210 yields

L A b c A bmin ( ) ( ) ( ( )) .= + − + − −2 2 2 (5.219)

We have written “max” and “min” on L in Equations 5.218 and 5.219 to indicate that these are the maximum and minimum values of L. We also note that Equations 5.214 and 5.216 are related by

sin 2 sin 21 2φ φ= − (5.220)

and Equations 5.215 and 5.217 by

cos 2 cos 21 2φ φ= − . (5.221)

2ф

(–2C)2 + (A − B)2

–2C

( A – B )

figuRe 5.5 Right triangle corresponding to Equation 5.212.

Stokes Polarization Parameters 87

We see that Equations 5.214 and 5.215 are satisfied by setting

φ φ π2 1

2= + . (5.222)

Thus, the maximum and minimum lengths, that is, the major and minor axes, are at ϕ1 and ϕ1 +90°, respectively, which is exactly what we would expect. We see from Equation 5.209 that

ρminmax

2 2=L

(5.223)

ρmaxmin

.2 2=L

(5.224)

The ratio of the square of the lengths of the major axis to the minor axis is defined to be

R = ρρ

max

min

2

2 (5.225)

so from Equations 5.218 and 5.219 we have

RA b c A bA b c A b

= + − + −+ + + −

( ) ( ) ( )( ) ( ) ( )

.22

2 2

2 2 (5.226)

We can now express Equation 5.226 in terms of the Stokes parameters from Equations 5.202 through 5.204 and we find that Equation 5.226 becomes

RS S SS S S

= − ++ +

0 12

22

0 12

22

. (5.227)

We have found the relation between the ratio of the lengths of the major and minor axes of the polarization ellipse and the Stokes parameters. This can be expressed directly by using Equations 5.223 and 5.224 and the Expressions 5.202 through 5.204, or as a ratio R given by Equation 5.227.

Not surprisingly there are other interesting relations between the Stokes parameters and the parameters of the polarization ellipse. These relations are fundamental to the development of the Poincaré sphere, so we shall discuss them in Chapter 11.

5.7 STokeS PaRameTeRS aNd The CoheReNCy maTRiX

We have demonstrated that the state of polarization is specified completely by the four Stokes param-eters S0, S1, S2, and S3. There is another representation in which the polarization is described by a 2 × 2 matrix known as the coherency matrix. Furthermore, there is a direct relationship between the elements of the coherency matrix and the Stokes parameters. This relationship, as well as the required mathematical background, is thoroughly discussed in the text by Born and Wolf [4]. We briefly discuss the coherency matrix here as it relates to the Stokes parameters.

Consider an optical field consisting of the components

E t E t ex xi t x( ) ( ) ( )= +

0ω δ (5.228)

E t E t ey yi t y( ) ( ) .( )= +

0ω δ (5.229)

88 Polarized Light, Third Edition

If we take the real part of these expressions; that is, let

E t E t ex xi t x( ) Re ( ) ( )= [ ]+

0ω δ (5.230)

E t E t ey yi t y( ) Re ( ) ( )= [ ]+

0ω δ (5.231)

then these are equivalent to Equations 5.1 and 5.2. The elements Jij of the coherency matrix J are defined to be

J E ET

E E dt i j x yij i jT

i jT

T

= = =→∞ −∫* *lim ( , , ).

12

(5.232)

It follows that

J Jxy yx= * (5.233)

and so the coherency matrix is Hermitian. The coherency matrix is defined to be the array

J =

=

J J

J J

E E E E

E E E Exx xy

yx yy

x x x y

y x y y

* *

* *

. (5.234)

The trace of this matrix; that is,

Tr J = + = +J J E E E Exx yy x x y y* * (5.235)

is equal to the total intensity of the light.There is a direct connection between the Stokes parameters and the elements of the coherency

matrix. The Stokes parameters for a quasi-monochromatic wave are defined to be (see Equations 5.28 through 5.31)

S E E E Ex x y y0 = +* * (5.236)

S E E E Ex x y y1 = −* * (5.237)

S E E E Ex y y x2 = +* * (5.238)

S i E E E Ex y y x3 = −( )* * , (5.239)

where the angular brackets are time averages. We see immediately from Equations 5.232 and 5.236 through 5.239 that

S J Jxx yy0 = + , (5.240)

S J Jxx yy1 = − , (5.241)

Stokes Polarization Parameters 89

S J Jxy yx2 = + , (5.242)

S i J Jxy yx3 = −( ). (5.243)

Equations 5.240 through 5.243 show that the Stokes parameters and the elements of the coherency matrix are linearly related. A specification of the wave in terms of the coherency matrix is in all respects equivalent to its specification in terms of the Stokes parameters.

There is a very simple way of describing the degree of polarization using the coherency matrix. From Schwarz’s inequality we have

A A dt A A dt A A dt A A dt i j x yi i j j i j i j* * * * , , .∫ ∫∫∫ ≥ = (5.244)

From the definition given by Equation 5.232 it follows that

J J J Jxx yy xy yx≥ (5.245)

or

J J J Jxx yy xy yx− ≥ 0. (5.246)

The equality sign clearly refers to completely polarized light, and the inequality to partially polar-ized light. Furthermore, we see that Equation 5.246 is the determinant of Equation 5.234 so that

det complete polarizationJ = ⇒0 (5.247)

det partial polarization.J > ⇒0 (5.248)

One can readily determine the coherency matrices for various states of polarized light using Equations 5.240 through 5.243. We easily find for unpolarized light that

J =

S0

2

1 0

0 1, (5.249)

for linearly horizontally polarized light

J =

S0

1 0

0 0, (5.250)

and for right circularly polarized light

J =−

S i

i0

2

1

1. (5.251)

The degree of polarization is readily found to be

P = −14

2

det( )

.J

JTr (5.252)

90 Polarized Light, Third Edition

The coherency matrix elements can also be introduced by considering the measurement of the polar-ization state of an optical beam. We recall that the intensity of a beam emerging from a retarder/polarizer combination is

I E E E E E E ex x y y x yiθ φ θ θ θ θφ, cos sin sin cos* * *( ) = + + +−2 2 EE E ex y

i* sin cos .φ θ θ (5.253)

The Stokes parameters were then found by expressing the sinusoidal terms in terms of the half-angle trigonometric formulas. If we had a quasi-monochromatic wave, then we could time-average the quadratic field terms and express Equation 5.253 as

I E E E E

E E e

x x y y

x yi

( , ) cos sin

sin cos

* *

*

θ φ θ θ

θ θφ

= +

+ +−

2 2

EE E ex yi* sin cosφ θ θ

(5.254)

or

I J J J e J exx yy yxi

xyi( , ) cos sin sin cosθ φ θ θ θ θφ φ= + + +−2 2 ssin cos ,θ θ (5.255)

where the Jij are defined to be

J E Eij i j= * , (5.256)

which are the coherency matrix elements.

5.8 STokeS PaRameTeRS aNd The Pauli maTRiCeS

There is a remarkable relation between the Stokes parameters and the coherency matrix. We can rearrange Equations 5.240 through 5.243 to obtain

JS S

xx = +0 1

2, (5.257)

JS S

yy = −0 1

2, (5.258)

JS iS

xy = −2 3

2, (5.259)

JS iS

yx = +2 3

2. (5.260)

The coherency matrix can then be expressed in terms of the Stokes vector elements as

J =

=+ −+ −

J J

J J

S S S iS

S iS Sxx xy

yx yy

12

0 1 2 3

2 3 0 SS1

. (5.261)

One can easily decompose Equation 5.261 into 2 × 2 matrices such that

J ==∑1

20

3

σ i i

i

S , (5.262)

Stokes Polarization Parameters 91

where

σ0

1 0

0 1=

, (5.263)

σ1

0 1

1 0=

, (5.264)

σ2

0

0=

−

i

i, (5.265)

σ3

1 0

0 1=

−

, (5.266)

which are the three Pauli spin matrices of quantum mechanics with the addition of the identity matrix, σ0. This connection between the coherency matrix, the Stokes parameters, and the Pauli spin matrices appears to have been first pointed out by Fano in 1954 [5]. What is even more sur-prising about the appearance of the Pauli spin matrices is that they were introduced into quantum mechanics by Pauli in order to describe the behavior of the spin of the electron, a particle. Indeed, in quantum mechanics the wave function that describes a pure state of polarization can be expanded in a complete set of orthonormal eigenfunctions; it has the same form for electromagnetic radiation and particles of spin 1/2 (e.g., the electron).

RefeReNCeS

1. Stokes, G. G., Trans. camb. Phil. Soc. 9 (1852): 399; Reprinted in Mathematical and Physical Papers, Vol. 3, 233, London: Cambridge University Press, 1901.

2. Chandrasekhar, S., Radiative Transfer, Oxford: Oxford University Press, 1950. 3. Collett, Ε., Am. J. Phys. 36 (1968): 713. 4. Born, M., and E. Wolf, Principles of Optics, 3rd ed., New York: Pergamon Press, 1965. 5. Fano, U., Phys. Rev. 93 (1954): 121.

93

6 Mueller Matrices for Polarizing Components

6.1 iNTRoduCTioN

In the previous chapters we have concerned ourselves with the fundamental properties of polarized light. In this chapter, we now turn our attention to the study of the interaction of polarized light with elements which can change the state of polarization and see that the matrix representation of the Stokes parameters leads to a very powerful mathematical tool for treating this interaction. In Figure 6.1, we show an incident beam, an interaction with a polarizing element, and the emerging beam. The incident beam is characterized by its Stokes parameters Si, where i = 0, 1, 2, 3. The inci-dent polarized beam interacts with the polarizing medium, and the emerging beam is characterized by a new set of Stokes parameters ′Si . We now assume that ′Si can be expressed as a linear combi-nation of the four Stokes parameters of the incident beam by the relations

′ = + + +S m S m S m S m S0 00 0 01 1 02 2 03 3, (6.1)

′ = + + +S m S m S m S m S1 10 0 11 1 12 2 13 3, (6.2)

′ = + + +S m S m S m S m S2 20 0 21 1 22 2 23 3, (6.3)

′ = + + +S m S m S m S m S3 30 0 31 1 32 2 33 3. (6.4)

In matrix form, these equations are written as

′′′′

=

S

S

S

S

m m m m

m m0

1

2

3

00 01 02 03

10 11 mm m

m m m m

m m m m

12 13

20 21 22 23

30 31 32 33

SS

S

S

S

0

1

2

3

(6.5)

or

′ = ⋅S M S, (6.6)

where S and S′ are the Stokes vectors and M is the 4 × 4 matrix known as the Mueller matrix. It was introduced by Hans Mueller during the early 1940s [1]. While Mueller appears to have based his 4 × 4 matrix on a paper by Perrin [2] and a still earlier paper by Soleillet [3], his name is firmly attached to it in the optical literature. Mueller’s important contribution was that he was apparently the first to describe polarizing components in terms of his Mueller matrices. Remarkably, Mueller never published his work on his matrices. Their appearance in the optical literature was due to

94 Polarized Light, Third Edition

others, such as Park [4], a graduate student of Mueller’s who published Mueller’s ideas along with his own contributions and others shortly after the end of the Second World War.

When an optical beam interacts with matter, its polarization state is almost always changed. In fact, this appears to be the rule rather than the exception. The polarization state can be changed by (1) changing the amplitudes of the components of the light, (2) changing the relative phase between orthogonal components, (3) changing the direction of the orthogonal field components, or (4) transferring energy from polarized states to the unpolarized state. An optical element that changes the orthogonal amplitudes unequally is called a polarizer or, more correctly, a diattenua-tor. Similarly, an optical device that introduces a phase shift between the orthogonal components is called a retarder; other names used for the same device are wave plate, compensator, or phase shifter. If the optical device rotates the orthogonal components of the beam through an angle θ as it propagates through the element, it is called a rotator. Finally, if energy in polarized states goes to unpolarized states, the element is a depolarizer.

It should be noted that diattenuation, retardance, and depolarization are the three fundamental properties of a polarization element that are encoded within the Mueller matrix. Rotation can occur through a physical rotation or a phase change, but it essentially results in a coordinate transform. While it is important to understand and have the tools to perform this function, rotation is not a property that can be extracted from a single experimental Mueller matrix, as can diattenuation, retardance, and depolarization (see Chapter 8).

These effects are easily understood by writing the transverse field components for a plane wave as

E z t E t zx x x, ( )( ) = − +0 cos ω κ δ (6.7)

E z t E t zy y y, ( ).( ) = − +0 cos ω κ δ (6.8)

Equations 6.7 and 6.8 can be changed by varying the amplitudes, E0x or E0y, or the phases, δx or δy, and finally, the directions of Ex(z, t) and Ey(z, t). The corresponding devices for causing these respec-tive changes are the diattenuator (polarizer), retarder, and rotator. The use of the names polarizer and retarder arose, historically, before the behavior of these polarizing elements was fully under-stood. The preferable names would be diattenuator for a polarizer and phase shifter for the retarder. All polarizing elements, diattenuator, retarder, rotator, and depolarizer, change the polarization state of an optical beam.

x

y

x’

y’

Incident beam

Emergent beamPolarization element

figuRe 6.1 Interaction of a polarized beam with a polarizing element.

Mueller Matrices for Polarizing Components 95

In the following sections, we derive the Mueller matrices for these polarizing elements. We then apply the Mueller matrix formalism to a number of problems of interest and see its tremendous utility.

6.2 muelleR maTRiX of a liNeaR diaTTeNuaToR (PolaRiZeR)

A diattenuator (polarizer) is an optical element that attenuates the orthogonal components of an optical beam unequally; that is, a diattenuator is an anisotropic attenuator; the two orthogonal trans-mission axes are designated px and py. This element is commonly known as a polarizer; the more recent, accurate, and descriptive term is diattenuator. Because of its historical and embedded use, we will make concessions to convention, and make free use of the term polarizer.

A polarizer is sometimes described also by the terms generator and analyzer to refer to its use and position in the optical system. If a polarizer is used to create polarized light, we call it a genera-tor. If it is used to “analyze” polarized light, by placing it immediately before the detection device, it is called an analyzer. If the orthogonal components of the incident beam are attenuated equally, then the polarizer becomes a neutral density filter. We now derive the Mueller matrix for a polarizer.

In Figure 6.2, a polarized beam is shown incident on a polarizer along with the emerging beam. The components of the incident beam are represented by Ex and Ey. After the beam emerges from the polarizer, the components are ′Ex and ′Ey , and they are parallel to the original axes. The fields are related by

′ = ≤ ≤E p E px x x x0 1 (6.9)

′ = ≤ ≤E p E py y y y0 1. (6.10)

The factors px and pY are the amplitude attenuation coefficients along orthogonal transmission axes. For no attenuation or perfect transmission along an orthogonal axis px (or py) = 1, whereas for com-plete attenuation px (or py) = 0. If one of the axes has an amplitude attenuation coefficient that is zero so that there is no transmission along this axis, the polarizer is said to have only a single transmis-sion axis.

The Stokes polarization parameters of the incident and emerging beams are, respectively,

S E E E Ex x y y0 = +* * (6.11)

S E E E Ex x y y1 = −* * (6.12)

Ey

E’x

E’y

px

py

figuRe 6.2 The Mueller matrix of a polarizer with attenuation coefficients px and py.

96 Polarized Light, Third Edition

S E E E Ex y y x2 = +* * (6.13)

S i E E E Ex y y x3 = −( )* * (6.14)

and

′ = ′ ′ + ′ ′S E E E Ex x y y0* * (6.15)

′ = ′ ′ − ′ ′S E E E Ex x y y1* * (6.16)

′ = ′ ′ + ′ ′S E E E Ex y y x2* * (6.17)

′ = ′ ′ − ′ ′S i E E E Ex y y x3 ( ).* * (6.18)

Substituting Equations 6.9 and 6.10 into Equations 6.15 through 6.18 and using Equations 6.11 through 6.14, we have

′′′′

=

+ −S

S

S

S

p p p px y x y0

1

2

3

2 2 2 2

12

0 0

pp p p p

p p

p p

x y x y

x y

x y

2 2 2 2 0 0

0 0 2 0

0 0 0 2

− +

S

S

S

S

0

1

2

3

. (6.19)

The 4 × 4 matrix in Equation 6.19 is written by itself as

M =

+ −− +1

2

0 0

0 0

0 0 2

2 2 2 2

2 2 2 2

p p p p

p p p p

p p

x y x y

x y x y

x y 00

0 0 0 2

0 1

p p

p

x y

x y

≤ ≤, . (6.20)

Equation 6.20 is the Mueller matrix for a polarizer with amplitude attenuation coefficients px and py. The quantities k px x= 2 and k py y= 2 are observables called the principal (intensity) transmittances. In general, the existence of the m33 term shows that the polarization of the emerging beam of light will be elliptically polarized. For a neutral density filter, px = py = p and Equation 6.20 becomes

M =

p2

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

, (6.21)

which is a unit diagonal matrix. Equation 6.21 shows that the polarization state is not changed by a neutral density filter, but the intensity of the incident beam is reduced by a factor of p2. This is the expected behavior of a neutral density filter, since it only affects the magnitude of the intensity and not the polarization state. According to Equation 6.21, the emerging intensity I′ is then

I p I′ = 2 , (6.22)

where I is the intensity in the incident beam.

Mueller Matrices for Polarizing Components 97

Equation 6.20 is the Mueller matrix for a polarizer that is described by unequal attenuations along the px and py axes. An ideal linear polarizer is one that has transmission along only one axis and no transmission along the orthogonal axis. This behavior can be described by first setting one of the attenuation coefficients, say py, to 0. Then Equation 6.20 reduces to

M =

px2

2

1 1 0 0

1 1 0 0

0 0 0 0

0 0 0 0

. (6.23)

Equation 6.23 is the Mueller matrix for an ideal linear polarizer that polarizes only along the x axis. It is most often called a linear horizontal polarizer, arbitrarily assigning the horizontal to the x direction. It would be a perfect linear polarizer if the transmission factor px was unity (px = 1). Thus, the Mueller matrix for an ideal perfect linear polarizer with its transmission axis in the x direction is

M =

12

1 1 0 0

1 1 0 0

0 0 0 0

0 0 0 0

. (6.24)

If the original beam is completely unpolarized, the maximum intensity of the emerging beam that can be obtained with a perfect ideal polarizer is only 50% of the original intensity. It is the price we pay for obtaining perfectly polarized light. If the original beam is perfectly horizontally polarized, there is no change in intensity. This element is called a linear polarizer because it affects a linearly polarized beam in a unique manner as we shall soon see.

In general, all linear polarizers are described by Equation 6.20. There is only one known natural material that comes close to approaching the perfect ideal polarizer described by Equation 6.24, and this is calcite. A synthetic material known as Polaroid is also used as a polarizer. Its performance is not as good as calcite, but its cost is very low in comparison with that of natural calcite polar-izers (e.g., a Glan–Thompson prism). Nevertheless, there are a few types of Polaroid that perform extremely well as “ideal” polarizers. We shall discuss the topic of calcite, Polaroid, and other polar-izers in the chapter on polarization optical elements.

For an ideal perfect linear polarizer in which the role of the transmission axes is reversed from that of our linear horizontal polarizer, that is, px = 0 and py = 1, then Equation 6.20 reduces to

M =

−−

12

1 1 0 0

1 1 0 0

0 0 0 0

0 0 0 0

, (6.25)

which is the Mueller matrix for a linear vertical polarizer.We can rewrite the Mueller matrix of a general linear polarizer, Equation 6.20, in terms of trigo-

nometric functions. This can be done by setting

p p px y2 2 2+ = (6.26)

98 Polarized Light, Third Edition

and defining px and py as

p p p px y= =cos sinγ γ . (6.27)

Substituting Equations 6.26 and 6.27 into Equation 6.20 yields

M =

p2

2

1 2 0 0

2 1 0 0

0 0 2 0

0 0 0 2

cos

cos

sin

sin

γγ

γγ

, (6.28)

where 0 ≤ γ ≤ 90°. For an ideal perfect linear polarizer p = 1. For a linear horizontal polarizer γ = 0, and for a linear vertical polarizer γ = 90°. The usefulness of the trigonometric form of the Mueller matrix in Equation 6.28 will appear later.

We can show that Equations 6.24 and 6.25 describe linear polarizers. Suppose we have an inci-dent beam of arbitrary intensity and polarization so that its Stokes vector is

S =

S

S

S

S

0

1

2

3

. (6.29)

We now form the product of Equation 6.24 or Equation 6.25 and Equation 6.29 and we can write

′′′′

=

±±

S

S

S

S

0

1

2

3

12

1 1 0 0

1 1 0 0

0 0 0 0

0 0 00 0

0

1

2

3

S

S

S

S

, (6.30)

which gives us the result

′′′′

= ±±

S

S

S

S

S S

0

1

2

3

0 112

1

1

0

0

( )

. (6.31)

Inspecting Equation 6.31, we see that the Stokes vector of the emerging beam is always linearly horizontally (+) or vertically (−) polarized. Thus an ideal linear polarizer always creates linearly polarized light regardless of the polarization state of the incident beam; however, note that because for real polarizers the factor 2pxpy in Equation 6.20 is never exactly zero, in practice there is no known perfect linear polarizer and all polarizers create elliptically polarized light. While the ellip-ticity may be small and, in fact, negligible, there is always some present.

The behavior of linear polarizers described above allows us to develop a test to determine if a polarizing element is actually a linear polarizer. The test to determine if we have a linear polarizer is shown in Figure 6.3. We assume that we have a linear polarizer and set its axis in the horizontal

Mueller Matrices for Polarizing Components 99

(H) direction. We then take another polarizer and set its axis in the vertical (V) direction as shown in the figure. The Stokes vector of the incident beam is S, and the Stokes vector of the beam emerg-ing from the first (horizontal) polarizer is

S M S′ = H . (6.32)

The beam now propagates to the second (vertical) polarizer, and the Stokes vector S″ of the emerg-ing beam is now

S M S M M S MS″ = ′ = =V V H , (6.33)

where we have used Equation 6.32 for S′. We see that M is the Mueller matrix of the combined vertical and linear polarizer; that is,

M M M= V H , (6.34)

where MH and MV are given by Equations 6.24 and 6.25, respectively. These results, Equations 6.33 and 6.34, show that we can relate the Stokes vector of the emerging beam to the incident beam by merely multiplying the Mueller matrix of each component and finding the resulting Mueller matrix. In general, the matrices do not commute. Commutation would imply that the order of the physi-cal polarization elements does not matter, and this is not the case as can easily be experimentally verified.

We now carry out the multiplication in Equation 6.34 and write, using Equations 6.24 and 6.25,

M =

−−

14

1 1 0 0

1 1 0 0

0 0 0 0

0 0 0 0

1 1 0 0

1 1 0 0

0 00 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

=

. (6.35)

We have obtained a null Mueller matrix and hence a null output intensity regardless of the polariza-tion state of the incident beam. The appearance of a null Mueller matrix (or intensity) occurs only when the linear polarizers are in the crossed polarizer configuration. Furthermore, the null Mueller matrix always arises whenever the polarizers are crossed, regardless of the angle of the transmission axis of the first polarizer.

Horizontal polarizerVertical polarizer

MH

MV

S

S’

S’’

figuRe 6.3 Testing for a linear polarizer.

100 Polarized Light, Third Edition

6.3 muelleR maTRiX of a liNeaR ReTaRdeR

A retarder is a polarizing element that changes the phase of the optical beam. Strictly speaking, its correct name is phase shifter. However, historical usage has led to the alternative names retarder, wave plate, and compensator. Retarders introduce a phase shift of ϕ between the orthogonal compo-nents of the incident field. For mathematical convenience, this phase shift ϕ is split evenly between orthogonal directions and so can be thought of as being accomplished by a phase shift of + ϕ/2 along the x axis and a phase shift of –ϕ/2 along the y axis (but note that for normal materials there is a net physical phase delay along both axes, the difference of which is ϕ). These axes of the retarder are referred to as the fast and slow axes, respectively. In Figure 6.4, we show the incident and emerging beam and the retarder. The components of the emerging beam are related to the incident beam by

′ = +E z t e E z txi

x( , ) ( , )/φ 2 (6.36)

′ = −E z t e E z tyi

y( , ) ( , )./φ 2 (6.37)

Referring again to the definition of the Stokes parameters in Equations 6.11 through 6.18 and sub-stituting Equations 6.36 and 6.37 into these equations, we find that

′ =S S0 0 , (6.38)

′ =S S1 1, (6.39)

′ = +S S S2 2 3cos sinφ φ, (6.40)

′ = − +S S S3 2 3sin cosφ φ. (6.41)

These Equations 6.38 through 6.41 can be written in matrix form as

′′′′

=

S

S

S

S

0

1

2

3

1 0 0 0

0 1 0 0

0 0 cos sinφ φ00 0

0

1

2

3−

sinφ φcos

S

S

S

S

. (6.42)

EX

Ey

E’X

E’y

+ /2

– /2

figuRe 6.4 Propagation of a polarized beam through a retarder.

Mueller Matrices for Polarizing Components 101

Note that for an ideal phase shifter (retarder) there is no loss in intensity; that is, ′ =S S0 0 .The Mueller matrix for a retarder with a phase shift ϕ with fast axis horizontal is, from

Equation 6.42,

M =

−

1 0 0 0

0 1 0 0

0 0

0 0

cos sin

sin cos

φ φφ φ

.. (6.43)

There are two special cases of Equation 6.43 which often appear in polarizing optics. These are the cases for quarter-wave retarders (ϕ = 90°, i.e., the phase of one component of the light is delayed with respect to the orthogonal component by one quarter wave) and half-wave retarders (ϕ = 180°, i.e., the phase of one component of the light is delayed with respect to the orthogonal component by one half wave). A retarder is naturally dependent on wavelength, although there are specially con-structed achromatic retarders that are slowly dependent on wavelength. We will discuss these topics in more detail in the chapter on polarization optical elements. For a quarter-wave retarder with its fast axis along the x axis, Equation 6.43 becomes

M =

−

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

. (6.44)

The quarter-wave retarder has the property that it transforms a linearly polarized beam with its axis oriented at + 45° or −45° relative to the fast axis of the retarder into a right or left circularly polar-ized beam. Consider the Stokes vector for a linearly polarized ± 45° beam

S =±

I0

1

0

1

0

. (6.45)

Multiplying Equation 6.45 by Equation 6.44 yields

′ =

S I0

1

0

0

1∓

, (6.46)

which is the Stokes vector for left or right circularly polarized light. The transformation of linearly polarized light to circularly polarized light is an important application of quarter-wave retarders. However, circularly polarized light is obtained only if the incident linearly polarized light is ori-ented at ±45° with respect to the retarder axes.

102 Polarized Light, Third Edition

On the other hand, if the incident light is right (left) circularly polarized light, then multiplying Equation 6.46 by Equation 6.44 yields

′ =

S I0

1

0

1

0

∓, (6.47)

which is the Stokes vector for linear −45° or + 45° polarized light. The quarter-wave retarder can be used to transform linearly polarized light to circularly polarized light or circularly polarized light to linearly polarized light.

The other important type of wave retarder is the half-wave retarder (ϕ = 180°). For this condition, Equation 6.43 reduces to, for a retarder with its fast axis along the x axis,

M =−

−

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

. (6.48)

A half-wave retarder is characterized by a diagonal matrix. The terms m22 = m33 = –1 reverse the ellipticity and orientation of the polarization state of the incident beam. Recall that the orientation angle ψ and the ellipticity angle χ are given in terms of the Stokes parameters as

tan2 2

1

ψ = SS

(6.49)

sin .2 3

0

χ = SS

(6.50)

Multiplying the Stokes vector Equation 6.29 by Equation 6.48 gives us

′ =

′′′′

=−−

S

S

S

S

S

S

S

S

S

0

1

2

3

0

1

2

3

, (6.51)

where

tan ,2 2

1

′ = ′′

ψ SS

(6.52)

sin .2 3

0

′ = ′′

χ SS

(6.53)

Using the Stokes vector elements from Equation 6.51 in Equations 6.52 and 6.53 yields

tan tan ,2 22

1

′ = − = −ψ ψSS

(6.54)

Mueller Matrices for Polarizing Components 103

sin sin ,2 23

0

′ = − = −χ χSS

(6.55)

and therefore,

′ = ° −ψ ψ90 , (6.56)

′ = ° +χ χ90 . (6.57)

Half-wave retarders also possess the property that they can rotate the polarization ellipse. This important property shall be discussed in Section 6.5.

6.4 muelleR maTRiX of a RoTaToR

Another way to change the polarization state of an optical field is to allow a beam to propagate through a polarizing element that rotates the orthogonal field components Εx(z, t) and Ey(z, t) through an angle θ. In order to derive the Mueller matrix for rotation, we consider Figure 6.5. The angle θ describes the rotation of Ex to ′Ex and of Ey to ′Ey . Similarly, the angle β is the angle between Ε and Ex. In the figure, the point Ρ is described in the ′Ex , ′Ey coordinate system by

′ = −E Ex cos( )β θ (6.58)

′ = −E Ey sin( ).β θ (6.59)

In the Ex, Ey coordinate system we have

E Ex = cos β (6.60)

E Ey = sin β. (6.61)

O

E’yEy

P

Ex

E’x

θ

β

E

figuRe 6.5 Rotation of the optical field components by a rotator.

104 Polarized Light, Third Edition

Expanding the trigonometric functions in Equations 6.58 and 6.59 gives

′ = +E Ex (cos cos sin sin )β θ β θ (6.62)

′ = −E Ey (sin cos sin cos ).β θ θ β (6.63)

Collecting terms in Equations 6.62 and 6.63 using Equations 6.60 and 6.61 then gives

′ = +E E Ex x ycos sinθ θ (6.64)

′ = − +E E Ey x ysin cos .θ θ (6.65)

Equations 6.64 and 6.65 are the amplitude equations for rotation. In order to find the Mueller matrix we form the Stokes parameters for Equations 6.64 and 6.65 as before and find the Mueller matrix for rotation to be

MRot ( )cos sin

sin cos2

1 0 0 0

0 2 2 0

0 2 2 0

0 0 0 1

θθ θθ θ

=−

. (6.66)

We note that a physical rotation of θ leads to the appearance of 2θ in the Mueller matrix, Equation 6.66, because we are working in the intensity domain; in the amplitude domain we would expect just θ.

Rotators are primarily used to change the orientation angle of the polarization ellipse. The ori-entation of the polarization ellipse of the incident beam is again given by Equation 6.49 and the emergent beam has the orientation given by Equation 6.52. Forming the product S′ = MRotS using Equation 6.66, we see that the orientation angle ψ′ is then

tansin cos

cos sin.2

2 22 2

1 2

1 2

′ = − ++

ψ θ θθ θ

S SS S

(6.67)

Equation 6.49 is now written as

S S2 1 tan 2= ψ. (6.68)

Substituting Equation 6.68 into 6.67, we find that

tan 2 tan 2 2ψ ψ θ′ = −( ), (6.69)

so

ψ ψ θ′ = − . (6.70)

Equation 6.70 shows that a rotator merely rotates the polarization ellipse of the incident beam; we can also show that the ellipticity remains unchanged. The sign of θ is negative in Equation 6.70 because the rotation is clockwise. If the rotation is counterclockwise, that is, if θ is replaced by −θ in Equation 6.66, then we find

ψ ψ θ′ = + . (6.71)

Mueller Matrices for Polarizing Components 105

In the derivation of the Mueller matrices for a polarizer, retarder, and rotator, we have assumed that the axes of these devices are aligned along the Ex and Ey (or x and y) axes. In practice, we find that the polarization elements are often rotated. Consequently, it is also necessary for us to know the form of the Mueller matrices for the rotated polarizing elements. We now consider this problem.

6.5 muelleR maTRiCeS foR RoTaTed PolaRiZiNg ComPoNeNTS

To derive the Mueller matrix for rotated polarizing components, we refer to Figure 6.6. The axes of the polarizing component are seen to be rotated through an angle θ to the x′ and y′ axes. We must, therefore, also consider the components of the incident beam along the x′ and y′ axes. In terms of the Stokes vector of the incident beam, S, we have

S M S′ = ( )2Rot θ , (6.72)

where MRot(2θ) is the Mueller matrix for rotation, Equation 6.66, and S′ is the Stokes vector of the beam whose axes are along x′ and y′.

The S′ beam now interacts with the polarizing element characterized by the Mueller matrix M. The Stokes vector S″ of the beam emerging from the rotated polarizing component is

S MS MM S″ = ′ = ( )Rot 2θ , (6.73)

where we have used Equation 6.72. Finally, we must take the components of the emerging beam along the original x and y axes as seen in Figure 6.6. This can be described by a counterclockwise rotation of S″ through −θ, back to the original x, y axes, so

′′′ = − ′′

= −[ ]

S M S

M MM S

Rot

Rot Rot

( )

( ) ( ) ,

2

2 2

θ

θ θ (6.74)

where MRot(−2θ) is, again, the Mueller matrix for rotation and S′″ is the Stokes vector of the emerg-ing beam. Equation 6.74 can be written as

′′′ = ( )S M S2θ , (6.75)

Ex

Ey

E’’’x

E’’’y

xSS’

y

Incident beam

Emergent beamRotated polarization element

S’’ S’’’

x’y’

M(2θ)

θ

figuRe 6.6 Derivation of the Mueller matrix for rotated polarizing components.

106 Polarized Light, Third Edition

where

M M MM2 2 2θ θ θ( ) = −( ) ( )Rot Rot . (6.76)

Equation 6.76 is the Mueller matrix of a rotated polarizing component.The rotated Mueller matrix expressed by Equation 6.76 appears often in the treatment of polarized

light. Of particular interest are the Mueller matrices for a rotated polarizer and a rotated retarder. The Mueller matrix for a rotated rotator is also interesting, but in a different way. We recall that a rotator rotates the polarization ellipse by an amount θ. If the rotator is now rotated through an angle α, then one discovers, using Equation 6.76, that M(2θ) = MRot(2θ); that is, the rotator is unaffected by a mechanical rotation. The polarization ellipse cannot be rotated by rotating a rotator! The rota-tion comes about only by the intrinsic behavior of the rotator. It is possible, however, to rotate the polarization ellipse mechanically by rotating a half-wave plate, as we shall soon demonstrate.

The Mueller matrix for a rotated polarizer is most conveniently found by expressing the Mueller matrix of a polarizer in angular form; namely, the form earlier expressed as

M =

p2

2

1 2 0 0

2 1 0 0

0 0 2 0

0 0 0 2

cos

cos

sin

sin

γγ

γγ

. (6.77)

Carrying out the matrix multiplication according to Equation 6.76 and using Equation 6.66, the Mueller matrix for a rotated polarizer is

M =+1

2

1 2 2 2 2 0

2 2 22

cos cos cos sin

cos cos cos

γ θ γ θγ θ θ ssin sin ( sin )sin cos

cos sin (

2 2 1 2 2 2 0

2 2

2γ θ γ θ θγ θ

−11 2 2 2 2 2 2 0

0 0 0

2 2− +sin )sin cos sin sin cos

si

γ θ θ θ γ θnn

.

2γ

(6.78)

In Equation 6.78 we have set p2 to unity. We note that γ = 0°, 45°, and 90° corresponds to a linear horizontal polarizer, a neutral density filter, and a linear vertical polarizer, respectively.

The most common form of Equation 6.78 is the Mueller matrix for an ideal linear horizontal polarizer (γ = 0°). For this value, Equation 6.78 reduces to

MP ( )

cos sin

cos cos sin cos2

12

1 2 2 0

2 2 2 2 02

θ

θ θθ θ θ θ

=ssin sin cos sin

.2 2 2 2 0

0 0 0 0

2θ θ θ θ

(6.79)

In Equation 6.79, we have written MP(2θ) to indicate that this is the Mueller matrix for a rotated ideal linear polarizer. The form of Equation 6.79 can be checked immediately by setting θ = 0 (no rotation). Upon doing this, we obtain the Mueller matrix of a linear horizontal polarizer

MP ( ) .012

1 1 0 0

1 1 0 0

0 0 0 0

0 0 0 0

° =

(6.80)

Mueller Matrices for Polarizing Components 107

One can readily see that for θ = 45°and 90°, Equation 6.79 reduces to the Mueller matrix for an ideal linear + 450 and vertical polarizer, respectively. The Mueller matrix for a rotated ideal linear polar-izer, Equation 6.79, appears often in the generation and analysis of polarized light.

We now determine the Mueller matrix for a rotated retarder or wave plate. We recall that the Mueller matrix for a retarder with phase shift ϕ is given by

MR =

−

1 0 0 0

0 1 0 0

0 0

0 0

cos sin

sin cos

φ φφ φ

. (6.81)

From Equation 6.76, the Mueller matrix for the rotated retarder Equation 6.81 is found to be

MR ( , )cos cos sin ( cos )sin

φ θθ φ θ φ

2

1 0 0 0

0 2 2 1 22 2

=+ − θθ θ φ θφ θ θ θ

cos sin sin

( cos )sin cos sin

2 2

0 1 2 2 22

−− + ccos cos sin cos

sin sin sin cos cos

φ θ φ θφ θ φ θ

2 2 2

0 2 2− φφ

. (6.82)

For θ = 0°, Equation 6.82 reduces to Equation 6.81 as expected. There is a particularly interesting form of Equation 6.82 for a phase shift of ϕ = 180°, a so-called half-wave retarder. For ϕ = 180°, Equation 6.82 reduces to

MR( , )cos sin

sin cos180 4

1 0 0 0

0 4 4 0

0 4 4 0

0 0

° θθ θθ θ

=−

00 1−

. (6.83)

Equation 6.83 looks very similar to the Mueller matrix for rotation Mrot, Equation 6.66, which we write as

MRot

cos=

−

1 0 0 0

1 2 2 0

0 2 2 0

0 0 0 1

θ θθ θ

sin

sin cos

. (6.84)

However, Equation 6.83 differs from Equation 6.84 in some essential ways. The first is the elliptic-ity. The Stokes vector of an incident beam is, as usual,

S =

S

S

S

S

0

1

2

3

. (6.85)

108 Polarized Light, Third Edition

Having a rotator act on an incident Stokes vector, that is, multiplying Equation 6.85 by Equation 6.84 yields the emergent Stokes vector

′ =+

− +

S

S

S S

S S

S

0

1 2

1 2

3

2 2

2 2

cos sin

sin cos

θ θθ θ

. (6.86)

The ellipticity angle χ′ is

sin sin ,2 23

0

3

0

′ = ′′

= =χ χSS

SS

(6.87)

and therefore the ellipticity is not changed under true rotation. Having a half-wave retarder act on an incident Stokes vector (i.e., multiplying Equation 6.85 by Equation 6.83) yields a Stokes vector

′ =+−

−

S

S

S S

S S

S

0

1 2

1 2

3

4 4

4 4

cos sin

sin cos

θ θθ θ

. (6.88)

The ellipticity angle χ′ is now

sin sin ,2 23

0

3

0

′ = ′′

= − = −χ χSS

SS

(6.89)

and this means that

χ χ′ = + °90 , (6.90)

so the ellipticity angle χ of the incident beam is advanced 90° by using a rotated half-wave retarder.

The next difference is for the orientation angle ψ′. For a rotator, Equation 6.84, the orientation angle ψ associated with the incident beam is given by

tan ,2 2

1

ψ = S

S (6.91)

so we immediately find from Equations 6.91 and 6.86 that

tansin cos sin coscos cos

22 2 2 22

2

1

′ = ′′

= −ψ ψ θ θ ψψ

S

S 22 2 22 22 2θ ψ θψ θψ θ+

= −−sin sin

sin( )cos( )

, (6.92)

consequently

ψ ψ θ′ = − . (6.93)

Mueller Matrices for Polarizing Components 109

Equation 6.93 shows that a mechanical rotation in θ increases ψ by the same amount and in the same direction (by definition, a clockwise rotation of θ is an increase). On the other hand, for a half-wave retarder the orientation angle ψ′ is given by the equation, using Equations 6.85 and 6.88,

tancos sin sin coscos cos sin

22 4 2 42 4 2

′ = −+

ψ ψ θ ψ θψ θ ψψ θ

θ ψθ ψsin

sin( )cos( )

,4

4 24 2

= −−

(6.94)

so

ψ θ ψ′ = −2 (6.95)

or

ψ ψ θ′ = − −( )2 . (6.96)

Comparing Equation 6.96 with Equation 6.93, we see that rotating the half-wave retarder clockwise causes ψ′ to rotate counterclockwise by an amount twice that of a rotator. Because the rotation of a half-wave retarder is opposite to a true rotator, it is called a pseudorotator.

To summarize explicitly, when a mechanical rotation of θ is made using a half-wave retarder (i.e., a pseudorotator) the polarization ellipse is rotated by 2θ and in a direction opposite to the direction of the mechanical rotation. For a true rotator undergoing a mechanical rotation of θ, the polarization ellipse is rotated by an amount θ and in the same direction as the rotation.

This discussion of rotation of half-wave retarders is more than academic, however. Very often, manufacturers sell half-wave retarders as polarization rotators. Strictly speaking, the belief that rotation will occur is quite correct. However, one must realize that the use of a half-wave retarder rather than a true rotator requires a mechanical mount with twice the resolution. That is, if we use a rotator in a mount with, say, 2′ of resolution, then in order to obtain the same resolution with a half-wave retarder, a mechanical mount with 1′ of resolution is required. The simple fact is that doubling the resolution of a mechanical mount can be very expensive in comparison with using a true rota-tor. The cost for doubling the resolution of a mechanical mount can easily double, whereas the cost increase between a quartz rotator and a half-wave retarder is usually much less. In general, if the objective is to rotate the polarization ellipse by a known fixed amount, it is better to use a rotator rather than a half-wave retarder.

A half-wave retarder is very useful as a rotator. Half-wave retarders can also be used to “reverse” the polarization state. In order to illustrate this behavior, consider that we have an incident beam that is right or left circularly polarized. Its Stokes vector is

S =

±

I0

1

0

0

1

. (6.97)

Multiplying Equation 6.97 by Equation 6.83 and setting θ = 0° yields

′ =

S I0

1

0

0

1∓

. (6.98)

110 Polarized Light, Third Edition

We see that we again obtain circularly polarized light but orthogonal to its original state; that is, right circularly polarized light is transformed to left circularly polarized light, and vice versa. Similarly, if we have incident linear + 45° polarized light, the emerging beam is linear −45° polar-ized light. It is this property of reversing the ellipticity and the orientation, manifested by the nega-tive sign in m22 and m33 that also makes half-wave plates very useful.

Finally, we consider the Mueller matrix of a rotated quarter-wave retarder. We set ϕ = 90° in Equation 6.83 and we have

MR( , )cos sin cos sin

sin90 2

1 0 0 0

0 2 2 2 2

0 2

2

° θθ θ θ θ

=−

θθ θ θ θθ θ

cos sin cos

sin cos

2 2 2

0 2 2 0

2

−

.. (6.99)

Consider that we have an incident linearly horizontally polarized beam, so its Stokes vector is (I0 = 1)

S =

1

1

0

0

. (6.100)

We multiply Equation 6.100 by Equation 6.99, and we find that the emergent Stokes vector S′ is

′ =

S

1

2

2 2

2

2cos

sin cos

sin

.θ

θ θθ

(6.101)

We see from Equation 6.101 that the orientation angle ψ′ and the ellipticity angle χ′ of the emerging beam are given by

tan 2 tan 2ψ θ′ = (6.102)

sin 2 sin 2χ θ′ = . (6.103)

The rotated quarter-wave plate has the useful property that it can be used to generate any desired orientation and ellipticity starting with an incident linearly horizontally polarized beam. However, we must choose to generate either a value of orientation or a value of ellipticity; we have no control over the other parameter. We also note that if we initially have right or left circularly polarized light the Stokes vector of the output beam is

′ =±

S

1

2

2

0

∓sin

cos,

θθ

(6.104)

which is the Stokes vector for linearly polarized light. While it is well known that a quarter-wave retarder can be used to create linearly polarized light, Equation 6.104 shows that an additional variation is possible by rotating the retarder (i.e., the orientation can be controlled).

Mueller Matrices for Polarizing Components 111

Equation 6.104 shows that, using a quarter-wave retarder, we can generate any desired orienta-tion or ellipticity of a beam, but not both. This raises the question of how we can generate an ellipti-cally polarized beam of any desired orientation and ellipticity regardless of the polarization state of an incident beam. We answer this in the last section of the chapter.

6.6 geNeRaTioN of elliPTiCally PolaRiZed lighT

In the previous section, we derived the Mueller matrices for a rotated polarizer and a rotated retarder. We now apply these matrices to the generation of an elliptically polarized beam of any desired orien-tation and ellipticity. Refer to Figure 6.7. In the figure, we have an incident beam of arbitrary polar-ization. The beam propagates first through an ideal polarizer rotated through an angle θ and then through a retarder with its fast axis along the x axis. The Stokes vector of the incident beam is

S =

S

S

S

S

0

1

2

3

. (6.105)

It is important that we consider the optical source to be arbitrarily polarized. At first sight, for example, we might wish to use unpolarized light or linearly polarized light. However, completely unpolarized light is surprisingly difficult to generate, and the requirement to generate ideal linearly polarized light calls for an excellent linear polarizer. We can avoid this problem if we consider that the incident beam is of unknown but arbitrary polarization. Our objective is to create an elliptically polarized beam of any desired ellipticity and orientation and which is totally independent of the polarization state of the incident beam.

The Mueller matrix of a rotated ideal linear polarizer is

MP ( )

cos sin

cos cos sin cos2

12

1 2 2 0

2 2 2 2 02

θ

θ θθ θ θ θ

=ssin sin cos sin

.2 2 2 2 0

0 0 0 0

2θ θ θ θ

(6.106)

Ex

Ey

Incident beam

Elliptically polarized beam

Rotated polarizer

θ

Retarder

+ /2

– /2

figuRe 6.7 The generation of elliptically polarized light.

112 Polarized Light, Third Edition

Multiplying Equation 6.105 by Equation 6.106 yields

′ = + +

S12

2 2

1

2

2

0

0 1 2( cos sin )cos

sinS S Sθ θ

θθ

. (6.107)

The Mueller matrix of the retarder of retardance ϕ (nonrotated) is

MR =

−

1 0 0 00 1 0 00 0

0 0

cos sin

sin cos

.φ φφ φ

(6.108)

Multiplying Equation 6.107 by Equation 6.108 then gives the Stokes vector of the beam emerging from the retarder as

′′ =

−

S I( )cos

cos sin

sin sin

,θθ

φ θφ θ

12

2

2

(6.109)

where

I S S S( ) ( cos sin ).θ θ θ= + +12

2 20 1 2 (6.110)

Equation 6.109 is the Stokes vector of an elliptically polarized beam. We find from Equation 6.109 that the orientation angle ψ (dropping the double prime) is

tan 2 cos tan 2ψ φ θ= , (6.111)

and the ellipticity angle χ is

sin 2 sin sin 2χ φ θ= − . (6.112)

We must now determine the θ and ϕ that will generate the desired values of ψ and χ. We divide Equation 6.111 by tan 2θ and Equation 6.112 by sin 2θ, square the results, and add. We obtain

cos 2 cos 2 cos 2θ χ ψ= ± . (6.113)

To determine the required phase shift ϕ, we divide Equation 6.112 by Equation 6.111 and obtain

sintan

tan cos .22

2χψ

φ θ= − (6.114)

Mueller Matrices for Polarizing Components 113

Solving for tan ϕ and using Equation 6.113, we find that

tantansin

.φ χψ

=∓ 22

(6.115)

Equations 6.113 and 6.115 define the angles θ and ϕ to which the polarizer and the retarder must be set in order to obtain the desired ellipticity and orientation angles χ and ψ.

We have shown that by using only two elements, a rotated ideal linear polarizer and a retarder, we can generate any state of elliptically polarized light. There is a final interesting fact about Equations 6.113 and 6.115. We write Equation 6.113 and 6.115 as the pair

cos 2 cos 2 cos 2θ χ ψ= ± (6.116)

tan sin tan .2 2χ ψ φ=∓ (6.117)

Equations 6.116 and 6.117 are recognized as equations arising from spherical trigonometry for a right spherical triangle. In Figure 6.8, we have drawn a right spherical triangle. The angle 2ψ (the orientation of the polarization ellipse) is plotted on the equator of a sphere, and the angle 2χ (the ellipticity of the polarization ellipse) is plotted on the longitude. If a great circle is drawn from point A to point b, the length of the arc Ab is given by Equation 6.116 and corresponds to 2θ as shown in the figure. Similarly, the phase ϕ is the angle between the arc Ab and the equator; its value is given by Equation 6.117. We see from Figure 6.8 that we can easily determine θ and ϕ by (1) measuring the length of the arc Ab and (2) measuring the angle between the arc Ab and the equator on a sphere.

The polarization Equations 6.116 and 6.117 are intimately associated with spherical trigonom-etry and a sphere. Furthermore, we recall from Section 5.3 that when the Stokes parameters were expressed in terms of the orientation angle and the ellipticity angle, they led directly to the Poincaré sphere. In fact, Equations 6.116 and 6.117 describe a spherical triangle that plots directly onto the Poincaré sphere. We see that even at this early stage in our study of polarized light, there is a strong connection between the equations of polarized light and its representation on a sphere. In fact, one of the most remarkable properties of polarized light is that there is such a close relation between

2θ

2ψ

2

A

B

C

figuRe 6.8 A right spherical triangle drawn on the surface of a sphere.

114 Polarized Light, Third Edition

these equations and the equations of spherical trigonometry. These relations will be discussed in depth in the chapter on the Poincaré sphere.

6.7 muelleR maTRiX of a dePolaRiZeR

A depolarizer transfers energy out of polarized states and into depolarized states. Mueller matrices have 16 degrees of freedom of which nine are associated with depolarization. The Mueller matrix for an ideal depolarizer is given by

1 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

. (6.118)

All polarized light is changed to unpolarized light through interaction with an ideal depolarizer. The Mueller matrix for a pure uniform partial depolarizer is given by

1 0 0 0

0 0 0

0 0 0

0 0 0

a

a

a

. (6.119)

The amount of polarization is uniformly reduced for all incident states. The Mueller matrix for a pure nonuniform partial depolarizer is given by

1 0 0 0

0 0 0

0 0 0

0 0 0

a

b

c

. (6.120)

Incident polarization states are reduced by an amount that is dependent on the incident state. It is not uncommon for experimental Mueller matrices for the reflected light from rough surfaces to take the form

1 0 0 0

0 0 0

0 0 0

0 0 0

a

a

a

−−

(6.121)

or

1 0 0 0

0 0 0

0 0 0

0 0 0

a

b

c

−−

. (6.122)

These results are the products of the Mueller matrix of a uniform or nonuniform partial depolarizer, and the Mueller matrix of a mirror, respectively.

Mueller Matrices for Polarizing Components 115

RefeReNCeS

1. Mueller, H., J. Opt. Soc. Am. 37 (1947): 110. 2. Perrin, F., J. chem. Phys. 10 (1942): 415. 3. Soleillet, P., Ann. Phys. 12, no. 10 (1929): 23. 4. Parke, III, N. G., Statistical Optics. II: Mueller Phenomenological Algebra, RLE TR-119, Research

Laboratory of Elect, at M.I.T., 1949.

117

7 Fresnel Equations: Derivation and Mueller Matrix Formulation

7.1 iNTRoduCTioN

The interaction of light beams with matter is described using the Fresnel equations for reflection and transmission. The fact that we must deal with polarized light becomes immediately apparent when we perform the Fresnel equations derivation. We shall see in this chapter that the mathematical for-malism we have developed to describe polarized light (i.e., the Stokes vectors and Mueller matrices) is ideally suited to the formulation of the Fresnel equations [1].

The Mueller matrices for reflection and refraction are complicated; however, there are three angles for which the Mueller matrices reduce to very simple forms. These special angles are (a) normal incidence, (b) the Brewster angle, and (3) an incident angle of 45º [2]. These three reduced matrix forms suggest methods of measuring the refractive index. These methods are discussed in some detail.

In a portion of the electromagnetic spectrum of interest to optical scientists and engineers, the thermal infrared, emission polarization becomes interesting and useful. We will briefly discuss considerations for polarized light in emission.

7.2 fReSNel eQuaTioNS foR RefleCTioN aNd TRaNSmiSSioN

In this section, we derive the Fresnel equations. Although this material can be found in many texts, it is useful and instructive to reproduce it here because it is so intimately tied to the polarization of light. Understanding the behavior of both the amplitude and phase of the components of light is essential to designing polarization components or analyzing optical system performance. We start with a review of concepts from electromagnetism.

7.2.1 definiTionS

Recall from electromagnetism that:

E is the electric field,B is the magnetic induction,D is the electric displacement,H is the magnetic field,ε0 is the permittivity of free space,ε is the permittivity,μ0 is the permeability of free space,μ is the permeability,

ε εε

χr = = +0

(1 ), (7.1)

118 Polarized Light, Third Edition

where εr is the relative permittivity or dielectric constant and χ is the electric susceptibility,

µ µµ

χr m= = +0

( ),1 (7.2)

and where μr is the relative permeability, and χ m is the magnetic susceptibility.Thus

ε ε ε ε χ= = +0 r 0 1( ) (7.3)

and

µ µ µ µ χ= = +0 0 1r m( ). (7.4)

Recall that (we use the International System of Units (abbreviated SI) here)

B H= µ (7.5)

and

D E= ε . (7.6)

Maxwell’s equations, where there are no free charges or currents, are

Ι ∇ ⋅ =D 0 (7.7)

ΙΙ ∇ ⋅ =B 0 (7.8)

ΙΙΙ ∇ × = − ∂∂

EBt

(7.9)

Ι ∇V × = ∂∂

HDt

(7.10)

7.2.2 boundaRy condiTionS

In order to complete our review of concepts from electromagnetism, we must recall the boundary conditions for the electric and magnetic field components. The integral form of Maxwell’s first equation (Equation 7.7) is

D A⋅ =∫∫ d 0 . (7.11)

This equation implies that, at the interface, the normal components on either side of the interface are equal; that is,

d dn n1 2= . (7.12)

Fresnel Equations: Derivation and Mueller Matrix Formulation 119

The integral form of Maxwell’s second equation (Equation 7.8), is

B A⋅ =∫∫ d 0 , (7.13)

which implies again that the normal components on either side of the interface are equal; that is,

b bn n1 2= . (7.14)

Invoking Ampere’s Law, we have,

H dl I⋅ =∫ (7.15)

which implies

H Ht t1 2= , (7.16)

i.e., the tangential component of H is continuous across the interface.Lastly,

E E⋅ = ∇ × ⋅ =∫∫∫ dl dA 0, (7.17)

which implies

E Et t1 2= ; (7.18)

that is, the tangential component of E is continuous across the interface.

7.2.3 deRivaTion of fReSnel equaTionS

We now have all the tools we need to derive Fresnel’s equations. Suppose we have a light beam intersecting an interface between two linear isotropic media. Part of the incident beam is reflected and part is refracted. The plane in which this interaction takes place is called the plane of incidence, and the polarization of light is defined by the direction of the electric field vector. There are two situ-ations that can occur. The electric field vector can either be perpendicular to the plane of incidence or parallel to the plane of incidence. We consider the perpendicular case first.

Case 1: E is perpendicular to the plane of incidence.This is the “s” polarization (from the German “senkrecht” for perpendicular) or σ polarization. This is also known as transverse electric, or TE, polarization (refer to Figure 7.1). Light travels from a medium with (real) index n1 and encounters an interface with a linear isotropic medium that has index n2. The angles of incidence (or reflection) and refraction are θi and θr, respectively.

In Figure 7.1, the y axis points into the plane of the paper consistent with the usual Cartesian coordinate system, and the electric field vector points out of the plane of the paper, consistent with the requirements of the cross product and the direction of energy flow. The electric field vector for the incident field is represented using the symbol E, whereas the fields for the reflected and

120 Polarized Light, Third Edition

transmitted components are represented by R and T, respectively. Using Maxwell’s third equation (Equation 7.9) we can write,

k E B.× = ω (7.19)

We can write this last equation as

Hk E= ×n

ωµ0

, (7.20)

where kn is the wave vector in the medium, and kn is

k an n= ω µ ε0 ˆ (7.21)

where an is a unit vector in the direction of the wave vector. Now we can write

Ha E a E= × = ×ω µ εωµ µ

ε

00 0

ˆ ˆn n (7.22)

or

Ha E= ׈

,n

η (7.23)

n1

n2

z

x

Es Rs

Bi Br

Bt

Ts

θr

θi θi

ki

kr

kt

figuRe 7.1 The plane of incidence for the transverse electric case.

Fresnel Equations: Derivation and Mueller Matrix Formulation 121

where

η µε ε

η η µε

η ε= = = =

0

0

andr

rn0

00

0

, (7.24)

where n is the refractive index and we have made the assumption that μr ≈ 1. This is the case for most dielectric materials of interest.

The unit vector in the directions of the incident, reflected, and transmitted wave vectors are

ˆ sin ˆ cos ˆa a ai i x i z= +θ θ (7.25)

ˆ sin ˆ cos ˆa a ar i x i z= −θ θ (7.26)

ˆ sin ˆ cos ˆ .a a at t x t z= +θ θ (7.27)

The magnetic field in each region is given by

Ha E

Ha E

Ha E

ii i

rr r

tt t= × = × = ׈ ˆ ˆ

,η η η1 1 2

(7.28)

and the electric field vectors tangential to the interface are

E a R a T as s y s s y s s yE R T= − = − = −ˆ ˆ ˆ . (7.29)

We can now write the magnetic field components as

Ha a

is i z s i xE E=

−+

sin ˆ cos ˆ,

θη

θη1 1

(7.30)

Ha a

rs i z s i xR R=

−−

sin ˆ cos ˆ,

θη

θη1 1

(7.31)

Ha a

ts r z s r xT T=

−+

sin ˆ cos ˆ.

θη

θη2 2

(7.32)

We know the tangential component of H is continuous, and we can find the tangential component by taking the dot product of each H with a x . We have, for the tangential components

H H Hi r ttan tan tan+ = (7.33)

or

E R T E Rs i s i s r s s rcos cos cos ( )cosθ

ηθ

ηθ

ηθ

η1

− = = +1 2 22

(7.34)

122 Polarized Light, Third Edition

using the fact that the tangential component of E is continuous (i.e., Es + Rs = Ts). We rearrange Equation 7.34 to obtain

E Rs i r s i rη θ η θ η θ η θ2 1 2 1cos cos cos cos ,−[ ] = +[ ] (7.35)

and now Fresnel’s equation for the reflection amplitude is

R Esi r

i rs= −

+η θ η θη θ η θ

2 1

2 1

cos coscos cos

. (7.36)

Using the relation in Equation 7.24 for each material region, we can express the reflection amplitude in terms of the refractive index and the angles as

Rn n

n nEs

i r

i rs= −

+1 2

1 2

cos coscos cos

.θ θθ θ

(7.37)

This last equation can be written, using Snell’s Law, n1 sin θi = n2 sin θr, to eliminate the dependence on index,

R Esi r

i rs= − −

+sin( )sin( )

.θ θθ θ

(7.38)

An expression for Fresnel’s equation for the transmission amplitude can be similarly derived and is

Tn

n nEs

i

i rs=

+2 1

1 2

cos

cos cos

θθ θ

(7.39)

or

T Esr i

i rs=

+2sin cossin( )

.θ θθ θ

(7.40)

Case 2: E is parallel to the plane of incidence.This is the “p” polarization (from the German “parallel” for parallel) or π polarization. This is also known as transverse magnetic, or TM, polarization. Refer to Figure 7.2. The derivation for the parallel reflection amplitude and transmission amplitude proceeds in a manner similarly to the perpendicular case, and Fresnel’s equations for the TM case are

Rn n

n nEp

i r

i rp= −

+2 1

2 1

cos coscos cos

θ θθ θ

(7.41)

or

R Epi r

i rp= −

+tan( )tan( )

,θ θθ θ

(7.42)

Fresnel Equations: Derivation and Mueller Matrix Formulation 123

and

Tn

n nEp

i

i rp=

+2 1

2 1

cos

cos cos

θθ θ

(7.43)

or

T Epr i

i r i rp=

+ −2sin cos

sin( )cos( ).

θ θθ θ θ θ

(7.44)

Figures 7.1 and 7.2 have been drawn as if light goes from a lower index medium to a higher index medium. This reflection condition is called an external reflection. Fresnel’s equations also apply if the light is in a higher index medium and encounters an interface with a lower index medium, a condition known as an internal reflection.

Before we show graphs of the reflection coefficients, there are two special angles we should con-sider. These are Brewster’s angle and the critical angle.

First, consider what happens to the amplitude reflection coefficient in Equation 7.42 when θi + θr sums to 90°. The amplitude reflection coefficient vanishes for light polarized parallel to the plane of incidence. The incidence angle for which this occurs is called Brewster’s angle. From Snell’s Law, we can relate Brewster’s angle to the refractive indices of the media with a very simple expression, that is,

θib

n

n= −tan .1 2

1

(7.45)

The other angle of importance is the critical angle. When we have an internal reflection, we can see from Snell’s Law that the transmitted light bends to ever larger angles as the incidence angle increases, and at some point the transmitted light leaves the higher index medium at a grazing angle.

x

n1

n2

EpRp

Bi

θi

θi θi

θr

Br

Bt

Tp

ki

kr

kt

z

figuRe 7.2 The plane of incidence for the transverse magnetic case.

124 Polarized Light, Third Edition

This is shown in Figure 7.3. The incidence angle at which this occurs is the critical angle. From Snell’s Law, n2 sin θi = n1 sin θr (writing the indices in reverse order to emphasize the light progres-sion from high [n2] to low [n1] index), and when θr = 90°,

sinθi

n

n= 1

2

(7.46)

or

θc

n

n= −sin ,1 1

2

(7.47)

where θc is the critical angle. For any incidence angle greater than the critical angle, there is no refracted ray and we have total internal reflection (TIR).

The amplitude reflection coefficients; that is,

rR

Ess

s

≡ (7.48)

and

rR

Epp

p

≡ (7.49)

and their absolute values for external reflection for n1 = 1 (air) and n2 = 1.5 (a typical value for glass in the visible spectrum) are plotted in Figure 7.4. Both the incident and reflected light have phases associated with them, and there may be a net phase change upon reflection. The phase changes for

x

n1

n2

θc

z

figuRe 7.3 The critical angle where the refracted light exits the surface at grazing incidence.

Fresnel Equations: Derivation and Mueller Matrix Formulation 125

external reflection are plotted in Figure 7.5. The amplitude reflection coefficients and their abso-lute values for the same indices for internal reflection are plotted in Figure 7.6. The phase changes for internal reflection are plotted in Figure 7.7. An important observation to make here is that the reflection remains total beyond the critical angle, but the phase change is a continuously changing function of incidence angle. The phase changes beyond the critical angle, that is, when the incidence angle is greater than the critical angle, are given by

tansin sin

cosφ θ θ

θs r c

r2

2 2

= − (7.50)

and

tansin sincos sin

,φ θ θ

θ θp r c

r c2

2 2

2= −

(7.51)

where ϕs and ϕp are the phase changes for the TE and TM cases, respectively. The reflected intensi-ties, that is, the square of the absolute value of the amplitude reflection coefficients, R 2 = r , for external and internal reflection are plotted in Figures 7.8 and 7.9, respectively.

The results in this section have assumed real indices of refraction for linear, isotropic materials. This may not always be the case; that is, the materials may be anisotropic and have complex indices of refraction, and in this case, the expressions for the reflection coefficients are not so simple. For example, the amplitude reflection coefficients for internal reflection at an isotropic to anisotropic interface as would be the case for some applications, for example, attenuated total reflection (see Deibler [3]; note that Deibler uses the definition for complex index, n n ik= + in contrast to the defi-nition used in this text, ˆ ( )n n i= −1 κ , and we quote Deibler’s forms for the coefficients here), are

rn k in k n n

n k is

x x x x

x x

= − + − −− +

2 212 2

1

2 2

2

2

sin cosθ θnn k n nx x − +1

2 21sin cosθ θ

(7.52)

–1.0

–0.8

–0.6

–0.4

–0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

Incidence angle (radians)

Am

plitu

de re

flect

ion

coeffi

cent

rs

|rs| |rp|

rp

figuRe 7.4 Amplitude reflection coefficients and their absolute values versus incidence angle for external reflection for n1 = 1 and n2 = 1.5.

126 Polarized Light, Third Edition

and

rn n k in k n n n k k i k

pz z z z y z y z=

− + − − − +12 2

12 22 sin ( (θ yy z z z

z z z z

n k n

n n k in k n n

+− + − +

))cos

sin (

θθ1

2 212 22 yy z y z y z z zn k k i k n k n− + +( ))cos

,θ

(7.53)

where nx, ny, and nz are the real parts of the complex indices of the anisotropic material, and kx, ky, and kz are the imaginary parts (in general, materials can have three principal indices). Anisotropic materials and their indices are covered in Chapter 21.

Before we go on to describe the reflection and transmission process in terms of Stokes param-eters and Mueller matrices we make note of two important points. First, the Stokes parameters must be defined appropriately for the field within and external to the dielectric medium. The first Stokes parameter represents the total intensity of the radiation and must correspond to a quantity known as

TE case

0

20

40

60

80

100

120

140

160

180(a)

(b)

Incidence angle (radians)

Incidence angle (radians)

Phas

e cha

nge (

degr

ees)

Phas

e cha

nge (

degr

ees)

TM case

0

20

40

60

80

100

120

140

160

180

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

figuRe 7.5 Phase changes for external reflection versus incidence angle for n1 = 1 and n2 = 1.5.

Fresnel Equations: Derivation and Mueller Matrix Formulation 127

the Poynting vector. This vector describes the flow of power of the propagating field components of the electromagnetic field. The Poynting vector is defined to be

S E HE = ×( ). (7.54)

In an isotropic dielectric medium, the time-averaged Poynting vector is

S E EE = •εr

2*. (7.55)

Second, the direction of the Poynting vector and the surface normal are different. This requires that the component of the Poynting vector in the direction of the surface normal must be taken. Consequently, a cosine factor must be introduced into the definition of the Stokes parameters. We will now establish the Mueller matrices for reflection and transmission at a dielectric interface.

7.3 muelleR maTRiCeS foR RefleCTioN aNd TRaNSmiSSioN aT aN aiR-dieleCTRiC iNTeRfaCe

The Stokes parameters for an incident field in air (n = 1) can be written

S E E E Es s p p0 = +( )* * (7.56)

S E E E Es s p p1 = −( )* * (7.57)

S E E E Es p p s2 = +( )* * (7.58)

S E E E Es p p s3 = −( )* * , (7.59)

–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

Incidence angle (radians)

Am

plitu

de re

flect

ion

coeffi

cien

trs

|rs|

rp

|rp|

figuRe 7.6 Amplitude reflection coefficients and their absolute values versus incidence angle for internal reflection for n1 = 1 and n2 = 1.5.

128 Polarized Light, Third Edition

where Es and Ep are the two orthogonal electric field components normal to the propagation direc-tion and are perpendicular and parallel to the plane of incidence, respectively. When we consider light that is described by this Stokes vector incident at angle θi on an air-dielectric interface, we must introduce a factor that is the cosine of the angle of incidence because this is the angle between the normal and the Poynting vector of the beam. The equations for the Stokes vector elements become

S E E E Ei s s p p0 = +( )cosθ * * (7.60)

S E E E Ei s s p p1 = −( )cosθ * * (7.61)

0

20

40

60

80

100

120

140

160

180

Incidence angle (radians)

Phas

e cha

nge (

degr

ees)

0

20

40

60

80

100

120

140

160

180

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

Incidence angle (radians)

Phas

e cha

nge (

degr

ees)

(b)

(a)

figuRe 7.7 Phase changes for internal reflection versus incidence angle for n2 = 1.5 and n1 = 1.

Fresnel Equations: Derivation and Mueller Matrix Formulation 129

S E E E Ei s p p s2 = +( )cosθ * * (7.62)

S i E E E Ei s p p s3 = −( )cosθ * * . (7.63)

The Stokes parameters for the reflected field are

S R R R RR i s s p p0 = +( )cosθ * * (7.64)

S R R R RR i s s p p1 = −( )cosθ * * (7.65)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5Incidence angle (radians)

Inte

nsity

refle

ctio

n

Rs

Rp

figuRe 7.8 Intensity reflection for external reflection versus incidence angle for n1 = 1 and n2 = 1.5.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5Incidence angle (radians)

Inte

nsity

refle

ctio

n

Rp

Rs

figuRe 7.9 Intensity reflection for internal reflection versus incidence angle for n2 = 1.5 and n1 = 1.

130 Polarized Light, Third Edition

S R R R RR i s p p s2 = +( )cosθ * * (7.66)

S i R R R RR i s p p s3 = −( )cosθ * * , (7.67)

where Rs and Rp are the electric fields of the reflected beam. Substituting the values of Rs and Rp from Equations 7.38 and 7.42 into Equations 7.64 through 7.67 and using Equations 7.60 through 7.63, the Stokes vector for the reflected beam SR is found to be related to the Stokes vector of the incident beam S by

S

S

S

S

R

R

R

R

0

1

2

3

212

=

×

−

+

tansin

θθ

ccos cos cos cos

cos cos c

2 2 2 2

2 2

0 0θ θ θ θθ θ

− + − +

− +

+ −− oos cos

cos cos

cos cos

2 2 0 0

0 0 2 0

0 0 0 2

θ θθ θ

θ

− +

+ −

+

+−

− θθ−

×

S

S

S

S

0

1

2

3

(7.68)

where θ± = θi ± θr. The Mueller matrix of a diattenuator is

M =

+ −− +1

2

0 0

0 0

0 0 2

2 2 2 2

2 2 2 2

p p p p

p p p p

p p

s p s p

s p s p

s p 00

0 0 0 2p ps p

(7.69)

where ps and pp are the absorption coefficients perpendicular and parallel to the plane of incidence. Comparing Equation 7.68 with Equation 7.69, we see that the matrix in Equation 7.68 corresponds to a Mueller matrix of a diattenuator.

The Stokes parameters for the transmitted field are

S n T T T TT r s s p p0 = +( )cosθ * * (7.70)

S n T T T TT r s s p p1 = −( )cosθ * * (7.71)

S n T T T TT r s p p s2 = +( )cosθ * * (7.72)

S n T T T TT r s p p s3 = −( )cosθ * * , (7.73)

Fresnel Equations: Derivation and Mueller Matrix Formulation 131

where Ts and Tp are the transmitted field components perpendicular and parallel to the plane of incidence and n is the index of the transmitting dielectric. Substituting the values of Ts and Tp from Equations 7.40 and 7.44 into Equations 7.70 through 7.73 and using Equations 7.60 through 7.63, the Stokes vector ST is found to be

S

S

S

S

T

T

T

T

i r

0

1

2

3

2 2

2

=+

sin sin

sin

θ θ

θ ccos

cos cos

cos cos

θ

θ θθ θ

−

− −

− −

( )

×

+ −− +

2

2 2

2 2

1 1 0 0

1 11 0 0

0 0 2 0

0 0 0 2

0

1

2

3

cos

cos

θθ

−

−

S

S

S

S

.

(7.74)

This result also corresponds to the Mueller matrix of a polarizer. It is straightforward to show from Equations 7.68 and 7.74 that

S S SR T0 0 0= + , (7.75)

as would be expected from the principle of the conservation of energy.Equation 7.68 shows that incident light that is completely polarized remains completely polar-

ized. If we have incident light that is unpolarized, then Equation 7.68 becomes

SR

R

R

R

R

S

S

S

S

=

=

−

+

0

1

2

3

12

tansin

θθ

×

+−

− +

− +

cos cos

cos cos

2 2

2 2

0

0

θ θθ θ

, (7.76)

and the degree of polarization is

PS

S= = −

+− +

− +

1

0

2 2

2 2

cos coscos cos

.θ θθ θ

(7.77)

The degree of polarization is less than or equal to 1, since S1 ≤ S0. The degree of polarization P = 1 when cosθ+ = 0, that is, when

cos cos .θ θ θ+ = +( ) =i r 0 (7.78)

This occurs when

θ θ πi r+ = =

290, (7.79)

132 Polarized Light, Third Edition

that is, when the sum of the incident angle and the refracted angle is 90°. This is the condition we discussed prior to Equation 7.45 where we defined Brewster’s angle in terms of the refractive indi-ces, and if we set cosθ + = 0 in Equation 7.76, we have

SR

R

R

R

R

i

S

S

S

S

b=

=

0

1

2

3

212

2

1

1

0

0

cos θ

. (7.80)

The reflected light is only linearly horizontally polarized, and the angle θib at which this occurs

is Brewster’s angle. Figure 7.10 is a plot of Equation 7.77, the degree of polarization P versus the incident angle θi, for a material with a refractive index of 1.50. For this glass, the Brewster angle is approximately 56.7°.

We can determine the intensity of the reflected light by examination of the first Stokes parameter from Equation 7.76. The intensity IR of the reflected beam is

IR =

+( )−

+− +

12

2

2 2tansin

cos cosθθ

θ θ . (7.81)

Figure 7.11 is a plot of the magnitude of the reflected intensity IR as a function of incident angle θi for a dielectric (glass) with a refractive index of 1.5. As the incidence angle increases, the reflected intensity increases, dramatically so as the angle of incidence approaches 90°. Thus, when sunlight reflects from smooth horizontal surfaces at these large angles, the “glare” becomes significant. At angles above the

0.9

1

0.7

0.8

0.6

0.4

0.5P

0.2

0.3

0

0.1

0 5 10 15 20 25 30 35 40 45 50θi (degrees)

55 60 65 70 75 80 85 90

figuRe 7.10 Plot of the degree of polarization P versus the incident angle θi for unpolarized light reflected from glass of refractive index 1.5.

Fresnel Equations: Derivation and Mueller Matrix Formulation 133

Brewster angle for that surface, polarized sunglasses are only partially effective because the reflected light is not completely polarized. At the Brewster angle, polarized sunglasses block all the reflected light, but the reflected intensity at the Brewster angle θib

by Equation 7.80 is only 7.9%.We can obtain from Equation 7.74 the Stokes vector for the transmitted beam where the incident

beam is again unpolarized; that is,

S

S

S

S

T

T

T

T

i r

0

1

2

3

2 2

2

=+

sin sin

sin

θ θ

θ ccos

cos

cos

θ

θθ

−

−

−

( )

+−

2

2

2

1

1

0

0

. (7.82)

The degree of polarization P of the transmitted beam is

P = −+

−

−

coscos

2

2

11

θθ

. (7.83)

We expect P to be less than 1. Figure 7.12 is a plot of the degree of polarization versus the incident angle for a material with index n = 1.50. The polarization of the transmitted light is small for small angles of incidence. It then increases to a maximum value of 0.385 at an incidence angle of 90°. Light can never become completely polarized by the transmission of unpolarized light through a single surface. Use of materials with larger refractive indices makes it possible to increase the degree of polarization as is shown in Figure 7.13. This plot shows the degree of polarization versus incidence angle for materials with refractive indices of n = 1.5, 2.5, and 3.5. There is a significant increase in the degree of polarization as n increases.

1.2

1

0.6

0.8

0.4

I R

0.2

00 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

θi (degrees)

figuRe 7.11 Plot of the intensity of a beam reflected by a dielectric of refractive index 1.5. The incident beam is unpolarized.

134 Polarized Light, Third Edition

0.35

0.4

0.3

0.25

0.15

0.2P

0.05

0.1

00 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

θi (degrees)

figuRe 7.12 Plot of the degree of polarization versus the incidence angle for incident unpolarized light transmitted through a single glass surface. The refractive index is 1.5.

0.9

1

0.7

0.8

0.6

0.5P

0.4n = 3.5

0.2

0.3n = 2.5

0

0.1

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

n = 1.5

θi (degrees)

figuRe 7.13 Plot of the degree of polarization versus the incidence angle for three refractive indices for an incident unpolarized beam transmitted through a single dielectric surface.

Fresnel Equations: Derivation and Mueller Matrix Formulation 135

The transmitted intensity IT is, from Equation 7.82,

ITi r=

( )+( )

+ −−

sin sin

sin coscos

2 2

21

22θ θ

θ θθ . (7.84)

At the Brewster angle θib, when the condition of Equation 7.79 holds, this becomes

IT ib b= +( )1

21 22sin θ . (7.85)

For glass of index 1.5 and a Brewster angle of 56.7°, the transmitted intensity is 92.1%. The corre-sponding intensity for the reflected beam is 7.9% as noted earlier. We have confirmed again that, as expressed in Equation 7.75, the sum of the reflected intensity and the transmitted intensity is 100%. We have plotted the intensity of Equation 7.84 for a dielectric with a refractive index of n = 1.5 as a function of the incidence angle in Figure 7.14. The transmission drops rapidly above 60° and goes to zero as the incidence angle approaches 90°.

We will consider some special cases in the next section where the Mueller matrices in Equations 7.68 and 7.74 simplify. We will then extend these results to dielectric and multiple plates.

7.4 SPeCial foRmS foR muelleR maTRiCeS foR RefleCTioN aNd TRaNSmiSSioN

There are three cases where the Mueller matrix for reflection by a dielectric simplifies. These sim-plifications occur at normal incidence θi = 0, at the Brewster angle θib

, and at θi = 45°. We will now consider these three cases and derive the corresponding Mueller matrices for transmission.

0.9

1

0.7

0.8

0.5

0.6

I T

0.3

0.4

0.1

0.2

00 10 20 30 40 50 60 70 80 90

θi (degrees)

figuRe 7.14 The intensity of a beam transmitted through a dielectric with a refractive index of 1.5 as a function of incidence angle. The incident beam is unpolarized.

136 Polarized Light, Third Edition

7.4.1 noRMal incidence

For the normal incidence case of a dielectric in air, we will use small angle approximations and express Snell’s Law as

θ θi rn (7.86)

since for θ1

sinθ θ . (7.87)

We can also write in the small angle approximation

cosθ1 (7.88)

tanθ θ θ θ− − = − i r (7.89)

sinθ θ θ θ+ + = + i r (7.90)

cosθ+ 1 (7.91)

cos .θ− 1 (7.92)

The Mueller matrix in Equation 7.68 then reduces to

M12

2 0 0 0

0 2 0 0

0 0 2 0

0 0 0 2

2θ θθ θ

i r

i r

−+

−

−

. (7.93)

Using the small angle form of Snell’s Law Equation 7.86 in Equation 7.93 results in

MR

n

n

12

11

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

2−+

−

−

(7.94)

which is the Mueller matrix for reflection at normal incidence. The negative sign in the matrix ele-ments m22 and m33 tells us that there is a 180° phase change on reflection.

In the case of transmission at normal incidence with small angle approximations, the Mueller matrix of Equation 7.74 becomes

M = ( )( )( )

+

2 2

2

2 0 0 0

0 2 0 0

0 0 2 0

0 0 0 2

2

θ θ

θi r

. (7.95)

Fresnel Equations: Derivation and Mueller Matrix Formulation 137

Using the small angle form of Snell’s Law Equation 7.86 in Equation 7.95 results in

MT

n

n=

+( )

4

1

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

2, (7.96)

which is the Mueller matrix for transmission at normal incidence. If we use the matrix MR of Equation 7.94 in Equation 7.68, the reflected intensity at normal

incidence is

In

nIR = −

+

11

2

0 (7.97)

and if we use the matrix MT of Equation 7.96 in Equation 7.74, the transmitted intensity at normal incidence is

In

nIT =

+( )4

12 0 , (7.98)

and these are the same forms we would find from the Fresnel equations. Adding Equations 7.97 and 7.98 gives us

I I IR T+ = 0 , (7.99)

as expected.

7.4.2 bRewSTeR angle

Consider the condition from Equation 7.79 (i.e., θi + θr = π/2 = 90°). Then

θ θ θ+ = + =i r 90 (7.100)

and

θ θ θ θ− = − = −i r i2 90. (7.101)

Using these expressions in Equation 7.68, we find that the Mueller matrix for reflection reduces to

MR ib b=

12

2

1 1 0 0

1 1 0 0

0 0 0 0

0 0 0 0

2cos θ (7.102)

where we have used the relation

sin cos2 90 2θ θi ib b

−( ) = − (7.103)

138 Polarized Light, Third Edition

and where we have included the subscript b to indicate the Brewster angle. The Mueller matrix has reduced to that of an ideal linear horizontal polarizer. At the Brewster angle, the reflected portion of an unpolarized or partially polarized beam will be completely polarized. There will be no reflection if the incident beam is vertically polarized.

At the interface between a dielectric in air, the expression for Brewster’s angle Equation 7.45 becomes,

tan .θibn= (7.104)

This expression says that the refractive index n of a dielectric can be obtained from a reflection measurement. If we can find Brewster’s angle, the angle at which the light is completely s polarized, then the determination of that angle allows us to know n.

The Mueller matrix for the part of the beam that transmits into the dielectric is, with appropriate substitutions in Equation 7.74,

MT b

i i

i

b b

b,

sin sin

sin sin=

+ −

−12

2 1 2 1 0 0

2 1

2 2

2 2

θ θ

θ 22 1 0 0

0 0 2 2 0

0 0 0 2 2

θ

θ

θ

i

i

i

b

b

b

+

sin

sin

.. (7.105)

This is still the matrix of a polarizer.

7.4.3 45° incidence

A third geometry where simplification of the Fresnel equations and the Mueller matrices occurs is the incidence angle of 45°. This appears to have been first noticed by Humphreys-Owen [4] only around 1960. We now derive the Mueller matrices for reflection and transmission at an incidence angle of 45°. The importance of the Mueller matrix for reflection at this angle of incidence is that it leads to another method for measuring the refractive index of an optical material. This method has a number of advantages over the normal incidence and Brewster angle methods.

At an incidence angle of θi = 45°, the Fresnel equations for Rs and Rp, Equations 7.38 and 7.42, reduce to

R Esr r

r rs= −

+

cos sincos sin

θ θθ θ

(7.106)

and

R Epr r

r rp= −

+

cos sincos sin

.θ θθ θ

2

(7.107)

We see that from Equation 7.67 and the definitions of the amplitude reflection coefficients in Equation 7.28 we have

r rs p2 = . (7.108)

We shall see that a corresponding relation exists between the orthogonal intensities Is and Ip.

Fresnel Equations: Derivation and Mueller Matrix Formulation 139

Using the condition that the incidence angle is 45° in Equations 7.64 through 7.67 and using Equations 7.106 and 7.107, we are led to the Mueller matrix for incident 45° light

MR ir

r

r

θ θ

θ

θ

=( ) = −

+( )45

1 2

1 2

1 2 0 0

22

sin

sin

sin

sin θθθ

θ

r

r

r

1 0 0

0 0 2 0

0 0 0 2

−−

cos

cos

. (7.109)

This is simplified, but still retains the form of a polarizer. Equation 7.109 suggests a simple way to determine the refractive index n of an optical material by reflection. Assume that we send a beam of s polarized light onto the surface with intensity I0. The Stokes vector is just

SS I=

0

1

1

0

0

. (7.110)

The outgoing Stokes vector will have an intensity

I Isr

r

= −+0

1 21 2

sinsin

.θθ

(7.111)

Now let us send in a beam of p polarized light with a Stokes vector

Sp I=−

0

1

1

0

0

. (7.112)

The intensity from the output Stokes vector is found from the product of Equations 7.112 and 7.109, that is,

I Ipr

r

= −+

0

21 21 2

sinsin

.θθ

(7.113)

Squaring Equation 7.111 and using Equation 7.113 gives us

I

I

I

Is p

0

2

0

= (7.114)

or

I

IIs

p

2

0= . (7.115)

140 Polarized Light, Third Edition

Expressions for the intensity reflection coefficients are

RssI

I=

0

arcsinθ (7.116)

and

RppI

I=

0

, (7.117)

and we have (using Equation 7.115)

R Rs p2 = , (7.118)

which is the analog of Equation 7.108 in the intensity domain.We can derive an expression for the refractive index in terms of Is and Ip. Dividing Equation 7.113

by Equation 7.111, we obtain

I

Ip

s

r

r

= −+

1 21 2

sinsin

.θθ

(7.119)

A little algebra yields

sin2θrs p

s p

I I

I I=

−+

(7.120)

and using the half angle formula from trigonometry yields

2sin cos .θ θr rs p

s p

I I

I I=

−+

(7.121)

We can factor Equation 7.121 as

2 2sin cos ,θ θr rs p s p

s p s p

I I I I

I I I I( )( ) =

−( ) +( )+ +

(7.122)

and this suggests that we can equate factors on the left- and right-hand sides such that

2 sinθrs p

s p

I I

I I( ) =

−( )+

(7.123)

2 cos .θrs p

s p

I I

I I( ) =

+( )+

(7.124)

Fresnel Equations: Derivation and Mueller Matrix Formulation 141

This factorization is satisfactory; when Is = Ip, sinθr = 0 and cosθr = 1. From Snell’s Law for an incidence angle of θi = 45°,

21

sin .θr n= (7.125)

Equating Equations 7.125 and 7.123 then yields

nI I

I Is p

s p

= +−

, (7.126)

and if we can measure the intensities of orthogonal components of polarized light reflected from the surface then we have another method by which we can find the index of refraction.

7.4.4 ToTal inTeRnal ReflecTion

Total internal reflection, or TIR, occurs when light propagates from a material with a larger index of refraction to one with a smaller index. In order to derive the Mueller matrix for TIR, we must first obtain the correct form of Fresnel’s equations for TIR. Figure 7.15 shows an optical beam propagating in an optically denser medium and being reflected at the dielectric-air interface. This diagram shows reflection and transmission below the critical angle, discussed earlier in this chap-ter. Figure 7.16 then shows internal reflections for the cases when the incidence angle is the critical angle and when the incidence angle is above the critical angle.

Snell’s Law for Figure 7.15 is now written

n i rsin sin .θ θ= (7.127)

Above the critical angle

n isin .θ > 1 (7.128)

Recall that the Fresnel amplitude reflection coefficients are

R Epi r

i rp= −( )

+( )tantan

θ θθ θ

(7.129)

n1 < n2

n1

n2

θ1

θ2

θ1

figuRe 7.15 Diagram of internal reflection below the critical angle.

142 Polarized Light, Third Edition

and

R Esi r

i rs= −( )

+( )sinsin

.θ θθ θ

(7.130)

Using the trigonometric sum and difference formulas in Equations 7.129 and 7.130 gives us

Rpi i r r

i i r

= −+

sin cos sin cossin cos sin cos

θ θ θ θθ θ θ θθr

pE (7.131)

and

Rsi r r i

i r r

= − ++

sin cos sin cossin cos sin c

θ θ θ θθ θ θ oos

.θi

sE (7.132)

Squaring Equation 7.127 and using sin2θ + cos2θ = 1, Snell’s Law can be written as

cos sin sin .θ θ θr i ii n n= − >2 2 1 1 (7.133)

If we now substitute Equation 7.133 into Equations 7.131 and 7.132 we obtain

Rin n

in nEp

i i

i ip= − −

+ −cos sin

cos sin

θ θθ θ

2 2

2 2

1

1 (7.134)

and

Rn i n

n i nEs

i i

i is= − −

+ −cos sin

cos sin.

θ θθ θ

2 2

2 2

1

1 (7.135)

θ2

θ2

θ1

θ1

θ1 > θc

θ1 = θc

θ1

θ1

n1

n2

n1

n2

figuRe 7.16 Diagram of internal reflection at the critical angle and above the critical angle.

Fresnel Equations: Derivation and Mueller Matrix Formulation 143

If we now let a = cosθi and b n n i= −2 2 1sin θ , then we can express Equation 7.134 as

ga ib

a ib= −

+, (7.136)

and see that gg* = 1. Thus, g can be expressed as

ga ib

a ibe ii

p pp= −

+= = −− δ δ δcos sin , (7.137)

where δp is the phase associated with Rp. Equating the real and imaginary parts in Equation 7.137 yields

cosδ p

a b

a b= −

+

2 2

2 2 (7.138)

and

sinδ p

ab

a b=

+22 2

(7.139)

and

tan .δ p

ab

a b=

−22 2

(7.140)

If we write sin δp and cosδp in terms of half-angle formulas, then,

tansin

cos

sin cos

c

δδδ

δ δ

pp

p

p p

= =

22 2

oos sin

.2 2

2 2

2 2

2δ δp p

ab

a b

−

=−

(7.141)

If we arbitrarily set sin δp/2 = b and cos δp/2 = a, then tan δp/2 = b/a and using a = cosθi and b n n i= −2 2 1sin θ , then

tansincos

.δ θ

θp i

i

n n

n212 2

= − (7.142)

In a similar manner we find

tansincos

.δ θ

θs i

i

n

n212 2

= − (7.143)

If δ = δs – δp, then an expression in terms of the difference of the phases is

tancos sin

sin.

δ θ θθ2

12 2

2= −i i

i

n

n (7.144)

144 Polarized Light, Third Edition

Equations 7.134 and 7.135 can now be written

R e Epi

pp= − δ (7.145)

and

R e Esi

ss= − δ . (7.146)

Using the Expressions 7.64 through 7.67 for the reflected Stokes vector elements in terms of the amplitude reflection coefficients, and relating these to the expressions for the input Stokes vector elements Equations 7.60 through 7.63, we can obtain the Mueller matrix for TIR as

MR =−

1 0 0 0

0 1 0 0

0 0

0 0

cos sin

sin cos

δ δδ δ

, (7.147)

where δ = δs – δp. This is the Mueller matrix for a retarder.It is instructive to evaluate the Fresnel rhomb, a prism invented by Fresnel around 1820 that

causes retardance and can be used to create circularly polarized light from linearly polarized light. Fresnel’s prism is shown in Figure 7.17 with a beam passing through it.

For a prism made of BK7 glass, the refractive index n at a wavelength of 6328 Å (He–Ne wave-length) is approximately 1.5151. For an angle of θi = 55.08°, the phase shift δ with the first TIR is δ1 = 45.00° and a second TIR produces an additional phase shift, δ2 = 45.00°. The net phase shift from the two TIRs is the product of the Mueller matrices for two retarders, each of which is repre-sented by Equation 7.147. The product is then

M =−

1 0 0 0

0 1 0 0

0 0

0 02 2

2 2

cos sin

sin cos

δ δδ δ

−

1 0 0 0

0 1 0 0

0 0

0 01 1

1 1

cos sin

sin cos

δ δδ δ

, (7.148)

55.08°

figuRe 7.17 The Fresnel rhomb.

Fresnel Equations: Derivation and Mueller Matrix Formulation 145

and multiplying out, we have

M =+( ) − +( )+

1 0 0 0

0 1 0 0

0 0

0 01 2 1 2

1 2

cos sin

sin

δ δ δ δδ δ(( ) +( )

cos

,

δ δ1 2

(7.149)

and for the Fresnel rhomb δ = δ1 + δ2 = 90°, so the Mueller matrix reduces to

M =−

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

. (7.150)

If we send a beam of linearly polarized light at 45° into the rhomb having the Stokes vector

S I=

0

1

0

1

0

, (7.151)

then the Stokes vector of the outgoing light is

S =

I0

1

0

0

1

; (7.152)

that is, the rhomb has produced right circularly polarized light.

7.5 emiSSioN PolaRiZaTioN

We have discussed polarization as a result of reflection to this point. In the thermal infrared wave-length region, polarization on emission becomes important [5]. We know from energy conservation that

R T A( , ) ( , ) ( , ) ,θ λ θ λ θ λ+ + = 1 (7.153)

where R( , )θ λ is the reflectance, T( , )θ λ is the transmittance, A( , )θ λ is the absorptance, λ is the wavelength, and θ is the angle from the normal to the surface. If we have a surface that does not transmit, then T( , )θ λ = 0 and

R A( , ) ( , ) .θ λ θ λ+ = 1 (7.154)

146 Polarized Light, Third Edition

From Kirchhoff’s Law, for bodies in thermal equilibrium, at some wavelength λ,

A E( ) ( );θ θ=

that is, the absorptance is equal to the emittance and we can rewrite Equation 7.154 as [6,7]

E R( ) ( ).θ θ= −1 (7.155)

Radiated emission from a surface can have two orthogonal components just as reflected and trans-mitted radiation, so that we can write

E Rp p( ) ( )θ θ= −1 (7.156)

E Rs s( ) ( )θ θ= −1 (7.157)

for the parallel and perpendicular components. If we plot the reflectance and emittance for both components, we obtain the graph shown in Figure 7.18 (using an arbitrary index for illustration). As before, the reflectance curves show that the component of polarized light that is reflected most strongly for all angles is the component perpendicular to the plane of incidence. For the emittance, the situation is completely inverted, and the component of polarized light most strongly emitted is that parallel to the plane of incidence. In the absence of reflected thermal radiation, the polariza-tion of the emitted radiation will predominate. When both are present, light received by a detec-tor will be a mixture of the two, and it may be difficult to distinguish the origin of the polarized light.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

Rp

Rs

Es

Ep

1.4

Emitt

ance

/refl

ecta

nce

Incidence angle (radians)

figuRe 7.18 Emission polarization.

Fresnel Equations: Derivation and Mueller Matrix Formulation 147

RefeReNCeS

1. Collett, E., Mueller-Stokes matrix formulation of Fresnel’s equations, Am. J. Phys. 39 (1971): 517–28. 2. Collett, E., Digital refractometry, Opt. commun. 63 (1987): 217–24. 3. Deibler, L. L., Infrared Polarimetry Using Attenuated Total Reflection, PhD dissertation, University of

Alabama in Huntsville, 2001. 4. Humphreys-Owen, S. P. F., Comparison of reflection methods for measuring optical constants without

polarimetric analysis, and proposal for new methods based on the Brewster angle, Proc. Phys. Soc. 77 (1961): 949.

5. Sandus, O., A review of emission polarization, Appl. Opt. 4 (1965): 1634–42. 6. Jordan, D. L., G. D. Lewis, and E. Jakeman, Emission polarization of roughened glass and aluminum

surfaces, Appl. Opt. 35 (1996): 3583–90. 7. Jordan, D. L., and G. D. Lewis, Measurements of the effect of surface roughness on the polarization state

of thermally emitted radiation, Opt. Lett. 19 (1994): 692–4.

149

8 Mathematics of the Mueller Matrix

8.1 iNTRoduCTioN

Mathematical development to better understand and describe the information contained in the Mueller matrix is given in this chapter. The experimental Mueller matrix can be a complicated function of polarization, depolarization, and noise. How do we separate the specific information we are interested in, for example, depolarization or retardance, from the measured Mueller matrix? When does an experimental matrix represent a physically realizable polarization element and when does it not? If it does not represent a physically realizable polarization element, how do we extract that information that will tell us about the equivalent physically realizable element? These are some of the questions we attempt to answer in this chapter.

Two algebraic systems have been developed for the solution of polarization problems in optics, the Jones formalism, and the Mueller–Stokes formalism. The Jones formalism is a natural conse-quence of the mathematical phase and amplitude description of light. The Mueller–Stokes formal-ism comes from experimental considerations of the intensity measurements of light.

R. C. Jones developed the Jones formalism in a series of papers published in the 1940s [1–3] and reprinted in a collection of historically significant papers on polarization [4]. The Jones formal-ism uses Jones vectors, two element vectors that describe the polarization state of light, and Jones matrices, 2 × 2 matrices that describe optical elements. The vectors are complex and describe the amplitude and phase of the light; that is,

JE

Et

t

tx

y

( ) =( )( )

(8.1)

is a time-dependent Jones vector where Ex, Ey are the x and y components of the electric field of light traveling along the z axis. The matrices are also complex and describe the effects of propaga-tion interactions in both amplitude and phase of optical elements on a light beam. The Jones matrix is of the form

J =

j j

j j11 12

21 22

, (8.2)

where the elements jij = aij + ibij are complex. The two elements of the Jones vector are orthogonal and typically represent the horizontal and vertical polarization states. The four elements of the Jones matrix make up the transfer function from the input to the output Jones vector. Since these elements are complex, the Jones matrix contains eight constants and has eight degrees of freedom corresponding to eight kinds of polarization behavior (e.g. Table 8.2). A physically realizable polar-ization element results from any Jones matrix; that is, there are no physical restrictions on the values of the Jones matrix elements. The Jones formalism is discussed in more detail in Chapter 10.

The Mueller formalism, already discussed in previous chapters but reviewed here, owes its name to Hans Mueller, who built upon the work of Stokes [5], Soleillet [6], and Perrin [7] to formalize

150 Polarized Light, Third Edition

polarization calculations based on intensity. This work, as Jones’s, was also done during the early 1940s but originally appeared in a now declassified report [8] and in a course of lectures at MIT in 1945–1946. As we have learned, the Mueller formalism uses the Stokes vector to represent the polarization state of light. The Mueller matrix is a 4 × 4 matrix of real numbers. There is redun-dancy built into the Mueller matrix, since only seven of its elements are independent if there is no depolarization in the optical system. In the most general case, the Mueller matrix can have 16 independent elements; however, not every 4 × 4 Mueller matrix represents a physically realizable polarizing element.

For each Jones matrix, there is a corresponding Mueller matrix. (A Jones matrix does not exist for every Mueller matrix, because Mueller matrices can contain information about depolarization, and this cannot be represented in a Jones matrix.) On conversion to a Mueller matrix, the Jones matrix absolute phase information is lost. A matrix with eight pieces of information is transformed to a matrix with seven pieces of information. Transformation equations for converting Jones matri-ces to Mueller matrices are given in Appendix D. The Mueller matrices can also be generated from equations. If it is true that

Tr M MT m( ) = 4 002 , (8.3)

then the Jones matrix is related to the Mueller matrix by [9]

M A J J A= ⊗( ) −* ,1 (8.4)

where ⊗ denotes the Kronecker product and A is

A =−

−

1 0 0 1

1 0 0 1

0 1 1 0

0 0i i

. (8.5)

The elements of the Mueller matrix can also be obtained from the relation

mij i j= ( )12

Tr J Jσσ σσ† , (8.6)

where J† is the Hermitian conjugate of J and the σ are the set of four 2 × 2 matrices that comprise the identity matrix and the Pauli matrices (see Section 8.3).

The Jones matrix cannot represent a depolarizer or depolarizing scatterer. The Mueller matrix can represent depolarizers and scatterers (see, for example, van de Hulst [10]). Since the Mueller matrix contains information on depolarization, the conversion of Mueller matrices to Jones matrices must discard depolarization information. There is no phase information in a Mueller matrix, and the conversion conserves seven degrees of freedom.

The Mueller formalism has two advantages for experimental work over the Jones formalism. The intensity is represented explicitly in the Mueller formalism, and depolarization can be included in the calculations. The Jones formalism is easier to use and more elegant for theoretical studies.

8.2 CoNSTRaiNTS oN The muelleR maTRiX

The issue of constraints on the Mueller matrix has been investigated by a number of research-ers (e.g., [11–16]). The fundamental requirement that Mueller matrices must meet in order to be

Mathematics of the Mueller Matrix 151

physically realizable is that they map physical incident Stokes vectors into physical resultant Stokes vectors. This recalls our requirement on Stokes vectors that the degree of polarization must always be less than or equal to one; that is,

PS S S

S= + +( ) ≤1

222

32

0

12

1. (8.7)

A well-known constraint on the Mueller matrix is the inequality [17]

Tr MMTij

i j

m m( ) = ≤=

∑ 2

0

3

0024

,

. (8.8)

The equal sign applies for nondepolarizing systems and the inequality otherwise.Many more constraints on Mueller matrix elements have been recorded. However, we shall not

attempt to list or even to discuss them further here. The reason for this is that they may be largely irrelevant when one is making measurements with real optical systems. The measured Mueller matrices are a mixture of pure (nondepolarizing) states, depolarization, and certainly noise (opti-cal and electronic). Is the magnitude of a particular Mueller matrix element due to diattenuation or retardance or is it really noise, or is it a mixture? If it is a mixture, what are the proportions? It is the responsibility of the experimenter to reduce noise sources as much as possible, determine the physi-cal realizability of his Mueller matrices, and if they are not physically realizable, find the closest physically realizable Mueller matrices; then the best possible estimate of the polarization properties of the sample can be extracted from the Mueller matrix through a matrix decomposition procedure. A method of finding the closest physically realizable Mueller matrix and a method of decomposing nondepolarizing and depolarizing Mueller matrices are discussed in the remaining sections of this chapter. These are very important and useful results; however, only so much can be done to reduce noise intrusion. A study was done [18] to follow error propagation in the process of finding the best estimates and it was found that the noise was reduced by one-third in nondepolarizing systems and reduced by one-tenth in depolarizing systems in going from the nonphysical matrix to the closest physically realizable matrix. The reduction is significant and worth doing, but no method can com-pletely eliminate measurement noise. We will give examples in Section 8.4.

8.3 eigeNVeCToR aNd eigeNValue aNalySiS

Cloude [19,20] has formulated a method to obtain polarization characteristics and answer the ques-tion of physical realizability. Any 2 × 2 matrix J (in particular, a Jones matrix) can be expressed as

J = ∑ ki i

i

σσ , (8.9)

where the σi are the Pauli matrices

σσ σσ σσ1 2 3

1 0

0 1

0 1

1 0

0

0=

−

=

=−

i

i, (8.10)

with the addition of the identity matrix

σσ0

1 0

0 1=

, (8.11)

152 Polarized Light, Third Edition

and the ki are complex coefficients given by

k Tri i= ⋅( )12

J σσ . (8.12)

The components of this vector also can be written

k j j0 11 22

12

= +( ) (8.13)

k j j1 11 22

12

= −( ) (8.14)

k j j2 12 21

12

= +( ) (8.15)

ki

j j3 12 212= −( ). (8.16)

Cloude introduces a 4 × 4 Hermitian “target coherency matrix” obtained from the tensor product of the k’s; that is,

T k kcT= ⊗ * . (8.17)

The elements of the Mueller matrix are given in terms of the Jones matrix as

m Trij i j= ( )12

J Jσσ σσ† (8.18)

and Cloude shows that this can also be written as

m Trij c i j= ( )+12 4T ηη , (8.19)

where the η are the 16 basis matrices for the group SU(4) [21]. The basis matrices are shown in Table 8.1.

The matrix Tc can be expressed as

Tc ij i jm= ⊗σσ σσ , (8.20)

where

Eij i j= ⊗σ σ (8.21)

Mathematics of the Mueller Matrix 153

are the Dirac matrices [22]. Tc can be written in the parametric form

A A c id H iG I iJ

c id b b E iF K iL

H iG E iF b

0

0

0

+ − + −+ + + −− − − bb M iN

I iJ K iL M iN A A

++ + − −

0

, (8.22)

where A through N are real numbers. If these real numbers are arranged into a 4 × 4 matrix where the ijth element is the expansion coefficient of the Dirac matrix E4i + j then the matrix

A b c N H L F I

c N A b E J G K

H L E J A b d M

I F K

0 0+ + + +− + + +− − − +− − GG M d A b− −

0 0

, (8.23)

is just the Mueller matrix when Tc is expressed in the Pauli base. The target coherency matrix is then obtained from the experimental Mueller matrix by solving for the real elements A through N. When this is done the elements of the coherency matrix are found to be

tm m m m

0000 11 22 33

2= + + +

(8.24)

Table 8.1basis matrices for the group Su(4)

η0 η1 η2 η3

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

0 1 0 0

1 0 0 0

0 0 0

0 0 0

i

i−

0 0 1 0

0 0 0

1 0 0 0

0 0 0

−

i

i

0 0 0 1

0 0 0

0 0 0

1 0 0 0

i

i−

η4 η5 η6 η7

0 1 0 0

1 0 0 0

0 0 0

0 0 0

−

i

i

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

−−

0 0 0

0 0 1 0

0 1 0 0

0 0 0

−

i

i

0 0 0

0 0 0 1

0 0 0

0 1 0 0

i

i−

η8 η9 η10 η11

0 0 1 0

0 0 0

1 0 0 0

0 0 0

i

i−

0 0 0

0 0 1 0

0 1 0 0

0 0 0

i

i−

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

−

−

0 0 0

0 0 0

0 0 0 1

0 0 1 0

−

i

i

η12 η13 η14 η15

0 0 0 1

0 0 0

0 0 0

1 0 0 0

−

i

i

0 0 0

0 0 0 1

0 0 0

0 1 0 0

−

i

i

0 0 0

0 0 0

0 0 0 1

0 0 1 0

i

i−

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

−−

154 Polarized Light, Third Edition

tm m i m m

0101 10 23 32

2= + − −( )

(8.25)

tm m i m m

0202 20 13 31

2= + + −( )

(8.26)

tm m i m m

0303 30 12 21

2= + − −( )

(8.27)

tm m i m m

1001 10 23 32

2= + + −( )

(8.28)

tm m m m

1100 11 22 33

2= + − −

(8.29)

tm m i m m

1212 21 03 30

2= + + −( )

(8.30)

tm m i m m

1313 31 02 20

2= + − −( )

(8.31)

tm m i m m

2002 20 13 31

2= + − −( )

(8.32)

tm m i m m

2112 21 03 30

2= + − −( )

(8.33)

tm m m m

2200 11 22 33

2= − + −

(8.34)

tm m i m m

2323 32 01 10

2= + + −( )

(8.35)

tm m i m m

3003 30 12 21

2= + + −( )

(8.36)

tm m i m m

3113 31 02 20

2= + + −( )

(8.37)

tm m i m m

3223 32 01 10

2= + − −( )

(8.38)

tm m m m

3300 11 22 33

2= − − +

. (8.39)

Mathematics of the Mueller Matrix 155

The eigensystem for the coherency matrix Tc can be found and provides the decomposition of Tc into four components; that is,

T T T T Tc c c c c= + + +λ λ λ λ1 1 2 2 3 3 4 4 , (8.40)

where the λ are the eigenvalues of Tc and

T k kci i iT= ⊗ * (8.41)

are the eigenvectors. The eigenvalues of Tc are real since Tc is Hermitian. The eigenvectors are in general complex. Each eigenvalue/eigenvector corresponds to a Jones matrix (and every Jones matrix corresponds to a physically realizable polarization element). The Jones matrix correspond-ing to the dominant eigenvalue is the matrix that describes the dominant polarizing action of the element. Extraction of this Jones matrix may be of interest for some applications; however, here the properties of the sample are most important.

These properties may be found with the realization that the eigenvector corresponding to the dominant eigenvalue is the quantity known as the C-vector [23]. The eigenvector components are the coefficients of the Pauli matrices in the decomposition of the Jones matrix; this is identical to the definition of the C-vector. The components of the C-vector give the information shown in Table 8.2.

Cloude has shown that for an experimental Mueller matrix to be physically realizable, the eigen-values of the corresponding coherency matrix must be nonnegative. The ratio of negative to positive eigenvalues is a quantitative measure of the realizability of the measured matrix. Further, a matrix that is not physically realizable can be “filtered,” or made realizable by subtracting the component corresponding to a negative eigenvalue from the coherency matrix. Calculation of a new Mueller matrix then yields one that most likely includes errors and scattering, but one that can be con-structed from real polarization components.

8.4 eXamPle eigeNVeCToR aNalySiS

In this section, a simple example of the calculations described in Section 8.3 is given. We will also give examples of the calculations to derive the closest physically realizable Mueller matrix from experimentally measured matrices.

Table 8.2meaning of the C-Vector Components

matrix Coefficient meaning

σ0 ρ0 Amplitude Absorption

σ0 ϕ0 Phase Phase

σ1 ρ1 Amplitude Linear diattenuation along axes

σ1 ϕ1 Phase Linear retardance along axes

σ2 ρ2 Amplitude Linear diattenuation 45°

σ2 ϕ2 Phase Linear retardance 45°

σ3 ρ3 Amplitude Circular diattenuation

σ3 ϕ3 Phase Circular retardance

156 Polarized Light, Third Edition

8.4.1 eigenvecToR analySiS

The Mueller matrix for a partial linear polarizer with principal intensity transmission coefficients k1 = 0.64 and k2 = 0.36 (i.e., principal amplitude transmission coefficients p1 = 0.8 and p2 = 0.6) along the principal axes and having an orientation θ = 0 is given by

0 50 0 14 0 0 0 0

0 14 0 50 0 0 0 0

0 0 0 0 0 48 0 0

0

. . . .

. . . .

. . . .

.00 0 0 0 0 0 48. . .

.

(8.42)

The equivalent Jones matrix is

0 8 0 0

0 0 0 6

. .

. ..

(8.43)

The Cloude coherency matrix is

0 98 0 14 0 0 0 0

0 14 0 02 0 0 0 0

0 0 0 0 0 0 0 0

0 0

. . . .

. . . .

. . . .

. 00 0 0 0 0 0. . .

.

(8.44)

There is only one nonzero eigenvalue of this matrix and it has a value of one. The eigenvector cor-responding to this eigenvalue is

0 9899

0 1414

0 000

0 000

.

.

.

.

,

(8.45)

where the second element of this vector is a measure of the linear diattenuation. Note that the terms corresponding to diattenuation at 45° and circular diattenuation are zero. Now suppose that the polarizer with the same principal transmission coefficients is rotated 40°. The Mueller matrix is

0 500000 0 024311 0 137873 0 000000

0 024311 0 4

. . . .

. . 880603 0 003420 0 000000

0 137873 0 003420 0 499

. .

. . . 3397 0 000000

0 000000 0 000000 0 000000 0 48000

.

. . . . 00

. (8.46)

The dominant eigenvalue of the corresponding target coherency matrix is approximately one, and the corresponding eigenvector is

0 9899

0 0246

0 1393

0 0002

.

.

.

.

.

i

(8.47)

Mathematics of the Mueller Matrix 157

With the rotation, the original linear polarization has coupled to polarization at 45° and circular polarization, and, in fact, the polarization at 45° is now the largest.

The linear diattenuation can now be calculated from (1) the original Mueller matrix, (2) the Jones matrix as found by Gerrard and Burch, and (3) the Cloude coherency matrix eigenvector. The linear diattenuation is given by

k k

k k1 2

1 2

64 3664 36

28−+

= −+

=. .. .

. . (8.48)

Calculation of the linear diattenuation from the Jones matrix derived directly from the Mueller matrix gives

r r

r r12

22

12

22

2 2

2 2

8 68 6

28−+

= −+

=. .. .

. . (8.49)

In the method of Cloude, the components of the eigenvector corresponding to the dominant eigen-value (i.e., the components of the C-vector) are given by

kr r

01 2

2= +( )

, (8.50)

and

kr r

11 2

2= −( )

, (8.51)

so that, solving for r1, r2, and calculating diattenuation, a value of 0.28 is again obtained.

8.4.2 noiSe

Let us now examine experimental Mueller matrices that have noise and are not likely to be physi-cally realizable, and convert these into the closest possible physically realizable Mueller matrix. We will follow a slightly different prescription [24] than that given above [20]. First, create the covari-ance matrix N for the experimental Mueller matrix M from the following equations, where we index from 1 to 4 in this subsection:

n m m m m11 11 22 12 21= + + + (8.52)

n n12 21= * (8.53)

n n13 31= * (8.54)

n n14 41= * (8.55)

n m m i m m21 13 23 14 24= + − +( ) (8.56)

n m m m m22 11 22 12 21= − − + (8.57)

158 Polarized Light, Third Edition

n n23 32= * (8.58)

n n24 42= * (8.59)

n m m i m m31 31 32 41 42= + + +( ) (8.60)

n m m i m m32 33 44 34 43= − + +( ) (8.61)

n m m m m33 11 22 12 21= − + − (8.62)

n n34 43= * (8.63)

n m m i m m41 33 44 34 43= + − −( ) (8.64)

n m m i m m42 31 32 41 42= − + −( ) (8.65)

n m m i m m43 13 23 14 24= − − −( ) (8.66)

n m m m m44 11 22 12 21= + − − . (8.67)

Since this results in a Hermitian matrix, the eigenvalues will be real and the eigenvectors orthogo-nal. Now find the eigenvalues and eigenvectors of this matrix, and form a diagonal matrix from the eigenvalues; that is,

ΛΛ =

λλ

λλ

1

2

3

4

0 0 0

0 0 0

0 0 0

0 0 0

. (8.68)

We now set any negative eigenvalues in Λ equal to zero because negative eigenvalues correspond to nonphysical components. Construct a matrix V composed of the eigenvectors of N and perform the similarity transform

ΓΓ ΛΛ= −V V 1, (8.69)

where Γ is the covariance matrix corresponding to the closest physical Mueller matrix to M. Finally construct the physical Mueller matrix using the linear transformation

′ = + − −m21

11 22 33 44

2γ γ γ γ

(8.70)

′ = + −m12 21 33 22γ γ γ (8.71)

Mathematics of the Mueller Matrix 159

′ = − −m22 11 22 12γ γ γ (8.72)

′ = − − −m11 11 22 12 212γ γ γ γ (8.73)

′ = +( )m Re13 21 43γ γ (8.74)

′ = ( ) − ′m Re m23 21 132γ (8.75)

′ = +( )m Re31 31 42γ γ (8.76)

′ = ( ) −m Re32 31 312γ γ (8.77)

′ = +( )m Re33 41 32γ γ (8.78)

′ = ( ) −m Re44 41 332γ γ (8.79)

′ = − +( )m Im14 21 43γ γ (8.80)

′ = ( ) + ′m Im m24 43 142γ (8.81)

′ = +( )m Im41 31 42γ γ (8.82)

′ = ( ) − ′m Im m42 31 412γ (8.83)

m Im43 41 32= +( )γ γ (8.84)

′ = ( ) − ′m Im m34 32 432γ . (8.85)

Let us now show numerical examples. The first example is an experimental calibration matrix for a rotating retarder polarimeter. The (normalized) matrix, which should ideally be the identity matrix, is

0 978 0 0 003 0 005

0 1 000 0 007 0 006

0 0 007 999

. . .

. . .

. .

−−−

− −

0 007

0 005 0 003 0 002 0 994

.

. . . .

. (8.86)

The eigenvalues of the corresponding covariance matrix are, written in vector form,

1 986 0 016 0 008 0 005. . . . .− − −[ ] (8.87)

160 Polarized Light, Third Edition

Three of these eigenvalues are negative so that the three corresponding eigenvalues must be removed (subtracted) from the diagonal matrix formed by the set of four eigenvalues. In this case, the filtered matrix is

0 993 0 0 001 0 005

0 0 993 0 007 0 004

0 002 0 007

. . .

. . .

. .

−00 993 0 002

0 005 0 004 0 001 0 993

. .

. . . .

−−

. (8.88)

The eigenvalue ratio, the ratio of the negative eigenvalue to the dominant eigenvalue in decibels, is a measure of the closeness to realizability. For this example the ratio of the largest negative eigenvalue to the dominant eigenvalue is approximately –21 dB. The original matrix was quite close to being physically realizable.

In a second example we have the case of a quartz plate that has its optic axis misaligned from the optical axis, inducing a small birefringence. The measured matrix was

1 000 0 019 0 021 0 130

0 024 0 731 0 726 0 00

. . . .

. . . .

−− − − 55

0 008 0 673 0 688 0 351

0 009 0 259 0 247 0

. . . .

. . . .

− −− − 9965

. (8.89)

The eigenvalues of the corresponding covariance matrix are

2 045 0 046 0 017 0 073. . . . ,− −[ ] (8.90)

and the eigenvalue ratio is approximately –14.5 dB. In this case there are two negative eigenvalues that must be subtracted. The filtered matrix becomes

1 045 0 021 0 019 0 093

0 024 0 725 0 718 0 00

. . . .

. . . .

−− − − 77

0 017 0 670 0 682 0 345

0 044 0 254 0 244 0

. . . .

. . . .

− −− − 9938

. (8.91)

Other approaches for obtaining physically realizable Mueller matrices from experimentally mea-sured matrices can be found in the literature [25,26].

8.5 The lu–ChiPmaN deComPoSiTioN

Given an experimental Mueller matrix, we would like to be able to separate the diattenuation, retardance, and depolarization. A number of researchers had addressed this issue, for example, [9,27] for nondepolarizing matrices. A general decomposition, a significant and extremely useful development, was only derived with the work of Lu and Chipman, based on the polar decomposition of nondepolarizing Mueller matrices by Gil and Bernabeu [9]. This polar decomposition, which we call the Lu–Chipman decomposition [28,29], allows a Mueller matrix to be decomposed into the product of the three factors diattenuation, retardance, and depolarization.

Mathematics of the Mueller Matrix 161

Let us first review the nondepolarizing factors of diattenuation and retardance in this context. Diattenuation changes the intensity transmittances of the incident polarization states. The diattenu-ation is defined as

dT TT T

≡ −+

max min

max min

, (8.92)

and takes values from 0 to 1. Eigenpolarizations are polarization states that are transmitted unchanged by an optical element except for a possible change in phase and intensity. A diattenuator has two eigenpolarizations. For example, a horizontal polarizer has the eigenpolarizations of hori-zontal polarization and vertical polarization. If the eigenpolarizations are orthogonal, the element is a homogeneous polarization element, and is inhomogeneous otherwise. The axis of diattenuation is along the direction of the eigenpolarization with the larger transmittance. Let this diattenuation axis be along the eigenpolarization described by the Stokes vector

1 11 2 3d d dT T

T( ) = ( ), ˆ ,D (8.93)

where

d d d12

22

32 1+ + = =ˆ .D (8.94)

Let us define a diattenuation vector

D D≡ =

=

d

dd

dd

dd

d

d

d

H

c

ˆ ,1

2

3

45 (8.95)

where dH is the horizontal diattenuation, d45 is the 45° linear diattenuation, and dc is the circular diattenuation. The linear diattenuation is defined as

d d dL H≡ +2452 , (8.96)

and the total diattenuation is

d d d d d dH c L c= + + = + =2452 2 2 2 D . (8.97)

The diattenuation vector provides a complete description of the diattenuation properties of a diattenuator.

The intensity transmittance can be written as the ratio of energies in the exiting to incident Stokes vector

TS

S

m S m S m S m S

S= ′ = + + +0

0

00 0 01 1 02 2 03 3

0

, (8.98)

where there is an intervening element with Mueller matrix M. The first row of the Mueller matrix completely determines the intensity transmittance. Equation 8.98 can be rewritten as

T mS

= + ⋅00

0

0

M S, (8.99)

162 Polarized Light, Third Edition

where the vectors are defined as M0 ≡ (m01,m02,m03) and S ≡ ( )s s s1 2 3, , . The maximum and mini-mum values of the dot product can be taken to be

S M M⋅ =0 0 0S , (8.100)

and

S M M⋅ = −0 0 0S , (8.101)

so that the maximum and minimum transmittances Tmax and Tmin are

T m m m mmax = + + +00 012

022

032 (8.102)

T m m m mmin .= − + +00 012

022

032 (8.103)

The normalized Stokes vectors associated with Tmax and Tmin are

ˆmaxS =

+ +

+ +

1

01

012

022

032

02

012

022

032

0

mm m m

mm m m

m 33

012

022

032m m m+ +

(8.104)

and

ˆminS =

−+ +

−+ +

1

01

012

022

032

02

012

022

032

mm m m

mm m m

−−+ +

m

m m m03

012

022

032

. (8.105)

The diattenuation of the Mueller matrix is

dT TT T m

m m m= −+

= + +max min

max min

,1

00012

022

032 (8.106)

and the axis of diattenuation is along the maximum transmittance and thus the direction of Smax . The axis of diattenuation is along the state Smax and the diattenuation vector of the Mueller matrix is then given by

D =

=

d

d

dm

m

m

m

H

c

4500

01

02

03

1 (8.107)

Mathematics of the Mueller Matrix 163

so that the first row of a Mueller matrix gives its diattenuation vector. The expressions for Smax and Smin can be written as

SDmax =

1 (8.108)

and

SDmin .=

−

1 (8.109)

Operational definitions for the components of the diattenuation vector are given by

T T

T T

m

mdH V

H VH

−+

= =01

00

(8.110)

T T

T T

m

md45 135

45 135

02

0045

−+

= = (8.111)

T T

T T

m

mdR L

R Lc

−+

= =03

00

, (8.112)

where TH is the transmittance for horizontally polarized light, TV is the transmittance for vertically polarized light, T45 is the transmittance for linear 45° polarized light, T135 is the transmittance for linear 135° polarized light, TR is the transmittance for right circularly polarized light, and TL is the transmittance for left circularly polarized light.

Now consider that we have incident unpolarized light; that is, only one element of the incident Stokes vector is nonzero. The exiting state is determined completely by the first column of the Mueller matrix. The property of changing completely unpolarized light to polarized light is called polarizance. The polarizance is given by

Pm

m m m= + +1

00102

202

302 , (8.113)

and can take values from 0 to 1. A normalized polarizance vector is given by

ˆ .P ≡

=

P

P

Pm

m

m

m

H

R

4500

10

20

30

1 (8.114)

The components of the polarizance vector are equal to the horizontal degree of polarization, 45° linear degree of polarization, and circular degree of polarization resulting from incident unpolar-ized light.

Retarders are phase changing devices and have constant intensity transmittance for any incident polarization state. Eigenpolarizations are defined for retarders according to the phase changes they

164 Polarized Light, Third Edition

produce. The component of light with leading phase has its eigenpolarization along the fast axis (see Chapters 21 and 23) of the retarder. Let us define a vector along this direction

1 11 2 3, , , , ,a a aT T T( ) = ( )R (8.115)

where

a a a12

22

32 1+ + = =R . (8.116)

The retardance vector and the fast axis are described by

R R≡ =

≡

R

Ra

Ra

Ra

R

R

R

H

c

ˆ ,1

2

3

45 (8.117)

where the components of R give the horizontal, 45° linear, and circular retardance components. The net linear retardance is

R R RL H= +2452 (8.118)

and the total retardance is

R R R R R RH c L c= + + = + =2452 2 2 2 R . (8.119)

Now that we have laid the groundwork for nondepolarizing Mueller matrices, let us consider the decomposition of these matrices. Nondepolarizing Mueller matrices can be written as the product of a retarder and diattenuator; that is,

M M M= R d , (8.120)

where MR is the Mueller matrix of a pure retarder and MD is the Mueller matrix of a pure diattenu-ator. A normalized Mueller matrix M can be written

M =

1 01 02 03

10 11 12 13

20 21 22 23

30 31

m m m

m m m m

m m m m

m m m332 33

1

m

T

=

D

P m, (8.121)

where the submatrix m is

m =

m m m

m m m

m m m

11 12 13

21 22 23

31 32 33

, (8.122)

Mathematics of the Mueller Matrix 165

and D and P are the diattenuation and polarizance vectors as given in Equations 8.107 and 8.114. The diattenuator Md is calculated from the first row of M, and Md

−1 can then be multiplied by M to obtain the retarder matrix M MMR d= −1 . The diattenuator matrix is given by

MD

D md

T

d

=

1, (8.123)

where

m I D DdTa b= + ×( )3 , (8.124)

and where I3 is the 3 × 3 identity matrix, and a and b are scalars derived from the norm of the diat-tenuation vector, that is,

d = D (8.125)

a d= −1 2 (8.126)

bd

d= − −1 1 2

2. (8.127)

Writing the diattenuator matrix out, we have

Md

m m m

m a bm bm m bm m

m bm=

+1 01 02 03

01 012

01 02 01 03

02 022 01 022

02 03

03 03 01 03 02 032

m a bm bm m

m bm m bm m a bm

++

, (8.128)

where

a m m m= − + +( )1 012

022

032 , (8.129)

and

bm m m

m m m= − − + +( )

+ +( )1 1 01

2022

032

012

022

032

. (8.130)

Md−1 is then given by

MD

D I

0

0 D Dd

T T

Ta a a− =

−−

++( ) ⋅( )

12

32

1 1 11

0

. (8.131)

The retarder matrix is

M0

0 mR

T

R

=

1, (8.132)

166 Polarized Light, Third Edition

where

m m P DRT

ab= − ⋅( )( )1

. (8.133)

The retarder matrix can be written explicitly as

MR a

a

m b m m m b m m m b m=

− ( ) − ( ) −1

0 0 0

0 11 10 01 12 10 02 13 100 03

21 20 01 22 20 02 23 200

m

m b m m m b m m m b m m

( )− ( ) − ( ) − 003

31 30 01 32 30 02 33 30 030

( )− ( ) − ( ) −m b m m m b m m m b m m(( )

. (8.134)

The total retardance R and the retardance vector can be found from the equations

Rtr

RR= = ( ) −

≤ ≤−

R

mcos 1 1

20 π, (8.135)

Rtr

RR= = − ( ) −

≤ ≤−

R

m2

12

21π π πcos and, (8.136)

R

M M

M M=

=( ) − ( )( ) − ( )

R

R

R

H

c

R R

R R45

23 32

31 133

12 21

2M MR R

RR( ) − ( )

( )sin

. (8.137)

The total retardance is then given explicitly as

Ra

m m m b m m m m m m= − + + − + +cos 1 12 11 22 33 10 01 20 02 30 03(( ) −

a , (8.138)

and the retardance vector is given by

R =

− − −( )− − −

m m b m m m m

m m b m m m

23 32 20 03 30 02

31 13 30 01 110 03

12 21 10 02 20 01

m

m m b m m m m

( )− − −( )

×− + + − + +

.

cos 1 12 11 22 33 10 01 20 02 30a

m m m b m m m m m m003

4 211 22 33 10 01 20

( ) −

− + + − +

a

a m m m b m m m m002 30 032+( ) −[ ]m m a

(8.139)

A pure (nonuniform) depolarizer can be represented by the matrix

1 0 0 0

0 0 0

0 0 0

0 0 0

a

b

c

, (8.140)

Mathematics of the Mueller Matrix 167

where a b c, , ≤ 1 . The principal depolarization factors are 1− a ,1− b , and 1− c , and these are measures of the depolarization of this depolarizer along its principal axes. The parameter Δ given by

∆ ∆≡ − + + ≤ ≤13

0 1a b c

, (8.141)

is the average of the depolarization factors, and this parameter is called the depolarization power of the depolarizer. An expression for a depolarizer can be written as

1 0

0 mm m

TT

∆∆ ∆

=, , (8.142)

where mΔ is a symmetric 3 × 3 submatrix. The eigenvalues of mΔ are the principal depolarization factors, and the eigenvectors are the three orthogonal principal axes. This last expression is not the complete description of a depolarizer, because it contains only six degrees of freedom when we require nine. The most general expression for a depolarizer can be written as

M0

P mm m∆

∆ ∆∆ ∆=

=1 T

T, , (8.143)

where PΔ is the polarizance vector, and with this expression we have the required nine degrees of freedom and no diattenuation or retardance. Thus we see that a depolarizer with a nonzero polari-zance may actually have polarizing properties according to our definition here.

Depolarizing Mueller matrices can be written as the product of the three factors of diattenuation, retardance, and depolarization; that is,

M M M M= ∆ R d , (8.144)

where MΔ is the depolarization, and this equation is the generalized polar decomposition for depo-larizing Mueller matrices. It is useful for the decomposition of experimental Mueller matrices to allow the depolarizing component to follow the nondepolarizing component. As in the nondepolar-izing case, we first find the matrix for the diattenuator. We then define a matrix M′ such that

′ −M = MM = M M1d R∆ . (8.145)

This expression can be written out as the product of the 2 × 2 matrices

M M0

P m

0

0 m

0

P m m∆∆ ∆ ∆ ∆

R

T T

R

T

R

=

=

1 1 1

=′

= ′1 0

P mM

T

∆

.

(8.146)

Let λ1, λ2, and λ3 be the eigenvalues of

′ ′( ) = ( ) =m m m m m m mT

R RT

∆ ∆ ∆2 . (8.147)

168 Polarized Light, Third Edition

We can obtain the relations

PP mD

∆ = −−1 2d

(8.148)

and

′ =m m m∆ R (8.149)

from Equation 8.145 to Equation 8.146.The eigenvalues of mΔ are then λ1 , λ2 , and λ3 . It should be pointed out that there is an

ambiguity in the signs of the eigenvalues [24]. The retarder submatrix mR is a three-dimensional rotation matrix and has positive determinant so that the sign of the determinant of m′ indicates the sign of the determinant of mΔ. The assumption that the eigenvalues all have the same sign is reason-able, especially since depolarization in measured systems is usually small and the eigenvalues are close to one. This assumption simplifies the expression for mΔ. An expression for mΔ is given by, from the Cayley–Hamilton theorem (a matrix is a root of its characteristic polynomial),

m m m I m m I∆ = ± ′ ′( ) + ′ ′( ) + −T Tκ κ κ2

1

1 3 , (8.150)

where

κ λ λ λ1 1 2 3= + + (8.151)

κ λ λ λ λ λ λ2 1 2 2 3 3 1= + + (8.152)

and

κ λ λ λ3 1 2 3= . (8.153)

The sign in front of the expression on the right-hand side in Equation 8.149 follows the sign of the determinant of m′. We can now find mR from the application of m∆

−1 to m′; that is,

m m m m m I m m mRT T= ′ = ± ′ ′( ) + ′ ′( ) ′ + ′−

−

∆1

1 3

1

2κ κ κ mm . (8.154)

The eigenvalues λ1, λ2, and λ3 can be found in terms of the original Mueller matrix elements by solving a cubic equation, but the expressions that result are long and complicated. It is more feasible to find the κ’s. We have

κ32= ( ) = ( ) = ′ ′( )( ) = ′( )det det det detm m m m m∆ ∆

T. (8.155)

Recall that M′ = M(Md)–1 has the form

′ =′

M0

P m

1 T

∆

, (8.156)

Mathematics of the Mueller Matrix 169

so that

κ31= ′( ) = ′( ) = ( ) ( ) = ( )−det det det det

detde

m M M MM

∆ ttdet

MM

∆( ) = ( )a4

. (8.157)

Let us define a τ1 and τ2 such that

τ λ λ λ12

1 2 3= [ ] = + +Tr m∆ (8.158)

and

τ κ λ λ λ λ λ λ2 32 2 1

1 2 1 3 2 3= ( ) = + +−Tr m∆ . (8.159)

Then κ1 satisfies the recursive equation

κ τ τ κ κ1 1 2 3 12= + + . (8.160)

This can be approximated by

κ τ τ κ τ1 1 2 3 12 2≈ + + . (8.161)

Since

κ κ τ2 12

112

= −[ ], (8.162)

we can use the approximation for κ1 to obtain the approximation for κ2

κ τ κ τ2 2 3 12≈ + . (8.163)

Expressions for τ1 and τ2 are given in terms of the original Mueller matrix elements and the elements of m∆

2

τ1 22

1

3

02

1

3

4

1 1= −

+= =

∑ ∑am m

ami j

i j

i

i

i,

,

, ,, , ,0 0

1

3 2

1

3

−

==

∑∑ m mi j j

ji

(8.164)

and

τ22

2 2 3 3 1 1 3 3 1 1 2 2 2 3= + + − +m m m m m m m∆ ∆ ∆ ∆ ∆ ∆ ∆, , , , , , ,

mm m∆ ∆1 3 1 2

2 2, ,

,+( ) (8.165)

where the elements of m∆2 are

m∆

∆ ∆ ∆

∆ ∆ ∆

∆ ∆

2

1 1 1 2 1 3

2 1 2 2 2 3

3 1 3

=

m m m

m m m

m m

, , ,

, , ,

, ,22 3 3m∆ ,

,

(8.166)

170 Polarized Light, Third Edition

where we note that m mi j j i∆ ∆, ,

= and

m∆i j am m m m

aik jk

k

i j,=

−

+

=∑1 1

21

3

0 0 4mm m m m m mi ik k

k

j jk k

k

0 0

1

3

0 0

1

3

−

−

= =∑ ∑ .. (8.167)

Then we can write

κ32 2 1

22 2 3 3 2 3 1 3 2 3 1

m∆

∆ ∆ ∆ ∆ ∆ ∆

( ) =

− −−

m m m m m m, , , , , ,22 3 3 1 2 2 3 2 2 2 3

1 3 2 3 1

m m m m m

m m m

∆ ∆ ∆ ∆ ∆

∆ ∆ ∆

, , , , ,

, , ,

−

−22 3 3 1 1 3 3 1 3 1 2 1 3 1 1 2

2m m m m m m m m∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆, , , , , , ,− −

,,

, , , , , , ,

3

1 2 2 3 2 2 2 3 1 2 1 3 1 1 2m m m m m m m m∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆− −

,, , , ,3 1 1 2 2 1 2

2m m m∆ ∆ ∆−

(8.168)

and the retarder rotation matrix is given by

m m m I m m mR = ′ = − + ( ) ′− −∆ ∆ ∆

1

1

232 11

κα β γκ 2 . (8.169)

If we can find approximations for the depolarizer eigenvalues λ1 , λ2 , and λ3 , then we can write an expression for m∆

−1 as

m I m m∆ ∆ ∆− −= − + ( )

1

1

232 2 11

κα β γκ , (8.170)

where

α λ λ λ λ λ λ λ λ λλ λ λ λ

= + +( ) + +( ) −+( ) +( )

1 2 3 1 2 3 1 2 3

1 2 1 3 λλ λ2 3

1+( ) + , (8.171)

βλ λ λ λ λ λ

=+( ) +( ) +( )

1

1 2 1 3 2 3

, (8.172)

and

γ λ λ λλ λ λ λ λ λ λ λ λ

= + +( )( ) +( ) +( ) +( )

1 2 3

1 2 3 1 2 1 3 2 3

. (8.173)

8.6 deComPoSiTioN oRdeR

Lu [28] has noted that the order of the constituent matrices in Equation 8.144 is not the only order that could have been chosen. In fact, there are six possibilities. They are

M M M M= −∆1 (Lu Chipman)R d1 1 (8.174)

M M M M= ∆2 d R2 2 (8.175)

Mathematics of the Mueller Matrix 171

M M M M= R d3 3 3∆ (8.176)

M M M M= d R4 4 ∆4 (8.177)

M M M M= R d5 5∆5 (8.178)

M M M M= d R6 6∆6 , (8.179)

where the decomposition in Equation 8.174 is that of Equation 8.144, which we used earlier. The basic difference in these choices is the order in which the matrices for the depolarizer and diattenu-ator appear. Decompositions 1, 2, and 5 have the depolarizer before the diattenuator and decomposi-tions 3, 4, and 6 have the diattenuator before the depolarizer. These are members of two different families of decompositions where the first family, of which the Lu–Chipman form is one, always leads to physical Mueller matrices. Members of the second family produce decompositions that may result in nonphysical results [30,31]. The first family has been called the “forward” decomposition, and the second family the “reverse” decomposition [31]. Procedures for obtaining physical results from the reverse decompositions have been described, and it has been suggested that if it is known that the depolarization occurs prior to the diattenuation, the reverse decomposition should be used [31], and vice versa. For experimental investigations on an unknown sample, the source of the depo-larization is generally not known.

Relationships between the forward decompositions can be established. Comparing the standard form in Equation 8.174 with the second ordering in Equation 8.175, we have that

M M M M M M M M∆ ∆2 1 2 1 2 1 1 1= = =, , ,R R d R d RT (8.180)

that is, the depolarizers and retarders are the same, and the diattenuators are related by a similarity transform. We can also perform a comparison of the standard form to that of the fifth ordering in Equation 8.178 and find that

M M M M M M M M∆ ∆5 1 1 1 5 1 5 1= = =TR R R R d d, , , (8.181)

where in this case, the retarders and diattenuators are the same and the depolarizers are related by a similarity transform.

8.7 deComPoSiTioN of dePolaRiZiNg maTRiCeS WiTh dePolaRiZaTioN SymmeTRy

Recall that the matrix for the Lu–Chipman depolarizer is given by

M0

P m∆∆ ∆

f

T

f

=

1, (8.182)

where we have added an “f” to the subscripts for forward decomposition. The depolarization matrix for reverse decomposition is given by

MD

0 m∆∆

∆r

T

r

=

1. (8.183)

172 Polarized Light, Third Edition

The forward decomposition MΔf results in a depolarizer with polarizance, and the reverse decomposi-tion MΔr results in a depolarizer with diattenuation. These depolarizers are quite asymmetric. However, in many cases, the depolarization produced results in a pure depolarizer of the (unnormalized) form

M∆d

d

d

d

d

=

0

1

2

3

0 0 0

0 0 0

0 0 0

0 0 0

, (8.184)

that is, a diagonal depolarizer matrix. With this form, it is possible to perform a symmetric decom-position [30] where the depolarizer is always between two sets of diattenuators and retarders, thus,

M M M M M M= d R d RT

d2 2 1 1∆ . (8.185)

As with the Lu–Chipman decomposition, there are alternate forms . In this case, there are four pos-sibilities where the other three are

M M M M M M= ′ ′R d d d RT

2 2 1 1∆ (8.186)

M M M M M M= ′d R d d RT

2 2 1 1∆ (8.187)

M M M M M M= ′R d d RT

d2 2 1 1∆ (8.188)

where we can show, using

M M M M IR RT

R RT

1 1 2 2= = (8.189)

′ =M M M Md RT

d R1 1 1 1 (8.190)

′ =M M M Md RT

d R2 2 2 2 (8.191)

that these alternate decomposition forms are equivalent. We will use the form in Equation 8.185 because it has a straightforward decomposition procedure to be described below.

Note that if the Mueller matrix is almost free of depolarization such that

M I∆d d≈ 0 (8.192)

then we have a matrix Mnd that is the nondepolarizing approximation to the weakly depolarizing matrix M. This approximation is then given by

M M M M Mnd = d d R RT

d0 2 2 1 1. (8.193)

The procedure for performing the symmetric decomposition consists of four steps. In the first step, solve the eigenvalue equations

M GMG S ST d( ) =1 02

1 (8.194)

MGM G S ST d( ) =2 02

2, (8.195)

Mathematics of the Mueller Matrix 173

where G is the Minkowski matrix

G =−

−−

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

, (8.196)

and where the eigenvectors are

SD

SD1

12

2

1 1=

=

. (8.197)

Only one of these needs to be solved, since we can obtain the other from the relationships

SMGSMGS

SM GSM GS2

1

1 01

2

2 0

= ( ) = ( )T

T, (8.198)

where (…)0 denotes the zeroth component of the vector. In the second step, construct the two diat-tenuator matrices Md1 and Md2 using the Equations 8.123 and 8.124. In the third step, calculate an intermediate matrix M′ given by

′ = − −M M MMd d21

11 , (8.199)

where the 3 × 3 block sub-matrix m′ is

′ =m m m mR d RT

2 1∆ . (8.200)

And, in the fourth step, perform a singular value decomposition on Equation 8.200 to obtain the retarder matrices MR1 and MR2 from mR1 and mR2 using Equation 8.132 and the diagonal depolarizer matrix MΔd from mΔd where

m∆d

d

d

d

d M=

= ′1

2

3

0 00

0 0

0 0

0 0

. (8.201)

This decomposition is valid when d1 <1 and d2 <1. Numerical examples are given in Reference [32].Note that two canonical forms have been proposed for depolarizer matrices [33], the diagonal

form that we have just discussed and shown in Equation 8.183, and a normalized nondiagonal form given by

M∆ndd

d

=

−

112

0 0

12

0 0 0

0 02

0

0 0 02

2

2

. (8.202)

174 Polarized Light, Third Edition

8.8 deComPoSiTioN uSiNg maTRiX RooTS

A scheme to decompose Mueller matrices using matrix roots and logarithms has been proposed [34]. This method is said to be order independent. If we compute the nth root of the Mueller matrix, that is,

V =

m m m m

m m m m

m m m m

m m

00 01 02 03

10 11 12 13

20 21 22 23

30 311 32 33m m

n , (8.203)

where n is large, then V may be close to the identity matrix, and V is decomposed as

V =

+ + +− − − + − +1 1 1 2 2 3 3

1 1 1 2 3 3 3 2 2

2

d e d e d e

d e q q r s r s

d −− − + − − +− + − + − −

e r s q q r s

d e r s r s q

2 3 3 1 1 3 1 1

3 3 2 2 1 1 1 1 qq2

. (8.204)

The three parameters d1, d2, and d3 are functions of the diattenuation, and r1, r2, and r3 are func-tions of the retardance. The remaining nine degrees of freedom of the Mueller matrix are described by the e’s, the s’s, and the q’s that are the depolarization parameters of the root matrix. The Mueller matrix for any one property may be found by raising the matrix with one of these parameters to the nth power. Note that the depolarization parameters are presumed to be distributed throughout the matrix in this method.

8.9 SummaRy

We have answered the questions posed at the beginning of this chapter. With the material presented here, we now have the tools to determine whether a Mueller matrix is physically realizable and we have a method to bring it to the closest physically realizable matrix. We can then separate the matrix into its constituent components of diattenuation, retardance, and depolarization. However, we must remember that noise, once introduced into the system, is impossible to remove entirely. The experimentalist must take prudent precautions to minimize the influence of errors peculiar to the system at hand.

RefeReNCeS

1. Jones, R. C., A new calculus for the treatment of optical systems: I. Description and discussion of the calculus, J. Opt. Soc. Am. 31 (1941): 488–93.

2. Jones, R. C., A new calculus for the treatment of optical systems: IV, J. Opt. Soc. Am. 32 (1942): 486–93.

3. Jones, R. C., A new calculus for the treatment of optical systems: V. A more general formulation, and description of another calculus, J. Opt. Soc. Am. 37 (1947): 107–10.

4. Swindell, W., Polarized Light in Optics, Stroudsberg, PA: Dowden, Hutchinson, & Ross, 1975. 5. Stokes, G. G., On the composition and resolution of streams of polarized light from different sources,

Trans. cambridge Phil. Soc. 9 (1852): 399. 6. Soleillet, P., Sur les parameters caracterisant la polarization partielle de la lumiere dans les phenomenes

de fluorescence, Ann. Phys. 12 (1929): 23. 7. Perrin, F., Polarization of light scattered by isotropic opalescent media, J. chem. Phys. 10 (1942): 415.

Mathematics of the Mueller Matrix 175

8. Mueller, H., Memorandum on the polarization optics of the photoelastic shutter, Report No. 2 of the OSRD project OEMsr-576, November 15, 1943.

9. Gil, J. J., and E. Bernabeu, Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix, Optik 76 (1986): 26–36.

10. van de Hulst, H. C., Light Scattering by Small Particles, New York: Dover Publications, 1981. 11. Anderson, D. G. M., and R. Barakat, Necessary and sufficient conditions for a Mueller matrix to be deriv-

able from a Jones matrix, J. Opt. Soc. Am. A. 11 (1994): 2305–19. 12. Brosseau, C., C. R. Givens, and A. B. Kostinski, Generalized trace condition on the Mueller-Jones polar-

ization matrix, J. Opt. Soc. Am. A 10 (1993): 2248–51. 13. Kostinski, A. B., C. R. Givens, and J. M. Kwiatkowski, Constraints on Mueller matrices of polarization

optics, Appl. Opt. 32, no. 9 (1993): 1646–51. 14. Givens, C. R., and A. B. Kostinski, A simple necessary and sufficient condition on physically realizable

Mueller matrices, J. Mod. Opt. 40, no. 3 (1993): 471–81. 15. Hovenier, J. W., H. C. van de Hulst, and C. V. M. van der Mee, Conditions for the elements of the scat-

tering matrix, Astron. Astrophys. 157 (1986): 301–10. 16. Barakat, R., Bilinear constraints between elements of the 4 × 4 Mueller-Jones transfer matrix of polariza-

tion theory, Opt. comm. 38, no. 3 (1981): 159–61. 17. Fry, E. S., and G. W. Kattawar, Relationships between elements of the Stokes matrix, Appl. Opt. 20

(1981): 2811–14. 18. Hayes, D. M., Private communication, 1996. 19. Cloude, S. R., Group theory and polarisation algebra, Optik 75, no. 1 (1986): 26–36. 20. Cloude, S. R., Conditions for the physical realisability of matrix operators in polarimetry, Proc. SPIE

1166 (1989): 177–85. 21. Cloude, S. R., Lie groups in electromagnetic wave propagation and scattering, J. Electro. Wav. Appl. 6

(1992): 947–74. 22. Arfken, G., Mathematical Methods for Physicists, 2nd ed., New York: Academic Press, 1970. 23. Chipman, R. A., Polarization Aberrations, PhD thesis, Tucson, AZ: University of Arizona, 1987, 27–52. 24. Hayes, D. M., Error propagation in decomposition of Mueller matrices, Proc. SPIE 3121 (1997):

112–23. 25. Aiello, A., G. Puentes, D. Voigt, and J. P. Woerdman, Maximum-likelihood estimation prevents unphysi-

cal Mueller matrices, http://arxiv.org/abs/physics/0508190v1, 2005. (Accessed on 26 August 2005) 26. Ahmad, J. E., and Y. Takakura, Estimation of physically realizable Mueller matrices from experiments

using global constrained optimization, Opt. Exp. 16 (2008): 14274–87. 27. Xing, Z.-F., On the deterministic and non-deterministic Mueller matrix, J. Mod. Opt. 39 (1992):

461–84. 28. Lu, S-Y., An Interpretation of Polarization Matrices, PhD dissertation, Dept. of Physics, University of

Alabama at Huntsville, 1995. 29. Lu, S-Y., and R. A. Chipman, Interpretation of Mueller matrices based on polar decomposition, J. Opt.

Soc. Am. A 13 (1996): 1106–13. 30. Morio, J., and F. Goudail, Influence of the order of diattenuator, retarder, and polarizer in polar decom-

position of Mueller matrices, Opt. Lett. 29 (2004): 2234–6. 31. Ossikovski, R., A. De Martino, and S. Guyot, Forward and reverse product decompositions of depolar-

izing Mueller matrices, Opt. Lett. 32 (2007): 689–91. 32. Ossikovski, R., Analysis of depolarizing Mueller matrices through a symmetric decomposition, J. Opt.

Soc. Am. A 26 (2009): 1109–18. 33. Ossikovski, R., Canonical forms of depolarizing Mueller matrices, J. Opt. Soc. Am. A 27 (2010):

123–30. 34. Chipman, R., Depolarization analyzed by matrix logarithms, in Frontiers in Optics, OSA Technical

Digest (CD), Paper FTuS1, Washington, DC: Optical Society of America, 2007.

177

9 Mueller Matrices for Dielectric Plates

9.1 iNTRoduCTioN

In Chapter 7, Fresnel’s equations for reflection and transmission of waves at an air-dielectric inter-face were cast in the form of Mueller matrices. In this chapter, we use these results to derive the Mueller matrices for dielectric plates. The study of dielectric plates is important because all materi-als of any practical importance are of finite thickness and so at least have upper and lower surfaces. Furthermore, dielectric plates always change the polarization state of a beam that is reflected or transmitted. This property can therefore be exploited to manipulate polarized and unpolarized light. For example, one application is that of a polarizer design using dielectric plates for the thermal infrared region of the spectrum. Wire grid polarizers of excellent quality are now available for this spectral region (see Chapter 23), but this was not always the case. There are no polarizer materials corresponding to calcite in the thermal infrared region, but materials such as germanium and sili-con, as well as others, do transmit very well in the infrared region. By making thin plates of these materials and then constructing a “pile of plates,” it is possible to create highly polarized light in the infrared. This arrangement therefore requires that the Mueller matrices for transmission play a more prominent role than the Mueller matrices for reflection.

In order to use the Mueller matrices to characterize a single plate or multiple plates, we must carry out matrix multiplications. The presence of off-diagonal terms of the Mueller matrices creates a considerable amount of work. On the other hand, we know that if we use diagonal matrices, the calculations are simplified; the product of diagonalized matrices leads to another diagonal matrix.

9.2 The diagoNal muelleR maTRiX aNd The aBCD PolaRiZaTioN maTRiX

When we apply the Mueller matrices to problems in which there are several polarizing elements, each of which is described by its own Mueller matrix, we soon discover that the appearance of the off-diagonal elements complicates the matrix multiplications. The multiplications would be greatly simplified if we were to use diagonalized forms of the Mueller matrices. In particular, the use of diagonalized matrices enables us to determine more easily the Mueller matrix raised to the mth power, Mm, an important problem when we must determine the transmission of a polarized beam through m dielectric plates.

In this chapter, we develop the diagonal Mueller matrices for a polarizer and a retarder. To reduce the amount of calculations, it is simpler to write a single matrix that simultaneously describes the behavior of a polarizer or a retarder or a combination of both. This simplified matrix is called the Abcd polarization matrix.

The Mueller matrix for a polarizer (unrotated) is

MP

s p s p

s p s p

s

p p p p

p p p p

p p=

+ −− +1

2

0 0

0 0

0 0 2

2 2 2 2

2 2 2 2

pp

s pp p

0

0 0 0 2

, (9.1)

178 Polarized Light, Third Edition

and the Mueller matrix for a retarder with fast axis horizontal is

MR =

−

1 0 0 0

0 1 0 0

0 0

0 0

cos sin

sin cos

φ φφ φ

, (9.2)

where ps and pp are the absorption coefficients of the polarizer along the s (or x) and p (or y) axes, respectively, and ϕ is the phase shift of the retarder.

The form of Equations 9.1 and 9.2 suggests that the matrices can be represented by a single matrix of the form

ΦΦ =

−

A b

b A

c d

d c

0 0

0 0

0 0

0 0

, (9.3)

which we call the Abcd polarization matrix. We see that for a polarizer

A p ps p= +( )12

2 2 (9.4)

b p ps p= −( )12

2 2 (9.5)

c p ps p= 12

2( ) (9.6)

d = 0 (9.7)

and for the retarder

A = 1 (9.8)

b = 0 (9.9)

c = cosφ (9.10)

d = sinφ. (9.11)

If we multiply Equation 9.1 by Equation 9.2, we see that we still obtain a matrix that can be repre-sented by an Abcd matrix; the matrix describes an absorbing retarder.

The matrix elements Abcd are not all independent; that is, there is a unique relationship between the elements. To find this relationship, we see that Equation 9.3 transforms the Stokes parameters of an incident beam Si to the Stokes parameters of an emerging beam ′Si so that we have

′ = +S AS bS0 0 1 (9.12)

Mueller Matrices for Dielectric Plates 179

′ = +S bS AS1 10 (9.13)

′ = +S cS dS2 2 3 (9.14)

′ = − +S dS cS3 2 3. (9.15)

We know that for completely polarized light the Stokes parameters of the incident beam are related by

S S S S02

12

22

32= + + (9.16)

and similarly,

′ = ′ + ′ + ′S S S S02

12

22

32 . (9.17)

Substituting the Stokes parameter expressions 9.12 through 9.15 into Equation 9.17 leads to

A b S S c d S S2 202

12 2 2

22

32−( ) −( ) = +( ) +( ). (9.18)

From Equation 9.16,

S S S S02

12

22

32− = + (9.19)

and making a substitution from Equation 9.19 into Equation 9.18 gives

A b c d S S2 2 2 202

12 0− − −( ) −( ) = , (9.20)

and so we may consider that

A b c d2 2 2 2= + + . (9.21)

We see that the elements of Equations 9.4 through 9.7 and Equations 9.8 through 9.11 both satisfy Equation 9.21. This is a very useful relation because it serves as a check when measuring Mueller matrix elements for the polarizer and retarder as described above.

The rotation of a polarizing device described by the Abcd matrix is given by the matrix equation

M M M= −( ) ( )2 2θ θΦΦ , (9.22)

which in its expanded form is

M =+ −( )

A b b

b A c A c

cos sin

cos cos sin

2 2 0

2 2 22 2 2

θ θθ θ θ ssin cos sin

sin sin cos sin

2 2 2

2 2 2

θ θ θθ θ θ

−−( )

d

b A c A 22 22 2 2

0 2 2

θ θ θθ θ

+−

c d

d d c

cos cos

sin cos

. (9.23)

180 Polarized Light, Third Edition

In carrying out the expansion of Equation 9.22, we used

M 2

1 0 0 0

0 2 2 0

0 2 2 0

0 0 0 1

θθ θθ θ

( ) =−

cos sin

sin cos

. (9.24)

We now find the diagonalized form of the Abcd matrix. This can be done using well-known meth-ods in matrix algebra. We first use Equation 9.3 as the eigenvector/eigenvalue equation

ΦΦS S= λ (9.25)

or

ΦΦ −( ) =λ S 0, (9.26)

where λ and S are the eigenvalues and the eigenvectors corresponding to Φ. In order to find the eigenvalues and the eigenvectors, the determinant of (Φ – λ) must be taken; that is,

A b

b A

c d

d c

−−

−− −

=

λλ

λλ

0 0

0 0

0 0

0 0

0. (9.27)

The determinant is easily expanded and leads to the secular equation

A b c d−( ) − −( ) + =2 2 2 2λ λ 0. (9.28)

The solution of Equation 9.28 yields the eigenvalues

λ1 = +A b (9.29)

λ2 = −A b (9.30)

λ3 = +c id (9.31)

λ4 = −c id. (9.32)

Substituting these eigenvalues into the determinant Equation 9.27, we find that the eigenvector cor-responding to each of the respective eigenvalues in Equations 9.29 through 9.32 is

S S1 2

2

1

1

0

0

1

2

1

1

0

0

= 1

=−

SS S3 41

2

0

0

1

1

2

0

0

1=

=

−

i i

. (9.33)

Mueller Matrices for Dielectric Plates 181

The factor 1 2/ has been introduced to normalize each of the eigenvectors.We now construct a new matrix K, called the modal matrix, whose columns are formed from the

eigenvectors in Equation 9.33 so that

K =−

−

1

2

1 1 0 0

1 1 0 0

0 0 1 1

0 0 i i

. (9.34)

The inverse matrix is easily found to be

K− =−

−

1 1

2

1 1 0 0

1 1 0 0

0 0 1

0 0 1

i

i

. (9.35)

We see that KK−1 = I, where I is the unit matrix. We now construct a diagonal matrix from each of the eigenvalues in Equations 9.29 through 9.32 and write

Md

A b

A b

c id

c id

=

+−

+−

0 0 0

0 0 0

0 0 0

0 0 0

. (9.36)

From Equation 9.4 to Equation 9.7, the diagonal Mueller matrix for a polarizer Md,P is then

Md P

s

p

s p

s p

p

p

p p

p p

, =

2

2

0 0 0

0 0 0

0 0 0

0 0 0

,, (9.37)

and from Equation 9.8 to Equation 9.11, the diagonal matrix for a retarder is

Md R i

i

e

e

, .=

−

1 0 0 0

0 1 0 0

0 0 0

0 0 0

φ

φ

(9.38)

A remarkable relation now emerges. From Equation 9.34 to Equation 9.36, we can see that

ΦΦK KM= d . (9.39)

Postmultiplying both sides of Equation 9.39 by K−1 we see that

ΦΦ = −KM Kd1, (9.40)

182 Polarized Light, Third Edition

or premultiplying

M K Kd = −1ΦΦ , (9.41)

where we have used KK–1 = I. We now square both sides of Equation 9.40 and find that

ΦΦ2 2 1= −KM Kd , (9.42)

which shows that Φm is obtained from

ΦΦmdm= −KM K 1. (9.43)

Thus, by finding the eigenvalues and the eigenvectors of Φ and then constructing the diagonal matrix and the modal matrix (and its inverse), the mth power of the Abcd matrix Φ can be found from Equation 9.43. Equation 9.40 also allows us to determine the diagonalized Abcd matrix Φ.

Equation 9.43 now enables us to find the mth power of the Abcd matrix Φ such that

ΦΦm

m

m

A b

b A

c d

d c

A b

=−

=

+( )

0 0

0 0

0 0

0 0

0 0

K

00

0 0 0

0 0 0

0 0 0

A b

c id

c id

m

m

m

−( )+( )

−( )

−K 1.

(9.44)

Carrying out the matrix multiplication using expressions for K and K-1, Equations 9.34 and 9.35, then yields

ΦΦm

m m m mA b A b A b A b

A=

+( ) + −( ) +( ) − −( ) +1

2

0 0

bb A b A b A b

c id

m m m m( ) − −( ) +( ) + −( ) +( )

0 0

0 0mm m m m

c id i c id i c id

i c id

+ −( ) − +( ) + −( ) +0 0 (( ) − −( ) +( ) + −( )

m m m mi c id c id c id

.

(9.45)

Using Equation 9.45, we readily find that the mth powers of the Mueller matrix of a polarizer and a retarder are, respectively,

MPm

sm

pm

sm

pm

sm

pm

sm

p

p p p p

p p p p=

+ −− +1

2

0 02 2 2 2

2 2 2 2mm

sm

pm

sm

pm

p p

p p

0 0

0 0 2 0

0 0 0 2

(9.46)

Mueller Matrices for Dielectric Plates 183

and

MRm

m m

m m

=

−

1 0 0 0

0 1 0 0

0 0

0 0

cos sin

sin cos

φ φφ φ

. (9.47)

The diagonalized Mueller matrices will play an essential role in the following section when we determine the Mueller matrices for single and multiple dielectric plates.

Before we conclude this section, we discuss another form of the Mueller matrix for a polarizer. We recall that the first two Stokes parameters, S0 and S1, are the sum and difference of the orthogo-nal intensities. The Stokes parameters can then be written as

S I Ix y0 = + (9.48)

S I Ix y1 = − (9.49)

S S2 2= (9.50)

S S3 3= , (9.51)

where

I E E I E Ex x x y y y= =∗ ∗ . (9.52)

We further define

I Ix = 0 (9.53)

I Iy = 1 (9.54)

S I2 2= (9.55)

S I3 3= . (9.56)

We can then relate S to I by

S

S

S

S

0

1

2

3

1 1 0 0

1 1 0 0

0 0 1 0

0 0 0 1

=−

I

I

I

I

0

1

2

3

, (9.57)

or I to S by

I

I

I

I

0

1

2

3

12

1 1 0 0

1 1 0 0

0 0 2 0

0 0 0 2

=−

S

S

S

S

0

1

2

3

. (9.58)

184 Polarized Light, Third Edition

The column matrix

I =

I

I

I

I

0

1

2

3

(9.59)

is called the intensity vector. The intensity vector is very useful because the 4 × 4 matrix that con-nects I to I′ is diagonalized, thus making the calculations simpler. To show that this is true, we can formally express Equations 9.57 and 9.58 as

S K I= A (9.60)

I K S= −A1 , (9.61)

where KA and K A−1 are defined by the 4 × 4 matrices in Equations 9.57 and 9.58, that is,

K KA A=−

=−

1 1 0 0

1 1 0 0

0 0 1 0

0 0 0 1

12

1 1 0 0

111 1 0 0

0 0 2 0

0 0 0 2

−

. (9.62)

The Mueller matrix M can be defined in terms of an incident Stokes vector S and an emerging Stokes vector S′,

′ =S MS. (9.63)

Similarly, we can define the intensity vector relationship

′ =I P I, (9.64)

where P is a 4 × 4 matrix.We now show that P is diagonal. We have from Equation 9.60 that

′ = ′S K IA . (9.65)

Substituting Equation 9.65 into Equation 9.63 along with Equation 9.60 gives

′ = ( )−I K M K IA A1 (9.66)

or, from Equation 9.64,

P K M K= −A A1 . (9.67)

Mueller Matrices for Dielectric Plates 185

We now show that for a polarizer, P is a diagonal matrix. The Mueller matrix for a polarizer in terms of the Abcd matrix elements can be written as

M =

A b

b A

c

c

0 0

0 0

0 0 0

0 0 0

. (9.68)

Substituting Equation 9.68 into Equation 9.67 and using KA and K A−1 from Equation 9.62, we readily

find that

P =

+−

A b

A b

c

c

0 0 0

0 0 0

0 0 0

0 0 0

. (9.69)

Thus, P is a diagonal polarizing matrix; it is equivalent to the diagonal Mueller matrix for a polar-izer. The diagonal form of the Mueller matrix was first used by Chandrasekhar [1] in his classic papers in radiative transfer in the late 1940s. It is called Chandrasekhar’s phase matrix in the litera-ture. In particular, for the Mueller matrix of a polarizer, we see that Equation 9.69 becomes

P =

p

p

p p

p p

s

p

s p

s p

2

2

0 0 0

0 0 0

0 0 0

0 0 0

, (9.70)

which is identical to the diagonalized Mueller matrix given by Equation 9.37. In Part IV of this book, we shall show that the Mueller matrix for scattering by an electron is proportional to

M p =

+ −− +1

2

1 0 0

1 0 0

0 0 2 0

2 2

2 2

cos sin

sin cos

cos

θ θθ θ

θ00 0 0 2cos

,

θ

(9.71)

where θ is the observation angle in spherical coordinates and is measured from the z axis (θ = 0°). Transforming Equation 9.71 to Chandrasekhar’s phase matrix, we find that

P =

cos

cos

cos

,

2 0 0 0

0 1 0 0

0 0 0

0 0 0

θ

θθ

(9.72)

which is the well-known representation for Chandrasekhar’s phase matrix for the scattering of polarized light by an electron.

186 Polarized Light, Third Edition

Not surprisingly, there are other interesting and useful transformations that can be developed. However, this development would take us too far from our original goal, which is to determine the Mueller matrices for single and multiple dielectric plates. We now apply the results in this section to the solution of that problem.

9.3 muelleR maTRiCeS foR SiNgle aNd mulTiPle dieleCTRiC PlaTeS

In the previous sections, Fresnel’s equations for reflection and transmission at an air-dielectric inter-face were cast into the form of Mueller matrices. In this section, we use these results to derive the Mueller matrices for dielectric plates. We first treat the problem of determining the Mueller matrix for a single dielectric plate. The formalism is then easily extended to multiple reflections within a single dielectric plate and then to a pile of m parallel transparent dielectric plates.

For the problem of transmission of a polarized beam through a single dielectric plate, the sim-plest treatment can be made by assuming a single transmission through the upper surface followed by another transmission through the lower surface. There are, of course, multiple reflections within the dielectric plates, and, strictly speaking, these should be taken into account. While this treatment of multiple internal reflections is straightforward, it turns out to be quite involved. In the treatment presented here, we choose to ignore these effects. The completely correct treatment is given in the papers quoted in the references at the end of this chapter. The difference between the exact results and the approximate results is quite small, and very good results are still obtained by ignoring the multiple internal reflections. Consequently, only the resulting expressions for multiple internal reflections are quoted. We shall also see that the use of the diagonalized Mueller matrices developed in the previous section greatly simplifies the treatment of all of these problems.

In Figure 9.1 a single dielectric plate is shown. The incident beam is described by the Stokes vec-tor S. Inspection of the figure shows that the Stokes vector S′ of the beam emerging from the lower side of the dielectric plate is related to S by the matrix relation

′ =S M ST2 , (9.73)

n

θi θi

θr

θr θr

θi

figuRe 9.1 Beam propagation through a single dielectric plate.

Mueller Matrices for Dielectric Plates 187

where MT is the Mueller matrix for transmission and is given by Equation 7.74 in Section 7.3. We easily see that, using Equation 7.74, MT

2 is then

MTi r2

2

2

4

12

2 2=( )

+ −

sin sin

sin cos

cos

θ θ

θ θ

θ−− −

− −

−

+ −− +

1 1 0 0

1 1 0 0

0 0 2 0

0

4

4 4

2

cos

cos cos

cos

θθ θ

θ00 0 2 2cos

,

θ−

(9.74)

where θi is the angle of incidence, θr is the angle of refraction, and θ± = θi ± θr. Equation 9.74 is the (transmission) Mueller matrix for a single dielectric plate. We can immediately extend this result to the transmission through m parallel dielectric plates by raising MT

2 to the mth power; that is, MTm2 .

The easiest way to do this is to transform Equation 9.74 to the diagonal form and raise the diagonal matrix to the mth power as described earlier. After this is done, we transform back to the Mueller matrix form and find that the Mueller matrix for transmission through m parallel dielectric plates is

MTm i r

m

22

212

2 2=( )

+ −

sin sin

sin cos

cos

θ θθ θ

44 4

4 4

1 1 0 0

1 1 0 0

0 0 2

m m

m m

θ θθ θ

− −

− −

+ −− +

cos

cos cos

coss

cos

.2

2

0

0 0 0 2

m

m

θθ

−

−

(9.75)

When m = 1, we have the result for a single dielectric plate. If the incident beam is unpolarized, the Stokes vector of a beam emerging from m parallel plates is, from Equation 9.75,

′ =( )

+ −

S12

2 22

2

4

sin sin

sin cos

cos

θ θ

θ θi r

m

m θθθ

−

−

+−

1

1

0

0

4cos.

m

(9.76)

The degree of polarization P of the emerging beam is then

Pm

m= −

+−

−

11

4

4

coscos

.θθ

(9.77)

In Figure 9.2, a plot of Equation 9.77 is shown for the degree of polarization as a function of the incident angle θi. The plot shows that at least six or eight parallel plates are required in order for the degree of polarization to approach unity at high angles of incidence. At normal incidence, the degree of polarization is always zero, regardless of the number of plates.

The use of parallel plates to create linearly polarized light appears very often outside the visible region of the spectrum. In the visible and near-infrared region ( < 2 μm), polaroid and calcite are available to create linearly polarized light. Above 2 μm, parallel plates made from other materials are an important practical way of creating linearly polarized light. Fortunately, natural materials such as germanium are available and can be used; germanium transmits more than 95% of the incident light up to 20 μm.

According to Equation 9.76, the intensity of the beam emerging from m parallel plates, IT is

ITi r

m

=( )

+

+ −

12

2 21

2

2sin sin

sin coscos

θ θ

θ θ44m θ−( ). (9.78)

Figure 9.3 shows a plot of Equation 9.78 for m dielectric plates.

188 Polarized Light, Third Edition

0.9

1

0.7

0.8m = 10

0.5

0.6

m = 8

m = 6

0.4m = 4

0.2

0.3Deg

ree o

f pol

ariz

atio

n n

P

m = 2

0

0.1

0.00 18.00 36.00 54.00Incidence angle (degrees)

72.00 90.00

figuRe 9.2 Plot of Equation 9.77, the degree of polarization P vs. incident angle and the number of parallel plates. The refractive index n is 1.5.

0.9

1

m=2

0.7

0.8 m = 4

0.5

0.6m = 8

m = 6

0.4

Inte

nsity

m = 10

0.2

0.3

0

0.1

0.00 18.00 36.00Incidence angle (degrees)

54.00 72.00 90.00

figuRe 9.3 The intensity of a beam emerging from m parallel plates as a function of the angle of incidence as in Equation 9.78. The refractive index is 1.5.

Mueller Matrices for Dielectric Plates 189

At the Brewster angle, the Mueller matrix for transmission through m dielectric plates is readily shown from the results given in Chapter 7 and Section 7.2 to be

MT bm

mi

mi

mi

b b

b,

sin sin

sin2

4 4

412

2 1 2 1 0 0

2=

+ −

−

θ θ

θ 11 2 1 0 0

0 0 2 2 0

0 0 0 2 2

4

2

2

sin

sin

sin

mi

mi

mi

b

b

b

θ

θ

θ

+

. (9.79)

For a single dielectric plate m = 1, Equation 9.79 reduces to

MT b

i i

i

b b

b,

sin sin

sin sin2

4 4

412

2 1 2 1 0 0

2 1=

+ −

−

θ θ

θ 44

2

2

2 1 0 0

0 0 2 2 0

0 0 0 2 2

θ

θ

θ

i

i

i

b

b

b

+

sin

sin

. (9.80)

If the incident beam is unpolarized, after passing through m parallel dielectric plates the Stokes vector for the transmitted beam will be

′ =

+

−

S12

2 1

2 1

0

0

4

4

sin

sin.

mi

mi

b

b

θ

θ (9.81)

The degree of polarization is then

Pm

i

mi

b

b

=−+

1 2

1 2

4

4

sin

sin.

θθ

(9.82)

A plot of Equation 9.82 is shown in Figure 9.4 for m dielectric plates. The intensity of the transmit-ted beam is given by S0 in Equation 9.81 and is

ITm

ib= +( )1

21 24sin .θ (9.83)

Equation 9.83 has been plotted in Figure 9.5.Several conclusions can be drawn from Figures 9.4 and 9.5. In Figure 9.4, there is a significant

increase in the degree of polarization up to m = 6. Figure 9.5, on the other hand, shows that the intensity decreases and then begins to “level off” for m = 6. Thus, these two figures show that after five or six parallel plates there is very little to be gained in using more plates to increase the degree of polarization and still maintain a “constant” intensity. In addition, the cost for making such large assemblies of dielectric plates, the materials, and mechanical alignment becomes considerable.

Dielectric plates can also rotate the orientation of the polarization ellipse. At first, this behavior may be surprising, but this is readily shown. Consider the situation when the incident beam is lin-

190 Polarized Light, Third Edition

0.9

1

0.7

0.8

n = 2.0

n = 2.5

0.6

0.4

0.5

n = 1.5

0.2

0.3Deg

ree o

f pol

ariz

atio

n P

0

0.1

1.00 2.00 3.00 4.00 5.00 6.00Number of dielectric plates

7.00 8.00 9.00 10.00

figuRe 9.4 Plot of the degree of polarization P vs. number of dielectric plates at the Brewster angle for refractive indices of 1.5, 2.0, and 2.5.

0.9

1

0.7

0.8 n = 1.5

0.5

0.6

n = 2.0

n = 2.5

0.4Tran

smiss

ion

0.2

0.3

0

0.1

1.00 2.00 3.00 4.00 5.00 6.00Number of dielectric plates

7.00 8.00 9.00 10.00

figuRe 9.5 Plot of the transmitted intensity of a beam propagating through m parallel plates at the Brewster angle θib

. The refractive indices are 1.5, 2.0, and 2.5.

Mueller Matrices for Dielectric Plates 191

ear + 45° polarized light. The normalized Stokes vector of the beam emerging from m dielectric plates is then, from Equation 9.79,

′ =

+

−

S12

2 1

2 1

2 2

0

4

4

2

sin

sin

sin

mi

mi

mi

b

b

b

θ

θ

θ

. (9.84)

The emerging light is still linearly polarized. However, the orientation angle ψ is

ψθ

θ=

−

−12

2 2

2 11

2

4tan

sin

sin.

mi

mi

b

b

(9.85)

We note that for m = 0 (no dielectric plates), the absolute magnitude of the angle of rotation is ψ = 45°, as expected. Figure 9.6 illustrates the change in the angle of rotation as the number of par-allel plates increases. For five parallel plates, the orientation angle rotates from + 45° to + 24.2°.

Equation 9.82 can also be expressed in terms of the refractive index, n. At the Brewster angle we have

tanθibn= (9.86)

and we can then write

sinθib

n

n=

+2 1 (9.87)

0 2 4 6 8 10

Ψ (m

)

π/4

m0

figuRe 9.6 Rotation of the polarization ellipse by m parallel dielectric plates according to Equation 9.85. The refractive index is 1.5.

192 Polarized Light, Third Edition

and

cosθib n=

+1

12 (9.88)

so

sin .22

12θib

n

n=

+ (9.89)

Substituting Equation 9.89 into Equation 9.84 yields

Pn n

n n

m

m= − +( )[ ]

+ +( )[ ]1 2 1

1 2 1

2 4

2 4

/

/. (9.90)

Equation 9.90 is a much-quoted result in the optical literature and optical handbooks. Figure 9.7 is a plot of Equation 9.90 in terms of m and n. As the inspection of Figure 9.7 shows, the curves are identical to those in Figure 9.4 except as to where the scale starts on the abscissa.

In the beginning of this section we pointed out that the Mueller matrix formalism can also be extended to the problem of including multiple reflections within a single dielectric plate as well as the multiple plates. G. G. Stokes was the first to consider this problem [2], and he showed that the degree of polarization for m parallel plates at the Brewster angle, with multiple reflections within the plates, is given by

Pm

m n n=

+ −( )[ ]2 12 2 2/

. (9.91)

n = 2.5n = 2.0

1

n = 1.5

Deg

ree o

f pol

ariz

atio

n P

0.00 1.00 2.00 3.00 4.00 5.00 6.00m

7.00 8.00 9.00 10.000

figuRe 9.7 Plot of the degree of polarization as a function of the number of parallel plates; multiple reflec-tions are ignored.

Mueller Matrices for Dielectric Plates 193

The derivation of Equation 9.91 along with similar expressions for completely and partially polar-ized light has been given by Collett [3], using the Jones matrix formalism (Chapter 10) and the Mueller matrix formalism. Equation 9.91 is plotted in Figure 9.8 as a function of m and n.

It is of interest to compare Equations 9.90 and 9.91. We have plotted these two equations for n = 1.5 in Figure 9.9. We see that the degree of polarization is very different. Starting with 0 parallel plates (i.e., the unpolarized light source by itself) we see that the degree of polarization is zero, as expected. As the number of parallel plates increases, the degree of polarization increases for both Equations 9.90 and 9.91. The curves diverge and the magnitudes differ by approximately a factor of two, so that for 10 parallel plates the degree of polarization is 0.93 for Equation 9.90 and 0.43 for Equation 9.91. For the case of multiple reflections represented by the lower curve, the degree of polarization is almost linear with a very shallow positive slope, and shows that each additional plate results in a small increase in the degree of polarization.

A final topic that we discuss is the use of a simpler notation for the Mueller matrices for reflec-tion and transmission by representing the matrix elements in terms of the Fresnel reflection and transmission coefficients. These coefficients are defined to be

ρ θθs

s

s

R

E=

=

−

+

2 2sinsin

(9.92)

ρ θθp

p

p

R

E=

=

−

+

2 2tantan

(9.93)

and

τ θθ

θθ

θs

r

i

s

s

i

r

rn T

E=

=cos

costantan

sin c2

2 oossin

sin sinsin

θθ

θ θθ

i i r

+ +

=

2

2

2 2 (9.94)

1

n = 2.5

n = 1.5

n = 2.0

Deg

ree o

f pol

ariz

atio

n P

0.0 1.0 2.0 3.0 4.0 5.0 6.0m

7.0 8.0 9.0 10.00

figuRe 9.8 Plot of the degree of polarization as a function of the number of parallel plates for the case where multiple reflections are included.

194 Polarized Light, Third Edition

τ θθ

θθ

θp

r

i

p

p

i

r

rn T

E=

=coscos

tantan

sin c2

2 oossin cos

sin sinsin co

θθ θ

θ θθ

i i r

+ − +

=

2

2

2 2ss

.2 θ−

(9.95)

One can readily show that the relations for Fresnel coefficients satisfy

ρ τs s 1+ = (9.96)

and

ρ τp p+ = 1. (9.97)

At the Brewster angle, written as θib, Fresnel’s reflection and transmission coefficients reduce to

ρ θs b ib, cos= 2 2 (9.98)

ρp b, = 0 (9.99)

τ θs b ib, sin= 2 2 (9.100)

τ p b, = 1. (9.101)

We see immediately that

ρ τs b s b, ,+ = 1 (9.102)

Multiple reflections ignored

1

Multiple reflections included

Deg

ree o

f pol

ariz

atio

n P

0.0 1.0 2.0 3.0 4.0 5.0m

6.0 7.0 8.0 9.0 10.00

figuRe 9.9 Degree of polarization for m parallel plates for n = 1.5. The upper curve corresponds to Equation 9.90, and the lower corresponds to Equation 9.91.

Mueller Matrices for Dielectric Plates 195

and

ρ τp b p b, ,+ = 1. (9.103)

Equations 9.102 and 9.103 are, of course, merely special cases of Equations 9.96 and 9.97.With these definitions the Mueller matrices for reflection and transmission can be written as

Mρ

ρ ρ ρ ρρ ρ ρ ρ

ρ ρ=

+ −− +

( )12

0 0

0 0

0 0 2 0

0

1 2

s p s p

s p s p

s p/

00 0 21 2ρ ρs p( )

/

(9.104)

and

Mτ

τ τ τ ττ τ τ τ

τ τ=

+ −− +

( )12

0 0

0 0

0 0 2 0

0

1 2

s p s p

s p s p

p s/

00 0 21 2τ τs p( )

/

. (9.105)

The reflection coefficients ρs and ρp, Equations 9.92 and 9.93, are plotted as a function of the inci-dent angle for a range of refractive indices in Figures 9.10 and 9.11. Similar plots are shown in Figures 9.12 and 9.13 for τs and τp, Equations 9.94 and 9.95.

In a similar manner, the reflection and transmission coefficients at the Brewster angle, Equations 9.98 through 9.101, are plotted as a function of the refractive index n in Figures 9.14 and 9.15.

1

0.8

0.9

0.6

0.7

ρ s(θ

i)

0.4

0.5

0.2

0.3

n = 2.0n = 2.5

0

0.1

0 10 20 30 40 50Incidence angle (degrees)

60 70 80 90

n = 1.5

figuRe 9.10 Plot of the Fresnel reflection coefficient ρs in Equation 9.92 as a function of incidence angle θi.

196 Polarized Light, Third Edition

1

0.8

0.9

0.6

0.7

0.4

0.5

0.2

0.3

n = 2.5

0

0.1

0 10 20 30 40 50Incidence angle (degrees)

60 70 80 90

n = 1.5n = 2.0

ρ s(θ

i)

figuRe 9.11 Plot of the Fresnel reflection coefficient ρp in Equation 9.93 as a function of incidence angle θi.

1

0.8

0.9 n = 1.5

n = 2.0

0.6

0.7

τ s

n = 2.5

0.4

0.5

0.2

0.3

0

0.1

0 10 20 30 40Incidence angle (degrees)

50 60 70 80 90

figuRe 9.12 Plot of the Fresnel transmission coefficient τs in Equation 9.94 as a function of incidence angle θi.

Mueller Matrices for Dielectric Plates 197

1 n = 1.5

0.8

n = 2.0

n = 2.5

0.6

τ p

0.4

0.2

00 10 20 30 40

Incidence angle (degrees)50 60 70 80 90

figuRe 9.13 Plot of the Fresnel transmission coefficient τp in Equation 9.95 as a function of incidence angle θi.

1

0.8

0.9

0.6

0.7

ρ s,B

0.4

0.5

0.2

0.3

0

0.1

1.5 1.6 1.7 1.8 1.9 2n

2.1 2.2 2.3 2.4 2.5

figuRe 9.14 Plot of the reflection coefficient ρs, b at the Brewster angle, Equation 9.98.

198 Polarized Light, Third Edition

The great value of the Fresnel coefficients is that their use leads to simpler forms for the Mueller matrices for reflection and transmission. For example, instead of the complicated matrix entries given above, we can write the diagonalized form of the Mueller matrices as

Mρ

ρρ

ρ ρ

ρ ρ

, /

/

d

s

p

s p

s p

= ( )( )

0 0 0

0 0 0

0 0 0

0 0 0

1 2

1 2

, (9.106)

and

Mτ

ττ

τ τ

τ τ

, /

/

d

s

p

s p

s p

= ( )( )

0 0 0

0 0 0

0 0 0

0 0 0

1 2

1 2

. (9.107)

For treating problems at angles other than the Brewster angle, it is much simpler to use either Equation 9.106 or Equation 9.107 rather than the earlier forms of the Mueller matrices, because the matrix elements ρs, ρp, τs, and τp are far easier to work with.

In this chapter, we have applied the Mueller matrix formalism to the problem of determining the change in the polarization of light by single and multiple dielectric plates. We have treated the problems in the simplest way by ignoring the thickness of the plates and multiple reflections within the plates. Consequently, the results are only approximately correct. Nevertheless, the results are still useful and allow us to predict the expected behavior of polarized light and its interaction with

1

0.8

0.9

0.6

0.7

τ s,B

0.4

0.5

0.2

0.3

0

0.1

1.5 1.6 1.7 1.8 1.9n2 2.1 2.2 2.3 2.4 2.5

figuRe 9.15 Plot of the transmission coefficient τs, b at the Brewster angle, Equation 9.100.

Mueller Matrices for Dielectric Plates 199

dielectric plates quite accurately. In particular, we have presented a number of formulas, much quoted in the optical literature and handbooks, which describe the degree of polarization for an incident unpolarized beam of light. These formulas describe the number of parallel plates required to obtain any degree of polarization. Additional discussion of the behavior of multiple plates can be found in Tuckerman [4].

RefeReNCeS

1. Chandrasekhar, S., Radiative Transfer, 24–34, New York: Dover Publications, 1960. 2. Stokes, G. G., On the intensity of the light reflected from or transmitted through a pile of plates, Proc.

Roy. Soc. (London) 41 (1862): 545–56. 3. Collett, E., Mueller-Stokes matrix formulation of Fresnel’s equations, Am. J. Phys. 39 (1971): 517–28. 4. Tuckerman, L. B., On the intensity of the light reflected from or transmitted through a pile of plates,

J. Opt. Soc. Am. 37 (1947): 818–9.

201

10 The Jones Matrix Formalism

10.1 iNTRoduCTioN

We have seen that the Stokes polarization parameters and the Mueller matrix formalism can be used to describe any state of polarization. In particular, if we are dealing with a single beam of polarized light, then the formalism of the Stokes parameters is completely capable of describing any polariza-tion state ranging from completely polarized light to completely unpolarized light. In addition, the formalism of the Stokes parameters can be used to describe the superposition of several polarized beams, provided that there is no amplitude or phase relation between them; that is, the beams are incoherent with respect to each other. This situation arises when optical beams are emitted from several independent sources and are then superposed.

However, there are experiments where several beams must be added and the beams are not independent of each other (e.g., beam superposition in interferometers). There we have a single optical source, the single beam is divided by a beam splitter and then the beams are “reunited” (i.e., superposed). Clearly there is an amplitude and phase relation between the beams. We see that we must deal with amplitudes and phases and superpose the amplitudes of each of the beams. After the amplitudes of the beam are superposed, the intensity of the combined beams is then found by taking the time average of the square of the total amplitude. If there were no amplitude or phase relations between the beams, then we would arrive at the same result as we obtained for the Stokes param-eters, however, if there is a relation between the amplitudes and the phases of the optical beams, an interference term will arise.

Of course, as pointed out earlier, the description of the polarizing behavior of the optical field in terms of amplitudes was one of the first great successes of the wave theory of light. The solution of the wave equation in terms of transverse components leads to elliptically polarized light and its degenerate linear and circular forms. On the basis of the amplitude results, many results could be understood (e.g., Young’s interference experiment and circularly polarized light). However, even using the amplitude formulation, numerous problems become difficult to treat, such as the propaga-tion of the field through several polarizing components. To facilitate the treatment of complicated polarization problems at the amplitude level, R. Clark Jones, in the early 1940s, developed a matrix formalism for treating these problems [1–8], now commonly called the Jones matrix formalism. It is most appropriately used when we must superpose amplitudes. The Jones formalism involves com-plex quantities contained in 2 × 1 column matrices, the Jones vector, and 2 × 2 matrices, the Jones matrices. At first sight it would seem that the use of the 2 × 2 matrices would be simpler than the use of the 4 × 4 Mueller matrices. Oddly enough, this is not the case. This is due primarily to the fact that even the matrix multiplication of several complex 2 × 2 matrices can be tedious. Furthermore, even after the complete matrix calculation has been carried out, additional steps are still required. For example, it is often necessary to separate the real and imaginary parts (e.g., Ex and Ey) and superpose the respective amplitudes. This can involve a considerable amount of effort. Another problem is that to find the intensity one must take the complex transpose of the Jones vector and then carry out the matrix multiplication between the complex transpose of the Jones vector and Jones vector itself. All this is done using complex quantities; the possibility of making a computational error is very real. While the 4 × 4 Mueller matrix formalism appears to be more complicated, all the entries are real quantities and there are typically many zero entries, as can be seen by inspect-ing the Mueller matrix for a polarizer, retarder, and rotator. This fact greatly simplifies the matrix multiplications, and of course, the Stokes vector is real.

202 Polarized Light, Third Edition

There are, nevertheless, many instances where the amplitudes must be added (superposed), and so the Jones matrix formalism must be used. There are many problems where either formalism can be used with success. As a general rule, the most appropriate choice of matrix method is to use the Jones formalism for amplitude superposition problems and the Mueller formalism for intensity superposition problems. Experience will usually indicate the best choice to make.

In this chapter, we develop the fundamental matrices for the Jones formalism along with its application to a number of problems.

10.2 The JoNeS VeCToR

The plane-wave components of the optical field in terms of complex quantities can be written as

E z t E i t kzx xx , exp( ( )( ) = − +0 ω δ (10.1)

E z t E i t kzy y y, exp( ( ).( ) = − +0 ω δ (10.2)

The propagator ωt – kz is now suppressed, so Equations 10.1 and 10.2 are then written as

E E ex xi x= 0δ (10.3)

E E ey yi y= 0δ . (10.4)

These last two equations can be arranged in a 2 × 1 column matrix E so that

E =

=

E

E

E e

E ex

y

xi

yi

x

y

0

0

δ

δ , (10.5)

and this is called the Jones column matrix or the Jones vector. The column matrix on the right-hand side of Equation 10.5 is the most general expression for the Jones vector, indicating elliptically polarized light.

In the Jones vector Equation 10.5, the maximum amplitudes E0x and E0y are real quantities. The presence of the exponent with imaginary arguments causes Ex and Ey to be complex quantities. Before we proceed to find the Jones vectors for various states of polarized light, we will discuss the normalization of the Jones vector, since it is customary to express the Jones vector in normalized form. The total intensity I of the optical field is given by

I E E E Ex x y y= +* *. (10.6)

Equation 10.6 can be obtained by the multiplication of two vectors, expressed as

I E EE

Ex yx

y

= ( )

* * . (10.7)

The row matrix E Ex y* *( ) is the complex transpose of the Jones vector E and is written E† so

that

I = ⋅E E† . (10.8)

The Jones Matrix Formalism 203

Carrying out the matrix multiplication of Equation 10.8 using Equation 10.5 yields

E E I Ex y02

02

02+ = = . (10.9)

It is customary to set E02 1= , whereupon we say that the Jones vector is normalized. The normal-

ized condition for Equation 10.7 can then be written as

E E† ⋅ = 1. (10.10)

Note that the Jones vector can only be used to describe completely polarized light. We now find the Jones vector for the following states of completely polarized light.

1. Linear horizontally polarized light. For this state Ey = 0, so Equation 10.5 becomes

E =

E exi x

0

0

δ

. (10.11)

From the normalization condition Equation 10.10, we see that E x02 1= . Thus, suppressing

ei xδ because it is unimodular, the normalized Jones vector for linearly horizontally polar-ized light is written

E =

1

0. (10.12)

In a similar manner the Jones vectors for the other well-known polarization states are easily found. 2. Linear vertically polarized light. Ex = 0, so E y0

2 1= and

E =

0

1. (10.13)

3. Linear + 45° polarized light. Ex = Ey, so 2 102E x = and

E =

1

2

1

1. (10.14)

4. Linear −45° polarized light. Ex = −Ey, so 2 102E x = and

E =−

1

2

1

1. (10.15)

5. Right-hand circularly polarized light. For this case E0x = E0y and δy – δx = + 90°. Then, 2 10

2E x = and we have

E =+

1

2

1

i. (10.16)

204 Polarized Light, Third Edition

6. Left-hand circularly polarized light. We again have E0x = E0y, but δy – δx = – 90°. The normalization condition gives 2 10

2E x = , and we have

E =−

1

2

1

i. (10.17)

Each of the Jones vectors Equations 10.12 through 10.17 satisfies the normalization condition Equation 10.10.

An additional property of the Jones vectors is the orthogonal or orthonormal property. Two vec-tors A and B are said to be orthogonal if A · B = 0 or, in complex notation, A B† ⋅  = 0. If this condi-tion is satisfied, we say that the Jones vectors are orthogonal. For example, for linear horizontally and vertically polarized light we find that

1 00

10( )

= , (10.18)

so that the states are orthogonal or, since we are using normalized vectors, orthonormal. Similarly, for right and left circularly polarized light

11

0+( )−

=ii

*. (10.19)

The orthonormal condition for two Jones vectors E1 and E2 is

E Ei j† ⋅ = 0. (10.20)

We see that the orthonormal condition Equation 10.20 and the normalizing condition Equation 10.10 can be written as the single equation

E Ei j ij i j† ⋅ = =δ , , ,1 2 (10.21)

where δij is the Kronecker delta and has the property

δij i j= =1 (10.22)

δij i j= ≠0 . (10.23)

In a manner analogous to the superposition of incoherent intensities or Stokes vectors, we can superpose coherent amplitudes (i.e., Jones vectors). For example, the Jones vector for horizontal polarization is EH and that for vertical polarization is EV, so

E EHx

i

Vy

i

E e

E e

x

y=

=

0

00

0δ

δ . (10.24)

Adding EH and EV gives

E E E= + =

H V

xi

yi

E e

E e

x

y

0

0

δ

δ , (10.25)

The Jones Matrix Formalism 205

which is the Jones vector for elliptically polarized light. Thus, superposing two orthogonal linear polarizations gives rise to elliptically polarized light. For example, if E0x = Ε0y and δy = δx then, from Equation 10.25, we can write

E =

E exi x

0

1

1δ , (10.26)

which is the Jones vector for linear +45° polarized light. Equation 10.26 could also be obtained by superposing Equations 10.12 and 10.13 so that

E E E= + =

+

=

H V

1

0

0

1

1

1, (10.27)

which, aside from the normalizing factor, is identical to Equation 10.14.As another example let us superpose left and right circularly polarized light of equal amplitudes.

Then, from Equations 10.16 and 10.17 we have

E =−

+

=

1

2

1 1

2

1 2

2

1

0i i, (10.28)

which, aside from the normalizing factor, is the Jones vector for linear horizontally polarized light Equation 10.12.

As a final Jones vector example, we show that elliptically polarized light can be obtained by superposing two opposite circularly polarized beams of unequal amplitudes. The Jones vectors for two circularly polarized beams of unequal amplitudes a and b can be represented by

E E+ −=+

=−

ai

bi

1 1and . (10.29)

According to the principle of superposition, the resultant Jones vector for Equation 10.29 is

E E E= + =+−( )

=

+ −

a b

i a b

E

Ex

y

. (10.30)

In component form Equation 10.30 is written as

E a bx = + (10.31)

E a b ey = −( ) i /2π . (10.32)

We now restore the propagator ωt – kz, and these equations are written as

E a b exi t kz= +( ) −( )ω (10.33)

E a b eyi t kz= −( ) − +( / ).ω π 2 (10.34)

206 Polarized Light, Third Edition

Taking the real part of Equations 10.33 and 10.34, we have

E z t a b t kzx ,( ) = +( ) −( )cos ω (10.35)

E z t a b t kz a b t kzy , cos( ) = −( ) − +

= −( ) −ω π ω

2sin(( ) (10.36)

Isolating the trigonometric functions in Equations 10.35 and 10.36, and squaring and adding both sides of the resulting equations yields

E z t

a b

E z t

a b

x y2

2

2

21

, ,.

( )+( )

+( )−( )

= (10.37)

Equation 10.37 is the equation of an ellipse whose major and minor axes lengths are a + b and a − b, respectively. Thus, the superposition of two oppositely circularly polarized beams of unequal magnitudes gives rise to a (nonrotated) ellipse with its locus vector moving in a counterclockwise direction.

10.3 JoNeS maTRiCeS foR The PolaRiZeR, ReTaRdeR, aNd RoTaToR

We now determine the matrix forms for polarizers (diattenuators), retarders (phase shifters), and rotators in the Jones matrix formalism. In order to do this, we assume that the components of a beam emerging from a polarizing element are linearly related to the components of the incident beam. This relation is written as

′ = +E j E j Ex xx x xy y (10.38)

′ = +E j E j Ey yx x yy y , (10.39)

where ′Ex and ′Ey are the components of the emerging beam, and Ex and Ey are the components of the incident beam. The quantities jik, where i, k = x, y, are the transforming factors. Equations 10.38 and 10.39 can be written in matrix form as

′′

=

E

E

j j

j j

E

Ex

y

xx xy

yx yy

x

y

, (10.40)

or

E′ = JE, (10.41)

where

J =

j j

j jxx xy

yx yy

. (10.42)

The 2 × 2 matrix J is called the Jones matrix. We now determine the Jones matrices for a polarizer, retarder, and rotator.

The Jones Matrix Formalism 207

A polarizer is characterized by the relations

E p Ex x x′ = (10.43)

′ =E p Ey y y , (10.44)

where 0 ≤ px,y ≤ 1.For complete transmission px,y = 1, and for complete attenuation px,y = 0. In terms of the Jones

vector, Equations 10.43 and 10.44 can be written as

′′

=

E

E

p

p

E

Ex

y

x

y

x

y

0

0, (10.45)

so the Jones matrix Equation 10.42 for a polarizer is

J px

yx y

p

pp=

≤ ≤0

00 1, . (10.46)

For an ideal linear horizontal polarizer there is complete transmission along the horizontal x axis and complete attenuation along the vertical y axis. This is expressed by px = 1 and py = 0, so that Equation 10.46 becomes

JPH =

1 0

0 0. (10.47)

Similarly, for a linear vertical polarizer, Equation 10.46 becomes

JPV =

0 0

0 1. (10.48)

If we wish to know the Jones matrix for a linear polarizer rotated through an angle θ, it is readily found by using the familiar rotation transformation,

J J J J′ = −( ) ( )θ θ , (10.49)

where J(θ) is the rotation matrix

J θθ θθ θ

( ) =−

cos sin

sin cos, (10.50)

and J is given by Equation 10.42. For a rotated linear polarizer represented by Equation 10.46 and rotated by angle θ we have from Equation 10.49 that

J′ =−

cos sin

sin cos

cos sθ θθ θ

θp

px

y

0

0

iin

sin cos.

θθ θ−

(10.51)

208 Polarized Light, Third Edition

Carrying out the matrix multiplication in Equation 10.51, we find that the Jones matrix for a rotated polarizer is

JPx y x y

x y

p p p p

p pθ

θ θ θ θ( ) =+ −( )

−( )cos sin sin cos2 2

ssin cos sin cos.

θ θ θ θp px y2 2+

(10.52)

For an ideal linear horizontal polarizer we can set px = 1 and py = 0 in Equation 10.52, so that the Jones matrix for a rotated linear horizontal polarizer is

JP ( )cos sin cos

sin cos sin.θ

θ θ θθ θ θ

=

2

2 (10.53)

The Jones matrix for a linear polarizer rotated through +45° is then seen from Equation 10.53 to be

JP 4512

1 1

1 1°( ) =

. (10.54)

If the linear polarizer is not ideal, then the Jones matrix for a polarizer Equation 10.46 at +45° is seen from Equation 10.52 to be

JPx y x y

x y x y

p p p p

p p p p45

12

°( ) =+ −− +

. (10.55)

We note that for θ = 0° and 90°, Equation 10.53 gives the Jones matrices for a linear horizontal and linear vertical polarizer, Equations 10.47 and 10.48, respectively.

Equation 10.52 also describes a neutral density (ND) filter. The condition for a ND filter is px = py = p, so Equation 10.52 reduces to

JND =

p1 0

0 1. (10.56)

Thus, JND is independent of rotation, and the amplitudes are equally attenuated by an amount p. This is, indeed, the behavior of a ND filter. The presence of the unit (diagonal) matrix in Equation 10.56 confirms that a ND filter does not affect the polarization state of the incident beam.

The next polarizing element of importance is the retarder. We can split the total phase shift ϕ that exists between the two orthogonally polarized components of the beam by assigning a phase advance of ϕ/2 to the fast (x) axis and a phase retardation of ϕ/2 along the slow (y) axis. This behav-ior is described by

′ = +E e Exi

xφ/2 (10.57)

′ = −E e Eyi

yφ/ ,2 (10.58)

where ′Ex and ′Ey are the components of the emerging beam and Ex and Ey are the components of the incident beam. We can immediately express Equations 10.57 and 10.58 in the Jones formalism as

J′ =′′

=

+

−

E

E

e

e

E

Ex

y

i

i

x

y

φ

φ

/

/

2

2

0

0

. (10.59)

The Jones Matrix Formalism 209

The Jones matrix for a retarder (phase shifter) is then

JR

i

i

e

eφ

φ

φ( ) =

+

−

/

/,

2

2

0

0 (10.60)

where ϕ is the total phase shift between the field components. The two most common types of com-mercial retarders are the quarter-wave retarder and the half-wave retarder. For these devices ϕ = 90° and 180°, respectively, and Equation 10.60 becomes

JR

i

ii

i

e

ee

e

λ π

ππ

π4

0

0

1 0

0

4

44

2

=

=− −

/

//

/

=−

ei

iπ /41 0

0 (10.61)

and

JR

i

i

e

e

i

ii

λ π

π2

0

0

0

0

12

2

=

=−

=−

/

/

00

0 1−

. (10.62)

The Jones matrix for a rotated retarder is found from Equation 10.50 to Equation 10.60 to be

JR

i i i ie e e eφ θ

θ θφ φ φ φ,

cos sin/ / / /

( ) =+ −(− −2 2 2 2 2 2 ))

−( ) +−

sin cos

sin cos sin/ / /

θ θθ θ θφ φ φe e ei i i2 2 2 2 ee i−

φ θ/ cos

.2 2

(10.63)

With the half-angle formulas, Equation 10.63 can also be written in the form

JR

i i

i

φ θ

φ φ θ φ θ

φ,

cos sin cos sin

sin sin

( ) =+

2 22

22

22θθ φ φ θcos sin cos

.

2 22−

i

(10.64)

For a quarter-wave retarder and a half-wave retarder, Equation 10.64 reduces to

JR

i i

i i

λ θθ θ

θ4

1

2 22

22

22

1

2 2

,cos sin

sin cos

=

+

− 22θ

(10.65)

and

JR iλ θ

θ θθ θ2

2 2

2 2,

cos sin

sin cos.

=

−

(10.66)

The factor i in Equation 10.66 is unimodular and can be suppressed. It is common, therefore, to write Equation 10.66 simply as

JR

λ θθ θθ θ2

2 2

2 2,

cos sin

sin cos.

=

−

(10.67)

210 Polarized Light, Third Edition

Inspecting Equation 10.67 we see that it is very similar to the matrix for rotation,

J θθ θθ θ

( ) =−

cos sin

sin cos. (10.68)

However, Equation 10.67 differs from Equation 10.68 in two ways. First, in Equation 10.67 we have 2θ rather than θ. Thus, a rotation of a retarder through θ rotates the polarization ellipse through 2θ. Second, a clockwise mechanical rotation θ in Equation 10.67 leads to a counterclockwise rotation of the polarization ellipse. In order to see this behavior clearly, consider that we have incident linear horizontally polarized light. Its Jones vector is

J =

Ex

0. (10.69)

The components of the beam emerging from a true rotator as in Equation 10.68 are then

′ =E Ex xcosθ (10.70)

′ = −E Ey xsin .θ (10.71)

The angle of rotation α is then

tansin

costan .α θ

θθ=

′′

= − = −( )E

Ey

x

(10.72)

In a similar manner, multiplying Equation 10.69 by Equation 10.67 leads to

′ =E Ex xcos2θ (10.73)

′ =E Ey xsin2θ (10.74)

so we now have

tansincos

tan .α θθ

θ=′′

= =E

Ey

x

22

2 (10.75)

Comparing Equation 10.75 with Equation 10.72, we see that the direction of rotation for a rotated retarder is opposite to the direction of true rotation. Equation 10.75 also shows that the angle of rotation is twice that of a true rotation. Because of this similar but analytically incorrect behavior of a rotated half-wave retarder, Equation 10.67 is called a pseudorotator. We note that an alterna-tive form of a half-wave retarder, which is the more common form, is given by factoring out i in Equation 10.62 or simply setting θ = 0° in Equation 10.67 so that

Jλ2

1 0

0 1

=

−

. (10.76)

The Jones Matrix Formalism 211

The final matrix of interest is the Jones matrix for a rotator. The defining equations are

′ = +E E Ex x ycos sinβ β (10.77)

′ = − +E E Ey x ysin cos ,β β (10.78)

where β is the angle of rotation. Equations 10.77 and 10.78 are written in matrix form as

J′ =′′

=−

E

E

E

Ex

y

x

y

cos sin

sin cos

β ββ β

(10.79)

so the Jones matrix for a rotator is

JROTsin

=−

cos sin

cos.

β ββ β

(10.80)

It is interesting to see the effect of rotating a true rotator. According to Equation 10.49, the rotation of a rotator, Equation 10.80, is given by

JROT θθ θθ θ

β β( ) =−

−

cos sin

sin cos

cos sin

sinββ βθ θθ θcos

cos sin

sin cos.

−

(10.81)

Carrying out the matrix multiplication in Equation 10.81 yields

J JROT ROTθβ ββ β

( ) =−

=cos sin

sin cos. (10.82)

We have the interesting result that the mechanical rotation of a rotator does not affect the rotation of the polarization ellipse. The polarization ellipse can only be rotated by an amount intrinsic to the rotator, which is the rotation angle β. We conclude that the only way to create a rotation of the polar-ization ellipse mechanically is to use a half-wave retarder placed in a mechanical rotating mount.

10.4 aPPliCaTioNS of The JoNeS VeCToR aNd JoNeS maTRiCeS

We now turn our attention to applying the results of Sections 10.2 and 10.3 to several problems of interest. One of the first problems is to determine the Jones vector and intensity for a beam emerging from a rotated linear polarizer. The Jones vector of the incident beam is

E =

E

Ex

y

. (10.83)

The Jones matrix of a rotated (ideal) linear polarizer was shown to be

JP θθ θ θ

θ θ θ( ) =

cos sin cos

sin cos sin.

2

2 (10.84)

212 Polarized Light, Third Edition

While it is straightforward to determine the Jones vector and the intensity of the emerging beam, it is of interest to restrict ourselves to the case where the incident beam is linearly horizontally polar-ized, so

E =

=

EEx

x0

1

0. (10.85)

Multiplying Equation 10.85 by Equation 10.84 yields

E′ =

E

Ex

x

cos

sin cos.

2 θθ θ

(10.86)

We now interpret the state of polarization of Equation 10.86. We can express Equation 10.86 as a Jones vector for elliptically polarized light,

′ =

Eae

be

i

i

x

y

δ

δ , (10.87)

where a and b are real. Equating Equations 10.86 and 10.87, we have

′ = =E E aex xi xcos2 θ δ (10.88)

′ = =E E bey xi ycos sin .θ θ δ (10.89)

Dividing Equation 10.89 by Equation 10.88 then gives

′′

= =

E

E

b

aey

x

isincos

,θθ

δ (10.90)

where δ = δy – δx. The real and imaginary parts of Equation 10.90 are

sincos

cosθθ

δ= b

a (10.91)

0 = ≠b

ab asin .δ (10.92)

We conclude immediately from Equation 10.92 that δ = 0°, so that Equation 10.91 is

b

a= sin

cos.

θθ

(10.93)

The polarization ellipse corresponding to Equation 10.87 is

x

a

y

b

xy

ab

2

2

2

222+ − =cos

sin .δ δ (10.94)

The Jones Matrix Formalism 213

For δ = 0°, Equation 10.94 reduces to

yb

ax x= = sin

cos.

θθ

(10.95)

Thus, the Jones vector Equation 10.86 describes a beam that is linearly polarized with a slope equal to

m = =tan tanα θ. (10.96)

The intensity of the emerging beam is

I E EE

Ex x

x

x

′ = ⋅ =E E† * *( cos sin cos )cos

sin c2

2

θ θ θθ

θ oos,

θ

(10.97)

so

I I′ = cos2θ, (10.98)

where I E Ex x= * . Equation 10.98 is Malus’s Law. It was discovered by Étienne Louis Malus while observing unpolarized light through a rotating calcite crystal. We recall he discovered that unpolar-ized light became partially polarized when it was reflected from a plate of glass. He found the form in Equation 10.98 solely from geometrical considerations.

We can continue to study this problem by allowing the beam emerging from the polarizer to be incident on a linear vertical polarizer. The Jones matrix for this vertical polarizer, setting θ = 90° in Equation 10.84, is

JP ( ) .900 0

0 1° =

(10.99)

The Jones vector of the beam emerging from this second linear polarizer, found by multiplying Equation 10.86 by Equation 10.99 is

′ =

E p xE cos sin ,θ θ0

1 (10.100)

and the intensity is immediately found to be

′ = −( )II

81 4cos ,θ (10.101)

where I E Ex x= * . As the second polarizer is rotated, a null intensity is observed at θ = 0°, 90°, 180°, and 270°. Equation 10.101 is, of course, the same as obtained using the Mueller–Stokes calculus.

We now apply the Jones formalism to several other problems of interest. In Section 14.6, we will use the method of Kent and Lawson to determine the Stokes parameters of an incident elliptically polarized beam. We can also treat the problem in the amplitude domain and apply the Kent–Lawson

214 Polarized Light, Third Edition

method to determine the phase and orientation of the beam. The incident beam can be written in the form

E =

cos

sin.

αα δei

(10.102)

The beam Equation 10.102 is incident on a retarder of arbitrary phase ϕ oriented at an angle θ. The phase and orientation of the retarder are now adjusted until circularly polarized light is obtained. This is detected by allowing the circularly polarized beam to be incident on a rotating linear polar-izer directly in front of the detector. Circular polarization is obtained when a constant intensity is detected. We can write this condition as

J J J E−( ) ( ) =

θ θRi

1

2

1. (10.103)

The column matrix on the right-hand side of Equation 10.103 is the Jones vector for right circularly polarized light; E is given by Equation 10.102 and J(θ) and JR are the Jones matrices for rotation and a retarder, respectively. We can find E in Equation 10.103 by multiplying through by J(θ), and so on. Carrying out this process, we arrive at

E =

=−+

cos

sin

cos sin

sin

αα

θ θθδ

φ

e

ie

iei

i

i

1

2 φφ θcos.

(10.104)

Equation 10.104 is easily checked because a retarder, even if rotated, does not affect the total inten-sity. Thus, it is easy to see that taking the complex transpose of Equation 10.104 for each Jones vec-tor and multiplying by its normal Jones vector gives unit intensity as required.

Dividing the second element by the first in each of the column matrices in Equation 10.104 we find

tansin coscos sin

.α θ θθ θ

δφ

φe

ie

iei

i

i= +

− (10.105)

Rationalizing the denominator in Equation 10.105, we easily find that

tansin cos cos

sin sin.α φ θ φ

θ φδe

ii = − ++

21 2

(10.106)

Equating real and imaginary parts in Equation 10.106 yields

tan cossin cossin sin

α δ φ θθ φ

= −+

21 2

(10.107)

tan sincos

sin sin.α δ φ

θ φ=

+1 2 (10.108)

Dividing Equation 10.107 by Equation 10.108, we obtain

cot tan cos .δ φ θ= − 2 (10.109)

The Jones Matrix Formalism 215

Squaring and adding Equations 10.107 and 10.108 then leads to

tansin sin

sin sin.α θ φ

θ φ= −

+1 2

1 2 (10.110)

Equation 10.110 can be rewritten by using the relations

cossin sinα θ φ= +1 2

2 (10.111)

sinsin sin

.α θ φ= −1 2

2 (10.112)

Squaring Equations 10.111 and 10.112 and subtracting, we find that

cos 2 sin 2 sinα θ φ= . (10.113)

We now write Equations 10.109 and 10.113 as the pair

cos 2 sin 2 sinα θ φ= (10.114)

cot tan cos 2δ φ θ= − . (10.115)

Equations 10.114 and 10.115 are the Kent–Lawson equations that will be derived using the Mueller–Stokes formalism in Section 14.6.

This treatment using the Jones formalism illustrates a very important point. At first glance the use of 2 × 2 rather than 4 × 4 matrices might lead us to believe that calculations are simpler with the Jones calculus. The example illustrated by the Kent–Lawson problem shows that this is not neces-sarily so. We see that even though it is relatively easy to solve for Ex and Ey, there is still a consid-erable amount of algebra to be carried out. Furthermore, because complex quantities are used, the chance of making a computational error is increased. Because the Mueller formalism contains only real quantities, it is actually easier to use; invariably, the algebra is considerably less. Experience usually indicates the preferable formalism to use to solve a problem.

These remarks can be illustrated further by considering another problem. Suppose we wish to create elliptically polarized light of arbitrary orientation and phase (α and δ) from, say, linear hori-zontally polarized light. This can be done by using only a Babinet–Soleil compensator (see Chapter 23) and adjusting its phase and orientation. For the purpose of comparison, we address this problem first by using the Mueller–Stokes formalism and then by using the Jones formalism.

In the Mueller–Stokes formalism, the problem is simply stated mathematically by

MR φ θθ φ

,cos cos si

2

1

1

0

0

1 0 0 0

0 22

( )

=+ nn cos sin cos sin sin

cos s

2 2 1 2 2 2

0 1

θ φ θ θ φ θφ

−( ) −−( ) iin cos sin cos cos sin cos

sin si

2 2 2 2 2

0

2 2θ θ θ φ θ φ θφ

+nn sin cos cos2 2

1

1

0

0θ φ θ φ−

=

1

2

2

2

cos

sin cos

sin sin

,α

α δα δ

(10.116)

216 Polarized Light, Third Edition

where MR(ϕ, 2θ) is the Mueller matrix of a rotated retarder (see Chapter 6). Carrying out the matrix multiplication in Equation 10.116 and equating Stokes vector elements, we have

cos 2 cos sin 2 cos 22 2θ φ θ α+ = (10.117)

1 cos sin 2 cos 2 sin 2 cos−( ) =φ θ θ α δ (10.118)

sin sin 2 sin 2 sinφ θ α δ= . (10.119)

We now solve these last equations for ϕ and θ. Equation 10.117 can be rewritten immediately as

1 cos sin 2 2 sin2 2−( ) =φ θ α. (10.120)

Dividing Equation 10.118 by Equation 10.120 then gives

cot 2 cot cosθ α δ= . (10.121)

Next, Equation 10.119 is divided by Equation 10.118 to obtain

cot cos tan ,φ θ δ2

2= (10.122)

where we have used the trigonometric half-angle formulas for ϕ. The cos 2θ term can be expressed in terms of α and δ. From Equation 10.121 we see that

coscos cos

cos sin2

1 2 2θ α δ

α δ=

− (10.123)

sinsin

cos sin.2

1 2 2θ α

α δ=

− (10.124)

We now substitute Equation 10.123 into Equation 10.122 and write the result along with Equation 10.121 as the pair

cot 2 cot cosθ α δ= (10.125)

cotcos sin

cos sin.

φ α δα δ2 1 2 2

=−

(10.126)

We can provide two simple numerical checks on Equations 10.125 and 10.126. We know that if we start with linear horizontally polarized light and wish to rotate the linearly polarized light to +45°, this can be done by rotating a half-wave retarder through +22.5°. We can show this formally by writing

1

0

1

0

1

2

2

2

=

cos

sin cos

sin sin

αα δα δ

. (10.127)

The Jones Matrix Formalism 217

We see that linear  +45° polarized light corresponds to 2α = 90° and δ = 0° in Equation 10.127. Substituting these conditions into Equations 10.125 and 10.126 yields

tan 2 1 andθ = (10.128)

tan ,φ2

= ∞ (10.129)

from which we immediately find that θ = 22.5° and ϕ = 180° as required. The other check on Equations 10.125 and 10.126 is to consider the conditions to create right circularly polarized light from linear horizontally polarized light. We know that a quarter-wave retarder whose fast axis is rotated by  +45° with respect to the polarizer axis will generate right circularly polarized light. Therefore, we write the equation appropriate to this case

1

0

0

1

1

2

2

2

=

cos

sin cos

sin sin

αα δα δ

, (10.130)

which is satisfied for 2α = 90° and δ = 90°. Substituting these conditions into Equations 10.125 and 10.126 gives

tan2 andθ = ∞ (10.131)

tan ,φ2

1= (10.132)

from which we see that we must set the Babinet–Soleil compensator to θ = 45° and ϕ = 90°, which is exactly what we would expect.

We now consider the same problem of the rotated Babinet–Soleil compensator using the Jones formalism. The mathematical statement for this problem is written as

JR ie( )

cos

sin,θ

αα δ

1

0

=

(10.133)

where JR(θ) is given by

JR

i

i

e

eθ

θ θθ θ

φ

φ( ) =

−

−

cos sin

sin cos

/

/

2

2

0

0 −

cos sin

sin cos.

θ θθ θ

(10.134)

Carrying out the matrix multiplication and equating terms, we find

e ei iφ φθ θ α/ /2 2 2 2cos sin cos+ =− (10.135)

e e ei i iφ φ δθ θ α/ / .2 2 sin cos sin−( ) =− (10.136)

218 Polarized Light, Third Edition

We first rewrite Equation 10.136 as

i eisin sin sin .φ θ α δ

22 = (10.137)

Next, we divide Equation 10.135 by Equation 10.137, group terms, then equate the real and imagi-nary terms and find that

cot 2 cot cosθ α δ= (10.138)

cot sin cot sin .φ θ α δ2

2= (10.139)

Replacing sin 2θ in Equation 10.139 with the expression from Equation 10.124, this pair of equa-tions can be written as

cot 2 cot cosθ α δ= (10.140)

cotcos sin

cos sin.

φ α δα δ2 1 2 2

=−

(10.141)

Equations 10.140 and 10.141 are identical to the results in Equations 10.125 and 10.126 obtained using the Mueller formalism. The reader will see that a considerable amount of increased effort is required to obtain these equations using the Jones formalism.

One of the fundamental problems continuously encountered in the field of polarized light is the determination of the orientation and ellipticity of an incident polarized beam. This can be done by analyzing the beam using a quarter-wave retarder and a linear polarizer, where both elements are capable of being rotated through the angles α and β, respectively. Thus, the Jones matrix for a quarter-wave retarder rotated by α and an ideal polarizer rotated by β using Equations 10.65 and 10.53, is

J J J= ( )

P Rβ λ α

4, , (10.142)

where

JR

i i

i i

λ αα α

α α41

2

1 2 2

2 1 2,

cos sin

sin cos

=

+−

(10.143)

and

JP ββ β β

β β β( ) =

cos cos sin

cos sin sin.

2

2 (10.144)

The matrix product Equation 10.142 is then written out as

J =

+1

2

1 22

2

cos cos sin

cos sin sin

cosβ β ββ β β

i αα αα α

i

i i

sin

sin cos.

2

2 1 2−

(10.145)

The Jones Matrix Formalism 219

The product in Equation 10.145 is a matrix from which it is clear that there is extinction at specific values of α and β. These extinction angles determine the ellipticity and orientation of the incident beam. Rather than giving a general solution of this problem, we consider a specific example.

Suppose we find that extinction occurs at α = 45° and β = 30°. Equations 10.143 and 10.144 become

JR

i

i

λ4

451

2

1

1, °

=

(10.146)

and

JP 3014

3 3

3 1°( ) =

. (10.147)

The product of these equations according to Equation 10.142 yields

J =+ +

+ +

1

4 2

3 3 3 3

3 3 1

i i

i i. (10.148)

Equation 10.148 describes the propagation of the incident beam first through the rotated quarter-wave retarder followed by a linear polarizer. The purpose of the rotated quarter-wave retarder is to transform the incident elliptically polarized beam to linearly polarized light. The linear polarizer is then rotated until a null intensity (i.e., extinction) occurs. This, incidentally, is the fundamental basis of ellipsometry. In order to have a null intensity, we must have from Equation 10.148 that

1

4 2

3 3 3 3

3 3 1

0

0

+ ++ +

=

i i

i i

E

Ex

y

.. (10.149)

Writing Equation 10.149 in component form gives

3 3 3 3 0+( ) + +( ) =i E i Ex y (10.150)

3 3 1 0+( ) + +( ) =i E i Ex y . (10.151)

We see that Equation 10.150 differs from Equation 10.151 only by a factor of 3 , so the equations are identical. We now solve Equation 10.151 for Ey/Ex and find that

E

Eiy

x

= − +

32

12

. (10.152)

This ratio Ey/Ex can be expressed as

E

E

a

bey

x

i=

δ (10.153)

220 Polarized Light, Third Edition

where a/b is real. Equating the real and imaginary parts in Equations 10.152 and 10.153, we have

a

bcosδ = − 3

2 (10.154)

a

bsin .δ = 1

2 (10.155)

Squaring Equations 10.154 and 10.155 and adding gives

a

b= ±1. (10.156)

Similarly, dividing Equation 10.155 by Equation 10.154 yields

δ = −

= − °−tan .1 1

330 (10.157)

Equation 10.156 tells us that the orthogonal amplitudes of the incident beam are equal, and Equation 10.157 tells us that the phase shift between the orthogonal components is −30°. The Jones vector of the original beam is then

E =

=

=±

°

E

E

a

be ex

yi i

1

2

1

2

130δ

, (10.158)

where we have introduced a factor of 1 2/ so that Equation 10.158 is normalized. In terms of the polarization ellipse, Equation 10.158 gives

x xy y2 2

2

31

2 2− + =

, (10.159)

which is the equation of a rotated ellipse. Equation 10.159 can be rotated to a nonrotated (standard) ellipse by using well-known equations of analytical geometry. The left-hand side of Equation 10.159 is of the form

Ax bxy cy2 22+ + . (10.160)

By using the well-known rotation equations, Equation 10.160 can be transformed to

a u b uv c v12

1 122+ + , (10.161)

where

a A b c12 2cos 2 sin cos sin= + +φ φ φ φ (10.162)

2 2 cos sin 21b b A c= − −( )φ φ (10.163)

c A b c12 2sin 2 sin cos cos= − +φ φ φ φ . (10.164)

The Jones Matrix Formalism 221

The “cross” term 2b1 will vanish, and the standard form of the ellipse is obtained for

cot .22

φ = −A c

b (10.165)

From Equation 10.159, we see that A = c = 1 and b = − 3 2/ . Thus, Equation 10.165 shows that the angle of rotation ϕ is −45°. Equations 10.162 and 10.164 then reduce to

a1

2 32

= − (10.166)

c1

2 32

= +. (10.167)

The ellipticity angle is seen to be

tan .χ = −+

= −+

2 32 3

1 3 21 3 2

//

(10.168)

Equation 10.168 can be reduced further by noting that cos 30 3 2° /= , and using the half-angle formulas we obtain

tancoscos

sincos

χ = −+

=1 301 30

2 152 15

2

2

°°

°°

(10.169)

so χ = 15°. Thus, Equation 10.159 describes an ellipse that is rotated 45° from the x axis. The axial length is L L c a2 1 1 1 2 3 2 3 3 7321/ / /= = +( ) −( ) = . .

The last problem that we consider is to show that a linear polarizer can be used to measure the major and minor axes of the polarization ellipse in standard form; that is, the major and minor axes of the ellipse are along the x and y axes, respectively. This is described by setting δ = 90° in Equation 10.102, so that

E =

=

cos

sin.

ααi

a

ib (10.170)

The amplitude equations corresponding to Equation 10.170 are

E tx = cos cosα ω (10.171)

E ty = sin sinα ω . (10.172)

We can eliminate ωt between Equations 10.171 and 10.172, so that

E

a

E

bx y2

2

2

21+ = , (10.173)

222 Polarized Light, Third Edition

where a = cos α and b = sin α. Thus, a and b are the lengths of the semimajor and semiminor axes of the polarization ellipse Equation 10.173.

We now return to the measurement of a and b. The Jones matrix of a rotated polarizer is

JP θθ θ θ

θ θ θ( ) =

cos sin cos

sin cos sin

2

2 (10.174)

so that the Jones vector of the emerging beam is, multiplying Equation 10.170 by Equation 10.174,

E′ =+

+

a ib

a ib

cos sin cos

sin cos sin

2

2

θ θ θθ θ θ

.. (10.175)

The intensity corresponding to Equation 10.175 is readily seen to be

I a bθ θ θ( ) = +2cos sin2 2 2 , (10.176)

where the prime on the intensity has been dropped. For the two angles θ = 0° and 90° Equation 10.176 gives

I a0° cos2 2( ) = = α (10.177)

I b9 ° sin2 20( ) = = α. (10.178)

We see that the square of the major and minor axes can be found by measuring the orthogonal inten-sities. It is usually convenient to express these last two equations as the ratio

ab

II

= ( )( )

090

°°

. (10.179)

Numerous problems using the Jones and Mueller matrices can be found in the references at the end of this Chapter [9–17]. In particular, Gerrard and Burch [13] treat a number of interesting problems.

10.5 JoNeS maTRiCeS foR homogeNeouS elliPTiCal PolaRiZeRS aNd ReTaRdeRS

We have described polarizers, retarders, circular polarizers, and so on, in terms of the Mueller and Jones matrices. In particular, we have pointed out that a linear polarizer and a circular polarizer derive their names from the fact that, regardless of the polarization state of the incident beam, the polarization state of the emerging beam is always linearly and circularly polarized, respectively. Let us look at this behavior more closely. The Jones matrix of a rotated linear polarizer is given by

JP θθ θ θ

θ θ θ( ) =

cos sin cos

sin cos sin.

2

2 (10.180)

The incident beam is represented by

E =

E

Ex

y

. (10.181)

The Jones Matrix Formalism 223

Multiplying Equation 10.181 by Equation 10.180 yields

E′ = +( )

E Ex ycos sincos

sin,θ θ

θθ

(10.182)

which is the Jones matrix for linearly polarized light. If θ = 0°, Equation 10.180 reduces to

JPH =

1 0

0 0. (10.183)

We observe that if the incident beam is elliptically polarized, multiplying Equation 10.181 by Equation 10.183 gives

E′ =

Ex

1

0 (10.184)

and we have obtained linearly polarized light. This can be written in normalized form as

E′ =

1

0. (10.185)

If we now try to transmit the orthogonal state; that is, linear vertically polarized light represented by the Jones vector

E =

0

1 (10.186)

we find from Equation 10.183 to Equation 10.186 that

E′ =

0

0 (10.187)

so that there is no emerging beam. This behavior of the horizontal linear polarizer Equation 10.183 can be summarized by writing

E′ =

=

1 0

0 0

1

01

1

0 (10.188)

E′ =

=

1 0

0 0

0

10

0

1. (10.189)

Written in this way, we see that the problem of transmission by a polarizer can be thought of in terms of an eigenvector/eigenvalue problem. We see that the eigenvectors of the 2 × 2 Jones matrix Equation 10.183 are

E

E

1

1

0

0

1

=

=

2

(10.190)

224 Polarized Light, Third Edition

and the corresponding eigenvalues are 1 and 0. A linear polarizer has the property that it transmits one of its eigenvectors perfectly and rejects the orthogonal eigenvector completely.

Let us now consider the same problem using a circular polarizer. We have seen that a circular polarizer can be constructed by using a linear polarizer set at  +45° followed by a quarter-wave retarder. The Jones matrix is

J =

12

1 1

i i. (10.191)

We now multiply Equation 10.181 by Equation 10.191 and obtain

E′ =+

=

E E

i ix y

2

1 1

2

1 (10.192)

in its normalized form. Thus, again, regardless of the polarization state of the incident beam, the emerging beam is always right circularly polarized. In the case of a linear polarizer, the transmis-sion of the orthogonal polarization state was completely blocked by the linear polarizer. Let us now see what happens when we try to transmit the orthogonal polarization state, that is, left circularly polarized light through the circular polarizer Equation 10.191. The Jones vector of the orthogonal state, left circularly polarized light, is

E =−

1

2

1

i. (10.193)

Multiplying Equation 10.193 by Equation 10.191, we find that the Jones vector of the emerging beam is

E′ = −

1

2 2

1i

i. (10.194)

The emerging beam is right circularly polarized. The circular polarizer Equation 10.191 does not block the left circularly polarized beam, and Equation 10.194 is not an eigenvector of Equation 10.191! The reason for this seemingly anomalous behavior, which is unlike the linear polarizer, is that the circular polarizer is constructed from a linear +45° polarizer and a quarter-wave retarder. This is not a homogeneous polarizing element. The true eigenvectors of Equation 10.191 are easily shown to be

E

E

1

2

1

2

1

1

1

2

1

=−

=

i

,

(10.195)

which are linear −45° and right circularly polarized light, respectively; the corresponding eigen-values are 0 and 1 + i. Consequently, Equation 10.191 does not describe a true “circular” polarizer. We would expect that a true circular polarizer would behave in a manner identical to that of the linear polarizer. That is, only one state of polarized light always emerges and this corresponds to

The Jones Matrix Formalism 225

one of the two eigenvectors. Furthermore, the other eigenvector is orthogonal to the transmitted eigenvector, but is completely blocked by the polarizing element, so that the eigenvalues are 1 and 0. A polarizing element that exhibits these two properties simultaneously is called homogeneous. We now wish to construct the homogeneous polarizing elements not only for circularly polarized light but also for the more general state, elliptically polarized light.

The key to solving this problem is to recall our earlier work on raising the matrix to the mth power. There we saw that the Mueller matrix could be diagonalized and that it was possible to repre-sent the Mueller matrix in terms of its eigenvalues, eigenvectors, and another matrix that we called the modal matrix. Let us now consider this problem again, this time using the Jones vector.

Let us represent the Jones vector of a beam by

E1 =

p

q. (10.196)

The orthogonal state is given by

E2 =−

q

p

*

*. (10.197)

The reader can easily prove that Equations 10.196 and 10.197 are orthogonal by applying the orthog-onality condition E E E E1 2 2 1 0† †⋅ = ⋅ = , where † represents complex transpose. We also know that the corresponding eigenvalues are λ1 and λ2. Earlier in this book, we saw that we could construct a new matrix K from the eigenvectors. We called this the modal matrix, and write it as

K =−

p q

q p

*

*. (10.198)

The inverse modal matrix K−1 is easily found to be

K− =+( ) −

1 1pp qq

p q

q p* *

* *

. (10.199)

It is easily shown that KK–1 = K–1K = I if we normalize (pp* + qq*) to 1.We saw earlier that there is a unique relationship between a matrix Φ(≡ J) and its eigenvalues

and eigenvectors, and this is expressed by

ΦΦ ΛΛK K= , (10.200)

where Λ is the diagonal eigenvalue matrix

ΛΛ =

λλ

1

2

0

0. (10.201)

We now solve Equation 10.200 for Φ to obtain

ΦΦ ΛΛ= −K K 1. (10.202)

226 Polarized Light, Third Edition

Equation 10.202 is a rather remarkable result because it shows that a matrix Φ can be constructed completely from its eigenvectors and eigenvalues. We can write Equation 10.202 as, replacing Φ by J,

J =+

−

−

1 0

01

2pp qq

p q

q p

p q

q p* *

* * *λλ

. (10.203)

Carrying out the multiplication yields

J =+

+ ( )( )

1 1 2 1 2

1 2 1pp qq

pp qq pq

qp* *

* * *

*

–

–

λ λ λ λλ λ λ qqq pp* *

.+

λ2

(10.204)

To check Equations 10.203 and 10.204, let us consider linearly polarized light. We know that its eigenvectors are

E E1 2

1

0

0

1=

=

(10.205)

and its eigenvalues are 1 and 0. The modal matrix K and its inverse K−1 are then

K K=

= −1 0

0 11. (10.206)

From Equation 10.203, we can write

J =

=

1 0

0 1

1 0

0 0

1 0

0 1

1 0

0 0,, (10.207)

which is the Jones matrix of a linear horizontal polarizer; this element is a homogeneous polarizing element.

Let us now construct a homogeneous right circular polarizer. The orthogonal eigenvectors are

E E1 2

1

2

1 1

2 1=

=

i

i. (10.208)

From Equation 10.203 to Equation 10.204, the Jones matrix for a right circular homogeneous polar-izer will be (p = 1, q = i)

J =

−−

=−1

2

1

1

1 0

0 0

1

1

12

1i

i

i

i

i

i 11

. (10.209)

We can check to see if Equation 10.209 is the Jones matrix for a homogeneous right circular polar-izer. First, we consider an elliptically polarized beam represented by

E =

E

Ex

y

. (10.210)

The Jones Matrix Formalism 227

We multiply Equation 10.210 by the right-hand side of Equation 10.209, and find that

E =−

E iE

ix y

2

1 (10.211)

so that only right circularly polarized light emerges, as required. Next, we take the products of the right-hand side of Equation 10.209 and the eigenvector for right circularly polarized light and the eigenvector for left circularly polarized light, and obtain the respective results

E′ =−

= ( )

=12

1

1

1

2

11

1

2

1 1

2

i

i i i

11

i

(10.212)

E′ =−

−

=

= 12

1

1

1

2

1 1

2 2

0

0

0

0

i

i i

, (10.213)

which is exactly what we require for a homogeneous right circular polarizer.We can now turn our attention to constructing a homogeneous elliptical polarizer. For conve-

nience, we describe this by the Jones vector

E1 =

p

q, (10.214)

and describe its orthogonal vector (eigenvector) by

E2 =−

q

p

*

*. (10.215)

From Equation 10.204 we then have, by setting λ1 = 1 and λ2 = 0,

J =

pp pq

qp qq

* *

* *. (10.216)

There are two other ways to represent an elliptical polarizer. The first is to write the incident Jones vector in the form

E1 =

a e

a ex

i

yi

x

y

φ

φ . (10.217)

The orthogonal state and eigenvalues are constructed as shown earlier in this section. Then, we eas-ily see from Equation 10.204 to Equation 10.217 that the Jones matrix for an elliptical polarizer is

J =

−

+

a a a e

a a e ax x y

i

x yi

y

2

2

φ

φ, (10.218)

where ϕ = ϕy – ϕx. The other representation of an elliptical polarizer can be obtained by using the Jones vector

E1 =

cos

sin

αα δei

(10.219)

228 Polarized Light, Third Edition

as the eigenvector. The orthogonal state and eigenvalues are again constructed as shown earlier. From Equation 10.204, we see that the Jones matrix for the elliptical polarizer is

J =−

−cos sin

sin cos

cα αα α

δ

δ

e

e

i

i

1 0

0 0

oos sin

sin cos

α αα α

δ

δ

e

e

i

i

−

−

(10.220)

or

J =

−

+

cos sin cos

sin cos sin.

2

2

α α αα α α

δ

δ

e

e

i

i (10.221)

The form expressed by Equation 10.221 enables us to determine the Jones matrix for any type of elliptical polarizer including, for example, a linear polarizer and a circular polarizer. For a linear horizontal polarizer, α = 0° and Equation 10.221 reduces to

J =

1 0

0 1, (10.222)

which is indeed the Jones matrix for a linear horizontal polarizer given earlier by Equation 10.47. Similarly, for α = 45° and δ = 90°, which are the conditions for right circularly polarized light, we see that Equation 10.221 reduces to

J =−

12

1

1

i

i, (10.223)

which is the Jones matrix for a homogeneous right circular polarizer, in agreement with Equation 10.209.

While we have considered only ideal polarizers, it is simple to extend this analysis to the general case where the polarizer is described by

J =

p

px

y

0

0 (10.224)

where px, py range between 0 and 1. For an ideal linear horizontal polarizer, px = 1 and py = 0. The terms used to describe nonideal behavior of polarizers are diattenuation or dichroism. We shall use the preferable diattenuation. For an explanation of the term dichroism and the origin of its usage, see Shurcliff [12]. Equation 10.224 describes a diattenuator. We see immediately that the eigenvalues of a diattenuator are px and py. Therefore, the Jones matrix for a nonideal (diattenuating) elliptical polarizer is

J =−

−cos sin

sin cos

α αα α

δ

δ

e

e

p

p

i

i

x

y

0

0 −

−cos sin

sin cos

α αα α

δ

δ

e

e

i

i (10.225)

or

J =+ −( )

−( )−p p p p e

p px y x y

i

x y

cos sin sin cos2 2α α α α δ

ssin cos sin cos.

α α α αδe p pix y

+ +

2 2

(10.226)

The Jones Matrix Formalism 229

Equation 10.226 enables us to describe any type of elliptical polarizer and is the most useful of all representations of homogeneous elliptical polarizers.

We now consider the other important polarizing element, the retarder, and we will show how to represent homogeneous linear, circular, and elliptical retarders. We begin this discussion by recall-ing that the Jones matrix for a retarder was given by

J =

+

−

e

e

i

i

φ

φ

/

/.

2

2

0

0 (10.227)

We now determine the eigenvectors and the eigenvalues of Equation 10.227. We do this by forming the familiar eigenvector/eigenvalue equation

e

e

p

q

i

i

+

−

−−

=φ

φ

λλ

/

/.

2

2

0

00 (10.228)

The eigenvalues are

λ λφ φ1

22

2= =+ −e ei i/ / (10.229)

and the corresponding eigenvectors are

E E1 2

1

0

0

1=

=

, (10.230)

which are the Jones vectors for linear horizontally and linear vertically polarized light, respectively. Thus, the respective eigenvector/eigenvalue equations are

e

ee

i

ii

+

−+

=

φ

φφ

/

//

2

22

0

0

1

0

1

0 (10.231)

and

e

ee

i

ii

+

−−

=

φ

φφ

/

//

2

22

0

0

0

1

0

1.. (10.232)

Because the eigenvectors of Equation 10.227 are orthogonal states of linear polarized light, the retarder is called a linear retarder. For an elliptical retarder we must obtain the same eigenvalues given by Equation 10.229. If the Jones vector for elliptically polarized light is of the form

E1 =

cos

sin

αα δei

(10.233)

then the Jones matrix for an elliptical retarder must be

J =−

− +

−

cos sin

sin cos

/α αα α

δ

δ

φe

e

e

e

i

i

i

i

2 0

0 φφ

δ

δ

α αα α/

cos sin

sin cos2

−

=

−e

e

e

i

i

iφφ φ φ φα α α α/ / / /cos sin sin cos2 2 2 2 2 2+ −( )− −e e ei i i ee

e e e e e

i

i i i i

−

− +−( ) +

δ

φ φ δ φα α α/ / /sin cos sin2 2 2 2 −−

iφ α/ cos

.2 2

(10.234)

230 Polarized Light, Third Edition

Equation 10.234 can be checked immediately by observing that for α = 0° (linear horizontally polarized light) it reduces to

J =

+

−

e

e

i

i

φ

φ

/

/,

2

2

0

0 (10.235)

which is the Jones matrix for a linear retarder Equation 10.227 as we expect.We can use Equation 10.234 to find, say, the Jones matrix for a homogeneous right circular

retarder using the familiar conditions of α = 45° and δ = 90°. Substituting these values in Equation 10.234 then gives

J =−

cos sin

sin cos

,

φ φ

φ φ2 2

2 2

(10.236)

which is the Jones matrix for a homogeneous right circular retarder.These results can be summarized by writing the Jones matrices for a homogeneous elliptical

polarizer and a homogeneous elliptical retarder as the pair

J =+ −( )

−( )−p p p p e

p px y x y

i

x y

cos sin sin cos2 2α α α α δ

ssin cos sin cosα α α αδe p pix y

+ +

2 2

(10.237)

J =+ −( )− −e e e ei i i iφ φ φ φα α α/ / / /cos sin sin c2 2 2 2 2 2 oos

sin cos sin/ / /

αα α

δ

φ φ δ φ

e

e e e e

i

i i i i

−

− +−( )2 2 2 2 αα αφ+

−e i / cos

.2 2

(10.238)

Shurcliff [12] and, more recently, Kliger, Lewis, and Randall [16], have tabulated the Jones matrices and the Mueller matrices for elliptical polarizers and retarders as well as their degenerate forms. All of their forms can, of course, be obtained from Equation 10.237 to Equation 10.238.

RefeReNCeS

1. Jones, R. C., A new calculus for the treatment of optical systems. I. Description and discussion of the calculus, J. Opt. Soc. Am. 31 (1941): 488–93.

2. Hurwitz, H., and R. C. Jones, A new calculus for the treatment of optical systems. II. Proof of three general equivalence theorems, J. Opt. Soc. Am. 31 (1941): 493–5.

3. Jones, R. C., A new calculus for the treatment of optical systems. III. The Sohncke Theory of optical activity, J. Opt. Soc. Am. 31 (1941): 500–3.

4. Jones, R. C., A new calculus for the treatment of optical systems. IV., J. Opt. Soc. Am. 32 (1942): 486–93.

5. Jones, R. C., A new calculus for the treatment of optical systems. V. A more general formulation, and description of another calculus, J. Opt. Soc. Am. 37 (1947): 107–10.

6. Jones, R. C., A new calculus for the treatment of optical systems. VI. Experimental determination of the matrix, J. Opt. Soc. Am. 37 (1947): 110–2.

7. Jones, R. C., A new calculus for the treatment of optical systems. VII. Properties of the N-matrices, J. Opt. Soc. Am. 38 (1948): 671–83.

8. Jones, R. C., A new calculus for the treatment of optical systems. VIII. Electromagnetic Theory, J. Opt. Soc. Am. 46 (1956): 126–31.

9. Hecht, Ε., and A. Zajac, Optics, Reading, MA: Addison-Wesley, 1974. 10. Azzam, R. M. A., and N. M. Bashara, Ellipsometry and Polarized Light, Amsterdam: North-Holland,

1977.

The Jones Matrix Formalism 231

11. Simmons, J. W., and M. J. Guttman, States, Waves, and Photons, Reading, MA: Addison-Wesley, 1970. 12. Shurcliff, W. Α., Polarized Light, Cambridge, MA: Harvard University Press, 1962. 13. Gerrard, A., and J. M. Burch, Introduction to Matrix Methods in Optics, London: Wiley, 1975. 14. Shurcliff, W. A., and S. S. Ballard, Polarized Light, New York: Van Nostrand, 1962. 15. Clarke, D., and J. F. Grainger, Polarized Light and Optical Measurement, Oxford: Pergamon Press,

1971. 16. Kliger, D. S., J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy, New York:

Academic Press, 1990. 17. Challener, W. A., and Τ. Α. Rinehart, Jones matrix analysis of magnetooptical media and read-back

systems, Appl. Opt. 26 (1987): 3974–80.

233

11 The Poincaré Sphere

11.1 iNTRoduCTioN

In the previous chapters, we have seen that the Mueller matrix formalism and the Jones matrix formalism enable us to treat many complex problems involving polarized light. However, the use of matrices only slowly made its way into physics and optics. In fact, before the advent of quantum mechanics in 1925, matrix algebra was rarely used. It is clear that matrix algebra greatly simplifies the treatment of many difficult problems. In polarized light, even the simplest problem of determin-ing the change in polarization state of a beam propagating through several polarizing elements becomes surprisingly difficult to do without matrices. Before the advent of matrices, only direct and very tedious algebraic methods were available. Consequently, other methods were sought to simplify these difficult calculations.

The need for simpler ways to carry out difficult calculations began in antiquity. The greatest of the Greek astronomers, Hipparchus (about 190 BC to about 120 BC), worked primarily on the island of Rhodes. He compiled a catalog of stars and also plotted the positions of these stars in terms of lat-itude and longitude (in astronomy, longitude and latitude are called right ascension and declination) on a large globe that we call the celestial sphere. In practice, transporting a large globe for use at dif-ferent locations is cumbersome. Therefore, he devised a method for projecting a three-dimensional sphere on to a two-dimensional plane. This type of projection is called a stereographic projection. It is still one of the most widely used projections and is particularly popular in astronomy. It has many interesting properties, foremost of which is that the longitudes and latitudes (right ascension and declination) continue to intersect each other at right angles on the plane as they do on the sphere. It appears that the stereographic projection was forgotten for many centuries and then rediscovered during the European Renaissance when the ancient writings of classical Greece and Rome were rediscovered. With the advent of the global exploration of the world by the European navigators and explorers there was a need for accurate charts, particularly charts that were mathematically correct. This need led not only to the rediscovery and use of the stereographic projection but also to the invention of new types of projections (e.g., the famous Mercator projection).

Henri Poincaré, a famous nineteenth–century French mathematician and physicist, discovered around 1890 that the polarization ellipse could be represented on a complex plane [1]. Further, he discovered that this plane could be projected onto a sphere in exactly the same manner as the stereo-graphic projection. In effect, he reversed the problem of classical antiquity, which was to project a sphere onto a plane. The sphere that Poincaré devised is extremely useful for dealing with polarized light problems and, appropriately, it is called the Poincaré sphere.

In 1892, Poincaré introduced his sphere in his text Traité de Lumierè. Before the advent of matri-ces and digital computers, it was extremely difficult to carry out calculations involving polarized light. As we have seen, as soon as we go beyond the polarization ellipse (e.g., the interaction of light with a retarder) the calculations become difficult. Poincaré showed that the use of his sphere enabled many of these difficulties to be overcome. In fact, Poincaré’s sphere not only simplifies many calcu-lations but also provides remarkable insight into the manner in which polarized light behaves in its interaction with polarizing elements.

While the Poincaré sphere became reasonably well known in the optical literature in the first half of the twentieth century, it was rarely used in the treatment of polarized light problems. This was probably due to the considerable mathematical effort required to understand its proper-ties. In fact, its use outside of France appears to have been virtually nonexistent until the 1930s.

234 Polarized Light, Third Edition

Ironically, the appreciation of its usefulness only came after the appearance of the Jones and Mueller matrix formalisms. The importance of the Poincaré sphere was finally established in the optical literature in the long review article by Ramachandran and Ramaseshan on crystal optics in 1961 [2].

The Poincaré sphere is still much discussed in the literature of polarized light. In larger part this is due to the fact that it is really surprising how simple it is to use once it is understood. In fact, despite its introduction nearly a century ago, new properties and applications of the Poincaré sphere are still being published and appearing in the optical literature. The two most interesting properties of the Poincaré sphere are (1) any point on the sphere corresponds to the three Stokes parameters S1, S2, and S3 for elliptically polarized light, and (2) the magnitude of the interaction of a polarized beam with an optical polarizing element corresponds to a rotation of the sphere; the final point describes the new set of Stokes parameters. In view of the continued application of the Poincaré sphere, we present a detailed discussion of it. This is followed by simple appli-cations of the sphere in describing the interaction of polarized light with a polarizer, retarder, and rotator. More complicated and involved applications of the Poincaré sphere are listed in the references.

11.2 TheoRy of The PoiNCaRÉ SPheRe

Consider a Cartesian coordinate system with axes x, y, z and let the direction of propagation of a monochromatic elliptically polarized beam of light be in the z direction. The equations of propaga-tion are described by

E z t E i t kzx x,( ) = −( )exp ω (11.1)

E z t E i t kzy y, ,( ) = −( )exp ω (11.2)

where Ex and Ey are the complex amplitudes

E E ix x x= ( )0 exp δ (11.3)

E E iy y y= ( )0 exp δ , (11.4)

and E0x and E0y are real quantities. We divide Equation 11.4 by Equation 11.2 and write

E

E

E

Ee

E

Ei

E

Ey

x

y

x

i y

x

y

x

= = +

0

0

0

0

0

0

δ δ δcos sin == +u iv, (11.5)

where δ = δy − δx and u and ν are orthogonal axes in the complex plane. On eliminating the propaga-tor in Equations 11.1 through 11.4, we obtain the familiar equation of the polarization ellipse,

E

E

E

E

E E

E Ex

x

y

y

x y

x y

2

02

2

02

0 0

22+ − =cos sin .δ δ (11.6)

We have shown in Section 4.2 that the maximum values of Ex and Ey are E0x and E0y. Equation 11.6 describes an ellipse inscribed in a rectangle of sides 2E0x and 2E0y. This is shown in Figure 11.1.

The Poincaré Sphere 235

In general, we recall, the axes of the ellipse are not necessarily along the x and y axes but are rotated, say, along some other axes x′ and y′. Thus, we can write the oscillation along x′ and y′ as

x a′ = cosφ (11.7)

y b′ = sin ,φ (11.8)

where ϕ = ωt – kz. The ellipticity e, which is the ratio of the minor axis to the major axis, is e = b / a. The orientation of the ellipse is given by the azimuth angle θ (0 ≤ θ ≤ 180°); this is the angle between the major axis and the positive x axis. From Figure 11.1, the angles ε and v are defined by the equations

tan ( )ε ε= ≤ ≤ba

0 90° (11.9)

tan ( ).vE

Evy

x

= ≤ ≤0

0

0 90° (11.10)

The sense of the ellipse or the direction of rotation of the light vector depends on δ; it is designated right or left according to whether sin δ is negative or positive. The sense will be indicated by the sign of the ratio of the principal axes. Thus, tan ε = +b/a or −b/a refers to left (counterclockwise) or right (clockwise) rotation, respectively.

By using the methods presented earlier (see Section 4.4), we see that the following relations exist with respect to the parameters of the polarization ellipse:

E E a bx y02

02 2 2+ = + (11.11)

E E a bx y02

02 2 2 2− = −( )cos θ (11.12)

O

y

x

2E0x

2E0y

ba

θ

εν

figuRe 11.1 Parameters of the polarization ellipse having amplitude components E0x and E0y along x and y axes, respectively. The angle v is related to E0x and E0y by tan v = E0y/E0x. The major and minor axes of the ellipse are 2a and 2b, and the ellipticity is e = b/a = tanε; the azimuth angle θ is with respect to the x axis.

236 Polarized Light, Third Edition

E E abx y0 0 sinδ = ± (11.13)

2 cos sin 22 2E E a bx y0 0 δ θ= −( ) . (11.14)

By adding and subtracting Equations 11.11 and 11.12, we can relate E0x and E0y to a, b, and θ. We find that

E a bx02 2 2 2 2= +cos sinθ θ (11.15)

E a by02 2 2 2 2= +sin cos .θ θ (11.16)

We see that when the polarization ellipse is not rotated, θ = 0°, and Equations 11.15 and 11.16 become

E a E bx y0 0= ± = ± , (11.17)

which is to be expected, as Figure 11.1 shows. The ellipticity is then seen to be

eb

a

E

Ey

x

= = 0

0

(11.18)

when θ = 0°.We can now obtain some interesting relations between the foregoing parameters. The first one can

be obtained by dividing Equation 11.14 by Equation 11.12. We obtain

tan cos .22 0 0

02

02

θ δ=−

E E

E Ex y

x y

(11.19)

Substituting Equation 11.10 into Equation 11.19 then yields

tantantan

cos .22

1 2θ δ=

−

v

v (11.20)

The factor in parentheses is equal to tan 2v. We then have

tan 2 tan 2 cosθ δ= v . (11.21)

The next important relationship is obtained by dividing Equation 11.13 by Equation 11.11, so that

±

+=

+ab

a b

E E

E Ex y

x y2 2

0 0

02

02

sin .δ (11.22)

Using both Equations 11.9 and 11.10, we find that Equation 11.22 becomes

± =sin 2 sin 2 sinε δv . (11.23)

The Poincaré Sphere 237

Another important relation is obtained by dividing Equation 11.12 by Equation 11.11. Then

E E

E E

a b

a bx y

x y

02

02

02

02

2 2

2 22

−+

= −+

cos .θ (11.24)

Substituting Equations 11.9 and 11.10 into Equation 11.24, we find that

cos 2 cos 2 cos 2v = ε θ. (11.25)

Equation 11.25 can be used to obtain still another relation. We divide Equation 11.14 by Equation 11.11 to obtain

2

20 0

02

02

2 2

2 2

E E

E E

a b

a bx y

x y

cossin .

δθ

+= −

+ (11.26)

Next, using Equations 11.9 and 11.10, we find that Equation 11.26 can be written as

sin 2 cos cos 2 sin 2v δ ε θ= . (11.27)

Equation 11.27 can be solved for cos 2ε by multiplying through by sin 2θ so that

sin 2 sin 2 cos cos 2 sin 2 cos 2 cos 22θ δ ε θ ε εv = = − ccos 22 θ (11.28)

or

cos 2 cos 2 cos 2 cos 2 sin 2 sin 2 coε ε θ θ θ= ( ) + v ssδ. (11.29)

We see that the term in parentheses is identical to Equation 11.25, so Equation 11.29 can be written as

cos 2 cos 2 cos 2 sin 2 sin 2 cosε θ θ δ= +v v . (11.30)

Equation 11.30 represents the law of cosines for sides from spherical trigonometry. Consequently, it represents our first hint or suggestion that the foregoing results can be related to a sphere. We shall not discuss Equation 11.30 at this time, but defer its discussion until we have developed some further relations.

Equation 11.30 can be used to find a final relation of importance. We divide Equation 11.23 by Equation 11.30 to obtain

± =+

tansin sin

cos cos sin sin cos.2

22 2 2

ε δθ θ δ

v

v v (11.31)

Dividing the numerator and the denominator of Equation 11.31 by sin2v cosδ yields

± =+

tantan

sin (cos cos ) / (sin cos ).2

2 2 2 2ε δ

θ θ δv v (11.32)

238 Polarized Light, Third Edition

We now observe that Equation 11.21 can be written as

cos 2 tan 2 sin 2 cosv vθ δ= (11.33)

and rearranging, we have

coscos tan

sin.δ θ= 2 2

2v

v (11.34)

Substituting Equation 11.34 into the second term in the denominator of Equation 11.32 yields the final relation

± =tan 2 sin 2 tanε θ δ. (11.35)

For convenience, we now collect relations Equations 11.21, 11.23, 11.25, 11.30, and 11.35 and write them as the set of relations

tan 2 tan 2 cosθ δ= v (11.36)

± =sin 2 sin 2 sinε δv (11.37)

cos 2 cos 2 cos 2v = ε θ (11.38)

cos 2 cos 2 cos 2 sin 2 sin 2 cosε θ θ δ= +v v (11.39)

± =tan 2 sin 2 tanε θ δ. (11.40)

These equations have very familiar forms. Indeed, they are well-known relations that appear in spherical trigonometry.

Figure 11.2 shows a spherical triangle formed by three great circle arcs, Ab, bc, and cA on a sphere. At the end of this section, the relations for a spherical triangle are derived by using vec-tor analysis. There it is shown that 10 relations exist for a so-called right spherical triangle. For an

B

A

Cc

a

b

figuRe 11.2 Spherical triangle on a sphere. The vertex angles are designated by A, b, c. The side opposite to each angle is represented by a, b, and c.

The Poincaré Sphere 239

oblique spherical triangle there exists, analogous to plane triangles, the law of sines and the law of cosines. With respect to the law of cosines, however, there is a law of cosines for the angles (upper-case letters) and a law of cosines for sides (lower case letters). Of particular interest are the following relations derived from Figure 11.2; that is,

cos cos cosc a b= (11.41)

sin sin sina c A= (11.42)

tan tan cosb c A= (11.43)

cos cos cos sin sin cosa b c b c A= + (11.44)

tan sin tana b A= . (11.45)

If we now compare these equations with Equations 11.36 through 11.40, we see that the equa-tions can be made completely compatible by constructing the right spherical triangle shown in Figure 11.3. If, for example, we equate the spherical triangles in Figures 11.2 and 11.3, we have

a b A= = =2 2ε θ δ . (11.46)

Substituting Equation 11.46 into Equation 11.41 gives

cos 2 cos 2 cos 2v = ε θ, (11.47)

which corresponds to Equation 11.38. In a similar manner, by substituting Equation 11.46 into the remaining Equations 11.42 through 11.45, we obtain Equations 11.36, 11.37, 11.39, and 11.40. We arrive at the very interesting result that the polarization ellipse on a plane can be transformed to a spherical triangle on a sphere. We shall return to these equations after we have discussed some further transformation properties of the rotated polarization ellipse in the complex plane.

The ratio Ey/Ex in Equation 11.5 defines the shape and orientation of the elliptical vibration given by Equation 11.6. This vibration may be represented by a point m on a plane in which the abscissa and ordinate are u and v, respectively. The diagram in the complex plane is shown in Figure 11.4.

From Equation 11.5 we have

uE

Ey

x

= 0

0

cosδ (11.48)

2ν

2ε

2θδ

figuRe 11.3 Right spherical triangle for the parameters of the polarization ellipse.

240 Polarized Light, Third Edition

vE

Ey

x

= 0

0

sin .δ (11.49)

The point m(u, v) is described by the radius Om and the angle δ. The angle δ is found from Equations 11.48 and 11.49 to be

tanδ = v

u (11.50)

or

δ = −tan .1 v

u (11.51)

Squaring Equations 11.48 and 11.49 and adding yields

u vE

Ey

x

2 2 0

0

2

2+ =

= ρ , (11.52)

which is the square of the distance from the origin to m. We see that we can also write Equation 11.52 as

u v u iv u ivE

E

E

Ey

x

y

x

2 2+ = + − =

=( )( )*

ρρ** =E

Ey

x

02

02

(11.53)

u

v

δo

m(u,v)

s t

P1

P2

P2

figuRe 11.4 Representation of elliptically polarized light by a point m on a plane; δ is the plane difference between the components of the ellipse.

The Poincaré Sphere 241

so

u ivE

Ey

x

+ = = ρ. (11.54)

Thus, the radius vector Om represents the ratio Ey/Ex, and the angle mOu represents the phase dif-ference δ. It is postulated that the polarization is left- or right-handed according to whether δ is between 0 and π or π and 2π, respectively.

We now show that Equation 11.53 can be expressed either in terms of the rotation angle θ or the ellipticity angle ε. To do this we have from Equation 11.52 that

u vE

Ey

x

2 2 0

0

2

2+ =

= ρ . (11.55)

We also have, from Equation 11.10

E

Evy

x

0

0

= tan . (11.56)

Squaring Equation 11.56 gives

E

Evy

x

02

02

2= tan (11.57)

and from trigonometry we can write

tantantan

22

1 2v

v

v=

− (11.58)

so that

tantan

tan.2 1

22

vv

v= − (11.59)

From Equation 11.36 we have

tan 2 tan 2 cosθ δ= v (11.60)

and substituting Equation 11.60 into Equation 11.59 gives

tantan

tancos .2 1

22

vv= −θ

δ (11.61)

Equating Equation 11.61 to Equations 11.57 and 11.55 we have

u v v

v

2 2 1 2 2

1 2 2

+ = − /

= −

(tan cos tan )

cot (tan cos

δ θ

θ δ)). (11.62)

242 Polarized Light, Third Edition

Substituting Equation 11.56 into Equation 11.62 and using Equation 11.48, we find that

u v u2 2 2 cot 2 1+ + − =θ 0. (11.63)

Thus, we have expressed u and ν in terms of the rotation angle θ of the polarization ellipse. It is also possible to find a similar relation to Equation 11.63 in terms of the ellipticity angle ε rather than θ. To show this we again use Equations 11.55, 11.56, and 11.58 to form

u vv

vv2 2 1

22

2+ = − tansin

cos . (11.64)

Substituting Equations 11.36 and 11.37 into Equation 11.64 then gives

u v v v2 2 1 2 2 2+ = +− csc cos .ε (11.65)

After replacing cos 2v with its half-angle equivalent and choosing the upper sign, we are led to

u v v2 2 2 csc 2 1+ − + =ε 0. (11.66)

Thus, we can describe Equation 11.55 terms of either θ or ε by

u v u2 2 2 cot 2 1+ + − =θ 0 (11.67)

u v v2 2 2 csc 2 1+ − + =ε 0. (11.68)

At this point it is useful to remember that the two most important parameters describing the polar-ization ellipse are the rotation angle θ and the ellipticity angle ε, as shown in Figure 11.1. Equations 11.67 and 11.68 describe the polarization ellipse in terms of each of these parameters.

Equations 11.67 and 11.68 are recognized as the equations of a circle. They can be rewritten in standard forms as

u v+( ) + = ( )cot 2 csc 22 2 2θ θ (11.69)

u v2 2 2csc 2 cot 2+ −( ) = ( )ε ε . (11.70)

Equation 11.69 describes, for a constant value of θ, a family of circles each of radius csc 2θ with centers at the point (−cot 2θ, 0). Similarly Equation 11.70 describes, for a constant value of ε, a fam-ily of circles each of radius cot 2θ and centers at the point (0, csc 2θ). The circles in the two systems are orthogonal to each other. To show this, we recall that if we have a function described by a dif-ferential equation of the form

M x y dx N x y dy, ,( ) + ( ) = 0 (11.71)

then the differential equation for the orthogonal trajectory is given by

N x y dx M x y dy, , .( ) − ( ) = 0 (11.72)

The Poincaré Sphere 243

We therefore consider Equation 11.67 and show that Equation 11.68 describes the orthogonal trajec-tory. We first differentiate Equation 11.67 to obtain

udu vdv du+ + =cot 2θ 0. (11.73)

We eliminate the constant parameter cot 2θ from Equation 11.73 by writing Equation 11.67 as

cot .21

2

2 2

θ = − −u v

u (11.74)

Substituting Equation 11.74 into Equation 11.73 and grouping terms, we obtain

1 22 2+ −( ) + =u v du uvdv 0. (11.75)

According to Equations 11.71 and 11.72, the trajectory orthogonal to Equation 11.75 must therefore be

2 1 2 2uvdu u v dv− + −( ) = 0. (11.76)

We now show that Equation 11.68 reduces to Equation 11.76. We differentiate Equation 11.68 to obtain

udu vdv dv+ − =csc 2ε 0. (11.77)

We eliminate the constant parameter csc 2ε by solving for csc 2ε in Equation 11.68 so that

csc .21

2

2 2

ε = + +u v

v (11.78)

We now substitute Equation 11.78 into Equation 11.77, group terms, and find that

2 1 2 2uvdu u v dv− + −( ) = 0. (11.79)

Comparing Equation 11.79 with Equation 11.76, we see that the equations are identical so that the trajectories are indeed orthogonal to each other. In Figure 11.5, we have plotted the family of circles for θ = 15° to 45° and for ε = 10° to 30°. We note that the circles intersect at m and that at this intersec-tion each circle has the same value of ρ and δ.

Each of the circles, Equations 11.69 and 11.70, has an interesting property. If v = 0, for example, then Equation 11.69 reduces to

u +( ) = ( )cot 2 csc 22θ θ 2

. (11.80)

Solving for u, we find that

u u= − =cot tan .θ θor (11.81)

Referring to Figure 11.4, these points occur at s and t and correspond to linearly polarized light in azimuth cot−1u and tan–1u, respectively; we also note from Equation 11.5 that because v = 0, we have δ = 0 so u = E0y/E0x. Similarly, if we set u = 0 in Equation 11.69, we find that v = ±1.

244 Polarized Light, Third Edition

Again referring to Equation 11.5, v = ±1 corresponds to E0y/E0x = 1 and δ = ±π/2; that is, right and left circularly polarized light. These points are plotted as P1 and P2 in Figure 11.4. Thus, the circle describes linearly polarized light along the u axis, circularly polarized light along the ν axis, and elliptically polarized light everywhere else in the u, v plane.

From these results we can now project the point m in the complex u, v plane onto a sphere, the Poincaré sphere. This is described in Section 11.3.

11.2.1 noTe on The deRivaTion of law of coSineS and law of SineS in SPheRical TRigonoMeTRy

In this section, we have used a number of formulas that originate from spherical trigonometry. The most important formulas are the law of cosines and the law of sines for spherical triangles and the formulas derived by setting one of the angles to 90° (a right angle). We derive these formulas by recall-ing the following vector identities:

A B C A C B A B C× ×( ) = ( ) − ( )· · (11.82)

( )A B C A C B B C A× × = ⋅( ) − ⋅( ) (11.83)

A B C D A C D B B C D A×( ) × ×( ) = ⋅ ×[ ] − ⋅ ×( )[ ]( (11.84)

A B C D A C B D A D B C×( ) ⋅ ×( ) = ⋅( ) ⋅( ) − ⋅( ) ⋅( ). (11.85)

u

v ε = 10°

ε = 20°

ε = 30°

θ = 15° θ = 30°θ = 45°

figuRe 11.5 Orthogonal circles of the polarization ellipse in the uv plane.

The Poincaré Sphere 245

The terms in brackets in Equation 11.84 are sometimes written as

A C D A B C⋅ ×( )[ ] = [ ], , (11.86)

B C D B C D⋅ ×( )[ ] = [ ], , . (11.87)

A spherical triangle is a three-sided figure drawn on the surface of a sphere as shown in Figure 11.6. The sides of a spherical triangle are required to be arcs of great circles. We recall that a great cir-cle is obtained by intersecting the sphere with a plane passing through its center. Two great circles always intersect at two distinct points, and their angle of intersection is defined to be the angle between their corresponding planes. This is equivalent to defining the angle to be equal to the plane angle between two lines tangent to the corresponding great circles at a point of intersection.

The magnitude of a side of a spherical triangle may be measured in two ways. Either we can take its arc length, or we can take the angle it subtends at the center of the sphere. These two methods give the same numerical result if the radius of the sphere is unity. We shall adopt the second of the two methods. In other words, if A, b, and c are the vertices of a spherical triangle with opposite sides a, b, and c, respectively, the numerical value of a will be taken to be the plane angle bOc, where O is the center of the sphere in Figure 11.6.

In the following derivations, we assume that the sphere has a radius R = 1 and the center of the sphere is at the origin. The unit vectors extending from the center to A, b, and c are α, β, and γ, respectively; the vertices are labeled in such a way that α, β, and γ are positively oriented.

We now refer to Figure 11.7. We introduce another set of unit vectors α′, β′, and γ′ extending from the origin and defined so that

αα ββ γγ γγ× = = ′sin c (11.88)

ββ γγ αα αα× = = ′sina (11.89)

a

bc

A

B C

α

β γ

O

figuRe 11.6 Fundamental angles and arcs on a sphere.

246 Polarized Light, Third Edition

γγ αα ββ ββ× = = ′sinb . (11.90)

In Figure 11.7 only α′ is shown. However, in Figure 11.8 all three unit vectors are shown. The unit vectors α′, β′, and γ′ determine a spherical triangle A′ b′ c′ called the polar triangle of Abc; this is shown in Figure 11.9. We now let the sides of the polar triangle be a′, b′, and c′. We see that b′ is a pole corresponding to the great circle joining A and c. Also, c′ is a pole corresponding to the

A’

A

B

C

a

b

c

Oγ

α

β

figuRe 11.7 The construction of a spherical triangle on the surface of a sphere.

a

ββ γ

α αb

c

A

BC

A

B C

γ

figuRe 11.8 Unit vectors within a unit sphere.

The Poincaré Sphere 247

great circle Ab. If these great circles are extended to intersect the side b′c′, we see that this side is composed of two overlapping segments, b′E and dc′, each of magnitude of 90°. Their common overlap has a magnitude A, so we see that

a A′ + = π (11.91)

b b′ + = π (11.92)

c c′ + = π. (11.93)

Equations 11.91 through 11.93 are useful for relating the angles of a spherical triangle to the sides of the corresponding polar triangle.

We now derive the law of cosines and law of sines for spherical trigonometry. In the identity Equation 11.85, we substitute α for A, β for B, α for C, and γ for D. Since α is a unit vector, we see that Equation 11.85 becomes

αα ββ αα γγ ββ γγ αα γγ ββ αα×( ) ⋅ ×( ) = ⋅ − ⋅( ) ⋅( ). (11.94)

In Figure 11.7, we have β · γ = cos α, α · β = cos c, and α · γ = cos b. Hence, the right-hand side of Equation 11.94 becomes

cos cos cosa b c− . (11.95)

From Equation 11.88 to Equation 11.90, we see that the left-hand side of Equation 11.94 becomes

sin sin sin sin bc b c′( ) ⋅ − ′( ) = − ′ ⋅ ′( )γγ ββ γγ ββ . (11.96)

a

bc

A

B C

A

B C

O

a

c b

DEA

figuRe 11.9 The polar triangle on a sphere.

248 Polarized Light, Third Edition

Now, just as γ · β is equal to cos a, we see from the polar triangle in Figures 11.8 and 11.9 that γ′ · β′ = cos a′. From Equation 11.91 cos a′ is cos(π − A), which equals − cos A. Thus, the left-hand side of Equation 11.94 equals

sin sin sinc b A. (11.97)

Equating the two sides we obtain the law of cosines for side a in the form

cos cos cos sin sin cosa b c b c A= + . (11.98)

We can, of course, imagine that Figure 11.7 is rotated so that the roles previously played by a, b, and c are now replaced by b, c, and a, respectively, so we can write law of cosines for sides b and c as

cos cos cos sin sin cosb c a c a b= + (11.99)

cos cos cos sin sin cosc a b a b c= + . (11.100)

Three other versions of the cosine law are obtained by applying the law of cosines to the polar tri-angle by merely changing a to a′, b to b’, and so on, according to Equation 11.91 through 11.93, so that we have the law of cosines for angles A, b, and c in the form

cos A = − cos b cos c + sin b sin c cos a (11.101)

cos b = − cos c cos A + sin c sin A cos b (11.102)

cos c = − cos A cos b + sin A sin b cos c. (11.103)

We now turn to the law of sines. Here, we make use of the identity in Equation 11.84. Replacing A by α, B by β, C by α, and D by γ, Equation 11.84 becomes

αα ββ αα γγ αα αα γγ ββ ββ αα γγ αα×( ) × ×( ) = ⋅ ×( )[ ] − ⋅ ×( )[ ] . (11.104)

From the relations given by Equations 11.88 and 11.90, the left-hand side of Equation 11.104 becomes

(sin ) ( sin ) sin sin ( )

sin si

c b b c

b

′ × − ′ = − ′ × ′

= −

γγ ββ γγ ββ

nn ( sin )

.

c a

b c A

− ′

= ( )

αα

ααsin sin sin

(11.105)

In this manner we obtain

sin sin sinb c A( ) = [ ]αα αα ββ γγ αα, , (11.106)

sin sin sinc a b( ) = [ ]ββ ββ γγ αα ββ, , (11.107)

sin sin sina b c( ) = [ ]γγ γγ αα ββ γγ, , . (11.108)

The Poincaré Sphere 249

We see from either Equation 11.86 or Equation 11.87 that [α, β, γ] = [β, γ, α] = [γ, α, β] and, hence, the left-hand sides of Equations 11.106 through 11.108 are all equal, and we can write

sin b sin c sin A = sin c sin a sin b, (11.109)

which yields

sinsin

sinsin

.b

b

a

A= (11.110)

Similarly, we obtain from Equation 11.106 to Equation 11.108 that

sinsin

sinsin

a

A

c

c= (11.111)

so that we can write the law of sines as

sinsin

sinsin

sinsin

,a

A

b

b

c

c= = (11.112)

From the law of cosines and the law of sines, we can derive the equations for a right spherical tri-angle. In order to derive these equations, we assume that the angle c is the right angle. The spherical right triangle is shown in Figure 11.10.

In Equation 11.100, we set c = 90°, and we have

cos cos cosc a b= . (11.113)

Similarly, from the law of sines Equation 11.112 we find that

sin a = sin c sin A (11.114)

b

c

a

A

B

C

figuRe 11.10 Arc length and angle relations for a right spherical triangle.

250 Polarized Light, Third Edition

sin b = sin c sin b. (11.115)

From the law of cosines for angles, setting c = 90°, we have

cos A = cos a sin b (11.116)

cos b = cos b sin A (11.117)

cos A cos b = sin A sin b cos c. (11.118)

We note that Equation 11.118 can also be derived by multiplying Equation 11.116 by Equation 11.117 and using Equation 11.113. Next, we divide Equation 11.114 by cos a so that

sincos

tan sinsincos

tan [cos sin ],

aa

a cAa

c b A

= =

=

= tan cosc b,

(11.119)

where we have used Equations 11.113 and 11.117. We see that we have found six relations. Further analysis shows that there are four more relations for a right spherical triangle, so that there are 10 relations altogether. We therefore find that, for a right spherical triangle, we have the following relations:

cos c = cos a cos b (11.120)

sin a = sin c sin A (11.121)

sin b = sin c sin b (11.122)

tan a = sin b tan A (11.123)

tan b = sin a tan b (11.124)

tan b = tan c cos A (11.125)

tan a = tan c cos b (11.126)

cos A = cos a sin b (11.127)

cos b = cos b sin A (11.128)

cos c = cot A cot b. (11.129)

These relations are important because they appear consistently in the study of polarized light.

11.3 PRoJeCTioN of The ComPleX PlaNe oNTo a SPheRe

We now consider the projection of the point m in the complex plane onto the surface of a sphere. This projection is shown in Figure 11.11. Specifically, the point m in the u, v plane is projected as point M on the sphere. A sphere of unit diameter (the radius r is equal to 1/2) is constructed such that

The Poincaré Sphere 251

point O is tangential to the u, v plane, and points in the plane on the u axis project onto the surface of the sphere by joining them to O′. The line OO′ is the diameter, and the points p1 and p2 project onto the poles P1 and P2 of the sphere. Then, by the principles of stereographic projection, the fam-ily of circles given by Equations 11.67 and 11.68 project into meridians of longitude and parallels of latitude, respectively. The point O′ (see Figure 11.11) is called the antipode of a sphere. If all the projected lines come from this point, the projection is called stereographic. The vector Om projects into the arc OM of length 2v, and the spherical angle MOO′ is δ. Thus, any point M on the sphere will, as does m on the plane, represent the state of polarization of light.

In order to find the relationship between the coordinates of M and the parameters of the light, that is, the ellipticity, azimuth, sense, and phase difference, it is necessary to determine the coordinates in terms of the angles θ and ε. We must, therefore, transform the coordinates of m on the u, v plane to M on the sphere. If the center of the sphere in Figure 11.11 is taken as the origin of the coordinate system, then the coordinates of m and O′ referenced to this origin must be found. The coordinates of m in terms of x, y, z are seen from the figure to be

x = − 12

(11.130)

OO

MR

δ

Q

T2θ

2ε

2ν

u

v

p2

p1

P2

P1m

x

yz

figuRe 11.11 Stereographic projection of the complex plane on a sphere. Elliptically polarized light is rep-resented by the points m on the plane and M on the sphere. The vector Om projects into the arc OM of length 2v; the angle δ projects into the spherical angle MOT . The latitude and longitude of M are 2ε and 2θ, respectively.

252 Polarized Light, Third Edition

y = tan v cos δ (11.131)

z = tan v sin δ. (11.132)

The coordinates of the point O′ are

′ =x12

(11.133)

y′ = 0 (11.134)

z′ = 0. (11.135)

The point m is projected along the straight line mMO′ onto M. That is, we must determine the coor-dinates of the straight line mMO′ and the point M on the sphere. The equation of the sphere is

x y z2 2 2

212

+ + =

. (11.136)

In order to find the equation of the straight line mMO′, we digress briefly and determine the gen-eral equation of a straight line in three-dimensional space. This is most easily done using vector analysis.

Consider Figure 11.12. A straight line is drawn through the point R0 and parallel to a constant vec-tor A. If the point R is also on the line, then the vector R − R0 is parallel to A. This is expressed by

(R − R0) × A = 0, (11.137)

which is the equation of a straight line. The fact that R − R0 is parallel to A may also be expressed by the vector equation

R − R0 = At, (11.138)

θ

O

RR0

R-R0

R1

A

R 1-R0

figuRe 11.12 Vector equation of a straight line in three-dimensional space.

The Poincaré Sphere 253

where t is a scalar. Thus, the equation of a straight line in parametric form is

R R A= + − < <0 t t∞ ∞. (11.139)

We can deduce the Cartesian form of Equation 11.139 by setting

R = xi + yj + zk (11.140)

R0 = x0i + y0j + z0k (11.141)

A = ai + bj + ck, (11.142)

where i, j, k are the Cartesian unit vectors. Thus, we have

x = x0 + at (11.143)

y = y0 + bt (11.144)

z = z0 + ct. (11.145)

Eliminating t in these equations, we find

x x

a

y y

b

z z

c

− = − = −0 0 0 . (11.146)

We now return to our original problem. We have

A = (l)i − (tan v cos δ)j − (tan v sin δ)k. (11.147)

Similarly, R0 is

R i j k0 = − + +12

(tan cos ) (tan sin ) .v vδ δ (11.148)

Thus, using Equations 11.140, 11.141, 11.142, 11.147, and 11.148, and from Equation 11.146 we find the relation

x y v

v

z v

v

+= −

−= −

−

12

1tan cos

tan costan sin

tan siδ

δδ

nn,

δ (11.149)

which is the equation of the line O’m.The coordinates (x, y, z) of M, the point of intersection of O’m and the sphere, are obtained by

simultaneously solving Equations 11.136 and 11.149. To do this, let us first solve for x. We write, from Equation 11.149,

z v x= −12

1 2(tan sin )( )δ (11.150)

y v x= −12

1 2(tan cos )( ).δ (11.151)

254 Polarized Light, Third Edition

We now substitute Equations 11.150 and 11.151 into Equation 11.136 and find that

4 tan 1 4 tan tan 12 2 2 2v x v x v+( ) − ( ) + −( ) = 0. (11.152)

The solutions of this quadratic equation are

x v= 12

212

cos , . (11.153)

In a similar manner the solutions for y and z are found to be

y v= 12

2 0sin cos ,δ (11.154)

z v= 12

2 0sin sin , .δ (11.155)

The first set of x, y, z coordinates of these solutions refers to the intersection of the straight line at M on the surface of the sphere. Thus, the coordinates of M are

M x y z v v v( , , ) cos , sin cos , sin sin=

12

212

212

2δ δ . (11.156)

The second set of coordinates of the solutions in Equations 11.153 through 11.155 describes the intersection of the line at the origin O′; that is, the antipode of the sphere, that is,

′ =

O x y z( , , ) , , .

12

0 0 (11.157)

We note that for v = 0, Equation 11.156 reduces to Equation 11.157. Using Equations 11.36 through 11.38, we can express the coordinates for M as

x v= =12

212

2 2cos cos cosε θ (11.158)

y v= =12

212

2 2sin cos cos sinδ ε θ (11.159)

z v= =12

212

2sin sin sin .δ ε (11.160)

Equations 11.158 through 11.160 have a familiar appearance. We recall that the orthogonal field components Ex and Ey are E0x exp(iδx) and E0y exp(iδy) from Equation 11.3 to Equation 11.4, respec-tively, where the propagator has been suppressed. The Stokes parameters using Ex and Ey are then defined in the usual way; that is,

S E E E Ex x y y0 = +* * (11.161)

The Poincaré Sphere 255

S E E E Ex x y y1 = −* * (11.162)

S E E E Ex y y x2 = +* * (11.163)

S i E E E Ex y y x3 = −( ).* * (11.164)

Substituting Equations 11.3 and 11.4 into these equations gives

S E Ex y0 02

02= + (11.165)

S E Ex y1 02

02= − (11.166)

S E Ex y2 0 02= cosδ (11.167)

S E Ex y3 0 02= sin ,δ (11.168)

where we have written δ = δy – δx. Recall from Equation 11.10 in Section 11.2 that

tan ( ).vE

Evy

x

= ≤ ≤0

0

0 90° (11.169)

We now set

S A b c0 = + =2 2 2 , (11.170)

where A = E0x and b = E0y and we construct the right triangle in Figure 11.13. We see immediately that the equations for the Stokes vectors can be rewritten in the form

S0 = C2 (11.171)

S1 = C2 cos 2v (11.172)

S2 = C2 sin 2v cos δ (11.173)

S3 = C2 sin 2v sinδ. (11.174)

Finally, we set c2 = 1/2 in these last equations, so that we have

S012

= (11.175)

S v112

2= cos (11.176)

S v212

2= sin cosδ (11.177)

256 Polarized Light, Third Edition

S v312

2= sin sin .δ (11.178)

We now compare Equations 11.176 through 11.178 with the coordinates of M in Equations 11.158 through 11.160 and we see that the equations for S1, S2, and S3, and x, y, and z are identical. Thus, the coordinates of the point M on the Poincaré sphere correspond exactly to the Stokes parameters S1, S2, and S3 of the optical beam and S0 corresponds to the radius of the sphere.

On the Poincaré sphere we see that for a unit intensity we can write the Stokes parameters as, using Equations 11.171 through 11.174 and Equations 11.158 through 11.160,

S0 = 1 (11.179)

S1 = cos 2ε cos 2θ (11.180)

S2 = cos 2ε sin 2θ (11.181)

S3 = sin 2ε, (11.182)

where ε and θ are the ellipticity and azimuth (rotation) of the polarized beam.In Figure 11.14 we have drawn the Poincaré sphere in terms of the Stokes parameters given in

Equations 11.179 through 11.182. The point M on the surface of the Poincaré sphere is described in terms of its latitude (2ε), where –π/2 ≤ 2 ε ≤ π/2, and its longitude (2θ), where –π ≤ 2θ ≤ π. We see immediately that for 2ε = 0, which corresponds to the equator on the Poincaré sphere, the equations for the Stokes vector parameters reduce to

S0 = 1 (11.183)

S1 = cos2θ (11.184)

S2 = sin 2θ (11.185)

S3 = 0. (11.186)

CB = C sin ν

A = C cos ν

ν

figuRe 11.13 Construction of a right triangle.

The Poincaré Sphere 257

These equations are the Stokes parameters for linearly polarized light oriented at an angle θ; for 2θ = 0, they reduce to the Stokes vector for linear horizontally polarized light, for 2θ = π/2 we find linear +45° polarized light, and for 2θ = π we have linear vertically polarized light. As we move counterclockwise on the equator, we pass through these different states of linearly polarized light.

If we now set 2θ = 0 so we move along the prime meridian (longitude), then Equations 11.179 through 11.182 reduce to

S0 = 1 (11.187)

S1 = cos 2ε (11.188)

S2 = 0 (11.189)

S3 = sin 2ε. (11.190)

These equations for the Stokes parameters can be recognized as elliptically polarized light for the polarization ellipse in its standard form. We see that if we start from the equator (2ε = 0) and move up in latitude then at the pole we have 2ε = π/2, and the equations result in parameters for right cir-cularly polarized light. Similarly, moving down from the equator at the lower pole 2ε = –π/2, and we have left circularly polarized light.

We can now summarize the major properties of the Poincaré sphere:

1. The coordinates of a point M on the Poincaré sphere are represented by latitude angle 2ε and longitude angle 2θ. A polarization state is described by P(2ε, 2θ).

2. The latitude 2ε = 0° corresponds to the equator and for this angle the Stokes vector is seen to reduce to the Stokes vector for linearly polarized light. Thus, linearly polarized light is always restricted to the equator. The angles 2θ = 0°, 90°, 180°, and 270° correspond

O O

P1

P2

M

R

δ

Q

T2θ

2ε

2ν

figuRe 11.14 The Poincaré sphere showing the representation of an elliptically polarized vibration by the point M. From the spherical triangle ΟΜΤ, the parameters of the vibration can be found. Points on the equa-tor OO′ represent linearly polarized light. The sense of rotation of the ellipse is left and right in the lower and upper hemispheres, respectively. The poles P1 and P2 represent right and left circularly polarized light, respectively.

258 Polarized Light, Third Edition

to polarization states linear horizontal, linear  +45°, linear vertical, and linear –45°, respectively.

3. The longitude 2θ = 0° corresponds to the prime meridian and for this angle the Stokes vector is seen to reduce to the Stokes vector for elliptically polarized light for a nonrotated polarization ellipse. We see that for 2ε = 0° we have linear horizontally polarized light, and as we move up along the prime meridian we pass from right elliptically polarized light to right circularly polarized light at 2ε = 90° (the north pole). Similarly, moving down the meridian from the equator we pass from left elliptically polarized light to left circularly polarized light at 2ε = –90° at the south pole.

4. The points along a given parallel represent ellipses of the same form (ellipticities) but dif-ferent orientations (azimuths).

5. The points on a given meridian represent vibrations of the same orientation (azimuth) whose eccentricity varies from 0 on the equator to ±1 at the north and south poles.

The real power of the Poincaré sphere is that it enables us to determine the state of polarization of an optical beam after it has propagated through a polarizing element or several polarizing elements without carrying out the calculations. In the following section, we apply the sphere to the problem of propagation of a polarized beam through (1) a polarizer, (2) a retarder, (3) a rotator, and (4) an elliptical polarizer consisting of a linear polarizer and a retarder.

11.4 aPPliCaTioNS of The PoiNCaRÉ SPheRe

In Section 11.1, we pointed out that the Poincaré sphere was introduced by Poincaré in order to treat the problem of determining the polarization state of an optical beam after it had propagated through a number of polarizing elements. Simply put, given the Stokes parameters of the input beam, the problem is to determine the Stokes parameters of the output beam after it has propagated through a polarizing element or several polarizing elements. In this section, we apply the Poincaré sphere to the problem of describing the effects of polarizing elements on an incident polarized beam. In order to understand this behavior, we first consider the problem using the Mueller matrix formalism, and then discuss the results in terms of the Poincaré sphere.

The Mueller matrix for an ideal linear polarizer rotated through an angle β is

MP = 12

1 2 2 0

2 2 2 2 0

2

2

cos sin

cos cos sin cos

sin

β ββ β β βββ β β βsin cos sin

.2 2 2 0

0 0 0 0

2

(11.191)

In the previous section, we saw that the Stokes vector for a beam of unit intensity written in terms of its ellipticity ε and its azimuth (orientation) θ is given by

S =

1

2 2

2 2

2

cos cos

cos sin

sin

.ε θε θ

ε

(11.192)

The incident beam is now represented by Equation 11.192 and is plotted as a point Ρ(2ε, 2θ) on the Poincaré sphere. The polarized beam now propagates through the rotated polarizer, and the

The Poincaré Sphere 259

Stokes vector of the emerging beam is found by multiplying Equation 11.192 by Equation 11.191 to obtain

′ = + −[ ]

S12

1 2 2

1

2

2

0

cos cos ( )cos

sinε β θ

ββ

. (11.193)

The Stokes vector of the emerging beam, aside from the intensity factor in Equation 11.193, can also be described as the general expression in terms of ellipticity and orientation as

′ =′ ′′ ′

′

S

1

2 2

2 2

2

cos cos

cos sin

sin

ε θε θ

ε

. (11.194)

The polarization state is described only by the parameters within the column matrix, and Equation 11.193 shows that regardless of the polarization state of the incident beam the polarization state of the emerging beam is a function only of β, the orientation angle of the linear polarizer. In order for the general expression for the emerging polarization state, given by Equation 11.194, to maintain consistency of form with the expression we have found in Equation 11.193 for the case of a rotated linear polarizer, we must have 2ε′ = 0 and θ′ = β. Thus, the ellipticity is zero and the point Ρ′(0, 2β) is always on the equator at the longitude β.

It is also possible to use the Poincaré sphere to obtain the intensity factor in Equation 11.193. For the Poincaré sphere, 2ε corresponds to the parallels and 2θ corresponds to the longitudes. Within the factor in Equation 11.193 we see that we have the factor (cos 2ε cos (2β – 2θ)). Recall that Equation 11.38 is

cos 2 cos 2 cos 2v = ε θ. (11.195)

Thus, the factor (cos 2ε cos (2β – 2θ)) is obtained by constructing a right spherical triangle on the Poincaré sphere. In order to determine the magnitude of the arc on a great circle (2v), we need only measure the length of the angle 2ε on the meridian (longitude) followed by measuring the length (2β − 2θ) on the equator (latitude). The length (2v) of the arc of the great circle is then measured from the initial point of the meridian to the final point along the equator. This factor is then added to 1 and the final result is divided by 2, as required by Equation 11.193. The Poincaré sphere, therefore, can also be used to determine the final intensity as well as the change in the polarization state.

The next case of interest is a retarder. The Mueller matrix for a linear retarder with retardance ϕ and with fast axis at 0º is given by

MR =

−

1 0 0 0

0 1 0 0

0 0

0 0

cos sin

sin cos

φ φφ φ

. (11.196)

260 Polarized Light, Third Edition

In order to determine the point P′ on the Poincaré sphere, we consider first the case where the inci-dent beam is linearly polarized. For linearly polarized light with its azimuth plane at an angle α, the Stokes vector is

S =

1

2

2

0

cos

sin.

αα

(11.197)

In terms of 2ε and 2θ (latitude and longitude), recall that we can express the Stokes vector as

S =

1

2 2

2

2

cos cos

cos sin

sin

.ε θε θ

ε

(11.198)

Equating the terms in Equations 11.197 and 11.198, we have

cos 2 cos 2 cos 2ε θ α= (11.199)

cos 2ε sin 2θ = sin 2α (11.200)

sin 2ε = 0. (11.201)

We immediately see from these equations that 2ε = 0 and 2θ = 2α and we can write the Stokes vec-tor in terms of θ as

S =

1

2

2

0

cos

sin.

θθ

(11.202)

The Stokes vector S′ of the emerging beam is found by multiplying Equation 11.202 by Equation 11.196 whereupon we find that

′ =

S

1

2

2

2

cos

cos sin

sin sin

.θ

φ θφ θ

(11.203)

The corresponding Stokes vector in terms of 2ε′ and 2θ′ is

′ =′ ′′ ′

′

S

1

2 2

2 2

2

cos cos

cos sin

sin

ε θε θ

ε

. (11.204)

The Poincaré Sphere 261

Equating the last elements of the Stokes vectors in Equations 11.203 and 11.204, and dividing the third by the second elements gives us

sin 2 sin 2 sinε θ φ′ = (11.205)

tan 2 tan 2θ θ φ′ = cos . (11.206)

Equations 11.205 and 11.206 can be expressed in terms of the right spherical triangle shown in Figure 11.15. The figure is constructed using the equations for a right spherical triangle given at the end of Section 11.2 (compare Figure 11.15 through Figure 11.10). Figure 11.15 shows how the retarder moves the initial point Ρ(2ε, 2θ) to Ρ′(2ε′, 2θ′) on the Poincaré sphere. To carry out the operations equivalent to the right spherical triangle, the following steps are performed:

1. Determine the initial point Ρ(2ε = 0, 2θ) on the equator and label it A. 2. Draw an angle at A from the equator of magnitude ϕ, the phase shift of the retarder. 3. Measure the arc length 2θ along the equator from A. Then draw this arc length from A to

b. The end of this arc corresponds to the point P′(2ε′, 2θ′). 4. The meridian 2ε′ is drawn down to the equator; this arc length corresponds to the elliptic-

ity angle 2ε′. The intersection of the meridian with the equator is the orientation angle 2θ′. Three cases are of special interest: linear horizontally polarized light, linear +45° polar-ized light, and linear vertically polarized light. We discuss each of these cases and their interaction with a retarder as they are described on the Poincaré sphere.

a. Linear horizontally polarized light. For this case 2α = 2θ = 0°. We see from Equations 11.205 and 11.206 that 2ε′ and 2θ′ are zero. Thus, the linear horizontally polarized light is unaffected by the retarder, and P is identical to P′.

b. Linear +45° light. Here, 2α = 2θ = π/2, and from Equation 11.205 to Equation 11.206 we have

sin 2 sinε φ′ = (11.207)

tan 2 ∞θ′ = . (11.208)

Thus, the arc length (the longitude or the meridian) is 2ε′ = ϕ and 2θ′ = π/2. We see that as ϕ increases, 2ε′ increases, so that when 2ε′ = π/2, which corresponds to right circularly polarized light, the arc length 2ε′ extends from the equator to the pole.

2θ

A

B

C

2θ

2ε

figuRe 11.15 Right spherical triangle for a retarder.

262 Polarized Light, Third Edition

c. Linear vertically polarized light. For this final case, 2α = 2θ = π. We see from Equation 11.205 that 2ε′ = 0; that is, P′ is on the equator. However, tan 2θ = −∞, so 2θ = –π. Thus, P′ is on the equator but diametrically opposite to P on the Poincaré sphere.

The Stokes vector confirms this behavior for these three cases, since we have from Equation 11.203 that the Stokes vector reduces to linear horizontally polarized light, linear +45° light, and linear vertically polarized light for 2θ = 0, π/2, and π, respectively.

We now consider the case where the incident light is elliptically polarized. In order to understand the behavior of the elliptically polarized light and the effect of a retarder on its polarization state in terms of the Poincaré sphere, we write the Stokes vector for the incident beam as

S =

S

S

S

S

0

1

2

3

. (11.209)

Multiplying Equation 11.209 by Equation 11.196 then gives

′ =+

− +

S

S

S

S S

S S

0

1

2 3

2 3

cos sin

sin cos

φ φφ φ

. (11.210)

The third and fourth elements, ′S2 , and ′S3 , describe rotation through the angle ϕ. To see this behavior more clearly, let us consider Equation 11.210 for a quarter-wave retarder (ϕ = π/2) and a half-wave retarder (ϕ = π). For these cases Equation 11.210 reduces to

′ =

−

=S

S

S

S

S

0

1

3

2

2, φ π

(11.211)

and

′ =−−

=S

S

S

S

S

0

1

2

3

, .φ π (11.212)

Let us now consider the Poincaré sphere in which we show the axes labeled as S1, S2, and S3. We see that, according to Equations 11.211 and 11.212, S1 remains invariant, but S2 → S3 → –S2 and S 3 → –S2 → –S3. As can be seen from Figure 11.16 for the Poincaré sphere, this corresponds to rotating the sphere around the S1 axis sequentially through π/2 and then again through another π/2 for a total rotation of π. Thus, the effect of the retarder can be expressed merely by rotating the Poincaré sphere around the S1 axis; the magnitude of the rotation is equal to the phase shift ϕ. It is this remarkably simple property of the Poincaré sphere that has led to its utility.

The Poincaré Sphere 263

In terms of the equation for 2ε′ and 2θ′, we can obtain these values by determining the Stokes vector S′ of the emerging beam by multiplying Equation 11.192 by Equation 11.196. The result is easily seen to be

′ =+

−

S

1

2 2

2 2 2

cos cos

cos cos sin sin sin

sin

ε θφ ε θ φ εφccos sin cos sin

.

2 2 2ε θ φ ε+

(11.213)

We immediately find by equating the elements of Equation 11.213 to Equation 11.204 that

tancos cos sin sin sin

cos cos2

2 2 22 2

′ = +θ φ ε θ φ εε θ

(11.214)

sin 2 sin cos 2 sin 2 cos sin 2ε φ ε θ φ ε′ = − + . (11.215)

In the Stokes vector Equation 11.213, the element S1 is recognized as the relation for a right spherical triangle as in Equation 11.120. The elements S2 and S3 are the relations for an oblique spherical triangle if the angle c shown in Figure 11.10 is an oblique angle. We can use the Poincaré sphere to obtain the orientation angle θ′ and the ellipticity angle χ′. For example, if we set a = 90° −2ε, b = ϕ, c = 90° – 2ε′, and c = 90° – 2θ in Equation 11.100 we obtain the ellipticity angle Equation 11.215 of the emerging beam; a similar set of angles leads to the orientation angle Equation 11.214.

2β

2

2

figuRe 11.16 Right spherical triangle for a linear polarizer-retarder combination.

264 Polarized Light, Third Edition

We now turn to the problem of describing the interaction of an elliptically polarized beam with a rotator using the Poincaré sphere. The Mueller matrix for a rotator is

Mrot =−

1 0 0 0

0 2 2 0

0 2 2 0

0 0 0 1

cos sin

sin cos

β ββ β

, (11.216)

where β is the angle of rotation. The Stokes vector of the emerging beam is found by multiplying Equation 11.209 by Equation 11.216 and we obtain

′ =+

− +

S

1

2 2

2 21 2

1 2

3

S S

S S

S

cos sin

sin cos

β ββ β

, (11.217)

or, to work in terms of ε and θ, multiplying Equation 11.192 by Equation 11.216 to obtain

′ =−−

S

1

2 2

2 2

2

cos cos ( )

cos sin ( )

sin

ε θ βε θ β

ε

. (11.218)

We see that from Equation 11.217 we have a rotation around the S3 axis, that is, starting from 2β = 0° and moving to 270° in increments of 90°, S1 → S2 → −S1, → −S2 and, similarly, S2 → − S1 → – S2 → S1. Thus, rotating the Poincaré sphere around the S3 axis by β transforms P(2ε, 2θ) to P′(2ε, 2(θ − β)); the ellipticity angle ε remains unchanged and only the orientation of the polariza-tion ellipse is changed.

To summarize, the rotation around the S1 axis describes the change in phase (i.e., propagation through a birefringent medium) and the rotation around the S3 axis describes the change in azimuth (i.e., propagation through an optically active medium).

The final problem we consider is the propagation of a polarized beam through a linear polarizer oriented at an angle θ to the x axis followed by a retarder with its fast axis along the x axis. For the linear polarizer, recall that we had from Equation 11.193 to Equation 11.194 that

′ = + −

S12

1 2 2

1

2

2

0

[ cos cos ( )]cos

sinε β θ

ββ

(11.219)

for a rotated polarizer, which can also be described in terms of ellipticity and orientation as

′ =′ ′′ ′

′

S

1

2 2

2 2

2

cos cos

cos sin

sin

ε θε θ

ε

. (11.220)

The Poincaré Sphere 265

We see that for these Stokes vectors to be equivalent that 2ε′ is zero and 2θ′ = 2β. The point P(2ε, 2θ), the incident beam, is moved along the equator through an angle β to the point P′(2ε′, 2θ′). Equivalently, we need only rotate the Poincaré sphere around its polar axis. Next, the beam propa-gates through the retarder. Starting with the Stokes vector of the polarizer Equation 11.219 and multiplying it by the Mueller matrix for a retarder Equation 11.196, we see that the Stokes vector is

′ =

S

1

2

2

2

cos

sin cos

sin sin

.β

β φβ φ

(11.221)

We now equate the elements in Equation 11.221 with Equation 11.220 to obtain

cos 2 cos 2 cos 2ε θ β′ ′ = (11.222)

cos 2 sin 2 sin 2 cosε θ β φ′ ′ = (11.223)

sin 2 sin 2 sinε β φ′ = . (11.224)

Alternatively, we can equate the elements to the Stokes vector representation of elliptically polar-ized light using χ′ and ψ′ notation for Equation 11.220; that is,

′ =′ ′′ ′

′

S

1

2 2

2 2

2

cos cos

cos sin

sin

χ ψχ ψ

χ

. (11.225)

Equating elements of Equation 11.221 with Equation 11.225 yields

cos 2 cos 2 cos 2χ ψ β′ ′ = (11.226)

cos 2 sin 2 sin 2 cosχ ψ β φ′ ′ = (11.227)

sin 2 sin 2 sinχ β φ′ = . (11.228)

We now divide Equation 11.228 by Equation 11.227 to obtain

tan 2 sin 2 tanχ ψ φ′ = ′ . (11.229)

Similarly, we divide Equation 11.227 by Equation 11.226 and find that

tan 2 tan 2 cosψ β φ′ = . (11.230)

We now collect Equations 11.226, 11.229, and 11.230 and write

cos 2 cos 2 cos 2χ ψ β′ ′ = (11.231)

266 Polarized Light, Third Edition

tan 2 sin 2 tanχ ψ φ′ = (11.232)

tan 2 tan 2 cosψ β φ′ = . (11.233)

Not surprisingly, Equations 11.231 through 11.233 correspond to Equations 11.120, 11.123, and 11.125, respectively. These equations are satisfied by the right spherical triangle in Figure 11.16. The arc 2β and the angle ϕ determine the magnitudes 2χ′ and 2ψ′. We see that all that is required to deter-mine these latter two angles is to rotate the sphere through an angle ϕ around the S1 axis and then to measure the arc length 2β. We note that the magnitude of the angle ϕ is then confirmed by the intersection of the arcs 2β and 2ψ′.

A number of further applications of the Poincaré sphere have been given in the optical literature. A very good introduction to some of the simplest aspects of the Poincaré sphere and certainly one of the clearest descriptions is found in Shurcliff [3]. An excellent and very detailed description, as well as a number of applications, has been given by Jerrard [4]; much of the material presented in this chapter is based on Jerrard’s excellent paper. Further applications have been considered by Ramaseshan and Ramachandran, who have also described the Poincaré sphere and its application in a very long and extensive review article entitled “Crystal Optics” in the Handbuch der Physik [5]. This is not an easy article to read, however, and requires much time and study to digest fully. Finally, E. A. West and colleagues have given an excellent discussion of the application of the Poincaré sphere to the design of a polarimeter to measure solar vector magnetic fields [6].

Remarkably, even though the Poincaré sphere was introduced a century ago, papers on the subject continue to appear. A paper of interest on a planar graphic representation of the state of polarization, a planar Poincaré chart, is given by Tedjojuwono, Hunter, and Ocheltree [7]. Finally, a very good review has been published by Boerner, Yan, and Xi [8] on polarized light, and includes other projections analogous to the stereographic projection (the mercator, the azimuthal, etc.).

RefeReNCeS

1. Poincaré, H., Chapter 12 in Théorie Mathematique de la Lumière, Vol. 2, Paris: Gauthiers-Villars, 1892. 2. Ramachandran, G. N., and S. Ramaseshan, Magneto-optic rotation in Birefringent media—application of

the Poincaré sphere, J. Opt. Soc. Am. 42 (1952): 49–52. 3. Shurcliff, W. Α., Polarized Light, Cambridge, MA: Harvard University Press, 1962. 4. Jerrard, H. G., Transmission of light through Birefringent and optically active media: The Poincaré

sphere, J. Opt. Soc. Am. 44 (1954): 634. 5. Ramachandran, G. N., and S. Ramaseshan, Crystal optics, in Encyclopedia of Physics, Vol. XXV/1,

Edited by S. Flügge, Berlin: Springer-Verlag, 1961. 6. West, Ε. Α., E. J. Reichman, M. J. Hagyard, and G. Α.Gary, Design of the polarimeter for the Solar

Activity Measurements (SAMEX vector magnetograph), Opt. Eng. 29 (1989): 131. 7. Tedjojuwono, K. K., W. W. Hunter, Jr., and S. L. Ocheltree, Planar Poincare chart: A planar graphic

representation of the state of light polarization, Appl. Opt. 28 (1989): 2614–22. 8. Boerner, W. M., W.-L. Yan, and A.-Q. Xi, Basic equations of radar polarimetry and its solutions: The

characteristic radar-polarization states for the coherent and partially polarized cases, Proc. SPIE 1317 (1990): 16–80.

267

12 Fresnel–Arago Interference Laws

12.1 iNTRoduCTioN

In his landmark paper of 1852 [1], Sir George Stokes developed the tools to put the interference laws of Fresnel and Arago on a mathematical footing. Stokes defined quantities A, B, C, and D that Chandrasekhar [2] almost 100 years later labeled I, Q, U, and V and called the Stokes parameters. Thus, the Mueller–Stokes formalism that allows us to deal with any type of polarized or unpolar-ized light traces its origins to the experiments of Fresnel and Arago [3] around 1817.

Fresnel and Arago knew of Young’s experiments, and Fresnel wanted to understand the process of interference, develop his own explanations, and perform his own experiments. The result of this was the Fresnel–Arago interference laws. The set of four laws that are generally stated are as follows:

1. Two beams of light that are linearly polarized in the same plane will interfere under the same conditions that interference is produced with unpolarized light, that is, two beams of polarized or unpolarized light must be from the same source.

2. Two beams of light that are linearly polarized in mutually perpendicular planes will not interfere under any circumstances.

3. Two beams of light obtained from unpolarized light and linearly polarized in perpendicu-lar planes will not interfere when they are brought into the same plane.

4. Two beams of light obtained from the same linearly polarized source, but perpendicularly polarized and then brought into the same plane, will interfere.

With an understanding of coherence, laws 3 and 4 become corollaries of law 2.The experiments of Fresnel and Arago are summarized in English in the out-of-print first and

second editions of a textbook by Wood [4]. In this chapter, we will review the Stokes vector and unpolarized light, summarize Young’s double slit experiment in order to illustrate interference with unpolarized light, and then describe experiments that demonstrate the laws of Fresnel and Arago.

12.2 STokeS VeCToR aNd uNPolaRiZed lighT

The Stokes parameters are defined as

S E E E E

S E E E E

S E E E E

S

x x y y

x x y y

x y y x

0

1

2

= +

= −

= +

* *

* *

* *

33 = −i E E i E Ex y y x* * ,

(12.1)

268 Polarized Light, Third Edition

where E represents the field amplitude of the source, and ⟨…⟩ denotes a time average. This is the same definition as in Chapter 5, but for our purpose in this chapter we ignore constants that are needed to put the Stokes vector in units of flux. For unpolarized light,

S E E E E

S

S

S

x x y y0

1

2

3

0

0

0

= +

=

=

=

* *

,

(12.2)

so we see that from the expression for S1 in Equations 12.1 and 12.2,

E E E Ex x y y* * ,= (12.3)

and from the expressions for S2 and S3

E E E Ex y y x* * .= = 0 (12.4)

The time average of the product of orthogonal components, having no constant phase relationship, is zero.

12.3 youNg’S double SliT eXPeRimeNT

The Young’s double slit experiment is illustrated in Figure 12.1. A source, represented here by an aperture in the left-most plane, provides light for the two slits of the middle plane. The resulting irradi-ance on the plane at the right-hand side of the figure at screen Σ is an interference pattern given by

I I= 420

2cosφ

, (12.5)

where ϕ is the net path length phase difference from the slits to a point on the screen and I0 is the irradiance from a single slit. The traditional derivation of this expression is found in many introduc-tory optics texts [5,6].

We approach Young’s experiment here using the language of polarized light [7]. The Stokes vec-tor of the unpolarized source is

S =

=

EE E Ex x

* *

1

0

0

0

1

1

0

0

12

+−

12

1

1

0

0

E Ey y* , (12.6)

where E E Ex y= = / 2 , and where we have resolved the unpolarized light into equal horizontal and vertical components. The Stokes vectors at slits 1 and 2 are

S112 1 1

12 1 1

1

1

0

0

1

1

0

0

=

+−

E E E Ex x y y* *

(12.7)

Fresnel–Arago Interference Laws 269

and

S212 2 2

12 2 2

1

1

0

0

1

1

0

0

=

+−

E E E Ex x y y* *

(12.8)

where we assume that the original energy is equally divided between the slits so that

E EE

x xx

1 2 2= = (12.9)

and

E EE

y yy

1 2 2= = . (12.10)

The fields at the screen Σ due to light from slit 1 are

E E i

E E i

x x

y y

∑

∑

= ( )

= ( )1 1

1 1

exp

exp

φ

φ (12.11)

and the fields at the screen Σ due to light from slit 2 are

E E i

E E i

x x

y y

∑

∑

= ( )

= ( )2 2

2 2

exp

exp

φ

φ , (12.12)

1

Σ

2

figuRe 12.1 Young’s double slit experiment.

270 Polarized Light, Third Edition

where ϕ1 and ϕ2 are the phases associated with the path lengths to an arbitrary point on the screen Σ from slit 1 and slit 2, respectively. Summing the fields, we have

E E i E ix x x∑ = ( ) + ( )1 1 2 2exp expφ φ (12.13)

and

E E i E iy y y∑ = ( ) + ( )1 1 2 2exp expφ φ , (12.14)

where we require E E E Ex x xds x1 2 2= = = / and E E E Ey y yds y1 2 2= = = / , where the subscripts xds and yds refer to the amplitudes at the double slit plane. We can rewrite the previous two equa-tions, factoring out exp (iϕ1), as

E E i ix xds∑ = + ( )( ) ( )1 1exp expφ φ (12.15)

and

E E i iy yds∑ = + ( )( ) ( )1 1exp exp ,φ φ (12.16)

where ϕ = ϕ2 – ϕ1. We will disregard the common phase factor exp(iϕ1) since we are interested in the result of phase differences. The Stokes vector at the screen is

S∑ ∑ ∑ ∑ ∑=

+−

12

12

1

1

0

0

1

1

0

0

E E E Ex x y y* *

. (12.17)

Using Equations 12.15 and 12.16 and the relationships E Exds x= / 2 and E Eyds y= / 2 we obtain

S∑ = +( )

+12

12

1

1

1

0

0

1E E E Ex x y y* *cosφ ++( ) −

cos ,φ

1

1

0

0

(12.18)

and recalling that we required E E Ex y= = / 2 , we have

S∑ = +

EE* cos12

1

0

0

0

φ (12.19)

Fresnel–Arago Interference Laws 271

or

S =

EE* cos .2

2

1

0

0

0

φ (12.20)

This is the Stokes vector of unpolarized light with an interference pattern and is an expression equivalent to Equation 12.5, the traditional double slit interference equation.

12.4 double SliT WiTh PaRallel PolaRiZeRS: The fiRST laW

We illustrate the first interference law of Fresnel and Arago with the experiment shown in Figure 12.2. This is Young’s double slit experiment with parallel ideal horizontal polarizers over the slits. The Stokes vector of light passing through the first slit is then

S112 1 1

1

1

0

0

=

E Ex x* , (12.21)

and the Stokes vector of light passing through the second slit is

S212 2 2

1

1

0

0

=

E Ex x* , (12.22)

1

Σ

2

figuRe 12.2 Young’s double slit experiment with parallel polarizers in front of slits.

272 Polarized Light, Third Edition

where again the fields at the slits are related to the source field by

E EE

x xx

1 2 2= = . (12.23)

The fields at the screen from the two slits, where again ϕ1 and ϕ2 are the phases associated with the path lengths to an arbitrary point on the screen Σ from slit 1 and slit 2, respectively, are

E E i

E E i

x x

x x

∑

∑

= ( )

= ( )1 1

2 2

exp

exp ,

φ

φ (12.24)

and we again sum these to obtain the total field at the screen

E E i E ix x x∑ = ( ) + ( )1 1 2 2exp exp ,φ φ (12.25)

where again the relationship of the fields at the double slit plane to the source fields is given by E E E Ex x xds x1 2 2= = = / . As before we have

E E i ix xds∑ = + ( )( ) ( )1 1exp exp ,φ φ (12.26)

where ϕ = ϕ2 – ϕ1 and where we will again disregard the common phase factor exp (iϕ1). The Stokes vector at the screen is

S∑ ∑ ∑=

PE Ex x

12

1

1

0

0

* , (12.27)

where the subscript P refers to parallel polarizers in front of the slits. Using Equation 12.26 and E Exds x= / 2 we now obtain

S∑ = +( )

PE Ex x

12

1

1

1

0

0

* cos .φ (12.28)

Finally, using the relationship E Ex = / 2 we have

S∑∗= +

PEE

14

1

1

0

0

cosφ (12.29)

Fresnel–Arago Interference Laws 273

or

S∑ =

PEE

12 2

1

1

0

0

2* cos .φ

(12.30)

This is the Stokes vector of horizontally polarized light with a double slit interference pattern at the screen. We have removed half of the light compared to the unpolarized double slit pattern by the use of polarizers, but the interference is still present.

12.5 double SliT WiTh PeRPeNdiCulaR PolaRiZeRS: The SeCoNd laW

We now repeat the double slit experiment with polarizers that are perpendicular to create conditions that illustrate the second interference law. Figure 12.3 shows Young’s experiment with a horizontal polarizer over one slit and a vertical polarizer over the other.

The Stokes vector at the first slit is

S112 1 1

1

1

0

0

=

E Ex x* , (12.31)

and the Stokes vector at the second slit is

S212 2 2

1

1

0

0

=−

E Ey y* . (12.32)

The fields at the screen are

E E i

E E i

x x

y y

∑

∑

= ( )

= ( )1 1

2 2

exp

exp ,

φ

φ (12.33)

and we cannot add these fields together because they are orthogonal components. The Stokes vector at the screen is then

S∑ ∑ ∑ ∑ ∑⊥=

+−

PE E E Ex x y y

12

12

1

1

0

0

1

1* *

00

0

(12.34)

274 Polarized Light, Third Edition

or

S∑ ⊥= ( )

+P

E E i Ex x12 1 1 1

12

2

1

1

0

0

* exp φ yy yE i2 2 22

1

1

0

0

* exp .φ( ) −

(12.35)

This can be expressed in terms of the field from the source as

S∑ ⊥= ( )

+P

EE i EE18 1

18

2

1

1

0

0

* *exp exφ pp .i2

1

1

0

0

2φ( ) −

(12.36)

We have unpolarized light at the screen that does not contain an interference pattern.

12.6 double SliT aNd The ThiRd laW

The third law requires that we start with unpolarized light, create perpendicular polarizations at the slits, and then resolve the result into the same plane of polarization at the screen. We should not find any interference according to the third law. This law and the fourth law to follow should be obvious from a consideration of the properties of coherence and polarization, but each is experimentally more complex to demonstrate and mathematically more complex to prove. Figure 12.4 is a repre-sentation of the experiment.

As in the last experiment in Figure 12.3, we have an unpolarized source and polarizers with per-pendicular axes in front of the two slits. For this experiment, we add a polarizer at the screen where

1

2

Σ

figuRe 12.3 Young’s double slit experiment with perpendicular polarizers in front of slits.

Fresnel–Arago Interference Laws 275

the axis of this polarizer is at some angle θ. As in the previous experiment, the Stokes vector at the first slit is

S112 1 1

1

1

0

0

=

E Ex x* , (12.37)

the Stokes vector at the second slit is

S212 2 2

1

1

0

0

=−

E Ey y* , (12.38)

and the fields at the screen before encountering the polarizer are

E E i

E E i

x x

y y

∑

∑

= ( )

= ( )1 1

2 2

exp

exp .

φ

φ (12.39)

The Jones matrix for an ideal linear polarizer at angle θ is

J θθ θ θ

θ θ θ( ) =

cos sin cos

sin cos sin,

2

2 (12.40)

and multiplying the Jones vector formed from the x and y field components of Equation 12.39 by the Jones matrix of Equation 12.40, we have field components after the polarizer and at the screen

1

2

Σ

figuRe 12.4 Young’s double slit experiment with perpendicular polarizers in front of slits and a polarizer at angle θ in front of the screen.

276 Polarized Light, Third Edition

E E i E i

E

x P x y

y

∑ = ( ) + ( )1 12

2 2exp cos exp sin cosφ θ φ θ θ

∑∑ = ( ) + ( )P x yE i E i1 1 2 22exp sin cos exp sin ,φ θ θ φ θ

(12.41)

where the subscript P indicates that this is the field with the polarizer at the screen. We must now find the new Stokes vector corresponding to these field components. Using Equation 12.1 and keep-ing in mind that the source is unpolarized, we obtain a Stokes vector

S =

12

1

2

2

0

EE*cos

sin.

θθ

(12.42)

By use of the polarizer, we have forced the polarization at the screen to be resolved into a single linear polarization. There is no dependence on phase and hence no interference.

12.7 double SliT aNd The fouRTh laW

The fourth law requires that we start with a linearly polarized source, create two orthogonally polarized sources at the slits, and finally resolve the result into the same plane of polarization. This process is illustrated in Figure 12.5 where we have placed a linear polarizer in front of the unpolar-ized source. We take the linear polarizer in front of the source to be at 45°.

The Mueller matrix for an ideal linear polarizer at 45° is

M4512

1 0 1 0

0 0 0 0

1 0 1 0

0 0 0 0

=

. (12.43)

11

2

Σ

figuRe 12.5 Young’s double slit experiment with polarizer at 45° from horizontal in front of unpolarized source, perpendicular polarizers in front of slits, and a polarizer at angle θ in front of the screen.

Fresnel–Arago Interference Laws 277

Let us take the unpolarized light from the source to be resolved into two orthogonal components at +45° and –45° this time instead of horizontal and vertical components as in Equation 12.6. We then have for the Stokes vector

S =

=

EE E Ex x

* *

1

0

0

0

1

0

1

0

12

+−

12

1

0

1

0

E Ey y* . (12.44)

Multiplying this by the Mueller matrix of the polarizer we have

S =

14

1

0

1

0

E Ex x* . (12.45)

The light emerging from the slits has Stokes vectors

S118

1

1

0

0

=

E Ex x* (12.46)

and

S218

1

1

0

0

=−

E Ex x* . (12.47)

The fields just in front of the screen are

E E i

E E i

x x

y x

∑

∑

= ( )

= ( )

exp

exp .

φ

φ

1

2

(12.48)

Using Equation 12.40 again to find the fields after the screen polarizer, we have

E E i E i

E

x P x x

y P

∑

∑

= ( ) + ( )exp cos exp sin cosφ θ φ θ θ12

2

== ( ) + ( )E i E ix xexp sin cos exp sin .φ θ θ φ θ1 22

(12.49)

The Stokes vector on the screen is

S = +( )

12

1 2

1

2

2

0

EE* sin coscos

sinθ φ

θθ

, (12.50)

278 Polarized Light, Third Edition

so that an interference pattern exists except when the polarizer at the screen is aligned with either of the polarizers at the slits.

RefeReNCeS

1. Stokes, G. G., On the composition and resolution of streams of polarized light from different sources, Trans. cambridge Phil. Soc. 9 (1852): 399–416.

2. Chandrasekhar, S., Radiative Transfer, Oxford: Oxford University Press, 1950. 3. Fresnel, A. J., L’Oeuvres complètes, Vol. I, Paris, Ministre de L’Instruction Publique, 1866. 4. Wood, R. W., Physical Optics, New York: The Macmillan, 1st ed., 1905; 2nd ed., 1914. 5. Jenkins, F. A., and H. E. White, Fundamentals of Optics, 3rd ed., New York: McGraw-Hill Book, 1957. 6. Hecht, E., and A. Zajac, Optics, Reading, MA: Addison-Wesley, 1974. 7. Collett, E., Mathematical formulation of the interference laws of Fresnel and Arago, Am. J. Phys. 39

(1971): 1483–95.

IIPart

Polarimetry

281

13 Introduction

The science of measuring polarization is known as polarimetry. In this second part of the book, we cover techniques in polarimetry. In Chapter 14, we describe methods of measuring the Stokes parameters; that is, these are methods to be used when we want to determine the polarization state of light, and in Chapter 15 we describe methods of measuring characteristics of polarizing ele-ments, that is, these are methods that result in the measurement of part or the entire Mueller matrix. Chapters 14 and 15 provide insight into general principles of polarimetry and include “null” tech-niques, which are those techniques accomplished with the observer’s eye as the detection device and that rely on the determination of the absence of light. The human eye is quite good at this, and these were the methods available before the age of photodetectors, multimeters, and computers. Chapters 14 and 15 may be skipped by those who are only interested in automated measurement techniques.

Chapters 16 and 17, on Stokes polarimetry and Mueller matrix polarimetry, respectively, describe polarimetry techniques with the assumption that measurements are to be done using modern detec-tors, electronics, and computer automation available to today’s experimental researcher.

As animals that form highly resolved images on a retina, humans naturally want to perceive the world in terms of images. Techniques in imaging polarimetry are covered in Chapter 18. The basic measurement principles as applied in Chapters 16 and 17 are still valid, but the requirements of obtaining an extended image and radiometric calibration make imaging polarimetry more dif-ficult. Historical background and a survey of imaging methods are given, as well as an extensive reference list.

Many of the techniques discussed up to this point are time sequential. Channeled polarimetry is an approach to collect data in “snapshot” fashion where everything is collected at a single point in time, and Chapter 19 is a description of this imaging data collection method. Channeled pola-rimetry can eliminate errors due to time-sequential operation and image misregistration, but it introduces other complications. Polarization information is encoded onto spectral or spatial carrier frequencies by using interference, an analogous process, for example, to amplitude modulation in the radio frequency domain.

283

14 Methods of Measuring Stokes Polarization Parameters

14.1 iNTRoduCTioN

We now turn our attention to the important problem of measuring the Stokes polarization parameters. Emphasis is on null, or manual methods, in this chapter although two methods requiring a detector are mentioned. A complete summary of modern automated methods is given in Chapter 16.

The first method for measuring the Stokes parameters is due to Stokes [1] and is probably the best known method; this method was discussed in Section 5.4. There are other methods for measuring the Stokes parameters, but we have refrained from discussing these methods until we introduced the Mueller matrices for a polarizer, a retarder, and a rotator. The Mueller matrix and Stokes vector formalism allows us to treat all of these measurement problems in a very simple and direct manner. While the problems could have been treated using the amplitude formulation, the use of the Mueller matrix formalism greatly simplifies the analysis.

In theory, the measurement of the Stokes parameters should be quite simple. In practice, there are difficulties. This is due primarily to the fact that while the measurement of S0, S1, and S2 is quite straightforward, the measurement of S3 is more difficult. In fact, as we pointed out, before the advent of optical detectors it was not even possible to measure the Stokes parameters using Stokes’s mea-surement method (Section 5.4). It is possible, however, to measure the Stokes parameter using the eye as a detector by using a so-called null method; this is discussed in Section 14.4. In this chapter, we discuss Stokes’s method along with other methods, which includes the circular polarizer method, the null-intensity method, the Fourier analysis method, and the method of Kent and Lawson.

14.2 ClaSSiCal meaSuRemeNT meThod: QuaRTeR-WaVe ReTaRdeR aNd PolaRiZeR meThod

The Mueller matrices for the polarizer (or diattenuator), retarder (or phase shifter), and rotator can now be used to analyze various methods for measuring the Stokes parameters. A number of meth-ods are known. We first consider the application of the Mueller matrices to the classical measure-ment of the Stokes polarization parameters using a quarter-wave retarder and a polarizer. This is the same problem that was treated in Section 5.4; it is the problem originally considered by Stokes [1]. The result is identical, of course, with that obtained by Stokes. However, the advantage of using the Mueller matrices is that a formal method can be used to treat not only this type of problem but other polarization problems as well.

The Stokes parameters can be measured as shown in Figure 14.1. An optical beam is charac-terized by its four Stokes parameters S0, S1, S2, and S3. The Stokes vector of this beam is repre-sented by

S =

S

S

S

S

0

1

2

3

. (14.1)

284 Polarized Light, Third Edition

The Mueller matrix of a retarder with its fast axis at 0° is

M =

−

1 0 0 0

0 1 0 0

0 0

0 0

cos sin

sin cos

φ φφ φ

.. (14.2)

The Stokes vector S′ of the beam emerging from the retarder is obtained by multiplication of Equations 14.2 and 14.1, so

′ =+

− +

S

S

S

S S

S S

0

1

2 3

2 3

cos sin

sin cos

φ φφ φ

. (14.3)

The Mueller matrix of an ideal linear polarizer with its transmission axis set at an angle θ is

M = 12

1 2 2 0

2 2 2 2 0

2

2

cos sin

cos cos sin cos

sin

θ θθ θ θ θθ ssin cos sin

.2 2 2 0

0 0 0 0

2θ θ θ

(14.4)

The Stokes vector S″ of the beam emerging from the linear polarizer is found by multiplication of Equation 14.3 by Equation 14.4. However, we are only interested in the intensity I″ that is the first Stokes parameter ′′S0 of the beam incident on the optical detector shown in Figure 14.1. Multiplying the first row of Equation 14.4 with Equation 14.3, we then find the intensity of the beam emerging from the quarter-wave retarder-polarizer combination to be

I S S S S( , ) [ cos sin cos sin sinθ φ θ θ φ θ= + + +12

2 2 20 1 2 3 φφ]. (14.5)

Equation 14.5 is Stokes’s famous intensity relation for the Stokes parameters. The Stokes param-eters are then found from the following conditions on θ and φ:

S I I0 0 0 90 0= +( , ) ( , ),° ° ° ° (14.6)

Ex

Ey

Incident beam

Detector planeLinear polarizer

θ

Retarder

– /2

+ /2

E x

E y

x

y

figuRe 14.1 Classical measurement of the Stokes parameters.

Methods of Measuring Stokes Polarization Parameters 285

S I I1 0 0 90 0= −( , ) ( , ),° ° ° ° (14.7)

S I S2 02 45 0= −( , ) ,° ° (14.8)

S I S3 02 45 90= −( , ) .° ° (14.9)

In practice, S0, S1, and S2 are easily measured by removing the quarter-wave retarder (ϕ = 90°) from the optical train. In order to measure S3, however, the retarder must be reinserted into the optical train with the linear polarizer set at θ = 45°. This immediately raises a problem because the retarder absorbs some optical energy. In order to obtain an accurate measurement of the Stokes parameters, the absorption factor must be introduced, ab initio, into the Mueller matrix for the retarder. The absorption factor that we write as p must be determined from a separate measurement and will then appear in Equations 14.5 and 14.6 through Equation 14.9. We can easily derive the Mueller matrix for an absorbing retarder as will now be shown.

The field components Ex and Ey of a beam emerging from an absorbing retarder in terms of the incident field components Ex and Ey are

′ = + −E E e ex xi xφ α/2 (14.10)

′ = − −E E e ey yi yφ α/ ,2 (14.11)

where αx and αy are the absorption coefficients. We can also express the exponential absorption fac-tors in Equations 14.10 and 14.11 as

p exx= −α (14.12)

p eyy= −α . (14.13)

Using Equations 14.10 through 14.13 in the defining equations for the Stokes parameters, we find that the Mueller matrix for an anisotropic absorbing retarder is

M =

+ −− +1

2

0 0

0 0

0 0 2

2 2 2 2

2 2 2 2

p p p p

p p p p

p p

x y x y

x y x y

x y ccos sin

sin cos

φ φφ φ

2

0 0 2 2

p p

p p p px y

x y x y−

. (14.14)

We see that an absorbing retarder behaves simultaneously as a polarizer and a retarder. If we use the angular representation for the polarizer behavior in Section 6.2, then we can write Equation 14.14 as

M = p2

2

1 2 0 0

2 1 0 0

0 0 2 2

0 0

cos

cos

sin cos sin sin

γ

γγ

φ φγ−−

sin sin sin cos

,

2 2γ φ γ φ

(14.15)

286 Polarized Light, Third Edition

where p p px y2 2 2+ = . We note that for γ = 45° we have an isotropic retarder; that is, the absorption is

equal along both axes. If p2 is also unity, then Equation 14.14 reduces to an ideal phase retarder.The intensity of the emerging beam I(θ,φ) is obtained by multiplying Equation 14.1 by Equation

14.15 and then by Equation 14.4, and the result is

I

pS S( , ) [( cos cos ) (cos cos )θ γ γφ θ θ= + + +

2

0 121 2 2 2 2

++ +(sin cos sin ) (sin sin sin ) ].2 2 2 22 3γ γφ θ φ θS S

(14.16)

If we were now to make all four intensity measurements with a quarter-wave retarder in the optical train, then Equation 14.16 would result for each of the combinations of θ and φ to

Sp

I I0 2

10 0 90 0= +[ ( , ) ( , )], (14.17)

Sp

I I1 2

10 0 90 0= −[ ( , ) ( , )], (14.18)

Sp

I S2 2 02

45 0= −( , ) , (14.19)

Sp

I S3 2 02

45 90= −( , ) . (14.20)

Thus, each of the intensities in Equations 14.17 through 14.20 are reduced by a factor p2, and this has no effect on the final value of the Stokes parameters with respect to each other. Furthermore, if we are interested in the ellipticity and the orientation, then we take ratios of the Stokes parameters S3/S0 and S2/S1 and the absorption factor p2 cancels out. However, this is not exactly the way the measurement is made. Usually the first three intensity measurements are made without the retarder present, so the first three parameters are measured according to Equations 14.6 through 14.8. The last measurement is done with a quarter-wave retarder in the optical train as in Equation 14.20, so the equations are

S I I0 0 0 90 0= +( , ) ( , ), (14.21)

S I I1 0 0 90 0= −( , ) ( , ), (14.22)

S I S2 02 45 0= −( , ) , (14.23)

Sp

I S3 2 02

45 90= −( , ) . (14.24)

Thus, Equation 14.24 shows that the absorption factor p2 enters in the measurement of the fourth Stokes parameters S3. It is therefore necessary to measure the absorption factor p2. The easiest way

Methods of Measuring Stokes Polarization Parameters 287

to do this is to place a linear polarizer between an optical source and a detector and measure the intensity; this is called I0. Next, the retarder with its fast axis in the horizontal x direction is inserted between the linear polarizer and the detector. The intensity is then measured with the polarizer generating linear horizontally and linear vertically polarized light. Dividing each of these measured intensities by I0 and adding the results gives p2. Thus, we see that the measurement of the first three Stokes parameters is very simple, but the measurement of the fourth parameter S3 requires a con-siderable amount of additional effort. It would therefore be preferable if a method could be devised whereby the absorption measurement could be eliminated. A method for doing this can be devised, and we now consider this method.

14.3 meaSuRemeNT of STokeS PaRameTeRS uSiNg a CiRCulaR PolaRiZeR

The problem of absorption by a retarder can be completely overcome by using a single polarizing element, the circular polarizer [2]. The beam is allowed to enter one side of the circular polarizer, whereby the first three parameters can be measured. The circular polarizer is then flipped 180°, and the final Stokes parameter is measured. A circular polarizer is made by cementing a quarter-wave retarder to a linear polarizer with its axis at 45° to the fast axis of the retarder, thereby ensuring that the retarder and polarizer axes are always fixed with respect to each other. Furthermore, because the same optical path is used in all four measurements, the problem of absorption vanishes; the four intensities are reduced by the same amount. The construction of a circular polarizer is illustrated in Figure 14.2.

The Mueller matrix for the polarizer-retarder combination is

M =

−

12

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

1 0 1 0

0 0 0 0

1 0 11 0

0 0 0 0

, (14.25)

and thus

M =

− −

12

1 0 1 0

0 0 0 0

0 0 0 0

1 0 1 0

. (14.26)

x

y

Linear polarizer at 45°

45°

Quarter-wave retarder

Fast axis

Slow axis

figuRe 14.2 Construction of a circular polarizer using a linear polarizer and a quarter-wave retarder.

288 Polarized Light, Third Edition

Equation 14.26 is the Mueller matrix of a circular polarizer. The reason for calling Equation 14.26 a circular polarizer is that regardless of the polarization state of the incident beam, the emerging beam is always circularly polarized. This is easily shown by assuming that the Stokes vector of an incident beam is of the form of Equation 14.1. Multiplication of Equation 14.1 by Equation 14.26 then yields

′ = +

−

S12

1

0

0

1

0 2( ) ,S S (14.27)

which is the Stokes vector for left circularly polarized light (LCP). Regardless of the polarization state of the incident beam, the output beam is always left circularly polarized, hence the name circular polarizer. Equation 14.26 defines a circular polarizer composed of a polarizer and quarter-wave retarder.

Next, consider that the quarter-wave retarder-polarizer combination is “flipped”; that is, the lin-ear polarizer now follows the quarter-wave retarder. The Mueller matrix for this combination is obtained with the Mueller matrices in Equation 14.25 interchanged; we note that the axis of the linear polarizer when it is flipped causes a sign change in the Mueller matrix (see Figure 14.2). Then we express the system as

M =

−

−

12

1 0 1 0

0 0 0 0

1 0 1 0

0 0 0 0

1 0 0 0

0 1 0 0

0 00 0 1

0 0 1 0−

, (14.28)

so

M =

−

−

12

1 0 0 1

0 0 0 0

1 0 0 1

0 0 0 0

. (14.29)

Equation 14.29 is the matrix of a linear polarizer. That Equation 14.29 is a linear polarizer can eas-ily be seen by multiplying Equation 14.1 by Equation 14.29 with the result

′ = −−

S12

1

0

1

0

0 3( ) ,S S (14.30)

which is the Stokes vector for linear −45° polarized light. Regardless of the polarization state of the incident beam, the final beam is always linear +45° polarized. It is of interest to note that in the case of the “circular” side of the polarizer configuration, Equation 14.27, the intensity varies only

Methods of Measuring Stokes Polarization Parameters 289

with the linear component, S2, in the incident beam. On the other hand, for the “linear” side of the polarizer, Equation 14.30, the intensity varies only with S3, the circular component in the incident beam.

The circular polarizer is now placed in a rotatable mount. We saw earlier in the book that the Mueller matrix for a rotated polarizing component, M, is given by the relation

M M M M( ) ( ) ( ),2 2 2θ θ θ= −R R (14.31)

where MR(2θ) is the rotation Mueller matrix

MR ( )cos sin

sin cos2

1 0 0 0

0 2 2 0

0 2 2 0

0 0 0 1

θθ θθ θ

=−

, (14.32)

and M(2θ) is the Mueller matrix of the rotated polarizing element. The Mueller matrix for the cir-cular polarizer with its axis rotated through an angle θ is then found by substituting Equation 14.26 into Equation 14.31. The result is

Mc ( )

sin cos

sin cos

212

1 2 2 0

0 0 0 0

0 0 0 0

1 2 2 0

θ

θ θ

θ θ

=

−

−

, (14.33)

where the subscript c refers to the fact that Equation 14.33 describes the circular side of the polar-izer combination. We see immediately that the Stokes vector emerging from the beam of the rotated circular polarizer is, using Equations 14.33 and 14.1,

Sc S S S= − +

−

12

2 2

1

0

0

1

0 1 2( sin cos ) .θ θ (14.34)

Thus, as the circular polarizer is rotated, the intensity varies but the polarization state remains unchanged as circular. We note again that the total intensity depends on S0 and on the linear com-ponents, S1 and S2, in the incident beam.

The Mueller matrix when the circular polarizer is flipped to its linear side is, from Equations 14.29 and 14.31,

ML ( )sin sin

cos cos2

12

1 0 0 1

2 0 0 2

2 0 0 2

0 0 0

θθ θθ θ

=

−−

−00

, (14.35)

290 Polarized Light, Third Edition

where the subscript L refers to the fact that Equation 14.35 describes the linear side of the polarizer combination. The Stokes vector of the beam emerging from the rotated linear side of the polarizer, multiplying Equations 14.35 and 14.1 is

SL S S= −−

12

1

2

2

0

0 3( )sin

cos.

θθ

(14.36)

Under a rotation of the circular polarizer on the linear side, Equation 14.36 shows that the polariza-tion is always linear. The total intensity is constant and depends on S0 and the circular component S3 in the incident beam.

The intensities detected on the circular and linear sides are, respectively, from Equation 14.34 to Equation 14.36,

I S S Sc ( ) ( sin cos )θ θ θ= − +12

2 20 1 2 (14.37)

I S SL ( ) ( ).θ = +12 0 3 (14.38)

The intensity on the linear side, Equation 14.38, is seen to be independent of the rotation angle of the polarizer. This fact allows a simple check when the measurement is being made. If the circular polarizer is rotated and the intensity does not vary, then one knows the measurement is being made on IL, the linear side.

In order to obtain the Stokes parameters, we first use the circular side of the polarizing element and rotate it to θ = 0º, 45º, and 90º, and then flip it to the linear side. The measured intensities are then

I S Sc ( ) ( ),012 0 2

= + (14.39)

I S Sc ( ) ( ),4512 0 1

= − (14.40)

I S Sc ( ) ( ),9012 0 2

= − (14.41)

I S SL ( ) ( ).012 0 3

= − (14.42)

The IL orientation value is conveniently taken to be θ = 0°. Solving these equations for the Stokes parameters yields

S I Ic c0 0 90= −( ) ( ), (14.43)

S S Ic1 0 2 45= − ( ), (14.44)

S I Ic c2 0 90= −( ) ( ), (14.45)

S S IL3 0 2 0= − ( ). (14.46)

Methods of Measuring Stokes Polarization Parameters 291

Equations 14.43 through 14.46 are similar to the classical equations for measuring the Stokes param-eters, Equations 14.6 through 14.9, but the intensity combinations are distinctly different. The use of a circular polarizer to measure the Stokes parameters is simple and accurate because (1) only a single rotating mount is used, (2) the polarizing beam propagates through the same optical path so that the problem of absorption losses can be ignored, and (3) the axes of the wave plate and polarizer are permanently fixed with respect to each other.

14.4 Null-iNTeNSiTy meThod

In previous sections, the Stokes parameters were expressed in terms of measured intensities. These measurement methods, however, are suitable only for use with quantitative detectors. We pointed out earlier that before the advent of solid-state detectors and photomultipliers, the only available detector was the human eye. It can only measure the presence of light or no light (a null intensity). It is possible, as we shall now show, to measure the Stokes parameters from the condition of a null-intensity state. This can be done by using a variable retarder followed by a linear polarizer in a rotatable mount. Devices are manufactured that can change the phase between the orthogonal components of an optical beam. They are called Babinet–Soleil compensators, and they are usually placed in a rotatable mount. Following the compensator is a linear polarizer that is also placed in a rotatable mount. This arrangement can be used to obtain a null intensity. In order to carry out the analysis, the reader is referred to Figure 14.3.

The Stokes vector of the incident beam to be measured is the general Stokes vector

S =

S

S

S

S

0

1

2

3

. (14.1)

The analysis is simplified considerably if the α, δ form of the Stokes vector derived in Section 5.3 is used; that is,

S =

I0

1

2

2

2

cos

sin cos

sin sin

.α

α δα δ

(14.47)

Ex

Ey

Incident beam

Rotatable linear polarizer

θ

Babinet–soleilcompensator

+ /2

– /2

E x

I(θ, )

E yx

y

figuRe 14.3 Null-intensity measurement of the Stokes parameters.

292 Polarized Light, Third Edition

The axis of the Babinet–Soleil compensator is set at 0°. The Stokes vector of the beam emerging from the compensator is found by multiplying the matrix of the nonrotated compensator (Section 6.3, Equation 6.43) with Equation 14.47 so that

′ =

−

S I0

1 0 0 0

0 1 0 0

0 0

0 0

cos sin

sin cos

φ φφ φ

1

2

2

2

cos

sin cos

sin sin

.α

α δα δ

(14.48)

Carrying out the matrix multiplication in Equation 14.48 and using the well-known trigonometric sum formulas, we readily find that

′ =−−

S I0

1

2

2

2

cos

sin cos( )

sin sin( )

αα δ φα δ φ

. (14.49)

Two important observations on Equation 14.49 can be made. The first is that Equation 14.49 can be transformed to linearly polarized light if ′S3 can be made to be equal to zero. This can be done by setting δ–ϕ = 0°. If we then analyze S′ with a linear polarizer, we see that a null intensity can be obtained by rotating the polarizer; at the null setting, we can then determine α. This method is the procedure that is almost always used to obtain a null intensity. The null-intensity method works because δ in Equation 14.48 is simply transformed to δ–ϕ in Equation 14.49 after the beam propa-gates through the compensator. For the moment, we shall retain the form of Equation 14.49 and not set δ–ϕ = 0°. The function of the Babinet–Soleil compensator in this case is to transform elliptically polarized light to linearly polarized light.

Next, the beam represented by Equation 14.49 is incident on a linear polarizer with its transmis-sion axis at an angle θ. The Stokes vector S″ of the beam emerging from the rotated polarizer is now

′′ =SI0

2

2

1 2 2 0

2 2 2 2 0

cos sin

cos cos sin cos

si

θ θθ θ θ θ

nn sin cos sin

cos

2 2 2 2 0

0 0 0 0

1

22θ θ θ θ

α

ssin cos( )

sin sin( )

,2

2

α δ φα δ φ

−−

(14.50)

where we have used the Mueller matrix of a rotated linear polarizer, Equation 6.79. We are inter-ested only in the intensity of the beam emerging from the rotated polarizer; that is, ′′=S I0 ( , )θ φ . Carrying out the matrix multiplication with the first row in the Mueller matrix and the Stokes vector in Equation 14.50 yields

II

( , ) [ cos cos sin sin cos( )]θ φ θ α θ α δ φ= + + −0

21 2 2 2 2 .. (14.51)

We now set δ–ϕ = 0° in Equation 14.51 and find

II

( , ) [ cos cos sin sin ],θ φ θ α θ α= + +0

21 2 2 2 2 (14.52)

Methods of Measuring Stokes Polarization Parameters 293

which reduces to

II

( , ) [ cos ( )].θ φ θ α= + −0

21 2 (14.53)

The linear polarizer is rotated until a null intensity is observed. At this angle, δ–α = π/2, and we have

I α π δ+

=

20, . (14.54)

The angles δ and α associated with the Stokes vector of the incident beam are thus found from the conditions

δ φ= (14.55)

α θ π= −2

. (14.56)

Equations 14.55 and 14.56 are the required relations between α and δ of the Stokes vector Equation 14.49 and ϕ and θ, the phase setting on the Babinet–Soleil compensator and the angle of rotation of the linear polarizer, respectively.

From the values obtained for α and δ, we can determine the corresponding values for the orienta-tion angle ψ and the ellipticity χ of the incident beam. We saw in Section 5.3 that ψ and χ could be expressed in terms of α and δ; that is,

tan tan cos2 2ψ α δ= (5.78)

sin sin sin .2 2χ α δ= (5.79)

Substituting Equations 14.55 and 14.56 into Equations 5.74 and 5.75, we see that ψ and χ can be expressed in the terms of the measured values of θ and φ; that is,

tan tan cos2 2ψ θ φ= (14.57)

sin sin sin .2 2χ θ φ= − (14.58)

Remarkably, Equations 14.57 and 14.58 are identical to Equations 5.74 and 5.75 in form. It is only necessary to take the measured values of θ and φ and insert them into Equations 14.57 and 14.58 to obtain ψ and χ. Equations 5.74 and 5.75 can be solved in turn for α and δ following the derivation given in Section 6.6, and we have

cos cos cos ,2 2 2α χ ψ= ± (14.59)

tantansin

.δ χψ

= 22

(14.60)

The procedure to find the null-intensity angles θ and ϕ is first to set the Babinet–Soleil compen-sator with its fast axis to 0º and its phase angle to 0º. The phase is then adjusted until the intensity is

294 Polarized Light, Third Edition

observed to be a minimum. At this point in the measurement, the intensity will not necessarily be zero, only a minimum, as we see from Equation 14.53. Next, the linear polarizer is rotated through an angle θ until a null intensity is observed; the setting at which this angle occurs is then measured. In theory, this completes the measurement. In practice, however, one finds that small adjustments in the phase of the compensator and rotation angle of the linear polarizer are almost always necessary to obtain a null intensity. Substituting the observed angular settings on the compensator and the polarizer into Equations 14.57 and 14.58, and Equations 14.59 and 14.60, we then find the Stokes vector Equation 14.47 of the incident beam. We note that Equation 14.47 is a normalized representa-tion of the Stokes vector if I0 is set to unity.

14.5 fouRieR aNalySiS uSiNg a RoTaTiNg QuaRTeR-WaVe ReTaRdeR

Another method for measuring the Stokes parameters is to allow a beam to propagate through a rotating quarter-wave retarder followed by a linear horizontal polarizer; the retarder rotates at an angular frequency of ω. This arrangement is shown in Figure 14.4.

The Stokes vector of the incident beam to be measured is

S =

S

S

S

S

0

1

2

3

. (14.1)

The Mueller matrix of the rotated quarter-wave retarder (Section 6.5) is

MR 90 2

1 0 0 0

0 2 2 2 2

0 2

2 ,

cos sin cos sin

sinθ

θ θ θ θ( ) =−

θθ θ θ θθ θ

cos sin cos

sin cos

2 2 2

0 2 2 0

2

−

,, (6.99)

and for a rotating retarder we set θ = ωt. Multiplying Equation 14.1 by Equation 6.99 yields

′ =+ −

S

S

S S S

S

0

12

2 3

1

2 2 2 2

2

cos sin cos sin

sin co

θ θ θ θθ ss sin cos

sin cos

2 2 2

2 22

23

1 2

θ θ θθ θ

+ +−

S S

S S

. (14.61)

Ex

Ey

Incident beam

Linear polarizer

ωt

Rotating retarder

+ /2– /2

E x

I(θ)

E yx

y

Detector

figuRe 14.4 Measurement of the Stokes parameters using a rotating quarter-wave retarder and a linear polarizer.

Methods of Measuring Stokes Polarization Parameters 295

The Mueller matrix of the linear horizontal polarizer is

M =

12

1 1 0 0

1 1 0 0

0 0 0 0

0 0 0 0

. (6.24)

The Stokes vector of the beam emerging from the rotating quarter-wave retarder-horizontal polar-izer combination is then found from Equation 14.61 to Equation 6.24 to be

′ = + + −S12

2 2 2 2

1

1

0

0

0 12

2 3( cos sin cos sin )S S S Sθ θ θ θ

. (14.62)

The intensity ′ =S I0 ( )θ is

I S S S S( ) ( cos sin cos sin ).θ θ θ θ θ= + + −12

2 2 2 20 12

2 3 (14.63)

Equation 14.63 can be rewritten as, using the trigonometric half-angle formulas,

I SS S S

S( ) cos sin sinθ θ θ θ= +

+ + −1

2 2 24

24 20

1 1 23

. (14.64)

Replacing θ with ωt, Equations 14.64 can be written as

I t A b t c t d t( ) [ sin cos sin ],ω ω ω ω= − + +12

2 4 4 (14.65)

where the coefficients A, b, c, and d, are

A SS= +0

1

2, (14.66)

b S= 3 , (14.67)

cS= 1

2, (14.68)

dS= 1

2. (14.69)

296 Polarized Light, Third Edition

Equations 14.65 through 14.69 describes a truncated Fourier series. It shows that we have a d.c. term (A), a double frequency term (b), and two quadruple frequency terms (C and d). The coefficients are found by carrying out a Fourier analysis of these equations. We easily find that (θ = ωt)

A I d= ∫1

0

2

πθ θ

π

( ) , (14.70)

b I d= ∫22

0

2

πθ θ θ

π

( )sin , (14.71)

c I d= ∫24

0

2

πθ θ θ

π

( )cos , (14.72)

d I d= ∫24

0

2

πθ θ θ

π

( )sin . (14.73)

Solving Equations 14.66 through 14.69 for the Stokes parameters gives

S A c0 = − , (14.74)

S c1 2= , (14.75)

S d2 2= , (14.76)

S b3 = . (14.77)

In practice, the quarter-wave retarder is placed in a fixed mount that can be rotated and driven by a stepper motor through Ν steps. Equation 14.65 then becomes, with ωt = nθ j (θj is the step size),

I A b n c n d nn j j j j( ) [ sin cos sin ],θ θ θ θ= − + +12

2 4 4 (14.78)

and

AN

I n j

n

N

==

∑2

1

( ),θ (14.79)

bN

I n nj

n

N

j==

∑42

1

( )sin ,θ θ (14.80)

cN

I n nj j

n

N

==

∑44

1

( )cos ,θ θ (14.81)

Methods of Measuring Stokes Polarization Parameters 297

dN

I n nj j

n

N

==

∑44

1

( )sin .θ θ (14.82)

As an example, consider the rotation of a quarter-wave retarder that makes a complete rotation in 16 steps, so Ν = 16. Then the step size is θj = 2π/N = 2π/16 = π/8. Equation 14.78 is then writ-ten as

A I nn

=

=∑1

8 81

16 π, (14.83)

b I n nn

=

=∑1

4 8 41

16 π πsin , (14.84)

c I n nn

=

=∑1

4 8 21

16 π πcos , (14.85)

d I n nn

=

=∑1

4 8 21

16 π πsin . (14.86)

Thus, the data array consists of 16 measured intensities, I1 through I16. We have written each intensity value as I(nπ/8) to indicate that the intensity is measured at intervals of π/8; we observe that when n = 16 we have I(2π) as expected. At each step, the intensity is stored to form A, the intensity is multiplied by sin (nπ/4) to form b, by cos (nπ/2)to form c, and by sin (nπ/2) to form d. The sums are then performed according to Equations 14.83 through 14.86, and we obtain A, b, c, and d. The Stokes parameters are then found from Equations 14.74 through 14.77 using these values.

14.6 meThod of keNT aNd laWSoN

In Section 14.4, we saw that the null-intensity condition could be used to determine the Stokes parameters and, hence, the polarization state of an optical beam. The null-intensity method remained the only practical way to measure the polarization state of an optical beam before the advent of photodetectors. It is fortunate that the eye is so sensitive to light and can easily detect its presence or absence. Had this not been the case, the progress made in polarized light would surely not have been as rapid as it was. One can obviously use a photodetector as well as the eye in the null-intensity method described in Section 14.4. However, the existence of photodetectors allows one to consider an extremely interesting and novel method for determining the polarization state of an optical beam.

In 1937, C. V. Kent and J. Lawson [3] proposed a new method for measuring the ellipticity and orientation of a polarized optical beam using a Babinet–Soleil compensator and a photomultiplier tube (PMT). They noted that it was obvious that a photomultiplier could simply replace the human eye as a detector and be used to determine the null condition. However, Kent and Lawson went beyond this and made several important observations. The first was that the use of the PMT could obviously overcome the problem of eye fatigue. They also noted that, in terms of sensitivity (at least

298 Polarized Light, Third Edition

in 1937) for weak illuminations, determining the null intensity was as difficult with a PMT as with the human eye. They observed that the PMT really operated best with full illumination. In fact, because the incident light at a particular wavelength is usually much greater than the laboratory illumination, the measurement could be done with the room lights on. They now noted that this property of the PMT could be exploited fully if the incident optical beam whose polarization was to be determined was transformed not to linearly polarized light but to circularly polarized light. By then analyzing the beam with a rotating linear polarizer, a constant intensity would be obtained when the condition of circularly polarized light was obtained or, as they said, “no modulation.” From this condition of “no modulation,” the ellipticity and orientation angles of the incident beam could then be determined. Interestingly, they detected the circularly polarized light by converting the optical signal to an audio signal and then used a headphone set to determine the constant-intensity condition.

It is worthwhile to study this method because it enables us to see how photodetectors provide an alternative method for measuring the Stokes parameters and how they can be used to their optimum; that is, in the measurement of polarized light at high intensities. The measurement is described by the experimental configuration in Figure 14.5. The Stokes vector of the incident elliptically polar-ized beam to be measured is represented by

S =

S

S

S

S

0

1

2

3

. (14.1)

The primary use of a Babinet–Soleil compensator is to create an arbitrary state of elliptically polar-ized light. This is accomplished by changing the phase and orientation of the incident beam. We recall from Section 6.5 that the Mueller matrix for a rotated retarder is

Mc ( , )cos cos sin ( cos )sin

φ γγ φ γ φ

2

1 0 0 0

0 2 2 1 22 2

=+ − γγ γ φ γφ γ γ γ

cos sin sin

( cos )sin cos sin

2 2

0 1 2 2 22

−− + ccos cos sin cos

sin sin sin cos cos

φ γ φ γφ γ φ γ

2 2 2

0 2 2− φφ

, (6.82)

Ex

Ey

Incident beam

Analyzer

θ

Babinet–soleilcompensator

+ /2

– /2

E x

E y

x

y

Photomultiplier

γ

figuRe 14.5 Measurement of the ellipticity and orientation of an elliptically polarized beam using a com-pensator and a photodetector.

Methods of Measuring Stokes Polarization Parameters 299

where γ is the angle that the fast axis makes with the horizontal x axis and ϕ is the phase shift. The Stokes vector of the beam emerging from the Babinet–Soleil compensator is then found by multiply-ing Equation 14.1 by Equation 6.82 with the result

′ =+ + −

S

S

S S0

12 2

22 2 1 2(cos cos sin ) ( cos )sin cγ φ γ φ γ oos sin sin

( cos )sin cos (si

2 2

1 2 23

1 2

γ φ γφ γ γ

−− +

S

S S nn cos cos ) sin cos

sin sin

2 23

1 2

2 2 2

2

γ φ γ φ γφ γ

+ +−

S

S S ssin cos cos

.

φ γ φ2 3+

S

(14.87)

Let us assume that we have elliptically polarized light incident on a rotating ideal linear polarizer. The normalized Stokes vector of the beam incident on the rotating linear polarizer is represented by

S =

1

2

2

2

cos

sin cos

sin sin

.α

α δα δ

(5.73)

Recall that the Mueller matrix of a rotating linear polarizer is

M = 12

1 2 2 0

2 2 2 2 0

2

2

cos sin

cos cos sin cos

sin

θ θθ θ θ θθ ssin cos sin

.2 2 2 0

0 0 0 0

2θ θ θ

(14.4)

The Stokes vector of the beam emerging from the rotating analyzer is found by multiplying Equation 5.73 by Equation 14.4 so that

′ = + +S12

1 2 2 2 2

1

2[ cos cos sin cos sin ]

cos

sinα θ α δ θ

θ22

0

θ

. (14.88)

We see that the intensity is modulated as the analyzer is rotated. If the intensity is to be independent of the rotation angle θ, then we must have

cos ,2 0α = (14.89)

sin cos .2 0α δ = (14.90)

We immediately see that Equations 14.89 and 14.90 are satisfied if 2α = 90° (or 270º) and δ = 90°. Substituting these values in Equation 5.73, we have

S =

1

0

0

1

, (14.91)

300 Polarized Light, Third Edition

which is the Stokes vector for right circularly polarized light. In order to obtain circularly polarized light, the Stokes parameters in Equation 14.87 must satisfy

the conditions

′ =S S0 0 , (14.92)

′ = + + −S S S1 1

2 222 2 1 2(cos cos sin ) ( cos )sin coγ φ γ φ γ ss (sin sin )

,

2 2

0

3γ φ γ−

=

S (14.93)

′ = − + +S S S2 1 2

2 21 2 2 2( cos )sin cos (sin cos cosφ γ γ γ φ 22 2

0

3γ φ γ) (sin cos )

,

+

=

S (14.94)

′ = − +S S S S3 1 2 32 2(sin sin ) (sin cos ) cos .φ γ φ γ φ (14.95)

We must now solve these equations for S1, S2, and S3 in terms of γ and ϕ (S0 is unaffected by the wave plate). While it is straightforward to solve Equations 14.92 through 14.95, the algebra is surprisingly tedious and complicated. Fortunately, the problem can be solved in another way because we know the transformation equation for describing a rotated compensator.

To solve this problem, we take the following approach. According to Figure 14.5, the Stokes vector of the beam S′ emerging from the compensator is related to the Stokes vector of the incident beam S by the equation

′ =S M Sc ( ) ,2γ (14.96)

where Mc(2γ) is given by Equation 6.82 above. We recall that Mc(2γ) is the rotated Mueller matrix for a retarder, so Equation 14.96 can also be written as

′ = −S M M M S[ ( ) ( )] ,2 2γ γc (14.97)

where

M( )cos sin

sin cos2

1 0 0 0

0 2 2 0

0 2 2 0

0 0 0 1

γγ γ

γ γ=

−

, (14.98)

and

Mc =

−

1 0 0 0

0 1 0 0

0 0

0 0

cos sin

sin cos

φ φφ φ

. (14.99)

Methods of Measuring Stokes Polarization Parameters 301

We now demand that our resultant Stokes vector represents right circularly polarized light and write Equation 14.97 as

′ = −

=

S M M M( ) ( )2 2

1

0

0

1

0

1

2

3

γ γc

S

S

S

S

. (14.100)

While we could immediately invert Equation 14.100 to find the Stokes vector of the incident beam, it is simplest to find S in steps. Multiplying both sides of Equation 14.100 by M(2γ), we have

M M Mc

S

S

S

S

( ) ( )2 2

1

0

0

1

0

1

2

3

γ γ

=

=

1

0

0

1

. (14.101)

Next, we multiply Equation 14.101 by Mc−1 to find

M M( )2

1

0

0

1

0

1

2

3

1γ

S

S

S

S

c

=

−

=−

1

0

sin

cos

.φ

φ

(14.102)

Finally, Equation 14.102 is multiplied by M(−2γ), and we have

S

S

S

S

0

1

2

3

2

1

0

= −−

M( )sin

cos

γφ

φ

=−

1

2

2

sin sin

cos sin

cos

γ φγ φ

φ

. (14.103)

We can check to see if Equation 14.103 is correct. We know that if ϕ = 0° (i.e., the retarder is not present) then the only way S′ can be right circularly polarized is if the incident beam S is right cir-cularly polarized. Substituting ϕ = 0° into Equation 14.103, we find that

S =

1

0

0

1

, (14.104)

which is the Stokes vector for right circularly polarized light.The numerical value of the Stokes parameters can be determined directly from Equation 14.103.

However, we can also express the Stokes parameters in terms of α and δ in Equation 5.73 or in terms

302 Polarized Light, Third Edition

of the orientation and ellipticity angles ψ and χ (Section 5.3). Thus, we can equate Equations 5.73 to 14.103 and write

S

S

S

S

0

1

2

3

1

2

2

2

=cos

sin cos

sin si

αα δα nn

sin sin

cos sin

cosδ

γ φγ φ

φ

=−

1

2

2

, (14.105)

or, in terms of the orientation and ellipticity angles,

S

S

S

S

0

1

2

3

1

2 2

2 2

=cos cos

cos sin

s

χ ψχ ψiin

sin sin

cos sin

cos2

1

2

2

χ

γ φγ φ

φ

=−

. (14.106)

We now solve for S in terms of the measured values of γ and ϕ. Let us first consider Equation 14.105 and equate the elements

cos sin sin ,2 2α γ φ= ± (14.107)

sin cos cos sin ,2 2α δ γ φ= (14.108)

sin sin cos .2α δ φ= ± (14.109)

In these equations, we have written ± to include left circularly polarized light. We divide Equation 14.108 by Equation 14.109 and find

cot cos tan .δ γ φ= ± 2 (14.110)

Similarly, we divide Equation 14.108 by Equation 14.107 and find

cos cot cot .δ γ α= ± 2 2 (14.111)

We can group the results by renumbering Equations 14.107, 14.110, and 14.111 and write

cos sin sin ,2 2α γ φ= ± (14.112)

cot cos tan ,δ γ φ= ± 2 (14.113)

cos cot cot .δ γ α= ± 2 2 (14.114)

Equations 14.112, 14.113, and 14.114 are the equations of Kent and Lawson. Thus, by measuring γ and ϕ, the angular rotation and phase shift of the Babinet–Soleil compensator, respectively, we can determine the azimuth α and phase δ of the incident beam.

Methods of Measuring Stokes Polarization Parameters 303

We also pointed out that we can use γ and ϕ to determine the ellipticity χ and orientation ψ of the incident beam from Equation 14.106. Equating terms in 14.106, we have

cos cos sin sin ,2 2 2χ ψ γ φ= ± (14.115)

cos sin cos sin ,2 2 2χ ψ γ φ= (14.116)

sin cos .2χ φ= ± (14.117)

Dividing Equation 14.116 by Equation 14.115, we find

tan cot .2 2ψ γ= ± (14.118)

Squaring Equations 14.115 and 14.116, adding, and taking the square root gives

cos sin .2χ φ= (14.119)

Dividing Equation 14.117 by 14.119 then gives

tan cot .2χ φ= ± (14.120)

We renumber Equations 14.118 and 14.120 as the pair

tan cot ,2 2ψ γ= ± (14.121)

tan cot .2χ φ= ± (14.122)

We can rewrite Equations 14.121 and 14.122 as

tan tan ( ),2 90 2ψ γ= ± − (14.123)

tan tan( ),2 90χ φ= ± − (14.124)

so that

ψ γ= −45 , (14.125)

χ φ= −452

. (14.126)

We can check Equations 14.125 and 14.126. We know that a linear +45° polarized beam of light is transformed to right circularly polarized light if we send it through a quarter-wave retarder. In terms of the incident beam, ψ = 45° and χ = 0°. Substituting these values in Equations 14.125 and 14.126, we find that γ = 0° and ϕ = 90° for the retarder. This is exactly what we would expect using a quarter-wave retarder with its fast axis in the x direction.

304 Polarized Light, Third Edition

While nulling techniques for determining the elliptical parameters are very common, we see that the method of Kent and Lawson provides a very interesting alternative. We emphasize that null-ing techniques were developed long before the appearance of photodetectors. Nulling techniques continue to be used because they are extremely sensitive and require, in principle, only an analyzer. Nevertheless, the method of Kent and Lawson has a number of advantages, foremost of which is that it can be used in ambient light and with high optical intensities. The method of Kent and Lawson requires the use of a Babinet–Soleil compensator and a rotatable polarizer. However, the novelty and potential of the method and its full exploitation of the quantitative nature of photodetectors should not be overlooked.

14.7 SimPle TeSTS To deTeRmiNe The STaTe of PolaRiZaTioN of aN oPTiCal beam

In the laboratory, we often have to determine if an optical beam is unpolarized, partially polar-ized, or completely polarized. If it is completely polarized, then we must determine if it is linearly, circularly, or elliptically polarized. In this section, we consider this problem. Stokes’s method for determining the Stokes parameters is a very simple and direct way of carrying out these tests (Section 5.4).

We recall that the polarization state can be measured using a linear polarizer and a quarter-wave retarder. If a polarizer made of calcite is used, then it transmits satisfactorily from 0.2 to 2.0 μm, more than adequate for visual work and into the near infrared. Quarter-wave retarders on the other hand, are designed to transmit at a single wavelength, for example, at the wave-length of He–Ne laser radiation at 0.6328 μm. Therefore, the quarter-wave retarder should be matched to the wavelength of the polarizing radiation. In Figure 14.6, we show the experimental configuration for determining the state of polarization. We emphasize that we are not trying to determine the Stokes parameters quantitatively but merely determining the polarization state of the light.

We recall from Section 14.2 that the intensity I(θ, φ) of the beam emerging from the retarder-polarizer combination shown in Figure 14.6 is

I S S S S( , ) [ cos cos sin sin sinθ φ θ φ θ φ= + + +12

2 2 20 1 2 3 θθ], (14.5)

Ex

Ey

Incident beam

Polarizer

θ

I(θ, )

Quarter-waveretarder

+45°

–45°

E x

E yx

y

figuRe 14.6 Experimental configuration to determine the state of polarization of an optical beam.

Methods of Measuring Stokes Polarization Parameters 305

where θ is the angle of rotation of the polarizer and ϕ is the phase shift of the retarder. In our tests, we shall set ϕ to 0° (no retarder in the optical train) or 90° (a quarter-wave retarder in the optical train). The respective intensities according to Equation 14.5 are then

I S S S( , ) [ cos sin ],θ θ θ012

2 20 1 2 = + + (14.127)

I S S S( , ) [ cos sin ].θ θ θ9012

2 20 1 3 = + + (14.128)

The first test we wish to perform is to determine if the light is unpolarized or completely polar-ized. In order to determine if it is unpolarized, the retarder is removed (ϕ = 0°), so we use Equation 14.127. The polarizer is now rotated through 180°. If the intensity remains constant throughout the rotation, then we must have

S S S1 2 and= = ≠0 00 . (14.129)

If the intensity varies so that Equation 14.129 is not satisfied, then we know that we do not have unpolarized light. If, however, the intensity remains constant, then we are still not certain if we have unpolarized light because the parameter S3 may be present. We must, therefore, test for its presence. The retarder is now reintroduced into the optical train, and we use Equation 14.128. The polarizer is now rotated. If the intensity remains constant, then

S S S1 3 and= = ≠0 00 . (14.130)

From Equation 14.129 to Equation 14.130 we see that Equation 14.5 becomes

I S( , ) ,θ φ = 12 0 (14.131)

which is the condition for unpolarized light.If neither Equation 14.129 nor Equation 14.130 are satisfied, we then assume that the light is

elliptically polarized; the case of partially polarized light is excluded for the moment. Before we test for elliptically polarized light, however, we test for linear or circular polarization. In order to test for linearly polarized light, the retarder is removed from the optical train and the intensity is again given by Equation 14.127. We recall that the Stokes vector for elliptically polarized light is

S =

1

2

2

2

cos

sin cos

sin sin

.α

α δα δ

(5.73)

Substituting S1 and S2 from Equation 5.73 into Equation 14.127 gives

I( , ) [ cos cos sin cos sin ].θ α θ α δ θ012

1 2 2 2 2° = + + (14.132)

306 Polarized Light, Third Edition

The polarizer is again rotated. If we obtain a null intensity, then we know that we have linearly polarized light because Equation 14.127 can only become a null if δ = 0° or 180º, a condition for linearly polarized light. For this condition we can write Equation 14.132 as

I( , ) [ cos( )],θ α θ012

1 2 2 = + − (14.133)

which can only be zero if the incident beam is linearly polarized light. However, if we do not obtain a null intensity, we can have elliptically polarized light or circularly polarized light. To test for these possibilities, the quarter-wave retarder is reintroduced into the optical train so that the intensity is again given by Equation 14.128. Now, if we have circularly polarized light, then S1 must be zero so Equation 14.128 will become

I S S( , ) [ sin ].θ θ9012

20 3 = + (14.134)

The polarizer is again rotated. If a null intensity is obtained, then we must have circularly polarized light. If, on the other hand, a null intensity is not obtained, then we must have a condition described by Equation 14.128, which is elliptically polarized light.

To summarize, if a null intensity is not obtained with either the polarizer by itself or with the combination of the polarizer and the quarter-wave retarder, then we must have elliptically polarized light. Thus, by using a polarizer quarter-wave retarder combination, we can test for the polarization states. The only state remaining is partially polarized light. If none of these tests described above is successful, we then assume that the incident beam is partially polarized.

If we are certain that the light is elliptically polarized, then we can consider Equation 14.5 fur-ther. We can express Equation 14.5 as

I S S S S( , ) [ cos ( cos sin )sin ]θ φ θ φ φ θ= + + +12

2 20 1 2 3 (14.135)

or

I A b c( , ) ,θ φ θ θcos 2 sin 2= + +[ ] (14.136)

where

AS= 0

2, (14.137)

bS= 1

2, (14.138)

cS S= +2 3

2cos sin

.φ φ

(14.139)

For an elliptically polarized beam given by Equation 5.73, we see that from Equation 14.137 to Equation 14.139

A = 12

, (14.140)

Methods of Measuring Stokes Polarization Parameters 307

b = cos

,2

2α

(14.141)

c = −cos( )sin.

φ δ α22

(14.142)

The intensity Equation 14.136 can then be written as

I = + + −12

1 2 2 2 2[ cos cos sin cos( )sin ].α θ α φ δ θ (14.143)

We can find the maximum and minimum intensities of Equation 14.143 by differentiating Equation 14.143 with respect to θ and setting dI(θ)/dθ = 0. The angles where the maximum and minimum intensities occur are then found to be

tansincos

.222

θ θθ

= = = −−

c

b

c

b (14.144)

Substituting Equation 14.144 into Equation 14.136, the corresponding maximum and minimum intensities are

I A b c(max) ,= + +( )2 2 (14.145)

I A b c(min) .= − +( )2 2 (14.146)

From Equation 14.140 to Equation 14.142, we see that we can then write Equations 14.145 and 14.146 as

I(max,min) cos sin cos ( ) .= ± + −( )[ ]12

1 2 22 2 2α α φ δ (14.147)

Let us now remove the retarder from the optical train so that ϕ = 0°; we then have only a linear polarizer that can be rotated through θ. In this case, Equation 14.147 reduces to

I(max,min) cos sin cos .= ± +( )[ ]12

1 2 22 2 2α α δ (14.148)

For linearly polarized light, δ = 0° or 180º, so Equation 5.73 becomes

S =±

1

2

2

0

cos

sin,

αα

(14.149)

and Equation 14.148 becomes

I(max,min) [ ] , .= ± =12

1 1 1 0 (14.150)

308 Polarized Light, Third Edition

Thus, linearly polarized light always gives a maximum intensity of unity and a minimum intensity of zero (null).

If we have circularly polarized light, δ = 90° or 270º and α = 45°, as is readily shown by inspect-ing Equation 5.73. For this condition Equation 14.148 reduces to

I(max,min) [ ] ,= ± =12

1 012

(14.151)

so the intensity is always constant and reduced to 1/2. We also see that if we have only the condition δ = 90° or 270º, then Equation 5.73 becomes

S =

±

1

2

0

2

cos

sin

,α

α

(14.152)

which is the Stokes vector of an ellipse in a standard form (i.e., unrotated). The corresponding intensity is, from Equation 14.147,

I(max,min) [ cos ].= ±12

1 2α (14.153)

Similarly, if α = ±45° and δ is not equal to either 90º or 270º, then Equation 5.73 becomes

S =

1

0

cos

sin

,δδ

(14.154)

and Equation 14.148 reduces to

I(max,min) [ cos ].= ±12

1 δ (14.155)

This final analysis confirms the earlier results given in the first part of this chapter. We see that if we rotate a linear polarizer and we observe a null intensity at two angles over a single rotation, we have linearly polarized light; if we observe a constant intensity, we have circularly polarized light; and, if we observe maximum and minimum (non-null) intensities, we have elliptically polarized light.

In Figures 14.7 and 14.8, we have plotted the intensity as a function of the rotation angle of the analyzer. Specifically, in Figure 14.7 we show the intensity for the condition where the parameters of the incident beam described by Equation 5.73 are α = π/6 (30°) and δ = π/3 (60°); the compensator is not in the wave train, so φ = 0. According to Equation 5.73, the Stokes vector is

S =

1

1 2

3 4

3 4

/

/

/

. (14.156)

Methods of Measuring Stokes Polarization Parameters 309

0.9

1

0.7

0.8

0.5

0.6

0.4

Inte

nsity

0.2

0.3

0

0.1

0.00 0.63 1.26 1.88 2.51 3.14θ (radians)

3.77 4.40 5.03 5.65 6.28

figuRe 14.7 Intensity plot of an elliptically polarized beam for α = π/6 and δ = π/3 .

0.9

1

0.7

0.8δ = π/4

δ = π/2

δ = 0

0.5

0.6

0.4

Inte

nsity

0.2

0.3

0

0.1

0.00 0.63 1.26 1.88 2.51 3.14θ (radians)

3.77 4.40 5.03 5.65 6.28

figuRe 14.8 Plot of the intensity for a linearly polarized beam, an elliptically polarized beam, and a cir-cularly polarized beam.

310 Polarized Light, Third Edition

The intensity expected for Equation 14.156 is seen from Equation 14.143 to be

I( ) cos sin .θ θ θ= + +

12

112

23

42 (14.157)

The plot of Equation 14.157 is given in Figure 14.7.We see from Equation 14.156 that the square root of the sum of the squares S1, S2, and S3 is equal

to unity as expected. Inspecting Figure 14.8, we see that there is a maximum intensity and a mini-mum intensity. Because there is no null intensity, we know that the light is elliptically polarized, which agrees with Equation 14.156.

In Figure 14.8 we consider an elliptically polarized beam such that α = π/4 and we have arbitrary phase δ. This beam is described by the Stokes vector given in Equation 14.154. The corresponding intensity for Equation 14.154, according to Equation 14.143, is

I = +12

1 2[ cos sin ].δ θ (14.158)

We now consider Equation 14.154 for δ = 0, π/4, and π/2. The Stokes vectors corresponding to these conditions are

S S( )0

1

0

1

0

4

1

0

1

21

2

=

=

π

=

Sπ2

1

0

0

1

. (14.159)

The Stokes vectors in Equation 14.159 correspond to linear +45° polarized light, elliptically polar-ized light, and right circularly polarized light. Inspection of Figure 14.8 shows the corresponding plot for the intensities given by Equation 14.158 for each of the Stokes vectors in Equation 14.159. The linearly polarized beam gives a null intensity, the elliptically polarized beam gives maximum and minimum intensities, and the circularly polarized beam yields a constant intensity of 0.5.

RefeReNCeS

1. Stokes, G. G., On the composition and resolution of streams of polarized light from different sources, Trans. camb. Phil. Soc. 9 (1852): 399–416. Reprinted in Mathematical and Physical Papers, Vol. 3, 233, London: Cambridge University Press, 1901.

2. Collett, Ε., Measurement of the four Stokes polarization parameters with a single circular polarizer, Opt. commun. 52 (1984): 77–80.

3. Kent, C. V., and J. Lawson, A photoelectric method for the determination of the parameters of elliptically polarized light, J. Opt. Soc. Am. 27 (1937): 117.

311

15 Measurement of the Characteristics of Polarizing Elements

15.1 iNTRoduCTioN

In the previous chapter, we described a number of methods for measuring and characterizing polar-ized light in terms of the Stokes polarization parameters. We now turn our attention to measuring the characteristics of the three major optical polarizing elements, the polarizer (diattenuator), retar-der, and rotator. As in the previous chapter, the emphasis is on null, or manual, methods. Automated methods are addressed in Chapter 17.

For a polarizer it is necessary to measure the attenuation coefficients of the orthogonal axes, for a retarder the relative phase shift, and for a rotator the angle of rotation. It is of practical importance to make these measurements. Before proceeding with any experiment in which polarizing elements are to be used, it is good practice to determine if they are performing according to their specifica-tions. This characterization is also necessary because over time polarizing components change, for example, the optical coatings deteriorate, and in the case of Polaroid, the material becomes discolored. In addition, one finds that, in spite of one’s best laboratory controls, quarter-wave and half-wave retarders, which operate at different wavelengths, become mixed up. Finally, the quality control of manufacturers of polarizing components is not perfect, and imperfect components are sold.

The characteristics of all three types of polarizing elements can be determined by using a pair of high-quality calcite polarizers that are placed in high-resolution angular mounts; the polarizing element being tested is placed between these two polarizers. A practical angular resolution is 0.1° (6′ of arc) or less. High-quality calcite polarizers and mounts are expensive, but in a laboratory where polarizing components are used continually their cost is well justified.

15.2 meaSuRemeNT of aTTeNuaTioN CoeffiCieNTS of a PolaRiZeR (diaTTeNuaToR)

A linear polarizer is characterized by its attenuation coefficients px and py along its orthogonal x and y axes. We now describe the experimental procedure for measuring these coefficients. The measurement configuration is shown in Figure 15.1. In the experiment, the polarizer to be tested is inserted between the two polarizers as shown. The reason for using two polarizers is that the same configuration can also be used to test retarders and rotators. Thus, we can have a single, permanent, test configuration for measuring all three types of polarizing components.

The Mueller matrix of a polarizer (diattenuator) with its axes along the x and y directions is

M p

x y x y

x y x y

x

p p p p

p p p p

p p=

+ −− +1

2

0 0

0 0

0 0 2

2 2 2 2

2 2 2 2

yy

x

x y

p p

p

y

0

0 0 0 2

0 1

≤ ≤, . (15.1)

312 Polarized Light, Third Edition

It is convenient to rewrite Equation 15.1 as

M p

A b

b A

c

c

=

0 0

0 0

0 0 0

0 0 0

, (15.2)

where

A p px y= +12

2 2( ), (15.3)

b p px y= −12

2 2( ), (15.4)

c p px y= 12

2( ). (15.5)

In practice, while we are interested only in determining px2 and py

2 , it is useful to measure pxpv as well, because a polarizer satisfies the relation

A b c2 2 2= + , (15.6)

as the reader can easily show from Equations 15.3, 15.4, and 15.5. Equation 15.6 serves as a useful check on the measurements. The optical source emits a beam characterized by a Stokes vector

S =

S

S

S

S

0

1

2

3

. (15.7)

x

y

Generating polarizer

Analyzing polarizerTest polarizer

x

y

I

y

x

figuRe 15.1 Experimental configuration to measure the attenuation coefficients px and py of a polarizer (diattenuator).

Measurement of the Characteristics of Polarizing Elements 313

In the measurement, the first polarizer, which is often called the generating polarizer, is set to +45°. The Stokes vector of the beam emerging from the generating polarizer is then

S =

I0

1

0

1

0

, (15.8)

where I0 = (1/2)(S0 + S2) is the intensity of the emerging beam. The Stokes vector of the beam emerg-ing from the test polarizer is found to be, after multiplying Equations 15.2 and 15.8,

′ =

S I

A

b

c0

0

. (15.9)

The polarizer before the optical detector is often called the analyzing polarizer or simply the ana-lyzer. The analyzer is mounted so that it can be rotated to an angle α. The Mueller matrix of the rotated analyzer is (see Chapter 6)

MA = 12

1 2 2 0

2 2 2 2 0

2

2

cos sin

cos cos sin cos

sin

α αα α α ααα α α αsin cos sin

.2 2 2 0

0 0 0 0

2

(15.10)

The Stokes vector of the beam incident on the optical detector, multiplying Equation 15.9 by Equation 15.10, is

′ = + +

SI

A b c0

22 2

1

2

2

0

( cos sin )cos

sinα α

αα

, (15.11)

and the intensity of the beam is

II

A b c( ) ( cos sin ).α α α= + +0

22 2 (15.12)

15.2.1 fiRST MeaSuReMenT MeThod

By rotating the analyzer to α = 0°, 45°, and 90°, Equation 15.12 yields the equations

II

A b( ) ( ),020° = + (15.13)

314 Polarized Light, Third Edition

II

A c( ) ( ),4520° = + (15.14)

II

A b( ) ( ).9020° = − (15.15)

Solving for A, b, and c, we find that

AI I

I= +( ) ( )

,0 90

0

° ° (15.16)

bI I

I= −( ) ( )

,0 90

0

° ° (15.17)

cI I I

I= − −2 45 0 90

0

( ) ( ) ( ),

° ° ° (15.18)

which are the desired relations. From Equations 15.3 to 15.4 we also see that

p A bx2 = + , (15.19)

p A by2 = − , (15.20)

so that we can write Equations 15.13 and 15.15 as

pIIx

2

0

2 0= ( ),

° (15.21)

pI

Iy2

0

2 90= ( ).

° (15.22)

Thus, it is only necessary to measure I(0°) and I(90°), the intensities in the x and y directions, respectively, to obtain px

2 and py2 . The intensity I0 of the beam emerging from the generating

polarizer is measured without the polarizer under test and the analyzer in the optical train. It is not necessary to measure c. Nevertheless, experience shows that the additional measurement of I(45°) enables one to use Equation 15.6 as a check on the measurements.

In order to determine px2 and py

2 in Equations 15.21 and 15.22, it is necessary to know I0. However, a relative measurement of p py x

2 2/ is just as useful. We divide Equation 15.20 by Equation 15.19 and we obtain

p

pII

y

x

2

2

900

= ( )( )

.°°

(15.23)

We see that this type of measurement does not require knowledge of I0. Thus, measuring I(0°) and I(90°) and forming the ratio yields the relative value of the absorption coefficients of the polarizer.

Measurement of the Characteristics of Polarizing Elements 315

In order to obtain A, b, and c and then px2 and py

2 in the method described above, an optical detector is required. However, the magnitude of px

2 and py2 , can also be obtained using a null-inten-

sity method. To show this, Equation 15.6 suggests that it can be satisfied by b and c of the form

b A= cos ,γ (15.24)

c A= sin .γ (15.25)

Substituting Equations 15.24 and 15.25 into Equation 15.12, we then have

II A

( ) [ cos( )],α α γ= + −0

21 2 (15.26)

and we can write

tan ,γ = c

b (15.27)

where Equation 15.27 has been obtained by dividing Equation 15.25 by Equation 15.24. We see that I(α) leads to a null intensity at

α γnull °= +90

2, (15.28)

where αnull is the angle at which the null is observed. Substituting Equation 15.28 into Equation 15.27 then yields

c

b= tan .2αnull (15.29)

Thus by determining γ from the null-intensity condition, we can find b/A and c/A from Equations 15.24 and 15.25. For convenience we set A = 1, and then from Equations 15.19 and 15.20 we have

p bx2 1= + , (15.30)

p by2 1= − , (15.31)

and b is found from the result of the measurement as in Equation 15.17.The ratio c/b in Equation 15.29 can also be used to determine the ratio py/px, which we can then

square to form p py x2 2/ . Substituting Equations 15.4 and 15.5 into Equation 15.29 gives

tan .22

2 2αnull =

−p p

p px y

x y

(15.32)

The form of Equation 15.32 suggests that we set

p p p px y= =cos sin ,β β (15.33)

316 Polarized Light, Third Edition

so

tansincos

tan222

2α ββ

βnull = = (15.34)

and

β α= null . (15.35)

This leads immediately to

p

py

x

= =tan tan ,β αnull (15.36)

or, using Equation 15.28

p

py

x

2

22

2=

cot .

γ (15.37)

Thus, the shift in the intensity in Equation 15.26 enables us to determine p py x2 2/ directly from γ.

We always assume p py x2 2 1/ ≤ . A neutral density filter is described by p px y

2 2= so that the range on p py x

2 2/ limits γ to

90 180° °≤ ≤γ . (15.38)

For p py x2 2 0/ = , corresponding an ideal polarizer, γ = 180°, whereas for p py x

2 2 1/ = , correspond-ing to a neutral density filter, γ = 90° as shown by Equation 15.37. We see that the closer the value of γ is to 180°, the better is the polarizer. As an example, for commercial Polaroid HN22 at 0.550 μm, p py x

2 2 0 0 0/ 2 1 / 48 4 2 16 6= × = ×− − . . so from Equation 15.37 we see that γ = 179.77° and αnull = 179.88° ; the nearness of γ to 180° shows that it is an excellent polarizing material.

15.2.2 Second MeaSuReMenT MeThod

The parameters A, b, and C can also be obtained by Fourier-analyzing Equation 15.12, assum-ing that the analyzing polarizer can be continuously rotated over a half or full cycle. Recall that Equation 15.12 is

II

A b c( ) cos sin .α α α= + +( )0

22 2 (15.39)

From the point of view of Fourier analysis, A describes a d.c. term, and Β and c describe second-harmonic terms. It is only necessary to integrate over half a cycle, that is, from 0 to π, in order to determine A, b, and c. We easily find that

AI

I d= ∫2

0 0πα α

π

( ) , (15.40)

Measurement of the Characteristics of Polarizing Elements 317

bI

I d= ∫42

0 0πα α α

π

( )cos , (15.41)

cI

I d= ∫42

0 0πα α α

π

( )sin . (15.42)

Throughout this analysis we have assumed that the axes of the polarizer being measured lie along the x and y directions. If this is not the case, then the polarizer under test should be rotated to its x and y axes in order to make the measurement. The simplest way to determine rotation angle θ is to remove the polarizer under test and rotate the generating polarizer to 0° and the analyzing polarizer to 90°.

15.2.3 ThiRd MeaSuReMenT MeThod

Finally, another method to determine A, b, and C is to place the test polarizer in a rotatable mount between polarizers in which the axes of both are in the y direction. The test polarizer is then rotated until a minimum intensity is observed from which A, b, and c can be found. The Stokes vector emerging from the generating polarizer is

S =−

I0

2

1

1

0

0

. (15.43)

The Mueller matrix of the test polarizer Equation 15.2 rotated by θ is

M =+ −

A b b

b A c A c

cos sin

cos cos sin ( )s

2 2 0

2 2 22 2

θ θθ θ θ iin cos

sin ( )sin cos sin co

2 2 0

2 2 2 22

θ θθ θ θ θb A c A c− + ss

.2 2 0

0 0 0 0

θ

(15.44)

The intensity of the beam emerging from the analyzing polarizer is

II

A c b A c( ) [( ) cos ( )cos ].θ θ θ= + − + −0 2

42 2 2 (15.45)

Equation 15.45 can be solved for its maximum and minimum values by differentiating I(θ) with respect to θ and setting dI(θ)/dθ = 0 to obtain

sin [ ( )cos ] .2 2 0θ θb A c− − = (15.46)

For the left-hand side of this equation to be zero, either

sin 2 0θ = (15.47)

or

cos .2θ =−b

A c (15.48)

318 Polarized Light, Third Edition

The solutions of Equation 15.47 are θ = 0° and 90°. The corresponding values of the intensities are then, from Equation 15.45

II

A b( ) [ ],020° = − (15.49)

II

A b( ) [ ].9020° = + (15.50)

The second Solution (15.48), on substitution into Equation 15.45, leads to I(θ) = 0. Thus, the mini-mum intensity is given by Equation 15.49 and the maximum intensity by Equation 15.50. Because both the generating and analyzing polarizers are in the y direction, this is exactly what one would expect. We also note in passing that at θ = 45°, Equation 15.45 reduces to

II

A c( ) [ ].4540° = + (15.51)

We can divide Equations 15.49 through 15.51 by I0 and then solve the equations for A, b, and C.We see that several methods can be used to determine the absorption coefficients of the orthogo-

nal axes of a polarizer. In the first method, we generate a linear +45° polarized beam and then rotate the analyzer to obtain A, b, and c of the polarizer being tested. This method requires a quantitative optical detector. However, if an optical detector is not available, it is still possible to determine A, b, and c by using the null-intensity method; that is, rotation of the analyzer until a null is observed leads to A, b, and c. On the other hand, if the analyzer can be mounted in a rotat-able mount that can be stepped (electronically), then a Fourier analysis of the signal can be made and we can again find A, b, and c. Finally, if the transmission axes of the generating and analyz-ing polarizers are parallel to one another, conveniently chosen to be in the y direction, and the test polarizer is rotated, then we can also determine A, b, and c by rotating the test polarizer to 0°, 45°, and 90°.

15.3 meaSuRemeNT of The PhaSe ShifT of a ReTaRdeR

There are numerous occasions when it is important to know the phase shift of a retarder. The most common types of retarders are quarter-wave and half-wave retarders. These two types are most often used to create circularly polarized light and to rotate or reverse the polarization ellipse, respectively.

Two methods can be used for measuring the phase shift using two linear polarizers following the experimental configuration given in the previous section.

In the first method a retarder is placed between the two linear polarizers mounted in the crossed position. Let us set the transmission axes of the first and second polarizers to be in the x and y direc-tions, respectively. By rotating the retarder, the direction (angle) of the fast axis is rotated and, as we shall soon see, the phase can be found. The second method is very similar to the first except that the fast axis of the retarder is rotated to 45°. In this position the phase can also be found. We now consider both methods.

15.3.1 fiRST MeThod

For the first method we refer to Figure 15.2. It is understood that the correct wavelength must be used; that is, if the retarder is specified for 6328 Å, then the optical source should emit this

Measurement of the Characteristics of Polarizing Elements 319

wavelength. In the visible domain calcite polarizers are best. However, high-quality polaroid is also satisfactory, but its optical bandpass is much more restricted. In Figure 15.2 the transmission axes of the polarizers (or diattenuators) are in the x (horizontal) and y (vertical) directions. The Mueller matrix for the retarder rotated through an angle θ is

M( , )cos cos sin ( cos )sin c

φ θθ φ θ φ θ

=+ −

1 0 0 0

0 2 2 1 22 2 oos sin sin

( cos )sin cos sin co

2 2

0 1 2 2 22

θ φ θφ θ θ θ

−− + ss cos sin cos

sin sin sin cos cos

φ θ φ θφ θ φ θ φ

2 2 2

0 2 2−

,

(15.52)

where the phase shift ϕ is to be determined. The Mueller matrix for an ideal linear polarizer is

Mx y, ,=

±±

12

1 1 0 0

1 1 0 0

0 0 0 0

0 0 0 0

(15.53)

where the plus sign corresponds to a horizontal polarizer and the minus sign to a vertical polarizer. The Mueller matrix for the arrangement of Figure 15.2 is then

M M M M= y x( , ) .φ θ (15.54)

Carrying out the matrix multiplication in Equation 15.54 using Equations 15.52 and 15.53 then yields

M = −( ) −( ) − −

1 1 48

1 1 0 0

1 1 0 0

0 0 0 0

0 0 0 0

cos cosφ θ

.

(15.55)

x

Generating polarizerAxis is horizontal

Analyzing polarizerAxis is vertical

Retarder

y

I

– /2

+ /2

figuRe 15.2 Closed polarizer method to measure the phase of a retarder.

320 Polarized Light, Third Edition

Equation 15.55 shows that the polarizing train behaves as a pseudopolarizer. The intensity of the optical beam on the detector is then

I I( , )cos cos

,θ φ φ θ= −( ) −( )0

1 1 44

(15.56)

where I0 is the intensity of the optical source.Equation 15.56 immediately allows us to determine the direction of the fast axis of the retarder.

When the retarder is inserted between the crossed polarizers, the intensity on the detector should be zero, according to Equation 15.56, at θ = 0°. If it is not zero, the retarder should be rotated until a null intensity is observed. After this angle has been found, the retarder is rotated 45° according to Equation 15.56 to obtain the maximum intensity. In order to determine ϕ, it is necessary to know I0. The easiest way to do this is to rotate the axis of the first polarizer so that it is parallel to the second and remove the retarder; the axes of both linear polarizers are then in the y direc-tion. The intensity Id on the detector is then (let us assume that unpolarized light enters the first polarizer)

II

d = 0

2, (15.57)

so Equation 15.56 can be written as

I II

d( , )( cos )( cos )

.θ φ φ θ= − −1 42

(15.58)

The retarder is now reinserted into the polarizing train. The maximum intensity occurs when the retarder is rotated to θ = 45°. At this angle Equation 15.58 is solved for ϕ, and we have

φ φ= −

−cos( , )

.1 145I

Id

° (15.59)

The disadvantage of using the crossed-polarizer method is that it requires that we know the intensity of the beam, I0, entering the polarizing train. This problem can be overcome by another method—that of rotating the analyzing polarizer and fixing the retarder at 45°. We now consider this second method.

15.3.2 Second MeThod

The experimental configuration is identical to the first method except that the analyzer can be rotated through an angle α. The Stokes vector of the beam emerging from the generating polarizer is (again let us assume that unpolarized light enters the generating polarizer)

S =

I0

2

1

1

0

0

. (15.60)

Measurement of the Characteristics of Polarizing Elements 321

Multiplication of Equation 15.60 by Equation 15.52 yields

′ =+

−S

I02 2

2

1

2 2

1 2 2

cos cos sin

( cos )sin cos

s

θ φ θφ θ θ

iin sin

.

φ θ2

(15.61)

We assume that the fast axis of the retarder is at θ = 0°. If it is not, the retarder should be adjusted to θ = 0° by using the crossed-polarizer procedure described in the first method; we note that at θ = 0°, Equation 15.61 reduces to

′ =

SI0

2

1

1

0

0

, (15.62)

so that the analyzing polarizer should give a null intensity when it is in the y direction. Assuming that the retarder’s fast axis is now properly adjusted, we rotate the retarder counterclockwise to θ = 45°. Then Equation 15.61 reduces to

′ =

SI0

2

1

0

cos

sin

.φ

φ

(15.63)

This is a Stokes vector for elliptically polarized light. The conditions ϕ = 90° and 180° correspond to right circularly polarized and linear vertically polarized light, respectively. We note that the lin-ear vertically polarized state arises because for ϕ = 180°, the retarder behaves as a pseudorotator. The Mueller matrix of the analyzing polarizer is

M( )

cos sin

cos cos sin cos

siφ

α αα α α α

= 12

1 2 2 0

2 2 2 2 02

nn sin cos sin.

2 2 2 2 0

0 0 0 0

2α α α α

(15.64)

The Stokes vector of the beam emerging from the analyzer is then

S = +

I0

41 2

1

2

2

0

( cos cos )cos

sin,φ α

αα

(15.65)

322 Polarized Light, Third Edition

so the intensity is

II

( , ) ( cos cos ).α φ φ α= +0

41 2 (15.66)

In order to find ϕ, Equation 15.66 is evaluated at α = 0° and 90°, and

II

04

10°, cos ,φ φ( ) = +( ) (15.67)

II

904

10°, cos .φ φ( ) = −( ) (15.68)

Equation 15.67 is divided by Equation 15.68 and solved for cosϕ to obtain

cos( , ) ( , )( , ) ( , )

.φ φ φφ φ

= −+

I II I

0 900 90

° °° °

(15.69)

We note that in this method the source intensity need not be known.We can also determine the direction of the fast axis of the retarder in a dynamic fashion. The

intensity of the beam emerging from the analyzer when it is in the y position is (see Equations 15.61 and 15.64)

II

y = − +( )[ ]0 2 2

41 2 2cos cos sin ,θ φ θ (15.70)

where θ is the angle of the fast axis measured from the horizontal axis. We now see that when the analyzer is in the x position that

II

x = + +( )[ ]0 2 2

41 2 2cos cos sin .θ φ θ (15.71)

Adding Equations 15.70 and 15.71 yields

I II

x y+ = 0

2. (15.72)

Next, subtracting Equations 15.70 from 15.71 yields

I II

x y− = +( )0 2 2

22 2cos cos sin .θ φ θ (15.73)

We see that when θ = 0, the sum and difference intensities of Equations 15.72 and 15.73 are equal. Thus, one can measure Ix and Iy continuously as the retarder is rotated and the analyzer is flipped between the horizontal and vertical directions until Equation 15.72 equals Equation 15.73. When this occurs, the amount of rotation that has taken place determines the magnitude of the rotation angle of the fast axis from the x axis.

Measurement of the Characteristics of Polarizing Elements 323

15.3.3 ThiRd MeThod

Finally, if a compensator is available, the phase shift can be measured as follows. Figure 15.3 shows the measurement method. The compensator is placed between the retarder under test and the ana-lyzer. The transmission axes of the generating and analyzing polarizers are set at +45° and +135°, that is, in the crossed position. The Stokes vector of the beam incident on the test retarder is

S =

I0

2

1

0

1

0

. (15.74)

The Mueller matrix of the test retarder is

M =

−

1 0 0 0

0 1 0 0

0 0

0 0

cos sin

sin cos

φ φφ φ

.. (15.75)

Multiplying Equation 15.74 by Equation 15.75 yields

S =

−

I0

2

1

0

cos

sin

.φφ

(15.76)

The Mueller matrix of a Babinet–Soleil compensator is

M =

−

1 0 0 0

0 1 0 0

0 0

0 0

cos sin

sin cos

∆ ∆∆ ∆

.. (15.77)

Linear polarizer at +45°Babinet–Soleilcompensator

Retarder

+ /2

– /2+45°

+∆/2

–∆/2

Linear polarizer at –45°

–45°

figuRe 15.3 Measurement of the phase shift of a wave plate using a Babinet–Soleil compensator.

324 Polarized Light, Third Edition

Multiplying Equation 15.76 by Equation 15.77 yields the Stokes vector of the beam incident on the linear −45° polarizer so that

S =+( )

− +( )

I0

2

1

0

cos

sin

.∆∆

φφ

(15.78)

Finally, the Mueller matrix for the ideal linear polarizer with its transmission axis at −45° ( +135°) is

M =

−

−

12

1 0 1 0

0 0 0 0

1 0 1 0

0 0 0 0

. (15.79)

Multiplying Equation 15.78 by the first row of Equation 15.79 gives the intensity on the detector

II

( ) [ cos( )].∆ ∆+ = − +φ φ0

41 (15.80)

We see that a null intensity is found at

∆ = −360° φ, (15.81)

from which we then find φ.There are still other methods to determine the phase of the retarder, and the techniques devel-

oped here can provide a useful starting point. However, the methods described here should suffice for most problems.

15.4 meaSuRemeNT of RoTaTioN aNgle of a RoTaToR

The final type of polarizing element that we wish to characterize is the rotator. The Mueller matrix of a rotator is

M =−

1 0 0 0

0 2 2 0

0 2 2 0

0 0 0 1

cos sin

sin cos

θ θθ θ

. (15.82)

15.4.1 fiRST MeThod

The angle θ can be determined by inserting the rotator between a pair of polarizers in which the generating polarizer is fixed in the y position and the analyzing polarizer can be rotated.

Measurement of the Characteristics of Polarizing Elements 325

This configuration is shown in Figure 15.4. The Stokes vector of the beam incident on the rotator is

S =−

I0

2

1

1

0

0

. (15.83)

The Stokes vector of the beam incident on the analyzer is then found by multiplying Equation 15.83 by Equation 15.82 to obtain

′ =−

SI0

2

1

2

2

0

cos

sin.

θθ

(15.84)

The Mueller matrix of the analyzer is

M = 12

1 2 2 0

2 2 2 2 0

2

2

cos sin

cos cos sin cos

sin

α αα α α αα ssin cos sin

.2 2 2 0

0 0 0 0

2α α α

(15.85)

The intensity of the beam emerging from the analyzer is then seen from the product of Equations 15.85 and 15.84 to be

II

( ) cos .α α θ= − +( )[ ]0

41 2 2 (15.86)

The analyzer is rotated and, according to Equation 15.86, a null intensity will be observed at

α θ= −180° , (15.87)

Generating polarizeraxis is vertical

Analyzing polarizerRotator

I

θα

figuRe 15.4 Measurement of the rotation angle θ of a rotator.

326 Polarized Light, Third Edition

or, simply,

θ α= −180° . (15.88)

15.4.2 Second MeThod

Another method for determining the angle θ is to rotate the generating polarizer sequentially to 0°, 45°, 90°, and 135°. The rotator and the analyzing polarizer are fixed with their axes in the horizontal direction. The intensities of the beam emerging from the analyzing polarizer for these four angles are then

II

04

1 20°( ) = +( )cos ,θ (15.89)

II

454

1 20°( ) = +( )sin ,θ (15.90)

II

904

1 20°( ) = −( )cos ,θ (15.91)

II

1354

1 20°( ) = −( )sin .θ (15.92)

Subtracting Equation 15.91 from Equation 15.89, and subtracting Equation 15.92 from Equation 15.90 yields

I

I I0

22 0 90

= ( ) − ( )cos ,θ ° ° (15.93)

I

I I0

245 135

= ( ) − ( )sin .θ ° ° (15.94)

Dividing Equation 15.94 by Equation 15.93 then yields the angle of rotation

θ = ( ) − ( )[ ] ( ) − ( )[ ] −tan .1 45 135 0 90I I I I° ° / ° ° (15.95)

In the null-intensity method an optical detector is not required, whereas in this second method a photodetector is needed. However, one soon discovers that even a null measurement can be improved by several orders of magnitude below the sensitivity of the eye by using an optical detector-amplifier combination.

As with the measurement of retarders, other configurations can be considered. However, the two methods described here should suffice for most problems.

327

16 Stokes Polarimetry

16.1 iNTRoduCTioN

In this chapter, we discuss methods of measuring (or creating) the Stokes vector, the real four-element entity that describes the state of polarization of a beam of light. The measurement process can be represented as

I AS= , (16.1)

where I is the vector of flux measurements as made by the detector, A is a matrix whose dimensions depend upon the number of measurements and whose elements depend on the optical system, and S is the incident Stokes vector. Since we want to determine the incident Stokes vector, we must invert Equation 16.1 so that S is given by

S A I= −1 . (16.2)

This system of equations is generated through a set of measurements and can be solved through Fourier or non-Fourier techniques. Both solution methods will be discussed in this chapter. The methods described in this chapter lend themselves to automated procedures.

A set of elements that analyzes a polarization state of incoming light is a polarization state ana-lyzer (PSA). A set of elements that generates a polarization state is a polarization state generator (PSG). The PSA and PSG are functionally depicted in Figure 16.1. All of the polarimeter types described in this chapter can be or have to be used with electronics and computers in order to auto-mate the data collection process.

A Stokes polarimeter is complete if it measures all four elements of the Stokes vector, and incom-plete if it measures less than four. We will describe several types of Stokes polarimeters in the remainder of the chapter. Rotating element polarimetry, oscillating element polarimetry, and phase modulation polarimetry are all methods that make a series of measurements over time to obtain the Stokes vector [1]. Other techniques, division of amplitude and division of wavefront polarimetry, described in the last section of the chapter, are designed to measure all four elements of the Stokes vector simultaneously.

16.2 RoTaTiNg elemeNT PolaRimeTRy

Stokes polarimeters that use rotating elements are shown in Figure 16.2. The elements shown are all linear retarders and polarizers (analyzers). The measured Stokes elements are shown in the box to the right of each diagram, where the large black dots indicate the Stokes components that are measured.

16.2.1 RoTaTing analyzeR PolaRiMeTeR

Shown in Figure 16.2a, the polarizer (analyzer) in this polarimeter rotates and produces a modulat-ing signal at the detector that is given by

Ia a b= + +0 2 2

2 22

22cos sinθ θ, (16.3)

328 Polarized Light, Third Edition

where θ is the azimuthal angle of the polarizer. The coefficients a0, a2, and b2 are the first three elements of the Stokes vector. At least three measurements must be made to determine the three measurable elements of the Stokes vector.

Equation 16.3 and subsequent expressions for the modulated signal in this chapter on Stokes polarimetry and the next chapter on Mueller matrix polarimetry are all derived from algebraic equations representing these polarimetric systems. For example, for the rotating analyzer polarim-eter, we have the equation

′′′′

=

S

S

S

S

0

1

2

3

12

1 cos2 sin2 0

cos2

θ θθθ θ θ θθ θ θ θ

cos sin2 cos2 0

sin2 sin2 cos sin 0

0 0

2

2

2

2 2

00 0

S

S

S

S

0

1

2

3

, (16.4)

where the input Stokes vector is multiplied by the Mueller matrix for a rotated ideal linear polarizer to obtain the (primed) output Stokes vector. We only need carry out the multiplication of the first row of the Mueller matrix with the input Stokes vector because we will be measuring the output signal I S= ′0 . Thus,

IS S S= + +0 1

2 2cos2

2sin22θ θ. (16.5)

Comparing this equation with Equation 16.3, we have the correspondence

S a S a S b0 0 1 2 2 2= = = . (16.6)

Polarization state analyzer with detector

Polarization state analyzer(a)

(b)

Detector

Polarization state generator with source

Polarization state generator

Source

figuRe 16.1 Functional diagrams of Stokes polarimetry.

Stokes Polarimetry 329

The coefficients have been purposely written as a’s and b’s to represent the modulated signal as a Fourier series where the fundamental frequency of modulation and its harmonics are the angle θ and its multiples. We will continue to do this for the polarimeters described in this chapter and the next.

16.2.2 RoTaTing analyzeR and fixed analyzeR PolaRiMeTeR

A fixed analyzer in front of the detector in this configuration shown in Figure 16.2b means that the detec-tor sees only one polarization, and any detector polarization sensitivity is made superfluous. A modu-lated signal composed of two frequencies is measured, and can be expressed as the Fourier series

Ia

a n b nn n

n

= + +( )=

∑02 2

1

2

414

2 2cos sinθ θ . (16.7)

The first three elements of the Stokes vector are

S a a

S a a a

S b b

0 0 4

1 2 0 4

2 2 4

23

2

0 4 2

= −

= − +( )

= +( )

,

,

. .

(16.8)

16.2.3 RoTaTing ReTaRdeR and fixed analyzeR PolaRiMeTeR

This is the basic complete Stokes polarimeter and is illustrated in Figure 16.2c. The detector observes only a single polarization, and the modulated signal is again composed of two frequencies. The signal is again expressed as a Fourier series

Ia

a n b nn n

n

= + +( )=

∑02 2

1

2

212

2 2cos sinθ θ (16.9)

where now the angle θ is the azimuthal angle of the retarder. If the retarder is quarter wave, the Stokes vector is given in terms of the Fourier coefficients as

S a a

S a

S b

S b

0 0 4

1 4

2 4

3 2

2

2

= −

=

=

=

,

,

,

.

(16.10)

16.2.4 RoTaTing ReTaRdeR and analyzeR PolaRiMeTeR

Both elements rotate in this polarimeter of Figure 16.2d. When the analyzer is rotated at three times the retarder angle and the retarder is quarter wave, the detected signal is given by

Ia

a n b nn n

n

= + +( )=

∑02 2

1

3

212

2 2cos sinθ θ , (16.11)

330 Polarized Light, Third Edition

Rotating analyzer

(a)

(b)

(c)

(d)

(e)

A

Detector

Rotating analyzer plus fixed analyzer

Detector

AA

Rotating retarder plus fixed analyzer R

Detector

A

Rotating retarder and rotating analyzerR

Detector

A

Rotating retarder, rotating analyzerand fixed analyzer

R

Detector

A A

figuRe 16.2 Rotating element polarimeters.

Stokes Polarimetry 331

where θ is the rotation angle of the retarder. The Stokes vector is

S a

S a a

S b b

S b

0 0

1 2 6

2 6 2

3 4

=

= +

= −

=

,

,

,

.

(16.12)

16.2.5 RoTaTing ReTaRdeR and analyzeR PluS fixed analyzeR PolaRiMeTeR

This case, combining the previous two cases and shown in Figure 16.2e, produces as many as nine harmonics in the detected signal when the analyzer is rotated by the factors 5/2, 5/3, or –3/2 times the retarder angle so that

Ia

a n b nn n

nn

= + +( )=≠

∑0

19

10

414

cos sinθ θ . (16.13)

The Stokes vector is given in terms of the Fourier coefficients, when the rotation factor is 5/2 and the fixed analyzer is at 0°, as

S a a

S a

S b

S b

0 0 4

1 1

2 1

3 3

2

2

= −

=

=

=

,

,

,

.

(16.14)

16.3 oSCillaTiNg elemeNT PolaRimeTRy

Oscillating element polarimeters rotate the polarization of light using some electro- or magneto-optical device such as a Faraday cell or a liquid crystal cell (see Chapter 21). If, for example, the plane of polarization is rotated by an angle θ in a Faraday cell, this has the effect of having mechani-cally rotated all subsequent elements by an angle –θ. The modulation is typically sinusoidal, which simulates an oscillating element, although a saw tooth signal could be used to drive the modulation to result in an equivalent to a synchronous rotation of the element. The advantages of oscillating element polarimeters include operation at high frequencies, and the absence of moving parts to dis-turb alignment. A disadvantage, when the modulation is sinusoidal, is the additional complication in the signal content. The azimuthal angles are sinusoids, and the detected intensity now contains an infinite number of harmonics whose amplitudes depend upon Bessel functions of the modulation amplitude. Oscillating element polarimeters derive harmonic content from the relationships (Bessel function expansions)

sin sin sinθ ω θ ωt J n tn

n

( ) = ( ) +( )[ ]+=

∞

∑2 2 12 1

0

, (16.15)

cos sin cosθ ω θ θ ωt J J n tn

n

( ) = ( ) + ( )=

∞

∑0 2

0

2 2 . (16.16)

332 Polarized Light, Third Edition

Experimentally, a lock-in amplifier is required for each detected frequency. Three oscillating ele-ment polarimeters are shown in Figure 16.3 and we describe these polarimeters in the following subsections.

16.3.1 oScillaTing analyzeR PolaRiMeTeR

The oscillating analyzer polarimeter (see Ref. [2]) is shown in Figure 16.3a. This polarimeter, like the rotating analyzer polarimeter, measures the first three components of the Stokes vector and hence is an incomplete polarimeter. The oscillating element produces an effective analyzer azimuth of

θ θ θ ω= +0 1 sin t, (16.17)

where the azimuth θ0 is determined by the mechanical azimuth of the fixed analyzer and/or a d.c. bias current in the Faraday cell, and θ1 is the amplitude of the sinusoidal optical rotation produced by the Faraday cell. Substituting Equation 16.17 into Equation 16.3 we have

Ia a b

ta= + + ( ) + −0 2 0 2 0

12

22 2

22

cos sincos sin

siθ θ θ ω nn cossin sin

2 22

20 2 01

θ θ θ ω+ ( )bt . (16.18)

Oscillating analyzer polarimeter

(a)

(b)

(c)

Detector

AOE

r

Oscillating retarder and fixed analyzer

Detector

A

–r+r

ROE OE

Oscillating retarder and analyzer

Detector

OE R2 AROE R1

r1 r2

figuRe 16.3 Oscillating element polarimeters.

Stokes Polarimetry 333

If we now use Equations 16.15 and 16.16 to replace cos (2θ1 sin ωt) and sin (2θ1 sin ωt), we have

Ia a b

J J= + +

( ) +0 2

02

0 0 1 2 12 22

22 2 2 2cos sinθ θ θ θ(( )[ ]

+ − +

(

cos

sin cos

2

22

22 2 22

02

0 1 1

ω

θ θ θ

t

a bJ ))[ ]sinω t ,

(16.19)

where we have neglected terms in frequency higher than 2ω.The zero frequency (d.c.), fundamental, and second harmonic of the detected signal are then

I Ja b

0 1 22

22

20 12

02

0( ) = + ( ) +

θ θ θcos sin ,,

sin cosI Ja bω θ θ θ( ) = ( ) − +

2 22

22

21 12

02

0

( ) = ( ) +

sin ,

cos sin

ω

ω θ θ θ

t

I Ja b

2 2 22

22

22 12

02

0

cos2ω t.

(16.20)

The d.c., fundamental, and second harmonic of the signal are detected synchronously, and the amplitude ratios are

η ω

η ω

ω

ω

= ( ) ( )

= ( ) ( )

I I

I I

/ ,

/ ,

0

2 02

(16.21)

and these are, using Equation 16.20,

ηθ θ θ

θω =

( ) − +

+ (

2 22

22

2

1 2

2 12

02

0

0 1

Ja b

J

sin cos

)) +

=( )

a b

Ja

20

20

2

2 12

22

22

2 22

cos sin,

co

θ θ

ηθ

ω

ss sin

cos si

22

2

1 22

22

02

0

0 12

02

θ θ

θ θ

+

+ ( ) +

b

Ja b

nn2 0θ

.

(16.22)

These last equations can be inverted to give the coefficients

aJ JJ2

2 1 0 2 1 1 0

1 1

2 2 2 22

= ( ) − ( )( )

η θ θ η θ θθ η

ω ωsin cos

22 0 1 2 1

22 1 0

2 2 2

2 2

ω

ω

θ θ

η θ θ η

J J

bJ

( ) − ( )[ ]

= − ( ) −

,

cos 22 1 1 0

1 1 2 0 1 2 1

2 22 2 2 2

ω

ω

θ θθ η θ θ

JJ J J

( )( ) ( ) − ( )[

sin

]] .

(16.23)

334 Polarized Light, Third Edition

If θ = 0° and 2θ1 = 137.8°, J0(2θ1) = 0, and the Stokes vector is given by

′ =

′ = ( )

′ = ( )

S I

SJ

SJ

0 0

12

2 1

21 1

2 2

2 2

,

,

,

ηθ

ηθ

ω

ω

(16.24)

where the primes indicate the output Stokes parameters.

16.3.2 oScillaTing ReTaRdeR wiTh fixed analyzeR PolaRiMeTeR

This polarimeter, the equivalent of the rotating retarder polarimeter, is shown in Figure 16.3b. As indicated in the figure, this is a complete Stokes polarimeter. A retarder is surrounded by two opti-cal rotators with equal and opposite rotations. For example, a quarter-wave retarder might have a Faraday cell on one side and an identical Faraday cell on the other side but connected to an electrical signal source of opposite polarity. A light beam passing through a linear retarder of retardance δ with fast axis azimuth θR and a linear polarizer (analyzer) of azimuth θA results in an output intensity corresponding to the first Stokes parameter of the emergent light; that is,

′ = + −( ) − −SS S

R A R R A00 1

2 22 2 2 2 2cos cos sin sinθ θ θ θ θ 22

22 2 2 22

θ δ

θ θ θ θ

R

R A R R

S

( )[ ]

+ −( ) +

cos

cos cos sisin nn cos

sin sin

2 2

22 23

θ θ δ

θ θ δ

A R

A R

S

−( )[ ]

+ −( )[ ].

(16.25)

If we assume that δ π= /2 and θA = 0, and I is the detected signal and k is a proportionality constant, then we have

kI S S S S SR R= +

+ + −0 1 1 2 3

12

12

412

4 2cos sin sinθ θ θθR (16.26)

or

kI R R R= + + −β β θ β θ β θ0 1 2 34 4 2cos sin sin , (16.27)

where

β0 0 112

= +S S , (16.28a)

β1 112

= S , (16.28b)

Stokes Polarimetry 335

β2 212

= S , (16.28c)

and

β3 3= −S . (16.28d)

The two optical rotators on either side of the retarder effectively oscillate the retarder azimuth and we have

θ θ θ ωR R R t= +0 1

sin , (16.29)

where θR0is the bias azimuth, and θR1

is the rotation amplitude. Using Equation 16.29 in Equation 16.27 and again making use of the Bessel functions expansions, we can obtain the Fourier ampli-tudes of the detected signal as

kI J Jdc R R R R= + ( )[ ]+β β θ θ β θ θ0 1 0 2 04 4 4 40 1 0 1

cos sin (( )[ ]+ ( )[ ]β θ θ3 02 20 1

sin R RJ , (16.30a)

kI J JR R R Rω β θ θ β θ θ= − ( )[ ]+ (1 1 2 12 4 4 2 4 40 1 0 1

sin cos ))[ ]+ ( )[ ]β θ θ3 12 2 20 1

cos R RJ , (16.30b)

kI J JR R R R2 1 2 2 22 4 4 2 4 40 1 0 1ω β θ θ β θ θ= ( )[ ]+ (cos sin ))[ ]+ ( )[ ]β θ θ3 22 2 2

0 1sin R RJ , (16.30c)

kI J JR R R R3 1 3 2 32 4 4 2 4 40 1 0 1ω β θ θ β θ θ= − ( )[ ]+sin cos (( )[ ]+ ( )[ ]β θ θ3 32 2 2

0 1cos R RJ . (16.30d)

In vector–matrix form, the last three equations are

k

I

I

I

JR R Rω

ω

ω

θ θ θ

2

3

12 4 4 2 40 1 0

=

− ( )sin cos JJ J

J

R R R

R R

1 1

2

4 2 2 2

2 4 4 21 0 1

0 1

θ θ θ

θ θ( ) ( )

( )cos

cos siin sin

sin

4 4 2 2 2

2 4 40 1 0 1

0

2 2

3

θ θ θ θ

θR R R R

R

J J

J

( ) ( )− θθ θ θ θ θR R R R RJ J

1 0 1 0 12 4 4 2 2 23 3( ) ( ) ( )

cos cos

βββ

1

2

3

. (16.31)

This equation can be solved for β1, β2, and β3 by inverting the 3 × 3 matrix. Equation 16.30a can then be used to find β0, and Equation 16.28 used to find the Stokes vector elements.

16.3.3 oScillaTing ReTaRdeR and analyzeR PolaRiMeTeR

The oscillating retarder and analyzer polarimeter is the generalization of oscillating element designs [3]. This polarimeter is shown in Figure 16.3c. A retarder is surrounded by two optical rotators as in the oscillating retarder and fixed analyzer polarimeter, but now the rotators produce rotations θr1

and θr2. The retarder is oriented at some angle θR and the linear polarizer is oriented

at some angle θP. With no optical rotators, the detected signal is given by

kI S S S SR R P R= + +( ) −( ) +0 1 2 32 2 2 2cos sin cos sinθ θ θ θ 22 2θ θP R−( ). (16.32)

Consider that the rotator R2 in Figure 16.3c is replaced by two equivalent rotators in series that have rotations –r1 and r1 + r2. The sum of these is r2 and we have not changed the resultant net rotation.

336 Polarized Light, Third Edition

The retarder is now surrounded by rotators with rotations r1 and –r1 and this is equivalent to the retarder in the new azimuth θR + r1. The rotator with rotation r1 + r2 rotates the polarizer azimuth to θP + r1 + r2. If we replace the angles in Equation 16.32 with the azimuthal angles resulting from the addition of the rotators, we have

kI S S r r

S

R P R= + +( ) − +( )

+

0 1 1 2

2

2 2 2 2 2

2

cos cos

sin

θ θ θ

θθ θ θ

θ θ

R P R

P R

r r

S r

+( ) − +( )

+ − +(

2 2 2 2

2 2 2

1 2

3 2

cos

sin ))

(16.33)

If we reference the angular coordinates to the azimuth of the polarizer, we can set θP = 0 and rewrite Equation 16.33 as

kI S S r r rR R= + −( ) − −0 1 1 2 1

12

4 2 2 4 2 2cos cos sin sinθ θ rr r r

S r rR

2 1 2

2 1 2

2 2

12

4 2 2

( ) + +( )[ ]

+ −( )

cos

sin cosθ ++ −( ) + +( )[ ]

−

cos sin sin

sin

4 2 2 2 2

2

1 2 1 2

3

θ

θ

R r r r r

S RR Rr rcos cos sin2 2 22 2−( )θ

(16.34)

Now consider that the rotators are oscillated at the same frequency and are either in phase or out of phase by π, then the rotations produced are given by

r tr1 1= θ ωsin (16.35a)

and

r tr2 2= θ ωsin . (16.35b)

We now can substitute the expressions of Equation 16.35 into Equation 16.34 and again use the Bessel function expansions of Equations 16.15 and 16.16 to obtain the equation

k nI MS= , (16.36)

where

I S=

=

I

I

I

S

S

Sn

ω

ω

ω

2

3

1

2

3

, (16.37)

and

M =− −( ) −( ) +sin cos4 2 2 4 2 21 21 2 1 2

θ θ θ θ θ θR r r R r rJ J J22 1

2

2 2 2 2 2

4 2 21 2 2

1

θ θ θ θθ θ θ

r r R r

R r

J

J

+( ) ( )−

cos

cos rr r r R r rJ J2 1 2 1 22 22 2 4 2 2 2( ) + +( ) −( ) −θ θ θ θ θsin sin22 2

4 2 2 4 2

2

3 2

2

1 2

θ θθ θ θ θ θ

R r

R r r R r

J

J J

( )− −( )sin cos

11 2 1 2 22 2 2 2 2 22 3−( ) + +( ) ( )

θ θ θ θ θr r r R rJ Jcos

.

(16.38)

Stokes Polarimetry 337

The zero frequency term is given by

I S S J Jdc R r r r r= + −( ) + +(0 1 0 012

4 2 2 2 21 2 1 2

cos θ θ θ θ θ ))[ ]

+ −( )[ ]−12

4 2 2 2 22 0 3 01 2S J S JR r r Rsin sinθ θ θ θ θrr2

( )[ ]. (16.39)

Sn is found by multiplying the signal vector I by the inverse of M and then S0 is obtained from Equation 16.39.

16.4 PhaSe modulaTioN PolaRimeTRy

Phase modulation polarimeters are shown in Figure 16.4. These polarimeters use devices that vary in retardance in response to an electrical signal. A common type of phase modulator is the photo-elastic modulator (see Chapter 21).

16.4.1 PhaSe ModulaToR and fixed analyzeR PolaRiMeTeR

This polarimeter, shown in Figure 16.4a, uses a single modulator with a fixed linear analyzer. The axes of the modulator and analyzer are inclined at 45° to each other.The detected signal is given by

IS

S S SA A= + +( ) ∆ + ∆01 2 32

12

2 2cos sin cos sinθ θ (16.40)

A single modulator with fixed analyzer

Detector

A

0°45°(a)

(b)

(c)

PM

A single modulator with fixed analyzer

Detector

A

45°90°

PM

Dual modulators with fixed analyzer

Detector

PM APM

0°0° 45°

figuRe 16.4 Phase modulation polarimeters.

338 Polarized Light, Third Edition

where θA is the azimuthal angle of the analyzer and Δ is the retardance of the modulator. The modu-lator retardance is

∆ = δ ωsin t, (16.41)

where ω is the frequency of modulation and δ is the magnitude of the modulation. The detected intensity is given by

II I

tI

t= + +0 1 2

2 2 22sin cosω ω . (16.42)

If δ = 137.8° [J0(δ) = 0 and θ = 0°] the Stokes vector is given by

S I

SI

J

SI

J

0 0

12

2

31

1

2

2

=

= ( )

= ( )

,

,δ

δ.

(16.43)

If the polarimeter elements are both rotated by 45° (see Figure 16.4b), we will measure the Stokes vector as

S I

SI

J

SI

J

0 0

22

2

31

1

2

2

=

= ( )

= ( )

,

,δ

δ.

(16.44)

16.4.2 dual PhaSe ModulaToR and fixed analyzeR PolaRiMeTeR

The dual phase modulator and fixed analyzer polarimeter is shown in Figure 16.4c. The first modu-lator (closest to the analyzer) is aligned 45° to the analyzer and has time-varying retardation

∆1 1 1= δ ωsin t. (16.45)

The second modulator, aligned to the analyzer axis, has time-varying retardation

∆2 2 2= δ ωsin t. (16.46)

All four Stokes parameters can be measured with this system. The signal is

I

S S S S= + ∆ + ∆ ∆ − ∆ ∆0 1 2 2 2 1 3 2 1

2 2 2 2cos sin sin sin cos

, (16.47)

Stokes Polarimetry 339

and if we demand that δ1 = δ2 = 137.8° then

II I t I I= + ± ±( ) + ±(0 1 2 2 2 1 3 2 1

22

2 22cos cos sinω ω ω ω ω ))t

2, (16.48)

and higher frequency terms. The Stokes vector is then given by

S I

SI

J

SI

J J

SI

0 0

11

2 2

22

1 1 1 2

3

2

2

=

= ( )

= ( ) ( )

= −

,

,

,

δ

δ δ

33

2 1 1 22J Jδ δ( ) ( ) .

(16.49)

16.5 TeChNiQueS iN SimulTaNeouS meaSuRemeNT of STokeS VeCToR elemeNTS

In the polarimetry techniques we have described in this chapter up to this point, all depend on a time sequential activity. That is, in rotating element polarimetry, polarizers and retarders are rotated and measurements are made at various angular positions of the elements; in oscillating element polarimetry, rotators are oscillated, and measurements are made at various points in the oscillation; in phase modulation polarimetry, measurements are made at various phase values in the modula-tion. We would like to be able to make all required measurements at the same time to ensure that time is not a factor in the result. In order to do this we can divide the wavefront spatially and make simultaneous measurements of different quantities at different points in space, or we can separate polarizations by dividing the amplitude of the wavefront. Polarimeters of these types generally have no moving parts.

16.5.1 diviSion of wavefRonT PolaRiMeTRy

Wavefront division relies on analyzing different parts of the wavefront with separate polarization elements. This has been done using a pair of boresighted cameras that were flown on the space shut-tle [4,5]. A linear polarizer was placed in front of each camera where the polarizers were orthogonal to each other. Chun et al. [6] have performed wavefront division polarimetry using a single infrared camera. Metal wire grid polarizers were formed on a substrate using microlithography in the pat-tern shown in Figure 16.5. This wire grid array was placed in front of the detector array so that light from different parts of object space passes through different polarization elements and on to different detectors. Each detector element of the infrared focal plane array has its own polarizer. These polarizers are linear polarizers at four different orientations, as shown in Figure 16.5, and the pattern is repeated up to the size of the array. There are no circular components measured and thus this is an incomplete polarimeter.

The advantage of this polarimetric measurement method is the simultaneous measurement of the Stokes vector elements available from the polarization element array. The reduction in resolu-tion of the detector by the number of different polarization elements and the spatial displacement of information within the polarization element pattern are disadvantages.

340 Polarized Light, Third Edition

16.5.2 diviSion of aMPliTude PolaRiMeTRy

In amplitude division polarimetry, the energy in the entire wavefront of the incident beam is split and analyzed before passing to detectors. The detectors should be spatially registered so that any detector element is looking at the same point in space as all other detector elements. This method can employ as few as two detectors with analysis of two orthogonally polarized components of light, or it can measure the complete Stokes vector using four detectors. There are a number of variations of division of amplitude polarimetry and we will describe several.

16.5.2.1 four-Channel Polarimeter using Polarizing beam SplittersA diagram of a four channel polarimeter [7] is shown in Figure 16.6. This polarimeter uses three polarizing beam splitters and two retarders. Readings are made at four detectors. The input Stokes vector is determined from the four detector measurements and use of a transfer Mueller matrix found during a calibration procedure. The polarizing beam splitters have transmissions of 80% and 20% for the parallel and perpendicular components. A quarter wave retarder before detectors 1 and 2 is oriented at 45° and the half wave retarder before detectors 3 and 4 is oriented at 22.5°.

The advantage of this system is the simultaneous measurement of all four Stokes components for each point in object space. Care must be taken to ensure spatial registration of the detectors and equalization of detector response. Two channel polarimeters [8] are substantially easer to construct.

16.5.2.2 azzam’s four-detector PhotopolarimeterAnother type of amplitude division complete Stokes polarimeter is the four-detector photopolarim-eter of Azzam [9,10]. A diagram of this polarimeter is shown in Figure 16.7.

In this four-detector polarimeter, a light beam strikes four detectors in sequence, as shown in Figure 16.7. Part of the light striking the first three is specularly reflected to the remaining detec-tors in the sequence, while the last detector absorbs substantially all the remaining light. The signal measured by each detector is proportional to the fraction of the light that it absorbs, and that frac-tion is a linear combination of the Stokes parameters. The light intensity measured by the detector

figuRe 16.5 Pattern of micropolarizers in a wavefront division polarimeter.

Stokes Polarimetry 341

HWR

QWRPBS

PBS

PBS

Detector 1

Detector 2

Detector 3

Detector 4

figuRe 16.6 A four channel polarimeter. PBS is a beam splitter, QWR is a quarter-wave retarder, and HWR is a half-wave retarder.

i3

D3D2

p2p1

α2

α1

p1

p0p0

D0

D1

i0

i2

i1

S0, S1, S2, S3

figuRe 16.7 Optical diagram of the four-detector photopolarimeter. (From Azzam, R. M. A., Opt. Lett., 10, 309–11, 1985. With permission from Optical Society of America.)

342 Polarized Light, Third Edition

is then linearly related to the input Stokes vector. The four detected signals are related to the input Stokes vector by

I A=

=

i

i

i

i

S

S

S

S

0

1

2

3

0

1

2

3

= AS, (16.50)

where A is a Mueller matrix of the instrument. The input Stokes vector is then obtained from

S A I1= − . (16.51)

In order to determine the Stokes vector uniquely, the instrument matrix must be nonsingular. We now derive this instrument matrix.

The Stokes vectors of the light reflected from the surfaces of the photodetectors D0, D1, and D2 are

S M S

S M R M S

S M R M R M S

00

11 1 0

22 2 1 1 0

( )

( )

( )

=

=

=

,

,

,

(16.52)

where S is the input Stokes vector,

Ml l

l

l

l l

r=

−−

1 2 0 0

2 1 0 0

0 0 2 2

cos

cos

sin cos sin

ψψ

ψ ψ∆ ll l

l l l l

sin

sin sin sin cos

∆∆ ∆0 0 2 2−

ψ ψ

, (16.53)

is the Mueller matrix of the lth detector, and

R ll l

l l

=−

1 0 0 0

0 2 2 0

0 2 2 0

0 0 0 1

cos sin

sin cos

α αα α

, (16.54)

is the rotation matrix describing the rotation of the plane of incidence between successive reflec-tions; rl is the reflectance of the lth detector for incident unpolarized or circularly polarized light and tan /ψ l

ipl sle r rl∆ = is the ratio of the complex reflection coefficients of the surface for polarizations

parallel and perpendicular to the local plane of incidence.Let us form a vector L composed of the first elements of the Stokes vectors S, S(0), S(1), and S(2),

that is, the elements that are proportional to the intensities. This can be accomplished by multiplying each of these Stokes vectors by the row vector:

ΓΓ = [ ]1 0 0 0 , (16.55)

Stokes Polarimetry 343

so that we have

L =

( )

( )

( )

S

S

S

S

0

00

01

02

. (16.56)

This vector L is linearly related to the input Stokes vector by

L FS= , (16.57)

where F is given in terms of its rows by

F

F

F

F

F

M

M R M

M R M R

0

1

2

3

0

1 1 0

2 2 1 1

=

=

ΓΓΓΓ

ΓΓΓΓ MM0

. (16.58)

The last three rows of this matrix are the first three rows of the matrices M0, M1R1M0, and M2R2M1R1M0. If we insert the appropriate forms of Equations 16.53 and 16.54 into Equation 16.58 we obtain the matrix

F =

1 0 0 0

0 010 11

20 21 22 23

30 31 32 33

f f

f f f f

f f f f

, (16.59)

where

f r10 0= ,

f r11 0 02= − cos ψ ,

f r r20 0 1 0 1 11 2 2 2= +( )cos cos cosψ ψ α ,

f r r21 0 1 0 1 12 2 2= − +( )cos cos cosψ ψ α ,

f r r22 0 1 0 1 12 2 2= − ( )sin cos cos sin0ψ ψ α∆ ,

f r r23 0 1 0 1 12 2 2= − ( )sin sin cos sin0ψ ψ α∆ ,

f r r r30 0 1 2 01 2= + +( cos cos2 cos2 cos2 cos21 1 1ψ ψ α ψ ψ22 2

0 2 1 2 0

cos2

cos2 cos2 cos2 cos2 cos2 sin

α

ψ ψ α α ψ+ − 22 cos cos2 sin2 sin2 ),1 1 2 1 2ψ ψ α α∆

344 Polarized Light, Third Edition

f r r r31 0 1 2 0 02 2= − + +(cos cos2 cos2 cos cos21 1ψ ψ α ψ ψ11 2 2

2 1 2 1

cos2 cos2

cos2 cos2 cos2 sin2 cos

ψ α

ψ α α ψ+ − ∆∆1 2 1 2cos2 sin2 sin2 ),ψ α α

f r r r32 0 1 2 0 0 1 1 02 2 2 2= − +(sin cos cos sin sin cψ ψ α ψ∆ oos cos sin cos

sin cos sin co

∆

∆

0 2 1 2

0 0 1

2 2 2

2 2

ψ α α

ψ ψ+ ss cos cos sin sin sin sin sin∆ ∆1 2 1 2 0 0 12 2 2 2 2ψ α α ψ ψ− ∆∆1 2 22 2cos sin ),ψ α

f r r r33 0 1 2 0 0 1 1 02 2 2 2= − +(sin sin cos sin sin sψ ψ α ψ∆ iin cos sin cos

sin cos sin si

∆

∆

0 2 1 2

0 0 1

2 2 2

2 2

ψ α α

ψ ψ+ nn cos sin sin sin sin cos cos∆ ∆ ∆1 2 2 0 0 1 12 2 2 2 2ψ α ψ ψ+ ψψ α α2 1 22 2cos sin ).(16.60)

The signal from each of the four detectors is proportional to the light absorbed by it. The light absorbed is the difference between the incident flux and the reflected flux; thus the signal from the first detector is the difference between the first two elements of the vector L in Equation 16.56 mul-tiplied by a proportionality constant that is dependent on the detector responsivity; the signal from the second detector is proportional to the difference between the second and third elements of the vector L; the signal from the third detector is proportional to the difference between the third and fourth elements of the vector L; and since the last detector is assumed to absorb the remaining light, the signal from this detector is proportional to the remaining flux. The signal from each detector is then expressed as

i k S S0 0 0 00= −( )( ) ,

i k S S1 1 00

01= −( )( ) ( ) ,

i k S S2 2 01

02= −( )( ) ( ) ,

i k S3 3 02= ( ). (16.61)

In matrix form, Equation 16.61 can be expressed as

I KDL= , (16.62)

where K is the detector responsivity matrix, L is the vector in Equation 16.56, and D is constructed so that it takes the difference between elements of the vector L; that is,

K =

k

k

k

k

0

1

2

3

0 0 0

0 0 0

0 0 0

0 0 0

(16.63)

and

D =

−−

−

1 1 0 0

0 1 1 0

0 0 1 1

0 0 0 1

. (16.64)

Stokes Polarimetry 345

Substituting Equation 16.57 into Equation 16.62 we obtain

I KDFS= , (16.65)

and we observe in comparing Equations 16.65 and 16.50 that the instrument matrix A is

A KDF= . (16.66)

We know K, D, and F from Equations 16.59, 16.63, and 16.64, and we have found the instrument matrix.

In order to compute A–1, A must be nonsingular and its determinant must be nonzero. We find the determinant from

det det det detA K D F= ( )( )( ), (16.67)

which becomes, when we make substitutions,

det A = −( )( )( )

×

k k k k r r r0 1 2 3 03

12

2 1 22 2sin sin

si

α α

nn cos sin cos cos sin20 0 1 1 2 12 2 2 2 2ψ ψ ψ ψ ψ( ) ∆ .

(16.68)

If any factor in this equation is zero, the determinant becomes zero. We can now make some observations about the conditions under which this can happen. The first term in parentheses is the product of the responsivities of the detectors. It is undesirable and unlikely that any of these are zero, but this might happen if a detector is not working. The next term in parentheses is a product of the reflectances of the first three detectors. If any of these are zero, light will not get to the fourth detector, and the system will not work. Again, this is a condition that is undesirable and unlikely. The third term in parentheses is a geometrical condition: these factors are nonzero as long as the planes of incidence of two successive reflections are not coincident or orthogonal. The detectors can be arranged so that this does not happen. The fourth term in parentheses van-ishes when

ψ π

ψ π

ψ π

ψ π

ψ π

0

1

0

1

2

02

02

4

4

4

=

=

=

=

=

,

,

.

,

,

,

,

(16.69)

The first two conditions in Equation 16.69 are equivalent to having the first two detectors as perfect linear polarizers. The last three conditions would require that the first three detectors reflect p and s polarizations equally or function as retarders. Since the detectors are designed to be absorbing elements and typical reflections from absorbing surfaces will not fulfill these

346 Polarized Light, Third Edition

conditions, they are unlikely. The last factor, sinΔ1, is the sine of the differential reflection phase shift at the second detector. A phase shift of 0 or π is usually associated with Fresnel reflections from nonabsorbing dielectrics. Again, we have absorbing detectors and this condition is not fulfilled.

Further details of polarimeter optimization, light path choice, spectral performance, and cali-bration are given in Azzam [10]. A fiber optic implementation of the four detector polarimeter is described in Bouzid et al. [11], and a corner cube configuration version of the polarimeter is dis-cussed in Liu and Azzam [12].

16.5.2.3 division of amplitude Polarimeters using gratingsA number of polarimeters based on division of amplitude using gratings have been proposed [13–16]. Diffraction gratings split a single incident light beam into multiple beams and introduce significant polarization [17]. Azzam has demonstrated a polarimeter based on conical diffraction [10]. This instrument is shown in Figure 16.8. An incident beam strikes a metal diffraction grating at an oblique incidence angle ϕ. The grating is positioned such that the lines of the gratings are at some arbitrary angle α to the plane of incidence, and this is the condition for conical diffraction. With this geometry, the diffraction efficiency is dependent on all elements of the Stokes vector, and thus this instrument is a complete polarimeter. A linear detector is placed at the location of each diffracted order to be detected. When four detectors are used, the same relationships apply to the grating polarimeter as in the four detector polarimeter; that is, the signal is linearly related to the incident Stokes vector by

I AS= , (16.70)

and we again invert the instrument matrix A to obtain the Stokes vector as in Equation 16.51, that is,

S A I= −1 . (16.51)

The derivation of the instrument matrix for this polarimeter follows the calibration procedures established for the four-detector polarimeter.

i

G

O

N

α2

10

D0

D1

D2

D3

i3

i0

i1

i2

–1

figuRe 16.8 Photopolarimeter using conical diffraction. (From Azzam, R. M. A., Appl. Opt., 31, 3574–6, 1992. With permission from Optical Society of America.)

Stokes Polarimetry 347

A polarimeter using a grating in the normal spectroscopic orientation, that is, in a planar diffrac-tion condition, has been designed and constructed [14]. This polarimeter is illustrated in Figure 16.9. Polarizers are placed in front of the detectors in this design in order to make the instrument sensi-tive to all Stokes parameters. Four detectors are used in four diffracted orders. At least two of the diffracted beams must have polarizers in order for this polarimeter to be complete. An instrument matrix is determined through a calibration process.

A 16-beam grating-based polarimeter has also been designed and demonstrated [16]. A proposed polarimeter using transmission gratings and four linear detector arrays is designed to measure spec-tral and polarization information simultaneously [15].

16.5.2.4 division of amplitude Polarimeter using a Parallel SlabA wavefront may be divided in amplitude using the multiple reflections obtained in a planar dielec-tric slab [18]. Figure 16.10 shows a polarimeter based on a parallel plane slab of material of index n1(λ). A coating of metal of complex index n2 – ik2 is placed on the bottom surface of the slab. A light beam incident on the slab at angle ϕ undergoes multiple reflections in the slab that results in a set of parallel and equally spaced outgoing beams. Linear polarizers are arranged in front of detec-tors in these beams with as many inclination angles of the transmission axes as there are detectors. The signal from the mth detector is then a linear combination of the elements of the Stokes vector, that is,

i a S mm mj j

j

= ==

∑0

3

0 1 2, , , , ,… (16.71)

where the mth vector am m m m ma a a a= [ ]0 1 2 3 is the first row of the Mueller matrix of the mth light path. If we limit the detectors to four, the output signal vector is related to the input Stokes vector by the equation we have seen before for division of amplitude polarimeters, that is,

QD

OEA

i–3

–1

+1

P2

P1

0

D1

P0

D0D2

–2 N –1

O

G

P3D3

i3

i2

i1

i0

figuRe 16.9 Photopolarimeter using planar diffraction. (From Azzam, R. M. A., Appl. Opt., 31, 3574–6, 1992. With permission from Optical Society of America.)

348 Polarized Light, Third Edition

I AS= . (16.70)

The matrix A is the instrument matrix determined through calibration, and, as in previous division of amplitude examples, an unknown Stokes vector is found from the equation

S A I= −1 . (16.51)

El-Saba, Azzam, and Abushagur [18] show that for a slab of fused silica coated with a layer of silver and operated at 633 nm, the preferred angle of incidence for maximum energy in the beams and maximum value of the determinant of the instrument matrix is around 80°.

16.6 oPTimiZaTioN of PolaRimeTeRS

To this point we have not discussed specific polarization element angular settings. We have made reference to the use of quarter-wave retarders, primarily because we can construct a complete Stokes polarimeter using the readily available quarter-wave retarder and linear polarizer. We now ask the question, are there measurement angles and values of retardance that will result in a more efficient and/or better polarimeter?

This question was first addressed with regard to the angular positions of the quarter wave retarder and linear polarizer in a rotating retarder and fixed analyzer polarimeter [19] and a rotating retarder, rotating analyzer polarimeter [20]. It was found in the first instance that angles of (–45°, 0°, 30°, 60°) or (–90°, –45°, 30°, 60°) resulted in the least sensitivity with regard to flux noise and rotation positional errors. In the second instance, if we let the rotation angle of the polarizer be θ and the rotation angle of the retarder be φ and define an α and β such that

d

P0 P1 P2 P3

10 2 3

D0 D1 D2 D3

i3

n0 = 1

n1(λ)

n2 – jk2

i2i1i0

figuRe 16.10 Parallel slab polarimeter. (From El-Saba, A. M., Azzam, R. M. A., and Abushagur, M. A. G., Opt. Lett., 21, 1709–11, 1996. With permission from Optical Society of America.)

Stokes Polarimetry 349

α φ

β θ φ

=

= −( )

2

2

,

, (16.72)

then an optimal set of α and β is

0 90 013

12011 1 , , , , ,[ ] −

−− −sin sin33

24013

1

−

−, , . sin (16.73)

If we allow both the measurement angles and retardance to take part in the optimization process for a rotating retarder polarimeter, we find that the optimal value of retardance is 0.36611 (≈132°) and the optimal retarder positions are either (±15.12°, ±51.69°) or (±74.88°, ±38.31°) where these angle pairs are complements of each other [21,22]. These values were found through numerical optimization described in the cited references where the optimal values offer the best signal-to-noise performance and least sensitivity to element misalignment. Figure 16.11 shows the locus of points on the Poincaré sphere for values of retardance of 45°, 90°, 132°, and 180°. The figure indi-cates that better global coverage of the sphere is made possible by using the retardance of 132°.

Figure 16.12 reinforces this intuition where the intersection of the curve for the retardance value 132° with the four retarder positions (±15.12°, ±51.69°) forms the corners of a regular tetrahedron inscribed in the Poincaré sphere, points as far as possible as one can make them on the surface of the sphere.

S3

S2

S1

90° 132°

45°

180°

figuRe 16.11 Locus of points on the Poincaré sphere for retardance values 45°, 90°, 132°, and 180° for a rotating retarder polarimeter. (From Sabatke, D. S., Descour, M. R., Dereniak, E. L., Sweatt, W. C., Kemme, S. A., and Phipps, G. S., Opt. Lett., 25, 802–4, 2000. With permission from Optical Society of America.)

350 Polarized Light, Third Edition

S1

S3

S2

90°

132°

figuRe 16.12 Curves for retardance values of 90° and 132° intersecting the retarder angles (±15.12°, ±51.69) to form the regular tetrahedron. (From Sabatke, D. S., Descour, M. R., Dereniak, E. L., Sweatt, W. C., Kemme, S. A., and Phipps, G. S., Opt. Lett., 25, 802–4, 2000. With permission from Optical Society of America.)

20

15

10

5Equa

lly w

eigh

ted

varia

nce (

EWV

)

4 6 8 10 12

Optimum for 4 measurements,quarter-wave retardance

1. Uniformly spaced angles, quarter-wave retardance2. Uniformly spaced angles, optimal wave retardance

3. Optimal angles repeated, optimal retardance

14 16 N

figuRe 16.13 Plots of a figure of merit versus number of measurements for several measurement methods. (From Sabatke, D. S., Descour, M. R., Dereniak, E. L., Sweatt, W. C., Kemme, S. A., and Phipps, G. S., Opt. Lett., 25, 802–4, 2000. With permission from Optical Society of America.)

Stokes Polarimetry 351

Figure 16.13 shows plots of a figure of merit for the rotating retarder fixed polarizer pola-rimeter versus number of measurements for the system with a quarter-wave retarder and an optimal retarder with both equally spaced angles and the optimal measurements angles. The results of this plot indicate that the optimal retarder with repeated optimal angles offers the best performance.

At this time, 132° retarders are not standard items from optical supply houses, and the improve-ment in performance gained by using these optimal elements may not be worth the cost and risk of ordering custom elements.

RefeReNCeS

1. Hauge, P. S., Recent developments in instrumentation in ellipsometry, Surface Sci. 96 (1980): 108–40.

2. Azzam, R. M. A., Oscillating-analyzer ellipsometer, Rev. Sci. Instrum. 47 (1976): 624–8. 3. Azzam, R. M. A., Photopolarimeter using two modulated optical rotators, Opt. Lett. 1 (1977):

181–3. 4. Whitehead, V. S., and K. Coulson, The space shuttle as a polarization observation platform, in Polarization

considerations for Optical Systems II, Proc. SPIE 1166, San Diego, CA, August 9–11, 1989; Edited by R. A. Chipman, 42–51, Bellingham, WA: SPIE, 1989.

5. Duggin, W. J., S. A. Israel, V. S. Whitehead, J. S. Myers, and D. R. Robertson, Use of polarization meth-ods in earth resources investigations, in Polarization considerations for Optical Systems II, Proc SPIE 1166, San Diego, CA, August 9–11, 1989; Edited by R. A. Chipman, 11–22, Bellingham, WA: SPIE, 1989.

6. Chun, C. S. L., D. L. Fleming, W. A. Harvey, and E. J. Torok, Polarization-sensitive infrared sensor for target discrimination, in Polarization: Measurement, Analysis, and Remote Sensing, Proc. SPIE 3121, San Diego, CA, July 30–August 1, 1997; Edited by D. H. Goldstein and R. A. Chipman, 55–62, Bellingham, WA: SPIE, 1997.

7. Gamiz, V. L., Performance of a four channel polarimeter with low light level detection, in Polarization: Measurement, Analysis, and Remote Sensing, Proc. SPIE 3121, San Diego, CA, July 30–August 1, 1997; Edited by D. H. Goldstein and R. A. Chipman, 35–46, Bellingham, WA: SPIE, 1997.

8. Wolff, L. B., Polarization camera for computer vision with a beam splitter, J. Opt. Soc. Am. A 11 (1994): 2935–45.

9. Azzam, R. M. A., Arrangement of four photodetectors for measuring the state of polarization of light, Opt. Lett. 10 (1985): 309–11.

10. Azzam, R. M. A., I. M. Elminyawi, and A. M. El-Saba, General analysis and optimization of the four-detector photopolarimeter, J. Opt. Soc. Am. A 5 (1988): 681–9.

11. Bouzid, A., M. A. G. Abushagur, A. El-Saba, and R. M. A. Azzam, Fiber-optic four-detector polarimeter, Opt. comm. 118 (1995): 329–34.

12. Liu, J., and R. M. A. Azzam, Corner-cube four-detector photopolarimeter, Optics & Laser Technol. 29 (1997): 233–8.

13. Azzam, R. M. A., Division-of-amplitude photopolarimeter based on conical diffraction from a metallic grating, Appl. Opt. 31 (1992): 3574–6.

14. Azzam, R. M. A., and K. A. Giardina, Photopolarimeter based on planar grating diffraction, J. Opt. Soc. Am. A 10 (1993): 1190–6.

15. Todorov, T., and L. Nikolova, Spectrophotopolarimeter: Fast simultaneous real-time measurement of light parameters, Opt. Lett. 17 (1992): 358–9.

16. Cui, Y., and R. M. A. Azzam, Calibration and testing of a sixteen-beam grating-based division-of-ampli-tude photopolarimeter, Rev. Sci. Instrum. 66 (1995): 5552–8.

17. Bennett, J. M., and H. E. Bennett, Polarization, in Handbook of Optics, Edited by W. G. Driscoll and W. Vaughan, New York: McGraw-Hill, 1978.

18. El-Saba, A. M., R. M. A. Azzam, and M. A. G. Abushagur, Parallel-slab division-of-amplitude photopo-larimeter, Opt. Lett. 21 (1996): 1709–11.

19. Ambirajan, A., and D. C. Look, Optimum angles for a polarimeter: part I, Opt. Eng. 34 (1995): 1651–5.

352 Polarized Light, Third Edition

20. Ambirajan, A., and D. C. Look, Optimum angles for a polarimeter: part II, Opt. Eng. 34 (1995): 1656–9.

21. Sabatke, D. S., M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, Optimization of retardance for a complete Stokes polarimeter, Opt. Lett. 25 (2000): 802–4.

22. Tyo, J. S., Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error, Appl. Opt. 41 (2002): 619–30.

353

17 Mueller Matrix Polarimetry

17.1 iNTRoduCTioN

The real 4 × 4 matrix that completely describes the polarization properties of a material in reflection or transmission is measured in Mueller matrix polarimetry. A Mueller matrix polarimeter is com-plete if all 16 of the elements are measured and incomplete otherwise. To be complete, a Mueller matrix polarimeter must have a complete polarization state analyzer (PSA) and a complete polariza-tion state generator (PSG). Figure 17.1 is a conceptual diagram of a Mueller matrix polarimeter.

The equation we wish to solve in Mueller matrix polarimetry is

I a a a a

•

•

•

• • • •

• • • •

• • • •

= =

aMp

1 2 3 4

m m m m

m m m m

m m m

11 12 13 14

21 22 23 24

31 32 333 34

41 42 43 44

1

2

3

4

m

m m m m

p

p

p

p

(17.1)

where M is the Mueller matrix to be measured, the vector p is the Stokes vector of the light entering the sample represented by M, the vector a is the first row of the polarization state analyzer Mueller matrix, and I is the signal from the detector. Note that the vector p is the product of the Stokes vec-tor of the source and the Mueller matrix of the polarization state generator, and only the first row of a is needed since the measured signal from the detector is the single value representing the first element of the output Stokes vector. We should measure at least 16 values of I with 16 settings of the polarization state generator and analyzer in order to obtain 16 equations in the 16 unknowns of the elements of the sample Mueller matrix. Very often, more than 16 measurements are made so that the matrix elements are overdetermined. Measurement methods using Fourier or non-Fourier data reduction techniques may be used. The methods described in this chapter lend themselves to automated procedures. Note that we have chosen to use indices from 1 to 4 in this chapter rather than 0 to 3.

In this chapter, we shall discuss a small selection of Mueller matrix polarimeters that have found practical use. This will serve to illustrate the variation in method and serve as examples for those contemplating measurement of Mueller matrices. Hauge [1] gives a more complete review of various types of incomplete and complete Mueller matrix polarimeters. We review practical examples of rotating element and phase modulating polarimeters. Another type, the four-detector polarimeter, is also reviewed in this chapter.

17.1.1 PolaRiMeTeR TyPeS

There are a number of different methods that have been devised to collect Mueller matrices. Many Mueller matrix polarimeters are either rotating-element polarimeters or phase-modulating polarim-eters. Rotating-element polarimeters use mechanical rotation of polarizers or retarders to achieve the desired measurements. Phase-modulating polarimeters use an electro-optical modulator to induce a time-varying retardation. Either of these polarimeter types may be complete or incomplete. Examples of different configurations of these two types are depicted in Figures 17.2 and 17.3 and show the Mueller matrix elements that are measured in each case (represented by the large dots).

354 Polarized Light, Third Edition

Source

Polarizationstate generator

Polarizationstate analyzer

Detector

Sample

figuRe 17.1 Conceptual diagram of a Mueller matrix polarimeter.

(a)

(b)

(c)

(d)

P A

Source Detector

S

AP

Source Detector

S

R

DetectorSource

S

P R A

SourceDetector

S

P R R A

figuRe 17.2 Rotating element polarimeters. P is a polarizer, A is an analyzer, R is a retarder, and S is the sample. Measured elements of the Mueller matrix are indicated by large dots.

Mueller Matrix Polarimetry 355

17.1.2 RoTaTing eleMenT PolaRiMeTeRS

Figure 17.2a shows a rotating polarizer-rotating analyzer polarimeter. When the polarizer is rotated by an angle θ and the analyzer by angle 3θ synchronously, the Fourier series representing the nor-malized intensity has the form (I0 is the source intensity)

I

I

aa k b kk k

k0

02 2

1

4

414

2 2= + +=

∑( cos sin )θ θ . (17.2)

The nine Fourier coefficients determine nine elements of the Mueller matrix

M =

⋅+ − + ⋅+ − ⋅

⋅ ⋅ ⋅ ⋅

a a b

a a a b b

b b b a a

0 2 2

6 4 8 4 8

6 4 8 4 8

. (17.3)

A rotating polarizer-rotating compensator plus fixed analyzer polarimeter is shown in Figure 17.2b. If the polarizer and retarder of this polarimeter are rotated synchronously in a 3:1 ratio, the normal-ized detected intensity can be expanded in the Fourier series; that is,

I

I

aa k b kk k

k0

02 2

1

7

414

2 2= + +=

∑( cos sin )θ θ . (17.4)

The 15 Fourier coefficients over determine the 12 elements of the Mueller matrix in the first three columns, so that

M =

−( ) − −( ) − +( ) ⋅+( )

a a a a a b b b

a a a b0 6 1 5 7 1 5 7

6 5 72 2 2 77 5

6 5 7 5 7

3 2 2

2 2 2

2 2 2

−( ) ⋅+( ) −( ) ⋅

− − − ⋅

b

b b b a a

b b a

. (17.5)

(a)

(b)

Source DetectorS

P PM PM A

Source DetectorS

P PM PM A

PM PM

0° 0° 0°0°45° 45°

0° 45° 90° 45°

figuRe 17.3 Phase modulating polarimeters. A phase modulator/phase modulator polarimeter is shown in (a); a dual phase modulator polarimeter is shown in (b). Measured elements of the Mueller matrix are indicated by large dots. P is a polarizer, A is an analyzer, PM is a phase modulator, and S is a sample.

356 Polarized Light, Third Edition

The polarimeter in Figure 17.2c determines the first three rows of the Mueller matrix. The last rotating element polarimeter in Figure 17.2d is the dual rotating-retarder polarimeter, and we will discuss this polarimeter in more detail in Section 17.2.

17.1.3 PhaSe-ModulaTing PolaRiMeTeRS

Two types of phase modulation polarimeters are shown in Figure 17.3. One has a single modula-tor on either side of the sample, and the other has a double modulator on either side. We describe the double modulator case in more detail later in this chapter. For the single modulator case, it can be shown that the detected intensity, when the modulator axes are inclined at 45° to each other, as shown in Figure 17.3a, is

II0

14

= ∆ ∆[ ] ∆

∆

1 0 cos sin

1

cos

0

sin

2 21

1

M

, (17.6)

where I0 is the source intensity and

cos cos sin

sin sin sin

i i

i i

∆

∆

= ( )

= ( )

δ ω

δ ω

i

i

t

t , (17.7)

and the subscripts 1 and 2 identify the first and second modulators. The detected signal is then given by

II

M M M M M0

14

= + + + +( cos sin cos co11 12 1 12 1 31 2 32∆ ∆ ∆ ss cos

sin cos sin

cos si

1 2

34 1 2 41 2

42 1

∆ ∆

∆ ∆ ∆

∆

+ +

+

M M

M nn sin sin ).2 44 1 2∆ ∆ ∆+ M

(17.8)

The frequencies ω and phases δ are chosen such that the nine matrix elements are measured by sequential or parallel phase-sensitive detection, that is, lock-in amplifiers.

One type of complete Mueller matrix polarimeter is represented in Figure 17.4. This is the dual rotating-retarder polarimeter [2]. It consists of a complete polarimeter as a PSG and a complete polarimeter as a PSA. The retarders are rotated and Fourier analysis is performed on the result-ing modulated signal to obtain the Mueller matrix of the sample. In Section 17.2, we will examine this polarimeter in more detail. This dual rotating-retarder method has been implemented as a nonimaging laser polarimeter in order to examine electro-optical samples in transmission [3]. An imaging version of this polarimeter has been constructed to obtain highly resolved polarimetric images of liquid crystal televisions [4] and electro-optic modulators [5]. This same method has been used in the construction of spectropolarimeters to evaluate samples in transmission and reflection [6,7]. In Section 17.3, we will discuss other types of Mueller matrix polarimeters. The polarimetric methods that were discussed in Chapter 15 were based primarily on manual methods. The methods described here are all automated and typically depend on computers to collect and process the information.

Mueller Matrix Polarimetry 357

17.2 dual RoTaTiNg ReTaRdeR PolaRimeTRy

This polarimeter configuration is based on a concept originally proposed by Azzam [2], elaborated on by Hauge [8], and by Goldstein [3], and has been used in spectropolarimetry as we shall see [6,7]. The technique also has been used with the sample in reflection to measure birefringence in the human eye at visible wavelengths [9–11]. We have shown in Figure 17.1 a functional block diagram of a general Mueller matrix polarimeter. The polarimeter has five sections, the source, the polar-izing optics, the sample, the analyzing optics, and the detector.

17.2.1 PolaRiMeTeR deScRiPTion

The polarizing optics consist of a fixed linear polarizer and a quarter wave retarder that rotates. The sample region is followed by the analyzing optics, which consist of a quarter wave retarder that rotates followed by a fixed linear polarizer. This is shown in Figure 17.4. One of the great advan-tages of this configuration is that the polarization sensitivity of the detector is not important because the orientation of the final polarizer is fixed.

The two retarders are rotated at different but harmonic rates, and this results in a modulation of the detected intensity. The Mueller matrix of the sample is found through a relationship between the Fourier coefficients of a series representing the modulation and the elements of the sample matrix.

The second retarder is rotated at least five times the rate of the first, and data might typically be collected for every 2°–6° of rotation of the first retarder. The stages are stopped completely after each incremental rotation, and an intensity reading is recorded. The resulting data set is a modulated waveform, which is then processed according to the algorithms we shall describe shortly.

The polarizing elements in the polarimeter are required to be aligned with respect to a common axis to start the measurements (this would typically be the axis of the polarized laser or the axis of the first polarizer if an unpolarized source is used). This alignment is done manually to try to mini-mize orientation errors, and the residual orientation errors are removed through a computational compensation method that we will describe.

17.2.2 MaTheMaTical develoPMenT: obTaining The MuelleR MaTRix

This polarimeter measures a signal that is modulated by rotating the retarders. The elements of the Mueller matrix are encoded on the modulated signal. The output signal is then Fourier analyzed to determine the Mueller matrix elements. The second retarder is rotated at a rate of five times that of the first. This generates 12 harmonic frequencies in the Fourier spectrum of the modulated intensity.

The Mueller matrix for the system is

P R MR P2 2 1 1( ) ( )θ θ , (17.9)

Source

SampleDetector

P1 R1 R2 P2

5θθ

figuRe 17.4 Dual rotating retarder polarimeter. P1 and P2 are polarizers, R1 and R2 are retarders.

358 Polarized Light, Third Edition

where P indicates a linear polarizer, R(θ) indicates an orientation-dependent retarder, and M is the sample and the matrix quantity to be determined. Mueller matrices are then substituted for a linear retarder with quarter wave retardation and a fast axis at θ and 5θ for R1 and R2, respectively; a hori-zontal linear polarizer for P2; a horizontal linear polarizer for P1; and a sample for M. The detected intensity is given by

I c= AMP, (17.10)

where P = R1P1S is the Stokes vector of light leaving the polarizing source (S is the Stokes vec-tor of the light from the source), A = P2R2 is the Mueller matrix of the analyzing optics, M is the Mueller matrix of the sample, and c is a proportionality constant obtained from the absolute inten-sity. Explicitly,

I c a p mi j ij

i j

==

∑,

,1

4

(17.11)

or

I c mij ij

i j

==

∑µ.

,1

4

(17.12)

where the ai are the elements of the first row of A, the pj are the elements of P, the mij are the ele-ments of the Mueller matrix M, and where

µ ij i ja p= . (17.13)

The order of matrix multiplication can be changed as shown above in going from Equation 17.10 to Equation 17.11 because we are only measuring intensity, that is, the first element of the Stokes vec-tor. Only the first row of the matrix A is involved in the calculation, that is,

a a a a m m m m1 2 3 4 11 12 13 1

i i i i

i i i i

i i i i

44

21 22 23 24

31 32 33 34

41 42 43 44

m m m m

m m m m

m m m m

=

p

p

p

p

I1

2

3

4

i

i

i

, (17.14)

and multiplying through, we have

I a m p m p m p m p

a m p m p

= + + +

+ +

1 11 1 12 2 13 3 14 4

2 21 1 22

( )

( 22 23 3 24 4

3 31 1 32 2 33 3 34 4

+ +

+ + + +

m p m p

a m p m p m p m p

)

( )

++ + + +

==

∑

a m p m p m p m p

mij ij

i j

4 41 1 42 2 43 3 44 4

1

4

( )

µ,

..

(17.15)

Mueller Matrix Polarimetry 359

When the rotation ratio is 5:1 the μij are given by

µ

µ θ

µ θ θ

µ θ

µ

11

122

13

14

1

2

2 2

2

=

=

=

=

,

cos ,

sin cos ,

sin ,

2212

222 2

23

10

2 10

2 2

=

=

=

cos ,

cos cos ,

sin cos

θ

µ θ θ

µ θ θθ θ

µ θ θ

µ θ θ

cos ,

sin cos ,

sin cos

2

242

31

10

2 10

10 10

=

= ,,

cos sin cos ,

sin cos sin

µ θ θ θ

µ θ θ32

2

33

2 10 10

2 2 1

=

= 00 10

2 10 10

10

34

41

θ θ

µ θ θ θ

µ θ

cos ,

sin sin cos ,

sin

=

= − ,,

cos sin ,

sin cos sin ,

µ θ θ

µ θ θ θ

µ

422

43

2 10

2 2 10

= −

= −

444 2 10= −sin sinθ θ.

(17.16)

These equations can be expanded in a Fourier series to yield the Fourier coefficients, which are functions of the Mueller matrix elements. The inversion of these relations gives the Mueller matrix elements in terms of the Fourier coefficients, that is,

m a a a a a

m a a a

m

11 0 2 8 10 12

12 2 8 12

13

2 2 2

2

= − + − +

= − −

=

,

,

bb b b

m b b b b b b b

m

2 8 12

14 1 11 1 9 1 9 11

2 2

2 2

+ −

= − = + = + −

,

,

221 8 10 12

22 8 12

23 8

2 2 2

4 4

4 4

= − + −

= +

= − +

a a a

m a a

m b b

,

,

112

24 9 11 9 11

31 8 10

4 4 2

2 2

,

( ),m b b b b

m b b

= − = = − +

= − + − 22

4 4

4 4

4 4

12

32 8 12

33 8 12

34 9

b

m b b

m a a

m a a

,

,

,

= +

= −

= = − 111 9 11

41 3 5 5 7 3 5 7

2

2 2

= −

= − = − + = − +

( ),

( )

a a

m b b b b b b b ,,

( ),

(

m b b b b

m a a a

42 3 7 3 7

43 3 7

4 4 2

4 4 2

= − = − = − +

= − = = − 33 7

44 4 6 6 42 2

+

= − = = −

a

m a a a a

),

( ).

(17.17)

360 Polarized Light, Third Edition

The 5:1 rotation ratio is not the only ratio that can be used to determine Mueller matrix elements, but it is the lowest ratio in which the expressions for the Fourier coefficients may be inverted to give the Mueller matrix elements.

Intensity values in the form of voltages are measured as the retarders are incrementally advanced such that the first retarder is rotated through 180°. The Fourier coefficients must be obtained from the measured intensity values. There are several methods of formulating the solution to this problem.

If the problem is formulated as

x Ia = , (17.18)

where I is a vector of 36 intensity values, a is the set of Fourier coefficients, and x is a 26 × 25 matrix where each row is of the form

( cos cos ... cos sin sin ... sin ),1 2 4 24 2 4 24θ θ θ θ θ θ

where the θ for each row represents the angle of the fast axis of the first retarder, then the solution is

a T T= −( )x x x I1 . (17.19)

(The minimum number of equations needed to solve for the coefficients uniquely is 25 so that the maximum rotation increment for the first retarder is 7.2°; for this example, 36 equations are obtained from 5° rotational increments through 180°.) This solution is equivalent to the least-squares solution [12]. In the least-squares formulation the expression for the instrument response is

I a a j b jj j

j

( ) ( cos sin ),θ θ θ= + +=

∑0

1

12

2 2 (17.20)

but the actual measurement Φ(θ) may be different from this value due to noise and/or error. The sum of the square of these differences may be formed, that is,

Φ( ) ( ) ( ),θ θl lI E a a a b b−[ ] =20 1 12 1 12, ,..., , ,...,∑∑ (17.21)

where E is a function of the coefficients and l is the subscript of the retarder angle. The values of the coefficients can now be found by taking the partial derivative of E with respect to the coefficients and setting these equal to zero, that is,

∂∂

= ∂∂

=E

a

E

bk k

0 0, . (17.22)

The expression becomes, for the derivative with respect to al,

Φ( ) ( cos sin )θ θ θl j l j l

j

a a j b j− + +

=∑0

1

12

2 2

× − =

=∑ ( cos )2 2 0

0

35

k l

l

θ . (17.23)

Mueller Matrix Polarimetry 361

Solving this system of 36 equations in 25 unknowns will give the least-squares solution for the coef-ficients, which is identical to the solution obtained from Equation 17.19.

17.2.3 ModulaTed inTenSiTy PaTTeRnS

Simulated modulated intensity patterns for no sample and various examples of ideal polarization elements are given in Figures 17.5 through 17.8. The abscissa represents measurement number in a sequence of 36 (corresponding to 5° increments over 180°) and the ordinate represents detector voltage, normalized to 0.5.

The quality of the measurement and the type of element in the sample position can be recognized by observation of the measured intensity modulation. For example, the pattern of a retarder with its

0

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20 25 30 35n

I

figuRe 17.5 Modulated intensity for no sample.

0

0.1

0.2

0.3

0.4

0.5

0 5 10 15 20 25 30 35n

I

figuRe 17.6 Modulated intensity for a linear horizontal polarizer.

362 Polarized Light, Third Edition

fast axis aligned and one with its slow axis aligned are immediately recognizable and differenti-ated. Good measurements yield modulated intensity patterns that are essentially identical to the simulations.

17.2.4 eRRoR coMPenSaTion

The true nature of the sample may be obscured by errors inherent in the polarimeter optical sys-tem. The Mueller matrix elements must be compensated for the known errors in retardance of the retarders and the errors caused by the inability to align the polarizing elements precisely. The fact that there are errors that cannot be eliminated through optical means led to an error analysis and a compensation procedure to be implemented during polarimeter data processing.

0

0.1

0.2

0.3

0.4

0.5

0 5 10 15 20 25 30 35n

I

figuRe 17.7 Modulated intensity for a linear vertical polarizer.

0

0.1

0.2

0.3

0.4

0.5

0 5 10 15 20 25 30 35n

I

figuRe 17.8 Modulated intensity for a half-wave plate at 45°.

Mueller Matrix Polarimetry 363

A summary of an error analysis of a dual-rotating retarder Mueller matrix polarimeter is presented in this section. The derivation of the compensated Mueller matrix elements using the small-angle approximation is documented in detail [13], and exact compensation equations for the Mueller matrix elements have been derived [14]. Errors in orientational alignment and errors caused by nonideal retardation elements are considered in these compensations. A compensation for imperfect retarda-tion elements is then made possible with the equations derived, and the equations permit a calibra-tion of the polarimeter for the azimuthal alignment of the polarization elements. A similar analysis was done earlier [8] for a dual rotating compensator ellipsometer, but that analysis did not include error in the last polarizer and did include errors caused by diattenuation in the retardation elements. Experimental experience with the polarimeter described here indicates that the deviation of the retarders from quarter wave is important compared with the diattenuation of the retarders [3].

In the error analysis, the effect of retardation associated with the polarizers, and polarization associated with the retarders, have not been included. It is also assumed that there are no angular errors associated with the stages that rotate the elements. It is only the relative orientations of the polarizers and retarders that are relevant, and the analysis is simplified by measuring all angles rela-tive to the angle of the polarization from the first polarizer. The errors are illustrated in Figure 17.9. The three polarization elements have errors associated with their initial azimuthal alignment with respect to the first polarizer. These are shown as ε3, ε4, and ε5 in Figure 17.9. In addition, one or both retarders may have retardances that differ from quarter wave. These are shown as δ1 and δ2 where ε1 and ε2 are the deviations from quarter wave in Figure 17.9. In general, both retarders will have different retardances and the three polarization elements will be slightly misaligned in azimuth.

The following calibration procedure is used. First, the polarimeter is operated with no sample, and Fourier coefficients obtained from the measured modulated intensity. Second, using error-com-pensation equations with matrix elements of the identity matrix inserted for the Mueller matrix ele-ments, errors in the element orientations and retardances are calculated. Third, in the routine use of the polarimeter, the systematic errors in the Fourier coefficients arising from the imperfections are compensated for by using the error-compensated equations with experimentally determined error values to obtain the error-compensated sample Mueller matrix elements as a function of measured Fourier coefficients.

With no sample in the polarimeter, the sample matrix is the identity matrix. Because all off-diagonal elements in the sample Mueller matrix are zero, all odd Fourier coefficients in Equation 17.20 become zero. Because the diagonal elements equal one, the coefficients of the 12th harmonic vanish also.

S

R2

P2

R1

P1

δ1 = ε1+ 90°

δ2 = ε2 + 90°ε3

ε4

ε5

figuRe 17.9 Significant error sources in the dual rotating retarder polarimeter.

364 Polarized Light, Third Edition

The Fourier coefficients are found to be functions of the errors, after we find the μ’s as in Equation 17.16 but this time as functions of errors. The Fourier coefficients are

a m m m0 11 3 12 4 5 21 3 4 5

12

14

14

218

2= + + +β β ε β β εcos cos mm m m

a

22 4 5 31 3 4 5 32

1

14

218

2

12

+ +

=

β ε β β εsin sin ,

sinδδ ε β δ ε ε β1 3 14 4 1 3 5 24 4214

2 214

sin sin sin cos sm m+ + iin sin sin ,

cos s

δ ε ε

β ε β

1 3 5 34

2 1 3 12 1

2 2

14

414

m

a m= + iin cos cos sin418

4 218

43 13 1 4 3 5 22 1 4ε β β ε ε β β εm m+ + 33 5 23

1 4 3 5 32 1 4

2

18

4 218

cos

cos sin sin

ε

β β ε ε β β

m

m+ + 44 2

18

18

3 5 33

3 1 2 3 42 1

ε ε

β δ α β

sin ,

sin sin sin

m

a m= − − δδ α

δ δ α

2 3 43

4 1 2 1 44

5

14

12

cos ,

sin sin cos ,

s

m

a m

a

= −

= iin sin sin sin ,

sin s

δ α β δ α

δ

2 5 41 3 2 5 42

6 1

14

14

m m

a

+

= iin cos ,

sin sin sin

δ α

β δ α β

2 2 44

7 1 2 4 42 1

18

18

m

a m= − + δδ α

β β α β β

2 4 43

8 1 2 9 22 33 1

116

116

cos ,

cos

m

a m m= +( ) + 22 9 32 23

9 2 1 6 24 2

18

18

sin ,

sin sin s

α

β δ α β

m m

a m

−( )

= + iin cos ,

cos cos

δ α

β α β β α

1 6 34

10 2 11 21 2 3 1

14

18

m

a m= + 11 22 2 11 31 2 3 11 32

11

14

18

18

m m m

a

+ +

= −

β α β β αsin sin ,

ββ δ α β δ α

β

2 1 7 24 2 1 7 34

12

18

116

sin sin sin cos ,m m

a

−

= 11 2 10 22 33 1 2 10 23 32

116

β α β β αcos sin ,m m m m−( ) + +( )

b

b m

0

1 1 3 14 4 1 3

0

12

214

2

=

= +

,

sin cos sin cos cosδ ε β δ ε 2214

2 2

14

5 24 4 1 3 5 34

2 1

ε β δ ε ε

β

m m

b

+

= −

sin cos sin ,

sinn cos cos cos414

418

4 23 12 1 3 13 1 4 3 5 2ε β ε β β ε εm m m+ + 33 1 4 3 5 22

1 4 3 5

14

4 2

18

4 2

−

+

β β ε ε

β β ε ε

sin cos

cos sin

m

mm m33 1 4 3 5 32

18

4 2− β β ε εsin sin ,

Mueller Matrix Polarimetry 365

b m m

b

3 1 2 3 42 1 2 3 43

4

18

18

= − +

=

β δ α β δ αsin cos sin sin ,

114

12

1

1 2 1 44

5 2 5 41

sin sin sin ,

sin cos

δ δ α

δ α

m

b m= − −44

14

3 2 5 42

6 1 2 2 44

β δ α

δ δ α

sin cos ,

sin sin sin ,

m

b m= −

bb m m

b

7 1 2 4 42 1 2 4 43

8

18

18

= − −β δ α β δ αsin cos sin sin ,

== − +( ) − −116

1161 2 9 22 33 1 2 9 23 3β β α β β αsin cosm m m m 22

9 2 1 6 24 2 1 6 318

18

( )

= − +

,

sin cos sin sinb m mβ δ α β δ α 44

10 2 11 21 2 3 11 22 214

18

14

,

sin sin cb m m= − − +β α β β α β oos cos ,

sin cos

α β β α

β δ α

11 31 2 3 11 32

11 2 1

18

18

m m

b

+

= 77 24 2 1 7 34

12 1 2 10

18

116

m m

b m

−

= −

β δ α

β β α

sin sin ,

sin 222 33 1 2 10 23 321

16−( ) + +( )m m mβ β αcos ,

(17.24)

where

β δ

β δ

β δ

β δ

1 1

2 2

3 1

4 2

1

1

1

1

= −

= −

= +

= +

cos ,

cos ,

cos ,

cos ,,

,

,

α ε ε ε

α ε ε ε

α ε ε

1 4 3 5

2 4 3 5

3 4 3

2 2 2

2 2 2

2 4

= − −

= + −

= − − 22

2 4 2

2 2

2 4 2

5

4 4 3 5

5 5 4

6 5 4

ε

α ε ε ε

α ε ε

α ε ε

,

,

,

= + −

= −

= − + εε

α ε ε ε

α ε ε ε α

α

3

7 5 4 3

8 5 4 3 6

9

2 4 2

2 4 2

4

,

,

,

= − −

= − + − = −

= εε ε ε

α ε ε ε

α ε ε

4 3 5

10 4 3 5

11 4 5

4 2

4 2 2

4 2

− −

= + −

= −

,

,

.

(17.25)

366 Polarized Light, Third Edition

These equations can be inverted for this case where there is no sample so that the sample Mueller matrix is the identity matrix, and we then solve for the errors in terms of the Fourier coefficients. The equations yield the errors as

ε

ε

31 8

8

1 10

10

4

14

14

=

−

− −tan tan ,b

a

b

a

==

−

+− − −1

212

14

1 2

2

1 6

6

1tan tan tanb

a

b

a

bb

a

b

a

b

8

8

1 10

10

51

14

12

−

=

−

−

tan ,

tanε 22

2

1 8

8

1 10

10

12

12a

b

a

b

a

+

−

− −tan tan

= −+

−

,

coscos coscos co

δ α αα1

1 10 9 8 11

10 9 8

a a

a a ss,

coscos cos

α

δ α ε ε

11

21 2 9 8 3 54 2

= − −( )− a a

a22 9 8 3 54 2cos cosα ε ε+ −( )

a

.

(17.26)

These values for the errors found from the calibration are now to be substituted back into the equa-tions for the Mueller matrix elements by using measured values of the Fourier coefficients with a sample in place, so that we have

ma a

m

441 2

4

1

6

2

43

4

8

= − +

=

sin sin cos cos,

δ δ α α

−− + + −a b a b

m

3 3 3 3 7 4 7 4

1 2

4

cos sin cos sin

sin,

α α α αβ δ

223 3 3 3 7 4 7 4

1 2

8= −+ + +a b a bsin cos sin cos

sin

α α α αβ δ

,,

cos sin,

sin

mm b

ma b

413 42 5

5 2

249 6 9

2

4

8

=−

−

=−

βα δ

α ccos sin cos

sin,

cos

α α αβ δ

α

6 11 7 11 7

2 1

3498

− +

=

a b

ma 66 9 6 11 7 11 7

2 1

144

+ − −

=−

b a b

m

sin cos sin

sin,

α α αβ δ

β ccos

cos sin

sin,

2

2

4

2

2

25 24 1

3 1

4 5 34

22

εε δ

β εm b m

m

+ −

==+ − −

16 8 9 12 10 8 9 12 10

1 2

a a b bcos cos sin sinα α α αβ β

,,

cos cos sin sinm

a a b b33

8 9 12 10 8 9 12 1016=− − +α α α α

ββ β

α α α

1 2

238 9 12 10 8 9 1216

,

sin sin cos cm

a a b b=

− + − + oos,

αβ β

10

1 2

Mueller Matrix Polarimetry 367

ma a b b

328 9 12 10 8 9 12 10

1

16=+ + +sin sin cos cosα α α α

β ββ

ε ε β β ε

2

122 3 2 3 1 4 5 2216 4 16 4 2

,

cos sin cosm

a b m=

− − −−

=+

β β εβ

ε ε

1 4 5 32

1

132 3 2

2

2

16 4 16 4

sin,

sin cos

m

ma b 33 1 4 5 23 1 4 5 33

1

2110

2 2

2

16

− −

=

β β ε β β εβ

cos sin,

m m

ma ccos sin

,α α β β

β

β β

11 10 11 2 3 22

2

312 3 32

16

2

− −

=−

b m

mm −− −( )

= −

16 16

2

41

2

10 11 10 11

2

11 0 3

b a

m a

cos sin,

α αβ

β mm m m12 4 5 21 3 4 5 22 4

1

22

1

42

1

22− − −β ε β β ε βcos cos sin εε β β ε5 31 3 4 5 32

1

42m m− sin .

(17.27)

17.2.5 oPTical PRoPeRTieS fRoM The MuelleR MaTRix

One objective of Mueller matrix polarimetry might be to obtain electro- and magneto-optic coef-ficients of crystals. The coefficients are derived from the Mueller matrices measured as a function of applied field strength. The method by which this derivation is accomplished is briefly summarized here [15].

The application of an electric field across a crystal produces an index change. Principal indices are obtained by solving an eigenvalue problem (see Chapter 21). For example, for a 43m cubic material with index n0 and with a field E perpendicular to the (110) plane, the index ellipsoid is

x y z

nr E yz zx

2 2 2

02 412 1

+ + + + =( ) . (17.28)

The eigenvalue problem is solved, and the roots of the secular equation are the new principal indices,

n n n r E

n n n r E

n n

x

y

z

'

'

'

,

,

.

= +

= −

=

012 0

341

012 0

341

0

(17.29)

The principal indices of the 43m cubic material for an electric field applied transversely and longi-tudinally are given by Namba [16].

The phase retardation accumulated by polarized light in traversing a medium with anisotropic properties is given by

Γ = −2π λ( ) /n n La b , (17.30)

where L is the medium thickness in the direction of propagation, λ is the wavelength of light, and na, nb are the indices experienced in two orthogonal directions perpendicular to the direction of

368 Polarized Light, Third Edition

propagation. In the longitudinal mode of operation, the electric field and propagation direction are both along the z axis. The refractive indices experienced by the light are in the plane containing the x and y principal axes. If the light polarization and crystal are aligned such that the polarization is 45° from either principal axis, the phase retardation will be

Γ = ′ − ′2π λ( ) /n n Ly x , (17.31)

where ′ny , ′nx are the (new) principal indices with the field applied. (For crystals with natural bire-fringence and no electric field, these indices may just be the principal indices.)

The phase delays for light polarized at 45° to the principal axes of the 43m material can now be calculated. The phase retardation for the 43m cubic material is

Γcubic /= 2 03

41π λn r EL . (17.32)

If the electric field is expressed in terms of electric potential and charge separation, that is, E = V/d, then the phase retardation is

Γcubiclong /= 2 0

341π λn r V , (17.33)

because the charge separation d is equal to the optical path through the crystal.The phase retardation for 43m cubic material in the transverse mode is also given by Equation

17.26. In the transverse mode the charge separation is not the same as the optical path so that when E is given as V/d, the phase delay is given as

Γcubictrans /= 2 0

341π λn r VL d . (17.34)

The cubic crystal described is expected to act as a linear retarder. The Mueller matrix formalism representation of a retarder with a fast axis at arbitrary orientation angle θ is

1 0 0 0

0 2 2 1 2 22 2cos sin cos ( cos )sin cos siθ θ δ δ θ θ+ − − nn sin

( cos )sin cos sin cos cos

2

0 1 2 2 2 22 2

θ δδ θ θ θ θ− + δδ θ δ

θ δ θ δ δcos sin

sin sin cos sin cos

2

0 2 2−

, (17.35)

where the retardance is δ. If the retarder fast axis is assumed to be at 0°, the matrix becomes, sub-stituting for δ the retardance of the crystal,

1 0 0 0

0 1 0 0

0 02 2

0 0

341

341cos sin

si

πλ

πλ

n r VL

dn r V

L

d

− nn cos2 23

413

41

πλ

πλ

n r VL

dn r V

L

d

. (17.36)

It is now clear that the electro-optic coefficient r42 can be obtained from the measured Mueller matrix.

Mueller Matrix Polarimetry 369

Note that for purposes of obtaining the electro-optic coefficient experimentally, the fast axis of an electro-optic crystal acting as an ideal retarder can be at any orientation. The (4,4) matrix ele-ment of the matrix for a retarder with the fast axis at angle θ is independent of fast-axis orientation, and the fast-axis orientation can be eliminated elsewhere by adding the (2,2) and (3,3) matrix ele-ments or squaring and adding elements in the fourth row and column. Given a measured Mueller matrix of a crystal, a known applied voltage, and a known refractive index, one can easily obtain the electro-optic coefficient r41.

17.2.6 MeaSuReMenTS

As an example of a calibration measurement and compensation, the ideal and measured Mueller matrices for a calibration (no sample) are, respectively,

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

,

0 998 0 026 0 019 0 002

0 002 0 976 0 030 0 009

0

. . . .

. . . .

−−

.. . . .

. . . .

007 0 033 0 966 0 002

0 002 0 004 0 002 1 000

−− −

.

The measured results, normalized to unity, are given without any error compensation. The mea-sured matrix is clearly recognizable as a noisy representation of the corresponding ideal matrix.

Error compensation may be demonstrated with the experimental calibration Mueller matrix. The source of the large error for the two middle elements of the diagonal is the retardance errors of the wave plates. Using calculated values for the errors and compensation by using the small-angle approximation error analysis as discussed above [13], one sees that the renormalized compensated Mueller matrix for no sample becomes

0 997 0 006 0 004 0 002

0 007 1 000 0 007 0 009

0

. . . .

. . . .

−−

.. . . .

. . . .

008 0 007 0 990 0 003

0 003 0 006 0 007 0 99

− −− − 88

.

Equations for the exact error compensation give slightly better results.

17.2.7 SPecTRoPolaRiMeTRy

Spectropolarimetry is the measurement of both spectral and polarization information. A spectropo-larimeter has been described [7] based on a Fourier transform infrared (FTIR) spectrometer with the dual rotating retarder polarimeter described previously. An optical diagram of this instrument, based on a Nicolet 6000 FTIR spectrometer, is given in Figure 17.10 and shows the complete pola-rimeter within the sample compartment. The spectrometer performs the normal spectral scanning, and after a scan period the dual rotating retarder changes to a new rotational position. This contin-ues, as described in the previous section, until all polarization information is collected. The data are

370 Polarized Light, Third Edition

then reduced to produce a Mueller matrix for each wavelength of the FTIR spectrometer scan. This spectropolarimeter has been used to analyze polarization properties of optical samples in reflection and transmission.

Spectropolarimetry requires polarization elements that are achromatic across a spectral region of the data collection. Polarizers that are achromatic are generally more readily available than achromatic retarders. For the infrared (2–25 μm), wire grid polarizers are achromatic over large ranges within this region, although their diattenuation performance is not generally as good as prism polarizers. Achromatic waveplates have been designed that are achromatic over wavelength ranges somewhat smaller than the polarizers, and these custom elements can be expensive and they have achromatic performance poorer than that of the polarizers. Fortunately, the compensation techniques described in the last section apply to this problem, and are used to great advantage to correct for the imperfect achromaticity of the retarders.

17.2.8 MeaSuReMenT MaTRix MeThod

An alternative to the Fourier method described above is the measurement matrix method (see Ref. [17]). Similar to Equation 17.14, we have

a a a a mq q q q, , , ,1 2 3 4 11

• • • •

• • • •

• • • •

mm m m

m m m m

m m m m

m m m

12 13 14

21 22 23 24

31 32 33 34

41 42 43 mm

p

p

p

p

q

q

q

q44

1

2

2

2

,

,

,

,

=

=•

•

•

==∑∑

I

a m s

q

q j j k q k

kj

, , ,

1

4

1

4

,, (17.37)

D1

M11

M6

M2

S1

S2

A1

M5

M3

M1

WLS

BSWL

BSIR

LD

WLD

M4

M7

Retarders

M10MF5

MF4

L1

Polarizers

figuRe 17.10 Optical diagram of a spectropolarimeter based on a Fourier-transform infrared spectrometer. L1 is a laser, S1 and S2 are sources, D1 is the detector, elements starting with M are mirrors, elements starting with BS are beam splitters, WLS and WLD are white light source and white light detector, LD is the laser detector, and BSIR is the infrared beam splitter.

Mueller Matrix Polarimetry 371

for the qth measurement at the qth position of the PSG and PSA. We now write the Mueller matrix as a 16 × 1 vector

M = …[ ]m m m m m mT

11 12 13 14 21 44 . (17.38)

We also define a 16 × 1 measurement vector for the qth measurement as

Wq q q q q q q

T

q

w w w w w w

a s

= …[ ]

=

, , , , , ,

,

11 12 13 14 21 44

1 qq q q q q q q q q q qa s a s a s a s a s, , , , , , , , , ,1 1 2 1 3 1 4 2 1 4… ,, .4[ ]T (17.39)

The qth measurement is then the dot product of M and W,

I a s a s a s a s aq q q q q q q q q q q= =•W M , , , , , , , , ,1 1 1 2 1 3 1 4 33 2 4 4

11

12

13

14

21

44

s a s

m

m

m

m

m

m

q q q, , ,…

[ ]

. (17.40)

We make a set of Q measurements so that we obtain a Q × 16 matrix where the qth row is the measurement vector Wq. The measurement equation relates the measurement vector I to the sample Mueller vector; that is,

I WM= =

=

−

I

I

I

w w w

Q

0

1

1

0 11 0 12 0 44

, , ,

ww w w

w w wQ Q Q

1 11 1 12 1 44

1 11 1 12 1 44

, , ,

, , ,

− − −

m

m

m

11

12

44

. (17.41)

If W contains 16 linearly independent columns, all 16 elements of the Mueller matrix can be deter-mined. If Q = 16, then the matrix inverse is unique and the Mueller matrix elements are determined from the data reduction equation

M W P= .1− (17.42)

If more than 16 measurements are made, which is usually the case, M is overdetermined, although now W may not have a unique inverse. The optimal polarimetric data reduction equation is equiva-lent to a least squares solution.

17.3 oTheR muelleR maTRiX PolaRimeTRy meThodS

Other polarimetric methods have been used to obtain Mueller matrices. We describe three of them in this section.

372 Polarized Light, Third Edition

17.3.1 ModulaToR-baSed MuelleR MaTRix PolaRiMeTeR

Another class of polarimeters has been designed using electro-optical modulators. Thompson, Bottiger, and Fry [18] describe a polarimeter for scattering measurements that uses four modula-tors. These modulators are Pockels cells made of potassium dideuterium phosphate (KD*P). A functional diagram of this four-modulator polarimeter is shown in Figure 17.11.

All elements of the Mueller matrix are measured simultaneously in this polarimeter. The polar-izers are aligned and fixed in position. The four Pockels cells are driven at four different frequen-cies. The normalized Stokes vector after the first polarizer is

S0

1

1

0

0

=

, (17.43)

so that the Stokes vector at the detector is

S (P M M FM M )Sf 2 4 3 2 1 0= I0 , (17.44)

where M1, M2, M3, and M4 are the modulator Mueller matrices, P2 is the second polarizer matrix, I0 is the initial intensity, and F is the sample matrix. The intensity at the detector is the first element of this vector and is given by

II

f f f ff = + + −011 12 1 13 1 2 14 12

( cos sin sin sin cδ δ δ δ oos cos cos cos

sin sin c

δ δ δ δ

δ δ

2 21 4 22 1 4

23 1 2

+ +

+

f f

f oos sin cos cos sin sin cδ δ δ δ δ δ4 24 1 2 4 31 3 4 32− + +f f f oos sin sin

sin sin sin sin

δ δ δ

δ δ δ δ

1 3 4

33 1 2 3 4 34+ −f f ssin cos sin sin cos

cos

δ δ δ δ δ δ

δ

1 2 3 4 41 3 4

42

+

+

f

f

sin

11 3 4 43 1 2 3 4 44cos sin sin sin cos sin sinδ δ δ δ δ δ+ −f f δδ δ δ δ1 2 3 4cos cos sin ).

(17.45)

where the fij are the elements of the sample matrix and δ1, δ2, δ3, and δ4 are the retardances of the four modulators. The retardances of the modulators are driven by oscillators at different frequencies so that they are

δ δ ωi oi it= cos , (17.46)

Laser

Polarizer Polarizer

Photodetector

Modulators Modulators

figuRe 17.11 Functional diagram of the four-modulator polarimeter.

Mueller Matrix Polarimetry 373

where δoi is the amplitude of the retardance of the ith retarder. The trigonometric functions in the oscillating retardances are expanded in terms of Bessel functions of the retardation amplitudes, these results are substituted into the expression for the intensity, and the Fourier expansion of the coefficients of the fij is taken. The primary frequencies at which each matrix element occurs are

0 2ω1 ω1±ω2 ω1±2ω2

(17.47)

2ω4 2ω1±2ω4 ω1±ω2±2ω4 ω1±2ω2±2ω4

ω3±ω4 2ω1±ω3±ω4 ω1±ω2±ω3±ω4 ω1±2ω2±ω3±ω4

2ω3±ω4 2ω1±2ω3±ω4 ω1±ω2±2ω3±ω4 ω1±2ω2±2ω3±ω4.

The modulation frequencies are chosen so that there are unique frequencies of signal correspond-ing to each matrix element. Lock-in amplifiers for these frequencies are used in the detector electronics.

Initial alignment of the modulators with the polarization direction is not perfect, and the fore-going analysis can be repeated with a constant retardation error for each modulator. This results in somewhat more complex expressions for the characteristic frequencies for the matrix elements. A calibration procedure minimizes the errors due to misalignment. Accuracy of 1% is said to be attainable with iterative calibration.

17.3.2 MuelleR MaTRix ScaTTeRoMeTeR

The scatter of light reflected from a surface into the sphere surrounding the point of incidence is measured in order to understand reflection properties of the surface. The effect of polarization in the reflection process can be measured with a Mueller matrix scatterometer, described by Schiff et al. [19]. The sample is mounted on a goniometer so that in-plane or out-of-plane measurements may be made. There are optics associated with the source (PSG) and receiver (PSA) that allow

Polarized laser

Half wave retarder

Quarter wave retarder

Aperture

Quarter wave retarder

Polarizer

Lens

Bandpass filter

Sample

Field stop

Detector

figuRe 17.12 Diagram of a mueller matrix scatterometer.

374 Polarized Light, Third Edition

complete polarization control, shown in Figure 17.12. The source optics consist of a linearly polar-ized laser source, a half-wave plate to control orientation of the linear polarization, and a quarter-wave retarder. The receiver optics consist of a quarter wave retarder and a linear polarizer.

The power measured by the detector is given by

Po = [r][M][s]Pi, (17.48)

where Pi is the input power from the laser, vector s is the (normalized) source optics Stokes vector, M is the sample matrix, and r is basically the top row of the Mueller matrix for the receiving optics. In order to measure M, the source optics are set so that six Stokes vectors are produced correspond-ing to the normalized Stokes vectors for linear horizontal, linear vertical, ±45° linear, and right and left circularly polarized light; that is, S1, S2, and S3 are set to ±1, one at a time. The PSA is set to these six polarization states for each of the six states of the PSG to produce 36 measurements. Expressing this in matrix form we have

[Po] = [R][M][S]Pi . (17.49)

where here R represents a 6 × 4 matrix, and S represents a 4 × 6 matrix.A calibration must be performed to compensate for errors, since there are multiple error sources

that will not allow the production of ideal polarization states. The R and S matrices above give 48 unknowns. A measurement is made with no sample to give 36 values of Po, and twelve more equa-tions are obtained from the quadrature relations associated with the over definition of the Stokes vectors. This comprises a system of 48 equations and 48 unknowns. Solving these produces the matrices [S] and [R], and now measurements of P0 can be made with a sample in place, and the matrix M can be calculated from

M R R R P S S S1

o[ ] = [ ] [ ]( ) [ ] [ ][ ] [ ][ ]( )

−T T T T

iP

− 1 1

. (17.50)

17.3.3 fouR-deTecToR PhoToPolaRiMeTeR

The four-detector photopolarimeter was described in Chapter 16. It is a complete Stokes polarim-eter. A Mueller matrix polarimeter is constructed by using a four-detector photopolarimeter as the PSA and a conventional polarizer-quarter wave retarder pair as a PSG. The polarizer is set at some fixed azimuth, and the output signal (a four element vector) from the four-detector photopolarimeter is recorded as a function of the azimuth of the fast axis of the quarter wave retarder. The signal is subject to Fourier analysis to yield a limited series whose vectorial coefficients determine the col-umns of the measured Mueller matrix.

Calibration of the instrument is required and takes place with no sample present. The optical ele-ments are aligned so that light is directed straight through. The fast axis of the quarter wave retarder is aligned with the fixed polarizer by adjusting it in small steps until S3 from the four-detector pho-topolarimeter is 0. After the light passes through the quarter wave plate, the Stokes vector is

S( )

cos

( ) cos cos

sin sinθ

θθ θ

θ=

+− + +

+

1 2

1 2 4

2

g

f g f

g f 44

2

2 20 1 1θθ

θ θ

sin

cos sin

= + + +S S S Sc s 22 24 4c scos sinθ θ+ S , (17.51)

Mueller Matrix Polarimetry 375

where θ is the retarder azimuth, and f and g are characteristic of the quarter wave retarder and where

S S0 1

1

1

0

0

0

0

=−

=

( )f

g

gc,

=

=

, ,S S1 2

0

0

1

0

0

0

s cg

f

=

, .S2

0

0

0

sf

(17.52)

The values of f and g are determined by a rotating quarter wave test [20]. The value of g is the diattenuation of the quarter-wave retarder, and 2f – 1 is the retardance error from quarter wave in radians.

The output vector of the four-detector polarimeter with a sample in position is

I AMS( ) ( )θ θ= (17.53)

where M is the sample Mueller matrix and A is the instrument calibration matrix [20]. Using S(θ) from Equation 17.52 gives a Fourier series for I(θ) of the same composition as S(θ) with vectorial coefficients given by

I A C C

I A C C

I A C

0 1 2

1 1 2

1 3

= + −[ ]

= +[ ]

=

M M

c M M

s

f

g

g

(1 ) ,

,

MM M

c M

s M

f

f

+[ ]

=

=

C

I AC

I AC

4

2 2

2 3

,

,

,

(17.54)

where C1M, C2M, C3M, and C4M are the columns of the Mueller matrix M. These columns are then given by

C A I21

21M cf= −( / ) ,

C A I31

21M sf= −( )/ ,

C A I C11

0 21M Mf= − −− ( ) ,

C A I C21

1 3M s Mg= −− . (17.55)

RefeReNCeS

1. Hauge, P. S., Recent development in instrumentation in ellipsometry, Surface Sci. 96 (1980): 108–40. 2. Azzam, R. M. A., Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single

detected signal, Opt. Lett. 2 (1978): 148–50. 3. Goldstein, D. H., Mueller matrix dual-rotating retarder polarimeter, Appl. Opt. 31 (1992): 6676–83. 4. Pezzaniti, J. L., S. C. McClain, R. A. Chipman, and S.-Y. Lu, Depolarization in liquid-crystal televisions,

Opt. Lett. 18 (1993): 2071–3.

376 Polarized Light, Third Edition

5. Sornsin, E. A., and R. A. Chipman, Electro-optic light modulator characterization using Mueller matrix imaging, Proc. SPIE 3121 (August 1997): 161–6.

6. Goldstein, D. H., and R. A. Chipman, Infrared Spectropolarimeter, U.S. Patent No. 5,045,701, September 3, 1991.

7. Goldstein, D. H., R. A. Chipman, and D. B. Chenault, Infrared spectropolarimetry, Opt. Eng. 28 (1989): 120–5.

8. Hauge, P. S., Mueller matrix ellipsometry with imperfect compensators, J. Opt. Soc. Am. 68 (1978): 1519–28.

9. klein Brink, H. B., Birefringence of the human crystalline lens in vivo, J. Opt. Soc. Am. A 8 (1991): 1788–93.

10. klein Brink, H. B., and van Blokland, G. J., Birefringence of the human foveal area assessed in vivo with Mueller matrix ellipsometry, J. Opt. Soc. Am. A. 5 (1988): 49–57.

11. van Blokland, G. J., Ellipsometry of the human retina in vivo: preservation of polarization, J. Opt. Soc. Am. A 2 (1985): 72–5.

12. Strang, G., Linear Algebra and Its Applications, 2nd ed., 112, New York: Academic, 1976. 13. Goldstein, D. H., and R. A. Chipman, Error analysis of Mueller matrix polarimeters, J. Opt. Soc. Am. 7

(1990): 693–700. 14. Chenault, D. B., J. L. Pezzaniti, and R. A. Chipman, Mueller matrix algorithms, Proc. SPIE 1746 (1992):

231–46. 15. Goldstein, D. H., R. A. Chipman, D. B. Chenault, and R. R. Hodgson, Infrared material properties mea-

surements with polarimetry and spectropolarimetry, Proc. SPIE 1307 (1990): 448–62. 16. Namba, C. S., Electro-optical effect of zincblende, J. Opt. Soc. Am 51 (1961): 76–9. 17. Chipman, R. A. Polarimetric Impulse Response, Proc. SPIE 1317, Polarimetry: Radar, Infrared, Visible,

Ultraviolet, and X-Ray, 223–41, May 1990. 18. Thompson, R. C., J. R. Bottiger, and E. S. Fry, Measurement of polarized light interactions via the

Mueller matrix, Appl. Opt. 19 (1980): 1323–32. 19. Schiff, T. C., J. C. Stover, D. R. Bjork, B. D. Swimley, D. J. Wilson, and M. E. Southwood, Mueller matrix

measurements with an out-of-plane polarimetric scatterometer, Proc. SPIE 1746 (1992): 295–306. 20. Azzam, R. M. A., and A. G. Lopez, Accurate calibration of the four-detector photopolarimeter with

imperfect polarizing optical elements, J. Opt. Soc. Am. A 6 (1989): 1513–21.

377

18 Techniques in Imaging Polarimetry*

18.1 iNTRoduCTioN

In the previous chapters in this section of the book, we have covered basic manual polarimetry techniques and a variety of polarimetry schemes that are best suited to be implemented with auto-mated systems. To this point, we have been concerned with a single signal and how we can collect and process polarimetric information. In remote sensing, we very often would like to have imagery because this is how we perceive the world, and because we would like to compare the polarization characteristics of different points in an image to differentiate materials and textures. In this chapter, we give an overview of imaging polarimetry.

The primary physical quantities associated with an optical field are the intensity, wavelength, coherence, and polarization. To obtain an image, we might measure some or all of these quanti-ties over a two-dimensional array of points in object space. Conventional panchromatic cameras measure only the intensity of optical radiation over some waveband of interest. Spectral imagers measure the intensity in a number of wave bands that can range from one or two (three is common for a color camera) through multispectral systems that measure on the order of 10 spectral chan-nels to hyperspectral systems that may measure 300 spectral channels or more. Spectral sensors tend to give us information about the distribution of material components in a scene. Polarimetry, on the other hand, seeks to measure information about the vector nature of the optical field across a scene. While the spectral information tells us about materials, polarization information tells us about surface features, shape, shading, and roughness. Polarization tends to provide information that is largely uncorrelated with spectral and intensity images, and thus has the potential to enhance many fields of optical metrology. Figure 18.1 shows one example of the ability of polarization to show enhanced contrast when there is little contrast in intensity imagery.

Imaging polarimetry is a special case of general polarimetry that is dedicated to mapping the state of polarization across a scene of interest. Applications of polarization imagery range from remote sensing to microscopy to industrial monitoring. All the concerns of general polarimetry apply; that is, a measurement method still has to be chosen and calibration must be performed, but now the additional issues associated with measuring a two-dimensional region in space exist. Sequential or simultaneous images must be registered, and we must know that the response of indi-vidual detectors is linear and, if multiple detectors are used, uniform in response with respect to all other detectors.

In this chapter, we provide a review of the progress that has been made specifically in the field of imaging optical polarimetry for remote sensing. Most of the work discussed here has been carried out over the past three decades. Our primary focus is on passive Stokes vector imagers, though we do discuss some of the work that has been done in active Mueller matrix imagers and polarization LIDAR. Where possible, we refer to the earliest source known to us, preferably from the reviewed scientific literature.

* This chapter is contributed by J. Scott Tyo, College of Optical Sciences, University of Arizona, Tucson, AZ; David B. Chenault, Polaris Sensor Technologies, Huntsville AL; Joseph A. Shaw, Electrical and Computer Engineering Department, Montana State University, Bozeman, MT and Dennis H. Goldstein.

378 Polarized Light, Third Edition

In Section 18.2, we give a brief historical perspective. Section 18.3 describes the phenomenol-ogy of imaging polarimetry, and Section 18.4 describes types of measurements and data reduc-tion techniques. We give general measurement strategies that have been used in Section 18.5, and a discussion of systems engineering issues in Section 18.6. Finally, conclusions are presented in Section 18.7.

18.2 hiSToRiCal PeRSPeCTiVe

Someone peering through a birefringent crystal and observing a pair of refracted polarized images probably did the earliest imaging polarimetry. There are two important early experiments by Arago and Fresnel [1], Arago [2], and Millikan [3,4] that are often reported as the earliest attempts at quantitative polarimetry. Arago performed a number of qualitative experiments involving polarized light, and was the first to observe the phenomena of optical activity and that emitted radiation is not always unpolarized. Millikan measured the linear polarization information from incandescent molten metals, and there were a number of subsequent studies that explored polarization of emitted radiation. Sandus [5] provides a thorough review of the physics and these early works.

To discuss imaging polarimetry in the modern quantitative sense, we must leap forward to the age of solid-state electronics. The earliest work known to us is contained in two originally

figuRe 18.1 (See color insert following page 394.) Visible picture of two pickup trucks in shade (top), long-wave IR intensity image (bottom left), and long-wave IR polarization image (bottom right). Strong con-trast in the polarization image shows advantages for enhanced target detection using imaging polarimetry. (Photo courtesy of Huey Anderson. With permission from Optical Society of America.)

Techniques in Imaging Polarimetry 379

classified government reports, the first by Johnson [6] in 1974 and the second by Chin-Bing [7] in 1976. The instrument described in these reports is a thermal infrared scanning camera that was modified by adding a second detector and a polarizing prism. A 1976 patent by Garlick, Steigmann, and Lamb [8] described a system that displayed a differential optical polarization image. The earliest publications describing imaging polarimetry in the visible are the papers by Walraven [9,10] where a linear polarizer was rotated in front of a film camera. The developed film was digitized, and linear Stokes vector elements calculated. Solomon [11] gave an early review of imaging polarimetry in 1981. Polarimetric sensors also have been used on manned and unmanned spacecraft. Pioneer 11 has the imaging photopolarimeter on board [12] and the space shuttle has carried dual film cameras [13] and later three-color digital cameras with polarization optics [14] operated by a mission specialist. These systems measured two or three components of linearly polarized light. Three cameras were used by Prosch, Hennings, and Raschke [15] to obtain the first three Stokes vector elements, and dual piezoelastic modulators were used by Stenflo and Povel [16] to measure the full Stokes vector. Pezzaniti and Chipman [17,18] developed a Mueller matrix imaging polarimeter that has been used to examine optical elements in transmission and reflection. There are many other examples. The sources cited are each early realizations of a par-ticular type of imaging instrument.

18.3 meaSuRemeNT CoNSideRaTioNS

The basic aspects of light that are typically measured in imaging scenarios are intensity, spectral content, coherence, and polarization. For passive imaging polarimetry it is often most convenient to represent the polarization information in terms of the Stokes vector, which is defined in terms of the time-averaged intensity. Implied is that the intensity measurement is made over some spec-tral range. The range could be broad or narrow, and the choice of spectral bands is discussed below.

18.3.1 SPecTRal conSideRaTionS

Spectral information usually tells the observer something about the molecular makeup of the mate-rials that compose a scene. Multispectral and hyperspectral imagers have been developed to exploit this class of information [19]. While there are exceptions, polarization information usually is a slowly varying function of wavelength [20–22], so it provides information that tends to be uncor-related with any spectral measurements that are made in a system.

When pursuing a particular application of imaging polarimeters, spectral considerations are among the first issues to be addressed. There are advantages and disadvantages in each spectral band as in intensity imaging both from the consideration of detection instrumentation as well as the phenomenology the user is trying to exploit. Imaging polarimeters typically are based on silicon in the visible (VIS) to near infrared (NIR), may use InGaAs in the short-wave infrared (SWIR), InSb in the midwave infrared (MWIR), and HgCdTe in the long-wave infrared (LWIR). The characteris-tics of these detector types that are considered when used in nonimaging systems apply to imaging polarimeters as well; that is, silicon-based imagers are inexpensive relative to IR systems, IR sys-tems often must be cooled but have day/night capability, and so on.

In terms of the phenomenology, polarization signatures in the visible and NIR part of the spec-trum are dominated by reflection. Thus, these signatures depend on an external source for illumi-nation, primarily the sun. The polarization has wide dynamic range and can show rapid spatial variation when imaging outdoor scenes. The measured polarization information is dependent on source-scene–sensor geometry and therefore can vary significantly depending on time-of-day or sensor location. In the MWIR, polarization signatures are a combination of both reflected and emit-ted radiation, which tend to cancel or reduce the overall degree of polarization. In the LWIR, the signatures are dominated by emission and can be very stable in time when scene temperatures are

380 Polarized Light, Third Edition

stable. Unfortunately, in the LWIR spatial resolution is reduced and cost and complexity of building a system are generally increased.

In outdoor measurements, the most rapid variations of polarization with wavelength result from atmospheric spectral absorption features [20]. In the VIS-NIR-SWIR, there is strong variation with atmospheric aerosol and cloud content [23]. The MWIR contains significant emitted and reflected terms, and LWIR scenes depend strongly on atmospheric water vapor. Some of the issues that arise for imaging polarimetry with respect to spectral regions are given in Table 18.1.

18.3.2 one-diMenSional PolaRiMeTeRS

The simplest possible use of polarimetry in imaging is to put a polarization analyzer in front of a camera and adjust the polarization state of this polarizer to maximize contrast between an object and its background. This is a common technique used in photography, for example, when taking a picture of an object against linearly polarized skylight. Similar techniques have been used in under-water imagery to mitigate the effect of scattering using both linear [24] and circular [25] polariza-tion analyzers with both unpolarized and polarized illumination. The light scattered by the medium may have a preferred polarization state owing to the polarization of the source and the illumination geometry. The general strategy is to select a polarization analyzer that is orthogonal to the polariza-tion state of the background light or light scattered by haze.

18.3.3 Two-diMenSional PolaRiMeTeRS

The natural extension of the one-dimensional polarization imager is a polarization difference imager that measures the intensity of light at two polarization states, then adds them to estimate S0 and subtracts them to estimate S1, S2, or S3, or some linear combination thereof. Simple two-dimensional imagers have shown applicability in a number of scenarios, but are most widely used in clutter rejection [22] and in mitigating the effects of random media [26–31]. The basic assumption in these cases is that there is a difference between the polarization properties of light coming from the background and the light coming from a target. In such cases, significant contrast enhancement can be obtained.

Table 18.1Polarization Phenomenology and effects from the Visible to the lWiR

advantages disadvantages

Visible, NIR, SWIRTypical signal: 1–60%Sensor resolution: > 1–2%

Sun is a strong source•High dynamic range of polarization •signaturesSensors cheaper, easier to build and •calibrate

Strongly dependent on geometry•High dynamic range of signatures•Inconsistent signatures•Small well size for FPAs limits •polarimetric resolutionNo night operation •

MWIRTypical signal: 0.1–25%Sensor resolution: > 0.2%

Good signatures for hot targets•Night operation•Large well sizes for FPA for better •sensitivity

Signatures combination of emissive and •reflectiveSensors require cooling•Sensors more expensive & difficult to •build & calibrate

LWIRTypical signal: 0.1–20%Sensor resolution: < 0.1%

Signatures dominated by emission•Less dynamic range for polarization •signaturesLarge well sizes for FPA for better •sensitivityNight operation •

Sensors require cooling•Sensors most expensive & difficult to •build & calibrate

Techniques in Imaging Polarimetry 381

Two-dimensional polarimetry has been used with both unpolarized [22,26] and polarized [25,27,28,32] illumination. Two-dimensional polarization discrimination has been widely used in scattering media, and has been shown to increase the range at which targets can be detected by a fac-tor of two to three [26,27]. When used with passive or quasi-passive systems, polarization imaging has been shown to penetrate as much as five to six photon mean-free paths into random media. For time-gated imagery, polarization can allow penetration to greater than 10 photon paths [28,33]. The improved performance of differential polarimetry over conventional imagery in scattering media can be directly attributed to the depolarizing effect of multiple scattering. This results in a spatially narrower point spread function for differential polarization imagery than for intensity imaging [34]. In time-gated imagery, there is a clear temporal dependence of the degree of polarization of scat-tered light that can be used to refine the time gate and mitigate the effect of scatterers [32,35].

18.3.4 ThRee-diMenSional PolaRiMeTeRS

The most common class of imaging polarimeter that has been developed is the linear polarization imager designed to measure S0, S1, and S2. In most passive imaging scenarios, there is very little, if any, expected circular polarization. Since the most complicated Stokes parameter to measure is S3, it is often omitted to reduce the cost of the imaging system. Probably the earliest well-known example of a full linear Stokes imaging polarimeter was reported by Walraven [9,10] who used linear polar-izers and photographic film. Other systems have been developed since then that perform full linear polarimetry in all regions of the optical spectrum.

When a fixed-position retarder of variable retardance is combined with a linear polarization analyzer, it is possible to create a 3D Stokes polarimeter that measures S0, S1, and S3 as discussed in Section 18.4.2. Such a system is sensitive to a linear polarization difference and a circular polar-ization difference, and systems such as these have been used for imaging in scattering media [25,27,32].

18.3.5 full STokeS PolaRiMeTeRS

In some applications, it is essential to measure all of the available polarization information. For a passive imaging system, this means that the full Stokes vector must be measured at every pixel in the scene. Solomon [11] provided one of the first early treatments that specifically addressed full Stokes imaging polarimetry in 1981. Since then, numerous systems have been built that can perform full Stokes imaging, and we review many of these systems below organized by the class of spectral imager and the techniques used to perform the measurement.

18.3.6 acTive iMaging PolaRiMeTeRS

Most of the polarimeters described in this chapter are passive imaging polarimeters that measure the state of polarization of light from an external source. However, it is appropriate to discuss some of the important advances in active systems that measure the Mueller matrix or some subset of the Mueller matrix. Similarly, we briefly discuss developments in polarization light-detection-and-ranging (lidar) systems, which record backscattered light from a pulsed laser in two or more polar-ization states as a function of range. In all active polarimeters, the source is known and controlled. The source may generate one or more states of polarization, and the detection system may sense two or more states of polarization. Partial or full measurement of the Stokes vector of the reflected light may be what the sensor is designed for, but in the most complete form of active imaging, the Mueller matrix for each pixel of the illuminated object is obtained. There are two primary forms of active imaging polarimeters. The first are those that create an entire scene in one image collection. The second are systems that scan pixel by pixel to create a scene, and possibly even a volumetric scene with range-gated data.

382 Polarized Light, Third Edition

18.3.6.1 mueller matrix and other active imaging SystemsPezzaniti and Chipman [18] and Chipman [36] describe Mueller matrix imaging polarimeters that are used to examine samples in transmission or in reflection. Dual rotating retarders are used in these instruments according to the scheme devised by Azzam [37]. Clémenceau, Breugnot, and Collot [38] operated a Mueller matrix imaging polarimeter in a monostatic configuration. They also used a dual rotating retarder system, but collected only 16 images, the minimum number of measurements needed to determine an arbitrary unknown Mueller matrix. All the systems dis-cussed so far use monochromatic sources. Le Hors, Hartemann, and Breugnot [39] showed a system using a white light source that was spectrally filtered prior to entering the CCD camera. A linear polarizer was placed in front of the source, and two linear polarization states were measured. In this way, images at three colors and two polarization states per color were obtained. Breugnot and Clémenceau [40] have set up a system based on Azzam’s dual rotating retarder configuration using a laser source in a monostatic configuration, but argue that a limited number of Mueller matrix ele-ments are important and these can be obtained with only two measurements. A diagram of such a system is shown in Figure 18.2. Réfrégier and Goudail [41] have also developed contrast parameters of polarization for active imagery and, with others, have looked at the problem of estimating the degree of polarization in active systems [42]. High-speed Mueller matrix imaging systems for labo-ratory samples have been described by Baba et al. [43] and by Wolfe and Chipman [44]. A different technique was introduced by Mujat, Baleine, and Dogariu [45] that uses interferometric methods with active imagery. If the direction within the Poincaré sphere across an image is uniform and is known or can be assumed, as is sometimes the case with active illumination, then the degree of polarization and retardance can be monitored in a single image.

18.3.6.2 lidar SystemsPolarization is also found to be useful in more traditional lidar remote sensing. For example, in atmospheric lidar, the presence of significant cross-polarized light relative to a linearly polarized transmitter can indicate the presence of ice in clouds or aspheric dust particles [46–49]. Similarly, polarized lidars have been used to distinguish biological agents from harmless aerosols by the use of a cross-polarized signal that indicates the presence of elongated spores. Polarized lidars also have

PSA

Camera

PSGSource

figuRe 18.2 Experimental setup of the active polarimetric imaging system. (With permission from Optical Society of America.)

Techniques in Imaging Polarimetry 383

been developed to measure Stokes parameters of backscattered light in studies of forest and Earth-surface properties [50,51] and to enhance contrast in the lidar detection of fish [52].

Polarization lidar systems typically employ linearly polarized laser transmitters that provide ranging from the round-trip transit time of a backscattered pulse. The polarization selectivity is typically built into the receiver, often using polarization beam splitters to send orthogonally polar-ized beams to two separate detectors for simultaneous detection of copolarized and cross-polarized scattering. Multiple telescopes can also be used to provide simultaneous measurement of the Stokes parameters of backscattered light [50]. Lidars also have been reported that use Pockels cells [47] or liquid crystal variable retarders [53] to vary the receiver polarization state electronically between laser pulses.

18.3.7 SPecTRoPolaRiMeTRic iMageRS

A spectropolarimetric imager allows measurement of polarization as a function of wavelength in an imaged scene. When it is not necessary to obtain spectral data rapidly or simultaneously, it is possible to combine a more traditional imaging polarimeter with a rotating filter wheel that selects predetermined spectral bands [54]. One example application where this kind of system finds use is the study of sky polarization, for which wide angular coverage and rapid polarization measure-ments are needed, but for which rapid spectral measurements may not be necessary [55–58]. This approach enjoys relatively simple data retrieval and spectral calibration, but is also slow (in spectral space) and requires moving parts, making it unsuitable for some applications where rapid spectral data are required. Lemke et al. [59] describe a system that uses a combination of rotating filters and polarizers to achieve time-sequential polarization images in an extremely wide wavelength range of 2–240 μm.

Loe and Duggin [60] described the use of a liquid-crystal tunable spectral filter to electronically tune across multiple 10 nm wide wavelength bands in a system that employed a rotating linear polar-izer and CCD camera to achieve three-Stokes-parameter spectropolarimetric imaging. This was developed as a prototype of a single channel for a four-channel system. Eventually the full system would employ four such systems with a stationary polarization element oriented to provide a full Stokes image at each wavelength band.

A faster, but still not simultaneous, method of achieving electronic spectral tuning in a spec-tropolarimeter is to use an acousto-optic tunable filter (AOTF) as a spectral tuning element. The separate ordinary-ray and extraordinary-ray beams from the AOTF can be used to generate two simultaneous images with orthogonal linear polarization. Alternatively, the AOTF can be combined with an external polarizing element (such as a variable retarder) to obtain time-sequential Stokes-vector images [61–64]. AOTF elements provide rapid spectral tuning with typical delay times of 10–20 ms. An active-spectropolarimeter variation of this approach was described by Prasad [65] using a simultaneously tuned AOTF receiver and tunable laser source.

Rather than obtaining multiple spatial dimensions simultaneously and spectral information over time, it is also possible to use one dimension of an imaging array to capture spectral data while using the other array dimension to record 1D spatial data. In this case, a full spectropolarimetric image is built up by spatially scanning the sensor’s field of view across the scene. For example, Tyo and Turner [21] used a polarimeter comprising two liquid crystal variable retarders and a fixed linear polarizer in combination with a monolithic Fourier transform interferometer to achieve line-scanned spectropolarimetric images of laboratory test objects. Jensen and Peterson [66] used a complementary strategy of feeding a grating spectrometer by an infrared liquid crystal for imaging polarimetry in the SWIR.

Several related schemes exist for obtaining simultaneous spectropolarimetric images with no moving parts and no temporal delay between spatial, spectral, or polarimetric data. The polarimet-ric strategy is channeled spectropolarimetry briefly discussed in Section 18.4.3, and discussed in more depth in Chapter 19. This snapshot imaging spectropolarimetry typically employs birefringent

384 Polarized Light, Third Edition

crystals [67,68] or holographic optical elements [69] to record fringe patterns from which spectropo-larimetric images can be retrieved through a variety of numerical inversion techniques. The obvious advantage is the simultaneous collection of all measured information, but the technique requires intensive computation and is not well suited to images with significant low-spatial-frequency or high-spectral-frequency content [69].

18.4 meaSuRemeNT STRaTegieS aNd daTa ReduCTioN TeChNiQueS

The Stokes vector cannot be measured directly. To create an image of a scene, several individ-ual measurements must be made and then combined to infer the Stokes vector. The measurement strategies can be broadly grouped into three categories: data reduction matrix (DRM) techniques, Fourier-based techniques, and channeled spectropolarimeters. The general principles of the first two types of polarimeters have been discussed in the chapters on Stokes polarimetry, and the prin-ciples used for the third will be addressed in detail in the next chapter on channeled polarimetry, but the basic material will be repeated briefly here.

18.4.1 daTa ReducTion MaTRix TechniqueS

The most straightforward method might be to measure four linearly polarized intensities through a linear analyzer oriented at 0°, 45°, 90°, and 135° and through a left and right circular analyzer. The elements could then be combined following the definition of the Stokes vector. However, the Stokes vector has only four degrees of freedom, and this strategy would entail six measurements. A method has been developed known as the DRM method [70] that describes the operation of a polarimeter designed to measure the Stokes vector.

A polarimeter is typically composed of a collection of retarders and polarizers that are cascaded to form a polarization state analyzer (PSA). In general there may be one or more retarders placed in front of a linear polarizer. The component Mueller matrices are multiplied together to form a general elliptical diattenuator Mueller matrix as [71]

MD

P I a a3d u

T

d dT

Td d

=− + − −( )

1

1 1 12 2. (18.1)

The three-element column vector D in Equation 18.1 is the diattenuation vector [71] that gives the location on the Poincaré sphere of the polarization state that passes the diattenuator with maximum intensity. The unit vector ad points in the direction of D, and d is the diattenuation of the diattenu-ator, defined as

dT T

T Tq r

q r

=−+

, (18.2)

where q and r are the two orthogonal states that are passed with maximum and minimum transmis-sion. When we consider ideal polarization optics, we typically have |D| = 1, and we can define a diattenuation Stokes vector as

S DdT T= [ ]1 . (18.3)

When the unknown incident Stokes vector Sin passes through the diattenuator, the resulting output Stokes vector is

S M Sout in= ⋅d . (18.4)

Techniques in Imaging Polarimetry 385

Since most photodetector elements are polarization insensitive, the output of the detector usually will be proportional to S0,out, which can be written in vector form as

S m S m S m SdT

d d d0 00 0 01 1 02, , , , , ,out in in in= ⋅ = + +S S 22 03 3, , , .in in+ m Sd (18.5)

Equation 18.5 has four unknowns, the input Stokes parameters, so to solve for these unknowns, we must build up a system of linear equations like Equation 18.5 using at least four different real-izations of the diattenuation Stokes vector in Equation 18.3. In matrix form, this system can be written as

X

S

=

=

( )S

S

S N

dT

01

02

0

1,

,

,

out

out

out

SS

S

S A SdT

dN T

2( )

( )

⋅ = ⋅

in in . (18.6)

The notation Sdi T( ) represents the ith realization of the diattenuation Stokes vector. In general, the

number of measurements N ≥ M, where M is the number of dimensions that will be reconstructed in the polarimeter. The matrix A in Equation 18.6 is referred to as the system matrix and its inverse is termed the DRM [70]. We can estimate the unknown input Stokes vector as

ˆ ,S A Xin = ⋅−1 (18.7)

where the hat indicates that Equation 18.7 is only providing an estimate. Sources of error could include noise in the measurement vector X and calibration measurements in determining the DRM. Clearly we need to be careful about the selection of Sd

i T( ) , as the condition number of the matrix A must be low enough so that the inversion process is well behaved. More details are provided on this issue in the section on polarimeter optimization below.

The DRM measurement strategy can be interpreted from a signal processing viewpoint [72]. Each of the entries in X can be thought of as a projection of the unknown input Stokes vector onto an analysis vector Sd

i T( ) . When N = M, the analysis vectors form a nonorthogonal basis in the conical space that is the allowed space of physically realizable Stokes vectors. When N > M, the analysis vectors form an over-determined basis, or frame [72]. As discussed in Section 18.6, use of a frame can enhance the robustness of the measurement process.

18.4.2 fouRieR ModulaTion TechniqueS

A common method of polarimetric measurement and data reduction is through the Fourier analysis of polarimetric signals. These methods were developed initially for Mueller and Jones matrix pola-rimeters for nonimaging measurement of polarized and partially polarized light [36,73,74] and for ellipsometric measurements [75–77]. They are readily generalized to spectral and imaging instru-ments [18,29,78,79].

In this approach, a series of images are acquired as the elements of the PSA are varied in a har-monic fashion. The polarization of the incident light is encoded onto the harmonics of the detected signal. The Stokes-vector elements of the incident light are then recovered from a Fourier transform of the measured data set. The Stokes vector is computed independently for each pixel.

386 Polarized Light, Third Edition

Consider a general polarimeter with incident light of unknown polarization and a PSA as shown schematically in Figure 18.3. A series of N intensity measurements Sn

0,out are made as in Subsection 18.4.1

X

S

=

=

( )S

S

S N

dT

01

02

0

1,

,

,

out

out

out

SS

S

S A SdT

dN T

2( )

( )

⋅ = ⋅

in in . (18.8)

Varying the polarization elements of the analyzer modulates the analyzed polarization states. A typical method of varying the polarization elements is by rotating some or all of the elements in discrete steps. If the angular increments of the polarization elements are constant, only discrete frequencies are generated in the detected intensity X whose elements are written as xn. The intensity xn is collected for the nth position of the polarization elements in the PSA. The detected signal can be written

xb

b k c kn k n k n

k

= + +=

∑0

12

( cos sin ),θ θ (18.9)

where the largest k is the highest frequency component in the signal and θn = nθ is proportional to the angular frequency of the polarization element. The polarization content of the scene being imaged is encoded onto the various frequencies of the detected signal, that is, the coefficients in the Fourier series expansion are functions of the incident Stokes vector. These relations are inverted to give the Stokes vector elements in terms of the Fourier coefficients. The coefficients are determined from the set of intensities by a discrete Fourier transform,

bN

x

bN

ink

Ni k

n

n

N

k n n n

0

0

11

2 2

=

=

=

=

−

∑ ,

cos cosπ θ(( )

=

=

=

−

=

−

∑∑n

N

n

N

k n ncN

ink

Ni

0

1

0

1

2 2

,

sin sinπ

kk n

n

N

n

N

θ( )=

−

=

−

∑∑0

1

0

1

,

(18.10)

Rotatingretarder

Polarizersensor head

Camera Interfaceinstrument controldata flow

Process andcontrol system

figuRe 18.3 (See color insert following page 394.) Polarimetric sensor using rotating polarization ele-ments. (With permission from Optical Society of America.)

Techniques in Imaging Polarimetry 387

where k is the harmonic, θn = nΔθ, and Δθ is the angular increment of the polarization elements. For N intensities, the coefficients for the K = N/2 harmonics are found. The step size of the rotation of the polarization element is determined by the number of measurements Δθ = 2π/N.

The highest harmonic K in the polarimetric signal is determined from the analytical expres-sion for the intensity written as a Fourier series. The minimum number of measurements, Nmin, required to calculate the dc term and all cosine and sine (real and imaginary) terms in the Fourier transform is Nmin = 2K + 1. It is often desirable to make more measurements than the minimum, or oversample, to help reduce the effects of noise. For oversampled data, the harmonics higher than the frequencies of the polarimetric signal are often used as a diagnostic tool to indicate sources of systematic error.

The Fourier analysis of polarimetric signals provides several significant advantages for data reduction. First, if the analytical form is readily derived via a system Mueller matrix expression, this data reduction method is straightforward and computationally fast. Second, the system Mueller matrix may be parameterized such that diattenuation and retardance values and orientation of the elements may be determined in calibration. Third, this method often encompasses instruments where the elements are rotated continuously. In this case, the angular increment used for discrete steps of the rotated elements is replaced by the angular increment at which the next data acquisi-tion is begun. Any motion of the element over the integration time of the sensor is compensated for in calibration. Fourth, the calculation of the discrete Fourier transform automatically gives a least-squares fit to the data. Finally, the discrete Fourier transform is a useful analytical tool for investigating many types of systematic error such as beam wander and linear drift. The suscepti-bility to harmful noise sources can be reduced through adjusting the parameters of measurements and the corresponding Fourier transform. More details of the effect of noise and errors on the measurements and ways to compensate or negate these effects are given by Goldstein and Chipman [80] and Chenault, Pezzaniti, and Chipman [81]. The chief disadvantage of this approach is that the system Mueller matrix must be well known. In practice, this requires that the polarizers are pure diattenuators and the retarders are pure retarders, that is, the polarizers contain no retardance and the retarders are not diattenuating.

18.4.3 channeled SPecTRoPolaRiMeTeRS

Most polarization techniques that rely on retarding elements have to go to great lengths to develop a waveplate that has uniform retardance across the spectral range of interest. Efforts have been made to develop achromatic retarders in the visible and IR. Achromatic retarders are commercially avail-able at visible wavelengths, but have also become available for infrared wavelengths [82]. When polarimetry and spectrometry are combined, the retardance can be calibrated wavelength-by-wave-length, reducing the problems associated with this effect [21].

Techniques have emerged that couple Fourier transform spectrometry with polarimetry in order to exploit the wavelength dependence of the retardance (in wavelengths) of high-order waveplates (see Chapter 19). We assume that a waveplate of thickness L can be described by an index of refrac-tion difference Δn = ne – no, where ne and no are the extraordinary and ordinary indices of refraction. We ignore dispersion in n over the wave band of interest for the purposes of this development. The phase difference induced between the radiation polarized parallel and perpendicular to the fast axis is given as

δ πλ

= ( )2 ∆n L. (18.11)

When the thickness of the waveplate is chosen so that δ >> 2π, then the retardance varies rapidly as a function of wavelength. The spectrum of the output intensity of the PSA is modulated in a

388 Polarized Light, Third Edition

manner analogous to Equation 18.9. When the spectrum of this signal is measured with a Fourier transform spectrometer, the spectrum is modulated in a manner that depends on the polarization state. If the retarders and PSA are designed carefully, and the spectrum of the incident signal is band limited, then spectrally distinct portions of the signal can be used to determine the Stokes parameters of interest.

It is essential that a Fourier transform spectrometer be used to make the spectral measure-ment. This is because the variation in retardance introduced by Equation 18.11 provides a sig-nal at spatial frequencies that correspond to wavelengths that are typically outside the spectral range of the detectors used. The method described by Oka and Kato [83] is not an imaging scheme. Sabatke et al. [69] developed a method to couple the channeled spectropolarimeter with a snapshot Fourier-based spectrometer to enable the instantaneous collection of spectral and polarimetric imagery information. This technique has been extended to several wavebands of interest [84].

Oka and Kaneko [85] introduced a novel and complimentary strategy for snapshot polarimetry when using monochromatic illumination. Whereas the channeled spectropolarimeter modulates the spectrum based on the polarization signature, the new method uses spatially varying thick retarders to spatially modulate the intensity image. The polarimetric features ideally can be reconstructed using a similar demodulation technique when the spatial Fourier spectrum of the scene is band limited. Chapter 19 contains many details.

18.5 geNeRal meaSuRemeNT STRaTegieS: imagiNg aRChiTeCTuRe foR iNTegRaTed PolaRimeTeRS

There are several different approaches for polarimetric detection. As with spectral imaging where multidimensional data is acquired, the data acquisition process is a study in compromises. By the very nature of measuring polarization, multiple images are required to even partially character-ize the polarization state of a scene. Since polarimetric data reduction manipulates the same pixel across multiple frames, any motion of the scene in the pixel between measurements results in arti-facts that have the potential to mask the true polarization signature. Ideally, two spatial dimen-sions are desired, but due to this temporal image registration issue, the images must be acquired simultaneously or as quickly as possible to minimize artifacts from platform or scene motion. The best solution for minimizing these artifacts is to acquire multiple images at the same time, but then the issue becomes spatial registration. Spatial registration of multiple images is complicated by the need to correct for both mechanical misalignment as well as optical misalignment arising from aberrations due to separate optical paths. Conceptually, the simplest way to measure the polariza-tion information is to use separate cameras with separate optics that are aligned to the same portion of the image (co-boresighted). Early imaging polarimeters did this with both film and electronic cameras [8,13] as well as scanning single-element photodetectors [86]. This strategy is difficult to execute properly, and has largely fallen out of favor. There are a number of integrated techniques that are used now. Tradeoffs among these methods, as well as issues of cost and difficulty of fabrica-tion and integration, are listed in Table 18.2.

18.5.1 diviSion of TiMe (doTP) PolaRiMeTeR

One commonly used approach is to rotate polarization elements in front of the camera system [10,14]. This approach is attractive because it is relatively straightforward in both system design and data reduction. However, the obvious drawback is that both the scene and platform must be stationary to avoid introducing interframe motion. Figure 18.3 shows an example of the common rotating retarder polarimeter. In this type of polarimeter, the rotation of the polarization elements causes a modulation of the polarized light incident on the focal plane from the scene, and the data can be reconstructed using the methods discussed in Section 18.4. Reducing the data on a pixel-

Techniques in Imaging Polarimetry 389

by-pixel basis produces Stokes images that can be used to produce images of the degree of linear polarization, degree of circular polarization, or other derived quantities such as orientation or ellipticity.

Most often the rotating element has been a polarizer. Only linear polarization states are detected in this approach. In addition, the rotation rate in previous attempts has either been too slow to achieve reasonable frame rates or the polarizer was moved in steps with the imagery acquired between movements. Even with published successes in continuously rotating the polarizer [79], arti-facts still remain if there is sufficient scene sensor movement during acquisition or if there is beam wander induced by the rotating element. Beam wander can result if there is a wedge in the rotating element or if the element wobbles in any sense. Nevertheless, if proper care is taken, the rotating element polarimeter can provide good results with a relatively small investment in hardware, design, and integration.

18.5.2 diviSion of aMPliTude PolaRiMeTeRS (doaMP)

Division of amplitude polarimeters (DoAmP) were first suggested and built by Garlick, Steigmann, and Lamb [8] for a two-channel system, then revived later for full Stokes polarimeters [87,88], and have since been exploited by a number of authors. Figure 18.4 shows a full-Stokes DoAmP pola-rimeter. This type of polarimeter consists of four separate focal plane arrays. The camera system consists of four separate cameras mounted such that a single objective lens is used in combination with a series of polarizing beam splitters, retarders, and relay lenses to produce a polarimetric image. Rigid mechanical mounts are used to support the cameras in positions facing the four cube assembly exits. The polarizing beam splitting cube assembly is used to balance the linear and circu-lar measurements. The cameras simultaneously capture the four images necessary for computing a complete Stokes image, thus eliminating false polarization effects due to scene changes during the collection process.

In this particular example [89], the polarimetric beam splitter assembly is designed to measure the complete Stokes vector. The beam splitting block includes three beam splitters, one 80/20 polar-izing beam splitting cube, two 50/50 polarizing beam splitting cubes, and a quarter-wave and half-wave retarder. Each path through the beam splitter block analyzes a different aspect of the incident

Table 18.2Comparison of imaging Polarimetry architectures

design featuresfabrication/integration

issues, Cost misregistration issues

Rotating element Robust•Relatively small•Not suitable for dynamic •scenes

Easiest to implement•Inexpensive•

Scene and platform motion•Beam wander not a problem •or removed in softwareMisregistration is linear•

Division of amplitude (multiple FPAs)

Simultaneous acquisition•Large system size•

High mechanical •flexibility and rigidity requiredExpensive•Large•

Must register multiple FPAs•Misregistration can be fixed•Can be nonlinear•

Division of aperture (single FPA)

Simultaneous acquisition•Smaller size•

Loss of spatial resolution•Expensive•

Fixed misregistration•Can be nonlinear•

Division of focal plane

Simultaneous acquisition•Small and rugged•Loss of spatial resolution•

Fabrication difficult•Alignment difficult•Very expensive•

IFOVs misregistered•Requires interpolation•Fixed registration•

Co-boresighted Simultaneous acquisition•Best used at long ranges•

Easy integration•Expensive•

Misregistration not as stable•

390 Polarized Light, Third Edition

polarization. This makes efficient use of the polarized light so that none of the light is absorbed or rejected. Furthermore, the analyzed polarization states are as nearly orthogonal as possible, and the analyzed states evenly span the possible incident polarization states.

As described in the first paragraph of Section 18.5, special care must be taken in alignment, and in practice mechanical alignment to the required tolerances just is not possible. Further, the many degrees of freedom in the relay lens sets result in different aberrations in each of the four channels. As a result, post-processing is required to coregister the four images. One of the chief disadvantages is the size of the system with the four focal planes and the breadboard required to rigidly mount the focal planes and their optics. When full spatial resolution is desired and size and cost of components is less of an issue, this approach is suitable.

18.5.3 diviSion of aPeRTuRe PolaRiMeTeR (doaP)

Figure 18.5 [90] shows an architecture that can both acquire all of the polarization data simulta-neously and ensure that the fields of view (FOVs) of all of the polarization channels are co-bore-sighted. The architecture uses a single focal plane array and a reimaging system to project multiple images onto a FPA that are accurately coaligned. This architecture has the advantage that once the optics are mechanically fixed, the alignment has been shown to be stable in time when compared to DoAmP polarimeters. The improved stability is likely due to the longer optical paths that are typically necessary in DoAmP systems, translating small changes into large deviations on the FPA. The architecture can be used as both a passive sensor (broadband illumination) and as an active monochromatic sensor. The primary disadvantages of the division of aperture polarimeter are the loss of spatial resolution (a factor of two in each linear dimension) and the volume and weight of the additional reimaging optics. In addition, matching transmission, apodization, magnification, and distortion between the channels are difficult, but can be accomplished. It should also be noted that

2°–13° , 2’- ∞zoom lens

1:1 zero distortionrelay lens

FPA

3-D translation,1-axis tilt

1-DTranslation

80/20 B.S.

PBSblock

2retarder

λ

λ4

retarder

figuRe 18.4 Division of amplitude polarimeter. The fourth camera is out of the plane of the page positioned above the PBS block after the quarter-wave retarder. (With permission from Optical Society of America.)

Techniques in Imaging Polarimetry 391

this strategy is more difficult to employ with coherent illumination due to coherent scattering and interference.

18.5.4 diviSion of focal Plane (dofP) aRRay PolaRiMeTeRS

Advances in (FPA) technology have led to the integration of micro-optical polarization elements directly onto the FPA [91,92]. Most DoFP systems that have been made to date are only sensitive to linear polarization, though some discussion of full Stokes DoFP systems has been raised [93]. An example system is shown in Figure 18.6. DoFP systems have been manufactured for imaging in all regions of the optical spectrum, including visible [94,95], SWIR [96], and LWIR [97]. Most DoFP systems have interlaced polarization super-pixels as shown in Figure 18.6, although some systems have been made where the polarization information is sampled on a line-by-line basis [94]. A typi-cal DoFP system will compute the Stokes vector at interpolation points in the FPA as indicated in Figure 18.6. DoFP systems necessarily trade off spatial resolution for polarization information, as a 2 × 2 (or larger) convolutional kernel must be applied to the image to estimate the Stokes vector at each point.

2×2 array ofFPA pixels

2×2 array ofmicropolarizers

SP1SP2

SP3 SP4 Incidentradiation

figuRe 18.6 (See color insert following page 394.) Division of focal plane polarimeter. (With permission from Optical Society of America.)

Polarimetricimagingoptics

Polarizationfilter/diffractive

optic array

FPA

Objective

Polaris building

figuRe 18.5 (See color insert following page 394.) Division of aperture polarimeter and a raw focal plane image showing the four polarization channels. The four channels are reduced to polarization products such as DoLP. For this specific case, the four images are linearly polarized at 0°, 45°, 90°, and 135°. (With permission from Optical Society of America.)

392 Polarized Light, Third Edition

DoFP systems have the significant advantage that all polarization measurements are made simul-taneously for every pixel in the scene. The component measurements that go into the Stokes vector estimation are by construction coming from nearby points in the scene. However, the DoFP system by definition has pixel-to-pixel registration error in computing the Stokes vectors. The instanta-neous fields of view (IFOVs) of adjacent pixels are (in principle) nonoverlapping, leading to exactly 1-pixel registration error. The IFOV error can be partially mitigated by intentionally defocusing the optical point spread function, and efforts have been made to minimize the information loss while simultaneously minimizing IFOV error through interpolation [97].

18.6 SySTem CoNSideRaTioNS

18.6.1 alignMenT and calibRaTion of iMaging PolaRiMeTeRS

Alignment and calibration are important issues in acquiring polarimetric imagery, and knowledge of system characteristics is essential to obtaining good data. The authors have been involved in many measurement campaigns that, while qualitatively useful to those taking the data, cannot be used for quantitative analysis because of poor calibration. The credibility of reported results is dependent upon documented calibration procedures.

Each polarimetric system will have its own calibration and alignment issues, and it is not possible to summarize all possible scenarios here, or offer a prescription for handling these issues. We review general items that must be addressed, and give examples of methods that have been used in the past to acquire data. System design will determine what methods might be relevant, and the designer must be ready to incorporate and justify the appropriate calibration and alignment procedures.

There are two calibration areas to be addressed, radiometric and polarimetric. The radiometric calibration determines dynamic range of the detectors, ensures that the detectors are operated in a linear response region (or if not, guarantees that the response is known and consistent), and ensures that the detectors are not operated near their saturation points. If a detector array is used, and this will more than likely be the case for modern imaging systems, the response of individual detector elements must be known. A correction procedure for nonuniformities must be in place (e.g., a nonuni-formity correction (NUC) lookup table) [97]. These issues are no different than those facing the user of a nonpolarimetric system and will not be covered here. However, radiometric calibration is even more important to the final result because of the potential for inaccurate radiometry among channels to couple into and invalidate the polarimetric result. Polarimetric measurement issues that may need to be addressed include polarimetric system calibration, optical element polarimetric uniformity, optical axis alignment and optical element rotational alignment, and polarization aberrations.

Examples of polarimetric system calibration techniques are given in Azzam, Elminyawi, and El-Saba [98] and Goldstein and Chipman [99]. The former contains a description of the calibration of the four-detector photopolarimeter where a simultaneous measurement of the Stokes vector is made. The latter concerns the calibration of a dual-rotating retarder Mueller matrix polarimeter using Fourier data reduction techniques where time-sequential measurements are made. While nei-ther of these are imaging systems, the approaches to calibration are instructive.

18.6.2 exPeRiMenTal deTeRMinaTion of daTa ReducTion MaTRix

The DRM described in Section 18.4.1 is estimated for an arbitrary polarimeter by measuring at least four linearly independent Stokes vectors that ideally form a maximum volume polyhedron inscribed inside the Poincaré sphere. When the number of measurements are exactly four, then the polyhedron is a regular tetrahedron [98]. The equation that describes this measurement is similar to Equation 18.7 given by

X A S= ⋅ , (18.12)

Techniques in Imaging Polarimetry 393

where X is a matrix of the measured Stokes vectors, S is a matrix with columns that are input Stokes vectors, and A is the instrument matrix. The instrument matrix is then empirically calculated as

A X S= ⋅ −1. (18.13)

Clearly the calibration Stokes vectors that make up S must be chosen so that the matrix is nonsin-gular. Measurements of unknown Stokes vectors may be made once the instrument matrix is estab-lished. This approach is described elsewhere as a general formulation [70].

18.6.3 calibRaTion of fouRieR-baSed RoTaTing ReTaRdeR SySTeMS

In the rotating retarder system referenced previously, the data reduction process is based on a Fourier analysis of the modulated signal that occurs when the retarders are rotated. Fourier coefficients are obtained as a function of Mueller matrix elements, and the equations are algebraically inverted. The Mueller matrix for the system is

M P R MR P2 2 1 1sys = , (18.14)

where M is the Mueller matrix of the sample, the R are the matrices for the retarders, and the P are matrices for the polarizers. The data collection process is allowed to proceed with no sample in place as a calibration. Since the Mueller matrix in the absence of a sample is the identity matrix, and expected and significant errors can be built into the data reduction equations (e.g., nonideal retarders and element orientation errors), these errors can be evaluated during calibration and used in sample data reduction. Note that for Mueller matrix systems, empty space or a high quality mir-ror are good calibration samples. For Stokes systems, one has to be careful in selection of calibra-tion elements such as polarizers or retarders, and assumptions about the quality or properties of particular polarizers or retarders must be made with caution. Proper polarimetric measurement and calibration procedures can be a process of building up a simultaneous calibration of the polarimeter and the test signature so that the quality of the measurements made is better than the quality of the individual elements used. This is the case in both of the techniques just described.

18.6.4 PolaRizaTion abeRRaTionS and iMage MiSalignMenT

Instantaneous field of view of the individual pixels may be an issue in polarization measurement, especially in imaging systems where the overall field of view of the system may be substantial [100]. This is an aspect of polarization aberrations [101,102], a subject that is independent of the classical wave-front aberrations more commonly studied in optics and that result from the differences cre-ated in amplitudes and phases as polarized light encounters interfaces.

Image registration, whether for sequential or simultaneous image collection, is a critical issue. Misregistration can result, for example, from separate focal planes that are not looking at the same region of space, or it can result from beam wander from a rotating element. Whatever the cause, Smith, Woodruff, and Howe [103] have suggested that images should be registered to 1/20th of a pixel for acceptable polarimetric results. Ideally, the alignment should be mechanical. In practice, achieving even a half pixel alignment mechanically can be difficult and software post-processing alignment is frequently necessary [104].

18.6.5 oPTiMizaTion

There have been significant research results published on the optimization of passive and active polarimeter systems using both Fourier and DRM techniques. For passive DRM-based polarimeters,

394 Polarized Light, Third Edition

the estimate of the unknown incident Stokes vector is as given in Equation 18.7. However, for real polarimeters, there are often many sources of uncertainty including, but not limited to, noise in the measurement process and calibration errors in determining the DRM. These error sources will be carried through the inversion of the DRM, leading to error in the reconstructed Stokes vector. Consider first the case of measurement noise. If the DRM for the polarimeter is A, but the measure-ment is

X A S n= ⋅ +in , (18.15)

where n is a noise vector, then we can define a measurement error

ε = − = ⋅−ˆ .S S A nin in

1 (18.16)

There are many potential metrics for quantifying the noise, and a detailed description of the tradeoffs is presented by Ambirajan and Look [105] and Sabatke et al. [106], but for the purposes of this discussion we will concentrate on the two-norm of the DRM. If we assume that the noise is independent and identically distributed from pixel to pixel, then the two-norm predicts the maxi-mum length of the error vector in the reconstructed Stokes images as

A AA n

nn2

12

12

2

2

2= =

⋅=−

−sup

sup,

εσN

(18.17)

where σ2 is the variance of the elements of the noise vector.It has been shown that minimization of the two-norm in Equation 18.17 means that the Sd

i T( ) in Equation 18.6 should be chosen to be maximally spaced on the surface of the Poincaré sphere [98] so as to inscribe the maximum volume polyhedron within the Poincaré sphere as shown in Figure 18.7. Satisfying the maximum volume condition also guarantees that the SNR is equal in each of the three polarization channels S1, S2, and S3 [72,107]. When N = 4 this is a tetrahedron, but for larger values of N the geometrical shape has the appropriate number of corners. The value of making more than four measurements is that the over-determined system in Equations 18.6 and 18.7 provides redundancy that increases the SNR by a factor of N [71,107,108] although reduction in noise can be made with similar reduction in spatiotemporal resolution by making four

s1

2ψ2

s0s2

s3

figuRe 18.7 (See color insert following page 394.) Optimal locations for measuring the polarization states are when the diattenuation vectors inscribe a regular tetrahedron inside the Poincaré sphere. (With permission from Optical Society of America.)

Techniques in Imaging Polarimetry 395

measurements and integrating for a longer time [108]. This optimization procedure was first car-ried out for rotating retarder polarimeters [105,108–110] with the result that the common rotating quarter-wave plate polarimeter is suboptimal to a system composed of a rotating 132° waveplate, and that the optimum rotation angles for the fast axis of the waveplate (independent of retardance) are at ±15.1° and ±51.6° with respect to the orientation of the linear polarization analyzer. Similar optimization studies have been performed for linear polarization [111], dual-rotating retarder [110], variable retardance [72,107], arbitrary-state polarimeters [97], and dual-rotating-retarder Mueller polarimeters [112]. A number of experimental validation studies exist that demonstrate the value of using an optimized polarimeter [21,106,108,110,113].

In addition to noise considerations, there are usually calibration errors associated with the exper-imental determination of the DRM A. In this case, we can recast Equation 18.6 as

X A S= +( ) ⋅∆ in , (18.18)

where Δ is the calibration error matrix. In this case, Equation 18.7 becomes

ε = ⋅−A S1∆ in , (18.19)

which tells us that the error is a function of the calibration error and the input Stokes vector [114]. Equation 18.19 implies that polarimeter optimization is not as straightforward as simply optimiz-ing the condition number of the DRM. The effects of Equation 18.19 are analyzed by Tyo [72] for a rotating retarder polarimeter, and found that the optimum polarimeter with respect to Equation 18.19 is not the optimum polarimeter with respect to Equation 18.17. For other classes of polarim-eters, Equation 18.19 can be used to find the best setting of parameters to provide minimum error.

18.7 SummaRy

The sensing of optical polarization information for remote sensing is an historically underused technique. In many applications, polarization phenomena are ignored, and the optical field is treated as scalar. While this can be reasonably accurate in many scenarios, it is clear that the ability to measure polarization information, especially across a scene, provides additional information that can be exploited. Passive and active imaging polarimetry are emerging techniques that promise to enhance many fields of optical metrology ranging from remote sensing to environmental sciences to industrial monitoring.

In this chapter, we have discussed many of the important developments in imaging polarimetry. We have attempted to survey the literature, and we have provided a general overview of most of the strategies that can be employed for various tasks. A subject such as imaging polarimetry would be adequately covered in a much longer work, but we refer the reader to the extensive reference list that provides much greater detail on the topics discussed herein. We have attempted to focus on first discussions of topics where possible, and have necessarily left out many relevant references through-out. However, an interested reader can follow through the references provided in order to get a more complete picture of the state of the literature.

Imaging polarimetry is a field rich with potential for future work. Advances are needed in sensor technology, data-retrieval and analysis algorithms, and applications. While much has been accom-plished, there is still a need for sensor systems with improved accuracy, precision, and stability. Particularly useful would be better and more complete quantification of these characteristics for currently existing and future systems. This implies a need for improved calibration techniques and more widely accepted and followed characterization procedures. For example, current technology makes the use of Mueller matrix images a practical way of characterizing polarization elements and systems. There is also a significant opportunity for creative ideas in dealing with the temporal and

396 Polarized Light, Third Edition

spatial tradeoffs that presently exist in imaging polarimetry. And, finally, as the technology advances, there is a great variety of applications waiting to be addressed with imaging polarimetry [115].

RefeReNCeS

1. Arago, D. F. J., and A. J. Fresnel, On the action of rays of polarized light upon each other, Ann. chim. Phys. 3 (1819): 288; Reprinted in The Wave Theory of Light, Memoires by Huygens, Young, and Fresnel, Edited by Henry Crew, New York: American Book Company, 1900.

2. Arago, D. F. J., L’action que les corps aimants et ceux qui ne le sont pas exercent les uns sur les autres Ann. chim. Phys. 1 (1824): 89.

3. Millikan, R. A., A study of the polarization of the light emitted by incandescent solid and liquid surfaces. I, Phys. Rev. 3 (1895): 81–99.

4. Millikan, R. A., A study of the polarization of the light emitted by incandescent solid and liquid surfaces. II, Phys. Rev. 3 (1895): 177–92.

5. Sandus, O., A review of emission polarization, Appl. Opt. 4 (1965): 1634–42. 6. Johnson, J. L., Infrared Polarization Signature Feasibility Tests, Rep. TR-EO-74-1, ADC001133, U.S.

Army Mobility Equipment Research and Development Center, 1974. 7. Chin-Bing, S. A., Infrared Polarization Signature Analysis, Rep. ADC008418, Defense Technical

Information Center, 1976. 8. Garlick, G. F. J., G. A. Steigmann, and W. E. Lamb, differential Optical Polarisation detectors, U.S.

Patent 3,992,571, November 16, 1976. 9. Walraven, R., Polarization imagery, in Optical Polarimetry: Instrumentation and Applications, Edited by

R. M. A. Azzam and D. L. Coffeen, Proc. SPIE 112 (1977): 164–7. 10. Walraven, R., Polarization imagery, Opt. Eng. 20 (1981): 14–8. 11. Solomon, J. E., Polarization imaging, Appl. Opt. 20 (1981): 1537–44. 12. Coffeen, D. L., Polarization and scattering characteristics in the atmospheres of Earth, Venus, and Jupiter,

J. Opt. Soc. Am. 69 (1979): 1051–64. 13. Coulson, K. L., V. S. Whitehead, and C. Campbell, Polarized Views of the Earth from Orbital Altitude, in

Ocean Optics VIII, Edited by M. A. Blizard, Proc. SPIE 637 (1986): 35–41. 14. Egan, W. G., W. R. Johnson, and V. S. Whitehead, Terrestrial polarization imagery obtained from the

space shuttle: characterization and interpretation, Appl. Opt. 30 (1991): 435–42. 15. Prosch, T., D. Hennings, and E. Raschke, Video polarimetry: A new imaging technique in atmospheric

science, Appl. Opt. 22 (1983): 1360–3. 16. Stenflo, J. O., and H. Povel, Astronomical polarimeter with 2-D detector arrays, Appl. Opt. 24 (1985):

3893–8. 17. Pezzaniti, J. L., and R. A. Chipman, Imaging polarimeters for optical metrology, in Polarimetry: Radar,

Infrared, Visible, Ultraviolet, and X-Ray, Edited by R. A. Chipman and J. W. Morris, Proc. SPIE 1317 (1990): 280–94.

18. Pezzaniti, J. L., and R. A. Chipman, Mueller matrix imaging polarimetry, Opt. Eng. 34 (1995): 1558–68.

19. Vane, G., and A. F. H. Goetz, Terrestrial imaging spectroscopy, Remote Sensing Environ. 24 (1988): 1–29.

20. Shaw, J. A., Degree of linear polarization in spectral radiances from water-viewing infrared polarimeters, Appl. Opt. 38 (1999): 3157–65.

21. Tyo, J. S., and T. S. Turner, Variable retardance, Fourier transform imaging spectropolarimeters for vis-ible spectrum remote sensing, Appl. Opt. 40 (2001): 1450–8.

22. Cheng, L. J., J. C. Mahoney, and G. Reyes, Target detection using an AOTF hyperspectral imager, in Optical Pattern Recognition V, Edited by D. P Casasent and T. Chao, Proc. SPIE 2237 (1994): 251–9.

23. Pust, N. J., and J. A. Shaw, Digital all-sky polarization imaging of partly cloudy skies, Appl. Opt. 47 (2008): H190–H198.

24. Duntley, S. Q., Underwater visibility and photography, in Optical Aspects of Oceanography, Edited by N. G. Jerlov, 138–49, New York: Academic Press, 1974.

25. Gilbert, G. D., and J. C. Pernicka. Improvement of underwater visibility by reduction of backscatter with a circular polarization technique, In Underwater Photo-Optics I, Edited by A. B. Dember, Proc. SPIE 7 (1966): A–III–1–A–III–10.

26. Tyo, J. S., M. P. Rowe, E. N. Pugh, and N. Engheta, Target detection in optically scattering media by polarization-difference imaging, Appl. Opt. 35 (1996): 1855–70.

Techniques in Imaging Polarimetry 397

27. Silverman, M. P., and W. Strange, Object delineation within turbid media by backscattering of phase modulated light, Opt. commun. 144 (1997): 7–11.

28. Demos, S. G., and R. R. Alfano, Optical polarization imaging, Appl. Opt. 36 (1997): 150–5. 29. Chenault, D. B., and J. L. Pezzaniti, Polarization imaging through scattering media, in Polarization

Measurement, Analysis, and Remote Sensing III, Edited by D. B. Chenault, M. J. Duggin, W. G. Egan, and D. H. Goldstein, Proc. SPIE 4133 (2000): 124–33.

30. Chang, P. C. Y., J. G. Walker, K. I. Hopcraft, B. Ablitt, and E. Jakeman, Polarization discrimination for active imaging in scattering media, Opt. commun. 159 (1999): 1–6.

31. Lewis, G. D., D. L. Jordan, and P. J. Roberts, Backscattering target detection in a turbid medium by polarization discrimination, Appl. Opt. 38 (1999): 3937–44.

32. Swartz, B. A., and J. D. Cummings, Laser range-gated underwater imaging including polarization dis-crimination, in Underwater Imaging, Photography, and Visibility, Edited by R. W. Spinrad, Proc. SPIE 1537 (1991): 42–56.

33. Demos, S. G., H. B. Radousky, and R. R. Alfano, Deep subsurface imaging in tissues using spectral and polarization filtering, Opt. Express 7 (2000): 23–28.

34. Tyo, J. S., Improvement of the point spread function in scattering media by polarization difference imag-ing, J. Opt. Soc. Am. A, 17 (2000): 1–10.

35. Demos, S. G., and R. R. Alfano, Temporal gating in highly scattering media by the degree of optical polarization, Opt. Lett. 21 (1996): 161–3.

36. Chipman, R. A., Polarization diversity active imaging, in Image Reconstruction and Restoration II, Edited by T. J. Schulz, Proc. SPIE 3170 (1997): 68–73.

37. Azzam, R. M. A., Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal, Opt. Lett. 2 (1978): 148–50.

38. Clémenceau, P., S. Breugnot, and L. Collot, Polarization diversity active imaging, in Laser Radar Technology and Applications III, Edited by G. W. Kamerman, Proc. SPIE 3380 (1998): 284–91.

39. Le Hors, L., P. Hartemann, and S. Breugnot, Multispectral polarization active imager in the visible band, in Laser Radar Technology and Applications V, Edited by G. W. Kamerman, U. N. Singh, C. Werner, and V. V. Molebny, Proc. SPIE 4035 (2000): 380–9.

40. Breugnot, S., and P. Clémenceau, Modeling and performances of a polarization active imager at lambda = 806 nm, Opt. Eng. 39 (2000): 2681–8.

41. Réfrégier, P., and F. Goudail, Invariant polarimetric contrast parameters of coherent light, J. Opt. Soc. Am. A 19 (2002): 1223–33.

42. Réfrégier, P., F. Goudail, and N. Roux, Estimation of the degree of polarization in active coherent imag-ery by using the natural representation, J. Opt. Soc. Am. A 21 (2004): 2292–300.

43. Baba, J. S., J.-R. Chung, A. H. DeLaughter, B. D. Cameron, and G. L. Cote, Development and calibra-tion of an automated Mueller matrix polarization imaging system, J. biomed. Opt. 7 (2002): 341–9.

44. Wolfe, J., and R. A. Chipman, High-speed imaging polarimeter, in Polarization Science and Remote Sensing, Edited by J. A. Shaw and J. S. Tyo, Proc. SPIE 5158 (2003): 24–32.

45. Mujat, M., E. Baleine, and A. Dogariu, Interferometric imaging polarimeter, J. Opt. Soc. Am. A 21 (2004): 2244–9.

46. Sakai, T., T. Nagai, M. Nakazato, Y. Mano, and T. Matsumura, Ice clouds and Asian dust studied with LIDAR measurements of particle extinction-to-backscatter ratio, particle depolarization, and water-vapor mixing ratio over Tsukuba, Appl. Opt. 42 (2003): 7103–16.

47. Intrieri, J. M., M. D. Shupe, T. Uttal, and B. J. McCarty, An annual cycle of Arctic cloud characteristics observed by radar and LIDAR at SHEBA, J. Geophys. Res. 107 (2002): SHE5-1–SHE-15.

48. Sassen, K., The Polarization LIDAR Technique for Cloud Research: A Review and Current Assessment, bull. Am. Meteorol. Soc. 72 (1991): 1848–66.

49. Reagan, J. A., M. P. McCormick, and J. D. Spinhirne, LIDAR sensing of aerosols and clouds in the tro-posphere and stratosphere, Proc. IEEE 77 (1989): 433–48.

50. Kalshoven, Jr., J. E., and P. W. Dabney, Remote sensing of the Earth’s surface with an airborne polarized laser, IEEE Trans. Geosci. Remote Sens. 31 (1993): 438–46.

51. Tan, S., and R. M. Narayanan, Design and performance of a multiwavelength airborne polarimetric LIDAR for vegetation remote sensing, Appl. Opt. 43 (2004): 2360–8.

52. Churnside, J. H., J. J. Wilson, and V. V. Tatarskii, Airborne LIDAR for fisheries applications, Opt. Eng. 40 (2001): 406–14.

53. Seldomridge, N. L., J. A. Shaw, and K. S. Repasky, Dual-polarization lidar using a liquid crystal variable retarder, Opt. Eng. 45 (2006): 106202.

398 Polarized Light, Third Edition

54. Bishop, K. P., H. D. McIntire, M. P. Fetrow, and L. McMackin, Multi-spectral polarimeter imaging in the visible to near IR, in Targets and backgrounds: characterization and Representation V, Edited by W. R. Watkins, D. Clement, and W. R. Reynolds, Proc. SPIE 3699 (1999): 49–57.

55. Voss, K., and Y. Liu, Polarized radiance distribution measurements of skylight 1: System description and characterization, Appl. Opt. 36 (1997): 6083–94.

56. North, J., and M. Duggin, Stokes vector imaging of the polarized sky-dome, Appl. Opt. 36 (1997): 723–30.

57. Horvath, G., A. Barta, J. Gal, B. Suhai, and O. Haiman, Ground-based full-sky imaging polarimetry of rapidly changing skies and its use for polarimetric cloud detection, Appl. Opt. 41 (2002): 543–59.

58. Pust, N. J., and J. A. Shaw, Dual-field imaging polarimeter using liquid crystal variable retarders, Appl. Opt. 45 (2006): 5470–8.

59. Lemke, D., F. Garzon, H. Gemuend, U. Groezinger, I. Heinrichsen, U. Klaas, W. Kraetschmer, et al., ISOPHOT: Far-infrared imaging, polarimetry, and spectrophotometry on Infrared Space Observatory, in Infrared Spaceborne Remote Sensing, Edited by M. S. Scholl, Proc. SPIE 2019 (1993): 28–33.

60. Loe, R. S., and M. J. Duggin, Hyperspectral imaging polarimeter design and calibration, in Polarization Analysis, Measurement, and Remote Sensing IV, Edited by D. H. Goldstein, D. B. Chenault, W. G. Egan, and M. J. Duggin, Proc. SPIE 4481 (2002): 195–205.

61. Glenar, D. A., J. J. Hillman, B. Saif, and J. Bergstralh, Acousto-optic imaging spectropolarimetry for remote sensing, Appl. Opt. 33 (1994): 7412–24.

62. Smith, W. M., and K. M. Smith, A polarimetric spectral imager using acousto-optic tunable filters, Exper. Astron. A 1 (1991): 329–43.

63. Denes, L. J., M. Gottlieb, B. Kaminsky, and P. Metes, AOTF polarization difference imaging, in 27th AIPR Workshop: Advances in computer-Assisted Recognition, Edited by R. J. Mericsko, Proc. SPIE 3584 (1999): 106–15.

64. Gupta, N., R. Dahmani, and S. Choy, Acousto-optic tunable filter based visible-to-near-infrared spec-tropolarimetric imager, Opt. Eng. 41 (2002): 1033–8.

65. Prasad, N. S., An acousto-optic tunable filter based active, long-wave IR spectrapolarimetric imager, in chemical and biological Standoff detection, Edited by J. O. Jensen and J.-M. Theriault, Proc. SPIE 5268 (2004): 104–12.

66. Jensen, G. L., and J. Q. Peterson, Hyperspectral imaging polarimeter in the infrared, in Infrared Spaceborne Remote Sensing VI, Edited by M. Strojnik and B. F. Andresen, Proc. SPIE 3437 (1998): 42–51.

67. Jones, S. H., F. J. Iannarilli, and P. L. Kebabian, Realization of quantitative-grade fieldable snapshot imaging spectropolarimeter, Opt. Express 12 (2004): 6559–73.

68. Iannarilli, F. J., J. A. Shaw, S. H. Jones, and H. E. Scott, Snapshot LWIR hyperspectral polarimet-ric imager for ocean surface sensing, in Polarization Analysis, Measurement, and Remote Sensing III, Edited by D. B. Chenault, M. J. Duggin, W. G. Egan, and D. H. Goldstein, Proc. SPIE 4133 (2000): 270–83.

69. Sabatke, D., A. Locke, E. L. Dereniak, M. Descour, J. Garcia, T. Hamilton, and R. W. McMillan, Snapshot imaging spectropolarimeter, Opt. Eng. 41 (2002): 1048–53.

70. Chipman, R. A., Polarimetry, Chapter 22 in Handbook of Optics, Vol. 2, New York: McGraw-Hill, 1995.

71. Lu, S.-Y., and R. A. Chipman, Interpretation of Mueller matrices based on the polar decomposition, J. Opt. Soc. Am. A 13 (1996): 1106–13.

72. Tyo, J. S., Design of optimal polarimeters: maximization of SNR and minimization of systematic errors, Appl. Opt. 41 (2002): 619–30.

73. Hauge, P. S., Mueller matrix ellipsometry with imperfect compensators, J. Opt. Soc. Am. A 68 (1978): 1519–28.

74. Azzam, R. M. A., A simple Fourier photopolarimeter with rotating polarizer and analyzer for measuring Jones and Mueller matrices, Opt. comm. 25 (1978): 137–40.

75. Hauge, P. S., and F. H. Dill, A rotating-compensator Fourier ellipsometer, Opt. comm. 14 (1975): 431–7.

76. Chen, L. Y., and D. W. Lynch, Scanning ellipsometer by rotating polarizer and analyzer, Appl. Opt. 26 (1987): 5221–8.

77. Aspnes, D. E., Photometric ellipsometer for measuring partially polarized light, J. Opt. Soc. Am. A 65 (1975): 1274–8.

78. Brezhna, S. Y., I. V. Berezhnyy, and M. Takashi, Dynamic photometric imaging polarizer-sample ana-lyzer polarimeter, J. Opt. Soc. Am. A. 18 (2001): 666–707.

Techniques in Imaging Polarimetry 399

79. Harchanko, J. S., D. B. Chenault, C. F. Farlow, and K. Spradley, Detecting a Surface Swimmer Using Long Wave Infrared Imaging Polarimetry, in Photonics for Port and Harbor Security, Edited by M. J. DeWeert and T. T. Saito, Proc. SPIE 5780 (2005).

80. Goldstein, D. H., and R. A. Chipman, An Error Analysis of a Mueller Matrix Polarimeter, J. Opt. Soc. Am. A 7 (1990): 693–700.

81. Chenault, D. B., J. L. Pezzaniti, and R. A. Chipman, Mueller matrix polarimeter algorithms, in Polarization Analysis and Measurement, Proc. SPIE 1746 (1992): 231–46.

82. Chenault, D. B., and R. A. Chipman, Infrared Birefringence Spectra for Cadmium Sulfide and Cadmium Selenide, Appl. Opt. 32 (1993): 4223–27.

83. Oka, K., and T. Kato, Spectroscopic polarimetry with a channeled spectrum, Opt. Lett. 24 (1999): 1475–7.

84. Hagan, N., A. Locke, D. Sabatke, E. Dereniak, and D. Sass, Methods and applications of snapshot spectropolarimetry, in Polarization: Measurement, Analysis, and Remote Sensing VII, Edited by D. H. Goldstein and D. B. Chenault, Proc. SPIE 5432 (2004): 167–74.

85. Oka, K., and T. Kaneko, Compact complete imaging polarimeter using birefringent wedge prisms, Opt. Express 11 (2003): 1510–9.

86. Halaijan, J., and H. Hallock, Principles and techniques of polarimetric mapping, Proceedings of the Eighth International Symposium on Remote Sensing of Environment, Vol. 1, 523–40, Rotterdam: ERIM, 1972.

87. Azzam, R. M. A., Arrangement of four photodetectors for measuring the state of polarization of light, Opt. Lett. 10 (1985): 309–11.

88. Barter, J. D., P. H. Y. Lee, and H. R. Thompson, Stokes Parameter Imaging of Scattering Surfaces, in Polarization: Measurement, Analysis, and Remote Sensing, Edited by D. H. Goldstein and R. A. Chipman, Proc. SPIE 3121 (1997): 314–20.

89. Farlow, C. A., D. B. Chenault, K. D. Spradley, M. G. Gulley, M. W. Jones, and C. M. Persons, Imaging Polarimeter Development and Application, in Polarization Analysis and Measurement IV, Edited by D. H. Goldstein, D. B. Chenault, W. Egan, and M. J. Duggin, Proc. SPIE 4819 (2001): 118–25.

90. Pezzaniti, J. L., and D. B. Chenault, A division of aperture MWIR imaging polarimeter, in Polarization Science and Remote Sensing II, Edited by J. A. Shaw and J. S. Tyo, Proc. SPIE 5888 (2005).

91. Rust, D. M., Integrated dual Imaging detector, U.S. Patent 5,438,414, August 1, 1995. 92. Andreau, A. G., and Z. K. Kalayjian, Polarization imaging: principles and integrated polarimeters, IEEE

Sens. J. 2 (2002): 566–76. 93. Nordin, G. P., J. T. Meier, P. C. Deguzman, and M. Jones, Diffractive optical element for Stokes vector

measurement with a focal plane array, in Polarization Measurement, Analysis, and Remote Sensing II, Edited by D. H. Goldstein and D. B. Chenault, Proc. SPIE 3754 (1999): 169–77.

94. Harnett, C. K., and H. G. Craighead, Liquid-crystal micropolarizer for polarization-difference imaging, Appl. Opt. 41 (2002): 1291–6.

95. Wolff, L. B., and A. G. Andreau, Polarization camera sensors, Image Vis. comput. 13 (1995): 497–510.

96. Chun, C. S. L., D. L. Fleming, and E. J. Torok, Polarization-sensitive, thermal imaging, in Automatic Object Recognition IV, Edited by F. A. Sadjadi, Proc. SPIE 2234 (1994): 275–86.

97. Boger, J. K., J. S. Tyo, B. M. Ratliff, M. P. Fetrow, W. Black, and R. Kumar, Modeling precision and accuracy of a LWIR microgrid array imaging polarimeter, in Polarization Science and Remote Sensing II, Edited by J. A. Shaw and J. S. Tyo, Proc. SPIE 5888 (2005).

98. Azzam, R. M. A., I. M. Elminyawi, and A. M. El-Saba, General analysis and optimization of the four-detector photopolarimeter, J. Opt. Soc. Am. A 5 (1988): 681–9.

99. Goldstein, D. H., and R. A. Chipman, Error analysis of a Mueller matrix polarimeter, J. Opt. Soc. Am. A 7 (1990): 693–700.

100. Chipman, R. A., Polarization analysis of optical systems, Opt. Eng. 28 (1989): 90–9. 101. McGuire, J. P., and R. A. Chipman, Polarization aberrations. 1. Rotationally symmetric optical systems,

Appl. Opt. 33 (1994): 5080–100. 102. McGuire, J. P., and R. A. Chipman, Polarization aberrations. 2. Tilted and decentered optical systems,

Appl. Opt. 33 (1994): 5101–7. 103. Smith, M. H., J. B. Woodruff, and J. D. Howe, Beam Wander Considerations in Imaging Polarimetry,

Proc. SPIE 3754 (1999): 50–4. 104. Persons, C. M., and D. B. Chenault, Automated registration of polarimetric imagery using Fourier

transform techniques, in Polarization: Measurement, Analysis, and Remote Sensing V, Edited by D. H. Goldstein and D. B. Chenault, Proc SPIE 4819 (2002): 107–17.

400 Polarized Light, Third Edition

105. Ambirajan, A., and D. C. Look, Optimum angles for a polarimeter: part I, Opt. Eng. 34 (1995): 1651–5. 106. Sabatke, D. S., A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and

G. S. Phipps, Figures of merit for complete Stokes polarimeters, in Polarization Analysis, Measurement, and Remote Sensing III, Edited by D. B. Chenault, M. J. Duggin, W. G. Egan, and D. H. Goldstein, Proc. SPIE 4133 (2000): 75–81.

107. Tyo, J. S., Noise equalization in Stokes parameter images obtained by use of variable retardance polarim-eters, Opt. Lett. 25 (2000): 1198–2000.

108. Sabatke, D. S., M. R. Descour, E. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, Optimization of retardance for a complete Stokes polarimeter, Opt. Lett. 25 (2000): 802–4.

109. Dlugunovich, V. A., V. N. Snopko, and A. V. Tsaruk, Minimizing the measurement error of the Stokes parameters when the setting angles of a phase plate are varied, J. Opt. Technol. 67 (2000): 797–800.

110. Ambirajan, A., and D. C. Look, Optimum angles for a polarimeter: part II, Opt. Eng. 34 (1995): 1656–9.

111. Tyo, J. S., Optimum linear combination strategy for a N-channel polarization sensitive vision or imaging system, J. Opt. Soc. Am. A 15 (1998): 359–66.

112. Smith, M. H., Optimization of a dual-rotating-retarder Mueller matrix polarimeter, Appl. Opt. 41 (2002): 2488–95.

113. Dlugunovich, V. A., V. N. Snopko, and A. V. Tsaruk, Analysis of a method for measuring polariza-tion characteristics with a Stokes polarimeter having a rotating phase plate, J. Opt. Technol. 68 (2001): 269–73.

114. Li, P., and J. S. Tyo, Experimental measurement of optimal polarimeter systems, in Polarization Science and Remote Sensing, Edited by J. A. Shaw and J. S. Tyo, Proc. SPIE 5158 (2003): 103–12.

115. Tyo, J. S., D. H. Goldstein, D. B. Chenault, and J. A. Shaw, Review of passive imaging polarimetry for remote sensing applications, Appl. Opt. 45 (2006): 5453–69.

401

19 Channeled Polarimetry for Snapshot Measurements*

19.1 iNTRoduCTioN

Temporal misregistration, or intensity differences between time-sequential measurements not induced by polarization, can be a significant source of error in certain applications. Such misreg-istration can be caused by motion of the platform or scene, and is therefore a particular concern in the field of remote sensing [1].

The Stokes parameters are defined by the addition or subtraction of at least two intensity mea-surements, such that

S =

=

+−−

S

S

S

S

I I

I I

I I

0

1

2

3

0 90

0 90

45 135

II IR L−

. (19.1)

If temporal scanning (e.g., a rotating retarder polarimeter) is used to measure the Stokes param-eters of a changing scene, then motion-based misregistration can occur between the measurements. Consequently, both polarimetric and motion-based intensity differences will appear as a signal after data reduction. An example of a motion-induced artifact from a rotating retarder polarimeter is illustrated in Figure 19.1. The moving sailboat creates the appearance of multiple targets in both circular and linear polarizations.

One method that avoids temporal misregistration is referred to as channeled polarimetry (CP). CP techniques make use of polarization interference in order to amplitude modulate the Stokes param-eters onto either spectral or spatial carrier frequencies. The use of interference can be beneficial in several respects when compared to a conventional polarimeter. For instance, in a conventional pola-rimeter, four intensity measurements must be taken (e.g., I0, I90, I135, and IR) for the calculation of a complete Stokes vector. Doing so requires these values be manipulated (added and subtracted from one another) within a computer during post processing. Conversely, CP enables the direct measure-ment of all four Stokes parameters simultaneously, by performing the addition and subtraction optically, through interference between four coherent beams. This is feasible because interference maintains the phase of each component within the complex amplitude, before the detector measures the intensity. Consequently, the amplitude and phase of the Stokes parameters are encoded within the amplitude and phase of the carrier frequency, enabling the magnitude and sign (or handedness) of the Stokes parameters to be extracted.

Another benefit of CP is realized through spectral or spatial registration. In a conventional pola-rimeter, image registration between the intensity measurements must be accomplished to within 1/20th of a pixel to achieve an accurate Stokes parameter reconstruction [2]. Otherwise, false polar-ization signatures can occur in the spectrum or scene. Again, CP resolves these concerns by its use of interference. Since a given Stokes parameter is calculated interferometrically and measured

* This chapter is contributed by Michael W. Kudenov, College of Optical Sciences, University of Arizona, Tucson, AZ and Dennis H. Goldstein.

402 Polarized Light, Third Edition

directly, image registration between several intensity measurements is unnecessary. Furthermore, since each Stokes parameter is modulated on coincident carrier frequencies, spatial or spectral registration between all the Stokes parameters is inherent. This significantly reduces the complex-ity of the Stokes vector calculation over conventional polarimeters. However, these benefits come at a tradeoff, typically to the spatial and spectral resolution of the sensor. Other issues, primarily sources of error in CP, are discussed later in this chapter.

19.2 ChaNNeled PolaRimeTRy

The fundamental concept of CP can be considered an analog to conventional amplitude modulation (AM) [3]. In AM, a time-dependent signal is mixed with a high frequency carrier such that

I t A b t Ut( ) = + ( ) +( )cos ,2π φ (19.2)

where A is the amplitude offset, b(t) is the signal, U is the carrier frequency, t is the time, and ϕ is the phase offset. The signal imparts an envelope to the carrier frequency, thus forming a new function, I(t). The AM process is depicted in Figure 19.2. Fourier transformation of the signal demonstrates that the frequency information contained within b(t) is shifted to frequencies centered around the carrier’s two Fourier components.

Naturally, this technique can be extended to many signals, such that

I t A b t n nUt nn

( ) = + ( ) + ( )[ ]∑ , cos .2π φ (19.3)

Consequently, AM can frequency-multiplex many unique signals, b(t, n), onto unique carrier fre-quencies. These are all contained within the same signal I(t). Assuming the input signals are band-limited, all information regarding the input can be recovered [4]. Recovery of b(t) is performed with a Fourier transformation and filtration, which will be described shortly in Section 19.2.3.

19.2.1 inTRoducTion To channeled SPecTRoPolaRiMeTRy

The basic concepts of AM are applied in the setup depicted in Figure 19.3 [5,6]. It consists of a generating polarizer (G), multiple-order uniaxial crystal retarder (R2), and analyzer (A) with orien-

DOCP DOLPIntensity(a) (b) (c)

figuRe 19.1 Rotating retarder Stokes polarimeter data of an outdoor scene. A sailboat, moving through the scene while the 16 measurements were collected, appears as a polarized signal. (a) Intensity image (S0), (b) degree of circular polarization (DOCP), and (c) degree of linear polarization (DOLP).

Channeled Polarimetry for Snapshot Measurements 403

0.6Fourier transform of B(t)

0 2 4 60

0.2

0.4

Am

p.

F

24

Signal

B(t) I(t)24

Amplitude modulatedcarrier frequency

–6 –4 –2Frequency (Hz)

0.3Fourier transform of I(t)

0 2 4 6 8 10–4–2

0

Am

plitu

de

0 2 4 6 8 10–4

–20

Am

p.–6 –4 –2 0 2 4 60

0.1

0.2

Am

p.

F

Time (s)

0.51 Carrier frequency

Time (s) Frequency (Hz)

0.30.40.5

Fourier transform of carrier

0 2 4 6 8 10–1

–0.50

Am

plitu

de

–6 –4 –2 0 2 4 60

0.10.2

Am

p.

Frequency (Hz)

F

Time (s)

figuRe 19.2 Diagram of AM. An arbitrary signal is mixed with a carrier frequency, forming a new signal I(t). Fourier transformation of I(t) shows the information contained within b(t) is now shifted to higher fre-quencies, centered around the carrier frequency’s two fourier components.

f

d2

f

1.4 1.6 1.8 2 2.2 2.4

R2G A

Spectrometerx

zy

y0.6

0.8

1Spectrum

45° 0°0° x0.2

0.4

Inte

nssit

y

Wave number (cm–1) × 104

s

figuRe 19.3 Basic channeled spectropolarimetry carrier frequency generator (i.e., a mixer) with a unity input signal created by the generator (G).

404 Polarized Light, Third Edition

tations of 0°, 45°, and 0°, respectively. If the retarder has a thickness d2, then the retardance between the fast and slow axes is

φ πλ2 2

2= bd , (19.4)

where λ is the free space wavelength and b is the birefringence of the crystal,

b n ne o= −( ), (19.5)

where ne, no are the crystal’s extraordinary and ordinary indices of refraction, respectively. The Mueller formalism for the configuration in Figure 19.3, assuming an unpolarized input that

is uniform over wavelength, is

Sout λ( ) =

14

1 1 0 0

1 1 0 0

0 0 0 0

0 0 0 0

1 0 0 0

0 ccos sin

sin cos

φ φ

φ φ

2 2

2 2

0

0 0 1 0

0 0

( ) ( )

− ( ) ( )

1 1 0 0

1 1 0 0

0 0 0 0

0 0 0 0

1

0

0

0

. (19.6)

The output is

Sout λ

φφ( ) =

+ ( )+ ( )

14

1

1

0

0

2

2

cos

cos. (19.7)

If the detector is insensitive to polarization (i.e., it is only responsive to the S0 Stokes parameter) then the intensity of the output is

I bdλ πλ

( ) = +

14

12

2cos . (19.8)

Since this is the response to a spectrally uniform input signal, only the carrier frequency component is present. The effective frequency of the carrier (U) is

U bd= 2 . (19.9)

Therefore, the carrier frequency increases with both retarder thickness and birefringence. The generation of this carrier frequency, U, can also be visualized by the measurement of a

multiple-order retarder on a Mueller matrix polarimeter. Such a measurement, performed on both a 4 and 8 mm thick sapphire retarder, is depicted in Figure 19.4a and b, respectively. Notable are the oscillations in the m22–m24, m32–m34, and m42–m44 components. These oscillations occur because the optical path difference (OPD) of the retarder is fixed. Consequently, the number of waves of retardance decreases with increasing wavelength. For instance, the maxima in the intensity versus wave number, depicted in Figure 19.3, are generated when the OPD of the retarder matches an inte-ger increment (n) of the wavelength, where n = OPD/λ. This means that the retarder behaves as a

Channeled Polarimetry for Snapshot Measurements 405

m11

m12

m13

1.07

1105

1.07

1105

–11.3

1.5

λ–1

1.3

1.5

λ–11.3

1.5

λ–11.3

1.5

λ–11.3

1.5

λ–11.3

1.5

λ–1

1.3

1.5

λ–11.3

1.5

λ

–11.3

1.5

λ–1

1.3

1.5

λ–11.3

1.5

λ–11.3

1.5

λ–11.3

1.5

λ–11.3

1.5

λ–1

1.3

1.5

λ–11.3

1.5

λ

–11.3

1.5

λ–1

1.3

1.5

λ–11.3

1.5

λ–11.3

1.5

λ–11.3

1.5

λ–11.3

1.5

λ–1

1.3

1.5

λ–11.3

1.5

λ

–11.3

1.5

λ–1

1.3

1.5

λ–11.3

1.5

λ–11.3

1.5

λ–11.3

1.5

λ–11.3

1.5

λ–11.3

1.5

λ–11.3

1.5

λ

1.07

1105

1.07

1105

m14

1.07

1105

m11

m12

m13

1.07

1105

1.07

1105

1.07

1105

m14

1.07

1105

m21

1.07

1105

m31

1.07

1105

m41

1.07

1105

m42

1.07

1105

m43

1.07

1105

m44

1.07

1105

m41

1.07

1105

m42

1.07

1105

m43

1.07

1105

m44

1.07

1105

m32

1.07

1105

m33

1.07

1105

m34

1.07

1105

m31

1.07

1105

m32

1.07

1105

m33

1.07

1105

m34

1.07

1105

m22

1.07

1105

m23

1.07

1105

m24

1.07

1105

m21

1.07

1105

m22

1.07

1105

m23

1.07

1105

m24

(a)

(b)

fig

uR

e 19

.4

Mea

sure

men

t of

an: (

a) 4

mm

thic

k an

d (b

) 8

mm

thic

k Sa

pphi

re r

etar

der

on a

Mue

ller

mat

rix

pola

rim

eter

. The

Mue

ller

mat

rix

com

pone

nts

are

depi

cted

ve

rsus

wav

elen

gth

(λ)

in μ

m.

406 Polarized Light, Third Edition

one-wave retardation plate, rotating the incident linearly polarized light by 180°. Since the analyzer is parallel to this polarization state, light passes unobstructed to the spectrometer, thus yielding a bright fringe. Conversely, the minima in the intensity versus wave number occur for half-integer increments of the wavelength, such that n + 1/2 = OPd/λ. These spectral locations now behave as a half-wave retardation plate, rotating the incident linearly polarized light by 90°. This orthogonal-ity between the polarized light exiting the retarder and the analyzer prevents its transmission to the spectrometer, resulting in a dark fringe. Consequently, maxima and minima in the m22–m24, m32–m34, and m42–m44 components of Figure 19.4 can be considered the effect of modulo changes through half-wave and integer-wave values of retardance. Furthermore, a retarder that has twice the thickness induces oscillations in the retardance that are twice as rapid, as observed when comparing the 4 and 8 mm thick samples.

The setup in Figure 19.3 can be used to produce a channeled spectropolarimeter (CS) that mea-sures two Stokes parameters. Removing the generating polarizer and using an arbitrary Stokes vec-tor as the input makes the Mueller formalism for this situation

I

T

λ( ) =

12

1

0

0

0

1 1 0 0

1 1 0 0

0 0 0 0

0 0 0 0

( ) ( )

− ( )

1 0 0 0

0 0

0 0 1 0

0

2 2

2

cos sin

sin

φ φ

φ 00 2

0

1

2

3cos φ

λλλλ( )

( )( )( )( )

S

S

S

S

, (19.10)

where the vector [1 0 0 0] describes the response of the detector and the superscript T indicates the transpose. Calculation of the Mueller matrices yields the intensity

I S S bd Sλ λ λ πλ

λ πλ

( ) = ( ) + ( )

+ ( )1

22 2

0 1 2 3cos sin bbd2

. (19.11)

Thus, with a single retarder and analyzer, it is possible to amplitude-modulate two of the Stokes parameters (S1 and S3) onto one output signal, while the offset contains the S0 component. Extraction and recovery of the Stokes parameters from I(λ) is overviewed in Section 19.2.3. These partial chan-neled spectropolarimeters, which do not measure a complete Stokes vector, have been described in the literature [7]. However, the system configuration required for measuring a complete Stokes vector will not be described until Section 19.3.

19.2.2 inTRoducTion To channeled iMaging PolaRiMeTRy

Channeled imaging polarimetry (CIP), in its original implementation, can be considered as a mono-chromatic extension of channeled spectropolarimetry (CS). Instead of generating carriers that mod-ulate the spectrum, CIP generates carriers that modulate the spatial image. In CIP, these spatial carrier frequencies are generated by creating a linearly varying retardance over a 2-dimensional (2-D) focal plane array (FPA). This can be done by placing a Wollaston prism directly onto an FPA [8], or by using a Savart plate [9], or Sagnac interferometer [10,11] in the pupil or collimated space of an imaging system.

If we begin by using the Wollaston prism implementation, then the CIP system (analogous to the CS implementation depicted previously in Figure 19.3) is illustrated in Figure 19.5 [12,13]. It consists of two cemented birefringent prisms, P3 and P4, with fast axis orientations of 45° and 135°, respectively. Each has an apex angle β and the prism combination is followed by an analyzer, ori-ented at 0°.

Channeled Polarimetry for Snapshot Measurements 407

Placing these prisms onto an FPA produces the imaging polarimeter depicted in Figure 19.6.In this configuration, the Wollaston formed by P3 and P4 is equivalent to retarder R2 in the previ-

ous setup, except that it varies in retardance as a function of spatial coordinates x and y. The func-tional form of the phases induced by the retardances of P3 and P4 are [14]

φ πλ

β φ πλ3 4

22

2= + ( ) +

=bd

dx

bdT

xTtan and ++ ( ) −

tan .β dxx

2 (19.12)

P3, 45°

β

A0°

dxdT

xy

x

dy

P4, 135°

figuRe 19.5 Prism polarimeter wedge set. The fast axes of prisms P3 and P4 are at 45° and 135°, respec-tively. The apex angle of both prisms is β while the prism’s side lengths are dx and dy. Lastly, the prism’s terminus thickness is dT.

Objective lens

Image plane

FPA

x

P3/P4 Ay

A

figuRe 19.6 The Wollaston prism and analyzer are placed onto an FPA. This makes the prism and ana-lyzer approximately coincident with the image plane.

408 Polarized Light, Third Edition

The intensity output for the CIP system can be calculated with the Mueller formalism for the prism configuration as

I x y

T

,( ) =

12

1

0

0

0

1 1 0 0

1 1 0 0

0 0 0 0

0 0 0 0

( ) − ( )1 0 0 0

0 0

0 0 1 0

0

4 4

4

cos sin

sin

φ φ

φ(( ) ( )

×( )

0

1 0 0 0

0 0

4

3 3

cos

cos sin

φ

φ φ(( )

− ( ) ( )

( )

0 0 1 0

0 03 3

0

sin cos

,

φ φ

S x y

SS x y

S x y

S x y

1

2

3

,

,

,

.( )( )( )

(19.13)

Expansion yields

I x y S S S, cos sin ,( ) = + ( ) + ( )[ ]12 0 1 34 3 34φ φ (19.14)

where

φ φ φ πλ

β34 3 4

22= −( ) = ( )b xtan , (19.15)

and the Stokes parameters are implicitly dependent on x and y. The optical path difference (OPD) of the prism can then be expressed as

OPD = ( )2b xtan .β (19.16)

A critical aspect pertaining to Equation 19.15 is that the dispersion term (λ–1), which generated the spectrally dependent carrier frequency in CS, is still present in CIP. Consequently, broadband illumination will create a continuum of spatial carrier frequencies spanning the wavelength dis-tribution of the source. This continuum will cause the carrier frequency’s visibility to change as a function of spatial location; areas with a large OPD, relative to the coherence length of the incident illumination, will suffer greater visibility loss compared to regions of small OPD. The low visibility carrier frequency that results from the dispersion means this method, in its original implementation, is restricted to near monochromatic illumination. This limitation can have detrimental effects on the signal-to-noise ratio for certain applications, especially in the thermal infrared [14]. Methods of circumventing this monochromatic constraint are described in Section 19.4.3.

19.2.3 calibRaTion algoRiThMS

To reconstruct the data in CS or CIP, the reference beam calibration technique is often implemented [15]. In this technique, sample data are calibrated by comparison to a set of reference data with a known Stokes parameter distribution.

19.2.3.1 CS CalibrationFirst, we consider the simplified CS instrument depicted previously in Figure 19.3. Rewriting Equation 19.14 in terms of wave number (σ = λ–1) yields

I S S bd S bdσ σ σ πσ σ πσ( ) = ( ) + ( ) ( ) + ( )12

2 20 1 2 3cos sin 22( )[ ]. (19.17)

Channeled Polarimetry for Snapshot Measurements 409

This implies that the intensity pattern must be uniformly sampled, which is an issue when the spec-trometer samples uniformly in wavelength. Inverse Fourier transformation of the intensity pattern yields the time domain signal

I I

S S bd

ν σν ν δ ν δ ν

( ) = ( )[ ] =( ) + ( ) ∗ +( ) + −

−F 1

0 1 212

12

bbd

i S bd bd

2

3 2 212

( )[ ]

− ( ) ∗ +( ) − −( )[ ]

ν δ ν δ ν

, (19.18)

where δ, with no subscript, is the Dirac delta function, ν is the Fourier transform variable of σ, S indicates the Fourier transformed Stokes parameters, and * indicates a convolution. This pro-duces three “channels” in the time domain. The process, thus far, is depicted with an example in Figure 19.7.

In Figure 19.7, retarder R2 is configured with an achromatic birefringence of b = 0.009 and thick-ness d2 = 3 mm. The incident Stokes parameters are modulated onto a carrier frequency by R2 and the analyzer. Fourier transformation yields three channels, where c1 and c1

* are located at OPD’s of + bd2 and –bd2 (±2.7E-3 cm), respectively, while c0 is un-modulated (0 cm). The contents of these channels in the time domain are depicted in Figure 19.8.

Extraction of the Stokes parameters involves applying a filter, H(ν – ν0), to isolate the desired channel. Two filters, one with ν0 = 0 and the other with ν0 = –bd2, enable extraction of channels c0 and c1, respectively. Forward Fourier transformation of each filtered channel partially isolates the Stokes parameters from the unknown sample, where

c I H S Hsample0, F F= ( ) ( )[ ] = ( ) ∗ ( )[ ]ν ν σ ν12 0 , (19.19)

c I H bd S iS esample1, F= ( ) −( )[ ] = ( ) + ( )( )ν ν σ σ2 1 314

−− ∗ −( )[ ]i bd H bd22

2π σ νF . (19.20)

S1S2

1

Incident Stokes parameters0.8

Channeled spectrum

R2 A

S3

1.4 1.6 1.8 2 2.2 2.4–0.5

00.5

Inte

nsity

(A.U

.)

1.4 1.6 1.8 2 2.2 2.40.20.40.6

Wave number (cm–1) × 104Wave number (cm–1) × 104

Inte

nsity

(A.U

.)

F–1

Time domain

Spectrometer

d2

S0(σ)S1(σ)S2(σ)

00.10.20.30.4

A.U

./cm

fs45° 0°

x

zyx

y

S3(σ)–5 0 50

OPD (cm) ×10–3

figuRe 19.7 Example of the simplified CS setup. d2 = 3 mm and the birefringence (b) is 0.009.

410 Polarized Light, Third Edition

Calibration of the sample data requires demodulation of the ei bd2 2π σ term in Equation 19.20, and is accomplished by measuring a known state of polarization. Measuring a known Stokes vector of

Sref σ σ σ σ σ( ) = ( ) ( ) ( ) (S S S Sref ref ref ref0 1 2 3, , , , ))[ ]T, (19.21)

produces a set of reference data, where the content of the channels is

c S Hreference0, 0,ref F= ( ) ∗ ( )[ ]12

σ ν , (19.22)

c S iS ereference ref refi b

1, = ( ) + ( )[ ] −14 1 3

2, ,σ σ π dd H bd2

2σ ν∗ −( )[ ]F . (19.23)

Reconstruction of the sample’s normalized Stokes parameters is achieved by

S c sample0 σ( ) = 0, , (19.24)

S

S

c

c

csample

reference

referen1

0

σσ

( )( ) = ℜ 1,

1,

0, cce

sample

ref ref

refc

S iS

S0,

1 3

0

, ,

,

σ σσ

( ) + ( )( )

, (19.25)

S

S

c

c

csample

reference

referen3

0

σσ

( )( ) = ℑ 1,

1,

0, cce

sample

ref ref

refc

S iS

S0,

1 3

0

, ,

,

σ σσ

( ) + ( )( )

, (19.26)

where ℜ and ℑ represent taking the real or imaginary part of the expression, respectively. It should be noted that, when generating a Stokes vector for calibration, it is not necessary to generate a known S3 state; for the partial CS of Figure 19.7, S1 is sufficient. In general, when using the reference

0.5Time domain (Abs. value)

12

0.3

0.4

14

14

(S1(ν) + iS3(ν))*δ(ν – Bd2)

S0(ν)

(S1(ν) – iS3(ν))*δ(ν + Bd2)

2

0.2A.U

./cm

C0

C10.1 C1*

–5 –4 –3 –2 –1 0 1 2 3 4 5x 10–3

0

OPD (cm)

figuRe 19.8 Time domain representation of the signal with the channel contents listed. Channels c1 and c1

* contain information regarding S1(σ) and S3(σ), while channel c0 contains S0(σ).

Channeled Polarimetry for Snapshot Measurements 411

beam calibration technique, the phase term ( )ei bd2 2π σ can be measured exclusively with known linear states in all CS and CIP implementations. This can simplify the calibration since wave plates with known retardance distributions are not needed to calibrate measurements in S3.

19.2.3.2 CiP CalibrationAll of the aforementioned calibration techniques for CS also extend to CIP, only now 2D Fourier transforms are required [8]. Taking the forward Fourier transform of Equation 19.14 yields

I I x y S Sξ η ξ η ξ η δ ξ ξ, , , , ,( ) = ( )[ ] = ( ) + ( ) ∗ +F12

140 1 0 ηη δ ξ ξ η

ξ η δ ξ ξ η δ ξ ξ

( ) + −( )[ ]

+ ( ) ∗ +( ) − −

0

3 0 0

14

,

, ,i S ,, ,η( )[ ]

(19.27)

where ξ and η are the Fourier transform variables for x and y, respectively and

ξλ

β0

2= ( )btan . (19.28)

An example of the CIP technique is depicted in Figure 19.9. Here, the simplified single Wollaston prism CIP sensor is depicted with various simulated input Stokes parameters.

The incident Stokes parameters are modulated onto a carrier frequency by prisms P3 and P4. Similar to CS, Fourier transformation of the FPA’s intensity yields three channels, where c1 and c1

* are located at spatial frequencies of 2bλ–1tan(β) and –2bλ–1tan(β) (±34.8 mm–1), respectively, while c0 is unmodulated (0 mm–1). The contents of these channels in the spatial frequency domain are depicted in Figure 19.10.

S0 FPA Intensity

0.40.60.81

y (m

m)

–0.2–0.1

00.1

y (m

m)

–0.2–0.1

0

0

0.5

0.2

Objective lens

l

F

x (mm)–0.2–0.1 0 0.1 0.2

0.2

S1–0.2–0.1 x (mm)

–0.2 –0.1 0 0.1 0.2

0.10.2

Fourier domain–60–40

0.5

–0.5

0 Image pIane

FPAS0(x,y)S1(x,y)S2(x,y)x (mm)

y (m

m)

–0.2–0.1 0 0.1 0.2

00.10.2

S3–0.2

η(m

m–1

) –200

2040

–0.5

0y

x

P3/P4 A

y (m

m) –0.1

00.10.2

(mm–1)–50 0 50

60

x (mm)–0.2–0.1 0 0.1 0.2

S3(x,y)

figuRe 19.9 Example of the simplified CIP setup. The prism apex angle β = 3°, birefringence b = 0.21, wavelength λ = 633 nm. The pixel size of the detector is 4.65 μm.

412 Polarized Light, Third Edition

Again, filtering the desired channel enables the Stokes parameters to be isolated. The filter in 2D becomes H(ξ – ξ0, η – η0), where the filter is centered at (ξ0, η0). Using two filters, one centered at the origin (0,0) and the other at [2bλ–1tan(β), 0], enables isolation of channels c0 and c1, respec-tively. In a procedure analogous to CS, inverse Fourier transformation of each filtered channel isolates the Stokes parameters, yielding

c I H S x y Hsample0, F F= ( ) ( )[ ] = ( ) ∗− −10

112

ξ η ξ η ξ, , , ,, ,η( )[ ] (19.29)

c I H bsample1, F= ( ) − ( )[ ] − −1 12ξ η ξ λ β η, tan ,

= ( ) − ( )[ ] − ( )14 1 3

4 1S x y iS x y ei b, , tanπ λ β xx H b∗ − ( )[ ] − −F 1 12ξ λ β ηtan , .

(19.30)

Again, the exponential phase factor in c1,sample must be demodulated by measuring a known Stokes vector,

Sref x y S x y S x y S x yref ref ref, , , ,, , ,( ) = ( ) ( ) ( )0 1 2 SS x yrefT

3, , .( )[ ] (19.31)

This produces a set of reference data, where

c S x y Hreference ref0, F= ( ) ∗ ( )[ ]−12 0

1, , ,ξ η (19.32)

c S x y iS x y ereference ref refi

1, = ( ) − ( )[ ]14 1 3, ,, , 44 1 11 2π λ β ξ λ β ηb x H b− ( ) − −∗ − ( )[ ] tan tan , .F (19.33)

12

C0

14

14

C1*C1

(S1( ,η) + iS3( ,η))* δ( + 0,η)

η (mm–1)

(mm–1)

(S1( ,η) – iS3( η))* δ( – 0,η)

(S0( ,η)

Fourier domain

0.1

0.080.060.04

100

–100–100 –80 –60 –40 –20 0 20 40 60 80 100

50

–50

0

figuRe 19.10 Fourier domain representation of the 2D intensity pattern with the channel contents listed. Channels c1 and c1

* contain information regarding S1 and S3, while channel c0 contains S0.

Channeled Polarimetry for Snapshot Measurements 413

Reconstruction of the sample’s normalized Stokes parameters is achieved by

S x y c sample0 , ,( ) = 0, (19.34)

S x y

S x y

c

c

csample

reference

ref1

0

,,

( )( ) = ℜ 1,

1,

0, eerence

sample

ref ref

c

S x y iS x y

S0,

1, 3,, ,

,

( ) − ( )0 rref x y,

,( )

(19.35)

S x y

S x y

c

c

csample

reference

ref3

0

,,

( )( ) = ℑ 1,

1,

0, eerence

sample

ref ref

c

S x y iS x y

S0,

1 3

0

, ,

,

, ,( ) − ( )rref x y,

.( )

(19.36)

19.3 ChaNNeled SPeCTRoPolaRimeTRy

Application of channeled spectropolarimetry (CS) to the measurement of the full four-element Stokes vector will be overviewed in this section. This begins with a discussion of CS when used with a dispersive spectrometer, followed by an overview of the technique when used with an inter-ferometer-based Fourier transform spectrometer (FTS).

19.3.1 cS wiTh a diSPeRSive SPecTRoMeTeR

Channeled spectropolarimetry is often implemented with a spectrometer that uses some form of dispersion, such as a diffraction grating or prism. A CS setup that is capable of measuring the complete Stokes vector is depicted in Figure 19.11, and is essentially the same system previously depicted in Figure 19.7 with an additional retarder [5]. The complete CS configuration consists of two multiple-order retarders, R1 and R2, with fast axes orientations of 0° and 45°, respectively. This is followed by an analyzer, oriented with its transmission axis at 0°, and the dispersive spectrometer.

R2 AR1

Spectrometerx

d2d1 Channeled spectrum

fs

45° 0°

zy

x

y

f

s

0.4

0.6

0.8

1.4 1.6 1.8 2 2.2 2.4

0.2

Wave number (cm–1) × 104

Inte

nsity

(A.U

.)

figuRe 19.11 Full channeled spectropolarimetry configuration. Two multiple-order retarders (R1 and R2) modulate all four Stokes parameters onto carrier frequencies. A dispersive spectrometer is used to measure the channeled spectrum.

414 Polarized Light, Third Edition

The intensity, measured by the spectrometer, can be calculated from the Mueller formalism by expansion of

I

T

λ( ) =

12

1

0

0

0

1 1 0 0

1 1 0 0

0 0 0 0

0 0 0 0

( ) ( )

− ( )

1 0 0 0

0 0

0 0 1 0

0

2 2

2

cos sin

sin

φ φ

φ 00

1 0 0 0

0 1 0 0

0 0

2

1

cos

cos sin

φ

φ

( )

×( ) − φφφ φ

λλ

1

1 1

0

1

0 0

( )( ) ( )

( )( )

sin cos

S

S

SS

S2

3

λλ

( )( )

,

(19.37)

where

φ πλ

φ πλ1 1 2 2

2 2= =bd bdand . (19.38)

Calculation of the Mueller matrices yields the intensity

I S S Sλ λ λ φ λ φ φ( ) = ( ) + ( ) ( ) + ( ) ( )12 0 1 2 2 1 2cos sin sin(( ) + ( ) ( ) ( )[ ]S3 1 2λ φ φcos sin . (19.39)

Re-expressing the intensity in wave number (σ), while expanding the sinusoidal terms, yields

I S S Sσ σ σ φ σ φ φ( ) = ( ) + ( ) ( ) + ( ) −12

12

140 1 2 2 1 2cos cos(( ) − +( )[ ]

+ ( ) +( ) − −

cos

sin sin

φ φ

σ φ φ φ φ

1 2

3 1 2 1 2

14

S (( )[ ].

(19.40)

Inverse Fourier transformation yields the time domain signal

I S S

S

ν ν ν δ ν φ δ ν φ( ) = ( ) + ( ) ∗ −( ) + +( )[ ]

+

12

14

18

0 1 2 2

2 νν δ ν φ φ δ ν φ φ δ ν φ φ δ ν φ( ) ∗ − +( ) + + −( ) − − −( ) − +1 2 1 2 1 2 1 ++( )[ ]

− ( ) ∗ − − +( ) + + −( ) +

φ

ν δ ν φ φ δ ν φ φ δ ν

2

3 1 2 1 2

18

i S −− −( ) − + +( )[ ]φ φ δ ν φ φ1 2 1 2 .

(19.41)

This produces seven channels within the time domain. These channels, with their content, are depicted in Figure 19.12a and b for retarder thickness ratios (d1:d2) of 1:2 and 3:1, respectively.

Extraction of the sample and reference channels, followed by a forward Fourier transformation, enables the reference beam calibration technique to be implemented for all four Stokes parameters,

S c sample0 σ( ) = 0, , (19.42)

Channeled Polarimetry for Snapshot Measurements 415

S

S

c

c

csample

reference

referen1

0

σσ

( )( ) = ℜ 1,

1,

0, cce

sample

ref

refc

S

S0,

1

0

,

,

,σσ

( )( )

(19.43)

S

S

c

c

csample

reference

referen2

0

σσ

( )( ) = ℜ 2,

2,

0, cce

sample

ref ref

refc

S iS

S0,

2 3

0

, ,

,

σ σσ

( ) − ( )( )

, (19.44)

S

S

c

c

csample

reference

referen3

0

σσ

( )( ) = ℑ 1,

1,

0, cce

sample

ref ref

refc

S iS

S0,

2 3

0

, ,

,

σ σσ

( ) − ( )( )

. (19.45)

Consequently, the complete spectrally dependent Stokes vector can be measured within a single integration of the spectrometer. However, dispersive spectrometers measure the spectral content of light directly in the frequency domain (λ–1). Conversely, an FTS, based on an interferometer, enables measurements to be taken in the time domain. This enables direct access to each of the seven channels as a function of the interferometer’s OPD.

19.3.2 fouRieR TRanSfoRM cS

Fourier transform spectrometers are generally implemented in the thermal infrared wavelengths due to the throughput (Jacquinot) and multiplex (Fellgett) advantages [16]. Additionally, direct measure-ment of the channels can be realized with these spectrometers. In an FTS, measurements are taken as a function of OPD to form an interferogram. Forward Fourier transformation of the interferogram produces a spectrum as a function of wave number (σ). The CP configuration is nearly identical to Figure 19.11, with the replacement of the spectrometer with an FTS as shown in Figure 19.13.

Using a Twyman–Green interferometer enables the functional form of the interferogram to be calculated as [17],

I S S SOPD FTS( ) = + ( )[ ] + +12

1 0 1 2 2 1cos cos( ) sin(φ φ φ ))sin( ) cos( )sin( ) ,φ φ φ σσ

σ

2 3 1 2

1

2

+[ ]∫ S d (19.46)

where σ1 and σ2 are the minimum and maximum wave numbers of integration, ϕ1 and ϕ2 are defined by Equation 19.38, and ϕFTS is the phase delay imparted by the interferometer, defined as

φ σFTS = 2π zF (19.47)

d1:d2 = 1:2 d1:d2 = 3:1(b)(a)

C0

18

(S2 + iS3) (S2 – iS3)

(–S2 – iS3)(–S2 + iS3) (–S2 + iS3) (S2 + iS3) (–S2 – iS3)(S2 – iS3)

2S0

18

2S0

C1

C0

C2

C1C2

18

4S1

4S1

18

18

18

4S1

4S1

18

18

0–4d2B–3d2B –2d2B –d2B d2B 2d2B 3d2B 4d2B

OPDOPD0–3d1B –1d1B –d1B d1B 2d1B 3d1B

figuRe 19.12 Channel locations, and their content, for thickness ratios (d1:d2) of: (a) 1:2 and (b) 3:1. The Stokes parameters are implicitly dependent upon ν.

416 Polarized Light, Third Edition

where zF is the OPD introduced by the FTS. Additionally, the Stokes parameters in Equation 19.46 are implicitly dependent upon wave number. Expansion of this equation yields

I zS S

F( ) = ( ) + +( ) +∫ 0 122 4

1

2

cos cos cosφ φ φσ

σ

FTS FTS φφ φ

φ φ φ φ φ

FTS

FTS FTS

−( )[ ]

+ − + +( ) − −

2

21 28

Scos cos 11 2 1 2 1 2−( ) + + −( ) + − +( )[ ]

+

φ φ φ φ φ φ φcos cosFTS FTS

S331 2 1 28

sin sin sinφ φ φ φ φ φ φFTS FTS FTS+ +( ) − − −( ) − + φφ φ φ φ φ σ1 2 1 2−( ) + − +( )[ ]sin .FTS d

(19.48)

This function is analogous to Equation 19.41. An illustration of an interferogram is depicted in Figure 19.14 for a thickness ratio of 1:2. Filtration and reconstruction of the channels is done in an identical fashion to the dispersion-based system. The exception is that the initial inverse Fourier transformation can be neglected, since it is achieved optically through the interferometer.

19.4 ChaNNeled imagiNg PolaRimeTRy

Similarly to the previous section, CIP, when applied to the measurement of the full Stokes vector, will be overviewed. This includes a discussion of CIP when implemented with a pair of opti-mized Wollaston prisms mounted to a FPA. Additionally, CIP implemented with a Savart plate in the pupil of an optical system, as well as some limitations to CIP and their solutions, will be discussed.

19.4.1 PRiSMaTic ciP

Similar to the full implementation of CS, an additional component, over the single Wollaston prism depicted in Figure 19.5, is needed to measure a full Stokes vector in CIP. This extra component consists of another Wollaston prism [8]. As depicted in Figure 19.15, the first pair of prisms, P1 and P2, varies in thickness along y with fast axis orientations of 0° and 90° with

R2 AR1

FTSx

fs

zy

y

d2d1

fs 100

200Channeled interferogram

45° 0°xf

–50 0 50–200–100

0

Inte

nsity

(A.U

.)

OPD (µm)

figuRe 19.13 Full channeled spectropolarimetry configuration where the spectrometer is replaced with an FTS (i.e., an interferometer). Intensity measurements are performed as a function of OPD.

Channeled Polarimetry for Snapshot Measurements 417

respect to the x axis, respectively. The second pair of prisms, P3 and P4, varies in thickness along x with fast axis orientations of 45° and 135°, respectively. An analyzer (A) follows the group with its transmission axis at 0°. This configuration produces a spatially varying retardance as a function of x and y.

When a polarized scene is imaged onto the prisms, interference fringes are developed. These fringes, combined with the spatial information of the scene, are either imaged by a FPA behind the prisms or are relayed onto the FPA, as illustrated in Figure 19.16a and b, respectively.

OPDmax

zF

4

cos( FTS + 1 + 2) + sin( FTS + 1 + 2)8 8S2– S3

cos( FTS + 1 + 2) + sin( FTS + 1 + 2)8 8S2– S3

S1

3cos( FTS + 1 – 2) –

cos( FTS – 2)4S1 cos( FTS + 2)

2S0 cos( FTS)

sin( FTS + 1 – 2)8 8S2+ S

3cos( FTS – 1 – 2) – sin( FTS – 1 – 2)8 8S2– S

figuRe 19.14 Depiction of a channeled interferogram. The horizontal axis is in units of distance (OPD), while the integration of each function over σ, from σ1 to σ2, is implicit.

A0°

dx

P3, 45°

β

β

dT

P1, 0°

xy

dy

P4, 135°P2, 90°

figuRe 19.15 Prism polarimeter wedge set. Extra prisms (P1 and P2) are included, to the simplified CIP configuration in Figure 19.5, to create a linearly varying retardance along y.

418 Polarized Light, Third Edition

With the use of Mueller formalism, the intensity pattern behind the analyzer can be calculated by [14]

I x y

T

,( ) =

12

1

0

0

0

1 1 0 0

1 1 0 0

0 0 0 0

0 0 0 0

( ) − ( )1 0 0 0

0 0

0 0 1 0

0

4 4

4

cos sin

sin

γ γ

γ(( ) ( )

( ) ( )

0

1 0 0 0

0 0

4

3 3

cos

cos sin

γ

γ γ00 0 1 0

0 0

1 0 0 0

0 1 0 0

3 3− ( ) ( )

×

sin cosγ γ

00 0

0 02 2

2 2

cos sin

sin cos

γ γγ γ

( ) ( )− ( ) ( )

( ) − ( )( )

1 0 0 0

0 1 0 0

0 0

0 01 1

1 1

cos sin

sin cos

γ γγ γ(( )

( )( )( )( )

S x y

S x y

S x y

S x y

0

1

2

3

,

,

,

,

,

(19.49)

where γ1 through γ4 are the spatially dependent retardances of the birefringent prisms, defined as

γ π

λβ γ1 2

22

x yb

dd

y x yTy, tan ,( ) = + ( ) +

(( ) = + ( ) −

( ) =

22

3

πλ

β

γ

bd

dy

x y

Tytan

,22

22

4

πλ

β γ πλ

bd

dx x y

bdT

xT+ ( ) +

( ) =tan , ++ ( ) −

tan .β dxx

2

(19.50)

This yields an intensity distribution of

I x y S x y S x y S x y, [ , , cos ,( ) = ( ) + ( ) −( ) + ( )1

2 0 1 3 4 2γ γ ssin sin

, cos sin

γ γ γ γ

γ γ

1 2 3 4

3 1 2

−( ) −( )

+ ( ) −( )S x y γγ γ3 4−( )].

(19.51)

Expansion of the sinusoidal terms and substituting for γ1 through γ4 yields

I x y S x y S x y b x, , , cos tan( ) = ( ) + ( ) ( )

12

12

220 1

πλ

β

+ ( ) ( ) −( )

−1

42

22

2S x y b y x, cos tan cosπ π

λβ

λ22

14

223

b y x

S x y b

tan

, sin

β

λ

( ) +( )

+ ( ) − πttan sin tanβ

λβ( ) −( )

+ ( ) +( )

y x b y x2

2π

.

(19.52)

Objective lens(b)(a) Relay lensObjective lens

Image plane FPAFPA

Image plane

Ax

y

x

y

AP1/P2 P3/P4 P3/P4P1/P2

figuRe 19.16 Optical configurations for the prismatic CIP: (a) FPA is placed directly behind the prisms and the analyzer and (b) a lens relays the intermediate image formed within the prisms and analyzer onto an FPA.

Channeled Polarimetry for Snapshot Measurements 419

Forward Fourier transformation yields the spatial frequency domain representation

I S Sξ η ξ η ξ η δ ξ ξ η δ ξ ξ, , , , ,( ) = ( )+ ( )∗ +( )+ −12

140 1 0 0 ηη

ξ η δ ξ ξ η η δ ξ ξ η η

( )[ ]

+ ( )∗ − +( )+ + −( )−18 2 0 0 0 0S , , , δδ ξ ξ η η δ ξ ξ η η

ξ η δ ξ

+ +( )− − −( )[ ]

+ ( )∗ − −

0 0 0 0

38

, ,

,i

S ξξ η η δ ξ ξ η η δ ξ ξ η η δ ξ ξ0 0 0 0 0 0 0, , , ,+( )+ + −( )+ + +( )− − ηη η−( )[ ]0 ,

(19.53)

where

ξλ

β ηλ

β0 0

2 2= ( ) = ( )b btan tan .and (19.54)

Again, as with the full implementation of CS, seven channels are present in the frequency domain, with the modulated Stokes parameters dependent upon the transform variables ξ and η. An illustra-tion of the prismatic CIP’s Fourier domain is depicted in Figure 19.17.

Filtration of channels c0, c1, and c2 from the reference and sample data enables reconstruction of the normalized Stokes parameters by

S x y c sample0 , ,( ) = 0, (19.55)

S x y

S x y

c

c

csample

reference

ref1

0

,,

( )( ) = ℜ 1,

1,

0, eerence

sample

ref

refc

S x y

S x y0,

1

0

,

,

,

,

( )( )

, (19.56)

1

2

1

1

8

1

1

4

C0

C2

C1

1

8

1

48

18 (–S2 + iS3)

( S2 – iS3 )

( S2 + iS3 )

(–S2 – iS3 )

Fourier domain

S1

S00.1

0.08

0.06

0.04

0.02

0100

–100–100 –80 –60 –40 –20 0

402060 80 100

50

–50

0

S1

η (mm–1)

ξ (mm–1)

figuRe 19.17 Fourier domain of the prismatic CIP with the channel contents indicated. Each Stokes parameter is implicitly dependent upon ξ and η.

420 Polarized Light, Third Edition

S x yS x y

c

c

csample

reference

ref2

0

,,

( )( ) = ℜ 2,

2,

0, eerence

sample

ref ref

c

S x y iS x y

S0,

2 3

0

, ,

,

, ,( ) − ( )rref x y,

,( )

(19.57)

S x yS x y

c

c

csample

reference

ref3

0

,,

( )( ) = ℑ 2,

2,

0, eerence

sample

ref ref

c

S x y iS x y

S0,

, 3,

0,

2 , ,( ) − ( )rref x y,

.( )

(19.58)

Consequently, all four spatially dependent Stokes parameters can be measured within a single inte-gration time.

However, the Mueller matrix model of the system implies the use of thin prisms; higher order effects, related to the rays propagating through the prisms, are neglected. These higher order effects cause the carrier frequency’s visibility to decrease. The visibility is defined as

VI I

I I= −

+max min

max min

, (19.59)

where Imax and Imin are the maximum and minimum intensities within the local fringe field. This reduction in visibility leads to lower signal-to-noise ratios, and is particularly prevalent in low f-num-ber systems. Measured data demonstrating the dependence on f-number is depicted in Figure 19.18.

More details regarding this effect can be obtained from Kudenov [18] and Luo et al. [19]. One method of avoiding this f-number dependence is to use an interferometer that does not operate in image space, as is observed when using a Savart plate in the pupil of an imaging system.

19.4.2 SavaRT PlaTe ciP

A Savart plate CIP can be used in lieu of a prismatic CIP to generate Stokes parameter carrier fre-quencies, and can enable the use of low f-number objective lenses [9,20]. The system layout for the Savart plate CIP is depicted in Figure 19.19. It consists of a 4F imaging system with two Savart plates (SP1 and SP2) in the collimated space. A half-wave plate (HWP), oriented at 22.5°, converts the lin-ear horizontal and vertical polarization states exiting SP1 into linear 45° and 135° polarization states, respectively. The Savart plates and HWP are then followed by a linear analyzer, oriented at 45°.

(S1) and (S2+iS3) carrier frequency visibility1

0.90.80.70.60.50.4Vi

sibili

ty

0.30.20.1

02.8 4.8 6.8 8.8 10.8

F-number12.8 14.8 16.8 18.8 20.8

S1

S2+iS3

figuRe 19.18 Carrier frequency visibility vs. f-number for a prismatic CIP with yttrium vanadate prisms (β = 2°, b ∼ 0.21) at λ = 633 nm with a spectral bandwidth of 3 nm.

Channeled Polarimetry for Snapshot Measurements 421

An illustration of the beam propagation through the two Savart plates and the half-wave plate is depicted in Figure 19.20. Four sheared beams are depicted exiting the system, which interfere at the image plane once transmitted through the analyzer and lens.

Calculation of the intensity pattern for the Savart plate CIP requires propagation of the incident polarization state through the system with scalar diffraction theory and Jones matrices [9,21]. The polarization state of the sample can be expressed as

Esamp =( )( )

( )

( )E x y e

E x y ex

i x y

yi x y

x

y

,

,

,

,

φ

φ . (19.60)

The incident polarization state on SP1 is proportional to the Fourier transform of the sample

E Einc sampξ ηξ ηξ η

,,

,.( ) = [ ] =

( )( )

FE

Ex

y

(19.61)

where Ex and Ey represent the Fourier transform of the sample’s x and y polarization components, while ξ and η are the transform variables for the x and y coordinates, respectively. Propagation through SP1 shears the orthogonally polarized components of the electric field by a distance of

x

y

AHWP

SP1 SP2

SampleObjective Reimaging

fo

FPA

fr

EA

EIEsamp

figuRe 19.19 Optical configurations for the savart plate CIP.

SP1

HWP (22.5°)

SP2

Einc

2t

y

ESP1

E

2t

xz

EHWP

ESP2

αα

αα αη

η

ξ

α

ξ

figuRe 19.20 Beam propagation through SP1, the HWP and SP2. Four sheared beams are present at the output of SP2.

422 Polarized Light, Third Edition

2 22 2

2 2α = −

+n n

n nte o

e o

. (19.62)

where 2α is the distance between the two beams in the x, y plane, t is the half thickness of the Savart plate, and ne, no are the extraordinary and ordinary indices of refraction, respectively [22]. The two beams exiting SP1 can be expressed by shifting the electric field’s components in ξ and η by α, so that

ESP1 =−( )

−( )

E

Ex

y

ξ α ηξ η α

,

,. (19.63)

Propagation through the HWP rotates the components by 45°, such that the field exiting the HWP is

E EHWP SP1=−

=−( ) + −(1

2

1 1

1 1

1

2

E Ex yξ α η ξ η α, , ))−( ) − −( )

E Ex yξ α η ξ η α, ,. (19.64)

Each of the two beams incident on SP2 are sheared, where x-polarized beams are sheared in ξ by –α and y-polarized beams are sheared in η by + α, yielding

ESP2 =−( ) + − −( )

− +( ) −1

2

2E E

E Ex y

x y

ξ α η ξ α η αξ α η α

, ,

, ξξ η,.( )

(19.65)

Transmission of these beams through the analyzer (EA), oriented at 45°, is calculated by

E EA SP2=

12

1 1

1 1, (19.66)

which yields

EA =−( ) + − −( ) + − +( ) −1

2 2

2E E E Ex y x yξ α η ξ α η α ξ α η α, , , ξξ ηξ α η ξ α η α ξ α η α

,

, , ,

( )−( ) + − −( ) + − +( ) −E E E Ex y x y2 ξξ η,

.( )

(19.67)

Notable is the common shift of the beams by α along ξ. Extracting this component as a convolution enables EA to be re-expressed as

EA =−( ) + −( ) + +( ) − +1

2 2

E E E Ex y x yξ α η ξ η α ξ η α ξ α η, , , ,(( )[ ] ∗ −( )−( ) + −( ) + +( )

δ ξ α ηξ α η ξ η α ξ η α

,

, , ,E E Ex y x −− +( )[ ] ∗ −( )

Ey ξ α η δ ξ α η, ,

. (19.68)

Transmission through the reimaging lens and propagation to the FPA (i.e., the focal point of fr) per-forms a Fourier transformation of EA as

F EA[ ] == =

−

ξλ

ηλ

λα λ

α

x

f

y

f

if

x

x

if

x

r r

r re E e,

2 2

2 2

π π

++ + −−

−

E e E e E e

E e

y

if

y

x

if

y

y

if

x

x

i

r r r

2 2 2π π πλ

αλ

αλ

α

22 2 2 2π π π πλ

αλ

αλ

αλf

x

y

if

y

x

if

y

y

ifr r r rE e E e E e+ + −

− αα

λα

x

if

x

o

o

e E

E

r

=

2

2 2

π

.

(19.69)

Channeled Polarimetry for Snapshot Measurements 423

where Ex and Ey are now implicitly dependent upon x and y. The absolute value squared yields the intensity at the FPA as

I x y E E, .*( ) = [ ] =F o oEA2 1

4 (19.70)

Calculation of this expression yields

I x y E E E E E E E Ex x y y x x y y, cos* * * *( ) = +( ) + −( )12

12

2πλλ

α

λα

fx y

E E E Ef

x

r

x y x yr

+( )

− +( ) 14

* * cos4π

+ +( )

+

14

14

E E E Ef

y

i E E

x y x yr

x y

* *

*

cos4πλ

α

−−( )

+ −( )E E

fx i E E E Ex y

rx y x y

* * *sin sin4πλ

α 14

44πλ

αf

yr

.

(19.71)

Substitution of the Stokes parameter definitions for the various combinations of Ex , Ex*, Ey , and

Ey* gives

I x y S x y S x yf

x yr

, , , cos( ) = ( ) + ( ) +( )

12

12

20 1

πλ

α

+ ( )

−

14 2S x y

fy

fx

r r

, cos cos4π 4πλ

αλ

α

+ ( )

+ 1

4 3S x yf

yf

xr r

, sin sin4π 4πλ

αλ

α

.

(19.72)

Hence, the intensity pattern on the FPA is nearly identical to that calculated for prismatic CIP as in Equation 19.52, but rotated by 45° in the x, y plane. The system’s calibration and the Stokes parameter reconstruction procedures are identical to prismatic CIP as in Equations 19.55 through 19.58, sans a shift in the Fourier filter’s positions to account for the 45° rotation. Through the use of the Savart plate CIP, the carrier frequency visibility is no longer limited to a minimum f-number. However, the overall signal is still limited by the narrow bandwidth required for the instrument.

19.4.3 diSPeRSion coMPenSaTion in ciP

A significant limitation to CIP involves the maximum spectral bandwidth of the technique. The coherence length of the illumination (lc), assuming a uniform spectral distribution of width Δλ centered at λ0, is [22]

lc = λλ02

∆. (19.73)

When the OPD in a CIP sensor is equal to the coherence length of the incident light, the carrier fre-quency visibility is zero, making it impossible to recover the modulated Stokes parameters. Given a fixed maximum OPD in a CIP sensor, then there is a maximum limit on the allowable spectral bandwidth before the carrier frequency is reduced to zero visibility. Generally, a decrease in the fringe visibility occurs for bandwidths Δλ exceeding 3 nm for λ0 of 633 nm. Consequently, resolv-ing this issue through dispersion compensation (DC) would enable larger spectral bandwidths to be utilized, leading to an enhancement in the signal-to-noise ratio of the reconstructed Stokes parameters.

424 Polarized Light, Third Edition

19.4.3.1 dC in Prismatic CiPCompensating the dispersion in the prismatic CIP has been demonstrated experimentally by Oka [23]. In this technique, the retardance characteristics of several birefringent uniaxial crystals were combined. Through optimization, the retardance of the crystal can be made to cancel the dis-persive (λ–1) term, a procedure analogous to the design of an achromatic crystal waveplate [24]. Consequently, implementing dispersion compensation in both of the original prism pairs (P1/P2 and P3/P4) significantly increases the spectral bandwidth. A depiction of the dispersion compen-sated prism pairs, using lithium niobate (LN) and alpha barium borate (αBBO) is provided in Figure 19.21.

Since the eight prisms required in the dispersion compensated prismatic CIP are considerably thicker than the four in the original prismatic CIP, resolving all of the carrier frequencies with an FPA-mounted prism set is not feasible. Consequently, an intermediate image and relay are neces-sary, as depicted in Figure 19.22.

These dispersion compensated prisms enabled experimental reconstructions for spectral band-widths up to 40 nm, yielding approximately one order of magnitude more bandwidth than the uncompensated prisms.

dx

P1, 0° P3, 90° P5, 45° P7, 135°A0°

xy

dy β1 β2

β1 β2

P2, 90°P4, 0°

P6, 135° P8, 45°

LN αBBO αBBOLN

figuRe 19.21 Broadband prism polarimeter wedge set. Each prism pair has an optimized wedge angle to enable cancellation of the dispersion. As is demonstrated in Oka et al. [23], β1 = 7.1° and β2 = 6.95° for a design wavelength of 554 nm.

Objective lens Relay lens

A

Image plane

FPA

y

x P1/P2(LN)

P5/P6(LN)

P7/P8(αBBO)

P3/P4(αBBO)

figuRe 19.22 Optical configuration for the dispersion compensated prismatic CIP. Due to the thickness of the wedge set, the intermediate image formed by the objective must be relayed onto the FPA.

Channeled Polarimetry for Snapshot Measurements 425

19.4.3.2 dC in Savart Plate CiPIt has been demonstrated that a standard polarization Sagnac interferometer can replace the Savart plate in CIP [10,25]. Analogous to the Savart plate CIP implementation, the carrier frequency in a Sagnac-based CIP sensor is

Ufr

Sagnac ∝ κλ

(19.74)

where κ is the shear of the Sagnac interferometer. Removal of the λ–1 term in the carrier frequency can be achieved by making the shear (S)

directly proportional to wavelength, such that S ∝ κλ. This can be implemented by using a modi-fied polarization Sagnac interferometer. By introducing blazed diffraction gratings into each arm of the Sagnac, the first-order shear becomes directly proportional to λ [11]. Thus, the dispersion term in the numerator effectively cancels with the term in the denominator. The general sys-tem configuration for this dispersion compensated polarization Sagnac interferometer (DCPSI) is depicted in Figure 19.23. It consists of a typical two-mirror polarization Sagnac interferometer, with mirrors M1 and M2, and a wire-grid beam splitter (WGBS), which preferentially reflects or transmits s or p polarized light, respectively. The system also contains two blazed gratings, G1 and G2.

The beam transmitted by the WGBS (now spectrally broadband) is diffracted by G1 into the +1 order. When these dispersed rays are incident on G2, the diffraction angle induced by G1 is removed. The rays emerge parallel to the optical axis, but are offset by a distance proportional to –λxo, where xo is some constant related to the DCPSI’s parameters. Conversely, the beam reflected by the WGBS is initially diffracted by G2. The dispersed rays are then diffracted to be parallel to the optical axis by G1, and exit the system offset by a distance proportional to +λxo. The shear is

Sm

da b c= + +( )2

λ, (19.75)

yp

M267.5°

67.5°

M1

WGBS

G1

G2

A

FPA

SDCPSIS (λ)

yx

zfr

RedGreenBlue

xp

figuRe 19.23 DCPSI with blazed diffraction gratings, G1 and G2, positioned at each output of the WGBS. Inclusion of the gratings generates a shear (SdcPSI) that is directly proportional to the wavelength.

426 Polarized Light, Third Edition

where d is the period of the grating, m is the diffraction order, and (a + b + c) is the cumulative distance between gratings G1 and G2. Such compensation has experimentally demonstrated spectral bandwidths exceeding 300 nm. More details regarding this system and its theory of operation are provided in [11].

Another method of compensating for the dispersion in the original Savart plate CIP technique is to make the focal length, fr, inversely proportional to wavelength. This kind of objective lens is commonly referred to as an Achromatic Fourier Transforming Lens (AFTL). Theoretical results of an AFTL have been demonstrated in Ref [26].

19.5 SouRCeS of eRRoR iN ChaNNeled PolaRimeTRy

Several error sources can be experienced in CP. These include reconstruction artifacts due to cross-talk between adjacent channels, temperature variations in the retarders, dichroism (i.e., diattenua-tion caused by absorption) in the crystal, nonuniform sampling, and dispersion within the retarders. A brief overview of these errors and their causes will be provided here. More detailed information is provided within the references.

19.5.1 ReconSTRucTion aRTifacTS (cS and ciP)

Crosstalk between adjacent channels occurs when the spectral or spatial features of the scene have high frequency content. Since the modulated Stokes parameters are not band-limited, this high frequency content is distributed across the Fourier domain, and consequently aliases into the neigh-boring channels. These aliasing effects appear as false polarimetric signatures after reconstruc-tion. A depiction of the aliasing problem is provided in Figure 19.24 for a FTS-based channeled spectropolarimeter. This demonstrates the summation of two nonband-limited Stokes parameters, based on

I S S dOPD FTS( ) = + ( )[ ] +[ ]∫12

1 0 1 2

1

2

cos cos( )φ φ σσ

σ

.. (19.76)

When the S0 and S1 components are overlaid, it is clear that some of the energy from the S0 compo-nent overlaps the S1 channels. Consequently, energy from S0 appears as modulated information in the reconstructed S1 data.

A significant reduction in aliasing artifacts can be realized by isolating the S0 component before reconstructing the Stokes parameters. This is done in FTS-based CS by implementing the false signature reduction technique (FSRT), which requires measurement and subtraction of the S 0 data

+ =

S0 S1 cos( 2)

I =

0 0 0OPD

(a) (b) (c)

OPD OPD

figuRe 19.24 In the time domain, CP preserves the centerburst, or S0 component (a), and incorporates the other Stokes parameters into displaced channels (b), because the collected data is a summation of each of these components (c), aliasing occurs when centerburst modulations extend into the displaced channels.

Channeled Polarimetry for Snapshot Measurements 427

from the total interferogram before reconstructing the Stokes parameters [27]. While two mea-surements are required to incorporate this technique, an added benefit is that a full resolution S0 spectrum is obtained, useful when isolating high frequency spectral details. Such a system could realize these benefits, while still remaining snapshot, through the use of two simultaneous mea-surements [7].

19.5.2 TeMPeRaTuRe vaRiaTionS (cS and ciP)

When a uniaxial crystal experiences a temperature change, the crystal will expand or contract based on its coefficient of thermal expansion (CTE). Consequently, the thickness of the retarder can increase or decrease depending on the environmental conditions. Additionally, the ordinary and extraordinary indices of refraction also have temperature dependencies [28]. Since the reference data can be measured at a different temperature than the sample data, thermal variations can be a significant concern when the reference beam calibration technique is implemented.

The simplified channeled spectropolarimetry system in Figure 19.3 can be used to demon-strate the effect that thermal expansion has on the calibration. The phase of retarder R2 can be modeled as

φ πσ2 22= ( ) ( )b T d T , (19.77)

where T is the temperature of the crystal. These temperature effects can be expressed, more generally, to change the phase ϕ2 to ϕ2 ± Δϕ. Assuming the reference (ϕ2) and sample (ϕ2 ± Δϕ) data are taken at temperatures of T1 and T2, respectively, then the reference and sample intensity distributions are

I S Sreference ref refσ σ σ φ( ) = ( ) + ( ) ( ) +12 20, 1, cos SS ref3, σ φ( ) ( )[ ]sin ,2 (19.78)

I S S Ssample σ σ σ φ φ σ( ) = ( ) + ( ) ±( ) + ( )12 0 1 2 3cos sin∆ φφ φ2 ±( )[ ]∆ . (19.79)

Inverse Fourier transformation of the reference and sample data, followed by filtration of the +ν component (channel c1) and a forward Fourier transformation, yields

c S iS ereference ref refi

1 1 321

4, , ,= ( ) + ( )[ ] −σ σ πσUU H2 1∗ ( )[ ]−F ν , (19.80)

c S iS esamplei U U

1 1 32 11

42

, = ( ) + ( )[ ] ∗− ±( ) −σ σ πσ ∆ F HH ν( )[ ], (19.81)

where U2 = σb(T1)d2(T1) and U2 ± ΔU = σb(T2)d2(T2). Applying the calibration procedure as in Equation 19.25 and 19.26 yields

S

S

S

SU

S

S1

0

1

0

3

0

2'

'cos

σσ

σσ

πσ σσ

( )( ) = ( )

( ) ( ) ± ( )(∆ )) ( )sin ,2πσ∆U (19.82)

S

S

S

SU

S

S3

0

3

0

1

0

2'

'cos

σσ

σσ

πσ σσ

( )( ) = ( )

( ) ( ) ( )(∆ ∓ )) ( )sin .2πσ∆U (19.83)

Consequently, the reconstructed Stokes parameters, ′S1 and ′S3 , have crosstalk that depends upon ΔU, or the difference between the carrier frequencies generated at temperatures of T1 and T2.

428 Polarized Light, Third Edition

To remedy concerns regarding temperature error, thermally stabilized retarders have been experimentally demonstrated with channeled spectropolarimetry [7]. These retarders combine more than one type of birefringent crystal in series, so that as the temperature changes, an increas-ing retardance in one material is canceled by a decreasing retardance in the second. Other methods of temperature compensation include a software-based alternative, referred to as the “self calibra-tion technique” [15]. In this single-measurement technique, which is implemented on the two-retarder CP implementation in Figure 19.11, the phase of retarder R2 is determined by extracting the phase of the S1 channel (c1 in Figure 19.12). This exclusively isolates ϕ2. However, since ϕ1 is combined with the measured Stokes parameter’s phase, ϕ23 = tan–1(S3/S2), the retarders’ thickness ratio (d1:d2) must be used to calculate ϕ1, such that ϕ1 = (d1/d2)ϕ2. Naturally, this technique assumes that both retarders are at the same temperature, and both are made of the same material.

19.5.3 dichRoiSM (cS and ciP)

Another error source for CP comes from dichroism [14,17]. In a crystal, differing amounts of absorp-tion between the ordinary and extraordinary axes can result in diattenuation. The general matrix for a linear diattenuator can be expressed as

Md

x y x y

x y x y

x

T T T T

T T T T

T T=

+( ) −( )−( ) +( )1

2

0 0

0 0

0 0 2 yy

x yT T

0

0 0 0 2

, (19.84)

where Tx and Ty are the intensity transmission coefficients along the x and y axes, respectively. The net result of this crystal absorption is that the polarization state changes as it propagates through the crystal. An example illustrating dichroism in yttrium vanadate (YVO4) crystal is depicted in Figure 19.25. Here, the absorption coefficient (α) for the ordinary and extraordinary axes is demon-strated. Notable is the difference in absorption between each axis spanning 3.6–5 μm.

If the dichroism is uncorrected, then several errors can be induced. These include

Absorption coefficient for YVO4

2.5αordinary

yαextraordinary

1.5

2

0.5

1α (c

m–1

)

3 3.5 4 4.5 50

Wave length (µm)

figuRe 19.25 YVO4 absorption coefficients spanning the mid wavelength infrared (MWIR) spectral region (3–5 μm) for the ordinary and extraordinary axes.

Channeled Polarimetry for Snapshot Measurements 429

1. Crosstalk between S1 and S0 in spatial or spectral regions where dichroism is present. 2. S2 and S3 experience error after normalization to S0. This error is a result of the crosstalk

from (1) above. 3. Direct modulation of S2 and S3 by the first retarder’s phase (R1) in CS, or the first pair of

prisms (P1 and P2) in CIP. In the 1:2 thickness ratio CS configuration, this can cause addi-tional aliasing into the innermost channels.

Removing the effect of dichroism, while still making use of the reference beam calibration tech-nique, requires that two reference datasets be taken. Additionally, the transmission of the ordinary and extraordinary axes of retarder R1 must be measured. From these reference data, the influence of the diattenuation can be removed from the sample data.

19.5.4 diSPeRSion (cS)

Birefringence dispersion in the crystal is an additional source of error in channeled spectropo-larimetry [29]. In previous derivations, it was implied that second and higher order terms in the retardance were negligible [30]. Neglecting the higher order terms means that the retardance varies linearly over wave number, yielding a spectral carrier that contains a single frequency. However, the higher order nonlinear dispersion terms produce a small continuum of carrier frequencies (i.e., a chirped carrier frequency).

To illustrate the effect dispersion has on the carrier frequency, a quartz retarder is simulated. The birefringence of quartz, at a temperature T (°C), can be expressed by

10 10 8 86410 0 107057 0 0013 3 2b n no eσ σ( ) = −( ) = + +. . . 99893 0 17175

10 1 900 1 01 0 2

4 2

3 2

σ σ

σ

−

− +( ) +

−

− −

.

. .T T (( ), (19.85)

where σ = λ–1, with λ in μm. The retardance is calculated by

φ πσ σ= ( )2 b d, (19.86)

where d is the thickness of the retarder. The retardance of a 5 mm thick quartz retarder at 25°C is depicted alongside its linear fit in Figure 19.26a, while its derivative is provided in Figure 19.26b.

650700

750Retardance of quartz vs. wave number

0.0215

0.022Derivative of quartz retardance vs. wave number

500550

600

650

0.02

0.0205

0.021

rad/

cm–1

1.4 1.6 1.8 2 2.2 2.4350400

450Reta

rdan

ce (r

ad)

QuartzFirst-order fit

1.4 1.6 1.8 2 2.2 2.4

0.019

0.0195

(a) (b)

x 104Wave number (cm–1) x 104Wave number (cm–1)

QuartzFirst-order fit

figuRe 19.26 (a) Retardance of a 5 mm thick quartz retarder, alongside its linear fit and (b) derivative of the two curves in (a) to clearly illustrate the nonlinearity.

430 Polarized Light, Third Edition

Simulating an output spectrum by modeling both an ideal and actual quartz retarder, between two parallel linear polarizers (Figure 19.3), enables the influence of the dispersion to be visual-ized. Taking the inverse Fourier transform of the output spectra for both cases is illustrated in Figure 19.27. Notable is the broadening of the carrier frequency when compared to the ideal case.

19.6 muelleR maTRiX ChaNNeled SPeCTRoPolaRimeTeRS

Channeled polarimetry has also been used to design snapshot Mueller matrix spectropolarimeters (MMSP) [31]. A typical MMSP consists of a polarization generator and analyzer, with the sample placed in between, as depicted in Figure 19.28. The generator has a similar configuration to the analyzer, only mirrored about the sample. The thicknesses of R1 through R4 are d1 through d4, respectively.

With a thickness ratio of d1:d2:d3:d4 of 1:2:5:10 (or a 1:5 ratio between d1:d3 and d2:d4), the system is capable of modulating the sample’s spectrally dependent Mueller matrix components (m00–m33) onto spectral carrier frequencies. The intensity measured by the spectrometer is

1 QuartzFirst-order fit Fourier transform

of intensity (Abs. value)

Fourier transform of intensity (Abs. value)

0.6

0.7

0.8

0.9

0.25

0.3

0.35

0.4

0.2

0.3

0.4

0.5A.U

.

0.05

0.1

0.15

0.2 A.U

.

–80 –60 –40 –20 0 20 40 60 800

0.1

OPD (µm)

OPD (µm)35 40 45 50 55 60 650

figuRe 19.27 Inverse Fourier transform of the spectrum generated when using an ideal first-order and an actual quartz retarder in the system as in Figure 19.3.

R4 AR3R2G R1

Generator Analyzer

A

Spectrometer

d4d3d1

Input spectrum

Sample

fs45° 0°

xx

yf

s

d2

0°fs

45°x

yf

s

zy

figuRe 19.28 System configuration of a channeled Mueller matrix spectropolarimeter. The fast axes ori-entations of R1 and R4 are 45° while R2 and R3 are 0°. The linear polarizers, G and A, both have their transmis-sion axes at 0°.

Channeled Polarimetry for Snapshot Measurements 431

IS

m m m minσσ

φ φ( ) =( )

+ + +, ( cos cos sin000 01 1 02 1 24

∆ 003 2 1 10 4

11 1 4 12

cos sin cos

cos cos sin

φ ϕ φ

φ φ

+

+ +

m

m m φφ φ φ φ φ φ φ1 2 4 13 1 2 4 20 3sin cos sin cos cos sin si+ +m m nn

cos sin sin sin sin sin s

φ

φ φ φ φ φ φ

4

21 1 3 4 22 1 2 3+ +m m iin sin cos sin sin

cos sin

φ φ φ φ φ

φ φ

4 23 1 2 3 4

30 3 4

+

−

m

m −− −m m31 1 3 4 32 1 2 3cos cos sin sin sin cos sinφ φ φ φ φ φ φ44

33 1 2 3 4−m sin cos cos sin ).φ φ φ φ

(19.87)

where ϕ1 through ϕ4 is the retardance of retarders R1 through R4, respectively, Sin,0 is the input spectrum, and the Mueller matrix components, m00 through m33, are implicitly dependent on wave number. Inverse Fourier transformation of the acquired intensity signal yields 37 channels contain-ing different information about the sample’s Mueller matrix. A depiction of the time domain for the MMSP is illustrated in Figure 19.29.

Calibration and reconstruction of the Mueller matrix components is accomplished in a nearly identical procedure to that depicted previously for CS, where each channel is filtered and demodu-lated. More details about this technique are provided by Hagen, Oka, and Dereniak [31], Otani et al. [32], and Dubreuil et al. [33,34].

19.7 ChaNNeled elliPSomeTeRS

A further application of CS is observed in ellipsometry for characterizing the properties of mate-rials and thin films [35]. By correcting many of the common errors associated with CS, espe-cially through the use of the self-calibration technique, precise and stable measurements have been experimentally demonstrated [36]. One configuration for a channeled ellipsometer is depicted in Figure 19.30.

This system has been experimentally verified against a rotating-compensator spectroscopic ellip-someter by measuring thin film samples with thicknesses spanning 3 to 4000 nm. By compensating for several error sources, the measurement stability of the film’s thickness, spanning an operating temperature range of 5°C to 45°C, was less than 0.11 nm. Further details regarding this experiment are provided by Okabe et al. [36].

Inverse Fourier transform of intensity

0.2

0.25

0.1

0.15A.U

.

0 100 200 3000

0.05

–300 –200 –100OPD (µm)

figuRe 19.29 Inverse Fourier transform of the intensity pattern from the spectrometer for the MMSP. A total of 37 channels are present, each containing differing Mueller matrix components.

432 Polarized Light, Third Edition

RefeReNCeS

1. Tyo, J. S., D. H. Goldstein, D. B. Chenault, and J. A. Shaw, Review of passive imaging polarimetry for remote sensing applications, Appl. Opt. 45 (2006): 5453–69.

2. Smith, M. H., J. B. Woodruff, and J. D. Howe, Beam wander considerations in imaging polarimetry, Proc. SPIE 3754 (1999): 50–4.

3. Sabatke, D. S., A. M. Locke, E. L. Dereniak, and R. W. McMillan, Linear operator theory of channeled spectropolarimetry, J. Opt. Soc. Am. A 22 (2005): 1567–76.

4. Carlson, A. B., P. B. Crilly, and J. C. Rutledge, communication Systems, New York: McGraw-Hill, 2002.

5. Oka, K., and T. Kato, Spectroscopic polarimetry with a channeled spectrum, Opt. Lett. 24 (1999): 1475–7.

6. Hagen, N., Snapshot Imaging Spectropolarimetry, Ph.D. Dissertation, Tucson, AZ: University of Arizona, 2007.

7. Snik, F., T. Karalidi, and C. U. Keller, Spectral modulation for full linear polarimetry, Appl. Opt. 48 (2009): 1337–46.

8. Oka, K., and T. Kaneko, Compact complete imaging polarimeter using birefringent wedge prisms, Opt. Express 11 (2003): 1510–9.

9. Oka, K., and N. Saito, Snapshot complete imaging polarimeter using Savart plates, Proc. SPIE 6295 (2006): 629508.

10. Mujat, M., E. Baleine, and A. Dogariu, Interferometric imaging polarimeter, J. Opt. Soc. Am. A 21 (2004): 2244–9.

11. Kudenov, M. W., M. E. L. Jungwirth, E. L. Dereniak, and G. R. Gerhart, White light Sagnac interferom-eter for snapshot linear polarimetric imaging, Opt. Express 17, no. 25 (2009), 22520–34.

12. Quan, C., P. Bryanston-Cross, and T. Judge, Photoelasticity stress analysis using carrier fringe and fast Fourier transform techniques, Opt. Lasers Eng. 18 (1993): 79–108.

13. Drobczynski, S., and H. Kasprzak, Application of space periodic variation of light polarization in imag-ing polarimetry, Appl. Opt. 44 (2005): 3160–6.

14. Kudenov, M. W., L. Pezzaniti, E. L. Dereniak, and G. R. Gerhart, Prismatic imaging polarimeter calibra-tion for the infrared spectral region, Opt. Express 16 (2008): 13720–37.

15. Taniguchi, A., K. Oka, H. Okabe, and M. Hayakawa, Stabilization of a channeled spectropolarimeter by self-calibration, Opt. Lett. 31 (2006): 3279–81.

16. Griffiths, P., and J. D. Haseth, Fourier Transform Infrared Spectrometry, New York: John Wiley & Sons, 1986.

17. Kudenov, M. W., N. A. Hagen, E. L. Dereniak, and G. R. Gerhart, Fourier transform channeled spec-tropolarimetry in the MWIR, Opt. Express 15 (2007): 12792–805.

18. Kudenov, M. W., Infrared Stokes Polarimetry and Spectropolarimetry, Ph.D. Dissertation, Tucson, AZ: University of Arizona, 2009.

19. Luo, H., K. Oka, N. Hagen, T. Tkaczyk, and E. Dereniak. Modeling and optimization for a prismatic snapshot imaging polarimeter, Appl. Opt. 45 (2006): 8400–9.

PL1

L4Emitter Receiver

Fiber

FiberR1

R2 L2 L3

A

DiffuserSpectrometer

Halogenlamp

0°fs

45°x

y

s 45°Samplef

figuRe 19.30 Schematic of an experimentally demonstrated channeled ellipsometer.

Channeled Polarimetry for Snapshot Measurements 433

20. Luo, H., K. Oka, E. DeHoog, M. Kudenov, J. Schwiegerling, and E. L. Dereniak, Compact and miniature snapshot imaging polarimeter, Appl. Opt. 47 (2008): 4413–7.

21. Luo, H., Snapshot Imaging Polarimeters using Spatial Modulation, Ph.D. Dissertation, Tucson, AZ: University of Arizona, 2008.

22. Malacara, D., Optical Shop Testing, New York: John Wiley, 1992. 23. Oka, K., T. Mizuno, Y. Bae, and A. Taniguchi, Improvement of imaging polarimetry using birefringent

prism pairs, Proc. SPIE in Polarization Science and Remote Sensing IV, Edited by J. A. Shaw and J. S. Tyo, Presentation only, 2009.

24. Beckers, J. M., Achromatic linear retarders, Appl. Opt. 10 (1971): 973–5. 25. Suda, R., N. Saito, and K. Oka, Imaging polarimetry by use of double Sagnac interferometers, in

Extended Abstracts of the 69th Autumn Meeting of the Japan Society of Applied Physics, 877, Tokyo: Japan Society of Applied Physics, 2008.

26. Oka, K., R. Suda, M. Ohnuki, D. Miller, and E. L. Dereniak, Snapshot imaging polarimeter for polychro-matic light using Savart plates and diffractive lenses, in Frontiers in Optics, OSA Technical digest (cd), Paper FThF4, Washington, DC: Optical Society of America, 2009.

27. Craven, J. M., M. W. Kudenov, and E. L. Dereniak, False signature reduction in infrared channeled spectropolarimetry, Proc. SPIE 7419 (2009): 741909.

28. Ballard, S. S., S. E. Brown, and J. S. Browder, Measurements of the thermal expansion of six optical materials, from room temperature to 250°C, Appl. Opt. 17 (1978): 1152–4.

29. Sabatke, D. S., A. M. Locke, E. L. Dereniak, and R. W. McMillan, Linear calibration and reconstruction techniques for channeled spectropolarimetry, Opt. Express 11 (2003): 2940–52.

30. Okabe, H., M. Hayakawa, H. Naito, A. Taniguchi, and K. Oka, Spectroscopic polarimetry using chan-neled spectroscopic polarization state generator (CSPSG), Opt. Express 15 (2007): 3093–109.

31. Hagen, N., K. Oka, and E. L. Dereniak, Snapshot Mueller matrix spectropolarimeter, Opt. Lett. 32 (2007): 2100–2.

32. Otani, Y., T. Wakayama, K. Oka, and N. Umeda, Spectroscopic Mueller matrix polarimeter using four-channeled spectra, Opt. commun. 281 (2008): 5725–30.

33. Dubreuil, M., S. Rivet, B. Le Jeune, and J. Cariou, Snapshot Mueller matrix polarimeter by wavelength polarization coding, Opt. Express 15 (2007): 13660–8.

34. Dubreuil, M., S. Rivet, B. Le Jeune, and J. Cariou, Systematic errors specific to a snapshot Mueller matrix polarimeter, Appl. Opt. 48 (2009): 1135–42.

35. Okabe, H., K. Matoba, M. Hayakawa, A. Taniguchi, K. Oka, H. Naito, and N. Nakatsuka, New configu-ration of channeled spectropolarimeter for snapshot polarimetric measurement of materials, Proc. SPIE 5878 (2005): 58780H.

36. Okabe, H., M. Hayakawa, J. Matoba, H. Naito, and K. Oka, Error-reduced channeled spectroscopic ellip-someter with palm-size sensing head, Rev. Sci. Instrum. 80 (2009): 083104.

IIIPart

Applications

437

20 Introduction

Polarized light and its applications appear in many branches of science and engineering. These include astrophysics (synchrotron radiation, solar physics, atmospheric scattering), chemistry (saccharimetry, optical activity, fluorescence polarization), microscopy (the polarizing micro-scope), and, of course, optics (polarization by reflection from glass and metals, liquid crystals, thin films, electro-optics, etc.). It is not practical to include all applications of polarized light in a single textbook. Therefore, in this third part, the discussion is restricted to several applications that are of notable importance and interest.

We begin with Chapter 21 “Crystal Optics.” The polarization of light was first discovered by Bartholinus while investigating the transmission of unpolarized light through a crystal of Iceland spar (calcite). It is a remarkable fact that in spite of all the research on materials over the last 300 years, very few natural or synthetic materials have been found that can be used to create and analyze polarized light. The crystals having the widest applications in the visible region of the spectrum are calcite, quartz, mica, and tourmaline. The optics of crystals can be quite complicated. Fortunately, commonly used materials such as calcite and quartz are uniaxial crystals and relatively easy to understand in terms of their polarizing behavior.

In Chapter 21, we also discuss very important applications where polarized light plays a key role, electro-optics and magneto-optics. Many crystals become anisotropic when subjected to an electric field or a magnetic field or both; the associated effects are called the electro-optical and magneto-optical effects, respectively. Of the two phenomena, in crystals the electro-optical effect is the more important, so we consider this effect in greater detail.

The polarization of light is changed when light is reflected from dielectric materials. The change in polarization also occurs when light is reflected (and transmitted) by metals and semiconductors. In Chapter 22, we discuss the optics of metals. In particular, we show that the optical constants of the metal can be determined by analyzing the polarization of the reflected light.

Chapter 23 is a summary of some of the most common polarization optical elements that are used in the practice of optics. One of these is Polaroid or sheet polarizer. For many years a syn-thetic material was sought that could create polarized light. This was finally accomplished with the invention of Polaroid by Edwin Land. Polaroid is a dichroic polarizer that creates polarized light by the differential absorption of an incident beam of light. For many applications, Polaroid is a useful substitute for calcite polarizers, which are very expensive. Because Polaroid is so widely used, its parameters and their measurement are presented and discussed in Section 23.2 with other types of polarizers.

In Chapter 24 we discuss one of the most important and elegant applications of polarized light, ellipsometry. The objective of ellipsometry is to measure the thickness, and real and imaginary refractive indices of thin films. We introduce the fundamental equation of ellipsometry and solve it by using the Stokes parameters and the Mueller matrices.

Finally, in Chapter 25, we briefly describe form birefringence and meanderline elements. These polarization-modifying devices have been made possible by advances in lithography techniques.

439

21 Crystal Optics

21.1 iNTRoduCTioN

We are fascinated by the infinite variety of shapes of crystalline water in the form of snowflakes, we take for granted crystals of sodium chloride that we add to our food, and we decorate our bodies with highly valued crystals of carbon. Most of the modern electronic devices we use today would not be possible without crystals of silicon and germanium. Crystals are pleasing and useful to us because they are materials that have order. An ideal crystal consists of an infinite three-dimensional structure that contains repeated structural units. The structural units may be single atoms, groups of atoms of single or multiple chemical elements, or many molecules. Crystals can be formed of organic as well as inorganic materials. Although crystals do have a high degree of order, they rarely have the same properties in all three dimensions. Even if they appear to have an isotropic structure, their properties can quickly be made anisotropic with the application of external electromagnetic or mechanical forces. For this reason, we must consider crystals to be anisotropic materials.

In this chapter, we discuss the interaction of light with anisotropic materials. An anisotropic material has properties (thermal, mechanical, electrical, optical, etc.) that are different in differ-ent directions. Most materials are anisotropic. This anisotropy results from the structure of the material, and our knowledge of the nature of that structure can help us to understand the optical properties.

The interaction of light with matter is a process that is dependent upon the geometrical relation-ships of the light and matter. By its very nature, light is asymmetrical. Considering light as a wave, it is a transverse oscillation in which the oscillating quantity, the electric field vector, is oriented in a particular direction in space perpendicular to the propagation direction. Light that crosses the boundary between two materials, isotropic or not, at any angle other than normal to the boundary, will produce an anisotropic result. The Fresnel equations illustrate this, as we saw in Chapter 7. Once light has crossed a boundary separating materials, it experiences the bulk properties of the material through which it is currently traversing, and we are concerned with the effects of those bulk properties on the light.

The study of anisotropy in materials is important to understanding the results of the interaction of light with matter. For example, the principle of operation of many solid state and liquid crys-tal spatial light modulators is based on polarization modulation. Modulation is accomplished by altering the refractive index of the modulator material, usually with an electric or magnetic field. Crystalline materials are an especially important class of modulator materials because of their use in electro-optics and in ruggedized or space-worthy systems, and also because of the potential for putting optical systems on integrated circuit chips.

We will briefly review the electromagnetics necessary to the understanding of anisotropic mate-rials, and show the source and form of the electro-optic tensor. We will discuss crystalline materials and their properties, and introduce the concept of the index ellipsoid. We will show how the appli-cation of electric and magnetic fields alters the properties of materials and give examples. Liquid crystals will be discussed as well.

A brief summary of electro-optic modulation modes using anisotropic materials concludes the chapter.

440 Polarized Light, Third Edition

21.2 ReVieW of CoNCePTS fRom eleCTRomagNeTiSm

Recall from electromagnetics [1–3] that the electric displacement vector D is given by (MKS units):

D E= εε , (21.1)

where ε is the permittivity and ε = εo(1 + χ), where εo is the permittivity of free space, χ is the elec-tric susceptibility, (1 + χ) is the dielectric constant, and n = +( )1

1 2χχ / is the index of refraction. The

electric displacement is also given by:

D E P= +εo , (21.2)

but

D E E= +( ) = +ε ε εo o o1 χχ χχE , (21.3)

so P, the polarization (also called the electric polarization or polarization density) is P = εoχE.The polarization arises because of the interaction of the electric field with bound charges. The

electric field can produce a polarization by inducing a dipole moment, that is, separating charges in a material, or by orienting molecules that possess a permanent dipole moment.

For an isotropic, linear medium,

P E= εoχχ , (21.4)

and χ is a scalar, but note that in

D E P= +εo , (21.5)

the vectors do not have to be in the same direction, and in fact in anisotropic media, E and P are not in the same direction (and so D and E are not in the same direction). Note that χ does not have to be a scalar nor is P necessarily related linearly to E. If the medium is linear but anisotropic,

P Ei o ij j= ∑ε χj

, (21.6)

where the χij are the elements of the susceptibility tensor; that is,

P

P

Po

1

2

3

11 12 13

21 22 23

31 32 3

= εχ χ χχ χ χχ χ χ 33

1

2

3

E

E

E

, (21.7)

and

d

d

d

E

E

Eo

1

2

3

1

2

3

1 0 0

0 1 0

0 0 1

=

ε

+

εo

χ χ χχ χ χχ χ χ

11 12 13

21 22 23

31 32 33

=+

+

E

E

E

o

1

2

3

11 12 13

21 22 23

3

1

1εχ χ χ

χ χ χχ 11 32 33

1

2

31χ χ+

E

E

E

,

(21.8)

Crystal Optics 441

where the vector indices 1, 2, 3 represent the three Cartesian directions. This can be written

d Ei ij j= ε , (21.9)

where

ε ε χij ij= +o 1( ) (21.10)

is variously called the dielectric tensor, or permittivity tensor, or dielectric permittivity tensor. Equations 21.9 and 21.10 use the Einstein summation convention, that is, whenever repeated indices occur, it is understood that the expression is to be summed over the repeated indices. This notation will be used throughout this chapter.

The dielectric tensor is symmetric and real (assuming the medium is homogeneous and nonab-sorbing) so that

ε εij ji= , (21.11)

and there are at most six independent elements.Note that for an isotropic medium with nonlinearity (which occurs with higher field strengths)

P E E Eo= + + +( )ε χ χ χ22

33 ... , (21.12)

where χ2, χ3, and so on, are the nonlinear terms.Returning to the discussion of a linear, homogeneous, anisotropic medium, the susceptibility

tensor

χ χ χχ χ χχ χ χ

χ χ χ11 12 13

21 22 23

31 32 33

11 12

=113

12 22 23

13 23 33

χ χ χχ χ χ

(21.13)

is symmetric so that we can always find a set of coordinate axes (that is, we can always rotate to an orientation) such that the off diagonal terms are zero and the tensor is diagonalized thus

′′

′

χχ

χ

11

22

33

0 0

0 0

0 0

. (21.14)

The coordinate axes for which this is true are called the principal axes, and these χ′ are the principal susceptibilities. The principal dielectric constants are given by

1 0 0

0 1 0

0 0 1

0 0

0 0

0 0

111

22

33

+

=χ

χχ

+++

+

=

χχ

χ

11

22

33

12

22

0 0

0 1 0

0 0 1

0 0

0 0

0 0

n

n

n332

,

(21.15)

where n1, n2, and n3 are the principal indices of refraction.

442 Polarized Light, Third Edition

21.3 CRySTalliNe maTeRialS aNd TheiR PRoPeRTieS

As we have seen above, the relationship between the displacement and the field is

d Ei ij j= ε , (21.16)

where εij is the dielectric tensor. The impermeability tensor ηij is defined as

η εij o ij= −( )εε 1 , (21.17)

where ɛ–1 is the inverse of the dielectric tensor. The principal indices of refraction, n1, n2, and n3 are related to the principal values of the impermeability tensor and the principal values of the permit-tivity tensor by

1 1 1

12

22

32n n nii

o

iijj

o

jjkk

o

kk

= = = = = =η εε

η εε

η εε

. (21.18)

The properties of the crystal change in response to the force from an externally applied electric field. In particular, the impermeability tensor is a function of the field. The electro-optic coefficients are defined by the expression for the expansion, in terms of the field, of the change of the imperme-ability tensor from zero field value, that is,

η η ηij ij ij ijk k ijkl k lr E s E E E n( ) ( ) ( ),E O n− ≡ = + +0 ∆ == …3 4, , , (21.19)

where ηij is a function of the applied field E, rijkare the linear, or Pockels electro-optic tensor coef-ficients, and the sijkl are the quadratic, or Kerr, electro-optic tensor coefficients. Terms higher than quadratic are typically small and are neglected.

Note that the values of the indices and the electro-optic tensor coefficients are dependent on the frequency of light passing through the material. Any given indices are specified at a particular frequency (or wavelength). Also note that the external applied fields may be static or alternating fields, and the values of the tensor coefficients are weakly dependent on the frequency of the applied fields. Generally, low and/or high frequency values of the tensor coefficients are given in tables. Low frequencies are those below the fundamental frequencies of the acoustic resonances of the sample, and high frequencies are those above. Operation of an electro-optic modulator subject to low (high) frequencies is sometimes described as being unclamped (clamped).

The linear electro-optic tensor is of third rank with 33 elements and the quadratic electro-optic tensor is of fourth rank with 34 elements; however, symmetry reduces the number of independent elements. If the medium is lossless and optically inactive:

ε• ij is a symmetric tensor, that is, εij = εji.η• ij is a symmetric tensor, that is, ηij = ηji.r• ijk has symmetry where coefficients with permuted first and second indices are equal, that is, rijk = rjik.s• ijkl has symmetry where coefficients with permuted first and second indices are equal and coefficients with permuted third and fourth coefficients are equal, that is, sijkl = sjikl and sijkl = sijlk.

Symmetry reduces the number of linear coefficients from 27 to 18, and reduces the number of quadratic coefficients from 81 to 36. The linear electro-optic coefficients are assigned two indices

Crystal Optics 443

so that they are rlk where l runs from 1 to 6 and k runs from 1 to 3. The quadratic coefficients are assigned two indices so that they become sij where i runs from 1 to 6 and j runs from 1 to 6. For a given crystal symmetry class, the form of the electro-optic tensor is known.

21.4 CRySTalS

Crystals are characterized by their lattice type and symmetry. There are 14 lattice types. As an example of three of these, a crystal that has a cubic structure can be simple cubic, face-centered cubic, or body-centered cubic.

There are 32 point groups corresponding to 32 different symmetries. For example, a cubic lat-tice has five types of symmetry. The symmetry is labeled with point group notation, and crystals are classified in this way. A complete discussion of crystals, lattice types, and point groups is outside the scope of the present work, and will not be given here; there are many excellent References [4–9]. Table 21.1 gives a summary of the lattice types and point groups, and shows how these relate to opti-cal symmetry and the form of the dielectric tensor.

In order to understand the notation and terminology of Table 21.1, some additional information is required that we now introduce. As we have seen in the previous sections, there are three principal indices of refraction. There are three types of materials; those for which the three principal indices are equal, those where two principal indices are equal and the third is different, and those where all three principal indices are different. We will discuss these three cases in more detail in the next sec-tion. The indices for the case where there are only two distinct values are named the ordinary index (no) and the extraordinary index (ne). These labels are applied for historical reasons [10]. Erasmus Bartholinus, a Danish mathematician, in 1669 discovered double refraction in calcite. If the calcite crystal, split along its natural cleavage planes, is placed on a typewritten sheet of paper, two images of the letters will be observed. If the crystal is then rotated about an axis perpendicular to the page, one of the two images of the letters will rotate about the other. Bartholinus named the light rays from the letters that do not rotate the ordinary rays, and the rays from the rotating letters he named the extraordinary rays, hence, the indices that produce these rays are named likewise. This explains the notation in the dielectric tensor for tetragonal, hexagonal, and trigonal crystals.

Let us consider such crystals in more detail. There is a plane in the material in which a single index would be measured in any direction. Light that is propagating in the direction normal to this plane with equal indices experiences the same refractive index for any polarization (orientation of the E vector). The direction for which this occurs is called the optic axis. Crystals that have one optic axis are called uniaxial crystals. Materials with three principal indices have two directions in which the E vector experiences a single refractive index. These materials have two optic axes and are called biaxial crystals. This will be more fully explained in the section on the index ellipsoid. Materials that have more than one principal index of refraction are called birefringent materials and are said to exhibit double refraction.

Crystals are composed of periodic arrays of atoms. The lattice of a crystal is a set of points in space. Sets of atoms that are identical in composition, arrangement, and orientation are attached to each lattice point. By translating the basic structure attached to the lattice point, we can fill space with the crystal. Define vectors a, b, and c that form three adjacent edges of a parallelepiped that spans the basic atomic structure. This parallelepiped is called a unit cell. We call the axes that lie along these vectors the crystal axes.

We would like to be able to describe a particular plane in a crystal, since crystals may be cut at any angle. The Miller indices are quantities that describe the orientation of planes in a crystal. The Miller indices are defined as follows: (1) Locate the intercepts of the plane on the crystal axes. These will be multiples of lattice point spacing. (2) Take the reciprocals of the intercepts and form the three small-est integers having the same ratio. For example, suppose we have a cubic crystal so that the crystal axes are the orthogonal Cartesian axes. Suppose further that the plane we want to describe intercepts the axes at the points 4, 3, and 2. The reciprocals of these intercepts are 1/4, 1/3, and 1/2. The Miller

444 Polarized Light, Third Edition

indices are then (3,4,6). This example serves to illustrate how the Miller indices are found, but it is more usual to encounter simpler crystal cuts. The same cubic crystal, if cut so that the intercepts are 1, ∞, ∞ (defining a plane parallel to the y, z plane in the usual Cartesian coordinates) has Miller indi-ces (1,0,0). Likewise, if the intercepts are 1, 1, ∞ (diagonal to two of the axes), the Miller indices are (1,1,0), and if the intercepts are 1, 1, 1 (diagonal to all three axes), the Miller indices are (1,1,1).

Two important electro-optic crystal types have the point group symbols 43m (this is a cubic crystal, e.g., CdTe and GaAs) and 42m (this is a tetragonal crystal, e.g., AgGaS2). The linear and quadratic electro-optic tensors for these two crystal types, as well as all the other linear and quadratic electro-optic coefficient tensors for all crystal symmetry classes, are given in Tables 21.2 and 21.3. Note from these tables that the linear electro-optic effect vanishes for crystals that retain symmetry under inversion, that is, centrosymmetric crystals, whereas the quadratic electro-optic effect never vanishes. For further discussion of this point, see Yariv and Yeh [11].

Table 21.1Crystal Types, Point groups, and the dielectric Tensors

Symmetry Crystal System Point group dielectric Tensor

Isotropic Cubic 43m

εε =

εo

n

n

n

2

2

2

0 0

0 0

0 0

432

m3

23

m3m

Uniaxial Tetragonal 4

εε =

εo

o

o

e

n

n

n

2

2

2

0 0

0 0

0 0

44/m

422

4mm

42m4/mmm

Hexagonal 6

6

6/m

622

6mm

6 2m

6/mmm

Trigonal 3

3

32

3m

3mBiaxial Triclinic 1

εε =

εo

n

n

n

12

22

32

0 0

0 0

0 0

1

Monoclinic 2

m

2/m

Orthorhombic 222

2mm

mmm

Source: From Yariv, A., and Yeh, P., Optical Waves in crystals, Wiley, New York, 1984.

Crystal Optics 445

Table 21.2linear electro-optic TensorsCentrosymmetric

1

2/mmmm4/m4/mmm

3

3m

6/m6/mmmm3m3m

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

Triclinic 1 r r r

r r r

r r r

r r r

r r r

11 12 13

21 22 23

31 32 33

41 42 43

51 52 553

61 62 63r r r

Monoclinic 2 (2 || x2) 0 0

0 0

0 0

0

0 0

0

12

22

32

41 43

52

61 63

r

r

r

r r

r

r r

2 (2 || x3) 0 0

0 0

0 0

0

0

0 0

13

23

33

41 42

51 52

63

r

r

r

r r

r r

r

m (m ⊥ x2) r r

r r

r r

r

r r

r

11 13

21 23

31 33

42

51 53

62

0

0

0

0 0

0

0 0

m (m ⊥ x3) r r

r r

r r

r

r

r r

11 12

21 22

31 32

43

53

61 62

0

0

0

0 0

0 0

0

(continued)

446 Polarized Light, Third Edition

Table 21.2 (continued)linear electro-optic TensorsOrthorhombic 222 0 0 0

0 0 0

0 0 0

0 0

0 0

0 0

41

52

63

r

r

r

2mm 0 0

0 0

0 0

0 0

0 0

0 0 0

13

23

33

42

51

r

r

r

r

r

Tetragonal 4 0 0

0 0

0 0

0

0

0 0 0

13

13

33

41 51

51 41

r

r

r

r r

r r−

4 0 0

0 0

0 0 0

0

0

0 0

13

13

41 51

51 41

63

r

r

r r

r r

r

−

−

422 0 0 0

0 0 0

0 0 0

0 0

0 0

0 0 0

41

41

r

r−

4mm 0 0

0 0

0 0

0 0

0 0

0 0 0

13

13

33

51

51

r

r

r

r

r

42m 2 1|| x( ) 0 0 0

0 0 0

0 0 0

0 0

0 0

0 0

41

41

63

r

r

r

Crystal Optics 447

Table 21.2 (continued)linear electro-optic TensorsTrigonal 3 r r r

r r r

r

r r

r r

r

11 22 13

11 22 13

33

41 51

51 41

2

0 0

0

0

−−

−− 22 11 0−

r

32 r

r

r

r

r

11

11

41

41

11

0 0

0 0

0 0 0

0 0

0 0

0 0

−

−−

3m (m ⊥ x1) 0

0

0 0

0 0

0 0

0 0

22 13

22 13

33

51

51

22

−

−

r r

r r

r

r

r

r

3m (m ⊥ x2) r r

r r

r

r

r

r

11 13

11 13

33

51

51

11

0

0

0 0

0 0

0 0

0 0

−

−

Hexagonal 6 0 0

0 0

0 0

0

0

0 0 0

13

13

33

41 51

51 41

r

r

r

r r

r r−

6mm 0 0

0 0

0 0

0 0

0 0

0 0 0

13

13

33

51

51

r

r

r

r

r

622 0 0 0

0 0 0

0 0 0

0 0

0 0

0 0 0

41

41

r

r−

(continued)

448 Polarized Light, Third Edition

21.4.1 index elliPSoid

Light propagating in anisotropic materials experiences a refractive index and a phase velocity that depends on the propagation direction, polarization state, and wavelength. The refractive index for propagation (for monochromatic light of some specified frequency) in an arbitrary direction (in Cartesian coordinates)

a i j k= + +x y zˆ ˆ ˆ (21.20)

Table 21.2 (continued)linear electro-optic Tensors

6 r r

r r

r r

11 22

11 22

22 11

0

0

0 0 0

0 0 0

0 0 0

0

−−

− −

6 2 1m m x⊥( ) 0 0

0 0

0 0 0

0 0 0

0 0 0

0 0

22

22

22

−

−

r

r

r

6 2 2m m x⊥( ) r

r

r

11

11

11

0 0

0 0

0 0 0

0 0 0

0 0 0

0 0

−

−

Cubic 43

23

m 0 0 0

0 0 0

0 0 0

0 0

0 0

0 0

41

41

41

r

r

r

432 0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

Source: From Yariv, A., and Yeh, P., Optical Waves in crystals, Wiley, New York, 1984.

Crystal Optics 449

Table 21.3Quadratic electro-optic TensorsTriclinic 1

1

s s s s s s

s s s s s s

s s s

11 12 13 14 15 16

21 22 23 24 25 26

31 32 333 34 35 36

41 42 43 44 45 46

51 52 53 54

s s s

s s s s s s

s s s s s555 56

61 62 63 64 65 66

s

s s s s s s

Monoclinic 2

2

m

m/

s s s s

s s s s

s s s s

11 12 13 15

21 22 23 25

31 32 33 35

0 0

0 0

0 0

0 00 0 0

0 0

0 0 0 0

44 46

51 52 53 55

64 66

s s

s s s s

s s

Orthorhombic 2

222

mm

mmm

s s s

s s s

s s s

s

11 12 13

21 22 23

31 32 33

44

0 0 0

0 0 0

0 0 0

0 0 0 0 00

0 0 0 0 0

0 0 0 0 055

66

s

s

Tetragonal 4

4

4 / m

s s s s

s s s s

s s s

11 12 13 16

12 11 13 16

31 31 33

0 0

0 0

0 0 0

0 0

−

00 0

0 0 0 0

0 0 0

44 45

45 44

61 61 66

s s

s s

s s s

−−

422

4

42

4

mm

m

mm/

s s s

s s s

s s s

s

11 12 13

12 11 13

31 31 33

44

0 0 0

0 0 0

0 0 0

0 0 0 0 00

0 0 0 0 0

0 0 0 0 044

66

s

s

Trigonal 3

3

s s s s s s

s s s s s s

s s

11 12 13 14 15 61

12 11 13 14 15 61

31

−− −

331 33

41 41 44 45 51

51 51 45 44

0 0 0

0

0

s

s s s s s

s s s s s

− −− − 441

61 61 15 1412 11 120s s s s s s− − −( )

32

3

3

m

m

s s s s

s s s s

s s s

s

11 12 13 14

12 11 13 14

13 13 33

4

0 0

0 0

0 0 0

−

11 41 44

44 41

1412 11 12

0 0 0

0 0 0 0

0 0 0 0

−

−( )

s s

s s

s s s

(continued)

450 Polarized Light, Third Edition

can be obtained from the index ellipsoid, a useful and lucid construct for visualization and determi-nation of the index. (Note that we now shift from indexing the Cartesian directions with numbers to using x, y, and z.) In the principal coordinate system the index ellipsoid is given by

xn

yn

znx y z

2

2

2

2

2

2+ + = 1 (21.21)

in the absence of an applied electric field. The lengths of the semimajor and semiminor axes of the ellipse formed by the intersection of this index ellipsoid and a plane normal to the propagation direction and passing through the center of the ellipsoid are the two principal indices of refraction

Table 21.3 (continued)Quadratic electro-optic TensorsHexagonal 6

6

6 / m

s s s s

s s s s

s s s

11 12 13 61

12 11 13 61

31 31 33

0 0

0 0

0 0 0

0 0

−

00 0

0 0 0 0

0 0 0

44 45

45 44

61 6112 11 12

s s

s s

s s s s

−− −( )

622

6

6 2

6

mm

m

mmm/

s s s

s s s

s s s

s

11 12 13

12 11 13

31 31 33

44

0 0 0

0 0 0

0 0 0

0 0 0 0 00

0 0 0 0 0

0 0 0 0 044

12 11 12

s

s s−( )

Cubic 23

3m

s s s

s s s

s s s

s

11 12 13

13 11 12

12 13 11

44

0 0 0

0 0 0

0 0 0

0 0 0 0 00

0 0 0 0 0

0 0 0 0 044

44

s

s

432

3

43

m m

m

s s s

s s s

s s s

s

11 12 12

12 11 12

12 12 11

44

0 0 0

0 0 0

0 0 0

0 0 0 0 00

0 0 0 0 0

0 0 0 0 044

44

s

s

Isotropics s s

s s s

s s s

s

11 12 12

12 11 12

12 12 11

12 1

0 0 0

0 0 0

0 0 0

0 0 0 11 12

12 11 12

12 11 12

0 0

0 0 0 0 0

0 0 0 0 0

−( )−( )

−( )

s

s s

s s

Source: From Yariv, A., and Yeh, P., Optical Waves in crystals, Wiley, New York, 1984.

Crystal Optics 451

for that propagation direction. Where there are three distinct principal indices, the crystal is defined as biaxial, and the above equation holds. If two of the three indices of the index ellipsoid are equal, the crystal is defined to be uniaxial and the equation for the index ellipsoid is

xn

yn

zno o e

2

2

2

2

2

2+ + = 1. (21.22)

Uniaxial materials are said to be uniaxial positive when no < ne and uniaxial negative when no > ne. When there is a single index for any direction in space, the crystal is isotropic and the equation for the ellipsoid becomes that for a sphere,

xn

yn

zn

2

2

2

2

2

2+ + = 1. (21.23)

The index ellipsoids for isotropic, uniaxial, and biaxial crystals are illustrated in Figure 21.1.Examples of isotropic materials are CdTe, NaCl, diamond, and GaAs. Examples of uniaxial

positive materials are quartz and ZnS. Materials that are uniaxial negative include calcite, LiNbO3, BaTiO3, and KDP (KH2PO4). Examples of biaxial materials are gypsum and mica.

21.4.2 naTuRal biRefRingence

Many materials have natural birefringence, that is, they are uniaxial or biaxial in their natural (absence of applied fields) state. These materials are often used in passive devices such as polarizers and retarders. Calcite is one of the most important naturally birefringent materials for optics, and is used in a variety of well known polarizers, that is, the Nichol, Wollaston, or Glan–Thompson prisms. As we shall see later, naturally isotropic materials can be made birefringent, and materials that have natural birefringence can be made to change that birefringence with the application of electromagnetic fields.

21.4.3 wave SuRface

There are two additional methods of depicting the effect of crystal anisotropy on light. Neither is as satisfying or useful to this author as the index ellipsoid; however, both will be mentioned for the sake of completeness and in order to facilitate understanding of those references that use these

nx

nx

nx nx

ny

ny

nyny

nz

nz

nz nz

Isotropic

Biaxial

Uniaxial negative

Uniaxial positive

nx = ny > nz

nx < ny < nznx = ny < nz

nx = ny = nz

figuRe 21.1 Index ellipsoids.

452 Polarized Light, Third Edition

models. They are most often used to explain birefringence, for example, in the operation of calcite-based devices [12–14].

The first of these is called the wave surface. As a light wave from a point source expands through space, it forms a surface that represents the wave front. This surface is comprised of points hav-ing equal phase. At a particular instant in time, the wave surface is a representation of the velocity surface of a wave expanding in the medium; it is a measure of the distance through which the wave has expanded from some point over some time period. Because the wave will have expanded further (faster) when experiencing a low refractive index and expanded less (slower) when experiencing high index, the size of the wave surface is inversely proportional to the index.

To illustrate the use of the wave surface, consider a uniaxial crystal. Recall that we have defined the optic axis of a uniaxial crystal as the direction in which the speed of propagation is independent of polarization. The optic axes for positive and negative uniaxial crystals are shown on the index ellipsoids in Figure 21.2, and the optic axes for a biaxial crystal are shown on the index ellipsoid in Figure 21.3.

Optic axisOptic axis

Uniaxial positive

Uniaxial negative

figuRe 21.2 Optic axis on index ellipsoid for uniaxial positive and uniaxial negative crystals.

Optic axis Optic axis

figuRe 21.3 Optic axes on the index ellipsoid for biaxial crystals are the normals to the two unique planes that cut the ellipsoid in perfect circles.

Crystal Optics 453

The wave surfaces are now shown in Figure 21.4 for both positive and negative uniaxial materi-als. The upper diagram for each pair shows the wave surface for polarization perpendicular to the optic axes (also perpendicular to the principal section through the ellipsoid), and the lower diagram shows the wave surface for polarization in the plane of the principal section. The index ellipsoid sur-faces are shown for reference. Similarly, cross sections of the wave surfaces for biaxial materials are shown in Figure 21.5. In all cases, polarization perpendicular to the plane of the page is indicated

Index ellipsoid surface

Wave surface for lightpolarized perpendicularto principal section

Wave surface for lightpolarized parallel toprincipal section

Uniaxial positive

Uniaxial negative

Optic axis

Optic axis

Index ellipsoid surface

figuRe 21.4 Wave surfaces for uniaxial positive and negative materials.

z y

y x

z

xx

figuRe 21.5 Wave surfaces for biaxial materials in principal planes.

454 Polarized Light, Third Edition

with solid circles along the rays, whereas polarization parallel to the plane of the page is shown with short double-headed arrows along the rays.

21.4.4 wavevecToR SuRface

A second method of depicting the effect of crystal anisotropy on light is the wavevector surface. The wavevector surface is a measure of the variation of the value of k, the wavevector, for different propagation directions and different polarizations. Recall that

kn

c= =2π

λω

, (21.24)

so k ∝ n. Wavevector surfaces for uniaxial crystals will then appear as shown in Figure 21.6. Compare these to the wave surfaces in Figure 21.4.

Wavevector surfaces for biaxial crystals are more complicated. Cross sections of the wavevector surface for a biaxial crystal where nx < ny < nz are shown in Figure 21.7. Compare these to the wave surfaces in Figure 21.5.

Optic axis

Positive uniaxial

Optic axis

kx

kz

Negative uniaxial

kx

kz

figuRe 21.6 Wavevector surfaces for positive and negative uniaxial crystals.

kx

kx

kz

ky

ky

kz

figuRe 21.7 Wavevector surface cross sections for biaxial crystals.

Crystal Optics 455

21.5 aPPliCaTioN of eleCTRiC fieldS: iNduCed biRefRiNgeNCe aNd PolaRiZaTioN modulaTioN

When fields are applied to materials, whether isotropic or anisotropic, birefringence can be induced or modified. This is the principle of a modulator; it is one of the most important optical devices, since it gives control over the phase and/or amplitude of light.

The alteration of the index ellipsoid of a crystal on application of an electric and/or magnetic field can be used to modulate the polarization state. The equation for the index ellipsoid of a crystal in an electric field is

ηij i jE x x( ) ,= 1 (21.25)

or

η ηij ij i jx x( ) .0 1+[ ] =∆ (21.26)

This equation can be written as

xn n

yn nx y

22

1

2

22

2

21 1 1 1+

+ +

∆ ∆

+ +

+

zn n

yzn

z

22

3

2

4

2

1 1

21

∆

∆ +

+

=2

12

11

5

2

6

2

xzn

xyn

∆ ∆ ,,

(21.27)

or

xn

r E s E s E E s E E s E Ex

k k k k2

2 1 12

14 2 3 15 3 1 16 1 21

2 2 2+ + + + +( ))+ + + + + +y

nr E s E s E E s E E s E

yk k k k

22 2 2

224 2 3 25 3 1 26 1

12 2 2 EE

zn

r E s E s E E s E Ez

k k k k

2

22 3 3

234 2 3 35 3 1

12 2

+ + + + + ++

+ + + +

2

2 2 2

36 1 2

4 42

44 2 3 45 3

s E E

yz r E s E s E E s E Ek k k k 11 46 1 2

5 52

54 2 3 55 3 1

2

2 2 2

+( )

+ + + +

s E E

zx r E s E s E E s E Ek k k k ++( )

+ + + + +

2

2 2 2

56 1 2

6 62

64 2 3 65 3 1

s E E

xy r E s E s E E s E Ek k k k 22 166 1 2s E E( ) = ,

(21.28)

where the Ek are components of the electric field along the principal axes and repeated indices are summed.

If the quadratic coefficients are assumed to be small and only the linear coefficients are retained, then

∆ 12

1

3

nr E

llk k

k

=

=∑ , (21.29)

456 Polarized Light, Third Edition

and k = 1, 2, 3 corresponds to the principal axes x, y, and z. The equation for the index ellipsoid becomes

xn

r E yn

r E zn

r Ex

k ky

k kz

k k2

2 12

2 22

2 31 1 1+( ) + +

+ +

+ ( )

+ ( ) + ( ) =

2

2 2 1

4

5 6

yz r E

zx r E xy r E

k k

k k k k . (21.30)

Suppose we have a cubic crystal of point group 43m, the symmetry group of such common materi-als as GaAs. Suppose further that the field is in the z direction. Then the index ellipsoid is

xn

yn

zn

r E xyz

2

2

2

2

2

2 412 1+ + + = . (21.31)

The applied electric field couples the x-polarized and y-polarized waves. If we make the coordinate transformation

x x y

y x y

= ′ − ′

= ′ − ′

cos sin

sin cos ,

45 45

45 45

° °

° ° (21.32)

and substitute these equations into the equation for the ellipsoid, the new equation for the ellipsoid becomes

′ +

+ ′ −

+ =x

nr E y

nr E

znz z

22 41

22 41

2

2

1 11, (21.33)

and we have eliminated the cross term. We want to obtain the new principal indices. The principal index will appear in Equation 21.33 as 1 2/ nx′ and must be equal to the quantity in the first parenthesis of the equation for the ellipsoid, that is,

1 12 2 41n n

r Ex

z′

= + . (21.34)

We can solve for nx′ so that Equation 21.34 becomes

n n n r Ex z' ( ) ./= +1 241

1 2 (21.35)

We assume n2r41Ez << 1 so that the term in parentheses in Equation 21.35 is approximated by

1 112

241

1 2 241+( ) ≅ −

n r E n r Ez z

/. (21.36)

The equations for the new principal indices are

n n n r E

n n n r E

n n

x z

y z

z

′

′

′

= −

= +

=

12

12

341

341

.

(21.37)

Crystal Optics 457

As a similar example for another important materials type, suppose we have a tetragonal (point group 42m) uniaxial crystal in a field along z. The index ellipsoid becomes

xn

yn

zn

r E xyo o e

z

2

2

2

2

2

2 632 1+ + + = . (21.38)

A coordinate rotation can be done to obtain the major axes of the new ellipsoid. In the present example, this yields the new ellipsoid

1 1

2 632

2 632

2

2nr E x

nr E y

zno

zo

ze

+

′ + −

′ +

= 1. (21.39)

As in the first example, the new and old z axes are the same, but the new x′ and y′ axes are 45° from the original x and y axes (see Figure 21.8).

The refractive indices along the new x and y axes are

′ = −

′ = +

n n n r E

n n n r E

x o o z

y o o z

12

12

363

363 .

(21.40)

Note that the quantity n3rE in these examples determines the change in refractive index. Part of this product, n3r, depends solely on inherent material properties, and is a figure of merit for electro-optical materials. Values for the linear and quadratic electro-optic coefficients for selected materials are given in Tables 21.4 and 21.5, along with values for n and, for linear materials, n3r. While much of the information from these tables is from Yariv and Yeh [11], materials tables are also to be found in Kaminow [5,15]. Original sources listed in these references should be consulted on materials of particular interest. Additional information on many of the materials listed here, including tables of refractive index versus wavelength and dispersion formulae, can be found in Tropf, Thomas, and Harris [16].

For light linearly polarized at 45°, the x and y components experience different refractive indices ′nx and ′n .y The birefringence is defined as the index difference ′ − ′n ny x. Since the phase velocities of

x

y

x′y′

45°

z, z′

figuRe 21.8 Rotated principal axes.

458 Polarized Light, Third Edition

Table 21.4linear electro-optic Coefficients

Substance SymmetryWavelength

(µm)

electro-optic Coefficients

rlk(10–12 m/V)indices of Refraction n3r (10–12 m/V)

CdTe 43m 1.0 r41 = 4.5 n = 2.84 103

3.39 r41 = 6.8

10.6 r41 = 6.8 n = 2.60 120

23.35 r41 = 5.47 n = 2.58 94

27.95 r41 = 5.04 n = 2.53 82

GaAs 43m 0.9 r41 = 1.1 n = 3.60 51

1.15 r41 = 1.43 n = 3.43 58

3.39 r41 = 1.24 n = 3.3 45

10.6 r41 = 1.51 n = 3.3 54

ZnSe 43m 0.548 r41 = 2.0 n = 2.66

0.633 r41a = 2.0 n = 2.60 35

10.6 r41 = 2.2 n = 2.39

ZnTe 43m 0.589 r41 = 4.51 n = 3.06

0.616 r41 = 4.27 n = 3.01

0.633 r41 = 4.04 n = 2.99 108

r41a = 4.3

0.690 r41 = 3.97 n = 2.93

3.41 r41 = 4.2 n = 2.70 83

10.6 r41 = 3.9 n = 2.70 77

Bi12SiO20 23 0.633 r41 = 5.0 n = 2.54 82

CdS 6mm 0.589 r51 = 3.7 no = 2.501

ne = 2.519

0.633 r51 = 1.6 no = 2.460

ne = 2.477

1.15 r31 = 3.1 no = 2.320

r33 = 3.2 ne = 2.336

r51 = 2.0

3.39 r13 = 3.5 no = 2.276

r33 = 2.9 ne = 2.292

r51 = 2.0

10.6 r13 = 2.45 no = 2.226

r33 = 2.75 ne = 2.239

r51 = 1.7

CdSe 6mm 3.39 r13a = 1.8 no = 2.452

r33 = 4.3 ne = 2.471

PLZTb

(Pb0.814La0.124Zr0.4Ti0.6O3)∞m 0.546 ne

3r33 – no3r13 = 2320 no = 2.55

LiNbO3 3m 0.633 r13 = 9.6 no = 2.286

r22 = 6.8 ne = 2.200

r33 = 30.9

r51 = 32.61.15 r22 = 5.4 no = 2.229

ne = 2.150

3.39 r22 = 3.1 no = 2.136

ne = 2.073

Crystal Optics 459

the x and y components are different, there is a phase retardation Γ (in radians) between the x and y components of E given by

Γ = ′ − ′( ) =ω πλc

n n d n r E dy x o z

2 363 , (21.41)

where d is the path length of light in the crystal. The electric field of the incident light beam is

E x y= +( )12

E ˆ ˆ . (21.42)

After transmission through the crystal, the electric field is

12

2 2E e ei iΓ Γ/ /ˆ ˆ .′ + ′( )−x y (21.43)

Table 21.4 (continued)linear electro-optic Coefficients

Substance SymmetryWavelength

(µm)

electro-optic Coefficients

rlk(10–12 m/V)indices of Refraction n3r (10–12 m/V)

LiTaO3 3m 0.633 r13 = 8.4 no = 2.176

r33 = 30.5 ne = 2.180

r22 = -0.2

3.39 r33 = 27 no = 2.060

r13 = 4.5 ne = 2.065

r51 = 15

r22 = 0.3

KDP (KH2PO4) 42m 0.546 r41 = 8.77 no = 1.5115

r63 = 10.3 ne = 1.4698

0.633 r41 = 8 no = 1.5074

r63 = 11 ne = 1.4669

3.39 r63 = 9.7

no3r63 = 33

ADP (NH4H2PO4) 42m 0.546 r41 = 23.76 no = 1.5079

r63 = 8.56 ne = 1.4683

0.633 r63 = 24.1

RbHSeO4c 0.633 13,540

BaTiO3 4mm 0.546 r51 = 1640 no = 2.437

ne = 2.365

KTN (KTaxNb1-xO3) 4mm 0.633 r51 = 8000 no = 2.318

ne = 2.277

AgGaS2 42m 0.633 r41 = 4.0 no = 2.553

r63 = 3.0 ne = 2.507

a These values are for clamped (high frequency field) operation.b PLZT is a compound of Pb, La, Zr, Ti, and O. Source: Haertling, G. H., and Land, C. E., J. Am. cer. Soc. 54, 1, 1971; Land,

C. E., Opt. Eng., 17, 317, 1978. The concentration ratio of Zr to Ti is most important to its electro-optic properties. In this case, the ratio is 40:60.

c Source: Salvestrini, J. P., Fontana, M. D., Aillerie, M., and Czapla, Z., Appl. Phys. Lett., 64, 1920, 1994.

460 Polarized Light, Third Edition

If the path length and birefringence are selected such that Γ = π, the modulated crystal acts as a half wave linear retarder and the transmitted light has field components

12

12

2 2 2 2E e e E e ei i i iπ π π π/ / / /ˆ ˆ ˆ ˆ′ + ′( ) = ′ − ′−x y x yy

x y

( )

= ′ − ′( )Eeiπ /

ˆ ˆ .2

2

(21.44)

The axis of linear polarization of the incident beam has been rotated by 90° by the phase retardation of π radians or one-half wavelength. The incident linear polarization state has been rotated into the orthogonal polarization state. An analyzer at the output end of the crystal aligned with the incident (or unmodulated) plane of polarization will block the modulated beam. For an arbitrary applied voltage producing a phase retardation of Γ, the analyzer transmits a fractional intensity cos2Γ. This is the principle of the Pockels cell.

Note that the form of the equations for the indices resulting from the application of a field is highly dependent upon the direction of the field in the crystal. For example, Table 21.6 gives the electro-optical properties of cubic 43m crystals when the field is perpendicular to three of the crystal planes. The new principal indices are obtained in general by solving an eigenvalue prob-lem. For example, for a hexagonal material with a field perpendicular to the (111) plane, the index ellipsoid is

1

31

31

213 2

213 2

233

nr E

xn

r Ey

nr E

o o e

+

+ +

+ +

332

32

312

51 51

+ + =z yzr

Ezxr

E, (21.45)

Table 21.5Quadratic electro-optic Coefficients

Substance SymmetryWavelength

(μm)electro-optic Coefficients

sij(10–18 m2/V2)index of

RefractionTemperature

(°C)

BaTiO3 m3m 0.633 s11 – s12 = 2290 n = 2.42 T > Tc(Tc = 120°C)

PLZTa ∞m 0.550 s33 – s13 = 26,000/n3 n = 2.450 Room temperature

KH2PO4(KDP) 4–2m 0.540 n s se

3 ( )33 13 31− =

n s so3 ( ) .31 11 13 5− =

n s so3 ( ) .12 11 8 9− =

n so3

66 3 0= .

no = 1.5115b

ne = 1.4698b

Room temperature

NH4H2PO4(ADP) 4–2m 0.540 n s se

3 ( )33 13 24− =

n s so3 ( ) .31 11 16 5− =

n s so3 ( ) .12 11 5 8− =

n so3

66 2=

no = 1.5266b

ne = 1.4808b

Room temperature

Source: From Yariv, A., and Yeh, P., Optical Waves in Crystals, Wiley, New York, 1984.a PLZT is a compound of Pb, La, Zr, Ti, and O [17,18]. The concentration ratio of Zr to Ti is most important to its electro-

optic properties. In this case, the ratio is 65:35.b At 0.546 μm.

Crystal Optics 461

and the eigenvalue problem is

13

02

3

01

32

3

23

2

213 51

213 51

51 5

nr E r E

nr E r E

r E r

o

o

+

+

112

33

2

31

3

1

En

r E

Vn

V

e

+

=′

. (21.46)

The secular equation is then

13

10

23

01

31

213

251

213

nr E

nr E

nr E

o

o

+

−

′

+

−

′′

+

−

′

nr E

r E r En

r Eno

251

51 512

332

23

23

23

13

1

= 0, (21.47)

and the roots of this equation are the new principal indices.

21.6 magNeTo-oPTiCS

When a magnetic field is applied to certain materials, the plane of incident linearly polarized light may be rotated in passage through the material. The magneto-optic effect linear with field strength is called the Faraday effect, and was discovered by Michael Faraday in 1845. A magneto-optic cell is illustrated in Figure 21.9. The field is set up so that the field lines are along the direction of the

Table 21.6electro-optic Properties of Cubic 4–3m Crystals

E field direction index ellipsoid Principal indices

E perpendicular to (001) plane

E E

E E

x y

z

= =

=

0

x y zn

r Exyo

2 2 2

2 412 1+ + + = ′ = +

′ = −

′ =

n n n r E

n n n r E

n n

x o o

y o o

z o

12

341

12

341

E perpendicular to (110) plane

E E E

E

x y

z

= =

=

2

0

x y zn

r E yz zxo

2 2 2

2 412 1+ + + +( ) = ′ = +

′ = −

′ =

n n n r E

n n n r E

n n

x o o

y o o

z o

12

341

12

341

E perpendicular to (111) plane

E E E Ex y z= = =

3

x y zn

r E yz zx xyo

2 2 2

2 4123

1+ + + + +( ) = ′ = +

′ = −

′ = −

n n n r E

n n n r E

n n

x o o

y o o

z o

12 3

341

12 3

341

13

nn r Eo3

41

Source: From Goldstein, D., Polarization Modulation in Infrared Electrooptic Materials, PhD Dissertation, University of Alabama in Huntsville, 1990.

462 Polarized Light, Third Edition

optical beam propagation. A linear polarizer allows light of one polarization into the cell. A second linear polarizer is used to analyze the result.

The Faraday effect is governed by the equation

θ =Vbd, (21.48)

where V is the Verdet constant, θ is the rotation angle of the electric field vector of the linearly polarized light, b is the applied field, and d is the path length in the medium. The rotatory power ρ, defined in degrees per unit path length, is given by

ρ = Vb. (21.49)

A table of Verdet constants for some common materials is given in Table 21.7. The material that is often used in commercial magneto-optic-based devices is some formulation of iron garnet. Data tabulations for metals, glasses, and crystals, including many iron garnet compositions, can be found in Chen [21]. The magneto-optic effect is the basis for magneto-optic memory devices, optical isola-tors, and spatial light modulators [22,23].

Other magneto-optic effects in addition to the Faraday effect include the Cotton–Mouton effect, the Voigt effect, and the Kerr magneto-optic effect. The Cotton–Mouton effect is a quadratic mag-neto-optic effect observed in liquids. The Voigt effect is similar to the Cotton–Mouton effect but is observed in vapors. The Kerr magneto-optic effect is observed when linearly polarized light is reflected from the face of either pole of a magnet. The reflected light becomes elliptically polarized.

PolarizerPolarizer

B

figuRe 21.9 Illustration of a setup to observe the Faraday effect.

Table 21.7Values of the Verdet Constant at λ = 5893 Å

material t (°C) Verdet Constant (deg/g⋅mm)

Water a 20 2.18 × 10–5

Air (λ = 5780 Å and 760 mm Hg)b 0 1.0 × 10–8

NaClb 16 6.0 × 10–5

Quartzb 20 2.8 × 10–5

CS2a 20 7.05 × 10–5

Pa 33 2.21 × 10–4

Glass, flinta 18 5.28 × 10–5

Glass, Crowna 18 2.68 × 10–5

Diamonda 20 2.0 × 10–5

a From Yariv, A., and Yeh, P., Optical Waves in crystals, Wiley, New York, 1984.b From Hecht, E., Optics, Addison-Wesley, Reading, MA, 1987.

Crystal Optics 463

21.7 liQuid CRySTalS

Liquid crystals are a class of substances that demonstrate that the premise that matter exists only in solid, liquid, and vapor (and plasma) phases is a simplification. Fluids, or liquids, generally are defined as the phase of matter that cannot maintain any degree of order in response to a mechanical stress. The molecules of a liquid have random orientations and the liquid is isotropic. In the period 1888–1890, Reinitzer, and separately Lehmann, observed that certain crystals of organic compounds exhibit behavior between the crystalline and liquid states [24]. As the temperature is raised, these crystals change to a fluid substance that retains the anisotropic behavior of a crystal. This type of liquid crystal is now classified as thermotropic because the transition is effected by a temperature change, and the intermediate state is referred to as a mesophase [25]. There are three types of meso-phases: smectic, nematic, and cholesteric. Smectic and nematic mesophases are often associated and occur in sequence as the temperature is raised. The term smectic derives from the Greek word for soap and is characterized by a more viscous material than the other mesophases. Nematic is from the Greek for thread and was named because the material exhibits a striated appearance (between crossed Polaroids). The cholesteric mesophase is a property of the cholesterol esters, hence the name.

Figure 21.10a illustrates the arrangement of molecules in the nematic mesophase. Although the centers of gravity of the molecules have no long-range order as crystals do, there is order in the orientations of the molecules [26]. They tend to be oriented parallel to a common axis indicated by the unit vector n.

The direction of n is arbitrary and is determined by some minor force such as the guiding effect of the walls of the container. There is no distinction between a positive and negative sign of n. If the molecules carry a dipole, there are equal numbers of dipoles pointing up as down. These molecules are not ferroelectric. The molecules are achiral, that is, they have no handedness, and there is no positional order of the molecules within the fluid. Nematic liquid crystals are optically uniaxial.

The temperature range over which the nematic mesophase exists varies with the chemical com-position and mixture of the organic compounds. The range is quite wide; for example, in one study of ultraviolet imaging with a liquid crystal light valve, four different nematic liquid crystals were used [27]. Two of these were MBBA [N-(p-methoxybenzylidene)-p-n butylaniline] with a nematic range of 17 to 43°C, and a proprietary material with a range of –20°C to 51°C.

There are many known electro-optical effects involving nematic liquid crystals [25,28,29]. Two of the more important are field-induced birefringence, also called deformation of aligned phases, and the twisted nematic effect, also called the Schadt–Helfrich effect.

n

Nematic order

Smectic order

Cholesteric order

(c)

(b)

(a)

n

n

n

n

n

figuRe 21.10 Schematic representation of liquid crystal order.

464 Polarized Light, Third Edition

An example of a twisted nematic cell is shown in Figure 21.11. Figure 21.11a shows the mol-ecule orientation in a liquid crystal cell, without and with an applied field. The liquid crystal material is placed between two electrodes. The liquid crystals at the cell wall align themselves in some direction parallel to the wall as a result of very minor influences. A cotton swab lightly stroked in one direction over the interior surface of the wall prior to cell assembly is enough to produce alignment of the liquid crystal [30]. The molecules align themselves with the direction of the rubbing. The electrodes are placed at 90° to each other with respect to the direction of rub-bing. The liquid crystal molecules twist from one cell wall to the other to match the alignments at the boundaries as illustrated, and light entering at one cell wall with its polarization vector aligned to the crystal axis will follow the twist and be rotated 90° by the time it exits the opposite cell wall. If the polarization vector is restricted with a polarizer on entry and an analyzer on exit, only the light with the 90° polarization twist will be passed through the cell. With a field applied between the cell walls, the molecules tend to orient themselves perpendicular to the cell walls, that is, along the field lines. Some molecules next to the cell walls remain parallel to their original orientation, but most of the molecules in the center of the cell align themselves parallel to the electric field, destroying the twist. At the proper strength, the electric field will cause all the light to be blocked by the analyzer.

Figure 21.11b shows a twisted nematic cell as might be found in a digital watch display, gas pump, or calculator. Light enters from the left. A linear polarizer is the first element of this device and is aligned so that its axis is along the left-hand liquid crystal cell wall alignment direction. With no field, the polarization of the light twists with the liquid crystal twist, 90° to the original orientation, passes through a second polarizer with its axis aligned to the right-hand liquid crystal cell wall alignment direction, and is reflected from a mirror. The light polarization twists back the way it came and leaves the cell. Regions of this liquid crystal device that are not activated by the applied field are bright. If the field is now applied, the light does not change polarization as it passes through the liquid crystal and will be absorbed by the second polarizer. No light returns from the mirror, and the areas of the cell that have been activated by the applied field are dark.

A twisted nematic cell has a voltage threshold below which the polarization vector is not affected due to the internal elastic forces. A device 10 μm thick might have a threshold voltage of 3 V [25].

No field(a)

(b)

Field

Molecule orientation in a liquid crystal cell, with no field and with field

Polarizer Polarizer Mirror

A typical nematic liquid crystal cell

figuRe 21.11 Liquid crystal cell operation.

Crystal Optics 465

Another important nematic electro-optic effect is field-induced birefringence or deformation of aligned phases. As with the twisted nematic cell configuration, the liquid crystal cell is placed between crossed polarizers. However, now the molecular axes are made to align perpendicular to the cell walls and thus parallel to the direction of light propagation. By using annealed SnO2 electrodes and materials of high purity, Schiekel and Fahrenschon [29] found that the molecules spontaneously align in this manner. Their cell worked well with 20 μm thick MBBA. The working material must be one having a negative dielectric anisotropy so that when an electric field is applied (normal to the cell electrodes) the molecules will tend to align themselves perpendicular to the electric field. The molecules at the cell walls tend to remain in their original orientation and the molecules within the central region will turn up to 90°; this is illustrated in Figure 21.12.

There is a threshold voltage typically in the 4–6 V range [25]. Above the threshold, the molecules begin to distort and become birefringent due to the anisotropy of the medium. Thus with no field, no light exits the cell; at threshold voltage, light begins to be divided into ordinary and extraordinary beams, and some light will exit the analyzer. The birefringence can also be observed with positive dielectric anisotropy when the molecules are aligned parallel to the electrodes at no field and both electrodes have the same orientation for nematic alignment. As the applied voltage is increased, the light transmission increases for crossed polarizers [25]. The hybrid field-effect liquid crystal light valve relies on a combination of the twisted nematic effect (for the off state) and induced birefrin-gence (for the on state) [31].

Smectic liquid crystals are more ordered than the nematics. The molecules are not only aligned, but they are also organized into layers, making a two-dimensional fluid. This is illustrated in Figure 21.10b. There are three types of smectics, A, B, and C. Smectic A is optically uniaxial. Smectic C is optically biaxial. Smectic B is most ordered, since there is order within layers. Smectic C, when chiral, is ferroelectric. Ferroelectric liquid crystals are known for their fast switching speed and bistability.

Cholesteric liquid crystal molecules are helical, and the fluid is chiral. There is no long range order, as in nematics, but the preferred orientation axis changes in direction through the extent of the liquid. Cholesteric order is illustrated in Figure 21.10c.

For more information on liquid crystals and an extensive bibliography, see Wu [32,33], and Khoo and Wu [34].

21.8 modulaTioN of lighT

We have seen that light modulators are composed of an electro- or magneto-optical material on which an electromagnetic field is imposed. Electro-optical modulators may be operated in a lon-gitudinal mode or in a transverse mode. In a longitudinal mode modulator, the electric field is imposed parallel to the light propagating through the material, and in a transverse mode modulator,

Applied field directionand

light propagationdirection

figuRe 21.12 Deformation of liquid crystal due to applied voltage.

466 Polarized Light, Third Edition

the electric field is imposed perpendicular to the direction of light propagation. Either mode may be used if the entire wavefront of the light is to be modulated equally. The longitudinal mode is more likely to be used if a spatial pattern is to be imposed on the modulation. The mode used will depend upon the material chosen for the modulator and the application.

Figure 21.13 shows the geometry of a longitudinal electro-optic modulator. The beam is normal to the face of the modulating material and parallel to the field imposed on the material. Electrodes of a material that is conductive yet transparent to the wavelength to be modulated are deposited on the faces through which the beam travels. This is the mode used for liquid crystal modulators.

Figure 21.14 shows the geometry of the transverse electro-optic modulator. The imposed field is perpendicular to the direction of light passage through the material. The electrodes do not need to be transparent to the beam. This is the mode used for modulators in laser beam cavities, for example, a CdTe modulator in a CO2 laser cavity.

21.9 PhoToelaSTiC modulaToRS

We have concentrated on electro- and magneto-optical properties and materials for most of this chapter. Acousto-optics can also be used to produce modulation of a beam of light via the photo-elastic effect. This effect can be described by

V

Transparent electrodes

Light beam

Eo crystal

figuRe 21.13 Longitudinal mode modulator.

Light

Eo crystal

Electrodes

V

figuRe 21.14 Transverse mode modulator.

Crystal Optics 467

∆ ∆ηijij

ijkl klnp S=

=1

2, (21.50)

where Δηij is the change in the optical impermeability tensor, Skl is the strain tensor, and pijk; is the strain-optic tensor. Note that this equation is very much like Equation 21.19, but now the change in refractive index is caused by the acoustic strain field. As in electro-optics, there are different forms of the strain-optic tensor for different crystal symmetries (the forms are identical to those of the quadratic electro-optic coefficients). A more detailed treatment of this subject is given in the excel-lent text by Yariv and Yeh [11].

Photoelastic modulators may be constructed of rectangular bars of a material such as fused silica or calcium flouride. A quartz piezoelectric transducer is cemented to the material. A fre-quency of oscillation is induced such that a standing acoustic wave of one half wavelength exists in the material. The frequency required is dependent on the length of the bar of material and the speed of sound in it. The result is an oscillating longitudinal strain and an oscillating retardance at the center of the bar. The magnitude of the oscillation is governed by the voltage to the trans-ducer, and photoelastic modulators are calibrated to, for example, a half wave of retardation over a range of wavelengths [35]. A photograph of a commercial photoelastic modulator is shown in Figure 21.15.

21.10 CoNCludiNg RemaRkS

The origins of the electro-optic tensor, the form of that tensor for various crystal types, and the val-ues of the tensor coefficients for specific materials have been discussed. The concepts of the index ellipsoid, the wave surface, and the wavevector surface were introduced. These are quantitative and qualitative models that aid in the understanding of the interaction of light with crystals. We have shown how the equation for the index ellipsoid is found when an external field is applied, and how expressions for the new principal indices of refraction are derived. Magneto-optics, liquid crystals, and elasto-optic modulators are described. The introductory concepts of constructing an electro-optic modulator were discussed.

While the basics of electro- and magneto-optics in bulk materials has been covered, there is a large body of knowledge dealing with related topics that cannot be covered here. A more detailed description of electro-optic modulators is covered in Yariv and Yeh [11]. Information on spatial

figuRe 21.15 (See color insert following page 394.) Photograph of a commercial photoelastic modulator. (Photo courtesy of D. H. Goldstein.)

468 Polarized Light, Third Edition

light modulators may be found in Efron [36]. Shen [37] describes the many aspects and applications of nonlinear optics, and current work in such areas as organic nonlinear materials can be found in SPIE Proceedings [38,39].

RefeReNCeS

1. Jackson, J. D., classical Electrodynamics, 2nd ed., New York: Wiley, 1975. 2. Lorrain, P., and D. Corson, Electromagnetic Fields and Waves, 2nd ed., New York: Freeman, 1970. 3. Reitz, J. R., and F. J. Milford, Foundations of Electromagnetic Theory, 2nd ed., Reading, MA: Addison-

Wesley, 1967. 4. Lovett, D. R., Tensor Properties of crystals, Bristol: Adam Hilger, 1989. 5. Kaminow, I. P., Linear electrooptic materials, in cRc Handbook of Laser Science and Technology,

Volume IV, Optical Materials, Part 2: Properties, Edited by M. J. Weber, Boca Raton, FL: CRC Press, 1986.

6. Nye, J. F., Physical Properties of crystals: Their Representation by Tensors and Matrices London: Oxford University Press, 1985.

7. Senechal, M., crystalline Symmetries: An Informal Mathematical Introduction, Bristol: Adam Hilger, 1990.

8. Wood, E. A., crystals and Light: An Introduction to Optical crystallography, Mineola, NY: Dover, 1977.

9. Kittel, C., Introduction to Solid State Physics, New York: Wiley, 1971. 10. Hecht, E., Optics, Reading, MA: Addison-Wesley, 1987. 11. Yariv, A., and P. Yeh, Optical Waves in crystals, New York: Wiley, 1984. 12. Jenkins, F. A., and H. E. White, Fundamentals of Optics, New York: McGraw-Hill, 1957. 13. Born, M., and E. Wolf, Principles of Optics, New York: Pergamon Press, 1975. 14. Klein, M. V., Optics, New York: Wiley, 1970. 15. Kaminow, I. P., An Introduction to Electrooptic devices, New York: Academic Press, 1974. 16. Tropf, W. J., M. E. Thomas, and T. J. Harris, Properties of Crystals and Glasses, Chapter 33 in Handbook

of Optics, Volume II, devices, Measurements, and Properties, 2nd ed., New York: McGraw-Hill, 1995. 17. Haertling, G. H., and C. E. Land, Hot-pressed (Pb,La)(Zr,Ti)O3 Ferroelectric Ceramics for Electrooptic

Applications, J. Am. cer. Soc. 54 (1971): 1. 18. Land, C. E., Optical information storage and spatial light modulation in PLZT ceramics, Opt. Eng. 17

(1978): 317. 19. Salvestrini, J. P., M. D. Fontana, M. Aillerie, and Z. Czapla, New material with strong electro-optic

effect: Rubidium hydrogen selenate (RbHSeO4), Appl. Phys. Lett. 64 (1994): 1920. 20. Goldstein, D., Polarization Modulation in Infrared Electrooptic Materials, PhD Dissertation, University

of Alabama in Huntsville, 1990. 21. Chen, D., Data tabulations, in cRc Handbook of Laser Science and Technology, Volume IV, Optical

Materials, Part 2: Properties, Edited by M. J. Weber, Boca Raton, FL: CRC Press, 1986. 22. Ross, W. E., D. Psaltis, and R. H. Anderson, Two-dimensional magneto-optic spatial light modulator for

signal processing, Opt. Eng. 22 (1983): 485. 23. Ross, W. E., K. M. Snapp, and R. H. Anderson, Fundamental characteristics of the Litton iron garnet

magneto-optic spatial light modulator, in Advances in Optical Information Processing, Proc. SPIE, 388 (1983).

24. Gray, G. W., Molecular Structure and the Properties of Liquid crystals, New York: Academic Press, 1962.

25. Priestley, E. B., P. J. Wojtowicz, and P. Sheng, P., Eds., Introduction to Liquid crystals, New York: Plenum Press, 1974.

26. De Gennes, P. G., The Physics of Liquid crystals, London: Oxford University Press, 1974. 27. Margerum, J. D., J. Nimoy, and S. Y. Wong, Reversible ultraviolet imaging with liquid crystals, Appl.

Phys. Lett. 17 (1970): 51. 28. Meier, G., H. Sackman, and F. Grabmaier, Applications of Liquid crystals, New York: Springer Verlag,

1975. 29. Schiekel, M. F., and K. Fahrenschon, Deformation of nematic liquid crystals with vertical orientation in

electrical fields, Appl. Phys. Lett. 19 (1971): 391. 30. Kahn, F. J., Electric-field-induced orientational deformation of nematic liquid crystals: tunable

birefringence, Appl. Phys. Lett. 20 (1972): 199.

Crystal Optics 469

31. Bleha, W. P., et al., Application of the liquid crystal light valve to real-time optical data processing, Opt. Eng. 17 (1978): 371.

32. Wu, S.-T., Liquid crystals, Chapter 14 in Handbook of Optics, Volume II, devices, Measurements, and Properties, 2nd ed., New York: McGraw-Hill, 1995.

33. Wu, S.-T., Nematic liquid crystals for active optics, in Optical Materials, A Series of Advances, Vol. 1, Edited by S. Musikant, New York: Marcel Dekker, 1990.

34. Khoo, I.-C., and S.-T. Wu, Optics and Nonlinear Optics of Liquid crystals, Hackensack, NJ: World Scientific, 1993.

35. Oakberg, T. C., Relative variation of stress-optic coefficient with wavelength in fused silica and calcium fluoride, in Polarization: Measurement, Analysis, and Remote Sensing II, Proc. SPIE 3754 (1999).

36. Efron, U., Ed., Spatial Light Modulator Technology, Materials, devices, and Applications, New York: Marcel Dekker, 1995.

37. Shen, Y. R., The Principles of Nonlinear Optics, New York: Wiley, 1984. 38. Kuzyk, M. G., Ed., Nonlinear optical properties of organic materials X, Proc. SPIE 3147 (1997). 39. Moehlmann, G. R., Ed., Nonlinear optical properties of organic materials IX, Proc. SPIE 2852 (1996).

471

22 Optics of Metals

22.1 iNTRoduCTioN

We have been concerned with the propagation of light in nonconducting media. We now turn our attention to describing the interaction of light with conducting materials such as metals and semi-conductors. Metals and semiconductors, absorbing media, are crystalline aggregates consisting of small crystals of random orientation. Unlike true crystals, they do not have repetitive structures throughout their entire forms.

The phenomenon of conductivity is associated with the appearance of heat; it is very often called Joule heat. It is a thermodynamically irreversible process in which electromagnetic energy is transformed to heat. As a result, the optical field within a conductor is attenuated. The very high conductivity exhibited by metals and semiconductors causes them to be practically opaque to light. The phenomenon of conduction and strong absorption corresponds to high reflectivity so that metallic surfaces act as excellent mirrors. In fact, up to the latter part of the nineteenth century, most large reflecting astronomical telescope mirrors were metallic. Eventually, metal mirrors were replaced with parabolic glass surfaces coated with silver, a material with a very high reflectivity. Unfortunately, silver oxidizes in a relatively short time with oxygen and sulfur compounds in the atmosphere and turns black, and consequently silver-coated mirrors must be recoated nearly every other year or so, a difficult, time-consuming, expensive process. This problem was finally solved by Strong [1] in the 1930s with his method of evaporating aluminum onto the surface of optical glass.

In the following sections, we shall not deal with the theory of metals. Rather, we shall concentrate on the phenomenological description of the interaction of polarized light with metallic surfaces. Therefore, in Section 22.2 we develop Maxwell’s equations for conducting media. We discover that for conducting media the refractive index becomes complex and has the form ˆ ( )n n i= −1 κ where n is the real refractive index and κ is the absorption index. Furthermore, Fresnel’s equations for reflec-tion and transmission continue to be valid for conducting (absorbing) media. However, because of the rapid attenuation of the optical field within an absorbing medium, Fresnel’s equations for trans-mission are not applicable. Using the complex refractive index, we develop Fresnel’s equations for reflection at normal incidence and describe them in terms of a quantity called the reflectivity. It is possible to develop Fresnel’s reflection equations for nonnormal incidence; however, the forms are very complicated, and so approximate forms are derived for the s and p polarizations. It is rather remarkable that the phenomenon of conductivity may be taken into account simply by introducing a complex index of refraction. A complete understanding of the significance of n and κ can only be understood on the basis of the dispersion theory of metals. However, experience does show that large values of reflectivity correspond to large values of κ.

In Sections 22.3 and 22.4, we discuss the measurement of the optical constants n and κ. A num-ber of methods have been developed over the past 100 years, nearly all of which are null-intensity methods. That is, n and κ are obtained from the null condition on the reflected intensity. The best-known null method is the principal angle of incidence/principal azimuthal angle method (Section 22.3). In this method, a beam of light is incident on the sample and the incidence angle is varied until an incidence angle is reached where a phase shift of π/2 occurs. The incidence angle where this takes place is known as the principal angle of incidence. An additional phase shift of π/2 is now introduced into the reflected light with a quarter-wave retarder. The condition of the principal angle of incidence and the quarter-wave shift and the introduction of the quarter-wave retarder, as we shall see, creates linearly polarized light. Analyzing the phase-shifted reflected light with a polarizer that

472 Polarized Light, Third Edition

is rotated around its azimuthal angle leads to a null intensity (at the principal azimuthal angle) from which n and κ can be determined.

Classical null methods were developed long before the advent of quantitative detectors, digi-tal voltmeters, and digital computers. Nulling methods are very valuable, but they have a serious drawback; the method requires a mechanical arm that must be rotated along with the azimuthal rotation of a Babinet–Soleil compensator and an analyzer until a null intensity is found. In addi-tion, a mechanical arm that yields scientifically useful readings is quite expensive. It is possible to overcome these drawbacks by reconsidering Fresnel’s equations for reflection at an incidence angle of 45° [2]. It is well known that Fresnel’s equations for reflection simplify at normal incidence and at the Brewster angle for nonabsorbing (dielectric) materials. Less well known is that Fresnel’s equations also simplify at an incidence angle of 45°. All of these simplifications were discussed in Chapter 7 assuming dielectric media. The simplifications at the incidence angle of 45° hold even for absorbing media. Therefore, in Section 22.4, we describe the measurement of an optically absorb-ing surface at an incidence angle of 45°. This method, called digital refractometry [3], overcomes the nulling problems and leads to equations to determine n and κ that can be solved on a digital computer by iteration.

22.2 maXWell’S eQuaTioNS foR abSoRbiNg media

We now solve Maxwell’s equations for a homogeneous isotropic medium described by a dielectric constant ε, a permeability μ, and a conductivity σ. Using the constitutive relations

D E= ε , (22.1)

B H= µ , (22.2)

j E= σ , (22.3)

Maxwell’s equations become, in MKSA units,

∇ × − ∂∂

=HE

Eε σt

, (22.4)

∇ × + ∂∂

=EHµt

0, (22.5)

∇ ⋅ =Eρε

, (22.6)

∇ ⋅ =H 0. (22.7)

These equations describe the propagation of the optical field within and at the boundary of a con-ducting medium. To find the equation for the propagation of the field E, we eliminate H between Equations 22.4 and 22.5. We take the curl of Equation 22.5 and substitute Equation 22.4 into the resulting equation to obtain

∇ × ∇ × + ∂∂

+ ∂∂

=( ) ( ) .EE Eµε µσ

2

20

t t (22.8)

Optics of Metals 473

Expanding the ∇ × (∇ × ) operator, we find that Equation 22.8 becomes

∇ = ∂∂

+ ∂∂

22

2E

E Eµε µσt t

. (22.9)

Equation 22.9 is the familiar wave equation modified by an additional term. From our knowledge of differential equations, the additional term described by ∂E/∂t corresponds to damping or attenu-ation of a wave. Equation 22.9 can be considered the damped or attenuated wave equation.

We proceed now with the solution of Equation 22.9. If the field is strictly monochromatic and of angular frequency ω so that E ≡ E(r,t) = E(r)exp(iωt), then substituting this form into Equation 22.9 yields

∇ ( ) = −( ) ( ) + ( ) ( )2 2 iE r E r E rµεω ω µσ , (22.10)

which can be written as

∇ = − −

2 2E r E r( ) ( ) ( ).µω ε σω

i (22.11)

In this form, Equation 22.11 is identical to the wave equation except that now the dielectric constant is complex, that is,

ε ε σω

= −

i , (22.12)

where ε is the real dielectric constant.The correspondence with nonconducting media is readily seen if ε is defined in terms of a com-

plex refractive index n (we set μ = 1 since we are not dealing with magnetic materials), so that

ε = ˆ .n2 (22.13)

We now express n in terms of the refractive index and the absorption of the medium. To find the form of n , which describes both the refractive and absorbing behavior of a propagating field, we first consider the intensity I(z) of the field after it has propagated a distance z. We know that the intensity is attenuated after a distance z has been traveled, so the intensity can be described by

I z I z( ) = −( )0exp α , (22.14)

where α is the attenuation or absorption coefficient. We wish to relate α to κ, the absorption or attenuation index. We first note that n is a dimensionless quantity, whereas we can see from Equation 22.14 that α has the dimension of inverse length. We can express αz as a dimensionless parameter by assuming that after a distance equal to a wavelength λ, the intensity has been reduced to

I Iλ πκ( ) = −( )0exp 4 . (22.15)

Equating the arguments of the exponents in Equations 22.14 and 22.15, we have

α πλ

κ κ=

=4

2k , (22.16)

474 Polarized Light, Third Edition

where k = 2π/λ is the wave number. Equation 22.14 can then be written as

I z I z( ) exp .= −

0

4πλ

κ (22.17)

From this result we can write the corresponding field E(z) as

E z E z( ) exp ,= −

0

2πλ

κ (22.18)

or

E z E k z( ) = −( )0exp κ . (22.19)

Thus, the field propagating in the z direction can be described by

E z E k z i t kz( ) = −( ) −( )[ ]0exp expκ ω . (22.20)

The argument of Equation 22.20 can be written as

i tk

z ik

z i tk

i zωω

κω

ωω

κ−

+

= − −

1. (22.21)

But k = ω/v = ωn/c, so the right-hand side of Equation 22.21 becomes

i tnc

i z i tnc

zω κ ω− −

= −

ˆ

,1 (22.22)

where

ˆ ( ).n n i= −1 κ (22.23)

Thus, the propagating field Equation 22.20 can be written in the form

E( ) expˆ

.z E i tnc

z= −

0 ω (22.24)

Equation 22.24 shows that conducting (i.e., absorbing) media lead to the same solutions as noncon-ducting media except that the real refractive index n is replaced by a complex refractive index n . Equation 22.23 relates the complex refractive index to the real refractive index and the absorption behavior of the medium, and will be used throughout the text.

We can relate n and κ to σ using Equations 22.12 through 22.23. We have

ε κ ε σω

= = − = −

ˆ ( ) ,n n i i2 2 21 (22.25)

which leads immediately to

n2 21−( ) =κ ε, (22.26)

Optics of Metals 475

and

nv

2

2 4κ σ

ωσπ

= = , (22.27)

where ν = ω/2π. We solve these equations to obtain

nv

2 2

212 4

= +

+

ε σ

πε , (22.28)

and

nv

2 2 2

212 4

κ ε σπ

ε= +

−

. (22.29)

Equations 22.28 and 22.29 are important because they enable us to relate the measured values of n and κ to the constants ε and σ of a metal or semiconductor. Because metals are opaque, it is not possible to measure these constants optically.

Since the wave equation for conducting media is identical to the wave equation for dielectrics, except for the appearance of a complex refractive index, we would expect the boundary conditions and all of their consequences to remain unchanged. This is indeed the case. Thus, Snell’s Law of refraction becomes:

sin sini rθ θ= ˆ ,n (22.30)

where the refractive index is now complex. Similarly, Fresnel’s Law of reflection and refraction con-tinue to be valid. Since optical measurements cannot be made with Fresnel’s refraction equations, only Fresnel’s reflection equations are of practical interest. We recall these equations are given by

R Esi r

i rs= − −

+sin( )sin( )

,θ θθ θ

(22.31)

R Epi r

i rp= −

+tan( )tan( )

.θ θθ θ

(22.32)

In Equations 22.31 and 22.32, θi is the angle of incidence and θr is the angle of refraction, and Rs, Rp, Es, and Ep have their usual meanings.

We now derive the equations for the reflected intensity using Equations 22.31 and 22.32. We consider Equation 22.31 first. We expand the trigonometric sum and difference terms, substitute sin sinθ θi rn= ˆ into the result, and find that

RE

nn

s

s

i r

i r

= −+

cos ˆ coscos ˆ cos

.θ θθ θ

(22.33)

We first use Equation 22.33 to obtain the reflectivity, that is, the normalized intensity at normal incidence. The reflectivity for the s polarization, Equation 22.33 is defined to be

Rss

s

RE

≡2

. (22.34)

476 Polarized Light, Third Edition

At normal incidence, θi = 0, so from Snell’s Law, Equation 22.30, θr = 0 and Equation 22.33 reduces to

RE

nn

s

s

= −+

11

ˆˆ

. (22.35)

Replacing n with the explicit form given by Equation 22.23 yields

RE

n inn in

s

s

= − ++ −

( )( )

.11

κκ

(22.36)

From the definition of the reflectivity Equation 22.34 we then see that Equation 22.36 yields

Rs

n nn n

= − ++ +

( ) ( )( ) ( )

.11

2 2

2 2

κκ

(22.37)

We observe that for nonabsorbing media (κ = 0), Equation 22.37 reduces to the well-known results for dielectrics. We also note that for this condition and for n = 1, the reflectivity is zero, as we would expect. In Figure 22.1, a plot of Equation 22.37 as a function of κ is shown. We see that for absorb-ing media with increasing κ, the reflectivity approaches unity. Thus, highly reflecting absorbing media (e.g., metals) are characterized by high values of κ.

In a similar manner, we can find the reflectivity for normal incidence for the p polarization, Equation 22.32. Equation 22.32 can be written as

R

Ep

p

i r i r

i r i

= − ++ −

sin( )cos( )sin( )cos(

θ θ θ θθ θ θ θrr )

. (22.38)

1.2

0.8

1

n = 2.0

0.6

n = 1.0

n = 1.5

0.4

Refle

ctan

cec

0.2

00 1 2 3 4 5 6 7 8 9 10

figuRe 22.1 Plot of the reflectivity as a function of κ. The refractive indices are n = 1.0, 1.5, and 2.0.

Optics of Metals 477

At normal incidence the cosine factor in Equation 22.38 is unity, and we are left with the same equa-tion for the s polarization, Equation 22.31, therefore

R Rp s= , (22.39)

and for normal incidence the reflectivity is the same for the s and p polarizations.We now derive the reflectivity equations for nonnormal incidence. We again begin with Equation

22.31 or, more conveniently, its expanded form, Equation 22.33, that is,

RE

nn

s

s

i r

i r

= −+

cos ˆ coscos ˆ cos

.θ θθ θ

(22.33)

Equation 22.33 is, of course, exact and can be used to obtain an exact expression for the reflectivity Rs, however, the result is quite complicated. We will derive an approximate equation, much quoted in the literature, for Rs that is sufficiently close to the exact result. We replace the factor cosθr by (1 – sin2θr)1/2 and use sin sinθ θi rn= ˆ . Then, Equation 22.33 becomes

RE

nn

s

s

i i

i i

= − −+ −

cos ˆ sincos ˆ sin

θ θθ θ

2 2

2 2.. (22.40)

Equation 22.40 can be approximated by noting that n i2 2sin θ , so Equation 22.40 can be writ-

ten as

RE

nn

s

s

i

i

= −+

cos ˆ

cos ˆ.

θθ

(22.41)

We now substitute Equation 22.23 into Equation 22.41 and group the terms into real and imaginary parts to obtain

RE

n inn in

s

p

i

i

= − ++ −

(cos )(cos )

.θ κθ κ

(22.42)

The reflectivity Rs is then

Rsi

i

n nn n

= − ++ −

( cos ) ( )( cos ) ( )

.θ κθ κ

2 2

2 2 (22.43)

We now develop a similar, approximate, equation for Rp. We first write Equation 22.32 as

RE

s

p

i r i r

i r i

= − ++ −

sin( )cos( )sin( )cos(

θ θ θ θθ θ θ θrr)

. (22.38)

The first factor is identical to Equation 22.31, so it can be replaced by its expanded form Equation 22.33, that is,

sin( )sin( )

cos ˆ coscos ˆ c

θ θθ θ

θ θθ

i r

i r

i r

i

nn

−+

= −+ oos

.θr

(22.44)

478 Polarized Light, Third Edition

The second factor in Equation 22.38 is now expanded, and again we use cosθr = (1 – sin2θr)1/2 and sin sinθ θi rn= ˆ to obtain

cos( )cos( )

cos ˆ sin sinθ θθ θ

θ θ θi r

i r

i i in+−

= − −2 2 2

ccos ˆ sin sin.

θ θ θi i in2 2 2− + (22.45)

Because ˆ sinn i2 2 θ , Equation 22.45 can approximated as

cos( )cos( )

ˆ cos sinˆ cos

θ θθ θ

θ θθ

i r

i r

i i

i

nn

+−

≅ −+

2

ssin.

2 θi

(22.46)

We now multiply Equation 22.44 by Equation 22.46 to obtain

R

Enn

nn

p

p

i

i

i i= −+

−cos ˆ

cos ˆˆ cos sinˆ co

θθ

θ θ2

ss sin.

θ θi i+

2

(22.47)

Carrying out the multiplication in Equation 22.47, we find that there is a sin2θi cos θi term. This term is always much smaller than the other terms and can be dropped. The remaining terms then lead to

R

En

np

p

i

i

= − ++

ˆ cosˆ cos

,θ

θ1

1 (22.48)

or

R

Enn

p

p

i

i

= − −+

cos / ˆ

cos / ˆ.

θθ

11

(22.49)

Replacing n by n(1 – iκ), grouping terms into real and imaginary parts, and ignoring the negative sign because it will vanish when we determine the reflectivity, gives

R

En inn in

p

p

i

i

= − −+ +

( / cos )( / cos )

.11

θ κθ κ

(22.50)

Multiplying Equation 22.50 by its complex conjugate, we obtain the reflectivity

Rpi

i

n nn n

= − ++ +

( / cos ) ( )( / cos ) ( )

.11

2 2

2 2

θ κθ κ

(22.51)

For convenience, we repeat the equation for Rs

Rsi

i

n nn n

= − ++ +

( cos ) ( )( cos ) ( )

.θ κθ κ

2 2

2 2 (22.43)

In Figures 22.2 through 22.5, plots are shown for the reflectivity as a function of the incidence angle θi of gold (Au), silver (Ag), copper (Cu), and platinum (Pt), using Equations 22.43 and 22.51. The values for n and κ are taken from Wood’s classic text Physical Optics [4].

In Figures 22.2 through 22.5, we observe that the p reflectivity has a minimum value. This mini-mum is called the pseudo-brewster angle minimum because, unlike the Brewster angle for dielectrics,

Optics of Metals 479

0.9

1

p

s

0.7

0.8

0.5

0.6

0.3

0.4Refle

ctan

ce

0.2

0

0.1

0 10 20 30 40 50 60 70 80 90θi

figuRe 22.3 Reflectance of silver (Ag) as a function of incidence angle. The refractive index and the absorption index are 0.18 and 20.2, respectively. The normal reflectance value is 0.951.

0.9

1

s

0.7

0.8p

0.5

0.6

0.3

0.4Refle

ctan

ce

0.2

0

0.1

0 10 20 30 40 50θi

60 70 80 90

figuRe 22.2 Reflectance of gold (Au) as a function of incidence angle. The refractive index and the absorp-tion index are 0.36 and 7.70, respectively. The normal reflectance value is 0.849.

480 Polarized Light, Third Edition

0.9

1

0.7

0.8 s

0.5

0.6 p

0.3

0.4Refle

ctan

ce

0.2

0

0.1

0 10 20 30 40 5 60 70 80 90θi

figuRe 22.4 Reflectance of copper (Cu) as a function of incidence angle. The refractive index and the absorption index are 0.64 and 4.08, respectively. The normal reflectance value is 0.731.

0.9

1qi

0.7

0.8 s

0.5

0.6 p

0.3

0.4Refle

ctan

ce

0.2

0

0.1

0 10 20 30 40 50 60 70 80 90θi

figuRe 22.5 Reflectance of platinum (Pt) as a function of incidence angle. The refractive index and the absorption index are 2.06 and 2.06, respectively. The normal reflectance value is 0.699.

Optics of Metals 481

the intensity does not go to zero for metals. Nevertheless, a technique based on this minimum has been used to determine n and κ. The interested reader is referred to the article by Potter [5].

We see that the refractive index can be less than unity for many metals. Born and Wolf [6] have shown that this is a natural consequence of the simple classical theory of the electron and the dis-persion theory. The theory provides a theoretical basis for the behavior of n and κ. Further details on the nature of metals and, in particular, the refractive index and the absorption index (n and κ) as it appears in the dispersion theory of metals can be found in the reference texts by Born and Wolf, and by Mott and Jones [7].

22.3 PRiNCiPal aNgle of iNCideNCe meaSuRemeNT of RefRaCTiVe iNdeX aNd abSoRPTioN iNdeX of oPTiCally abSoRbiNg maTeRialS

In the previous section, we saw that optically absorbing materials are characterized by a real refrac-tive index n and an absorption index κ. Because these constants describe the behavior and perfor-mance of optical materials such as metals and semiconductors, it is very important to know these constants over the entire optical spectrum.

Methods have been developed to measure the optical constants. One of the best known is the principal angle of incidence method. The basic idea is as follows. Incident + 45° linearly polarized light is reflected from an optically absorbing material. In general, the reflected light is elliptically polarized; the corresponding polarization ellipse is in nonstandard form. The angle of incidence of the incident beam is now changed until a phase shift of 90° is observed in the reflected beam. The incident angle where this takes place is called the principal angle of inci-dence. Its significance is that, at this angle, the polarization ellipse for the reflected beam is now in its standard form. From this condition, relatively simple equations can then be found for n and κ. Because the polarization ellipse is now in its standard form, the orthogonal field compo-nents are parallel and perpendicular to the plane of incidence. The reflected beam is now passed through a quarter-wave retarder. The beam of light that emerges is linearly polarized with its azimuth angle at an unknown angle. The beam then passes through an analyzing polarizer that is rotated until a null intensity is found. The angle at which this null takes place is called the prin-cipal azimuth angle. From the measurement of the principal angle of incidence and the principal azimuth angle, the optical constants n and κ can then be determined. In Figure 22.6, we show the measurement configuration.

i i

Sample

Linear polarizer at +45°

Source

Quarter-wave retarder

Analyzer

figuRe 22.6 Measurement of the principal angle of incidence and the principal azimuth angle.

482 Polarized Light, Third Edition

To derive the equations for n and κ, we begin with Fresnel’s reflection equations for absorbing media, that is,

R Esi r

i rs= − −

+sin( )sin( )

,θ θθ θ

(22.31)

R Epi r

i rp= −

+tan( )tan( )

.θ θθ θ

(22.32)

The angle θr is now complex, so the ratios Rp /Ep and Rs /Es are also complex. Thus, the amplitude and phase change on being reflected from optically absorbing media. Incident polarized light will, in general, become elliptically polarized on reflection from an optically absorbing medium. We now let ϕp and ϕs be the phase changes, and let ρp and ρs be the absolute values of the reflection coef-ficients rp and rs. Then we can write

rR

Eip

p

pp p= = ρ φexp( ), (22.52)

rRE

iss

ss s= = ρ φexp( ). (22.53)

Equations 22.52 and 22.53 can be transformed to the Stokes parameters. The Stokes parameters for the incident beam are

S E E E Ei s s p p0 = +cos ( ),* *θ (22.54)

S E E E Ei s s p p1 = −cos ( ),* *θ (22.55)

S E E E Ei s p p s2 = +cos ( ),* *θ (22.56)

S i E E E Ei s p p s3 = −cos ( ).* *θ (22.57)

Similarly, the Stokes parameters for the reflected beam are defined as

′ = +S R R R Ri s s p p0 cos ( ),* *θ (22.58)

′ = −S R R R Ri s s p p1 cos ( ),* *θ (22.59)

′ = +S R R R Ri s p p s2 cos ( ),* *θ (22.60)

′ = −S i R R R Ri s p p s3 cos ( ).* *θ (22.61)

Using Equation 22.52 through Equation 22.61, we obtain the relationship between the Stokes param-eters for the incident and reflected beams as

′′′′

=

+ −S

S

S

S

s p s p0

1

2

3

2 2 2 2

12

0 0ρ ρ ρ ρρρ ρ ρ ρ

ρ ρ ρ ρρ

s p s p

s p s p

s

2 2 2 2 0 0

0 0 2 2

0 0 2

− +∆ ∆

−cos sin

ρρ ρ ρp s p

S

S

S

Ssin cos∆ ∆

2

0

1

2

3

, (22.62)

Optics of Metals 483

where Δ = ϕs – ϕp.We now allow the incident light to be + 45° linearly polarized so that Ep = Es. Furthermore, we

introduce an azimuthal angle α (generally complex) for the reflected light, which is defined by

tancos( )cos( )

exp( ),α θ θθ θ

= = − −+

= ∆RR

P is

p

i r

i r

(22.63)

where we have used Equation 22.31 and Equation 22.32, and P is real and we write it as

P = tan ψ, (22.64)

where ψ is called the azimuthal angle. From Equation 22.52 to Equation 22.53, we also see that

P s

ps p= ∆ = −ρ

ρφ φ . (22.65)

We note that α is real in the following two cases:

1. For normal incidence, that is, θi = 0; from Equation 22.63, we see that Ρ = 1 and Δ = π. 2. For grazing incidence, that is, θi = π/2; from Equation 22.63 we see that Ρ = 1 and Δ = 0.

Between these two extreme values there exists an angle θi , called the principal angle of incidence for which Δ = π/2. Let us now see the consequences of obtaining this condition. We first write Equation 22.65 as

ρ ρs p= P . (22.66)

Substituting Equation 22.66 into Equation 22.62, we obtain the Stokes vector of the reflected light to be

′′′′

=

+ − −−

S

S

S

S

P P

p

0

1

2

3

2

2 2

2

1 1 0 0

ρ( )

(( ) ( )

cos sin

sin cos

1 1 0 0

0 0 2 2

0 0 2 2

2 2− +∆ ∆

− ∆

P P

P P

P P ∆∆

S

S

S

S

0

1

2

3

. (22.67)

For incident + 45° linearly polarized light, the Stokes vector is

S

S

S

S

I

0

1

2

3

0

1

0

1

0

=

. (22.68)

Substituting Equation 22.68 into Equation 22.67, we find the Stokes vector of the reflected light to be

′′′′

=

+− −

S

S

S

S

I

P

Pp

0

1

2

3

20

2

2

2

1

1

2

ρ ( )

PP

P

cos

sin

.∆

− ∆

2

(22.69)

484 Polarized Light, Third Edition

The ellipticity angle χ is

χ = ′′

= − ∆

+

− −12

12

21

1 3

0

12

sin sinsin

.SS

PP

(22.70)

Similarly, the orientation angle ψ is

ψ = − ∆−

−12

21

12

tancos

.P

P (22.71)

We see that χ is greatest when Δ = π/2, but then ψ = 0, that is, the polarization ellipse correspond-ing to Equation 22.69 is in its standard, nonrotated, form. For Δ = π/2, the Stokes vector, Equation 22.69, becomes

′′′′

=

+− −

S

S

S

S

I

P

Pp

0

1

2

3

20

2

2

2

1

1

0

ρ ( )

−−

2P

, (22.72)

and χ and ψ of the polarization ellipse corresponding to Equation 22.72 are

χ = ′′

= −

+

− −12

12

21

1 3

0

12

sin sin ,SS

PP

(22.73)

ψ = ′′

=−1

201 2

1

tan .SS

(22.74)

We must now transform the elliptically polarized light described by the Stokes vector Equation 22.72 to linearly polarized light. A quarter-wave retarder can be used to transform elliptically polarized light to linearly polarized light. The Mueller matrix for a quarter-wave retarder oriented at 0° is

M =

−

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

. (22.75)

Multiplying Equation 22.72 by Equation 22.75 yields

′′′′

=

+− −

−

S

S

S

S

I

P

Pp

0

1

2

3

20

2

2

2

1

1ρ ( )

22

0

P

, (22.76)

Optics of Metals 485

which is the Stokes vector for linearly polarized light. The Mueller matrix for a linear polarizer at an angle β is

M = 12

1 2 2 0

2 2 2 2 0

2

2

cos sin

cos cos sin cos

sin

β ββ β β ββ ssin cos sin

.2 2 2 0

0 0 0 0

2β β β

(22.77)

Multiplying Equation 22.76 by Equation 22.77, we obtain the intensity of the beam emerging from the analyzing polarizer as, ignoring the constant factor in Equation 22.76,

I P P P( ) ,β β β= +( ) − −( ) −1 1 cos2 2 sin22 2 (22.78)

or

I A b cβ β β( ) = − −cos2 sin2 , (22.79)

where A = 1 + P2, b = 1 – P2, and c = 2P.Equation 22.79 is now written as

I AbA

cA

( ) cos sin .β β β= − −

1 2 2 (22.80)

We set

cos ,γ = bA

(22.81)

sin ,γ = cA

(22.82)

so that Equation 22.80 can now be written as

I Aβ γ β( ) = − −( )[ ]1 2cos , (22.83)

and

γ =

=

−

− −tan tan .1 12

21

cb

PP

(22.84)

A null intensity for Equation 22.83 is obtained when

γ β β γ= =2 or2

. (22.85)

486 Polarized Light, Third Edition

The azimuthal angle where the null intensity occurs is called the principal azimuthal angle ψ . We can relate Equation 22.85 to the principal azimuthal angle ψ. We recall from Equation 22.64 that

P = tan ψ. (22.64)

Substituting Equation 22.64 into Equation 22.84, we find that

γ ψ β=

= =− −tan tan [tan ] ,1 1 2 2

cb

(22.86)

or

β ψ= . (22.87)

The magnitude of P is then

P s

p

= =ρρ

ψtan . (22.88)

It is possible to obtain the same results by irradiating the sample surface with circularly polarized light rather than linearly polarized light. We remove the quarter-wave retarder from the analyzing arm (see Figure 22.6) and insert it between the +45° linear polarizer and the optical sample in the generating arm. The Stokes vector of the beam emerging from the linear polarizer in the generating arm is

S

S

S

S

I

0

1

2

3

0

1

0

1

0

=

. (22.89)

Multiplying Equation 22.89 by the Mueller matrix for a quarter-wave retarder oriented at 0°, Equation 22.75, we obtain the Stokes vector for left circularly polarized light, that is,

S

S

S

S

I

0

1

2

3

0

1

0

0

1

=

−

. (22.90)

The Stokes vector Equation 22.90 is now used in the equation for the light reflected from the sample, Equation 22.67, whereupon the Stokes vector of the reflected beam is found to be

′′′′

=

+− −−

S

S

S

S

I

P

Pp

0

1

2

3

20

2

2

2

1

1ρ ( )

22

2

P

P

sin

cos

.∆

− ∆

(22.91)

Optics of Metals 487

At the principal angle of incidence, Δ = π/2, so Equation 22.91 reduces to

′′′′

=

+− −

−

S

S

S

S

I

P

Pp

0

1

2

3

20

2

2

2

1

1ρ ( )

22

0

P

, (22.92)

which is identical to the Stokes vector found in Equation 22.76. Thus, the quarter-wave retarder can be inserted into either the generating or analyzing arm, because the phase shift of π/2 can be gener-ated before or after the reflection from the optical surface.

We also point out that a quarter-wave retarder can also transform elliptically polarized light to linearly polarized light if the polarization ellipse is in its standard form. However, the orientation angle is different from 45°. To see this clearly, let us represent the Stokes vector of elliptically polar-ized light in its “standard” form, that is,

S

S

S

S

A

b

d

0

1

2

3

0

=

. (22.93)

The Mueller matrix of a quarter-wave retarder is

M =

−

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

. (22.75)

Multiplying Equation 22.93 by Equation 22.75 yields

S

S

S

S

A

b

d

0

1

2

3 0

=

, (22.94)

which is a Stokes vector for linearly polarized light. However, the polarization ellipse is now ori-ented at an angle ψ given by

ψ =

−12

1tan .db

(22.95)

We now relate the principal angle of incidence θ πi ( / )∆ = 2 and the principal azimuthal angle ψ to the optical constants n and κ. We recall that

tancos( )cos( )

exp( ),α θ θθ θ

= = − −+

= ∆RR

P is

p

i r

i r

(22.63)

488 Polarized Light, Third Edition

P s

ps p= ∆ = −ρ

ρφ φ . (22.65)

We expand Equation 22.63 so that

P i i r i r

i r

exp( )cos cos sin sincos cos s

∆ = − +−

θ θ θ θθ θ iin sin

tan tantan tan

.θ θ

θ θθ θi r

i r

i r

= +−

11

(22.96)

Solving Equation 22.96 for tan θi tan θr and using Snell’s Law gives

11

+ ∆− ∆

= − = −P iP i i r

iexp( )exp( )

tan tantan sinθ θ θ θii

in sin.

2 2− θ (22.97)

At the principal angle of incidence, Δ = π/2. Furthermore, sin2 2θi n and may be disregarded. Equation 22.97 becomes

11 1

+−

= −−

iPiP n i

i isin tan( )

,θ θ

κ (22.98)

where θi is the principal angle. Multiplying Equation 22.98 by its complex conjugate leads imme-diately to

n i i1 2+ =κ θ θsin tan . (22.99)

Equation 22.99 serves as a very useful check on n and κ.We now establish the relations between n and κ and θi and ψ . First, we invert Equation 22.98

and obtain

11

1−+

= − −iPiP

n i

i i

( )sin tan

.κ

θ θ (22.100)

Next, we replace P by tanψ. Multiplying the numerator and denominator of the left-hand side of Equation 22.100 by 1 tan−( )i ψ and equating real and imaginary parts, we find that

n

i isin tantantan

,θ θ

ψψ

= − −+

11

2

2 (22.101)

n

i i

κθ θ

ψψsin tan

tantan

.= −+2

1

2

2 (22.102)

The right-hand sides of Equations 22.101 and 22.102 reduce to cos2ψ and sin2ψ , respectively. This leads immediately to

n i i= −sin tan cos ,θ θ ψ2 (22.103)

Optics of Metals 489

κ ψ= tan .2 (22.104)

We can substitute Equations 22.103 and 22.104 into the expression for the index, Equation 22.23 and find that

ˆ sin tan exp( ).n ii i= − −θ θ ψ2 (22.105)

Thus, by measuring the principal angle of incidence θi , and the principal azimuthal angle ψ , we can determine n and κ from Equations 22.103 and 22.104, respectively.

In the present formulation of relating n and κ to θi and ψ , the term sin2θi has been neglected. Interestingly, as pointed out by Wood [4], the inclusion of sin2θi leads to the same equations.

Further information on the principal angle of incidence method is given in the textbooks by Born and Wolf [6], Wood [4], and Longhurst [8]. For example, Wood also describes the application of the method to the measurement of optical materials in the ultraviolet region of the optical spectrum.

22.4 meaSuRemeNT of RefRaCTiVe iNdeX aNd abSoRPTioN iNdeX aT aN iNCideNT aNgle of 45°

In the previous section, we saw that the principal angle of incidence method can be used to obtain the optical constants n and κ. We also pointed out that there is another method known as the pseudo-Brewster angle method; this method is described by Potter [5]. The classical Brewster angle method, we recall, leads to a null intensity at the Brewster angle for dielectrics. For absorbing opti-cal materials, however, one can show that a minimum intensity is obtained instead; this is indicated in Figures 22.2 through 22.5. From a measurement of the minimum intensity, n and κ can then be found. The Brewster angle method is very useful. However, a broad minimum is obtained, and this limits the accuracy of the results to only two or three decimal places at most.

All of these classical methods are based on a null or minimum intensity condition and reading of mechanical dials; consequently, these methods can be called optomechanical; that is, only optics and mechanical components are used to determine n and κ. While these methods have long been the standard means for determining n and κ, they have a number of disadvantages. The first and most serious is that a mechanical arm must be used and moved to find the appropriate angle, for example, the principal angle of incidence or the Brewster angle. Very often, apparatus to do this, such as a divided circle, is not readily available. Furthermore, a mechanical divided circle is quite expensive. Another disadvantage is that it is time consuming to move the mechanical arm and search for a null or minimum intensity. In addition, automating the movement of a mechanical arm is difficult and expensive. Finally, these measurement methods do not utilize to any significant extent the develop-ments made in electronics and optical detectors in recent years.

Ideally, it would be preferable if n and κ could be measured without any mechanical movement whatsoever, especially with respect to moving a mechanical arm. This can indeed be done by irra-diating the optical surface at an incident angle of 45°. At this angle, Fresnel’s equations reduce to relatively simple forms, and the measurement of the reflected intensity can be easily made with an optical detector and a digital voltmeter. Mechanical fixed mounts are, of course, still necessary, but there are no major mechanical movements. Furthermore, the required mechanical and optical components are nearly always available in any modern optical laboratory. In addition, because the angles involved are 45°, and the components are aligned perpendicularly to each other, these mea-surements are easily carried out on an optical table. Finally, a digital voltmeter capable of reading to, say, 5½ digits, is relatively inexpensive. In this method, therefore, the optical constants are derived by using only quantitative detectors and readings on a digital voltmeter rather than a mechanical dial. In fact, the four Stokes parameters must be measured, but these measurements are made at

490 Polarized Light, Third Edition

settings of 0°, 45°, and 90°, so that searching for a null is not required. Consequently, this measure-ment method can be called optoelectronic. It has been called digital refractometry [3]. Therefore, we consider Fresnel’s equations for reflection at an incident angle of 45°. From the measurement of the Stokes parameters of the reflected beam, n and κ can then be determined. We now derive the equations that relate the Stokes parameters to n and κ at an incident angle of 45°.

Figure 22.7 shows the incident field components Ep and Es, and the reflected field components Rp and Rs. For absorbing optical materials, Fresnel’s reflection equations continue to hold, so

R Esi r

i rs= − −

+sin( )sin( )

,θ θθ θ

(22.31)

R Epi r

i rp= −

+tan( )tan( )

.θ θθ θ

(22.32)

Absorbing optical media are characterized by a complex refractive index n of the form

ˆ .n n= −( )1 iκ (22.23)

When θI = 45°, a relatively simple form of Fresnel’s equations emerges. Snell’s Law of refraction continues to be valid for media described by Equation 22.23, so we have

sinsin

ˆ.θ θ

ri

n= (22.106)

Equation 22.106 can be expressed in terms of cos θr as

cosˆ sin

ˆ.θ θ

rin

n= −2 2

(22.107)

i i

r

Ep

Es

Rp

Rs

Ts

Tp

figuRe 22.7 Optical field components for the incident and reflected fields.

Optics of Metals 491

For an incident angle of 45°, Equation 22.106 becomes

sinˆ

,θr n= 1

2 (22.108)

and Equation 22.107 becomes

cosˆ

ˆ.θr

nn

= −2 12

2

(22.109)

For an incident angle of 45°, Equations 22.31 and 22.32 reduce to

R Esr r

r rs= − −

+

cos sincos sin

,θ θθ θ

(22.110)

R Epr r

r rp= −

+

cos sincos sin

.θ θθ θ

2

(22.111)

Replacing the cosine and sine terms using Equations 22.108 and 22.109, we can write Equation 22.110 as

Rnn

Es s= − − −− +

2 1 12 1 1

2

2

ˆˆ

. (22.112)

In a similar manner the equation for Rp, Equation 22.111 becomes

Rnn

Ep p= − −− +

2 1 12 1 1

2

2

2ˆˆ

. (22.113)

We now set

2 12ˆ exp( ).n a ib A i− = − = − φ (22.114)

Equations 22.112 and 22.113 can then be written, using Equation 22.114, as

Ra iba ib

Es s= − − −+ −

( )( )

,11

(22.115)

Ra iba ib

Ep p= − −+ −

( )( )

.11

2

(22.116)

Equations 22.112 and 22.113 can be written also in terms of A and ϕ, as in Equation 22.114. Straightforward substitution gives

RA iA i

Es s= − −+ −

11

exp( )exp( )

,φφ

(22.117)

492 Polarized Light, Third Edition

RA iA i

Ep p= − −+ −

11

2exp( )exp( )

.φφ

(22.118)

From Equation 22.114 we have

A a b2 2 2= + , (22.119)

φ =

−tan .1 ba

(22.120)

A and ϕ can also be expressed in terms of n and κ. Using the left and center expressions from Equation 22.114, and substituting Equation 22.23 for n and then squaring both sides of the equa-tion leads to

( ) ( ) .2 2 1 4 22 2 2 2 2 2n n i n a b i ab− − − = − − ( )κ κ (22.121)

Equating real and imaginary terms we have

a b n n2 2 2 2 22 2 1− = − −κ , (22.122)

ab n= 2 2κ. (22.123)

We can also find an expression for a2 + b2. We take, from Equation 22.114, the complex conjugate of

a ib n− = −2 12ˆ , (22.124)

as

a ib n ik+ = + −2 1 12 2( ) . (22.125)

We also have from Equation 22.124 that

a ib n ik− = − −2 1 12 2( ) . (22.126)

Multiplying Equation 22.125 by Equation 22.126 gives us

A a b n n2 2 2 4 2 2 2 24 1 4 1 1= + = + − − +( ) ( ) .κ κ (22.127)

Adding and subtracting Equation 22.122 and Equation 22.127 yields

a n n n n2 4 2 2 2 2 2 2 212

4 1 4 1 1 2 2 1= + − − + + − −[ ]( ) ( ) ( )κ κ κ ,, (22.128)

b n n n n2 4 2 2 2 2 2 2 212

4 1 4 1 1 2 2 1= + − − + − − −[ ]( ) ( ) ( )κ κ κ .. (22.129)

Optics of Metals 493

Then, from Equations 22.120, 22.128, and 22.129, we see that

φ κ κ κ= + − − + − − −( )−tan( ) ( )1

4 2 2 2 2 2 2 24 1 4 1 1 2 2 1n n n n44 1 4 1 1 2 2 14 2 2 2 2 2 2 2n n n n( ) ( )+ − − + + − −( )

κ κ κ

11 2/

. (22.130)

For nonabsorbing materials, κ = 0, so Equations 22.127 and 22.130 reduce to

A a n2 2 22 1 0= = − =and φ , (22.131)

as expected.We must now transform the amplitude equations 22.117 and 22.118 to intensity equations, and

from these derive the Stokes polarization parameters. We defined the Stokes parameters of the incident and reflected beams in Equations 22.54 through 22.61. Substituting Equations 22.117 and 22.118 into Equation 22.58 through Equation 22.61 and using Equation 22.54 through Equation 22.57 yields

′′′′

= − ++

S

S

S

S

A AA

0

1

2

3

21 21 2

cos( co

φss )

cos

cos

s

φ

φφ

+

×

++

− −

A

A A

A A

A A

2 2

2

2

2

1 2 0 0

2 1 0 0

0 0 1 2 iin

sin

φφ0 0 2 1 2

0

1

2

3A A

S

S

S

S−

.

(22.132)

The 4 × 4 matrix is the Mueller matrix for optically absorbing materials at an incident angle of 45°. The presence of the off-diagonal terms in the upper and lower parts of the matrix shows that optically absorbing materials simultaneously change the amplitude and phase of the incident beam. To determine n and κ, we measure A and ϕ and solve Equations 22.127 and 22.130 for n and κ by iteration. It is straightforward to show that Equation 22.132 reduces to the equation for dielectrics by setting κ = 0, that is,

′′′′

= −+

S

S

S

S

r

r

0

1

2

3

1 21 2

sin( sin )

θθ 22

1 2 0 0

2 1 0 0

0 0 2 0

0 0 0 2

sin

sin

cos

cos

θθ

θθ

r

r

r

r

−−

S

S

S

S

0

1

2

3

. (22.133)

We can derive an important relation between the intensity of an incident beam, I0, and the orthogo-nal intensities of the reflected beam, Is and Ip. Consider that we irradiate the surface of an optically absorbing material with a linear vertically polarized beam of intensity I0; we call this the p polar-ized beam, and its Stokes vector is

S

S

S

S

I

0

1

2

3

0

1

1

0

0

=−

. (22.134)

494 Polarized Light, Third Edition

Multiplying Equation 22.132 with Equation 22.134 substituted for the incident Stokes vector gives

I IA AA Ap = + −

+ +0

2 2

2 2

1 21 2

( cos )( cos )

.φφ

(22.135)

Next, we irradiate the optical surface with a linearly horizontally polarized beam; we call this the s polarized beam. Its Stokes vector is

S

S

S

S

I

0

1

2

3

0

1

1

0

0

=

. (22.136)

Multiplying Equation 22.132 with Equation 22.136 substituted for the incident Stokes vector yields

I IA AA As = + −

+ +0

2

2

1 21 2

( cos )( cos )

.φφ

(22.137)

From Equation 22.135 to Equation 22.137, we find that

I I Is p2

0= . (22.138)

Equation 22.138 is a fundamental relation. It is the intensity analog of the amplitude relation

R Rs p2 = . (22.139)

Equation 22.138 shows that it is incorrect to square Equation 22.139 in order to obtain Equation 22.138; the correct relation includes the source intensity I0. From an experimental point of view, Equation 22.138 is very useful because it shows that by measuring the orthogonal intensities of the reflected beam, the source intensity can be directly monitored or measured. Similarly, if I0 is known, then Equation 22.138 serves as a useful check on the measurement of Is and Ip.

We now turn to the measurement of A and ϕ in Equation 22.132. Let us irradiate the optical sur-face with an optical beam of intensity I0 that is right circularly polarized. The Stokes vector of the incident beam is then

S

S

S

S

I

0

1

2

3

0

1

0

0

1

=

. (22.140)

Performing the multiplication in Equation 22.132 with Equation 22.140 substituted for the incident Stokes vector, we find that the Stokes vector of the reflected beam is

′′′′

= − ++

S

S

S

S

IA AA

0

1

2

3

0

21 21 2

cos(

φccos )

cos

sinφφφ+

+

−−

A

A

A

A

A

2 2

2

2

1

2

2

1

.. (22.141)

Optics of Metals 495

The reflected beam is elliptically polarized. We can determine the quantities A2 and ϕ directly from measuring the Stokes parameters. Dividing ′S3 by ′S0 , we find that

AS SS S

2 0 3

0 3

= ′ − ′′ + ′

. (22.142)

Dividing ′S2 by ′S1 gives

φ = − ′′

−tan .1 2

1

SS

(22.143)

We also see that the ellipticity angle is

χ

κ

= −+

= − + −

−

−

12

11

12

1 4 1 4

12

2

14 2 2

sin

sin( )

AA

n nnn n

2 2

4 2 2 2 2

1 11 4 1 4 1 1

( )( ) ( )

.− +

+ + − − +

κκ κ

(22.144)

The orientation angle ψ is, using Equation 22.130,

ψ φ κ κ= − = − + − − + − −−

214

4 1 4 1 1 2 212 2 2 2 2 2

tan( ) ( ) (n n n nn

n n n n

2 2

2 2 2 2 2 2 2 2

14 1 4 1 1 2 2 1

κκ κ κ

−+ − − + + − −

)( ) ( ) ( ))

./

1 2

(22.145)

For the condition where we have no absorption (κ = 0), χ and ψ become

χ = −

−12

112

2sin ,

nn

(22.146)

ψ = 0, (22.147)

as expected. To determine A2 and ψ uniquely, we must measure all four Stokes parameters. Before we describe an experimental configuration for carrying out the measurement, we relate the above equations to another commonly used representation, the reflection coefficients representation.

The reflection coefficients are defined [3] to be

rRE

ess

ss

i s= = ρ γ , (22.148)

rR

Eep

p

pp

is p

p= = = −ρ γ γ γγ . (22.149)

From the definitions of the Stokes parameters given by Equations 22.54 through 22.61, the ampli-tude equations 22.148 and 22.149 are found to transform as

′′′′

=

+ −S

S

S

S

s p s p0

1

2

3

2 2 2 2

12

0 0ρ ρ ρ ρρρ ρ ρ ρ

ρ ρ γ ρ ρ γρ

s p s p

s p s p

s

2 2 2 2 0 0

0 0 2 2

0 0 2

− +

−cos sin

ρρ γ ρ ρ γp s p

S

S

S

Ssin cos2

0

1

2

3

. (22.150)

496 Polarized Light, Third Edition

We can relate the coefficients in Equation 22.150 to A and ψ in Equation 22.141 by irradiating the surface with right circularly polarized light. The respective Stokes parameters of the reflected beam are

12

1 21 2

2 22

2 2ρ ρ φ

φs p

A AA A

+[ ] = + −+ +

cos( cos )

(( ),1 2+ A (22.151)

12

1 21 2

2 22

2 2ρ ρ φ

φs p

A AA A

−[ ] = + −+ +

cos( cos )

(( cos ),2A φ (22.152)

ρ ρ γ φφs p

A AA A

sincos

( cos )(= + −

+ +

−1 21 2

22

2 2AAsin ),φ (22.153)

ρ ρ γ φφs p

A AA A

coscos

( cos )(= + −

+ +

−1 21 2

12

2 2AA2). (22.154)

Adding Equations 22.151 and 22.152 gives

ρ φφs

sII

A AA A

2

0

2

2

1 21 2

= = + −+ +

coscos

, (22.155)

and subtracting Equation 22.152 from Equation 22.151 gives

ρ φφp

pI

IA AA A

2

0

2 2

2 2

1 21 2

= = + −+ +

( cos )( cos )

. (22.156)

The relation for γ in terms of A and ϕ is then obtained by dividing Equation 22.153 by Equation 22.154 so that

tan sin .γ φ= −−

21 2

AA

(22.157)

We see that the reflection coefficients in Equations 22.155 and 22.156 are identical to the ratio of the orthogonal intensities, Equations 22.135 and 22.137, of the reflected beam. We also see from Equations 22.155 and 22.156 that

ρ ρs p4 4= , (22.158)

in agreement with our previous observations.Figure 22.8 shows a block diagram of the experimental configuration for measuring n and κ. In

this measurement, a He–Ne laser of wavelength 6328 Å is used as the optical source. The optical beam emerging from the laser is expanded and collimated; this creates a plane wave. In addition, an improved signal-to-noise ratio is obtained by chopping the beam. The frequency at which the beam is chopped is then used as a reference signal for a lock-in amplifier. The circular polarizer before the sample creates a circularly polarized beam that then irradiates the optical surface at an incident angle of 45°. The autocollimator is used to align the optical surface of the sample being measured

Optics of Metals 497

to exactly 45°. The reflected beam is then analyzed by a circular polarizer in order to obtain the four Stokes parameters in accordance with polarimetry techniques described previously. The beam that emerges from the circular polarizer in the analyzing path is incident on a silicon detector. The chopped voltage signal is then fed to the lock-in amplifier along with the reference signal. The lock-in amplifier consists, essentially, of a phase-sensitive detector along with an RC network to smooth the output d.c. (analog) voltage. The d.c. voltage is then converted into a digital voltage by a digital voltmeter, for example, a 5-1/2 digit voltmeter. The 5½ means that the minimum voltage that can be displayed or read is five digits after the decimal point. The 1/2 term means that the number to the left of the decimal point can vary from 0 to 1 for an average of (1 + 0)/2 = 1/2. If we have a volt-age greater than 1.99999 V, then the maximum displayed reading can only be read to four decimal places, for example, 2.1732 V. The digital output is read by a digital computer, and the values of A2 and ϕ are then calculated from Equations 22.142 and 22.143, respectively.

The optical constants n and κ are calculated from the values of A2 and ϕ in Equations 22.127 and 22.130. For convenience we repeat the equations here as

A a b n n2 2 2 4 2 2 2 24 1 4 1 1= + = + − − +( ) ( ) ,κ κ (22.127)

φ κ κ κ= + − − + − − −−tan( ) ( ) ( )1

4 2 2 2 2 2 2 24 1 4 1 1 2 2 1n n n n44 1 4 1 1 2 2 14 2 2 2 2 2 2 2n n n n( ) ( ) ( )+ − − + + − −

κ κ κ

11 2/

. (22.130)

To determine n and κ, we first estimate these values. This is most easily done from the plots of Equations 22.127 and 22.130 in Figures 22.9 and 22.10.

Laser

Chopper

Linear polarizer

Linear polarizer

Quarter-wave retarder

Sample

Autocollimato

r

Silicon detector

Reference signalLock-in

amplifier

5-1/2 digit digital voltmeter

Computer

45°

45°

figuRe 22.8 Experimental configuration for measuring the Stokes parameters and the optical constants n and κ of an optically absorbing material.

498 Polarized Light, Third Edition

Inspecting Figures 22.9 and 22.10, we observe that small values of κ yield small values of A2 and relatively small phase shifts ϕ. This information is very useful for determining the approximate value of the complex refractive index.

Let us now consider two examples of the determination of n and κ. In the first example, a sample is measured, and its normalized Stokes vector is

0

20

40

60

80

100

120

140

0 0.5 1 1.5 2 2.5 3

A2

n = 2.5n = 2.0

n = 1.5

n = 1.0

figuRe 22.9 Plot of Equation 22.127, A2 vs. κ for varying values of n.

70

80

60

40

50 n = 1.0

30

ф n = 2.5

20

0

10

0 0.5 1 1.5κ

2 2.5 3

figuRe 22.10 Plot of Equation 22.130, ϕ vs. κ for varying values of n.

Optics of Metals 499

S

S

S

S

0

1

2

3

1 000

0 365

0 246

0 898

=−−

.

.

.

.

. (22.159)

We find then from Equations 22.159, 22.142 to Equation 22.143 that

A2 18 6 8,= . 0 (22.160)

and

φ = °33 979 .. (22.161)

Using these values in Equations 22.135 and 22.137, we find that the ratio of the orthogonal intensi-ties is

I

Ip

s

= 0 837. . (22.162)

This value provides a final check on the measurement. To obtain a seed value for the complex refractive index, we construct Table 22.1, using Equations 22.127 and 22.130. Inspection of the table shows that, as n increases, the first entry of A2 and ϕ, for each value of n, increases and decreases, respectively. Thus, we need only match the pair of A2 and ϕ that is closest to the actual value. In this case, the desired values are A2 = 18.608 and ϕ = 33.979°. The closest pair in the table matching these are A2 = 15.046 and ϕ = 30.182, and the corresponding values of n and κ are 2.5 and 0.5. Thus, the seed complex refractive index is chosen to be

ˆ . . .n i= −( )2 5 1 50 (22.163)

Table 22.1 was constructed for small values of κ. If, for example, large values of A2 and ϕ were found, this would indicate that a new table should be constructed from Equations 22.127 to Equation 22.130, starting with values of, say, n = 0.5 and κ = 5.0, and so on. For the present example, we now iterate Equations 22.127 and 22.130 around n = 2.50 and κ = 0.50, and we find that for A2 = 18.608 and ϕ = 33.979°, the complex refractive index is represented by

ˆ . . .n i= −( )2 679 1 57450 0 (22.164)

We can use this result to find the reflectivity of this surface at normal incidence. We recall from Section 22.2 that the reflectivity for any incident polarization is

R = − ++ +

( ) ( )( ) ( )

.n nn n

11

2 2

2 2

κκ

(22.165)

Substituting the above values of n and κ into Equation 22.165, we find that

R = 32 6. %, (22.166)

500 Polarized Light, Third Edition

which shows that this optical material is a very poor reflector.We now consider another example. The normalized Stokes vector for this example is

S

S

S

S

0

1

2

3

1 000

0 053

0 160

0 986

=

−

.

.

.

.

. (22.167)

We immediately find that

A2 139 845,= . (22.168)

φ = 71 672°.. (22.169)

Table 22.1initial Values for determining n and κ from equations 22.127 and 22.130

n κ a2 ϕ

0.5 0.5 0.800 68.954

0.5 1.0 1.414 67.815

0.5 1.5 2.211 68.254

0.5 2.0 3.202 68.954

1 0.5 2.062 42.393

1 1.0 4.123 57.961

1 1.5 6.946 64.113

1 2.0 10.630 67.007

1.5 0.5 5.088 33.833

1.5 1.0 9.055 54.216

1.5 1.5 15.038 62.774

1.5 2.0 23.114 66.465

2 0.5 9.434 31.280

2 1.0 16.032 52.752

2 1.5 26.401 62.244

2 2.0 40.608 66.256

2.5 0.5 15.046 30.182

2.5 1.0 25.020 50.052

2.5 1.5 41.020 61.987

2.5 2.0 63.105 66.156

3 0.5 21.915 29.608

3 1.0 36.014 51.667

3 1.5 58.892 61.844

3 2.0 90.604 66.701

Optics of Metals 501

From Figure 22.9 to Figure 22.10, we see that the very large values of A2 and ϕ indicate a relatively large value for κ. We again construct a seed table as before, and we find that the appropriate seed value for the complex refractive index is

ˆ . .n i= −( )0 5 1 15 (22.170)

Proceeding as before, we obtain

ˆ . . .n i= −( )0 65 1 12 78 (22.171)

For this sample, we find that the reflectivity R is

R = 96 4. %. (22.172)

This large value of R shows that this sample is an excellent reflector.Thus we see that Fresnel’s equations for an incidence angle of 45° enable us to determine n and κ

by taking advantage of all of the developments of modern electronics and computers. In particular, this method is readily automated. While the simplest measurement configuration has been shown in Figure 22.8, more complicated configurations that simplify the measurements, such as a dual-beam configuration to measure Is and Ip simultaneously, can be conceived.

The measurement of the refractive index and the absorption index of materials is critical to the development of modern optical materials (e.g., fiber-optic glass, metals and metal alloys, and semi-conductors). In this and previous sections, we have dealt with determining the optical constants that are inherent to the material itself. In practice, this means that the material and, in particular, the optical surface must be free of any other substances resting on the surface (e.g., a thin film).

The problem of considering the effects of a thin film on an optical surface appears to have been first studied by Drude about 1890. He was probably initially interested in characterizing these thin films in terms of their optical properties. However, as he advanced in his investigations, he came to realize that the subject was far from simple and required substantial effort. In fact, the fundamental equations could not be solved until the advent of digital computers. In order to determine n and κ for thin films as well as the substrates, he developed a method that has come to be known as ellip-sometry. As time developed, further applications were found, for example, the measurement of thin films deposited on optical lenses in order to improve their optical performance. In Chapter 24, we consider the fundamentals of ellipsometry.

RefeReNCeS

1. Strong, J., Procedures in Applied Optics, New York: Marcel Dekker, 1989. 2. Humphreys-Owen, S. P. F., Comparison of reflection methods for measuring optical constants without

polarimetric analysis, and proposal for new methods based on the Brewster angle, Proc. Phys. Soc. (London) 77 (1961): 941.

3. Collett, Ε., Digital refractometry, Opt. commun. 63 (1987): 217–24. 4. Wood, R. W., Physical Optics, 3rd ed., Washington, DC: Optical Society of America, 1988. 5. Potter, R. F., Optical Properties of Solids, Edited by S. Nudelman and S. S. Mitra, New York: Plenum

Press, 1969. 6. Born, M., and E. Wolf, Principles of Optics, 5th ed., New York: Pergamon Press, 1975. 7. Mott, N. F., and H. Jones, The Theory of the Properties of Metals and Alloys, Mineola, NY: Dover,

1958. 8. Longhurst, R. S., Geometrical and Physical Optics, 2nd ed., New York: Wiley, 1967.

503

23 Polarization Optical Elements

23.1 iNTRoduCTioN

A polarization optical element is any optical element that modifies the state of polarization of a light beam. Polarizers, retarders, rotators, and depolarizers are all polarization optical elements, and we will discuss their properties in this chapter. The many references on polarization elements, and catalogs and specifications from manufacturers are good sources of information. We include here a survey of elements, and brief descriptions so that the reader has at least a basic understanding of the range of available polarization elements.

23.2 PolaRiZeRS

A polarizer is an optical element that is designed to produce polarized light of a specific state inde-pendent of the incident state. The desired state might be linear, circular, or elliptically polarized, and an optical element designed to produce one of these states is a linear, circular, or elliptical polarizer. Polarization elements are based on polarization by absorption, refraction, and reflec-tion. Since this list describes most of the things that can happen when light interacts with matter, the appearance of polarized light should not be surprising. We will cover polarization by all these methods in the following sections.

23.2.1 abSoRPTion PolaRizeRS: PolaRoid

Polaroid is a material invented in 1928 by Edwin Land, who was then a 19-year-old student at Harvard University. (The generic name for Polaroid, sheet polarizer, applies to a polarizer whose thickness normal to the direction of propagation of light is much smaller than the width.) Land used aligned microcrystals of herapathite in a transparent medium of index similar to the crystal-line material. Herapathite is a crystalline material discovered about 1852 by the English medical researcher William Bird Herapath. Herapath had been feeding quinine to dogs, and the substance that came to be known as herapathite crystallized out of the dogs’ urine. Crystals of herapathite tend to be needle-shaped and the principal absorption axis is parallel to the long axis of the crystal. Land was reading David Brewster’s book on kaleidoscopes and noticed the reference to herapathite. (Land’s paper [1] on his development of Polaroid makes fascinating reading and should be read by all optics students.) Inspired by Brewster, Land reduced crystals of herapathite to small size, aligned them, and placed them in a solution of cellulose acetate. This first absorption polarizer is known as J-sheet.

Sheet polarizer, operating on the principle of differential absorption along orthogonal axes, is also known as dichroic polarizer. This is because the unequal absorptions also happen to be spec-trally dependent, that is, linearly polarized light transmitted through a sample of Polaroid along one axis appears to be a different color from linearly polarized light transmitted along the orthogonal axis.

The types of sheet polarizer typically available are molecular polarizers, that is, they consist of transparent polymers that contain molecules that have been aligned and stained with a dichroic dye. The absorption takes place along the long axis of the molecules, and the transmission axis is perpendicular to this. H-sheet, K-sheet, and L-sheet are of this type, with H-sheet being the most common. Sheet polarizers can be made in large sizes (several square feet) for both the visible and

504 Polarized Light, Third Edition

near infrared, and is an extremely important material, because, unlike calcite, it is inexpensive. Polaroid material can be laminated between glass plates, and the performance of these polarizers is extremely good.

We now derive equations that describe sheet polarizer properties; the equations are equally applicable to any type of polarizer. Suppose we have a light source that is passed through an ideal polarizer with its transmission axis at some angle α from a reference. The output of the ideal polarizer then passes through a sheet polarizer with its transmission axis oriented at an angle θ with respect to a reference, as shown in Figure 23.1. The Mueller matrix of this last polarizer is

M pol

A b b

b A cθ

θ θθ θ θ( ) =

+cos sin

cos cos sin

2 2 0

2 2 22 2 AA c

b A c A

−−

( )( )

sin cos

sin sin cos sin

2 2 0

2 2 2 2

θ θθ θ θ 22 2 0

0 0 0

2θ θ+

c

c

cos, (23.1)

where

Ap p

bp p

c p px y x yx y=

+=

−=

2 2 2 2

2 2, (23.2)

and where the quantities px and py are the absorption coefficients of the orthogonal optical axes, and 0 ≤ px, py ≤ 1. The Stokes vector of the beam emerging from the ideal polarizer with its transmission axis at angle α is

S =

I0

1

2

2

0

cos

sin,

αα

(23.3)

Axis of ideal polarizer

α

Source

Axis of polarizer to be measured

θ

figuRe 23.1 Measurement configuration for characterizing a single polarizer.

Polarization Optical Elements 505

where I0 is the intensity of the beam. The light intensity emerging from the sheet polarizer is found from multiplying Equation 23.3 by Equation 23.1 where we obtain

I I A bθ α θ α, cos .( ) = + −( )[ ]0 2 (23.4)

The maximum intensity occurs at θ = α and is

I I A b I pxmax .= +[ ] =0 02 (23.5)

The minimum intensity occurs at θ = α + π/2 and is

I I A b I pymin .= −[ ] =0 02 (23.6)

A linear polarizer has two transmittance parameters: the major principal transmittance k1 and the minor principal transmittance k2. The parameter k1 is defined as the ratio of the transmitted intensity to the incident intensity when the incident beam is linearly polarized in that vibration azimuth that maximizes the transmittance. Similarly, the ratio obtained when the transmittance is a minimum is k2. Thus

kI

IA b px1

0

2= = + =max , (23.7)

kI

IA b py2

0

2= = − =min . (23.8)

The ratio k1/k2 is represented by Rt and is called the principal transmittance ratio. Rt of a high quality polarizer may be as large as 105. The reciprocal of Rt is called the extinction ratio, and is often quoted as a figure of merit for polarizers. The extinction ratio should be a small num-ber and the transmittance ratio a large number; if this is not the case, the term at hand is being misused.

Because the principal transmittance can vary over several orders of magnitude, it is common to express k1 and k2 in terms of logarithms. Specifically, k1 and k2 are defined in terms of the minor and major principal densities, d1 and d2,

dk

dk1 10

12 10

2

1 1=

=

log log , (23.9)

or

k kd d1 210 101 2= =− − . (23.10)

Dividing k1 by k2 yields

Rtd= 10 , (23.11)

where d = d2 – d1 is the density difference or dichroitance.

506 Polarized Light, Third Edition

The average of the principal transmittances is called the total transmittance kt so that

kk k p p

Atx y= + =

+=1 2

2 2

2 2. (23.12)

The parameter kt is the ratio of the transmitted intensity to incident beam intensity when the inci-dent beam is unpolarized (multiply a Stokes vector for unpolarized light by the matrix in Equation 23.1). Furthermore, we see that kt is an intrinsic constant of the polarizer and does not depend on the polarization of the incident beam, as is the case with k1 and k2.

Figure 23.1 shows the measurement of k1 and k2 of a single polarizer. We assumed that we had a source of perfectly polarized light from an ideal (or nearly ideal) polarizer. Another way to deter-mine k1 and k2 is to measure a pair of identical polarizers and use an unpolarized light source. This method requires an extremely good source of unpolarized light. It turns out to be surprisingly dif-ficult to obtain a perfectly unpolarized light source. Nearly every optical source has some elliptical polarization associated with it, that is, the emitted light is partially polarized to some degree. One reason this is so is because a reflection from nearly every type of surface, even one that is rough, creates polarized light. Assuming we can produce a light source that is sufficiently unpolarized as to lead to meaningful data, the parameters k1 and k2 can, in principle, be determined from a pair of identical polarizers. Figure 23.2 illustrates the experiment.

Let us assume we can align the polarization axes. From Equation 26.1, the Stokes vector result-ing from the passage of unpolarized light through the two aligned polarizers is

A b

b A

c

c

A b

b A

c

0 0

0 0

0 0 0

0 0 0

0 0

0 0

0 0 0

0 0 0

cc

I

I

A b

Ab

=

+0

0

2 2

0

0

0

2

0

00

. (23.13)

The intensity for the beam emerging from the polarizer pair is

I p I A b( ) = +( )02 2 . (23.14)

Unpolarized source

Axis of first polarizer

Axis of second polarizer

figuRe 23.2 Measurement of k1 and k2 of identical polarizers.

Polarization Optical Elements 507

This may be written

I pk k

I( ) = +12

22

02. (23.15)

We now rotate the polarizer closest to the unpolarized source through 90°. The Stokes vector of the beam emerging from the polarizer pair is now

A b

b A

c

c

A b

b A

c

0 0

0 0

0 0 0

0 0 0

0 0

0 0

0 0 0

0

−−

00 0

0

0

0

0

0

0

0

2 2

c

I

I

A b

=

−

00

. (23.16)

The intensity from the crossed pair, I(s), is

I s I A b( ) = −( )02 2 , (23.17)

and this may be written

I s k k( ) = 1 2 . (23.18)

Now let the ratio of intensities I(p)/I0 when the polarizers are aligned be H0 and let the ratio of inten-sities I(s)/I0 when the polarizers are perpendicular be H90. Then we can write

Hk k

A b012

22

2 2

2= + = +( ), (23.19)

and

H k k A b90 1 22 2= = −( ). (23.20)

Multiplying Equations 23.19 and 23.20 by 2 and adding gives

2 2 20 90 12

1 2 22H H k k k k+ = + + . (23.21)

Taking the square root, we have

2 0 90 1 2

12H H k k+( ) = + . (23.22)

Multiplying Equations 23.19 and 23.20 by 2, subtracting, and taking the square root, we have

2 0 90 1 2

12H H k k−( ) = − . (23.23)

Now we solve for k1 and k2 by adding and subtracting Equations 23.22 and 23.23, and we obtain

k H H H H1 0 90 0 90

22

12

12= +( ) + −( )

, (23.24)

508 Polarized Light, Third Edition

k H H H H2 0 90 0 90

22

12

12= +( ) − −( )

. (23.25)

The principal transmittance ratio can now be expressed in terms of H0 and H90, that is,

Rk

k

H H H H

H Ht = =

+( ) + −( )

+( )1

2

0 90 0 90

0 90

12

12

12 −− −( )

H H0 90

12

. (23.26)

Thus if we have a perfect unpolarized light source and we can be assured of aligning the polarizers parallel and perpendicular to each other, we can determine the transmittance parameters k1 and k2 of a polarizer when they are arranged in a pair. However, as has been pointed out, it is very difficult to produce perfectly unpolarized light. It is much easier if a known high quality polarizer is used to produce linearly polarized light and the measurement of k1 and k2 follows the measurement method illustrated in Figure 23.1.

Suppose we cannot align the two polarizer axes perfectly. If one of the polarizers is rotated from the horizontal axis by angle θ, then we have the situation shown in Figure 23.3.

The Stokes vector of the beam emerging from the first polarizer is

I

A b

b A

c

c

0

0 0

0 0

0 0 0

0 0 0

1

0

0

0

=

I

A

b0

0

0

. (23.27)

The second polarizer is represented by Equation 23.1, and so the beam intensity emerging from the second polarizer is

I I A bθ θ( ) = +[ ]02 2 2cos . (23.28)

Using a trigonometric identity, this can be written as

I I A b bθ θ( ) = −( ) +[ ]02 2 2 22 cos . (23.29)

Unpolarized source

Axis of first polarizer

Axis of second polarizer

θ

figuRe 23.3 Nonaligned identical linear polarizers.

Polarization Optical Elements 509

Equation 23.29 can be expressed in terms of H0 and H90, that is,

HI

IH H Hθ θ θ( ) = ( ) = + −( )

090 0 90

2cos . (23.30)

Equation 23.30 is a generalization of Malus’s Law for nonideal polarizers. This relation is usually expressed for an ideal polarizer so that A2 = b2 = 1/4, H0 = 2A2, and H90 = 0 so that

H θ θ( ) = 12

2cos . (23.31)

We now apply data to these results. In Figure 23.4, the spectral curves of different types of Polaroid sheet are shown with the values of k1 and k2. In Table 23.1, values of H0 and H90 are listed for the sheet Polaroids HN-22, HN-32, and HN-38 over the visible wavelength range. From this table we can construct Table 23.2 and determine the corresponding principal transmittances. We see from Table 23.2 that HN-22 has the largest principal transmittance ratio in comparison with HN-32 and HN-38, consequently it is the best polaroid polarizer. Calcite polarizers typically have a principal transmittance ratio of 1 × 106 from 300 to 2000 nm. This is three times better than Polaroid HN-22 at its best value. Nevertheless, in view of the lower cost of sheet polarizer, it is useful in many applications.

23.2.2 abSoRPTion PolaRizeRS: PolaRcoR

Polarcor is an absorption polarizer consisting of elongated silver particles in glass. This polarizer, developed commercially by Corning, has been produced with transmittance ratios of 10,000 in the near infrared. The polarizing ability of silver in glass was observed in the late 1960s [3], and high transmittance ratios polarizers were developed in the late 1980s [4]. Because these polarizers depend on a resonance phenomenon, performance is strongly dependent on wavelength, but they can be engineered to operate with good performance over broad wavelength regions centered on near infrared wavelengths from 800 to 1500 nm.

0.0001

0.001

0.01

0.1

1

HN-32

HN-32 k1

HN-38

HN-38 HN-22

k

0.000001

0.00001

400 450 500 550 600 650 700 750

Prin

cipa

l tra

nsm

ittan

ces

Wavelength (nanometers)

k2HN-22

k2k2

figuRe 23.4 Curves of k1 and k2 for three grades of HN polarizer.

510 Polarized Light, Third Edition

23.2.3 wiRe gRid PolaRizeRS

A wire grid is a planar array of parallel wires. It is similar to the sheet polarizer in that the transmit-ted light is polarized perpendicularly to the wires. Light polarized parallel to the wires is reflected instead of absorbed as in sheet polarizer. To be an effective polarizer, the wavelength should be longer than the spacing between wires. For practical reasons, wire grids are usually placed on a substrate. Until relatively recently, they have been typically manufactured for the infrared region of the spectrum ( >2 μm) because the small grid spacing required for shorter wavelengths has been difficult to produce. Grid spacing for these infrared polarizers are normally 0.5 μm or greater, although smaller spacings have been fabricated. With technological improvements in grid fabrica-tion techniques, grids with wires of width 0.065 μm or less have been produced. These grids are useful into the near infrared and visible [5,6]. Photomicrographs of wire grid polarizers composed of 0.065 aluminum wires are given in Figure 23.5.

Since reflection loss and absorption reduce the transmittance ratio of wire grids, an antireflec-tion coating is often applied to the substrate. The quality of this coating and its achromaticity are important factors in the overall performance of wire grids. Commercial wire grid polarizers have

Table 23.1Parallel-Pair h0 and Crossed-Pair Transmittance h90 of hN Polarizers

hN-22 hN-32 hN-38

Wavelength(nm) H0 H90 H0 H90 H0 H90

400 0.02 0.0000002 0.0674 0.000559 0.2141 0.0264

450 0.10 0.000002 0.185 0.000187 0.3014 0.0103

500 0.15 0.000001 0.243 0.000043 0.3303 0.0015

550 0.12 0.000001 0.225 0.000032 0.314 0.00004

600 0.09 0.000001 0.204 0.000025 0.2864 0.000017

650 0.11 0.000001 0.213 0.000024 0.2917 0.000015

700 0.17 0.000002 0.256 0.000026 0.3199 0.000015

750 0.24 0.000007 0.318 0.000031 0.3606 0.0003

Source: Data for HN-22 from Shurcliff, W. A., Polarized Light—Production and Use, Harvard University Press, Cambridge, MA, 1962; data for HN-32 and HN-38 courtesy of American Polarizers, Inc.

Table 23.2Principal Transmittances of hN-22, hN-32, and hN-38

rt

Wavelength (nm) hN-22 hN-32 hN-38

400 21,000 241 16

450 1,50,000 1,979 59

500 2,75,000 11,198 440

550 2,40,000 14,241 15700

600 2,15,000 16,126 33694

650 2,35,000 17,750 38131

700 1,96,667 19,768 41817

750 69,000 20,319 2404

Polarization Optical Elements 511

transmittance ratios of 20–10,000. More information on wire grids is given in Bennett and Bennett [7], Bennett [8], and the cited patents [5,6,9].

23.2.4 PlaSMonic lenSeS aS ciRculaR PolaRizeRS

Microstructures in the form of Archimedean spiral slots have been etched into metal films to form circular polarization filters [10]. These are illustrated in Figure 23.6. They are called plasmonic lenses because they rely on the interaction of surface plasmons with photons to either focus or defo-cus the light of a certain circular polarization. Left-handed spirals focus light of right-hand circular polarization into a spot, whereas a right-handed spiral will defocus the light and disperse it so that there is no focal spot. A transmission ratio of 100 is said to be achievable.

0001 100000X 25kv 100nm

0003 40000X 5kv 100nm

(a)

(a)

figuRe 23.5 Photomicrographs of wire grid polarizers: (a) side view and (b) top down view. (Courtesy of MOXTEK, Inc.)

512 Polarized Light, Third Edition

23.2.5 PolaRizaTion by RefRacTion (PRiSM PolaRizeRS)

Crystal prism polarizers use total internal reflection at internal interfaces to separate the polarized components. There are many designs of prism polarizers, and we will not cover all of these here. The reader should consult the excellent article by Bennett and Bennett [7] for a comprehensive treatment.

The basis of most prism polarizers is the use of a birefringent material, as described in Chapter 21. We illustrate the phenomenon of double refraction with the following example of the construction of a Nicol polarizing prism. We know that calcite has a large birefringence. (Calcite, the crystalline form of limestone, marble, and chalk, occurs naturally. It has not been grown artificially in anything but very small sizes. It can be used in prism polarizers for wavelengths from 0.25 to 2.7 μm.) If the propagation is not perpendicular to the direction of the optic axis, the ordinary and extraordinary rays separate. Each of these rays is linearly polarized. A Nicol prism is a polarizing prism con-structed so that one of the linear polarized beams is rejected and the other is transmitted through the prism unaltered. It was the first polarizing prism ever constructed (1828). However, it is now obsolete and has been replaced by other prisms, such as the Glan–Thompson, Glan–Taylor, Rochon, and Wollaston prisms. These new designs have become more popular because they are optically superior; for example, the light is nearly uniformly polarized over the field of view, whereas it is not for the Nicol prism. The Glan–Thompson type has the highest reported transmittance ratio [7].

In a Nicol prism, a flawless piece of calcite is split so as to produce an elongated cleavage rhomb about three times as long as it is broad. The end faces, which naturally meet the edges at angles of 70°53′, are ground so that the angles become 68° (this allows the field-of-view angle to be increased); apparently, this practice of trimming was started by Nicol himself. Figure 23.7 shows the construc-tion of the Nicol prism. The calcite is sawed diagonally and at right angles to the ground and pol-ished end faces. The halves are cemented together with Canada balsam, and the sides of the prism

mag 10 000x

detETD

WD5.1 mm

tilt0°

HV5.00kv

10/15/200911:26:11 Am

5 μm

figuRe 23.6 Spiral slots in metal film forming left- and right-handed circular polarization filters. (Courtesy Qiwen Zhan, University of Dayton, Ohio.)

Polarization Optical Elements 513

are covered with an opaque, light absorbing coating. The refractive index of the Canada balsam is 1.54, a value intermediate to the ordinary (no = 1.6584) and extraordinary (ne = 1.4864) refractive indices of the calcite. Its purpose is to deflect the ordinary ray (by total internal reflection) out of the prism and to allow the extraordinary ray to be transmitted through the prism.

We now compute the angles. The limiting angle for the ordinary ray is determined from Snell’s Law. At 5893 Å, the critical angle θ2 for total internal reflection at the calcite–balsam interface is obtained from

1 6583 1 54 902. sin . sin ,θ = ° (23.32)

so that θ2 = 68.28°. The cut is normal to the entrance face of the prism, so that the angle of refraction θr1

at the entrance face is 90° – 68.28° = 21.72°. The angle of incidence is then obtained from

sin . sin . ,θi11 6583 21 72= ° (23.33)

so that the angle of incidence is θi137 88= . °. Since the entrance face makes an angle of 68° with

the longitudinal axis of the prism, the normal to the entrance face is 90° – 68° = 22 with respect to the longitudinal axis. The limiting angle at which the ordinary ray is totally reflected at the balsam results in a half field angle of θ1 = 37.88° – 22° = 15.88°. A similar computation is required for the limiting angle for the extraordinary ray at which total reflection does not occur. The refractive index for the extraordinary ray is a function of the angle (let us call it ϕ) between the wave normal and the optic axis. Using the same procedure as before (but not shown in Figure 23.7), we have

′ = − ′θ θ2 901

° r , and the critical angle at the calcite/balsam interface is obtained from

sin cos.

.901 54

1 1°− ′( ) = ′ =θ θ

φr r n

(23.34)

The index of refraction nϕ of the extraordinary wave traveling in a uniaxial crystal at an angle ϕ with the optic axis is given by

1

2

2

2

2

2n n ne oφ

φ φ= +sin cos. (23.35)

68°70°53′

Extraordinaryray

(a)

Ordinary ray

(b)

θ1

θi1

θr1 θ2

figuRe 23.7 Diagram of a Nicol prism: (a) longitudinal section and (b) cross section.

514 Polarized Light, Third Edition

For our Nicol prism

φ = ′+ ′r1 41 44° , (23.36)

and Equation 23.35 becomes

cos

.

sin . cos2

2

2

2

21 1 1

1 54

41 73θ θ θr r

e

r

n

′=

′ + °( )+

′ ++ °( )41 732

..

no

(23.37)

This transcendental equation is easily solved using a computer, and ′θr1 is found to be 7.44° and θ1

′ is 11.61°, using the values of the indices for λ = 5893Å. The semi field angle is 10.39° and the total field angle is 20.78°.

The cross section of the Nicol prism is also shown in Figure 23.7. Only the extraordinary ray emerges and the plane of vibration is parallel to the short diagonal of the rhombohedron, so that the direction of polarization is obvious. The corners of the prism are sometimes cut, making the direc-tion of polarization more difficult to discern.

Six other types of polarization prisms are shown in Figure 23.8. All these diagrams are for prisms of calcite. The optic axes are represented by the short lines with double-headed arrows if aligned with the surface of the prism, and by dots if the optic axis is perpendicular to the surface. The extraordi-nary ray is labeled with an e, and the ordinary ray with an o where the directions of polarization are shown with the short lines or dots along the ray. The Rochon prism in Figure 23.8a deviates the e ray whereas the o ray continues straight through. The Wollaston prism in Figure 23.8b produces a devia-tion of both rays. The deviation is larger than in the Rochon prisms, and the Wollaston prism might be used when both beams will be used. The Senarmont prism in Figure 23.8c is similar to the Rochon except that the polarizations of the resultant rays are reversed. In the Glan–Taylor and Glan–Foucault prisms of Figure 23.8d and e, the calcite axes are perpendicular to the direction of propagation, but in the first case the axis is vertical whereas in the second case the axis is horizontal. Both of these prisms have air gaps between the two halves of the prism. The polarizations in the transmitted and reflected rays are reversed. The final example is the Glan–Thompson prism in Figure 23.8f. This prism is like the Glan–Foucault in that the axes are perpendicular to the propagation direction and are horizontal. Instead of an air gap, there is a gap filled with an optical cement, which might have been Canada bal-sam (a natural material) in older prisms but is likely some synthetic material in newer devices.

23.2.6 PolaRizaTion by ReflecTion

One has only to examine plots of the Fresnel equations, as described in Chapter 7, to see that polar-ization will almost always occur on reflection. Polarizers that depend on reflection are usually com-posed of plates oriented near the Brewster angle. Because sheet and prism polarizers do not operate in the infrared and ultraviolet, reflection polarizers are sometimes used in these regions. Brewster angle polarizers are necessarily sensitive to incidence angle and are physically long devices because Brewster angles can be large, especially in the infrared where materials with high indices are used. A diagram of a pile-of-plates polarizer is shown in Figure 23.9, and Figure 23.10 is a photo of an infrared polarizer based on this principal.

23.3 ReTaRdeRS

A retarder is an optical element that produces a specific phase difference between two orthogo-nal components of incident polarized light. A retarder can be in prism form, called a rhomb, or it can be in the form of a window or plate, called a waveplate. Waveplates can be zero order, that is, the net phase difference is actually the specified retardance, or multiorder, in which case the phase difference can be a multiple, sometimes large, of the specified retardance. Retarders are also

Polarization Optical Elements 515

sometimes called compensators, and can be made variable, for example, the Babinet-Soleil com-pensator. Retarders may be designed for single wavelengths, or be designed to operate over larger spectral regions, that is, achromatic retarders.

23.3.1 biRefRingenT ReTaRdeRS

The properties of isotropic, uniaxial, and biaxial optical materials were discussed in Chapter 21. We can obtain from that discussion that the phase retardation of linearly polarized light in going through a uniaxial crystal with its optic axis parallel to the faces of the crystal is

Γ = −( )2πλ

d n ne o , (23.38)

o

e

o

e

o

e

(a) Rochon prism

(b) Wollaston prism

(c) Senarmont prism

figuRe 23.8 Representational diagrams of (a) the Rochon prism, (b) the Wollaston prism, (c) the Senarmont prism, (d) the Glan–Taylor prism, (e) the Glan–Foucault prism, and (f) the Glan–Thompson prism.

516 Polarized Light, Third Edition

when the polarization is at an angle with the optic axis. The optical path difference experienced by the two components is d(ne – no) and the birefringence is (ne – no). These quantities are all positive for positive uniaxial materials, that is, materials with ne > no. The component of the light experienc-ing the refractive index ne is parallel with the optic axis while the component experiencing the index no is perpendicular to the optic axis. The slow axis is the direction in the material in which light experiences the higher index ne, that is, for the positive uniaxial material, the direction of the optic axis. The fast axis is the direction in the material in which light experiences the lower index, no. It is the fast axis that is usually marked with a line on commercial waveplates. The foregoing discussion is the same for negative uniaxial material with the positions of ne and no interchanged.

The most common commercial retarders are quarter wave and half wave, that is, where there are π/2 and π net phase differences between components, respectively. The quarter-wave retarder produces circular polarization when the azimuth of the (linearly polarized) incident light is 45° to

o

e

Air gap

o

e

Air gap

o

e

Optical cement

(d) Glan–Taylor prism

(f) Glan–Thompson prism

(e) Glan–Foucault prism

figuRe 23.8 (Continued) Representational diagrams of (a) the Rochon prism, (b) the Wollaston prism, (c) the Senarmont prism, (d) the Glan–Taylor prism, (e) the Glan–Foucault prism, and (f) the Glan–Thompson prism.

Polarization Optical Elements 517

the fast axis. The half-wave retarder produces linearly polarized light rotated by an angle 2θ when the azimuth of the (linearly polarized) incident light is at an angle θ with respect to the fast axis of the half wave retarder.

As we have seen above, the net retardance is an extensive property of the retarder; that is, the retardance increases with path length through the retarder. When the net retardation for a retarder reaches the minimum net value desired for the element, then that retarder is known as a single-order retarder (sometimes called a zero-order retarder). Although many materials have small birefrin-gence, some (calcite) have large values of birefringence (see Table 23.3). Birefringence is, like index, a function of wavelength. A single order retarder may not be possible because it would be too thin to be practical. A retarder called a first order retarder may be constructed by joining two pieces of material such that the fast axis of one piece is aligned with the slow axis of the other. The thicknesses of the pieces of material are adjusted so that the difference in the thicknesses of the two pieces is equal to the thickness of a single order retarder. The retardation can be found from the equation

Γ = −( ) −( )21 2

πλ

d d n ne o , (23.39)

where d1 and d2 are the thicknesses.

figuRe 23.9 Diagram of a pile-of-plates polarizer.

figuRe 23.10 (See color insert following page 394.) A photograph of a pile-of-plates polarizer for the infrared. (Photo courtesy of D. H. Goldstein.)

518 Polarized Light, Third Edition

A multiple order retarder is a retarder of thickness such that its net retardation is an integral number of wavelength plus the desired fractional retardance, for example, 5λ/4, 3λ/2, and so on. Multiple order retarders may be less expensive than single order retarders, but they are sensitive to temperature and incidence angle.

23.3.2 vaRiable ReTaRdeRS

Retarders have been constructed of movable elements in order to produce variable retardance. Two of the most common designs based on movable wedges are the Babinet and Soleil (also variously called Babinet–Soleil, Soleil–Babinet, or Soleil–Bravais) compensators, shown in Figure 23.11. The term compensator is used for these elements because they are often used to allow adjustable com-pensation of retardance originating in a sample under test.

The Babinet compensator, shown in Figure 23.11a, consists of two wedges of a (uniaxial) bire-fringent material (e.g., quartz). The bottom wedge is fixed while the top wedge slides over the bot-tom by means of a micrometer. The optic axes of both wedges are parallel to the outer faces of the wedge pair, but are perpendicular to one another. At any particular location across the face of the Babinet compensator, the net retardation is

Γ = −( ) −( )21 2

πλ

d d n ne o , (23.40)

where d1 and d2 are the thicknesses at that location. If monochromatic polarized light oriented at 45° to one of the optic axes is incident on the Babinet compensator, one component of the light becomes the extraordinary component and the other is the ordinary component in the first wedge. When the light enters the second wedge, the components exchange places, that is, the extraordinary becomes the ordinary and vice versa. An analyzer whose azimuth is perpendicular to the original polariza-tion can be placed behind the compensator to show the effect of the retardations. Everywhere there is zero or a multiple of 2π phase difference there will be a dark band. When the upper wedge is translated, the bands shift. These bands indicate the disadvantage of the Babinet compensator—a desired retardance only occurs along these parallel bands.

The Soleil compensator, shown in Figure 23.11b consists of two wedges with parallel optic axes followed by a plane parallel quartz prism with its optic axis perpendicular to the wedge axes. The top wedge is the only moving part again. The advantage to this design is that the retardance is

Table 23.3birefringence for optical materials at 589.3 nm

material birefringence (ne–no)

Positive Uniaxial crystals

Ice (H2O) 0.004

Quartz (SiO2) 0.009

Zircon (ZrSiO4) 0.045

Rutile (TiO2) 0.287

Negative Uniaxial crystals

Beryl [Be3Al2(SiO3)6] –0.006

Sodium nitrate (NaNO3) –0.248

Calcite (CaCO3) –0.172

Sapphire (Al2O3) –0.008

Polarization Optical Elements 519

uniform over the whole field where the wedges overlap. A photograph of a commercial Soleil–Babinet compensator is given in Figure 23.12.

Jerrard [11] gives a review of these and many other compensator designs.

23.3.3 achRoMaTic ReTaRdeRS

The most common type of retarder is the waveplate, as described above, a plane parallel plate of birefringent material, with the crystal axis oriented perpendicular to the propagation direction of light. As the wavelength varies, the retardance of the zero order waveplate must also vary, unless by coincidence the birefringence was linearly proportional to wavelength. Since this does not occur in practice, the waveplate is only approximately quarter wave (or whatever retardance it is designed for) for a small wavelength range. For higher order waveplates, m = 3, 5, …, the effective wavelength range for quarter wave retardance is even smaller.

The achromatic range of waveplates can be enlarged by assembling combinations of waveplates of birefringent materials [7]. This method has been common in the visible region; however, in the infrared the very properties required to construct such a device are the properties to be measured polarimetrically, and there are not an abundance of data available to readily design high perfor-mance devices of this kind. Nevertheless, an infrared achromatic waveplate has been designed [4] using a combination of two plates. This retarder has a theoretical retardance variation of about 20° over the 3–11 μm range.

OA OA

OA

OA OA

figuRe 23.11 Diagrams of (a) Babinet compensator and (b) Soleil compensator where OA is the optic axis.

figuRe 23.12 (See color insert following page 394.) Photograph of a Soleil–Babinet compensator. (Photo courtesy of D. H. Goldstein.)

520 Polarized Light, Third Edition

A second class of achromatic retardation element is the total internal reflection prism. Here a specific phase shift between the s and p components of light (linear retardance) occurs on total internal reflection. This retardance depends on the refractive index, which varies slowly with wave-length. But since this retardance is independent of any thickness, unlike the waveplate, the variation of retardance with wavelength is greatly reduced relative to the waveplate. A common configuration for retarding prisms is the Fresnel rhomb, with the diagram of this device in Figure 23.13 and a photograph in Figure 23.14. These figures show a Fresnel rhomb designed for the visible spectrum. The nearly achromatic behavior of this retarder is the desired property; however, the Fresnel rhomb has the disadvantages of being long with large beam offset. In an application where the retarder must be rotated, any beam offset is unacceptable. A quarter-wave Fresnel rhomb for the infrared, made of ZnSe and having a clear aperture of x inches, has a beam offset of 1.7x inches and a length of 3.7x inches.

23.3.3.1 infrared achromatic RetarderFigure 23.15 shows a prism retarder that was designed for no beam deviation. This design includes two total internal reflections and an air–metal reflection. Similar prisms have been designed previ-ously, but special design considerations for the infrared make this prism retarder unique. Previous designs for the visible have included a solid prism with similar shape to the retarder in Figure 23.15 but with no air space [12], and a set of confronting rhombs called the double Fresnel rhomb. The latter design includes four total internal reflections. These designs are not appropriate for the infrared.

54.7

figuRe 23.13 Fresnel rhomb.

figuRe 23.14 (See color insert following page 394.) Photograph of a quarter-wave Fresnel rhomb. (Photo courtesy of D. H. Goldstein.)

Polarization Optical Elements 521

The prism design relies on the fact that there are substantial phase shifts between the s and p components of polarized light at the points of total internal reflection (TIR). The phase changes of s and p components on TIR are given by the formulas [13]

δ φφs

nn

prism = −( )−211

2 212

tansin

cos, (23.41)

and

δ φφp

n nprism = −( )−211

2 212

tansincos

, (23.42)

where ϕ is the angle of incidence and n is the index of refraction of the prism material. The lin-ear retardance associated with the total internal reflection is the net phase shift between the two components

∆prism prism prism= −δ δp s . (23.43)

In addition there are phase shifts on reflection from the metal given by [7]

δ ηηs

os

os

b

a bmetal =

− +( )−tan ,1

2 2 2

2 (23.44)

δη

ηpop

op

d

c dmetal =

−+ −

−tan ,12 2 2

2 (23.45)

where

η η θos = 0 0cos , (23.46)

η ηθ00

0p =

cos, (23.47)

a b n k n n k2 212

12

02 2

02

12

122

12+ = − −( ) + sin ,θ (23.48)

Metal Mirror

figuRe 23.15 Infrared achromatic prism retarder design.

522 Polarized Light, Third Edition

c dn k

a b2 2 1

212 2

2 2+ = +( )

+( ) , (23.49)

ba b n k n= +( ) − − −( )

2 212

12

02 2

0

2 2

12sin,

θ (23.50)

d bn

a b= −

+

1 0

2 20

2 2

sin,

θ (23.51)

and where n0 is the refractive index of the incident medium, θ0 is the angle of incidence, and n1 and k1 are, respectively, the index of refraction and extinction index for the metal mirror. The linear retardance associated with the metal mirror is the net phase shift between the s and p components

∆metal metal metal= −δ δp s . (23.52)

The net retardance for the two TIRs and the metal reflection is then

δ = +2∆ ∆prism metal . (23.53)

The indices of refraction of materials that transmit well in the infrared are higher than indices of materials for the visible. Indices for infrared materials are generally greater than 2.0, where indices for materials for the visible are in the range 1.4–1.7. The higher indices for the infrared result in greater phase shifts between s and p components for a given incidence angle than would occur for the visible. Prism retarder designs for the infrared that have more than two TIRs soon become impractically large as the size of the clear aperture goes up or the desired retardance goes down. The length of a solid prism retarder of the shape of Figure 23.15 is governed by the equation

Lada=

−( )tan,

90 θ (23.54)

where da is the clear aperture and θ is the angle of incidence for the first TIR. The theoretical mini-mum length of the two-prism design for a clear aperture of 0.5 inches and a retardance of a quarter wave is 2.1 inches. The minimum length for the same retardance and clear aperture in a three TIR design is 4.5 inches.

Materials that are homogeneous (materials with natural birefringence are unacceptable) and good infrared transmitters must be used for such a device. Suitable materials include zinc selenide, zinc sulfide, germanium, arsenic trisulfide glass, and gallium arsenide. Metals that may be used for the mirror include gold, silver, copper, lead, or aluminum, with gold being preferable because of its excellent reflective properties in the infrared and its resistance to corrosion.

Beam angles at the entry and exit points of the two prism arrangement are designed to be at normal incidence to minimize Fresnel diattenuation. Figure 23.16 shows the theoretical phase shift versus wavelength for this design. For zinc selenide prisms and a gold mirror at the angles shown, the retardation is very close to a quarter of a wavelength over the 3–14 μm band. (The angles were computed to give a retardance of 90° near 10 μm.) Table 2.4 gives numerical values of the phase shift along with indices of zinc selenide and gold. The indices for gold are from Ordal et al. [14]

Polarization Optical Elements 523

and the indices for ZnSe are from Wolfe and Zissis [15]. The requirement of a nearly achromatic retarder with no beam deviation is satisfied, although the disadvantage of the length of the device remains (actual length is dependent on the clear aperture desired).

23.3.3.2 achromatic Waveplate RetardersAs we have seen, waveplates are made of birefringent materials and the retardance is given by

Γ = −( )2πλ

n n de o . (23.55)

The retardance is explicitly inversely proportional to wavelength. If the value of the birefringence

∆n n ne o= −( ), (23.56)

for some material was directly proportional to wavelength then achromatic waveplates could be made from the material. This condition is not normally satisfied in nature.

Plates made up of two or three individual plates have been designed that are reasonably achromatic [7]. If we consider a plate made of two materials a and b having thicknesses da

100

95

90

85

803.0 4.0 5.0 6.0 7.0 8.0 10.0 12.0 14.0

Wavelength (micrometers)

Reta

rdan

ce (d

egre

es)

figuRe 23.16 Theoretical retardance of achromatic prism retarder in the infrared.

Table 23.4Numerical data for achromatic Retarder

Wavelength (µm) ZnSe index gold index (n) gold index (k) Total Phase Shift

3 2.440 0.704 21.8 88.39

4 2.435 1.25 29.0 89.03

5 2.432 1.95 36.2 89.42

6 2.438 2.79 43.4 89.66

7 2.423 3.79 50.5 89.81

8 2.418 4.93 57.6 89.91

10 2.407 7.62 71.5 90.02

12 2.394 10.8 85.2 90.04

14 2.378 14.5 98.6 89.98

524 Polarized Light, Third Edition

and db and wish to make the retardance equal at two wavelengths λ1 and λ2, we can write the equations

N n d n da a b bλ1 1 1= +∆ ∆ , (23.57)

N n d n da a b bλ2 2 2= +∆ ∆ , (23.58)

where N is the retardance we require in waves, that is, 1/4, 1/2, and so on, and the subscripts on the birefringence Δn designates the wavelength and material. Solving the equations for da and db we have

dN n n

n n n nab b

a b b a

= −( )−

λ λ1 2 2 1

1 2 1 2

∆ ∆∆ ∆ ∆ ∆

, (23.59)

and

dN n n

n n n nba a

a b b a

= −( )−

λ λ2 1 1 2

1 2 1 2

∆ ∆∆ ∆ ∆ ∆

. (23.60)

The optimization of the design is facilitated by changing the thickness of one of the plates and the ratio of the thicknesses. There will generally be an extremum in the retardance function in the wavelength region of interest. A good achromatic design will have the extremum near the middle of the region. Changing the ratio of the ratio of the thicknesses shifts the position of the extremum. Changing the thickness of one of the plates changes the overall retardance value.

There are important practical considerations for compound plate design. For example, it may not be possible to fabricate plates that are too thin, or they may result in warped elements; and, plates that are thick will be more sensitive to angular variation of the incident light. Precision of alignment of the plates in a multiplate design is extremely important, and misalignments will result in oscil-lation of retardance. A compound waveplate for the infrared mentioned earlier is composed of two plates of CdS and CdSe with fast axes oriented perpendicularly [16]. This design calls for a CdS plate about 1.3 mm thick followed by a CdSe plate about 1 mm thick. The theoretical achromaticity over the 3–11 μm wavelength region is 90° ± 20° although measurements indicate somewhat better performance [17]. The useful wavelength range of these achromatic waveplates is often determined by the design of the antireflection coatings.

23.4 RoTaToRS

Rotation of the plane of polarization can occur through optical activity, the Faraday effect, and by the action of liquid crystals.

23.4.1 oPTical acTiviTy

Arago first observed optical activity in quartz in 1811. During propagation of light though a material, a rotation of the plane of polarization occurs that is proportional to the thickness of the material and also depends on wavelength. There are many substances that exhibit optical activity, notably quartz and sugar solutions (e.g., place a bottle of corn syrup between crossed

Polarization Optical Elements 525

polarizers!). Many organic molecules can exist as stereoisomers, that is, a molecule of the same chemical formula is constructed such that it either rotates light to the right or to the left. Since these molecules can have drastically different effects when taken internally, it has become important to distinguish and separate them when producing pharmaceuticals. Natural sugar is dextrorotatory, meaning it rotates to the right; amino acids are generally levorotatory, rotating to the left.

Optical activity can be explained in terms of left and right circularly polarized waves and the refractive indices that these waves experience. The rotatory power of an optically active medium is

ρ πλ

= −( ),

n nL R (23.61)

in degrees per centimeter, where nL is the index for left circularly polarized light, and nR is the index for right circularly polarized light.

The rotation angle is

δ πλ

= −( ).

n n dL R (23.62)

Suppose we have a linearly polarized wave entering an optically active medium. The linearly polar-ized wave can be represented as a sum of circular components. Using the Jones formalism,

1

0

=−

+

12

1 12

1

i i. (23.63)

We have written the linear polarized light as a sum of left circular and right circular components. After traveling a distance d through the medium, the Jones vector will be

12

1 12

1

12

−

+

=

ie

iei n d i n dL R2 2π λ π λ/ /

eei

ei n n d i n n dR L R L2 2 2 2π λ π λ( ) ( )+ − −

−

+/ /1 1

iiei n n dR L

−2 2π λ( ) / .

(23.64)

Let,

ψ πλ

= +22

( )n n dR L , (23.65)

and

δ πλ

= −22

( )n n dL R . (23.66)

526 Polarized Light, Third Edition

Substituting these values into the right-hand side of Equation 23.64 gives

ei

ei

e ei i i iψ δ δ ψ12

1 12

11

−

+

=− 2212

e e

i e e

i i

i i

δ δ

δ δ

+( )

− −( )

−

−

=

eiψδδ

cos

sin, (23.67)

which is a linearly polarized wave whose polarization has been rotated by δ.

23.4.2 faRaday RoTaTion

The Faraday effect has been described in Chapter 21. Faraday rotation can be used as the basis for optical isolators. Consider a Faraday rotator between two polarizers that have their axes at 45°. Suppose that the Faraday rotator is such that it rotates the incident light by 45°. It then should pass through the second polarizer since the light polarization and the polarizer axis are aligned. Any light returning through the Faraday rotator is rotated an additional 45° and will be blocked by the first polarizer. In this way, very high isolation, up to 90 dB [18], is possible. Rotation in devices based on optical activity and liquid crystals retrace the rotation direction and cannot be used for isolation. Faraday rotation is the basis for spatial light modulators, optical memory, and optical crossbar switches.

23.4.3 liquid cRySTalS

A basic description of liquid crystals has been given in Chapter 21. Liquid crystal cells of various types can be configured to act as polarization rotators. The rotation is electrically controllable, and may be continuous or binary. For a detailed treatment of liquid crystals, see Khoo and Wu [19].

23.5 dePolaRiZeRS

A depolarizer reduces the degree of polarization. We recall that the degree of polarization is given by

PS S S

S= + +1

222

32

0

. (23.68)

An ideal depolarizer produces a beam of unpolarized light regardless of the initial polarization state, so that the goal of an ideal depolarizer is to reduce P to 0. The Mueller matrix for an ideal depolarizer is

1 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

. (23.69)

A partial depolarizer (or pseudodepolarizer) reduces the degree of polarization. It could reduce one, two, or all three of the Stokes vector components by varying amounts, and there are many possibili-ties [20]. Examples of depolarizers in an everyday environment include waxed paper and projection screens. Integrating spheres have been shown to function as excellent depolarizers [21]. A sample of

Polarization Optical Elements 527

Spectralon is also an excellent depolarizer over a wide wavelength range as evidenced by the Mueller matrix measurement result shown in Figure 23.17. Commercial depolarizers are offered that are based on producing a variable phase shift across their apertures. These rely on obtaining a randomized mix of polarization states over the beam width. A small beam will defeat this depolarization scheme.

RefeReNCeS

1. Land, E. H., Some aspects on the development of sheet polarizers, J. Opt. Soc. Am. 41 (1951): 957–63.

2. Shurcliff, W. A., Polarized Light—Production and Use, Cambridge, MA: Harvard University Press, 1962.

3. Stookey, S. D., and R. J. Araujo, Selective polarization of light due to absorption by small elongated silver particles in glass, Appl. Opt. 7 (1968): 777–79.

4. Chenault, D. B., and R. A. Chipman, Infrared spectropolarimetry, in Polarization considerations for Optical Systems II, Edited by R. A. Chipman, Proc. SPIE 1166 (1989).

5. Perkins, R. T., D. P. Hansen, E. W. Gardner, J. M. Thorne, and A. A. Robbins, Broadband wire grid polarizer for the visible spectrum, U.S. Patent 6,112,103, September 19, 2000.

1

0.5

00.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9

1

0.5

0

–0.5

–1

1

0.5

0

–0.5

–1

1

0.5

0

–0.5

–1

1

0.5

0

–0.5

–1

1

0.5

0

–0.5

–1

m00 m01Spectrallon99527_45 135

m02 m03

m13m12m10 m11

m20 m21 m22 m23

m32 m33m30 m31

0.5 0.6 0.7 0.8 0.9

0.5 0.6 0.7 0.8 0.90.5 0.6 0.7 0.8 0.90.5 0.6 0.7 0.8 0.90.5 0.6 0.7 0.8 0.9

0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9

0.5 0.6 0.7 0.8 0.90.5 0.6 0.7 0.8 0.90.5 0.6 0.7 0.8 0.90.5 0.6 0.7 0.8 0.9

1

0.5

0

–0.5

–1

1

0.5

0

–0.5

–1

1

0.5

0

–0.5

–1

1

0.5

0

–0.5

–1

1

0.5

0

–0.5

–1

1

0.5

0

–0.5

–1

1

0.5

0

–0.5

–1

1

0.5

0

–0.5

–1

1

0.5

0

–0.5

–1

1

0.5

0

–0.5

–1

figuRe 23.17 Measured Mueller matrix of a sample of spectralon SRM-99 in reflection over the 0.5–1.0 μm wavelength range. (Data courtesy of D. H. Goldstein.)

528 Polarized Light, Third Edition

6. Perkins, R. T., E. W. Gardner, and D. P. Hansen, Imbedded wire grid polarizer for the visible spectrum, U.S. Patent 6,288,840, September 11, 2001.

7. Bennett, J. M., and H. E. Bennett, Polarization, in Handbook of Optics, Edited by W. G. Driscoll and W. Vaughan, New York: McGraw-Hill, 1978, 1–164.

8. Bennett, J. M., Polarization, in Handbook of Optics, Edited by M. Bass, New York: McGraw-Hill, 1995, 1–30.

9. Chipman, R. A., and D. B. Chenault, Infrared achromatic retarder, U.S. Patent 4,961,634, October 9, 1990.

10. Yang, Y., W. Chen, R. L. Nelson, and Q. Zhan, Miniature circular polarization analyzer with spiral plas-monic lens, Opt. Lett. 34, no. 20 (2009), 3047–9.

11. Jerrard, H. G., Optical compensators for measurement of elliptical polarization, J. Opt. Soc. Am. 38 (1948): 35–59.

12. Clapham, P. B., M. J. Downs, and R. J. King, Some applications of thin films to polarization devices, Appl. Opt. 8 (1969): 1965–74.

13. Jenkins, F. A., and H. E. White, Fundamentals of Optics, New York: McGraw-Hill, 1957. 14. Ordal, M. A., L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. J. Alexander, and C. A. Ward, Optical

properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared, Appl. Opt. 22 (1983): 1099–119.

15. Wolfe, W. L., and G. J. Zissis, The Infrared Handbook, Washington, DC: Office of Naval Research, 1978.

16. Chenault, D. B., Achromatic retarder design study, Report No. NRC-TR-96-075, Suffolk, VA: Nichols Research Corporation, 1996.

17. Chenault, D. B., Infrared Spectropolarimetry, Ph.D. Dissertation, Huntsville, AL: University of Alabama, 1992.

18. Saleh, B. E. A., and M. C. Teich, Fundamentals of Photonics, New York: John Wiley, 1991. 19. Khoo, I.-C., and S.-T. Wu, Optics and Nonlinear Optics of Liquid crystals, Singapore: World Scientific,

1993. 20. Chipman, R. A., Depolarization, in Polarization: Measurement, Analysis, and Remote Sensing II, Edited

by D. H. Goldstein and D. B. Chenault, 14–20, Proc. SPIE 3754 (1999). 21. McClain, S. C., C. L. Bartlett, J. L. Pezzaniti, and R. A. Chipman, Depolarization measurements of an

integrating sphere, Appl. Opt. 34 (1995): 152–4.

529

24 Ellipsometry

24.1 iNTRoduCTioN

One of the most important applications of polarized light is the measurement of the complex refrac-tive index and thickness of thin films. A field of optics has been developed to do this and has come to be known as ellipsometry. In its broadest sense, ellipsometry is the art of measuring and analyz-ing the elliptical polarization of light. The name appears to have been given in 1945 by Alexandre Rothen [1], one of the pioneers in the field. However, the field of ellipsometry has become much more restrictive, so that now it almost always applies to the measurement of the complex refractive index and thickness of thin films. In its most fundamental form, it is an optical method for mea-suring the optical parameters of a thin film by analyzing the reflected polarized light. The optical parameters are the refractive index n, the extinction coefficient κ, and the thickness d of a thin film deposited on a substrate. The optical procedure for determining these parameters is done in a very particular manner, and it is this manner that has come to be known as ellipsometry. The fundamen-tal concepts of ellipsometry are quite simple and straightforward; however, we shall see that this seeming simplicity is deceptive. Nevertheless, the method is very elegant.

The fact that a thin film on a substrate could significantly change the measured characteristics of an optical material, for example, a micro thin coating of oil on water, came apparently as a surprise to nineteenth-century optical physicists. The great Lord Rayleigh admitted as much when he was experimenting with the surface viscosity of liquids and said:

“Having proved that the superficial viscosity of water was due to a greasy contamination whose thick-ness might be much less than one-millionth of a millimetre, I too hastily concluded that films of such extraordinary tenuity were unlikely to be of optical importance until prompted by a remark of Sir G. Stokes, I made an actual estimate of the effect to be expected.”

At about the time that Rayleigh was investigating the optical properties of light reflected from the surface of liquids, Drude was investigating the optical properties of light reflected from solids. In two fundamental articles [2,3] published in 1889 and 1890, he laid the foundations for ellipsom-etry. As we have pointed out many times, at that time the only optical detector was the human eye, which has only a capability of measuring a null-intensity condition. Drude cleverly exploited this very limited quantitative condition of the human eye to determine the optical parameters of a thin film. He recognized that an optical material such as a metal behaves simultaneously as a polarizer and a phase shifter so that, in general, light reflected from the optical surface of a metal is elliptically polarized. Analysis shows that by adjusting the amplitude and the phase of the incident beam it is possible to transform the reflected elliptically polarized light to linearly polarized light. Drude did this by inserting a polarizer and a compensator (retarder) between the optical source and the sample.

By setting the compensator with its fast axis at 45°, and rotating the polarizer through an angle P, the reflected elliptically polarized light could be transformed to linearly polarized light. The reflected linearly polarized light was then analyzed by another linear polarizer (the analyzer) by rotating it through an angle Q until a null intensity was observed. Analysis showed that these angles could be used to determine the ellipsometric parameters ψ and Δ, which described the change in amplitude and phase in the reflected wave. Further analysis based on Fresnel’s reflection equations could then relate ψ and Δ to n, κ, and d. The elegance of the method will become apparent when this analysis is presented in the following sections.

530 Polarized Light, Third Edition

Ellipsometry can be used to determine the optical constants of a reflecting material or the optical constants and thickness of the film deposited on an optical substrate. It has a number of advantages over other methods for determining the optical constants. Among these are its applicability to the measurement of strongly absorbing materials, the simplicity of the measurement method, and the ease of the sample preparation. In addition, it is nondestructive and requires only a very small sample size. For studying the properties of surface films, its directness, sensitivity, and simplicity are without parallel. Also, ellipsometry can be applied to the measurement of surface films whose thickness ranges from monatomic dimensions to micrometers. Throughout this range, the index of refraction n of a film can be determined and, for absorbing film media, the extinction coefficient κ can be determined as well.

Ellipsometry can be conveniently divided into two parts. The first is the measurement technique for determining ψ and Δ. The second is the theory required to relate the optical parameters of the thin film to the measured values of ψ and Δ. Throughout this chapter, we use the formalism of the Stokes parameters and the Mueller matrices to derive some important results. We begin by deriving the fundamental equation of ellipsometry, that is, the equation relating ψ and Δ to n, κ, and d.

24.2 fuNdameNTal eQuaTioN of ClaSSiCal elliPSomeTRy

In this section, we derive an equation that relates the amplitude and phase of the incident and reflected beams from a thin film, the so-called ellipsometric parameters, to the complex refractive index and the thickness of the film. The equation is called the fundamental equation of ellipsometry. To derive this equation, we consider Figure 24.1. In the figure, Ep and Es are the incident field com-ponents parallel (p) and perpendicular (s) to the plane of paper. Similarly, Rp and Rs are the parallel and perpendicular reflected field components. For the incident field components we can write

E E ep pi p= 0α , (24.1)

E E es si s= 0α . (24.2)

Substrate

Filmd

n2

n0

n1

κ2

κ0 = 0

κ1

Ep

Es

Rp

Rs

figuRe 24.1 Reflection of an incident beam by an optical film of thickness d with a refractive index n1 and an extinction coefficient κ1.

Ellipsometry 531

A similar pair of equations can also be written for the reflected field, namely,

R R ep pi p= 0β , (24.3)

R R es si s= 0β . (24.4)

In Equations 24.1 through 24.4, the propagation factor, ωt – κz, has been suppressed. Measurements have shown that Rp,s is directly related to Ep,s, and, in general for optically absorbing materials, the incident field will be attenuated and undergo a phase shift. In order to describe this behavior we introduce complex reflection coefficients, ρp and ρs defined by

R Ep p p= ρ , (24.5)

R Es s= ρ s , (24.6)

or, in general,

ρmm

m

RE

m p s= = , . (24.7)

Substituting Equations 24.1 through 24.4 into Equation 24.7 then yields

ρ β αm

m

m

iRE

e m p sm m= ( ) =−0

0

( ) , . (24.8)

We define a complex relative amplitude attenuation as

ρ ρρ

β α= = ( ) −p

s

p p

s s

iR E

E Re0 0

0 0

//

( ) , (24.9)

where α = αp – αs and β = βp – βs. The quantities α and β describe the phase before and after reflec-tion, respectively. Traditionally, the factors in Equation 24.9 are written in terms of the tangent of the angle ψ, that is,

tan ,ψ = R E

E Rp p

s s

0 0

0 0

//

(24.10)

and a phase angle

∆ = − = − − −β α β β α α( ) ( ).p s p s (24.11)

From Equation 24.10 to Equation 24.11, we can then express Equation 24.9 as

ρ ψ= tan ei∆. (24.12)

Thus, ellipsometry involves the measurement of the change in the amplitude ratio, expressed in terms of tan ψ, and Δ, the change in phase. The quantities ψ and Δ are functions of the optical con-stants of the medium, the thin film and the substrate, the wavelength of light, the angle of incidence,

532 Polarized Light, Third Edition

and, for an optical film deposited on a substrate, its thickness. With these factors in mind we now express Equation 24.12 as

ρ ψ κ= = ( )tan e f n di∆ , , . (24.13)

Equation 24.13 is called the fundamental equation of ellipsometry. Ideally, by measuring ψ and Δ, the quantities n, κ, and d can be determined. In Equation 24.13, ρ has been expressed in terms of a general functional form, f(n, κ, d). Later we will derive the specific form of f(n, κ, d) for a thin film deposited on a substrate.

Equation 24.13 shows that the basic problem of ellipsometry is to measure ψ and Δ and relate it to f(n, κ, d). In the next section, we develop the equations for measuring ψ and Δ. In the following section, we relate these measurements to f(n, κ, d). We shall soon see that the form of Equation 24.13 is deceptively simple and that considerable effort is needed to solve it.

24.3 ClaSSiCal meaSuRemeNT of The elliPSomeTRiC PaRameTeRS PSi (ψ) aNd delTa (Δ)

In this section, we describe the classical measurement of ψ and Δ in the fundamental equation of ellipsometry, Equation 24.13. This is done by using a polarizer and compensator before the sample, and a polarizer after the sample. The objective of the present analysis is to relate the angu-lar settings on the polarizers and the compensator to ψ and Δ. Figure 24.2 shows the experimental configuration.

We first determine the Mueller matrix of the combination of the linear polarizer and the compen-sator in the generating arm. The linear polarizer can be rotated to any angle P. The compensator, on the other hand, has its fast axis fixed at 45°, but its phase ϕ can be varied from 0° to 360°. The Mueller matrices for the polarizer and compensator are then

Mpol ( )

cos sin

cos cos cos sinP

P P

P P P P= 1

2

1 2 2 0

2 2 2 22 00

2 2 2 2 0

0 0 0 0

2sin cos sin sin,

P P P P

(24.14)

i i

Substrate

Linear polarizer

Source

Analyzer

Compensator(quarter-wave retarder)

Film

Detector

figuRe 24.2 Experimental configuration to measure ψ and Δ of an optical sample.

Ellipsometry 533

and

Mcomp °( )cos sin

sin cos

+ =

−

45

1 0 0 0

0 0

0 0 1 0

0 0

φ φ

φ φ

. (24.15)

The Mueller matrix for the polarizer–compensator combination, Equations 24.14 and 24.15, is

M M MPSG comp pol= ( ) ( )φ P , (24.16)

and so

MPSG = 12

1 2 2 0

2 22

cos sin

cos cos cos cos cos c

P P

P Pφ φ φ oos sin

sin cos sin sin

sin cos

2 2 0

2 2 2 2 0

2

2

P P

P P P P

P− φ −− −

sin cos sin cos sin

,

φ φ2 2 2 2 0P P P

(24.17)

where PSG stands for polarization state generator. The Stokes vector of the beam incident on the polarizer–compensator combination is represented by its most general form

S =

S

S

S

S

0

1

2

3

. (24.18)

Multiplying Equation 24.18 by Equation 24.17, we obtain the Stokes vector of the beam incident on the samples as

′ =

′′′′

= + +S

S

S

S

S

S S P S

0

1

2

3

0 1

12

2( cos 22 2

1

2

2

2

sin )cos cos

sin

sin sin

PP

P

P

φ

φ−

, (24.19)

which is a Stokes vector for elliptically polarized light. The orientation angle Ψ of the beam is defined by

tan tantancos

,22Ψ = Pφ

(24.20)

and, similarly, the ellipticity angle χ is defined by

sin 2 sin cos 2χ φ= − P. (24.21)

534 Polarized Light, Third Edition

(Note that in standard notation, the Greek letter psi is used for both orientation of the ellipse and an ellipsometric parameter. We use capital psi here for the orientation angle.) Thus, by varying Ρ and ϕ we can generate any state of elliptically polarized light. We now write Equation 24.19 as

S =

−

IP

P

P

0

1

2

2

2

cos cos

sin

sin cos

,φ

φ

(24.22)

and drop the primes on the Stokes vector.The phase shift between the components emerging from the polarizer–compensator pair, accord-

ing to the relations derived in Section 24.2, is expressed in terms of an angle α. The Stokes param-eters of the beam incident on the sample can then be written in terms of field components as seen from Equation 24.1 to Equation 24.2 as

S E E E E E Es s p p s p0 02

02= + = +* * , (24.23)

S E E E E E Es s p p s p1 02

02= − = −* * , (24.24)

S E E E E E Es p p s s p2 0 02= + =* * cos ,α (24.25)

S i E E E E E Es p p s s p3 0 02= − =( ) sin .* * α (24.26)

The phase shift α is seen from Equation 24.22 to Equation 24.26 to be

tansincos

sin cossin

.α αα

φ= = = −SS

PP

3

2

22

(24.27)

Now,

sin 2 9 cos 2P P−( ) = −0° , (24.28)

cos 2 9 sin 2P P−( ) =0° . (24.29)

Substituting Equations 24.28 and 24.29 into Equation 24.27 then yields

tan sin tan 2 9α φ= −( )P 0° . (24.30)

Thus, the phase α of the beam emerging from the polarizer–compensator combination can be var-ied by adjusting the phase shift ϕ of the compensator and the polarizer orientation angle P. In par-ticular, if we have a quarter-wave retarder so that ϕ = 90°, then from Equation 24.30, α = 2P – 90°. By rotating the polarizer angle from P = 0 (α = –90°) to P = 90° (α = 90°), the total phase change is 180°. In terms of the Stokes vector S, Equation 24.22, for ϕ = 90° we then have

S =

−

IP

P

0

1

0

2

2

sin

cos

. (24.31)

Ellipsometry 535

Equation 24.31 is the Stokes vector for elliptically polarized light; its orientation angle Ψ is always 45°. However, according to Equation 24.31, the ellipticity angles corresponding to P = 0°, 45°, and 90°, are χ = –45°, 0°, and +45°, and the respective Stokes vectors are 1,0,0,−1, 1,0,1,0, and 1,0,0, +1; these vectors correspond to left circularly polarized light, linear +45° polarized light, and right circularly polarized light, respectively. By rotating the polarizer from 0° to 90°, we can generate any state of elliptically polarized light ranging from left circularly polarized light to right circularly polarized light.

The ratio of the amplitudes Ep and Es of the beam emerging from the polarizer–compensator pair can be defined in terms of an angle L as

tan .LE

Ep

s

= (24.32)

From Equation 24.22 to Equation 24.24, we have

SS

E E E E

E E E EPs s p p

s s p p

1

0

2= −+

=* *

* *cos cos ,φ (24.33)

or

11

2−+

=( )( )( )( )

cos cos .* *

* *

E E E E

E E E EPp s p s

p s p s

/ // /

φ (24.34)

Because tan L is real, Equation 24.32 can be expressed as

(tan ) tan .**

*L

E

ELp

s

= = (24.35)

Thus, Equation 24.34 can be written with the aid of Equation 24.35 as

11

11

2

2

2 2

2 2

−+

= −+

tantan

(sin ) (cos )(sin ) (cos

LL

L LL

// LL

L L P)

cos sin cos cos ,= − =2 2 2φ (24.36)

or

cos 2 cos cos 2L P= − φ . (24.37)

We note that if S1 is defined as the negative of Equation 24.24, that is,

S E E E Ep p s s1 = −* *, (24.38)

then Equation 24.37 becomes

cos 2 cos cos 2L P= − φ , (24.39)

which is the form usually given in ellipsometry. Thus, again, by varying ϕ and P, the angle L can be selected. For circularly polarized light, Es = Ep, so L = 45° from Equation 24.23, and cos 2L = 0. For linearly horizontally polarized light, Ep = 0, L = 0, and cos 2L = 1. Finally, for linearly vertically polarized light, Es = 0, L = 90°, and cos 2L = –1.

536 Polarized Light, Third Edition

Equations 24.30 and 24.37 appear very often in ellipsometry and so are rewritten here together as the pair

tan sin tan 2 9α φ= −( )P 0° , (24.30)

cos 2 cos cos 2L P= − φ . (24.37)

We emphasize that Equations 24.30 and 24.37 relate the amplitude and phase of the optical beam inci-dent on the sample to the value of the compensator phase ϕ and the polarizer angle P, respectively.

The procedure for measuring ψ and Δ consists of rotating the generating polarizer and the ana-lyzing polarizer until the reflected beam is extinguished. Because the compensator is fixed with its fast axis at 45°, only two polarizing elements rather than three must be adjusted. The Stokes vector of the reflected light is

′ =

′ + ′′ − ′′ ′′

S

E E

E E

E E

E

s p

s p

s p

s

02

02

02

02

0 0

0

2

2

cosβ′′

E p0 sin

,

β

(24.38)

where β, using the notation in Section 24.2, is the phase associated with the reflected beam. To obtain linearly polarized light, sin β in Equation 24.38 must be zero, and there are two values of β that satisfy this requirement, β = 0°, 180°. The Stokes vector S′ in Equation 24.38 then becomes

′ =

′ + ′′ − ′

± ′ ′

S

E E

E E

E E

s p

s p

s p

02

02

02

02

0 02

0

. (24.39)

The condition on β then transforms Equation 24.11 to

∆ = − = − =β α α β( ),0° (24.40)

or

∆ = − =180 180° °α β( ). (24.41)

The angles of the polarizer in the generating arm corresponding to Equations 24.40 and 24.41 can be written as P0 and ′P0 , respectively. We have

tan sin tan 2 9α φ= −( )P0 0° , (24.42)

cos 2 cos cos2L P0 0= − φ , (24.43)

and

tan sin tan( ),′ = ′ −α φ 2 2700P ° (24.44)

cos cos cos .2 20 0′ = − ′L Pφ (24.45)

Ellipsometry 537

The linearly polarized reflected beam will be extinguished when the analyzer angles corresponding to P0 and ′P0 are A0 and ′A0, respectively. This leads immediately to the form for tan ψ, Equation 24.10,

tan .ψ = =R

R

R

REE

p

s

p

s

s

p

0

0

0

0

(24.46)

Substituting Equation 24.32 into Equation 24.46, we have

tan cot ,ψ = R

RLp

s

0

00 (24.47)

where we have used the measurement value L0. We also see that

tan( ) ,− =AR

Rp

s0

0

0

(24.48)

(the angle –A0 is opposite to P0). Then, using Equation 24.48, Equation 24.47 becomes

tan cot tanψ = −( )L A0 0 , (24.49)

for the polarizer–analyzer pair settings of P0 and A0. Similarly, for the pair ′P0 and ′A0, we have

tan cot tan .ψ = ′ ′L A0 0 (24.50)

From Equation 24.42 to Equation 24.44, we see that

′ = ±P P0 0 90°, (24.51)

and

′ = ±A A0 0 90°. (24.52)

Using Equations 24.51 and 24.52, and setting Equation 24.49 equal to Equation 24.50 yields

cot tan ,′ =L L0 0 (24.53)

so that multiplying Equations 24.49 and 24.50 gives

tan tan( ) tan( ).20 0ψ = ′ −A A (24.54)

Equation 24.54 shows that tan ψ can be determined by measuring ′A0 and A0, the angular settings on the analyzer. Similarly, the phase shift Δ can be obtained from Equation 24.40 to Equation 24.43 or Equations 24.44 and 24.45.

For the special case where ϕ = 90°, a quarter-wave retarder, the equations relating ψ and Δ simplify. From Equations 24.42 and 24.43, we have

∆ = − = ′ −2 90 2 2700 0P P° °, (24.55)

538 Polarized Light, Third Edition

from Equation 24.43 to Equation 24.45 we have

′ =L L0 0 , (24.56)

and from Equation 24.49 to Equation 24.50 we have

− = ′A A0 0. (24.57)

If a Babinet–Soleil compensator is used, then the phase shift ϕ can be set to 90°, and A0, ′A0, P0, and ′P0 can be used to give tan ψ and Δ, Equations 24.54 and 24.55, respectively, that is,

tan tan tan ( ),2 20

20ψ = = − ′A A (24.58)

so

ψ = = − ′A A0 0 , (24.59)

and

∆ = − = ′ −2 90 2 2700 0P P° °. (24.60)

In order to select the correct equations for calculating Δ and ψ from a pair of extinction settings, it is necessary to establish whether the settings correspond to the condition Δ′ = –Δ or Δ′ = Δ +180°. This is accomplished by observing that, although Δ may have any value between 0° and 360°, ψ is limited to values between 0° and 90°. From this fact, the sign of the analyzer extinction setting, according to ψ = − = ′A A0 0, determines whether the setting corresponds to the primed or unprimed case.

The relations presented above describe the measurement formulation of ellipsometry. The for-mulation rests on the conditions required to obtain a null intensity, that is, linearly polarized light will be obtained for reflected light if sinβ = 0° or 180°. From this condition, one works backward to find the corresponding values of Ρ and A and then ψ and Δ.

There are other configurations and formulations of ellipsometry. One of the most interesting has been given by Holmes and Feucht [4]. Their formulation is particularly valuable because it leads to a single expression for the complex reflectivity ρ in terms of the polarizer angles P and A. We des-ignate the analyzing polarizer angle by A. Moreover, it includes the imperfections of the compensa-tor with its fast axis at an angle c. This formulation was used by F. L. McCrackin, one of the first researchers to use digital computers to solve the ellipsometric equations, in the early 1960s [5].

We recall that ρ of an optical surface is related to the ellipsometric parameters ψ and Δ by

ρ ψ= tan ei∆ . (24.12)

We assume the same ellipsometric measurement configuration as before, that is, an ideal polarizer and a compensator in the generating arm, and an ideal polarizer in the analyzing arm. The transmis-sion axes of the generating and analyzing polarizers are at P and A, respectively. The compensator is considered to be slightly absorbing, and its fast axis is at an angle c. Lastly, the beam incident on the generating polarizer is assumed to be linearly horizontally polarized with unit amplitude. We use the Jones formalism to carry out the calculations. The Jones matrix for the incident beam is

Jinc =

1

0. (24.61)

Ellipsometry 539

The Jones matrix of a rotated linear polarizer is

Jpol =−

cos sin

sin cos

cos sP P

P P

P1 0

0 1

iin

sin cos

cos sin cos

sin cos si

P

P P

P P P

P P

−

=2

nn.

2 P

(24.62)

Multiplying Equation 24.61 by Equation 24.62 then gives

J =

coscos

sin.P

P

P (24.63)

The term cos P is an amplitude factor that can be ignored, and so the Jones matrix of the beam incident on the compensator is

J =

cos

sin.

P

P (24.64)

The Jones matrix for an ideal compensator is

Jcomp =

e

e

i

i

x

y

φ

φ

0

0. (24.65)

If there is also absorption along each of the axes, then the Jones matrix Equation 24.65 can be rewritten as

Jcomp =

a e

a ex

i

yi

x

y

φ

φ

0

0, (24.66)

where 0 ≤ ax,y <1. We see that we can now write Equation 24.66 as

Jcomp =

1 0

0 ac

, (24.67)

where ac = (ay/ax)exp(iϕ) and ϕ = ϕy = ϕx. The variable ac is called the absorption ratio of the com-pensator, and we have ignored the factor a ex

i xφ that would have been outside of the matrix in Equation 24.67. The Jones matrix of the compensator, Equation 24.67, with its fast axis rotated to an angle c is

Jcomp =−

cos sin

sin cos

cosc c

c c ac

1 0

0

cc c

c c

c a c ac c

sin

sin cos

cos sin ( )si

−

=+ −2 2 1 nn cos

( )sin cos sin cos.

c c

a c c c a cc c1 2 2− +

(24.68)

540 Polarized Light, Third Edition

Multiplying Equation 24.64 by Equation 24.68, the Jones matrix of the beam incident on the optical sample is

J =− + −− −

cos cos( ) sin sin( )

sin cos( )

c c P a c c P

c c P ac

cc c c Pcos sin( ).

−

(24.69)

We must now determine the Jones matrix of the optical sample. By definition, the reflected beam is related to the incident beam by

R Ep p p= ρ , (24.5)

R Es s s= ρ , (24.6)

where ρp and ρs are the complex reflection coefficients for the parallel and perpendicular components, respectively. The Jones matrix of the sample is then seen from Equation 24.5 to Equation 24.6 to be

Jsamp =

ρρ

p

s

0

0. (24.70)

The complex relative amplitude attenuation ρ in Equation 24.12 is defined by

ρ ρρ

= p

s

, (24.9)

so Equation 24.70 can be written as

Jsamp =

ρ 0

0 1, (24.71)

where we have ignored the factor ρs because it will drop out of our final equation, which is a ratio. The Jones matrix of the beam incident on the analyzing polarizer is now seen from multiplying Equation 24.69 by Equation 24.71 to be

J =− + −

−ρ [cos cos( ) sin sin( )]

sin cos(

c c P a c c P

c c Pc

)) cos sin( ).

− −

=

a c c P

E

Ec

x

y

(24.72)

Equation 24.72 shows that the reflected light is, in general, elliptically polarized. However, if c, P, and ac are adjusted so that the reflected light is linearly polarized, then the azimuthal angle γ of the linearly polarized light is

tansin cos( ) cos sin( )[cos cos(

γρ

= = − − −E

Ec c P a c c P

cy

x

c

cc P a c c Pc− + −) sin sin( )]. (24.73)

The linearly polarized light, Equation 24.73, is now analyzed by the analyzer. We know that if the analyzer is rotated through 90° from γ, we will obtain a null intensity. Thus, we have

A = +γ 90°, (24.74)

Ellipsometry 541

so

A = −γ 90°. (24.75)

Taking the tangent of both sides of Equation 24.75 yields

tantan

.γ = −1A

(24.76)

Solving now for ρ in Equation 24.73 using Equation 24.76 and factoring out [cosc cos(c –P)] from numerator and denominator yields

ρ = + −− −

tan [tan tan( )]tan tan( )

,A c a P c

a c P cc

c 1 (24.77)

where we have expressed Equation 24.77 with the argument P – c rather than c – P, as is customary in ellipsometry.

Equation 24.77 enables us to determine ρ from A, P, c and knowledge of ac. As an example of Equation 24.77, suppose that we use a perfect quarter-wave retarder so that a ic = −( )1 . Furthermore, suppose that Ρ is measured to be 60°, c = 30°, and A = 45°. Substituting these values into Equation 24.77, we find that

ρ =− −3 2 3

5

i. (24.78)

Equating Equations 24.78 through Equation 24.12 we find that

ψ = ( ) =−tan .1 35

37 8°, (24.79)

∆ = ( ) =−tan 2 63 41 . °. (24.80)

Because Equation 24.77 is so easy to use, it is probably the simplest way to determine the ellipso-metric parameters ψ and Δ from ρ.

As we mentioned, other ellipsometric configurations can be conceived, for example, placing the compensator in the analyzing arm. For a variety of reasons, the most popular configuration is the one discussed here. Further information on the measurement of the ellipsometric parameters can be found in the references.

24.4 SoluTioN of The fuNdameNTal eQuaTioN of elliPSomeTRy

We now turn to the problem of finding a specific form for f(n, κ, d), the right-hand side of the ellip-sometric equation, and then the solution of the fundamental equation of ellipsometry. The model proposed by Drude, and the one that has been used with great success, is that of a homogeneous thin film superposed on a substrate. An optical beam is then incident on the thin film and undergoes mul-tiple reflections within the film. From knowledge of the polarization state of the incident and reflected beams, the refractive index, extinction coefficient, and thickness of the film can be determined.

In order to solve this problem, several related problems must be addressed. The first is to deter-mine the relation between the refractive indices of two different media and the complex relative

542 Polarized Light, Third Edition

amplitude attenuation ρ. In Figure 24.3, we show the oblique reflection and transmission of a plane wave incident on a boundary.

Fresnel’s equations for the reflection coefficients rp and rs can be written as (see Chapter 7)

rn nn n

pi r

i r

= −+

2 1

2 1

cos coscos cos

,θ θθ θ

(24.81)

rn nn n

si r

i r

= −+

1 2

1 2

cos coscos cos

.θ θθ θ

(24.82)

The complex relative amplitude attenuation ρ is defined to be

ρ = r

rp

s

. (24.83)

Substituting Equations 24.81 and 24.82 into Equation 24.83 gives

ρ θ θθ θ

θ θ= = −+( ) +r

rxx

xp

s

i r

i r

i rcos coscos cos

cos coscoss cos

,θ θi rx−( ) (24.84)

where x = n2/n1. The refractive angle θr can be eliminated from Equation 24.84 by using Snell’s Law, which we write as

sinsin

,θ θr

i

x= (24.85)

so Equation 24.84 can then be rewritten as

ρ θ θθ θ

= = − −+ −

r

rx xx x

p

s

i i

i i

2 2 2

2 2 2

cos sincos sin

coss sincos sin

.θ θθ θ

i i

i i

xx

+ −− −

2 2

2 2 (24.86)

θi

θr

Medium 1Medium 2

n1

n2

figuRe 24.3 Oblique reflection and transmission of a plane wave at the planar interface between two semi-infinite media 1 and 2.

Ellipsometry 543

We now set

a b xi i= = −cos sin ,θ θand 2 2 (24.87)

and let,

U a x b= −2 2 2, (24.88)

V ab x= −( )1 2 , (24.89)

so that with these substitutions, Equation 24.86 becomes

ρ = +−

U VU V

. (24.90)

Setting f = U/V, we solve Equations 24.88 and 24.89 for x2 and find that

xf

ii2 2

2

21= +

sintan

.θ θ (24.91)

Equation 24.90 can be solved for f in terms of ρ, and we find that

f = +−

11

ρρ

. (24.92)

Finally, from x = n2/n1 and Equation 24.92, we see that Equation 24.91 then becomes

nn

i i2

1

22

1 2

111

= + −+

sin tan ,/

θ ρρ

θ (24.93)

which is the desired relation between n2, n1, and ρ. A slightly different form of Equation 24.93 can be written by replacing tan θi with (sinθi/cosθi). A little bit of further algebra then leads to

nn

i i2

12

21 2

14

1= −

+

tan( )

sin ./

θ ρρ

θ (24.94)

The elimination of the refractive angle θr is advantageous from a computational point of view because it is easier to evaluate Equation 24.93 or Equation 24.94 in terms of ρ rather than a com-plex angle.

We recall that, for materials with a real refractive index n, Fresnel’s reflection coefficient at the Brewster angle θib is rp = 0, so ρ = 0. We then see that Equation 24.94 reduces to

tan .θib

n

n= 2

1

(24.95)

For a medium such as air, whose refractive index is practically equal to 1, Equation 24.95 becomes

n ib= tanθ , (24.96)

which is the usual form of Brewster’s Law.

544 Polarized Light, Third Edition

It is of interest to solve Equation 24.94 for ρ and then investigate the behavior of ρ as a function of the incidence angle θi. Solving Equation 24.94 for ρ leads to a quadratic equation in ρ whose solution is

ρ = − + ± + −( ) ( ),

x y x y xx

2 2

(24.97)

where, when we set n2 = n and n1 = 1,

xn i= −2 2

2tan

,θ

(24.98)

y i

i

= sincos

.4

2

θθ

(24.99)

The positive value of the square root is chosen in Equation 24.97 because, as we shall see, this cor-rectly describes the behavior of ρ. For an incidence angle of θi = 0, Equations 24.97 through 24.99 become

xn

y= = = −2

20 1ρ . (24.100)

The negative value of ρ shows that at normal incidence there is a 180° phase shift between the inci-dent and reflected waves. For the Brewster angle, we find that

x yn

n= =

+=0

10

4

2ρ . (24.101)

The determination of ρ at an incidence angle of θi = 90° can be found from the limiting value as θi → 90°. First, Equation 24.97 is written as

ρ = − +( ) + +( ) −1 1 12y

xyx

. (24.102)

For large values of θi we see that Equations 24.98 and 24.99 can be written as

x i≅ − tan,

2

2θ

(24.103)

y i

i

= sincos

,4

2

θθ

(24.104)

so

yx

i= −2 2sin .θ (24.105)

Equation 24.102 then becomes

ρ θ θ= − − + − −( sin ) ( sin ) .1 2 1 2 12 2 2i i

(24.106)

In the limit as θi → 90°we see that ρ → 1, so we have

x y→ −∞ → ∞ →ρ 1. (24.107)

Ellipsometry 545

This behavior is confirmed in Figures 24.4 and 24.5. In the first figure, a plot is made of ρ(θi) versus θi. We see that ρ = –1 at θi = 0°, ρ = 0 at θ θi ib= (the Brewster angle), and ρ = 1 at θi = 90°. Similarly, in Figure 24.5, a plot is made of the absolute magnitude of ρ(θi).

In terms of measurable quantities, the reflectances Rp and Rs are of practical importance and are defined by

Rp pr= 2, (24.108)

Rs sr= 2, (24.109)

0.8

1

0.4

0.6

0.2

–0.2

–0.4

0ρ

–0.6

–1

–0.8

0 10 20 30 40 50θi (degrees)

60 70 80 90

figuRe 24.4 Plot of the complex relative amplitude attenuation ρ, Equation 24.97 as a function of incident angle θi.

0.9

1

0.7

0.8

0.6

0.4

0.2

0.3

0.5ρ

0

0.1

0 10 20 30 40 50 60 70 80 90θi (degrees)

figuRe 24.5 Plot of the absolute magnitude of ρ, Equation 24.97, as a function of θi.

546 Polarized Light, Third Edition

which gives the fraction of the total intensity of an incident plane wave that appears in the reflected wave for the p and s polarizations.

At this point it is of interest to use Equation 24.94 to determine the complex refractive index of a material for a specific angle of incidence. We see that at an incidence angle of 45° and for n1 = 1, Equation 24.94 reduces to the simple form

n22

2

2

11

= ++

ρρ( )

. (24.110)

The complex relative amplitude attenuation ρ can be written as

ρ = +a ib. (24.111)

Substituting Equations 24.111 into 24.110 and grouping terms into real and imaginary parts yields

ˆ ( ) ( )

,

ncE dF i cF dE

E F

A ib

22

2 2= + − −

+

= − (24.112)

where we now include the caret above the n to denote it as a complex quantity, and

AcE dFE F

= ++2 2

, (24.113)

bcF dEE F

= −+2 2

, (24.114)

and

c a b= + −1 2 2, (24.115)

d ab= 2 , (24.116)

E a b= +( ) −1 2 2, (24.117)

F b a= +( )2 1 . (24.118)

We recall that n2 is complex and defined in terms of its real refractive index n and extinction coef-ficient κ as

ˆ .n n i2 1= −( )κ (24.119)

We can now find n and κ in terms of A and b by equating the square of Equation 24.119 to Equation 24.112, that is,

ˆ ( ) .n n i A ib22 2 21= − = −κ (24.120)

Ellipsometry 547

Expanding Equation 24.120 and equating the real and imaginary parts yields

n n A2 2 2− =κ , (24.121)

2 2n bκ = . (24.122)

Equations 24.121 and 24.122 then lead to a quadratic equation in n2

n Anb4 2

2

40− − = , (24.123)

whose solution is

nA A b2

2 2

2= ± +

, (24.124)

and for κ, Equation 24.122,

κ =± +

bA A b2 2

. (24.125)

For real refractive indices, b must be zero, so we must choose the positive sign in Equation 24.124 and we have

n A2 = , (24.126)

κ = 0. (24.127)

We can now consider a specific example. In Section 24.3, we saw that ellipsometric measure-ments on a material led to a value for ρ of

ρ =− −3 2 3

5

i. (24.128)

From Equation 24.111 to Equation 24.125, the complex refractive index n2 is then found to be

ˆ . . .n i2 3953 1 4641= −( )0 0 (24.129)

Equation 24.94 is very important because, in practice, thin films are deposited on substrates and the complex refractive index of the substrate, written n2, must be known in order to characterize the thin film.

In the problem described, we have assumed that the incident beam propagates in medium 0 and is reflected and transmitted at the interface of medium 0 and 1. We can denote the reflection and transmission coefficients at the interface by r01 and t01; by convention, the order of the subscripts denotes that the beam is traveling from the medium represented by the first subscript (0) to the medium represented by the second subscript (1). If the incident beam is propagating in medium 1, and is reflected and transmitted at the interface of medium 0, then the reflection and transmission coefficients are denoted by r10 and t10, respectively. It is necessary to know the relation between these coefficients. A direct way to do this is to interchange n1 and n2 in Fresnel’s reflection and transmission equations. Another method, due to Stokes, is not only elegant but very novel and is

548 Polarized Light, Third Edition

given in Section 24.4.1. If the ambient medium is designated by 0 and the film by 1, then the rela-tions are found

r r1 10 0= − , (24.130)

t t r01 10 0121= − . (24.131)

With this background, we can now consider a specific form for f(n, κ, d) for the thin film deposited on a substrate and very often called the ambient-film-substrate (AFS) system. This system is shown in Figure 24.6. The film has parallel-plane boundaries of thickness d and is sandwiched between semi-infinite ambient and substrate media. The three media are all homogeneous and optically isotropic with complex refractive indices n0, n1, and n2. In most cases, the ambient medium is transparent and n0 is real.

In the figure, the incident beam is seen to undergo multiple reflections and transmissions at the interfaces between the ambient and the thin film, and the thin film and the substrate. We know that there will be destructive or constructive interference for these multiple reflections. The interfer-ence will take place constructively if the phase shift between each of the adjacent beams from the thin film into the ambient medium differs by 2π radians. In order to proceed with the problem, it is necessary to determine the relation between the phase shift between each of the adjacent beams and the film thickness. Figure 24.7 shows the geometry of the path difference between two adja-cent beams.

In Figure 24.7, the path lengths between the two adjacent beams are bd′ and bc + cd. The opti-cal path difference is Δl, so the phase difference is δ = kΔl or,

δ = + − ′k n bc cd bd[ ( ) ], (24.132)

Ambient

Film

Substrate

d

φ0

φ2

φ1

figuRe 24.6 Oblique reflection and transmission of a plane wave by an ambient (0)–film (l) substrate (2) system with parallel-plane boundaries. The film thickness is d, ϕ0 is the angle of incidence in the ambient medium, and ϕ1 and ϕ2 are the angles of refraction in the film and the substrate, respectively.

Ellipsometry 549

where k = 2π/λ and λ is the free-space wavelength of the incident light. We see that

bc cdd

r

= =cos

,θ

(24.133)

bd bd n bdi r′ = =sin ( )sin ,θ θ (24.134)

and

bdd

rr= 2

cossin .

θθ (24.135)

Substituting Equations 24.133 through 24.135 into Equation 24.132 yields

δ πλ

θ= 4 ndrcos . (24.136)

In Figure 24.6, we see that we replace θr by ϕ1, so we have

δ πλ

φ= 41

ndcos . (24.137)

If δ = 2π, then there is constructive interference between the adjacent beams, that is, the waves are in phase with one another. Similarly, if δ = π, there is destructive interference, so the waves are completely out of phase with one another.

Equation 24.137 is readily expressed in terms of the incident angle ϕ0. From Snell’s Law, we see that Equation 24.137 can be written as

δ πλ

φ= −412

02 2

01 2d

n n( sin ) ./ (24.138)

We must now add all the contributions of the beams contributing to the total reflected beam. For the moment, we ignore the polarizations s and p; they will be restored later. If the incident field is written as E0, then we see that the first four beams are

E r E1 01 0= , (24.139)

d

θi

θr

E0E1

E2

B

C

D

D′

figuRe 24.7 Geometry of the path difference between two adjacent beams on reflection at oblique inci-dence by front and back surfaces of a thin film.

550 Polarized Light, Third Edition

E t t r e Ei2 01 10 12 0= − δ , (24.140)

E t t r r e Ei3 01 10 10 12

2 20= − δ , (24.141)

E t t r r e Ei4 01 10 10

2122 3

0= − δ , (24.142)

so the total field Ε is

E r E t t r e E t t r r e E ti i= + + +− −01 0 01 10 12 0 01 10 10 12

2 20 0

δ δ11 10 10

2122 3

0t r r e Ei− δ . (24.143)

We can write all the terms after the first term r0lE0 for Ν beams as

t t r e r r e r r e ri i i N01 10 12 10 12 10

2122 2

101− − −+ + + +δ δ δ[ −− − − −112

1 1r eN i N( ) ].δ (24.144)

The terms within the brackets can be written as

S x x x N= + + + + −1 2 1 , (24.145)

where

x r r e i= −10 12

δ . (24.146)

Equation 24.145 is a geometric sum. The solution is readily obtained by multiplying Equation 24.145 through by x, that is,

xS x x x x N= + + + +2 3 , (24.147)

and then subtracting Equation 24.147 from Equation 24.145 to obtain

Sxx

N

= −−

11

. (24.148)

The factor x is always less than 1, so that for an infinite number of beams, N → ∞ and the limiting value of S in Equation 24.148 is

Sx

=−1

1. (24.149)

We see that Equation 24.143 becomes

r rt t r e

r r e

i

i= +

−

−

−0101 10 12

10 121

δ

δ, (24.150)

or

rr r e

r r e

i

i= +

+

−

−01 12

01 121

δ

δ, (24.151)

where r = E/E0 and we have used Stokes’s relations r10 = –r01 and t t r01 10 0121= − . We observe that

Stokes’s relations are extremely important because they not only enable us to determine the correct

Ellipsometry 551

signs between the coefficients, but they also allow us to express R in terms of r01 and r12 only, the reflection coefficients for the ambient–film (0-1) interface and the film–substrate (1-2) interface, respectively.

Equation 24.151 is valid when the incident wave is linearly polarized either parallel (p) or per-pendicular (s) to the plane of incidence. We may express the complex reflection coefficients as, add-ing the subscripts for the polarization components,

ρδ

δpp p

i

p pi

r r e

r r e= +

+

−

−01 12

01 121, (24.152)

ρδ

δss s

i

s si

r r er r e

= ++

−

−01 12

01 121, (24.153)

where δ is the same for the p and s polarizations and is given by Equation 24.138. The Fresnel reflec-tion coefficients at the 0–1 and 1–2 interfaces for the p and s polarizations are now

rn nn n

p011 0 0 1

1 0 0 1

= −+

cos coscos cos

,φ φφ φ

(24.154)

rn nn n

p122 1 1 2

2 1 1 2

= −+

cos coscos cos

,φ φφ φ

(24.155)

and

rn nn n

s010 0 1 1

0 0 1 1

= −+

cos coscos cos

,φ φφ φ

(24.156)

rn nn n

s121 1 2 2

1 1 2 2

= −+

cos coscos cos

.φ φφ φ

(24.157)

The three angles ϕ0, ϕ1, and ϕ2 between the directions of propagation of the plane waves in media 0, 1, and 2, and the normal to the film boundaries are related by Snell’s Law, that is,

n n n0 0 1 1 2 2sin sin sin .φ φ φ= = (24.158)

Using Equations 24.154 through 24.158, all the quantities can be found for determining the reflec-tion coefficients r01p, r12p, and r01s, r12s.

We can consider an example of the calculation of these coefficients. For simplicity, so that we can see how a calculation of this type is carried through, let us consider that we have media that are characterized only by real refractive indices, for example, a thin-film dielectric deposited on a glass substrate. Let the ambient medium be represented by air, so that the refractive index is n0 = 1 and the film and substrate refractive indices are n1 = 1.5 and n2 = 2.0, respectively. Further, let the incident angle be ϕ0 = 30°. We then find from Snell’s Law, Equation 24.158, that

φ0 30= °, (24.159)

φ1 19 4712= . °, (24.160)

φ2 14 4775= . °. (24.161)

552 Polarized Light, Third Edition

We now substitute these values along with the corresponding refractive indices into Equations 24.154 through 24.157 and find that

r p01 0 5916= . , (24.162)

r p12 0 6682= . , (24.163)

r s01 0 2679= . , (24.164)

r s12 0 4776= . . (24.165)

Inspecting Equations 24.152 and 24.153, we see that there are only two unknown quantities, the complex amplitude reflection coefficient ρp (or ρs) and δ. If we measure either ρp or ρs, we can deter-mine δ; in practice, we actually measure |ρp|2 and |ρs|2.

The usual problem is to determine the thickness of the thin film d, that is, to determine δ. We can readily determine δ if all the coefficients are real. For example, we can rewrite Equation 24.152 as

ρδ

δp

i

i

a beabe

= ++

−

−1, (24.166)

where

a r b rp p= =01 12 . (24.167)

Multiplying Equation 24.166 by its complex conjugate then gives

ρ δδp

a b aba b ab

22 2

2 2

21 2

= + ++ +

coscos

. (24.168)

Equation 24.168 is readily solved for δ so that

δ ρρ

= + − +−( )

−cos( ) ( )

.12 2 2 2 2

2

12 1

a b a b

abp

p

(24.169)

Thus, by measuring |ρp|2 and with knowledge of a and b from Equation 24.167, we can determine δ and, from Equation 24.138, the film thickness d.

While the above equations are useful, they do not describe the fundamental equation of ellip-sometry. To obtain this equation, we must introduce ρ, which is equal to the ratio of ρp and ρs; that is, dividing Equation 24.152 by Equation 24.153, we have

ρ ρρ

ψδ

δ= = = +

+

−

−p

s

i p pi

p pi

er r e

r r etan ∆ 01 12

01 121 +

+( )−

−

1 01 12

01 12

r r er r e

s si

s si

δ

δ. (24.170)

Equation 24.170 is the fundamental equation of ellipsometry. The right-hand side is the specific form of f(n, κ, d). However, we now see that f(n, k, d) is a very complicated function. In actuality, it relates the measured ellipsometric angles ψ and Δ to the optical properties of a three-phase system, that is, the (complex) refractive indices of the ambient medium ( n0), the film (n1), the substrate (n2), the film thickness (d1) for given values of the vacuum wavelength (λ) of the ellipsometer light beam, and

Ellipsometry 553

the angle of incidence (ϕ0) in the ambient medium; the subscript 1 on d indicates that it is the thick-ness of the film associated with the medium n1. The equation can now be written symbolically as

ρ ψ φ λ= =tan ( ˆ , ˆ , ˆ , , , ).e f n n n di∆0 1 2 1 0 (24.171)

Equation 24.171 may be broken into two real equations for ψ and Δ, that is,

ψ φ λ= −tan ( ˆ , ˆ , ˆ , , , ) ,10 1 2 1 0f n n n d (24.172)

∆ = arg[ ( ˆ , ˆ , ˆ , , , )],f n n n d0 1 2 1 0φ λ (24.173)

where |ρ| and arg ρ are the absolute value and argument (angle of the complex function ρ), respectively.

Azzam and Bashara [6] have correctly stated that “although the function ρ may appear from Equation 24.170 to be deceptively simple, it is, in reality, quite complicated and can be handled satisfactorily only by a digital computer.” In fact, the solution of Equation 24.170 had to wait until the development of digital computers in the 1950s and 1960s. Inspection of Equation 24.171 shows that ρ is, in general, explicitly dependent on nine real arguments; the real and imaginary parts of the three complex refractive indices n0, n1, n2, the film thickness d, the angle of incidence ϕ0, and the wavelength. Not surprisingly, the solution of Equation 24.170 must be obtained in a piecemeal fash-ion following the same development given above for real refractive indices (and, therefore, reflection coefficients). Here, however, the numerical solution is greatly complicated because the reflection coefficients are now complex. Fortunately, computer programs have been developed that enable the complex refractive indices to be determined as well.

In practice, the refractive indices of the ambient medium, thin film, and substrate are very often known, and the quantity of interest is the thickness of the film. The thickness of the thin film, d1, can be found in the following way. We write Equations 24.152 and 24.153 as

ρpa bX

abX= +

+1, (24.174)

ρsc dX

cdX= +

+1, (24.175)

where a, b, c, and d are the complex coefficients in Equations 24.152 and 24.153 and

X e i= − δ . (24.176)

From Equation 24.170, we then have

ρ = + ++ +

( )( )( )( )

,a bX a cdX

abX c dX1 (24.177)

where

( , ) ( , ),a b r rp p= 01 12 (24.178)

( , ) ( , ).c d r rs s= 01 12 (24.179)

554 Polarized Light, Third Edition

Carrying out the multiplication in Equation 24.177, we then find that

ρ = + ++ +

A bX cXd EX FX

2

2, (24.180)

where

A r p= 01 , (24.181)

b r r r rp p s s= +12 01 10 12 , (24.182)

c r r rp s s= 12 01 12 , (24.183)

d r s= 01 , (24.184)

E r r r rs p p s= +12 01 12 01 , (24.185)

F r r rp p s= 01 12 12 . (24.186)

Equation 24.180 can now be written as the quadratic equation

a X a X a22

1 0 0+ + = , (24.187)

where

a F c r r r rp s p s2 12 12 01 01= − = −( )ρ ρ , (24.188)

a E b r r r r r r r rs p p s p p s s1 12 01 12 01 12 01 01 12= − = +( ) − +(ρ ρ )), (24.189)

a d A r rs p0 01 01= − = −ρ ρ . (24.190)

The two solutions of Equation 24.187 are

Xa a a a

a1

1 12

2 0

2

42

= − + −, (24.191)

Xa a a a

a2

1 12

2 0

2

42

= − − −. (24.192)

Thus, we have found a formal solution to the problem. To solve for X1 and X2, we substitute the val-ues of a2, a1, and a0 from Equations 24.188 through 24.190 into Equations 24.191 and 24.192. The result is the complex number

X U iV1 2, .= ± (24.193)

We recall from Equation 24.176 to Equation 24.137 that X1,2 is

Xin d

1 21 1 14

, expcos

.= −( )π φλ

(24.194)

Ellipsometry 555

and furthermore,

n n n1 1 12

0 02 1 2cos ( sin ) ./φ φ= −[ ] (24.195)

Substituting Equation 24.195 into Equation 24.194 yields

Xi n n d

U iV1 212

0 02 1 2

14,

/

exp( sin )

.= − −[ ]

= ±π φ

λ (24.196)

This can be rewritten still further by setting

d n n= −[ ]−λ φ2 1

20 0

2 1 2( sin ) ,/ (24.197)

so that

X idd

1 2 2, exp ,= − ( )

π (24.198)

where we have dropped the subscript 1 on d. We need only iterate d until X1,2 is equal to the right-hand side of Equation 24.196. In order to do this, however, the square roots in Equations 24.191 and 24.192 must first be converted to Cartesian form. We briefly review this process. We express the square roots in Equations 24.191 and 24.192 as

a ib x iy

cei

+ = +

= θ. (24.199)

We square both sides and equate the real and imaginary terms and find that

c a2 2cos ,θ = (24.200)

c b2 2sin .θ = (24.201)

Squaring and adding both sides of Equations 24.200 and 24.201, and solving for c then leads to

c a b= +( )2 2 1 4/ . (24.202)

Next, we divide Equation 24.201 by Equation 24.200 to obtain

tan2θ = ba

. (24.203)

Using the trigonometric identity

tan 2tantan

θ θθ

=−2

1 2, (24.204)

556 Polarized Light, Third Edition

and equating the right sides of Equations 24.203 and 24.204 results in a quadratic equation of the form

b a btan tan2 2 0θ θ+ − = . (24.205)

The solutions are

tan 1θ = − + +a a bb

2 2

, (24.206)

tan 2θ = − − +a a bb

2 2

. (24.207)

We restrict the angle θ to the positive quadrant, so we use the first solution. Constructing the famil-iar right triangle from Equation 24.206, we then find that

sin( )

,/θ = − + ++ − +[ ]a a b

a b a a b

2 2

2 2 2 2 1 22

(24.208)

cos( )

./θ =+ − +[ ]

b

a b a a b2 2 2 2 2 1 2 (24.209)

As an example of the usage of the formulas, we convert 4 3+ i to Cartesian coordinates, that is,

4 3+ = +i x iy. (24.210)

We see that a = 4 and b = 3, and readily find that

4 3 5310

110

12

3+ = +

= +[ ]i i i . (24.211)

The equality is readily checked by squaring both sides of Equation 24.211. The Cartesian form of the square root in Equation 24.191 (or Equation 24.192) is now added (or subtracted) from −a in the numerator. We now have Cartesian forms in the numerator and the denominator. We can then write

Xm ino ip

mo pn i no mpo p

U iV

1 2

2 2

,

.

= ++

= +( ) + +( )

+= +

(24.212)

We can express U + iV in complex polar coordinates and write Equation 24.212 as

exp exp( ),− ( )

= + = −i

dd

U iV A i2π α (24.213)

Ellipsometry 557

where A and α are real quantities and we have

A U V= +2 2 , (24.214)

α = −( )tanV

U. (24.215)

We take the natural logarithm of both sides of Equation 24.213 and obtain

− ( ) = −idd

A i2π α1n , (24.216)

so

dd

i A= +[ ]2π

α 1n , (24.217)

where

d n n= −[ ]−λ φ2 1

20 0

2 1 2( sin ) ./ (24.197)

If n1 is real, then Equation 24.217 can be iterated by using a range of values from d = 0 to d = d until the correct value is found. We also observe that if n1 and n0 are real, there is no imaginary part because d must be real.

If n1 is complex, Equation 24.217 should first be squared and the result separated into its real and imaginary parts. When this is done, we find that

d n n A2 2 202 2

0

22 21

4−( ) −[ ] = ( ) − ( )[ ]κ φ λ

παsin ,ln (24.218)

d n A2 2κ α= − 1n . (24.219)

If n and κ are known, then Equation 24.218 can be iterated until the solution is found. The result can then be checked by using Equation 24.219. However, if both d and the optical constants n and κ are not known, then both equations can be iterated by using a range of values for d, n, and κ until the equations are satisfied. It is clear that this process is tedious at best, but is readily carried out on a digital computer. One can see that it is a time-consuming process even to write a computer program in order to evaluate the appropriate ellipsometric equations presented here. Fortunately, computer programs have been written and are available from manufacturers of ellipsometers.

Archer has carried out a well-known computer solution for the evaluation of ψ and Δ for a trans-parent film on a substrate of a single crystal of silicon [7]. He solved the above equations and made a Cartesian plot of ψ and Δ as shown in Figure 24.8. The constants used in the evaluation were an angle of incidence of 70.00°, a wavelength of 5461 Å, and a complex index of refraction for silicon of 4.050 − 0.028i. Each curve in Figure 24.8 is the locus of points of increasing thickness for a film of fixed index of refraction. The arrows show the direction of increasing thickness, and the underlined numbers are the indices of refraction of the films. A thickness scale is marked off on each curve in 20° increments in δ. The phase shift is denoted by δ, which is measured in degrees and given by

δλ

φ= °( ) −[ ]36012 2 1 2d n sin ,/ (24.220)

558 Polarized Light, Third Edition

and may be used to convert from degrees to angstroms. The δ scales for all of the curves have a common origin at 0°, which is the point ( , )∆ ψ for a film-free silicon surface. The quantities Δ and ψ are cyclic functions of thickness, and the curves repeat periodically with every 180° change in δ. For example, for a film index of refraction of 1.5, the period is 2430 Å.

A significant property of the dependence of Δ and ψ on the index of refraction of the film is that, for all practical cases, no two curves overlap or intersect. Consequently, each point in the plane corresponds to a unique value for the index of refraction of the film. Strictly speaking, curves for very low and very high indices of refraction do intersect, but the extreme values are seldom, if ever, encountered. Although it is an academic point, as the index of refraction becomes indefinitely large, the corresponding curve coincides with the curve marked 100. Only the position of the δ scale on the curve shifts with increasing index of refraction.

The property of uniqueness allows the determination of the thickness and index of refraction of an unknown transparent film from a single measurement of Δ and ψ. Figure 24.8 constitutes a nomogram for translating the measurement into thickness and index of refraction.

To summarize, in the previous section, equations were developed to measure the ellipsometric parameters ψ and Δ; the measurement of these two parameters allows us to determine ρ. In this sec-tion, the appropriate equations were solved to determine the thickness d and the optical constants n and κ from a knowledge of ρ. Specifically, this is accomplished by determining the complex reflectivities, Equations 24.154 through 24.157 along with Equation 24.158. With these values, the quadratic equation for X, Equations 24.187 and 24.176, are solved, where a2, a1, and a0 are given by Equations 24.188 through 24.190. X is an exponential function for d, the thickness of the thin film, and by further algebraic manipulation is determined by using either Equation 24.217 or Equations 24.218 and 24.219.

360

3.0

3.5 5 1.21.1 1.3 1.4 1.481000

0°

20°

40° 60°

80°

160°

140° 120°

100°

2.5 2.0 1.8 1.6 1.5320

280

240

200

160

120

80

40

00 10 20 30 40 50 60 70 80 90

, Degrees

∆, D

egre

es

figuRe 24.8 The dependence of Δ and ψ on the properties of transparent films on silicon. The parameter plotted is the index of refraction of the film (underlined numbers). The thickness scale is marked off in 20° increments in δ. The thickness is given by 15.17 δ/(n2 – 0.8830)1/2 Å. (From Archer, R. J., J. Opt. Soc. Am. 52, 970–7, 1962. With permission from Optical Society of America.)

Ellipsometry 559

Ellipsometry has received wide attention for the past 40 years. The subject has been best described by Azzam and Bashara [6], and their text contains a wealth of information as well as numerous references. In addition, they also treat in detail and with much mathematical skill the subject of polarized light, especially as it relates to ellipsometry. Because of the wide range and applications of ellipsometry, the reader will find the references of great interest. The introduction to ellipsometry presented here should provide the interested reader with the background to read and understand the papers listed in the References [8–14].

24.4.1 STokeS’S TReaTMenT of ReflecTion and RefRacTion aT an inTeRface

In the above discussion, the reflection and transmission coefficients were used in the derivation of the equations of ellipsometry. In particular, it is necessary to know the reflection coefficients for a beam traveling from one medium to another, and vice versa. This problem appears to have first been treated by Stokes. In this section, we derive these relations. A very clear discussion of this derivation has been given by Hecht and Zajac, and we follow their treatment closely [15].

Suppose we have an incident wave of amplitude E0i incident on the planar interface separating two dielectric media as shown in Figure 24.9a. Since r and t are the fractional reflected and trans-mitted amplitudes, respectively (and where ni = n1 and nt = n2), we have E0r = rE0i and E0t = tE0i. Fermat’s principle also allows reversibility, that is, a wave’s direction of propagation can be reversed with the one proviso that there is no energy dissipation (absorption). In the language of physics, one speaks of time-reversal invariance; that is, if a process occurs, the reversed process can also occur.

In Figure 24.9c two incident waves of amplitude rE0i, and tE0i, are shown. A portion of the wave whose amplitude is tE0i is both reflected and transmitted at the interface. Without making any assumptions, let r′ and t′ be the amplitude reflection and transmission coefficients for a wave incident from below (i.e., ni = n2 and nt = n1). Consequently, the reflected portion is r′tE0i, while that transmit-ted is t′tE0i. Similarly, the incoming wave whose amplitude is rE0i splits into segments of amplitude rrE0i and trE0i. If the configuration of Figure 24.9c is to be identical with that of Figure 24.9b, we must have

tt E rrE E′ + =0 0 0 , (24.221)

trE r tE0 0 0+ ′ = , (24.222)

and therefore

tt r′ = −1 2, (24.223)

(a)

n2

n1

E0i rE0i

tE0i

θ1 θ1

θ2

(b) E0i

θ1θ1

θ2

n2

n1

rE0i

tE0i

(c)

θ1θ1

θ2

n2

n1

tE0i

rrE0i

t'tE0irE0i

trE0i

r'tE0i

figuRe 24.9 Stokes analysis of reflection and refraction at an interface.

560 Polarized Light, Third Edition

and

′ = −r r, (24.224)

which are Stokes’s relations used in the main body of the text but written as r′ = r10, r = r01t′ = t10, and t = t01. In their derivation, Hecht and Zajac point out some other subtleties with respect to Stokes’s treatment, and the reader is referred to their text for further discussion.

24.5 fuRTheR deVeloPmeNTS iN elliPSomeTRy: muelleR maTRiX RePReSeNTaTioN of 𝞇 aNd ∆

The foundations of ellipsometry were developed primarily by Paul Drude around 1890. At that time, the types of optical sources available were extremely limited. Furthermore, it was only possible to measure ψ and Δ using the human eye as a detector, and this is only possible using a null-intensity condition. For these reasons, ellipsometry and its mathematical representation was developed under very restrictive conditions, that is, those that presumed constant optical sources that allowed the settings and mechanical dial movements for the generating and analyzing polarizers to be moved relatively slowly until the null-intensity condition was found. In other words, classical ellipsometry can only be done under conditions in which the optical source and the sample (thin film) do not change and there is a considerable amount of time available to make the required measurements.

If we use optical sources of very short duration (e.g., pulsed lasers) or the sample is continually changing (e.g., the continuous deposition of an optical film onto a substrate), then clearly the classi-cal formulation of the measurement process is inadequate. The concepts of representing the optical surface in terms of ψ and Δ are still valid, but a different procedure must be developed for measur-ing these quantities. Ideally, it would be useful to develop a formulation of ellipsometry that is valid regardless of the behavior of the optical source and the type of optical detector. This can be done by reformulating the equations of ellipsometry in terms of the Abcd Mueller matrix and the Stokes polarization parameters. In this final section, we develop this matrix and solve for ψ and Δ in terms of the Stokes parameters.

Consider that we have an optical beam incident on an optical surface. The Stokes vector of the incident beam is

S E E E Es s p p0 = +∗ ∗ , (24.225)

S E E E Es s p p1 = −∗ ∗ , (24.226)

S E E E Es p p s2 = +∗ ∗, (24.227)

S i E E E Es p p s3 = −( )∗ ∗ . (24.228)

Similarly, the Stokes vector of the reflected beam is

′ = +S R R R Rs s p p0∗ ∗ , (24.229)

′ = −S R R R Rs s p p1∗ ∗ , (24.230)

′ = +S R R R Rs p p s2∗ ∗, (24.231)

′ = −( )S i R R R Rs p p s3∗ ∗ . (24.232)

Ellipsometry 561

We saw earlier that the complex reflection coefficients are defined by

ρss

s

RE

= , (24.233)

ρpp

p

R

E= , (24.234)

or

R Ep p p= ρ , (24.5)

R Es s s= ρ . (24.6)

Substituting Equations 24.5 and 24.6 into the equations for the reflected Stokes parameters, Equations 24.229 through 24.232 yields

′ = ( ) + ( )S E E E Es s s s p p p p0 ρ ρ ρ ρ∗ ∗ ∗ ∗ , (24.235)

′ = ( ) − ( )S E E E Es s s s p p p p1 ρ ρ ρ ρ∗ ∗ ∗ ∗ , (24.236)

′ = ( ) + ( )S E E E Es p s p p s p s2 ρ ρ ρ ρ∗ ∗ ∗ ∗, (24.236)

′ = ( ) − ( )[ ]S i E E E Es p s p p s p s3 ρ ρ ρ ρ∗ ∗ ∗ ∗ . (24.238)

We have Equations 24.225 through 24.228 for the input Stokes vector and Equations 24.235–24.238 for the output Stokes vector. The complete equation, with the resulting Mueller matrix, is

′′′′

=

+S

S

S

S

s s p p s s0

1

2

3

12

ρ ρ ρ ρ ρ ρ∗ ∗ ∗ −−− +

+

ρ ρρ ρ ρ ρ ρ ρ ρ ρ

ρ ρ ρ

p p

s s p p s s p p

s p p

∗

∗ ∗ ∗ ∗

∗

0 0

0 0

0 0 ρρ ρ ρ ρ ρρ ρ ρ ρ ρ ρ ρ

s s p p s

s p p s s p p

i

i

∗ ∗ ∗

∗ ∗ ∗

− −( )−( ) +0 0 ρρs

S

S

S

S∗

0

1

2

3

. (24.239)

The matrix has the familiar form of the Abcd matrix. We also saw that

tan//

ψ = R R

E Ep s

p s

0 0

0 0

, (24.240)

∆ = −β α, (24.241)

and

ρ ρρ

ψ= = ∆p

s

ietan . (24.242)

This last relation can be written as

ρ ρ ψp sie= ∆tan . (24.243)

562 Polarized Light, Third Edition

Substituting Equation 24.243 into Equation 24.239, we find that

′′′′

=

+ −S

S

S

S

s s

0

1

2

3

2

2

1 1

ρ ρψ

∗

tan tan22

2 2

0 0

1 1 0 0

0 0 2 2

0 0

ψψ ψ

ψ ψ− +

∆ ∆tan tan

tan tancos sin

−− ∆ ∆

2 2

0

1

2

3tan tanψ ψsin cos

S

S

S

S

. (24.244)

Equation 24.156 represents ψ and Δ in terms of the Abcd Mueller matrix. The matrix can be used regardless of the duration of the optical source, that is, with both c.w. and pulsed optical sources. Because of this general formulation of ψ and Δ, Equation 24.244 is of fundamental importance to ellipsometry. Equation 24.244 can be used to determine ψ and Δ using a specific polarization state of the incident beam. For example, consider an incident beam that is right circularly polarized so that its Stokes vector is

S =

I0

1

0

0

1

. (24.245)

Multiplication of Equation 24.245 with Equation 24.244 then yields the Stokes vector for the reflected beam

′ =

′′′′

=

+

S

S

S

S

S

Is s

0

1

2

3

0

2

1

ρ ρψ

∗

tan2

11−∆∆

tan

tan

tan

2ψψψ

sin

cos

. (24.246)

Solving Equation 24.246 for ψ and Δ in terms of the reflected Stokes parameters, we find that

tanψ = ′ − ′′ + ′

S SS S

0 1

0 1

1 2/

, (24.247)

tan ∆ = ′′

SS

2

3

. (24.248)

We see that by measuring each of the four Stokes parameters of the reflected beam, we can deter-mine ψ and Δ. In forming Equations 24.247 and 24.248, we see that the factor ( )*ρ ρs s I0 2/ cancels out; therefore, we can simply drop the factor ρ ρs sI*

0, but we retain the 1/2 since this allows us to represent a polarizer and retarder in their standard forms. The Abcd or Mueller matrix for ellip-sometry then becomes

M =

+ −− +1

2

1 1 0 0

1 1 0 0

0 0 2

2 2

2 2

tan tan

tan tan

tan

ψ ψψ ψ

ψ ccos sin

sin cos

∆ ∆− ∆ ∆

2

0 0 2 2

tan

tan tan

ψψ ψ

. (24.249)

Ellipsometry 563

The form of Equation 24.249 for an ideal polarizer and an ideal compensator is easily found. For a perfect polarizer, there is no phase shift, so Δ = 0 and Equation 24.249 is written as

Mpol

tan tan

tan tan

t=

+ −− +1

2

1 1 0 0

1 1 0 0

0 0 2

2 2

2 2

ψ ψψ ψ

aan

tan

ψψ

0

0 0 0 2

. (24.250)

Equation 24.250 is another representation of a linear polarizer. As an example of Equation 24.250, an ideal linear horizontal polarizer is described by

Mpol =

12

1 1 0 0

1 1 0 0

0 0 0 0

0 0 0 0

. (24.251)

Comparing Equation 24.251 with 24.250, we see that we must have tan ψ = 0 if these are to be equivalent. According to the definition given by Equation 24.240, this is exactly what we would expect if there were no R0P component but only an R0S component. Similarly, the Mueller matrix for a perfect compensator is

Mcomp =

−

1 0 0 0

0 1 0 0

0 0

0 0

cos sin

sin cos

φ φφ φ

. (24.252)

Comparing Equation 24.252 with Equation 24.249, we see that we must have

M =∆ ∆

− ∆ ∆

1 0 0 0

0 1 0 0

0 0

0 0

cos sin

sin cos

,, (24.253)

and tan2ψ = 1; Equation 24.253 shows that the emerging beam is unattenuated, and the magnitude of the reflected beam is unchanged from the incident beam. This also is the behavior expected of a perfect phase-shifting material. From Equations 24.252 and 24.253, we see that Δ = ϕ as expected.

Let us now determine ψ and Δ in Equation 24.249 by generating an elliptically polarized beam as before using a linear polarizer at angle P and a quarter-wave retarder fixed at +45°. The Stokes vector of the beam incident on the sample is

S =

−

IP

P

0

1

0

2

2

sin

cos

. (24.254)

564 Polarized Light, Third Edition

Multiplying Equation 24.254 by Equation 24.249, we find that the reflected Stokes vector is

′ =

+−

− ∆( )S

I

P0

2

2

1

1

2 2

2 2

tan

tan

tan

tan

2

ψψ

ψψ

sin

cos PP − ∆( )

. (24.255)

This is, of course, the Stokes vector of elliptically polarized light. In order for the reflected light to be linearly polarized, we must have

cos .2 0P − ∆( ) = (24.256)

Equation 24.256 is satisfied if the generating linear polarizer is set to

2 901P − ∆ = °, (24.257)

or

2 902P − ∆ = − °, (24.258)

and solving Equations 24.257 and 24.258 for Δ gives

∆ = −2 901P °, (24.259)

∆ = +2 902P °, (24.260)

so

∆ = + = −2 90 2 902 1P P° °. (24.261)

Equation 24.261 is recognized as the condition that was obtained previously on the measurement of Δ when the problem was treated following the classical formulation in Section 24.3. Subtracting Equation 24.260 from Equation 24.259, we find that

P P2 1 90= − °. (24.262)

We note that for the condition Equation 24.262, the reflected Stokes vector becomes

′ =

+−±

SI0

2

2

1

1

2

tan

tan

tan

0

2

ψψ

ψ, (24.263)

where the + and − sign refers to Equations 24.259 and 24.260, respectively.In order to find tan ψ, or ψ, we now consider the null-intensity condition created by using an

analyzing linear polarizer. The Mueller matrix of the analyzer, oriented at angle Q, is

Ellipsometry 565

M = 12

1 2 2 0

2 2 2 2 0

2

2

cos sin

cos cos cos sin

sin

Q Q

Q Q Q Q

Q ssin cos sin.

2 2 2 0

0 0 0 0

2Q Q Q

(24.264)

We now assume that the angle P has been adjusted so that the reflected beam has become linearly polarized and is represented by Equation 24.263. The intensity of the beam emerging from the ana-lyzer is obtained by multiplying Equation 24.263 by Equation 24.264 and writing the first Stokes parameter as

I QI

Q Qψ ψ ψ ψ, cos sin( ) = +( )+ −( ) ±[ ]0

41 1 2 2 2tan tan tan2 2 ,, (24.265)

where the + sign refers to the P1 condition and the − sign refers to the P2 condition. The null inten-sity conditions for Q1 and Q2 corresponding to P1 and P2 are

I Q Q Qψ ψ ψ ψ1 1 10 1 1 2 2 2, cos sin( ) = = +( )+ −( ) +tan tan tan2 211, (24.266)

I Q Q Qψ ψ ψ ψ1 2 20 1 1 2 2 2, cos sin( ) = = +( )+ −( ) −tan tan tan2 222. (24.267)

Subtracting Equation 24.267 from Equation 24.266 gives

1 2 2 2 2 2 021 2 1 2−( ) −[ ]+ +[ ] =tan tanψ ψcos cos sin sinQ Q Q Q .. (24.268)

Equation 24.268 can only be satisfied if

cos cos ,2 2 01 2Q Q− = (24.269)

and

sin sin .2 2 01 2Q Q+ = (24.270)

Squaring Equations 24.269 and 24.270 and adding the results yields

cos cos sin sin ,2 2 2 2 11 2 1 2Q Q Q Q− = (24.271)

or

cos .2 2 11 2Q Q+( ) = (24.272)

For this to be true, we must have

Q Q2 1= − , (24.273)

or

Q Q2 190= −° , (24.274)

which are exactly the conditions found earlier for the analyzer.

566 Polarized Light, Third Edition

With knowledge of Q1 (or Q2) we can now solve for tan ψ and ψ. We see that Equation 24.266 (or Equation 24.267) can be rearranged as the quadratic equation

1 2 2 2 1 2 012

1 1−( ) + + +( ) =cos sin cos .Q Q Qtan tanψ ψ (24.275)

Equation 24.275 can be solved to obtain

tanψ = −−

sincos

,2

1 21

1

QQ

(24.276)

which reduces to

tanψ = −cot .Q1 (24.277)

The tangent and cotangent functions in Equation 24.277 can be rewritten in terms of their sine and cosine functions. With some algebra and use of an angle difference relation, we obtain

cos ,ψ −( ) =Q1 0 (24.278)

and we finally have

ψ = − = −90 2701 1° °Q Q . (24.279)

Equations 24.279 and 24.261 are of fundamental importance, so they are rewritten here as the pair

∆ = + = −2 90 2 902 1P P° °, (24.261)

ψ = − = −90 2701 1° °Q Q . (24.279)

In the foregoing analysis, the angular settings on the polarizer and the compensator in the gen-erating arm were made so that linearly polarized light rather than elliptically polarized light was reflected from the optical sample. Let us now assume that these adjustments are not carried out first, but that we wish to determine the conditions on the settings such that the intensity of the beam emerging from the analyzer is a minimum, which in this case is zero (the null condition). The inten-sity of the beam is found by multiplying Equations 24.255 and 24.264, so that we have

I P QI

Qψ ψ ψ ψ, , , cos sin∆( ) = +( )+ −( ) +0 2

41 1 2 2tan tan tan2 22 2P Q− ∆( )[ ]sin . (24.280)

The minimum intensity is found from the conditions

∂ ∆( )

∂=I P Q

Pψ, , ,

,0 (24.281)

∂ ∆( )

∂=I P Q

Qψ, , ,

.0 (24.282)

Differentiating Equation 24.280 according to Equation 24.281 leads immediately to

cos ,2 0P − ∆( ) = (24.283)

Ellipsometry 567

which is exactly the same result we obtained in Equation 24.256, that is,

2 90 270P − ∆ = ° °., (24.284)

Next, Equation 24.280 is differentiated according to Equation 24.282 and we find that

tanψ = − +( )= −1 2

2cos

sincot ,

QQ

Q (24.285)

which is identical to Equation 24.277.We see that we can obtain all the previous conditions derived in Section 24.3 relating Δ and ψ

to P and Q. We emphasize that with quantitative optical detectors, the optical surface can be irra-diated, for example, with right circularly polarized light, whereupon the measurement of all four Stokes parameters can then yield Δ and ψ, Equations 24.247 and 24.248.

This concludes our discussion of ellipsometry. We have seen that the Stokes polarization param-eters and the Mueller matrix allow us not only to obtain easily the formulas of classical ellipsom-etry, as was done in previous sections, but to reformulate the subject in a very general way, namely, representing an optical surface in terms of the Abcd, or Mueller, matrix.

RefeReNCeS

1. Rothen, A., The ellipsometer, an apparatus to measure thicknesses of thin surface films, Rev. Sci. Instrum. 16 (1945): 26–30.

2. Drude, P., Ueber Oberflächenschichten I. Theil, Ann. d. Physik und chemie 36 (1889): 532. 3. Drude, P., Bestimmung der optischer Constanten der Metalle, Ann. d. Physik und chemie 39 (1890):

481–554. 4. Holmes, D. A., and D. L. Feucht, Formulas for using wave plates in ellipsometry, J. Opt. Soc. Am. 57

(1967): 466–71. 5. McCrackin, F. L., E. Passaglia, R. R. Stromberg, and H. L. Steinberg, Measurement of the thickness and

refractive index of very thin films and the optical properties of surfaces by ellipsometry, J. Res. Natl. bur. Std. 67 A (1963): 363.

6. Azzam, R. M. A., and N. M. Bashara, Ellipsometry and Polarized Light, Amsterdam: North-Holland, 1977.

7. Archer, R. J., Determination of the properties of films on silicon by the method of ellipsometry, J. Opt. Soc. Am. 52 (1962): 970–7.

8. Hauge, P. S., R. H. Muller, and C. G. Smith, Conventions and formulas for using the Mueller-Stokes calculus in ellipsometry, Surf. Sci. 96 (1980): 81–107.

9. Hauge, P. S., Recent developments in instrumentation in ellipsometry, Surf. Sci. 96 (1980): 108–40. 10. Aspnes, D. E., Optimizing precision of rotating-analyzer ellipsometers, J. Opt. Soc. Am. 6 (1974):

639–46. 11. Aspnes, D. E., and A. A. Studna, Geometrically exact ellipsometer alignment, Appl. Opt. 10 (1971):

1024–30. 12. Moritani, Α., Y. Okuda, H. Kubo, and J. Nakai, High-speed retardation modulation ellipsometer, Appl.

Opt. 22 (1983): 2429–36. 13. Collett, E., Determination of the ellipsometric characteristics of optical surfaces using nanosecond laser

pulses, Surf. Sci. 96 (1980): 156–67. 14. Jellison, Jr., G. E., and D. H. Lowndes, Time-resolved ellipsometry, Appl. Opt. 24 (1985): 2948–55. 15. Hecht, Ε., and A. Zajac, Optics, Reading, MA: Addison-Wesley, 1974.

569

25 Form Birefringence and Meanderline Retarders

25.1 iNTRoduCTioN

In this chapter, we very briefly describe two alternative methods of creating optical elements that produce effective retardances, form birefringent retarders and meanderline retarders. Form birefrin-gence comes about from the anisotropic but ordered arrangement of isotropic materials where scale of the materials in at least one dimension is small relative to wavelength [1]. Theoretical descrip-tion of form birefringence is accomplished using electromagnetic theory and solid state physics. Meanderline elements rely on geometric arrangements of metallic wires on transparent substrates, and design considerations are based on electromagnetic principles, similar to that for antennas in the radio frequency domain.

25.2 foRm biRefRiNgeNCe

An effective birefringence can occur from ordered arrangements of isotropic materials if the fea-ture size is small compared to the wavelength. An example of such an arrangement is shown in Figure 25.1. Here we have alternate layers of thin parallel plates of two different isotropic materials of indices of refraction n1 and n2 of thicknesses d1 and d2, respectively. A unit of this structure is a pair of layers of period Λ = d1 + d2. We can define the fractions of the period that each material occupies as

fd

fd

f11

22

11= = = −Λ Λ

. (25.1)

When the thicknesses of the layers are small compared to the wavelength, that is,

Λλ λ

= + <d d1 2 0 1. , (25.2)

then formulas for effective dielectric constants can be derived [1] based on electric fields and dis-placements parallel and perpendicular to the plates. The index of refraction corresponding to the dielectric constant parallel to the plane of the plates is

n f n f n0 1 12

2 22

12= +( ) , (25.3)

and the index corresponding to the dielectric constant perpendicular to the plane of the plates is

nf

nfn

e = +

−1

12

2

22

12. (25.4)

570 Polarized Light, Third Edition

The birefringence, which can be shown to be negative, is

∆ = −n n ne o , (25.5)

so that this structure is effectively a negative uniaxial material where the optic axis is perpendicular to the plates as shown in Figure 25.1. A practical example of the fabrication of this type of form birefringent retarder is given by Kitagawa and Tateda [2] where 50 periods of SiO2 and Ta2O5 were sputtered on a SiO2 substrate. A birefringence of 0.13 was achieved.

In order for the planar dielectric to exhibit birefringence, the optical beam must enter the form birefringent element at some angle to the optic axis so that the substrate is tilted in the optical system. Other designs allow the beam to be normal to the form birefringent elements. In this case, one-dimensional surface relief gratings are formed on a planar surface [3,4] as in Figure 25.2. These gratings have a period that is much smaller than the wavelength, and diffraction orders greater than zero are evanescent. Period, depth of the structure, duty cycle, material, and shape of the surface relief structure are all variables for the design process.

Modeling and analysis of form birefringent elements can be accomplished using effective-me-dium theory [5] or rigorous coupled-wave analysis [6–9], with the latter probably preferred in most cases, since it produces the exact solution of Maxwell’s equations for electromagnetic diffraction by the grating structures.

25.3 meaNdeRliNe elemeNTS

Metallic patterns known as meanderline wave plates have been known for some time as retarders in radio frequency applications. With advances in lithography techniques, meanderline elements have

Optic axis

Substrate

n1n2d1

d2

figuRe 25.1 Form birefringent retarder composed of thin parallel plates.

Form Birefringence and Meanderline Retarders 571

been more recently fabricated for the thermal infrared. In form, the elements appear to be classical Greek design elements. An example of a meanderline structure designed to be a 90° retarder for the 8–12 μm region [10] is shown in Figure 25.3. The structures were fabricated of gold on silicon. Meanderline retarders have been shown to be fairly achromatic (±3°) over a range of 2 μm [11], and have consistent performance over angles of incidence from 0° to 60°. The design of these elements

Λ << λ

figuRe 25.2 Surface relief grating.

pw = 0.9 µm

ph = 0.8 µm

1.45 µm

w = 0.6 µm

figuRe 25.3 Example of meanderline geometry.

572 Polarized Light, Third Edition

relies on a design program called Periodic Method of Moments [12,13] supplemented with mea-sured frequency-dependent permittivity and conductivity for the metallic and dielectric layers [10]. Along the meanderline axis they act as inductors while perpendicular to the axis they act as capaci-tors. Many more details on meanderline elements are given in Tharp [14].

RefeReNCeS

1. Born, M., and E. Wolf, Principles of Optics, Oxford: Pergamon, 1975. 2. Kitagawa, M., and M. Tateda, Form birefringence of SiO2/Ta2O5 periodic multilayers, Appl. Opt. 24

(1985): 3359–62. 3. Richter, I., P.-C. Sun, F. Xu, and Y. Fainman, Design considerations of form birefringent microstructures,

Appl. Opt. 34 (1995): 2421–9. 4. Xu, F., R.-C. Tyan, P.-C. Sun, Y. Fainman, C.-C. Cheng, and A. Scherer, Fabrication, modeling, and char-

acterization of form-birefringent nanostructures, Opt. Lett. 20 (1995): 2457–9. 5. Campbell, G., and R. K. Kostuk, Effective-medium theory of sinusoidally modulated volume holograms,

J. Opt. Soc. Am. A 12 (1995): 1113–7. 6. Moharam, M. G., and T. K. Gaylord, Diffraction analysis of dielectric surface-relief gratings, J. Opt. Soc.

Am. 72 (1982): 1385–92. 7. Moharam, M. G., and T. K. Gaylord, Rigorous coupled-wave analysis of grating diffraction—E-mode

polarization and losses, J. Opt. Soc. Am. 73 (1983): 451–5. 8. Moharam, M. G., E. B. Grann, D. A. Pommet, and T. K. Gaylord, Formulation for stable and efficient

implementation of the rigorous coupled-wave analysis of binary gratings, J. Opt. Soc. Am. A 12 (1995): 1068–76.

9. Moharam, M. G., D. A. Pommet, E. B. Grann, and T. K. Gaylord, Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: Enhanced transmittance matrix approach, J. Opt. Soc. Am. A 12 (1995): 1077–86.

10. Tharp, J. S., J. Alda, and G. D. Boreman, Off-axis behavior of an infrared meander-line waveplate, Opt. Lett. 32 (2007): 2852–4.

11. Tharp, J. S., J. M. Lopez-Alonso, J. C. Ginn, C. F. Middleton, B. A. Lail, B. A. Munk, and G. D. Boreman, Demonstration of a single-layer meanderline phase retarder at infrared, Opt. Lett. 31 (2006): 2687–9.

12. Munk, B. A., Frequency Selective Surfaces, Theory and design, New York: Wiley-Interscience, 2000. 13. Munk, B. A., Finite Antenna Arrays and FSS, New York: Wiley-IEEE Press, 2003. 14. Tharp, J. S., design and demonstration of Meanderline Retarders at Infrared Frequencies, PhD

Dissertation, Orlando, FL: University of Central Florida, 2007.

IVPart

Classical and Quantum Theory of Radiation by Accelerating Charges

575

26 Introduction to Classical and Quantum Theory of Radiation by Accelerating Charges

In Part I of this book, we dealt with the polarization of the optical field and the phenomenological interaction of polarized light with optical components, that is, polarizers, retarders, depolarizers, and rotators. All this was accomplished with only the classical theory of light. By the mid-nineteenth century, Fresnel’s theory of light was a complete triumph. The final acceptance of the wave theory took place when Stokes showed that the Fresnel–Arago interference laws could also be explained and understood on the basis of classical optics. Most importantly, Stokes showed that unpolarized light and partially polarized light were completely compatible with the wave theory of light. Thus, polarized light played an essential role in the acceptance of this theory. We shall now see how polar-ized light was again to play a crucial role in the acceptance of an entirely new theory of the optical field, Maxwell’s theory of the electrodynamic field.

In spite of all of the successes of Fresnel’s theory, there was an important problem that classical optics could not treat. We saw earlier that the classical optical field was described by the wave equation. This equation, however, says nothing about the source of the optical field. In 1865, James Clerk Maxwell introduced a totally new and unexpected theory of light [1]. Maxwell’s new theory was difficult to understand because it arose not from the description of optical phenomena but from a remarkable synthesis of the laws of the electromagnetic field. This theory was summarized by expressing all of the known behavior of the electromagnetic field in the form of four differential equations. In these equations, a source term existed in the form of a current j(r, t) along with a new term postulated by Maxwell, namely, the displacement current, ∂D(r, t)/∂t.

After Maxwell had formulated his equations, he proceeded to solve them. He was completely surprised at his results. First, when either the magnetic or electric field was eliminated between the equations, he discovered that in free space the electromagnetic field was described by the wave equation of classical optics. The next result surprised him even more. It appeared that the electro-magnetic field propagated at the same speed as light. This led him to speculate that, perhaps, the optical field and the electromagnetic field were actually manifestations of the same disturbance, being different only in their frequency.

Maxwell died in 1879. Nearly 10 years later, Heinrich Hertz (1888) carried out a set of very sophisticated and brilliant experiments and confirmed Maxwell’s theory [2]. In spite of Hertz’s verification, however, Maxwell’s theory was not immediately adopted by the optics community for several reasons. Perhaps the most obvious reason was that Hertz confirmed Maxwell’s theory not at optical wavelengths but at millimeter wavelengths. For the optical community, this was not enough. In order for optical scientists to accept Maxwell’s theory, it would have to be proved at optical wave-lengths. Another reason for the slow acceptance of Maxwell’s theory was that for 30 years after the publication of Maxwell’s theory in 1865 nothing had been found that could clearly differentiate between the classical wave theory and Maxwell’s theory. Nothing had appeared in optics that was not known or understood using Fresnel’s theory; no one yet understood exactly what fluorescence

576 Polarized Light, Third Edition

or the photoelectric effect was. There was, however, one very slim difference between the two theories. Maxwell’s theory, in contrast to Fresnel’s theory, showed that in free space only transverse waves existed. It was this very slim difference that sustained the “Maxwellians” for several decades. A third important reason why Maxwell’s theory was not readily embraced by the optics community was that a considerable effort had to be expended to study electromagnetism—a nonoptical sub-ject—in order to understand fundamental optical phenomena. Furthermore, as students to this day know, a fair degree of mathematical training is required to understand and manipulate Maxwell’s equations (this was especially true before the advent of vector analysis). It was, therefore, very understandable why the optics community was reluctant to abandon a theory that explained every-thing in a far simpler way and accounted for all the known observations.

In 1896, less than a decade after Hertz’s experiments, two events took place that overthrew Fresnel’s elastic theory of light and led to the complete acceptance of Maxwell’s theory. The first was the discovery of the electron, the long-sought source of the optical field, and the second was the splitting of unpolarized spectral lines that became polarized when an electron was placed in a magnetic field (the Lorentz–Zeeman effect). In this part of the book, we shall see how polarized light played a crucial role in the acceptance of Maxwell’s theory. We shall use the Stokes parameters to describe the radiation by accelerating electrons and see how the Stokes parameters and the Stokes vector take on a surprising new role in all of this. In the final chapter of this part, we shall show that the Stokes vector can be used to describe both classical and quantum radiating systems, thereby providing a single description of radiation phenomena.

RefeReNCeS

1. Maxwell, J. C., A Treatise on Electricity and Magnetism, 3rd ed., Oxford: Clarendon Press, 1892. 2. Hertz, H., Electric Waves, London: Macmillan, 1893.

577

27 Maxwell’s Equations for Electromagnetic Fields

Maxwell’s equations describe the basic laws of the electromagnetic field. Over the 40 years preced-ing Maxwell’s enunciation of his equations (1865), the four fundamental laws describing the electro-magnetic field had been discovered. They are known as Ampère’s Law, Faraday’s Law, Coulomb’s Law, and the magnetic continuity law. These four laws were cast by Maxwell, and further refined by his successors, into four differential equations:

∇∇ × = ∂∂

H jD

+t

, (27.1)

∇∇ × − ∂∂

EB

=t

, (27.2)

∇∇⋅ =D ρ, (27.3)

∇∇⋅ =B 0. (27.4)

These are Maxwell’s famous equations for fields and sources in macroscopic media: E and H are the instantaneous electric and magnetic fields, D and B are the displacement vector and the magnetic induction vector, and j and ρ are the current and the charge density, respectively. We note that Equation 27.1 without the term ∂D/∂t is Ampère’s Law; the second term in Equation 27.1 was added by Maxwell and is called the displacement current. A very thorough and elegant discussion of Maxwell’s equations is given in the text classical Electrodynamics by J. D. Jackson, and the reader will find the required background to Maxwell’s equations there [1].

When Maxwell first arrived at his equations, the term ∂D/∂t was not present. He added this term because he observed that Equation 27.1 did not satisfy the continuity equation. To see that the addi-tion of this term leads to the continuity equation, we take the divergence of both sides of Equation 27.1, thus

∇∇ ∇∇ ∇∇ ∇∇⋅ × = ⋅ + ∂∂

⋅[ ] ( ) ( ).H j Dt

(27.5)

The divergence of the curl is zero, the left-hand side is zero, and we have

( ) ( ) .∇∇ ∇∇⋅ + ∂∂

⋅ =j Dt

0 (27.6)

Next, we substitute Equation 27.3 into Equation 27.6 and find that

∇∇⋅ + ∂∂

=jρt

0, (27.7)

578 Polarized Light, Third Edition

or

∇∇⋅ + ∂∂

=jρt

0, (27.8)

which is the continuity equation. Equation 27.8 states that the divergence of the current (∇ ⋅ j) is equal to the time rate of change of the creation of charge (−∂ρ/∂t). What Maxwell saw, as Jackson has pointed out, was that the continuity equation could be converted into a vanishing divergence by using Coulomb’s Law, Equation 27.3. Thus Equation 27.3 could only be satisfied if

∇∇ ∇∇⋅ + ∂∂

= ⋅ + ∂∂( ) =j jDρ

t t0. (27.9)

Maxwell replaced j in Ampère’s Law by its generalization, and arrived at a new type of current for the electromagnetic field, that is,

jj jjDD→ + ∂∂t

, (27.10)

for time-dependent fields. The additional term ∂D/∂t in Equation 27.10 is called the displacement current.

Maxwell’s equations form the basis for describing all electromagnetic phenomena. When com-bined with the Lorentz force equation (which shall be discussed shortly) and Newton’s second law of motion, these equations provide a complete description of the classical dynamics of interacting charged particles and electromagnetic fields. For macroscopic media the dynamical response of the aggregates of atoms is summarized in the constitutive relations that connect D and j with E, and H with B; that is, D = εE, j = σE, and B = μH, respectively, for an isotropic, permeable, conducting dielectric.

We can now solve Maxwell’s equations. The result is remarkable and was the primary reason for Maxwell’s belief in the validity of his equations. In order to do this, we first use the constitutive relations:

D E= ε , (27.11)

B H= µ . (27.12)

Equations 27.11 and 27.12 are substituted into Equations 27.1 and 27.2, respectively, to obtain

∇∇ × = + ∂∂

H jEεt

, (27.13)

∇∇ × = − ∂∂

EHµt

. (27.14)

Next, we take the curl of both sides of Equation 27.14 so that

∇∇ ∇∇ ∇∇× ×( ) = − ∂∂

×( )E Hµt

. (27.15)

Maxwell’s Equations for Electromagnetic Fields 579

We can eliminate ∇ × H in Equation 27.15 by using Equation 27.13, and find that

∇∇ ∇∇× × = −∂∂

+ ∂∂( )( ) ,E jEµ ε

t t (27.16)

so

∇∇ ∇∇× × = − ∂∂

− ∂∂

( ) .Ej Eµ µεt t

2

2 (27.17)

From vector analysis we can write the expression

∇∇ ∇∇ ∇∇ ∇∇ ∇∇× × = ⋅ −E E E( ) .2 (27.18)

Equation 27.17 then reduces to

∇∇ ∇∇ ∇∇( ) .⋅ − = − ∂∂

− ∂∂

E Ej E2

2

2µ µε

t t (27.19)

Finally, if there are no free charges then ρ = 0 and Equation 27.3 becomes

∇∇ ∇∇⋅ = ⋅ =D Eε 0,

or

∇∇⋅ =E 0. (27.20)

Thus, Equation 27.19 can be written as

∇∇22

2E

E j− ∂∂

= − ∂∂

µε µt t

. (27.21)

Inspection of Equation 27.21 quickly reveals that, if there are no currents so that j = 0, Equation 27.21 becomes

∇∇22

2E

E= ∂∂

µεt

, (27.22)

which is the wave equation of classical optics. We have found that the electric field E propagates exactly according to the classical wave equation. Furthermore, if we write Equation 27.22 as

∇∇22

2

11

EE= ∂

∂/µε t, (27.23)

580 Polarized Light, Third Edition

then we have

∇∇22

2

2

1E

E= ∂∂v t

, (27.24)

where v2 = c2. The propagation of the electromagnetic field is not only governed by the wave equa-tion but propagates at the speed of light. It was this result that led Maxwell to the belief that the electromagnetic field and the optical field were one and the same.

Maxwell’s equations showed that the wave equation for optics, if his theory was correct, was no longer a hypothesis but rested on firm experimental and theoretical foundations.

The association of the electromagnetic field with light was only a speculation on Maxwell’s part. In fact, there was only a single bit of evidence for its support initially. We saw that in a vacuum we have

∇∇⋅ =E 0. (27.25)

It is easy to show that the solution of Maxwell’s equation gives rise to an electric field whose form is

E E= ⋅ −0ei t( ),kk rr ω (27.26)

where

E u u u= + +E E Ex x y y z z , (27.27)

E u u u0 0 0 0= + +E E Ex x y y z z , (27.28)

k u u u= + +k k kx x y y z z , (27.29)

r u u u= + +x y zx y z , (27.30)

k r⋅ = + +k x k y k zx y z . (27.31)

Substituting Equation 27.26 into Equation 27.25 quickly leads to the relation

k E⋅ = 0, (27.32)

where we have used the remaining equations in Equations 27.27 through 27.31 to obtain Equation 27.32. The wave vector is k and is in the direction of propagation of the field, E. Equation 27.32 is the condition for orthogonality between k and E. Thus, if the direction of propagation is taken along the z axis, we can only have field components along the x and y axes; that is, the field in free space is transverse. This is exactly what is observed in the Fresnel–Arago interference equa-tions. In Maxwell’s theory this result is an immediate consequence of his equations, whereas in Fresnel’s theory it is a defect. This fact was the only known difference between Maxwell’s theory and Fresnel’s theory when Maxwell’s theory appeared in 1865. For most of the scientific community and especially the optics community this was not a sufficient reason to overthrow the highly suc-cessful Fresnel theory. Much more evidence would be needed to do this.

Maxwell’s Equations for Electromagnetic Fields 581

Maxwell’s equations differ from the classical wave equation in another very important respect, however. The right-hand term in Equation 27.21 is something very new. It describes the source of the electromagnetic field or the optical field. Maxwell’s theory now describes not only the propagation of the field but also enables one to say something about the source of these fields, something that no one had been able to say with certainty before Maxwell. According to Equation 27.21 the field E arises from the term ∂j/∂t, the time rate of change of the current. This can be interpreted by noting that the current can be written as

j v= e , (27.33)

where e is the charge and v is the velocity of the charge. Substituting Equation 27.33 into Equation 27.21, we have

∇∇22EE− ∂

∂= ∂

∂=µε µ µ

2

2te

te

vv . (27.34)

The term ∂v/∂t is obviously an acceleration. Thus, the field arises from accelerating charges. In 1865, no one knew of the existence of actual charges let alone accelerating charges, and certainly no one knew how to generate or control accelerating charges. In other words, the term (μe)∂ν/∂t in 1865 was superfluous, and so we are left just with the classical wave equation in optics,

∇∇22EE− ∂

∂=µε

2

20

t. (27.35)

We arrive at the same result from Maxwell’s equations, after a considerable amount of effort, as we do by introducing Equation 27.35 as a hypothesis or deriving it from mechanics. This differ-ence is especially distinctive when we recall that it takes only a page to obtain the identical result from classical mechanics! Aside from the existence of the transverse waves and the source term in Equation 27.21, there was very little motivation to replace the highly successful Fresnel theory with Maxwell’s theory. The only difference between the two theories was that in Fresnel’s theory the wave equation was the starting point, whereas Maxwell’s theory led up to it.

Gradually, however, the nature of the source term began to become clearer. New developments in physics, for example, Lorentz’s theory of the electron (1892), led physicists to search for the source of the optical field. Thus, Equation 27.21 became a fundamental equation of interest. Because it plays such an important role in the discussion of the optical field, Equation 27.21 is also known as the radiation equation, a name that will soon be justified. In general, Equation 27.21 has the form of the inhomogeneous wave equation.

The solution of the radiation equation can be obtained by a technique called the Green’s function method. This is a very elegant and powerful method for solving differential equations in general. However, it is quite involved and requires a considerable amount of mathematical background. Consequently, in order not to detract from our discussions on polarized light, we refer the reader to its solution by Jackson [1]. Here, we merely state the result. Using the Green’s function method, the solution of the radiation equation in the form given by Equation 27.34 is found to be

E rn

n v v( , ) ( ) ,te

c k R= × − ×

4 0

2 3πε (27.36)

where

k = − ⋅1 n v, (27.37)

582 Polarized Light, Third Edition

and n = R/R is a unit vector directed from the position of the charge to the observation point. The geometry of the moving charge is shown in Figure 27.1.

In the next chapter we determine the field components of the radiated field for Equation 27.36 in terms of the accelerating charges.

RefeReNCe

1. Jackson, J. D., classical Electrodynamics, New York: Wiley, 1962.

v(t')

n(t')R(t')

e

P

O

r(t') x

figuRe 27.1 Radiating field coordinates arising from an accelerating charge; P is the observation point.

583

28 The Classical Radiation Field

28.1 field ComPoNeNTS of The RadiaTioN field

Equation 27.36 is valid for any acceleration of the electron. However, it is convenient to describe Equation 27.36 in two different regimes, namely, for nonrelativistic speeds ( / 1)v c and for rela-tivistic speeds v/ 1c( ). The field emitted by an accelerating charge observed in a reference frame where the velocity is much less than the speed of light: that is, the nonrelativistic regime, is seen from Equation 27.36 to reduce to

E r u u v( , ) ( ) ,tec R

= ( ) × ×[ ]4 0

2πε (28.1)

where E(r, t) is the field vector of the radiated field measured from the origin, e is the charge, c is the speed of light, R is the distance from the charge to the observer, u = R/R is the unit vector directed from the position of the charge to the observation point, and v is the acceleration vector of the charge. The relation between the vectors X and u is shown in Figure 28.1.

To apply Equation 28.1, we consider the radiated electric field E in spherical coordinates. Since the field is transverse, we can write

E u u= +E Eθ θ φ φ , (28.2)

where uθ and uϕ are unit vectors in the θ and ϕ directions, respectively. Because we are relatively far from the source, we can take u to be directed from the origin and write u = ur, where ur is the radial unit vector directed from the origin. The triple vector product in Equation 28.1 can then be expanded and written as

u u v u u v vr r r r× × = ⋅ −( ) ( ) . (28.3)

For many problems of interest it is preferable to express the acceleration of the charge v in Cartesian coordinates,

v u u u= + +x y zx y z , (28.4)

where the double dot refers to the second derivative with respect to time. The unit vectors u in spherical and Cartesian coordinates are shown later to be related by

u u u ur x y z= + +sin cos sin sin cos ,θ φ θ φ θ (28.5)

u u u uθ θ φ θ φ θ= + −cos cos cos sin sin ,x y z (28.6)

u u uφ φ φ= − +sin cos ,x y (28.7)

584 Polarized Light, Third Edition

or

u u u ux r= + −sin cos cos cos sin ,θ φ θ φ φθ φ (28.8)

u u u uy r= + +sin sin cos sin cos ,θ φ θ φ φθ φ (28.9)

u u uz r= −cos sin .θ θ θ (28.10)

Using these relations, we readily find that Equation 28.3 expands to

u u v v ur r x y( ) ( cos cos cos sin⋅ − = − + − θ θ φ θ φ zz x ysin ) ( sin cos ).θ φ φφ+ − +u (28.11)

We see that ur is not present in Equation 28.11, so the field components are indeed transverse to the direction of the propagation ur.

An immediate simplification in Equation 28.11 can be made by noting that we shall only be interested in problems that are symmetric in ϕ, and we can conveniently take ϕ = 0. Then from Equations 28.1, 28.2, and 28.11, the transverse field components of the radiation field are found to be

Eec R

x zθ πεθ θ= −

4 02

[ cos sin ], (28.12)

Eec R

yφ πε=

4 02

[ ]. (28.13)

Equations 28.12 and 28.13 are the desired relations between the transverse radiation field compo-nents, Eθ, and Eϕ, and the accelerating charge described by x y z, , .and We note that Eθ, Eϕ, and θ refer to the observer’s coordinate system, and x y z, , and refer to the charge’s coordinate system.

Because we are interested in field quantities that are actually measured, namely the Stokes parameters, in spherical coordinates the Stokes parameters are defined by

S E E E E0 = +φ φ θ θ* *, (28.14)

S E E E E1 = −φ φ θ θ* *, (28.15)

O

Pe

x

n

r(t’)

R(t’)

figuRe 28.1 Vector relation for a moving charge and the radiation field.

The Classical Radiation Field 585

S E E E E2 = +φ θ θ φ* *, (28.16)

S i E E E E3 = −( ),* *φ θ θ φ (28.17)

where i = −1. While it is certainly possible to substitute Equations 28.12 and 28.13 directly into the equations for the Stokes parameters and express them in terms of the acceleration, it is simpler to break the problem into two parts. Namely, we first determine the acceleration and the field com-ponents and then form the Stokes parameters according to Equations 28.14 through 28.17.

28.2 RelaTioN beTWeeN uNiT VeCToR iN SPheRiCal CooRdiNaTeS aNd CaRTeSiaN CooRdiNaTeS

We derive the relation between the vector in a spherical coordinate system and a Cartesian coordi-nate system.

The rectangular coordinates x, y, z are expressed in terms of spherical coordinates r, θ, ϕ by the equations

x x r y y r z z r= = =( , , ) ( , , ) ( , , ).θ φ θ φ θ φ (28.18)

Conversely, these equations can be expressed so that r, θ, ϕ can be written in terms of x, y, z. Any point with coordinates (x, y, z) has corresponding coordinates (r, θ, ϕ). We assume that the correspondence is unique. If a particle moves from a point P in such a way that θ and ϕ are held constant and only r varies, a curve in space is generated. We call this curve the r curve. Similarly, two other coordinate curves, the θ curve and the ϕ curve, are determined at each point as shown in Figure 28.2. If only one coordinate is held constant, we determine successively three surfaces pass-ing through a point in space, these surfaces intersecting in the coordinate curves. It is generally convenient to choose the new coordinates in such a way that the coordinate curves are mutually perpendicular to each other at each point in space. Such coordinates are called orthogonal curvi-linear coordinates.

x

z

y

ur

uθ

uφ

P

figuRe 28.2 Determination of the r, θ, and ϕ curves in space.

586 Polarized Light, Third Edition

Let r represent the position vector of a point P in space. Then

r i j k= + +x y z . (28.19)

From Figure 28.2 we see that a vector vr tangent to the r curve at P is given by

vr r= ∂

∂= ∂

∂( ) ⋅( )r sdsdrr

r , (28.20)

where sr is the arc length along the r curve. Since ∂r/∂sr is a unit vector (this ratio is the vector chord length Δr to the arc length Δsr such that in the limit as Δsr → 0 the ratio is 1), we can write Equation 28.20 as

v ur r rh= , (28.21)

where ur is the unit vector tangent to the r curves in the direction of increasing arc length. From Equation 28.21 we see then that hr = dsr/dr is the length of vr.

Considering now the other coordinates, we write

v u v u v ur r rh h h= = =θ θ θ φ φ φ , (28.22)

so Equation 28.21 can be simply written as

v uk k kh k r= = , , ,θ φ (28.23)

where uk(k = r, θ, ϕ) is the unit vector tangent to the uk curve. Furthermore, we see from Equation 28.20 that

hdsdr r

rr= = ∂

∂r

, (28.24)

hdsd

θθ

θ θ= = ∂

∂r

, (28.25)

hds

dφ

φ

φ φ= = ∂

∂r

. (28.26)

These equations can be written in differential form as

ds h dr ds h d ds h dr r= = =θ θ φ φθ φ. (28.27)

We see that hr, hθ, hϕ are scale factors, giving the ratios of differential distances to the differentials of the coordinate parameters. The calculations of vk from Equation 28.22 leads to the determination of the scale factors from hk = ⎮vk⎮ and the unit vector from uk = vk/hk.

We now apply these results to determining the unit vectors for a spherical coordinate system. In Figure 28.3 we show a spherical coordinate system with unit vectors ur, uθ, and uϕ. The angles θ

The Classical Radiation Field 587

and ϕ are called the polar and azimuthal angles, respectively. We see from the figure that x, y, and z can be expressed in terms of r, θ, and ϕ by

x r y r z r= = =sin cos sin sin cos .θ φ θ φ θ (28.28)

Substituting Equation 28.28 into Equation 28.19, the position vector r becomes

r i j k= + +( sin cos ) ( sin sin ) ( cos ) .r r rθ φ θ φ θ (28.29)

From Equation 28.20 we find that

vr

i j krr

= ∂∂

= + +sin cos sin sin cos ,θ φ θ φ θ (28.30)

vr

i j kθ θθ φ θ φ θ= ∂

∂= + −r r rcos cos cos sin sin , (28.31)

vr

i jφ φθ φ θ φ= ∂

∂= − +r rsin sin sin cos . (28.32)

The scale factors are, from Equations 28.24 to 28.26,

hr

r = ∂∂

=r1, (28.33)

h rθ θ= ∂

∂=r

, (28.34)

h rφ φθ= ∂

∂=r

sin , (28.35)

x

z

y

ur

uφ

uθ

P

x

y

z

rO

θ

φ

figuRe 28.3 Unit vectors for a spherical coordinate system.

588 Polarized Light, Third Edition

Finally, from Equation 28.30 through 28.35 the unit vectors are

uv

i j krr

rh= = + +sin cos sin sin cos ,θ φ θ φ θ (28.36)

uv

i j kθθ

θθ φ θ φ θ= = + −

hcos cos cos sin sin , (28.37)

uv

i jφφ

φφ φ= = − +

hsin cos , (28.38)

which corresponds to the results given by Equations 28.5 through 28.7 where the notation for the unit vectors has changed from ux, uy, uz to i, j, k.

We can easily check the direction of the unit vectors shown in Figure 28.3 by considering Equations 28.36 through 28.38 at, say, θ = 0° and ϕ = 90°. For this condition, these equations reduce to

u kr = , (28.39)

u jθ = , (28.40)

u iφ = − , (28.41)

which is exactly what we would expect according to Figure 28.3.An excellent discussion of the fundamentals of vector analysis can be found in the text by

Hildebrand [1] given in the references at the end of this chapter. The material presented here was adapted from Chapter 6 of that reference.

28.3 RelaTioN beTWeeN PoyNTiNg VeCToR aNd STokeS PaRameTeRS

Before we proceed to use the Stokes parameters to describe the field radiated by accelerating charges, it is useful to see how the Stokes parameters are related to the Poynting vector and Larmor’s radia-tion formula in classical electrodynamics.

In the discussion of Young’s interference experiment in Chapter 12, it was pointed out that two ideas were borrowed from mechanics. The first was the wave equation. Its solution alone, however, was found to be insufficient to arrive at a mathematical description of the observed interference fringes. In order to describe these fringes, another concept was borrowed from mechanics, namely, energy. Describing the optical field in terms of energy did lead to results in complete agreement with the observed fringes with respect to their brightness and spacing. However, the wave equation and the intensity formulation were only accepted as hypotheses. In particular, it was not at all clear why the quadratic averaging of the amplitudes of the optical field led to the correct results. In short, neither aspect of the optical field had a theoretical basis.

With the introduction of Maxwell’s equations, which were a mathematical formulation of the fundamental laws of the electromagnetic field, it was possible to show that these two hypothe-ses were a direct consequence of Maxwell’s theory. The first success was provided by Maxwell himself, who showed that the wave equation of optics arose directly from his field equations. In addition, he was surprised that his wave equation showed that the waves were propagating with the speed of light. The other hypothesis—that the energy is found by taking time averages of the

The Classical Radiation Field 589

quadratic field components—was also shown around 1885 by Poynting to be a direct consequence of Maxwell’s equations. We now show this by returning to Maxwell’s equations where we now write those Equations 27.1 through 27.4 for free space as

∇∇ × = − ∂∂

E µ Ht

, (28.42)

∇∇ × = ∂∂

HEεt

, (28.43)

∇∇⋅ =E 0, (28.44)

∇∇⋅ =B 0, (28.45)

and where we have also used the constitutive Equations 27.11 and 27.12. First, we take the scalar product of Equation 28.42 and H so that we have

H E HH⋅ × = − ⋅ ∂

∂∇∇ µ

t. (28.46)

Next, we take the scalar product of Equation 28.43 and E so that we have

E H EE⋅ × = ⋅ ∂

∂∇∇ ε

t. (28.47)

We now subtract Equation 28.47 from Equation 28.46 and obtain

H E E H HH

EE⋅ × − ⋅ × = − ⋅ ∂

∂− ⋅ ∂

∂∇∇ ∇∇ µ ε

t t. (28.48)

The left-hand side of Equation 28.48 is recognized as the identity

∇∇ ∇∇ ∇∇⋅ × = ⋅ × − ⋅ ×( ) ( ) ( ).E H H E E H (28.49)

The terms on the right-hand side of Equation 28.48 can be written as

HH

H H⋅ ∂∂

= ∂∂

⋅t t

12

( ) (28.50)

and

EE

E E⋅ ∂∂

= ∂∂

⋅t t

12

( ). (28.51)

Then, using Equations 28.49 through 28.51, Equation 28.48 can be written as

∇∇⋅ × + ∂∂

⋅ + ⋅

=( )

( ) ( ).E H

H H E Et

µ ε2

0 (28.52)

590 Polarized Light, Third Edition

Inspection of Equation 28.52 shows that it is identical in form to the continuity equation for current and charge, that is,

∇∇⋅ + ∂∂

=jρt

0. (28.53)

In Equation 28.53 j is the current. We can write the corresponding term for current in Equation 28.52, where we define a quantity S, as

S E H= ×( ), (28.54)

and this vector S is known as the Poynting vector. The second term in Equation 28.52 is interpreted as the time derivative of the sum of the electrostatic and magnetic energy densities. The assumption is now made that this sum represents the total electromagnetic energy even for time-varying fields, so the energy density w is

wH E= +µ ε2 2

2, (28.55)

where

H H⋅ = H 2 (28.56)

and

E E⋅ = E2. (28.57)

Equation 28.52 can now be written as

∇∇⋅ + ∂∂

=Swt

0. (28.58)

S in this equation is analogous to the flow of charge j in the continuity Equation 28.53. Furthermore, if we write Equation 28.58 as

∇∇⋅ = − ∂∂

Swt

, (28.59)

then we can see that the physical meaning of Equation 28.59 is that the decrease in the time rate of change of electromagnetic energy within a volume is equal to the flow of energy out of the volume. Thus, Equation 28.59 is a conservation statement for energy.

We now consider the Poynting vector of Equation 28.54 further. In free space the solution of Maxwell’s equations yields the plane-wave solutions

E r k r( , ) ,( )t E ei t= ⋅ −0

ω (28.60)

H r k r( , ) .( )t H ei t= ⋅ −0

ω (28.61)

The Classical Radiation Field 591

We can use Equation 28.42 to relate E to H

∇∇ × = − ∂∂

EHµ0t

. (28.62)

For the left-hand side of Equation 28.62 we have, using Equation 28.60,

∇∇ ∇∇× = × = ×⋅ −E k Ek r[ ]( )E e ii t0

ω , (28.63)

where we have used the vector identity

∇∇ ∇∇ ∇∇× = × + ×( ) .φ φ φa a a (28.64)

Similarly, for the right-hand side we have

− ∂∂

=µ ωµ0 0H

Ht

i . (28.65)

After some algebra, Equation 28.42 becomes

n EH× =

cε0

, (28.66)

since c = 1

0 0ε µ , k = ω/c, and where

nk=k

. (28.67)

The vector n is the direction of propagation of S. Equation 28.66 shows that n, E, and H are per-pendicular to one another. Thus, if n is in the direction of propagation, then E and H are perpen-dicular to n, so that they are in a plane transverse to the direction of propagation. We now substitute Equation 28.66 into Equation 28.54 and we have

S E n E= × ×cε0[ ( )]. (28.68)

From the vector identity

a b c a c b a b c× × = ⋅ − ⋅( ) ( ) ( ) , (28.69)

we see that Equation 28.68 reduces to

S E E n= ⋅cε0( ) . (28.70)

In Cartesian coordinates the quadratic term in Equation 28.70 is written out as

E E⋅ = +E E E Ex x y y. (28.71)

592 Polarized Light, Third Edition

Thus, Maxwell’s theory leads to quadratic terms, which we associate with the flow of energy.For more than 20 years after Maxwell’s enunciation of his theory in 1865, physicists constantly

sought to arrive at other well-known results from his theory, for example, Snell’s Law of refrac-tion, or Fresnel’s equations for reflection and transmission at an interface. Not only were these fundamental formulas found but their derivations led to new insights into the nature of the optical field. Nevertheless, this did not give rise to acceptance of this theory. An experiment would have to be undertaken that could only be explained by Maxwell’s theory. Only then would his theory be accepted.

If we express E and H in complex terms, then the time-averaged flux of energy is given by the real part of the complex Poynting vector, so

S = ×12

( ),*E H (28.72)

where the brackets ⟨ ⟩ indicate a time averaged quantity.From Equations 28.66 and 28.67 we have

n E H× =* *, (28.73)

and substituting Equation 28.73 into Equation 28.72 leads immediately to

S c= ⋅12

0ε ( ) .*E E n (28.74)

Thus, Maxwell’s theory justifies the use of writing the energy in the light, or the flux, as

I E E E Ex x y y= +* *, (28.75)

for the time-averaged intensity of the optical field.In spherical coordinates the field is written as

E = +E Eθ θ φ φu u , (28.76)

so the Poynting vector Equation 28.74 becomes

Sc

E E E E= +εφ φ θ θ

0

2( ) .* * n (28.77)

The quantity within parentheses is the total intensity of the radiation field, that is, the Stokes param-eter S0. Thus, the Poynting vector is directly proportional to the first Stokes parameter.

Another quantity of interest is the power radiated per unit solid angle, written as

dPd

cR

Ω= ⋅ε0 2

2( ) .*E E (28.78)

We saw that the field radiated by accelerating charges is given by

E n n v= × ×ec R4 0

2πε[ ( )]. (28.79)

The Classical Radiation Field 593

Expanding Equation 28.79 by the vector triple product gives us

E n n v v= ⋅ −ec R4 0

2πε[ ( ) ]. (28.80)

We denote

n v n v⋅ = cos ,Θ (28.81)

where Θ is the angle between n and v. Using Equations 28.80 and 28.81, we then find Equation 28.78 becomes

dPd

eΩ

Θ= 2 2v sin . (28.82)

We saw that the components of the field radiated by accelerating charges is given by

Eec R

x y zθ πεθ φ θ φ θ= + −

4 02

( cos cos cos sin sin ) (28.83)

and

Eec R

x yφ πεφ φ= − +

4 02

( sin cos ). (28.84)

The total radiated power over the sphere is given by integrating Equation 28.78 over a solid angle so that

Pc

E E E E R d d= +∫ ∫ε θ θ φπ π

φ φ θ θ0

0

2

0

2

2( ) sin .* * (28.85)

We easily find that

0

2

0

22

202 4

2416

π π

φ φ θ θ φ ππ ε∫ ∫ = +( ) sin (*E E R d d

ec

x y 2) (28.86)

and

0

2

0

22

202 4

43 16

π π

θ θ θ θ φ ππ ε∫ ∫ =( ) sin

( )(*E E R d d

ec

x 22 2 24+ + y z ). (28.87)

Adding Equations 28.86 and 28.87 yields

0

2

0

22

02

43 4

π π

φ φ θ θ θ θ φπε∫ ∫ + =( ) sin* *E E E E R d de

c44

2( ),r (28.88)

where

r u u u= + +x y zx y z . (28.89)

594 Polarized Light, Third Edition

Substituting Equation 28.88 into Equation 28.85 yields the power radiated by an accelerating charge

Pe

c= 2

3 4

2

03

2

πεr . (28.90)

Equation 28.90 was first derived by J. J. Larmor in 1900 and, consequently, is known as Larmor’s radiation formula.

The material presented in this chapter shows how Maxwell’s equations led to the Poynting vector and then to the relation for the power radiated by the acceleration of an electron expressed by Larmor’s radiation formula. As we shall see in the next chapter, we can now apply these results to obtain the polarization of the radiation emitted by accelerating electrons. Very detailed discussions of Maxwell’s equations and the radiation by accelerating electrons are given in the texts by Jackson [2] and Stratton [3].

RefeReNCeS

1. Hildebrand, F. B., Advanced calculus for Engineers, Englewood Cliffs, NJ: Prentice-Hall, 1949. 2. Jackson, J. D., classical Electrodynamics, New York: Wiley, 1962. 3. Stratton, J. A., Electromagnetic Theory, New York: McGraw-Hill, 1941.

595

29 Radiation Emitted by Accelerating Charges

29.1 STokeS VeCToR foR a liNeaRly oSCillaTiNg ChaRge

We have shown in Chapter 28 how Maxwell’s equations gave rise to the equations of the radiation field and the power emitted by an accelerating electron. We now discuss the polarization of the radiation emitted by specific electron configurations, for example, bound charges and charges mov-ing in circular and elliptical paths.

At the beginning of the nineteenth century, the nature of electric charges was not fully under-stood. The long-sought source of the optical field was finally found when the electron was discovered by J. J. Thomson in 1897. A year after Thomson’s discovery, Pieter Zeeman performed a remarkable experiment by placing radiating atoms in a constant magnetic field. He thereupon discovered that the original single spectral line was split into multiple spectral lines.

Shortly thereafter, Hendrik Lorentz heard of Zeeman’s results. Using Maxwell’s theory and his electron theory, Lorentz then treated this problem. Lorentz’s calculations predicted that the spectral lines should not only be split but also completely polarized. On Lorentz’s suggestions, Zeeman then performed further measurements and completely confirmed the predictions in all respects. It was only after the work of Zeeman and Lorentz (for which they received the Nobel Prize in phys-ics for 1902) that Maxwell’s theory was accepted and Fresnel’s theory of light replaced. We should emphasize that the polarization predictions of the spectral lines played a key part in understanding these experiments. This prediction, more than any other factor, was one of the major reasons for the acceptance of Maxwell’s theory into optics.

In this chapter, we build up to the experiment of Zeeman and the theory of Lorentz. We do this by first applying the Stokes parameters to a number of classical radiation problems. These are the radiation emitted by (1) a charge oscillating along an axis, (2) an ensemble of randomly oriented oscillating charges, (3) a charge moving in a circle, (4) a charge moving in an ellipse, and (5) a charge moving in a magnetic field. In the following chapter we then consider the problem of a randomly oriented oscillating charge moving in a constant magnetic field—the Lorentz–Zeeman effect.

We consider a bound charge oscillating along the z axis as shown in Figure 29.1. The motion of the charge is described by

d zdt

z2

2 02 0+ =ω . (29.1)

The solution of Equation 29.1 is

z t z t( ) = ( ) +0 0cos( ),ω α (29.2)

where z(0) is the maximum amplitude and α is an arbitrary phase constant. Because we shall be using the complex form of the Stokes parameters, we write Equation 29.2 as

z t z ei t( ) ( ) ,( )= +0 0ω α (29.3)

596 Polarized Light, Third Edition

where it is understood that by taking the real part of Equation 29.3, we recover Equation 29.2; that is,

Re cosz t z t( )[ ] = ( ) +0 0( ).ω α (29.4)

The radiation field equations from Chapter 28 are

Eec R

x zθ πεθ θ= −

4 02

[ cos sin ] (29.5)

Eec R

yφ πε=

4 02

[ ]. (29.6)

Recall that these equations refer to the observation being made in the x, z plane, that is, at ϕ = 0. The angle θ is the polar angle in the observer’s reference frame.

Performing the differentiation of Equation 29.3 to obtain z , we have

z z ei t= − +ω ω α02 0 0( ) .( ) (29.7)

Substituting Equation 29.7 into Equation 29.5 yields

Eec R

z ei tθ

ω α

πεω θ= +

40

02 0

2 0[ ( )sin ]( ) (29.8)

and

E f = 0. (29.9)

xy

z

θ

O

e

figuRe 29.1 Motion of a linear oscillating charge.

Radiation Emitted by Accelerating Charges 597

The Stokes parameters are defined in a spherical coordinate system to be

S E E E E0 = +φ φ θ θ* *, (29.10)

S E E E E1 = −φ φ θ θ* *, (29.11)

S E E E E2 = +φ θ θ φ* *, (29.12)

S i E E E E3 = −( ).* *φ θ θ φ (29.13)

Substituting Equations 29.8 and 29.9 into these equations yields

Sez

c R0

02

2

04 20

4= ( )( )

sin ,πε

ω θ (29.14)

Sez

c R1

02

2

04 20

4= −( )( )

sin ,πε

ω θ (29.15)

S2 = 0, (29.16)

S3 = 0. (29.17)

We now arrange these last equations in Stokes vector form,

Sez

c R=

−

( )sin

04

1

1

0

00

2

2

204

πεθω

. (29.18)

Equation 29.18 shows that the observed radiation is always linearly vertically polarized light at a frequency ω0, the fundamental frequency of oscillation of the bound charge. Furthermore, when we observe the radiation parallel to the z axis (θ = 0°), the intensity is zero. Observing the radiation per-pendicular to the z axis (θ = 90°), we note that the intensity is a maximum. This behavior is shown in Figure 29.2. In order to plot the intensity behavior as a function of θ, we set

I( ) .θ θ= sin2 (29.19)

In terms of x(θ) and z(θ) we then have

x I( ) ( ) ,θ θ θ θ θ= =sin sin sin2 (29.20)

z I( ) ( ) .θ θ θ θ θ= =cos sin cos2 (29.21)

The term ez(0) in Equation 29.18 is recognized as a dipole moment. A characteristic of dipole radia-tion is the presence of the sin2θ term shown in Equation 29.18. Hence, Equation 29.18 describes the Stokes vector of a dipole radiation field. This type of field is very important because it appears in many types of radiation problems in physics and engineering. Finally, we note that a linearly

598 Polarized Light, Third Edition

oscillating charge gives rise to linearly polarized light. Thus, the state of polarization is a manifesta-tion of the fundamental motion of the electron. This observation will be confirmed for other types of radiating systems.

29.2 STokeS VeCToR foR aN eNSemble of RaNdomly oRieNTed oSCillaTiNg ChaRgeS

In the previous section, we considered the radiation field emitted by a charge or electron oscillating with an angular frequency ω0 about an origin. Toward the end of the nineteenth century a model was proposed for the atom in which an oscillating electron was bound to a positively charged atom. The electron was believed to be negative (from work with free electrons in gases and chemical experiments). The assumption was made that the electron was attracted to the positively charged atom, and the force on the electron was described by Hooke’s Law, namely, –kr. This model was used by Lorentz to solve a number of longstanding problems, for example, the relation between the refractive index and the wavelength, the so-called dispersion relation. The motion of the electron was described by the simple force equation

m kr r= − , (29.22)

or

r r+ =ω02 0, (29.23)

where m is the mass of the electron, k is the restoring force constant, and the angular frequency is ω0

2 = k m/ . We saw in Part I of this book that the nature of unpolarized light was not well understood throughout most of the nineteenth century. We shall now show that this simple model for the motion of the electron within the atom leads to the correct Stokes vector for unpolarized light.

The treatment of this problem can be considered to be among the first successful applications of Maxwell’s equations in optics. This simple atomic model provides a physical basis for the source

0.4

0.5

0.3

0.1

0.2

–0.1

0–1.00 –0.80 –0.60 –0.40 –0.20 0.00 0.20 0.40 0.60 0.80 1.00

z(θ)

–0.3

–0.2

–0.5x (θ)

–0.4

figuRe 29.2 Plot of the intensity behavior of a dipole radiation field.

Radiation Emitted by Accelerating Charges 599

term in Maxwell’s equations. The model leads to the appearance of unpolarized light, a quantity that was a complete mystery up to the time of the electron. Thus, an ensemble of oscillating charges bound to a positive nucleus and randomly oriented gives rise to unpolarized light.

We now determine the Stokes vector of an ensemble of randomly oriented, bound, charged oscil-lators moving through the origin. This problem is treated by first considering the field emitted by a single charge oriented at the polar angle α and the azimuthal angle β in the reference frame of the charge. An ensemble average is then taken by integrating the radiated field over the solid angle sinα dα dβ. The diagram describing the motion of a single charge is given in Figure 29.3.

The equations of motion of the charged particle can be written immediately from Figure 29.3 and are

x t A ei t( ) sin sin ,= α β ω0 (29.24)

y t A ei t( ) sin sin ,= α β ω0 (29.25)

z t A ei t( ) cos ,= α ω0 (29.26)

where ω0 is the angular frequency of natural oscillation. Differentiating these equations twice with respect to time gives

x t A ei t( ) sin cos ,= −ω α β ω02 0 (29.27)

y t A ei t( ) sin sin ,= −ω α β ω02 0 (29.28)

z t A ei t( ) cos .= −ω α ω02 0 (29.29)

Substituting Equations 29.27 through 29.29 into the radiation field equations, we find that

EeA e

c R

i t

θ

ωωπε

α β θ α θ= − −02

02

0

4(sin cos cos cos sin ), (29.30)

x

y

z

A

O

α

β

e

figuRe 29.3 Instantaneous motion of an ensemble of oscillating charges.

600 Polarized Light, Third Edition

EeA e

c R

i t

φ

ωωπε

α β= − 02

02

0

4(sin sin ), (29.31)

where θ is the observer’s viewing angle measured from the z axis.The Stokes parameters were defined in terms of the electric field components in Equations 29.10

through 29.13, and if we substitute Equations 29.30 and 29.31 into those equations, we find that the Stokes parameters are given by

S c0

2

= +

−

[

sin cos cos co

sin sin sin cos cos2 2 2 2 2α β α β θ

α α β ss sin cos sin ],θ θ α θ+ 2 2

(29.32)

S c1

2 2 2 2 2sin sin sin cos cos= −

+

[

sin cos cos co

α β α β θ

α α β2 ss sin cos sin ],θ θ α θ− 2 2

(29.33)

S c222 sin sin cos cos cos sin sin sin= −[ ( ],α β β θ α α β θ (29.34)

S3 = 0, (29.35)

where

ceA

c R= ( )4 0

2

2

04

πεω . (29.36)

The fact that S3 is zero in Equation 29.35 shows that the emitted radiation is always linearly polar-ized as we would expect from a model in which the electron only undergoes linear motion.

In order to describe an ensemble of randomly oriented charges, we integrate the Stokes vector elements S0, S1, S2, and S3 over the solid angle sinα dα dβ

⟨ ⟩ = ∫∫ ( )sin ,α α βππ

d d00

2

(29.37)

where ⟨…⟩ is the ensemble average and (…) represents Equations 29.32 through 29.35. Carrying out the integration and forming the Stokes vector, we find that

S =

83 4

1

0

0

00

2

2

04π

πεωeA

c R, (29.38)

which is the Stokes vector for unpolarized light. This is exactly what is observed from natural light sources. Note that the polarization state is always independent of the observer’s viewing angle θ; the observed light always appears to be unpolarized. Thus, this simple model explains the appearance of unpolarized light from optical sources. Unpolarized light can only arise from an ensemble of ran-domly oriented accelerating charges, which can be the case for bound electrons. Electrons moving at a constant velocity, even if the motion is random, cannot give rise to unpolarized light.

Radiation Emitted by Accelerating Charges 601

This simple atomic model received further support when it was used by Lorentz to explain the Lorentz–Zeeman effect, the effect that is observed in the radiation field emitted by a bound electron moving in a constant magnetic field. We emphasize that the motion of a free accelerating electron gives rise to a different result, as we shall see.

29.2.1 noTe on uSe of hooke’S law foR a SiMPle aToMic SySTeM

At first glance the use of Hooke’s Law to describe the motion of a negative electron bound to a posi-tive charge in the nucleus within an atom may appear to be quite arbitrary. The use of Hooke’s Law is based, however, on the following simple atomic model.

The force of attraction between two opposite but equal charges e separated by a distance r is given by

Fe e

rr= + −( )( ),

4 02πε

u (29.39)

where ur is a unit radius vector. The positive charge is located at the origin of a spherical coordinate system.

We now assume that the positive charge is distributed over a sphere of volume V and radius r, so the charge density ρ is

ρπ

= + = +eV

er4 33 /

, (29.40)

or

+ =er4

3

3πρ. (29.41)

Substituting Equation 29.41 into Equation 29.39 gives

F r= −k , (29.42)

where r = rur, and k − eρ/3ε0. Equation 29.42 is Hooke’s Law. Thus, on the basis of this very simple atomic model, the motion of the electron is expected to undergo simple harmonic motion.

29.3 STokeS VeCToR foR a ChaRge RoTaTiNg iN a CiRCle

We now continue with our application of the Stokes parameters to describe radiation problems. In this section we turn our attention to the determination of the field radiated by a charge moving in a circle. This is shown in Figure 29.4. The coordinates of the charge are

x t a t( ) = cosω0 , (29.43)

y t a t( ) = sinω0 , (29.44)

z t( ) = 0. (29.45)

The position of the charge describes a counterclockwise motion as it moves in a circle of radius a.

602 Polarized Light, Third Edition

To express the complex form of the Stokes parameters, the coordinates x(t), y(t), and z(t) must also be expressed in complex form. We have

e t i ti tω ω ω00 0= +cos sin . (29.46)

The real part of Equation 29.46 is cos ω0t. We can also express sin ω0t in terms of the real part of Equation 29.46, Re , by multiplying Equation 29.46 by −i. We see that

Re cos ,e ti tω ω00 = (29.47)

Re sin .− =e ti tω ω00 (29.48)

In complex notation, Equations 29.43 and 29.44 become

x t aei t( ) ,= ω0 (29.49)

y t iaei t( ) ,= − ω0 (29.50)

and the acceleration is then

x t a ei t( ) ,= − ω ω02 0 (29.51)

y t ia ei t( ) .= + ω ω02 0 (29.52)

Substituting Equations 29.51 and 29.52 into the radiation field Equations 29.5 and 29.6 we find that

Eec R

a ei tθ

ω

πεω θ= −

4 02 0

2 0[ cos ], (29.53)

x

y

z

a

eω0t

figuRe 29.4 Motion of a charge moving counterclockwise in a circle of radius a in the x, y plane with an angular frequency ω0.

Radiation Emitted by Accelerating Charges 603

Eec R

ia ei tφ

ω

πεω=

4 02 0

2 0[ ]. (29.54)

Using Equations 29.53 and 29.54 in the expressions for the Stokes parameters, Equations 29.10 through 29.13, and forming the Stokes vector, we obtain

Sea

c R=

+−

4

1

1

0

20

2

2

04

2

2

πεω

θθ

θ

cos

cos

cos

. (29.55)

Equation 29.55 is the Stokes vector for elliptically polarized light. Thus, we see that the radiation is elliptically polarized and is characterized by a frequency ω0, the frequency of rotation of the elec-tron. Furthermore, we see that we have the factor ea in Equation 29.55, the familiar expression for the dipole moment. We observe that Equation 29.55 shows that the orientation angle

ψ = ( )−12

1 2

1

tan ,SS

(29.56)

ψ of the polarization ellipse is always zero. Similarly, the ellipticity angle χ is

χ = ( )−12

1 3

0

sin ,SS

(29.57)

so from Equation 29.55 we have

χ θθ

=+( )−1

22

11

2sin

coscos

. (29.58)

The ellipticity angle is a function of the observation angle θ. We see that for θ = 0°, that is, we view the rotating electron from a position on the z axis, Equation 29.58 becomes χ = 45°. The Stokes vec-tor Equation 29.55 reduces to

S =

24

1

0

0

10

2

2

04ea

c Rπεω , (29.59)

and we observe right circularly polarized light. If we now view the rotating electron perpendicular to the z axis, that is, θ = 90°, we find that χ = 0° and we observe linearly horizontally polarized light. The corresponding Stokes vector is

S =

eac R4

1

1

0

00

2

2

04

πεω . (29.60)

These results agree with our earlier observation that the polarization of the emitted radiation is a manifestation of the motion of the charge. Thus, if we look along the z axis we would see an electron

604 Polarized Light, Third Edition

moving counterclockwise in a circle, so we observe right circularly polarized light. If we look per-pendicular to the z axis, the electron appears to behave as a linear oscillator and we observe linearly horizontally polarized light, in agreement with our earlier conclusion. The linear polarization is to be expected because if we view the motion of the charge as it moves in a circle at θ = 90° it appears to move from left to right and then from right to left, identical to the behavior of a linear oscillator described in Section 29.1. Finally, for θ = 180° we see that Equation 29.58 becomes χ = −45°, so we observe left circularly polarized light.

We can also observe that Equation 29.55 satisfies the equality

S S S S02

12

22

32= + + . (29.61)

The equals sign shows that the emitted radiation is always completely polarized. Furthermore, the degree of polarization is independent of the observation angle θ.

29.4 STokeS VeCToR foR a ChaRge moViNg iN aN elliPSe

It is of interest to consider the case where an electron moves in an elliptical orbit. The equations of motion are

x t a t( ) = cosω0 , (29.62)

y t b t( ) = sinω0 , (29.63)

where a and b are the semimajor and semiminor axes lengths, respectively. In complex notation Equations 29.62 and 29.63 become

x t aei t( ) ,= ω0 (29.64)

y t ibei t( ) .= − ω0 (29.65)

The acceleration is then

x t a ei t( ) ,= − ω ω02 0 (29.66)

y t ib ei t( ) .= ω ω02 0 (29.67)

Again using the radiation field Equations 29.5 and 29.6, the radiated fields are found to be

Ee

c Re ai t

θωω

πεθ= ( ) −0

2

024

0 [ cos ], (29.68)

Ee

c Re ibi t

φωω

πε= ( )0

2

024

0 [ ]. (29.69)

We now form the Stokes vector using Equations 29.68 and 29.69 in the equations for the Stokes vec-tor Equations 29.10 through 29.13 and find that

S =

+−e

c R

b a

b a

ab

4 0

20

2

2

04

2 2 2

2 2 2

πεω

θθ

cos

cos

ccos

.

θ

(29.70)

Radiation Emitted by Accelerating Charges 605

Equation 29.70 is the Stokes vector for elliptically polarized light. We see immediately that if a = b then Equation 29.70 reduces to the Stokes vector for an electron moving in a circle. The orientation angle ψ of the polarization ellipse is seen from Equation 29.70 to be 0°. The ellipticity angle χ is

χ θθ

=+( )−1

2212 2

sincos

cos.

abb a

(29.71)

There are two cases of interest. The radiation is always elliptically polarized except when the observation point is along the z axis, that is, when θ = 90°. We see that for θ = 0°, Equation 29.70 reduces to

S =

+−

ec R

b a

b a

ab

4 0

20

2

2

04

2 2

2 2

πεω

, (29.72)

which is the Stokes vector for elliptically polarized light. When θ = 90°, Equation 29.70 reduces to

S =

ec R

b4

1

1

0

00

2

2

04 2

πεω , (29.73)

which is the Stokes vector for linear horizontally polarized light. Again, this is perfectly under-standable, because at this angle the moving charge appears to be oscillating in a straight line as it moves in its elliptical path.

The Stokes vectors derived here will reappear when we discuss the Lorentz–Zeeman effect.

607

30 Radiation of an Accelerating Charge in the Electromagnetic Field

30.1 moTioN of a ChaRge iN aN eleCTRomagNeTiC field

In previous chapters, the Stokes vectors were determined for charges moving in a linear, circular, or elliptical path. At first sight the examples chosen appear to have been made on the basis of sim-plicity. However, the examples were chosen because charged particles actually move in these paths in an electromagnetic field; that is, the examples are based on physical reality. In this section we show from Lorentz’s force equation that, in an electromagnetic field, charged particles follow linear and circular paths. In the following section we determine the Stokes vectors corresponding to these physical configurations.

The reason for treating the motion of a charge in this chapter as well as in the previous chapter is that the material prepares us to understand and describe the Lorentz–Zeeman effect. Another reason for discussing the motion of charged particles in the electromagnetic field is that it has many important applications. Many physical devices of importance to science, technology, and medicine are based on our understanding of the fundamental motion of charged particles. In particle physics these include the cyclotron, betatron (invented by Don Kerst, one of the author’s professors at the University of Wisconsin–Madison), and synchrotron, and in microwave physics the magnetron and traveling-wave tubes. While these devices will not be discussed here, the mathematical analysis presented is the basis for describing all of them. Our primary interest is to describe the motion of charges as they apply to atomic and molecular systems and to determine the intensity and polariza-tion of the emitted radiation.

In this chapter, we treat the motion of a charged particle in three specific configurations of the electromagnetic field: (1) the acceleration of a charge in an electric field; (2) the acceleration of a charge in a magnetic field; and (3) the acceleration of a charge in perpendicular electric and mag-netic fields. In particular, the motion of a charged particle in perpendicular electric and magnetic fields is extremely interesting not only from the standpoint of its practical importance but because the paths taken by the charged particle are quite beautiful and remarkable. Much of this material is taught in courses on plasma physics, since trajectories and containment of particles are so impor-tant to that field.

In an electromagnetic field, the motion of a charged particle is governed by the Lorentz force equation:

F E v B= + ×q[ ( )], (30.1)

where q is the magnitude of the charge, E is the applied electric field, B is the applied magnetic field, and v is the velocity of the charge. The background to the Lorentz force equation and the phenom-enon of the radiation of accelerating charges can be found in the texts given in the references [1–8]. The text by G. P. Harnwell [3] on electricity and magnetism is especially clear and illuminating.

608 Polarized Light, Third Edition

30.1.1 MoTion of an elecTRon in a conSTanT elecTRic field

The first and simplest example of the motion of an electron in an electromagnetic field is for a charge moving in a constant electric field. The field is directed along the z axis and is of strength E0. The vector representation for the general electric field E is

E u u u= + +E E Ex x y y z z . (30.2)

Since the electric field is directed only in the z direction, Ex = Ey = 0, so

E u u= =E Ez z z0 . (30.3)

For simplicity, the motion of the electron is restricted to the x, z plane and is initially moving with a velocity v0 at an angle α from the z axis. This is shown in Figure 30.1.

Because there is no magnetic field, the Lorentz force Equation 30.1 reduces to

m er E= − , (30.4)

where m is the mass of the electron. In component form Equation 30.4 is

mx = 0, (30.5)

my = 0, (30.6)

mz eE eEz = − = − 0. (30.7)

At the initial time t = 0 the electron is assumed to be at the origin of the coordinate system, so

x z( ) ( ) .0 0 0= = (30.8)

xy

z

V0

O

E0

e

α

figuRe 30.1 Motion of an electron in the x, z plane in a constant electric field directed along the z axis.

Radiation of an Accelerating Charge in the Electromagnetic Field 609

Similarly, the velocity at the initial time is assumed to be

x v vx( ) sin ,0 0= = α (30.9)

z v vz( ) cos .0 0= = α (30.10)

There is no force in the y direction, so Equation 30.6 can be ignored. We integrate Equations 30.5 and 30.7 and find

x t c( ) ,= 1 (30.11)

z teE tm

c( ) ,= − − +02 (30.12)

where c1 and c2 are constants of integration. From the initial conditions, c1 and c2 are easily found, and the specific solutions of Equations 30.11 and 30.12 are

x t v( ) sin ,= 0 α (30.13)

z teE tm

v( ) cos .= − +00 α (30.14)

Integrating Equations 30.13 and 30.14 yields

x t v t( ) sin ,= 0 α (30.15)

z teE t

mv t( ) cos ,= − +0

2

02α (30.16)

where the constants of integration are found from Equation 30.8 to be zero. We can eliminate t between Equations 30.15 and 30.16 to obtain

z teE

mvx x( )

sin(cot ) ,= −

+0

02 2

2

2 αα (30.17)

which is the equation of a parabola. The path is shown in Figure 30.2.Inspecting Equation 30.17 we see that if α = 0 then z(t) = ∞. That is, the electron moves in a

straight line along the z axis starting from the origin and intercepts the z axis at infinity. However, if α is not zero, then we can determine the positions x(t) where the electron intercepts the z axis by setting z(t) = 0 in Equation 30.17. On doing this, the intercepts are found to occur at

x t( ) ,= 0 (30.18)

x tmveE

( ) sin .= 02

0

2α (30.19)

610 Polarized Light, Third Edition

The first value corresponds to our initial condition x(0) = z(0) = 0. Equation 30.19 shows that the maximum value of x is attained by setting α = 45°, so

xmveEmax .= 0

2

0

(30.20)

This result is not at all surprising, since Equation 30.17 is identical in form to the equation for describing a projectile moving in a constant gravitational field. Finally, the maximum value of z is found from Equation 30.17 to be

z tmveE

( ) sin ,=

12

202

α (30.21)

or

z xmax max ,= 12

(30.22)

where we have used Equation 30.19.

30.1.2 MoTion of a chaRged PaRTicle in a conSTanT MagneTic field

We now consider the motion of an electron moving in a constant magnetic field. The coordinate sys-tem is shown in Figure 30.3. In the figure, B is the magnetic field directed in the positive z direction. Defining the charge on an electron as q = −e, the Lorentz force Equation 30.1 then reduces to

F v B= − ×e( ). (30.23)

Equation 30.23 can be expressed as the differential equation

m er v B= − ×( ), (30.24)

xy

z v

O

E0

αe

figuRe 30.2 Parabolic path of an electron in a constant electric field.

Radiation of an Accelerating Charge in the Electromagnetic Field 611

where m and r are the mass and acceleration vector of the charged particle, respectively. In com-ponent form Equation 30.24 is

mx e x = − ×( ) ,v B (30.25)

my e y = − ×( ) ,v B (30.26)

mz e z = − ×( ) .v B (30.27)

The vector product v × B can be expressed as a determinant

v B

u u u

× =x y z

x y z

x y z

b b b

, (30.28)

where ux, uy, and uz, are the unit vectors pointing in the positive x, y, and z directions, respectively, and the velocities have been expressed as x y, , and z . The constant magnetic field is directed only along z, so bz = b and bx = by = 0. Substituting these values into the determinant, the three compo-nents of the force in Equations 30.25 through 30.27 become

mx e yb = − ( ), (30.29)

my e xb = − −( ), (30.30)

mz = 0. (30.31)

Equation 30.31 is of no interest because the motion along z is not influenced by the magnetic field. The equations of motion are then

xebm

y= −, (30.32)

x

y

z

O

B

e

figuRe 30.3 Motion of an electron in a constant magnetic field.

612 Polarized Light, Third Edition

yebm

x= . (30.33)

Equations 30.32 and 30.33 can be written as a single equation by introducing the complex variable ζ(t) defined to be

ζ( ) ( ) ( ).t x t iy t= + (30.34)

Differentiating Equation 30.34 with respect to time, we have

ζ = +x iy, (30.35)

ζ = +x iy. (30.36)

Multiplying Equation 30.33 by i and adding this result to Equation 30.32 and using Equation 30.35 multiplied by i leads to

ζ ζ− =iebm

0. (30.37)

The solution of Equation 30.37 is readily found by assuming a solution of the form

ζ ω( ) .t e t= (30.38)

Substituting Equation 30.38 into Equation 30.37 we find that

ω ω ω( ) ,− =i c 0 (30.39)

where ωc = eB/m is the frequency of rotation, known as the cyclotron frequency.Equation 30.39 is called the auxiliary or characteristic equation of Equation 30.37, and from

Equation 30.39 the roots are ω = 0, iωc. The general solution of Equation 30.37 can be written immediately as

ζ ω( ) ,t c c ei tc= +1 2 (30.40)

where c1 and c2 are constants of integration. To provide a specific solution for Equation 30.37, assume that initially the charge is at the origin and moves along the x axis with a velocity v0. We have

x y( ) ( ) ,0 0 0 0= = (30.41)

x v y( ) ( ) ,0 0 00= = (30.42)

which can be expressed in terms of Equations 30.34 and 30.35 as

ζ( ) ( ) ( ) ,0 0 0 0= + =x iy (30.43)

Radiation of an Accelerating Charge in the Electromagnetic Field 613

ζ( ) ( ) ( ) .0 0 0 0= + =x iy v (30.44)

This leads immediately to

c c1 2= − , (30.45)

civ

c2

0=ω

, (30.46)

so the specific solution of Equation 30.40 is

ζω

ω( ) ( ).tiv

ec

i tc= − −0 1 (30.47)

Taking the real and imaginary part of Equation 30.47 then yields

x tv

tc

c( ) sin ,= 0

ωω (30.48)

y tv

tc

c( ) ( cos ),= − −0 1ω

ω (30.49)

or

x tv

tc

c( ) sin ,= 0

ωω (30.50)

y tv v

tc c

c( ) cos .+ =0 0

ω ωω (30.51)

Squaring and adding Equations 30.50 and 30.51 gives us

x yv v

c c

2 02

02

+ +

=

ω ω

, (30.52)

which is an equation of a circle with radius v0/ωc and center at x = 0 and y = −v0/ωc.Equation 30.52 shows that in a constant magnetic field a charged particle does indeed move in a

circle, and Equations 30.50 and 30.51 describe a charged particle moving in a clockwise direction as viewed along the positive axis toward the origin. Equation 30.52 is of great historical and scientific interest, because it is the basis for one of the first methods used to measure the ratio e/m and the instrument known as the mass spectrometer. To see how this measurement is made, we note that since the electron moves in a circle, Equation 30.52 can be solved for the condition where it crosses the y axis, at x = 0. We see from Equation 30.52 that this occurs at

y = 0, (30.53)

614 Polarized Light, Third Edition

yv

c

= −2 0

ω. (30.54)

We note that Equation 30.54 is twice the radius ρ (ρ = ν0/ωc). This is to be expected because the charged particle moves in a circle. Since ωc = eb/m, we can solve Equation 30.54 for e/m to find that

em

vyb

= −

2 0 . (30.55)

The initial velocity v0 is known from equating the kinetic energy of the electron with the voltage applied to the charged particle as it enters the chamber of the mass spectrometer. The magnitude of y where the charged particle is intercepted (x = 0) is measured. Finally, the strength of the magnetic field B is measured with a magnetic flux meter. Consequently, all the quantities on the right side of Equation 30.55 are known, so the ratio e/m can then be found. The value of this ratio found in this manner agrees with those of other methods.

30.1.3 MoTion of an elecTRon in a cRoSSed elecTRic and MagneTic field

The final case of interest is that of an electron that moves in a constant magnetic field directed along the z axis and in a constant electric field directed along the y axis. The electric and magnetic fields are perpendicular to each other. This configuration is shown in Figure 30.4.

For this case, Lorentz’s force Equation 30.1 in component form becomes

mx e yb = − ( ), (30.56)

my eE e xb = − + ( ), (30.57)

mz = 0. (30.58)

x

y

z

O

B

e

E

figuRe 30.4 Motion of an electron in a crossed electric and magnetic field.

Radiation of an Accelerating Charge in the Electromagnetic Field 615

From Equations 30.34, 30.35, and 30.36, these equations can be written as a single equation

ζ ω ζ− = −iieEmc , (30.59)

where ωc = eb/m. Equation 30.59 is easily solved by noting that if we multiply by e i tc− ω , then Equation 30.59 can be rewritten as

ddt

eieEm

ei t i tc c( ) .− −= −

ω ωζ (30.60)

Straightforward integration of Equation 30.60 yields

ζω ω

ω=

−

+eE

mt

ice c

c c

i tc12, (30.61)

where c1 and c2 are constants of integration. We choose the initial conditions to be

x y( ) ( ) ,0 0 0 0= = (30.62)

x v y( ) ( ) .0 0 00= = (30.63)

The specific solution of Equation 30.61 is

ζ φ φ φ= + − +a ib b( cos ) sin ,1 (30.64)

where

φ ω= ct, (30.65)

aeE

m c

=ω2

, (30.66)

bv eE m c

c

= −0 / ωω

. (30.67)

Equating the real and imaginary parts of Equations 30.64 and 30.34, we have

x a bφ φ φ( ) = + sin , (30.68)

and

y bφ φ( ) = −( )1 cos . (30.69)

Equations 30.68 and 30.69 are well known from analytical geometry and describe a general cycloid or trochoid. The trochoidal path is a prolate cycloid, cycloid, or curtate cycloid, depending on

616 Polarized Light, Third Edition

whether a < b, a = b, or a > b, respectively. We can easily understand the meaning of this result. First, we note that if the applied electric field Ε were not present then Equations 30.68 and 30.69 would reduce to the equations for a circle of radius b, so the electron moves along a circular path. However, an electric field in the y direction forces the electron to move in the same direction con-tinuously as the electron moves in the circular path due to the magnetic field. Consequently the path is stretched, so the circle becomes a general cycloid or trochoid. This stretching factor is represented by the term αϕ in Equation 30.68. We note that Equation 30.68 shows ϕ = 0 corresponds to the ori-gin. Thus, ϕ is measured from the origin and increases in a clockwise motion.

We can easily find the maximum and minimum values of x(ϕ) and y(ϕ) over a single cycle of ϕ. The maximum and minimum values of y(ϕ) are simply 0 and 2b and occur at ϕ = 0 and π, respec-tively. For x(ϕ) the situation is more complicated. From Equation 30.68, the angles where the mini-mum and maximum values of x(ϕ) occur are

φ = ± −

−tan .12 2b aa

(30.70)

The negative sign refers to the minimum value of x(ϕ), and the positive sign refers to the maximum value of x(ϕ). The corresponding maximum and minimum values of x(ϕ) are then found to be

x a b ab a

ab a( , ) tan .= ± −

± −−1

2 22 2 (30.71)

If we set b = 1 in Equations 30.70 and 30.71, we have

φ = ± −

−tan ,121 a

a (30.72)

x a aa

aa( ) tan .= ± −

± −−1

221

1 (30.73)

Equation 30.73 shows that x(a) is imaginary for a > 1; that is, a maximum and a minimum do not exist. This behavior is confirmed in Figures 30.13 and 30.14 for a = 1.25 and a = 1.5.

Equation 30.72 ranges from a = 0 to 1; for a = 0 (no applied electric field), ϕ = π/2 and 3π/2 (or −π/2). This is exactly what we would expect for a circular path. Following the conventional nota-tion, the path of the electron moves counterclockwise, so π/2 is the angle at the maximum point and 3π/2 (−π/2) corresponds to the angle at the minimum point. Figure 30.5 shows the change in ϕ(a) as the electric field (or equivalently, the value of a) increases. The upper curve corresponds to the positive sign of the argument in Equation 30.72, and the lower curve corresponds to the negative sign. We see that at a = 1 the maximum and minimum values converge. The point of convergence corresponds to a cycloid. This behavior is confirmed by the curve for x(a) in the figure for a = 1, as we shall soon see.

The maximum and minimum points of the (prolate) cycloid are given by Equation 30.73. We see immediately that for a = 0, we have x(0) = ±1. This, of course, applies to a circle. For 0 < a < 1, we have a prolate cycloid. For a cycloid, a = 1, and Equation 30.73 gives x(1) = 0 and π; that is, the maximum and minimum points coincide. This behavior is also confirmed for the plot of x(a) versus a at the value where a = 1. In Figure 30.6, we have plotted the change in the maximum and mini-mum values of x(a) as a increases from 0 to 1. The upper curve corresponds to the positive sign in Equation 30.73, and the lower curve corresponds to the negative sign.

Radiation of an Accelerating Charge in the Electromagnetic Field 617

It is of interest to determine the points on the x axis where the electron path intersects or is tan-gent to the x axis. This is found by setting y = 0 in Equation 30.69. We see that this is satisfied by ϕ = 0 or ϕ = 2π. Setting b = 1 in Equation 30.68, the points of intersection on the x axis are given by x = 0 and x = 2πa; the point x = 0 and y = 0, we recall, is the position of the electron at the ini-tial time t = 0. Thus, setting b = 1 in Equation 30.68, the initial and final positions of the electron for a = 0 are at x(i) = 0 and x( f) = 0, which is the case for a circle. For the other extreme obtained by setting a = 1, the initial and final intersections are 0 and 2π, respectively. As the magnitude of the electric field increases, the final point of intersection on the x axis increases. In addition, as a increases, the prolate cycloid advances so that for a = 0 (a circle) the midpoint of the path is at x = 0, and for a = 1 the midpoint is at x = π.

1.43

0.93

0.43

–0.07 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

φ(a)

(rad

ians

)

–0.57

–1.57a

–1.07

figuRe 30.5 Plot of the angle ϕ(a), Equation 30.72, for the maximum and minimum points as the electric field (or equivalently, a) increases from 0 to 1.

1.57

1.05

0.52

0.000.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90 0.1 1

x(a)

(rad

ians

)

–0.52

–1.05

–1.57a

figuRe 30.6 Plot of the maximum and minimum values of x(ϕ) written as x(a), Equation 30.73, as the electric field (or equivalently, a) increases from 0 to 1.

618 Polarized Light, Third Edition

We now plot the evolution of the trochoid as the electric field E(a) increases. The equations used are Equations 30.68 and 30.69 with b = 1,

x aφ φ φ( ) = + sin , (30.74)

y φ φ( ) = −1 cos . (30.75)

It is of interest to plot Equation 30.74 from ϕ = 0 to 2π for a = 0, 0.25, 0.50, 0.75, and 1.0. Figure 30.7 is a plot of the evolution of x(ϕ) from a pure sinusoid for a = 0 to a cycloid for a = 1.

The most significant feature of Figure 30.7 is that the maxima shift to the right as a increases. This behavior continues until a = 1, whereupon the maximum point virtually disappears. Similarly, the minima shift to the left, so that at a = 1 the minimum point virtually disappears. This behavior is later confirmed for a = 1, a cycloid.

The paths of the electrons are specifically shown in Figures 30.8 through 30.15. The curves are plotted over a single cycle of ϕ (0–2π). For these values, Equation 30.72 shows that the path intersects the x axis at 0 and 2πa. We select a to be 0, 0.25, 0.5, …, 1.5. The corresponding inter-sections of the path on the x axis are then (0, 0), (0, π/2), (0, π), …, (0, 3π). With these values of a, Figures 30.8 through 30.15 show the evolutionary change in the path. Figure 30.15 shows the path of the electron as it moves over four cycles.

30.2 STokeS VeCToRS foR RadiaTioN emiTTed by aCCeleRaTiNg ChaRgeS

We now determine the Stokes vectors for the radiation emitted by the accelerating charges undergo-ing the motions described in the previous section, the motion of an electron in a constant electric field, the motion of an electron in a constant magnetic field, and the motion of the electron in a crossed electric and magnetic field.

The components of the radiation field in spherical coordinates were shown in Chapter 28 to be

Eec R

x zθ πεθ θ= −

4 02

[ cos sin ], (30.76)

6

4

5

3

a = 1

a = 0.75

2

x(φ)

a = 0.5

1 a = 0.25

–1

0

a = 0.0

0 φ 2π

figuRe 30.7 Plot of x(ϕ), Equation 30.74, for a = 0 to 1.

Radiation of an Accelerating Charge in the Electromagnetic Field 619

and

Eec R

yφ πε=

4 02

[ ]. (30.77)

These equations refer to the observation being made in the x, z plane, that is, at ϕ = 0. The angle θ is the polar angle in the observer’s reference frame. Recall that the Stokes parameters of the radiation field are defined by

S E E E E0 = +φ φ θ θ* *, (30.78)

0.8

1

0.4

0.6

0

0.2

–0.2

x(φ)

y(φ)

–0.6

–0.4

–1

–0.8

0.00 0.50 1.00 1.50 2.00

figuRe 30.8 The trochoidal path of an electron, a = 0 (a circle).

1.4

1.2

1

0.6

0.8

x(φ)

y(φ)

0.4

0

0.2

0.00 0.50 1.00 1.50 2.00

figuRe 30.9 The trochoidal path of an electron, a = 0.25.

620 Polarized Light, Third Edition

S E E E E1 = −φ φ θ θ* *, (30.79)

S E E E E2 = +φ θ θ φ* *, (30.80)

S i E E E E3 = −( ).* *φ θ θ φ (30.81)

In the following three cases, we represent the emitted radiation and its polarization in the form of Stokes vectors.

3

2.5

2

1.5x(φ)

y(φ)

0.5

1

00.00 0.50 1.00 1.50 2.00

figuRe 30.10 The trochoidal path of an electron, a = 0.5.

4.5

3.5

4

2.5

3

2

1.5

x(φ)

1

0

0.5

0.00 0.50 1.00 1.50 2.00y(φ)

figuRe 30.11 The trochoidal path of an electron, a = 0.75.

Radiation of an Accelerating Charge in the Electromagnetic Field 621

30.2.1 STokeS vecToR foR a chaRge Moving in an elecTRic field

The path of the charge moving in a constant electric field in the x, z plane was found to be

x t v t( ) = 0 sinα, (30.82)

z teE t

mv t( ) cos .= − +0

2

02α (30.83)

We see that the accelerations of the charge in the x and z directions are then

x t( ) ,= 0 (30.84)

6

5

3

4x(

φ)

2

1

00.00 0.50 1.00 1.50 2.00

y(φ)

figuRe 30.12 The trochoidal path of an electron, a = 1.0.

7

6

4

5

x(φ)

y(φ)

3

2

0

1

0.00 0.50 1.00 1.50 2.00

figuRe 30.13 The trochoidal path of an electron, a = 1.25.

622 Polarized Light, Third Edition

z teEm

( ) .= − 0 (30.85)

Substituting these expressions for the accelerations into Equations 30.76 and 30.77 yields

Ee E

mc Rθ πεθ=

20

024

sin , (30.86)

Eφ = 0, (30.87)

9

7

8

5

6

3

4

x(φ)

2

0

1

0.00 0.50 1.00 1.50 2.00y(φ)

figuRe 30.14 The trochoidal path of an electron, a = 1.5.

6

5

4

2

3x(φ)

1

00.00 0.50 1.00 1.50 2.00

y(φ)

figuRe 30.15 The trochoidal path of an electron over four cycles, a = 0.25.

Radiation of an Accelerating Charge in the Electromagnetic Field 623

and we immediately find, using these expressions in the equations for the Stokes vector parameters Equations 30.78 through 30.81 that the Stokes vector is

S =

−

e Emc R

20

02

2

2

4

1

1

0

0

πεθsin . (30.88)

Equation 30.88 shows that the emitted radiation is linearly vertically polarized. It also shows the accelerating electron emits the familiar dipole radiation pattern described by sin2 θ, so the intensity observed along the z axis is zero (θ = 0) and is a maximum when viewed along the x axis (θ = π/2).

Before we finish the discussion of Equation 30.88, there is another point of interest that should be noted. We observe that in Equation 30.88 there is a constant factor of e2/4πε0mc2. What is the physical meaning of this factor? The answer can be obtained by recalling that the electric field E outside of an electron is given by

E u= er r4 0

2πε, (30.89)

where r is the distance from the center of the electron and ur is the unit radius vector. We now imag-ine the electron has a radius a and compute the work that must be done to move another (positive) charge of the same magnitude from the surface of this electron to infinity. The total work, or energy, required to do this is

W e da

= − ⋅∞

∫ E r, (30.90)

where dr is drur. Substituting Equation 30.89 into Equation 30.90 gives

We dr

re

aa= =

∞

∫2

02

2

04 4πε πε. (30.91)

We now equate Equation 30.91 to the rest mass of the electron mc2 and find that

ae

mc=

2

024πε

. (30.92)

Thus the factor e2/4πε0mc2 is the classical radius of the electron. The value of a is readily calcu-lated from the values e = 1.60 × 10−19 C, m = 9.11 × 10−31 kg, and c = 2.997 × 108 m/sec to obtain a value for a of 2.82 × 10−15 m. We see that the radius of the electron is extremely small. The factor e2/4πε0mc2 appears repeatedly in radiation problems. It will appear again later when we consider the problem where radiation is incident on an electron and is then reemitted, that is, the scattering of radiation by an electron.

30.2.2 STokeS vecToR foR a chaRge acceleRaTing in a conSTanT MagneTic field

In the previous section we saw that the path described by an electron moving in a constant magnetic field is given by the equations

x tv

tc

c( ) sin ,= 0

ωω (30.93)

624 Polarized Light, Third Edition

y tv

tc

c( ) ( cos ),= − −0 1ω

ω (30.94)

where v0 is the initial velocity and ωc = eb/m is the cyclotron frequency. Using the exponential representation

Re cos ,e ti tc

cω ω = (30.95)

Re sin ,− =ie ti tc

cω ω (30.96)

we can write

x t ieci tc( ) ,= −( )α ω (30.97)

y t eci tc( ) ,+ = ( )α α ω (30.98)

where

αωc

c

v= 0 . (30.99)

The accelerations x t( ) and y t( ) are then

x t i ec ci tc( ) ,= α ω ω2 (30.100)

y t ec ci tc( ) ,= −α ω ω2 (30.101)

and the radiation field components become

Eie

c Rec c i tcθ

ωα ωπε

θ= − 2

024

cos , (30.102)

Ee

c Rec c i tcφ

ωα ωπε

=2

024

. (30.103)

From the definition of the Stokes parameters in Equations 30.78 through 30.81, the Stokes vector is

S =

+−

ec Rc

c

απε

ω

θθ

θ4

1

1

0

20

2

2

4

2

2

cos

cos

cos

, (30.104)

which is the Stokes vector for elliptically polarized light radiating at the same frequency as the cyclotron frequency ωc. Thus, the Stokes vector found earlier for a charge moving in a circle

Radiation of an Accelerating Charge in the Electromagnetic Field 625

is based on physical reality. We see that Equation 30.104 reduces to right circularly polarized light, linearly horizontally polarized light, and left circularly polarized light for θ = 0, π/2, and π, respectively.

30.2.3 STokeS vecToR foR a chaRge Moving in a cRoSSed elecTRic and MagneTic field

The path of the electron was seen to be a trochoid described by the equations

x a bφ φ φ( ) = + sin , (30.105)

y bφ φ( ) = −( )1 cos , (30.106)

where

φ ω= ct, (30.107)

aeE

m c

=ω2

, (30.108)

bv eE m c

c

= −0 /.

ωω

(30.109)

Differentiating Equations 30.105 and 30.106 twice with respect to time and making use of the expressions in Equations 30.95 and 30.96 then gives

x t ib eci tc( ) ,= ω ω2 (30.110)

y t b eci tc( ) ,= ω ω2 (30.111)

and we find that the Stokes vector is

S =

+−

b c2 4

2

2

1

1

0

2

ω

θθ

θ

cos

cos

cos

, (30.112)

which, again, is the Stokes vector for elliptically polarized light.With this material behind us we now turn our attention to the Lorentz–Zeeman effect and see

how the role of polarized light led to the acceptance of Maxwell’s electrodynamic theory in optics.

RefeReNCeS

1. Jackson, J. D., classical Electrodynamics, New York: John Wiley, 1962. 2. Sommerfeld, Α., Lectures on Theoretical Physics, Vols. I–V, New York: Academic Press, 1952. 3. Harnwell, G. P., Principles of Electricity and Electromagnetism, New York: McGraw-Hill, 1949. 4. Humphries, Jr., S., charged Particle beams, New York: John Wiley, 1990.

626 Polarized Light, Third Edition

5. Hutter, R. C. E., and S. W. Harrison, beam and Wave Electronics in Microwave Tubes, Princeton, NJ: D. Van Nostrand, 1960.

6. Panofsky, W. K. H., and M. Phillips, classical Electricity and Magnetism, Reading, MA: Addison-Wesley, 1955.

7. Goldstein, H., classical Mechanics, Reading, MA: Addison-Wesley, 1950. 8. Corben, H. C., and P. Stehle, classical Mechanics, New York: John Wiley, 1957.

627

31 The Classical Zeeman Effect

31.1 hiSToRiCal iNTRoduCTioN

In 1846, Michael Faraday discovered that by placing a block of heavy lead glass between the poles of an electromagnet and passing a linearly polarized beam through the block in the direction of the lines of force, the plane of polarization of the linearly polarized beam was rotated by the magnetic medium; this is called the Faraday effect. He established that there was a link between electromag-netism and light. It was this discovery that stimulated Maxwell to begin to think of the relation between the electromagnetic field and the optical field.

Faraday was very skillful at inverting questions in physics. In 1819, Oersted discovered that a current gives rise to a magnetic field. Faraday then asked the inverse question: how can a magnetic field give rise to a current? After many years of experimentation Faraday discovered that a chang-ing magnetic field rather than a steady magnetic field generates a current (Faraday’s Law). In the Faraday effect, Faraday had shown that a magnetic medium affects the polarization of light as it propagates through the medium. Faraday now asked the question: how, if at all, does the magnetic field affect the source of light itself? To answer this question, he placed a sodium flame between the poles of a large electromagnet and observed the D lines of the sodium radiation when the magnetic field was on and when it was off. By 1862, and after many attempts, he was still unable to convince himself that any change resulted in the appearance of the lines, a circumstance that we now know was due to the insufficient resolving power of his spectroscope.

In 1896, Pieter Zeeman, using a more powerful magnet and an improved spectroscope, repeated Faraday’s experiment. This time there was success. He established that the D lines were broadened when a constant magnetic field was applied. Lorentz heard of Zeeman’s discovery and quickly developed a theory to explain the phenomenon.

We have pointed out that, even with the success of Hertz’s experiments in 1888, Maxwell’s the-ory was still not accepted by the optics community because Hertz had carried out his experiments not at optical frequencies but at microwave frequencies. For Maxwell’s theory to be accepted by the optical community it would be necessary to prove the theory at optical frequencies (wavelengths); that is, an optical source that could be characterized in terms of a current would have to be created. There was nothing in Fresnel’s wave theory that enabled this to be done. Lorentz recognized that at long last an optical source could be created that could be understood in terms of the simple electron theory (sodium has only a single electron in its outer shell). Therefore, he used the simple model of the sodium atom in which an electron was bound to the nucleus and its motion governed by Hooke’s Law. With this model he then discovered that Zeeman’s line broadening should actually consist of two or even three spectral lines. Furthermore, using Maxwell’s theory he was able to predict that the lines would be linearly, circularly, or elliptically polarized in a completely predictable manner. Lorentz communicated his theoretical conclusions to Zeeman, who investigated the edges of his broadened lines and confirmed Lorentz’s predictions in all respects.

Lorentz’s spectacular predictions with respect to the splitting, intensity, and polarization of the spectral lines led to the complete acceptance of Maxwell’s theory. Especially impressive were the polarization predictions, because they were very complicated. It was virtually impossible without Maxwell’s theory and the electron theory even remotely to understand the polarization behavior of the spectral lines. Thus, polarization played a critical role in the acceptance of Maxwell’s theory. In 1902, Zeeman and Lorentz shared the Nobel Prize in physics for their work. The prize was given not just for their discovery of and understanding of the Zeeman effect but, even more importantly,

628 Polarized Light, Third Edition

for the verification of Maxwell’s theory at optical wavelengths. It is important to recognize that Lorentz’s contribution was of critical importance. Zeeman discovered that the D lines of the sodium were broadened, not split. Because Lorentz predicted that the spectral lines would be split, further experiments were conducted and the splitting was observed. Soon after Zeeman’s discovery, how-ever, it was discovered that additional spectral lines appeared. In fact, just as quickly as Lorentz’s theory was accepted, it was discovered that it was inadequate to explain the appearance of the numerous spectral lines. The explanation would only come with the advent of quantum mechanics in 1925.

The Zeeman effect and the Faraday effect belong to a class of optical phenomena that are called magneto-optical effects. In this chapter, we analyze the Zeeman effect in terms of the Stokes vector. We shall see that the Stokes vector takes on a new and very interesting interpretation. In Chapter 33, we describe the Faraday effect along with other related phenomena in terms of the Mueller matrices.

31.2 moTioN of a bouNd ChaRge iN a CoNSTaNT magNeTiC field

To describe the Zeeman effect and determine the Stokes vector of the emitted radiation, it is neces-sary to analyze the motion of a bound electron in a constant magnetic field, that is, determine x(t), y(t), z(t) of the electron and then the corresponding accelerations. The model proposed by Lorentz to describe the Zeeman effect was a charge bound to the nucleus of an atom and oscillating with an amplitude A through the origin. The motion is shown in Figure 31.1; χ is the polar angle and ψ is the azimuthal angle. In particular, the angle ψ describes the projection of OP onto the x, y plane. The significance of emphasizing this will appear shortly.

The equation of motion of the bound electron in the magnetic field is governed by the Lorentz force equation

m k er r v B+ = ×– [ ], (31.1)

xy

z

A

O

Bχ

ψ

P

figuRe 31.1 Motion of bound charge in a constant magnetic field; χ is the polar angle and ψ is the azi-muthal angle. In particular, the angle ψ describes the projection of OP on to the x, y plane.

The Classical Zeeman Effect 629

where m is the mass of the electron, kr is the restoring force (Hooke’s Law), v is the velocity of the electron, and B is the strength of the applied magnetic field. In component form, Equation 31.1 can be written

mx kx e x+ = − ×[ ] ,v B (31.2)

my ky e y+ = − ×[ ] ,v B (31.3)

mz kz e z+ = − ×[ ] .v B (31.4)

We saw in the previous chapter that for a constant magnetic field directed along the positive z axis (B = buz), these last equations become

mx kx e yb + = − [ ], (31.5)

my ky e xb + = − −[ ], (31.6)

mz kz+ = 0. (31.7)

These equations can be rewritten further as

x xebm

y+ = −

ω0

2 , (31.8)

y yebm

x+ = −

ω0

2 , (31.9)

z z+ =ω02 0, (31.10)

where ω0 = k m/ is the natural frequency of the charge oscillating along the line OP.Equation 31.10 can be solved immediately. We assume a solution of the form z(t) = eωt. Then, the

auxiliary equation for Equation 31.10 is

ω ω202 0+ = (31.11)

and

ω ω= ±i 0. (31.12)

The general solution of Equation 31.10 is then

z t c e c ei t i t( ) .= + −1 2

0 0ω ω (31.13)

To find a specific solution, the constants c1 and c2 must be found from the initial conditions on z(0) and z (0). From Figure 31.1, we see that when the charge is at Ρ we have

z A0( ) = cosχ, (31.14)

630 Polarized Light, Third Edition

z( ) .0 0= (31.15)

Using Equations 31.14 and 31.15, we find the solution for Equation 31.13 to be

z t A t( ) = cos cosχ ω0 . (31.16)

Next, we solve Equations 31.8 and 31.9. We again introduce the complex variable

ζ = +x iy. (31.17)

In the same manner as in the previous chapters, Equations 31.8 and 31.9 can be written as a single equation

ζ ζ ω ζ+ −

+ =ieb

m 02 0. (31.18)

Again, assuming a solution of the form z(t) = eω t, the solution of the auxiliary equation is

ω ω=

± −

i

ebm

iebm2 20

2

2 1 2/

. (31.19)

The term (eb/2m)2 in Equation 31.19 is orders of magnitude smaller than ω02 , so Equation 31.19 can

be written as

ω ω ω± = ±( )i L 0 , (31.20)

where

ωL

ebm

=2

. (31.21)

The frequency ωL is known from the Larmor precession frequency; the reason for the term preces-sion will soon become clear. The solution of Equation 31.18 is then

z t c e c ei t i t( ) ,= ++ −1 2

ω ω (31.22)

where ω±is given by Equation 31.20.To obtain a specific solution of Equation 31.22, we must again use the initial conditions. From

Figure 31.1, we see that

x A0( ) = sin cosχ ψ, (31.23)

y A0( ) = sin sinχ ψ, (31.24)

so

ζ χ ψ0 0 0( ) = ( ) + ( ) =x iy A isin exp( ), (31.25)

ζ( ) .0 0= (31.26)

The Classical Zeeman Effect 631

After a little algebraic manipulation we find that the conditions in Equations 31.25 and 31.26 lead to the specific relations for x(t) and y(t)

x tA

t t tL L L( )sin

[ cos( )cos sin(= + + +χω

ω ψ ω ω ω ψ ω0

0 0 ))sin ],ω0t (31.27)

y tA

t t tL L L( )sin

[ sin( )cos cos(= + − +χω

ω ψ ω ω ω ψ ω0

0 0 ))sin ].ω0t (31.28)

Because the Larmor frequency is much smaller than the fundamental oscillation frequency of the bound electron, that is, ωL << ω0, the second terms in Equations 31.27 and 31.28 can be dropped. The equations of motion for x(t), y(t), and z(t) are then simply

x t A t tL( ) sin cos( )cos ,= +χ ψ ω ω0 (31.29)

y t A t tL( ) sin sin( )cos ,= +χ ψ ω ω0 (31.30)

z t A t( ) cos cos .= χ ω0 (31.31)

In this set of equations, we have also included an expression for z(t) from Equation 31.16. We see that ωLt, the angle of precession, is coupled only with ψ and is completely independent of χ. To show this precessional behavior we deliberately chose to show ψ in Figure 31.1. The angle ψ is completely arbitrary and is symmetric around the z axis. We could have chosen its value immediately to be zero; however, to demonstrate clearly that ωLt is restricted to the x, y plane, we chose to include ψ in the formulation. We therefore see from Equations 31.29 and 31.30 that, as time increases, the fac-tor ψ increases by ωLt. Thus, while the bound charge is oscillating to and fro along the radius OP, there is a simultaneous counterclockwise rotation in the x, y plane. This motion is called precession, and we see that ωLt is the angle of precession. The precession caused by the presence of the mag-netic field is very often called the Larmor precession, after Joseph Larmor, who, around 1900, first pointed out this behavior of an electron in a magnetic field.

The angle ψ is arbitrary, so we can conveniently set ψ = 0 in Equations 31.29 and 31.30. The equations then become

x t A t tL( ) sin cos cos ,= χ ω ω0 (31.32)

y t A t tL( ) sin sin cos ,= χ ω ω0 (31.33)

z t A t( ) cos cos .= χ ω0 (31.34)

Using these equations, we can write

r t x t y t z t A t2 2 2 2 2 20( ) ( ) ( ) ( ) cos .= + + = ω (31.35)

This result is completely expected because the radial motion is due only to the natural oscillation of the electron. The magnetic field has no effect on this radial motion, and, indeed, we see that there is no contribution.

632 Polarized Light, Third Edition

Equations 31.32, 31.33, and 31.34 are the fundamental equations that describe the path of the bound electron. The accelerations can then be obtained as in the following section. However, we consider Equations 31.32 and 31.33 further. If we plot these equations, we can follow the preces-sional motion of the bound electron as it oscillates along OP. Equations 31.32 and 31.33 give rise to a remarkably beautiful pattern called a petal plot. Physically, the electron oscillates very rapidly along the radius OP while the magnetic field forces the electron to move relatively slowly counterclock-wise in the x, y plane. Normally, ωL << ω0 where ωL ≃ ω0 /107. Thus, the electron oscillates about 10 million times through the origin during one precessional revolution. Clearly, this is a practical impossibility to illustrate or plot. However, if we artificially take ωL to be close to ω0, we can dem-onstrate the precessional behavior and still lose none of our physical insight. To show this behavior we first arbitrarily set the factor A sin χ to unity. Then, using the well-known trigonometric sum and difference formulas, Equations 31.32 and 31.33 can be written as

x t t tL L( ) [cos( ) cos( ) ],= + + −12 0 0ω ω ω ω (31.36)

y t t tL L( ) [sin( ) sin( ) ].= + − −12 0 0ω ω ω ω (31.37)

We now set

θ ω θ ω0 0= =t tL Land , (31.38)

so that Equations 31.36 and 31.37 become

x L L( ) [cos( ) cos( )],θ θ θ θ θ0 0 012

= + + − (31.39)

y L L( ) [sin( ) sin( )].θ θ θ θ θ0 0 012

= + − − (31.40)

To plot the precessional motion, we set θL = θ0/p, where p can take on any integer value. Equations 31.39 and 31.40 can then be written as

xp

pp

p( ) cos cos ,θ θ θ0 0 0

12

1 1= +

+ −

(31.41)

yp

pp

p( ) sin sin .θ θ θ0 0 0

12

1 1= +

− −

(31.42)

As a first example, we set ωL = ω0/5 so θL = 0.2θ0. In Figure 31.2, Equations 31.41 and 31.42 have been plotted over 360° for θL = 0.2θ0 (in which time the electron makes 5 × 360 = 1800 radial oscil-lations, which is equivalent to θ taking on values from 0 to 1800°). The figure shows that the elec-tron describes five petals over a single precessional cycle. The actual path and direction taken by the electron can be followed by starting at the origin, proceeding counterclockwise around the petal that is bisected by the x axis (so that the interior of the petal is always on the electron’s left-hand side), and continuing around the petal in the third quadrant, and so on.

The Classical Zeeman Effect 633

One can readily consider other values of ωL. In Figures 31.3 through 31.6, other petal diagrams are shown for four additional values of ωL: ω0, ω0/2, ω0/4, and ω0/8. The result shows a proportional increase in the number of petals and reveals a very beautiful pattern for the precessional motion of the bound electron.

Equations 31.39 and 31.40 can be transformed in an interesting manner by a rotational transfor-mation. The transformation is defined as

′ = +x x ycos sin ,θ θ (31.43)

–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

1

–1 –0.5 0 0.5 1

x(θ)

y(θ)

figuRe 31.2 Petal diagram for a precessing electron where ωL = ω0/5, θL = θ0/5.

–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

1

–1 –0.5 0 0.5 1y(θ)

x(θ)

figuRe 31.3 Petal diagram for a precessing electron; ωL = ω0, θL = θ0.

634 Polarized Light, Third Edition

′ = − +y x ysin cos ,θ θ (31.44)

where θ is the angle of rotation. We now substitute Equations 31.39 and 31.40 into Equations 31.43 and 31.44, group terms, and find that

′ = + ′ + − ′x12 0 0[cos( ) cos( )],θ θ θ θ (31.45)

′ = + ′ − − ′y12 0 0[sin( ) sin( )],θ θ θ θ (31.46)

–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

1

–1 –0.5 0 0.5 1

x(θ)

y(θ)

figuRe 31.4 Petal diagram for a precessing electron; ωL = ω0/2, θL = θ0/2.

–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

1

–1 –0.5 0 0.5 1

x(θ)

y(θ)

figuRe 31.5 Petal diagram for a precessing electron; ωL = ω0/4, θL = θ0/4.

The Classical Zeeman Effect 635

where

′ = −θ θ θL . (31.47)

Inspecting Equations 31.45 and 31.46, we see that the equations are identical in form with Equations 31.39 and 31.40, that is, under a rotation of coordinates x and y are invariant. In a (weak) magnetic field, Equations 31.45 and 31.46 show that the equations of motion with respect to axes rotating with an angular velocity ωL are the same as those in a nonrotating system when B is zero. This result is known as Larmor’s theorem. Equations 31.45 and 31.46 allow us to describe x′ and y′ in a very simple way. If we set θ = θL − θ0, then θ′ = θ0 and Equations 31.45 and 31.46 reduce to

′ = +x12

1 2 0[ cos ],θ (31.48)

′ =y12

2 0sin .θ (31.49)

Thus, in the primed coordinate system only θ0, the natural oscillation angle, appears. The angle θ0 can be eliminated and we find that

( ) ( ) ,′ − + ′ =x y1 2 1 22 2 2/ / (31.50)

which is a circle of unit diameter with intercepts on the x′ axis at 0 and 1.A final observation can be made. The petal diagrams for precession based on Equations 31.39

and 31.40 and shown in the figures appear to be remarkably similar to the rose diagrams that arise in analytical geometry, described by the equation

ρ θ= =cos , , , ,k k N1 2… (31.51)

–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

1

–1 –0.5 0 0.5 1y(θ)

x(θ)

figuRe 31.6 Petal diagram for a precessing electron; ωL = ω0/8, θL = θ0/8.

636 Polarized Light, Third Edition

where there are 2N petals if N is even and N petals if N is odd. We can express Equation 31.51 in terms of x and y from

x = ρ θcos , (31.52)

y = ρ θsin , (31.53)

so that

x k k k= = + + −cos cos [cos( ) cos( ) ],θ θ θ θ12

1 1 (31.54)

y k k k= = + − −cos sin [sin( ) sin( ) ],θ θ θ θ12

1 1 (31.55)

where we have used the sum and difference formulas for the cosine and sine functions.We can show that the precession Equations 31.39 and 31.40 reduce to either Equation 31.51 or

Equations 31.54 and 31.55 by writing them as

x p q= +12

[cos cos ], (31.56)

y p q= −12

[sin sin ], (31.57)

where

p L= +θ θ0 , (31.58)

q L= −θ θ0 . (31.59)

Equations 31.56 and 31.57 can be transformed to polar coordinates by squaring and adding so that

ρ2 2 2 12

1= + = + +x y p q[ cos( )]. (31.60)

We now set θ0 to

θ θ θ0 1 2= = =k k k NL , , , ,… (31.61)

so that

p k k NL= + = + =θ θ θ0 1 1 2( ) , , , ,… (31.62)

q k k NL= − = − =θ θ θ0 1 1 2( ) , , , ,… (31.63)

and

p q k+ = 2 θ. (31.64)

The Classical Zeeman Effect 637

Substituting Equations 31.62 and 31.63 into Equations 31.56 and 31.57 yields

x k k= + + −12

1 1[cos( ) cos( ) ],θ θ (31.65)

y k k= + − −12

1 1[sin( ) sin( ) ],θ θ (31.66)

and substituting Equation 31.64 into Equation 31.60 yields,

ρ θ θ2 212

1 2= + =[ cos ] cosk k (31.67)

or

ρ θ= =cos , , , .k k N1 2… (31.68)

We see that Equation 31.68 is the well-known rose equation of analytical geometry. Thus, the rose equation describes the phenomenon of the precession of a bound electron in a magnetic field, an interesting fact that does not appear to be pointed out in courses on analytical geometry.

31.3 STokeS VeCToR foR The ZeemaN effeCT

We now determine the Stokes vector for the Zeeman effect. Recall that we had the equations that describe the path of the oscillating electron bound to an atom as

x t A t tL( ) sin cos cos ,= χ ω ω0 (31.69)

y t A t tL( ) sin sin cos ,= χ ω ω0 (31.70)

z t A t( ) cos cos ,= χ ω0 (31.71)

where

ωL

ebm

=2

. (31.72)

These equations can be represented in complex form by first rewriting them using the trigonometric identities for sums and differences, so that

x tA

t t( ) sin (cos cos ),= ++ −2χ ω ω (31.73)

y tA

t t( ) sin (sin sin ),= −+ −2χ ω ω (31.74)

z t A t( ) cos cos ,= χ ω0 (31.75)

638 Polarized Light, Third Edition

where

ω ω ω± = ±0 L . (31.76)

Replacing the trigonometric functions with exponentials as we have done before, we have

x tA

i t i t( ) sin [exp( ) exp( )],= ++ −2χ ω ω (31.77)

y t iA

i t i t( ) sin [exp( ) exp( )],= −

−+ −2

χ ω ω (31.78)

z x A i t( ) cos exp( ).= χ ω0 (31.79)

Twofold differentiation of Equations 31.77, 31.78, and 31.79 with respect to time yields

x tA

i t i t( ) sin [ exp( ) exp( )],= − ++ + − −22 2χ ω ω ω ω (31.80)

y t iA

i t i t( ) sin [ exp( ) exp( )],= −+ + − −22 2χ ω ω ω ω (31.81)

z t A i t( ) ( cos ) exp( ).= − χ ω ω02

0 (31.82)

The radiation field equations are

Eec R

x t z tθ πεθ θ= −

4 02

[ ( )cos ( )sin ], (31.83)

Eec R

y tφ πε=

4 02

[ ( )]. (31.84)

Substituting Equations 31.80, 31.81, and 31.82 into Equations 31.83 and 31.84 yields

E

eAc R

i t iθ πεχ θ ω ω ω= ++ + −8 0

22 2[sin cos exp( ) exp( ωω

ω χ θ ω

−

+

t

i t

)

cos sin exp( )]2 02

0

(31.85)

and

Eie A

c Ri t i tφ

χπε

ω ω ω ω= −+ + − −sin

exp( ) exp(8 0

22 2 )). (31.86)

The Stokes parameters are defined in spherical coordinates to be (see Chapter 28):

S E E E E0 = +φ φ θ θ* *, (31.87)

The Classical Zeeman Effect 639

S E E E E1 = −φ φ θ θ* *, (31.88)

S E E E E2 = +φ θ θ φ* *, (31.89)

S i E E E E3 = −( ).* *φ θ θ φ (31.90)

We now form the quadratic field products of Equations 31.85 and 31.86 according to the Stokes parameter equations, drop all cross-product terms, and average χ over a sphere of unit radius. Finally, we group terms and find that the Stokes vector for the classical Zeeman effect is

S =

+ + ++ −

eAc R8

23

143

02

2

4 4 204

πε

ω ω θ ω( )( cos ) siin

( )sin sin

(

2

4 4 204 2

4

23

43

0

43

θ

ω ω θ ω θ

ω ω

− + +

−

+ −

+ −44 )cos

.

θ

(31.91)

The form of Equation 31.91 suggests that we can decompose the column matrix according to fre-quency. We choose to decompose Equation 31.91 into column matrices in terms of frequencies ω−, ω0, and ω+. We now do this and find that

S =

+−

−

−23 8

1

0

20

2

2

4

2

2eAc Rπε

ω

θθ

θ

cos

sin

cos

+

+ω

θθ

ω04

2

2

2

2

0

0

sin

sin++

+−

4

2

2

1

0

2

cos

sin

cos

θθ

θ

. (31.92)

This decomposition indicates that we should observe three spectral lines at frequency ω–, ω0, and ω+ with different polarizations. This is exactly what is observed in a spectroscope. We see that the Stokes vectors associated with ω– and ω+ correspond to elliptically polarized light with their polar-ization ellipses oriented at 90° and of opposite ellipticity. Similarly, the Stokes vector associated with the ω0 spectral line is always linearly horizontally polarized. In Figure 31.7, we represent the spectral lines corresponding to Equation 31.92 as they would be observed in a spectroscope.

By describing the Zeeman effect in terms of the Stokes vector, we have obtained a mathematical formulation that corresponds exactly to the observed spectrum. Furthermore, the column matrix that we know as the Stokes vector contains all of the information that can be measured, the fre-quency (wavelength), intensity, and polarization. In this way we have extended the usefulness of the Stokes vector.

Originally the Stokes parameters were introduced to obtain a formulation of the optical field whereby the polarization could be measured in terms of the intensity, a measurable quantity. The Stokes vector was then constructed and introduced to facilitate the mathematical analyses of polar-ized light via the Mueller–Stokes formalism. The Stokes vector now takes on another meaning; it can be used to represent the observed spectral lines. In a sense, we have finally reached a goal enunciated first by Werner Heisenberg (1925) in his formulation of quantum mechanics [1] and later, for optics, by Emil Wolf (1954)—the description of atomic and optical phenomena in terms of observables [2].

We see from Equation 31.92 that the ellipticity angle is a function of the observation angle θ. In Figure 31.8, a plot is made of the ellipticity angle versus θ. We observe that from θ = 0° (viewing

640 Polarized Light, Third Edition

down along the magnetic field) to θ = 180° (viewing up along the magnetic field) there is a reversal in the ellipticity.

Equation 31.92 reduces to special forms when the radiation is observed parallel to the magnetic field (θ = 0°) and perpendicular to the magnetic field (θ = 90°). For θ = 0°, we see from Equation 31.92 that the Stokes vector associated with the ω0 column matrix vanishes, and only the Stokes vectors associated with ω− and ω+ remain. We then have

S =

−

+− +43 8

1

0

0

1

1

02

2

4 4eAc Rπε

ω ω00

0

1

. (31.93)

figuRe 31.7 The Zeeman effect observed in a spectroscope.

0 50 100 150(θ)

(degrees)

ω+ ω–

–π/4

π/4

figuRe 31.8 Plot of the ellipticity angle χ(θ) versus the viewing angle θ of the spectral lines associated with the ω− and ω+ frequencies in Equation 31.92.

The Classical Zeeman Effect 641

Thus, we observe two spectral lines at frequencies ω− and ω+ that are left and right circularly polar-ized, respectively. The intensities are equal; the magnitudes of the frequencies ω± are nearly equal. The observation of only two spectral lines when observed parallel to the magnetic field is some-times called the longitudinal Zeeman effect. Figure 31.9 corresponds to Equation 31.93 as viewed in a spectroscope.

Next we consider the case where the radiation is observed perpendicular to the magnetic field (θ = 90°). Equation 31.92 now reduces to

S =

−

+−23 8

1

1

0

0

20

2

2

404eA

c Rπεω ω

11

1

0

0

1

1

0

0

4

+−

+ω

. (31.94)

Three components (spectral lines) are observed at frequencies ω−, ω0, and ω+. The spectral lines observed at ω− and ω+ are linearly vertically polarized, and the spectral line at ω0 is linearly hori-zontally polarized. Furthermore, we see that the intensity of the center spectral line of frequency ω0 is twice that of the lines at ω− and ω+. The observation of the Zeeman effect perpendicular to the magnetic field is sometimes called the transverse Zeeman effect or the Zeeman triplet. The appear-ance of the spectra corresponding to Equation 31.94 is shown in Figure 31.10.

It is of interest to determine the form of the Stokes vector Equation 31.92 when the applied magnetic field is removed. When Β = 0 we have ω− = ω+ = ω0, and Equations 31.92 and 31.94 both reduce to

S =

83 8

1

0

0

00

2

2

04eA

c Rπεω , (31.95)

which is the Stokes vector for unpolarized light, and we observe a single spectral line radiating at the frequency ω0, the natural frequency of oscillation of the bound electron. This is exactly what

figuRe 31.9 The longitudinal Zeeman effect. The spectral lines observed in a spectroscope for the Zeeman effect parallel to the magnetic field (θ = 0°).

642 Polarized Light, Third Edition

we would expect for an electron oscillating randomly about the nucleus of an atom, and we would therefore observe the spectral line represented in Figure 31.11.

Additional and background material on these topics may be found in the references [3–11]. In the next chapter we extend the observable formulation to a description of the intensity and polarization of the radiation emitted by relativistically moving electrons. In Chapter 34, we use the Stokes vec-tors to describe the emission of radiation by quantized atomic systems.

RefeReNCeS

1. Heisenberg, W., Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, Z. Phys. 33 (1925): 879–93.

2. Wolf, E., Optics in terms of observable quantities, Il Nuovo cimento 12 (1954): 884–8. 3. Collett, Ε., The description of polarization in classical physics, Am. J. Phys. 36 (1968): 713–25. 4. McMaster, W. H., Polarization and the Stokes parameters, Am. J. Phys. 22 (1954): 351–62.

– +0

figuRe 31.10 The transverse Zeeman effect. The spectral lines observed in a spectroscope for the Zeeman effect perpendicular to the magnetic field (θ = 90°).

ω0

figuRe 31.11 The Zeeman effect with the magnetic field removed. A single unpolarized spectral line is observed radiating at a frequency ω0.

The Classical Zeeman Effect 643

5. Jackson, J. D., classical Electrodynamics, New York: Wiley, 1962. 6. Sommerfeld, Α., Lectures on Theoretical Physics, Vols. I–V, New York: Academic Press, 1952. 7. Born, M., and E. Wolf, Principles of Optics, 3rd ed., New York: Pergamon Press, 1965. 8. Wood, R. W., Physical Optics, 3rd ed., Washington, DC: Optical Society of America, 1988. 9. Strong, J., concepts of classical Optics, San Francisco, CA: Freeman, 1959. 10. Jenkins, F. S., and Η. Ε. White, Fundamentals of Optics, New York: McGraw-Hill, 1957. 11. Stone, J. M., Radiation and Optics, New York: McGraw-Hill, 1963.

645

32 Further Applications of the Classical Radiation Theory

32.1 RelaTiViSTiC RadiaTioN aNd The STokeS VeCToR foR a liNeaR oSCillaToR

In previous chapters we have considered the emission of radiation by nonrelativistic moving par-ticles. In particular, we determined the Stokes parameters for particles moving in linear or curvi-linear paths. In this chapter we reconsider these problems in the relativistic regime. It is customary to describe the velocity of the charge relative to the speed of light using the ratio defined by the parameter β = v/c. (Note that we will be using this ratio in vector and magnitude form, i.e., β = |β| = |v|/c = v/c, where appropriate.)

For a linearly oscillating charge we saw that the emitted radiation was linearly polarized and its intensity dependence varied as sin2 θ. This result was derived for the nonrelativistic regime (|β| << 1). We now consider the same problem, using the relativistic form of the radiation field. Before we can do this, however, we must first show that for the relativistic regime (|β| ∼ 1) the radia-tion field continues to consist only of transverse components, Eθ and Eφ, and the radial or longitudi-nal electric component Er is zero. If this is true, then we can continue to use the same definition of the Stokes parameters for a spherical radiation field.

The relativistic radiated field has been shown by Jackson [1] to be

E xn

n( , ) ( ) ,te

c R= × − ×

4 0

2 3πε κββ ββ

ret

(32.1)

where

κ = − ⋅1 n ββ. (32.2)

The brackets […]ret mean that the field is to be evaluated at an earlier or retarded time, t′ = t – R(t′)/c, where R/c is just the time of propagation of the disturbance from one point to the other. Furthermore, cβ is the instantaneous velocity of the particle, cβ

. is the instantaneous acceleration, and n = R/R.

The quantity κ → 1 for nonrelativistic motion. For relativistic motion, the fields depend on the velocity as well as the acceleration. Consequently, the angular distribution is more complicated.

In Figure 32.1 we show the relations among the coordinates given in Equation 32.1.Recall that the Poynting vector SE is given by, where we use SE to distinguish the Poynting vector

from the Stokes vector S,

S E nE = 12 0

2cε . (32.3)

Thus, we can write, using Equation 32.1,

[ ] [( ) ]S n n nEre

⋅ = × − ×

ec R

2

20

3 6 2

2

321

π ε κββ ββ

tt

. (32.4)

646 Polarized Light, Third Edition

There are two types of relativistic effects present. The first is the effect of the specific spatial relationship between β and β

., which determines the detailed angular distribution. The other is

a general relativistic effect arising from the transformation from the rest frame of the particle to the observer’s frame and manifesting itself by the presence of the factor κ in the denomina-tor of Equation 32.4. For ultra-relativistic particles, the latter effect dominates the whole angular distribution.

In Equation 32.4, SE · n is the energy per unit area per unit time detected at an observation point at time t due to radiation emitted by the charge at time t′ = t – R(t′)/c. To calculate the energy radi-ated during a finite period of acceleration, say from t′ = T1 to t′ = T2 , we write

W dtdtdt

dtt T

t T

t T= ⋅ = ⋅

′′

′=

′=

= ∫[ ] ( )S n S nE ret E1

2

1++

= +

∫ R T c

t T R T c

( )

( )

.1

2 2

(32.5)

The quantity (SE · n)(dt/dt′) is the power radiated per unit area in terms of the charge’s own time. We have defined

′ = − ′t t

R tc( )

. (32.6)

Furthermore, as Jackson [1] has also shown,

κ = + ′′

11c

dR tdt

( ). (32.7)

Differentiating Equation 32.6 yields

dtdt′

= κ. (32.8)

The power radiated per unit solid angle is

dP t

dR

dtdt

R( )

( ) .′ = ⋅

′= ⋅

Ω2 2S n S nE Eκ (32.9)

v(t′)

n(t′) R(t′)

e

P

O

r(t′) x

figuRe 32.1 Coordinate relations for an accelerating electron. P is the observation point and O is the origin.

Further Applications of the Classical Radiation Theory 647

These results show that we will obtain a set of Stokes parameters consistent with Equation 32.9 by defining the Stokes parameters as

S c R E E E E0 021

2= +∗ ∗ε κ φ φ θ θ[ ], (32.10)

S c R E E E E1 021

2= −∗ ∗ε κ φ φ θ θ[ ], (32.11)

S c R E E E E2 021

2= +∗ ∗ε κ φ θ θ φ[ ], (32.12)

S c R i E E E E3 021

2= −∗ ∗ε κ φ θ θ φ[ ( )], (32.13)

where the electric field E(x, t) is calculated from Equation 32.1.Before we proceed to apply these results to various problems of interest, we must demonstrate

that the definition of the Stokes parameters is valid for relativistic motion. That is, the field is trans-verse and there is no longitudinal component (Er = 0). We thus write Equation 32.1 as

E xn n n

( , )[ ( )] [ ( )]

tec R

= × × − × ×

4 0

2 3πε κ

ββ ββ ββ

ret

. (32.14)

Because the unit vector n is practically in the same direction as ur, Equation 32.14 is rewritten as

E r u u u( , ) [ ( )] [ ( )]tec R r r r= × × − × ×

4 02 3πε κ

ββ ββ ββ .. (32.15)

Recall that the triple vector product is

a b c b a c c a b× ×( ) = ( ) − ( )· · , (32.16)

so Equation 32.15 can be rewritten as

E r u u u u u( , ) [ ( ) ( ) (tec R r r r r r= ⋅ − ⋅ −

4 02 3πε κ

ββ ββ ββ ⋅⋅ + ⋅ ββ ββ ββ) ( )].ur (32.17)

In spherical coordinates the field E(r, t) is

E r u u u, .t E E Er r( ) = + +θ θ φ φ (32.18)

Taking the dot product of both sides of Equation 32.17 with ur and using Equation 32.18, we see that

Er r r r r r r= ⋅ − ⋅ − ⋅ ⋅ + ⋅( ) ( ) ( )( ) ( )(u u u u u u ββ ββ ββ ββ ββ ⋅⋅ =ββ) ,0 (32.19)

648 Polarized Light, Third Edition

so the longitudinal (radial) component is zero. Thus, the radiated field is always transverse in both the nonrelativistic and relativistic regimes. Hence, the Stokes parameters definition for spherical coordinates continues to be valid.

The components Eθ and Eϕ are readily found for the relativistic regime. We have

ββ = + + x y zc

i j k, (32.20)

ββ = + +x y z

ci j k

. (32.21)

The Cartesian unit vectors in Equations 32.20 and 32.21 can be replaced with the unit vectors in spherical coordinates, that is,

i u usin cos= +θ θ θr , (32.22)

j u= ϕ , (32.23)

k u ucos sin= −θ θ θr . (32.24)

In these equations, the azimuthal angle has been set to zero because we assume that we always have symmetry around the z axis. Substituting Equations 32.22, 32.23, and 32.24 into Equations 32.20 and 32.21 yields

c x z y x zrββ = + + + −( sin cos ) ( cos sin ) θ θ θ θφ θu u u ,, (32.25)

c x z y x zr ββ = + + + −( sin cos ) ( cos sθ θ θφu u iin ) .θ θu (32.26)

The transverse components Eθ and Eϕ are then

Ee

c Rx z

xz xzθ πε κ

θ θ= − − − −4 0

2 3( cos sin )

cc

, (32.27)

Ee

c Ry

yx yx yzφ πε κ

θ= − − − + +4 0

2 3 ( )sin ( yz

c)cos

.θ

(32.28)

The second terms in square brackets in Equations 32.27 and 32.28 are the relativistic contributions. For β and β ≪ 1, Equations 32.27 and 32.28 reduce to the nonrelativistic forms used in previous chapters.

We now apply these results to determining the radiation and the polarization emitted by charges undergoing linear and circular motion. In the following sections, we treat synchrotron radiation and the motion of a charge moving in a dielectric medium (Čerenkov radiation). In the final section we deal with the scattering of radiation by electric charges.

Further Applications of the Classical Radiation Theory 649

For a linear charge that is accelerating along the z axis, β and β. are parallel, so

ββ ββ× = 0. (32.29)

Equation 32.1 then reduces to

E x n n( , ) [ ( )],tec R

= × ×4 0

2 3πε κββ (32.30)

or

E ru u v

,( )

.tec R

r r( ) = × ×

4 0

2 3πε κ

(32.31)

According to Equations 32.15 and 32.18, the field components of Equation 32.31 are

Eec k R

x zθ πεθ θ= −

4 02 3

[ cos sin ], (32.32)

Eec k R

yφ πε=

4 02 3

. (32.33)

From the definition of the Stokes parameters Equations 32.10 through 32.13, we then find that the Stokes vector for the relativistic accelerating charge (from Equations 32.32, 32.33, and 32.2) is

S =−

e zc

2 2

20

3

2

532 1

1

1

0

0

π ε

θβ θsin

( cos )

, (32.34)

where n ⋅ β = β cos θ. We see immediately that the radiation is linearly horizontally polarized as in the nonrelativistic case.

The intensity of the radiation field is seen from Equation 32.34 to be

I I( , )sin

( cos ),θ β θ

β θ=

−

0

2

51 (32.35)

where I e z c02 2 2

0332= / π ε . For the nonrelativistic case, β → 0, and Equation 32.35 reduces to

I Iθ θ( ) = sin20 , (32.36)

which is the well-known dipole radiation distribution. The minimum intensity is at θ = 0°, and the maximum intensity is at θ = 90°.

650 Polarized Light, Third Edition

Equation 32.35, on the other hand, shows that the maximum intensity shifts toward the z axis as β increases. To determine the positions of the maximum and minimum, we differentiate Equation 32.35 with respect to θ, set the result equal to zero, and find that

sinθ = 0, (32.37)

3 cos 2cos 52β θ θ β+ − = 0. (32.38)

The solution of the quadratic Equation 32.38 is

cos ,θ ββ

= + −15 1 13

2

(32.39)

where we have taken the positive root because of the requirement that |cos θ| ≤ 1. For small values of β, Equation 32.39 reduces to

cos ,θ β

52

(32.40)

so that for β = 0 the angle θ is 90° as before. For extreme relativistic motion β ≃ 1, and Equation 32.40 then reduces to

cos 1,θ (32.41)

so θ ≃ 0°. We see that the maximum intensity has moved from 0 = 90° (β = 1) to θ = 0° (β ≅ 1); that is, the direction of the maximum intensity moves toward the charge moving along the z axis.

In Figure 32.2, the intensity contours for various values of β are plotted. The contours clearly show the shift of the maximum intensity toward the z axis for increasing β. In the figure, the charge

1.6

1.1

z(θ)

y(θ)

0.6

0.1

–0.4–2 –1.5 –1 –0.5 0 0.5 1 1.5 2

β = 0

β = 0.2

β = 0.4

figuRe 32.2 Intensity distribution of a relativistic moving charge for β = 0, 0.2, and 0.4.

Further Applications of the Classical Radiation Theory 651

is moving up the z axis from the origin, and the horizontal axis corresponds to the y direction. To make the plot, we equated I(θ) with ρ, so

y θ ρ θ( ) = sin , (32.42)

z θ ρ θ( ) = cos , (32.43)

where

ρ θ θβ θ

= =−

I( )sin

( cos ).

2

51 (32.44)

We see that, as β increases, the familiar sin2 θ distribution becomes lobe-like, a characteristic behavior of relativistically moving charges.

The formulation we have derived is readily extended to an oscillating charge. The motion of a charge undergoing linear oscillation is described by

z t z ei t( ) .= 00ω (32.45)

In vector form Equation 32.45 can be written as

z t z t z ezi t

z( ) ( ) .= =u u00ω (32.46)

Furthermore, using β notation, β = ż/c, we can express the velocity and acceleration in vector form as

ββ ββ= = zc

zcz zu u . (32.47)

We see immediately that

ββ ββ× = 0, (32.48)

which is identical to Equation 32.29. Hence, we have the same equations for an oscillating charge as for a unidirectional relativistically moving charge. We easily find that the corresponding Stokes vector is

S =

−

12 4 1

1

1

00

02 2

5 04

cez

cε πθ

β θωsin

( cos )

00

. (32.49)

The radiation appears at the same frequency as the frequency of oscillation. With respect to the intensity distribution, we now have radiation also appearing below the z = 0 axis because the charge

652 Polarized Light, Third Edition

is oscillating above and below the x, y plane. The intensity pattern is identical to the unidirectional case but is now symmetrical with respect to the x, y plane. In Figure 32.3 we show a plot of the intensity contour for β = 0.4.

32.2 RelaTiviSTic MoTion of a chaRge Moving in a ciRcle: SynchRoTRon RadiaTion

In the previous section, we dealt with the relativistic motion of charges moving in a straight line and with the intensity and polarization of the emitted radiation. This type of radiation is emit-ted by electrons accelerated in linear accelerators. We have determined the radiation emitted by nonrelativistic charges moving in circular paths as well. In particular, we saw that a charge moves in a circular path when a constant magnetic field is applied to a region in which the free charge is moving.

In this section, we consider the radiation emitted by relativistically moving charges in a con-stant magnetic field. The radiation emitted from highly relativistic charges is known as synchrotron radiation, after its discovery in the operation of the synchrotron. A charge moving in a circle of radius a is shown in Figure 32.4.

The coordinates of the electron are

x t a t y t a t( ) cos ( ) sin .= =ω ω (32.50)

Using the familiar complex notation, we can express Equation 32.50 as

x t ae y t iaei t i t( ) ( ) ,= = −ω ω (32.51)

x t i ae y t a ei t i t( ) ( ) ,= =ω ωω ω (32.52)

x t a e y t ia ei t i t( ) ( ) .= − =ω ωω ω2 2 (32.53)

z(θ)

y(θ)

1.5

0

2

1

0.5

–0.5

0

–1.5

–1

–2–2 –1.5 –1 –0.5 0 0.5 1 1.5 2

figuRe 32.3 Intensity contours for a relativistic oscillating charge (β = 0.4).

Further Applications of the Classical Radiation Theory 653

For the nonrelativistic case we saw that ω, the cyclotron frequency, was given by

ω = ebm

, (32.54)

where e is the magnitude of the charge, b is the strength of the applied magnetic field, m is the mass of the charge, and c is the speed of light in free space. We can obtain the corresponding form for ω for relativistic motion by merely replacing m in Equation 32.54, the rest mass, with the relativistic mass m by the substitution

mm→

−( ).

1 2 1 2β (32.55)

Thus, Equation 32.54 becomes

ω β= −ebm

( ) .1 2 1 2 (32.56)

The frequency ω in Equation 32.56 is now called the synchrotron frequency.To find the Stokes vector of the emitted radiation, we recall from Section 32.1 that the relativistic

field components are

Ee

c Rx z

xz xzθ πε κ

θ θ= − − − −4 0

2 3( cos sin )

cc

, (32.57)

Ee

c Ry

yx yx yxφ πε κ

θ= − − − + +4 0

2 3 ( )sin ( yx

c)cos

.θ

(32.58)

Because there is no motion in the z direction, Equations 32.57 and 32.58 reduce to

Ee

c Rxθ πε κ

θ= − [ ]4 0

2 3cos , (32.59)

z

y

xωt

aO

figuRe 32.4 Motion of a relativistic charge moving in a circle of radius a in the x, y plane with an angular frequency ω.

654 Polarized Light, Third Edition

Ee

c Ry

yx yxcφ πε κ

θ= − − −

4 0

2 3 ( )sin

. (32.60)

Substituting Equations 32.52 and 32.53 into Equations 32.59 and 32.60, we find

Ee

c Raθ πε κ

ω θ= −4 0

2 32[ cos ], (32.61)

Ee

c Ria

acφ πε κ

ω ω θ= − −

4 0

2 32

2 3

sin , (32.62)

where we have suppressed the exponential time factor eiωt. From the definition of the Stokes param-eters given in Section 32.1, we then find that the Stokes vector for synchrotron radiation is

S =−

+ +− −e

a

2 4 4

2 5

2 2 2

2

1

1

1β ωβ θ

θ β θβ

( cos )

cos sin

( )siin

sin

cos

,2

2

2

θβ θ

θ−

(32.63)

where we emphasize that θ is the observer’s angle measured from the z axis. Equation 32.63 shows that synchrotron radiation is, in general, elliptically polarized. The Stokes vector Equation 32.63 is easily shown to be correct because the matrix elements satisfy the equality

S S S S02

12

22

32= + + . (32.64)

We saw earlier when dealing with the motion of a charge moving in a circle for the nonrelativistic case that the Stokes vector reduces to simpler (degenerate) forms. A similar situation arises with relativistically moving charges. Thus, when we observe the radiation at θ = 0°, the Stokes vector Equation 32.63 reduces to

S =−

−

21

1

0

0

1

2 4 4

2 5

ea

β ωβ( )

, (32.65)

which is the Stokes vector for left circularly polarized light. For θ = π/2, the Stokes vector is

S =

+− −

e ac

2 2 4

4

2

2

1

1

2

0

ωββ

β( )

. (32.66)

At this observation angle, the radiation is linearly polarized. Finally, at θ = π, we see that the radia-tion is right circularly polarized.

Further Applications of the Classical Radiation Theory 655

For β ≪ 1, the nonrelativistic regime, Equation 32.63 reduces to

S =

+−

−

e ac

2 2 4

4

2

2

1

0

2

ωθ

θ

θ

cos

sin

cos

, (32.67)

where ω = eb/m = ωc is the cyclotron frequency. This is very similar to the Stokes vector we found in Section 18.3 for a charge rotating in the x, y plane.

We now examine the intensity, orientation angle, and ellipticity of the polarization ellipse for synchrotron radiation expressed by Equation 32.63. The intensity of the radiation field, I(θ), can be written from Equation 32.63 as

Ie

ac( )

( ) ( cos sin( cos

θ ω β θ β θβ θ

= − + +−

2 4

2

2 2 2 2 21 11 ))

,5

(32.68)

where we have set ω = ωc(1 – β2)1/2. The presence of the factor (1 – β cosθ)5 in the denominator of Equation 32.68 shows that a lobe-like structure will again emerge. The orientation angle ψ and the ellipticity angle χ are

ψ ββ θ

= −−

−12

21

12

tan( )sin

(32.69)

and

χ θθ β θ

= −+ +

−12

21

12 2 2

sincos

cos sin. (32.70)

In Figure 32.5, the intensity Equation 32.68 has been plotted as a function of the observation angle θ for β = 0, 0.1, and 0.2. We see that for β = 0 (i.e., the nonrelativistic radiation pattern), the

5

6

4

5

2

3

z(θ)

y(θ)

1

2β = 0

β = 0.1

β = 0.2

–1

0

–2–3 –2 –1 0 1 2 3

figuRe 32.5 Relativistic intensity contours for β = 0.0, 0.1, and 0.2.

656 Polarized Light, Third Edition

intensity contour follows a bubble-like distribution. However, as β increases, the bubble-like con-tour becomes lobe-like. This behavior is further emphasized in Figures 32.6 and 32.7. Figure 32.6 shows Equation 32.68 for β = 0.3, 0.4, and 0.5, and Figure 32.7 shows Equation 32.49 for β = 0.6, 0.7, and 0.8.

In Figure 32.8, we have plotted the logarithm of the intensity I(θ) from θ = 0° to 180° for β = 0 to 0.9 in steps of 0.3. For β equal to 0.99 (not plotted), we have I(0°)/I(90°) = 2 × 109, which results in an extraordinarily narrow beam.

In order to plot the orientation angle ψ, Equation 32.69 as a function of θ, we note that for β = 0.0 and 1.0, ψ = 0 and −π/4, respectively. In Figure 32.9, we plot ψ as a function of θ where the contours correspond to β = 0.0, 0.1, …, 1.0.

35

26

z(θ)

y(θ)

17

8 β = 0.3

β = 0.4

β = 0.5

–1–10 –5 0 5 10

figuRe 32.6 Relativistic intensity contours for β = 0.3, 0.4, and 0.5.

800

600

700

500

300

400z(θ)

y(θ)

200

0

100

–120 –70 –20 30 80

β = 0.6

β = 0.7

β = 0.8

figuRe 32.7 Relativistic intensity contours for β = 0.6, 0.7, and 0.8.

Further Applications of the Classical Radiation Theory 657

In Figure 32.10, the ellipticity angle χ, Equation 32.70, is plotted for β = 0 to 1.0 over a range of θ = 0° to 180°. For the extreme relativistic case Equation 32.70 becomes

χ θ= − −12

1sin (cos ). (32.71)

It is straightforward to show that Equation 32.71 can be rewritten in the form of an equation for a straight line

χ θ= −2

45°, (32.72)

4

2

3

1

0

I(θ)

θ

β = 0

β = 0.3

β = 0.6

β = 0.9

–2

–1

–30 30 60 90 120 150 180

figuRe 32.8 Logarithmic plot of the intensity for β = 0.0 through 0.9.

β = 0.3

β = 0.5

β = 0.7

β = 0.9

β = 0.2

β = 0.1

0 30 60 90 120 150 180

π/4

ψ(θ)

0

θ

figuRe 32.9 Orientation angle ψ for synchrotron radiation.

658 Polarized Light, Third Edition

and this behavior is confirmed in Figure 32.10. We see that Figure 32.10 shows that the ellipticity varies from χ = −45° (a circle) at θ = 0° to χ = 45° (a counterclockwise circle) at θ = 180°.

Finally, it is of interest to compare the Stokes vector for β = 0 and for β = 1. The Stokes vec-tors are

S =

+

=K

1

0

2

0

2

2

cos

sin

cos

,

θθ

θ

β (32.73)

and

′ = ′

=S K

1

12sin

sin cos

cos

.θ

θ θθ

β (32.74)

where Κ and K′ are constants (see Equation 32.63). For θ = 0° and 90°, S and S′ become

S S=

′ = ′

=2

1

0

0

1

1

0

0

1

K K θ 00, (32.75)

S S=

′ = ′

=K K

1

1

0

0

1

1

0

0

9θ 00°. (32.76)

π/4

β = 0

β = 1

–π/40 30 60 90 120 150 180

χ(θ)

θ

figuRe 32.10 Ellipticity angle χ for synchrotron radiation.

Further Applications of the Classical Radiation Theory 659

Thus, in the extreme cases of β = 0 and β = 1, the Stokes vectors, that is, the polarization states, are identical! However, between these two extremes the polarization states are very different.

Synchrotron radiation was first observed in the operation of synchrotrons. However, many astro-nomical objects emit synchrotron radiation, and it has been associated with sunspots, the crab neb-ula in the constellation of Taurus, and radiation from Jupiter. Numerous papers and discussions of synchrotron radiation have appeared in the literature, and further information can be found in the references.

32.3 ČeReNkoV effeCT

A charged particle in uniform motion and traveling in a straight line in free space does not radiate. However, if the particle is moving with a constant velocity through a material medium, it can radi-ate if its velocity is greater than the phase velocity of light in the medium. This radiation is called Čerenkov radiation, after its discoverer, Pavel Čerenkov (1937). According to the great German phys-icist Arnold Sommerfeld [2], the problem of the emission of radiation by charged particles moving in an optical medium characterized by a refractive index n was studied as early as the beginning of the last century. The emission of Čerenkov radiation is a cooperative phenomenon involving a large number of atoms of the medium whose electrons are accelerated by the fields of the passing particle and so emit radiation. Because of the collective aspects of the process, it is convenient to use the mac-roscopic concept of a dielectric constant ε rather than the detailed properties of individual atoms.

Our primary concern in this section is to determine the polarization of Čerenkov radiation. The mathematical background as well as additional information on the Čerenkov effect can be found in Jackson’s text on classical electrodynamics [1]. Here, we shall determine the radiated field E(x, t) for the Čerenkov effect, whereupon we will then find the Stokes parameters.

A qualitative explanation of the Čerenkov effect can be obtained by considering the fields of the fast particle in the dielectric medium as a function of time. The medium is characterized by a refractive index n, so the phase velocity of the light is c/n, where c is the speed of light in a vacuum. The particle velocity is denoted by v. In order to understand the Čerenkov effect, it is not necessary to include the refractive index in the analysis, however. As an initial value, we set n = 1. At the end of the analysis we shall see the significance of n.

If we have a charged particle that is stationary but capable of emitting spherical waves, then after the passage of time t the waves are described by

x y z r t ct2 2 2 2 2+ + = ( ) = ( ) . (32.77)

If the charge is moving along the positive x axis with a velocity v, then the coordinate x is replaced by x − vt, so Equation 32.77 becomes

x vt y z ct−( ) + + = ( )2 2 2 2. (32.78)

We can consider the form of Equation 32.78 in the x, y plane by setting z = 0, so the two-dimensional representation of the spherical wave is

x vt y ct−( ) + = ( )2 2 2. (32.79)

Recall that β = v/c. Furthermore, for convenience we set c = 1, so Equation 32.79 becomes

x t y t−( ) + =β 2 2 2. (32.80)

660 Polarized Light, Third Edition

The intercepts of the spherical wave on the x axis are found by setting y = 0 in Equation 32.80. Then,

x t± = ±( )β 1 . (32.81)

The intercept of the leading edge of the spherical wave front is then

x t+ = +( )β 1 (32.82)

and, similarly, the intercept of the trailing edge of the spherical wave front is

x t− = −( )β 1 . (32.83)

The maximum and minimum values of the spherical wave along the y axis are found from the con-dition dy/dx = 0. From Equation 32.80 we can then show that the maximum and minimum values of y occur at

x t= β . (32.84)

This result is to be expected for a wave source propagating with a velocity ν = β. The corresponding maximum and minimum values of y are then found from Equation 32.80 to be

y t± = ± . (32.85)

Since the radius of the spherical wave front is r(t) = ct = (1)t, this, too, is to be expected. We see that at t = 0 both x and y = 0 correspond to the particle’s position (x – βt) at the origin. The phase velocity vp of the spherical wave is determined from r(t) = ct and vp = dr(t)/dt = c( = 1).

Solving Equation 32.80 for y(t) we have

y t t x t( ) ( ) .= ± − −2 2β (32.86)

It is of interest to plot Equation 32.86 for β = 0, 0.5, and 1.0. We see from Equation 32.83 that for β = 1 we have x– = 0; that is, the trailing edges of the spherical wave fronts coincide. In Figures 32.11 to 32.13 we have made plots of Equation 32.86 for β = 0, 0.5, and 1.0. However, to describe the expansion of the spherical wave with the passage of time as the particle moves, the coordinates of the x axis have been reversed. The largest circle corresponds to 4 seconds and appears first, followed by decreasing circles for 3, 2, and 1 second. For completeness we have included a plot for β = 0. The plot for β = 1, Figure 32.13, confirms that when the particle is moving with the speed of light, the trailing edges, which are shown as the leading edges in the plot, coincide.

Figure 32.13 is especially interesting because it shows that the wave fronts only coincide for β = 1. The question now arises, what happens when β > 1? To answer this question we return to Equation 32.86. We observe that y(t) is imaginary if

x t+ > +( )β 1 (32.87)

and

x t− > −( )β 1 . (32.88)

Further Applications of the Classical Radiation Theory 661

If we now choose, say, β = 1.5, then Equations 32.87 and 32.88 become

x t+ > 2 5. (32.89)

and

x t− > 0. .5 (32.90)

Equation 32.90 is especially interesting. We see from the Condition 32.83 that for β = 0, x is always less than 0 and for β = 1 it is exactly 0. However, Equation 32.90 now shows that if the speed of the particle exceeds the speed of light then there is a reversal of sign. However, so long as x− is less than 0.5t, y(t) is real, so the wave can propagate! In Figures 32.14 and 32.15, we show this behavior for β = 1.5 and β = 2.5.

If we now observe Figures 32.11 to 32.13 we see that the spherical wave fronts do not interfere for 0 ≤ β ≤ 1. Furthermore, we observe from Figures 32.14 and 32.15 that if we extend a straight line from the origin through the tangents of the spherical wave fronts, then a new wave front appears that is linear. This behavior is exactly what is observed when a boat moves quickly through water. It should be clearly understood that for β < 1 or β > 1 spherical waves are always generated. However, for β < 1 the waves cannot interfere whereas for β > 1 the waves can interfere. Furthermore, this reinforcement of waves for β > 1 appears suddenly as soon as this condition appears. Hence, we experience a shock, and so the straight line or tangent line is called a shock wave. The appearance

–4

–3

–2

–1

0

1

2

3

4

–4–2024

y

x

figuRe 32.11 Propagation of a spherical wave for a stationary particle (β = 0).

662 Polarized Light, Third Edition

of this shock wave does not occur because there is a sudden change in the medium (the medium is unaffected), but because the waves, which were previously noninterfering (β ≤ 1), now interfere (β > 1). In Figure 32.16, we have drawn the straight line from the origin through the tangents of the spheres.

The tangents line in Figure 32.16 is called a wake. The normal to the wake makes an angle θc, which is called the critical angle. Referring to the figure, we see that it can be expressed as

cos .θc

cv

= (32.91)

In free space a particle cannot propagate equal to or faster than the speed of light. However, in an optical medium the phase velocity of the light is less than c and is given by c/n. Thus, if a particle moves with a speed greater than c/n, it will generate an interference phenomenon exactly in the same manner as we have been describing. This behavior was first observed by Čerenkov, and con-sequently in optics the phenomenon is called the Čerenkov effect, and the emitted radiation is called Čerenkov radiation. Furthermore, the critical angle θc is called the Čerenkov angle and the shock wave is in the direction given by θc.

The Čerenkov radiation is characterized by a cone. Its most important application is to measure the velocity of fast particles. The Čerenkov angle θc is measured by moving a detector such that the maximum intensity is observed, and ν can then be immediately found.

–4

–3

–2

–1

0

1

2

3

4

–2–10123456

y

x

figuRe 32.12 Propagation of a spherical wave for a particle moving with a velocity β = 0.5.

Further Applications of the Classical Radiation Theory 663

–4

–3

–2

–1

0

1

2

3

4

012345678

y

x

figuRe 32.13 Propagation of a spherical wave for a particle moving with a velocity β = 1.0.

0–4

–3

–2

–1

0

1

2

3

4

12345678910

y

x

figuRe 32.14 Propagation of a spherical wave for a particle moving with a velocity β = 1.5.

664 Polarized Light, Third Edition

With this background, we now determine the intensity and polarization of the Čerenkov radia-tion. Our analysis draws heavily on Jackson’s treatment [1] of the Čerenkov effect and classical radiation in general.

We restate the first two equations of the chapter. The electric field emitted by an accelerating charge is given by

E xn

n( , ) ( ) ,ret

te

c R= × − ×

4 0

2 3πε κββ ββ (32.92)

–4

–3

–2

–1

0

1

2

3

4

02468101214

y

x

figuRe 32.15 Propagation of a spherical wave for a particle moving with a velocity β = 2.5.

4

2

3

1

y

x

vt

ct

–1

0

–2

–4

–3

0246810

θc

figuRe 32.16 Construction of the tangent line for β = 1.5.

Further Applications of the Classical Radiation Theory 665

where […]ret means that the quantity in the brackets is to be evaluated at the retarded time t′ = t – R(t′)/c. The quantity κ is given by

κ = − ⋅1 n ββ, (32.93)

where cβ is the instantaneous velocity of the particle, cβ. is the instantaneous acceleration, and

n = R/R. The quantity κ → 1 for nonrelativistic motion. See Figure 32.1 for the relations among the coordinates. The instantaneous energy flux is given by the Poynting vector

S E HE = ×[ ] (32.94)

or

S E nE = 12 0

2cε . (32.95)

The power radiated per unit solid angle is then

dPd

c RΩ

= 12 0

2ε E . (32.96)

The total energy radiated per unit solid angle is the time integral of Equation 32.96,

dWd

c R t dtΩ

=−∞

∞

∫12 0

2

2

ε E x( , ) . (32.97)

Equation 32.97 describes the radiation of energy in the time domain. A similar expression can be obtained in the temporal frequency domain (Parseval’s theorem), and Equation 32.97 can be expressed as

dWd

c R dΩ

=−∞

∞

∫12 0

2 2ε ω ωE x( , ) . (32.98)

We now introduce the Fourier transform pair

E x E x( , ) ( , ) ,ωπ

ω=−∞

∞

∫12

t e dti t (32.99)

E x E x( , ) ( , ) .t e di t= −

−∞

∞

∫12π

ω ωω (32.100)

By substituting Equation 32.92, the electric field of an accelerated charge, into Equation 32.99, we obtain a general expression for the energy radiated per unit solid angle per unit frequency interval in terms of an integral over the trajectory of the particle. Thus, we find that

E xn

n( , ) ( ) ωπε π κ

= × − ×

−∞

∞

∫ec R4 20

3 3ββ ββ

rett

e dti tω . (32.101)

666 Polarized Light, Third Edition

We now change the variable of integration from t to t′ by using the relation between the retarded time t′ and the observer’s time t,

′ + ′ =tR t

ct

( ), (32.102)

and we have

E xn

n( , ) ( ) ωπε π κ

= × − ×

−∞

∞

∫ec R

ei

4 203 2

ββ ββ ωω( ( ) ) .′+ ′ ′t R t c dt (32.103)

In obtaining Equation 32.103, we have used the relation from Equation 32.102 that dt = κdt′. We also observe that transforming to t′ in Equation 32.103 requires that the “ret” be dropped because the integral is no longer being evaluated at the retarded time. Since the observation point is assumed to be far away from the region where the acceleration occurs, the unit vector n is sensibly constant in time. Furthermore, referring to Figure 32.1, the distance R(t′) can be approximated as

R t x′( ) − n r· . (32.104)

Substituting this relation into Equation 32.103, we then have

E xn

n( , ) ( ) ωπε π κ

= × − ×

−∞

∞

∫ec R

ei

4 203 2

ββ ββ ωω( ( ) ) ,t t c dt− ⋅ ′ ′n r (32.105)

where x is the distance from the origin O to the observation point Ρ, r(t′) is the position of the particle relative to O as shown in Figure 32.17, and where we have neglected the unimodular phase factor.

The integral in Equation 32.105 can be simplified further. It can be shown that the factor within the integrand of Equation 32.105 can be rewritten as

nn n n× − × =

′× ×( ) ( )

.ββ ββ ββ

κ κ2

ddt

(32.106)

R(t′)eP

O

n(t′) xr(t′)

figuRe 32.17 Coordinate relations for a moving charge.

Further Applications of the Classical Radiation Theory 667

Making this substitution in Equation 32.105, we have

E xn n n( , )

( ) (ωπε π κ

ω=′

× × −∞

∞− ⋅∫e

c Rd

dtei t

4 203

ββ rr( ) ) .′ ′t c dt (32.107)

We note that d/dt′ = (dt/dt′)(d/dt′) = κ(d/dt′), and we use this and then drop the prime on the final dt in Equation 32.107. Thus, Equation 32.107 becomes

E x n n n r( , ) ( ) ( ( )ωπε π

ω= × ×−∞

∞− ⋅∫e

c Rd ei t t

4 203

ββ cc). (32.108)

Equation 32.108 can now be integrated by parts to obtain

E x n n v n r( , ) [ ( )] ( ( )ω ωπε π

ω= × ×−∞

∞− ⋅∫e

c Rei t t

4 204

cc dt) . (32.109)

For a nonpermeable medium the correct fields and energy radiated for a particle moving in free space with a velocity ν > c require that at the end of the calculation we make the replacement

cc

ee→ →

ε ε. (32.110)

Equation 32.109 then becomes

E x n n v n r( , ) [ ( )] (ω ωεπε π

ω= × ×−∞

∞− ⋅∫e

c Rei t

1 2

044 2

(( ) ) .t c dtε1 2 (32.111)

To describe the Čerenkov effect, we have a charged particle moving in a straight line whose motion is described by

r vt( ) = t. (32.112)

Since the velocity is constant, the triple vector product in Equation 32.111 can be factored out and we have

E x n n v n v( , ) [ ( )] (ω ωεπε π

ω ε= × × − ⋅ec R

ei t1 2

04

1

4 21 2 cc dt) .

−∞

∞

∫ (32.113)

The integral is a Dirac delta function, and we have

E x n n v( , ) [ ( )] cosω εε π

δ ε θ= × × −

ec R

vc

1 2

04

1 2

2 21 , (32.114)

668 Polarized Light, Third Edition

where θ is measured relative to the velocity v. The delta function only leads to a nonzero result when its argument is zero, that is, when

cos ,θβεc = 1

1 2 (32.115)

which is the condition we found earlier for the emission of radiation at θc, the critical or Čerenkov angle; the radiation is emitted only at the Čerenkov angle.

The significance of the delta function in Equation 32.115 is that the field, that is, the total energy radiated per unit frequency interval, is infinite. This infinity occurs because the particle has been moving through the medium forever. To obtain a meaningful result, we assume that the particle passes through a slab of dielectric in a time interval 2T. Then the infinite integral in Equation 32.113 is replaced by

ωπ

ωπ

ω ε βω ε

211

1 21 2e dt

T Ti t c

T

T( ) sin ( cos− ⋅

−∫ = −n v θθω ε β θ

)( cos )

.[ ]

−T 1 1 2 (32.116)

For the moment we shall represent the right-hand side of Equation 32.116 by f(ω, T) and write Equation 32.114 as

E x n n v( , ) [ ( )] ( , ).ω εε π

ω= × ×ec R

f T1 2

042 2

(32.117)

To find the Stokes parameters for Equation 32.117, we must expand the triple vector product. The vector n can be set to ur, the unit vector in the radial direction. Then

u u v u v u vr r r r× × = ( ) −( ) · . (32.118)

As before, we express the velocity in Cartesian coordinates as

v i j k= + + x y z , (32.119)

where i, j, and k, are unit vectors in the x, y, and z directions, respectively. We now express i, j, and k in spherical coordinates. Assume that we have symmetry around the

z axis, so we can arbitrarily take ϕ = 0°. Then, the unit vectors in Cartesian coordinates are related to the unit vectors in spherical coordinates ur, uθ, and uϕ, by

i u u= +sin cosθ θ θr , (32.120)

j u= φ , (32.121)

k u u= −cos sinθ θ θr . (32.122)

Then v in Equation 32.119 becomes

v u u u= + + − +r x z x z y( sin cos ) ( cos sin ) . θ θ θ θθ φ (32.123)

Further Applications of the Classical Radiation Theory 669

Substituting Equation 32.123 into the right-hand side of Equation 32.118 yields

u u v u ur r x z y× × = − − −( ) ( cos sin ) ,θ φθ θ (32.124)

which shows that the field is transverse to the direction of propagation ur. We now replace the triple vector product in Equation 32.117 by Equation 32.124, so that we have

E x u( , ) ( , )[ ( cos sinω εε π

ω θ θθ= − −ec R

f T x z1 2

042 2

)) ].− y uφ (32.125)

The vector E(x, ω) can be expressed in terms of its spherical coordinates so that

E x u u u( , ) .ω θ θ φ φ= + +E E Er r (32.126)

Equating the right-hand sides of Equations 32.125 and 32.126, we have

Ee

c Rf T x zθ

εε π

ω θ θ= − −1 2

042 2

( , )( cos sin ), (32.127)

Ee

c Rf T yφ

εε π

ω= − 1 2

042 2

( , )[ ]. (32.128)

Let us now assume that the charge is moving along the z axis with a velocity ż = cβ, so x y= = 0. Then, Equations 32.127 and 32.128 reduce to

Ee

c Rf Tθ

ε βε π

ω θ=1 2

032 2

( , )sin , (32.129)

Eφ = 0. (32.130)

The Stokes polarization parameters are defined as usual by

S E E E E0 = +∗ ∗φ φ θ θ , (32.131)

S E E E E1 = −∗ ∗φ φ θ θ , (32.132)

S E E E E2 = +∗ ∗φ θ θ φ , (32.133)

S i E E E E3 = −∗ ∗( ).φ θ θ φ (32.134)

670 Polarized Light, Third Edition

Substituting Equations 32.129 and 32.130 into these equations and forming the Stokes vector, we find that

S =−

ec

f T2 2

02 6

2 2

8

1

1

0

0

εβπε

ω θ( , )sin .. (32.135)

Čerenkov radiation is linearly vertically polarized.Finally, we can integrate S in Equation 32.135 over the solid angle dΩ. On doing this we find that

the Stokes vector for Čerenkov radiation is

S =−

e f Tc

c2 2 2

02 62

1

1

0

0

tan ( , ).

θ ωε

(32.136)

Further information on the Čerenkov effect can be found in the texts by Jackson [1] and Sommerfeld [2], as well as in the references listed in Jackson’s text.

32.4 ThomSoN aNd Rayleigh SCaTTeRiNg

Maxwell’s original purpose for developing his theory of the electromagnetic field was to encom-pass all the known phenomena of electromagnetism into a fundamental set of equations. It came as a surprise to Maxwell (and his contemporaries) that his differential equations led to waves propagating with the speed of light. After the work of Hertz and Lorentz and Zeeman the only conclusion that could be drawn was that Maxwell’s theory was a unifying theory between the electromagnetic field and the optical field. Furthermore, the phenomena were one and the same in both disciplines, the major difference being the wavelength (or frequency). Electromagnetic phenomena were associated with low frequencies, and optical phenomena were associated with high frequencies.

Maxwell’s theory, when coupled with Lorentz’s theory of the electron, led not only to the correct description of the seemingly complex Lorentz–Zeeman effect, but also to a very good understanding of the phenomenon of dispersion. Lorentz’s electron theory was able to provide a description of dispersion that led to a complete understanding of Cauchy’s simple empirical relation between the refractive index and the wavelength. This result was another triumph for Maxwell’s theory.

But there was still another application for Maxwell’s theory that was totally unexpected, the phenomenon known as scattering. It is not clear at all that Maxwell’s theory can be applied to this phenomenon, but it can and does lead to results in complete agreement with experiments. The phe-nomenon of scattering is described within Maxwell’s theory as follows: An incident field consist-ing of transverse components impinges on a free electron. The electron will be accelerated and so emits radiation; that is, it reradiates the incident radiation. If the electron is bound to a nucleus so that it is oscillating about the nucleus with a fundamental frequency, then it, too, is found to scatter or reradiate the incident radiation. The reradiation or scattering takes place in an extremely short time (nanoseconds or less). Remarkably, one discovers that the scattered radiation exhibits two distinct characteristics. The first is that there is a change in the polarization state between the incident and scattered radiation in which the degree of polarization varies with the observer’s

Further Applications of the Classical Radiation Theory 671

viewing angle. This behavior is very different from the Lorentz–Zeeman effect. There we saw that the polarization state changed as the observation angle varied, but the degree of polarization remained the same, and, in fact, is unity. The other notable difference is that the incident radiation field propagates along one axis and, ideally, can only be observed along this axis. The scattered radiation, on the other hand, is observed to exist not only along the axis but away from the axis as well. Characteristically, the maximum intensity of the scattered radiation is observed along the axis of the incident radiation, and the minimum intensity perpendicular to the direction of the propagation of the incident beam. However, unlike the behavior of dipole radiation, the intensity does not go to zero anywhere in the observed scattered radiation field. Maxwell’s theory, along with Lorentz’s electron theory, completely accounts for this behavior. We now treat the problem of scattering and present the results in terms of the Stokes parameters. The scattering behavior is represented by the Mueller matrix.

We first determine the Stokes parameters for the scattering of electromagnetic waves by a so-called free electron located at the origin of a Cartesian coordinate system. This is illustrated in Figure 32.18. The incident field is represented by E(z, t) and propagates in the z direction. The motion of a free electron is then described by

m er E= − , (32.137)

or, in component form,

xe

mE tx= −

( ), (32.138)

ye

mE ty= −

( ), (32.139)

where m is the mass of the electron, e is the charge, and Ex(t) and Ey(t) are the transverse components of the incident field. The incident field components can be written as

E t E ex xi t x( ) ,( )= +

0ω δ (32.140)

E t E ey yi t y( ) .( )= +

0ω δ (32.141)

z

P

e y

x

EyEx

Incident beam

θ

φ

figuRe 32.18 Scattering of incident radiation by a free electron.

672 Polarized Light, Third Edition

Equations 32.138 and 32.139 can be written from Equations 32.140 and 32.141 as

xe

mE e ex

i i tx= −0

δ ω , (32.142)

ye

mE e ey

i i ty= −0

δ ω . (32.143)

The accelerations are now known, so we can substitute these results directly into the equations for the radiated field in spherical coordinates,

Eec R

x zθ πεθ θ= −

4 02

[ cos sin ], (32.144)

Eec R

yφ πε=

4 02

[ ], (32.145)

and obtain

Eemc R

E e exi i txθδ ω

πεθ= − 2

02 04

[ ]cos , (32.146)

and

Eemc R

E e eyi i ty

φδ ω

πε= − 2

02 04

[ ]. (32.147)

The Stokes vector S′ is computed in the usual way using the equations

S E E E E0 = +φ φ θ θ* *, (32.148)

S E E E E1 = −φ φ θ θ* *, (32.149)

S E E E E2 = +φ θ θ φ* *, (32.150)

S i E E E E3 = −( ),* *φ θ θ φ (32.151)

and we obtain

′ =

′′′′

=

S

S

S

S

S

emc R

0

1

2

3

2

02

12 4πε

+ ++ +2

02

12

02

12

1

1

2

S S

S S

S

( cos ) sin

sin ( cos )

θ θθ θ

22

32

cos

cos

,θθS

(32.152)

Further Applications of the Classical Radiation Theory 673

where S0, etc., are the Stokes parameters for the incident plane wave Equations 32.140 and 32.141.

If we compose a linear algebraic equation that expresses the Stokes vector of the scattered field in terms of the incident field, we find that the Mueller matrix for the scattering process is

M =

++1

2 4

1 0 0

12

02

2

2 2

2emc Rπε

θ θθ

cos sin

sin cos22 0 0

0 0 2 0

0 0 0 2

θθ

θcos

cos

.

(32.153)

We see that Equation 32.153 corresponds to the Mueller matrix of a polarizer. This type of scatter-ing by a free charge is known as Thomson scattering and is applicable to the scattering of X rays by electrons and gamma rays by protons. As we saw earlier, the term e2/4πε0mc2 is the classical electron radius r0. We observe that the radius enters Equation 32.153 as a squared quantity. Thus, the scattered intensity is proportional to the area of the electron.

Two other observations can be made. First, according to Equation 32.152, the scattered intensity is

I S S( ) [ ( cos ) sin ],θ θ θ= + +12

102

12 (32.154)

where, for convenience, we have set the factor containing the physical constants to unity. We see immediately that the magnitude of the scattered radiation depends on the contribution of the lin-ear polarization, S1, of the incident beam. To plot Equation 32.154, we use the normalized Stokes parameters and set S0 to unity. We see that the two extremes for Equation 32.154 are for linearly polarized light (S1 = −1 and S1 = 1) and at the midpoint we have unpolarized light (S1 = 0). The cor-responding intensities are

I S( ) [ cos ] ( ),θ θ= + = −12

1 2 11 (32.155)

I S( ) [ cos ] ( ),θ θ= + =12

1 021 (32.156)

I S( ) [ ] ( ).θ = =12

2 11 (32.157)

We see that there is a significant change in the intensity over this range of polarization. In Figure 32.19 we have plotted Equation 32.154 by setting S0 = 1 and varying S1 = −1 to 1 in steps of 0.5 over a range of θ = 0°–360°. The inner lobe corresponds to S1 = −1, and the outer lobe to S1 = 1. We note that for S1 = 0 we obtain a peanut-like lobe.

The second observation is that we can express the scattering in terms of the scattering cross-section. This is defined by

dd

σΩ

= energy radiated/unit time/unit solid anngleincident energy/unit area/unit time

. (32.158)

674 Polarized Light, Third Edition

From Equations 32.158 and 32.152 we see that the ratio of the scattered to incident Stokes param-eters is the differential cross-section expressed by

dd

emc

S SS

σπε

θ θΩ

=

+ +12 4

12

02

20

21

2

0

[ cos ] sin. (32.159)

For the case of incident unpolarized light, Equation 32.158 reduces to

dd

emc

σπε

θΩ

=

+1

2 41

2

02

2

2( cos ). (32.160)

Equation 32.158 is known as Thomson’s formula for scattering by free charges. The total cross-section is defined to be

σ σπ

T

dd

d= ∫ ΩΩ

4. (32.161)

Integrating Equation 32.160 over the solid angle according to Equation 32.161, the total cross-sec-tion for the free electron is

σ ππεT

emc

=

83 4

2

02

2

. (32.162)

1.5

2

1

S1 = 1

0

0.5

S1 = –1S1 = 0

S1 = 0.5

–0.5

0z(θ)

x(θ)

S1 = –0.5

–1

–2

–1.5

–2 –1.5 –1 –0.5 0 0.5 1 1.5 2

figuRe 32.19 Intensity contours for scattering by a free electron for incident linearly polarized light from linear vertically polarized light (innermost contour) to linear horizontally polarized light (outermost contour) in steps of 0.5.

Further Applications of the Classical Radiation Theory 675

The Thomson cross section is equal to 0.665 × 10−28 m2 for electrons. The unit of length, e2/4πε0mc2 = 2.82 × 10–15m, is the classical electron radius, because a classical distribution of charge totaling the electronic charge must have a radius of this order if its electrostatic self-energy is equal to the electron mass. Finally, we note that classical Thomson scattering is valid only at low frequencies. The quantum effects become important when the frequency ω becomes comparable to mc2/ℏ, that is, when the photon energy ℏω is comparable with, or larger than, the particle’s rest energy mc2.

Another quantity of interest is the degree of polarization. According to Equation 32.152, this depends on both the polarization of the incident radiation and the observer’s viewing angle. For example, from Equation 32.152 we see that if we have linearly horizontally polarized light, S0 = S1 and S2 = S3 = 0, the scattered radiation is also linearly horizontally polarized and the degree of polarization is unity. However, if the incident radiation is unpolarized light, the Stokes vector is I0, 0, 0, 0, and the Stokes vector of the scattered radiation is

S =

+

12 4

1

0

0

2

02

2

0

2emc R

Iπε

θθ

cos

sin

. (32.163)

Equation 32.163 shows that the scattered radiation is, in general, partially polarized and the degree of polarization is

DOP =+sin

cos.

2

21θ

θ (32.164)

We see that for θ = 0° (so-called on-axis scattering) the degree of polarization DOP is zero, whereas for θ = 90° (off-axis scattering) the degree of polarization is unity. This behavior in the degree of polarization is characteristic of all types of scattering. In Figure 32.20 we have plotted Equation 32.164 as a function of the angle of scattering.

0.9

1

0.7

0.8

0.5

0.6

P

0.4

0.2

0.3

0

0.1

0 20 40 60 80θ (degrees)

100 120 140 160 180

figuRe 32.20 The degree of polarization DOP for scattering of unpolarized light by a free electron.

676 Polarized Light, Third Edition

We now consider the scattering from a bound charge. The equation of motion is

m k er r E+ = − , (32.165)

or, in component form,

x xe

mEx+ = −ω0

2 , (32.166)

y ye

mEy+ = −ω0

2 , (32.167)

z z+ =ω02 0, (32.168)

where ω0 = (k/m)1/2 and the incident field is again propagating along the z axis and consists of the transverse components Ex(t) and Ey(t). We first consider the solution of Equation 32.166. In order to solve this equation we know that the solution is

x t x t x tc p( ) = ( ) + ( ), (32.169)

where xc(t) is the complementary solution and xp(t) is the particular solution. Using the notation

dddt

≡ , (32.170)

we can write Equation 32.166 as

( ) ( ) ( )d x t R t202+ =ω (32.171)

where

R te

mE t

em

E e ex xi i tx( ) ( )= −

= −

0

δ ω (32.172)

and ω is the frequency of the incident light. By using the well-known methods of differential equa-tions for solving nonhomogeneous equations, we obtain the general solution

x t c e c e c ei t i t i t( ) ,= + +−1 2 3

0 0ω ω ω (32.173)

where c1, c2, and c3 are arbitrary constants. By substituting Equation 32.173 into Equation 32.166, we readily find that c3 is

ce

mE ex

i x3 2

02 0=

−( ),

ω ωδ (32.174)

Further Applications of the Classical Radiation Theory 677

so the solution of Equation 32.166 is

x t c c ee

mE e ei t i t

xi i tx( )

( ).= + +

−−

1 2 202 0

0 0ω ω δ ω

ω ω (32.175)

The first two terms in Equation 32.175 describe the natural oscillation of the bound electron and are not involved in the scattering process. The last term in Equation 32.175 is the term that arises from the interaction of the incident field E with the bound electron and describes the scattering process. Hence, the scattering term is

x te

mE e ex

i i tx( )( )

.=−ω ω

δ ω2

02 0 (32.176)

Similarly, for Equation 32.167, we have

y te

mE e ey

i i ty( )( )

.=−ω ω

δ ω2

02 0 (32.177)

The x and y accelerations of the bound electron are, from Equation 32.176 to Equation 32.177

x te

mE tx( )

( )( ),= −

−ω

ω ω

2

202

(32.178)

y t

em

E ty( )( )

( ),= −−ω

ω ω

2

202

(32.179)

where

E t E ex xi t i x( ) ,= +

0ω δ (32.180)

E t E ey yi t i y( ) .= +

0ω δ (32.181)

The radiation field components, that is, scattered field components, are

E e c R xθ πε θ= −( )[ cos ],4 02 (32.182)

E e c R yφ πε= −( )[ ].4 02 (32.183)

Substituting Equations 32.178 and 32.179 into Equations 32.182 and 32.183, respectively, and form-ing the Stokes parameters, we find that the Stokes vector of the scattered radiation is

S =−

+ +12 4

12

02 2

02

2

4

02

em c R

S

πε ω ωω

θ

( )

( cos ) SS

S S

S

S

12

02

12

2

3

1

2

2

sin

sin ( cos )

cos

cos

θθ θ

θθ

+ +

. (32.184)

678 Polarized Light, Third Edition

The result is very similar to the one we obtained for scattering by a free electron. In fact, if we set ω0 = 0 in Equation 32.184 (the free-electron condition), we obtain the same Stokes vector given by Equation 32.152. For a bound electron, however, we have an important difference. While the polarization behavior is identical, we see that the scattered intensity is now proportional to ω4 or (2πc/λ)4, that is, to the inverse fourth power of the wavelength. This shows that as the wavelength of light decreases, for example, from the red region to the blue region of the visible spectrum, the intensity of the scattered light increases. This accounts for the blue sky; the sky is blue because of the scattering by bound electrons. This behavior was first explained by Lord Rayleigh in the latter part of the nineteenth century. Consequently, the scattering process associated with ω4 (or 1/λ4) is called Rayleigh scattering.

Scattering phenomena play an important role not only in optics but, especially, in nuclear phys-ics. The ideas developed here are readily extended to particle scattering, and the interested reader can find further discussions of other aspects of scattering in the References [1–6].

RefeReNCeS

1. Jackson, J. D., classical Electrodynamics, New York: John Wiley, 1962. 2. Sommerfeld, Α., Lectures on Theoretical Physics, Vols. I–V, New York: Academic Press, 1952. 3. Strong, J., concepts of classical Optics, San Francisco: Freeman, 1959. 4. Stone, J. M., Radiation and Optics, New York: McGraw-Hill, 1963. 5. Blatt, J. M., and V. F. Weisskopf, Theoretical Nuclear Physics, New York: John Wiley, 1952. 6. Heitier, W., Quantum Theory of Radiation, 3rd ed., Oxford: Oxford University Press, 1954.

679

33 The Stokes Parameters and Mueller Matrices for Optical Activity and Faraday Rotation

33.1 iNTRoduCTioN

In 1811, Arago discovered that the plane of polarization of linearly polarized light was rotated when a beam of light propagated through quartz in a direction parallel to its optic axis. This property of quartz is called optical activity. Shortly afterward, in 1815, Biot discovered (quite by accident) that many liquids and solutions are also optically active. Among these are sugars, albumens, and fruit acids, to name a few. In particular, the rotation of the plane of polarization as the beam travels through a sugar solution can be used to measure its concentration. The measurement of the rotation in sugar solutions is a widely used method and is called saccharimetry. Furthermore, polarization measuring instruments used to measure the rotation are called saccharimeters.

The rotation of the optical field occurs because optical activity is a manifestation of an unsym-metrical isotropic medium; that is, the molecules lack not only a center of symmetry but also a plane of symmetry. Molecules of this type are called enantiomorphic since they cannot be brought into coincidence with their mirror image. Because this rotation takes place naturally, the rotation associ-ated with optically active media is called natural rotation.

In this chapter, we shall only discuss the optical activity associated with liquids and solutions, and the phenomenon of Faraday rotation in transparent media and plasmas. Optical activity in crys-tals was covered in Chapter 21.

Biot discovered that the rotation was proportional to concentration and path length. Specifically, for an optically active liquid or for a solution of an optically active substance such as sugar in an inactive solvent, the specific rotation or rotary power γ is defined as the rotation produced by a 10 cm column of liquid containing 1 g of active substance per cubic centimeter of solution. For a solution containing m g/cm3 the rotation for a path length l is given by

θ γ= ml10

, (33.1)

or, in terms of the rotary power γ,

γ θ= 10ml

. (33.2)

The product of the specific rotation and the molecular weight of the active substance is known as the molecular rotation.

In 1845, after many unsuccessful attempts, Faraday discovered that the plane of polarization was also rotated when a beam of light propagates through a medium subjected to a strong magnetic field. Still later, Kerr discovered that very strong electric fields rotate the plane of polarization. These effects are called either magneto-optical or electro-optical. The magneto-optical effect discovered

680 Polarized Light, Third Edition

by Faraday took place when lead glass was subjected to a relatively strong magnetic field, and this effect has since become known as the Faraday effect. It was through this discovery that a connection between electromagnetism and light was first made.

The Faraday effect occurs when an optical field propagates through a transparent medium along the direction of the magnetic field. This phenomenon is strongly reminiscent of the rotation that occurs in an optically active uniaxial crystal when the propagation is along its optical axis, dis-cussed previously in Chapter 21.

The magnitude of the rotation angle θ for the Faraday effect is given by

θ = VHl, (33.3)

where H is the magnetic intensity, l is the path length in the medium, and V is a constant called Verdet’s constant, a constant that depends weakly on frequency and temperature. In Equation 33.3, H can be replaced by b, the magnetic field strength. If b is in gauss, l in centimeters, and θ in minutes of arc, then Verdet’s constant measured with yellow sodium light is typically about 10−5 for gases under standard conditions, and about 10−2 for transparent liquids and solids. Verdet’s constant becomes much larger for ferromagnetic solids or colloidal suspensions of ferromagnetic particles.

The theory of the Faraday effect can be easily worked out for a gas by using the Lorentz theory of the bound electron. This analysis is described very nicely in the text by Stone [1]. However, our interest here is to derive the Mueller matrices that explicitly describe the rotation of the polarization ellipse for optically active liquids and the Faraday effect. Therefore, we derive the Mueller matrices using Maxwell’s equations along with the necessary additions from Lorentz’s theory.

In addition to the Faraday effect observed through rotation of the polarization ellipse in a trans-parent medium, we can easily extend the analysis to Faraday rotation in plasma (a mixture of charged particles).

There is an important difference between natural rotation and Faraday rotation (magneto-optical rotation), however. In the Faraday effect, the medium is levorotatory for propagation in the direction of the magnetic field and dextrorotatory for propagation in the opposite direction. If at the end of the path l the light ray is reflected back along the same path, then the natural rotation is canceled while the magnetic rotation is doubled. We shall see that this magnetic rotation effect is because for the return path, not only are the forward and backward wave vectors interchanged, but the signs of the arguments of the wave equation solutions are also interchanged. The result is that the vector direc-tion of a positive rotation is opposite to the direction of the magnetic field. Because of this, Faraday was able to amplify his very minute rotation effect by repeated back-and-forth reflections. He was then able to observe the effect in spite of the relatively weak magnetic field that he used.

33.2 oPTiCal aCTiViTy

In optically active media there are no free charges or currents. Furthermore, the permeability of the medium is, for all practical purposes, unity, so B = H. Maxwell’s equations then become

∇ × = − ∂∂

EHt

, (33.4)

∇ × = ∂∂

HDt

, (33.5)

∇ ⋅ =D 0, (33.6)

The Stokes Parameters and Mueller Matrices for Optical Activity and Faraday Rotation 681

∇ ⋅ =B 0. (33.7)

Eliminating H between Equations 33.4 and 33.5 leads to

∇ × ∇ × = − ∂∂

∂∂

( )E

E12c t t

(33.8)

or

∇ ∇ ⋅ − ∇ =( )E E E22

2

ωc

, (33.9)

where we have assumed a sinusoidal time dependence for the fields.In an optically active medium the relation between D and E is

D = ε ⋅ E, (33.10)

where ε is a tensor whose form is

εε =−

−−

ε α αα ε αα α ε

x z y

z y x

y x z

i i

i i

i i

. (33.11)

The parameters εx, εy, and εz correspond to real (on-axis) components of the refractive index and αx, αy, and αz correspond to imaginary (off-axis) components of the refractive index. For isotropic media the diagonal elements are equal, so we have

ε ε εx y z n= = = 2, (33.12)

where n is the refractive index. The vector quantity α can be expressed as

αα =

,b

λs (33.13)

where b is a constant (actually a pseudoscalar) of the medium, λ is the wavelength, and s is a unit vector in the direction of propagation equal to k/k. We thus can write Equation 33.10 as

D E k E= + ×nik

2 β( ), (33.14)

where β = b/λ. From Equation 33.6 we have

∇ ⋅ = ⋅ =D k Di 0. (33.15)

682 Polarized Light, Third Edition

Taking the scalar product of k with D in Equation 33.14, we see that

k D k E⋅ = ⋅ =n2 0; (33.16)

that is, the displacement vector and the electric vector are both perpendicular to the propagation vector k. This is quite important, since the formation of the Stokes parameters requires that the direction of energy flow (along k) and the direction of the fields be perpendicular.

Using Equations 33.6 and 33.14, Equation 33.9 now becomes, replacing k/k by s,

∇ = − + ×2 2 2E E s Eω β( ).n i (33.17)

From the symmetry of this equation we see that we can take the direction of propagation to be along any arbitrary axis. We choose this to be the z axis, so Equation 33.17 then reduces to

∂∂

= − +2

22 2 2E

zn E i Ex

x yω ω β , (33.18)

∂∂

= − −2

22 2 2

E

zn E i Ey

y xω ω β . (33.19)

The equation for Ez is trivial and need not be considered further. We assume that we have plane waves of the form

E E ex xi ik zx z= −

0δ , (33.20)

E E ey yi ik zy z= −

0δ , (33.21)

and substitute Equations 33.20 and 33.21 into Equations 33.18 and 33.19 and obtain

k n E i Ez x y2 2 2 2 0−( ) + =ω ω β , (33.22)

− + −( ) =i E k n Ex z yω β ω2 2 2 2 0. (33.23)

This pair of equations can have a nontrivial solution only if their determinant vanishes, that is,

k n i

i k nz

z

2 2 2 2

2 2 2 20

−− −

=ω ω β

ω β ω, (33.24)

so that the solution of Equation 33.24 is

k k nz2

02 2= ±( ).β (33.25)

Because we are interested in the propagation along the positive z axis, we take only the positive root of Equation 33.25, so that

′ = +k k nz 02 1 2( ) ,/β (33.26)

The Stokes Parameters and Mueller Matrices for Optical Activity and Faraday Rotation 683

′′= −k k nz 02 1 2( ) ./β (33.27)

Substituting Equation 33.26 into Equation 33.22, we find that

′ = − ′E iEy x , (33.28)

while substitution of Equation 33.27 into Equation 33.22 yields

′′= ′′E iEy x . (33.29)

For the single primed wave field we can write

E i j i j= ′ + ′ = ′ + ′′ ′ − ′E E E e E e ex y xi

yi ik zx y z( ) .0 0

δ δ (33.30)

Now from Equation 33.28 we see that

′ = ′E Ex y0 0 (33.31)

and

′ = ′ +δ δ πx y 2

. (33.32)

Hence, we can write Equation 33.30 as

′ = ′ − ′′ ′ − ′E i j( ) .E e E e exi

xi ik zx x z

0 0δ δ (33.33)

In a similar manner the double-primed wave field is found to be

′′ = ′′ + ′′′′ ′′ − ′′E i j( ) .E e E e exi

xi ik zx x z

0 0δ δ (33.34)

To simplify notation let ′ = ′ = ′′ =E E E Ex x x0 01 1 0 02, ,δ δ , and ′′ =δ δx 2 . Then the fields are

E i j1 01 011 1 1= −( ) ,E e iE e ei i ik zδ δ (33.35)

E i j2 02 011 2 2= +( ) ,E e iE e ei i ik zδ δ (33.36)

where k kz1 = ′ and k kz2 = ′′ . We now add the x and y components of Equations 33.35 and 33.36 and obtain

E E e E exi k z i k z= ++ +

01 021 1 2 2( ) ( ) ,δ δ (33.37)

E i E e E eyi k z i k z= − −[ ]+ +

01 021 1 2 2( ) ( ) .δ δ (33.38)

684 Polarized Light, Third Edition

The Stokes parameters at any point z in the medium are defined to be

S z E z E z E z E zx x y y0( ) ( ) ( ) ( ) ( ),* *= + (33.39)

S z E z E z E z E zx y yx1( ) ( ) ( ) ( ) ( ),* *= − (33.40)

S z E z E z E z E zx y y x2( ) ( ) ( ) ( ) ( ),* *= + (33.41)

S z i E z E z E z E zx y y x3( ) ( ) ( ) ( ) ( ) .* *= −[ ] (33.42)

Straightforward substitution of Equations 33.37 and 33.38 into these equations gives us

S z E E0 012

0222( ) ( ),= + (33.43)

S z E E kz1 01 024( ) cos( ),= +δ (33.44)

S z E E kz2 01 024( ) sin( ),= +δ (33.45)

S z E E3 012

0222( ) ( ),= − (33.46)

where δ = δ2 = δ1 and k = k2 – k1. We can find the incident Stokes parameters by considering the Stokes parameters at z = 0. They are

S E E0 012

0220 2( ) ( ),= + (33.47)

S E E1 01 020 4( ) cos ,= δ (33.48)

S E E2 01 020 4( ) sin ,= δ (33.49)

S E E3 012

0220 2( ) ( ).= − (33.50)

We now expand Equations 33.43 through 33.46 using the familiar trigonometric identities and find that

S z E E0 012

0222( ) ( ),= + (33.51)

S z E E kz E E kz1 01 02 01 024 4( ) ( cos )cos ( sin )sin= −δ δ ,, (33.52)

S z E E kz E E kz2 01 02 01 024 4( ) ( sin )cos ( cos )sin= +δ δ ,, (33.53)

S z E E3 012

0222( ) ( ),= − (33.54)

The Stokes Parameters and Mueller Matrices for Optical Activity and Faraday Rotation 685

which can now be written in terms of the incident Stokes parameters as given by Equations 33.47 through 33.50, so that

S z S0 0 0( ) ( ),= (33.55)

S z S kz S kz1 1 20 0( ) ( )cos ( )sin ,= − (33.56)

S z S kz S kz2 1 20 0( ) ( )sin ( )cos ,= + (33.57)

S z S3 3 0( ) ( ),= (33.58)

or in matrix form,

S z

S z

S z

S z

kz0

1

2

3

1 0 0 0

0

( )

( )

( )

( )

cos

=−−

sin

sin cos

( )

kz

kz kz

S

S0

0 0

0 0 0 1

00

1(( )

( )

( )

.0

0

02

3

S

S

(33.59)

Thus, the optically active medium is characterized by a Mueller matrix whose form corresponds to a rotator. The expression for k in Equation 33.59 can be rewritten using Equations 33.26 and 33.27 as

k k k k k k n k nz z= − = ′′− ′ = + − −2 1 02 1 2

02 1 2( ) ( ) ./ /β β (33.60)

Since β ≪ n2, Equation 33.60 can be approximated as

kkn

0β . (33.61)

The degree of polarization at any point in the medium is defined to be

dOP zS z S z S z

S z( )

( ) ( ) ( )( )

./

= + +[ ]12

22

32 1 2

0

(33.62)

On substituting Equations 33.55 through 33.58 into Equation 33.62, we find that

dOP zS S S

SdOP( )

( ) ( ) ( )( )

(/

= + +[ ] =12

22

32 1 2

0

0 0 00

0)); (33.63)

that is, the degree of polarization does not change as the optical beam propagates through the medium.

The ellipticity of the optical beam is given by

sin ( )( )

( ) ( ) ( ).

/2 3

12

22

32 1 2

χ zS z

S z S z S z=

+ +[ ] (33.64)

686 Polarized Light, Third Edition

Substituting Equation 33.55 into Equation 33.58 then shows that the ellipticity is

sin ( )( )

( ) ( ) ( )sin

/2

0

0 0 03

12

22

32 1 2

χ zS

S S S=

+ +[ ]= 22 0χ( ), (33.65)

so the ellipticity is unaffected by the medium. Finally, the orientation angle of the polarization ellipse is given by

tan ( )( )( )

( )sin ( )cos2

0 02

1

1 2

1

ψ zS zS z

S kz S kzS

= = +(( )cos ( )sin

.0 02kz S kz−

(33.66)

When the incident beam is linearly vertically or horizontally polarized, the respective Stokes vec-tors are

( , , , ) ( , , , ),1 1 0 0 1 1 0 0− and (33.67)

so S1(0) = ±1, S2(0) = 0, and Equation 33.66 reduces to

tan ( ) tan ,2ψ z kz= ± (33.68)

so that

ψ β πβλ

( ) .z kzk

nz

nz= ± = ±

= ±

12 2

0 (33.69)

Thus, the orientation angle ψ(z) is proportional to the distance traveled by the beam through the optically active medium and inversely proportional to wavelength, in agreement with experimental observations. We can now simply equate Equation 33.69 with Equation 33.1 and express the phenomenon in terms of observable quantities of the rotation, medium, and light. As a result, we see that Maxwell’s equations completely account for the behavior of the optical activity.

Before we conclude this section, one question should still be answered. In Section 33.1 we pointed out that for natural rotation the polarization of the beam is unaffected by the optically active medium when it is reflected back through the medium. To study this problem, we consider Figure 33.1. The Mueller matrix of the optically active medium is, from Equation 33.59

M( )cos sin

sin coskz

kz kz

kz kz=

−

1 0 0 0

0 0

0 0

0 0 0 1

. (33.70)

For a reflected beam we must replace z by −z and k by −k. We again obtain Equation 33.70. From a physical point of view we must obtain the same Mueller matrix regardless of the direction of

The Stokes Parameters and Mueller Matrices for Optical Activity and Faraday Rotation 687

propagation of the beam, otherwise we would have a preferential direction! The Mueller matrix for a perfect reflector is

MR =−

−

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

. (33.71)

From Figure 33.1, the Mueller matrix for propagation through the medium, reflection, and propaga-tion back through the medium, is

M M M M=

=−

( ) ( )

cos sin

sin cos

kz kz

kz kz

kz kz

R

1 0 0 0

0 0

0 00

0 0 0 1

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

−−

−

1 0 0 0

0 0

0 0

0 0 0 1

cos sin

sin cos

kz kz

kz kz

=−

−

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

.

(33.72)

With this simple example, Equation 33.72 shows that the forward and backward propagation, as well as polarization of the beam, are completely unaffected by the presence of the optically active medium.

33.3 faRaday RoTaTioN iN a TRaNSPaReNT medium

Natural rotation of the plane of polarization was first observed in quartz by Arago in 1811. With the development of electromagnetism, physicists began to investigate the effects of the magnetic field on materials and, in particular, the possible relationship between electromagnetism and light. In 1845, Michael Faraday discovered that when a linearly polarized wave is propagating in a dielec-tric medium parallel to a static magnetic field, the plane of polarization rotates. This phenomenon is known as the Faraday effect. The behavior is similar to that taking place in optically active media. However, there is an important difference. If, at the end of a path l the radiation is reflected

Mirror

Incident beam

Reflected beamOptically active medium

Z

figuRe 33.1 Reflection of a polarized beam propagating through an optically active medium.

688 Polarized Light, Third Edition

backward, then the rotation in optically active media is opposite to the original direction and cancels out; this was shown at the end of the previous section. For the magnetic case, however, the angle of rotation is doubled. This behavior, along with some other important observations, will be shown at the end of this section.

In the present problem, we take the direction of the magnetic field to be along the z axis. In addi-tion, the plane waves are propagating along the z axis, and the directions of the electric (optical) vibrations are along the x and y axes. In such a medium (transparent, isotropic, and nonconducting) the displacement current vector is

D E P= +ε0 , (33.73)

where P is the polarization vector (this vector refers to the electric polarizability of the material) and is related to the position vector r of the electron by

P r= −Ne . (33.74)

From Maxwell’s Equations 33.4 and 33.5 then become

∇ × = −E Hiω , (33.75)

∇ × = +( )H E Piω ε0 . (33.76)

Eliminating H between Equations 33.75 and 33.76, we find that

∇ + = −2 20

2E E Pω ε ω , (33.77)

or, in component form,

∇ + = −2 20

2E E Px x xω ε ω , (33.78)

∇ + = −2E E Py y yω ε ω20

2 . (33.79)

The position of the electron can readily be found from the Lorentz force equation to be

ζ ω ω ω± ±=

−

em

EeH

m2

02∓ , (33.80)

where

ζ± = ±x iy, (33.81)

E E iEx y± = ± . (33.82)

The polarization vector is then expressed as

P Ne P iPx y± ±= = ±ζ . (33.83)

The Stokes Parameters and Mueller Matrices for Optical Activity and Faraday Rotation 689

Solving for Px and Py, we find that

P AE ibEx x y= + , (33.84)

P AE ibEy y x= + , (33.85)

where

ANem

eHm

= − −( ) −

−2

202 2

02 2

2 1

( ) ,ω ω ω ω ω (33.86)

bNe H

meH

m= −( ) −

−3

202 2

2 1

ω ω ω ω. (33.87)

With Px and Py now known, Equations 33.78 and 33.79 become

∂∂

+ + ( ) + ( ) =2

22

02 2 0

Ez

E A E i b Exx x yω ε ω ω , (33.88)

∂∂

+ + ( ) + ( ) =2

22

02 2 0

E

zE A E i b Ey

y y xω ε ω ω . (33.89)

Since we are assuming that there is propagation only along the z axis, we can rewrite Equations 33.88 and 33.89 as

− + +( ) + ( ) =k A E i b Ex y2 2

02 2 0ω ε ω ω , (33.90)

− + +( ) + ( ) =k A E i b Ey x2 2

02 2 0ω ε ω ω . (33.91)

If we now compare Equations 33.88 and 33.89 with Equations 33.18 and 33.19, we see that the forms of the equations are identical. We can proceed directly with writing the Mueller rotation matrix and the remaining relations. In addition, we find the wave number for the propagating waves to be

′ ′′ = −− ±

kNe m

eH m,

//

( ) /,ω

ω ω ω1

2

202

1 2

(33.92)

where the single and double primes correspond to the (+) and (−) solutions in Equation 33.92, respectively. The orientation angle for linearly polarized radiation is then determined from Equation 33.66 to be

ψ = ′′ − ′( )12

k k z. (33.93)

690 Polarized Light, Third Edition

Since the second term under the square root in Equation 33.92 is small compared with unity, we easily find that

′′ − ′−

k kNem

H

2 3 2

2 202

ωω ω

, (33.94)

so the orientation angle of the radiation is

ψ ωω ω

Ne Hz

mVHz

3 2

2 202−( ) = , (33.95)

where Verdet’s constant V is

VNe

m=

−( )3 2

2 202

ωω ω

. (33.96)

We see that the Mueller matrix for the Faraday effect is

M( )cos sin

sin cosz

VHz VHz

VHz VHz=

− 1 0 0 0

0 0

0 0

0 0 0 1

, (33.97)

and it is evident that the rotation Equation 33.95 is proportional to the path length in agreement with the experimental observation.

Before concluding, let us again consider the problem where the beam propagates through the magneto-optical medium and is reflected back toward the optical source. For convenience, we replace VHz with θ and we write Equation 33.97 as

M z( ) =−

1 0 0 0

0 0

0 0

0 0 0 1

cos sin

sin cos

θ θθ θ

. (33.98)

For a reflected beam we must replace z by −z; however, VH is unaffected. Unlike natural rotation, in the Faraday effect we have superposed an asymmetry in the problem with the unidirectional mag-netic field. Thus, θ transforms to −θ, and the Mueller matrix M(z) for the beam propagating back to the source becomes

M −( ) =−

z

1 0 0 0

0 0

0 0

0 0 0 1

cos sin

sin cos

θ θθ θ

. (33.99)

The Stokes Parameters and Mueller Matrices for Optical Activity and Faraday Rotation 691

Recall that the Mueller matrix for a reflector (mirror) is

MR =−

−

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

, (33.100)

and therefore the Mueller matrix for propagation through the medium, reflection, and propagation back through the medium, is

M M M M= −( ) ( )

=−

z zR

1 0 0 0

0 0

0 0

0 0 0 1

cos sin

sin cos

θ θθ θ

−−

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

11 0 0 0

0 0

0 0

0 0 0 1

cos sin

sin cos

θ θθ θ

−

=

11 0 0 0

0 2 2 0

0 2 2 0

0 0 0 1

cos sin

sin cos

θ θθ θ

−− −

−

.

(33.101)

Since θ is arbitrary, we can replace θ by −θ in Equation 33.101 without changing its meaning, and we then have

M =−

−

1 0 0 0

0 2 2 0

0 2 2 0

0 0 0 1

cos sin

sin cos

θ θθ θ

. (33.102)

Equation 33.102 is recognized as the Mueller matrix for a pseudorotator. That is, the rotation as well as the ellipticity are opposite to the behavior of a true rotator. Thus, unlike natural rotation, the angle of rotation is doubled upon reflection, so that for m reflections, 2θ in Equation 33.102 is replaced by 2mθ, and a relatively large rotation angle can then be measured.

33.4 faRaday RoTaTioN iN a PlaSma

While we have used Maxwell’s equations to describe the propagation and polarization of light in optical media, the fact is that Maxwell’s equations are universally applicable. In this section, we briefly wish to show that the phenomenon of Faraday rotation appears when waves propagate in plasmas. Plasmas are gaseous matter consisting of charged particles. They appear not only in the laboratory but throughout the universe.

In a plasma, the fields are again described by Maxwell’s equations that we write here as

∇ × = −E Hiω , (33.103)

∇ × = ⋅H Eiω εε , (33.104)

692 Polarized Light, Third Edition

where ε is the plasma dielectric tensor. For a plasma having a static magnetic field along the z axis we have H = Hk, and the tensor ε is then found to be (see Bekefi [2])

εε = −

ε εε ε

ε

xx xy

xy xx

zz

0

0

0 0

, (33.105)

where

ε εω

ω ωxx yyp

g

= = −−

12

2 2, (33.106)

ε εω ω

ω ω ωxy yxg p

g

i= − =

−−( )

2

2 2, (33.107)

εωωzz

p= −12

2, (33.108)

and

ωεp

em

22

0

= = plasma frequency, (33.109)

ωg

eHm

= = electron gyrofrequency. (33.110)

Eliminating H between Equations 33.103 and 33.104 gives

∇ ∇ ⋅( ) − ∇ =2 2E E Eω εε . (33.111)

We now consider the wave to be propagating along the z axis, that is, in the direction of the static magnetic field. For this case, it is not difficult to show that ∇ ⋅ E = 0. Equation 33.111 then reduces to

∂∂

= − +[ ]2

22E

zE Ex

xx x xy yω ε ε , (33.112)

∂∂

= − −[ ]2

22

E

zE Ey

xx y xy xω ε ε . (33.113)

These equations are similar in form to Equations 33.88 and 33.89, and we solve for the wavenumber solutions in the same way and find that the wavenumbers for individual waves are

′ = +( )k ixx xyω ε ε 1 2/, (33.114)

The Stokes Parameters and Mueller Matrices for Optical Activity and Faraday Rotation 693

′′ = +( )k ixx xyω ε ε 1 2/, (33.115)

so

′ − ′′−( )k k g p

g

ω ω

ω ω

2

2 2, (33.116)

and the angle of rotation is

ψω ωω ω

= ′ − ′′

=

−( )k k

zzg p

g2 2

2

2 2. (33.117)

For a plasma we obtain the rotation Mueller matrix

M =−

1 0 0 0

0 0

0 0

0 0 0 1

cos sin

sin cos

ψ ψψ ψ

.. (33.118)

The subject of optical activity and magneto-optical phenomena is vast. Many of the details of par-ticular aspects as well as general treatments of the subject can be found in the references [1–8].

RefeReNCeS

1. Stone, J. M., Radiation and Optics, New York: McGraw-Hill, 1963. 2. Bekefi, G., Radiation Processes in Plasmas, New York: Wiley, 1966. 3. Condon, E. U., Theories of optical rotatory power, Rev. Mod. Phys. 9 (1937): 432–57. 4. Born, M., Optik, Berlin: Springer Verlag, 1933. 5. Sommerfeld, A., Lectures on Theoretical Physics, Vols. I–V, New York: Academic Press, 1952. 6. Papas, C. H., Theory of Electromagnetic Wave Propagation, New York: McGraw-Hill, 1965. 7. Wood, R. W., Physical Optics, 3rd ed., Washington, DC: Optical Society of America, 1988. 8. Jenkins, F. S., and H. E. White, Fundamentals of Optics, New York: McGraw-Hill, 1957.

695

34 Stokes Parameters for Quantum Systems

34.1 iNTRoduCTioN

In previous chapters we saw that classical radiating systems could be represented in terms of the Stokes parameters and the Stokes vector. In addition, we saw that the representation of spectral lines in terms of the Stokes vector enabled us to arrive at a formulation of spectral lines that cor-responds exactly to spectroscopic observations in terms of the polarization, frequency, and bright-ness. Specifically, when this formulation was applied to describing the motion of a bound electron moving in a constant magnetic field, there was complete agreement between the Maxwell-Lorentz theory and Zeeman’s experimental observations. Thus, by the end of the nineteenth century the combination of Maxwell’s theory of radiation, represented by Maxwell’s equations, and the Lorentz theory of the atom appeared to completely explain optical and electromagnetic phenomena. This triumph of the new theory was short-lived, however.

The simple fact was that while the electrodynamic theory explained the appearance of spectral lines in terms of polarization, frequency, and brightness in specific cases, there was still a very seri-ous problem. Spectroscopic observations actually showed that even for the simplest element, that is, ionized hydrogen gas, there was a multiplicity of spectral lines. Furthermore, as the elements increased in atomic number, the number of spectral lines for each element greatly increased. For example, the spectrum of iron showed hundreds of lines whose intensities and frequencies appeared to be totally irregular. In spite of the best efforts of nineteenth-century theoreticians, no theory was ever devised within classical physics that could account for the number and position of the spectral lines.

Nevertheless, the fact that the Lorentz–Zeeman effect was completely explained by the electro-dynamic theory clearly showed that in many ways the theory was on the right track. One must not forget that Lorentz’s theory not only predicted the polarizations and the frequencies of the spectral lines, but even showed that the brightness of the central line in the “three line linear spectrum (θ = 90°)” would be twice as bright as the outer lines. It was this quasi-success that was so puzzling for such a long time.

Intense efforts were carried out for the first 25 years of the twentieth century on this problem of the multiplicity of spectral lines. The first real breakthrough was by Niels Bohr [1] in a paper pub-lished in 1913. Using Planck’s quantum ideas (1900) and the Rutherford model of an atom (1911) in which an electron revolved around a nucleus, Bohr was able to predict with great accuracy the spec-trum of ionized hydrogen gas. A shortcoming of this model, however, was that even though the elec-tron revolved in a circular orbit it did not appear to radiate, in violation of classical electrodynamics; we saw earlier that a charged particle moving in a circular orbit radiates. According to Bohr’s model the atomic system radiated only when the electron dropped to a lower orbit; the phenomenon of absorption corresponded to the electron moving to a higher orbit. In spite of the difficulty with the Bohr model of hydrogen, it worked successfully. It was natural to try to treat the next element, the two-electron helium atom, in the same way. The attempt was unsuccessful.

Finally, in 1925, Werner Heisenberg published a new theory of the atom [2] that has since come to be known as quantum mechanics. This theory was a radical departure from classical physics. In this theory, Heisenberg avoided all attempts to introduce those quantities that are not subject to experimental observation, for example, the motion of an electron moving in an orbit.

696 Polarized Light, Third Edition

In its simplest form he constructed a theory in which only observables appeared. In the case of spectral lines this was, of course, the polarization, frequency, and brightness. This approach was considered even then to be extremely novel. By now, however, physicists had long forgotten that a similar approach had been taken nearly 75 years earlier by Stokes [3]. The reader will recall that to describe unpolarized light Stokes had abandoned a model based on amplitudes (nonobservables) and succeeded by using an intensity formulation (observables). Heisenberg applied his new theory to determining the energy levels of the harmonic oscillator and was delighted when he arrived at the formula En = ћω(n + 1/2). The significance of this result was that for the first time the factor of 1/2 arose directly out of the theory and not as a factor to be added to obtain the right result. Heisenberg noted at the end of his paper, however, that his formulation might be difficult to apply even to the simplest of problems such as the hydrogen atom because of the very formidable math-ematical complexities.

At the same time that Heisenberg was working, an entirely different approach was being taken by another physicist, Erwin Schrödinger. Using an idea put forth in a thesis by Louis de Broglie, he developed a new equation to describe quantum systems. This new equation was a partial dif-ferential equation, which has since come to be known as Schrödinger’s wave equation. On applying his equation to a number of outstanding problems such as the harmonic oscillator, he also arrived at the same result for the energy as Heisenberg. Remarkably, Schrödinger’s formulation of quantum mechanics was totally different from Heisenberg’s. His formulation, unlike Heisenberg’s, used the pictorial representation of electrons moving in orbits in a wavelike motion, an idea proposed by de Broglie.

The question then arose, how could two seemingly different theories arrive at the same results? The answer was provided by Schrödinger. He discovered that Heisenberg’s quantum mechanics, which was now being called quantum matrix mechanics and his wave mechanics were mathemati-cally identical. In a very remarkable result, Schrödinger showed that Heisenberg’s matrix elements could be obtained by simply integrating the absolute magnitude squared of his wave equation solu-tion multiplied by the variable over the volume of space. This result is extremely important for our present problem because it provides the mechanism for calculating the variables x y z, , and in our radiation equation.

We saw that the radiation equations for Eθ and Eϕ were proportional to the acceleration com-ponents x y z, , and . To obtain the corresponding equations for quantum mechanical radiating systems, we must calculate these quantities using the rules of quantum mechanics. In Section 34.4 we transform the radiation equations so that they also describe the radiation emitted by quantum systems. In Section 34.5 we determine the Stokes vectors for several quantized systems. We therefore see that we can describe both classical and quantum radiating systems by using the Stokes vector.

Before we carry this out, however, we will describe some relationships between classical and quantum radiation fields.

34.2 RelaTioN beTWeeN STokeS PolaRiZaTioN PaRameTeRS aNd QuaNTum meChaNiCal deNSiTy maTRiX

In quantum mechanics the treatment of partially polarized light and the polarization of the radiation emitted by quantum mechanical systems appear to be very different from the classical methods. In classical optics, the radiation field is described in terms of the polarization ellipse and amplitudes. On the other hand, in quantum optics the radiation field is described in terms of density matrices. Furthermore, the polarization of the radiation emitted by quantum systems is described in terms of intensities and selection rules rather than the familiar amplitude and phase relations of the optical field. Let us examine the descriptions of polarization in classical and quantum mechanical terms. We start with a historical review and then present the mathematics for the quantum mechanical treatment.

Stokes Parameters for Quantum Systems 697

It is a remarkable fact that after the appearance of Stokes’s paper in 1852 [3] and introduction of his parameters, they were practically forgotten for nearly a century! It appears that only in France was the significance of his work fully appreciated. After the publication of Stokes’s paper, Verdet expounded upon them (1862). It appears that the Stokes parameters were thereafter known to French students of optics, for example, Henri Poincaré (ca. 1890) and Paul Soleillet (1927) [4]. The Stokes parameters did not reappear in any publication in the English-speaking world until 1942, in a paper by Francis Perrin [5]. Perrin was the son of the Nobel laureate Jean Perrin. Both father and son fled to the United States after the fall of France in June 1940. Jean Perrin was a scientist of international standing, and he also appears to have been a very active voice against fascism in prewar France. Had both father and son remained in France, they would have very probably been killed during the occupation.

Perrin’s 1942 paper is very important because: (1) he reintroduced the Stokes parameters to the English-speaking world, (2) he presented the relation between the Stokes parameters for a beam of light that underwent rotation or was phase shifted, (3) he showed the connection between the Stokes parameters and the wave statistics of John von Neumann [6], and (4) he derived conditions on the Mueller matrix elements for scattering (the Mueller matrix had not been named at that date). Perrin also stated that Soleillet (1927) had pointed out that only a linear relation could exist between the Stokes parameter for an incident beam (Si) and the transmitted (or scattered) beam ( ′Si ). According to Perrin, the argument for a linear relation was a direct consequence of the superposition of the Stokes parameters for m independent beams; only a linear relation would satisfy this requirement. This is discussed further in this section. The impact of his paper did not appear for several years because of its publication during the Second World War. As a result, even by 1945 the Stokes param-eters were still not generally known.

The question of the relation between the classical and quantum representation of the radiation field only appears to have arisen after the rediscovery of Stokes’s 1852 paper and the Stokes param-eters by the Nobel laureate Subrahmanyan Chandrasekhar in 1947 while writing his fundamental papers on radiative transfer [7–9]. Chandrasekhar’s astrophysical research was well known, and consequently, his papers were immediately read by the scientific community.

Shortly after the appearance of Chandrasekhar’s radiative transfer papers, Ugo Fano showed that the Stokes parameters are a very suitable analytical tool for treating problems of polarization in both classical optics and quantum mechanics [10–12]. He appears to have been the first to give a quantum mechanical description of the electromagnetic field in terms of the Stokes parameters; he also used the formalism of the Stokes parameters to determine the Mueller matrix for Compton scattering. Fano also noted that the reason for the successful application of the Stokes parameters to the quantum theoretical treatment of electromagnetic radiation problems is that they are the observ-able quantities of phenomenological optics.

The appearance of the Stokes parameters of classical optics in quantum physics appears to have come as a surprise at the time. The reason for their appearance was pointed out by Falkoff and MacDonald [13] shortly after the publication of Fano’s first paper. In classical and quantum optics, the representations of completely (i.e., elliptically) polarized light are identical (this was also first pointed out by Perrin) and can be written as

ψ ψ ψ= +c c1 1 2 2. (34.1)

However, the classical and quantum interpretations of this equation are quite different. In classi-cal optics, ψ1 and ψ2 represent perpendicular unit vectors, and the resultant polarization vector ψ for a beam is characterized by the complex amplitudes c1 and c2. The absolute magnitude squared of these coefficients then yields the intensities |c1|2 and |c2|2 that one would measure through an analyzer in the direction of ψ1 and ψ2. In the quantum interpretation, ψ1 and ψ2 represent orthogonal polarization states for a photon, but now |c1|2 and |c2|2 yield the relative probabilities

698 Polarized Light, Third Edition

for a single photon to pass through an analyzer that admits only quanta in the states ψ1 and ψ2, respectively.

In both interpretations, the polarization of the beam, or photon, is completely determined by the complex amplitudes c1 and c2. In terms of these quantities, one can define a 2 × 2 matrix with elements

ρij i jc c i j= =* , , .1 2 (34.2)

In quantum mechanics, an arbitrary wave equation can be expanded into any desired complete set of orthonormal eigenfunctions, that is,

ψ ψ= ∑ci i

i

, (34.3)

and then,

ψ ψψ ψ ψ2 = = ∑* * *.c ci j i j

ij

(34.4)

From the expansion coefficients we can form a matrix ρ by the rule

ρρij i jc c i j= =* , , .1 2 (34.5)

According to Equation 34.1, we can then express Equation 34.5 in a 2 × 2 matrix

ρρ =

ρ ρρ ρ

11 12

21 22

. (34.6)

The matrix ρ is known as the density matrix and has a number of interesting properties; it is usually associated with von Neumann (1927) and Landau. First, we note that ρii i ic c= * gives the probability of finding the system in the state characterized by the eigenfunction ψi. If we consider the ψ func-tion as being normalized, then

ψψ τ ψ ψ τ ρ ρ* * * .d c c d c ci j i j i i

iij

= = = + =∑∫∑∫ 11 22 1 (34.7)

Thus, the sum of the diagonal matrix elements is 1. The process of summing these elements is known as taking the trace of the matrix, and is written as Tr(…), so we have

Tr(ρ) = 1. (34.8)

If we measure some variable F in the system described by ψ, the result is given by

F F d c Fc d c c Ei i j j

ij

i j ij

ij

= = =∫∑∫ ∑ψ ψ τ ψ ψ τ* * * * , (34.9)

Stokes Parameters for Quantum Systems 699

where the matrix Fij is defined by the formula

F dij i j= ∫ψ ψ τF * . (34.10)

However,

Fij ij ii

i

ρ =∑ ( ) ,Fρρ (34.11)

therefore,

⟨ ⟩ = ∑F F( )ρρ ii

i

(34.12)

or

⟨ ⟩ =F FTr( ).ρρ (34.13)

Thus, the expectation value of F, that is, ⟨F⟩, which is the average value we would expect to obtain from a large number of measurements of F, is determined by taking the trace of the matrix product of F and ρ.

In classical statistical mechanics, the density function ρ(p, q) in phase space, where p and q are the momentum and the position, respectively, is normalized by the condition

ρρ( )p q p q 1, ,∫ =d d (34.14)

and the average value of a variable is given by

F , .= ∫F p q p qρρ( ) d d (34.15)

We see immediately that a similar role is played by the density matrix in quantum mechanics by comparing Equations 34.7 and 34.13 with Equations 34.14 and 34.15.

The polarization of electromagnetic radiation can be described by the vibration of the elec-tric vector. For a complete description, the field may be represented by two independent beams of orthogonal polarizations. That is, the electric vector can be represented by

E e e= +c c1 1 2 2, (34.16)

where e1 and e2 are two orthogonal unit vectors and c1 and c2, which are in general complex, describe the amplitude and phase of the two vibrations. From the two expansion coefficients in Equation 34.16, we can form a 2 × 2 density matrix. Furthermore, from the viewpoint of quantum mechanics, the equation analogous to Equation 34.16 is given by

ψ ψ ψ= +c c1 1 2 2. (34.17)

700 Polarized Light, Third Edition

We now consider the representation of an optical beam in terms of its density matrix. An optical beam can be represented by

E e e= +E1 1 2 2E , (34.18)

where

E a t1 1 1cos= +( ),ω δ (34.19)

E a t2 2 2cos= +( ).ω δ (34.20)

In complex notation, Equations 34.19 and 34.20 are written as

E a i t1 1 1exp= +( )ω δ , (34.21)

E a i t2 2exp= +2 ( ).ω δ (34.22)

Let a1 = cos θ, a2 = sin θ, and δ = δ2 – δ1. Equation 34.18 can then be expressed as

E e e= +−cos sini1 2θ θδe , (34.23)

so we have

c1icos e= −θ δ , (34.24)

c2 sin= θ. (34.25)

The density matrix is now explicitly written out as

ρρ =

=

ρ ρρ ρ

11 12

21 22

1 1 1 2

2 1 2 2

c c c c

c c c c

* *

* *

=

−cos cos sin

sin cos sin

2

2

θ θ θθ θ θ

δ

δ

e

e

i

i . (34.26)

Complete polarization can be described by writing Equation 34.1 in terms of a single eigenfunction for each of the two orthogonal states, that is,

ψ ψ= c1 1 (34.27)

or

ψ ψ= c2 2, (34.28)

where ψi refers to a state of pure polarization. The corresponding density matrices are then

ρρ11 1 0

0 0

1 0

0 0=

=

c c*

(34.29)

Stokes Parameters for Quantum Systems 701

and

ρρ22 2

0 0

0

0 0

0 1=

=

c c*

, (34.30)

where we have set c c1 1* and c c2 2

* equal to 1 to represent a beam of unit intensity.We can use Equations 34.29 and 34.30 to obtain the density matrix for unpolarized light. Since

an unpolarized beam may be considered to be the incoherent superposition of two polarized beams with equal intensity, if we add Equations 34.29 and 34.30, the density matrix is

ρρU =

12

1 0

0 1, (34.31)

The factor 1/2 has been introduced because the normalization condition requires that the trace of the density matrix be unity. Equation 34.31 can also be obtained from Equation 34.26 by averaging the angles θ and δ over π and 2π, respectively.

In general, a beam will have an arbitrary degree of polarization, and we can characterize such a beam by the incoherent superposition of an unpolarized beam and a totally polarized beam. From Equation 34.36, the polarized contribution is described by

ρρP

c c c c

c c c c=

1 1 1 2

2 1 2 2

* *

* *. (34.32)

The density matrix for a beam with arbitrary polarization can then be written in the form

ρρ =

+

U Pc c c c

c c c c

1 0

0 11 1 1 2

2 1 2 2

* *

* *, (34.33)

where U and P are the factors to be determined. In particular, P is the degree of polarization; it is a real quantity and its range is 0 ≤ P ≤ 1. We now note the following three cases:

1. If 0 < P < 1, then the beam is partially polarized. 2. If P = 0, then the beam is unpolarized. 3. If P = 1, then the beam is totally polarized.

For P = 0, we know that

ρρU =

12

1 0

0 1. (34.34)

and U = 1/2. For P = 1, the density matrix is given by Equation 34.32, so U = 0. We can now easily determine the explicit relation between U and P by writing

U aP b= + . (34.35)

702 Polarized Light, Third Edition

From the condition on U and P just given we find that b = 1/2 and a = −b, so the explicit form of Equation 34.35 is

U P= − +12

12

. (34.36)

Thus Equation 34.33 becomes

ρρ = −

+

12

11 0

0 11 1 1 2

2 1 2 2

( )* *

* *P P

c c c c

c c c c

. (34.37)

Equation 34.37 is the density matrix for a beam of arbitrary polarization.By the proper choice of pure states of polarization ψi, the part of the density matrix representing

total polarization can be written in one of the forms given by Equations 34.27 and 34.28. Therefore, we may write the general density matrix as

ρρ = −

+

=+

−1

2

1 0

0 1

1 0

0 0

12

1 0

0 1( )1 P P

P

P

. (34.38)

Hence, any intensity measurement made in relation to these pure states will yield the eigenvalues

I P+ = +12

1( ), (34.39)

I P− = −12

1( ). (34.40)

Classical optics requires that to determine experimentally the state of polarization of an optical beam four measurements must be made. The optical field in classical optics is described by

E e e= +E1 1 2 2E , (34.41)

where

E a i t1 1 1exp= +( ),ω δ (34.42)

E a í t2 2 2exp= +( ).ω δ (34.43)

In quantum optics the optical field is described by

ψ ψ ψ= +c c1 1 2 2. (34.44)

Comparing c1 and c2 in Equation 34.44 with E1 and E2 in Equations 34.42 and 34.43 suggests that we set

c a i t1 1 1exp= +( ),ω δ (34.45)

Stokes Parameters for Quantum Systems 703

c a i t2 2exp= +2 ( ).ω δ (34.46)

We now define the Stokes polarization parameters for a beam to be

S c c c c0 1 1 2 2= +* * , (34.47)

S c c c c1 1 1 2 2= −* * , (34.48)

S c c c c2 1 2 2 1= +* * , (34.49)

S i c c c c3 1 2 2 1= −( ).* * (34.50)

We now substitute Equations 34.45 and 34.46 into these expressions for the Stokes parameters and find that

S a a0 12

22= + , (34.51)

S a a1 12

22= − , (34.52)

S a a2 1 22 cos= δ, (34.53)

S a a3 1 22 sin= δ. (34.54)

We see that these equations are exactly the classical Stokes parameters, with a1 and a2 replacing, for example, E0x and E0y as previously used in this text. Expressing Equations 34.51 through 34.54 in terms of the density matrix elements, ρ11 1 1= c c*, and so on, the Stokes parameters are linearly related to the density matrix elements by

S0 = +ρ ρ11 22 , (34.55)

S1 11 22= −ρ ρ , (34.56)

S2 12 21= +ρ ρ , (34.57)

S i3 12 21= −( ).ρ ρ (34.58)

The Stokes parameters are linear combinations of the elements of the 2 × 2 density matrix.Let us express Equation 34.55 using the symbol I for the intensity and the remaining parameters

of the beam by P1, P2, and P3, so that

I = +ρ ρ11 22 , (34.59)

P1 11 22= −ρ ρ , (34.60)

704 Polarized Light, Third Edition

P2 12 21= +ρ ρ , (34.61)

P i3 12 21= −( )ρ ρ . (34.62)

We can write the density matrix Equation 34.26 in terms of I, P1, P2, and P3 as

ρρ =

=+ −+ −

ρ ρρ ρ

11 12

21 22

1 2 3

2 3 1

12

1

1

P P iP

P iP P

, (34.63)

where we have set I = 1. From the point of view of measurement, both the classical and quantum theories yield the same results; however, the interpretations, as pointed out above, are completely different.

We also recall that the Stokes parameters satisfy the condition

I P P P212

22

32≥ + + . (34.64)

Substituting expressions for I, P1, P2, and P3 into Equation 34.64, we find that

det 11 22 12 21ρρ( ) = − ≥ρ ρ ρ ρ 0, (34.65)

where “det” signifies determinant. Similarly, the degree of polarization P is given by

P = − ++

( ).

ρ ρ ρ ρρ ρ

11 222

12 21

11 22

4 (34.66)

There is one further point that we wish to make. The wave function ψ can be expanded in a complete set of orthonormal eigenfunctions. For electromagnetic radiation this consists only of the terms

ψ ψ ψ= +c c1 1 2 2. (34.67)

The wave functions describing pure states may be chosen to be of the form

ψ ψ1 2

1

0

0

1=

=

and . (34.68)

Substituting Equation 34.68 into Equation 34.67, we have

ψ =

c

c1

2

. (34.69)

Using this wave function leads to the following expressions for the expectation values (see Equation 34.9) of the unit matrix and the Pauli spin matrices

I c cc

cc c c c= = ( )

= +11 0

0 11 2

1

21 1 2

* * *22* , (34.70)

Stokes Parameters for Quantum Systems 705

P c cc

cc cz1 1 2

1

21 1

1 0

0 1= = ( )

−

= −σ * * * cc c2 2* , (34.71)

P c cc

cc c cx2 1 2

1

21 2

0 1

1 0= = ( )

= +σ * * *22 1c* , (34.72)

P c ci

i

c

ci c cy3 1 2

1

21 2

0

0= = ( ) −

=σ * * ( ** *).− c c2 1 (34.73)

We see that the terms on the right-hand side of these equations are exactly the Stokes polarization parameters. The Pauli spin matrices are usually associated with particles of spin 1/2, for example, the electron. However, for both the electromagnetic radiation field and for particles of spin ½, the wave function can be expanded in a complete set of orthonormal eigenfunctions consisting of only two terms, Equation 34.1. Thus, the quantum mechanical expectation values correspond exactly to observables.

Further information on the quantum mechanical density matrices and the application of the Stokes parameters to quantum problems, for example, Compton scattering, can be found in the numerous papers cited in the references.

34.3 NoTe oN PeRRiN’S iNTRoduCTioN of STokeS PaRameTeRS, The deNSiTy maTRiX, aNd liNeaRiTy of muelleR maTRiX elemeNTS

It is worthwhile to discuss Perrin’s observations further. It is rather remarkable that he discussed the Stokes polarization parameters and their relationship to the Poincaré sphere without any intro-duction or background. While the Stokes parameters appear to have been known by French optical physicists, the only English-speaking references to them are in the papers of Lord Rayleigh [14] and a textbook by Walker [15]. Walker’s textbook is remarkably well written, but does not appear to have had a wide circulation. It was in this book, incidentally, that Chandrasekhar found the Stokes polarization parameters and recognized that they could be used to incorporate the phenomenon of polarization in the radiative transfer equations.

As is often the case, because Perrin’s paper was one of the first papers on the Stokes parameters, his presentation serves as a very good introduction to the subject. Furthermore, he briefly described their relation to the quantum mechanical density matrix. For completely polarized monochromatic light, the optical vibrations may be represented along the two rectangular axes as

E a t1 1 1cos= +( ),ω δ (34.74)

E a t2 2 2cos= +( ),ω δ (34.75)

where a1 and a2 are the maximum amplitudes, and δ1 and δ2 are the phases. The phase difference between these components is

δ δ δ= −2 1, (34.76)

and the total intensity of the vibration is

I a a= +12

22 . (34.77)

706 Polarized Light, Third Edition

In nature, light is not strictly monochromatic. Furthermore, as we have seen, because of the rapid vibrations of the optical field only mean values can be measured. To analyze polarized light, we must use analyzers, that is, polarizers (with transmission factors k1 and k2 along the axes) and phase shifters (with phase shifts of η1, and η2 along the fast and slow axes, respectively). These analyzers then yield the mean intensity of a vibration Ea obtained as a linear combination, with given changes in phase, of the two components E1 and E2 of the initial vibration as

E k a t k a ta = + + + + +1 1 1 1 2 2 2 2cos cos( ) ( ).ω δ η ω δ η (34.78)

We note that this form is identical to the quantum mechanical form given by Equation 34.1. The mean intensity of Equation 34.78 is then

I k k a a k k a aa = +( ) +( ) + −( ) −( )[1

2 12

22

12

22

12

22

12

22

++ − ( )+ −2 2 21 2 1 2 1 2 1 2 1 2k k a a k kcos( ) cos sin(η η δ η η )) sin

.

2 1 2a a δ( )] (34.79)

We can write the terms within parentheses as

S a a0 12

22= + , (34.80)

S a a1 12

22= − , (34.81)

S a a2 1 22= cos ,δ (34.82)

S a a3 1 22= sin ,δ (34.83)

where ⟨…⟩ refers to the mean or average value, and S0, S1, S2, and S3 are the four Stokes parameters of the optical beam. Equation 34.79 can then be rewritten as

I k k S k k S k ka = + + − + −12

212

22

0 12

22

1 1 2 1 2( ) ( ) cos(η η )) sin( ) .S k k S2 1 2 1 2 32+ − ][ η η (34.84)

As we have seen, by choosing different combinations of a1 and a2, and η1 and η2, we can determine S0, S1, S2, and S3. Equation 34.84 is essentially the equation first derived by Stokes.

The method used by Stokes to characterize a state of polarization may be generalized and con-nected with the wave statistics of von Neumann. Consider a system of m harmonic oscillations of the same frequency subjected to small random perturbations. This may be represented by the complex expression

E P i tk k= exp ω , (34.85)

where

P p ik k k= exp φ , (34.86)

Stokes Parameters for Quantum Systems 707

and the modulus pk and the argument ϕk vary slowly over time in comparison with the period of oscillation but quickly with respect to the period of measurement. Suppose we can measure the mean intensity of an oscillation E linearly dependent on these oscillations, that is,

E c E c P i tk k k k

kk

= = ∑∑ exp( ),ω (34.87)

where

c c ik k k= ( )exp η . (34.88)

The mean intensity corresponding to Equation 34.87 is then

EE c c P Pk l k l

kl

∗ = ∑ * . (34.89)

The mean intensity depends on the particular oscillations involving only the von Neumann matrix elements (the density matrix), that is,

ρkl k lP P= * . (34.90)

The knowledge of these matrix elements determines all that we can know about the oscillations by such measurements. Since this matrix is Hermitian, we can set

ρ µ ρ γ σkk k kl kl kli k= = + ≠( ),1 (34.91)

where μk, γkl = γlk, and σkl = −σkl are real quantities. The diagonal terms μk are the mean intensities of the oscillations

µ k kp= 2 , (34.92)

and the other terms give the correlations between the oscillations

γ φ φkl k l k lp p= −cos( ) , (34.93)

σ φ φkl k l k lp p= −sin( ) . (34.94)

While Perrin did not explicitly show the relation of the Stokes parameters to the density matrix, it is clear as we have shown that only an additional step is required to do this.

Before we conclude, we wish to investigate the issue of linearity. Perrin noted that Soleillet first pointed out that when a beam of light passes through some optical arrangement, or, more generally, produces a secondary beam of light, the intensity and the state of polarization of the emergent beam are functions of those of the incident beam. If two independent incident beams are superposed, the

708 Polarized Light, Third Edition

new emergent beam will be, if the process is linear, the superposition without interference of the two emergent beams corresponding to the separate incident beams. Consequently, in such a linear process, from the additivity properties of the Stokes parameters, the parameters ′ ′ ′ ′S S S S0 1 2 3, , , ,and which define the polarization of the emergent beam must be homogenous linear functions of the parameters S0, S1, S2, and S3, corresponding to the incident beam; the 16 coefficients of these linear functions will completely characterize the corresponding optical phenomenon.

Perrin offers this statement without proof. We can easily show that from Stokes’s law of additiv-ity of independent beams that the relationship between ′S0 and S0, and so on, must be linear.

Let us assume a functional relation between ′ ′S S0 1, , and so on, such that

′ =S f S S S S0 0 1 2 3( , , , ), (34.95)

′ =S f S S S S1 0 1 2 3( , , , ), (34.96)

′ =S f S S S S2 0 1 2 3( , , , ), (34.97)

′ =S f S S S S3 0 1 2 3( , , , ). (34.98)

To determine the explicit form of this functional relationship, consider only ′S0. Furthermore, assume that ′S0 is simply related to S0 only by

′ = ( )S f S0 0 . (34.99)

For two independent incident beams with intensities I1 and I2 (where we now make the replacement I = S0, etc., so that we can index the incident and emergent beams without confusion with the Stokes parameter subscripts), the corresponding emergent beams ′ ′I I1 2and are functionally related to the incident beams by

′ =I f I1 1( ), (34.100)

′ =I f I2 2( ). (34.101)

Both equations must have the same functional form. From Stokes’s Law of additivity, we can then write

′ + ′ = = +I I I f I f I1 2 1 2( ) ( ). (34.102)

Adding ′ ′I I1 2and , the total intensity I must also be a function of I1 + I2 by Stokes’s law of additivity. Thus, we have from Equation 34.102

f I f I f I I1 2 1 2( ) + ( ) = +( ). (34.103)

Equation 34.103 is a functional equation. The equation can be solved for f(I) by expanding f(I1), f(I2), and f(I1 + I2) in a series so that

f I a a I a I( ) ,1 0 1 1 2 12= + + + (34.104)

Stokes Parameters for Quantum Systems 709

f I a a I a I( ) ,2 0 1 2 2 22= + + + (34.105)

f I I a a I I a I I( ) ( ) ( ) ,1 2 0 1 1 2 2 1 22+ = + + + + + (34.106)

so

f I f I a a I I a I I

a

( ) ( ) ( ) ( )1 2 0 1 1 2 2 12

22

0

2+ = + + + + +

= +

aa I I a I I1 1 2 2 1 22+( ) + +( ) +… .

(34.107)

The left- and right-hand sides of Equation 34.107 are only consistent with Stokes’s law of additivity for the linear terms, that is, a0 = 0, a1 ≠ 0, a2 = 0, and so on, so the solution of Equation 34.103 is

f I a I1 1 1( ) = , (34.108)

f I a I2 1 2( ) = , (34.109)

f I I a I I1 2 1 1 2+( ) = +( ). (34.110)

Thus, f (I) is linearly related to I; f (I) must be linear if Stokes’s law of additivity is to apply simulta-neously to I1 and I2, and ′ ′I I1 2and . We can therefore relate ′S0 to S0, S1, S2, and S3 by a linear relation of the form:

′ = = + + +S f S S S S a S b S c S d S0 0 1 2 3 1 0 1 1 1 2 1 3( , , , ) , (34.111)

and similar relations for ′ ′ ′S S S1 2 3, , and . Thus, the Stokes vectors are related by 16 coefficients aik.As examples of this linear relationship, Perrin noted that, for a light beam rotated through an

angle ξ around its direction of propagation, for instance, by passing through a crystal plate with simple rotatory power, we have

′ =S S0 0 , (34.112)

′ = −S S S1 1 22 2cos( ) sin( ) ,ξ ξ (34.113)

′ = +S S S2 1 22 2sin( ) cos( ) ,ξ ξ (34.114)

′ =S S3 3. (34.115)

Similarly, when there is a difference in phase ϕ introduced between the components of the vibration along the axes, for instance, by birefringent crystals with axes parallel to the reference axes, then

′ =S S0 0 , (34.116)

710 Polarized Light, Third Edition

′ =S S1 1, (22.117)

′ = −S S S2 2 3cos sin ,φ φ (34.118)

′ = +S S S3 2 3sin cos .φ φ (34.119)

Perrin then determined the number of nonzero (independent) coefficients aik for different media. These included: (1) symmetrical media, where there are eight independent coefficients, (2) the scat-tering of light by an asymmetrical isotropic medium, where there are 10, (3) forward axial scat-tering, where there are five, (4) forward axial scattering for a symmetric medium, where there are three, (5) backward scattering by an asymmetrical medium, where there are four, and (6) scattering by identical spherical particles without mirror symmetry, where there are five.

Perrin’s paper is actually quite remarkable because so many of the topics that he discussed have become the basis of much research. Even to this day there is much to learn from it.

34.4 RadiaTioN eQuaTioNS foR QuaNTum meChaNiCal SySTemS

We now turn to the problem of determining the polarization of radiation emitted by atomic and molecular systems. We assume that the reader has been exposed to the rudimentary ideas and methods of quantum mechanics, particularly Schrödinger’s wave equation and Heisenberg’s matrix mechanics.

Experimental evidence of atomic and molecular systems has shown that a dynamical system in an excited state may spontaneously go to a state of lower energy, where the transition is accompa-nied by the emission of energy in the form of radiation. In quantum mechanics, the interaction of matter and radiation is allowed from the beginning so that we start with a dynamical system:

atom + radiation. (34.120)

Every energy value of the system described by Equation 34.120 can be interpreted as a possible energy of the atom alone plus a possible energy of the radiation alone plus a small interaction energy, so that it is still possible to speak of the energy levels of the atom itself. If we start with a system, Equation 34.120, at t = 0 in a state that can be described roughly as

atom in an excited state n + no radiation, (34.121)

we find at a subsequent time t the system may have gone over into a state described by

atom in an excited state m + radiation, (34.122)

which has the same total energy as the initial state in Equation 34.121, although the energy of the atom itself is now smaller. Whether or not the transition Equation 34.121 → Equation 34.122 will actually occur, or the precise instant at which it takes place, if it does take place, cannot be inferred from the information that at t = 0 the system is certainly in the state given by Equation 34.121. In other words, an excited atom may jump spontaneously into a state of lower energy and in the process emit radiation.

To obtain the radiation equations suitable for describing quantum systems, two requirements must be met. The first is the Bohr frequency condition, which states that a spontaneous transi-tion of a dynamical system from an energy state of energy En to an energy state of lower energy

Stokes Parameters for Quantum Systems 711

Em is accompanied by the emission of radiation of spectroscopic frequency ωn→m given by the formula

ωn mn mE E→ =

−1

( ), (34.123)

where ℏ is Planck’s constant divided by 2π.The other requirement is that the transition probability An→m for a spontaneous quantum jump of

a one-dimensional dynamical system from an energy state n to an energy state m of lower energy is, to a high degree of approximation, given by the formula

Ae

c hx dxn m n m n m→ →= ∫

2

03

32

3πεω ψ ψ* , (34.124)

where e is the electric charge and c is the speed of light. The transition probability An→m for a spon-taneous quantum jump from the nth to the mth energy state is seen to be proportional to the square of the absolute magnitude of the expectation value of the variable x; that is, the quantity within the absolute magnitude signs is ⟨x⟩. Equation 34.124 shows that to determine ⟨x⟩ we must also know the eigenfunction ψ of the atomic system. The expectation value of x is then found by carrying out the required integration.

The Bohr frequency condition and the transition probability allow us to proceed from the clas-sical radiation equations to the equations required to describe the radiation emitted by quantum systems.

According to classical electrodynamics, the radiation field components, in spherical coordinates, emitted by an accelerating charge are given by

Eec R

x zθ πεθ θ= −

4 02

[ cos sin ], (34.125)

Eec R

yφ πε=

4 02

[ ]. (34.126)

Quantum theory recognized early that these equations were essentially correct. They could also be used to describe the radiation emitted by atomic systems; however, new rules were needed to calcu-late x y z, , and . Thus, we retain the classical radiation Equations 34.125 and 34.126, but we replace x y z, , and by their quantum mechanical equivalents.

To derive the appropriate form of Equations 34.125 and 34.126 suitable for quantum mechani-cal systems, we use Bohr’s correspondence principle along with the frequency condition given by Equation 34.123. Bohr’s correspondence principle states that in the limit of large quantum numbers, quantum mechanics reduces to classical physics. We recall that the energy emitted by an oscillator of moment p = er is

Ic

= 16 0

3

2

πεp . (34.127)

Each quantum state n has two neighboring states, one above and one below, which for large quan-tum numbers differ by the same amount of energy ℏωnm. Hence, if we replace p by the matrix

712 Polarized Light, Third Edition

element pnm, we must at the same time multiply Equation 34.127 by two so that the radiation emitted per unit time is

Ic

pe

crnm nm nm= =1

3 303

2 2

03

4 2

πε πεω . (34.128)

We see that the transition probability is simply the intensity of radiation emitted per unit time. Thus, dividing Equation 34.128 by ωnm gives the transition probability stated in Equation 34.124. The quantity rnm can now be calculated according to the rules of wave mechanics, that is,

r t t dnm n m

V

= ∫Ψ Ψ( ) ( ) ,*r r r r, , (34.129)

where r is the radius vector from the nucleus to the field point, Ψm(r, t) and Ψn(r, t) are the Schrödinger wave functions for the mth and nth states of the quantum system, the asterisk denotes the complex conjugate, dr is the differential volume element, and V is the volume of integration.

In quantum mechanics, rnm is calculated from Equation 34.129. We now assume that by a twofold differentiation of Equation 34.129 with respect to time, we can transform the classical r to the quan-tum mechanical form rnm. Thus, according to Bohr’s correspondence principle, x is transformed to xnm, and so on, that is,

x xnm→ , (34.130)

y ynm→ , (34.131)

z znm→ . (34.132)

We now write Equation 34.129 in component form:

x t x t dnm n m

V

= ∫Ψ Ψ( , ) ( ) ,*r r r, (34.133)

y t y t dnm n m

V

= ∫Ψ Ψ( , ) ( ) ,*r r r, (34.134)

z t z t dnm n m

V

= ∫Ψ Ψ( , ) ( ) .*r r r, (34.135)

The wave functions Ψm(r, t) and Ψn(r, t) can be written as

Ψm mi tt e m( , ) ( ) ,r r= ψ ω (34.136)

Ψn ni tt e n( , ) ( ) ,r r= ψ ω (34.137)

Stokes Parameters for Quantum Systems 713

where ωmn = 2πfmn. Substituting Equations 34.136 and 34.137 into Equations 34.133, 34.134, and 34.135 and then differentiating the result twice with respect to time yields

x e t x dnm n mi

n m

V

n m= − − − ∫( ) ( ) ( ) ,( ) *ω ω ψ ψω ω2 r r r (34.138)

y e t y dnm n mi

n m

V

n m= − − − ∫( ) ( ) ( ) ,( ) *ω ω ψ ψω ω2 r r r (34.139)

z e t z dnm n mi

n m

V

n m= − − − ∫( ) ( ) ( ) .( ) *ω ω ψ ψω ω2 r r r (34.140)

It is easily proved that the integrals in these equations vanish for all states of an atom for n = m, so the derivative of the dipole moment vanishes and, accordingly, the emitted radiation also vanishes; that is, a stationary state does not radiate. This explains the fact—unintelligible from the standpoint of Bohr’s theory—that an electron revolving around the nucleus, which according to the classical laws ought to emit radiation of the same frequency as the revolution, can continue to revolve in its orbit without radiating.

Returning now to the classical radiation equations 34.125 and 34.126, we see that the corre-sponding equations are, using Equations 34.130, 34.131, and 34.132,

Eec R

x znm nmθ πεθ θ= −

4 02

[ cos sin ], (34.141)

Eec R

ynmφ πε=

4 02

[ ], (34.142)

where x y znm nm nm, , and are calculated according to Equations 34.138, 34.139, and 34.140, respectively.The Schrödinger wave function ψ(r) is found by solving Schrödinger’s time independent wave

equation

∇ + − =22

20ψ ψ( ) ( ) ( ) ,r r

mE V

(34.143)

where ∇2 is the Laplacian operator; in Cartesian coordinates it is

∇ ≡ ∂∂

+ ∂∂

+ ∂∂

22

2

2

2

2

2x y z. (34.144)

The quantities Ε and V are the total energy and potential energy, respectively, m is the mass of the particle, and ℏ = h / 2π is Planck’s constant divided by 2π.

Not surprisingly, Schrödinger’s equation 34.143, is extremely difficult to solve. Fortunately, sev-eral simple problems can be solved exactly, and these can be used to demonstrate the manner in which the quantum radiation equations 34.141 and 34.142, and the Stokes parameters can be used. We now consider these problems.

714 Polarized Light, Third Edition

34.5 STokeS VeCToRS foR QuaNTum meChaNiCal SySTemS

In this section we determine the Stokes vectors for several quantum systems of interest [16]. The problems we select are chosen because the mathematics is relatively simple. Nevertheless, the exam-ples presented are sufficiently detailed so that they clearly illustrate the difference between the classical and quantum representations. This is especially true with respect to the so-called selection rules as well as the representation of emission and absorption spectra. The examples presented are: (1) a particle in an infinite potential well, (2) a one-dimensional harmonic oscillator, and (3) a rigid rotator restricted to rotating in the x, y plane. We make no attempt to develop the solutions to these problems, but merely present the wave function and then determine the expectation values of the coordinates. The details of these problems are quite complicated, and the reader is referred to any of the numerous texts on quantum mechanics given in the References [17–22].

34.5.1 PaRTicle in an infiniTe PoTenTial well

The simplest quantum system is that of the motion of a particle in an infinite potential well of width extending from 0 to L. We assume the motion is along the z axis, so Schrödinger’s equation for the system is

− =2 2

22md z

dzE z

ψ ψ( )( ), (34.145)

and vanishes outside of the region. The normalized eigenfunctions are

ψ πn z

Ln zL

z L( ) sin ,/

=

≤ ≤2

01 2

(34.146)

and the corresponding energy is

EmL

n nn =

=π2 2

22

21 2 3

…, , , . (34.147)

Since the motion is only along the z axis, we need only evaluate znm. Thus,

z z z z dzL

n zL

zm z

Lnm n m

L

= =

∫ ψ ψ π π* ( ) ( ) sin sin

0

2

∫ dz

L

0. (34.148)

Straightforward evaluation of this integral yields

zLnm

n mn mnm =

−+8

2 2 2 2π ( )( ),odd (34.149)

= =Ln m

2( ), (34.150)

= ( )otherwise0 . (34.151)

Stokes Parameters for Quantum Systems 715

Equations 34.150 and 34.151 are of no interest because ωnm describes a nonradiating condition and the field components are zero for znm = 0. Equation 34.149 is known as the selection rule for a quantum transition. Emission and absorption of radiation only take place in discrete amounts. The result is that there will be an infinite number of discrete spectral lines in the observed spectrum.

The field amplitudes are

EeL

cnm

n mnmθ π εω θ=

−

23

02

22 2 2( )

sin , (34.152)

Eφ = 0, (34.153)

where we have set R to unity. We now form the Stokes parameters and then the Stokes vector in the usual way and obtain

S =

−

23

02

2

2 42 2 2

2eL

cnm

n mnmπ εθ ωsin

( )

1

1

0

0

. (34.154)

This is the Stokes vector for linearly horizontally polarized light. We also have the familiar dipole radiation angular factor sin2 θ. We can observe either absorption or emission spectra, depending on whether we have a transition from a lower energy level to an upper energy level, or from an upper to a lower level, respectively. For the absorption case, the spectrum that would be observed is obtained by considering all possible combinations of n and m subject to the condition that n + m is odd. Thus, for example, for a maximum number of five we have

S =

2 23

1

1

0

0

30

2

2

2124

2

4

eLcπ ε

θ ωsin

, ,ω ω144

2

4 2344

15

1

1

0

0

622

45

1

1

0

0

. (34.155)

Similarly, for the emission spectrum we would observe

S =

2 23

1

1

0

0

30

2

2

2214

2

4

eLcπ ε

θ ωsin

, ,ω ω414

2

4 3244

15

1

1

0

0

622

45

1

1

0

0

. (34.156)

The intensity of the emission lines are in the ratio

ω ω ω214

2

4 414

2

4 324

2

4

23

415

65

: : . (34.157)

716 Polarized Light, Third Edition

Using the Bohr frequency condition and Equation 34.147, we can write ωnm as

ω πnm

n mE EmL

n m= − = −

22

2 2

2( ). (34.158)

Thus, the ratio of the intensities of the emission lines are 22:42:62 or 1:4:9, showing that the transition 3 → 2 is the most intense.

34.5.2 one-diMenSional haRMonic oScillaToR

The potential V(z) of a one-dimensional harmonic oscillator is V(z) = z2/2. Schrödinger’s equation then becomes

− + =2 2

2

2 2

2 2md z

dzm z

z E zψ ω ψ ψ( )

( ) ( ). (34.159)

The normalized solutions are

ψ ωπ

ωn

n

zn

m m z( )

( !)exp

/

/

/

=

−

−22

2

1 2

1 2 2

HH

mz nn

20 1 2

1 2

=

/

, , , (34.160)

where Hn(u) are the Hermite polynomials. The corresponding energy levels are

E nn = +

12ω, (34.161)

where ω2 = k/m. The expectation value of z is readily found to be

zm

nn nnm =

+

→ +ω

1 2 1 21

21

/ /

,absorption (34.162)

=

→ −m

nn n

ω

1 2 1 2

21

/ /

,emission (34.163)

= ( )otherwise0 . (34.164)

The field components for the emitted and absorbed fields are then

Eec m

nn nθ πε ω

θ ω= −

+

+4

120

2

1 2

12

1 2

/

,

/

sin

, (34.165)

Eφ = 0, (34.166)

Stokes Parameters for Quantum Systems 717

and

Eec m

nn nθ πε ω

θ ω= −

−4 20

2

1 2

12

1 2

/

,

/

sin

, (34.167)

Eφ = 0. (34.168)

The Stokes vectors for the absorption and emission spectra are then

S =

+

+

ec m

nn n

2

202 4

21

4

161

2

1

π ε ω

θ ωsin ,

11

0

0

, (34.169)

S =

−

ec m

nn n

2

202 4

21

4

16 2

1

1

0

π ε ω

θ ωsin ,

00

. (34.170)

Equations 34.169 and 34.170 show that for both absorption and emission spectra, the radiation is linearly horizontally polarized, and again we have the familiar sin2 θ angular dependence of dipole radiation. To obtain the observed spectral lines we take n = 0, 1, 2, 3, …, for the absorption spec-trum and n = 1, 2, 3, ..., for the emission spectrum. We then obtain a series of spectral lines similar to Equation 34.158. With respect to the intensities of the spectral lines for, say, n = 5, the ratio of intensities is 1:2:3:4:5:6, showing that the strongest transition is 6 → 5 for emission and 5 → 6 for absorption.

34.5.3 Rigid RoTaToR

The ideal diatomic molecule is represented by a rigid rotator; that is, a molecule can be represented by two atoms with masses m1 and m2 rigidly connected where the distance between them is a con-stant. If there are no forces acting on the rotator, the potential may be set to zero, and the radial distance r to unity. Schrödinger’s equation for this case is then

(sin ) sin sinθθ

θ ψθ

θ ψφ

− −∂∂

∂∂

+ ( ) ∂

∂+1 2 1 2

2

2IE2

0

=ψ , (34.171)

where I is the moment of inertia. The solution of Schrödinger’s equation 34.171 is then

ψ θ φ θ φl m l m l m m, , ,( , ) ( ),= = ( )± ± ±Y Θ Φ (34.172)

where l ≥ |m|. The energy levels are given by

EI

I l l=

+ =2

21 0 1 2 3( ) , , , , ... . (34.173)

718 Polarized Light, Third Edition

A very important and illustrative example is the case where the motion of the rotator is restricted to the x, y plane. For this case, the polar angle θ = π/2 and Equation 34.171 reduces to

dd

IE2

2 2

2ψφ

ψ= −

, (34.174)

with the solutions

ψ φ π φ= = ± =±−Φ m im m( ) ( ) exp( ) , , , ./2 1 2 31 2 … (34.175)

Equation 34.175 can also be obtained from Equation 34.172 by evaluating the associated Legendre polynomial at θ = π/2. The energy levels for Equation 34.174 are found to be

EI

m m=

=

…2

1 2 32 , , , . (34.176)

We now calculate the Stokes vector corresponding to Equation 34.174. Since we are assuming that ϕ is measured positively in the x, y plane, the z component vanishes, and we need only calculate xnm and ynm. The coordinates x and y are related to ϕ by

x a= cosφ, (34.177)

y a= sin ,φ (34.178)

where a is the radius of the rigid rotator. We now calculate the expectation values

x x da

in im dnm n m= = −∫ ψ ψ φπ

φ φ φ φπ

* exp( )cos exp( )0

2

2 00

2

0

2

41

4

π

π

πφ φ

π

∫

∫= − − − + − −ai n m d

ai nexp[ ( ) ] exp[ ( mm d+∫ 1

0

2

) ] .φ φπ

(34.179)

In this last expression, the first integral vanishes except for m = n − 1, while the second integral van-ishes except for m = n + 1; we then have the selection rule that Δm = ±1. Evaluation of the integrals in Equation 34.179 then gives

xa

m m, .± = +1 2 (34.180)

In a similar manner, we find that

yaim m, .± = ±1 2

(34.181)

Stokes Parameters for Quantum Systems 719

The amplitudes for the absorbed and emitted fields are

Eea

c m mθ πεω θ= −

+8 0

2 12

, cos , (34.182)

Eea

i c m mφ πεω= −

+8 0

2 12

, , (34.183)

and

Eea

c m mθ πεω θ= −

−8 0

2 12

, cos , (34.184)

Eea

i c m mφ πεω=

−8 0

2 12

, , (34.185)

respectively. The Stokes vectors using these expressions for Eθ and Eϕ are then readily found to be

S =

+−

−

+ea

c m m8

1

0

20

2

2

14

2

2

πεω

θθ

θ

,

cos

sin

cos

, (34.186)

and

S =

+−

−ea

c m m8

1

0

20

2

2

14

2

2

πεω

θθ

θ

,

cos

sin

cos

. (34.187)

In general, we see that for both the absorption and emission spectra the spectral lines are ellipti-cally polarized and of opposite ellipticity. As usual, if the radiation is observed parallel to the z axis (θ = 0°), then Equations 34.186 and 34.187 reduce to

S =

−

+28

1

0

0

10

2

2

14ea

c m mπεω , , (34.188)

and

S =

−28

1

0

0

10

2

2

14ea

c m mπεω , , (34.189)

720 Polarized Light, Third Edition

which are the Stokes vectors for left and right circularly polarized light. For θ = 90°, Equations 34.186 and 34.187 reduce to

S =

−

+ea

c m m8

1

1

0

00

2

2

14

πεω , , (34.190)

and

S =

−

−ea

c m m8

1

1

0

00

2

2

14

πεω , , (34.191)

which are the Stokes vectors for linearly vertically polarized light.Inspection of Equations 34.190 and 34.191 shows that the Stokes vectors, aside from the fre-

quency ωm,m±1 are identical to the classical result. Thus, the quantum behavior expressed by Planck’s constant is nowhere to be seen in the spectrum! This result is very different from the result for the linear harmonic oscillator where Planck’s constant ℏ appears in the intensity. It was this peculiar behavior of the spectra that made their interpretation so difficult for such a long time. For some problems, for example, the linear oscillator, the quantum behavior appeared in the spectral intensity, and for other problems, for example, the rigid rotator, it did not. The reason for the disappearance of Planck’s constant could usually be traced to the fact that it actually appeared in both the denomina-tor and numerator of many problems and simply canceled out. In all cases, using Bohr’s correspon-dence principle, in the limit of large quantum numbers ℏ always canceled out of the final result.

We have seen that the Stokes vector can be used to represent both classical and quantum radia-tion phenomena. Before we conclude, a final word must be said about the influence of the selection rules on the polarization state. The reader is sometimes led to believe that the selection rule itself is the cause for the appearance of either linear, circular, or elliptical polarization. This is not quite correct. We recall that the field equations emitted by an accelerating charge are

Eec R

x zθ πεθ θ= −

4 02

[ cos sin ], (34.192)

Eec R

yφ πε=

4 02

[ ]. (34.193)

We have seen that we can replace x y z, , and by their quantum mechanical equivalents, that is,

x xnm nm→ −ω2 , (34.194)

y ynm nm→ −ω2 , (34.195)

z znm nm→ −ω2 , (34.196)

Stokes Parameters for Quantum Systems 721

so that Equations 34.192 and 34.193 become

E

ec R

x znm nm nmθ πεω θ θ= −

−

4 02

2 [ cos sin ], (34.197)

Eec R

ynm nmφ πεω= −

4 0

22 . (34.198)

If only a single Cartesian variable remains in these equations, then we have linearly polarized light. If two variables appear, for example, xnm and ynm, then we obtain elliptically or circularly polarized light. However, if the selection rule is such that either xnm or ynm were to vanish, then we would obtain linearly polarized light regardless of the presence of the angular factor. In other words, in classical physics the angular factor dominates the state of polarization emitted by the radiation. In quantum mechanics, the fact that either xnm or ynm can vanish and thus give rise to linearly polarized light shows that the role of the selection rule is equally significant in the polarization of the emitted or absorbed radiation.

Numerous other problems can easily be treated with the methods discussed here, such as the rigid rotator in three dimensions and the Zeeman effect (see, e.g., Collett [16]). We refer the reader to the numerous texts on quantum mechanics for further examples and applications.

RefeReNCeS

1. Bohr, N. I. On the constitution of atoms and molecules, Phil. Mag. S. 6, no. 26 (1913): 1–25. 2. Heisenberg, W., Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen,

Z. Phys. 33 (1925): 879–93. 3. Stokes, G. G., Trans. camb. Phil. Soc. 9 (1852): 399; Reprinted in Mathematical and Physical Papers,

Vol. 3, 233, London: Cambridge University Press, 1901. 4. Soleillet, P., Sur les parametres caracterisant la polarisation partielle de le lumière dans les phenomenes

de fluorescence, Ann. Phys. 12 (1929): 23. 5. Perrin, F., Polarization of light scattered by isotropic opalescent media, J. chem. Phys. 10 (1942):

415–27. 6. von Neumann, J., Mathematische Begründung der Quantenmechanik, Göttinger Nachr. 24 (1927): 273. 7. Chandrasekhar, S., On the radiative equilibrium of a stellar atmosphere. XI, Astrophys. J. 104 (1946):

110–32. 8. Chandrasekhar, S., On the radiative equilibrium of a stellar atmosphere. XXI, Astrophys. J. 105 (1947):

152–216. 9. Chandrasekhar, S., Radiative Transfer, Mineola, NY: Dover, 1960. 10. Fano, U. J., Remarks on the classical and quantum-mechanical treatment of partial polarization, J. Opt.

Soc. Am. 39 (1949): 859–63. 11. Fano, U. Note on quantum effects in optics, J. Opt. Soc, Am. 41 (1951): 58–9. 12. Fano, U. A Stokes-parameter technique for the treatment of polarization in quantum mechanics, Phys.

Rev. 93 (1954): 121–3. 13. Falkoff, D. L., and J. E. MacDonald, J. Opt. Soc. Am. 41 (1951): 861. 14. Lord Rayleigh, Scientific Papers, 6 Vols., London: Cambridge University Press, 1899–1920. 15. Walker, J., The Analytical Theory of Light, Cambridge: Cambridge University Press, 1904. 16. Collett, E., Stokes parameters for quantum systems, Am. J. Phys. 38 (1970): 563–74. 17. Schiff, L. I., Quantum Mechanics, 2nd ed., New York: McGraw-Hill, 1955. 18. Rojansky, V. B., Introductory Quantum Mechanics, Englewood Cliffs, NJ: Prentice-Hall, 1938. 19. Ruark, A. E., and H. C. Urey, Atoms, Molecules and Quanta, Mineola, NY: Dover, 1964. 20. French, A. P., Principles of Modern Physics, New York: Wiley, 1958. 21. Dirac, P. A. M., Quantum Mechanics, 3rd ed., Oxford: Clarendon Press, 1947. 22. Pauling, L., and E. B. Wilson, Introduction to Quantum Mechanics, New York: McGraw-Hill, 1935.

723

Appendix A: Conventions in Polarized Light

In August of 1968, a Symposium on Recent Developments in Ellipsometry was held at the University of Nebraska with optical scientists concerned with ellipsometry in attendance. One of the topics discussed was definitions and conventions in ellipsometry, and a paper was later published that summarized the discussion [1]. This was and remains an important topic since it relates to polarized light. A choice of signs and definitions can propagate throughout a mathematical development, and result in, for example, left-handedness for one choice being right-handedness of another. Workers in fields as diverse as astronomy, chemistry, optical science, and radio engineering, all have need to make these choices in dealing with electromagnetic radiation, so that the choice, whatever it may be, must at least be defined and consistent in order for mutual understanding.

Muller [1] made recommendations as a result of the meeting. His summary, as well as the work of Clarke [2,3] and Bennett [4] should be read by the student of polarized light. In most cases, the recommended conventions are the mostly widely used, at least among optical scientists. The two most relevant for general use in polarized light are the time dependence factor, defined as

ei tω (A.1)

and the complex index, defined by Bennett [4] as

n n ik= − . (A.2)

Bennett notes in Muller (there is a record of the discussion that took place) that he does not feel strongly about using Equation A.2 rather than

n n i= −( )1 κ , (A.3)

and only uses Equation A.2 because it is more directly related to the absorption coefficient α. The choice of the sign in the exponent of the time dependence factor of Equation A.1 determines signs in the Jones and Mueller matrices. The sign convention normally followed in optics texts for the Jones and Mueller matrices, and used to obtain the matrices in Appendix C, is based on the positive sign in the time dependence factor as in Equation A.1.

The sign of the exponent in the time dependence factor in Born and Wolf [5] and in Jackson [6] is negative, and Bennett [4] reminds us that the usual form in quantum mechanics and solid state physics is

e i t− ω , (A.4)

and consequently the complex index is

n n ik= + , (A.5)

but these forms are in the minority and not as expedient for optics and electrical engineering.

724 Appendix A: Conventions in Polarized Light

The other preferred conventions are: for the relative amplitude attenuation on reflection,

tan ;ψ =r

rp

s

(A.6)

for the relative phase change on reflection,

∆ = −δ δp s; (A.7)

and for the complex relative amplitude attenuation on reflection,

ρ ψ= =r

rep

s

itan .∆ (A.8)

Note that Born and Wolf have Equation A.6 inverted and a negative sign in the exponent of Equation A.8. Preferred terms and symbols, as suggested by Muller and Bennett, with regard to the complex index are;

n = refractive index,

  κ = absorption or attenuation index,

  k = nκ = extinction index or extinction coefficient,

and

  α = absorption coefficient.

A note on the convention for handedness is in order. The convention is explained in Clarke [3] and Shurcliff [7]. Right-handedness is associated with a positive sign and a right-hand helix, where the helix represents the tip of the electric vector. For an observer looking at an oncoming optical beam, right circular polarization is that represented by a circle traversed in the clockwise sense. This is Shurcliff’s sectional pattern, and is an end view of the snapshot picture [3]. The snapshot picture requires consideration of the right-hand helix, which has a handedness independent of the observer’s viewpoint. Visualize this nonrotating helix passing through any plane perpendicular to the beam path. The point at which the helix pierces this plane, as perceived by the observer, advances in a clockwise fashion as the helix advances with time along the beam path without rotation. As Clarke states, it is erroneous to consider that the helix screws its way through space.

RefeReNCeS

1. Muller, R. H., Definitions and conventions in ellipsometry, Surf. Sci. 16 (1969): 14–33. 2. Clarke, D., Nomenclature of polarized light: Linear polarization, Appl. Opt. 13 (1974): 3–5. 3. Clarke, D., Nomenclature of polarized light: Elliptical polarization, Appl. Opt. 13 (1974): 222–4. 4. Bennett, J. M., and H. E. Bennett, Polarization, Section 10 in Handbook of Optics, Edited by W. G.

Driscoll and W. Vaughan, New York: McGraw-Hill, 1978. 5. Born, M., and E. Wolf, Principles of Optics, 5th ed., Pergamon Press, Oxford, 1975. 6. Jackson, J. D., classical Electrodynamics, Wiley, New York, 1962. 7. Shurcliff, W. A., Polarized Light, Cambridge, MA: Harvard University Press, 1962.

725

Appendix B: Jones and Stokes Vectors

Normalized Jones Vectors Normalized Stokes Vectors

Linear horizontally polarized light1

0

11

1

0

0

(B.1)

Linear vertically polarized light0

1

1

−11

0

0

(B.2)

Linear 45° polarized light1

2

1

1

1

0

1

0

(B.3)

Linear 45° polarized light− 1

2

1

1

1

0

1

0

−

−

(B.4)

Right circularly polarized light1

2

1

−

i

11

0

0

1

(B.5)

Left circularly polarized light1

2

1

1

0

i

00

1−

(B.6)

727

Appendix C: Jones and Mueller Matrices

JoNeS maTRiCeS

Jones matrix for free space1 0

0 1

(C.1)

Jones matrix for an isotropic absorbingmaterial whose transmittannce is 2p

p

p

0

0

(C.2)

Jones matrix for linear polarizer at 0°1 0

0 0

(C.3)

Jones matrix for linear polarizer at 90°0 0

0 11

(C.4)

Jones matrix for linear polarizer at 45°12

1 11

1 1

(C.5)

Jones matrix for a right circular polarizer112

1

1

i

i−

(C.6)

Jones matrix for a left circular polarizer122

1

1

−

i

i (C.7)

Jones matrix for a linear retarder at angle θθ θ θ

θ θ θ

cos cos sin

cos sin sin

2

2

(C.8)

Jones matrix for linear retarderwith fast axis at angle andretardation

θδ

δei ccos sin ( )sin cos

( )sin cos

2 2 1

1

θ θ θ θθ θ

δ

δ

+ −−

e

e e

i

i iiδ θ θsin cos2 2+

(C.9)

Jones matrix for quarter-wave linearretarder with fast axis att 0°

e

e

i

i

π

π

/

/

4

4

0

0 −

(C.10)

728 Appendix C: Jones and Mueller Matrices

Jones matrix for half-wave retarderwith fast axis at 45°°

0 1

1 0

(C.11)

muelleR maTRiCeS

Mueller matrix for free space

1 0 0 0

0 1 0 0

0 0 1 0

0 0 00 1

(C.12)

Mueller matrix for an isotropic absorbingmaterial whose transmittance iis k

k

k

k

k

0 0 0

0 0 0

0 0 0

0 0 0

(C.13)

Mueller matrix for a linearpolarizer at angle θ

θ θθ θ θ θ1

2

1 2 0

2 2 2 22

cos sin

cos cos cos sin 00

2 2 2 2 0

0 0 0 1

2sin cos sin sinθ θ θ θ

(C.14)

Mueller matrix for a horizontal linear polarrizer12

1 1 0 0

1 1 0 0

0 0 0 0

0 0 0 0

(C.15)

Mueller matrix for a vertical linear polarizzer12

1 1 0 0

1 1 0 0

0 0 0 0

0 0 0 0

−−

(C.16)

Mueller matrix for a linear polarizer at 45°°12

1 0 1 0

0 0 0 0

1 0 1 0

0 0 0 0

(C.17)

Mueller matrix for a right circular polarizeer12

1 0 0 1

0 0 0 0

0 0 0 0

1 0 0 1

(C.18)

Appendix C: Jones and Mueller Matrices 729

Mueller matrix for a left circular polarizerr12

1 0 0 1

0 0 0 0

0 0 0 0

1 0 0 1

−

−

(C.19)

Mueller matrix for a linear retarder with faast axis at angle and retardationθ δ

1 0 0 0

0 ccos sin cos ( cos )sin cos sin s2 22 2 1 2 2 2θ θ δ δ θ θ θ+ − − iin

( cos )sin cos sin cos cos cos

δδ θ θ θ θ δ0 1 2 2 2 22 2− + 22

0 2 2

θ δθ δ θ δ δ

sin

sin sin cos sin cos−

(C.20)

Linear quarter-wave retarder with fast axis at 0°

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0−

(C.21)

Linear quarter-wave retarder with fast axis at 90°

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

−

(C.22)

Linear quarter-wave retarder with fast axis at 45°

1 0 0 0

0 0 0 1

0 0 1 0

0 1 0 0

−

(C.23)

Linear half-wave retarder with fast axis at 0°, 90°

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

−−

(C.24)

Linear half-wave retarder with fast axis at ±45°

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

−

−

(C.25)

730 Appendix C: Jones and Mueller Matrices

Mueller matrix for ideal depolarizer

1 0 0 0

0 0 0 00

0 0 0 0

0 0 0 0

(C.26)

Mueller matrix for uniform partial depolarizzer

1 0 0 0

0 0 0

0 0 0

0 0 0

a

a

a

(C.27)

Mueller matrix for nonuniform partial depolaarizer

1 0 0 0

0 0 0

0 0 0

0 0 0

a

b

c

(C.28)

731

Appendix D: Relationships between the Jones and Mueller Matrix Elements

Mueller matrix elements in terms of Jones matrix elements:

m j j j j j j j j11 11 11 12 12 21 21 22 22 2= + + +( )* * * * / (D.1)

m j j j j j j j j21 11 11 21 21 12 12 22 22 2= + − −( )* * * * / (D.2)

m j j j j j j j j13 12 11 22 21 11 12 21 22 2= + + +( ) )* * * * / (D.3)

m i j j j j j j j j14 12 11 22 21 11 12 21 22 2= + − −( )* * * * / (D.4)

m j j j j j j j j21 11 11 12 12 21 21 22 22 2= + − −( )* * * * / (D.5)

m j j j j j j j j22 11 11 21 21 12 12 22 22 2= − − +( )* * * * / (D.6)

m j j j j j j j j23 11 12 12 11 21 22 22 21 2= + − −( )* * * * / (D.7)

m i j j j j j j j j24 12 11 21 22 22 21 11 12 2= + − −( )* * * * / (D.8)

m j j j j j j j j31 11 22 21 11 12 22 22 12 2= + + +( )* * * * / (D.9)

m j j j j j j j j32 11 21 21 11 12 22 22 12 2= + − −( )* * * * / (D.10)

m j j j j j j j j33 11 22 12 21 21 12 22 11 2= + + +( )* * * * / (D.11)

m i j j j j j j j j34 11 22 12 21 21 12 22 11 2= − + − +( )* * * * / (D.12)

m i j j j j j j j j41 11 21 12 22 21 11 22 12 2= + − −( )* * * / (D.13)

m i j j j j j j j j42 11 21 12 22 21 11 22 12 2= − − +( )* * * * / (D.14)

m i j j j j j j j j43 11 22 12 21 21 12 22 11 2= + − −( )* * * * / (D.15)

m j j j j j j j j44 11 22 12 21 21 12 22 11 2= − − +( )* * * * / (D.16)

732 Appendix D: Relationships between the Jones and Mueller Matrix Elements

Expressing the Jones matrix elements in polar form (i.e., j rei= θ ) the Jones matrix elements in terms of the Mueller matrix elements are:

r m m m m11 11 12 21 221 2

/2= + + +( )[ ] / (D.17)

r m m m m12 11 12 21 221 2

/2= − + −( )[ ] / (D.18)

r m m m m21 11 12 21 221 2

/2= + − −( )[ ] / (D.19)

r m m m m22 11 12 21 221 2

/2= − − +( )[ ] / (D.20)

cos( )( )

[( ) ( )θ θ11 12

13 23

11 212

12 22

− = ++ − +

m mm m m m 22 1 2] /

(D.21)

sin( )( )

[( ) ( )θ θ11 12

14 24

11 212

12 22

− = ++ − +

m mm m m m ]] /1 2

(D.22)

cos( )( )

[( ) ( )θ θ21 11

31 32

11 122

21 22

− = ++ − +

m mm m m m 22 1 2] /

(D.23)

sin( )( )

[( ) ( )θ θ21 11

41 42

11 122

21 22

− = ++ − +

m mm m m m 22 1 2] /

(D.24)

cos( )( )

[( ) ( )θ θ11 22

33 44

11 222

21 12

− = ++ − +

m mm m m m 22 1 2] /

(D.25)

sin( )( )

[( ) ( )θ θ22 11

43 34

11 222

21 12

− = −+ − +

m mm m m m 22 1 2] /

(D.26)

733

Appendix E: Vector Representation of the Optical Field: Application to Optical ActivityWe have emphasized the Stokes vector and Jones matrix formulation for polarized light. However, polarized light was first represented by another formulation introduced by Fresnel and called the vector representation for polarized light. This representation is still much used, and for the sake of completeness we discuss it. This formulation was introduced by Fresnel to describe the remarkable phenomenon of optical activity in which the plane of polarization of a linearly polarized beam is rotated as the optical field propagates through an optically active medium. Fresnel’s mathematical description of this phenomenon was a brilliant success. After we have discussed the vector represen-tation, we shall apply it to describe the propagation of light through an optically active medium.

For a plane wave propagating in the z direction, the components of the optical field in the x, y plane are

E z t E kz tx x x, ( )cos( ) = − +0 ω δ (E.1)

E z t E kz ty y y, ( ).cos( ) = − +0 ω δ (E.2)

Eliminating the propagator kz – ωt between Equations E.l and E.2 yields

E z t

E

E z t

E

E z t E z tx

x

y

y

x y2

02

2

02

2( , ) ( , ) ( , ) ( , )co+ −

sssin .

δδ

E Ex y0 0

2= (E.3)

The Stokes vector corresponding to Equations E.l and E.2 is

S =

+−

E E

E E

E E

E E

x y

x y

x y

x y

02

02

02

02

0 0

0 0

2

2

cos

sin

δδ

. (E.4)

In the x, y plane we construct the vector

E i jz t E z t E z tx y, ) ( , , ,=( ) + ( ) (E.5)

where i and j are unit vectors in the x and y directions, respectively. Substituting Equations E.1 and E.2 into Equation E.5 gives us

E i jz t E kz t E kz tx x y y, ( ) ( )( ) = − + + − +cos cos0 0ω δ ω δ .. (E.6)

734 Appendix E: Vector Representation of the Optical Field: Application to Optical Activity

We can also express the optical field in terms of complex quantities by writing

E z t E kz t E i kz tx x x x, ( ) [ (cos Re exp( ) = − + = −0 0ω δ ω ++ δx )] (E.7)

E z t E kz t E i kz ty y y y y( , ) ( ) = − + = − +0 0cos Re expω δ ω δ(( )[ ], (E.8)

where Re… means the real part is to be taken. In complex quantities, Equation E.6 can be written as

E i jz t E i E ix x y y, ( ) ( ) .( ) = +0 0exp expδ δ (E.9)

In Equation E.9 we have factored out and then suppressed the exponential propagator [exp i (kz – ωt)], since it vanishes when the intensity is formed. Further, factoring out the term exp(iδx) in Equation E.9, we can write

E i j( , ) ( ) ,z t E E ix y= +0 0 exp δ (E.10)

where δ = δy – δx. The exponential propagator [exp i (kz – ωt)] is now restored in Equation E.10 and the real part taken, resulting in

E i jz t E kz t E kz tx y, ( ) ( ) .( ) = − + − +0 0cos expω ω δ (E.11)

Equation E.11 is the vector representation for elliptically polarized light. There are two special forms of Equation E.11. The first is for δ = 0° or 180°, which leads to linearly polarized light at an angle ψ (see Equation E.3). If either E0y or E0x is zero, we have linear horizontally polarized light or linear vertically polarized light, respectively. For linearly polarized light, Equation E.11 reduces to

E i jz t E E kz tx y, ( ),( ) = ±( ) −0 0 cos ω (E.12)

where ± corresponds to δ = 0° and 180°, respectively. The corresponding Stokes vector is seen from Equation E.4 to be

S =

+−

±

E E

E E

E E

x y

x y

x y

02

02

02

02

0 02

0

. (E.13)

The orientation angle ψ of the linearly polarized light is

tan .222

1

0 0

02

02

ψ = =±

−SS

E E

E Ex y

x y

(E.14)

From trigonometric half-angle formulas, we find that

tan ,ψ =E

Ey

x

0

0

(E.15)

Appendix E: Vector Representation of the Optical Field: Application to Optical Activity 735

which is exactly what we would expect from inspection of Equation E.12.The other special form of Equation E.11 is for δ = –90° or 90°, whereupon the polarization

ellipse reduces to the standard form of an ellipse. This reduces further to the equation of a circle if E0x = E0y = E0. For δ = –90°, Equation E.11 reduces to

E i jz t E kz t kz t,( ) = −( ) + −( )[ ]0 cos sinω ω (E.16)

and for δ = 90°

E i j( , ) [ ( ) ].z t E kz t kz t= −( ) − −0 cos sinω ω (E.17)

The behavior of Equations E.16 and E.17 is readily seen by considering the equations at z = 0 and then allowing ωt to take on the values 0–2π radians in intervals of π/2. One readily sees that Equation E.16 describes a vector E(z, t) that rotates clockwise at an angular frequency of ω. Consequently, Equation E.16 is said to describe left circularly polarized light. Similarly, in Equation E.17, E(z, t) rotates counterclockwise as the wave propagates toward the viewer and, therefore, we have right circularly polarized light.

Equations E.16 and E.17 lead to a very interesting observation. If we label E(z, t) in Equations E.16 and E.17 as El(z, t) and Er(z, t), respectively, and add the two equations, we see that

E E i il r xz t z t E t kz E z t, ( , ) ( ) ( , ) .( ) + = − =2 cos0 ω (E.18)

This equation tells us that a linearly polarized wave can be synthesized from two oppositely polar-ized circular waves of equal amplitude. This property played a key role in enabling Fresnel to describe the propagation of a beam in an optically active medium. The vector representation intro-duced by Fresnel revealed for the first time the mathematical existence of circularly polarized light; before Fresnel, no one suspected its existence. Before we conclude, another important property of the vector formulation must be discussed.

Elliptically polarized light can be decomposed into two orthogonal polarized states (coherent decomposition). We consider the form of the polarization ellipse that can be represented in terms of (1) linearly ±45° polarized light and (2) right and left circularly polarized light. We decompose an elliptically polarized beam into linear ±45° states of arbitrary real amplitudes A and b and write Equation E.10 as

E i j i j i jz t E E i A b A bx y, ( )( ) = + ( ) = +( ) + − = +exp0 0 δ (( ) + −( )i jA b . (E.19)

Taking the vector dot product of the left- and right-hand sides of Equation E.19 with i and then j and equating terms yields

E A bx0 = + (E.20)

E e A byi

0δ = − . (E.21)

Because A and b are real quantities, the left-hand side of Equation E.21 can be real only for δ = 0° or 180°. Thus, Equations E.20 and E.21 become

E A bx0 = + (E.22)

736 Appendix E: Vector Representation of the Optical Field: Application to Optical Activity

± = −E A by0 , (E.23)

which leads immediately to

AE Ex y=

±0 0

2 (E.24)

bE Ex y= 0 0

2

∓. (E.25)

We see that elliptically polarized light cannot be represented by linear ±45° polarization states. The only state that can be represented in terms of linear ±45° light is linear horizontally polarized light. This is readily seen by writing

EE E E E

xx x x x

00 0 0 0

2 2 2 2i i i j=

+

+

−

= + + −

j

i j i jE Ex x0 0

2 2[ ] [ ].

(E.26)

We see that the right-hand side of Equation E.26 consists of linear ±45° polarized components of equal amplitudes.

It is also possible to express linearly polarized light, E0xi, in terms of right and left circularly polarized light of equal amplitudes. We can write, using complex quantities,

EE E

iE

iE

xx x x x

00 0 0 0

2 2 2 2i i i j=

+

+

−

= + + −

j

i j i jE

iE

ix x0 0

2 2[ ] [ ].

(E.27)

We see that the last expression in Equation E.27 describes two oppositely circularly polarized beams of equal amplitudes.

We now represent elliptically polarized light in terms of right and left circularly polarized light of real amplitudes A and b. We express Equation E.10 as

E i j i j i jz t E E i A i b i

A

x y, ( )( ) = + = +( ) + −( )

= +

0 0 exp δ

bb i A b( ) + −( )i j.

(E.28)

We then find that

E A bx0 = + (E.29)

E e i A byi

0δ = −( ). (E.30)

Appendix E: Vector Representation of the Optical Field: Application to Optical Activity 737

We see immediately that for δ = ±90°, Equations E.29 and E.30 become

E A bx0 = + (E.31)

± = −( )E A by0 (E.32)

so the right-hand side of Equation E.28 then becomes

E i jz t E iEx y, .( ) = ±0 0 (E.33)

Equation E.33 is the vector representation of the standard form of the polarization ellipse. For convenience we only consider the + value of Equation E.32 so that the amplitudes (i.e., the radii) of the circles are

AE Ex y=

±0 0

2 (E.34)

bE Ex y=

−0 0

2. (E.35)

The condition δ = ±90° restricts the polarization ellipse to the standard form of the ellipse (see Equation E.2); that is,

E z t

E

E z t

Ex

x

y

y

2

02

2

02

1( , ) ( , )

.+ = (E.36)

Thus, only the nonrotated form of the polarization ellipse can be represented by right and left cir-cularly polarized light of unequal amplitudes, A and b, Equations E.34 and E.35. In Figure E.l, we show elliptically polarized light as the superposition of right (R) and left (L) circularly polarized light. We can determine the points where the circles (RCP) and (LCP) intersect the polarization ellipse. We write Equation E.36 as

x

A by

A b

2

2

2

21

( ) ( )++

−= (E.37)

A

B

figuRe e.1 Superposition of left and right circularly polarized light of unequal amplitudes to form ellipti-cally polarized light.

738 Appendix E: Vector Representation of the Optical Field: Application to Optical Activity

and the RCP and LCP circles as

x y A2 2 2+ = (E.38)

x y b2 2 2+ = , (E.39)

where we have set Ex = x and Ey = y. Straightforward algebra shows the points of intersection (xR, yR) for the RCP circle are

xA b A b

AR = ± + −2

2, (E.40)

yA b A b

AR = ± − +2

2, (E.41)

and the points of intersection (xL, yL) for the LCP circle are

xA b b A

bL = ± + −2

2, (E.42)

yA b b A

bL = ± − +2

2. (E.43)

Equations E.40 through E.43 can be confirmed by squaring and adding Equations E.40 and E.41 and, similarly, Equations E.42 and E.43. We then find that

x y AR R2 2 2+ = (E.44)

x y bL L2 2 2+ = (E.45)

as expected.As a numerical example of these results, consider that we have an ellipse where A = 3 and b = 1.

From Equations E.40 through E.43, we have

xR = ±2 53

(E.46)

yR = ± 5 (E.47)

and the points of intersection (xL, yL) for the LCP circle are

x iL = ±2 (E.48)

yL = ± 5. (E.49)

As we can see from Figure E.1, the RCP circle intersects the polarization ellipse, whereas the exis-tence of the imaginary number in Equation E.48 shows that there is no intersection for the LCP circle.

Appendix E: Vector Representation of the Optical Field: Application to Optical Activity 739

We will now use these results to analyze the problem of the propagation of an optical beam through an optically active medium. Before we do this, however, we provide some historical and physical background to the phenomenon of optical activity.

Optical activity was discovered in 1811 by Arago, when he observed that the plane of vibration of a beam of linearly polarized light underwent a continuous rotation as it propagated along the optic axis of quartz. Shortly thereafter, Biot (1774–1862) discovered this same effect in vaporous and liq-uid forms of various substances, such as the distilled oils of turpentine and lemon and solutions of sugar and camphor. Any material that causes the E field of an incident linear plane wave to appear to rotate is said to be optically active. Moreover, Biot discovered that the rotation could be left- or right-handed. If the plane of vibration appears to revolve counterclockwise, the substance is said to be dextrorotatory or d-rotatory (Latin for right, dextro). On the other hand, if E rotates clockwise, it is said to be levorotatory or l-rotatory (Latin for left, levo).

The English astronomer and physicist Sir John Herschel (1792–1871), son of Sir William Herschel, the discoverer of the planet Uranus, recognized that the d-rotatory and l-rotatory behavior in quartz actually corresponded to two different crystallographic structures. Although the molecules (SiO2) are identical, crystal quartz can be either right- or left-handed, depending on the arrangement of these molecules. In fact, careful inspection shows that there are two forms of the crystals, and they are the same in all respects except that one is the mirror image of the other; they are said to be enantiomorphs of each other. All transparent enantiomorphic structures are optically active.

In 1825, Fresnel, without addressing himself to the actual mechanism of optical activity, pro-posed a remarkable solution. Since an incident linear wave can be represented as a superposition of right-circular and left-circular states, he suggested that these two forms of circularly polarized light propagate at different speeds in an optically active medium. An active material shows circular bire-fringence; that is, it possesses two indices of refraction, one for the right-circular state, nR, and one for the left-circular state, nL. In propagating through an optically active medium, the two circular waves get out of phase and the resultant linear wave appears to rotate. We can see this behavior by considering this phenomenon analytically for an incident beam that is elliptically polarized; linearly polarized light is then a degenerate case.

In Figure E.2, we show an incident elliptically polarized beam entering an optically active medium with field components Ex and Ey. After the beam has propagated through the medium, the field components are ′Ex and ′Ey .

Fresnel suggested that in an optically active medium, a right circularly polarized beam propa-gates with a wave number kR and a left circularly polarized beam propagates with a different wave

Optically active medium

Ex

Ey

E x

E y

figuRe e.2 Field components of an incident elliptically polarized beam propagating through an optically active medium.

740 Appendix E: Vector Representation of the Optical Field: Application to Optical Activity

number kL. In order to treat this problem analytically, we consider the decomposition of Ex(z, t) and Ey(z, t) separately. Furthermore, we suppress the factor ωt in the equations because the time varia-tion plays no role in the final equations.

For the Ex(z) component, we can write this in terms of circular components as

E i jRxx

R RzE

k z k z( ) [cos( ) sin( ) ],= −2

(E.50)

E i jLxx

L RzE

k z k z( ) [cos( ) sin( ) ].= +2

(E.51)

Adding Equations E.50 and E.51 we see that, at z = 0,

E E iRx Lx xE0 0( ) + ( ) = , (E.52)

which shows that Equations E.50 and E.51 represent the x component of the incident field. Similarly, for the Ey(z) component we can write

E i jRyy

R RzE

k z k z( ) [sin( ) cos( ) ]= +2

(E.53)

E i jLyy

L LzE

k z k z( ) [ sin( ) cos( ) ].= − +2

(E.54)

Adding Equations E.53 and E.54, we see that at z = 0

E E jRy Ly yxE0 0( ) + ( ) = , (E.55)

so Equations E.53 and E.54 correspond to the y component of the incident field. The total field E′ (z) in the optically active medium is

′ = ′ + ′ + = + + +E i j E E E E( ) .z E Ex y Rx Lx Ry Ly (E.56)

Substituting Equations E.50, E.51, E.53, and E.54 into Equation E.56, we have

′ = + + +E i( ) [cos cos ] [sin sinzE

k z k zE

k z k zxR L

yR L2 2

]]

[sin sin ] [cos si

+ − − + +jE

k z k zE

k zxR L

yR2 2

nn ] ,k zL

(E.57)

and we see that

′ = + + +E zE

k z k zE

k z k zxx

R Ly

R L( ) [cos cos ] [sin sin2 2

]], (E.58)

′ = − − + +E zE

k z k zE

k z kyx

R Ly

R L( ) [sin sin ] [cos cos2 2

zz]. (E.59)

Appendix E: Vector Representation of the Optical Field: Application to Optical Activity 741

Equations E.58 and E.59 can be simplified by rewriting the terms [cos kRz + cos kLz] and [sin kRz – sin kLz]. Let

ak k zR L= +( )

2 (E.60)

bk k zR L= −( )

2 (E.61)

so

k z a bR = + (E.62)

k z a bL = − (E.63)

and [cos kRz + cos kLz] and [sin kRz – sin kLz] then become

cos cos cos cosk z k z a b a bR L+ = +( ) + −( ) (E.64)

sin sin sin sink z k z a b a bR L− = +( ) − −( ). (E.65)

Using the familiar sum and difference formulas for the cosine and sine terms of the right-hand sides of Equations E.64 and E.65 along with Equations E.60 and E.61, we find that

cos cos cos( )

cos( )

k z k zk k z k k

R LR L R L+ = +

−2

2zz

2

(E.66)

sin sin cos( )

sin( )

k z k zk k z k k

R LR L R L− = +

−2

2zz

2

. (E.67)

The term cos ( )k k zR L+

2 in Equations E.66 and E.67 plays no role in the final equations and

can be dropped. Substituting the remaining cosine and sine term in Equations E.66 and E.67 into Equations E.58 and E.59, we finally obtain

′ = − + −E z

E k k z E k k zx

x R L y R L( ) cos( )

sin( )

,2 2 2 2

(E.68)

′ = − + −E z

E k k z E k k zy

x R L y R L( ) sin( )

cos( )

.2 2 2 2

(E.69)

We see that Equations E.68 and E.69 are the equations for rotation of Ex and Ey. We can write Equations E.68 and E.69 in terms of the Stokes vector and the Mueller matrix as

′′′′

=−

S

S

S

S

0

1

2

3

1 0 0 0

0 2 2 0

0

cos sinβ βssin cos2 2 0

0 0 0 0

0

1

2

3

β β

S

S

S

S

, (E.70)

742 Appendix E: Vector Representation of the Optical Field: Application to Optical Activity

where

β = −( ).

k k zR L

2 (E.71)

The angle of rotation β can be expressed in terms of the refractive indices nR and nL of the medium and the wavelength λ of the incident beam by writing

k k nn

R RR= =0

2πλ

(E.72)

k k nn

L LL= =0

2πλ

(E.73)

and k0 = 2π/λ. If nR ≤ nL, the medium is d-rotatory, and if nR ≥ nL, the medium is l-rotatory. Substituting Equations E.72 and E.73 into Equation E.71, we then have

β πλ

= −( ).

n n zR L (E.74)

The quantity β/d is called the specific rotatory power. For quartz, it is found to be 21.7°/mm for sodium light, from which it follows that |nR – nL| = 7.1 × 10–5. The small difference in the refrac-tive indices shows that at an optical interface, the two oppositely circularly polarized beams will be very difficult to separate. Fresnel was able to show the existence of the circular components and separate them by an ingenious construction of a composite prism consisting of R- and L-quartz, as shown in Figure E.3. He reasoned that since the two components traveled with different velocities they should be refracted by different amounts at an oblique interface. In the prism, the separation is increased at each interface. This occurs because the right-handed circular component is faster in the R-quartz and slower in the L-quartz. The reverse is true for the left-handed component. The former component is bent down and the latter up, the angular separation increasing at each oblique inter-face. If the two images of a linearly polarized source are observed through the compound prism and then examined with a linear polarizer, the respective intensities are unaltered when the polarizer is rotated. Thus, the beams must be circularly polarized.

The subject of optical activity is extremely important. In the field of biochemistry a remarkable behavior is observed. When organic molecules are synthesized in the laboratory, an equal number of d- and l-isomers are produced, with the result that the mixture is optically inactive. One might expect in nature that equal amounts of d- and l-stereoisomers would exist. This is by no means the case. Natural sugar (sucrose, C12H22O6) always appears in the d-rotatory form, regardless of where it is grown or whether it is extracted from sugar cane or sugar beets. Moreover, the sugar dextrose,

R R

L

figuRe e.3 Fresnel’s construction of a composite prism consisting of R-quartz and L-quartz to demon-strate optical activity and the existence of circularly polarized light.

Appendix E: Vector Representation of the Optical Field: Application to Optical Activity 743

or d-glucose (C6H12O11), is the most important carbohydrate in human metabolism. Evidently, living cells can distinguish in a manner not yet fully understood between l- and d-molecules.

One of the earliest applications of optical activity was in the sugar industry, where the angle of rotation was used as a measure of the quality of the sugar (measured by saccharimetry). In recent years, optical activity has become very important in other branches of chemistry. For example, the artificial sweetener aspartame and the pain reducer ibuprofen are optically active. In the pharmaceu-tical industry, it has been estimated that approximately 500 out of the nearly 1300 commonly used drugs are optically active. The difference between the l- and d-forms can, it is believed, lead to very undesirable consequences. For example, it is believed that the optically active sedative drug thalido-mide, when given in the l-form, acts as a sedative, but the d-form is the cause of birth defects.

Interest in optical activity has increased greatly in recent years. Several sources are listed in the references. Of special interest is the stimulating article by Applequist [1], which describes the early investigations of optical activity by Biot, Fresnel, and Pasteur, as well as recent investigations, and provides a long list of related references.

RefeReNCe

1. Applequist, J., American Scientist, 75, 59 (1987).

745

BibliographybookS PRimaRily abouT PolaRiZaTioN

1. Azzam, R. M. A., and N. M. Bashara, Ellipsometry and Polarized Light, Amsterdam: North-Holland, 1977.

2. Brosseau, C., Fundamentals of Polarized light: A Statistical Optics Approach, New York: Wiley, 1998. 3. Clarke, D., and J. F. Grainger, Polarized Light and Optical Measurement, Oxford: Pergamon Press,

1971. 4. Huard, S., Polarization of Light, New York: Wiley, 1997. 5. Shurcliff, W., Polarized Light, London: Oxford University Press, 1962.

ColleCTed PaPeRS

Swindell, W., Polarized Light in Optics, Stroudsburg, PA: Dowden, Hutchinson, & Ross, 1975.

haNdbook ChaPTeRS

1. Azzam, R. M. A., “Ellipsometry,” Chapter 27 in Handbook of Optics, 2nd ed., Vol. 2, Edited by M. Bass, New York: McGraw-Hill, 1994.

2. Bennett, J. M., and H. E. Bennett, “Polarization,” Chapter 10 in Handbook of Optics, Edited by W. G. Driscoll and W. Vaughan, New York: McGraw-Hill, 1980.

3. Bennett, J. M., “Polarizers,” Chapter 3 in Handbook of Optics, 2nd ed., Vol. 2, Edited by M. Bass, New York: McGraw-Hill, 1994.

4. Chipman, R. Α., “Polarimetry,” Chapter 22 in Handbook of Optics, 2nd ed., Vol. 2, Edited by M. Bass, New York: McGraw-Hill, 1994.

iNTRoduCToRy oPTiCS TeXTS

1. Hecht, Ε., Optics, Reading, MA: Addison-Wesley, 1987. 2. Jenkins, F. A., and Η. Ε. White, Fundamentals of Optics, New York: McGraw-Hill, 1976.

miSCellaNeouS

1. Können, G. P., Polarized Light in Nature, Cambridge: Cambridge University Press, 1985. 2. Horvath, G., and D. Varju, Polarized Light in Animal Vision: Polarization Patterns in Nature, Berlin:

Springer, 2004. 3. Humphreys, W. J., Physics of the Air, Mineola, NY: Dover, 1964. 4. Yariv, A., and P. Yeh, Optical Waves in crystals, New York: Wiley, 1984.

oPTiCS bookS WiTh maTeRial oN PolaRiZaTioN

1. Born, M., and E. Wolf, Principles of Optics, New York: Pergamon Press, 1980. 2. O’Neill, E. L., Introduction to Statistical Optics, Reading, MA: Addison-Wesley, 1963. 3. Saleh, B., and M. Teich, Fundamentals of Photonics, New York: Wiley-Interscience, 1991. 4. van de Hülst, H. G., Light Scattering by Small Particles, New York: Dover, 1981.

PolaRiZaTioN maThemaTiCS

1. Gerrard, A., and J. M. Burch, Introduction to Matrix Methods in Optics, London: Wiley, 1975. 2. Theocaris, P. S., and E. E. Gdoutos, Matrix Theory of Photoelasticity, Berlin: Springer-Verlag, 1979.

746 Bibliography

RemoTe SeNSiNg

1. Egan, W. G., Ed., Optical Remote Sensing: Science and Technology, New York: Marcel Dekker, 2003. 2. Gehrels, T., Ed., Planets, Stars and Nebulae Studied with Photopolarimetry, Tucson, AZ: University of

Arizona Press, 1974. 3. Schott, J. R., Fundamentals of Polarimetric Remote Sensing, Bellingham, WA: SPIE Press, 2009.

SPie PRoCeediNgS

1. Volume 7672, Polarization: Measurement, Analysis, and Remote Sensing IX, eds. D. H. Goldstein and D. B. Chenault, May 2010.

2. Volume 7461, Polarization Science and Remote Sensing IV, eds. J. A. Shaw and J. S. Tyo, August 2009. 3. Volume 6972, Polarization: Measurement, Analysis, and Remote Sensing VIII, eds. D. H. Goldstein and

D. B. Chenault, May 2008. 4. Volume 6682, Polarization Science and Remote Sensing III, eds. J. A. Shaw and J. S. Tyo, September

2007. 5. Volume 6240, Polarization: Measurement, Analysis, and Remote Sensing VI, eds. D. H. Goldstein and D.

B. Chenault, May 2006. 6. Volume 5888, Polarization Science and Remote Sensing II, eds. J. A. Shaw and J. S. Tyo, August 2005. 7. Volume 5432, Polarization Measurement, Analysis, and Remote Sensing VI, eds. D. H. Goldstein and D.

B. Chenault, July 2004. 8. Volume 4819, Polarization Measurement, Analysis, and Applications V, eds. D. H. Goldstein and D. B.

Chenault, July 2002. 9. Volume 4481, Polarization Analysis, Measurement, and Remote Sensing IV, eds, D. H. Goldstein, D. B.

Chenault, W. G. Egan, and M. J. Duggin, August 2001. 10. Volume 4133, Polarization Analysis, Measurement, and Remote Sensing III, eds. D. B. Chenault, M. J.

Duggin, W. G. Egan, and D. H. Goldstein, August 2000. 11. Volume 3754, Polarization: Measurement, Analysis, and Remote Sensing II, eds. D. H. Goldstein and D.

B. Chenault, July 1999. 12. Volume 3121, Polarization: Measurement, Analysis, and Remote Sensing, eds, D. H. Goldstein and D. B.

Chenault, August 1997. 13. Volume 2873, International Symposium on Polarization Analysis and Applications to device Technology,

eds. T. Yoshizawa and H. Yokota, August 1996. 14. Volume 2265, Polarization Analysis and Measurement II, eds. D. H. Goldstein and D. B. Chenault,

August 1994. 15. Volume 1747, Polarization and Remote Sensing, ed. W. G. Egan, July 1990. 16. Volume 1746, Polarization Analysis and Measurement, eds. D. H. Goldstein and R. A. Chipman, July

1992. 17. Volume 1317, Polarimetry: Radar, Infrared, Visible, Ultraviolet, and X-Ray, eds. R. A. Chipman and J.

W. Morris, October 1990. 18. Volume 1166, Polarization considerations in Optical Systems II, ed. R. A. Chipman, August 1989. 19. Volume 891, Polarization considerations in Optical Systems, ed. R. A. Chipman, January 1988. 20. MS23, Selected Papers on Polarization, ed. B. H. Billings, November 1990.

Figure 1.1 The double image seen through a calcite crystal. (Photo courtesy of D. H. Goldstein.)

(a) (b) (c)

Figure 1.4 Images of an automobile in a field; (a) black and white photograph, (b) linear ±45° polariza-tion encoded in pseudocolor, and (c) degree of polarization encoded in pseudocolor. (Photos courtesy of D.H. Goldstein.)

Figure 1.6 Camphor between crossed polarizers. (Photo courtesy of D. H. Goldstein.)

Figure 1.5 Mica between crossed polarizers. (Photo courtesy of D. H. Goldstein.)

(a) (b) (c)

Figure 1.7 Corn syrup (a) between parallel polarizers, (b) between polarizers at 45° to one another, and (c) between crossed polarizers. (Photo courtesy of D.H. Goldstein.)

(a) (b)

Figure 1.8 View from vehicle as seen (a) without polarized sunglasses and (b) with polarized sunglasses. (Photo courtesy of D. H. Goldstein.)

Original image(horizontal pol)

1800h0 = +2.7°

1840h0 = –7.1°

1800h0 = +2.7°

1850h0 = –9.6°

1900h0 = –12.0°

1910h0 = –14.4°

1840h0 = –7.1°

1850h0 = –9.6°

1900h0 = –12.0°

1910h0 = –14.4°

1840h0 = –7.1°

1850h0 = –9.6°

1900h0 = –12.0°

1910h0 = –14.4°

0%

50%

100%

1810h0 = +0.5°

1820h0 = –2.2°

1830h0 = –4.7°

1810h0 = +0.5°

1820h0 = –2.2°

1830h0 = –4.7°

1800h0 = +2.7°

1810h0 = +0.5°

1820h0 = –2.2°

1830h0 = –4.7°

045

90

% polarization

e-vector angle

Figure 2.3 Linear polarization of the sky at sunset and during evening twilight. (From Cronin, T. W., Warrant, E. J., and Greiner, B., Appl. Opt., 45, 5582–9, 2006. With permission from Optical Society of America.) Full-sky images were acquired using a Nikon Coolpix 5700 digital camera with a fisheye lens attach-ment having a linear polarizing filter mounted between the lens and the camera itself. Data were acquired on September 15, 2004 at Lizard Island, Australia. Sunset was at 1814 local time. Each image is labeled with the local time at which data were acquired (at 10 minute intervals) as well as the solar elevation, h0. In all images, the zenith is in the center, north is to the top, and east to the left. The top set of images are original digital pho-tographs acquired when the polarizing filter was oriented east–west (indicated by the double-headed arrow), emphasizing the dark band of north–south electric vector orientation. The middle set shows linear polariza-tion in percentages in pseudocolor. The bottom set indicates electric vector angle, also in pseudocolor as coded in the key to the right. Note that clouds appear in some of the images and are particularly noticeable at 1850.

Figure 2.5 Representation of the rainbow with tangential polarization indicated with the arrows.

(a) (b)

Figure 2.6 Photographs of a rainbow. In (a) the rainbow polarization is aligned with the polarization axis of the polarizer. A secondary bow, as well as supernumerary bows, are evident. In (b) the rainbow polarization is perpendicular to the polarization axis of the polarizer and the rainbow almost disappears. (Photos courtesy of D. H. Goldstein.)

Figure 2.8 Solar disk with (a) polarimetric image and (b) white light image.

No polarizer Right circular polarizer in front of camera

(a) (b)

Figure 2.9 Photographs of Plusiotis resplendens (b) with and (a) without a polarizer. A photograph of the scarab with a left circular polarizer in front of the camera would appear as in (a). (Photos courtesy of D. H. Goldstein.)

No polarizer

(a) (b)

Right circular polarizer in front of camera.

Figure 2.10 Photographs of Plusiotis gloriosa (b) with and (a) without a polarizer. A photograph of the scarab with a left circular polarizer in front of the camera would appear as in (a). (Photos courtesy of D. H. Goldstein.)

No polarizer Right circular polarizerin front of camera

(a) (b)

Figure 2.11 Photographs of Plusiotis clypealis (b) with and (a) without a polarizer. A photograph of the scarab with a left circular polarizer in front of the camera would appear as in (a). This scarab looks as though it were made of silver. With the polarizer, much of the light is lost, although, like Plusiotis resplendens, this scarab has no black backing layer in its cuticle so it does not have the dramatic loss of color as does Plusiotis gloriosa. (Photos courtesy of D. H. Goldstein.)

Figure 2.18 Sepioteuthis lessoniana, or Bigfin Reef Squid, photographs in natural light (top) and a pseudo-color polarization image (bottom) with color scale to right indicating degree of linear polarization from 0 to 100%. (Photographs courtesy of Tsyr-Huei Chiou, University of Queensland, Australia.)

(a) (b)

Figure 2.19 (a) Sepia plangon, or Mourning Cuttlefish, photo from the side in natural light (top) and a pseudocolor polarization image (bottom) with color scale to right indicating degree of linear polarization from 0 to 100%. (Photos courtesy of Tsyr-Huei Chiou, University of Queensland, Australia.) (b) Sepia officinalis, or Common Cuttlefish, photo from the front in natural light (top) and a pseudocolor polarization image (bottom). (Adapted from the Journal of Experimental Biologists, Cover Photo, Vol. 210 (20), 2007. With permission from The Company of Biologists.)

Odontodactylus cultrifer malelinear polarizer 210 degrees

Odontodactylus cultrifer malelinear polarizer 240 degrees

Odontodactylus cultrifer malelinear polarizer 330 degrees

Odontodactylus cultrifer malelinear polarizer 360 degrees

Odontodactylus cultrifer malelinear polarizer 300 degrees

Odontodactylus cultrifer malelinear polarizer 270 degrees

Figure 2.20 Odontodactylus cultrifer as seen through a linear polarizer rotated through 150º. (Photos courtesy of Roy Caldwell, University of California, Berkeley.)

The stomatopod crustacean odontodactylus cultrifer

The sail of O.cultrifer in transmitted light The sail of O.cultrifer in transmitted light

L_CPL L_CPL

R_CPL R_CPL

Figure 2.21 A natural-color photograph of the stomatopod crustacean Odontodactylus cultrifer show-ing the prominent sail-like keel on the telson (the posterior segment). The photographs in the lower panels show the keel from both the right and left sides as seen in transmitted light and photographed through lin-ear and circular polarizers, as indicated by the double-headed arrows (electric vector orientation of linear polarization) or R-CPL and L-CPL for right and left circular polarization, respectively. Note that the keel preferentially transmits horizontally polarized light when seen from either side, but that it transmits cir-cularly polarized light of opposite handedness on each side. (Photos courtesy of Roy Caldwell, University of California, Berkeley.)

Figure 18.1 Visible picture of two pickup trucks in shade (top), long-wave IR intensity image (bottom left), and long-wave IR polarization image (bottom right). Strong contrast in the polarization image shows advantages for enhanced target detection using imaging polarimetry. (Photo courtesy of Huey Anderson. With permission from Optical Society of America.)

Rotatingretarder

Polarizersensor head

Camera Interfaceinstrument controldata flow

Process andcontrol system

Figure 18.3 Polarimetric sensor using rotating polarization elements. (With permission from Optical Society of America.)

Polarimetricimagingoptics

Polarizationfilter/diffractive

optic array

FPA

Objective

Polaris building

Figure 18.5 Division of aperture polarimeter and a raw focal plane image showing the four polarization channels. The four channels are reduced to polarization products such as DoLP. For this specific case, the four images are linearly polarized at 0°, 45°, 90°, and 135°. (With permission from Optical Society of America.)

2×2 array ofFPA pixels

2×2 array ofmicropolarizers

SP1SP2

SPSP3 SP4 Incidentradiation

Figure 18.6 Division of focal plane polarimeter. (With permission from Optical Society of America.)

Figure 21.15 Photograph of a commercial photoelastic modulator. (Photo courtesy of D. H. Goldstein.)

s1

2ψ2

s0s2

s3

Figure 18.7 Optimal locations for measuring the polarization states are when the diattenuation vec-tors inscribe a regular tetrahedron inside the Poincaré sphere. (With permission from Optical Society of America.)

Figure 23.10 A photograph of a pile-of-plates polarizer for the infrared. (Photo courtesy of D. H. Goldstein.)

Figure 23.12 Photograph of a Soleil–Babinet compensator. (Photo courtesy of D. H. Goldstein.)

Figure 23.14 Photograph of a quarter-wave Fresnel rhomb. (Photo courtesy of D. H. Goldstein.)