Hermann A. Haus

Noiseand QuantumOpticalMeasurements

A-

Advanced Texts in PhysicsThis program of advanced texts covers a broad spectrum of topics which are ofcurrent and emerging interest in physics. Each book provides a comprehensive andyet accessible introduction to a field at the forefront of modern research. As such,these texts are intended for senior undergraduate and graduate students at the MSand PhD level; however, research scientists seeking an introduction to particularareas of physics will also benefit from the titles in this collection.

Hermann A. Haus

Electromagnetic Noiseand Quantum OpticalMeasurements

With 151 Figures and 117 Problemswith 41 Selected Solutions

Solutions Manual for Instructors on RequestDirectly from Springer-Verlag

Springer

Professor Hermann A. HausMassachusetts Institute of TechnologyDepartment of Electrical Engineeringand Computer SciencesVassar Street 50, Office 36-345Cambridge, MA 02139, USAE-mail: haus@mit.edu

Library of Congress Cataloging-in-Publication Data

Haus, Hermann A.Electromagnetic noise and quantum optical measurements / Hermann Haus.

p. cm. -- (Advanced texts in physics, ISSN 1439-2674)Includes bibliographical references and index.ISBN 3540652728 (hc.: alk. paper)1. Electronic circuits--Noise. 2. Electromagnetic noise--Measurement. 3. Quantum

optics--Measurement. 4. Optoelectronic devices--Noise. 5. Interference (Light) I. Title.II. Series.

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99-045237

ISSN 1439-2674

ISBN 3-540-65272-8 Springer-Verlag Berlin Heidelberg New York

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Preface

Throughout my professional career I have been fascinated by problems involv-ing electrical noise. In this book I would like to describe aspects of electricalnoise somewhat in the manner of a Russian matryoshka doll, in which eachshell contains a different doll, alluding to deeper and deeper meanings hiddeninside as outer appearances are peeled away.

Let us look at some dictionary definitions of noise. Surprisingly, the originof the word in the English language is unknown. The Oxford Universal Dic-tionary (1955) has the following definition: "Noise. 1. loud outcry, clamouror shouting; din or disturbance; common talk, rumour, evil report, scandal -1734. A loud or harsh sound of any kind; a din ... An agreeable or melodioussound. Now rare, ME. A company or band of musicians."

This is not a helpful definition of the technical meaning of noise. The Sup-plement to the Oxford English Dictionary (1989) lists the following: "Noise.7. In scientific use, a collective term (used without the indefinite article) for:fluctuations or disturbances (usu. irregular) which are not part of a wantedsignal, or which interfere with its intelligibility or usefulness."

The last definition is an appropriate one and relates to the work of Prof.Norbert Wiener who developed the mathematics of statistical functions inthe 1930s and 1940s. To this day I am awed by the power of mathemati-cal prediction of averages of outcomes of statistically fluctuating quantities.These predictions extend to the theory of and experiments on noise.

Let us look at the interpretation in other languages of the word used forthe technical term "noise".

In German Rauschen: rush, rustle, murmur, roar, thunder, (poet.)sough.

In Russian sham: noise, hubbub, uproar; vetra, voln: sound ofwind, waves.

In French bruit: noise, din, racket, uproar, commotion, clamor;(fig.) tumult, sedition; fame, renown, reputation; beau-coup de bruit pour rien, much ado about nothing.

In Italian rumore: noise, din, clamor, outcry, uproar; rumor.

It is interesting how different languages attach different meanings to noise.The German and Russian origins are onomatopoetic, simulating the sound of

VI Preface

rushing water or rustling of leaves, and do not necessarily possess the conno-tation of unpleasantness. The French and Italian words have more abstractmeanings. Surprisingly, in French, it describes characteristics of persons whostick out, are famous. In Italian it is clearly related to the word "rumor". Theetymology of the word "noise" is a glimpse of the complexity and subtlety ofthe meanings attached to words by different cultures. In the world of physicsand technology, noise is equally multifaceted.

A fascinating fact is that the ear is adjusted to have the highest al-lowed sensitivity without being disturbed by one of the fundamental sourcesof noise, thermal noise. Thermal noise is the agitation experienced by themolecules in gases, liquids, and solids at all temperatures above absolutezero (on the Kelvin scale). The molecules of air bounce around and hit theeardrums in a continuous pelting "rain" of particles. If the ear were sensitiveto that bombardment, one would hear a continuous hissing noise comparableto that of the noise of a radio tuned between stations with the volume turnedup. A simple computation finds that the power impinging upon the ear fromthis thermal noise is of the order of 0.3 x 10-12 W, a third of the thresholdof hearing [1], a rather remarkable fact.

Many of us have experienced the strange sensation that is produced whena large shell is held to the ear. Popularly this is known as "hearing the ocean".In fact, this effect is due to the noise of the air particles impinging upon theear, enhanced by the shell acting as a resonator. Thus, even a normal ear canhear the air particles impinging upon the ear when the effect is enhanced bysome means. Later in this book we shall learn how resonators enhance thespectrum of noise near their resonance frequency.

My interest in noise, reflected in the content of this book, was and ismainly in electrical and optical noise. It is not hard to understand the ori-gin of electrical noise, at least the one related to the agitation of particles.Particles with charge are surrounded by fields which, in turn, produce chargeaccumulation (of opposite sign) in surrounding electrodes. As the particlesbounce around when driven by thermal effects or quantum effects, the chargesin the electrodes are dragged along and produce spurious currents, noise cur-rents.

Electrical communications engineers worry about noise because they haveto discern signals in the presence of such background noise. In all casesin which the background noise is worrisome, the signals are weak so thatamplifiers are needed to raise their power to detectable levels. Amplifiersadd noise of their own to the background noise. The ultimate source of low-frequency (including microwave) amplifier noise is the "graininess of the elec-trical charge". This fact was recognized in its full significance by Schottky inhis classic paper in 1918 [2]. 1 quote from Schottky (my English translation):

Preface VII

Cascading of vacuum tube amplifiers has made possible in recentyears the detection and measurement of alternating currents ofexceedingly small amplitude. Many technical tasks have therebyrealized a sudden benefit, but also a new field of research has beenopened up. The new amplifying circuits have the same impact onelectrical studies as the microscope has had for optics. Because noclear limit has appeared to date on the achievable amplification,one could hope to advance to the infinitesimally small by propershielding, interference-free layouts, etc. of the amplifying circuits;the dream of "hearing the grass grow" has appeared achievable tomankind.

This is an allusion by Schottky to the sensory power ascribed by the brothersGrimm fairy tales to particularly endowed individuals. In the sequel he showsthat the dream will not come true and I quote:

The first insurmountable obstacle is provided, remarkably, by thesize of the elementary quantum of electricity (the charge of theelectron).

Schottky wrote his paper a decade before the formulation of the uncer-tainty principle of Heisenberg. Some of the noise generated in amplifiers andrecognized by Schottky can be controlled. The amplifiers can be cooled orrefrigerated. The shot noise can be reduced by utilizing the mutual repulsionamong the negatively charged electrons. Schottky was careful to point out inhis paper that, with the current densities achievable in his day, such repulsioncould be ignored. In the intervening 75 years a great deal has happened andthis research led to the development of ultra-low-noise amplifiers.

The fundamental limit of the noise performance of amplifiers is ultimatelydetermined by quantum mechanics. This was the reason why I studied opticalamplification, at frequencies at which the quantum effects of the electromag-netic field are observable, and at which quantum effects are, fundamentally,responsible for the noise performance of optical amplifiers. This very prop-erty of optical amplifiers makes them ideal models of quantum measurementapparatus and permits study of the theory of quantum measurement with theaid of simple optical measurement devices. This book thus spans the rangefrom microwave propagation and amplification to optical propagation andamplification, all the way to issues of the theory of quantum measurement.

A book based on the work of 45 years clearly rests on collaborationwith many individuals. Among those I should mention with gratitude arethe late Prof. Richard B. Adler, Charles Freed, Dr. James Mullen, Prof. Y.Yamamoto, Dr. J. P. Gordon, and many past and present students. Amongthese, credit goes to Patrick Chou, John Fini, Leaf Jiang, Thomas Murphy,Steve Patterson, Michael Watts, William Wong, and Charles Yu for the care-ful reading of the manuscript that led to many corrections and suggestionsfor improvements.

VIII Preface

Research cannot do without financial support. Much of the early workwas done with general funding by the Joint Services Electronics Program ofthe Research Laboratory of Electronics. More recently, as the funding becamemore program-specific, credit goes to the Office of Naval Research and Dr.Herschel S. Pilloff, who encouraged the research on squeezed-state generation,and Dr. Howard R. Schlossberg and the Air Force Office of Scientific Research,who funded the work on long-distance fiber communications.

I gratefully acknowledge the work by Ms. Mary Aldridge and Ms. CindyKopf, who typed the manuscript with exemplary patience and attention todetail. Ms. Cindy Kopf redrew and finished most of the figures in final form. Iexpress my appreciation for the careful and thorough editing by Copy EditorMs. Christine Tsorpatzidis.

Cambridge, Massachusetts Hermann A. HausJuly 2000

Contents

Introduction .................................................. 1

1. Maxwell's Equations, Power, and Energy ................. 111.1 Maxwell's Field Equations .............................. 111.2 Poynting's Theorem .................................... 151.3 Energy and Power Relations and Symmetry of the Tensor E . 171.4 Uniqueness Theorem ................................... 221.5 The Complex Maxwell's Equations ....................... 231.6 Operations with Complex Vectors ....................... 251.7 The Complex Poynting Theorem ........................ 281.8 The Reciprocity Theorem ............................... 331.9 Summary ............................................. 34Problems .................................................. 35Solutions .................................................. 37

2. Waveguides and Resonators .............................. 39

2.1 The Fundamental Equationsof Homogeneous Isotropic Waveguides .................... 39

2.2 Transverse Electromagnetic Waves ....................... 442.3 Transverse Magnetic Waves ............................. 47

2.4 Transverse Electric Waves .............................. 53

2.4.1 Mode Expansions ................................ 562.5 Energy, Power, and Energy Velocity ...................... 59

2.5.1 The Energy Theorem ............................ 592.5.2 Energy Velocity and Group Velocity ............... 602.5.3 Energy Relations for Waveguide Modes ............. 612.5.4 A Perturbation Example ......................... 62

2.6 The Modes of a Closed Cavity ........................... 642.7 Real Character of Eigenvalues and Orthogonality of Modes . 672.8 Electromagnetic Field Inside a Closed Cavity with Sources .. 722.9 Analysis of Open Cavity ................................ 742.10 Open Cavity with Single Input .......................... 77

2.10.1 The Resonator and the Energy Theorem ............ 78

X Contents

2.10.2 Perturbation Theory and the Generic Formof the Impedance Expression ...................... 79

2.11 Reciprocal Multiports .................................. 832.12 Simple Model of Resonator ............................. 84

2.13 Coupling Between Two Resonators ....................... 882.14 Summary ............................................. 91

Problems .................................................. 92

Solutions .................................................. 95

3. Diffraction, Dielectric Waveguides, Optical Fibers,and the Kerr Effect ....................................... 993.1 Free-Space Propagation and Diffraction ................... 1003.2 Modes in a Cylindrical Piecewise Uniform Dielectric ....... 1063.3 Approximate Approach ................................. 1093.4 Perturbation Theory ................................... 1133.5 Propagation Along a Dispersive Fiber .................... 1133.6 Solution of the Dispersion Equation for a Gaussian Pulse ... 1153.7 Propagation of a Polarized Wave

in an Isotropic Kerr Medium ............................ 117

3.7.1 Circular Polarization ............................. 1193.8 Summary ............................................. 120

Problems .................................................. 120

Solutions .................................................. 123

4. Shot Noise and Thermal Noise ............................ 1274.1 The Spectrum of Shot Noise ............................ 1284.2 The Probability Distribution of Shot Noise Events ......... 1344.3 Thermal Noise in Waveguides and Transmission Lines ...... 1364.4 The Noise of a Lossless Resonator ....................... 1404.5 The Noise of a Lossy Resonator ......................... 1434.6 Langevin Sources in a Waveguide with Loss ............... 1444.7 Lossy Linear Multiports at Thermal Equilibrium .......... 1464.8 The Probability Distribution of Photons

at Thermal Equilibrium ................................ 1504.9 Gaussian Amplitude Distribution

of Thermal Excitations ................................. 1524.10 Summary ............................................. 154Problems .................................................. 155

Solutions .................................................. 156

5. Linear Noisy Multiports .................................. 1575.1 Available and Exchangeable Power from a Source .......... 1595.2 The Stationary Values of the Power Delivered

by a Noisy Multiport and the Characteristic Noise Matrix .. 160

Contents XI

5.3 The Characteristic Noise Matrixin the Admittance RepresentationApplied to a Field Effect Transistor ...................... 166

5.4 Transformations of the Characteristic Noise Matrix ........ 1685.5 Simplified Generic Forms of the Characteristic Noise Matrix. 1725.6 Noise Measure of an Amplifier ........................... 175

5.6.1 Exchangeable Power ............................. 1755.6.2 Noise Figure .................................... 1765.6.3 Exchangeable Power Gain ........................ 1775.6.4 The Noise Measure and Its Optimum Value ......... 179

5.7 The Noise Measure in Terms of Incidentand Reflected Waves ................................... 1815.7.1 The Exchangeable Power Gain .................... 1835.7.2 Excess Noise Figure .............................. 184

5.8 Realization of Optimum Noise Performance ............... 1855.9 Cascading of Amplifiers ................................ 1895.10 Summary ............................................. 190

Problems .................................................. 192

Solutions .................................................. 193

6. Quantum Theory of Waveguides and Resonators .......... 1976.1 Quantum Theory of the Harmonic Oscillator .............. 1986.2 Annihilation and Creation Operators ..................... 2036.3 Coherent States of the Electric Field ..................... 2056.4 Commutator Brackets, Heisenberg's Uncertainty Principle

and Noise ............................................. 2096.5 Quantum Theory of an Open Resonator .................. 2116.6 Quantization of Excitations on a Single-Mode Waveguide ... 2156.7 Quantum Theory of Waveguides with Loss ................ 2176.8 The Quantum Noise of an Amplifier

with a Perfectly Inverted Medium ....................... 2206.9 The Quantum Noise

of an Imperfectly Inverted Amplifier Medium .............. 2236.10 Noise in a Fiber with Loss Compensated by Gain .......... 2266.11 The Lossy Resonator and the Laser Below Threshold ....... 2296.12 Summary ............................................. 237

Problems .................................................. 238

Solutions .................................................. 239

7. Classical and Quantum Analysisof Phase-Insensitive Systems .............................. 2417.1 Renormalization of the Creation and Annihilation Operators 2427.2 Linear Lossless Multiports

in the Classical and Quantum Domains ................... 243

XII Contents

7.3 Comparison of the Schrodinger and Heisenberg Formulationsof Lossless Linear Multiports ............................ 248

7.4 The Schrodinger Formulation and Entangled States ........ 2517.5 Transformation of Coherent States ....................... 2547.6 Characteristic Functions and Probability Distributions ..... 256

7.6.1 Coherent State .................................. 2567.6.2 Bose-Einstein Distribution ........................ 258

7.7 Two-Dimensional Characteristic Functionsand the Wigner Distribution ............................ 259

7.8 The Schrodinger Cat State and Its Wigner Distribution .... 2637.9 Passive and Active Multiports ........................... 2677.10 Optimum Noise Measure of a Quantum Network........... 2727.11 Summary ............................................. 276Problems .................................................. 277Solutions .................................................. 278

8. Detection ................................................. 281

8.1 Classical Description of Shot Noiseand Heterodyne Detection .............................. 282

8.2 Balanced Detection .................................... 2858.3 Quantum Description of Direct Detection ................. 2888.4 Quantum Theory of Balanced Heterodyne Detection ....... 2908.5 Linearized Analysis of Heterodyne Detection .............. 2928.6 Heterodyne Detection of a Multimodal Signal ............. 2958.7 Heterodyne Detection with Finite Response Time

of Detector ........................................... 296

8.8 The Noise Penalty of a Simultaneous Measurementof Two Noncommuting Observables ...................... 298

8.9 Summary ............................................. 300Problems .................................................. 301

Solutions .................................................. 302

9. Photon Probability Distributions and Bit-Error Rateof a Channel with Optical Preamplification ............... 3059.1 Moment Generating Functions .......................... 305

9.1.1 Poisson Distribution ............................. 3089.1.2 Bose-Einstein Distribution ........................ 3089.1.3 Composite Processes ............................. 309

9.2 Statistics of Attenuation ................................ 3119.3 Statistics of Optical Preamplification with Perfect Inversion. 3149.4 Statistics of Optical Preamplification

with Incomplete Inversion .............................. 3209.5 Bit-Error Rate with Optical Preamplification .............. 324

9.5.1 Narrow-Band Filter, Polarized Signal, and Noise ..... 3249.5.2 Broadband Filter, Unpolarized Signal .............. 327

Contents XIII

9.6 Negentropy and Information ............................ 3309.7 The Noise Figure of Optical Amplifiers ................... 3339.8 Summary ............................................. 339

Problems .................................................. 340

Solutions .................................................. 342

10. Solitons and Long-Distance Fiber Communications ....... 34510.1 The Nonlinear Schrodinger Equation ..................... 34610.2 The First-Order Soliton ................................ 34810.3 Properties of Solitons .................................. 35210.4 Perturbation Theory of Solitons ......................... 35410.5 Amplifier Noise and the Gordon-Haus Effect .............. 35710.6 Control Filters ........................................ 36110.7 Erbium-Doped Fiber Amplifiers and the Effect

of Lumped Gain ....................................... 365

10.8 Polarization ........................................... 36710.9 Continuum Generation by Soliton Perturbation ............ 37010.10 Summary ............................................. 374

Problems .................................................. 376

Solutions .................................................. 377

11. Phase-Sensitive Amplification and Squeezing ............. 37911.1 Classical Analysis of Parametric Amplification ............. 38011.2 Quantum Analysis of Parametric Amplification ............ 38311.3 The Nondegenerate Parametric Amplifier as a Model

of a Linear Phase-Insensitive Amplifier ................... 38611.4 Classical Analysis

of Degenerate Parametric Amplifier ...................... 38711.5 Quantum Analysis

of Degenerate Parametric Amplifier ...................... 39011.6 Squeezed Vacuum and Its Homodyne Detection ........... 39311.7 Phase Measurement with Squeezed Vacuum ............... 39511.8 The Laser Resonator Above Threshold ................... 39811.9 The Fluctuations of the Photon Number .................. 40311.10 The Schawlow-Townes Linewidth ........................ 40611.11 Squeezed Radiation from an Ideal Laser .................. 40811.12 Summary ............................................. 412Problems .................................................. 413Solutions .................................................. 414

12. Squeezing in Fibers ....................................... 41712.1 Quantization of Nonlinear Waveguide .................... 41812.2 The x Representation of Operators ...................... 42012.3 The Quantized Equation of Motion of the Kerr Effect

in the x Representation ................................. 422

XIV Contents

12.4 Squeezing ............................................. 424

12.5 Generation of Squeezed Vacuumwith a Nonlinear Interferometer ......................... 427

12.6 Squeezing Experiment .................................. 432

12.7 Guided-Acoustic-Wave Brillouin Scattering ............... 43412.8 Phase Measurement Below the Shot Noise Level ........... 43612.9 Generation of Schrodinger Cat State via Kerr Effect ....... 440

12.10 Summary ............................................. 442

Problems .................................................. 442

Solutions .................................................. 443

13. Quantum Theory of Solitons and Squeezing ............... 44513.1 The Hamiltonian and Equations of Motion

of a Dispersive Waveguide .............................. 44613.2 The Quantized Nonlinear Schrodinger Equation

and Its Linearization ................................... 44913.3 Soliton Perturbations Projected by the Adjoint ............ 45313.4 Renormalization of the Soliton Operators ................. 45713.5 Measurement of Operators .............................. 46113.6 Phase Measurement with Soliton-like Pulses ............... 46213.7 Soliton Squeezing in a Fiber ............................ 46513.8 Summary ............................................. 469Problems .................................................. 471Solutions .................................................. 472

14. Quantum Nondemolition Measurements and the "Collapse"of the Wave Function ..................................... 47314.1 General Properties of a QND Measurement ............... 47514.2 A QND Measurement of Photon Number ................. 47514.3 "Which Path" Experiment .............................. 48114.4 The "Collapse" of the Density Matrix .................... 48414.5 Two Quantum Nondemolition Measurements in Cascade .... 49014.6 The Schrodinger Cat Thought Experiment ................ 49314.7 Summary ............................................. 497Problems .................................................. 498Solutions .................................................. 499

Epilogue ...................................................... 503

Appendices ................................................... 505A.1 Phase Velocity and Group Velocity of a Gaussian Beam .... 505A.2 The Hermite Gaussians and Their Defining Equation ....... 506

A.2.1 The Defining Equation of Hermite Gaussians ........ 506A.2.2 Orthogonality Property of Hermite Gaussian Modes.. 507

Contents XV

A.2.3 The Generating Function and Convolutionsof Hermite Gaussians ............................ 508

A.3 Recursion Relations of Bessel Functions .................. 512A.4 Brief Review of Statistical Function Theory ............... 513A.5 The Different Normalizations of Field Amplitudes

and of Annihilation Operators ........................... 515A.5.1 Normalization of Classical Field Amplitudes ........ 515A.5.2 Normalization of Quantum Operators .............. 516

A.6 Two Alternative Expressions for the Nyquist Source ....... 517A.7 Wave Functions and Operators in the n Representation ..... 518A.8 Heisenberg's Uncertainty Principle ....................... 523A.9 The Quantized Open-Resonator Equations ................ 524A.10 Density Matrix and Characteristic Functions .............. 527

A.10.1 Example 1. Density Matrix of Bose-Einstein State ... 528A.10.2 Example 2. Density Matrix of Coherent State ....... 528

All Photon States and Beam Splitters ....................... 529A.12 The Baker-Hausdorff Theorem .......................... 530

A.12.1 Theorem 1 ...................................... 530

A.12.2 Theorem 2 ...................................... 531

A.12.3 Matrix Form of Theorem 1 ....................... 531A.12.4 Matrix Form of Theorem 2 ....................... 532

A.13 The Wigner Function of Position and Momentum .......... 533A.14 The Spectrum of Non-Return-to-Zero Messages ............ 535A. 15 Various Transforms of Hyperbolic Secants ................ 538A. 16 The Noise Sources Derived from a Lossless Multiport

with Suppressed Terminals .............................. 541A. 17 The Noise Sources of an Active System

Derived from Suppression of Ports ....................... 542A. 18 The Translation Operator and the Transformation

of Coherent States from the 3 Representationto the x Representation ................................ 543

A. 19 The Heisenberg Equation in the Presence of Dispersion ..... 544A.20 Gaussian Distributions and Their e-1/2 Loci .............. 544

References .................................................... 549

Index ......................................................... 555

Introduction

Quantitatively, the noise of a linear amplifier can be described as the noisepower added by the amplifier to the signal power in the process of signalamplification. It has been found convenient to refer both the noise power andthe signal power to the input of the amplifier, before amplification, becausethen one can make a direct comparison between the amplifier noise and thethermal noise that accompanies the signal. We have gone so far as to ex-press the noise ascribed to the amplifier in Kelvin, namely, in terms of thethermal power that would be emitted by a thermal source if it were at thistemperature.

In the 1950s, Penzias and Wilson were readying a microwave antenna forsatellite communications using the latest in ultra-low-noise amplifiers. Theypointed their antenna in various directions of the sky, away from the highemitters of noise such as the sun and some interstellar radio sources, andfound a background noise that could not be accounted for by the noise in theamplifier. They had discovered the 3.5 K background radiation of interstellarspace. (This discovery decided in favor of the big-bang theory of the origin ofthe universe over a rival cosmological theory.) The background noise observedby Penzias and Wilson and quoted in the book The First Three Minutes bySteven Weinberg [3] is roughly 1/100 of room temperature. They had to havean excellent understanding of the noise in their receiver to attribute the slightdiscrepancy in the observed noise power from the output of their amplifierto an unknown source of noise. Professor Bernard Burke of the MIT physicsdepartment was made aware of their discovery and brought them into contactwith Prof. R. H. Dicke of Princeton, who had indicated that the backgroundtemperature of the universe should be of this magnitude if the universe indeedstarted from the initial big bang in a very small volume and expanded eversince. One may understand this in a somewhat simplified form as a decreaseof the frequency and energy density of the original high-temperature, high-frequency electromagnetic waves as they extended over a larger and largervolume. The same would happen to the sound frequency and energy in anorgan pipe in which the ends were moved continually farther and fartherapart.

It is indeed remarkable that a purely technical accomplishment - thedesign of low-noise amplifiers, the construction of a satellite communications

2 Introduction

link, and a very good understanding of the noise in amplifiers - has providedthe evidence for one of the theories of the origins of the universe. The existenceof the background radiation is now well established. The number 3.5 K hasbeen modified to 2.76 K.

At the very same time as these developments were taking place some of uspractitioners were asking ourselves whether there are any fundamental lowerbounds to the noise performance of an amplifier. Offhand, one might expectthat the minimum amount of noise added to the signal could not be lowerthan the thermal background noise associated with the temperature at whichthe amplifier operates. But this is not the case. There is ample evidence thatamplifiers can do better. Indeed, refrigerators produce locally lower tempera-tures than the environment in which they operate and amplifiers can performthe same feat. Further, truly super-deluxe amplifiers include refrigeration tohelp them reduce their noise. It looks as if there is no lower limit to the noiseof an amplifier, if one is willing to pay the price of the refrigeration. Eventhe shot noise, which is fundamental under random emission, can be reducedby active control, at low frequencies. As the frequencies become higher andhigher, such control becomes not only physically more difficult, but impossi-ble in a more fundamental way. The intrinsic noise has a fundamental lowerbound and that fundamental bound is of quantum mechanical origin. Thenoise of fundamental origin is proportional to the frequency of the amplifier.What makes laser noise so interesting is that it is truly fundamental; becauseof its enormously high level it is detectable. Before we bring up this point inmore detail, let us return to noise radiation, namely the kind of radiation leftover by the big bang.

Whereas it is rather clear that bouncing charged particles cause noise, whyshould there be an excitation of free space? The reason for its existence isthe following. Free space can transmit electromagnetic radiation. Thermallyagitated charged particles excite electromagnetic radiation. The radiationin turn can transfer its energy to the particles. Thus, free space containingcharged particles at any temperature must contain radiation. This radiationhas a very specific intensity if it is at thermal equilibrium with the thermallyagitated particles, gaining as much energy per unit time from the chargedparticles owing to their radiation as it is losing energy per unit time to thecharged particles. This radiation obeys laws very similar to the acoustic ra-diation caused by thermal noise.

An electromagnetic mode of frequency v can carry energy only in unitsof hv, where h is Planck's constant; h = 6.626 x 10-34 J S. Quantum effectspredominate over thermal effects when

by > kT, (0.1)

where k is Boltzmann's constant, k = 1.38 x 10-23 J/K. For T = 290 K, roomtemperature, the crossover occurs in the far-infrared regime at a frequencyv = 6 x 1012 Hz, that is, much higher than conventional microwave frequen-cies. At frequencies below the limit imposed by (0.1), shot noise, thermal

Introduction 3

noise and related sources of noise predominate, at higher frequencies quan-tum noise is predominant. Quantum noise has its origin in the graininess ofelectromagnetic radiation, somewhat as shot noise has its origin in the graini-ness of electric charge. According to quantum theory, electromagnetic energyis a phenomenon that can be both particle-like and wave-like, the principle ofduality. Each particle, i.e. each photon, carries an energy hv, this energy be-ing higher the higher the frequency v. For a given amount of power received,the number of particles received decreases with increasing frequency, makingtheir graininess more noticeable. For this very reason, amplifiers of opticalradiation are much noisier than amplifiers of microwave or lower-frequencyradiation.

In 1973 A. Hasegawa and G. Tappert at Bell Telephone Laboratories sug-gested [4] that optical fibers could propagate solitons. An optical fiber madeof silicon dioxide glass is dispersive in that the velocities of travel of sinusoidaloptical waves of different wavelengths are different. It is nonlinear owing tothe so-called Kerr effect: the index of refraction of the optical material de-pends upon the intensity of the optical wave. This effect is named after JohnKerr, like Maxwell a Scot. (It turns out that W. C. Roentgen of X-ray famealso discovered the effect, but Kerr published first.)

Optical pulses that maintain their shape as they propagate (solitons) canform in glass fibers if the dispersion and Kerr effect balance. The Kerr effectis called positive if the index increases with increasing intensity, negative ifit decreases with increasing intensity. The dispersion is called positive if thevelocity increases with wavelength A, negative if it changes in the opposite di-rection. The Kerr effect in glass is positive. Negative dispersion and a positiveKerr effect can balance each other to allow for soliton propagation. Hence,to see solitons in fibers one must excite them at wavelengths at which silicondioxide has negative dispersion. This is the case for wavelengths longer than1.3 µm (although fiber dispersion can be affected by core-cladding design).Optical fibers have one other remarkable property: at a wavelength of 1.5 µmthey have extremely low loss; they are extraordinarily transparent. Light atthis wavelength loses only a few percent of its power when propagating over a1 km fiber. For this reason, optical fibers are a particularly felicitous mediumfor signal propagation.

It was the stability of the soliton pulses that motivated Hasegawa in 1984to propose long-distance optical communications using soliton pulses [5]. Thesignal would be digital, made up of pulses (solitons) and empty time intervals,symbolizing a string of ones and zeros. Over a trans-Atlantic distance of 4800km, the optical signal would have to be amplified to compensate for the loss.

At the present time, most practical amplifiers for fiber transmission aremade of rare-earth-doped fibers (the rare earth being erbium) "pumped" bya source at a wavelength in an absorption band of the dopant. The opticalpumping is done by light from an optical source, a laser with photons of en-ergy hvr,. The dopant atoms (erbium in the case of the fiber) absorb the pump

4 Introduction

photons and are excited to higher-lying energy levels which decay rapidly andnonradiatively to the upper laser level. When an atom in the upper laser levelis stimulated by signal photons of energy hv, the atom makes a transitionfrom the upper laser level to the lower laser level, emitting a photon. Thisso-called stimulated emission increases the signal, i.e. amplifies it.

Stimulated emission is not the only radiation emitted by the excitedatoms. As already pointed out by Einstein, an excited atom eventually decaysradiatively to a lower-lying level by spontaneous emission even in the absenceof stimulating radiation. This emission is independent of the stimulated emis-sion. It masks the signal and is experienced as "noise" after detection.

At the time of Hasegawa's proposal, long-distance optical signal trans-mission was more complicated: the signal (pulse or no pulse) was detected,regenerated and reemitted in so-called "repeaters" spaced every 100 km or so.In this way the intervening loss was compensated but, equally importantly,the noise added to the signal by random disturbances was removed. Digitalsignals transmitted via repeaters were thus particularly immune to noise. Onedisadvantage of this robust scheme of communications in transoceanic cabletransmission is that, once the cable has been laid, the format of transmissioncannot be changed, because the repeaters are designed to handle only oneparticular format. Hasegawa's bold move would do away with repeaters andreplace them with simple optical amplifiers. Once a cable of this type is in-stalled, it is not tied to a particular signaling format. The pulse rate could bechanged at the transmission end and the receiver at the reception end, butno changes would have to be made in the cable and amplifier "pods" at thebottom of the ocean.

The implementation of Hasegawa's idea took some time. The first ques-tion was whether the solitons propagating along a fiber would be sufficientlyimmune to the spontaneous-emission noise "added" in the optical amplifiers.In 1984, while on sabbatical at AT&T Bell Laboratories, the author, with J.P. Gordon, showed [6] that the noise in the amplifiers would change the carrierwavelength of the solitons in a random way. Since the speed of the solitons isa function of the carrier wavelength, the arrival time of the pulses would ac-quire a random component; the solitons may end up in the wrong time slots,causing errors [6]. This effect is now known as the Gordon-Haus effect. Withthe parameters of the fiber proposed by Hasegawa, his "repeaterless" schemecould not have spanned the Atlantic. The analysis clearly demonstrated thedependence of the effect on the parameters of the fiber. But with a redesignof the fiber, the Atlantic could be spanned!

L. F. Mollenauer and his group at AT&T Bell Laboratories [7] made pio-neering experiments in which they verified many of the predicted propertiesof soliton propagation. Since a fiber 4800 km long would cost of the orderof $100 million, they used a loop of the order of 100 km in length, withthree amplifiers, in which they launched a pseudorandom sequence of soli-tons (ones) and empty intervals (zeros) and recirculated them as many times

Introduction 5

as they wished, thus simulating long distance propagation. They confirmedthe Gordon-Haus effect.

Noise is a familiar phenomenon accompanying any measurement. Thenumerical values of the quantity measured differ from measurement to mea-surement. In undertaking a measurement, the experimentalist starts from theassumption that a sequence of measurements on identically prepared systemswill arrive at a set of outcomes that will have an average, the value of whichwill be identified with the average value of the quantity measured. (This as-sumes of course that the measurement is not distorting the average value asoften happens when the measurement apparatus is nonlinear.) Measurementsin quantum theory fit into this general view of measurement. The ideal ap-paratus of quantum measurements does not have nonlinear distortions; theaverage value of the measurements on an observable is indeed its expecta-tion value. The individual outcomes of the measurements, in general, exhibitscatter, just as they do for a classical signal in the presence of noise

Bell of "Bell's inequality" fame was disturbed by the interpretation ofa quantum measurement, in particular by the von Neumann postulate bywhich every measurement projects the wave function of the observable intoan eigenstate of the measurement apparatus [8]. He saw the postulate as agraft onto the standard quantum description. He considered quantum theoryincomplete, like Einstein before him, but in a different sense. As an exampleof a complete theory, he cited Maxwell's theory of electromagnetism. Theequations that describe the electromagnetic field also contain in them therules for the measurement of the field. In contrast, the von Neumann postulatehas to be invoked in interpreting the outcome of a quantum measurement.

In the last chapter in this book, we attack the problem of quantum mea-surements in the optical domain, since quantum formalisms for optical ap-paratus will be well developed at that point. We shall discuss "quantumnondemolition" (QND) measurements that leave the measured observableunchanged. A QND measurement can be used to "derive" the von Neumannpostulate through the study of two QND measurements in cascade. One canshow that the conditional probability of measuring the same value of an ob-servable in the second setup as in the first can be made unity through properdesign of the apparatus. We consider this a direct derivation from quantummechanics of the von Neumann postulate, in response to Bell's criticism.

Bell was questioning the placement of the boundary between the quan-tum and classical domains [9]: "Now nobody knows where the boundarybetween the classical and quantum domain is situated." We shall argue thatthe boundary can be placed in most situations by virtue of the nature of allmeasurement apparatus. A measurement apparatus has to deliver a resultthat can be interpreted classically [10], such as the position of the needle ofa meter or a trace on a scope. For this to be possible, the measurement ap-paratus, even though described quantum mechanically, must have lost, at itsoutput, quantum coherences that have no interpretation in terms of positive

6 Introduction

probabilities. This is the point of Zurek [11] and others [12-14], who haveshown that macroscopic systems lose coherence extremely rapidly.

It is appropriate that the subject of noise should lead us to ask somefundamental questions in quantum theory. Quantum theory predicts the be-havior of an ensemble of identically prepared systems. The statistical theoryof noise does likewise. The fluctuations in the observations made on a quan-tum system can be, and should be, interpreted as noise. It is, in this writer'sopinion, futile to search for a means to predict the outcome of one single mea-surement. Statistical mechanics makes only probabilistic predictions abouta system, because of a lack of complete knowledge of the system's initialconditions. Quantum mechanics raises the lack of knowledge of the initialconditions to the level of a principle. Hence the statistical character of thedescription of nature by quantum mechanics is unavoidable.

At the outset, a disclaimer is in order. This book is not a synopsis of theexcellent work on electrical noise, optical communications, squeezed states,and quantum measurement that has appeared in the literature. Instead, itis a personal account of the author's and his coworkers' work over a careerspanning 45 years. Such an account has a certain logical consistency that hasdidactic merit, a feature that would be sacrificed if an attempt had been madeto include the excellent work of other authors in such a way as to do it justice.For the same reason, the literature citations will be found to be deficient. Yetthe author hopes that despite these deficiencies, and maybe even on accountof them, the reader will find this to be a coherent presentation from a personalpoint of view of a very fascinating field.

The first three chapters provide the background necessary to understandthe basic concepts used in the remainder of the book: power flow, electro-magnetic energy, group velocity, and group velocity dispersion; modes inwaveguides and resonators; resonators as multiports and their impedancematrix and scattering-matrix description; and single-mode fibers, the opticalKerr effect, and polarization coupling in fibers. Most concepts and laws willbe familiar to the reader. The first three chapters thus serve mainly as aconvenient reference for the later developments.

Chapter 4 derives the probability distribution for the carriers of a cur-rent exhibiting shot noise and arrives at the spectrum of the current. Next,the thermal noise on a transmission line is derived from the equipartitiontheorem. From this analysis of a reversible (lossless) system it is possible,surprisingly, to derive Nyquist's theorem that describes the emission of noisefrom a resistor, an irreversible process. The noise associated with linear lossat thermal equilibrium calls for the introduction of Langevin noise sources.Finally, we derive the probability distribution of photons on a waveguide(one-dimensional system) at thermal equilibrium, the so called Bose-Einsteindistribution.

With the background developed in Chap. 4 we enter the discussion ofclassical noise in passive and active multiports. If the multiports are lin-

Introduction 7

ear, their noise can be described fully by associated Langevin sources. Atthermal equilibrium, these possess some very simple properties. In particu-lar, the spectral density matrix, appropriately weighted, forms the so-calledcharacteristic noise matrix. For a passive network at thermal equilibrium, thismatrix is proportional to the identity matrix. In the more general case of alinear passive network not at equilibrium, or a linear active network, such asa linear amplifier, the characteristic noise matrix contains all the informationnecessary to evaluate the optimum noise performance of the network, thenoise performance that leads to the maximum signal-to-noise ratio at largegain. This optimum noise performance is described, alternatively, as the min-imum excess noise figure at large gain, or the minimum noise measure. Theoptimization is studied with the simple example of a microwave field effecttransistor (FET).

Chapter 6 develops the background for the treatment of quantum noise.The electromagnetic field is expressed in terms of a superposition of modeswhose amplitudes obey simple-harmonic-oscillator equations. The field isquantized by quantization of the harmonic-oscillator amplitudes. The quan-tum noise of a laser oscillator below threshold is derived. The Heisenbergdescription of operator evolution is adhered to, in which the operators evolvein time. Langevin operator noise sources are introduced in the equations forpassive and active waveguides (an example of the latter is erbium-doped-fiberamplifiers). The role of the noise sources is to ensure conservation of com-mutators, which are a fundamental attribute of the modes in the waveguide.The noise of a typical fiber amplifier is derived. Through much of the text,the quantum noise will appear additive to the "classical" c-number signal.Laser amplifiers are well described in this way. However, in general, the quan-tum noise is not represented so simply. The Wigner function is the quantumequivalent of a probability distribution. In contrast to a classical probabilitydistribution, the Wigner function is not positive definite. In order to gaina better understanding of peculiar forms of quantum noise, we study theWigner distribution as applied to a so-called Schrodinger cat state, a quan-tum state of macroscopic character. This analysis is followed up in Chap. 7by the quantum description of linear multiports. The formalism is presentedin the Heisenberg representation, which displays the correspondence with theclassical network description. The Schrodinger representation, in which thewave functions, rather than the operators, evolve in time, is introduced and acomparison between the two descriptions is made. The concept of entangledstates is introduced. A strong analogy is found to exist between the classicalcharacteristic noise matrix and its quantum counterpart. It is found that thecommutator relations determine the characteristic noise matrix of a quantumnetwork. This is the manifestation of a fundamental law, first explicitly statedby Arthurs and Kelly [151, that requires all linear phase-insensitive amplifiersto add noise to the amplified signal, if the amplification is phase-insensitve.

8 Introduction

Chapter 8 analyzes detection of microwave signals and optical signals. Theformer can be treated classically; the latter require a quantum description.Direct, homodyne, and heterodyne detection are described. The latter twoprovide gain. Heterodyne detection provides phase-insensitive gain and thusbehaves like any other linear amplifier that must add noise to the signal.Homodyne detection is phase-sensitive and it is found that, in principle, itneed not add noise to the signal.

Chapter 9 looks in detail at high-bit-rate optical-communication detec-tion via optical preamplification followed by direct detection. In the process,we find the full photon probability distributions for ideal amplifiers as wellas for the practical case of an erbium-doped-fiber amplifier. The analysis isbased on a quantum description of amplifiers developed by J. A. Mullen andthe author in 1962 [16]. The statistics of the photodetector current are deter-mined by the photon statistics, from which the bit-error rate is derived. Theminimum number of photons per pulse required for a bit-error rate of 10-9is determined. The analysis is backed up by recently obtained experimentaldata from Lucent Technologies, Bell Laboratories. Engineering practice hasintroduced a definition of a so-called noise figure for the characterization ofthe noise performance of optical amplifiers. This definition is in conflict withthe definition of the noise figure used for the description of low-frequency andmicrowave amplifiers as standardized by the Institute of Electrical and Elec-tronic Engineers. In concluding the chapter we construct a definition that isconsistent with the IEEE definition [17].

Chapter 10 studies soliton propagation along optical fibers. Solitons pos-sess particle-like properties as well as wave-like properties: one may assignto them position and momentum, and amplitude and phase. In the quantumtheory of solitons, these four excitations are quantized in the same way asthey are quantized for particles on one hand and waves on the other hand.The perturbation theory of solitons is established and from it we derive thetiming jitter of solitons in long-distance propagation, which is the main sourceof error in a long-haul soliton communication system. Means of controllingthis effect are described. We show that periodically amplified solitons shedso-called continuum that limits the allowed spacing between amplifiers. Inlong-distance communications, the noise added by the amplifiers is always solarge that the system operates at a power level much larger than that of theminimum photon number derived in Chap. 9.

Chapter 11 treats phase-sensitive amplification. One important exampleis the laser above threshold, in which a fluctuation component in phase withthe signal sees a different amplification from the one seen by a fluctuation inquadrature with the signal. The Schawlow-Townes linewidth [18] is derived.Next, we turn to parametric amplification. This amplification is producedvia a pump excitation of a medium with a so-called second-order nonlinear-ity, a nonlinearity with a response that is quadratic in the exciting fields.The amplification can be nondegenerate or degenerate. In the former case,

Introduction 9

the amplification is closely analogous to linear phase-insensitive amplification.Degenerate parametric amplification is phase-sensitive and thus need not addnoise to the signal. In the quantum description of such an amplifier we findthat it produces so-called squeezed states: the quantum noise in one phasewith respect to the "pump" is amplified, and the quantum noise in quadra-ture is attenuated. Degenerate parametric amplifiers can produce "squeezedvacuum". We show how squeezed vacuum can be used in an interferometerto improve the signal-to-noise ratio of a phase measurement.

Squeezed vacuum can also be produced by a third-order nonlinearity, suchas the optical Kerr effect. Fibers are particularly convenient for the use ofthe Kerr nonlinearity because of their small mode volume and small loss. Thetheory of the generation of squeezed vacuum in a fiber loop is presented inChap. 12. Experiments are described that have generated squeezed vacuum,leading to a reduction of noise by 5.1 dB below shot noise. Further, a phasemeasurement is described that used the squeezed vacuum so generated for animproved signal-to-noise ratio. Chapter 13 discusses the squeezing of solitons.Solitons behave as particles and waves as outlined in Chap. 10. The squeezingthat can be achieved can address both the particle and the wave nature ofthe soliton.

The last chapter takes up the issue of the theory of quantum measure-ment using optical measurements as an example. At this point, we can usethe formalism developed in the book to present a full quantum analysis ofthe measurement process. We take the point of view that physical realitycan be assigned to an observable only with a full description of the mea-surement apparatus, which in turn is a quantum system obeying quantumlaws. Further, we go through the analysis of a quantum measurement and theevolution of the density matrix of the observable as it proceeds through themeasurement apparatus. We show that the reduced density matrix obtainedby tracing the density matrix over the measurement apparatus "collapses"into diagonal form, an observation consistent with, yet different from, the vonNeumann postulate of the collapse of the wave function of the observable intoan eigenstate of the measurement apparatus. Pursuing this point further, weanalyze the effect of a cascade of two measurements of the photon number ofa signal. We show that with proper design of the measurement apparatus, theconditional probability of observing m photons in the second measurement ifn photons have been measured in the first approaches a Kronecker delta, bn,m.This is again consistent with, yet somewhat different from, the von Neumannpostulate that the measurement apparatus projects the state of the observ-able into an eigenstate of the measurement apparatus. Finally we addressthe Schrodinger cat paradox, using an optical realization of the measurementapparatus, and show that the cat does not end up in a superposition state of"dead" and "alive."

1. Maxwell's Equations, Power, and Energy

This book is about fluctuations of the electromagnetic field at microwaveand optical frequencies. The fluctuations take place in microwave and opticalstructures. Hence a study of electromagnetic-field fluctuations requires theterminology and analytic description of structures excited by microwave oroptical sources. The equipartition theorem of statistical mechanics used inChap. 4 in the derivation of Nyquist's theorem is formulated in terms of en-ergy. Hence, in the application of the equipartition theorem, an understandingof the concept of energy is necessary. When media are present, the mediumstores energy as well. The excitation of a mode of the electromagnetic field,as discussed in Chap. 2, involves both the energy of the electromagnetic fieldand the energy in the excited medium.

We start with Maxwell's equations, which characterize electromagneticfields at all frequencies. Media are described by constitutive laws which mustobey certain constraints if the medium is to be conservative (lossless). Suchmedia store energy when excited by an electromagnetic field. Poynting's the-orem relates the temporal rate of change of stored-energy density to thedivergence of the power flow. The characterization of dispersive media isstraightforward in the complex formulation, with frequency-dependent sus-ceptibilities. The energy density in the medium involves the susceptibilitytensor and its derivative with respect to frequency. Finally, we look at thereciprocity theorem, which provides relations among the scattering coeffi-cients of a multiport network. The chapter contains topics from [19-24].

1.1 Maxwell's Field Equations

The first two of Maxwell's equations, in their familiar differential form, relatethe curl of the electric field E to the time rate of change of the magnetic fluxdensity B, and the curl of the magnetic field H to the sum of the electriccurrent density J and the time rate of change of the displacement flux densityD.

Faraday's law is

V x E5i_ *

12 1. Maxwell's Equations, Power, and Energy

Ampere's law is

V xH=J+ 6t-. (1.2)One may take the fields E and H as the fundamental fields, and the vectorsB and D as the hybrid fields that contain both the fundamental fields andproperties of the medium. Alternately, one may define E and B as funda-mental and consider D and H as hybrid. The former point of view is thatof the so called Chu formulation; the latter is more widely accepted by thephysics community. It has been shown [19] that the two points of view givethe same physical answers and thus one is free to choose either. The differencebetween the two formulations is hardly noticeable in a discussion of station-ary media. However, when moving media and forces are taken into account,the difference is both profound and subtle. While the issue involved does notaffect the discussion in the remainder of this book, the author neverthelesstakes the opportunity to discuss some of its aspects, since it played an impor-tant role in his research in the 1960s, and the way the issue was eventuallyresolved is typical of any fundamental research. Professor L. J. Chu modeledmagnetization by representing magnetic dipoles by two magnetic charges ofequal magnitude and opposite sign. In this way, a perfect analogy was es-tablished between polarizable and magnetizable media. The formulation ofmoving dielectric media, as developed by Panofsky and Phillips [20], couldbe applied to moving magnetic media in a way that was consistent with rela-tivity. Further, this point of view established an analogy between the electricfield E and the polarization density P on one hand, and the magnetic-fieldintensity H and the magnetization density M on the other hand. Soon af-ter the publication of this approach in a textbook on electromagnetism [21],the approach was criticized by Tellegen [22]. He pointed out that magneticdipoles ought to be represented by circulating currents, because such cur-rents are the sources of magnetism at the fundamental level. More seriously,the force on a circulating current was shown to be different from that on amagnetic dipole in the presence of time-varying electric fields. It turned outthat the difference between the force on a magnetic dipole and the force ona current loop with the same dipole moment as found by Tellegen was small,involving relativistic terms. However, if there were such a difference, the re-placement of magnetic dipoles by magnetic charge pairs would be flawed.The argument seemed valid at the time. It led Prof. P. Penfield and the au-thor to study the problem more carefully. We assumed that Chu's approachwas valid, and that there must exist a subtle error in Tellegen's derivationof the force on a magnetic dipole formed from a current loop. This "hunch"proved correct. It turned out that a magnetic dipole made up of a currentloop in a self-consistent way, such as a current flowing in a superconductingwire loop, undergoes changes in a time-varying electric field, changes thatwere omitted by Tellegen. The charges induced by the electric field createcurrents when the field is time-varying. These currents, when exposed to the

1.1 Maxwell's Field Equations 13

magnetic field, are acted upon by a force that cancels the critical term foundby Tellegen [22]. The force on a magnetic dipole made up of two magneticcharges or of a circulating current was indeed the same, except that in thecase of the current model relativistic effects had to be included in the restframe of the loop, because there is motion in the rest frame of the loop. Thus,Chu's model was not only correct, but much simpler, since it did not needto consider relativistic issues in the rest frame of the magnetic dipole. A fullaccount of this investigation is presented in [19]. As happens so often, relatedwork went on at the same time, resulting in publications by Shockley andJames [24] and Coleman and van Vleck [25].

Returning to the discussion at hand, we shall opt for Chu's approach, inwhich E and H are considered fundamental field quantities, whereas D andB are hybrid quantities containing the polarization and magnetization of themedium. In addition to Faraday's law (1.1) and Ampere's law (1.2), whichrelate the curl of the electric and magnetic fields to their vector sources,we have the two Maxwell's equations which relate E and H to their scalarsources by two divergence relationships.

Gauss's law for the electric field is

(1.3)

where p is the charge density other than the polarization charge density.Gauss's law for the magnetic field is

(1.4)

The equation of continuity

v.J=_ap

is a consequence of (1.2) and (1.3). The vectors and scalars appearing in (1.1)-(1.5) are, in general, all functions of time and space. We use rationalized mksunits. The electric field E is given in V/m; H is given in A/m. A convenientunit for the magnetic flux density B is V s/rn 2, the current density J is givenin A/m2.

The medium acts as a source of electromagnetic fields via its polarizationdensity P and magnetization density M:

D = E0E + P , (1.6)

B = µ0(H + M) . (1.7)

Equations (1.1)-(1.7) by themselves do not yet determine the fields. In addi-tion one has to know the relations between M and H, and between P and E,and the relation between the fields and the current density J. These are the

14 1. Maxwell's Equations, Power, and Energy

so-called constitutive relations. Once the constitutive relations are availablethe set of equations is complete and the equations can be solved subject toappropriate boundary conditions.

In the case of a linear anisotropic dielectric medium, the polarization Pis related to the electric field by linear equations:

Px = Eo(XxxEx + XxyE''y + XxzEz) , (1.8a)

Py = fo(XyxEx + XyyEy + XyzEz) , (1.8b)

Pz = fo(XzxEx + XzyEy + XzzEz) (1.8c)

These three equations are written succinctly in tensor notation:

P = EoXe E. (1.9)

It is convenient to combine the constitutive law (1.9) with the definitionof the displacement flux density (1.4) and write it in the form

D = E E (1.10)

with E defined as the dielectric tensor

Eo (1 + Xe) , (1.11)

where 1 is the identity tensor. The dielectric permeability tensor E is sym-metric, as will be proved later.

Analogous relations may be written between the magnetization M andthe magnetic field intensity H. Since there is symmetry between polarizationeffects and magnetization effects in the Chu formulation, it is easy to treatmagnetization effects by analogy. One writes for the magnetic field

(1.12)

where µ is the permeability tensor. At optical frequencies, magnetic effectsare generally negligible, except in the case of the Faraday effect.

In the special case of an isotropic medium, the tensors µ and E reduce toscalars p and f times the identity tensor. Finally, in the absence of any matterthe constants f and p assume particular values, which are worth remembering

1 x 10_9Asfo =mhos/m,361r Vm

µo = 47r x10-7 V s

A m = ohm s/m.

The product of fo and µo has a fundamental significance:

1/iofo = 2 s2/m2T2

(1.13)

(1.14)

(1.15)

1.2 Poynting's Theorem 15

where c is the light velocity in free space. The value of eo is adjusted toprovide the correct value of the speed of light; it changes as the speed of lightis determined more and more accurately.

If the only currents in the medium considered are due to conduction andif the medium is linear, we have the simple relation for the current density J

J=oE, (1.16)where a is the conductivity of the medium in mho/m. This is the field-theoretical form of Ohm's law. A form of Ohm's law more general than (1.16)applies to anisotropic linear conducting media. In such media the currentdensity J and field E are related by a tensor relation analogous to (1.9):

(1.17)

where v is a tensor. In general, Q is not symmetric. However, in Sect. 1.3 weshall show that Q must be a symmetric tensor if the material is resistive inthe true sense of the word.

Equations (1.1)-(1.7) in conjunction with (1.10), (1.12), and (1.17) aresufficient to find the electromagnetic field in a linear medium, provided properboundary conditions are stated.

Before concluding this section, we note that Maxwell's equations are time-reversible if they do not contain a conduction current J and there is no freecharge p. Indeed, suppose we have found a solution E(r, t) and H(r, t) toMaxwell's equations (1.1) and (1.2), with the constitutive laws (1.10) and(1.12) determining D(r,t) and B(r,t). Then, if we switch from t to -t,from E(r,t) to E(r, -t), H(r,t) to -H(r, -t), D(r,t) to D(r, -t), and-B(r, t) to -B(r, -t), it is easy to verify that (1.1) and (1.2) are obeyedautomatically, along with the constitutive laws (1.10) and (1.12). The newsolution is called the time-reversed solution. It is obtained from the evolutionof the forward-running solution as if the movie reel on which the evolution isrecorded were run backwards. The B and H fields are, of course, reversed.

1.2 Poynting's Theorem

In radiation problems or in problems of electromagnetic propagation, we areoften interested in the transmission of power from one region of space toanother. It is, therefore, important to clarify all concepts relating to powerand energy. Poynting's theorem accomplishes this. Poynting's theorem is amathematical identity which can be endowed with profound physical signif-icance. We start with Maxwell's equation (1.1) and dot-multiply by H. Wetake (1.2) and dot-multiply by E. Subtracting the two relations and makinguse of a well-known vector identity, we obtain

=0. (1.18)

16 1. Maxwell's Equations, Power, and Energy

Equation (1.18) is the differential form of Poynting's theorem. Integratingover a volume V, bounded by a surface S, we obtain

\(f E H IdV=O.s v

(1.19)

In (1.19) we have made use of Gauss's theorem. Equation (1.19) is the inte-gral form of Poynting's theorem. Let us turn to an interpretation of (1.19).The integral fv E E. JdV is the power imparted to the current flow J insidethe volume V. This power may be consumed in the ohmic loss of the materialwithin which the current flows; or, for example, if the current is due to aflow of electrons in free space, the power goes into the time rate of increaseof the kinetic energy of the electrons. The second volume integral in (1.19)is interpreted as the power that is needed to change the electric and mag-netic fields. Part of it may be used up in the magnetization or polarizationprocesses, the rest goes into storage. With the integral f E JdV interpretedas the power imparted to the current flow and the last integral in (1.19) asthe power needed to change the fields in the medium, there is only one inter-pretation for the first term in (1.19) on the basis of the principle of energyconservation. The integral f E x H dS over the surface enclosing the volumemust be the electromagnetic power flow out of the volume. Indeed, from theprinciple of energy conservation we have to postulate that

(a) the power flowing out of the volume, through the surface enclosing thevolume,

(b) the power imparted to the current flow, and(c) the power that goes into the changes of the fields in the medium (and

vacuum where there is no medium)

should all add up to zero. One may attach the meaning of density of electro-magnetic power flow to the vector E x H, often denoted by S, the so-calledPoynting vector. The second volume integral in (1.19) can be separated intoa field part and a material part, using (1.6) and (1.7):

f(E.+H.)dV

= dtf (oE2 + 2µ0H2 I dV (1.20)

JJJ

\+Jv(E at

where we have replaced the partial time derivative 8/at by d/dt, since thevolume integral is independent of r. The first part of the right-hand side,

1.3 Energy, Power and Symmetry 17

involving the time derivative of Ze0E2 + Zµ0H2, can be considered to be therate of change of the energy stored in the electric and magnetic fields, andthe second part the rates at which energy is imparted to the polarization andmagnetization. Whether the energy imparted to the polarization is stored ornot depends upon whether E dP integrated from a value P = 0 to a valuep = P is independent of the path of integration in P space. Indeed, considerthe energy imparted to P per unit volume. If P = 0 at t = -oo and P = Pat t, we have

ftdtE aP = E dP .

oo fit

lp

If P returns to zero at t = t', then

rtdtE aP = E(P) dP

J0

(1.21)

(1.22)

where the last expression is an integral over a closed contour in P space, withE treated as a function of P. If the integral fP E dP is independent of thepath of integration in P space, then f E dP = 0 and no energy has beenconsumed in raising P from zero to some value P and returning it back tozero. In this case, the integral f E dP can be interpreted as energy storedin the polarization. Analogous statements can be made about the magneticcontribution H d(µ0M).

In a linear medium, it is more convenient to add the field part of theimparted-energy differential, d(2 e0E2 ), to the polarization part, E E. dP, iden-tifying the total-energy differential, dWe, with

(1.23)

In the next section we shall take advantage of this identification.The physical conclusions drawn from Poynting's theorem will enable us to

evaluate the electromagnetic power that passes through a given cross sectionin space, say the cross section of a waveguide. However, Poynting's theorem,as a mathematical identity, can be used for purposes other than the evaluationof power flow. An illustration of one of these applications is the so-calleduniqueness theorem of Sect. 1.4.

1.3 Energy and Power Relationsand Symmetry of the Tensor E

In Sect. 1.1 we introduced the dielectric tensor and the magnetic permeabil-ity tensor as descriptive of the response of a linear medium. These tensorsmust obey symmetry and positive-definiteness conditions imposed by energy

18 1. Maxwell's Equations, Power, and Energy

considerations that follow from Poynting's theorem, derived in the precedingsection. From Poynting's theorem we know that the energy per unit volumesupplied to the field and polarizable medium is

fDWe

0(1.24)

In the above integral, the electric field is considered a function of D. Theenergy is obtained as a line integral of a field E in the space of D.,, Dy, andD. Hence, the energy is naturally a function of the displacement density D.In the case of a linear medium, however, it is more convenient to use E asthe independent variable. When the constitutive relation (1.10) between Dand E is introduced, we obtain

We thus have for the electric energy density, (1.25),

(1.25)

fEWe E dE. (1.26)

0

The integral (1.26) is best visualized by considering it as a line integral in aspace within which the three components of the electric field are used as thecoordinates (see Fig. 1.1). Now suppose that we apply an electric field to thedielectric material and then remove it. In doing so we obtain for the integral(1.26)

(1.27)

where the contour integral is carried out over a closed path in the space ofE. The contour integral (1.27) must be zero. The problem is identical to theproblem of defining a conservative force field F(r) in the three-dimensionalspace r(x, y, z). If the contour integral fC F dr over any closed contour Cvanishes, then the force field is conservative. By Stokes' theorem, the contourintegral can be converted into a surface integral over a surface S spanningthe contour C

curlic swhere the curl is given in Cartesian coordinates by

8FZ 8Fy8y 8z

curl F=VxF= OFy--FZaz ax

aFy aFF8x 8y

1.3 Energy, Power and Symmetry 19

Ez

dE

E= E

E-space

EY

Fig. 1.1. Integration path in E space

Since the integral vanishes over any arbitrary contour, a conservative fieldhas to be curl-free. This analogy can be used to obtain constraints on thetensor E. The argument is cast into the space of coordinates E,;, E., and E..The "force field" is

(E . E)x = ExxEx + EyxEy + EzxEz 1

(E . E)y = ExyEx + EyyEy + EzyEz ,

(E . E)z = ExzEx + EyzEy + EzzEz .

(1.28)

This "force field" has to be curl-free in the Cartesian "space" of E, wherethe partial derivatives are with respect to E.,, Ey, and E,:

(1.29)

It follows from (1.29) that

Eyz = Ezy , (1.30a)

Exz = Ezx , (1.30b)

Eyx = Exy . (1.30c)

The E tensor must be symmetric.Next, we turn to the evaluation of the energy. We note that for a sym-

metric E tensor the order in the multiplication

is immaterial. But, since

d(E. . E) = dE E E+E. dE,

20 1. Maxwell's Equations, Power, and Energy

we have

(1.31)

Using the above expression, we can find immediately for the stored-energydensity

WeEE dE=- /

E(1.32)

Since the stored-energy density must be a positive quantity for any field E,the elements of E have to form a positive-definite matrix. A matrix is positive-definite if all determinants of the principal minors of the matrix are positive.In particular,

EXX > 0, Eyy > 0, and EZZ > 0

is necessary but not sufficient.The preceding proof started from the postulate that the integral (1.26)

carried out over a closed contour must yield zero so that the medium returnsall the energy supplied to it in a process which starts with zero field andends up with zero field. In fact, an integral over a closed contour must alwaysyield zero if we do not permit the medium to generate power. Indeed, if theintegral happened to come out positive when the contour was followed inone sense, indicating power consumption, then reversal of the sense wouldresult in a negative value, i.e. energy generation. Hence, the contour integralmust yield zero for all passive media. But, then, the medium is dissipation-free. Therefore, one may state unequivocally that a linear dielectric whichresponds instantaneously to the field, as in (1.10), is dissipation-free.

In the special case of an isotropic medium, where the tensor E can bereplaced by a scalar E (or rather by the identity tensor multiplied by thescalar e), (1.32) reduces to

We = 2EE2 . (1.33)

In a very similar manner one can arrive at the conclusion that the per-meability tensor µ is symmetric and that linear materials fulfilling (1.12) arelossless, and one can obtain the expression for the magnetic energy storedper unit volume:

W,,,, B0

(1.34),

is the energy supplied by the magnetic field in order to produce the magneticflux density B. The similarity of (1.34) and (1.24) shows that all mathemat-ical steps performed in connection with the treatment of a linear dielectricmedium are applicable to linear magnetic media. For the density of magneticenergy storage in a linear medium, we have

1.3 Energy, Power and Symmetry 21

Wn= (1.35)

As was found in the case of a dielectric medium, the elements of µ have toform a positive definite matrix. Again, for an isotropic medium (1.35) reducesto

W,,, = 2µH2. (1.36)

Finally, consider briefly the power dissipated in a conducting mediumcharacterized by (1.17). The power per unit volume P is

Only the symmetric part of the conductivity tensor contributes to the powerdissipation. Indeed, it is easy to show that for an antisymmetric tensor, Q(a),

E. (a).E=0.If the medium is passive, the power must always be dissipated (and notgenerated), and P must always be positive, regardless of the applied field E.Accordingly, the elements of the symmetric Q tensor must form a positivedefinite matrix.

The Poynting theorem (1.19) was stated generally, and no assumptionabout the linearity of the medium had been made. If we introduce (1.32) and(1.34), we have

aD M = l aE E (E E)at at 2 at

and

aB aH 1 aHat H µ at 2 at (H µ H)

Introducing these two expressions into (1.19), we have for a linear dielectricmedium

2(E E+H H)dV=0.(1.37)

In an isotropic medium within which E reduces to scalars, (1.37) assumes theform

i(ExH).dS+ J E JdV + dt f(fE2 + pH2)dV = 0. (1.38)

In free space, in the absence of currents, J = 0, E = Eo, µ = µo, and (1.37)reduces to

i E x H dS + dt J 2 (EoE2 +µ0H2) dV = 0 . (1.39)

22 1. Maxwell's Equations, Power, and Energy

1.4 Uniqueness Theorem

In the analysis of electromagnetic fields it is necessary to know what intitialconditions and what boundary conditions are necessary to determine thefields. It is also of interest to know whether a set of initial and boundaryconditions determines the fields uniquely. Energy conservation theorems ortheir generalizations often serve to provide the proof of uniqueness. In thissection we use Poynting's theorem to determine the necessary and sufficientboundary conditions and initial conditions to describe the evolution of a fielduniquely.

Consider a volume V enclosed by the surface S. The volume is assumedto be filled with a linear medium characterized by (1.10), (1.12), and (1.17).The quantities E, µ, and v may be functions of position. Suppose that at thetime t = 0 the magnetic field and the electric field are completely specifiedthroughout the volume V. Assume further that for all time the tangential Efield is specified over the part S' of the surface S, and the tangential H fieldis specified over the remaining part S". The uniqueness theorem then statesthat the E and H fields through the entire volume are specified uniquelythrough all time by these initial and boundary conditions.

The best way of proving the theorem is to suppose that it is not fulfilled.When this supposition leads to a contradiction, the proof is accomplished.Thus, suppose that, for given initial E and H fields throughout the volume,and for tangential E and H fields over the surface given for all time, two dif-ferent solutions exist inside the volume. We denote the two different solutionsby the subscripts 1 and 2. Since Maxwell's equations in the presence of lin-ear materials are linear, the difference of the two solutions is also a solution.Thus, consider the difference solution

Hd = H1 - H2 , (1.40)

Ed=E1- E2,with

Hd(t = 0) = Ed(t = 0) = 0

and

(1.41)

(1.42)

nxEd=O on S', nxHd=O on S" forallt. (1.43)The difference field must fulfill Poynting's theorem, (1.37), applied to thevolume enclosed by the surface S:

Ed x Hd dS + IV Ed Ed dVsd

IV

1 (1.44)dt 2(EdeEd+HdµHd)dV=0 .

1.5 The Complex Maxwell's Equations 23

The surface integral in (1.44) vanishes for all time by virtue of (1.43),and the volume integrals vanish at t = 0 by virtue of (1.42). The volumeintegral has the form of an energy storage of the difference solution, a positivedefinite quantity since the matrices of E and µ are positive definite (Sect. 1.3).Since the initial energy storage of the difference solution is equal to zero att = 0, the time derivative of the second volume integral in (1.44) can onlybe positive (or zero). The first volume integral in (1.44) can only be positive(or zero). It follows that the E field and H field of the difference solutionmust remain zero through all time. Therefore, the original solutions 1 and 2,by assumption different, must actually be identical. The uniqueness theoremis proved. Once a solution of Maxwell's equations is obtained for a linearmedium which fulfills the initial conditions and the boundary conditions overall time, one can conclude from the uniqueness theorem that the solutionobtained is the only possible solution.

1.5 The Complex Maxwell's Equations

In the study of electromagnetic processes in linear media, processes withsinusoidal time variation at one single (angular) frequency w are of particu-lar importance. The reason for this is the following. Microwave and opticalfrequencies are extremely high. Any modulation of a carrier is usually at afrequency low compared with the carrier frequency. Thus, in most cases, amodulated microwave or optical process can be treated as a slow successionof steady states, each at one single frequency. More generally, even if theprocess cannot be treated as a slow succession of steady states, any arbi-trary time-dependent process can be treated as a superposition of sinusoidalprocesses by Fourier analysis.

In a linear medium a steady-state excitation at a single frequency w pro-duces responses that are all at the same frequency. A field vector dependssinusoidally upon time if all three of its orthogonal coordinates are sinu-soidally time dependent. The three components of a vector are scalars. Theuse of complex scalars for sinusoidally time-varying scalars is well known.The following treatment of complex vectors is based on this knowledge.

Thus, suppose that we write the electric and magnetic fields in complexform:

E(r, t) = Re(E e-;wt) = 2 (E e-'wt + E*e+'wt) (1.45)

B(r, t) = Re(B a-;wt) = 2 (B a-iwt + B*e+iwt) , (1.46)

where the asterisk indicates the complex conjugate. Let us introduce theexpressions for E and B into Maxwell's equation (1.1). We obtain

24 1. Maxwell's Equations, Power, and Energy

V X (Be-'w' + E*e+'wt) = iw(Be-"O' - B*e+iwt) . (1.47)Equation (1.47) must apply at an arbitrary time. Setting the time to t = 0,we obtain

V x (E+E*)=iw(B-B*). (1.48)

Setting wt = --7r/2, we obtain

V x (iE - iE*) = iw(iB + iB*) . (1.49)

Dividing (1.49) by i and adding the result to (1.48), we finally have

V x E = iwB . (1.50)

In (1.50), the time does not enter. This equation is an equation for functionsof space only. The introduction of complex notation has thus enabled usto separate out the time dependence and obtain equations involving spatialdependence only. Thus far we have indicated the complex fields E and B,which are functions of r, by an overbar. Henceforth we shall dispense withthis special notation. It will be obvious from the context whether the fieldsare real and time-dependent or complex and time-independent.

In a similar manner we obtain for all Maxwell's equations

VxE=iwB, (1.51)

VxH=J - iwD, (1.52)

(1.53)

V.D=p, (1.54)

(1.55)

(1.56)

V.J=iwp. (1.57)

The quantities in (1.51)-(1.57) are complex vector or scalar quantities andare functions of space only.

The complex form of Maxwell's equations can treat dispersive media ina simple way that is not possible with the real, time-dependent form of

1.6 Operations with Complex Vectors 25

Maxwell's equations. The polarization of dispersive polarizable media is re-lated to the electric field by a differential equation in time. Complex notationin the Fourier transform domain replaces differential equations in time withalgebraic equations with frequency-dependent coefficients. For an instanta-neous response, the polarization is related to the electric field by a suscep-tibility tensor x as shown in (1.9). In a dispersive dielectric medium, thedielectric susceptibility simply becomes a function of frequency, Xe = Xe(w):

(1.58)

The dielectric tensor f becomes frequency-dependent through the definition(1.11), E = E(w). The same holds for a dispersive magnetic medium; themagnetic suceptibility tensor becomes frequency-dependent, Xm = Xm (w)The magnetization density is given by

M = Xm(w) H . (1.59)

The magnetic permeability tensor µ also becomes frequency dependent, µ =µ(w)

In Sect. 1.1 we mentioned the time reversibility of Maxwell's equations intheir real, time-dependent form, in the absence of free charges and conductioncurrent. Time reversibility can also be extracted from the complex form ofMaxwell's equations. Replacing w by -w effectively turns the time evolutionaround. This reversal of the sign of frequency leaves (1.51), (1.52), (1.55), and(1.56) unchanged if E*, D*, -H*, and -B* are accepted as the new fieldsolutions, and the susceptibility and permeability tensors obey the relation

Xe(w) = Xe(-w)Xm(w) = Xm(-w) .

(1.60a)

(1.60b)

The relations (1.60a) and (1.60b) are the consequence of the fact thatP, M, E, and H are real, time-dependent vectors. For this condition to hold

P* (-W) = Xe(-w) E* (-w) = Xe(-w) . E(w) = Xe(w) - E(w) .Since E(w) can be adjusted arbitrarily, it follows that xe(-w) = Xe(w)

Another aspect of time reversibility is of importance. Note that -B*replacing B implies also the reversal of any d.c. magnetic field present. Ifthis is not done, the field solutions are not time-reversible. This is the casein the Faraday effect.

1.6 Operations with Complex Vectors

In order to get a better understanding of what is involved in complex-vectoroperations, we shall study a few special cases. As an example, consider thedot product of a complex vector E with itself. Splitting the complex vectorinto its real and imaginary parts, we can write

26 1. Maxwell's Equations, Power, and Energy

E E = [Re(E) + i Im(E)] [Re(E) + i Im(E)](1.61)

= Re(E) Re(E) - Im(E) Im(E) + 2i Re(E) Im(E)

Equation (1.61) indicates an interesting feature of complex vectors. It is quitepossible for the dot product of a complex vector with itself to be equal tozero without the vector itself being zero. (This feature should be contrastedwith a dot product of a real vector with itself. If this dot product turns outto be zero, one must conclude that the vector itself is a zero vector.) Indeed,looking at (