Carousel Interferometer and its Applications in Precise ...prr.hec.gov.pk/jspui/bitstream/123456789/8475/1/... · Thesis Submission Approval This is to certify that the work contained

Sumara Ashraf

2017

Department of Physics and Applied Mathematics

Pakistan Institute of Engineering and Applied Sciences

Nilore, Islamabad, Pakistan

Carousel Interferometer and its

Applications in Precise Phase

Measurement

This page intentionally left blank.

Reviewers and Examiners

Foreign Reviewers

1. Dr. Harry Moseley

Department of Medical Physics, University of Dundee, DDI 9SY, Scotland UK

[email protected]

2.

2. Dr. Tariq Hasan Gilani

Department of Physics, Millersville University, Millersville, USA

[email protected]

4.

Thesis Examiners

1. Dr. Farhan Saif

Department of Physics, QAU, Islamabad

[email protected]

2. Dr. Javed Anwar

Department of Physics, CIIT, Islamabad.

[email protected]

3. Dr. Muhammad Nawaz

NILOP, Islamabad

[email protected]

Head of the Department (Name): Dr. Shahid Qamar

Signature with Date: _________________________________

mailto:[email protected]





Author’s Declaration

I Sumara Ashraf, Registration Number 07-7-1-051-2010, Department of Physics

and Applied Mathematics, hereby declare that my PhD Thesis Titled “Carousel

Interferometer and its Applications in Precise Phase Measurement “is my own work

and has not been submitted previously by me or anybody else for taking any degree

from Pakistan Institute of Engineering and Applied Sciences (PIEAS) or any other

university / institute in the country / world.

At any time if my statement is found to be incorrect (even after my graduation),

PIEAS has the right to withdraw my PhD degree.

Student Name: Sumara Ashraf Signature: ----------------------------

Certificate of PhD Thesis Approval

We, the following PIEAS Examination Committee, hereby state our full approval of

the thesis submitted by the above student in partial fulfillment of the requirements for

the degree of Doctor of Philosophy.

Examiners Name, Designation & Address Signature

Internal Examiner 1

Prof. Dr. Farhan Saif

Department of Physics, QAU, Islamabad.

Internal Examiner 2

Prof. Dr. Javed Anwar

Department of Physics, CIIT, Islamabad.

Internal Examiner 3

Dr. Muhammad Nawaz

NILOP, Islamabad.

Supervisor

Dr. Masroor Ikram

DPAM, PIEAS.

Co-Supervisor

Dr. Ghazanfar Hussain

NILOP, Islamabad.

Dean Research PIEAS

Dr. Mutawarra Hussain

DCIS, PIEAS, Islamabad.

Thesis Submission Approval

This is to certify that the work contained in this thesis entitled Carousel

Interferometer and its Applications in Precise Phase Measurement, was carried

out by Sumara Ashraf, and in my opinion, it is fully adequate, in scope and quality,

for the degree of Ph.D. Furthermore, it is hereby approved for submission of review

and thesis defense.

Supervisor: _____________________

Name: Dr. Masroor Ikram

Date: 27 July 2017

Place: PIEAS, Islamabad.

Co-Supervisor: __________________

Name: Dr. Ghazanfar Hussain

Date: 27 July 2017


Head, Department of Physics and Applied Mathematics: ___________________

Name: Dr. Shahid Qamar

Date: 27 July 2017


Sumara Ashraf

Submitted in partial fulfillment of the requirements

for the degree of Ph.D.

2017

Department of Physics and Applied Mathematics

Pakistan Institute of Engineering and Applied Sciences

Nilore, Islamabad, Pakistan

Carousel Interferometer and its

Applications in Precise Phase

Measurement

ii

Dedicated to

My Parents,

and sister Capt. Shumaila Ashraf

my inspirations

iii

Acknowledgments

First and foremost, I would like to articulate my heartedly thanks to Almighty Allah

for His blessings in achieving my goals. He has given me the courage, strength and

patience during the completion of this research work. Having sincere teachers and

cooperative friends during my PhD studies are all the blessings of Allah Almighty.

Next, I would like to pay my deepest gratitude to my supervisor Dr. Masroor

Ikram whose dedication, enthusiasm, and devotion to work always inspired me. He

has been a constant source of motivation and inspiration for me throughout my Ph.D

research. I always found him generous in sharing his knowledge and wisdom.

Working with him was a great learning experience. His kind attitude really made the

difference. I am really thankful to him for his consistent guidance and support during

my research work.

I would like to express my sincere gratitude to my co-supervisor

Dr. Ghazanfar Hussain for his constant support, guidance and motivation. It would

never have been possible for me to take this work to completion without his incredible

support and encouragement. At many stages in the course of this research project I

benefited from his advice, particularly when exploring new ideas. His positive

outlook and confidence in my research inspired me and gave me confidence. His

careful editing contributed enormously to the production of this thesis.

I would also like to pay my gratitude to Dr I. Alex Vitkin, who provided me

the opportunity to work under his kind supervision at University of Toronto, during

my six months research under IRSIP program of HEC. He has given me full support

in pursuing my scientific interests along with editing of manuscript and being so

readily available. I am also very grateful to Adam Gribble, Andrew Weatherbee and

Maqsood Bandesha for their cooperation during my research training.

I am also very grateful to Dr. Shahid Qamar, Dr. Ahmat Khurshid, Dr. Asloob

A. Mudassar, Dr. Yousaf Hamza, Dr. Sikandar Majid Mirza , Dr. Afshan Irshad,

Dr. Samina Jahandad, Dr. Kashif Sabeeh, Dr. Iftikhar Ahmed, Dr. Hafeezullah Janjua,

iv

and Dr. Manzoor Ahmed for their advice, guidance and invaluable comments during

my PhD. I would also appreciate my friends, particularly, Ms. Quratulain Safdar, Ms.

Sidra Nasir Qureshi, Ms. Attia Gul, Ms Saira Bibi, Ms. Farzana, Ms. Sidra Aslam,

Ms. Fatima Javed, Ms. Ammara Maryam, Ms. Saira Akhter, Ms. Sidra Afreen, Mr.

Yasir Ali, Mr. Sadiq Nawaz Khan and Mr. Safdar Ali for their cooperative and

encouraging behavior throughout my stay at PIEAS.

Most importantly, I am thankful to my family members for their unconditional

support and unmatchable love throughout my educational career. They helped me a

lot to reach this stage in my life.

Finally, I would like to thank Higher Education Commission of Pakistan for

their financial support under Indigenous Ph.D Fellowship Program for 5000 Scholars

and International Research Support Initiative Program (IRSIP).

v

Declaration of Originality

I hereby declare that the work contained in this thesis and the intellectual content of

this thesis are the product of my own work. This thesis has not been previously

published in any form nor does it contain any verbatim of the published resources

which could be treated as infringement of the international copyright law. I also

declare that I do understand the terms ‘copyright’ and ‘plagiarism’ and that in case of

any copyright violation or plagiarism found in this work, I will be held fully

responsible of the consequences of any such violation.

__________________

(Sumara Ashraf)

27 July 2017

PIEAS, Islamabad.

vi

Copyrights Statement

The entire contents of this thesis entitled Carousel Interferometer and its

Applications in Precise Phase Measurement by Sumara Ashraf are an intellectual

property of Pakistan Institute of Engineering & Applied Sciences (PIEAS). No

portion of the thesis should be reproduced without obtaining explicit permission from

PIEAS.

vii

Table of Contents

Dedicated to ................................................................................................................... ii

Acknowledgments........................................................................................................ iii

Declaration of Originality .............................................................................................. v

Copyrights Statement .................................................................................................... vi

Table of Contents ......................................................................................................... vii

List of Figures ................................................................................................................ x

List of Tables .............................................................................................................. xiv

Abstract ........................................................................................................................ xv

List of Publications ..................................................................................................... xvi

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

1.1 Historical Perspective of Light ......................................................................... 1

1.2 Types of Interferometers .................................................................................. 3

1.2.1 Michelson Interferometer ........................................................................ 4

1.2.2 Fizeau Interferometer .............................................................................. 5

1.2.3 Mach Zehnder Interferometer ................................................................. 5

1.2.4 Carousel Interferometer ........................................................................... 6

1.2.5 Sagnac Interferometer ............................................................................. 7

1.2.6 Rayleigh Interferometer .......................................................................... 8

1.3 Phase Modulation (PM) .................................................................................... 9

1.3.1 Generation of Phase Modulation ............................................................. 9

1.4 Application of Phase Modulator in Polarimetry ............................................. 12

1.5 Thesis Overview ............................................................................................. 13

2 Light Polarization................................................................................................... 14

2.1 Introduction .................................................................................................... 14

viii

2.2 Types of Polarization ...................................................................................... 15

2.3 Graphical Representation of Polarized Light ................................................. 16

2.3.1 Polarization Ellipse ................................................................................ 16

2.3.2 Poincare Sphere ..................................................................................... 18

2.4 Mathematical Formalism for Light Polarization ............................................ 19

2.4.1 Stokes Calculus ..................................................................................... 19

2.4.2 Mueller Calculus ................................................................................... 21

2.5 Mueller Matrix Decomposition ...................................................................... 22

2.5.1 Lu-Chipman Decomposition Method .................................................... 22

2.5.2 Decomposition of Non-depolarizing Mueller Matrix ........................... 27

2.5.3 Decomposition of Depolarizing Mueller Matrix ................................... 27

3 Carousel Interferometer to find Angle and Axis of Rotation ................................ 31

3.1 Introduction .................................................................................................... 31

3.2 Theory 32

3.2.1 Calculation of Residual and Nonlinearity in OPD ................................ 35

3.2.2 Calculation of Retrieved Angle of Rotation .......................................... 36

3.3 Parametric Compensation Method ................................................................. 37

3.3.1 Case -1 Without Glass Plates ................................................................ 37

3.3.2 Case -2 With Glass Plates ..................................................................... 39

3.4 Results and Discussion ................................................................................... 41

3.4.1 Case-1 Without Glass Plates ................................................................. 41

3.4.2 Case-2 With Glass Plates ...................................................................... 44

3.5 Summary ........................................................................................................ 47

4 Carousel Interferometer as a Phase Modulator ...................................................... 48

4.1 Introduction .................................................................................................... 48

4.2 Method ............................................................................................................ 49

4.2.1 Scheme-1 ............................................................................................... 49

ix

4.2.2 Scheme-2 ............................................................................................... 51


4.4 Summary ........................................................................................................ 56

5 Phase Modulator based Polarimeter to find Optical Rotation ............................... 57

5.1 Introduction .................................................................................................... 57

5.2 Material and Method ...................................................................................... 59

5.2.1 Sample Preparation ................................................................................ 59

5.2.2 Input Polarization States ........................................................................ 60

5.2.3 Experimental Setup ............................................................................... 62

5.2.4 Polar Decomposition Method ................................................................ 62

5.2.5 Calculation of Error and Sensitivity in Optical Rotation ...................... 64


5.3.1 Case-1 Determination of Optical Rotation without Scattering Media (μs

= 0/cm) ................................................................................................. 65

5.3.2 Case-2 Determination of Optical Rotation with Scattering Media (μs =

21.5/cm) ............................................................................................... 71

5.4 Summary ........................................................................................................ 76

6 Conclusion and Future Work ................................................................................. 77

7 References .............................................................................................................. 78

x

List of Figures

Figure 1-1 Schematic layout of Michelson interferometer where BS: beam splitter

M1& M2: two orthogonal mirrors .................................................................................. 4

Figure 1-2 Experimental layout of Fizeau interferometer: BS: beam splitter L: optical

lens to collimate the two beam to the two reflecting surfaces ....................................... 5

Figure 1-3 Optical setup for Mach Zehnder interferometer: BS1: first beam splitter

BS2: second beam splitter: M1 & M2: two reflecting mirrors ........................................ 6

Figure 1-4 A schematic representation of carousel interferometer, 𝛉: incident angle of

the beam, 𝛂: angle of rotation about the axis ‘o’, β: angle of the mirrors M3 & M4, γ:

angle of the mirrors M1 & M2, z: distance of the left side of beam from the axis of

rotation ‘o’ ..................................................................................................................... 7

Figure 1-5 Optical layout for Sagnac interferometer: BS: beam splitter M1, M2 & M3:

optical mirrors ................................................................................................................ 8

Figure 1-6 Optical setup for Rayleigh interferometer: L1: first lens used for

collimation L2: second lens used to recombine the beam at screen (focus point):

n1&n2: index of refraction of liquid/gas in the glass plate 1& 2 respectively ................ 9

Figure 1-7 A schematic representation of polarization Michelson interferometer, PBS:

polarization beam splitter, R1 & R2: quarter wave plates ............................................ 11

Figure 1-8 A schematic representation of photo-elastic modulator f: sinusoidal

modulation frequency, d & L: thickness and length of the modulator ........................ 12

Figure 2-1 A schematic representation of polarization ellipse, ψ: azimuth angle, ε:

ellipticity, Eoy & Eox: amplitude of the electric field vectors ....................................... 17

Figure 2-2 A schematic representation of Poincare sphere ......................................... 19

Figure 3-1 A schematic diagram of generalized six mirrored carousel interferometer,

where GP1 and GP2 is the glass plates, BS is the beam splitter and o (x, y) is the

arbitrary axis of rotation............................................................................................... 32

xi

Figure 3-2 Plot of change in OPD without glass plates as a function of rotation angle

...................................................................................................................................... 42

Figure 3-3 Plot of residual OPD at an optimized values without glass plates as a

function of angle of rotation; a, b, c, & d represent rotation ranges of ± 5o, ± 10

o ±

15o and ± 20

o respectively. .......................................................................................... 43

Figure 3-4 Plot of nonlinearity without glass plates at an optimized values as a


o ±

15o and ± 20

o respectively. .......................................................................................... 43

Figure 3-5 Plot of error in retrieved angle as a function of angle of rotation without

glass plates; a, b, c, & d, represent rotation ranges of ± 5o, ± 10

o ± 15

o and ± 20

o

respectively. ................................................................................................................. 44

Figure 3-6 Plot of change in OPD with glass plates as a function of rotation angle ... 44

Figure 3-7 Plot of residual OPD at optimized value with glass plates as a function of

angle of rotation; a, b, c, & d represent rotation ranges of ± 5o, ± 10

o ± 15

o and ± 20

o

respectively. ................................................................................................................. 45

Figure 3-8 Plot of nonlinearity with glass plates at optimized values as a function of

angle of rotation; a, b, c, & d represent rotation ranges of ± 5o, ± 10

o ± 15

o and ± 20

o

respectively. ................................................................................................................. 45

Figure 3-9 Plot of error in retrieved angle with glass plates as a function of angle of

rotation; a, b, c, & d, represent rotation ranges of ± 5o, ± 10

o ± 15

o and ± 20

o

respectively. ................................................................................................................. 46

Figure 4-1 Experimental layout of scheme-1 where s-polarized beams is passing

through glass plate (GP) kept at an initial angle of 5o BS: beam splitter, PBS:

polarization beam splitter, A: analyzer, MRS: motorized rotational stage, GP: glass

plate and L is the lens................................................................................................... 50

Figure 4-2 A schematic diagram of an interferometric setup where s- and p-polarized

beams are passing through glass plate (GP) kept at an initial angle of 5o. The p-

polarized beam is making a small angle with the s polarized beam. BS: beam splitter,

xii

PBS: polarization beam splitter, A: analyzer, MRS: motorized rotational stage and L

is the lens...................................................................................................................... 51

Figure 4-3 Plot of intensity modulation as a function of angle of rotation when single

s-polarized beam passed through the glass plate; (a) experimental plot, (b) theoretical

plot ............................................................................................................................... 54

Figure 4-4 Plot of intensity modulation as a function of angle of rotation when both s-

and p-polarized beams pass through the glass plate: (a) experimental plot: (b)

theoretical plot ............................................................................................................. 55

Figure 4-5 Plot of error in ∆𝐎𝐏𝐃 vs angle of rotation recorded for uncertainty of

+0.01o added to the initial values of φ1 and φ2 ............................................................ 56

Figure 5-1 Plot of optimum and rotated optimum input polarizations states plotted on

the Poincare sphere.(A) Optimum Stokes vectors (B) Rotated optimum Stoke vectors

Column (i) gives 3D views; column (ii) shows the front face of the Poincare sphere;

column (iii) shows the back face ................................................................................. 61

Figure 5-2 Plot of random input polarizations states plotted on Poincare sphere. (C)

Random1 (D) Random2 (E) Random3 Column (i) gives 3D views; column (ii) shows

the front face of the Poincare sphere; column (iii) shows the back face ..................... 61

Figure 5-3 Experimental scheme of Dual PEM polarimeteric system where PSG:

polarization state generator consists of a polarizer and QWP, PSA: polarization state

analyzer made up of two PEM (at 0o and 45

o above the horizontal) and a linear

polarizer (at 22.5oabove the horizontal), fc: chopper frequency where as f1 and f2 are

the modulation frequencies of two PEM ..................................................................... 63

Figure 5-4 Plot of optical rotation as a function of glucose concentrations without

scattering media (μs=0/cm) .......................................................................................... 65

Figure 5-5 Plot of error in the determination of optical rotation of 1.0 M glucose

without scattering media ( μs=0/cm) ............................................................................ 67





xiii



Figure 5-9 Plot of sensitivity in the determination of optical rotation of 1.0 M glucose

without scattering media ( μs=0 /cm). .......................................................................... 69

Figure 5-10 Plot of sensitivity in the determination of optical rotation of 0.5 M

glucose without scattering media ( μs=0 /cm) .............................................................. 69





Figure 5-13 Plot of optical rotation as a function of glucose concentrations with

scattering media (μs=21.5/cm) ..................................................................................... 71

Figure 5-14 Plot of error in the determination of optical rotation of 1.0 M glucose with

scattering media ( μs=21.5 /cm) ................................................................................... 72

Figure 5-15 Plot of error in the determination of optical rotation of 0.5 M glucose with

scattering media ( μs=21.5 /cm) ................................................................................... 73


with scattering media ( μs=21.5 /cm) ........................................................................... 73


with scattering media ( μs=21.5 /cm) ........................................................................... 74


glucose with scattering media ( μs=21.5 /cm) .............................................................. 74







xiv

List of Tables

Table 2-1 Types of Polarization ................................................................................... 16

Table 2-2 Stokes vector for different polarization states ............................................. 21

Table 3-1 Given parameters ......................................................................................... 35

Table 3-2 Optimized parameters without glass plates ................................................. 39

Table 3-3 Optimized parameters with glass plates ...................................................... 41

Table 3-4 Maximum residual OPD, nonlinearity and error in retrieved angle of

rotation without glass plates......................................................................................... 43


rotation with glass plates .............................................................................................. 46

Table 4-1 Parameters used for simulation in scheme-1 and scheme-2 ........................ 53

Table 5-1 Input polarization parameters ...................................................................... 62

Table 5-2 Optical rotation (OR) values without scattering media ............................... 66

Table 5-3 Optical rotation (OR) values with scattering media .................................... 72

xv

Abstract

Carousel interferometer is based on the measurement of optical path difference

generated via a rotating assembly of mirror system. However, the inherent

nonlinearity in the existing carousel interferometers limits the sensitivity and

resolution of rotation angle. In the present work, a generalized six mirrored carousel

interferometer with and without glass plate is optimized to address the nonlinearity.

The overall maximum nonlinearity and error in retrieved angle of rotation are

improved by three orders of magnitude for a rotation up to ± 20o. The significant

improvement of nonlinearity in six mirrored carousel interferometers enhance the

resolution of spectrometers, gas analyzing systems and rotational metrological

systems.

A glass plate based carousel interferometer is simulated and implemented

experimentally for optical phase measurement. The relative phase shift is introduced

between the orthogonally polarized beams when passed through a glass plate mounted

on a rotational stage with a resolution of 2.18 arcsec. The measured sensitivity of the

designed phase modulator is 13.8o per fringe which leads to the measurement of ≈

22,789th part of a wavelength.

A high frequency phase modulator based polarimeter is used for the

determination of optical rotation. Various concentrations of glucose are probed with

five different input Stokes vectors and the corresponding Mueller matrices are

determined. Using polar decomposition method, the optical rotation with and without

scattering media are measured as 0.862o

M⁄ and 0.766o

M⁄ respectively. The

obtained results for optimized input polarization states are in good agreement with the

theoretical model. Therefore, the proposed scheme may have the potential for

quantitative analysis of any chiral samples.

xvi

List of Publications

Thesis Publications

1. S. Ashraf, G. Hussain and M. Ikram, “Large angular range carousel

interferometer for spectroscopic applications” J. Optics and Spectroscopy, vol.

118, no. 5, pp. 829–833, 2015.

2. S. Ashraf, A. Gribble, A.Vitkin, I. Ahmad, G. Hussain and M. Ikram,

“Measurements of optical rotation of glucose with dual Photo-elastic

modulator: An application towards glucose monitoring” (Submitted).

3. S. Ashraf, S. Nawaz, G. Hussain and M. Ikram, “Polarization based carousel

interferometer as a phase modulator for optical sensing” (Ready to submit).

Other Publications

1. I Ahmad, M Ahmad, K Khan, S. Ashraf S Ahmad, M. Ikram, “Ex vivo

characterization of normal and adenocarcinoma colon samples by Mueller

matrix polarimetry” J. Biomed. Opt., vol. 20, no. 5, pp. 056012, 2015

Chapter 1: Introduction

1

1 Introduction

1.1 Historical Perspective of Light

Light has remained a fascinating topic for scientists in the field of mathematics and

physics extended from ancient times to the present day. In about 300 B.C Euclid

wrote Optica in which he proposed that light travels in straight line and mentioned

about the law of reflection. According to Euclid surroundings can be seen as if the

rays emerge from the eyes of the observer and fall on the object. He also gave an

important relationship between the apparent sizes of the objects and the angles of the

rays subtend at the eyes. The knowledge about reflection and refraction etc. was

advanced further with the description of Claudius Ptolemy (90-168), where he

explained the laws of refraction based on the observation of light passage through the

outer-atmosphere. He suggested an important relation between the angle of refraction

and the angle of incidence [1]. Subsequently, Al-Haytham 965-1039) made

tremendous contribution in the field of optics and light by introducing the use of

different geometrical mirrors i.e. spherical and parabolic. He explained the spherical

aberration in detail. Moreover, for the first time, he explained the vision in his famous

book ‘Kitab-al-Manazir” by saying that sight is due to light entering into the eyes

after reflection from the object rather than emerging from the eyes. In continuation to

“vision” explanation he introduced pinhole camera to the scientific community. It was

based on the passage of light through pinhole which falls on the screen thereby

forming an inverted image of an object [2]–[4]. Following Greeks, European

scientists Grossetest in 1220 and Roger Bacon about 50 years later contributed in the

field of optics and light by introducing geometry in the study of the light. Importantly

Roger Bacon mentioned the finiteness of the speed of light and proposed the use of

convex lenses for defective eyesight. He also described the phenomena of multicolor

formation observed in the sky just after the rain fall [5]–[8].

Different surrounding phenomena’s based on light interference were mostly

understood until 13th century. The origin of physiological description of human eyes

was explained first time by Leonardo da Vinci, a known artist and scientist [9].

Thereafter, Johannes Kepler (1571-1630), gave famous relation of light intensity


2

inverse square law. His attribution to the complete description of image formation on

retina, long-sightedness and short-sightedness are well known to the scientist

community. He also contributed in the formation of convergent/divergent lens

microscopes and telescopes. Kepler found total internal reflection but was unable to

find a satisfactory relationship between the incidence and refraction angles [5]. Light

traverses through different medium with different speed based on the medium “index

of refraction” which was first explained by Willebrord van Roijen Snell (1580-1626),

also known as Snell’s law [10]. A long known nature of light as wave was

theoretically supported with the description of diffraction of light by Fanscesco Maria

Grimaldi (1618-1663). In late 17th century, Issac Newton’s contribution

revolutionized the science of optics by constructing first reflecting telescope, a

solution to chromatic aberration. Physical optics stone was laid by Newton while he

presented his conclusion of sunlight splitting after passing through prism into

different colors. For the first time, he coined the particle nature of light based on the

fact that light traverses in straight line[11], [12]. In almost the same time, Christiaan

Huygens (1629-1695) explained in detail the propagation characteristics of light,

including double refraction in calcite discovered by Bartholinus. His theory “wave

nature of light” broke the spell of Newton’s particle theory of light. Also, Thomas

Young (1773-1829) continued with the wave nature of light and described different

phenomena i.e. interference, diffraction etc. A famous experimental evidence of wave

nature of light was demonstrated with young’s double slit set-up [13].

A different perspective of light called as “polarized light” was discovered by

Etienne Louis Malus (1755-1812) while sitting in a room and observing the light

entering from the window. In advancement to this knowledge David Brewster (1781-

1868) added a famous relationship for light waves. He observed that a polarized light

may be achieved by passing the ray through a transparent surface so that the refracted

ray makes an angle of 90° with the reflected ray. Later, it was named in the honor of

Brewster [14]. A nature of polarized light was then studied by Dominique Francois

Jean Argo (1786-1853) in projection perspective and he discovered transverse nature

of light by conducting experiment on polarization with his collaborator Augustin-Jean

Fresnel. This helped to resolve the debate that either light existed as particle or as a

wave. In the same time Poisson coined the experiment intending to support Newton’s


3

particle theory of light. A famous notation Poisson Spot was introduced to the French

Academy of Sciences [15].

In 18th Century, James Clerk Maxwell laid the stone of modern theory of

Electricity and Magnetism by presenting famous four equations with the conclusion

that the light is electromagnetic in nature. In these equations he also mentioned about

the constant speed of light in vacuum [16], [17]. Following Maxwell theory of

electromagnetism, in the early 20th century, Einstein put forward great theory of

“wave-particle duality of light” [18], [19]. He also published his famous observation

about the relative motion in 1905, where he suggested that the speed of light remains

constant irrespective of the relative motion of the observer. He has also added in his

credit, the first conceive of “Laser” by introducing theory of stimulation emission of

light. Later on with the discovery of Laser in 1959, scientific community had better

light source for the advancement of modern optics [20].

A long known phenomena of interference of light has also made significant

advances in the field of Optical Interferometry with the availability of precise light

source. Many interferometer were then designed and are still in development phases

which include Fizeau, Michelson, Mach-Zehnder, Rayleigh, Fabri-Perot, Sagnac and

Carousel interferometer. They all work on the same principle and are extensively used

in optical metrology, velocimetry, optometry, polarimetry and spectroscopy [21]–

[26]. Specifically interferometers have applications in optical testing, surface

profiling, remote sensing, rotation sensing, detection of gravitational waves,

characterization of polarized light and measurement of wavelength, refractive indices,

particle velocities, vibration amplitude, temperature distribution in plasmas and

angular diameter of stars [27]–[37].

1.2 Types of Interferometers

Interferometers may be divided into two classes known as amplitude division and

wave front division interferometers. The former class is the one, in which partial

reflectors or beam splitters are used to divide the amplitude of the incident beam into

two separate paths and recombine them i.e., the Michelson, Mach-Zehnder , Fizeau,

Sagnac and carousel interferometers whereas in the later one apertures are commonly

used to isolate portion of the primary wave front in order to obtain interference i.e.,


4

Rayleigh interferometer [38].The brief description of these interferometers are given

below.

1.2.1 Michelson Interferometer

The Michelson interferometer was introduced in 1887 in the "Michelson-Morley

Experiment", in response to a long discussion on the existence of Ether [39]. The

interferometer consists of two mirrors and a beam splitter (Figure 1-1). The incoming

light beam splits into two arms of the interferometer while on their return after

reflection both recombines at the same beam splitter where fringes are achieved as a

result of their interference. The interferometer has contributed towards the

development of special theory of relativity, optical coherence tomography, stellar

interferometry and spectroscopy [40]–[45].

Figure 1-1 Schematic layout of Michelson interferometer where BS: beam

splitter M1& M2: two orthogonal mirrors

For the first time, the gravitational wave was observed on September 14, 2015

based on laser interferometer gravitational wave observatory (LIGO) having two 4-

km arms of the Michelson interferometer with movable end mirrors [31]. The

M2

Screen

Laser source

M1

M2

BS

https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment

https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment


5

modified form of Michelson interferometer is called Twyman-Green interferometer in

which the two beams have a small relative angle between them.

1.2.2 Fizeau Interferometer

This type of interferometer consist of a light source, pinhole, beam splitter, lens and

two reflecting surfaces (Figure 1-2). Light after passing through the pinhole incident

on beam splitter whereby two collimated beams are directed on the reference and test

surfaces.

Figure 1-2 Experimental layout of Fizeau interferometer: BS: beam splitter L:

optical lens to collimate the two beam to the two reflecting surfaces

Thin layer of air between the two surfaces causes interference and fringes are

obtained at the screen. Such type of interferometer is commonly used for optical

testing to check the quality of optical component and as a guide for the manufacturer

of the optical component [27].

1.2.3 Mach Zehnder Interferometer

The interferometer uses laser as an input source and consist of two mirrors and two

beam splitters (Figure 1-3). The first beam splitter is used to split the incident light

into two arms of the interferometer while the second one is used to recombine them.

The fringe spacing is controlled by changing the angle between interfering beams and

their lateral separation.

Well known application of this type of interferometer are the measurement of

pressure, density, heat transfer and temperature changes in gases [46]–[48].

BS

Test surface

Reference surface

Imaging system

Laser source

Pinhole


6

Figure 1-3 Optical setup for Mach Zehnder interferometer: BS1: first beam

splitter BS2: second beam splitter: M1 & M2: two reflecting mirrors

1.2.4 Carousel Interferometer

The carousel interferometer (CI) is proposed by Kauppinen (Figure 1-4). It is made up

of a beam splitter and five plane mirrors M1-M5. The beam splitter and end mirror M5

are static while the four mirrors M1-M4 rotates about an axis of rotation of ‘o’. The

directions of the mirrors M1 and M2 are determined by the angle γ while angle β

determines the direction of mirror M3 and M4. The incident light falls at mirror M5

perpendicularly and obey the following conditions

2γ + 2β = θ.

Let the distance between M1 and M2 be the R and the one between M3, M4 and

M5 be the D as shown in Figure (1-4). The incident light beam falls on the beam

splitter at an angle θ thereby transmitted and reflected beam pass through two arms of

the interferometer.

The optical path difference generated due to rotation of the carousel about axis

of rotation o is given as

(OPD)CI = [R{cot(β + α) − cot(β − α)} − 2 cot(β + α) z(−α)…+ 2 cot(β − α) z(+α)], (1-1)

where as (OPD)CI is the optical path difference due to carousel interferometer.

z(−α) =

R2 cot(2β + 2α) − D(cos(α) + sin (α)tan (β + α))

tan(β + α) + cot(2β + 2α) ,

Screen

M1

M2 Path 2

Path 1

Laser source

BS1

BS2


7

z(+α) =

R2 cot(2β − 2α) − D(cos(α) − sin (α)tan (β − α))

tan(β − α) + cot(2β − 2α).

Figure 1-4 A schematic representation of carousel interferometer, 𝛉: incident

angle of the beam, 𝛂: angle of rotation about the axis ‘o’, β: angle of the mirrors

M3 & M4, γ: angle of the mirrors M1 & M2, z: distance of the left side of beam

from the axis of rotation ‘o’

The optical path difference x of the carousel interferometer as expressed in

Equation (1-1) is linear only for small range of rotation. For a given parameters the

nonlinearity values for ±5o, ±10

o angle of rotation are 3x10−1 and 2 respectively. It

is commonly known as swinging interferometer and commercially available in

GASMET DXTM-4000 gas analyzer system and many spectrometers. Carousal

interferometer are also employed for precise phase measurement.

1.2.5 Sagnac Interferometer

It consists of a light source, a beam splitter and three mirrors (Figure 1-5). Incoming

light after passing through beam splitter divides into two beams which traverse along

BS

Laser

source Detector

M1M2

M3M4

M5

R

D

o

a

b

cz


8

the closed loop but in opposite directions. The interferometer is simple and easy to

construct even with broadband light sources. As a result of rotation take place at an

angle θ normal to the plane of the interferometer with angular velocity ω causes a

phase between the two beams and given as

∅ =8πωΑ cos(θ)

λ c,

where Α be the area, λ is the wavelength and c be the speed of light.

The sensitivity of Sagnac interferometer is determined by area covered by the

closed loop multiplied by the number of cycles and there is a possibility to obtain a

sensitivity value which could determine the rotation of Earth about its axis [49]. It is

commonly used in inertial guidance system, ring laser gyroscope and many other

optical systems.

Figure 1-5 Optical layout for Sagnac interferometer: BS: beam splitter M1, M2 &

M3: optical mirrors

1.2.6 Rayleigh Interferometer

The Rayleigh interferometer uses wave front division to produce two beams from the

same light source, it consists of two lenses and pair of compensating glass plates

(Figure 1-6). Light after passing through slit made collimated by a lens. The

Laser source

Screen

M1

M2M3

Path 1

Path 2

BS


9

collimated beams are then passed through glass plates of different refractive indexes

and interference pattern are collected at the screen.

The interferometer is simple and commonly used to measure refractive index

of gases, molecular weight and concentration of molecules with high accuracy [50].

Figure 1-6 Optical setup for Rayleigh interferometer: L1: first lens used for

collimation L2: second lens used to recombine the beam at screen (focus point):

n1&n2: index of refraction of liquid/gas in the glass plate 1& 2 respectively

In all these interferometers phase shift, amplitude, frequency and phase

modulation are the key features of the optical signals which determine the interference

pattern. The mathematical descriptions of these are given as.

1.3 Phase Modulation (PM)

Phase modulation is an optical technique where carrier waves phase variations are

used to transmit information. In an interferometer, phase modulation is achieved with

phase shifters such as moving mirrors, rotating a grating, tilting a glass plate, rotating

half wave plates and rotating analyzers. These phase shifters are placed in one arm of

the interferometer and their movement can be controlled through piezoelectric

transducers (PZT). As a result of this movement, optical path difference (OPD) is

generated between the reference and the test arm of the optical beam and

corresponding phase difference is calculated [51]. The accuracy of these methods are

limited due to inherent characteristics of the PZT such as nonlinearities, hysteresis,

creep and thermal drift [52].

1.3.1 Generation of Phase Modulation

There are many techniques to generate phase modulation i.e., Phase shifting

interferometry, electro-optic, photo-elastic and liquid crystal variable retarder

techniques [53]–[56]. The device used to control the phase modulation is called phase

modulator.

Source

GlassPlate 1

Glass Plate 2

Focus

point

SlitL2

L1

n1

n2


10

1.3.1.1 Interferometric Modulator (IM)

In this type of modulator the basic principle is a splitting of incident light beam into

two orthogonally polarized beams. The phase difference of known phase shift is

created by placing the phase shifters in the reference arm which is recorded in terms

of intensity at the detector.

The simplest form of interferometric modulator is a polarization Michelson

interferometer. It consist of polarizing beam splitter (PBS), two optical isotropic

reflecting mirrors M1 , M2 and two quarter wave retarders R1 , R2 or equivalent 45o

Faraday rotator as shown in Figure (1-7). The PBS separates and recombines the two

orthogonally polarized beams. The incident light (containing both s and p polarized

beam) is traversed through the two arms of interferometer .The double passage of s

and p polarized beam through R1 and R2 changes their polarization by 90o.

The two orthogonal polarized beams then recombine at PBS and the output

intensity is measured. The polarization states are same as the incident light except in

90o azimuthal rotation. By the movement of one of two mirrors through a distance d,

the phase difference can be calculated is

∆∅ =4πnd

λ, (1-2)

where ∆∅ is the phase difference also named as retardance, n is the refractive index, d

is the distance moved by the mirror and λ is the wavelength of laser source.

Polarization modulation is achieved by using piezoelectric vibrating mirror

and the resulting sinusoidal retardance modulation is given by

Δ = Δm cos(ωt), (1-3)

where Δm is the modulation of any desired amplitude and ω is the modulated

frequency. The interferometer can be used to study the polarization states of the light

beam and can be a part of Mueller matrix polarimeter [32].


11

Figure 1-7 A schematic representation of polarization Michelson interferometer,

PBS: polarization beam splitter, R1 & R2: quarter wave plates

1.3.1.2 Photo-elastic Modulator (PEM)

In such type of modulator stress induces birefringence in optical medium thereby to

generate phase shift between orthogonally polarized beams. An optical medium could

be a transparent glass plate made of quartz crystal used to vary or modulate the

polarization states of light.

Photo-elastic modulators are made up of isotropic crystal and consist of fused silica

bar which becomes birefringent as a result of applied stress. The sinusoidal varied

stress is applied by attaching a piezo electric crystal at the end of the bar. This results

in time varying phase shift and is given as

∆∅ =2πd

λ f(t), (1-4)

f(t) = (nx − ny),

where d is the modulator thickness, nx & ny be the refractive index in the x and y

direction and f(t) be the sinusoidal modulating frequency. The device behaves like a

Screen

Laser source

M1

M2

PBS

R1

R2


12

phase retarder due to time varying sinusoidal modulation which varies at a frequency

of several thousand kHz [57]. Therefore, incident light at 45o entering the photo-

elastic modulator will create a phase difference of delta relative to x and y direction as

shown in Figure (1-8).

Figure 1-8 A schematic representation of photo-elastic modulator f: sinusoidal

modulation frequency, d & L: thickness and length of the modulator

1.4 Application of Phase Modulator in Polarimetry

Polarimetry is one of the basic means to investigate the light-matter interaction.

Complete description of polarized light is described by measuring four vectors called

as Stokes vectors. The Stokes vectors are geometrically represented on the “Poincare

sphere” introduced by Poincarè. Further, Mueller formulated his calculus based on the

work of Soleillet who reported that Stokes vectors are transformed linearly while

Perrin’s showed that the linear relations can be put in to a matrix form. Thus Mueller

matrix is a 4 x 4 matrix consisting of sixteen elements that are the signature of the

sample under test. For a complete description of the optical properties of the sample,

Mueller matrix of the sample is calculated and optical properties of the sample are

extracted using polar decomposition method [14]. Polarimeter consists of two parts. A

Linearly

polarized light

Fused Quartz

f = 50 kHz

Piezo-electric

Transducer

Modulator

d

L


13

set of elements that generates polarization states is called polarization state generator

(PSG) and the other one which analyzes the polarized light is called polarization

states analyzer (PSA). Polarimeters have a wide range of applications in bio photonics

and tissue engineering. However, the heterogeneity of biological tissues cause

multiple scattering of light and hence randomizing its polarization, which causes loss

of information about the polarization properties of the tissue [58].

As depolarization is dominant in biological tissue, conventional polarimeters

are impractical for the precise measurement of polarization properties. In fact

minimum of twenty-four measurements are required to construct Mueller matrix of

the biological tissues. A promising method to increase accuracy is to use a Mueller

matrix polarimeter based on phase modulator [59].

1.5 Thesis Overview

Chapter 1 includes the introduction and thesis overview while in chapter 2, theory and

mathematical formalism of polarized light is discussed. Chapter 3 deals with the

analysis of six mirrored carousel interferometer for the determination of angle and

axis of rotation. In chapter 4, two configurations of polarization carousel

interferometer as a phase modulator are proposed for phase measurement and their

experimental results are reported. In chapter 5, biomedical application of phase

modulator based polarimeter for the determination of optical rotation has been

presented. In chapter6, conclusion is drawn on the basis of results of this project and

future work is recommended.

Chapter 2: Light Polarization

14

2 Light Polarization

2.1 Introduction

Polarization is an important property of electromagnetic waves which arises due to

their transverse nature and is described by the patterns of electric field vector at a

fixed point in space as a function of time [14]. It can be classified into three major

groups; linear, circular and elliptical based on the magnitude of two field’s component

(E0x & E0y) and relative phase difference δ.

Many mathematical formalism have been proposed for the representation of

polarized light. The important and frequently used method is called Stokes parameters

in which intensity measurements are employed to describe total, elliptical as well as

un-polarized light beam [60]. The other one is coherency formalism introduced by

Wiener and Wolf in which partially polarized light is described by 2 x 2 coherency

matrix [61]. Importantly, coherency matrices are similar to Stokes parameters.

Another is Jones formalism which offered great simplicity in describing totally

polarized light and is applicable to homogeneous medium instead of depolarizing

medium (non-homogeneous) in which light coherency changed due to scattering [62]–

[65]. In order to resolve this problem Stokes-Mueller calculus is perhaps the most

suited one [66]–[70].

In Stokes-Mueller calculus; input Stokes vector is incident on investigated

sample and output Stokes vector are measured through a transformation matrix called

as Mueller matrix. Typically, all the polarimetric properties of incident polarized light

in its interaction with investigated sample are encoded in Mueller matrix. Many

decomposition techniques have been introduced to decouple the individual

polarimetric properties. Lu-Chipman decomposition is a widely used method now a

days [71]. In this method, Mueller matrix is decomposed into a product of three

matrices; a di-attenuator matrix followed by a retarder matrix and then followed by a

depolarizer matrix.


15

2.2 Types of Polarization

Light is an electromagnetic wave in which electric and magnetic fields oscillates

perpendicular to each other and to the direction of propagation of wave. Polarization

of light refers to the configurations traced by the tip of electric field vector as a

function of time in space.

Let light with field vector E(z, t) is travelling in z-direction then its

components oscillating along x and y direction are given as

Ex(z, t) = E0x cos(kz − ωt + δx) or E0xei(kz−ωt+δx), (2-1)

Ey(z, t) = E0y cos(kz − ωt + δy) or E0yei(kz−ωt+δy), (2-2)

where Ex(z, t) and Ey(z, t) are the components of electric field vectors in x and y

direction respectively, E0x and E0y are the complex amplitudes, k =2π

λ is the wave

number, λ is the wavelength, z is the distance covered in time t, ω is the angular

frequency of the beam and δx & δy are the phases in the x and y direction

respectively. Equation (2-1) and (2-2) can be written in more compact form as given

under

E(z) = [Ex

Ey] = [

E0xeiδx

E0yeiδy

] = [E0x

E0yeiδ],

where δ = δy − δx is the phase difference between the orthogonal components of the

electric field and is assumed to remain constant at all points in space. Therefore the

spatial variation of the electric field vector can be ignored and thus we get

E = [E0x

E0yeiδ]. (2-3)

This is the representation of Jones vector used to represent only totally polarized light

beam [62].

Polarized light is said to be linear, circular and elliptical based on the

magnitude of two field’s component (E0x & E0y) and relative phase difference δ . This

is summarized in the Table (2-1).


16

Table 2-1 Types of Polarization

Characteristic

of Polarized

Light

Symbol Criteria for

Amplitude

Criteria for

Relative

Phase

Linear horizontal ⟺ E0y= 0 N/A

Linear vertical ⇕ E0x= 0 N/A

Linear +45o ⤢ E0x = E0y δ = 0

Linear −45o ⤡ E0x = E0y δ = π

Right circular ⥀ E0x = E0y δ =π

2

Left circular ⥁ E0x = E0y δ = −π

2

2.3 Graphical Representation of Polarized Light

2.3.1 Polarization Ellipse

It is the description of a well-defined geometrical shape that is traced by the tip of

electric field vector along its propagation direction (Figure 2-1). For a fully polarized

light beam it can be described by manipulating Equation (2-1) & (2-2) the resulting

equation becomes as under

[Ex

E0x]2

+ [Ey

E0y]

2

− 2ExEy

E0xE0ycos(δy − δx) = sin2(δy − δx), (2-4)

[Ex

E0x]2

+ [Ey

E0y]

2

− 2ExEy

E0xE0ycos δ = sin2δ. (2-5)

The above equation is known as equation of ellipse and is fully describing the

polarized light, such an ellipse is called polarization ellipse. The major and minor axis

of the ellipse are described by the orthogonal components of electric field vector as

shown in Figure (2-1). Nature of polarized light (circular, linear or elliptical) is

determined by selecting particular values of two amplitudes (E0x & E0y) and relative

phase shift δ as expressed in Equation (2-5). However, the product term in Equation

(2-5) is an indication that ellipse is rotated through angle Ψ with respect to x-axis

where Ψ → (0,π

2). The rotational angle ψ of the polarization ellipse, also called as


17

azimuth, which is defined as to be the angle subtended by the major axis of ellipse

with the x-axis as shown in Figure (2-1). Mathematically

Ψ = tan−1 (E0y

E0x). (2-6)

Figure 2-1 A schematic representation of polarization ellipse, ψ: azimuth angle,

ε: ellipticity, Eoy & Eox: amplitude of the electric field vectors

Ellipticity (ε) is another parameter of the ellipse which measures the nature of

polarization states (circular, linear or elliptical) and also specifies the shape and

orientation of the polarization ellipse. Mathematically

ε = ±tan−1 (b

a), (2-7)

where ε → (−π

4,

π

4), b & a are the semi-minor and semi-major axis of the ellipse, ±

sign shows direction of polarization either to be left or right. When semi-minor axis of

the ellipse is zero (b=0) then ε = 0 and light is said to be linearly polarized whereas

when both semi-minor and semi-major axis of the ellipse are equal (a=b) then ε = 1

a

b

E

Eoy

X-axis

Y-axis

Minor-axisMajor-axis

Eox


18

the light is said to circularly polarized. For any other combination of a & b the light

will be ellipticaly polarized.

The other important parameter which describes the shape of ellipse and is

related with the ellipticity is called eccentricity. Mathematically

χ = √(1 − ε2), (2-8)

where χ → (0,1). For ε = 1, the eccentricity will be zero (χ = 0) and shape of the

ellipse will be circular whereas for ε = 0 the eccentricity will be equal to one (χ = 1),

ellipse will be converted to line and hence light will be linearly polarized. For (χ > 0)

it will represent an ellipse which goes thinner as χ increases [14].

It is noteworthy that the polarization ellipse is useful only for the description

of fully polarized light and not for un-polarized or partially polarized light. These

shortcomings are addressed with another versatile 3D graphical tool called as

Poincare sphere.

2.3.2 Poincare Sphere

Poincare sphere is a graphical tool where any polarization state of light (totally,

partially & un-polarized) can be represented on a sphere with unit radius centered at

the Cartesian coordinate’s axis. The Cartesian coordinate’s axis for Poincare sphere

are the normalized Stokes parameter (S1, S2, S3) of the represented polarized light. It

is also useful for the description of changes in polarization states resulting from the

interaction with an optical medium or device [14].

On Poincare sphere fully polarized light beams are represented on the surface

of the sphere, partially polarized light beams within the sphere and un-polarized light

beams at the origin. The upper and lower poles correspond to right and left circular

polarized states and the equator shows linear polarization states. The upper and lower

half hemispheres correspond to right and left elliptical polarized states (RE &LE).


19

Figure 2-2 A schematic representation of Poincare sphere

2.4 Mathematical Formalism for Light Polarization

2.4.1 Stokes Calculus

In 1852, Sir George Stokes introduced four parameters based on measured intensity of

polarized light named as Stokes parameters [60]. These are arranged in a 4 x 1 column

vector and useful for the characterization of total, partial and un-polarized states.

Mathematically Stokes vector can be derived by taking the time average of

Equation (2-5) and given as

[< Ex(t) >

E0x]2

+ [< Ey(t) >

E0y]

2

− 2< Ex(t) >< Ey(t) >

E0xE0ycos δ

= sin2δ, (2-9)

where < > is the time average taken over the whole ensemble and δ = δy(t) − δx(t)

is the relative phase shift. After doing some mathematical operations, Equation (2-9)

becomes

S3

S2

S1

LH

LV

LPLB

RC

LC

S3

S1

S2

RE

LE


20

(E0x2 + E0y

2 )2

= (E0x2 − E0y

2 )2+ (2E0xE0ycos δ)

2

+ (2E0xE0ysin δ)2,

(2-10)

S02 = S1

2 + S22 + S3

2,

where S0, S1, S2 and S3 are the observable polarization intensities and named as

Stokes parameters. Stokes vectors can be re- written as

S = [

S0

S1

S2

S3

] =

[

E0x2 + E0y

2

E0x2 − E0y

2

2E0xE0ycos δ

2 E0xE0ysin δ]

= [

I0o + I90o

I0o − I90o

I+45o − I−45o

IRC − ILC

]. (2-11)

In the above Equation (2-11) the first parameter represents the total intensity of light,

the second parameter S1 represents the difference in linear horizontal and vertical

polarization, S2 shows intensity difference between linear +45o and linear −45

o

polarization states and S3 indicates the intensity difference between left circular (LC)

and right circular (RC) polarization states.

Another important parameters which deals with the partial or un-polarized

beams in Stokes formalism is called degree of polarization (DOP) and defined as

DOP =√S1

2 + S22 + S3

2

S0 =

Ipol

Itot , 0 ≤ DOP ≤ 1, (2-12)

where DOP = 1 for total polarized light, DOP = 0 for un-polarized light and 0 <

DOP < 1 for partial polarized beam. Ipol and Itot stands for intensity of polarized

light and total light respectively.

Similarly, the relationship between degree of linear and circular polarization in

terms of Stokes vectors can be given as

DOLP =√S1

2 + S22

S0 , 0 ≤ DOLP ≤ 1, (2-13)

DOCP =S3

S0 , 0 ≤ DOCP ≤ 1. (2-14)

For various polarization states the Stokes vectors are summarized in Table (2-

2) and given below.


21

Table 2-2 Stokes vector for different polarization states

Polarization state Symbol Stokes parameters

Linear horizontal ⟺

Linear vertical ⇕

Linear +45o ⤢

Linear – 45o ⤡

Left circular ⥀

Right circular ⥁

Unpolarized ∗

2.4.2 Mueller Calculus

The Stokes vector S = [S0 S1 S2 S3]T completely describes the various

polarization states of light as explained in the above section. Whereas the interaction

of polarized light with the investigated sample is described by 4 x 4 matrix known as

Mueller matrix. Mathematically, Mueller matrix is represented by the following

expression,

Sout = MSinp , (2-15)

where Sout and Sinp represents output and input Stokes vectors respectively and M is

the corresponding Muller matrix as described below

Sout = [

m00 m01 m02 m03

m10 m11 m12 m13

m20 m21 m22 m23

m30 m31 m32 m33

] [

S0

S1

S2

S3

], (2-16)

M = [

m00 m01 m02 m03

m10 m11 m12 m13

m20 m21 m22 m23

m30 m31 m32 m33

], (2-17)

where (m00 …m33) denotes the elements of Mueller matrix while the first and second

number in the subscript represents the row and column number respectively. The

Mueller matrix M can be re-written as

M =1

4[ HH + HV + VH + VV HH + HV − VH − VV 2(PH + PV) − 4m11 2(RH + RV) − 4m11

HH − HV + VH − VV HH − HV + VH − VV 2(PH − PV) − 4m21 2(RH − RV) − 4m21

HP − HB + VP − VB HP − HB − VP + VB 2(PP − PB) − 4m31 2(RP − RB) − 4m31

HR − HL + VR − VL HR − HL − VR + VL 2(PR − PL) − 4m41 2(RR − RL) − 4m41 ]

.

(2-18)

1 1 0 0T

1 1 0 0T

1 0 1 0T

1 0 1 0T

1 0 0 1T

1 0 0 1T

1 0 0 0T


22

The above equation describes the elements of the Mueller matrix M determined

through various combination of linear horizontal (H), linear vertical (V), linear+45o

(P), linear −45o (B), right circular (R) and left circular (L) polarization states and

each measurement is represented by two letters. The first letter denotes incident

polarization states while the second one represents output polarization states.

It is to be noted that all the sixteen elements of the Muller matrix correspond

to the sixteen polarimetric properties of the sample which are real and dimensionless.

It means that all the polarimetric properties of sample under consideration are folded

in 4 x 4 Mueller matrix. Therefore, in order to extract all the individual polarization

properties of the sample such as depolarization, retardance and diattenuation the

decomposition of the measured Mueller matrix is required.

2.5 Mueller Matrix Decomposition

The physical interpretation of polarized light interaction with the sample is of great

importance as all the polarimetric properties of the sample under test are folded into

it. Thereby to unfold these polarization properties it should be interpreted by

decomposing it algebraically. There are two types of decomposition method named as

sum and product decomposition method. In sum decomposition method, depolarizing

Mueller matrix is decomposed into a sum of four non-depolarizing Muller matrices

[72] while Mueller matrix is decomposed into the product of three elementary

polarization components in product decomposition method. The product

decomposition method, also called as Lu-Chipman decomposition method, is a widely

used method for the extraction of individual polarimetric properties of the optical

medium. In this project we have used Lu-Chipman decomposition method for the

determination of optical rotation [14], [71].

2.5.1 Lu-Chipman Decomposition Method

In this method an arbitrary Mueller matrix Equation (2-17) is decomposed into a

product of three constitute matrices

where M∆, MR and MD are the depolarizer, retarder and diattenuation matrices

respectively. However, on the basis of the degree of polarization (Equation (2-12))

M = M∆MRMD, (2-19)


23

experimentally measured Mueller matrix of the sample is classified into two types

called as non-depolarizing Mueller matrix and depolarizing Mueller matrix

respectively. In order to determine these matrices prior information about retardance,

depolarization, polarizance and diattenuation are essential.

2.5.1.1 Di-attenuation

Di-attenuation can be determined from variations in the intensity transmittance of

incident polarized beam after interacting with the sample and defined as

where the value of di-attenuation (D) can be 0 or 1 while Tmax and Tmin be the

maximum and minimum transmitted intensity. From Mueller matrix M Equation (2-

17), the di-attenuation vector is obtained directly from the first row elements by

neglecting m00 and given as

where DH, D45 and DR be the linear horizontal, linear 45o and circular diattenuation

respectively. The magnitude of the diattenuation vector is

The total (DT), linear (DL) and circular diattenuation DC are defined as

D =Tmax − Tmin

Tmax + Tmin, (2-20)

D =1

m00[

m01

m02

m03

] = [

DH

D45

DC

], (2-21)

D =1

m00 √m01

2 + m022 + m03

2 = √DH2 + D45

2 + DC2. (2-22)

DT =1

m00 √m01

2 + m022 + m03

2, (2-23)

DL =1

m00 √m01

2 + m022, (2-24)

DC =m03

m00. (2-25)


24

Generally, the diattenuation Mueller matrix MD of a diattenuator calculated

from the first row of M Equation (2-17) can be written as

where D is the 3 x 1 diaatenaution vector expressed in Equation (2-21) and mD is the

sub-matrix of MD obtained from di-attenuation vector D as given under

where I3 is the 3 x 3 identity matrix while a & b be the scalar calculated from

magnitude of D Equation (2-21) and given as

2.5.1.2 Polarizance

The polarizance vector describes the changes in incident un-polarized light to

polarized light and define by the first column of Mueller matrix M Equation (2-17) by

omitting m00.

where P be the polarizance vector, PH, P45 and PC be the linear horizontal, linear

45oand circular polarization arising from incident un-polarized beam. The magnitude

of total, linear and circular polarizance are defined as

MD = [1 D T

D mD

], (2-26)

mD = aI3 + b(D . D T), (2-27)

a = √(1 − D2),

b =1 − √(1 − D2)

D2.

P =1

m00[

m10

m20

m30

] = [PH

P45

PC

], (2-28)

PT =1

m00 √m10

2 + m202 + m30

2, (2-29)

PL =1

m00 √m10

2 + m202,

(2-30)


25

2.5.1.3 Retardance

The phase change of incident polarized beam while interacting with sample is defined

as retardance. In retardance, the intensity of transmitted beam remains constant.

Unlike polarizance and diattenuation it is not directly measured from the Mueller

matrix. Mathematically

The magnitude of total, and linear retardance are calculated as

Typically, the Mueller matrix for a retarder can be written as

where 0 be the 3 x 1 null vector and mR be the 3 x 3 sub-matrix of MR obtained by

striking out first row and first column of MR.

The elements of mR can be determine using following relation

(mR)ij = δij cos R + aiaj (1 − cos R) + ∑ϵijk sin R , i, j = 1,2,3.

3

k=1

Conversely, the magnitude of retardance vector and direction of its fast axis

can be found from the retarder Mueller matrix MR and give as

PC =m30

m00. (2-31)

R = [RH

R45

RC

]. (2-32)

RT = √RH2 + R45

2 + RC2, (2-33)

RL = √RH2 + R45

2. (2-34)

MR = (1 0 T

0 mR

), (2-35)

R = cos−1 [

tr(MR)

2− 1], (2-36)


26

where ϵijk is the Levi-Civita tensor.

2.5.1.4 Depolarizance

Depolarization (∆) is caused by the scattering of light passing through a polarizing

medium. The depolarization matrix for a depolarizer with zero di-attenuation and

retardance is represented as a diagonal depolarizer matrix

where a, b and c denotes depolarization factors of depolarizer along its principal axis.

The average depolarization capability of a depolarizer called as depolarization power

is defined as

The most general expression for a depolarizer with polarizance can be written as

where m∆ is the 3 x 3 symmetric matrix obtained by omitting first row and first

column of depolarizer matrix M∆, P∆ represents polarizance vector as give under

where m be the 3 x 3 sub matrix obtained from Equation (2-17).

By knowing the above terminologies we would proceed with decomposition of

non-depolarizing and depolarizing Mueller matrix respectively. These are explained

as given below.

ai = 1

2 sin R ∑ ϵijk

3

j,k=1

(mR)jk , (2-37)

M∆ = [

1 0 0 00 a 0 00 0 b 00 0 0 c

], |a|, |b|, |c| ≤ 1, (2-38)

∆≡ 1 −|a| + |b| + |c|

3 , 0 ≤ ∆≤ 1. (2-39)

M∆ = [1 0 T

P∆ m∆

], (2-40)

P∆ =

P − mD

1 − D2, (2-41)


27

2.5.2 Decomposition of Non-depolarizing Mueller Matrix

The experimental determined Mueller matrix will be non-depolarizing when incident

polarized beam maintains its polarization (Equation (2-12)) after interacting with the

sample and depolarizing matrices will not be included in the decomposition Equation

(2-19). Thereby, retardence MR and di-attenuation MD matrices are the prominent

effects in the subsequent analysis of the investigated samples. The Mueller matrix for

non-depolarizing medium can be written as.

where MR is the Mueller matrix of retarder (Equation (2-35)) and MD is the Mueller

matrix for a di-attenuator (Equation (2-26).

In order to determine MD, the diattenuation vector is directly calculated from

Equation (2-21) then MD is found with the help of Equation (2-26). Further,

singularity of MD is checked. If the MD is non-singular |MD| ≠ 0 then Mueller matrix

of retarder can be obtained as

If MD is singular |MD| = 0 then infinite many solution of MR will exist and we select

the solution which gives minimum retardance called as minimum retardance

principle.

2.5.3 Decomposition of Depolarizing Mueller Matrix

In this case obtained Mueller matrix for an optical medium will be depolarizing, that

is, incident polarized light will not maintain its polarization and polarizing matrices

M∆ will be non-zero and involved in the subsequent analysis. Therefore, the Mueller

matrix for depolarizing medium can be written as a product of three constituent

matrices

where M∆, MR and MD are the depolarizer, retarder and diattenuation matrices

respectively. Further, to determine each individual matrix, MD can be determined

M = MRMD , (2-42)

MR = MD−1 M . (2-43)

M = M∆MRMD , (2-44)


28

from Equation (2-26), and check whether the diattenuation matrix is singular or not.

Consider the case when Mueller matrix of diattenuator is nonsingular| MD| ≠ 0.

Therefore Mueller matrix Equation (2-44) will be proceeded with an intermediate

matrix M′

where the M′ is free from diattenuation effects and only contains depolarizing and

retardance effects given as

M′= M∆MR

= [1 0 T

P ∆ m∆

] [1 0 T

0 mR

] = [1 0 T

P ∆ m∆mR

] = [1 0 T

P ∆ m′], (2-47)

where m′ is a 3x3 sub matrix of M′ and can be determined as

In Equation (2-48) the sub matrix m∆ can be determined as

m∆ = κ[m′(m′)T + (√λ1λ2 + √λ2λ3 + √λ3λ1 )I]−1

× [(√λ1 + √λ2 + √λ3 )m′(m′)T + √λ1λ2λ3 I],

(2-49)

where λ1 , λ2 and λ3 are the eigen values of matrix m′(m′)T, and κ =±1 is the same

as that of the determinant of m′. From Equation (2-48) and (2-49), mR can be

calculated as

Thus by knowing mR and m∆ the retardance matrix MR and depolarizing Mueller

matrix M∆ can be found respectively.

Now consider the case when Mueller matrix of diattenuator is singular

| MD| = 0, thereby sub matrix m′ as shown in Equation (2-48) will also be singular

thus singular value decomposition method on m′ is used.

M′= MMD

−1, (2-45)

M′= M∆MR

, (2-46)

m′ = m∆mR. (2-48)

mR = m∆−1m′. (2-50)


29

where V, D and UT are unitary matrix, diagonal matrix and conjugate transpose of V.

The Equation (2-48) can be re-written as

m′ = (v1, v2, v3) diag (√λ1 , √λ2 , √λ3) (u1, u2, u3), (2-52)

where vj and uj denotes column of V and U for (j = 1,2,3). From the above Equation

(2-52) it can be shown that

For situation when m′ is singular, the following three cases are considered:

1. If all the eigen values of m′ are zero (λ1 = λ2 = λ3 = 0), the retardance and

depolarizer matrix becomes

M∆ = [1 0 T

P ∆ m∆

], and MR = I. (2-55)

2. If λ1 ≠ 0 and λ2 = λ3 = 0 then m′ is

√λ1 v uT = (√λ1 v v

T) mR,

m∆ = √λ1 v vT =

m′(m′T)

√tr[m′(m′T)]. (2-56)

3. If λ3 = 0 and λ1 = λ2 ≠ 0, then

m∆ = (√λ1 + √λ2) [m′(m′T) + √λ1λ2 I]

−1m′(m′T), (2-57)

mR = ±(v1 u1T + u2

T +v1 × v2

|v1 × v2|+

(u1 × u2)T

|u1 × u2|). (2-58)

By using values of m∆ and mR, M∆ and MR can be calculated.

In the coming chapters, carousel interferometer uses to find angle and axis of

rotation and to address nonlinearity will be discussed. Furthermore, optimized glass

m′ = VDUT, (2-51)

m∆ = ±(√λ1v1 v1T, √λ2v2 v2

T, √λ3v3 v3T), (2-53)

mR = ±(v1 u1T, v2 u2

T, v3 u3T). (2-54)


30

plates based CI experimentally demonstrated for phase modulation and commercially

available phase modulator used for the determination of optical rotation is discussed.

Chapter 3: Carousel Interferometer to find Angle and Axis of Rotation

31

3 Carousel Interferometer to find Angle and

Axis of Rotation

3.1 Introduction

The measurements of the angle and axis of rotation in optical interferometry has a

wide range of applications in the field of metrology, optical sensing, optical signal

processing, biophotonics, high resolution spectroscopy and polarimetry [73]. Several

interferometers including the one proposed in this project has been reported in the

literature [74]–[79]. The class of swinging interferometer also called as the carousel

interferometer (CI) was designed by Kaupunin in 1995, in which rotational movement

is used to create the optical path difference (OPD) [80]. It also overcomes the tilt and

shearing problems as were in the Michelson interferometer. Nonlinearity in OPD is a

major issue in CI which constrained the rotation to a small angle. Currently, CI is

used in commercially available gas analyzing system and spectrometers[26], [30].

In the present work, a generalized six mirrored CI is proposed and its

nonlinearity is addressed [81]. In this configuration, a glass plate with optimized

thickness and refractive index is inserted in both arms of the interferometer. The

optimization of the glass plate is done through parametric compensation method

[81]–[84]. The OPD generated as a result of rotation (up to ±20o) has greatly reduced

nonlinearity. The maximum nonlinearity for ±5o, ±10

o, ±15

o and ±20

o rotation is

5.40 x 10−7, 8.53 x 10−6, 4.18 x 10−5 and 1.26 x 10−4 respectively for a particular

set of parameters. The interferometer can be used to find both angle and axis of

rotation simultaneously. It is noteworthy that the significant reduction in nonlinearity

with the insertion of the glass plates opens a new area in optics and spectroscopy. The

basic structure of the interferometer makes it useful for high resolution Fourier

transform spectroscopic applications. Other important applications can be for

rotational stages, seismic pickups and calibration of meteorological standards.


32

In this chapter, we will discuss in detail the theory and mathematical model of

the proposed interferometeric setup. Moreover, the parametric compensation method,

residual in OPD, nonlineairty in OPD and simulation results are also discussed.

3.2 Theory

The optical setup consists of six mirrors M1-M6, a beam splitter (BS) and a pair of

optimized glass plates as shown in Figure 3-1. The BS and end-mirrors (M5-M6) are

fixed while mirrors M1-M4 are mounted on a disc that rotates about an axis o(x, y).

Figure 3-1 A schematic diagram of generalized six mirrored carousel

interferometer, where GP1 and GP2 is the glass plates, BS is the beam splitter

and o (x, y) is the arbitrary axis of rotation


33

The axis of rotation o(x, y) is set perpendicular to the plane of the paper. The

incident light beam from He-Ne laser makes an angle θ1with BS. The transmitted

beam from BS propagates towards the left arms of the interferometer, first incident at

M4, M3 and finally at right angle on the end mirror M5. After reflection from mirror

M5, the reflected beam traverses back the same path and reaches at BS.

Similarly, the reflected beam from BS propagates towards the right arms of the

interferometer, first incident at M1, M2 and finally at right angle on end mirror M6

from where it traverses back the same path and reaches at BS. The two beams reaches

simultaneously at the BS thus interfere and fringes are obtained. The end-mirrors M5,

M6 makes an angle α with vertical line a1a5, which gives the interferometer an extra

flexibility if beam angle is changed. The five mirror interferometer can also be

accomplished by choosing an appropriate value of θ1or even of β.

By rotating assembly of mirrors M1-M4 in anti-clockwise direction the OPD in

the left hand arm decreases while it increases in right hand arms and vice versa. OPDs

in right and left hand sides of the interferometer are given as

OPDR(θr) = −2a1a3 sin(θ1 + θ2 + 2β + θr) sin(2β)…

+ 2a1a5 sin(θ1 + 2β + θr) sin(2β)…

−4x sin(2β) sin(θr

2)sin (θ1 + 2β +

θr

2)…

−4y sin(2β) sin (θr

2) cos (θ1 + 2β +

θr

2), (3-1)

OPDL(θr) = −2a1a3 sin(θ1 + θ2 + 2β − θr) sin(2β)

+ 2a1a5 sin(θ1 + 2β − θr) sin(2β)

−4x sin(2β) sin(θr

2)sin (θ1 + 2β −

θr

2)…

−4y sin(2β) sin (θr

2) cos (θ1 + 2β −

θr

2), (3-2)

The change in optical path difference as a function of rotation angle is


34

∆OPD(θr) = −4a1a3 cos(θ1 + θ2 + 2β) sin(2β) sin(θr)…

+ 4a1a5 cos(θ1 + 2β) sin(2β) sin(θr)…

−4x cos(θ1 + 2β) sin(2β)(1 − cos (θr))…

−4y cos(θ1 + 2β) sin(2β) sin(θr), (3-3)

where ∆OPD(θr) represents the net change in optical path difference without glass

plates which varies nonlinearly with angle of rotation θr. Whereas β is the angle

between mirrors M1-M2 and M3-M4, θ2 is the angle between vertical line a1a5 and

mirrors M2, θ1 is the incident angle of the beam and a1a3 is the length of the mirror

M4. The x and y are the generalised coordinates of the axis of rotation o(x, y). The

change in OPD is linear only for a few degrees of rotation. As the angle of rotation

increases, it deviates significantly from the linear behavior called nonlinearity.

A pair of transparent glass plates with thickness d and refractive index n is

placed at an angle θ in both arms of the interferometer in order to overcome the

nonlinearity in OPD. The change in OPD due to glass plate is

δgp(θr) = +d [n − cos(θ − γ1(θr) + θr)

cos(γ1(θr))…

−n − cos(θ − γ2(θr) − θr)

cos(γ2(θr))]

2

λ ,

(3-4)

γ1(θr) = sin−1 [1

nsin(θ + θr)] ,

γ2(θr) = sin−1 [1

nsin(θ − θr)] ,

where γ1(θr) and γ2(θr) are the angle of refraction within the glass plates, λ is the

wavelength of He-Ne laser beam, d and n are the optimized thickness and refractive

index. The glass plate OPD varies nonlinearly with angle of rotation θr. The resultant

OPD of the interferometer

∆OPDgp(θr) = [∆OPD(θr) + δgp(θr) ], (3-5)


35

∆OPDgp(θr) = [−4a1a3 cos(θ1 + θ2 + 2β) sin(2β) sin…

+ 4a1a5 cos(θ1 + 2β) sin(2β) sin(θr)…

− 4x cos(θ1 + 2β) sin(2β)(1 − cos(θr))…

− 4y cos(θ1 + 2β) sin(2β) sin(θr)…

+ d [n − cos(θ1 − γ1(θr) + θr)

cos(γ1(θr))…

−n − cos(θ2 − γ2(θr) − θr)

cos(γ2(θr))]]

2

λ,

(3-6)

where ∆OPDgp(θr) represents the change in optical path difference with glass plates

Thus, the resultant OPD is the superposition of two nonlinear OPD, made almost

linear through optimization. The optimization is done by employing parametric

compensation method. By putting values from Equation (3-3) and Equation (3-4) into

Equation (3-5), the combined OPD obtained at the output of the interferometer is

The parameters used for simualtion are shown in Table (3-1).

Table 3-1 Given parameters

Given parameters Symbol Values

Wavelength of light λ 632.8 х 10-6

mm

Dimension of mirror M3, M2 a1a3 200 mm

Distance between the mirror M2 and M6 a1a5 160 mm

Refractive index n 1.84489

Initial angle of laser beam θ1 45o

Angle between a1a5 and mirror M2 θ2 110o

Angle between the-mirrors M1 & M2 and

M3 & M4 β 40

o

3.2.1 Calculation of Residual and Nonlinearity in OPD

The OPD with glass and without glass plates as expressed in Equation (3-6) and (3-3)

is not perfectly linear. There is a small deviation from ideal linear behavior called

residual OPD. The residual OPD without glass plates is σ (θr) can be expressed in

terms of the number of fringes and is

σ (θr) = [2

λ(∆OPD(θr)) − mθr], (3-7)


36

in the above Equation (3-7), σ (θr) shows residual OPD without glass plates,

2

λ∆OPD(θr) is the actual number of fringes while mθr represents the ideal number of

fringes and m indicates number of fringes per unit angle of rotation. The Equation (3-

7) can be re-written as

σ (θr) =2

λ[∆OPD(θr) − (mθr)

λ

2].

The residual OPD is used to find nonlinearity in OPD. The amount of

nonlinearity in OPD without glass plates is

L(θr) = (σ (θr)

mθr) , (3-8)

the nonlinearity L(θr) is directly proportional to residual OPD and inversely related

with mθr the ideal number of fringes. Nonlinearity also illustrates the sensitivity of an

interferometer in terms of rotational angle.

A similar expression of residual OPD and nonlinearity in OPD with glass

plates

σ (θr)gp =2

λ[∆OPDgp(θr) − (mθr)

λ

2], (3-9)

L(θr)gp = ( σ (θr)gp

mθr) . (3-10)

To reduce the amount of nonlinearity (L(θr)gp), residual OPD should be

minimized. For minimization of residual OPD with and without glass plates,

optimized values of x, y and d are used employing parametric compensation method.

3.2.2 Calculation of Retrieved Angle of Rotation

The retrieved angle of rotation is determined by dividing the change in OPD

with m (λ

2) where m represents the number of fringes per unit angle of rotation and λ

is the wavelength of the laser beam. The retrieved angle of rotation without glass plate

is

θret =∆OPD(θr)

m(λ2)

, (3-11)


37

where ∆OPD(θr) is the change in OPD without glass plates as expressed in Equation

(3-3).

The retrieved angle of rotation with glass plates is

θretgp = (

∆OPDgp(θr)

m(λ2)

) , (3-12)

where ∆OPDgp(θr) is the change in OPD with glass plates as shown in Equation (3-

6).

3.3 Parametric Compensation Method

3.3.1 Case -1 Without Glass Plates

In this method, the optimized values of x & y can be obtained for which the overall

residual OPD as expressed in Equation (3-7) could be minimized [84] and is as

∑(σ (i))2

θr

i=1

= min∑[2

λ[∆OPD(i) − (mi)

λ

2]]

2

,

n

i=1

∑(σ (i))2

θr

i=1

= min∑4

λ2[[∆OPD(i) − (mi)

λ

2]]

2

.

θr

i=1

(3-13)

By putting the values of ∆OPD(i), from Equation (3-3) in the above equation, it takes

the form as given by

∑(σ (i))2

θr

i=1

= min∑4

λ2[[Aix + Biy + Di]]

2

,

θr

i=1

(3-14)

where Ai & Bi are the coefficient of x & y, and Di is a constant term.

Ai = −4 cos(θ1 + 2β) sin(2β)(1 − cos(i)),

Bi = −4y cos(θ1 + 2β) sin(2β) sin(i),

Di = −(4a1a3 cos(θ1 + θ2 + 2β) sin(2β) sin(i)…

+ 4a1a5 cos(θ1 + 2β) sin(2β) sin(i) − miλ

2).


38

For minimization of the residual OPD, partial derivative with respect to x & y of

Equation (3-14) are set to zero and is

∂

∂x∑(σ (i))

2

θr

i=1

= 0,

∂

∂y∑(σ (i))

2

θr

i=1

= 0.

Substituting values of ∑ (σ (i))2θr

i=1 from Equation (3-14) in the above equations

∂

∂x[[Aix + Biy + Di]]

2= 0, (3-15)

∂

∂y[[Aix + Biy + Di]] = 0. (3-16)

Equation (3-15) and (3-16) leads to simultaneous linear equations,

x∑(Ai)2

θr

i=1

+ y∑(AiBi)

θr

i=1

= −∑(AiDi)

θr

i=1

, (3-17)

x∑(BiAi)

θr

i=1

+ y∑(Bi)2

θr

i=1

= −∑(BiDi)

θr

i=1

. (3-18)

Equation (3-17) and (3-18) can be written in matrix form

[ ∑(Ai)

2

θr

i=1

∑(BiAi)

θr

i=1

∑(AiBi)

θr

i=1

∑(Bi)2

θr

i=1 ]

[ x

y]

= −

[ ∑(AiDi)

θr

i=1

∑(BiDi)

θr

i=1 ]

,

[ x

y]

= −

[ ∑(Ai)

2

θr

i=1

∑(BiAi)

θr

i=1

∑(AiBi)

θr

i=1

∑(Bi)2

θr

i=1 ] −1

[ ∑(AiDi)

θr

i=1

∑(BiDi)

θr

i=1 ]

. (3-19)


39

By using Equation (3-19) optimized values of x & y are calculated based on

simulations in MATLAB and shown below in Table 3-2.

Table 3-2 Optimized parameters without glass plates

Angle of rotation Optimized value of x

(mm)

Optimized value of y

(mm)

-5o to +5

o 0.00000 20.348

-10o to +10

o 0.00000 20.275

-15o to +15

o 0.00000 20.155

-20o to +20

o 0.00000 19.993

3.3.2 Case -2 With Glass Plates

In this method, the optimized values of x, y and d can be obtained for which the

overall residual OPD as expressed in Equation (3-9) could be minimized and is as

∑(σ (i))2

θr

i=1

= min∑[2

λ[ ∆OPDgp(i) − (mi)

λ

2]]

2

,

n

i=1

∑(σ (i))2

θr

i=1

= min∑4

λ2[[ ∆OPDgp(i) − (mi)

λ

2]]

2

.

θr

i=1

(3-20)

By putting the values of ∆OPDgp, from Equation (3-6) in the above equation, it takes

the form as given by

∑(σ (i))2

θr

i=1

= min∑4

λ2[[Aix + Biy + Cid + Di]]

2

,

θr

i=1

(3-21)

where Ai,Bi, Ci are the coefficient of x, y, d and Di is a constant term.

Ai = −4 cos(θ1 + 2β) sin(2β)(1 − cos(i)),

Bi = −4y cos(θ1 + 2β) sin(2β) sin(i),

Ci = d

[ n − cos(θ1 − γ1(i) + i)

cos(γ1(i))−

n − cos(θ2 − γ2(i) − i)

cos(γ2(i)) ]

,


40

Di = −(4a1a3 cos(θ1 + θ2 + 2β) sin(2β) sin(i)…

+ 4a1a5 cos(θ1 + 2β) sin(2β) sin(i) − miλ

2).

For minimization of the residual in OPD, partial derivative with respect to x, y and d

of Equation (3-21) are set to zero and is

∂

∂x∑(σ (i))

2

θr

i=1

= 0,

∂

∂y∑(σ (i))

2

θr

i=1

= 0,

∂

∂d∑(σ (i))

2

θr

i=1

= 0.

Substituting values of ∑ (σ (i))2θr

i=1 from Equation (3-21) in the above equations

∂

∂x[[Aix + Biy + Cid + Di]]

2= 0, (3-22)

∂

∂y[[Aix + Biy + Cid + Di]] = 0, (3-23)

∂

∂d[[Aix + Biy + Cid + Di]] = 0. (3-24)

Equation (3-22), (3-23) and (3-24), leads to three simultaneous linear equations,

x∑(Ai)2

θr

i=1

+ y∑(AiBi)

θr

i=1

+ d∑(AiCi)

θr

i=1

= −∑(AiDi)

θr

i=1

, (3-25)

x∑(BiAi)

θr

i=1

+ y∑(Bi)2

θr

i=1

+ d∑(BiCi)

θr

i=1

= −∑(BiDi)

θr

i=1

, (3-26)

x∑(CiAj)

θr

i=1

+ y∑(CiBi)

θr

i=1

+ d∑(Ci)2

θr

i=1

= −∑(CiDi)

θr

i=1

. (3-27)

Equation (3-25), (3-26), and (3-27), can be written in matrix form


41

[ ∑(Ai)

2

θr

i=1

∑(BiAi)

θr

i=1

∑(CiAi)

θr

i=1

∑(AiBi)

θr

i=1

∑(Bi)2

θr

i=1

∑(CiBi)

θr

i=1

∑(AiCi)

θr

i=1

∑(BiCi)

θr

i=1

∑(Ci)2

θr

i=1 ]

[ x

y

d]

= −

[ ∑(AiDi)

θr

i=1

∑(BiDi)

θr

i=1

∑(CiDi)

θr

i=1 ]

,

where as

[ x

y

d]

= −

[ ∑(Ai)

2

θr

i=1

∑(BiAi)

θr

i=1

∑(CiAi)

θr

i=1

∑(AiBi)

θr

i=1

∑(Bi)2

θr

i=1

∑(CiBi)

θr

i=1

∑(AiCi)

θr

i=1

∑(BiCi)

θr

i=1

∑(Ci)2

θr

i=1 ] −1

[ ∑(AiDi)

θr

i=1

∑(BiDi)

θr

i=1

∑(CiDi)

θr

i=1 ]

. (3-28)

By using Equation (3-28) optimized values of x, y and d are calculated based

on simulations in MATLAB and shown below in Table 3-3.

Table 3-3 Optimized parameters with glass plates

Angle of

rotation

Optimized value of x

(mm)

Optimized value of y

(mm)

Optimized value of d

(mm)

-5o to +5

o 0.00000 20.348 77.877

-10o to +10

o 0.00000 20.275 77.338

-15o to +15

o 0.00000 20.155 76.475

-20o to +20

o 0.00000 19.993 75.337

3.4 Results and Discussion

3.4.1 Case-1 Without Glass Plates

The known and optimized parameters used for simulation in MATLAB are

summarized in Table 3-1 and 3-2. The change in OPD without glass plates is

expressed in Equation (3-3) and the simulated result as a function of rotation angle is

shown in Figure 3-2. The OPD is not perfectly linear for ±5o,±10

o, ±15

o and ±20

o.


42

However, there is a very small deviation from the ideal straight line, namely as

residual OPD, which we have used to find nonlinearity in OPD. Both residual OPD

and nonlinearity in OPD without glass plates are calculated by using Equation (3-7)

and (3-8) while their simulated results are shown in Figure 3-3 and Figure 3-4. For a

rotation range of ±20o the maximum value of residual OPD is 2335 fringes while

nonlinearity in OPD is 1.25 x 10−2. The values for the other ranges are shown in

Table 3-4.

Figure 3-2 Plot of change in OPD without glass plates as a function of rotation

angle

Figure 3-5 shows the plot of error in retrieved angle calculated by subtracting

the angle of rotation from retrieved angle of rotation expressed in Equation (3-27).

The maximum value of error in retrieved angle is 1.5 x 10−1 for ±20o rotation as

shown in Table 3-4. The maximum residual OPD, nonlinearity and error in retrieved

angle of rotation without glass plates for all ranges of rotation are summarized in

Table 3-4 as given under.


43

Figure 3-3 Plot of residual OPD at an optimized values without glass plates as a


o

± 15o and ± 20

o respectively.

Figure 3-4 Plot of nonlinearity without glass plates at an optimized values as a


o

± 15o and ± 20

o respectively.


rotation without glass plates

Range of

rotation

Maximum residual

OPD (fringes) Maximum

nonlinearity

Maximum error in

retrieved angle of rotation

-5o to +5

o 37 7.7 х 10

-4 2.4 x 10

-3

-10o to +10

o 295 6.20 х 10

-3 19 x 10

-3

-15o to +15

o 991 7.02 х 10

-3 66 x 10

-3

-20o to +20

o 2335 1.25 х 10

-2 158 x 10

-3


44

Figure 3-5 Plot of error in retrieved angle as a function of angle of rotation

without glass plates; a, b, c, & d, represent rotation ranges of ± 5o, ± 10

o ± 15

o

and ± 20o respectively.

3.4.2 Case-2 With Glass Plates

The change in OPD with glass plates is expressed in Equation (3-6) and the simulated

result as a function of rotation angle is shown below.

Figure 3-6 Plot of change in OPD with glass plates as a function of rotation angle

The OPD is relatively linear for ±5o,±10o, ±15

o and ±20

o rotation ranges.

The very small deviation of OPD from linear behavior called the residual OPD is


45

calculated using Equation (3-9) and the result is shown in Figure 3-8. The nonlinearity

is calculated using Equation (3-10) and simulated result is shown in Figure 3-9.

Figure 3-7 Plot of residual OPD at optimized value with glass plates as a function

of angle of rotation; a, b, c, & d represent rotation ranges of ± 5o, ± 10

o ± 15

o


Figure 3-8 Plot of nonlinearity with glass plates at optimized values as a function

of angle of rotation; a, b, c, & d represent rotation ranges of ± 5o, ± 10

o ± 15

o


It is evident from Figure 3-7 the residual OPD with glass plates shows

reduction of two to three orders of magnitude as a result of superposition of two

nonlinear function (OPD of glass plates and OPD of the interferometer) that makes

them linear. However, there is significant deviation for large of rotation. Thus, the


46

insertion of glass plates improves linearity of the interferometer as shown in Figure 3-

9.

Figure 3-9 Plot of error in retrieved angle with glass plates as a function of angle

of rotation; a, b, c, & d, represent rotation ranges of ± 5o, ± 10

o ± 15

o and ± 20

o

respectively.

Figure 3-9 shows the plot of error in retrieved angle, calculated by subtracting

the angle of rotation from retrieved angle of rotation expressed in Equation (3-12).

The maximum value of error in retrieved angle is 1.1 x 10−3 for ±20o rotation as

shown in Table 3-5.

The maximum residual OPD, nonlinearity and error in retrieved angle of

rotation with glass plates for all ranges of rotation are summarized in Table 3-5 as

given under.


rotation with glass plates

Range of

rotation

Maximum residual

OPD (fringes) Maximum

nonlinearity

Maximum error in

retrieved angle of rotation

-5o to +5

o 0.018 5.40 х 10

-7 1.2 x 10-6

-10o to +10

o 0.58 8.53 х 10

-6 39 x 10

-6

-15o to +15

o 4.20 4.18 х 10

-5 283 x 10

-6

-20o to +20

o 16.56 1.26 х 10

-4 1121 x 10

-6


47

Consequently, the axis of rotation with and without glass plates can be

determined by putting two values of retrieved angles from Equation (3-11) and (3-12)

in Equation (3-6) and Equation (3-3).

By comparing the values of residual OPD, nonlinearity in OPD and error in

retrieved angle of rotation with and without glass plates from Table 3-4 and 3-5, it is

concluded that they are two to three orders of magnitude smaller in the presence of

glass plates and this is the maximum limit achieved in literature. The significant large

reduction of nonlinearity in a large range of rotation with glass plates, thus generates

large OPD and makes interferometer suitable for spectroscopic applications.

3.5 Summary

In this chapter, we have introduced a technique for the precise measurements of the

angle and axis of rotation based on six mirrored carousel interferometer as shown in

Figure (3-1). A time varying optical path difference (OPD) is generated as a function

of rotation by revolving assembly of mirrors M1-M4 that are mounted on a disc. The

maximum value of nonlinearity in OPD without glass plates is 1.25 x 10-2

for ±20o

rotation. The nonlinearity in OPD is further improved by inserting a pair of optimized

glass plates in both arms of the interferometer. The combined OPD is made almost

linear through parametric compensation method. The maximum value of nonlinearity

in OPD achieved with glass plates is 1.26 x 10-4

for ±20o rotation. The significant

decrease in nonlinearity with glass plates opens up the new world for high resolution

Fourier transform spectroscopic applications. In conclusion, the main advantage of the

proposed setup is simple geometry, compact form and cost-effective as it is built of

only plane mirrors.

.

Chapter 4: Carousel Interferometer as a Phase Modulator

48

4 Carousel Interferometer as a Phase Modulator

4.1 Introduction

Polarization based phase shifting interferometry is a widely used method for optical

testing, surface characterization and optical sensing [85]. Many interferometers such

as radial polarization, point diffraction, shearing, scatter plate sagnac, heterodyne and

polarization Michelson have been reported earlier [86]–[90]. In all these

interferometers incident light is split into orthogonally polarized beams (s- and p-

polarized) through polarization beam splitter (PBS). The phase delayed between s-

and p-polarized beam is generated in different ways i.e., by rotating a polarizer, tilting

a glass plate, by translating a mirror or grating with a piezo electric transducer [32],

[91]–[95]. These optical interferometers have many advantages but also face

challenges such as mechanical stability, complex geometry, high cost and quick

response to changes in polarization states. There is still a need to develop more

versatile interferometer to overcome the above mentioned problems for phase

measurement.

In the present work, we designed and studied two schemes for optical phase

measurement based on carousel interferometer. The phase delay between orthogonally

polarized beams was created using a glass plate mounted on a rotational stage with

resolution 2.18 arc sec. In scheme-1, the glass plate was placed in the path of s-

polarized beam and rotated through 1o while in scheme-2 it was placed in the path of

both s-and p-polarized beams and rotated through 30o. A relative phase shift of more

than 9 fringes was obtained in scheme-1 and less than two fringes in scheme-2. The

observed difference in relative phase shift was due to geometry of s-and p-polarized

beams. Therefore, the angle dependent sensitivity of scheme-1 is more compare to

scheme-2. Thus scheme-2 seems good for the measurements of polarization

parameters.


49

4.2 Method

In this work, two schemes of polarization based phase shifting carousel interferometer

were designed. A linearly polarized light at 45o from a He-Ne laser source was used

as an input source. The basic principle involved is splitting of linearly polarized beam

into orthogonal polarizations (s- and p-polarized) using a polarizing beam splitter

(PBS) and then introducing a phase shift between two beams through a glass plate. A

quartz strip of thickness 5.85 mm and refractive index 1.4570 was used as glass plate

which was mounted on a rotational stage with resolution of 2.18 arc sec. In scheme-1,

the glass plate was placed in the path of s-polarized beam only while in scheme-2 the

glass plate was placed in both s- and p-polarized beams path. In both schemes, the

two returned beams after reflection from the end mirrors passed through the beam

splitter (BS) and analyzer. Finally, the interference between the beams resolved along

the transmission axis of the analyzer was achieved and recorded through a computer

program. However, rotation of the glass plate will change the differential phase

between the two orthogonally polarized beams in both schemes. The varying phase

change thus generates different polarization states at the output.

4.2.1 Scheme-1

In this configuration input light at 45o from a He-Ne laser source passes through a

Faraday isolator to avoid back reflection of the beam from entering back into the laser

as shown in Figure4-1. After this the beam goes into beam splitter and then pass

through a polarizing beam splitter which generates two beams with orthogonal

polarizations (s-and p-polarized). The transmitted beam (p-polarized) falls on the

mirror M1 while the reflected beam (s-polarized) falls on the mirror M2 after passing

through glass plate. The glass plate is placed at an initial angle of 5o. Upon reflection

from the mirror M1 and M2 the two beams are re-combine at BS and pass through the

analyzer. The rotation of the glass plate introduces a phase shift between the two

beams. The optical path difference OPD(θr) due to rotation of the glass plate is

OPD1(θr) = 2d [n − cos(φ1 − α)

cos(α)−

n − cos(φ1 − β(θr) + θr)

cos(β(θr))] , (4-1)


50

where OPD1(θr) is the change in OPD for scheme-1, φ1 is the incidence angle at the

glass plate ( 5o), β(θr) and α are the angle of refraction inside the glass plate with

and without rotation, θr is the angle of rotation, d is the thickness and n is the

refractive index of the glass plate.

Figure 4-1 Experimental layout of scheme-1 where s-polarized beams is passing

through glass plate (GP) kept at an initial angle of 5o BS: beam splitter, PBS:

polarization beam splitter, A: analyzer, MRS: motorized rotational stage, GP:

glass plate and L is the lens

The phase difference is given by

α = sin−1 [1

nsin(φ1)] ,

β(θr) = sin−1 [1

nsin(φ1 + θr)],


51

where λ is the wavelength of the laser source.

The total intensity of s-and p-polarized , beams at the output of the analyzer is

where θ is the angle of analyzer, I1 and I2 are the intensities of s-and p-polarized

beams respectively, ∅1(θr) defines the phase difference between s-and p-polarized

beams.

4.2.2 Scheme-2

Figure 4-2 shows an experimental setup of scheme-2 where two orthogonally

polarized light beams pass through the rotating glass plate which is placed at an angle

of 5o.

Figure 4-2 A schematic diagram of an interferometric setup where s- and p-

polarized beams are passing through glass plate (GP) kept at an initial angle of

5o. The p-polarized beam is making a small angle with the s polarized beam. BS:

beam splitter, PBS: polarization beam splitter, A: analyzer, MRS: motorized

rotational stage and L is the lens

∅1(θr) =2π

λ (OPD1(θr)) , (4-2)

ID1 = (I1 + I2 + 2√I1I2 cos (∅1 (θr)) cos2(θ), (4-3)


52

The two beams in the horizontal plane are at very small angle (0.03)o so that

when they pass through the same glass plate a differential phase shift is generated.

These beams after reflection from the end mirrors M2 and M3 again pass through the

glass plate to introduce a similar amount of phase shift between them.

Finally, the two returned beams pass through the beam splitter and analyzer.

At the output of the analyzer interference occurs between the electric components

resolved along the transmission axis of the polarizer. Rotation of the glass plate

changes the differential phase between the two orthogonally polarized beams. The

varying phase change generates different polarization states at the output. The glass

plate is mounted on a rotational stage with a resolution of 2.18 arc sec. With such a

resolution a phase shift of several thousands of wavelength can be introduced and

fringes produced are recorded at the detector.

The expression of the relative OPD change is given as

where

in the above equations, OPDs(θr) and OPDp(θr) are the relative OPD due to s-and p-

polarized beams, θr is the angle of rotation , d is the thickness and n is the refractive

index of the glass plate. φ1 is the incidence angle at the glass plate for the reference

beam, while φ2 defines the incidence angle at the glass plate for the slightly tilted

beam. β(θr) & γ(θr) are the angle of refraction inside the glass plate for the

reference and tilted beam respectively. The angle between the two beams is defined

OPDs(θr) = 2d (n − cos(φ1 − α)

cos(α) −

n − cos(φ1 − β(θr) + θr)

cos(β(θr))), (4-4)

OPDp(θr) = 2d (n − cos(φ2 − α)

cos( α)−

n − cos(φ2 − γ(θr) + θr)

cos(γ(θr))−), (4-5)

α = sin−1 [1

nsin(φ1)] ,

β(θr) = sin−1 [1

nsin(φ1 + θr)] ,

γ(θr) = sin−1 [1

nsin(φ2 + θr)],


53

by δ = (φ1−φ2). Thus subtracting Equation (4-4) and Equation (4-5), we achieve

relative OPD between the two beams as

where ∆OPD(θr) is the change in OPD for scheme-2. The resulting phase difference

is

The phase difference for the current scheme ∅2(θr) is recorded by a detection system

and a computer in the form of fringes.

The total intensity of s-and p-polarized interfering beams at the detector is [2]

where θ is the angle of analyzer, I1and I2 are the intensities of s- and p- polarized

beams respectively and ∅2(θr) defines the phase difference between s-and p-

polarized beams.

The theoretical simulation of intensity recorded at the detector in scheme-1

and 2 (as expressed by Equation (4-3) & (4-8)) carried out in MATLAB using

parameters as shown in Table4-1.

Table 4-1 Parameters used for simulation in scheme-1 and scheme-2

Parameters Symbols Scheme-1 Scheme-2

Thickness of glass plate d 5.85 mm 5.85 mm

Refractive index of glass plate n 1.4570 1.4570

Angle of rotation θr 1o 30

o

Angle between s- and p-polarized beams δ NA 0.03o

Initial angle of s- polarized beam φ1 5o 5

o

Initial angle of p-polarized beam φ2 NA 5.03o


Experimentally and theoretically generated intensity plots as a function of angle of

rotation for scheme-1 are shown in Figure 4-3(a) and 4-3(b). The theoretical

simulation of intensity as expressed in Equation (4-3) was carried out in MATLAB. A

quartz strip of thickness 5.85 mm and refractive index 1.4570 is used as glass plate.

∆OPD(θr) = OPDs(θr) − OPDp(θr), (4-6)

∅2(θr) =2π

λ(∆OPD(θr)). (4-7)

ID2 = (I1 + I2 + 2√I1I2 cos(∅2(θr))) cos2(θ), (4-8)


54

The data of Figure 4-3 (a) & 4-3 (b) was recorded when initial angle of the beam was

5o with the normal to the glass plate. More than eight fringes were moved against a

rotation of 1o.

Figure 4-3 Plot of intensity modulation as a function of angle of rotation when

single s-polarized beam passed through the glass plate; (a) experimental plot, (b)

theoretical plot

The differential phase change as expressed in Equation (4-7) was used to find

theoretically generated intensity for scheme-2. Figure 4-4 (a) and 4-4 (b) shows the

experimentally and theoretically generated intensity plots for scheme-2 as a function

of angle of rotation. The data of Figure 4-4 (a) and 4-4 (b) was recorded when two

beams were at an initial angle of 5o and 5.03

o with the normal to the glass plates

respectively. Less than two fringes were moved for a rotation of 30o.

More than eight fringes have been obtained when single beam passes through

the glass plate against a rotation of 1o while less than two fringes were observed for

scheme-2. This is due to less angle dependent sensitivity of the scheme 2 contrary to

scheme 1.


55

Figure 4-4 Plot of intensity modulation as a function of angle of rotation when

both s- and p-polarized beams pass through the glass plate: (a) experimental

plot: (b) theoretical plot

On the basis of above result it is concluded that experimental scheme-2 can be

used as a characterization of polarization parameters. The sinusoidal modulation of

any desired phase can be achieved by varying the phase sinusoidally,

where t is time, ∅(t) is the sinusoidal phase modulation ,∅0 is the required phase and

ω is the modulation frequency. Thus carousel phase modulator can be used as an

economical alternative of photo-elastic modulator.

Effect of uncertainty in the initial angles φ1 and φ2 were also simulated. An

uncertainty of +0.01o was added to the initial values of φ1 and φ2 and it is found that

uncertainty in initial angle has very small effect on the change in OPD and phase. The

uncertainty/error in ∆OPD is plotted as number of fringes recorded and shown in

Figure. (4-5). However, our system is sensitive to the angle δ = (φ1−φ2).

∅(t) = ∅0 cos(ωt), (4-9)


56

Figure 4-5 Plot of error in ∆𝐎𝐏𝐃 vs angle of rotation recorded for uncertainty of

+0.01o added to the initial values of φ1 and φ2

4.4 Summary

We have proposed and demonstrated two schemes for the polarization based phase

shifting carousel interferometer. In both schemes, phase difference was generated

between the two orthogonally polarized beams using a glass plate mounted on a

rotational stage with resolution of 2.18 arc sec. In scheme-1, the glass plate was

placed in the path of s-polarized beam only while in scheme-2 it was placed in the

path of both s- and p-polarized beam. In Scheme-2 a small relative phase shift

equivalent to 1.8 fringes are obtained between the two beams against a rotation of

30o while more than eight fringes are obtained in scheme 1 against rotation 1

o.This

is due to the fact that scheme-2 has less angle dependence sensitivity unlike scheme-1.

A controllable small relative phase shift as a result of large angle of rotation makes

this technique suitable for the characterization of polarization parameters.

Chapter 5: Phase Modulator based Polarimeter to find Optical Rotation

57

5 Phase Modulator based Polarimeter to find

Optical Rotation

5.1 Introduction

Diabetes mellitus is a common chronic systemic disease associated with the failure of

either production or proper maintenance of insulin, a hormone which regulates

glucose levels in the body. If improperly managed long term side-effects include heart

disease, blurred vision, slow healing and kidney failure [96]. Therefore, frequent

surveillance of blood glucose levels in the body for proper glucose therapies is of

paramount importance. The conventional methods for monitoring glucose levels

include haemoglobin A1c tests, fasting plasma glucose test and oral glucose tolerance

test [97]–[99]. These methods are based on the finger-pricking for collection of blood

and have low patient compliance due to their invasive nature. Many diabetic patients

fail to monitor their blood levels as frequently as recommended due to these invasive

method. A non-invasive diagnostic technique for glucose monitoring would eliminate

the need for painful blood extraction while hopefully improving patient compliance

and correspondingly lowering the risk of long term side effects that has severe impact

on the quality of life [98], [100].

A significant amount of research into the development of non-invasive glucose

monitoring techniques has been done in several fields. Optical methods are non-

invasive in nature, therefore, many approaches being investigated for the development

of optical technology for the monitoring of glucose levels. These optical methods

include metabolic heat conformation studies, optical absorption and scattering

methods, near and mid-infrared spectroscopy, Raman spectroscopy, photo acoustic

spectroscopy, spectrophotometer, optical coherence tomography (OCT) and single-

beam Z-scan technique [101]–[108]. Clinically none of these techniques has been

approved for monitoring glucose level. Moreover, the accuracy in the measurements

is still under investigation though, for some techniques it is demonstrated in

acceptable range. Therefore, tremendous amount of work is required to improve the


58

existing modalities and also needed either for improvement of these existing

modalities or development of new methods.

Polarized light contains very rich functional and morphological information

about biological tissues and may be used as a non-invasive diagnostic modality.

Particularly, tissue polarimetry provides information about the most prominent

polarization properties such as depolarization, birefringence, diattenuation and optical

activity [109]–[113]. Pierangelo et al investigated polarimetric response of uterine

cervix samples both in vivo and ex-vivo [114]. Jacques et al. have developed portable

polarization camera for imaging of superficial skin tissues and guiding surgical

excision of skin cancers [115]

The Mueller matrix of a sample contains information about all of its

polarimetric properties [66], [116]–[119]. Thus, Mueller matrix polarimetry is being

investigated for use in a wide range of biomedical applications including glucose

monitoring, early cancer and precancerous conditions, healthy and cancerous human

cell characterization etc [120]–[122]. Glucose has a chiral (asymmetric) structure

which causes circular retardance (optical rotation or optical activity). This

characteristic is due to the difference in refractive index for right and left circularly

polarized light causing a phase shift between them. Many researchers have worked to

use the Muller matrix polarimetry to extract optical rotation from turbid samples

containing glucose [123]–[125]. Ghosh et al. utilized Muller Matrix polarimetry in the

forward detection geometry to detect reasonable estimates for optical rotation of

sucrose sample[116]. Firdous et al. measured the glucose concentrations in turbid

media [126]. Pham et al. measured the optical rotation angle (173o M⁄ ) for glucose

(C6H12O6) sample containing polystyrene sphere (PST) suspension [121].

Mathematically,

Rc = [α]λT LC, (5-1)

where [α]λT, is the specific rotation of the sample at a temperature T and wavelength λ,

L is the photon path length in the sample and C is the concentration of the turbid

sample.


59

However, the heterogeneity of biological tissues causes multiple scattering of

light thereby randomizing its polarization. This depolarization results in a loss of

information about the polarization properties of the tissue. In other words, detected

signal is a combination of the rotation of polarized light by glucose molecules and

scattering by biological tissues. Depolarization is dominant in biological tissue, and

the optical rotation from glucose must be extracted from this highly depolarized

background. Thus, the precise measurement of glucose concentrations in biological

tissue is a challenge and requires a very sensitive polarimeter with high accuracy and

minimal sensitivity to measure signal to noise ratio. A promising method is to use a

Mueller matrix polarimeter based on photo elastic modulators (PEMs). The dual PEM

polarimetry system combines polarization modulation with phase-sensitive

synchronous detection to accurately measure all four Stokes parameters within about

5milli-sec, with no mechanically moving parts [127]–[129][48], [107], [108]. Recent

studies have shown that dual PEM can determine the Mueller matrix of turbid media

more precisely when a specific set of input polarizations states is used to probe the

sample e.g. cube on the Poincare sphere [130].

In the present work, we used a dual PEM polarimetry system to determine

optical rotation of glucose in turbid phantoms (aqueous solutions of glucose mixed

with polystyrene spheres suspensions). The samples with different glucose

concentration were interrogated with five different sets of input polarization states

called as the optimum, rotated-optimum, random-1, random-2 and random-3. From

the Muller matrix of each sample we obtained optical rotation values for each states.

We found that the value of optical rotation of glucose with optimum set of input

polarization states was more accurate and precise compared to other states. Thus dual-

PEM polarimetry system with optimum input polarization states seems good to be

used as a diagnostic tool for non-invasive blood glucose detection.

5.2 Material and Method

5.2.1 Sample Preparation

Glucose C6H12O6 is a monosaccharide sugar and the principal source of energy in our

body. Due to its chiral nature it rotates linear polarized light either clockwise or

anticlockwise. We used D+ glucose (Sigma-Aldrich Inc.) as a chiral media and PST


60

sphere suspension with diameter of 0.96 μm as a scattering media. Samples of varying

D-glucose concentration (1.0 M, 0.5 M, 0.25 M, and 0.125 M) with and without

scattering media were prepared and analyzed in the forward detection geometery.

5.2.2 Input Polarization States

Polarized light is fully described by four measureable quantities grouped into 4x1

vector called stoke vectors and is

S = [

IQUV

],

where I represents the total intensity of the light beam, Q; intensity difference between

0oand 90

o polarization states, U; intensity difference between +45

o and −45

o

polarization states and V; intensity difference between right and left circular polarized

light. The change in polarization states of light in its interaction with any material can

be represented by a matrix equation

Sout = MSin, (5-2)

where Sin represent input polarization states of incident beam, while Sout represents

change in polarization states of light when it interacts with material

M = Sout ( Sin)−1, (5-3)

for n > 4 least square best fit on M is given as

M = Sout ( Sin)+, (5-4)

where the Moore-Penrose pseudo inverse of Sin is

( Sin)+ = ( Sin)T [Sin( Sin)T] −1. (5-5)

Five different input polarization states (optimum, rotated optimum, random-1,

random-2 and random-3) were used to probe the sample under test. These states form

a cube, when plotted on the Poincare sphere. On the Poincare sphere the linear

polarizations states are represented on the equator while circular ones on the poles.

The phase error regions are shown with red circle where Q, U, or V = 0. A robust

Mueller matrix with Dual PEM can be determined with optimum input polarization


61

states [131]. These states form the vertices of a cube when plotted on the Poincare and

lies away from the phase-error regions. The rotated-optimum Stoke vectors also form

a cube but their vertices lies into the phase-error regions as shown in Figure 5-1.

Figure 5-1 Plot of optimum and rotated optimum input polarizations states

plotted on the Poincare sphere.(A) Optimum Stokes vectors (B) Rotated

optimum Stoke vectors Column (i) gives 3D views; column (ii) shows the front

face of the Poincare sphere; column (iii) shows the back face

Figure 5-2 Plot of random input polarizations states plotted on Poincare sphere.

(C) Random1 (D) Random2 (E) Random3 Column (i) gives 3D views; column (ii)

shows the front face of the Poincare sphere; column (iii) shows the back face


62

Table 5-1 Input polarization parameters

Input Stokes vectors Mean separation angle Stokes element near

phase error region

Optimum 102.3o 0

Rotated-optimum 102.6o 8

Random-1 94.2o 2

Random-2 97.8o 3

Random-3 90.4o 5

5.2.3 Experimental Setup

A schematic diagram of the experimental setup is shown in Figure 5-3. A Laser beam

(660 nm, 937 Hz) is chopped for intensity modulation. It then passes through a linear

polarizer and quarter wave plate (mounted on a computer controlled rotating stage)

which constitute the polarization state generator (PSG) and determine the input

polarization state that impinges on the sample. After interacting with sample, light

transmitted in the forward direction geometry is collected with a lens and passes

through two photo elastic modulators (PEM) and a fixed linear polarizer at 22.5o.

These PEMs and polarizer constitute the polarization state analyzer (PSA). PEMs are

resonant devices with a time-varying birefringence. The first PEM (oriented at 0o to

the horizontal) modulates at 47 kHz and the second (45o to the horizontal) at 42 kHz.

The peak linear retardance for both PEMs was set at 2.045 rad. After passing through

the PSA the light hits an avalanche photo-detector and the intensity is converted to an

electrical signal which passes through three lock-in amplifiers. The first one uses a

reference from the optical chopper to recover the overall intensity of the signal. The

second one measures Stoke vector V at f1 of first PEM and Q at 2f1 of the same PEM,

whereas the third one measure Stoke vector U using reference from second PEM at

2f2. Finally the values measured by the all amplifier are passed to computer which

controls the overall system.

5.2.4 Polar Decomposition Method

Polar decomposition method is a robust mathematical tool to determine polarization

properties of any medium. Five sets of input Stokes vectors called as optimum,

rotated-optimum, random-1, random-2 and random-3 were shine on the samples.


63

Thereafter, the corresponding sets of output Stokes vectors were measured. Based on

the input and output Stokes vectors a Muller matrix was formulated (as shown in

Equation (5-4)) which was further processed by polar decomposition method for the

extraction of optical rotation of samples. In this method, any arbitrary Muller matrix

can be decomposed into a product of three constitute matrices given as under

M = M∆MRMD , (5-6)

M∆ = Muller matrix of a depolarizer

MR = Muller matrix of a retarder

MD = Muller matrix of a diattenuator

The optical rotation found is

Rc = tan−1 ((MR32

−MR23)

(MR22+ MR33

)), (5-7)

where MRi,j denotes the ith and jth element of the retarder matrix.

Figure 5-3Experimental scheme of Dual PEM polarimeteric system where PSG:

polarization state generator consists of a polarizer and QWP, PSA: polarization

state analyzer made up of two PEM (at 0o and 45

o above the horizontal) and a

linear polarizer (at 22.5oabove the horizontal), fc: chopper frequency where as f1

and f2 are the modulation frequencies of two PEM


64

5.2.5 Calculation of Error and Sensitivity in Optical

Rotation

To calculate error and sensitivity in optical rotation five input Stokes vectors were

used. Each consist of 8 input states, Smnin (m =1 to 8), and was repeatedly measured 10

times (n = 1 to 10) with dual PEM, for a total 80 input Stokes vectors. The sample

was than probed with all of these input Stokes vectors Smnin and the corresponding

output Stokes vectors Smnout were measured. Thus each data set contained a total of 160

Stokes vectors measurements, 80 input Stokes vectors (S1,1in to S8,10

in ) and 80 out Stokes

vectors (S1,1out to S8,10

out ). The average Muller matrix M(0) was calculated using

Equation (5-4) by taking the average of eight input vectors S(avg)in =

[s(1,avg)in … . . s(8,avg)

in ] and corresponding output Stokes vectors S(avg)out =

[s(1,avg)out … . . s(8,avg)

out ]. Afterwards, optical rotation value, Rc(0), was calculated

employing polar decomposition method on the average Muller matrix M(0).

Further, we calculated 160 “erroneous” Muller matrices M(z) (for z = 1:160)

to check the stability of average Muller matrix M(0). Thus employing polar

decomposition method on M(z), optical rotation values Rc(z) were calculated.

The difference in optical rotation values is given as

δ(Rc) = Rc(z) − Rc(0),

and the sensitivity in optical rotation is

Sensitivity in optical rotation = (∑ < δ(Rc) >160

z=1

160). (5-8)

The error in the measurement of the optical rotation and difference in optical rotation

values were calculated as given under

∆(Rc) = Rc(z) − Rc,

where Rc the theoretical calculated optical rotation as shown in Equation (5-1), error

in optical rotation is

Error in optical rotation = (∑ <∆(Rc)>

160z=1

160). (5-9)


65

The error in optical rotation is the measure of agreement of experimentally calculated

optical rotation with the theoretical ones where as sensitivity in optical rotation shows

the reproducibility of experimental calculated optical rotation values.


5.3.1 Case-1 Determination of Optical Rotation without

Scattering Media (μs = 0/cm)

Figure (5-4) shows the plot of optical rotation values calculated for five input

polarization states against different glucose concentrations. Straight line represents the

optical rotation values calculated using Equation (5-1). It is evident from the graph

that for different glucose concentrations optical rotation increases linearly with

concentrations.

Figure 5-4 Plot of optical rotation as a function of glucose concentrations without

scattering media (μs=0/cm)


66

The theoretical and experimentally measured optical rotation values without

scattering media are shown in Table 5-2.

Table 5-2 Optical rotation (OR) values without scattering media

Conc. of

samples

(Mole)

Theoretical

value of OR

(Deg)

Experimental value of optical rotation

for input Stokes vectors (Deg)

Optimum Rotated

optimum Rnd-1 Rnd-2 Rnd-3

1.0 M 0.7535 0.7664 0.4193 0.7831 0.7718 0.9258

0.5 M 0.3817 0.3786 0.2091 0.3637 0.3763 0.3936

0.25 M 0.1909 0.1638 0.1715 0.1096 0.2018 0.026

0.125 M 0.0954 0.0875 0.1193 0.0304 0.0065 0.0268

Figure (5-5) to Figure (5-8) shows the plot of error in the value of optical

rotation as determined by Equation (5-9). The error is minimum for optimum Stokes

vectors whereas it is higher for the other states. This is due to the fact that optimum

Stokes vectors form the vertices of a cube when plotted on the Poincare sphere and

avoid the phase error regions. Therefore, the optimum input polarization states are

reliable to determine optical properties of the sample under consideration. However,

high error in optical rotation for the rotated optimum states is attributed to its vectors

which approaching the phase error regions when plotted on the Poincare sphere.

These phase errors are due to 0o or 180

o phase difference in signal and PEM

reference frequency which causes noise fluctuations. The results of the random Stokes

vectors can be arbitrary as they have vectors that are generated using random

polarization states generator.

The sensitivity as expressed in Equation (5-8) are plotted for different samples

against each set of input polarization states and shown in Figures (5-9) to (5-12). It is

notable that the sensitivity in optical rotation is also minimum for optimum set while

it’s on higher end for random 3 among random Stokes vectors. Thus optimum Stokes

vectors with Dual PEM is capable to measure optical rotation with high accuracy and

precision.

(

a)


67


without scattering media ( μs=0/cm)




68



Figure 5-8 Plot of error in the determination of optical rotation of 0.125 M

glucose without scattering media ( μs=0/cm)


69


glucose without scattering media ( μs=0 /cm).


glucose without scattering media ( μs=0 /cm)


70






71

5.3.2 Case-2 Determination of Optical Rotation with

Scattering Media (μs = 21.5/cm)

Figure (5-13) represents the plot of optical rotation against different glucose

concentration. The optical rotation value obtained for optimum Stokes vector is

(0.85o M⁄ ) which is close to theoretical values in contrary to the other states. The

increase in optical rotation for scattering medium (15% M⁄ ) is attributed to the

increase in path length which causes multiple scattering due to the addition of

polystyrene microsphere suspension with particle size of 0.96 μm.

Figure 5-13 Plot of optical rotation as a function of glucose concentrations with

scattering media (μs=21.5/cm)

Therefore, it is inferred that a robust Muller matrix can be obtained with

optimum input polarization states that results in precise and accurate measurement of

optical properties of the sample. The theoretical and experimentally calculated optical

rotation values with scattering media are shown in Table 5-3.


72

Table 5-3 Optical rotation (OR) values with scattering media

Conc. of

samples

(Mole)

Theoretical

value of OR

(Deg)

Experimental value of optical rotation

for input Stokes vectors (Deg)

Optimum Rotated

optimum Rnd-1 Rnd-2 Rnd-3

1.0 M 0.867 0.8620 0.6298 0.8691 0.8878 1.252

0.5 M 0.433 0.4183 0.1057 0.2683 0.4031 0.4585

0.25 M 0.216 0.2354 0.2258 0.5354 0.0981 0.3142

0.125 M 0.108 0.1214 0.0825 0.0139 0.3382 0.6026

Figures (5-14) to (5-21) shows error and sensitivity in optical rotation for

scattering samples. From the graph it is evident that error and sensitivity in optical

rotation are least for optimum Stokes vectors in contrast to the other whereas random-

2 is the most accurate one among random Stokes vectors.


with scattering media ( μs=21.5 /cm)


73


with scattering media ( μs=21.5 /cm)


glucose with scattering media ( μs=21.5 /cm)


74






75






76



5.4 Summary

In this chapter, we used a precise low noise Mueller matrix polarimeter based on dual

photo elastic modulator for the measurements of the optical rotation of glucose with

and without scattering phantom in the forward detection geometry. The optical

rotation value with and without scattering media for optimum Stokes vectors were

(0.85o

M⁄ ) and (0.76o

M⁄ ) respectively. Whereas the error and sensitivity in optical

rotation were ((1.1 × 10−3)o

M⁄ ), ((2.2 × 10−3)o

M⁄ ) and ((1.0 × 10−3)o

M⁄ )

and ((2.8 × 10−3)o

M⁄ ). These values are much smaller as compared to other

polarization states. Therefore, the lowest error and sensitivity values for the optimum

states build confidence toward measurements of optical rotation at clinical level of

glucose. The difference in error and sensitivity in optical rotation for all states were

significant as determined by two tailed unpaired T test (p < 0.05). However the values

of optical rotation for optimum states was more accurate and highly precise.

Therefore optimum input polarization states with Dual PEM has a potential to be used

as a polarimetric plate form for monitoring of glucose which is a challenge. This also

seems to be a promising method to measure other optical properties such as

birefringence, depolarization etc. with optimum input polarization states.

Chapter 6: Conclusion and Future Work

77

6 Conclusion and Future Work

In this thesis, the existing nonlinearity in carousel interferometer was addressed with

the efficient generalized six mirrored carousel interferometer. This was significantly

improved with the insertion of optimized glass plates in two arms of the

interferometer. The maximum value of nonlinearity in optical path difference attained

with glass plates was 1.26 x 10−4 for ± 20o rotation. Carousel interferometer was

also used as a phase modulator device for precise phase measurement in optical

sensing. Furthermore, phase modulator was used to measure optical rotation of

glucose in-vitro. Where with a particular set of input polarization states optical

rotation values were achieved with high accuracy and better precision.

In future work, it is recommended to design a carousel interferometer which

would be independent of origin and would also have practical implementation. Based

on the said interferometer a real time phase modulator may be proposed. Ultimately

this may lead to the development of carousel interferometer based polarimeter for

biomedical applications.

78

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