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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
h.moseley@dundee.ac.uk
2.
2. Dr. Tariq Hasan Gilani
Department of Physics, Millersville University, Millersville, USA
tariq.gilani@millersville.edu
4.
Thesis Examiners
1. Dr. Farhan Saif
Department of Physics, QAU, Islamabad
farhan.saif@qau.edu.pk
2. Dr. Javed Anwar
Department of Physics, CIIT, Islamabad.
drjavedanwar@comsats.edu.pk
3. Dr. Muhammad Nawaz
NILOP, Islamabad
mnawaz58@gmail.com
Head of the Department (Name): Dr. Shahid Qamar
Signature with Date: _________________________________
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
Place: PIEAS, Islamabad.
Head, Department of Physics and Applied Mathematics: ___________________
Name: Dr. Shahid Qamar
Date: 27 July 2017
Place: PIEAS, Islamabad.
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
4.3 Results and Discussion ................................................................................... 53
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 Results and Discussion ................................................................................... 65
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
function of angle of rotation; a, b, c, & d represent rotation ranges of ± 5o, ± 10
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
Figure 5-6 Plot of error in the determination of optical rotation of 0.5 M glucose
without scattering media ( μs=0/cm) ............................................................................ 67
Figure 5-7 Plot of error in the determination of optical rotation of 0.25 M glucose
without scattering media ( μs=0/cm) ............................................................................ 68
xiii
Figure 5-8 Plot of error in the determination of optical rotation of 0.125 M glucose
without scattering media ( μs=0/cm) ............................................................................ 68
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 5-11 Plot of sensitivity in the determination of optical rotation of 0.25 M
glucose without scattering media ( μs=0 /cm) .............................................................. 70
Figure 5-12 Plot of sensitivity in the determination of optical rotation of 0.125 M
glucose without scattering media ( μs=0 /cm) .............................................................. 70
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
Figure 5-16 Plot of error in the determination of optical rotation of 0.25 M glucose
with scattering media ( μs=21.5 /cm) ........................................................................... 73
Figure 5-17 Plot of error in the determination of optical rotation of 0.125 M glucose
with scattering media ( μs=21.5 /cm) ........................................................................... 74
Figure 5-18 Plot of sensitivity in the determination of optical rotation of 1.0 M
glucose with scattering media ( μs=21.5 /cm) .............................................................. 74
Figure 5-19 Plot of sensitivity in the determination of optical rotation of 0.5 M
glucose with scattering media ( μs=21.5 /cm) .............................................................. 75
Figure 5-20 Plot of sensitivity in the determination of optical rotation of 0.25 M
glucose with scattering media ( μs=21.5 /cm) .............................................................. 75
Figure 5-21 Plot of sensitivity in the determination of optical rotation of 0.125 M
glucose with scattering media ( μs=21.5 /cm) .............................................................. 76
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
Table 3-5 Maximum residual OPD, nonlinearity and error in retrieved angle of
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
Chapter 1: Introduction
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
Chapter 1: Introduction
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.,
Chapter 1: Introduction
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
Chapter 1: Introduction
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
Chapter 1: Introduction
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
Chapter 1: Introduction
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
Chapter 1: Introduction
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
Chapter 1: Introduction
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
Chapter 1: Introduction
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].
Chapter 1: Introduction
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
Chapter 1: Introduction
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
Chapter 1: Introduction
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.
Chapter 2: Light Polarization
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).
Chapter 2: Light Polarization
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
Chapter 2: Light Polarization
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
Chapter 2: Light Polarization
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).
Chapter 2: Light Polarization
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
Chapter 2: Light Polarization
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.
Chapter 2: Light Polarization
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
Chapter 2: Light Polarization
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)
Chapter 2: Light Polarization
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)
Chapter 2: Light Polarization
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)
Chapter 2: Light Polarization
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)
Chapter 2: Light Polarization
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)
Chapter 2: Light Polarization
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)
Chapter 2: Light Polarization
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)
Chapter 2: Light Polarization
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)
Chapter 2: Light Polarization
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.
Chapter 3: Carousel Interferometer to find Angle and Axis of Rotation
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
Chapter 3: Carousel Interferometer to find Angle and 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
Chapter 3: Carousel Interferometer to find Angle and Axis of Rotation
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)
Chapter 3: Carousel Interferometer to find Angle and Axis of Rotation
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)
Chapter 3: Carousel Interferometer to find Angle and Axis of Rotation
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)
Chapter 3: Carousel Interferometer to find Angle and Axis of Rotation
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).
Chapter 3: Carousel Interferometer to find Angle and Axis of Rotation
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)
Chapter 3: Carousel Interferometer to find Angle and Axis of Rotation
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)) ]
,
Chapter 3: Carousel Interferometer to find Angle and Axis of Rotation
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
Chapter 3: Carousel Interferometer to find Angle and Axis of Rotation
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.
Chapter 3: Carousel Interferometer to find Angle and Axis of Rotation
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.
Chapter 3: Carousel Interferometer to find Angle and Axis of Rotation
43
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.
Figure 3-4 Plot of nonlinearity without glass plates at an optimized values as a
function of angle of rotation; a, b, c, & d represent rotation ranges of ± 5o, ± 10
o
± 15o and ± 20
o respectively.
Table 3-4 Maximum residual OPD, nonlinearity and error in retrieved angle of
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
Chapter 3: Carousel Interferometer to find Angle and Axis of Rotation
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
Chapter 3: Carousel Interferometer to find Angle and Axis of Rotation
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
and ± 20o respectively.
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 ± 20o respectively.
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
Chapter 3: Carousel Interferometer to find Angle and Axis of Rotation
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.
Table 3-5 Maximum residual OPD, nonlinearity and error in retrieved angle of
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
Chapter 3: Carousel Interferometer to find Angle and Axis of Rotation
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.
Chapter 4: Carousel Interferometer as a Phase Modulator
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)
Chapter 4: Carousel Interferometer as a Phase Modulator
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)],
Chapter 4: Carousel Interferometer as a Phase Modulator
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)
Chapter 4: Carousel Interferometer as a Phase Modulator
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)],
Chapter 4: Carousel Interferometer as a Phase Modulator
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
4.3 Results and Discussion
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)
Chapter 4: Carousel Interferometer as a Phase Modulator
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.
Chapter 4: Carousel Interferometer as a Phase Modulator
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)
Chapter 4: Carousel Interferometer as a Phase Modulator
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
Chapter 5: Phase Modulator based Polarimeter to find Optical Rotation
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.
Chapter 5: Phase Modulator based Polarimeter to find Optical Rotation
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
Chapter 5: Phase Modulator based Polarimeter to find Optical Rotation
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
Chapter 5: Phase Modulator based Polarimeter to find Optical Rotation
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
Chapter 5: Phase Modulator based Polarimeter to find Optical Rotation
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.
Chapter 5: Phase Modulator based Polarimeter to find Optical Rotation
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
Chapter 5: Phase Modulator based Polarimeter to find Optical Rotation
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)
Chapter 5: Phase Modulator based Polarimeter to find Optical Rotation
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 Results and Discussion
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)
Chapter 5: Phase Modulator based Polarimeter to find Optical Rotation
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)
Chapter 5: Phase Modulator based Polarimeter to find Optical Rotation
67
Figure 5-5 Plot of error in the determination of optical rotation of 1.0 M glucose
without scattering media ( μs=0/cm)
Figure 5-6 Plot of error in the determination of optical rotation of 0.5 M glucose
without scattering media ( μs=0/cm)
Chapter 5: Phase Modulator based Polarimeter to find Optical Rotation
68
Figure 5-7 Plot of error in the determination of optical rotation of 0.25 M glucose
without scattering media ( μs=0/cm)
Figure 5-8 Plot of error in the determination of optical rotation of 0.125 M
glucose without scattering media ( μs=0/cm)
Chapter 5: Phase Modulator based Polarimeter to find Optical Rotation
69
Figure 5-9 Plot of sensitivity in the determination of optical rotation of 1.0 M
glucose without scattering media ( μs=0 /cm).
Figure 5-10 Plot of sensitivity in the determination of optical rotation of 0.5 M
glucose without scattering media ( μs=0 /cm)
Chapter 5: Phase Modulator based Polarimeter to find Optical Rotation
70
Figure 5-11 Plot of sensitivity in the determination of optical rotation of 0.25 M
glucose without scattering media ( μs=0 /cm)
Figure 5-12 Plot of sensitivity in the determination of optical rotation of 0.125 M
glucose without scattering media ( μs=0 /cm)
Chapter 5: Phase Modulator based Polarimeter to find Optical Rotation
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.
Chapter 5: Phase Modulator based Polarimeter to find Optical Rotation
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.
Figure 5-14 Plot of error in the determination of optical rotation of 1.0 M glucose
with scattering media ( μs=21.5 /cm)
Chapter 5: Phase Modulator based Polarimeter to find Optical Rotation
73
Figure 5-15 Plot of error in the determination of optical rotation of 0.5 M glucose
with scattering media ( μs=21.5 /cm)
Figure 5-16 Plot of error in the determination of optical rotation of 0.25 M
glucose with scattering media ( μs=21.5 /cm)
Chapter 5: Phase Modulator based Polarimeter to find Optical Rotation
74
Figure 5-17 Plot of error in the determination of optical rotation of 0.125 M
glucose with scattering media ( μs=21.5 /cm)
Figure 5-18 Plot of sensitivity in the determination of optical rotation of 1.0 M
glucose with scattering media ( μs=21.5 /cm)
Chapter 5: Phase Modulator based Polarimeter to find Optical Rotation
75
Figure 5-19 Plot of sensitivity in the determination of optical rotation of 0.5 M
glucose with scattering media ( μs=21.5 /cm)
Figure 5-20 Plot of sensitivity in the determination of optical rotation of 0.25 M
glucose with scattering media ( μs=21.5 /cm)
Chapter 5: Phase Modulator based Polarimeter to find Optical Rotation
76
Figure 5-21 Plot of sensitivity in the determination of optical rotation of 0.125 M
glucose with scattering media ( μs=21.5 /cm)
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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