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TitlePolarization of Light With Applications in Optical Fibers (SPIE Tutorial Texts Vol. TT90)
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LanguageEnglish
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Table of Contents
                            Contents
Preface
Acronyms and Abbreviations
1 Introduction
2 Maxwell’s Equations and Plane Wave Solutions
3 Basic Concepts of Polarization
4 Double Refraction and Applications
5 Jones Vector Representation of Polarized Light
6 The Stokes Parameters Representation
7 Poincaré Sphere Representation of Polarized Light
8 Propagation and Polarization Characteristics of
Single-Mode Fibers
9 Birefringence in Optical Fibers: Applications
10 Polarization Mode Dispersion in Optical Fibers
Index
                        
Document Text Contents
Page 123

106 Chapter 6






Figure 6.1 Schematic representation of a general SOP in Stokes subspace.


2 2 2
1 2 3

0

4
0.8

5

S S S
p

S

 
    .


Thus, the given Stokes vector can be written as [Eq. (6.38)]


5 0.8 5 0.2 5 1 1

4 4 0 1 0
4

0 0 0 0 0

0 0 0 0 0

          
         
            
         
         
         

,

which means that the intensity of the given beam is 5 units, out of which 80% is
linear horizontally polarized and 20% is unpolarized.

6.4 Determination of Stokes Vectors

As suggested by Eq. (6.3), the Stokes vector of a given light beam can be
determined first by measuring its intensity directly and then measuring its
intensity after passing it through three different polarizers—namely, a linear
horizontal, a linear +45-deg, and a right-circular polarizer. The measured
intensities will be 0 H 45 RCP, , , andI I I I respectively, from which the Stokes

parameters 0 1 2 3, , andS S S S  can easily be calculated using Eq. (6.3). We
demonstrate this using the following simple example.

Example 6.2 Determine the Stokes parameters of a linearly polarized light
oriented at an angle  with the x axis.


Solution: Let the intensity of the given beam be I0. It can easily be shown that if
this beam is passed separately through a linear horizontal polarizer, a linear +45-

Page 124

The Stokes Parameters Representation 107

deg polarizer, and a right-circular polarizer, the intensities H 45 RCP, , andI I I of
the output beam obtained in the three cases will be given by




2
H 0

2 0
45 0

0
RCP

cos ,

cos ( / 4 ) (1 sin 2 ),
2

.
2

I I

I
I I

I
I



 



    



(6.43)


Using Eq. (6.3), the Stokes parameters will be given as





0 0

2
1 H 0 0 0 0

2 45 0 0 0 0

3 RCP 0 0 0

,

2 2 cos cos 2 ,

2 (1 sin 2 ) sin 2 ,

2 0.

S I

S I I I I I

S I I I I I

S I I I I

 
 



    
     
    

(6.44)


Thus, the normalized Stokes vector of a linearly polarized light oriented at an
angle  with the x axis is given by




1

cos 2

sin 2

0

S



 
 
 
 
 
 

, (6.45)


which is the normalized form of Eq. (6.15). Similarly, Stokes parameters of
other polarization states can also be determined.

6.5 Mueller Matrices
Like Jones vectors, Stokes vectors of the input and output states of polarization
corresponding to a polarizing device can also be related by a square matrix (4 × 4
in this case), which is known as a Mueller matrix, after Hans Mueller, who
devised this method in 1943. According to this method, if Sand S are the Stokes
vectors of the input and output states of polarization of a polarizing device, they
are related as

M S S, (6.46)

where M, known as the Mueller matrix, is a 4 × 4 matrix of the form

Page 246

Arun Kumar received his M.Sc. and Ph.D. degrees
in physics from the Indian Institute of Technology
Delhi (IITD), in 1972 and 1976, respectively. Since
1977, he has been on the faculty of the physics
department at IITD, where he has been a professor
since 1995. He has been a visiting scientist at the
Technical University of Hamburg, Germany (1980–
1981); the Opto-electronic Group, Strathclyde
University, Glasgow, UK (1988); the National
Institute of Standards and Technology, Boulder,
Colorado (1993–1994); the University of Nice, France
(1996); and the University of Jean Monnet, Saint

Etienne, France (several times). He has authored/coauthored more than 90
research papers in international journals and has supervised/co-supervised ten
Ph.D. theses in the area of fiber and integrated optics.
A perturbation method for the analysis of rectangular-core waveguides,
developed by Kumar and coworkers, is now known as the “Kumar method” in
the literature. Kumar is a recipient of research fellowships from the Alexander
von Humboldt Foundation of Germany (1980–1981) and the Indian National
Science Academy (INSA) (1990–1992). His research interests are in the fields of
optical waveguides, fiber and integrated optic devices, polarization mode
dispersion, and plasmonic waveguides.



Ajoy Ghatak recently retired as a professor of
physics from the Indian Institute of Technology
Delhi (IITD). He obtained his M.Sc. degree from
Delhi University and his Ph.D. degree from Cornell
University. Professor Ghatak has held visiting
positions in many universities in the United States,
Europe, Hong Kong, Australia, and Singapore. His
areas of research are fiber optics and quantum
mechanics, and he has written several books in
these areas. The first edition of his book Optics
(McGraw-Hill, 2009) has been translated into
Chinese and Persian, and his book Inhomogeneous

Optical Waveguides (Plenum Press, 1977) (coauthored by Professor M. S.
Sodha) has been translated into Russian and Chinese. Some of his other books
are Optical Electronics (Cambridge University Press, 1989) (coauthored by K.
Thyagarajan), Introduction to Fiber Optics (Cambridge University Press, 1998)
(coauthored by K. Thyagarajan), Quantum Mechanics (Macmillan India, 2004
and Kluwer, 2004) (coauthored by S. Lokanathan), and Mathematical Physics
(Macmillan India, 1995) (coauthored by I. C. Goyal and S. J. Chua).
Professor Ghatak is the recipient of several awards, including the 2008 SPIE
Educator Award in recognition of “his unparalleled global contributions to the
field of fiber optics research, and his tireless dedication to optics education
worldwide and throughout the developing world, in particular” and the 2003
Optical Society of America Esther Hoffman Beller Award in recognition of “his

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