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Spie Press Book

Polarization of Light with Applications in Optical Fibers
Author(s): Arun Kumar; Ajoy Ghatak
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Book Description

This book covers the basic concepts and methods involved in polarization of light, and features important methods of analysis such as Jones matrices, Stokes parameters, and Poincaré sphere. It provides the background needed to understand the workings of, and to design, various photonic devices, including Faraday rotators, inline fiber optic components such as polarizers, wave plates, and polarization controllers, and polarimetric sensors such as fiber optic current sensors. Birefringence and the phenomenon of polarization mode dispersion (PMD) in single-mode fibers are also covered. The discussion of concepts is succinct, and the presentation of methods includes concrete examples, making the book an ideal text for students and a useful resource for engineers.


Book Details

Date Published: 25 January 2011
Pages: 246
ISBN: 9780819482150
Volume: TT90

Table of Contents
SHOW Table of Contents | HIDE Table of Contents
Preface /xiii
Acronyms and Abbreviations /xv
1 Introduction /1
2 Maxwell's Equations and Plane Wave Solutions /3
2.1 Introduction /3
2.2 Maxwell's Equations and the Wave Equations in an Isotropic
     Dielectric /3
2.3 Plane Waves in a Homogenous Dielectric /7
2.4 The Poynting Vector /13
Bibliography /17
3 Basic Concepts of Polarization /19
3.1 Introduction /19
3.2 Various Polarization States /19
    3.2.1 Linear polarization /19
    3.2.2 Circular polarization /21
        3.2.2.1 Left-circular polarization (LCP) /22
        3.2.2.2 Right-circular polarization (RCP) /22
    3.2.3 Elliptical polarization /23
3.3 Superposition of Two Orthogonal Linear Polarizations /24
    3.3.1 Method for obtaining the polarization state /28
3.4 Retarders or Wave Plates /35
3.5 Polarizers /39
    3.5.1 Producing linearly polarized light /39
        3.5.1.1 Using a Polaroid /39
        3.5.1.2 Using reflection /39
        3.5.1.3 Using double refraction /40
        3.5.1.4 Linear polarization and Malus' law /41
    3.5.2 Producing circularly polarized light /41
    3.5.3 Producing elliptically polarized light /42
3.6 Analysis of Polarized Light /42
Bibliography /45
4 Double Refraction and Applications /47
4.1 Introduction /47
4.2 Anisotropic Media /47
4.3 Plane Wave Propagation in Anisotropic Media /49
    4.3.1 Polarization eigenmodes /51
        4.3.1.1 Wave propagation along principal axes /52
        4.3.1.2 Wave propagation in a uniaxial medium: arbitrary
        direction /53
4.4 Ray Velocity and Ray Refractive Index /56
    4.4.1 Ray surfaces /58
4.5 Index Ellipsoid /60
    4.5.1 Uniaxial medium /61
4.6 Refraction in a uniaxial medium /62
    4.6.1 Normal incidence /62
    4.6.2 Oblique incidence /65
4.7 Polarization Components Using Double Refraction /66
    4.7.1 Linear polarizer /66
        4.7.1.1 The Nichol prism /66
    4.7.2 Quarter-wave plates and half-wave plates /67
    4.7.3 The Ronchon prism /67
    4.7.4 The Wollaston prism /69
4.8 Circular Birefringence or Optical Activity /70
References /73
Bibliography /73
5 Jones Vector Representation of Polarized Light /75
5.1 Introduction /75
5.2 Jones Vectors /75
    5.2.1 Normalized form of Jones vectors /76
        5.2.1.1 Linear polarization /77
        5.2.1.2 Circular polarization /77
        5.2.1.3 Elliptical polarization with major and minor axes along the
        x and y directions /78
        5.2.1.4 General elliptical polarization /80
5.3 Jones Matrices /82
    5.3.1 Linear polarizer /82
    5.3.2 Linear retarder /83
5.4 Jones Vectors in Terms of the Circular Basis Vectors /88
    5.4.1 Jones matrix of an ideal circular polarizer /90
    5.4.2 Jones matrix of a circularly polarized medium (Faraday
    rotator) /92
5.5 Jones Vectors in Terms of the Elliptical Basis Vectors /94
    5.5.1 Jones matrix of an ideal elliptical polarizer and retarder /95
Bibliography /96
6 Stokes Parameters Representation /97
6.1 Introduction /97
6.2 The Stokes Parameters /97
    6.2.1 In terms of amplitudes and phases of x and y components /98
    6.2.2 In terms of complex amplitudes of x and y components /99
6.3 Stokes Vectors /100
    6.3.1 Completely polarized light /100
    6.3.2 Unpolarized light /104
    6.3.3 Partially polarized light /104
6.4 Determination of Stokes Vectors /106
6.5 Mueller Matrices /107
6.6 Determination of Mueller Matrices /108
    6.6.1 Mueller matrix of a linear polarizer /109
    6.6.2 Mueller matrix of a circular polarizer /112
    6.6.3 Mueller matrix of a linear retarder /116
    6.6.4 Mueller matrix of a rotator /119
Bibliography /119
7 Poincaré Sphere Representation of Polarized Light /121
7.1 Introduction /121
7.2 Various Polarization States /121
7.3 Poincaré Sphere Representation /122
    7.3.1 A polarizer and a birefringent medium /125
7.4 Basic Properties of Poincaré Sphere Representation /125
    7.4.1 Effect of a QWP/HWP on a linear SOP /127
    7.4.2 Effect of a QWP/HWP on a circular SOP /128
    7.4.3 Effect of a QWP/HWP on an elliptical SOP /129
    7.4.4 An ideal circular polarizer/analyzer /130
    7.4.5 An ideal elliptical polarizer/analyzer /130
7.5 Poincaré Sphere and Stokes Parameters /134
Bibliography /135
8 Propagation and Polarization Characteristics of Single-Mode    Fibers /137
8.1 Introduction /137
8.2 Attenuation in Optical Fibers /138
8.3 Modes of a Step-Index Fiber /139
    8.3.1 Linearly polarized (LP) modes /140
    8.3.2 Cutoff V values of LPlm modes /144
8.4 Single-Mode Fiber /145
    8.4.1 Modal field pattern of the fundamental mode /146
    8.4.2 Spot size of the fundamental mode /147
8.5 Pulse Dispersion in Single-Mode Optical Fibers /148
    8.5.1 Material dispersion /148
    8.5.2 Waveguide dispersion /152
    8.5.3 Dispersion-shifted fibers /154
8.6 Polarization Characteristics of Single-Mode Fibers /155
    8.6.1 Exact vector modes of a step-index fiber /156/
        8.6.1.1 Eigenvalue equations of HE and EH modes for Δ<<1 /159
        8.6.1.2 Cutoff conditions of the various modes /160
    8.6.2 Origins of birefringence in optical fibers /161
        8.6.2.1 Core ellipticity /161
        8.6.2.2 Lateral stress /163
        8.6.2.3 Bending /164
        8.6.2.4 Twists /164
        8.6.2.5 Magnetic field /165
        8.6.2.6 Metal layer near the fiber core /166
References /166
Bibliography /167
9 Birefringence in Optical Fibers: Applications /169
9.1 Introduction /169
9.2 Polarization-Maintaining Fibers /170
    9.2.1 High-birefringence (Hi-Bi) fibers /170
        9.2.1.1 Elliptical-core fibers /171
            9.2.1.1.1 Geometrical birefringence Bg /172
            9.2.1.1.2 Equivalent rectangular waveguide mode /173
            9.2.1.1.3 Stress birefringence Bs /174
        9.2.1.2 Side-pit and side-tunnel fibers /176
        9.2.1.3 Stress-induced fibers /177
        9.2.1.4 Circularly birefringent fibers /179
    9.2.2 Single-polarization single-mode (SPSM) fibers /180
        9.2.2.1 Fibers with different cutoffs for the two
        modes /180
        9.2.2.2 Single-guided polarized mode /181
        9.2.2.3 High differential leakage loss between the two
        modes /181
            9.2.2.3.1 By bending a Hi-Bi fiber /181
            9.2.2.3.2 Using fibers made with birefringent material /183
9.3 Applications of Birefringence in Optical Fibers /183
    9.3.1 Applications requiring a stable polarization state /184
        9.3.1.1 Coupling to integrated optical circuits /184
        9.3.1.2 Interferometric sensors /184
        9.3.1.3 Coherent communication systems /184
    9.3.2 In-line fiber optic devices and components /185
        9.3.2.1 Two-mode elliptical-core fiber sensors /185
        9.3.2.2 Dispersion compensator using a two-mode elliptical-core
        fiber /186
        9.3.2.3 Fiber optic polarization beamsplitter /187
    9.3.3 Fiber optic devices using controlled birefringence in
    SSMFs /189
        9.3.3.1 Zero-birefringence optical fiber holder /189
        9.3.3.2 In-line fiber optic wave plates /191
        9.3.3.3 All-fiber polarization controller /192
        9.3.3.4 Fiber optic current sensor /192
        9.3.3.5 In-line fiber optic polarizer /194
9.4 Surface Plasmon Polaritons and Devices /195
References /196
10 Polarization Mode Dispersion in Optical Fibers /201
10.1 Introduction /201
10.2 PMD in Short-Length and High-Birefringence Fibers /201
10.3 PMD in Long-Length Fibers /204
10.4 Theoretical Modeling of PMD /206
    10.4.1 PMD vector /206
    10.4.2 Birefringence vector /207
    10.4.3 Dynamical equation for the PMD vector /209
    10.4.4 Concatenation model: an alternative approach /210
        10.4.4.1 Jones matrix eigenanalysis /211
10.5 PMD Measuring Techniques /212
    10.5.1 Time-domain techniques /213
    10.5.2 Frequency-domain techniques /213
    10.5.3 Wavelength-scanning method /214
10.6 PMD Mitigation Techniques /217
    10.6.1 Low-PMD fibers/218
    10.6.2 PMD compensators /218
References /222
Index /225

Preface

In the recent past, the polarization phenomena associated with light waves have become extremely important in many areas of photonics. For example, many important polarization-based devices have been developed, including Faraday rotators, in-line fiber optic components such as polarizers, wave plates, and polarization controllers, and sensors such as fiber optic current sensors and fiber optic gyroscopes. In order to understand the workings of such photonic devices and to improve their design, sound knowledge of the basic concepts involved in polarization is required.

Furthermore, in optical communication systems, polarization mode dispersion has become an extremely important issue, particularly for very high-bit-rate (>10 Gb) systems. Polarization mode dispersion arises because of random birefringence present in a practical optical fiber. The birefringence that causes polarization mode dispersion in optical fibers may be linear, circular, or, in general, elliptical. In order to understand the nature of polarization mode dispersion and to control or reduce it, one must know how the various types of birefringent media affect the polarization state of the guided light while propagating through an optical fiber. Thus, it has become almost essential for most engineers (working in the general area of photonics) to have a basic knowledge of the polarization phenomena, associated concepts, and basic methods of analysis such as Jones matrices, Stokes parameters, and Poincaré Spheres.

In this book, our aim is to provide in one source all of the basic concepts and methods involved in the understanding and design of various photonic devices, keeping the discussions as succinct as possible. Poincaré Sphere representation of polarized light is a very important method that is not discussed in sufficient detail in most of the literature. Therefore, we have included several numerical examples to make the method very clear. This book works through all of the steps using many examples; therefore, even undergraduate students should be able to follow along without much difficulty.

We have been teaching various aspects of polarization to our undergraduate students and to our master's students at the Indian Institute of Technology Delhi (IITD). This book has grown out of the lecture notes that we have prepared over the last 25 years. We have also used this material in several short courses organized at IITD and at other institutions.

We thank our colleagues in the physics department of IITD for many helpful discussions—in particular, we thank Profs. B. D. Gupta, B. P. Pal, A. Sharma, M. R. Shenoy, and K. Thyagarajan, and Dr. Ravi Varshney for research collaboration and useful discussions that have helped us in improving the presentation. We are also thankful to our research students Ms. Triranjita Srivastava and Mr. Saurabh Mani Tripathi, who helped us at various stages through their suggestions, in carrying out some of the calculations, and in creating some of the diagrams. Finally, we are grateful to our wives, Shobhita and Gopa, for their patience and understanding.

Arun Kumar
Ajoy Ghatak
New Delhi
January 2011


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