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Polarized Light and the Mueller Matrix ApproachSERIES IN OPTICS AND
OPTOELECTRONICS
Series Editors: E Roy Pike, Kings College, London, UK Robert G W
Brown, University of California, Irvine, USA
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José J. Gil Pérez Razvigor Ossikovski
Polarized Light and the
Mueller Matrix Approach
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vii
Contents
1 Polarized electromagnetic waves 1 1.1 Introduction: Nature of
polarized electromagnetic waves 1 1.2 Polarization ellipse 3
1.3 Analytic signal representation and the Jones vector 6 1.4
Coherency matrix and Stokes vector 10
1.4.1 2D coherency matrix 10 1.4.2 Stokes vector 11
1.5 2D space–time and space–frequency representations of coherence
and polarization 16 1.5.1 2D representations of coherence and
polarization 16
1.5.1.1 Mutual coherence matrix 17 1.5.1.2 Space–time two-point
Stokes vector 18 1.5.1.3 Cross-spectral density matrix 18 1.5.1.4
Space–frequency two-point Stokes vector 19
1.5.2 Measures of the degree of coherence of 2D electromagnetic
fields 20 1.5.2.1 Complex degree of coherence 20 1.5.2.2 Complex
degree of mutual polarization 21 1.5.2.3 Intrinsic degrees of
coherence 22 1.5.2.4 Electromagnetic degree of coherence 24 1.5.2.5
Overall degree of coherence 25
1.5.3 Cross-spectral purity and coherence–polarization purity 26
1.6 Poincaré sphere 29 1.7 Polarimetric interpretation of the Pauli
matrices 31 1.8 Intrinsic coherency matrix 32 1.9 Polarimetric
purity 36
1.9.1 Concept of polarimetric purity 36 1.9.2 Components of purity
of a 2D state of polarization 39 1.9.3 Degree of mutual coherence
and polarimetric purity 40 1.9.4 Polarization entropy 43
1.10 Composition and decomposition of 2D states of polarization 44
1.10.1 Coherent composition and decomposition of 2D pure states 44
1.10.2 Incoherent composition and decomposition of 2D mixed states
44
1.11 Classification of 2D states of polarization 46 1.12 Invariant
quantities of a 2D polarization state 46 1.13 Quantum description
of 2D states of polarization 46 1.14 Summary 49
2 Three-dimensional states of polarization 51 2.1 Introduction 51
2.2 3D Jones vector 51
viii Contents
2.3 3D Coherency matrix 53 2.4 3D Stokes parameters 54 2.5
Composition and decomposition of 3D states of polarization 56
2.5.1 Coherent composition of 3D pure states 57 2.5.2 Arbitrary
decomposition of 3D states 57 2.5.3 Spectral decomposition of 3D
states 58 2.5.4 Characteristic decomposition of 3D states 58 2.5.5
Polarimetric subtraction 59
2.6 3D space–time and space–frequency representations of coherence
and polarization 60 2.6.1 3D representations of coherence and
polarization 60 2.6.2 Measures of the 3D degree of coherence of
electromagnetic fields 62
2.6.2.1 Intrinsic degrees of coherence 63 2.6.2.2 Electromagnetic
degree of coherence 64 2.6.2.3 Overall space–frequency degree of
coherence 64
2.7 Intrinsic 3D coherency matrix 65 2.8 Intrinsic 3D Stokes
parameters 67
2.8.1 Intrinsic Stokes parameters for 2D states embedded into the
3D representation 69
2.9 3D polarimetric purity 70 2.9.1 Norms in the spaces of 3D
coherency matrices and Stokes parameter
matrices 70 2.9.2 Degree of polarimetric purity 71 2.9.3 Components
of purity of a 3D state of polarization 73 2.9.4 Indices of
polarimetric purity 74 2.9.5 3D purity space 76 2.9.6 Degrees of
mutual coherence of a 3D polarization state 78 2.9.7 3D
polarization entropy 79
2.10 Interpretation of the coherency matrix for 3D polarization
states 80 2.10.1 Pure states (rank R = 1) 80
2.10.1.1 Linearly polarized pure states (r = 1, t = 1) 81 2.10.1.2
Pure states with arbitrary polarization ellipse and nonzero
ellipticity (r = 1, t = 2) 82 2.10.2 Mixed states with rank R = 2
82
2.10.2.1 Mixed states with fixed direction of propagation (r = 2, t
= 2 ⇒ pd = 1) 83
2.10.2.2 Mixed states with rank r = 2 and fluctuating direction of
propagation (r = 2, t = 3 ⇒ pd < 1) 86
2.10.3 Mixed states with rank R = 3 89 2.10.3.1 Arbitrary
decomposition 89 2.10.3.2 Characteristic decomposition 90
2.10.4 Classification of 3D polarization states 92 2.11 Invariant
quantities of a 3D polarization state 92 2.12 Quantum formulation
for 3D polarization states 92 2.13 Summary 97
3 Nondepolarizing media 99 3.1 Introduction 99 3.2 Basic
polarimetric interaction: Jones calculus 102
3.2.1 Jones matrix 102 3.2.2 Jones algebra and its physical
interpretation 103
3.2.2.1 Product of Jones matrices 104 3.2.2.2 Product of a Jones
matrix and a scalar 104
Contents ix
3.2.2.3 Determinant and norms of a Jones matrix 104 3.2.2.4 Inverse
of a Jones matrix 105 3.2.2.5 Additive composition of Jones
matrices 105
3.2.3 Reciprocity in Jones matrices 106 3.2.4 Changes of reference
frame and rotated Jones matrices 106
3.3 Pure Mueller matrices 107 3.3.1 Concept of pure Mueller matrix
107 3.3.2 Block form of a Mueller matrix 111 3.3.3 Reciprocity
properties of pure Mueller matrices 111 3.3.4 Passivity condition
for pure Mueller matrices 112 3.3.5 Algebraic operations with pure
Mueller matrices and their physical
interpretation 113 3.3.5.1 Product of pure Mueller matrices 113
3.3.5.2 Product of a pure Mueller matrix and a nonnegative scalar
113 3.3.5.3 Determinant and norms of a pure Mueller matrix 113
3.3.5.4 Inverse of a pure Mueller matrix 114 3.3.5.5 Additive
composition of Mueller matrices 115
3.3.6 Changes of reference frame and rotated Mueller matrices
115 3.4 Singular states of polarization 116 3.5 Normality and
degeneracy of Jones and Mueller matrices 118
3.5.1 Normal operators 118 3.5.2 Nonnormal operators 119 3.5.3
Degenerate operators 120
3.6 Summary 122 4 Nondepolarizing media: Retarders, diattenuators,
and serial decompositions 123
4.1 Introduction 123 4.2 Retarders 123
4.2.1 Jones matrices of retarders 124 4.2.1.1 Elliptic retarder 125
4.2.1.2 Elliptic retarder oriented at 0° 125 4.2.1.3 Circular
retarder and rotator 125 4.2.1.4 Linear retarder 126 4.2.1.5
Horizontal linear retarder 126 4.2.1.6 Pseudorotator 127 4.2.1.7
Operational form of the Jones matrix of a retarder 127 4.2.1.8
Exponential form of the Jones matrix of a retarder 128 4.2.1.9
Jones matrix of a serial combination of retarders 128
4.2.2 Mueller matrices of retarders 129 4.2.2.1 Retardance vector
and components of retardance 129 4.2.2.2 Mueller matrix of a
rotator 131 4.2.2.3 Horizontal linear retarder 132 4.2.2.4
Operational form of the Mueller matrix of a retarder 132 4.2.2.5
Eigenvalues and eigenstates of the Mueller matrix of a retarder 132
4.2.2.6 Elliptic retarder oriented at 0° 133 4.2.2.7 Circular
retarder 133 4.2.2.8 Linear retarder 133 4.2.2.9 Pseudorotator 134
4.2.2.10 Mueller matrix of a serial combination of retarders 134
4.2.2.11 Euler parameterization of the Mueller matrix of a retarder
134
4.2.3 Equivalence theorems for serial combinations
of retarders 135 4.3 Diattenuators 135
x Contents
4.3.1 Jones matrices of diattenuators 137 4.3.1.1 Elliptic
diattenuator 137 4.3.1.2 Elliptic diattenuator oriented at 0° 138
4.3.1.3 Circular diattenuator 138 4.3.1.4 Linear diattenuator 139
4.3.1.5 Horizontal linear diattenuator 139 4.3.1.6 Operational form
of the Jones matrix of a normal diattenuator 139 4.3.1.7
Exponential form of the Jones matrix of a diattenuator 139 4.3.1.8
Serial combination of diattenuators 140 4.3.1.9 Diattenuating
retarder 141
4.3.2 Mueller matrices of diattenuators 142 4.3.2.1 Components of
diattenuation 142 4.3.2.2 Horizontal linear diattenuator 143
4.3.2.3 Operational form of the Mueller matrix of a normal
diattenuator 143 4.3.2.4 Eigenvalues and eigenstates of the Mueller
matrix of a normal
diattenuator 144 4.3.2.5 Elliptic diattenuator oriented at 0° 144
4.3.2.6 Circular diattenuator 145 4.3.2.7 Linear diattenuator 145
4.3.2.8 Horizontal linear diattenuator 146 4.3.2.9 Serial
combination of diattenuators 146 4.3.2.10 Diattenuating retarder
146
4.3.3 Equivalence theorems for serial decompositions of normal
diattenuators 147 4.4 Other mathematical representations of the
polarimetric properties of
nondepolarizing systems 147 4.4.1 Pure covariance matrix 148 4.4.2
Covariance vector 148 4.4.3 Jones operator 149 4.4.4 The scattering
matrix: Sinclair matrix and Kennaugh matrix 149
4.5 Polar decomposition of a nondepolarizing system 151 4.5.1
Application of the polar decomposition to an experimental example
154
4.6 General serial decomposition of a nondepolarizing system
155 4.7 Dual linear retarder transformation of a nondepolarizing
system 157 4.8 Constitutive vectors of a nondepolarizing
Mueller matrix 158 4.9 Invariant polarimetric quantities of a
nondepolarizing Mueller matrix 159 4.10 Particular forms of
nondepolarizing Mueller matrices 161
4.10.1 Normal pure Mueller matrices 161 4.10.2 Nonnormal pure
Mueller matrices 162 4.10.3 Degenerate pure Mueller matrices 162
4.10.4 Singular pure Mueller matrices 162
4.10.4.1 Nonnormal elliptic polarizer 163 4.11 Summary 164
5 The concept of Mueller matrix 167 5.1 Introduction 167 5.2 The
concept of Mueller matrix 168 5.3 Covariance and coherency matrices
associated with a Mueller matrix 170 5.4 Changes of reference frame
and rotated Mueller matrices 175 5.5 Characterization of Mueller
matrices: Covariance criterion 175
5.5.1 Covariance criterion 175 5.5.2 Explicit algebraic formulation
of the covariance criterion 176
5.6 Normal form of a Mueller matrix 177
Contents xi
5.6.1 Type-I canonical Mueller matrix 178 5.6.2 Type-II canonical
Mueller matrix 180 5.6.3 Covariance characterization of Mueller
matrices through their normal
form 183 5.7 Reciprocity properties of Mueller matrices 184 5.8
Passivity constraints for Mueller matrices 185 5.9 Vectorial
partitioned expression of a Mueller matrix 187 5.10 Spectral and
characteristic decompositions of a Mueller matrix 187
5.10.1 Spectral decomposition 187 5.10.2 Characteristic
decomposition 188
5.11 Polarimetric purity of a Mueller matrix 189 5.11.1 Norms of
the covariance, coherency, and Mueller matrices 189 5.11.2 Purity
criterion 190 5.11.3 Depolarization index and depolarizance 190
5.11.4 Polarization entropy 193 5.11.5 Lorentz depolarization
indices 194 5.11.6 Other overall measures of depolarization
195
5.11.6.1 Average degree of depolarization 195 5.11.6.2
Depolarization power 195 5.11.6.3 Scalar metric Qf(M) 196
5.12 Summary 196 6 Physical quantities in a Mueller matrix
199
6.1 Introduction 199 6.2 Components of purity of a Mueller matrix
199
6.2.1 Average intensity coefficient 200 6.2.2 Diattenuation 200
6.2.3 Reciprocal diattenuation 201 6.2.4 Polarizance 202 6.2.5
Reciprocal polarizance 203 6.2.6 Degree of polarizance 203 6.2.7
Components of diattenuation and polarizance 204 6.2.8 Degree of
spherical purity 204 6.2.9 Physical significance of the components
of purity 206
6.2.9.1 Components of purity of polarizers and analyzers 206
6.2.9.2 Components of purity of the canonical depolarizers
207
6.3 Indices of polarimetric purity 208 6.4 Invariant quantities of
a Mueller matrix 211
6.4.1 Dual retarder transformation 211 6.4.2 Single retarder
transformation 212 6.4.3 Dual rotation transformation 213 6.4.4
Single rotation transformation 214
6.5 Purity space 215 6.5.1 Purity space for the components of
purity 215 6.5.2 Classification of Mueller matrices according to
the values of the
components of purity 216 6.5.3 Purity space for the indices of
polarimetric purity 219 6.5.4 Purity regions in the space of
components of purity 220
6.6 Anisotropy coefficients of a Mueller matrix 221 6.7 From a
nondepolarizing to a depolarizing Mueller matrix 225
6.7.1 Synthesis of a type-I Mueller matrix 227 6.7.2 Synthesis of a
type-II Mueller matrix 228
xii Contents
6.7.3 On the reference pure Mueller matrix 232 6.7.4 Depolarization
synthesis 232
6.8 Summary 233 7 Parallel decompositions of Mueller matrices
235
7.1 Introduction 235 7.2 Additive composition of Mueller matrices
235 7.3 Arbitrary decomposition of a Mueller matrix 237
7.3.1 Application of the arbitrary decomposition to an experimental
example 239 7.4 On the rank of the covariance matrix of a parallel
composition 240 7.5 Characteristic decomposition of a Mueller
matrix 241
7.5.1 Application of the characteristic decomposition to an
experimental example 243
7.6 Polarimetric subtraction of Mueller matrices 245 7.6.1
Condition for polarimetric subtractability 245 7.6.2 Polarimetric
subtraction of a pure component 246 7.6.3 Polarimetric subtraction
of a pure component from a rank-two mixture 247 7.6.4
Polarimetric subtraction of a depolarizing component 249
7.7 Passivity constraints 250 7.8 Optimum filtering of measured
Mueller matrices 251 7.9 Summary 254
8 Serial decompositions of depolarizing Mueller matrices 257
8.1 Introduction 257 8.2 Generalized polar decomposition 257
8.2.1 Forward decomposition of a nonsingular Mueller matrix 259
8.2.2 Forward decomposition of a singular Mueller matrix 260 8.2.3
Reverse decomposition of a Mueller matrix 262
8.3 Symmetric decomposition 262 8.3.1 Symmetric decomposition of a
type-I Mueller matrix 263
8.3.1.1 N ≠ 0 and N′ ≠ 0 263 8.3.1.2 N ≠ 0 and N′ = 0 267 8.3.1.3 N
= 0 and N′ ≠ 0 267 8.3.1.4 N = N′ = 0 267
8.3.2 Symmetric decomposition of a type-II Mueller matrix 268 8.3.3
Synthetic view of the symmetric decomposition procedure 270
8.4 Passivity constraints in serial decompositions of depolarizing
Mueller matrices 271 8.4.1 Passivity constraints in the Lu–Chipman
decomposition 271 8.4.2 Passivity constraints in the symmetric
decomposition 272
8.5 Invariant-equivalent Mueller matrices 274 8.5.1
Invariant-equivalent transformations 274 8.5.2 Reduced forms of a
Mueller matrix 275 8.5.3 Invariant-equivalent transformation
induced by the symmetric
decomposition 276 8.5.4 Kernel form of a Mueller matrix 276
8.6 Arrow decomposition of a Mueller matrix 277 8.6.1 Arrow form of
a Mueller matrix 277 8.6.2 Characterization of Mueller matrices
through the arrow form 279
8.6.2.1 Characterization of nonpolarizing Mueller matrices 280
8.6.2.2 Characterization of symmetric Mueller matrices 280 8.6.2.3
Characterization of a Mueller matrix through its reduced form
281
8.7 Singular Mueller matrices 282
Contents xiii
8.7.1 Depolarizing polarizer 282 8.7.2 Depolarizing analyzer 283
8.7.3 Pure polarizer 283 8.7.4 Singular depolarizer 283
8.8 Serial-parallel decompositions 284 8.8.1 Serial-parallel
decomposition of a type-I Mueller matrix 284 8.8.2
Serial-parallel decomposition of a type-II Mueller matrix
287
8.9 Summary 290 9 Differential Jones and Mueller matrices 291
9.1 Introduction 291 9.2 Differential Jones matrices and elementary
polarization properties 291
9.2.1 Evolution equation for continuous media 291 9.2.2 Definition
of the elementary polarization properties 292 9.2.3 Elementary
polarization properties and Jones matrices of homogeneous
media 296 9.2.4 Extraction of the elementary polarization
properties from the Jones
matrix 298 9.3 Differential Mueller matrices 300
9.3.1 Differential Mueller matrix of a nondepolarizing medium 300
9.3.2 Mueller matrix of a homogeneous nondepolarizing medium
302 9.3.3 Extraction of the elementary polarization properties from
a
nondepolarizing Mueller matrix 303 9.4 Differential decomposition
of Mueller matrices 305
9.4.1 Differential Mueller matrix of a depolarizing medium 305
9.4.2 Existence and multiplicity of the Mueller matrix logarithm
307 9.4.3 Local physical realizability 311 9.4.4 Algebraic
structure of the differential Mueller matrix formalism 314 9.4.5
Relation of the differential decomposition to the product
decompositions of Mueller matrices 316 9.5 Differential Mueller
matrix of a homogeneous depolarizing medium 319
9.5.1 Differential Mueller matrix of a fluctuating homogeneous
medium 319 9.5.2 Statistical and geometrical interpretation of the
differential Mueller matrix 322 9.5.3 Interpretation of canonical,
general, and rotationally invariant depolarizers 326 9.5.4 Relation
between the differential Mueller matrix and the Mueller
matrix
logarithm 329 9.6 Summary 330
10 Geometric representation of Mueller matrices 333 10.1
Introduction 333 10.2 P-image and I-image of a Mueller matrix 334
10.3 Representative ellipsoids of a Mueller matrix 336 10.4
Ellipsoids associated with some special Mueller matrices 339
10.4.1 Intrinsic depolarizer 340 10.4.2 Type-I canonical
depolarizer followed by a retarder 340 10.4.3 Type-I canonical
depolarizer followed by a normal diattenuator 341 10.4.4 Type-II
canonical depolarizer 341 10.4.5 Type-II canonical depolarizer
followed by a retarder 342 10.4.6 Type-II canonical depolarizer
followed by a normal diattenuator 342
10.5 Characteristic ellipsoids of a depolarizing Mueller matrix 342
10.5.1 Characteristic ellipsoids of a type-I Mueller matrix
343
10.5.1.1 N ≠ 0 and N′ ≠ 0 (D1 < 1, D2 < 1) 343
xiv Contents
10.5.1.2 N ≠ 0 and N′ = 0 (D1 = 1, D2 < 1) 345 10.5.1.3 N = 0
and N′ ≠ 0 (D1 < 1, D2 = 1) 345 10.5.1.4 D1 = D2 = 1
346
10.5.2 Characteristic ellipsoids of a type-II Mueller matrix 346
10.6 Intrinsic ellipsoids of a Mueller matrix 347
10.6.1 Intrinsic ellipsoid of a Mueller matrix with D < 1 and P
< 1 348 10.6.2 Intrinsic ellipsoid of a depolarizing analyzer
349 10.6.3 Intrinsic ellipsoid of a depolarizing polarizer
349
10.7 Topological properties of the characteristic ellipsoids 349
10.7.1 Polarizers and analyzers 350
10.7.1.1 Polarizers 350 10.7.1.2 Analyzers 350 10.7.1.3 Pure
polarizers 350
10.7.2 rank H = 1 350 10.7.3 rank H = 2 350
10.7.3.1 Type-I 350 10.7.3.2 Type-II 351
10.7.4 rank H = 3 351 10.7.4.1 Type-I 351 10.7.4.2 Type-II
351
10.7.5 rank H = 4 351 10.8 Five-vector representation 352 10.9
Geometric view of depolarization, diattenuation and polarizance
354
10.9.1 Depolarization 354 10.9.2 Polarizance and diattenuation
(dichroism) 355 10.9.3 Retardance (birefringence) 355
10.10 Geometric representation of nondepolarizing Mueller matrices
355 10.10.1 Two-vector representation 355 10.10.2 Ellipsoid of a
nondepolarizing Mueller matrix 355
10.11 Experimental examples 359 10.12 Summary 361
References 363
Index 375
xv
Preface
Polarization is a fundamental property of electromagnetic waves.
Profound knowledge of it, in terms of both mathematical formulation
and physical interpretation, is required in various fields of
classical and quantum physics. Polarimetry, which most generally
refers to the various procedures and methods for the measure- ment
and analysis of physical properties related to polarization and its
transformation by the effect of material media, actually covers a
large, rapidly increasing range of applications in science and
technology: astronomy and astrophysics, atmospheric and
environmental studies, remote sensing, photonics and nanophotonics,
fiber optic telecommunications, chemical engineering, medicine and
biology, materials science, the optics industry, plasma physics,
liquid crystal display devices, thin films and layered media,
microwave devices, quantum teleportation, and so forth.
The main objective of this book is to integrate, in a comprehensive
and coherent way, the basic concepts of the polarization phenomena
from the double point of view of the states of polarization of
electromagnetic waves and the transformations of these states by
the action of material media. Recent decades have been
characterized by successive contributions that constitute today the
consolidated basis of our knowledge of the field, and that deserve
to be put together and described in a detailed monographic
treatise. Despite the indubi- table interest of a number of
nonlinear polarization phenomena, the subjects dealt with here are
focused only on linear effects, covering most of the practical
situations in polarimetry.
For several years, the works on near-field phenomena and
nanotechnologies have motivated the study and characterization of
three-dimensional states of polarization of electromagnetic waves,
beyond the con- ventional two-dimensional approaches. Thus,
fundamental concepts such as Stokes parameters, polarization
matrix, degree of polarization, and intrinsic angular momentum have
to be appropriately extended to three- dimensional formulations.
Two-dimensional polarization states are characterized by the fact
that the electric field of the wave evolves in a fixed plane, while
in general, three-dimensional states require consideration of the
three components of the electric field of the wave regardless of
the reference frame considered. Chapters 1 and 2 are respectively
devoted to the analysis of the basic concepts related to two- and
three-dimensional polarization states, including their mathematical
representation through appropriate structures such as the Stokes
vector and the polarization matrix, as well as to the study of some
quantities derived from these struc- tures that are useful in
practice. Further, the space–time and space–frequency formulations
combining the concepts of polarization and coherence of
electromagnetic waves are studied and interpreted in light of the
recent contributions to this fundamental branch of physical
optics.
One of the concepts that plays a central role in the content of
this book is that of the coherency matrix associated, respectively,
with electromagnetic waves and media. It has the structure of a
two-dimensional (for two-dimensional polarization states),
three-dimensional (for three-dimensional polarization states), or
four-dimensional (for material media) statistical covariance
matrix. Moreover, its scope is not limited to the optical
frequencies only, but can be applied to any kind of electromagnetic
radiation, as well as to the linear transformation of the state of
polarization of the radiation resulting from its interaction with
material media.
The action of material media, linearly transforming the state of
polarization of the electromagnetic waves interacting with them, is
studied in several consecutive chapters. The nondepolarizing linear
transformations of the states of polarization are formulated in
Chapter 3 through the Jones and Mueller–Jones approaches, including
the physical interpretation of the mathematical operations in those
spaces, as well as such fun- damental concepts as passivity (the
physical restriction of not amplifying the intensity of the
interacting electromagnetic waves) and reciprocity (the behavior of
the medium when input and output electromagnetic beams are
interchanged).
xvi Preface
Chapter 4 is dedicated to the particular forms and properties of
the Jones and Mueller matrices associated with different basic
types of nondepolarizing media, such as retarders (linear,
circular, or elliptic), diattenua- tors (linear, circular, and
elliptic), and pseudorotators. The main approaches for the serial
decomposition of the Jones and Mueller matrices associated with
nondepolarizing media are also analyzed and interpreted. These
serial decompositions correspond to equivalent systems composed of
cascades of simple components that consecutively exert their effect
on the polarization state of the interacting electromagnetic
wave.
The mathematical representation of the polarimetric action of
depolarizing systems requires consider- ation of the concept of the
general Mueller matrix, that is, a 4 × 4 real matrix
whose structure corresponds to a physically realizable linear
transformation of the Stokes parameters of the incoming
electromagnetic wave into the Stokes parameters of the outgoing
wave. Thus, Chapter 5 deals with the foundations of the notion of
the Mueller matrix, applicable to both depolarizing and
nondepolarizing interactions, and rely- ing on well-defined
statistical mixtures of basic nondepolarizing interactions. Most
generally, the Mueller matrix represents the total information one
is able to obtain from the interaction of polarized radiation with
a linear-response medium. In particular, its 16 real elements are
the complete phenomenological descriptors of a polarized light
scattering experiment. As a result of the statistical picture for
physical realizability, a fun- damental characteristic property of
a Mueller matrix is that it has an associated Hermitian matrix with
the structure and properties of a covariance matrix.
Concepts such as passivity, reciprocity, and polarimetric purity
(which refers to the measure of how close a given system is to a
nondepolarizing one), as well as certain decompositions related to
the essential alge- braic structure of Mueller matrices, are also
considered and generalized throughout respective sections of
Chapter 5. In particular, the so-called normal form (together with
the associated symmetric decomposition) of a Mueller matrix leads
to the key concepts of type-I and type-II Mueller matrices, which
have distinct physical and mathematical natures and cover, in a
complementary way, the entire set of Mueller matrices. In order to
be comprehensible to readers coming from different fields, Chapter
5 also reviews certain alternative conven- tions that are commonly
used in remote sensing and synthetic aperture radar polarimetry,
together with the rules for conversion between alternative
definitions.
The content of Chapter 6 goes more deeply into the analysis of the
physical parameters associated with Mueller matrices, such as
diattenuation, polarizance, indices of polarimetric purity,
polarization entropy, and anisotropy coefficients, as well as
the physical quantities that are invariant under certain types of
transfor- mations. The last section of Chapter 6 is devoted to the
synthesis of an arbitrary depolarizing Mueller matrix M through the
smooth and continuous transformation of a reference pure Mueller
matrix associated with M, thus providing a comprehensive view of
the nature of the properties of depolarizing systems.
As a direct consequence of the statistical origin of the concept of
the Mueller matrix, any given depolar- izing system can be
conceived as a parallel combination of nondepolarizing components;
that is, any given Mueller matrix can be expressed, in a variety of
ways, as a sum of nondepolarizing Mueller matrices whose respective
weights sum to 1. This arbitrary decomposition, as well as other
interesting parallel decomposi- tions, is described and interpreted
throughout Chapter 7, which also deals with the closely related
concept of polarimetric subtraction of a given component from the
parallel combination.
With the ready availability of Mueller matrix polarimeters and the
continuously increasing complexity of the materials and media under
investigation, the interpretation of an experimentally obtained
Mueller matrix in terms of physical properties of the medium is of
ever-growing importance today. Indeed, in numer- ous practical
cases, it is not possible to directly relate the polarimetric
response of the medium to its ele- mentary properties (dichroism,
birefringence, optical activity, etc.) through rigorous
electromagnetic theory modeling. An alternative approach to this
problem is algebraically decomposing the experimental Mueller
matrix into simpler components without any explicit reference to an
electromagnetic model. The various matrix decomposition approaches
at hand can be grouped into two classes, serial (or product) and
parallel (or sum) decompositions.
A powerful tool for the analysis of any—experimental, theoretical,
or simulated—depolarizing Mueller matrix M, complementary to the
various parallel decompositions, is provided by the serial
decomposi- tions of M in terms of ordered products of particularly
simple Mueller matrices. They represent a gener- ally depolarizing
Mueller matrix as a product of the Mueller matrices of basic
optical components such as
Preface xvii
diattenuators, retarders, and canonical depolarizers. The potential
benefit of applying the algebraic approach to experimentally
obtained Mueller matrices, for example, from biological samples, is
twofold. First, the alge- braic methodology is universal, in
contrast to modeling the optical response of the sample. That is,
alge- braic decompositions are applicable to any experimental
Mueller matrix, whether an electromagnetic model describing the
medium under investigation exists or not. This allows the
experimentalist to obtain immediate physical information on the
sample—through the representation of the latter as a chain of
elementary optical components—even in the absence of any optical
model or (most often) when the latter is either too complex or not
accurate enough. The second advantage of the algebraic approach
stems from the standard representation of every Mueller matrix in a
canonical form playing the role of an optical equivalent system
having the same polarimetric response as the medium represented by
the original matrix. The equivalent system approach thus makes it
possible to perform a formal comparison, in terms of polarimetric
properties, of Mueller matri- ces of various physical origins. The
serial decompositions are described, interpreted, and compared in
the respective sections of Chapter 8, which also includes a
detailed and comprehensive analysis of the important subset of
singular Mueller matrices.
The early contributions of Jones to the differential formulation of
the Jones matrices, which was later translated by Azzam to
nondepolarizing Mueller matrices, has been the subject of renewed
interest in recent years. A series of works have extended this
formal framework, particularly relevant for the description of con-
tinuous media, to the general case of depolarizing systems. The
differential approach, based on the physical picture of
continuously distributed polarization and depolarization
components, parallels and complements the product decomposition
approach whereby depolarization is modeled as a spatially localized
“lump” phenomenon. It allows one to characterize a continuous
depolarizing medium in terms of six elementary polarization
properties and a 3 × 3 complex covariance matrix
describing the depolarization. Within the framework of the
statistical interpretation of the differential formalism, the
polarization and depolariza- tion components of the differential
Mueller matrix are identified physically with the mean values and
the variances–covariances of the fluctuating polarization
properties. The general formalism, as well as the spe- cific
algebraic quantities and notions related to the concept of
differential Mueller matrix, such as the six elementary
polarization properties, is introduced, analyzed, and physically
interpreted in Chapter 9.
The intricate structure of Mueller matrices has motivated, for a
long time, the necessity of developing appropriate geometric
representations of the polarimetric properties of material media.
The most useful approaches providing a geometric viewpoint on the
polarization effects exhibited by different types of media are
presented and discussed in a systematic way in Chapter 10. In
particular, it is shown that the definition of an appropriate set
of characteristic ellipsoids provides a meaningful, visual, and
readily interpretable rep- resentation of any depolarizing Mueller
matrix. The geometric counterparts of the principal polarization
properties of media, such as depolarization,
polarizance–diattenuation (dichroism), and retardance (bire-
fringence), are also analyzed. Chapter 10 also includes additional
useful approaches, such as the two-vector representation of a
nondepolarizing system.
In summary, this monograph on the polarization properties of both
electromagnetic waves and material media provides a general and
unified view, as well as a detailed description, analysis, and
interpretation, of the fundamental concepts and main approaches
underlying this field of physical optics. The presentation of the
theoretical and mathematical foundations is combined with
appropriate physical interpretations, illustra- tive figures, and a
number of selected practical examples taken from experiments. The
authors believe that this book will be useful not only to beginners
wanting to become acquainted with the fundamentals, but also to
confirmed workers in the field looking to grasp the topic more
deeply.
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xix
Acknowledgments
The authors are deeply indebted to Drs. Ignacio San José and Enric
García-Caurel for their permanent sup- port and contributions on
topics dealt with in this book. They would also like to express
their gratitude to and acknowledge Dr. Alfredo Luis for his careful
review and valuable suggestions and comments on Chapters 1 and
2. The authors thank the excellent cooperation received from the
staff of Taylor & Francis and Deanta Global Publishing
Services. In particular, they appreciate the professional advice
and wise suggestions of Luna Han about the structure of this book,
and the careful editing support provided by Michelle van
Kampen.
Razvigor Ossikovski is grateful for the permanent support of his
colleagues, as well as for the creative atmosphere of the
Laboratory of Physics of Interfaces and Thin Films and Ecole
Polytechnique, without which his contribution to this book would
have been impossible.
With a special thought and appreciation for our colleague Dr.
Antonello De Martino, who should be remembered as one of the
pioneers of experimental Mueller polarimetry and its biomedical
applications.
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xxi
Authors
José Jorge Gil was born in Zaragoza, Spain, in 1956. He received
his master of science degree in physics from the University of
Zaragoza, Spain, in 1979. He was a research student from 1980 to
1983 at the University of Zaragoza and received his PhD in phys-
ics from there in 1983. During his PhD, supervised by Professor
Eusebio Bernabéu, he developed an original dual-rotating-retarder
absolute Mueller polarimeter and introduced some new concepts, such
as the depolarization index. He has been a pro- fessor at the
University of Zaragoza since 1987, where he has led a large number
of R&D projects in physics, as well as in e-learning
technologies and methodologies. He was also the general manager of
the R&D Department of the Spanish company BGL (1991–1996),
where he led the development of wireless systems for
interactive
meetings, which earned the Tecnova award from the Spanish Industry
Ministry in 1993. He was the recipient of the G. G. Stokes Award
2013 from the International Society for Optics and Photonics (SPIE)
in recognition of his “groundbreaking collection of rigorous
mathematical descriptions of polarization that are used widely to
interpret experimental data.”
Razvigor Ossikovski was born in Rousse, Bulgaria, in 1967. He
received his engineer’s degree in electronics from the Technical
University of Rousse, Bulgaria, in 1991. He was an international
program student (X88, PEI) at Ecole Polytechnique, Palaiseau,
France, wherefrom he received his MSc (1992) and PhD (1995) degrees
in physics. He then held R&D engineer and team leader positions
at the companies HORIBA Jobin Yvon, Corning, Inc., and HighWave
Optical before taking his current academic position as, first,
assistant professor (2003) and, after his habilitation, associate
pro- fessor (2010) at Ecole Polytechnique. His current research
interests are the theory of polarimetry (Mueller matrix algebra)
and experimental tip-enhanced Raman spec- troscopy. He is the
leader of the fundamental polarimetry and Raman spectroscopy
activities of the Applied Optics and Polarimetry group of the
Laboratory of Physics of Interfaces and Thin Films, Ecole
Polytechnique.
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1
1
1.1 INTRODUCTION: NATURE OF POLARIZED
ELECTROMAGNETIC WAVES
Due to the specific nature of electromagnetic waves, the electric
field of the wave evolves in time at any given point r in space.
The complete description of the electromagnetic wave at point r
requires the knowledge of four field vectors, namely, the electric
field strength (or intensity) E, the electric displacement density
(electric induction) D, the magnetic field strength (or intensity)
H, and the magnetic flux density (magnetic induction) B.
In general, the microscopic forces exerted by the electric field of
the wave on matter are much larger than the forces produced by the
magnetic field, and so the temporal evolution of the electric field
is chosen as representative of the property called polarization.
Since the indicated four field vectors are related through the
Maxwell equations, together with the constitutive relations for the
considered medium, their temporal evolution can be determined from
that of E.
When, in particular, the electromagnetic wave propagates in an
isotropic medium, the electric and mag- netic strengths are tangent
to the wavefront at the considered point r (Figure 1.1).
Nevertheless, in the case of an anisotropic medium, it is D and not
E what is tangent to the direction of propagation at point r. Thus,
unlike in the case of isotropic media, for anisotropic media, the
temporal evolution of D is usually taken as the representative of
the polarization of the electromagnetic wave (Cloude 2009, p. 13),
even though we shall use the symbol E without distinction in the
mathematical formulation of polarization.
As we have indicated in the introduction, in spite of the fact that
we sometimes use the term light, in gen- eral, unless otherwise
indicated, the contents of this book are applicable to the whole
range of frequencies of the electromagnetic spectrum. Whenever
appropriate, we include particular indications for some examples
that are relative to specific spectral ranges.
Electromagnetic waves are produced by accelerated charges as well
as by any kind of nuclear (e.g., gamma radiation), atomic, and
molecular emissions, so that the electromagnetic wave can be
considered to be com- posed of a very high number of more or less
independent contributions. An electromagnetic wave is said to be
monochromatic in the ideal case where its spectral width is zero,
and hence its coherence time τ is infinite. The end point of the
electric field vector of a monochromatic wave describes a fixed and
stable ellipse (the polarization ellipse), and thus this ideal
situation corresponds to the coherent superposition of
contributions with the same frequency ν; the wave is said to be
totally polarized. In a great variety of real situations, the
spectral width Δν is very narrow compared to the mean frequency ν
of the wave, which then is said to be quasi-monochromatic. In all
situations in practice, the coherence time τ = 1/Δν (i.e., a
measure of the time during which the polarization ellipse is
stable) is much larger than the mean natural period T0 1= ν of the
wave. The third temporal scale to be considered is the measurement
time T. In general, polychromatic waves behave as monochromatic for
time intervals shorter than the coherence time τ so that, given a
measurement time T, and leaving aside temporal intensity
fluctuations, the following cases can be distinguished:
1. During time T, the shape of the polarization ellipse remains
constant, so that the electromagnetic wave is said to be totally
polarized (Figure 1.2).
2. The shape of the polarization ellipse changes during the time T,
but it remains inside a certain fixed plane Π, which in turn is
tangent to the wavefront at point r, so that the direction k
perpendicular to Π can be considered a stable and well-defined
direction of propagation (Figure 1.3). Then, by taking a reference
frame that includes the direction k as the Z axis, the electric
field of the wave is completely
2 Polarized electromagnetic waves
determined by two components along two orthogonal reference axes XY
contained in Π. Hereafter, we shall refer to this kind of
polarization states as two-dimensional (2D) states.
a. When the shape of the polarization ellipse changes, but not in a
totally random manner, a mean polarization ellipse can be
identified and the wave is said to be 2D partially polarized.
b. When the shape of the polarization pattern evolves fully
randomly during T, the wave is said to be 2D unpolarized.
3. The plane containing the polarization ellipse changes during
time T, so that the description of the electric field of the wave
requires the consideration of its three nonzero components,
regardless of the reference frame considered. That is, unlike in
the case of 2D states, for these three-dimensional (3D)
polarization states, it is not possible to determine a stable
direction of propagation of the wave at point r.
E × H
∏
Figure 1.1 When the electromagnetic wave propagates in an isotropic
medium, both electric and magnetic strength vectors are tangent to
the wavefront at the considered point r.
T
t
Π
Figure 1.2 Despite intensity fluctuations, totally polarized 2D
states are characterized by the fact that the shape of the
polarization ellipse remains constant during the measurement time
T.
T
t
Π
Figure 1.3 Partially polarized 2D states: the shape of the
polarization ellipse changes during the measure- ment time, while
the direction of propagation is stable.
1.2 Polarization ellipse 3
In general, the measurement time T (i.e., the response time of the
detector) in an optical experiment is much longer than the
coherence time. Typical values of the indicated time intervals in
the optical range are the following: T0 ≅ 10–15 s, 10–9 s ≤ τ ≤
10–4 s, and T > 10–4s (Loudon 1983; Mandel and Wolf 1995;
Brosseau 1998). Moreover, for the microwave range (used in radar
polarimetry), T is much shorter than τ. In practice, the bandwidth
is much smaller that the central frequency, and thus for most
purposes, the assumption of quasi-monochromaticity is justified
and, except where otherwise indicated, the electromagnetic waves
dealt with in the book will be considered
quasi-monochromatic.
This chapter is devoted to 2D states of polarization, that is,
those whose electric field evolves into a plane Π that is constant
in time, so that the direction of propagation is fixed along the
reference axis Z orthogonal to Π (Figure 1.4). The general case of
3D states of polarization will be dealt with in Chapter 2.
1.2 POLARIZATION ELLIPSE
Let us consider the temporal evolution of the electric field E of a
quasi-monochromatic electromagnetic wave that, at a given point r,
can be considered 2D in the sense that there exists a plane Π in
which E lies regardless of the time instant considered, and take
the direction perpendicular to Π as the Z axis of the XYZ reference
frame (Figure 1.4). The components of the electric field can be
expressed as
E z t A t k z t t
E z t A t k z t t
x x x
y y y
ω β (1.1)
where k ,ω are the respective mean values of the wavevector length
k = 2π/λ0 (λ0 being the wavelength for the vacuum) and the angular
frequency, ω = 2πν (ν being the natural frequency); βx, βy are the
respective phases; and Ax, Ay are the respective amplitudes. Note
that, obviously, the cosine functions in Equation 1.1 can be
replaced by sine ones by adding π/2 to the phases. Moreover, the
change of signs of the arguments of the cosine functions pre-
serves the interpretation of Ex(z, t) and Ey(z, t) as the
components of the electric field of a wave traveling in the posi-
tive Z direction, while the choice ± +( )k z tω would correspond to
a wave traveling in the negative Z direction.
The polarization of the wave is a concept linked to measurable
quantities given by averages over mea- surement times that involve
a very high number of cycles, so that the common harmonic
dependence on ω βt tx+ ( ) can be removed. Moreover, the value z =
0 can be taken as the reference, in such a manner that the state of
polarization, at the point r, is defined from the variables
E t A t E t A t tx x y y( ) = ( ) ( ) = ( ) ( )( )cos δ (1.2)
with δ(t) ≡ βy(t) − βx(t).
EYEX
Figure 1.4 In the case of waves with arbitrary form, the
polarization is referred to the local reference frame constituted
by a pair of axes XY lying in the local plane Π tangent to the
wavefront at the considered point r, together with an axis Z
orthogonal to Π that determines the local direction of
propagation.
4 Polarized electromagnetic waves
To analyze the locus traced out by the end point of the electric
field E of the wave, let us observe that its components always
satisfy the following ellipse equation:
E t A t
E t A t
x
x
y
y
( )( ) = ( )( )cos sinδ δ2 (1.3)
The variables Ex(t), Ey(t) fluctuate slowly in comparison to the
mean natural period T0, and therefore the shape of the ellipse
remains constant within time intervals shorter than the coherence
time (Figure 1.5). Despite the fact that an observation time equal
to or greater than T0 is required for the end point of the electric
field vector to draw a complete ellipse, the mentioned slow
fluctuation is the reason why the ellipse defined by
Equation 1.3) is usually termed the instantaneous polarization
ellipse.
Depending on the nature and characteristics of the field
fluctuations, the mentioned cases of totally polar- ized, partially
polarized, and unpolarized states can be distinguished, provided
that the plane containing the polarization ellipse is constant
during the measurement time.
Common light sources, such as the sun, light bulbs, or flames, emit
unpolarized light. In general, the interaction of light (and of
other kinds of electromagnetic waves) with matter via scattering,
transmission, refraction, and reflection produces or modifies the
state of polarization to some extent.
Unpolarized light is also called natural light. Certain artificial
light sources such as lasers or radar anten- nas typically emit
polarized light. Other natural or artificial electromagnetic waves
outside the optical range (which covers infrared, visible, and
ultraviolet), for example, x-ray, microwaves, and radio waves, are
com- monly emitted with a certain degree of polarization, and the
states of polarization change through some kinds of interaction
with matter.
In the ideal case of monochromatic waves, the variables Ax, Ay, and
δ are constant in time, and hence the shape and the size of the
polarization ellipse are fixed.
The instantaneous intensity
I t A t A tx y( ) ≡ ( ) + ( )2 2 (1.4)
is a measure of the power density flux of the wave in the direction
k orthogonal to the plane containing the polarization ellipse and
is referred to time intervals longer than the mean natural period
and much shorter than the coherence time. Therefore, leaving aside
the theoretical interest of considering the instantaneous
intensity, measurable intensities refer to an average of the
fluctuations of I(t) during the measurement time.
It should be noted that while the ratio Ay(t)/Ax(t), as well as the
phase difference δ(t), can remain constant (totally polarized
states), the instantaneous intensity fluctuates (Figure 1.3) so
that regardless of the constancy or not of the shape of the
polarization ellipse, its size varies during a typical measurement
time, resulting in an average size that is proportional to the
corresponding average intensity
I A t A t A t A t a ax y x y x y= ( ) + ( ) = ( ) + ( ) = +2 2 2 2
2 2 (1.5)
where ⟨ ⟩ indicates time average over the measurement time and a A
tx x 2 2≡ ( ) , a A ty y
2 2≡ ( ) .
t
Π
Figure 1.5 The field variables fluctuate slowly in comparison to
the mean natural period T0, and therefore the shape of the ellipse
remains constant during time intervals shorter than the coherence
time τ.
1.2 Polarization ellipse 5
Thus, a totally polarized state is fully described by the
characteristic parameters of the polarization ellipse, namely, the
intensity I (average size), the ratio ay/ax ≡ tan α (0 ≤ α ≤ π/2),
and the phase shift δ (0 ≤ δ ≤ 2π). Alternatively, the polarization
ellipse can be characterized through I together with the azimuth φ
(0 ≤ φ < π) and the ellipticity angle χ (−π/4 ≤ χ ≤ π/4) (Figure
1.6), which are related to ax, ay, α, and δ by the equations
tan cos tan cos sin sin sin2 2
2 2 2
= = +
x y 22α δsin (1.6)
The polarization ellipse is represented for different values of δ
in Figure 1.7.
Values δ = 0, π correspond to linearly polarized states (χ = 0),
regardless of the value of α. Values δ = π/2, 3π/2 correspond to
states whose polarization ellipse has its semiaxes aligned along
the
reference axes XY, or to right-handed and left-handed circularly
polarized states, respectively.
Y0
X0
Y
X
a
b
χ
Figure 1.6 The polarization ellipse is characterized by I = a2 + b2
together with the azimuth φ (0 ≤ φ < π) and the ellipticity
angle χ (−π/4 ≤ χ≤ π/4); a and b are the semiaxes of the
polarization ellipse.
δ = 0 δ = π/4
= π/2 χ = 0
χ = 0 = 0
= π/4 χ = 0
= indet χ = π/4
= –π/4 χ = –π/8
= π/4 χ = –π/8
= –π/4 χ = 0
1 2
Figure 1.7 Values δ = 0, π correspond to linearly polarized states
(χ = 0). Values δ = π/2, 3π/2 correspond to states whose
polarization ellipse has its semiaxes aligned along the reference
axes XY, or to right-handed and left-handed circularly polarized
states, respectively. Intermediate values of δ correspond to states
with χ ≠ 0. Positive and negative values of the ellipticity angle
correspond, respectively, to right-handed and left-handed elliptic
polarized states. Right-handed and left-handed circular polarized
states correspond, respectively, to the particular combined values
(δ = π/2, α = π/4) and (δ = 3π/2, α = π/4). For a fixed value of α,
the area of the ellipse is maximum when δ = π/2 or δ = 3π/2, and it
is zero when δ = 0 or δ = π (linearly polarized states).
6 Polarized electromagnetic waves
Intermediate values of δ correspond to states with χ ≠ 0. Positive
and negative values of the ellipticity angle correspond,
respectively, to right-handed and left-handed elliptic polarized
states. Right-handed and left-handed circular polarized states
correspond, respectively, to the particular combined values (δ =
π/2, α = π/4) and (δ = 3π/2, α = π/4).
Note that in this book we assume the common convention that
right-handed states correspond to clockwise-handedness with respect
to an observer toward whom the wave travels (i.e., the wave
propagates toward the reader in Figure 1.7).
The ellipticity is defined as the ratio ±b/a = tan χ, where b and a
are the respective minor and major axes of the polarization
ellipse, the plus and minus signs corresponding, respectively, to
right-handed and left- handed states. The eccentricity of the
polarization ellipse is given by 1 2− tan χ .
The area A of the polarization ellipse of a totally polarized state
is
A a a Ix y= =π δ π χsin sin 4
22 2 (1.7)
so that, as expected, the normalized area  ≡ A/I2 has its maximum
value Âmax = π/4 for circularly polarized states, and its minimum
Âmin = 0 for linearly polarized states.
Even though the concept of polarization applies in principle to the
evolution of the electric field of the wave at a given point r, it
is straightforward to extend this concept to all points distributed
on a certain region of a common wavefront and sharing a common
state of polarization. In these cases of polarization states
relative to waves with arbitrary form, the polarization is referred
to the local reference frame constituted by a pair of axes XY lying
in the local plane Π tangent to the wavefront surface at the
considered point r, together with an axis Z orthogonal to Π, which
determines the local direction of propagation at point r (Figure
1.4).
1.3 ANALYTIC SIGNAL REPRESENTATION AND THE JONES VECTOR
Following the common practice in polarization optics, and under the
assumption of quasi-monochromatic- ity, it is very advantageous to
use the analytic signal representation where the components of the
fields are described through their respective complex variables.
For a detailed description and rigorous foundation of the analytic
signal representation of the wavefield, we redirect the reader to
Wolf (1959) and Brosseau (1998).
The stochastic analytic signal representations ηx(t), ηy(t) of the
two mutually orthogonal components of the electric field are
formulated as
η
η
i u t t
t E t iE t
x( ) = ( ) + ( ) = ( )
(1.8)
where u t kz t( ) ≡ − ω , and E tx ( ), E ty ( ) are the stochastic
Hilbert transforms of the real components of the electric field
(Wolf 1959)
E t E t t t
dt k x yk k( ) =
′( ) ′ −
′ =( ) −∞
∞
, (1.9)
The real field variables Ek(t) are zero-mean, and we assume that
they are also temporally stationary (at least in the wide sense)
random processes obeying the same statistics and having the same
power spectrum (ν). The Hilbert transforms E tk ( ) obey the same
statistics and also have the same power spectrum (ν) as the real
components, so that the analytic signals ηx(t), ηy(t) are also
zero-mean, temporally stationary (at least in the wide sense)
random processes with a common power spectrum given by 4(ν) for ν
> 0 and zero for ν < 0.
1.3 Analytic signal representation and the Jones vector
7
x x x
y y y
ω β (1.10)
The variables ηx(t) and ηy(t) can be arranged as the components of
the following 2 × 1 complex vector:
η η η
y i t t
(1.11)
Measurable quantities as, for instance, the Stokes parameters
(Stokes 1852; Fano 1953), which will be con- sidered later, involve
necessarily time or ensemble averaging of second-order products
such as η ηi jt t( ) ( )∗ (or even higher-order products) taken at
a fixed point z so that, as it has been done for the real
representation in Equation 1.1, the global phase factor can be
removed in the description of polarization states in terms of
observables. Consequently, the instantaneous Jones vector is
defined as
ee t A t e
A t e x
2 (1.12)
Note that ε(t) is defined up to a nonmeasurable global phase factor
eiφ. In the general case of a polychro- matic wave, the
instantaneous Jones vector has slow time dependence with respect to
the coherence time, so that for time intervals shorter than the
coherence time, the polarization ellipse can be considered
constant. For time intervals larger than the coherence time of the
electromagnetic wave, the instantaneous Jones vector can vary,
resulting in partial polarization.
The instantaneous Jones vector includes all measurable information
relative to the temporal evolution of the electric field. As for
the polarization ellipse and for the intensity, ε(t) is called
instantaneous in the sense that the possible time dependence of the
amplitudes and relative phase is considered.
Let us now consider the particular case where the quantities
Ay(t)/Ax(t) and δ(t) remain constant in time, and consequently, the
shape of the polarization ellipse remains fixed during the
measurement time. The cor- responding state of polarization is
described by means of the Jones vector (Jones 1941),
ee ≡
2 (1.13)
where ax and ay are respectively given by the averages a Ax x 2 2≡
and a Ay y
2 2≡ of the respective square of the amplitudes Ax(t) and Ay(t)
over the measurement time T.
Leaving aside a global phase factor, the Jones vector can also be
expressed in terms of the intensity I, the azimuth φ, and the
ellipticity angle χ as follows:
ee = − +
cos sin si
nn cos
i (1.14)
which can be interpreted in the following manner (from right to
left): the rightmost vector, a function of χ, represents the Jones
vector of an elliptic state whose semiaxes are aligned with the
reference laboratory axes
8 Polarized electromagnetic waves
X and Y; the matrix, a function of φ, is a rotation matrix that
rotates the said Jones vector by the angle φ; and the scalar factor
I represents the overall amplitude (i.e., the square root of the
intensity of the state).
Thus, a totally polarized state is fully described by its
corresponding Jones vector ε, which provides com- plete information
about the characteristic quantities of the polarization ellipse, as
well as the intensity. The definition (1.13) of the Jones vector is
consistent with the fact that total polarization is compatible with
inten- sity fluctuations. In fact, totally polarized waves maintain
the azimuth and ellipticity of the polarization ellipse fixed,
whereas the size of the ellipse fluctuates, resulting in a mean
intensity over the measurement time. Moreover, slow time variations
of the Jones vector with respect to the measurement time can be
repre- sented by this model (Gil 2007).
Jones vectors have been defined in Equation 1.13 with respect
to a XY reference frame in plane Π (Figure 1.3) in such a
manner that a generic Jones vector ε can be written as
ee = + ≡
ε εx x y y x ye e e e
1 0
0 1 (1.15)
where the basis vectors ex and ey represent respective linearly
polarized states whose electric fields lie along the axes X and
Y.
′ = ( ) ( ) = −
cos sin sin cos (1.16)
where the orthogonal matrix Q corresponds to a proper
counterclockwise rotation, by the angle θ about the axis Z, from
the original reference frame XY to the new axes X′Y′ (Figure
1.8).
Moreover, in general, any pair of complex vectors (e1, e2)
satisfying
e e e e e e1 2 1 1 2 20 1† † †= = = (1.17)
where the superscript † denotes conjugate transpose, constitutes a
generalized orthonormal basis. Thus, pairs of mutually orthogonal
linear, elliptical, or circular states can be used as generalized
reference bases by trans- forming the canonical basis (ex, ey)
through unitary transformations like
′ = =( )−ee eeU U U† 1 (1.18)
Y
X'Y'
Xθ
Figure 1.8 A change of coordinate frame from XY to X′Y′ for the
representation of Jones vectors is performed to an orthogonal
transformation of the form ε′ = Q(θ)ε, where Q corresponds to a
proper counterclockwise rotation by the angle θ, around the axis Z,
from the original reference frame XY to the new axes X′Y′.
1.3 Analytic signal representation and the Jones vector
9
Particularly interesting alternative bases are the linear +45° and
linear –45° (e+π/4, e−π/4) defined by the basis vectors
e e U Q+ −≡
(1.19)
and the right-handed and left-handed circular (er, el), defined by
the basis vectors
e e Ur li i i i ≡
≡
−
=
−
1 1 2
1 1 2
1 1 (1.20)
Despite the fact that, unless otherwise stated, in this book the
polarization states are described with respect to the basis (ex,
ey), the generic notation ε = ε1e1 + ε2e2 is used in order to
indicate the validity of the mathematical expressions regardless of
the particular basis chosen.
Generalized bases containing complex components are very useful for
some purposes, for example, rep- resenting a pure state as a
coherent superposition of a right-handed and a left-handed
circularly polarized state. However, such types of generalized
bases involving imaginary parameters, while being algebraically
acceptable, are not physically realizable as laboratory reference
frames. In fact, only orthogonal transforma- tions of the form ei′
= Qei (i = x, y), where Q is a 2 × 2 orthogonal matrix (i.e., a
unitary matrix that is real), are admissible for generating
physically realizable laboratory reference frames.
The scalar product of two Jones vectors μ and ν is defined as
mm nn† ,= ( )
µ ν µ ν1 2 1
2 1 1 2 2 (1.21)
and the squared absolute value |ε|2 = ε†ε = I of a given Jones
vector ε is precisely the intensity of the corre- sponding state of
polarization. Two pure states represented by respective Jones
vectors μ and ν are said to be orthogonal when μ†ν = 0. Moreover,
the product of a Jones vector ε by a complex number t produces a
new Jones vector ε′ = tε.
To complete this brief survey of the algebraic properties of Jones
vectors, let us now consider the coherent superposition, at a given
point r, of two totally polarized waves whose respective
polarization ellipses lie in a common plane Π (Figure 1.9).
The composed wave at point r is totally polarized, and its Jones
vector is given by the addition of the Jones vectors of the
mutually coherent components
ee ee ee= +1 2 (1.22)
Due to the very definition of the Jones vector, it cannot represent
partially polarized states. The use of Jones vectors is restricted
to totally polarized (or pure) states. In the case of partially
polarized states, the
Coherent
Y Π
X Z
Y Π
X Z
Y Π
Figure 1.9 Coherent superposition, at a given point r, of two
totally polarized waves whose respective polar- ization ellipses
lie in a common plane Π. The composed wave at point r is totally
polarized, and its Jones vector is given by the addition of the
Jones vectors of the mutually coherent components.
10 Polarized electromagnetic waves
azimuth or the ellipticity of the polarization ellipse varies
during the measurement time and a different math- ematical
description, different from that used in the Jones approach, is
necessary in order to take into account all parameters that
characterize completely the state of polarization.
1.4 COHERENCY MATRIX AND STOKES VECTOR
The analytic signals of the components of the electric field of the
wave are zero-mean variables that can be considered ergodic
stochastic processes whose complete statistical description,
equivalent to the bivariate joint probability distribution function
for the two real components of the electric field of the wave,
requires in general the knowledge of all their n-order moments. In
the particular case of waves with a Gaussian spectral profile, as
is the case of thermal light, the second-order moments are
sufficient (Brosseau 1998). Nevertheless, there are important cases
in practice where higher-order moments play an important role,
especially for radi- ation emitted by certain artificial sources,
as well as in the quantum domain. In the second-order approach,
polarization refers to the second-order moments of the zero-mean
analytic signals (ε1, ε2) at a given fixed point in space.
A proper description of the second-order polarization properties of
electromagnetic waves relies on the concept of the coherency
matrix. This mathematical formulation is applicable regardless of
the particular spectral range of the electromagnetic spectrum
considered.
1.4.1 2D coherency matrix
The 2D coherency matrix (or polarization matrix) Φ (Wiener 1930;
Wolf 1959; Barakat 1963), is defined as
FF ee ee= ( )⊗ ( ) = ( ) ( ) ( ) ( )
(1.23)
where ε is the instantaneous Jones vector whose two components are
the analytic signals of the electric field of the wave, ⊗ stands
for the Kronecker product, and the brackets indicate time averaging
over the measure- ment time
x t T
0
(1.24)
As a result of this definition, Φ is a 2 × 2 covariance matrix
(i.e., a positive semidefinite Hermitian matrix) that contains all
second-order measurable information about the 2D state of
polarization (includ- ing intensity). Under the assumption that the
stochastic processes (ε1, ε2) are stationary and ergodic, the
brackets can alternatively be considered ensemble averaging of
ε⊗ε†, where ε(t) in Equation 1.23 are simple
realizations.
The statistical definition of Φ as a covariance matrix entails the
fact that its two eigenvalues are nonnega- tive. These constraints
constitute a complete set of necessary and sufficient conditions
for a Hermitian matrix Φ to be a coherency matrix, that is, to
represent a particular 2D state of polarization of an
electromagnetic wave at a given point in space.
The elements ij (i.j = 1, 2) of Φ can be written as follows in
terms of the corresponding standard deviations σ1, σ2 and the
complex degree of mutual coherence μ:
FF =
1 2
1.4 Coherency matrix and Stokes vector 11
where
σ φ ε σ φ ε µ φ σ σ
φ φ φ
≡ = ≡ = = =( ) ( )t t (1.26)
For some purposes, it is useful to consider the normalized
coherency matrix (or polarization density matrix)
ˆ tr
≡ (1.27)
which in turn can be interpreted as the density matrix containing
complete information about the popula- tions and coherences of the
polarization states (Fano 1953, 1957).
Coherency matrices inherit, as an underlying reference basis, the
generalized reference basis e1, e2 used for representing the
analytic signals constitutive of the two components of Jones
vectors. Unless otherwise stated, Φ will be considered as described
with respect to the underlying canonical basis constituted by the
orthonormal set of column vectors (1, 0)T, (0, 1)T (where the
superscript T stands for transposition). Moreover, regardless of
the underlying reference basis considered, the coherency matrix Φ
can be expressed as a linear expansion, with real coefficients, on
the following matrix basis constituted by the three Pauli matrices
plus the identity matrix
ss ss ss ss0 1 2 3 1 0 0 1
1 0 0 1
0 1 1 0
i (1.28)
Note that the notations σ1, σ2 (plain letters) are used for the
variances in Equation 1.25 and σi (bold letters) are used for
the Pauli matrices (in order to preserve the common notations used
in related works), but this should not lead to confusion because σi
are matrices, while the variances σ1, σ2 are scalar
quantities.
These well-known linearly independent matrices σi have interesting
properties as hermiticity ss ssi i= † and trace-orthogonality
tr(σiσj) = 2δij (δij being the Kronecker delta) and satisfy
ssi
2 2= I , (I2 being the 2 × 2 identity
matrix). Therefore, σi are also unitary and, except for σ0, are
traceless.
1.4.2 StokeS vector
As mentioned above, it is straightforward to show that Φ always
admits the following linear expansion (Falkoff and Macdonald 1951;
Fano 1953):
FF ss= = ∑1
s ii i= ( ) =( )tr FFss 0 1 2 3, , , (1.30)
or, in the explicit form,
s t t t t
s t t
1 11 22 1 1
= + = ( ) ( ) + ( ) ( )
= − = ( ) ( ) −
∗ ∗
∗
2 12 21 1 2 2 1
3 12 21
s i
φ φ ε ε ε ε
φ φ == ( ) ( ) − ( ) ( )( )∗ ∗i t t t tε ε ε ε1 2 2 1 (1.31)
12 Polarized electromagnetic waves
The quantities s0, s1, s2, s3 are the so-called Stokes parameters
(Stokes 1852) and constitute a complete set of measurable
parameters, which allow for expressing Φ in the following
manner:
FF = + − + −
s s s i s s i s s s
(1.32)
The nonnegativity of Φ (i.e., the positive semidefiniteness of Φ)
entails the following constraints, which constitute a pair of
necessary and sufficient conditions for a Hermitian matrix to be a
covariance matrix:
tr detFF FF= ≥ = − − − ≥s s s s s0 0 2
1 2
2 2
3 20 4 0 (1.33)
Consequently, any set of four parameters s0, s1, s2, s3 satisfying
conditions (1.33) can be considered a physi- cally realizable set
of Stokes parameters. Even though the indicated notation is
commonly used in many related works, it is important to warn the
reader that the Stokes parameters are also frequently noted as I,
Q, U, V.
The Stokes parameters are usually arranged as a 4 × 1 Stokes vector
s:
s ≡
0
1
2
3
(1.34)
Let us note now that the relation between Φ and s can also be
expressed as
s =Ljj (1.35a)
L = −
−
1 0 0 1 1 0 0 1 0 1 1 0 0 0i i
(1.35b)
and the coherency vector φ is defined as the column vector whose
components are the elements of the coher- ency matrix arranged in
the following manner:
jj ≡
ee ee (1.35c)
When appropriate, Stokes vectors and other column vectors are
expressed in the horizontal notation as s ≡ (s0, s1, s2, s3)T.
Equation 1.31 shows that, obviously, the information contained
in s (or φ) is completely equiv- alent to that provided by Φ.
It should be noted that the term vector is used here in a very wide
sense as referring to s as the indicated 4-tuple. The
multiplication of a Stokes vector s by a real scalar c produces a
Stokes vector s′ = cs = sc if and only if c ≥ 0. The resultant
Stokes vector s′ represents the same state of polarization as s up
to a positive scale factor that only affects the intensity, I(s′) =
cI(s). Moreover, in general, the addition of two Stokes vectors s1
and s2 only has physical meaning in the form s = s1 + s2, and not
as a subtraction s1 − s2. In fact, the addition
1.4 Coherency matrix and Stokes vector 13
represents the incoherent superposition of two 2D states whose
respective polarization ellipses lie in a com- mon plane. The
intensity of the resultant Stokes vector is given by the sum of the
intensities of the superposed states I(s) = (s1) + (s2). However,
there are situations where the subtraction can be physically
admissible; if we consider the superposition represented by s = s1
+ s2, then the Stokes vector s1 can be considered the result of the
polarimetric subtraction s1 = s − s2 in the sense that s2 is a
Stokes vector that, added to s1, gives a Stokes vector s = s1 + s2,
which represents the incoherent superposition of s1 and s2. The
polarimetric subtraction is a relevant concept in polarimetry that
will be dealt with in later sections.
Obviously, since negative intensities do not have physical meaning,
given a Stokes vector s ≠ 0, it does not have an inverse Stokes
vector with respect to the + operation. Consequently, the set of
Stokes vectors
s s s s s s s s s T
0 1 2 3 0 0 2
1 2
2 2
3 20, , , ,( ) ≥ ≥ + +{ }, together with the product (.) by a
nonnegative scalar and the sum (+),
constitutes a semiring algebraic structure, and not a vector space.
Even though the use of generalized bases for the representation of
Jones vectors is not unusual, Stokes vec-
tors are always considered to refer to an underlying real basis
(ex, ey), that is, to a laboratory reference frame XYZ, Z being the
direction of propagation.
′ = ( ) ( ) ≡ −
θ θ θ θ
1 0 0 0 0 2 2 0 0 2 2 0 0 0 0 1
cos sin sin cos
(1.36)
where the orthogonal matrix MG(θ) corresponds to a proper
counterclockwise rotation by the angle θ about the axis Z, from the
original X reference axis to X′.
A Stokes vector sp satisfying
s Gs Gp T
p p p p ps s s s= − − − = ≡ − − −( )0 2
1 2
2 2
3 2 0 1 1 1 1diag , , , (1.37)
corresponds to a totally polarized state and is said to be a
totally polarized or pure Stokes vector. The matrix G represents
the Minkowskian metric.
Two pure Stokes vectors s1, s2 are said to be mutually orthogonal
when their corresponding Jones vectors ε1, ε2 are mutually
orthogonal ee ee1 2 0† =( ), so that the mutual orthogonality of
s1, s2 is expressed by the fact that the scalar product of s1 and
s2 is zero, s s1 2 0T = . In other words, a pure Stokes vector s1 =
(s0, s1, s2, s3)T is said to be orthogonal to another pure Stokes
vector s2 when s Gs2 1 0 1 2 3
T T T s s s s= = − − −( ), , , .
Multiplication by a positive scalar and additive compositions
translate directly from the space of Stokes vectors to the space of
2D coherency matrices, and consequently, both formalisms are
completely equivalent with regard to their physical
interpretation.
A Stokes vector can always be expressed as
s = + +( ) + − + +( )s s s s s s s s s s T T
1 2
2 2
3 2
2 2
3 2 0 0 0, , , , , , (1.38)
so that s can be interpreted as an incoherent superposition of a
pure state (first addend, hereafter called the characteristic
component) and an unpolarized state (hereafter called the 2D
unpolarized component). The characteristic component defines
the average polarization ellipse (or characteristic polarization
ellipse) of the whole state s. Furthermore, s can be
parameterized as
s =
2
sin
χ
where
I = s0 is the intensity, or power density flux through the
reference plane Π containing the polarization ellipse.
P ≡ + +s s s s1 2
2 2
3 2
0 is the degree of polarization of the 2D state represented by s. P
is a dimensionless quantity whose values are restricted to 0 ≤ P ≤
1. The maximum P = 1 corresponds to totally polarized states that
therefore can also be represented by respective Jones vectors.
Intermediate values 0 < P < 1 correspond to partially
polarized states; the higher the value of the degree of
polarization P, the higher is the correlation (or mutual coherence)
of the field components. The minimum P = 0 corresponds to
unpolarized states, that is, to states with a completely random
temporal distribution of the polarization ellipse or, in other
words, to states with zero correlation between the field
components.
The azimuth φ, with 0 ≤ φ < π, is that of the direction of the
major semiaxis of the characteristic polar- ization ellipse with
respect to the given reference axis X.
The ellipticity angle χ, with −π/4 ≤ χ ≤ π/4, is that of the
characteristic polarization ellipse.
The above parameterization provides an interpretation of 2D states
of polarization in terms of meaningful physical quantities. By
taking into account the above analysis, the Stokes parameters can
also be interpreted as follows:
s0 is the intensity, given by the sum of the intensities associated
with the components of the electric field with respect to any
orthonormal generalized basis: s I I I I I I I Ix y r l0 45 45 1 2=
+ = + = + = ++ ° − ° e e .
s1 is the difference between the respective intensities
corresponding to the components of the electric field with respect
to the canonical basis (ex, ey) (see Equation 1.15), s1 = Ix
−Iy. s1 = 1 (φ = 0, χ = 0) for lin- early x-polarized states; s1 =
−1 (φ = π/2, χ = 0) for linearly y-polarized states. A simple
procedure for the measurement of the parameter s1 of a plane wave
consists of two consecutive intensity measurements by placing a
linear polarizer (usually called analyzer) at 0° and 90° before the
detector.
s2 is the difference between the respective intensities
corresponding to the components of the elec- tric field with
respect to the basis (e+π/4, e−π/4) (see Equation 1.19), s2 =
I+π/4 − I−π/4. s2 = 1 (φ = π/4, χ = 0) for linearly +45°-polarized
states; s2 = −1 (φ = 3π/4, χ = 0) for linearly –45°-polarized
states. A simple procedure for the measurement of the parameter s2
of a plane wave consists of two consecutive intensity measurements
by placing a linear analyzer at +45° and –45° before the
detector.
s3 is the difference between the respective intensities
corresponding to the components of the elec- tric field with
respect to the basis (er, el) (see Equation 1.20), s3 = Ir−Il.
s3 = 1 (χ = π/4) for right-handed circular polarized states; s3 =
−1 (χ = −π/4) for left-handed circularly polarized states. A simple
pro- cedure for the measurement of the parameter s3 of a plane wave
consists of two consecutive intensity measurements by placing a
right-circular analyzer and a left-circular analyzer before the
detector. Note that a circular polarizer (analyzer) can be achieved
by the serial combination of a linear total polarizer and a
quarter-wave plate whose respective eigenaxes make an angle of
45°.
At this point, it is worth summarizing some preliminary conclusions
derived from the above analysis:
Two-dimensional unpolarized states entail the equality of the
intensities associated with the respec- tive pair of orthogonal
components of the electric field with respect to the bases (ex,
ey), 0 = s1 = Iy − Ix; (e+45°, e−45°), 0 = s2 = I+45° −I−45°, and
(er, el), 0 = s3 = Ir−Il. In fact, the fulfillment of these three
equalities implies necessarily the equality I1 = I2 of the
intensities I1, I2 associated with the respective pair of orthog-
onal components of the electric field with respect to any
generalized orthonormal basis e1, e2. As pointed out in the seminal
works of Stokes (1852) and Verdet (1869), the indicated invariance
is an essential and characteristic property of unpolarized states.
There are various random distributions that correspond to
unpolarized waves. As Ellis and Dogariu (2004a) have shown, the
measurement of the correlations of the Stokes parameters allows for
distinguishing between the different types of unpolarized
states.
Any 2D polarization state s can be considered the result of the
incoherent superposition of the characteristic component s p
Ts s s s s s≡ + +( , , , )1 2
2 2
3 2
Ts s s s≡ − + +( , , , )0 1 2
2 2
3 2 0 0 0 . That is, a 2D state represented by a given Stokes
vector s ≡ ( , , , )I s s s T
1 2 3
1.4 Coherency matrix and Stokes vector 15
is polarimetrically indistinguishable from an incoherent
combination of two states propagating in the same direction,
namely, a pure state sp with intensity I s s s Ip = + + =1
2 2 2
with intensity I I s s s Iu = − + + = −1 2
2 2
3 2 1( )P . For pure states that are characterized by the equality
P = 1,
the total intensity is associated with the pure contribution
(characteristic component) and the shape of the polarization
ellipse is constant for time intervals larger than the measurement
time. For 2D unpo- larized states that propagate in a well-defined
direction and satisfy the equality P = 0, the total intensity is
associated with the unpolarized contribution where the shape of the
polarization ellipse fluctuates in a completely random manner
during the measurement time.
The degree of polarization P is just the ratio of the intensity I s
s s Ip = + + =1 2
2 2
3 2 P of the characteristic
component to the intensity I = Ip + Iu of the entire state
(Figure 1.10). Thus, P is a dimensionless and nonnegative
quantity limited by the double inequality 0 ≤ P ≤ 1. Moreover, P is
invariant with respect to any rotation of the underlying reference
frame XY about the direction of propagation Z. Furthermore, from a
more general point of view, P is invariant with respect to any
change of the generalized underly- ing reference basis (e1, e2),
that is, with respect to any unitary transformation of the basis
vectors (e1, e2).
An alternative formulation of the polarimetric purity of a 2D state
of polarization is given by the randomness, or degree of
depolarization, defined as D ≡ s GsT I= ( )2 2tr trFF FF = −1 P2 ,
which is a measure of the randomness of the polarization ellipse.
Obviously, D2 + P2 = 1, and therefore 0 ≤ D ≤ 1; D = 0 for totally
polarized states, while D = 1 for unpolarized states. Note that the
quantity s GsT I= 2 2D (which is intensity dependent and has the
same dimension as I2) was called the mean randomness by Barakat
(1987a).
The characteristic component determines the corresponding
characteristic polarization ellipse with semiaxes
a s s s s s b s s s s s= + + + +( ) = + + − +( )1 2
1 21
A s I= =π π χ 4 4
23 2 2 2 2P sin (1.41)
s3 is a measure of twice the magnitude n of the angular momentum n
of the state s (Figure 1.11). The vector n lies along the axis
Z (i.e., along the direction of propagation at the point r
considered). Since the Stokes vector s associated with a given
state of polarization is scaled by the intensity I = s0, the
normalized angular momentum n k≡ ( )s s3 0/2 (where k is the unit
vector along the positive Z direction) provides an appropriate way
to represent this property regardless of the value of I. Therefore,
the scalar value n s s≡ 3 02 of the normalized angular momentum is
restricted by − ≤ ≤1 2 1 2n . Thus, n = +1 2 corresponds to
right-handed circularly polarized pure states, n = −1 2 corresponds
to left-handed circularly polarized pure states, while the minimum
n = 0 is reached when s3 = 0, that is, for states whose
characteristic polarization ellipse has zero ellipticity, including
the particular cases of linearly polarized
Ip
Y
Z
Π
Figure 1.10 The degree of polarization P of a 2D state is defined
as the ratio of the intensity Ip of the totally polarized component
to the intensity I = Ip + Iu of the whole state.
16 Polarized electromagnetic waves
pure states, as well as unpolarized states. For states whose
characteristic polarization ellipse has positive ellipticity (i.e.,
with right-handedness), n is parallel to k, while n is antiparallel
to k for states whose characteristic polarization ellipse has
negative ellipticity (i.e., left-handedness). Since the state of
polar- ization refers to a given point r in space, all previous
comments about the angular momentum refer to the spin or intrinsic
angular momentum, despite the possibility of considering a complete
or partial spatial region of the wavefront and its associated
orbital angular momentum (Gori et al. 1998). Thus, the total
angular momentum of the electromagnetic wave can often be separated
into two parts, the spin angular momentum (wave polarization) and
the orbital angular momentum, which is determined by the spatial
variation in intensity and phase (Van Enk and Nienhuis 1992; Gori
et al. 1998). The orbital angu- lar momentum can, in turn, be
further decomposed into (1) an origin-independent angular momentum
that is associated with the helical or twisted properties of the
shape of the wavefront (internal orbital angular momentum) and (2)
an origin-dependent angular momentum given by the vector product of
the position vector of the center of the electromagnetic beam and
its total linear momentum (external orbital angular
momentum).
1.5 2D SPACE–TIME AND SPACE–FREQUENCY REPRESENTATIONS OF COHERENCE
AND POLARIZATION
Electromagnetic waves exhibit randomness due to the random
fluctuations associated with the spontaneous or stimulated emission
of photons by matter, as well as to random fluctuations in the
propagation medium. The degree of correlation of the emission
processes caused by myriads of atoms or molecules of the source
material located closely to each other leads to a certai