1. Vectorial Propagation of Light ............................... 1
1.1. Maxwell's Equations and Free-Space Solutions ............... 2
1.2. The Vector and Scalar Potentials ........................... 8
1.3. Time-Harmonic Solutions ................................... 10
1.4. Classical Description of Polarization ..................... 12
1.4.1. Stokes Vectors, Jones and Muller Matrices .......... 17
1.4.2. The Poincare Sphere ................................ 20
1.5. Partial Polarization ...................................... 22
1.5.1. Coherently Polarized Waves ......................... 24
1.5.2. Incoherently Depolarized Waves ..................... 28
1.5.3. Pseudo-Depolarized Waves ........................... 31
1.5.4. A Heterogeneous Ray Bundle ......................... 33
References ..................................................... 36
2. The Spin-Vector Calculus of Polarization .................... 37
2.1. Motivation ................................................ 37
2.2. Vectors, Length, and Direction ............................ 39
2.2.1. Bra and Ket Vectors ................................ 39
2.2.2. Length and Inner Products .......................... 41
2.2.3. Projectors and Outer Products ...................... 42
2.2.4. Orthonormal Basis .................................. 43
2.3. General Vector Transformations ............................ 44
2.3.1. Operator Relations ................................. 44
2.4. Eigenstates, Hermitian and Unitary Operators .............. 46
2.4.1. Hermitian Operators ................................ 47
2.4.2. Unitary Operators .................................. 48
2.4.3. Connection between Hermitian and Unitary
Matrices ........................................... 49
2.4.4. Similarity Transforms .............................. 49
2.4.5. Construction of General Unitary Matrix ............. 50
2.4.6. Group Properties of SU(2) .......................... 51
2.5. Vectors Cast in Jones and Stokes Spaces ................... 52
2.5.1. Complete Measurement of the Polarization Ellipse ... 52
2.5.2. Pauli Spin Matrices ................................ 54
2.5.3. The Pauli Spin Vector .............................. 55
2.5.4. Spin-Vector Identities ............................. 56
2.5.5. Conservation of Length ............................. 58
2.5.6. Orthogonal Polarization States ..................... 59
2.5.7. Non-Orthogonal Polarization States ................. 60
2.5.8. Pauli Spin Operators ............................... 61
2.6. Equivalent Unitary Transformations ........................ 63
2.6.1. Group Properties of SU(2) and 0(3) ................. 65
2.6.2. Matrix Entries of R in a Fixed Coordinate System ... 66
2.6.3. Vector Expression of R in a Local Coordinate
System ............................................. 67
2.6.4. Select Vector Identities ........................... 70
2.6.5. Euler Rotations .................................... 71
2.6.6. Some Relevant Transformation Applications .......... 72
References ..................................................... 78
3. Interaction of Light and Dielectric Media ................... 79
3.1. Introduction of Media Terms into Maxwell's Equations ...... 80
3.2. Constitutive Relation Tensors ............................. 85
3.3. The kDB System ............................................ 87
3.4. The Lorentz Force ......................................... 90
3.5. Isotropic Materials ....................................... 90
3.5.1. Permittivity of Isotropic Materials ................ 91
3.5.2. Propagation in Isotropic Materials ................. 94
3.5.3. Refraction at an Interface ......................... 96
3.5.4. Reflection and Transmission for ТЕ Waves ........... 96
3.5.5. Reflection and Transmission for TM Waves ........... 99
3.5.6. Total Internal Reflection ......................... 101
3.6. Birefringent Materials ................................... 105
3.6.1. Propagation in Uniaxial Materials ................. 106
3.6.2. Refraction at an Interface ........................ 112
3.6.3. Total Internal Reflection ......................... 117
3.6.4. Polarization Transformation ....................... 120
3.7. Gyrotropic Materials ..................................... 122
3.7.1. Magnetic Material Classes ......................... 123
3.7.2. Permittivity of Diamagnetic Materials ............. 124
3.7.3. Propagation in Gyrotropic Materials ............... 126
3.7.4. Faraday Rotation .................................. 129
3.7.5. The Verdet Constant ............................... 132
3.7.6. Faraday Rotation in Ferrous Materials ............. 133
3.8. Optically Active Materials ............................... 135
3.8.1. Propagation in Bi-Isotropic Media ................. 138
References .................................................... 142
4. Elements and Basic Combinations ............................ 143
4.1. Wavelength-Division Multiplexed Frequency Grid ........... 143
4.2. Properties of Select Materials ........................... 146
4.2.1. Isotropic Glass Materials ......................... 146
4.2.2. Birefringent Crystals ............................. 147
4.2.3. Iron Garnets ...................................... 150
4.2.4. Packaging Alloys .................................. 153
4.3. Fabry-Perot and Gires-Tournois Interferometers ........... 154
4.3.1. Fabry-Perot Response .............................. 157
4.3.2. Gires-Tournois Response ........................... 161
4.4. Temperature Dependence of Select Birefringent Crystals ... 163
4.4.1. Experimental Setup ................................ 163
4.4.2. Quadratic Temperature-Dependence Model ............ 166
4.4.3. Association of Resonant Peak Shift With
Temperature Coefficients .......................... 167
4.4.4. Group Index and Thermal-Optic Coefficients ........ 168
4.4.5. Passive Temperature Compensation .................. 170
4.5. Compound Crystals For Off-Axis Delay ..................... 173
4.6. Polarization Retarders ................................... 178
4.6.1. Half-Wave and Quarter-Wave Waveplates ............. 179
4.6.2. Birefringent Waveplate Technologies ............... 182
4.6.3. Waveplate Combinations ............................ 184
4.6.4. Elementary Polarization Control ................... 191
4.6.5. TIR Polarization Retarders ........................ 196
4.7. Single and Compound Prisms ............................... 198
4.7.1. Wollaston and Rochon Prisms ....................... 199
4.7.2. Kaifa Prism ....................................... 202
4.7.3. Shirasaki Prism ................................... 204
References .................................................... 208
5. Collimator Technologies .................................... 211
5.1. Collimator Assemblies .................................... 213
5.2. Gaussian Optics .......................................... 219
5.2.1. q Transformation and ABCD Matrices ................ 224
5.2.2. ABCD Ray Tracing .................................. 227
5.2.3. Action of a Single Lens ........................... 228
5.2.4. Action of a GRIN Lens ............................. 230
5.2.5. Some Limitations of the ABCD Matrix ............... 232
5.3. Select Collimators Analyzed with the ABCD Matrix ......... 234
5.4. Fiber-to-Fiber Coupling by a Lens Pair ................... 239
5.4.1. Coupling Coefficients ............................. 242
References .................................................... 245
6. Isolators .................................................. 247
6.1. Polarizing Isolator ...................................... 247
6.2. Comparison of Lens Systems ............................... 252
6.3. Deflection-Type Isolators ................................ 254
6.4. Displacement-Type Isolators .............................. 259
6.5. Two-Stage Isolators ...................................... 263
6.6. PMD-Compensated Isolators ................................ 266
References .................................................... 271
7. Circulators ................................................ 273
7.1. Polarizing Circulator .................................... 274
7.2. Historical Development ................................... 277
7.3. Displacement Circulators ................................. 279
7.4. Deflection Circulators ................................... 284
7.5. Summary .................................................. 294
References .................................................... 295
8. Properties of PDL and PMD .................................. 297
8.1. Polarization-Dependent Loss .............................. 298
8.1.1. Definitions ....................................... 299
8.1.2. Change of Polarization State ...................... 304
8.1.3. Repolarization .................................... 306
8.1.4. PDL Evolution Equations ........................... 308
8.2. Polarization-Mode Dispersion ............................. 312
8.2.1. A PMD Primer ...................................... 313
8.2.2. Fundamental Derivations ........................... 327
8.2.3. Connection Between Jones and Stokes Space ......... 330
8.2.4. Concatenation Rules for PMD ....................... 333
8.2.5. PMD Evolution Equations ........................... 338
8.2.6. Time-Domain Representation ........................ 342
8.2.7. Fourier Analysis of the DGD Spectrum .............. 364
8.3. Combined Effects of PMD and PDL .......................... 371
8.3.1. Frequency-Dependence of the Polarization State .... 372
8.3.2. Non-Orthogonality of PSP's ........................ 374
8.3.3. PMD and PDL Evolution Equations ................... 376
8.3.4. Separation of PMD and PDL ......................... 378
References .................................................... 381
9. Statistical Properties of Polarization in Fiber ............ 385
9.1. Polarization Evolution Model ............................. 388
9.1.1. Random Birefringent Orientation ................... 389
9.1.2. Random Component Birefringence .................... 391
9.2. Polarization Diffusion ................................... 392
9.3. RMS Differential-Group Delay Evolution ................... 397
9.4. PMD Statistics ........................................... 399
9.4.1. Probability Densities ............................. 401
9.4.2. Autocorrelation Functions ......................... 408
9.4.3. Mean-DGD Measurement Uncertainty .................. 414
9.4.4. Discrete Waveplate Model .......................... 417
9.4.5. Karhunen-Loeve Expansion of Brownian Motion ....... 419
9.5. PDL Statistics ........................................... 422
References .................................................... 425
10.Review of Polarization Test and Measurement ................ 429
10.1.SOP Measurement .......................................... 430
10.2.PDL Measurement .......................................... 432
10.3.PMD Measurement .......................................... 436
10.3.1.Mean DGD Measurement .............................. 438
10.3.2.PMD Vector Measurement ............................ 440
10.3.3.Polarization OTDR ................................. 450
10.4.Programmable PMD Sources ................................. 451
10.4.1.Sources of DGD and Depolarization ................. 454
10.4.2.ECHO Sources ...................................... 463
10.5.Receiver Performance Validation .......................... 478
References .................................................... 483
A. Addition of Multiple Coherent Waves ........................ 491
В. Select Magnetic Field Profiles ............................. 493
References .................................................... 496
С. Efficient Calculation of PMD Spectra ....................... 497
D. Multidimensional Gaussian Deviates ......................... 505
Index ......................................................... 509
|
