Table of contents for Nonlinear fiber optics / Govind P. Agrawal.


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1 Introduction                                                         1
.1   Historical Perspective . . .  . . ..    .   . .   .            1
1.2  Fiber Characteristics  . ..  . .  . .  . .  . .  .  .  . . .
1.2.1  Material and Fabrication . .  . . . . ..                4
1.212  Fiber Losses  .  .    .   . . . .  .      . .  . .      5
1.2.3  Chromatic Dispersion      .       . . .  . .            6
1.2.4  Polarization-Mode Dispersion .  . .  .  . .    .       1
1.3  Fiber Nonlinearities ..      . .  .      . .        .    . .  13
1.3.1  Nonlinear Refraction     .    .     . .  . .           14
1.3.2  Stimulated Inelastic Scattering  . .    ...            15
1.3.3  Importance of Nonlinear Effects .        .     . .     17
1.4  Overview      ..  .  .  .  .  .. .  .. .       .      ....    18
Problems    .. .                                   ...       ...   20
References .                                                 .  . ..  .  21
2  Pulse Propagation in Fibers                                        25
2.1  Maxwell's Equations .. .                                . . ..  25
22   Fiber Modes .......                 .                 . ..    27
2.2. 1  Eigenvalue Equation . . . .  . . . .         .   . .  28
2.2.2  Single-Mode Condition . .  . . . . ..         .   .    29
2.2,3  Characteristics of the Fundamental Mode     . .     .  30
2.3  Pulse-Propagation Equation . . . . . ...     .          . . .  31
2.3.1  Nonlinear Pulse Propagation  ..           .      ..     32
2.3.2  Higher-Order Nonlinear Effects       . .      .  . .    36
2.4  Numerical Methods .. . . .            . . . .   .       . .   41
2.4.   Split-Step Fourier Method .                   . ...    41
2.4.2  Finite-Difference Methods . ...             .  .  .    45
Problem s     .  .  ..... ...  .. . .   .. .  .  .                 46
References . . ..                .  .   .                          47
3  Group-Velocity Dispersion                                              51
3.1  Different Propagation Regimes.           . . .                    51
3.2  Dispersion-Induced Pulse Broadening              . . .            53
3.2.1  Gaussian Pulses .  . ........   . .       . . .            54
3 2.2  Chirped Gaussian Pulses  .      . .     .                  56
3.2.3  Hyperbolic Secant Pulses      .        . . .               58
3.2.4  Super-Gaussian Pulses  . .           . . .  . .  .  . .    58
3.2.5  Experimental Results .  .  . . . .                ..       61
3.3  Third-Order Dispersion .  .   .                . .        ..      62
3.3,1  Evolution of Chirped Gaussian Pulses .  . .   .  .   .     63
3 3.2  Broadening Factor .   ...                     .  ..    .   65
3.3.3  ArbitrarShap Pulses . . .    .    . . . . .  . . .  .      67
3,3.4  Ultrashort-Pulse Measurements   . . . . . .         . .    69
3.4  Dispersion Management ..      .   . . . . .  . . .  . .   . .     71
3.4.1  GVD-Induced Limitations .           .               ...    71
3.4.2  Dispersion Compensation               .       .    . . .   73
3.4.3  Compensation of Third-Order Dispersion              . .    74
Problem s                 .  .  .  .  . .  .. .  .  .  . .  .  .  .  .  .  76
References    .       .     .                                          77
4  Self-Phase Modulation                                                  79
4.1  SPM-Jnduced Spectral Changes .                                    79
411    Nonlinear Phase Shift . . . . ..                           80
4.1.2  Changes in Pulse Spectra . . .   ... . .                   82
4. 13  Effect of Pulse Shape and Initial Chirp .    . .8 . .  .   85
4.1.4  Effect of Partial Coherence .  .  .           . . . .  .   87
4.2  Effect of Group-Velocity Dispersion .  ...                   .    89
4.2.1  Pulse Evolution .        ... ..                            90
42. 2  Broadening Facto  . .            . .       . .  . . . .    91
4.2.3  Optical Wave Breaking .              .       .         .   94
4.2.4  Experimental Results  .            . .            . .  .   97
4.2.5  Effect of Third-Order Dispersion      . .    . . .  . .    98
4.2.6  SPM Effects in Fiber Amplifiers  .. . . .             .   100
4.3  Semianalytic Techniques  ..              .                 . .  .  102
4.3.1  Moment Method .                              . .  .       102
4.3.2  Variational Method  ....  . .  .  .     . . . . .  .  .    03
4..3   Specific Analytic Solutions .   .                  .      104
4.4 Hiiher-Order Nonlinear Effects     . . . . .. .            . .    106
4.4.1  Self-Steepenilng ... . .    .                     . . .   107
4.4.2  Effect of GVD on Optical Shocks .               .   .     1 09
4. 43  Intrapulse Raman Scatlernl g .  .         .      .   . .   1
Problems    ..     . . . .. .             .  .                . .     114
References .       . .                       .      .           .  .   16
: Optical Solitons                                                  120
51   M odulation Instability        ........ . .          .   .  120
5,1.1  Linear Stability  Analysis  .  . .  .  . .  .  .  .  .  .  .  .  .  .  .  .   121
5.1.2  Gain  Spectrum  .  .  . .  .  .  . . . . ..      ..   122
5.1 3  Experimental Results.  ..... .....            . ..    124
5.1.4  Ultrashort Pulse Generation  ..... . . .        .125
5..5  Impact on Lightwave Systems   .     .   . .  . . . . . . . .127
.2   Fiber Solitons     .  .  .  ..       . .  .  .  .  .  .  .  .  .   .  1 29
5.2.1  Inverse Scattering  Method  . .........  .  . . .  . .  130
5.2.2  Fundamental Soliton  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  132
5.2.3  Higher-Order Solitons  . . . .  .  . ........     .   134
5.2.4  Experimental Confirmation . . . .  .                  136
5.2.5  Soliton  Stability  .  ........  ...                .  ..   137
5.3  Other Types of Solitons  .  . .  .  . . . . . . . . .  .  . .  . . .  140
5.3.1  Dark  Solitons  .  .  .         . .  .  .  . .  .  . . .  .  140
53. 2  Dispersion-Managed Solitons . .                    . 144
5. 3  Bistable Solitons  . . .  .  .  . . . . .  . . . . .  .  .  144
5.4  Perturbation ofSolitons . ..  .... .                   .  .  146
5.4.1  Perturbation Methods .. . ....         .  . . . . . ..  146
5.4.2  Fiber Losses   ......     .  .  ...  .        .  ..   147
5.4.3  Soliton Amplification ... ..               .  .       149
5.4.4  Soliton Interaction  ...  .......    . .     . .. .   152
5.5  Higher-Order Effects ...........                       .  . 156
5.5 1  Moment Equations for Pulse Parameters .. . 156
5.5.2  Third-Order Dispersion ........        ...            158
S5.5.  Self-Steepening  .  . .   . a  . .  . .  .            160
.15.4  intrapulse Raman Scattering . . . ........       ..   162
5.55   Propagation of Fernosecond Pulses              . . .  167
Problem s  . . . .  .  . ..     ...  . . .. . . . . . . . . .  .  .   .   .   169
References                         .  .  .  .  .  .  .  .  .  .  . .  170
6 Polarization Effects                                              177
6.1  Nonlinear Birefrin ence  ..  ......   ...  .   .  .  .   .   177
6 1.1  Origin of Nonlinear Birefringencer   .          . .   178
6.1.2  Coupled-Mode Equations  . ... .  . ..  . .    . . ..  180
S.1.3  Elliptically  Birefringent Fibers  .. . .  .    . ..   181
6.2  Nonlinear Phase Shift . .  . .  . ..  . .  .  .   .  . .  . . .  182
6.2. 1  Nondispersive XPM      1.r........                .  182
6.2.2  Optical Kerr Effect  .  .  .  . .  .  .  .  .  ........  183
6.2.3  Pulse Shaping  .        .              . . . .  .  .  . .  187
6.3    olution of Polarization State  . . . . .  . . . . . . .    189
.3.1   Analtic Solution  .                 . . . . . . .     189
6.3.2  Poincare-Sphere Representation  . .  . .  . . . .  . . 191
6. .   Polarization Instabilit  .  .     . . .        .   .  194
6.. 4  Polarization Chaos                        .. . . . .  196
6.4  Vector Modulation  Instability  .  . . . . .   .   . . .  .  . . .  . . . 197
6.4.1  Low-Birefringence Fibers .  .  . . . .  . ..    .    197
6.4.2  High-Birefringence Fibers                      ..    200
6.4.3  isotropic Fibers ... .           . .   .    .  . . .  202
6.4.4  Experimental Results. ..                        .    203
6.5  Birefringence and Solitons .....                           9206
6.5.1  Low-Birefringence Fibers           . . .  .    . . 206
6.52  High-Birefringence Fibers           . .               207
6.5.3  Soliton-Dragging Logic Gates  . . . . . .  .  .    .  211
6.5.4  Vector Solitons  ...................        ..  . .  .  212
6.6  Random Birefringence . . ...                               213
6.6.1  Polarization-Mode Dispersion . .  . .          . .   2114
6.6.2  Vector Form of the NLS Equation . .            . .   215
6.6.3  Effects of PMD on Solitons . .               ..      216
Problems   ....        .  . .  .... .. ..220
References                        ...... .. ... .. .. . . ..    22,1
7  Cross-Phase Modulation                                          226
7.1 XPM-Induced Nonlinear Coupling  ..        .     . .     .   227
"7.1.1  Nonlinear Refractive Index . . . .         .       227
7.1.2  Coupled NLS Equations             . .  . . . . .     228
.2  XPM-Induced Modulation Instability ...                      229
7.21   Linear Stability Analysis  . .           .     . .   229
7.2.2  Experimental Results.  .    . . .  .  .  . .  . . . . .  232
"73  XPM-Paired Solitons  . . . . . . .                         233
7.3.1  Bright-Dark Soliton Pair ... . . . . .  . .         233
7.3.2  Bright-Gray Soliton Pair . . . .  .  .          .   234
7.33  Periodic Solutions . .        . . . . . . .      .    235
7.3.4  Multiple Coupled NLS Equations .    . . . .237
7.4  Spectral and Temporal Effects . . . . . .                  238
7.4.1  Asymmetric Spectral Broadening  . .                 239
7.4.2  Asymmetric Temporal Changes . . . . .        .       244
7.4.3  Higher-Order Nonlinear Effects .. . . . .      .     247
7.5  Applications of XPM  . . . . . . . . .. .              . .248
7.5.1  XPM-Induced Pulse Compression ..      .   . .   .   248
7.5.2  XPM-lnduced Optical Switching . . . , . . .  . .     251
7.5.3  XPM-Induced Nonreciprocity . . . . . .       .      252
7.6  Polarization Effects  ..... .                         . . .  254
7.6.1  Vector Theory ofXPM              . .         . . . . 254
7.62  Polarization Evolution . .      .     . .   . . .    255
.6.3  Polarization-Dependent Spectral Broadening   . .     257
7.64  Pulse Trapping and Compression  .                    260
7.6.5  XPM-Induced Wave Breaking  .                 .       262
7.   XPM Effects in Birefringent Fibers . .  . . .. .264
7.7.1  Fibers with Low Birefringence  . . .  . .  . . .  .  264
7.7.2  Fibers with High Birefringence            .  . . .  267
Problems .    . ...... .... . . . .                           . 268
R et.f rences  .  .  .  .  .  .  .  .  .  .  .  .  .  .. . ... . .  .  .  .  .  . .  . .  .  .  .  .  27 0/
8  Stimulated Raman Scattering                                      274
. 1  Basic Concepts  . .... .         .                       .  274
8.1.1  Raman-Gain Spectrum     .    .       . .   .   . .   275
8.1.2  Raman Threshold.            . . . . . . .   . . . . . .  276
8.1.3  Coupled Amplitude Equations  .  .                    279
b.1.4  Effect of Four-Wave Mixing     .                .    281
8 2  Quasi-Continuous SRS  . . . . . . . . . . . . . . . . .283
8.2.1  Single-Pass Raman Generation . .  . . . . . . .      283
"8.2.2  Raman Fiber Lasers ... ,.       ....               2385
82.3   Raman Fiber Amplifiers  . . .                        288
8.2.4  Raman-Induced Crosstalk .   . . . ...             . 292
S.3  SRS with Short Pump Pulses .  .   ...      . ...     ..     294
8.3.1  Pulse-Propagation Equations . .  .. ..      .   .    294
8.3.2  Nondispersive Case .......   .. 295
83.3   Effects ofGVD . .                                .   297
8.3.4  Experimental Results  .  .  .. . . . . . . .    . . .  300
8.3.5  Synchronously Pumped Raman Lasers. . . . . . .  . . .  304
8.3.6  Short-Pulse Raman Amplification . .  . . . . . . . . . . 305
8.4  Soliton Effects ..          .    . .. . .             .  .  306
84.    Raman  Solitons  . ....    .   .  .  .  .  .  .  ..........  306
8.4.2  Raman Soliton Lasers  . . .... . ..                  311
8.4.3  Soliton-Effect Pulse Compression . . . . . .  ... . ..  313
8.5  Polarization Effects           ...      . .    .     . .  . 315
8.5.1  Vector Theory of Raman Amplification    . . . . . . .  315
8.5.2  PMD Effects on Raman Amplification .  . .            319
Problem s  ..        .   .   .. .       .   .  .  .  . .  . .   .   . .  . 321
References  .   .  .  .  .  .  .  .  .  .  .  .  .  .  .  . .   .  322
9 Stimulated Brillouin Scattering                                   329
9.1 Basic Concepts                   . . .. 329
9.1.1  Physical Process              .  . . .  .  .  .    .  330
9 ..2  Brillouin-Gain Spectrum  . . . .   . . . . . . . .  .  0
9.2  Quasi-CW SBS  . . . .   . .              . .   . . . . . . . .  333
9.21   Brillouin  Threshold  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . .  .  .  333
9.2.2  Polarization Effects  . .  .  .  . .  .  . .  .  . . .  334
"9.2.3  Techniques for Controlling the SBS Threshold  . . .   335
9.2.4  Experimental Results .. . . . ..          .        . 338
9.3  Brillouin Fiber Amplifiers ..  . .  ...                     340
9.3.1  Gain Saturation  .. . . . . . . . .              . .  341
9.3.2  Amplifier Design and Applications . . . .            342
9.4  SBS Dynamics     ......... .. . ..                          344
9.4.1  Coupled Amplitude Equations . . .  ......      . . . . 345
9.4.2  SBS with Q-Switched Pulses . .          . .          346
9.4.3  SBS-Induced Index Changes ..........             .   350
9.4.4  Relaxation  Oscillations  .  .  .  .  .  .  .  .  .   .  .  .  .  . .  . .  .. 352
9.4.5  Modulation Instability and Chaos . .  . . . . .  . . . . 354
9.5  Brillouin Fiber Lasers ...  . . . . .  . ... .. . . .       . 356
95.1   CW Operation  ..        . . ..... 356
9.5.2  Pulsed Operation  .  .  .  .  .   . .  .  . . . .. .  .  360
Problems                    .   . .   .  .  .  .  .    .  .  .  .  .  362
References .               .   . . .                     . . .363
10 Four-Wave Mixing                                                   368
10.1 Origin of Four-Wave Mixing   .. .             . .             368
10.2 Theory of Four-Wave Mixing         . . . . .  .     . . .  . 370
10.2.1 Coupled Amplitude Equations   . . . . . . . .  .  . .  371
10.2.2 Approximate Solution ..                          . .  ..  371
10.2.3 Effect of Phase Matching          . . . . . . .  . . . . . 373
10.2.4 Ultrafast Four-Wave Mixing . .    .  . . . . .         374
10.3 Phase-Matching Techniques . . .                               376
10.3.1 Physical Mechanisms .  . . . . . .  .       .          376
10.3.2 Phase Matching in Multimode Fibers   .     .    .      377
10.3.3 Phase Matching in Single-Mode Fibers             . .   380
10.3.4 Phase Matching in Birefringent Fibers           .      383
10.4 Parametric Amplification  .. .  .. . . .    .  .              387
10.4.1 Review of Early Work           . . . ...   ...         387
10.4.2 Gain Spectrum and Its Bandwidth         . . . . . .  . 389
10.4.3 Single-Pump Configuration  .  ..            .   . . . 391
10.4.4 Dual-Pump Configuration         . . . . . .  . . . . . . 394
10.4.5 Effects of Pump Depletion ..                       . .  399
10.5  Polarization Effects  .  . ...............   .  .  . .  .  .  .  ..  401
10.5.1 Vector Theory of Four-Wave Mixing .. .  .   . .   .    401
10.5.2 Polarization Dependence of Parametric Gain   .  .  ..  403
10.5.3 Linearly and Circularly Polarized Pumps  .  . .  .  . . 405
10.5.4 Effect of Residual Fiber Birefringence   . .    . . . . 408
10.6 Applications of Four-Wave Mixing .                 .    .     411
10.6.1  Parametric Oscillators  .  . . .       .    .  . . . .  412
10.6.2  Ultrafast Signal Processing  .  . . .  .  . . . ....  . .  .  413
10.6.3 Quantum Noise and Correlation . .. .       . . .. .    415
Problems  . .  .            . . .  . .  . . . . . . . . .  .       417
References    .    .  .           .  . .  . .  .  .  .  .  .  .  .  418
11 Highly Nonlinear Fibers                                            424
11.1  Nonlinear Parameter  .     .   . .  . .  . . . . . . .       424
11.1.1 Units and Values ofn  . .. ....                        425
11.1.2 SPM-Based Techniques   .    ..        . . .    .   .   426
11.1.3 XPM-Based Technique I4 . ......                    . . 429
111..4 FWM-Based Technique   ........ ...         .   ....    430
11.1.5 Variations in n2) Values  .                    . . . . 431
11.2 Fibers with Silica Cladding  ...           ...                434
1 1  Tapered Fibers with Air Cladding  . . . . . . . . .  . . . .  436
1   M1.4  icrostructured Fibers    .... .          .   . . .    440
1 .5  No -Silica  Fibers  . . .  .  .  .  .  .  .  .  .  .  .  .  .  . .  .  444
Problem s  ..       .       .  .  .  .  .  .   . .  .  .  .  . . . .  .  448
References  .    .  .  .  .      .  .  . ...          ..  .  .  .  .  449
12 Novel Nonlinear Phenomena                                       453
12.1  Intrapulse Rarnan Scattering  . . . .  .  . .  . .  . . .  453
12.1.1 Enihanced RIFS and Wavelength Tuning.  ..     ..    454
12.1.2 Nonsolitonic Radiation  .  . . . .  ...         .   457
12 .13 Effects of Birefringence . .  . .           .       459
12.1 4 Suppression of Raman-Induced Frequency Shifts  . .  461
12.2  Four-Wave Mixing      .  .  . .. .  .  . ..      . .      464
S22 1 FWM in Highly Nonlinear Fibers  . . .            ..  464
12.2.2 Effects of Fiber Birefringence  . ... .     .  .    467
12.3 Supercontinuum Generation      . . . . . .                 469
12.3.1 Pumping with Picosecond Pulses              . .     470
12.3.2 Continuous-Wave Pumping  .  .  . .    .     .  .    474
12.3.3 Pumping with Femtosecond Pulses  .           . . .  475
12.4 Temporal and Spectral Evolution .. .  ..          .. 477
12.4 .1 Numerical Modeling of Supercontinuum  . . . . . . .  477
12.4.2 Soliton Fission and Nonsolitonic Radiation . . . . . .  480
12.4.3 Effects of Cross-Phase Modulation  . . . .   . .  .  484
12.4.4  Polarization Effects . .  . .  .  . . . .  . .  .  .  88
12.4.5 Coherence Properties of a Supercontinuum  . .  .  ..  492
12.5 Harmonic Generation .  .   . .    . . . . . .  .     .     495
12.5.1 Second-Harmonic Generation . . . .  . . .   . . . .. 495
12.5.2 Third-Harmonic Generation   .    . . . .       . .  502



Library of Congress subject headings for this publication: Fiber optics, Nonlinear optics