ISBN: 3-540-66314-2
TITLE: Optical Solitons: Theoretical Challenges and Industrial Perspectives
AUTHOR: Zakharov, Vladimir E.; Wabnitz, Stefan (Eds.)
TOC:

LECTURE 1
Fundamentals of Optical Soliton Theory in Fibers by A. Hasegawa
1. Introduction 1
2. Electromagnetic waves in dielectric materials 1
2.1 Polarization effects 1
2.2 Plane electromagnetic waves in dielectric materials 3
2.3 Kerr effect and Kerr coefficient 5
2.4 Dielectric waveguides 6
3. Envelope of electromagnetic wave in dielectric materials 10
3.1 Lightwave envelope in fibers  derivation of nonlinear Schrdinger equation 10
3.2 Evolution of the wave packet due to the group velocity dispersion 12
3.3 Evolution of wave packet due to the nonlinearity 14
3.4 Lax theorem 15
3.5 The soliton solution of the nonlinear Schrdinger equation 15
4. Ultrafast communication based on optical solitons 16
5. Conclusion 18
LECTURE 2
Hamiltonian Theory of Bcklund Transformation
by V.G. Marikhin and A.B. Shabat
1. Introduction 19
2. Lattice equations 22
3. Canonical Bcklund transformations 25
4. First integrals 27
LECTURE 3
Stability of Solitons
by E.A. Kuznetsov
1. Introduction 31
2. Lyapunov stability 33
2.1 Nonlinear Schrdinger equation 33
2.2 The three-wave system 36
2.3 Soliton solutions of the 3-wave system 39
2.4 Nonlinear stability 41
3. Linear stability 43
3.1 Linear stability for 1D NLS solitons 43
3.2 Solitons for the FF-SH interaction 46
LECTURE 4
Chaotic Dynamics of Optical Solitons
by F.Kh. Abdullaev
1. Introduction 51
2. Variational approach to solitons dynamics in random media 52
2.1 Optical solitons in media with fluctuating quadratic potential 53
2.2 Spatial soliton in array with fluctuating parameters 54
2.3 A random Kepler problem 55
3. Inverse scattering transform technique for solitons in random media 56
3.1 Single soliton propagation in random media 56
3.2 Interaction of optical solitons in random media 59
4. Conclusion 61
LECTURE 5
Variationalism and Empirio-Criticism. (Exact and Variational Approaches to Fibre Optics Equations)
by A.V. Mikhailov
1. Introduction 63
2. Variational approach 64
3. What is wrong with the Variational approach 68
LECTURE 6
Propagation of Optical Pulses in Nonlinear Systems with Varying Dispersion
by V.E. Zakharov
1. Introduction 73
2. Basic model 74
3. Effective Hamiltonian 77
4. Monochromatic wave and its stability 81
5. Weak dispersion management 82
6. Streng dispersion management (SDM) 85
7. Solitons and their stability 87
LECTURE 7
Dispersion-Managed Solitons
by S.K. Turitsyn, N.J. Doran, J.H.B. Nijhof, V.K. Mezentsev, T. Schfer and W. Forysiak
1. Introduction 91
2. Basic equations 94
3. Linear solution and qualitative description of DM solitons 96
4. DM pulse evolution over one period 98
4.1 Root-mean-square momentum equations 98
4.2 Power enhancement 105
4.3 How to find the DM soliton numerically 106
5. A path-average theory of DM solitons in the time domain 106
6. Path-averaged equations in the spectral domain 110
7. Conclusions 112
LECTURE 8
Dispersion-Managed Solitons: Applications to Terabit/s Transmission over Transoceanic Distances
by T. Georges
Introduction 117
Modelling 119
Single pulse propagation 119
Perturbation 125
Experiments 135
Set-up 135
Spectrum evolution 136
Phase diagram 136
System margin 136
Comparison to a soliton transmission system with constant dispersion 139
Narrow band 1.02 Tbit/s (51  20 Gbit/s) soliton DWDM transmission over 1000 km of Standard fibre 140
Conclusion 141
LECTURE 9
Nonlinear Pulses in Ultra-Fast Optical Communications
by V. Cautaerts, Y. Kodama, A. Maruta and H. Sugahara
1. Introduction 147
2. The DM solitons 148
2.1 The Lagrangian method 149
2.2 Hermite-Gaussian ansatz 151
3. The DM solitons in WDM 153
3.1 Mechanism of frequency shift for DM soliton 153
3.2 Optimal allocation of amplifier 156
3.3 Statistical analysis of collision induced timing jitter 158
4. NRZ pulse propagation 161
4.1 The NLS-Whitham equations 163
4.2 Control of NRZ pulse 165
LECTURE 10
Soliton Wavelength-Division-Multiplexing System: From Numerical Design to Recirculating Loop Experiments 
by J.-P. Hamaide, B. Biotteau, F. Pitel and E. Desurvire
1. Introduction 171
2. Soliton transmission over dispersion-managed systems 173
3. Results from the analytical/basic numerical tool 175
4. Results from the numerical tool 177
5. Results from the experimental tool 178
6. Conclusion 181
LECTURE 11
Mode-Locked Fiber Ring Lasers and Fiber Ring Memories
by H.A. Haus
1. Introduction 183
2. The passively mode-locked fiber ring laser and the master equation 184
3. Harmonic modelocking and the makings of an all-optical memory 190
4. The first order soliton 199
5. Perturbation theory of solitons 200
6. The stretched pulse fiber ring laser 208
LECTURE 12
Modulational Instabilities in Passive Cavities: Theory and Experiment
by M. Haelterman and S. Coen
1. Introduction 215
2. Basic properties of the nonlinear fiber resonator 217
3. The effects of dispersion: Theory 222
3.1 cw-MI and the MI-induced up-switching process 223
3.2 Period-doubling MI 225 
4. Experimental results 226
4.1 Period-doubling MI 227
4.2 cw-MI and the MI-induced up-switching process 229
5. Conclusion 230
LECTURE 13
Recent Developments in the Theory of Optical Gap Solitons
by S. Trillo, C. Conti, A. de Rossi and G. Assanto
1. Introduction 233
2. Coupled-mode models 234
3. Stability 236
4. Quadratic gap solitons 242
5. Conclusions 246
LECTURE 14
Vector Modulational Instabilities and Soliton Experiments
by G. Millot, S. Pitois, E. Seve, P. Tchofo Dinda, P. Grelu, S. Wabnitz, M. Haelterman and S. Trillo
1. Introduction 249
2. Observation of vector MI for normal dispersion 250
2.1 High-birefringence fiber 250
2.2 Low-birefringence fiber 251
2.3 Bimodal fiber 252
3. MI gain spectra from linear stability analysis 252
3.1 High-birefringence fiber 253
3.2 Low-birefringence fiber 253
3.3 Bimodal fiber 254
4. Induced vector MI and soliton generation 255
4.1 High-birefringence fiber 255
4.2 Low-birefringence fiber 258
4.3 Bimodal fiber 260
5. Conclusions 262
LECTURE 15
Transient Raman Amplification
by J. Leon and A.V. Mikhailov
1. Introduction 265
2. Derivation of the SRS system 269
3. Steady state regime 273
4. Transient SRS: A complete solution 274
5. Raman soliton generation 276
6. Stokes phase flips and the Raman spike 277
7. The Raman spike in the time domain 278
8. Conclusion 280
LECTURE 16
Self-Structuration of Three-Wave Dissipative Solitons in CW-Pumped Optical Cavities
by C. Montes, A. Picozzi and M. Haelterman
1. Introduction 283
2. Three-wave model 284
3. Two-wave adiabatic approximation 286
4. Self-pulsing in a cavity 289
LECTURE 17
The Description of the Ultrashort Pulse Propagation in Non-Linear Media Under Quasi-Resonance Condition
by A.I. Maimitsov
1. Introduction 293
2. Maxwell-Bloch, RMB, and SVEPA equations 295
3. Solution of the Bloch equation 297
4. Scalar wave equations 299
4.1 Non-linear wave equation 299
4.2 Unidirectional non-linear wave (mKdV equation) 302
4.3 Non-linear wave in SVEPA 303
5. Vector waves 304
5.1 Generalised Maxwell-Bloch equations 305
5.2 Solution of the generalised Bloch equations 305
5.3 Vector non-linear wave equation 307
5.4 Unidirectional vector non-linear waves 308
5.5 Polarised quasi-monochromatic non-linear wave (vector NLS equation) 309
6. Conclusion 310
LECTURE 18
Bright Spatial Soliton Interactions
by G.I. Stegeman and M. Segev
1. Introduction 313
2. Coherent interactions: Basic theoretical properties 316
2.1 Kerr nonlinearities 317
2.2 Saturating nonlinearities 321
3. Coherent interactions: Experiments 322
4. Incoherent soliton interactions 324
5. Full 3D soliton interactions 326
6. Anisotropie soliton interactions 329
7. Summary 330
LECTURE 19
Spatial Solitons in Saturating Nonlinear Materials
by B. Luther-Davies, V. Tikhonenko, J. Christou, W. Krolikowski, Y. Kivshar and N. Akmediev
1. Introduction 335
2. Dark and bright spatial solitons 338
3. Saturating nonlinearities 340
4. Experimental demonstrations 341
5. Conclusions 346
LECTURE 20
Discrete Solitons in Nonlinear Waveguide Arrays
by F. Lederer and J.S. Aitchison
1. Introduction 349
2. Basic properties ofwaveguide arrays 352
2.1 Evolution equations 352
2.2 Linear properties  "discrete diffraction" 353
2.3 Nonlinear properties  modulational instability 354
3. Discrete Solitons 355
3.1 Moderately localized bright solitons  basic properties 355
3.2 Moderately localized bright solitons  self-trapping and switching 358
3.3 Strongly localized discrete solitons  properties and stability 359
4. Further studies 360
5. Experiments in nonlinear waveguide arrays 361
6. Conclusions 364
LECTURE 21
Solitons in Cavities with Quadratic Nonlinearities
by W.E. Torruellas, P.S. Jian, S. Trillo, M. Haelterman, U. Peschel and F. Lederer
Introduction 367
1. The case of quadratic nonlinearities 368
2. Why cavities? 369
3. Multidimensional spatial solitons in optical cavities 370
4. Optical bullets in nonlinear optical cavities 370
5. Temporal solitons in singly resonant optical parametric oscillators 371
6. Conclusion 379
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