Solitons. Nonlinear pulses and beams

Transcription

1 Solitons Nonlinear pulses and beams Nail N. Akhmediev and Adrian Ankiewicz Optical Sciences Centre The Australian National University Canberra Australia m CHAPMAN & HALL London Weinheim New York Tokyo Melbourne Madras

2 Foreword Acknowledgements xi xv 1 Basic equations l 1.1 Stationary beams in homogeneous nonlinear media Nonlinear terms Pulse propagation in nonlinear media Dispersive effects in waveguides Pulse propagation in single-mode fibers Birefringent fibers Nonlinear guided waves The state of polarization of monochromatic optical waves Nonlinear couplers Nonlinear fiber arrays Passively mode-locked lasers with fast saturable absorbers Soliton-based optical transmission lines Soliton X-junctions 24 2 The nonlinear Schrödinger equation One-soliton solution Scaling transformation Galilean transformation Conserved quantities Continuity equations Non-interaction with radiation Basics of integrability Solution of the initial value problem Darboux transformations Solution of the linearized NLSE Solution of the linear problem related to the NLSE 40 3 Exact solutions Special ansatz 43

3 3.2 Reduction to a finite-dimensional dynamical system Solutions of the dynamical system The case of complex roots Reduction of the number of parameters Special cases Modulation instability Geometric interpretation of the solution Evolution of spectral components Rational solution Periodic solutions Solitons on a finite background Stationary periodic waves Higher-order solutions Second-order solution Multi-soliton solutions Breathers Modulation instability with two pairs of initial side-bands 70 Non-Kerr-law nonlinearities Stationary solutions Some examples Saturable media Stability of solitary waves Solitons for parabolic nonlinearity law Internal friction between solitons 83 Normal dispersion regime General form of the solution Particular cases Periodic solutions Stationary periodic solution Collision of two dark solitons Excitation of pairs of dark solitons using symmetric initial conditions Black soliton Relation between NLSE solutions in normal and anomalous dispersion regimes Grey soliton phase 'Darker than black' solitons 108 Multiple-port linear devices made from solitons General principles Composite waveguides made from solitons Refractive index profile of./v-soliton solution 116

4 6.4 'Modes' of the composite waveguides Power in each waveguide Transmission coefficients Example 1: soliton X-junction Example 2: 3 x 3 switch Deviations from optimal case X-junctions based on dark soliton collisions 127 Nonlinear pulses in birefringent media Symmetries and conserved quantities Approximation of low birefringence Transformation to circularly polarized components Stationary solutions (linearly polarized solitons) Elliptically polarized solitons Energy-dispersion diagram Hamiltonian versus energy diagram Stokes parameters Dynamic solitons Approximation of the average profile Comparison with c.w. beams Linear and nonlinear beat lengths Analysis of the system Instability of the fast soliton Radiation of energy from the soliton Numerical examples Approximation for long soliton period Transformation to rotating frame Multi-soliton solutions The role of the difference in group velocities Transformation to different frequencies Stationary solutions in the presence of group velocity delay Soliton states with locked phase and group velocities 164 Pulses in nonlinear couplers Couplers with Kerr-type nonlinearity Stationary soliton states Asymmetric states The energy-dispersion diagram The Hamiltonian versus energy diagram Stokes parameter formalism Stability of soliton states Radiation of small-amplitude waves by solitons Linear and nonlinear beat lengths Numerical examples 179

5 8.11 Switching Arbitrary initial conditions Non-Kerr-law anomalous dispersion couplers Dissimilar cores Solitons with time offsets Multi-core nonlinear fiber arrays n-core nonlinear fiber arrays Soliton states in three-core couplers The influence of higher-order dispersion on solitons Soliton renormalization Radiation frequency Transition radiation Relation to Cherenkov radiation Radiation intensity Radiated energy The spectral shift of the soliton Negative fourth-order dispersion and zero third-order dispersion Solitons with oscillating tails Bound states of solitons with oscillating tails Hamiltonian versus energy diagram Stability criterion for bound states Interactions of solitons with oscillating tails Beam dynamics Stationary (in time) solutions of the (2+l)-dimensional problem Radially symmetric solutions Stability of the ground state Examples of exact solutions Collapse of optical beams Beyond the paraxial approximation Radially symmetric solutions for the case of saturable nonlinearity Loss of cylindrical symmetry Optical bullets Longitudinal modulation instability of self-trapped beam Planar nonlinear guided waves Nonlinear waves in a layered medium Waves on a single boundary of a nonlinear medium Nonlinear waves in the three-layer symmetric waveguide 249

6 12.4 Power-dispersion diagram Lateral field shift in nonlinear medium Stability analysis Perturbation function Three-layer structure Complex eigenvalues Particle analogy Analogy with soliton states in couplers Amplification of nonlinear guided waves Nonlinear pulses in presence of gain, loss and spectral filtering Complex quintic Ginzburg-Landau equation Perturbative approach Special ansatz Solitons of the cubic CGLE Solitons of the quintic CGLE Systems without spectral filtering Stability of solutions with fixed amplitude Regions in the parameter space where stable pulses exist Reduction to a set of ordinary differential equations Composite pulses Moving pulses Coexistence of pulses Interaction of moving and stationary pulses Soliton bound states Soliton interactions Concluding remarks 308 References 311 Index 333 ix