- •1 Introduction
- •1.1 Historical Survey
- •1.2 Patterns in Nonlinear Optical Resonators
- •1.2.1 Localized Structures: Vortices and Solitons
- •1.2.2 Extended Patterns
- •1.3.3 Optical Feedback Loops
- •1.4 The Contents of this Book
- •References
- •2 Order Parameter Equations for Lasers
- •2.1 Model of a Laser
- •2.2 Linear Stability Analysis
- •2.3 Derivation of the Laser Order Parameter Equation
- •2.3.1 Adiabatic Elimination
- •2.3.2 Multiple-Scale Expansion
- •References
- •3 Order Parameter Equations for Other Nonlinear Resonators
- •3.1 Optical Parametric Oscillators
- •3.2.1 Linear Stability Analysis
- •3.2.2 Scales
- •3.2.3 Derivation of the OPE
- •3.3.1 Linear Stability Analysis
- •3.3.2 Scales
- •3.3.3 Derivation of the OPE
- •3.4 The Order Parameter Equation for Photorefractive Oscillators
- •3.4.1 Description and Model
- •3.4.2 Adiabatic Elimination and Operator Inversion
- •3.5 Phenomenological Derivation of Order Parameter Equations
- •References
- •4.1 Hydrodynamic Form
- •4.2 Optical Vortices
- •4.2.3 Intermediate Cases
- •4.3 Vortex Interactions
- •References
- •5.2 Domains of Tilted Waves
- •5.3 Square Vortex Lattices
- •References
- •6 Resonators with Curved Mirrors
- •6.1 Weakly Curved Mirrors
- •6.2 Mode Expansion
- •6.2.1 Circling Vortices
- •6.2.2 Locking of Transverse Modes
- •6.3 Degenerate Resonators
- •References
- •7 The Restless Vortex
- •7.1 The Model
- •7.2 Single Vortex
- •7.3 Vortex Lattices
- •7.3.2 Parallel translation of a vortex lattice
- •7.4.1 Mode Expansion
- •7.4.2 Phase-Insensitive Modes
- •7.4.3 Phase-Sensitive Modes
- •References
- •8 Domains and Spatial Solitons
- •8.1 Subcritical Versus Supercritical Systems
- •8.2 Mechanisms Allowing Soliton Formation
- •8.2.1 Supercritical Hopf Bifurcation
- •8.2.2 Subcritical Hopf Bifurcation
- •8.3 Amplitude and Phase Domains
- •8.4 Amplitude and Phase Spatial Solitons
- •References
- •9 Subcritical Solitons I: Saturable Absorber
- •9.1 Model and Order Parameter Equation
- •9.2 Amplitude Domains and Spatial Solitons
- •9.3 Numerical Simulations
- •9.3.1 Soliton Formation
- •9.4 Experiments
- •References
- •10.2 Spatial Solitons
- •10.2.1 One-Dimensional Case
- •10.2.2 Two-Dimensional Case
- •References
- •11 Phase Domains and Phase Solitons
- •11.2 Phase Domains
- •11.3 Dynamics of Domain Boundaries
- •11.3.1 Variational Approach
- •11.4 Phase Solitons
- •11.5 Nonmonotonically Decaying Fronts
- •11.7 Domain Boundaries and Image Processing
- •References
- •12 Turing Patterns in Nonlinear Optics
- •12.1 The Turing Mechanism in Nonlinear Optics
- •12.2.1 General Case
- •12.2.2 Laser with Saturable Absorber
- •12.3.1 Turing Instability in a DOPO
- •12.3.2 Stochastic Patterns
- •References
- •13 Three-Dimensional Patterns
- •13.1 The Synchronously Pumped DOPO
- •13.1.1 Order Parameter Equation
- •13.3 The Nondegenerate OPO
- •13.4 Conclusions
- •13.4.1 Tunability of a System with a Broad Gain Band
- •13.4.2 Analogy Between 2D and 3D Cases
- •References
- •14 Patterns and Noise
- •14.1 Noise in Condensates
- •14.1.1 Spatio-Temporal Noise Spectra
- •14.1.2 Numerical Results
- •14.1.3 Consequences
- •14.2 Noisy Stripes
- •14.2.1 Spatio-Temporal Noise Spectra
- •14.2.2 Stochastic Drifts
- •14.2.3 Consequences
- •References
8 Domains and Spatial Solitons
This chapter is devoted to the description of the general properties of a particular class of solutions of extended nonlinear systems, namely spatial solitons, or localized structures. In contrast to extended patterns, such as rolls or hexagons, spatial solitons are solutions where the field is inhomogeneous only in a localized region of the space and is homogeneous in the rest of the space.
The stability of spatial solitons is due to a balance between linear and nonlinear e ects. In transverse nonlinear optics, these balancing e ects are usually the spreading caused by di raction and the compression (self-focusing) caused by a focusing nonlinearity.
Spatial solitons have been theoretically predicted for a variety of nonlinear optical cavities [1]. In this chapter we present some general concepts concerning spatial domains and solitons, and the next three chapters are devoted to the implementation of these ideas in concrete optical systems.
The study of optical solitons is relevant not only from a fundamental viewpoint but also because of their potential applications in information processing technology. Such practical applications will be discussed in Chaps. 9 and 11.
8.1 Subcritical Versus Supercritical Systems
The existence of spatial solitons is closely related to the character of the bifurcation from the nonlasing (trivial) to the lasing regime. It is useful to review here some basic concepts of bifurcation theory [2].
A bifurcation is supercritical if the transition from one solution to another, obtained by varying the control parameter (the pump intensity in the optical case), is continuous, and the solutions connect at the bifurcation point. Equivalently, an infinitesimal variation of the control parameter leads to an infinitesimal change in the amplitude of the solution in the case of a supercritical bifurcation. Examples of a supercritical bifurcation are shown in Figs. 8.1a,c. A supercritical bifurcation corresponds to a phase transition of type II. In contrast, a bifurcation is subcritical if, at the bifurcation point, the amplitudes of the solutions di er by a finite quantity (i.e. they are disconnected). In this case, both solutions may coexist below the bifurcation
K. Stali¯unas and V. J. S´anchez-Morcillo (Eds.):
Transverse Patterns in Nonlinear Optical Resonators, STMP 183, 117–124 (2003)c Springer-Verlag Berlin Heidelberg 2003