Parametric spatial solitary waves due to type II second-harmonic generation

Analysis of mixing between waves at different carrier frequencies is a fundamental problem of nonlinear optics.1 Resonant interaction is especially important since it leads to an efficient energy conversion. In an optical medium without the center of symmetry, the lowest-order nonlinear response is quadratic, leading to three-photon interactions. Resonant parametric mixing among three waves at different carrier frequencies occurs when the resonance condition v3 5 v1 1 v2 is fulfilled. This includes both parametric upconversion (or sum-frequency generation, when the frequency v3 is generated), and downconversion (or difference-frequency generation, when either v1 or v2 is generated). A particular case of this three-wave interaction, formally described by the similar system of equations, is known as type II secondharmonic generation (SHG), for which the frequencies are degenerate (i.e., v1 5 v2) but the polarizations of the fields are different. In a degenerate case (when the two waves at the fundamental frequency are identical), such a resonant interaction reduces to two-wave mixing or type I SHG. Three-wave parametric interaction also appears in different branches of physics (see, e.g., Refs. 2–4 and references therein) involving waves of a different nature. When the group velocities (for a temporal problem) or the directions of stationary energy flows (for a spatial problem) of the interacting waves are essentially different, dispersive or diffractive wave broadening is negligible with respect to walk-off and nonlinear self-modulation or self-focusing. In this limit, the equations describing three-wave parametric interactions are known to be integrable.2 In this regime, soliton features in the nondegenerate (three-wave mixing) and degenerate (twowave mixing) cases turn out to be qualitatively different. The former leads to bright solitons, which are relevant in type II SHG experiments (see, e.g., Ref. 5), whereas the spectral problem for the latter becomes singular,6 and