Four-wave solitons in Bragg cross-gratings

Abstract

A temporal-domain model for a two-dimensional (2D) waveguide with two transversal Bragg gratings (BGs) is introduced. In this medium, four waves are coupled linearly by the Bragg reflections, and nonlinearly by XPM and four-wave mixing. The same model in the spatial domain applies to a 3D photonic crystal. The subject of the work is (effectively) 1D solitons in this system. Equations which govern their dynamics also describe the evolution of two circular polarizations in a fibre BG. Depending on the relative strength of the two gratings, the system may or may not have a gap in its spectrum, and it always has 'semi-gaps', where embedded solitons (ES) may be found. In the case when the propagation constant k (frustration parameter) is zero, the system admits two reductions, symmetric (S) and anti-symmetric (anti-S), to two-wave systems, which are equivalent to the standard fibre-BG model. These systems give rise to exact solutions, which, in terms of the full system, may be gap solitons (GS) or ES. Asymmetric (aS) solitons for k = 0, and solutions generated by the continuation of the S, anti-S and aS solitons to , are found by means of the variational approximation and in a numerical form. In the semi-gaps, these solutions may be both ES and quasi-solitons (QS) with nonvanishing tails. Stability of the solitons is first investigated by means of the Vakhitov–Kolokolov (VK) criterion, which predicts no instability for the extended solution families of the S and anti-S type. For the aS solitons, the VK criterion predicts a nontrivial stability border inside the bandgap, the location of which is controlled by k: at k = 0 the entire bandgap is VK-stable, while for large k the area of the VK stability shrinks to nothing. This intra-gap stability border is a new feature of GS models. Direct simulations demonstrate that, at k = 0, the S and anti-S reductions are always stable against perturbations violating the reduction, hence the stability of the exact S and anti-S solitons can be predicted on the basis of known results for the standard fibre-BG model. Continuing these solutions in k and testing their stability by simulations, we conclude that the solution branches stemming from the stable and unstable exact solitons at k = 0 always remain, respectively, stable or unstable. As a result, the extended soliton family of the S type is chiefly (but not exactly) stable inside the bandgap and unstable in the semi-gaps, while the opposite is true for the extended family of the anti-S type solutions. The aS solitons which are predicted to be VK-unstable quickly decay indeed, while their VK-stable counterparts survive for a long time, but are finally destroyed by oscillatory perturbations, which the VK criterion does not take into consideration. Moving solitons are constructed too, and a limit velocity is found for them.