Abstract

In a recent investigation of the displacement suffered by two spatial solitons approaching each other some general coupled equations were used^1-3. They describe the complete nonlinear interaction between two spatial solitons, with the same frequency and polarisation, located in a pair of, identical, coupled slab waveguides (1 and 2) and have the form, where propagation is along the z-axis, beam confinement by the nonlinearity is along the x-axis, modal confinement is along the y-axis, k = 3-i, i = 1,2, ψ_i and ψ_k are normalised slowly varying amplitudes of the electric fields in the guides, ν represents the inter-guide coupling and μ₁, μ₂ are nonlinear coefficients. The first point to make is that these coupled equations go far beyond the classical work of Jensen⁴ and will only return to it⁵ provided that the last four terms are neglected. Indeed, some of the usually neglected terms are larger than the terms that are, normally, retained and, therefore, have a crucial influence on the stability and the threshold powers of the spatial solitons. The second point is that a full variational solution of the coupled equations can be made to yield a mathematical, analytical, nonlinear stability analysis, generating switching formulae with tractable forms for the stability edges, or thresholds. Even within a Jensen model, the results to be presented here are entirely new and it should be stressed that all the parameters ν, μ₁ and μ₂ are required for inter-guide interactions. A Jensen formalism, on the other hand, involving ν and only one other parameter μ, is adequate for intra-guide polarisation interactions. The objectives here are (a) to present a mathematical and numerical solution of the general coupled equations that will account for energy exchange between spatial solitons (b) to generate a theory of stability for solitons hovering, one above the other, in the respective guides (c) to investigate the possibilities for modulation instability.

© 1993 Optical Society of America