Theory of light-beam propagation at nonlinear interfaces. II. Multiple-particle and multiple-interface extensions

The theory presented in the preceding paper is extended to account for incident optical beam breakup into multiple self-focused channels and to deal with multiple reflection and transmission at multiple interfaces. Beam breakup is explained by examining the decomposition of the channel in one medium, into its soliton and radiation components after it has crossed into the new medium. A formula is derived which gives the criterion for the number of channels appearing as a consequence of breakup. This formula also provides analytic expressions for the individual self-focused channel powers and the amount of radiation generated. An important observation here is that the amount of radiation generated at the interface shrinks rapidly as a function of increasing channel number N. Each new component generated can be treated as a separate equivalent particle moving in its own equivalent potential. The theory of the preceding paper [Aceves, Moloney, and Newell, Phys. Rev. A 39, 1809 (1989)] can therefore be applied directly to show that low-power channels generated in the breakup will suffer reflection while higher-power channels will undergo transmission. An added ingredient to allow for mutual-channel interaction is the soliton-collision formula. The multiple-interface extension of the single-interface problem results from patching individual single-interface equivalent potentials together. The theory is illustrated with two applications: (i) a nonlinear version of a directional coupler requiring just two interfaces and (ii) trapping of a channel at an interface by a ramped linear refractive index.