Kerr-type nonlinearities in lightguides: coupled-mode field representations

Abstract

Third-order nonlinearities in optical waveguides have received considerable recent attention,1with particular interest in the possibility of devices for all-optical signal processing. In the small-intensity limit (linear case), field decomposition for longitudinally invariant lightguides allows for independently propagating modes, which are wave equation solutions after separation of transverse and longitudinal parts.2,3For intensity-dependent refractive indices, the illuminated guide is no longer longitudinally invariant, and thus this separation is not permitted. However, as is possible for tapered guides, we can describe the field as a combination oflocal modes,which couple as the light propagates. As local modes we can use modes of the linear-index guide. Alternatively, we may use solutions of the nonlinear wave equation (NLWE) for the transverse field dependence.4For approximate monomode propagation this can provide a more natural field representation in particular. However, these modes are only solutions of the transverse NLWE if considered in isolation; i.e., each nonlinear mode modifies the nonlinear index differently. For few-mode propagation, although a coupled nonlinear-local-mode approach may involve fewer local modes depending on the structure, increased complexity in the formalism results when non-orthogonality of the nonlinear modes is accounted for. In contrast, the linear-local-mode treatment has the attraction of simplicity. In Figs. 1-3 we consider the nonlinearly coupled linear-guide local-mode field representation and convergence thereof given initial bimodal excitation, assuming a Kerr-type nonlinear index and slowly varying approximations.3When we adopt the notation of Ref.3with difference between the effective indices of the two initially excited modesδneff,the Kerr coefficientn21,core half-widthρ, andPref=ρδneff/n2,I, forP/Pref<1, the two-mode representation is highly accurate for describing the nonlinear phase shifting and small nonlinear coupling. AsP/Pref≈ 1 is approached, the field evolution is well described in terms of 3 or 4 local modes. ForP/Pref≥2, "violent pulsations"2are observed, and the local-mode representation may not necessarily be uniformly convergent. These results may be compared with field evolution obtained by using the beam-propagation method.