Abstract

Localized stationary envelope solitary wave solutions of the wave equation for a nonlinear dispersive optical medium are derived, both in one and higher dimensions. This is achieved by starting out from a heuristically derived modified Klein-Gordon equation, where the cubically nonlinear term represents a contribution from the nonlinear self-induced polarization of the medium. For the 1-D case, the solution is of the form A sechKx cosω_ct, where the amplitude A and the width (proportional to 1/K) are related to the nonlinear dispersive properties of the medium, with denoting the temporal carrier frequency. For the higher-dimensional case, however, no exact analytic solutions exist; but stationary radially symmetric wave shapes, obtainable by numerical methods, are similar to those derived in connection with the analysis of optical self-focusing. Experimental conditions required to generate such stationary wave shapes, recently observed using water waves, are also discussed. Finally, since spatially stationary solutions must be produced by the interaction of two contradirected propagating waves, a possible application lies in the area of convolution/correlation of two optical signals in a cubically nonlinear environment.

© 1985 Optical Society of America