The paraxial approximation is applied to calculate the propagation characteristics of cw beams possessing Gaussian spatial profiles in nonlinear optical waveguides. For powers below the critical power for self-focusing in a homogeneous medium, we find that the waveguide effect dominates, i.e., the beam becomes trapped and the spot size varies sinusoidally. Above the critical power, the propagation is dominated by nonlinearity; the beam becomes unstable and displays self-focusing analogous to that in a homogeneous medium. Mode mixing is defined in terms of the nonlinearity-induced mixing of the initially excited modes at the guide face, and explicit expressions are obtained for the mode mixing for special cases. This mixing vanishes only if the spot size is a constant, a condition that can be satisfied only for special values of the parameters in longitudinally homogeneous, lossless guides.
Bernard Bendow, Peter D. Gianino, and Narkis Tzoar, "Theory of continuous-wave beam propagation in nonlinear optical waveguides," J. Opt. Soc. Am. 71, 656-663 (1981)