Theory of Nonlinear Beam Propagation in Optical Waveguides

We analyze the effect of transverse inhomogeneity on nonlinear beam propagation in a dielectric medium. Specifically, we consider the propagation of both CW beams and pulses in optical waveguides possessing a real nonlinear refractive index of the form n=n₁ (¯r,_ω) + n₂|E|2. The CW problem is treated within the paraxial approximation, for the case of a Gaussian beam incident on-axis. For powers lower than the homogeneous medium critical power, waveguiding dominates, and the beam focal parameter, although altered quantitatively, continues to vary sinusoidally as a function of distance as in the linear waveguide case, with a spectral period independent of the nonlinearity. Above the critical power, however, waveguiding is superceded and nonlinearity dominates. The beam becomes unstable, and displays oscillatory focussing in a fashion which is very similar to self-focussing in homogeneous media. Our pulse propagation studies employ a rather different starting point, based on the slowly varying envelope approximation and involving an averaging over the transverse coordinates. Our principal objective is to determine the conditions for undistorted pulse propagation, i.e., the existence of optical solitons. We obtain the equations governing the existence of solitons and find that they differ significantly from those for the homogeneous medium case. In particular, while “bright” soliton propagation is restricted to the anomolous dispersion regime in homogeneous media, in waveguides it is possible to propagate “bright” solitons in regimes of normal dispersion as well.