Nonlinear Schrödinger equation for optical media with quadratic nonlinearity

Wave propagation in optical media with strong dispersion and weak quadratic nonlinearity is analyzed using the method of multiple scales. This method shows that the evolution of the envelope for a single nondepleted pump wave is described by the nonlinear Schr\"odinger equation. Hence various self-modulation effects, due to an effective intensity-dependent refractive index, are possible to observe in materials with quadratic nonlinearity. That is, materials that are known to generate χ⁽²⁾ wave processes may also support, for example, soliton propagation. Physical conditions and numerical examples are given for observing solitons, self-defocusing, and spectral broadening. Other self-modulation effects are also discussed as well.