Abstract
This work develops approximate solutions for the electric field associated with a Gaussian beam moving through a material with a transverse variation in the refractive index. The solutions rely on a Lagrangian problem formulation whereby the path taken by the beam is governed by an ordinary differential equation and the intensity is governed by a continuity equation. Quadratic, exponential, and quartic transverse variations in refractive index are considered. The first (quadratic) case can be handled analytically while the latter two (quartic, exponential) are handled using a new, semi-analytical approach. In each case the approximations are compared to the standard numerical solution.
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