Abstract

The problem of computing the ray trajectory between points a and b, given the refractive-index distribution, is complicated by our ignorance of the initial ray direction at point a, which is needed in defining the path intercepting the endpoint b when numerically integrating the ray equation. It is shown that this ray-linking problem can be avoided by transforming the ray equation into an implicit integral equation for the true ray path that satisfies the given boundary conditions. The integral equation can be solved for the true path by the method of successive approximations. Simulations suggest that this iterative scheme often converges rapidly to the true path. An explicit expression for a ray path, obeying the boundary conditions, is also derived that is correct to first order in the refractive-index perturbation. This path provides an excellent approximation to the true path when the refractive-index perturbation is small and becomes increasingly better as the perturbation goes to zero. This first-order path can also be used as the first guess in the iterative scheme.

© 1987 Optical Society of America

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