Differential ray tracing in inhomogeneous media

Differential ray tracing determines an optical system’s first-order properties by finding the first-order changes in the configuration of an exiting ray in terms of changes in that ray’s initial configuration. When one or more of the elements of a system is inhomogeneous, the only established procedure for carrying out a first-order analysis of a general ray uses relatively inefficient finite differences. To trace a ray through an inhomogeneous medium, one must, in general, numerically integrate an ordinary differential equation, and Runge–Kutta schemes are well suited to this application. We present an extension of standard Runge–Kutta schemes that gives exact derivatives of the numerically approximated rays.