Ray tracing based on the Wigner representation of optical fields

Abstract

An optical ray is defined by its distance to the optical axis of an object, an image, or a refracting surface, and an angle giving its direction of propagation, that is, by a 2-dimensional space vector and a 2-dimensional angular frequency. Since we use spherical segments, the angular frequency is related to the angle between the ray and the normal to the spherical segment at the incident point. The Wigner representation of optical fields is a 4-dimensional space-frequency function, defined after choosing appropriate scaled space and frequency variables that are linked to space variables and angular frequencies. The effect of diffraction on the Wigner distribution of an optical field is deduced from a scalar theory of diffraction and provides a link between scaled space and frequency variables on an emitter and on a receiver at a given distance. By changing scaled variables back into physical ones, we obtain a ray tracing method which is based on a scalar theory of diffraction. An explicit example and an application to optical resonators illustrate the method.