Complex Rays with an Application to Gaussian Beams

TLDR

Rays are used to construct fields by determining the field in the region z > 0 from data specified on the plane z = 0. The method applies complex rays to a Gaussian field on z = 0, yielding a Gaussian beam while relying solely on complex rays. The ray method reproduces stationary‑phase results, is simpler, and works even when no integral representation of the field exists.

Abstract

The use of rays to construct fields is illustrated by finding the field in the region z>0 when the field is given on the plane z = 0. This construction is valid for complex rays as well as real ones. The method is applied to a gaussian field in the plane z = 0, in which case a gaussian beam results. The calculation involves only complex rays. Exactly the same results are also obtained by applying the method of stationary phase to an integral representation of the field. However, the ray method is simpler than the stationary-phase method, and it is also applicable to problems for which the stationary-phase method cannot be used because no integral representation of the field is known.