Reflection and Transmission of Beams at a Dielectric Interface

Abstract

When complex values are assigned to the source coordinates in the expressions for the fields radiated by a line or point source in a homogeneous medium, the resulting fields have the form of a two- or three-dimensional Gaussian beam. This fact may be utilized to develop results for beam propagation and scattering in inhomogeneous regions from corresponding results for point or line source fields. After justifying the analytic continuation procedure to derive rigorous integral solutions for the (two-dimensional) beam fields reflected by and transmitted through a plane dielectric interface, asymptotic results are obtained by the saddle point method; it is found that the analytic continuation to complex source coordinates can be performed directly on the asymptotic expressions for reflected, refracted and lateral wave constituents of ordinary line source fields. Interpretation of the reflected fields reveals that they incorporate directly the lateral beam shift, a phenomenon usually regarded as a diffraction effect. When the beam is incident from the denser medium at an angle greater than the critical angle for total reflection, calculation of the evanescent transmitted fields requires special care since the corresponding results for an ordinary line source have not been well developed. A “local inhomogeneous plane wave” approximation is derived for the transmitted field in this case, together with physically meaningful restrictions on the range of validity of the approximation. These restrictions, and their interpretation, which have not previously been reported, are of importance in approximating the evanescent fields produced by total reflection in more general configurations not amenable to rigorous analysis.