Reflection and transmission of beams at a curved interface

Abstract

Gaussian-beam reflection and transmission in the presence of a convex circular boundary separating two different dielectrics is analyzed by the complex ray method whereby rays are traced in a complex coordinate space from the complex source point to the complex extension of the interface and then to the real observation point. The search for the complex parameters describing the ray connecting the source and the observer can be numerically time consuming. Much more efficient are paraxial approximations that express the field as perturbations about its values on the axis of maximum strength. Such paraxial approximations are explored here, using on-axis complex ray and even on-axis real ray solutions with corrections accounting for differences in phase, ray reflection and transmission coefficients, and ray divergence coefficients. Extensive numerical comparisons establish a reasonable range of validity of both paraxial approximations, except in regions characterizing the transition to total reflection, if that can occur.