Evaluation of Beam Fields Reflected at a Plane Interface

The complex ray method is employed to calculate the fields of a Gaussian beam reflected at a plane interface between two different media. A lateral wave is included in the region of total reflection. To simplify the calculation, a paraxial approximation is employed when the observer is located near the beam axis. These calculations all fail when the beam axis and observer simultaneously approach the critical angle, the criterion of failure being a parameter called the numerical distance. The correct solution in the transition region is generated by a parabolic cylinder transition function. Detailed numerical comparisons with a reference solution generated by numerical integration of the exact spectral integral representation provide an assessment of the validity of the various formulations and approximations.