The phenomena associated with the reflection and refraction of light have been widely studied. As the level of dimensionality of the model used in the analysis increases, so does the variety of interesting phenomena that emerge from the analysis. For example, plane-wave calculations give rise to Brewster angle and total internal reflection effects. If a finite two-dimensional slab beam is considered, Goos-Hänchen, focal, and angular shifts become apparent. As the problem is generalized to higher dimensions, possibly including the temporal dimension, additional phenomena may be expected as well. This paper begins by defining the terms and general notation used with the interface problem by reviewing plane-wave (infinite field) solutions. This is followed by a literature review of some two-dimensional results, including a discussion of the Goos-Hänchen, focal, and angular shift. Then numerical solutions to the full three-dimensional problem of finite beams (under a paraxial approximation) are presented using modern visualization techniques. The numerical results provide an intuitive understanding of the interface phenomena from a new perspective, graphically underscoring the difference between the geometrical ray model of reflection and refraction and the complicated field interaction that actually occurs. Finally, the numerical model is extended to include results for the nonlinear interface.

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