Reflection and refraction of an arbitrary wave at a plane interface separating two uniaxial crystals

Exact solutions are obtained for the reflected and transmitted fields resulting when an arbitrary electromagnetic field is incident on a plane interface separating two uniaxial media. The orientation of the optical axis of each medium is arbitrary. The solutions are in the form of two triple integrals, one of which is a superposition of ordinary plane waves and the other a superposition of extraordinary plane waves. In the special case in which the incident wave is a single plane wave, the reflected and transmitted fields each reduce to a superposition of a single ordinary and a single extraordinary plane wave. The general results include, for that case, the reflection and transmission coefficients and the relations connecting the wave-propagation vectors and the Poynting vectors of the reflected and transmitted plane waves with the corresponding vectors of the incident plane wave. The expressions for the reflection and transmission coefficients are complicated in general but simplify greatly in the special cases in which the incident plane wave produces only a single reflected plane wave and a single transmitted plane wave. The conditions required of the two media and the incident plane wave for these special cases to result are presented. Also, the general results are specialized to the simple case in which the optical axis of each medium is normal to the interface. Finally, asymptotic approximations for each monochromatic spectral component of the exact solutions are derived which are valid at distances from the interface large compared to the wavelength of the radiation.