Abstract
We present an effective method for calculating phase-matching conditions in biaxial crystals, especially for nonlinear orthorhombic crystals. Exploiting the angle definition introduced by Japanese mathematician Kodaira Kunihiko, we deduce the angular relations in geometry and obtain the expressions of refractive indices depending on angular orientation of wave vector k and optical axis angle. Then, we directly calculate the phase-matching conditions with BIBO crystal in spontaneous parametric down-conversion (SPDC) process and gain the optimum phase matching schemes for the type I and type II. On its basis, we discuss the angular gradients of the pump and emission wave refractive index near the exact phase matching direction and compare the SPDC with double-frequency process in geometrical relations of the refractive index ellipsoids. This method based on angle-dependent refractive index can be applied to three-wave interactions. It is convenient to calculate the phase matching parameters without solving the quadratic Fresnel equations.
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