Exact solutions for vector phase matching conditions in nonlinear uniaxial crystals

The strongly transcendental equations of vector phase matching are transformed into a fourth order polynomial equation that admits analytical solution. The real roots of this equation provide the optical axis orientations that are useful for efficient down-conversion in nonlinear uniaxial crystals. The production of entangled photon pairs is discussed in both collinear and non-collinear configurations of the spontaneous parametric down conversion (SPDC) process. Degenerate and non-degenerate cases are also distinguished. As a practical example, SPDC processes of type-I and type-II are studied for beta-barium-borate (BBO) crystals. The predictions are in very good agreement with experimental measurements already reported in the literature, and include theoretical results of other authors as particular cases. Some properties that seem to be exclusive to BBO crystals are reported, the experimental verification of the latter would allow a better characterization of these crystals.