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The strongly transcendental equations of vector phase matching aretransformed into a fourth order polynomial equation that admits analyticalsolution. The real roots of this equation provide the optical axis orientationsthat are useful for efficient down-conversion in nonlinear uniaxial crystals.The production of entangled photon pairs is discussed in both collinear andnon-collinear configurations of the spontaneous parametric down conversion(SPDC) process. Degenerate and non-degenerate cases are also distinguished. Asa practical example, SPDC processes of type-I and type-II are studied forbeta-barium-borate (BBO) crystals. The predictions are in very good agreementwith experimental measurements already reported in the literature, and includetheoretical results of other authors as particular cases. Some properties thatseem to be exclusive to BBO crystals are reported, the experimentalverification of the latter would allow a better characterization of thesecrystals.

Exact solutions for vector phase matching conditions in nonlinear uniaxial crystals