  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.
