Calculation of effective nonlinear coefficient in BIBO for spontaneous parametric down conversion

We theoretically investigate the effective nonlinear coefficient d _eff of spontaneous parametric down-conversion (SPDC) process in biaxial nonlinear crystal BiB₃O₆ (BIBO). We solve the Fresnel equations in the collinear degenerate down-converted configuration, and get the relations of phase-matching angles. Considering the double phase-matching, we compute the phase-matching angles with different wavelength. Furthermore, we simplify the expression of effective nonlinear coefficient out of the principal planes, and numerically calculate it varying with the angle and wavelength. These results indicate that there are three effective phase-matching schemes for SPDC which can be used to construct a system to generate entangled photon pairs.