Optimizing Spontaneous Parametric Down-Conversion Through Wave Convergence Geometry in β-BaB₂O₄

Four testable predictions for enhancing SPDC photon pair generation in BBO crystals, derived from treating the crystal as a wave convergence cavity. (1) Circularly polarized pumping is predicted to enhance the pair rate by 1.5× over linear polarization by maintaining coupling to the B₃O₆ ring's three-fold screw symmetry — the standard prediction is 0.5×, making the test decisive with a 3× difference between predictions. (2) Möbius cavity resonances spaced 0.010 nm apart should produce periodic modulation in the coincidence rate when scanned with a narrow-linewidth laser. (3) The phase matching angle (28.82°) equals half the atomic-scale bipyramid angle (57.2°) formed by the Ba²⁺ spacing and B₃O₆ ring radius. (4) Molybdenum oxide crystals with three-fold symmetry should outperform BBO because the Mo–O mass ratio is a perfect fifth (3:2) to 0.05% accuracy — 30× closer than BBO's B–O bond (1.33%). A harmonic analysis of 11 known nonlinear crystals confirms BBO ranks first in frequency consonance, explaining its empirical dominance. The primary test requires only a quarter-wave plate added to a standard SPDC apparatus.