Quantum Theory of Spontaneous Parametric Scattering of Intense Light

A quantum theory is developed for the spontaneous optical parametric process in nonlinear, anisotropic, and dispersive media; the process is treated directly as a scattering problem rather than a problem involving equivalent coupled oscillators. Following the general procedure in quantum field theory, a general expression is obtained for the differential extinction coefficient, which contains all the information on the intensity of the spontaneous emission in the parametric scattering process. The tuning characteristics, beam divergence, and spectral properties of the spontaneous emission for colinear and noncolinear interactions are described in detail and illustrated with numerical examples on such standard nonlinear optical crystals as LiNb${\mathrm{O}}_3$, N${\mathrm{H}}_4$ ${\mathrm{H}}_2$P${\mathrm{O}}_4$, and K${\mathrm{H}}_2$P${\mathrm{O}}_4$ under various pumping conditions. Explicit formulas for the intensity of the spontaneous emission are also obtained. The spontaneous intensity calculated using the formulas given is shown to be in complete agreement with known experimental data on N${\mathrm{H}}_4$ ${\mathrm{H}}_2$P${\mathrm{O}}_4$ pumped at 3472 Å.