Focusing dependence of the efficiency of a singly resonant optical parametric oscillator

Dependence of the conversion efficiency of a singly resonant optical parametric oscillator on the focusing parameters of Gaussian-shaped pump and signal beams is determined here by the Green's function method. The electric fields of the pump and the idler beams are expressed as even- and add-powered series, respectively, in the second-order nonlinear optical coefficient, d. It is shown that the gain in power of the signal beam can be expressed as a even-powered series in d. The lowest-order term in this series, quadratic in d, is proportional to a double integral, the next-higher-order term is a quadruple integral, and the third term is a six-dimensional integral. The integrals are expressed in a form such that reduction to the case of no diffraction (collimated beams) is straightforward. These results provide, for the first time to the author's knowledge, analytical expressions for the values of the (near-threshold) conversion efficiency in the presence of arbitrary focusing of the pump and the signal beams.