In the case of three parametrically interacting beams of light, one beam (the pump beam) is usually known. In addition, it is often assumed that one or both of the other two beams are bounded by resonators, which simplifies the problem considerably. We present a variational method which can be used when the above assumption is not fulfilled, and we apply this method to the theory of the backward-wave parametric oscillator. Diffraction determines the operating point of such an oscillator, and also the angular part of the complex eigenvalue (which is just the phase mismatch Δ<i>k</i>) has a discrete spectrum.
ERIK W. ASLAKSEN, "Theory of Parametric Interaction between Three Diffracting Beams of Light," J. Opt. Soc. Am. 61, 320-324 (1971)