Threshold of the optical backward-wave parametric oscillator

We present a discussion of the threshold condition for the optical backward-wave parametric oscillator, taking into account diffraction due to the finite transverse extent of the fields, and using three transverse modes of both the forward- and backward-wave fields. The coupled differential equations are solved numerically, and the threshold is obtained by minimizing the pump field with respect to the parameters of the forward- and backward-wave fields. Denoting the confocal parameters by
b_{1}, b_{2}
, and b
3
for the backward, forward, and pump waves, respectively, and if the length of the crystal is 1, we find that for
1/b_{3} \geq 2
, we may set
b_{1} = b_{2} =b_{3}
. For most purposes, the phase-matching condition may be taken as
k_{2} = k_{1} + k_{3}
. Also, when calculating the threshold, it is adequate to consider only the two lowest order transverse modes of the forward-and backward-wave fields.