Threshold of the optical backward-wave parametric oscillator

Abstract

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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b_{1}, b_{2}</tex> , and b <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</inf> for the backward, forward, and pump waves, respectively, and if the length of the crystal is 1, we find that for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/b_{3} \geq 2</tex> , we may set <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b_{1} = b_{2} =b_{3}</tex> . For most purposes, the phase-matching condition may be taken as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k_{2} = k_{1} + k_{3}</tex> . Also, when calculating the threshold, it is adequate to consider only the two lowest order transverse modes of the forward-and backward-wave fields.