The effects of focusing in the three-frequency parametric upconverter

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Abstract
This paper extends the theory of parametric upconversion to describe the mixing of two focused optical beams with unequal confocal parameters. The results of this paper have greatest applicability when focusing of one input beam is limited by physical constraints, e.g., damage. The effects of phase matching, walk‐off, and diffraction are treated herein. The analysis shows that if the beams have fixed confocal parameters b1 and b2, optimum mixing generally occurs with b2≠b1. If both confocal parameters can be arbitrarily chosen, maximum efficiency occurs with b1=b2= (crystal length)/2.84 in the absence of Poynting vector walk‐off, and with b1≠b2 in the presence of walk‐off. For weak focusing upconversion efficiency is written in a closed form which shows explicitly the effect of walk‐off.