Theory of second-harmonic generation with high-conversion efficiency

We analyze second-harmonic generation (SHG) with focused pump beams in the high-conversion regime where pump depletion is significant. Numerical solutions of the coupled equations for pump and harmonic fields are obtained for the cylindrically symmetric case of collinear phase matching. The optimum values of phase mismatch and confocal parameters σ and
b
found by Boyd and Kleinman are very nearly independent of pump power to above 95 percent conversion. When phase mismatch is not optimized, the best confocal parameter depends upon the input power. The conversion efficiency versus input power is determined numerically for optimum
\sigma, b
, and approximate analytical formulas are given for this relation.