Theory of Resonant Optical Second-Harmonic Generation from a Focused Gaussian Beam

Optical second-harmonic generation (SHG) from a focused beam is studied using the methods and results developed in the previous paper. We consider specifically SHG from a Gaussian beam in a uniaxial medium such as lithium niobate, where phase matching is possible perpendicular to the optic axis, so that it is possible to neglect double refraction. An analytic expression for the single-harmonic output is given for the limiting cases: (a) when the nonlinear medium lies in the near fields of both the single-harmonic resonator and the Gaussian fundamental beam; (b) when the nonlinear medium lies in the far field of both these. For the general case, numerically computed curves are given for the dependence of the output on phase matching, focus position, and focusing. The optimum value of the focusing parameter $\frac{l}{z_0}$ (where $l$ is the length of the nonlinear crystal and $z_0$ is ½ the confocal parameter of the fundamental beam) for conversion into the Gaussian single-harmonic mode is $\frac{l}{z_0}=5.65\mathrm{}$.