General analysis of type I second-harmonic generation with elliptical Gaussian beams

The optimal focusing arrangement for second-harmonic generation is a compromise between the focal intensity and the depth of focus (confocal parameter) within the crystal. In the absence of birefringence (noncritical phasematching), the fundamental beam should have a circular cross section and a confocal parameter of 0.35 of the crystal’s length.1 Angle-tuned (critical) phase matching causes the cylindrical symmetry of harmonic generation to be lost, and elliptical (or anamorphic) focusing into the crystal has been shown to be advantageous. In this paper we present a theoretical analysis of type I phase matching and conclude that the harmonic power can be enhanced by as much as 30%. The less common type II phase matching, in which the fundamental is split between ordinary and extraordinary polarizations, requires a rather different analysis if these components are collinear2; a proposed noncollinear scheme is, however, equivalent to the type I arrangement considered here.3 Elliptical focusing will be especially useful for the frequency doubling of diode lasers,4 for not only is the beam profile usually elliptical but any enhancement cavities are likely to be astigmatic. To match such a geometry efficiently, or at least to be allowed a degree of insensitivity, will be useful indeed. Such considerations will be yet more important when frequency-doubling or -mixing stages are cascaded, as with frequency tripling5 and quadrupling.6,7 Our results will also be of particular relevance to second-harmonic generation by use of Brewsterangled crystals that render a circular incident beam elliptical. For high-power lasers as well, if the optimal circular focus leads to crystal damage8 or to thermally induced dephasing, elliptical focusing may provide a solution. Cylindrical focusing was first considered for secondharmonic generation by Volosov and Nilov,9 Volosov,10