Optical Second-Harmonic Generation Using a Focused Gaussian Laser Beam

Abstract

This paper reports analytical and experimental results on optical second-harmonic generation in the focus of the lowest order transverse mode of a cw gas laser beam. The results of the calculation are explained in physical terms and are confirmed by experiments carried out in crystals of ammonium dihydrogen phosphate (ADP). The dependence of the second-harmonic power generated in a negatively birefringent crystal upon the crystal double-refraction angle and the divergence, or diffraction, of the focused beam is obtained. There are found to be four distinct asymptotic regions, determined by the ratios of the characteristic lengths $z_R$ and $l_{\alpha }$ to the crystal length $l$, where the Rayleigh range of the focused beam $z_R$ characterizes the focus, and $l_{\alpha }$ is characteristic of the crystal double-refraction angle $\alpha$ and the laser beam focal spot size $w_0$. Proceeding from weak focusing to strong focusing (or in the direction of decreasing $w_0$), the second-harmonic power in the four regions varies as $\frac{l^2}{{w_0}^2}$, $\frac{l}{\alpha w_0}$, $\frac{w_0}{\alpha }$, and ${w_0}^2$, respectively. There is an optimum degree of focusing, determined only by the crystal length, for which a maximum amount of second-harmonic power is generated. This degree of focusing corresponds to $z_R=\frac{l}{\pi }$, and the corresponding power which is generated depends upon both $l$ and $\alpha$. Optimum focusing in a crystal of ADP 1 cm long yields about 400 times more second-harmonic power than the collimated laser beam. The excellent agreement between analysis and experiment allows the accurate measurement of optical nonlinearities using focused beams. The results for the general case of a crystal anywhere along the focused beam are also presented. Interpretation of them shows that the limiting of second-harmonic generation by double refraction is determined by beam divergence, not beam radius.