The effects of phase matching method and of crystal symmetry on the polar dependence of third-order non-linear optical polarization

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Abstract
The theory of the phase matching technique of Giordmaine and Maker et al. has been extended to include four wave third-order interactions. Three different experimental situations are considered, one with the three lower frequency waves polarized orthogonally to the fourth, one with two orthogonal to the fourth and one with one orthogonal to the fourth. The equations relating the phase matching angle in uniaxial crystals to the crystal birefringence and the frequencies of the four waves are given. These can be solved to give a cone of phase matched directions about the crystal optic axis. The effect of the symmetry properties of the third-order polarizability tensor on the strength of the phase matched output signal as a function of phase matching angle theta and azimuthal angle phi is evaluated for uniaxial crystals. Expressions are also given for cubic and isotropic groups although in these classes phase matching is, of course, not possible by the birefringence method. The form of the third-order polarizability tensor is tabulated assuming Kleinman's symmetry conjecture to be valid.