Parametric Interaction of Focused Gaussian Light Beams

A theoretical study is presented on the optimization of second harmonic generation (SHG) and parametric generation (PG) by a laser beam in a uniaxial nonlinear crystal. Numerically computed curves show the dependence of the SHG power, and the reciprocal of the PG threshold power, on the parameter l/b, where l is the optical path length in the crystal and b is the confocal parameter (determined by the focal length of the focusing lens and the minimum radius of the laser beam, assumed to be in the TEM₀₀ mode of an optical resonator). The calculations take full account of diffraction and double refraction. In the absence of double refraction, the optimum focusing condition is found to be l/b=2.84. For PG the optimization of the crystal length l is also discussed, and curves are given showing the dependence of the threshold on l for the case in which signal and idler have the same losses. It is shown that the computed functions are also relevant to the mixing of two Gaussian beams and to parametric amplification. Pump depletion is neglected. Appendices are provided on (1) the theory of Gaussian extraordinary beams and the extension of the theory to cover both positive and negative birefringent crystals, (2) the general definition of nonlinear coefficients, (3) the effective nonlinear coefficient, and (4) details of the computations. The theory of the PG threshold is applied to tellurium and LiNbO₃. On the basis of reasonable assumptions about the losses, a PG threshold of 1.0 W is obtained for a pump at 10.6 μ in Te. The optimum length is found to be l=0.14 cm. For LiNbO₃ of length l=1 cm the threshold is 22 mW at 0.5147 μ. Also calculated is the quantum efficiency for up‐conversion in HgS from 10.6 to 0.6729 μ using the 0.6328 μ He–Ne laser.