Influence of a linear inhomogeneity of the refractive index of nonlinear crystals on second-harmonic generation

An analysis is given of the influence of a linear inhomogeneity of the refractive index n, represented by the gradient ∂n/∂r, on second-harmonic generation. The angle between the direction of ∂n/∂r and the normal to a crystal (the crystal is assumed to be cut along the phase-matching direction) is not restricted in any way. An expression is given for the dependence of the second-harmonic power P₂ on the detuning of the wave vectors of the fundamental and the second-harmonic radiation. An analysis is made of the limiting cases of a medium with a linear inhomogeneity of the refractive index; these cases correspond to nonlinear media with longitudinal or transverse inhomogeneities and to a homogeneous medium. The results are given of numerical calculations of P₂, and of the width and shape of the phase-matching curves. An experimental investigation of the influence of a linear inhomogeneity on second-harmonic generation is shown to be in good agreement with the calculated results.