Magnitude and dispersion of Kleinman forbidden nonlinear optical coefficients

Starting with a perturbation expansion for the Kleinman forbidden nonlinear optical coefficient d_{ijk^{F}} and for Miller's \Delta_{ijk^{F}} , and making several approximations, we arrive at a simple result for the ratio of forbidden to allowed mixing nonlinearities ( \omega_{1} + \omega_{2} = \omega_{3} ), namely \Delta_{ijk^{F}}/\Delta_{ijk^{A}} \propto (\omega_{3}^{2} + 2\omega_{1}\omega_{2}) . For second-harmonic generation (SHG) this can be expressed as \Delta_{ijk^{F}}/\Delta_{ijk^{A}} \simeq (\omega/\chi)(\partial\chi/\partial\omega) , which clearly shows the close connection between \Delta_{ijk^{F}} and the linear dispersion. These expressions are shown to give good agreement with literature experimental values, as well as for our measurements on TeO 2 for various input frequencies ω 1 and ω 2 (i.e., \omega_{3} = 1.88, 2.33, 2.82, 3.50 , and 3.76 eV).