Magnitude and dispersion of Kleinman forbidden nonlinear optical coefficients

Starting with a perturbation expansion for the Kleinman forbidden nonlinear optical coefficient <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d_{ijkF}</tex> and for Miller's <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ_{ijkF}</tex> , and making several approximations, we arrive at a simple result for the ratio of forbidden to allowed mixing nonlinearities ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ω₁ + ω₂ = ω₃</tex> ), namely <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ_{ijkF}/Δ_{ijkA} ∝ (ω₃² + 2ω₁ω₂)</tex> . For second-harmonic generation (SHG) this can be expressed as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ_{ijkF}/Δ_{ijkA} (ω/χ)(∂χ/∂ω)</tex> , which clearly shows the close connection between <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ_{ijkF}</tex> and the linear dispersion. These expressions are shown to give good agreement with literature experimental values, as well as for our measurements on TeO <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> for various input frequencies ω <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> and ω <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> (i.e., <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ω₃ = 1.88, 2.33, 2.82, 3.50</tex> , and 3.76 eV).