On dispersion of nonlinear optical susceptibility for HHG in crystals

Abstract

The Bloembergen-Armstrong formula [1] for the first nonlinear susceptibility (χ) of a molecular crystal in terms of hyperpolarizabilities of individual molecules is generalized to any order of nonlinearity. It accounts for dispersion and is presented in a similar way as the original one: χ^(k)(k) = c'(k)ξ^(k)C(1,1,…,1), the superscript is the order of nonlinearity, lower by 1 than the tensor rank, c is the local field mixed tensor, C is the symmetrized k^th power of the latter. The arguments are the frequency multiplicities (C depends on k arguments). A graphic technique is developed for construction of the tensor ξ^(k) from molecular hyperpolarizabilities of ranks, not exceeding k. The diagram technique is based on the root tree graphs approach and allows finding all parameters of the constituting elements, including the numerical coefficients and frequency multiplicities of all tensors. A recurrent formula for ξ^(k) is also given.