International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 |
International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 182-183
Section 1.7.2.1.2. Linear and nonlinear susceptibilities^{a}Institut Néel CNRS Université Joseph Fourier, 25 rue des Martyrs, BP 166, 38042 Grenoble Cedex 9, France, and ^{b}Laboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France |
Whereas the polarization response has been expressed so far in the time domain, in which causality and time invariance are most naturally expressed, Fourier transformation into the frequency domain permits further simplification of the equations given above and the introduction of the susceptibility tensors according to the following derivation.
The direct and inverse Fourier transforms of the field are defined aswhere as E(t) is real.
By substitution of (1.7.2.15) in (1.7.2.7),where
In these equations, to satisfy the energy conservation condition that will be generalized in the following. In order to ensure convergence of χ^{(1)}, ω has to be taken in the upper half plane of the complex plane. The reality of R^{(1)} implies that .
Substitution of (1.7.2.15) in (1.7.2.12) yieldsorwithand . Frequencies ω_{1} and ω_{2} must be in the upper half of the complex plane to ensure convergence. Reality of R^{(2)} implies . is invariant under the interchange of the (α, ω_{1}) and (β, ω_{2}) pairs.