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International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D. ch. 1.7, p. 179
Section 1.7.2.1.1.2. Quadratic response
a
Laboratoire de Spectrométrie Physique, Université Joseph Fourier, 140 avenue de la Physique, BP 87, 38 402 Saint-Martin-d'Hères, France, and bLaboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France |
The most general expression for P(2)(t) which is quadratic in E(t) is![[{\bf P}^{(2)}(t)=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_1\,\,{\rm d}\tau_2\,\,T^{(2)}(t,\tau_1,\tau_2)\cdot{\bf E}(\tau_1)\otimes{\bf E}(\tau_2)\eqno(1.7.2.8)]](/teximages/dach1o7/dach1o7fd8.svg)
or in Cartesian notation![[P^{(2)}_{\mu}(t)=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_1\,\,{\rm d}\tau_2\,\,T^{(2)}_{\mu\alpha\beta}(t,\tau_1,\tau_2)E_{\alpha}(\tau_1)E_{\beta}(\tau_2).\eqno(1.7.2.9)]](/teximages/dach1o7/dach1o7fd9.svg)
It can easily be proved by decomposition of T(2) into symmetric and antisymmetric parts and permutation of dummy variables (α, τ1) and (β, τ2), that T(2) can be reduced to its symmetric part, satisfying![[T^{(2)}_{\mu\alpha\beta}(t,\tau_1,\tau_2)=T^{(2)}_{\mu\alpha\beta}(t,\tau_2,\tau_1).\eqno(1.7.2.10)]](/teximages/dach1o7/dach1o7fd10.svg)
From time invariance![[\displaylines{\hfill T^{(2)}(t,\tau_1,\tau_2)=R^{(2)}(t-\tau_1,t-\tau_2),\hfill(1.7.2.11)\cr{\bf P}^{(2)}(t)=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_1\;{\rm d}\tau_2\;R^{(2)}(t-\tau_1,t-\tau_2)\cdot{\bf E}(\tau_1)\otimes{\bf E}(\tau_2),\cr {\bf P}^{(2)}(t)=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_1\;{\rm d}\tau_2\;R^{(2)}(\tau_1,\tau_2)\cdot{\bf E}(t-\tau_1)\otimes{\bf E}(t-\tau_2).\cr\hfill(1.7.2.12)}%fd1.7.2.12]](/teximages/dach1o7/dach1o7fd11.svg)
Causality demands that R(2)(τ1, τ2) cancels for either τ1 or τ2 negative while R(2) is real. Intrinsic permutation symmetry implies that Rμαβ(2)(τ1, τ2) is invariant by interchange of (α, τ1) and (β, τ2) pairs.
