International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.7, p. 182

Section 1.7.2.1.1.2. Quadratic response

B. Boulangera* and J. Zyssb

aInstitut Néel CNRS Université Joseph Fourier, 25 rue des Martyrs, BP 166, 38042 Grenoble Cedex 9, France, and bLaboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France
Correspondence e-mail:  benoit.boulanger@grenoble.cnrs.fr

1.7.2.1.1.2. Quadratic response

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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)]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)]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)]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]Causality demands that R(2)1, τ2) cancels for either τ1 or τ2 negative while R(2) is real. Intrinsic permutation symmetry implies that Rμαβ(2)1, τ2) is invariant by interchange of (α, τ1) and (β, τ2) pairs.








































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