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

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

Section Linear and nonlinear responses

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: Linear and nonlinear responses

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Let us first consider the first-order linear response in ([link] and ([link]: the most general possible linear relation between P(t) and E(t) is[{\bf P}^{(1)}(t)=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau\;T^{(1)}(t, \tau)\cdot{\bf E}(\tau),\eqno(]where T(1) is a rank-two tensor, or in Cartesian index notation[P_{\mu}^{(1)}(t)=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau\;T_{\mu\alpha}^{(1)}(t, \tau)E_{\alpha}(\tau).\eqno(]Applying the time-invariance assumption to ([link] leads to[\eqalignno{{\bf P}^{(1)}(t+t_0)&=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau\,\,T^{(1)}(t+t_0,\tau)\cdot{\bf E}(\tau) &\cr &=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau\,\,T^{(1)}(t, \tau+t_0)\cdot{\bf E}(\tau) &\cr &=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau'\,\,T^{(1)}(t, \tau'-t_0)\cdot{\bf E}(\tau'), &(}]hence [T^{(1)}(t+t_0,\tau)=T^{(1)}(t, \tau - t_0)] or, setting [t=0] and [t_0=t],[T^{(1)}(t,\tau)=T^{(1)}(0,\tau-t)=R^{(1)}(t-\tau),\eqno(]where R(1) is a rank-two tensor referred to as the linear polarization response function, which depends only on the time difference [t-\tau]. Substitution in ([link] leads to[\eqalignno{{\bf P}^{(1)}(t)&=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau\,\,R^{(1)}(t-\tau){\bf E}(\tau) &\cr &=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau\,\,R^{(1)}(\tau){\bf E}(t-\tau). &(}]R(1) can be viewed as the tensorial analogue of the linear impulse function in electric circuit theory. The causality principle imposes that R(1)(τ) should vanish for [\tau\,\lt\,0] so that P(1)(t) at time t will depend only on polarizing field values before t. R(1), P(1) and E are real functions of time. 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(]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(]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(]From time invariance[\displaylines{\hfill T^{(2)}(t,\tau_1,\tau_2)=R^{(2)}(t-\tau_1,t-\tau_2),\hfill(\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(}%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. Higher-order response

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The nth order polarization can be expressed in terms of the ([n+1])-rank tensor [T^{(n)}(t,\tau_1,\tau_2,\ldots,\tau_n)] as[\eqalignno{{\bf P}^{(n)}(t) &=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_1\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_2\ldots\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_n\,\,T^{(n)}(t,\tau_1,\tau_2,\ldots,\tau_n) &\cr &\quad\cdot{\bf E}(\tau_1)\otimes{\bf E}(\tau_2)\otimes\ldots\otimes{\bf E}(\tau_n). &(}]

For similar reasons to those previously stated, it is sufficient to consider the symmetric part of T(n) with respect to the n! permutations of the n pairs (α1, τ1), (α2, τ2) [\ldots]n, τn). The T(n) tensor will then exhibit intrinsic permutation symmetry at the nth order. Time-invariance considerations will then allow the introduction of the ([n+1])th-rank real tensor R(n), which generalizes the previously introduced R operators:[\eqalignno{{\bf P}^{(n)}_{\mu}(t)&=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_1\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_2\ldots\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_n\,\,R^{(n)}_{\mu\alpha_1\alpha_2\ldots\alpha_n}(\tau_1,\tau_2,\ldots\tau_n)&\cr&\quad \times E_{\alpha_1}(t-\tau_1)E_{\alpha_2}(t-\tau_2)\ldots E_{\alpha_n}(t-\tau_n).&(}]R(n) cancels when one of the τi's is negative and is invariant under any of the n! permutations of the (αi, τi) pairs.

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