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

International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 182-183

Section 1.7.2.1.2. Linear and nonlinear susceptibilities

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

1.7.2.1.2. Linear and nonlinear susceptibilities

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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 as[\eqalignno{{\bf E}(t) &=\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\omega\,\,{\bf E}(\omega)\exp(-i\omega t)&(1.7.2.15)\cr {\bf E}(\omega) &=(1/2\pi)\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}t\,\,{\bf E}(t)\exp(i\omega t),&(1.7.2.16)}%fd1.7.2.16]where [{\bf E}(\omega)^*={\bf E}(-\omega)] as E(t) is real.

1.7.2.1.2.1. Linear susceptibility

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By substitution of (1.7.2.15)[link] in (1.7.2.7)[link],[\eqalignno{{\bf P}^{(1)}(t)&=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\omega\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau\,\,R^{(1)}(\tau)\cdot{\bf E}(\omega)\exp[-i\omega(t-\tau)]&\cr {\bf P}^{(1)}(t)&=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\omega\,\,\chi^{(1)}(-\omega_{\sigma}\semi\omega){\bf E}(\omega)\exp(-i\omega_{\sigma}t),&\cr&&(1.7.2.17)}]where[\chi^{(1)}(-\omega_{\sigma}\semi\omega)=\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau\,\,R^{(1)}(\tau)\exp(i\omega\tau).]

In these equations, [\omega_{\sigma}=\omega] 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 [\chi^{(1)}(-\omega_{\sigma};\omega)^*= \chi^{(1)}(\omega_{\sigma}^*;-\omega^*)].

1.7.2.1.2.2. Second-order susceptibility

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Substitution of (1.7.2.15)[link] in (1.7.2.12)[link] yields[\eqalignno{{\bf P}^{(2)}(t)&=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\omega_1\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\omega_2\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_1\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_2\,\,R^{(2)}(\tau_1,\tau_2)&\cr&\quad\cdot{\bf E}(\omega_1)\otimes{\bf E}(\omega_2)\exp\{-i[\omega_1(t-\tau_1)+\omega_2(t-\tau_2)]\}&\cr&&(1.7.2.18)}]or[\eqalignno{{\bf P}^{(2)}(t)&=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\omega_1\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\omega_2\,\,\chi^{(2)}(-\omega_\sigma\semi\omega_1,\omega_2)\cdot{\bf E}(\omega_1)\otimes{\bf E}(\omega_2)&\cr&\quad\times\exp(-i\omega_\sigma t)&(1.7.2.19)}]with[\eqalign{\chi^{(2)}(-\omega_\sigma\semi\omega_1,\omega_2)&=\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_1\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_2\,\,R^{(2)}(\tau_1,\tau_2)\cr&\quad \times \exp[i(\omega_1\tau_1+\omega_2\tau_2)]}]and [\omega_\sigma=\omega_1+\omega_2]. Frequencies ω1 and ω2 must be in the upper half of the complex plane to ensure convergence. Reality of R(2) implies [\chi^{(2)}(-\omega_\sigma;\omega_1,\omega_2)^* =] [\chi^{(2)}(\omega_\sigma^*;-\omega_1^*,-\omega_2^*)]. [\chi^{(2)}_{\mu\alpha\beta}(-\omega_\sigma;\omega_1,\omega_2)] is invariant under the interchange of the (α, ω1) and (β, ω2) pairs.

1.7.2.1.2.3. nth-order susceptibility

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Substitution of (1.7.2.15)[link] in (1.7.2.14)[link] provides[\eqalignno{{\bf P}^{(n)}(t)&=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\omega_1\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\omega_2\ldots\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\omega_n\,\,\chi^{(n)}(-\omega_\sigma\semi\omega_1,\omega_2,\ldots\omega_n)&\cr&\quad\cdot{\bf E}(\omega_1)\otimes{\bf E}(\omega_2)\otimes\ldots\otimes{\bf E}(\omega_n)\exp(-i\omega_\sigma t)&\cr&&(1.7.2.20)}]where[\eqalignno{&\chi^{(n)}(-\omega_\sigma\semi\omega_1,\omega_2,\ldots,\omega_n)&\cr&\quad=\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)}(\tau_1,\tau_2,\ldots,\tau_n)\exp\big(i\textstyle \sum \limits_{j=1}^{n}\omega_j\tau_j\big)&\cr&&(1.7.2.21)}]and [\omega_\sigma=\omega_1+\omega_2+\ldots+\omega_n].

All frequencies must lie in the upper half complex plane and reality of χ(n) imposes[\chi^{(n)}(-\omega_\sigma;\omega_1,\omega_2,\ldots,\omega_n)^*=\chi^{(n)}(\omega_\sigma^*;-\omega_1^*,-\omega_2^*,\ldots,-\omega_n^*).\eqno(1.7.2.22)]Intrinsic permutation symmetry implies that [\chi^{(n)}_{\mu\alpha_1\alpha_2\ldots\alpha_n}(-\omega_\sigma;] [\omega_1,\omega_2,\ldots,\omega_n)] is invariant with respect to the n! permutations of the (αi, ωi) pairs.








































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