Linear and nonlinear responses

Boulanger, B.; Zyss, J.

doi:10.1107/97809553602060000634

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

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International Tables for Crystallography (2006). Vol. D. ch. 1.7, p. 179

Section 1.7.2.1.1. Linear and nonlinear responses

B. Boulanger^a ^* and J. Zyss^b

^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
Correspondence e-mail: benoitb@satie-bourgogne.fr

1.7.2.1.1. Linear and nonlinear responses

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1.7.2.1.1.1. Linear response

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Let us first consider the first-order linear response in (1.7.2.1) and (1.7.2.2): the most general possible linear relation between P(t) and E(t) is $[{\bf P}^{(1)}(t)=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau\;T^{(1)}(t, \tau)\cdot{\bf E}(\tau),\eqno(1.7.2.3)]$ where T⁽¹⁾ is a rank-two tensor, or in Cartesian index notation $[P_{\mu}^{(1)}(t)=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau\;T_{\mu\alpha}^{(1)}(t, \tau)E_{\alpha}(\tau).\eqno(1.7.2.4)]$ Applying the time-invariance assumption to (1.7.2.4) leads to $[\eqalignno{{\bf P}^{(1)}(t+t_0)&=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau\,\,T^{(1)}(t+t_0,\tau)\cdot{\bf E}(\tau) &\cr &=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau\,\,T^{(1)}(t, \tau+t_0)\cdot{\bf E}(\tau) &\cr &=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau'\,\,T^{(1)}(t, \tau'-t_0)\cdot{\bf E}(\tau'), &(1.7.2.5)}]$ hence $[T^{(1)}(t+t_0,\tau)=T^{(1)}(t, \tau - t_0)]$ or, setting [t=0] and [t_0=t] , $[T^{(1)}(t,\tau)=T^{(1)}(0,\tau-t)=R^{(1)}(t-\tau),\eqno(1.7.2.6)]$ 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 (1.7.2.5) leads to $[\eqalignno{{\bf P}^{(1)}(t)&=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau\,\,R^{(1)}(t-\tau){\bf E}(\tau) &\cr &=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau\,\,R^{(1)}(\tau){\bf E}(t-\tau). &(1.7.2.7)}]$ 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.

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

1.7.2.1.1.3. 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 $[\eqalignno{{\bf P}^{(n)}(t) &=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_1\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_2\ldots\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_n\,\,T^{(n)}(t,\tau_1,\tau_2,\ldots,\tau_n) &\cr &\quad\cdot{\bf E}(\tau_1)\otimes{\bf E}(\tau_2)\otimes\ldots\otimes{\bf E}(\tau_n). &(1.7.2.13)}]$

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) $[\ldots]$ (α_n, τ_n). 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: $[\eqalignno{{\bf P}^{(n)}_{\mu}(t)&=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_1\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_2\ldots\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_n\,\,R^{(n)}_{\mu\alpha_1\alpha_2\ldots\alpha_n}(\tau_1,\tau_2,\ldots\tau_n)&\cr&\quad \times E_{\alpha_1}(t-\tau_1)E_{\alpha_2}(t-\tau_2)\ldots E_{\alpha_n}(t-\tau_n).&(1.7.2.14)}]$ R⁽ⁿ⁾ cancels when one of the τ_i's is negative and is invariant under any of the n! permutations of the (α_i, τ_i) pairs.

References

International Tables for Crystallography (2006). Vol. D. ch. 1.7, p. 179