Second-order susceptibility

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. 180

Section 1.7.2.1.2.2. Second-order susceptibility

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

1.7.2.1.2.2. Second-order susceptibility

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Substitution of (1.7.2.15) in (1.7.2.12) 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.

References

International Tables for Crystallography (2006). Vol. D. ch. 1.7, p. 180