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. 183-184

Section 1.7.2.1.4.1. Classical convention

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

1.7.2.1.4.1. Classical convention

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Insertion of (1.7.2.26)[link] in (1.7.2.25)[link] together with permutation symmetry provides[\eqalignno{P_\mu^{(n)}(\omega_\sigma)&=\varepsilon_o\textstyle \sum \limits_{\alpha_1\alpha_2\ldots\alpha_n}\textstyle \sum \limits_{\omega}K(-\omega_\sigma\semi\omega_1,\omega_2,\ldots,\omega_n)&\cr&\quad\times\chi^{(n)}_{\mu\alpha_1\alpha_2\ldots\alpha_n}(-\omega_\sigma\semi\omega_1,\omega_2,\ldots,\omega_n)&\cr&\quad\times E_{\alpha_1}(\omega_1)E_{\alpha_2}(\omega_2)\ldots E_{\alpha_n}(\omega_n),&(1.7.2.29)}]where the summation over ω stands for all distinguishable permutation of [\omega_1,\omega_2,\ldots,\omega_n], K being a numerical factor given by[K(-\omega_\sigma;\omega_1,\omega_2,\ldots,\omega_n)=2^{s+m-n}p,\eqno(1.7.2.30)]where p is the number of distinct permutations of [\omega_1,\omega_2,\ldots,\omega_n], n is the order of the nonlinear process, m is the number of d.c. fields (e.g. corresponding to [\omega_\iota=0]) within the n frequencies and [s=0] when [\omega_\sigma=0], otherwise [s=1]. For example, in the absence of a d.c. field and when the ωi's are different, [K=2^{s-n}n!].

The K factor allows the avoidance of discontinuous jumps in magnitude of the [\chi^{(n)}_{\mu\alpha_1\alpha_2\ldots\alpha_n}] elements when some frequencies are equal or tend to zero, which is not the case for the other conventions (Shen, 1984[link]).

The induced nonlinear polarization is often expressed in terms of a tensor d(n) by replacing χ(n) in (1.7.2.29)[link] by[\chi^{(n)}=2^{-s-m+n}d^{(n)}.\eqno(1.7.2.31)]Table 1.7.2.1[link] summarizes the most common classical nonlinear phenomena, following the notations defined above. Then, according to Table 1.7.2.1[link], the nth harmonic generation induced nonlinear polarization is written[\eqalignno{P_\mu^{(2)}(n\omega)&=\varepsilon_o\textstyle \sum \limits_{\alpha_1\alpha_2\ldots\alpha_n}{}2^{n-1}\chi^{(n)}_{\mu\alpha_1\alpha_2\ldots\alpha_n}(-n\omega\semi\omega,\omega,\ldots,\omega)&\cr&\quad\times E_{\alpha_1}(\omega)E_{\alpha_2}(\omega)\ldots E_{\alpha_n}(\omega).&(1.7.2.32)}]The [E_{\alpha_i}] are the components of the total electric field E(ω).

Table 1.7.2.1| top | pdf |
The most common nonlinear effects and the corresponding susceptibility tensors in the frequency domain

ProcessOrder n[-\omega_\sigma; \omega_1,\omega_2,\ldots,\omega_n]K
Linear absorption 1 [-\omega;\omega] 1
Optical rectification 2 [0;-\omega,\omega] 1/2
Linear electro-optic effect 2 [-\omega;\omega,0] 2
Second harmonic generation 2 [-2\omega;\omega,\omega] 1/2
Three-wave mixing 2 [-\omega_3;\omega_1,\omega_2] 1
D.c. Kerr effect 3 [-\omega;\omega,0,0] 3
D.c. induced second harmonic generation 3 [-2\omega;\omega,\omega,0] 3/2
Third harmonic generation 3 [-3\omega;\omega,\omega,\omega] 1/4
Four-wave mixing 3 [-\omega_4;\omega_1,\omega_2,\omega_3] 3/2
Coherent anti-Stokes Raman scattering 3 [-\omega_{\rm as};\omega_p,-\omega_p,-\omega_s] 3/4
Intensity-dependent refractive index 3 [-\omega;\omega,-\omega,\omega] 3/4
nth harmonic generation n [-n\omega;\omega,\omega,\ldots,\omega] [2^{1-n}]

References

Shen, Y. R. (1984). The principles of nonlinear optics. New York: Wiley.








































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