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

International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 183-184

Section Conventions for 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: Conventions for nonlinear susceptibilities

| top | pdf | Classical convention

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Insertion of ([link] in ([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),&(}]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(]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 ([link] by[\chi^{(n)}=2^{-s-m+n}d^{(n)}.\eqno(]Table[link] summarizes the most common classical nonlinear phenomena, following the notations defined above. Then, according to Table[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).&(}]The [E_{\alpha_i}] are the components of the total electric field E(ω).

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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}] Convention used in this chapter

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The K convention described above is often used, but may lead to errors in cases where two of the interacting waves have the same frequency but different polarization states. Indeed, as demonstrated in Chapter 1.6[link] and recalled in Section 1.7.3[link], a direction of propagation in an anisotropic crystal allows in the general case two different directions of polarization of the electric field vector, written E+ and E. Then any nonlinear coupling in this medium occurs necessarily between these eigen modes at the frequencies concerned.

Because of the possible non-degeneracy with respect to the direction of polarization of the electric fields at the same frequency, it is suitable to consider a harmonic generation process, second harmonic generation (SHG) or third harmonic generation (THG) for example, like any other non-degenerated interaction. We do so for the rest of this chapter. Then all terms derived from the permutation of the fields with the same frequency are taken into account in the expression of the induced nonlinear polarization and the K factor in equation ([link] disappears: hence, in the general case, the induced nonlinear polarization is written[\eqalignno{P_\mu^{(n)}(\omega_\sigma) &=\varepsilon_o\textstyle \sum \limits_{\alpha_1,\ldots,\alpha_n}\chi^{(n)}_{\mu\alpha_1\ldots\alpha_n}(-\omega_\sigma\semi\omega_1,\ldots,\omega_n)&\cr&\quad\times E_{\alpha_1}^{\pm}(\omega_1)\ldots E_{\alpha_n}^{\pm}(\omega_n), &(}]where [+] and − refer to the eigen polarization modes.

According to ([link], the nth harmonic generation induced polarization is expressed as[\eqalignno{P_\mu^{(n)}(n\omega) &=\varepsilon_o\textstyle \sum \limits_{\alpha_1,\ldots,\alpha_n}\chi^{(n)}_{\mu\alpha_1\ldots\alpha_n}(-n\omega\semi\omega,\ldots,\omega)&\cr&\quad\times E_{\alpha_1}^{\pm}(\omega_1)\ldots E_{\alpha_n}^{\pm}(\omega_n).&(}]For example, in the particular case of SHG where the two waves at ω have different directions of polarization E+(ω) and E(ω) and where the only nonzero [\chi^{(2)}_{yij}] coefficients are [\chi_{yxz}] and [\chi_{yzx}], ([link] gives[\eqalignno{P_y^{(2)}(2\omega) &=\varepsilon_o[\chi_{yxz}(-2\omega\semi\omega,\omega)E_x^+(\omega)E_z^-(\omega) &\cr &\quad +\chi_{yzx}(-2\omega\semi\omega,\omega)E_z^+(\omega)E_x^-(\omega)].&\cr&&(}]The two field component products are equal only if the two eigen modes are the same, i.e. [+] or −.

According to ([link] and ([link], we note that [\chi^{(n)}_{\mu\alpha_1\ldots\alpha_n}(-\omega_\sigma;] [\omega_1,\ldots,\omega_n)] changes smoothly to [\chi^{(n)}_{\mu\alpha_1\ldots\alpha_n}(-n\omega;] [\omega,\ldots,\omega)] when all the [\omega_1,\ldots\omega_n] approach continuously the same value ω.


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

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