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

International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 181-186

Section 1.7.2. Origin and symmetry of optical nonlinearities

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:

1.7.2. Origin and symmetry of optical nonlinearities

| top | pdf | Induced polarization and susceptibility

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The macroscopic electronic polarization of a unit volume of the material system is classically expanded in a Taylor power series of the applied electric field E, according to Bloembergen (1965[link]):[{\bf P}={\bf P}_0+\varepsilon_o(\chi^{(1)}\cdot{\bf E}+\chi^{(2)}\cdot{\bf E}^2+\ldots+\chi^{(n)}\cdot{\bf E}^n+\ldots),\eqno(]where χ(n) is a tensor of rank [n+1], En is a shorthand abbreviation for the nth order tensor product [{\bf E}] [\otimes] [{\bf E}] [\otimes\ldots\otimes] [{\bf E}] [=\otimes^n\,\,{\bf E}] and the dot stands for the contraction of the last n indices of the tensor χ(n) with the full En tensor. More details on tensor algebra can be found in Chapter 1.1[link] and in Schwartz (1981[link]).

A more compact expression for ([link] is[{\bf P}={\bf P}_0+{\bf P}_1(t)+{\bf P}_2(t)+\ldots+{\bf P}_n(t)+\ldots, \eqno(]where P0 represents the static polarization and Pn represents the nth order polarization. The properties of the linear and nonlinear responses will be assumed in the following to comply with time invariance and locality. In other words, time displacement of the applied fields will lead to a corresponding time displacement of the induced polarizations and the polarization effects are assumed to occur at the site of the polarizing field with no remote interactions. In the following, we shall refer to the classical formalism and related notations developed in Butcher (1965[link]) and Butcher & Cotter (1990[link]).

Tensorial expressions will be formulated within the Cartesian formalism and subsequent multiple lower index notation. The alternative irreducible tensor representation, as initially implemented in the domain of nonlinear optics by Jerphagnon et al. (1978[link]) and more recently revived by Brasselet & Zyss (1998[link]) in the realm of molecular-engineering studies, is particularly advantageous for connecting the nonlinear hyperpolarizabilities of microscopic (e.g. molecular) building blocks of molecular materials to the macroscopic (e.g. crystalline) susceptibility level. Such considerations fall beyond the scope of the present chapter, which concentrates mainly on the crystalline level, regardless of the microscopic origin of phenomena. Linear and nonlinear responses

| top | pdf | Linear response

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Let us first consider the first-order linear response in ([link] and ([link]: 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(]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(]Applying the time-invariance assumption to ([link] 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'), &(}]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(]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 ([link] 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). &(}]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. 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(]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(]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(]From time invariance[\displaylines{\hfill T^{(2)}(t,\tau_1,\tau_2)=R^{(2)}(t-\tau_1,t-\tau_2),\hfill(\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(}%fd1.7.2.12]Causality demands that R(2)1, τ2) cancels for either τ1 or τ2 negative while R(2) is real. Intrinsic permutation symmetry implies that Rμαβ(2)1, τ2) is invariant by interchange of (α, τ1) and (β, τ2) pairs. 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). &(}]

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]n, τn). 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).&(}]R(n) cancels when one of the τi's is negative and is invariant under any of the n! permutations of the (αi, τi) pairs. 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)&(\cr {\bf E}(\omega) &=(1/2\pi)\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}t\,\,{\bf E}(t)\exp(i\omega t),&(}%fd1.7.2.16]where [{\bf E}(\omega)^*={\bf E}(-\omega)] as E(t) is real. Linear susceptibility

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By substitution of ([link] in ([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&&(}]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^*)]. Second-order susceptibility

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Substitution of ([link] in ([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&&(}]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)&(}]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. nth-order susceptibility

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Substitution of ([link] in ([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&&(}]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&&(}]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(]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. Superposition of monochromatic waves

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Optical fields are often superpositions of monochromatic waves which, due to spectral discretization, will introduce considerable simplifications in previous expressions such as ([link] relating the induced polarization to a continuous spectral distribution of polarizing field amplitudes.

The Fourier transform of the induced polarization is given by[{\bf P}^{(n)}(\omega)=(1/2\pi)\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}t\;{\bf P}^{(n)}(t)\exp(i\omega t).\eqno(]Replacing P(n)(t) by its expression as from ([link] and applying the well known identity[(1/2\pi)\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}t\;\exp[i(\omega-\omega_\sigma)t]=\delta(\omega-\omega_\sigma)\eqno(]leads to[\eqalignno{{\bf P}^{(n)}(\omega)&=\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\times {\bf E}(\omega_1){\bf E}(\omega_2)\ldots {\bf E}(\omega_n)\delta(\omega-\omega_\sigma).&(}]

In practical cases where the applied field is a superposition of monochromatic waves[{\bf E}(t)=(1/2)\textstyle \sum \limits_{\omega'}[E_{\omega'}\exp(-i\omega't)+E_{-\omega'}\exp(i\omega't)]\eqno(]with [E_{-\omega'}=E_{\omega'}^*]. By Fourier transformation of ([link][{\bf E}(\omega)=(1/2)\textstyle \sum \limits_{\omega'}[E_{\omega'}\delta(\omega-\omega')+E_{-\omega'}\delta(\omega+\omega')].\eqno(]The optical intensity for a wave at frequency [\omega'] is related to the squared field amplitude by[I_{\omega'}=\varepsilon_o c n(\omega')\langle {\bf E}^2(t)\rangle_t=\textstyle{1\over 2}\varepsilon_ocn(\omega')|E_{\omega'}|^2.\eqno(]The averaging as represented above by brackets is performed over a time cycle and [n(\omega')] is the index of refraction at frequency [\omega']. 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. Symmetry properties

| top | pdf | Intrinsic permutation symmetry

| top | pdf | ABDP and Kleinman symmetries

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Intrinsic permutation symmetry, as already discussed, imposes the condition that the nth order susceptibility [\chi^{(n)}_{\mu\alpha_1\alpha_2\ldots\alpha_n}(-\omega_\sigma;] [\omega_1,\omega_2,\ldots,\omega_n)] be invariant under the [n!] permutations of the ([\alpha_i,\omega_i]) pairs as a result of time invariance and causality. Furthermore, the overall permutation symmetry, i.e. the invariance over the [(n+1)!] permutations of the ([\alpha_i,\omega_i]) and ([\mu,-\omega_\sigma]) pairs, may be valid when all the optical frequencies occuring in the susceptibility and combinations of these appearing in the denominators of quantum expressions are far removed from the transitions, making the medium transparent at these frequencies. This property is termed ABDP symmetry, from the initials of the authors of the pioneering article by Armstrong et al. (1962[link]).

Let us consider as an application the quantum expression of the quadratic susceptibility (with damping factors neglected), the derivation of which being beyond the scope of this chapter, but which can be found in nonlinear optics treatises dealing with microscopic interactions, such as in Boyd (1992[link]):[\eqalignno{&\chi^{(2)}_{\mu\alpha\beta}(-\omega_\sigma\semi\omega_1,\omega_2)&\cr&\quad ={Ne^3 \over \varepsilon_o^2\hbar^2}S_T\displaystyle\sum\limits_{abc}\rho_o(a){r_{ab}^{\mu}r_{bc}^\alpha r_{ca}^{\beta}\over (\Omega_{ba}-\omega_1-\omega_2)(\Omega_{ca}-\omega_1)},&\cr&&(}]where N is the number of microscopic units (e.g. molecules in the case of organic crystals) per unit volume, a, b and c are the eigen states of the system, Ωba and Ωca are transition energies, [r_{ab}^\mu] is the μ component of the transition dipole connecting states a and b, and [\rho_o(a)] is the population of level a as given by the corresponding diagonal term of the density operator. ST is the summation operator over the six permutations of the ([\mu, -\omega_\sigma]), ([\alpha, \omega_1]), ([\beta, \omega_2]). Provided all frequencies at the denominator are much smaller than the transition frequencies Ωba and Ωca, the optical frequencies [-\omega_\sigma], [\omega_1], [\omega_2] can be permuted without significant variation of the susceptibility. It follows correspondingly that the susceptibility is invariant with respect to the permutation of Cartesian indices appearing only in the numerator of ([link], regardless of frequency. This property, which can be generalized to higher-order susceptibilities, is known as Kleinman symmetry. Its validity can help reduce the number of non-vanishing terms in the susceptibility, as will be shown later. Manley–Rowe relations

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An important consequence of overall permutation symmetry is the Manley–Rowe power relations, which account for energy exchange between electromagnetic waves in a purely reactive (e.g. non-dissipative) medium. Calling Wi the power input at frequency ωi into a unit volume of a dielectric polarizable medium,[W_i=\left\langle {\bf E}(t)\cdot{{\rm d}{\bf P} \over {\rm d}t}(t)\right\rangle,\eqno(]where the averaging is performed over a cycle and[\eqalignno{{\bf E}(t)&=Re[E_{\omega_i}\exp(-j\omega_i t)]&\cr {\bf P}(t)&=Re[P_{\omega_i}\exp(-j\omega_i t)].&(}]The following expressions can be derived straightforwardly:[W_i=\textstyle{1 \over 2}\omega_i \,Re(iE_{\omega_i}\cdot P_{\omega_i})=\textstyle{1 \over 2}\omega_i \,Im(E_{\omega_i}^* \cdot P_{\omega_i}).\eqno(]Introducing the quadratic induced polarization P(2), Manley–Rowe relations for sum-frequency generation state[{W_1 \over \omega_1}={W_2 \over \omega_2}=-{W_3 \over \omega_3}.\eqno(]Since [\omega_1+\omega_2=\omega_3], ([link] leads to an energy conservation condition, namely [W_3+W_1+W_2=0], which expresses that the power generated at ω3 is equal to the sum of the powers lost at ω1 and ω2.

A quantum mechanical interpretation of these expressions in terms of photon fusion or splitting can be given, remembering that [W_i/\hbar\omega_i] is precisely the number of photons generated or annihilated per unit volume in unit time in the course of the nonlinear interactions. Contracted notation for susceptibility tensors

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The tensors [\chi^{(2)}_{\mu\alpha\beta}(-2\omega;\omega,\omega)] or [d^{(2)}_{\mu\alpha\beta}(-2\omega;\omega,\omega)] are invariant with respect to (α, β) permutation as a consequence of the intrinsic permutation symmetry. Independently, it is not possible to distinguish the coefficients [\chi^{(2)}_{ijk}(-2\omega;\omega,\omega)] and [\chi^{(2)}_{ikj}(-2\omega;\omega,\omega)] by SHG experiments, even if the two fundamental waves have different directions of polarization.

Therefore, these third-rank tensors can be represented in contracted form as [3\times 6] matrices [\chi_{\mu m}(-2\omega;\omega,\omega)] and [d_{\mu m}(-2\omega;\omega,\omega)], where the suffix m runs over the six possible (α, β) Cartesian index pairs according to the classical convention of contraction:[\eqalign{\hbox{for }\mu\hbox{: } &x\rightarrow 1\quad y\rightarrow 2\quad z\rightarrow 3\hfill\cr \hbox{for }m\hbox{: } &xx\rightarrow 1\quad yy\rightarrow 2\quad zz\rightarrow 3\quad yz=zy\rightarrow 4\hfill\cr& xz=zx\rightarrow 5\quad xy=yx\rightarrow 6.\hfill}]The 27 elements of [\chi^{(2)}_{\mu\alpha\beta}(-2\omega;\omega,\omega)] are then reduced to 18 in the [\chi_{\mu m}] contracted tensor notation (see Section[link] ).

For example, ([link] can be written[\eqalignno{P_y^{(2)}(2\omega)&=\varepsilon_o\chi_{25}(-2\omega\semi\omega,\omega)[e_x^+(\omega){\bf E}^+(\omega)e_z^-(\omega){\bf E}^-(\omega)&\cr&\quad +e_z^+(\omega){\bf E}^+(\omega)e_x^-(\omega){\bf E}^-(\omega)].&(}]The same considerations can be applied to THG. Then the 81 elements of [\chi^{(3)}_{\mu\alpha\beta\gamma}(-3\omega;\omega,\omega,\omega)] can be reduced to 30 in the [\chi_{\mu m}] contracted tensor notation with the following contraction convention:[\eqalign{\hbox{for }\mu\hbox{: } &x\rightarrow 1\quad y\rightarrow 2\quad z\rightarrow 3\hfill\cr \hbox{for }m\hbox{: } &xxx\rightarrow 1\quad yyy\rightarrow 2\quad zzz\rightarrow 3\quad yzz \rightarrow 4\quad yyz\rightarrow 5\hfill\cr& xzz\rightarrow 6\quad xxz\rightarrow 7\quad xyy\rightarrow 8\quad xxy\rightarrow 9\quad xyz\rightarrow 0.\hfill}]If Kleinman symmetry holds, the contracted tensor can be further extended beyond SHG and THG to any other processes where all the frequencies are different. Implications of spatial symmetry on the susceptibility tensors

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Centrosymmetry is the most detrimental crystalline symmetry constraint that will fully cancel all odd-rank tensors such as the d(2) [or χ(2)] susceptibilities. Intermediate situations, corresponding to noncentrosymmetric crystalline point groups, will reduce the number of nonzero coefficients without fully depleting the tensors.

Tables[link] to[link][link][link] detail, for each crystal point group, the remaining nonzero χ(2) and χ(3) coefficients and the eventual connections between them. χ(2) and χ(3) are expressed in the principal axes x, y and z of the second-rank χ(1) tensor. ([x,y,z]) is usually called the optical frame; it is linked to the crystallographical frame by the standard conventions given in Chapter 1.6[link] .

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Nonzero χ(2) coefficients and equalities between them in the general case

Symmetry classχ(2) nonzero elements
C1 (1) All 27 elements are independent and nonzero
C2 (2) (twofold axis parallel to z) [xyz], [xzy], [xxz], [xzx], [yyz], [yzy], [yxz], [yzx], [zxx], [zyy], [zzz], [zxy], [zyx]
Cs (m) (mirror perpendicular to z) [xxx], [xyy], [xzz], [xxy], [xyx], [yxx], [yyy], [yzz], [yxy], [yyx], [zyz], [zzy], [zxz], [zzx]
C2v (mm2) (twofold axis parallel to z) [xzx], [xxz], [yyz], [yzy], [zxx], [zyy], [zzz]
D2 (222) [xyz], [xzy], [yzx], [yxz], [zxy], [zyx]
C4 (4) [xyz=-yxz], [xzy=-yzx], [xzx=yzy], [xxz=yyz], [zxx=zyy], [zzz], [zxy=-zyx]
S4 ([\bar 4]) [xyz=yxz], [xzy=yzx], [xzx=-yzy], [xxz=-yyz], [zxx=-zyy], [zxy=zyx]
D4 (422) [xyz=-yxz], [xzy=-yzx], [zxy=-zyx]
C4v (4mm) [xzx=yzy], [xxz=yyz], [zxx=zyy], [zzz]
D2d ([\bar 4 2 m]) [xyz=yxz], [xzy=yzx], [zxy=zyx]
C6 (6) [xyz=-yxz], [xzy=-yzx], [xzx=yzy], [xxz=yyz], [zxx=zyy], [zzz], [zxy=-zyx]
C3h ([\bar 6]) [xxx=-xyy=-yxy=-yyx], [yyy=-yxx=] [-xyx=-xxy]
D6 (622) [xyz=-yxz], [xzy=-yzx], [zxy=-zyx]
C6v (6mm) [xzx=yzy], [xxz=yyz], [zxx=zyy], [zzz]
D3h ([\bar 6 2 m]) (mirror perpendicular to x) [yyy=-yxx=-xxy=-xyx]
C3 (3) [xxx=-xyy=-yyx=-yxy], [xyz=-yxz], [xzy=-yzx], [xzx=yzy], [xxz=yyz], [yyy=-yxx=-xxy=-xyx], [zxx=zyy], [zzz], [zxy=-zyx]
D3 (32) [xxx=-xyy=-yyx=-yxy], [xyz=-yxz], [xzy=-yzx], [zxy=-zyx]
C3v (3m) (mirror perpendicular to x) [yyy=-yxx=-xxy=-xyx], [xzx=yzy], [xxz=yyz], [zxx=zyy, zzz]
T (23), Td ([\bar 4 3 m]) [xyz=xzy=yzx=yxz=zxy=zyx]
O (432) [xyz=-xzy=yzx=-yxz=zxy=-zyx]

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Nonzero χ(2) coefficients and equalities between them under the Kleinman symmetry assumption

Symmetry classIndependent nonzero χ(2) elements under Kleinman symmetry
C1 (1) [xxx], [xyy=yxy=yyx], [xzz=zxz=zzx], [xyz=xzy=yxz=yzx=zxy=zyx], [xxz=xzx=zxx], [xxy=xyx=yxx], [yyy], [yzz=zyz=zzy], [yyz=] [yzy=] [zyy], [zzz]
C2 (2) (twofold axis parallel to z) [xyz=xzy=yxz=yzx=zxy=zyx], [xxz=xzx=zxx], [yyz=yzy=zyy], [zzz]
Cs (m) (mirror perpendicular to z) [xxx], [xyy=yxy=yyx], [xzz=zxz=zzx], [xxy=xyx=yxx], [yyy], [yzz=zyz=zzy]
C2v (mm2) (twofold axis parallel to z) [xzx=xxz=zxx], [yyz=yzy=zyy], [zzz]
D2 (222) [xyz=xzy=yzx=yxz=zxy=zyx]
C4 (4) [xzx=xxz=zxx=yzy=yyz=zyy], [zzz]
S4 ([\bar 4]) [xyz=xzy=yzx=yzx=zxy=zyx], [xzx=] [xxz=] [zxx=] [-yzy=] [-yyz=] [-zyy]
D4 (422) All elements are nil
C4v (4mm) [xzx=xxz=zxx=yyz=yzy=zyy], [zzz]
D2d ([\bar 4 2 m]) [xyz=xzy=yzx=yxz=zxy=zyx]
C6 (6) [xzx=xxz=zxx=yyz=yzy=zyy], [zzz]
C3h ([\bar 6]) [xxx=-xyy=-yxy=-yyx], [yyy=] [-yxx=] [-xyx=] [-xxy]
D6 (622) All elements are nil
C6v (6mm) [xzx=xxz=zxx=yyz=yzy=zyy], [zzz]
D3h ([\bar 6 2 m]) (mirror perpendicular to x) [yyy=-yxx=-xxy=-xyx]
C3 (3) [xxx=-xyy=-yyx=-yxy], [xzx] [=] [xxz] [=] [zxx] [=] [yyz] [=] [yzy] [=] [zyy], [yyy] [=] [-yxx] [=] [-xxy] [=] [-xyx], [zzz]
D3 (32) [xxx=-xyy=-yyx=-yxy]
C3v (3m) (mirror perpendicular to x) [yyy=-yxx=-xxy=-xyx], [xzx] [=] [xxz] [=] [zxx] [=] [yyz] [=] [yzy] [=] [zyy], [zzz]
T (23), Td ([\bar 4 3 m]) [xyz=xzy=yzx=yxz=zxy=zyx]
O (432) All elements are nil

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Nonzero χ(3) coefficients and equalities between them in the general case

Symmetry classχ(3) nonzero elements
C1 (1), Ci ([\bar 1]) All 81 elements are independent and nonzero
Cs (m), C2 (2), C2h [\left(2 \over m\right)] (twofold axis parallel to z) [xxxx], [xyyy], [xyzz], [xzyz], [xzzy], [xxzz], [xzxz], [xzzx], [xxyy], [xyxy], [xyyx], [xxxy], [xxyx], [xyxx], [yxxx], [yyyy], [yyzz], [yzyz], [yzzy], [yxzz], [yzxz], [yzzx], [yxyy], [yyxy], [yyyx], [yxxy], [yxyx], [yyxx], [zzzz], [zyyz], [zyzy], [zzyy], [zxxz], [zxzx], [zzxx], [zxyz], [zxzy], [zyxz], [zzxy], [zyzx], [zzyx]
C2v (mm2), D2 (222), D2h (mmm) (twofold axis parallel to z) [xxxx], [xxzz], [xzxz], [xzzx], [xxyy], [xyxy], [xyyx], [yyyy], [yyzz], [yzyz], [yzzy], [yxxy], [yxyx], [yyxx], [zzzz], [zyyz], [zyzy], [zzyy], [zxxz], [zxzx], [zzxx]
S4 ([\bar 4]), C4 (4), C4h [\left(4\over m\right)] [xxxx=yyyy], [xyyy=-yxxx], [xyzz=-yxzz], [xzyz=-yzxz], [xzzy=-yzzx], [xxzz=yyzz], [xzxz=yzyz], [xzzx=yzzy], [xxyy=yyxx], [xyxy=yxyx], [xyyx=yxxy], [xxxy=-yyyx], [xxyx=-yyxy], [xyxx=-yxyy], [zzzz], [zyyz=zxxz], [zyzy=zxzx], [zzyy=zzxx], [zxyz=-zyxz], [zxzy=-zyzx], [zzxy=-zzyx]
C4v (4mm), D2d ([\bar 4 2 m]), D4 (422), D4h [\left({4 \over m}mm\right)] [xxxx=yyyy], [xxzz=yyzz], [xzxz=yzyz], [xzzx=yzzy], [xxyy=yyxx], [xyxy=yxyx], [xyyx=yxxy], [zzzz], [zyyz=zxxz], [zyzy=zxzx], [zzyy=zzxx]
C3h ([\bar 6]), C6 (6), C6h [\left(6\over m\right)] [xxxx=yyyy=xxyy+xyxy+xyyx], [xyyy=xxxy+xxyx+xyxx=-yxxx], [xyzz=-yxzz], [xzyz=-yzxz], [xzzy=-yzzx], [xxzz=yyzz], [xzxz=yzyz], [xzzx=yzzy], [xxyy=yyxx], [xyxy=yxyx], [xyyx=yxxy], [xxxy=-yyyx], [xxyx=-yyxy], [xyxx=-yxyy], [zzzz], [zyyz=zxxz], [zyzy=zxzx], [zzyy=zzxx], [zxyz=-zyxz], [zxzy=-zyzx], [zzxy=-zzyx]
C6v (6mm), D3h ([\bar 6 2 m]), D6 (622), D6h [\left({6 \over m}mm\right)] [xxxx=yyyy=xxyy+xyxy+xyyx], [xxzz=yyzz], [xzxz=yzyz], [xzzx=yzzy], [xxyy=yyxx], [xyxy=yxyx], [xyyx=yxxy], [zzzz], [zyyz=zxxz], [zyzy=zxzx], [zzyy=zzxx]
C3 (3), C3i ([\bar 3]) [xxxx=yyyy=xxyy+xyxy+xyyx], [xyyy=xxxy+xxyx+xyxx=-yxxx], [xyzz=-yxzz], [xzyz=-yzxz], [xzzy=-yzzx], [xyyz=yxyz=yyxz=-xxxz], [xyzy=yyzx=yxzy=-xxzx], [xzyy=yzxy=yzyx=-xzxx], [xxzz=yyzz], [xzxz=yzyz], [xzzx=yzzy], [xxyy=yyxx], [xyxy=yxyx], [xyyx=yxxy], [xxxy=-yyyx], [xxyx=-yyxy], [xyxx=-yxyy], [xxyz] [=] [xyxz] [=] [yxxz] [=] [-yyyz], [xxzy] [=] [xyzx] [=] [yxzx] [=] [-yyzy], [xzxy] [=] [xzyx] [=] [yzxx] [=] [-yzyy], [-zxxx] [=] [zxyy] [=] [zyxy] [=] [zyyx], [-zyyy] [=] [zxxy] [=] [zxyx] [=] [zyxx], [zzzz], [zyyz=zxxz], [zyzy=zxzx], [zzyy=zzxx], [zxyz=-zyxz], [zxzy=-zyzx], [zzxy=-zzyx]
C3v (3m), D3 (32), D3d ([\bar 3 m]) (mirror perpendicular to x) (twofold axis parallel to x) [xxxx=yyyy=xxyy+xyxy+xyyx], [xxzz=yyzz], [xzxz=yzyz], [xzzx=yzzy], [xxyy=yyxx], [xyxy=yxyx], [xyyx=yxxy], [xxyz=xyxz=yxxz=-yyyz], [xxzy=xyzx=yxzx=-yyzy], [xzxy=xzyx=yzxx=-yzyy], [-zyyy] [=] [zxxy] [=] [zxyx] [=] [zyxx], [zzzz], [zyyz=zxxz], [zyzy=zxzx], [zzyy=zzxx]
T (23), Th (m3) [xxxx=yyyy=zzzz], [xxzz=yyxx=zzyy], [xzxz=yxyx=zyzy], [xzzx=yxxy=zyyz], [xxyy=yyzz=zzxx], [xyxy] [=] [yzyz] [=] [zxzx], [xyyx] [=] [yzzy] [=] [zxxz]
Td ([\bar 4 3 m]), O (432), Oh (m3m) [xxxx=yyyy=zzzz], [xxzz=xxyy=yyzz=yyxx=zzyy=zzxx], [xzxz=xyxy=yzyz=yxyx=zyzy=zxzx], [xzzx=xyyx=yzzy=yxxy=zyyz=zxxz]

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Nonzero χ(3) coefficients and equalities between them under the Kleinman symmetry assumption

Symmetry classIndependent nonzero elements of χ(3) under Kleinman symmetry
C1 (1), Ci ([\bar 1]) [xxxx], [xyyy=yxyy =yyxy=yyyx], [xzzz=zxzz=zzxz=zzzx], [xyzz] [=] [xzyz] [=] [xzzy] [=] [yxzz] [=] [yzxz] [=] [yzzx] [=] [zxyz] [=] [zxzy] [=] [zyxz] [=] [zyzx] [=] [zzxy] [=] [zzyx], [xyyz] [=] [xyzy] [=] [xzyy] [=] [yxyz] [=] [yxzy] [=] [yyxz] [=] [yyzx] [=] [yzxy] [=] [yzyx] [=] [zxyy] [=] [zyxy] [=] [zyyx], [xxzz] [=] [xzxz] [=] [xzzx] [=] [zxxz] [=] [zxzx] [=] [zzxx], [xxxz] [=] [xxzx] [=] [xzxx] [=] [zxxx], [xxyy] [=] [xyxy] [=] [xyyx] [=] [yxxy] [=] [yxyx] [=] [yyxx], [xxxy=xxyx=xyxx=yxxx], [xxyz] [=] [xxzy] [=] [xyxz] [=] [xyzx] [=] [xzxy] [=] [xzyx] [=] [yxxz] [=] [yxzx] [=] [yzxx] [=] [zxxy] [=] [zxyx] [=] [zyxx], [yyyy], [yzzz=zyzz=zzyz=zzzy], [yyzz] [=] [yzyz] [=] [yzzy] [=] [zyyz] [=] [zyzy] [=] [zzyy], [yyyz] [=] [yyzy] [=] [yzyy] [=] [zyyy], [zzzz]
Cs (m), C2 (2), C2h [\left(2\over m\right)] (twofold axis parallel to z) [xxxx], [xyyy=yxyy=yyxy=yyyx], [xyzz] [=] [xzyz] [=] [xzzy] [=] [yxzz] [=] [yzxz] [=] [yzzx] [=] [zxyz] [=] [zxzy] [=] [zyxz] [=] [zyzx] [=] [zzxy] [=] [zzyx], [xxzz] [=] [xzxz] [=] [xzzx] [=] [zxxz] [=] [zxzx] [=] [zzxx], [xxyy] [=] [xyxy] [=] [xyyx] [=] [yxxy] [=] [yxyx] [=] [yyxx], [xxxy=xxyx=xyxx=yxxx], [yyyy], [yyzz] [=] [yzyz] [=] [yzzy] [=] [zyyz] [=] [zyzy] [=] [zzyy], [zzzz]
C2v (mm2), D2 (222), D2h (mmm) (twofold axis parallel to z) [xxxx], [xxzz] [=] [xzxz] [=] [xzzx] [=] [zxxz] [=] [zxzx] [=] [zzxx], [xxyy] [=] [xyxy] [=] [xyyx] [=] [yxxy] [=] [yxyx] [=] [yyxx], [yyyy], [yyzz] [=] [yzyz] [=] [yzzy] [=] [zyyz] [=] [zyzy] [=] [zzyy], [zzzz]
S4 ([\bar 4]), C4 (4), C4h [\left(4\over m\right)] [xxxx=yyyy], [xyyy] [=] [yxyy] [=] [yyxy] [=] [yyyx] [=] [-xxxy] [=] [-xxyx] [=] [-xyxx] [=] [-yxxx], [xxzz] [=] [xzxz] [=] [xzzx] [=] [yyzz] [=] [yzyz] [=] [yzzy] [=] [zyyz] [=] [zyzy] [=] [zzyy] [=] [zxxz] [=] [zxzx] [=] [zzxx], [xxyy] [=] [xyxy] [=] [xyyx] [=] [yxxy] [=] [yxyx] [=] [yyxx], [zzzz]
C4v (4mm), D2d ([\bar 4 2 m]), D4 (422), D4h [\left({4 \over m}mm\right)] [xxxx=yyyy], [xxzz] [=] [xzxz] [=] [xzzx] [=] [yyzz] [=] [yzyz] [=] [yzzy] [=] [zyyz] [=] [zyzy] [=] [zzyy] [=] [zxxz] [=] [zxzx] [=] [zzxx], [xxyy] [=] [xyxy] [=] [xyyx] [=] [yxxy] [=] [yxyx] [=] [yyxx], [zzzz]
C3h ([\bar 6]), C6 (6), C6h [\,\left(6\over m\right)], C6v (6mm), D3h ([\bar 6 2 m]), D6 (622), D6h [\left({6\over m}mm\right)] [xxxx=yyyy=xxyy+xyxy+xyyx], [xxzz] [=] [xzxz] [=] [xzzx] [=] [yyzz] [=] [yzyz] [=] [yzzy] [=] [zyyz] [=] [zyzy] [=] [zzyy] [=] [zxxz] [=] [zxzx] [=] [zzxx], [xxyy] [=] [xyxy] [=] [xyyx] [=] [yxxy] [=] [yxyx] [=] [yyxx], [zzzz]
C3 (3), C3i ([\bar 3]) [xxxx=yyyy=xxyy+xyxy+xyyx], [xyyz] [=] [xyzy] [=] [xzyy] [=] [-xxxz] [=] [-xxzx] [=] [-xzxx] [=] [yxyz] [=] [yxzy] [=] [yyxz] [=] [yyzx] [=] [yzxy] [=] [yzyx] [=] [-zxxx] [=] [zxyy] [=] [ zyxy] [=] [zyyx], [xxzz] [=] [xzxz] [=] [xzzx] [=] [yyzz] [=] [yzyz] [=] [yzzy] [=] [zyyz] [=] [zyzy] [=] [zzyy] [=] [zxxz] [=] [zxzx] [=] [zzxx], [xxyy] [=] [xyxy] [=] [xyyx] [=] [yxxy] [=] [yxyx] [=] [yyxx], [xxyz] [=] [xxzy] [=] [xyxz] [=] [xyzx] [=] [xzxy] [=] [xzyx] [=] [-yyyz] [=] [-yyzy] [=] [-yzyy] [=] [yxxz] [=] [yxzx] [=] [yzxx] [=] [-zyyy] [=] [zxxy] [=] [zxyx] [=] [zyxx], [zzzz]
C3v (3m), D3 (32), D3d ([\bar 3 m]) (mirror perpendicular to x) (twofold axis parallel to x) [xxxx=yyyy=xxyy+xyxy+xyyx], [xxzz] [=] [xzxz] [=] [xzzx] [=] [yyzz] [=] [yzyz] [=] [yzzy] [=] [zyyz] [=] [zyzy] [=] [zzyy] [=] [zxxz] [=] [zxzx] [=] [zzxx], [xxyy] [=] [xyxy] [=] [xyyx] [=] [yxxy] [=] [yxyx] [=] [yyxx], [xxyz] [=] [xxzy] [=] [xyxz] [=] [xyzx] [=] [xzxy] [=] [xzyx] [=] [-yyyz] [=] [-yyzy] [=] [-yzyy] [=] [yxxz] [=] [yxzx] [=] [yzxx] [=] [-zyyy] [=] [zxxy] [=] [zxyx] [=] [zyxx], [zzzz]
T (23), Th (m3), Td ([\bar 4 3 m]), O (432), Oh (m3m) [xxxx=yyyy=zzzz], [xxzz] [=] [xzxz] [=] [xzzx] [=] [xxyy] [=] [xyxy] [=] [xyyx] [=] [yyzz] [=] [yzyz] [=] [yzzy] [=] [yyxx] [=] [yxyx] [=] [yxxy] [=] [zzyy] [=] [zyzy] [=] [zyyz] [=] [zzxx] [=] [zxzx] [=] [zxxz]


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