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

International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 181-184

Section Induced polarization and susceptibility

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: 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.


Bloembergen, N. (1965). Nonlinear optics. New York: Benjamin.
Brasselet, S. & Zyss, J. (1998). Multipolar molecules and multipolar fields: probing and controlling the tensorial nature of nonlinear molecular media. J. Opt. Soc. Am. B, 15, 257–288.
Butcher, P. N. (1965). Nonlinear optical phenomena. Bulletin 200, Engineering Experiment Station, Ohio State University, USA.
Butcher, P. N. & Cotter, D. (1990). The elements of nonlinear optics. Cambridge series in modern optics. Cambridge University Press.
Jerphagnon, J., Chemla, D. S. & Bonneville, R. (1978). The description of physical properties of condensed matter using irreducible tensors. Adv. Phys. 27, 609–650.
Schwartz, L. (1981). Les tenseurs. Paris: Hermann.
Shen, Y. R. (1984). The principles of nonlinear optics. New York: Wiley.

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