International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 
International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 181186
Section 1.7.2. Origin and symmetry of optical nonlinearities^{a}Institut Néel CNRS Université Joseph Fourier, 25 rue des Martyrs, BP 166, 38042 Grenoble Cedex 9, France, and ^{b}Laboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France 
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):where χ^{(n)} is a tensor of rank , E^{n} is a shorthand abbreviation for the nth order tensor product and the dot stands for the contraction of the last n indices of the tensor χ^{(n)} with the full E^{n} tensor. More details on tensor algebra can be found in Chapter 1.1 and in Schwartz (1981).
A more compact expression for (1.7.2.1) iswhere P_{0} represents the static polarization and P_{n} 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) and Butcher & Cotter (1990).
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) and more recently revived by Brasselet & Zyss (1998) in the realm of molecularengineering 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.
Let us first consider the firstorder linear response in (1.7.2.1) and (1.7.2.2): the most general possible linear relation between P(t) and E(t) iswhere T^{(1)} is a ranktwo tensor, or in Cartesian index notationApplying the timeinvariance assumption to (1.7.2.4) leads tohence or, setting and ,where R^{(1)} is a ranktwo tensor referred to as the linear polarization response function, which depends only on the time difference . Substitution in (1.7.2.5) leads toR^{(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 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.
The most general expression for P^{(2)}(t) which is quadratic in E(t) isor in Cartesian notationIt can easily be proved by decomposition of T^{(2)} into symmetric and antisymmetric parts and permutation of dummy variables (α, τ_{1}) and (β, τ_{2}), that T^{(2)} can be reduced to its symmetric part, satisfyingFrom time invarianceCausality demands that R^{(2)}(τ_{1}, τ_{2}) cancels for either τ_{1} or τ_{2} negative while R^{(2)} is real. Intrinsic permutation symmetry implies that R_{μαβ}^{(2)}(τ_{1}, τ_{2}) is invariant by interchange of (α, τ_{1}) and (β, τ_{2}) pairs.
The nth order polarization can be expressed in terms of the ()rank tensor as
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 (α_{1}, τ_{1}), (α_{2}, τ_{2}) (α_{n}, τ_{n}). The T^{(n) }tensor will then exhibit intrinsic permutation symmetry at the nth order. Timeinvariance considerations will then allow the introduction of the ()thrank real tensor R^{(n)}, which generalizes the previously introduced R operators:R^{(n)} cancels when one of the τ_{i}'s is negative and is invariant under any of the n! permutations of the (α_{i}, τ_{i}) pairs.
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 aswhere as E(t) is real.
By substitution of (1.7.2.15) in (1.7.2.7),where
In these equations, 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 .
Substitution of (1.7.2.15) in (1.7.2.12) yieldsorwithand . Frequencies ω_{1} and ω_{2} must be in the upper half of the complex plane to ensure convergence. Reality of R^{(2)} implies . is invariant under the interchange of the (α, ω_{1}) and (β, ω_{2}) pairs.
Optical fields are often superpositions of monochromatic waves which, due to spectral discretization, will introduce considerable simplifications in previous expressions such as (1.7.2.20) relating the induced polarization to a continuous spectral distribution of polarizing field amplitudes.
The Fourier transform of the induced polarization is given byReplacing P^{(n)}(t) by its expression as from (1.7.2.20) and applying the well known identityleads to
In practical cases where the applied field is a superposition of monochromatic waveswith . By Fourier transformation of (1.7.2.26)The optical intensity for a wave at frequency is related to the squared field amplitude byThe averaging as represented above by brackets is performed over a time cycle and is the index of refraction at frequency .
Insertion of (1.7.2.26) in (1.7.2.25) together with permutation symmetry provideswhere the summation over ω stands for all distinguishable permutation of , K being a numerical factor given bywhere p is the number of distinct permutations of , n is the order of the nonlinear process, m is the number of d.c. fields (e.g. corresponding to ) within the n frequencies and when , otherwise . For example, in the absence of a d.c. field and when the ω_{i}'s are different, .
The K factor allows the avoidance of discontinuous jumps in magnitude of the elements when some frequencies are equal or tend to zero, which is not the case for the other conventions (Shen, 1984).
The induced nonlinear polarization is often expressed in terms of a tensor d^{(n)} by replacing χ^{(n)} in (1.7.2.29) byTable 1.7.2.1 summarizes the most common classical nonlinear phenomena, following the notations defined above. Then, according to Table 1.7.2.1, the nth harmonic generation induced nonlinear polarization is writtenThe are the components of the total electric field E(ω).

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 and recalled in Section 1.7.3, 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 nondegeneracy 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 nondegenerated 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 (1.7.2.29) disappears: hence, in the general case, the induced nonlinear polarization is writtenwhere and − refer to the eigen polarization modes.
According to (1.7.2.33), the nth harmonic generation induced polarization is expressed asFor 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 coefficients are and , (1.7.2.34) givesThe two field component products are equal only if the two eigen modes are the same, i.e. or −.
According to (1.7.2.33) and (1.7.2.34), we note that changes smoothly to when all the approach continuously the same value ω.
Intrinsic permutation symmetry, as already discussed, imposes the condition that the nth order susceptibility be invariant under the permutations of the () pairs as a result of time invariance and causality. Furthermore, the overall permutation symmetry, i.e. the invariance over the permutations of the () and () 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).
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):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, Ω_{ba} and Ω_{ca} are transition energies, is the μ component of the transition dipole connecting states a and b, and is the population of level a as given by the corresponding diagonal term of the density operator. S_{T} is the summation operator over the six permutations of the (), (), (). Provided all frequencies at the denominator are much smaller than the transition frequencies Ω_{ba} and Ω_{ca}, the optical frequencies , , 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 (1.7.2.36), regardless of frequency. This property, which can be generalized to higherorder susceptibilities, is known as Kleinman symmetry. Its validity can help reduce the number of nonvanishing terms in the susceptibility, as will be shown later.
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. nondissipative) medium. Calling W_{i} the power input at frequency ω_{i} into a unit volume of a dielectric polarizable medium,where the averaging is performed over a cycle andThe following expressions can be derived straightforwardly:Introducing the quadratic induced polarization P^{(2)}, Manley–Rowe relations for sumfrequency generation stateSince , (1.7.2.40) leads to an energy conservation condition, namely , which expresses that the power generated at ω_{3} is equal to the sum of the powers lost at ω_{1} and ω_{2}.
A quantum mechanical interpretation of these expressions in terms of photon fusion or splitting can be given, remembering that is precisely the number of photons generated or annihilated per unit volume in unit time in the course of the nonlinear interactions.
The tensors or 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 and by SHG experiments, even if the two fundamental waves have different directions of polarization.
Therefore, these thirdrank tensors can be represented in contracted form as matrices and , where the suffix m runs over the six possible (α, β) Cartesian index pairs according to the classical convention of contraction:The 27 elements of are then reduced to 18 in the contracted tensor notation (see Section 1.1.4.10 ).
For example, (1.7.2.35) can be writtenThe same considerations can be applied to THG. Then the 81 elements of can be reduced to 30 in the contracted tensor notation with the following contraction convention: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.
Centrosymmetry is the most detrimental crystalline symmetry constraint that will fully cancel all oddrank 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 1.7.2.2 to 1.7.2.5 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 secondrank χ^{(1)} tensor. () is usually called the optical frame; it is linked to the crystallographical frame by the standard conventions given in Chapter 1.6 .




References
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