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. 181184
Section 1.7.2.1. Induced polarization and susceptibility^{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.
References
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.