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. 183184
Section 1.7.2.1.4. Conventions for nonlinear susceptibilities^{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 
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
Shen, Y. R. (1984). The principles of nonlinear optics. New York: Wiley.