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. 197199
Section 1.7.3.2.4.2. Uniaxial class^{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 fieldtensor components are calculated from (1.7.3.11) and (1.7.3.12). The phasematching case is the only one considered here: according to Tables 1.7.3.1 and 1.7.3.2, the allowed configurations of polarization of threewave and fourwave interactions, respectively, are the 2o.e (two ordinary and one extraordinary waves), the 2e.o and the 3o.e, 3e.o, 2o.2e.
Tables 1.7.3.7 and 1.7.3.8 give, respectively, the matrix representations of the threewave interactions (eoo), (oee) and of the fourwave (oeee), (eooo), (ooee) interactions for any direction of propagation in the general case where all the frequencies are different. In this situation, the number of independent components of the field tensors are: 7 for 2o.e, 12 for 2e.o, 9 for 3o.e, 28 for 3e.o and 16 for 2o.2e. Note that the increase of the number of ordinary waves leads to an enhancement of symmetry of the field tensors.


If there are equalities between frequencies, the field tensors oee, oeee and ooee become totally symmetric in the Cartesian indices relative to the extraordinary waves and the tensors eoo and eooo remain unchanged.
Table 1.7.3.9 gives the fieldtensor components specifically nil in the principal planes of uniaxial and biaxial crystals. The nil components for the other configurations of polarization are obtained by permutation of the Cartesian indices and the corresponding polarizations.

From Tables 1.7.3.7 and 1.7.3.8, it is possible to deduce all the other 2e.o interactions (eeo), (eoe), the 2o.e interactions (ooe), (oeo), the 3o.e interactions (oooe), (oeoo), (ooeo), the 3e.o interactions (eoee), (eeoe), (eeeo) and the 2o.2e interactions (oeoe), (eoeo), (eeoo), (oeeo), (eooe). The corresponding interactions and types are given in Tables 1.7.3.1 and 1.7.3.2. According to (1.7.3.31) and (1.7.3.33), the magnitudes of two permutated components are equal if the permutation of polarizations are associated with the corresponding frequencies. For example, according to Table 1.7.3.2, two permutated fieldtensor components have the same magnitude for permutation between the following 3o.e interactions:
The contraction of the field tensor and the uniaxial dielectric susceptibility tensor of corresponding order, given in Tables 1.7.2.2 to 1.7.2.5, is nil for the following uniaxial crystal classes and configurations of polarization: D_{4} and D_{6} for 2o.e, C_{4v} and C_{6v} for 2e.o, D_{6}, D_{6h}, D_{3h} and C_{6v} for 3o.e and 3e.o. Thus, even if phasematching directions exist, the effective coefficient in these situations is nil, which forbids the interactions considered (Boulanger & Marnier, 1991; Boulanger et al., 1993). The number of forbidden crystal classes is greater under the Kleinman approximation. The forbidden crystal classes have been determined for the particular case of third harmonic generation assuming Kleinman conjecture and without consideration of the field tensor (Midwinter & Warner, 1965).
References
Boulanger, B., Fève, J. P. & Marnier, G. (1993). Field factor formalism for the study of the tensorial symmetry of the fourwave non linear optical parametric interactions in uniaxial and biaxial crystal classes. Phys. Rev. E, 48(6), 4730–4751.Boulanger, B. & Marnier, G. (1991). Field factor calculation for the study of the relationships between all the threewave non linear optical interactions in uniaxial and biaxial crystals. J. Phys. Condens. Matter, 3, 8327–8350.
Midwinter, J. E. & Warner, J. (1965). The effects of phase matching method and of crystal symmetry on the polar dependence of thirdorder nonlinear optical polarization. J. Appl. Phys. 16, 1667–1674.