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. 192195
Section 1.7.3.2.2.3. Biaxial crystals^{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 situation of biaxial crystals is more complicated, because the two sheets that must intersect are both elliptical in several cases. For a given interaction, all the phasematching directions generate a complicated cone which joins two directions in the principal planes; the possible loci a, b, c, d are shown on the stereographic projection given in Fig. 1.7.3.5.

Stereographic projection on the optical frame of the possible loci of phasematching directions in the principal planes of a biaxial crystal. 
The basic inequalities of normal dispersion (1.7.3.7) forbid collinear phase matching for all the directions of propagation located between two optic axes at the two frequencies concerned.
Tables 1.7.3.5 and 1.7.3.6 give, respectively, the inequalities that determine collinear phase matching in the principal planes for the three types of threewave SFG and for the seven types of fourwave SFG.


The inequalities in Table 1.7.3.5 show that a phasematching cone which would join the directions a and d is not possible for any type of interaction, because the corresponding inequalities have an opposite sense. It is the same for a hypothetical cone joining b and c.
The existence of typeII or typeIII SFG phase matching imposes the existence of type I, because the inequalities relative to type I are always satisfied whenever type II or type III exists. However, type I can exist even if type II or type III is not allowed. A typeI phasematched SFG in area c forbids phasematching directions in area b for typeII and typeIII SFG. The exclusion is the same between d and a. The consideration of all the possible combinations of the inequalities of Table 1.7.3.5 leads to 84 possible classes of phasematching cones for both positive and negative biaxial crystals (Fève et al., 1993; Fève, 1994). There are 14 classes for second harmonic generation (SHG) which correspond to the degenerated case () (Hobden, 1967).
The coexistence of the different types of fourwave phase matching is limited as for the threewave case: a cone joining a and d or b and c is impossible for typeI SFG. Type I in area d forbids the six other types in a. The same restriction exists between c and b. Types II, III, IV, V^{4}, VI^{4} and VII^{4} cannot exist without type I; other restrictions concern the relations between types II, III, IV and types V^{4}, VI^{4}, VII^{4} (Fève, 1994). The counting of the classes of fourwave phasematching cones obtained from all the possible combinations of the inequalities of Table 1.7.3.6 is complex and it has not yet been done.
For reasons explained later, it can be interesting to consider a noncollinear interaction. In this case, the projection of the vectorial phasematching relation (1.7.3.26) on the wavevector of highest frequency leads towhere is the angle between and , with for a threewave interaction and for a fourwave interaction. The phasematching angles () can be expressed as a function of the different () by the projection of (1.7.3.26) on the three principal axes of the optical frame.
The configurations of polarization allowing noncollinear phase matching are the same as for collinear phase matching. Furthermore, noncollinear phase matching exists only if collinear phase matching is allowed; the converse is not true (Fève, 1994). Note that collinear or noncollinear phasematching conditions are rarely satisfied over the entire transparency range of the crystal.
References
Fève, J. P. (1994). Existence et symétrie des interactions à 3 et 4 photons dans les cristaux anisotropes. Méthodes de mesure des paramètres affectant les couplages à 3 ondes: étude de KTP et isotypes. PhD Dissertation, Université de Nancy I, France.Fève, J. P., Boulanger, B. & Marnier, G. (1993). Calculation and classification of the direction loci for collinear types I, II and III phasematching of threewave non linear optical parametric interactions in uniaxial and biaxial acentric crystals. Optics Comm. 99, 284–302.
Hobden, M. V. (1967). Phasematched second harmonic generation in biaxial crystals. J. Appl. Phys. 38, 4365–4372.