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. 191195
Section 1.7.3.2.2. Phase matching^{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 transfer of energy between the waves is maximum for , which defines phase matching: the energy flow does not alternate in sign and the generated field grows continuously. Note that a condition relative to the phases Φ(ω_{i}, Z) also exists: the work of P^{NL}(ω_{i}, Z) on E(ω_{i}, Z) is maximum if these two waves are π/2 out of phase, that is to say if , where ; thus in the case of phase matching, the phase relation is (Armstrong et al., 1962). The complete initial phase matching is necessarily achieved when at least one wave among all the interacting waves is not incident but is generated inside the nonlinear crystal: in this case, its initial phase is locked on the good one. Phase matching is usually realized by the matching of the refractive indices using birefringence of anisotropic media as it is studied here. From the point of view of the quantum theory of light, the phase matching of the waves corresponds to the total photonmomentum conservation i.e.with for a threephoton interaction and for a fourphoton interaction.
According to (1.7.3.4), the phasematching condition (1.7.3.26) is expressed as a function of the refractive indices in the direction of propagation considered (); for an interaction where the γ wavevectors are collinear, it is writtenwith(1.7.3.28) is the relation of the energy conservation.
The efficiency of a nonlinear crystal directly depends on the existence of phasematching directions. We shall see by considering in detail the effective coefficient that phase matching is a necessary but insufficient condition for the best expression of the nonlinear optical properties.
In an hypothetical nondispersive medium [], (1.7.3.27) is always verified for each of the eigen refractive indices n^{+} or n^{−}; then any direction of propagation is a phasematching direction. In a dispersive medium, phase matching can be achieved only if the direction of propagation has a birefringence which compensates the dispersion. Except for a propagation along the optic axis, there are two possible values, n^{+} and n^{−} given by (1.7.3.6), for each of the three or four refractive indices involved in the phasematching relations, that is to say 2^{3} or 2^{4 }possible combinations of refractive indices for a threewave or a fourwave process, respectively.
For a threewave process, only three combinations among the 2^{3} are compatible with the dispersion in frequency (1.7.3.7) and with the momentum and energy conservations (1.7.3.27) and (1.7.3.28). Thus the phase matching of a threewave interaction is allowed for three configurations of polarization given in Table 1.7.3.1.

The designation of the type of phase matching, I, II or III, is defined according to the polarization states at the frequencies which are added or subtracted. Type I characterizes interactions for which these two waves are identically polarized; the two corresponding polarizations are different for types II and III. Note that each phasematching relation corresponds to one sumfrequency generation SFG () and two differencefrequency generation processes, DFG () and DFG (). Types II and III are equivalent for SHG because .
For a fourwave process, only seven combinations of refractive indices allow phase matching in the case of normal dispersion; they are given in Table 1.7.3.2 with the corresponding configurations of polarization and types of SFG and DFG.

The convention of designation of the types is the same as for threewave interactions for the situations where one polarization state is different from the three others, leading to the types I, II, III and IV. The criterion corresponding to type I cannot be applied to the three other phasematching relations where two waves have the same polarization state, different from the two others. In this case, it is convenient to refer to each phasematching relation by the same roman numeral, but with a different index: V^{i}, VI^{i} and VII^{i}, with the index corresponding to the index of the frequency generated by the SFG or DFG. For THG (), types II, III and IV are equivalent, and so are types V^{4}, VI^{4} and VII^{4}.
The index surface allows the geometrical determination of the phasematching directions, which depend on the relative ellipticity of the internal (−) and external (+) sheets divided by the corresponding wavelengths: according to Tables 1.7.3.1 and 1.7.3.2 the directions are given by the intersection of the internal sheet of the lowest wavelength with a linear combination of the internal and external sheets at the other frequencies . The existence and loci of these intersections depend on specific inequalities between the principal refractive indices at the different wavelengths. Note that independently of phasematching considerations, normal dispersion and energy conservation impose with .
There is no possibility of collinear phase matching in a dispersive cubic crystal because of the absence of birefringence. In a hypothetical nondispersive anaxial crystal, the 2^{3} threewave and 2^{4} fourwave phasematching configurations would be allowed in any direction of propagation.
The configurations of polarization in terms of ordinary and extraordinary waves depend on the optic sign of the phasematching direction with the convention given in Section 1.7.3.1: Tables 1.7.3.1 and 1.7.3.2 must be read by substituting (+, −) by (e, o) for a positive crystal and by (o, e) for a negative one.
Because of the symmetry of the index surface, all the phasematching directions for a given type describe a cone with the optic axis as a revolution axis. Note that the previous comment on the anaxial class is valid for a propagation along the optic axis ().
Fig. 1.7.3.4 shows the example of negative uniaxial crystals () like βBaB_{2}O_{4 }(BBO) and KH_{2}PO_{4} (KDP).
From Fig. 1.7.3.4, it clearly appears that the intersection of the sheets is possible only if with for a threewave process and for a fourwave one. The same considerations can be made for the positive sign and for all the other types of phase matching. There are different situations of inequalities allowing zero, one or several types: Table 1.7.3.3 gives the five possible situations for the threewave interactions and Table 1.7.3.4 the 19 situations for the fourwave processes.


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
Armstrong, J. A., Bloembergen, N., Ducuing, J. & Pershan, P. (1962). Interactions between light waves in a nonlinear dielectric. Phys. Rev. 127, 1918–1939.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.