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. 190199
Section 1.7.3.2. Equations of propagation of threewave and fourwave interactions^{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 nonlinear crystals considered here are homogeneous, lossless, nonconducting, without optical activity, nonmagnetic and are optically anisotropic. The nonlinear regime allows interactions between γ waves with different circular frequencies . The Fourier component of the polarization vector at ω_{i} is , where is the nonlinear polarization corresponding to the orders of the power series greater than 1 defined in Section 1.7.2.
Thus the propagation equation of each interacting wave ω_{i} is (Bloembergen, 1965)The γ propagation equations are coupled by :
The complex conjugates come from the relation .
We consider the plane wave, (1.7.3.3), as a solution of (1.7.3.19), and we assume that all the interacting waves propagate in the same direction Z. Each linearly polarized plane wave corresponds to an eigen mode E^{+} or E^{−} defined above. For the usual case of beams with a finite transversal profile and when Z is along a direction where the doublerefraction angles can be nonzero, i.e. out of the principal axes of the index surface, it is necessary to specify a frame for each interacting wave in order to calculate the corresponding powers as a function of Z: the coordinates linked to the wave at ω_{i} are written (), which can be relative to the mode (+) or (−). The systems are then linked by the doublerefraction angles ρ: according to Fig. 1.7.3.1, we have for two waves (+) with , and for two waves (−) with .
The presence of in equations (1.7.3.19) leads to a variation of the γ amplitudes E(ω_{i}) with Z. In order to establish the equations of evolution of the wave amplitudes, we assume that their variations are small over one wavelength λ_{i}, which is usually true. Thus we can stateThis is called the slowly varying envelope approximation.
Stating (1.7.3.20), the wave equation (1.7.3.19) for a forward propagation of a plane wave leads toWe choose the optical frame () for the calculation of all the scalar products , the electric susceptibility tensors being known in this frame.
For a threewave interaction, (1.7.3.21) leads towith , , and , called the phase mismatch. We take by convention .
If ABDP relations, defined in Section 1.7.2.2.1, are verified, then the three tensorial contractions in equations (1.7.3.22) are equal to the same quantity, which we write , where is called the effective coefficient:The same considerations lead to the same kind of equations for a fourwave interaction:The conventions of notation are the same as previously and the phase mismatch is . The effective coefficient isExpressions (1.7.3.23) for and (1.7.3.25) for can be condensed by introducing adequate third and fourthrank tensors to be contracted, respectively, with and . For example, or , and similar expressions. By substituting (1.7.3.8) in (1.7.3.22), we obtain the derivatives of Manley–Rowe relations (1.7.2.40) for a threewave mixing, where is the Z photon flow. Identically with (1.7.3.24), we have for a fourwave mixing.
In the general case, the nonlinear polarization wave and the generated wave travel at different phase velocities, and , respectively, because of the frequency dispersion of the refractive indices in the crystal. Then the work per unit time W(ω_{i}), given in (1.7.2.39), which is done on the generated wave E(ω_{i}, Z) by the nonlinear polarization P^{NL}(ω_{i}, Z), alternates in sign for each phase shift of π during the Zpropagation, which leads to a reversal of the energy flow (Bloembergen, 1965). The length leading to the phase shift of π is called the coherence length, , where Δk is the phase mismatch given by (1.7.3.22) or (1.7.3.24).
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.
When index matching is not allowed, it is possible to increase the energy of the generated wave continuously during the propagation by introducing a periodic change in the sign of the nonlinear electric susceptibility, which leads to a periodic reset of π between the waves (Armstrong et al., 1962). This method is called quasi phase matching (QPM). The transfer of energy between the nonlinear polarization and the generated electric field never alternates if the reset is made at each coherence length. In this case and for a threewave SFG, the nonlinear polarization sequence is the following:
QPM devices are a recent development and are increasingly being considered for applications (Fejer et al., 1992). The nonlinear medium can be formed by the bonding of thin wafers alternately rotated by π; this has been done for GaAs (Gordon et al., 1993). For ferroelectric crystals, it is possible to form periodic reversing of the spontaneous polarization in the same sample by proton or ionexchange techniques, or by applying an electric field, which leads to periodically poled (pp) materials like ppLiNbO_{3} or ppKTiOPO_{4} (Myers et al., 1995; Karlsson & Laurell, 1997; Rosenman et al., 1998).
Quasi phase matching offers three main advantages when compared with phase matching: it may be used for any configuration of polarization of the interacting waves, which allows us to use the largest coefficient of the tensor, as explained in the following section; QPM can be achieved over the entire transparency range of the crystal, since the periodicity can be adjusted; and, finally, double refraction and its harmful effect on the nonlinear efficiency can be avoided because QPM can be realized in the principal plane of a uniaxial crystal or in the principal axes of biaxial crystals. Nevertheless, there are limitations due to the difficulty in fabricating the corresponding materials: diffusionbonded GaAs has strong reflection losses and periodic patterns of ppKTP or ppLN can only be written over a thickness that does not exceed 3 mm, which limits the input energy.
The refractive indices and their dispersion in frequency determine the existence and loci of the phasematching directions, and so impose the direction of the unit electric field vectors of the interacting waves according to (1.7.3.9). The effective coefficient, given by (1.7.3.23) and (1.7.3.25), depends in part on the linear optical properties via the field tensor, which is the tensor product of the interacting unit electric field vectors (Boulanger, 1989; Boulanger & Marnier, 1991; Boulanger et al., 1993; Zyss, 1993). Indeed, the effective coefficient is the contraction between the field tensor and the electric susceptibility tensor of corresponding order:
Each corresponds to a given eigen electric field vector.
The components of the field tensor are trigonometric functions of the direction of propagation.
Particular relations exist between fieldtensor components of SFG and DFG which are valid for any direction of propagation. Indeed, from (1.7.3.31) and (1.7.3.33), it is obvious that the fieldtensor components remain unchanged by concomitant permutations of the electric field vectors at the different frequencies and the corresponding Cartesian indices (Boulanger & Marnier, 1991; Boulanger et al., 1993):andwhere e_{i} is the unit electric field vector at ω_{i}.
For a given interaction, the symmetry of the field tensor is governed by the vectorial properties of the electric fields, detailed in Section 1.7.3.1. This symmetry is then characteristic of both the optical class and the direction of propagation. These properties lead to four kinds of relations between the fieldtensor components described later (Boulanger & Marnier, 1991; Boulanger et al., 1993). Because of their interest for phase matching, we consider only the uniaxial and biaxial classes.
(a) The number of zero components varies with the direction of propagation according to the existence of nil electric field vector components. The only case where all the components are nonzero concerns any direction of propagation out of the principal planes in biaxial crystals.
(b) The orthogonality relation (1.7.3.10) between any ordinary and extraordinary waves propagating in the same direction leads to specific relations independent of the direction of propagation. For example, the field tensor of an (eooo) configuration of polarization (one extraordinary wave relative to the first Cartesian index and three ordinary waves relative to the three other indices) verifies , with i and j equal to x or y; the combination of these three relations leads to , and . In a biaxial crystal, this kind of relation does not exist out of the principal planes.
(c) The fact that the direction of the ordinary electric field vectors in uniaxial crystals does not depend on the frequency, (1.7.3.11), leads to symmetry in the Cartesian indices relative to the ordinary waves. These relations can be redundant in comparison with certain orthogonality relations and are valid for any direction of propagation in uniaxial crystals. It is also the case for biaxial crystals, but only in the principal planes xz and yz. In the xy plane of biaxial crystals, the ordinary wave, (1.7.3.15), has a walkoff angle which depends on the frequency, and the extraordinary wave, (1.7.3.16), has no walkoff angle: then the field tensor is symmetric in the Cartesian indices relative to the extraordinary waves. The walkoff angles of ordinary and extraordinary waves are nil along the principal axes of the index surface of biaxial and uniaxial crystals and so everywhere in the xy plane of uniaxial crystals. Thus, any field tensor associated with these directions of propagation is symmetric in the Cartesian indices relative to both the ordinary and extraordinary waves.
(d) Equalities between frequencies can create new symmetries: the field tensors of the uniaxial class for any direction of propagation and of the biaxial class in only the principal planes xz and yz become symmetric in the Cartesian indices relative to the extraordinary waves at the same frequency; in the xy plane of a biaxial crystal, this symmetry concerns the indices relative to the ordinary waves. Equalities between frequencies are the only situations for which the field tensors are partly symmetric out of the principal planes of a biaxial crystal: the symmetry concerns the indices relative to the waves (+) with identical frequencies; it is the same for the waves (−): for example, , , and so on.
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).
The symmetry of the biaxial field tensors is the same as for the uniaxial class, though only for a propagation in the principal planes xz and yz; the associated matrix representations are given in Tables 1.7.3.7 and 1.7.3.8, and the nil components are listed in Table 1.7.3.9. Because of the change of optic sign from either side of the optic axis, the field tensors of the interactions for which the phasematching cone joins areas b and a or a and c, given in Fig. 1.7.3.5, change from one area to another: for example, the field tensor (eoee) becomes an (oeoo) and so the solicited components of the electric susceptibility tensor are not the same.
The nonzero fieldtensor components for a propagation in the xy plane of a biaxial crystal are: , , for (eoo); , for (oee); , , , for (eooo); , for (oeee); , , for (ooee). The nonzero components for the other configurations of polarization are obtained by the associated permutations of the Cartesian indices and the corresponding polarizations.
The field tensors are not symmetric for a propagation out of the principal planes in the general case where all the frequencies are different: in this case there are 27 independent components for the threewave interactions and 81 for the fourwave interactions, and so all the electric susceptibility tensor components are solicited.
As phase matching imposes the directions of the electric fields of the interacting waves, it also determines the field tensor and hence the effective coefficient. Thus there is no possibility of choice of the coefficients, since a given type of phase matching is considered. In general, the largest coefficients of polar crystals, i.e. , are implicated at a very low level when phase matching is achieved, because the corresponding field tensor, i.e. , is often weak (Boulanger et al., 1997). In contrast, QPM authorizes the coupling between three waves polarized along the z axis, which leads to an effective coefficient which is purely , i.e. , where the numerical factor comes from the periodic character of the rectangular function of modulation (Fejer et al., 1992).
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