International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 190-199

## Section 1.7.3.2. Equations of propagation of three-wave and four-wave interactions

B. Boulangera* and J. Zyssb

aInstitut Néel CNRS Université Joseph Fourier, 25 rue des Martyrs, BP 166, 38042 Grenoble Cedex 9, France, and bLaboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France
Correspondence e-mail:  benoit.boulanger@grenoble.cnrs.fr

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#### 1.7.3.2.1. Coupled electric fields amplitudes equations

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The nonlinear crystals considered here are homogeneous, lossless, non-conducting, without optical activity, non-magnetic and are optically anisotropic. The nonlinear regime allows interactions between γ waves with different circular frequencies . The Fourier component of the polarization vector at ωi 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 ωi is (Bloembergen, 1965)The γ propagation equations are coupled by :

 (1) for a three-wave interaction, γ = 3, (2) for a four-wave interaction

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 double-refraction 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 ωi are written (), which can be relative to the mode (+) or (−). The systems are then linked by the double-refraction 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 Ei) with Z. In order to establish the equations of evolution of the wave amplitudes, we assume that their variations are small over one wavelength λi, 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 three-wave 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 four-wave 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 fourth-rank 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 three-wave mixing, where is the Z photon flow. Identically with (1.7.3.24), we have for a four-wave 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 Wi), given in (1.7.2.39), which is done on the generated wave Ei, Z) by the nonlinear polarization PNLi, Z), alternates in sign for each phase shift of π during the Z-propagation, 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).

#### 1.7.3.2.2. Phase matching

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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(ωi, Z) also exists: the work of PNLi, Z) on Ei, 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 photon-momentum conservation i.e.with for a three-photon interaction and for a four-photon interaction.

According to (1.7.3.4), the phase-matching 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 phase-matching 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 non-dispersive 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 phase-matching 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 phase-matching relations, that is to say 23 or 24 possible combinations of refractive indices for a three-wave or a four-wave process, respectively.

For a three-wave process, only three combinations among the 23 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 three-wave interaction is allowed for three configurations of polarization given in Table 1.7.3.1.

 Table 1.7.3.1| top | pdf | Correspondence between the phase-matching relations, the configurations of polarization and the types according to the sum- and difference-frequency generation processes SFG (), DFG () and DFG ()
 are the unit electric field vectors relative to the refractive indices in the phase-matching direction (Boulanger & Marnier, 1991).
Phase-matching relationsConfigurations of polarizationTypes of interaction
ω3ω1ω2SFG (ω3)DFG (ω1)DFG (ω2)
e e+ e+ I II III
e e e+ II III I
e e+ e III I II

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 phase-matching relation corresponds to one sum-frequency generation SFG () and two difference-frequency generation processes, DFG () and DFG (). Types II and III are equivalent for SHG because .

For a four-wave 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.

 Table 1.7.3.2| top | pdf | Correspondence between the phase-matching relations, the configurations of polarization and the types according to SFG (), DFG (), DFG () and DFG () (Boulanger et al., 1993)
Phase-matching relationsConfigurations of polarizationTypes of interaction
ω4ω1ω2ω3SFG (ω4)DFG (ω1)DFG (ω2)DFG (ω3)
e e+ e+ e+ I II III IV
e e e e+ II III IV I
e e e+ e III IV I II
e e+ e e IV I II IV
e e e+ e+ V4 V1 V2 V3
e e+ e e+ VI4 VI1 VI2 VI3
e e+ e+ e VII4 VII1 VII2 VII3

The convention of designation of the types is the same as for three-wave 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 phase-matching relations where two waves have the same polarization state, different from the two others. In this case, it is convenient to refer to each phase-matching relation by the same roman numeral, but with a different index: Vi, VIi and VIIi, 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 V4, VI4 and VII4.

The index surface allows the geometrical determination of the phase-matching 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 phase-matching considerations, normal dispersion and energy conservation impose with .

#### 1.7.3.2.2.1. Cubic crystals

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There is no possibility of collinear phase matching in a dispersive cubic crystal because of the absence of birefringence. In a hypothetical non-dispersive anaxial crystal, the 23 three-wave and 24 four-wave phase-matching configurations would be allowed in any direction of propagation.

#### 1.7.3.2.2.2. Uniaxial crystals

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The configurations of polarization in terms of ordinary and extraordinary waves depend on the optic sign of the phase-matching 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 phase-matching 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 β-BaB2O4 (BBO) and KH2PO4 (KDP).

 Figure 1.7.3.4 | top | pdf |Index surface sections in a plane containing the optic axis z of a negative uniaxial crystal allowing collinear type-I phase matching for SFG (), , or for SFG (), . is the corresponding phase-matching direction.

From Fig. 1.7.3.4, it clearly appears that the intersection of the sheets is possible only if with for a three-wave process and for a four-wave 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 three-wave interactions and Table 1.7.3.4 the 19 situations for the four-wave processes.

 Table 1.7.3.3| top | pdf | Classes of refractive-index inequalities for collinear phase matching of three-wave interactions in positive and negative uniaxial crystals
 Types I, II and III refer to SFG; the types of the corresponding DFG are given in Table 1.7.3.1 (Fève et al., 1993).
Positive sign ()Negative sign ()Types of SFG
I, II, III
I, II
I, III
I
None
 Table 1.7.3.4| top | pdf | Classes of refractive-index inequalities for collinear phase matching of four-wave interactions in positive () and negative () uniaxial crystals with
 If this inequality is not verified, no phase matching is allowed. The types of phase matching refer to SFG; the types of the corresponding DFG are given in Table 1.7.3.2 (Fève, 1994).
Positive sign ()Negative sign ()Types of SFG
I
I, V4
I, VI4
I, VII4
I, V4, VI4
I, II, V4, VI4
I, V4, VII4
I, III, V4, VII4
I, VI4, VII4
I, IV, VI4, VII4
I, V4, VI4, VII4
I, II, V4, VI4, VII4
I, III, V4, VI4, VII4
I, IV, V4, VI4, VII4
I, II, III, V4, VI4, VII4
I, II, IV, V4, VI4, VII4
I, III, IV, V4, VI4, VII4
All

#### 1.7.3.2.2.3. Biaxial crystals

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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 phase-matching 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.

 Figure 1.7.3.5 | top | pdf |Stereographic projection on the optical frame of the possible loci of phase-matching 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 three-wave SFG and for the seven types of four-wave SFG.

 Table 1.7.3.5| top | pdf | Refractive-index conditions that determine collinear phase-matching loci in the principal planes of positive and negative biaxial crystals for three-wave SFG
 a, b, c, d refer to the areas given in Fig. 1.7.3.5. The types corresponding to the different DFGs are given in Table 1.7.3.1 (Fève et al., 1993).
Types of SFGPhase-matching loci in the principal planesInequalities determining three-wave collinear phase matching in biaxial crystals
Positive biaxial crystalNegative biaxial crystal
Type I a
b
c
d
Type II a
b
c
c*
d
d*
Type III a
b
c
c*
d
d*
Conditions c, d are applied if
Conditions c*, d* are applied if
 Table 1.7.3.6| top | pdf | Refractive-index conditions that determine collinear phase-matching loci in the principal planes of positive and negative biaxial crystals for four-wave SFG
 The types corresponding to the different DFGs are given in Table 1.7.3.2 (Boulanger et al., 1993). (a) SFG type I.
Phase-matching loci in the principal planesInequalities determining four-wave collinear phase matching in biaxial crystals
Positive signNegative sign
a
b
c
d
 (b) SFG type II (), SFG type III (), SFG type IV ().
Phase-matching loci in the principal planesInequalities determining four-wave collinear phase matching in biaxial crystals
Positive signNegative sign
a
b
c
c*
d
d*
SFG type II ; SFG type III ; SFG type IV
Conditions c, d are applied if
Conditions c*, d* are applied if
 (c) SFG type V4 (), SFG type VI4 (), SFG type VII4 ().
Phase-matching loci in the principal planesInequalities determining four-wave collinear phase matching in biaxial crystals
Positive signNegative sign
a
b
c
c**
d
d**
SFG type V4, (); SFG type VI4 () ; SFG type VII4 ()
Conditions c′, d′ are applied if
Conditions c**, d** are applied if

The inequalities in Table 1.7.3.5 show that a phase-matching 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 type-II or type-III 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 type-I phase-matched SFG in area c forbids phase-matching directions in area b for type-II and type-III 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 phase-matching 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 four-wave phase matching is limited as for the three-wave case: a cone joining a and d or b and c is impossible for type-I 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, V4, VI4 and VII4 cannot exist without type I; other restrictions concern the relations between types II, III, IV and types V4, VI4, VII4 (Fève, 1994). The counting of the classes of four-wave phase-matching 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 non-collinear interaction. In this case, the projection of the vectorial phase-matching relation (1.7.3.26) on the wavevector of highest frequency leads towhere is the angle between and , with for a three-wave interaction and for a four-wave interaction. The phase-matching 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 non-collinear phase matching are the same as for collinear phase matching. Furthermore, non-collinear phase matching exists only if collinear phase matching is allowed; the converse is not true (Fève, 1994). Note that collinear or non-collinear phase-matching conditions are rarely satisfied over the entire transparency range of the crystal.

#### 1.7.3.2.3. Quasi phase matching

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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 three-wave SFG, the nonlinear polarization sequence is the following:

 (i) from 0 to Lc, ; (ii) from Lc to 2Lc, , which is equivalent to .

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 ion-exchange techniques, or by applying an electric field, which leads to periodically poled (pp) materials like ppLiNbO3 or ppKTiOPO4 (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: diffusion-bonded 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.

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#### 1.7.3.2.4.1. Definitions and symmetry properties

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The refractive indices and their dispersion in frequency determine the existence and loci of the phase-matching 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:

 (i) For three-wave mixing,with where correspond to for SFG ( ); to for DFG (); and to for DFG (). (ii) For four-wave mixing,withwhere correspond to for SFG ( ); to for DFG ( ); to for DFG ( ); and to for DFG ( ).

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 field-tensor 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 field-tensor 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 ei is the unit electric field vector at ωi.

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 field-tensor 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 walk-off angle which depends on the frequency, and the extraordinary wave, (1.7.3.16), has no walk-off angle: then the field tensor is symmetric in the Cartesian indices relative to the extraordinary waves. The walk-off 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.

#### 1.7.3.2.4.2. Uniaxial class

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The field-tensor components are calculated from (1.7.3.11) and (1.7.3.12). The phase-matching 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 three-wave and four-wave 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 three-wave interactions (eoo), (oee) and of the four-wave (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.

 Table 1.7.3.7| top | pdf | Matrix representations of the (oee) and (eoo) field tensors of the uniaxial class and of the biaxial class in the principal planes xz and yz, with (Boulanger & Marnier, 1991)
InteractionsThree-rank field tensors
Type eoo
SFG (ω3) type I < 0
DFG (ω1) type I > 0
DFG (ω2) type I > 0
Type oee
SFG (ω3) type I > 0
DFG (ω1) type I < 0
DFG (ω2) type I < 0
 Table 1.7.3.8| top | pdf | Matrix representations of the (oeee), (eooo) and (ooee) field tensors of the uniaxial class and of the biaxial class in the principal planes xz and yz, with (Boulanger et al., 1993)
InteractionsFour-rank field tensors
Type oeee
SFG(ω4) type I > 0
DFG (ω1) type I < 0
DFG (ω2) type I < 0
DFG (ω3) type I < 0
Type eooo
SFG (ω4) type I < 0
DFG (ω1) type I > 0
DFG (ω2) type I > 0
DFG (ω3) type I > 0
Type ooee
SFG (ω4) type V4 > 0
DFG (ω1) type V1 > 0
DFG (ω2) type V2 > 0
DFG (ω3) type V3 > 0

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 field-tensor 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.

 Table 1.7.3.9| top | pdf | Field-tensor components specifically nil in the principal planes of uniaxial and biaxial crystals for three-wave and four-wave interactions
 .
Configurations of polarizationNil field-tensor components
(xy) plane(xz) plane(yz) plane

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 field-tensor components have the same magnitude for permutation between the following 3o.e interactions:

 (i) (eooo) SFG (ω4) type I < 0 and the three (oeoo) interactions, DFG (ω1) type II < 0, DFG (ω2) type III < 0, DFG (ω3) type IV < 0; (ii) the three (oooe) interactions, SFG (ω4) type II > 0, DFG (ω1) type III > 0, DFG (ω2) type IV > 0 and (eooo) DFG (ω3) type I > 0; (iii) the two (ooeo) interactions SFG (ω4) type III > 0, DFG (ω1) type IV > 0, (eooo) DFG (ω2) type I > 0, and (oooe) DFG (ω3) type II > 0; (iv) (oeoo) SFG (ω4) type IV > 0, (eooo) DFG (ω1) type I > 0, and the two interactions (ooeo) DFG (ω2) type II > 0, DFG (ω3) type III > 0.

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: D4 and D6 for 2o.e, C4v and C6v for 2e.o, D6, D6h, D3h and C6v for 3o.e and 3e.o. Thus, even if phase-matching 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).

#### 1.7.3.2.4.3. Biaxial class

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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 phase-matching 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 field-tensor 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 three-wave interactions and 81 for the four-wave 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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