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. 197-199

Section 1.7.3.2.4.2. Uniaxial class

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

1.7.3.2.4.2. Uniaxial class

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The field-tensor components are calculated from (1.7.3.11)[link] and (1.7.3.12)[link]. The phase-matching case is the only one considered here: according to Tables 1.7.3.1[link] and 1.7.3.2[link], 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[link] and 1.7.3.8[link] 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 [\omega_1\ne \omega_2] (Boulanger & Marnier, 1991[link])

[Scheme scheme1]

InteractionsThree-rank [F_{ijk}(\theta,\varphi)] field tensors
Type eoo [Scheme scheme2]
SFG (ω3) type I < 0
DFG (ω1) type I > 0
DFG (ω2) type I > 0
Type oee [Scheme scheme3]
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 [\omega_1\ne\omega_2\ne\omega_3] (Boulanger et al., 1993[link])

[Scheme scheme4]

InteractionsFour-rank [F_{ijkl}(\theta,\varphi)] field tensors
Type oeee [Scheme scheme5]
SFG(ω4) type I > 0
DFG (ω1) type I < 0
DFG (ω2) type I < 0
DFG (ω3) type I < 0
Type eooo [Scheme scheme6]
SFG (ω4) type I < 0
DFG (ω1) type I > 0
DFG (ω2) type I > 0
DFG (ω3) type I > 0
Type ooee [Scheme scheme7]
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[link] 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

[(i,j,k) = x, y \hbox{ or } z].

Configurations of polarizationNil field-tensor components
(xy) plane(xz) plane(yz) plane
[e o o] [F_{xjk}= 0; F_{yjk}= 0] [F_{ixk}= F_{ijx}= 0 ] [F_{iyk}= F_{ijy} = 0 ]
    [F_{yjk}= 0] [F_{xjk}= 0]
[o e e] [F_{ixk}= F_{ijx}= 0] [F_{iyk}= F_{ijy}= 0] [F_{ixk}= F_{ijx}= 0]
  [F_{iyk}= F_{ijy}= 0] [F_{xik}= 0] [F_{yjk}= 0]
[e o o o] [F_{xjkl}= 0; F_{yjkl}= 0] [F_{ixkl}= F_{ijxl}= F_{ijkx}= 0] [F_{iykl}= F_{ijyl}= F_{ijky}= 0]
    [F_{yjkl}= 0] [F_{xjkl}= 0]
[o e e e] [F_{ixkl}= F_{ijxl}= F_{ijkx}= 0] [F_{iykl}= F_{ijyl}= F_{ijky}= 0] [F_{ixkl}= F_{ijxl}= F_{ijkx}= 0]
  [F_{iykl}= F_{ijyl}= F_{ijky}= 0] [F_{xjkl}= 0 ] [F_{yjkl}= 0]
[o o e e] [F_{ijxl}= F_{ijkx}= 0] [F_{xjkl}= F_{ixkl}= 0] [F_{yjkl}= F_{iykl}= 0]
  [F_{ijyl}= F_{ijky}= 0] [F_{ijyl}= F_{ijky}= 0] [F_{ijxl}= F_{ijkx}= 0]

From Tables 1.7.3.7[link] and 1.7.3.8[link], 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[link] and 1.7.3.2[link]. According to (1.7.3.31)[link] and (1.7.3.33)[link], 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[link], 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[link] to 1.7.2.5[link][link][link], 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[link]; Boulanger et al., 1993[link]). 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[link]).

References

Boulanger, B., Fève, J. P. & Marnier, G. (1993). Field factor formalism for the study of the tensorial symmetry of the four-wave 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 three-wave 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 third-order non-linear optical polarization. J. Appl. Phys. 16, 1667–1674.








































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