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

International Tables for Crystallography (2013). Vol. D, ch. 1.7, p. 199

Section Biaxial 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: 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[link] and[link], and the nil components are listed in Table[link]. 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.[link], 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: [F_{zxx}], [F_{zyy}], [F_{zxy}\ne F_{zyx}] for (eoo); [F_{xzz}], [F_{yzz}] for (oee); [F_{zxxx}], [F_{zyyy}], [F_{zxyy}\ne F_{zyxy}\ne F_{zyyx}], [F_{zxxy}\ne F_{zxyx} \ne F_{zyxx}] for (eooo); [F_{xzzz}], [F_{yzzz}] for (oeee); [F_{xyzz}\ne F_{yxzz}], [F_{xxzz}], [F_{yyzz}] 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 [\chi^{(2)}] coefficients, since a given type of phase matching is considered. In general, the largest coefficients of polar crystals, i.e. [\chi_{zzz}], are implicated at a very low level when phase matching is achieved, because the corresponding field tensor, i.e. [F_{zzz}], is often weak (Boulanger et al., 1997[link]). In contrast, QPM authorizes the coupling between three waves polarized along the z axis, which leads to an effective coefficient which is purely [\chi_{zzz}], i.e. [\chi_{\rm eff}=(2/\pi)\chi_{zzz}], where the numerical factor comes from the periodic character of the rectangular function of modulation (Fejer et al., 1992[link]).


Boulanger, B., Fève, J. P., Marnier, G., Bonnin, C., Villeval, P. & Zondy, J. J. (1997). Absolute measurement of quadratic nonlinearities from phase-matched second-harmonic-generation in a single crystal cut as a sphere. J. Opt. Soc. Am. B, 14, 1380–1386.
Fejer, M. M., Magel, G. A., Jundt, D. H. & Byer, R. L. (1992). Quasi-phase-matched second harmonic generation: tuning and tolerances. IEEE J. Quantum Electron. 28(11), 2631–2653.

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