Definitions and symmetry properties

Boulanger, B.; Zyss, J.

doi:10.1107/97809553602060000634

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

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International Tables for Crystallography (2006). Vol. D. ch. 1.7, pp. 193-194

Section 1.7.3.2.4.1. Definitions and symmetry properties

B. Boulanger^a ^* and J. Zyss^b

^a Laboratoire de Spectrométrie Physique, Université Joseph Fourier, 140 avenue de la Physique, BP 87, 38 402 Saint-Martin-d'Hères, France, and ^bLaboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France
Correspondence e-mail: benoitb@satie-bourgogne.fr

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, $[\eqalignno{ \chi _{\rm eff}^{(2)}(\omega _a, \omega _b, \omega _c, \theta, \varphi) &= \textstyle\sum\limits_{ijk}{\chi _{ijk}}(\omega _a)F_{ijk}(\omega _a, \omega _b, \omega _c, \theta, \varphi) &\cr&= \chi ^{(2)}(\omega _a)\cdot F^{(2)}(\omega _a, \omega _b, \omega _c, \theta, \varphi), &\cr&&(1.7.3.30)}]$ with $[F^{(2)}(\omega _a, \omega _b, \omega _c, \theta, \varphi) = {\bf e}(\omega _a, \theta, \varphi) \otimes {\bf e}(\omega _b, \theta, \varphi) \otimes {\bf e}(\omega _c, \theta, \varphi), \eqno(1.7.3.31)]$ where $[\omega_a, \omega_b, \omega_c]$ correspond to $[\omega_3, \omega_1, \omega_2]$ for SFG ( $[\omega_3 =]$ $[ \omega_1+ \omega_2]$ ); to $[\omega_1, \omega_3, \omega_2]$ for DFG ( $[\omega_1= \omega_3- \omega_2]$ ); and to $[\omega_2, \omega_3, \omega_1]$ for DFG ( $[\omega_2= \omega_3- \omega_1]$ ).
(ii) For four-wave mixing, $[\eqalignno{ \chi _{\rm eff}^{(3)}(\omega _a, \omega _b, \omega _c, \omega _d, \theta, \varphi) &= \textstyle\sum\limits_{ijkl}{\chi _{ijkl}}(\omega _a)F_{ijkl}(\omega _a, \omega _b, \omega _c, \omega _d, \theta, \varphi) & \cr &= \chi ^{(3)}(\omega _a)\cdot F^{(3)}(\omega _a, \omega _b, \omega _c, \omega _d, \theta, \varphi), &\cr&&(1.7.3.32)}]$ with $[\displaylines{F^{(3)}(\omega _a, \omega _b, \omega _c, \omega _d \theta, \varphi) \hfill\cr\quad= {\bf e}(\omega _a, \theta, \varphi) \otimes {\bf e}(\omega _b, \theta, \varphi) \otimes {\bf e}(\omega _c, \theta, \varphi) \otimes {\bf e}(\omega _d, \theta, \varphi),\hfill\cr\hfill(1.7.3.33)}]$ where $[\omega_a, \omega_b, \omega_c, \omega_d]$ correspond to $[\omega_4, \omega_1, \omega_2,\omega_3]$ for SFG ( $[\omega_4 =]$ $[ \omega_1+ \omega_2 + \omega_3]$ ); to $[\omega_1, \omega_4, \omega_2,\omega_3]$ for DFG ( $[\omega_1 =]$ $[ \omega_4 - \omega_2 - \omega_3]$ ); to $[\omega_2, \omega_4, \omega_1, \omega_3]$ for DFG ( $[\omega_2 =]$ $[ \omega_4 - \omega_1-\omega_3]$ ); and to $[\omega_3,]$ $[ \omega_4, ]$ $[\omega_1,]$ $[\omega_2]$ for DFG ( $[\omega_3 =]$ $[ \omega_4 - \omega_1 - \omega_2]$ ).

Each $[{\bf e}(\omega_i,\theta,\varphi)]$ 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): $[\eqalignno{F_{ijk}^{{\bf e}_3{\bf e}_1{\bf e}_2}(\omega_3=\omega_1+\omega_2) &=F_{jik}^{{\bf e}_1{\bf e}_3{\bf e}_2}(\omega_1=\omega_3-\omega_2)&\cr &=F_{kij}^{{\bf e}_2{\bf e}_3{\bf e}_1}(\omega_2=\omega_3-\omega_1)&(1.7.3.34)}]$ and $[\eqalignno{&F_{ijkl}^{{\bf e}_4{\bf e}_1{\bf e}_2{\bf e}_3}(\omega_4=\omega_1+\omega_2+\omega_3)&\cr&=F_{jikl}^{{\bf e}_1{\bf e}_4{\bf e}_2{\bf e}_3}(\omega_1=\omega_4-\omega_2-\omega_3)&\cr&=F_{kijl}^{{\bf e}_2{\bf e}_4{\bf e}_1{\bf e}_3}(\omega_2=\omega_4-\omega_1-\omega_3)&\cr&=F_{lijk}^{{\bf e}_3{\bf e}_4{\bf e}_1{\bf e}_2}(\omega_3=\omega_4-\omega_1-\omega_2),&\cr&&(1.7.3.35)}]$ where 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 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 $[F_{xxij} + F_{yyij}]$ $[(+ \,\,F_{zzij} = 0)=]$ $[ F_{xixj} + F_{yiyj}]$ $[ (+\,\, F_{zizj} = 0)=]$ $[F_{xijx} + F_{yijy}]$ $[ (+ \,\,F_{zijz} = 0)=]$ , with i and j equal to x or y; the combination of these three relations leads to $[F_{xxxx}=]$ $[-F_{yyxx}=]$ $[ -F_{yxyx}=]$ $[-F_{yxxy}]$ , $[F_{yyyy}=]$ $[-F_{xxyy}=]$ $[-F_{xyxy}=]$ $[-F_{xyyx}]$ and $[F_{yxyy}=]$ $[F_{yyxy}=]$ $[F_{yyyx}=]$ $[ -F_{xyxx}=]$ $[F_{xxyx} =]$ $[-F_{xxxy}]$ . 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, $[F_{ijk}^{-++}(2\omega=]$ $[\omega+\omega)=]$ $[F_{ikj}^{-++}(2\omega=]$ $[\omega+\omega)]$ , $[F_{ijkl}^{-++-}(\omega_4=]$ $[\omega+\omega+\omega_3)=]$ $[F_{ikjl}^{-++-}(\omega_4=]$ $[\omega+\omega+\omega_3)]$ , $[F_{ijkl}^{---+}(\omega_4=]$ $[\omega+\omega+\omega_3)=]$ $[F_{ikjl}^{---+}(\omega_4=]$ $[\omega+\omega+\omega_3)]$ and so on.

References

Boulanger, B. (1989). Synthèse en flux et étude des propriétés optiques cristallines linéaires et non linéaires par la méthode de la sphère de KTiOPO₄ et des nouveaux composés isotypes et solutions solides de formule générale (K,Rb,Cs)TiO(P,As)O₄. PhD Dissertation, Université de Nancy I, France.Google Scholar

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.Google Scholar

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.Google Scholar

Zyss, J. (1993). Molecular engineering implications of rotational invariance in quadratic nonlinear optics: from dipolar to octupolar molecules and materials. J. Chem. Phys. 98(9), 6583–6599.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 1.7, pp. 193-194