Contracted notation for susceptibility tensors

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, p. 182

Section 1.7.2.2.1.3. Contracted notation for susceptibility tensors

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

1.7.2.2.1.3. Contracted notation for susceptibility tensors

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The tensors $[\chi^{(2)}_{\mu\alpha\beta}(-2\omega;\omega,\omega)]$ or $[d^{(2)}_{\mu\alpha\beta}(-2\omega;\omega,\omega)]$ are invariant with respect to (α, β) permutation as a consequence of the intrinsic permutation symmetry. Independently, it is not possible to distinguish the coefficients $[\chi^{(2)}_{ijk}(-2\omega;\omega,\omega)]$ and $[\chi^{(2)}_{ikj}(-2\omega;\omega,\omega)]$ by SHG experiments, even if the two fundamental waves have different directions of polarization.

Therefore, these third-rank tensors can be represented in contracted form as $[3\times 6]$ matrices $[\chi_{\mu m}(-2\omega;\omega,\omega)]$ and $[d_{\mu m}(-2\omega;\omega,\omega)]$ , where the suffix m runs over the six possible (α, β) Cartesian index pairs according to the classical convention of contraction: $[\eqalign{\hbox{for }\mu\hbox{: } &x\rightarrow 1\quad y\rightarrow 2\quad z\rightarrow 3\hfill\cr \hbox{for }m\hbox{: } &xx\rightarrow 1\quad yy\rightarrow 2\quad zz\rightarrow 3\quad yz=zy\rightarrow 4\hfill\cr& xz=zx\rightarrow 5\quad xy=yx\rightarrow 6.\hfill}]$ The 27 elements of $[\chi^{(2)}_{\mu\alpha\beta}(-2\omega;\omega,\omega)]$ are then reduced to 18 in the $[\chi_{\mu m}]$ contracted tensor notation (see Section 1.1.4.10 ).

For example, (1.7.2.35) can be written $[\eqalignno{P_y^{(2)}(2\omega)&=\varepsilon_o\chi_{25}(-2\omega\semi\omega,\omega)[e_x^+(\omega){\bf E}^+(\omega)e_z^-(\omega){\bf E}^-(\omega)&\cr&\quad +e_z^+(\omega){\bf E}^+(\omega)e_x^-(\omega){\bf E}^-(\omega)].&(1.7.2.41)}]$ The same considerations can be applied to THG. Then the 81 elements of $[\chi^{(3)}_{\mu\alpha\beta\gamma}(-3\omega;\omega,\omega,\omega)]$ can be reduced to 30 in the $[\chi_{\mu m}]$ contracted tensor notation with the following contraction convention: $[\eqalign{\hbox{for }\mu\hbox{: } &x\rightarrow 1\quad y\rightarrow 2\quad z\rightarrow 3\hfill\cr \hbox{for }m\hbox{: } &xxx\rightarrow 1\quad yyy\rightarrow 2\quad zzz\rightarrow 3\quad yzz \rightarrow 4\quad yyz\rightarrow 5\hfill\cr& xzz\rightarrow 6\quad xxz\rightarrow 7\quad xyy\rightarrow 8\quad xxy\rightarrow 9\quad xyz\rightarrow 0.\hfill}]$ If Kleinman symmetry holds, the contracted tensor can be further extended beyond SHG and THG to any other processes where all the frequencies are different.

References

International Tables for Crystallography (2006). Vol. D. ch. 1.7, p. 182