Coupling of light with elastic waves

Vacher, R.; Courtens, E.

doi:10.1107/97809553602060000641

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. 2.4, p. 330

Section 2.4.3. Coupling of light with elastic waves

R. Vacher^a ^* and E. Courtens^a

^a Laboratoire des Verres, Université Montpellier 2, Case 069, Place Eugène Bataillon, 34095 Montpellier CEDEX, France
Correspondence e-mail: rene.vacher@ldv.univ-montp2.fr

2.4.3. Coupling of light with elastic waves

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2.4.3.1. Direct coupling to displacements

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The change in the relative optical dielectric tensor $[\boldkappa]$ produced by an elastic wave is usually expressed in terms of the strain, using the Pockels piezo-optic tensor p, as $[(\Delta \kappa ^{- 1})_{ij} = p_{ijk\ell }S_{k\ell }. \eqno (2.4.3.1)]$ The elastic wave should, however, be characterized by both strain S and rotation A (Nelson & Lax, 1971; see also Section 1.3.1.3 ): $[A_{[{k\ell }]} = {\textstyle{1 \over 2}}\left({{{\partial u_k }\over {\partial x_\ell }}- {{\partial u_\ell }\over {\partial x_k }}}\right). \eqno (2.4.3.2)]$ The square brackets on the left-hand side are there to emphasize that the component is antisymmetric upon interchange of the indices, $[A_{[{k\ell }]} = - A_{[{\ell k}]}]$ . For birefringent crystals, the rotations induce a change of the local $[\boldkappa]$ in the laboratory frame. In this case, (2.4.3.1) must be replaced by $[(\Delta \kappa ^{- 1})_{ij} = p'_{ijk\ell }{{\partial u_k }\over {\partial x_\ell }}, \eqno (2.4.3.3)]$ where $[{\bf p}']$ is the new piezo-optic tensor given by $[p'_{ijk\ell } = p_{ijk\ell } + p_{ij [{k\ell } ]}. \eqno (2.4.3.4)]$ One finds for the rotational part $[p_{ij [{k\ell } ]} = {\textstyle{1 \over 2}} [{ ({\kappa ^{- 1}} )_{i\ell }\delta _{kj} + ({\kappa ^{- 1}} )_{\ell j}\delta _{ik} - ({\kappa ^{- 1}} )_{ik}\delta _{\ell j} - ({\kappa ^{- 1}} )_{kj}\delta _{i\ell }} ]. \eqno (2.4.3.5)]$ If the principal axes of the dielectric tensor coincide with the crystallographic axes, this gives $[p_{ij [{k\ell } ]} = {\textstyle{1 \over 2}} ({\delta _{i\ell }\delta _{kj} - \delta _{ik}\delta _{\ell j}} ) ({1/n_i^2 - 1/n_j^2 } ). \eqno (2.4.3.6)]$ This is the expression used in this chapter, as monoclinic and triclinic groups are not listed in the tables below.

For the calculation of the Brillouin scattering, it is more convenient to use $[({\Delta \kappa })_{mn} = - \kappa _{mi}\kappa _{nj}p'_{ijk\ell }{{\partial u_k }\over {\partial x_\ell }}, \eqno (2.4.3.7)]$ which is valid for small $[\Delta \kappa]$ .

2.4.3.2. Coupling via the electro-optic effect

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Piezoelectric media also exhibit an electro-optic effect linear in the applied electric field or in the field-induced crystal polarization. This effect is described in terms of the third-rank electro-optic tensor r defined by $[({\Delta \kappa ^{- 1}})_{ij} = r_{ijm}E_m. \eqno (2.4.3.8)]$ Using the same approach as in (2.4.2.10), for long waves $[E{_m}]$ can be expressed in terms of $[S_{k\ell }]$ , and (2.4.3.8) leads to an effective Pockels tensor $[{\bf p}{^e}]$ accounting for both the piezo-optic and the electro-optic effects: $[p_{ijk\ell }^e = p_{ijk\ell } - {{r_{ijm}e_{nk\ell }\hat Q_m \hat Q_n }\over {\varepsilon _{gh}\hat Q_g \hat Q_h }}. \eqno (2.4.3.9)]$ The total change in the inverse dielectric tensor is then $[({\Delta \kappa ^{- 1}})_{ij} = ({p_{ijk\ell }^e + p_{ij[{k\ell }]}}){{\partial u_k }\over {\partial x_\ell }}= p'_{ijk\ell }{{\partial u_k }\over {\partial x_\ell }}. \eqno (2.4.3.10)]$ The same equation (2.4.3.7) applies.

References

Nelson, D. F. & Lax, M. (1971). Theory of photoelastic interaction. Phys. Rev. B, 3, 2778–2794.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 2.4, p. 330