Linear electro-optic effect

Glazer, A. M.; Cox, K. G.

doi:10.1107/97809553602060000633

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. 1.6, pp. 172-173

Section 1.6.6. Linear electro-optic effect

A. M. Glazer^a ^* and K. G. Cox^b ^‡

^a Department of Physics, University of Oxford, Parks Roads, Oxford OX1 3PU, England, and ^bDepartment of Earth Sciences, University of Oxford, Parks Roads, Oxford OX1 3PR, England
Correspondence e-mail: glazer@physics.ox.ac.uk

1.6.6. Linear electro-optic effect

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The linear electro-optic effect, given by $[P_i^\omega = \varepsilon_o \chi_{ijk}E_j^\omega E_k^0]$ , is conventionally expressed in terms of the change in dielectric impermeability caused by imposition of a static electric field on the crystal. Thus one may write the linear electro-optic effect as $[\Delta\eta_{ij} = r_{ijk}E_k^0, \eqno (1.6.6.1)]$ where the coefficients $[ r_{ijk}]$ form the so-called linear electro-optic tensor. These have identical symmetry with the piezoelectric tensor and so obey the same rules (see Table 1.6.6.1). Like the piezoelectric tensor, there is a maximum of 18 independent coefficients (triclinic case) (see Section 1.1.4.10.3 ). However, unlike in piezoelectricity, in using the Voigt contracted notation there are two major differences:

(1) In writing the electro-optic tensor components as $[r_{ij}]$ , the first suffix refers to the column number and the second suffix is the row number.

Table 1.6.6.1 | top | pdf |
Symmetry constraints (see Section 1.1.4.10 ) on the linear electro-optic tensor $[r_{ij}]$ (contracted notation)

Triclinic	Monoclinic		Orthorhombic
Point group 1	Point group 2 ( $[2 \parallel x_{2}]$ )	Point group m ( $[m \perp x_{2}]$ )	Point group 222
$[\,\,\pmatrix{ r_{11} & r_{12} & r_{13}\cr r_{21} & r_{22} & r_{23}\cr r_{31} & r_{32} & r_{33}\cr r_{41} & r_{42} & r_{43}\cr r_{51} & r_{52} & r_{53}\cr r_{61} & r_{62} & r_{63} }]$	$[\pmatrix{ 0 & r_{12} & 0\cr 0 & r_{22} & 0\cr 0 & r_{32} & 0\cr r_{41} & 0 & r_{43}\cr 0 & r_{52} & 0\cr r_{61} & 0 & r_{63} } ]$	$[\pmatrix{ r_{11} & 0 & r_{13}\cr r_{21} & 0 & r_{23}\cr r_{31} & 0 & r_{33}\cr 0 & r_{42} & 0\cr r_{51} & 0 & r_{53}\cr 0 & r_{62} & 0 } ]$	$[\pmatrix{ 0 & 0 & 0\cr 0 & 0 & 0\cr 0 & 0 & 0\cr r_{41} & 0 & 0\cr 0 & r_{52} & 0\cr 0 & 0 & r_{63} } ]$
	Point group 2 ( $[2\parallel x_3]$ )	Point group m ( $[m \perp x_3]$ )	Point group mm2
	$[\pmatrix{ 0 & 0 & r_{13}\cr 0 & 0 & r_{23}\cr 0 & 0 & r_{33}\cr r_{41} & r_{42} & 0\cr r_{51} & r_{52} & 0\cr 0 & 0 & r_{63} }]$	$[\pmatrix{ r_{11} & r_{12} & 0\cr r_{21} & r_{22} & 0\cr r_{31} & r_{32} & 0\cr 0 & 0 & r_{43}\cr 0 & 0 & r_{53}\cr r_{61} & r_{62} & 0 } ]$	$[\pmatrix{ 0 & 0 & r_{13}\cr 0 & 0 & r_{23}\cr 0 & 0 & r_{33}\cr 0 & r_{42} & 0\cr r_{51} & 0 & 0\cr 0 & 0 & 0 }]$

Tetragonal		Trigonal
Point group 4	Point group $[\bar 4]$	Point group 3	Point group 32
$[\,\,\pmatrix{ 0 & 0 & r_{13}\cr 0 & 0 & r_{13}\cr 0 & 0 & r_{33}\cr r_{41} & r_{51} & 0\cr r_{51} & -r_{41} & 0\cr 0 & 0 & 0 }]$	$[\pmatrix{ 0 & 0 & r_{13}\cr 0 & 0 & -r_{13}\cr 0 & 0 & 0\cr r_{41} & -r_{51} & 0\cr r_{51} & r_{41} & 0\cr 0 & 0 & r_{63} }]$	$[\pmatrix{ r_{11} & -r_{22} & r_{13}\cr -r_{11} & r_{22} & r_{13}\cr 0 & 0 & r_{33}\cr r_{41} & r_{51} & 0\cr r_{51} & -r_{41} & 0\cr -r_{22} & -r_{11} & 0 } ]$	$[\pmatrix{ r_{11} & 0 & 0\cr -r_{11} & 0 & 0\cr 0 & 0 & 0\cr r_{41} & 0 & 0\cr 0 & -r_{41} & 0\cr 0 & -r_{11} & 0 }]$
Point group $[\bar 42m]$	Point group 422	Point group 3m1 ( $[m \perp x_1]$ )	Point group 31m ( $[m \perp x_2]$ )
$[\,\,\pmatrix{ 0 & 0 & 0\cr 0 & 0 & 0\cr 0 & 0 & 0\cr r_{41} & & 0\cr & r_{41} & 0\cr 0 & 0 & r_{63} }]$	$[\pmatrix{ 0 & 0 & 0\cr 0 & 0 & 0\cr 0 & 0 & 0\cr r_{41} & 0 & 0\cr 0 & -r_{41} & 0\cr 0 & 0 & 0 }]$	$[\pmatrix{ 0 & -r_{22} & r_{13}\cr 0 & r_{22} & r_{13}\cr 0 & 0 & r_{33}\cr 0 & r_{51} & 0\cr r_{51} & 0 & 0\cr -r_{22} & 0 & 0 }]$	$[\pmatrix{ r_{11} & 0 & r_{13}\cr -r_{11} & 0 & r_{13}\cr 0 & 0 & r_{33}\cr 0 & r_{51} & 0\cr r_{51} & 0 & 0\cr 0 & -r_{11} & 0 } ]$
Point group 4mm
$[\pmatrix{ 0 & 0 & r_{13}\cr 0 & 0 & r_{13}\cr 0 & 0 & r_{33}\cr 0 & r_{51} & 0\cr r_{51} & 0 & 0\cr 0 & 0 & 0 }]$

Hexagonal			Cubic
Point group 6	Point group 6mm	Point group 622	Point groups $[\bar 43m]$ , 23
$[\,\,\pmatrix{ 0 & 0 & r_{13}\cr 0 & 0 & r_{13}\cr 0 & 0 & r_{33}\cr r_{41} & r_{51} & 0\cr r_{51} & -r_{41} & 0\cr 0 & 0 & 0 } ]$	$[\pmatrix{ 0 & 0 & r_{13}\cr 0 & 0 & r_{13}\cr 0 & 0 & r_{33}\cr 0 & r_{51} & 0\cr r_{51} & 0 & 0\cr 0 & 0 & 0 }]$	$[\pmatrix{ 0 & 0 & 0\cr 0 & 0 & 0\cr 0 & 0 & 0\cr r_{41} & 0 & 0\cr 0 & -r_{41} & 0\cr 0 & 0 & 0 }]$	$[\pmatrix{ 0 & 0 & 0\cr 0 & 0 & 0\cr 0 & 0 & 0\cr r_{41} & 0 & 0\cr 0 & r_{41} & 0\cr 0 & 0 & r_{41} }]$
Point group $[\bar 6]$	Point group $[\bar 6m2]$ ( $[m \perp x_1]$ )	Point group $[\bar 62m]$ ( $[m \perp x_2]$ )	Point group 432
$[\,\,\pmatrix{ r_{11} & -r_{22} & 0\cr -r_{11} & r_{22} & 0\cr 0 & 0 & 0\cr 0 & 0 & 0\cr 0 & 0 & 0\cr -r_{22} & -r_{11} & 0 }]$	$[\pmatrix{ 0 & -r_{22} & 0\cr 0 & r_{22} & 0\cr 0 & 0 & 0\cr 0 & 0 & 0\cr 0 & 0 & 0\cr -r_{22} & 0 & 0 }]$	$[\pmatrix{ r_{11} & 0 & 0\cr -r_{11} & 0 & 0\cr 0 & 0 & 0\cr 0 & 0 & 0\cr 0 & 0 & 0\cr 0 & -r_{11} & 0 }]$	$[\pmatrix{ 0 & 0 & 0\cr 0 & 0 & 0\cr 0 & 0 & 0\cr 0 & 0 & 0\cr 0 & 0 & 0\cr 0 & 0 & 0 }]$

(2) There are no factors of 1/2 or 2.

Typical values of linear electro-optic coefficients are around 10⁻¹² mV⁻¹.

1.6.6.1. Primary and secondary effects

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In considering the electro-optic effect, it is necessary to bear in mind that, in addition to the primary effect of changing the refractive index, the applied electric field may also cause a strain in the crystal via the converse piezoelectric effect, and this can then change the refractive index, as a secondary effect, through the elasto-optic effect . Both these effects, which are of comparable magnitude in practice, will occur if the crystal is free. However, if the crystal is mechanically clamped, it is not possible to induce any strain, and in this case therefore only the primary electro-optic effect is seen. In practice, the free and clamped behaviour can be investigated by measuring the linear birefringence when applying electric fields of varying frequencies. When the electric field is static or of low frequency, the effect is measured at constant stress, so that both primary and secondary effects are measured together. For electric fields at frequencies above the natural mechanical resonance of the crystal, the strains are very small, and in this case only the primary effect is measured.

1.6.6.2. Example of LiNbO₃

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In order to understand how tensors can be used in calculating the optical changes induced by an applied electric field, it is instructive to take a particular example and work out the change in refractive index for a given electric field. LiNbO₃ is the most widely used electro-optic material in industry and so this forms a useful example for calculation purposes. This material crystallizes in point group [3m] , for which the electro-optic tensor has the form (for the effect of symmetry, see Section 1.1.4.10 ) (with [x_1] perpendicular to m) $[ \pmatrix{ 0&-r_{22}& r_{13}\cr 0&r_{22}&r_{13}\cr 0&0&r_{33}\cr 0&r_{51}&0\cr r_{51}&0&0\cr -r_{22}&0&0}, \eqno (1.6.6.2)]$ with r₁₃ = 9.6, r₂₂ = 6.8, r₃₃ = 30.9 and r₅₁ = 32.5 × 10⁻¹² mV⁻¹, under the normal measuring conditions where the crystal is unclamped.

Calculation using dielectric impermeability tensor . Suppose, for example, a static electric field [E_3^0] is imposed along the [x_3] axis. One can then write $[\pmatrix{ \Delta\eta_1\cr \Delta\eta_2\cr \Delta\eta_3\cr \Delta\eta_4\cr \Delta\eta_5\cr \Delta\eta_6\cr } = \pmatrix{ 0&-r_{22}& r_{13}\cr 0&r_{22}&r_{13}\cr 0&0&r_{33}\cr 0&r_{51}&0\cr r_{51}&0&0\cr -r_{22}&0&0 }\pmatrix{ 0\cr 0\cr E_3^0} = \pmatrix{ r_{13}E_3^0\cr r_{13}E_3^0\cr r_{33}E_3^0\cr 0\cr 0\cr 0\cr }. \eqno (1.6.6.3)]$ Thus $[\eqalignno{\Delta\eta_1 & = r_{13}E^0 = \Delta\eta_2 &\cr \Delta\eta_3 & = r_{33}E^0 &\cr \Delta\eta_4 & = \Delta\eta_5 = \Delta\eta_6 = 0. & (1.6.6.4)}]$ Since the original indicatrix of LiNbO₃ before application of the field is uniaxial, $[\eqalignno{\eta_1 & = {1\over n_o^2} = \eta_2 &\cr \eta_3 & = {1\over n_e^2}, & (1.6.6.5)}]$ and so differentiating, the following are obtained: $[\eqalignno{\Delta\eta_1 & = \Delta\eta_2 = - {2\over n_o^3}\Delta n_o &\cr \Delta\eta_3 & = - {2\over n_e^3}\Delta n_e. & (1.6.6.6)}]$ Thus, the induced changes in refractive index are given by $[\eqalignno{\Delta n_1 & = \Delta n_2 = - {n_o^3\over 2} r_{13}E_3^0&\cr \Delta n_3 & = - {n_e^3\over 2 }r_{33}E_3^0. &(1.6.6.7)}]$ It can be seen from this that the effect is simply to change the refractive indices by deforming the indicatrix, but maintain the uniaxial symmetry of the crystal. Note that if light is now propagated along, say, [x_1] , the observed linear birefringence is given by $[ (n_e - n_o) - {\textstyle{1\over 2}} (n_e^3r_{33} - n_o^3r_{13}) E_3^0 .\eqno (1.6.6.8)]$

If, on the other hand, the electric field [E_2^0] is applied along [x_2] , i.e. within the mirror plane, one finds $[\eqalignno{\Delta\eta_1 & = - r_{22}E_2^0&\cr \Delta\eta_2 & = + r_{22}E_2^0&\cr \Delta\eta_4 & = + r_{51}E_2^0&\cr \Delta\eta_3 & = \Delta\eta_5 = \Delta\eta_6 = 0.& (1.6.6.9)}]$ Diagonalization of the matrix $[ \pmatrix{ \Delta\eta_1 & 0&0 \cr 0&\Delta\eta_2 & \Delta\eta_4 \cr 0&\Delta\eta_4 &\Delta\eta_3 }\eqno (1.6.6.10)]$ containing these terms gives three eigenvalue solutions for the changes in dielectric impermeabilities: $[\displaylines{\quad(1) \hfill -r_{22}E_2^0 \hfill\cr \quad(2) \hfill {r_{22}+(r_{22}^2+4r_{51}^2)^{1/2}\over 2} E_2^0 \hfill\cr \quad(3) \hfill {r_{22}-(r_{22}^2+4r_{51}^2)^{1/2}\over 2} E_2^0. \hfill \cr\hfill(1.6.6.11)}]$ On calculating the eigenfunctions, it is found that solution (1) lies along [x_1] , thus representing a change in the value of the indicatrix axis in this direction. Solutions (2) and (3) give the other two axes of the indicatrix: these are different in length, but mutually perpendicular, and lie in the [x_2 x_3] plane. Thus a biaxial indicatrix is formed with one refractive index fixed along [x_1] and the other two in the plane perpendicular. The effect of having the electric field imposed within the mirror plane is thus to remove the threefold axis in point group 3m and to form the point subgroup m (Fig. 1.6.6.1).

Figure 1.6.6.1 | top | pdf |

(a) Symmetry elements of point group 3m. (b) Symmetry elements after field applied along [x_2] . (c) Effect on circular section of uniaxial indicatrix.

Relationship between linear electro-optic coefficients $[r_{ijk}]$ and the susceptibility tensor $[\chi_{ijk}^{(2)}]$ . It is instructive to repeat the above calculation using the normal susceptibility tensor and equation (1.6.3.14). Consider, again, a static electric field along [x_3] and light propagating along [x_1] . As before, the only coefficients that need to be considered with the static field along [x_3] are $[\chi_{113} = \chi_{223}]$ and $[\chi_{333}]$ . Equation (1.6.3.14) can then be written as $[\displaylines{ \pmatrix{\varepsilon_1+\varepsilon_o\chi_{13}E_3^0+n^2&0&0\cr 0&\varepsilon_1+\varepsilon_o\chi_{13}E_3^0&0\cr 0&0&\varepsilon_3+\varepsilon_o\chi_{33}E_3^0} \pmatrix{ E_1\cr E_2\cr E_3}\cr\hfill\quad = \pmatrix{ n^2&0&0\cr 0&n^2&0\cr 0&0&n^2 }\pmatrix{ E_1\cr E_2\cr E_3},\hfill (1.6.6.12)}]$ where for simplicity the Voigt notation has been used. The first line of the matrix equation gives $[(\varepsilon_1+\varepsilon_o\chi_{13}E_3^0+n^2) E_1 = n^2E_1. \eqno (1.6.6.13)]$ Since only a transverse electric field is relevant for an optical wave (plasma waves are not considered here), it can be assumed that the longitudinal field [ E_1 = 0] . The remaining two equations can be solved by forming the determinantal equation $[ \left| \matrix{\varepsilon_1+\varepsilon_o\chi_{13}E_3^0-n^2&0\cr 0&\varepsilon_3+\varepsilon_o\chi_{33}E_3^0-n^2 }\right| = 0, \eqno (1.6.6.14)]$ which leads to the results $[ n_1^2 = \varepsilon_1+\varepsilon_o\chi_{13}E_3^0 \;\hbox{ and } \; n_2^2 = \varepsilon_3+\varepsilon_o\chi_{33}E_3^0. \eqno (1.6.6.15)]$ Thus $[ n_1^2 = n_o^2+\varepsilon_o\chi_{13}E_3^0 \;\hbox{ and } n_2^2 = n_e^2+\varepsilon_o\chi_{33}E_3^0, \eqno (1.6.6.16)]$ and so $[(n_1 - n_o)(n_1 + n_o) = \varepsilon_o\chi_{13}E_3^0 \;\hbox{ and }\; (n_2- n_e)(n_2 + n_e) = \varepsilon_o\chi_{33}E_3^0, \eqno(1.6.6.17)]$ and since $[n_1 \simeq n_o]$ and $[n_2 \simeq n_e]$ , $[n_1 - n_o = {\varepsilon_o\chi_{13}E_3^0\over 2n_o} \;\hbox{ and }\; n_2- n_e = {\varepsilon_o\chi_{33}E_3^0\over 2n_e}. \eqno(1.6.6.18)]$ Subtracting these two results, the induced birefringence is found: $[(n_e - n_o) - {\textstyle{1\over 2}}\left({\varepsilon_o\chi_{33}\over n_e} - {\varepsilon_o\chi_{13}\over n_o}\right) E_3^0 .\eqno(1.6.6.19)]$ Comparing with the equation (1.6.6.8) calculated for the linear electro-optic coefficients, $[(n_e - n_o) - {\textstyle{1\over 2}}\left(n_e^3r_{33} - n_o^3r_{13}\right) E_3^0, \eqno (1.6.6.20)]$ one finds the following relationships between the linear electro-optic coefficients and the susceptibilities $[\chi^{(2)}]$ : $[ r_{13} = {\varepsilon_o\chi_{13}\over n_o^4} \;\hbox{ and } \; r_{33} = {\varepsilon_o\chi_{33}\over n_e^4}. \eqno(1.6.6.21)]$