International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 
International Tables for Crystallography (2013). Vol. D, ch. 1.6, pp. 175176
Section 1.6.6. Linear electrooptic effect^{a}Department of Physics, University of Oxford, Parks Roads, Oxford OX1 3PU, England, and ^{b}Department of Earth Sciences, University of Oxford, Parks Roads, Oxford OX1 3PR, England 
The linear electrooptic effect, given by , 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 electrooptic effect as where the coefficients form the socalled linear electrooptic 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:

Typical values of linear electrooptic coefficients are around 10^{−12} mV^{−1}.
In considering the electrooptic 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 elastooptic 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 electrooptic 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.
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. LiNbO_{3} is the most widely used electrooptic material in industry and so this forms a useful example for calculation purposes. This material crystallizes in point group , for which the electrooptic tensor has the form (for the effect of symmetry, see Section 1.1.4.10 ) (with perpendicular to m) with r_{13} = 9.6, r_{22} = 6.8, r_{33} = 30.9 and r_{51} = 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 is imposed along the axis. One can then write Thus Since the original indicatrix of LiNbO_{3} before application of the field is uniaxial, and so differentiating, the following are obtained:Thus, the induced changes in refractive index are given by 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, , the observed linear birefringence is given by
If, on the other hand, the electric field is applied along , i.e. within the mirror plane, one finds Diagonalization of the matrix containing these terms gives three eigenvalue solutions for the changes in dielectric impermeabilities: On calculating the eigenfunctions, it is found that solution (1) lies along , 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 plane. Thus a biaxial indicatrix is formed with one refractive index fixed along 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).

(a) Symmetry elements of point group 3m. (b) Symmetry elements after field applied along . (c) Effect on circular section of uniaxial indicatrix. 
Relationship between linear electrooptic coefficients and the susceptibility tensor . 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 and light propagating along . As before, the only coefficients that need to be considered with the static field along are and . Equation (1.6.3.14) can then be written as where for simplicity the Voigt notation has been used. The first line of the matrix equation gives 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 . The remaining two equations can be solved by forming the determinantal equation which leads to the results Thus and so and since and , Subtracting these two results, the induced birefringence is found: Comparing with the equation (1.6.6.8) calculated for the linear electrooptic coefficients, one finds the following relationships between the linear electrooptic coefficients and the susceptibilities :