International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. D, ch. 1.6, pp. 173176
Section 1.6.7. The linear photoelastic 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 photoelastic (or piezooptic) effect (Narasimhamurty, 1981) is given by , and, like the electrooptic effect, it can be discussed in terms of the change in dielectric impermeability caused by a static (or lowfrequency) field, in this case a stress, applied to the crystal. This can be written in the form The coefficients form a fourthrank tensor known as the linear piezooptic tensor. Typically, the piezooptic coefficients are of the order of 10^{−12} m^{2} N^{−1}. It is, however, more usual to express the effect as an elastooptic effect by making use of the relationship between stress and strain (see Section 1.3.3.2 ), thus where the are the elastic stiffness coefficients. Therefore equation (1.6.7.2) can be rewritten in the form or, in contracted notation, where, for convenience, the superscript 0 has been dropped, the elastic strain being considered as essentially static or of low frequency compared with the natural mechanical resonances of the crystal. The are coefficients that form the linear elastooptic (or strainoptic) tensor (Table 1.6.7.1). Note that these coefficients are dimensionless, and typically of order 10^{−1}, showing that the change to the optical indicatrix is roughly onetenth of the strain.

The elastooptic effect can arise in several ways. The most obvious way is through application of an external stress, applied to the surfaces of the crystal. However, strains, and hence changes to the refractive indices, can arise in a crystal through other ways that are less obvious. Thus, it is a common finding that crystals can be twinned, and thus the boundary between twin domains, which corresponds to a mismatch between the crystal structures either side of the domain boundary, will exhibit a strain. Such a crystal, when viewed between crossed polars under a microscope will produce birefringence colours that will highlight the contrast between the domains. This is known as strain birefringence. Similarly, when a crystal undergoes a phase transition involving a change in crystal system, a socalled ferroelastic transition, there will be a change in strain owing to the difference in unitcell shapes. Hence there will be a corresponding change in the optical indicatrix. Often the phase transition is one going from a hightemperature optically isotropic section to a lowtemperature optically anisotropic section. In this case, the hightemperature section has no internal strain, but the lowtemperature phase acquires a strain, which is often called the spontaneous strain (by analogy with the term spontaneous polarization in ferroelectrics).
As an example of the calculation of the relationship between spontaneous strain and linear birefringence, consider the hightemperature phase transition of the well known perovskite BaTiO_{3}. This substance undergoes a transition at around 403 K on cooling from its hightemperature phase to the roomtemperature phase. The phase is both ferroelectric and ferroelastic. In this tetragonal phase, there is a small distortion of the unit cell along [001] and a contraction along compared with the unit cell of the hightemperature cubic phase, and so the roomtemperature phase can be expected to have a uniaxial optical indicatrix.
The elastooptic tensor for the phase is (Table 1.6.7.1) Consider the lowtemperature tetragonal phase to arise as a small distortion of this cubic phase, with a spontaneous strain given by the lattice parameters of the tetragonal phase:Therefore, the equations (1.6.7.4) for the dielectric impermeability in terms of the spontaneous strain component are given in matrix form as so that By analogy with equations (1.6.6.5) and (1.6.6.6), the induced changes in refractive index are then where is the refractive index of the cubic phase. Thus the birefringence in the tetragonal phase as seen by light travelling along is given by Thus a direct connection is made between the birefringence of the tetragonal phase of BaTiO_{3} and its lattice parameters via the spontaneous strain. As in the case of the linear electrooptic effect, the calculation can be repeated using equation (1.6.3.14) with the susceptibilities and to yield the relationship
The acoustooptic effect (Sapriel, 1976) is really a variant of the elastooptic effect, in that the strain field is created by the passage of a sound wave through the crystal. If this wave has frequency , the resulting polarization in the presence of a light wave of frequency is given by , where . However, since the soundwave frequency is very small compared with that of the light, to all intents and purposes the change in frequency of the light field can be ignored. The effect then of the sound wave is to produce within an acoustooptic crystal a spatially modulated change in refractive index: a beam of light can then be diffracted by this spatial modulation, the resulting optical diffraction pattern thus changing with the changing sound signal. Acoustooptic materials therefore can be used as transducers for converting sound signals into optical signals for transmission down optical fibres in communications systems. Consider, for instance, a sound wave propagating along the [110] direction in gallium arsenide (GaAs), which crystallizes in point group . Suppose that this sound wave is longitudinally polarized. With respect to the cube axes, this corresponds to an oscillatory shear strain , where is a distance along the [110] direction (Fig. 1.6.7.1). Then one can write or in contracted notation From Table 1.6.7.1, it seen that the change in dielectric impermeability tensor is since all other components are zero. This means that the original spherical indicatrix of the cubic crystal has been distorted to form a biaxial indicatrix whose axes oscillate in length according to thus forming an optical grating of spatial periodicity given by the term. In gallium arsenide, at a wavelength of light equal to 1.15 µm, , and . It is convenient to define a figure of merit for acoustooptic materials (Yariv & Yeh, 1983) given by where v is the velocity of the sound wave and d is the density of the solid. For gallium arsenide, d = 5340 kg m^{−3}, and for a sound wave propagating as above v = 5.15 m s^{−1}. At the wavelength λ= 1.15 µm, n = 3.37, and so it is found that M = 104. In practice, figures of merits can range from less than 0.001 up to as high as 4400 in the case of Te, and so the value for gallium arsenide makes it potentially useful as an acoustooptic material for infrared signals.
References
Narasimhamurty, T. S. (1981). Photoelastic and electrooptic properties of crystals. New York: Plenum.Sapriel, J. (1976). Acoustooptics. Chichester: Wiley.
Yariv, A. & Yeh, P. (1983). Optical waves in crystals. New York: Wiley.