Light transmitted by an anisotropic material
(See also the TLP "Introduction to Anisotropy")
When monochromatic light is incident on the polariser, only the component of light with an electric vector parallel to the axis of the polariser will be allowed to pass through. When the plane polarised light arrives at the specimen it is refracted and, if the material of the specimen is anisotropic, it is split into two separate waves, one vibrating parallel to one permitted vibration direction and the other wave parallel to the other (orthogonal) permitted vibration direction. The velocities of these waves will be determined by the relevant refractive indices, which will be different for the two directions and therefore the waves will become progressively out of phase as they pass through the material. The phase difference can alternatively be expressed in terms of the optical path difference, the distance that progressively separates points on the two waves that coincided initially. Upon emerging, the two waves recombine; however the exact way they recombine will depend on the phase difference between, which depends on the difference between the two refractive indices, the birefringence, Δn, and the distance travelled by the light through the specimen. In general the resultant wave will have a component of its electric vector parallel to the analyser direction.
Plane polarised light has its electric vector vibrating along one direction, the polariser direction. When a material is orientated so that one of the permitted vibration direction lies parallel to the polariser direction, the light travels through the specimen without change in its polarisation state and therefore emerges from the specimen with its electric vector still parallel to the polariser direction and so perpendicular the the analyser direction. This light will not pass through the analyser. These settings are known as extinction positions and produce isoclinic fringes, fringes which occur wherever either principal stress direction coincides with the polariser direction.
The transmitted intensity will also be zero when the optical path difference is an integral number of wavelengths (the phase difference is an integral multiple of 2π). In this case, the beams recombine to give a beam with the same polarisation state as the incident beam, i.e. with the electric vector parallel to the polariser direction, and hence the transmitted intensity is zero.
We have seen that an applied stress can result in a change in the refractive index of a transparent substance. If a general system of stresses is applied in a plane, the optical birefringence, Δn, produced will be proportional to the difference, Δσ between the two principal stresses in the plane. We can define the stress-optical coefficient C, such that
|Δn| = CΔσ
For a sample of uniform thickness, regions in which Δσ [or equivalently (σP - σQ)] is constant show the same interference colour when viewed between crossed polars. Contours of constant principal stress difference are therefore observed as isochromatic lines. In order to determine the directions of the principal stress it is necessary to use isoclinic lines as these dark fringes occur whenever the direction of either principal stress aligns parallel to the analyser or polariser direction.