Exact and linearized refractive index stress-dependence in anisotropic photoelastic crystals

For the permittivity tensor of photoelastic anisotropic crystals we obtain the exact non-linear dependence on the Cauchy stress tensor. We obtain the same result for its square root whose principal components, the crystal principal refractive index, are the starting point for any photoelastic analysis of transparent crystals. From these exact results then we obtain, in a total general manner, the linearized expressions to within higher-order terms in the stress tensor for both the permittivity tensor and its square root. We finish by showing some relavant examples of both non-linear and linearized relations for optically isotropic, uniaxial and biaxial crystals.