  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.
