as N = n – 1, often scaled as either N = (n – 1) or N = (n – 1); the multiplication factors are used because the refractive index for air, n deviates from unity by at most a few parts per ten thousand. Molar refractivity, on the other hand, is a measure of the total polarizability of a mole of a substance and can be calculated from the refractive index as where ρ is the density, and M is the molar mass. Nonscalar, nonlinear, or nonhomogeneous refraction So far, we have assumed that refraction is given by linear equations involving a spatially constant, scalar refractive index. These assumptions can break down in different ways, to be described in the following subsections. Birefringence In some materials, the refractive index depends on the polarization and propagation direction of the light. This is called birefringence or optical anisotropy. In the simplest form, uniaxial birefringence, there is only one special direction in the material. This axis is known as the optical axis of the material. Light with linear polarization perpendicular to this axis will experience an ordinary refractive index no while light polarized in parallel will experience an extraordinary refractive index ne. The birefringence of the material is the difference between these indices of refraction, Δn = ne − no. Light propagating in the direction of the optical axis will not be affected by the birefringence since the refractive index will be no independent of polarization. For other propagation directions the light will split into two linearly polarized beams. For light traveling perpendicularly to the optical axis the beams will have the same direction. This can be used to change the polarization direction of linearly polarized light or to convert between linear, circular, and elliptical polarizations with waveplates. Many crystals are naturally birefringent, but isotropic materials such as plastics and glass can also often be made birefringent by introducing a preferred direction through, e.g., an external force or electric field. This effect is called photoelasticity, and can be used to reveal stresses in structures. The birefringent material is placed between crossed polarizers. A change in birefringence alters the polarization and thereby the fraction of light that is transmitted through the second polarizer. In the more general case of trirefringent materials described by the field of crystal optics, the dielectric constant is a rank-2 tensor (a 3 by 3 matrix). In this case