How does index of refraction have direction?

I found it in my textbook that there are some quantities like the elastic spring constant and index of refraction that have magnitude as well as direction, but are not called 'vectors'. I wonder how they have directions - I can somewhat make some sense of the spring constant but the index of refraction part sounds ambiguous.

• You need to show some context. A reference or quote from the text, maybe. – nasu May 29 '17 at 13:12
• In some anisotropic crystals, the speed of light in the crystal changes with direction. Since index of refraction is $n = c/v$, so does $n$. The index of refraction is a tensor. See en.wikipedia.org/wiki/Crystal_optics. – mmesser314 May 29 '17 at 13:49

1 Answer

Almost certainly what's being talked about here is the refractive index of an anisotropic material. In this case, there are preferred directions in the material in question and the refractive index, i.e. the reciprocal of the phase velocity of the wave, depends in general both on the wave's direction and polarization.

In an isotropic medium, the electric (and possibly, but very seldom, magnetic) constants are no longer scalars. For example, the electric constant can be a homogeneous linear map - the dielectric tensor described by a symmetric $3\times3$ matrix $\boldsymbol{\epsilon}$ - that maps the electric field to the electric displacement:

$$\vec{D} = \boldsymbol{\epsilon} \vec{E}$$

and the electric field and displacement are not in the same directions.