EM waves in anisotropic materials

For a monochromatic plane wave in a material v = 1/(εμ)½ = c/n.  The index of refraction is n = c(εμ)½.  For most dielectric materials μ ~ μ0, and n = (ε/ε0)½ .

What determines the permittivity ε for a material?

For a linear, isotropic, homogeneous material we have

P = ε0χeED = ε0E + P = ε0 (1 + χe)E = εE.

So the permittivity ε is determined by how easily a material can be polarized.

horizontal rule

Anisotropic materials:

Assume that in a material the electron moves in a potential that has a local minimum and that near this minimum can be modeled as a anisotropic harmonic oscillator potential.  In a simple model in two dimensions, we may picture the electron in a square box, connected by strong springs to the right and left walls and by weak springs to the top and bottom walls.

 

Let a force of magnitude Fh = F' pointing towards the right produce a displacement of magnitude d and a dipole moment of magnitude -qed.
 

Let a force of magnitude Fv = F' pointing upwards produce a displacement of magnitude 2d and a dipole moment of magnitude -2qed.

If we choose the coordinate system shown in the figure below, and apply a force F = Fh + Fv in the x-direction, then this force produces a displacement Δx = 3d/√2 in the x-direction and a displacement Δy = d/√2 in the y-direction.  The dipole moment and the polarization therefore have a x- and a y-component.   If the force F is due to an electric field E, then P and E are not parallel vectors.

 

The electromagnetic wave produced by the oscillation dipoles can be viewed as a superposition of two waves, polarized along the two different symmetry axes.  These two component propagate with different speed c/n, because the index of refraction is different for the different components.  The angle of refraction depends on n, so the angle of refraction will be different for the different components.