The electric and magnetic polarization states for plane waves in arbitrary linear crystals, in which each of
D
and
B
is coupled to both of
E
and
H
, can be characterized by their typical singularities in direction space: degeneracies, where two refractive index eigenvalues coincide;
C
e
and
C
m
points, where the electric or magnetic field is circularly polarized; and
L
e
and
L
m
lines, where either field is linearly polarized. The well-known 4×4 matrix formalism, expressed in terms of the stereographic projection of directions, enables extensive numerical and visual exploration of the singularities in the general case (which involves 65 crystal parameters), incorporating bianisotropy, natural and Faraday optical activity, and absorption, as well as special cases where one or more effect is absent. For crystals whose anisotropy is weak but which are otherwise general, an unusual perturbation theory leads to a powerful 2×2 formalism capturing all the essential singularity phenomena, including the principal feature of the general case, namely the separation between the electric and magnetic singularities.
Computer generated optical volume elements by additive manufacturing