Birefringence
Birefringence (or double refraction) happens when light is passed through a special material and it splits into two different rays of light. Calcite crystals and boron nitride are two good examples of materials that cause birefringence. Just as a prism splits light into a large number of colors, birefringence splits light into two images of the same color. You see this as a "double image".
Birefringence Media
A calcite crystal laid upon a graph paper with blue lines showing the double refraction
Comparison of positive and negative birefringence : In positive birefringence (figure 1), the ordinary ray (p-polarisation in this case w.r.t. magenta-coloured plane of incidence), perpendicular to optic axis A is the fast ray (F) while the extraordinary ray (s-polarisation in this case and parallel to optic axis A) is the slow ray (S). In negative birefringence (figure 2), it is the reverse.
Color pattern of a plastic box with "frozen in" mechanical stress placed between two crossed polarizers
Birefringent rutile observed in different polarizations using a rotating polarizer (or analyzer)
Surface of the allowed k vectors for a fixed frequency for a biaxial crystal (see eq. 7).
Other websites
Media related to Birefringence at Wikimedia Commons