Anisotropy and Birefringence
Anisotropy: A difference in a physical property (absorbance, refractive index, density, etc) for a given material when measured along different axes. (As opposed to Isotropy, or homogeneity in all directions).

Birefringence: Property of some crystalline materials (e.g., calcite) where the anisotropy is exhibited as a change of refractive index between light rays vibrating in different planes. Another name for this is "Double Refraction". A subset of birefringence, when the anisotropy is wavelength dependent (a measure of dispersion), is called Dichroism. Literally "Two Colors", a dichroic material emits a plane polarized beam of light of a specific wavelength when impinged upon by a beam of white light. An example is the mineral Tourmaline that absorbs all wavelengths but green. See this page for a description.

Birefringence number: The numerical difference between the two refractive indices of a birefringent substance. For example Calcite has a Refractive index(o-ray) = 1.658 and RI (e-ray) = 1.486, so the birefringence number = 0.172. Birefringence numbers are diagnostic tools for identifying unknown substances.

A: When a beam of non-polarized light passes into a crystal of calcite the vibrational plane determines its velocity. Since refraction is a function of velocity, a beam of light will be decomposed into two beams that refract at different angles. One plane of vibration will realize a higher refractive index and refract according to Snell's law. This is called the "Ordinary-Ray". Another beam, vibrating at right angles to the O-Ray, will realize an lower refractive index and bend at a lower angle. This beam is called the "Extraordinary-Ray", or E-Ray.

Thus, two spatially separated rays will exit the birefringent object. The rays will be Plane Polarized, and will be vibrating orthogonal to each other. In addition, the slow ray will lag in phase relative to the fast ray. When they emerge from the birefringent crystal, they may interact via wave interference which results in the production of Interference Colors (depending on the optical path of the material). The colors directly relate to the wave Retardation. The Michel-Lévy Color Chart relates color, retardation, and birefringence.

Below are refractive indices of the O- and E-rays of common birefringent materials. Note that some are "+ve" and some "-ve" in their delta n ("Birefringence number). (Chart From Wikipedia). B: Creating a beam of plane polarized light. The Nicol Prism.

The first device used to produce a beam of polarized light was invented by William Nicol (1770–1851). It consists of a piece of calcite, rhombohedral in shape, and cut at 68° (a, above). The crystal is then bisected, one half inverted and glued back onto the first half. Canada Balsam (n=1.555) was used originally to glue the two halves together.

When non-polarized light impinges on the crystal, the resulting slow, or O-Ray bends at an angle that results in total reflection at the interface between the two halves. The faster, less refracted E-ray passes through the interface and exits the opposite end of the crystal as a beam of plane polarized light. Nicol prisms were common at the later half or the 19th Century.

Modern polarizers are made of iodine crystals embedded in a plastic (PVA) sheet ("H-Sheet"). When the sheet is stretched, the crystals align. This Polaroid Filter produces plane polarized light in a manner similar to the Nicol prism. Light that vibrates perpendicular to the crystal orientation pass through the sheet. Light that vibrates parallel to the crystals interacts with the atomic structure of the crystals and is absorbed.

A Nicol prism (above) was used in this polarized light microscope from 1880.  Click image to see more