In double refraction, light enters a crystal the optical properties of which differ along two or more of the crystal axes. What is observed depends on the angle of the beam with respect to the entrant face. Double refraction was first observed in 1669 by Erasmus Bartholin in experiments with Iceland spar crystal and elucidated in 1690 by Huygens.

If a beam of light is made to enter an Iceland spar crystal at right angles to a face, it persists in the crystal as a single beam perpendicular to the face and emerges as a single beam through an opposite parallel face. If the exit face is at an angle not perpendicular to the beam, however, the emergent beam is split into two beams at different angles, called the ordinary and extraordinary rays, and they are usually of different intensities. Clearly, any beam that enters an Iceland spar crystal perpendicular to its face and emerges perpendicular to another face is of changed character—although superficially it may not appear to be changed. Dependent on the relative intensities and the phase relationship of its electric components (i.e., their phase shift), the beam is described as either elliptically or circularly polarized. There are other ways of producing partially polarized, plane-polarized, and elliptically (as well as circularly) polarized light, but these examples illustrate the phenomena adequately.

Polarization of an electromagnetic wave can be shown mathematically to relate to the space-time relationship of the electromagnetic-field vector (conventionally taken as the electric vector, a quantity representing the magnitude and direction of the electric field) as the wave travels. If the field vector maintains a fixed direction, the wave is said to be plane-polarized, the plane of polarization being the one that contains the propagation direction and the electric vector. In the case of elliptic polarization, the field vector generates an ellipse in a plane perpendicular to the propagation direction as the wave proceeds. Circular polarization is a special case of elliptic polarization in which the so-described ellipse degenerates into a circle.

An easy way to produce circularly polarized light is by passage of the light perpendicularly through a thin crystal, as, for example, mica. The mica sample is so selected that the path difference for the ordinary and the extraordinary rays is one-quarter the wavelength of the single-wavelength, or monochromatic, light used. Such a crystal is called a quarter-wave plate, and the reality of the circular polarization is shown by the fact that, when the quarter-wave plate is suitably suspended and irradiated, a small torque—that is, twisting force—can be shown to be exerted on it. Thus, the action of the crystal on the light wave is to polarize it; the related action of the light on the crystal is to produce a torque about its axis.

The ratio of the intensity of the reflected light to that of the incident light is called the reflection coefficient. This quantitative measure of reflection depends on the angles of incidence and refraction, or the refractive index, and also on the nature of polarization.

It can be shown that the reflection coefficient at any angle of incidence is greater for polarization perpendicular to the plane of incidence than for polarization in the plane of incidence. As a result, if unpolarized light is incident at a plane surface separating two media, reflected light will be partially polarized perpendicular to the plane of incidence, and refracted light will be partially polarized in the plane of incidence. An exceptional case is the Brewster angle, which is such that the sum of the angles of incidence and refraction is 90°. When that happens, the reflection coefficient for polarization in the plane of incidence equals zero. Thus, at the Brewster angle, the reflected light is wholly polarized perpendicular to the plane of incidence. At an air-glass interface, the Brewster angle is approximately 56°, for which the reflection coefficient for perpendicular polarization is 14 percent. Another extremely important angle for refraction is the critical angle of incidence when light passes from a denser medium to a rarer one. It is that angle for which the angle of refraction is 90° (in this case the angle of refraction is greater than the angle of incidence). For angles of incidence greater than the critical angle there is no refracted ray; the light is totally reflected internally. For a glass-to-air interface the critical angle has a value 41°48′.

Dispersion

The variation of the refractive index with frequency is called dispersion. It is this property of a prism that effects the colour separation, or dispersion, of white light. An equation that connects the refractive index with frequency is called a dispersion relation. For visible light the index of refraction increases slightly with frequency, a phenomenon termed normal dispersion. The degree of refraction depends on the refractive index. The increased bending of violet light over red by a glass prism is therefore the result of normal dispersion. If experiments are done, however, with light having a frequency close to the natural electron frequency, some strange effects appear. When the radiation frequency is slightly greater, for example, the index of refraction becomes less than unity (<1) and decreases with increasing frequency; the latter phenomenon is called anomalous dispersion. A refractive index less than unity refers correctly to the fact that the speed of light in the medium at that frequency is greater than the speed of light in vacuum. The velocity referred to, however, is the phase velocity or the velocity with which the sine-wave peaks are propagated. The propagation velocity of an actual signal or the group velocity is always less than the speed of light in vacuum. Therefore, relativity theory is not violated. An example is shown in Figure 3, in which a light source is initially pointed in the direction A. The source rotates in such a way that the velocity of the light image moves from D to E with a velocity v approximating c. Thus, the phase velocity with which the image moves from A to B is greater than c, but the relativity principle is not violated because the velocity of transmission of matter or energy does not exceed the velocity of light.