Polarization by Double Refraction

Solids can be classified on the basis of internal structure. Those in which the atoms are arranged in a specific order are called crystalline; the NaCl structure is just one example of a crystalline solid. Those solids in which the atoms are distributed randomly are called amorphous. When light travels through an amorphous material, such as glass, it travels with a speed that is the same in all directions. That is, glass has a single index of refraction. In certain crystalline materials, however, such as calcite and quartz, the speed of light is not the same in all directions. Such materials are characterized by two indices of refraction. Hence, they are often referred to as doublerefracting or birefringen t materials.

Upon entering a calcite crystal, unpolarized light splits into two plane-polarized rays that travel with different velocities, corresponding to two angles of refraction, as shown in Figure 3.9. The two rays are polarized in two mutually perpendicular directions, as indicated by the dots and arrows. One ray, called the ordinary (O) ray, is characterized by an index of refraction n_o that is the same in all directions. This means that if one could place a point source of light inside the crystal, as in Figure 3.10, the ordinary waves would spread out from the source as spheres.

Figure 3.9. Unpolarized light incident on a calcite crystal splits into an ordinary (O) ray and an extraordinary (E) ray. These two rays are polarized in mutually perpendicular directions. (Drawing not to scale.)

Figure 3.10. A point source S inside a double-refracting crystal produces a spherical wave front corresponding to the ordinary ray and an elliptical wave front corresponding to the extraordinary ray. The two waves propagate with the same velocity along the optic axis.

The second plane-polarized ray, called the extraordinary (E) ray, travels with different speeds in different directions and hence is characterized by an index of refraction n_E that varies with the direction of propagation. Consider again the point source within a birefringent material, as in Figure 3.10. The source sends out an extraordinary wave having wave fronts that are elliptical in cross section. Note from Figure 3.9 that there is one direction, called the optic axis, along which the ordinary and extraordinary rays have the same speed, corresponding to the direction for which n_O = n_E. The difference in speed for the two rays is a maximum in the direction perpendicular to the optic axis. For example, in calcite, n_O = 1.658 at a wavelength of 589.3nm, and n_E varies from 1.658 along the optic axis to 1.486 perpendicular to the optic axis.

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