In crystal optics, the index ellipsoid (also known as the optical indicatrix or sometimes as the dielectric ellipsoid) is a geometric construction which concisely represents the refractive indices and associated polarizations of light, as functions of the orientation of the wavefront, in a doubly-refractive crystal (provided that the crystal does not exhibit optical rotation). When this ellipsoid is cut through its center by a plane parallel to the wavefront, the resulting intersection (called a central section or diametral section) is an ellipse whose major and minor semiaxes have lengths equal to the two refractive indices for that orientation of the wavefront, and have the directions of the respective polarizations as expressed by the electric displacement vector D. The principal semiaxes of the index ellipsoid are called the principal refractive indices. It follows from the sectioning procedure that each principal semiaxis of the ellipsoid is generally not the refractive index for propagation in the direction of that semiaxis, but rather the refractive index for propagation perpendicular to that semiaxis, with the D vector parallel to that semiaxis (and parallel to the wavefront). Thus the direction of propagation (normal to the wavefront) to which each principal refractive index applies is in the plane perpendicular to the associated principal semiaxis.
Terminology The index ellipsoid is not to be confused with the index surface, whose radius vector (from the origin) in any direction is indeed the refractive index for propagation in that direction; for a birefringent medium, the index surface is the two-sheeted surface whose two radius vectors in any direction have lengths equal to the major and minor semiaxes of the diametral section of the index ellipsoid by a plane normal to that direction. If we let n a , n b , n c {\displaystyle n_{\text{a}},n_{\text{b}},n_{\text{c}}} denote the principal semiaxes of the index ellipsoid, and choose a Cartesian coordinate system in which these semiaxes are respectively in the x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} directions, the equation of the index ellipsoid is
If the index ellipsoid is triaxial (meaning that its principal semiaxes are all unequal), there are two cutting planes for which the diametral section reduces to a circle. For wavefronts parallel to these planes, all polarizations are permitted and have the same refractive index, hence the same wave speed. The directions normal to these two planes—that is, the directions of a single wave speed for all polarizations—are called the binormal axes or optic axes, and the medium is therefore said to be biaxial. Thus, paradoxically, if the index ellipsoid of a medium is triaxial, the medium itself is called biaxial. If two of the principal semiaxes of the index ellipsoid are equal (in which case their common length is called the ordinary index, and the third length the extraordinary index), the ellipsoid reduces to a spheroid (ellipsoid of revolution), and the two optic axes merge, so that the medium is said to be uniaxial. As the index ellipsoid reduces to a spheroid, the two-sheeted index surface constructed therefrom reduces to a sphere and a spheroid touching at opposite ends of their common axis, which is parallel to that of the index ellipsoid; but the principal axes of the spheroidal index ellipsoid and the spheroidal sheet of the index surface are interchanged. In the well-known case of calcite, for example, the index ellipsoid is an oblate spheroid, so that one sheet of the index surface is a sphere touching that oblate spheroid at the equator, while the other sheet of the index surface is a prolate spheroid touching the sphere at the poles, with an equatorial radius (extraordinary index) equal to the polar radius of the oblate spheroidal index ellipsoid. If all three principal semi-axes of the index ellipsoid are equal, it reduces to a sphere: all diametral sections of the index ellipsoid are circular, whence all polarizations are permitted for all directions of propagation, with the same refractive index for all directions, and the index surface merges with the (spherical) index ellipsoid; in short, the medium is optically isotropic. Cubic crystals exhibit this property as well as amorphous transparent media such as glass and water.
History A surface analogous to the index ellipsoid can be defined for the wave speed (normal to the wavefront) instead of the refractive index. Let n denote the length of the radius vector from the origin to a general point on the index ellipsoid. Then dividing equation (1) by n2 gives
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