Index ellipsoid

In crystal optics, the index ellipsoid (also known as the optical indicatrix^[1] or sometimes as the dielectric ellipsoid^[2]) 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.^[3] The principal semiaxes of the index ellipsoid are called the principal refractive indices.^[4]

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.