of Fresnel's ellipsoid, and which MacCullagh later called the "ellipsoid of indices". In 1891, Lazarus Fletcher called this ellipsoid the optical indicatrix. Electromagnetic interpretation Deriving the index ellipsoid and its generating property from electromagnetic theory is non-trivial. Given the index ellipsoid, however, we can easily relate its parameters to the electromagnetic properties of the medium. The speed of light in a vacuum is where and are respectively the magnetic permeability and the electric permittivity of the vacuum. For a transparent material medium, we can still reasonably assume that the magnetic permeability is (especially at optical frequencies), but must be replaced by where is the relative permittivity (also called the dielectric constant), so that the wave speed becomes Dividing by , we obtain the refractive index: This derivation treats as a scalar, which is valid in an isotropic medium. In an anisotropic medium, the result holds only for those combinations of propagation direction and polarization which avoid the anisotropy—that is, for those cases in which the electric displacement vector is parallel to the electric field vector , as in an isotropic medium. In view of the symmetry of the index ellipsoid, these must be the cases in which is in the direction of one of the axes. So, denoting the relative permittivities in the , , and directions by (the so-called principal dielectric constants), and recalling that denote the refractive indices for these directions of , we must have indicating that the semiaxes of the index ellipsoid are the square roots of the principal dielectric constants. Substituting these expressions into (), we obtain the equation of the index ellipsoid in the alternative form which explains why it is also called the dielectric ellipsoid. See also Birefringence Complex refractive index Crystal optics D-DIA Mathematical descriptions of opacity. Notes References Bibliography M. Born and E. Wolf, 2002, Principles of Optics, 7th Ed., Cambridge University Press, 1999 (reprinted with corrections, 2002), . A. Fresnel (ed. H. de Senarmont, E. Verdet, and L. Fresnel), 1868, Oeuvres complètes d'Augustin Fresnel, Paris: Imprimerie Impériale (3 vols., 1866–70), vol. 2 (1868). F.A. Jenkins and H.E. White, 1976, Fundamentals of Optics, 4th Ed., New York: McGraw-Hill, . L.D. Landau and E.M. Lifshitz (tr. J.B. Sykes & J.S. Bell), 1960, Electrodynamics of Continuous Media (vol. 8 of Course of Theoretical Physics), London: Pergamon Press. A. Yariv and P. Yeh, 1984, Optical Waves in Crystals: Propagation and control of laser radiation,