Dispersion relations, stored energy and group velocity for anisotropic electromagnetic media

Abstract

Abstract.The Hermitian and skew-Hermitian components of the susceptibility matrix of a general linear electromagnetic medium are represented as Hilbert transforms of each other. These so-called dispersion relations lead to a priori inequalities which must be satisfied by the susceptibility of a passive medium in a frequency interval in which the medium is lossless. One such inequality states that the stored energy density for a given E(«) and H(ai) is always greater than in free space. This is also verified directly from the usual gyrotropic susceptibilities of ferrites and plasmas.The group velocity for an eigenwave or mode of a structure with one, two or three independent translational symmetry vectors is shown to be, in general, an average Poynting vector divided by an average stored energy density. This formula is then combined with the above inequality for the stored energy density to show that the magnitude of the group velocity is less than c, the velocity of light in free space.I. Introduction. A main point of the present paper is to extend some results, known for isotropic media [1], to the case of general anisotropic media. Moreover, some of the discussion, for example that relating to group velocity, is novel even when specialized to the isotropic case.For simplicity as well as for generality the discussion is phrased for a medium where As in the isotropic case, a consequence of causality and time invariance is that the susceptibility, 7(0>) -y0, satisfies an integral equation with (Cauchy) Kernel 1/(jV(x -«)), where the range of integration of x is from -co to + co. One can separate this integral equation into its real and imaginary components. The resulting equations [1], [5], called dispersion relations, state that the real and imaginary components of the susceptibility matrix, in their dependence upon frequency, are Hilbert transforms of each other. However, in the anisotropic case one has the additional possibility of separating the integral equation into its Hermitian and skew-Hermitian components. This leads to the new dispersion relations stated in Sec. III. These are particularly significant with regard