Electromagnetic Waves in Variable Media

Abstract

Two methods are explained to exactly solve Maxwell's equations where permittivity, permeability, and conductivity may vary in space. In the constitutive relations, retardation is regarded. If the material properties depend but on one coordinate, general solutions are derived. If the properties depend on two coordinates, geometrically restricted solutions are obtained. Applications to graded reflectors, especially to dielectric mirrors, to filters, polarizers, and to waveguides, plain and cylindrical, are indicated. New foundations for the design of optical instruments, which are centered around an axis, and for the design of invisibility cloaks, plain and spherical, are proposed. The variability of material properties makes possible effects which cannot happen in constant media, e.g. stopping the flux of electromagnetic energy without loss. As a consequence, spherical devices can be constructed which bind electromagnetic waves. Exact Solutions of Maxwell's EquationsThis work originated from a scrutiny for the foundations of quantum mechanics. Do the equations, which Schrödinger considered as the description of the propagation of light, rest upon Maxwell's equations? The answer is no. The study produced analytic methods for the exact solution of Maxwell's equations even if permittivity, permeability, and conductivity vary in space. These methods should be useful to everyone concerned with electromagnetic fields. Thus readers interested in basic physics might read Section 1.1, whereas technicians may begin with Section 1.2. The Foundations of Quantum MechanicsLet us trace the way Schrödinger walked to find the Schrödinger equation. He thought that the propagation of light, if it is construed as propagation of particles, is best described by the eikonal equationSurfaces of equal eikonal s(r) are perpendicular to the light rays everywhere in the space described by the vector of location r. n(r) is the index of refraction. Schrödinger compared this with the Helmholtz equationwhich he considered as the best description of waves. ω denotes the frequency of that wave and c is the velocity of light. The meaning of ψ is not known. Next Schrödinger remembered that there is an eikonal equation for massive particles, too, the Hamilton-Jacobi equationThe mechanical eikonal S(r) has a similar meaning as in ray optics, but its dimension is different. Hence Schrödinger deduced from a comparison of (1) and (3) a mechanical index of refraction n(r) = c 2 h 2 ω 2m(E −V (r)) .The factor in front of 2m(E − V (r)) is an adjustable constant to get dimensions right. That it is related c 2012 Verlag der Zeitschrift für Naturforschung, Tübingen ·