Electromagnetic Waves in Random Media: A Supersymmetric Approach

A general method is set up, which casts problems of electromagnetic waves in random media into a systematic formalism akin to that of supersymmetric quantum field theory, by analogy with electrons in disordered metals. The characteristic functional of the field is related to that of the medium by means of a diagrammatic expansion; for a linear medium, the vertices are the cumulants of the permittivity, conductivity and permeability. Among the auxiliary fields introduced to account for the field equations, fermionic ones can be eliminated by discarding closed loops in the diagrams. A matrix Green function relates the expectation value of the field to electric and magnetic monopole sources; its general structure and properties are reviewed. The versality of the approach, which allows us to take advantage of perturbative and variational techniques drawn from quantum field theory or statistical mechanics, is illustrated by examples: electromagnetic response of a weakly disordered medium, representation of correlations and of energy dissipation by means of four-leg diagrams. The coupling induced by disorder between long-and short-range effects, between transverse and longitudinal parts of the wave, is accounted for. The convergence of the expansions can be improved by making them self-consistent, and also by characterizing th medium by the statistics of its polarizability rather than that of its permittivity, which partly accounts for screening or depolarization. The resulting expansions are of Padé type. Bruggeman's approximation is recovered in the low-frequency, weak-disorder limit; correction are evaluated. An intricate structure for the electromagnetic response, involving several poles and a transition line, arises in the high-frequency, weak-disorder limit