Hamiltonian treatment of the electromagnetic field in dispersive and absorptive structured media

Abstract

We introduce a Hamiltonian formulation of electromagnetic fields in dispersive and absorptive structured media of arbitrary dimensionality; the Kramers-Kronig relations are satisfied by construction. Our method is based on an identification of the photonic component of the polariton modes of the system. Although the medium degrees of freedom are introduced in an oscillator model, only the susceptibility of the medium appears in the derived eigenvalue equation for the polaritons; the theory is applicable to both classical and quantum optics. A discrete polariton spectrum is obtained in the transparent regime below the absorption cutoff frequency, and the normalization condition contains the material dispersion in a simple way. In the absorptive regime, a continuous polariton spectrum is obtained. The expressions for the full electromagnetic field of the system can be written in terms of the modes of a limiting, nondispersive, nonabsorptive system, so the theory is well suited to studying the effect of dispersion or absorption on photonic dispersion relations and mode structure. Available codes for dispersive photonic modes can easily be leveraged to obtain polariton modes in both the transparent and absorptive regimes.