Electromagnetic Energy, Absorption, and Casimir Forces. Inhomogeneous Dielectric Media

A general, exact formula is derived for the expectation value of the electromagnetic energy density of an inhomogeneous absorbing and dispersive dielectric medium in thermal equilibrium, assuming that the medium is well approximated as a continuum. From this formula we obtain the formal expression for the Casimir force density. Unlike most previous approaches to Casimir effects in which absorption is either ignored or admitted implicitly through the required analytic properties of the permittivity, we include dissipation explicitly via the coupling of each dipole oscillator of the medium to a reservoir of harmonic oscillators. We obtain the energy density and the Casimir force density as a consequence of the van der Waals interactions of the oscillators and also from Poyntings theorem.

Electromagnetic Energy, Absorption, and Casimir Forces. I. Uniform Dielectric Media in Thermal Equilibrium

The derivation of Casimir forces between dielectrics can be simplified by ignoring absorption, calculating energy changes due to displacements of the dielectrics, and only then admitting absorption by allowing permittivities to be complex. As a first step towards a better understanding of this situation we consider in this paper the model of a dielectric as a collection of oscillators, each of which is coupled to a reservoir giving rise to damping and Langevin forces on the oscillators and a noise polarization acting as a source of a fluctuating electromagnetic (EM) field in the dielectric. The model leads naturally to expressions for the quantized EM fields that are consistent with those obtained by different approaches, and also results in a fluctuation-dissipation relation between the noise polarization and the imaginary part of the permittivity; comparison with the Rytov fluctuation-dissipation relation employed in the well-known Lifshitz theory for the van der Waals (or Casimir) force shows that the Lifshitz theory is actually a classical stochastic electrodynamical theory. The approximate classical expression for the energy density in a band of frequencies at which absorption in a dielectric is negligible is shown to be exact as a spectral thermal equilibrium expectation value in the quantum-electrodynamical theory. Our main result is the derivation of an expression for the QED energy density of a uniform dispersive, absorbing media in thermal equilibrium. The spectral density of the energy is found to have the same form with or without absorption. We also show how the fluctuation-dissipation theorem ensures a detailed balance of energy exchange between the (absorbing) medium, the reservoir and the EM field in thermal equilibrium.