Perturbative Roughness Corrections to Electromagnetic Casimir Energies

Perturbative corrections to the Casimir free energy due to macroscopic roughness of dielectric interfaces are obtained in the framework of an effective low-energy field theory. It describes the interaction of electromagnetic fields with materials whose plasma frequency $\omega_p$ determines the low-energy scale. The na\"ive perturbative expansion of the single-interface scattering matrix in the variance of the profile is sensitive to short wavelength components of the roughness correlation function. We introduce generalized counter terms that subtract and correct these high-momentum contributions to the loop expansion. To leading order the counter terms are determined by the phenomenological plasmon model. The latter is found to be consistent with the low-energy description. The proximity force approximation is recovered in the limit of long correlation length and gives the upper limit for the roughness correction to the Casimir force. The renormalized low-energy theory is insensitive to the high-momentum behavior of the roughness correlation function. Predictions of the improved theory are compared with those of the unrenormalized model and with experiment. The Casimir interaction of interfaces with low levels of roughness is found to be well reproduced by that of flat parallel plates with the measured reflection coefficients at a distance that is slightly less than the mean separation of the rough surfaces.