Bounds for the group velocity of electromagnetic signals in two phase materials

Abstract

The energy associated with an electromagnetic signal travels in general with the group velocity. We consider the group velocity of a composite and ask the question whether the group velocity in the composite can be higher than in either of the constituent phases. Here we show that it can, and by a large factor. The key point is that the group velocity depends on both the refractive index and the dispersion. By combining one phase with high refractive index and low dispersion with another phase with low refractive index and high dispersion, the composite can be made to exhibit comparatively low refractive index and low dispersion and hence a large group velocity. In particular this can be realized when the dispersion relations of both phases are described by a Lorentzian model and one phase is close to resonance. The ‘speed-up’ is largest in a laminate microgeometry, but can be made large also in isotropic microstructures, described by a Maxwell–Garnett model. These geometries attain the bounds on the speed-up that we derive. The group velocity can also be smaller in the composite than in the phases and we derive bounds for the possible ‘slow-down’. These bounds are attained by similar geometries as those that realize the optimal bounds for the speed-up.