Average light velocities in periodic media

Abstract

Electromagnetic Bloch modes are used to describe the field distribution of light in periodic media that cannot be adequately approximated by effective macroscopic media. These modes explicitly take into account the spatial modulation of the medium and therefore contain the full physical information at any specific location in the medium. For instance, the propagation velocity of light can be determined locally, and it is not an invariant of space, as it is often implicitly assumed when definitions such as that of the group velocity vgr=dω/dk are used (where ω is the angular frequency and k is the Bloch index of a monochromatic mode). Spatially invariant light velocities can only be expected if the medium is assumed to show an effective behavior similar to a homogeneous material (where a plane-wave ansatz would be more appropriate). This inevitably leads to the question: what exactly is dω/dk of a Bloch mode, if it is not the group velocity? The answer is the average group velocity. This is not a trivial observation, and it has to be taken into account, for instance, when the enhancement of nonlinear effects induced by slow light is estimated. The example of a Kerr nonlinearity is studied, and we show formally that using the average group velocity can lead to an underestimation of the effect. Furthermore, this article critically reviews the concepts of energy and phase velocity. In particular, the different interpretations of phase velocity that exist in the literature are unified using a generic definition of the quantity.