Group velocity in lossy periodic structured media

In lossless periodic media, the concept of group velocity is fundamental to the study of propagation dynamics. When spatially averaged, the group velocity is numerically equivalent to energy velocity, defined as the ratio of energy flux to energy density of modal fields. However, in lossy media, energy velocity diverges from group velocity. Here, we define a modal field velocity which remains equal to the complex modal group velocity in homogeneous and periodic media. The definition extends to the more general situation of modal fields that exhibit spatial or temporal decay due to lossy elements or Bragg reflection effects. Our simple expression relies on a generalization of the concepts of energy flux and density. Numerical examples, such as a two-dimensional square array of silver rods in vacuum, are provided to confirm the result. Examples demonstrate how the dispersion relation of the periodic structure, the properties of its modes, and their group velocities change markedly in lossy media.