Group velocity in lossy periodic structured media

Abstract

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.

Speed-of-light limitations in passive linear media

“…B3c the fact mentioned in Sec. IV on achieving the upper bound ||v e || = c that the condition (31) being true for h at a single frequency in a transparency window implies that (31) and (32) (in the distributional sense as defined in Ref. [16, p. 54] or as in Secs.…”

“…And, moreover, we also prove the fact mentioned in Sec. IV on achieving the upper bound ||v e || = c that the condition (31) being true for h at a single frequency in a transparency window implies that (31) and (32) are true for all frequencies.…”

“…For lossy media, Glasgow et al [8] employed similar passivity assumptions in order to obtain positivity of the dissipated energy and hence subluminal energy velocity. All of these previous results, however, were for energy velocity defined pointwise in space, or equivalently for light propagation in a homogeneous medium (or possibly a homogenized approximation to an inhomogeneous "metamaterial"), but a more general setting is that of periodic media in which Bloch waves propagate with a well-defined group and energy velocity [11,[26][27][28][29][30][31][32]36]. Joannopoulos et al [11] proved ||v e || c in this periodic setting, but only under the assumptions of isotropic, dispersionless, linear media with real permittivity and permeability bounded below by one.…”

Suppression of narrow-band transparency in a metasurface induced by a strongly enhanced electric field

“…The electromagnetic energy density increases with decreasing energy velocity [18] and therefore the interaction between the electromagnetic wave and the nonlinear medium is enhanced.…”

Microplasma generation by slow microwave in an electromagnetically induced transparency-like metasurface

“…A strongly enhanced electric field is induced in the gaps of the cut-wire resonators owing to the enhancement of the electromagnetic energy density associated with a large group index. 27,28 Assuming that γ ′ ≪ γ 0 is satisfied, which implies that the radiative coupling between the two resonators is strong, the group index and local electric field enhancement at ω = ω 0 are maximized when ω H − ω L ≈ √ 2γ 0 γ ′ . The geometrical parameters shown in Fig.…”