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.

Figure 1

Unit cell selected to simplify the calculation of $\mathcal{F}$ to a 1D calculation in a structure with 2D periodicity. Both (a) and (b) are appropriate choices for unit cells applying to a mode occupying the $\Gamma$-M segment of a square lattice. Vertex coordinates are shown, where $d$ is the lattice parameter. The vector $\mathit{n̂}$ is normal to the chosen interface of the unit cell, where $\mathcal{F}$ is evaluated. The appropriate distance for (27) is $l$.

Figure 2

Boundary condition between semi-infinite free space and a semi-infinite 2D photonic crystal. A Bloch mode described by $\mathit{k}$ is excited by a plane wave described by ${\mathit{k}}_0$. Boundary conditions ensure that the parallel component of ${\mathit{k}}_{0\parallel }$ is conserved from the plane wave to the Bloch mode, determining ${\mathit{k}}_{\parallel }$.

Figure 3

Field quantities of a 1D photonic crystal composed of two materials $a$ and $b$, with dielectric constants ${\epsilon }_a=4+4i$ and ${\epsilon }_b=1$ and lengths $l_a=0.5$ and $l_b=0.5$, with modes at $\omega /c=0.45$. Shown are the magnitude (blue solid) and phase (green dash dot) of $E$ (a) and $E^{†}$ (b) fields and the real (blue solid) and imaginary parts (green dash dot) of the integrands of $\mathcal{N}$ (c) and $\mathcal{F}$ (d) given by (29) and (28).

Figure 4

Band diagram of a 2D periodic structure consisting of a square array dielectric cylinders in vacuum. The index of the cylinders is $n=3$, and the radius is $a=0.3d$, where $d$ is the lattice parameter.

Figure 5

Comparison for the $\Gamma$-$X$ segment of the group velocity as calculated from Fig. 4 (lines) and the adjoint field velocity calculated from the modal fields (dots). Blue solid and green dash dot lines denote the real and imaginary parts, respectively. Shown are the first and second bands, where $v_g$ is real, and the band gap, where $v_g$ is purely imaginary.

Figure 6

Band structure of a square array of silver rods in vacuum, with $a=0.3d$, in the $\Gamma$-$X$ direction and loss neglected. Around the plasma frequency, at $d/\lambda \approx 0.87$, a series of flat bands is observed.

Figure 7

Same as Fig. 6, but loss is not neglected. Shown are the three least lossy modes.

Figure 8

Comparison of the real (blue solid) and imaginary (green dash dot) parts of group velocity (solid lines) and adjoint field velocity (dots) in the $\Gamma$-$X$ segment for the bands of Fig. 7. Only the least lossy mode is shown.