Surface-plasmon polaritons in a lattice of metal cylinders

Plasmon modes of a two-dimensional lattice of long conducting circular wires are investigated by using an embedding technique to solve Maxwell’s equations rigorously. The frequency-dependent density of states is calculated for various values of the wave vector and the filling fraction. At low filling fractions, collective modes are all found to accumulate at the surface-plasmon frequency ${\omega }_p∕\sqrt{2}$, ${\omega }_p$ being the bulk plasmon frequency. As the filling fraction increases, the interference between the electromagnetic fields generated by localized surface-plasmon polaritons leads to the presence of new resonances, whose frequency strongly depends on the interparticle separation. For touching wires, a number of multipole resonances fill the spectral range between dipole resonances, as occurs in the case of a three-dimensional packing of metal spheres.

Figure 1

Density of states $n_I\left(\omega \right)$ for $E$-polarized electromagnetic waves in a square array of Drude cylinders in vacuum, with $\tilde{r}=0.5$ $(f=0.02)$. Different curves correspond to increasing values of ${\tilde{k}}_x$, from 0.001 to 0.1 (with ${\tilde{k}}_y=0$).

Figure 2

$E$-polarization band structure as a function of ${\tilde{k}}_x$ (with ${\tilde{k}}_y=0$), for cylinders of radius $\tilde{r}=0.5,$ 2.0, and $\pi$ (touching cylinders); the corresponding filling fractions $f$ are 0.02, 0.32, and 0.79, respectively. The solid circles represent the band structure that we have obtained from the peaks in our calculated density of states $n_I\left(\omega \right)$. The solid lines correspond to a homogeneous electron gas with the reduced plasma frequency $\sqrt{f}{\omega }_p$. The dashed line is the light line, $\omega =ck$.

Figure 3

Density of states $n_I\left(\omega \right)$ for $H$-polarized electromagnetic waves in a square array of Drude cylinders in vacuum, with $\tilde{r}=1.0$ $(f=0.08)$, $m_{\max }=4$, ${\tilde{k}}_y=0$, and varying ${\tilde{k}}_x$: 0.001 (thick solid line), 0.05 (thin solid line), and 0.1 (dotted line).

Figure 4

Density of states $n_I\left(\omega \right)$ for $H$-polarized electromagnetic waves in a square array of Drude cylinders in vacuum, with $\tilde{r}=1.0$ $(f=0.08)$, wave vector $\stackrel{\mathrm{̃}}{\mathbf{k}}=(0.001,0)$, and varying $m_{\max }$: 1 (thin solid line), 2 (stippled line), 3 (dotted line), and 4 (thick solid line).

Figure 5

Density of states $n_I\left(\omega \right)$ for $H$-polarized electromagnetic waves in a square array of Drude cylinders in vacuum, with $\tilde{r}=2.62$ $(f=0.55)$, wave vector $\stackrel{\mathrm{̃}}{\mathbf{k}}=(0.001,0)$, and varying $m_{\max }$: 1 (thin solid line), 6 (stippled line), 10 (dotted line), and 12 (thick solid line).

Figure 6

Density of states $n_I\left(\omega \right)$ for $H$-polarized electromagnetic waves in a square array of touching Drude cylinders in vacuum, with $\tilde{r}=\pi$ $(f=0.79)$, wave vector $\stackrel{\mathrm{̃}}{\mathbf{k}}=(0.001,0)$, and varying $m_{\max }$: 1 (thin solid line), 6 (stippled line), 10 (dotted line), and 12 (thick solid line).

Figure 7

$H$-polarization band structure as a function of ${\tilde{k}}_x$ (with ${\tilde{k}}_y=0$, i.e., along the $\overline{\Gamma }\overline{M}$ direction), for cylinders of radius $\tilde{r}=1.0$ $(f=0.08)$. The solid lines and circles represent the band structure (converged in $m_{\max }$) that we have obtained from the peaks in our calculated density of states $n_I\left(\omega \right)$. The dotted lines represent the $\omega \left(k\right)$ curves derived from Eq. (2) with the MG effective dielectric function ${ϵ}_{\mathrm{eff}}\left(\omega \right)$ of Eq. (12).

Figure 8

As in Fig. 7, but now for cylinders of radius $\tilde{r}=2.0$ $(f=0.32)$. This is converged in $m_{\max }$.

Figure 9

As in Fig. 7, but now for touching cylinders of radius $\tilde{r}=\pi$ $(f=0.79)$. $m_{\max }=12$, and although the band edges have converged, the individual bands have not.