Light wave states in quasiperiodic metallic structures

We investigate the light wave states in the octagonal and decagonal quasiperiodic metallic structures by considering their respective approximants at different orders. The mechanisms underlying the light wave behaviors are studied in relation to various structure parameters and configurations. We show that the formation of the first passbands, that delimit the photonic band gaps and determine the plasma gaps, involves only the lowest frequency resonance modes inside the fat tiles, and that light localization occurs due to resonances in high symmetry local centers as well as in the fragments of such centers, formed by the skinny tiles. The structure filling rate affects the localized state frequencies relative to the first passbands, as well as the plasma frequency levels, by modulating the frequency levels of the resonance modes and the widths of the passbands. The results of this study can be generalized to other metallic quasiperiodic and related structures.

Figure 1

The $1/1$ [(a)] and $3/2$ [(b)] approximants of the octagonal QP structure, and the (3/2,5/3) [(c)] and (8/5,13/8) [(d)] approximants of the decagonal QP structure. The lattices nodes are represented by circles and the unit cells delimited by dashed lines. The fat and skinny tiles for each structure family are also outlined (solid lines).

Figure 2

The band diagrams for the 1/1 [(a)] and 3/2 [(b)] octagonal approximants, and the (3/2,5/3) [(c)] and (8/5,13/8) [(d)] decagonal approximants. Photonic states inside the photonic band gaps are labeled from $a$ to $g$, two almost degenerate states inside the second passband for the (8/5,13/8) decagonal approximant, $h$ and $h^{\prime }$, are also indicated in (d).

Figure 3

Electric field distributions at $\Gamma$ point for the lowest [(a) and (c)] and the highest [(b) and (d)] branches of the first passbands for respectively the 1/1 octagonal and the (3/2,5/3) decagonal approximants, as well as those resulted from a Gaussian source with center frequency ${\omega }_b$ (Table ) and a frequency width of 0.2 ($\omega a/2\pi c$) in respectively the 3/2 octagonal [(e)] and the (8/5,13/8) decagonal [(f)] approximants. The fat tiles for each structure family are outlined by solid lines, and the unit cells by dashed lines. The “+/$-$” signs indicate the field polarities.

Figure 4

A square [(a)] and a $2\pi /5$ rhombic [(b)] tile, both formed by four metallic cylinders at their vertices. The electric field distributions corresponding to the $T{M}_{11}$ resonance modes inside these tiles are also shown. The “$+/-$” signs indicate the field polarities.

Figure 5

Electric field patterns for the localized states inside the photonic band gaps ($a$ to $g$), as well as two almost degenerate states in the second passband of the (8/5,13/8) decagonal structure ($h$ and $h^{\prime }$). The patterns are lettered following the band labeling in Fig. 2. The “+/$-$” signs indicate the field polarities.

Figure 6

Normalized electric field distributions for modes $b$ [(a)] and $g$ [(b)] (Fig. 5), in slices passing through the ring center along the $x$ (solid line) and $x^{\prime }$ (at $\pi /8$) (dashed line) axes [(a)] and along the $x$ (solid line) and $y$ (dashed line) axes [(b)], respectively. The vertical dashed-dotted lines represent the cylindrical scatterer axes. The scatterer cross sections along the $x$ axis are delimited by the dotted lines. Various length parameters are also presented.

Figure 7

The removal of the center scatterer [indicated by the arrow in (a)] in the 5-fold local center leads to a defect [(b)] in the (3/2,5/3) decagonal approximant, that sustains a local resonance mode [electric field patterns in (b)], with its frequency level in the plasma gap, labeled as $i$ [(c)]. The “+/$-$” signs indicate the field polarities.

Figure 8

The lower and upper edges of the first passband, ${\omega }_1$ and ${\omega }_2$, the fat-tile resonance mode frequencies ${\omega }_{T{M}_{11}}$, the localized mode frequencies in the maximum symmetry local centers ${\omega }_b$, ${\omega }_{f\left(f^{\prime }\right)}$ and ${\omega }_g$, ${\omega }_{h\left(h^{\prime }\right)}$, together with the lower edges of the second passbands ${\omega }_3$, as functions of the scatterer radius $r$ for respectively the 3/2 octagonal [(a)] and the (8/5,13/8) decagonal [(b)] approximants.

Figure 9

The photonic band gap and the upper edge of the plasma gap, together with localized state $a$, obtained for the 1/1 octagonal approximant for a dispersive metal with ${\omega }_p=10(\omega a/2\pi c)$ (“$+$”), compared to those in Fig. 2a (“$·$”). For clarity, only the branches at the gap edges are plotted.