Tight-binding photonic bands in metallophotonic waveguide networks and flat bands in kagome lattices

We propose that we can realize “tight-binding photonic bands” in metallophotonic waveguide networks, where the photonic bound states localized around the crossings of a network form a tight-binding band. The formation of bound states at the crossings is distinct from the conventional bound states at defects or virtual bound states in photonic crystals, but comes from a photonic counterpart of the zero-point states in wave mechanics. Model calculations show that the low-lying photon dispersions are indeed described accurately by the tight-binding model. To exemplify how we can exploit the tight-binding analogy for designing of photonic bands, we propose a “flat photonic band” in the kagome network, in which the photonic flat band is shown to arise with group velocities that can be as small as 1/1000 times the velocity of light in vacuum.

Figure 1

(Color online) Metallophotonic waveguide networks are schematically shown: (a) a three-dimensional cubic lattice, (b) a two-dimensional square lattice, and (c) a two-dimensional kagome lattice. In each case, light is confined in the shaded regions, while the rest of the space is filled with a light-prohibiting material such as a metal. For two-dimensional cases as in (b) and (c), the system is assumed to have a large extension in the $z$ direction.

Figure 2

(Color online) Calculated photonic band structure for the bottom band for a square-lattice network [Fig. 1b] for two values of $a$. The error bars come from restricting the number of plane waves to 7225. Curves represent tight-binding fits of the bands with nearest-neighbor transfers (dashed lines) or with transfers extending to further neighbors (solid lines). Inset depicts low-lying bands for $a=2.25$.

Figure 3

(Color online) The best-fit “hopping integrals” for the photonic tight-binding band plotted against the lattice constant $a$ for the nearest-neighbor transfer $\left(t\right)$ and further-neighbor ones $\left(t_{n1},t_{n2}\right)$ as depicted in the inset. Errors in the fitting are smaller than the size of the symbols.

Figure 4

(Color online) Eigenmodes in the bottom band at $\Gamma$ point for (a) $a=2.0$ and (b) $a=3.0$. The boundaries of the waveguide are shown as dotted lines. Modes are normalized in each unit cell.

Figure 5

(Color online) The lowest three photonic bands calculated numerically for the kagome network shown in Fig. 1c for $a=4.0$. Inset depicts a wider energy range.

Figure 6

(Color online) Absolute value of the eigenmodes in the flat (third) band in a kagome network [Fig. 1c] with $a=4.0$ for several points (a)–(c) in the first Brillouin zone as shown in the inset. (d) is for the sixth band at point (c) in the Brillouin zone. Modes are normalized in each unit cell.