Flat Bands from Diffraction in Periodic Systems

Abstract

Periodic photonic structures enable precise control over the light-matter interaction through band structure engineering. Certain lattice geometries exhibit dispersionless flat bands, characterized by vanishing group velocity and diverging density of states, which present unique opportunities for applications such as slow light, nonlinear optical processes and controlling photoluminescence. However, thus far, flat bands have not been reported in systems where the lattice sites are radiatively coupled over a long range. Here we show that lattices consisting of superposed equispaced one dimensional chains exhibit flat bands with a purely diffractive origin, with the energies and angles of the flat bands controlled by the geometrical parameters of the lattice and the unit cell. The flat bands extend over all angles, can have linewidths on the order of a few nanometers, and are linearly polarized. We experimentally observe flat bands at predicted energies in lattices of gold nanoparticles at near-infrared frequencies using Fourier spectroscopy. Our results provide a general and efficient design strategy for lattices with flat, polarized dispersions for applications such as flat-band lasing, enhancing light-matter interaction, and controlling the emission or absorption of electromagnetic radiation over a wide spectral range.

Figures (13)

• Figure 1: Schematic figure of the various means of achieving photonic flat bands. Panel (a) depicts an example of the class of flat-band lattices we propose: a rectangular chain lattice, realized here by gold nanocylinders, that diffracts incident white light preferentially into a narrow-linewidth flat band of a specified polarization. The gray plane is the plane of incidence ($yz$ plane) and here the incident light is $s$-polarized (in the $x$ direction). Panel (b) presents an experimentally observed flat band in such a geometry, discussed further in Section \ref{['main:s:rectangular_cross']}. The in-plane momentum $k_y$ corresponds to the angle of light emission/absorption by the lattice. The color scale from blue (minimum) to yellow (maximum) shows the extinction of the array in TM polarization. Panel (c) shows a tight-binding emulating system; its top right part depicts the lattice structure of the Lieb lattice, indicating its bipartite division into $A$ (squares) and $B$ sites (circles). The dashed rectangle shows the unit cell of the lattice. The red and blue colors indicate the amplitudes of the wavefunction at the $B$ sites surrounding a single $A$ site. Hopping to the $A$ site is inhibited by destructive interference. The bottom part shows the dispersions of the three energy bands, including one exactly flat band, of the Hubbard model of the Lieb lattice. Panel (d) depicts counterpropagating guided modes (red arrows) in a metasurface--waveguide structure. The metasurface couples the odd and even modes of the waveguide, and the interference of the counterpropagating modes flattens the dispersion munley2023visible. In some metasurface--waveguide systems, the flattening emerges via the index contrast between the guided mode and the surrounding medium Eyvazi2025amedalor2023high (not depicted here).
• Figure 2: Band structure (a), and the structure factor in the reciprocal lattice (b), of a rectangular lattice with $a_y = 5a_x$. (a) Flat bands form from the narrowing of the first Brillouin zone (1BZ) in the $k_y$ direction and the repetition of the mainly TM-polarized modes. At larger angles, more TE polarization mixes in. The inset shows the lattice structure and the unit cell shaded in gray. (b) The structure factor $S(\mathbf{k})$ has its maxima (black dots) at the sites of the reciprocal lattice and is zero elsewhere. The gray shaded area indicates the 1BZ; also the reciprocal lattice vectors $\mathbf{b}_1$ and $\mathbf{b}_2$ are indicated. The horizontal, gray lines in panel (b) are guides for the eye to highlight the linelike features in the structure factor. The distance between the lines is $|\mathbf{b}_1|$---this determines the energy of the flat band. Note that the axes in panel (b) are not to scale; the 1BZ is five times wider in the $k_x$ direction than in the $k_y$ direction. The minor ticks on the $k_x$ axis in panel (b) correspond to the width of the 1BZ in the $k_y$ direction.
• Figure 3: Calculated band structure [panel (a)] and extinction spectrum [panel (b)] of a rectangular lattice of gold nanoparticles with diameter 110nm and height 50nm with lattice constants $a_x = 580nm$, $a_y = 2930nm$ in an index-symmetric environment with refractive index $n = 1.52$. Sample fabrication (Appendix \ref{['main:s:sample_fabrication']}) and experimental setup and procedures (Appendix \ref{['main:s:experimental_setup']}) are described in the Appendix. The scale bar of panel (a) shows the magnitude of the reciprocal lattice vector (RLV) in the $k_y$ direction. In panel (b), the color scale from blue (minimum) to yellow (maximum) corresponds to extinction averaged over five measurements. The red bar estimates the linewidth of the flat band as 8meV (5nm).
• Figure 4: Panel (a) shows a schematic of a chain lattice consisting of two chains. Lattice sites are indicated by black dots, and the unit cell of the lattice is shaded in gray. The structure factor of the lattice computed from the definition in Eq. \ref{['main:e:def_S']} is shown in panel (b), with darker color indicating a greater value. The blue and orange lines are the lines defined by Eq. \ref{['main:e:flat_band_lines']} for $l = 0$ for the angles $\theta_1$ and $\theta_2$.
• Figure 5: (a) Calculated dispersion and structure factor (inset) of a $200µm \times 200µm$ rectangular chain lattice with $N_x = 13$, $a_x = 580nm$, $\theta_x = 0^\circ$ and $N_y = 7$, $a_y = 1160nm$, $\theta_y = 90^\circ$ [see inset in panel (b)] of gold nanocylinders with diameter 130nm and height 50nm in an index-symmetric background with the refractive index $n = 1.52$. The calculated dispersion only shows modes with the structure factor $S \geq 0.1$ and with TE-polarization fraction $p_\text{TE} \leq 0.4$. The color scale of the inset has been explained in the main text. The axes have been slightly displaced for clarity. The flat band is formed by the modes on the top and bottom horizontal rows of peaks (green circles). These modes correspond to the diffraction orders with $q_1 = \pm N_x$ or equivalently, $k_x = \pm 2 \pi / a_x$. (b) Experimentally measured extinction spectrum of the same lattice. The extinction spectrum is filtered to show only TM polarization. The linewidth of the flat band indicated by the red bar is 6meV (3.7nm), corresponding to a Q-factor of 235. In the color bar in (b), yellow denotes maximum and blue minimum extinction.