Far-field radiation of bulk, edge and corner eigenmodes from a finite 2D Su-Schrieffer-Heeger plasmonic lattice

TL;DR

This work develops an eigenmode analysis based on a coupled electric-dipole model to dissect the far-field radiation of bulk, edge, and corner modes in a finite 2D SSH plasmonic lattice with out-of-plane dipolar resonances. By constructing effective dispersion bands from finite-array eigenmodes and examining extinction, the authors show that bulk Γ-point out-of-plane modes are dark (transversality), while symmetry breaking yields high-Q antisymmetric d_xy modes and radiative edge/corner states via open channels. Edge states radiate along the boundary and remain accessible in the far field as the array grows, whereas corner states stay radiative due to their 0D localization, providing natively open emission channels even without losses. Overall, the work demonstrates how controlled symmetry breaking in multipartite lattices tailors radiation patterns and Q-factors, enabling design of robust, topological plasmonic metasurfaces for nanoscale light control.

Abstract

Subwavelength arrays of plasmonic nanoparticles allow us to control the behaviour of light at the nanoscale. Here, we develop an eigenmode analysis, employing a coupled electromagnetic dipole formalism, which permits us to isolate the contribution to the far-field radiation of each array mode. Specifically, we calculate the far-field radiation patterns by bulk, edge and corner out-of-plane eigenmodes in a finite 2D Su-Schrieffer-Heeger (SSH) array of plasmonic nanoparticles with out-of-plane dipolar resonances. The breaking of symmetries in multipartite unit cells is exploited to tailor the optical properties and far-field radiation of the resonant modes. We prove that the antisymmetric modes are darker and have higher Q-factors than their symmetric counterparts. Also, the out-of-plane nature of the dipolar resonances imposes that all bulk $Γ$-modes are dark, while corner and edge states need extra in-plane symmetries to cancel the far-field radiation; radiation patterns in turn become more complex and concentrated along the array plane with increasing array size.

Figures (8)

• Figure 1: Plasmonic 2D SSH array. (a) Scheme for a plasmonic 2D SSH array and symmetry breaking at the boundary, leading to corner and edge states. (b) Scheme for isolating out-of-plane array modes by using elongated nanoparticles with major axis perpendicular to the lattice. (c) Dispersion bands of a $15\times 15$ plasmonic 2D SSH array with particle radius $a = 10$ nm, lattice constant $d = 150$ nm, shrinking/expanding factor $\beta=1.6$, localized surface plasmon resonance frequency $\hbar\omega_{sp} =2.50$ eV, and optical losses $\hbar\gamma = 1$ meV. Red, green and blue dotted lines represent corner $C_{1-4}$, edge $E_{1-4}$ and bulk $B_{1-4}$ bands. White dashed lines are the light lines $k = k_\parallel$. The colormap represents the optical extinction cross section of the array $\sigma_{ext}$. As an inset we plot the light cone for $\omega = \omega_{sp}$ (white circle) inside the first Brillouin zone and the $k$-path $\Gamma X M \Gamma$ (red line). The modes which are outside the light cone $k \geq k_\parallel$ are evanescent and therefore dark in the far-field, while the modes inside the light lines $k \leq k_\parallel$ are radiative.
• Figure 2: Far-field radiation by bulk mode $B_4$ ($s$ symmetry) $\Gamma$ mode of a finite array of plasmonic nanoparticles depending on the number of unit cells. (a) Radiation pattern $|\textbf{E}_R^T(\textbf{k},\omega,r)|^2$ of the $B_1$ mode of a single unit cell ($N_c = 1$). The radiation pattern is very similar to the one from a single dipole as the dipoles in the unit cell are in phase. (b) Real part of the eigenmode $\Re(\textbf{P})$ of the $B_4$$\Gamma$ mode with $N_c = 1$. (c) Radiation pattern of the $B_4$$\Gamma$ mode of a $10\times10$ unit cells array. (d) Real part of the eigenmode of the $B_1$$\Gamma$ mode of the $10\times 10$ array. (e) Radiation pattern $|\textbf{E}_R^T(\textbf{k},\omega,r)|^2$ of the $B_4$ mode of a $20\times20$ unit cells array. As we see, the main lobe of the pattern is around $\Gamma$, with sidelobes due to diffraction by the edges. The far-field radiation becomes less and less intense with the number of unit cells, and would eventually vanish for $N_c\rightarrow \infty$. (f) Real part of the eigenmode $\Re(\textbf{P})$ of the $B_4$$\Gamma$ of a finite array of $20\times20$ unit cells. As the number of unit cells increases, the eigenmode of the finite array tend to the bulk eigenmode $B_4$ of the periodic array, at least far from the edges.
• Figure 3: Far-field radiation by bulk mode $B_1$ ($d_{xy}$ symmetry) $\Gamma$ mode of a finite array of plasmonic nanoparticles depending on the number of unit cells. (a) Radiation pattern $|\textbf{E}_R^T(\textbf{k},\omega,r)|^2$ of the $B_1$ mode of a single unit cell ($N_c = 1$). The radiation pattern has four lobes centered at $k_x = \pm k_y$ due to the $d_{xy}$ symmetry. (b) Real part of the eigenmode $\Re(\textbf{P})$ of the $B_1$$\Gamma$ mode with $N_c = 1$. (c) Radiation pattern of the $B_1$$\Gamma$ mode of a $10\times10$ unit cells array, with a 125x zoom compared with the radiation pattern from the unit cell. (d) Real part of the eigenmode of the $B_1$$\Gamma$ mode of the $10\times 10$ array. (e) Radiation pattern $|\textbf{E}_R^T(\textbf{k},\omega,r)|^2$ of the $B_1$ mode of a $20\times20$ unit cells array. As we see, the main lobe of the pattern is around $\Gamma$, with sidelobes due to diffraction by the edges. The far-field radiation decays very fast with the array size. It eventually vanishes for $N_c\rightarrow\infty$ (f) Real part of the eigenmode $\Re(\textbf{P})$ of the $B_1$$\Gamma$ of a finite array of $20\times20$ unit cells. As the number of unit cells increases, the eigenmode of the finite array tend to the bulk eigenmode $B_1$ of the periodic array, at least far from the edges.
• Figure 4: Far-field radiation by bulk mode $E_3$ ($s$ symmetry) $\Gamma$ mode of a finite array of plasmonic nanoparticles depending on the number of unit cells. (a) Radiation pattern $|\textbf{E}_R^T(\textbf{k},\omega,r)|^2$ of the $B_1$ mode of a $2\times 2$ unit cells array, the smallest square lattice with edge state. The radiation pattern is very similar to the one from a single dipole as all the dipoles oscillate in phase. (b) Real part of the eigenmode $\Re(\textbf{P})$ of the $E_4$$\Gamma$ mode of a $2\times2$ unit cells array. (c) Radiation pattern of the $E_4$$\Gamma$ mode of a $10\times10$ unit cells array. (d) Real part of the eigenmode of the $B_1$$\Gamma$ mode of the $10\times 10$ array. (e) Radiation pattern $|\textbf{E}_R^T(\textbf{k},\omega,r)|^2$ of the $B_1$ mode of a $20\times20$ unit cells array. As we see, the main lobe of the pattern is around $\Gamma$, with sidelobes due to diffraction by the edges. The far-field radiation lobes at $k_x=0$ and $k_y = 0$ do not decrease with size, actually they get more intense. This means that even when the $E_4$ mode is dark at $\Gamma$, this is not a q-BIC mode. (f) Real part of the eigenmode $\Re(\textbf{P})$ of the $B_1$$\Gamma$ of a finite array of $20\times20$ unit cells. As the number of unit cells increases, the eigenmode of the finite array tend to the bulk eigenmode of the periodic array, at least far from the edges.
• Figure 5: Far-field radiation by edge mode $E_2$ ($d_{xy}$ symmetry) $\Gamma$ mode of a finite array of plasmonic nanoparticles depending on the number of unit cells. (a) Radiation pattern $|\textbf{E}_R^T(\textbf{k},\omega,r)|^2$ of the $E_2$ mode of a $2\times 2$ unit cell array, the smallest square array with edge states. The radiation pattern is four-lobbed due to the $d_{xy}$ symmetry.(b) Real part of the eigenmode $\Re(\textbf{P})$ of the $B_1$$\Gamma$ mode with $N_c = 1$. (c) Radiation pattern of the $B_1$$\Gamma$ mode of a $10\times10$ unit cells array. (d) Real part of the eigenmode of the $E_2$$\Gamma$ mode of the $10\times 10$ array. (e) Radiation pattern $|\textbf{E}_R^T(\textbf{k},\omega,r)|^2$ of the $B_1$ mode of a $20\times20$ unit cells array. As we see, the main lobe of the pattern is around $\Gamma$, with sidelobes due to diffraction by the edges. The far-field radiation gets less and less intense with the number , and would eventually vanish for $N_c$ (f) Real part of the eigenmode $\Re(\textbf{P})$ of the $B_1$$\Gamma$ of a finite array of $20\times20$ unit cells. As the number of unit cells increases, the eigenmode of the finite array tend to the bulk eigenmode of the periodic array, at least far from the edges.