Subwavelength Topological Edge States in Optically Resonant Dielectric Structures

We suggest a novel type of photonic topological edge states in zigzag arrays of dielectric nanoparticles based on optically induced magnetic Mie resonances. We verify our general concept by the proof-of-principle microwave experiments with dielectric spherical particles, and demonstrate, experimentally, the ability to control the subwavelength topologically protected electromagnetic edge modes by changing the polarization of the incident wave.

Figure 1

Artist’s view of (a) topologically nontrivial zigzag array and (b) topologically trivial linear chain of nanoparticles. (c),(d) Numerically calculated electric field distributions in both structures under the excitation by the plane wave with the direction of magnetic field ${\mathit{H}}_0$ shown in the graphs. (e),(f) Energy spectrum and parity of the winding number $P\equiv W\mathrm{mod}2$ of the zigzag chain calculated as functions of the bond angle $\theta$. Red dashed line indicates the region when the edge states are formed. Calculated at $t_{\parallel }=4$, $t_{\perp }=-1$, $E_0=0$ for $N=60$ particles.

Figure 2

(a) Scattering efficiency calculated by Mie theory, and (b) measured forward scattering of a dielectric sphere.

Figure 3

Experimental results. (a)–(d) Measured electric field intensity at the frequency of the magnetic quadrupole resonance of a single sphere for different incident wave polarizations $\varphi =0°,90°,-45°$, and 45°, respectively, indicated in the panels by arrows. (e),(f) Electric field intensity measured at the 1 mm distance from the surface of particles for (e) $\varphi =0°$ and (f) $\varphi =90°$ polarizations. (a)–(d) Results for odd number of particles. (e)–(f) Results for even number of particles. White scale bar represents the incident wavelength.

Figure 4

(a) Extinction cross section spectrum for a single particle (red solid curve), and for the zigzag array at $\varphi =45°$ (blue dashed curve) and $\varphi =0°$ (black solid curve). (b) Dependence of the electric field near the particle center on the particle’s number for $\varphi =45°$. (c)–(f) Near-field intensity at 1 mm above the first and last spheres for incident wave polarization $\varphi =0°,90°,-45°$, and 45°, respectively. Solid lines correspond to the analytical results and dashed lines to the results of numerical simulations. The insets correspond to the electric field intensity at the quadrupole magnetic resonance of the single particle (marked by a dashed black curve).