Subwavelength waveguides composed of dielectric nanoparticles

We study waveguiding of the electromagnetic energy below the diffraction limit with arrays of dielectric nanoparticles through the excitation of both electric and magnetic Mie resonances. We analyze the dispersion characteristics of such coupled-resonator optical waveguides by means of the coupled-dipole approximation and then verify the validity of the coupled-dipole model by comparing the results with direct numerical simulations. We reveal that a chain of silicon nanoparticles with realistic material losses can guide light for the distances exceeding several tens of micrometers, which is significantly better than the guiding by any plasmonic waveguide with the propagation distances less than 1 $\mu$m. We verify the main concept and our theoretical findings experimentally at microwaves for an array of ceramic particles.

Figure 1

(a) A chain of dielectric (with permittivity ${\varepsilon }_{sp}$) particles with radius $R$ and period $a$, located in the host medium with permittivity ${\varepsilon }_h$. (b) Schematic of all different eigenmodes in a chain of nanoparticles.

Figure 2

(a),(b) Dispersion diagrams of infinite chains of lossless spherical silicon nanoparticles and uncoupled chains of electric and magnetic dipoles (only real solutions of dispersion equations for both transverse and longitudinal polarizations are shown) for period of (a) 200 nm and (b) 140 nm. Solid black curves (TM and TE) show solutions of Eq. (1), dashed curves (MD and ED) show solutions of Eqs. (5), solid red and green curves (LM and LE, respectively) show solutions of Eqs. (2) and (3). Eigenmodes, numerically calculated with the MPB package, are shown with black squares. The oblique gray line is the light line. Horizontal dashed blue lines indicate the positions of MD and ED resonance frequencies. (c),(d) Numerically calculated transmission spectra of a chain of six silicon spheres with periods (c) $a=200$ mm and (d) $a=140$ nm. (e)–(l) Electric field distributions in the corresponding modes. The operational range of normalized frequenices $k_{h}a/\pi$ lies within an optical spectral range for the chosen values of period.

Figure 3

Structure of complex modes for the chain of (a),(b) lossless and (c),(d) lossy silicon spheres with period of 200 nm. Solid black lines show solutions of Eq. (1), short-dashed red and long-dashed green lines show dispersion curves of the chains of purely electric and purely magnetic dipoles, respectively, described by Eqs. (5).

Figure 4

Propagation distance $z_0$ as a function of the period $a$ of the chain for three different radii of localization $r_{\mathrm{loc}}$ for transversely polarized modes. Red solid curves correspond to the “magnetic” TM modes; green dashed curves correspond to the “electric” TE modes.

Figure 5

Comparison between experimental data (solid red curves) and numerically calculated transmission spectra (dashed blue curves) for a chain of six ceramic spheres with radius $R_c=7.5$ mm, ${\varepsilon }_c=15.4$, and periods (a) $a=21.4$ mm and (b) $a=15$ mm.