Dielectric Metamaterials with Toroidal Dipolar Response

Toroidal multipoles are the terms missing in the standard multipole expansion; they are usually overlooked due to their relatively weak coupling to the electromagnetic fields. Here, we propose and theoretically study all-dielectric metamaterials of a special class that represent a simple electromagnetic system supporting toroidal dipolar excitations in the THz part of the spectrum. We show that resonant transmission and reflection of such metamaterials is dominated by toroidal dipole scattering, the neglect of which would result in a misunderstanding interpretation of the metamaterials’ macroscopic response. Because of the unique field configuration of the toroidal mode, the proposed metamaterials could serve as a platform for sensing or enhancement of light absorption and optical nonlinearities.

Popular Summary

A toroidal dipole is a rare yet fundamental electromagnetic excitation produced by electrical currents that circulate in space as if they were moving on the surface of a torus along its meridians. Although the importance of toroidal excitations has already been recognized in particle, atomic, and solid-state physics, these excitations are extremely weak and typically remain unnoticed in the frame of classical electrodynamics. Only recently have advances in metamaterial research enabled the observation of such excitations in specially engineered artificial metamaterials (toroidal metamaterials) composed of properly arranged metallic resonators. We show how to eliminate Ohmic loss, one of the major drawbacks of existing toroidal metamaterials. We also propose a novel class of all-dielectric metamaterials that exhibit an almost lossless resonant toroidal dipolar response in the terahertz part of the spectrum.

A novel, low-loss, dielectric metamaterial exhibiting toroidal dipolar response can be obtained via a periodic arrangement of clusters formed by four high-index ${\mathrm{LiTaO}}_3$ dielectric cylinders with dimensions measured in micrometers. These cylinders form the basic building blocks (metamolecules) of the proposed toroidal dielectric metamaterial. Near-field coupling occurs among the cylinders, which are separated by less than their radius. The toroidal response forms from the poloidal displacement currents circulating in each cylinder. We conduct a theoretical investigation of the properties of such metamaterials; we find that not accounting for the toroidal dipole moment results in unphysical transmissions above 100%.

Owing to the unique topology of the toroidal excitation, toroidal metamaterials may be employed as a platform for terahertz sensing or enhancement of light absorption and excitation of nonlinearities and as components for testing of the Aharonov-Bohn effect.

Figure 1

A fragment of an all-dielectric metamaterial slab supporting toroidal dipolar excitation. Its metamolecule is composed of four closely spaced infinitely long high-index dielectric cylinders. The dashed box indicates the unit cell of the metamaterial, which is periodic along the $x$ axis only. Purple arrows show displacement currents $\mathbf{j}$ induced by the vertically polarized plane wave, red arrows show magnetic dipole moments of the constituent dielectric cylinders $\mathbf{m}$, and the green arrow represents the net toroidal dipole moment of the metamolecule $\mathbf{T}$.

Figure 2

Cross section of the dielectric metamolecule supporting toroidal dipolar excitation. The dimensions are $R=8\mu \mathrm{m}$, $a=18\mu \mathrm{m}$, and $d=58\mu \mathrm{m}$.

Figure 3

(a)–(c) Calculated distributions of the corresponding electric field ($z$ component ${\mathbf{E}}_z$), magnetic field (absolute value $|\mathbf{H}|$), and amplitude of the displacement current $\mathbf{j}$ ($z$ component) induced in the metamolecule at 1.89 THz. Arrows show instantaneous directions in the magnetic-field distribution.

Figure 4

(a) Transmission $T$ (red lines) and reflection $R$ (black lines) spectra calculated for the metamaterial slab composed of four-cylinder clusters periodically placed along the $x$ axis, as shown in Fig. 2. Solid curves correspond to the results of the CST Microwave Studio simulations. The dashed curves are obtained based on the multipole decomposition of the displacement current data, which includes both the standard multipole terms (up to the magnetic octupole) and toroidal multipoles (up to the toroidal quadrupole). The dotted curves are obtained based on the multipole decomposition missing the toroidal multipoles. (b) Contributions of the six strongest multipolar excitations to the reflection of the metamaterial array. The gray curve marked “All multipoles” gives the normalized total power scattered by the multipoles and corresponds to the reflection of the metamaterial slab, shown by the dashed black curve in (a). It is obtained by coherent (amplitude and phase) summation of all multipole contributions. The log scale in the $y$ axis in (b) is chosen so as to reveal more clearly the contribution of the quadrupole terms as well.

Figure 5

(a) Transmission $T$ spectra of the metamaterial slab calculated with the CST Microwave Studio for obliquely incident $E$-polarized plane waves (the electric vector of the waves is set parallel to the axes of the cylinders) with the step of 10 degrees. The white bars mark the frequency of the toroidal mode. (b) Relative strength of four leading standard multiples and a toroidal dipole compared for various angles of incidence at the frequency of the toroidal mode. (c) Magnetic near-field distributions calculated for angles of incidence $\theta =0°$, 40°, 80° at the frequency of the toroidal mode [corresponding to the white bars in (a)].

Figure 6

Transmission $T$ (red line) and reflection $R$ (black line) spectra of the metamaterial slab calculated with the CST Microwave Studio for normally incident $\mathbf{H}$-polarized plane waves. (The magnetic vector of the waves is set parallel to the axes of the cylinders.) The insets show the magnetic near-field distributions calculated at 2.13 and 2.32 THz.

Figure 7

(a),(b) Calculated distributions of the absolute values of the (a) electric field ($z$ component) and (b) magnetic field induced in the metamolecule at 1.95 THz. (c) The distribution of the displacement current $\mathbf{j}$ within the dielectric rods corresponds to the magnetic quadrupole response induced in the metamolecule.

Figure 8

(a) Transmission $T$ (red lines) and reflection $R$ (black lines) spectra calculated for the metamaterial slab composed of four-cylinder clusters periodically placed as described in the text. The solid curves correspond to the results of the CST Microwave Studio simulations for $400\text{−}\mu \mathrm{m}$-in-height cylinders. The dashed curves are obtained for the case of infinite-in-height cylinders. (b) Individual contributions of the six strongest multipolar excitations to the normalized power radiated (scattered) by our metamaterial array of finite-height cylinder clusters near its toroidal resonance.

Figure 9

(a)–(c) Calculated distributions of the corresponding electric field ($z$ component ${\mathbf{E}}_z$), magnetic field (absolute value $|\mathbf{H}|$), and amplitude of the displacement current $\mathbf{j}$ induced in the metamolecule at 1.87 THz. The arrows show instantaneous directions in the magnetic-field distribution.