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Abstract
Electromagnetic toroidal excitations open up a new avenue for strong light-matter interactions. Although toroidal dipole resonances (TDRs) based on artificial meta-molecules were reported intensely, the TDRs supported in a single dielectric particle remain largely unknown. In this work, we show that an all-dielectric sub-wavelength torus can support a dominant TDR. The magnetic field can be enhanced greatly, and it shows a “vortex-like” configuration in the torus, confirming the toroidal excitation. The evolutions of the TDRs due to the geometrical parameters, dielectric permittivity, and polarization are discussed. It is found that the toroidal excitation is achieved mainly for TM polarization, while the anapole state is uncovered for TE polarization. This work suggests a new strategy for toroidal excitations based on a simple dielectric resonator.
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1. Introduction
The strong interaction of light and small particles is of great interest in various applications, such as biological/chemical sensors [1], nanoantennas [2], optical manipulations [3,4] and nonlinear meta-optics [5]. The light scattering process can be interpreted as the re-emission of light due to the polarized charge-current in scatterers. In the framework of classical electrodynamics, the distributions of polarized charge-current are decomposed into three families of dynamic multipoles [6], i.e., electric multipoles, magnetic multipoles and toroidal multipoles. The Mie resonances resulting from the excitations of dipolar modes play a crucial role in sub-wavelength photonics. In natural materials, the electric dipolar response is much stronger than the magnetic and toroidal ones. For instance, the optical scattering of a small metallic particle is dominated by the electric dipole resonances, due to the localized plasmonic resonances [1]. Since most natural materials have no magnetic response (µ=1), the magnetic dipole resonance is much more difficult to be obtained. Fortunately, strong magnetic dipole resonances can be realized in the artificial split-ring resonators (SRRs) [7,8]. One drawback of the SRRs is the saturation effect at high frequencies (e.g., optical frequencies), where resonant magnetic responses decrease enormously [9]. Nonetheless, the plasmonic clusters [10,11] and high-refractive-index materials [2,12] are suitable substitutions of SRRs at high frequencies. In addition to electric and magnetic dipoles, the toroidal dipole is of great interest due to its unique feature. The toroidal dipole moment is induced by oscillating poloidal current, where the current flows on the surface (along the meridians) of a torus [6]. The radiation patterns from toroidal and electric dipoles in the far-field are indistinguishable [13]. However, the charge-current configurations of the toroidal dipoles are completely different from the electrical ones. Moreover, the destructive interference between the toroidal and electric dipoles can produce the anapole state (a peculiar non-radiating source) [14–16].
For a sub-wavelength object, the radiation from a toroidal dipole is generally weaker than that from electric and magnetic dipoles. Thus, one long-standing goal in the past was to enhance toroidal excitations. In 2010, the first direct evidence of toroidal dipole resonances (TDRs) was observed in the SRR metamaterials [17]. Due to the near-field coupling of SRRs, toroidal dipole modes can be formed by a closed loop (head-to-tail) of magnetic dipoles. After that, asymmetric double-bar metamaterials [18], metallic cavities [19,20], plasmonic spherical clusters [21,22], and other meta-structures [23–30] were proposed to enhance the toroidal radiation. Compared to its metallic counterparts, high-refractive-index materials (without Ohmic losses) exhibit the feature of low dissipation loss, which is a good alternative for toroidal excitations. In 2015, A. A. Basharin et al. showed that the TDRs could be realized by all-dielectric meta-materials [31]. The unit cell of their meta-structures consists of four infinite-long cylinders. Magnetic dipole resonances can be supported in an individual cylinder, and the coupling between cylinders gives rise to TDRs. However, dielectric cylinders with infinite length are difficult to implement, and those with finite high (i.e., disks) are desirable in experiments. By breaking the symmetry of positions, toroidal dipole modes can be supported in clusters of all-dielectric disks [32,33]. Moreover, the phenomena of bound states in the continuum (BIC) due to the toroidal excitations were reported in all-dielectric disks [34,35] and other all-dielectric nanostructures [36–38].
Although TDRs due to the coupling of multiple meta-molecules were reported intensely, a dominant TDR supported in a single particle remains largely unknown. Since the geometry of a torus consists with the toroidal electric and magnetic fields, it should be a high-efficient resonator for toroidal excitations. In this work, we show that a dominant TDR can be supported by an all-dielectric sub-wavelength torus. The magnetic field can be enhanced significantly and it shows a “vortex-like” configuration in the torus, confirming the toroidal excitation. Moreover, the toroidal dipole and the electric dipole interfere constructively at the TDR, leading to a high-Q scattering spectrum. The properties of TDRs due to the geometrical parameters, dielectric permittivity, and polarization are also studied.
2. Theory of multipole radiation
Here, scattering cross section and multipole radiation from the dielectric torus are obtained numerically by the full-wave electromagnetic simulation software. The general expressions of multipoles are given as follows [6]:
Electric dipole moment:
Magnetic dipole moment:
Toroidal dipole moment:
Electric quadrupole moment:
Magnetic quadrupole moment:
The total radiation power from the electric and toroidal dipoles is written as:
where k0 = ω/c is the wavevector, the cancelling of the radiation can be realized when P = −ik0T, leading to the anapole state [14–16] and non-trivial transparency [39,40].3. Results and discussions
Figure 1 schematically shows the dielectric torus under study, known as a “doughnut” structure. Two geometrical parameters, i.e., a and b, represent the major radius and minor radius, respectively. The incident plane wave is propagating along the y-direction, and its magnetic field is expressed as $H = {H_0}{e^{i{k_0}y}}{\hat{e}_x}$, where H0 represents the incident amplitude. Mathematically, the geometric topology of a torus (genus g = 1) is nontrivial, which differs from that of a sphere or a disk (g = 0) [41]. Let us begin by comparing the scattering features between toroidal and spherical particles.
The scattering cross section Csca and multipole contributions for a dielectric sphere and a torus are shown in Figs. 2(a) and 2(b), respectively. Here, the permittivity we applied is as high as 60 (e.g., ceramics [42] or water [43] in the microwave frequency). The geometrical parameters are set in centimeters so that the Mie resonances appear in the microwave band. The results indicate that the scattering spectrum of the dielectric sphere resembles the case of a silicon sphere [44]. The first three Mie resonances are attributed to the excitations of the magnetic dipole, electric dipole and magnetic quadrupole, respectively. However, the toroidal dipole excitation is low in a dielectric sphere. Compared with a dielectric sphere, the scattering spectrum of a torus shows different features, where the second and third resonances become the magnetic quadrupole and toroidal dipole (see Fig. 2(b)). The magnetic quadrupole, i.e., the third Mie resonance in the sphere, has been promoted to the second one for a dielectric torus. Remarkably, a dominant TDR supported in the dielectric torus is revealed. The toroidal and electrical dipoles interference constructively, resulting in a high-Q resonance (also referred as “super-radiation” [45]). Note that the size of the torus (∼6.0 cm) is much smaller than the incident wavelength (∼20.4 cm) at the TDR.
Fig. 2. Scattering cross section and the multipole contributions for topological trivial and nontrivial scatterers. (a) A dielectric sphere with ε=60 and a radius of 3 cm; (b) a dielectric torus with ε =60, a = 2 cm and b = 1 cm. Here, the contributions from P (electric dipole), T (toroidal dipole), M (magnetic dipole), Qe (electric quadrupole), Qm (magnetic quadrupole) and the interference term P + ik0T are presented. The dash lines represent the total scattering cross sections.
Figure 3(a) exhibits the magnetic field of the first resonances generated from the magnetic dipole. Interestingly, the distribution of the magnetic field looks like a crescent, and the absolute value has been enhanced more than 20 times, compared with the incident one. In addition, the hot-spot of the magnetic resonance for a dielectric torus differs from the case of a dielectric sphere (located at the center [44]). The distribution diagram for the second Mie resonance is given in Fig. 3(b). Clearly, the magnetic field distribution confirms the resonance generated from the magnetic quadrupole. Moreover, the maximum absolute value of the magnetic field has been enhanced over 80 times due to the high Q-factor of the magnetic quadrupole. The distribution diagram for the third resonance is given in Fig. 3(c), which corresponds to the toroidal excitation. Remarkably, the magnetic field is mainly confined within the torus, and it shows a vortex-like configuration. Again, the magnetic field can be significantly enhanced for the TDR, and the absolute value at the hot-spot is over two times of that from magnetic dipole resonance.
Fig. 3. The magnetic field distributions for the first three Mie resonances of the dielectric torus. (a) Magnetic dipole resonance; (b) Magnetic quadrupole resonance; (c) TDR. The top and button panels represent the absolute values and plowing vector of the magnetic field, respectively. The arrow in (a) indicates the incident wavevector. Here, the field diagrams in (a-c) lie in the x-y plane (i.e., plane z = 0).
The evolutions of the TDRs via geometrical parameters are presented in Figs. 4(a) and 4(b). Firstly, we keep the major radius a be constant while changing the minor radius b. It is found that the scattering spectra have similar shapes for the first three Mie resonances. As the parameter b increases from 0.8 cm to 1.2 cm, the resonant frequencies undergo a redshift (see the Fig. 4(a)). Meanwhile, the second Mie resonance (magnetic quadrupole) stays away from the first one (magnetic dipole), while getting closer to the third one (TDRs). In addition to the first three Mie resonances, we also see other high-order Mie resonances, which are out of interest in this work. Figure 4(b) summarizes the trend of TDRs, where the resonant frequencies decrease almost linearly with parameter b.
Fig. 4. Evolution of scattering cross section with the changes of the geometrical parameters. (a) Scattering cross section changes with minor radius b, where the major radius a = 2 cm is fixed. The arrows indicate the TDRs. (b) The frequencies of TDRs as a function of radius b. (c) Scattering cross section as a function of radius a, where the minor radius b =1 cm is fixed. (d) The frequencies of TDRs via major radius a.
By contrast, the evolution of TDRs via the major radius a shows different features (see Fig. 4(c)). The shapes of scattering spectra change dramatically when the major radius increases from 1.5 cm to 3.0 cm. For a =1.5 cm, the resonant peak from the magnetic quadrupole (the second resonance) almost overlaps with that from the toroidal dipole (the third resonance), whereas it stays away as the radius a increases further. Thus, a moderate major radius is preferred to observe the isolated TDR. Figure 4(d) shows the evolution of the TDRs as a function of major radius. The result indicates that resonant frequencies decline slightly at the beginning, and stay largely constant with increasing the major radius.
Geometrically, the dielectric torus turns to an infinite-long cylinder when the major radius a increases to infinite. As a result, the TDRs (think about the diagram with “head-to-tail” magnetic dipoles) are determined by the magnetic dipole resonance of an infinite-long cylinder. For a normal incidence, the analytical form of magnetic response in a dielectric cylinder can be obtained by the Mie theory [46,47]:
Figure 5(a) shows the scattering spectra of the dielectric torus under different permittivity of the material. The results indicate that the TDRs can be supported in a dielectric torus for a vast choice of permittivity. The resonant frequencies for the magnetic dipole, magnetic quadrupole and toroidal dipole undergo a redshift as the permittivity increases from 40 to 80. The scattering spectra resulting from the toroidal excitation are sharp when the permittivity is large (e.g., ε=80, 70, 60). As the permittivity decreases to 40, the TDR becomes flatter, indicating a low Q-factor. Figure 5(b) shows the resonant frequencies and Q-factor of the toroidal excitation as a function of permittivity. It is found that the Q-factor tends to increase with the permittivity, and it reaches the magnitude of about 220 for the permittivity of 80. Such a high Q-factor provides a possibility for sensing applications.
Fig. 5. (a) Scattering cross section of a dielectric torus under different values of permittivity. (b) The resonant frequencies and Q-factor of TDRs as a function of dielectric permittivity. The geometry parameters are fixed with a = 2 cm and b = 1 cm.
The toroidal dipole excitations depend strongly on the polarization of incident plane waves. Here, the TM (TE) polarization is defined for incident magnetic (electric) field perpendicular to the z-y plane. The wavevector lies in the z-y plane, and its angle between the y-axis makes an incident angle θ. Figure 6(a) shows the Csca for the TM polarization under different incident angles. The results suggest that the TDRs can be supported for vast incident angles. The strong TDRs can be clarified and should be detectable by the experiments, even though their amplitudes decrease with increasing the incident angle. The toroidal Mie resonance disappears as the incident angle increases to 90°. By contrast, the magnetic dipole resonance increases with the incident angle.
Fig. 6. Scattering cross section of a dielectric torus for (a) TM polarization and (b) TE polarization. The scattering spectra at 90° for TM and TE polarizations are the same. Multipole radiation powers for TE polarization under incident angles of (c) 0°, (d) 45°, and (e) 90°. The point marked by the star in (b) and the arrow in (c) indicate the anapole state. The geometric parameters and refractive index are kept the same as those in Fig. 2(b).
On the other side, the Mie resonances for TE polarization show complicated spectra with incident angles (see Fig. 6(b)). Unlike the case of TM polarization, the TDRs disappear for TE wave incidence. Remarkably, there is scattering transparency (marked by the star) for incident angle θ=0°, and the corresponding scattering cross section reduces to near 5 cm2. However, the performance of transparency degrades considerably as θ increase to 45° and 90°. The multipole radiation with incident angles θ=0°, 45° and 90° are given in Figs. 6(c), 6(d) and (6)(e). For angle 0°, the first three Mie resonances are respectively attributed to the excitations of the electric dipole, electric quadrupole, and magnetic dipole modes. Undoubtedly, the scattering transparency is attributed to the anapole state (∼1.44 GHz), resulting from the destructive interference between the electric and toroidal dipoles, as shown in Fig. 6(c). Not only the radiation power from the electric dipole is reduced, but also those from other conventional multipoles are significantly suppressed at anapole state. For incident angle 45°, several Mie resonances can be found. The first Mie resonance is attributed to the overlapping of electric and magnetic dipole resonances. Meanwhile, electric and magnetic quadrupoles are also presented. As the angle increases to 90°, the dominant Mie resonances for the two peaks are caused by the magnetic dipole modes.
Finally, we show that the TDRs in the microwave can be scaled to higher frequencies. Here, we take the terahertz (THz) wave as an example. The dielectric torus consists of LiTaO3, and its permittivity can be given by the Lorentz model [31]:
Fig. 7. TDRs at the THz frequencies. The excitations are (a) a plane wave for TM polarization and (b) an electrical dipole placed at the center of the torus. The dashed line in (a) represents the total scattering cross section. The inset in (b) shows the magnetic-field distribution at the TDR.
In addition to the plane wave, the TDRs can also be excited by other sources (e.g., an electrical dipole [45] or a polarized beam [48]). Here, a point electric dipole (polarized along the z-direction) is placed at the center of the dielectric torus, and the radiation powers from the multipoles are shown in Fig. 7(b). It is illustrated that only the toroidal and electric dipoles are excited, while other multipoles are completely suppressed. It is worth mentioning that the suppressed multipole moments can be excited if the position of the dipole excitation is deviated from the center of the system. The magnetic-field distribution at the frequency of TDR is shown in the inset of Fig. 7(b). As expected, a “vortex-like” configuration can be found for the electrical dipole excitation.
4. Conclusion
In summary, the Mie scattering of an all-dielectric and high-permittivity torus is numerically investigated. In addition to conventional multipoles, a strong toroidal dipole excitation is revealed in the dielectric torus. The magnetic field can be enhanced significantly and it shows a “vortex-like” configuration, confirming the toroidal excitation. Moreover, the toroidal dipole and the electric dipole interfere constructively at TDRs, leading to high-Q spectra. The evolutions of the TDRs due to the geometrical parameters, dielectric permittivity, and polarization are discussed. Although the TDRs disappear for TE polarization, an interesting anapole state is revealed at specific frequencies. Experimentally, the toroidal structure and its array can be fabricated by state-of-the-art 3D prints. Therefore, we believe that the strong TDRs in an all-dielectric torus (as a building block) can find promising applications in photonics metadevices.
Funding
National Natural Science Foundation of China (11804288).
Acknowledgments
L.X.G. is supported by Nanhu Scholars Program for Young Scholars of XYNU.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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