## Abstract

Due to metal losses in plasmonic metamaterials, high-refractive-index dielectrics are promising to improve optical performances of their metallic counterparts. In this paper, a LiTaO_{3} microtube metamaterial is numerically investigated to explore the toroidal dipolar resonance based on the multipole expansion theory. The local field strength probed on the central axis of the microtube is greatly enhanced for the toroidal dipolar mode, forming a strong hot spot concentrated in the deep-subwavelength scale. Furthermore, we also show the influences of geometrical parameter on the quality (Q) factor of the toroidal mode. The high Q factor and strongly concentrated field strength in the toroidal microtube metamaterial offer application potentials such as sensing, energy havesting, particle trapping, and nonlinear optical effects.

© 2015 Optical Society of America

## 1. Introduction

The classical multipole expansion primarily includes two families of multipoles, i.e., electric and magnetic dipoles, quadrupoles, octupoles, and so on [1–3], which was theoretically incomplete until the concept of toroidal multipoles was introduced to this expansion system. In 1957, Zel’dovich firstly proposed the toroidal dipole, characterized by a vortex distribution of magnetic dipoles, to explain the parity violation in the weak interaction force [4,5]. As the third family of electromagnetic multipoles, a toroidal dipole is inherently different from conventional electromagnetic dipoles (i.e., electric and magnetic dipoles) in that a toroidal dipole typically arises from a current flowing on the surface of a torus along its meridian [6–10], while an electric or a magnetic dipole is resulted from a pair of opposite charges or a current loop, respectively. Static electric polarization in natural materials can generate static toroidal dipole, which is interesting due to the intriguing feature of simultaneous violation of both space-inversion and time-reversal symmetries [1,7,11]. For such static toroidal phenomenon, there were many works focused on naturally occurring media, such as DNA condensates and multiferroics [12–15]. On the other hand, dynamic current/field may produce dynamic toroidal dipole excitation [8,16,17]. However, an optical toroidal-dipole response in natural materials, if any, is much weaker than other conventional multipolar responses in the optical range, such as electric, magnetic dipoles, or even their high-order multipoles, and thus was unfortunately out of general attentions.

Metamaterials are composed of artificially constructed, periodically arranged, sub-wavelength structures with unit size much smaller than the operating wavelength [18,19]. Most importantly of all, metamaterials own various novel electromagnetic properties unattainable in natural media, such as left-handed electromagnetic behavior [20], super imaging [21], electromagnetically induced transparency [22], perfect absorber [23], and cloaking [24]. Therefore, it has attracted extensive attentions in the science community. On the other hand, due to the subwavelength resonant feature, numerous physical effects that are normally weak in naturally occurring materials can be significantly enhanced, e.g., fluorescence, particle trapping, and the one we focused here, toroidal dipolar response. In 2007, the metamaterial of a 3D-array of toroidal solenoids was put forward by K. Marinov *et al*. to theoretically explore the toroidal dipolar response [6]. After that, based on the multipole expansion theory, serval toroidal metastructures [7,16,25–28] were proposed to explore the toroidal dipole resonance. For example, toroidal metamaterials of four split-ring resonators were demonstrated, from microwave [1] to optical frequencies [17], by suppressing the electric and magnetic multipolar resonances while enhancing the toroidal dipolar response. Even so, the optical ohmic-damping loss is, unfortuately, still a common issue in various plasmonic metamaterials by noble metals (Ag and Au) [7,25–27].

To avoid the loss issue in plasmonic metamaterials, all-dielectric metamaterials are competent due to their low-loss responses [29–32]. Generally, alternative dielectric materials with high refractive indices are obtainable for different frequency ranges. While a ferroelectric is of a good choice for microwave response and a semiconductor is suitable for optical range, the polaritonic LiTaO_{3} material is preferred to obtain a terahertz response. More recently, a metastructure consisting of four LiTaO_{3} solid cylinders as a cluster was investigated, where the dominant toroidal dipolar response was verified by this composite structure [8]. In this work, a simplified polaritonic LiTaO_{3} microtube is proposed to explore the dominant toroidal dipolar response in the terahertz regime. Inherently, the origination of toroidal response in the microtube shares a similar mutual coupling nature with the four-cylinder metastructure [8]. For the latter, the mutual coupling comes from neighboring four cylinders, while for the former, it comes from excitation of different parts of tube (thinking about that a microtube is constructed by numerous cylinders, rather than just four of them).

According to the numerical results shown in this work, a low-loss toroidal resonance can be obtained with a high quality (Q) factor as well as a strong local field enhancement in the deep-subwavelength scale. These results indicate promising application potentials in improving the sensing capability, energy havesting, particle trapping, and nonlinear optical effects, based on the high-Q-factor toroidal mode with an enhanced hot spot concentrated centrally along the axis of the microtube. Considering the extensive application backgrounds based on microtubes, such a toroidal microtube is convenient for experimental explorations with certain functional purposes of application. For fabrications, it will be complicated to drill holes in the long LiTaO_{3} cylinders by top-down techniques, but we expect that it may be chemically synthesized by bottom-up self-assembly method.

## 2. Numerical model for the microtube metamaterial

The proposed metamaterial structure comprises a periodic array of infinitely-long straight LiTaO_{3} microtubes as shown in Fig. 1(a). The inner and outer radii as shown in Fig. 1(b) are ${R}_{1}=5\text{\hspace{0.17em}}\text{\mu m}$ and ${R}_{2}=25\text{\hspace{0.17em}}\text{\mu m}$, respectively. In this work, the polaritonic LiTaO_{3} material is considered numerically with the Lorentz-type dispersion described as [8,30,33]

*z*-direction and has a

*y*-polarization (along the microtube). For numerical calculations, full-wave simulations based on the finite-element method were performed [34].

## 3. Results and discussions

To analysis the far-field characteristic of the polaritonic LiTaO_{3} microtube, the transmittance, reflectance, and absorbance spectra are shown in Fig. 2(a). Obviously, there is a resonant dip in the transmittance spectrum at 1.33 THz, corresponding to a weak absorbance about 5.5% (due to the limited imaginary part of permittivity for the polaritonic LiTaO_{3} material). This dielectric-based resonant mode, so-called Mie resonance, can be verified to be a toroidal dipolar response by calculating the decomposed radiated powers according to the general multipole scattering theory [9, 17]:

*c*is the speed of light in the vacuum, $\stackrel{\rightharpoonup}{r}$ is the distance vector from the origin to point (

*x*,

*y*,

*z*) in a Cartesian coordinate system, and $\alpha ,\beta =x,\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}z$. Therefore, the decomposed far-field scattered power by these multipole moments can be written as ${I}_{P}=2{\omega}^{4}{\left|\stackrel{\rightharpoonup}{P}\right|}^{2}/3{c}^{3}$, ${I}_{M}=2{\omega}^{4}{\left|\stackrel{\rightharpoonup}{M}\right|}^{2}/3{c}^{3}$, ${I}_{T}=2{\omega}^{6}{\left|\stackrel{\rightharpoonup}{T}\right|}^{2}/3{c}^{5}$, ${I}_{Q}^{e}={\omega}^{6}{{\displaystyle \sum \left|{Q}_{\alpha \beta}\right|}}^{2}/5{c}^{5}$, and ${I}_{Q}^{m}={\omega}^{6}{{\displaystyle \sum \left|{M}_{\alpha \beta}\right|}}^{2}/40{c}^{5}$. As shown in Fig. 2(b), the calculated scattered powers manifest that the toroidal dipolar response is the dominant resonant mode by suppressing the electric dipole and other multipolar components at a broad frequency range from 1.30 to 1.45 THz. Over this whole resonant frequency band, the decomposed power for the toroidal dipolar moment is approximately more than one order stronger than other multipole moments and four times higher than total contribution of all multipoles except the toroidal dipole itself, especially around 1.33 THz. It should be mentioned that the simplified microtube metastructure exhibits a blue shift for the toroidal response as compared with a same-size four-cylinder metastructure proposed in [8]. Specifically, for a microtube with an outer radius of $8\text{\hspace{0.17em}}\text{\mu m}$ and an inner radius of $3\text{\hspace{0.17em}}\text{\mu m}$, the toroidal dipole resonance can be found at 4.97 THz, in contrast to the toroidal frequency at 1.89 THz in [8].

For a straightforward analysis of the origination of the toroidal dipolar response, the resonant mode is visualized in terms of the displacement current density [Fig. 3(a)]. It can be found that the antiparallel displacement currents confined to the outer and inner side-walls of the microtube are formed and thus result in a closed magnetic vortex and a centrally concentrated hot-spot field as shown in Figs. 3(b) and 3(c), respectively. This local-field distribution is a unique characteristic that confirms the toroidal dipole mode in addition to the multipolar scattered powers shown in Fig. 2(b) [8,25]. From the *E*_{y} magnitude curve probed along the *x*-direction diametral path [see the lower panel in Fig. 3(c)], it is obvious that the hot spot has a deep-subwavelength concentration area with a radius about $10\text{\hspace{0.17em}}\text{\mu m}$, which is confined by the magnetic vortex and basically attributed to the high refractive index of the polaritonic LiTaO_{3} material (corresponding to a permittivity of 41.4 at the considered terahertz regime). Such a deep-subwavelength field concentration is estimated to be only one eleventh of the free-space wavelength, far beyond the one-fifth amount claimed in the toroidal four-solid-cylinder cluster [8]. Another unique property for this hot spot is its extension characteristic along the dimension of microtube axis. In addition, the electric-field amplitude probed at the central axis of the microtube [Fig. 3(d)] shows a high Q factor about 20, which can even be improved in further by modulating the inner radius of the microtube (shown below). This toroidal resonant characteristic in the microtube metastructure manifests itself in terahertz application capabilities such as energy harvesting, particle trapping, and nonlinear optical phenomena preferring a significant enhancement of local-field strength. Besides, we also verify that microtubes with finite lengths can support such toroidal response as well. For example, the toroidal dipole resonance in a $30\text{-\mu m-length}$ microtube is found at 1.40 THz, exhibiting a slight blue shift as compared to the infinite case.

In order to investigate the modulation characteristic of geometrical parameter on the toroidal resonant response, Fig. 4(a) illustrates the frequency dependence of the toroidal dipolar response on inner radius *R*_{1}. For ${R}_{1}=0\text{\hspace{0.17em}}\text{\mu m}$ while keeping the outer radius ${R}_{2}=25\text{\hspace{0.17em}}\text{\mu m}$, the toroidal dipolar resonance occurs at 1.24 THz and has relatively low Q factor and hot-spot intensity. However, with increasing the inner radius ${R}_{1}$, the toroidal dipolar frequency will experience a blue shift. Meanwhile, it is interesting to find that the Q factor will be greatly enlarged up to 118 for ${R}_{1}=8\text{\hspace{0.17em}}\text{\mu m}$ and the probed central hot-spot intensity can be significantly enhanced as well [Fig. 4(b)]. It is noted that such a Q value is the highest one than those claimed by other toroidal metamaterials with high Q factors [1,26]. Intuitively, this improved Q factor with increasing the inner radius is a consequence of the squeezing local magnetic vortex into a thinner microtube structure. However, it does not mean that a further increase of *R*_{1} would constantly lead to enhancements of the hot-spot intensity and the Q factor. In fact, it is found numerically that features of the toroidal dipolar mode cannot be excited in a dominant sense in thin microtubes, because the magnetic vortex will not always be well behaved in a thin microtube for ${R}_{1}$ approaching ${R}_{2}$.

## 4. Summary

In this paper, a polaritonic LiTaO_{3} microtube metamaterial has been proposed and numerically simulated to investigate the toroidal dipolar response in the terahertz frequency region. The numerical results show that a dominant toroidal dipolar response can be obtained in a broad frequency range for the simplified microtube metastructure, which is theoretically verified by the multipole scattering theory. In addition, the influences of geometrical parameter on the toroidal dipolar response in terms of the transmission spectrum, Q factor, and local-field enhancement capability were also explored. It is found that, with increasing the inner radius of microtube, the toroidal dipolar resonance not only experiences a blue shift but also obtains a high Q-factor performance. Meanwhile, a strong hot spot that extends along the axis dimension of the microtube can be squeezed into deep-subwavelength scales in the other two dimensions. These demonstrated characteristics associated with the toroidal response in the proposed microtube metamaterial can provide potential applications in sensing, energy harvesting, particle trapping, nonlinear optical effects, and so on.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) (Nos. 11174051, 11374049, and 11511140278), and Natural Science Foundation of Jiangsu Province of China (BK20131283).

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