1. Introduction

Optical nano-resonator is an important platform for efficient light-matter interactions at the nanoscale. Recently, all-dielectric nanostructures become potential choices to obtain high Q-factor resonances due to their low dissipative losses at optical frequencies [1]. Major contributors to optical resonances include electric dipole, magnetic dipole and toroidal dipole. Owing to its relatively weak response and distinct characteristics, toroidal dipole has gained widespread attention since recent years [2,3]. At present, investigations about toroidal dipole responses in dielectric nanostructures mainly focus on the microwave [4], THz [5,6], visible [7] and near-infrared regions [8–10]. For instance, Basharin theoretically studied toroidal resonances of all-dielectric nanostructures in the THz region of the spectrum [6]. Xu et al. used all-dielectric resonant metasurfaces based on trimers to experimentally demonstrate toroidal dipole modes in the microwave range [4]. Sayanskiy et al. investigated the toroidal dipole response of all-dielectric metalattices in the visible region [7]. Recently, we also experimentally demonstrated multiple toroidal responses at the near-infrared wavelengths [8]. While explorations of ultraviolet resonators supporting the strong toroidal dipole are rare, because the sizes of required nanostructures are relatively small which adds more challenges to the fabrication. With the rapid development of nanofabrication, nanostructures operating at the ultraviolet region become possible, greatly promoting researches in ultraviolet spectroscopy, such as ultraviolet LED, sterilization and so on. In addition, some new routes may also be proposed to realize ultraviolet resonators with high performance, such as designing nanodevices available for fabrication or exploring novel theories in resonance excitation.

Recently, symmetry-protected BICs have also drawn extensive attention in nanophotonics owing to their unique optical properties. Symmetry-protected BICs are states that remain perfectly localized in dynamic systems even though they coexist with a continuous spectrum of radiation due to the symmetry mismatch between the bound state and free space [11–15]. In photonic systems, early reports of the symmetry-protected BIC at $\Gamma$-point were done by Shipman et al. [16] and Bulgakov et al. [17] in optical waveguides. Then Plotnik et al. presented the first experimental observation of symmetry-protected BIC in an optical waveguide array structure [18]. When eigenmodes of dielectric nanostructures exhibit intrinsic toroidal dipolar characters and infinite lifetime, BICs with strong toroidal dipole responses can be produced and build great local field enhancement within nanostructures. Lately, He et al. theoretically investigated toroidal dipole responses governed by symmetry-protected BICs in the near-infrared wavelengths via introduction of in-plane asymmetry into double nanodisks [19]. Then Kupriianov et al. experimentally demonstrated symmetry-protected BICs supporting toroidal dipole responses in an out-of-plane symmetric array at microwave wavelengths [20]. These works take full advantage of the nonradiative characteristic of BIC to facilitate the excitation of high Q-factor toroidal resonances. As a distinct optical mode, researches about BIC have covered areas from THz, microwave to near-infrared regions [21–32]. While resonances governed by the symmetry-protected BIC are always polarization-dependent, because the in-plane symmetry perturbation required for their excitation always breaks the ${\rm C_4}$ rotational symmetry of the structures. Recently, researches about polarization-independent resonances governed by BICs have gained more attention due to their flexible applications. Yu et al. introduced round eccentric penetrating holes to unit super-cells consisting of four nanodisks and theoretically analyzed their polarization-insensitive responses [33]. Sayanskiy et al. experimentally investigated the trapped mode excitation in a polarization-insensitive metasurface in the microwave frequency range, whose unit super-cell is constructed by particularly arranging four nanodisks [34]. Lately, Overvig et al. theoretically analyzed the polarization dependence for each symmetry mode of square or hexagonal photonic crystal lattices using group theory principles [35]. Further researches on this topic are still necessary to motivate novel devices with polarization-independent responses.

In this work, we theoretically investigate polarization-independent Fano resonances supporting toroidal dipole responses governed by BIC in the ultraviolet region. First, we use an asymmetric nanohole array to explore the excitation of BIC and characteristics of produced Fano resonances. Then, we verify that toroidal dipole contributes most to these Fano resonances through multipole decomposition calculation and field analysis. Transmissions under different incident polarization also demonstrate the polarization-independence of our design. Finally, we evaluate the enhancement factor of electric field within this nanostructure. Our work offers a way for realization of novel devices with polarization-independent responses.

2. Fano resonance governed by symmetry-protected BIC

Via breaking the in-plane symmetry of nanostructures, symmetry-protected BICs are perturbed and then transformed to quasi-BICs, which can radiate to the external continuum and produce Fano resonances with high Q-factors [12–14]. Herein, we propose a nanohole array composed of unit cells with four asymmetric nanoholes to excite Fano resonances in the ultraviolet region. As shown in Fig. 1(a), four nanoholes are etched off under a lattice constant $P$ of 480 nm from a ${\rm Si_3N_4}$ slab with a thickness of 500 nm. Radiuses of nanoholes lying at the two diagonals are represented by $r_1$ and $r_2$, respectively, with $r_2$ fixed at 70 nm. Through varying $r_1$ from 70 nm, mirror symmetry along in-plane coordinates can be broken, compensating the symmetry mismatch of BIC and incident polarization and thus exciting sharp Fano resonances. Distances between nanoholes along $x$ and $y$ axes are all 240 nm. We choose ${\rm Si_3N_4}$ as the building material of these nanostructures because it has relatively low absorption at ultraviolet wavelengths. A ${\rm SiO_2}$ substrate is placed under the nanohole array. Optical properties of this structure are calculated using a finite difference time domain (FDTD)-based commercial software (Lumerical FDTD Solutions), where periodical boundary conditions are set in the $x$ and $y$ directions and perfectly matched layers are set in the $z$ direction. A plane wave propagated along the $z$ axis is normally incident on the structure. With the excitation of $x$-polarized incident light, the calculated transmission spectra at different $r_1$ are illustrated in Fig. 1(b). Three modes are manifested in the transmission, marked as I, II, III, respectively. When $r_1$ increases from 58 nm to 75 nm, mode I can be spotted all the time, only with a slight blue-shift of the position resulting from the decreased effective refractive index of the nanostructure [36]. While for modes II and III, these resonances become sharper when $r_1$ approaches $r_2$ from a smaller value. When $r_1$ reaches 70 nm, equaling to $r_2$, modes II and III vanish, which means no leaky energy from the bound state to the free space and demonstrates the existence of two symmetry-protected BICs. Continuing increasing $r_1$, these two modes emerge again and grow wider as the difference between radiuses gets larger. That is because deviating $r_1$ from $r_2$ perturbs the in-plane symmetry of this nanostructure and transforms the above two symmetry-protected BICs to two quasi-BICs lying at different wavelengths. Then two obvious asymmetric Fano resonances are produced, originating from the interference between discrete states supported by the nanohole array and the continuum free-space radiation. We fit resonance curves of modes II and III by the classical Fano formula [8,37,38]:

 

Fig. 1. (a) Schematic geometry of the unit cell of nanostructure. (b) Simulated transmission spectra when $r_1$ changes from 58 nm to 75 nm. Three modes are marked by I, II and III, respectively. (c), (d) Fano fittings of modes II and III for $r_1$= 58 nm. The dashed curves are simulation results, and the solid curves are Fano fittings.

Download Full Size | PDF

3. Toroidal dipole responses

Through multipole decomposition under the Cartesian coordinate, the toroidal excitation can be distinguished and its moment can be expressed by the formula $\vec T=\frac {1}{10c}\int {[(\vec r \cdot \vec j)\vec r - 2{r^2}\vec j]{d^3}r}$, where $\vec {j}=-i\omega \epsilon _{0} (n^2-1) \vec {E}$ is the displacement current, $c$ and $\omega$ are the speed and angular frequency of light, respectively. The electric field is calculated by FDTD method and then the extracted data are employed to evaluate the scattered powers of different multipoles (see detailed descriptions in [8,14]). Therefore, we perform far-field multipole decomposition when $r_1$= 58 nm to evaluate the contribution from toroidal dipole quantitatively. Figures 2(a)–2(c) depict scattered power from electric dipole, magnetic dipole, toroidal dipole, electric quadrupole and magnetic quadrupole, respectively, with only the most contributive component of each multipole drawn on. For mode I, toroidal dipole along the $x$ axis plays a dominant role in a relatively large wavelength range around the resonance and reaches a maximum at the resonant wavelength 325.34 nm. For modes II and III, toroidal dipole along the $x$ axis is significantly enhanced near the resonance with other multipole components dramatically suppressed, producing their narrow lineshapes. These dominant contributions of toroidal dipole can also be verified in electric field distributions at resonant wavelengths in Figs. 2(d)–2(f). Two circular electric field loops are formed with opposite orientation in the $x$-$y$ plane near each nanohole, giving rise to magnetic dipole oscillation along the $z$ axis. Accompanying with the second contributive multipole, magnetic dipole along the $y$ axis, which mainly contributes by the $y$-polarized magnetic field of incident light, a circular magnetic field generates in the $y$-$z$ plane and thus excites toroidal dipole along the $x$ axis. In addition, the electric intensity within the unit cell is also calculated at resonant wavelengths of modes I, II and III. Obvious field enhancement is spotted in the ${\rm Si_3N_4}$ material, especially located near the electric vortex. Mode I also shows an asymmetric field distribution at nanoholes with different radiuses, with stronger field enhancement collected at four sidelobes around the smaller nanoholes.

 

Fig. 2. (a)-(c) Scattered powers of multipole components for modes I, II and III. The $x$-directed toroidal dipole is dominant. (d)-(f) The electric field distributions of modes I, II and III in the $x$-$y$ plane. The arrows represent the direction of the electric field, and the color scale represents the corresponding intensity distribution.

Download Full Size | PDF

4. Characteristic of polarization-independence

Most nanostructures governed by symmetry-protected BICs are polarization-dependent, which can only excite strong Fano resonances at a specific incident polarization [13]. While our design exhibits the same responses under incident light with different polarizations, which is attributed to its rotational symmetry. This property can provide our design with robust responses under diverse incident configurations. To demonstrate this characteristic, we calculate the transmissions of this nanostructure under varied polarization angles when $r_1$= 58 nm. As shown in Fig. 3(a), the polarization angle $\theta$ is defined as the angle between the polarization direction of incident electric field and the $x$ axis. Marked with I, II, III in Fig. 3(c), the transmission coefficients and positions of these three modes almost keep unchanged when polarization angles vary from 0 to 90 degrees, demonstrating the polarization-independent characteristic of our design. Several cross-sections at polarization angles of 0, 30, 70 and 90 degrees are shown in Fig. 3(b), with shaded areas highlighting these three modes. The shapes, positions and extinction ratios of these resonances are almost the same.

 

Fig. 3. (a) Sketch of the polarization angle $\theta$. (b)(c) Transmissions under different polarization angles. I, II and III denote the three modes supported by our design.

Download Full Size | PDF

5. Enhancement factor

Fano resonances can lead to huge local field enhancement within the nanostructrues [39–41], which have great potentials in applications such as light emission [42,43], harmonic generation [44,45] etc. Herein, we also evaluate the ability of our design to trap the electric field inside the structures when $r_1$= 58 nm. The enhanced electric field intensity is calculated through integration within the nanostructures, and then it is normalized with respect to the incident intensity in the same volume. Calculating the enhancement factor ($EF$) of the electric field [46]:

 

Fig. 4. Enhancement factors of the local electric field in the nanostructures of modes I, II and III.

Download Full Size | PDF

6. Conclusion

In conclusion, we propose a nanostructure with strong polarization-independent toroidal resonances governed by the symmetry-protected BIC. A nanohole array etched off from the ${\rm Si_3N_4}$ slab composed of units of four asymmetric nanoholes is employed to excite three strong Fano resonances. Through breaking the in-plane symmetry, the radiation channel between the BIC and the free space can be built, producing two sharp Fano resonances in the transmission. Multipole decomposition and field distribution are carried out as supplement for each other to discuss the excitation of these Fano resonances and indicate the dominant role of toroidal dipole. Then we demonstrate that our design supports polarization-independent responses given by its symmetry. Finally, we evaluate its capacity of light confinement by calculating the enhancement factors of these three modes, which reach the maximum at resonant wavelengths. Our work may facilitate the investigation of polarization-independent resonators with good performance.