## Abstract

We design and numerically investigate the high Q-factor, high modulation depth, and multiple Fano resonances based on a periodical all-dielectric asymmetric nanorod dimer in the near-infrared regime. It is demonstrated that, due to the excitation of the subradiant hybrid modes, five sharp Fano resonances can be achieved by breaking the symmetry of the dimer and can be flexibly tuned by varying the geometrical parameters. All five Fano resonances have a narrow line width, the maximal Q-factor exceeds 9700, and even the minimal Q-factor also reaches about 1090 in magnitude. Particularly, the modulation depth can reach nearly 100%. In addition, the maximal figure of merit reaches 5045. Considering the narrow line-width and significant near-field enhancement, five Fano resonances with large modulation depths in the proposed array are useful for lasing, nonlinear optics, and multiwavelength biosensing.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Engineering high Q-factor responses in metamaterials is generally associated with the low loss rate and large local field enhancement and offer surprisingly rich physics, spanning many areas of research including plasmonic lasers [1], nonlinear optics [2], optical sensors [3,4] and quantum optics [5]. Fano resonance, characterized by asymmetric spectral line profiles, provides an effective way to realize the high Q-factor in metamaterials and has attracted great attention [6–10]. Fano resonances are generally attributed to the interaction between a superradiant (highly radiative) mode and a subradiant (poorly radiative) mode [9,11]. Unfortunately, Fano resonance in plasmonic metamaterials typically suffers from low Q-factor due to the ohmic losses.

All-dielectric metamaterials can eliminate the ohmic losses and provide a promising alternative to achieve higher Q-factor Fano resonance due to the lower radiation loss of series of magnetic responses in the dielectric Fano structures [12–15]. Recently, it has shown that the excited toroidal dipole (TD) resonance in felicitously designed all-dielectric metamaterial can be pictured as radiating fields generated by a solenoid bent into a torus and can be used to achieving a high Q-factor Fano resonance by taking advantage of the weak free-space coupling [16–20]. In addition, the toroidal resonance in metamaterials is highly relevant to the engineering of a type of essentially non-radiating anapole mode [18,21–24], which arises from the destructive interference of the toroidal and electric dipole moments in the far-field at all angles. Due to non-radiating nature and efficient field confinement, the higher Q-factor and larger near-field enhancements can be obtained with excitation of anapole modes. Although nonradiative losses can be weak for dielectric *resonators*, strong radiative losses are still challenges to further enlarge enlarge the Q-factor [25–27]. In addition, breaking the symmetry transversely in the direction perpendicular to a metamaterial effectively controls appearance of the high Q-factor Fano resonances in metasurfaces associated with the bound states in the continuum [25,28,29]. Recently, the collective oscillation mediated by near-field interaction between the unit cells in metamaterials has been proposed to effectively suppress radiation loss [15,30–32]. Due to the sharp surface lattice resonance in array, it has been shown that extremely high Q-factor can be realized for dielectric metasurface based on the weak non-radiative losses and suppressed radiative losses [33]. However, the modulation depths, defined as the transmission intensity differences between the Fano peaks and the Fano dips T_{peak} − T_{dip}, always decrease in the presence of non-radiative losses with the increasing Q-factors [34].

Compared to a single Fano resonance, multiple Fano resonances can be adjusted simultaneously at several different spectral positions and have been reported in the past few years. Liu et al. reported that tunability of the modulation depth of multiple Fano resonance from the plasmonic heptamer clusters [35]. Arash Ahmadivand et al. showed that multiple coil-type Fano resonances in all-dielectric antisymmetric quadrumers [36]. Xia et al. showed that multiple Fano resonances in symmetry breaking silicon gives rise to 3 orders of magnitude enhancement [37]. However, there are two shortcomings in these reported multiple Fano resonances, which make them not suitable in practical applications, i.e., (1) the Q factor in multiple Fano resonances underpinning most devices are limited to rather small values and (2) the modulation depths, that is, one or more modulation depths of the multiple Fano resonances are small in the spectra. So far, there are few studies that can realize more than four Fano resonances with high Q-factors and high modulation depths at the same time.

In this letter, we design the periodic paired nanorods and realize multiple Fano resonances with high Q-factor and large modulation depths by introducing the unique degree of freedom beyond the in-plane symmetry. The maximal Q-factor of multiple Fano resonances exceeds 9700 and even the minimal Q-factor also reaches about 1090 in magnitude. Most importantly, the modulation depth of each Fano resonance reaches nearly 100% in the near-infrared regime. Considering the narrow linewidth with large near-field enhancement, we theoretically demonstrate that the refractive index sensitivity exceeds 361 nm/RIU and the maximal figure of merit (FoM = (*Δλ*/*Δn*)/line width) reaches 5045. Moreover, the spectral positions and modulation depths of the multiple Fano resonances can be flexibly tuned and controlled by varying the geometrical parameters. Five Fano resonances with high Q-factor, large modulation depth and strong field enhancement simultaneously make the array promising for multiwavelength biomedical sensing and nonlinear optics.

## 2. Structures description

The unit cell of proposed paired nanorods array is placed over silica substrate and shown in Figs. 1(a) and 1(b). The lattice constants along the x and y directions are *p _{x}* =

*p*= 670 nm, respectively. The two rods in a unit cell have the same length and width

_{y}*a*, and have the different height

*h*,

_{1}*h*The parameter

_{2}.*δ*= |

*h*–

_{1}*h*| is introduced to specify structural asymmetry of unit cell. The gap between two nanorods is denoted as

_{2}*g*. In our simulations, the proposed array is illuminated by

*y*-polarized (electric field

*E*along the

*y*axis) plane waves, as illustrated in Fig. 1. And the array is immersed in water with refractive index n = 1.33. Numerical simulations are conducted using finite element and finite-difference time-domain methods. In our simulations, the experimentally measured dielectric function is utilized for silicon and silica [38]. The proposed array can be fabricated by the following method: the paired holes are etched with the focused ion beam etching (FIB) in the PMMA (poly (methyl methacrylate)). Differing from the processing method of silicon dimer [39,40], such holes have the different height to introduce the structural asymmetry by controlling the etching process. The nanostructured PMMA was covered with the sufficient thick silicon by the ion beam sputtering coating so that the thick film can form a plane [41]. Then such silicon film is etched via reactive ion etching. Subsequent the silicon dimer is transferred to the silica substrate and the PMMA is removed by a lift-off process in acetone.

## 3. Simulation results and discussions

Figure 2(a) shows the transmission spectrum of a symmetric paired nanorods array with *δ *= 0 nm. The parameters width *a*, height *h _{1}*, gap

*g*, and period

*p*have initially been set to be 280 nm, 565 nm, 35 nm, and 670 nm, respectively. In the symmetric design, it is shown that there are two Fano peaks occur at the wavelengths of 1232.7 nm and 1410.9 nm, marked as the M1 and TD resonances. And the transmission intensity at two resonances reaches nearly 100%. To quantitatively evaluate the contributions of multipoles in forming the resonant responses, the cartesian multipole moment analysis are applied and the radiating powers of the induced electric dipole (

*Py*), magnetic dipole (

*Mx*), toroidal dipole (

*Ty*), electric quadrupole (

*Qyz*) and magnetic quadrupole (

*Mxz*) were calculated according to the the current density ${\boldsymbol J} = -i{\omega}{{\varepsilon }_{0}}{(}{{n}^{2}}{-1){\boldsymbol E}({\boldsymbol r})}$ by integrating spatially distributed fields in nanorods [20]. As shown in Fig. 2(b), in addition to conventional multipole resonances, there is very strong contribution of the toroidal dipole excitation at 1232.7 nm and 1410.9 nm, which is, actually, dominant over the whole spectrum. The scattering of the magnetic quadrupole

*Mxz*shows a similar frequency dependence on the toroidal dipole Ty; however, the radiating intensity of the toroidal dipole

*T**y*is larger than that of the

*Mxz*. We note that the radiating powers of components of the

*Py*and

*Qyz*are lower in comparison with the three mainly contributed multipoles. Once we introduce the asymmetry

*δ*= 40 nm (

*h*= 565 nm and

_{1}*h*= 525 nm), besides the resonances related to the previously discussed M1 and TD resonances at 1222.3 nm and 1406.9 nm, three additional resonances at 1293.5 nm (M2), 1312.4 nm (M3) and 1378.7 nm (M4) appear in the transmitted spectrum, as shown in Fig. 2(c). Breaking the symmetry transversely in the direction perpendicular to a metamaterial effectively control appearance of the high Q-factor Fano resonances. And these are associated with the physics of bound states in the continuum. [42]. Remarkably, such resonances produce the resonant spectral line having the Fano feature with narrow line widths and large modulation depths. In addition, one can see that as the fundamental resonances of the nanorods the M1 and TD modes have redshift with increasing

_{2}*δ*. In order to improve contrast, the spectrum (

*δ*= 0) has also been shown with blue line in Fig. 2(c).

The Fano spectral feature of the proposed array can be reproduced accurately by a Fano model [43]. The transmittance spectrum in Fig. 2(a) can be fitted to a Fano line shape given by: ${\textrm{T}_{\textrm{Fano}}}\textrm{=}{\left|{{\textrm{a}_\textrm{1}}\textrm{+i}{\textrm{a}_\textrm{2}}\textrm{+}\frac{\textrm{b}}{{{\omega -}{{\omega }_\textrm{0}}{+i \gamma }}}} \right|^\textrm{2}},$ where ${\textrm{a}_\textrm{1}}$, ${\textrm{a}_\textrm{2}}$ and $\textrm{b}$ are constant numbers, ${{\omega }_\textrm{0}}$ is the oscillation frequency, and ${\gamma }$ is the damping factor. It can be seen clearly that the analytical derivation can reproduce well the result attained from the simulation, as shown in the inset of Fig. 2(a). Furthermore, the Q-factor of the spectral response can be extracted by such model and calculated by Q = (${{\omega }_{0}}/{2\gamma }$). The extracted Q-factors as a function of the asymmetry parameter *δ* are shown in Fig. 2(d). One can see that the Q-factors of the M1 and TD resonances are almost unaffected by asymmetry parameter *δ*. However, it is evident that the Q-factors of the M2, M3, and M4 resonances depend crucially on the *δ* and increase by decreasing the *δ*, which defines the degree of the introduced asymmetry. When *δ *= 40 nm, the corresponding Q-factors of the M1, M2, M3, M4 and TD resonances reach 2697 (${{\omega }_\textrm{0}}$ = 1.014 eV and ${\gamma}\,$ 0.1880×10^{−3} eV), 1094 (${{\omega }_\textrm{0}}$ = 0.9587 eV and ${\gamma }$ = 0.4380×10^{−3} eV), 4723 (${{\omega }_\textrm{0}}$ = 0.9447 eV and ${\gamma}$ 0.1000×10^{−3} eV), 6575 (${{\omega }_\textrm{0}}$ = 0.8993 eV and ${\gamma }$ = 0.0550×10^{−3} eV), and 1546 (${{\omega }_\textrm{0}}$ = 0.8815 eV and ${\gamma }$ = 0.2850×10^{−3} eV), respectively. In particular, the Q-factors of M3 and M4 resonances reach 8277 and 9766 when *δ *= 25 nm, respectively. Further, it can be seen that multiple Fano resonances all have large modulation depths. The modulation depths reach nearly 100% at M1 and TD resonances, while at M2, M3 and M4 the modulation depths also reaches 92%, 69% and 93%, respectively. Comparing with the Q factors in the range of a few hundreds in ref. [37], the quality factor has been increased by at least 35 times, and the modulation depth is also significantly better than the modulation depth in ref. [37]. Therefore, breaking symmetry can effectively excite multiple Fano resonances with the high Q-factors and high modulation depths while keeping the remaining dimensions of the dimers unchanged.

In order to further illustrate the subradiant properties related to multiple Fano resonances, the magnetic field enhancement |*H*/*H _{0}*| as well as the field vector distributions on different center cross sections of the asymmetric array (

*δ*= 40 nm) are represented in Fig. 3. One can see that the magnetic field can be firmly constrained within the nanorods and no radiation energy transmits outward at 1406.9 nm, where electric field in the x − y plane indicate two peculiar reversed swirls and the magnetic field in the x − z plane forms a swirl, indicating a typical TD feature [17,33,44,45], as shown in Fig. 3. The electric field at 1312.4 nm and 1378.7 nm forms two swirls rotating in the same direction in the

*x*−

*y*plane, while the magnetic field in the two nanorods has the same direction and is almost linearly polarized along the

*z*axis in the

*x*−

*z*plane. As a result, the resonances at 1312.4 nm and 1378.7 nm can radiate like two parallel magnetic dipoles oriented along the

*z*axis. According to the magnetic field vector distributions at 1293.5 nm, the magnetic field in the two nanorods oriented along the

*x*axis and have opposite direction in the

*x*−

*y*plane, and the magnetic field in single nanorod forms two reversed swirls in the

*x*−

*z*plane, indicating a magnetic quadrupole resonance at 1293.5 nm. The magnetic field of a single nanorod at 1222.3 nm have opposite direction oriented along the

*x*axis in the

*x*−

*y*plane or the

*z*axis in the

*x*−

*z*plane, as shown in Fig. 3(a) and 3(b), which proves the presence of a magnetic quadrupole resonance in a single nanorod and provides a coupling pathway between magnetic quadrupole resonances in two nanorods through magnetic field interaction, ultimately leading to Fano resonance at 1222.3 nm. In addition, thanks to the high quality factors of Fano resonances, the maximum magnetic field can be enhanced by more than 75 times, comparable to plasmonic nanostructures.

In the following, we investigate transmission spectra of the asymmetric paired nanorods array at different geometric parameters shown in Fig. 4. Other parameters are the same as that used in Fig. 2(c). As shown in Figs. 4, the M1 and TD resonances are virtually unaffected by the parameters *p*, *g* and *δ*. In Fig. 4(a), the M2 and M3 are not sensitive to the parameters *p* and only generate a slight redshift, however, the M4 is sensitive to the array periodicity *p*. In Fig. 4(b), it can be seen clearly that the all Fano peaks are very sensitive to width *a* and experience a significant redshift as the width increases, altogether resulting in the easily traceable tuning characteristics of Fano resonances. It can be found that the reduced interaction with the increasing gap *g* causes a slight redshift of the M2 and M3, as shown in Fig. 4(c), while the M4 can be readily tuned and undergoes an obvious blueshift. Meanwhile, the M4 has a relatively large redshift compared to the M3 and is enhanced with the weaken interaction. As shown in Fig. 4(d), the resonant frequencies of Fano resonances remain nearly unchanged with increasing asymmetry because the symmetry breaking along the *z* axis of the dimer does not change the excitation energy of the resonances. Meanwhile, the line-width of the M2 is clearly broadened with increasing asymmetry, and the corresponding Q-factor dramatically decreases. The reduction of Q-factor with increasing *δ* is due to the increase of radiation loss.

Due to high Q-factor and large field enhancement of multiple Fano resonances, one can expect that the proposed nanorod dimer array can be used for sensing applications. The spectral shift per refractive index (RI) unit (RIU) and FoM are two good indicators for the local sensitivity. Figure 5(a) shows the shift in transmission spectra of the asymmetric array (*δ *= 40 nm), whose surface is covered by water with a refractive index from 1.32 to 1.34. It can be seen that a clear redshift of the multiple Fano resonances is visible as the refractive index increases. The spectral shift sensitivity of the M1, M2, M3, M4 and TD resonances reaches 117 nm/RIU, 131 nm/RIU, 361 nm/RIU, 344 nm/RIU and 162 nm/RIU, respectively. The extracted line width from the Fano model is 0.2331 nm, 0.5431 nm, 0.1240 nm, 0.0682 nm and 0.3534 nm, and the corresponding FoM reaches 502, 241, 2912, 5045 and 459, respectively. Although the sensitivity 361 nm/RIU is even less than sensitivity demonstrated experimentally for plasmonic sensors, the largest FoM exceeds 5000 due to the high quality factor [46]. The differences in the sensitivity of multiple Fano resonances are mainly due to the different near-field distributions. Thus, the proposed array offers a good platform to design high-performance multichannel bio-sensing devices.

## 4. Conclusion

In conclusion, we have shown that multiple Fano resonances with high Q-factor and large modulation depth can be realized by a periodical all-dielectric asymmetric nanorod dimer in the near-infrared regime. Due to the formation of the subradiant hybrid modes by breaking the symmetry of dimer in array, five Fano resonances (M1, M2, M3, M4 and TD) all have the high Q-factors, the maximal Q-factor exceeds 9700 and even the minimal Q-factor also reaches 1090. The higher Q-factor can be obtained by adjusting the asymmetry. And the modulation depths reach nearly 100% at M1 and TD, while at M2, M3 and M4 the modulation depths also reach 92%, 69% and 93%, respectively. Moreover, the Q-factors and modulation depths can be flexibly tuned by varying the geometrical parameters. In addition, five Fano resonances with the strongly enhanced magnetic field make the proposed array suitable for refractive index sensing. The bulk refractive index sensitivity reaches 361 nm/RIU, the maximal FoM reaches 5045 and can be further improved by reducing the asymmetry. Such sharp multiple Fano resonances, high Q-factors, large modulation depths and FoM values supported by the proposed design make it more adaptable to numerous potential applications ranging from nonlinear optics and multiwavelength biosensing to the realization of new types of optical modulation and low-loss slow-light devices.

## Funding

Dalian Polytechnic University (71600160); National Natural Science Foundation of China (11647102, 12785604).

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