## Abstract

A triple-band perfect plasmonic metamaterial absorber based on a metal/insulator/metal (MIM) structure is designed. A new freedom through tuning the thicknesses of each ring structures is introduced to realize a quasi-three-dimensional perfect absorber at three extinction wavelengths by using the finite difference time domain method. The physical machine is explained by the time domain field analyses and the coupled mode theory. The characteristics of the absorber make our proposed strategy applicable for the design of more general multiband and broadband perfect absorbers. In addition, these perfect absorbing metamaterials are found to exhibit excellent performance in refractive index sensing.

© 2016 Optical Society of America

## 1. Introduction

Due to the unique abilities of confining light to the nanometer-scale and enhancing electromagnetic fields, the plasmonic nanostructures have been widely used in light absorption in visible and infrared wavelength range [1]. Among all these structures, three layered metal/dielectric/metal (MIM) nanostructure absorbers have attracted particular interest due to their excellent performance to absorb almost all incident radiations [2–7]. In this structure, each metal layer provides an electrical response and the surface currents excited by the electrical response are found to be anti-parallel in the top and bottom metal layers thus forming the excellent coupled mode which favors the perfect absorption [8]. Although with high absorptivity, most of the existing MIM absorbers are typically optimized only for single-band which limits their potential applications in spectroscopic detection and other multi-color optical applications [9, 10]. For these multiband or broadband applications, the on-demand design of multiple resonances with excellent absorbability is highly desirable. So far, a few works have addressed the realization of two-color or three-color perfect absorption [11–17]. These works can be categorized into two types: One is based on supercell design [11, 13–17], in which each cell of the supercell structure is responsible for one targeted resonance meanwhile it is optimized for high absorption, and as a whole structure, the supercell exhibits multiple resonances with extremely high absorptions. Nevertheless, the very low area filling factor of each sub-cell resonance structure over the whole super-cell requires that each resonance structure should be somewhat similar in size so that their resonances are close to each other. What's more, the supercell approach requires larger computational capacity. The other type is based on higher order resonances [12], in which higher-frequency resonance is realized in accompany with the fundamental one. Unfortunately, in this case, it is challenging to realize perfect absorption for multiple order of resonances simultaneously. Furthermore, the design of both these reported approaches suffer from a common challenge that the field profiles of the multiple resonances overlap largely with each other. As a result, independent adjustment of each resonance remains very difficult and often requires sophisticated simulation trials. Therefore, it is anticipated that a more straightforward and generalized strategy can be developed to guide the spectroscopic design of multiband metamaterials.

In this work, we numerically design a MIM metamaterial nanostructure consisting of three nested closed-rings on a thin dielectric spacer with a continuous bottom Au mirror film to realize a triple-band perfect absorber (TBPA). Each resonance of our proposed novel quasi-3D has excellent correlation with associated geometrical parameters (mainly ring height) and excellent tunability for each resonant frequency. Though only tri-band perfect absorption is demonstrated here, our approach is also applicable for more general multicolor (>3) perfect absorption, and in wider frequency region, viz, infrared range. For practical applications, our approach can be combined with reported work such as the super-cell structure and this combination will bring tremendous freedom in realizing applicable perfect absorbing metamaterials. In addition, we make use of the strong enhancement of the electromagnetic field in the gap between the ring structures and examine the refractive index infrared sensing performance. It is found that excellent performance can be realized for all three resonances and the maximum figure-of-merit (FoM) can reach >2 times higher than the planar MIM structure reported previously in the same range [18].

## 2. Structure design and simulations

The schematic of the unit cell of the proposed triple-band infrared absorber is shown in Fig. 1(a). Three nested rings metamaterial array is periodically arranged on the metallic ground plane with a periodicity of $\text{P}={\text{P}}_{\text{x}}={\text{P}}_{\text{y}}=3.2\text{\mu m}$. The width of the ring and the gap between rings are both fixed at w$=$g$=$100 nm. The structure consists of three nested rings with optimized length and height, including outer ring (L_{1}$=1.8\text{\mu m}$, H_{1}$=0.11\text{\mu m}$), middle ring (L_{2}$=1.4\text{\mu m}$, H_{2}$=0.26\text{\mu m}$) and inner ring (L_{3}$=1\text{\mu m}$, H_{3}$=0.56\text{\mu m}$). The dielectric layer with a thickness of 400 nm is embedded between the metal-ring array and the 100 nm thickness Au. All the parameters of the rings are carefully optimized to realize multiple perfect absorption. The whole structures are illuminated by a plane wave impacting normally with the E-field polarized along the x-axis. All the simulation results including the absorbance and reflectance spectra of the structure are calculated by using the finite difference time domain (FDTD) method. The frequency-dependent complex dielectric constants of gold are chosen to be typical Drude model with a plasma frequency of${\text{\omega}}_{\text{p}}=1.37\times {10}^{16}\text{rad}/\text{s}$ and a damping constant of ${\text{\gamma}}_{\text{c}}=39.47\times {10}^{12}\text{rad}/\text{s}$ [19]. The relative permittivity of the dielectric spacer is taken to be ${\epsilon}_{d}=9.4$.

The bottom Au is a continuous metallic layer with the thickness much larger than the skin depth in the infrared region. Therefore, this metal layer will block all incident light leading the transmissivity $T(\omega )$ nearly to zero in the working region of the TBPA, as shown in Fig. 1(b). Another function of the thick gold mirror is to interact wsith the upper rings to form electric and magnetic dipoles which couple the incident electromagnetic energy inside the structure [20, 21]. Thus the absorptivity $A(\omega )$ could be calculated by Eq. (1), where $R(\omega )$ represent the reflectivity from the TBPA.

## 3. Simulation results and discussion

The Fig. 1(b) presents the typical absorption spectra which has been optimized to realize TBPA. There are three distinct absorption peaks located at$7.3\text{\mu m}$, $11.2\text{\mu m}$ and $16.4\text{\mu m}$ with the absorptivity of 99.99%, 99.91% and 99.99%, respectively, implying perfect absorption at three wavelengths. In order to clarify the nature of these three absorption peaks, we simulated the absorption spectra of three different single ring structures. The optimized geometries of the TBPA in this work are: ring 1: L_{1}$=1.8\text{\mu m}$, H_{1}$=0.11\text{\mu m}$; ring 2: L_{2}$=1.4\text{\mu m}$, H_{2}$=0.26\text{\mu m}$; ring 3: L_{3}$=1\text{\mu m}$, H_{3}$=0.56\text{\mu m}$. As we can see from Fig. 2(a), the orange, blue and green curves represent the absorptivity of the unit cell contains solely outer ring (ring 1), middle ring (ring 2) and inner ring (ring 3) structure, with the absorption peaks located at $=16.4\text{\mu m}$ (peak 1), $=12\text{\mu m}$ (peak 2) and $8.5\text{\mu m}$ (peak 3). This implies that the ring shape resonance, or termed as local surface plasmon, can be easily tuned by adjusting its length. When the ring 1 is fixed at certain height, approximately the corresponding resonant frequency of peak 1 (${f}_{P\text{eak}\text{1}}$) is inversely proportional to the side length L_{1} following Eq. (2) [22],

The Fig. 3 depicts the electric field intensity and the induced surface current distribution maps in the near-field (at the cross-section of the surface of the dielectric and the cross-section across the ring center along x-axis) at three extinction peaks. From Fig. 3, it is evident that the electric field profiles of each resonance have good correspondence with each ring structure. In the dielectric spacer region, the maximum field for each resonance appears consistently underneath the corresponding ring structure. While in gap regions between neighboring rings, the electric field is dramatically enhanced for the associated resonance. It is interesting to note that the strong electric field enhancement of one resonance occurs only for the gap between associated ring and its neighboring outer ring structure. This agrees well with the observed peak shift in Fig. 2. A straightforward qualitative understanding to this is that the inner electric field is screened by the metal ring itself while the electric field skin effect makes its coupling to outer ring being dominant. In Figs. 3(g)-3(i), the magnetic field profiles demonstrate clearly how the field surrounds the ring structure. We can find large magnetic field shows up consistently in the gap region and the current distributions in Figs. 3(j)-3(l) also show antiparallel currents in neighboring rings. This implies that the eigenmodes of peak 2 and peak 3 originate mainly from hybrid oscillations of neighboring rings (L_{2} and L_{3}) due to their strong capacitive coupling and mutual inductance, but not from individual ring resonator.

It’s noteworthy to point out that, the near-field enhancement is extremely strong in the vicinity of the rings and particularly in the small gap region. Quantitatively strong enhancement with peak 1: ${E}_{max}{}^{2}/{E}_{0}^{2}=500$, peak 2: ${E}_{max}{}^{2}/{E}_{0}^{2}=1000$, peak 3: ${E}_{max}{}^{2}/{E}_{0}^{2}=1500$) can be obtained. Besides, the presence of the electric field hot spots in the air side (not in the dielectric spacer region as in Ref [24].) suggests its advantage in refractive index sensing as will be seen as following.

In order to demonstrate the essential role of ring heights for the perfect absorption, we show the systematic data with varying the representative middle ring height (H_{2}), without losing generality. Figure 4(a) depicts the absorption spectra with increasing H_{2} from 0.1 $\text{\mu m}$ to 0.7 $\text{\mu m}$ while keeping H_{1} = 0.11 $\text{\mu m}$ and H_{3} = 0.56 $\text{\mu m}$. It is obvious that the absorption at both peak 2 and peak 3 first increase and then decrease with increasing H_{2} and, interestingly, there exists an optimum and common height value for ring 2 to realize maximum absorption at both peak wavelengths. (peak 1 is almost unaffected by H_{2} and therefore omitted in Fig. 4(a)).

Noting that perfect absorbing resonant modes are orthogonal to each other and well-separated in frequencies and spatially, we can analyze the spectrum with the couple mode theory (CMT) [25, 26]. According to CMT, the (complex) reflection coefficient $r$ at frequencies around a particular resonance (${\omega}_{0}$) of such model is

where ${\tau}_{a}$ and ${\tau}_{r}$ denote the lifetimes of the resonance due to absorption inside theresonator structure and radiation leakage to the far field, respectively. Indeed, we find that the absorbing bands exhibit excellent Lorentzian line shapes and can be well fitted with Eq. (3). This validates the CMT being implemented here with Eq. (3) which requires single mode [27, 28]. Because the insulator is lossless in our structure, the absorption inside the resonator structure is only from the intrinsic ohmic loss of the metal. For this model, the physical property is fully determined by two dimensionless parameters the absorption factor (${\mathcal{Q}}_{a}={\omega}_{0}{\tau}_{a}/2$) and radiative quality factor${\mathcal{Q}}_{r}={\omega}_{0}{\tau}_{a}/2$) of the system. A simple calculation reveals that most recently reported behaviors of MIM structures [29–31] can be easily reproduced by tuning these two parameters. Aiming to better understand the tuning machine of these two parameters, we also calculated them in three single ring structure and the nested-ring structure at same height of 110 nm (including two rings and three rings). As for the single ring structure (inner ring and middle ring), the calculate result shows ${\mathcal{Q}}_{r}>{\mathcal{Q}}_{a}$; as for the nested-ring structure, the peak 2 and peak 3 are both exhibit ${\mathcal{Q}}_{r}>{\mathcal{Q}}_{a}$. This is caused by the coupling with the outer ring.Which means that, for the same height nested-ring structure, the absorption inside the resonator structure is higher than the radiation leakage which limit to achieve the perfect absorption in high frequency resonance, thus breaking the balance between them. This explanation can also be verified by the simulation and calculation results. As shown in Figs. 4(b) and 4(d), with the H_{2}increasing from 0.22$\text{\mu m}$ to 0.3 $\text{\mu m}$, there exists a threshold condition (H

_{2}= 0.26 $\text{\mu m}$) lead to the match of ohmic loss and radiation loss, i.e., ${\mathcal{Q}}_{r}>{\mathcal{Q}}_{a}$ for both peaks. With increasing H

_{2}in this region, peak 3 undergoes a transition from overdumping region (${\mathcal{Q}}_{r}>{\mathcal{Q}}_{a}$) to underdumping one (${\mathcal{Q}}_{r}>{\mathcal{Q}}_{a}$), while for peak 2, it transited from underdamping to overdamping. From the results presented in Fig. 4, we conclude that both peaks reach perfect absorption condition when crossing the critical dumping condition (${\mathcal{Q}}_{r}>{\mathcal{Q}}_{a}$). From Figs. 4(b) and 4(d), it can be seen that ${\mathcal{Q}}_{r}$ can be largely tuned by sweeping H

_{2}. With increasing H

_{2}, ${\mathcal{Q}}_{r}$ for Peak 3 decreases while for peak 2 its increases, both the tendency suggesting the smaller height difference between neighboring rings will lead to larger radiation loss change for the associated resonance. So, we can only speculate that with well optimizing the height of each ring, ${\mathcal{Q}}_{r}$ and ${\mathcal{Q}}_{a}$ can be well matched to achieve perfect absorption.

We can also distinguish the overdamping and underdamping regions by checking the variation range $\Delta \varphi $ of the reflection phase ($\varphi $) in the absorption frequency domain. As depicted in Figs. 4(c) and 4(e), for both two peaks, in the underdamping region, $\Delta \varphi $ can cover the full 360° range with $\varphi $ undergoing a continuous −180° to 180° variation as frequency crosses the resonance. In contrast, in the overdamping region, the variation of $\varphi $ only occupies a small range less than 180°. The existence of the phase singularity between overdamping and underdamping regions make sure the perfect absorption being realized in our triple-ring structure. The coincidence that both peaks reach perfect absorption at same middle ring height (H_{2}) suggests that our proposed method is highly valuable for multiband perfect absorber design. This is made possible because heights of ring 1 and ring 3 are optimized beforehand here and practically the spectroscopic design of multiband perfectly absorbing metamaterials can be realized by optimizating the height of each ring from the outer to the inner one by one.

## 4. Infrared sensing

The sensitivity of refractive index sensors depends critically on the intensity of the local electric field and the overlap of the hot spots with the analyte [7, 18, 29]. The advantages of our proposed structure fulfill this criteria while immersed in into target sensing media and therefore a high sensitivity can be expected. With the lossless analyte of different refractive index deposited on the top surface of the TBPA, as shown in Fig. 5(a), the top of the analyte layer is assumed to be 100 nm higher than top plane of the inner ring, with a total thickness of $0.66\text{\mu m}$.

As shown in Fig. 5(b), the absorption spectra of the metasurface when coating analyte layer with different refractive index (n), varying from 1 to 1.7, all the three resonant wavelengths are highly sensitive, although different plasmonic modes lead to different resonant wavelengths response. The simulated data in Fig. 5(b) represents the resonance wavelengths shift from $7.3\text{\mu m}$, $11.2\text{\mu m}$ and $16.4\text{\mu m}$ in case of n $=$ 1 to $10.1\text{\mu m}$, $14.3\text{\mu m}$ and $19.3\text{\mu m}$ in case of n $=$1.7 for peak 1 to peak 3, respectively. The reason for these red-shift can be explained from an effective capacitance increase, since increasing the refractive index of the analyte effectively increase the capacitance of resonant structure. It is obviously to see that a substantial movement in the peak position occurs for all three peaks. Meanwhile the absorption intensities drop especially for peak 3, partially due to the strongest electrical field enhancement.

In Fig. 6, we numerically calculate the peak wavelength, the full width of half maximum (FWHM), the sensitivity, and the FoM for the identical range of variation in the refractive index shown as the plots in Fig. 5(b). The sensitivities of the resonator calculated, defined as a function of the change in resonant wavelength over the change in refractive index in units of $\text{\mu m/RIU}$.The sensitivity is then divided by the FWHM of the absorption of the resonant peak to determine the FoM of the resonator array, shown as the Eq. (4).

It is noteworthy that, even the extinction peaks generated from different resonant mode as we discussed above, the tendency of all the calculated parameters almost vary samely of a linear function when increasing the refractive index as Fig. 5, as the refractive index increase, the peak wavelength, the FWHM, the sensitivity is increase while the FoM is decrease for all the three resonant peaks. Even the sensitivity of three resonant peaks has little difference, the peak 1 exhibit the highest FoM (about 5.78) for localized surface plasmon resonance (LSPR) sensor due to the lowest FWHM (about 0.33$\text{\mu m}$) compared with other Peaks. We can infer that, if we keep adding inner ring structure, the added resonance can achieve a better FoM even than peak 1. And the sensitivity improvement can be mainly assigned to an increase of the sensitivity in the metal region between rings.

Moreover, unlike conventional MIM structure which contains a flat top metallic layer, our structure has an bigger overlap volume of the hot spots and the analyte, which is accessible to the environmental change, in turn making the TBPA structure additional advantage for index sensing application. As a result, our proposed structure exhibit a FoM enhancement suggests that this plasmonic mode is more suitable for sensing. So despite its comparably narrow plasmon resonance, this TBPA could be used not only for very sensitive refractive sensing, but also for enhancing most monolayer sensitivity, rendering it a highly desirable structure.

## 5. Summary

In conclusion, a triple-band infrared absorber has been designed by utilizing a quasi-3D MIM nanostructure for highly sensitive refractive index application. By tuning the heights of the rings, nearly 100% absorption peaks at three resonance frequencies in infrared region can be achieved. CMT analyses indicate that changing the heights of the rings can balance the radiative and intrinsic Ohmic losses of the metamaterial structure and eventually lead to tri-band critical dumping and perfect absorption. The small-gap-induced strong electrical field enhancement, refractive index dependence, linear response, high sensitivity, and most attractively, an open cavity that is accessible to the external environment, make the TBPA an attractive refractive index sensor. A highest FoM of 5.78 is achieved for the refractive index sensing. Our work provide valuable guidance for the future developments of more advanced metamaterial absorber based devices for many applications, such as chemical sensing, field-enhanced spectroscopy and photonic applications.

## Funding

National Natural Science Foundation of China (NSFC) (No. 11427807, 11304040, 11634012, 11674070).

## References and links

**1. **J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. **9**(3), 193–204 (2010). [CrossRef] [PubMed]

**2. **J. Le Perchec, Y. Desieres, N. Rochat, and R. Espiau de Lamaestre, “Subwavelength optical absorber with an integrated photon sorter,” Appl. Phys. Lett. **100**(11), 113305 (2012). [CrossRef]

**3. **S. Chen, H. Cheng, H. Yang, J. Li, X. Duan, C. Gu, and J. Tian, “Polarization insensitive and omnidirectional broadband near perfect planar metamaterial absorber in the near infrared regime,” Appl. Phys. Lett. **99**(25), 253104 (2011). [CrossRef]

**4. **K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. **2**, 517 (2011). [CrossRef] [PubMed]

**5. **J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, “High performance optical absorber based on a plasmonic metamaterial,” Appl. Phys. Lett. **96**(25), 251104 (2010). [CrossRef]

**6. **J. Hao, L. Zhou, and M. Qiu, “Nearly total absorption of light and heat generation by plasmonic metamaterials,” Phys. Rev. B **83**(16), 165107 (2011). [CrossRef]

**7. **N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. **10**(7), 2342–2348 (2010). [CrossRef] [PubMed]

**8. **Y. Cui, J. Xu, K. Hung Fung, Y. Jin, A. Kumar, S. He, and N. X. Fang, “A thin film broadband absorber based on multi-sized nanoantennas,” Appl. Phys. Lett. **99**(25), 253101 (2011). [CrossRef]

**9. **G. Huang, J. Yang, P. Bhattacharya, G. Ariyawansa, and A. G. U. Perera, “A multicolor quantum dot intersublevel detector with photoresponse in the terahertz range,” Appl. Phys. Lett. **92**(1), 011117 (2008). [CrossRef]

**10. **A. J. Schain, R. A. Hill, and J. Grutzendler, “Label-free in vivo imaging of myelinated axons in health and disease with spectral confocal reflectance microscopy,” Nat. Med. **20**(4), 443–449 (2014). [CrossRef] [PubMed]

**11. **C. Koechlin, P. Bouchon, F. Pardo, J. Jaeck, X. Lafosse, J. L. Pelouard, and R. Haïdar, “Total routing and absorption of photons in dual color plasmonic antennas,” Appl. Phys. Lett. **99**(24), 241104 (2011). [CrossRef]

**12. **F. Mao, J. Xie, S. Xiao, S. Komiyama, W. Lu, L. Zhou, and Z. An, “Plasmonic light harvesting for multicolor infrared thermal detection,” Opt. Express **21**(1), 295–304 (2013). [CrossRef] [PubMed]

**13. **X. Liu, T. Tyler, T. Starr, A. F. Starr, N. M. Jokerst, and W. J. Padilla, “Taming the blackbody with infrared metamaterials as selective thermal emitters,” Phys. Rev. Lett. **107**(4), 045901 (2011). [CrossRef] [PubMed]

**14. **B. Zhang, Y. Zhao, Q. Hao, B. Kiraly, I. C. Khoo, S. Chen, and T. J. Huang, “Polarization-independent dual-band infrared perfect absorber based on a metal-dielectric-metal elliptical nanodisk array,” Opt. Express **19**(16), 15221–15228 (2011). [CrossRef] [PubMed]

**15. **J. Hendrickson, J. Guo, B. Zhang, W. Buchwald, and R. Soref, “Wideband perfect light absorber at midwave infrared using multiplexed metal structures,” Opt. Lett. **37**(3), 371–373 (2012). [CrossRef] [PubMed]

**16. **Z. H. Jiang, S. Yun, F. Toor, D. H. Werner, and T. S. Mayer, “Conformal dual-band near-perfectly absorbing mid-infrared metamaterial coating,” ACS Nano **5**(6), 4641–4647 (2011). [CrossRef] [PubMed]

**17. **P. Bouchon, C. Koechlin, F. Pardo, R. Haïdar, and J. L. Pelouard, “Wideband omnidirectional infrared absorber with a patchwork of plasmonic nanoantennas,” Opt. Lett. **37**(6), 1038–1040 (2012). [CrossRef] [PubMed]

**18. **L. Cong, S. Tan, R. Yahiaoui, F. Yan, W. Zhang, and R. Singh, “Experimental demonstration of ultrasensitive sensing with terahertz metamaterial absorbers: A comparison with the metasurfaces,” Appl. Phys. Lett. **106**(3), 031107 (2015). [CrossRef]

**19. **M. Amin, M. Farhat, and H. Baǧcı, “A dynamically reconfigurable Fano metamaterial through graphene tuning for switching and sensing applications,” Sci. Rep. **3**, 2105 (2013). [CrossRef] [PubMed]

**20. **H.-T. Chen, “Interference theory of metamaterial perfect absorbers,” Opt. Express **20**(7), 7165–7172 (2012). [CrossRef] [PubMed]

**21. **C. Wu, B. Neuner, G. Shvets, J. John, A. Milder, B. Zollars, and S. Savoy, “Large-area wide-angle spectrally selective plasmonic absorber,” Phys. Rev. B **84**(7), 075102 (2011). [CrossRef]

**22. **Y. Ye, Yi Jin, and S. He, “Omnidirectional polarization-insensitive and broadband thin absorber in the terahertz regime,” J. Opt. Soc. Am. B. **27**(3), 498–504 (2010).

**23. **X. Shen, T. J. Cui, J. Zhao, H. F. Ma, W. X. Jiang, and H. Li, “Polarization-independent wide-angle triple-band metamaterial absorber,” Opt. Express **19**(10), 9401–9407 (2011). [CrossRef] [PubMed]

**24. **R. Adato, A. Artar, S. Erramilli, and H. Altug, “Engineered absorption enhancement and induced transparency in coupled molecular and plasmonic resonator systems,” Nano Lett. **13**(6), 2584–2591 (2013). [CrossRef] [PubMed]

**25. **Z. Miao, Q. Wu, X. Li, Q. He, K. Ding, Z. An, Y. Zhang, and L. Zhou, “Widely tunable terahertz phase modulation with gate-controlled graphene metasurfaces,” Phys. Rev. X **5**(15), 2160–3308 (2015).

**26. **C. Qu, S. Ma, J. Hao, M. Qiu, X. Li, S. Xiao, Z. Miao, N. Dai, Q. He, S. Sun, and L. Zhou, “Tailor the Functionalities of Metasurfaces Based on a Complete Phase Diagram,” Phys. Rev. Lett. **115**(23), 235503 (2015). [CrossRef] [PubMed]

**27. **S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A **20**(3), 569–572 (2003). [CrossRef] [PubMed]

**28. **N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. **8**(9), 758–762 (2009). [CrossRef] [PubMed]

**29. **W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE. J. Quantium. Elect. **40**(10), 1511–1518 (2004). [CrossRef]

**30. **S. Collin, “Nanostructure arrays in free-space: optical properties and applications,” Rep. Prog. Phys. **77**(12), 126402 (2014). [CrossRef] [PubMed]

**31. **Y. Shen, J. Zhou, T. Liu, Y. Tao, R. Jiang, M. Liu, G. Xiao, J. Zhu, Z. K. Zhou, X. Wang, C. Jin, and J. Wang, “Plasmonic gold mushroom arrays with refractive index sensing figures of merit approaching the theoretical limit,” Nat. Commun. **4**, 2381 (2013). [CrossRef] [PubMed]