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Abstract
With the rapid developments in compact devices, the multi-function and reconfigurability of nanostructures are highly appreciated, while still very challenging. A majority of devices are usually mono-functional or hard to switch between different functions in one design. In this paper, we proposed graphene-wrapped core-shell nanowires to realize real-time reconfigurable sensors and nanoantenna by tuning the Fermi energies of graphene layers at the surfaces of core and shell, respectively. Owing to the electromagnetic coupling between the two graphene layer, two corresponding Fano resonances of scattering can arise in the Terahertz spectrum, which arises from the interference of bright modes and dark modes. Around the Fano resonances, the scattering can be considerably resonant (as an antenna) or suppressed (as a sensor). Interestingly, the field distributions are distinct at the suppressed scattering states for the two Fano resonances. The presented reconfigurable nanostructures may offer promising potentials for integrated and multi-functional electromagnetic control such as dynamic sensing and emission.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The asymmetric Fano resonance first discovered by Ugo Fano in the quantum mechanics study of the self-ionization state of atoms in 1961 [1] has received extensive attention in recent years due to its typical spectral characteristics. Fanno resonance is caused by the constructive and destructive interference with continuous spectrum and narrow discrete resonance [2–5]. Recently, researchers have been designing many nanostructures to generate Fano profiles facilitated by their potential applications and the progress of modern technology of micro-fabrication. According to the current research results, there are mainly several nanostructures, such as broken symmetry nanostructures [6], clusters of nanoparticles [1] and metal nanoparticles array [7]. However, these structures are limited in dynamic modulating Fano resonance.
Due to the inherent sensitivity of Fano resonance, we can consider designing sensors and nanoantennas. Then we propose an effective and convenient nanostructure for modulating Fano resonance: the dielectric-graphene-dielectric-graphene structure. It is a core-shell structure consisted of two layers of medium and two layers of graphene. Graphene is a two-dimensional atomic thin carbon material with unique and unique electro-optical properties. And it has attracted enormous research attention in recent years [8]. In far-infrared and THz region of frequencies, graphene has the characteristics of optoelectronic devices, and it can form the surface current at low loss rates [3]. According to this characteristic of graphene, the frequency range we are interested in is the terahertz region. There are two resonant and two cloaking modes in this frequency region. And each pair of cloaking mode and resonant mode would be coupled at one frequency and then a narrowband Fano-like response is obtained when the aspect ratio is very small or large. We find that each layer of graphene dominates one pair of resonant and cloaking mode. What's more, graphene can be used to dynamically control Fano response in plasmonic structures at the near-infrared region of frequencies. And it can alter the position and lifetime of Fano resonances due to the tunability of its plasmons enhances the interaction between sub-radiation modes [9,10]. Y. C. Jun et al. have proposed dielectric resonators, which are made of semiconductor cylinder arrays and block pairs. And they find that this system could significantly tune the Fano resonances spectrum [11]. This property provides a great way to modulate Fano resonance for us, that is, we can consider applying applied voltage to manipulate the Fermi energy of the two layers of graphene to realize the tunable Fano-like response. To understand the mechanisms of Fano-like response better [4], we show the direction and magnitude of the near-field energy flux around Fano resonance. And we find that the near-field energy flux can be reversed in the vicinity of Fano resonance. In addition, we find that the local fields at two cloaking frequencies are different. Then we propose a new scheme to design the cloaking sensor and energy concentrator. For this device, some researchers have proposed implementer scheme [12–15]. However, the scheme they proposed is stringent on the size of the structure. The size of structure has to be continuously changed to achieve the effect of cloaking sensor and energy concentrator. While it is still challenging to be entirely accurate for the size of the nanostructure, and changing its size optionally is even harder. The scheme we proposed avoids this problem. We can manipulate the Fermi energy of the two layers of graphene to realize the switch between two cloaking states without changing other parameters of structure. Then the switch between functions of cloaking sensor and energy concentrator is realized.
2. Theoretical model and methods
2.1 Near-field and far-field behavior based on full-wave electromagnetic scattering theory
Consider the geometry shown in Fig. 1. A dielectric-graphene-dielectric-graphene multilayer core-shell nanostructure is illuminated by the TM-polarized plane wave, which is surrounded by free space. In this model, the radius of the dielectric core is $a$ and the relative permittivity constant is ${\varepsilon _c}$, the thickness of middle dielectric layers is $b - a$ and the relative permittivity constant is ${\varepsilon _s}$.
Fig. 1. (a) Schematic of dielectric-graphene-dielectric-graphene multilayer nano-shells structure. The radii of the inner and outer cylindrical are $a$ and $b$, respectively. (b) The cross-sectional view of Fig. 1(a). ${\varepsilon _c}$, ${\varepsilon _s}$ and ${\varepsilon _h}$ are the relative permittivity’s of the inner, outer dielectric layers and background medium, respectively. The chemical potential of the inner and outer graphene layers can be manipulated through applied voltages ${V_{g1}}$ and ${V_{g2}}$. The plane wave is polarized (electric field) along $y$ axis and is propagating along $x$ axis.
The incident TM-polarized plane wave along the $x$ axis, where the magnetic field is perpendicular to the x-y plane, and the electric field is parallel to $y$ axis. By applying the separation of variables, we obtain the governing equation for the magnetic field $({H_z})$ by assuming the temporal dependence ${e^{ - i\omega t}}$,
For simplicity, we assume the nanostructure to be nonmagnetic, i.e., the relative permeability of the medium is ${\mu _c} = {\mu _s} = {\mu _h} = 1$. The simplified form of graphene linear conductivity is ${\sigma _j} = i{e^2}{E_{Fj}}/\pi {\hbar ^2}(\omega + i/{\tau _j})$ [16] ($j = 1,2$ represent the Fermi energy of the inner and outer graphene layers). The scattering coefficients can be simplified as
In addition, coefficients ${A_m}$, ${B_m}$ and ${C_m}$ have similar forms. The scattering efficiencies [17] are expressed as
2.2 Derivations in the quasistatic limit
In our work, the diameter of the Nano cylinder is much smaller than the incident wavelength, so we can apply the quasistatic limit to acquire the scattering coefficients. In the quasistatic limit, the Laplace equation ${\nabla ^2}{\phi _{c,\;s,\;o}} = 0$ is satisfied, where the $c,\;s,\;o$ represent the electric potentials of the inner dielectric layers, outer dielectric layers and background medium, respectively. The local electric potentials can be written as:
3. Results and discussion
Different from conventional Fano response [18,19], our graphene-wrapped core-shell nanowires show two Fano responses by the interaction between resonant mode and cloaking mode. As shown in Fig. 2, these two Fano responses arise from the two graphene layers, respectively. Next we will focus on the scattering properties of this system. In the lossless limit considered here, cloaking modes arise when the numerator of scattering coefficient ${D_m}$ corresponding to the minimal scattering, and resonant modes arise when the denominator of scattering coefficient ${D_m}$ corresponding to the maximal scattering. The scattering property of this system is shown in Fig. 3(a) and 3(b). It can be seen that there are two resonant and two cloaking modes in the THz region of frequencies. For the first pair of cloaking mode and resonant mode, cloaking mode can be achieved for shorter wavelengths, and resonant mode can be achieved for longer wavelengths. When the aspect ratio $\eta = a/b$ is very small (i.e. $\eta = 0.1$), it corresponds to very small distances between layers of graphene, in which case cloaking mode and resonant mode would be coupled at the same condition, and then a narrowband Fano-like response is obtained. While the other pair of cloaking mode and resonant mode, in turn, were coupled at one frequency when the aspect ratio $\eta = a/b$ is very large (i.e. $\eta = 0.7$), it corresponds to very large distances between layers of graphene [3–5].
Fig. 2. The scattering property under different graphene layers of this system. Without graphene layers(black line), only inner graphene layer(red line), only outer graphene layer(green line) and there are two layers of graphene inside and outside(blue). Other parameters are $b = 250\;nm,{\kern 1pt} {\kern 1pt} \eta = 0.5,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\varepsilon _c} = {\varepsilon _s} = 3.9,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\varepsilon _h} = 1,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\tau _1} = {\tau _2} = 6.5ps,{\kern 1pt} {\kern 1pt} {\kern 1pt} {E_{F1}} = {E_{F2}} = 0.5\;eV$.
Fig. 3. (a) and (b): The scattering section of dielectric-graphene-dielectric-graphene multilayer nano-shells structure with different frequency, aspect ratio and chemical potential of the inner and outer graphene layers, (a) ${E_{F1}} = {E_{F2}} = 0.2{\kern 1pt} \;eV$, (b) ${E_{F1}} = {E_{F2}} = 0.5\;eV$. Figure 3(c) and (d): the local field distribution at points A and B of this model. (c) $\eta = 0.7,\;f = 5.7\;THz$ and (d) $\eta = 0.7,\;f = 16.2\;THz$. Other parameters are $b = 250\;nm,{\varepsilon _c} = {\varepsilon _s} = 3.9,{\varepsilon _h} = 1,{\tau _1} = {\tau _2} = 6.5ps$. The dotted circles represent the boundaries of the particles.
Compared with conventional metal, graphene conductivity ${\sigma _n}$ could be tuned by changing Fermi energy. And the carrier concentration of graphene increases with the increase of Fermi energy, indicating that the metallic properties of graphene are gradually enhanced [4,10]. Then the scattering property of this system would be changed by changing Fermi level of graphene layers. In detail, if we change simultaneously the Fermi energy of the inner and outer graphene layers, both the two groups of resonant and cloaking mode will exhibit the blue-shift. [See Fig. 3(a) and 3(b)]. For very small aspect ratio ($\eta = 0.1$), the second set of resonant and cloaking mode undergoes the blue-shift, and the first set of resonant and cloaking mode don’t move basically with increasing the Fermi energy of the inner graphene layer, accompanied by the narrowband Fano-like curve. Inversely, with the Fermi energy of the outer graphene layer increases from $0.2\;eV$ to $0.5\;eV$, the first set of resonant and cloaking mode undergo blue-shift, and the second set of resonant and cloaking mode don’t move basically. It is evident that the first set of resonant and cloaking mode is mainly dominated by the outer graphene layer and the inner graphene layer mainly affects the other set of resonant and cloaking mode. In addition, it’s different from what was previously mentioned with the increase of the aspect ratio. For large aspect ratio ($\eta = 0.7$), corresponding to a thin shell [5], the interaction between the two layers of graphene will be enhanced. Hence each pair of cloaking mode and resonant mode would be affected by two layers of graphene. In addition, the above conclusion is further explained by the near-field distributions of this system [20–22]. The local field pattern can be understood by using hybridization modle [23], which describes bonding hybridization and anti-bonding hybridization owing to the interaction of sphere and cavity plasmonic modes. For bonding hybridization, the same kinds of charge are distributed on the inner and outer graphene layers. And for anti-bonding hybridization, the different kinds of charge is distributed on the inner and outer graphene layers [24–25]. As is shown in Fig. 3(c), the local field pattern corresponding to the electric field distribution at point A. In this case, the local electric fields in the shell are enhanced due to anti-bonding hybridization. While for the electric field distribution at point B, the local electric fields resonance in the core and near the outer and inner interface are enhanced observably due to bonding hybridization is shown in Fig. 3(d).
In the vicinity of Fano resonance, near-field energy flux can be reversed [26,27]. Saddle points and vortex are two kinds of optical singularities. Saddle-vortex pairs would be formed and the energy flux around them would be modified dramatically when these points get close [26–27]. To understand the mechanisms of Fano-like response better [2], the Poynting vector ${\boldsymbol S}$ plots for the dielectric-graphene-dielectric-graphene structure are displayed in Fig. 4, which shows the direction and magnitude of the electromagnetic energy of this structure. Figure 4(a) illustrates the narrowband Fano-like curve for different Fermi energy. And we found that the positions of the peaks and dips of the Fano response can be changed by adjusting the Fermi energy of graphene layers. For the narrowband Fano-like curve of ${\mu _{c1}} = {\mu _{c2}} = 0.5\;eV$, when frequencies below Fano resonant frequency, the energy flux in the inner and outer dielectric layers is counterclockwise at point A as shown in Fig. 4(b), where the incident frequency $f = 7\;THz$. But at point B, this is above Fano resonant frequency and the incident frequency $f = 10\;THz$ in Fig. 4(c), the situation is opposite. We change ${\mu _{c1}}$ and ${\mu _{c2}}$ form $0.5\;eV$ to $0.8\;eV$ by keeping the incident frequency $f = 10\;THz$. We find that the near-field energy flux can be totally reversed by comparing Fig. 4(c) with Fig. 4(d). Therefore, we can change the distribution of near-field energy flux by adjusting the Fermi energy of the graphene layers at the same incident frequency.
Fig. 4. (a) the scattering efficiency of the dielectric-graphene-dielectric-graphene structure as a function of the Fermi level of the inner and outer graphene layers, with ${E_{F1}} = {E_{F2}} = 0.5\;eV$ (red line), and ${E_{F1}} = {E_{F2}} = 0.8\;eV$ (green line). And Poynting vector $S$ distribution, lines of Poynting vector $S$, and singular points for (b) ${E_{F1}} = {E_{F2}} = 0.5\;eV$, the incident frequency $f = 7\;THz$. (c) ${E_{F1}} = {E_{F2}} = 0.5\;eV$, the incident frequency $f = 10THz$ and (d) ${E_{F1}} = {E_{F2}} = 0.8\;eV$, the incident frequency $f = 10\;THz$. Other parameters are $b = 250\;nm,\eta = 0.5,{\varepsilon _c} = {\varepsilon _s} = 3.9,{\varepsilon _h} = 1,{\tau _1} = {\tau _2} = 6.5ps$.
Figure 5(a) reveals the scattering efficiency for various Fermi energy of the inner and outer graphene layers. It can be seen that there are also two resonant and two cloaking modes. The first pair of resonant and cloaking modes in Fig. 5(a) corresponds to the second pair of resonant and cloaking modes in Fig. 3(a) and 3(b). Similarly, another couple of resonant and cloaking modes in Fig. 4(a) corresponds to the first pair of resonant and cloaking modes in Fig. 3(a) and 3(b). We select one point in each of cloaking modes arbitrarily. And we find that two different cloaking modes could be overlapped by simply changing the Fermi energy of the inner and outer graphene layers. Meanwhile Fano parameter ‘q’ is very important for the asymmetrical shape of Fano response, which associated with the sensitivity of system. Then we obtain the q parameter of the fist Fano response and the second Fano response by fitting Fano formula $F(\Omega ) = {\sigma _0}{({\Omega + q} )^2}/({{\Omega ^2}\textrm{ + }1} )$ respectively, where q is the asymmetry parameter, $\Omega = ({\omega - {\omega_0}} )/({\gamma /2} )$, ${\omega _0}$ is the resonant frequency of the peak, $\gamma$ is the width of the Fano spectral curves and ${\sigma _0}$ the normalized scattering [1,20]. As shown in Fig. 5(b), the first Fano response shows more obvious asymmetric profile than the second one($|{{q_\textrm{I}}} |$ is closer to 1 than $|{{q_{\textrm{II}}}} |$ [2]). It means small perturbation of frequency at the fist Fano response can cause fast swop of resonant and cloaking mode, which indicating high sensitivity.
Fig. 5. (a) The scattering efficiency for various chemical potential of the inner and outer graphene layers. And (b) ${E_{F1}} = 0.98\;eV,{E_{F2}} = 0.53\;eV$ (red line) and ${E_{F1}} = 0.19\;eV,{E_{F2}} = 0.03\;eV$ (green line). The first Fano response and the second Fano response are labeled with $\textrm{I}$ and $\textrm{II}$ in corresponding colors. The fitting results (black dotted lines) of the first Fano response and the second Fano response in Fig. 5(b) with ${E_{F1}} = 0.98\;eV,{E_{F2}} = 0.53\;eV$. $|{{q_\textrm{I}}} |$ and $|{{q_{\textrm{II}}}} |$ are the Fano parameter of the first Fano response and the second Fano response, respectively. (c) The local electric fields with ${E_{F1}} = 0.98\;eV,{E_{F2}} = 0.53\;eV$, and (d) ${E_{F1}} = 0.19\;eV,{E_{F2}} = 0.03\;eV$. Other parameters are $b = 250\;nm,$ $\eta = 0.5,$ ${\varepsilon _c} = {\varepsilon _s} = 3.9,$ ${\varepsilon _h} = 1,$ ${\tau _1} = {\tau _2} = 6.5ps$ and the incident frequency $f = 10\;THz$.
The corresponding field distributions at the two different cloaking modes have obvious differences, as shown in Fig. 5(c) and 5(d). Compared with Fig. 5(c), the electric field in Fig. 5(d) has a significant enhancement in the central region. As a consequence, we can consider designing a cloaking sensor and electric concentrator [12–15]. For a cloaking state (such as Fig. 5(c)), the external fields outside the device are undisturbed and the near-field is negotiable. For the state of electric concentrator (such as Fig. 5(d)), the electric field is enhanced, but the external fields are not disturbed. Meanwhile, the device keeps invisible in the far-field. We can see from Fig. 5(c) and 5(d) that the intensity of electric fields in central region varies significantly only by changing the Fermi energy of the two layers of graphene without changing other parameters of the device. The Fermi energy of the two layers of graphene can be manipulated through an applied voltage on the graphene layers.
Fig. 6. The numerical simulation results on Fig. 4(b) and Fig. 5(d). (a) The distribution of Poynting vector $S$, where the direction of arrow represents the direction of $S$. (b) The distribution of the local electric field. The distribution of colors in the figures indicates the numerical size of $S$ and local electric field. The corresponding parameters are the same as those in Fig. 4(b) and Fig. 5(d).
Finally, in order to show that our research results are correct and reliable, we numerically simulate the two cases in Fig. 4(b) and Fig. 5(d) by using the finite-element-method solver in COMSOL Multiphysics. As shown in Fig. 6, our theoretical results are in good agreement with numerical simulations.
4. Conclusions
In conclusion, we have presented a theoretical investigation on scattering property in the dielectric-graphene-dielectric-graphene multilayer core-shell nanostructure. We find that there are two resonant and two cloaking states in the THz region of frequencies. In addition, each pair of cloaking mode and resonant mode would be coupled to a narrowband Fano-like response at specific aspect ratio. The dependence of Fano-like response on the Fermi energy of two graphene layers and the distance between two graphene layers is analyzed in detail. We found that the interaction between the two layers of graphene would be enhanced as the shell medium becomes thinner. Moreover, we displayed the Poynting vector for this structure to understand the mechanisms of Fano-like response better. Our theoretical results can be used to explore the cloaking sensor and the electric concentrator based on the characteristics of significant differences in the central region under two cloaking states. The dielectric-graphene-dielectric-graphene multilayer core-shell nanostructure may provide promising opportunities for designing high sensitivity devices and understanding the tunable mechanisms of a plasmonic graphene structure.
Funding
National Natural Science Foundation of China (11504252, 11774252); Collaborative Innovation Center of Suzhou Nano Science and Technology; Qinglan Project of Jiangsu Province of China (BRA2015353); China Postdoctoral Science Foundation (2018M630596); Priority Academic Program Development of Jiangsu Higher Education Institutions.
Acknowledgments
All the authors are grateful to Professor Roman E. Noskov for his advice on the figures in this paper and Kai Zhang for his knowledge and help in this work.
Disclosures
The authors declare no conflicts of interest.
References
1. B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonant in plasmonic Nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010). [CrossRef]
2. A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010). [CrossRef]
3. M. Naserpour, C. J. Zapata-Rodriguez, S. M. Vukovic, H. Pashaeiadl, and M. R. Belic, “Tunable invisibility cloaking by using isolated graphene-coated nanowires and dimers,” Sci. Rep. 7(1), 12186 (2017). [CrossRef]
4. X. He, F. Lin, F. Liu, and W. Shi, “Terahertz tunable graphene Fano resonance,” Nanotechnology 27(48), 485202 (2016). [CrossRef]
5. C. Argyropoulos, P. Chen, F. Monticone, G. D. Aguanno, and A. Alù, “Nonlinear Plasmonic Cloaks to Realize Giant All-Optical Scattering Switching,” Phys. Rev. Lett. 108(26), 263905 (2012). [CrossRef]
6. F. Hao, Y. Sonnefraud, P. V. Dorpe, S. A. Maier, N. J. Halas, and P. Nordlander, “Symmetry Breaking in Plasmonic Nanocavities: Subradiant LSPR Sensing and a Tunable Fano Resonance,” Nano Lett. 8(11), 3983–3988 (2008). [CrossRef]
7. V. G. Kravets, F. Schedin, and A. N. Grigorenko, “Extremely narrow plasmon resonances based on diffraction coupling of localized plasmons in arrays of metallic nanoparticles,” Phys. Rev. Lett. 101(8), 087403 (2008). [CrossRef]
8. T. Zhao, M. Hu, R. Zhong, X. Chen, P. Zhang, S. Gong, C. Zhang, and S. Liu, “Plasmon modes of circular cylindrical double-layer graphene,” Opt. Express 24(18), 20461 (2016). [CrossRef]
9. D. A. Smirnova, A. E. Miroshnichenko, Y. S. Kivshar, and A. B. Khanikaev, “Tunable nonlinear graphene metasurfaces,” Phys. Rev. B 92(16), 161406 (2015). [CrossRef]
10. N. K. Emani, T. Chung, A. V. Kildishev, V. M. Shalaev, Y. P. Chen, and A. Boltasseva, “Electrical Modulation of Fano Resonance in Plasmonic Nanostructures Using Graphene,” Nano Lett. 14(1), 78–82 (2014). [CrossRef]
11. E. Lee, I. C. Seo, S. C. Lim, H. Y. Jeong, and Y. C. Jun, “Active switching and tuning of sharp Fano resonances in the mid-infrared spectral region,” Opt. Express 24(22), 25684 (2016). [CrossRef]
12. T. Han, H. Ye, Y. Luo, S. P. Yeo, J. Teng, S. Zhang, and C. W. Qiu, “Manipulating DC currents with bilayer bulk natural materials,” Adv. Mater. 26(21), 3478–3483 (2014). [CrossRef]
13. T. Han, X. Bai, J. T. L. Thong, B. Li, and C. Qiu, “Full Control and Manipulation of Heat Signatures: Cloaking, Camouflage and Thermal Metamaterials,” Adv. Mater. 26(11), 1731–1734 (2014). [CrossRef]
14. T. Yang, X. Bai, D. Gao, L. Wu, B. Li, J. T. L. Thong, and C. Qiu, “Invisible Sensors: Simultaneous Sensing and Camouflaging in Multiphysical Fields,” Adv. Mater. 27(47), 7752–7758 (2015). [CrossRef]
15. T. H. Han, J. J. Zhao, and T. Yuan, “Theoretical realization of an ultra-efficient thermal-energy harvesting cell made of natural materials,” Energy Environ. Sci. 6(12), 3537–3541 (2013). [CrossRef]
16. K. Zhang, Y. Huang, A. E. Miroshnichenko, and L. Gao, “Tunable Optical Bistability and Tristability in Nonlinear Graphene-Wrapped Nanospheres,” J. Phys. Chem. C 121(21), 11804–11810 (2017). [CrossRef]
17. K. Zhang and L. Gao, “Optical bistability in graphene-wrapped dielectric nanowires,” Opt. Express 25(12), 13747 (2017). [CrossRef]
18. Y. Huang and L. Gao, “Tunable Fano resonances and enhanced optical bistability in composites of coated cylinders due to nonlocality,” Phys. Rev. B 93(23), 235439 (2016). [CrossRef]
19. H. L. Chen and L. Gao, “Tunablity of the unconventional Fano resonances in coated nanowires with radial anisotropy,” Opt. Express 21(20), 23619 (2013). [CrossRef]
20. D. Gao, L. Gao, A. Novitsky, H. Chen, and B. Luk’Yanchuk, “Topological effects in anisotropy- induced nano-fano resonance of a cylinder,” Opt. Lett. 40(17), 4162–4165 (2015). [CrossRef]
21. Y. Xu, A. E. Miroshnichenko, and A. S. Desyatnikov, “Optical vortices at Fano resonances,” Opt. Lett. 37(23), 4985–4987 (2012). [CrossRef]
22. B. S. Luk Yanchuk, A. E. Miroshnichenko, and Y. S. Kivshar, “Fano resonances and topological optics: an interplay of far- and near-field interference phenomena,” J. Opt. 15(7), 073001 (2013). [CrossRef]
23. E. Prodan, “A Hybridization Model for the Plasmon Response of Complex Nanostructures,” Science 302(5644), 419–422 (2003). [CrossRef]
24. J. Zhu, J. Li, and J. Zhao, “The Effect of Dielectric Coating on the Local Electric Field Enhancement of Au-Ag Core-Shell Nanoparticles,” Plasmonics 10(1), 1–8 (2015). [CrossRef]
25. Z. Jian, L. Jian-jun, and Z. Jun-wu, “Tuning the Dipolar Plasmon Hybridization of Multishell Metal-Dielectric Nanostructure: Gold Nanosphere in a Gold Nanoshell,” Plasmonics 6(3), 527–534 (2011). [CrossRef]
26. Y. Huang and L. Gao, “Equivalent Permittivity and Permeability and Multiple Fano Resonances for Nonlocal Metallic Nanowires,” J. Phys. Chem. C 117(37), 19203–19211 (2013). [CrossRef]
27. W. J. Yu, P. J. Ma, H. Sun, L. Gao, and R. E. Noskov, “Optical tristability and ultrafast Fano switching in nonlinear magnetoplasmonic nanoparticles,” Phys. Rev. B 97(7), 075436 (2018). [CrossRef]
