⇦Chinese Physics Letters, 2020, Vol. 37, No. 3, Article code 037101 Bright-Dark Mode Coupling Model of Plasmons * Jing Zhang (张静)1,2, Yong-Gang Xu (徐永刚)1,3, Jian-Xin Zhang (张建鑫)1, Lu-Lu Guan (关璐璐)1, Yong-Fang Li (李永放)1** Affiliations 1School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710119 2School of Electronic Engineering, Xi'an Shiyou University, Xi'an 710065 3School of Science, Xi'an University of Posts and Telecommunications, Xi'an 710121 Received 19 November 2019, online 22 February 2020 *Supported by the National Natural Science Foundation of China under Grant No. 11474191, and the Natural Science Foundation of Shaanxi Province under Grant No. 2018JQ1050.
**Corresponding author. Email: yfli@snnu.edu.cn Citation Text: Zhang J, Xu Y G, Zhang J X, Guan L L and Li Y F et al 2020 Chin. Phys. Lett. 37 037101    Abstract We propose a coupling model to describe the interaction between the bright and dark modes of the plasmons of a dimer composed of two orthogonal gold nano-rods (GNRs), referred to as the BDMC model. This model shows that the eigen-frequencies of the coupled plasmons are governed by Coulomb potential and electrostatic potential. With the BDMC model, the behaviors of the coupling coefficient and the frequency offset, which is a new parameter introduced here, are revealed. Meanwhile, the asymmetric behavior of two eigen-frequencies related to gap of two GNRs is explained. Using the harmonic oscillator model and the coupled parameters obtained by the BDMC model, the bright mode absorption spectra of the dimer are calculated and the results agree with the numerical simulation. DOI:10.1088/0256-307X/37/3/037101 PACS:71.45.Gm, 78.67.Bf, 75.25.Dk © 2020 Chinese Physics Society Article Text The Fano effect and like-electromagnetic induced transparency (EIT) effect have become popular topics in research of the interaction of metal nano-elements.[1–23] The coupling process of plasmons plays an important role in both the Fano effect and the EIT effect. The coupling model of harmonic oscillators is mostly used in research of metal nano-element coupling. For example, Zhang et al. studied the like-EIT effect in trimetric plasmons using linear Lorenz oscillator coupling mode.[1] Alzar et al. investigated the EIT effect using the coupling model of harmonic oscillator.[24] Novotny explored the splitting of energy states of atoms and light fields in a strongly coupling process in quantum physics using a harmonic oscillator coupling model.[25] Andrea et al. theoretically and experimentally demonstrated that a simple oscillator model is provided to predict and to fit the far-field scattering of the plasmons.[26] Cheng et al. analytically described destructive interference for the tunable plasmonically induced transparency (PIT) with the coupling Lorentz oscillator model.[27] Jorge et al. investigated the maximum near-field enhancements for the localized plasmons of metallic nanoparticles and nanostructures, and the results came directly from the physics of a driven and damped harmonic oscillator.[28] Lassiter et al. explained intuitively Fano resonance of nanodisk clusters with a coupling oscillator picture.[29] In the above-mentioned literature, we see that the harmonic oscillator model cannot obtain the physical factors that affect the coupling coefficients, and it cannot be used to describe the relationship between interaction and distance. In terms of plasmons interactions, Davis and Gómez presented a simple algebraic approach for modeling localized surface plasmons.[30] Their approach is derived from an electrostatic formalism and is appropriate for near-field interactions associated with localized surface plasmons at optical frequencies. In view of the coupling asynchronism of the plasmons related to the coupling phase, we introduced a phase factor in the coupling coefficient and revealed the generation mechanism of the asymmetric of the absorption spectrum after coupling.[31,32] In this Letter, we numerically simulate the behaviors of the two absorption peaks of the dimer related to the change of the gap distance between two gold nano-rods (GNRs) with the finite element method. One of the absorption peaks related to the change of the gap distance between two GNRs approaches a straight line, and the other approaches a hyperbolic line. These two curves show the asymmetric behaviors. To explain the coupling processes between the bright and the dark mode, based on the idea of Ref. [30] and the plasmons theory, here we present a new coupling model, called the BDMC model. In Ref. [30], the authors showed clearly that the near field dominates the interaction between closely spaced metal structures and described approximately localized surface plasmons with an electrostatic method. Following their results, we show that the Coulomb potential and the electrostatic potential govern the plasmons interaction and establish the relation between the eigen-frequencies of the plasmons and the distance of the nano-elements. Using the BDMC model, we explain the asymmetric behavior of two eigen-frequencies relative to gaps of the nano-rods with the aid of the two parameters; i.e., the coupling coefficient and the frequency offset. The absorption spectra are simulated and the results agree with the numerical simulation. We improve the model in Refs. [1,31,32] by introducing a new parameter named as the frequency offset $\Delta {\it\Omega}$, which describes that the coupling process of the plasmons results in an offset of the resonant frequency of the plasmons. The Lorenz oscillator model reads $$\begin{alignat}{1} \!\!\!\!\!\!\!\begin{pmatrix} {\delta +\Delta {\it\Omega} +i\gamma }& {\kappa e^{i\phi /2}}\\ {\kappa e^{i\phi /2}}& {\delta +\Delta {\it\Omega} +\Delta \omega }\end{pmatrix} \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} E_{0}\\ 0 \end{pmatrix} ,~~ \tag {1} \end{alignat} $$ where $E_{0}$ is the amplitude of the light field; $\Delta \omega =\omega_{20} -\omega_{10}$ is the frequency difference between bright mode and dark mode ($\omega_{10}$ and $\omega_{20}$ are the resonant frequencies of the bright mode and dark mode, respectively), it is a very small value and satisfies the relation $\Delta \omega \ll \omega_{10,20}$; $\delta =\omega_{10} -\omega \ll \omega_{10}$ with $\omega$ being the frequency of the incident light field; $\gamma$ is the decay coefficient of the bright mode. The decay coefficient of the dark mode due to small can be negligible. Here, $\kappa$ is the coupling coefficient and $\phi$ is the coupling phase. $\Delta {\it\Omega}$ is the frequency offset because of the coupling of the plasmons. The coupling coefficient and the frequency offset will simultaneously approach to zero when coupling process disappears. The quantities $\lambda_{1,2} =({\delta +\omega_{1,2} })$ are the eigen-values of the matrix $\begin{pmatrix} {\delta +\Delta {\it\Omega} +i\gamma } & {\kappa e^{i\phi /2}} \\ {\kappa e^{i\phi /2}} & {\delta +\Delta {\it\Omega} +\Delta \omega } \end{pmatrix}$; $\omega_{1,2} =({2\Delta {\it\Omega} +\Delta \omega \mp ({\alpha +i{\it\Gamma}_{1,2} })})/2$ are the eigen-frequencies, where ${\it\Gamma}_{1,2} =\beta \mp \gamma$ is the decay factors ($\alpha ={\rm Re}({\sqrt {(2\kappa e^{i\phi /2})^{2}+({i\gamma -\Delta \omega })^{2}} })$, $\beta ={\rm Im}({\sqrt {(2\kappa e^{i\phi /2})^{2}+({i\gamma -\Delta \omega })^{2}} })$). If we only focus on the resonant frequencies of the absorption spectra and take $\delta =\omega_{10} -\omega =0$ as the coordinate's origin, then the resonant frequencies of the absorption spectra are accordant with the eigen-values real part of the coupling matrix $G$, which is the function of the offset and coupling coefficient. Therefore, we suppose that the phase and the decay factor approach to zero; i.e., $\phi =\gamma =0$. Finally, the real part of the eigen-values and the eigen-frequencies are equivalent and can be written as $$ \omega_{1,2}^{0} =\Delta {\it\Omega} +\frac{\Delta \omega }{2}\mp \frac{\sqrt {4\kappa^{2}+\Delta \omega^{2}} }{2}.~~ \tag {2} $$ Equation (2) depicts the influence on the resonant frequency of the bright mode that is the result of the coupling process of the plasmons. As mentioned in the literature,[30] the near field interaction between closely spaced metal structures is dominated by an electrostatic field in the nano-size. Based on these ideas, we consider that the bright mode is a detectable plasmon, its charge density is $\rho_{e} ({x'})$, which is located at the neighborhood of the plasmon produced by dark mode. In that region, there is a certain distribution of the positive charge density $e\sigma_{i} (x)$ and the negative charge density $e\sigma_{e} (x)$ in the thermal equilibrium. Therefore, the electric potential produced by all charges satisfies the Poisson equation. $$ \varepsilon_{0} \nabla^{2}\phi \left(x \right)=-Ze\sigma_{i} \left(x \right)+e\sigma_{e} \left(x \right)+\rho_{e} \left({x'} \right).~~ \tag {3} $$ Under the action of electric potential $\phi (x)$, the charges density satisfies the Boltzmann distribution at thermal equilibrium; i.e.,$\sigma_{e} (x)=\sigma_{e0} e^{e\phi (x)/kT}$, where $\sigma_{e0}$ is charge density for $\phi (x)=0$, $T$ is the temperature of the electron gas. If $kT\gg \,\,e\phi (x)$, $\sigma_{e} (x)=\sigma_{e0} e^{e\phi (x)/kT}$ can be simplified to $\sigma_{e} \approx \sigma_{e0} ({1+\frac{e\phi }{kT}})$. Taking into account the electrical neutrality of plasma—i.e., $Z\sigma_{i} (x)=\sigma_{e} (x)$—Eq. (3) can be rewritten as $$ \Big({\nabla^{2}-\frac{1}{l_{\rm s}^{2} }} \Big)\phi (x)=\frac{\rho_{e} ({x'})}{\varepsilon_{0}},~~~ l_{\rm s}^{2} =\frac{\varepsilon_{0} kT}{e^{2}\sigma_{e0}},~~ \tag {4} $$ where $l_{\rm s}$ is called the shielding length. The $\sigma_{e} \approx \sigma_{e0} ({1+\frac{e\phi }{kT}})$ reflects that the electric field changes the charge distribution under thermal equilibrium, and Eq. (3) reflects that the change of the charge density then affects the electric field. This is the reason why we introduce the frequency offset $\Delta {\it\Omega}$. The solution to Eq. (4) can be written as $$ \phi \left(x \right)=\int {\frac{\rho_{e} \left({x'} \right)}{4\pi \varepsilon_{0} \left| {x-x'} \right|}e^{-\left| {x-x'} \right|/l_{\rm s} }dV'} .~~ \tag {5} $$ The relationship $| {\boldsymbol{x-x}'} |\ll l_{\rm s}$ is satisfied in near field interaction region. Therefore, the exponential item can be expressed as a series and retained to the first three terms; i.e., $$ \exp \Big({-\frac{| {x-x'} |}{l_{\rm s} }}\Big)\approx 1-\frac{| {x-x'} |}{l_{\rm s} }+\frac{1}{2}\Big({\frac{| {x-x'} |}{l_{\rm s} }}\Big)^{2}. $$ We suppose that the charges density $\rho_{e} ({x'})$ in the small nano-volume $\Delta V$ is a uniform distribution—i.e., $\rho_{e} ({x'})\Delta {\rm V=Q}$—and then we obtain the expression of the electronic potential as $$\begin{alignat}{1} \!\!\!\!\!\!\!\phi (x)\!=\!\phi_{0}\!+\!\frac{Q}{4\pi \varepsilon_{0} }\Big({\frac{1}{\left| {x-x'} \right|}\!-\!\frac{1}{l_{\rm s} }} \Big)\!+\!\frac{Q}{8\pi \varepsilon_{0} l_{{\rm s}}^{2} }\left| {x-x'} \right|.~~ \tag {6} \end{alignat} $$ The first term is the zero electric potential, the second term is the Coulomb potential, and the third term is the electrostatic potential. The electronic potential energy of the two coupled modes is expressed as $$\begin{align} \Delta W=\,&\frac{1}{2}Q\phi (x)=\frac{1}{2}Q\Big[\phi_{0} +\frac{Q}{4\pi \varepsilon_{0} }\Big({\frac{1}{| {x-x'} |}-\frac{1}{l_{\rm s} }} \Big)\\ &+\frac{Q}{8\pi \varepsilon_{0} l_{{\rm s}}^{2} }| {x-x'} | \Big]. \end{align} $$ $\Delta W=N\hbar {\it\Omega}$ is the absorption energy of the bright mode because of the coupling process of the plasmons, where $N$ is the number of plasmons in the nano-element and ${\it\Omega}$ is the resonant frequency of the plasmons. These two energy expressions are equal, therefore we can obtain the resonant frequencies of the bright mode, as follows: $$ {\it\Omega} \!=\!{\it\Omega}^{0}\!+\!\frac{Q^{2}}{8\pi \varepsilon_{0} N\hbar }\Big({\frac{1}{\left| {x-x'} \right|}\!-\!\frac{1}{l_{\rm s} }} \Big)\!+\!\frac{Q^{2}}{16\pi \varepsilon_{0} l_{{\rm s}}^{2} N\hbar }\left| {x\!-\!x'} \right|.~~ \tag {7} $$ Using some parameters to simplify Eq. (7) we have $$\begin{align} {\it\Omega}_{i} \left(g \right)=\,&\Big({{\it\Omega}^{0}-\frac{Q^{2}}{8\pi \varepsilon_{0} l_{\rm s} N\hbar }} \Big)+\frac{Q^{2}}{8\pi \varepsilon_{0} N\hbar }\Big({\frac{1}{\left| {x-x'} \right|}} \Big)\\ &+\frac{Q^{2}}{16\pi \varepsilon_{0} l_{{\rm s}}^{2} N\hbar }\left| {x-x'} \right| \\ =\,&\Big({{\it\Omega}^{0}-\frac{Q^{2}}{16\pi \varepsilon_{0} l_{\rm s} N\hbar }} \Big)+\frac{Q^{2}}{8\pi \varepsilon_{0} N\hbar }\Big({\frac{1}{g+l_{i} }} \Big)\\ &+\frac{Q^{2}}{16\pi \varepsilon_{0} l_{{\rm s}}^{2} N\hbar }g \\ =\,&{\it\Omega}^{i0}+\frac{d_{i1} }{l_{i} +g}+d_{i2} g ,~~ \tag {8} \end{align} $$ where ${x-x'}={l_{i} +g}$ is the center–center distance between two coupling plasmons, $g$ is the gap of the GNRs. The symbol $i$ represents to different modes for the micro-nano system. The first term ${\it\Omega}^{i0}$ (THz) of Eq. (8) is the constant term which dominates the initial frequency of the eigen-frequencies; $d_{i1}$ (THz$\cdot$nm) is the coefficients of the Coulomb potential, $d_{i2}$ (THz$\cdot$nm) is the coefficients of the electrostatic potential. Since Eq. (8) depicts the influence on the resonant frequencies of the interaction between the bright and dark modes of the plasmons, it is equivalent to the real part of the eigen-frequencies of Eq. (2) and can be written as $$ \begin{array}{l} {\it\Omega}_{1, 2} \left(g \right)=\omega_{1,2}^{0} =\Delta {\it\Omega} +\frac{\Delta \omega }{2}\mp \frac{\sqrt {4\kappa^{2}+\Delta \omega^{2}} }{2}. \\ \end{array}~~ \tag {9} $$ From Eq. (9) we obtain the relation between the two-mode interaction of the plasmons and the resonant frequencies of the plasmons. To study the coupling process of the plasmons, we design a dimer structure formed by two gold nano-rods (GNRs) on the glass substrate, as shown in Fig. 1. The thickness and the width of GNRs are both 20 nm, and $L_{1}$ and $L_{2}$ are the lengths of the vertical and horizontal GNRs, respectively, while $g$ is the gap between the vertical and horizontal GNRs. The glass substrate has a dissipation coefficient of 0.0001 and a refractive index of 1.45. The structure is periodically arranged in the $xy$ plane. We take a one unit structure of the dimer in our calculations (unit size $a=b=430$ nm, as shown in Fig. 1(a)). The glass substrate does not affect the electromagnetic characteristics of the metal dimer, but the absorption spectra detected through the glass substrate will show to a small shift.
cpl-37-3-037101-fig1.png
Fig. 1. (a) Schematic diagram of the dimer of GNRs in the glass substrate, where the dashed lines show one unit ($a = b = 430$ nm). (b) An amplified schematic diagram of one unit structure. The incident light field of the plane wave acting on the dimer is along the $z$-axis, and the polarization direction of the light field is along the $y$-axis and perpendicular to the horizontal GNR.
The absorption characteristics, the electric field and the charge distribution of the dimer are simulated numerically using the finite element method and the refractive index parameters of metal materials provided in the literature.[33] The absorption is obtained by calculating heat loss. Perfectly matched layers are implemented in the following numerical simulation,[34–36] which ensures that the absorption will be perfect. The incident direction of the light field is along the $z$-axis, and the linear polarization direction of the light field is along the $y$-axis. To study the characteristics of the absorption spectra of the dimer, the absorption spectra of the dimer in the different geometry configurations are investigated systematically and calculated numerically by changing the gap distance between two GNRs, as shown in Figs. 2 and 3. The first column of Fig. 2 is the absorption spectra of the dimer related to the gap changes between the GNRs simulated numerically using the finite element method. The lengths of two GNRs are equal; i.e., $L_{1}=L_{2}=100$ nm. The geometry configuration of the dimer in the first row of Fig. 2, as shown in the inset of Fig. 2(d), is that the horizontal GNR is upward by 20 nm away from the center of the vertical GNR. For the middle row of Fig. 2, the horizontal GNR is off the center position of the vertical GNR by 30 nm, as shown in the inset of Fig. 2(h). For the bottom row in Fig. 2, the horizontal GNR is located at the top edge of the vertical GNR, as shown in the inset of Fig. 2(l). To compare the results with the first column of Fig. 2, the $y$-axis of the second and third columns in Fig. 2 are taken as the gap between the two GNRs. For the equal lengths of two GNRs ($L_{1}=L_{2}$), the resonant frequencies of two GNRs are also equal. This means that $\omega_{20} =\omega_{10}$ and $\Delta \omega =0$. Equation (9) is simplified to $$\begin{align} &{\it\Omega}_{i} \left({{\rm g}} \right)=\Delta {\it\Omega} \mp \kappa ={\it\Omega}^{i0}+\frac{d_{i1} }{l_{i} +g}+d_{i2} g,~~(i=1,2) \\ &{\frac{{\it\Omega}_{1} +{\it\Omega}_{2} }{2}=\Delta {\it\Omega} },~~ {\frac{{\it\Omega}_{2} -{\it\Omega}_{1} }{2}=\kappa }.~~ \tag {10} \end{align} $$ The hollow and solid circle curves in the second column (b, f, j) of Fig. 2 show the behaviors of frequencies of two peaks in the first column (a, e, i) of Fig. 2 related to the gap changes of the GNRs, respectively. These numerical results are consistent with Eq. (10) as marked by the red and blue curves in the second column (b, f, j) of Fig. 2. Similarly, the third column (c, g, k) of Fig. 2 shows the behaviors of the parameters $\Delta {\it\Omega}$ and $\kappa$ of the BDMC model related to the gap changes of the GNRs. When the gap becomes greater, the coupling between the bright and the dark mode disappears, and the coupling coefficient $\kappa$ and the frequency offset $\Delta {\it\Omega}$ both approach to the zero point of the $x$-axis, as shown in the third column (c, g, k) of Fig. 2. The results satisfy the initial hypothesis. The dominant reason of the asymmetrical evolutions of the resonant peaks with the gap of the GNRs is that two parameters of the coupling coefficient $\kappa$ and the frequency offset $\Delta {\it\Omega}$ have different evolution behaviors. The last column (d, h, l) of Fig. 2 shows the absorption spectra calculated by Eq. (7) of the literature,[32] the coupling coefficient $\kappa$ and the frequency offset $\Delta {\it\Omega}$ for a fixed gap ($g=20$ nm and 30 nm) calculated by Eq. (10). The results agree with the numerical simulation. The first column of Fig. 3 is the absorption spectra of the dimer with the gap change of the GNRs simulated numerically by using the finite element method for the case of $L_{1} \ne L_{2}$ ($L_{1}=100$ nm, $L_{2}=80$, 120 nm). The intensity of the right-hand absorption peak is evidently smaller than that of the left-hand one for $L_{2}=80$ nm in the dimer in Fig. 3(a); i.e., the decay rate of the right peak is greater than that of the left one. However, for $L_{2}=120$ nm, the intensity of the left-hand absorption peak is evidently smaller than that of the right-hand one in Fig. 3(e), and the decay rate of the left-hand peak is greater than that of the right-hand one. This indicates that the absorption spectra of the dimer for the different horizontal GNRs are different.
cpl-37-3-037101-fig2.png
Fig. 2. First column (a, e, i) shows the absorption spectra of the dimer, simulated numerically using the finite element method for different GNRs' gap. The hollow and solid circle curves in the second column (b, f, j) show that the frequencies of the two absorption peaks evolve with the change of GNRs' gap, and the results with the BDMC model are indicated by the red and blue solid curves. Third column (c, g, k) indicates the evolutions of $\Delta {\it\Omega}$ and $\kappa$ with the change of GNRs' gap. The $x$-axis zero points of the second and the third columns are both located at $\omega_{10}$; i.e., $\omega_{10} -\omega =0$. The geometry configurations of the dimer in each row are indicated in the last column, as shown in the insets (d, h, l). The last column (d, h, l) indicates the comparison of the results between the numerical simulation and the harmonic oscillator model, the calculation parameters are obtained by the BDMC model.
For the different lengths of the two GNRs ($L_{1} \ne L_{2}$), the resonant frequencies of the bright mode and dark mode are also different; i.e., $\omega_{20} =\omega_{10} +\Delta \omega$. The red and blue curves in the second column (b, f) in Fig. 3 are obtained using Eq. (9). When the GNRs gap ($g > 45$ nm) is greater and $L_{1}=100$ nm, $L_{2}=80$ nm ($L_{1} >L_{2}$, $\Delta \omega >0$), the real part of the eigen-frequencies is $\omega_{1}^{0} =0$ and $\omega_{2}^{0} =\Delta \omega$. Similarly, for the case $L_{1}=100$ nm and $L_{2}=120$ nm ($L_{1} < L_{2}$, $\Delta \omega < 0$), the real part of the eigen-frequencies is $\omega_{1}^{0} =-\Delta \omega$ and $\omega_{2}^{0} =0$. The results are shown in Figs. 3(f) and 3(g). The last column (d, h) shows a comparison of the numerical simulation and the harmonic oscillator model,[32] the calculation parameters are obtained by the BDMC model. The two eigenmodes are generated because of the coupling process of the bright and dark modes. This is equivalent to the energy levels of an artificial atom that are split into dressed states by a strong coupling field, which are called the eigenmodes of the dimer. Then, the destructive interference effect can occur between two eigenmodes,[32] as shown in Figs. 2 and 3. The two eigenmodes can also be understood such that the coupling process between the vertical and the horizontal GNRs results in the plasmon hybridization, which are a higher energy antibonding and a lower energy bonding resonance, respectively. The antibonding and the bonding modes reflect the different coupling ways of two GNRs, one of which is dominated by the electrostatic potential and the other is dominated by the Coulomb potential. Therefore, they are the different evolution characters for the two eigenmodes.
cpl-37-3-037101-fig3.png
Fig. 3. The first column is the absorption spectra of the dimers simulated numerically using the finite element method for different GNRs' gap. The hollow and solid circle curves in the second column (b, f) show that the frequencies of the two absorption peaks evolve with the change of GNRs' gap, and the red and blue solid curves are obtained by the BDMC model. The third column (c, g) indicates the evolutions of $\Delta {\it\Omega}$ and $\kappa$ related to the GNRs' gap. The $x$-axis zero points of the second and third columns are both located at $\omega_{10}$, i.e., $\omega_{10} -\omega =0$. The geometry configurations of the dimer on each row are indicated in the last column, as shown in the insets (d, h). The last column (d, h) indicates the comparison of the results between the numerical simulation and the harmonic oscillator model, the calculation parameters are obtained by the BDMC model.
In summary, a novel BDMC model is developed to describe and explain the coupling behaviors of plasmons, which dominates the behaviors of the real parts of the eigen-frequencies $\omega_{1, 2}^{2}$. The BDMC model reveals the evolution of eigenmodes related to gap of two GNRs, which is confirmed by two parameters; i.e., the coupling coefficient and the frequency offset. Similarly, the BDMC model also reveals the behaviors of the coupling coefficient and the frequency offset. The frequency difference $\Delta \omega$ between the bright mode and dark mode, which is hard to detect in the experiment, can also be obtained using the BDMC model. References Plasmon-Induced Transparency in MetamaterialsFano resonances in nanoscale structuresSymmetry Breaking in Plasmonic Nanocavities: Subradiant LSPR Sensing and a Tunable Fano ResonanceEnhanced sensing performance by the plasmonic analog of electromagnetically induced transparency in active metamaterialsFabrication of planar metamaterials with sharp and strong electromagnetically induced transparency-like characteristics at wavelengths around 820 nmPhase-Coupled Plasmon-Induced TransparencyDispersionless slow light in MIM waveguide based on a plasmonic analogue of electromagnetically induced transparencyFano resonances in dipole-quadrupole plasmon coupling nanorod dimersThe Fano resonance in plasmonic nanostructures and metamaterialsFanoshells: Nanoparticles with Built-in Fano ResonancesPronounced Fano Resonance in Single Gold Split Nanodisks with 15 nm Split Gaps for Intensive Second Harmonic GenerationPlasmonic analogue of electromagnetically induced transparency at the Drude damping limitMultiple Fano Resonances in Symmetry-Breaking Silicon Metasurface for Manipulating Light EmissionStrong Fano Resonance Excited in an Array of Nanoparticle-in-Ring Nanostructures for Dual Plasmonic Sensor ApplicationsPlasmonically induced transparent magnetic resonance in a metallic metamaterial composed of asymmetric double barsTunable plasmon-induced transparency in plasmonic metamaterial composed of three identical ringsDynamically tunable plasmonically induced transparency in sinusoidally curved and planar graphene layersAchieving a High Q-Factor and Tunable Slow-Light via Classical Electromagnetically Induced Transparency (Cl-EIT) in MetamaterialsSignatures of Strong Coupling on Nanoparticles: Revealing Absorption Anticrossing by Tuning the Dielectric EnvironmentSelective bright and dark mode excitation in coupled nanoantennasFano-like resonance emerging from magnetic and electric plasmon mode coupling in small arrays of gold particlesSpoof surface plasmon polaritons in terahertz transmission through subwavelength hole arrays analyzed by coupled oscillator modelGeneralized analytical model based on harmonic coupling for hybrid plasmonic modes: comparison with numerical and experimental resultsClassical analog of electromagnetically induced transparencyStrong coupling, energy splitting, and level crossings: A classical perspectiveMechanisms of Fano Resonances in Coupled Plasmonic SystemsDynamically tunable plasmonically induced transparency in periodically patterned graphene nanostripsOn the Energy Shift between Near-Field and Far-Field Peak Intensities in Localized Plasmon SystemsDesigning and Deconstructing the Fano Lineshape in Plasmonic NanoclustersColloquium : An algebraic model of localized surface plasmons and their interactionsExtended coupled Lorentz oscillator model and analogue of electromagnetically induced transparency in coupled plasmonic structuresOptical Constants of the Noble MetalsResonant tunneling of surface plasmon-polaritonsA perfectly matched layer for the absorption of electromagnetic wavesA new look at the perfectly matched layer (PML) concept for the reflectionless absorption of electromagnetic waves
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