Open Access
Add to BibSonomy
Abstract
In plasmonic Fano resonance, the interaction between a discrete plasmonic mode and a continuum of plasmonic mode gives rise to an asymmetric line shape in the scattering or absorption spectrum, enabling a wide range of applications such as sensing, switching, and slow light devices. Here, we establish a theoretical solution in the framework of temporal coupled-mode theory (TCMT) to study the three-dimensional (3D) and two-dimensional (2D) Fano resonances induced by strong coupling between metal hole (MH) and split ring resonator (SRR) array. We first separately analyze the transmission spectra of the MH array and SRR array under different polarized light excitation. We further investigate the electromagnetic field and charge density distribution corresponding to the resonant modes at the peak or valley wavelength of the transmission spectrum and figure out the electric/magnetic dipole feature of these resonance modes. We then establish a theoretical solution by TCMT for Fano resonances arising from the coupling of these modes. The calculated transmission spectrum is closely matching with the numerically simulated transmission spectrum for these Fano resonances in the MH-SRR array, which effectively elucidates that the asymmetry of the Fano resonances is caused by the coupling between bright and dark plasmonic modes involved in the two structures. Our results can help to understand the profound physics in such coupled plasmonic systems.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The fundamental criterion for a Fano resonance is the interference between a spectrally overlapping broad resonance or continuum state and a narrow discrete resonance state [1,2], which have been reported in metamaterials, plasmons and photonic crystals [2–7]. Plasmonic hybridization in sub-wavelength systems has recently revealed unconventional electromagnetic (EM) properties, which result from the combination of strongly radiative bright modes (super-radiant) and spectrally sharp dark plasmonic modes (sub-radiant) [3]. The interference of a narrow dark mode and a broad bright mode is a common mechanism for observing Fano resonance in the two-dimensional (2D) (i. e. coupling in one plane) [8–16] and three-dimensional (3D) (i. e. coupling in two vertical planes) plasmonic systems [17–20]. In this case, the narrow dark mode acts as a “discrete” state, while the broad bright mode acts as a “continuum” state. The structure supports a multipolar-like dark mode and a strong dipolar bright mode, which together give rise to the Fano line shape in the extinction cross-section. Owing to its strong sensitivity to the local environment and changes in geometry, the plasmonic Fano resonance in periodic metal nanostructures can be applied for sensing [21], switching [22], slow light devices [23]and so on. The mode coupling theories are developed such as the plasmon hybridization model [24], coupled-mode theory under a generalized tight-binding approximation [25–27], a formal theoretical framework from first principles [28] and time-domain coupled mode theory (TCMT) [29] to analyze the underlying physics between the mode coupling. Fan and coworkers [30] theoretically demonstrated Fano resonance spectra for the first time using a fully dielectric silicon-based coupled cavity and employed the classical transfer matrix method for transmission spectrum analysis by using TCMT. Due to the ease of understanding of the transfer matrix method, this analysis method has been widely used to date [31–38]. Yet, this method is not suitable for visually representing the influence of mode-mode coupling effects on Fano resonance.
Cui et al. developed a nano-origami technology based on focused-ion-beam (FIB) patterning and folding and built 3D metamaterials composed of an array of planar metal-hole (MH) coupled with vertical split ring resonator (SRR) [17]. The experiments demonstrated that these 3D metamaterials can realize strong 3D plasmon Fano resonances. Based on this method, Tian et al. further fabricated more complicated 3D metamaterials to realize multiple Fano resonances [18]. Later on, Liu et al. analyzed the current direction within the whole 3D microstructure and provide a brief explanation to the coupling between the MH and SRR structures [19]. Yet, this work only presents the overall electromagnetic picture of the Fano resonance modes, but does not disclose the physical origin of these resonance modes using the methodologies established in the framework of standard Fano resonance physics. In this paper, we wish to conduct a deeper physical analysis of the plasmonic Fano resonances in these MH-SRR array metamaterials in the framework of Fano resonance physics, by disclosing what is the bright mode and the dark mode, and how they interact and couple with each other to create Fano resonance. To this aim, we first separately analyze the transmission spectra of the MH array and SRR array under different polarized light excitation. We further investigate the electromagnetic field and charge density distribution corresponding to the resonant modes at the peak or valley wavelength of the transmission spectrum. Based on these, we deduce that the Fano resonances observed in the MH-SRR array are attributed to the coupling between electric dipole and electric dipole modes, or electric dipole and magnetic dipole modes. Furthermore, we employ the TCMT to construct a detailed theoretical solution for the Fano resonances, which effectively elucidates that the asymmetry of the Fano resonances is caused by the coupling between the two structures and between broad bright mode and narrow dark modes. The theoretical calculation matches closely with the simulated transmission spectrum, confirming the accuracy and validity of the coupling model.
2. Simulation and discussion
The unit cell of the 3D Fano resonance structure studied in the work is shown in Fig. 1(a) in which the vertical SRR is connected to one edge of a metallic hole. The free-standing structure is completely composed of Au without any substrate. Additionally, the unit cell of the 2D MH-SRR array is shown in Fig. 1(b) in which the SRR is connected to one edge of the hole and parallel to the MH. The structural parameters of the 3D and 2D structures are b = 3 µm, a = 2 µm, t = 80 nm, l=1 µm, h = 1.2 µm and w = 0.32 µm, respectively. In order to further analyze the coupling between the MH and SRR, we first separately numerically simulate the transmission spectra of the MH array without the SRR structure under x-polarized and y-polarized light excitation and the SRR array without the MH structure under x-polarized, y-polarized and z-polarized light excitation in the mid-infrared wavelength region by finite element method. The results are shown in Fig. 1(c) and Fig. 1(d), respectively. Due to the central symmetry of the metallic hole, the transmission spectra under x-polarized and y-polarized light excitation are identical and can be considered as a symmetrical Lorentzian line shape, where the peak wavelength of the spectra is at 3.4 µm.
Fig. 1. (a) Schematic of the unit cell of a 3D metamaterial in which the vertical SRR is connected to one edge of a metallic hole. (b) Schematic of the unit cell of a 2D metamaterial in which SRR is parallel to a metallic hole. (c) Simulated transmission spectra of the MH array without SRR structure under y-polarized light excitation in the mid-infrared wavelength region. (d) Simulated transmission spectra of the SRR array without MH structure under x-polarized, y-polarized and z-polarized light excitation in the mid-infrared wavelength region. (e) Simulated transmission spectra of 2D MH-SRR array under x-polarized and y-polarized light excitation in the mid-infrared wavelength region. (d) Simulated transmission spectra of 3D MH-SRR array under x-polarized and y-polarized light excitation in the mid-infrared wavelength region. Structural parameters: b = 3 µm, a = 2 µm, t = 80 nm, $l$=1 µm, h = 1.2 µm, w = 0.32 µm.
The transmission spectra of the SRR array under x-polarized light excitation can be considered as a symmetrical anti-Lorentzian line shape, while the minimum value of the spectrum occurs at a wavelength of 5.6 µm. The transmission spectra of the SRR array under z-polarized light excitation has two minimum values at 3.5 µm and 5.6 µm respectively in the mid-infrared wavelength region. Each resonance spectra can be considered as a symmetrical anti-Lorentzian line shape. When the y-polarized light is incident, the SRR array is not excited out of the resonance mode, and the incident light just directly transmits out as shown in Fig. 1(d). The transmission spectra of MH array is much wider than that of SRR array for each resonance mode. Furthermore, the simulated transmission spectra of 2D MH-SRR array under x-polarized and y-polarized light excitation in the mid-infrared wavelength region are shown in Fig. 1(e). The interference between the narrow dark mode in Fig. 1(d) and the broad bright mode in Fig. 1(c) leads to a Fano resonance for the 2D MH-SRR array under x-polarized light excitation, which manifests as an asymmetric line shape in the transmission spectrum, while the 2D MH-SRR array do not exhibit Fano resonances under y-polarized light excitation.
The simulated transmission spectra of 3D MH-SRR array under x-polarized and y-polarized light excitation in the mid-infrared wavelength region are shown in Fig. 1(f). The interference between the narrow dark mode Fig. 1(d) and the broad bright mode in Fig. 1(c) leads to two Fano resonances for the 3D MH-SRR array under y-polarized light excitation, which manifests as an asymmetric line shape in the transmission spectrum, while the 3D MH-SRR array do not exhibit Fano resonances under x-polarized light excitation.
To further illustrate the resonance mode at a wavelength of 3.4 µm in the MH array under y-polarized light excitation, the charge density distribution and field distribution within one unit cell of the MH array at a wavelength of 3.4 µm is simulated and shown in Fig. 2. The positive and negative charges shown in Fig. 2(a) are distributed separately at the edge of the hole along the y direction. Furthermore, the electric field distribution shown in Fig. 2(b) is a dipole distribution while the electric field direction is along the y-axis direction, in parallel to the incident light polarization direction. Similarly, when the MH array is excited by an x-polarized light, the positive and negative charges at the resonance wavelength of 3.4 µm are distributed along the x-axis. The electric field direction is also along the x-axis, and the electric field intensity exhibits a dipole distribution along the x-axis. The charge density distribution and field distribution both demonstrate that the MH array exhibits an electric dipole mode [39] when excited by x- or y-polarized light, providing the continuum state with the broad resonance spectrum.
Fig. 2. (a) The charge density distribution and (b) the electric field strength and direction distribution at the top surface of one unit cell of the MH array when illuminated by incident light polarized in the y-direction propagating in the z-direction, showing the feature of an electric dipole. The black arrow direction is the electric field direction.
According to Ref. [17], the distribution of electric field direction for the MH array under x-polarized and y-polarized light excitation reveals that when under x-polarized light excitation, the distribution of the electric field direction is x-polarized light at the location of the SRR. However, when under y-polarized light excitation, the distribution of the electric field direction is z-polarized light at the location of the SRR. There is no resonant mode observed when the SRR is under y-polarized light excitation, as seen from the transmission spectrum of the SRR shown in Fig. 1(d). However, two resonances valleys appear when the SRR is excited by z-polarized light. The wavelength of these resonances aligns with the two 3D-Fano resonances observed in Fig. 1(f) when the MH-SRR array is excited by y-polarized light. This indicates that these two Fano resonances arise not from the SRR be excited by y-polarized light but rather from being excited by z-polarized light, resulting in the coupling of the two resonant modes of the SRR array with the resonant mode of the MH array, leading to Fano resonances.
To further illustrate the resonance modes in SRR array under x-polarized, y-polarized and z-polarized light excitation, the charge density distribution and field distribution within one unit cell of SRR array is simulated and shown in Fig. 3. Figure 3(a) shows the charge density distribution of the resonance mode at a wavelength of 3.5 µm under z-polarized light excitation, where the positive and negative charges are separately distributed at the two ends of the SRR along the z direction. The electric current direction shown in Fig. 3(a) is along the z direction. Furthermore, the electric field distribution shown in Fig. 3(b) is a dipole distribution when the electric field direction is along the z-axis direction confirming to the incident light polarization direction. The charge density distribution and field distribution demonstrate that the resonance mode is an electric dipole mode [39]. Due to its narrow resonant spectral response at the wavelength of 3.5 µm and being a transmission valley, the energy circulates within the SRR. The resonant mode at the wavelength of 3.5 µm in the SRR provides a dark mode.
Fig. 3. (a) The charge density distribution with polarization directions indicated by black arrows and (b) the electric field strength and direction distribution at the top surface of one unit cell of SRR array when illuminated by incident light polarized in the z-direction at the wavelength of 3.6 µm, showing the feature of an electric dipole. (c) The charge density distribution with polarization directions indicated by black arrows and (d) the magnetic field strength and direction distribution at the top surface of one unit cell of SRR array when illuminated by incident light polarized in the z-direction at the wavelength of 5.6 µm. (e) The charge density distribution with polarization directions indicated by black arrows and (f) the magnetic field strength and direction distribution at the top surface of one unit cell of SRR array when illuminated by incident light polarized in the x-direction at the wavelength of 5.6 µm.
Figure 3(c) shows the charge density distribution of the resonance mode at a wavelength of 5.6 µm under z-polarized light excitation, where the positive and negative charges are separately distributed at the two ends of the SRR along the x direction. The electric current direction illustrated by arrows shown in Fig. 3(c) is a current loop flowing in the xz plane. Thus, the magnetic field direction is oriented along the y-axis. Furthermore, the corresponding distribution of magnetic field (H) with both strength and direction are calculated and plotted in Fig. 3(d) for further illustration. As shown in Fig. 3(d), the magnetic field direction is pointing to the y-axis direction, while the magnetic field intensity and direction show a magnetic dipole feature. The fact that magnetic field has a dipole distribution and electric current direction shows a current loop flowing clearly indicates the occurrence of magnetic dipole resonance mode excited by the incident electromagnetic wave at the wavelength of 5.6 µm. Figure 3(e) and Fig. 3(f) show the charge density distribution and magnetic field strength and direction respectively of the resonance mode at a wavelength of 5.6 µm under x-polarized light excitation, which have similar features with the resonance mode at a wavelength of 5.6 µm in the SRR array under z-polarized light excitation. Due to its narrow resonant spectral response at 5.6 µm wavelength and being a transmission valley, the energy circulates within the SRR. Therefore, the resonant mode at the 5.6 µm as shown in Fig. 3(e) and Fig. 3(f) in the SRR array are magnetic dipole resonance mode, and it provides the dark mode.
By integrating the charge distributions of MH and SRR under the different polarizations, we can further analyze the distinct resonance curves observed for MH-SRR under x and y-polarized excitation. When the MH-SRR is excited by x-polarized light, the SRR contacts with two polarity types of charges of the MH (similar to an electric quadrupole), while when the MH-SRR is excited by y-polarized light, the SRR contacts with only one polarity type of charge of MH. The latter charge distribution of the coupling between MH and SRR is stronger than that of the former (similar to an electric quadrupole). Consequently, compared to the x-polarized excitation, a more pronounced Fano resonance occurs under y-polarized excitation of MH-SRR.
3. Theory
In order to provide a better explanation for the Fano resonance resulting from the coupling between the resonant modes of the MH and SRR, we propose a theoretical solution using the TCMT to systematically analyze the Fano resonances discussed in the above sections. Figure 4 is the schematic of an optical resonator system coupled with two physical ports. ${S_{1 + }}$ and ${S_{2 + }}$ are the amplitudes of the incoming waves in the ports of the resonator, respectively, while ${S_{1 - }}$ and ${S_{2 - }}$ are the amplitudes of the outcoming waves in the ports of the resonator, respectively.
We first calculate the mode field amplitude and transmission spectrum of the MH array by the TCMT as following Eq. (1), where ${\omega _m}$ is the center frequency of the resonance, ${\gamma _{0\textrm{m}}}$ is the intrinsic loss rate due to for the material absorption of MH structure, ${\gamma _\textrm{m}}$ is the external leakage rate due to the coupling of the resonance to the outgoing wave, respectively. The amplitude ${a_m}\; $ is normalized such that ${|a |^2}$ corresponds to the energy inside the resonator. The resonant mode is excited by the incoming waves from port 1 and part of the energy exits the resonant cavity, while part of the energy is directly reflected out. We assume that ${S_{2 + }} = 0$ and there are no incoming waves from port 2$.$ The resonant cavity is symmetrical at both ends, and the inherent losses of the waveguide are ignored. According to energy conservation and TCMT [31,40], we can get
In the frequency domain, the mode magnitude can be described as $\frac{{d{a_m}}}{{dt}} ={-} \mathrm{j\omega }{a_m}$. Therefore, through Fourier transformation, the amplitude ${a_m}\textrm{}$ of the resonant mode and the transmission coefficient can be determined as
The transmission spectrum is of Lorentzian line shape, and the full width at half maximum (FWHM) is 2$({{\gamma_{0\textrm{m}}} + 2{\gamma_m}} )$. Figure 5(a) shows the calculated transmission spectrum by the TCMT solution using Eq. (3), and it is in good agreement with the simulation spectrum of a metallic hole under y-polarized light excitation. The fitting parameter in Eq. (3) is ${\omega _m} = 5.54 \times {10^{14}}$ Hz, ${\gamma _{0\textrm{m}}} = 2 \times {10^{12}}$ Hz and ${\gamma _\textrm{m}} = 8.5 \times {10^{13}}$ Hz, respectively. The fitting parameters are obtained from the numerical simulation.
Fig. 5. The calculated transmission spectrum by TCMT theory compared with the simulation spectrum of the MH array (a) without SRR structure or (d) with SRR structure under y-polarized light excitation. The calculated transmission spectrum by TCMT theory compared with the simulation spectrum of the SRR array without MH structure under (c) z-polarized or (e) x-polarized light excitation. The calculated transmission spectrum by TCMT theory is compared with the simulation spectrum of a (b) 3D MH-SRR array or (f) 2D MH-SRR array under x-polarized light excitation.
We then calculate the mode field and transmission spectrum of the SRR array by the TCMT as following Eq. (4), where ${\omega _s}$ is the center frequency of the resonance, ${\gamma _{0\textrm{s}}}$ is the intrinsic loss rate due to for the material absorption of SRR, ${\gamma _\textrm{s}}$ is the external leakage rate due to the coupling of the resonance to the outgoing wave, respectively. The amplitude ${a_s}\textrm{}$ is normalized such that ${|a |^2}$ corresponds to the energy inside the resonator. The resonant mode is excited by the incoming waves and part of the energy exits the resonant cavity, while part of the energy directly transmits out. We assume that ${S_{2 + }} = 0$ and there are no incoming waves from port 2. The resonant cavity is symmetrical at both ends, and the inherent losses of the waveguide are ignored. According to energy conservation and TCMT [31,40], they can be expressed as:
In the frequency domain, the mode magnitude can be described as $\frac{{d{a_s}}}{{dt}} ={-} \mathrm{j\omega }{a_s}$. Therefore, the amplitude ${a_s}\textrm{}$ of the resonant mode and the transmission coefficient can be determined as
The transmission spectrum is anti-Lorentzian line type, and the FWHM is 2$({{\gamma_0} + 2{\gamma_s}} )$, which is much smaller than 2$({{\gamma_{0\textrm{m}}} + 2{\gamma_m}} )$. Figures 5(c) and 5(e) show the calculated transmission spectrum by the TCMT solution using Eq. (6), it is in good agreement with the simulation spectrum of the SRR array without MH structure under z-polarized and x-polarized light excitation, respectively. The fitting parameters are obtained from the simulation. The fitting parameters in Eq. (6) for the dark mode 1 at resonance wavelength of 5.6 µm are ${\omega _s} = 3.36 \times {10^{14}}$ Hz, ${\gamma _{0\textrm{s}}} = 6 \times {10^{12}}$ Hz, and ${\gamma _\textrm{s}} = 5 \times {10^{12}}$ Hz, respectively. The fitting parameter in Eq. (6) for dark mode 2 at resonance wavelength of 3.5 µm is ${\omega _s} = 5.38 \times {10^{14}}$ Hz, ${\gamma _{0\textrm{s}}} = 0$ Hz, and ${\gamma _\textrm{s}} = 1 \times {10^{13}}$ Hz, respectively. The fitting parameters in Eq. (6) of dark mode 3 of the SRR array without MH structure under z-polarized light excitation at the resonance wavelength of 5.6 µm are ${\omega _s} = 3.36 \times {10^{14}}$ Hz, ${\gamma _{0\textrm{s}}} = 6 \times {10^{12}}$ Hz, and ${\gamma _\textrm{s}} = 2 \times {10^{12}}$ Hz, respectively.
We then calculate the mode field and the transmission spectrum of the MH-SRR array based on the mode field amplitude as discussed before. We have found by simulation that x-direction (or y-direction) and z-direction components of electromagnetic waves of far field of MH are 2∼3 times smaller than the component in the y-direction (or x-direction) when the MH is excited by the y-polarized (or x- polarized) light. Moreover, we have found by simulation that y-direction and z-direction components of electromagnetic waves of far field of SRR are 2∼3 times smaller than the component in the x-direction when the SRR is excited by the x-polarized or z-polarized light. Considering the complex and extensive nature of theoretical calculations involving the coupling of vector fields and to simplify the theoretical derivation and due to these far field behaviors, we have approximated the coupling between MH and SRR by the scalar fields. When the resonance modes of the MH and SRR are in coupling, the energy of the new coupling mode field per unit frequency interval is considered to be the square of the superposition of the two mode amplitudes as following Eq. (7). Here ${e_m}$ and ${e_s}$ are the superposition coefficient, ${e_m} = b\textrm{exp}({ - i{\varphi_0}} )$ and ${e_s} = c\textrm{exp}({ - i({{\varphi_0} + \Delta \varphi } )} )$, b and c are arbitrary constant. The parameter $\Delta \varphi $ represents the phase difference between the MH and SRR when they are in coupling. We find
In Eq. (7), the first term represents the isolated effect of the MH, the second term represents the isolated effect of the SRR, while the parameters b and c respectively represent the contribution levels of the individual effects of the MH and SRR. The third and fourth terms in Eq. (7) represent the interaction effects between the MH and SRR, while the parameter $\Delta \varphi $ represents the phase difference between the MH and SRR when they are in coupling. The spectral lines of the third and fourth terms exhibit an asymmetric profile. Therefore, the Fano resonance is generated by the contribution of three parts, namely the individual influence of the MH, the individual influence of the SRR, and the interaction of the MH and SRR. Each component contributes to the overall spectral response, leading to the observed Fano resonance phenomenon. Figures 5(b), 5(d) and 5(f) show the comparison of calculated transmission spectrum by the TCMT solution using Eq. (7) and the simulation spectrum of the 3D MH-SRR array under x-polarized and y-polarized light excitation and a 2D MH-SRR array under x-polarized light excitation, respectively. The theoretically solved transmission spectrum as displayed in Figs. 5(b), 5(d) and 5(f) show a good agreement with the transmission spectrum obtained from numerical simulations.
The fitting parameters ${\gamma _{0\textrm{m}}} + 2{\gamma _m}$, ${\gamma _{0\textrm{s}}} + 2{\gamma _s}$, ${\omega _m}$ and ${\omega _s}$ are obtained from the individual MH array or SRR array simulation, while the other fitting parameters b, c and $\Delta \varphi $ are adjusted to make the transmission spectrum of simulation and TCMT theory to match closely. The 3D Fano resonance dip at the wavelength of 5.6 µm as shown in Fig. 5(b) is the result of the coupling of the bright mode and dark mode 3 under x-polarized light excitation, where the fitting parameters in Eq. (7) are b = 2.2, c = 0.1, and $\Delta \varphi = \pi /6$. It can be observed that the 3D MH-SRR array do not exhibit Fano resonance under x-polarized light excitation. This is also supported by the fitted values of parameters b and c, where the parameter b and c respectively represent the contribution levels of the individual effects of the MH and SRR. The small value of c indicates that the individual contribution of the SRR is minimal. Consequently, the interaction term between the MH and SRR is also small, leading to the absence of Fano resonance in this configuration. However, as shown in Fig. 5(f), significant Fano resonance can be observed in the case of the 2D MH-SRR array under x-polarized light excitation. The 2D Fano resonance dip at the wavelength of 5.4 µm as shown in Fig. 5(f) is the result of the coupling of the bright mode and dark mode 3 under x-polarized light excitation, where the fitting parameters in Eq. (7) are b = 1.5, c = 0.35, and $\Delta \varphi = 2\pi /9$. The value of c is relatively large compared to the case of 3D MH-SRR array, indicating that the individual effect of the SRR and the interaction term between the MH and SRR are much stronger. Thus, when the 2D MH-SRR array is under the indication of x-polarized light, a Fano resonance can be observed at a wavelength of 5.6 µm in Fig. 5(f).
The 3D Fano resonance dip at the wavelength of 3.6 µm as shown in Fig. 5(d) is the result of the coupling of the bright mode and dark mode 1 under y-polarized light excitation, where the fitting parameters in Eq. (7) are b = 1.6, c = 0.4, and $\Delta \varphi = 10\pi /13$. When the resonant frequencies of the bright and dark modes are close to each other, the third term in Eq. (7), which represents the interaction term between MH and SRR, no longer exhibits an asymmetric spectrum but rather a Lorentzian line shape. However, the fourth term in Eq. (7) still remains as an asymmetric spectrum. Additionally, the value of c is relatively large compared to the case of x-polarized incident light, indicating that the individual effect of the SRR and the interaction term between the MH and SRR are stronger. Thus, under the indication of y-polarized light, a Fano resonance can be observed at a wavelength of 3.6 µm in Fig. 5(d). Moreover, Eq. (7) is not the exact solution for the 3D Fano resonance. It is merely a theoretical solution by superimposing the amplitudes of the bright mode and dark mode within the framework of TCMT, which does have certain limitations dealing with the coupling of the resonant modes that have closer wavelengths. Therefore, the Fano resonance fitting curves at short wavelength shown in Fig. 5(d) deviate somewhat from the simulation results. The 3D Fano resonance dip at the wavelength of 5.6 µm as shown in Fig. 5(d) is the result of the coupling of the bright mode and dark mode 3 under y-polarized light excitation, where the fitting parameters in Eq. (7) are b = 2.5, c = 0.9, and $\Delta \varphi = \pi /2$. When $\Delta \varphi = \pi /2$, the third term in Eq. (7) becomes zero, and the interaction term is mainly provided by the fourth term in Eq. (7). Since c = 0.9, it implies that the individual contribution of the SRR in the second term of Eq. (7) and the interaction term between the MH and SRR are much larger compared to the cases shown in Fig. 5(b) and Fig. 5(f) with the resonance occurring at 5.6 µm. Because the mutual coupling between the MH and SRR is quite strong, a much more pronounced Fano resonance happens at 5.6 µm compared to the cases shown in Fig. 5(b) and Fig. 5(f) that also happen at 5.6 µm.
4. Conclusion
In summary, we have systematically investigated the physics underlying the plasmonic Fano resonance induced by the coupling between MH and SRR arranged in 2D and 3D microstructures. Based on the simulation results of the transmission spectra for MH array and SRR array under different polarized light excitation, we figure out the resonance modes of these two arrays, and extract all the crucial parameters of the Lorentzian or anti-Lorentian line shape. We further investigate the electromagnetic field and charge density distribution corresponding to these resonant modes at the peak or valley wavelength of the transmission spectrum, which manifests as either electric or magnetic dipole resonant mode. We have found that the Fano resonances in 3D or 2D MH-SRR array is due to the coupling of the bright mode provided by the MH array and the dark mode provided by the SRR array. We then establish a theoretical solution for such dark-bright-mode coupled systems based on the TCMT, which effectively elucidates that the asymmetry of the Fano resonance is caused by the coupling between the two structures and the coupling strength of two structures is responsible for different asymmetric degrees of Fano resonances. The theoretical solution of transmission spectrum closely matches with the numerically simulated transmission spectrum, confirming its accuracy and validity. Our findings provide a deeper physical understanding on such complex systems, and pave the way to engineer the coupling of various Lorentzian resonant mode to create Fano resonance with diversified features.
Funding
Guangdong Innovative and Entrepreneurial Research Team Program Innovative and Entrepreneurial Talents; National Natural Science Foundation of China (11974119); National Key Research and Development Program of China (2018YFA 0306200).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Reference
1. U. Fano, “Effects of Configuration Interaction on Intensities and Phase Shifts,” Phys. Rev. 124(6), 1866–1878 (1961). [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. B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010). [CrossRef]
4. M. Limonov, M. Rybin, A. Poddubny, and Y. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017). [CrossRef]
5. W. J. Chen, J. C. W. Lee, J. W. Dong, C. W. Qiu, and H. Z. Wang, “Fano resonance of three-dimensional spiral photonic crystals: Paradoxical transmission and polarization gap,” Appl. Phys. Lett. 98(8), 081116 (2011). [CrossRef]
6. C. Wu, N. Arju, G. Kelp, J. Fan, J. Dominguez, E. Gonzales, E. Tutuc, I. Brener, and G. Shvets, “Spectrally selective chiral silicon metasurfaces based on infrared Fano resonances,” Nat. Commun. 5(1), 3892 (2014). [CrossRef]
7. K.C. Smith, A. Olafsson, X. Hu, S. Quillin, J. C. Idrobo, R. Collette, P. Rack, J. Camden, and D. Masiello, “Direct observation of infrared plasmonic Fano antiresonances by a nanoscale electron probe,” Phys. Rev. Lett. 123(17), 177401 (2019). [CrossRef]
8. C. Wu, A. Khanikaev, R. Adato, N. Arju, A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat. Mater. 11(1), 69–75 (2012). [CrossRef]
9. S. Simoncelli, Y. Li, E. Cort’es, and S. A. Maier, “Imaging Plasmon Hybridization of Fano Resonances via Hot-Electron-Mediated Absorption Mapping,” Nano Lett. 18(6), 3400–3406 (2018). [CrossRef]
10. P. Alonso-Gonzalez, M. Schnell, P. Sarriugarte, H. Sobhani, C. Wu, N. Arju, A. Khanikaev, F. Golmar, P. Albella, L. Arzubiaga, F. Casanova, L. Hueso, P. Nordlander, G. Shvets, and R. Hillenbrand, “Real-Space Mapping of Fano Interference in Plasmonic Metamolecules,” Nano Lett. 11(9), 3922–3926 (2011). [CrossRef]
11. H. Lu, X. Liu, D. Mao, and G. Wang, “Plasmonic nanosensor based on Fano resonance in waveguide-coupled resonators,” Opt. Lett. 37(18), 3780–3782 (2012). [CrossRef]
12. M. Wang, A. Krasnok, T. Zhang, L. Scarabelli, H. Liu, Z. Wu, L. Liz-Marzán, M. Terrones, A. Alù, and Y. Zheng, “Tunable Fano Resonance and Plasmon–Exciton Coupling in Single Au Nanotriangles on Monolayer WS2 at Room Temperature,” Adv. Mater. 30(22), 1705779 (2018). [CrossRef]
13. N. Dabidian, I. N. Kholmanov, A. B. Khanikaev, K. Tatar, S. Trendafilov, S. H. Mousavi, C. W. Magnuson, R. S. Ruoff, and G. Shvets, “Electrical switching of infrared light using graphene integration with plasmonic Fano resonant metasurfaces,” ACS Photonics 2(2), 216–227 (2015). [CrossRef]
14. F. Hao, Y. Sonnefraud, P. V. Dorpe, S. Maier, N. 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]
15. S. P. Zhang, K. Bao, N. Halas, H. X. Xu, and P. Nordlander, “Substrate-Induced Fano Resonances of a Plasmonic Nanocube: A Route to Increased-Sensitivity Localized Surface Plasmon Resonance Sensors Revealed,” Nano Lett. 11(4), 1657–1663 (2011). [CrossRef]
16. S. Zhang, D. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-Induced Transparency in Metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008). [CrossRef]
17. A. Cui, Z. Liu, J. F. Li, T. Shen, X. Xia, Z. Y. Li, Z. Gong, H. Li, B. Wang, J. Li, H. Yang, W. Li, and C. Gu, “Directly patterned self-supporting plasmonic ‘nanograter’ structures with unusual Fano resonances,” Light: Sci. Appl. 4(7), e308 (2015). [CrossRef]
18. X. Tian, Z. G. Liu, H. Lin, B. Jia, Z. Y. Li, and J. F. Li, “Five-fold plasmonic Fano resonances with giant bisignate circular dichroism,” Nanoscale 10(35), 16630–16637 (2018). [CrossRef]
19. Z. G. Liu, Z. Liu, J. Li, W. Li, J. F. Li, C. Gu, and Z. Y. Li, “3D conductive coupling for efficient generation of prominent Fano resonances in metamaterials,” Sci. Rep. 6(1), 27817 (2016). [CrossRef]
20. Z. Liu, J. F. Li, Z. Liu, W. Li, J. Li, C. Gu, and Z. Y. Li, “Fano resonance Rabi splitting of surface plasmons,” Sci. Rep. 7(1), 8010 (2017). [CrossRef]
21. Y. Zhan, D. Y. Lei, X. Li, and S. A. Maier, “Plasmonic Fano resonances in nanohole quadrumers for ultra-sensitive refractive index sensing,” Nanoscale 6(9), 4705–4715 (2014). [CrossRef]
22. W. S. Chang, J. B. Lassiter, P. Swanglap, H. Sobhani, S. Khatua, P. Nordlander, N. Halas, and S. Link, “A plasmonic Fano switch,” Nano Lett. 12(9), 4977–4982 (2012). [CrossRef]
23. C. Jiang, L. Jiang, H. Yu, Y. Cui, X. Li, and G. Chen, “Fano resonance and slow light in hybrid optomechanics mediated by a two-level system,” Phys. Rev. A 96(5), 053821 (2017). [CrossRef]
24. E. Prodan, C. Radloff, N. J. Halas, and P. nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003). [CrossRef]
25. B. Xi, H. Xu, S. Xiao, and L. Zhou, “Theory of coupling in dispersive photonic systems,” Phys. Rev. B 83(16), 165115 (2011). [CrossRef]
26. H. Xu, Q. He, S. Xiao, B. Xi, J. Hao, and L. Zhou, “Tight-binding analysis of coupling effects in metamaterials,” J. Appl. Phys. 109(2), 023103 (2011). [CrossRef]
27. B. Xi, M. Qiu, S. Xiao, H. Xu, and L. Zhou, “Effective model for plasmonic coupling: A rigorous derivation,” Phys. Rev. B 89(3), 035110 (2014). [CrossRef]
28. J. Lin, M. Qiu, X. Zhang, H. Guo, Q. Cai, S. Xiao, Q. He, and L. Zhou, “Tailoring the lineshapes of coupled plasmonic systems based on a theory derived from first principles,” Light: Sci. Appl. 9(1), 158 (2020). [CrossRef]
29. H. A. Haus and W. Huang, “Coupled-Mode Theory,” Proc. IEEE 79(10), 1505–1518 (1991). [CrossRef]
30. S. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80(6), 908–910 (2002). [CrossRef]
31. 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]
32. W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004). [CrossRef]
33. Z. Ruan and S. Fan, “Temporal Coupled-Mode Theory for Fano Resonance in Light Scattering by a Single Obstacle,” J. Phys. Chem. C 114(16), 7324–7329 (2010). [CrossRef]
34. Q. Li, T. Wang, Y. Su, M. Yan, and M. Qiu, “Coupled mode theory analysis of mode-splitting in coupled cavity system,” Opt. Express 18(8), 8367–8382 (2010). [CrossRef]
35. Z. Ruan and S. Fan, “Temporal coupled-mode theory for light scattering by an arbitrarily shaped object supporting a single resonance,” Phys. Rev. A 85(4), 043828 (2012). [CrossRef]
36. Z. Li, Q. Zhu, Y. Wang, and S. Xie, “Ultrasensitive optical reflectivity in annular nanohole array on Photonic crystal slab based on bound states in the continuum,” Sci. Rep. 8(1), 12455 (2018). [CrossRef]
37. S. Bakhti, A. Tishchenko, and X. Zambrana-Puyalto, “Fano-like resonance emerging from magnetic and electric plasmon mode coupling in small arrays of gold particles,” Sci. Rep. 6(1), 32061 (2016). [CrossRef]
38. W. Huang, J. Lin, M. Qiu, T. Liu, Q. He, S. Xiao, and L. Zhou, “A complete phase diagram for dark-bright coupled plasmonic systems: applicability of Fano’s formula,” Nanophotonics 9(10), 3251–3262 (2020). [CrossRef]
39. Q. R. Deng, J. F. Chen, L. Long, B. Q. Chen, H. K. Yu, and Z. Y. Li, “Silicon cuboid nanoantenna with simultaneous large Purcell factor for electric dipole, magnetic dipole and electric quadrupole emission,” Opto-Electron. Adv. 5(2), 210024 (2022). [CrossRef]
40. H. A. Haus, “Waves and Fields in Optoelectronics,” New Jersey, Prentice Hall (1984).
