Double Fano resonances excited in a compact structure by introducing a defect

Surface plasmon polaritons (SPPs) are considered to be the most promising candidates for the implementation of highly integrated optical circuits, because of their ability to overcome the diffraction limit of light [1]. This collective phenomenon is of enormous interest for a variety of exciting phenomena [2]. As a fundamental resonant effect, Fano resonances are naturally analyzed. Fano resonances result from the destructive interference between the superradiant (bright) and subradiant (dark) modes, depending on the modes coupling strength. Those modes overlap spectrally and spatially with the transmissions spectrum features characteristic narrow and asymmetric line shape. Fano resonances can be tuned by optimizing the structure geometry and strengthening the plasmon coupling [2]. The narrower line width of Fano resonances is believed to be very sensitive to the dielectric environment variation and can be applied in biological and chemical sensing [3,4], surface-enhanced Raman scattering (SERS) [5] and slow-light devices [6]. Therefore, Fano resonance has been theoretically and experimentally studied in metamaterials [7–9], plasmonic waveguide-cavity structure [10–12], nanostructure dimers [13–15] systems. The resonance appearances become more important and have drawn extraordinary interests [16,17]. However, most researchers keep their interests on the generation of single-mechanism Fano resonance in various plasmonic systems. When two nanostrips are placed in close proximity, the incident light can couple into plasmonics through near-field interactions and give rise to a hybridized phenomenon [18–22]. Fano resonances can be formed via the symmetry breaking, which can produce plasmonic coupling [23]. In addition, using a unit-cell structure side-coupled with detuned resonators, the Fano resonances are also theoretically predicted [24–28] and even experimentally demonstrated [29]. However, very few numerical investigations with analytical description have been performed systematically on the Fano resonances, which are meanwhile excited by different mechanisms amazingly involving the quadrupole as superradiant (bright) mode to excite other mode, although this is very meaningful for the manipulation of SPPs in the near-infrared domain.

In this letter, we will demonstrate that Fano resonances based on two different mechanisms can be achieved in two coupled silver nanostrips where one is defective. Numerical results from finite-difference time-domain (FDTD) simulations will show that the mode hybridization between the dipole resonances supported by the two detuned nanostrips gives rise to the right-side Fano resonance, while the left-side high-Q Fano resonance is derived from the destructive interference between the quadrupole resonances supported by the two nanostrips where the defective one is superradiant and the complete one is subradiant. The evolution of the Fano resonances can be attributed to the coupling distance and structural parameters of the defect. More interestingly, multiple Fano resonances will be achieved by simply adjusting the polarization of the electric field rather than the conventional method, i.e. by adding more resonance structures. These results may provide a new horizon in the design of plasmonic devices based on Fano resonance.

The investigated resonance system is simply composed of two nanostrips side by side with a defect in the left or right at silver nanostrip 1. (To facilitate the following description, we use the "1" and "2" to represent as reference the label of fig. 1(a).) Otherwise, the structure can be approximated as a U-shape split-ring resonator (SRR) and a complete nanostrip although the defect of the nanostrip is small compared with the actual U-shape SRR. Notably, the location of the defect can be changed. If we place the defect at different positions in nanostrip 1 together with the nanostrip 2 sitting, the nanostructure also possesses a similar Fano resonance line shape. Figure 1(b) schematically depicts the kind of structure arrangement method. In this letter, we mainly investigate the case of fig. 1(a). The main parameters of the nanostrips are the length $(l)$, width $(w)$ and the thickness $(d)$, the coupling distance $(g)$, and the width, length and the thickness of the defect (s, t and d, respectively). These two nanostrips are placed on a substrate that the period is p and refractive index is set to be 1. The corresponding geometric parameters in fig. 1(a) are given as follows: $l = 300\ \text{nm}$, $w = 100\ \text{nm}$, $s = 30\ \text{nm}$, $t = 90\ \text{nm}$, $d = 30\ \text{nm}$, $g = 100\ \text{nm}$ and $p = 500\ \text{nm}$. The incident plane wave illumination is taken to be along the backward z-direction with the polarization along the y-direction $(\theta = 90^{\circ})$ in the simulations. To perform our numerical calculations, the computational domain is truncated by perfectly matched layers (PMLs) in the z-direction and the periodical boundry in the x-direction and y-direction.

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Figure 2(a) shows that the transmission of single nanostrip 1 (approximate U-shape SRR) supports both of the superradiant (bright) dipole and quadrupole modes. Herein, the fig. 2(c) quadrupole mode at 536 nm and the fig. 2(d) dipole mode at 1008 nm are clearly identified, which are directly excited under the normal illumination with polarization along the y-axis. Figure 2(b) presents the transmission of the complete silver nanostrip 2 where only a dipole mode is emerged at 906 nm, as the field distributions shown in fig. 2(e). Importantly, both of the dipole and quadrupole modes can be directly excited in the nanostrip 1 by introducing the defect. It is easy to get that Fano resonance is governed by the introduction of the defect because of the different transmission distributions and propagation characteristics of these two nanostrips in the structure.

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The Fano resonances can arise from significant involvement of higher order multipole modes. Hence, the interference of the nanostrip 1 with the complete silver nanostrip 2 results in the appearance of apparent Fano resonance in the transmission. Figure 3(a) exhibits the transmission for the structure of silver nanostrips. The single nanostrip 1 (approximate U-shape SRR) transmission is denoted by red line, while the single nanostrip 2 is denoted by black line, and the blue line represents coupled transmission spectra. It is observed that two narrow asymmetric profiles together with a broad Lorentzian-like profile emerge in the coupled transmission spectrum where three dips are located at 555, 804 and 1106 nm, respectively. To better learn the physical origin of these plasmon resonance dips, the electric-field distributions of E_z at the dips are displayed in figs. 3(b)–(d). Here, the defective nanostrip 1 (approximate U-shape SRR) supports both of the superradiant (bright) dipole and quadrupole modes, but the complete nanostrip 2 supports a bright dipole mode. It should be noted that the quadrupole mode of the nanostrip 2 is not directly excited by the plan wave [30]. The superradiant (bright) mode is excited with the nanostrip electric resonance, while the nanostrip 2 quadrupole mode is excited due to the coupling from the nanostrip 1 (approximate U-shape SRR) bright quadrupole mode.

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Next, we explore theoretically the transmission spectra of the whole system in fig. 1(a). The left-side high-Q Fano resonance dip b around 555 nm is originated from the "bright-dark" coupling between superradiant and subradiant modes. Here, the "bright-dark" modes are excited in turn through the pathway of $|I\rangle\to|B\rangle1\to|D\rangle2\to|B\rangle1$, where $|I\rangle$ is the incident excitation source, $|B\rangle1$ is the bright quadrupole mode in the nanostrip 1 which is aligned with the excitation polarization and acted as resonant nanostrip, $|D\rangle2$ is the dark quadrupole mode in the complete nanostrip 2 [31]. Since the excitation light is normally incident with polarization along the y-axis, the quadrupole in the defective nanostrip 1 can be excited and exhibits as a superradiant (bright) mode, while the quadrupole in the complete nanostrip 2 cannot be excited directly by the light and exhibits as a subradiant (dark) mode. Thus, the defective nanostrip 1 quadrupole as a bright mode excites the complete nanostrip 2 dark quadrupole mode forming the Fano resonance. Moreover, the Fano resonance dip b based on the couplings between quadrupole modes is very interesting and sharp. We calculated that the Q-factor of the Fano resonance b is 62, which is larger than 20 [7] and 50 [32]. Thus, the proposed structure may have potential applications for nanosensor in highly integrated circuits. In the proposed structure, due to the existence of the defect, the complete nanostrip 2 quadrupole mode is excited. Therefore, the defect plays important roles in exciting the Fano resonances in our designed structure. It is observed that there are two strong electric-field distributions located in the region of the nanostrips and two strong quadrupole modes are in-phase at 555 nm. The asymmetric Fano resonance b in the transmission is therefore attributed to the interaction of the superradiant and subradiant modes. Thus, the bright quadrupole mode of the nanostrip 1 coupled with the dark quadrupole mode of the nanostrip 2 as one resonance pathway form a Fano resonance b dip in the plasmonic resonator, while the other resonance pathway is dipole modes of two nanostrips directly excited by the incident excitation source. It can be inferred that the Fano resonance can be caused by destructive interference of superradiant (bright) mode resonance and subradiant (dark) mode resonance in the system [30]. When the defect was placed in the left of silver nanostrip 1, as shown in fig. 1(b), the nanostructure also possesses a similar Fano resonance line shape. The asymmetric Fano resonance dip f in the transmission is also attributed to the interaction of the superradiant and subradiant modes. The differences between two arrangement methods of defect are quadrupole mode opposite phase and the different resonance strength due to the different position of the defect.

Otherwise, a broad Lorentzian-like profile c around 804 nm is further discussed. It is evident that there is an intriguing phenomenon of superposition beats in the plasmon resonance, when the defective nanostrip 1 dipole mode interacts with the complete nanostrip 2 dipole mode. It corresponds to a broad bright mode due to the radiative damping, which those two nanostrips have similar damping rates, as shown in figs. 2(d), (e). The electric-field distribution at 804 nm (fig. 3(c)) clearly demonstrates the above analysis. By coupling of the silver nanostrip pairs, the broad dip wavelength exhibits a blue-shift. At the resonance position, the spectral line shapes show more symmetry likely. It can be regarded as a dipole mode on the whole system. When the defect was placed as the fig. 1(b), it can form resonance dip h. The right-side Fano resonance dip d at 1106 nm is also detailed analyzed. Both the nanostrip 1 and nanostrip 2 support in-phase dipole mode at 1008 and 906 nm, respectively, as shown in figs. 2(d), (e). However, for the total coupled system, the electric-field distribution of the nanostrip 1 and nanostrip 2 are out-of-phase, as shown in fig. 3(d). The two nanostrips are coupled strongly, and the coupling induces large field enhancement of nanostrips. Thus, it can be explained that the Fano resonance d is caused by mode hybridization between the dipole resonances supported by the two detuned nanostrips. (The Fano resonance k is caused by mode hybridization between the dipole resonances supported by the two detuned nanostrips.) Moreover, the electric-field distributions can be viewed as a quadrupole mode of the entire system. The reason why the red-shift of Fano resonance dip b and Fano resonance dip d is the hybridization between the defective nanostrip and the complete nanostrip [33]. And a modification of the circumstance in or around the nanostrips affects the resonant mode: its resonance wavelength will change compared with that of the single nanostrip. These two kinds of red-shifts are different because the left-side Fano resonance dip b red-shift is originated from the defective nanostrips quadrupole mode wavelength shifts, while the right-side Fano resonance dip d red-shifts is attributed to the dipole modes wavelengths of two nanostrips shift. The nanostructure possesses great meanings that double different Fano resonances are simultaneously excited at one system.

Next, we investigated the transmission properties of the proposed structure. Figure 4(a) shows the calculated transmission spectra for different coupling distance g between the nanostrip pairs. It is obvious that both the Fano resonance dips (A and C) are proportional to the coupling distance g. Specially, we can find that the resonance dips A and C increase with the coupling distance g increasing, which may attribute the radiative damping rates decreases when the coupling distance g increases. With the gap g increasing from $g = 50$ to 100 nm, the dip C shows a blue-shift of about 36 nm (from 1143 to 1107 nm) and the dip A also shows a blue-shift of about 15 nm (from 570 to 555 nm). According to the simulation results, the Fano resonance wavelengths can be easily manipulated by adjusting the coupling distance g between the silver nanostrips.

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Successively, we investigate the transmission spectra of the plasmonic resonator are calculated by varying the defect length $(t)$ and width $(s)$ when the other parameters of the system are fixed as same as fig. 1(a). By increasing the defect length t, the strength of the Fano resonance dip C decreased at first and then increased with exhibiting a red-shift. The red-shift of Fano resonance dip C can be explained as resulting from the degree of nanostrip detuning where the degree is influenced by the variation of the defect length t. It is noted that the slight variations of the length has little influence on the Fano resonance A and the broad Lorentzian-like resonance B, as shown in fig. 4(b). In fig. 4(c), it is observed that the Fano resonance dip C wavelength of the profile increases linearly with s (the slope being about 33.4) while the Fano resonance dip A of the narrow Fano profile increases more and more slowly with s. Therefore, one can simply manipulate the Fano resonance wavelength by changing the defect length and the width.

To further probe the characterstics, the transmission spectra are calculated by varying the electric-field polarization angle when the other parameters of the system remain unchanged. It is observed that multiple Fano resonances together with a broad Lorentzian-like profile emerge in the transmission spectrum. Moreover, the Fano resonance transmission dips, corresponding to the resonant wavelengths, exhibit different dependences on electric-field polarization angle, as shown in fig. 4(d). There is high-order resonance mode Fano dip existing in the left of the transmission spectrum with the variation of the electric-field polarization angle from 15° to 75°. Until accessing to 90°, the higher modes vanish and the line shape becomes a dip. Based on the above calculated results and analysis, the Fano resonance dip A of the transmission spectrum is excited by the "bright-dark" coupling, whose resonance frequency is a constant value, while the Fano resonance dip C is formed by the detuning mode, whose resonance phase can shift to left or right, thus, there is high-order resonance mode Fano dip existing in the left of the transmission spectrum. So, the proposed Fano structure in fig. 3(a) is flexible and can be easily extended to a multiple Fano resonances system by changing the electric-field polarization angle, as shown in fig. 4(d). These behaviors of the Fano resonances accord well with the analysis. Thus, the utilization of the electric polarization opens up a possibility for designing high-performance plasmonic devices.

Therefore, the resonances and transmission windows of the hybrid structures can be tuned by varying the parameters of systems. Moreover, the defect is introduced to produce a small structure break approximated as a U-shape SRR, which can result in that both of the "bright-dark" coupling and the detuning modes Fano resonance dips are excited in the compact structure. Undoubtedly, the structure is full of charm owing to the two different-types Fano resonance dips.

In conclusion, the transmission characteristics of the proposed structure, which consists of two silver nanostrips and a defect, are analyzed and investigated. Simulation results show that by introducing a defect, which makes the silver nanostrip 1 support bright dipole and quadrupole modes, two asymmetric Fano dips emerge in the transmission spectrum, and then the mechanisms are significantly perturbed. Different-types Fano resonances are achieved at a nanostructure. Otherwise, the Fano line shape can be easily tuned by changing the parameters of the structure. In addition, multiple Fano resonances are achieved by varying the electric-field polarization angle. The utilization of a defect creates a path for realizing different-mechanisms Fano resonance in the plasmonic resonance system.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 61505052, 61176116, 11074069).