1. Introduction

Fano resonance is characterized by a distinct asymmetric line shape, which originates from the constructive and destructive interference of a narrow discrete resonance with a broad spectral line [1–3]. This type of resonance was originally studied in atomic and quantum systems, such as quantum dots, nanowires, and tunnel junctions [2–6]. Over the past few years, a number of nanoscale classical oscillator systems enabled by plasmonic nanostructures have been shown to support the Fano resonance, such as the metallic arrays [7–10], dolmen-type slab arrangements [11, 12], non-concentric ring/disk cavity [13–15], and finite clusters (or oligomers) of plasmonic nanoparticles [16–21]. Besides in plasmonic nanostructures, the Fano resonances were also observed in metamaterials and many works were based on the merging of plasmonics and metamaterials [22–26]. Recently, Fano phase resonances using grating structures have been attracting significant interest [27,28], and many applications to filtering and antennas have been discussed [29–31]. The fundamental principle for all these phenomena is based on the interference between the spectrally overlapping broad resonance and the narrow discrete resonance [3]. Therefore, the Fano resonance is very sharp, large and highly sensitive to the local environment, which can be used in sensing applications [32–34]. Recently, such multiple Fano resonances have also gained much attention [35].

In the optical region, the metallic nanostructures with localized surface plasmon (LSP) resonances are ideal systems for realizing classical oscillators at nanoscale [36,37]. However, the surface plasmons (SPs) including surface plasmon polaritons (SPPs) and LSPs cannot be realized at lower frequencies due to the lack of negative permittivity [38]. To solve the problem, spoof (or designer) SPPs which can imitate the natural SPP properties have been reported in the microwave and terahertz frequencies [39–41], in which the metal surfaces are drilled with sub-wavelength grooves or holes to achieve the confinement ability [34,35]. Recently, ultrathin spoof SPP structures have been proposed, which were fabricated on nearly zero-thickness metallic surfaces, and hence can be easily integrated [42–45]. On the other hand, spoof LSPs were firstly presented in 2012 using plasmonic metamaterials [46]. In this way, all the capabilities of natural LSPs in the optical regime can be transferred to lower frequencies. More recently, the spoof LSPs on an ultrathin corrugated metallic disk have been observed experimentally, which demonstrate the multipolar LSP behaviors [47].

In this work, we focus on the study of multiple Fano resonances arising from a side-by-side arrangement of two asymmetrically ultrathin spoof LSP structures. We have illustrated that the spoof LSPs are strongly dependent on the disk size and identified the origin of spoof LSPs. We show that for spoof LSPs, only the bright dipole mode can be directly excited by normal incident waves. As the incidence direction is parallel to the LSP structure, however, several narrow multipolar resonances are generated, which can be seen as dark modes in this system [36, 48]. When we put two appropriately-sized particles together with normal incidence of electromagnetic waves, strong coupling occurs between the two particles since the near fields of the bright dipole modes are overlapped and dark multipolar modes are excited. Then an asymmetric multiple Fano lineshape arises from the destructive interference of narrow discrete resonances with a broad spectral line. The dependence of the Fano lineshape on the polarization of incident waves is examined. We also experimentally demonstrate the validity of the proposed design and observe the multiple Fano resonances in the microwave frequency, which are supported by the numerical predictions. These multiple Fano resonances could be exploited for new opportunities for plasmonic devices in the microwave and terahertz regimes.

2. Spoof LSPs and dark modes

We first study an ultrathin spoof LSP particle. In Fig. 1(a) we schematically illustrate the geometry of an ultrathin spoof LSP particle, which is composed of an inner metallic disk of radius r = 2mm surrounded by N = 60 periodically radical metallic grooves with the periodicity d. Hence the LSP particle is in fact an ultrathin textured metallic disk, which is based on a 1mm-thick substrate with dielectric constant 5 and loss tangent 0.02. The grooves’ width is a = 0.5d and thickness of the textured disk is 0.018mm. As outer radius R varies from 8.5 to 9.5, 10.5, 11.5mm, the extinction cross sections (ECSs) of the LSP particle under grazing incidence are calculated using the commercial software, CST Microwave Studio 2012, which are the sums of absorption cross sections (ACSs) and scattering cross sections (SCSs, or radar cross sections, RCSs), as shown in Fig. 1(b). In fact, spoof LSPs arise from standing surface waves on the periodically grooves of textured metallic disks [47]. Hence, the spoof LSPs frequencies are conformed to the dispersion relations of spoof SPPs, whose asymptote frequency is mainly controlled by the groove depth. In order to gain a deeper insight into their relationship, we present the simulated dispersion curves for the similar ultrathin corrugated metallic structure with d = 1.1mm, a = 0.55mm, r = 2mm and different lengths of R shown in Fig. 1(c). It is clearly observed that the resonance-shift property of LSPs with respect to disk outer radius R actually results from the SPP-dispersion change of the corresponding grooves (inset of Fig. 1(c)) with different length. When the length R of groove increases from 8.5 mm to 11.5mm, the asymptote frequency of dispersion curves decreases from 6.3GHz to 4.5GHz, and redshifts of the spoof LSPs frequencies are observed, as plotted in Figs. 1(b) and 1(c). The more important thing is that the third resonances of spoof LSPs (in Fig. 1(b)) approximate the corresponding asymptote frequencies. Noting the reason for the variance between them is because the radial widths of grooves in LSPs are not same. For spoof SPPs, the more closes to asymptote frequency the work frequency is, the slower and more tightly confined to the corrugated metal structure the propagation is, as shown in Fig. 1(c). As a consequence, the intensity of the corresponding spoof LSPs is weaker, as shown in Fig. 1(b).

 

Fig. 1 The schematic picture of the ultrathin spoof LSP disk and the resonance properties. (a) An ultrathin spoof LSP structure, which is based on a thin dielectric substrate. (b) The calculated ECS spectra of textured metallic disks with different outer radius (R = 8.5 mm, 9.5 mm, 10.5 mm and 11.5mm). (c) Dispersion curves of spoof SPPs for the corrugated strips (see inset) with different length (R = 8.5 mm, 9.5 mm, 10.5 mm and 11.5mm). (d) The calculated ECS spectra for grazing and normal incidences, in which the green line corresponds to the grazing incidence, and the red line corresponds to the normal incidence.

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And then, we study the structure, which is illuminated by electromagnetic waves from two directions separately. Both polarizations of incidences are parallel to the X axis, as shown in Fig. 1(a). We illustrate the simulation results of ECSs in Fig. 1(d), in which the red and green lines correspond to illuminations at the normal incidence and grazing incidence, respectively. It is obvious that the ECS responses of two different incidences are quite distinct. For the grazing incidence from the left to the right, multiple extinction peaks (green line) are found in the ECS spectrum, indicating that multipolar resonances are excited.

In order to verify these high-order LSPs, we show the z component of near electric fields on an observation plane that is 0.5 mm above the textured disk at three resonance peaks (I, II, III) in Figs. 2(a)–2(c), in which the red and blue colors indicate the positive and negative values, respectively. Note that the color bars in these figures and following demonstrations have been adjusted to show the mode patterns clearly. The patterns in Figs. 2(a)–2(c) present the dipolar, quadrupolar and hexapolar resonances, corresponding to the natural LSP behaviors. However, when the structure is excited by plane waves propagating along the –Z direction, the ECS spectrum has only one peak (IV). To visualize the resonance mode, Fig. 2(d) plots the simulated near-field distribution excited by the normal incidence on the same observation plane, showing the unique dipolar resonance.

 

Fig. 2 The simulation results of near-electric-field distributions (a-c) correspond to the dipole mode, quadrupole mode and hexapole mode, which is illuminated by grazing incidence. (d) corresponds to the dipole mode which is illuminated by normal incidence.

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Comparing such two incidence situations, we notice that multipolar resonances can only be excited by grazing-incident plane waves. This is because the multipolar resonances of spoof LSPs are actually standing surface waves on the textured metallic disk. Imagine that a short pulse of grazing incident wave impinges on its side and then propagates along the groove. When the disk circumference equals integer wavelengths in grooves approximately, strong resonances appear and form quadrupolar resonance, hexapolar resonance, etc. On the contrary, the normal incident field cannot produce the same physics, and hence the multipolar resonances are very weak. Such multipolar resonances are called as “dark mode” for normal incidence, because only the dipolar resonance appears. Compared to the ECS of the dipolar resonance which is broad in spectrum, the multipolar resonances are very sharp. It is remarkable that this property observed on ultrathin spoof LSPs are much akin to the gold nano ring in the optical frequency due to the retardation [36, 48].

3. Fano resonances

To further control the lineshapes of spoof LSPs, Fano resonances are introduced to modify the lineshape of broad resonance by interacting with sharp resonances, which are typically the dark modes. For this purpose, we place two asymmetrically metallic disks in close proximity, as illustrated in Fig. 3(a), in which R₁ = 10.5mm, R₂ = 8.5mm, r₁ = r₂ = 2mm, g = 8mm and other parameters are set in Fig. 1(a). For normal incidence, the bright mode (i.e., dipolar resonance) of the smaller disk exerts a force on the dark modes (quadrupolar and hexapolar resonances) of the larger disk by coupling the two LSPs. Since the dipolar resonance is much broader than multipolar resonances, the Fano-resonance conditions are satisfied when they occur in the same spectral range.

 

Fig. 3 The schematic picture of the ultrathin spoof LSP dimer and the multiple Fano resonances. (a) The configuration of the LSP dimer. (b) The calculated ECS spectra for the LSP dimer and individual LSP disks. The black solid line indicates the case of LSP dimer, in which the dips are marked as M1 and M2. The red dotted line corresponds to the big disk, which is illuminated by grazing incidence. The blue dashed line corresponds to the small disk, which is illuminated by normal incidence. (c) Numerical simulation results of the near-electric-field distributions at the resonant frequencies M1 and M2.

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The structure is excited by a plane wave propagating along the –Z direction with the electric field polarized in the Y direction. Figure 3(b) illustrates the calculation result of the ECS spectra for the dimer (black solid line). The resulting asymmetric Fano lineshape at around 5GHz is clearly observed. The spectra display two dips at 4.77 and 5.2 GHz due to the interaction between bright (dipole) mode of the small disk and dark (multipole) modes of the big disk. To find out the link between the multiple Fano resonance lineshape and the spectral position of resonances of individual disks, Fig. 3(b) also shows the ECS spectra for the small disk (blue dashed line) under the normal incidence and for big disk (red dotted line) under the grazing incidence. Comparing the three spectra, it is clearly observed that the first ECS peak of dimer (the black line) is mainly affected by the dipolar resonance of the big disk (red dotted line), and no interference occurs from 3 to 4 GHz. We remark that there are slightly spectral shifts in the dimer spectra with respect to dipolar resonances of individual disks, due to the hybridization between two adjacent disks.

We observe that the multiple Fano lineshape occurs from 4GHz to 6GHz, which is the spectral overlap between the broad dipole mode and the narrow multiple modes. More importantly, the spectral positions of dips are in good agreements with the multipolar resonances (red dotted line). That is to say, the multiple Fano resonance of the dimer structure originates from the destructive interference between the bright dipole mode of the smaller particle and the dark multipole modes of the larger particle. To further reveal the nature of such resonances, we numerically calculate the Ez near-field distributions at the dips M1 and M2, as shown in Fig. 3(c). It is apparent that the dipole mode of the small disk and the quadrupole and hexapole modes of the big disk are coupled to each other. It follows that the dark modes of the big disk are excited by the bright mode of the small disk, and destructive interferences happen between them.

Then we examine the dependence of the extinction spectra on the polarization of incidence. We assume that the polarization direction along Y axis is 0°, and rotate the polarization direction clockwise with 45° steps in the XOY plane. We focus on the Fano spectrum band (4-6 GHz) and the simulation results are illustrated in Fig. 4(a). For the electric-field vector parallel to the Y axis (0°), the maximum electric field of the dipole mode is located on the left and right sides of the small disk, and hence the coupling strength between two disks is the strongest. As a consequence, the multiple Fano resonances are the most obvious. It is evident that the dips’ depths (M1 and M2) decrease with the increase of the polarization angle, because the coupling strength is reduced. When the electric-field vector is along with the X axis (90°), the multiple Fano resonances are weak, since the electric field of the dipolar resonance is weak in the gap. In this case, the excitation intensity to the dark modes is weak, and hence weakly destructive interference happens.

 

Fig. 4 The simulated and measured ECS spectra for different polarizations of the incident waves. (a) The simulated results with the polarization direction of 0° (blue), 45° (black), and 90° (red). (b) The measured results with the polarization direction of 0° (blue), 45° (black), and 90° (red). The inset of (b) presents a fabricated sample.

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In order to further investigate the dependence behavior on the polarization of incident wave, we fabricate a set of asymmetric disks which are conformal to the simulation condition, as shown in the inset of Fig. 4(b). To measure the extinction spectra, we use two horn antennas to transmit and receive electromagnetic waves, which are connected to the vector network analyzer (Agilent N5230C) by low-loss coaxial cables. The time domain gating technology in the vector network analyzer is applied for the transmission measurement to minimize the interference. We place the sample in the middle of two horn antennas. The experimental results are presented in Fig. 4(b), which have reasonable agreements to simulations, demonstrating how the experimental spectra changes with the polarization rotation. The small differences in dip frequencies and intensity observed in experiments are most likely due to the mismatching tolerance.

4. Conclusion

In conclusion, we have shown that the plasmonic resonances (bright and dark modes) of the ultrathin spoof LSP structure (corrugated metallic disk) are strongly dependent on size and incident directions. The multiple Fano lineshape are observed in the combination of two LSP disks. We have numerically and experimentally demonstrated that the Fano lineshape depends very sensitively on the polarization directions of incident waves. This structure has potential for a number of applications including LSP resonance sensors, biological detection and so on in the microwave and terahertz frequencies. An additional application that will be studied is free space filters. In this application, stop band can be modified by structure and polarization directions.