High-order localized spoof surface plasmon resonances and experimental verifications

We theoretically demonstrated and experimentally verified high-order radial spoof localized surface plasmon resonances supported by textured metal particles. Through an effective medium theory and exact numerical simulations, we show the emergence of these geometrically-originated electromagnetic modes at microwave frequencies. The occurrence of high-order radial spoof plasmon resonances is experimentally verified in ultrathin disks. Their spectral and near-field properties are characterized experimentally, showing an excellent agreement with theoretical predictions. Our findings shed light into the nature of spoof localized surface plasmons, and open the way to the design of broadband plasmonic devices able to operate at very different frequency regimes.

although 1D spoof plasmon structures were firstly associated with positive effective permittivity parameters 33 , it has been shown recently that there are also significant discrepancies between spoof SP and dielectric resonators 34 . There exist, though, a common feature shared by dielectric and spoof SPP waveguides. Both support high-order modes, associated with Fabry-Perot-like cavity resonances within the plane normal to the direction of propagation. Several theoretical studies have predicted the existence of these high-order spoof SPPs [32][33][35][36] , which have been also recently reported experimentally 37 .
In this article, we introduce and verify experimentally the emergence of high-order radial spoof LSPs in periodically corrugated metal particles. These modes resemble the optical whispering gallery modes 38 sustained by dielectric resonators, whose high Q-factor characteristics are currently being exploited in a wide range of technologies 39,40 . We provide analytical and numerical insights into the emergence of these high-order radial resonances for different azimuthal dependencies (hexa-, octo-, deca-, dodeca-, tetradeca-, and hexadeca-pole modes). Finally, we design, fabricate and characterize an ultrathin corrugated mm-sized spoof plasmon disk. Our microwave experiments demonstrate that this structure supports secondorder radial octopole and decapole LSPs.

Results
Theory and analysis. First, we investigate theoretically the LSPs supported by the textured 2D perfectly conducting cylinder (with radius R) shown in Fig. 1 30,41 . The structure consists of an inner core of radius r surrounded by periodic array of grooves of pitch d 5 2pR/ N (where N is the number of grooves), height h 5 R 2 r, and width a. For simplicity, we assume that the dielectric constant filling grooves is n 5 1. The system is excited under TM plane wave illumination. Using Mie scattering and an effective medium approach (see the method section), an analytical expression for the scattering cross section (SCS) of the corrugated cylinder can be expressed as s4 The spoof LSP resonances supported by the structure are given by the singularities in the Mie expansion coefficients, b n (where n labels the azimuthal characteristics of the LSP mode). The vanishing condition for the denominator of the n-coefficient can be expressed as where J n () and Y n () are Bessel functions of the first and second kind, respectively and H (1) n ()~J n ()ziY n (). We can gain physical insight into Equation (1) by taking the limit k 0 R, k 0 r ? 1, in which the asymptotic form of the Bessel functions can be used. In this limit, Equation (1) can be rewritten as which identifying the right hand side with the propagating wavevector, k 5 n/R (note that the azimuthal index n gives the number of modal wavelengths which fit within the cylinder perimeter), recovers the spoof SPP dispersion relation in a 1D array of grooves 17 . Despite their approximate character (the effective medium approximation is only valid in the limit a, d = l), Equation (2) establishes a clear analogy between spoof SPs in corrugated flat surfaces and spoof LSPs in textured cylinders. Note that the structures we consider here do not fulfil the conditions which led to Equation (2). However, an equivalent relation can be obtained from Equation (1) in the most realistic configuration, k 0 R ? 1, k 0 r = 1 (small core radius), having Figure 1a depicts the scattering cross section (SCS) for a corrugated cylinder of outer radius R 5 25 mm versus the incident frequency and h/R 5 (R 2 r)/R for a groove width a 5 0.5d. Note that the cross sections are normalized to the structure physical cross section. The contour plot was calculated using the effective medium approach described above, and the blue dots render the resonant condition obtained from Equation (3). The spectra show a prominent scattering maximum band which redshifts with increasing groove depth. This band comprises several LSP resonances 30,41 , as reflected by our analytical predictions. Importantly, Fig. 1a shows another set of SCS peaks for large groove depths and high frequencies (green dots). The effective medium field maps evaluated at the various SCS maxima for r 5 2 mm are rendered in Fig. 2. These results show that the modes within each scattering band share the same radial characteristics, but present different azimuthal properties. Importantly, whereas the LSPs in the first band have been investigated in the past 30,41 , to the best of our knowledge, the high frequency resonances, whose fields exhibit a node along the groove depth, had not been observed yet. It is the aim of this work to characterize in detail these high, second order radial modes. Figure 1a indicates that in order to design a spoof SP resonator supporting high-order radial LSPs at lower frequencies, deep grooves must be carved at the cylinder surface. Thus, we fix the outer and inner radii to R 5 25 mm and r 5 2 mm. Fig. 1b illustrates how, within the effective medium approximation, the ratio a/d affects slightly the scattering properties of the structure, yielding a small redshift of the LSP resonances with increasing groove width. In the following, we set a/d 5 0.5 for a feasible, three-dimensional full In order to retain the EM properties of the previous 2D textured cylinders, we consider thick disks with a thickness-to-radius aspect ratio L/R 5 10. The extinction cross section (ECS) of a representative textured cylinder, simulated using the full-wave commercial software CST Microwave Studio, is shown in Fig. 3 (black solid line). The parameters of the structure are in which the parameters are chosen as R 5 25 mm, r 5 2 mm, L 5 250 mm, N 5 60 (d 5 2.6 mm), and a 5 0.5d 5 1.3 mm (see inset of Fig. 3). From the simulation results, we observe that the 3D ECS contains two well-separated bands as well, both of which have multiple peaks. As discussed above, we expect the lower (higher) extinction band to encompass spoof LSP modes presenting zero (one) field nodes along the radial direction inside the grooves. The first (second) set of modes are labelled as 1 to 5 (6-9). The SCS obtained from our effective mode theory for a textured 2D cylinder with the same geometric parameters is plotted in red dashed line. Note that whereas the theoretical SCS has been normalized to the 2D cylinder diameter 2R, the numerical ECS value has been normalized to the disk side area 2RL. Remarkably, both normalized spectra are in excellent agreement (note that the absorption cross section of both structures vanishes, as we are using a perfect conducting model for the metal permittivity).
To further confirm the nature of the spoof LSP modes governing the EM response of 3D our textured metal disk, Fig. 4 shows the near electric field patterns for the 9 cross section maxima identified in Fig. 3. Specifically, the z-component of the electric field is plotted within the x-y plane 1.5 mm above the disk upper surface. In all panels, the colour scale has been saturated to show clearly the mode profile, ranging from red (positive) to blue (negative). Fig. 4 demonstrates that the corrugated disks supports two kinds of LSP modes. The first five LSPs (panels a-e) do not present radial nodes, and correspond to fundamental dipole to decapole modes. These were evaluated at the frequencies labelled as 1 to 5 in Fig. 3, and have been reported earlier in theory 30 and experiments 31 . Importantly, the dipole (a) and quadrupole (b) LSPs are resonances are not evident in the ECS spectrum (see Fig. 3), which is consistent to what observed in Ref. 30. Equation (3) sheds light into this observation. It is straightforward to show that it cannot be satisfied for n 5 1 (dipole) and n 5 2 (quadrupole) for the geometric parameters of the structure. Thus, the spectral shoulders labelled as 1 and 2 in Fig. 2 emerge due to a maximum, rather than a divergence, in the corresponding Mie scattering coefficient.
The last four electric field maps in Fig. 4 (panels f-i) correspond to the cross section maxima 5 to 9 in Fig. 3 and show azimuthal characteristics corresponding to dodeca to octadecapole modes. However, all these field profiles present a radial node along the grooves, and can be identified as second-order radial spoof LSPs. It is noteworthy that all these patterns are in accordance with the field distributions based on effective medium theoretical results shown in Fig. 2. Whereas the first set of resonances could be related to the LSPs sustained by metallic nanoparticles, the emergence of these highorder radial modes indicates that the analogy with optical whispering gallery modes in dielectric resonators 38 is more appropriate.
Recently, it has been reported that fundamental spoof LSP resonances (whose electric field does not exhibit radial nodes) are significantly altered when the disk thickness is reduced to values much    smaller than R 31 . In the ultrathin limit, the dipole and quadrupole resonances became clearly apparent (note that this cannot be reproduced by Equation (3), as it is valid only for large L). In Fig. 5, the dependence of second-order radial spoof LSPs on the disk thickness is analysed. This figure shows that, for the geometric parameters considered in Fig. 3, reducing the disk thickness from L 5 25 mm to L 5 0.018 mm the ECS maxima experience a moderate red-shift, while the most apparent resonance shifts from a second-order radial dodeca to a decapole LSP mode. This observation can be understood through the analogy that Equation (2) establishes between spoof LSPs and spoof SPPs 26 (see Fig. 6). In accordance with the previous study 34 , we find that that localized spoof plasmon resonances can present different physical features in 2D and 3D systems. Specifically, the spectral ordering of magnetic and electric modes can change. Figure 7 shows the ECS spectrum between 7.5 and 10 GHz for an ultrathin corrugated metallic disk with nearly zero thickness (see inset), with geometric parameters R 5 25 mm, r 5 2 mm, L 5 0.018 mm, N 5 60, d 5 2.61 mm, and a 5 0.5d. The emergence of second-order radial spoof LSP resonances translates into various extinction maxima. In order to reveal the nature of the LSPs, we simulate the near electric fields at the frequencies labelled as 1 to 6, and shown in the right panels of Fig. 7. We clearly observe secondorder radial hexa-, octo-, deca-, dodeca-, tetradeca-, and hexadecapole spoof LSP modes. Note that the lowest second-order radial modes (hexa-and octopole), labelled as 1 and 2, are not apparent in the extinction spectrum, in a similar way as it was observed for thick disks in Figs. 3 and 4. Fig. 7 shows that, as we already mentioned, the most intense spoof second-order radial LSP in thin disks is a decapole mode, whereas it was a dodecapole mode in the case of thick cylinders.   Experimental results. We fabricate an ultrathin (L 5 0.018 mm) corrugated cooper disk for the sake of experimental verification of the second-order radial LSPs predicted by theory and numerical calculations. The structure is shown in the Fig. 8a, whose geometric parameters are chosen to be the same as in Fig. 7 (r 5 2 mm, R 5 25 mm, L 5 0.018 mm, N 5 60, a 5 0.5d). The cooper disk is etched on a 0.2 mm-thick substrate with dielectric constant 3.5 and loss tangent 0.02. In order to investigate the influence of the dielectric substrate on the high-order radial spoof LSPs supported by the system, we performed CST Microwave Studio simulations in which both cooper and dielectric losses were considered to match the experimental conditions. The system is excited using a linearly polarized plane wave. The numerical ECS spectrum is plotted in black solid line in Fig. 8a. The comparison with Fig. 7 (without substrate) shows that the effect of the dielectric substrate is a slight redshifting of the spoof LSP frequencies. On the other hand, the introduction of dielectric and metal losses translates into the degradation of the spoof LSPs quality factor. Thus, the dominant, low frequency, resonance peaks (1-3 in Fig. 7) have been broaden, and the sharp, high frequency modes (4-6 in Fig. 7) have vanished. Fig. 8b render the simulated near electric-field patterns at the frequencies indicated in panel (a). As expected, the modes 1 and 2 can be clearly identified with second-order radial octopole and decapole spoof LSP resonances, respectively.
To demonstrate the predicted ECS spectrum experimentally, we measured the near field response of the fabricated LSP structure, as presented in Fig. 8a (the red line). Fig. 8a shows the excellent agreement between experiment and simulations, and the measured spectrum exhibits extinction peaks at 7.75 GHz and 7.96 GHz very close to the numerical spoof LSP resonances. Figure 8c renders the measured near-field maps at 7.75 GHz and 7.96 GHz, which are the resonant frequencies identified from the measured spectrum. These maps are in very good agreement with the full-wave simulations in panel (b), which demonstrates experimentally the emergence of second-order radial spoof octopole and decapole LSPs in our corrugated cooper disk sample.

Conclusion
In summary, we have introduced the concept of high-order radial spoof localized surface plasmon resonances. Using an effective medium theory, we have provided analytical insight into the properties of these electromagnetic modes in corrugated perfect metal cylinders. Through numerical simulations, we have showed their emergence in thick and ultrathin disks at microwave frequencies. Finally, we have verified experimentally the occurrence of second-order radial octoand decapole spoof plasmon resonances, showing an excellent agreement between numerical predictions and measurements for both spectral and near-field characteristics. Our findings shed light into the actual nature of spoof localized surface plasmons, and open the way to the design of plasmonic broadband devices, which exploiting high-order radial resonances, can be able to operate efficiently at very different frequency ranges (from the microwave to the terahertz regime).

Methods
Experimental setup and measurement. The near electric-field response of the spoof LSP sample is measured by means of two monopole antennas. First, a transmitter antenna placed 3 mm away from the disk plane is used to excite the spoof LSP modes. Second, a receiving antenna, able to move freely within the plane 1.5 mm above the sample, is used to probe the near-field response of the system, as illustrated in Fig. 9. Both monopole antennae are connected to an Agilent vector network analyser.
To map the near spoof LSP electric-field distributions experimentally, we make use a metamaterial-based Vivaldi antenna 42 to produce quasi-plane waves in the ultrathin corrugated disk plane. A coaxial probe is used to record the z-component of the electric field (out of the disk plane) at different locations within a plane 1.5 mm above the sample surface. The experimental setup is shown in Fig. 10.