1. INTRODUCTION

Periodically corrugated metallic films or structured slabs [1, 2, 3, 4, 5, 6, 7, 8] have been studied for decades due to their potential for excitation and interaction of surface plasmon polaritons (SPPs). Recently, intensive studies on these structures have resumed, aiming at novel plasmonic devices including plasmonic bandgap lasers [8, 9, 10], plasmonic Bragg mirrors [11, 12, 13], optical switches [14], as well as enhanced Raman and photoluminescent emission [15, 16]. Due to the coupling between the SPP modes on both surfaces of structured metallic films or slabs and the variety of structural possibilities, the optical responses and band structures are diverse and complicated. In addition, enhanced transmission is often accompanied by the excitation of the SPP modes. Similar to the enhanced transmission in periodic holes or slit arrays perforated in metal films, it was attributed to the coupling of the SPP with localized plasmon modes and resonant tunnelling [17, 18, 19].

In this report we investigate numerically the optical response of a vertical metallic meander structure in vacuum. It is similar to a metal slab with conformal corrugation on both surfaces [6, 7, 20, 21], but rather with a rectangular profile [22, 23]. The structure is more general and simpler for nanofabrication. First we study the plasmonic band structure of the metallic meanders and their corresponding transmittance. Then we use a transmission line (TL) circuit model to interpret both the mode behavior and the enhanced transmittance, which provides an alternative physical interpretation behind the enhanced transmission and mode interactions.

It has been demonstrated that electronic circuit models can be applied to metamaterials in the optical domain [24, 25, 26, 27, 28, 29]. TL circuit models turn out to be more powerful to analyze nanostructured metamaterials or structured metallic surfaces [30, 31, 32]. Using TL circuit models, the tangential magnetic response of the metallic structures can be separated from the tangential electric response [33]. Only via this kind of circuit model can the optical response of metallic structures be reproduced [32], and therefore more physical insight is obtained.

We find that the two lowest coupled SPP modes, which are identified as short-range SPP (SRSPP) and long-range SPP (LRSPP), respectively, can be described by magnetic and electric resonators in a TL model. The parameters of the lumped elements are obtained by directly fitting the response of the circuit model to the numerical results of the structure, whose physical meanings are justified by quasi-static or radiative models with respect to structural parameters. In addition, the effective permeability and effective permittivity of the slabs can be obtained directly from the impedance or admittance of the circuit through the relations $Z^{\prime }=Z∕l=j\omega {\mu }_0{\mu }_r$ and $Y^{\prime }=Y∕l=j\omega {\epsilon }_0{\epsilon }_r$, in which l is assumed to be the effective length of the slab. It might be improper to assign effective material parameters to such a thin slab, because the parameters might be changed when several identical layers are stacked together. Especially, these parameters are connected to an effective length that is difficult to define. However, the effective length does not play a role when ${\mu }_r$ and ${\epsilon }_r=0$, where the enhanced transmission normally occurs. The enhanced transmission may be well explained from the ${\mu }_r$ or ${\epsilon }_r$ near-zero or equivalent near-zero index point of view, providing simple and straightforward insight. Additionally, through the TL analysis we achieve an alternative explanation for the double Fano-shaped responses [34]. This perspective may help to optimize the structure for different applications.

2. NUMERICAL SIMULATION MODELS AND ANALYSIS METHOD

The metallic meander structure under study is composed of an one-dimensional conformal corrugated metallic film with periodic rectangular ridge profiles, as shown in Fig. 1a . It is characterized by periodicity $P_x$, metal thickness d, and meander depth D. Inverse symmetry along the propagation direction is achieved by setting $W_r=P_x∕2-d$, which is crucial to obtain a strong coupling between the SPP modes on the two interfaces and large enhanced transmission. A fixed film thickness of $30\mathrm{nm}$ is used throughout the report. The structure is placed in vacuum without the loss of generality. In previous studies, this structure was used to produce an effective negative permeability [20, 23]. It is similar to metallic slabs with two corrugated surfaces with a π phase difference and an equal filling factor [5, 8]. However, those structures are more difficult to fabricate when compared with the meanders. The spectra of the structures are calculated by two numerical methods to cross-check and complement each other. One is the commercial Maxwell solver CST Microwave Studio, and the other one is a Fourier modal method (FMM) [35] improved by factorization rules [36] and adaptive spatial resolution [37]. We use silver as the metal due to its low loss at visible frequencies. In order to study the mode behavior in a larger spectral range, a Drude model is used to describe its dielectric dispersion, although the Johnson–Christy dispersion resembles the real metal properties more [38]. In our Drude model, the plasma frequency is $1.37\times {10}^{16}\mathrm{rad}∕\mathrm{s}$ and the scattering frequency is $8.5\times {10}^{13}\mathrm{rad}∕\mathrm{s}$, which are fitted values from experiments [39]. In the second method, 61 Floquet–Bloch modes were considered, although fewer modes are sufficient for a good convergence. Only the zeroth diffraction order is presented in the spectra. In Fig. 1b, transmittance spectra of an Ag meander calculated at normal incidence using both methods are compared. Below the Rayleigh frequency, both methods deliver the same spectra. However, some spurious resonances beyond this frequency show up in the solid curve due to numerical inaccuracies of CST Microwave Studio.

3. BAND STRUCTURES OF METALLIC MEANDERS

To excite SPP modes in the corrugated metallic surfaces, an E field component perpendicular to the ridge is needed. In this report we investigate only the relationship of the modes versus the parallel wave vector $K_x$, namely with TM polarized light, and change the polar angle θ in the $xz$ plane, as shown in Fig. 1. Typical dispersion diagrams with respect to extinction, absorption, and transmission versus the parallel wave vector $K_x∕K_g$ of silver meanders with $P_x=200$ and $P_x=400\mathrm{nm}$ are shown in Figs. 2a, 2b, 2c, 2d, 2e, 2f , respectively. The extinction is defined as $-\ln \left(T\right)$, in which T is the transmittance. $K_g$ is the reciprocal lattice vector defined as $K_g=2\pi ∕P_x$. Due to the grating scattering, the momentum of the SPPs will be changed according to the relation [15]

Figures 2a, 2b, 2c show the dispersion relation for the meander with $P_x=200\mathrm{nm}$. The extinction spectra in Fig. 2a demonstrate that two relatively flat bands with large frequency distance occur. At $K_x=0$, the SRSPP is located around $700\mathrm{THz}$ and the LRSPP around $1000\mathrm{THz}$. The back-fold branch of the LRSPP is almost flat due to the scattering of the lattice vector $-K_g$, therefore having a high optical density of states [40]. From the absorption spectra shown in Fig. 2b the LRSPP mode is difficult to recognize due to its weak absorption. With the increase of $K_x$, an increased transmittance up to 90% is observed as shown in Fig. 2c, which behaves similarly as weakly coupled SPPs [5]. The larger frequency distance between the two modes (SRSPP and LRSPP) would be very interesting for realizing plasmonic bandgap lasers, in which the probability to couple the spontaneous emissions from a source into the two modes simultaneously is reduced [10]. Furthermore, at normal incidence both the LRSPP and the SRSPP modes are far away from the grating diffraction mode, revealing that they are localized plasmonic resonances.

Figures 2d, 2e, 2f show the corresponding spectra for $P_x=400\mathrm{nm}$. Compared to the dispersion diagram for $P_x=200\mathrm{nm}$, both LRSPP and SRSPP modes are redshifted and the frequency difference between the two modes at $K_x=0$ is decreased. Due to the increased periodicity and decreased Bragg vector, the modes are more dispersive, and more diffraction orders show up in the light cone. In Figs. 2d, 2e intramode and intermode crossings are observed [7, 8]. At $K_x=0$, intramode band gaps are formed at $650\mathrm{THZ}$ and $750\mathrm{THz}$, respectively, due to the existence of the $2K_g$ Fourier component of the grating [7, 41]. At $K_x>0$ around $800\mathrm{THz}$, intermode interaction between the SRSPP scattered by $+2K_g$ and the LRSPP scattered by $-K_g$ also leads to a bandgap. No intramode interaction is observed between the SRSPP branches scattered by $-K_g$ and $+2K_g$ at $K_x∕K_g=0.5$. Figure 2f shows the transmittance spectra versus $K_x$. With the increase of $K_x$, the transmittance is increased instead of being decreased [5], while the passband is getting narrower when the dispersion is approaching the grating diffraction line. In order to study the spectra in detail, selected transmittance curves are replotted in Figs. 3a, 3b for the two periodicities, respectively. From Fig. 3a for $P_x=200\mathrm{nm}$, we find that the spectra are getting narrower with incident angle from the high frequency side only, namely from the LRSPP mode side. This can be attributed to the cutoff of the Rayleigh frequency, which is induced by the diffraction of the grating. The LRSPP resonance above this line cannot be excited, which makes the high-frequency side of the passband profile sharper. Similarly, the narrowing of the passband at $P_x=400\mathrm{nm}$ can be interpreted in the same way, as we can see from Fig. 3b.

Figures 3a, 3b show a possibility of how to apply our design for various frequency filters. At a smaller periodicity than $P_x=200\mathrm{nm}$ [Fig. 3a], the frequency of the passband is less dependent on the incident angle of light, especially when this angle is larger than $40\text{\hspace{0.17em}degrees}$. It is similar to a radome filter in electrical engineering designed for antennas [42]. Through optimizing the structural parameters, a simple radome filter can be realized [43]. For the meanders at a larger periodicity than $P_x=400\mathrm{nm}$ [see Fig. 3b], the transmission frequency is selected through changing the incident angle. The passband is getting narrower with almost unchanged transmission amplitude for increased angles, which could result in a frequency filter for a broadband spectrum.

Although both modes are excited through the coupling of the grating with the SPP modes, they behave differently upon a change of their structural parameters. Figure 4a shows the extinction spectra of the meander with varied meander depth D at fixed $P_x=400\mathrm{nm}$ and $d=30\mathrm{nm}$. We see that the SRSPP branch in (a) can be downshifted from $580\mathrm{THz}\text{to}460\mathrm{THz}$ when D is changed from $20\mathrm{nm}\text{to}80\mathrm{nm}$ without changing the frequency of the LRSPP mode. This behavior confirms that the SRSPP mode is strongly localized in the meander loop. It also provides another possibility to move the SRSPP mode away from the LRSPP mode for realizing a plasmonic bandgap laser. The transmittance spectra as a function of meander depth are plotted in Fig. 4b. We see that the enhanced transmission is related to the interaction of the two plasmonic resonances. With the increase of D, the frequency distance between the two modes is increased and the interaction is decreased. This leads to a decreased transmission, as will be further elucidated in the following section.

Contrary to the behavior upon a change of the meander depth, both SPP modes are redshifted with the increase of the periodicity, as shown in Fig. 5 . At larger periodicity the SRSPP mode approaches the LRSPP mode, and both approach the SPP modes in thin metallic films. On the other hand the LRSPP resonance is clipped by the grating diffraction line, and this leads to a narrowed passband. However, the interaction between the two modes remains strong, which leads to a larger enhanced transmission in a broad parameter range, as shown in Fig. 5b. The transmittance achieves its maximum around $P_x=400\mathrm{nm}$. At larger D (for instance $80\mathrm{nm}$—results not shown here), the LRSPP mode maintains the same behavior as in Fig. 5a, but the SRSPP mode shifts more gradually with $P_x$, which again indicates a stronger mode localization in the loop.

4. TRANSMISSION LINE ANALYSIS OF MEANDER STRUCTURES

As shown above, the excited SPP modes and the transmittance of the meander change strongly, not only with $K_x$ but also with meander depth D and periodicity $P_x$ when varied within a range that is accessible through standard nanofabrication (also with metal thickness, but it is fixed in this study). Furthermore, the change is not monotonous. In this section, we try to interpret these properties using a TL equivalent circuit model, which gives a novel explanation for the phenomena of the enhanced transmission that we discussed above. In addition, the observed Fano line shapes can also be explained from the response of the TL circuit model [34].

Figure 6a shows the schematic of a meander together with its circuit elements. Figure 6b shows its TL circuit model loaded with frequency-independent L and C elements, which can be constructed using the method introduced in [32]. Resistance R is implemented to describe optical losses. The slab is viewed as a dimensionless impedance sheet perpendicular to the k direction [33] with series and shunt elements. The circuit elements are arranged along a symmetric and reciprocal two-port Π type TL (also equivalent to a T type TL circuit). The length of the TL section is taken as $l=D+d$, i.e., the physical length of the slab. In the circuit model, $L_1{C}_1$ form a magnetic resonator [44] and $L_2{C}_2$ form an electric resonator due to their position in the TL [33, 45]. The two resonators represent the SRSPP mode and the LRSPP modes, respectively, as will be discussed in the following. The shunt inductance $L_3$ represents the unstructured metallic film when the frequency is smaller than the effective plasma frequency [46]. By fitting the numerical curves of the meander structure using TL equations [47], the proper circuit parameters can be obtained [32]. Figure 7 shows the reflectance/transmittance spectra and the phase spectra of the scattering parameters $S_{11}$ and $S_{21}$ (or equivalently, $S_{22}$ and $S_{12}$ due to the reciprocity and symmetry of the structure) from numerical simulations (solid curves) and TL calculations (dashed curves) of a meander with $P_x=500\mathrm{nm}$ and $D=40\mathrm{nm}$ at normal incidence. These spectra are typical for Ag meanders, among which the transmittance has a passband with two sharp edges. In principle, the circuit model is also valid for inclined incidence, although the resonances are redshifted due to the weakened electric field coupling. Correspondingly, the circuit parameters are also changed. However, at a larger incident angle, the LRSPP mode might be suppressed by the diffracted light line and makes the fitting procedure difficult, as can be seen from Fig. 2d.

Fitted circuit parameters are listed in Table 1 . Good agreement is obtained not only for the transmittance and reflectance spectra but also for the phase spectra. We have to stress that by using other combinations of circuit elements, one might also be able to obtain the same transmittance and reflectance spectra, but would not be able to obtain the same phase spectra simultaneously. Evidently, the metallic meander can be well represented by the circuit model. Once the inverse symmetry of the meander along the propagation direction is broken, for instance in the presence of a substrate, the circuit parameters might be position dependent along the k direction. In this case a distributed circuit model has to be used in which the circuit parameters cannot be uniquely determined.

To gain more physical insight into the line shapes of the meander response, the transmittance spectra of the TL model shown in Fig. 6b with different combinations of these circuit elements are calculated using the parameters listed in Table 1 for $D=40\mathrm{nm}$. As shown in Fig. 8a , the magnetic resonator $\left(L_1{C}_1\right)$ induces a stop band resonance and $L_3$ induces a flat transmission spectrum, which is typical for inductive metallic films. The combination of both results in an asymmetrical Fano line-shaped enhanced transmittance with a dip at the low frequency side. Similarly, as shown in Fig. 8b, the electric resonator $\left(L_2{C}_2\right)$ also induces a stop band resonance. Although its combination with $L_3$ results in an asymmetrical resonance, the dip is at the high-frequency side. Both line shapes can be interpreted by an interference between resonant and nonresonant processes [34]. In the meander the nonresonant response originates from the continuous metallic film. Such a localized plasmonic resonance-induced transparency is also one possible explanation of the macroscopic mechanism for enhanced transmittance with perforated metallic films [48]. They also display a Fano line shape. Finally, when the two resonators are combined together with $L_3$, a typical transmittance line shape of meander structures accompanied by two minima can be obtained, as shown in Fig. 8c.

An interesting observation is the stronger absorption by the magnetic resonance, as indicated in Figs. 8a, 8b, although the ohmic resistance of the two resonances is almost the same as can be seen in Table 1. In fact, the absorption induced by the magnetic resonance alone is not stronger than that of the electric resonance alone. However, the combination with $L_3$ leads to a higher absorption from the magnetic resonance than that from the electric resonance due to the different interaction processes.

The circuit model can be further understood by analyzing field distributions. Figures 9a, 9b, 9c, 9d, 9e, 9f show the distributions of the electric field in the $xz$ plane, the magnetic field $H_y$, and the current density at the magnetic and electric resonance frequencies $482\mathrm{THz}$ and $585\mathrm{THz}$, respectively, for the meander shown in Fig. 7. At $482\mathrm{THz}$, the instantaneous charge distribution manifests that this mode originates from the symmetric SPP mode. Due to the presence of the meander corrugation, the electric field oscillates mainly between the loop edges of A and B. Through the coupling of the magnetic field to the loop, a circular current is induced as shown in Figs. 9c, 9e, revealing that it has the property of a magnetic resonance. This clarifies the origin of $L_1$. The potential difference of the electric field along $k_z$ [in Fig. 9a] elucidates the existence of the series capacitance $C_1$. It stems from the instantaneous charge distribution of the induced current. The series inductance $L_1$ and the series capacitance $C_1$ are combined in parallel in the TL and, therefore, lead to a magnetic resonance [32, 44]. At $585\mathrm{THz}$, the electric field oscillates mainly between the edge points B and C at the top of the meander [Fig. 9b], and the electrically induced current also oscillates mainly between B and C [Fig. 9f]. This is a typical behavior of cut wires and can be described by the shunt $L_2$ and $C_2$ combined in series in TL [32]. On the other hand, $L_3$ originates from the background plasmon oscillation as shown by the current distribution in Fig. 9g at a lower frequency away from any resonances. To see whether the circuit parameters have a physical meaning and to understand the enhanced transmittance, relationships of the circuit parameters with the structural parameters are discussed below.

4A. Variation of Meander Depth

First the meander depth D is varied with fixed $P_x=400\mathrm{nm}$, and the dependence of the circuit parameters is shown in Fig. 10 .

For the capacitances shown in Figs. 10a, 10c, $C_1$ and $C_2$ are the parameters obtained by fitting the numerical results, while $C_{1\mathrm{qs}}$ and $C_{2\mathrm{qs}}$ are the results from the quasi-static equations for a classic parallel-plate capacitor

The inductances are shown in Figs. 10b, 10d. Both $L_1$ and $L_3$ increase with D, and the relationship is becoming linear at larger D. A quasi-static picture can still be applied to the linear part, because the inductance of a metal loop is proportional to its loop area. $L_3$ has a similar dependence on D as does $L_1$. This could be simply the consequence of the increased metal area when the meander is unfolded into a planar metal film. Contrary to $L_1$, $L_2$ decreases with D oppositely to the quasi-static relation. In order to quantitatively characterize $L_2$, the radiative inductance of the meander is calculated using an analytical formula for planar SRRs given by Meyrath et al. in [27]. Without any modification, the meander dimensions with $l_1=P_x∕2$, $l_2=D+d∕2$, and a network factor of 0.4 were used. The calculated radiative inductance $L_r$ is plotted in (d) as dashed curve. We conclude that $L_2$ can be reproduced by the radiative model very well.

4B. Variation of Meander Period

The meander period $P_x$ was varied with fixed $D=40\mathrm{nm}$. Figure 11 shows the relationship of the circuit parameters with $P_x$. In contrast to the dependence on D, $C_1$ increases with the period while $C_2$ decreases. On the other hand, $L_1$ decreases with $P_x$ while $L_2$ increases. $L_3$ first changes little but decreases for larger $P_x$. To check whether the capacitances can still be described by the quasi-static formula, the same fitting parameters $(G_1,G_2,d_c,D_c)$ are put into Eq. (2) and Eq. (3). The calculated $C_{1\mathrm{qs}}$ (solid square) and $C_{2\mathrm{qs}}$ (solid circle) are shown in Figs. 11a, 11c, respectively. Evidently, $C_2$ can be well understood from the quasi-static picture, while $C_1$ deviates from $C_{1\mathrm{qs}}$ substantially, especially at larger $P_x$. The reason for this might stem from the fact that at larger $P_x$ the LRSPP modes approach the grating diffraction, which influences the line shape of the response and makes the fitting for the circuit parameters difficult. $L_r$ is calculated based on the same input parameters as for the D variation, and the results are plotted as dashed curve in Fig. 11d. Similarly, $L_2$ can be well described by the radiative inductance.

5. DISCUSSION

From the analysis above we conclude that the interaction of the magnetic and electric resonances induced by localized SPPs results in an enhanced transmission in structured metallic films. This phenomenon can be understood from a simple near-zero index perspective. To elucidate this point, effective permeability and permittivity of meanders with $P_x=500\mathrm{nm}$ and different D are calculated using the fitted parameters of the circuit elements listed in Table 1 and the TL homogenization method [32]. This method delivers physical effective material parameters for impedance sheets [32], in contrast to the retrieval procedure from scattering parameters [49]. In fact, it is an alternative description for electric and magnetic polarizabilities [45], and it turns out to be very useful in understanding the enhanced transmission in the meanders.

Figures 12a, 13a show the calculated effective permittivity ${\epsilon }_r$ and permeability ${\mu }_r$ homogenized in the physical length of $l=D+d$ with different D. Figures 12b, 13b show the corresponding effective index, which is defined as $n_{\mathrm{eff}}=\sqrt{{\epsilon }_r{\mu }_r}$. A negative index might be obtained from this calculation. However, it is not relevant for our present discussion and, therefore, only the absolute values of the real part $\left|\mathrm{Re}\left(n_{\mathrm{eff}}\right)\right|$ are shown in the figures. Both effective parameters have Lorentzian responses. However, due to the background plasmon from the continuous metallic film $\left(L_3\right)$, ${\epsilon }_r$ becomes negative at the low-frequency side. Enhanced transmission occurs in the region where both $\mathrm{Re}\left({\mu }_r\right)$ and $\mathrm{Re}\left({\epsilon }_r\right)$ are near zero (hatched region). In Fig. 12a, the two transmission peaks are corresponding to the frequencies at which both material parameters are zero, respectively. Consequently, in the same region the effective index is near zero as we can infer from Fig. 12b. For the structure with $D=80\mathrm{nm}$ shown in Fig. 13a, although enhanced transmission still occurs, the ${\epsilon }_r=0$ frequency is lower than the ${\mu }_r=0$ frequency. This results in an overlap of the ${\epsilon }_r$ near-zero region with the strong negative region of the ${\mu }_r$ (and, therefore, with the strong absorption region, which is not shown here [32]). Although $\mathrm{Re}\left(n_{\mathrm{eff}}\right)$ is still near zero [Fig. 13b], the absorption is larger and the transmission amplitude is lower than that by the meander with $D=40\mathrm{nm}$. In addition, only one peak can be seen, although it is induced by two resonances. Therefore, both the shape and the amplitude of the transmission depend on the coupling and oscillation strength of both resonators. At optimized structural parameters, high transmittance with sharp cutoffs at both sides can be obtained, as we have seen in Fig. 4 and Fig. 5.

From this model, some of the observed phenomena can be well understood. For instance, for the dispersion diagram shown in Fig. 2, the coupling of the electric field to the meander is decreased (x component) with increased incident angle of light or increased $K_x$, while that of the magnetic field into the meander loop does not change. This changes the overlap region of the magnetic and electric resonances and, therefore, the amplitude and shape of the enhanced transmission. This is the same as when changing D or $P_x$. Growing evanescent waves might be obtained in stacked frequency-selective surfaces [50, 51]. Using the TL circuit model and the geometry-related circuit elements, one could design a subwavelength imaging system based on meanders.

The application potential of near-zero index materials have been studied for a long time. Directional emission through such a material can be expected [52, 53]. In earlier work, such a metallic meander structure has been used to enhance the emission and quantum efficiency of GaAs LEDs [54, 55]. At the same time, a directional beam was obtained [54, 55, 56, 57]. Further physical understanding concerning a directional beam is still needed.

6. CONCLUSION

In this report we have studied the optical properties of a vertical metallic meander structure that exhibits rich features in the band diagrams and could be interesting for many applications. The structure can be especially well interpreted using a TL circuit model in which the lowest SRSPP mode is viewed as a magnetic resonance, and the lowest LRSPP mode is viewed as an electric resonance. The enhanced transmittance is well understood from the near-zero index point of view. Through varying the structural parameters, the amplitude, and line shape of the transmission are varied, depending on the incidence angle of the light, the meander depth, and the periodicity. This variation is the consequence of the varied overlap and strength of the magnetic and electric resonances. The double Fano line shapes are also well interpreted by the TL model, which would be an extension to the studies in [34]. Different functional filters can be designed, depending on specific applications. The structure is well suited for nanofabrication. The TL model provides us a way to retrieve the circuit parameters for frequency-selected surfaces, which could become a very powerful tool for designing subwavelength imaging systems.

ACKNOWLEDGMENTS

We thank T. Meyrath for many helpful discussions and acknowledge the support from Deutsche Forschungsgemeinschaft (DFG) grant FOR557, Bundesministerium für Bildung und Forschung (BMBF) grant 3D Metamat 13N10146, and Landesstifung Baden Württemberg (BW) Project OPTIM.

Table 1. Circuit parameters of the TL model for meanders at $P_x=400\mathrm{nm}$ and $d=30\mathrm{nm}$

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Fig. 1 (a) Schematic of a metallic meander with two unit cells in vacuum composed of a corrugated metallic film with a rectangular ridge profile. The structure is characterized by periodicity $P_x$, metal thickness d, and meander depth D. The groove width $W_r$ is set to be $P_x∕2-d$. (b) Transmittance spectra of an Ag meander at normal incidence simulated using time-domain transient solver in CST (solid curve) and using FMM (dashed curve).

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Fig. 2 Dispersion diagram of Ag meanders with $D=40\mathrm{nm}$, $d=30\mathrm{nm}$, and different periodicities plotted in extinction, absorption, and transmittance spectra calculated using FMM. (a)–(c) $P_x=200\mathrm{nm}$; (d)–(f) $P_x=400\mathrm{nm}$. The light dashed lines are diffracted light lines and the dark dashed lines represent the propagating light in free space. The solid lines show the right border of the first Brillouin zone at $K_x=K_g∕2$.

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Fig. 3 Transmittance versus frequency calculated using FMM at different incident angles for meanders with (a) $P_x=200\mathrm{nm}$ and (b) $P_x=400\mathrm{nm}$. The arrows show the position of the Rayleigh frequency induced by the diffraction of the grating [34].

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Fig. 4 (a) Extinction and (b) transmittance spectra of an Ag meander with $P_x=400\mathrm{nm}$, $d=30\mathrm{nm}$, and varied meander depth at normal incidence calculated using FMM.

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Fig. 5 (a) Extinction and (b) transmittance spectra of an Ag meander with $D=40\mathrm{nm}$, $d=30\mathrm{nm}$, and varied periodicity calculated using FMM.

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Fig. 6 (a) Schematic of a metallic meander structure to show the origin of the elements. The dashed arrows show the current flow responsible for the formation of $L_3$. (b) TL circuit model for the meander with the polarization shown in (a). The light gray TL sections represent the free space.

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Fig. 7 Comparison of reflectance/transmittance spectra and phase spectra calculated from the TL circuit model with those from the numerical simulation (CST) for a meander with $P_x=500\mathrm{nm}$ and $D=40\mathrm{nm}$ at normal incidence. (a) Reflectance, transmittance, and absorption spectra; (b) Phase $S_{11}$ and $S_{21}$ spectra. Dashed curves are from the TL model, solid curves are from the numerical simulation, and the short dashed curve in (a) is absorption. MR, magnetic resonance; ER, electric resonance.

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Fig. 8 Transmittance and absorption spectra from the TL circuit model with selective combination of circuit elements: (a) with the combination of the magnetic resonator and $L_3$; (b) with the combination of the electric resonator and $L_3$; (c) combining both the magnetic and electric resonators with $L_3$.

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Fig. 9 (a),(b) Instantaneous electric field, (c),(d) normalized magnetic field, and (e),(f) current density distributions of a meander with $P_x=500\mathrm{nm}$, $D=40\mathrm{nm}$, and $d=30\mathrm{nm}$ at the magnetic resonance ($482\mathrm{THz}$, left column) and electric resonance ($585\mathrm{THz}$, right column), respectively. (g) Normalized current density at the lower frequency side of the magnetic resonance (at $200\mathrm{THz}$, for instance). Results are taken from CST.

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Fig. 10 Dependence of circuit parameters $C_1$, $C_2$, $L_1$, $L_2$, and $L_3$ on meander depth D. Other parameters are fixed with $P_x=400\mathrm{nm}$. Solid square curve in (a) and solid circle curve in (c) are results from the quasi-static model [Eqs. (2, 3)]. The dashed curve in (d) represents the radiative model for SRRs.

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Fig. 11 Dependence of circuit parameters ($C_1$, $L_1$, $C_2$, $L_2$, and $L_3$) on meander period $P_x$ with fixed $D=40\mathrm{nm}$. Solid-square curve in (a) and solid-circle curve in (c) are calculated results using the quasi-static model [Eqs. (2, 3)]. The dashed curve in (d) is from the radiative model for SRRs.

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Fig. 12 (a) Effective material parameters and (b) effective index and transmittance versus frequency for Ag meanders with $P_x=500\mathrm{nm}$, $d=30\mathrm{nm}$, and $D=40\mathrm{nm}$.

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Fig. 13 (a) Effective material parameters and (b) effective index and transmittance versus frequency (b) for Ag meanders with $P_x=500\mathrm{nm}$, $d=30\mathrm{nm}$, and $D=80\mathrm{nm}$.

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