1. Introduction

Surface plasmon polaritons (SPPs) are propagating surface modes on the interface of metal and dielectrics [1] in visible and near-infrared wavelengths [2]. Since the fields decay exponentially in pace with distance away from the interface while propagating along the interface, the SPP modes exhibit highly confined property. Meanwhile, SPPs have much shorter working wavelength, which provide a brand new approach to overcome the diffraction limit that restricts the minimum size of optical devices. In the past ten years, significant progresses have been achieved in such a growing field due to its unique properties, ranging from the super-resolution imaging [3,4], SPP circuits [5–7], electromagnetically induced transparency (EIT) [8–10], environmental detection [11], to the optical enhancements [12,13] and other optical phenomena [14–16].

However, the SPP modes cannot be excited and supported in the microwave or terahertz wavelengths because of the perfectly electric conductor (PEC) behavior of metals. Consequently, plasmonic metamaterials have been proposed to transform the traditional Sommerfeld or Zenneck surface waves [17]–longer wavelength surface electromagnetic responses with worse confinement to the surface–into the highly confined surface waves, which are called as designer SPPs or spoof SPPs [18–20]. Recently, a series of researches have been conducted on the propagation of spoof SPPs through plasmonic metamaterials with periodic subwavelength grooves, holes and dimmers [18–27]. Among the many plasmonic metamaterials, an ultrathin corrugated metallic strip structure with nearly zero thickness is very promising [21,22], which can support conformal spoof SPPs on curved surfaces and guide the SPP modes to long distance with small bending loss in a wide frequency range. More importantly, based on the ultrathin corrugated metallic strip, a high-efficiency conversion from the conventional guided waves to SPPs was achieved in broadband through a transition with gradient grooves [25], which makes it possible to realize combined devices and circuits of guided waves and SPPs.

In this work, we propose a method to control the rejections of spoof SPPs, which is required in some functional plasmonic devices and systems. When metamaterial particles are introduced in the vicinity of the ultrathin corrugated metallic strip, they will excite strong and tight coupling to the SPP waveguide and make the original surface impedance mismatch near the resonant frequencies, resulting in rejections of SPP modes in the propagating band. Since the resonant frequencies of metamaterial particles with different scales can be tuned nearly independently, we can control the propagation or rejection of SPPs. This feature is very important for filtering SPP waves in the plasmonic circuits and systems.

2. Meliorated spoof SPP waveguide

To begin with we shall first briefly introduce a meliorated corrugated metallic strip on which spoof SPP modes are supported. Figure 1(a) illustrates a schematic configuration of the high-efficiency spoof SPP waveguide, which consists of three parts: an energy transition section, a mode conversion and momentum matching section, and a SPP transmission line. Figures 1(b)-1(d) show details of such parts, respectively, marked as Part I, II and III. The golden area represents metal, and the gray area stands for dielectric substrate. Here, we choose F4B as the substrate with thickness of 0.5 mm and relative permittivity of 2.65(1 + i0.003).

 

Fig. 1 (a) The top view of spoof SPP waveguide, which is divided into three comparatively independent regions (I, II and III). (b) Part I: the CPW region with dimensions of wth (the width of inner conductor) and S_c (the width of slot lines). (c) Part II: the convertor from CPW to spoof SPP modes. $P_0\left(x_0,y_0\right)$and $P_1\left(x_1,y_1\right)$ are starting and ending points of the exponential slot line. The transition corrugated spoof SPP waveguide is described by d1-d7, the gradient depths of grooves. (d) Double cycles of Part III, in which D, G and S are the cycle, depth and width of grooves, respectively.

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Part I, as is magnified in Fig. 1(b), is a coplanar waveguide (CPW) to obtain the input signal from coaxial cable via SMA connectors. The dimensions S_c and wth labeled in this figure are designed to achieve 50 Ω input impedance, in which S_c = 0.15 mm is the width of the symmetrical slots, and wth = 3 mm is the width of the inner conductor line. As shown in Fig. 1(c), Part II is the convertor that comprises two symmetrically exponential slot lines and an array of seven distinctive grooves with their depth deepening by degrees. The exponential slot lines work as exciting source of spoof SPP modes, which are determined by the classical exponential equation hailing from the Vivaldi antenna [28], a traditional broadband planar antenna. Based on the analysis of Vivaldi antenna, we surely confirm that such structure can provide high-efficient excitation among a wide frequency band. The equation of the upper part of the Vivaldi slot is written as:$y=C_1{e}^{\alpha x}+C_2$, where$C_1=\frac{y_1-y_0}{e^{\alpha x_1}-e^{\alpha x_0}}$, $C_2=\frac{y_0{e}^{\alpha x_1}-y_1{e}^{\alpha x_0}}{e^{\alpha x_1}-e^{\alpha x_0}}$, and the exponent parameter is $\alpha =0.015$. $P_0\left(x_0,y_0\right)$ and $P_1\left(x_1,y_1\right)$ marked in Fig. 1(c) are the coordinates of the starting and ending points of the slot line.

For the corrugated spoof SPP waveguide, Part III, we settle two improvements to realize the miniaturization in the y direction and achieve high transmission efficiency, compared to the earlier proposals [21–23, 25]. Firstly, we set the grooves to be protruded out of the inner conductor of CPW, instead of inlaying them into it. In such a circumstance, the grooves could capture the incident energy efficiently since they avoid extra radiation losses coming from the energy leakage at the terminal of Vivaldi slot. Meanwhile, since the propagating frequency is restricted by the grooves’ depth, such design is an effective solution to avoid oversize width of the waveguide reported in Ref. 25 when the designed frequency drops. Secondly, we design the ceiling of each groove to be beveled. Since the electric field coming out of the Vivaldi slot has a spherical-like wave front, such beveled grooves could adapt the electric field perpendicular to the metallic strip and lead to higher coupling efficiency. Figure 1(d) gives the sketch of double unit cells of the spoof SPP waveguide. The cycle of grooves is set as D = 6 mm, while the width and depth are S = 2 mm and G = 4 mm, respectively. To confirm momentum matching from CPW to plasmonic metamaterials, we discuss the dispersion curves of spoof SPPs with different groove depths, using the eigen-mode solver of commercial software, the CST Microwave Studio. From Fig. 2(a), as the depth decreases, we notice that the cut-off frequency of spoof SPPs raises and the momentum k approaches to $k_0$, the momentum of CPW. Thus, momentum matching is realized during the gradient SPP waveguide. The depths of these seven grooves, labeled as d1 - d7 in Fig. 1(c), change from 0.5 mm to 3.5 mm with equal step.

 

Fig. 2 (a) The dispersion curves of spoof SPPs for different groove depths. (b) The near-field sketch of the spoof SPPs.

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3 Design of multiple-frequency rejections

From full-wave numerical simulations, we notice that the electric field near the SPP waveguide has both x and y components, as shown in Fig. 2(b). Hence it has potential to excite strong responses of electrically resonant metamaterial particles, which can be used to control the propagations and rejections of SPPs in the propagating band. Here we construct two rows of electric-field-coupled-LC (ELC) [29] particles on each side of the meliorated spoof SPP waveguide. Figures 3(a) and 3(b) illustrate the photograph and partially enlarged details of a fabricated sample. Since the y component of the near-field distribution stands a leading role, the ELC particles are set with their equivalent capacitors of two parallel metal lines along the x direction to achieve the strong resonances. As the resonant property of ELCs exhibits a sudden change in the impedance spectrum while the rest impedance relative to background material keeps unity within a wide frequency band, such design could provide a perfect absorption peak at the resonant frequency point as well as highly transparent transmission among the rest wide-frequency band. In the other word, such a resonance makes a rejection of SPP waves within the wide propagating band. On the other hand, the distance between ELCs and the corrugated waveguide (the distance from edge to edge) shown in Fig. 3 (b) would evidently influence the transmission. According to the exponential decaying property shown in Fig. 2(b), the field intensity weakens rapidly while the distance is enlarged. Thus, the performance of rejection degrades along with the enlargement of the distance. Here, an appropriate distance is chosen to be Dgap = 1.05mm.

 

Fig. 3 (a) The photograph of a fabricated sample for double-frequency rejections. (b) The schematic configuration of ELC array in the DFR SPP waveguide.

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To better understand the relationship between the rejecting property and ELC parameters, we study the SPP transmissions while the sizes of ELCs are scaled by a coefficient k, the scaling factor. An example of the calculated impedance relative to background material is illustrated in Fig. 4(a) based on the effective medium theory [30], from which we observe that the relative impedance keeps unity deviating from the resonant frequency. For a single-frequency rejection SPP transmission system, the sizes of ELCs are set to be identical and the propagation properties of SPPs are simulated under a series of ELC’s scaling factors. The basic transmission efficiency of the spoof SPP waveguide without ELCs is shown as the black line in Fig. 4(b), demonstrating excellent propagations of SPPs in a wide frequency band with a cutoff at 10.8 GHz. In the same figure, simulated transmission coefficients of the spoof SPP waveguides with three differently scaled ELCs are also given, in which the red, pink, and blue lines correspond to scaling parameters k = 0.7, 0.8, and 1, respectively. Each of these results corroborates that the rejected frequency alters with the change of ELC’s size, and large narrow-absorption peak below −15dB is achieved without breaking the transmission efficiency in other frequencies. Besides, the electromagnetic fields near ELCs at the resonant frequencies are confined inside and near the particles, and hence there is little influence between adjacent ELC cells. This feature makes it much convenient to constitute a multiple rejections of spoof SPP modes, which can serve as narrow-stop-band filters.

 

Fig. 4 (a) The calculated relative impedance of ELCs on the DFR SPP waveguide. (b) The transmission coefficients of the DFR SPP waveguide with different ELC scaling factors, in which the black line stands for the spoof SPP waveguide without ELCs, while the red, pink, and blue lines represent the DFR SPP wavguides with scaling parameters k = 0.7, k = 0.8, and k = 1, respectively.

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Owing to the maneuverability of distributing dimensions of each single ELC, a double-frequency rejection (DFR) SPP waveguide is fabricated with two ELC scaling coefficients $k_1=0.7$ and $k_2=0.9$ (see Figs. 3(a) and 3(b)). The whole device is symmetrically designed for continuity in propagation. Four middle ELCs work with scaling factor $k_2=0.9$ while the bilateral four are scaled with $k_1=0.7$. To confirm the insulation of ELCs, the near-field distributions at the propagating frequency and two resonant frequencies are compared in Fig. 5. The field distribution at 8 GHz in propagating band, as shown in Fig. 5(a), agrees with that of a spoof SPP waveguide shown in Fig. 2(b). However, at resonant frequencies, Figs. 5(b) and 5(c) demonstrate a good support of isolations. At 7.65 GHz, the fields mainly oscillate around the larger ELCs, as seen in Fig. 5(b); at 9.47 GHz, the fields exist only around the smaller ones, as shown in Fig. 5(c). No distinct energy exists in the propagating space or ELCs with different scales.

 

Fig. 5 The near-field sketches of the DFR SPP waveguide at (a) 8 GHz, (b) 7.65 GHz, and (c) 9.47 GHz.

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In experiments, two coaxial cables with 50 Ω port impedances are connected to both ends of the DFR SPP waveguide sample via SMA connectors. Figure 6 illustrates the experimental results of transmission coefficients (S21) in various cases and their comparison with simulation results. Although, there exists a small discrepancy between simulated and measured curves, which is caused by mismachining tolerance and loss of the coaxial cable during the experiment, the results show clearly significant rejections. From Fig. 6(a), we experimentally confirm that the proposed DFR waveguide provides excellent absorptions at 7.65 and 9.47 GHz within the propagating band (red lines), while its transmissions in the rest frequencies are nearly the same as those of a SPP waveguide without ELC particles (black lines). From Fig. 6(b), we notice that the experimental results of the DFR waveguide (red lines) have excellent agreements to full-wave numerical simulations (black lines). On the basis of tunable property of resonances, the multiple-frequency rejections can be easily achieved.

 

Fig. 6 The simulated and measured transmission coefficients (S21) and reflection coefficients (S11). (a) The measurement results of spoof SPP waveguide without ELC particles (black lines) and DFR SPP waveguide (red lines). (b) The simulation (black lines) and measurement (red lines) results of DFR SPP waveguide.

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In order to exhibit an intuitionistic view of the operation principle, we discuss the simulated near-electric-field distributions of the DFR SPP waveguide at the two rejection frequencies and two other reference frequencies within the propagating band, as displayed in Figs. 7(a)-7(d). At the rejection frequencies 7.65 and 9.47 GHz, we notice that SPP fields are cut off at the places where the corresponding ELCs are resonant, as demonstrated in Figs. 7(b) and 7(d). However, at non-resonant frequencies (7.9 and 9.1 GHz), the SPP fields are propagating through the ELC particles to the other end, as shown in Figs. 7(a) and 7(c). These phenomena have also been verified by measured near-electric fields illustrated in Figs. 7(e)-7(h), which have good agreements with the corresponding simulation results. Therefore, this provides an intuitionistic proof to corroborate a double ultra-narrow band-stop filtering performance.

 

Fig. 7 The near-electric-field distributions of DFR waveguide at different frequencies. (a-d) The simulated results at (a) 7.9GHz, (b) 7.65GHz, (c) 9.1GHz, and (d) 9.47GHz. (e-h) The measured results at (e) 7.9GHz, (f) 7.65GHz, (g) 9.1GHz, and (h) 9.47GHz.

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4 Design of broadband frequency rejections

The strong resonance of ELC particle is unsuitable for broadband applications since its resonant modes always work in a narrow frequency band. In this section, we propose to realize broadband frequency rejections by using the nonlinear impedance property. For this purpose, the ELC particles are designed and arranged on the bottom surface of the SPP waveguide, as illustrated in Fig. 8(a) and 8(b). According to the boundary condition of such ELCs, a broadband mismatching appears for impedance, and hence a broadband rejection of SPP modes would be achieved. Figure 8(a) shows the photographs of top and bottom views of a fabricated sample, and Fig. 8(b) gives the schematic configuration of the ELC array, in which the contour dashed lines mark the location of the SPP waveguide on the top surface. ELC particles’ equivalent capacitors are aligned with grooves along z coordinate. So that the cycle of ELCs along x coordinate is the same with that of grooves D = 6mm. Simultaneously, two rows of ELC particles are separated by a distance Dw = 10mm (the distance from center to center), which would influence the bandwidth of the rejection. Since the field intensity near the top of the grooves reaches the maximum, the coupling between ELC particles and the SPP waveguide weakens and the beginning frequency of the rejection would shift to higher frequency, when ELCs are moved towards the bottom of the grooves or away from the waveguide. Besides, the coupling between particles weakens and the bandwidth of the rejection narrows when the thickness of the substrate rises. Also, the SPP modes’ cut-off frequency would shift to a higher frequency.

 

Fig. 8 (a) Photographs of top and bottom surfaces of the broadband-frequency rejection waveguide. (b) The detailed arrangement of ELCs. (c) The simulated relative impedance of ELCs on the broadband-frequency rejection waveguide. (d) The simulated (black lines) and measured (red lines) transmission coefficients (S21) and reflection coefficients (S11).

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The calculated relative impedance of the ELC particle is demonstrated in Fig. 8(c). From the figure we clearly observe a remarkable mismatch of impedance in the frequency band from 6.8 to 8.1 GHz. To realize a broadband rejection, here we have chosen the groove depth to be 3.8 mm, which leads to a cut-off frequency at 9.6 GHz. The simulation and experimental results of the transmission coefficients (S21) and reflection coefficients (S11) are illustrated in Fig. 8(d). We clearly observe a rejection bandwidth of 1.3 GHz with the SPP-mode transmissions below −20 dB. Both results clarify that a broadband reflective-type spoof SPP filter is constructed. Finally, the near-electric-field distributions of SPP modes in the propagating and rejection bands are presented in Fig. 9. Unlike the absorption-type multiple-frequency-rejection SPP waveguide, the electric fields near the broadband-rejection waveguide oscillate as standing waves in the stop band, as shown in Fig. 9(b), implying a total reflection along the SPP waveguide.

 

Fig. 9 The near-field sketches of the broadband-frequency rejection SPP waveguide in the propagating band (a) and the rejection band (b).

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5. Conclusions

Utilizing the electrically resonant and impedance mismatching properties of the ELC metamaterial, we have constructed absorption-type ultra-narrow multiple-frequency- rejection SPP waveguides and reflection-type broadband-frequency-rejection SPP waveguides, which can serve as good narrow and broadband filters for the spoof SPP waves in the microwave and terahertz frequencies. Both numerical simulations and experimental results are provided to verify the above phenomena. By modulating the scaling factor of the ELC particle, we can realize tunable frequency-rejection devices in the future.