Properties of Transmission and Leaky Modes in a Plasmonic Waveguide Constructed by Periodic Subwavelength Metallic Hollow Blocks

Based on the concept of low-frequency spoof surface plasmon polaritons (spoof SPPs), a kind of leaky mode is proposed in a waveguide made of a subwavelength metal-block array with open slots. Numerical results reveal that a new transmission mode is found in the periodic subwavelength metal open blocks. This modal field is located inside the interior of a hollow block compared with that in a solid metal block array. The dispersion curve shows that such a new SPPs mode has a negative slope, crossing the light line, and then going into a zone of leaky mode at higher frequencies. The leaky mode has a wider frequency bandwidth, and this can lead to a radiation scanning angle of 53° together with high radiation efficiency. Based on the individual characteristics exhibited by a frequency-dependent radiation pattern for the present leaky mode, the waveguide structure can have potential applications such as frequency dividers and demultiplexers. Experimental verification of such a leaky mode at microwave has been performed, and the experimental results are found to be consistent with the theoretical analysis.

Scientific RepoRts | 5:14461 | DOi: 10.1038/srep14461 in detail in ref. 7. The geometry-controlled SPPs, which are referred to as the spoof SPPs (SSPPs), were first experimentally verified at microwave by taking advantage of subwavelength rectangular metal holes 8 . A periodic subwavelength corrugated metal surface with an EM field confined tightly on it has been widely investigated for THz applications. For instance, in order to transmit THz signals, a novel kind of cylindrical metal wires with transmission properties controlled by geometric parameters of lattice constant was proposed 9,10 . A new technique for highly efficient conversion of surface-plasmonlike modes to spatial radiated modes has been suggested based on spoof surface plasmon polariton emitters 11 .
As far as the new waveguide structure supporting spoof SPPs is concerned, a one-dimensional (1D) periodic Domino plasmonic waveguide was theoretically analyzed in THz regime in ref. 12 and then experimentally verified in ref. 13. Another kind of subwavelength periodic V-shaped channel waveguide with low bending loss was also reported 14 . These THz subwavelength periodic structures can be more flexible than before in design of new metallic waveguide devices. Besides, a kind of microwave subwavelength periodically corrugated metal wires covered by dielectric layer with high dielectric constant was studied 15 , and 1D Domino array was also investigated at microwave 16,17 .
In the literature, most studies, in which the properties of SSPPs in a periodic Domino structure have been widely theoretically and experimentally investigated, focused on the analysis of field distribution and dispersion characteristics 16,17 . In addition, accurate S-parameter properties are also essential for a waveguide device. For example, a new kind of SSPPs based on metallic UV channel waveguide (with a structure of alternating arrangement of U-and V-shaped grooves) in microwave regime was proposed in ref. 18, where the authors measured the S-parameters by vector network analyzer. For a 1D Domino array structure, the transmission properties can be measured through a waveguide converter technique 19,20 . A possible application of such SSPPs is that they can provide a low crosstalk transmission line in microwave integrated circuits [21][22][23] . For example, a microstrip line, which has a structure of corrugated gradient grooves, can serve as a potential device of broadband molecular sensing 24 . Other interesting applications include ultra-thin SSPPs with CPW feed structures 25,26 and SSPPs-assisted deep-subwavelength negative-index waveguiding technique 27 .
Less attention has been paid to the properties of leaky modes in aforementioned works. Very recently, Cui et al. have suggested a new scheme for controllably manipulating electromagnetic radiation by holographic metasurfaces (a structure of subwavelength metallic patches on grounded dielectric substrates), where some interesting topics relevant to leaky waves have been investigated both theoretically and experimentally 28,29 . The hybrid metasurface that is composed of planar metamaterial and holographic metasurface has also been developed aiming at simultaneous controls of propagating waves and surface waves (including leaky-surface waves) 30 . Now in this work, we shall propose a new kind of leaky modes radiation based on the concept of low-frequency SSPPs in a periodic subwavelength structure. By tuning the geometric parameters, the corresponding dispersion can be obtained and therefore the desired transmission and radiation properties can be achieved. The dispersion curves will be calculated by making use of the commercial software, COMSOL. We shall consider a THz metallic waveguide constructed by periodic subwavelength metallic blocks with open slots. It will be found that an additional SPPs mode exists in such periodic block array. The corresponding dispersion curve lies within the light cone and thus corresponds to a leaky mode. Compared with a transmission mode, the propagation constant of the leaky mode has a complex form, k z = β + jα. This means there is radiation loss under the assumption that the metal is identified with a perfect electric conductor (PEC). An experiment will be carried out to confirm the theoretical results. Figure 1. Waveguide based on a periodic subwavelength metallic block array supporting spoof SPPs, in which the width is L, the period is d, the adjacent spacing is a, the height is h, and the waveguide lateral width is w. In a single subwavelength metallic hollow block, the bottom length is a 1 , the mouth length is a 2 , the aperture height is h 1 , and the opening angle is θ.

Results
Transmission properties of a THz plasmonic waveguide. The proposed plasmonic waveguide made of a periodic subwavelength metallic bevel hollow block array structure, of which the geometric parameters are shown in Fig. 1. For simplicity, we assume the metal to be a PEC in our simulation. This is a reasonable approximation at microwave and THz 1,2 . The plasmonic waveguide with a periodic metallic block array will be simulated by the finite element method (FEM), from which the dispersion curve, the field distribution and the transmission coefficient can be obtained.
The dispersion curves (f vs. β) of the plasmonic waveguide are displayed in Fig. 2(a). Here, β is restricted in the first Brillouin zone of β π ≤ /d. For the periodic metallic hollow block array, the geometric parameters are given by d = 100 μ m, a = 0.5d, L = 50 μ m, h = 40 μ m, a 2 = 10 μ m, a 3 = 30 μ m, h 1 = 20 μ m, and θ = 53.4°. We shall focus on the fundamental mode and the leaky mode of this plasmonic waveguide. The fundamental mode (as a guided mode) locates on the right side of the light line. The dispersion curve of the solid block array is also given for the purpose of comparison. For the period metallic solid block array (SBA) with d = 100 μ m, a = 0.5d, L = 50 μ m, and h = 40 μ m, the cutoff frequency f sc is 0.9338 THz, the asymptotic frequency f ss (β = π/d) is 1.124 THz, and the working bandwidth is 0.1902 THz, as can be seen in the red dotted curve. For the period metallic hollow block array, the cutoff frequency f 1c of the fundamental mode is 0.928 THz, the asymptotic frequency f 1s (β = π/d) is 1.1645 THz, and the working bandwidth is 0.237 THz, as shown in the black solid curve. With the same geometric parameters, the hollow block array and the solid block array both have a similar asymptotic frequency, but the hollow block array has a wider transmission band due to smaller cutoff frequency. Except for the above-mentioned guided mode, there exists a new leaky mode for the hollow block array in Fig  2 for the two waveguides). The result in Fig. 2(c) is also confirmed in Fig. 2(d), i.e., the radiation efficiency decreases significantly in the leakage regime when the metallic block width changes from L = 50 μ m to L = 100 μ m. In the guiding regime, however, the electromagnetic field is confined more tightly in the periodic structure of metallic blocks of L = 100 μ m than that of L = 50 μ m.
The magnetic field distribution in a unit cell at four typical frequencies is presented in Fig. 3. In Fig. 3(a) we show the field profile of the fundamental mode at the asymptotic frequency of f 1s = 1.1645 THz (β = π/d) for L = 50 μ m. It can be seen that most of the EM field spreads out of the block, much similar to the case in solid block, implying that these two structures have almost the same asymptotic frequency. In Fig. 3(b), we have distribution of the second mode at the asymptotic frequency of f 2s = 1.2927 THz (β = π/d), in which most of the EM field is localized inside the block. In addition, the EM field is uniformly distributed both inside and outside the block at f = 1.00256 THz (β = 0.7π/d) for the fundamental mode, while, for the second mode, most of the EM field is distributed inside the block at f = 1.3606 THz (β = 0.7π/d). In Fig. 4, we plot the magnetic near-field distributions of the plasmonic waveguides at β = 0.7π/d, for L = 50 μ m and 100 μ m, respectively. In Fig. 4(a) is the magnetic near-field distribution of the fundamental mode at f = 1.00256 THz for L = 50 μ m. It can be clearly seen that the EM field is highly confined near the surface of the plasmonic waveguide. In Fig. 4(b), we show the near-field distribution of the leaky mode at f = 1.3606 THz for L = 50 μ m. The EM field is input at the right end and it can be radiated from the waveguide into the free space at a specific angle. Fig. 4(c) is the magnetic near-field distribution at  However, the structure with L = 100 μ m has exhibited an effect of strong field confinement. Fig. 4(d) is the magnetic near-field distribution of the leaky mode at f = 1.33531 THz for L = 100 μ m. Shown in Fig. 5(a-d) is the far-field distribution of the leaky mode at frequencies 1.3058 THz, 1.3326 THz, 1.3538 THz and 1.3708 THz, respectively. The corresponding radiation angles are φ = 286°, φ = 303°, φ = 312° and φ = 320°, respectively, for L = 50 μ m. Since the present plasmonic waveguide is a subwavelength periodic structure, there must be a number of space harmonics with phase constant β n . When β < k n 0 , the electromagnetic field will be radiated from the metallic periodic structure to the free space. It is evident that the radiation direction, which can be evaluated by φ β = / − k sin n n 1 0 in ref. 31, scans as the frequency changes.
In the above analysis, the metal is assumed to be PEC, so that there is no loss for the transmission mode. In the realistic case, however, there exists a metallic loss due to the surface current in metal with a finite conductivity. We will study the metallic loss using an aluminum waveguide. The lattice constant of our periodic waveguide is d = 100 μ m, and so the waveguide can work in the THz band. The metallic loss can be calculated with the perturbation method 22 , from which we have , where p d is the total metallic loss in a periodic unit and p f is the total transmission power. The propagation length is thus given by The normalized propagation length (L m /λ ) as a function of frequency for the fundamental mode is shown in Fig. 6, where the normalized propagation length of the solid block array is also plotted in the dotted line for the purpose of comparison. At low frequencies, the propagation length is large for the solid block array compared with the hollow block array, because there is a significant metallic loss for the latter, in which more EM field is confined near the metal surface. The transmission lengths of these two cases decrease as the frequency increases and become zero at the asymptotic frequency. In addition, at a certain frequency (e.g., at 1.067 THz), both curves (for propagation length in hollow and solid block array) have an intersection for L = 50 μ m.

Experimental verification of plasmonic waveguide in microwave band.
Since it is difficult to fabricate a waveguide at THz, as an alternative by scaling down the frequency, we design and fabricate a  metallic structure in Fig. 7 with aluminum to verify the characteristics of a plasmonic waveguide in the microwave regime. At each end of the waveguide, there is a transition region of 50 mm in length and the height of the periodic metallic hollow block array is gradually increased. As a result, the field distribution from the rectangular waveguide is gradually transformed into the spoof SPPs field distribution along the transition region, reducing the reflection in the junction between the plasmonic waveguide and the feeding waveguide. Here, h = 4 mm, d = 10 mm, a = 0.5d, L = 5 mm, w = 50 mm, and the length of the plasmonic waveguide t = 375 mm. Some other geometric parameters are h 1 = 2 mm, a 1 = 3 mm, and a 2 = 1 mm.
The simulated dispersion curves at microwave are given in Fig. 8(a). It can be found that there are two modes for such a plasmonic waveguide. The results agree with those at THz. For the fundamental mode plotted in the solid black curve, there is a cutoff frequency at f 3c = 9.3328 GHz and an asymptotic frequency at f 3s = 11.6524 GHz. Thus, the transmission bandwidth is 2.3195 GHz. For the comparison purpose, the guided mode of solid block array is plotted in the dotted curve from which we find a cutoff frequency, f 4c = 9.873 GHz, an asymptotic frequency f 4s = 11.503 GHz, and consequently the transmission bandwidth of 1.63041 GHz, smaller than that of the hollow block array. The measured dispersion curve of such a mode is indicated by the triangle marks in Fig. 8(a). For the second mode of the hollow block array in the dashed curve, the asymptotic frequency is f 5s = 12.959 GHz, and the dispersion curve intersects with light line at f = 13.15 GHz and then enters into the leaky mode zone until f = 14.443 GHz. The propagation constant is complex, i.e., k z = β + jα for a leaky mode. In Fig. 8(b), we show the imaginary part α/k 0 as a function of frequency. It is found that α/k 0 = 0.00659 at 13.2 GHz, 0.02459 at 13.611 GHz, and afterwards drops to 0.00798 at f = 14.2 GHz. In addition, this structure has a maximum leakage radiation efficiency at f = 13.611 GHz.
To study the transmission properties, an aluminum waveguide with a length of 375 mm is fabricated and measured by Vector Network Analyzer (VNA). Fig. 9(a) displays the S parameters for the case with geometric parameters: h = 4 mm, d = 10 mm, a = 0.5d, L = 5 mm and w = 50 mm. It can be seen that S 21 is 0.358 at 8 GHz, and then it increases up to the maximum value of 0.843 at 10.38 GHz, decreases to 0.507 at 11.62 GHz, and drops sharply to 0.068 at 11.66 GHz. On the other hand, S 11 is 0.1774 at 8 GHz, and is relatively large in the band gap between 11.66 GHz and 12.96 GHz. In the leakage zone between 13.11 GHz and 14.5 GHz, S 11 is smaller than 0.5 while S 21 is smaller than 0.2, and hence + S S 11 21 has a much smaller value 0.155 at the frequency of 14.02 GHz. Hence, the experimental results for the main characteristics of guided mode and leaky mode agree well with the theoretical simulation. Fig. 9(b) presents the far field radiation pattern for L = 5.0 mm. At 13.2 GHz, the radiation direction of main beam is 285°, and the half power beam width (HPBW) is Δ φ = 16°. At the frequency 13.8 GHz, the radiation angle is 315°, and the HPBW is Δ φ = 5°, and at 14.3 GHz, the radiation angle is 339° and the HPBW is Δ φ = 5°. In Fig. 9(c) is the leakage radiation efficiency for the two waveguides. The confinement effect in the solid block array is also plotted in the dotted curve for comparison. Apparently it is a guided mode between 8 GHz and 11.5 GHz, in which the leakage radiation efficiency of bevel hollow block waveguide is smaller than that of the solid block waveguide. This is due to the fact that the EM field is highly confined for the additional hollow block in that region. From 13.15 GHz to 14.443 GHz, the bevel hollow block waveguide enters into the leaky mode zone, where the leakage radiation efficiency is much higher than that in the solid block waveguide, agreeing with the theoretical results. It is noteworthy that the leakage radiation efficiency is 97.8% at 14 GHz.
From the previous analysis, the plasmonic waveguide with these geometric parameters has an effect of high confinement of the modal field. As a result, the EM field from the input can be efficiently guided to the output by such a metallic hollow block array. On the other hand, such a hollow block array has an additional leaky radiation mode, which gives rise to frequency scanning radiation. The slight difference between the experimental results and the theoretical simulation is caused by the fabrication precision and the metallic loss.

Discussion
In conclusion, we have numerically and experimentally studied the properties of transmission and leaky modes in a plasmonic waveguide constructed by periodic subwavelength metallic hollow blocks. The structure can support spoof SPPs whose modal field is highly confined to the structure surface. It is emphasized that, in addition to the transmission mode, there is an extra leaky mode induced by the metallic open apertures, and hence the leakage mode can lead to the effect of frequency scanning radiation. Our analysis indicates that the transmission bandwidth and leaky radiation efficiency can be controllably manipulated via SSPPs waveguide geometric parameter adjustment. Such a kind of waveguide structures can be employed in high power THz, microwave transmission or high directive radiation systems.

Method
The dispersion curves, radiation efficiency and the ohmic losses of the periodic waveguide structures were calculated by commercial FEM software (COMSOL). The near field distribution and far field radiation pattern of the waveguides were simulated by CST Microwave Studio. The aluminum waveguides at microwave frequencies were fabricated by a computer numerical control (CNC) machine. The performance of the fabricated waveguides such as transmission bandwidth, leaky radiation efficiency and far field radiation pattern were tested through experiments.