Continuous leaky-wave scanning using periodically modulated spoof plasmonic waveguide

The plasmonic waveguide made of uniform corrugated metallic strip can support and guide spoof surface plasmon polaritons (SSPPs) with high confinements. Here, we propose periodically-modulated plasmonic waveguide composed of non-uniform corrugated metallic strip to convert SSPPs to radiating waves, in which the main beam of radiations can steer continuously as the frequency changes. To increase the radiation efficiency of the periodically-modulated plasmonic waveguide at the broadside, an asymmetrical plasmonic waveguide is further presented to reduce the reflections and realize continuous leaky-wave scanning. Both numerical simulations and experimental results show that the radiation efficiency can be improved greatly and the main beam of leaky-wave radiations can steer from the backward quadrant to the forward quadrant, passing through the broadside direction, which generally is difficult to be realized by the common leaky-wave antennas.

transmission line (TL) can achieve high efficient radiation at broadside under the balanced condition, which requires complicated sub-wavelength unit cells to construct artificial series capacitance and shunt inductance for balancing series and shunt TL impedances 26,27 . More recently, M. Memarian and G. V. Eleftheriades proposed a kind of Dirac Leaky-wave antenna (DLWA) made up of a photonic crystal to make high efficient radiation at broadside due to the closed Γ -point bandgap of the Dirac PC 28 , which is designed based on a dielectric-loaded rectangular waveguide. More recently, a one-dimensional (1D) planar corrugated surface with infinite thickness and a cylindrical corrugated surface have been proposed to radiate surface waves 29 , but these bulky structures are not only difficult to be integrated with conventional circuits but also helpless to improve the radiation efficiency at broadside.
In this paper, we present periodically-modulated plasmonic waveguides composed of two-side corrugated metallic grooves to convert SSPPs to radiating waves. The periodic modulation introduces infinite space harmonics k N (N = 0, ± 1, ± 2, … ), in which the space harmonics with negative N can be fast with k N < k 0 under suitable conditions. In practice, the N = − 1 space harmonic is usually chosen to obtain single radiation beam, which can be achieved by modulating the groove depths of the plasmonic waveguide periodically. The simulation results show that the periodically-modulated plasmonic waveguide can convert SSPPs to radiating waves, in which the radiation pattern along azimuth is omnidirectional, while the radiation pattern along elevation can be steered as the frequency changes. However, we find that the radiation efficiency is reduced dramatically near the broadside radiation, which is due to large reflections of energy back to the feeding line 23 . To improve the radiation efficiency at the broadside direction, we simply stagger the periodically modulated grooves on both sides of the plasmonic waveguide with a phase displacement of π /2 along the propagating direction, so that the reflections generated by each pair unit cells on both sides are cancelled to each other. Different from the symmetrical case, the radiation pattern of the asymmetrical plasmonic waveguide is not omnidirectional along the azimuth, which is due to the enhancement and cancellation of radiating energies in forward and backward radiations, respectively. As frequency changes, the leaky-wave radiation can steer in continuous way, from the backward quadrant to broadside, then through the broadside to the forward quadrant. The measurement results have good agreements with numerical simulations, which have potential applications in integrated plasmonic circuits and antennas.

Results
Theoretical analysis. The SPPs are bounded on the boundary of two different dielectrics, which propagate parallel along the interface and decay exponentially in the direction vertical to the interface, as demonstrated in Fig. 1(a). According to the Maxwell Equations, the electromagnetic fields in the upper space should satisfy the following equations For lossless media, k z = β z is the propagating constant along the + z direction (β z is a positive real number), and k x = − jα x is the propagating constant along the + x direction (α x is also a positive real number). The relationship between the k z and k x can be written as = − k k k z x 0 2 2 , in which k 0 is the wavenumber in free space. According to Eq. (1), the surface impedance along the + z direction is calculated as where η 0 is the wave impedance in free space. Hence the relationship between the propagating constant along the surface (k z ) and surface impedance (η surf ) along the + z direction is obtained z surf 0 2 0 2 SSPPs at lower frequencies have similar characteristics to SPPs in the optical frequency. The corrugated metallic strip is a typical plasmonic waveguide, which can support and propagate SSPPs in the microwave and terahertz frequencies. Figure 1(b) shows the dispersion curves of one unit cell of the corrugated metallic strip by varying groove depth, illustrating that the wavenumbers k z along the propagating direction become larger as the groove depth (h) increases with k z > k 0 . The plasmonic waveguide is designed by using printed circuit board (PCB) of F4B with relative permittivity of 2.65 and loss tangent of 0.001, and the thickness of the substrate is chosen as t = 0.5 mm. The dimensions of unit cell shown in Fig. 1(b) are H = 5 mm, a = 1.13 mm, p = 2.825 mm, and varied h. Generally, SSPPs are slow waves (k z > k 0 ) bounded on the surface of the plasmonic waveguide tightly, which cannot be radiated into the free space. In order to convert SSPPs to radiating waves, we modulated the surface impedance of the plasmonic waveguide sinusoidally by changing the depths of metal grooves obeying the following equation in which X s is average surface reactance, M is degree of modulation, and A is modulation period. The model of the sinusoidally-modulated plasmonic waveguide is demonstrated in Fig. 1(c). The modulated plasmonic waveguide can introduce infinite space harmonics 18 , whose phase constant k N can be calculated as where k 0 is the phase constant in free space, and n is an effective surface refractive index (n ≥ 1). Equation (5) shows that the space harmonics are all slow waves for N ≥ 0 with k N > k 0 . But once N ≤ − 1, the space harmonics can be fast with |k N | < k 0 under suitable conditions. According to Eq. 5, if N ≤ − 2 space harmonics are designed to be fast and radiated, then N = − 1 space harmonic usually is avoidable to be fast and radiated. Hence, in order to obtain a single radiated beam, N = − 1 space harmonic is usually chosen in practice. We assume that the angle between the radiation beam and the + z direction is θ, as shown in Fig. 1(c). Hence the relationship between k −1 and θ can be written as To make N = − 1 space harmonic be fast wave, the condition of |k −1 | < k 0 must be guaranteed, and hence the range of effective surface refractive index n as determined as where λ 0 is the wavelength at the working frequency f 0 . According to Eqs (3) and (4), the relationship between n and X s is set up as Hence the range of X s is also fixed according to Eq. (7). Once the parameters X s and A are determined, the modulation parameter M should be chosen to satisfy the range of η surf based on Eq. (4). From Eq. (3), the surface impedance of each unit cell can be calculated as The range of k z can be determined by changing the groove depth (h) of unit cell from the minimum value to the maximum value at the working frequency. Then the range of η surf can be determined from Eq. (9). According to the unit cell shown in Fig. 1(b), the surface impedance η surf of the proposed plasmonic waveguide can cover the range from 766 to 130 when h varies from 4.5 mm to 0.5 mm at 9.3 GHz.
Design, simulation and experiment. Figure 1(c) shows a schematic of the designed sinusoidallymodulated plasmonic waveguide, which is composed of two coplanar waveguides (CPWs), two matching transitions, and a sinusoidally-modulated metallic strip. CPWs in two ports are designed with 50 Ω impedance, and two matching transitions between CPWs and the sinusoidally-modulated metallic strip are designed to make a smooth conversion between the spatial waves (supported by CPWs) and SSPPs (supported by the sinusoidally-modulated metallic strip) 11 . PCB used in our design is chosen as F4B with the thickness of 0.5 mm, and the thickness of the printed copper film is 0.018 mm.
According to Eq. (6), the direction of the radiation beam is designable by choosing different A and n. Here, we design a sinusoidally-modulated plasmonic waveguide arbitrarily to radiate the spatial waves to the direction of θ = 55° from the + z axis at 9.3 GHz. In our design, the modulation period A is chosen as 33.9 mm with 10 periods, in which each period (p = 2.825 mm) contains 12 symmetrical metallic grooves with different groove depths h, as shown in Fig. 1(b). Once the period A and radiation angle θ are fixed, then the effective surface refractive index can be calculated as n = 1.52 according to Eq. (6). Hence we choose X s = 1.1η 0 according to Eq. (8). Furthermore, the surface impedance of plasmonic waveguide can be modulated from min(η surf ) = 130 to max(η surf ) = 766 at 9.3 GHz, when the groove depth h is increased from 0.5 mm to 4.5 mm, as the red solid line demonstrated in Fig. 1(b). As a result, from Eq. (4), we choose M = 0.68 so that the surface impedance of the modulated plasmonic waveguide can cover the range of the minimum and maximum η surf as much as possible to achieve the high-efficiency radiation. Hence the final groove depths (h) of the sinusoidally-modulated plasmonic waveguide are determined according to the relationship between the surface impedance η surf and the groove depth h, as the red solid line shown in Fig. 1(b). Figure 2(a) presents the near-field distribution of E z by using the commercial software, CST Microwave Studio, in the yoz plane with x = 0, which clearly shows that the energy of surface waves is decreased gradually and radiated to the free space with an angle of θ = 55° when SSPPs propagate along the + z direction. Figure 2(b) shows the near-field distribution of E z in the xoy plane with z = 205 mm, illustrating that the radiation beam along azimuth is nearly omnidirectional around the plasmonic waveguide. The simulated far-field radiation patterns are given in Fig. 2(c-e) at 8.7 GHz, 9.3 GHz, and 9.9 GHz with radiation angles of 63.9°, 55.9° and 43.7°, respectively.
The final fabricated sinusoidally-modulated plasmonic waveguide is demonstrated in Fig. 3(a), whose measurement S parameters are presented in Fig. 3(b). Both the transmission coefficient S 21 and reflection coefficient S 11 are lower than − 10 dB from 8.4 GHz to 10 GHz, which means that the power is radiated to the free space efficiently. We can evaluate the radiation efficiency from the S parameters by 1 − |S 11 | 2 − |S 21 | 2 due to the low loss of the plasmonic waveguide at microwave frequencies. According to Fig. 3(b), the simulated radiation efficiency can be calculated with a minimal of 90% from 8.4 GHz to 10 GHz and maximum of 95% from 9 GHz to 10 GHz. The measured radiation efficiency also can be calculated with 95% from 8.8 GHz to 10 GHz, but which becomes worse with 86% from 8.4 GHz to 8.8 GHz. The worse radiation efficiency in measurement composed to the simulation is caused directly by the large measured S 11 , which may be due to the mismatch between the 50 Ω coaxial SMA connectors and CPWs on the two terminals of the plasmonic waveguide in measurement, because the full-wave simulations are carried out by using wave-port setup directly instead of using SMA connectors. Figures 3(c-e) show the measured far-field patterns of E plane in both ϕ = 0 (xoz) and ϕ = 90° (yoz) planes, and the patterns in Scientific RepoRts | 6:29600 | DOI: 10.1038/srep29600 both planes have good agreements to each other with main beams directing to 62.9°, 54.3° and 42.7° at 8.7 GHz, 9.3 GHz and 9.9 GHz, respectively. The measured gains and radiation angles at different frequencies are demonstrated in Fig. 3(f), which show that the measured gain is about 11.4 dBi and the radiation angle can be steered from 67.1° to 42.7° as the frequency changes from 8.4 GHz to 9.9 GHz.
However, the radiation efficiency near the broadside direction (θ = 90°) is decreased dramatically for the periodic structures with large voltage standing wave ratio (VSWR), and the energy in this region is mostly reflected back to the feeding source rather than being radiated, which is called as "open stop band" region 23 . In order to improve the radiation efficiency at broadside, we propose an asymmetrical plasmonic waveguide to cancel the reflections, as shown in Fig. 4(a), in which the periodically modulated grooves on both sides of the corrugated strip have a λ/4 offset along the propagation direction. In this way, the reflections generated by upper and lower grooves can be cancelled with each other to obtain highly-efficiency radiation at the broadside direction.
The dimensions of the asymmetrical plasmonic waveguide are A = 22.6 mm, a = 1.13 mm, 2H = 10 mm, P = 2.825 mm, X s = η 0 and M = 0.88. The upper and lower modulated grooves have λ/4 offset along the propagation direction with different groove depths in one period of h 1 = 4.3 mm, h 2 = 3.8 mm, h 3 = 2.2 and h 4 = 0.1 mm, as illustrated in Fig. 4(b). The full-wave simulation results of three-dimensional (3D) radiation patterns at 8.5 GHz, 9.3 GHz and 9.8 GHz are demonstrated in Fig. 4(c-e), respectively, which show that the main lobe can steer from backward quadrant (from 8.5 GHz to 9.3 GHz) to forward quadrant (from 9.3 GHz to 9.8 GHz), through the broadside (at 9.3 GHz) exactly. However, we notice that the radiation patterns are not omnidirectional along azimuth any more, in which the power is mostly radiated to the + y direction but reduced in the − y direction. The reason is that the periodically modulated grooves have λ/4 offset along the propagating direction, leading to π /2 phase difference on the same cross section of plasmonic waveguide. Hence the phase difference on any interface of corrugated metallic strip in + y and − y sides will be δψ π π λ ≈ ± ⋅ . As a result, if the δ ψ + y → 0, then δ ψ − y → π , and the power will be superposed in the + y direction and cancelled in the − y direction to form directional radiation patterns.
The measurement results show that both reflection coefficient (S 11 ) and transmission efficient (S 21 ) are lower than − 10 dB from 8.5 GHz to 9.8 GHz, and the radiation efficiency is much larger than that of the symmetrical structure, as green line shown in Fig. 5(a). The measured far-field radiation patterns of E plane in the plane of ϕ = 90° (the yoz plane) are shown in Fig. 5(b), which clearly show that the main lobe of radiation directed to 100°, 90° and 82° at 8.5 GHz, 9.3 GHz and 9.8 GHz, respectively. The continuously scanning ability of leaky wave is demonstrated in Fig. 5(c), by changing the frequency from 8.5 GHz to 9.8 GHz, the main lobe of radiation can steer from the backward quadrant (100° at 8.5 GHz) to the forward quadrant (82° at 9.8 GHz), through the broadside direction (90° at 9.3 GHz) exactly, and the radiation efficiency is about 95% at broadside. The measured gains and radiation angles at different frequencies are demonstrated in Fig. 5(d), which show that the measured gain is about 13.5 dBi and the radiation angle can steer from 100° to 82° as the frequency changes from 8.5 GHz to 9.8 GHz. We notice that the gain is nearly unchanged in whole operating frequency region.

Discussion
We presented a method to convert SSPPs to radiating waves using periodically-modulated plasmonic waveguide (i.e., corrugated metallic strip), whose surface impedance is modulated sinusoidally by changing the depths of corrugation grooves. The periodic modulation of the plasmonic waveguide introduces infinite space harmonics k N (N = 0, ± 1, ± 2, … ), in which the negative N corresponds to radiating fast waves under suitable conditions. In this paper, we choose N = − 1 space harmonic to obtain single radiation beam. The periodically-modulated plasmonic waveguide is connected to CPW with 50 Ω impedance at two ports by matching transitions, which can make the plasmonic waveguide be easily integrated to conventional microwave circuits. We have fabricated and measured the sinusoidally-modulated plasmonic waveguide at microwave frequencies, and the measured results have good agreements with the numerical simulations. For symmetrically two-side plasmonic waveguide, we show that SSPPs are converted to radiating waves efficiently with omnidirectional radiation pattern in the azimuth direction while the radiation beam in the elevation direction is steered as the frequency changes. In order to improve the radiation efficiency at the broadside, we further designed an asymmetrical plasmonic waveguide with sinusoidal modulation, in which the metal grooves on both sides have λ/4 offset to each other. The measured (a) The measured S parameters, which show that the reflection is reduced greatly by using asymmetrically plasmonic waveguide. (b) The measured far-field radiation patterns of E plane, which show that the main lobes are directed to 100°, 90° and 82° at 8.5 GHz, 9.3 GHz and 9.8 GHz, respectively. (c) The main lobe can steer from backward to forward, through broadside (90° at 9.3 GHz) continuously as frequency changes from 8.5 GHz to 9.8 GHz. (d) The measured gains and directions at different frequencies.
results showed that the radiation efficiency is increased greatly by using the asymmetrical plasmonic waveguide, and the radiation pattern along azimuth is not omnidirectional any more. Both simulation and experiment results have demonstrated that the radiation beams can be steered from the backward quadrant to the frontward quadrant, passing through the broadside, which show good performance of the designed leaky-wave antenna.