I. INTRODUCTION

Surface plasmon polaritons (SPPs) have attracted consid- erable interest due to potential applications such as antenna ^[1], filter ^[2], sensor ^[3], transmission line ^[4], etc, since Hendry et. al. proposed the SPPs structure using sub- wavelength periodic structures such as grooves and holes ^[5]. Among the potential applications, transmission line based SPPs have been studied extensively as one of the most promising candidate for high-speed interconnects ^[4,6].

Transmission lines (TLs) based on SPPs have been pro- posed by using sub-wavelength periodic structures such as grooves and holes ^[7], which are called spoof SPPs. Due to the geometry dependency of the spoof surface plasmon excitation, the spoof surface plasmon propagation at microwave frequencies can be realized with implementations such as rectangular grooves ^[2], T-grooves ^[4], I-grooves ^[8], pence-shaped grooves ^[9]. However, the operating frequency of the spoof SPP based TL is limited by the groove height, making it challenging to realize a high frequency TL for millimeter-wave operation using conventional photolithography or machining processes.

In this letter, we report on the high frequency spoof SPP TL implemented with low-cost fabrication. A significant correlation between groove shapes and field confinement have been thoroughly investigated, which eventually affects the cut- off frequency of spoof SPP based transmission line.

II. DESIGN AND METHOD

Fig. 1 shows the structures of the proposed spoof SPP transmission lines (TLs). The structures consist of a metal strip and a metallic ground plane printed on a dielectric substrate. The top view of TL structures with the various shapes of the metal strips including bar, inverted trapezoid and trapezoid, respectively, as shown in Fig. 1(a)-(c). The proposed spoof SPP TLs can be divided into two regions of the conversion section with length of l1(I and III regions) and the SPP TL with length of l2(II region). The metal strips are etched onto an Arlon 25 substrate of height (h) 0.76 mm, with a relative permittivity (${\epsilon}$r ) of 3.38 and a loss tangent (${\delta}$) of 0.018 by using a simple and low cost milling machining method, fabricated structures are shown in Fig. 1(d)-(g).

Fig. 1. The top-view of the proposed spoof SPP transmission line with various shapes: (a) bar; (b) inverted trapezoid; (c) trapezoid. The fabrication image of (d) the transition region; (e)-(g) transmission line region with different groove shapes magnified at the corresponding region illustrated in Fig. 1(a)-(c).

The proposed structures are designed to operate with transmission characteristics having high cut-off frequency, with the structure dimensions given by l1= 13.75 mm, l2= 21.01 mm, d = 8.47 mm, g = 1.32 mm, p = 1.375 mm, i = 0.385 mm, w = 0.385 mm, and v = 0.99 mm, according to Fig. 1.

We use the commercial software Computer Simulation Technology (CST) Microwave Studio with eigenmode solver and frequency domain solver to calculate the dispersion di- agram and evaluate the transmission characteristic of the proposed spoof TL. Furthermore, to verify the design, the scattering parameter of the fabricated samples are measured using a network analyzer.

III. RESULT AND DISCUSSION

Fig. 2 depicts the dispersion curves of the spoof TLs with different groove shapes. The free space light dispersion, or light line is the frequency-wave number dispersion relation of free space photons, which represented as $\Omega =\mathrm{ck}_{0}~ $ (${\Omega}$ is the angular frequency, k₀ is the wavenumber, and c is the speed of light in vacuum). Meanwhile, the frequency dispersion data is collected using the CST Eigenmode Solver ^[10]. The detailed simulation setup is shown in Ref. ^[11]. As seen in Fig. 2, the dispersion curves exhibit the spoof SPP-like behavior, which gradually deviate from the light-line and then asymptotically approach different cut-off frequencies. It is clear that the asymptotic frequency of the trapezoid-shaped TL > the asymptotic frequency of the bar-shaped TL > the asymptotic frequency of the inverted trapezoid-shaped TL.

Fig. 2. Dispersion diagram of the proposed spoof TL unit cell.

The Fig. 3 shows the electric field energy density for the proposed TLs with different groove shapes. It is clear that comparing with the bar-shaped TL, the trapezoid-shaped TL can enhance the field confinement while the inverted trapezoid- shaped TL has reduced field confinement. This observation shows that the enhanced field confinement are strongly cor- related to the reduced asymptotic frequency of the spoof TL. The spoof SPP structure with lower asymptotic frequency exhibits tighter field confinement ^[8].

Fig. 3. (a) Simulated electric field energy density on the surface at 28 GHz of bar-shaped; (b) inverted trapezoid-shaped; (c) trapezoid-shaped spoof TL unit cell.

To verify the transmission characteristics of the proposed spoof TL, the simulated and measured reflection coefficient (S11) and transmission coefficient (S21) are investigated as shown in Fig. 4. The simulated cut-off frequencies (-3 dB) of the proposed TLs with bar, inverted trapezoid, and trapezoid groove shapes are 35.9~GHz, 30.6 GHz, and 40.1 GHz. Com- pared with simulated results, the measured cut-off frequency of the proposed TLs trend to increase due to the fabrication error. These observations confirm that the spoof TL with tighter field confinement realized the lower cut-off frequency. This is consistent with the theoretical prediction as shown in Fig. 2. Furthermore, the measured loss of the proposed TLs increased at higher frequencies above 10 GHz compared with the simulated loss of the TLs. This can be explained by the increase of loss tangent of the substrate with increasing the frequency to higher region (above 10 GHz). The measured insertion loss per unit length at 28 GHz for the bar-shaped, inverted trapezoid-shaped, and trapezoid-shaped TLs is 0.084 dB/mm, 0.081 dB/mm and 0.074 dB/mm, respectively.

Fig. 4. Simulated and measured S-parameter of the proposed spoof TL with different groove shapes: (a) bar; (b) inverted trapezoid; (c) trapezoid.

To visualize the high frequency operation of the proposed spoof SPP TLs, the electric-field distributions at various frequencies of 10 GHz, 28 GHz, and 40 GHz are simulated along the propagation direction, as depicted in Fig. 5. The proposed spoof SSP TLs allow propagation with high efficiency at high frequencies, inside the usable bandwidth. Furthermore, the stop-band characteristic at frequencies above the cut-off frequency can be seen in Fig. 5(c) and (i).

Fig. 5. Simulated electric field distributions 0.15 mm under the structure surface at different frequencies of 10 GHz, 28 GHz, and 40 GHz for the proposed spoof SSP TLs with (a)-(c) bar-shaped grooves; (d)-(f) inverted trapezoid-shaped grooves; (g)-(i) trapezoid-shaped grooves, respectively.

To gain insights into the effect of groove shapes on field confinement and transmission characteristics, the simulated electric field distribution at the cross section of the spoof SPP TLs and normal microstrip line at 28 GHz have been investigated as shown in Fig. 6. The electric field of the proposed spoof SPP TLs occurs denser around the surface than that of the normal microstrip line. Furthermore, to quantitatively analyze the confinement phenomena of the proposed spoof SPP TLs and normal microstrip line, the distributions of electric field amplitudes in the cross-section located 0.15 mm below the surface, on the surface, and 0.15 mm above the surface, along the y-axis direction are gathered as illustrated in Fig. 7. As seen in Fig. 7, the magnitude of electric field exponential decay along the y-axis direction, indicating the occurring of the confinement effect of SPP modes. The electric field peak value of the proposed spoof SPP TLs is higher than that of the normal microstrip line that further confirms the stronger electric field confinement of spoof SPP TLs. Furthermore, the electric field distribution curve shows that the electric field concentrates mainly at the outer side of the near top edge grooves of the spoof SPP TLs. Higher electric field peak and stronger field confinement occurs in the inverted trapezoid- shaped spoof SPP TLs, resulting in reduced cut-off frequency due to the capacitance increase in the LC resonant circuit. Meanwhile, the peaks of the trapezoid-shaped TLs in this region are lower than the bar-shape TL, indicating the lower field confinement and higher cut-off frequency.

Finally, the performance of our structure is compared with existing spoof SPP TL structures. Table 1 exhibits the spoof SPP TL properties in terms of shape structure, operating frequency, total length, and insertion loss. As seen in Table 1, the proposed structure achieves the high frequency and compact total length while keeping a moderate insertion loss.

Fig. 6. The simulated cross section electric-field distributions at 28 GHz in the groove center of the proposed spoof SPP TLs with (a) bar-shaped grooves, inverted trapezoid-shaped grooves; (c) trapezoid-shaped grooves; (d) normal microstrip line.

Fig. 7. Simulated cross-section electric field distributions on the YZ plane of the spoof TL structures located: (a) 0.15 mm below the spoof surface, on the spoof surface; (c) 0.15 mm above the spoof surface.

IV. CONCLUSION

A high frequency TL based on spoof surface plasmon transmission is proposed. The transmission characteristic of the proposed TLs is investigated using a numerical method, which is also verified by measurement result, indicating that the proposed spoof SPP TLs can operate up to above 28 GHz. By tailoring the different groove shapes of bar, inverted trapezoid and trapezoid which arranged periodicity along the TL, the proposed TL can realize higher field confinement or higher cut-off frequency. Compared with the bar-shaped TL, the inverted trapezoid-shaped TL exhibited enhanced field confinement. Meanwhile, the cut-off frequency of the proposed TL using the trapezoid shape significantly increased up to 40 GHz. Due to excellent features such as low-cost, ease of fabrication, and good operating characteristics, this design is a promising candidate for millimeter-wave devices.