1. Introduction

Surface plasmon polaritons (SPPs) are surface electromagnetic waves propagating along the interface between metal and dielectric in optical frequency regime [1,2]. Because of its unique subwavelength confinement and near field enhancement properties, SPPs have been applied to miniaturized photonic circuits [3,4], sensing and detection [5,6], and near-field imaging [7], etc. However, because metal behaves a perfect electric conductor (PEC) property, the natural SPPs do not exist in the microwave and terahertz regions [8]. To achieve subwavelength confinement at microwave and terahertz frequencies, the concept of spoof SPPs (SSPPs) or designer SPPs was proposed to mimic the electromagnetic properties of SPPs at optical frequencies [9,10]. Based on this concept, various bulk plasmonic waveguides such as textured metal surfaces, ring corrugated metal transmission lines, and domino waveguides [11–13] have been designed and investigated. Recently, planar plasmonic waveguides with single-side corrugations, symmetrical and staggered double-side corrugations, and the L-shaped, folded, and spiral stubs [8,14–17] have attracted increasing attention owing to their advantages of stronger subwavelength confinement and smaller size compared with the bulk plasmonic waveguides. The dispersion characteristics of the bulk and planar plasmonic waveguides can be manipulated by the geometric parameters of the corrugations or grooves. Besides, lumped elements like resistors, capacitors, and varactors [18–20] can also be introduced into the planar plasmonic waveguides to realize additional control of the SSPP waveguiding properties. By using the planar plasmonic waveguides, many microwave and terahertz integrated plasmonic devices like conversions, filters, dividers, and antennas have also been designed [17,21–24].

On the other hand, it is very important to further reduce size and interference in the development of highly integrated circuits and systems. Although shielding boxes [25–27], absorbing materials [28], and differential microstrips [29] are used to reduce the crosstalk and noise interference from the surrounding environment or between adjacent transmission lines, the volume of the devices is inevitably increased. Taking advantage of the subwavelength confinement of the SSPPs, the planar plasmonic waveguide can act as a promising alternative to solve this problem [15,17,30]. However, most of the proposed planar structures usually require complicate and bulky conversion structures with flaring grounds of the coplanar waveguide for achieving smooth mode matching between the guided waves and SSPPs, which restricts the application of the concept of the SSPPs in the design of compact devices. How to further reduce crosstalk and simultaneously achieve miniaturization of the planar plasmonic waveguides remains a challenge.

In this paper, we propose a novel variety of V-shaped microstrips for highly efficient and strongly confined SSPPs propagation in microwave region. First, we study the dispersion characteristics of the proposed V-shaped microstrip and find that the asymptotic frequency can be significantly reduced by adding the folded stub without increasing the lateral dimension of the microstrip. Because the V-shaped microstrip is compatible with conventional microstrip, a broadband transition is designed by directly connecting a tapered microstrip and an array of graded height V-shaped units to achieve smooth impedance matching and momentum matching. The size of this conversion is greatly reduced compared to the conventional mode conversion structures in other typical grooved SSPP waveguides. Then, we investigate coupling characteristics of the parallel and symmetrically arranged V-shaped microstrips. Compared with conventional parallel microstrips with a separation of 2.8 mm, the proposed V-shaped microstrips have significantly lower coupling effects in the whole range of 0 - 10 GHz, even for the parallel V-shaped microstrips with 2 mm inner-overlapping. The proposed V-shaped microstrips have excellent properties of strong field confinement, flexible dispersion manipulation, low coupling, and small size, which may have great potential in achieving device miniaturization and noise interference suppression in highly microwave integrated circuits and systems.

2. Dispersion characteristics of the V-shaped microstrip units

The proposed V-shaped microstrip is composed of a V-shaped metal strip separated from a ground plane by a flexible dielectric substrate, as shown in Fig. 1. For the V-shaped metal strip, the period length p = 5 mm, the height h = 8 mm, the strip width w = 0.5 mm, the thickness of the metal strip and ground plane t = 0.018 mm, and the thickness of dielectric substrate t_s = 0.508 mm. The dielectric substrate is assumed to be Rogers 5880 with the relative permittivity of 2.2 and the loss tangent of 0.0009, and the metal material is copper with the conductivity of 5.8 × 10⁷ S/m. First, we use the eigenmode solver based on the finite element method to study the dispersion characteristics of the fundamental SSPP mode on the V-shaped microstrip units. Figure 2(a) shows that the dispersion curves of the V-shaped microstrip with and without ground plane distinctly deviate away from the light line (black line). It is worth noting that the proposed V-shaped microstrip unit with a metal ground has an even lower asymptotic frequency than the corresponding structure without metal ground, implying stronger subwavelength field confinement capacity. Meanwhile, the V-shaped microstrip, as a double-conductor structure, is compatible with conventional microstrip and active chips in microwave or terahertz integrated devices and circuits. Furthermore, the dependence of the dispersion relations on the geometrical parameters of the V-shaped microstrip is also investigated. The dependence of the dispersion relations on the metal strip thickness t, the dielectric substrate width a, the period length p, the height h, and strip width w are shown in Figs. 2(b)-2(f), respectively, while keeping other parameters unchanged. It is clearly observed that the width w and thickness t of the metal strip, and the width a of the dielectric substrate have a little effect on the asymptotic frequency, especially the thickness t. The dispersion curves are almost unchanged with the asymptotic frequencies around 7.7 GHz when a is reduced from 24 mm to 16 mm, while the asymptotic frequency slightly increases to 8.14 GHz as a further decreases to 12 mm. On the contrary, the period length p and height h have a significant impact on the asymptotic frequencies of the dispersion curves, especially the height h. When the height h increases from 4 mm to 10 mm, the asymptotic frequency decreases from 11.52 to 6.38 GHz. Therefore, the asymptotic frequency of the proposed V-shaped microstrip is mainly dictated by the effective length of the strip relying on the p and h. Considering these dispersion relations and the available metal layer thickness t on a Rogers 5880 substrate, we set a = 16 mm and t = 0.018 mm in this work.

 

Fig. 1 Schematic diagram of the proposed V-shaped microstrip. The geometric parameters of the unit structure are set as follows: the period length p, the height h, the strip width w, the thickness of the V-shaped metal strip t, the width and thickness of the dielectric substrate are a and t_s, respectively.

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Fig. 2 The dispersion relations for the fundamental SSPP mode of the proposed V-shaped microstrip units. (a) The dispersion relations of the light line, the V-shaped microstrip units with and without ground, where p = 5 mm, h = 8 mm, w = 0.5 mm, a = 16 mm, t = 0.018 mm, and t_s = 0.508 mm. (b)-(f) are the dependence of the dispersion relations on the geometric parameters t, a, w, p, and h of the V-shaped microstrips, respectively, while keeping other parameters unchanged.

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To further manipulate the dispersion relations and achieve stronger confinement, we study the dispersion relations of the proposed V-shaped microstrip with different additional stubs. Since the asymptotic frequency is mainly determined by the effective length of the strip, a lower asymptotic frequency can be obtained by introducing additional stub to increase the effective length of the strip while maintaining the lateral size of the unit structure unchanged. Figures 3(a)-3(c) demonstrate the schematic diagrams of the proposed V-shaped microstrip unit with a single horizontal stub, a folded stub, and a spiral stub, respectively. The simulated dispersion relations of the V-shaped strip with different stub lengths are compared in Fig. 3(d). It is clear that the asymptotic frequencies of the dispersion curves can be significantly reduced from 6.67 to 2.25 GHz as the stub length increases from 0 to 21.96 mm. Compared with the original V-shaped microstrip, the asymptotic frequencies of V-shaped microstrip with spiral stub is reduced by 66.27%, implying better confinement performance, without increasing the size of the unit structure. Therefore, introducing additional stub structures is an effective route to further improving the field confinement performance and structure minimization level of the proposed V-shaped microstrip.

 

Fig. 3 (a)-(c) are schematic diagrams for the proposed V-shaped SSPPs microstrip with a horizontal stub, a folded stub, and a spiral stub, respectively, where p = 7 mm, h = 8 mm, w = 0.5 mm, a = 16 mm, t = 0.018 mm, and t_s = 0.508 mm. (b) The dispersion relations of the proposed V-shaped SSPPs microstrip unit with different length stubs.

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3. Propagation characteristics of the V-shaped microstrips

To achieve smooth impedance and momentum matching between the SSPPs and the quasi-TEM waves of the microstrip, broadband high-efficient V-shaped microstrip - microstrip transitions are employed in the design of microstrip - V-shaped microstrip - microstrip whole structure, as shown in Fig. 0.4. For example, the whole structure A is composed of three parts, a microstrip (MS), a V-shaped MS, and a transition between V-shaped MS and MS. In the MS part, the characteristic impedance, length, and width of the microstrip are set as Z₀ = 50 Ω, L₁ = 5 mm, and w₁ = 1.516 mm, respectively. In the V-shaped MS part, the total length of the V-shaped microstrip L₃ = 120 mm, which consists of 24 periodic V-shaped microstrip unit cells with p = 5 mm, h = 8 mm, and w = 0.9 mm. In the transition part, the length L₂ = 55 mm, consisting of a tapered microstrip with the width decreasing from w₁ to w for the impedance conversion and four graded-height V-shaped microstrip units with the step of 2 mm for the momentum conversion as expected from Fig. 2(f). Similarly, with the same transition structure, two whole structures B and C based on the V-shaped microstrip with additional stubs are shown in Figs. 4(b) and 4(c), where the length of the stubs l_a = 3.11 mm and l_ab = 5.17 mm while keeping other parameters the same as Fig. 4(a). The fabricated prototypes with SMA connectors of these structures are displayed in Figs. 4(d)-4(f), respectively.

 

Fig. 4 (a)-(c) Schematic diagrams of the whole structures A, B, and C, respectively, where p = 5 mm, h = 8 mm, and w = 0.9 mm, l_a = 3.11 mm, l_ab = 5.17 mm. (d)-(f) Fabricated whole structures A, B, and C, respectively.

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To demonstrate the propagation properties of the proposed V-shaped microstrips, we simulate and measure the above-mentioned whole structures with and without additional stubs. The simulated and measured scattering parameters (S-parameters) of the whole structure A (Fig. 4(a)) are shown in Fig. 5(a). As expected from the dispersion characteristics, the whole structure acts as a wideband lowpass filter with a steep roll-off. High-efficient transmission performance is achieved with the reflection coefficient (S₁₁) less than −10 dB and the transmission coefficient (S₂₁) higher than −3 dB in the passband from 0 to 10.3 GHz. Unlike the single-conductor SSPP waveguide that cannot support SSPPs at low frequencies [22,31,32], the proposed V-shaped microstrip can maintain excellent transmission performance at extremely low frequency (even at 0 GHz) owing to its double-conductor structure. Similarly, the simulated and measured S-parameters for the structures B and C are presented in Figs. 5(b) and 5(c), respectively. It should be noted that the measured results are in good agreement with simulation results, and some slight differences are caused by the impedance mismatching in the SMA soldering and fabrication tolerance. Clearly, the cutoff frequency of the frequency response can be flexibly adjusted by the length of the additional stub. As shown in Fig. 5(d), by introducing the additional stubs, the cutoff frequencies of the structure B with the stub l_a = 3.11 mm and the structure C with the stub l_ab = 5.17 mm are significantly reduced to 6.8 and 5.7 GHz, respectively. Furthermore, Figs. 6(a)-6(c) show the simulated electric field magnitude (|E|) distribution at 5 GHz on the xoy plane cut at the top surface of the Rogers 5880. It is clear that smooth mode conversions from the quasi-TEM waves of the microstrip to the tightly confined SSPPs of the proposed V-shaped microstrips with and without additional stubs are achieved in all cases. In addition, Fig. 6(d) shows that the electric field confinement of the structures is improved with the increasing the length of the additional stubs. Therefore, the proposed V-shaped microstrips possess good propagation characteristics with low loss, tight field confinement. By employing the additional stubs, the cutoff frequency and confinement of SSPPs for the proposed V-shaped microstrip can be significantly adjusted without increasing the lateral dimensions, which is of great significance in miniaturization and signal interference suppression.

 

Fig. 5 (a)-(c) Simulated (Sim.) and measured (Meas.) S-parameters of the whole structure A, B, and C, respectively. (d) Influence of the stub length on cut-off frequency of the proposed structures.

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Fig. 6 (a)-(c) Simulated electric field magnitude (|E|) distributions propagating along the x-direction at 5 GHz on the xoy plane cut at the top surface of the Rogers 5880. (d) Comparison of the |E| along the cut line along z-direction at the dot positions shown in (a)-(c).

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4. Coupling characteristics of the V-shaped microstrips

To demonstrate the potentials of crosstalk and interference suppression in the highly integrated circuit and systems, the coupling characteristics of the proposed V-shaped microstrips are further studied. Figures 7(a)-7(c) depict the schematic diagrams of the couplers based on symmetrically arranged V-shaped microstrips, parallel arranged V-shaped microstrips, and parallel microstrips, respectively. The separations of V-shaped microstrips is set as w_s2 = w_p2 = 1 mm, w_ms = 2.8 mm, and other dimensions are shown in the caption of Fig. 7. The distributions of the electric field component E_z of these three structures, located at 1 mm above the microstrip on the xoy plane at 10 GHz, are displayed in Figs. 7(d)-7(f). Obviously, for the parallel and symmetrically arranged V-shaped microstrips, the coupling effects are extremely low with almost all of the incident microwaves from port 1 propagating to the port 2 while only negligible microwaves coupled to the port 4. However, for the parallel microstrips, there are obvious microwaves coupling to the port 4 when port 1 is excited. To quantitatively demonstrate these effects, the simulated and measured S-parameters of these couplers based on the symmetrically arranged V-shaped microstrip, parallel arranged V-shaped microstrips, and parallel microstrips are shown in Figs. 8(a)-8(c), respectively, and the comparison of S₄₁ among these structures are also presented in Fig. 8(d). Clearly, the measured results are in good agreement with the simulations. It is found that the S₄₁ of the symmetrically and parallel arranged V-shaped microstrips with a separation of 1 mm is far lower than - 20 dB (except for the range of 10-11.5 GHz around the cut-off frequency), which is much lower than the parallel microstrips with a separation of 2.8 mm in the whole frequency band.

 

Fig. 7 (a)-(c) Schematic diagrams of the three couplers based on symmetrically arranged V-shaped microstrips, parallel arranged V-shaped microstrips, and parallel microstrips, respectively, where l_s1 = 22.4 mm, w_s1 = 2.8 mm, w_s2 = 1 mm, l_s2 = 195 mm, w_p1 = 10.81 mm, w_p2 = 1 mm, and w_ms = w_s1 = 2.8 mm, while keeping other parameters consistent with Fig. 4(a). (d), (e), and (f) are the E_z distributions at 1 mm above the microstrips on the xoy plane at 10 GHz.

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Fig. 8 (a)-(c) Simulated and measured S parameters of the couplers based on the symmetrically arranged V-shaped microstrips, parallel arranged V-shaped microstrips, and parallel microstrips. (d) Comparison of S₄₁ parameters of the three structures mentioned above.

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To further demonstrate the low coupling characteristics of the proposed V-shaped microstrips, we investigate the coupler based on the parallel arranged V-shaped microstrips with reduced the separations. Owing to the unique structure, the separation w_p2 can be a negative value when both V-shaped microstrips overlap partially. Figure 9(a) shows the S₄₁ of the coupler with different values w_p2 of 1, 0, −2, and −3 mm, which are compared with the coupler based on parallel microstrips. The results show that the proposed parallel arranged V-shaped microstrips with w_p2 of 0 and −2 mm have lower coupling effects than that of the parallel microstrips with a separation of 2.80 mm in the frequency ranging from 0 to 10 GHz. Notably, even if the overlapping is further increased to 3 mm (w_p2 = −3 mm, corresponding V-shaped metal strip vertical distance of 1.49 mm), the S₄₁ of V-shaped microstrips is still comparable to that of the parallel microstrips with a separation of 2.80 mm, where the V-shaped microstrips show much lower S₄₁ around 8 GHz. Furthermore, Fig. 9(a) also depict two extreme points A (weak coupling point) and B (strong coupling point). For example, the coupler with w_p = −2 mm has a point A with S₄₁ lower than −45.76 dB at 7.65 GHz and a point B with the S₄₁ as high as −4.85 dB at 10.95 GHz (corresponding S21 as low as −23.742 dB), as shown in Fig. 9(b). Figures 9(c) and 9(d) show the normalized E_z distributions at 1 mm above the coupler on the xoy plane at 7.65 and 10.95 GHz, respectively. It is clear that there is almost no field is coupled to port 4 at point A, showing extremely low coupling effects of the structure. However, it should be noted that the coupling effect increases obviously when operating at point B, where a lot of fields are coupled to port 4. In other words, the structure may act as an efficient narrow-band coupler around point B. Therefore, the effective operating frequency band of proposed V-shaped microstrip should below point B to maintain low coupling effects.

 

Fig. 9 (a) Simulated S₄₁ of the coupler based on parallel V-shaped microstrips with different separations (w_p2). (b) Simulated S-parameter of the coupler based on parallel V-shaped microstrips with w_p2 = −2 mm. (c) and (d) are the E_z distributions at 1 mm above the coupler with w_p2 = −2 mm at 7.65 GHz and 10.95 GHz, respectively.

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5. Conclusion

In this work, we propose a novel variety of V-shaped microstrips for high-efficient and low coupling SSPPs propagation. The results show that the dispersion characteristics can be significantly engineered by tuning the parameters of the V-shaped microstrip unit. Introducing additional stubs is an effective method to reduce the asymptotic frequency and increase the ability of field confinement without increasing the lateral dimension. This V-shaped microstrip is compatible with conventional microstrips and active chips, and its broadband transition can be directly formed by connecting tapered microstrip without the need for complicated and bulky flaring grounds in other single conductor SSPP waveguides. Due to the strong field confinement, the coupling between the V-shaped microstrips is much smaller than the coupling between the microstrips. This work opens a new door for microwave plasmonic waveguide design, which may have great applications in microwave integrated devices and circuits.