1. Introduction

The control of velocities of light and electromagnetic waves has tremendous potential applications in both optical and microwave regimes. For example, in the optical frequency, slow light may be used in optical delays, optical modulation and switching, and all-optical buffers [1]. In the past decade, many studies have been conducted to trap the light using various approaches, such as the stimulated Raman scattering [2], electromagnetically induced transparency [3], photonic crystal [4], and tapered left-handed material structure [5], including the experiment illustration of broadband “trapped rainbow” based on tapered nano-waveguide [6].

Recently, plasmonic structures have been considered in trapping electromagnetic waves by introducing solid-state materials into the nanostructures [7–9], owing to the fact that surface plasmon polaritons (SPPs) have strong abilities to confine electromagnetic waves in the subwavelength scales [10]. It has been demonstrated that broadband SPP waves are trapped by the plasmonic graded structures in both visible and telecommunication frequencies [9, 11, 12]. Up to now, several architectures have been developed for efficiently trapping electromagnetic waves, such as the surface graded metallic grating [9], the insulator-metal-insulator waveguide taper [13], the metal-insulator-metal waveguide taper [14, 15], the insulator-negative-index-insulator structure [16, 17], and most recently the hyperbolic metamaterial structure [18]. These efforts aim to facilitate applications of the intriguing slow-wave behaviors.

In microwave and terahertz frequencies, the structured metal surfaces formed by generating periodic arrays of subwavelength grooves, holes, and blocks on metal surfaces have been used to guide spoof SPPs [19–23]. In the terahertz regime, grating waveguides were suggested to slow down the electromagnetic waves, which are composed of two parallel metallic plates with periodic corrugations on inner boundaries [24,25]. In the microwave regime, a broadband plasmonic graded waveguide was presented to realize slow waves [26]. However, all above spoof SPP systems have relatively large volumes with inherent three-dimensional geometries. To achieve more compact spoof SPP waveguide, a novel corrugated metallic structure with nearly zero thickness has been proposed [27], which has been used to realize various SPP devices in the microwave and terahertz frequencies [28–36].

In this work, we propose a simple method to trap broadband spoof SPP waves on ultrathin corrugated metallic strips in the microwave frequency. Here, we design non-uniform corrugations on ultrathin metallic strip with gradient groove depths, so that slow SPP waves are guided the SPP waves and the group velocity are slowed down gradually and then reflected at pre-designed positions when the frequency varies [17,18]. We design and fabricate the ultrathin gradient-corrugation metallic strip on a very thin dielectric film, and both numerical simulations and measurement results validate the efficient trapping of SPP waves in a broad frequency band from 9 to 14 GHz.

2. Design and analysis of slow-wave plasmonic waveguide

We start to investigate the dispersion characteristics of the ultrathin corrugated metallic strip. As shown in Fig. 1(a), we consider a non-uniform ultrathin corrugated metallic strip with gradient-depth groove, in which p represent the period, a is the groove width, and h is the strip width. The groove depth d changes gradiently to build up the slow-wave system. Using the eigen-mode solver of commercial software, CST Microwave Studio, we obtain the dispersive relations for the electromagnetic waves propagating along the corrugated metallic strips with different groove depths, as illustrated in Fig. 2(a), which are much similar to the dispersion properties of SPPs in the optical frequency. Here, we set p = 5mm, h = 8mm, a = 2mm, and the thickness of the metallic strip is t = 0.018 mm, which is 0.0006 wavelength at 10 GHz. When the groove depth d increases from 4mm to 7mm, we clearly notice that the dispersion curve has more deviation from the light line and the cutoff frequency becomes lower, which makes stronger confinement of SPP waves.

 

Fig. 1 Ultrathin non-uniform corrugated metallic strip that can trap SPP waves in broadband. (a) The sketch of the ultrathin non-uniform corrugated metallic strip with gradient groove depth. Here, the period p = 5mm, the strip width h = 8mm, the groove width a = 2mm, and the groove depth d increases from 4mm to 7mm with step Δd = 0.075mm. The length of the whole metallic strip is 200mm with 40 periods. (b) The photography of the ultrathin non-uniform corrugated metallic strip. Here, the thickness of the metallic strip is 0.018mm, which is printed on a very thin and flexible dielectric film with thickness of 0.0255mm and dielectric constant Ɛ = 3.0.

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Fig. 2 The dispersion property and group velocity of the ultrathin corrugated metallic strip. (a) Dispersion curves of the ultrathin corrugated metallic strip at different groove depths varying from 4mm to 7mm at a step of 1mm. in which the black curve is the light line. The circles show the dispersion dots of the structure when slot depth d = 4mm, and the copper is treated as lossy metal with conductivity $\sigma =5.8\times {10}^{7}s/m$. (b) Group velocities vg of the spoof SPPs on the ultrathin corrugated metallic strip at different depths. With the increase of depth, the group velocity decreases and drops quickly at different positions for different frequencies.

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We remark that the metallic strip is treated as a perfect electrical conductor (PEC) in calculations due to its highly electric conductivity in the microwave regime [37]. In calculating the dispersion of the metallic strips, we did not consider the metal loss. In the microwave frequency, the conductivity of metal is very high, and can be treated as perfect electrical conductor (PEC). To verify this issue, we have calculated the dispersion curves of the same structure when the copper is treated as lossy metal with conductivity $\sigma =5.8\times {10}^{7}s/m$ using the commercial software HFSS. The calculated results (circles) in the microwave frequency are illustrated in Fig. 2(a), from which we observe that these two curves almost overlap with little shifts. Hence the metal can be treated as PEC in the microwave regime. In the calculation, the parameters of the unit structure are taken as: p = h = 5mm, d = 4mm, a = 2mm, and the thickness of the strip is 0.018mm.

Based on the definition of group velocity ν_g = dω/dk, in which k is the propagating factor and ω is the radian frequency, we can calculate the group velocity ν_g of the SPP waves from the dispersion relations. Figure 2(b) demonstrates the group velocities of spoof SPPs on the ultrathin corrugated metallic strip at different depths. As the groove depth increases, the group velocity decreases. This is because the ultrathin metallic strip with gradient grooves can couple the incident waves gradually into the spoof SPPs with very low group velocity. When the SPP wave propagates on the corrugated strip, its velocity will decrease gradually along the propagation direction. By properly designing the groove depths, Fig. 2(b) shows that the group velocity can even reach zero at specific locations for different frequencies. From Fig. 2(b), we clearly see the apparent reductions of group velocities from the light speed in free space at 9GHz, 9.5 GHz, 10GHz, 10.5GHz, 11GHz, 11.5GHz and 12 GHz. At such frequencies, the SPP waves are reflected at the locations with the groove depths d = 6.7mm, 6.3mm, 5.8mm, 5.5mm, 5.2mm, 4.9mm, and 4.6 mm, respectively.

To demonstrate the above slow-wave properties directly, we perform full-wave simulations on the surface electric fields along a gradiently-corrugated metallic strip, which has the length of 200mm with 40 periods. The metallic strip is excited by an electric monopole with unity current that is parallel to the left edge. The dominant mode of the electric monopole has the zeroth-order transverse magnetic (TM) pattern, which is matched to the principal mode of the corrugated strip. As a consequence, the SPP waves are efficiently generated on the corrugated metallic strip by the monopole. Using CST Microwave Studio, the simulated electric-field distributions on an observation plane that is 0.5 mm above the metallic strip are presented in Fig. 3, in which the left panels (a-e) indicate the real parts of transient fields, while the right panels (f-j) represent the magnitude distributions of electric fields. From Fig. 3, we clearly notice that the SPP waves are trapped on the ultrathin corrugated metallic strip at different locations in different frequencies. To read the locations accurately, we plot the electric-field distributions along the central line on the strip, as illustrated in Fig. 5(a). From the simulation results, we clearly see that the magnitude of spoof SPPs drops to zero after propagating along the x direction at distances of 175mm, 147mm, 123mm, 93mm, and 77mm for the frequencies 9GHz, 9.5GHz, 10GHz, 10.5GHz, and 11GHz, respectively.

 

Fig. 3 Numerical simulation results of the spoof SPPs propagating on the ultrathin corrugated metallic strip. The left panels (a-e) and right panels (f-j) represent the transient-field and magnitude distributions of electric fields at different frequencies 9.0, 9.5, 10.0, 10.5, and 11.0 GHz, respectively.

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In comparison with the earlier calculations from group velocities, we should calculate the propagation distance of SPPs corresponding to groove depth. From the geometry configuration shown in Fig. 1, the propagation distance x and groove depth d have the following relation x = 200 × (d-4)/3mm. Hence the groove depths d = 6.7mm, 6.3mm, 5.8mm, 5.5mm, and 5.2mm correspond to propagation distances x = 180mm, 146mm, 120mm, 99mm, and 78mm for the frequencies 9GHz, 9.5GHz, 10GHz, 10.5GHz, and 11GHz, which have excellent agreements to the full-wave simulations.

3. Experiment verification

As experimental verifications, we perform a series of measurements at different frequencies. The experiment sample was fabricated using the standard printed circuit board process on a flexible copper clad laminate, which is composed of a polyimide layer (12.5 microns), an epoxy adhesive layer (13 microns), and an electrolytic copper clad sheet (18 microns). Hence the total thickness of the ultrathin sample is 43.5 microns, including the dielectric film. The fabricated ultrathin corrugated metallic strip with gradient grooves is illustrated in Fig. 1(b), which is flexible and foldable, and thus supporting conformal surface plasmons²⁷. In experiments, a home-made near-field mapping system is used to measure the near electric fields, which includes a vector network analyzer (Agilent N5230C) and two monopole antennas to excite and receive SPP fields. The measured electric-field distributions on the same observation plane as that in simulations are demonstrated in Fig. 4. Both measured transient fields and amplitudes of electric fields agree excellently to numerical simulations shown in Fig. 3.

 

Fig. 4 Experiment results of the spoof SPPs propagating on the ultrathin corrugated metallic strip. The left panels (a-e) and right panels (f-j) represent the transient-field and magnitude distributions of electric fields at different frequencies 9.0, 9.5, 10.0, 10.5, and 11.0 GHz, respectively.

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Similarly, to seek the reflected locations of SPP waves on the corrugated metal strip at different frequencies, we extract the measured field distributions on the central line from Fig. 4, and the results are given in Fig. 5(b). Comparing with Fig. 5(a), good matching is observed between the measured and simulated results. In experiments, the spoof SPPs reflect at x = 173mm, 141mm, 118mm, 95mm, and 75mm in frequencies of 9GHz, 9.5GHz, 10GHz, 10.5GHz, and 11GHz, respectively. Therefore, the three propagation distances of SPPs on corrugated metal strip from theoretical calculations, full-wave simulations, and experiments are in excellent agreements, showing the good performance in trapping SPP waves.

 

Fig. 5 The numerical simulation and experiment results of the |Ez| distributions along the x axis for different frequencies. (a) Numerical simulations. (b) Experiment results.

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4. Conclusion

In summary, we have proposed a very compact and simple broadband slow-wave plasmonic waveguide, which is printed an ultrathin and flexible dielectric film. Both numerical simulations and measurement results show that broadband SPP waves are efficiently trapped on ultrathin corrugated metallic strips with good behaviors. The design method can be directly extended to the terahertz frequency to realize terahertz slow waves. Due to the simple, ultrathin, and flexible features, the proposed method and design provide a good candidate of special slow-wave devices to interact with moving electrons and acoustic waves that have much smaller velocities in both microwave and terahertz regimes.