1. Introduction

A dielectric/metal interface supports surface plasmon polaritons (SPPs) with transverse magnetic (TM) polarization in visible or near infrared regime, due to the coupling between external photons and plasmons in metals [1–3]. SPPs were shown to exhibit many fascinating applications, such as plasmon waveguiding [4–6], extraordinary optical transmission (EOT) [7–9], and surface-enhanced sensing [10–12]. In low frequency domains (e.g., microwave and terahertz regimes) where metals often behave like perfectly electric conductors (PEC), conventional SPPs do not exist but geometrically induced surface waves (SWs) (also called spoof-SPP or designer-SPP) [13–18] can be defined in structured metallic surfaces. Such a concept significantly expanded our abilities to tailor the surface electromagnetic (EM) modes of metallic systems at low frequencies, leading to many interesting applications [16–18]. Currently, popular design schemes for spoof-SPP systems include cutting periodic holes or slits onto a metallic surface. However, some inherent limitations exist in such schemes. For example, one has to fill the holes with high-index materials in some systems [13], and the thickness of the entire system must be at least ~$\lambda \text{/4}$ in some other systems [16,17], both of which are inconvenient for practical applications. Moreover, the generated spoof-SPPs in such systems typically exhibit TM polarization only.

Here, we propose an alternative approach to tailor the SW properties of a metallic surface. Instead of cutting holes and slits onto a metal surface, we put an ultra-thin anisotropic metamaterial (MTM) layer on a continuous metallic surface, and show that such a thin capping layer can significantly modify the SW properties of the entire system, leading to many interesting new physics. Our paper is organized as follows. We first analytically studied the rich SW behaviors of the proposed model system in Sec. 2. The interesting SW characteristics discovered include: 1) the system now simultaneously supports SWs with both TM and transverse-electric (TE) polarizations; 2) the excited SWs exhibit hybridized polarization characteristics in some cases; and 3) a SW gap can open within some particular propagation direction range. In Sec. 3, we designed one possible structure to realize the model system and then fabricated a sample based on the design, and further employed both microwave experiments and full-wave simulations to verify the fascinating SW behaviors predicted in Sec. 2. After presenting some potential applications of our proposed system in Sec. 4, we concluded our paper in the last section.

2. Theoretical analysis on the model system

We start from studying a model system as shown in Fig. 1(a), which consists of an anisotropic homogeneous MTM layer with a dispersive relative permeability tensor ${\overleftrightarrow{\mu }}_{\text{2}}$ (with diagonal elements ${\mu }_{xx}$, ${\mu }_{yy}$, ${\mu }_{zz}$) and a relative permittivity ${\epsilon }_2$, put on top of a plasmonic metal (with ${\epsilon }_3<<-1$,${\mu }_3=1$). For definiteness, we assumed that the material and geometrical parameters were given by $d=1.3\text{mm}$,${\mu }_{xx}=1+\text{60}/(\text{13}{\text{.35}}^2-f^2)$, ${\mu }_{yy}=1+\text{4}2/(\text{10}{\text{.2}}^2-f^2)$, ${\mu }_{zz}=1$, ${\epsilon }_2=1$ with $f$ being the linear frequency measured by GHz, and studied the SW characteristics of the model system. These parameters are chosen to make the model correspond to a realistic MTM system, as will be discussed in Sec. 3. For ${\overrightarrow{k}}_{\text{sw}}$ (wave vector of the SW) parallel to one of the lateral principle axes of present anisotropic system, the problem of solving the SW dispersion can be easily handled by a standard $2\times 2$ transfer-matrix-method (TMM) for an isotropic multiplayer system. Things become much more complicated when ${\overrightarrow{k}}_{\text{sw}}$ is not along any lateral principle axis, where we have to reply on the $4\times 4$ TMM developed in [19,20]. Specifically, we can employ the $4\times 4$ TMM to study the reflection properties of the system under illuminations of input waves with a fixed frequency and different parallel wavevectors,

 

Fig. 1 (a) Geometry of the model system studied in this paper. SW band structures of the model system for (b)$\phi =0^{\circ }$, (c)$\phi ={30}^{\circ }$, (d)$\phi ={60}^{\circ }$, and (e)$\phi ={90}^{\circ }$, calculated by the $4\times 4$ TMM based on the double-layer model shown in (a).Two dashed lines denote two resonance frequencies 10.2 GHz and 13.35 GHz. (f) SW bandgap width as a function of the azimuth angle $\phi$, calculated by the $4\times 4$ TMM.

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The computed SW dispersion curves are shown in Figs. 1(b)-1(e), for four different propagation directions characterized by the azimuth angle $\phi$ which is the angle between ${\overrightarrow{k}}_{\text{sw}}$ and $\widehat{x}$ [Fig. 1(a)]. We note that these SW dispersion curves exhibit very intriguing characteristics. For the case of $\phi =0^{\circ }$, a TM-polarized SW branch appears at low frequency regime and saturates at ~10.2 GHz, exactly mimicking the behaviors of natural SPP of a plasmonic metal in visible domain. The effective plasmonic resonance frequency for such a SW branch is just the magnetic resonance frequency at which ${\mu }_{yy}$ of the top MTM layer diverges. The physics is that a TM-SW mode traveling along $\widehat{x}$ direction (with $\overrightarrow{H}\left|\right|\widehat{y},\overrightarrow{E}\in xz$) can “see” this magnetic resonance, and as ${\mu }_{yy}\to +\infty$, the refraction index inside the MTM layer approaches infinity making $k_{\text{sw}}\to +\infty$. On the other hand, a series of TE-SW curves appear within the frequency range of $13.35\text{GHz}<f<15.2\text{GHz}$, where ${\mu }_{xx}$ varies from $-\infty$ to $\text{0}$. The physics is that those TE-SW modes with ${\overrightarrow{k}}_{\text{sw}}\left|\right|\widehat{x}$ (with $\overrightarrow{E}\left|\right|\widehat{y},\overrightarrow{H}\in xz$) can “see” ${\mu }_{xx}$ of the MTM, and when $-\infty <{\mu }_{xx}<\text{0}$, such modes (with TE polarizations) can be trapped on the air/MTM interface. The inherent physics underlying these two SW branches are basically the same as those already discussed in [21] for an (in-plane) isotropic model. In fact, when ${\overrightarrow{k}}_{\text{sw}}$ is along one of the lateral principle axes, the cross reflection terms (i.e., $r_{sp},r_{ps}$) vanish and thus the problem [Eq. (1)] reduces to previously studied isotropic model in which the SW modes exhibit well-defined polarization characteristics. However, here the in-plane anisotropy introduced a new effect to separate these two branches in frequency domain, leaving a complete SW band gap within 10.2 GHz – 13.35 GHz. We now turn to study the case of $\phi ={90}^{\circ }$ corresponding to ${\overrightarrow{k}}_{\text{sw}}\left|\right|\widehat{y}$. There are still two SW branches in the figure, but now the TM-SW branch appears at high frequency region while the TE-SW branch appears at low frequency region, due to the fact that a TM (TE) SW mode can “see” ${\mu }_{xx}({\mu }_{yy})$ in this case. As the result, the SW band gap is closed.

More interesting features appear when ${\overrightarrow{k}}_{\text{sw}}$ is not parallel to any of the principle axes. In such general cases, the cross reflection coefficients (i.e., $r_{sp},r_{ps}$) do not vanish, and thus the generated SW modes do not exhibit well-defined polarization characteristics (such as pure TE or TM) but rather exhibit hybridized features. In addition, the key features of these SW curves continuously evolve from the $\phi =0^{\circ }$ case to the $\phi ={90}^{\circ }$ case. We employed the $4\times 4$ TMM to calculate the frequency bandwidth of the SW gap as a function of the azimuth angle. As shown in Fig. 1(f), the bandwidth of the SW gap gradually shrinks as $\phi$ increases and is completely closed at $\phi \text{~}\text{5}{5}^{\circ }$. Therefore, at a frequency within [10.2 GHz - 13.35 GHz], our system only allow SWs to propagate along directions within a certain $\phi$ range, showing that the in-plane anisotropy plays a crucial role to control the SW propagations. We note that these fascinating behaviors can be easily tuned by modifying the material properties (i.e., ${\mu }_{xx},{\mu }_{yy}$) of the capping MTM layers, and they can be realized in any frequency domain as long as the requested capping MTM layer can be practically designed.

3. Simulations and experiments on realistic structures

To realize the model system studied in Sec. 2 and demonstrate its rich SW properties, we designed a MTM structure consisting of a periodic array of (anisotropic) metallic crosses on a metal-backed dielectric slab. As shown in Fig. 2(a), the structural parameters are carefully designed to take the values of $P_x=10\text{mm}$,$P_y=8\text{mm}$,$L_x=8\text{mm}$,$L_y=6\text{mm}$,$W=1\text{mm}$, and $t=1.2\text{mm}$. The thicknesses of the top metallic layer and the dielectric slab are respectively 0.1 mm and 1.2 mm, and the relative permittivity of the dielectric slab is ${\epsilon }_r=3.6$. We note that the thickness of the entire system (1.3 mm) is much thinner than the working wavelength ($~\text{3}0\text{mm}$) so that such systems are typically called “meta-surfaces”, which have been found to exhibit numerous applications in practice [21–23]. A single metallic cross possesses electric resonances for distinct polarizations. When a metal plate is added, near-field couplings between the metallic crosses and the ground plane generate electric currents flowing oppositely on two metal layers, leading to magnetic resonances at specific frequencies for distinct polarizations. In fact, it has been demonstrated in [21] that such a meta-surface can be perfectly modeled by the double-layer effective-medium system depicted in Fig. 1(a). Via selecting appropriate geometrical parameters (see above), we found that the presently designed system can well represent the model system studied in Fig. 1 with the desired material/geometrical parameters. To demonstrate this point, we computed the reflection-phase spectra for the designed anisotropic meta-surface by finite-difference-time-domain (FDTD) simulations [24], and compared the simulated results with those obtained on the model system in Figs. 3(a) and 3(b) for two incident polarizations. In our simulations, we modeled the metal as a perfect electric conductor, which is reasonable in microwave regime. Excellent agreement between two calculated spectra demonstrates that the designed structure can indeed well represent the model system depicted in Fig. 1(a). We note that the reflection phase spectrum is the only relevant physical property to be considered here, since the system does not allow any transmission but always perfectly reflect EM waves.

 

Fig. 2 (a) A unit cell of the designed structure. (b) A top-view picture of part of fabricated sample. (c) Schematics of the experimental setup (not to scale). Yellow color in (a) and (c) denotes the areas occupied by metal.

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Fig. 3 Reflection-phase spectra of the meta-surface for normally incident waves with polarizations $\overrightarrow{E}\parallel \widehat{x}$(a) and $\overrightarrow{E}\parallel \widehat{y}$ (b), obtained by measurements (stars), FDTD simulations (lines) on realistic system, and TMM calculations on model system (circles).

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We then fabricated a sample based on the theoretical design [see picture shown in Fig. 2(b)], and experimentally measured the reflection phase spectra of the sample for two incident polarizations. The metal adopted in fabricating the sample is copper. In our experiments, we shined the meta-surface by a horn placed 50 cm away from the sample, and used another identical horn to receive the reflected signal. Two horns were connected to a vector-field analyzer (Agilent E8362C PNA) so that both the amplitudes and phases of the reflected signals can be obtained. Solid stars are the measured reflection-phase spectra for the sample, which are in good agreement with both simulations and model calculations. The slight differences between experiments and simulations are possibly due to the fact that the measurements were performed at a small incident angle ${\text{~5}}^{°}$, while FDTD simulations assumed strict normal incidence condition.

Encouraged by the good agreements between theory and experiment as shown in Fig. 3, we next employed both simulations and experiments to study the SW dispersion curves of the designed/fabricated structure. Consider the $\phi =0^{\circ }$case first. Solid circles (stars) in Fig. 4(a) are the computed dispersion curves of SWs for TM (TE) polarization on the designed system, obtained by numerical simulations based on a finite-element-method (FEM) [25]. Then we experimentally characterized the SW dispersions on the realistic structure, employing the setup schematically depicted in Fig. 2(c). Following the method described in [26,27], we put a metallic razor blade at a distance 2 mm away from the sample, and then shined it by a horn to excite near-field signals through the slit. The excited SW then propagates from left to right if the surface supports a SW mode. A monopole antenna was used as a probe to measure the electric field distribution on the surface. Both the horn (source) and the monopole antenna (detector) were connected to a vector-field analyzer (Agilent E8362C PNA) so that the amplitude and phase information of the electric field can be easily obtained. From the distribution of $\mathrm{Re}(E_z)$ on the surface, one can easily identify the wavelength of the generated SW, and in turn, the wave vector $k_{\text{sw}}$ of the SW. Inset to Fig. 4(a) shows the measured $\mathrm{Re}(E_z)$ pattern for the excitation frequency 9.0 GHz, from which we get $k_{\text{sw}}=0.31\times (2\pi /P_x)$. The experimental data thus obtained are shown in the same figure as open circles. However, such an experimental technique has its own limitations. First, it can only be used to excite the SW of TM polarization, so that we can only measure the TM-SW dispersion curve. Second, the signal is too weak when $k_{\text{sw}}$ exceeds a certain critical value, making it difficult to unambiguously detect the “flat” part of the SW dispersion (near the Brilluion zone boundary). The latter effect is possibly due to the finite size of the sample, which caused reflections at the boundaries leading to standing SW waves. Nevertheless, we note that available experimental data fits perfectly with the TM-SW dispersion curve obtained by full-wave simulations.

 

Fig. 4 (a) SW dispersions on the designed/fabricated system obtained by measurements (open circles) and FEM simulations (blue solid circles and red solid stars) for $\phi \text{=}{0}^{\circ }$. Inset depicts the normalized $\mathrm{Re}(E_z)$ distribution measured on the sample at the excitation frequency 9.0 GHz. (b) FDTD simulated SW transmission spectra with different polarizations for the designed meta-surface in the case of $\phi \text{=}{0}^{\circ }$.

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To further illustrate the interesting polarization-dependent SW properties of the meta-surface, we employed FDTD simulations [24] to study the transmission spectra of SWs on the meta-surface. Following the method described in [28], in our FDTD simulations, we put a monopole (dipole) antenna perpendicular (parallel) to the meta-surface (sized $100\text{mm}\times 96\text{mm}$) to launch the TM (TE) polarized SWs, and then used another identical monopole (dipole) antenna to collect the received signals at 100 mm away from the source. Perfect absorption boundary conditions were placed around the sample to kill the backward scatterings from the sample edges. Figure 4(b) shows the calculated SW transmission spectra for both TE and TM polarizations. Obviously, a TE-SW transmission band appears at ~13.0 GHz while a TM-SW transmission band appears around ~10.0 GHz. These features are in good agreement with the SW dispersion characteristics as shown in Fig. 4(a). We note that only a small portion of energy radiated from the source antenna can be coupled into the SW and only another small portion of SW energy can be collected by the receiver antenna. Therefore, the computed transmittance shown in Figs. 4-5 is typically low.

 

Fig. 5 (a) SW dispersions on the designed/fabricated system obtained by measurements (open circles) and FEM simulations (blue solid circles and red solid stars) for $\phi \text{=9}{0}^{\circ }$. Inset depicts the normalized $\mathrm{Re}(E_z)$ distribution measured on the sample at the excitation frequency 10.0 GHz. (b) FDTD simulated SW transmission spectra with different polarizations for the designed meta-surface in the case of $\phi \text{=9}{0}^{\circ }$.

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The same characterization techniques have been applied to study the SW properties in the case of $\phi ={90}^{\circ }$. The simulated and experimental results are shown in Figs. 5(a) and 5(b). In sharp contrast to the $\phi =0^{\circ }$ case, here the TM-SW band persists to ~13.0 GHz, while a sharp TE-SW band appears around ~10.0 GHz. Microwave experiments were performed to study the TM-SW dispersion in such a case, and again the obtained data are in good agreement with full-wave simulations.

The simulated/measured SW properties on the designed/fabricated sample are not only consistent with each other, but also in reasonable agreement with the TMM calculations on the ideal model system (Fig. 1(a)) as presented in Sec. 2. However, we note that FEM-calculated TE-SW curves based on the realistic system exhibited red shifts as compared to the model calculations on the ideal system, especially for the high-frequency branch as shown in Fig. 4(a). Simulations revealed that the magnetic resonant frequency of the realistic structure may down-shift at large incidence angles under illuminations of TE-polarized plane waves, due to the coupling between adjacent resonant units. We believe that this can explain the red-shifts of the TE-SW branches compared to the model calculations (on the ideal system) which completely overlooked such an effect.

Figures 4-5, performed on two particular $\phi$ cases, already implied that the SW transmissions exhibit anisotropic properties. Taking the working frequency as 10.5 GHz, we further employed microwave experiments to directly characterize this fascinating behavior. Using the blade coupler to excite a SW signal on the meta-surface [Fig. 2(c)], we put a monopole antenna at a lateral distance 60 mm away from the source to measure the strength of the SW signal. By rotating the sample only yet keeping all others (source, blade, receiver, etc.) fixed, we obtained the SW transmission as a function of propagating direction $\phi$. Appropriate field average was taken over a certain area around the receiver to suppress the strong local field variations on the realistic sample. The measured SW transmission pattern is depicted in Fig. 6 as open circles. Obviously, the launched SW can only propagate on the meta-surface along two certain angle ranges around the $\pm \widehat{y}$ axes, which is consistent with theoretical predictions as shown in Fig. 1. We further performed FEM simulations on realistic system to directly study the anisotropic transmission properties of SWs. In our simulations, we vertically put a 6 mm-long monopole antenna near the top surface of the realistic system to launch the SW signal at the working frequency 10.5 GHz, and then computed the strength of ${\left|E_z\right|}^2$ on a circle with a radius 20 mm centered at the source position. The simulated ${\left|E_z\right|}^2$ pattern is shown in Fig. 6 as solid line. We also denoted the critical angles predicted by TMM calculations on model system as two magenta dashed lines in Fig. 6. Comparing these three results, we found that while FEM simulations well reproduced the key features of the experimental results, the simulated pattern is obviously broader than the measured one. The discrepancies are caused by the fact that the FEM simulations adopted a smaller sample ($\text{5}0\text{mm}\times \text{5}6\text{mm}$) and a different SW excitation strategy (monopole antenna excitation), due to the limitations in computational resources. Nevertheless, the reasonable agreement among model, simulations, and experiments firmly demonstrates that the in-plane anisotropy adds an additional and important freedom to manipulate the SW propagations in such a meta-surface.

 

Fig. 6 Normalized patterns of SW transmission on the meta-surface, obtained by experiments (blue open circles) using blade-coupling technique and FEM simulations (red solid lines) using monopole-antenna excitation on the realistic system. Magenta dashed lines denote the critical angles predicted by the TMM on model system. Here the working frequency is set as 10.5 GHz.

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4. Potential applications

The fascinating SW properties discovered above offer the proposed system many interesting applications, particularly in building planar EM devices/circuits with subwavelength lateral dimensions. Previous studies only utilized the strongly modulated TM-polarized SWs in these types of systems [16,17]. In contrast, present meta-surfaces support both TE and TM SWs and the in-plane anisotropy further enriches the SW properties by adding the angular degree of freedom to control the SW propagations. As an illustration, here we propose two simple devices utilizing the highly directional transmission properties of the TE-polarized SW (i.e., lower SW band in Fig. 5(a)).

The first example is a TE-SW planar waveguide working at 10.3 GHz, which consists of an array of metallic crosses (with the same geometrical parameters as in Fig. 2(a)) printed on a dielectric spacer backed by a PEC ground plane. It has been shown in Fig. 5(b) that such system supports TE-polarized SWs propagating along y direction. In our FDTD simulations, we excited the planar waveguide from its upper end, and then simulated the stabilized field distributions on the whole system. Figure 7(a) depicts the stabilized $\mathrm{Re}(E_x)$ distribution on a plane near the top surface of planar waveguide, which clearly shows that such strongly confined SW signals can be transported efficiently from the upper end to the lower end of the waveguide.

 

Fig. 7 FDTD simulated $\mathrm{Re}(E_x)$ patterns (in logarithmic scale) on two EM planar devices: (a) a line planar waveguide and (b) a SW coupler. Here the working frequency is 10.3 GHz and the arrows represent the propagation directions.

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The second example is a TE-SW coupler consisting of two identical waveguides placed parallel to each other separated by a distance S, as shown in Fig. 7(b). Using the same technique to excite the TE-SW from the upper end of the right waveguide, we found that the generated SW can propagate along the right waveguide and then couple to the left one and finally reach the lower end of the left waveguide, as shown by the field pattern recorded in Fig. 7(b). FDTD calculations reveal that the coupling efficiency of such a TE-SW coupler is about 40%, which sensitively depends on the separation S between two waveguides and other geometrical parameters. Such a coupling coefficient can be significantly enhanced by geometrical optimization, which is out of the scope of present work. Compared to TM-polarized SWs which have been widely realized in experiments [17,29], the TE-polarized SWs in our system exhibit higher photonic density of states [Figs. 4-5]. However, they are also difficult to measure experimentally. Compared to the conventional microstrip waveguide working on the quasi-TEM modes guided inside the inner dielectric region [30], our waveguide can confine the SWs much more tightly so that the wave leaking problem can be solved. Lots of other applications could be expected, thanks to the rich SW properties of such kind of anisotropic meta-surfaces. We believe that realizing these effects are interesting and challenging future projects.

5. Conclusions

To summarize, we demonstrated that adding an ultra-thin anisotropic MTM layer to a metallic surface can dramatically change the SW properties of the original system, leading to several new features of the SW characteristics which can be utilized to design certain planar EM circuit/devices. We designed and fabricated a realistic system, and successfully verified several key theoretical predictions on such a system, through both full-wave simulations and microwave experiments. We emphasize once again that the key motivation of our paper is to find a new way to control the SW properties of a plasmonic metal, rather than to create a new kind of SW which does not exist in literature. Nevertheless, some of the SWs discovered on such a capped system (e.g., the SWs with TE or mixed polarizations) do show fascinating characteristics which have not been fully understood in standard textbook based on a conventional metallic surface [31]. These new characteristics offer our system many potential applications, some of which have been discussed here but many more can be expected in the future.