1. Introduction

Since the surface plasmon polaritons (SPPs) are not constrained by the optical diffraction limit, it is hoped that they could enable the construction of compact optical components, having applications in on-chip integration of optical circuits, surface or interface technology, and data storage, etc [1,2]. While most SPP studies were focused on the visible and infrared frequency ranges, an increasing attention has been paid to the spoof SPPs [3] in the terahertz (THz) frequency in recent years on both planar [4–9] and cylindrically shaped [10–16] structures. An experiment in the microwave frequency has been conducted to verify such spoof SPPs [17]. To achieve full potential applications, it is vital that SPPs could be effectively excited and guided within subwavelength dimensions.

It is important to find the effective methods for ensuring that the generated SPPs only travel in the desired directions. Recently, efficient unidirectional nanoslit couplers for surface plasmon have been proposed to generate a unique propagation direction for SPPs [18]. A bidirectional subwavelength frequency splitter operating in the THz domain was then presented in theory based on a single tapered slit [19,20] to confine and guide the electromagnetic (EM) waves at different frequencies in the predetermined opposite directions. The splitting of surface EM waves has been investigated experimentally in the microwave frequency [21] and at visible frequencies [22]. The commonly employed SPP-coupling structures include prisms, apertures, and metallic gratings [6, 7]. For the prisms and gratings structures, the incident light is a significant source of noise. The backside illumination of subwavelength apertures [18–25] in optically-thick metal films can eliminate this problem. However, the use of free-space optics to guide and manipulate THz beams requires advanced experience with optical techniques and the sample of interest must provide direct line-of-sight access [26].

In this paper, we propose a novel method to excite the spoof SPPs by using a cylindrical wire, generating a bidirectional frequency splitter, which can be used for several applications such as the biosensing and optical components. Such a method could also be applied to design the “trapped rainbow” structure [27, 28] in the microwave or THz frequencies. The coupling mechanism and design approaches are presented. The finite integral time domain (FIT) method is used to model the bidirectional splitter excited by the cylindrical wire. Experiments are conducted in the microwave frequency, and the simulated and measured results have very good agreements.

2. Design principles

The proposed frequency-splitter structure is illustrated in Fig. 1 (a) , which consists of two grooved plates with slot depth h, slot width w, periodicity p, and a coaxial feedline whose inner conductor is extended to excite the spoof SPPs on the gratings. The length and depth of the whole aluminum plate are L and H, respectively. The dispersion curves for a one-dimensional (1D) groove array (the length Lof the structure is infinitely long) can be generated by using the following formula [29]

 

Fig. 1 (a) The sketch of the bidirectional surface wave splitter excited by a cylindrical wire; (b) Dispersion curves calculated for $w=2mm$and $p=5mm$with groove depths $h=5mm$and$h=11mm$. The solid curve and the solid circles correspond to the analytical solutions from Eq. (1) and the calculated results with the eigenmode solver of CST MWS, when the lengthLof the structure is infinite. The dashed curve shows the calculated results when the lengthLof the structure is 10$mm$. (c) and (d) provide the eigenmode frequencies with different structure lengths, when $\beta =\pi /p$. The groove depths in (c) and (d) are$h=5mm$and$h=11mm$, respectively.

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We define the depths of the right and left structures as $h_r$ and $h_l$, the cutoff frequencies for $h_r$ and $h_l$ as $f_r$ and $f_l$, and the wave frequency as f, in which $h_r>h_l$ and $f_r<f_l$. The splitting mechanism is described as below. If f is close to $f_r$, the wave could be strongly confined on the right surface and should be weakly confined on the left surface. On the other hand, if $f>f_r$ and is close to $f_l$, the wave should be confined well by the left structure, but cannot be coupled into the right. The coupling mechanism from the cylindrical wire to two surface gratings is to match the field built up by the launching device to the field of the surface wave as much as possible. The launching device can be considered as a field transformer which converts the field of a waveguide into that of surface wave. The efficiency will be greater if the field built up by the launching device has a good agreement to that of surface wave. Based on the principle, a launching device was proposed to excite the surface wave on the cylindrical conductors with high efficiency [30]. Recently, a radially polarized surface wave was excited with high coupling efficiency through the use of a radially symmetric photoconductive antenna which generates a largely radially polarized terahertz beam [26]. For finite-sized slots the TEM mode is the principal mode if the slot width is much less than the free-space wavelength [19, 31] and the TEM mode is the dominant mode in the coaxial line. Since the field built up by the coaxial cable matches the field of the spoof SPPs on the gratings, we believe that it is feasible to launch spoof SPPs on the gratings based on the coaxial line depicted in Fig. 1 (a), where the inner conductor of the coaxial cable extends above the surface to excite the desired wave on the surface gratings. A similar concept has appeared in the design of surface-wave antenna in the microwave frequency [31]. The bidirectional surface wave splitter excited by a cylindrical wire has been verified by using the FIT method and experiments later.

The lengthLof the grating structure proposed in the paper is finite. To design the surface gratings, the dispersive relation for the EM waves propagating on the surface of the grating with finite lengthL is needed. However, Eq. (1) is derived based on an assumption that the length Lof the grating structure is infinitely large. Since there is no analytical formula for finite-length metallic grating structures, we need to use numerical methods to analyze the dispersive relation of the finite-length grating structure. The eigenmode solver of the CST Microwave Studio (CST MWS) is used to do it. Only a unit cell of the grating structure is needed in the simulation. The periodicityp and slot widthw of the grating are 5$mm$and 2$mm$. The length Lof the splitter is set to 10$mm$and the thickness Hof the metal is 40$mm$. The gratings with groove depths 5$mm$and 11$mm$ are calculated, respectively. Firstly, we have calculated the dispersion curves of 1D groove array and compared those with the analytical solutions generated from Eq. (1), as shown in Fig. 1 (b). The solid circles and the solid curves correspond to the calculated results and the analytical results, respectively. The agreements are perfect. Then the dispersion curves of the finite-length grating structure are calculated, shown as dashed curves in Fig. 1(b). It can be seen that the dispersion curve of the grating with $L=10mm$ is slightly lower than that of the 1D grating. At last, the eigenmode frequencies of the grating with different grating lengths are calculated, when$\beta =\pi /p$. The length of the grating varies from 2$mm$ to 40$mm$. The results for$h=5mm$and $h=11mm$are shown in Fig. 1 (c) and (d), respectively. It can be seen that the eigenmode frequencies increase as the length Lof the grating becomes large and approaches the eigenmode frequency for 1D grating. The discrepancy is larger for the smaller groove depthh. Since the discrepancies are slight overall, it is acceptable and convenient to use the dispersion curves obtained from Eq. (1) in the design of the gratings with the length$L\ge 10mm$.

3. Modeling and experimental verification

The time-domain solver of CST MWS which is based on he FIT method has been used to model the bidirectional surface wave splitter excited by the cylindrical wire in the microwave frequencies, and the simulation parameters are shown in Fig. 2 (a) . The depths h of the gratings are 5$mm$ on the left side, and 11$mm$ on the right side. The periodicity p and widthw of the gratings on both sides are 5$mm$ and 2$mm$. The length L and the thickness Hof the splitter are 10$mm$ and 40$mm$, respectively. The coaxial cable is SFT-50-3. The extended height $h_f$ of the inner conductor is 8$mm$. As a comparison, a control structure consisting of two smooth aluminum plates and a coaxial feedline shown in Fig. 2 (b) is also analyzed. The electric fields on the xz-plane which is 2$mm$ away from the metal structure are recorded and illustrated in Figs. 3 (a), (c) and (e) .

 

Fig. 2 (a) The sketch of the bidirectional splitter consisting of two grooved aluminum plates. The coaxial feedline is SFT-50-3 cable. The inner conductors of the cable are extended 8$mm$. (b) The control structure with two smooth aluminum plates on different sides of the cable. (c) The photograph of the field mapping experiment setup. The coaxial detecting probe is mounted onto the stationary upper plate which is lifted now. The lower plate is mounted to two computer-controlled linear translation stages, enabling a scanning area of 20$cm$by 20$cm$with a resolution of 1$mm$. The detecting probe is SFT-50-1 cable. The inner conductor of the probe is extended 2$mm$and bended 90 degrees in order to sample the z-component of the electric fields.

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Fig. 3 Results obtained from the FIT simulations and experiments. (a)-(f) 2D distributions of the z-component electric fields on the xz-plane which is 2$mm$away from the structure, and 1D distributions of the z-component electric fields along the line (the white dashed line in (a)) which is 2$mm$above the surface of the structure, respectively. The structure in (a) and (b) consists of two smooth aluminum plates and the observed frequency is 10GHz. The structure in (c)-(f) consists of two grooved aluminum plates. The width (w) and periodicity (p) of the gratings are 2$mm$and 5$mm$, respectively. The depths (h) of the left-hand side and right-hand side gratings are 5$mm$ and 11$mm$. The observed frequency in (c) and (d) is 5GHz, and the observed frequency in (e) and (f) is 10GHz. The results in (a), (c) and (e) are obtained from FIT simulations, while the results in (b), (d) and (f) are obtained from experiments. The thicknessH and lengthL of the metal are 40$mm$and 10$mm$, respectively.

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The samples are fabricated for experiments. The experimental platform used here is similar to the 2D mapping apparatus in Ref [32], as shown in Fig. 2 (c). The sample is placed on the bottom plate of the experimental platform to achieve the automated scanning. The bottom plate is attached to a pair of computer controlled linear stages, whose scanning range is 200$mm$ by 200$mm$ with a resolution of 1$mm$. The Vector Network Analyzer (VNA, Agilent N5230C) provides the microwave source signal and the detection of return signal. A custom Lab-View program coordinates the motion of stages. The transmitted-signal ($S_{21}$) data are stored as complex values in matrices that can be plotted as intensity maps. The coaxial cable (SFT-50-3) connected to VNA is used as the feeding source. Another coaxial cable (SFT-50-1) is used as a detecting probe to sample the electric fields on the xz-plane 2$mm$ away from the metal structure. The probe is mounted onto the stationary upper plate to eliminate the influence to the fields of the structure. The inner conductor of the probe is extended 2$mm$ and bended 90 degrees in order to sample the z-component of the electric fields. The measured results are demonstrated in Figs. 3 (b), (d) and (f).

The simulation and measurement results of the control structure at 10 GHz are shown in Figs. 3 (a) and (b), respectively, in which both 2D distributions of the z-component electric field on the xz-plane and 1D field distributions on a line which is 2$mm$above the metal surface along the z axis are presented. It can be seen that the EM waves cannot be confined on the metal surface, and the measured and simulated results have good agreements. From Figs. 3 (c)-(f), we clearly observe that the EM waves at 5GHz and 10GHz are guided towards different sides of the grating structure, and the simulated and measured field distributions agree very well.

Considering that the design in the microwave domain could be scaled down to the THz domain based on similar theoretical principles, we have simulated the bidirectional splitter for the THz surface plasmons excited by a cylindrical wire. In the THz domain, the coaxial cable is not suitable for a waveguide because of the high attenuation and dispersion. Instead, the cylindrical wire waveguide proposed by Wang and Mittleman [10] could be used directly. In practice, a radially polarized surface wave on the cylindrical wire can be excited with high coupling efficiency through the use of a radially symmetric photoconductive antenna [26]. In the simulation, we put an outer shielding conductor at the bottom of the cylindrical wire. The outer conductor and the wire construct a short coaxial cable to introduce the radially polarized surface waves. The height, depth, and outer diameter of the outer conductor are 10 μm, 5 μm, and 200 μm, respectively. The diameter d of the cylindrical wire is set to 80 μm. The extended height$h_f$of the wire above the grating surface is 80 μm. The periodicityp and slot widthw of the gratings on both sides are 50 μm and 20 μm. The depths hof the gratings on the left and right sides are 50 μm and 110 μm, respectively. The length L and the thickness Hof the splitter are 100 μm and 400 μm, and the simulation region is chosen as 3700μm ×300μm ×1480μm. The whole structure is surrounded by perfectly matched layer absorbers. The field distributions on the y=0 plane which passes the center of the cylindrical wire and the 1D distributions of electric field intensity on the line 2 μm above the surface grating are demonstrated in Fig. 4 . From Fig. 4(a), we observe that most of the EM waves at 0.5 THz are guided towards the right-hand side gratings. From Fig. 4(b), it can be seen that nearly all EM waves at 1 THz are guided towards the left-hand side gratings. The field distributions of the bidirectional frequency splitter in the THz domain could be measured referring to the experimental setup in Ref [26].

 

Fig. 4 The FIT simulation results of the bidirectional surface wave splitter in the THz frequencies: The 2D field distributions on the $y=0$plane and the optical intensity (${\left|E\right|}^2$) distribution on the line 2$\mu m$above the surface of the structure. The observed frequencies are 0.5 THz in (a) and 1.0 THz in (b), respectively. The lengthLof the metal splitter is 100$\mu m$, and the outer diameterdof the metal wire and the shielding conductor on the bottom are 80$\mu m$ and 200$\mu m$, respectively.

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4. Conclusions

In conclusions, we propose a bidirectional surface wave splitter excited by a cylindrical wire waveguide. The novelty of the proposed structure is the coupling mechanism from the cylindrical wire to the surface gratings. The FIT simulations and experiments are conducted to verify the splitter in the microwave frequency. We have shown that most EM waves at different frequencies are guided towards different directions along the gratings structures placed around the cylindrical wire. The measurement results have good agreements to the simulation results. The proposed bidirectional surface wave splitter excited by a cylindrical wire could be extended to the THz frequencies.