Abstract

Strongly confined surface waves can be achieved on periodically structured metal surfaces and are known as spoof surface plasmon polaritons (SPPs). In this work, several terahertz SPP devices based on curved waveguides are demonstrated. The transmittance and bending loss of 90-degree curved spoof SPP waveguides with a radius of curvature ranging from 200 to 2300 µm are investigated to identify the regime for high transmission. A commutator is designed and experimentally demonstrated. Furthermore, coupling equations are derived and verified for efficient coupling between bend-straight waveguides and between bend-bend waveguides. The results will be of great value for future integrated terahertz plasmonic systems.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Surface plasmon polaritons (SPPs) are localized surface waves propagating along a metal-dielectric interface with exponential decay in both directions perpendicular to the interface [13]. In the microwave and terahertz (THz) regimes, high confinement of SPPs at flat metal surfaces cannot be achieved due to the fact that metals behave close to perfect conductors [4,5]. Ever since Pendry et al. proposed that perforated conducting surfaces can support confined electromagnetic waves, which are called spoof SPPs [68], various structures have been reported to support spoof SPP modes in the low frequency range, such as holes in a metal plane [9,10], pillars on a metal surface [11,12], and structured thin metallic strips [1317]. Among the proposed spoof SPP waveguides, various groove or pillar structures providing high field confinement have been proposed [1822] and demonstrated by near field detection techniques in THz regime [23,24]. Meanwhile, different spoof SPP functional devices such as amplifiers, wavelength filters, and sharp bends, have also been reported [2530]. Nevertheless, it is still a great challenge to design and realize compact, low bending loss and high-precision THz integrated circuits, and this is further complicated by the fact that any practical integrated system may have curved connecting sections.

Bending losses have been studied experimentally in photonic crystals [3133], silicon [34], silicon-on-insulators [35], and plasmonic nanowire waveguides [36]. Theoretical analyses for V-shaped grooves [19], metallic wedges [37], and metallic pillars with dielectric cladding [38] have also been conducted. In addition, couplings between parallel waveguides [3941] as well as between nonparallel waveguides [4245] have been studied. However, directly quantifying the bending losses of curved THz spoof SPPs in the near field still remains challenging. Furthermore, to our knowledge, none of the previous works has studied the coupling between curved spoof SPP waveguides, which can reveal important features of power exchange between them and is thereby of considerable interest for practical applications.

In this paper, we report on the design and actualization of several THz plasmonic devices based on 90-degree curved spoof SPP waveguides that are composed of periodic metallic rectangular pillars. The structure is similar to the domino-shaped waveguides in [24], where the guiding properties of straight waveguides and a variety of devices such as S-bend, Y-splitter, and directional couplers are reported. However, as stated above, the bending loss and coupling properties of curve SPP waveguides are the focus of this work. Hence, the transmittance and bending loss of such 90-degree waveguide bends as a function of the radius of curvature are first characterized. Based on this, a commutator is designed and fabricated. The commutator is demonstrated to accomplish wave transmittance successfully with a fiber-optic scanning near-field terahertz microscopy (SNTM) system [46]. In addition, coupling equations for bend-straight waveguides and bend-bend waveguides are derived and validated. The results can help to optimize the spacing and radii of the two waveguides, and thus control the output power ratio between them, which is instructive for developing more robust and complex THz functional plasmonic devices.

2. Results and discussions

Figure 1(a) shows the schematic of the waveguide structure with a row of periodic metal pillars standing on a metal surface and resembling a chain of domino pieces [22,24]. The properties of its guided modes are mainly determined by the geometric parameters defining the domino structure, i.e., pillar width w, length l, height h, and period p, as shown in the lower right inset of Fig. 1(a). Here, we choose the parameters as w = 120 µm, l = 50 µm, h = 80 µm, and p = 100 µm.

 figure: Fig. 1.

Fig. 1. Structure and dispersion relation. (a) Schematic diagram and geometric parameters of the structure (the red arrow depicts the propagation direction of SPPs). The lower left inset shows the SEM image of the fabricated arc-shaped curved hole array excitation region as well as the fan-shaped coupling region. The lower right inset shows the fabricated waveguide structure. (b) The simulated dispersion relation (red line) of SPP mode for one row of metal pillars.

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A special phase-matching technique is needed to couple free-space THz radiation to excite the SPPs. This is because the propagation constant of the SPPs is greater than the wave vector in the dielectric, resulting in a mismatch between them [1]. Periodic subwavelength hole arrays on a thin metal are often used to excite SPPs. The periodic hole array provides additional wave vector components for the incident wave, so that the wave vector matching condition can be satisfied [47,48]. In this paper, we use an arc-shaped curved hole array to achieve high coupling efficiency [46], as shown in the lower left inset of Fig. 1(a). The innermost and outermost radii of the curved hole arrays are 2220 µm and 3820 µm, respectively, and the width of each array is 40 µm wide. The center angle of the curved holes is 60° and the period along the radial direction is 400 µm. The curved hole arrays are divided into smaller holes by metal strips with a spacing of 50 µm and a center angle of 5°. In addition, the centers of all curved holes coincide with the first pillar of the waveguide to obtain high coupling efficiency. The excited SPPs are coupled to the waveguide through a fan-shaped funnel structure consisting of pillars with decreasing periods [24,49].

To eliminate the interference with free-space THz waves, a linearly polarized THz wave with its polarization direction parallel to the x-axis is focused from the bottom side of the sample to the hole array in the z-direction.

One unit cell is calculated using the eigenmode solver of the commercial software CST Microwave Studio to evaluate the dispersion property of the waveguide. In numerical modeling, one unit cell of the periodic structure is considered using periodic boundary conditions and the phase in φ is varied from 0° to 180° (with a step of 10°) along the x direction of propagation. The values of kx are obtained based on kx = φ×π/(180×p). The bottom of the waveguide and the pillars are modeled as perfect electrical conductors, which is applicable for metals in the microwave and THz regimes [2224]. Figure 1(b) presents the simulated dispersion curve of the propagating SPP mode, which approaches the light line for low frequencies and reaches a horizontal frequency limit at the edge (kx =π/p) of the first Brillouin zone, typical of a plasmonic character. As depicted in the figure, in the first Brillouin region, as the frequency increases from ∼ 0.4 THz to ∼ 0.7 THz, the kx of the SPP mode becomes larger than that of the light line (the dashed black line), which indicates that the waveguide has good field confinement. In this dispersion relation of the SPP mode, we define the frequency at which the dispersion curve of the propagating SPP mode is separated from the light line as the band-edge frequency, and the frequency at which the group velocity approaches zero as the cutoff frequency.

A two-step process is employed to fabricate high-quality waveguide structures. First, optical lithography and deep reactive ion etching techniques are used to obtain the basic waveguide structure on a 4-inch silicon wafer. In the second process, a 200-nm-thick gold film metallization of the chips is then conducted in a gold sputter coater. The thickness of gold is selected based on the penetration depth of the THz wave in the metal. The insets of Fig. 1(a) show the scanning electron microscopy (SEM) images of the fabricated excitation area, the funnel structure, and the domino waveguide section.

A fiber-coupled SNTM is used to characterize the near-field distribution above the waveguide surface, as described in detail elsewhere [22,23,46]. THz waves are generated by a femtosecond fiber laser operating at 1550 nm and detected by a near-field probe based on low-temperature-grown GaAs. The sample is located between the probe and a THz lens. In order to excite SPPs effectively, THz waves are irradiated vertically on the excitation region of the sample through a THz lens with a focal length of 100 mm. As the excited SPPs propagate along the waveguide, the probe collects the electrical field information at a distance of 100 µm above the sample surface. The probe is mounted on a 2-dimensional translation detector and moved in the x- and y-directions in steps of 200 µm.

2.1 Bending loss of curved spoof SPP waveguides

Bends, which are used for connecting one waveguide with another, are very important in integrated optical circuits [50]. In order to study the bending loss of curved waveguides, 90-degree curved bends with different radii R = 200, 900, 1600, and 2300 µm are designed and fabricated [Figs. 2(a)–2(d)]. The metallic pillars at the curved portion are set to rotate around the center with an angle of θ to maintain periodicity, where θ = (180×p)/(R×π). In all cases, the structures contain up to 70 periods, and the straight parts of the channel are slightly adjusted in the bends in order to conform to the curved geometry. All the components are designed and simulated using the time domain solver of CST. As the source, a plane wave is irradiated vertically on the excitation region of the sample from the bottom side to excite the SPPs. Figures 2(e)–2(h) show the simulated normalized power |Ez|2 distributions with a scanning area of 5 mm × 5 mm at 0.56 THz for different bends. Clearly, a large bending radius enables higher transmission. Figures 2(i)–2(l) show the corresponding measured images, which are in good agreement with the simulations.

 figure: Fig. 2.

Fig. 2. Performance of bends and waveguide components. (a)-(d) SEM images of structures corresponding to R = 200, 900, 1600, 2300 µm, respectively. (e)-(h) Simulated and (i)-(l) experimental results for normalized power |Ez|2 distributions corresponding to (a)-(d) in a horizontal plane slightly above (at 100 µm) the surface of each structure at 0.56 THz. (m) Optical image of commutator. (n) Simulated and (o) experimental results for normalized power |Ez|2 distributions corresponding to the commutator in (m).

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In order to achieve two-dimensional integrated optical circuits, a commutator composed of two mirrored 90° sharp bends as shown in the optical image in Fig. 2(m) is designed and simulated. The commutator is composed of two mirrored 90° sharp bends, and the length of the straight part is 2000 µm. Based on the above studies, it is found that a large bending radius is more helpful for transmission of the surface wave. Considering the compactness of the structure and also acceptable loss, we use a 90° sharp bend with R = 1600 µm as a basic element for the commutator. The simulated and experimental results for the normalized power |Ez|2 distributions corresponding to the commutator at 0.56 THz are presented in Figs. 2(n) and 2(o), respectively. As can be seen, the SPPs transmit along the waveguide and there is almost no overflow at the bends. Therefore, the component based on 90° sharp bends is expected to accomplish wave transmission successfully.

The numerical transmittance spectra for the four 90-degree curved bends with different radii R = 200, 900, 1600, and 2300 µm are shown in Fig. 3(a), where a general trend of decreasing transmittance with a decreasing radius of curvature can be observed, and the highest transmittance of 64% is observed in the case of R = 2300 µm at 0.56 THz. The red dashed arrows in Fig. 3(a) indicate the band-edge and cutoff frequencies predicted by CST simulations. It is important to note that as the band approaches its cutoff frequency, the spoof SPP mode becomes more localized and sensitive, and the numerical result is less accurate due to the limitation of calculation accuracy.

 figure: Fig. 3.

Fig. 3. Performance of bends. (a) Transmittance spectra and (b) bending losses as a function of frequency for 90-degree curved bends with different R values from 200 to 2300 µm.

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The bending losses as a function of frequency for four radii of bend R = 200, 900, 1600, and 2300 µm are presented in Fig. 3(b). In all four cases, the losses are high at low frequencies but decrease as frequency increases. At low frequencies, the penetration depth of the electric field in space is large, resulting in a large bending loss. As the frequency increases, the field confinement is improved and the bending loss decreases. The loss is calculated as ratio of the power coupled to the fundamental mode of the output waveguide to the input power: Loss = 10log (PInput/POutput). Since metal is assumed to be lossless, the only source of loss is radiation. Therefore, the bending loss can be obtained by calculating the difference between the insertion loss of the structure and that of a straight waveguide with the same length. The power is obtained by integrating the longitudinal component of the Poynting vector along surface planes perpendicular to the input and exit sides of the waveguide. The modal size δ, defined here as the transverse dimension where the electric field amplitude has fallen to one tenth of its maximum value [19,37], is δ = 375 µm = 0.5 λ for f = 0.4 THz and δ = 225 µm = 0.41 λ for f = 0.55 THz. The two integrating regions have the same dimensions of 800 µm × 800 µm, which are larger than the maximum modal size to ensure the correct calculation of the power distribution.

In Fig. 4, the numerical and measured bending losses for each bend radius at 0.56 THz are plotted as a function of R, where the measured bending loss can be as small as 1.55 dB for the case of maximum R = 2300 µm but increases as R is reduced, being 6.45 dB for R = 200 µm. In the measurement, most of the energy of SPPs is localized on the waveguide surface such that the electric field component Ez is much larger than other components, and thus the intensity of Ez can approximately represent the energy flux density. Consequently, the power of SPPs is obtained by integrating the simulated normalized power |Ez|2 near the input and output ports of the waveguides. Experimental results are normalized by the transmitted power measured in the straight section to eliminate the influence of propagation loss, which is about 1.06 dB/mm. A deterioration of the measured loss values compared to simulation can be attributed to the fabrication error and actual metallic loss neglected in simulation.

 figure: Fig. 4.

Fig. 4. Simulated (red circles) and measured (black circles) bending losses as a function of R at 0.56 THz.

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2.2 Coupling characteristics of curved spoof SPP waveguides

Curved waveguide forms an essential part of an integrated system and therefore its coupling with another waveguide is of great concern from an application point of view. In this subsection, we further discuss the coupling characteristics of curved spoof SPP waveguides.

The structure diagrams of two systems under consideration are shown in Figs. 5(a) and 5(b), for bend-straight waveguides and bend-bend waveguides, respectively. There are two sections for each system, and the unit size and material for all waveguides are the same. The first part comprises two parallel straight waveguides similar to those discussed in the previous subsection. The gap between them is g and the length of the parallel section is L. Following the first part, for the bend-straight waveguides in Fig. 5(a), a 90° curved waveguide with radius R and a straight waveguide are connected respectively to the two straight waveguides to separate the waves. For the bend-bend waveguides in Fig. 5(b), two 90° curved waveguides are connected to the first part. In the second curved part, Cartesian coordinate systems are established along the tangential (x-axis) and radial (y-axis) directions of the curved waveguide connected to waveguide 1, as shown in Figs. 5(a) and 5(b).

 figure: Fig. 5.

Fig. 5. Curved waveguide geometries and analysis. Structures of (a) bend-straight waveguides and (b) bend-bend waveguides. (c) Dispersion relation of even and odd supemodes for waveguides with varying g. The left inset shows normalized electric component (EZ) distributions in the yz cross section for even (upper left) and odd modes (lower left) supported by two parallel waveguides with g = 80 µm at 0.62 THz. The inset on the right indicates the electric field at g = 280 µm. (d) The difference in kx between two supermodes as a function of g from 80 to 280 µm at 0.62 THz.

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The coupling between two waveguides is affected by the distance between them. In the curved portion, the distance between the two waveguides increases with the propagation distance of the SPPs. To investigate the effect of changes of g on the coupling of the two waveguides, the supermode approach is applied for analysis. When two identical waveguides are close to each other, there are two supermodes supported by the entire structure. In Fig. 5(c), the propagation constants of the even mode (kxe) and the odd mode (kxo) as a function of frequency for different values of g varied from 80 to 280 µm are displayed. The left inset shows the normalized electric distributions (Ez) of the two modes in the yz cross section with g = 80 µm at 0.62 THz. The difference in propagation constants kx between these two supermodes at the same frequency will lead to a phase difference. After the SPPs have propagated a certain distance, the accumulated phase difference reaches a value of π, and the mode power will be shifted completely from one waveguide to the other in the case of identical waveguide geometries. The length corresponding to this π phase shift is defined as the coupling length, which can be expressed as Lc =π/(kxe - kxo). As can be seen from Fig. 5(d), the difference in kx between the two supermodes decreases as g increases and reduces almost to zero when g = 280 µm at 0.62 THz. At this point, we will assume that there is no coupling between the two waveguides, and this distance between them is defined as the maximum coupling distance gmax. The electric field distribution for this case is shown in the right inset of Fig. 5(c).

The line connecting the farthest coupling point to the center of the curved waveguide has an angle with respect to the y-axis, and the angles are marked as αi and αj in Fig. 5(a) and 5(b), respectively. When αi and αj are small, the curved waveguides can be approximated as inclined waveguides making an angle with the x-axis. For the bend-straight waveguides and bend-bend waveguides, the angles can be obtained from simple geometry, respectively, as:

$$\begin{array}{l} {\theta _\textrm{i}} = \frac{{{\alpha _\textrm{i}}}}{2} = \frac{1}{2}\textrm{arccos}\frac{{R - ({{g_{\max }} - g} )}}{R} \\ {\theta _\textrm{j}} = \frac{{{\alpha _\textrm{j}}}}{2} = \frac{1}{2}\textrm{arccos}\frac{{2R - ({{g_{\max }} - g} )}}{{2R}}. \end{array}$$
Therefore, the distances between the two waveguides at arbitrary abscissa values xi and xj for the two systems can be written as:
$$\begin{array}{l} {g_\textrm{i}} = g + {x_\textrm{i}}\textrm{tan}{\theta _\textrm{i}} \\ {g_\textrm{j}} = g + 2{x_\textrm{j}}\textrm{tan}{\theta _\textrm{j}}. \end{array}$$
By substituting (2) into the coupling coefficient between parallel optical waveguides [44], the coupling coefficients of the two systems can be expressed as:
$${K_\textrm{i}} = {K_\textrm{0}}\exp ({ - p{x_\textrm{i}}\textrm{tan}{\theta_\textrm{i}}} ),$$
$${K_\textrm{j}} = {K_\textrm{0}}\exp ({ - 2p{x_\textrm{j}}\textrm{tan}{\theta_\textrm{j}}} ),$$
where ${p^2} = {\beta ^2} - n_0^2{k^2}$, β is the propagation constant of a single waveguide and $k = 2\pi /\lambda$. K0 is the coupling coefficient between two parallel optical waveguides with a distance g, according to the coupled-mode theory:
$${L_\textrm{c}} = \frac{\pi }{{{k_{xe}} - {k_{xo}}}} = \frac{\pi }{{2{K_\textrm{0}}}},$$
$${K_0} = \Delta k\textrm{ = }\frac{{{k_{xe}} - {k_{xo}}}}{2}.$$
For bend-straight waveguides, by substituting (3) into the coupling equations for two parallel waveguides, the coupling equations can be expressed as:
$$\begin{array}{l} \frac{{\textrm{d}{A_1}}}{{\textrm{d}x}} ={-} \textrm{j}{K_\textrm{0}}{A_2}\exp ({ - p{x_\textrm{i}}\textrm{tan}{\theta_\textrm{i}}} )\\ \frac{{\textrm{d}{A_2}}}{{\textrm{d}x}} ={-} \textrm{j}{K_\textrm{0}}{A_1}\exp ({ - p{x_\textrm{i}}\textrm{tan}{\theta_\textrm{i}}} ), \end{array}$$
where A1 and A2 are the field amplitudes. The output powers from waveguides 1 and 2 are, respectively:
$$\begin{array}{l} {P_{\textrm{i1}}} = {\cos ^2}\left\{ {\frac{{{K_\textrm{0}}}}{{p\tan {\theta_\textrm{i}}}}[{1 - \exp ({ - p{x_\textrm{i}}\tan {\theta_\textrm{i}}} )} ]} \right\}\\ {P_{\textrm{i2}}} = \textrm{si}{\textrm{n}^2}\left\{ {\frac{{{K_\textrm{0}}}}{{p\tan {\theta_\textrm{i}}}}[{1 - \exp ({ - p{x_\textrm{i}}\tan {\theta_\textrm{i}}} )} ]} \right\}. \end{array}$$
Similarly, for bend-bend waveguides, the output powers from waveguides 1 and 2 are, respectively:
$$\begin{array}{l} {P_{\textrm{j1}}} = {\cos ^2}\left\{ {\frac{{{K_\textrm{0}}}}{{2p\tan {\theta_\textrm{j}}}}[{1 - \exp ({ - 2p{x_\textrm{j}}\tan {\theta_\textrm{j}}} )} ]} \right\}\\ {P_{\textrm{j2}}} = \textrm{si}{\textrm{n}^2}\left\{ {\frac{{{K_\textrm{0}}}}{{2p\tan {\theta_\textrm{j}}}}[{1 - \exp ({ - 2p{x_\textrm{j}}\tan {\theta_\textrm{j}}} )} ]} \right\}. \end{array}$$
According to the above formulas, for two-coupled curved waveguide systems with L = 0 µm, g = 80 µm and R = 2300 µm, the normalized output powers from waveguides 1 and 2 as a function of xi and xj at 0.62 THz can be obtained as in Figs. 6(a) and 6(c). The red dotted lines represent the values of the normalized output powers from the two waveguides when the propagation distances of the SPPs are large enough in the two systems. At this point, for the bend-straight waveguides, as can be seen from Fig. 6(a), 30% of the energy in waveguide 1 is coupled into waveguide 2, and the remaining 70% of the energy remains to propagate in waveguide 1. For the bend-bend waveguides, the maximum coupling distance can be achieved by a shorter propagation distance of the SPPs, resulting in only 16% of the energy being coupled into waveguide 2, as shown in Fig. 6(c).

 figure: Fig. 6.

Fig. 6. Normalized powers for (a) bend-straight waveguides and (c) bend-bend waveguides as a function of propagation distance at 0.62 THz. The red dotted lines indicate the output powers from the curved portion when the propagation distances of the SPPs are large enough. (b) and (d) The normalized output powers of two parallel waveguides as a function of L. In all cases, the black solid and red dashed lines represent the normalized powers from waveguides 1 and 2, respectively.

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In order to verify the above theoretical analysis, the two systems discussed above are simulated. Waveguide 1 is connected to the SPP source. Since part of the energy in waveguide 1 is coupled into waveguide 2 through the curved portion, suitable lengths of the parallel sections Li and Lj for the two systems are needed to ensure that the SPPs can be completely coupled into waveguide 2. The normalized output powers of two parallel waveguides as a function of L are shown in Figs. 6(b) and 6(d), where the red dotted lines are consistent with those in Figs. 6(a) and 6(c), respectively, from which Li and Lj can be obtained. The section lengths L are calculated to be 780 µm for the bend-straight waveguides and 910 µm for the bend-bend waveguides, which are set to be equal to the distances of eight and nine metal pillars, respectively.

The bend-straight waveguides and bend-bend waveguides are fabricated as illustrated in Figs. 7(a) and 7(b) to verify the numerical analysis. Figures 7(c) and 7(d) show the simulated normalized power |Ez|2 distributions at 0.62 THz for the two directional couplers, respectively. Obviously, the SPPs in waveguide 1 are completely coupled into waveguide 2, which indicates that the directional couplers based on 90° sharp bends can accomplish successful wave coupling. Figures 7(e) and 7(f) show the corresponding measured images, which are in good agreement with the simulations. In Figs. 7(g) and 7(h), the simulated and experimental cross-sectional normalized power distributions are displayed at the end of the coupling regions (line x = 4 mm) as indicated by the dotted lines in the corresponding figures of Figs. 7(c)–7(f) and also the values of the x-coordinates. For the recorded cross-sectional field distributions, the fields are strongly concentrated in waveguides 2 near y = -0.5 mm for bend-straight waveguides and y = -0.7 mm for bend-bend waveguides, and the field amplitudes decay quickly to almost zero away from waveguides 1. The excellent coupling performance of this component is thus fully corroborated.

 figure: Fig. 7.

Fig. 7. Performance of directional couplers. (a) and (b) SEM images of structures corresponding to bend-straight waveguides and bend-bend waveguides, respectively. (c) and (d) Simulated and (e) and (f) experimental results for normalized power |Ez|2 distributions corresponding to (a) and (b) in a horizontal plane slightly above (at 100 µm) the surface of each structure at 0.62 THz. (g) and (h) Simulated and experimental cross-sectional normalized power distributions at the end of the coupling regions (line x = 4 mm) in (c)-(f).

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3. Conclusion

In conclusion, the transmittance, bending loss, and coupling performance of curved THz spoof SPP waveguide are investigated both in simulation and experiment. In the case of curved waveguides with a radius from 200 to 2300 µm, bending losses of 1.55 dB are obtained at 0.56 THz when the radius is 2300 µm. Based on this, a commutator is demonstrated to be able to accomplish wave transmission successfully. Furthermore, couplings of bend-straight waveguide and bend-bend waveguide are analyzed by coupled mode theory and verified by experiments. Complete directional power coupling is shown to be realized in a system composed of the correct sections of a straight waveguide and a curved waveguide. We believe that the results achieved here will be very promising for routing THz waves at planar surfaces and form important components in future integrated plasmonic systems.

Funding

National Key Research and Development Program of China (2017YFA0701004); National Natural Science Foundation of China (61605143, 61722509, 61735012, 61871212, 61875150, 61935015); Tianjin Municipal Fund for Distinguished Young Scholars (18JCJQJC45600); King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) (OSR-2016-CRG5-2950).

Disclosures

The authors declare no conflicts of interest.

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19. A. I. Fernández-Domínguez, E. Moreno, L. Martín-Moreno, and F. J. García-Vidal, “Guiding terahertz waves along subwavelength channels,” Phys. Rev. B 79(23), 233104 (2009). [CrossRef]  

20. A. I. Fernández-Domínguez, E. Moreno, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz wedge plasmon polaritons,” Opt. Lett. 34(13), 2063–2065 (2009). [CrossRef]  

21. G. Kumar, S. Li, M. M. Jadidi, and T. E. Murphy, “Terahertz surface plasmon waveguide based on a one-dimensional array of silicon pillars,” New J. Phys. 15(8), 085031 (2013). [CrossRef]  

22. D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Dominguez, F. J. García-Vidal, L. Martín-Moreno, and E. Moreno, “Domino plasmons for subwavelength terahertz circuitry,” Opt. Express 18(2), 754–764 (2010). [CrossRef]  

23. Y. Zhang, S. Li, Q. Xu, C. Tian, J. Gu, Y. Li, Z. Tian, C. Ouyang, J. Han, and W. Zhang, “Terahertz surface plasmon polariton waveguiding with periodic metallic cylinders,” Opt. Express 25(13), 14397–14405 (2017). [CrossRef]  

24. Y. Zhang, Y. Xu, C. Tian, Q. Xu, X. Zhang, Y. Li, X. Zhang, J. Han, and W. Zhang, “Terahertz spoof surface-plasmon-polariton subwavelength waveguide,” Photonics Res. 6(1), 18–23 (2018). [CrossRef]  

25. J. Yin, J. Ren, H. Zhang, B. Pan, and T. Cui, “Broadband frequency-selective spoof surface plasmon polaritons on ultrathin metallic structure,” Sci. Rep. 5(1), 8165 (2015). [CrossRef]  

26. X. Gao, L. Zhou, and T. Cui, “Odd-mode surface plasmon polaritons supported by complementary plasmonic metamaterial,” Sci. Rep. 5(1), 9250 (2015). [CrossRef]  

27. Z. Li, B. Xu, L. Liu, J. Xu, C. Chen, C. Gu, and Y. Zhou, “Localized spoof surface plasmons based on closed subwavelength high contrast gratings: concept and microwave-regime realizations,” Sci. Rep. 6(1), 27158 (2016). [CrossRef]  

28. Y. Zhou and B. Yang, “Planar spoof plasmonic ultra-wideband filter based on low-loss and compact terahertz waveguide corrugated with dumbbell grooves,” Appl. Opt. 54(14), 4529–4533 (2015). [CrossRef]  

29. L. Ye, W. Zhang, B. K. Ofori-Okai, W. Li, J. Zhuo, G. Cai, and Q. Liu, “Super subwavelength guiding and rejecting of terahertz spoof SPPs enabled by planar plasmonic waveguides and notch filters based on spiral-shaped units,” J. Lightwave Technol. 36(20), 4988–4994 (2018). [CrossRef]  

30. L. Ye, Y. Chen, K. Xu, W. Li, Q. Liu, and Y. Zhang, “Substrate integrated plasmonic waveguide for microwave bandpass filter applications,” IEEE Access 7, 75957–75964 (2019). [CrossRef]  

31. M. H. Shih, W. J. Kim, W. Kuang, J. R. Cao, S. J. Choi, J. D. O’Brien, and P. D. Dapkus, “Experimental characterization of the reflectance of 60° waveguide bends in photonic crystal waveguides,” Appl. Phys. Lett. 86(19), 191104 (2005). [CrossRef]  

32. M. Ayre, T. J. Karle, L. Wu, T. Davies, and T. F. Krauss, “Experimental verification of numerically optimized photonic crystal injector, Y-splitter, and bend,” IEEE J. Sel. Area. Comm. 23(7), 1390–1395 (2005). [CrossRef]  

33. I. Ntakis, P. Pottier, and R. M. De La Rue, “Optimization of transmittance properties of two-dimensional photonic crystal channel waveguide bends through local lattice deformation,” J. Appl. Phys. 96(1), 12–18 (2004). [CrossRef]  

34. M. Lipson, “Guiding, modulating, and emitting light on silicon-challenges and opportunities,” J. Lightwave Technol. 23(12), 4222–4238 (2005). [CrossRef]  

35. Y. A. Vlasov and S. J. Mcnab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express 12(8), 1622–1631 (2004). [CrossRef]  

36. D. J. Dikken, M. Spasenović, E. Verhagen, D. van Oosten, and L. K. Kuipers, “Characterization of bending losses for curved plasmonic nanowire waveguides,” Opt. Express 18(15), 16112–16119 (2010). [CrossRef]  

37. Z. Gao, X. Zhang, and L. Shen, “Wedge mode of spoof surface plasmon polaritons at terahertz frequencies,” J. Appl. Phys. 108(11), 113104 (2010). [CrossRef]  

38. M. A. K. Moghaddam and M. Ahmadi-Boroujeni, “Design of a hybrid spoof plasmonic sub-terahertz waveguide with low bending loss in a broad frequency band,” Opt. Express 25(6), 6860–6873 (2017). [CrossRef]  

39. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE Photonics Technol. Lett. 9(9), 919–933 (1973). [CrossRef]  

40. H. F. Taylor and A. Yariv, “Guided wave optics,” Proc. IEEE 62(8), 1044–1060 (1974). [CrossRef]  

41. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3(5), 1135–1146 (1985). [CrossRef]  

42. M. Matsuhara and A. Watanabe, “Coupling of curved transmittance lines, and application to optical directional couplers,” J. Opt. Soc. Am. 65(2), 163–168 (1975). [CrossRef]  

43. C. Wang, Q. Li, G. Liu, G. Jin, and X. Xu, “The calculation of switching power of symmetric and asymmetric nonlinear directional couplers with variable coupling coefficient,” IEEE Photonics Technol. Lett. 16(10), 2248–2250 (2004). [CrossRef]  

44. Y. Jia and Y. Hao, “Power exchange between two nonparallel waveguides,” Acta Photon. Sin. 34(6), 852–856 (2005). (In Chinese).

45. H. Liang, S. Shi, and L. Ma, “Coupled-mode theory of nonparallel optical waveguides,” J. Lightwave Technol. 25(8), 2233–2235 (2007). [CrossRef]  

46. M. Yuan, Y. Li, Y. Lu, Y. Zhang, Z. Zhang, X. Zhang, X. Zhang, J. Han, and W. Zhang, “High-performance and compact broadband terahertz plasmonic waveguide intersection,” Nanophotonics 8(10), 1811–1819 (2019). [CrossRef]  

47. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmittance through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998). [CrossRef]  

48. L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5(7), 1399–1402 (2005). [CrossRef]  

49. L. Liu, Z. Li, B. Xu, C. Gu, C. Chen, P. Ning, J. Yan, and X. Chen, “High-efficiency transition between rectangular waveguide and domino plasmonic waveguide,” AIP Adv. 5(2), 027105 (2015). [CrossRef]  

50. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (3rd Edition) (John Wiley & Sons, 2019).

References

  • View by:

  1. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer Berlin, 2007).
  2. T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85(24), 5833–5835 (2004).
    [Crossref]
  3. K. Tanaka and M. Tanaka, “Simulations of Nanometric Optical Circuits: Open-Type Surface Plasmon Polariton Gap Waveguide,” Appl. Phys. Lett. 42(6A), L585–L588 (2003).
    [Crossref]
  4. T. I. Jeon and D. Grischkowsky, “THz Zenneck surface wave (THz surface plasmon) propagation on a metal sheet,” Appl. Phys. Lett. 88(6), 061113 (2006).
    [Crossref]
  5. L. Shen, X. Chen, and T. J. Yang, “Terahertz surface plasmon polaritons on periodically corrugated metal surfaces,” Opt. Express 16(5), 3326–3333 (2008).
    [Crossref]
  6. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
    [Crossref]
  7. F. J. Garcia-Vidal, L. Martín-Moreno, and J. B. Pendry, “Surfaces with holes in them: New plasmonic metamaterials,” J. Opt. A: Pure Appl. Opt. 7(2), S97–S101 (2005).
    [Crossref]
  8. C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics 2(3), 175–179 (2008).
    [Crossref]
  9. W. Zhu, A. Agrawal, and A. Nahata, “Planar plasmonic terahertz guided-wave devices,” Opt. Express 16(9), 6216–6226 (2008).
    [Crossref]
  10. L. Tian, Z. Zhang, J. Liu, K. Zhou, Y. Gao, and S. Liu, “Compact spoof surface plasmon polaritons waveguide drilled with L-shaped grooves,” Opt. Express 24(25), 28693–28703 (2016).
    [Crossref]
  11. S. Li, M. M. Jadidi, T. E. Murphy, and G. Kumar, “Terahertz surface plasmon polaritons on a semiconductor surface structured with periodic V-grooves,” Opt. Express 21(6), 7041–7049 (2013).
    [Crossref]
  12. D. Martin-Cano, O. Quevedo-Teruel, E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, “Waveguided spoof surface plasmons with deep-subwavelength lateral confinement,” Opt. Lett. 36(23), 4635–4637 (2011).
    [Crossref]
  13. X. Shen and T. Cui, “Planar plasmonic metamaterial on a thin film with nearly zero thickness,” Appl. Phys. Lett. 102(21), 211909 (2013).
    [Crossref]
  14. Y. Zhang, P. Zhang, and Z. Han, “One-dimensional spoof surface plasmon structures for planar terahertz photonic integration,” J. Lightwave Technol. 33(18), 3796–3800 (2015).
    [Crossref]
  15. L. Ye, Y. Xiao, Y. Liu, L. Zhang, G. Cai, and Q. Liu, “Strongly confined spoof surface slasmon solaritons waveguiding enabled by planar staggered plasmonic waveguides,” Sci. Rep. 6(1), 38528 (2016).
    [Crossref]
  16. L. Ye, Y. Xiao, N. Liu, Z. Song, W. Zhang, and Q. Liu, “Plasmonic waveguide with folded stubs for highly confined terahertz propagation and concentration,” Opt. Express 25(2), 898–906 (2017).
    [Crossref]
  17. L. Ye, H. Feng, G. Cai, Y. Zhang, B. Yan, and Q. Liu, “High-efficient and low-coupling spoof surface plasmon polaritons enabled by V-shaped microstrips,” Opt. Express 27(16), 22088–22099 (2019).
    [Crossref]
  18. S. A. Maier, S. R. Andrews, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett. 97(17), 176805 (2006).
    [Crossref]
  19. A. I. Fernández-Domínguez, E. Moreno, L. Martín-Moreno, and F. J. García-Vidal, “Guiding terahertz waves along subwavelength channels,” Phys. Rev. B 79(23), 233104 (2009).
    [Crossref]
  20. A. I. Fernández-Domínguez, E. Moreno, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz wedge plasmon polaritons,” Opt. Lett. 34(13), 2063–2065 (2009).
    [Crossref]
  21. G. Kumar, S. Li, M. M. Jadidi, and T. E. Murphy, “Terahertz surface plasmon waveguide based on a one-dimensional array of silicon pillars,” New J. Phys. 15(8), 085031 (2013).
    [Crossref]
  22. D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Dominguez, F. J. García-Vidal, L. Martín-Moreno, and E. Moreno, “Domino plasmons for subwavelength terahertz circuitry,” Opt. Express 18(2), 754–764 (2010).
    [Crossref]
  23. Y. Zhang, S. Li, Q. Xu, C. Tian, J. Gu, Y. Li, Z. Tian, C. Ouyang, J. Han, and W. Zhang, “Terahertz surface plasmon polariton waveguiding with periodic metallic cylinders,” Opt. Express 25(13), 14397–14405 (2017).
    [Crossref]
  24. Y. Zhang, Y. Xu, C. Tian, Q. Xu, X. Zhang, Y. Li, X. Zhang, J. Han, and W. Zhang, “Terahertz spoof surface-plasmon-polariton subwavelength waveguide,” Photonics Res. 6(1), 18–23 (2018).
    [Crossref]
  25. J. Yin, J. Ren, H. Zhang, B. Pan, and T. Cui, “Broadband frequency-selective spoof surface plasmon polaritons on ultrathin metallic structure,” Sci. Rep. 5(1), 8165 (2015).
    [Crossref]
  26. X. Gao, L. Zhou, and T. Cui, “Odd-mode surface plasmon polaritons supported by complementary plasmonic metamaterial,” Sci. Rep. 5(1), 9250 (2015).
    [Crossref]
  27. Z. Li, B. Xu, L. Liu, J. Xu, C. Chen, C. Gu, and Y. Zhou, “Localized spoof surface plasmons based on closed subwavelength high contrast gratings: concept and microwave-regime realizations,” Sci. Rep. 6(1), 27158 (2016).
    [Crossref]
  28. Y. Zhou and B. Yang, “Planar spoof plasmonic ultra-wideband filter based on low-loss and compact terahertz waveguide corrugated with dumbbell grooves,” Appl. Opt. 54(14), 4529–4533 (2015).
    [Crossref]
  29. L. Ye, W. Zhang, B. K. Ofori-Okai, W. Li, J. Zhuo, G. Cai, and Q. Liu, “Super subwavelength guiding and rejecting of terahertz spoof SPPs enabled by planar plasmonic waveguides and notch filters based on spiral-shaped units,” J. Lightwave Technol. 36(20), 4988–4994 (2018).
    [Crossref]
  30. L. Ye, Y. Chen, K. Xu, W. Li, Q. Liu, and Y. Zhang, “Substrate integrated plasmonic waveguide for microwave bandpass filter applications,” IEEE Access 7, 75957–75964 (2019).
    [Crossref]
  31. M. H. Shih, W. J. Kim, W. Kuang, J. R. Cao, S. J. Choi, J. D. O’Brien, and P. D. Dapkus, “Experimental characterization of the reflectance of 60° waveguide bends in photonic crystal waveguides,” Appl. Phys. Lett. 86(19), 191104 (2005).
    [Crossref]
  32. M. Ayre, T. J. Karle, L. Wu, T. Davies, and T. F. Krauss, “Experimental verification of numerically optimized photonic crystal injector, Y-splitter, and bend,” IEEE J. Sel. Area. Comm. 23(7), 1390–1395 (2005).
    [Crossref]
  33. I. Ntakis, P. Pottier, and R. M. De La Rue, “Optimization of transmittance properties of two-dimensional photonic crystal channel waveguide bends through local lattice deformation,” J. Appl. Phys. 96(1), 12–18 (2004).
    [Crossref]
  34. M. Lipson, “Guiding, modulating, and emitting light on silicon-challenges and opportunities,” J. Lightwave Technol. 23(12), 4222–4238 (2005).
    [Crossref]
  35. Y. A. Vlasov and S. J. Mcnab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express 12(8), 1622–1631 (2004).
    [Crossref]
  36. D. J. Dikken, M. Spasenović, E. Verhagen, D. van Oosten, and L. K. Kuipers, “Characterization of bending losses for curved plasmonic nanowire waveguides,” Opt. Express 18(15), 16112–16119 (2010).
    [Crossref]
  37. Z. Gao, X. Zhang, and L. Shen, “Wedge mode of spoof surface plasmon polaritons at terahertz frequencies,” J. Appl. Phys. 108(11), 113104 (2010).
    [Crossref]
  38. M. A. K. Moghaddam and M. Ahmadi-Boroujeni, “Design of a hybrid spoof plasmonic sub-terahertz waveguide with low bending loss in a broad frequency band,” Opt. Express 25(6), 6860–6873 (2017).
    [Crossref]
  39. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE Photonics Technol. Lett. 9(9), 919–933 (1973).
    [Crossref]
  40. H. F. Taylor and A. Yariv, “Guided wave optics,” Proc. IEEE 62(8), 1044–1060 (1974).
    [Crossref]
  41. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3(5), 1135–1146 (1985).
    [Crossref]
  42. M. Matsuhara and A. Watanabe, “Coupling of curved transmittance lines, and application to optical directional couplers,” J. Opt. Soc. Am. 65(2), 163–168 (1975).
    [Crossref]
  43. C. Wang, Q. Li, G. Liu, G. Jin, and X. Xu, “The calculation of switching power of symmetric and asymmetric nonlinear directional couplers with variable coupling coefficient,” IEEE Photonics Technol. Lett. 16(10), 2248–2250 (2004).
    [Crossref]
  44. Y. Jia and Y. Hao, “Power exchange between two nonparallel waveguides,” Acta Photon. Sin. 34(6), 852–856 (2005). (In Chinese).
  45. H. Liang, S. Shi, and L. Ma, “Coupled-mode theory of nonparallel optical waveguides,” J. Lightwave Technol. 25(8), 2233–2235 (2007).
    [Crossref]
  46. M. Yuan, Y. Li, Y. Lu, Y. Zhang, Z. Zhang, X. Zhang, X. Zhang, J. Han, and W. Zhang, “High-performance and compact broadband terahertz plasmonic waveguide intersection,” Nanophotonics 8(10), 1811–1819 (2019).
    [Crossref]
  47. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmittance through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
    [Crossref]
  48. L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5(7), 1399–1402 (2005).
    [Crossref]
  49. L. Liu, Z. Li, B. Xu, C. Gu, C. Chen, P. Ning, J. Yan, and X. Chen, “High-efficiency transition between rectangular waveguide and domino plasmonic waveguide,” AIP Adv. 5(2), 027105 (2015).
    [Crossref]
  50. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (3rd Edition) (John Wiley & Sons, 2019).

2019 (3)

L. Ye, H. Feng, G. Cai, Y. Zhang, B. Yan, and Q. Liu, “High-efficient and low-coupling spoof surface plasmon polaritons enabled by V-shaped microstrips,” Opt. Express 27(16), 22088–22099 (2019).
[Crossref]

L. Ye, Y. Chen, K. Xu, W. Li, Q. Liu, and Y. Zhang, “Substrate integrated plasmonic waveguide for microwave bandpass filter applications,” IEEE Access 7, 75957–75964 (2019).
[Crossref]

M. Yuan, Y. Li, Y. Lu, Y. Zhang, Z. Zhang, X. Zhang, X. Zhang, J. Han, and W. Zhang, “High-performance and compact broadband terahertz plasmonic waveguide intersection,” Nanophotonics 8(10), 1811–1819 (2019).
[Crossref]

2018 (2)

2017 (3)

2016 (3)

L. Ye, Y. Xiao, Y. Liu, L. Zhang, G. Cai, and Q. Liu, “Strongly confined spoof surface slasmon solaritons waveguiding enabled by planar staggered plasmonic waveguides,” Sci. Rep. 6(1), 38528 (2016).
[Crossref]

L. Tian, Z. Zhang, J. Liu, K. Zhou, Y. Gao, and S. Liu, “Compact spoof surface plasmon polaritons waveguide drilled with L-shaped grooves,” Opt. Express 24(25), 28693–28703 (2016).
[Crossref]

Z. Li, B. Xu, L. Liu, J. Xu, C. Chen, C. Gu, and Y. Zhou, “Localized spoof surface plasmons based on closed subwavelength high contrast gratings: concept and microwave-regime realizations,” Sci. Rep. 6(1), 27158 (2016).
[Crossref]

2015 (5)

Y. Zhou and B. Yang, “Planar spoof plasmonic ultra-wideband filter based on low-loss and compact terahertz waveguide corrugated with dumbbell grooves,” Appl. Opt. 54(14), 4529–4533 (2015).
[Crossref]

J. Yin, J. Ren, H. Zhang, B. Pan, and T. Cui, “Broadband frequency-selective spoof surface plasmon polaritons on ultrathin metallic structure,” Sci. Rep. 5(1), 8165 (2015).
[Crossref]

X. Gao, L. Zhou, and T. Cui, “Odd-mode surface plasmon polaritons supported by complementary plasmonic metamaterial,” Sci. Rep. 5(1), 9250 (2015).
[Crossref]

Y. Zhang, P. Zhang, and Z. Han, “One-dimensional spoof surface plasmon structures for planar terahertz photonic integration,” J. Lightwave Technol. 33(18), 3796–3800 (2015).
[Crossref]

L. Liu, Z. Li, B. Xu, C. Gu, C. Chen, P. Ning, J. Yan, and X. Chen, “High-efficiency transition between rectangular waveguide and domino plasmonic waveguide,” AIP Adv. 5(2), 027105 (2015).
[Crossref]

2013 (3)

X. Shen and T. Cui, “Planar plasmonic metamaterial on a thin film with nearly zero thickness,” Appl. Phys. Lett. 102(21), 211909 (2013).
[Crossref]

S. Li, M. M. Jadidi, T. E. Murphy, and G. Kumar, “Terahertz surface plasmon polaritons on a semiconductor surface structured with periodic V-grooves,” Opt. Express 21(6), 7041–7049 (2013).
[Crossref]

G. Kumar, S. Li, M. M. Jadidi, and T. E. Murphy, “Terahertz surface plasmon waveguide based on a one-dimensional array of silicon pillars,” New J. Phys. 15(8), 085031 (2013).
[Crossref]

2011 (1)

2010 (3)

2009 (2)

A. I. Fernández-Domínguez, E. Moreno, L. Martín-Moreno, and F. J. García-Vidal, “Guiding terahertz waves along subwavelength channels,” Phys. Rev. B 79(23), 233104 (2009).
[Crossref]

A. I. Fernández-Domínguez, E. Moreno, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz wedge plasmon polaritons,” Opt. Lett. 34(13), 2063–2065 (2009).
[Crossref]

2008 (3)

L. Shen, X. Chen, and T. J. Yang, “Terahertz surface plasmon polaritons on periodically corrugated metal surfaces,” Opt. Express 16(5), 3326–3333 (2008).
[Crossref]

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics 2(3), 175–179 (2008).
[Crossref]

W. Zhu, A. Agrawal, and A. Nahata, “Planar plasmonic terahertz guided-wave devices,” Opt. Express 16(9), 6216–6226 (2008).
[Crossref]

2007 (1)

2006 (2)

T. I. Jeon and D. Grischkowsky, “THz Zenneck surface wave (THz surface plasmon) propagation on a metal sheet,” Appl. Phys. Lett. 88(6), 061113 (2006).
[Crossref]

S. A. Maier, S. R. Andrews, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett. 97(17), 176805 (2006).
[Crossref]

2005 (6)

F. J. Garcia-Vidal, L. Martín-Moreno, and J. B. Pendry, “Surfaces with holes in them: New plasmonic metamaterials,” J. Opt. A: Pure Appl. Opt. 7(2), S97–S101 (2005).
[Crossref]

M. Lipson, “Guiding, modulating, and emitting light on silicon-challenges and opportunities,” J. Lightwave Technol. 23(12), 4222–4238 (2005).
[Crossref]

M. H. Shih, W. J. Kim, W. Kuang, J. R. Cao, S. J. Choi, J. D. O’Brien, and P. D. Dapkus, “Experimental characterization of the reflectance of 60° waveguide bends in photonic crystal waveguides,” Appl. Phys. Lett. 86(19), 191104 (2005).
[Crossref]

M. Ayre, T. J. Karle, L. Wu, T. Davies, and T. F. Krauss, “Experimental verification of numerically optimized photonic crystal injector, Y-splitter, and bend,” IEEE J. Sel. Area. Comm. 23(7), 1390–1395 (2005).
[Crossref]

Y. Jia and Y. Hao, “Power exchange between two nonparallel waveguides,” Acta Photon. Sin. 34(6), 852–856 (2005). (In Chinese).

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5(7), 1399–1402 (2005).
[Crossref]

2004 (5)

C. Wang, Q. Li, G. Liu, G. Jin, and X. Xu, “The calculation of switching power of symmetric and asymmetric nonlinear directional couplers with variable coupling coefficient,” IEEE Photonics Technol. Lett. 16(10), 2248–2250 (2004).
[Crossref]

I. Ntakis, P. Pottier, and R. M. De La Rue, “Optimization of transmittance properties of two-dimensional photonic crystal channel waveguide bends through local lattice deformation,” J. Appl. Phys. 96(1), 12–18 (2004).
[Crossref]

Y. A. Vlasov and S. J. Mcnab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express 12(8), 1622–1631 (2004).
[Crossref]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[Crossref]

T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85(24), 5833–5835 (2004).
[Crossref]

2003 (1)

K. Tanaka and M. Tanaka, “Simulations of Nanometric Optical Circuits: Open-Type Surface Plasmon Polariton Gap Waveguide,” Appl. Phys. Lett. 42(6A), L585–L588 (2003).
[Crossref]

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmittance through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[Crossref]

1985 (1)

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3(5), 1135–1146 (1985).
[Crossref]

1975 (1)

1974 (1)

H. F. Taylor and A. Yariv, “Guided wave optics,” Proc. IEEE 62(8), 1044–1060 (1974).
[Crossref]

1973 (1)

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L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5(7), 1399–1402 (2005).
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L. Liu, Z. Li, B. Xu, C. Gu, C. Chen, P. Ning, J. Yan, and X. Chen, “High-efficiency transition between rectangular waveguide and domino plasmonic waveguide,” AIP Adv. 5(2), 027105 (2015).
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J. Yin, J. Ren, H. Zhang, B. Pan, and T. Cui, “Broadband frequency-selective spoof surface plasmon polaritons on ultrathin metallic structure,” Sci. Rep. 5(1), 8165 (2015).
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X. Gao, L. Zhou, and T. Cui, “Odd-mode surface plasmon polaritons supported by complementary plasmonic metamaterial,” Sci. Rep. 5(1), 9250 (2015).
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X. Shen and T. Cui, “Planar plasmonic metamaterial on a thin film with nearly zero thickness,” Appl. Phys. Lett. 102(21), 211909 (2013).
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M. H. Shih, W. J. Kim, W. Kuang, J. R. Cao, S. J. Choi, J. D. O’Brien, and P. D. Dapkus, “Experimental characterization of the reflectance of 60° waveguide bends in photonic crystal waveguides,” Appl. Phys. Lett. 86(19), 191104 (2005).
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I. Ntakis, P. Pottier, and R. M. De La Rue, “Optimization of transmittance properties of two-dimensional photonic crystal channel waveguide bends through local lattice deformation,” J. Appl. Phys. 96(1), 12–18 (2004).
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A. I. Fernández-Domínguez, E. Moreno, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz wedge plasmon polaritons,” Opt. Lett. 34(13), 2063–2065 (2009).
[Crossref]

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J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
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D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Dominguez, F. J. García-Vidal, L. Martín-Moreno, and E. Moreno, “Domino plasmons for subwavelength terahertz circuitry,” Opt. Express 18(2), 754–764 (2010).
[Crossref]

A. I. Fernández-Domínguez, E. Moreno, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz wedge plasmon polaritons,” Opt. Lett. 34(13), 2063–2065 (2009).
[Crossref]

A. I. Fernández-Domínguez, E. Moreno, L. Martín-Moreno, and F. J. García-Vidal, “Guiding terahertz waves along subwavelength channels,” Phys. Rev. B 79(23), 233104 (2009).
[Crossref]

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics 2(3), 175–179 (2008).
[Crossref]

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L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5(7), 1399–1402 (2005).
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M. Ayre, T. J. Karle, L. Wu, T. Davies, and T. F. Krauss, “Experimental verification of numerically optimized photonic crystal injector, Y-splitter, and bend,” IEEE J. Sel. Area. Comm. 23(7), 1390–1395 (2005).
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S. Li, M. M. Jadidi, T. E. Murphy, and G. Kumar, “Terahertz surface plasmon polaritons on a semiconductor surface structured with periodic V-grooves,” Opt. Express 21(6), 7041–7049 (2013).
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T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmittance through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
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C. Wang, Q. Li, G. Liu, G. Jin, and X. Xu, “The calculation of switching power of symmetric and asymmetric nonlinear directional couplers with variable coupling coefficient,” IEEE Photonics Technol. Lett. 16(10), 2248–2250 (2004).
[Crossref]

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Li, W.

Li, Y.

M. Yuan, Y. Li, Y. Lu, Y. Zhang, Z. Zhang, X. Zhang, X. Zhang, J. Han, and W. Zhang, “High-performance and compact broadband terahertz plasmonic waveguide intersection,” Nanophotonics 8(10), 1811–1819 (2019).
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Z. Li, B. Xu, L. Liu, J. Xu, C. Chen, C. Gu, and Y. Zhou, “Localized spoof surface plasmons based on closed subwavelength high contrast gratings: concept and microwave-regime realizations,” Sci. Rep. 6(1), 27158 (2016).
[Crossref]

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[Crossref]

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Z. Li, B. Xu, L. Liu, J. Xu, C. Chen, C. Gu, and Y. Zhou, “Localized spoof surface plasmons based on closed subwavelength high contrast gratings: concept and microwave-regime realizations,” Sci. Rep. 6(1), 27158 (2016).
[Crossref]

L. Liu, Z. Li, B. Xu, C. Gu, C. Chen, P. Ning, J. Yan, and X. Chen, “High-efficiency transition between rectangular waveguide and domino plasmonic waveguide,” AIP Adv. 5(2), 027105 (2015).
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M. Yuan, Y. Li, Y. Lu, Y. Zhang, Z. Zhang, X. Zhang, X. Zhang, J. Han, and W. Zhang, “High-performance and compact broadband terahertz plasmonic waveguide intersection,” Nanophotonics 8(10), 1811–1819 (2019).
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[Crossref]

S. A. Maier, S. R. Andrews, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett. 97(17), 176805 (2006).
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D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Dominguez, F. J. García-Vidal, L. Martín-Moreno, and E. Moreno, “Domino plasmons for subwavelength terahertz circuitry,” Opt. Express 18(2), 754–764 (2010).
[Crossref]

A. I. Fernández-Domínguez, E. Moreno, L. Martín-Moreno, and F. J. García-Vidal, “Guiding terahertz waves along subwavelength channels,” Phys. Rev. B 79(23), 233104 (2009).
[Crossref]

A. I. Fernández-Domínguez, E. Moreno, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz wedge plasmon polaritons,” Opt. Lett. 34(13), 2063–2065 (2009).
[Crossref]

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics 2(3), 175–179 (2008).
[Crossref]

S. A. Maier, S. R. Andrews, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett. 97(17), 176805 (2006).
[Crossref]

F. J. Garcia-Vidal, L. Martín-Moreno, and J. B. Pendry, “Surfaces with holes in them: New plasmonic metamaterials,” J. Opt. A: Pure Appl. Opt. 7(2), S97–S101 (2005).
[Crossref]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
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Mcnab, S. J.

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Moreno, E.

Murphy, T. E.

G. Kumar, S. Li, M. M. Jadidi, and T. E. Murphy, “Terahertz surface plasmon waveguide based on a one-dimensional array of silicon pillars,” New J. Phys. 15(8), 085031 (2013).
[Crossref]

S. Li, M. M. Jadidi, T. E. Murphy, and G. Kumar, “Terahertz surface plasmon polaritons on a semiconductor surface structured with periodic V-grooves,” Opt. Express 21(6), 7041–7049 (2013).
[Crossref]

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Nesterov, M. L.

Nikolajsen, T.

T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85(24), 5833–5835 (2004).
[Crossref]

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L. Liu, Z. Li, B. Xu, C. Gu, C. Chen, P. Ning, J. Yan, and X. Chen, “High-efficiency transition between rectangular waveguide and domino plasmonic waveguide,” AIP Adv. 5(2), 027105 (2015).
[Crossref]

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I. Ntakis, P. Pottier, and R. M. De La Rue, “Optimization of transmittance properties of two-dimensional photonic crystal channel waveguide bends through local lattice deformation,” J. Appl. Phys. 96(1), 12–18 (2004).
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M. H. Shih, W. J. Kim, W. Kuang, J. R. Cao, S. J. Choi, J. D. O’Brien, and P. D. Dapkus, “Experimental characterization of the reflectance of 60° waveguide bends in photonic crystal waveguides,” Appl. Phys. Lett. 86(19), 191104 (2005).
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Ouyang, C.

Pan, B.

J. Yin, J. Ren, H. Zhang, B. Pan, and T. Cui, “Broadband frequency-selective spoof surface plasmon polaritons on ultrathin metallic structure,” Sci. Rep. 5(1), 8165 (2015).
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L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5(7), 1399–1402 (2005).
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F. J. Garcia-Vidal, L. Martín-Moreno, and J. B. Pendry, “Surfaces with holes in them: New plasmonic metamaterials,” J. Opt. A: Pure Appl. Opt. 7(2), S97–S101 (2005).
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J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
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I. Ntakis, P. Pottier, and R. M. De La Rue, “Optimization of transmittance properties of two-dimensional photonic crystal channel waveguide bends through local lattice deformation,” J. Appl. Phys. 96(1), 12–18 (2004).
[Crossref]

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Ren, J.

J. Yin, J. Ren, H. Zhang, B. Pan, and T. Cui, “Broadband frequency-selective spoof surface plasmon polaritons on ultrathin metallic structure,” Sci. Rep. 5(1), 8165 (2015).
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Z. Gao, X. Zhang, and L. Shen, “Wedge mode of spoof surface plasmon polaritons at terahertz frequencies,” J. Appl. Phys. 108(11), 113104 (2010).
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X. Shen and T. Cui, “Planar plasmonic metamaterial on a thin film with nearly zero thickness,” Appl. Phys. Lett. 102(21), 211909 (2013).
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Shih, M. H.

M. H. Shih, W. J. Kim, W. Kuang, J. R. Cao, S. J. Choi, J. D. O’Brien, and P. D. Dapkus, “Experimental characterization of the reflectance of 60° waveguide bends in photonic crystal waveguides,” Appl. Phys. Lett. 86(19), 191104 (2005).
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Y. Zhang, Y. Xu, C. Tian, Q. Xu, X. Zhang, Y. Li, X. Zhang, J. Han, and W. Zhang, “Terahertz spoof surface-plasmon-polariton subwavelength waveguide,” Photonics Res. 6(1), 18–23 (2018).
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Y. Zhang, S. Li, Q. Xu, C. Tian, J. Gu, Y. Li, Z. Tian, C. Ouyang, J. Han, and W. Zhang, “Terahertz surface plasmon polariton waveguiding with periodic metallic cylinders,” Opt. Express 25(13), 14397–14405 (2017).
[Crossref]

Tian, L.

Tian, Z.

van Oosten, D.

Verhagen, E.

Vlasko-Vlasov, V. K.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5(7), 1399–1402 (2005).
[Crossref]

Vlasov, Y. A.

Wang, C.

C. Wang, Q. Li, G. Liu, G. Jin, and X. Xu, “The calculation of switching power of symmetric and asymmetric nonlinear directional couplers with variable coupling coefficient,” IEEE Photonics Technol. Lett. 16(10), 2248–2250 (2004).
[Crossref]

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Welp, U.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5(7), 1399–1402 (2005).
[Crossref]

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L. Ye, Y. Xiao, N. Liu, Z. Song, W. Zhang, and Q. Liu, “Plasmonic waveguide with folded stubs for highly confined terahertz propagation and concentration,” Opt. Express 25(2), 898–906 (2017).
[Crossref]

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[Crossref]

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Z. Li, B. Xu, L. Liu, J. Xu, C. Chen, C. Gu, and Y. Zhou, “Localized spoof surface plasmons based on closed subwavelength high contrast gratings: concept and microwave-regime realizations,” Sci. Rep. 6(1), 27158 (2016).
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[Crossref]

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J. Yin, J. Ren, H. Zhang, B. Pan, and T. Cui, “Broadband frequency-selective spoof surface plasmon polaritons on ultrathin metallic structure,” Sci. Rep. 5(1), 8165 (2015).
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L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5(7), 1399–1402 (2005).
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Zhang, W.

Zhang, X.

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M. Yuan, Y. Li, Y. Lu, Y. Zhang, Z. Zhang, X. Zhang, X. Zhang, J. Han, and W. Zhang, “High-performance and compact broadband terahertz plasmonic waveguide intersection,” Nanophotonics 8(10), 1811–1819 (2019).
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L. Ye, H. Feng, G. Cai, Y. Zhang, B. Yan, and Q. Liu, “High-efficient and low-coupling spoof surface plasmon polaritons enabled by V-shaped microstrips,” Opt. Express 27(16), 22088–22099 (2019).
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Zhou, L.

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Z. Li, B. Xu, L. Liu, J. Xu, C. Chen, C. Gu, and Y. Zhou, “Localized spoof surface plasmons based on closed subwavelength high contrast gratings: concept and microwave-regime realizations,” Sci. Rep. 6(1), 27158 (2016).
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IEEE J. Sel. Area. Comm. (1)

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IEEE Photonics Technol. Lett. (2)

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Nanophotonics (1)

M. Yuan, Y. Li, Y. Lu, Y. Zhang, Z. Zhang, X. Zhang, X. Zhang, J. Han, and W. Zhang, “High-performance and compact broadband terahertz plasmonic waveguide intersection,” Nanophotonics 8(10), 1811–1819 (2019).
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Nat. Photonics (1)

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Nature (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmittance through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
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[Crossref]

L. Ye, Y. Xiao, N. Liu, Z. Song, W. Zhang, and Q. Liu, “Plasmonic waveguide with folded stubs for highly confined terahertz propagation and concentration,” Opt. Express 25(2), 898–906 (2017).
[Crossref]

L. Ye, H. Feng, G. Cai, Y. Zhang, B. Yan, and Q. Liu, “High-efficient and low-coupling spoof surface plasmon polaritons enabled by V-shaped microstrips,” Opt. Express 27(16), 22088–22099 (2019).
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W. Zhu, A. Agrawal, and A. Nahata, “Planar plasmonic terahertz guided-wave devices,” Opt. Express 16(9), 6216–6226 (2008).
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L. Tian, Z. Zhang, J. Liu, K. Zhou, Y. Gao, and S. Liu, “Compact spoof surface plasmon polaritons waveguide drilled with L-shaped grooves,” Opt. Express 24(25), 28693–28703 (2016).
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S. Li, M. M. Jadidi, T. E. Murphy, and G. Kumar, “Terahertz surface plasmon polaritons on a semiconductor surface structured with periodic V-grooves,” Opt. Express 21(6), 7041–7049 (2013).
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Phys. Rev. B (1)

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L. Ye, Y. Xiao, Y. Liu, L. Zhang, G. Cai, and Q. Liu, “Strongly confined spoof surface slasmon solaritons waveguiding enabled by planar staggered plasmonic waveguides,” Sci. Rep. 6(1), 38528 (2016).
[Crossref]

J. Yin, J. Ren, H. Zhang, B. Pan, and T. Cui, “Broadband frequency-selective spoof surface plasmon polaritons on ultrathin metallic structure,” Sci. Rep. 5(1), 8165 (2015).
[Crossref]

X. Gao, L. Zhou, and T. Cui, “Odd-mode surface plasmon polaritons supported by complementary plasmonic metamaterial,” Sci. Rep. 5(1), 9250 (2015).
[Crossref]

Z. Li, B. Xu, L. Liu, J. Xu, C. Chen, C. Gu, and Y. Zhou, “Localized spoof surface plasmons based on closed subwavelength high contrast gratings: concept and microwave-regime realizations,” Sci. Rep. 6(1), 27158 (2016).
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Figures (7)

Fig. 1.
Fig. 1. Structure and dispersion relation. (a) Schematic diagram and geometric parameters of the structure (the red arrow depicts the propagation direction of SPPs). The lower left inset shows the SEM image of the fabricated arc-shaped curved hole array excitation region as well as the fan-shaped coupling region. The lower right inset shows the fabricated waveguide structure. (b) The simulated dispersion relation (red line) of SPP mode for one row of metal pillars.
Fig. 2.
Fig. 2. Performance of bends and waveguide components. (a)-(d) SEM images of structures corresponding to R = 200, 900, 1600, 2300 µm, respectively. (e)-(h) Simulated and (i)-(l) experimental results for normalized power |Ez|2 distributions corresponding to (a)-(d) in a horizontal plane slightly above (at 100 µm) the surface of each structure at 0.56 THz. (m) Optical image of commutator. (n) Simulated and (o) experimental results for normalized power |Ez|2 distributions corresponding to the commutator in (m).
Fig. 3.
Fig. 3. Performance of bends. (a) Transmittance spectra and (b) bending losses as a function of frequency for 90-degree curved bends with different R values from 200 to 2300 µm.
Fig. 4.
Fig. 4. Simulated (red circles) and measured (black circles) bending losses as a function of R at 0.56 THz.
Fig. 5.
Fig. 5. Curved waveguide geometries and analysis. Structures of (a) bend-straight waveguides and (b) bend-bend waveguides. (c) Dispersion relation of even and odd supemodes for waveguides with varying g. The left inset shows normalized electric component (EZ) distributions in the yz cross section for even (upper left) and odd modes (lower left) supported by two parallel waveguides with g = 80 µm at 0.62 THz. The inset on the right indicates the electric field at g = 280 µm. (d) The difference in kx between two supermodes as a function of g from 80 to 280 µm at 0.62 THz.
Fig. 6.
Fig. 6. Normalized powers for (a) bend-straight waveguides and (c) bend-bend waveguides as a function of propagation distance at 0.62 THz. The red dotted lines indicate the output powers from the curved portion when the propagation distances of the SPPs are large enough. (b) and (d) The normalized output powers of two parallel waveguides as a function of L. In all cases, the black solid and red dashed lines represent the normalized powers from waveguides 1 and 2, respectively.
Fig. 7.
Fig. 7. Performance of directional couplers. (a) and (b) SEM images of structures corresponding to bend-straight waveguides and bend-bend waveguides, respectively. (c) and (d) Simulated and (e) and (f) experimental results for normalized power |Ez|2 distributions corresponding to (a) and (b) in a horizontal plane slightly above (at 100 µm) the surface of each structure at 0.62 THz. (g) and (h) Simulated and experimental cross-sectional normalized power distributions at the end of the coupling regions (line x = 4 mm) in (c)-(f).

Equations (9)

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θ i = α i 2 = 1 2 arccos R ( g max g ) R θ j = α j 2 = 1 2 arccos 2 R ( g max g ) 2 R .
g i = g + x i tan θ i g j = g + 2 x j tan θ j .
K i = K 0 exp ( p x i tan θ i ) ,
K j = K 0 exp ( 2 p x j tan θ j ) ,
L c = π k x e k x o = π 2 K 0 ,
K 0 = Δ k  =  k x e k x o 2 .
d A 1 d x = j K 0 A 2 exp ( p x i tan θ i ) d A 2 d x = j K 0 A 1 exp ( p x i tan θ i ) ,
P i1 = cos 2 { K 0 p tan θ i [ 1 exp ( p x i tan θ i ) ] } P i2 = si n 2 { K 0 p tan θ i [ 1 exp ( p x i tan θ i ) ] } .
P j1 = cos 2 { K 0 2 p tan θ j [ 1 exp ( 2 p x j tan θ j ) ] } P j2 = si n 2 { K 0 2 p tan θ j [ 1 exp ( 2 p x j tan θ j ) ] } .

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