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We proposed a novel planar terahertz (THz) plasmonic waveguide with folded stub arrays to achieve excellent terahertz propagation performance with tight field confinement and compact size based on the concept of spoof surface plasmon polaritons (spoof SPPs). It is found that the waveguide propagation characteristics can be directly manipulated by increasing the length of the folded stubs without increasing its lateral dimension, which exhibits much lower asymptotic frequency of the dispersion relation and even tighter terahertz field confinement than conventional plasmonic waveguides with rectangular stub arrays. Based on this waveguiding scheme, a terahertz concentrator with gradual step-length folded stubs is proposed to achieve high terahertz field enhancement, and an enhancement factor greater than 20 is demonstrated. This work offers a new perspective on very confined terahertz propagation and concentration, which may have promising potential applications in various integrated terahertz plasmonic circuits and devices, terahertz sensing and terahertz nonlinear optics.
© 2017 Optical Society of America
1. Introduction
Surface plasmon polaritons (SPPs) sustained at the interface between a conductor and a dielectric have the ability to propagate subwavelength spatially confined electromagnetic wave at the optical frequencies [1–3]. Due to their extraordinary characteristics, SPPs offer an efficient solution to overcome the diffraction limit and demonstrate great potential applications in ultra-compact photonic circuits and devices, near-field optics, nano-imaging, optical sensing, detecting, etc [4–8]. In order to introduce such advantages to terahertz regime, various plasmonic waveguides such as metal wire waveguides, planar Goubau lines, waged waveguides, and parallel-plate waveguides have been investigated in recent years [9–13]. However, since the operating frequency is much lower than the metal’s plasma frequency, terahertz SPPs are usually weakly bounded on the waveguide’s surface, which results in large radiation and bend loss and, in turn, severely limits the efficiency and viability of such concepts [14]. Recently, it is demonstrated that bulk metal plasmonic waveguides tailored with subwavelength one-dimensional grooves or two-dimensional holes can efficiently support tightly confined microwave and terahertz spoof SPPs, which can resemble the behavior of SPPs at optical frequencies [14–17]. To further reduce the dimension and ease the complexity in the fabrication process, periodic corrugated planar plasmonic waveguides have attracted more and more attentions [18–21]. The typical structures of the planar plasmonic waveguides are composed of a single rectangular stub array or dual rectangular stub arrays attached to the ultrathin metallic strip printed on the flexible dielectric substrates, such as conformal, symmetrical and staggered plasmonic waveguides. It is demonstrated that the dispersion characteristics of these waveguides can be tuned by the lateral height of the rectangular stubs, which offers an efficient route to manipulating the spoof SPP propagation properties at different frequencies. Based on the spoof SPP concept, various promising planar plasmonic circuits and devices including filters, splitters, antennas, amplifiers, and sensors have been developed for the microwave and terahertz applications [22–26]. However, because the asymptotic frequencies of the spoof SPPs are usually reciprocal to the lateral height of the stubs, there is still a contradiction between field confinement and waveguide’s lateral dimension. In an attempt to meet the compelling need for efficient and compact terahertz plasmonic waveguides, investigation of novel planar plasmonic waveguides with lower asymptotic frequency, higher field confinement, and smaller structure dimension has sparked keen interests, which has become a hot research focus in terahertz science and technology.
In this paper, a novel plasmonic waveguide with folded stubs is proposed for excellent terahertz propagation and concentration performance with tight field confinement and compact size. The structure consists of two arrays of metallic folded stubs symmetrically attached to the metallic strip of a planar Goubau line. Compared to the conventional planar plasmonic waveguides with rectangular stubs, the proposed waveguide’s dispersion relation of the spoof SPP mode exhibits much lower asymptotic frequency, implying much tighter terahertz field confinement. As one of the most exciting characteristics, the dispersion relation of this waveguide can be directly manipulated by increasing the length of the folded stubs without increasing its lateral dimension. Based on this waveguide structure, a terahertz concentrator with step-length folded stubs under same lateral dimension is presented, and an increase up to 21 times terahertz field amplitude enhancement is achieved. This work offers a new perspective on very confined terahertz propagation and concentration, which may have promising potential applications in various integrated terahertz plasmonic circuits and devices, terahertz sensing and terahertz nonlinear optics.
The paper is organized as follows. Section 2 presents the dispersion characteristics of the fundamental spoof SPP mode for the proposed plasmonic waveguide with folded stubs. Subsequently, Section 3 demonstrates the terahertz concentrator with step-length folded stubs. Finally, conclusions are drawn in Section 4.
2. Dispersion characteristics of the plasmonic waveguides with folded stubs
The proposed planar terahertz plasmonic waveguide is composed of dual arrays of metallic folded stubs symmetrically attached to a central ultrathin metallic strip printed on a dielectric substrate. Here, we begin by considering a plasmonic waveguide with periodic L-shape-folded stubs in the big folded structure family. The schematic configuration of this waveguide unit is illustrated in Fig. 1(c), which can be regarded as a variation from a conventional corrugated planar plasmonic waveguide Fig. 1(a) by tailoring most of the rectangular stub area or Fig. 1(b) by adding an extended cap stub part parallel to the center metallic strip. We assume the metallic strip is copper with the conductivity of 5.8 × 107 S/m and the thickness of 1 µm, and the dielectric substrate is Rogers 5880 with the relative dielectric constant (εr) of 2.2, the loss tangent of 0.0009, and the thickness of 10 µm. The detailed geometrical parameters are illustrated in Fig. 1(c), where the period is d, the width of the center strip is b, the base-width of folded stub is g, the parallel length and lateral height of the 90° rotated letter L are a and h, respectively. It is found that the asymptotic frequency of the fundamental spoof SPP mode is mainly determined by the height of the stubs for the conventional plasmonic waveguide due to the lowest order resonance [18]. While, for the proposed plasmonic waveguides with folded stubs, the asymptotic frequency is mainly dictated by the total length of the folded stubs, which can be approximated as the effective stub height for the conventional plasmonic waveguides. In this case, the sum of the length of the L-shape-folded stub, a + h, plays a significant role in defining the asymptotic frequency, while other geometrical parameters like g and b play a minor role. Note that, by introducing the folded stubs, the proposed plasmonic waveguide shown in Fig. 1(c) adds more tunable geometrical parameters, resulting in more design flexibility compared to the conventional ones. More importantly, we can improve the propagation properties of the spoof SPPs with tighter confinement by increasing the length of the folded stub without changing the unit dimension of the waveguide.
Fig. 1 Configurations of different waveguide units: (a) conventional plasmonic waveguides with wide rectangular stubs, (b) conventional plasmonic waveguides with narrow rectangular stubs and (c) the proposed planar plasmonic waveguide with L-shape-folded stubs, where d is the period of the units, b is the width of the center metallic strip, g is the base-width of folded stub, a and h are the parallel length and the lateral height of an L-shape folded stubs, respectively.
The dispersion characteristics of the fundamental spoof SPP mode for the proposed plasmonic waveguides with L-shape-folded stubs are studied and compared with the conventional plasmonic waveguides with rectangular stubs. In the numerical simulations, we use the eigenmode solver based on the finite element method to simulate the dispersion curves by calculating the eigenfrequencies frequencies of the waveguide unit structures with different phase shifts between the periodic boundaries in x-direction. Because the eigenmode Solver does not support open boundaries and lossy metal materials, the boundaries in y and z directions are all set as PEC at enough large distance (over 10λ0 at the asymptotic frequency) from the unit structure to approximately simulate the real space and the copper is treated as PEC in these calculations. As shown in Fig. 2(a), the dispersion curves of the proposed waveguides with folded stubs are calculated for different values of a and h, where the curves with solid marks result from a = 6 µm, 8 µm, 10 µm while keeping h = 9 µm fixed, the curves with hollow marks result from h = 7 µm, 9 µm, 11 µm while keeping a = 14 µm fixed, and d = 20 µm, b = 4 µm, g = 1 µm for all curves. Obviously, all the dispersion curves deviate far away from the light line, which resembles SPPs in the optical regime. As observed in this figure, the asymptotic frequencies are dependent on the total length (a + h) of the folded stubs. When a + h increases from 15 µm to 25 µm, the asymptotic frequency decreases from 3.3 THz to 2.17 THz. In other words, the longer total length (a + h) of the folded stub is, the lower asymptotic frequency becomes, indicating even tighter spoof SPP field confinement. In order to gain insight into how the stub shapes affect the dispersion characteristics, we compare the dispersion curves among the three different waveguides shown in Fig. 1. The parallel length and the lateral height of the stub are chosen as a = 14 µm, h = 9 µm, respectively, while keeping other geometrical parameters all the same as Fig. 2(a) for three unit models. As illustrated in Fig. 2(b), the asymptotic frequencies for the model in Figs. 1(a), 1(b) and 1(c) are 4.2 THz, 3.57 THz and 2.38 THz, respectively. It is remarkable that the dispersion curve of the proposed structure has much lower asymptotic frequency, which means the larger wave number and stronger field confinement at the same operating frequency. When the lateral heights of the conventional spoof SPP waveguides in Figs. 1(a) and 1(b) are adjusted to h′ = 20 µm, their dispersion curves (as depicted by the orange line with hollow triangle and blue line with solid triangle) become close to the proposed waveguide (h = 9 µm) dispersion curve (as depicted by the red dotted line with solid circle) with the same asymptotic frequency of 2.38 THz. In this case, the total lateral width (2h′ + b) of the conventional waveguides with rectangular stubs is up to 44 µm, while the total lateral width (2h + b) of the proposed one is only 22 µm, which means the proposed waveguide with folded stubs can achieve about 50% size decrease compared with the conventional ones. Therefore, the proposed waveguides with folded stubs can be used to build more compact planar plasmonic devices and circuits at terahertz frequencies.
Fig. 2 (a) Dispersion curves for the fundamental SPP mode of the proposed waveguides with different values of (a, h) are displayed, where the geometrical parameters are set as d = 20 µm, t = 1 µm, g = 1 µm, b = 4 µm. The total length (a + h) of the folded stubs increases gradually from 15 µm to 25 µm, leading to the reduce of the asymptotic frequency from 3.3 THz to 2.17 THz. (b) The comparison among the three kinds of waveguides with same geometrical parameters as in Fig. 2(a) and set a = 14 µm. When the heights of the conventional waveguides are first set as h = 9 µm, the proposed waveguide demonstrates much lower asymptotic frequency than the conventional ones. Then the heights of conventional waveguides are adjusted to h′ = 20 µm, their dispersion curves become close to the proposed one with h = 9 µm, which implies that the proposed waveguide can realize about 50% size decrease.
Next, we further study the dispersion characteristics of the proposed waveguides with different folded stub structures. Since the asymptotic frequency is mainly dictated by the effective length of the folded stubs for the fundamental spoof SPP mode resonance, we may obtain even lower asymptotic frequency by extending longer folded stub part or introducing various folded transformations to increase such effective resonance length while maintaining the lateral size and waveguide unit dimension unchanged. For example, as shown in the insets in Fig. 3, two new plasmonic waveguides with different stub structures: double-L-shape-folded stubs (model B) and triple-L-shape-folded stubs (model C) are presented and compared with plasmonic waveguide with single-L-shape-folded stubs (model A). The geometrical parameters of the common parts of the three models are exactly the same as the Fig. 2(b) with a = 14 µm and h = 9 µm, while h1 = 6 µm and a1 = 10 µm for the model B, and h2 = 4 µm and a2 = 11 µm for the model C. The dispersion curves are displayed in Fig. 3, where the blue curve with hollow rectangle, the green curve with solid circle and the red curve with hollow triangle illustrate the dispersion curves for model A, model B, and model C, respectively. It is found that the asymptotic frequencies drop drastically from 2.38 THz (model A) to much lower frequency at 1.7 THz (model B) and even lower frequency at 1.57 THz (model C) under the same waveguide unit dimension, resulting in much stronger subwavelength field confinement in the latter two models. Clearly, the dispersion characteristics can be easily manipulated by the proposed waveguides with different folded stubs, which can be directly predicted from the effective spoof SPP mode resonance length change. It is important to note that the tunability of the spoof SPP dispersion relation enables the possibility of deep subwavelength field concentration in the proposed plasmonic waveguides with gradual step-length folded stubs under the same lateral waveguide dimension.
Fig. 3 Dispersion curves for the proposed plasmonic waveguides with different folded stub structures shown in the insets, which are the waveguides with single-L-shape-folded stubs (model A), double-L-shape-folded stubs (model B) and triple-L-shape-folded stubs (model C). The geometrical parameters: h = 9 µm, a = 14 µm, h1 = 6 µm, a1 = 10 µm, h2 = 4 µm, and a2 = 11 µm.
3. Terahertz concentrator with step-length folded stubs
Apart from the dispersion characteristics, the subwavelength confinement and concentration of the spoof SPPs are another two of the most important characteristics of the proposed plasmonic waveguides. To intuitively view these characteristics, we use the frequency domain simulations to study the field distributions of the waveguides with multi-unit structures. In these simulations, open boundaries are applied in all directions to simulate the real space and avoid spurious reflections instead of the periodic and PEC boundaries in the eigenmode simulations, and a waveguide port is assigned to the left-side of such waveguides as an excitation source. Meanwhile, the copper and Rogers 5880 are treat as lossy materials. Here, we first compare the electric field distributions of terahertz spoof SPPs for the conventional plasmonic waveguides with rectangular stubs and the proposed plasmonic waveguide with L-shape-folded stubs. All the three waveguides are composed of 16 periods of units with a total length of 320 µm, and the parameters of the unit dimensions are the same as those presented above, as well as using a = 14 µm, h = 9 µm and g = 1 µm for both the rectangular stubs and folded stubs. Figure 4 displays the simulated electric field amplitude (|E|) distributions for the plasmonic waveguides with wide rectangular stubs, narrow rectangular stubs and L-shape-folded stubs at 2.2 THz, where Figs. 4(a), 4(b) and 4(c) are the normalized electric field distributions on the xoy plane with z = 0.5 µm (cut in the middle of the copper strips), Figs. 4(d), 4(e) and 4(f) are the normalized electric field distributions on the cross-sectional yoz plane (cut along the vertical lines shown in the Figs. 4(a), 4(b) and 4(c)), and Figs. 4(h), 4(i) and 4(j) are the electric field amplitudes along the horizontal lines cut on the stub tips shown in the Figs. 4(d), 4(e) and 4(f), respectively. It is observed that all the waveguides exhibit tight subwavelength confinement characteristics of the terahertz spoof SPPs. And the electric field distributions are particularly strong on the stub tips. It is remarkable that the proposed waveguide demonstrates huge stronger confinement with higher electric field amplitude value and that decay sharply away from metallic strip compared with the conventional waveguides. Such phenomenon mainly results from the wave vector at 2.2 THz for the proposed waveguide is much larger than the conventional ones, as shown in Fig. 2(b), which is expected from our previous discussion. This comparison approves that the plasmonic waveguide with folded stubs exhibits a fascinating confinement performance.
Fig. 4 The electric field amplitude distributions of terahertz spoof SPPs for the plasmonic waveguides with wide rectangular stubs (a), (d), & (h), narrow rectangular stubs (b), (e), & (i), and L-shape-folded stubs (c), (f), & (j) at 2.2 THz, where (a), (b) and (c) are the normalized electric field distributions on the xoy plane with z = 0.5 µm (cut in the middle of the copper strips), (d), (e) and (f) are the normalized electric field distributions on the cross-sectional yoz plane (cut along the vertical lines shown in (a), (b) and (c)), and (h), (i) and (j) are the electric field amplitudes along the horizontal cut lines shown in the (d), (e) and (f), respectively.
As discussed above, the dispersion characteristics of the proposed waveguide can be directly manipulated without changing its lateral dimension. Based on this waveguiding scheme, a terahertz concentrator using the proposed waveguide with gradual step-length folded stubs is proposed and investigated to further validate the concentration characteristics. This concentrator is also composed of 16 periods of units with gradually increased step-length folded stubs, and the total length is 320 µm. The parameters of the unit dimensions are as same as those presented above, and the length of the folded-stub from the narrow rectangular stub at the beginning to the double-L-shape-folded stubs at the end is gradually increased. The concentrator is divided into three regions, where the first 6 unit elements belong to Region I, the second 5 unit elements belong to Region II, and the last 5 unit elements belong to Region III. As shown in Fig. 5(a), when propagating along this concentrator, the terahertz spoof SPPs at 1.6 THz become increasingly confined to metallic strip. Figure 5(b) shows the normalized wave vector kx/k0 along the whole concentrator, which is also divided into three regions corresponding to the concentrator. The larger kx/k0 is, the lower asymptotic frequency becomes, resulting in higher terahertz confinement and concentration. Therefore, we can concentrate terahertz wave along this waveguide device to achieve high terahertz field enhancement. It is clearly observed from the terahertz spoof SPP field distributions at 1.6 THz on the yoz plane with x = 60 µm, 180 µm, and 300 µm in Fig. 5(c). To intuitively view the field enhancement characteristics, the normalized peak electric field Ep/Ep0 along the concentrator in the x-direction is presented in Fig. 5(d), where Ep (x) and Ep0 are the sampled maximum electric amplitude values from a series of cross-sectional field distribution plots at position x and at the beginning of the concentrator at 1.6 THz. It should be pointed out that the normalized peak electric field Ep/Ep0 along x direction will fluctuate greatly due to the longitudinal heterogeneity in this waveguide structure. Here, we just simply connected the sampled points to provide a general field enhancement trend prediction for the proposed terahertz concentrator. As shown in this figure, it is found that the normalized peak field increases gradually during region I and region II, and increases obviously faster in region III, which is expected from Fig. 5(b). A field enhancement factor of up to 21 times of terahertz spoof SPPs is achieved at the end of this concentrator. Note that, tight subwavelength terahertz concentration with higher enhancement factor can be obtained by optimizing the length and the shape of the concentrator’s folded stubs, which may have extensive potential in nonlinear terahertz phenomena and terahertz sensing applications.
Fig. 5 The properties of terahertz spoof SPPs propagating along the proposed concentrator: (a) the normalized electric field amplitude distributions on the xoy plane with z = 0.5 µm at 1.6 THz, (b) the normalized wave vector kx/k0, (c) the electric field distributions on the yoz plane with x = 60, 180, and 300 µm at 1.6 THz, and (d) the normalized peak electric field Ep/Ep0. The concentrator is composed of 16 periods of units with gradually increased step-length folded stubs, and the total length is of 320 µm. The parameters of the unit dimensions are as same as those presented above, and the stub length is gradually increased from the narrow rectangular stub at the beginning to double-L-shape-folded stubs at the end.
4. Conclusion
In this work, an efficient terahertz plasmonic waveguide with folded stub arrays is proposed and analyzed, based on the concept of spoof SPPs. The investigated results show that the proposed waveguide has an excellent terahertz propagation performance with tight field confinement and compact size. It is found that the propagation characteristics of the guided spoof SPPs can be directly manipulated by increasing the length of the folded stubs without increasing the waveguide’s lateral dimension. Compared to the conventional plasmonic waveguides with rectangular stub arrays, the proposed waveguide exhibits much lower asymptotic frequency of the dispersion relation and much tighter terahertz field confinement. Based on this waveguide structure, a terahertz concentrator with gradual step-length folded stubs is proposed to achieve high terahertz field enhancement. The investigated results show that an enhancement factor of up to 21 times of spoof SPP electric field amplitude is obtained at the end of a 16-unit step-length folded stub concentrator. Such highly confined terahertz propagation and concentration characteristics can be potentially applied in the exploration of various integrated terahertz plasmonic circuits and devices, terahertz sensing and terahertz nonlinear optics. The spoof SPP waveguiding scheme of the proposed waveguide can be easily extended to microwave and inferred regimes.
Funding
National Natural Science Foundation of China (NSFC) (61601393, 11501481, 11504305); Natural Science Foundation of Fujian Province of China (2016J01321); Natural Science Foundation of Guangdong Province of China (2015A030310009, 2016A030310372).
Acknowledgments
The authors thank Dr. Xiong Xu, Dr. Xiaofan Yang, and Prof. Fei Shen for fruitful discussions.
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