1. Introduction

Surface plasmons (SPs) [1,2] are mixed excited state in which free electrons bounded at the interface between metal and dielectric interact with external photons at optical frequencies, which include surface plasmon polarizations (SPPs) in infinite metal surfaces [3,4], and localized surface plasmons (LSPs) in finite metallic particles [5]. In the past decades, SPs have grown into a hot research field due to their exotic optical properties about deep subwavelength field confinement and field enhancement. Therefore, a series of study based on SPs modes has been proposed gradually, such as optical antennas [6,7], surface enhanced Raman scattering [8,9], and plasmonic nanofocusing [10]. However, the metallic will behave kin to perfect electric conductors (PEC) while the frequency of the electromagnetic wave is reduced to the microwave and terahertz regions, which means the metal will do not supports the existence of the SPs modes. In order to achieve the multifarious applications in low frequencies, plasmonic metamaterials have been proposed which named spoof SPPs [11–13]. It is noticeable: the electromagnetic properties of spoof SPPs strongly depend on the engineer geometric parameters, and Pors et al have proposed a periodic textured closed metal surface structure which can supports the spoof LSPs with similar performance of LSPs in optical region firstly [14]. Thence, a series of models have been proposed to check the spoof LSPs modes, such as closed subwavelength high contrast gratings [15], ultrathin fan-shaped metallic strips [16], and ultrathin metallic disks [17]. In addition, structures with metallic spiral also have been demonstrated that supports both electric spoof LSPs and magnetic spoof LSPs in experimentally and theoretically [18]. With the development of the spoof LSPs about theories and technologies, SPs hybridization between adjacent structured particles is study to find the many novel properties at low frequencies [19]. For instance, near field energy transport by a spoof plasmonic meta-dimer [20], corrugated metal disks produce multiple Fano [21], and the metal spiral structure to realize magnetic coupling in deep subwavelength scale [22].

In the microwave and terahertz regions, metal surfaces have been designed to generate various periodic subwavelength holes, grooves, and blocks to guide propagation of spoof SPPs [23–25]. Such as, the grating structures have been proposed to make the electromagnetic waves slowdown, which consist of two parallel metal bands with corrugations strips decorated [26]. Most spoof SPPs structure depend on three-dimensional metal planes, metal thickness constraint (order of λ/4) and larger modal size due to the large unit cell [27], which limits the possibility of practical application. In order to get more applicable spoof SPPs waveguide, in 2013 Cui et al proposed a kind of ultrathin corrugated metallic structure [28], which perform well in structural curve, and rotate. Based on these excellent characteristics, various spoof SPPs modes optical devices have been manufacture in the microwave and terahertz regions [29]. Recently, a subwavelength metal spiral structure localization of the spoof LSPs has been proposed to control the spoof SPPs transmissions by the electromagnetic coupling between the combined systems [30–34].

In this work, we propose a combined structure, which contains a textured closed surface with defect units and a wave vector matching corrugated metallic strips, can realizes multiband modulation of spoof SPPs transmission by tuning defect scales of textured closed surface in terahertz region. We observe that the electromagnetic response of the textured closed surface can be motivated though the spoof SPPs, meanwhile, the defect unit of the spoof localized plasmonic structure can effective captures the spoof SPP transmissions. Furthermore, due to the multiple defect units with different dimensions can trap spoof SPPs at different frequency positions, we further design a textured closed surface structure with multiple defect units to modulate the transmissions of the spoof SPPs. The results provide an efficiently approach for freely tunable the combined system about spoof LSPs - SPPs modal transmissions in terahertz region.

2. Results and discussion

2.1 Electromagnetic property of textured closed surface structure with defect units

We first study a two-dimension (2D) textured prefect electric conductor (PEC) cylinder structure that has a defect unit as show in Fig. 1(a). We can find the structure have two different colors which the yellow represent the metal and it can equivalent to PEC, and the blue represent dielectric substance and the refractive index is 4.2 which close to the germanium. The structural constitute are a cylinder with an outer radius R which inset by textured radial grooves, and depth of the grooves are h = R – r, width is a, and the periodicity about grooves d = 2πR/N, where the r and N are inner radius and the total number of grooves. In addition, we made a defect unit in one of the grooves as show in Fig. 1(b) and marked it as ${R_d}$ which in order to have a detailed description we magnifying the red region of Fig. 1(a). In the following, we chose the structural parameters as N = 30, r = 10um, R = 28um, and a = 0.8d. Noticeably, the structural grooves depth h or the refractive index n of the blue zone should satisfy ${\omega _p} \approx \pi c/({2hn} )$ so that the structure can keep dipole and quadrupole modes [35–39].

 

Fig. 1. (a) A 2D corrugated cylinder structure with a defect unit and the depth of the defect unit can be freely tailored. (b) The enlarging schematic of the red triangle area in (a). The structural outer radius is R, inner radius is r, the groove width about corrugated cylinder is a, and the structural periodicity is d, besides, the depth of the defect unit is R_d. The whole structure is exposed to air.

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In order to investigate the unknown characteristics in the texture structure with defect unit, we consider a beam of the transverse magnetic (TM) polarized plane wave through this texture structure. The field and electromagnetic scattering have been calculated in terahertz region through the finite element method. As we can see in Fig. 2(a), we calculated the textured cylinder structural normalized scattering cross-section (SCS) normalized to the diameter 2R, and the whole structure is exposed to air. In order to explore the influence of different depths R_d on the SCS spectrum, we calculated three different SCS in Fig. 2(a-c) which gradually increasing R_d from 22um to 24um and keep other structural parameters invariant, respectively. As we can see, the SCS spectrums have three peaks in different frequency positions. However, the causes of the formation about the peaks in the blue and red regions are quite different. The two peaks in blue region are electric dipole and electric quadrupole which we named by “ed” and “eq”, respectively. The peak in red region is generated by the defect unit, due to the defect unit trapped the electromagnetic wave of the spoof SPPs transmission. It is obvious find that when we change the structural parameter ${R_d}$ from 22um to 24um alone, the SCS spectrum just moves peak “1” from right to left. It demonstrated that the different structural dimensions of defect unit can regulation peak “1” in the SCS spectrum while keep other peaks constantly. This good phenomenon means that we can freely modulate the structural capture frequency by tailoring the size R_d within limits. Furthermore, we show the structural near field distribution H_z about those three peaks in the SCS spectrum in Fig. 2(d-f) while R_d = 23um and keep other parameters, and they located at 1.163THz, 1.187THz, and 1.329THz from left to right, respectively. We can find the electromagnetic wave have been well localized within the defect unit of the textured cylinder structure in Fig. 2(f).

 

Fig. 2. (a)-(c) Calculated the normalized SCS spectrum of the textured cylinder structure as show in Fig. 1, and the parameter R_d is vary from 18.5um to 19.5um. The different background colors represent the diverse excite modes. (d)-(f) The field distributions Hz about the SCS spectrum in (b) denoted by “ed”, “eq”, and “1”.

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2.2 Controlling transmission of the combined structure

Following, we study a 2D spoof SPPs waveguide as shown in Fig. 3(a) that consists of one single side corrugated metal strips. In order to meet the structural requirements and extract the spoof SPPs wave mode of the corrugated metal strips, we designed two-transition sections structure for smooth conversion to get the spoof SPPs wave mode in THz. As we can see in Fig. 3(a), the structure designed a flaring metal line and gradient metal corrugations. In this paper, the structural geometrical parameters are l₁ = 100um, l₂ = 600um, l₃ = 1230um, and the width about the metal flaring and the metal corrugated band are W = 150um and H = 100um, the distance from the flaring and the metal band is g = 5um. Furthermore, the gradient grooves have different depth which h₁ = 5um, h₂ = 10um, h₃ = 15um, h₄ = 20um, h₅ = 25um, h₆ = 30um, h₇ = 35um and h = 40um, and the whole structure is exposed to air. Then, in order to investigate the structural propagation characteristic about the spoof SPPs mode, we calculate the dispersion relation about the spoof SPPs waveguide in Fig. 3(b). The blue solid line and the black dotted line represent velocity of metal strips structure and light in air, which conform to the propagation constant k_x and k₀, respectively. We can clearly find that the blue line is low than the black line in higher frequencies, but in lower frequencies the line of k_x is close to the k₀, it represent that the electromagnetic wave have extend many wavelengths into air, due to the metal strips structure confinement the spoof SPPs mode is weaker relative to higher frequencies. Consider all possible influences we finally focus on the structural spectrum from 1.1THz to 1.4THz, so that the structure will have a good transmission. The inset figure in Fig. 3(b) is the enlarged of the blue dashed box in Fig. 3(a), as we can see the period of the metal strips is p = 50um and the distance between two metal strips is a = 30um. In the Fig. 3(c-d) we show the structural near field distribution H_z and E_y at 1.3THz so that visually demonstrate the performance of structure propagation. We can clearly find that distribution of magnetic field in Fig. 3(c), and we can find electromagnetic waves are effectively bound to the propagation surface in Fig. 3(d).

 

Fig. 3. (a) The schematic picture about the metal strips structure, the structural length of each part is l₁ = 100um, l₂ = 600um, l₃ = 1230um, and the width about the metal flaring and the metal corrugated band are W = 150um and H = 100um, the distance from the flaring and the metal band is g = 5um. The gradient grooves have different depth which h₁ = 5um, h₂ = 10um, h₃ = 15um, h₄ = 20um, h₅ = 25um, h₆ = 30um, h₇ = 35um and h = 40um. (b) The blue solid line metal strips structural dispersion cure, and the black dashed line represent the light line. The inset figure show the slit width a = 30um, and the periodicity d = 50um. (c)-(d) We show the metal strips structural field distributions H_z and E_y at 1.3THz.

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In the combined system of the spoof SPPs and spoof LSPs, we put the textured cylinder with a defect unit structure on the metal strips which the distance d = 30um in order to explore the characteristic in the combined system as show in Fig. 4(a) of the inset figure. The other parameters remain unchanged. We simulated the transmission coefficient about the combined system as we can see in Fig. 4(a), which the red solid line represent the transmission about the metal strips without the textured cylinder structure. The structural transmission coefficient is about -5 dB, which means that 95% energy have been constraint during this frequencies range. The black solid line in Fig. 4(a) represents the transmission coefficient about the combined system of the spoof SPPs and spoof LSPs. We can find that there are three valleys in the transmission spectrum due to added the textured cylinder structure. It is worth noting that the formation of the three valleys is due to different physical mechanisms, which the valleys “ed” and “eq” are caused due to the coupling of the combined structure. While the excited spoof SPPs waveguide propagation through the textured cylinder structure, it will be coupled with the structural electric dipole and electric quadrupole. Specially, we pay our attention to the third valley “1” of the transmission spectrum, which this valley caused by the defect unit of the textured cylinder structure. Due to the defect unit in the textured cylindrical structure can trapped the spoof SPPs into the deep subwavelength defect unit. In order to further explore the influence of the rotation of the defect unit, we rotated three different angles for comparison. As we can see, while we rotated the defect unit for 45°and -45°, the filtering effect decreases when compared with rotated 0°, because the coupling coefficient decreases when the distance increases. And the transmission spectrum is red-shifted, and the formant is shifted due to the tilt of the defect element. Therefore, as the rotation angle increases, the coupling effect will gradually decrease to zero and no longer have good filtering effect. To show the transmission mode of the combined system, we show the near field distribution ${H_z}$ about the three valleys in the transmission spectrum in Fig. 4(b-d). We can find that the spoof SPPs waveguide have been cut-off while the electromagnetic waves travel through the textured cylinder structure at 1.163THz, 1.187THz, and 1.329THz from left to right, respectively. We further show the modes within the red dotted line for better visibility in Fig. 4(e-g). Contrast the coupling modes of the combined system about the valleys Fig. 4(e-f) with the textured cylinder structural electromagnetic resonance as show in Fig. 2(d-e). We can observe that the textured cylinder structural electric dipole and electric quadrupole modes have been motivated by the spoof SPPs waveguide in the system. Moreover, in Fig. 4(g) we find that the wave have been trapped in the defect unit, it means the defect unit can tuned the spoof SPPs waveguide mode effectively. Based on this result, we may find a most effective way to modulate the wave propagation between the spoof SPPs mode and spoof LSPs mode.

 

Fig. 4. (a) Simulated the transmission coefficients about the spoof SPPs and spoof LSPs combination system, the inset schematic represent the system. The red solid line represents the transmission spectrum without defect unit while the black solid line represents the transmission spectrum of the whole combined system. The inset transmission spectrums explore the influence of the rotation of the defect unit. (b)-(d) We describe the filed distributions H_z of the combined system to show the transmission effect at three valleys. (e)-(g) We enlarge the modes in the red dashed box in (b)-(d) for better visibility, the modes about the combined system can be clearly observed at the corresponding frequency.

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We have discussed that the textured cylinder structure with multiple defect units can trapped multiple different frequency bands light in to the defect units. In order to find the detailed relationship between the structural defect units and the location of the capture frequency, we calculated the normalized SCS spectrum about the textured cylinder structure with different dimensis defect units. The textured cylinder structural geometrical parameters are N = 30, r = 10um, R = 33um, and a = 0.8d, for better presentation we choose the structural outside radius R = 33um and inside radius r = 10um, we designed three defect units which have different depth so that to explore the depth factors. As we can see inset figure in the Fig. 5(a), we enlarged the design of the defect units for better visibility. And the depth ${R_d}$ of the defect units are 24um, 22um, and 20um which conform to the label “1”, “2”, and “3” in Fig. 5(a), respectively. Observed the normalized SCS spectrum in Fig. 5(a) we can find five peaks, and exclude those caused by the electric dipole and electric quadrupole modes of “ed” and “eq”. We find three peaks caused by the structural defect units as show in the blue dotted box in Fig. 5(a). Through the normalized SCS spectrum and the structural defect units, we can find that R_d = 24um correspond to peak “1” at 1.15THz, R_d = 22um correspond to peak “2” at 1.24THz, and R_d = 20um correspond to peak “3” at 1.33THz. It represent that the different dimensions defect units can trapped different frequencies waves, and compared to the shorter depth defect unit, the longer depth defect unit ${R_d}$. can capture waves at lower frequencies in a range. The result may provide a good way for us to design a device to multiple modulate in the combined system. So that we put the textured cylinder structure which have multiple defect units above the metal strips, and the distance from the textured cylinder structure and the metal strips is d = 30um. The schematic diagram we can find in the inset figure in the Fig. 5(b). Then we calculated the transmission spectrum of the combined system as show in Fig. 5(b). We observed the transmission spectrum find that there are three valleys in the spectrum. It caused by the trapped through the three defect units. To prove the point, we further show the near field distribution H_z of the textured cylinder structure in the combined system as show in Fig. 5(c). The spoof SPPs waveguide has been trapped by the defect units at different frequencies. As we can see the spoof SPPs mode is trapped into the depth R_d = 24um at 1.15THz, trapped into the depth R_d = 22um at 1.24THz, and trapped into the depth R_d = 20um at 1.33THz in Fig. 5(c) from left to right, respectively. Moreover, we further show the combined structural transmission filed distributions H_z in Fig. 5(d), so that we can clearly observed the defect units effectively multiband modulated the spoof SPPs transmission.

 

Fig. 5. (a) Calculated the normalized SCS spectrum of the textured cylinder structure which have three defect units, the inset figure show the schematic about the spoof LSPs structure. (b) The transmission coefficients about the combined system which we designed three defect units in the system to realize multi-frequency control, the inset figure show the combined system which have three defect units. (c) Further gives the filed distributions H_z under the trapped frequencies points in (b). (d) The transmission filed distributions H_z of the combined structure under the trapped frequencies points.

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3. Conclusions

In summary, we have proposed combined structure about spoof SPPs and spoof LSPs, which the spoof SPPs waveguide can be freely tuned by multiple defect units in the textured cylinder structure. We numerically show that the trapped spoof LSPs resonances can be excited by the spoof SPPs modes at terahertz frequencies. Furthermore, the number of trapped frequencies points and the position of the trapped frequencies points can be freely modulate by changing the number and tailor the dimension of the defect units. Utilizing this combined structure can easily modulate the spoof SPPs due to the structural simple, and flexible. So that, we can easily tailor the parameter of the combined structure, and adjust the electrical and magnetic positions to achieve Fano and electromagnetically induced transparency (EIT). The higher modulate efficiency of the combined system about spoof SPPs and spoof LSPs may provide potential applications for plasmonic sensors, circuitry and filters in terahertz region.