1. Introduction

Guiding terahertz (THz) waves is a challenge for the applications in communications, sensing, as well as imaging due to the high absorption from water vapor [1–3]. Also, most THz systems are based on bulky free-space optics, which needs precise alignment and servicing.

To address these issues, one of the solutions, as distinctly demonstrated at microwave and IR bands, is to replace the free-space THz waves with an optical waveguide [4]. The waveguide, for example, optical fiber, is a structure that supports waves guiding with minimal loss of energy, which has been exploited as sensing and imaging probes and widely used in communication technology. By using THz waveguides, THz waves can be easily trapped in the subwavelength region, offering tightly confinement of the electromagnetic waves to the structure beyond the Rayleigh range [4]. Besides, the high confinement mode could be explored to enhance the light-matter interaction, which is beneficial for THz sensing applications, specifically for label-free and non-destructive detection [5,6]. However, a primary bottleneck for THz waveguides is the lack of suitable material [4,7–11]. Thus, selecting an appropriate material for the waveguide is in urgent need. To this end, different waveguides based on dielectric and metal have been proposed [12–29]. Numerous researches are focused on dielectric waveguides perhaps due to the giant variety of available materials and the greater flexibility in waveguide designing [12–23]. Typical examples of dielectric waveguides such as solid-core, hollow-core, and porous-core waveguides are proposed. For instance, a hollow-core (or pipe) waveguide based on Teflon has been demonstrated, which realizes a loss less than 0.02 cm⁻¹ from 0.34 to 0.53 THz [21 ].

Different from dielectric, metal exhibit huge conductivity in THz range [27–29]. Metallic waveguides are considered as a great candidate for THz guiding because the metal can be regarded as a perfect electronic conductor (PEC). The metal-air device with low-loss and low-dispersion was intensively studied for waveguiding [27]. In contrast to conventional plasmonic waveguides with additional couplers, free-standing photonic crystals (PCs) are attractive alternatives that offer further advantages of compact size [30–39]. Owing to the existing surface plasmon polaritons (SPPs), metallic photonic crystals enable to confine THz waves in the sub-wavelength scale [38,39]. A photonic crystal based on metal rod array has been experimentally demonstrated in 0.1–1 THz, which shows a low waveguide propagation loss of 0.03 cm⁻¹ [38]. By optimizing structural geometry, the loss and field confinement of MRAs can be improved [39]. However, it is still a challenge to fabricate a miniature THz photonic crystal device. Conventional electron beam lithography and direct laser writing are limited in their capabilities to produce structures in miniature. The procedures of lithography are complicated and time-consuming, which need an additional mask and etch chemical materials [38,40]. As a result, the experimental work of geometry-dependent transmission properties of metallic photonic crystals has not been fully explored. Fortunately, a relatively simple fabrication of 3D printing with low-cost and timeless has been proposed. 3D printers based on stereolithography (SLA) and digital light processing (DLP) with high accuracy are widely used to fabricate THz waveguides and free-standing structures [41–43].

In this work, we demonstrated one metallic photonic crystal composed of metal rod arrays (MRAs) fabricated by 3D printers based on digital light processing (DLP). Such waveguides exhibit a broad bandgap and two resonant modes of the fundamental and high-order modes. The high-order mode shows higher field confinement and more sensitive to the geometry changes in contrast to the fundamental mode. We can break the symmetry of MRAs by changing the alignment of metal rods, for example, increases or decreases the metal rod interspace. With the increase of the interspace of metal rods, the spectral positions, bandwidths, as well as the transmittances of high-order modes can be optimized, while the symmetry of MRA is broken. In the first section, several symmetric MRAs are demonstrated to study the transmission properties of THz waves. The second section mentioned the simulated and measured results for asymmetric MRAs. Experimental and simulated results showing that fine-tuning in the alignment of metal rods will lead to a great change in the transmission of high-order modes. These findings suggest that the transportation efficiency of THz waves through an MRA is tunable by breaking the structural symmetry.

2. Configuration of 3D printed metal rod arrays

The MRA model configuration is schematically depicted in Fig. 1(a). The MRA consists of a periodically arranged uniform metal rod of 9 lines and 5 layers. The metal rod has an identified height h and diameter D. The period of the MRA (Λ) is determined from the rod diameter D and the air interspace G. The period between rods in the X- and Y-axis are defined as Λ_x and Λ_y. The top-view photos of fabricated MRAs shown in Fig. 1(b). The schematic diagram of the experimental system is shown in Fig. 1(c), where the focused beam size is about 2 mm. In the edge-coupled configuration, the 2 mm-high MRAs are sufficiently thick because the focused THz wave spot is nearly diffraction limited [38–40]. The fabrication process of the device is described as follows. Firstly, a resin (405 nm UV-resin) structure is fabricated using a 3D printer (ANYCUBIC Photon, UV-LED-405 nm) with a transverse resolution of 47 µm and a longitudinal resolution of 1.25 µm (i.e., along the structure height). After the resin structure is developed in the 99% ethanol and solidified by a UV LED, the solid resin rod array is obtained. The solid resin rod array is then coated with a silver metal thickness of about 200 nm, using a sputter coating system, to obtain the metal rod array (MRA). A sputter coating system (CFS-4EP-LL) at room temperature is used for the metal coating. The sample is fixed on the target (target substrate distance is about 8 cm)). As for the sputtering conditions, the sputtering power of 300 W is applied for silver (1 nm s⁻¹). The thickness of the sputtered material deposited on the sample is directly proportional to the sputtering time. In this condition, the sputtering time is set as 213 s. Therefore, considering the error, the coated silver thickness is about 200 nm. The silver thickness is larger than the skin depth of 1 THz wave (around 100 nm). Simulations of the structures are mainly performed by using the finite-difference time-domain (FDTD) method. In the simulation, the perfectly matched layer is applied in the X-, Y- and Z-directions. The mesh size in X-, Y- and Z-directions is respectively set as 0.01 mm, 0.01 mm, and 0.02 mm.

 

Fig. 1. (a) The configuration of 3D printed metal rod arrays. (b) The microscopic photos for experimental samples. (c) The schematic diagram of the experimental system, where the focused beam size is about 2 mm.

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3. Results and discussion

3.1 Transmission properties of symmetric MRAs

In this section, we investigated the transmission properties of symmetric MRAs. The Fig. 10 in Appendix A presented the measured structural parameter of four samples. Noted that the period in X- and Y- direction is not consistent (${\mathrm{\Lambda }_x} \ne {\mathrm{\Lambda }_y}$) because of the low accuracy of the 3D printer, which suggests that these MRAs are not fully symmetric waveguides. The waveforms of the transmitted THz waves along the metal rod array at edge-coupling can be reliably measured by a terahertz time-domain spectroscopy (THz-TDS) based on Ti: Sapphire laser (780 nm). Figures 2(a-d) show the transmission spectra for samples A, B, C, and D, respectively. One noticeable Bragg reflection band occurs in the experimental and simulated transmission spectra, which is also found in the photonic crystal [24]. Indicated that 5 layers of MRA behave as a THz photonic crystal. Two primary transmission bands are separating by the 1st order Bragg bandgap, in comparison to the higher-frequency transmission bands having the lowest transmittance. The experimental results agree well with that of simulated results using FDTD in which the little discrepancy between experimental and simulated transmission spectra results from the un-uniform rods and rough metal surface. It is noted that the difference between experimental and simulation results in high-frequency regions comes from the imperfectness in the metal coating process. Means that the thickness of coated metal is not uniform in MRAs. In other words, the MRA having a rough metal surface. As a result, the scattering loss of the rod array is high in high-frequency regions. But in the simulation, the surface of the metal rod is perfect, which is different from the experimental sample.

 

Fig. 2. The experimental and simulated transmission spectra for symmetric MRAs.

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In fact, the widths and positions of the photonic bandgaps are determined by the geometrical parameters of the MRA waveguide. The center frequency of Bragg bandgap is calculated by the equation f_B=mC/2n_effΛ, where C, n_eff, and Λ are the light speed in vacuum, effective refractive index, and the period of metallic photonic crystal, respectively. The center frequency of bandgap is tightly correlated with the period in the propagation direction, where the period in propagation, in this case, is termed as ${\mathrm{\Lambda }_y}$. With increasing ${\mathrm{\Lambda }_y}$, the center frequency of bandgap shifts to lower frequencies, where the simulated and theoretical results agree well with that of experimental (Fig. 3(a)). It is noted that the bandgap position can be easily tuned by varying the period of MRA. The bandwidth of bandgap for various samples is shown in Fig. 3(b). Sample B realizes a sizeable photonic bandgap due to the lower porosity (period/diameter), the maximum bandgap equals to 90 GHz, corresponding to 44% bandgap ratio in the vicinity of 0.196 THz. The bandgap ratio of this compact metal rod array-based waveguide is larger than that of 3D printed hollow-core waveguides having a diameter of 20 mm [43].

 

Fig. 3. The Bragg bandgap center frequency (a) and bandgap width (b) with the changing of ${\mathrm{\Lambda }_y}$.

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In order to investigate the resonant behavior with the number of stacked layers, 1-,2-,3-, and 4-layer MRA with the same geometry have been studied for comparative analysis. The transmission spectra for different layers symmetric MRAs are shown in Fig. 4(a). With increasing MRA layers, the bandgap becomes obvious. For example, the transmittance at 0.2 THz is lower than 0.01 when the layer number is five. Compared to the lower-frequency band, the transmittance of high-frequency transmission band around 0.388 THz decreases. The first and second band peak respectively at 0.121 and 0.257 THz exhibits unapparent transmittance changes. The fundamental mode is at 0.121 THz, modes at 0.257 THz and 0.388 THz are both high-order modes. Indicated that the high-order mode around 0.388 THz has larger scattering loss compared to 0.121 THz and 0.257 THz. Besides, the oscillation transmittance occurs at the first and second transmission band peak, which originates from the un-stable propagation modes [33]. Therefore, the mode at 0.121 and 0.257 THz can be regarded as waveguide modes having high waveguiding performance. To further understand the origin of the spectral characteristics, we simulated the dispersion relation as well as the 2D spectrum for MRAs, the results are depicted in Fig. 4(b). In this calculation, we only consider three modes for sample B because the transmittance is low when the frequency larger than 0.5 THz. The 1st order Bragg bandgap is located between mode 1 and mode 2, whose center frequency equals to 0.196 THz. The inset figures are electric field distribution in a unit cell for modes 1, 2, and 3 at 0.136 THz, 0.219 THz, and 0.395 THz, respectively. The mode 1 at 0.136 THz exhibits stronger field confinement, opposing to that of mode 3 at 0.395 THz having a weak field on the metallic rod surface. As a result, this weak field confinement shows low transmittance in the transmission spectra. The simulated 3D spectrum result confirms that the transmittance around 0.388 THz is lower than 0.2, which is different from that of the 2D spectrum.

 

Fig. 4. (a) The simulated transmission spectra for different layers MRAs. (b) The dispersion relation of TM modes and the 2D and 3D MRA transmission spectra for sample B. The inset is the electric field distribution of three modes at 0.136 THz, 0.219 THz, and 0.395 THz.

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Figure 5(a-f) shown the electric field distribution for sample B at 0.121 and 0.257 THz. At 0.121 THz, corresponding to the low-frequency band peak, the THz field does not propagate along the center of MRA (see Fig. 5(a)), opposite to that of 0.257 THz in Fig. 5(d). It means that the peak at 0.121 THz behaves as a radiation mode rather than the tightly confined mode. The field distribution at the top MRA agrees well with that of the field distribution of mode 1 in Fig. 4(b) (the inset figure), where the strong field is confined between metal rods in the X-direction. As illustrated in Fig. 5(c), the strong electric field is located at the top MRA, where the maximum field is confined on the rod surface. For the spectral peak of the high-frequency band at 0.257 THz, most of the fields are confined in the center of MRAs, which is depicted in Fig. 5(d). Figure 5(f) demonstrated the field distribution at the output of MRAs. 0.257 THz wave is tightly confined in the center of MRA, where the strong field has concentrated in the gap between two rods (−1 mm $< $X$< $ 1 mm). Furthermore, the field along the Z-axis direction is confined inside MRA and shows a multi-node field pattern. Compared to the fundamental mode at 0.121 THz, the high-order mode at 0.257 THz shows higher field confinement.

 

Fig. 5. The electric field distribution at 0.121 THz in X-Y (a), Z-Y (b), and Z-X (c) cut plane for sample B. The electric field distribution at 0.257 THz in X-Y (d), Z-Y (e), and Z-X (f) cut plane for sample B.

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3.2 Transmission properties of asymmetric MRAs

In this section, we investigated the transmission properties of MRA by changing the alignment of metal rods, for example, increasing or decreasing the metal rod interspace (G_x). The Fig. 11 in Appendix B shows the measured structural parameters for several MRAs with different ${\mathrm{\Lambda }_x}$. Only $\; {\mathrm{\Lambda }_x}$ values larger than 0.582 mm are fabricated due to the limited accuracy of the 3D printer. The corresponding microscopic photos of fabricated samples are shown in Appendix B. The experimental and simulated spectra for samples E, F, G, H, I, and J are shown in Fig. 6(a-f). Experimental and calculated results proven that a change in the interspace of metal rods leads to an obvious variation in the transmission of high-order modes. The experimental results show a good agreement with that of simulated results, where the small discrepancy between experimental and simulated transmission spectra results from the un-uniform rods and rough metal surface. With increasing G_x, from 0.252 mm to 0.815 mm, the transmission properties of MRAs show different variations. For instance, sample E realizes one noticeable bandgap width of 60 GHz with a center frequency of 0.22 THz because of the narrow air interspace (G_x=0.252 mm). Compared to the sample A, the high-frequency band of sample J shows a transmission dip at 0.33 THz (Fig. 6(f)). It means that larger G_x MRA results in two transmission bandgaps. These results suggest that the number of transmission bands can be manipulated by changing G_x.

 

Fig. 6. The experimental and simulated transmission spectra for asymmetric MRAs.

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To further know the relation between the center frequency of 1st bandgap and the metal rod interspace G_x, we summarized these results and shown in Fig. 7. Experiment results show good agreement with Bragg theory and simulation results. The small discrepancy of bandgap center frequency between the FDTD and Bragg theory results from the different structural boundary and input THz beam sizes. In FDTD, the input THz beam approximating to the rod length about 2 mm. But for Bragg theory, the boundary is 2D periodic boundary with an infinite rod length, and the input beam considers as a plane wave cover the asymmetric MRA cross-section area. The experimental result agrees well with that of simulation because the input beam in FDTD is a Gaussian beam approximating THz beam in the experiment (the red-circle dot and black line). The center frequency of Bragg bandgap is tightly correlated with the air interspace of MRA in the propagation direction ($\mathrm{\Lambda }$_y) rather than that of G_x. Results proved that the center frequency of 1st bandgap has no obvious shifts as the G_x increases. For instance, the center bandgap frequency of sample F (G_x = 0.815 mm) is 0.25 THz, which is calculated by equation of f_B=C/2Λ_y.

 

Fig. 7. The center frequency of 1st bandgap with the varying of G_x and $\varLambda $_y.

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Figure 8 shows the counter map of the calculated transmission spectra for MRAs with various G_x. In simulation, the rod diameter, height, and G_y are fixed as 0.3, 2.0, and 0.3 mm, respectively. An obvious bandgap can be found in 0.2-0.3 THz, whose bandwidth becomes narrower as the G_x increases. It indicates that the larger G_x weaken the destructive interference effect among the metal rods. Besides, the 1st pass-band shifts to the higher frequencies, and its transmittance are increased. The transmittance of the 1st pass-band peak is proportional to G_x, which is consistent with that of one PPWG that depends on its hollow core space [28,29]. Conversely, the high-order modes in the second pass-band of 0.3-0.4 THz exhibit a redshift with the increased G_x. Noted that the cut-off frequencies of the high-order mode are inversely proportional to the G_x, which can be expressed as ν_c=mC/2n_effG_x, where ν_c, m, C, and n_eff are the cutoff frequency, number of TM waveguide mode, light speed in vacuum, and effective waveguide refractive index, respectively. This phenomenon can also be observed from the experimental results in Fig. 6. For instance, for 0.252 mm G_x-MRA (Fig. 6(a)), the spectral peak locates at 0.366 THz. When G_x up to 0.815 mm, the frequency of the spectral peak is reduced from 0.366 to 0.293 THz (see Fig. 6(f)). Meanwhile, the transmittance of that spectral peak decreases from 0.72 to 0.23. The waveguide-mode bandwidths based on the full width at half maximum (FWHM) and the corresponding peak transmittance that depends on G_x are further observed. For the increased G_x, the 2nd band is shifted to the low-frequencies; its FWHM and transmittance are greatly decreased, which agrees well with experimental results in Fig. 6. These results show that the high distinction of lateral confinement and waveguide transmittance at the 2nd TM mode spectrum is consequently determined by the asymmetric interspace of G_x. However, in contrast to the 2nd band, the transmittance of the 3rd band is improved for the G_x increment. For example, as shown in Fig. 6(f), the transmittance of the 3rd band peak is larger than 60% when the G_x interspace is 0.815 mm, but the transmittance of 0.293 THz is lower than 30%.

 

Fig. 8. The counter map of calculated transmission spectra for MRAs with different G_x (0.1-0.9 mm), where the rod diameter, height, and G_y are, respectively, 0.3 mm, 2.0 mm, and 0.3 mm.

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Figure 9 presented the G_x-dependent modal field distributions of asymmetric MRAs at the corresponding spectral peak. Here, G_x=0.252 mm and G_x=0.815 mm asymmetric MRAs are select as examples to demonstrate. For 0.252 mm-G_x MRAs, the 0.168 THz wave corresponding to the fundamental mode is tightly confined at the top MRA (see Fig. 9(a)). It suggests that low-frequency THz waves can be confined and to be guided in a narrow interspace of MRA. This result coincides with that of the single metal slit [28]. As shown in Fig. 9(b), the 0.366 THz wave is tightly bound on the metal rod surface and located inside the interspace channels, which shows higher field confinement in the transverse directions comparing with that of 0.166 THz. A weak field with loose confinement can be found at 0.522 THz (Fig. 9(c)). It exhibits that the higher-order mode (3rd mode) cannot be confined and to be guided by the MRA with a certain length. Indicated that symmetric MRA with long length cannot sustain higher-order TM mode to propagate due to the high impedance [38,39]. As shown in Fig. 9(d), the field of 0.170 THz is confined inside MRA for a large G_x interspace of 0.815 mm. Noted that larger G_x achieves stronger confinement and higher transmittance of fundamental modes. It agrees well with the trend that the spectral peak transmittance of the fundamental mode increases for the G_x increment. Compared to G_x=0.252 mm, the 0.293 THz field is loosely concentrated inside 0.815 mm-G_x MRA. This means that the larger G_x interspace reduces the 2nd TM mode transmittance. In other words, the largest G_x MRA exhausted the MRA waveguide performance. To sustain the 2nd TM mode for long propagation length, in this case, the G_x interspace around 0.252 mm is the best value for the MRA waveguide. In Fig. 6(f), both experimental and simulated results demonstrated that the larger G_x interspace leads to an enhanced 3rd TM mode. With increasing G_x, the field confinement of the 3rd TM mode has been improved. The transmittance of the 3rd TM mode is larger than 60% when the G_x interspace is 0.815 mm. As shown in Fig. 9(f), the 0.424 THz field is concentrated in the center of asymmetric MRAs.

 

Fig. 9. (a), (b) and (c) are the electric field distribution respectively at 0.168, 0.366, and 0.522 THz for 0.252 mm-G_x MRAs. (d), (e) and (f) are the electric field distribution respectively at 0.170, 0.293, and 0.424 THz for 0.815 mm-G_x MRAs.

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4. Conclusions

In conclusion, a series of 3D printed MRAs with various metal rod interspaces have been investigated in detail by FDTD simulations and THz-TDS measurements. Results show that the spectral positions, bandwidths, and transmittances of TM modes can be manipulated by breaking the structural symmetry via the interspace (G_x) between metal rods: to increase or decrease. With the rise of the interspace G_x, from 0.252 to 0.815 mm, the fundamental mode shifts to higher frequencies, and its transmittance is increased. In contrast to the 2nd mode, the transmittance of the 3rd mode is efficiently improved for the G_x increment. Thus, by tuning the air interspace, the transportation efficiency of THz waves through an MRA can be improved. We analyzed the transmission properties of symmetric and asymmetric MRAs both experimentally and numerically, which suggests that the geometry-dependent waveguide performance of the MRA enables the MRA structure to be flexible for THz applications in communications and sensing.

Appendix A

To study the transmission properties of symmetric MRA, we fabricated four samples by 3 D printer. The printed resin rod array is a three-dimensional (3D) structure, whose surface is not uniform. As a result, the surface of metal coated MRA is also un-uniform. In Fig. 10, the value is the measured structural parameters using a microscope. It means that the resolution of the microscope can reach to 0.001 mm, which can be changed by adjusting the magnification of the microscope. For the 3D printer, the resolution in X-Y is 0.047 mm, which means that the one slice (or one layer) is 0.047 mm. One is measurement resolution, and another is fabrication resolution. Figure 10 presented the measured structural parameter of four samples. Noted that the period in X- and Y- direction is not the same (${\mathrm{\Lambda }_x} \ne {\mathrm{\Lambda }_y}$) because of the low accuracy of the 3D printer.

 

Fig. 10. The measured structural parameters of symmetric MRAs. The microscopic photos for corresponding MRAs.

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Appendix B

The figure in Appendix B shows the measured structural parameters for several asymmetric MRAs with various G_x (or Λ_x). Only Λ_x values larger than 0.582 mm are fabricated due to the limited accuracy of the 3D printer. The corresponding microscopic photos of fabricated samples are also shown in Fig. 11.

 

Fig. 11. The measured structural parameters of asymmetric MRAs. The microscopic photos for corresponding MRAs.

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Funding

Shanghai Normal University (SK202010); Shanghai Pujiang Program (20PJ1412200); Science and Technology Commission of Shanghai Municipality (17142200100, 18590780100, 19590746000); Shanghai Municipal Education Commission (2019-01-07-00-02-E00032); Japan Society for the Promotion of Science (16K17525); China Scholarship Council (201606890003).

Acknowledgements

D. Liu organized and wrote the paper. D. Liu and S. Zhao performed fabrication and simulation. T. Hattori and B. You provided the metal coating and fabrication equipment. B. You, Sheng-Syong Jhuo, and Shuan Chou completed the measurement. Ja-Yu Lu provided the measurement system.

Disclosures

The authors declare no conflicts of interest.

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