Terahertz (THz) waves refer to electromagnetic waves with a frequency range of 0.1–10 THz, corresponding to a free-space wavelength range of 30 µm to 3 mm. Owing to their unique characteristics of penetration and low single-photon energy, THz waves have potential applications in various fields such as 6G communication, security inspection, accelerator physics, and biomedicine^[1–4]. However, the large size of THz optical elements and the high absorption by water vapor limit their direct transmission in the atmosphere with flexibility and low loss^[1]. Developing low-loss THz waveguides is regarded as a viable solution for addressing these challenges and facilitating practical applications of THz waves^[5]. Hollow-core fiber (HCF) is a promising candidate for low-loss THz transmission by confining electromagnetic waves within a low-loss air core. Research on HCFs has witnessed significant progress in the near- and mid-infrared regions, where the transmission loss (TL) of silica HCFs is quite close to the limit of solid-core optical fiber, i.e., 0.16 dB/km^[6]. It is regrettable that traditional materials like silica, ZBLAN, and chalcogenide exhibit high absorption in the THz region^[1]. Therefore, reducing the TL of HCFs with innovative structure designs and suitable materials while ensuring large-scale fabrication feasibility are critical requirements for developing THz fibers.

In recent years, the THz antiresonant fiber (THz-ARF) has emerged as a promising candidate for low-loss THz transmission by extending the antiresonant effect in mid-infrared HCFs^[7–13]. In general, THz-ARFs are composed of thick cladding and inner thin-wall structures. Among these fibers and manufacturing techniques, the proposal of a hollow-core tube fiber (TF) aims to simplify the THz fiber structure^[14]. It removes the cladding of ARF and retains only one dielectric layer. Although the THz-TF sacrifices certain features that help reduce the TL, it has been numerically and experimentally verified that it follows the antiresonant reflecting optical waveguide (ARROW) principle^[15]. Additionally, according to scaling laws, confinement loss (CL) is proportional to the ratio of wavelength to air core radius^[16,17]. Therefore, THz-TFs are designed with a large core area compensating for increased CL due to lack of cladding. Lai et al. proposed a THz-TF with a large hollow core and a single-layer wall acting as a Fabry–Perot etalon, achieving a TL of <0.02dB/m^[14,18]. They have discovered that a commercially available Teflon THz fiber with a diameter of 9 mm and a wall thickness of 0.5 mm exhibits a TL of 0.02 dB/m at 0.38 THz and a bending loss of 0.065 dB/m at 0.368 THz with a bending radius of 60 cm^[14,15,18], which are significantly lower than most similarly sized THz fibers reported^[7,9,12,19]. Subsequently, Zhong et al. fabricated an ultrathin-wall TOPAS THz fiber achieving a large transmission bandwidth exceeding 2 THz^[20]. Furthermore, various low-loss polymer-based THz fibers have been proposed^[19,21–24]. Based on the extrusion technology, hollow-core THz-ARFs with solid struts have been fabricated and investigated^[9–11,25]. This technique enables efficient production of large-scale THz fibers made from low-loss polymers like copolymers of cycloolefin (COC) plastics, which are widely recognized as ideal materials for THz devices. However, further exploration is needed to develop structure designs compatible with the extrusion technique. It is promising to fabricate nodeless THz-TF combined with negative curvature by extrusion, which is well recognized to reduce the TL^[26,27].

A major challenge in the development of THz fiber lies in the limited availability of fabrication methods that can ensure low-loss transmission^[1]. To address this issue, we present a TOPAS COC nodeless negative curvature THz tube fiber (NC-TF) in this paper, encompassing its design, fabrication, and experimental tests. The NC-TF is optimized by analyzing various boundary shapes with negative curvature, and a structure comprising six elliptical elements that can be easily fabricated using an extrusion mold is selected. According to the simulation results, the TL spectra of NC-TF are found to be identical to those of half-ring fibers (HRFs), exhibiting significantly lower values compared to the circular TF. In comparison to well-known HCFs such as ARF, HRF, porous fiber, or Bragg fiber, NC-TF eliminates the presence of air cavities in the cladding, making the fabrication easy and feasible while retaining TL at the same level as ARF or HRF. However, it should be noted that the single-layer configuration of NC-TF may not possess sufficient robustness for applications requiring thin walls. To the best of our knowledge, this study represents the first endeavor to fabricate THz waveguides with negative curvature using the extrusion method. To experimentally verify this advantage, the NC-TF and the reference TF are fabricated using a commercial extruder and specially designed molds. The measured TL for NC-TF is significantly lower than that of TF with similar parameters, thus confirming the advantage of this cladding-free, nodeless NC-TF structure.

THz TFs are categorized as leaky-mode waveguides because the corresponding light-guiding mechanism follows the ARROW model^[15]. In general, TL is the sum of CL and material loss (ML). Therefore, the TL of THz TF can be obtained from the formula calculating CL when the absorption coefficient α of the constructing material is considered^[19,22], i.e.^[9], TL=40π·Im(neff)(ln10)·λ(dB/m),(1)where λ is the operating wavelength and Im(neff) is the imaginary part of the effective refractive index for the calculated mode. TL equals CL when the absorption coefficient α is set to 0. For THz NC-TF, CL depends on the antiresonant condition and the negative curvature effect. For the latter, destructive interference caused by the negative curvature boundary suppresses the far-field radiation and helps decrease the CL, as shown in previous computational and experimental studies^[28]. In addition, the number of dielectric layers^[26] and the thickness of both the air layer^[16] and the constructing material strongly influence the TL.

According to the ARROW model^[15,16], the antiresonant conditions for the wall thickness tmat and the air layer thickness tair could be written as tmat=(m−0.5)·λ2·n2−Re2(neff),(2)tair=(m−0.5)·λ2·1−Re2(neff),(3)where m is a positive integer, λ is the operating wavelength, n represents the refractive index of the constructing material, and Re(neff) is the real part of the effective refractive index for the calculated mode. Eqs. (2) and (3) indicate that tair contributes to the ARROW effect and constitutes a prominent characteristic of multilayer fibers.

The COC polymers exhibit low absorption characteristics as well as a constant refractive index within the entire THz band, and they are considered among the most promising materials for THz devices^[29,30]. Here, TOPAS COC is considered as the constructing material, with a refractive index of n=1.531+j·4.77×10−4 (0.5–5 THz), where the imaginary part is calculated from the absorption coefficient (α=0.2cm−1 at 1.0 THz)^[29,31]. To determine the fiber transmission properties, the finite-element method of the electromagnetic field is adopted using the commercial software, COMSOL Multiphysics. The perfectly matched layer (PML) and mesh size are optimized to ensure calculation accuracy^[32]. The mesh size in the constructing material is set to λ/6, where λ is the operating wavelength. The PML is set to 2 mm, which is more than 4 times the maximum wavelength (0.7 THz, 0.43 mm) calculated.

A three-interface model is employed to elucidate the design ideas, as shown in Fig. 1. This model consists of a 0.194-mm-thick dielectric layer (which satisfies the antiresonant condition at 1 THz) and an air layer of variable thickness (tair). Two structures are considered: circular air core boundary and inverted circular air core boundary, where the material absorption coefficients α are 0, 0.2, and 2.5cm−1. When the absorption coefficient α is 0 (green line or black line), a variation in TL with tair is observed, which arises from the antiresonant effect of the air layer, which represents a prominent difference between multilayer fibers and single-layer fibers. According to the ARROW model, a corresponding tair(antiresonant) of 2.94 mm can be predicted by Eq. (3), as shown in Fig. 1(a). For a specific air-core boundary shape (circle or inverted circle) in Fig. 1(a), the TL curves gradually flatten with increasing α, indicating a weakened antiresonant effect of the air layer. The TL variation on tair is expressed by presenting the differential values for curves in Figs. 1(a) and 1(b). Independent of α, negatively curved structures exhibit smaller differential values over a wider range of tair compared to circular structures, suggesting that negative curvature mitigates the antiresonant effect in the air layer. Based on the attenuated antiresonant effect of the air layer, it could be concluded that both ML and negative curvature contribute toward reducing the difference between single-layer and multilayer fibers regarding TL.

In order to simplify the structure of THz fiber while ensuring its transmission performance, a configuration of THz NC-TF that integrates negative curvature and single-layer design is proposed, as shown in Fig. 2. The aspect ratio (C=a/b) is employed to quantify the curvature C of each negative curvature arc. The circular arc is quantified through the length of g (separation between two adjacent negative curvature arcs), which depends on the number of negative curvature arcs (denoted by M) and the core radius r. In addition, h=ri/r is introduced to evaluate the fiber volume, where ri is the inner radius of the entire fiber. The thickness of the tube wall, including circular and negatively curved arcs, is defined as t. It can be observed from Fig. 2 that there is no air layer structure presented in the cladding. Besides, the nodeless nature of NC-TF eliminates problems due to Fano resonances that may be introduced from the connecting nodes^[33].

The simulation results of the TL for NC-TF at 1.04 THz are shown in Fig. 3, including different parameter combinations of M and C. The values of r and t are set to 4.5 and 0.2 mm, respectively. The size parameters of h=1.5 and h=1.725 are considered, corresponding to Figs. 3(a) and 3(b), respectively. The interval is 0.02 for C and 1 for M. ML is taken into consideration, as the refractive index of tube wall is set to n=1.531+j·4.77×10−4. Based on different values of C, M, and TL, Figs. 3(a) and 3(b) can be divided into four regions: Region I is characterized by small C and high TL; Region II is characterized by optimized C, M, and the lowest TL; Region III is characterized by proper C, small M, and medium TL; and Region IV is characterized by large C and medium or high TL. For the optimized combination of C and M, which is approximately 1.3≤c≤2.1, M≥7 in Fig. 3(a) and 1.4≤c≤2.6, M≥5 in Fig. 3(b), the TL can reach the minimum level of around 0.004 dB/m. This parameter range is colored in blue or deep blue and denoted by Region II in Fig. 3. The TL of 0.004 dB/m is an extremely low value for THz fiber; it could be attributed to the following factors. First, the low value of TL at 1.04 THz is obtained with a centimeter-scale NC-TF (r=4.5mm). The large air core area is beneficial for decreasing TL, which is the main advantage of TFs. Second, a wide range of C and M is considered to get an optimized result of the negative curvature effect on the NC-TF. Third, the thickness of the tube wall in the calculation is set to 0.2 mm, which is close to the limit of extrusion manufacturing. In practice, the wall thickness may be thicker to meet the mechanical strength. In other divided regions, TL increases when M is smaller than 7 or 5, denoted by Region III in Figs. 3(a) and 3(b), respectively. When C is too large or small, denoted by Region I and Region IV, the low TL cannot be found with a relatively wide variation span of C or M.

The loss characteristics of NC-TF in Fig. 3 can be explained by analyzing the electric field distributions. Figures 4(a)–4(d) depict contour lines of the electric field, with each image containing a total of 500 contour lines and covering an intensity range of 60 dB. For structures with relatively small values of C in Region I, as shown in Fig. 4(a), significant leakage occurs through the inwardly curved arcs, regardless of the element number M or the size parameter h. In addition, there is a remarkable increase in both CL and ML owing to the coupling of the electric field into the tube wall of NC-TF. The TL basically reaches 0.01 dB/m in this circumstance. In Region II, the electric field leakage through negative curvature arcs is mitigated, and the mode coupling into the wall is minimized, resulting in the lowest CL and ML, as shown in Fig. 4(b). As the separation g increases (M decreases), heavy leakage occurs through the circular arc boundary, leading to a significant increase in TL, corresponding to Fig. 4(c). In Region IV, as shown in Fig. 4(d), a large value of C results in strong mode coupling between the air core and the tube wall, causing severe leakage and high TL.

Considering the practical fabrication of the extrusion mold, the NC-TF design with geometrical parameters of r=4.5mm, h=1.725, C=1.7, M=6, and t=0.5mm is selected to demonstrate its low-loss advantage. Figure 5(a) compares the TL spectra (0.7–1.6 THz) of TF, NC-TF, and HRF. The latter two share identical parameters, while TF has a core diameter of 9 mm and a wall thickness of 0.5 mm. Their shared material refractive index is 1.531+4.77·j. NC-TF exhibits a TL of 0.0096 dB/m at 1.2 THz, which is 6 times lower than that of TF (blue line), 0.057 dB/m. Notably, TF, NC-TF, and HRF share similar propagation constant β due to their identical core radius. When manufactured using the extrusion method, TF and NC-TF exhibit similar characteristics in terms of the extrusion mold design and production process, while NC-TF shows a significantly lower TL than TF. In Fig. 5(a), NC-TF presents a TL curve identical to that of HRF, indicating that simplifying the THz fiber structure from HRF to NC-TF does not result in an increase in TL. This implies that the intricate process involved in HRF extrusion mold processing can be substantially simplified without compromising the TL performance.

There is a trade-off between fiber size and TL in optical fibers. Increasing the fiber radius would result in a decrease in TL. Understanding the intricate relationship between the fiber size and TL serves as a valuable reference for comprehending variations in core radius during practical fabrication. Figure 5(b) portrays the TL variation with the core radius for TF, NC-TF, and HRF, keeping the wall thickness at 0.5 mm. As depicted in Fig. 5(b), the TL curves for NC-TF and HRF exhibit an overlap and are significantly lower than that of TF. As the core radius r increases from 2.5 to 6.5 mm, the TL of NC-TF gradually decreases in inverse proportion to r3. The scaling laws whereby the CLs of antiresonant fibers decrease with respect to the ratio of the core radius r to the operating wavelength λ have been widely studied^[28]. Poletti’s numerical results demonstrated a 1/r4 dependence for CL in the case of single circular layer fibers^[16], and Bird confirmed through numerical calculations that CL followed a dependence of (λ/r)N+3 in the model of antiresonant fibers consisting of concentric regions of air and glass, where N is the number of glass layers^[17]. Based on this research, the fitting curves for TF, NC-TF, and HRF in Fig. 6(b) comply with the loss dependence of (1/r)N+3. For TF, TL follows an inverse proportion to 1/r3.85, consistent with the 1/r4 scaling law of attenuation when N equals 1 for single-layer fibers^[34]. The deviation from 4 to 3.85 may be attributed to the effect of ML. On the other hand, for NC-TF, TL parallels 1/r3.04, while HRF exhibits a near-identical TL dependence of 1/r3.07. As the air core boundary transitions into negative curvature, the TL dependence on r shifts drastically from 1/r4 to 1/r3. The similar TL performance of NC-TF and HRF potentially allows substitution of HRF with NC-TF through extrusion-based manufacturing techniques.

Figure 5(c) plots the TL variation on wall thickness t for TF, NC-TF, and HRF at 0.92 THz (r=4.5mm). Independent of t, NC-TF and HRF show TLs one order lower than TF due to the negative curvature. At t=0.1 or 0.2 mm, the proportion of ML in TL is not significant enough to make the discrepancy between CLs of HRF (1.53×10−4dB/m) and NC-TF (1.11×10−3dB/m) unobservable, and there are a few differences between TLs for NC-TF and HRF. As t increases, the proportion of ML in TL gradually increases, thereby rendering the disparity in CL negligible. The requirement for mechanical strength necessitates a t larger than 0.1 mm, making NC-TF a viable alternative to HRF in such scenarios.

Figure 5(d) illustrates the variation of TL on bending radius Rc for TF, NC-TF, and HRF at 0.92 THz (r=4.5mm). An equivalent index profile is used for bent fibers^[3], neq(x,y)=n·exp(xRc),(4)where n and neq(x,y) represent the refractive index for straight and bent fiber, respectively. x stands for the bending direction (horizontal) and Rc means the bending radius, with Rc→∞ indicating a scenario of a straight fiber.

In Fig. 5(d), for a bent fiber situation, NC-TF and HRF exhibit TLs lower than TF due to the negative curvature, while HRF shows a loss peak caused by the cladding airy mode and core mode within an Rc range from 64 to 76 cm, shown in the inset in Fig. 5(d). This does not happen in NC-TF, since the air layer cladding is removed; thus NC-TF shows a lower TL than HRF, as Rc<130cm. Without this mode coupling, the curve for HRF gets close to that of NC-TF as Rc increases (two curves coincide as Rc reaches 135 cm).

The proposed TOPAS COC fiber is fabricated using a commercial horizontal extruder, which serves as the essential equipment for fiber production. The production process is illustrated in Fig. 6(a). Through a hopper, the TOPAS COC polymer pellets are fed into the extruder screw, where they undergo a shearing and heating process to enhance their fluidity. As shown in Figs. 6(a) and 6(b), a specially designed mold with an NC-TF outline is mounted at the end of the screw. The air gaps in the exit face are 0.5-mm thick. The NC-TF design with r=4.5mm, h=1.725, C=1.7, and M=6 is chosen for ease of manufacture. It allows the mold to constrain the material flow through its exit face in the regions where the NC-TF shape is designed to be. During the fabrication, the extrusion screw rotation speed is set to 200 r/min, and the heating temperature is set to 170°C. As shown in Fig. 6(b), the extruded material successfully maintains its intended shape throughout the fabrication process. Additionally, a slight swelling phenomenon is observed, which can be attributed to the inherent elasticity of the polymer.

The mold for extruding NC-TF is illustrated in Fig. 7. It consists of two stainless steel components: the mold core and the outer body. The melt is extruded through air gaps between these components to form NC-TF. The hickey on the left side of the outer body serves as a temperature sensor port, enabling precise control of the entire mold temperature at 170°C during fabrication. To ensure that the negative boundary closely matches the intended shape, no traction is applied, as even minimal pulling would result in flattening of the negative boundary. The output rate is controlled to a few centimeters per minute.

The fabricated samples are shown in Fig. 8. Figure 8(a) presents an NC-TF sample with a length of 36 cm, constrained by the dimensions of the table accommodating the extrusion setup. For comparison, a TF with a similar air core diameter and wall thickness is fabricated. The average air core diameters are 7.90 and 8.12 mm for TF and NC-TF, respectively. The average wall thicknesses for TF and NC-TF are 0.732 and 0.734 mm, respectively. The measured curvature values for NC-TF approximate C=1.86, which closely aligns with the designed value of 1.7. The notable achievement lies in the smooth surface of our samples. In contrast to previous works where 3D printed nozzles (or molds) exhibited rough surfaces^[11,35], the mold undergoes meticulous machining processes, resulting in a superior surface quality for the fibers. In addition, by utilizing a standard extrusion production line equipped with a higher power motor and a more precise molding control system, the potential yield of NC-TF could be elevated to 20 m/s. Due to manufacturing deviations in the extrusion mold and deformation process, there exists a maximal fluctuation of approximately 0.2 mm in the fabricated TF and NC-TF wall thicknesses. To minimize the manufacturing deviations and achieve a smaller thickness as well as a reduced air core diameter, future work will focus on mold modification and implementation of traction or other fiber-shaping schemes.

A THz time-domain spectrometer (THz-TDS) is utilized to investigate fiber transmission in the THz range. The experimental setup and measuring method employed in this study closely resemble those used previously for measuring TLs of a pentagram THz HCF, a double pentagon nested HCF, and a six-ring HRF^[19,31,36]. The femtosecond (fs) laser generates 800-nm pulses with a duration of 40 fs and a repetition rate of 80 MHz. The fs-pulse is split into two beams using a polarization beam splitter (PBS), where one beam serves as the pump to emit THz waves while the other acts as the probe. A delay stage that consists of two mirrors and a motorized stage is used to scan the delay time of the THz pulse. The THz pulse waveform is measured by detecting the differential signal as a function of delay time. To ensure consistent light coupling in all experiments, the fiber samples are securely held within the setup using two irises featuring an aperture that matches the air core size of the samples. The detector is a differential photodiode for detecting the intensity disparity between the two polarization probe beams produced by a Wollaston prism. Modulation and amplification of the probe signal are achieved using a chopper and lock-in amplifier. To reduce water vapor absorption, nitrogen gas (N2) is introduced to maintain the humidity below 5%.

The frequency domain signals can be acquired through the fast Fourier transform (FFT) of the time-domain signals, enabling resolution of both phase φ and signal intensity I as a function of the frequencies. By comparing the phase delay Δφ and the signal intensity reduction with a reference, the refractive index neff and the TL of the sample can be determined. The effective refractive index is calculated as^[22,29]neff=Δφ·cconstω·L+nref,(5)where Δφ indicates the phase difference between the sample and the reference measurements, cconst is the speed of light, ω is the angular frequency, L denotes the sample length, which is kept at 67 mm, and nref is assigned a value of 1 in this study.

The TL can be calculated using the equation T(ω)=Psam(ω)/Pref(ω), where Psam(ω) represents the power spectrum of the sample and Pref(ω) represents that of the reference signal^[35]. Therefore, the TL could be estimated as TL=−10lgT(ω)/L, where L is the fiber length, kept at 67 mm for both TF and NC-TF. The calculated neff and TL spectra are shown in Figs. 9(a) and 9(b).

Following the mentioned procedures, Figs. 9(a) and 9(b) present the neff and TL of the TF and NC-TF samples, respectively. The inset in Fig. 9(a) plots the fluctuations in neff within the frequency range of 1.0 to 1.8 THz. The neff values of TF and NC-TF in Fig. 9(a) exhibit an increase from 0.998 and 0.999 to 1.0 within the frequency range of 0.4 to 1.8 THz, respectively. In the case of NC-TF, gray dotted lines are used to mark dips occurring at frequencies of approximately 0.54, 0.75, 1.08, 1.32, 1.56, and 1.68 THz, which are associated with the high loss peaks shown in Fig. 9(b). These dips result from avoided crossings between the core mode and modes supported by the tube wall, according to the ARROW model^[22]. The dips at approximately 0.6, 1.0, and 1.56 THz are observed in the TF, accompanied by the corresponding high loss peaks depicted in Fig. 9(b). The discrepancies in TL values and resonant peak positions between the measured data and simulated TLs in Fig. 5(a) may be attributed to imperfections introduced during the fabrication process. As shown in Fig. 8, both NC-TF and TF samples exhibit variations in wall thickness, which significantly impact the resonant frequencies and TLs due to their strong correlation with wall thickness^[37]. The TF sample possesses a smaller core diameter compared to NC-TF, resulting in higher TLs due to reduced neff values. However, the presence of a negatively curved boundary is another contributing factor leading to lower TL for NC-TF when compared with that of TF. The NC-TF sample demonstrates a TL of 0.2 dB/m at 0.6 THz, 0.33 dB/m at 0.84 THz, 2.32 dB/m at 1.14 THz, 3.07 dB/m at 1.41 THz, and 2.66 dB/m at 1.62 THz, which is comparatively lower than those in recent studies^[9,11,35,38]. Therefore, NC-TF holds great potential for short-haul communication, as it can retain half of the transmitted signal after propagating 1 m within its transmission bands.

Figure 10(a) shows a practical mismatch between the mold core and the outer body, causing uneven distribution of tube wall thickness due to tilting the mold core in a certain direction. This leads to variations in loss characteristics, as depicted in Fig. 10(b), where multiple randomly distributed resonant points and antiresonant points can be observed within a TL range from 0.2 to 10 dB/m. Notably, there is an evident antiresonant point at 0.92 THz and a resonant point at 1.08 THz. The simulation may provide an explanation for the discrepancy between the measured TL spectra and the ideal one.

In this study, the transmission performance of the proposed NC-TF is numerically and experimentally investigated. In the fiber design, NC-TF combines negative curvature and a single-layer structure to reduce its TL and the structure complexity. By removing the air layer, HRF can be simplified to NC-TF. Numerical results show that HRF and NC-TF exhibit nearly identical TLs, which are significantly lower than those of TF with the same geometrical parameters. This enables simplification of the extrusion mold while maintaining the TL performance.

The fabrication process involves the utilization of an extruder and a specially designed mold. This is the first attempt to fabricate THz fibers with negative curvature using the extrusion method. The fabricated samples consistently maintain their intended shape, while exhibiting impeccably smooth surfaces. Note that the mold used for NC-TF has a much simpler structure compared to those utilized in previous studies for fabricating THz HCFs^[11,35].

The transmission performance of NC-TF is conducted using a THz-TDS. NC-TF shows low TLs below 3 dB/m in the transmission bands, with the minimum TL of 0.2 dB/m at 0.6 THz. The experimental data confirm the feasibility and practicability of incorporating negative curvature into THz fiber using the extrusion method. It should be noted that there are imperfections in the obtained results, which could be attributed to the wall thickness fluctuation in the fabrication process. To minimize the manufacturing deviations and achieve a smaller wall thickness as well as a reduced air core diameter, future work will focus on mold modification and implementation of traction or other fiber-shaping schemes. We believe that the simplicity and practicality of NC-TF will enable its application in various transmission or sensing scenarios within the THz range.