Terahertz (THz) band (0.1–10 THz) is the next frontier for ultra-high-speed communication systems. Currently, most of communications research in this spectral range is focused on wireless systems, while waveguide/fiber-based links have been less explored. Although free space communications have several advantages such as convenience in mobility for the end user, as well as easier multi-device interconnectivity in simple environments, fiber-based communications provide superior performance in certain short-range communication applications such as multi-device connectivity in complex geometrical environments (ex., intra-vehicle connectivity) and secure communications with low probability of eavesdropping, as well as secure signal delivery to hard-to-reach or highly protected environments. In this work, we present an in-depth experimental and numerical study of the short-range THz communications links that use subwavelength dielectric fibers for information transmission and define the main challenges and trade-offs in the link implementation. Particularly, we use air or foam-cladded polypropylene-core subwavelength dielectric THz fibers of various diameters (0.57–1.75 mm) to study link performance as a function of the link length of up to , and data bit rates of up to 6 Gbps at the carrier frequency of 128 GHz (2.34 mm wavelength). We find that depending on the fiber diameter, the quality of the transmitted signal is mostly limited either by the modal propagation loss or by the fiber velocity dispersion (GVD). An error-free transmission over 10 m is achieved for the bit rate of 4 Gbps using the fiber of smaller 0.57 mm diameter. Furthermore, since the fields of subwavelength fibers are weakly confined and extend deep into the air cladding, we study the modal field extent outside of the fiber core, as well as fiber bending loss. Finally, the power budget of the rod-in-air subwavelength THz fiber-based links is compared to that of free space communication links, and we demonstrate that fiber links offer an excellent solution for various short-range applications.
© 2020 Chinese Laser Press
The terahertz (THz) frequency spectrum (0.1–10 THz) holds high promises for many applications that include communications , imaging , sensing , and spectroscopy . In communications, in order to meet the bandwidth demand set by the next generation of wireless systems, a shift in the carrier frequency toward the THz band is unavoidable [4,5]. THz communications have been already demonstrated in the context free space wireless links that profit from the presence of several low-/modest-loss atmospheric transmission windows . Although there are many advantages of wireless communications including convenience in mobility for the end user, ease in scaling up the network, flexibility of device interconnectivity, etc., they also possess many challenges. Particularly, due to high directionality of the THz beams, THz wireless links are known for their high sensitivity to alignment errors, thus requiring careful positioning of the transmitters and receivers . Moreover, reliable communications in non-static environments (ex., between moving objects) require complex beam steering solutions. The situation is further exacerbated in geometrically complex environments (such as inside vehicles and buildings), where highly complex channel modeling is required. Moreover, free space links have higher chances of eavesdropping, thereby increasing the risks for secure communications . Finally, atmospheric weather conditions such as rain, snow, and fog, play a major role in affecting the performance and reliability of the wireless THz links. In view of these limitations of wireless THz communications, short-range THz fiber links () can offer an alternative solution as THz fibers present a closed highly controlled propagation environment, can span complex geometrical paths, and can offer reliable coupling to receiver and transmitter for both static and dynamic applications. One interesting area of application for THz fiber links is in reliable onboard connectivity and intra-vehicle communications for military and civil transportation. In Fig. 1, we show the schematic of various communication modalities within and between the airborne vehicles, as well as place of THz fibers in such applications. For example, a high-speed THz data link with a moving vehicle can be established using a tracking ground station or another vehicle. A high power and high gain transmitter antenna can be used for such a long distance communication . Given high directionality of the THz signals, multiple antenna modules have to be installed on the vehicle surface to cover several possible directions of communication. Next, received THz signals have to be demodulated and interpreted using expensive and environmentally sensitive signal processing units, which should be preferably located deep inside the vehicle.
This type of scenario in which THz signals are detected using multiple antennas and then are relayed over the complex geometrical paths to a central processing unit can profit greatly from flexible THz fiber links. Another scenario is using THz fiber links for reliable delivery of high-speed data through partially blocked or geometrically complex areas, which is of importance for hard-to-reach or highly protected environments such as enclosures for aggressive environments (ex., bio-enclosures) and protected structures (ex., shelters and bunkers), as well as for intra- and inter-device THz communications in which different parts of the same system can be conveniently linked using flexible fibers. Finally, THz fiber links can be used as a backup solution for short-range wireless communications in case of sudden deterioration of the atmospheric conditions, which can be of particular importance for places with harsh weather.
In designing an efficient THz fiber communication link, the fiber parameters such as transmission loss, bend loss, dispersion, coupling strength, and ease of handling play a significant role. Furthermore, the degree of complexity in the fiber fabrication process determines the cost and commercialization opportunities. While the fiber loss and coupling strength limit the communication link distance, the maximum achievable bit rate can also be limited by the fiber dispersion. Therefore, low transmission loss and low dispersion are the primary concerns for the THz fiber designs. We start by reviewing several types of existing THz fibers. The choice of waveguide material is one of the key factors in achieving THz guidance with low loss and low dispersion. In the case of metallic waveguides, the finite conductivity of metallic layers leads to ohmic losses, whereas in dielectric waveguides the loss is mainly due to material absorption. Independently of the materials used, longer THz waveguides (over 1 m) are frequently designed to use modes predominantly guided in the low-loss dry air region. Most recently in Ref. , bare metal wires in air were proposed as open waveguides for 5G communication applications; however, such waveguides suffer from high coupling losses and difficulty in mechanical handling due to wire waveguide open structure. Low-loss, low-dispersion, and efficient coupling can be achieved using two-wire plasmonic THz waveguides; however, longer (over 1 m) two-wire waveguides are inconvenient in practice due to challenges in packaging and handling . This is because in the two-wire plasmonic waveguides, the air gap between the two metallic wires should be precisely maintained along the whole fiber length, which is difficult to achieve in long fiber links. While encapsulating the two metallic wires within a porous dielectric cladding using fiber drawing offers a solution to the mechanical stability and handling problem, this also leads to additional losses and dispersion due to coupling of a plasmonic mode to the dielectric cladding .
Alternatively, by selecting proper dielectric materials with low absorption loss [Teflon, polyethylene, polypropylene (PP), cyclic olefin copolymer, to name a few] and engineering the waveguide structure to expel the mode into the low-loss dry air region, highly efficient THz waveguides can be demonstrated [13–31]. In general, dielectric THz waveguides or polymer microwave fibers (PMFs) fall under one of the three main categories: hollow core waveguides [anti-resonant reflecting optical waveguides (ARROWs) or photonic bandgap (PBG) waveguides] [13–19], porous core waveguides [that use total internal reflection (TIR) or PBG guidance] [20–24], and solid core waveguides (TIR guidance) [25–28]. In the hollow core dielectric tube fibers, the finite thickness of a thin tubular cladding determines the bandwidth of the low-loss ARROW guidance regime, while such waveguides are generally multimode and can support many core and cladding modes. By increasing the size of the hollow core, the propagation loss can be further minimized (in expense to beam quality) as guided modes propagate almost completely in the low-loss air core . Although the cladding modes can be suppressed and the transmission bandwidth can be improved by introducing lossy tubing material such as PMMA, in such fibers one still excites multiple core modes, which becomes problematic for long link THz communications due to intermode dispersion and intermode interference . Similarly, in the hollow core THz fibers, by arranging alternative layers of high and low refractive index (RI) cladding materials (Bragg fibers) or by introducing judiciously designed (periodic or aperiodic) arrays of air inclusions in the cladding (PBG fibers), the loss and the transmission bandwidth can be improved when compared with the tube-based ARROW fibers [3,14,16,19,32–35]. Moreover, effectively single mode regime can be achieved in long sections of such fibers, which can significantly reduce effective fiber dispersion. Additionally, low loss and low dispersion can be achieved when using spatially variable dense arrays of subwavelength air holes both in the core and cladding regions (porous fibers) [20,21]. Apart from the circular and hexagonal porous structure , honey-comb  and rectangular  porous geometries were introduced for which the dispersion is comparable to the THz microwires (subwavelength rod-in-air fibers). Similarly, using dispersion flattened porous THz waveguides, an ultra-wide transmission bandwidth () , which is comparable to the free space THz wireless channel, can be achieved for short distances. Finally, porous fibers with graded density of pores have been demonstrated to significantly decrease intermode dispersion when operated in the multimode regime . Both hollow core and porous fiber, however, are challenging to fabricate as they rely on precise arrangements of air inclusions in a polymer, glass, or crystalline matrix . Fabrication of most of such fibers involves drawing under pressure thermo-polymer or glass-based preforms with drilled or 3D printed air inclusions. Achieving and maintaining target porosity throughout the length of the fiber requires careful calibration and monitoring of the entire drawing process, which is often challenging due to small dimensions of structured preforms used in such drawings. Recently, an alternative method for the fabrication of THz microstructured and PBG fibers was detailed in Ref. , where over meter-long monocrystalline sapphire fibers were grown directly from the melt using structured dies. Alternatively, such fibers could be 3D printed directly using infinity 3D printing techniques, which was recently demonstrated in Ref. , in which authors continuously printed several meters of a wagon-wheel highly porous ARROW fiber using PP.
In solid core THz fibers, the transmission bandwidth is much larger than that in the hollow core fibers as the propagation mechanism is TIR. However, the transmission loss in such fibers is much higher and is generally comparable to the absorption loss of the fiber material. In order to minimize the transmission loss, one usually resorts to either subwavelength core dielectric fibers that are simple rod-in-air fibers or rod-in-foam fibers [26,41] or small solid core photonic crystal fibers (PCFs) with porous claddings . The rod-in-air/foam THz fibers with subwavelength size cores offer low loss and low dispersion guidance as a large fraction of the modal fields in such waveguides is guided in the low-loss air or foam regions [26–28,41]. In such fibers, scattering from inhomogeneities along the fiber length such as diameter variation, micro- and macrobending, and material density variation, is the dominant loss mechanisms due to weak confinement in the fiber core . Scattering loss in such fibers can be somewhat mitigated by increasing the fiber diameter and realizing better confinement in the fiber core in the expense of the increased losses due to materials absorption. By choosing low-loss plastics for the core materials, as we show in the following, a good compromise can be found, and multi-meter THz fiber links can be realized. Therefore, despite of all these challenges, rod-in-air/foam subwavelength fibers are a simple, but reliable platform for enabling various short-range THz communication applications. Furthermore, such fibers can be used to fabricate non-trivial signal processing components, such as directional couplers, power dividers, and bandpass filters, capable of real-time THz signal processing, which potentially allows building complete transmission/signal processing subsystems using the same base technology [44–46].
As discussed earlier, although simple rod-in-air subwavelength fibers are easy to fabricate and potentially offer low propagation loss and low dispersion, mechanical manipulation of such fibers and their integration into systems are problematic due to significant extent of the modal fields into air . Therefore, for practical applications, such fibers have to be encapsulated in such a way as not to significantly affect the weakly core-bound guided modes, while allowing direct mechanical handling of the fibers. One solution to this problem is a wagon-wheel structure in which solid core is suspended in air using several deeply subwavelength bridges . Fabrication of such fibers is, however, challenging due to complexity of the fiber cross section. Alternatively, solid subwavelength cores can be encapsulated using low-loss, low-RI () dielectric foams, resulting in what we call throughout the paper rod-in-foam fibers (see Fig. 2). In practical terms, high porosity low-loss material foams with deeply subwavelength air cells are virtually indistinguishable from the uniform air cladding, while providing the necessary mechanical stability and ease of manipulation [11,48]. Although, in principle, the dielectric foams can contribute to additional propagation losses, the effect is negligible for short distances (several meters) and lower THz frequencies (below 200 GHz) for many of the common foams.
Now, we briefly review some of the recent demonstration of THz communications using THz fibers [49–55]. In Ref. , an error-free transmission of 7.6 Gbps and 1.5 Gbps data over the link distance of 8 m and 15 m has been demonstrated using a hollow core waveguide made of Teflon at the carrier frequency of 120 GHz. The modal loss of the waveguide was 2.5 dB/m. The maximum bit rate at a longer link length here was mainly limited by the propagation loss. Recently, a 1 m expanded porous polytetrafluoroethylene THz fiber link has been established together with the resonant tunneling diode integrated with photonic crystal waveguide for 10 Gbps and uncompressed 4K video transmission . For completeness, we note that one promising approach for high-speed THz interconnects at short distances (millimeters to few centimeters) for on-chip and inter-chip communications is based on silicon as the core material as such waveguides feature low transmission loss, high RI contrast, and strong confinement, as well as mature fabrication technology . For example, in Ref. , an ortho-mode sub-THz interconnect channel for planar chip-to-chip communications using silicon dielectric waveguide has been investigated. By engineering the waveguide structure, silicon-based photonic crystal waveguides were proposed and demonstrated for efficient data transmission [59–61]. Moreover, by functionalizing silicon with other materials such as graphene, active components such as THz waveguide-based modulators can be fabricated . Additionally, advanced shaped growth techniques [35,63] can be used to extend the use of low-loss, high RI monocrystalline materials such as silicon or sapphire to longer THz transmission lengths up to 1 m; however, application of such waveguides in THz communications has not yet been reported. In all these works, however, no in-depth analysis was presented as to the key reasons for the limitations in the transmission length and maximal bit rate in such fiber links. While, the most obvious reason for such limitations is often claimed to be fiber losses, in our following analysis, we conclude that fiber dispersion is another major factor that is often overlooked. In fact, we find that depending on the fiber design and operation frequency, one can be in the power-limited or dispersion-limited regime even in the case of short several-meter-long fiber links. From the practical point of view, in the power-limited regime, transmission is possible up to the highest communication bit rate supported by the available hardware even when approaching the maximal link length when signal strength becomes comparable to the noise level. In this regime, the eye diagram opening collapses along a single vertical direction, while no significant degradation in its overall form (skewedness) is observed. In the power-limited regime, negative effects of the modal loss dominate over those due to modal dispersion. In contrast, in the dispersion-dominated regime, even for short fiber links when signal strength is significantly larger than the noise level, one cannot achieve maximal bit rate as allowed by the hardware. In this case, the eye diagram shows significant asymmetry and shape contortion due to modal group velocity dispersion and resultant pulse spreading. In the dispersion-limited regime, negative effects of the modal dispersion dominate over those due to modal loss.
In this work, we aim at deeper understanding of the limitations of the fiber link quality posed by the combined effects of the modal loss and dispersion. Without the loss of generality, we concentrate on a pure system of rod-in-air dielectric THz subwavelength fiber for short-range () communication links with up to 6 Gbps data speeds. In fact, the rod-in-air fiber can be used in further studies as a performance benchmark for the more practical fibers such as rod-in-foam and suspended core fibers. In the following, we fix the carrier frequency at 128 GHz, while using fibers of various diameters to realize power-limited or dispersion-limited transmission regimes. The dielectric fibers are made of low-loss PP material with three different diameters of 1.75 mm, 0.93 mm, and 0.57 mm. Both theoretical and experimental studies are then carried out, and a comparative analysis of the two is presented. Experiments were conducted using a photonics-based THz communication system reported in Refs. [64,65]. We then demonstrate that the limitation in the error-free link distance is mainly due to the modal loss for the 1.75-m-diameter fiber, while for the 0.93 mm and 0.57 mm diameter fibers, the link distance is limited due to modal dispersion. By optimizing the decision threshold, an error-free -long link at 4 Gbps is achieved with the 0.57-mm-diameter fiber, while the argument is made for over 10 Gbps fiber links with over 10 m length when designing the fiber to operate near zero dispersion frequency (ZDF). Furthermore, study of the bending losses of the rod-in-air fibers is presented, in which we conclude that even relatively tight bends of sub-10-cm radius can be well tolerated by such fibers. Finally, the power budget of the fiber-based link is compared with that of the free space links, and the case is made for the strong potential of the rod-in-air fibers in short-range communications. To the best of our knowledge, this is the first comprehensive study of all the major limiting factors and comparative advantages that relate to design and operation of short-range fiber-assisted THz communications links.
2. THEORY OF ROD-IN-AIR DIELECTRIC THz FIBERS
Many polymers possess almost constant RI and low absorption losses at lower THz frequencies (). PP, in particular, has one of the lowest losses over the wide THz frequency range ( below 1 THz) [66–68]. Moreover, this material is compatible with 3D printing using the cost-effective fused deposition modeling (FDM) technique that opens many exciting opportunities in design and manufacturing of various 3D patterned bulk optical components and photonic integrated circuits. Due to the importance of PP material for THz application, in our studies we therefore used PP filaments of three different diameters (, 0.93 mm, and 0.57 mm) as rod-in-air fibers. The filaments having smaller diameters (0.93 mm and 0.57 mm) were extruded using an FDM printer. Optical characterization of the fibers was then carried out using an in-house photonics-based THz communication system detailed in Refs. [64,65] that operates at 128 GHz carrier frequency. Complex RI of PP was measured using THz-continuous wave spectroscopy system (see Appendix A.1). Mode analysis of the rod-in-air fibers was carried out using commercial finite element software COMSOL Multiphysics. The goal of this work is to establish limiting factors in transmission of high bit rate data streams over long distances; therefore, modal loss, group velocity dispersion, coupling efficiency, and bending losses are the key parameters to model.
A. Effective Index, Modal Losses, and Excitation Efficiency
The normalized electric field distribution of the fundamental mode (normalized to 1 W of carrying power) for the PP fibers of different diameters at 128 GHz is shown in Figs. 3(a)–3(c). The normalized power fraction of the fundamental mode contained within the aperture of a variable diameter centered around the fiber is presented in Fig. 3(d). Clearly, for the fibers of larger diameter, the modal field is mostly confined within the fiber, while for smaller diameters the modal fields are strongly present in the low-loss air cladding, which is also the reason for the lower absorption losses of smaller diameter fibers. In Fig. 3(e), we present the effective refractive indices of the fundamental fiber guided modes as a function of the fiber core diameter at 128 GHz operation frequency, and the corresponding modal absorption losses (in straight fibers) are presented in Fig. 3(f). At the carrier frequency of 128 GHz, the fiber operates in a single mode regime up to the fiber diameter of 1.63 mm. For a 1.75 mm fiber, one expects excitation of three modes (, , and ), while in practice, excitation of and does not happen as such modes are incompatible by symmetry with the mode of a WR-6 waveguide (THz emitter waveguide flange) that is centered with respect to the fiber input facet. Thus, all the waveguides used in this work are effectively single mode.
We next study excitation efficiency of the fiber fundamental modes using external THz sources. Generally, the excitation efficiency is maximized when the size of the source field distribution is comparable to that of the fiber mode. In our experiments, the subwavelength fibers are butt coupled to the conical horn antenna that is connected to the WR-6 waveguide flange of the THz emitter. The horn antenna is a mode converter with the tapered structure that converts the fundamental mode of a rectangular waveguide to the Gaussian-like mode at the output. Coupling efficiency can be further optimized by properly positioning the fiber input end inside of the horn antenna. While many coupling scenarios have been considered in the literature [56,69,70,71], they all essentially result in a similar coupling efficiency as achieved by a simple free space coupler that uses a single plano–convex lens. The schematic of such a coupler is shown in the inset of Fig. 4(a). By optimizing the lens parameters (focal length, diameter) so as to match the size of the focused Gaussian beam with that of the fiber mode (or alternatively by matching the numerical apertures of the lens with that of the fiber), one can optimize excitation efficiency of the fiber guided modes.
Next, we use the transfer matrix theory and a mode matching technique to estimate excitation efficiency of the fiber fundamental mode with the focused linearly polarized Gaussian beam generated by such a coupler/taper at the input facet of a fiber [72,73]. In Fig. 4(a), the excitation efficiency (by power) of the fundamental mode for rod-in-air fibers of three different diameters is presented as a function of the Gaussian beam diameter (). The maximum excitation efficiency and the corresponding Gaussian beam diameter for all three fibers at 128 GHz carrier frequency are summarized in Table 1. Similarly, by using the optimized Gaussian beam size, the fundamental mode excitation efficiency is calculated as a function of frequency, which is presented in Fig. 4(b). The bandwidth of the excitation efficiency is one of the critical parameters that determines the maximum channel capacity. Even if we disregard the frequency dependent sensitivity of the receiver and assume dispersionless transmission, the maximal channel bandwidth in the fibers is still limited by the frequency dependent transmission loss and excitation efficiency. Therefore, maximizing the THz communication link bandwidth is a very challenging problem as it depends on a large number of factors from emitter/receiver hardware to the fiber design. In the particular case of our fibers, as seen from Fig. 4(b), over 90% excitation efficiency of the fiber fundamental mode with the bandwidth of at least 20 GHz is achieved for 1.75 mm and 0.93 mm fibers, whereas for 0.57 mm fiber, the bandwidth of with the excitation efficiency of 80% is achieved. Although the available transmission bandwidth (in terms of excitation efficiency) is sufficient for the transmission of greater than 10 Gbps data (ASK modulation), the maximal bit rate is limited by the group velocity dispersion, which is discussed in Section 2.C.
From the data presented earlier in this section, we can now estimate the maximal fiber link distance given a power budget that is typical for our optics-based THz communication system (for the moment we ignore modal dispersion and bending loss). Thus, received power at the end of the fiber link can be estimated using the following expression:4), and is the modal propagation loss by power. The modal loss for all three fibers is obtained from the numerical simulation [see Fig. 3(f)] as 2.2 dB/m, 0.62 dB/m, and 0.01 dB/m for 1.75 mm fiber, 0.93 mm fiber, and 0.57 mm fiber, respectively. The transmitter power is (, which is used in our experiments), and the received power after propagation along the distance is shown in Fig. 5.
The signal loss level for the error-free transmission of data with the bit rate of 6 Gbps is found to be (10 μW), while the absolute noise level below which transmission is impossible is found to be at (0.4 μW) (see Appendix A.4). From these measurements and from Fig. 5, we can estimate maximal distances of the fiber links capable of error-free 6 Gbps data transmission, which are 5 m for the 1.75 mm fiber and 20 m for the 0.97 mm fiber. Note that transmission with errors is possible up to in 1.75 mm fiber and 43 m in 0.97 mm fiber. Finally, we note that while for 0.57 mm fiber the power budget considerations limit transmission distances to range, in practice, the dispersion and bending loss result in much shorter fiber link distances of several tens of meters.
B. Bending Losses
The bending losses of the guided modes of the rod-in-air THz fibers are calculated using both numerical simulation and analytical approximations. The numerical simulations were carried out using the 2D axis-symmetric model in COMSOL Multiphysics. Here, the radiating wave propagates in the azimuthal direction, , and the electric field is expressed as
Bending losses of the fiber fundamental mode in dB/m are computed for the bending radii in the range of 4–30 mm. The calculated and fitted values of the bending loss are presented on the logarithmic scale in Fig. 6(a) as solid and dashed curves for X and Y polarizations, respectively. In the inset of Fig. 6, the normalized electric field profile of the bent fundamental mode for the bending radius of 3 cm is presented. For 1.75 mm fiber, the bending loss becomes smaller than the modal absorption loss in a straight fiber () for bending radii as small as or larger. At the same time, for the 0.93 mm fiber bending radius should be larger than in order for the bending loss to be smaller than the modal absorption loss (). We note that optical performance of both fibers (1.75 mm and 0.93 mm) is quite robust with respect to bending as in many practical applications bending radii superior to 2–10 cm can be easily accommodated; however, using fibers of smaller diameters becomes challenging. This is due to the fact that for smaller core diameters, the fundamental mode is only weakly confined to the fiber core leading to higher radiation losses due to bending. For example, as seen from the inset of Fig. 6 for the 0.57 mm diameter fiber at 3 cm bending radius, the leaky mode of a bend has a much stronger contribution of the radiation continuum to its modal field distribution when compared to the leaky modes of the fibers of larger core diameters. Therefore, while straight 0.57-mm-diameter fiber features very low absorption loss , at the same time it is strongly affected by the radiation loss due to micro- and macrobending.3). The propagation constants of the fundamental modes at 128 GHz for all three fibers are obtained from numerical simulations using COMSOL. Estimated bending losses using analytical expression Eq. (3) are presented in Fig. 6(a) as dotted lines and show reasonable correspondence with the losses of the bent leaky modes described earlier.
C. Modal Group Velocity Dispersion and Maximal Bit Rate Estimation
Next, we study modal group velocity dispersion and maximum error-free bit rate for the three fibers assuming a 10-m-long fiber link. The link length of 10 m is chosen to be long enough to be of practical importance, while making sure that all the fibers have no more than 25 dB loss (by power) over the link distance. In general, the maximum bit rate in the communication link is limited by the pulse dispersion and propagation loss. The dispersion parameters and are the second- and third-order derivate of the propagation constant () with respect to the angular frequency , which is used to characterize the degree of pulse broadening in fibers.
Particularly, considering only the second-order modal dispersion, the maximum bit rate (for ASK modulation) supported by the fiber of length can be estimated using Eq. (4), which is derived by requiring that of the power of the broadened pulse form still remains within the time slot allocated to logical “1” . In Fig. 7(a), the of all the three fibers is presented along with their single mode cutoff frequency . We see that dispersion of the 0.93 mm fiber  is much higher than those of the 1.75 mm and 0.57 mm fibers at 128 GHz carrier frequency, thus significantly limiting maximal bit rates for the 0.93 mm fiber. Moreover, dispersion of the 1.75 mm fiber is near zero at 128 GHz, thus promising very high bit rates at this carrier frequency. The estimated bit rates for different fibers and different carrier frequencies are presented in Fig. 7(b). At 128 GHz and a 10-m-long fiber link, the maximum error-free bit rates of 4.7 Gbps and 1.2 Gbps are predicted for fibers of 0.57 mm and 0.93 mm diameter,
It is important to mention that 1.75-mm-diameter fiber has a ZDF (at which ) in the immediate vicinity of 128 GHz carrier frequency used in our experiments. The maximum error-free bit rate supported by such a fiber at ZDF is, thus, limited by the third-order dispersion , and can be estimated using Eq. (5) . Applying Eq. (5) to all the three fibers operating at their respective ZDFs and assuming a link distance of 10 m, we estimate for in the 10–20 Gbps range (see Table 2). We thus conclude that, by optimizing the fiber diameter to achieve low dispersion at a given carrier frequency, 10 m fiber links can be realized with rod-in-air fibers capable of over error-free bit rates per channel, which is sufficient for many practical THz fiber-based communication applications,
Finally, we note that dispersion of a bent fiber can be different from that of a straight fiber, especially at smaller bending radii. In Fig. 6(b), for example, we show dispersion of the fundamental X-polarized leaky mode of a bent 1.75 mm fiber. At large bending radii, dispersion approaches zero (that of a straight fiber), while at bending radii smaller than several cm it grows rapidly and can become as large as . Additionally, ZDW can shift to a somewhat different value in bent fibers, which should be considered when designing fiber links.
3. EXPERIMENTAL CHARACTERIZATION OF THE ROD-IN-AIR SUBWAVELENGTH FIBERS
A. THz Communication System, Fiber Holding Method, and Principal Measurement Challenges
The experimental characterization of the fibers was carried out using an in-house photonics-based THz communication system reported earlier in Refs. [64,65]. The schematic and the experimental setup of the THz communication system are shown in Figs. 8(a) and 8(b), respectively. Particularly, two independently tunable distributed feedback lasers (TOPTICAPhotonics) operating in the infrared C-band with slightly different center frequencies are used to optically drive the photomixer. The laser beams are combined using a 3 dB coupler and are intensity modulated (ASK) using an external electro-optic Mach–Zehnder modulator. A baseband signal source of pseudo random bit sequence (PRBS) with a varying bit rate from 1 Gbps to 6 Gbps and pattern length of is used. The modulated laser beams are amplified using an erbium-doped fiber amplifier (EDFA) and injected into the waveguide coupled uni-traveling-carrier-photodiode (UTC-PD) photomixer (NTT Electronics) for THz generation. In the receiver section, a zero bias Schottky diode (WR8.0 ZBD-F) is used to detect and demodulate the baseband signal. The baseband signal is then amplified using a high gain low noise amplifier (LNA), and the bit error rate (BER) is measured using the test equipment (Anritsu-MP2100B).
The largest rod-in-air fiber used in our experiments is the 1.75 mm fiber, which is a commercial 3D printing PP filament (Verbatim) with a diameter of . The 0.93 mm fiber was fabricated by reducing the diameter of the 1.75 mm PP filament using a 3D printer (Raise3D Pro2). The temperature of the extruder was set to 220°C, and a 1 mm nozzle was used for the extrusion. A motorized spinning tool is used to precisely control the diameter of the filament that is extruded from the nozzle by adjusting the drawing speed. Thus, fabricated fiber had diameter along the 8 m length. Similarly, a 10-m-long fiber is fabricated by using the same 1 mm nozzle at increased drawing speed. While characterizing fiber links, a stable and consistent fiber coupling must be used. This is of particular importance for subwavelength fibers that can have significant modal presence in the air cladding. In our experiment, we used butt coupling with the fiber ends judiciously positioned inside the horn antenna. To counter the weight of the free-hanging fiber, the fiber is held tightly using an arrangement of two fisherman’s knots made of thin threads and positioned at both emitter and detector ends [see inset of Fig. 8(a)]. Additionally, rod-in-air fibers do not preserve the in-coupled light polarization state as the light propagates along the fiber. The birefringence is caused by local imperfections in the rod shape (ellipticity, for example), as well as material property fluctuations, which alter the polarization state of the propagating field. Stochastic polarization rotation becomes an issue for fiber links longer than several meters and must be considered while aligning the fiber at the detector side. The optimal alignment is achieved by rotating the fiber end at the detector side until a maximum signal amplitude is recorded.
B. Measuring Fiber Propagation Loss Using Cutback Technique
The modal propagation loss of the PP fiber is studied experimentally using the THz communication system and a cutback method, which is then compared to the theoretical values [see Fig. 3(f)]. Without the loss of generality, here we detail only the measurement for the 1.75 mm fiber, while smaller diameter fibers can be characterized in a similar manner. First, the detector electronics are calibrated for direct power estimation from the eye pattern as discussed in the Appendix A.3. Second, a 1-m-long 1.75 mm fiber is butt coupled at both emitter and detector antenna by direct insertion into the horn antenna linked to the WR-6 hollow rectangular waveguide flange. By fixing the DC bias voltage of the emitter to , the infrared optical power is increased using EDFA, and the eye pattern for 1 Gbps is recorded using the test equipment. When the emitter photocurrent reaches 4 mA, the eye pattern starts clipping indicating that maximum threshold of the oscilloscope is reached. Therefore, in the following modal loss measurement, the emitter photocurrent is fixed to 4 mA, which corresponds to the THz power of (). Next, the 1.75 mm fiber of length 5 m is butt coupled at both emitter and detector antenna. The eye amplitudes for three bit rates (1 Gbps, 3 Gbps, and 6 Gbps) were recorded. Then, the fiber is cut back from the detector side with a step of 0.5 m, and the eye amplitude is recorded until the fiber reaches 1 m as shown in Fig. 9(a). At each cutback length, the fiber is carefully inserted into the detector horn antenna, while the position of the fiber tip is minimally adjusted to achieve the largest opening in the eye diagram. The recorded eye amplitude is then fitted using the form . In Fig. 9(b), the estimated THz power is presented in logarithmic scale. It is observed that the modal loss of 1 Gbps, 3 Gbps, and 6 Gbps is , , and , respectively, which agrees well with the theoretically estimated absorption loss.
C. Modal Field Extent in the Air
As mentioned earlier, subwavelength fibers can have significant modal presence in the air cladding. In practice, one has to choose the size of fiber cladding (foam, for example) in such a way as to incapsulate most of the modal field. Experimentally, to measure the extent of modal field into air, we place the fiber in the center of a circular metallic aperture of the variable diameter (1–25 mm). The eye amplitude for 1 Gbps bit rate is then recorded as a function of the aperture diameter, from which the fraction of the received THz power is estimated [see Fig. 10(a)] by normalizing with respect to the corresponding value measured for the fully open aperture. From this measurement, we conclude that 90% of guided power for the 1.75 mm and 0.93 mm fibers is contained within the circles of and , respectively, and thus define characteristic sizes of the claddings to be used with practical fiber designs. At the same time, the corresponding size for the 0.57 mm fiber is theoretically estimated to be and could not be measured directly due to small size of the used aperture. At the same time, the measured results agree well with the numerical simulations [see Fig. 3(d)].
As mentioned earlier, one of the major challenges posed by the rod-in-air subwavelength fibers is the difficulty in their handling to significant presence of the modal field in the air cladding. To counter this problem, and to enable easy handling and manipulation of such fibers in practical installations, one can insert the subwavelength THz fiber core, for example, in a circular/square-shaped low-loss foam that features RI close to that of air. In our experiments, we realized some of such fibers using polystyrene foams with RI of 1.0104 and losses . In Figs. 10(b)–10(d), we present photographs of several THz subwavelength fibers with fiber diameters of 1.75 mm, 0.93 mm, and 0.57 mm surrounded by the polystyrene foam cladding of size 5 mm, 6 mm, and 45 mm, respectively, which are chosen from Fig. 10(a) to guarantee that 90% of the modal power is confined within the cladding.
4. BIT ERROR RATE MEASUREMENTS
The BER measurements were carried out to study the fiber link performance under the laboratory environment. In the measurements, the emitter photocurrent was set to 7.5 mA, which corresponds to the THz power of (). A non-return to zero (NRZ) PRBS with the bit rates between 1 Gbps and 6 Gbps and a pattern length of was used as a baseband signal. For the target BER of (error-free detection), the duration of a single measurement was . Furthermore, the decision threshold is optimized so that insertion error (digital 0 is mistaken as digital 1) and omission error (digital 1 is mistaken as digital 0) are approximately the same.
A. BER Measurement for the 1.75 mm and 0.93 mm Fibers at 8 m Link Length
First, we identify the maximal fiber length of the 1.75 mm fiber for BER measurements in our system. For that, we consider the eye pattern and observe that beyond 8 m of fiber length, the eye amplitude becomes comparable to the 0 and 1 noise levels, thus resulting in impractical BER values. Therefore, the maximum link length is fixed at 8 m. Similar to the modal loss measurement, an 8-meter-long 1.75 mm fiber is butt coupled to the emitter and detector units using fisherman’s knots to hold the fiber in place. The BER measurement is carried out for the 1.75 mm fiber by varying the bit rate from 1 Gbps to 6 Gbps. At each bit rate, the decision threshold is optimized so that both insertion and omission errors are the same.
The total BER of the 8 m link is presented in Fig. 11 (blue solid line). Similarly, for the purpose of comparison, the BER measurements of the 8-m-long 0.93 mm fiber link is carried out. In our measurements, we did not observe any error (for optimized decision threshold) for the bit rates below 2.4 Gbps, within the measurement duration. At the same time, we were not able to measure BER for the bit rates beyond 2.8 Gbps due to high group velocity dispersion of the guided mode. The BER recorded for the bit rates between 2.4 Gbps and 2.8 Gbps is presented in Fig. 11 (red solid line). The inset in Fig. 11 shows the recorded eye patterns for both the 1.75 mm and 0.93 mm fibers.
From the eye patterns, we can judge the effects of modal loss and modal dispersion on the fiber link performance. Thus, in the case of 1.75 mm fiber, the eye amplitude is much smaller than that for the 0.93 mm fiber indicating that 1.75 mm fiber link performance is limited by the modal absorption loss. The error-free operation for the 1.75 mm fiber is observed for link lengths shorter than 5 m (input THz power is ) and bit rates of up to 6 Gbps [currently limited by the THz communication system, and the input THz power is ()]. At the same time, even at the link distance of 8 m, the measured BER of is well below the forward error correction (FEC) limit (). As the eye patterns for the 1.75 mm fiber stay relatively symmetric even for longer fiber links (8 m) and at high bit rates (6 Gbps), we believe that such a fiber can support at least 9 Gbps up to 10 m (as predicted theoretically), by compensating modal absorption losses with higher input powers [above ()]. In contrast, for the case of 0.93 mm 8-m-long fiber, although its absorption loss is much lower than that of the 1.75 mm fiber due to much higher group velocity dispersion in such fibers , the maximal bit rate is limited to only 3 Gbps. This is also confirmed by the shapes of the eye patterns for the 0.93 mm fiber that show significant shape degradation at higher bit rates, while also featuring almost 10 times higher powers of the received signals in the case of the 1.75 mm fiber.
B. BER Measurement for the 0.57 mm Fiber at 10 m Link Length
We now consider the 0.53 mm fiber that is theoretically predicted to have a very small absorption loss and a relatively small value of dispersion . As predicted theoretically, an error-free transmission of up to 4.7 Gbps can be achieved using 0.57 mm 10-m-long straight fiber link. Experimentally, we use the same arrangement as discussed earlier, and then conduct BER measurements for data bit rates between 1 Gbps and 6 Gbps and a fiber length of 10 m. Error-free transmission with an optimized decision threshold is observed up to 4 Gbps (measured in steps of 1 Gbps) as shown in Fig. 13, which is in good agreement with theoretical predictions. The inset in Fig. 12 presents the eye patterns for 1, 2, and 6 Gbps bit rates. Although propagation loss of the 0.57 mm straight fiber is much lower than those of the 1.75 mm and 0.93 mm fibers, due to considerable modal diameter of the 0.57 mm fiber (), a significant portion of the modal power is cut by the receiving horn antenna of 10.8-mm-diameter aperture.
C. BER Measurement for the 1.75 mm Fiber and a 90° Bend
The effect of bending on performance of the 8-m-long 1.75 mm fiber is studied using a 90° bend of 6.5 cm bending radius. The BER measurement (see Fig. 13) was carried out in a configuration similar to that of the straight fiber detailed earlier, while the fiber bend was realized by suspending the fiber using several knots and holders. For the chosen bending radius, we observe only a small increase in the measured BER of a bent fiber compared to that of a straight fiber, which is consistent with theoretical observation that bending loss of the 1.75 mm fiber at 6.5 cm bending radius is much smaller than the modal absorption loss, while dispersion of the mode of a bend at this bending radius is also small . We note that performing a reliable measurement of the bending loss using subwavelength rod-in-air fibers is difficult as such fibers cannot be conveniently handled due to significant extent of the modal fields in air. Therefore, we defer a more detailed experimental study of the bending loss to future works, which will be conducted using rod-in-foam fibers placed into predesigned bending molds.
5. POWER BUDGET COMPARISON OF THE ROD-IN-AIR THz FIBER LINKS WITH THE FREE SPACE COMMUNICATION LINK
In this section, we highlight the advantages of fiber-based communication links by comparing them with the free space communications assuming a simple ASK modulation scheme. In free space optics, the power received at a distance from the source is given by77]. The received THz power using the fiber link can be estimated using Eq. (1). As a value for (assuming that ), the aperture sizes of a standard horn antenna (10.8 mm) and a lens (2” optics: 50.8 mm) were considered. In free space communication, the received power is mainly limited by the divergence of the propagating beam.
By using a large-area lens or parabolic reflector antenna, it is possible to collect most of the transmitted energy. However, using large collecting optics is not favorable in many space-limited applications. For the emitter power of 0 dBm (1 mW), the received power after distance for both fiber and free space links is shown in Fig. 14. From this figure, we see that at shorter distances, fiber-based links are superior to free space beams in terms of received power due to fast divergence of the THz beams, while at longer distances, free space links are generally superior to fiber links due to smaller atmospheric losses compared to absorption losses of the fiber materials. From the figure, we see that when using a relatively small horn antenna (aperture size of 10.8 mm) at both emitter and detector sides, performance of the 1.75 mm fiber-based communication link is superior to that of a free space link in terms of the received power for the link lengths of up to 24 m. However, at this link distance, the received THz power is well below the noise floor () for both free space and 1.75 mm fiber-based links. Moreover, even when using relatively bulky 2-inch optics (lens) for collimating and collecting the THz beam in free space, the performance of 1.75 mm fiber is still superior to a free space link of up to 8 m in length. By reducing the fiber diameter, the modal loss can be further reduced, thereby increasing the link distance. Alternatively, the noise floor and the minimal power required for the error-free transmission can be further improved as they depend on many factors including the type of modulation, responsivity of the detector, bandwidth and gain of LNA, etc. Although the link distance can be improved by using subwavelength fibers of smaller diameters or by better system design, the maximal achievable bit rate in the fiber links is still much lower compared to the free space THz communication links as fibers feature much higher group velocity dispersion than free space. The maximal bit rate in the free space THz links can be estimated by considering the dispersion of dry air  .
In Table 3, we use both theoretical and experimental data (pertaining to our system) to summarize some estimates for the power budget and maximal bit rates of the fiber-based and free space THz communication links using the ASK modulation mechanism. Here we use the experimentally found as a minimal signal level to achieve the error-free transmission at 6 Gbps bit rates; moreover, since the ZDF for the 1.75 mm fiber is 128 GHz, third-order dispersion is used to estimate the maximal bit rate, while for the 0.93 mm and 0.57 mm fibers, we use instead. It is clear from Table 3 that, while the fiber-based communication links offer higher transmitted powers at shorter distances, nevertheless, they also consistently underperform in terms of the maximal achievable data rates compared to the free space links due to higher dispersion. To increase data transmission rates in fiber links, the modal dispersion (as well as modal loss) can be reduced by reducing the fiber diameter to deep subwavelength sizes; however, it comes with a higher sensitivity to bending and larger mode diameters. Alternatively, dispersion compensation techniques can be used to compensate for the fiber-link dispersion as we have recently demonstrated using strong hollow core waveguide Bragg grating ; however, such devices tend to have limited operation bandwidth, and further studies are necessary to establish feasibility of this approach. Alternatively, by using higher order modulation schemes such as orthogonal frequency division multiplexing (OFDM), the effect of dispersion can be minimized due to smaller bandwidth required at each carrier frequency . This could potentially increase the bit rate several folds in longer THz fiber links.
In this work, we presented a comprehensive theoretical and experimental study of simple, yet practical dielectric rod-in-air/foam THz fibers in view of their potential applications in short-range THz communication applications. The THz fibers under study were made of PP and featured three different core diameters of 1.75 mm, 0.93 mm, and 0.57 mm. Furthermore, THz communication link performance was characterized with fibers of lengths 8 m and 10 m as a function of the variable data bit rates from 1 Gbps to 6 Gbps at the carrier frequency of 128 GHz. Our main conclusion was that, depending on the fiber diameter, the communication links were operating either in the power-limited or dispersion-limited regime. Thus, the 1.75 mm fiber featured loss and zero dispersion at the carrier frequency, and it could carry the highest bit rate of 6 Gbps up to the maximal distance of 8 m only limited by the fiber absorption loss, while error-free transmission with such a fiber was observed up to 5 m link length. Modal field extent of the core-guided mode into air cladding was only several mm deep due to relatively strong confinement of the modal field in the fiber core. As a result, the 1.75 mm fiber was also well tolerant to bending with virtually no degradation in the link performance when inserting a 90° tight bend of 6.5 cm radius. Further encapsulation of the fiber with polystyrene foam of sub-1-cm diameter makes such a fiber an excellent candidate for practical short-range THz communication links due to its ease of handling and installation, as well as good optical properties and tolerance to perturbations such as bending. Similarly, the 0.57 mm straight fiber featured a very low absorption loss and a relatively small dispersion of . The resultant performance was similar to that of a 1.75 mm fiber; however, maximal link length was rather limited by dispersion than by the modal loss. As a result, error-free transmission was realized for a 10 m link with up to 4 Gbps data rates, while signal strength was considerably higher than the noise level. One of the major disadvantages of this fiber is high sensitivity to bending and several-cm-deep penetration of the modal fields into the air cladding, thus making even the foam-cladded fibers inflexible and somewhat difficult to handle. Finally, the 0.93 mm fiber, while featuring relatively small absorption loss of , also featured relatively high dispersion of , thus significantly limiting the maximal supported bit rate even for a good signal strength. Aa a result, a maximal bit rate of only 2.4 Gbps was demonstrated for an 8 m fiber link. Finally, we compared the THz fiber communication links with free space links that use relatively small focusing optics (up to 5 cm diameter) and concluded that in this case fiber links are generally more efficient in terms of the power budget for short-range communications up to several tens of meters. Fiber links are also more reliable and easier to install, maintain, and reconfigure than free space links, especially in complex communication environments (on-board communications, for example). At the same time, free space communications outperform fiber-based links in terms of maximal bit rate as air features significantly lower dispersion than fiber.
1. Complex Refractive Index Measurement of the Polypropylene Filament
The RI of the PP fiber is measured using the continuous wave (CW) THz spectroscopy system [78,80]. The schematic of the experimental setup is shown in Fig. 15, which is briefly explained as follows. The setup has two distributed feedback (DFB) lasers with slightly different center wavelengths and uniform power ( each) operating in the telecom region. A 50:50 coupler combines and splits the two wavelengths equally into the emitter and detector arm, respectively. Two single mode polarization maintaining fibers wounded with piezo actuators, which stretch in the opposite directions, were connected to both the arms for the measurement of phase. The symmetric arrangement of the fiber stretchers provides the additional path delay as well as uniform disturbance due to any variation in the external environment. The path lengths between the emitter and detector arms were balanced to have a flat phase. The generated THz waves, which are the frequency difference of the two lasers, were modulated using the bias voltage for lock-in detection. The THz beams were collimated and focused into the detector using the parabolic mirrors. The generated photocurrent in the detector was recorded along with the phase information as a function of frequency. The sample was kept in the collimated THz path for the RI measurement.
The PP sample for RI measurement was prepared as follows. A circular slab of PP material was fabricated by melting the PP fiber in a crucible at a temperature of 240°C for 35 min followed by 20 min of cooling. The densities of the PP fiber and the fabricated slab after melting were and , respectively, with the difference of only . Therefore, it is concluded that the melting process does not affect the density of the material, and no significant amount of air inclusions were introduced.
Three slabs with slightly different thicknesses were fabricated and then polished on both sides to get a smooth surface. A cutback measurement technique was used to measure the real part of the RI using the CW THz system. Both the amplitude and phase were recorded after removing each slab. The unwrapped phase for different sample (PP) thickness is shown in Fig. 16(a). Using the phase information, the real part of the RI of PP is extracted using Eq. (A1), where is the frequency dependent real part of RI, is the measured phase difference due to the presence of a sample, and is the speed of light. The RI of PP was measured as 1.485 in the measured frequency range, which is shown in Fig. 16(b),
Although the real part of the RI of PP was measured using the CW THz spectroscopy system, extracting the imaginary part accurately requires thick sample (∼in meters) due to very low PP material loss. Therefore, we used the modified THz communication system for the loss measurement. The instrumental setups for both the CW THz spectroscopy system and the THz communication are similar (TOPTICA Photonics) except for the additional data modulation unit and detector in the communication system.
In spectroscopy, a heterodyning detection scheme is used, whereas in the communication system, a direct detection scheme using a Schottky detector is employed. To measure the PP loss, the data modulation unit was disabled; however, the bias to the emitter antenna was modulated at lower frequency (12 kHz) for lock-in detection. In the detector section, a trans-impedance amplifier was placed after the LNA where the voltage is amplified and converted to photocurrent, which is proportional to the received THz power. In order to measure the loss of the PP fiber, we used the 1.75 mm fiber, and a similar fiber holding arrangement was used as for the BER measurement, as well as a metallic aperture at the detector side that was closed around the fiber. By keeping the input coupling undisturbed, the fiber was cut from the detector side during the cutback measurement. The photocurrents of two fiber lengths (5 m and 1 m) were recorded from 100 GHz to 150 GHz as shown in Fig. 17(a). The fiber loss was then estimated using Eq. (A2) [see Fig. 17(b)],A2) one can now extract the bulk absorption loss of the PP material by dividing Eq. (A2) by a certain frequency dependent normalization function . Such a function can be computed numerically by assuming a certain frequency independent bulk absorption , then finding numerically the corresponding absorption loss of a 1.75 mm rod-in-air fiber made of such a material, and then defining . Such a function is universal as long as bulk loss used in simulations is small. Thus, extracted bulk absorption loss [black curve in Fig. 17(b)] can then be fitted using the second-order polynomial [Eq. (A3)], where is the frequency in THz,
Finally, bulk absorption loss of PP at 128 GHz is estimated to be 2.36 dB/m.
The dispersion of THz fibers can be measured using both CW and time-domain THz spectroscopy systems as detailed in Refs. [15,78,81]. In Ref. , it was demonstrated that for subwavelength fibers in particular, experimentally measured dispersion agrees well with the theoretical estimations; therefore, in our work, only the theoretically calculated dispersion values were used to estimate the maximal bit rate for all the three fibers.
2. Selection of the THz Carrier Frequency
The choice of carrier frequency is determined by the output THz power and responsivity of the detector antenna. In Fig. 18(a), the THz power versus frequency measured using a calibrated powermeter (PM3-Erickson powermeter, Virginia Diodes) is presented. Similarly, the corresponding developed DC voltage (without data modulation) was recorded [see Fig. 18(b)] using the oscilloscope (Anritsu-MP2100B). Both these measurements were carried out independently by keeping the DC bias voltage () and photocurrent (7.5 mA) of the emitter antenna constant. A higher DC voltage was recorded for the frequency of 130 GHz. However, we observe a better eye pattern at 128 GHz, which was also verified by doing the frequency sweep from 125 GHz to 135 GHz.
3. Calibration of Detector Electronics for Direct THz Power Estimation
The calibrated calorimetric THz powermeter is standard when measuring the absolute power over the broad frequency range. In communications, such THz powermeters are integrated with hollow rectangular metallic waveguides for direct coupling with an electronic source. However, for fiber-based links, the absolute power measurement is challenging due to fluctuations caused by coupling consistency issues between the fiber and a waveguide flange. In fact, we found that using horn antenna coupled to a Schottky diode as a detector allows efficient and more consistent optimization of the coupling conditions at the fiber/detector side. Therefore, we first calibrated the detector antenna for direct power measurement using the received digital signal in the communication system. To calibrate the zero bias Schottky diode (ZBD) detector for direct power measurement, we proceed as follows. The data modulation unit in the communication system is disabled for the purpose of power measurement. First, the THz output power was measured using calorimetric THz powermeter by fixing the output frequency at 128 GHz and the DC bias voltage to . The infrared optical input power to the emitter antenna was varied, which is controlled in terms of the photocurrent. The measured THz power as a function of the photocurrent is shown in Fig. 19(a).
Second, the THz emitter is butt coupled to the ZBD by removing the horn antenna attached to it. Similar to the above measurement, the emitter photocurrent was varied, which is proportional to the THz output power, and the developed DC voltage (without the LNA) in the ZBD was measured by the oscilloscope with a load resistance of 50 Ω. The developed DC voltage as a function of emitter photocurrent is shown in Fig. 19(b). Then, the photocurrent in Figs. 19(a) and 19(b) was replaced with the THz power as shown in Fig. 19(c). The measurement in Fig. 19(c) was fitted using the second-order polynomial,19(b)] is recorded when there is no data modulation, which is shown in Fig. 19(d) as a blue solid line. It means that the measured DC voltage is the same as logical 1 of the modulated signals (for binary modulation). In our experiments, the digital logical 1 is represented as , and logical 0 is represented as . Therefore, measuring the eye amplitude and dividing it by 2 is equivalent to the DC voltage measured when there is no modulation as shown in Fig. 19(d) (red line and green line). The developed DC voltage in the ZBD is referred to here as mean DC voltage recorded when the data modulation unit is disabled. We see a small deviation between the mean DC voltage and eye amplitude/2, which could be due to the amplifier noise and minor discrepancy in estimating the eye amplitude. To summarize the calibration process, the measured eye amplitude is divided by and used in Eq. (A4) to estimate the received THz power.
4. Noise Floor and Error-Free Detection Using THz Communication System
The following experiment has been carried out to measure the effect of signal level on the BER in our communication system and in order to characterize the noise floor and minimal signal power necessary for error-free detection (). In the THz receiver unit, we have used the LNA with the bandwidth of 3 GHz. Therefore, the bit rate in the communication system is limited to the maximum of 6 Gbps (ASK modulation). In this measurement, the emitter antenna was butt coupled to the ZBD. The output from the LNA was connected to the test equipment (oscilloscope and BER tester). The DC bias voltage to the emitter antenna was set to . The emitter photocurrent was varied by increasing the infrared optical power to vary the THz signal power in the system. The eye pattern and BER with optimized decision threshold for the bit rate of 6 Gbps were then recorded. Since the emitter and detector were butt coupled, the received THz power is considered the same as the transmitted THz power, which is extracted from Fig. 19(a).
From Fig. 20, for the received THz power of (10 μW) and higher, we did not observe any errors within the measurement duration for the optimized decision threshold. Therefore, the signal level of for 6 Gbps is defined as a minimal signal power required for the error-free transmission in our THz communication system.
The noise floor of our THz communication system is measured by decreasing the emitter power, while the mean and standard deviations of one level and zero level of the 6 Gbps data were recorded from the eye pattern. The noise floor is considered as the THz power at which the distance between the mean one and zero levels equals the average of standard deviations of the one and zero levels. In Fig. 21, we show the eye pattern of the 6 Gbps data recorded for various emitter powers. The noise floor for 6 Gbps data in our system was .
5. Modal Properties of Rod-in-Foam Fibers
The RI of highly porous foams (polystyrene foam, for example) is close to that of air. Therefore, the modal propagation properties of fibers with foam cladding (such as mode size and dispersion) are similar to those of fibers with air cladding. In Fig. 22, we show normalized electric field profiles of the fundamental modes of the rod-in-foam fibers of different core diameters, which look very similar to those of air-clad fibers shown in Fig. 3. The main effect of the foam claddings is in the additional transmission losses due to foam material absorption and scattering on the foam microstructure. Therefore, by replacing the air cladding of rod-in-air fibers with a foam, one should mostly observe changes in the modal propagation loss. Thus, considering, for example, the experimentally measured RI and loss of a polystyrene foam in the THz region as 1.0104 and 1 dB/m (see Ref. ), the total loss of the fundamental mode of the straight rod-in-foam fiber is then computed as 2.4 dB/m, 1.45 dB/m, and 1.01 dB/m for the fiber diameters of 1.75 mm, 0.93 mm, and 0.57 mm, respectively. These losses should be compared to those of the air-clad fibers, which are 2.2 dB/m, 0.6 dB/m, and 0.01 dB/m for the same fiber diameters. As propagation of the fundamental mode in the 1.75 mm fiber is mostly inside of the PP core, the effect of the foam cladding is minimal, thus resulting in only 0.2 dB/m increase in the modal loss compared to the air-clad fiber. In contrast, for the smaller diameter fibers, a significant part of the mode propagates outside of the fiber core and in the foam cladding, which can potentially lead to significant increase in the modal losses and reduction of the maximal error-free link distance if the foam losses are important.
The Canada Research Chairs I program and the Canada Foundation for Innovation grant (Project No. 34633) in Ubiquitous THz Photonics of Prof. Maksim Skorobogatiy.
We thank technician Mr. Jean-Paul Levesque for his assistance.
The authors declare no conflicts of interest.
1. T. Nagatsuma, G. Ducournau, and C. C. Renaud, “Advances in terahertz communications accelerated by photonics,” Nat. Photonics 10, 371–379 (2016). [CrossRef]
2. H. Guerboukha, K. Nallappan, and M. Skorobogatiy, “Toward real-time terahertz imaging,” Adv. Opt. Photon. 10, 843–938 (2018). [CrossRef]
3. J. Li, K. Nallappan, H. Guerboukha, and M. Skorobogatiy, “3D printed hollow core terahertz Bragg waveguides with defect layers for surface sensing applications,” Opt. Express 25, 4126–4144 (2017). [CrossRef]
4. Y. Niu, Y. Li, D. Jin, L. Su, and A. V. Vasilakos, “A survey of millimeter wave communications (mmWave) for 5G: opportunities and challenges,” Wireless Netw. 21, 2657–2676 (2015). [CrossRef]
5. J. Zhang, P. Tang, L. Tian, Z. Hu, T. Wang, and H. Wang, “6–100 GHz research progress and challenges from a channel perspective for fifth generation (5G) and future wireless communication,” Sci. China Inf. Sci. 60, 080301 (2017). [CrossRef]
6. P. H. Siegel, “Terahertz technology,” IEEE Trans. Microwave Theory Tech. 50, 910–928 (2002). [CrossRef]
7. Z. Chen, X. Ma, B. Zhang, Y. Zhang, Z. Niu, N. Kuang, W. Chen, L. Li, and S. Li, “A survey on terahertz communications,” China Commun. 16, 1–35 (2019). [CrossRef]
8. J. Ma, R. Shrestha, J. Adelberg, C.-Y. Yeh, Z. Hossain, E. Knightly, J. M. Jornet, and D. M. Mittleman, “Security and eavesdropping in terahertz wireless links,” Nature 563, 89–93 (2018). [CrossRef]
9. Q. Wu, C. Lin, B. Lu, L. Miao, X. Hao, Z. Wang, Y. Jiang, W. Lei, X. Den, and H. Chen, “A 21 km 5 Gbps real time wireless communication system at 0.14 THz,” in 42nd International Conference on Infrared, Millimeter, and Terahertz Waves (IRMMW-THz) (IEEE, 2017), pp. 1–2.
10. S. Galli, J. Liu, and G. Zhang, “Bare metal wires as open waveguides, with applications to 5G,” in IEEE International Conference on Communications (ICC) (IEEE, 2018), pp. 1–6.
11. A. Markov, H. Guerboukha, and M. Skorobogatiy, “Hybrid metal wire-dielectric terahertz waveguides: challenges and opportunities,” J. Opt. Soc. Am. B 31, 2587–2600 (2014). [CrossRef]
12. A. Markov and M. Skorobogatiy, “Two-wire terahertz fibers with porous dielectric support,” Opt. Express 21, 12728–12743 (2013). [CrossRef]
13. C.-H. Lai, B. You, J.-Y. Lu, T.-A. Liu, J.-L. Peng, and C.-K. Sun, “Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding,” Opt. Express 18, 309–322 (2010). [CrossRef]
14. A. Dupuis, K. Stoeffler, B. Ung, C. Dubois, and M. Skorobogatiy, “Transmission measurements of hollow-core THz Bragg fibers,” J. Opt. Soc. Am. B 28, 896–907 (2011). [CrossRef]
15. H. Bao, K. Nielsen, O. Bang, and P. U. Jepsen, “Dielectric tube waveguides with absorptive cladding for broadband, low-dispersion and low loss THz guiding,” Sci. Rep. 5, 7620 (2015). [CrossRef]
16. T. Ma, H. Guerboukha, M. Girard, A. D. Squires, R. A. Lewis, and M. Skorobogatiy, “3D printed hollow-core terahertz optical waveguides with hyperuniform disordered dielectric reflectors,” Adv. Opt. Mater. 4, 2085–2094 (2016). [CrossRef]
17. A. Cruz, C. Cordeiro, and M. J. F. Franco, “3D printed hollow-core terahertz fibers,” Fibers 6, 43 (2018). [CrossRef]
18. S. Yang, X. Sheng, G. Zhao, Y. Wang, and Y. Yu, “Novel pentagram THz hollow core anti-resonant fiber using a 3D printer,” J. Infrared Millimeter Terahertz Waves 40, 720–730 (2019). [CrossRef]
19. R.-J. Yu, B. Zhang, Y.-Q. Zhang, C.-Q. Wu, Z.-G. Tian, and X.-Z. Bai, “Proposal for ultralow loss hollow-core plastic Bragg fiber with cobweb-structured cladding for terahertz waveguiding,” IEEE Photon. Technol. Lett. 19, 910–912 (2007). [CrossRef]
20. A. Hassani, A. Dupuis, and M. Skorobogatiy, “Low loss porous terahertz fibers containing multiple subwavelength holes,” Appl. Phys. Lett. 92, 071101 (2008). [CrossRef]
21. A. Hassani, A. Dupuis, and M. Skorobogatiy, “Porous polymer fibers for low-loss terahertz guiding,” Opt. Express 16, 6340–6351 (2008). [CrossRef]
22. S. Atakaramians, S. Afshar, B. M. Fischer, D. Abbott, and T. M. Monro, “Porous fibers: a novel approach to low loss THz waveguides,” Opt. Express 16, 8845–8854 (2008). [CrossRef]
23. A. Dupuis, J.-F. Allard, D. Morris, K. Stoeffler, C. Dubois, and M. Skorobogatiy, “Fabrication and THz loss measurements of porous subwavelength fibers using a directional coupler method,” Opt. Express 17, 8012–8028 (2009). [CrossRef]
24. M. I. Hasan, S. A. Razzak, G. Hasanuzzaman, and M. S. Habib, “Ultra-low material loss and dispersion flattened fiber for THz transmission,” IEEE Photon. Technol. Lett. 26, 2372–2375 (2014). [CrossRef]
25. K. Nielsen, H. K. Rasmussen, A. J. Adam, P. C. Planken, O. Bang, and P. U. Jepsen, “Bendable, low-loss Topas fibers for the terahertz frequency range,” Opt. Express 17, 8592–8601 (2009). [CrossRef]
26. L.-J. Chen, H.-W. Chen, T.-F. Kao, J.-Y. Lu, and C.-K. Sun, “Low-loss subwavelength plastic fiber for terahertz waveguiding,” Opt. Lett. 31, 308–310 (2006). [CrossRef]
27. M. Roze, B. Ung, A. Mazhorova, M. Walther, and M. Skorobogatiy, “Suspended core subwavelength fibers: towards practical designs for low-loss terahertz guidance,” Opt. Express 19, 9127–9138 (2011). [CrossRef]
28. H. Li, H. Xiao, J. Yuan, W. Wang, B. Yin, B. Wu, and Z. Han, “Terahertz polarization-maintaining subwavelength dielectric waveguides,” J. Opt. 20, 125602 (2018). [CrossRef]
29. S. Atakaramians, S. Afshar, T. M. Monro, and D. Abbott, “Terahertz dielectric waveguides,” Adv. Opt. Photon. 5, 169–215 (2013). [CrossRef]
30. G. Humbert, “Optical fibers in terahertz domain,” in Handbook of Optical Fibers (2019), pp. 1–49.
31. A. Barh, B. P. Pal, G. P. Agrawal, R. K. Varshney, and B. A. Rahman, “Specialty fibers for terahertz generation and transmission: a review,” IEEE J. Sel. Top. Quantum Electron. 22, 365–379 (2015). [CrossRef]
32. B. Ung, A. Mazhorova, A. Dupuis, M. Rozé, and M. Skorobogatiy, “Polymer microstructured optical fibers for terahertz wave guiding,” Opt. Express 19, B848–B861 (2011). [CrossRef]
33. A. Argyros, “Microstructures in polymer fibres for optical fibres, THz waveguides, and fibre-based metamaterials,” Int. Scholarly Res. Not. 2013, 785162 (2013). [CrossRef]
34. J.-Y. Lu, C.-P. Yu, H.-C. Chang, H.-W. Chen, Y.-T. Li, C.-L. Pan, and C.-K. Sun, “Terahertz air-core microstructure fiber,” Appl. Phys. Lett. 92, 064105 (2008). [CrossRef]
35. G. M. Katyba, K. I. Zaytsev, N. V. Chernomyrdin, I. A. Shikunova, G. A. Komandin, V. B. Anzin, S. P. Lebedev, I. E. Spektor, V. E. Karasik, and S. O. Yurchenko, “Sapphire photonic crystal waveguides for terahertz sensing in aggressive environments,” Adv. Opt. Mater. 6, 1800573 (2018). [CrossRef]
36. K. Nielsen, H. K. Rasmussen, P. U. Jepsen, and O. Bang, “Porous-core honeycomb bandgap THz fiber,” Opt. Lett. 36, 666–668 (2011). [CrossRef]
37. S. Atakaramians, S. Afshar, H. Ebendorff-Heidepriem, M. Nagel, B. M. Fischer, D. Abbott, and T. M. Monro, “THz porous fibers: design, fabrication and experimental characterization,” Opt. Express 17, 14053–14062 (2009). [CrossRef]
38. M. S. Islam, J. Sultana, S. Rana, M. R. Islam, M. Faisal, S. F. Kaijage, and D. Abbott, “Extremely low material loss and dispersion flattened TOPAS based circular porous fiber for long distance terahertz wave transmission,” Opt. Fiber Technol. 34, 6–11 (2017). [CrossRef]
39. T. Ma, A. Markov, L. Wang, and M. Skorobogatiy, “Graded index porous optical fibers-dispersion management in terahertz range,” Opt. Express 23, 7856–7869 (2015). [CrossRef]
40. G. Xu, K. Nallappan, Y. Cao, and M. Skorobogatiy, “Fabrication of low loss and near zero dispersion suspended core polypropylene fibers for terahertz communications using infinity 3D printing technique,” engrXiv 239j4 [CrossRef] (2020).
41. M. De Wit, Y. Zhang, and P. Reynaert, “Analysis and design of a foam-cladded PMF link with phase tuning in 28-nm CMOS,” IEEE J. Solid-State Circuits 54, 1960–1969 (2019). [CrossRef]
42. H. Han, H. Park, M. Cho, and J. Kim, “Terahertz pulse propagation in a plastic photonic crystal fiber,” Appl. Phys. Lett. 80, 2634–2636 (2002). [CrossRef]
43. H.-W. Chen, Y.-T. Li, C.-L. Pan, J.-L. Kuo, J.-Y. Lu, L.-J. Chen, and C.-K. Sun, “Investigation on spectral loss characteristics of subwavelength terahertz fibers,” Opt. Lett. 32, 1017–1019 (2007). [CrossRef]
44. H.-W. Chen, C.-M. Chiu, C.-H. Lai, J.-L. Kuo, P.-J. Chiang, Y.-J. Hwang, H.-C. Chang, and C.-K. Sun, “Subwavelength dielectric-fiber-based THz coupler,” J. Lightwave Technol. 27, 1489–1495 (2009). [CrossRef]
45. H. Li, S. Atakaramians, J. Yuan, H. Xiao, W. Wang, Y. Li, B. Wu, and Z. Han, “Terahertz polarization-maintaining subwavelength filters,” Opt. Express 26, 25617–25629 (2018). [CrossRef]
46. J. Ma, M. Weidenbach, R. Guo, M. Koch, and D. Mittleman, “Communications with THz waves: switching data between two waveguides,” J. Infrared Millimeter Terahertz Waves 38, 1316–1320 (2017). [CrossRef]
47. M. Taherkhani, R. Sadeghzadeh, J. Taiber, J. Ornik, and M. Koch, “The effect of humidity and temperature on dielectric fibre-bound THz transmission,” J. Infrared Millimeter Terahertz Waves 40, 1092–1102 (2019). [CrossRef]
48. H. Guerboukha, G. Yan, O. Skorobogata, and M. Skorobogatiy, “Silk foam terahertz waveguides,” Adv. Opt. Mater. 2, 1181–1192 (2014). [CrossRef]
49. K. Nallappan, H. Guerboukha, Y. Cao, C. Nerguizian, and M. Skorobogatiy, “Experimental demonstration of 5 Gbps data transmission using long subwavelength fiber at 140 GHz,” in IEEE Radio and Wireless Symposium (RWS) (IEEE, 2019), pp. 1–4.
50. K. Nallappan, C. Nerguizian, H. Guerboukha, M. Skorobogatiy, and Y. Cao, “High bitrate data transmission using polypropylene fiber in terahertz frequency range,” in International Workshop on Antenna Technology (iWAT) (IEEE, 2019), pp. 81–83.
51. K. Nallappan, C. Nerguizian, H. Guerboukha, Y. Cao, and M. Skorobogatiy, “Signal transmission using solid core terahertz waveguide,” in The 8th International Conference on Optical Terahertz Science and Technology (2019), paper Tu-P-23.
52. F. Voineau, C. Dehos, B. Martineau, M. Sié, M. Perchicot, N. H. Nguyen, A. Ghiotto, and E. Kerhervé, “A 12 Gb/s 64QAM and OFDM compatible millimeter-wave communication link using a novel plastic waveguide design,” in IEEE Radio and Wireless Symposium (RWS) (IEEE, 2018), pp. 250–252.
53. W. Volkaerts, N. Van Thienen, and P. Reynaert, “10.2 An FSK plastic waveguide communication link in 40 nm CMOS,” in IEEE International Solid-State Circuits Conference (IEEE, 2015), pp. 1–3.
54. N. Van Thienen, Y. Zhang, M. De Wit, and P. Reynaert, “An 18 Gbps polymer microwave fiber (PMF) communication link in 40 nm CMOS,” in ESSCIRC Conference: 42nd European Solid-State Circuits Conference (IEEE, 2016), pp. 483–486.
55. K. Nallappan, Y. Cao, G. Xu, H. Guerboukha, C. Nerguizian, and M. Skorobogatiy, “Increasing reliability of terahertz communication links using onboard fiber connectivity,” in 10th Annual Computing and Communication Workshop and Conference (CCWC) (IEEE, 2020), pp. 1065–1070.
56. X. Yu, Y. Hosoda, T. Miyamoto, K. Obata, J.-Y. Kim, M. Fujita, and T. Nagatsuma, “Terahertz fibre transmission link using resonant tunnelling diodes integrated with photonic-crystal waveguides,” Electron. Lett. 55, 398–400 (2019). [CrossRef]
57. Q. J. Gu, “THz interconnect: the last centimeter communication,” IEEE Commun. Mag. 53, 206–215 (2015). [CrossRef]
58. B. Yu, Y. Ye, X. Ding, Y. Liu, Z. Xu, X. Liu, and Q. J. Gu, “Ortho-mode sub-THz interconnect channel for planar chip-to-chip communications,” IEEE Trans. Microwave Theory Tech. 66, 1864–1873 (2018). [CrossRef]
59. W. Withayachumnankul, M. Fujita, and T. Nagatsuma, “Integrated silicon photonic crystals toward terahertz communications,” Adv. Opt. Mater. 6, 1800401 (2018). [CrossRef]
60. X. Yu, M. Sugeta, Y. Yamagami, M. Fujita, and T. Nagatsuma, “Simultaneous low-loss and low-dispersion in a photonic-crystal waveguide for terahertz communications,” Appl. Phys. Express 12, 012005 (2019). [CrossRef]
61. K. Tsuruda, M. Fujita, and T. Nagatsuma, “Extremely low-loss terahertz waveguide based on silicon photonic-crystal slab,” Opt. Express 23, 31977–31990 (2015). [CrossRef]
62. A. Locatelli, G. E. Town, and C. D. Angelis, “Graphene-based terahertz waveguide modulators,” IEEE Trans. Terahertz Sci. Technol. 5, 351–357 (2015). [CrossRef]
63. K. I. Zaytsev, G. M. Katyba, N. V. Chernomyrdin, I. N. Dolganova, A. S. Kucheryavenko, A. N. Rossolenko, V. V. Tuchin, V. N. Kurlov, and M. Skorobogatiy, “Overcoming the Abbe diffraction limit using a bundle of metal-coated high-refractive-index sapphire optical fibers,” Adv. Opt. Mater. 8, 2000307 (2020). [CrossRef]
64. K. Nallappan, H. Guerboukha, C. Nerguizian, and M. Skorobogatiy, “Live streaming of uncompressed 4K video using terahertz wireless links,” in IEEE International Conference on Communications (ICC) (IEEE, 2018), pp. 1–7.
65. K. Nallappan, H. Guerboukha, C. Nerguizian, and M. Skorobogatiy, “Live streaming of uncompressed HD and 4K videos using terahertz wireless links,” IEEE Access 6, 58030–58042 (2018). [CrossRef]
66. R. Ortuño, C. García-Meca, and A. Martínez, “Terahertz metamaterials on flexible polypropylene substrate,” Plasmonics 9, 1143–1147 (2014). [CrossRef]
67. Y.-S. Jin, G.-J. Kim, and S.-G. Jeon, “Terahertz dielectric properties of polymers,” J. Korean Phys. Soc. 49, 513–517 (2006).
68. M. Afsar, “Dielectric measurements of common polymers at millimeter wavelength,” in IEEE MTT-S International Microwave Symposium Digest (IEEE, 1985), pp. 439–442.
69. M. Navarro-Cia, J. Wu, H. Liu, and O. Mitrofanov, “Generation of radially-polarized terahertz pulses for coupling into coaxial waveguides,” Sci. Rep. 6, 38926 (2016). [CrossRef]
70. Y. Yang, A. Shutler, and D. Grischkowsky, “Measurement of the transmission of the atmosphere from 0.2 to 2 THz,” Opt. Express 19, 8830–8838 (2011). [CrossRef]
71. J.-F. Roux, F. Aquistapace, F. Garet, L. Duvillaret, and J.-L. Coutaz, “Grating-assisted coupling of terahertz waves into a dielectric waveguide studied by terahertz time-domain spectroscopy,” Appl. Opt. 41, 6507–6513 (2002). [CrossRef]
72. M. Skorobogatiy, Nanostructured and Subwavelength Waveguides: Fundamentals and Applications (Wiley, 2012).
73. M. Skorobogatiy and J. Yang, Fundamentals of Photonic Crystal Guiding (Cambridge University, 2009).
74. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66, 216–220 (1976). [CrossRef]
75. R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. 43, 899–909 (2007). [CrossRef]
76. G. P. Agrawal, Lightwave Technology: Telecommunication Systems (Wiley, 2005).
77. Y. Yang, A. Shutler, and D. Grischkowsky, “Measurement of the transmission of the atmosphere from 0.2 to 2 THz,” Opt. Express 19, 8830–8838 (2011). [CrossRef]
78. T. Ma, K. Nallapan, H. Guerboukha, and M. Skorobogatiy, “Analog signal processing in the terahertz communication links using waveguide Bragg gratings: example of dispersion compensation,” Opt. Express 25, 11009–11026 (2017). [CrossRef]
79. J. Y. Suen, M. T. Fang, S. P. Denny, and P. M. Lubin, “Modeling of terabit geostationary terahertz satellite links from globally dry locations,” IEEE Trans. Terahertz Sci. Technol. 5, 299–313 (2015). [CrossRef]
80. A. Roggenbuck, K. Thirunavukkuarasu, H. Schmitz, J. Marx, A. Deninger, I. C. Mayorga, R. Güsten, J. Hemberger, and M. Grüninger, “Using a fiber stretcher as a fast phase modulator in a continuous wave terahertz spectrometer,” J. Opt. Soc. Am. B 29, 614–620 (2012). [CrossRef]
81. B. You, J.-Y. Lu, T.-A. Liu, J.-L. Peng, and C.-L. Pan, “Subwavelength plastic wire terahertz time-domain spectroscopy,” Appl. Phys. Lett. 96, 051105 (2010). [CrossRef]