Silver-Coated Teflon Tubes for Waveguiding at 1 – 2 THz

Realization of single-mode low-loss waveguides for 1.0 – 2.0 THz remains a chal-lenging problem due to large absorption in most dielectrics and ohmic losses in metals. To address this problem, we investigate dielectric-lined hollow metallic waveguides fabricated by coating 1-mm diameter 38- μ m-thick polytetrafluoroethylene tubes with silver. These waveguides support a hybrid HE 11 mode, which exhibits low attenuation and low dispersion. Quasi-single-mode propagation is achieved in the band of 1.0 – 1.6 THz, in which the hybrid HE 11 mode is supported by the waveguide. In this band, the experimentally measured loss is ~20 dB/m (~0.046 cm − 1 ), whereas the numerically computed loss is ~7 dB/m (~0.016 cm − 1 ). The difference is attributed to additional losses in the dielectric layer, which can be reduced by using alternative polymers.

The dielectric-lined HMWs belong to the family of oversized waveguides [10,11], which were developed to solve the problem of increasing ohmic losses of single-mode coaxial and rectangular waveguides at millimeter-wave frequencies. By introducing a dielectric coating on the inner metallic walls of an oversized waveguide (as a rule of thumb,~λ 0 /4 thick), the waveguide dominant mode becomes the linearly polarized hybrid HE 11 mode [1][2][3][4][5][11][12][13][14][15][16]. This HE 11 mode is advantageous because it suffers low losses and it couples efficiently to freespace linearly polarized Gaussian beams [11,17]. It also can be coupled to the widely used TE 10 mode of the rectangular waveguide using a relatively simple conversion [11].
The attenuation of the HE 11 mode falls rapidly with the waveguide diameter [2,3,12,18,19]. The diameter to wavelength ratio (d/λ 0 ) of~4.4 or larger (assuming a low-lossdielectric layer) reduces the loss below that of the fundamental TE 11 mode of a silver-only waveguide [18]. For dielectrics with losses, the required diameter to wavelength ratio increases.
Despite the oversized waveguide cross-section, the multimode nature is suppressed in dielectric-lined HMWs as the other modes, e.g., the TE 11 , TM 11 , and TE 12 modes, suffer a significant increase in their attenuation coefficient. This effect is known as self-filtering [11]. The dielectric-lined HMWs as a result can enable quasi-single-mode propagation with transmission loss potentially lower than that of the fundamental modes in the rectangular waveguide (TE 10 ) and in the cylindrical metallic waveguide (TE 11 ) with the fractional bandwidth larger than 50 %.
To realize a polymer-lined HMW with transmission losses <10 dB/m and self-filtering properties for the 0.5-1.5 THz range, the polymer thickness must be between~30 and 120 μm and d>0.9 mm. Depositing uniform polymer layers of such thickness remains a challenge for the dynamic liquid phase fabrication technique [20] used for the 1.5-5.0 THz range. Alternative methods, such as mandrel-winding [5], mechanical-assembling of dielectric and metallic cylinders [11,13,14], and high-temperature bonding [21] do not offer a solution because the minimum diameter d for these methods is limited to~3 mm, which allows undesirable higher-order modes, and fabrication tolerances are excessive.
A way to circumvent the obstacle of fabricating dielectric-lined HMWs for 0.5 THz<v<1.5 THz may be found by metallizing the outer surface of polymer tubes [19], which are cheaply and easily available in a wide range of thicknesses and diameters. Among the large catalogue of polymer tubes, those made of polytetrafluoroethylene (PTFE), commonly known by its trade name Teflon, may be good candidates. PTFE was indeed used for the original millimeterwave dielectric-lined HMWs [11] and it is widely used for many millimeter-wave devices because of its low absorption [22,23]. Although the absorption of PTFE is not the lowest among polymers and it increases with frequency, this material can serve to test properties of the waveguides fabricated by this method and to establish a benchmark for future investigations with low-loss polymers.
To answer the question whether commercial PTFE tubes are suitable for fabrication of the high-performance dielectric-lined HMW for 0.5 THz <v <1.5 THz band, we characterize transmission of THz pulses through PTFE tubes coated with silver experimentally. The extracted transmission loss properties are compared with eigenmode simulations.
PTFE tubes of 1 mm bore diameter and~38-μm-thick walls are chosen for this study. According to initial analytical work, the fundamental HE 11 mode for a waveguide with such dimensions (d/λ 0~4 .4) should have attenuation~10 dB/m, similar to the fundamental TE 11 mode of a silver-only waveguide within the transmission band centered at 1.3-1.4 THz. The silver-only waveguide with the same diameter however is multimode for the whole spectral window studied here.
We fabricate flexible waveguides by coating the PTFE tube with silver. The quasi-singlemode (HE 11 ) propagation is observed after~283 mm of propagation in the waveguide. We find the transmission loss of~20 dB/m within a wide transmission band (~1.0 to~1.6 THz) experimentally using the cut-back loss method and the THz time-domain spectroscopy system. The numerical simulations show evolution of mode profiles for the supported modes and their attenuation with frequency, and explain the quasi-single-mode (HE 11 ) propagation. Comparison of the experimental and simulation results suggests that dielectric losses in PTFE used in this study are larger than nominal. Nevertheless, this waveguide concept realized with other lower-loss dielectric tube can lead to quasi-single-mode waveguides outperforming cylindrical metallic waveguides of similar diameter.

Fabrication and experimental setup 2.1 Fabrication
The silver-coated PTFE hollow waveguide is fabricated with a three-step procedure [19]. First, the outer surface of the 1-mm-bore diameter flexible PTFE tube (Zeus Inc. [24]) is etched using a commercial etchant, FluorEtch©, from Acton Technologies. As the surface energy of PTFE is exceptionally low, this step is necessary to ensure subsequent adhesion of the silver film. Second, a palladium-based sensitizing solution is used to further treat the tube surface. Finally, the silver layer is formed via dynamic liquid phase deposition of ammonia-complexed silver ion and dextrose solutions.
Thickness of the PTFE layer is verified using the transmission spectrum of the waveguide in the infrared region. Figure 1 shows the spectrum obtained using a Tensor Bruker 37 FTIR spectrometer and liquid-nitrogen-cooled mercury cadmium telluride detector in the 5.5-7 μm range for a 283-mm waveguide. Taking the average interference peak spacing and assuming a refractive index of 1.35, the film thickness is determined to be approximately 35 μm, close to the expected thickness of 38 μm [19].

Experimental setup
We characterize the waveguide samples using the near-field THz time-domain spectroscopy method (shown in Section 4). Near-field mapping of the terahertz field at the waveguide output allows us to optimize coupling from the incident terahertz pulse and determine the spatial mode profiles. The cut-back loss measurement method also avoids potential errors due to the variation of the coupling coefficient with frequency [4,25,26].
The waveguide part of the near-field mode spectroscopy and imaging system [4,27,28] is shown in detail in Fig. 2a-c. THz pulses are generated by horizontally polarized (E x ) 100 fs pulses from a Ti:Sapphire mode-locked laser through optical rectification in a 1-mm-thick ZnTe crystal. The generated E x -polarized THz beam is then collimated by a 3-mm-diameter hyper hemispherical lens. The waveguide sample is placed at a distance of~500 μm away from the lens, see Fig. 2b. The output of the waveguide is mounted on an automated xy translation stage for mapping the waveguide modes and it is positioned within~1 mm from the integrated terahertz near-field probe, as shown in Fig. 2c. The near-field probe consists of a low-temperature-grown GaAs photoconductive detector sensitive to the horizontal polarization, E x , and a 10 μm×10 μm sub-wavelength aperture [26,29]. In order to optimize coupling of the incident terahertz beam to the fundamental hybrid waveguide mode and to minimize the excitation of higher-order modes, the xy-position of the input waveguide facet is adjusted while monitoring the output field pattern. The input and the output of the waveguide are enclosed

Numerical modeling
We first review the eigenmodes supported by the PTFE-lined HMW to provide a basis for the analysis of the experimental results.

Transmission losses and field distribution inside the waveguide
In order to model the experiment accurately, we calculate the eigenmodes and the corresponding eigenvalues (dispersion diagram) of the PTFE-lined HMW numerically with the 2D eigenmode solver of CST Microwave Studio TM . The effective dispersive silver conductivity model and two sets of absorption data for PTFE are used (Fig. 3a). The first set represents tabulated dielectric properties of PTFE [22,23]. In the second set, we assume that the imaginary values of the tabulated data are increased by a factor of 4. We note that such absorption above the nominal PTFE absorption is comparable to the absorption values reported for other polymers [30,31]. A detailed description of the simulation can be found in Appendix: Eigenmode numerical analysis.
The attenuation coefficient curves for the first four modes that are most likely to be excited under the on-center illumination in our experimental system are shown in Fig. 3b. Electric field profiles of these modes are shown in Fig. 4. At very low frequencies, the PTFE film is electrically negligible, and thus, the PTFE-lined HMW supports similar modes as a silver-only HMW [25,[32][33][34]. Hence, we denote the discussed modes in the case of the PTFE-lined HMW as TE 11 * , TM 11 * , TE 12 * , and TM 12 * . Fig. 3 a Material properties [the real part of the PTFE permittivity is ε=2.074]. b Numerically computed attenuation for the fundamental modes: first (magenta), second (violet), third (cyan), and fourth (green); solid and dotted lines account for nominal PTFE according to tabulated data ε and the lossy PTFE with 4ε, respectively Attenuation of the TE 11 * mode, which has the lowest cut-off frequency mode (magenta line in Fig. 3b), increases drastically with frequency. Near its cut-off frequency, the mode profile is identical to the TE 11 mode of the HMW. At high frequencies, the TE 11 * mode evolves into a field distribution highly localized in the PTFE layer (Fig. 4), resulting into a fast increase in its attenuation coefficient (Fig. 3b) [34]. Above 0.75 THz, the higher-order modes show significantly lower attenuation.
The second mode (TM 11 * , violet line in Fig. 3b) suffers some increase in attenuation within the lowest part of the spectral window. However, at~0.75 THz there is an inflection point and the attenuation drops, despite the increase in material absorption (Fig. 3a). A transmission band is formed for this mode within 0.9-1.7 THz with the minimum attenuation of~6.6 dB/m (~0.015 cm −1 ). Within this band, waves travel as the HE 11 1 mode (Fig. 4), where the superscript denotes the order of the transmission band. The hybrid HE 11 1 mode has negligible field at the walls, and therefore the dielectric and conductive losses are minimized. We note that the other modes have significantly higher attenuation within this frequency range, i.e., self-filtering is expected. For ν>1.7 THz, the attenuation of this second mode suddenly rises as a result of field concentration in the top and bottom air-dielectric interface (Fig. 4).
The third mode (TE 12 * , cyan line in Fig 3b) has two distinct transmission bands: 0.6 THz <ν<1.2 THz and for ν>2.2 THz. The TE 12 * mode maintains the field distribution almost uniformly until the absorption band centered at~1.9 THz. Hence, its attenuation coefficient does not show any sudden change at low frequencies. In the second band, the mode transforms into the HE 11 -like mode (Fig. 4). To differentiate it from the lower-frequency HE 11 mode, we will use the following notation: HE 11 2 .
Finally, the attenuation of the fourth mode (TM 12 * , green line in Fig. 3b) decreases gradually with frequency, displaying relatively low attenuation for ν>2.2 THz. The field distribution remains almost constant, which explains the gradual changes observed in its attenuation coefficient.  For both hybrid HE 11 modes, the transmission bands become narrower and the attenuation increases when higher dielectric losses are considered in the modeling [4,18]. Specifically, the HE 11 1 mode transmission band is limited between 1.0 and 1.6 THz with a minimum attenuation of~16.2 dB/m (~0.037 cm −1 ).

Impact of bending and metal model in transmission losses
We investigate robustness of the effective single-mode HE 11 1 propagation against bending by computing the total attenuation as a function of the bending radii with Lumerical MODE Solutions (further details can be found in Appendix: Eigenmode numerical analysis). Losses due to bending in the plane parallel to the polarization (xz) are shown in Fig. 5a, and in the plane perpendicular to the polarization (yz) in Fig. 5b for the HE 11 1 , TE 12 * , and TM 12 * modes. Although the total attenuation follows different trends for each mode, it is evident that the effective single-mode HE 11 1 propagation endures within the spectral window of interest (1.0-1.6 THz) for bending radii larger than 25 mm.
We estimate the effect of metallic losses within the transmission band numerically by considering four models for the metallic walls. The minimum loss value of 3.2 dB/m is found for perfect electric conductor, 5.3 dB/m for bulk silver (σ Ag =6.3×10 7 S/m) and 6.6 and 7.0 dB/m for silver with surface roughness of 60 and 100 nm (RMS), respectively. These results indicate that surface roughness has a small effect on the overall transmission loss.

Dispersion: modal group delay
The numerically computed eigenvalues allow us to construct the expected modal group delay for the two sample lengths used in this study: 109 and 283 mm. These results are plotted in Figs. 6a and 7a, where the absorption band is clearly marked as a gray rectangular box.
By inspection of Figs. 6a and 7a, it is evident that the high-frequency HE 11 2 mode arrives earlier than the HE 11 1 mode. According to the numerically computed group delays, the hybrid HE 11 2 mode is the only mode present with the spectral components above 2.2 THz for t<5.5 ps in the launch waveguide and for t<17.9 ps in the 283 mm long waveguide. Within the intervals Fig. 5 Numerically computed attenuation for the second (violet), third (cyan), and fourth (green) modes for bending radii from infinite (i.e., straight waveguide) to 25 mm: a xz-plane of bending, and b yz plane of bending. To discern the evolution of the attenuation with the bending radii, the modal loss for the straight waveguide is shown by solid black lines. Insets in each panel show the bent waveguide along with the HE 11 (ν=1.3 THz) mode profile for a bend radius of 25 mm 5.5-10 ps and 18-27 ps for the two waveguides, respectively, the HE 11 1 mode is the only mode present. At later times, the TM 12 * mode (green line) with frequency components above 2 THz is also present. Since this TM mode has relative low loss at high frequencies, signs of multimode propagation may be expected experimentally in the waveforms for time delays beyond~10 and 27 ps for the two waveguides, respectively. In frequency domain, we then expect spectra relatively clear of mode interference peaks within the transmission band 1.0-1.6 THz. Although in principle multimode propagation is expected within this band, the large attenuation of the TE 12 * and TM 12 * modes results in quasi-single-mode (HE 11 1 ) propagation. The comparison of the modal group delay for each length exposes an issue for the calculation of the transmission spectrum. For the longer waveguide, the higher-order TE 12 * mode (cyan line) arrives after the measured interval of 140 ps for ν<1.0 THz. The recorded waveform therefore is missing the frequency components from 0.75 to 1.0 THz for the 283- (green) modes in the launch waveguide. The gray rectangle represents the absorption band that defines the border between the HE 11 1 and the HE 11 2 mode band. b Waveform detected on the waveguide axis. c Power spectral density of the measured waveform shown right above b. The numerically computed group delay for the TM 11 * (dotted violet) and the TM 12 * modes (dotted green) are superimposed to facilitate the comparison. d, e Normalized near-field |E x | 2 profiles at the waveguide output at 2.73 (d) and 8.73 ps (e). The yellow circle outlines the waveguide walls to guide the eye mm-long waveguide. If the input terahertz beam is coupled initially to the TM 11 * mode and the TE 12 * mode, the calculation of the transmission spectrum overestimates losses within the 0.75-1.0 THz frequency span. Indeed, excitation of both modes is likely at low frequencies since both modes have similar overlap integrals with the Gaussian beam. A similar argument is applied to ν<0.75 THz, for the TE 11 * and TM 12 * modes. Therefore, for ν<1.0 THz, the experimentally measured losses can only provide an upper bound.

Near-field time-domain measurements and discussion
The output waveform and E-field xy-maps are measured for the launch waveguide followed by the same set of experiments for the 283-mm-long waveguide, which is constructed by Fig. 7 a Numerically computed group delay for the TE 11 * (magenta), TM 11 * (violet), TE 12 * (cyan), and TM 12 * (green) modes in the 283 mm long waveguide. The gray rectangle represents the absorption band that defines the border between the HE 11 1 and the HE 11 2 mode band. b Waveform detected on the waveguide axis. c Power spectral density of the measured waveform shown right above b. The numerically computed group delay for the TM 11 * (dotted violet) and the TM 12 * modes (dotted green) are superimposed to facilitate the comparison. d, e Normalized near-field |E x | 2 profiles at the waveguide output at 11.73 (d) and 24.6 ps (e). The yellow circle outlines the waveguide walls to guide the eye attaching an additional segment of 174 mm to the launch waveguide. This approach allows us to determine the transmission loss produced by the 174-mm segment.
The waveforms detected in the center of the waveguide output for the launch and joined samples are shown in Figs. 6b and 7b, respectively. Two pulses can be resolved in both waveforms. Using the modeling results, we could tentatively assign them to the HE 11 2 and HE 11 1 modes. Indeed, the intensity xy-map at the peaks of the leading and second transmitted pulses reveals HE 11 mode profile for both pulses (Figs. 5d, e and 6d, e), confirming the assumption.
Signs of multimode interference are also noticeable for the second pulse. Namely, the coherent oscillations in the pulse are interrupted at~10 and~27 ps for the launch and 283-mmlong waveguide, respectively. This is also in agreement with the numerical results.
The mode interference can be observed in the spectrograms, i.e., time-frequency-maps, computed for each waveform (Figs. 6c and 7c). The waveform is divided into 28 Hammingwindowed segments with 90 % overlap. At earlier time delays, only the spectral component at around 2.4 THz, which is linked to the HE 11 2 mode, is observed. The strongest signal is at 1.1-1.6 THz in both spectrograms. The HE 11 1 mode is expected in this range. Low dispersion can be seen near the band center. The lack of signal around 1.9 THz is linked to the absorption band.
In both spectrograms, the HE 11 1 mode (violet line) forms a tail due to large dispersion for v <1.1 THz, when it changes to the TM 11 * mode. The TM 12 * mode (green line) also yields another tail for 1.3 THz<v<2.5 THz. For the 283-mm-long waveguide, the trace associated to the TM 12 * mode is almost absent, suggesting that quasi-single HE 11 mode propagation is obtained after 283 mm within the HE 11 1 transmission band. The waveguide transmission spectrum is calculated by taking the ratio of the power spectra (Fig. 8a). Figure 8b presents the attenuation in decibels per meter, α dB . A signal-to-noise ratio above 10 is achieved within a frequency span from 0.5 to 2.75 THz (spectral window highlighted by the orange box in Fig. 8a). This spectral region includes the fundamental transmission band, the absorption band and the low frequency part of the second transmission band [4,5,19,[32][33][34].
In the transmission band, 1.0 THz<ν<1.6 THz, where the hybrid mode HE 11 is fully developed, losses decrease to~17 dB/m (~0.039 cm −1 ) at approximately 1.3 THz. To put this value of attenuation in context, it is worth looking at commercial single-mode hollow waveguides. For instance, the rectangular waveguide WR-0.65 used by Virginia Diodes Inc. to cover the bandwidth 1.1-1.7 THz (similar to the transmission band here) has an estimated waveguide loss between 258 and 406 dB/m at the low and high end of the band, respectively [35], which is hundreds of decibels higher than our experimentally reported waveguide loss. The experimental spectrum is consistent with the prediction that the transmission loss is minimal for this waveguide at~1-2 THz. The spectrum quantitatively matches the transmission band from the simulation with dielectric absorption larger than the nominal used before (i.e., 4ε Teflon ″ ). The rapid oscillations in this band over the baseline are caused by multimode propagation involving the dominant HE 11 1 and the TM 12 * modes. The more pronounced rapid oscillations observed for ν>1.6 THz in both waveform spectra (Fig. 8a) as well as in the transmission spectrum (Fig. 8b) suggest multimode propagation, but with a less defined dominant mode. In this range, the HE 11 and TM 12 * modes travel with similar attenuation (see Fig. 8b), and thus, there is not a distinct dominant mode producing a clear baseline. This is in agreement with the spectrograms and the modeling.

Comparison with the literature
Different fabrication techniques have been used for dielectric-lined HMWs. For v>0.5 THz, mandrel-wiring [5] and dynamic liquid phase deposition [20] can produce waveguides with reduced diameters required for suppressing higher-order modes and for waveguide flexibility. The reported characteristics of such waveguides are summarized in Table 1, and they will be discussed afterwards.
These two techniques, however, have not produced waveguides with d < 3 mm (Fig. 9). There is still a niche for fabrication dielectric-lined HWMs with radii above the d/λ 0 =4 line, where they outperform metal-only waveguides of the same diameter. The results presented here show that the coating of polymer tubes gives access to such restricted area. Table 1 shows the transmission loss of different terahertz-dielectric-lined HMWs reported in the literature that operate in the spectral window of interest for this paper together with our initial findings with PTFE for the lowest part of the terahertz-gap, i.e., 0.5 THz. The attenuation for our previous PTFE-lined HMW was lower than that reported here despite its smaller d/λ 0 . This is explained by the significantly lower absorption of PTFE (Fig. 3a) at ν<1 THz, the higher effective conductivity of Ag and the lower influence of the roughness.
HMWs coated with polystyrene (PS), polyethylene (PE), or cyclic olefin copolymer (COP) have shown very low attenuation when the d/λ 0 is larger than 15. Specifically, attenuations one order of magnitude lower than the attenuation here reported were measured. This is, in principle, expected given the rapid drop undergone by the attenuation with the diameter size [2,3,12,18,19]. Indeed, according to our estimations, experimental transmission loss below 2 dB is achievable for Ag/PTFE waveguides provided larger diameters are used. Nevertheless, given the exceptional low absorption of COP at terahertz frequencies, Ag/PTFE HMWs should not outperform Ag/COP HMWs in terms of transmission loss. However, the range of COP tubes with different diameter and size is not yet as extensive as PTFE tubes, which limit the flexibility of the fabrication approach presented.

Conclusion
We characterize for the first time PTFE-lined hollow metallic waveguides operating at 1.0-2.0 THz experimentally using near-field time-domain measurements and numerically using eigenmode calculations. We report effective quasi-single-mode propagation for the 1.0<ν<1.6 THz (i.e., HE 11 1 mode band) with a baseline transmission loss 20 dB/m (~0.046 cm −1 ) and the multimode propagation for ν>1.6 THz.  The fabrication approach employed here to produce a quasi-single-mode dielectric-lined hollow waveguide holds promise for the 1.0-2.0 THz range owing to its simplicity (coating commercially available tubes) and because the use of alternative fabrication techniques remains elusive. The proposed approach is universal and can be applied to any other polymer provided the suitable etching agent is used such that a silver film can be effectively deposited. Therefore, it is necessary to investigate other polymers with known lower absorption values in the region of interest such as COP.
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