1. Introduction

Technology utilizing THz radiation has increased dramatically in recent times, with demonstrations such as the realization of metamaterials and negative index [1] at these frequencies. From a more practical aspect, unguided free-space THz radiation utilized in non-destructive imaging or diagnostics is one example of an application that can potentially tap into a vast market. Beyond the in situ use of THz radiation for material characterization [2], there exists a demand for flexible THz transportation using waveguides and as such, a growing interest in the study of THz waveguides made of dielectrics and metals has been observed over the last decade. Dielectric waveguides such as polymer ribbons [3] and wires [4], sapphire fibers [5], photonic crystal slabs [6] and dielectric-core photonic crystal fibers (PCF) [7,8] have all been demonstrated, but are ultimately limited by the material absorption of the dielectric used. In the pursuit of an improvement beyond these, research has been directed towards waveguides that allow transmission unhindered by dielectric losses, with the THz radiation propagating essentially in air.

THz guidance in air can be achieved by using hollow metal waveguides, such as metallic tubes [9], dielectric-lined hollow metal waveguides [10] and parallel plates [11], but is not limited to this approach. Hollow-core dielectric PCF waveguides such as Bragg fibers [12,13] and kagome fibers [14] offer an alternative, as does propagation on the surfaces of bare wires [15] and metal sheets [16]. The transverse electromagnetic mode (TEM) and the Sommerfeld-wave mode have been demonstrated to propagate along the air-metal interface of the latter two examples, with minimal losses and low dispersion. The propagating TEM modes however do suffer to some extent from diffraction losses due to the incomplete confinement of the guide [17], in comparison to the modes in the other enclosed metallic waveguides such as hollow tubes. Meanwhile the Sommerfeld-wave mode is inherently susceptible to bend losses, with the bare wires also being constantly exposed to perturbations and being difficult to handle [18].

To improve the usefulness of wire-based waveguides, examples consisting of embedding wires in Styrofoam and using a plastic-coated cable have been reported [19]. Also, a recently reported metallic-grating hollow waveguide [20] was fabricated from sheets of copper lines transferred onto a polymer substrate via photolithography techniques before being rolled on a cylindrical model and fused by thermal processing methods. However the fabrication techniques reported [19,20] seem to limit the available options of wire dimensions and waveguide materials. In this paper we investigate the propagation of THz pulses in a hybrid metal-dielectric air-core fiber with embedded indium wires. The complexities of waveguide construction can be reduced by using PCF fabrication technology; we are able to create a hollow guiding region that is unhindered by the supporting material of the wires without the expense of constrained molds. The advantages of exploiting this fabrication method are that the spatial distance between the wires can be controlled with ease and precision, and the inclusion of thinner metal wires becomes possible since the drawn fibers are not prohibited by manual mounting and fragile supports. The use of a suitable dielectric host can also enhance the fiber flexibility. Indeed, this fabrication method of co-drawing polymers with indium has already been demonstrated to be a propitious option in manufacturing metamaterials alongside the more intricate methods of photolithography [21].

In this work, we report on the fabrication of these hybrid fibers and their characterization using THz time domain spectroscopy (TDS). We investigate configurations with two and four metal wires surrounding the hollow core, and report on numerical studies of such designs for comparison. We demonstrate that THz pulses can be guided with low loss in such hollow-core fibers with metallic inclusions.

2. Method of experiments, fiber fabrication and numerical simulations

A standard free-space THz-TDS setup as in Fig. 1(a) was used to characterize the fibers, where both the THz generator and detector consisted of photoconductive antennas. The THz beam was collimated and focused using specially designed symmetric-pass (s-p) lenses [22] made of ultrahigh molecular weight polyethylene (UHMWPE), one of the low-loss polymers widely used for THz radiation, instead of the commonly used off-axis parabolic mirrors. A pair of s-p lenses with a focal length f = 75 mm (NA is 0.33) was used for coupling the THz pulses to and from the fibers. The fibers themselves were made by drilling a preform made of Zeonex [8], another polymer with low loss in the THz region, and drawing it to the desired dimensions. Molten indium was inserted via suction into the selected (two or four) holes to form the conducting wires. Indium was chosen because of its low melting temperature (156.6 °C). The fibers investigated had a core diameter of approximately 2 mm measured as the distance between the edges of the parallel wires. The wires had an average diameter of 1.0 mm along the semi-minor axis and 1.8 mm along the semi-major axis. An image of the fiber endface with two wires is shown in Fig. 1(b).

 

Fig. 1 (a) Schematic of THz-TDS setup using photoconductive antennas as a THz emitter and a THz detector to characterize the fibers. The THz emission is modulated with 40V amplitude at 28 kHz. The THz electric field direction is out of the page. (b) Micrograph of the fabricated two-wire fiber sample with core diameter of about 2 mm. The indium wires are located at the left- and right-hand side of the hollow core and are seen as optically silvertone in this micrograph. The four-wire fiber samples have all cladding holes filled with Indium.

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The lengths of fiber used were 5 to 10 cm for the two-wire configuration, and 5 to 8 cm for the four-wire configuration. The Fourier transform of the recorded electric field of the reference (without the fiber sample) and that through the fibers were used to collate the spectral information for the fibers in each case. The power attenuation constant was determined from the ratio of the absolute square of the complex modulus of the transformed data for different fiber lengths. The coupling to fibers for samples of different length was assumed to be constant, as the average of tens of scans was used for each of the fiber measurements. The phase of the radiation transmitted through the fibers was compared to the reference scan to determine the mode effective phase indices.

The numerical investigation of the fiber properties was conducted using the finite-difference-frequency-domain solver employed by MODE [23]. The simulations used an image of the transverse cross-section of the fiber, with the appropriate material data, i.e. the refractive index and attenuation constant of Zeonex [8], and the complex permittivity of indium [24]. Alternatively, indium was treated as a perfect conductor.

3. Fiber characteristics

3.1 Mode propagation in two-wire and four-wire configurations

The measured temporal signals of the reference and through the two-wire and four- wire fibers are plotted in Figs. 2(a) and 2(b). For the two-wire configuration, the electric-field polarization was oriented in the plane of the wires. It is seen that the centers of the pulses from the different fibers are only displaced by a few ps relative to the reference. This delay is expected as the group index of the propagating modes will be greater than 1, and indeed was calculated to range between ~1.01 at higher frequencies, and 1.06 at lower frequencies in the range of interest (0.6 to 1.1 THz), as shown in Fig. 2(c). The chirping observed in the fiber output pulses in Figs. 2(a) and 2(b) is also in qualitative agreement with this decrease in group index at higher frequencies.

 

Fig. 2 Measured temporal signals through several lengths of (a) two-wire and (b) four wire fibers. Inset in (a) is the input reference pulse. Vertical dashed cyan lines indicate the position of the amplitude peak, to highlight time delays from 1 to 2 ps for the different fiber lengths. All temporal signals through the fibers are on the same scale. (c) The calculated group index for the fibers shows a group index is close to that of air (n = 1.0).

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The mode guided through the fibers in the frequency range investigated is the HE₁₁-like mode. The scanned mode profiles are shown in Figs. 3(a) and 3(b) for each fiber, in agreement with the simulation results, and furthermore the simulated mode fields show non-zero E_z and H_z components (Figs. 3(c) and 3(d)), confirming the HE₁₁-like behavior. The measured mode profile is taken from raster scanning the output end of the fiber with a metal pinhole with a diameter of 0.8 mm in the focal plane of the s-p lens. Our numerical simulations indicate that higher order modes show markedly higher loss. The high losses of the other higher order modes and the better mode-matching efficiency with the Gaussian-profile input beam contribute to the fundamental mode being the only mode observed. We also note that the annular TE₀₁ mode, the lowest loss mode found in circular hollow metallic waveguides [25] and also present in dielectric-lined hollow metallic waveguides [10], did not emerge in the simulation results of either the two-wire or four-wire fibers, nor was any evidence of it observed experimentally. The discontinuity of the fiber boundary conditions in the azimuthal direction is thought to prohibit the manifestation of such azimuthally polarized modes. We confirmed in simulations that in an ideal, thin dielectric-lined hollow metal tube with the same parameters used in this work, the TE₀₁ mode is indeed supported in the core. However, introducing an azimuthal discontinuity in the metal akin to the fibers presented in this work caused this mode to no longer be supported.

 

Fig. 3 Experimentally observed mode field intensity (normalized, linear plot) at 0.8 THz from a 5 cm length of (a) two-wire and (b) four-wire fiber. In the two-wire fiber in (a) the metal wires are aligned in the x-direction. The white vectors show the simulation results of magnitude and direction of the mode electric-field components obtained from MODE. The mode simulation plots at 1.2 THz for (c) H_z and (d) E_z components for the two-wire fiber show the hybrid mode characteristics, having non-zero longitudinal components. The simulated mode at (e) 0.65 THz and (f) 1.2 THz for the two-wire fiber shows some energy leaked into the cladding holes for the lower frequency, compared to the near-round shape at higher frequency. In all (c-f) plots, the color map is linear and the metal is in the left and right side of the hollow core. (g) The full-width-half-maximum (FWHM) of the modes as calculated from the simulations for the two-wire (blue) and four-wire (red-filled triangle) fibers. The FWHM experimental data are obtained from the mode scan of the two-wire (black-hollow square and black-filled circle) and the four-wire (black-hollow triangle) configurations. The FWHM for the four-wire configuration has one-standard deviation (black vertical line). The average FWHM for the two-wire configuration is calculated separately along the axes with and without wires.

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At low frequencies the spatial distribution of the modes become increasingly distinct as differences develop in the two-wire fiber along the axes that have/lack the wires as shown by the simulated mode in Figs. 3(e) and 3(f). The two-wire and four-wire fibers do not differ much in the shape of the mode at high frequencies, clearly seen in Fig. 3(f). In Fig. 3(g), the full-width-half-maximum (FWHM) of the Poynting vector in the direction of propagation (P_z) is plotted for each fiber. The simulated mode areas from both fibers were fitted with an elliptical function to approximate the mode widths – for circular shaped modes, a single width was returned. The directions of the axes of the ellipses were found to coincide with the positions of the wires, as expected. With the four-wire fiber, the mode size was almost identical along both axes, and this was averaged and plotted as red-filled triangles in Fig. 3(g). In the two-wire configuration, the mode is elliptical (plotted as blue-empty circles and filled squares in Fig. 3(g)), most pronounced at low frequencies. At these lower frequencies/longer wavelengths there is some leakage into the air holes adjacent to the core, as well as the mode re-shaping (contracting) in the plane of the two wires. The former does not greatly influence the mode shape as observed in Fig. 3(g) (blue-empty circles), where the mode FWHM is seen to remain approximately constant. The latter effect is more prominent, also seen in Fig. 3(g) (filled squares). Figure 3(e) shows that as the central lobe of the mode reshaped in the plane of the wires, two additional lobes emerge concentrated on the thin dielectric layer on the metal surface. The wire separation in Fig. 3(e) is approximately 4λ(at 0.65 THz), and this weak additional concentration of field near the metal surface is reminiscent of surface plasmon modes. Such modes are prone to some radiation losses at randomly uneven surfaces [26] along the fiber axis. From the scanned mode profiles, we calculated the mode widths with respect to the axis that have/lack metal wires, and the measurement give similar curves to the calculated FWHMs from the simulation. The discrepancies of the values from the measured and simulated FWHMs are attributed to the variance in the measured fiber core diameters. We note that the asymmetry in the mode shape is not extreme, with the mode size differing by up to only 20% along the two axes. However, this will cause the mode-matching to the input Gaussian beam to be slightly reduced in the two-wire fiber compared to the four-wire configuration.

3.2 Attenuation constants

Figures 4(a) and 4(b) shows the power absorption coefficient determined from the experiments, along with the simulated values for the two-wire and four-wire fibers. The experimentally measured average power attenuation coefficients were 0.3 cm⁻¹ for the two-wire and 0.5 cm⁻¹ for the four-wire configurations averaged over several lengths of fiber. Overall, the experimentally determined data show good agreement with the values obtained from the simulations. The simulated attenuation was not affected when the indium was replaced with a perfect electric conductor, indicating that metallic losses were not significant. However, scattering loss due to the metal is not computed in this simulation procedure. Discrepancies between the measured loss values and the simulated values originate from defects in the fiber samples used in the experiments such as non-uniformities along the core walls and irregularities of the indium surfaces.

 

Fig. 4 Loss coefficients as measured (grey) and as simulated (red line) for (a) two-wire and (b) four-wire fibers. The plotted vertical thin (grey) lines represent one standard deviation of measurement values. The green regions in both plots indicate the frequencies where the simulations gave no solutions to the eigenvalue problem formulation employed by MODE.

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The simulation data also show a trend of decreasing attenuation with higher frequencies. From our simulations, we observed that the mode field extent at the core-cladding boundaries is substantially weaker at higher frequencies. This behavior is clearly seen from the plots of the E-field in Figs. 3(e) and 3(f) for the two-wire fiber. In the cladding the E-field will experience a higher attenuation, and can also more easily leak into cladding modes. At higher frequencies, where the fields do not extent further outward, radiation loss is minimized thus reducing the mode attenuation. The anomalous attenuation behavior is also reported in [17] and discussed in [27] for another class of mode, the TE₁ mode, in the parallel-plates configuration. Due to certain symmetries, other conducting waveguides can also exhibit similar loss behavior for a particular mode, the TE₀₁ mode [28,29]. We note that the numerical simulations show low-frequency “cut-offs” for the two-wire and the four-wire configuration, at 0.57 THz and 0.64 THz respectively, below which solutions for the HE₁₁-like mode were not found. The experimental data however show slightly lower cut-off frequencies, at around 0.5 and 0.6 THz, below which the loss increases. We associate this onset of higher loss with the HE₁₁-like mode escaping from the core. We believe that this hybrid fiber low-frequency mode cut-off is related to the increase of the mode field diameter in the hollow core as this causes more coupling to the cladding modes.

The effect of the two-wire orientation with respect to the input beam electric field direction was also investigated in the experiment. The results presented thus far had the field oriented in the plane of the wires, and hence perpendicular to the metallic interface. Using a fiber sample of 8 cm in length, the fiber was rotated azimuthally so that the incident electric field polarization was either parallel or perpendicular to the two-wire plane. The comparison of the transmission spectra is shown in Fig. 5. In both cases, the same effective low-frequency cut-off was observed, around 0.5 THz, and the propagating mode was the HE₁₁-like mode as discussed in Section 3.1. The transmitted power for the polarization perpendicular to the wire plane was reduced compared to the polarization parallel to the wire plane. This and related effects such as birefringence in the fibers will be investigated in more detail at a later instance.

 

Fig. 5 Normalized transmission spectrum of 8 cm two-wire fiber with different wire plane orientation with respect to the electric field polarization as indicated. The indium wires are depicted as the grey ellipses, while the air region is the filled white space. The grey area indicates the noise floor of the measurement.

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3.3 Effective phase index and dispersion parameter

The effective phase indices for both fibers shown in Fig. 6 indicate that they are close to the air index. The phase details extracted from the measurement data are corrected by integer multiples of 2π due to the undetermined phase shift at very low frequencies. It is also noted that the phase extracted from the measurement are not immune to additional random phase noise, especially at the frequency range where the measured mode signal-to-noise ratio is low. The second derivative of the phase index is related to the group velocity dispersion (GVD), β₂ parameter. We adopt the use β₂ parameters commonly practiced in the optics regime to describe pulses broadening to reflect the significance of ultra-wideband propagation of THz waves. This value is calculated from the measured phase index data and plotted for both of the fibers, shown in Figs. 7(a) and 7(b). The values extracted from the measurements show rapid fluctuation, where the artifacts influence the meaningful data. We therefore applied a moving average filter to the data presented here whilst at the same time preserving the pattern of the data.

 

Fig. 6 Phase refractive indices as measured (grey) and as simulated (red line) for (a) two-wire and (b) four-wire fibers. The vertical lines indicate one standard deviation of the measured data. The green regions in both plots indicate the frequencies where the simulations gave no solutions to the eigenvalue problem formulation employed by MODE.

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Fig. 7 The group velocity dispersion (GVD) parameter determined from the measured data (grey) and the as calculated from the simulation (red line) for (a) two-wire and (b) four-wire fibers. The green regions in both plots indicate the frequencies for which the simulations give no solutions to the eigenvalue problem formulation employed by MODE.

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For the two-wire configuration, the experimentally determined data give on average a dispersion magnitude of the order of 5 psTHz⁻¹cm⁻¹ within the frequency range of 0.65 to 1.0 THz; the four-wire configuration has less than 5 psTHz⁻¹cm⁻¹ between 0.7 to 0.95 THz. For comparison, an explicit β₂ value of 8 psTHz⁻¹cm⁻¹ is reported in [10] for dielectric waveguides with metallic inclusions, and the data obtained here strongly indicate comparable low dispersion propagation achieved with the two-wire and four-wire configurations. We note that qualitative comparison with the data obtained from other works on metallic inclusion waveguides (e.g [9,11]. among others) shows similar trend of dispersion behavior: steeper rise near the cut-off frequencies and flatter curves towards higher frequencies. Simulation data for these fibers suggest that the two-wire configuration is a better candidate than the four-wire configuration in managing lower dispersion pulse propagation, as it is seen in Fig. 7 where an average of zero dispersion is feasible from 0.6 to 1.05 THz. Nonetheless, both fibers exhibit calculated β₂ values of less than 10 psTHz⁻¹cm⁻¹ in the frequency range of 0.6 to 1.05 THz. From the perspective of guided modes in a hollow core fiber, we note that this is a marked improvement in comparison to that obtained from some all-dielectric PCFs [14].

4. Conclusions and outlook

We demonstrated THz guidance in an air core fiber with two or four embedded indium wires. The average measured power loss was 0.3 cm⁻¹ for the two-wire fiber, and 0.5 cm⁻¹ for the four-wire fiber. The measured data are in good agreement with simulations. The dominant guided mode is a HE₁₁-like mode. We observed experimentally that both polarizations; parallel and perpendicular to the wire plane in the two-wire case, are guided and with similar low-frequency cut-offs, albeit a reduced transmission amplitude was observed in the latter. It is also noted that the TE₀₁ mode was not observed either in experiments or simulations. Also, no higher order modes were observed experimentally.

Comparing the two-wire and four-wire configurations, we note several trends in the fibers. First, the attenuation constants for both of the fibers decrease with higher frequencies and above 0.85 THz there is no significant difference in the loss figures, remaining at approximately 0.2 cm⁻¹. Secondly, the simulation results show close-to-zero dispersion propagation is feasible in the two-wire configuration from 0.65 to 1.05 THz. The four-wire configuration fiber exhibits slightly higher β₂ parameter, however still not exceeding 10 psTHz⁻¹cm⁻¹ over the same frequency range. The two-wire configuration gave a narrowed spatial mode distribution in the plane of the wires as the wavelength increased; this variation however was small at higher frequencies. In contrast, the mode shape in the four-wire fiber did not vary with frequency, resulting in different coupling efficiencies for the two fibers.

In the context of the performance of our metallic inclusion fibers, these fibers offer an approximate two-fold reduction in the loss compared to hollow multimode metallic tubes [30] and have comparable loss to manually mounted parallel-plate configurations [11]. For waveguides with approximate dimensions as the fibers presented here, they offer at least three times reduction in loss compared to broadband air-core kagome fibers [14] within the same frequency range. Improvement of the metallic inclusion fiber designs and fabrication is anticipated to enable low loss HE₁₁ mode guidance as demonstrated in [7,8,20].

In conclusion, we demonstrated THz guidance through a hollow-core dielectric fiber with metallic wire inclusions. We found that the mode diffraction losses and radiation losses in open boundary waveguides such as parallel-plates and dual wires configurations can be eliminated in the proposed hollow-core fibers with embedded metallic wires. Work towards reducing the current loss values of these fibers should include improving the fabrication techniques to ensure smoother metal surface contact to the dielectric film by the core region as well as maintaining uniformity of the shape of the metal wires. Future work will also concentrate on the birefringence of the two-wire configuration.