1. Introduction

In order to enable various stand-off and remote THz applications¹ a considerable amount of research has been focused on developing low-loss waveguides and fibers as a key technology for the flexible and convenient delivery of THz light. Given relatively high material losses of the dielectric and metallic materials in THz region it seems that the only viable solution for designing low-loss THz waveguides is through maximization of the fraction of power guided in low-loss gas regions. The simplest waveguide that operates on this principle is a subwavelength wire featuring large evanescent fields that extend far beyond the waveguide core and into the surrounding gaseous cladding. Both the metallic² and dielectric³ subwavelength fibers have been demonstrated to have propagation losses on the order of 0.01 cm⁻¹ in the vicinity of 0.3 THz, which are among the lowest losses reported to date. Metallic wires, in particular, have been shown to support Sommerfeld plasmons^{2, 4, 5} offering a low dispersion, low propagation loss single-mode regime over a wide range of frequencies. However, the radially polarized Sommerfeld mode is difficult to excite and specialized sources are required to increase the coupling efficiency.⁶ Recent efforts on metallic wires have demonstrated the use of a dual-wire suspended in air waveguide,⁸ which supports a more easily excitable TEM mode at the expense of a more complicated waveguide geometry that requires maintaining a constant sub-millimeter distance between the two wires. On the other hand, dielectric subwavelength wires^{3, 9–11} feature a low-loss single-mode regime with a linearly-polarized HE₁₁ fundamental mode which is very easy to excite with a gaussian-like beam of a THz source. Main disadvantage of the dielectric wires is a relatively high dispersion, and a limited operational bandwidth due to onset of large absorption losses at higher frequencies. Moreover, both types of subwavelength fiber feature extremely delocalized modes (as large as several mm in diameter), which are susceptible to high bending loss and strong proximity cross-talk with the environment. In some implementations, strong mode delocalization can be used for a benefit to enable, for example, non-destructive cut-back-like measurements of the fiber propagating losses using an evanescent directional coupler technique.^{10, 12} Another potential benefit of the subwavelength waveguides is in THz optical sensing which relies on a strong presence of the modal evanescent fields in the analyte.¹³ While bending loss of the subwavelength fibers has been shown to be very high,5 the deflection loss^{7, 11} (micro bending at a single point point) can be relatively small provided that the deflection angle is also small (< 2°). The group of Sun et al. has exploited low-loss fiber deflections to make a THz fiber-based imaging system^{10, 11} and a near-field microscope.¹⁴

In this paper, we report several fabrication techniques, as well as spectral transmission and propagation loss measurements of subwavelength plastic wires with various porous and non-porous transverse geometries. Our original theoretical investigations^{15, 16} have predicted that addition of an array of subwavelength holes into the crossection of a subwavelength fiber increases the fraction of power guided in the air, thus leading to a lower absorption loss. Moreover, we have demonstrated that for identical absorption loss a porous fiber will have a larger diameter than a non-porous one, thus concentrating more light within the porous fiber core. This not only dramatically reduces the proximity cross-talk of the fiber mode with the environment but also leads to a much smaller bending loss for porous fibers compared to the non-porous fibers. Finally, we have demonstrated that porous fibers with outer diameters larger than the wavelength of light can be designed, while still operating in a single mode regime and exhibiting very low propagation and bending loss.¹⁶ To design such fibers one has to use very high porosity and highly subwavelength material veins. Our findings were later reproduced by another group.¹⁷ One of the practical benefits for the use of porous fibers compared to the non-porous fibers of the same propagation loss is the much larger size of porous fibers,¹⁶ which simplifies their handling. We have recently published experimental results where porous fibers were fabricated and their spectrally averaged transmission losses were characterized using a novel Directional Coupler Method.¹² In that work, however, spectral information was inferred indirectly as a broadband THz source and a bolometer detector were used to perform cut-back-like measurements of the fiber losses. Moreover, our original fiber fabrication strategies resulted in fibers with relatively low porosities below 40%.

In this paper, we present a comparative study of the spectrally resolved loss measurements of porous and non-porous subwavelength fibers performed using a novel THz-TDS (Time Domain Spectroscopy) setup of adjustable optical path length specifically designed for fiber measurements. Moreover, we report a novel fabrication method using microstructured mold casting which allowed us to fabricate fibers of very high 86% porosity. Although both porous and non-porous fibers of the same diameter show very low propagation losses below 0.02 cm⁻¹, we find, however, that the porous fibers exhibit a much wider spectral transmission window and enable transmission at higher frequencies compared to the non-porous fibers. We then show that the typical bell-shaped transmission spectra of the subwavelengths fibers can be very well explained by the onset of material absorption loss at higher frequencies due to strong confinement of the modal fields in the material region of the fiber, as well as onset of a strong coupling loss at lower frequencies due to mismatch between the modal field diameter and the size of the gaussian-like beam of a THz source.

The paper is organized as follows. Section 2 presents a variable optical path length THz-TDS setup that can accommodate fibers up to 50 cm in length. Section 3 details principles of operation of a porous subwavelength fiber, and description of the two methods for the fabrication of such fibers. Section 4 presents spectral transmission and loss measurements. Section 5 develops theoretical justification of the observed transmission spectra and measured losses. The final section concludes with a discussion on fabrication difficulties and the effect of fiber imperfections on the transmission through subwavelength fibers.

 

Fig. 1. Tunable THz-TDS setup for waveguide transmission measurements. a) Schematic of setup. E:Emitter, D:Detector, PM:Parabolic Mirror, BS:Beam Splitter, FM: Flat Mirror, b) Source spectrum (red) and background noise level (blue). There are traces of water vapor (black) despite efforts to purge with a nitrogen atmosphere. c), d) Photographs of a setup for different positions of the mirror assembly that allows to either perform measurements of a point sample c) or to accommodate a waveguide up to 50 cm in length d).

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2. Measurement setup

The standard THz-TDS setup, using off-axis parabolic mirrors in a focal-point-to-focal-point configuration, was developed to measure small point samples not extended fiber samples. In order to conveniently measure the transmission spectra of THz waveguides, an easily reconfigurable setup was needed in order to accommodate waveguides of different length. Schematic of a solution that was developed in our group is presented in Fig. 1(a). It features two sets of rails with mirror assemblies to allow convenient adjustment of a THz optical path (rail 2), and to allow insertion of a THz fiber (rail 1). A fixed parabolic mirror focuses the THz radiation into the waveguide, the in-coupling plane is therefore fixed. The output of a waveguide is placed at the focal point of another parabolic mirror which can be displaced along the rail 1 together with a flat mirror. Light collected and collimated by the parabolic mirror is then redirected towards the fixed detector with a flat mirror. In our setup, waveguides of length up to 50 cm can be measured by simply translating the position of these two mirrors along the rail 1. Two low temperature grown GaAs photo-conductive antennae from Menlo Systems GmbH were used as both emitter and detector. These were pumped by a frequency-doubled C-fiber laser, also from Menlo Systems GmbH. The emitter pump beam was chopped and the detected signal was measured with a Stanford Research Systems lock-in amplifier. The various waveguides studied were held in place with 3-axis positioning mounts, and the entire assembly was housed in a nitrogen purged cage to reduce the effects of water vapor. Figure 1(b) presents the spectrum of the source as well as the background noise level. The system had an amplitude dynamic range of more than 20 dB. Although much effort was expended trying to properly purge the cage with dry nitrogen, some residual water vapor absorption lines were still detected in the spectrum (see dashed black lines for reference).

Note that rail 2 is an additional delay line used to compensate for the change in the optical path generally introduced by the addition of a waveguide. Rail 2 assembly is especially important when measuring long waveguides that support modes of refractive indices which are significantly different from 1.0. In such a case, the difference in THz path lengths between a setup with a fiber in it and an empty setup (reference) could become so large that the maximal pulse delay of a computer controlled variable delay line would be insufficient to compensate it. Interestingly, in the case of subwavelength fibers the refractive index of a guided mode is very close to 1.0. Therefore, the additional pulse delay introduced by the waveguide (as compared to an empty setup) is easily compensated by the variable delay line. Therefore, in all the measurements reported in this paper the mirrors on rail 2 were fixed, and therefore, the physical length between a source and a detector (denoted as L_path2) was fixed. Following a standard modeling procedure¹⁸ the waveguide transmission T can be described as:

where E_waveguide(ω) is the modal field coming out of a waveguide of length L_w, and E_reference(ω) is the reference signal measured in the absence of the waveguide at the position of the coupling plane. E_source(ω) is the spectrum of the source, whereas η is the transfer function describing transmission through the setup in the absence of a waveguide. C_in and C_out are respectively the input and output coupling coefficients with respect to the waveguide, while α and n_eff are the power propagation loss and effective index of the mode guided by the waveguide. Note that the transmission through the waveguide induces a time delay of $\frac{(n_{\mathrm{eff}}-1)L_{\mathrm{w}}}{c}$ with respect to a reference THz pulse propagating through an empty setup. This delay is small in the case of subwavelength fibers since n_eff ~ 1 for the modes of such waveguides. A comparison of the amplitude transmission of fiber segments of different length (cutback method) can be used to calculate the power attenuation coefficient α,

where the superscript (j) of E^(j)_reference(ω) is used to indicate that the reference spectrum for segment L_j is measured immediately after the measurement of E_waveguide(ω, L_j). Note that because of the power fluctuations of the source and the long time required to cut and re-align the fiber segments, |E⁽²⁾_source(ω)| ≠ |E⁽¹⁾_source(ω)|. Thus, using the ratio of transmission amplitudes takes into account any fluctuations in |E_source(ω)| which may occur between the times when segments L₁ and L₂ are measured.

3. Porous subwavelength fibers

3.1. Principles of operation

 

Fig. 2. Schematics of various subwavelength fibers and their fabrication techniques. a)-c) Poynting vector distributions across fiber cross-sections for subwavelength fibers featuring 0, 1, and 7 holes, respectively. The outer diameter of all the fibers is 400µm, the diameter of all the holes is 100µm, and the frequency is 0.3 THz (λ = 1000µm). Schematics of the d) sacrificial polymer technique and e) microstructured molding technique for fabricating porous subwavelength fibers.

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The simplest subwavelength dielectric fiber consists of a solid dielectric wire acting as a fiber core, surrounded by the low refractive index low loss cladding such as dry air (see Fig. 2(a)). At wavelengths larger than the fiber diameter, this simple fiber with a step-index refractive index profile supports modes that extend into the air cladding over distances that can be many times larger than the fiber diameter. It is due to significant presence of a mode in the low loss cladding that the effective modal loss due to material absorption can be made very small. As an improvement to this design, Nagel et al. proposed adding a subwavelength hole to a THz subwavelength fiber.¹⁹ As can be seen in Fig. 2(b), in this design the fraction of power guided in the air increases because of the higher field concentration within the hole. This is due to the continuity of the electric flux density normal to the dielectric interfaces. Particularly, the electric field at the dielectric/air interface shows a discontinuous jump to higher values inside of the air hole because the hole has a much lower refractive index compared to that of a fiber material.²⁰ As a further improvement, adding more holes within the subwavelength fiber (Fig. 2(c)) was recently proposed by our group as a means of further reducing the fiber absorption loss by forcing a greater portion of light to guide in the dry low-loss gas filling the porous fiber core.^{15, 16}

Theoretical simulations have predicted that the porous fibers have lower absorption loss than non-porous fibers of the same diameter. This is due not only to a higher fraction of power in the air cladding but also because of a significant fraction of power within the air holes of the core. It is straightforward to understand that the porous fiber has a lower index contrast than the non-porous fiber. As a result, the evanescent field of the porous fiber extends farther into the air cladding (as shown in Figs. 2(a) and 2(c)) and the lower field confinement leads to higher bending loss for the porous fiber, when comparing fibers of same diameter. However, it has been shown that the situation is reversed when comparing fibers of same absorption. The porous fiber has a lower absorption loss than a non porous fiber of same diameter. Thus, for the same absorption loss as a non-porous fiber, the porous fiber will have a larger diameter. In such a case, it is possible to show that despite the lower index contrast the field of the large diameter porous fiber is more confined when compared to that of the smaller diameter non-porous fiber. Thus it is the porous fiber that has lower bending loss in this case.¹⁶

It is worth mentioning that although periodic arrays of holes are considered and fabricated, the subwavelength fibers presented in this paper are not photonic crystal fibers. Although bandgap guidance in holey fibers has been attempted for guiding THz radiation,²¹ the size of the holes in the subwavelength fibers are such that the guidance mechanism remains total internal reflection.

3.2. Fiber fabrication

Several fiber fabrication techniques were explored in order to fabricate porous subwavelength fibers. The main challenge for the fabrication of such fibers remains preserving a highly porous thin-walled structure during the fiber drawing process. All the samples presented in this paper were made from low-density polyethylene (LDPE) preforms. Non-porous fibers were fabricated from the same PE material to serve as a comparison to the optical properties of porous fibers. Non-porous fiber preforms were made by fusing PE granules within a tube to form a solid cylinder that was subsequently drawn into a fiber.

3.2.1. Sacrificial polymer technique

The first method that was developed for porous fiber fabrication is a Sacrificial Polymer Technique, and has been described in detail earlier.¹² Schematics of the fabrication steps are presented in Fig. 2(d). First, the rods of a sacrificial polymer (polymethyl-methacrylate (PMMA) in our case) are arranged in a hexagonal pattern without touching. Then the rest of a preform is filled with polyethylene (PE) granules. The sacrificial material is chosen to have a significantly higher glass transition temperature. This allows melting the low viscosity PE polymer granules that consequently fill out the space between the sacrificial rods. Thus fabricated preform was then drawn into a fiber at 210°C. Finally, the air holes in a fiber are revealed by dissolving the sacrificial PMMA rods in tetrahydrafuran (THF) without affecting the rest of the fiber. The final fiber retains the same geometry as the preform, but with air holes in place of sacrificial rods. The presence of PMMA prevents collapse of the holes that otherwise would have happened if they were left empty. The main advantage of the sacrificial polymer technique is that drawing of porous structures is greatly facilitated as hole collapse is completely prevented during fabrication. The main disadvantage of this method is that a postprocessing step of removing the sacrificial polymer is required. Although we present a fiber with 35% porosity by area, it should be noted that this technique is very versatile and much higher porosities can easily be achieved by incorporating more sacrificial polymer into the preform (larger PMMA rods, for instance).

3.2.2. Microstructured molding technique

Another method that was developed for the fabrication of porous (and, in general, microstructured fibers) uses casting of a fiber preform in a microstructured mold. The resultant preform features air holes which have to be pressurized during drawing to prevent hole collapse. Schematic of the fabrication steps is presented in Fig. 2(e). A porous cylindrical preform, featuring 7 holes running its entire length, was fabricated by melting PE granules and solidifying the melt within a microstructured glass mold. The microstructured mold was prepared by first aligning thin-walled quartz capillaries in a hexagonal arrangement using special alignment rigs, followed by the fusion of such aligned capillaries to a circular quartz plate with the aid of a propane torch. The ends of the capillaries were sealed to prevent PE from filling the capillaries during the molding process. The assembly of glass capillaries was then inserted in and fused to the bottom of a large diameter quartz tube. The other end of the tube is left open to allow placement of the polymer granules on top of the microstructured mold. The tube was then filled with granules and placed into a furnace. After melting the polymer, the polymer melt was transferred by gravity into the microstructured mold region. Upon cooling, most of the glass mold could be removed by simply pulling it from the polymer preform. Any residual glass was dissolved in hydrofluoric acid, which had no effect on the PE preform. A porous fiber was subsequently drawn while pressurizing the preform holes. In addition to preventing hole collapse during drawing, a sufficiently large air pressure could inflate the holes and greatly increase the fiber porosity. Note that molding has enabled the fabrication of a preform that does not have the interstitial holes which are present in preforms fabricated by tube stacking. The pressurization step is thus facilitated. The molding technique in combination with hole pressurization allowed us to produce fibers of very higher porosity (as high as 86%).

4. Transmission and loss measurements

The transmission spectra of the subwavelength fibers were measured using the THz-TDS setup described in section 2. The cutback method was used in order to evaluate modal propagation loss. During measurements the fibers were held straight by knotted threads¹² and the fiber ends were aligned with respect to pre-installed aligning apertures using 3-axis mounts. These apertures marked the locations of focal points of the input and output off-axis parabolic mirrors. Proper alignment with respect to these apertures insured consistent coupling into and out of the fibers. Figures 3(a) and (e) present cross-sections of the measured subwavelength fibers. A porous fiber made using the sacrificial polymer technique (designated as PE/PMMA fiber in Fig. 3(c)) features 35% porosity, has an outer diameter of d = 450 µm, hole size d_h = 100 µm, and a hole diameter to the hole-to-hole pitch ratio d_h/Λ = 0.60. In turn, a porous fiber made with the microstructured mold casting technique (designated as molded PE fiber in Fig. 3(g)) features an outer diameter d = 775 µm and 86% porosity (equivalent to the hole diameter to the pitch ratio d_h/Λ = 0.93). To our knowledge, this is currently the highest reported porosity for a subwavelength fiber. Diameters of the corresponding non-porous fibers are 445 µm and 695 µm, respectively.

 

Fig. 3. Transmission and loss measurements of porous and non-porous subwavelength PE fibers. Left column: small diameter fibers, Right column: large diameter fibers. The data for the porous fibers is in red and data for the non-porous fibers is in blue. Fiber diameters and measured segments lengths are indicated in the legends. Photos a) and e) show measured fiber cross-sections; b) and f) Normalized amplitude transmission; c) and g) Power propagation loss calculated from the transmission spectra using cutback technique; d) and h) Upper bound on propagation loss given by normalized (per unit of unit of length) total loss.

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In Figs. 3(b) and (f) we present the measured amplitude transmission spectra of the porous and non-porous fiber segments, up to 38.5cm in length. For each fiber diameter, two segments of different length are presented. The lengths of the corresponding fiber segments are indicated in the figure legends. In Figs. 3(c) and (g) we present the power propagation loss of the subwavelengths fibers, as calculated using equation (4) and the transmission spectra shown in Figs. 3(b) and (f). We would like to note that due to the short lengths of the studied fibers it was difficult to cut them while keeping the input end fixed. Therefore, during cutback measurements we had to remove the fibers segments, cut them, and then place them back and re-align them for every measurement, thus incurring small alignment errors.²² In Figs. 3(c) and (g) in dotted curves we also indicate error bars associated with our cutback loss measurements. From Figs. 3(c) and (g) we find that the propagation loss minima of the ~ 450 µm diameter fibers are 0.024 ± 0.004 cm⁻¹ at 0.190 THz for the non-porous fiber, and 0.022 ± 0.014 cm⁻¹ at 0.234 THz for the porous fiber; for the ~700 µm diameter fibers we find losses of 0.040 ± 0.008 cm⁻¹ at 0.117 THz for the non-porous fiber, and 0.012 ± 0.010 cm⁻¹ at 0.249 THz for the porous fiber. Note that transmission minima of the porous fibers are located at higher frequencies than those of the non-porous fibers of the same diameter. Moreover, spectral bandwidths of the porous fibers are larger than those for the non-porous fiber of the same diameter. Both of these observations are consistent with our prior theoretical work.^{15, 16}

Finally, in Figs. 3(d) and (h) we take the transmission spectra of Figs. 3(b) and (f) and plot them in the form of total loss normalized with respect to the fiber segment length (−2ln(|T|)/L). Such curves offer two interesting pieces of information. First, at high frequencies, where the light is highly confined within the core, we get an estimate for the bulk absorption loss of the fiber material. Second, if we assume 100% coupling efficiency then the normalized total loss curve gives us an upper bound value for the propagation loss. The propagation loss cannot be higher than this upper bound, otherwise the total loss would have been higher, and the propagation loss will actually be lower if the coupling efficiency is not 100%. Thus, if we consider the values at the normalized total loss minima, which correspond to the transmission maxima, then we deduce α ≤ 0.02cm⁻¹. This is in good agreement with the cutback measurements.

Interestingly, results shown in Figs. 3(b) and (f) also suggest that a much larger diameter fiber of higher porosity can have similar transmission spectra to that exhibited by a smaller diameter fiber of lower porosity. This is in agreement with our prior theoretical simulations¹⁶ where we have demonstrated that by increasing fiber porosity, fibers of diameter much larger than the wavelength of light can be designed to still guide in a single mode low-loss regime.

Note that although theory predicts lower losses for the porous fibers, we are unable to distinguish between the already very small losses of the porous and non-porous fibers within the error of our experimental setup. As we demonstrate in the following section, the transmission peak (bell-shaped transmission curve) results from the balance between absorption loss (at high frequencies), scattering loss (at low frequencies),⁹ and frequency dependent coupling loss.

5. Theoretical modeling

In order to better understand the transmission spectra and propagation loss results presented in Fig. 3, we carried out vectorial simulations of the optical properties of the fundamental mode propagating in the subwavelength fibers studied in this work. It should be noted that whereas the small diameter porous fiber (PE/PMMA fiber) retained a hexagonal arrangement of the holes, the large diameter porous fiber (molded PE fiber) had such a high porosity that it seems to be better approximated by a tube with a subwavelength-thick wall. To model the non porous fibers, an 86% porous fiber, and a 35% porous fiber, we therefore considered fundamental modes of a circular rod, a thin circular tube, and a 7 hole fiber, respectively. To model a non-porous rod fiber and a thin-wall-tube fiber we have used a transfer matrix code²³ to find the fundamental HE₁₁ mode of both circularly-symmetric fibers. In the case of a 7 hole porous fiber we have used a vectorial finite element code to find fiber modes.

In order to properly calculate the modal absorption losses the experimental value of the polyethylene bulk absorption loss α_PE must first be estimated. In the literature there are several widely different reported losses for bulk PE, ranging from ~ 0.1cm⁻¹ in Ref. 22 to ~ 2cm⁻¹ in Ref. 23 at frequencies around 0.2 THz. The exact value depends greatly on the fabrication process of the polymer. In our case, THz-TDS transmission measurements through a 1 mm thick film of the used LDPE yielded a refractive index n_PE = 1.56, however the absorption loss was too low to yield an accurate measurement. Nevertheless, the total loss measurements of the non-porous plastic fibers (Fig. 3(d)) at high frequencies suggest that the bulk loss of the LDPE polymer used in our work is α_PE = 0.2 cm⁻¹. These values were assumed for the calculations presented in this paper.

 

Fig. 4. Modal losses of the porous and non-porous fibers as a function of the fiber geometry parameters (d_F = fiber diameter, P = porosity). First row: non-porous subwavelength fiber; second row: subwavelength fiber with 7 air holes; third row: subwavelength fiber with one air hole. First column: schematics of the fiber geometries; second column: fundamental modal attenuation loss as a function of frequency; third column: power coupling coefficient between a gaussian beam and the fundamental mode.

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Figure 4 presents the simulation results for each of the three geometries. The first column presents schematics of the fiber geometries. The second column presents the power absorption loss of the fundamental mode as a function of frequency. Figure 4(b) presents absorption loss of the fundamental mode of a non-porous fiber; in the figure, different curves correspond to the different fiber diameters. Figures 4(e) and (h) present modal losses of the porous fibers of diameters corresponding to those used in the experiment; in the figures, different curves correspond to the different values of the fiber porosity.

To vary fiber porosity, in both cases one simply varies the size of the air holes (porosity is defined as the ratio of the total area of the air holes to the total area of the fiber core). Behavior of the loss curves for various values of the fiber geometry parameters is easy to understand. Consider, for example, Figure 4(b). At a given frequency, choosing the fibers of smaller diameters results in a fundamental mode which is strongly delocalized, extending beyond the lossy fiber core and into the low-loss cladding. Therefore, at a fixed frequency, fibers of smaller diameters will exhibit smaller absorption loss due to stronger modal presence in the low-loss cladding. Inversely, for a fiber of fixed diameter, when decreasing the frequency of operation, the fundamental mode dispersion relation approaches the cladding light line. In other words, at lower frequencies the effective refractive index of the fundamental core mode approaches that of the cladding material. This results in a stronger modal presence of the guided mode in the low-loss cladding, and as a consequence, in the reduction of the modal absorption loss at lower frequencies. Consider now Figs 4(e) and (h). There, one observes decrease in the fiber absorption loss for higher fiber porosities. This is easy to rationalize by noting that higher fiber porosities lead to a lower refractive index contrast between the porous fiber core and a gaseous cladding. Lower refractive index contrast, in turn, means higher modal presence in the low-loss, low refractive index material (pores and cladding), hence explaining lower absorption losses of fibers featuring higher porosities.

Consider now the coupling losses from the gaussian-like THz beam into the fundamental mode of porous and non-porous fibers. The third column of Fig. 4 presents the power coupling efficiency between the linearly-polarized gaussian beam of a THz source and the fundamental mode of fibers under consideration. Using the knife-edge technique, the beam at the focal plane of a parabolic mirror was measured to have a gaussian power distribution with a full-width-half-maximum (FWHM) of 2.3 mm at 0.2 THz. This value was roughly independent of the operation frequency due to the use of low-dispersion parabolic mirrors in the THz-TDS setup. Note that the calculated coupling coefficients are strongly frequency dependent due to strong dependence of the modal diameters on the frequency of operation. In fact, for all the fibers there exists a frequency of optimal coupling from the gaussian beam into the fiber fundamental mode at which the beam size and the modal diameter are matched. Notably, frequency of the optimal coupling into porous fibers is always higher than that of the non-porous fibers of the same diameter, which is due to stronger modal delocalization in the porous fibers compared to non-porous fibers of same diameter.

Experimentally measured transmission spectra through subwavelength fibers all feature bell-like profiles with frequencies of the transmission maxima defined by the competition between several loss mechanisms such as coupling, material absorption, and scattering on fiber imperfections.^{9, 12} Particularly, at higher frequencies, both the absorption loss and coupling loss increase substantially leading to a decrease in the fiber transmission (see Figs. 3(c) and (g), and Figs. 4 (b) and (c), for example). At lower frequencies, it is the increased coupling loss and scattering loss that lead to a decrease in the fiber transmission. Thus, at the frequency of strongest transmission, fiber mode extends sufficiently into the porous region and into the gaseous cladding to lower absorption loss, however still remaining relatively confined to the fiber core to avoid too much scattering losses on the fiber defects. As the onset of strong absorption and coupling losses for porous fibers happen at higher frequencies than those for the non-porous fibers of the same diameter (compare Figs. 4(e) and (f) to 4(b) and (c)), it is not surprising that experimentally measured transmission curves for the porous fibers are shifted to the higher frequencies with respect to those of the non-porous fibers (see Figs. 3(b) and (f)).

 

Fig. 5. Theoretical calculations of the dispersion parameter of non-porous (a) and porous (b) fibers. The effective index curves of the non-porous and porous fibers were taken from the simulations presented in Figs. 3.b) and 3.e), respectively. Decrease of the fiber diameter and increase of the porosity result in the reduction of dispersion. Comparison of the time scans of ~450 µm diameter porous (red curve) and non-porous (blue curve) fibers confirms that dispersion is smaller in porous fibers because the length of the dispersed THz pulse is shorter (envelope is decaying faster). The porous fiber time scan is offset vertically and the reference pulse of the source is scaled a factor 1/40 for clarity.

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Moreover, it is well known that such step-index fibers have large waveguide dispersions. Figure 5 presents some theoretical calculations of the dispersion parameter β₂ = (1/2πc)(f · ∂²n_eff/∂f² + 2 · ∂n_eff/∂f). We calculate the dispersion parameter using the effective propagation index, n_eff, obtained from the simulations shown in Figs. 3.b) and 3.e). Figures 5(a) and 5(b) indicate that a reduction of the fiber diameter or an increase of the porosity will decrease the fiber dispersion (in addition to the absorption) because a higher fraction of power is guided in the air. As a result, the n_eff(f) curve is closer to n_air and will vary more slowly as a function of frequency. Furthermore, a porous fiber will have lower dispersion than a non-porous fiber of same diameter. This is corroborated by the experimental time scans shown in Fig. 5(c) where the transmitted THz pulses of 450 µm porous and non-porous fibers are compared. Although the experimental values of β₂ were not calculated, it is clear from Fig. 5(c) that the porous fiber has lower dispersion since the envelope of the THz pulse is decaying more rapidly than that of the non-porous fiber.

 

Fig. 6. Theoretical fit of transmission spectra through large and small diameter subwavelength fibers. The theoretical fits (solid lines) take into account coupling loss (dotted lines) and absorption loss (dashed lines) contributions but neglect scattering losses. The calculations assumed α_mat = 0.2cm⁻¹, a porosity of 35% for the small diameter fiber, and a porosity of 72% for the large diameter fiber. Optimal fits for the non-porous fibers were found for diameters slightly different from the experimentally measured diameters.

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Finally, in Fig. 6 we present fits of the experimentally measured transmission curves using theoretically calculated absorption and coupling losses. The theoretical fits for the fiber transmission (solid lines) are calculated using equation (3) by taking into account coupling loss (dotted lines) and absorption loss (dashed lines) contributions (see Fig. 4). Even without taking scattering loss into account we find a good correspondence between the theoretical and experimental curves. Scattering losses (not considered in this work), would further narrow the transmission peaks on the low frequency side. Although the porosity of the large diameter fiber was estimated to be 86% from the photo of the fiber cross-section, we find that a porosity of 72% yielded the best fit when approximating this fiber as a thin-wall tube. The only fitting parameter for the non-porous fibers was the fiber diameter, which is reasonable considering the diameter fluctuations in the experimental fibers. The coupling coefficients were not measured experimentally and although the use of experimental values for α(ω) and |C(ω)|² in equation (3) could have been a self-consistency test of the cutback measurement, the decent fits obtained with certain reasonable theoretical assumptions is seen as a further validation of the theoretical simulations of the absorption and coupling losses.

In Fig. 6, the greater widths of the porous fiber transmission spectra are attributed in part to the slower frequency dependence of the absorption loss curves and in part to the broader coupling loss curves. The latter being due to the slower variation of the modal size as a function of frequency for the fibers with lower refractive index contrast (porous fibers versus non-porous fibers). Finally, we would like to note that although the absorption loss of a porous fiber is predicted to be much lower than that of a non-porous fiber of the same diameter, we find experimentally that the lowest measured losses of the porous and non-porous fibers are quite similar and on the order of 0.01 cm⁻¹ – 0.02 cm⁻¹. We attribute this finding to the limit on the lowest obtainable loss set by the scattering loss on various fiber imperfections such as dust on a fiber surface, bulk material impurities, fiber diameter fluctuations, fluctuations in the porosity microstructure, micro- and macro-bending, etc.

6. Discussion and conclusion

In this work we have demonstrated low-loss THz transmission using various subwavelength polyethylene fibers. Solid core 40cm-long subwavelength fibers of diameters 445 µm and 695 µm were demonstrated to guide in the 0.14-0.21 THz and in the 0.06-0.14 THz spectral regions respectively with losses as low as 0.02 cm⁻¹. Addition of porosity to the subwavelength rod-in-the-air dielectric fibers has been shown to shift the propagation loss minimum to higher frequencies and to broaden the fiber transmission spectra. Thus, porous fibers of diameters similar to those of the non-porous fibers were demonstrated to guide in the 0.16-0.31 THz (fiber of 450 µm diameter with 35% porosity) and in the 0.18-0.30 THz (fiber of 775 µm diameter with 86% porosity) spectral ranges. Interestingly, these results also suggest that a much larger diameter fiber of higher porosity can have a transmission spectrum similar to that exhibited by a smaller diameter fiber of lower porosity. We, therefore, believe that highly porous fibers can enable single mode transmission at higher frequencies beyond 0.4 THz, while still exhibiting very efficient coupling to the gaussian-like several mm-diameter beam of a THz source.

In this paper we have also presented two techniques for the fabrication of highly porous fibers. Both techniques start with a fabrication of a microstructured mold which is then used to cast a THz fiber preform by melting the polyethylene polymer and filling the mold with it. In the sacrificial polymer technique the mold is made from a sacrificial plastic which is drawn together with a PE polymer into a microstructured fiber; the sacrificial polymer is then removed from the drawn fiber by dissolving it with an organic solvent. The main advantage of this method is that a complex air microstructure can be created as hole collapse during drawing is not an issue. The main limitation of the sacrificial polymer technique is the necessity of a post-processing step to remove the plastic of a mold. Another fiber fabrication strategy presented in this paper is based on a microstructured mold casting method where the material of the mold is quartz that has to be removed before fiber drawing. The main advantage of this method is that very high precision microstructured molds can be created using standard glasswork and that the mold does not change shape during the casting process. The main disadvantage is that air microstructure has to be pressurized during drawing to prevent its collapse due to surface tension effects.