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Abstract: We report on the characterization of 3D-printed hollow core Terahertz waveguides with metal wire inclusions over a frequency range of 0.2-1.0 THz using standard THz time-domain spectroscopy. We observe single-mode broadband THz propagation in these waveguides, and measure the loss coefficient and the mode effective phase index. Our measurement data agree well with predicted values obtained from numerical simulations.
© 2014 Optical Society of America
1. Introduction
The terahertz (THz) frequency range is continuing to find applications ranging from biophysics, optics, and other areas [1–3]. The non-invasive and non-ionizing properties of THz radiation combined with unique spectroscopic signatures of most materials in the THz region offer significant advantages in non-destructive material characterizations [4,5] and biomedical label-free diagnostics [6,7]. Advances in THz applications are challenging due to the lack of suitable materials for THz optical components as well as THz waveguides. Guiding broadband THz pulses using metal and/or dielectric waveguides has been demonstrated with numerous waveguide designs: parallel metal plates [8,9], metal wires and tubes [10–14], photonic crystal polymer fibers [15–17], porous fibers [18,19] among others have been promoted as low-loss and low-dispersion THz waveguides. Among these, THz propagation in the waveguides with air-core shows notable flexibility in that the influence of material losses is minimized and therefore the waveguide designs can be made independent of material selection [20–22].
The level of complications in fabricating these waveguides increases with complex structures, particularly when it involves metallic inclusions as one of the waveguide boundaries. Fortunately, it is possible nowadays to manufacture suitable designs of THz waveguides (and components) using the increasingly advanced 3D printing technology. This fabrication method offers robust and high-quality patterning as well as affordability and provides opportunities in versatile designs [23]. On the other hand, access to large-scale fabrication facilities such as drawing/extrusion machines will enable the production of even smaller dimension of THz waveguides than those achievable with current 3D printing machines [17,18,24]. Co-drawing of metal and polymer/glass has been reported to experience difficulties from maintaining the quality and integrity of the multicomponent of the drawn waveguides due to the mismatch in melting temperatures of the metal and polymer/glass [25–28]. In this respect, this process of fabrication of metallic-inclusion waveguides is limited to a selection of compatible materials.
Inspired by the low-loss guidance of HE11-like mode in our previous hollow waveguides with metallic components fabricated using a drawing tower [29], we report on our work on the design of THz waveguides manufactured from a commercially available 3D printer. We fabricated hollow core waveguides with several air holes in the cladding, and later attached metal wires to these air holes. Different numbers and configuration of metal wire inclusions are investigated, and the characterizations of these prototypes are conducted using a THz time-domain spectroscopy (THz-TDS) setup. This paper is organized as follows: Section 2 will describe the design and fabrication of hollow core THz waveguides with metal wire inclusions, the measurements method using our THz-TDS and the simulation details, Section 3 will outline and discuss our experimental and simulation results on the mode characteristics, attenuation coefficients and dispersion properties of the waveguides, and Section 4 will summarize our report.
2. 3D fabrication details, experimental method, and numerical simulation
Figure 1(a) shows our waveguide design in which the polymer tube (shaded in blue color) contains 12 holes surrounding the hollow core. The fabricated polymer cladding not only provides mechanical support for the metal wires, but also introduces boundary conditions that enable mode guidance in the air-core. We fabricated the prototypes using a UV curable polymer, identified as VisiJet® M3 with a Projet 3500HD Plus 3D printer [30] with specifications of X-Y resolution of 375 DPI (equivalent to 68 μm linewidth) and 790 DPI in Z-resolution (equivalent to 32 μm linewidth). Our waveguide polymer tube prototypes were measured to have 85 µm in X-Y resolution and 60 µm in Z-resolution. The fabricated waveguides have 3 and 4 mm core diameters and the core is surrounded by the air holes with diameters of 350 μm (Fig. 1(a)). The waveguides are measured using a vernier as well as examined using a microscope camera to evaluate the accuracy of the dimensions. This simple design allows several wires configurations as the copper wires will be manually attached into selected holes in different configurations. Figure 1(b)-1(d) shows the fabricated waveguide in which two, three and four copper wires are inserted into elected holes.
Fig. 1 The cross-section of (a) ideal waveguide design (without the metallic components) used in the simulation. The micrograph of the fabricated 3 mm core diameter waveguides with (b) two-wire, (c) three-wire in isosceles and (d) four-wire configurations. In (b-d), the 1mm-thick polymer cladding surrounds the 3mm diameter air core, and copper wires have been inserted in the appropriate positions. (e) Measured transmission spectra in logarithmic scale of the reference and the UV curable polymer used in the 3D fabrication of the waveguides. The noise floor of the THz-TDS setup is indicated in the cyan region, about 10−6 below the peak amplitude of the system at 0.35 THz. (f) Refractive index as measured (dashed line) and extrapolated (solid line) for the UV curable polymer. (g) Attenuation constant as measured (dashed line) and extrapolated (solid line) for the UV curable polymer.
The standard THz-TDS setup, as described in our previous work [29], is used to characterize the UV curable polymer and our waveguides. We used the specially designed symmetric-pass lenses [31] with a focal length of 75 mm (NA = 0.33) to achieve a good mode-match with the fundamental mode excited in the waveguide. The reference signals are taken without the waveguide sample and the lenses in focus. The waveguide lengths that we usmeasured in our experiments are 50 mm and 100 mm. The UV curable polymer material used in the 3D printer is characterized using a 1-cm cube placed in the parallel THz beam path of our THz-TDS setup. The experimentally determined attenuation coefficients and refractive indexes are calculated from comparing the amplitude and phase spectrum of the Fourier-transformed temporal waveform of the sample and the reference.
Our measurements of the UV curable polymer result in a refractive index of nTHz = 1.50 ± 0.02 and an average loss coefficient of 7.56 cm−1 at 0.3 THz. The material shows very low transmission above 0.6 THz. This causes the determination of the refractive index and loss coefficient for frequencies above 0.6 THz to be inaccessible due to the low signal-to-noise ratio. For the numerical simulations of our waveguides, the data for refractive index and loss coefficient of the polymer are extrapolated for higher frequencies beyond 0.6 THz, and there is no free parameter included in the simulations. The numerical simulations of our waveguides are performed using the finite difference frequency domain (FDFD) solver employed by MODE Lumerical Solutions [32]. The simulation results are obtained by using the ideal cross-section of the waveguide (see Fig. 1(a)), and while we assume that the metal wire components are perfect electric conductor (PEC), we also find that there is no difference in the simulation results when the actual value for the conductivity of copper is used. The imperfections in the fabrication of the waveguide are not taken into account in the simulations result.
3. Results and discussion
Temporal evolution and mode profile distribution
Figure 2(b)-2(c) show the measured THz pulses for 50 and 100 mm length, respectively, for two-wire configurations in the 3 mm and 4 mm core diameter waveguides. The temporal waveforms of the other wire configurations also have qualitatively similar features as in Figs. 2(b)-2(c). Compared to the reference signal (Fig. 2(a)), we observed a time lag of 1 picosecond (ps) for 50 mm and 2 ps for 100 mm length in the 3 mm core diameter waveguides, and a time delay of 0.8 ps for 50 mm and 1.6 ps for 100 mm length in the 4 mm core diameter waveguide. The time delay is very similar for other wire configurations. The small time delay introduced between the temporal waveforms of the reference and the waveguide indicates that the pulses are travelling mainly in the air core of the waveguide. An indication of the dispersion effect can be qualitatively seen in the spreading of the envelope of the temporal waveform of the waveguides when compared to that of the reference waveform. A quantitative discussion of the waveguide dispersion will follow in the next subsections. We also observe that the measured waveform amplitudes are reduced in the smaller core diameter waveguides when compared to that of the larger core diameter waveguides. In our experiments, we observe no additional pulses trailing behind the main pulse (as shown in Fig. 2): an indication that higher order modes that might be guided in the waveguides are not excited in our experiments.
Fig. 2 (a) Typical reference THz waveform as measured. The measured THz waveform for two-wire configuration in (b) 3 mm core diameter waveguide and (c) 4 mm core diameter waveguide. In (b-c), the dashed-dot line is measured for 50 mm length while the solid line is measured for 100 mm length of waveguides. A time delay of 1 ps is observed in the 50 mm length while it is 2 ps in 100 mm length of 3 mm core diameter waveguides when compared to the reference waveform. For the 4 mm core diameter waveguides, the time delay is 0.8 ps for 50 mm length and 1.6 ps for 100 mm length.
The simulation results in Figs. 3(a)-3(c) show the fundamental mode with HE11 characteristics. The simulated mode profiles at 0.7 THz are shown for two-, three- and four-wire configuration of the 3 mm core diameter waveguides in Figs. 3(a)-3(c), respectively. We numerically calculated the full-width at 1/e2 of the maximum amplitude of the simulated mode intensity profiles of these waveguides as a function of frequency and we will refer to this as w hereon. To achieve this, we perform a spatial-elliptic function fitting with the 1/e2 of the simulated mode field intensity profile and this returns the size of the major and minor axes of the fitted ellipse. The values of w of the simulated mode profile for our waveguides show very slight ellipticity in all wire configurations and are plotted in Figs. 4(a)-4(c) and Figs. 4(e)-4(g). The direction of the w-major axis coincides with the mode electric field polarization (along the x-axis in Fig. 3(a)-3(c)). As can be qualitatively seen in Fig. 4, the mode area in the 4 mm core diameter waveguide is larger than that in the 3 mm core diameter waveguide. We identify that the calculated w-minor and w-major of the mode profile in the two- and three-wire configurations of the 3 mm core diameter waveguide remains unchanged within 0.3-1.0 THz range (Figs. 4(a)-4(b)). Similar behavior is also observed in the calculated w of the two- and three-wire configurations of the 4 mm core diameter waveguide (Figs. 4(e)-4(f)). In the two- and three-wire configurations of the 3 mm core diameter waveguide, there is less than 7 percent difference in w-major and w-minor within the 0.3-1 THz range. In the two- and three-wire configurations of the 4 mm core diameter waveguide, the percentage difference is calculated to be less than 4% for the 0.3-1 THz range.
Fig. 3 The calculated mode intensity, Pz, profile at 0.7 THz in linear scale (a) two-wire (b), three-wire (isosceles) and (c) four-wire configurations in a 3 mm core diameter waveguide. The white line indicates the dielectric-air boundary of the waveguide core and the wires are represented as the solid circles. The direction of the electric field polarization is chosen to be along the x-axis, as indicated in the inset of (a).
Fig. 4 (a,b,c) The calculated w-major and w-minor of the simulated mode in two-wire, three-wire and four-wire configuration, respectively, for the 3 mm core diameter waveguide. (d) The normalized mode intensity, Pz, profile of four-wire configuration in 3 mm core diameter waveguide at 0.3 THz (dotted line), 0.6 THz (dashed line) and 1 THz (solid line) along the w-major axis. The vertical cyan lines indicate the position of the metal-air interface. The solid black horizontal line is the 1/e2 threshold. The plot shows increasing coupling between waveguide mode and plasmonic mode with low frequencies. (e,f,g) The calculated w-major and w-minor of the simulated mode in two-wire, three-wire and four-wire configuration respectively, for the 4 mm core diameter waveguide. (h) The normalized mode intensity, Pz, profile of four-wire configuration in 4 mm core diameter waveguide at 0.3 THz (dotted line), 0.6 THz (dashed line) and 1 THz (solid line) along the w-major axis. The vertical cyan lines indicate the positions of the metal-air interface. The solid black horizontal line is the 1/e2 threshold. The plot shows increasing coupling between waveguide mode and plasmonic mode with low frequencies.
We observe noticeable changes in the calculated w for the simulated mode profiles in the four-wire configuration for both 3 mm and 4 mm core diameter waveguides (see Figs. 4(c) and 4(g)). First, the percentage difference between the w-major and w-minor is larger in comparison to that of the two- and three-wire configurations for an identical core diameter waveguide. At 1 THz, the percentage difference is calculated to be 7% for the 3 mm core diameter waveguide, and 6% for the 4 mm core diameter waveguide. This value increases to more than 30% for the 3 mm core diameter waveguide and up to 10% for the 4 mm core diameter waveguide at 0.3 THz. Our simulation results suggest that while the mode-matching characteristics of the two- and three-wire configuration waveguides are very similar, it is expected that mode-matching in the four-wire configuration waveguide differs from that of two- and three-wire configurations waveguides because of the mode large ellipticity, especially in the smaller core diameter waveguide in the low frequency region. The experimental coupling efficiency obtained from these waveguides is discussed in the following subsection.
We also highlight an interesting field behavior in our simulation results of these waveguides that we observe at long wavelengths: the presence of surface bound modes at the metal-air interface. The mode exists along the wire pair where its plane is in the same direction as that of the mode electric field polarization. This effect—the field enhancement near the metal-air interface—is not present in designs with high-index dielectric (e.g. Silicon, nTHz~3.4) wire inclusions of the UV curable polymer cladding. In our waveguides, the amplitude of the surface bound modes coupled with the fundamental mode increases with lower frequencies, and this is plotted in Figs. 4(d) and 4(h). This leads to an overall increasingly larger full-width as the field amplitude exceeds 1/e2 field intensity threshold. As seen in Figs. 4(c) and 4(g), the w-major calculated for the four-wire configuration waveguides show drastic mode field reshaping with the field coupling from HE11 mode into the surface bound mode: the frequencies at which the coupled surface-bound mode and HE11 mode fields exceeds the 1/e2 field intensity threshold are 0.3 THz and 0.5 THz for the 3 mm and 4 mm core diameter waveguides, respectively (see Figs. 4(d) and 4(h)).
Attenuation coefficient and coupling efficiency
Figure 5 shows the loss coefficients for the two-, three- and four-wire configurations for 3 and 4 mm core diameter waveguides, obtained from experiments and simulations for the frequency range of 0.25-1.0 THz. As mentioned in Section 2, the UV curable material used as the waveguide tube has very low transmission above 0.6 THz yet we experimentally observe guidance in the printed waveguide within the frequency bandwidth of our THz-TDS setup. This demonstrates an advantage as THz guidance in the air-core region of the waveguide enables broadband transmission un-inhibited by the cladding material attenuation. The measured loss coefficients are averaged over several independent scans taken over different setup conditions.
Fig. 5 The measured (dotted line) and simulated (solid line) loss coefficients for hollow core waveguides with wire inclusions. The attenuation coefficient for (a) two-wire, (b) three-wire and (c) four-wire configuration in a 3 mm core diameter waveguide. The attenuation coefficient for (d) two-wire, (e) three-wire and (f) four-wire configuration in a 4 mm core diameter waveguide. The shaded blue region indicates the frequency region in which the signal-to-noise ratio is low for the measurement data.
We note that the overall attenuation coefficients decrease with larger core diameter in all wire configurations. This trend is consistent with the observed loss characteristics found in hollow core photonic crystal fibers reported for infrared [33], kagome fibers for visible and THz regions [22,34] as well as THz metallo-dielectric waveguides [35]. All of the waveguides show similar trend in that the loss coefficient increases rapidly closer to the low frequency cutoff region. This is the region where the signal-to-noise ratio of the measured signal drops to the noise floor. Below these frequencies, our measurement data show no consistent features. For the simulation results, a numerically stable mode profile ceases to exist below the cutoff frequency. This frequency depends on the core diameter, and it is 0.3 THz for the 3 mm core diameter waveguide and 0.2 THz for the 4 mm core diameter waveguide. It is interesting that these cutoff frequencies also are unaffected by the number of metal wires surrounding the core. Overall, Fig. 5 shows a very good agreement between the experimental results and the simulation data for all our waveguides. The modulation in the loss coefficients for the waveguides as predicted by the simulations are also observed and matched in the measured data, and the amplitude of the modulation in the loss curves increases with lower frequencies, as seen in Fig. 5. The origin of these modulations is thought to be the result of anti-resonant effect associated with the polymer cladding of the waveguide. The strong resonance effects have been reported in thin dielectric tubes [36]. In our simulation runs, we also observe the existence of higher order hybrid modes in the air-core in all of the waveguides with all wire configurations. These higher order modes register higher losses at all frequencies as well as higher cutoff frequencies in comparison to those of the fundamental mode. Experimentally, we are not able to detect these higher order modes as a result of mode-mismatch with the input beam which is assumed to have a Gaussian-shaped profile.
In the 0.7-1.0 THz frequency range, the loss coefficient for the 3 mm core diameter waveguides is measured to be about 0.2 cm−1. The loss coefficient increases up to 1 cm−1 at 0.3 THz of the low-frequency end of the spectrum. Meanwhile, the attenuation curves between 0.4 and 1.0 THz for the 4 mm core diameter waveguide exhibit overall lower loss values in comparison to those of the 3 mm core diameter waveguide with loss coefficient of about 0.1 cm−1 for higher frequencies. In the low frequency region, below 0.4 THz, the magnitude of the measured loss increases to about 1 cm−1. These features and trends in the measured loss coefficient match closely that of the simulated results. Our simulations results (not shown here) also indicate that the loss curves do not differ between using a 100 μm wire diameter and using a 350 μm wire diameter in the four-wire configuration of the 3 mm core diameter waveguide within the 0.3-1.0 THz range. There is no spectral shift in the high-loss peaks of the attenuation curves with different metal wire diameters. Using our measurement data, we are also able to determine the overall coupling efficiency achieved at both ends of the waveguide. We calculate that we achieve average coupling efficiencies of about 60% for the two-wire, 62% for the three-wire and 50% for the four-wire configuration for the 3 mm core diameter waveguides. For the 4 mm core diameter waveguides, we achieve average coupling efficiencies of about 58% for the two-wire, 60% for the three-wire and 56% for the four-wire configuration. These experimentally achieved coupling efficiencies are within reasonable presumption based on the simulation results of the mode distribution discussed in the previous subsection.
The good agreement between the experimental data and the simulation results motivates us to numerically investigate the effects of metal wire inclusions in bend waveguides. The 3D printing method allows for fabrication of rigid bend waveguide design of any curvature as opposed to manual force used to form the curved section. In our simulations of the two-wire configuration with 4 mm core diameter, we set the curvature plane along the plane of the parallel wires and varied the radius of curvature (Rc). For this waveguide, we find that the fundamental mode exists up to a cutoff frequency of 0.3 THz for Rc = 50 mm and Rc = 100 mm. This is in contrast to the cutoff frequency of 0.2 THz in a similar but straight waveguide. The center of the fundamental mode profile distribution is seen to shift closer to the one of the wires, and a reduced overall mode diameter in comparison to the straight waveguide within 0.3-1.0 THz range. The simulations predicted an increase in the loss coefficient of the mode in bend waveguides at all frequencies. However, an even higher loss is evident for the mode guided in a curved waveguide in which the curvature plane is perpendicular to the two-wire plane. Experimental work on curved waveguides is currently ongoing for other configurations.
Effective phase index and group velocity dispersion β2 parameter
Figure 6 shows the measured and simulated phase index of the fundamental mode in the 3 and 4 mm core diameter waveguides for the two-, three-, and four-wire configurations. The phase details obtained from the measured data are corrected for phase jumps at very low frequencies by adding multiple integers of 2π. This free parameter only allows us to shift the measured phase index along the vertical axis in Fig. 6 to match the simulation results. The agreement between the measured and the simulated data is very good for the 0.5-1.0 THz frequency range in all of our waveguides. We observed that the phase index decreases more rapidly for the smaller core diameter. Notice that the frequency range on Fig. 6 is different for the 3 mm and 4 mm core diameter waveguides. Within the 0.5-1.0 THz frequency region, the simulated phase index of the mode predicts a gradual increase with frequency and this behavior was also observed in the effective phase index data extracted from the measurement. Overall, we are able to measure effective phase index in the range of 0.985 to 1.0 as predicted by the simulation data. The trend and features in the measured and simulated phase index of the fundamental mode of these waveguides are observed to be independent of the number of wires in the waveguide.
Fig. 6 The plot of measured (dotted line) and simulated (solid line) effective phase index for waveguides. The phase index of (a) two-wire, (b) three-wire and (c) four-wire configurations in 3 mm core diameter waveguides. The phase index of (d) two-wire, (e) three-wire and (f) four-wire configurations in 4 mm core diameter waveguides. The shaded blue region indicates the frequency region in which the signal-to-noise ratio is low for the measurement data.
Next, we calculate the group velocity dispersion parameter using the effective phase index, represented by β2. This parameter is the second derivative of β with respect to ω, where ω is the angular frequency and β is the propagation constant of the fundamental mode (β = k0/neff, k0 is the vacuum propagation constant and neff is the mode phase index). The β2 curves for the experimental and simulation data are shown in Fig. 7. The values of experimentally determined β2 for the two-, three- and four-wire configurations of 4 mm core diameter waveguide in the high frequency region, 0.4-1.0 THz, exhibit close-to-zero dispersion characteristics, while those of 3 mm core diameter waveguide show small β2 parameters (smaller than |5| ps.THz−1.cm−1) within 0.5-1.0 THz range. The experimentally determined data are in the same order of magnitude as predicted by the simulation data. An increasing dispersive effect is observed in the low frequency band (i.e. long wavelengths) close to the cutoff frequencies at 0.3 and 0.2 THz for 3 mm and 4 mm core diameter waveguides, respectively. This leads to temporal waveform distortions, and is directly observed in the measured THz pulses of the waveguide samples, as seen in Fig. 2. Overall, the experimentally determined data are seen to follow the features predicted by the simulation results where modulation of the β2 parameter is a prominent feature. The magnitudes of the modulation observed in the experimental data does not exactly match that of the simulated data, however, and we attribute this discrepancy to the amplification of the noise originating from the measurement data.
Fig. 7 The plot of experimentally determined (dotted line) and simulated (solid line) group velocity dispersion β2 parameter, for different waveguides. The β2 parameter for (a) two-wire, (b) three-wire and (c) four-wire configurations in 3 mm core diameter waveguides. The β2 parameter for (d) two-wire, (e) three-wire and (f) four-wire configuration in 4 mm core diameter waveguides. The shaded blue region indicates the frequency region in which the signal-to-noise ratio is low for the measurement data. The horizontal dashed line indicates β2 = 0 as a guide to the eye.
4. Conclusion
We report on the THz pulse propagation in the hollow core of 3D printed waveguides made of a UV curable polymer with metallic inclusion for the 0.2-1.0 THz range. These waveguides are characterized using a standard THz-TDS setup with specially designed THz lenses, and numerically simulated using the fully-vectorial FDFD method. We observe single-mode broadband THz propagation for two-, three- and four-wire configurations with 3 mm and 4 mm core diameters. The fundamental mode has HE11-mode characteristics, and we do not observe the existence of higher order modes in our measurements. Our numerical simulations show that there is increasing coupling between the HE11 mode field of the waveguide and the surface bound mode field on the metal components in the lower frequency region.
We observe that the waveguide with larger core diameter has an overall lower attenuation coefficient than that of the waveguide with smaller core diameter. These waveguides also exhibit low frequency cutoff at 0.3 THz for the 3 mm core diameter waveguides and 0.2 THz for the 4 mm core diameter waveguides. For frequencies from 0.4 to 1 THz, the waveguide losses are measured to be in the range of 0.1 to 0.4 cm−1 for the 3 mm core diameter waveguides, and they are in the range of 0.05 to 0.4 cm−1 for the 4 mm core diameter waveguides. We highlight that the measured loss of these waveguides are small in comparison to the UV curable polymer material used to fabricate the waveguides, and this is made possible since the guided mode is tightly confined in the air core region of the waveguides.
The phase details obtained from the THz-TDS measurement allow us to calculate the effective phase index of the guided mode in the waveguides. The mode has an effective phase index below unity, a strong indication that the mode is essentially guided in the hollow core region of the waveguide. Using the measured effective phase index, we obtain the experimentally determined group velocity dispersion β2 parameter for the waveguides. Within the frequency range of 0.4-1.0 THz, low dispersion characteristics are observed for all of the waveguides. We calculate that the β2 parameter is smaller than |5| ps.THz−1.cm−1 for the 3 mm core diameter waveguide, whilst it is close-to-zero dispersion for the 4 mm core diameter waveguides.
The attenuation coefficient, effective phase index, and the β2 parameter obtained from the measurements show very good agreement with the values obtained from simulations within the whole bandwidth of our THz-TDS setup. This should allow us to numerically investigate other waveguide designs that can be used as robust and highly performing waveguides produced by fast and inexpensive 3D printer machines. The hollow core waveguides allow for flexibility in the material used to fabricate the waveguides as it does not influence the waveguide loss. Issues arising from thermal incompatibility in the fabrication from the metal components and the polymer cladding could be avoided via this fabrication method. Finally, 3D printing machines can also be foreseen as a potential apparatus to produce other interconnected THz components such as curved sections, waveguide-splitters and wave-plates.
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