Open Access
Add to BibSonomy
Abstract
Single-ring hollow-core photonic crystal fibers, consisting of a ring of one or two thin-walled glass capillaries surrounding a central hollow core, hold great promise for use in optical communications and beam delivery, and are already being successfully exploited for extreme pulse compression and efficient wavelength conversion in gases. However, achieving low loss over long (km) lengths requires highly accurate maintenance of the microstructure—a major fabrication challenge. In certain applications, for example adiabatic mode transformers, it is advantageous to taper the fibers, but no technique exists for measuring the delicate and complex microstructure without first cleaving the taper at several positions along its length. In this Letter, we present a simple non-destructive optical method for measuring the diameter of individual capillaries. Based on recording the spectrum scattered from whispering gallery modes excited in the capillary walls, the technique is highly robust, allowing real-time measurement of fiber structure during the draw with sub-micron accuracy.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The external diameter of optical fibers can be measured to sub-micrometer accuracy in real-time during fiber drawing using commercially available instruments that scan a laser beam across the fiber and measure the extent of the shadow behind it. A number of more precise techniques, suitable for standard step-index fibers, have been reported. The fiber can be illuminated with broadband (or tunable single-line laser) light and the scattered spectrum monitored for the presence of resonant features [1, 2]; comparison with Mie theory then permits the external diameter of the fiber to be determined. Whispering gallery mode (WGM) resonances, excited by wavelength-tunable laser light incident perpendicular to the fiber axis, can be used to perform nm-level studies of uniformity in outer diameter [3]. Light from the guided mode in one tapered fiber can be coupled into WGMs in a second tapered fiber placed at right-angles to it, enabling nm resolution of the external diameter [4].
Whereas for standard step-index fibers it is sufficient to measure the external fiber diameter to ensure uniformity along the fiber length, this is not the case for photonic crystal fibers (PCFs) [5], when small changes in drawing temperature or internal air-pressure can significantly disturb the hollow microstructure, leaving the external diameter practically unperturbed [6]. As a result, a long-standing challenge in drawing microstructured fibers with a uniformity matching that of standard fibers has been how to monitor the PCF structure in real time during the draw. A related problem occurs when tapering PCFs, for although one may taper the outer fiber diameter to a desired shape, there is no guarantee that the fiber microstructure scales isomorphically. Furthermore, short of cleaving the tapered fiber, no method has been reported that allows precise non-destructive measurement of its microstructure. In certain cases it is advantageous to be able to measure the orientation of the internal fiber structure noninvasively, so as to ensure that the fiber is absolutely straight without a twist, or conversely to measure the amount of twist frozen into the fiber during fiber drawing. Any method allowing non-invasive probing of the internal microstructure is therefore of great potential interest.
A number of approaches have been reported in the past. Doppler-assisted tomography using side-scattered light can be used to detect the position of air-holes in microstructured fibers, but shows low accuracy in determining air-hole diameter, requires fast rotation (∼10 Hz) of the fiber relative to the illumination-detection system, and is very sensitive to misalignment of the rotation axis relative to the fiber axis [7]. Tomographic imaging with visible light requires the fiber to be filled with liquid [8]. Although x-ray tomography non-invasively provides sub-micron resolution 3D images and allows detection of defects within the fiber, a scan of a fiber takes several minutes [9]—much too slow for real-time characterization during fiber drawing. Analysis of the diffraction pattern from a coherently side-illuminated fiber shows potential for real-time measurements during fiber drawing [10], but has only been demonstrated for simple single-hole capillary fibers, because the diffraction pattern becomes too complex for fast analysis when the microstructure is complex. Recently, a method based on spectral analysis of Fabry-Pérot interference was used to determine the hollow-core core diameter with nanometer-level resolution, but required the cladding holes to be selectively filled with liquid [11], making it unsuitable for real-time measurements.
We recently reported for the first time the use of WGM-spectroscopy for real-time probing of the internal structure of single-ring hollow-core photonic crystal fiber (SR-PCF) during fiber drawing [12]. SR-PCF can offer remarkably low transmission loss [13], is a critical component in novel gas-based pulse compression systems [14], and in helically twisted form can provide strong circular dichroism [15]. We reported that WGMs in individual cladding capillaries can be selectively excited through the fiber cladding without need for complicated optics. We here study in detail how the technique works, and report measurements showing that the diameter of a cladding capillary can be monitored, during fiber drawing and in real-time, with sub-micron-scale transverse resolution and cm-scale axial resolution. We also report post-drawing measurement of all the cladding capillaries in a SR-PCF, which is particularly relevant for the characterization of tapered SR-PCFs [16].
2. Principles and theory of WGM-spectroscopy
The geometry of SR-PCF permits WGMs to be excited in individual capillaries simply by illuminating the side of the fiber with a collimated beam (Fig. 1). A ray incident at a particular point P will refract so that it encounters the point Q where one of the capillaries is fused to the thicker jacket capillary. For an inner to outer jacket diameter ratio of ${{47.6} \mathord{\left/ {\vphantom {{47.6} {80}}} \right.} {80}} = 0.595$ (example parameters used also for Fig. 2), when a capillary is placed at ${\theta _{\textrm{cap}}} = 60^\circ $, the angle between the incident ray and the normal to the glass-air interface at Q is 49.8°, greater than the critical angle $\arcsin ({{1 \mathord{\left/ {\vphantom {1 {{n_\textrm{s}}}}} \right.} {{n_\textrm{s}}}}} )$, which for a silica index of ${n_\textrm{s}} = 1.44$ is 43.9°. As a result the ray undergoes total internal reflection at Q, creating conditions for excitation of a WGM. As indicated in Fig. 1, light that has travelled around the capillary wall exits at the same angle as light reflected at the glass/air boundary. After refraction at the outer boundary of the jacket, the resulting light ray exits the fiber at an angle θcap to the dashed line joining the fiber axis to the point Q.
Fig. 1. (a) Sketch of the ray path that excites a WGM in a capillary at θcap = 60°. Gray indicates glass and white indicates hollow regions of the SR-PCF. (b) Close-up of the ray paths at the point where the capillary is fused to the jacket tube, illustrating the different coupling mechanisms. A: evanescent tunneling, B: ray escapes, C: direct excitation, D: evanescent tunneling.
Fig. 2. FDTD snap-shots of the fields in the SR-PCF at three different times for the same geometry as in Fig. 1. The colors (red for positive and blue for negative) represent the amplitude of the magnetic field pointing along the fiber axis. (a) The pulse wavefronts propagate through the jacket tube towards the capillary. (b) The pulse travels around the capillary. (c) The pulse reaches the point Q and partly couples out, a small fraction continuing around the capillary. Also see Visualization 1.
To better understand this interaction, we used commercially available finite-difference-time-domain (FDTD) software (Lumerical) to model excitation by pulses with bandwidth extending from 300 nm to 1200 nm and duration ∼10 fs (note that for numerical efficiency the pulses are not transform limited). The silica refractive index was kept constant at 1.44 in the simulations. The results are shown in Fig. 2 and the accompanying multimedia file. The FDTD simulations make it possible to observe the temporal evolution of the complex optical fields, the local optical wavelength and group velocity, and the effects of refraction, diffraction and interference.
Straightforward analysis shows that adjacent WGMs are spaced spectrally by:
Fig. 3. Phase (solid) and group (dashed) indices for the $m = 0$ (blue) and $m = 1$ (red) modes of an air-clad planar waveguide 300 nm thick made from silica glass for a fixed silica index of 1.44 (used in the FDTD modelling). The gray curves were calculated with the dispersion of silica included. The group index can be related directly to the frequency or wavelength spacing between adjacent WGMs using the expressions in Eq. (1). The $m = 1$ mode cuts off at 636 nm. The values of group index used to calculate the capillary diameter in Fig. 4 are ${n_{\textrm{G}0}} = 1.485$ at longer wavelength ($m = 0$ mode) and ${n_{\textrm{G1}}} = 1.555$ at shorter wavelength ($m = 1$ mode).
The spectrum of the light coupling out of the capillary (Fig. 2(c)), calculated by FDTD simulations, is plotted in Fig. 4(a). The frequency resolution used was ∼0.3 THz. To obtain a clear WGM spectrum, temporal apodization was used to filter out the portion of the pulse that does not couple into the capillary, and which therefore arrives earlier at the simulated detector than the light travelling around the capillary (note that the experiment detects light that does not couple into the capillary, adding noise to the measured WGM spectrum). The calculated spectrum has two bands of WGM resonances, one extending from ∼420 to 1000 nm, and the other from ∼420 nm to below 300 nm (outside the range considered in this work). The simulations show that in the longer wavelength band the single-lobed $m = 0$ mode is excited in the capillary walls (not surprising since no higher order modes are supported at wavelengths >636 nm), whereas in the shorter wavelength band the $m = 1$ mode is predominantly excited.
Fig. 4. (a) Spectrum of the light emerging from the capillary (dc = 14 µm, t = 300 nm) for the configuration in Fig. 1, simulated by FDTD modelling for fixed silica index of 1.44. Fundamental ($m = 0$) waveguide modes are excited in the 420-1200 nm band (blue) and $m = 1$ modes are excited in the 300-420 nm band (red). (b) Fourier transform of S(ν) for the two bands, plotted against capillary diameter, as explained in the text.
The optical spectrum, evaluated as a function of optical frequency, can be written in the approximate form $A(\omega - {\omega _0}){\cos ^2}(k\omega /2)$ where A is a slowly varying envelope function that peaks at $\omega = {\omega _0}$ and $\delta \omega = 2{\pi \mathord{\left/ {\vphantom {\pi k}} \right.} k}$ is the spacing between adjacent WGMs. Taking the Fourier transform of the experimental data yields a peak at $\tau = k$, where τ is the independent variable (units of time) in the Fourier domain. Using Eq. (1) the capillary diameter can then be calculated as:
where nGm is evaluated at the peak of the spectral band. Carrying out this procedure for the 420-1200 nm band, which peaks at ∼650 nm, a good fit is found for ${n_{\textrm{G}m}} = {n_{\textrm{G}0}} = 1.485$. Similarly, for the short wavelength band, which peaks at ∼320 nm, a good fit is found for ${n_{\textrm{G}m}} = {n_{\textrm{G1}}} = 1.555$. The Fourier transforms of the FDTD data in Fig. 4(b), scaled following Eq. (2), show peaks at dc = 14 µm, in agreement with the diameter used in the simulations.The origin of the two spectral bands may be understood by reference to the ray picture in Fig. 1(b). Depending on where an incoming ray strikes, light will either evanescently tunnel into, or directly excite, a waveguide mode in the capillary wall. The efficiency of excitation is likely to be highest when the component of wavevector parallel to the capillary wall matches the propagation constant of the guided mode, although this is unlikely to be a strict condition. For the geometry in Fig. 1(a), the effective phase index of the incoming light, tangential to the capillary wall at the point Q, is 1.10, a value that depends (weakly) on the thickness of the jacket tube. WGMs will be most efficiently excited when the phase index of a waveguide mode equals this value. In Fig. 3 we see that this condition is matched at ∼1100 nm for the m = 0 mode and ∼370 nm for the $m = 1$ mode. For the $m = 0$ mode, very little light is seen in the FDTD spectrum at 1100 nm, which we attribute to bend-loss in the curved capillary walls at longer wavelengths. The short-wavelength edge of the $m = 0$ spectrum is caused by increasing phase mismatch (Fig. 3).
3. Real-time measurements during fiber drawing
The set-up is sketched in Fig. 5. A multimode optical fiber (600 µm core diameter), terminated with a collimating lens, is used to deliver light from a Xe lamp (Ocean Optics HPX-2000, 35 W power, 3 dB emission band ∼400 to ∼1000 nm) to the drawing tower. A polarizer orients the electric field perpendicular to the fiber axis. A second length of the multimode fiber, placed ∼1 cm from the fiber, is used to collect scattering light from the SR-PCF at an angle of ∼60° to the illuminating beam and deliver it to a fast spectrometer (Ocean Optics HR4000), interfaced to a PC for data analysis. To allow easy optimization, the optical components are mounted on a rotation stage centered on the SR-PCF axis.
Fig. 5. (a) Sketch of the measurement set-up. LP: linear polarizer. (b) Example of measured spectrum. (c) Fourier transform of the experimentally measured spectrum, scaled using Eq. (2) and ${n_{\textrm{G}0}} = 1.414$.
A LabVIEW program was written to take a Fourier transform of each spectrum in real-time, locate its peak, calculate the capillary diameter using Eq. (2), and display and log the results. The lower spectral brightness of the Xe lamp below 400 nm and above 1000 nm, and the fall-off in spectrometer sensitivity at shorter wavelength, meant that only the $m = 0$ region was observed in practice, with the result that a single scaling factor nG0 was sufficient for calculating the capillary diameter. We found that excitation and detection of WGMs was so efficient that a fast integration time was possible (typically 70 ms), and that the measurement could be made under normal room illumination. No special focusing of the excitation and scattered light was needed, and the measurement was robust against transverse vibrations of the fiber, although slow rotation of the fiber during drawing (commonly seen) caused the measured signal to periodically disappear and reappear—in itself a useful feature since with appropriate feedback it could be used to draw perfectly twist-free fiber.
To illustrate the power of the technique, we carried out a continuous series of measurements at a draw speed of 32.3 m/min (axial resolution ∼4 cm for 70 ms sampling time) while increasing the pressure (N2 gas) inside the capillaries step-wise from 75 mbar to 130 mbar. The results are shown in Fig. 6, revealing the temporal dynamics of capillary expansion after each pressure step, until at 130 mbar the capillaries touch each other, causing the WGM signal to disappear. Fiber samples were collected during the draw and examined in a scanning electron microscope. The perimeter of the (not always perfectly circular) capillaries was extracted from the resulting SEMs (using ImageJ software [18]) and converted to an effective diameter. The results are included in Fig. 6 (red data-points). There is very good agreement between the two sets of data, showing that the WGM technique can accurately measure the capillary diameter in real time, avoiding the need for destructive and time-consuming imaging after the draw.
Fig. 6. Capillary diameter measured during fiber drawing with the WGM technique (blue) using the scaling factor ${n_{\textrm{G}0}} = 1.484$. The vertical lines indicate the times at which the pressure applied to the capillaries was changed. The insets show SEM images made post-drawing, and the SEM-measured diameter is shown by red bars, the length indicating the standard deviation between the 5 capillaries.
4. Structural characterization post-draw
With multiple illumination and detection angles, or by rotating the system around the fiber or using scanning optics, several capillaries can be monitored simultaneously. To illustrate this, we carried out measurements on an already-drawn SF-PCF, mounting a 60 cm length in two rotation stages and rotating the fiber through 360° while recording the WGM signal. The SR-PCF was a 12 capillary fiber with two missing capillaries (see SEM in Fig. 7(a)), and Fourier transforms of the measured WGM spectra, plotted versus dc and angle, are shown in Fig. 7(b) for ${n_{\textrm{G}0}} = 1.55$ (calculated for $t = 370\textrm{ nm}$ at wavelength 615 nm, silica dispersion included). Each capillary yields a V-shaped signal as the fiber is rotated because the angle of incidence on the probed capillary changes, causing dephasing. The tip of the V occurs at diameters that agree very well with the SEM values.
Fig. 7. (a) SEM of the fiber microstructure. The effective diameter of each capillary, estimated from the SEM, is written in. Comparison with (b) shows that each capillary can be identified with good agreement. (b) An example of the scattered signals detected during a post-drawing measurement in which a SR-PCF was rotated. The angle was scanned in steps of 0.1° and the scaling factor ${n_{G0}} = 1.55$ (valid for $t = 370\textrm{ nm}$, $\lambda \sim 615\textrm{ nm}$) was used.
In contrast to measurements on the fiber drawing tower, the axial resolution on drawn fibers is not limited by the product of spectrometer integration time and drawing speed, but only by the beam width at the capillary. In Fig. 8 we show the diameter of a single capillary measured by scanning the WGM optics (step size 5 mm, beam width 3 mm) along the fiber. To check the repeatability of the measurement, a second scan was made after removing the fiber and replacing it the opposite way around. To reduce the influence of noise, the diameter was estimated to coincide with the vertex of a parabolic fit to the V-shaped signal originating from the selected capillary. The mean of the deviation in measured capillary diameter was only 22 nm, or 0.16% of the average value.
Fig. 8. Diameter of a single capillary along a 550 mm length of SR-PCF, obtained by finding the vertex of a parabola fitted to the V-shaped signals (Fig. 7(b)). The fiber was scanned in 5 mm steps. Two measurements are shown, the second being made after reversing the fiber in the set-up.
5. Discussion
The minimum detectable change in capillary diameter is related to the bandwidth of the detected spectrum, which is approximately proportional to the inverse width of the peak in the Fourier transform:
The precision of diameter measurement depends on having an accurate value for nGm, Eq. (2), which in turn requires an estimate of the capillary wall thickness in the drawn fiber. Since in the steady-state the mass flow rate of glass is conserved during the draw, the wall thickness and average capillary diameter in the fiber (t2 and dc2) can be related to the values in the preform (t1 and dc1) by:
where Vf and Vd are the feed and draw speeds. Knowing dc1, t1, Vf and Vd, Eq. (4) can be used to estimate matching values of dc2 and t2, the ratio of which, dc2/t2, can be altered by applying differential pressure to the capillaries. With an accurate value for t2, nGm can be calculated and used to scale the Fourier transform in the measurements of dc2. Iterating this procedure will yield more and more accurate on-line measurements of dc2 and t2.Even without these refinements, real-time diameter measurements using a fixed value of nG0 (Fig. 6) showed less than 0.4 µm (<3%) deviations from SEMs. We attribute these small differences to non-uniformities in wall thickness around the circumference, as well as the finite length of the fused contact between capillary and jacket (Fig. 1(b)), which makes it difficult to estimate the optical path-length around the capillary. Since the spectral spacing between WGMs depends only on this path-length, small elliptical or other distortions in capillary shape do not noticeably change the results.
Although the reported simulations and experiments focus on SR-PCF, preliminary FDTD simulations suggest that the WGM technique would be useful for real-time quality control during drawing of more complex single-ring structures such as nested [19] or split-capillary [20] fibers. As briefly explained in Fig. 9, two nested capillaries show four peaks in the Fourier transform, whereas the split-capillary structure shows two peaks, all predictable from the geometrical parameters. Although the appearance of these additional peaks complicates interpretation of the Fourier transform, for simple structures it is straightforward to extract the geometrical parameters.
Fig. 9. (a) FDTD analysis of a nested two-capillary structure (inset) [19] yields a four-peaked Fourier transform, corresponding to resonator diameters dc1, dc2, ${d_{\textrm{c}1}} + {d_{\textrm{c}2}}$, and ${d_{\textrm{c}1}} - {d_{\textrm{c}2}}$ (dashed vertical lines). The latter resonance occurs due to spectral interference between light travelling in the inner and outer capillaries. (b) A split capillary structure (inset) [20] yields peaks corresponding to equivalent resonator diameters dc1 and ${{{d_{\textrm{c}1}}({1 + {\pi \mathord{\left/ {\vphantom {\pi 2}} \right.} 2}} )} \mathord{\left/ {\vphantom {{{d_{\textrm{c}1}}({1 + {\pi \mathord{\left/ {\vphantom {\pi 2}} \right.} 2}} )} \pi }} \right.} \pi }$ (dashed vertical lines).
6. Conclusions
WGM spectroscopy is a powerful technique for real-time measurement of the relatively simple microstructure in SR-PCFs to sub-micron accuracy. Since it can be used for continuous monitoring during the fiber draw, and could be incorporated in a feedback system, it could be the key to drawing long lengths of high quality SR-PCF for applications in, e.g., telecommunications. Operating with an inexpensive Xe lamp, the technique can be used in existing fiber drawing facilities without any concern for laser safety or need for laboratory modifications. By rotating the fiber relative to the detector, or using multiple illumination and detection directions, one could probe the WGM spectrum of multiple resonators. Since the value of nGm will be affected by the index of a gas or liquid filling the capillaries, the WGM technique could be used for sensing, using just a lamp and a spectrometer. We note that the surrounding jacket tube of SR-PCF directs rays from a collimated incident beam into the capillaries, resulting in efficient excitation of WGMs without need for focusing optics. The technique is also useful for non-destructive post-drawing analysis, for example of tapered SR-PCFs, and if required can be used to measure the frozen-in twist of a fiber.
Funding
Max-Planck-Gesellschaft.
References
1. J. F. Owen, P. W. Barber, B. J. Messinger, and R. K. Chang, “Determination of optical-fiber diameter from resonances in the elastic-scattering spectrum,” Opt. Lett. 6(6), 272–274 (1981). [CrossRef]
2. A. Ashkin, J. M. Dziedzic, and R. H. Stolen, “Outer diameter measurement of low birefringence optical fibers by a new resonant backscatter technique,” Appl. Opt. 20(13), 2299–2303 (1981). [CrossRef]
3. A. W. Poon, R. K. Chang, and J. A. Lock, “Spiral morphology-dependent resonances in an optical fiber: effects of fiber tilt and focused Gaussian beam illumination,” Opt. Lett. 23(14), 1105–1107 (1998). [CrossRef]
4. T. A. Birks, J. C. Knight, and T. E. Dimmick, “High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment,” IEEE Photonics Technol. Lett. 12(2), 182–183 (2000). [CrossRef]
5. P. St.J. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003). [CrossRef]
6. G. T. Jasion, J. R. Hayes, N. V. Wheeler, Y. Chen, T. D. Bradley, D. J. Richardson, and F. Poletti, “Fabrication of tubular anti-resonant hollow core fibers: modelling, draw dynamics and process optimization,” Opt. Express 27(15), 20567–20582 (2019). [CrossRef]
7. A. Stefani, M. H. Frosz, T. G. Euser, G. K. L. Wong, and P. St.J. Russell, “Real-time Doppler-assisted tomography of microstructured fibers by side-scattering,” Opt. Express 22(21), 25570–25579 (2014). [CrossRef]
8. W. Gorski and W. Osten, “Tomographic imaging of photonic crystal fibers,” Opt. Lett. 32(14), 1977–1979 (2007). [CrossRef]
9. S. R. Sandoghchi, G. T. Jasion, N. V. Wheeler, S. Jain, Z. Lian, J. P. Wooler, R. P. Boardman, N. Baddela, Y. Chen, J. Hayes, E. N. Fokoua, T. Bradley, D. R. Gray, S. M. Mousavi, M. Petrovich, F. Poletti, and D. J. Richardson, “X-ray tomography for structural analysis of microstructured and multimaterial optical fibers and preforms,” Opt. Express 22(21), 26181–26192 (2014). [CrossRef]
10. S. Schmidt, T. Tiess, S. Schrotter, A. Schwuchow, M. Jäger, H. Bartelt, A. Tünnermann, and H. Gross, “Noninvasive characterization of optical fibers,” Opt. Lett. 42(23), 4946–4949 (2017). [CrossRef]
11. X. B. Xu, X. Y. Wang, T. T. Zhu, F. Y. Gao, and N. F. Song, “Nondestructive determination of the core size of a hollow-core photonic bandgap fiber using Fabry-Perot interference,” Opt. Lett. 43(13), 3045–3048 (2018). [CrossRef]
12. M. H. Frosz, R. Pennetta, M. T. Enders, G. Ahmed, and P. St.J. Russell, “Non-invasive real-time characterization of hollow-core photonic crystal fibres using whispering gallery mode spectroscopy,” in 2019 Conference on Lasers and Electro-Optics and Europe & European Quantum Electronics Conference, Centre Munich, Germany, 23 June 2019, CJ-3.6.
13. B. Debord, A. Amsanpally, M. Chafer, A. Baz, M. Maurel, J. M. Blondy, E. Hugonnot, F. Scol, L. Vincetti, F. Gérôme, and F. Benabid, “Ultralow transmission loss in inhibited-coupling guiding hollow fibers,” Optica 4(2), 209–217 (2017). [CrossRef]
14. U. Elu, M. Baudisch, H. Pires, F. Tani, M. H. Frosz, F. Köttig, A. Ermolov, P. St. J. Russell, and J. Biegert, “High average power and single-cycle pulses from a mid-IR optical parametric chirped pulse amplifier,” Optica 4(9), 1024–1029 (2017). [CrossRef]
15. P. Roth, Y. Chen, M. C. Günendi, R. Beravat, N. N. Edavalath, M. H. Frosz, G. Ahmed, G. K. L. Wong, and P. St. J. Russell, “Strong circular dichroism for the HE11 mode in twisted single-ring hollow-core photonic crystal fiber,” Optica 5(10), 1315–1321 (2018). [CrossRef]
16. R. Pennetta, M. T. Enders, M. H. Frosz, F. Tani, and P. St.J. Russell, “Fabrication and non-destructive characterization of tapered single-ring hollow-core photonic crystal fiber,” APL Photonics 4(5), 056105 (2019). [CrossRef]
17. H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer, 1975) pp. 13–81.
18. C. T. Rueden, J. Schindelin, M. C. Hiner, B. E. DeZonia, A. E. Walter, E. T. Arena, and K. W. Eliceiri, “ImageJ2: ImageJ for the next generation of scientific image data,” BMC Bioinformatics 18(1), 529 (2017). [CrossRef]
19. T. D. Bradley, J. R. Hayes, Y. Chen, G. T. Jasion, S. R. Sandoghchi, R. Slavik, E. N. Fokoua, S. Bawn, H. Sakr, I. A. Davidson, A. Taranta, J. P. Thomas, M. N. Petrovich, D. J. Richardson, and F. Poletti, “Record low-loss 1.3 dB/km data transmitting antiresonant hollow core fibre,” in 2018 European Conference on Optical Communication (ECOC), pp. 1–3 (2018). [CrossRef]
20. S.-F. Gao, Y.-F. Wang, W. Ding, D.-L. Jiang, S. Gu, X. Zhang, and P. Wang, “Hollow-core conjoined-tube negative-curvature fibre with ultralow loss,” Nat. Commun. 9(1), 2828 (2018). [CrossRef]
