1. Introduction

Anti-resonant hollow-core fibers (AR-HCFs, or ARFs in short) [1–4] have recently attracted great attention because they can offer flexibly designed wide transmission windows [5], high optical damage thresholds [6,7], effective single-mode operation [8], low group velocity dispersion [9], low attenuation comparable to silica solid-core fiber [10], and high resilience to environmental perturbations [11–13]. Since the first demonstration of a geometry consisting only of a single ring of tubes in the cladding (referred to as SR-ARF herein) [14], a variety of tubular ARF structures [15–20] have been extensively investigated. Through continuous technological advancement, recent years have witnessed the realization of many novel optical functionalities in tubular ARFs, whose inner geometry play decisive roles.

To achieve the desired optical properties, the microstructure geometry of ARFs has to be controlled very accurately during fabrication, e.g., by adjusting the drawing parameters such as furnace temperature, gas pressure, fiber drawing tension and speed. To adjust these drawing parameters, monitoring of the fiber microstructure is required. The current solution is to manually cut a fiber sample mid-draw and inspect its cross-section using either an optical or electron microscope. However, this method not only is destructive, therefore causing severe fiber waste, but also suffers with low efficiency and low drawing speed. A non-invasive, fast, and complete characterization of a tubular ARF is highly desired.

Tomographic imaging, which allows the reconstruction of a 3D view, naturally has the virtues of non-destructive and arbitrary geometry characterization. Techniques based on phase [21,22], amplitude [23], and Doppler-frequency-shift [24] measurements of micro-structured optical fibers have been reported, however, requiring to fill air-holes with oil of a refractive index close to silica [21,22], a sophisticated (but bulky) X-ray source [23], or keeping the rotation axis and speed invariant [24]. Moreover, the multidirectional data acquisition nature of a tomographic method is associated with the time-consuming attribute, making it difficult to real-time monitor a fiber during draw. Another issue of tomography is the resolution of the reconstructed image, which in general compromises with the operating wavelength and the scan duration [25].

To resolve this dilemma, fast analysis of diffraction pattern under side illumination of a coherent beam is resorted to [26]. However, so far, this method is only valid for capillary optical fibers having a single center hole. As the complexity of the fiber internal structure increases, the diffraction pattern becomes too complicated to analyze. In line with this research, the micro-structured optical fibers for non-destructive characterization gradually evolve to the simpler ones such as tubular ARFs, for which a whispering gallery mode spectroscopic method has demonstrated real-time measurement of cladding tube diameters with sub-micron accuracy [27].

Nevertheless, to specify the optical properties of a tubular ARF, more geometry information, including hollow core diameter, glass wall thickness, azimuthal positions of cladding tubes, and gap distances, are needed. In this work, we fully exploit the potential of side-scattering spectroscopy by proposing a four-ray interference model based on simple geometric optics. All the structural parameters of a SR-ARF can be experimentally measured or estimated in a non-invasive and fast fashion. V-shaped interference peaks with the rotation of the SR-ARF are clarified by our model and utilized to retrieve group delay differences with high accuracy. Additionally, in order to acquire the wall thickness of a cladding tube, the polarization and the transverse mode dependences can be employed.

2. Principles of four-ray interference

Figure 1(a) plots a cross-sectional diagram of a SR-ARF. The jacket tube is assumed to have an ideally annular shape with the outer and the inner radii referred to as R and r [see Fig. 1(b)], respectively. Under illumination of a collimated beam, perpendicular to the fiber axis, light can find four paths to traverse the SR-ARF and leave at the same scattering angle (φ). The four ray-paths are described as follows [Figs. 1(b) and 1(c)]:

 

Fig. 1. (a) Sketch of the four rays that are scattered by a SR-ARF to the same detection angle φ. (b) Paths of Rays 1, 2 and 4. (c) Close-up of the path of Ray 3.

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Ray 1 is the result of one reflection at the outer surface of the fiber;

Ray 2 goes through two refractions at the outer interface and one reflection at the inner interface of the jacket to point at the detection direction;

Ray 3 also experiences two refractions at the outer interface, whilst at the inner interface of the jacket, it excites guided modes in a cladding tube, travels around the circumference, and exits from the other end. Because of the fusion of the two glass tubes, the entry and the exit points of the guided wave stay apart from each other (i.e., the opening angle of the fused segment γ ≠ 0), and considering diffraction effects, the emergent angle of Ray 3 (α₃′) can slightly differ with the incident angle (α₃) [see Fig. 1(c)]. Note that, albeit the light propagation around the cladding tube is not suit to using geometric optics, we still adopt the name of ‘Ray 3’ for simplicity to describe the whole light path inside the air, the jacket, and the cladding tube;

Ray 4, which lies on the other side of the jacket tube, undergoes two refractions at the outer surface of the fiber. Depending on the inner radius of the jacket (r) and the detection angle (φ), Ray 4 could be blocked.

As displayed in Fig. 1(b), the optical paths of Rays 1, 2, and 4 share a common symmetry axis (i.e., the blue dashed-dotted line), whereas Ray 3 does not comply with this symmetry, therefore allowing a nonzero deviation angle θ [as defined in Fig. 1(c)].

2.1 Interference of Rays 1, 2, and 4

If the shape of the fiber is an ideal circle, the optical path-length difference between Ray 1 and Ray 4 is only relevant to the outer radius (R). The phase difference (Δϕ₁₄) consists of the contributions in air and in silica,

Once Δϕ₁₄ is known, Eq. (1) can give the quantity of R.

Similarly, the phase difference between Ray 2 and Ray 4 can be expressed as

And, the Snell’s Law of Ray 2 yields

Since R, α₁, and α₄ have been obtained from Eqs. (1) and (2), the inner radius of the jacket (r) and the incident angle of Ray 2 (α₂) can be calculated from Eqs. (3) and (4), given that Δϕ₂₄ is known. The quantity r can also be calculated from the phase difference between Ray 1 and Ray 2 (Δϕ₁₂).

Regarding measurement, interferometry in general revolves around the interference of component wavelets which have been subject to different phase shifts due to propagation. The SR-ARF structure in Fig. 1 can be seen as an interferometer, which splits light into four spatially separated tones. Because of coherent superposition, the group delay difference between any two rays (τ_jk) can be inferred from the Fourier transform (FT) of the measured side scattering spectrum, and the relationship between group delay difference and phase difference is

2.2 Interference of Rays 2 and 3

In order to measure the diameter of a cladding tube (d), the optical path-length of Ray 3 needs to be clarified. As shown in Fig. 1(c), both the deviation angle (θ) of the cladding tube from the symmetry axis (i.e., the blue dashed-dotted line) and the angle of fused segment (γ) need to be counted.

Splitting the full optical path of Ray 3 into three segments (i.e., in air, in bulk silica, and in the cladding tube), we can write the phase difference between Rays 2 and 3 as

In Eq. (7), the values of R, r, α₁, and α₂ have been acquired from Eqs. (1)–(5). With regard to the angles α₃ and α₃′, the Snell’s Law gives

For simplicity, the optical path of Ray 3 around the cladding tube can be approximated by the propagation of a guided wave in a planar slab. The effect of the bend can be neglected provided t/d ${\ll} $ 1, where t is the wall thickness of the cladding tube. Furthermore, since the modal indices of guided waves of different orders and polarizations can be quickly calculated in the slab form using standard planar waveguide theory [29], this offers a convenient means to estimate the quantity t, which will be discussed below.

3. Experiment setup

To test the feasibility of the above model, two SR-ARF samples (ARF #1 and ARF #2) were fabricated in-house. Figure 2 shows the scanning electron microscope (SEM) images with the measured parameters of R = 62.0 ∼ 64.0 µm, r = 15.9 ∼ 16.8 µm, d = 7.0 ∼ 7.7 µm, γ = 52.5 ∼ 63.0°, t = 0.34 ∼ 0.38 µm for ARF #1 and R = 85.8 ∼ 86.0 µm, r = 21.9 ∼ 22.0 µm, d = 11.0 ∼ 11.2 µm, γ = 54.1 ∼ 60.2°, t = 0.78 ∼ 0.83 µm for ARF #2. The resolution of our SEM facility is approximately 20 nm.

 

Fig. 2. SEM images of a 7-tube SR-ARF (ARF #1) and a 6-tube SR-ARF (ARF #2).

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3.1 Side-scattering spectrum measurement

Figure 3 shows the flow chart of side-scattering spectrum measurement and analysis process. In the spectrum acquisition part [Fig. 3(a)], a supercontinuum source (YSL Photonics, SC-5) spliced to an endlessly single-mode photonic crystal fiber launches broadband light (spanning from 470 to 2400 nm) to side-illuminate the tested fiber, which is vertically mounted on a precision rotation stage (Thorlabs, PRM1Z8). The illumination light is collimated by a reflective concave parabolic mirror (LBTEK, MPM0515-90-AG) to eliminate spherical and chromatic aberration over the whole spectrum range. A Glan-Taylor calcite polarizer (Thorlabs, GT10) orients the polarization. In the detection end, an optical fiber with a core diameter of 50 µm collects light at a certain angle. A fast spectrometer (Ocean Optics QEPRO, with the working wavelength of 178 - 973 nm, the spectrum resolution of 6.87 nm, and the A/D resolution of 18-bit) records the interference spectra. Compared to the setup in Ref. [27], which consists of a Xe lamp, a refractive convex lens, and a collection fiber with the core diameter of 600 µm, the adoption of a perfectly-collimated single-mode white-light source and a collection fiber of a much smaller core in Fig. 3(a) allows us to exploit the full potential of side-scattering spectroscopy with a high angle resolving capability and a high excitation-detection efficiency. With a sufficient signal-to-noise ratio, a typical integration time of a scattering spectrum in our setup can be less than 8 ms (one order of magnitude less than [27]) with the collection fiber placed ∼5 cm from the sample. By rotating the tested fiber over the full angle of 360° (in steps of 0.5°), all the cladding tubes inside a SR-ARF can be measured. The full reconstruction time except the time for fiber rotation and data processing is approximately 6 s. In the future, given that we obtain a higher-power-density supercontinuum source and a faster spectrometer, the acquisition time of a single spectrum could drop to far less than 8 ms.

 

Fig. 3. Flow chart of side-scattering spectrum measurement and analysis. (a) Experimental setup. SC, super-continuum. (b)-(e) Data-processing of exemplar spectra acquired from ARF #1.

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In the data processing and analysis part [Figs. 3(b)–3(e)], the optical power spectrum, I(ν), recorded at each rotation angle Θ, which contains the information of relative phase differences of different rays due to their coherent superposition, was firstly filtered by a rectangular window (e.g., from 550 to 950 nm) to get rid of the noise far away from the center of spectrum. Then, the spectrum was Fourier transformed to the time domain, $\mathcal{F}$[I(ν)], to visualize the interference peaks. As highlighted in Fig. 3(c), for ARF #1, the key interference peaks between the four rays (S_jk with j, k = 1-4) can be clearly identified by comparing the calculated results (the red dashed lines) based on SEM measured geometry, thus validating high efficiency of our measurement setup. In this FT curve, the group delay differences [τ_jk in Eq. (5)] can be read out. By rotating the tested fiber [Fig. 3(d)], the S_jk spectrum can be expanded to the full angle of Θ. And, the locations of the interference peaks (τ) can be plotted versus the rotation angle Θ [Fig. 3(d)]. All these data processing along with spectrum acquisition are automated using a LabVIEW script.

3.2 Dual-polarization measurement

The single-modedness of the illumination source and the ideal beam collimation across a broad spectrum allow us to conduct dual-polarization comparative measurement by simply rotating the linear polarizer. Under the two polarizations, parallel (s-) and perpendicular (p-) to the fiber axis [see Fig. 3(a)], the intensities of transmission and reflection along the four ray paths differ greatly [30]. Utilizing Fresnel equations, Fig. 4(a) plots the calculated transmittance of Rays 1, 2, and 4 through ARF #1 (with R = 63.0 µm and r = 16.3 µm) as a function of the detection angle φ. In calculation, the refractive index of silica is set to be 1.46 at the wavelength of 550 nm. It is seen that Ray 2 and Ray 4 can be well preserved in the detection end for both polarizations, whilst Ray 1 can only be detected in the s-polarization because the incident angle α₁ = (π-φ)/2 is close to the Brewster angle. Figure 4(b) plots the measured FT curves over all the rotation angels for ARF #1. It is manifest that all the interferences associated with Ray 1 are absent in the p-polarization.

 

Fig. 4. (a) Calculated transmittances of the light rays through ARF #1 versus the detection angle in the two polarizations. (b) Measured FT curves across all the rotation angels for ARF #1 and in the two polarizations when the detection angle φ = 70°.

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4. Characterization results

4.1 Radii of jacket tube (R and r)

To measure the outer diameter of a fiber, standard practice on any drawing tower simply uses commercially available detectors [31], which can perceive the shadow cast of a fiber with sub-micron accuracy by side illumination. However, for the integrity of technique, we still start our study from outer diameter measurement. Under the s-polarization, Fig. 5(a) plots the measured group delay difference τ₁₄ (between Ray 1 and Ray 4) for ARF #1 and ARF #2 at φ = 70° across the full rotation angle. In the τ₁₄ trace of ARF #1, it is found that the measured result varies periodically with the rotation angle. We attribute this to the slightly elliptical shape of the outer boundary of ARF #1. After carefully inspecting the SEM image in Fig. 2, the half of major and the minor axes are ∼64.0 and ∼62.0 µm, respectively. Incorporating this elliptical shape into a new calculation (with φ = 70°) yields the two τ₁₄’s as labelled by the dashed lines in Fig. 5(a), which validates again the accuracy of our measurement.

 

Fig. 5. (a) Measured τ₁₄ as a function of the rotation angle Θ for ARF #1 (magenta) and ARF #2 (green) when φ = 70°. The two dashed lines are the calculated results, taking into account the elliptical outer boundary of ARF #1 with the half of major and the minor axes being 64 µm and 62 µm, respectively. In calculation of Case 1 and Case 2, either the major or the minor axis is in line with the symmetry axis in Fig. 1. (b) FT curves of S₁₄ with use of different rectangular window functions. The black dashed line is a FT of a rectangular function from 550 to 950 nm.

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The calculation errors of R arise from two origins: one is the inaccurate measurement of group delay difference, the other is the varied refractive index of silica n_silica over the operating wavelengths. We consider the uncertainty of group delay difference (Δτ) in terms of the full width at half maximum (FWHM) of the interference peaks [e.g., S₁₄ in Fig. 3(c)]. In general, this uncertainty is caused by two factors: (1) the limited bandwidth of the optical spectrum, and (2) the chromatic dispersion of the refractive index [32]. The red line in Fig. 5(b) shows the close-up of the S₁₄ peak in Fig. 3(c), which reads Δτ ≈ 6 fs (corresponding to an uncertainty of the path-length in silica of ∼1.2 µm). If the rectangular window function (before FT) is shrunken to the wavelength range of 650-850 nm, the uncertainty of group delay difference increases to Δτ ≈ 9 fs [the blue line in Fig. 5(b)]. Whereas, directly making FT to a rectangular function of from 550 to 950 nm yields Δτ ≈ 4 fs [the black dashed line in Fig. 5(b)]. This implies that the chromatic dispersion effect in silica enlarges the FWHM of an interference peak by at most 2 fs (or one third), therefore does not need to be taken into account in our FT calculation. On the other hand, the different values of n_silica (across the wavelength range of 550-950 nm) indeed influence the estimation to the length of a certain optical path, so that cause measurement error. The average outer radii of ARF #1 and ARF #2 are evaluated to be 63.3 ± 0.6 µm and 86.4 ± 0.8 µm, respectively, by Eqs. (1) and (2), agreeing roughly with the SEM measured values (62.0 ∼ 64.0 µm and 85.8 ∼ 86.0 µm, respectively).

As proposed in the subsection 2.1, a group delay difference τ₂₄ can help to derive the inner radius (r) from R. Figure 6 shows the τ₂₄ of ARF #1 and ARF #2 as a function of the rotation angle. The average inner radii of these two fibers are calculated to be 16.7 ± 0.8 µm and 22.7 ± 0.9 µm, respectively, by Eqs. (3) and (4), agreeing roughly with the SEM measured values (15.9 ∼ 16.8 µm and 21.9 ∼ 22.0 µm, respectively). Here, the calculation error of R has been added into that of r, because the calculation of the latter needs the knowledge of the former.

 

Fig. 6. Measured τ₂₄ as a function of the rotation angle Θ for ARF #1 and ARF #2 when φ = 70°. Inset: sketches of Ray 2 going through the reflection point outside or inside the fused segment of a cladding tube and the jacket.

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An interesting feature is observed in the τ₂₄ curve of ARF #2 (the green line in Fig. 6). Every time Ray 2 travels through the fused segment of the jacket and one cladding tube, the measured τ₂₄ decreases slightly. This can be explained by the fact that the fusion between two glass tubes pushes the glass-air interface inward (i.e., r’ < r as delineated in the inset of Fig. 6). A slightly elongated path of Ray 2 results in a reduced τ₂₄. Accordingly, the angular positions of the six cladding tubes of ARF #2 can be clearly observed in Fig. 6. The derived values of r - r’ (0.54 ∼ 0.69 µm) and the widths of the fused segments (corresponding to their opening angles γ of 30.0° ∼54.9°) both agree roughly with the SEM measurement (in Fig. 2). In ARF #1, this feature of reduced τ₂₄ at specific Θ has not been observed, probably because the thicknesses of the cladding tubes of ARF #1 are too thin that the difference between r’ and r cannot be discriminated.

4.2 Diameter of cladding tube (d)

To calculate the diameter of a cladding tube, the group delay difference between τ₂₃ is needed. According to Eqs. (6)–(8), even excluding the ellipticity of all the relevant tubes, the value of τ₂₃ still depends non-trivially on the rotation angle Θ [see Fig. 7(a)].

 

Fig. 7. (a) Schematic of path-length variation of Ray 3 with Θ. (b) An example of the measured FT curves over a span of rotation angles for ARF #1 and at φ = 70°. FWHM is measured as the full width at half maximum of a curve in the temporal domain. (c) Measured and fitted τ₂₃ with the variation of angle θ. (d) Derived and SEM measured diameters of all the cladding tubes. (e) Calculation errors of d with and without parabolic fit of τ₂₃.

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In our measurement, e.g., that shown in Fig. 7(b), the peaks of S₂₃ exhibits a V-shaped curve (with the variation of Θ). The minimum τ₂₃ appears at the deviation angle (of the cladding tube from the symmetry axis) θ = 0 [27,33], where the optical path-length of Ray 3 [Fig. 1(c)] reaches to its minimum. Applying a parabolic fit to the measured τ₂₃’s at different Θ’s [Fig. 7(c)], the vertex value can be read out accurately. The diameters of the cladding tubes in ARF #1 and ARF #2 can then be calculated utilizing the method in subsection 2.2. As shown in Fig. 7(d), the discrepancies between the calculated and the SEM measured d’s are less than 0.12 µm for all the 13 cladding tubes. The adoption of the parabolic fit remarkably reduces the calculation error of d’s to less than 0.17 µm as indicated in Fig. 7(e), where the cyan rectangles represent the calculation from a single S₂₃ curve at one rotation angle Θ. As discussed in Fig. 5(b), the FWHM of a S₂₃ curve is limited by some intrinsic factors and will bring about a big deviation like that in calculation of R and r. The inclusion of different S₂₃ curves at various Θ angles increase the amount of information and improve the calculation precision of d by 3-4 fold [Fig. 7(e)]. Note that in our calculation, a priori γ (measured from the SEM image) and n_eff have been used. And, the variations of R and r in different azimuthal angles, which have been embodied in their calculation errors (the last subsection), will negligibly influence the calculation of d.

After securing all the parameters of R, r and d, we can derive the core diameter (with the preliminary resolution of around 2 µm), the azimuthal angle of individual cladding tube, and the gap distance, therefore learn all the structural information of the cross-section of an ARF.

4.3 Thickness of cladding tube (t)

To estimate t, both the polarization and the transverse mode dependences of the optical path-length of Ray 3 may be exploited. For simplicity, the glass wall of a cladding tube is modelled as an ideal air-clad planar waveguide [27], and its thickness t is relevant to the difference between τ₂₃ of two polarizations or two modes at one wavelength. Subsection 3.2 has described the principle of dual-polarization spectroscopic measurement. As illustrated in Fig. 8(a), a s- or p-polarized incident beam can excite TE or TM guided mode inside the glass wall, respectively. The group delay difference between Rays 2 and 3 under the two polarizations can be approximately written as

 

Fig. 8. (a) Sketch of exciting a TE(TM) mode in a cladding tube by Ray 3. (b) Measured τ₂₃ and τ₂₄ of ARF #1 (magenta) under the two polarizations. (c) SEM image of ARF #3. (d) Calculated n_g^(TE)/n_g^(TM) versus t of an air-clad planar silica waveguide at λ = 700 nm (blue). The pinkish, greenish, and yellowish bands refer to the SEM measured glass wall thicknesses in ARF #1, ARF #2, and ARF #3, respectively. Inset: measured τ₂₃ (as Θ) of one cladding tube in ARF #2 (green) and in ARF #3 (brown) under the two polarizations.

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We fabricate a new fiber sample of ARF #3 with the SEM measured glass wall thickness of 0.24∼0.31 µm [Fig. 8(c)]. The blue curve in Fig. 8(d) plots the calculated ratio of n_g^(TE)/n_g^(TM) as a function of the thickness of a planer silica waveguide at λ = 700 nm. We carried out dual-polarization measurements of τ₂₃ for ARF #2 and ARF #3, and the insets of Fig. 8(d) show the results of one cladding tube of the two ARFs. An interesting measurement result of n_g^(TE)/n_g^(TM) < 1 for ARFs #1 and #2 and n_g^(TE)/n_g^(TM) > 1 for ARF #3 is manifest, coinciding well with the calculation. Note that in Fig. 8(d), the horizontal dashed lines represent the average values of the ratios of τ₂₃^(s)/τ₂₃^(p) for all the cladding tubes. Limited by the measurement precision of τ₂₃, the deviation of these ratios stays 0.01 ∼ 0.03.

In addition to the polarization dependence, the dependence of group delay difference on the order of guided mode can also infer t. As shown in Fig. 9(a), two interference peaks of S₃₄ (between Ray 3 and Ray 4) appear in the measurement of ARF #2 under the p-polarization. Calculation reveals that the TM modes at play are the fundamental (TM₀) and the first higher-order (TM₁) modes with n_g^(TM0) < n_g^(TM1). Figure 9(b) plots the calculated n_g’s at λ = 700 nm. When t > 0.58 µm, the n_g^(TM1) of a planar slab silica waveguide surpasses n_g^(TM0). However, the measurement precision of the ratio of these two τ₃₄’s in Fig. 9(a) is far from accurate estimation of t. The influences of the diameter of the cladding tube on corresponding τ’s at different wavelengths lie out of the scope of this study.

 

Fig. 9. (a) Measured FT curves of S₃₄ over a span of rotation angle for ARF #2, at φ = 60°, and in the p-polarization. (b) Calculated group indices of the TM₀ and the TM₁ modes in an air-clad planar silica waveguide at λ = 700 nm.

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The accurate measurement of parameter t can resort to the conservation of the ratio of the cross-sectional areas of the jacket and the cladding tubes along the drawn fiber [34], which can be expressed as

5. Conclusion

In conclusion, a four-ray interference model in combination with white-light side-scattering spectroscopy exhibit great potentials for non-invasive, fast, and complete characterization of SR-ARFs. We demonstrate that by measuring interference spectra, the outer and the inner radii of the jacket tube can be retrieved with sub-micron accuracy. By fitting the interference peaks acquired at different rotation angles of the SR-ARF (i.e., exploiting the V-shaped characteristic), the measurement precision of the diameters of cladding tubes can be increased by 3-4 fold compared with the case without fitting. By optimizing the illumination light source and the collimator, the integration time of a single spectrum drops to less than 8 ms, which is the smallest option of our spectrometer. Real-time and high signal-to-noise ratio acquisition of all the interference peaks between the four rays is therefore assured. Additionally, both the polarization and the transverse mode dependences are discernible in interference spectra and can be used to estimate glass wall thickness. Since the experimental system is compact without need for focusing optics, it can be easily incorporated in the existing fiber drawing facilities. By adding more ray-paths, our geometry characterization approach should be of applicability to other types of tubular ARFs [27,33], such as nested ARFs [17] and semi-tube ARFs [20].