Open Access
Add to BibSonomy
Abstract
We examine experimentally the influence of the fiber inelastic twist on polarimetric sensitivity to hydrostatic pressure and pressure sensitivity in a Rayleigh-scattering-based optical frequency-domain reflectometer (OFDR) for highly birefringent side-hole fibers. The fibers were drawn from the same preform with different spin pitches varying from 5 mm to 200 mm. We also demonstrate that the sensitivities of spun fiber can be estimated analytically based on its spin pitch, measured birefringence, and the sensitivity of the corresponding non-twisted fiber, showing good agreement with the experimental results. We proved that polarimetric sensitivity to pressure decreases with the shorter spin pitches, while distributed pressure sensitivity decreases for the one polarization eigenmode and increases for the second polarization eigenmode. Therefore, the spun-fibers can operate well as the polarimetric sensors of other physical parameters inducing circular birefringence even under varying pressure. Moreover, they can be used to obtain the desired difference between sensitivities for both polarization eigenmodes
Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
1. Introduction
Fiber optic pressure sensing is of great importance in the gas and oil industry and in geotechnical and aerospace applications [1,2]. Some specialty birefringent optical fibers are known to feature enhanced pressure sensitivities [3–19]. The greater the mechanical asymmetry of the fiber, the greater the asymmetry in the distribution of pressure-induced stresses in the region of the core. As a consequence, the fiber is characterized by a greater differentiation of responses of both polarization modes, and thus higher polarimetric sensitivity to pressure Kp. The enhanced pressure sensitivity of such fibers has been investigated since the late 1980s. Xie et al. [3] studied side-hole fibers (SHFs), in which a high sensitivity value is achieved by two air channels parallel to the core. The author analyzed a fiber with a circular core and found that the Kp sensitivity of the SHFs is dependent on the size of the holes and wall thickness of either side of the fiber—the larger the side hole and the closer to the core, the higher the pressure sensitivity that can be achieved [3,4]. Depending on the size and location of the hole, the polarimetric sensitivity to pressure in SHFs reaches the value of 30 rad/MPa × m at 1550 nm [5], −82 rad/MPa × m at 830 nm [6] and can reach the value of 110 rad/MPa × m [3] or even over −150 rad/MPa × m [7] at 633 nm. Furthermore, the sensitivity of SHFs is much higher than that of fibers with an elliptical core (0.5 rad/MPa × m) and an asymmetric stress distribution, such as a bow-tie fiber or panda fiber (8 rad/MPa × m in bow-tie fibers at 850 nm [8]). In [7,9] a comprehensive theoretical and experimental analysis of SHFs with an elliptical core was performed. The authors analyzed the influence of the core dimensions, the shape and dimensions of the holes, and the distance between the holes and the core on the pressure sensitivity [7]. Depending on the orientation of the axis of the core ellipse (SHFs of type P or type K), the authors showed that the sign of the pressure sensitivity may be different. Moreover, the significant ellipticity of the core makes the fiber internal birefringence sufficiently large to avoid the cancelation by the pressure-induced birefringence [10]. A similar observation was published in [11]. The authors compared two microstructured fibers (MOFs), which were made of comparable preforms but fabricated under different drawing conditions. In both preforms, the doped area was circular, but due to the strongly asymmetric cladding microstructure, it was deformed during fiber drawing. As the MOF2 fiber was drawn, the pressure was increased and the drawing speed was changed to enlarge the air holes surrounding the core. This resulted in the opposite orientation of the core polarization axes in relation to MOF1. As a result, for optical Bragg gratings inscribed in the examined fibers, the pressure sensitivity of the Bragg peak separation was negative for MOF1 and positive for MOF2, with values corresponding to polarimetric pressure sensitivities of −56 rad/MPa × m and 126 rad/MPa × m, respectively, at 1550 nm wavelength. In recent years, many different geometries of MOFs have also been proposed for pressure sensing, from fibers with two larger holes adjacent to the core −7.5 rad/MPa × m [12], to fibers with different arrangements and sizes of air holes in the microstructured cladding −43 rad/MPa × m [13], −9 rad/MPa × m [14], −9.5 rad/MPa × m [15], −62.4 rad/MPa × m, and −40.1 rad/MPa × m [16] at 1550 nm. In the literature, one can also find fibers combining the construction of SHFs with the microstructure region near the core that have a pressure sensitivity of −11 rad/MPa × m [17], −25.6 rad/MPa × m [18], and −93 rad/MPa × m [19] at 1550 nm.
The first documented distributed fiber optic sensor and the idea of measuring polarization changes along the fiber for sensing pressure date back to 1980. The idea of measuring pressure or transverse force through the birefringence it induces was later exploited and studied in several papers, using not only Rayleigh but also Brillouin scattering [20–24]. One of the first publications about a distributed pressure measurement using an SHF was published by the aforementioned A. J. Rogers [10]. Due to the large air holes in the fiber, its birefringence is very sensitive to pressure changes, which translates into a frequency response of 1.59 GHz/MPa (1.486 GHz/MPa and −0.104 GHz/MPa for the slow and fast axis, respectively) [25]. Several years later, Chen et al. [26] studied polarization-maintaining MOF and SHF fibers interrogated with a polarization optical frequency-domain reflectometer (POFDR). They found differential pressure sensitivities between the slow and fast polarization axes of the tested fibers of 3.48 µε/MPa (equivalent to −0.522 GHz/MPa) and 12.5 µε/MPa (equivalent to −1.874 GHz/MPa) over a range of 0 to13.8 MPa for the two types of fiber, respectively [24,26]. In [27], Mikhailov et al. characterized two highly birefringent “butterfly” microstructured fibers using a frequency-scanned phase-OTDR. The measured differential pressure sensitivities for the fibers were −2.19 GHz/MPa and −0.954 GHz/MPa. Similar results, −2.072 GHz/MPa and −1.332 GHz/MPa, were obtained for the construction of other MOFs [16].
In this work, we analyze the influence of the twist on the pressure-sensing characteristics of SHFs with an elliptical core of type P. In recent years, spun fibers have found several interesting applications that include the generation of modes with orbital angular momentum [28] and single-mode operating fiber lasers [29]. The effect of fiber elastic twist to polarimetric sensitivity to pressure and inelastic twist on polarimetric sensitivities to torsion and temperature were also described in [30,31]. It was also shown that the local birefringence and external force can be measured in spun highly birefringent fiber in distributed measurements [32]; however, the detailed impact of inelastic twist on distributed measurements, particularly for changing pressure, is shown, to the best of our knowledge, for the first time in this study. We examined six SHFs with spin pitches equal to 200, 100, 50, 30, 10, and 5 mm and one non-spun SHF. Fibers with different spin pitches were drawn from the same preform rotating with different angular velocities. Both experimental results and analytical predictions show that the polarimetric sensitivity to hydrostatic pressure increases with the twisting period and is the highest for the non-spun fiber. The difference in the response of both polarization modes in distributed pressure measurements is the greatest for the non-spun fiber and becomes smaller the more the fiber is twisted. However, while the distributed sensitivity for one polarization eigenmode decreases, it increases for the second. For fibers with large values of circular birefringence, the shortest pitches, the responses for the two polarization modes are almost the same.
2. Pressure sensitivities measurements
Sensitivity to hydrostatic pressure was characterized for non-spun SHF whose cross-section is shown in Fig. 1(a),(b), and corresponding spun SHFs with spin pitches Λ ranging from 200 mm to 5 mm. The geometrical properties of the fibers are as follows: core size 6 µm × 9.7 µm, air-hole diameter 12 µm, and the wall between the core and an air-hole width 1.2 µm. The birefringence of these fibers was measured using a spectral interferometry method combined with the point-force method and presented earlier in [33].
Fig. 1. Cross-section of the non-spun SHF used in the sensitivity measurements (a), (b) and phase birefringence measured for corresponding SHFs with different spin pitches (c).
In spun fibers with a centrally located elliptical core, the fundamental eigenmodes are generally elliptically polarized and the birefringence can be expressed in cartesian coordinates as:
where Δnl is the birefringence for non-spun fiber, and Δnc is the circular birefringence dependent on twist pitch Λ and wavelength λ as follows:2.1 Polarimetric sensitivity to pressure
Polarimetric sensitivity to hydrostatic pressure Kp depends on fiber birefringence and can be expressed as [12]:
where L – length of the fiber exposed to pressure changes, Δϕ – the phase difference between polarization eigenmodes. The setup used for the measuring the polarimetric sensitivity of the SHFs to hydrostatic pressure is shown in Fig. 2. The light from a supercontinuum source was passed through a linear polarizer with transmission axis set at 45° with respect to the examined fiber axes and coupled to the fiber using microscopic objective MO1. The part of the coated fiber of length L was exposed to hydrostatic pressure changing in the range 0–7 MPa. The oil was forced into the pressure chamber slowly to avoid undesired temperature variations. The linear polarizer P2, with transmission axis also set at 45° with respect to the examined fiber axes, enabled the observation of the spectral interference fringes for polarization eigenmodes on optical spectrum analyzer OSA and measurement of the displacement of the fringes with the change of hydrostatic pressure. The phase difference between the polarization eigenmodes for successive wavelengths where minima of intensity occur can be expressed as: where q is the interference order. The interference order cannot be strictly determined for each minimum from single interferogram; however, to obtain Kp, Eq. (3), we only need to know the change of Δϕ with the change of pressure, not its exact value. Therefore we followed the displacement of successive spectral fringes without losing interference order unambiguity, estimated Δ(Δϕ) after each pressure change, fitted the dependence of Δ(Δϕ) upon Δp with linear function and according to Eq. (3) calculated polarimetric sensitivity. Moreover, assuming that the hydrostatic pressure impacts only fiber birefringence (more specifically, the linear component of birefringence), not the fiber length, and using Eq. (3) we can also express the polarimetric sensitivity Kp,S for spun fiber similarly as was done for temperature in [31] as follows:Fig. 2. Experimental setup for polarimetric sensitivity measurements based on spectral interferometry. SC source – supercontinuum source, OSA – optical spectrum analyzer, P1–2 – linear polarizers, MO1–2 – microscopic objectives, HPC – hydrostatic pressure chamber, FUT – fiber under test.
Fig. 3. Measurements of polarimetric sensitivity to hydrostatic pressure. Interference fringes registered for the fiber of Λ = 50 mm under two pressures for spectral ranges 800–1300 nm (top) and 1300–1600 nm (bottom) (a), change of the phase difference between interfering modes upon pressure for two wavelengths for the fiber of Λ = 50 mm (b), and polarimetric sensitivity of spun side-hole fibers measured (solid) and calculated with respect to the sensitivity of non-spun (NS) fiber based on their birefringence (dashed) (c).
2.2 Pressure sensitivity in Rayleigh-scattering-based OFDR
The setup for the distributed sensitivity of spun SHF measurements is shown in Fig. 4. A LUNA OFDR system (model OBR 4600 of a spatial resolution 0.1 mm) is equipped with a tunable laser source and collects Rayleigh backscattered light from the fiber path. To control the input polarization we used SMF fiber placed in a polarization controller (PC) and observed a signal transmitted through an FUT and passed through the quarter-wave plate (Q) and the linear polarizer (P) to the OSA. After adjusting the PC to enable the single polarization eigenmode to propagate in FUT, we added refractive-index matching gel to the fiber end tip to avoid strong light reflection. Then, we performed the measurements of frequency shift for the backscattered light upon the pressure change, since the pressure sensitivity in Rayleigh scattering-based OFDR can be expressed as:
where Δν is the frequency shift caused by pressure change and ν is the mean optical frequency of the LUNA tunable laser. Figure 5 shows the signals registered for both polarizations for non-spun fiber and the fiber with the shortest spin pitch (5 mm). The pressure chamber was sealed with the epoxy glue, leading to increased stresses at the fiber’s ends and causing sharp peaks at the edges of the measured signals. The signal obtained from the part of the fiber placed in the pressure chamber was averaged and the dependence of the averaged Δν upon Δp fitted with a linear function. Similarly we examined the second polarization eigenmode and repeated the procedure for all the spun fibers. The distributed sensitivity obtained from the measurements for all the investigated fibers and both polarization eigenmodes is shown in Fig. 6.Fig. 4. Experimental setup for distributed pressure sensitivity measurements. LUNA OFDR – optical frequency-domain reflectometer, SMF – single-mode fiber, PC – polarization controller, Q – quarter-wave plate, MG – refractive-index matching gel. The part of the setup in the dashed frame was used for PC adjustment, not in the distributed sensitivity measurements.
Fig. 5. Frequency shift registered as a function of fiber distance with LUNA OFDR for the Δp = 6 MPa for the polarization eigenmodes of the fiber placed in pressure chamber with Λ = ∞ (a) and Λ = 5 mm (b). Dependence of averaged spectral shift upon pressure change for the polarization eigenmodes of the fibers with Λ = ∞ and Λ = 5 mm; error bars represent standard deviations (c).
Fig. 6. Pressure sensitivity measured in Rayleigh scattering-based OFDR (dotted) and calculated (line) for polarization eigenmodes dependent on fiber twist rate.
The spun fiber exposed to hydrostatic pressure can be regarded as the perturbed fiber, since the pressure-induced stresses affect fiber birefringence [34]. Despite the axes of the spun fiber rotate with respect to the cartesian coordinates, it was recently shown in [35] that if the perturbation rotates with the fiber axes, it can be described with the same permittivity change tensor for spun and non-spun fiber, as:
While substituting the above equations into Eq. (8) and making the assumption that the difference in mode effective indices between spun and non-spun fiber is insignificant, we find that the distributed sensitivity for elliptically polarized modes can be expressed as superposition of the sensitivities for the non-spun fiber:
The results of the above calculations are shown in Fig. 6 and are in relatively good agreement with the measurements. Both of them indicate that the distributed sensitivity Kp is negative for both polarizations, the absolute value of Kp decays for polarization connected with the x-azimuth and increases for polarization with the y-azimuth, and the difference between the distributed sensitivities for two polarization eigenmodes decreases for the higher twist rate.
3. Conclusions
In this study we analyzed the influence of the inelastic twist on the pressure-sensing characteristics of SHFs with an elliptical core of type P. The fibers we used were drawn from the same preform with spin pitches varying from 5 mm to 200 mm. We investigated polarimetric sensitivity of the fibers as well as the pressure sensitivity for two polarization eigenmodes separately in Rayleigh-scattering-based OFDR.
We showed experimentally that polarimetric sensitivity to pressure in the examined spun side-hole fibers decreases with the shortening of spin pitch. The experimental results, obtained in wide spectral range, agree well with the analytical predictions based on the non-spun fiber sensitivity and the linear and circular components of spun fiber birefringence. It is worth noting that the polarimetric sensitivity is negative and depends on the wavelength, so the acceptable pressure changes would be limited.
Distributed measurements for two polarization eigenmodes with the ellipticity angle dependent on spin pitch showed the decrease of sensitivity for one polarization (from −5.3 × 10−6 1/MPa for non-spun fiber to −3.6 × 10−6 1/MPa for the fiber with 5 mm spin pitch) and increase of sensitivity for another polarization for higher twist rates (from −1.9 × 10−6 1/MPa for non-spun fiber to −3.5 × 10−6 1/MPa for the fiber with 5 mm spin pitch). Thus, the difference between distributed sensitivities also decreases. While exposure to the pressure can be represented as a permittivity tensor perturbation, we showed that the sensitivity of a particular eigenmode for spun fiber can be estimated as the superposition of the sensitivities for the eigenmodes of the non-spun fiber and depends on the eigenmode ellipticity angle. The analytical study confirmed the trend from the experimental results. However, the certain mismatch can be observed. The probable reasons for that are the assumptions in calculations or uncertainties in experiments. In calculations we primarily differentiate twisted and non-twisted fiber by its spin pitch, while Ref. [33] indicates that the linear components of birefringence are not perfectly the same, which suggests some differences in the cross-sections of the fibers. On the other hand, even minor imperfections in polarization adjustment would influence the results obtained experimentally. Additionally, it is important to note that the values of distributed sensitivities for examined fibers are relatively small which makes them more vulnerable to perturbations.
From the results obtained in this work we can assume that the spun-fibers can operate well as the sensors of other physical parameters inducing circular birefringence, such as torsion or magnetic field, not only in varying temperature [31] but also in varying pressure, while their sensitivity to pressure can also be lowered. Moreover, they can be used to control the relations between distributed sensitivities for both polarizations eigenmodes.
Funding
Narodowe Centrum Nauki (Opus 18, DEC-2019/35/B/ST7/04135).
Disclosures
The authors declare no conflict of interests.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. H. H. Zhu, B. Shi, and C. C. Zhang, “FBG-based monitoring of geohazards: Current status and trends,” Sensors 17(3), 452 (2017). [CrossRef]
2. L. Zou, X. Bao, F. Ravet, and L. Chen, “Distributed Brillouin fiber sensor for detecting pipeline buckling in an energy pipe under internal pressure,” Appl. Opt. 45(14), 3372–3377 (2006). [CrossRef]
3. H. M. Xie, Ph. Dabkiewicz, R. Ulrich, and K. Okamoto, “Side-hole fiber for fiber-optic pressure sensing,” Opt. Lett. 11(5), 333–335 (1986). [CrossRef]
4. J. R. Clowes, S. Syngellakis, and M. N. Zervas, “Pressure sensitivity of side-hole optical fiber sensors,” IEEE Photonics Technol. Lett. 10(6), 857–859 (1998). [CrossRef]
5. T. Chen, R. Chen, C. Jewart, B. Zhang, K. Cook, J. Canning, and K. P. Chen, “Regenerated gratings in air-hole microstructured fibers for high-temperature pressure sensing,” Opt. Lett. 36(18), 3542–3544 (2011). [CrossRef]
6. E. Chmielewska, W. Urbańczyk, and W. J. Bock, “Measurement of pressure and temperature sensitivities of a Bragg grating imprinted in a highly birefringent side-hole fiber,” Appl. Opt. 42(31), 6284–6291 (2003). [CrossRef]
7. J. Wojcik, W. Urbanczyk, W. J. Bock, B. Janoszczyk, P. Mergo, M. Makara, K. Poturaj, and W. Spytek, “Prototype of the side-hole HB optical fiber,” Proc. SPIE 3731, 88–93 (1999). [CrossRef]
8. W. Urbanczyk, T. Martynkien, and W. J. Bock, “Dispersion effects in elliptical-core highly birefringent fibers,” Appl. Opt. 40(12), 1911–1920 (2001). [CrossRef]
9. W. J. Bock, W. Urbanczyk, and J. Wojcik, “Characterization of elliptical-core side-hole fibers for interferometric pressure sensing,” Proc. SPIE 2341, 152–1159 (1994). [CrossRef]
10. A. J. Rogers, S.V. Shatalin, and S.E. Kanellopoulos, “Distributed measurement of fluid pressure via optical-fibre backscatter polarimetry,” Proc. SPIE 5855, 230–233 (2005). [CrossRef]
11. S. Sulejmani, C. Sonnenfeld, T. Geernaert, P. Mergo, M. Makara, K. Poturaj, K. Skorupski, T. Martynkien, G. Statkiewicz-Barabach, J. Olszewski, W. Urbanczyk, C. Caucheteur, K. Chah, P. Megret, H. Terryn, J. V. Roosbroeck, F. Berghmans, and H. Thienpont, “Control Over the Pressure Sensitivity of Bragg Grating-Based Sensors in Highly Birefringent Microstructured Optical Fibers,” IEEE Photonics Technol. Lett. 24(6), 527–529 (2012). [CrossRef]
12. G. Statkiewicz, T. Martynkien, and W. Urbanczyk, “Measurements of modal birefringence and polarymetric sensitivity of the birefringent holey fiber to hydrostatic pressure and strain,” Opt. Commun. 241(4-6), 339–348 (2004). [CrossRef]
13. T. Martynkien, G. Statkiewicz-Barabach, J. Olszewski, J. Wojcik, P. Mergo, T. Geernaert, C. Sonnenfeld, A. Anuszkiewicz, M. K. Szczurowski, K. Tarnowski, M. Makara, K. Skorupski, J. Klimek, K. Poturaj, W. Urbanczyk, T. Nasilowski, F. Berghmans, and H. Thienpont, “Highly birefringent microstructured fibers with enhanced sensitivity to hydrostatic pressure,” Opt. Express 18(14), 15113–15121 (2010). [CrossRef]
14. T. Martynkien, M. Szpulak, G. Statkiewicz, G. Golojuch, J. Olszewski, W. Urbanczyk, J. Wojcik, P. Mergo, M. Makara, T. Nasilowski, F. Berghmans, and H. Thienpont, “Measurements of sensitivity to hydrostatic pressure and temperature in highly birefringent photonic crystal fibers,” Opt. Quantum Electron. 39(4-6), 481–489 (2007). [CrossRef]
15. M. Szpulak, T. Martynkien, and W. Urbanczyk, “Effects of hydrostatic pressure on phase and group modal birefringence in microstructured holey fibers,” Appl. Opt. 43(24), 4739 (2004). [CrossRef]
16. S. Mikhailov, A. Matthes, J. Bierlich, J. Kobelke, K. Wondraczek, F. Berghmans, and T. Geernaert, “Highly birefringent microstructured optical fiber for distributed hydrostatic pressure sensing with sub-bar resolution,” Opt. Express 30(11), 19961–19973 (2022). [CrossRef]
17. J. H. Osório, J. G. Hayashi, Y. A. V. Espinel, M. A. R. Franco, M. V. Andrés, and C. M. B. Cordeiro, “Photonic-crystal fiber-based pressure sensor for dual environment monitoring,” Appl. Opt. 53(17), 3668–3672 (2014). [CrossRef]
18. A. Anuszkiewicz, T. Martynkien, J. Olszewski, P. Mergo, and W. Urbańczyk, “Polarimetric sensitivity to hydrostatic pressure and temperature in a side-hole fiber with squeezed microstructure,” J. Opt. 17(12), 125609 (2015). [CrossRef]
19. C. Wu, J. Li, X. Feng, B. Guan, and H. Tam, “Side-hole photonic crystal fiber with ultrahigh polarimetric pressure sensitivity,” J. Lightwave Technol. 29(7), 943–948 (2011). [CrossRef]
20. A. J. Rogers, “Polarisation optical time domain reflectometry,” Electron. Lett. 16(13), 489–490 (1980). [CrossRef]
21. A. J. Rogers, “Polarization-optical time domain reflectometry: a technique for the measurement of field distributions,” Appl. Opt. 20(6), 1060–1074 (1981). [CrossRef]
22. L. Palmieri and L. Schenato, “Distributed optical fiber sensing based on rayleigh scattering,” Open Opt. J. 7(1), 104–127 (2013). [CrossRef]
23. L. Schenato, A. Galtarossa, A. Pasuto, and L. Palmieri, “Distributed optical fiber pressure sensors,” Opt. Fiber Technol. 58, 102239 (2020). [CrossRef]
24. T. Feng, J. Zhou, Y. Shang, X. Chen, and X. S. Yao, “Distributed transverse-force sensing along a single-mode fiber using polarization-analyzing OFDR,” Opt. Express 28(21), 31253–31271 (2020). [CrossRef]
25. L. Zhang, Z. Yang, Ł. Szostkiewicz, K. Markiewicz, S. Mikhailov, T. Geernaert, E. Rochat, and L. Thévenaz, “Long distance distributed pressure sensing based on frequency-scanned phase-sensitive optical time-domain reflectometry,” Opt. Express 29(13), 20487–20497 (2021). [CrossRef]
26. T. Chen, Q. Wang, R. Chen, B. Zhang, C. Jewart, K.P. Chen, M. Maklad, and P.R. Swinehart, “Distributed high-temperature pressure sensing using air-hole microstructural fibers,” Opt. Lett. 37(6), 1064 (2012). [CrossRef]
27. S. Mikhailov, L. Zhang, T. Geernaert, F. Berghmans, and L. Thevenaz, “Distributed Hydrostatic Pressure Measurement Using Phase-OTDR in a Highly Birefringent Photonic Crystal Fiber,” J. Lightwave Technol. 37(18), 4496–4500 (2019). [CrossRef]
28. C. N. Alexeyev, A. N. Alexeyev, B. P. Lapin, G. Milione, and M. A. Yavorsky, “Spin-orbit-interaction-induced generation of optical vortices in multihelicoidal fibers,” Phys. Rev. A 88(6), 063814 (2013). [CrossRef]
29. P. Wang, L. J. Cooper, J. K. Sahu, and W. A. Clarkson, “Efficient single-mode operation of a cladding-pumped ytterbium-doped helical-core fiber laser,” Opt. Lett. 31(2), 226–228 (2006). [CrossRef]
30. T. R. Wolinski and W. J. Bock, “Simultaneous twist and pressure effects in highly-birefringent single-mode bow-tie fibers,” J. Lightwave Technol. 11(3), 389–394 (1993). [CrossRef]
31. D. Kowal, G. Statkiewicz-Barabach, M. Bernas, M. Napiorkowski, M. Makara, L. Czyzewska, P. Mergo, and W. Urbanczyk, “Polarimetric sensitivity to torsion in spun highly birefringent fibers,” Sensors 19(7), 1639 (2019). [CrossRef]
32. Z. Ding, C. Wang, K. Liu, Y. Liu, G. Xu, J. Jiang, Y. Guo, and T. Liu, “Distributed measurements of external force induced local birefringence in spun highly birefringent optical fibers using polarimetric OFDR,” Opt. Express 27(2), 951–964 (2019). [CrossRef]
33. D. Kowal, G. Statkiewicz-Barabach, M. Napiorkowski, M. Makara, K. Poturaj, P. Mergo, and Waclaw Urbanczyk, “Measurement of birefringence and ellipticity of polarization eigenmodes in spun highly birefringent fibers using spectra interferometry and lateral point-force method,” Opt. Express 26(26), 34185–34199 (2018). [CrossRef]
34. A. W. Snyder and J. D. Love, Optical Wave-Guide Theory, 1st ed. (Chapman & Hall, 1983).
35. M. Napiorkowski and W. Urbanczyk, “Rigorous modeling of twisted anisotropic optical fibers with transformation optics formalism,” Opt. Express 29(10), 15199–15216 (2021). [CrossRef]
36. M. Aerssens, A. Gusarov, B. Brichard, V. Massaut, P. Megret, and M. Wuilpart, “Faraday effect based optical fiber current sensor for tokamaks,” 2nd International Conference on Advancements in Nuclear Instrumentation, Measurement Methods and their Applications (IEEE, 2011), pp. 1–6.
