Abstract

The paper presents a novel three-dimensional quasi-continuous shape sensor based on an FBG array inscribed by femtosecond laser pulses into a 7-core optical fiber with a polyimide protective coating. The measured bending sensitivity of individual FBGs ranges from 0.046 nm/m−1 to 0.049 nm/m−1. It is shown that the sensor allows for reconstructing 2- and 3-dimensional shapes with high accuracy. Due to the high value of the core aperture and individual calibration of each FBG we were able to measure the smallest reported bending radii down to 2.6 mm with a record accuracy of ∼1%. Moreover, we investigate the magnitude of the errors of curves reconstruction and errors associated with measurement of curvature radii in the range from 2.6 to 500 mm. The main factors affecting the accuracy of measurements are also discussed. The temperature resistance of both the inscribed FBG structures and of the protective coating, along with the high mechanical strength of the polyimide, makes it possible to use the sensor in harsh environments or in medical and composite material applications.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (2)

F. Khan, A. Denasi, D. Barrera, J. Madrigal, S. Sales, and S. Misra, “Multi-Core Optical Fibers With Bragg Gratings as Shape Sensor for Flexible Medical Instruments,” IEEE Sens. J. 19(14), 5878–5884 (2019).
[Crossref]

A. Wolf, A. Dostovalov, K. Bronnikov, and S. Babin, “Arrays of fiber Bragg gratings selectively inscribed in different cores of 7-core spun optical fiber by IR femtosecond laser pulses,” Opt. Express 27(10), 13978–13990 (2019).
[Crossref]

2018 (3)

2017 (4)

V. Budinski and D. Donlagic, “Fiber-Optic Sensors for Measurements of Torsion, Twist and Rotation: A Review,” Sensors 17(3), 443 (2017).
[Crossref]

I. Gasulla, D. Barrera, J. Hervás, and S. Sales, “Spatial Division Multiplexed Microwave Signal processing by selective grating inscription in homogeneous multicore fibers,” Sci. Rep. 7(1), 41727 (2017).
[Crossref]

P. S. Westbrook, T. Kremp, K. S. Feder, W. Ko, E. M. Monberg, H. Wu, D. A. Simoff, T. F. Taunay, and R. M. Ortiz, “Continuous Multicore Optical Fiber Grating Arrays for Distributed Sensing Applications,” J. Lightwave Technol. 35(6), 1248–1252 (2017).
[Crossref]

O. N. Egorova, M. E. Belkin, D. A. Klushnik, S. G. Zhuravlev, M. S. Astapovich, and S. L. Semojnov, “Microwave signal delay line based on multicore optical fiber,” Phys. Wave Phen. 25(4), 289–292 (2017).
[Crossref]

2016 (2)

V. V. Shishkin, V. S. Terentyev, D. S. Kharenko, A. V. Dostovalov, A. A. Wolf, V. A. Simonov, M. Y. Fedotov, A. M. Shienok, I. S. Shelemba, and S. A. Babin, “Experimental Method of Temperature and Strain Discrimination in Polymer Composite Material by Embedded Fiber-Optic Sensors Based on Femtosecond-Inscribed FBGs,” J. Sens. 2016, 1–6 (2016).
[Crossref]

A. V. Dostovalov, A. A. Wolf, A. V. Parygin, V. E. Zyubin, and S. A. Babin, “Femtosecond point-by-point inscription of Bragg gratings by drawing a coated fiber through ferrule,” Opt. Express 24(15), 16232–16237 (2016).
[Crossref]

2015 (1)

2014 (2)

2013 (2)

2012 (1)

2008 (3)

A. A. Stolov, D. A. Simoff, and J. Li, “Thermal Stability of Specialty Optical Fibers,” J. Lightwave Technol. 26(20), 3443–3451 (2008).
[Crossref]

J. Canning, “Fibre gratings and devices for sensors and lasers,” Laser Photonics Rev. 2(4), 275–289 (2008).
[Crossref]

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
[Crossref]

2007 (1)

2006 (1)

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast Processes for Bulk Modification of Transparent Materials,” MRS Bull. 31(8), 620–625 (2006).
[Crossref]

2005 (2)

B. J. Soller, D. K. Gifford, M. S. Wolfe, and M. E. Froggatt, “High resolution optical frequency domain reflectometry for characterization of components and assemblies,” Opt. Express 13(2), 666–674 (2005).
[Crossref]

A. Martinez, I. Y. Khrushchev, and I. Bennion, “Thermal properties of fibre Bragg gratings inscribed point-by-point by infrared femtosecond laser,” Electron. Lett. 41(4), 176–178 (2005).
[Crossref]

2002 (1)

1978 (1)

Aitchison, J. S.

Albert, J.

J. Albert, L.-Y. Shao, and C. Caucheteur, “Tilted fiber Bragg grating sensors,” Laser Photonics Rev. 7(1), 83–108 (2013).
[Crossref]

Astapovich, M. S.

O. N. Egorova, M. E. Belkin, D. A. Klushnik, S. G. Zhuravlev, M. S. Astapovich, and S. L. Semojnov, “Microwave signal delay line based on multicore optical fiber,” Phys. Wave Phen. 25(4), 289–292 (2017).
[Crossref]

Babin, S.

Babin, S. A.

V. V. Shishkin, V. S. Terentyev, D. S. Kharenko, A. V. Dostovalov, A. A. Wolf, V. A. Simonov, M. Y. Fedotov, A. M. Shienok, I. S. Shelemba, and S. A. Babin, “Experimental Method of Temperature and Strain Discrimination in Polymer Composite Material by Embedded Fiber-Optic Sensors Based on Femtosecond-Inscribed FBGs,” J. Sens. 2016, 1–6 (2016).
[Crossref]

A. V. Dostovalov, A. A. Wolf, A. V. Parygin, V. E. Zyubin, and S. A. Babin, “Femtosecond point-by-point inscription of Bragg gratings by drawing a coated fiber through ferrule,” Opt. Express 24(15), 16232–16237 (2016).
[Crossref]

Barrera, D.

F. Khan, A. Denasi, D. Barrera, J. Madrigal, S. Sales, and S. Misra, “Multi-Core Optical Fibers With Bragg Gratings as Shape Sensor for Flexible Medical Instruments,” IEEE Sens. J. 19(14), 5878–5884 (2019).
[Crossref]

I. Gasulla, D. Barrera, J. Hervás, and S. Sales, “Spatial Division Multiplexed Microwave Signal processing by selective grating inscription in homogeneous multicore fibers,” Sci. Rep. 7(1), 41727 (2017).
[Crossref]

Belkin, M. E.

O. N. Egorova, M. E. Belkin, D. A. Klushnik, S. G. Zhuravlev, M. S. Astapovich, and S. L. Semojnov, “Microwave signal delay line based on multicore optical fiber,” Phys. Wave Phen. 25(4), 289–292 (2017).
[Crossref]

Bennion, I.

Y. Lai, K. Zhou, K. Sugden, and I. Bennion, “Point-by-point inscription of first-order fiber Bragg grating for C-band applications,” Opt. Express 15(26), 18318–18325 (2007).
[Crossref]

A. Martinez, I. Y. Khrushchev, and I. Bennion, “Thermal properties of fibre Bragg gratings inscribed point-by-point by infrared femtosecond laser,” Electron. Lett. 41(4), 176–178 (2005).
[Crossref]

Beresna, M.

Bernier, M.

Borrelli, N. F.

Boukenter, A.

Brambilla, G.

Bronnikov, K.

Budinski, V.

V. Budinski and D. Donlagic, “Fiber-Optic Sensors for Measurements of Torsion, Twist and Rotation: A Review,” Sensors 17(3), 443 (2017).
[Crossref]

Butler, G.

S. Klute, R. Duncan, R. Fielder, G. Butler, J. Mabe, A. Sang, R. Seeley, and M. Raum, “Fiber-Optic Shape Sensing and Distributed Strain Measurements on a Morphing Chevron,” in 44th AIAA Aerospace Sciences Meeting and Exhibit, Aerospace Sciences Meetings (American Institute of Aeronautics and Astronautics, 2006).

Butter, C. D.

Cannas, M.

Canning, J.

J. Canning, “Fibre gratings and devices for sensors and lasers,” Laser Photonics Rev. 2(4), 275–289 (2008).
[Crossref]

Carrier, J.

Caucheteur, C.

J. Albert, L.-Y. Shao, and C. Caucheteur, “Tilted fiber Bragg grating sensors,” Laser Photonics Rev. 7(1), 83–108 (2013).
[Crossref]

Coulas, D.

Denasi, A.

F. Khan, A. Denasi, D. Barrera, J. Madrigal, S. Sales, and S. Misra, “Multi-Core Optical Fibers With Bragg Gratings as Shape Sensor for Flexible Medical Instruments,” IEEE Sens. J. 19(14), 5878–5884 (2019).
[Crossref]

Ding, H.

Donko, A.

Donlagic, D.

V. Budinski and D. Donlagic, “Fiber-Optic Sensors for Measurements of Torsion, Twist and Rotation: A Review,” Sensors 17(3), 443 (2017).
[Crossref]

Dostovalov, A.

Dostovalov, A. V.

A. V. Dostovalov, A. A. Wolf, A. V. Parygin, V. E. Zyubin, and S. A. Babin, “Femtosecond point-by-point inscription of Bragg gratings by drawing a coated fiber through ferrule,” Opt. Express 24(15), 16232–16237 (2016).
[Crossref]

V. V. Shishkin, V. S. Terentyev, D. S. Kharenko, A. V. Dostovalov, A. A. Wolf, V. A. Simonov, M. Y. Fedotov, A. M. Shienok, I. S. Shelemba, and S. A. Babin, “Experimental Method of Temperature and Strain Discrimination in Polymer Composite Material by Embedded Fiber-Optic Sensors Based on Femtosecond-Inscribed FBGs,” J. Sens. 2016, 1–6 (2016).
[Crossref]

Duncan, R.

S. Klute, R. Duncan, R. Fielder, G. Butler, J. Mabe, A. Sang, R. Seeley, and M. Raum, “Fiber-Optic Shape Sensing and Distributed Strain Measurements on a Morphing Chevron,” in 44th AIAA Aerospace Sciences Meeting and Exhibit, Aerospace Sciences Meetings (American Institute of Aeronautics and Astronautics, 2006).

Dyer, R. S.

L. Huang, R. S. Dyer, R. J. Lago, A. A. Stolov, and J. Li, “Mechanical properties of polyimide coated optical fibers at elevated temperatures,” in Optical Fibers and Sensors for Medical Diagnostics and Treatment Applications XVI (International Society for Optics and Photonics, 2016), 9702, p. 97020Y.

Egorova, O. N.

O. N. Egorova, M. E. Belkin, D. A. Klushnik, S. G. Zhuravlev, M. S. Astapovich, and S. L. Semojnov, “Microwave signal delay line based on multicore optical fiber,” Phys. Wave Phen. 25(4), 289–292 (2017).
[Crossref]

Feder, K. S.

Fedotov, M. Y.

V. V. Shishkin, V. S. Terentyev, D. S. Kharenko, A. V. Dostovalov, A. A. Wolf, V. A. Simonov, M. Y. Fedotov, A. M. Shienok, I. S. Shelemba, and S. A. Babin, “Experimental Method of Temperature and Strain Discrimination in Polymer Composite Material by Embedded Fiber-Optic Sensors Based on Femtosecond-Inscribed FBGs,” J. Sens. 2016, 1–6 (2016).
[Crossref]

Fernandes, L. A.

Fielder, R.

S. Klute, R. Duncan, R. Fielder, G. Butler, J. Mabe, A. Sang, R. Seeley, and M. Raum, “Fiber-Optic Shape Sensing and Distributed Strain Measurements on a Morphing Chevron,” in 44th AIAA Aerospace Sciences Meeting and Exhibit, Aerospace Sciences Meetings (American Institute of Aeronautics and Astronautics, 2006).

Fomenko, A. T.

A. T. Fomenko and A. S. Mishchenko, A Short Course in Differential Geometry and Topology (Cambridge Scientific Publishers, 2009).

Froggatt, M. E.

Fukuchi, K.

K. Satori, K. Fukuchi, Y. Kurosawa, A. Hongo, and N. Takeda, “Polyimide-coated small-diameter optical fiber sensors for embedding in composite laminate structures,” in Smart Structures and Materials 2001: Sensory Phenomena and Measurement Instrumentation for Smart Structures and Materials (International Society for Optics and Photonics, 2001), 4328, pp. 285–294.

Gasulla, I.

I. Gasulla, D. Barrera, J. Hervás, and S. Sales, “Spatial Division Multiplexed Microwave Signal processing by selective grating inscription in homogeneous multicore fibers,” Sci. Rep. 7(1), 41727 (2017).
[Crossref]

Gattass, R. R.

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
[Crossref]

Gifford, D. K.

Girard, S.

Grenier, J. R.

Grobnic, D.

Haque, M.

Hayes, J.

Herman, P. R.

Hervás, J.

I. Gasulla, D. Barrera, J. Hervás, and S. Sales, “Spatial Division Multiplexed Microwave Signal processing by selective grating inscription in homogeneous multicore fibers,” Sci. Rep. 7(1), 41727 (2017).
[Crossref]

Hnatovsky, C.

Hocker, G. B.

Holschemacher, K.

M. Weisbrich and K. Holschemacher, “Comparison between different fiber coatings and adhesives on steel surfaces for distributed optical strain measurements based on Rayleigh backscattering,” J. Sens. Sens. Syst. 7(2), 601–608 (2018).
[Crossref]

Hongo, A.

K. Satori, K. Fukuchi, Y. Kurosawa, A. Hongo, and N. Takeda, “Polyimide-coated small-diameter optical fiber sensors for embedding in composite laminate structures,” in Smart Structures and Materials 2001: Sensory Phenomena and Measurement Instrumentation for Smart Structures and Materials (International Society for Optics and Photonics, 2001), 4328, pp. 285–294.

Huang, L.

L. Huang, R. S. Dyer, R. J. Lago, A. A. Stolov, and J. Li, “Mechanical properties of polyimide coated optical fibers at elevated temperatures,” in Optical Fibers and Sensors for Medical Diagnostics and Treatment Applications XVI (International Society for Optics and Photonics, 2016), 9702, p. 97020Y.

Itoh, K.

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast Processes for Bulk Modification of Transparent Materials,” MRS Bull. 31(8), 620–625 (2006).
[Crossref]

Ji, M.

S.-Y. Yang and M. Ji, “Chapter 3 - Polyimide Matrices for Carbon Fiber Composites,” in Advanced Polyimide Materials, S.-Y. Yang, ed. (Elsevier, 2018), pp. 93–136.

Jung, Y.

Khan, F.

F. Khan, A. Denasi, D. Barrera, J. Madrigal, S. Sales, and S. Misra, “Multi-Core Optical Fibers With Bragg Gratings as Shape Sensor for Flexible Medical Instruments,” IEEE Sens. J. 19(14), 5878–5884 (2019).
[Crossref]

Kharenko, D. S.

V. V. Shishkin, V. S. Terentyev, D. S. Kharenko, A. V. Dostovalov, A. A. Wolf, V. A. Simonov, M. Y. Fedotov, A. M. Shienok, I. S. Shelemba, and S. A. Babin, “Experimental Method of Temperature and Strain Discrimination in Polymer Composite Material by Embedded Fiber-Optic Sensors Based on Femtosecond-Inscribed FBGs,” J. Sens. 2016, 1–6 (2016).
[Crossref]

Khrushchev, I. Y.

A. Martinez, I. Y. Khrushchev, and I. Bennion, “Thermal properties of fibre Bragg gratings inscribed point-by-point by infrared femtosecond laser,” Electron. Lett. 41(4), 176–178 (2005).
[Crossref]

Klushnik, D. A.

O. N. Egorova, M. E. Belkin, D. A. Klushnik, S. G. Zhuravlev, M. S. Astapovich, and S. L. Semojnov, “Microwave signal delay line based on multicore optical fiber,” Phys. Wave Phen. 25(4), 289–292 (2017).
[Crossref]

Klute, S.

S. Klute, R. Duncan, R. Fielder, G. Butler, J. Mabe, A. Sang, R. Seeley, and M. Raum, “Fiber-Optic Shape Sensing and Distributed Strain Measurements on a Morphing Chevron,” in 44th AIAA Aerospace Sciences Meeting and Exhibit, Aerospace Sciences Meetings (American Institute of Aeronautics and Astronautics, 2006).

Ko, W.

Kremp, T.

Kurosawa, Y.

K. Satori, K. Fukuchi, Y. Kurosawa, A. Hongo, and N. Takeda, “Polyimide-coated small-diameter optical fiber sensors for embedding in composite laminate structures,” in Smart Structures and Materials 2001: Sensory Phenomena and Measurement Instrumentation for Smart Structures and Materials (International Society for Optics and Photonics, 2001), 4328, pp. 285–294.

Lago, R. J.

L. Huang, R. S. Dyer, R. J. Lago, A. A. Stolov, and J. Li, “Mechanical properties of polyimide coated optical fibers at elevated temperatures,” in Optical Fibers and Sensors for Medical Diagnostics and Treatment Applications XVI (International Society for Optics and Photonics, 2016), 9702, p. 97020Y.

Lai, Y.

Lee, K. K. C.

Li, J.

A. A. Stolov, D. A. Simoff, and J. Li, “Thermal Stability of Specialty Optical Fibers,” J. Lightwave Technol. 26(20), 3443–3451 (2008).
[Crossref]

L. Huang, R. S. Dyer, R. J. Lago, A. A. Stolov, and J. Li, “Mechanical properties of polyimide coated optical fibers at elevated temperatures,” in Optical Fibers and Sensors for Medical Diagnostics and Treatment Applications XVI (International Society for Optics and Photonics, 2016), 9702, p. 97020Y.

Lu, P.

Mabe, J.

S. Klute, R. Duncan, R. Fielder, G. Butler, J. Mabe, A. Sang, R. Seeley, and M. Raum, “Fiber-Optic Shape Sensing and Distributed Strain Measurements on a Morphing Chevron,” in 44th AIAA Aerospace Sciences Meeting and Exhibit, Aerospace Sciences Meetings (American Institute of Aeronautics and Astronautics, 2006).

Macé, J.-R.

Madrigal, J.

F. Khan, A. Denasi, D. Barrera, J. Madrigal, S. Sales, and S. Misra, “Multi-Core Optical Fibers With Bragg Gratings as Shape Sensor for Flexible Medical Instruments,” IEEE Sens. J. 19(14), 5878–5884 (2019).
[Crossref]

Marcandella, C.

Mariampillai, A.

Marin, E.

Martinez, A.

A. Martinez, I. Y. Khrushchev, and I. Bennion, “Thermal properties of fibre Bragg gratings inscribed point-by-point by infrared femtosecond laser,” Electron. Lett. 41(4), 176–178 (2005).
[Crossref]

Mazur, E.

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
[Crossref]

Mihailov, S. J.

Mishchenko, A. S.

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S. Klute, R. Duncan, R. Fielder, G. Butler, J. Mabe, A. Sang, R. Seeley, and M. Raum, “Fiber-Optic Shape Sensing and Distributed Strain Measurements on a Morphing Chevron,” in 44th AIAA Aerospace Sciences Meeting and Exhibit, Aerospace Sciences Meetings (American Institute of Aeronautics and Astronautics, 2006).

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Appl. Opt. (1)

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J. Opt. Soc. Am. B (1)

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J. Sens. Sens. Syst. (1)

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K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast Processes for Bulk Modification of Transparent Materials,” MRS Bull. 31(8), 620–625 (2006).
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Nat. Photonics (1)

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
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Opt. Lett. (3)

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Sci. Rep. (1)

I. Gasulla, D. Barrera, J. Hervás, and S. Sales, “Spatial Division Multiplexed Microwave Signal processing by selective grating inscription in homogeneous multicore fibers,” Sci. Rep. 7(1), 41727 (2017).
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Other (5)

A. T. Fomenko and A. S. Mishchenko, A Short Course in Differential Geometry and Topology (Cambridge Scientific Publishers, 2009).

L. Huang, R. S. Dyer, R. J. Lago, A. A. Stolov, and J. Li, “Mechanical properties of polyimide coated optical fibers at elevated temperatures,” in Optical Fibers and Sensors for Medical Diagnostics and Treatment Applications XVI (International Society for Optics and Photonics, 2016), 9702, p. 97020Y.

S.-Y. Yang and M. Ji, “Chapter 3 - Polyimide Matrices for Carbon Fiber Composites,” in Advanced Polyimide Materials, S.-Y. Yang, ed. (Elsevier, 2018), pp. 93–136.

K. Satori, K. Fukuchi, Y. Kurosawa, A. Hongo, and N. Takeda, “Polyimide-coated small-diameter optical fiber sensors for embedding in composite laminate structures,” in Smart Structures and Materials 2001: Sensory Phenomena and Measurement Instrumentation for Smart Structures and Materials (International Society for Optics and Photonics, 2001), 4328, pp. 285–294.

S. Klute, R. Duncan, R. Fielder, G. Butler, J. Mabe, A. Sang, R. Seeley, and M. Raum, “Fiber-Optic Shape Sensing and Distributed Strain Measurements on a Morphing Chevron,” in 44th AIAA Aerospace Sciences Meeting and Exhibit, Aerospace Sciences Meetings (American Institute of Aeronautics and Astronautics, 2006).

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Figures (11)

Fig. 1.
Fig. 1. Multi-core fiber cross-section sketch, and relevant parameters for curvature calculation.
Fig. 2.
Fig. 2. Schematic representation of the FBG array inscribed in 7-core optical fiber through polyimide protective coating: (a) 3D representation, and (b) transverse cross section of the used MCF.
Fig. 3.
Fig. 3. Reflection/transmission spectra of the FBG array measured for one of the cores of the 7-core optical fiber.
Fig. 4.
Fig. 4. Interrogation scheme of shape sensor based on a FBG array in a 7-core optical fiber.
Fig. 5.
Fig. 5. (a) Dependence of the FBG wavelength on the angle of rotation of the MCF measured for node 1 with a radius of curvature of 10 mm. (b) Maximum FBG wavelength shift measured for the sinusoidal fit curve of the side core 1 in the node 1 versus curvature.
Fig. 6.
Fig. 6. Photos of the MCF with an inscribed FBG array of length L = 72 mm, and calculated curves (red dashed line): (a) a loop-like curve, (b) an s-shaped curve. Blue crosses indicate FBG nodes. In Fig. 6(a), the fiber crossing the loop was used to align the sensor with the plane of the table.
Fig. 7.
Fig. 7. Reconstructed 3D curves and ground truth curves: (a) projection onto xz plane, (b) yz plane, (c) xy plane; (d) 3D view. С1: Dspiral = 25 mm, hspiral = 21 mm; С2: Dspiral = 29.5 mm, hspiral = 49 mm; С3: Dspiral = 38.1 mm, hspiral = 10 mm; С4: Dspiral = 66 mm, hspiral = 49 mm.
Fig. 8.
Fig. 8. (a) absolute reconstruction error, (b) error per unit length versus arc length.
Fig. 9.
Fig. 9. FBG spectra at different radii of curvature for the central core (a) and for one of the side cores (b) subjected to a maximum bending strain.
Fig. 10.
Fig. 10. The relative measurement error vs radius of curvature (dots), the estimation of influence of Δλp on relative error according to Eq. (8) (filled area).
Fig. 11.
Fig. 11. (a) FBG spectra before and after the annealing process at room temperature, (b) resonant wavelength detuning depending on time at 330 °C.

Tables (4)

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Table 1. Parameters of the used polyimide-coated 7-core optical fiber

Tables Icon

Table 2. Geometric parameters of the spirals and errors of shape recovery

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Table 3. Results of small radii of curvature measurement

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Table 4. Results of large radii of curvature measurement

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

κ i = 1 r c ( ε i cos α i n x + ε i sin α i n y ) ,
d λ λ = C ε + Ω d T ,
ε i = 1 C i Δ λ i λ i B ,
κ = 2 N r c ( i = 1 N ε i cos α i n x + i = 1 N ε i sin α i n y ) .
{ r ( s ) = T ( s ) T ( s ) = κ ( s ) N ( s ) N ( s ) = τ ( s ) B ( s ) κ ( s ) T ( s ) B ( s ) = τ ( s ) N ( s ) .
C i = k i r c λ i B .
δ abs ( s ) = | | r meas ( s ) r ref ( s ) | | and δ rel ( s ) = δ abs ( s ) / s ,
δ R R = | R R | R = | k k ± R Δ λ p / Δ λ p 2 2 1 | .

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