Abstract

We discuss a measurement method that aims to determine the radial distribution of the photoelastic constant C in an optical fiber. This method uses the measurement of the retardance profile of a transversely illuminated fiber as a function of applied tensile load and requires the computation of the inverse Abel transform of this retardance profile. We focus on the influence of the measurement error on the obtained values for C. The results suggest that C may not be constant across the fiber and that the mean absolute value of C is slightly larger for glass fibers than for bulk fused silica. This can, for example, influence the accuracy with which one is able to predict the response of optical fiber sensors used for measuring mechanical loads.

© 2013 Optical Society of America

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  1. J. F. Doyle and J. W. Philips, Manual on Experimental Stress Analysis, 5th ed. (Society for Experimental Mechanics, 1989), pp. 136–149.
  2. G. L. Cloud, Optical Methods of Engineering Analysis (Cambridge University, 1998).
  3. E. Hecht, Optics (Addison Wesley, 2002).
  4. T. Geernaert, G. Luyckx, and E. Voet, “Transversal load sensing with fiber Bragg gratings in microstructured optical fibers,” IEEE Photon. Technol. Lett. 21, 6–8 (2009).
    [CrossRef]
  5. A. D. Kersey and M. A. Davis, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
    [CrossRef]
  6. K. Peeters, Polymer Optical Fibre Sensors: A Review (IOP, 2011).
  7. A. Othonos and K. Kalli, Fiber Bragg Gratings: Fundamentals and Applications in Telecommunications and Sensing (Artech House, 1999).
  8. L. N. Glass Stress Summer School, “Photoelasticity of glass,” Tallinn, Germany, 2011.
  9. N. Lagakos and R. Mohr, “Stress optic coefficient and stress profile in optical fibers,” Appl. Opt. 20, 2309–2313 (1981).
    [CrossRef]
  10. A. J. Barlow and D. N. Payne, “The stress-optic effect in optical fibers,” IEEE J. Quantum Electron. 19, 834–839 (1983).
    [CrossRef]
  11. COMSOL, “COMSOL Multiphysics,” www.comsol.com/products/multiphysics .
  12. W. Primak, “Photoelastic constants of vitreous silica and its elastic coefficient of refractive index,” J. Appl. Phys. 30, 779–788 (1959).
    [CrossRef]
  13. A. Bertholds and B. Dändliker, “Determination of the individual strain-optic coefficients in single-mode optical fibers,” J. Lightwave Technol. 6, 17–20 (1988).
    [CrossRef]
  14. A. Bertholds and R. Dändliker, “Deformation of single-mode optical fibers under longitudinal stress,” J. Lightwave Technol. 5, 895–900 (1987).
    [CrossRef]
  15. P. L. Chu and T. Whitbread, “Measurement of stresses in optical fiber and preform,” Appl. Opt. 21, 4241–4245 (1982).
    [CrossRef]
  16. M. P. Varnham, S. B. Poole, and D. N. Payne, “Thermal stress measurements in optical-fibre preforms using preform-profiling techniques,” Electron. Lett. 20, 1034–1035 (1984).
    [CrossRef]
  17. D. J. Webb, K. Kali, W. Urbanzyck, and F. Berghmans, “Photonics skins for optical sensing, White Paper 2,” 2010, www.phosfos.eu .
  18. A. D. Yablon, “Multi-wavelength optical fiber refractive index profiling bu spatially resolved Fourier transform spectroscopy,” J. Lightwave Technol. 28, 360–364 (2010).
    [CrossRef]
  19. A. D. Yablon, “New transverse techniques for characterizing high-power optical fibers,” Opt. Eng. 50, 111603 (2011).
    [CrossRef]
  20. J. Hecht, Understanding Fiber Optic (Prentice Hall, 2005).
  21. K. J. Gadvik, Optical Metrology, 3rd ed. (Wiley, 2002).
  22. M. Kalal and K. Nugent, “Abel inversion using fast Fourier transforms,” Appl. Opt. 27, 1956–1959 (1988).
    [CrossRef]
  23. K. Tatekura, “Determination of the index profile of optical fibers from transverse interferograms using Fourier theory,” Appl. Opt. 22, 460–463 (1983).
    [CrossRef]
  24. M. R. Hutsel and C. Montarou, “Algorithm performance in the determination of the refractive-index profile of optical fibers,” Appl. Opt. 47, 760–767 (2008).
    [CrossRef]
  25. A. Kuske and G. Robertson, Photoelastic Stress Analysis (Wiley, 1974).
  26. H. Poritsky, “Analysis of thermal stresses in sealed cylinders and the effect of viscous flow during anneal,” Physics 5, 406–411 (1934).
    [CrossRef]
  27. C. C. Montarou, “Two-wave-plate compensator method for full-field retardation measurements,” Appl. Opt. 45, 271–280 (2006).
    [CrossRef]
  28. C. C. Montarou, “Residual stress profiles in optical fibers determined by the two-wave-plate-compensator method,” Opt. Commun. 265, 29–32 (2006).
    [CrossRef]
  29. Thorlabs, “Single mode fiber: 633/680  nm,” http://www.thorlabs.de/Thorcat/12600/SM600-SpecSheet.pdf .
  30. W. K. Fester and M. W. Davidson, “The de Sénarmont compensator,” http://www.olympusmicro.com/primer/techniques/polarized/desenarmontcompensator.html .
  31. P. A. Vicharelli, “Iterative method for computing the inverse Abel transform,” Appl. Phys. Lett. 50, 557–559 (1987).
    [CrossRef]

2011

A. D. Yablon, “New transverse techniques for characterizing high-power optical fibers,” Opt. Eng. 50, 111603 (2011).
[CrossRef]

2010

2009

T. Geernaert, G. Luyckx, and E. Voet, “Transversal load sensing with fiber Bragg gratings in microstructured optical fibers,” IEEE Photon. Technol. Lett. 21, 6–8 (2009).
[CrossRef]

2008

2006

C. C. Montarou, “Two-wave-plate compensator method for full-field retardation measurements,” Appl. Opt. 45, 271–280 (2006).
[CrossRef]

C. C. Montarou, “Residual stress profiles in optical fibers determined by the two-wave-plate-compensator method,” Opt. Commun. 265, 29–32 (2006).
[CrossRef]

1997

A. D. Kersey and M. A. Davis, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

1988

A. Bertholds and B. Dändliker, “Determination of the individual strain-optic coefficients in single-mode optical fibers,” J. Lightwave Technol. 6, 17–20 (1988).
[CrossRef]

M. Kalal and K. Nugent, “Abel inversion using fast Fourier transforms,” Appl. Opt. 27, 1956–1959 (1988).
[CrossRef]

1987

P. A. Vicharelli, “Iterative method for computing the inverse Abel transform,” Appl. Phys. Lett. 50, 557–559 (1987).
[CrossRef]

A. Bertholds and R. Dändliker, “Deformation of single-mode optical fibers under longitudinal stress,” J. Lightwave Technol. 5, 895–900 (1987).
[CrossRef]

1984

M. P. Varnham, S. B. Poole, and D. N. Payne, “Thermal stress measurements in optical-fibre preforms using preform-profiling techniques,” Electron. Lett. 20, 1034–1035 (1984).
[CrossRef]

1983

A. J. Barlow and D. N. Payne, “The stress-optic effect in optical fibers,” IEEE J. Quantum Electron. 19, 834–839 (1983).
[CrossRef]

K. Tatekura, “Determination of the index profile of optical fibers from transverse interferograms using Fourier theory,” Appl. Opt. 22, 460–463 (1983).
[CrossRef]

1982

1981

1959

W. Primak, “Photoelastic constants of vitreous silica and its elastic coefficient of refractive index,” J. Appl. Phys. 30, 779–788 (1959).
[CrossRef]

1934

H. Poritsky, “Analysis of thermal stresses in sealed cylinders and the effect of viscous flow during anneal,” Physics 5, 406–411 (1934).
[CrossRef]

Barlow, A. J.

A. J. Barlow and D. N. Payne, “The stress-optic effect in optical fibers,” IEEE J. Quantum Electron. 19, 834–839 (1983).
[CrossRef]

Bertholds, A.

A. Bertholds and B. Dändliker, “Determination of the individual strain-optic coefficients in single-mode optical fibers,” J. Lightwave Technol. 6, 17–20 (1988).
[CrossRef]

A. Bertholds and R. Dändliker, “Deformation of single-mode optical fibers under longitudinal stress,” J. Lightwave Technol. 5, 895–900 (1987).
[CrossRef]

Chu, P. L.

Cloud, G. L.

G. L. Cloud, Optical Methods of Engineering Analysis (Cambridge University, 1998).

Dändliker, B.

A. Bertholds and B. Dändliker, “Determination of the individual strain-optic coefficients in single-mode optical fibers,” J. Lightwave Technol. 6, 17–20 (1988).
[CrossRef]

Dändliker, R.

A. Bertholds and R. Dändliker, “Deformation of single-mode optical fibers under longitudinal stress,” J. Lightwave Technol. 5, 895–900 (1987).
[CrossRef]

Davis, M. A.

A. D. Kersey and M. A. Davis, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Doyle, J. F.

J. F. Doyle and J. W. Philips, Manual on Experimental Stress Analysis, 5th ed. (Society for Experimental Mechanics, 1989), pp. 136–149.

Gadvik, K. J.

K. J. Gadvik, Optical Metrology, 3rd ed. (Wiley, 2002).

Geernaert, T.

T. Geernaert, G. Luyckx, and E. Voet, “Transversal load sensing with fiber Bragg gratings in microstructured optical fibers,” IEEE Photon. Technol. Lett. 21, 6–8 (2009).
[CrossRef]

Hecht, E.

E. Hecht, Optics (Addison Wesley, 2002).

Hecht, J.

J. Hecht, Understanding Fiber Optic (Prentice Hall, 2005).

Hutsel, M. R.

Kalal, M.

Kalli, K.

A. Othonos and K. Kalli, Fiber Bragg Gratings: Fundamentals and Applications in Telecommunications and Sensing (Artech House, 1999).

Kersey, A. D.

A. D. Kersey and M. A. Davis, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Kuske, A.

A. Kuske and G. Robertson, Photoelastic Stress Analysis (Wiley, 1974).

Lagakos, N.

Luyckx, G.

T. Geernaert, G. Luyckx, and E. Voet, “Transversal load sensing with fiber Bragg gratings in microstructured optical fibers,” IEEE Photon. Technol. Lett. 21, 6–8 (2009).
[CrossRef]

Mohr, R.

Montarou, C.

Montarou, C. C.

C. C. Montarou, “Two-wave-plate compensator method for full-field retardation measurements,” Appl. Opt. 45, 271–280 (2006).
[CrossRef]

C. C. Montarou, “Residual stress profiles in optical fibers determined by the two-wave-plate-compensator method,” Opt. Commun. 265, 29–32 (2006).
[CrossRef]

Nugent, K.

Othonos, A.

A. Othonos and K. Kalli, Fiber Bragg Gratings: Fundamentals and Applications in Telecommunications and Sensing (Artech House, 1999).

Payne, D. N.

M. P. Varnham, S. B. Poole, and D. N. Payne, “Thermal stress measurements in optical-fibre preforms using preform-profiling techniques,” Electron. Lett. 20, 1034–1035 (1984).
[CrossRef]

A. J. Barlow and D. N. Payne, “The stress-optic effect in optical fibers,” IEEE J. Quantum Electron. 19, 834–839 (1983).
[CrossRef]

Peeters, K.

K. Peeters, Polymer Optical Fibre Sensors: A Review (IOP, 2011).

Philips, J. W.

J. F. Doyle and J. W. Philips, Manual on Experimental Stress Analysis, 5th ed. (Society for Experimental Mechanics, 1989), pp. 136–149.

Poole, S. B.

M. P. Varnham, S. B. Poole, and D. N. Payne, “Thermal stress measurements in optical-fibre preforms using preform-profiling techniques,” Electron. Lett. 20, 1034–1035 (1984).
[CrossRef]

Poritsky, H.

H. Poritsky, “Analysis of thermal stresses in sealed cylinders and the effect of viscous flow during anneal,” Physics 5, 406–411 (1934).
[CrossRef]

Primak, W.

W. Primak, “Photoelastic constants of vitreous silica and its elastic coefficient of refractive index,” J. Appl. Phys. 30, 779–788 (1959).
[CrossRef]

Robertson, G.

A. Kuske and G. Robertson, Photoelastic Stress Analysis (Wiley, 1974).

Tatekura, K.

Varnham, M. P.

M. P. Varnham, S. B. Poole, and D. N. Payne, “Thermal stress measurements in optical-fibre preforms using preform-profiling techniques,” Electron. Lett. 20, 1034–1035 (1984).
[CrossRef]

Vicharelli, P. A.

P. A. Vicharelli, “Iterative method for computing the inverse Abel transform,” Appl. Phys. Lett. 50, 557–559 (1987).
[CrossRef]

Voet, E.

T. Geernaert, G. Luyckx, and E. Voet, “Transversal load sensing with fiber Bragg gratings in microstructured optical fibers,” IEEE Photon. Technol. Lett. 21, 6–8 (2009).
[CrossRef]

Whitbread, T.

Yablon, A. D.

Appl. Opt.

Appl. Phys. Lett.

P. A. Vicharelli, “Iterative method for computing the inverse Abel transform,” Appl. Phys. Lett. 50, 557–559 (1987).
[CrossRef]

Electron. Lett.

M. P. Varnham, S. B. Poole, and D. N. Payne, “Thermal stress measurements in optical-fibre preforms using preform-profiling techniques,” Electron. Lett. 20, 1034–1035 (1984).
[CrossRef]

IEEE J. Quantum Electron.

A. J. Barlow and D. N. Payne, “The stress-optic effect in optical fibers,” IEEE J. Quantum Electron. 19, 834–839 (1983).
[CrossRef]

IEEE Photon. Technol. Lett.

T. Geernaert, G. Luyckx, and E. Voet, “Transversal load sensing with fiber Bragg gratings in microstructured optical fibers,” IEEE Photon. Technol. Lett. 21, 6–8 (2009).
[CrossRef]

J. Appl. Phys.

W. Primak, “Photoelastic constants of vitreous silica and its elastic coefficient of refractive index,” J. Appl. Phys. 30, 779–788 (1959).
[CrossRef]

J. Lightwave Technol.

A. Bertholds and B. Dändliker, “Determination of the individual strain-optic coefficients in single-mode optical fibers,” J. Lightwave Technol. 6, 17–20 (1988).
[CrossRef]

A. Bertholds and R. Dändliker, “Deformation of single-mode optical fibers under longitudinal stress,” J. Lightwave Technol. 5, 895–900 (1987).
[CrossRef]

A. D. Yablon, “Multi-wavelength optical fiber refractive index profiling bu spatially resolved Fourier transform spectroscopy,” J. Lightwave Technol. 28, 360–364 (2010).
[CrossRef]

A. D. Kersey and M. A. Davis, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Opt. Commun.

C. C. Montarou, “Residual stress profiles in optical fibers determined by the two-wave-plate-compensator method,” Opt. Commun. 265, 29–32 (2006).
[CrossRef]

Opt. Eng.

A. D. Yablon, “New transverse techniques for characterizing high-power optical fibers,” Opt. Eng. 50, 111603 (2011).
[CrossRef]

Physics

H. Poritsky, “Analysis of thermal stresses in sealed cylinders and the effect of viscous flow during anneal,” Physics 5, 406–411 (1934).
[CrossRef]

Other

Thorlabs, “Single mode fiber: 633/680  nm,” http://www.thorlabs.de/Thorcat/12600/SM600-SpecSheet.pdf .

W. K. Fester and M. W. Davidson, “The de Sénarmont compensator,” http://www.olympusmicro.com/primer/techniques/polarized/desenarmontcompensator.html .

A. Kuske and G. Robertson, Photoelastic Stress Analysis (Wiley, 1974).

J. Hecht, Understanding Fiber Optic (Prentice Hall, 2005).

K. J. Gadvik, Optical Metrology, 3rd ed. (Wiley, 2002).

D. J. Webb, K. Kali, W. Urbanzyck, and F. Berghmans, “Photonics skins for optical sensing, White Paper 2,” 2010, www.phosfos.eu .

K. Peeters, Polymer Optical Fibre Sensors: A Review (IOP, 2011).

A. Othonos and K. Kalli, Fiber Bragg Gratings: Fundamentals and Applications in Telecommunications and Sensing (Artech House, 1999).

L. N. Glass Stress Summer School, “Photoelasticity of glass,” Tallinn, Germany, 2011.

COMSOL, “COMSOL Multiphysics,” www.comsol.com/products/multiphysics .

J. F. Doyle and J. W. Philips, Manual on Experimental Stress Analysis, 5th ed. (Society for Experimental Mechanics, 1989), pp. 136–149.

G. L. Cloud, Optical Methods of Engineering Analysis (Cambridge University, 1998).

E. Hecht, Optics (Addison Wesley, 2002).

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Figures (13)

Fig. 1.
Fig. 1.

Variation of the reflected Bragg wavelength change ΔλB for a maximal error of ±15% on C1 and C2, with C=C1C2. The applied vertical line load is 2N/mm.

Fig. 2.
Fig. 2.

Illustration of an optical fiber transversely illuminated (left) and resulting retardance profile R(y) (right). b is the radius of the fiber, σr and σθ are respectively the radial and angular principal axes of stress. The z axis is taken along the fiber length with a direction exiting the page.

Fig. 3.
Fig. 3.

Polarization microscope setup to measure the full-field retardance profile using the Sénarmont compensation method. To obtain a controlled tensile stress, a predefined axial load is applied to the fiber using an external loading system illustrated in Fig. 5.

Fig. 4.
Fig. 4.

Retardance map measured for a tensile stress of 50 MPa. The gray-level scale is given in meters. The maximum retardance is 31nm. This corresponds to 1/20th of the wavelength (λ=633nm).

Fig. 5.
Fig. 5.

Scheme of the loading system for the determination of the photoelastic constant of the fiber under test.

Fig. 6.
Fig. 6.

Retardance in one fiber portion for increasing axial tensile stress. Each profile is computed as an average of five independent measurements.

Fig. 7.
Fig. 7.

Elliptical shape E(y) and retardance R(y) of one fiber section.

Fig. 8.
Fig. 8.

Comparison between the analytical and the numerical inverse Abel transform of the ellipse without added noise. The amount of Fourier coefficient k equals 270 and the amount of radial points P is 136.

Fig. 9.
Fig. 9.

Inverse Abel transform of the noisy ellipse for k=10, k=15, k=25, and k=40. The variance of the signal is 0.5×109. A high value of k generates additional fluctuations due to the decomposition of the original signal in its Fourier series.

Fig. 10.
Fig. 10.

Minimum error as a function of the number of radial points P (top) and as a function of the chosen amount of Fourier coefficients k (bottom). The mean error is taken over the interval r=[0,b/2].

Fig. 11.
Fig. 11.

σz·C as a function of axial stress. The regression coefficient is the photoelastic constant C which is indicated in the legend. The bars indicate the maximum spreading of the measurement data around the fitted value.

Fig. 12.
Fig. 12.

Radial distribution of the product σz·C in a fiber portion as a function of increasing load. The amount of Fourier coefficients k is 35, and the number of radial points P is 300.

Fig. 13.
Fig. 13.

Radial distribution of the photoelastic constant C(r) in one fiber section for k=35.

Tables (1)

Tables Icon

Table 1. (k, P) Combinations for Minimal RMS Error on the Inverse Abel Transform

Equations (10)

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

{neff,x=nx0+C1σx+C2(σy+σz)neff,y=ny0+C1σy+C2(σx+σz)neff,z=nz0+C1σz+C2(σx+σy),
R(y)=Cb2y2b2y2(σzσy)dx=Cb2y2b2y2σzdx.
R(y)=2Cybσz(r)drr2y2.
σz(r)=1πCrbdR(y)/dyy2r2dy.
σz(r)·C=1πrbdR(y)/dyy2r2dy.
R(y)=θA180λθAin[Deg].
f(r)=π2bCk=1akk2π01ρ2(t2+ρ2)1/2sin(kπt2+ρ2)dt,
R(y)E(y)=A21+y2B2.
1πrByABB2y2(y2r2)1/2dy=12BA.
σz(r)·C=12max(abs(R(y)))b,

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