Abstract

Multiple-beam Fizeau fringes in transmission have been applied to the study of the nonlinear stress–strain relationship due to bending in the cladding of single-mode optical fibers. The present study yields a relation between the variation of refractive indices of bent single-mode fibers, represented by the fringe shift, and the nonlinear radial change of Young’s modulus along the fiber cross section. Experimentally, the study confirms the nonlinear asymmetric stress–strain relation across the fiber cross section. This relation is due to the asymmetric distribution of the compression and tensile stresses over the fiber cross section rather than to the shift in the centroid (neutral axis).

© 2001 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  41. H. El-Hennawi, F. El-Diasty, O. Meshrif, “Interferometric determination of the refractive index of optical fiber cladding and an examination of its homogeneity.” J. Appl. Phys. 62, 4931–4933 (1987).
    [CrossRef]
  42. N. Barakat, H. A. El-Hennawi, F. El-Diasty, “Multiple-beam interference fringes applied to GRIN optical fiber,” J. Appl. Phys. 27, 5090–5094 (1988).
  43. N. Barakat, H. A. El-Hennawi, H. E. Sobeah, “Multiple-beam interferometric studies on optical fibers,” Pure Appl. Opt. 2, 419–428 (1993).
    [CrossRef]
  44. F. El-Diasty, “Interferometric determination of induced birefringence due to bending in single-mode optical fibers,” J. Opt. A Pure Appl. Opt. 1, 197–200 (1999).
    [CrossRef]
  45. F. El-Diasty, “Interferometric characterization of single-mode optical fibers,” J. Appl. Phys. 87, 3254–3257 (2000).
    [CrossRef]
  46. F. El-Diasty, “Multiple-beam interferometric determination of Poisson’s ratio and strain distribution profiles along the cross section of bent single-mode optical fibers,” Appl. Opt. 39, 3197–3201 (2000).
    [CrossRef]
  47. A. Bertholds, R. Dandliker, “Determination of the individual strain-optic coefficients in single-mode optical fibers,” J. Lightwave Technol. 6, 17–20 (1988).
    [CrossRef]

2000

1999

F. El-Diasty, “Interferometric determination of induced birefringence due to bending in single-mode optical fibers,” J. Opt. A Pure Appl. Opt. 1, 197–200 (1999).
[CrossRef]

K. Tsujikawa, K. Arakawa, K. Yoshida, “Reflection of light caused by sharp bends in optical fiber,” IEICE Trans. Electron. E82-C, 2105–2107 (1999).

1998

1997

L. Foustin, G. Martini, “Bend loss in single-mode fibers,” J. Lightwave Technol. 15, 671–679 (1997).
[CrossRef]

1995

1993

E. Suhir, “Effect of the nonlinear stress–strain relationship on the maximum stress in silica fibers subjected to two-point bending,” Appl. Opt. 32, 1567–1572 (1993).
[CrossRef] [PubMed]

N. Barakat, H. A. El-Hennawi, H. E. Sobeah, “Multiple-beam interferometric studies on optical fibers,” Pure Appl. Opt. 2, 419–428 (1993).
[CrossRef]

1992

W. Griffioen, “Effect on nonlinear elasticity on measured fatigue data and lifetime estimations of optical fibers,” J. Am. Ceram. Soc. 75, 2692–2696 (1992).
[CrossRef]

1991

D. R. Roberts, E. Cuellar, J. E. Ritter, T. H. Service, “Design requirements for optical fibers in small radii bends,” J. Mater. Sci. 26, 3197–3201 (1991).
[CrossRef]

C. R. Kurkjian, D. Inniss, “Understanding mechanical properties of lightguides: a commentary,” Opt. Eng. 30, 681–689 (1991).
[CrossRef]

F. P. Kapron, H. H. Yuce, “Theory and measurement for predicting stressed fiber lifetime,” Opt. Eng. 30, 700–708 (1991).
[CrossRef]

G. S. Glaesemann, S. T. Gulati, “Design methodology for the mechanical reliability of optical fiber,” Opt. Eng. 30, 709–715 (1991).
[CrossRef]

1990

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written grating,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

1988

N. Barakat, H. A. El-Hennawi, F. El-Diasty, “Multiple-beam interference fringes applied to GRIN optical fiber,” J. Appl. Phys. 27, 5090–5094 (1988).

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

1987

H. El-Hennawi, F. El-Diasty, O. Meshrif, “Interferometric determination of the refractive index of optical fiber cladding and an examination of its homogeneity.” J. Appl. Phys. 62, 4931–4933 (1987).
[CrossRef]

M. J. Matthewson, C. R. Kurkjian, “Static fatigue of optical fibers in bending,” J. Am. Ceram. Soc. 70, 662–668 (1987).
[CrossRef]

H. Gerbel, J. Herskowitz, “Effect of strain in periodically deformed single-mode optical fibers,” Appl. Opt. 26, 2155–2158 (1987).
[CrossRef]

1986

M. J. Matthewson, C. R. Kurkjian, S. T. Gulati, “Strength measurement of optical fibers by bending,” J. Am. Ceram. Soc. 69, 815–821 (1986).
[CrossRef]

1985

1984

H. F. Taylor, “Bending effects in optical fibers,” J. Lightwave Technol. LT-2, 617–622 (1984).
[CrossRef]

P. C. Bouten, H. M. Wagemanns, “Double mardrel: a modified technique for studying static fatigue of optical fibers,” Electron. Lett. 20, 280–281 (1984).
[CrossRef]

S. F. Cowap, S. D. Brown, “Static fatigue testing of a hermetically sealed optical fiber,” Am. Ceram. Soc. Bull. 63, 495–497 (1984).

1983

C. R. Kurkjian, U. C. Paek, “Single-valued strength of ‘perfect’ silica fibers,” Appl. Phys. Lett. 42, 251–253 (1983).
[CrossRef]

1982

S. Sakaguchi, Y. Sawaki, Y. Abe, T. Kawasaki, “Delayed failure in silica glass,” J. Mater. Sci. 17, 2878–2886 (1982).
[CrossRef]

1980

J. T. Krause, “Zero stress strength reduction and transitions in static fatigue of fused silica fiber lightguides,” J. Non-Cryst. Solids 38–39, 497–502 (1980).
[CrossRef]

A. M. Smith, “Birefringence induced by bends and twists in single-mode optical fiber,” Appl. Opt. 19, 2606–2611 (1980).
[CrossRef] [PubMed]

P. L. Key, A. Fox, E. O. Fuchs, “Mechanical reliability of optical fibers,” J. Non-Cryst. Solids 38–39, 463–468 (1980).
[CrossRef]

R. Ulrich, S. C. Rashleigh, W. Eichoff, “Bending-induced birefringence in single-mode fibers,” Opt. Lett. 5, 273–275 (1980).
[CrossRef] [PubMed]

1979

T. J. Krause, L. R. Testardi, R. N. Thurston, “Deviations from linearity in the dependence of elongation upon force for fibers of simple glass formers and of glass optical lightguides,” Phys. Chem. Glasses 20, 135–139 (1979).

K. Jurgenson, “Dispersion minimum of monomode fibers,” Appl. Opt. 18, 1259–1261 (1979).
[CrossRef]

1978

1976

1975

1974

D. B. Keck, “Observation of externally controlled mode coupling in optical waveguides,” Proc. IEEE 62, 649–650 (1974).
[CrossRef]

1971

J. E. Ritter, C. L. Sherburne, “Dynamic and static failure of silicate glasses,” J. Am. Ceram. Soc. 54, 601–605 (1971).
[CrossRef]

1964

F. P. Mallinder, B. A. Procter, “Elastic constants of fused silica as a function of large tensile strain,” Phys. Chem. Glasses 5, 91–103 (1964).

Abe, Y.

S. Sakaguchi, Y. Sawaki, Y. Abe, T. Kawasaki, “Delayed failure in silica glass,” J. Mater. Sci. 17, 2878–2886 (1982).
[CrossRef]

Arakawa, K.

K. Tsujikawa, K. Arakawa, K. Yoshida, “Reflection of light caused by sharp bends in optical fiber,” IEICE Trans. Electron. E82-C, 2105–2107 (1999).

Barakat, N.

N. Barakat, H. A. El-Hennawi, H. E. Sobeah, “Multiple-beam interferometric studies on optical fibers,” Pure Appl. Opt. 2, 419–428 (1993).
[CrossRef]

N. Barakat, H. A. El-Hennawi, F. El-Diasty, “Multiple-beam interference fringes applied to GRIN optical fiber,” J. Appl. Phys. 27, 5090–5094 (1988).

N. Barakat, A. A. Hamza, A. S. Goneid, “Multiple-beam interference fringes applied to GRIN optical waveguides to determine fiber characteristics,” Appl. Opt. 24, 4383–4386 (1985).
[CrossRef] [PubMed]

Bertholds, A.

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

Bilodeau, F.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written grating,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Bouten, P. C.

P. C. Bouten, H. M. Wagemanns, “Double mardrel: a modified technique for studying static fatigue of optical fibers,” Electron. Lett. 20, 280–281 (1984).
[CrossRef]

Brown, D. A.

Brown, S. D.

S. F. Cowap, S. D. Brown, “Static fatigue testing of a hermetically sealed optical fiber,” Am. Ceram. Soc. Bull. 63, 495–497 (1984).

Chernikov, S. V.

S. V. Chernikov, F. Koch, J. R. Taylor, L. Gruner-Nielsen, “Measurement of the effect of bending on dispersion in dispersion-compensating fibers,” in Proceedings of OFC’98, Optical Fiber Communication Conference and Exhibition (Optical Society of America, Washington, D.C., 1998), pp. 23–24.

Cowap, S. F.

S. F. Cowap, S. D. Brown, “Static fatigue testing of a hermetically sealed optical fiber,” Am. Ceram. Soc. Bull. 63, 495–497 (1984).

Cuellar, E.

D. R. Roberts, E. Cuellar, J. E. Ritter, T. H. Service, “Design requirements for optical fibers in small radii bends,” J. Mater. Sci. 26, 3197–3201 (1991).
[CrossRef]

Dandliker, R.

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

Eichoff, W.

El-Diasty, F.

F. El-Diasty, “Interferometric characterization of single-mode optical fibers,” J. Appl. Phys. 87, 3254–3257 (2000).
[CrossRef]

F. El-Diasty, “Multiple-beam interferometric determination of Poisson’s ratio and strain distribution profiles along the cross section of bent single-mode optical fibers,” Appl. Opt. 39, 3197–3201 (2000).
[CrossRef]

F. El-Diasty, “Interferometric determination of induced birefringence due to bending in single-mode optical fibers,” J. Opt. A Pure Appl. Opt. 1, 197–200 (1999).
[CrossRef]

N. Barakat, H. A. El-Hennawi, F. El-Diasty, “Multiple-beam interference fringes applied to GRIN optical fiber,” J. Appl. Phys. 27, 5090–5094 (1988).

H. El-Hennawi, F. El-Diasty, O. Meshrif, “Interferometric determination of the refractive index of optical fiber cladding and an examination of its homogeneity.” J. Appl. Phys. 62, 4931–4933 (1987).
[CrossRef]

El-Hennawi, H.

H. El-Hennawi, F. El-Diasty, O. Meshrif, “Interferometric determination of the refractive index of optical fiber cladding and an examination of its homogeneity.” J. Appl. Phys. 62, 4931–4933 (1987).
[CrossRef]

El-Hennawi, H. A.

N. Barakat, H. A. El-Hennawi, H. E. Sobeah, “Multiple-beam interferometric studies on optical fibers,” Pure Appl. Opt. 2, 419–428 (1993).
[CrossRef]

N. Barakat, H. A. El-Hennawi, F. El-Diasty, “Multiple-beam interference fringes applied to GRIN optical fiber,” J. Appl. Phys. 27, 5090–5094 (1988).

Foustin, L.

L. Foustin, G. Martini, “Bend loss in single-mode fibers,” J. Lightwave Technol. 15, 671–679 (1997).
[CrossRef]

Fox, A.

P. L. Key, A. Fox, E. O. Fuchs, “Mechanical reliability of optical fibers,” J. Non-Cryst. Solids 38–39, 463–468 (1980).
[CrossRef]

Fuchs, E. O.

P. L. Key, A. Fox, E. O. Fuchs, “Mechanical reliability of optical fibers,” J. Non-Cryst. Solids 38–39, 463–468 (1980).
[CrossRef]

Gambling, W. A.

Gerbel, H.

Glaesemann, G. S.

G. S. Glaesemann, S. T. Gulati, “Design methodology for the mechanical reliability of optical fiber,” Opt. Eng. 30, 709–715 (1991).
[CrossRef]

Glasemann, G. S.

G. S. Glasemann, S. T. Gulati, J. D. Helfinstine, “Effect of strain and surface composition on Young’s modulus of optical fibers,” in Proceedings of the 11th Optical Fiber Communication Conference, Vol. 1 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), p. 26.

Goneid, A. S.

Griffioen, W.

W. Griffioen, “Effect on nonlinear elasticity on measured fatigue data and lifetime estimations of optical fibers,” J. Am. Ceram. Soc. 75, 2692–2696 (1992).
[CrossRef]

Gruner-Nielsen, L.

S. V. Chernikov, F. Koch, J. R. Taylor, L. Gruner-Nielsen, “Measurement of the effect of bending on dispersion in dispersion-compensating fibers,” in Proceedings of OFC’98, Optical Fiber Communication Conference and Exhibition (Optical Society of America, Washington, D.C., 1998), pp. 23–24.

Gulati, S. T.

G. S. Glaesemann, S. T. Gulati, “Design methodology for the mechanical reliability of optical fiber,” Opt. Eng. 30, 709–715 (1991).
[CrossRef]

M. J. Matthewson, C. R. Kurkjian, S. T. Gulati, “Strength measurement of optical fibers by bending,” J. Am. Ceram. Soc. 69, 815–821 (1986).
[CrossRef]

G. S. Glasemann, S. T. Gulati, J. D. Helfinstine, “Effect of strain and surface composition on Young’s modulus of optical fibers,” in Proceedings of the 11th Optical Fiber Communication Conference, Vol. 1 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), p. 26.

Hamza, A. A.

Helfinstine, J. D.

G. S. Glasemann, S. T. Gulati, J. D. Helfinstine, “Effect of strain and surface composition on Young’s modulus of optical fibers,” in Proceedings of the 11th Optical Fiber Communication Conference, Vol. 1 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), p. 26.

Herskowitz, J.

Hill, K. O.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written grating,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Inniss, D.

C. R. Kurkjian, D. Inniss, “Understanding mechanical properties of lightguides: a commentary,” Opt. Eng. 30, 681–689 (1991).
[CrossRef]

Johnson, D. C.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written grating,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Jurgenson, K.

Kapron, F. P.

F. P. Kapron, H. H. Yuce, “Theory and measurement for predicting stressed fiber lifetime,” Opt. Eng. 30, 700–708 (1991).
[CrossRef]

Kawakami, S.

Kawasaki, T.

S. Sakaguchi, Y. Sawaki, Y. Abe, T. Kawasaki, “Delayed failure in silica glass,” J. Mater. Sci. 17, 2878–2886 (1982).
[CrossRef]

Keck, D. B.

D. B. Keck, “Observation of externally controlled mode coupling in optical waveguides,” Proc. IEEE 62, 649–650 (1974).
[CrossRef]

Key, P. L.

P. L. Key, A. Fox, E. O. Fuchs, “Mechanical reliability of optical fibers,” J. Non-Cryst. Solids 38–39, 463–468 (1980).
[CrossRef]

Koch, F.

S. V. Chernikov, F. Koch, J. R. Taylor, L. Gruner-Nielsen, “Measurement of the effect of bending on dispersion in dispersion-compensating fibers,” in Proceedings of OFC’98, Optical Fiber Communication Conference and Exhibition (Optical Society of America, Washington, D.C., 1998), pp. 23–24.

Krause, J. T.

J. T. Krause, “Zero stress strength reduction and transitions in static fatigue of fused silica fiber lightguides,” J. Non-Cryst. Solids 38–39, 497–502 (1980).
[CrossRef]

Krause, T. J.

T. J. Krause, L. R. Testardi, R. N. Thurston, “Deviations from linearity in the dependence of elongation upon force for fibers of simple glass formers and of glass optical lightguides,” Phys. Chem. Glasses 20, 135–139 (1979).

Kurkjian, C. R.

C. R. Kurkjian, D. Inniss, “Understanding mechanical properties of lightguides: a commentary,” Opt. Eng. 30, 681–689 (1991).
[CrossRef]

M. J. Matthewson, C. R. Kurkjian, “Static fatigue of optical fibers in bending,” J. Am. Ceram. Soc. 70, 662–668 (1987).
[CrossRef]

M. J. Matthewson, C. R. Kurkjian, S. T. Gulati, “Strength measurement of optical fibers by bending,” J. Am. Ceram. Soc. 69, 815–821 (1986).
[CrossRef]

C. R. Kurkjian, U. C. Paek, “Single-valued strength of ‘perfect’ silica fibers,” Appl. Phys. Lett. 42, 251–253 (1983).
[CrossRef]

Lee, B. H.

Mallinder, F. P.

F. P. Mallinder, B. A. Procter, “Elastic constants of fused silica as a function of large tensile strain,” Phys. Chem. Glasses 5, 91–103 (1964).

Malo, B.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written grating,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Marcuse, D.

Martini, G.

L. Foustin, G. Martini, “Bend loss in single-mode fibers,” J. Lightwave Technol. 15, 671–679 (1997).
[CrossRef]

Matsumura, H.

Matthewson, M. J.

M. J. Matthewson, C. R. Kurkjian, “Static fatigue of optical fibers in bending,” J. Am. Ceram. Soc. 70, 662–668 (1987).
[CrossRef]

M. J. Matthewson, C. R. Kurkjian, S. T. Gulati, “Strength measurement of optical fibers by bending,” J. Am. Ceram. Soc. 69, 815–821 (1986).
[CrossRef]

Maurer, R. D.

R. D. Maurer, “Strength of optical fibers,” Appl. Phys. Lett. 27, 220–224 (1975).
[CrossRef]

Meshrif, O.

H. El-Hennawi, F. El-Diasty, O. Meshrif, “Interferometric determination of the refractive index of optical fiber cladding and an examination of its homogeneity.” J. Appl. Phys. 62, 4931–4933 (1987).
[CrossRef]

Morse, T. F.

Nagano, K.

Nishida, S.

Nishii, J.

Paek, U. C.

C. R. Kurkjian, U. C. Paek, “Single-valued strength of ‘perfect’ silica fibers,” Appl. Phys. Lett. 42, 251–253 (1983).
[CrossRef]

Payne, D. N.

Procter, B. A.

F. P. Mallinder, B. A. Procter, “Elastic constants of fused silica as a function of large tensile strain,” Phys. Chem. Glasses 5, 91–103 (1964).

Rashleigh, S. C.

Reinhart, L. J.

Ritter, J. E.

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D. R. Roberts, E. Cuellar, J. E. Ritter, T. H. Service, “Design requirements for optical fibers in small radii bends,” J. Mater. Sci. 26, 3197–3201 (1991).
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J. E. Ritter, C. L. Sherburne, “Dynamic and static failure of silicate glasses,” J. Am. Ceram. Soc. 54, 601–605 (1971).
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T. J. Krause, L. R. Testardi, R. N. Thurston, “Deviations from linearity in the dependence of elongation upon force for fibers of simple glass formers and of glass optical lightguides,” Phys. Chem. Glasses 20, 135–139 (1979).

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[CrossRef]

P. C. Bouten, H. M. Wagemanns, “Double mardrel: a modified technique for studying static fatigue of optical fibers,” Electron. Lett. 20, 280–281 (1984).
[CrossRef]

Fiber Integr. Opt.

J. E. Ritter, “Probability of fatigue failure in glass fibers,” Fiber Integr. Opt. 1, 387–399 (1978).
[CrossRef]

IEICE Trans. Electron.

K. Tsujikawa, K. Arakawa, K. Yoshida, “Reflection of light caused by sharp bends in optical fiber,” IEICE Trans. Electron. E82-C, 2105–2107 (1999).

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D. R. Roberts, E. Cuellar, J. E. Ritter, T. H. Service, “Design requirements for optical fibers in small radii bends,” J. Mater. Sci. 26, 3197–3201 (1991).
[CrossRef]

S. Sakaguchi, Y. Sawaki, Y. Abe, T. Kawasaki, “Delayed failure in silica glass,” J. Mater. Sci. 17, 2878–2886 (1982).
[CrossRef]

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Other

K. A. Varshneya, Fundamentals of Inorganic Glasses, 1st ed. (Academic, New York, 1994).

S. Timoshenko, Strength of Materials, 1st ed. (Van Nostrand, New York, 1980), pp. 189–249.

G. S. Glasemann, S. T. Gulati, J. D. Helfinstine, “Effect of strain and surface composition on Young’s modulus of optical fibers,” in Proceedings of the 11th Optical Fiber Communication Conference, Vol. 1 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), p. 26.

S. V. Chernikov, F. Koch, J. R. Taylor, L. Gruner-Nielsen, “Measurement of the effect of bending on dispersion in dispersion-compensating fibers,” in Proceedings of OFC’98, Optical Fiber Communication Conference and Exhibition (Optical Society of America, Washington, D.C., 1998), pp. 23–24.

L. S. Srubshchik, “Strength measurement of optical fibers by bending,” in Optical Wireless Communications, E. J. Korevaas, ed. Proc. SPIE3532, 114–121 (1998).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of a multiple-beam Fizeau fringe crossing a bent single-mode optical fiber perpendicular to its axis. The light is linearly polarized in a direction perpendicular to the fiber axis.

Fig. 2
Fig. 2

Microinterferogram showing multiple-beam Fizeau fringes crossing the cladding of a bent single-mode optical fiber immersed in a matching liquid; R=12 mm.

Fig. 3
Fig. 3

Microinterferograms showing the separate induced birefringence components: (a) component parallel to the fiber axis (unshifted component), (b) component perpendicular to the fiber axis (shifted component).

Fig. 4
Fig. 4

Evaluated values of the radial Young’s modulus for a bent single-mode fiber for each value of x across the fiber cross section for different radii of curvature. (a) Evaluated values of the radial Young’s modulus due to expansion (x>0), (b) evaluated values of the radial Young’s modulus due to compression (x<0).

Fig. 5
Fig. 5

Maximum value of changes for Young’s modulus versus different radii of curvature in the cladding compressive and tensile regions.

Equations (6)

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

δn=n-n
δn=z(x)λM4Δz(r2-x2)-1/2,
ϵx=zλM(r2-x2)-1/22n3Δz(ρ11+ρ12)-xρ12RM(ρ11+ρ12),
σ=E0ϵ1+12αϵ,
E=dσdϵ=E0(1+αϵ).
Ex=E01+zαλM(r2-x2)-1/22n3Δz(ρ11+ρ12)-xαρ12RM(ρ11+ρ12).

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