Abstract

A finite deformation theory of elasticity and a theory of nonlinear photoelasticity are applied to describe the wavelength shifts of cladding-mode resonance in corrugated long-period fiber gratings under torsion. The deformation of fiber is found by use of the Murnaghan model of a solid elastic body. The quadratic photoelastic effect that is proportional to the second-order displacement gradient is investigated and compared with the classical photoelastic effect. The electromagnetic field in the twisted corrugated structure is presented as a superposition of circularly polarized modes of the etched fiber section. The wavelength shift is found to be proportional to the square of the twist angle. As predicted by our theory, a wavelength shift of the same nature has been found in a conventionally photoinduced long-period fiber grating.

© 2003 Optical Society of America

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References

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  1. A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
    [CrossRef]
  2. V. Bhatia, A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21, 692–694 (1996).
    [CrossRef] [PubMed]
  3. C. Y. Lin, L. A. Wang, G. W. Chern, “Corrugated long-period fiber gratings as strain, torsion, and bending sensors,” J. Lightwave Technol. 19, 1159–1168 (2001).
    [CrossRef]
  4. A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
    [CrossRef] [PubMed]
  5. C. E. Chryssou, “Gain-equalizing filters for wavelength division multiplexing optical communication systems: a comparison of notch and long-period grating filters for integrated optoelectronics,” Opt. Commun. 184, 375–384 (2000).
    [CrossRef]
  6. G. W. Chern, L. A. Wang, C. Y. Lin, “Transfer-matrix approach based on modal analysis for modeling corrugated long-period fiber gratings,” Appl. Opt. 40, 4476–4486 (2001).
    [CrossRef]
  7. Y. Liu, L. Zhang, I. Bennion, “Fibre optic load sensors with high transverse strain sensitivity based on long-period gratings in B/Ge co-doped fibre,” Electron. Lett. 35, 661–663 (1999).
    [CrossRef]
  8. Y. Liu, L. Zhang, J. A. R. Williams, I. Bennion, “Optical bend sensor based on measurement of resonance mode splitting of long-period fiber grating,” IEEE Photon. Technol. Lett. 12, 531–533 (2000).
    [CrossRef]
  9. B. H. Lee, Y. Liu, S. B. Lee, S. S. Choi, J. N. Jang, “Displacements of the resonant peaks of a long-period fiber grating induced by a change of ambient refractive index,” Opt. Lett. 22, 1769–1771 (1997).
    [CrossRef]
  10. S. Kim, Y. Jeong, S. Kim, J. Kwon, N. Park, B. Lee, “Control of the characteristics of a long-period grating by cladding etching,” Appl. Opt. 39, 2038–2042 (2000).
    [CrossRef]
  11. L. A. Wang, C. Y. Lin, G. W. Chern, “A torsion sensor made of a corrugated long period fibre grating,” Meas. Sci. Technol. 12, 793–799 (2001).
    [CrossRef]
  12. C. Y. Lin, G. W. Chern, L. A. Wang, “Periodical corrugated structure for forming sampled fiber Bragg grating and long-period fiber grating with tunable coupling strength,” J. Lightwave Technol. 19, 1212–1220 (2001).
    [CrossRef]
  13. C. Y. Lin, Lon A. Wang, “A wavelength- and loss-tunable band-rejection filter based on corrugated long-period fiber grating,” IEEE Photon. Technol. Lett. 13, 332–334 (2001).
    [CrossRef]
  14. C. Y. Lin, G. W. Chern, L. A. Wang, “Periodical corrugated structure for forming sampled fiber Bragg grating and long-period fiber grating with tunable coupling strength,” J. Lightwave Technol. 19, 1212–1220 (2001).
    [CrossRef]
  15. R. J. Atkin, N. Fox, An Introduction to the Theory of Elasticity (Longmans Green, New York, 1980).
  16. F. D. Murnaghan, Finite Deformation of an Elastic Solid (Wiley, New York, 1951).
  17. T. S. Narasimhamurty, Photoelastic and Electro-Optic Properties of Crystals (Plenum, New York, 1981).
    [CrossRef]
  18. D. F. Nelson, M. Lax, “Theory of photoelastic interaction,” Phys. Rev. B 3, 2778–2794 (1971).
    [CrossRef]
  19. K. Vedam, R. Srinivasan, “Non-linear piezo-optics,” Acta Crystallogr. 22, 630–634 (1967).
    [CrossRef]
  20. G. N. Burlak, V. V. Grimal’skii, K. Ishkabulov, “Dynamics of acoustoelectromagnetic solitons as a manifestation of higher nonlinearity,” Phys. Solid State 39, 987–990 (1997).
    [CrossRef]
  21. K. Vedam, E. D. D. Schmidt, R. Roy, “Nonlinear variation of refractive index of vitreous silica with pressure to 7 Kbars,” J. Am. Ceram. Soc. 49, 531–535 (1971).
    [CrossRef]
  22. A. M. Smith, “Birefringence induced by bends and twists in single-mode optical fiber,” Appl. Opt. 19, 2606–2611 (1980).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  24. A. I. Lurie, Nonlinear Theory of Elasticity (Elsevier, Amsterdam, 1990).

2001 (6)

2000 (3)

C. E. Chryssou, “Gain-equalizing filters for wavelength division multiplexing optical communication systems: a comparison of notch and long-period grating filters for integrated optoelectronics,” Opt. Commun. 184, 375–384 (2000).
[CrossRef]

Y. Liu, L. Zhang, J. A. R. Williams, I. Bennion, “Optical bend sensor based on measurement of resonance mode splitting of long-period fiber grating,” IEEE Photon. Technol. Lett. 12, 531–533 (2000).
[CrossRef]

S. Kim, Y. Jeong, S. Kim, J. Kwon, N. Park, B. Lee, “Control of the characteristics of a long-period grating by cladding etching,” Appl. Opt. 39, 2038–2042 (2000).
[CrossRef]

1999 (1)

Y. Liu, L. Zhang, I. Bennion, “Fibre optic load sensors with high transverse strain sensitivity based on long-period gratings in B/Ge co-doped fibre,” Electron. Lett. 35, 661–663 (1999).
[CrossRef]

1997 (2)

G. N. Burlak, V. V. Grimal’skii, K. Ishkabulov, “Dynamics of acoustoelectromagnetic solitons as a manifestation of higher nonlinearity,” Phys. Solid State 39, 987–990 (1997).
[CrossRef]

B. H. Lee, Y. Liu, S. B. Lee, S. S. Choi, J. N. Jang, “Displacements of the resonant peaks of a long-period fiber grating induced by a change of ambient refractive index,” Opt. Lett. 22, 1769–1771 (1997).
[CrossRef]

1996 (3)

1980 (1)

1979 (1)

1971 (2)

K. Vedam, E. D. D. Schmidt, R. Roy, “Nonlinear variation of refractive index of vitreous silica with pressure to 7 Kbars,” J. Am. Ceram. Soc. 49, 531–535 (1971).
[CrossRef]

D. F. Nelson, M. Lax, “Theory of photoelastic interaction,” Phys. Rev. B 3, 2778–2794 (1971).
[CrossRef]

1967 (1)

K. Vedam, R. Srinivasan, “Non-linear piezo-optics,” Acta Crystallogr. 22, 630–634 (1967).
[CrossRef]

Atkin, R. J.

R. J. Atkin, N. Fox, An Introduction to the Theory of Elasticity (Longmans Green, New York, 1980).

Bennion, I.

Y. Liu, L. Zhang, J. A. R. Williams, I. Bennion, “Optical bend sensor based on measurement of resonance mode splitting of long-period fiber grating,” IEEE Photon. Technol. Lett. 12, 531–533 (2000).
[CrossRef]

Y. Liu, L. Zhang, I. Bennion, “Fibre optic load sensors with high transverse strain sensitivity based on long-period gratings in B/Ge co-doped fibre,” Electron. Lett. 35, 661–663 (1999).
[CrossRef]

Bergano, N. S.

Bhatia, V.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

V. Bhatia, A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21, 692–694 (1996).
[CrossRef] [PubMed]

Burlak, G. N.

G. N. Burlak, V. V. Grimal’skii, K. Ishkabulov, “Dynamics of acoustoelectromagnetic solitons as a manifestation of higher nonlinearity,” Phys. Solid State 39, 987–990 (1997).
[CrossRef]

Chern, G. W.

Choi, S. S.

Chryssou, C. E.

C. E. Chryssou, “Gain-equalizing filters for wavelength division multiplexing optical communication systems: a comparison of notch and long-period grating filters for integrated optoelectronics,” Opt. Commun. 184, 375–384 (2000).
[CrossRef]

Davidson, C. R.

Erdogan, T.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Fox, N.

R. J. Atkin, N. Fox, An Introduction to the Theory of Elasticity (Longmans Green, New York, 1980).

Grimal’skii, V. V.

G. N. Burlak, V. V. Grimal’skii, K. Ishkabulov, “Dynamics of acoustoelectromagnetic solitons as a manifestation of higher nonlinearity,” Phys. Solid State 39, 987–990 (1997).
[CrossRef]

Ishkabulov, K.

G. N. Burlak, V. V. Grimal’skii, K. Ishkabulov, “Dynamics of acoustoelectromagnetic solitons as a manifestation of higher nonlinearity,” Phys. Solid State 39, 987–990 (1997).
[CrossRef]

Jang, J. N.

Jeong, Y.

Judkins, J. B.

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
[CrossRef] [PubMed]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Kim, S.

Kwon, J.

Lax, M.

D. F. Nelson, M. Lax, “Theory of photoelastic interaction,” Phys. Rev. B 3, 2778–2794 (1971).
[CrossRef]

Lee, B.

Lee, B. H.

Lee, S. B.

Lemaire, P. J.

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
[CrossRef] [PubMed]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Lin, C. Y.

Liu, Y.

Y. Liu, L. Zhang, J. A. R. Williams, I. Bennion, “Optical bend sensor based on measurement of resonance mode splitting of long-period fiber grating,” IEEE Photon. Technol. Lett. 12, 531–533 (2000).
[CrossRef]

Y. Liu, L. Zhang, I. Bennion, “Fibre optic load sensors with high transverse strain sensitivity based on long-period gratings in B/Ge co-doped fibre,” Electron. Lett. 35, 661–663 (1999).
[CrossRef]

B. H. Lee, Y. Liu, S. B. Lee, S. S. Choi, J. N. Jang, “Displacements of the resonant peaks of a long-period fiber grating induced by a change of ambient refractive index,” Opt. Lett. 22, 1769–1771 (1997).
[CrossRef]

Lurie, A. I.

A. I. Lurie, Nonlinear Theory of Elasticity (Elsevier, Amsterdam, 1990).

Murnaghan, F. D.

F. D. Murnaghan, Finite Deformation of an Elastic Solid (Wiley, New York, 1951).

Narasimhamurty, T. S.

T. S. Narasimhamurty, Photoelastic and Electro-Optic Properties of Crystals (Plenum, New York, 1981).
[CrossRef]

Nelson, D. F.

D. F. Nelson, M. Lax, “Theory of photoelastic interaction,” Phys. Rev. B 3, 2778–2794 (1971).
[CrossRef]

Park, N.

Pedrazzani, J. R.

Roy, R.

K. Vedam, E. D. D. Schmidt, R. Roy, “Nonlinear variation of refractive index of vitreous silica with pressure to 7 Kbars,” J. Am. Ceram. Soc. 49, 531–535 (1971).
[CrossRef]

Schmidt, E. D. D.

K. Vedam, E. D. D. Schmidt, R. Roy, “Nonlinear variation of refractive index of vitreous silica with pressure to 7 Kbars,” J. Am. Ceram. Soc. 49, 531–535 (1971).
[CrossRef]

Simon, A.

Sipe, J. E.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Smith, A. M.

Srinivasan, R.

K. Vedam, R. Srinivasan, “Non-linear piezo-optics,” Acta Crystallogr. 22, 630–634 (1967).
[CrossRef]

Ulrich, R.

Vedam, K.

K. Vedam, E. D. D. Schmidt, R. Roy, “Nonlinear variation of refractive index of vitreous silica with pressure to 7 Kbars,” J. Am. Ceram. Soc. 49, 531–535 (1971).
[CrossRef]

K. Vedam, R. Srinivasan, “Non-linear piezo-optics,” Acta Crystallogr. 22, 630–634 (1967).
[CrossRef]

Vengsarkar, A. M.

Wang, L. A.

Wang, Lon A.

C. Y. Lin, Lon A. Wang, “A wavelength- and loss-tunable band-rejection filter based on corrugated long-period fiber grating,” IEEE Photon. Technol. Lett. 13, 332–334 (2001).
[CrossRef]

Williams, J. A. R.

Y. Liu, L. Zhang, J. A. R. Williams, I. Bennion, “Optical bend sensor based on measurement of resonance mode splitting of long-period fiber grating,” IEEE Photon. Technol. Lett. 12, 531–533 (2000).
[CrossRef]

Zhang, L.

Y. Liu, L. Zhang, J. A. R. Williams, I. Bennion, “Optical bend sensor based on measurement of resonance mode splitting of long-period fiber grating,” IEEE Photon. Technol. Lett. 12, 531–533 (2000).
[CrossRef]

Y. Liu, L. Zhang, I. Bennion, “Fibre optic load sensors with high transverse strain sensitivity based on long-period gratings in B/Ge co-doped fibre,” Electron. Lett. 35, 661–663 (1999).
[CrossRef]

Acta Crystallogr. (1)

K. Vedam, R. Srinivasan, “Non-linear piezo-optics,” Acta Crystallogr. 22, 630–634 (1967).
[CrossRef]

Appl. Opt. (4)

Electron. Lett. (1)

Y. Liu, L. Zhang, I. Bennion, “Fibre optic load sensors with high transverse strain sensitivity based on long-period gratings in B/Ge co-doped fibre,” Electron. Lett. 35, 661–663 (1999).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

Y. Liu, L. Zhang, J. A. R. Williams, I. Bennion, “Optical bend sensor based on measurement of resonance mode splitting of long-period fiber grating,” IEEE Photon. Technol. Lett. 12, 531–533 (2000).
[CrossRef]

C. Y. Lin, Lon A. Wang, “A wavelength- and loss-tunable band-rejection filter based on corrugated long-period fiber grating,” IEEE Photon. Technol. Lett. 13, 332–334 (2001).
[CrossRef]

J. Am. Ceram. Soc. (1)

K. Vedam, E. D. D. Schmidt, R. Roy, “Nonlinear variation of refractive index of vitreous silica with pressure to 7 Kbars,” J. Am. Ceram. Soc. 49, 531–535 (1971).
[CrossRef]

J. Lightwave Technol. (4)

Meas. Sci. Technol. (1)

L. A. Wang, C. Y. Lin, G. W. Chern, “A torsion sensor made of a corrugated long period fibre grating,” Meas. Sci. Technol. 12, 793–799 (2001).
[CrossRef]

Opt. Commun. (1)

C. E. Chryssou, “Gain-equalizing filters for wavelength division multiplexing optical communication systems: a comparison of notch and long-period grating filters for integrated optoelectronics,” Opt. Commun. 184, 375–384 (2000).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. B (1)

D. F. Nelson, M. Lax, “Theory of photoelastic interaction,” Phys. Rev. B 3, 2778–2794 (1971).
[CrossRef]

Phys. Solid State (1)

G. N. Burlak, V. V. Grimal’skii, K. Ishkabulov, “Dynamics of acoustoelectromagnetic solitons as a manifestation of higher nonlinearity,” Phys. Solid State 39, 987–990 (1997).
[CrossRef]

Other (4)

R. J. Atkin, N. Fox, An Introduction to the Theory of Elasticity (Longmans Green, New York, 1980).

F. D. Murnaghan, Finite Deformation of an Elastic Solid (Wiley, New York, 1951).

T. S. Narasimhamurty, Photoelastic and Electro-Optic Properties of Crystals (Plenum, New York, 1981).
[CrossRef]

A. I. Lurie, Nonlinear Theory of Elasticity (Elsevier, Amsterdam, 1990).

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Figures (5)

Fig. 1
Fig. 1

Schematic structure of a CLPFG.

Fig. 2
Fig. 2

Energy fluxes of the first (solid curves) and the second (dashed curves) cladding modes as functions of radius at 1300-nm wavelength for a dispersion-shifted fiber that is (a) unetched and (b) etched with a 26.5-µm radius. The energy flux over the whole fiber cross section was normalized to unity.

Fig. 3
Fig. 3

Phase-matching curves for the unetched (solid curves) and the etched (dashed curves) sections of the CLPFGs for several cladding modes.

Fig. 4
Fig. 4

Dependence of resonance WS on the applied twist rate: points, measured data; curve, best-fit parabola.

Fig. 5
Fig. 5

(a) Typical spectrum of the PLPFG made in a photosensitive step-index fiber. (b) Dependencies of WS for three cladding-mode resonances after a specific twist was applied.

Tables (1)

Tables Icon

Table 1 Parameters of Parabolic Functions that Describe WS and Nonlinear Photoelastic Constants for CLPFG and PLPFG

Equations (28)

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

τe=ϕle+reru4lu+Lb+1kreru4Lp-1,
τe/τu=ru/re4.
η=12JTJ-I,
r=r+τ2ν, θ=θ+τz, z=z+τ2s, ν=CR2r+Fr3, s=DR2z, F=2μ-λ-m+34n8λ+2μ,
2λ+μC+λD+116λ+2μ×20λμ+24μ2+4μm+2λ+μn=0, 2λC+λ+2μD+116λ+2μ×24λμ+16μ2+8μm+3λn=0.
urr=ηrr=τ2dν/dr, uθθ=ηθθ=τ2ν/r, uzz=τ2ds/dz, ηzz=τ2ds/dz+r2/2, uθz=τr, ηθz=ηzθ=τr/2,
ξij-ξij0=pijklηkl,
εij-εij0=-εim0εnj0pmnklηkl,
εij=εij0+εij1+εij2=εij0+αijklukl+γijklmnuklumn,
εij=εij0+qijklηkl+sijklmnηklηmn,
εij=εij0+12qijkl+qijlkukl+14sijklmn+sijlkmn+sijklnm+sijlknm+2qijkmδlnuklumn.
γxxxxxx, γxxxxyy, γxxxyxy, γxxxyyx, γxxyxyx, γxxyyyy, γxxyyzz, γxxyzyz, γxxyzzy, γxyxxxy, γxyxxyx, γxyzzxy, γxyxzyz, γxyxzzy, γxyzxyz, γxyzxzy,
γxyzxzy=γxxyyzz-γxxxxyy-γxyxzyz -2γxyxzzy+γxxxxxx/2-γxxyyyy/2, γxxxyyx=γxxyyyy-γxxyyzz-γxxxyxy +2γxyxzyz+2γxyxzzy, γxyzxyz=γxyxzzy, γxxyzyz=γxxxyxy-2γxyxzyz, γxyxxxy=γxxxxyy/2-γxxyyzz/2+γxyxzzy+γxyzxzy, γxxyxyx=γxxyzyz+2γxyzxzy, γxyxxyx=γxyxxxy+γxyxzyz-γxyzxzy, γxyyzzy=γxxyyyy-γxxyyzz-γxxyzyz, γxyzzxy=γxyxxyx-γxyxzyz-γxyzxyz.
αxxxx=-ε2p11, αxxyy=-ε2p12, αxyxy=-ε2p11-p12/2, p11=pxxxx, p12=pxxyy,
εij2=γijyzyzuθzuθz, εrr2=γxxyzyzuθz2, εθθ2=γxxxyxyuθz2, εzz2=γxxyxyxuθz2,
εrr=ε0+εrrnl=ε0+τ2R2Ar+r2Br+r2γ1, εθθ=ε0+εθθnl=ε0+τ2R2Aθ+r2Bθ+r2γ2, εzz=ε0+εzznl=ε0+τ2R2Az+r2Bz+r2γ3, εθz=εzθ=-12p11-p12τrε2,
Ar=Aθ=-ε2p11C+p12C+p12D, Az=-ε2p11D+2p12C, Br=-ε23p11+p12F, Bθ=-ε2p11+3p12F, Bz=-4ε2p12F,
ψ=-εp11-p12τ/2.
βj±0=βj0±ψ/2ε,
E±=Ex±iEy.
κi±j±=k04ε0μ0Ei±*ΔεEj±dA, Δεkl=εklnl in the etched section, Δεkl=0,0<r<reε-1δkl,re<r<ruin the unetched section,
βi±0-βj±0=2π/Λ0,
βi±-βj±=2π/Λ,
Δβi=κii.
Δλ=-Δβi-Δβjdβi/dλ-dβj/dλ-1.
κii=k024βilele+lu×R2τ2Ar+Aθ+Br+Bθ+γ1+γ2Ii, Ii=βi2k0ε0/μ00rer3R2Ei*Eidr.
Δλ=Mτ2, M=gk0R2Ico-Iclεdβco/dλ-dβcl/dλlele+lu, g=ε2p11+p12F-γ1+γ2/4.
λ=15.872 GPa, μ=31.261 GPa, l=ν1/2+ν2, m=ν2+2ν3, n=4ν3, ν1=54±13 GPa, ν2=93±8 GPa, ν3=-11±3 GPa, p11=0.113, p12=0.252.

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