Abstract

The first-order paraxial approximation is used to obtain the distributions of the electric and magnetic fields for the core and cladding hybrid fiber modes. The coupling coefficients of these modes are found for fibers subject to twist. The longitudinal electric field component determines the mode coupling in twisted fibers. It is shown that in the first-order paraxial approximation the cladding hybrid modes propagating in a twisted fiber rotate along the direction of the twist at the same rate as the core mode, independently of the azimuthal and radial mode numbers. Four hybrid modes constituting one linearly polarized mode have different longitudinal components, and the corresponding cladding-mode resonances of a long-period fiber grating undergo different shifts owing to different mode self-coupling coefficients. This results in the removal of mode degeneracy and splitting of resonances of long-period gratings in twisted fibers.

© 2005 Optical Society of America

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References

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  1. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  2. D. L. A. Tjaden, “First-order correction to ‘weak-guidance’ approximation in fibre optics theory,” Philips J. Res. 33(1/2), 103–112 (1978).
  3. C. Tsao, Optical Fibre Waveguide Analysis (Oxford, New York, 1992).
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    [CrossRef]
  5. R. Ulrich, A. Simon, “Polarization optics of twisted single-mode fibers,” Appl. Opt. 18, 2241–2251 (1979).
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    [CrossRef] [PubMed]
  7. V. Bhatia, A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21, 692–694 (1996).
    [CrossRef] [PubMed]
  8. 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]
  9. 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]
  10. 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]
  11. 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 Photonics Technol. Lett. 12, 531–533 (2000).
    [CrossRef]
  12. O. V. Ivanov, L. A. Wang, “Wavelength shifts of cladding modes resonance in corrugated long-period fiber gratings under torsion,” Appl. Opt. 42, 2264–2272 (2003).
    [CrossRef] [PubMed]
  13. O. V. Ivanov, “Wavelength shift and split of cladding mode resonances in microbend long-period fiber gratings under torsion,” Opt. Commun. 232, 159–166 (2004).
    [CrossRef]
  14. H. Kogelnik, “Theory of optical waveguides,” in Guided-Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990).
  15. A. Bertholds, R. Danliker, “Determination of strain-optic coefficients in single-mode optical fibers,” J. Lightwave Technol. 6, 17–20 (1988).
    [CrossRef]

2004

O. V. Ivanov, “Wavelength shift and split of cladding mode resonances in microbend long-period fiber gratings under torsion,” Opt. Commun. 232, 159–166 (2004).
[CrossRef]

2003

2000

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 Photonics Technol. Lett. 12, 531–533 (2000).
[CrossRef]

1999

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

1996

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

1988

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

1980

1979

1978

D. L. A. Tjaden, “First-order correction to ‘weak-guidance’ approximation in fibre optics theory,” Philips J. Res. 33(1/2), 103–112 (1978).

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 Photonics 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]

Bertholds, A.

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

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]

Choi, S. S.

Danliker, R.

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

Erdogan, T.

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997).
[CrossRef]

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]

Ivanov, O. V.

O. V. Ivanov, “Wavelength shift and split of cladding mode resonances in microbend long-period fiber gratings under torsion,” Opt. Commun. 232, 159–166 (2004).
[CrossRef]

O. V. Ivanov, L. A. Wang, “Wavelength shifts of cladding modes resonance in corrugated long-period fiber gratings under torsion,” Appl. Opt. 42, 2264–2272 (2003).
[CrossRef] [PubMed]

Jang, J. N.

Judkins, J. B.

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]

Kogelnik, H.

H. Kogelnik, “Theory of optical waveguides,” in Guided-Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990).

Lee, B. H.

Lee, S. B.

Lemaire, P. J.

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]

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 Photonics 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]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

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.

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Tjaden, D. L. A.

D. L. A. Tjaden, “First-order correction to ‘weak-guidance’ approximation in fibre optics theory,” Philips J. Res. 33(1/2), 103–112 (1978).

Tsao, C.

C. Tsao, Optical Fibre Waveguide Analysis (Oxford, New York, 1992).

Ulrich, R.

Vengsarkar, A. M.

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

Wang, L. A.

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 Photonics 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 Photonics 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]

Appl. Opt.

Electron. Lett.

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 Photonics Technol. Lett.

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 Photonics Technol. Lett. 12, 531–533 (2000).
[CrossRef]

J. Lightwave Technol.

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]

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

J. Opt. Soc. Am. A

Opt. Commun.

O. V. Ivanov, “Wavelength shift and split of cladding mode resonances in microbend long-period fiber gratings under torsion,” Opt. Commun. 232, 159–166 (2004).
[CrossRef]

Opt. Lett.

Philips J. Res.

D. L. A. Tjaden, “First-order correction to ‘weak-guidance’ approximation in fibre optics theory,” Philips J. Res. 33(1/2), 103–112 (1978).

Other

C. Tsao, Optical Fibre Waveguide Analysis (Oxford, New York, 1992).

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

H. Kogelnik, “Theory of optical waveguides,” in Guided-Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990).

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Figures (4)

Fig. 1
Fig. 1

Comparison between the distributions of (a) transverse and (b) longitudinal electric field components calculated exactly and in the paraxial approximation for the HE 1 , 10 mode.

Fig. 2
Fig. 2

Overlapping integrals for various hybrid modes calculated exactly (solid curves and symbols) and in the paraxial approximation (dashed lines) versus the radial mode number for the cases of (a) self-coupling and (b) intermode coupling.

Fig. 3
Fig. 3

Scheme of resonances in LPFGs with (a) symmetric grating, (b) antisymmetric grating, (c) asymmetric grating.

Fig. 4
Fig. 4

Wavelength shifts of cladding mode resonances in an LPFG subject to twist with a rate of 10 rad/cm for coupling between the following modes: (1) HE 1 , 1 ( co ) and HE 1 , m , (2) HE - 1 , 1 ( co ) and HE - 1 , m , (3) HE 1 , 1 ( co ) and HE 2 , m , (4) HE - 1 , 1 ( co ) and TM 0 , m , (5) HE 1 , 1 ( co ) and TM 0 , m , (6) HE - 1 , 1 ( co ) and HE - 2 , m .

Equations (68)

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ˆ ( r ) = ( r ) + Δ ˆ ( r ) .
E ( r ,   ϕ ) = v , m A vm E vm ( r ) exp [ i ( β vm z - ω t + v ϕ ) ] ,
H ( r ,   ϕ ) = v , m A vm H vm ( r ) exp [ i ( β vm z - ω t + v ϕ ) ] ,
E vm ( r ) = E vm ( r ) e r + E vm ( ϕ ) e ϕ + E vm ( z ) e z ,
H vm ( r ) = H vm ( r ) e r + H vm ( ϕ ) e ϕ + H vm ( z ) e z ,
HE vm : E ( r ) = E λ m ( r ) , E ( ϕ ) = i   sign ( v ) E λ m ( r ) ,
H ( r ) = - sign ( v )   i β λ m ω μ 0   E λ m ( r ) ,
H ( ϕ ) = β λ m ω μ 0   E λ m ( r ) , λ = | v | - 1 ;
EH vm : E ( r ) = E λ m ( r ) , E ( ϕ ) = - i   sign ( v ) E λ m ( r ) ,
H ( r ) = i   sign ( v )   β λ m ω μ 0   E λ m ( r ) ,
H ( ϕ ) = β λ m ω μ 0   E λ m ( r ) , λ = | v | + 1 ;
TE 0 m : E ( r ) = 0 , E ( ϕ ) = iE 1 m ( r ) ,
H ( r ) = - i β 1 m ω μ 0   E 1 m ( r ) , H ( ϕ ) = 0 ;
TM 0 m : E ( r ) = E 1 m ( r ) , E ( ϕ ) = 0 ,
H ( r ) = 0 , H ( ϕ ) = β 1 m ω μ 0   E 1 m ( r ) ;
E λ m + ( 1 / r ) E λ m + ( k 0 2 ( r ) - β λ m 2 - λ 2 / r 2 ) E λ m = 0 .
HE vm : E ( z ) = i β λ m E λ m - λ r   E λ m ,
H ( z ) = - i   sign ( v )   β λ m ω μ 0   E ( z ) , λ = | v | - 1 ;
EH vm : E ( z ) = i β λ m E λ m + λ r   E λ m ,
H ( z ) = i   sign ( v )   β λ m ω μ 0   E ( z ) , λ = | v | + 1 ;
TE 0 m : E ( z ) = 0 , H ( z ) = 1 ω μ 0 E 1 m + 1 r   E 1 m ;
TM 0 m : E ( z ) = i β 1 m E 1 m + 1 r   E 1 m , H ( z ) = 0 .
Δ ˆ = τ p 44 2 0 0 0 0 0 - r 0 - r 0 ,
κ pq = ω 0 4 E p * ( r ,   ϕ ) Δ ˆ E q ( r ,   ϕ ) d S .
κ pq = ω 0 2 0 E p * ( r ) Δ ˆ E q ( r ) π r d r .
κ pq = - ω 0 2   τ p 44 cl 2 0 r cl ( E p ( ϕ ) E q ( z ) * + E p ( z ) E q ( ϕ ) * ) π r 2 d r = k 0 2 2 β p τ β q   p 44 cl 2 J pq ,
J pq = - β p β q ω μ 0 0 r cl ( E p ( ϕ ) E q ( z ) * + E p ( z ) E q ( ϕ ) * ) π r 2 d r .
J pq = sign ( v ) ω μ 0 0 r cl 1 r   ( λ p β q + λ q β p ) E p E q + β p E p E q + β q E q E p π r 2 d r .
J pq = - sign ( v ) ω μ 0 0 r cl 1 r   ( λ p β q + λ q β p ) E p E q + β p E p E q - β q E q E p π r 2 d r ,
J pq = - β q ω μ 0 0 r cl [ E p E q + E p E q r ] π r d r ,
J pq = 0 .
β q 2 ω μ 0 E p E q * d S = δ pq .
E p E q 2 π r d r = ω μ 0 β p   δ pq .
( f g + fg ) r 2 d r = - 2 fgr d r ,
HE p and HE q , EH p and EH q : J pq = v δ mn ,
HE p and EH q : J pq = - v δ mn + sign ( v )   β q ω μ 0 ( E q E p - E p E q ) π r 2 d r ,
TE p and TM q : J pq = - 1 2 δ mn - β q ω μ 0 E p E q π r 2 d r ,
TE p and TE q , TM p and TM q : J pq = 0 .
J pp = v .
J ( HE 11 ( co ) ,   HE 11 ( co ) ) = 1.0009 .
Δ β p = κ pp .
Δ β vm = τ p 44 cl v / 2
LP λ , m = HE λ + 1 , m + HE - ( λ + 1 ) , m + EH λ - 1 , m + EH - ( λ - 1 ) , m ,
LP 0 , m = HE 1 , m + HE - 1 , m ,
LP 1 , m ( a ) = HE 2 , m + HE - 2 , m + TE 0 , m ,
LP 1 , m ( b ) = HE 2 , m + HE - 2 , m + TM 0 , m .
E ( r ,   ϕ ) = A vm E vm ( r ) × exp { i [ β vm z - ω t + v ( ϕ + τ p 44 cl z / 2 ) ] } .
ψ = - p 44 cl τ / 2 .
ψ p = - ( 1 + Δ J p / v ) p 44 cl τ / 2 ,
Δ J p = J pp - v .
E ( r ) ( r ,   ϕ ,   z ) = 4 E λ m ( r ) cos [ λ ( ϕ - ψ λ m z ) ] × cos ( ϕ - ψ λ m z ) exp [ i ( β λ m z - ω t ) ] ,
E ( ϕ ) ( r ,   ϕ ,   z ) = - 4 E λ m ( r ) cos [ λ ( ϕ - ψ λ m z ) ] × sin ( ϕ - ψ λ m z ) exp [ i ( β λ m z - ω t ) ] ,
ψ λ m = - 1 2   p 44 cl τ 1 + Δ J λ + 1 , m + Δ J λ - 1 , m 2 λ ,
ψ λ m = - 1 2   p 44 cl τ 1 + Δ J λ + 1 , m - Δ J λ - 1 , m 2 .
Δ ψ λ = ψ λ - ψ λ = - 1 4 λ   p 44 cl τ [ ( λ + 1 ) Δ J λ - 1 , m - ( λ - 1 ) Δ J λ + 1 , m ] .
Δ ψ λ m z r = π / λ .
β p + Δ β p - β q - Δ β q = 2 π / Λ ,
Δ λ = - ( Δ β q - Δ β p ) d β q d λ - d β p d λ - 1 .
δ β = β ( HE 1 , 1 ( co ) ) + Δ β ( HE 1 , 1 ( co ) ) - β ( HE 1 , m ) - Δ β ( HE 1 , m ) ,
δ β = β ( HE - 1 , 1 ( co ) ) + Δ β ( HE - 1 , 1 ( co ) ) - β ( HE - 1 , m ) - Δ β ( HE - 1 , m ) .
δ β 1 - δ β 2 = δ β 1 - δ β 2 = τ p 44 cl .
δ β 1 - δ β 2 = δ β 1 - δ β 2 = ρ τ p 44 cl .
symmetric grating :
δ β - δ β = τ p 44 cl [ Δ J ( HE 1 , m ) - Δ J ( HE 1 , 1 ( co ) ) ] ,
antisymmetric grating :
δ β - δ β = τ p 44 cl [ Δ J ( HE 2 , m ) / 2 - Δ J ( HE 1 , 1 ( co ) ) ] ,
asymmetric grating :
δ β - δ β = τ p 44 cl [ Δ J ( HE ρ + 1 , m ) / 2 + Δ J ( EH ρ - 1 , m ) / 2 - Δ J ( HE 1 , 1 ( co ) ) ] .

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