Abstract

A coupled-mode analysis is presented for a bent single-mode birefringent fiber. The coupling coefficients are given in closed forms. The effects of bending on the birefringence, the polarization dispersion, and the mode coupling in a bent birefringent fiber are discussed. Comparisons with other authors' work are also made by reducing our results to the case of a bent, isotropic, single-mode fiber.

© 1987 Optical Society of America

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References

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  1. T. Okoshi, in Digest of Conference on Integrated Optics and Optical Fiber Communication (Optical Society of America, Washington, D.C., 1984), paper TUB1.
  2. D. A. Steinberg, T. G. Giallorenzi, Appl. Opt. 15, 2440 (1976).
    [CrossRef] [PubMed]
  3. G. B. Hocker, Appl. Opt. 18, 1445 (1979).
    [CrossRef] [PubMed]
  4. R. Ulrich, M. Johnson, Opt. Lett. 4, 152 (1979).
    [CrossRef] [PubMed]
  5. I. P. Kaminow, IEEE J. Quantum. Electron. QE-1715 (1981).
    [CrossRef]
  6. B. Y. Kim, H. J. Shaw, IEEE Spectrum 23 (3), 54 (1986).
  7. J. Sakai, S. Machida, T. Kimura, Opt. Lett. 6, 496 (1981).
    [CrossRef] [PubMed]
  8. D. Marcuse, Bell Syst. Tech. J. 54, 985 (1975).
  9. S. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1951).
  10. J. F. Nye, Physical Properties of Crystals (Oxford U.Press, Oxford, 1957).
  11. R. Ulrich, S. C. Rashleigh, W. Eickhoff, Opt. Lett. 5, 273 (1980).
    [CrossRef] [PubMed]
  12. J. Sakai, T. Kimura, IEEE J. Quantum Electron. QE-17, 1041 (1981).
    [CrossRef]

1986 (1)

B. Y. Kim, H. J. Shaw, IEEE Spectrum 23 (3), 54 (1986).

1981 (3)

J. Sakai, T. Kimura, IEEE J. Quantum Electron. QE-17, 1041 (1981).
[CrossRef]

I. P. Kaminow, IEEE J. Quantum. Electron. QE-1715 (1981).
[CrossRef]

J. Sakai, S. Machida, T. Kimura, Opt. Lett. 6, 496 (1981).
[CrossRef] [PubMed]

1980 (1)

1979 (2)

1976 (1)

1975 (1)

D. Marcuse, Bell Syst. Tech. J. 54, 985 (1975).

Eickhoff, W.

Giallorenzi, T. G.

Goodier, J. N.

S. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1951).

Hocker, G. B.

Johnson, M.

Kaminow, I. P.

I. P. Kaminow, IEEE J. Quantum. Electron. QE-1715 (1981).
[CrossRef]

Kim, B. Y.

B. Y. Kim, H. J. Shaw, IEEE Spectrum 23 (3), 54 (1986).

Kimura, T.

J. Sakai, S. Machida, T. Kimura, Opt. Lett. 6, 496 (1981).
[CrossRef] [PubMed]

J. Sakai, T. Kimura, IEEE J. Quantum Electron. QE-17, 1041 (1981).
[CrossRef]

Machida, S.

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 54, 985 (1975).

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Oxford U.Press, Oxford, 1957).

Okoshi, T.

T. Okoshi, in Digest of Conference on Integrated Optics and Optical Fiber Communication (Optical Society of America, Washington, D.C., 1984), paper TUB1.

Rashleigh, S. C.

Sakai, J.

J. Sakai, T. Kimura, IEEE J. Quantum Electron. QE-17, 1041 (1981).
[CrossRef]

J. Sakai, S. Machida, T. Kimura, Opt. Lett. 6, 496 (1981).
[CrossRef] [PubMed]

Shaw, H. J.

B. Y. Kim, H. J. Shaw, IEEE Spectrum 23 (3), 54 (1986).

Steinberg, D. A.

Timoshenko, S.

S. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1951).

Ulrich, R.

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

D. Marcuse, Bell Syst. Tech. J. 54, 985 (1975).

IEEE J. Quantum Electron. (1)

J. Sakai, T. Kimura, IEEE J. Quantum Electron. QE-17, 1041 (1981).
[CrossRef]

IEEE J. Quantum. Electron. (1)

I. P. Kaminow, IEEE J. Quantum. Electron. QE-1715 (1981).
[CrossRef]

IEEE Spectrum (1)

B. Y. Kim, H. J. Shaw, IEEE Spectrum 23 (3), 54 (1986).

Opt. Lett. (3)

Other (3)

S. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1951).

J. F. Nye, Physical Properties of Crystals (Oxford U.Press, Oxford, 1957).

T. Okoshi, in Digest of Conference on Integrated Optics and Optical Fiber Communication (Optical Society of America, Washington, D.C., 1984), paper TUB1.

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Figures (4)

Fig. 1
Fig. 1

Bending of a highly birefringent optical fiber. The bending plane makes an angle θb with respect to the xz plane determined by the principal directions.

Fig. 2
Fig. 2

Cross-coupled power versus bending angle θb with (δβb/δβ0) as a parameter. (a) (δβ0z = 2π), which corresponds to a bending length of just one beat length of the unperturbed birefringent fiber Lb = 2π/δβ0. (b) (δβ0z = π), which is equivalent to having a bending length that is one half of the beat length of the original fiber.

Fig. 3
Fig. 3

The coupling function H2(V) versus the normalized frequency V. Comparisons of the LP approximation, the Gaussian approximation, and the corresponding coupling function H(V) in Ref. 12.

Fig. 4
Fig. 4

The polarization-dispersion function Hd(V) versus the normalized frequency V. The LP approximation, the Gaussian approximation, and the corresponding Hd(V) in Ref. 12 are compared.

Equations (9)

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d a 1 ( z ) d z = j K 11 a 1 ( z ) j K 12 a 2 ( z ) , d a 2 ( z ) d z = j K 21 a 1 ( z ) j K 22 a 2 ( z ) ,
K 11 = β x + ω 0 + e 1 * [ Δ ] e 1 d x d y , K 22 = β y + ω 0 + e 2 * [ Δ ] e 2 d x d y , K 12 = K 21 * = ω 0 + e 1 * [ Δ ] e 2 d x d y ,
K 11 = β x + ½ k 0 E ( b / R ) 2 { [ ( C 1 cos 2 θ b + C 2 sin 2 θ b ) C 2 ] + ( a / b ) 2 [ C 2 ½ ( C 1 cos 2 θ b + C 2 sin 2 θ b ) ] H 2 ( V ) } , K 22 = β y + ½ k 0 E ( b / R ) 2 { [ ( C 2 cos 2 θ b + C 1 sin 2 θ b ) C 2 ] + ( a / b ) 2 [ C 2 ½ ( C 2 cos 2 θ b + C 1 sin 2 θ b ) ] H 2 ( V ) } , K 12 = K 21 * = ½ k 0 E C ( b / R ) 2 [ 1 1 / 12 ( a / b ) 2 H 2 ( V ) ] × sin θ b cos θ b ,
H 2 ( V ) = 2 + 4 W 2 V 2 [ J 0 ( U ) U J 1 ( U ) J 0 2 ( U ) J 1 2 ( U ) ] + 4 1 V 2 [ U 2 W 2 U 2 ]
H 2 ( V ) = 6 r 0 2 a 2 [ 1 exp ( a 2 r 0 2 ) ]
δ β = [ ( δ β 0 ) 2 + 2 δ β 0 δ β b cos 2 θ b + ( δ β b ) 2 ] 1 / 2 , δ β 0 = β x β y , δ β b = ½ k 0 E C ( b / R ) 2 [ 1 1 / 12 ( a / b ) 2 H 2 ( V ) ] .
δ τ g = ( δ β 0 + δ β b cos 2 θ b δ β ) δ τ g 0 + ( δ β 0 cos 2 θ b + δ β b δ β ) δ τ g b , δ τ g 0 = 1 υ 0 d δ β 0 d k 0 , δ τ g b = 1 υ 0 ½ E C ( b / R ) 2 [ 1 1 / 12 ( a / b ) 2 H d ( V ) ] , H d ( V ) = H 2 ( V ) + V d H 2 ( V ) d V ,
P 2 ( z ) = | a 2 ( z ) | 2 = q 2 2 sin 2 ( ½ δ β z ) , q 2 = ( δ β b / δ β ) sin 2 θ b .
δ β = δ β b , δ τ g = δ τ g b , P 2 ( z ) = 0 ,

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