Abstract

Accurate forms for the generalized LP1m mode in a uniform circular-core curved fiber are given. We show that each generalized LP1m mode is composed of four linearly polarized partial fields. We also show that, when the propagation constants of HE2m, TM0m, and TE0m modes are degenerate, there are four linearly polarized modes for each generalized LP1m mode.

© 1998 Optical Society of America

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References

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  3. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  4. C Pask, ed., Special issue on Optical Waveguide Modeling, Opt. Quantum Electron. 26 (1994).
  5. S. J. Garth, C. Pask, “Polarization rotation in nonlinear bimodal optical fibers,” J. Lightwave Technol. 8, 129–137 (1990).
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  6. J. Blake, “Polarization behavior of axially strained two-mode fibers,” Opt. Lett. 17, 589–591 (1992).
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  9. F. Farahi, D. J. Webb, J. D. C. Jones, D. A. Jackson, “Simultaneous measurement of temperature and strain: cross-sensitivity considerations,” J. Lightwave Technol. 8, 138–142 (1990).
    [CrossRef]
  10. K. Kusano, A. Ankiewicz, S. V. Burke, “Accurate expressions for the LP modes in a uniform circular-core curved fiber,” Appl. Opt. 35, 2041–2047 (1996).
    [CrossRef] [PubMed]
  11. K. Kusano, S. Nishida, “Analysis of anisotropic single-mode fibers,” Electro. Commun. Jpn. 64, 26–33 (1981).
    [CrossRef]

1996 (1)

1994 (2)

1992 (2)

1990 (2)

F. Farahi, D. J. Webb, J. D. C. Jones, D. A. Jackson, “Simultaneous measurement of temperature and strain: cross-sensitivity considerations,” J. Lightwave Technol. 8, 138–142 (1990).
[CrossRef]

S. J. Garth, C. Pask, “Polarization rotation in nonlinear bimodal optical fibers,” J. Lightwave Technol. 8, 129–137 (1990).
[CrossRef]

1981 (1)

K. Kusano, S. Nishida, “Analysis of anisotropic single-mode fibers,” Electro. Commun. Jpn. 64, 26–33 (1981).
[CrossRef]

1978 (1)

1971 (1)

Ankiewicz, A.

Blake, J.

Burke, S. V.

Carneiro, S. R. M.

Carrara, S. L. A.

Castro, F. A.

Covington, C. E.

Farahi, F.

F. Farahi, D. J. Webb, J. D. C. Jones, D. A. Jackson, “Simultaneous measurement of temperature and strain: cross-sensitivity considerations,” J. Lightwave Technol. 8, 138–142 (1990).
[CrossRef]

Garth, S. J.

S. J. Garth, C. Pask, “Polarization rotation in nonlinear bimodal optical fibers,” J. Lightwave Technol. 8, 129–137 (1990).
[CrossRef]

Gloge, D.

Jackson, D. A.

F. Farahi, D. J. Webb, J. D. C. Jones, D. A. Jackson, “Simultaneous measurement of temperature and strain: cross-sensitivity considerations,” J. Lightwave Technol. 8, 138–142 (1990).
[CrossRef]

Jones, J. D. C.

F. Farahi, D. J. Webb, J. D. C. Jones, D. A. Jackson, “Simultaneous measurement of temperature and strain: cross-sensitivity considerations,” J. Lightwave Technol. 8, 138–142 (1990).
[CrossRef]

Kusano, K.

Lisboa, O.

Love, J. D.

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

Nishida, S.

K. Kusano, S. Nishida, “Analysis of anisotropic single-mode fibers,” Electro. Commun. Jpn. 64, 26–33 (1981).
[CrossRef]

Pask, C.

S. J. Garth, C. Pask, “Polarization rotation in nonlinear bimodal optical fibers,” J. Lightwave Technol. 8, 129–137 (1990).
[CrossRef]

Snyder, A. W.

A. W. Snyder, W. R. Young, “Modes of optical waveguides,” J. Opt. Soc. Am. 68, 297–307 (1978).
[CrossRef]

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

Webb, D. J.

F. Farahi, D. J. Webb, J. D. C. Jones, D. A. Jackson, “Simultaneous measurement of temperature and strain: cross-sensitivity considerations,” J. Lightwave Technol. 8, 138–142 (1990).
[CrossRef]

Young, W. R.

Appl. Opt. (2)

Electro. Commun. Jpn. (1)

K. Kusano, S. Nishida, “Analysis of anisotropic single-mode fibers,” Electro. Commun. Jpn. 64, 26–33 (1981).
[CrossRef]

J. Lightwave Technol. (2)

F. Farahi, D. J. Webb, J. D. C. Jones, D. A. Jackson, “Simultaneous measurement of temperature and strain: cross-sensitivity considerations,” J. Lightwave Technol. 8, 138–142 (1990).
[CrossRef]

S. J. Garth, C. Pask, “Polarization rotation in nonlinear bimodal optical fibers,” J. Lightwave Technol. 8, 129–137 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (3)

Opt. Quantum Electron. (1)

C Pask, ed., Special issue on Optical Waveguide Modeling, Opt. Quantum Electron. 26 (1994).

Other (1)

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

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Figures (4)

Fig. 1
Fig. 1

Curved fiber positioned on the xy plane and coordinates (ρ, θ, v), (r, ϕ, z), and (x, y, z): solid arrow, E field; dashed arrow, H field.

Fig. 2
Fig. 2

Four partial electric fields of the generalized LP11 mode in a curved fiber: solid arrow, E field; dashed arrow, H field.

Fig. 3
Fig. 3

Definitions of the polarization angle (ψpol) and the directional angle (ψdir) for linearly polarized modes: solid arrow, E field; dashed arrow, H field.

Fig. 4
Fig. 4

Transverse electric fields (E rz ) of four linearly polarized modes for the LP11 mode when l 1 = l 5 = l 6, in a uniformly curved fiber: solid arrow, E rz ; dashed arrow, H rz .

Tables (1)

Tables Icon

Table 1 Polarization Angles (ψpol) and Directional Angles (ψdir) for Linearly Polarized LP1m Modesa

Equations (28)

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E 5 = u jkNJ 1 u ρ i ρ + uJ 0 u ρ i v 2 A 5 exp - jh 5 v , H 5 = j kNu / η J 1 u ρ i θ 2 A 5 exp - jh 5 v
E 5 = w jkNK 1 w ρ i ρ - wK 0 w ρ i v 2 B 5 exp - jh 5 v , H 5 = j kNw / η K 1 w ρ i θ 2 B 5 exp - jh 5 v
E 6 = - jkNuJ 1 u ρ i θ 2 A 6 exp - jh 6 v , H 6 = u / η jkNJ 1 u ρ i ρ + uJ 0 u ρ i v 2 A 6 exp - jh 6 v
E 6 = - jkNwK 1 w ρ i θ 2 B 6 exp - jh 6 v , H 6 = w / η jkNK 1 w ρ i ρ - wK 0 w ρ i v × 2 B 6 exp - jh 6 v
E 5 = u jkNJ 1 u ρ sin θ i r + uJ 0 u ρ i ϕ + jkNJ 1 u ρ cos θ i z 2 A 5 exp - jl 5 ϕ , H 5 = j kNu / η J 1 u ρ cos θ i r - sin θ i z × 2 A 5 exp - jl 5 ϕ
E 5 = w jkNK 1 w ρ sin θ i r - wK 0 w ρ i ϕ + jkNK 1 w ρ cos θ i z 2 B 5 exp - jl 5 ϕ , H 5 = j kNw / η K 1 w ρ cos θ i r - sin θ i z × 2 B 5 exp - jl 5 ϕ
E 6 = jkNuJ 1 u ρ - cos θ i r + sin θ i z 2 A 6 exp - jl 6 ϕ , H 6 = u / η jkNJ 1 u ρ sin θ i r + uJ 0 u ρ i ϕ + jkNJ 1 u ρ cos θ i z 2 A 6 exp - jl 6 ϕ
E 6 = jkNwK 1 w ρ - cos θ i r + sin θ i z × 2 B 6 exp - jl 6 ϕ , H 6 = w / η jkNK 1 w ρ sin θ i r - wK 0 w ρ i ϕ + jkNK 1 w ρ cos θ i z 2 B 6 exp - jl 6 ϕ
E HE = E rz core + u 2 J 2 u ρ cos 2 θ - ψ 1 i ϕ × 2 A 1 exp - jl 1 ϕ = jkNuJ 1 u ρ sin θ cos ψ 1 i r - cos θ cos ψ 1 i z - jkNuJ 1 u ρ cos θ sin ψ 1 i r + sin θ sin ψ 1 i z + u 2 J 2 u ρ cos 2 θ - ψ 1 i ϕ 2 A 1 exp - jl 1 ϕ , H HE = - 1 / η E rz core xi ϕ + u 2 / η J 2 u ρ sin 2 θ - ψ 1 i ϕ × 2 A 1 exp - jl 1 ϕ
E HE = E rz clad - w 2 K 2 w ρ cos 2 θ - ψ 1 i ϕ × 2 B 1 exp - jl 1 ϕ = - jkNwK 1 w ρ sin θ cos ψ 1 i r - cos θ cos ψ 1 i z + jkNwK 1 w ρ cos θ sin ψ 1 i r + sin θ sin ψ 1 i z - w 2 K 2 w ρ cos 2 θ - ψ 1 i ϕ 2 B 1 exp - jl 1 ϕ , H HE = - 1 / η E rz clad xi ϕ - w 2 / η K 2 w ρ sin 2 θ - ψ 1 i ϕ × 2 B 1 exp - jl 1 ϕ
E = E HE + E 5 + E 6 ,     H = H HE + H 5 + H 6 .
exp - jl 1 ϕ = 1 / 2 exp - jl 1 ϕ + exp - jl 5 ϕ + exp - jl 1 ϕ - exp - jl 5 ϕ , exp - jl 1 ϕ = 1 / 2 exp - jl 1 ϕ + exp - jl 6 ϕ + exp - jl 1 ϕ - exp - jl 6 ϕ , exp - jl 5 ϕ = 1 / 2 exp - jl 1 ϕ + exp - jl 5 ϕ - exp - jl 1 ϕ - exp - jl 5 ϕ , exp - jl 6 ϕ = 1 / 2 exp - jl 1 ϕ + exp - jl 6 ϕ - exp - jl 1 ϕ - exp - jl 6 ϕ .
E a 1 = j 2 kNuJ 1 u ρ sin θ i r ,   E a 2 = j 2 kNuJ 1 u ρ cos θ i z , E b 1 = j 2 kNuJ 1 u ρ cos θ i r ,   E b 2 = j 2 kNuJ 1 u ρ sin θ i z , E a ϕ 1 = 2 u 2 J 2 u ρ cos 2 θ cos ψ 1 A 1 + J 0 u ρ A 5 i ϕ , E a ϕ 2 = 2 u 2 J 2 u ρ cos 2 θ cos ψ 1 A 1 - J 0 u ρ A 5 i ϕ , E b ϕ 1 = E b ϕ 2 = 2 u 2 J 2 u ρ sin 2 θ sin ψ 1 A 1 i ϕ , H a ϕ 1 = H a ϕ 2 = 2 u 2 / η J 2 u ρ sin 2 θ cos ψ 1 A 1 i ϕ , H b ϕ 1 = 2 u 2 / η J 2 u ρ cos 2 θ sin ψ 1 A 1 - J 0 u ρ A 6 i ϕ , H b ϕ 2 = 2 u 2 / η J 2 u ρ cos 2 θ sin ψ 1 A 1 + J 0 u ρ A 6 i ϕ ,
E c 1 = j 2 kNwK 1 w ρ sin θ i r ,   E c 2 = j 2 kNwK 1 w ρ cos θ i z , E d 1 = j 2 kNwK 1 w ρ cos θ i r ,   E d 2 = j 2 kNwK 1 w ρ sin θ i z , E c ϕ 1 = 2 w 2 K 2 w ρ cos 2 θ cos ψ 1 B 1 + K 0 w ρ B 5 i ϕ , E c ϕ 2 = 2 w 2 K 2 w ρ cos 2 θ cos ψ 1 B 1 - K 0 w ρ B 5 i ϕ , E d ϕ 1 = E d ϕ 2 = 2 w 2 K 2 w ρ sin 2 θ sin ψ 1 B 1 i ϕ , H c ϕ 1 = H c ϕ 2 = 2 w 2 / η K 2 w ρ sin 2 θ cos ψ 1 B 1 i ϕ , H d ϕ 1 = 2 w 2 / η K 2 w ρ cos 2 θ sin ψ 1 B 1 - K 0 u ρ B 6 i ϕ , H d ϕ 2 = 2 w 2 / η K 2 w ρ cos 2 θ sin ψ 1 B 1 + K 0 w ρ B 6 i ϕ .
E = E a 1 f a 1 - E a 2 f a 2 + E a ϕ 1 cos l 1 - l 5 ϕ / 2 - j E a ϕ 2 sin l 1 - l 5 ϕ / 2 exp - j l 1 + l 5 ϕ / 2 - E b 1 f b 1 + E b 2 f b 2 - E b ϕ 1 cos l 1 - l 6 ϕ / 2 + j E b ϕ 2 sin l 1 - l 6 ϕ / 2 exp - j l 1 + l 6 ϕ / 2 , H = - 1 / η E a 1 f a 1 - E a 2 f a 2 xi ϕ - H a ϕ 1 cos l 1 - l 5 ϕ / 2 + j H a ϕ 2 sin l 1 - l 5 ϕ / 2 exp - j l 1 + l 5 ϕ / 2 + 1 / η E b 1 f b 1 + E b 2 f b 2 xi ϕ - H b ϕ 1 cos l 1 - l 6 ϕ / 2 + j H b ϕ 2 sin l 1 - l 6 ϕ / 2 exp - j l 1 + l 6 ϕ / 2
E = - E c 1 f c 1 - E c 2 f c 2 - E c ϕ 1 cos l 1 - l 5 ϕ / 2 + j E c ϕ 2 sin l 1 - l 5 ϕ / 2 exp - j l 1 + l 5 ϕ / 2 + E d 1 f d 1 + E d 2 f d 2 - E d ϕ 1 cos l 1 - l 6 ϕ / 2 + j E d ϕ 2 sin l 1 - l 6 ϕ / 2 exp - j l 1 + l 6 ϕ / 2 , H = 1 / η E c 1 f c 1 - E c 2 f c 2 xi ϕ - H c ϕ 1 cos l 1 - l 5 ϕ / 2 + j H c ϕ 2 sin l 1 - l 5 ϕ / 2 exp - j l 1 + l 5 ϕ / 2 - 1 / η E d 1 f d 1 + E d 2 f d 2 xi ϕ - H d ϕ 1 × cos l 1 - l 6 ϕ / 2 + j H d ϕ 2 sin l 1 - l 6 ϕ / 2 exp - j l 1 + l 6 ϕ / 2
f a 1 = A 1 cos ψ 1 + A 5 cos l 1 - l 5 ϕ / 2 - j A 1 cos ψ 1 - A 5 sin l 1 - l 5 ϕ / 2 , f a 2 = A 1 cos ψ 1 - A 5 cos l 1 - l 5 ϕ / 2 - j A 1 cos ψ 1 + A 5 sin l 1 - l 5 ϕ / 2 , f b 1 = A 1 sin ψ 1 + A 6 cos l 1 - l 6 ϕ / 2 - j A 1 sin ψ 1 - A 6 sin l 1 - l 6 ϕ / 2 , f b 2 = A 1 sin ψ 1 - A 6 cos l 1 - l 6 ϕ / 2 - j A 1 sin ψ 1 + A 6 sin l 1 - l 6 ϕ / 2 , f c 1 = B 1 cos ψ 1 - B 5 cos l 1 - l 5 ϕ / 2 - j B 1 cos ψ 1 + B 5 sin l 1 - l 5 ϕ / 2 , f c 2 = B 1 cos ψ 1 + B 5 cos l 1 - l 5 ϕ / 2 - j B 1 cos ψ 1 - B 5 sin l 1 - l 5 ϕ / 2 , f d 1 = B 1 sin ψ 1 - B 6 cos l 1 - l 6 ϕ / 2 - j B 1 sin ψ 1 + B 6 sin l 1 - l 6 ϕ / 2 , f d 2 = B 1 sin ψ 1 + B 6 cos l 1 - l 6 ϕ / 2 - j B 1 sin ψ 1 - B 6 sin l 1 - l 6 ϕ / 2 .
E = E rz + E ϕ exp - jl 1 ϕ , H = - 1 / η E rz xi ϕ + H ϕ exp - jl 1 ϕ ,
E rz = j 2 kNuJ 1 u ρ sin θ A 1 cos ψ 1 + A 5 i r - A 1 sin ψ 1 - A 6 i z - j 2 kNuJ 1 u ρ cos θ A 1 sin ψ 1 + A 6 i r + A 1 cos ψ 1 - A 5 i z , E ϕ = E a ϕ 1 + E b ϕ 1 = 2 u 2 J 2 u ρ cos 2 θ - ψ 1 A 1 + J 0 u ρ A 5 i ϕ , H ϕ = H a ϕ 1 - H b ϕ 1 = 2 u 2 J 2 u ρ sin 2 θ - ψ 1 A 1 + J 0 u ρ A 6 i ϕ
E rz = - j 2 kNwK 1 w ρ sin θ B 1 cos ψ 1 - B 5 i r - B 1 sin ψ 1 + B 6 i z + j 2 kNwK 1 w ρ cos θ B 1 sin ψ 1 - B 6 i r + B 1 cos ψ 1 + B 5 i z , E ϕ = - E c ϕ 1 - E d ϕ 1 = - 2 w 2 K 2 w ρ cos 2 θ - ψ 1 B 1 + K 0 w ρ B 5 i ϕ , H ϕ = - H c ϕ 1 + H d ϕ 1 = - 2 w 2 / η K 2 w ρ sin 2 θ - ψ 1 B 1 + K 0 w ρ B 6 i ϕ
E rz = j 4 kNuJ 1 ur sin θ - ψ dir × sin ψ pol i r + cos ψ pol i z A 1
E rz = - j 4 kNwK 1 wr sin θ - ψ dir × sin ψ pol i r + cos ψ pol i z B 1
x = R + ρ   sin θ cos v / R , y = R + ρ   sin θ sin v / R , z = ρ   cos θ ,
x = r   cos ϕ ,     y = r   sin ϕ ,     z = z ,
G r = sin θ G ρ + cos θ G θ , G ϕ = G v , G z = cos θ G ρ - sin θ G θ .
E = jkNuJ 1 u ρ sin θ cos ψ 1 i r - jkNuJ 1 u ρ cos θ cos ψ 1 i z + u 2 J 2 u ρ cos 2 θ cos ψ 1 i ϕ 2 A 1 exp - jl 1 ϕ + jkNuJ 1 u ρ sin θ i r + jkNuJ 1 u ρ cos θ i z + u 2 J 0 u ρ i ϕ 2 A 5 exp - jl 5 ϕ + - jkNuJ 1 u ρ cos θ sin ψ 1 i r - jkNuJ 1 u ρ sin θ sin ψ 1 i z + u 2 J 2 u ρ sin 2 θ sin ψ 1 i ϕ 2 A 1 exp - jl 1 ϕ + - jkNuJ 1 u ρ cos θ i r + jkNuJ 1 u ρ sin θ i z 2 A 6 exp - jl 6 ϕ
E = jkNuJ 1 u ρ sin θ A 1 cos ψ 1 + A 5 i r - jkNuJ 1 u ρ cos θ A 1 cos ψ 1 - A 5 i z + u 2 J 2 u ρ cos 2 θ cos ψ 1 A 1 + J 0 u ρ A 5 i ϕ × exp - jl 1 ϕ + exp - jl 5 ϕ + jkNuJ 1 u ρ sin θ A 1 cos ψ 1 - A 5 i r - jkNuJ 1 u ρ cos θ A 1 cos ψ 1 + A 5 i z + u 2 J 2 u ρ cos 2 θ cos ψ 1 A 1 - J 0 u ρ A 5 i ϕ × exp - jl 1 ϕ - exp - jl 5 ϕ + - jkNuJ 1 u ρ cos θ A 1 sin ψ 1 + A 6 i r - jkNuJ 1 u ρ sin θ A 1 sin ψ 1 - A 6 i z + u 2 J 2 u ρ sin 2 θ sin ψ 1 A 1 i ϕ × exp - jl 1 ϕ + exp - jl 6 ϕ + - jkNuJ 1 u ρ cos θ A 1 sin ψ 1 - A 6 i r - jkNuJ 1 u ρ sin θ A 1 sin ψ 1 + A 6 i z + u 2 J 2 u ρ sin 2 θ sin ψ 1 A 1 i ϕ × exp - jl 1 ϕ - exp - jl 6 ϕ .
exp - jl 1 ϕ + exp - jl 5 ϕ = 2   cos l 1 - l 5 ϕ / 2 × exp - j l 1 + l 5 ϕ / 2 , exp - jl 1 ϕ - exp - jl 5 ϕ = - j 2   sin l 1 - l 5 ϕ / 2 × exp - j l 1 + l 5 ϕ / 2 ,

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