Abstract

The effects of periodic axis deformations on propagation in multimode optical fibers with single-mode excitation are investigated numerically and experimentally. The numerical study, based on ray theory, deals with helical rays in the presence of sinusoidal axis deformations for various shapes of index profile. The corresponding experimental observations and results, carried out on tubular modes, confirm the existence of resonance effects between the helical ray period and the fiber axis deformation. This technique permits the observation of mode-to-mode power transfer and provides a sensitive tool to investigate the mode-coupling mechanism in optical fibers.

© 1983 Optical Society of America

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References

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  1. J. N. Fields, “Attenuation of a parabolic index fiber with periodic bends,” Appl. Phys. Lett. 36, 10 (1980).
    [Crossref]
  2. J. Arnaud and M. Rousseau, “Ray theory of randomly bent multimode optical fibers,” Opt. Lett. 3, 63–65 (1978).
    [Crossref] [PubMed]
  3. J. Arnaud, “Application des techniques Hamiltoniennes aux fibres multimodales,” Ann. Télécom. 32, 135–143 (1977).
  4. P. Facq, P. Fournet, and J. Arnaud, “Observation of tubular modes in multimode graded-index optical fibers,” Electron. Lett. 16, 648 (1980).
    [Crossref]
  5. P. Facq and J. Arnaud, “Tubular mode excitation in graded-index multimode fibers” in Proceedings of Photon 80, Quartz et Silice (Pithiviers, France, 1980).
  6. J. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

1980 (2)

J. N. Fields, “Attenuation of a parabolic index fiber with periodic bends,” Appl. Phys. Lett. 36, 10 (1980).
[Crossref]

P. Facq, P. Fournet, and J. Arnaud, “Observation of tubular modes in multimode graded-index optical fibers,” Electron. Lett. 16, 648 (1980).
[Crossref]

1978 (1)

1977 (1)

J. Arnaud, “Application des techniques Hamiltoniennes aux fibres multimodales,” Ann. Télécom. 32, 135–143 (1977).

Arnaud, J.

P. Facq, P. Fournet, and J. Arnaud, “Observation of tubular modes in multimode graded-index optical fibers,” Electron. Lett. 16, 648 (1980).
[Crossref]

J. Arnaud and M. Rousseau, “Ray theory of randomly bent multimode optical fibers,” Opt. Lett. 3, 63–65 (1978).
[Crossref] [PubMed]

J. Arnaud, “Application des techniques Hamiltoniennes aux fibres multimodales,” Ann. Télécom. 32, 135–143 (1977).

P. Facq and J. Arnaud, “Tubular mode excitation in graded-index multimode fibers” in Proceedings of Photon 80, Quartz et Silice (Pithiviers, France, 1980).

J. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

Facq, P.

P. Facq, P. Fournet, and J. Arnaud, “Observation of tubular modes in multimode graded-index optical fibers,” Electron. Lett. 16, 648 (1980).
[Crossref]

P. Facq and J. Arnaud, “Tubular mode excitation in graded-index multimode fibers” in Proceedings of Photon 80, Quartz et Silice (Pithiviers, France, 1980).

Fields, J. N.

J. N. Fields, “Attenuation of a parabolic index fiber with periodic bends,” Appl. Phys. Lett. 36, 10 (1980).
[Crossref]

Fournet, P.

P. Facq, P. Fournet, and J. Arnaud, “Observation of tubular modes in multimode graded-index optical fibers,” Electron. Lett. 16, 648 (1980).
[Crossref]

Rousseau, M.

Ann. Télécom. (1)

J. Arnaud, “Application des techniques Hamiltoniennes aux fibres multimodales,” Ann. Télécom. 32, 135–143 (1977).

Appl. Phys. Lett. (1)

J. N. Fields, “Attenuation of a parabolic index fiber with periodic bends,” Appl. Phys. Lett. 36, 10 (1980).
[Crossref]

Electron. Lett. (1)

P. Facq, P. Fournet, and J. Arnaud, “Observation of tubular modes in multimode graded-index optical fibers,” Electron. Lett. 16, 648 (1980).
[Crossref]

Opt. Lett. (1)

Other (2)

P. Facq and J. Arnaud, “Tubular mode excitation in graded-index multimode fibers” in Proceedings of Photon 80, Quartz et Silice (Pithiviers, France, 1980).

J. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

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Figures (16)

Fig. 1
Fig. 1

a, Ray trajectory in a linear-profile graded-index fiber (κ = 0.5), subject to sinusoidal axis deformation, with amplitude A0 = 0.5 μm in part AB. The ray is launched under such conditions that it would be helical in the absence of microbends. For clarity, the fiber representation has been compressed 30 times following the z axis. rc = 26 μm, μ = 4, rm = 5.1 μm, N.A. = 0.25. b, Cross-section projection of the part of the ray traveling in the undeformed region BC following immediately the microbend region AB.

Fig. 2
Fig. 2

Cross-section projection of the ray trajectories for different values of the perturbation-period to ray-period ratio Λ/p (with A0 = 5.10−3μm, κ = 1, μ = 4, rc = 26 μm, rm = 6.4 μm, γ = /2, and m = 0, 1, 2, 3, 4). a, Λ/p =0.5; b, Λ/p = 1; c, Λ/p = 1.5.

Fig. 3
Fig. 3

Trajectory projection broadening parameter w versus perturbation-length to resonant-period ratio L/p for two index profiles: parabolic (κ = 1, rm = 6.4 μm; solid line) and linear with μ = 4 and rc = 26 μm (κ = 0.5, rm = 5.1 μm; dotted–dashed line), with various deformation amplitudes A0: a, 1.25 × 10−3μm; b, 1.25 × 10−2μm; c, 1.25 × 10−1μm.

Fig. 4
Fig. 4

a, μ = 2 tubular mode at the output end of a 10-m-long quadratic-index profile fiber. rm = 4 μm. b, Index profile of the corresponding fiber (rc = 26 μm, N.A. = 0.25).

Fig. 5
Fig. 5

a, μ = 13 tubular mode at the output end of a 10-m-long linear-profile fiber rm = 10.5 μm. b, Index profile of the corresponding fiber (rc = 21 μm, N.A. = 0.21).

Fig. 6
Fig. 6

a, μ = 4 tubular mode at the output end of a 10-m-long fiber, rm = 5.5 μm. b, Index profile of the corresponding fiber (rc = 26 μm, N.A. = 0.17).

Fig. 7
Fig. 7

a, Photograph of the exit end of an undulated profile fiber showing the appearance of a μ = 5 tubular mode at rm 16.4 μm when a μ = 2 tubular mode has been launched at rm = 4.6 μm. b, Undulating index profile of the fiber (rc = 24 μm, N.A. = 0.11).

Fig. 8
Fig. 8

Fiber-deformation device. a, Side view; b, top view.

Fig. 9
Fig. 9

Circular mask for μ = 2 tubular-mode blockage. The power flowing through the area rs < r < rc is collected by a photodetector.

Fig. 10
Fig. 10

Amount of power Pext coupled to high-order modes and flowing outside the blocking screen of Fig. 9. Quadratic-index-profile fiber (see Fig. 4b) under μ = 2 tubular-mode injection subject to periodic microbends of period Λ and mean amplitude ā. Total deformation force F = 0.1 N.

Fig. 11
Fig. 11

As in Fig. 10 for a deformation force F = 0.25 N.

Fig. 12
Fig. 12

Field configurations at the output end of a 10-m-long fiber under tubular-mode launching (μ = 2), subject to a periodic deformation (period Λ, mean amplitude ā) over a 100-mm length. The μ = 2 ray period in that quadratic-index-profile fiber is p = 1 mm. F = 0.25 N. a, Λ/p = 0.6, ā = 10−2μm; b, Λ/p = 1, ā ≃ 0.1 μm; c, Λ/p = 1.2, ā ≃ 0.25 μm.

Fig. 13
Fig. 13

Amount of power Pext coupled to high-order modes and flowing outside the blocking screen of Fig. 9. Slightly undulating index-profile fiber (see Fig. 6b) under μ = 2 tubular-mode launching, subject to periodic microbends of period Λ and mean amplitude ā.

Fig. 14
Fig. 14

Field configurations at the output end of a 10-m-long fiber under tubular-mode launching (μ = 4) subject to a periodic deformation (period Λ) over a 100-mm length. The μ = 4 ray period in that slightly undulating index profile is p = 0.69 mm. a, Λ/p = 0.70, ā ≃ 1 × 10−2μm; b, Λ/p = 1.06, ā ≃ 1.5 × 10−2μm; c, Λ/p = 2.57, ā ≃ 0.6 μm.

Fig. 15
Fig. 15

Sketch of the modified potential function U′(r) = [n0n(r)]/n0 + (½)(μ/k0r)2 for μ = 2 for the strongly undulating index profile of Fig. 7b.

Fig. 16
Fig. 16

Field configurations at the output end of a 10-m-long fiber under tubular-mode launching (μ = 2) subject to a periodic deformation (period Λ) (ā = 0.1 μm) over a 100-mm length. The μ = 2 ray period in that strongly undulating index profile is p = 0.96 mm: a, Λ/p ≠ 1; b, Λ/p = 1.

Equations (20)

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d 2 x / d z 2 = - U ( x , y ) / x + C x ( z ) ,
d 2 y / d z 2 = - U ( x , y ) / y + C y ( z ) ,
U ( x , y ) = 1 - n ( x , y ) / n 0 ,
μ = k 0 ( x y ˚ - y x ˚ ) ,
x ˚ = d x / d z ,             y ˚ = d y / d z ,
ψ μ ( r , ϕ , z ) = f ( r ) cos μ ϕ exp ( i β z ) ,
A ( z ) = A 0 sin ( 2 π z / Λ ) ,
C x ( z ) = d 2 A ( z ) / d z 2 ,             C y ( z ) = 0.
C x ( z ) = - ( 4 π 2 Λ 0 / Λ 2 ) sin ( 2 π z / Λ ) .
x 0 = r m cos γ ,
y 0 = r m sin γ ,
x ˚ 0 = R d sin γ ,
y ˚ 0 = - R d cos γ ,
R d = - μ / k 0 r m .
w = r max - r min ,
μ 2 / k 0 2 r m 3 = d U ( r m ) / d r m ,
U ( r ) = U ( r ) + ( ½ ) ( μ 0 / k 0 r ) 2
Λ = Λ 0 / cos α .
ā = F Λ 4 / ( 3 π E l D 4 ) ,
ā = 4 × 10 - 3 μ m .