Abstract

The power-flow equation is approximated by the Fokker-Planck equation that is further transformed into a stochastic differential (Langevin) equation, resulting in an efficient method for the estimation of the state of mode coupling along step-index optical fibers caused by their intrinsic perturbation effects. The inherently stochastic nature of these effects is thus fully recognized mathematically. The numerical integration is based on the computer-simulated Langevin force. The solution matches the solution of the power-flow equation reported previously. Conceptually important steps of this work include (i) the expression of the power-flow equation in a form of the diffusion equation that is known to represent the solution of the stochastic differential equation describing processes with random perturbations and (ii) the recognition that mode coupling in multimode optical fibers is caused by random perturbations.

© 2002 Optical Society of America

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References

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  1. A. F. Garito, J. Wang, R. Gao, “Effects of random perturbations in plastic optical fibers,” Science 281, 962–967 (1998).
    [Crossref] [PubMed]
  2. D. Gloge, “Optical power flow in multimode fibers,” Bell Syst. Tech. J. 51, 1767–1783 (1972).
    [Crossref]
  3. W. A. Gambling, D. N. Payne, H. Matsumura, “Mode conversion coefficients in optical fibers,” Appl. Opt. 14, 1538–1542 (1975).
    [Crossref] [PubMed]
  4. M. Rousseau, L. Jeunhomme, “Numerical solution of the coupled-power equation in step index optical fibers,” IEEE Trans. Microwave Theory Tech. 25, 577–585 (1977).
    [Crossref]
  5. A. Djordjevich, S. Savović, “Investigation of mode coupling in step index plastic optical fibers using the power flow equation,” IEEE Photon. Technol. Lett. 12, 1489–1491 (2000).
    [Crossref]
  6. M. Eve, J. H. Hannay, “Ray theory and random mode coupling in an optical fibre waveguide. I.,” Opt. Quantum Electron. 8, 503–508 (1976).
    [Crossref]
  7. L. Jeunhomme, M. Fraise, J. P. Pocholle, “Propagation model for long step-index optical fibers,” Appl. Opt. 15, 3040–3046 (1976).
    [Crossref] [PubMed]
  8. H. Risken, The Fokker-Planck Equation (Springer-Verlag, Berlin, 1989).
    [Crossref]
  9. G. Jiang, R. F. Shi, A. F. Garito, “Mode coupling and equilibrium mode distribution conditions in plastic optical fibers,” IEEE Photon. Technol. Lett. 9, 1128–1130 (1997).
    [Crossref]

2000 (1)

A. Djordjevich, S. Savović, “Investigation of mode coupling in step index plastic optical fibers using the power flow equation,” IEEE Photon. Technol. Lett. 12, 1489–1491 (2000).
[Crossref]

1998 (1)

A. F. Garito, J. Wang, R. Gao, “Effects of random perturbations in plastic optical fibers,” Science 281, 962–967 (1998).
[Crossref] [PubMed]

1997 (1)

G. Jiang, R. F. Shi, A. F. Garito, “Mode coupling and equilibrium mode distribution conditions in plastic optical fibers,” IEEE Photon. Technol. Lett. 9, 1128–1130 (1997).
[Crossref]

1977 (1)

M. Rousseau, L. Jeunhomme, “Numerical solution of the coupled-power equation in step index optical fibers,” IEEE Trans. Microwave Theory Tech. 25, 577–585 (1977).
[Crossref]

1976 (2)

M. Eve, J. H. Hannay, “Ray theory and random mode coupling in an optical fibre waveguide. I.,” Opt. Quantum Electron. 8, 503–508 (1976).
[Crossref]

L. Jeunhomme, M. Fraise, J. P. Pocholle, “Propagation model for long step-index optical fibers,” Appl. Opt. 15, 3040–3046 (1976).
[Crossref] [PubMed]

1975 (1)

1972 (1)

D. Gloge, “Optical power flow in multimode fibers,” Bell Syst. Tech. J. 51, 1767–1783 (1972).
[Crossref]

Djordjevich, A.

A. Djordjevich, S. Savović, “Investigation of mode coupling in step index plastic optical fibers using the power flow equation,” IEEE Photon. Technol. Lett. 12, 1489–1491 (2000).
[Crossref]

Eve, M.

M. Eve, J. H. Hannay, “Ray theory and random mode coupling in an optical fibre waveguide. I.,” Opt. Quantum Electron. 8, 503–508 (1976).
[Crossref]

Fraise, M.

Gambling, W. A.

Gao, R.

A. F. Garito, J. Wang, R. Gao, “Effects of random perturbations in plastic optical fibers,” Science 281, 962–967 (1998).
[Crossref] [PubMed]

Garito, A. F.

A. F. Garito, J. Wang, R. Gao, “Effects of random perturbations in plastic optical fibers,” Science 281, 962–967 (1998).
[Crossref] [PubMed]

G. Jiang, R. F. Shi, A. F. Garito, “Mode coupling and equilibrium mode distribution conditions in plastic optical fibers,” IEEE Photon. Technol. Lett. 9, 1128–1130 (1997).
[Crossref]

Gloge, D.

D. Gloge, “Optical power flow in multimode fibers,” Bell Syst. Tech. J. 51, 1767–1783 (1972).
[Crossref]

Hannay, J. H.

M. Eve, J. H. Hannay, “Ray theory and random mode coupling in an optical fibre waveguide. I.,” Opt. Quantum Electron. 8, 503–508 (1976).
[Crossref]

Jeunhomme, L.

M. Rousseau, L. Jeunhomme, “Numerical solution of the coupled-power equation in step index optical fibers,” IEEE Trans. Microwave Theory Tech. 25, 577–585 (1977).
[Crossref]

L. Jeunhomme, M. Fraise, J. P. Pocholle, “Propagation model for long step-index optical fibers,” Appl. Opt. 15, 3040–3046 (1976).
[Crossref] [PubMed]

Jiang, G.

G. Jiang, R. F. Shi, A. F. Garito, “Mode coupling and equilibrium mode distribution conditions in plastic optical fibers,” IEEE Photon. Technol. Lett. 9, 1128–1130 (1997).
[Crossref]

Matsumura, H.

Payne, D. N.

Pocholle, J. P.

Risken, H.

H. Risken, The Fokker-Planck Equation (Springer-Verlag, Berlin, 1989).
[Crossref]

Rousseau, M.

M. Rousseau, L. Jeunhomme, “Numerical solution of the coupled-power equation in step index optical fibers,” IEEE Trans. Microwave Theory Tech. 25, 577–585 (1977).
[Crossref]

Savovic, S.

A. Djordjevich, S. Savović, “Investigation of mode coupling in step index plastic optical fibers using the power flow equation,” IEEE Photon. Technol. Lett. 12, 1489–1491 (2000).
[Crossref]

Shi, R. F.

G. Jiang, R. F. Shi, A. F. Garito, “Mode coupling and equilibrium mode distribution conditions in plastic optical fibers,” IEEE Photon. Technol. Lett. 9, 1128–1130 (1997).
[Crossref]

Wang, J.

A. F. Garito, J. Wang, R. Gao, “Effects of random perturbations in plastic optical fibers,” Science 281, 962–967 (1998).
[Crossref] [PubMed]

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

D. Gloge, “Optical power flow in multimode fibers,” Bell Syst. Tech. J. 51, 1767–1783 (1972).
[Crossref]

IEEE Photon. Technol. Lett. (2)

A. Djordjevich, S. Savović, “Investigation of mode coupling in step index plastic optical fibers using the power flow equation,” IEEE Photon. Technol. Lett. 12, 1489–1491 (2000).
[Crossref]

G. Jiang, R. F. Shi, A. F. Garito, “Mode coupling and equilibrium mode distribution conditions in plastic optical fibers,” IEEE Photon. Technol. Lett. 9, 1128–1130 (1997).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

M. Rousseau, L. Jeunhomme, “Numerical solution of the coupled-power equation in step index optical fibers,” IEEE Trans. Microwave Theory Tech. 25, 577–585 (1977).
[Crossref]

Opt. Quantum Electron. (1)

M. Eve, J. H. Hannay, “Ray theory and random mode coupling in an optical fibre waveguide. I.,” Opt. Quantum Electron. 8, 503–508 (1976).
[Crossref]

Science (1)

A. F. Garito, J. Wang, R. Gao, “Effects of random perturbations in plastic optical fibers,” Science 281, 962–967 (1998).
[Crossref] [PubMed]

Other (1)

H. Risken, The Fokker-Planck Equation (Springer-Verlag, Berlin, 1989).
[Crossref]

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Figures (2)

Fig. 1
Fig. 1

Normalized probability distribution calculated with the Fokker-Planck equation for SI plastic optical fiber of length (a) z = 5 m and (b) z = 20 m for input angles θ0 = 0° (solid curve), 5° (dashed curve), 10° (dotted curve) and 15° (dashed-dotted curve) with FWHM = 2°; and using the Langevin equation for input angles θ0 = 0° (open circles), 5° (open squares), 10° (down triangles) and 15° (up triangles).

Fig. 2
Fig. 2

Normalized output angular power distribution calculated with the power-flow equation for SI plastic optical fiber of length (a) z = 5 m and (b) z = 20 m for input angles θ0 = 0° (solid curve), 5° (dashed curve), 10° (dotted curve) and 15° (dashed-dotted curve) with FWHM = 2°.

Equations (14)

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

Pθ, zz=-αθPθ, z+Dθθθ Pθ, zθ,
Pθ, zz=DθPθ, zθ+D 2Pθ, zθ2.
Pθ, zz=-V Pθ, zθ+D 2Pθ, zθ2,
V=1/Mr=1M Vr,
dθdz=h+gΓz,
Γz=0 ΓzΓz=2δz-z,
dθdz=V+DΓz.
dθdz=DΓz.
zn=kn; k=zfN; n=1, 2, , N
θz+k-θz=zz+k Vdz+Dzz+k ΓzdzVk+DΓˆz.
ΓˆΓˆ=b2ω2z=2b2
ΓˆΓˆ=zz+kdz zz+k ΓzΓzdz=2 zz+kdz×zz+k δz-zdz=2k.
θn+1=θn+Vk+Dkωn,
θn+1=Dkωn.

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