Abstract

We propose to characterize optical power transmission in step-index plastic optical fibers by estimating fiber diffusion and attenuation as functions of the propagation angle. We assume that power flow is described by Glogeś differential equation and find a global solution that was fitted to experimental far field patterns registered using a CCD camera as a function of fiber length. The diffusion and attenuation functions obtained describe completely the fiber behavior and thus, along with the power flow equation, can be used to predict the optical power distribution for any condition.

© 2006 Optical Society of America

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References

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  1. G. Jiang, R. F. Shi, and A. F. Garito, "Mode coupling and equilibrium more distribution conditions in plastic optical fibers," IEEE Photon. Technol. Lett. 9,1128-1130 (1997).
    [CrossRef]
  2. W. A. Gambling, D. N. Payne, and H. Matsumura, "Mode conversion coefficients in Optical Fibers," Appl. Opt. 15, 1538-1542 (1975).
    [CrossRef]
  3. J. Zubía, G. Durana, G. Aldabaldetreku, J. Arrúe, M. A. Losada, and M. López-Higuera, "New method to calculate mode conversion coefficients in SI multimode optical fibres," J. Lightwave Technol. 21, 776-781 (2003).
    [CrossRef]
  4. M. A. Losada, I. Garcés, J. Mateo, I. Salinas, J. Lou and J. Zubía "Mode coupling contribution to radiation losses in curvatures for high and low numerical aperture plastic optical fibres," J. Lightwave Technol. 20, 1160-1164 (2002).
    [CrossRef]
  5. R. Olshansky, and S. M. Oaks, "Differential mode attenuation measurements in graded-index fibers," Appl. Opt. 17, 1830-1835 (1978).
    [CrossRef] [PubMed]
  6. T. Ishigure, M. Kano and Y. Koike, "Which is a more serious factor to the bandwidth of GI POF: differential mode attenuation or mode coupling?," J. Lightwave Technol. 18, 959-965 (2000).
    [CrossRef]
  7. S. E. Golowich, W. White, W. A. Reed, and E. Knudsen, "Quantitative estimates of mode coupling and differential modal attenuation in perfluorinated graded-index plastic optical fiber," J. Lightwave Technol. 21, 111-121 (2003).
    [CrossRef]
  8. D. Gloge, "Optical power flow in multimode fibers," Bell Syst. Tech. J. 51, 1767-1783 (1972).
  9. M. Rousseau, and L. Jeunhomme, "Numerical solution of the coupled-power equation in step-index optical fibers," IEEE Trans. Microwave Theory Technol. 25, 577-585 (1977).
    [CrossRef]
  10. L. Jeunhomme, M. Fraise, and J. P. Pocholle, "Propagation model for long step-index optical fibers," Appl. Opt. 15, 3040-3046 (1976).
    [CrossRef] [PubMed]
  11. A. Djordjevich, and S. Savovic, "Investigation of mode coupling in step index plastic optical fibers using the power flow Equation," IEEE Photon. Technol. Lett. 12, 1489-1491 (2000).
    [CrossRef]
  12. A. Djordjevich, and S. Savovic, "Numerical solution of the power flow equation in step-index plastic optical fibers," J. Opt. Soc. Am. B 21, 1437-1438 (2004).
    [CrossRef]
  13. S. Savovic, and A. Djordjevich, "Optical power flow in plastic-clad silica fibers," Appl. Opt. 41, 7588-7591 (2002).
    [CrossRef]
  14. N. Hashizume, E. Okugaki, S. Suyama, and M. Tatsutsuke, "Far field pattern measurement of POF in the presence of speckle noise," in Proceedings of the International Conference on Plastic Optical Fibers and Application, XII ed., Seattle, USA (2003).
  15. M. A. Losada, J Mateo, D. Espinosa, I. Garcés and J. Zubia, "Characterisation of the far field pattern for plastic optical fibres," in Procceedings of the International Conference on Plastic Optic Fibres and Application, XIII ed., Nuremberg, Germany, (2004), pp. 458-465.
  16. R. D. Skeel, and M. Berzins, "A Method for the Spatial Discretization of Parabolic Equations in One Space Variable," SIAM J. Sci. Stat. Comp. 11, 1-32 (1990).
    [CrossRef]
  17. R. M. Lewis, and V. Torczon, "Pattern Search Algorithms for Bound Constrained Minimization," SIAM J. on Optimization 9, 1082-1099 (1999).
    [CrossRef]
  18. M. A. Losada, J. Mateo, I. Garcés, J. Zubía, J. A. Casao, and P. Pérez-Vela, "Analysis of strained plastic optical fibres," IEEE Photon. Technol. Lett. 16, 1513-1515 (2004).
    [CrossRef]

2004 (2)

A. Djordjevich, and S. Savovic, "Numerical solution of the power flow equation in step-index plastic optical fibers," J. Opt. Soc. Am. B 21, 1437-1438 (2004).
[CrossRef]

M. A. Losada, J. Mateo, I. Garcés, J. Zubía, J. A. Casao, and P. Pérez-Vela, "Analysis of strained plastic optical fibres," IEEE Photon. Technol. Lett. 16, 1513-1515 (2004).
[CrossRef]

2003 (2)

2002 (2)

2000 (2)

T. Ishigure, M. Kano and Y. Koike, "Which is a more serious factor to the bandwidth of GI POF: differential mode attenuation or mode coupling?," J. Lightwave Technol. 18, 959-965 (2000).
[CrossRef]

A. Djordjevich, and S. Savovic, "Investigation of mode coupling in step index plastic optical fibers using the power flow Equation," IEEE Photon. Technol. Lett. 12, 1489-1491 (2000).
[CrossRef]

1999 (1)

R. M. Lewis, and V. Torczon, "Pattern Search Algorithms for Bound Constrained Minimization," SIAM J. on Optimization 9, 1082-1099 (1999).
[CrossRef]

1997 (1)

G. Jiang, R. F. Shi, and A. F. Garito, "Mode coupling and equilibrium more distribution conditions in plastic optical fibers," IEEE Photon. Technol. Lett. 9,1128-1130 (1997).
[CrossRef]

1990 (1)

R. D. Skeel, and M. Berzins, "A Method for the Spatial Discretization of Parabolic Equations in One Space Variable," SIAM J. Sci. Stat. Comp. 11, 1-32 (1990).
[CrossRef]

1978 (1)

1977 (1)

M. Rousseau, and L. Jeunhomme, "Numerical solution of the coupled-power equation in step-index optical fibers," IEEE Trans. Microwave Theory Technol. 25, 577-585 (1977).
[CrossRef]

1976 (1)

1975 (1)

W. A. Gambling, D. N. Payne, and H. Matsumura, "Mode conversion coefficients in Optical Fibers," Appl. Opt. 15, 1538-1542 (1975).
[CrossRef]

1972 (1)

D. Gloge, "Optical power flow in multimode fibers," Bell Syst. Tech. J. 51, 1767-1783 (1972).

Aldabaldetreku, G.

Arrúe, J.

Berzins, M.

R. D. Skeel, and M. Berzins, "A Method for the Spatial Discretization of Parabolic Equations in One Space Variable," SIAM J. Sci. Stat. Comp. 11, 1-32 (1990).
[CrossRef]

Casao, J. A.

M. A. Losada, J. Mateo, I. Garcés, J. Zubía, J. A. Casao, and P. Pérez-Vela, "Analysis of strained plastic optical fibres," IEEE Photon. Technol. Lett. 16, 1513-1515 (2004).
[CrossRef]

Djordjevich, A.

Durana, G.

Fraise, M.

Gambling, W. A.

W. A. Gambling, D. N. Payne, and H. Matsumura, "Mode conversion coefficients in Optical Fibers," Appl. Opt. 15, 1538-1542 (1975).
[CrossRef]

Garcés, I.

M. A. Losada, J. Mateo, I. Garcés, J. Zubía, J. A. Casao, and P. Pérez-Vela, "Analysis of strained plastic optical fibres," IEEE Photon. Technol. Lett. 16, 1513-1515 (2004).
[CrossRef]

M. A. Losada, I. Garcés, J. Mateo, I. Salinas, J. Lou and J. Zubía "Mode coupling contribution to radiation losses in curvatures for high and low numerical aperture plastic optical fibres," J. Lightwave Technol. 20, 1160-1164 (2002).
[CrossRef]

Garito, A. F.

G. Jiang, R. F. Shi, and A. F. Garito, "Mode coupling and equilibrium more 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).

Golowich, S. E.

Ishigure, T.

Jeunhomme, L.

M. Rousseau, and L. Jeunhomme, "Numerical solution of the coupled-power equation in step-index optical fibers," IEEE Trans. Microwave Theory Technol. 25, 577-585 (1977).
[CrossRef]

L. Jeunhomme, M. Fraise, and 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, and A. F. Garito, "Mode coupling and equilibrium more distribution conditions in plastic optical fibers," IEEE Photon. Technol. Lett. 9,1128-1130 (1997).
[CrossRef]

Kano, M.

Knudsen, E.

Koike, Y.

Lewis, R. M.

R. M. Lewis, and V. Torczon, "Pattern Search Algorithms for Bound Constrained Minimization," SIAM J. on Optimization 9, 1082-1099 (1999).
[CrossRef]

López-Higuera, M.

Losada, M. A.

Lou, J.

Mateo, J.

M. A. Losada, J. Mateo, I. Garcés, J. Zubía, J. A. Casao, and P. Pérez-Vela, "Analysis of strained plastic optical fibres," IEEE Photon. Technol. Lett. 16, 1513-1515 (2004).
[CrossRef]

M. A. Losada, I. Garcés, J. Mateo, I. Salinas, J. Lou and J. Zubía "Mode coupling contribution to radiation losses in curvatures for high and low numerical aperture plastic optical fibres," J. Lightwave Technol. 20, 1160-1164 (2002).
[CrossRef]

Matsumura, H.

W. A. Gambling, D. N. Payne, and H. Matsumura, "Mode conversion coefficients in Optical Fibers," Appl. Opt. 15, 1538-1542 (1975).
[CrossRef]

Oaks, S. M.

Olshansky, R.

Payne, D. N.

W. A. Gambling, D. N. Payne, and H. Matsumura, "Mode conversion coefficients in Optical Fibers," Appl. Opt. 15, 1538-1542 (1975).
[CrossRef]

Pérez-Vela, P.

M. A. Losada, J. Mateo, I. Garcés, J. Zubía, J. A. Casao, and P. Pérez-Vela, "Analysis of strained plastic optical fibres," IEEE Photon. Technol. Lett. 16, 1513-1515 (2004).
[CrossRef]

Pocholle, J. P.

Reed, W. A.

Rousseau, M.

M. Rousseau, and L. Jeunhomme, "Numerical solution of the coupled-power equation in step-index optical fibers," IEEE Trans. Microwave Theory Technol. 25, 577-585 (1977).
[CrossRef]

Salinas, I.

Savovic, S.

Shi, R. F.

G. Jiang, R. F. Shi, and A. F. Garito, "Mode coupling and equilibrium more distribution conditions in plastic optical fibers," IEEE Photon. Technol. Lett. 9,1128-1130 (1997).
[CrossRef]

Skeel, R. D.

R. D. Skeel, and M. Berzins, "A Method for the Spatial Discretization of Parabolic Equations in One Space Variable," SIAM J. Sci. Stat. Comp. 11, 1-32 (1990).
[CrossRef]

Torczon, V.

R. M. Lewis, and V. Torczon, "Pattern Search Algorithms for Bound Constrained Minimization," SIAM J. on Optimization 9, 1082-1099 (1999).
[CrossRef]

White, W.

Zubía, J.

Appl. Opt. (4)

Bell Syst. Tech. J. (1)

D. Gloge, "Optical power flow in multimode fibers," Bell Syst. Tech. J. 51, 1767-1783 (1972).

IEEE Photon. Technol. Lett. (3)

A. Djordjevich, and S. Savovic, "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, and A. F. Garito, "Mode coupling and equilibrium more distribution conditions in plastic optical fibers," IEEE Photon. Technol. Lett. 9,1128-1130 (1997).
[CrossRef]

M. A. Losada, J. Mateo, I. Garcés, J. Zubía, J. A. Casao, and P. Pérez-Vela, "Analysis of strained plastic optical fibres," IEEE Photon. Technol. Lett. 16, 1513-1515 (2004).
[CrossRef]

IEEE Trans. Microwave Theory Technol. (1)

M. Rousseau, and L. Jeunhomme, "Numerical solution of the coupled-power equation in step-index optical fibers," IEEE Trans. Microwave Theory Technol. 25, 577-585 (1977).
[CrossRef]

J. Lightwave Technol. (4)

J. Opt. Soc. Am. B (1)

SIAM J. on Optimization (1)

R. M. Lewis, and V. Torczon, "Pattern Search Algorithms for Bound Constrained Minimization," SIAM J. on Optimization 9, 1082-1099 (1999).
[CrossRef]

SIAM J. Sci. Stat. Comp. (1)

R. D. Skeel, and M. Berzins, "A Method for the Spatial Discretization of Parabolic Equations in One Space Variable," SIAM J. Sci. Stat. Comp. 11, 1-32 (1990).
[CrossRef]

Other (2)

N. Hashizume, E. Okugaki, S. Suyama, and M. Tatsutsuke, "Far field pattern measurement of POF in the presence of speckle noise," in Proceedings of the International Conference on Plastic Optical Fibers and Application, XII ed., Seattle, USA (2003).

M. A. Losada, J Mateo, D. Espinosa, I. Garcés and J. Zubia, "Characterisation of the far field pattern for plastic optical fibres," in Procceedings of the International Conference on Plastic Optic Fibres and Application, XIII ed., Nuremberg, Germany, (2004), pp. 458-465.

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Figures (5)

Fig. 1.
Fig. 1.

(a). Scheme of the set up. (b) Screen of the custom program to acquire the FFP and radial profiles. (c) Picture of the set up showing the FFP over the screen.

Fig. 2.
Fig. 2.

Images of the far field pattern for the GH fiber at 10m, 50m and 150m acquired with the CCD and their corresponding calculated radial profiles.

Fig. 3.
Fig. 3.

Fig. 3. Total power versus length and SSD radial profiles for the three tested fibers: GH data is shown as circles and (blue) lines; HFB as squares and (green) lines, and PGU as diamonds and (red) lines. In Fig. 3(b) data symbols represent the raw data from the radial profiles and the lines give the best-fit to Eq. (6).

Fig. 4.
Fig. 4.

Experimental radial profiles (blue lines) for the GH fiber at 10, 30, 50, 75, 100 and 150m are shown in both graphs along with those predicted by both models (red lines).

Fig. 5.
Fig. 5.

Diffusion and attenuation functions for the three fibers obtained by modeling diffusion with the sigmoid function given by Eq. (9). GH fiber results are shown as blue solid lines, HFB fiber results as green dashed lines and PGU fiber results as red dash-dotted lines.

Tables (2)

Tables Icon

Table 1. Attenuation γ and QN (θ) parameters for the best fits to the SSD of the three fibers

Tables Icon

Table 2. Parameters for the constant diffusion and sigmoid diffusion functions that minimizes the error between experimental and model-predicted far field profiles.

Equations (7)

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P ( θ , z ) z = α ( θ ) P ( θ , z ) + 1 θ θ ( θ D ( θ ) P ( θ , z ) θ ) .
P SSD = Q ( θ ) e γ z .
α ( θ ) = γ + 1 Q ( θ ) θ θ ( θ D ( θ ) Q ( θ ) θ ) .
P ( θ , z ) θ θ = 0 = 0 P ( θ π 2 , z ) = 0 .
P T ( z ) = Ω f P ( θ , z ) d Ω = 0 2 π d φ 0 π 2 sin ( θ ) P ( θ , z ) d θ = 2 π 0 π 2 sin ( θ ) P ( θ , z ) d θ .
Q N ( θ ) = ( 1 + e σ 1 2 θ 1 2 ) ( 1 + e σ 2 2 θ 2 2 ) ( 1 + e σ 1 2 ( θ 1 2 θ 2 ) ) ( 1 + e σ 2 2 ( θ 2 2 θ 2 ) ) .
D ( θ ) = D 0 + D 1 1 + D 2 e σ d 2 θ 2 ,

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