Abstract

We present a method to obtain the frequency response of step index (SI) plastic optical fibers (POFs) based on the power flow equation generalized to incorporate the temporal dimension where the fibre diffusion and attenuation are functions of the propagation angle. To solve this equation we propose a fast implementation of the finite-difference method in matrix form. Our method is validated by comparing model predictions to experimental data. In addition, the model provides the space-time evolution of the angular power distribution when it is transmitted throughout the fibre which gives a detailed picture of the POFs capabilities for information transmission. Model predictions show that angular diffusion has a strong impact on temporal pulse widening with propagation.

© 2009 Optical Society of America

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References

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  1. D. Kalymnios, "Squeezing more bandwidth into high NA POF," in 8th Intl. Conf. on Plastic Optical Fibres and Applications, pp. 18-24 (1999).
  2. J. Mateo, M. A. Losada, I. Garcés, J. Arrúe, J. Zubia, and D. Kalymnios, "High NA POF dependence of bandwidth on fibre length," in 12th Intl. Conf. on Plastic Optical Fibres and Applications, pp. 123-126 (2003).
  3. J. Mateo, M. A. Losada, and I. Garcés, "Global characterization of optical power propagation in step-index plastic optical fibers," Opt. Express 14, 9028-9035 (2006).
    [CrossRef] [PubMed]
  4. M. A. Losada, J. Mateo, I. Garcés, and J. Zubia, "Estimation of the attenuation and diffusion functions in plastic optical fibers from experimental far field patterns," in 15th Intl. Conf. on Plastic Optical Fibres and Applications, pp. 336-341 (2006).
  5. D. Gloge, "Impulse response of clad optical multimode fibers," Bell Syst. Tech. J. 52, 801-816 (1973).
  6. F. Breyer, N. Hanik, S. Lee, and S. Randel, POF Modelling: Theory, Measurement and Application, C. A. Bunge, H. Poisel (Ed.), chap. Getting the Impulse Response of SI-POF by Solving the Time-Dependent Power-Flow Equation using the Crank-Nicholson Scheme (Verlag Books on Demand GmbH, Norderstedt, 2007).
  7. M. Rousseau and L. Jeunhomme, "Numerical solution of coupled-power equation in step-index optical fibers," IEEE Trans. Microwave Theory Tech. 25, 577-585 (1977).
    [CrossRef]
  8. J. Mateo, M. A. Losada, J. J. Martínez-Muro, I. Garcés, and J. Zubia, "Bandwidth measurement in POF based on general purpose equipment," in 14th Intl. Conf. on Plastic Optical Fibres and Applications, pp. 53-56 (2005).
  9. M. A. Losada, J. Mateo, D. Espinosa, I. Garcés, and J. Zubia, "Characterisation of the far field pattern for plastic optical fibres," in 13th Intl. Conf. on Plastic Optical Fibres and Applications, pp. 458-465 (2004).
  10. O. Ziemann, J. Krauser, P. E. Zamzow, and W. Daum, POF Handbook, 2nd ed. (Springer, 2008).
  11. M. A. Losada, J. Mateo, and L. Serena, "Analysis of Propagation Properties of Step Index Plastic Optical Fibers at Non-Stationary Conditions," in 16th Intl. Conf. on Plastic Optical Fibres and Applications, pp. 299-302 (2007).
  12. P. Heredia, J. Mateo, and M. A. Losada, "Transmission capabilities of large-core GI-POF based on BER measurements," in 16th Intl. Conf. on Plastic Optical Fibres and Applications, pp. 307-310 (2007).
  13. M. A. Losada, J. Mateo, J. J. Martínez, and A. López, "SI-POF frequency response obtained by solving the power flow equation," in 17th Intl. Conf. on Plastic Optical Fibres and Applications (2008).

2006 (1)

1977 (1)

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

1973 (1)

D. Gloge, "Impulse response of clad optical multimode fibers," Bell Syst. Tech. J. 52, 801-816 (1973).

Garcés, I.

Gloge, D.

D. Gloge, "Impulse response of clad optical multimode fibers," Bell Syst. Tech. J. 52, 801-816 (1973).

Jeunhomme, L.

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

Losada, M. A.

Mateo, J.

Rousseau, M.

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

Bell Syst. Tech. J. (1)

D. Gloge, "Impulse response of clad optical multimode fibers," Bell Syst. Tech. J. 52, 801-816 (1973).

IEEE Trans. Microwave Theory Tech. (1)

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

Opt. Express (1)

Other (10)

M. A. Losada, J. Mateo, I. Garcés, and J. Zubia, "Estimation of the attenuation and diffusion functions in plastic optical fibers from experimental far field patterns," in 15th Intl. Conf. on Plastic Optical Fibres and Applications, pp. 336-341 (2006).

F. Breyer, N. Hanik, S. Lee, and S. Randel, POF Modelling: Theory, Measurement and Application, C. A. Bunge, H. Poisel (Ed.), chap. Getting the Impulse Response of SI-POF by Solving the Time-Dependent Power-Flow Equation using the Crank-Nicholson Scheme (Verlag Books on Demand GmbH, Norderstedt, 2007).

J. Mateo, M. A. Losada, J. J. Martínez-Muro, I. Garcés, and J. Zubia, "Bandwidth measurement in POF based on general purpose equipment," in 14th Intl. Conf. on Plastic Optical Fibres and Applications, pp. 53-56 (2005).

M. A. Losada, J. Mateo, D. Espinosa, I. Garcés, and J. Zubia, "Characterisation of the far field pattern for plastic optical fibres," in 13th Intl. Conf. on Plastic Optical Fibres and Applications, pp. 458-465 (2004).

O. Ziemann, J. Krauser, P. E. Zamzow, and W. Daum, POF Handbook, 2nd ed. (Springer, 2008).

M. A. Losada, J. Mateo, and L. Serena, "Analysis of Propagation Properties of Step Index Plastic Optical Fibers at Non-Stationary Conditions," in 16th Intl. Conf. on Plastic Optical Fibres and Applications, pp. 299-302 (2007).

P. Heredia, J. Mateo, and M. A. Losada, "Transmission capabilities of large-core GI-POF based on BER measurements," in 16th Intl. Conf. on Plastic Optical Fibres and Applications, pp. 307-310 (2007).

M. A. Losada, J. Mateo, J. J. Martínez, and A. López, "SI-POF frequency response obtained by solving the power flow equation," in 17th Intl. Conf. on Plastic Optical Fibres and Applications (2008).

D. Kalymnios, "Squeezing more bandwidth into high NA POF," in 8th Intl. Conf. on Plastic Optical Fibres and Applications, pp. 18-24 (1999).

J. Mateo, M. A. Losada, I. Garcés, J. Arrúe, J. Zubia, and D. Kalymnios, "High NA POF dependence of bandwidth on fibre length," in 12th Intl. Conf. on Plastic Optical Fibres and Applications, pp. 123-126 (2003).

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Figures (4)

Fig. 1.
Fig. 1.

Frequency responses for one of the samples of the HFB fiber at three lengths (15m on the left, 50m in the center, and 100m on the right). Experimental measurements are shown in black dots. Gaussian functions with the same bandwidths as the corresponding experimental curve are shown as red lines. Model predictions without the injection matrix are shown as green lines, while the blue lines show the predictions with injection matrix.

Fig. 2.
Fig. 2.

Image representation of the injection matrix for the GH, HFB and PGU fibers (left, center and right graphs) obtained from our previous measurements in [11]. Horizontal and vertical axis are input and output angles respectively.

Fig. 3.
Fig. 3.

Bandwidth versus fibre length for the three fibre types: GH on the left, HFB in the centre and PGU on the right. Symbols of different colours represent data for different samples of the same fibre.The blue lines show the model predictions extended up to 750 m.

Fig. 4.
Fig. 4.

The graph on the upper left is the image representation of the space-time power distribution at the output of 150 m of the PGU fiber. Below, the lower leftmost graph shows the overall pulse spread obtained as the integral of power over output angle. The power integral over time renders the radial profile of the FFP shown on the upper rightmost graph.

Equations (23)

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P ( θ , z , t ) z + t z P ( θ , z , t ) t = α ( θ ) P ( θ , z , t ) + 1 θ θ ( θ · d ( θ ) · P ( θ , z , t ) θ )
P ( θ , z , t ) z = α ( θ ) P ( θ , z , t ) n c cos θ · P ( θ , z , t ) t + 1 θ θ ( θ · d ( θ ) · P ( θ , z , t ) θ ) .
P ( θ , z , ω ) z = ( α ( θ ) + n c cos θ · j ω ) p ( θ , z , ω ) + 1 θ θ ( θ · d ( θ ) · P ( θ , z , ω ) θ ) ,
p ( θ , z + Δ z , ω ) = ( 1 ( α ( θ ) + n c cos θ · j ω ) z ) p ( θ , z , ω )
+ Δ z 2 · Δ θ ( d ( θ ) θ + d ( θ ) ) ( p ( θ + Δ θ , z , ω ) p ( θ Δ θ , z , ω ) )
2 d ( θ ) Δ z Δ θ 2 p ( θ , z , ω )
+ d ( θ ) Δ z Δ θ 2 ( p ( θ + Δ θ , z , ω ) + p ( θ Δ θ , z , ω ) ) .
p ( z 2 , ω ) = ( A ( ω ) + D ) m · p ( z 1 , ω ) ,
A k , k ( ω ) 1 Δ z · α ( k · Δ θ ) Δ z · n c cos ( k · Δθ ) · j ω
A k , k ( ω ) = exp ( Δ z · α ( k · Δ θ ) Δ z · n c cos ( k · Δθ ) · j ω ) ,
D k , k 1 = ( d ( k · Δ θ ) 1 2 d ( k · Δ θ ) k 1 2 d ( k · Δ θ ) Δ θ ) Δ z Δ θ 2
D k , k = 2 d ( k · Δ θ ) Δ z Δ θ 2
D k , k + 1 = ( d ( k · Δ θ ) + 1 2 d ( k · Δ θ ) k + 1 2 d ( k · Δ θ ) Δ θ ) Δ z Δ θ 2 .
1 θ θ ∂θ ( θ · d ( θ ) · P ( θ , z , t ) θ ) = d ' ( θ ) P ( θ , z ) θ + d ( θ ) θ θ θ ( θ · P ( θ , z ) θ )
lim θ 0 ( 1 θ θ θ ( θ · d ( θ ) · P ( θ , z , t ) θ ) ) = d ( 0 ) P ( θ , z ) θ | θ = 0 + 2 d ( 0 ) 2 p ( θ , z ) θ θ | θ = 0
lim θ 0 ( 1 θ θ θ ( θ · d ( θ ) · P ( θ , z , t ) θ ) ) = d ' ( 0 ) p ( 0 + Δ θ , z ) p ( 0 Δ θ , z ) 2 · Δ θ
+ 2 d ( 0 ) p ( 0 + Δ θ , z ) 2 p ( 0 , z ) + p ( 0 Δ θ , z ) Δ θ 2 .
lim θ 0 ( 1 θ θ θ ( θ · d ( θ ) · P ( θ , z , t ) θ ) ) = 4 d ( 0 ) p ( 0 + Δ θ , z ) p ( 0 , z ) Δ θ 2 ,
D 0,0 = 4 d ( 0 ) Δ z Δ θ 2 D 0,1 = 4 d ( 0 ) Δ z Δ θ 2 .
D N , N 1 = 2 d ( N ) Δ z Δ θ 2 D N , N = 2 d ( N ) Δ z Δ θ 2 .
H ( L , ω ) = 0 π / 2 sin ( θ ) p ( θ , L , ω ) d θ .
p ( z = 0 + ) = J · p ( z = 0 ) .
δ ( θ ) = L n c ( 1 cos ( θ ) 1 ) ,

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