## Global characterization of optical power propagation in step-index plastic optical fibers

Optics Express, Vol. 14, Issue 20, pp. 9028-9035 (2006)

http://dx.doi.org/10.1364/OE.14.009028

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### 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

## 1. Introduction

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]

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]

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]

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]

*θ*). The diffusion equation can be solved analytically only in some particular cases [1

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]

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]

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]

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]

*D*(

*θ*), and

*α*(

*θ*), respectively, and for which a global solution is found.

*D*(

*θ*), with a few free parameters whose values are determined by minimizing the error between the model predictions and the experimental FFPs. Even more, we make no assumptions over the angular attenuation,

*α*(

*θ*), which is calculated directly from the SSD and from the estimated diffusion function. We found that our estimated functions

*D*(

*θ*) and

*α*(

*θ*) jointly with Gloge’s power flow equation are able to reproduce the measured FFPs for the three fibers.

## 2. Theoretical modeling

*P*(

*θ, z*), in a multimode fiber as a function of fiber length (

*z*) and of propagation inner angle with respect to fiber axis (

*θ*) is the following:

*α*(

*θ*) and by next-neighbor power diffusion described by

*D*(

*θ*). For large

*z*values, when the angular power distribution has reached its SSD, the solution of the equation can be expressed as the product of two functions of independent variables:

*Q*(

*θ*) describes the shape of the SSD profile that depends only on the propagation angle, while the dependence on fiber length

*z*is given by a decreasing exponential function which accounts for the power decrease due to the fiber attenuationγ. Introducing this solution into Eq. (1), an equation can be obtained relating:

*α*(

*θ*),

*Q*(

*θ*) and

*D*(

*θ*). Thus,

*α*(

*θ*)can be expressed in terms of the others as follows:

*γ,Q*(

*θ*)and

*D*(

*θ*) are known. On the other hand, although fiber diffusion has usually been modeled by a constant value in POFs [9

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]

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]

*θ*=0 and its physical meaning is that there is no diffusion for the straight rays. The second condition indicates that there is no power propagating at high angles and, in most works, it was imposed for angles greater than the critical angle [11

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]

## 3. Experimental method to acquire FFP images and radial profiles

## 4. Fiber characterization method and results

*γ,Q*(

*θ*) and

*D*(

*θ*) are known. Thus, in the first subsection we describe how we estimate

*γ*directly from our experimental data. In the second subsection, the experimental SSDs are fitted by a sigmoid-like function of the squared propagation angle to obtain an analytical form of

*Q*(

*θ*) and its derivatives. In the last subsection, we estimate fiber diffusion by modeling

*D*(

*θ*)both by a constant and a sigmoid function.

### 4.1 Estimation of attenuation γ

### 4.2 Bi-sigmoid fits to the steady state profiles

*Q*(

*θ*) and for its derivatives. Thus, the normalized SSDs for the three fibers were fitted by a product of two sigmoid functions of the squared inner angle, given by:

### 4.3 Determination of the diffusion function

*D*(

*θ*), we solve numerically Eq. (1) [16

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]

*D*(

*θ*). The final estimate of

*D*(

*θ*) is the one that minimize the RMSE between the measured FFPs and the model predictions. We use a direct search pattern method [17

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

*D*(

*θ*) is introduced in Eq. (3) to calculate the corresponding attenuation function

*α*(

*θ*).

*D*as it is usually assumed [9

_{c}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]

**21**, 1437–1438 (2004). [CrossRef]

*D*and the RMSE for the best fit for the three fibers. Fig. 4(a) shows the radial profiles for the GH fiber at several lengths. The comparison between the measured and the predicted profiles shows how, in the intermediate lengths, the power at small angles is underestimated, while at larger angles power it is overestimated suggesting that diffusion should be higher at small angles than at larger ones. This fact justifies our proposal to model fiber diffusion by a function of propagation angle. We chose a sigmoid function of the squared inner angle because of its mathematical properties, although other similar function could be used. This function is given by the following expression:

_{c}*D*

_{0},

*D*

_{1},

*D*

_{2}, and

*σ*are free parameters of our model. The value of this function at small angles tends to

_{d}*D*

_{0}+

*D*

_{1}/(1+

*D*

_{2}) while for larger angles tends to

*D*

_{0}. The position and magnitude of the slope are governed jointly by

*D*

_{2}and

*σ*. The model predictions for the GH fiber are shown in Fig. 4(b) displaying a considerably better agreement than was found previously for a constant diffusion model.

_{d}### 5. Discussion

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

**25**, 577–585 (1977). [CrossRef]

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]

*γ*, although there are also some ripples that could be explained by the different behavior of the fibers under the curvatures of the reel [4

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]

*γ*is quite higher than its nominal attenuation and also this fiber exhibits the highest diffusion. This behavior suggests that the tested PGU sample was probably strained. In fact, it was shown [18

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]

## 5. Conclusion

## Acknowledgments

## References and links

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. |

2. | W. A. Gambling, D. N. Payne, and H. Matsumura, “Mode conversion coefficients in Optical Fibers,” Appl. Opt. |

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. |

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. |

5. | R. Olshansky and S. M. Oaks, “Differential mode attenuation measurements in graded-index fibers,” Appl. Opt. |

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. |

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. |

8. | D. Gloge, “Optical power flow in multimode fibers,” Bell Syst. Tech. J. |

9. | M. Rousseau and L. Jeunhomme, “Numerical solution of the coupled-power equation in step-index optical fibers,” IEEE Trans. Microwave Theory Technol. |

10. | L. Jeunhomme, M. Fraise, and J. P. Pocholle, “Propagation model for long step-index optical fibers,” Appl. Opt. |

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. | A. Djordjevich and S. Savovic, “Numerical solution of the power flow equation in step-index plastic optical fibers,” J. Opt. Soc. Am. B |

13. | S. Savovic and A. Djordjevich, “Optical power flow in plastic-clad silica fibers,” Appl. Opt. |

14. | N. Hashizume, E. Okugaki, S. Suyama, and M. Tatsutsuke, “Far field pattern measurement of POF in the presence of speckle noise,” in |

15. | M. A. Losada, J Mateo, D. Espinosa, I. Garcés, and J. Zubia, “Characterisation of the far field pattern for plastic optical fibres,” in |

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. |

17. | R. M. Lewis and V. Torczon, “Pattern Search Algorithms for Bound Constrained Minimization,” SIAM J. on Optimization |

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. |

**OCIS Codes**

(060.0060) Fiber optics and optical communications : Fiber optics and optical communications

(060.2270) Fiber optics and optical communications : Fiber characterization

(060.2300) Fiber optics and optical communications : Fiber measurements

(060.2310) Fiber optics and optical communications : Fiber optics

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: May 31, 2006

Revised Manuscript: September 6, 2006

Manuscript Accepted: September 11, 2006

Published: October 2, 2006

**Citation**

Javier Mateo, M. Angeles Losada, Ignacio Garcés, and Joseba Zubia, "Global characterization of optical power propagation in step-index plastic optical fibers," Opt. Express **14**, 9028-9035 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9028

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### References

- 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]
- W. A. Gambling, D. N. Payne, and H. Matsumura, "Mode conversion coefficients in Optical Fibers," Appl. Opt. 15, 1538-1542 (1975). [CrossRef]
- 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]
- 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]
- R. Olshansky, and S. M. Oaks, "Differential mode attenuation measurements in graded-index fibers," Appl. Opt. 17, 1830-1835 (1978). [CrossRef] [PubMed]
- 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]
- 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]
- D. Gloge, "Optical power flow in multimode fibers," Bell Syst. Tech. J. 51, 1767-1783 (1972).
- 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]
- 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]
- 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]
- S. Savovic, and A. Djordjevich, "Optical power flow in plastic-clad silica fibers," Appl. Opt. 41, 7588-7591 (2002). [CrossRef]
- 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.
- 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]
- R. M. Lewis, and V. Torczon, "Pattern Search Algorithms for Bound Constrained Minimization," SIAM J. on Optimization 9, 1082-1099 (1999). [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]

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