## Analytical theory for polarization mode dispersion of spun and twisted fiber

Optics Express, Vol. 11, Issue 19, pp. 2403-2410 (2003)

http://dx.doi.org/10.1364/OE.11.002403

Acrobat PDF (168 KB)

### Abstract

We derive an analytical expression for differential group delay of spun and twisted fibers, which should provide valuable guidance for optimization of such parameters to produce low polarization mode dispersion fiber.

© 2003 Optical Society of America

## 1. Introduction

2. B. W. Hakki, “Polarization mode dispersion compensation by phase diversity detection,” IEEE Photon. Technol. Lett. , **9**, 121–123 (1997). [CrossRef]

4. M. J. Li and D. A. Nolan, “Fiber spin-profile designs for producing fibers with low polarization mode dispersion,” Opt. Lett. , **23**, 1659–1661 (1998). [CrossRef]

5. R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, and D. H. O. Bebbington, “Theoretical analysis and measurement of effects of fiber twist on polarization mode dispersion of optical fibers,” Electron. Lett. , **31**, 1772–1773 (1995). [CrossRef]

4. M. J. Li and D. A. Nolan, “Fiber spin-profile designs for producing fibers with low polarization mode dispersion,” Opt. Lett. , **23**, 1659–1661 (1998). [CrossRef]

5. R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, and D. H. O. Bebbington, “Theoretical analysis and measurement of effects of fiber twist on polarization mode dispersion of optical fibers,” Electron. Lett. , **31**, 1772–1773 (1995). [CrossRef]

*et al.*[6] derived an approximate analytical expression of PMD for spun fibers by using the coupled-mode theory, and they ignored twist. However, twist differs from spin inasmuch as it introduces an additional circular birefringence on fibers owing to the photoelastic effect. Beginning with the dynamic polarization dispersion vector (PDV) equation [7], an analytical expression for PMD of fiber with arbitrary spin and twist rates is derived based on perturbation theory. To test the validity of our analytical model, we compared the analytical results with the numerical simulations using various spin and twist profiles. The results show that a maximum PMD reduction can be achieved with optimal periodic spin profiles. In this case, the PMD along the fiber is periodic. For fiber twisting, a particular twist rate exists that can maximally reduce PMD for a certain amount of fiber birefringence and length. However, if the twist rate were large, the PMD would increase linearly.

## 2. Theoretical Background

*z*represents the position along the fiber,

*ω*represents the angular frequency, and

*z*. Note that Eq. (1) is written in a fixed reference frame.

*z*,

*ω*) of an unspun fiber with just linear birefringence can be assumed to be

*z*independent with magnitude

*β*

_{l}(

*ω*) and can be expressed as [

*β*

_{l}(

*ω*), 0, 0]. From Eq. (1) we can easily obtain the analytical result of the PDV of a linear birefringence fiber:

*z*,

*ω*)=[

*β*

_{ω}

*z*,0, 0], where

*β*

_{l}(

*ω*) with respect to

*ω*. When frequency dependence properties of the mode field are omitted,

*β*

_{ω}can be approximated by

*z*,

*ω*), i.e., the DGD Δ

*τ*along fibers, can be expressed as Δ

*τ*=

*β*

_{ω}

*z*. It is straightforward to observe that DGD of unspun fibers grows linearly with the fiber length.

*a*(

*z*) and externally applied twist rate

*γ*are added to a fiber with initial linear birefringence

*β*

_{l}, the LBV can be expressed as [8]

*g*is the rotation coefficient. It is worthwhile to note that twist differs from spin because spin is applied to the fiber by oscillation during the drawing operation, whereas twist is applied when the fiber is already solidified and cool, which results in an additional circular birefringence

*gγ*. The LBV can be simplified by rotation of the reference frame (on the Poincaré sphere) at the rate of 2

*a*′(

*z*)+2

*γ*. This introduces an additional apparent circular birefringence

*β*

_{rc}=2

*a*′(

*z*)+2

*γ*. The LBV is transformed as follows:

*gω*denotes differentiation of

*g*with respect to

*ω*and Ω

_{k}(

*k*=1, 2, 3) are the components of

*β*

_{l}≪1, i.e., assuming the intrinsic linear birefringence is a small parameter, the perturbation theory can be applied to solve Eq. (4) approximately. Here we omit the mathematical details and present just the analytical results as

*V*

^{T}represents the transpose of the

*V*matrix. From the results it is straightforward to observe that the spin does not contribute to Ω

_{3}, which is determined only by twist and thus grows linearly. Consequently, we can easily obtain the DGD of the spun and twisted fiber as

*φ*=

*β*

_{ω}/

*C*(

*t*). Eq. (6) describes the DGD evolution along the fiber and allows us to have insight into the physical spin and twist of fiber. Note that the DGD of the spun and twisted fiber is determined mainly by these three parameters, namely, the initial linear birefringence, the twist rate, and the spin profile, which includes spin magnitude and the period of the alternating spinning process.

## 3. Results and Discussion

*a*

_{0},

*p*represent the spin magnitude and the period, respectively, and

*r*indicates the asymmetrical properties of the triangular spin profile. It is assumed that the fiber has an initial beat length of 15 m, rotation coefficient

*g*is set to be 0.14, and

*g*

_{ω}=0.09

*g*/

*ω*. The excellent agreement between them confirms the validity of our approximate analytical model. As twist and spin are two main techniques to produce low PMD fibers, we prefer to study the effects of twist and spin on PMD reduction.

### 3.1 PMD Reduction by Twist

*A*is simplified as

*A*=(

*g*-2)

*γz*. Fig. 2 shows the variation of DGD versus the external twist rate for two fiber samples with different initial linear birefringence. We also compare our results based on the perturbation theory with the analytical solution derived by Schuh

*et al.*[5

5. R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, and D. H. O. Bebbington, “Theoretical analysis and measurement of effects of fiber twist on polarization mode dispersion of optical fibers,” Electron. Lett. , **31**, 1772–1773 (1995). [CrossRef]

*g*

_{ω}. Second, DGD decreases with the increase of twist rate at low twist rates because mode coupling was strengthened between two polarizations. At high twist rates, DGD increases linearly with the twist rate because of twist-induced circular birefringence. This instability puts the twist method at a disadvantage because DGD could increase immediately when twist is a little larger than the best twist for the minimum PMD condition. Finally, we found that, when the twist rate is low, our results are inexact and differ significantly from that of Schuh

*et al.*compared with that in a high twist case. In contrast, the difference also increases with the initial linear birefringence because our analytical model is based on the perturbation theory and tends to be valid only when

*β*

_{l}is small. With the improvement of fiber manufacturing techniques in recent years, most intrinisic birefringence of fibers is small enough to meet this requirement.

### 3.2 PMD Reduction by Spin

*n*·

*p*can be written as

*i*·

*A*)

*dt*=0 i.e., the phase-matching condition is satisfied. Taking the triangular spin function described in Eq. (8) as an example, we can obtain fiber DGD from Eq. (9):

*erf*is the error function defined as

*L*(

*n*). Figure 3 illustrates the evolution of DGD along the fiber with an initial beat length of 15 m and a triangular spin profile with different sets of parameters. We found that a periodic DGD can be achieved with optimum spinning parameters or else the DGD grows linearly with the fiber length.

*a*,

*r*) for optimal PMD reduction is shown in Fig. 4. We can observe that there are several couples of (

*a*

_{0},

*r*)that satisfy the phase-matching conditions for maximum PMD reduction. In addition, the optimum condition is determined mainly by the spin magnitude and is insensitive to the value of parameter

*r*. Galtarossa

*et al.*[9

9. A. Galtarossa, L. Palmieri, and A. Pizzinat, “Optimized spinning design for low PMD fibers: an analytical approach,” J. Lightwave Technol. , **19**, 1502–1512 (2001). [CrossRef]

*et al.*are incorrect because spin profile

*A*does not have the odd harmonic properties as reported in their paper. We also believe that the constraint conditions for spin periods reported in Ref. 9

9. A. Galtarossa, L. Palmieri, and A. Pizzinat, “Optimized spinning design for low PMD fibers: an analytical approach,” J. Lightwave Technol. , **19**, 1502–1512 (2001). [CrossRef]

*RF*=|

*L*(1)|. Figure 5 illustrates variation of the PMD RF with respect to spin magnitude and parameter

*r*. We note that the RF generally decreases gradually when the spin magnitude increases, although there are some oscillations embedded in the dominant decreasing trend.

## 4. Conclusion

## Acknowledgments

## References and links

1. | M. Karlsson, “Polarization mode dispersion mitigation performance of various approaches,” in |

2. | B. W. Hakki, “Polarization mode dispersion compensation by phase diversity detection,” IEEE Photon. Technol. Lett. , |

3. | F. Roy, C. Francia, F. Bruyere, and D. Penninckx, “A simple dynamic polarization mode dispersion compensator,” in |

4. | M. J. Li and D. A. Nolan, “Fiber spin-profile designs for producing fibers with low polarization mode dispersion,” Opt. Lett. , |

5. | R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, and D. H. O. Bebbington, “Theoretical analysis and measurement of effects of fiber twist on polarization mode dispersion of optical fibers,” Electron. Lett. , |

6. | X. Chen, M. Li, and D. A. Nolan, “Analytical results for polarization mode dispersion of spun fibers,” in |

7. | C. D. Poole, J. H. Winters, and J. A. Nagel, “Dynamical equation for polarization dispersion,” Opt. Lett. , |

8. | R. E. Schuh, A. Altuncu, X. Shan, and A. S. Siddiqui, “Measurement and theoretical modeling of polarization mode dispersion in distributed erbium doped fibers,” in European Conference on Optical Communication, Edinburgh, U.K., 1997, Vol. 3, pp. 203–206. |

9. | A. Galtarossa, L. Palmieri, and A. Pizzinat, “Optimized spinning design for low PMD fibers: an analytical approach,” J. Lightwave Technol. , |

**OCIS Codes**

(060.2400) Fiber optics and optical communications : Fiber properties

(260.5430) Physical optics : Polarization

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 2, 2003

Revised Manuscript: September 9, 2003

Published: September 22, 2003

**Citation**

Muguang Wang, Tangjun Li, and Shuisheng Jian, "Analytical theory for polarization mode dispersion of spun and twisted fiber," Opt. Express **11**, 2403-2410 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-19-2403

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### References

- M. Karlsson, �??Polarization mode dispersion mitigation performance of various approaches,�?? in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), paper WI1, pp. 231�??232.
- B. W. Hakki, �??Polarization mode dispersion compensation by phase diversity detection,�?? IEEE Photon. Technol. Lett. 9, 121�??123 (1997). [CrossRef]
- F. Roy, C. Francia, F. Bruyere, and D. Penninckx, �??A simple dynamic polarization mode dispersion compensator,�?? in Optical Fiber Communication Conference, (Optical Society of America, Washington, D.C., 1999), TuS4-1, pp. 275 �??278.
- M. J. Li and D. A. Nolan, �??Fiber spin-profile designs for producing fibers with low polarization mode dispersion,�?? Opt. Lett. 23, 1659�??1661 (1998). [CrossRef]
- R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, and D. H. O. Bebbington, �??Theoretical analysis and measurement of effects of fiber twist on polarization mode dispersion of optical fibers,�?? Electron. Lett. 31, 1772�??1773 (1995). [CrossRef]
- X. Chen, M. Li, and D. A. Nolan, �??Analytical results for polarization mode dispersion of spun fibers,�?? in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), paper Th1, pp. 448�??449.
- C. D. Poole, J. H. Winters, and J. A. Nagel, �??Dynamical equation for polarization dispersion,�?? Opt. Lett. 16, 372�??374 (1991). [CrossRef] [PubMed]
- R. E. Schuh, A. Altuncu, X. Shan, and A. S. Siddiqui, �??Measurement and theoretical modeling of polarization mode dispersion in distributed erbium doped fibers,�?? in European Conference on Optical Communication, Edinburgh, U.K., 1997, Vol. 3, pp. 203�??206.
- A. Galtarossa, L. Palmieri, and A. Pizzinat, �??Optimized spinning design for low PMD fibers: an analytical approach,�?? J. Lightwave Technol. 19, 1502�??1512 (2001). [CrossRef]

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