2118 OPTICS LETTERS / Vol. 29, No. 18 / September 15, 2004

Analytical solution of polarization mode dispersion fortriangular spun fibers

Grégory Bouquet, Louis-Anne de Montmorillon, and Pascale Nouchi

Alcatel Cable, Conflans Sainte Honorine, 53 Rue Jean Broutin, 78703 Conflans Cedex, France

Received April 14, 2004

An analytical solution for the differential group delay of a fiber spun according to a triangular function isderived from concatenation of Jones matrices for a fiber length equal to N 3 T , where T is the spinningperiod and N is an integer. This solution holds for any value of linear deterministic birefringence Db ofamplitude A and period T of the triangular spinning function. We use the solution to emphasize the effectof birefringence on the efficiency of the spinning function. © 2004 Optical Society of America

OCIS codes: 060.2280, 060.2310.

Polarization mode dispersion (PMD) is a key fibercharacteristic that can strongly degrade transmissionperformance if not properly controlled. Over theyears manufacturers have improved their manufactur-ing processes to guarantee low PMD. Fiber spinningduring drawing has been proposed.1 It is one of themost eff icient ways to achieve low PMD. The peri-odic spin effect on PMD has been modeled by severalauthors, using various methods.2 – 7 In those papersanalytical expressions for the differential group delay(DGD) of spun fiber were presented for many spinningfunctions in both the so-called short- and long-periodassumptions. In this Letter we choose a specificspinning function, i.e., a triangular spinning function,to derive an exact analytical formula for the DGD thatis valid for all the birefringence and spin-parametervalues. This analytical solution allows us to link theshort- and long-period assumptions, which is of majorimportance when one considers the birefringence ofcurrent fibers: low birefringence (typically under1 m21) for transmission fibers but medium to highbirefringence for components fibers, such as disper-sion-compensating fibers8 or photonic crystal f ibers.9

We develop this analytical solution by consideringthe fiber as a succession of sections that are uniformlyspun in clockwise and counterclockwise directions.We first develop an analytical expression for the Jonesmatrix for one period of the spinning function and useit to derive the Jones matrix for N periods. We thenuse this final expression to determine the analyticalexpressions of the PMD vector and its norm, theDGD. Finally, we discuss the analytical expressionof the DGD.

To derive the PMD vector expression we use Jonesformalism and the correspondence between the Jonesand Stokes spaces.10 In Jones formalism a birefrin-gent element is described by the Jones matrix10,11

U cos wI 2 jn sin w, where I is the 2 3 2 identitymatrix and n n ? s is a 2 3 2 matrix such thatn ? s n1s1 1 n2s2 1 n3s3; s s1e1 1 s2e2 1 s3e3and n n1e1 1 n2e2 1 n3e3 are, respectively, thePauli spin vector with Pauli matrix components andthe direction of the proper axes in the Stokes space ofthe birefringent element; e1,e2, e3 is a basis in theStokes space; and 2w is the polarization eigenmode

0146-9592/04/182118-03$15.00/0

phase delay. Matrix U can be put in the exponentialform10: U exp2jnw. A linear birefringent fibersubject to a uniform spin between position z0 and z isrepresented by the following Jones matrix12:

U z0, z exp2js3uz0

3 exp∑2j Dbs1 2 2ts3

z0 2 z2

∏

3 exp js3uz , (1)

where Db is the birefringence, t is the spin rate, t duzdz, and uz is the periodic spinning function:

uz

(2AzT 0 , z , t22A1 2 zT T2 , z , T

. (2)

A and T are, respectively, the maximum amplitude(in radians) and the period of the spinning function(in meters). The Jones matrix for the f iber locatedbetween positions 0 and T , U T , 0, is the product ofthe Jones matrix of the first half-period, U T2, 0, andthe Jones matrix of the second half-period, U T ,T2,in the following order: U T , 0 U T ,T2U T2, 0,with t 2AT between 0 and T2 and t 22ATbetween T2 and T . Using Eq. (1) and the propertiesof Pauli matrices, we can express U T , 0 in the follow-ing way:

U T , 0 cos dI 2 jed sin d , (3)

where 2d and ed are, respectively, the phase delay andthe proper axis in the Stokes space of the equivalentbirefringent element located between 0 and T . Wecalculate

cos d 1 2 2µTDb4

D

∂2

sin2 D , (4)

sin d 2µTDb

4

∂jsinc Dj

µcos2 D 1

A2

D2 sin2 D

∂12, (5)

ed TDb

2sin21 dsinc 2De1 1 A sinc2 De2 , (6)

D

∑µTDb

4

∂2

1 A2∏12

. (7)

© 2004 Optical Society of America

September 15, 2004 / Vol. 29, No. 18 / OPTICS LETTERS 2119

Here 2D is the phase delay of the equivalent birefrin-gent element situated between 0 and T2 and sinc isthe sinus cardinal function.

The Jones matrix for N periods is given by the prod-uct for k 1 to N of Jones matrix (3). Since matrix (3)is valid for all k, we find that

U NT , 0 cos NdI 2 j ed sin Nd . (8)

As the frequency is changing, Jones matrixU NT , 0 will cause the output polarization stateto evolve. At each frequency this evolution can bedescribed on the Poincaré sphere by a rotation aroundPMD vector VNT . This vector is linked to Jonesmatrix U NT , 0 by the relation10,11 1/2 VNT j≠vU NT , 0U NT , 021, where v is the frequencyand ≠v is the derivative with respect to v. WithEq. (8), the PMD vector and the DGD at the end of Nperiods reads as10

VNT 2N≠vded 2 2 sin2Nd≠ved 3 ed

1 sin2Nd≠ved , (9)

DGDNT 2N2≠vd2 1 j≠vedj2sin2 Nd12. (10)

Equations (9) and (10), obtained for an integer multipleof a period, can be generalized to any length as soon asthe fiber is sufficiently long, i.e., for large N , that wecan neglect the DGD contribution over one period infront of the accumulated DGD. By calculating ≠ved

with Eq. (6) and ≠vd with Eqs. (4) and (5), we find anexplicit expression of the two terms appearing underthe square root in Eq. (10):

j≠vedj2sin2 Nd

µTDb4

D

∂2µAD

∂2

31 2 sinc 2D2∑

cos2 D 1

µAD

∂2

sin2 D

∏2

3TDt2

16sin2 Nd , (11)

N2≠vd2

µAD

∂4

∑sinc D 1

µTDb4

A

∂2

cos D

∏2

∑cos2 D 1

µAD

∂2

sin2 D

∏

3TDt2

4N2, (12)

where Dt ≠vDb is the local DGD. Note that wehave validated our analytical DGD expression by useof numerical calculations based on PMD concatenationrules.13

We note that the key parameters of the analyticalDGD expression are A and TDb4, which play a sym-metrical role, and D. We represent the evolution ofDGD with A for a short f iber (N 1) in Fig. 1 anda long fiber (N 100) in Fig. 2. Low, medium, andhigh birefringence levels are considered here (Db 1;12 and 56 m21). Period T is chosen to be of the or-der of the beat lengths of standard telecommunicationsfibers [from short (1-m) to long (20-m) beat lengths]

with a fixed value of 5 m. All the results shown arenormalized by the DGD of the unspun fiber (NTDt).

Looking at Figs. 1 and 2, one can observe theclassical DGD behavior of spun fiber2 – 7: the DGDdecreases globally as A increases, and this effect is allthe more important when Db is low; DGD presents anoscillatory behavior with the minima depth increas-ing with decreasing Db or increasing N . We alsoobserve that the global upper envelope of the DGDevolution is nearly the same whatever the N value.This last result leads us to f ind a simple expressionfor the DGD. To do so we neglect the sinusoidal termand look for an upper envelope of the linear term. Wefind that

DGDenv NT NTDt

"""A2 1 B4

2D4

√√√1 1

Ω1 2

µ1 2

A4

D4

∂

3

∑1 2

A2 2 B42

A2 1 B42

∏æ12!!!###

12, (13)

with R AD and B TDb4. Note that the pa-rameter DGDenv is also represented in Figs. 1 and

Fig. 1. DGD [Eq. (10)] and DGDenv [Eq. (13)] versus A forN 1; T 5 m; and Db 56, 12, and 1 m21 from the topto the bottom curve.

Fig. 2. DGD [Eq. (10)] and DGDenv [Eq. (13)] versus A forN 100; T 5 m; and Db 56, 12, and 1 m21 from thetop to the bottom curve.

2120 OPTICS LETTERS / Vol. 29, No. 18 / September 15, 2004

2. By combining this simple formula with the morecomplex Eqs. (11) and (12), it is easier to discuss theeffect of the two key parameters A and B. Moreover,the random birefringence evolution in optical fibersmoothes out the phase-matching behavior of theDGD,5,7 which emphasizes the interest in an enve-lope description. Below, we discuss the formulas[Eqs. (10) and (13)] for low, strong, and arbitrarybirefringence.

One can see in Figs. 1 and 2 that the DGD evolu-tion for the low Db case (1 m21) is almost identical forN 1 and N 100. This means that, for these val-ues of T , Db, and A, the DGD is independent of thenumber of periods, N . In fact, when A is much largerthan B, we can derive a simple expression for DGDfrom Eqs. (10)– (12) by neglecting B in front of A inthe expression of D. From a physical point of view,this corresponds to the case in which the number offiber turns over one beat length is high. We then findthat Eq. (11) has a square dependence in BA and canbe neglected in front of Eq. (12), which has a sinus car-dinal dependence on spin amplitude A and is indepen-dent of period T and of birefringence Db. The DGDis then simply given by

DGDNT NTDt

Çsin AA

Ç. (14)

Note that this dependence of Eq. (14) on A could beobtained by use of a short-period assumption andanalytical expressions developed elsewhere.2 – 7 Fi-nally, using Eq. (13), we find that the envelope canbe simply written as NTDtA, in agreement withexpression (14).

For high Db values, which correspond to AB ,, 1for typical spinning parameters A and T , we can derivea simple expression for the envelope of the DGD:

DGDenv NT NTDt

∑1 2

µAB

∂2∏. (15)

It is interesting to note that expression (15) is similarto that given in the long-period assumption7 for a si-nusoidal spinning function. As for the sinusoidal spin-

ning5,7 function, we observe that triangular spinning isineffective here in reducing the DGD.

Finally, in all the considered cases represented inFigs. 1 and 2, that is, whatever the birefringence andA values, we find that the complete Eq. (13) can be wellapproximated by

DGDenv NT NTDt

pA2 1 B4

D2. (16)

One can see with expression (16) that the spinningfunction efficiency decreases rapidly when B becomesgreater than A.

To conclude, we have found an analytical solutionfor the DGD of a symmetric triangular spun fiber inthe deterministic birefringence regime for f iber lengthequal to an entire number of periods. We showed thatthe exact DGD expression depends in a complex way onDb, A, and T and that the global behavior of the DGDdoes not depend on the number of periods, N . Usingan envelope description, we have found a simplifiedexpression for the DGD and used it to study the globalbehavior of the DGD versus A and Db.

G. Bouquet’s e-mail address is gregory.bouquet@alcatel.fr.

References

1. A. J. Barlow, Appl. Opt. 20, 2962 (1981).2. A. Galtarossa, J. Lightwave Technol. 19, 1502 (2001).3. M. J. Li, Opt. Lett. 23, 1659 (1998).4. X. Chen, Opt. Lett. 27, 294 (2002).5. A. Galtarossa, Opt. Lett. 27, 692 (2002).6. D. A. Nolan, J. Lightwave Technol. 22, 1066 (2004).7. A. Pizzinat, Opt. Lett. 28, 390 (2003).8. P. Nouchi, in Proceedings of the 21st European Confer-

ence on Optical Communication (ECOC’95) (Interuni-versity MicroElectronics Center, Gent, Belgium, 1995),p. 389.

9. A. Peyrilloux, J. Lightwave Technol. 21, 536 (2003).10. J. P. Gordon, Proc. Natl. Acad. Sci. USA 97, 4541

(2000).11. M. Karlsson, Opt. Lett. 23, 688 (1998).12. P. P. Finet, Optik 84, 169 (1990).13. N. Gisin, Opt. Commun. 89, 316 (1992).