Research Article Analytical Approach to Polarization Mode ...Research Article Analytical Approach to...

Research ArticleAnalytical Approach to Polarization Mode Dispersion inLinearly Spun Fiber with Birefringence

Vinod K. Mishra

US Army Research Laboratory, Aberdeen, MD 21005, USA

Correspondence should be addressed to Vinod K. Mishra; [email protected]

Received 30 October 2015; Accepted 4 January 2016

Academic Editor: Gang-Ding Peng

Copyright © 2016 Vinod K. Mishra. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The behavior of Polarization Mode Dispersion (PMD) in spun optical fiber is a topic of great interest in optical networking. Earlierwork in this area has focused more on approximate or numerical solutions. In this paper we present analytical results for PMDin spun fibers with triangular spin profile function. It is found that in some parameter ranges the analytical results differ from theapproximations.

1. Introduction

The Polarization Mode Dispersion (PMD) is a well-knownphenomenon in optical fibers and its role in the propagationof light pulse in various kinds of optical fibers has been asubject of intensive investigation [1–6] in the past. Its physicalorigin lies in the birefringence property of an optical fiberso that the orthogonal modes of the light electromagneticwave acquire different propagation speeds resulting in a phasedifference between them. The optical fiber at granular levelis nonhomogeneous and also has other defects accumulatedduring the manufacturing process. Due to these issues, thebirefringence in a physical fiber becomes random as pointedout by Foschini and Poole in [7]. In addition, Menyuk andWai [8] have also considered the nonlinear effects arisingfrom higher order susceptibility that also becomes importantunder certain physical conditions.

Sometime ago, Wang et al. [1] derived expressions for theDifferential Group Delay (DGD) of a randomly birefringentfiber in the Fixed Modulus Model (FMM) in which theDGD has both modulus and the phase. The FMM assumesthat the modulus of the birefringence vector is a randomvariable. They presented analytical results with the followingassumptions: (i) the spin function is periodic (a sine function)and (ii) the periodicity length (𝑝) of the fiber is much smallerthan the fiber correlation length (𝐿

𝐹) or 𝑝 ≪ 𝐿

𝐹. Later

they also generalized the FMM and presented the RandomModulus Model (RMM), which includes the randomness in

the direction of the birefringence vector. But then the RMMequations could only be solved numerically.

The present work is a contribution to the analyticalcalculations within FMM and so is only valid for a short fiberdistance. This limitation arises because beyond that distancethe birefringence randomness [7] becomes dominant. In thepresent work the full implications of the FMM have beenexplored under the following conditions: (i) The 𝑝 ≪ 𝐿

𝐹

approximation has been relaxed, (ii) a nonzero twist has beenincluded, and (iii) the periodic spin rate has been replacedwith a constant spin rate. We give the analytical solutions ofthe exact FMM equations under these conditions and alsopresent some numerical results based on them showing theeffect of different physical conditions.The analytical methodsare those applicable to the coupled mode theory calculationsadapted to the optical fibers [9].

2. Theoretical Analysis

2.1.TheModel with Periodic Spin Function. The starting pointis the well-known vector equation describing the change inthe Jones local electric field vector �⃗�(𝜔, 𝑧) with the angularfrequency 𝜔 and distance 𝑧 along a twisted fiber. Consider

[[

[

𝑑𝐴1 (𝑧)

𝑑𝑧𝑑𝐴2 (𝑧)

𝑑𝑧

]]

]

=𝑖

2(Δ𝛽) [

0 𝑒2𝑖Θ(𝑧)

𝑒−2𝑖Θ(𝑧)

0][

𝐴1 (𝑧)

𝐴2 (𝑧)

] . (1)

Hindawi Publishing CorporationInternational Journal of OpticsVolume 2016, Article ID 9753151, 9 pageshttp://dx.doi.org/10.1155/2016/9753151

2 International Journal of Optics

I

s

III

II

3𝜋/2

Θ(s)

𝜋/2

2𝜋𝜋0

Figure 1: The 3-segment approximation to the periodic sine function.

Here Δ𝛽(𝜔) is the birefringence and

Θ (𝑧) =𝛼0

𝜂sin (𝜂𝑧) (2)

is the periodic spin profile function with spin magnitude 𝛼0

and angular frequency of spatial modulation 𝜂.The boundary conditions are

𝐴1 (0) = 1,

𝑑𝐴1 (0)

𝑑𝑧= 0,

(3a)

𝐴2 (0) = 0,

𝑑𝐴2 (0)

𝑑𝑧= 𝑖 (

Δ𝛽

2) .

(3b)

Let 𝑠 = 𝜂𝑧 be a dimensionless variable. We use (𝑑/𝑑𝑧) =𝜂(𝑑/𝑑𝑠) to rewrite (1). Consider

[𝐴1𝑠 (𝑠)

𝐴2𝑠 (𝑠)

] = 𝑖𝑎 [0 𝑒

2𝑖𝑐 sin 𝑠

𝑒−2𝑖𝑐 sin 𝑠

0][

𝐴1 (𝑠)

𝐴2 (𝑠)

] . (4)

The subscripts denote differentiation (𝐴1𝑠= 𝑑𝐴1𝑠/𝑑𝑠, 𝐴

2𝑠=

𝑑𝐴2𝑠/𝑑𝑠). Also, 𝑎 = (Δ𝛽/2𝜂) and 𝑐 = (𝛼

0/𝜂) are dimension-

less constants.We express all parameters in terms of the lengths given

as beat length (𝐿𝐵

= 2𝜋/Δ𝛽), spin period (Λ = 2𝜋/𝜂),and coupling length (𝑙

0= 2𝜋/𝛼

0). Then we can write 𝑎 =

Λ/2𝐿𝐵, 𝑐 = 𝐿

𝐵/𝑙0.

The new boundary conditions are

𝐴1 (0) = 1,

𝐴1𝑠 (0) = 0,

(5a)

𝐴2 (0) = 0,

𝐴2𝑠 (0) = 𝑖𝑎.

(5b)

These equations ((1) or equivalently (4)) do not have analyti-cal solutions.

In the perturbative approximation (see Appendix B),an analytical result has been derived earlier [1]. In thepresent work we derive analytic solutions by replacing thesine function by linear segments and compare them to theperturbative solutions for the same segments.

2.2. Linear Segment Approximation to the PeriodicSpin Function: Analytical Solutions for the JonesAmplitude Equations

The Model. The periods of the straight line segments shownin Figure 1 approximate the periodic sine function. Here asingle period with 3-segment approximation is shown inFigure 1.

The field amplitudes for a given segment satisfy thefollowing equations:

[

[

𝐴1𝑠

(𝑚)(𝑠)

𝐴2𝑠

(𝑚)(𝑠)

]

]

= 𝑖𝑎[

[

0 𝑒2𝑖𝜃𝑚(𝑠)

𝑒−2𝑖𝜃𝑚(𝑠)

0

]

]

[

[

𝐴1

(𝑚)(𝑠)

𝐴2

(𝑚)(𝑠)

]

]

. (6)

The superscript and subscript 𝑚 both indicate the segmentsfor which the coupled equations hold.The limits of segmentsare given below.

We require that the endpoints of 𝜃𝑚(𝑠) should be the same

as that of the sine-function spin profile Θ(𝑠)|spin = 𝑐 sin 𝑠for all segments. Define �̃� = (2c/𝜋) so that the endpointconditions for segments hold.

For n = 1, Segment I (0 ≤ 𝑠 ≤ 𝜋/2),

𝜃1 (𝑠) = �̃�𝑠,

Θ (𝑠 = 0)|spin = 0 = 𝜃1 (𝑠 = 0) ,

Θ (𝑠 =𝜋

2)spin

= 𝑐 = 𝜃1(𝑠 =

𝜋

2) .

(7)

International Journal of Optics 3

For n = 2, Segment II (𝜋/2 ≤ 𝑠 ≤ 3𝜋/2),

𝜃2 (𝑠) = −�̃�𝑠 + 2𝑐,

Θ (𝑠 =𝜋

2)spin

= 𝑐 = 𝜃2(𝑠 =

𝜋

2) ,

Θ(𝑠 =3𝜋

2)spin

= −𝑐 = 𝜃2(𝑠 =

3𝜋

2) .

(8)

For n = 3, Segment III (3𝜋/2 ≤ 𝑠 ≤ 2𝜋),

𝜃3 (𝑠) = �̃�𝑠 − 4𝑐,

Θ (𝑠 =3𝜋

2)spin

= −𝑐 = 𝜃3(𝑠 =

3𝜋

2) ,

Θ (𝑠 = 2𝜋)|spin = 0 = 𝜃3 (𝑠 = 2𝜋) .

(9)

The General 𝑚-Segment Solutions. The solutions for the 𝑚thsegment have the following general form:

[

[

𝑒−𝑖𝜃𝑚(𝑠)𝐴1

(𝑚)(𝑠)

𝑖𝑎𝑒𝑖𝜃𝑚(𝑠)𝐴2

(𝑚)(𝑠)

]

]

=[[

[

𝑎1

(𝑚)+ 𝑖𝑏1

(𝑚)𝑎2

(𝑚)+ 𝑖𝑏2

(𝑚)

{−𝜃𝑚/𝑠

𝑏1

(𝑚)+ 𝑞𝑚𝑎2

(𝑚)+ 𝑖 (𝜃𝑚/𝑠

𝑎1

(𝑚)+ 𝑞𝑚𝑏2

(𝑚))} {− (𝜃

𝑚/𝑠𝑏2

(𝑚)+ 𝑞𝑚𝑎1

(𝑚)) + 𝑖 (𝜃

𝑚/𝑠𝑎2

(𝑚)− 𝑞𝑚𝑏1

(𝑚))}

]]

]

[cos 𝑞𝑚𝑠

sin 𝑞𝑚𝑠]

(10)

with

𝑞𝑚

2= 𝑎2+ 𝜃𝑚

2(𝑠) ,

𝜃𝑚/𝑠

=𝑑𝜃𝑚 (𝑠)

𝑑𝑠.

(11)

The exact solutions for the coupled equations in one segmentare related to those in the previous adjacent segment by thefollowing chain-relations among the coefficients.

Define 𝑢 = (𝑞𝑚−1

/𝑞𝑚), V = (𝜃

𝑚/𝑠− 𝜃𝑚−1/𝑠

)/𝑞𝑚, and then

the chain-relations are given by

[[[[[[

[

𝑎1

(𝑚)

𝑎2

(𝑚)

𝑏1

(𝑚)

𝑏2

(𝑚)

]]]]]]

]

=

{{{{{

{{{{{

{

[[[[[

[

𝑡1𝑡3

0 0

𝑡2𝑡4

0 0

0 0 𝑡1𝑡3

0 0 𝑡2𝑡4

]]]]]

]

+ 𝑢

[[[[[

[

𝑡4

−𝑡2

0 0

−𝑡3

𝑡1

0 0

0 0 𝑡4

−𝑡2

0 0 −𝑡3

𝑡1

]]]]]

]

+ V[[[[[

[

0 0 −𝑡2−𝑡4

0 0 𝑡1

𝑡3

𝑡2

𝑡4

0 0

−𝑡1−𝑡3

0 0

]]]]]

]

}}}}}

}}}}}

}

[[[[[[

[

𝑎1

(𝑚−1)

𝑎2

(𝑚−1)

𝑏1

(𝑚−1)

𝑏2

(𝑚−1)

]]]]]]

]

. (12)

Here the matrix elements are

𝑡1= cos 𝑞

𝑚−1𝑠𝑚−1

cos 𝑞𝑚𝑠𝑚−1

,

𝑡2= cos 𝑞


sin 𝑞𝑚𝑠𝑚−1

,

𝑡3= sin 𝑞


cos 𝑞𝑚𝑠𝑚−1

,

𝑡4= sin 𝑞


sin 𝑞𝑚𝑠𝑚−1

.

(13)

The matrix chain-relations can be written compactly byexpressing the 4 × 4 matrices as outer products (denoted bythe symbol ⊗) of two 2 × 2matrices as

[[[[[[

[

𝑎1

(𝑚)

𝑎2

(𝑚)

𝑏1

(𝑚)

𝑏2

(𝑚)

]]]]]]

]

= {([𝑡1𝑡3

𝑡2𝑡4

] + 𝑢[𝑡4

−𝑡2

−𝑡3

𝑡1

]) ⊗ [1 0

0 1] + V[

𝑡2

𝑡4

−𝑡1−𝑡3

] ⊗ [0 −1

1 0]}

[[[[[[

[

𝑎1

(𝑚−1)

𝑎2

(𝑚−1)

𝑏1

(𝑚−1)

𝑏2

(𝑚−1)

]]]]]]

]

. (14)


1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00.00 1.00 2.00 3.00 4.00 5.00 6.00

PMD

chan

ge fa

ctor

Dimensionless distance, s

PCF versus s

PCF, exactPCF, pert

Figure 2: The PCF curves for a perturbative limit with Λ = 1 and 𝐿𝐵= 12.

2.3. Calculation of PMD Correction Factor (PCF). The sumof squares of the 𝜔-differentiated amplitudes is similar topower and can be calculated by the following expressionusing expressions from Appendix A:

𝐴1𝜔

(𝑚)(𝑠)

2

+𝐴2𝜔

(𝑚)(𝑠)

2

(𝑎𝜔/𝑞)2

= (1

2) [(1 − 𝑛

2)

⋅ {(𝑝1

(𝑚))2

+ (𝑝2

(𝑚))2

+ (𝑝3

(𝑚))2

+ (𝑝4

(𝑚))2

}

+ (𝑝5

(𝑚))2

+ (𝑝6

(𝑚))2

+ (𝑝7

(𝑚))2

+ (𝑝8

(𝑚))2

]

+ (1

2) [(1 − 𝑛

2)

⋅ {(𝑝1

(𝑚))2

+ (𝑝2

(𝑚))2

− (𝑝3

(𝑚))2

− (𝑝4

(𝑚))2

}

+ (𝑝5

(𝑚))2

+ (𝑝6

(𝑚))2

− (𝑝7

(𝑚))2

− (𝑝8

(𝑚))2

]

⋅ cos 2𝑞𝑠 + [(1 − 𝑛2) {𝑝1

(𝑚)𝑝3

(𝑚)+ 𝑝2

(𝑚)𝑝4

(𝑚)}

+ 𝑝5

(𝑚)𝑝7

(𝑚)+ 𝑝6

(𝑚)𝑝8

(𝑚)] sin 2𝑞𝑠.

(15)

Here𝑚 (= 1, 2, 3) refers to segments in sequential manner.For calculating the normalized PCF we need a similar

expression for unspun-fiber given below:

[𝐴1𝜔 (𝑠)

2+𝐴2𝜔 (𝑠)

2]unspun-fiber

(𝑎𝜔/𝑞)2

= (𝑞𝑠)2. (16)

Then the expression for the PCF becomes

PCF(𝑚) (𝑠)

= [

[

𝐴1𝜔

(𝑚)(𝑠)

2

+𝐴2𝜔

(𝑚)(𝑠)

2

[𝐴1𝜔 (𝑠)

2+𝐴2𝜔 (𝑠)

2]unspun-fiber

]

]

1/2

.

(17)

The LHS is a function of parameters 𝑛 and 𝑞 and argument 𝑠.In general the expressions are quiet complicated, but for thefirst segment, the PCF is easily calculated and is given by

PCF(1) (𝑠) = √1 − 𝑛2 {1 − (sin 𝑞𝑠𝑞𝑠

)

2

}. (18)

3. Numerical Results

Thephysical constants ((Δ𝛽, 𝛼0, 𝜂) or equivalently (𝐿

𝐵, 𝑙0, Λ))

and the parameters (𝑛, 𝑞) appearing in the PCF expressionsare related by

𝑞 = (2Λ

𝜋𝑙0

)[1 + (𝜋𝑙0

4𝐿𝐵

)]

1/2

,

𝑛 = [1 + (𝜋𝑙0

4𝐿𝐵

)]

−1/2

.

(19)

We show results for sets of parameters in two extremelimits to emphasize the difference between the exact andperturbative calculations.

The Small-𝑞 Limit (Λ < 𝐿𝐵). In this limit two sets of

parameters were chosen to get small-𝑞-values (less than 1).This corresponds to beat length being much larger than thespin period.

The resulting plots are given in Figures 2 and 3.It is seen that the curves in Figure 2 for exact and per-

turbative calculations for small-𝑞 approximation are almostidentical.

The curves in Figure 3 for exact and perturbative calcula-tions are almost identical. Note that after 𝑠 = 5 the two curvesstart diverging a little.

The Large-𝑞 Limit (Λ > 𝐿𝐵). In this limit two sets of

parameters were chosen to get large-𝑞-values (much largerthan 1). This corresponds to beat length being smaller thanspin period.

The resulting plots are given in Figures 4 and 5.


1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0PM

D ch

ange

fact

or0.00 1.00 2.00 3.00 4.00 5.00 6.00

PCF versus s

PCF, exactPCF, pert


Figure 3: The PCF curve for a perturbative limit with Λ = 1 and 𝐿𝐵= 5.

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00.00 1.00 2.00 3.00 4.00 5.00 6.00

PMD

chan

ge fa

ctor

PCF versus s

PCF, exactPCF, pert


Figure 4: The PCF curve for a nonperturbative limit with Λ = 5 and 𝐿𝐵= 1.

0.00 1.00 2.00 3.00 4.00 5.00 6.00


1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

PMD

chan

ge fa

ctor

PCF, exact

PCF versus s

PCF, pert

Figure 5: The PCF curves for a nonperturbative limit with Λ = 12 and 𝐿𝐵= 1.

The top and bottom curves in Figure 4 show exactand perturbative calculations, respectively. It is seen thatperturbative approximation underestimates the PCF in thisregime. The two start diverging significantly for value of 𝑠a little less than 1.

The top and bottom curves in Figure 5 show exact andperturbative calculations, respectively. It is seen that pertur-bative approximation underestimates the PCF in this regime.The two start diverging significantly for value of 𝑠 a littlebeyond zero.


Table 1: PCF versus 𝑧 plots.

Parameters: Λ, 𝐿𝐵, 𝑙0

(in meters) Values (𝑛, 𝑞) Comments

(1, 12, 1) (0.9978, 0.6379) Λ ≪ 𝐿𝐵(1, 5, 1) (0.9879, 0.6444) Λ < 𝐿𝐵

Table 2: PCF versus 𝑧 plots.

Parameters: Λ, 𝐿𝐵, 𝑙0

(in meters) Values (𝑛, 𝑞) Comments

(5, 1, 1) (0.7864, 4.0475)Λ > 𝐿

𝐵(physical

nonperturbativelimit)

(12, 1, 1) (0.7864, 9.7139)Λ ≫ 𝐿

𝐵(physical

very nonperturbativelimit)

4. Conclusions

Thesine-function spin profile can be approximated in generalby any number of segments. In this work a 3-segmentapproximation was chosen and analytical results for thePCF function were obtained. The PCF calculations werealso repeated under the assumptions of the perturbativeapproximationmade in [1]. As expected, it was shown that theperturbative approximation has limited validity compared toan exact calculation.

The 3-segment approximation given here can be extendedto any number of segments for the spin function. Theanalytical results become very complicated very soon but theywill approach the exact results as the number of segmentsincreases. The method is also generalizable to an arbitraryspin function,which can be approximated by linear segments.This applies to almost all practically realizable spin functions.The exact analytic expressions for segment-approximatedspin function and approximate numerical calculation ofthe exact spin function should complement one another toenhance our understanding of the underlying physics (Tables1 and 2).

Appendix

A. Exact Calculation for Segments

A.1. The Specific 3-Segment Solutions. The details about solu-tions for 3 segments follow.

Segment I (0 ≤ 𝑠 ≤ 𝜋/2).The equations are

[

[

𝐴1𝑠

(1)(𝑠)

𝐴2𝑠

(1)(𝑠)

]

]

= 𝑖𝑎[

[

0 𝑒2𝑖𝜃1(𝑠)

𝑒−2𝑖𝜃1(𝑠)

0

]

]

[

[

𝐴1

(1)(𝑠)

𝐴2

(1)(𝑠)

]

]

. (A.1)

The boundary conditions are

[𝐴1

(1)(𝑠 = 0)] = 1,

[𝐴1𝑠

(1)(𝑠 = 0)] = 0,

(A.2a)

[𝐴2

(1)(𝑠 = 0)] = 0,

[𝐴2𝑠

(1)(𝑠 = 0)] = 𝑖𝑎.

(A.2b)

Let

𝑛 = (�̃�

𝑞) = [1 + (

𝜋𝑙0

4𝐿𝐵

)]

−1/2

, (A.3)

and then the analytical solutions are similar to those given inSection 2.2. Consider

[

[

𝑒−𝑖�̃�𝑠

𝐴1

(1)(𝑠)

(𝑞

𝑎) 𝑒𝑖�̃�𝑠𝐴2

(1)(𝑠)

]

]

= [1 −𝑖𝑛

0 𝑖] [

cos 𝑞𝑠sin 𝑞𝑠

] . (A.4)

Comparison with general expression gives the followingcoefficients:

𝑎1

(1)= 1,

𝑏1

(1)= 0,

𝑎2

(1)= 0,

𝑏2

(1)= −𝑛.

(A.5)

For calculating PCF, the amplitudes have to be differentiatedwith respect to𝜔, which will be denoted by subscript𝜔. Someuseful relations needed for this are

𝑑

𝑑𝜔(𝑎

𝑞) = 𝑛

2(𝑎𝜔

𝑞) ,

𝑎𝜔=

𝑑𝑎

𝑑𝜔=

𝛾

2𝜂, 𝛾 =

𝑑 (Δ𝛽)

𝑑𝜔,

𝑛𝜔= −𝑛(

𝑎

𝑞)(

𝑎𝜔

𝑞) ,

𝑞𝜔= 𝑎(

𝑎𝜔

𝑞) .

(A.6)

Then we can write

[[

[

(𝑞

𝑎) 𝑒−𝑖�̃�𝑠

𝐴1𝜔

(1)(𝑠)

𝑒𝑖�̃�𝑠𝐴2𝜔

(1)(𝑠)

]]

]

= (𝑎𝜔

𝑞)[

𝑝1

(1)+ 𝑖𝑝2

(1)𝑝3

(1)+ 𝑖𝑝4

(1)

𝑝5

(1)+ 𝑖𝑝6

(1)𝑝7

(1)+ 𝑖𝑝8

(1)][


] ,

𝑝1

(1)= 0,

𝑝2

(1)= −𝑛𝑞𝑠,

𝑝3

(1)= −𝑞𝑠,


𝑝4

(1)= 𝑛,

𝑝5

(1)= 0,

𝑝6

(1)= (1 − 𝑛

2) 𝑞𝑠,

𝑝7

(1)= 0,

𝑝8

(1)= 𝑛2.

(A.7)

Some interesting relations are found as

Δ𝛽 = (4𝜋𝑞

Λ)√1 − 𝑛2,

𝑧 = (Λ

2𝜋) 𝑠,

𝛼0= (

2𝜋2𝑞2

Λ)𝑛√1 − 𝑛2.

(A.8)

Segment II (𝜋/2 ≤ 𝑠 ≤ 3𝜋/2).The equations are

[𝐴1𝑠

(2)(𝑠)

𝐴2𝑠

(2)(𝑠)

] = 𝑖𝑎 [0 𝑒

2𝑖𝜃2(𝑠)


0][

𝐴1

(2)(𝑠)

𝐴2

(2)(𝑠)

] . (A.9)


[𝐴1

(1)(𝑠 =

𝜋

2)] = [𝐴

1

(2)(𝑠 =

𝜋

2)] ,

[𝐴1𝑠

(1)(𝑠 =

𝜋

2)] = [𝐴

1𝑠

(2)(𝑠 =

𝜋

2)] .

(A.10)

Similar expressions exist for 𝐴2

(2)(𝑠). Using the chain-

relations with 𝑛 = 2, the analytical solutions are obtained:

[[

[

𝑒−𝑖(−�̃�𝑠+2𝑐)

𝐴1

(2)(𝑠)

(𝑞

𝑎) 𝑒𝑖(−�̃�𝑠+2𝑐)

𝐴2

(2)(𝑠)

]]

]

= [1 − 𝑛2+ 𝑛2 cos𝜋𝑞 − 𝑖𝑛 sin𝜋𝑞 𝑛 (𝑛 sin𝜋𝑞 + 𝑖 cos𝜋𝑞)

−𝑛 (1 − cos𝜋𝑞) 𝑛 sin𝜋𝑞 + 𝑖] [


] . (A.11)

The 𝜔-differentiated amplitudes are found as

[

[

(𝑞

𝑎) 𝑒−𝑖(�̃�𝑠−2𝑐)

𝐴1𝜔

(2)(𝑠)

𝑒𝑖(�̃�𝑠−2𝑐)

𝐴2𝜔

(2)(𝑠)

]

]

= (𝑎𝜔

𝑞)

⋅ [𝑝1

(2)+ 𝑖𝑝2

(2)𝑝3

(2)+ 𝑖𝑝4

(2)

𝑝5

(2)+ 𝑖𝑝6

(2)𝑝7

(2)+ 𝑖𝑝8

(2)][


] ,

𝑝1

(2)= 𝑛2{2 (1 − cos𝜋𝑞) − 𝜋𝑞 sin𝜋𝑞 + 𝑞𝑠 sin𝜋𝑞} ,

𝑝2

(2)= 𝑛 {sin𝜋𝑞 − 𝜋𝑞 cos𝜋𝑞 + 𝑞𝑠 cos𝜋𝑞} ,

𝑝3

(2)= 𝑛2(−2 sin𝜋𝑞 + 𝜋𝑞 cos𝜋𝑞) − (1 − 𝑛2

+ 𝑛2 cos𝜋𝑞) 𝑞𝑠,

𝑝4

(2)= 𝑛 {− (cos𝜋𝑞 + 𝜋𝑞 sin𝜋𝑞) + 𝑞𝑠 sin𝜋𝑞} ,

𝑝5

(2)= 𝑛 {(1 − 2𝑛

2) (1 − cos𝜋𝑞)

− (1 − 𝑛2) 𝜋𝑞 sin𝜋𝑞 + (1 − 𝑛2) 𝑞𝑠 sin𝜋𝑞} ,

𝑝6

(2)= (1 − 𝑛

2) 𝑞𝑠,

𝑝7

(2)= 𝑛 {− (1 − 2𝑛

2) sin𝜋𝑞 + (1 − 𝑛2) 𝜋𝑞 cos𝜋𝑞

+ (1 − 𝑛2) (1 − cos𝜋𝑞) 𝑞𝑠} ,

𝑝8

(2)= 𝑛2.

(A.12)

Segment III (3𝜋/2 ≤ 𝑠 ≤ 2𝜋).The equations are

[𝐴1𝑠

(3)(𝑠)

𝐴2𝑠

(3)(𝑠)

] = 𝑖𝑎 [0 𝑒

2𝑖𝜃3(𝑠)


0][

𝐴1

(3)(𝑠)

𝐴2

(3)(𝑠)

] . (A.13)


[𝐴1

(2)(𝑠 =

3𝜋

2)] = [𝐴

1

(3)(𝑠 =

3𝜋

2)] ,

[𝐴1𝑠

(2)(𝑠 =

3𝜋

2)] = [𝐴

1𝑠

(3)(𝑠 =

3𝜋

2)] .

(A.14)

Similar expressions exist for 𝐴2

(3)(𝑠). Using the chain-

relations with 𝑛 = 3, the analytical solutions are obtained:

[[

[

𝑒−𝑖(�̃�𝑠−4𝑐)

𝐴1

(3)(𝑠)

(𝑞

𝑎) 𝑒𝑖(�̃�𝑠−4𝑐)

𝐴2

(3)(𝑠)

]]

]

= [

[

1 − 𝑛2+ 𝑛2 cos𝜋𝑞 + 𝑖𝑛 {𝑛2 sin 2𝜋𝑞 + (1 − 𝑛2) (sin 3𝜋𝑞 − sin𝜋𝑞)} 𝑛2 sin 2𝜋𝑞 − 𝑖𝑛 {𝑛2 cos 2𝜋𝑞 + (1 − 𝑛2) (1 + cos 3𝜋𝑞 − cos𝜋𝑞)}

[𝑛 (cos𝜋𝑞 − cos 3𝜋𝑞) + 𝑖𝑛2 (sin 3𝜋𝑞 − sin 2𝜋𝑞 − sin𝜋𝑞)] [𝑛 (sin𝜋𝑞 − sin 3𝜋𝑞) + 𝑖 {1 − 𝑛2 + 𝑛2 (cos𝜋𝑞 + cos 2𝜋𝑞 − cos 3𝜋𝑞)}]]

]

[cos 𝑞𝑠sin 𝑞𝑠

] .

(A.15)


The 𝜔-differentiated amplitudes are found as

[

[

(𝑞

𝑎) 𝑒−𝑖(�̃�𝑠−4𝑐)

𝐴1𝜔

(3)(𝑠)

𝑒𝑖(�̃�𝑠−4𝑐)

𝐴2𝜔

(3)(𝑠)

]

]

= (𝑎𝜔

𝑞)

⋅ [

[

𝑝1

(3)+ 𝑖𝑝2

(3)𝑝3

(3)+ 𝑖𝑝4

(3)

𝑝5

(3)+ 𝑖𝑝6

(3)𝑝7

(3)+ 𝑖𝑝8

(3)

]

]

[cos 𝑞𝑠sin 𝑞𝑠

] ,

𝑝1

(3)= 2𝑛2(1 − cos 2𝜋𝑞 − 𝜋𝑞 sin 2𝜋𝑞) + 𝑛2

⋅ 𝑞𝑠 sin 2𝜋𝑞,

𝑝2

(3)= 𝑛 [−3𝑛

2 sin 2𝜋𝑞 − (1 − 3𝑛2) (sin 3𝜋𝑞

− sin𝜋𝑞) + 𝜋𝑞 {2𝑛2 cos 2𝜋𝑞

+ (1 − 𝑛2) (3 cos 3𝜋𝑞 − cos𝜋𝑞)} − {𝑛2 cos 2𝜋𝑞

+ (1 − 𝑛2) (1 − cos𝜋𝑞 + cos 3𝜋𝑞)} 𝑞𝑠] ,

𝑝3

(3)= −2𝑛

2(sin 2𝜋𝑞 − 𝜋𝑞 cos 2𝜋𝑞) − (1 − 𝑛2 + 𝑛2

⋅ cos 2𝜋𝑞) 𝑞𝑠,

𝑝4

(3)= 𝑛 [3𝑛

2 cos 2𝜋𝑞 + (1 − 3𝑛2) (1 − cos𝜋𝑞

+ cos 3𝜋𝑞) + 𝜋𝑞 {2𝑛2 sin 2𝜋𝑞

+ (1 − 𝑛2) (3 sin 3𝜋𝑞 − sin𝜋𝑞)} − {𝑛2 sin 2𝜋𝑞

+ (1 − 𝑛2) (sin 3𝜋𝑞 − sin𝜋𝑞)} 𝑞𝑠] ,

𝑝5

(3)= 𝑛 (1 − 2𝑛

2) (cos 3𝜋𝑞 − cos𝜋𝑞) + 𝑛 (1 − 𝑛2)

⋅ (sin 3𝜋𝑞 − sin𝜋𝑞) 𝜋𝑞 + 𝑛 (1 − 𝑛2) (sin𝜋𝑞

− sin 3𝜋𝑞) 𝑞𝑠,

𝑝6

(3)= (1 − 𝑛

2) 𝑞𝑠 + 𝑛

2[(2 − 3𝑛

2) (sin𝜋𝑞

+ sin 2𝜋𝑞 − sin 3𝜋𝑞) + (1 − 𝑛2) (3 cos 3𝜋𝑞

− 2 cos 2𝜋𝑞 − cos𝜋𝑞) 𝜋𝑞 − (1 − 𝑛2) (1 − cos𝜋𝑞

− cos 2𝜋𝑞 + cos 3𝜋𝑞) 𝑞𝑠] ,

𝑝7

(3)= 𝑛 (1 − 2𝑛

2) (sin 3𝜋𝑞 − sin𝜋𝑞) + 𝑛 (1 − 𝑛2)

⋅ (cos𝜋𝑞 − 3 cos 3𝜋𝑞) 𝜋𝑞 + 𝑛 (1 − 𝑛2) (cos 3𝜋𝑞

− cos𝜋𝑞) 𝑞𝑠,

𝑝8

(3)= 𝑛2+ 𝑛2[(2 − 3𝑛

2) (1 − cos𝜋𝑞 − cos 2𝜋𝑞

+ cos 3𝜋𝑞) + (1 − 𝑛2) (3 sin 3𝜋𝑞 − 2 sin 2𝜋𝑞

− sin𝜋𝑞) 𝜋𝑞

+ (1 − 𝑛2) (sin𝜋𝑞 + sin 2𝜋𝑞 − sin 3𝜋𝑞) 𝑞𝑠 ] .

(A.16)

B. Perturbative Calculation for Segments

The perturbative approach is based on the following assump-tions:

(i) The coupling between the polarization states is sosmall that the equations become decoupled.

(ii) The top component is constant (𝐴1

(𝑚)= 1, 𝑚 =

1, 2, 3) and only the second component changes.(iii) The boundary conditions remain unchanged.

Under these assumptions the dimensionless constant 𝑞becomes �̃�, which is related to the physical lengths as

�̃� =2

𝜋(Λ

𝑙0

) . (B.1)

The new equations and their solutions take the followingform.

Segment I (0 ≤ 𝑠 ≤ 𝜋/2). Perturbative equations are asfollows:

[

[

𝐴1𝑠

(1)(𝑠)

𝐴2𝑠

(1)(𝑠)

]

]

= 𝑖𝑎[

[

0 𝑒2𝑖�̃�𝑠

𝑒−2𝑖�̃�𝑠

0

]

]

[1

0] . (B.2)

Solutions are as follows:

𝐴2

(1)(𝑠) = (

𝑎

�̃�) 𝑖𝑒−𝑖�̃�𝑠 sin �̃�𝑠. (B.3)

The sum of squares of the 𝜔-differentiated amplitudes is asfollows:

(

𝐴1𝜔

(1)(𝑠)

2

+𝐴2𝜔

(1)(𝑠)

2

(𝑎𝜔/�̃�)2

)

pert

=1

2(1 − cos 2�̃�𝑠)

= sin2�̃�𝑠.

(B.4)

So

PCF(1) (𝑠)pert

=[[

[

(𝐴1𝜔

(1)(𝑠)

2

+𝐴2𝜔

(1)(𝑠)

2

)pert

[𝐴1𝜔 (𝑠)

2+𝐴2𝜔 (𝑠)

2]unspun-fiber

]]

]

1/2

=sin �̃�𝑠�̃�𝑠

.

(B.5)


Segment II (𝜋/2 ≤ 𝑠 ≤ 3𝜋/2). Perturbative equations are asfollows:

[

[

𝐴1𝑠

(2)(𝑠)

𝐴2𝑠

(2)(𝑠)

]

]

= 𝑖𝑎[

[

0 𝑒2𝑖(−�̃�𝑠+2𝑐)

𝑒−2𝑖(−�̃�𝑠+2𝑐)

0

]

]

[1

0] . (B.6)


𝐴2

(2)(𝑠) = 𝑒

𝑖(�̃�𝑠−2𝑐)(𝑎

�̃�)

⋅ [− (1 − cos 2𝑐) cos �̃�𝑠 + (sin 2𝑐 + 𝑖) sin �̃�𝑠] .(B.7)


(𝐴1𝜔

(2)(𝑠)

2

+𝐴2𝜔

(2)(𝑠)

2

)pert

=1

2(𝑎𝜔

�̃�)

2

{(3 − 2 cos 2𝑐) + (cos 4𝑐 − 2 cos 2𝑐) cos 2�̃�𝑠 + (sin 4𝑐 − 2 sin 2𝑐) sin 2�̃�𝑠} . (B.8)

Expression for PCF is obtained as before.

Segment III (3𝜋/2 ≤ 𝑠 ≤ 2𝜋). Perturbative equations are asfollows:

[𝐴1𝑠

(3)(𝑠)

𝐴2𝑠

(3)(𝑠)

] = 𝑖𝑎 [0 𝑒

2𝑖(�̃�𝑠−4𝑐)

𝑒−2𝑖(�̃�𝑠−4𝑐)

0][

1

0] . (B.9)


𝐴2

(3)= 𝑒𝑖(−�̃�𝑠+4𝑐)

(𝑎

�̃�)

⋅ [{(−1 + cos 2𝑐 + cos 4𝑐 − cos 6𝑐)

+ 𝑖 (− sin 2𝑐 − sin 4𝑐 + sin 6𝑐)} cos �̃�𝑠

+ {(sin 2𝑐 + sin 4𝑐 − sin 6𝑐)

+ 𝑖 (1 + cos 2𝑐 + cos 4𝑐 − cos 6𝑐)} sin �̃�𝑠] .(B.10)


(𝐴1𝜔

(3)(𝑠)

2

+𝐴2𝜔

(3)(𝑠)

2

)pert

=1

2(𝑎𝜔

�̃�)

2

{(5 − 4 cos 4𝑐) + (2 cos 10𝑐 − cos 8𝑐 − 2 cos 6𝑐) cos 2�̃�𝑠 + (2 sin 10𝑐 − sin 8𝑐 − 2 sin 6𝑐) sin 2�̃�𝑠} .(B.11)

The PCF can be calculated as before.

Competing Interests

The author declares that he has no competing interests.

Acknowledgments

The author thanks Nick Frigo (formerly at AT&T Labsand now at United States Naval Academy) for getting himinterested in this topic.

References

[1] M. Wang, T. Li, and S. Jian, “Analytical theory for polarizationmode dispersion of spun and twisted fiber,” Optics Express, vol.11, no. 19, pp. 2403–2410, 2003.

[2] A. Pizzinat, B. S. Marks, L. Palmieri, C. R. Menyuk, and A.Gastarossa, “Influence of the model for random birefringenceon the differential group delay of periodically spun fibers,” IEEEPhotonics Technology Letters, vol. 15, no. 6, pp. 819–821, 2003.

[3] A. Galtarossa, L. Palmieri, A. Pizzinat, B. S. Marks, and C. R.Menyuk, “An analytical formula for the mean differential group

delay of randomly birefringent spunfibers,” Journal of LightwaveTechnology, vol. 21, no. 7, pp. 1635–1643, 2003.

[4] A.Galtarossa, L. Palmieri, andA. Pizzinat, “Optimized spinningdesign for low PMD fibers: an analytical approach,” Journal ofLightwave Technology, vol. 19, no. 10, pp. 1502–1512, 2001.

[5] P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion,decorrelation, and diffusion in optical fibers with randomlyvarying birefringence,” Journal of Lightwave Technology, vol. 14,no. 2, pp. 148–157, 1996.

[6] C. R. Menyuk and P. K. A. Wai, “Polarization evolutionand dispersion in fibers with spatially varying birefringence,”Journal of the Optical Society of America B, vol. 11, no. 7, p. 1288,1994.

[7] G. J. Foschini and C. D. Poole, “Statistical theory of polarizationdispersion in single mode fibers,” Journal of Lightwave Technol-ogy, vol. 9, no. 11, pp. 1439–1456, 1991.

[8] C. R. Menyuk and P. K. A. Wai, “Elimination of nonlinearpolarization rotation in twisted fibers,” Journal of the OpticalSociety of America B, vol. 11, no. 7, pp. 1305–1309, 1994.

[9] N. J. Frigo, “A generalized geometrical representation coupledmode theory,” IEEE Journal of Quantum Electronics, vol. QE-22,no. 11, pp. 2131–2140, 1986.

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