Post on 26-Jan-2021
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; vkmishr@gmail.com
Received 30 October 2015; Accepted 4 January 2016
Academic Editor: Gang-Ding Peng
Copyright Β© 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 [1β6] 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 (π) of the fiber is much smallerthan the fiber correlation length (πΏ
πΉ) or π βͺ πΏ
πΉ. 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 π βͺ πΏ
πΉ
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 οΏ½βοΏ½(π, π§) with the angularfrequency π and distance π§ along a twisted fiber. Consider
[[
[
ππ΄1 (π§)
ππ§ππ΄2 (π§)
ππ§
]]
]
=π
2(Ξπ½) [
0 π2πΞ(π§)
πβ2πΞ(π§)
0][
π΄1 (π§)
π΄2 (π§)
] . (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
Ξ(s)
π/2
2ππ0
Figure 1: The 3-segment approximation to the periodic sine function.
Here Ξπ½(π) is the birefringence and
Ξ (π§) =πΌ0
πsin (ππ§) (2)
is the periodic spin profile function with spin magnitude πΌ0
and angular frequency of spatial modulation π.The boundary conditions are
π΄1 (0) = 1,
ππ΄1 (0)
ππ§= 0,
(3a)
π΄2 (0) = 0,
ππ΄2 (0)
ππ§= π (
Ξπ½
2) .
(3b)
Let π = ππ§ be a dimensionless variable. We use (π/ππ§) =π(π/ππ ) to rewrite (1). Consider
[π΄1π (π )
π΄2π (π )
] = ππ [0 π
2ππ sin π
πβ2ππ sin π
0][
π΄1 (π )
π΄2 (π )
] . (4)
The subscripts denote differentiation (π΄1π = ππ΄1π /ππ , π΄
2π =
ππ΄2π /ππ ). Also, π = (Ξπ½/2π) and π = (πΌ
0/π) are dimension-
less constants.We express all parameters in terms of the lengths given
as beat length (πΏπ΅
= 2π/Ξπ½), spin period (Ξ = 2π/π),and coupling length (π
0= 2π/πΌ
0). Then we can write π =
Ξ/2πΏπ΅, π = πΏ
π΅/π0.
The new boundary conditions are
π΄1 (0) = 1,
π΄1π (0) = 0,
(5a)
π΄2 (0) = 0,
π΄2π (0) = ππ.
(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:
[
[
π΄1π
(π)(π )
π΄2π
(π)(π )
]
]
= ππ[
[
0 π2πππ(π )
πβ2πππ(π )
0
]
]
[
[
π΄1
(π)(π )
π΄2
(π)(π )
]
]
. (6)
The superscript and subscript π both indicate the segmentsfor which the coupled equations hold.The limits of segmentsare given below.
We require that the endpoints of ππ(π ) should be the same
as that of the sine-function spin profile Ξ(π )|spin = π sin π for all segments. Define οΏ½ΜοΏ½ = (2c/π) so that the endpointconditions for segments hold.
For n = 1, Segment I (0 β€ π β€ π/2),
π1 (π ) = οΏ½ΜοΏ½π ,
Ξ (π = 0)|spin = 0 = π1 (π = 0) ,
Ξ (π =π
2)spin
= π = π1(π =
π
2) .
(7)
International Journal of Optics 3
For n = 2, Segment II (π/2 β€ π β€ 3π/2),
π2 (π ) = βοΏ½ΜοΏ½π + 2π,
Ξ (π =π
2)spin
= π = π2(π =
π
2) ,
Ξ(π =3π
2)spin
= βπ = π2(π =
3π
2) .
(8)
For n = 3, Segment III (3π/2 β€ π β€ 2π),
π3 (π ) = οΏ½ΜοΏ½π β 4π,
Ξ (π =3π
2)spin
= βπ = π3(π =
3π
2) ,
Ξ (π = 2π)|spin = 0 = π3 (π = 2π) .
(9)
The General π-Segment Solutions. The solutions for the πthsegment have the following general form:
[
[
πβπππ(π )π΄1
(π)(π )
ππππππ(π )π΄2
(π)(π )
]
]
=[[
[
π1
(π)+ ππ1
(π)π2
(π)+ ππ2
(π)
{βππ/π
π1
(π)+ πππ2
(π)+ π (ππ/π
π1
(π)+ πππ2
(π))} {β (π
π/π π2
(π)+ πππ1
(π)) + π (π
π/π π2
(π)β πππ1
(π))}
]]
]
[cos πππ
sin πππ ]
(10)
with
ππ
2= π2+ ππ
2(π ) ,
ππ/π
=πππ (π )
ππ .
(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 π’ = (ππβ1
/ππ), V = (π
π/π β ππβ1/π
)/ππ, and then
the chain-relations are given by
[[[[[[
[
π1
(π)
π2
(π)
π1
(π)
π2
(π)
]]]]]]
]
=
{{{{{
{{{{{
{
[[[[[
[
π‘1π‘3
0 0
π‘2π‘4
0 0
0 0 π‘1π‘3
0 0 π‘2π‘4
]]]]]
]
+ π’
[[[[[
[
π‘4
βπ‘2
0 0
βπ‘3
π‘1
0 0
0 0 π‘4
βπ‘2
0 0 βπ‘3
π‘1
]]]]]
]
+ V[[[[[
[
0 0 βπ‘2βπ‘4
0 0 π‘1
π‘3
π‘2
π‘4
0 0
βπ‘1βπ‘3
0 0
]]]]]
]
}}}}}
}}}}}
}
[[[[[[
[
π1
(πβ1)
π2
(πβ1)
π1
(πβ1)
π2
(πβ1)
]]]]]]
]
. (12)
Here the matrix elements are
π‘1= cos π
πβ1π πβ1
cos πππ πβ1
,
π‘2= cos π
πβ1π πβ1
sin πππ πβ1
,
π‘3= sin π
πβ1π πβ1
cos πππ πβ1
,
π‘4= sin π
πβ1π πβ1
sin πππ πβ1
.
(13)
The matrix chain-relations can be written compactly byexpressing the 4 Γ 4 matrices as outer products (denoted bythe symbol β) of two 2 Γ 2matrices as
[[[[[[
[
π1
(π)
π2
(π)
π1
(π)
π2
(π)
]]]]]]
]
= {([π‘1π‘3
π‘2π‘4
] + π’[π‘4
βπ‘2
βπ‘3
π‘1
]) β [1 0
0 1] + V[
π‘2
π‘4
βπ‘1βπ‘3
] β [0 β1
1 0]}
[[[[[[
[
π1
(πβ1)
π2
(πβ1)
π1
(πβ1)
π2
(πβ1)
]]]]]]
]
. (14)
4 International Journal of Optics
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 Ξ = 1 and πΏπ΅= 12.
2.3. Calculation of PMD Correction Factor (PCF). The sumof squares of the π-differentiated amplitudes is similar topower and can be calculated by the following expressionusing expressions from Appendix A:
π΄1π
(π)(π )
2
+π΄2π
(π)(π )
2
(ππ/π)2
= (1
2) [(1 β π
2)
β {(π1
(π))2
+ (π2
(π))2
+ (π3
(π))2
+ (π4
(π))2
}
+ (π5
(π))2
+ (π6
(π))2
+ (π7
(π))2
+ (π8
(π))2
]
+ (1
2) [(1 β π
2)
β {(π1
(π))2
+ (π2
(π))2
β (π3
(π))2
β (π4
(π))2
}
+ (π5
(π))2
+ (π6
(π))2
β (π7
(π))2
β (π8
(π))2
]
β cos 2ππ + [(1 β π2) {π1
(π)π3
(π)+ π2
(π)π4
(π)}
+ π5
(π)π7
(π)+ π6
(π)π8
(π)] sin 2ππ .
(15)
Hereπ (= 1, 2, 3) refers to segments in sequential manner.For calculating the normalized PCF we need a similar
expression for unspun-fiber given below:
[π΄1π (π )
2+π΄2π (π )
2]unspun-fiber
(ππ/π)2
= (ππ )2. (16)
Then the expression for the PCF becomes
PCF(π) (π )
= [
[
π΄1π
(π)(π )
2
+π΄2π
(π)(π )
2
[π΄1π (π )
2+π΄2π (π )
2]unspun-fiber
]
]
1/2
.
(17)
The LHS is a function of parameters π and π and argument π .In general the expressions are quiet complicated, but for thefirst segment, the PCF is easily calculated and is given by
PCF(1) (π ) = β1 β π2 {1 β (sin ππ ππ
)
2
}. (18)
3. Numerical Results
Thephysical constants ((Ξπ½, πΌ0, π) or equivalently (πΏ
π΅, π0, Ξ))
and the parameters (π, π) appearing in the PCF expressionsare related by
π = (2Ξ
ππ0
)[1 + (ππ0
4πΏπ΅
)]
1/2
,
π = [1 + (ππ0
4πΏπ΅
)]
β1/2
.
(19)
We show results for sets of parameters in two extremelimits to emphasize the difference between the exact andperturbative calculations.
The Small-π Limit (Ξ < πΏπ΅). In this limit two sets of
parameters were chosen to get small-π-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-π approximation are almostidentical.
The curves in Figure 3 for exact and perturbative calcula-tions are almost identical. Note that after π = 5 the two curvesstart diverging a little.
The Large-π Limit (Ξ > πΏπ΅). In this limit two sets of
parameters were chosen to get large-π-values (much largerthan 1). This corresponds to beat length being smaller thanspin period.
The resulting plots are given in Figures 4 and 5.
International Journal of Optics 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
Dimensionless distance, s
Figure 3: The PCF curve for a perturbative limit with Ξ = 1 and πΏπ΅= 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
Dimensionless distance, s
Figure 4: The PCF curve for a nonperturbative limit with Ξ = 5 and πΏπ΅= 1.
0.00 1.00 2.00 3.00 4.00 5.00 6.00
Dimensionless distance, s
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 Ξ = 12 and πΏπ΅= 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 π a 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 π a littlebeyond zero.
6 International Journal of Optics
Table 1: PCF versus π§ plots.
Parameters: Ξ, πΏπ΅, π0
(in meters) Values (π, π) Comments
(1, 12, 1) (0.9978, 0.6379) Ξ βͺ πΏπ΅(1, 5, 1) (0.9879, 0.6444) Ξ < πΏπ΅
Table 2: PCF versus π§ plots.
Parameters: Ξ, πΏπ΅, π0
(in meters) Values (π, π) Comments
(5, 1, 1) (0.7864, 4.0475)Ξ > πΏ
π΅(physical
nonperturbativelimit)
(12, 1, 1) (0.7864, 9.7139)Ξ β« πΏ
π΅(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 β€ π β€ π/2).The equations are
[
[
π΄1π
(1)(π )
π΄2π
(1)(π )
]
]
= ππ[
[
0 π2ππ1(π )
πβ2ππ1(π )
0
]
]
[
[
π΄1
(1)(π )
π΄2
(1)(π )
]
]
. (A.1)
The boundary conditions are
[π΄1
(1)(π = 0)] = 1,
[π΄1π
(1)(π = 0)] = 0,
(A.2a)
[π΄2
(1)(π = 0)] = 0,
[π΄2π
(1)(π = 0)] = ππ.
(A.2b)
Let
π = (οΏ½ΜοΏ½
π) = [1 + (
ππ0
4πΏπ΅
)]
β1/2
, (A.3)
and then the analytical solutions are similar to those given inSection 2.2. Consider
[
[
πβποΏ½ΜοΏ½π
π΄1
(1)(π )
(π
π) πποΏ½ΜοΏ½π π΄2
(1)(π )
]
]
= [1 βππ
0 π] [
cos ππ sin ππ
] . (A.4)
Comparison with general expression gives the followingcoefficients:
π1
(1)= 1,
π1
(1)= 0,
π2
(1)= 0,
π2
(1)= βπ.
(A.5)
For calculating PCF, the amplitudes have to be differentiatedwith respect toπ, which will be denoted by subscriptπ. Someuseful relations needed for this are
π
ππ(π
π) = π
2(ππ
π) ,
ππ=
ππ
ππ=
πΎ
2π, πΎ =
π (Ξπ½)
ππ,
ππ= βπ(
π
π)(
ππ
π) ,
ππ= π(
ππ
π) .
(A.6)
Then we can write
[[
[
(π
π) πβποΏ½ΜοΏ½π
π΄1π
(1)(π )
πποΏ½ΜοΏ½π π΄2π
(1)(π )
]]
]
= (ππ
π)[
π1
(1)+ ππ2
(1)π3
(1)+ ππ4
(1)
π5
(1)+ ππ6
(1)π7
(1)+ ππ8
(1)][
cos ππ sin ππ
] ,
π1
(1)= 0,
π2
(1)= βπππ ,
π3
(1)= βππ ,
International Journal of Optics 7
π4
(1)= π,
π5
(1)= 0,
π6
(1)= (1 β π
2) ππ ,
π7
(1)= 0,
π8
(1)= π2.
(A.7)
Some interesting relations are found as
Ξπ½ = (4ππ
Ξ)β1 β π2,
π§ = (Ξ
2π) π ,
πΌ0= (
2π2π2
Ξ)πβ1 β π2.
(A.8)
Segment II (π/2 β€ π β€ 3π/2).The equations are
[π΄1π
(2)(π )
π΄2π
(2)(π )
] = ππ [0 π
2ππ2(π )
πβ2ππ2(π )
0][
π΄1
(2)(π )
π΄2
(2)(π )
] . (A.9)
The boundary conditions are
[π΄1
(1)(π =
π
2)] = [π΄
1
(2)(π =
π
2)] ,
[π΄1π
(1)(π =
π
2)] = [π΄
1π
(2)(π =
π
2)] .
(A.10)
Similar expressions exist for π΄2
(2)(π ). Using the chain-
relations with π = 2, the analytical solutions are obtained:
[[
[
πβπ(βοΏ½ΜοΏ½π +2π)
π΄1
(2)(π )
(π
π) ππ(βοΏ½ΜοΏ½π +2π)
π΄2
(2)(π )
]]
]
= [1 β π2+ π2 cosππ β ππ sinππ π (π sinππ + π cosππ)
βπ (1 β cosππ) π sinππ + π] [
cos ππ sin ππ
] . (A.11)
The π-differentiated amplitudes are found as
[
[
(π
π) πβπ(οΏ½ΜοΏ½π β2π)
π΄1π
(2)(π )
ππ(οΏ½ΜοΏ½π β2π)
π΄2π
(2)(π )
]
]
= (ππ
π)
β [π1
(2)+ ππ2
(2)π3
(2)+ ππ4
(2)
π5
(2)+ ππ6
(2)π7
(2)+ ππ8
(2)][
cos ππ sin ππ
] ,
π1
(2)= π2{2 (1 β cosππ) β ππ sinππ + ππ sinππ} ,
π2
(2)= π {sinππ β ππ cosππ + ππ cosππ} ,
π3
(2)= π2(β2 sinππ + ππ cosππ) β (1 β π2
+ π2 cosππ) ππ ,
π4
(2)= π {β (cosππ + ππ sinππ) + ππ sinππ} ,
π5
(2)= π {(1 β 2π
2) (1 β cosππ)
β (1 β π2) ππ sinππ + (1 β π2) ππ sinππ} ,
π6
(2)= (1 β π
2) ππ ,
π7
(2)= π {β (1 β 2π
2) sinππ + (1 β π2) ππ cosππ
+ (1 β π2) (1 β cosππ) ππ } ,
π8
(2)= π2.
(A.12)
Segment III (3π/2 β€ π β€ 2π).The equations are
[π΄1π
(3)(π )
π΄2π
(3)(π )
] = ππ [0 π
2ππ3(π )
πβ2ππ3(π )
0][
π΄1
(3)(π )
π΄2
(3)(π )
] . (A.13)
The boundary conditions are
[π΄1
(2)(π =
3π
2)] = [π΄
1
(3)(π =
3π
2)] ,
[π΄1π
(2)(π =
3π
2)] = [π΄
1π
(3)(π =
3π
2)] .
(A.14)
Similar expressions exist for π΄2
(3)(π ). Using the chain-
relations with π = 3, the analytical solutions are obtained:
[[
[
πβπ(οΏ½ΜοΏ½π β4π)
π΄1
(3)(π )
(π
π) ππ(οΏ½ΜοΏ½π β4π)
π΄2
(3)(π )
]]
]
= [
[
1 β π2+ π2 cosππ + ππ {π2 sin 2ππ + (1 β π2) (sin 3ππ β sinππ)} π2 sin 2ππ β ππ {π2 cos 2ππ + (1 β π2) (1 + cos 3ππ β cosππ)}
[π (cosππ β cos 3ππ) + ππ2 (sin 3ππ β sin 2ππ β sinππ)] [π (sinππ β sin 3ππ) + π {1 β π2 + π2 (cosππ + cos 2ππ β cos 3ππ)}]]
]
[cos ππ sin ππ
] .
(A.15)
8 International Journal of Optics
The π-differentiated amplitudes are found as
[
[
(π
π) πβπ(οΏ½ΜοΏ½π β4π)
π΄1π
(3)(π )
ππ(οΏ½ΜοΏ½π β4π)
π΄2π
(3)(π )
]
]
= (ππ
π)
β [
[
π1
(3)+ ππ2
(3)π3
(3)+ ππ4
(3)
π5
(3)+ ππ6
(3)π7
(3)+ ππ8
(3)
]
]
[cos ππ sin ππ
] ,
π1
(3)= 2π2(1 β cos 2ππ β ππ sin 2ππ) + π2
β ππ sin 2ππ,
π2
(3)= π [β3π
2 sin 2ππ β (1 β 3π2) (sin 3ππ
β sinππ) + ππ {2π2 cos 2ππ
+ (1 β π2) (3 cos 3ππ β cosππ)} β {π2 cos 2ππ
+ (1 β π2) (1 β cosππ + cos 3ππ)} ππ ] ,
π3
(3)= β2π
2(sin 2ππ β ππ cos 2ππ) β (1 β π2 + π2
β cos 2ππ) ππ ,
π4
(3)= π [3π
2 cos 2ππ + (1 β 3π2) (1 β cosππ
+ cos 3ππ) + ππ {2π2 sin 2ππ
+ (1 β π2) (3 sin 3ππ β sinππ)} β {π2 sin 2ππ
+ (1 β π2) (sin 3ππ β sinππ)} ππ ] ,
π5
(3)= π (1 β 2π
2) (cos 3ππ β cosππ) + π (1 β π2)
β (sin 3ππ β sinππ) ππ + π (1 β π2) (sinππ
β sin 3ππ) ππ ,
π6
(3)= (1 β π
2) ππ + π
2[(2 β 3π
2) (sinππ
+ sin 2ππ β sin 3ππ) + (1 β π2) (3 cos 3ππ
β 2 cos 2ππ β cosππ) ππ β (1 β π2) (1 β cosππ
β cos 2ππ + cos 3ππ) ππ ] ,
π7
(3)= π (1 β 2π
2) (sin 3ππ β sinππ) + π (1 β π2)
β (cosππ β 3 cos 3ππ) ππ + π (1 β π2) (cos 3ππ
β cosππ) ππ ,
π8
(3)= π2+ π2[(2 β 3π
2) (1 β cosππ β cos 2ππ
+ cos 3ππ) + (1 β π2) (3 sin 3ππ β 2 sin 2ππ
β sinππ) ππ
+ (1 β π2) (sinππ + sin 2ππ β sin 3ππ) ππ ] .
(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 (π΄1
(π)= 1, π =
1, 2, 3) and only the second component changes.(iii) The boundary conditions remain unchanged.
Under these assumptions the dimensionless constant πbecomes οΏ½ΜοΏ½, which is related to the physical lengths as
οΏ½ΜοΏ½ =2
π(Ξ
π0
) . (B.1)
The new equations and their solutions take the followingform.
Segment I (0 β€ π β€ π/2). Perturbative equations are asfollows:
[
[
π΄1π
(1)(π )
π΄2π
(1)(π )
]
]
= ππ[
[
0 π2ποΏ½ΜοΏ½π
πβ2ποΏ½ΜοΏ½π
0
]
]
[1
0] . (B.2)
Solutions are as follows:
π΄2
(1)(π ) = (
π
οΏ½ΜοΏ½) ππβποΏ½ΜοΏ½π sin οΏ½ΜοΏ½π . (B.3)
The sum of squares of the π-differentiated amplitudes is asfollows:
(
π΄1π
(1)(π )
2
+π΄2π
(1)(π )
2
(ππ/οΏ½ΜοΏ½)2
)
pert
=1
2(1 β cos 2οΏ½ΜοΏ½π )
= sin2οΏ½ΜοΏ½π .
(B.4)
So
PCF(1) (π )pert
=[[
[
(π΄1π
(1)(π )
2
+π΄2π
(1)(π )
2
)pert
[π΄1π (π )
2+π΄2π (π )
2]unspun-fiber
]]
]
1/2
=sin οΏ½ΜοΏ½π οΏ½ΜοΏ½π
.
(B.5)
International Journal of Optics 9
Segment II (π/2 β€ π β€ 3π/2). Perturbative equations are asfollows:
[
[
π΄1π
(2)(π )
π΄2π
(2)(π )
]
]
= ππ[
[
0 π2π(βοΏ½ΜοΏ½π +2π)
πβ2π(βοΏ½ΜοΏ½π +2π)
0
]
]
[1
0] . (B.6)
Solutions are as follows:
π΄2
(2)(π ) = π
π(οΏ½ΜοΏ½π β2π)(π
οΏ½ΜοΏ½)
β [β (1 β cos 2π) cos οΏ½ΜοΏ½π + (sin 2π + π) sin οΏ½ΜοΏ½π ] .(B.7)
The sum of squares of the π-differentiated amplitudes is asfollows:
(π΄1π
(2)(π )
2
+π΄2π
(2)(π )
2
)pert
=1
2(ππ
οΏ½ΜοΏ½)
2
{(3 β 2 cos 2π) + (cos 4π β 2 cos 2π) cos 2οΏ½ΜοΏ½π + (sin 4π β 2 sin 2π) sin 2οΏ½ΜοΏ½π } . (B.8)
Expression for PCF is obtained as before.
Segment III (3π/2 β€ π β€ 2π). Perturbative equations are asfollows:
[π΄1π
(3)(π )
π΄2π
(3)(π )
] = ππ [0 π
2π(οΏ½ΜοΏ½π β4π)
πβ2π(οΏ½ΜοΏ½π β4π)
0][
1
0] . (B.9)
Solutions are as follows:
π΄2
(3)= ππ(βοΏ½ΜοΏ½π +4π)
(π
οΏ½ΜοΏ½)
β [{(β1 + cos 2π + cos 4π β cos 6π)
+ π (β sin 2π β sin 4π + sin 6π)} cos οΏ½ΜοΏ½π
+ {(sin 2π + sin 4π β sin 6π)
+ π (1 + cos 2π + cos 4π β cos 6π)} sin οΏ½ΜοΏ½π ] .(B.10)
The sum of squares of the π-differentiated amplitudes is asfollows:
(π΄1π
(3)(π )
2
+π΄2π
(3)(π )
2
)pert
=1
2(ππ
οΏ½ΜοΏ½)
2
{(5 β 4 cos 4π) + (2 cos 10π β cos 8π β 2 cos 6π) cos 2οΏ½ΜοΏ½π + (2 sin 10π β sin 8π β 2 sin 6π) sin 2οΏ½ΜοΏ½π } .(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, βAnalytical theory for polarizationmode dispersion of spun and twisted fiber,β Optics Express, vol.11, no. 19, pp. 2403β2410, 2003.
[2] A. Pizzinat, B. S. Marks, L. Palmieri, C. R. Menyuk, and A.Gastarossa, βInfluence of the model for random birefringenceon the differential group delay of periodically spun fibers,β IEEEPhotonics Technology Letters, vol. 15, no. 6, pp. 819β821, 2003.
[3] A. Galtarossa, L. Palmieri, A. Pizzinat, B. S. Marks, and C. R.Menyuk, βAn analytical formula for the mean differential group
delay of randomly birefringent spunfibers,β Journal of LightwaveTechnology, vol. 21, no. 7, pp. 1635β1643, 2003.
[4] A.Galtarossa, L. Palmieri, andA. Pizzinat, βOptimized spinningdesign for low PMD fibers: an analytical approach,β Journal ofLightwave Technology, vol. 19, no. 10, pp. 1502β1512, 2001.
[5] P. K. A. Wai and C. R. Menyuk, βPolarization mode dispersion,decorrelation, and diffusion in optical fibers with randomlyvarying birefringence,β Journal of Lightwave Technology, vol. 14,no. 2, pp. 148β157, 1996.
[6] C. R. Menyuk and P. K. A. Wai, βPolarization evolutionand dispersion in fibers with spatially varying birefringence,βJournal of the Optical Society of America B, vol. 11, no. 7, p. 1288,1994.
[7] G. J. Foschini and C. D. Poole, βStatistical theory of polarizationdispersion in single mode fibers,β Journal of Lightwave Technol-ogy, vol. 9, no. 11, pp. 1439β1456, 1991.
[8] C. R. Menyuk and P. K. A. Wai, βElimination of nonlinearpolarization rotation in twisted fibers,β Journal of the OpticalSociety of America B, vol. 11, no. 7, pp. 1305β1309, 1994.
[9] N. J. Frigo, βA generalized geometrical representation coupledmode theory,β IEEE Journal of Quantum Electronics, vol. QE-22,no. 11, pp. 2131β2140, 1986.
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