# Analytical Approach to Polarization Mode Dispersion in Linearly Spun Fiber with Birefringence

Authors: Vinod K. Mishra
Journal: International Journal of Optics
 (2016)

---

## Abstract

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


---

## Body

## 1. Introduction

The Polarization Mode Dispersion (PMD) is a well-known phenomenon in optical fibers and its role in the propagation of light pulse in various kinds of optical fibers has been a subject of intensive investigation [1–6] in the past. Its physical origin lies in the birefringence property of an optical fiber so that the orthogonal modes of the light electromagnetic wave acquire different propagation speeds resulting in a phase difference between them. The optical fiber at granular level is nonhomogeneous and also has other defects accumulated during the manufacturing process. Due to these issues, the birefringence in a physical fiber becomes random as pointed out by Foschini and Poole in [7]. In addition, Menyuk and Wai [8] have also considered the nonlinear effects arising from higher order susceptibility that also becomes important under certain physical conditions.Sometime ago, Wang et al. [1] derived expressions for the Differential Group Delay (DGD) of a randomly birefringent fiber in the Fixed Modulus Model (FMM) in which the DGD has both modulus and the phase. The FMM assumes that the modulus of the birefringence vector is a random variable. They presented analytical results with the following assumptions: (i) the spin function is periodic (a sine function) and (ii) the periodicity length (p) of the fiber is much smaller than the fiber correlation length (LF) or p≪LF. Later they also generalized the FMM and presented the Random Modulus Model (RMM), which includes the randomness in the direction of the birefringence vector. But then the RMM equations could only be solved numerically.The present work is a contribution to the analytical calculations within FMM and so is only valid for a short fiber distance. This limitation arises because beyond that distance the birefringence randomness [7] becomes dominant. In the present work the full implications of the FMM have been explored under the following conditions: (i) The p≪LF approximation has been relaxed, (ii) a nonzero twist has been included, and (iii) the periodic spin rate has been replaced with a constant spin rate. We give the analytical solutions of the exact FMM equations under these conditions and also present some numerical results based on them showing the effect of different physical conditions. The analytical methods are those applicable to the coupled mode theory calculations adapted to the optical fibers [9].


## 2. Theoretical Analysis

### 2.1. The Model with Periodic Spin Function

The starting point is the well-known vector equation describing the change in the Jones local electric field vectorA→(ω,z) with the angular frequency ω and distance z along a twisted fiber. Consider(1)dA1zdzdA2zdz=i2Δβ0e2iΘze-2iΘz0A1zA2z.Here Δβ(ω) is the birefringence and (2)Θz=α0ηsin⁡ηzis the periodic spin profile function with spin magnitude α0 and angular frequency of spatial modulation η.The boundary conditions are(3a)A10=1,dA10dz=0,(3b)A20=0,dA20dz=iΔβ2.Let s=ηz be a dimensionless variable. We use (d/dz)=η(d/ds) to rewrite (1). Consider(4)A1ssA2ss=ia0e2icsin⁡se-2icsin⁡s0A1sA2s.The subscripts denote differentiation (A1s=dA1s/ds,A2s=dA2s/ds). Also, a=(Δβ/2η) and c=(α0/η) are dimensionless 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 (l0=2π/α0). Then we can write a=Λ/2LB,c=LB/l0.The new boundary conditions are(5a)A10=1,A1s0=0,(5b)A20=0,A2s0=ia.These equations ((1) or equivalently (4)) do not have analytical solutions.In the perturbative approximation (see AppendixB), an analytical result has been derived earlier [1]. In the present work we derive analytic solutions by replacing the sine function by linear segments and compare them to the perturbative solutions for the same segments.


### 2.2. Linear Segment Approximation to the Periodic Spin Function: Analytical Solutions for the Jones Amplitude Equations

The Model. The periods of the straight line segments shown in Figure 1 approximate the periodic sine function. Here a single period with 3-segment approximation is shown in Figure 1.Figure 1
The 3-segment approximation to the periodic sine function.The field amplitudes for a given segment satisfy the following equations:(6)A1smsA2sms=ia0e2iθmse-2iθms0A1msA2ms.The superscript and subscript m both indicate the segments for which the coupled equations hold. The limits of segments are given below.We require that the endpoints ofθm(s) should be the same as that of the sine-function spin profile Θsspin=csin⁡s for all segments. Define c~=(2c/π) so that the endpoint conditions for segments hold.Forn = 1,Segment I(0≤s≤π/2),(7)θ1s=c~s,Θs=0spin=0=θ1s=0,Θs=π2spin=c=θ1s=π2.Forn = 2,Segment II(π/2≤s≤3π/2),(8)θ2s=-c~s+2c,Θs=π2spin=c=θ2s=π2,Θs=3π2spin=-c=θ2s=3π2.Forn = 3,Segment III(3π/2≤s≤2π),(9)θ3s=c~s-4c,Θs=3π2spin=-c=θ3s=3π2,Θs=2πspin=0=θ3s=2π.The General
m
-Segment Solutions. The solutions for the mth segment have the following general form:(10)e-iθmsA1msiaeiθmsA2ms=a1m+ib1ma2m+ib2m-θm/sb1m+qma2m+iθm/sa1m+qmb2m-θm/sb2m+qma1m+iθm/sa2m-qmb1mcos⁡qmssin⁡qmswith(11)qm2=a2+θm2s,θm/s=dθmsds.The exact solutions for the coupled equations in one segment are related to those in the previous adjacent segment by the following chain-relations among the coefficients.Defineu=(qm-1/qm),v=(θm/s-θm-1/s)/qm, and then the chain-relations are given by(12)a1ma2mb1mb2m=t1t300t2t40000t1t300t2t4+ut4-t200-t3t10000t4-t200-t3t1+v00-t2-t400t1t3t2t400-t1-t300a1m-1a2m-1b1m-1b2m-1.Here the matrix elements are (13)t1=cos⁡qm-1sm-1cos⁡qmsm-1,t2=cos⁡qm-1sm-1sin⁡qmsm-1,t3=sin⁡qm-1sm-1cos⁡qmsm-1,t4=sin⁡qm-1sm-1sin⁡qmsm-1.The matrix chain-relations can be written compactly by expressing the 4×4 matrices as outer products (denoted by the symbol ⊗) of two 2×2 matrices as(14)a1ma2mb1mb2m=t1t3t2t4+ut4-t2-t3t1⊗1001+vt2t4-t1-t3⊗0-110a1m-1a2m-1b1m-1b2m-1.


### 2.3. Calculation of PMD Correction Factor (PCF)

The sum of squares of theω-differentiated amplitudes is similar to power and can be calculated by the following expression using expressions from Appendix A:(15)A1ωms2+A2ωms2aω/q2=121-n2p1m2+p2m2+p3m2+p4m2+p5m2+p6m2+p7m2+p8m2+121-n2p1m2+p2m2-p3m2-p4m2+p5m2+p6m2-p7m2-p8m2cos⁡2qs+1-n2p1mp3m+p2mp4m+p5mp7m+p6mp8msin⁡2qs.Here m(=1,2,3) refers to segments in sequential manner.For calculating the normalized PCF we need a similar expression for unspun-fiber given below:(16)A1ωs2+A2ωs2unspun-fiberaω/q2=qs2.Then the expression for the PCF becomes(17)PCFms=A1ωms2+A2ωms2A1ωs2+A2ωs2unspun-fiber1/2.The LHS is a function of parameters n and q and argument s. In general the expressions are quiet complicated, but for the first segment, the PCF is easily calculated and is given by(18)PCF1s=1-n21-sin⁡qsqs2.


## 2.1. The Model with Periodic Spin Function

The starting point is the well-known vector equation describing the change in the Jones local electric field vectorA→(ω,z) with the angular frequency ω and distance z along a twisted fiber. Consider(1)dA1zdzdA2zdz=i2Δβ0e2iΘze-2iΘz0A1zA2z.Here Δβ(ω) is the birefringence and (2)Θz=α0ηsin⁡ηzis the periodic spin profile function with spin magnitude α0 and angular frequency of spatial modulation η.The boundary conditions are(3a)A10=1,dA10dz=0,(3b)A20=0,dA20dz=iΔβ2.Let s=ηz be a dimensionless variable. We use (d/dz)=η(d/ds) to rewrite (1). Consider(4)A1ssA2ss=ia0e2icsin⁡se-2icsin⁡s0A1sA2s.The subscripts denote differentiation (A1s=dA1s/ds,A2s=dA2s/ds). Also, a=(Δβ/2η) and c=(α0/η) are dimensionless 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 (l0=2π/α0). Then we can write a=Λ/2LB,c=LB/l0.The new boundary conditions are(5a)A10=1,A1s0=0,(5b)A20=0,A2s0=ia.These equations ((1) or equivalently (4)) do not have analytical solutions.In the perturbative approximation (see AppendixB), an analytical result has been derived earlier [1]. In the present work we derive analytic solutions by replacing the sine function by linear segments and compare them to the perturbative solutions for the same segments.


## 2.2. Linear Segment Approximation to the Periodic Spin Function: Analytical Solutions for the Jones Amplitude Equations

The Model. The periods of the straight line segments shown in Figure 1 approximate the periodic sine function. Here a single period with 3-segment approximation is shown in Figure 1.Figure 1
The 3-segment approximation to the periodic sine function.The field amplitudes for a given segment satisfy the following equations:(6)A1smsA2sms=ia0e2iθmse-2iθms0A1msA2ms.The superscript and subscript m both indicate the segments for which the coupled equations hold. The limits of segments are given below.We require that the endpoints ofθm(s) should be the same as that of the sine-function spin profile Θsspin=csin⁡s for all segments. Define c~=(2c/π) so that the endpoint conditions for segments hold.Forn = 1,Segment I(0≤s≤π/2),(7)θ1s=c~s,Θs=0spin=0=θ1s=0,Θs=π2spin=c=θ1s=π2.Forn = 2,Segment II(π/2≤s≤3π/2),(8)θ2s=-c~s+2c,Θs=π2spin=c=θ2s=π2,Θs=3π2spin=-c=θ2s=3π2.Forn = 3,Segment III(3π/2≤s≤2π),(9)θ3s=c~s-4c,Θs=3π2spin=-c=θ3s=3π2,Θs=2πspin=0=θ3s=2π.The General
m
-Segment Solutions. The solutions for the mth segment have the following general form:(10)e-iθmsA1msiaeiθmsA2ms=a1m+ib1ma2m+ib2m-θm/sb1m+qma2m+iθm/sa1m+qmb2m-θm/sb2m+qma1m+iθm/sa2m-qmb1mcos⁡qmssin⁡qmswith(11)qm2=a2+θm2s,θm/s=dθmsds.The exact solutions for the coupled equations in one segment are related to those in the previous adjacent segment by the following chain-relations among the coefficients.Defineu=(qm-1/qm),v=(θm/s-θm-1/s)/qm, and then the chain-relations are given by(12)a1ma2mb1mb2m=t1t300t2t40000t1t300t2t4+ut4-t200-t3t10000t4-t200-t3t1+v00-t2-t400t1t3t2t400-t1-t300a1m-1a2m-1b1m-1b2m-1.Here the matrix elements are (13)t1=cos⁡qm-1sm-1cos⁡qmsm-1,t2=cos⁡qm-1sm-1sin⁡qmsm-1,t3=sin⁡qm-1sm-1cos⁡qmsm-1,t4=sin⁡qm-1sm-1sin⁡qmsm-1.The matrix chain-relations can be written compactly by expressing the 4×4 matrices as outer products (denoted by the symbol ⊗) of two 2×2 matrices as(14)a1ma2mb1mb2m=t1t3t2t4+ut4-t2-t3t1⊗1001+vt2t4-t1-t3⊗0-110a1m-1a2m-1b1m-1b2m-1.


## 2.3. Calculation of PMD Correction Factor (PCF)

The sum of squares of theω-differentiated amplitudes is similar to power and can be calculated by the following expression using expressions from Appendix A:(15)A1ωms2+A2ωms2aω/q2=121-n2p1m2+p2m2+p3m2+p4m2+p5m2+p6m2+p7m2+p8m2+121-n2p1m2+p2m2-p3m2-p4m2+p5m2+p6m2-p7m2-p8m2cos⁡2qs+1-n2p1mp3m+p2mp4m+p5mp7m+p6mp8msin⁡2qs.Here m(=1,2,3) refers to segments in sequential manner.For calculating the normalized PCF we need a similar expression for unspun-fiber given below:(16)A1ωs2+A2ωs2unspun-fiberaω/q2=qs2.Then the expression for the PCF becomes(17)PCFms=A1ωms2+A2ωms2A1ωs2+A2ωs2unspun-fiber1/2.The LHS is a function of parameters n and q and argument s. In general the expressions are quiet complicated, but for the first segment, the PCF is easily calculated and is given by(18)PCF1s=1-n21-sin⁡qsqs2.


## 3. Numerical Results

The physical constants ((Δβ,α0,η) or equivalently (LB,l0,Λ)) and the parameters (n,q) appearing in the PCF expressions are related by(19)q=2Λπl01+πl04LB1/2,n=1+πl04LB-1/2.We show results for sets of parameters in two extreme limits to emphasize the difference between the exact and perturbative 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 the spin period.The resulting plots are given in Figures2 and 3.Figure 2
The PCF curves for a perturbative limit withΛ=1 and LB=12.Figure 3
The PCF curve for a perturbative limit withΛ=1 and LB=5.It is seen that the curves in Figure2 for exact and perturbative calculations for small-q approximation are almost identical.The curves in Figure3 for exact and perturbative calculations are almost identical. Note that after s=5 the two curves start diverging a little.The Large-
q
Limit (
Λ
>
L
B
). In this limit two sets of parameters were chosen to get large-q-values (much larger than 1). This corresponds to beat length being smaller than spin period.The resulting plots are given in Figures4 and 5.Figure 4
The PCF curve for a nonperturbative limit withΛ=5 and LB=1.Figure 5
The PCF curves for a nonperturbative limit withΛ=12 and LB=1.The top and bottom curves in Figure4 show exact and perturbative calculations, respectively. It is seen that perturbative approximation underestimates the PCF in this regime. The two start diverging significantly for value of s a little less than 1.The top and bottom curves in Figure5 show exact and perturbative calculations, respectively. It is seen that perturbative approximation underestimates the PCF in this regime. The two start diverging significantly for value of s a little beyond zero.


## 4. Conclusions

The sine-function spin profile can be approximated in general by any number of segments. In this work a 3-segment approximation was chosen and analytical results for the PCF function were obtained. The PCF calculations were also repeated under the assumptions of the perturbative approximation made in [1]. As expected, it was shown that the perturbative approximation has limited validity compared to an exact calculation.The 3-segment approximation given here can be extended to any number of segments for the spin function. The analytical results become very complicated very soon but they will approach the exact results as the number of segments increases. The method is also generalizable to an arbitrary spin function, which can be approximated by linear segments. This applies to almost all practically realizable spin functions. The exact analytic expressions for segment-approximated spin function and approximate numerical calculation of the exact spin function should complement one another to enhance our understanding of the underlying physics (Tables1 and 2).Table 1
PCF versusz plots.
Parameters:Λ,LB,l0  (in meters)
Values (n,q)
Comments
(1,12,1)
(0.9978, 0.6379)
Λ
≪
L
B
(1,5,1)
(0.9879, 0.6444)
Λ
<
L
BTable 2
PCF versusz plots.
Parameters:Λ,LB,l0  (in meters)
Values (n,q)
Comments
(5,1,1)
(0.7864, 4.0475)
Λ
>
L
B   (physical nonperturbative limit)
(12,1,1)
(0.7864, 9.7139)
Λ
≫
L
B (physical very nonperturbative limit)



---

*Source: 9753151-2016-05-23.xml*