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

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1 International Optics Volume 216, Article ID , 9 pages Research Article Analytical Approach to Polarization Mode Dispersion in Linearly Spun Fiber with Birefringence Vinod K. Mishra US Army Research Laboratory, Aberdeen, MD 215, USA Correspondence should be addressed to Vinod K. Mishra; vkmishr@gmail.com Received 3 October 215; Accepted 4 January 216 Academic Editor: Gang-Ding Peng Copyright 216 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. 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. 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 (L F )orp L F.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 L F 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 arethoseapplicabletothecoupledmodetheorycalculations adapted to the optical fibers 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 vector A(ω, z) with the angular frequency ω and distance z along a twisted fiber. Consider da 1 (z) dz da 2 (z) dz = i e2iθ(z) (Δβ) 2 e 2iΘ(z) A 1 (z). (1) A 2 (z)

2 2 International Optics Θ(s) π/2 I s π II 2π III 3π/2 Figure 1: The 3-segment approximation to the periodic sine function. Here Δβ(ω) is the birefringence and Θ (z) = α η sin (ηz) (2) is the periodic spin profile function with spin magnitude α and angular frequency of spatial modulation η. The boundary conditions are A 1 () =1, da 1 () =, dz A 2 () =, da 2 () dz (3a) =i( Δβ 2 ). (3b) Let s = ηz be a dimensionless variable. We use (d/dz) = η(d/ds) to rewrite (1). Consider A 2ic sin s 1s (s) A 2s (s) =ia e A 1 (s). (4) e 2ic sin s A 2 (s) The subscripts denote differentiation (A 1s =da 1s /ds, A 2s = da 2s /ds). Also, a= (Δβ/2η) and c=(α /η) are dimensionless constants. We express all parameters in terms of the lengths given as beat length (L B = 2π/Δβ), spin period (Λ = 2π/η), andcouplinglength(l = 2π/α ). Then we can write a = Λ/2L B, c = L B /l. The new boundary conditions are A 1 () =1, A 1s () =, A 2 () =, A 2s () =ia. (5a) (5b) These equations ((1) or equivalently (4)) do not have analytical solutions. In the perturbative approximation (see Appendix B), 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 Linear Segment Approximation to the Periodic Spin Function: Analytical Solutions for the Jones Amplitude Equations The Model. Theperiodsofthestraightlinesegmentsshown in Figure 1 approximate the periodic sine function. Here a single period with 3-segment approximation is shown in Figure 1. The field amplitudes for a given segment satisfy the following equations: A 1s (s) =ia A 2s (s) e 2iθ m(s) A 1 (s). (6) A 2 (s) e 2iθ m(s) 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 of θ m (s) should be the same as that of the sine-function spin profile Θ(s) spin = csin s for all segments. Define c = (2c/π) so that the endpoint conditions for segments hold. For n =1,Segment I ( s π/2), θ 1 (s) = cs, Θ (s =) spin ==θ 1 (s=), Θ(s= π 2 ) =c=θ 1 (s = π spin 2 ). (7)

3 International Optics 3 For n =2,Segment II (π/2 s 3π/2), For n =3,Segment III (3π/2 s 2π), θ 2 (s) = cs + 2c, θ 3 (s) = cs 4c, Θ(s= 3π 2 ) = c=θ 3 (s = 3π spin 2 ), (9) Θ(s= π 2 ) =c=θ 2 (s = π spin 2 ), (8) Θ (s =2π) spin ==θ 3 (s=2π). Θ(s= 3π 2 ) = c=θ 2 (s = 3π spin 2 ). The General m-segment Solutions. Thesolutionsforthemth segment have the following general form: e iθm(s) A 1 (s) iae iθm(s) A 2 (s) = a 1 +ib 1 a 2 +ib 2 { θ m/s b 1 +q m a 2 +i(θ m/s a 1 +q m b cos q ms 2 )} { (θ m/s b 2 +q m a 1 )+i(θ m/s a 2 q m b 1 )} sin q m s (1) with q m 2 =a 2 +θ m 2 (s), θ m/s = dθ m (s). ds (11) The exact solutions for the coupled equations in one segment arerelatedtothoseinthepreviousadjacentsegmentbythe following chain-relations among the coefficients. Define u=(q m 1 /q m ), V =(θ m/s θ m 1/s )/q m,andthen the chain-relations are given by a 1 a 2 b 1 b 2 t 1 t 3 t 4 t 2 t 2 t 4 { t = 2 t 4 t t { 1 t +u 3 t 1 t 3 t 4 t + V 1 t 3 } 2 t 2 t 4 } { t 2 t 4 t 3 t 1 t 1 t 3 } a 1 a 2 b 1 b 2. (12) Herethematrixelementsare t 1 = cos q m 1 s m 1 cos q m s m 1, t 2 = cos q m 1 s m 1 sin q m s m 1, t 3 = sin q m 1 s m 1 cos q m s m 1, t 4 = sin q m 1 s m 1 sin q m s m 1. (13) The matrix chain-relations can be written compactly by expressing the 4 4matrices as outer products (denoted by the symbol )of two 2 2matrices as a 1 a 2 b 1 b 2 ={( t 1 t 3 t 2 t 4 +u t 4 t 2 t 3 t 1 ) 1 1 +V t 2 t 4 1 t 1 t 3 1 } a 1 a 2 b 1 b 2. (14)

4 4 International Optics PMD change factor 1 PCF versus s PCF, pert PCF, exact Dimensionless distance, s Figure 2: The PCF curves for a perturbative limit with Λ=1and L B = Calculation of PMD Correction Factor (PCF). The sum of squares of the ω-differentiated amplitudes is similar to power and can be calculated by the following expression using expressions from Appendix A: A 1ω 2 (s) + A 2ω 2 (s) (a ω /q) 2 =( 1 2 )(1 n2 ) {(p 1 ) 2 +(p 2 ) 2 +(p 3 ) 2 +(p 4 ) 2 } +(p 5 ) 2 +(p 6 ) 2 +(p 7 ) 2 +(p 8 ) 2 +( 1 2 )(1 n2 ) {(p 1 ) 2 +(p 2 ) 2 (p 3 ) 2 (p 4 ) 2 } +(p 5 ) 2 +(p 6 ) 2 (p 7 ) 2 (p 8 ) 2 cos 2qs + (1 n 2 ){p 1 p 3 +p 2 p 4 } +p 5 p 7 +p 6 p 8 sin 2qs. (15) 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: A 1ω (s) 2 + A 2ω (s) 2 unspun-fiber (a ω /q) 2 =(qs) 2. (16) ThentheexpressionforthePCFbecomes 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 PCF (1) (s) = 1 n 2 {1 ( 3. Numerical Results 2 sin qs ) }. (18) qs The physical constants ((Δβ, α,η)or equivalently (L B,l,Λ)) and the parameters (n, q) appearing in the PCF expressions are related by q=( 2Λ )1+( πl 1/2 ), πl 4L B n=1+( πl 1/2 ). 4L B (19) 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 < L B ). 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 Figures 2 and 3. It is seen that the curves in Figure 2 for exact and perturbative calculations for small-q approximation are almost identical. The curves in Figure 3 for exact and perturbative calculations are almost identical. Note that after s=5the two curves start diverging a little. PCF (s) = A 1ω 2 (s) + A 2ω 2 (s) A 1ω (s) 2 + A 2ω (s) 2 unspun-fiber 1/2. (17) 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 Figures 4 and 5.

5 International Optics 5 PMD change factor 1 PCF versus s PCF, pert PCF, exact Dimensionless distance, s Figure 3: The PCF curve for a perturbative limit with Λ=1and L B =5. PMD change factor PCF versus s PCF, exact PCF, pert Dimensionless distance, s Figure 4: The PCF curve for a nonperturbative limit with Λ=5and L B =1. PMD change factor 1.9 PCF versus s Dimensionless distance, s PCF, exact PCF, pert Figure 5: The PCF curves for a nonperturbative limit with Λ=12and L B =1. The top and bottom curves in Figure 4 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 alittlelessthan1. The top and bottom curves in Figure 5 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 alittle beyond zero.

6 6 International Optics Parameters: Λ, L B,l (in meters) Table 1: PCF versus z plots. Values (n, q) Comments (1, 12, 1) (.9978,.6379) Λ L B (1, 5, 1) (.9879,.6444) Λ<L B The boundary conditions are A (1) 1 (s=) =1, A (1) 1s (s=) =, A (1) 2 (s=) =, A (1) 2s (s=) =ia. (A.2a) (A.2b) Parameters: Λ, L B,l (in meters) Table 2: PCF versus z plots. Values (n, q) (5, 1, 1) (.7864, 4.475) (12, 1, 1) (.7864, ) 4. Conclusions Comments Λ>L B (physical nonperturbative limit) Λ L B (physical very nonperturbative limit) 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 spinfunction,whichcanbeapproximatedbylinearsegments. 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 (Tables 1and2). Let n=( c 1/2 q )=1+(πl ), (A.3) 4L B and then the analytical solutions are similar to those given in Section 2.2. Consider e i cs A (1) 1 (s) ( q )ei cs A (1) a 2 (s) = 1 in qs cos i sin qs. (A.4) Comparison with general expression gives the following coefficients: a 1 (1) =1, b 1 (1) =, a 2 (1) =, b 2 (1) = n. (A.5) For calculating PCF, the amplitudes have to be differentiated with respect to ω, which will be denoted by subscript ω.some useful relations needed for this are d dω (a q )=n2 ( a ω q ), n ω = n( a q )(a ω q ), q ω =a( a ω q ). a ω = da dω = γ d(δβ), γ = 2η dω, (A.6) Appendix A. Exact Calculation for Segments A.1. The Specific 3-Segment Solutions. The details about solutions for 3 segments follow. Segment I ( s π/2).the equations are A (1) 1s (s) =ia A (1) 2s (s) e 2iθ 1(s) A (1) 1 (s). (A.1) A (1) 2 (s) e 2iθ 1(s) Thenwecanwrite ( q )e i cs A (1) a 1ω (s) e i cs A (1) 2ω (s) =( a ω q )p 1 (1) (1) +ip 2 p (1) (1) 5 +ip 6 p (1) 1 =, p 2 (1) = nqs, p 3 (1) = qs, p 3 (1) +ip 4 (1) p 7 (1) +ip 8 (1) cos qs sin qs,

7 International Optics 7 p (1) 4 =n, p (1) 5 =, p (1) 6 =(1 n 2 )qs, p (1) 7 =, p (1) 8 =n 2. (A.7) Segment II (π/2 s 3π/2).The equations are A 1s (2) (s) A 2s (2) (s) 2iθ e 2 (s) =ia e 2iθ 2(s) The boundary conditions are A 1 (2) (s) A 2 (2) (s). (A.9) Some interesting relations are found as Δβ = ( 4πq Λ ) 1 n 2, z=( Λ 2π )s, α =( 2π2 q 2 Λ )n 1 n 2. (A.8) A 1 (1) (s = π 2 ) = A 1(2) (s = π 2 ), A (1) 1s (s = π 2 ) = A (2) 1s (s = π (A.1) 2 ). Similar expressions exist for A 2 (2) (s). Using the chainrelations with n=2, the analytical solutions are obtained: e i( cs+2c) A (2) 1 (s) ( q = 1 n2 +n 2 cos πq in sin πq n (n sin πq + i cos πq) cos qs )ei( cs+2c) A (2) a 2 (s) n (1 cos πq) n sin πq + i sin qs. (A.11) The ω-differentiated amplitudes are found as ( q )e i( cs 2c) A (2) a 1ω (s) e i( cs 2c) A (2) 2ω (s) =( a ω q ) p 1 (2) (2) +ip 2 p (2) (2) 3 +ip 4 qs cos p (2) (2) 5 +ip 6 p (2) (2) 7 +ip 8 sin qs, p (2) 1 =n 2 {2 (1 cos πq) πq sin πq + qs sin πq}, p (2) 2 =n{sin πq πq cos πq + qs cos πq}, p (2) 3 =n 2 ( 2 sin πq + πq cos πq) (1 n 2 +n 2 cos πq) qs, +(1 n 2 )(1 cos πq) qs}, p 8 (2) =n 2. Segment III (3π/2 s 2π). The equations are A 1s (3) (s) A 2s (3) (s) 2iθ e 3 (s) =ia e 2iθ 3(s) The boundary conditions are A 1 (3) (s) A 2 (3) (s). (A.12) (A.13) p (2) 4 = n { (cos πq + πq sin πq) + qs sin πq}, p (2) 5 =n{(1 2n 2 )(1 cos πq) (1 n 2 )πqsin πq + (1 n 2 )qssin πq}, p (2) 6 =(1 n 2 )qs, p (2) 7 =n{ (1 2n 2 ) sin πq + (1 n 2 )πqcos πq A (2) 1 (s = 3π 2 ) = A1(3) (s = 3π 2 ), A (2) 1s (s = 3π 2 ) = A (3) 1s (s = 3π (A.14) 2 ). Similar expressions exist for A 2 (3) (s). Using the chainrelations with n=3, the analytical solutions are obtained: e i( cs 4c) A (3) 1 (s) ( q )ei( cs 4c) A (3) a 2 (s) 1 n 2 +n 2 cos πq + in {n 2 sin 2πq + (1 n 2 )(sin 3πq sin πq)} = n(cos πq cos 3πq) + in 2 (sin 3πq sin 2πq sin πq) n 2 sin 2πq in {n 2 cos 2πq + (1 n 2 )(1+cos 3πq cos πq)} cos qs n(sin πq sin 3πq)+i{1 n 2 +n 2 (cos πq + cos 2πq cos 3πq)} sin qs. (A.15)

8 8 International Optics The ω-differentiated amplitudes are found as ( q )e i( cs 4c) A (3) a 1ω (s) e i( cs 4c) A (3) 2ω (s) =( a ω q ) p (3) (3) 1 +ip 2 p (3) (3) 3 +ip 4 cos qs p (3) (3) 5 +ip 6 p (3) (3) 7 +ip 8 sin qs, p 1 (3) =2n 2 (1 cos 2πq πq sin 2πq) + n 2 qssin 2πq, p 2 (3) =n 3n 2 sin 2πq (1 3n 2 )(sin 3πq sin πq) + πq {2n 2 cos 2πq +(1 n 2 )(3cos 3πq cos πq)} {n 2 cos 2πq +(1 n 2 )(1 cos πq + cos 3πq)} qs, + cos 3πq) + (1 n 2 )(3sin 3πq 2 sin 2πq sin πq) πq + (1 n 2 )(sin πq + sin 2πq sin 3πq) qs. B. Perturbative Calculation for Segments (A.16) The perturbative approach is based on the following assumptions: (i) The coupling between the polarization states is so small that the equations become decoupled. (ii) The top component is constant (A 1 = 1, m = 1, 2, 3) and only the second component changes. (iii) The boundary conditions remain unchanged. Under these assumptions the dimensionless constant q becomes c, which is related to the physical lengths as p 3 (3) = 2n 2 (sin 2πq πq cos 2πq) (1 n 2 +n 2 c = 2 π (Λ l ). (B.1) cos 2πq) qs, p (3) 4 = n 3n 2 cos 2πq + (1 3n 2 )(1 cos πq + cos 3πq) + πq {2n 2 sin 2πq +(1 n 2 )(3sin 3πq sin πq)} {n 2 sin 2πq +(1 n 2 )(sin 3πq sin πq)} qs, p (3) 5 =n(1 2n 2 )(cos 3πq cos πq) + n (1 n 2 ) (sin 3πq sin πq) πq + n (1 n 2 )(sin πq sin 3πq) qs, The new equations and their solutions take the following form. Segment I ( s π/2). Perturbative equations are as A (1) 1s (s) e 2i cs =ia 1. (B.2) A (1) 2s (s) e 2i cs Solutions are as A 2 (1) (s) =( a c )ie i cs sin cs. (B.3) The sum of squares of the ω-differentiated amplitudes is as p (3) 6 =(1 n 2 )qs+n 2 (2 3n 2 )(sin πq + sin 2πq sin 3πq) + (1 n 2 )(3cos 3πq 2cos 2πq cos πq) πq (1 n 2 )(1 cos πq cos 2πq + cos 3πq) qs, p (3) 7 =n(1 2n 2 )(sin 3πq sin πq) + n (1 n 2 ) (cos πq 3 cos 3πq)πq+n(1 n 2 )(cos 3πq cos πq) qs, p (3) 8 =n 2 +n 2 (2 3n 2 ) (1 cos πq cos 2πq So ( A 1ω (1) 2 (s) + A 2ω (1) 2 (s) (a ω / c) 2 ) = sin 2 cs. PCF (1) (s) pert pert = 1 (1 cos 2 cs) 2 = ( A 1ω (1) 2 (s) + A 2ω (1) 2 (s) )pert A 1ω (s) 2 + A 2ω (s) 2 unspun-fiber = sin cs. cs 1/2 (B.4) (B.5)

9 International Optics 9 Segment II (π/2 s 3π/2). Perturbative equations are as A (2) 1s (s) e 2i( cs+2c) =ia 1. (B.6) A (2) 2s (s) e 2i( cs+2c) Solutions are as A 2 (2) (s) =e i( cs 2c) ( a c ) (1 cos 2c) cos cs + (sin 2c + i) sin cs. (B.7) The sum of squares of the ω-differentiated amplitudes is as ( A (2) 2 1ω (s) + A (2) 2 2ω (s) ) = 1 pert 2 (a ω c )2 {(3 2cos 2c) + (cos 4c 2 cos 2c) cos 2 cs + (sin 4c 2 sin 2c) sin 2 cs}. (B.8) Expression for PCF is obtained as before. Segment III (3π/2 s 2π). Perturbative equations are as A 1s (3) 2i( cs 4c) (s) A (3) 2s (s) =ia e e 2i( cs 4c) 1. (B.9) Solutions are as +i( sin 2c sin 4c + sin 6c)} cos cs + {(sin 2c + sin 4c sin 6c) +i(1+cos 2c + cos 4c cos 6c)} sin cs. (B.1) A 2 (3) =e i( cs+4c) ( a c ) {( 1 + cos 2c + cos 4c cos 6c) The sum of squares of the ω-differentiated amplitudes is as ( A (3) 2 1ω (s) + A (3) 2 2ω (s) )pert = 1 2 (a ω c )2 {(5 4cos 4c) + (2 cos 1c cos 8c 2 cos 6c) cos 2 cs + (2 sin 1c sin 8c 2 sin 6c) sin 2 cs}. (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 Labs and now at United States Naval Academy) for getting him interested in this topic. References 1 M. Wang, T. Li, and S. Jian, Analytical theory for polarization mode dispersion of spun and twisted fiber, Optics Express,vol. 11,no.19,pp ,23. 2 A. Pizzinat, B. S. Marks, L. Palmieri, C. R. Menyuk, and A. Gastarossa, Influence of the model for random birefringence on the differential group delay of periodically spun fibers, IEEE Photonics Technology Letters,vol.15,no.6,pp ,23. 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 spun fibers, Lightwave Technology,vol.21,no.7,pp ,23. 4 A. Galtarossa, L. Palmieri, and A. Pizzinat, Optimized spinning design for low PMD fibers: an analytical approach, Lightwave Technology,vol.19,no.1,pp ,21. 5 P.K.A.WaiandC.R.Menyuk, Polarizationmodedispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence, Lightwave Technology, vol. 14, no. 2, pp , C. R. Menyuk and P. K. A. Wai, Polarization evolution and dispersion in fibers with spatially varying birefringence, the Optical Society of America B,vol.11,no.7,p.1288, G. J. Foschini and C. D. Poole, Statistical theory of polarization dispersion in single mode fibers, Lightwave Technology,vol.9,no.11,pp , C. R. Menyuk and P. K. A. Wai, Elimination of nonlinear polarization rotation in twisted fibers, the Optical Society of America B,vol.11,no.7,pp , N. J. Frigo, A generalized geometrical representation coupled mode theory, IEEE Quantum Electronics,vol.QE-22, no. 11, pp , 1986.

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