INTERNATIONAL JOURNAL O MICROWAVE AND OPTICAL TECHNOLOGY, 176 VOL.5 NO.3 MAY 1 Polarization Mode Disersion Mitigation through Sun ibers Dowluru Ravi Kumar*, Dr.. Prabhakara Rao * Lecturer in ECE, Deartment of Technical Education, A.P, Hyderabad, India Tel: +91 9444978; E-mail: dowlururavi_kumar@yahoo.com Professor, Deartment of ECE, Jawaharlal Nehru Technological University, Kakinada, India Tel: +91 9618545555; E-mail: drbr@rediffmail.com Abstract- Polarization mode disersion (PMD) in single mode fibers is one of the main limiting factors for high seed otical communication system esecially for systems with bit rates of 4- Gbs and beyond. PMD occurs because of the existence of the birefringence, which is due to slight ellitical nature of the core and asymmetric stresses along the fibers. Thus roagating light is slit into two local olarization modes that travel with different velocities. The resulting difference in roagation times between the two olarizationmodes is known as differential grou delay (DGD). Sinning the fiber during the drawing rocess reduces the differential grou delay and hence the olarization mode disersion. This article resents the numerical simulation of olarization maintaining fibers and sun fibers. Simulation results show that the sun fibers have less differential grou delay comared to the un-sun fibers and thus reducing the effect of olarization mode disersion in single mode fibers. Index Te- irefringence, eat length, Correlation length, Differential grou delay, Polarization mode disersion, Sun fibers I. INTRODUCTION A single mode fiber suorts only the fundamental mode HE 11. In general, an otical wave of arbitrary olarization can be reresented as the linear suerosition of two orthogonally olarized HE 11 modes. or an ideal fiber formed as a cylindrically symmetric waveguide, the two HE 11 modes are degenerate since they roagate with the same roerties. However, random birefringence in real otical fibers removes the degeneracy, resulting in two distinct HE 11 olarization modes with distinct hase velocities, v x and v y and grou velocities, v gx and v gx. In the time domain, PMD leads to ulse broadening which in the lowest order yields a time delay T between the two HE11modes that is termed DGD [1]. PMD can also be viewed in the frequency domain, in which the difference of hase velocities, v v x - v y, results in the rotation of the olarization along the fiber length. Due to the random nature of the fiber birefringence, olarization mode disersion is stochastic and hence a far more comlex henomenon than the chromatic disersion. irst, the advent of the otical amlifier has dramatically increased the transmission distance, resulting in a non-negligible ulse distortion due to the accumulation of PMD along fiber length. urther, the develoments of transmitter and receiver technologies have increased the bit rates in otical communication systems. To satisfy the ever-increasing demand for communication caacity, otical systems are migrating from the current 1-Gb/s to 4-Gb/s or higher. In such high bit rate and long haul communication systems, PMD effects are the main limiting factors. Consider a birefringent fiber that suorts two olarization modes with roagation constants of β1 and β. The difference between the roagation constant of two models is defined as the fiber birefringence β, given by β β β1. It is π related to the beat length as L. The β hysical meaning of beat length is that the state IJMOT-9-11-45 1 ISRAMT
INTERNATIONAL JOURNAL O MICROWAVE AND OPTICAL TECHNOLOGY, 177 VOL.5 NO.3 MAY 1 of light that is reroduced after traveling a distance of L. The PMD, γ of a fiber at a osition z along the fiber is defined as the deferential grou delay (DGD) between the two olarization modes in a unit length and is given by γ d β (1) d where is the angular frequency of light. or a uniform birefringent fiber, the total DGD is roortional to the fiber length L and is given asτ L γ II. IER IRERINGENCE MODELLING iber birefringence is the hysical origin of olarization mode disersion and can be fully described by the local, frequency deendent, birefringence vector defined as b, [ β (, β (, )]/ () ( 1 z The quantities β1(, and β (, are the roagation constants of monochromatic light olarized along or orthogonal to the rincial birefringence axis. Given the random birefringence vector b ( z ) b (, z ) of an otical fiber, the evolution of the PMD vector with distance z is determined according to the dynamical PMD equation τ b + b τ z z (3) iber birefringence results from various erturbations that act on a single mode fiber. These erturbations can be classified as intrinsic and extrinsic erturbations []. Intrinsic erturbations, such as the noncircular core and nonsymmetrical stress fields in the core region, occur during the manufacturing rocess. A noncircular core gives rise to geometric birefringence, whereas a nonsymmetrical stress field creates stress birefringence. iber birefringence also results from external forces, such as lateral stress of the fiber, fiber bending, and fiber twisting. Due to the dynamic nature of the extrinsic forces, the resulting birefringence changes unredictably with time. In other words, both the intrinsic and extrinsic erturbations exist unavoidably in real otical fibers, and the magnitude and orientation of the birefringence tyically varies with time, distance along the fiber, and among fiber samles. To facilitate the study of PMD henomena, a statistical model of fiber birefringence was first advanced by oschini and Poole, which we term the oschini-poole model. In this model, the olarization-deendent roerties of a single mode fiber are reresented by a real 3 3 rotation matrix which is a roduct of two matrix comonents: a length indeendent background comonent and a small length deendent erturbation. This model redicts that otical fibers have nearly fixed birefringence axes, varying only slightly but raidly along the fiber length. Later Wai and Menyuk roosed another two additional birefringent models [3], namely the fixed and random modulus models (MM and RMM). The MM assumes a randomly varying birefringence axis but fixed birefringence strength, while the RMM as sums that both quantities vary randomly. In the MM, the birefringence strength b and the secific grou delay er unit length b at carrier frequency are assumed constant along the fiber, and are π determined by the beat length L, as b, b and b. The rate of change of the birefringence orientation θ is modeled by a white noise rocess g θ (, L dθ / dz g ( (4) θ with g θ ( and g θ ( g θ ( z ) σ θ δ ( z z ) and the variance σ / θ L is given by b () b( b cos[ θ ( θ ()] IJMOT-9-11-45 1 ISRAMT
INTERNATIONAL JOURNAL O MICROWAVE AND OPTICAL TECHNOLOGY, 178 VOL.5 NO.3 MAY 1 z b ex( ) (5) L where L is the fiber elation length. In the RMM, the birefringence vector obeys the Langevin rocess db ab( + g(, (6) dz where g( is an isotroic Gaussian white noise vector with g (, g ( g( z ) σ δ ( z z ) (7) and has the solution z b( b() ex( a + dz g( z ) ex( az a (8) Here a 1/ L and σ b / L are determined by imosing the conditions b() b( b () ex( z / ) and L lim z b ( b where b π / L, if we denote b( n) b(, n) and b ( n) b (, n) as two random series, we can calculate b( n) b ( n) (9) The root mean square DGD is defined asτ τ, the root mean square DGD of olarization-maintaining fibers (PM fibers) is τ PM π z (1) L The esonding exression for the root mean square DGD for un-sun fibers is τ * π [zl L + L ex( z / L ) L III. POLARIZATION MODE DISPERSION MITIGATION THROUGH SPUN IERS ] (11) 1/ Polarization mode disersion (PMD) mitigation can be achieved through two aroaches. The first is to comensate the PMD in already installed systems. ecause PMD fluctuates in time, the comensation must be alied in-site and dynamically, greatly comlicating the develoment of comensators and adding substantially to their exense. The second aroach is to design low-pmd fibers. One rocedure for manufacturing low PMD fibers involves sinning the fiber during the fiber drawing rocess. Generally, the fiber is rotated with tractors, which induces a similar rotation of the fiber's birefringence axis along the fiber length. This sinning can be alied at either a constant or a variable rate. The latter rocedure aears to reduce PMD more than the former. Sinning was imlemented in fiber manufacturing about a decade ago to reduce the PMD of transmission fibers. Exerimentally, the PMD coefficient was reduced from 1.-.1 s / km to.5-.1 s / km, which is almost an exact agreement with the results of this aer. A. Two Models for the irefringence: The evolution along the fiber of the olarization disersion vector β ( is governed by the dynamical equation Ω( β ( + β ( Ω( z (1) where β( is the local birefringence vector. or standard telecommunication fibers β( is the random function of z, whose roerties may be described by the aroriate statistical models. β ( b( ˆ( β, where b( the modulus and ˆ β ( z ).The direction we indicate the IJMOT-9-11-45 1 ISRAMT
INTERNATIONAL JOURNAL O MICROWAVE AND OPTICAL TECHNOLOGY, 179 VOL.5 NO.3 MAY 1 derivative of b( asb W, and in our simulation we assume that b( ) b( b w ( (13) The second Wai Menyuk model is the RMM, which describes β 1( and β ( as indeendent Langevin rocesses dβ i dz ρβ ( + ση ( (14) i i1, ; where η 1( and η ( are indeendent Gaussian white noise rocesses. As a consequence, the modulus b ( is a Rayleigh distributed random variable. The arameters ρ and σ define the statistical roerties of the birefringence, in articular, the elation length is given by L 1/ ρ and the beat length is given 1 / by L π / b π ρ / σ π / σ β. Note that the new arameter σ β is related to the statistical roerties of the modulus of h h birefringence by the relation b h!(σ ). urthermore, in the long-length regime, the mean DGD becomes [4] 16zL zl (15) 8 π τ ( b ( 3π L 3 According to the MM, the local birefringence T vector is β ( z, b( (cos θ (,sin(,) where θ ( is a Wiener rocess, i.e., it obeys the equation dθ / dz σ η(. In this exression, σ is a constant arameter, and η ( is a white noise rocess According to this model, the birefringence elation length and the beat length are given by L 1/(σ ) and L π / b resectively. Once L, L, and α ( are fixed, the statistical roerties of the DGD are comletely defined by changing the erturbation conditions. We cannot change the sin rofile, but we might change L and L, and i β when this haens the PMD statistical roerties are modified. This means that DGD values obtained by changing the erturbation conditions belong to the same statistical ensemble only if the birefringence arameters L and L do not change.. Constant Sinning: Here we consider the first order PMD. The first result we resent concerns the asymtotic evolution of the mean square DGD of a constantly sun randomly birefringent fiber which is given by the asymtotic exression [5] τ b b 4α z 1 + 4 σ σ b (16) where is the otical frequency and, for convenience, we introduced the birefringence strength b π / L, the sin rateα π /, and the square diffusion coefficient σ 1/(L ). The most imortant consequence of this result is that the mean square DGD of a constantly sun fiber grows asymtotically at the same seed as that of an un-sun fiber with equal birefringence arameters. This also means that, as the fiber length increases, the relative DGD reduction due to the sin decreases, and for z the sin has no effects at all. This behavior is fundamentally different from that of eriodically sun fibers, in which the rate of growth of τ is affected by the sin, and the sin effectiveness does not deend on fiber length.however, real fibers have finite lengths; hence the constant sin still has beneficial effects in reducing the DGD. C. Sin Induced Reduction actor (SIR): The sin effect on fiber DGD can be conveniently described by the SIR, and is defined as [6] τ SIR (17) τ unsun IJMOT-9-11-45 1 ISRAMT
INTERNATIONAL JOURNAL O MICROWAVE AND OPTICAL TECHNOLOGY, 18 VOL.5 NO.3 MAY 1 Here the term τ is the root-mean square DGD of sun fibers and τ unsun is that of standard fiber, varies between and 1. or constant sun the SIR is defined below is urely deends on beat length L. 1 1 4α SIR (1 + ) (18) σ Z b whereσ 1/ L, α π /, b π / L. IV. PERIODICALY SPUN IERS The asymtotic inefficiency of the constant sin rofiles can be intuitively derived also from the analysis of eriodically sun fibers. or simlicity, consider a sinusoidal sin rofile A( sin(πz / s ) ; with esonding sin-rate α ( A ( π ( A / s ) cos(πz / s ). If we now increase s and A so that the ratio A / s remains constant, for s the sin rate may be aroximated as a constant function α ( π ( A / s ) π /. ecause of this the behavior of constant sun fibers may be the same as that of a eriodically sun fiber with a very long sin eriod and roer sin amlitude. Indeed, it is shown that as the sin eriod s increases, the effectiveness of the sin decreases and the SIR tends to become unity. In this section, we introduce the exression for the mean DGD of a randomly birefringent eriodically sun fiber. We obtain the result using the MM model of birefringence with the short-eriod assumtion. A. Within the Short-Period Assumtion: Let us consider a randomly birefringent fiber with a local birefringence vector that is modeled using the MM, If that fiber is sun according to a eriodic function A ( of eriod, Here we choose a sinusoidal sin function [4,7,8] A( Z) A Sin(πz / ) with eriod, the SIR is C S SIR (19) C + Where C and S are defined as C S γ ex( Cos[ A( t) A( t ] dtdu () σ γ ex( Sin[ A( t) A( t σ ] dtdu (1) where γ σ /[1 ex( σ )], σ 1/ L, we remark that the exressions given in (19) deend only on the ratio / L.. or L is infinite: Exression for SIR in the case L may be simlified. If L >>, then in both integrals (,1), we may set ex( σ 1, and γ 1. As a consequence, one finds S, and, therefore, the SIR simlifies to, SIR C where C can be rearranged as 1 C cos A( du 1 + sin A( du () This secific case of olarization-maintaining fibers has been analyzed also by Chen et al., using a different aroach and yielding the same result. This result exlains the numerical observation that sin functions otimized for a olarization-maintaining fiber allow quasiotimal DGD reduction, even when the fiber birefringence evolves randomly. V. SIMULATIONS AND RESULTS Numerical simulations are carried out to analyze the reduction in grou velocity disersion for different fibers. ig.1 shows the evolution of mean square DGD of un-sun, olarization maintaining and sun fibers. It can be observed that sun fibers rovide minimum mean square DGD when comared to the other two fibers. IJMOT-9-11-45 1 ISRAMT
INTERNATIONAL JOURNAL O MICROWAVE AND OPTICAL TECHNOLOGY, 181 VOL.5 NO.3 MAY 1 ig.1. Mean square DGD as a function of distance with L 1.5 m L 8 m and 3 with the RMM. ig.4. SIR as a function of sin amlitude A, for different values of ρ 1,.1 ig.. SIR as a function of elation length L for different beat length L is.1,.85, and 1.5m ig.5. SIR as a function of sin amlitude A varying between to 1, for sin eriod 4 ig.3. SIR function of elation length L, for different sin eriods 3, 5, and 15 ig.6. SIR as a function of sin amlitude A varying between to, for sin eriod 4 IJMOT-9-11-45 1 ISRAMT
INTERNATIONAL JOURNAL O MICROWAVE AND OPTICAL TECHNOLOGY, 18 VOL.5 NO.3 MAY 1 ig. and ig.3 deicts the variation of SIR as a function of elation length for different values of beat lengths and sin eriods resectively. rom ig. it can be seen that SIR imroves when the beat length is small comared to the elation length for a given sin eriod and ig.3 shows how SIR imroves with increase in sin eriod for a given beat length. The variation of SIR with the arameter ρ is given in ig.4. inally ig.5 and 6 illustrates deendence of SIR on sin amlitude and sin eriod together. VI. CONCLUSION In this article we reorted an extensive study on the fiber birefringence and Polarization Mode Disersion mitigation through sun fibers. The theoretical characterization of various asects of PMD has been studied. We reresented birefringence by MM model and comared DGDs for un-sun fibers, PM fibers and sun fibers. We observed that the delay is less in sun fibers than normal fibers and PM fibers. The sin effect on fiber DGD is conveniently described by the SIR in both constant sun fibers and eriodically sun fibers. It is observed that in constant sun fibers, the SIR mainly deends on beat length L for a given sin eriod. In eriodically sun fibers we made roer assumtions like (i) short eriod assumtion in which the value of is less than elation length which mainly deends on the factor ρ ( / L ). (ii) L to be infinite, by this assumtion, we notice that for different values of sin amlitude A, we get the desired characteristics between to 5. rom the above observations we conclude that we get less SIR in eriodically sun fibers when comared to constant sun fibers. Delay of Periodically Sun ibers, IEEE Photonics Technology Letters, vol. 15, NO. 6, June.3. [4] Andrea Galtarossa, Member, IEEE, Luca Palmieri, Anna Pizzinat,rian S. Marks,and Curtis R. Menyuk, An Analytical ormula for the Mean Differential Grou Delay of Randomly irefringent Sun ibers IEEE Journal of light wave technology, VOL. 1, NO. 7, JULY 3. [5] Anna Pizzinat, Luca Palmieri, rian S. Marks, Curtis R. Menyuk AnalyticalTreatment of Randomly irefringent Periodically Sun ibers Journal of light wavetechnology, IEEE,VOL. 1, NO. 1, December 3 [6] Andrea Galtarossa, Senior Member, IEEE, Yongmin Jung, Myoung J.Kim, yeong H. Lee, Effects of Sin Inaccuracy on PMD Reduction in Sun ibers IEEE Journal of light wave technology, vol. 3, no. 1, December 5. [7] Luca Palmieri, Member, IEEE Polarization Proerties of Sun Single Mode ibers invited aer to IEEE Journal of light wave technology,vol. 4, NO. 11,November 6 [8] Andrea Galtarossa, Member, IEEE, Paola Griggio, Luca Palmieri, and Anna Pizzinat irst- and Second-Order PMD Statistical Proerties of Constantly Sun Randomly irefringent ibers IEEE Journal of light wave technology, vol., no. 4 Aril 4 REERENCES [1] G. P. Agrawal, Nonlinear iber Otics, 3rd Ed, Academic ress, 1. [] DJaffer K. Mynbaev, Lowell L. Scheiner, iber Otic Communications technology (Pearson education) Asia [3] Anna Pizzinat, rian S. Marks, Luca Palmieri, Curtis R. Menyuk Influence of the Model for Random irefringence on the Differential grou IJMOT-9-11-45 1 ISRAMT