3 Conference on Information Sciences and Systems, The Johns Hopkins University, March 12 14, 3 Efficient method for obtainin parameters of stable pulse in ratin compensated dispersion-manaed communication systems K.Nakkeeran,Y.H.C.Kwan andp.k.a.wai Photonics Research Centre and Department of Electronic and Information Enineerin The Hon Kon Polytechnic University Hun Hom, Kowloon Hon Kon e-mail: ennaks@polyu.edu.hk Abstract We obtain a simple and fast method to estimate the enery of the stable soliton solutions for a iven bit rate in a dispersion-manaed soliton communication system utilizin chirped fiber ratins for dispersion compensation. The estimates are in ood areement with the simulation results. I. Introduction Recently, dispersion manaement becomes a key component in hih speed lon-haul optical communication systems. Amon the many different dispersion compensatin methods, the use of chirped fiber ratins (CFGs) is an effective one because of the lare lumped dispersion iven by a short ratin lenth. Also, CFGs can compensate hiher order dispersion, have low insertion loss and no nonlinear effects. It has been shown that solitons exist in dispersion-manaed (DM) systems utilizin CFGs for dispersion compensation [1, 2]. There are no analytical method to determine the soliton solutions in DM systems compensated by chirped fiber ratins. Numerical averain method which can determine the periodic stable soliton solutions in a iven DM system and initial pulse enery have been developed [3]. Similar to classical solitons, the widths of the solitons iven by the alorithm depend on the input pulse eneries only which are conserved by the numerical alorithm. We are, however, interested in the inverse problem, i.e., for a iven DM system and transmission rate (hence the pulse width), we want to determine the eneries of the DM solitons. Of course, we can use the numerical averain alorithm and interpolation to find the pulse enery for a particular transmission rate. The process, however,canbetimeconsumin. In this work, we present a useful and efficient procedure to obtain the Gaussian pulse width and enery of the DM soliton solution in a iven ratin compensated dispersion-manaed communication system [4]. We will layout the procedure to obtain the pulse parameters in DM systems. We also show that our results are in ood areement with those obtained usin our semi-analytical method [3]. II. Variational analysis Pulse propaation in DM systems is overned by the nonlinear Schrödiner equation i q z β(z) 2 q 2 t + 2 γ q 2 q =, (1) where q is the envelope of electric field, z is the normalized transmission distance, and t is the normalized time. The parameter β(z) is the coefficient of roup velocity dispersion (GVD) alon the transmission distance and γ represents the nonlinear Kerr coefficient. The GVD parameter β(z) = β for z (n+1/2)l, where n is an inteer and L is the lenth of periodic dispersion function called dispersion map. The ratins are located at z =(n+1/2)l and their actions are iven by the transfer function F (ω) such that q out(z, ω) =F (ω) q in(z, ω), where ω is the anular frequency, q in and q out are the pulse spectra before and after the ratins. The CFGs have roup delay ripples which is the result of imperfections in the ratin manufacturin processes. The ripple period of roup delay ripples in chirped fiber ratins can be as small as picometer which is much shorter than the sinal bandwidth in our study. The effect of roup delay ripples is to modify the ratin dispersion. When the sinal bandwidth is larer than the ripple period, the effect of roup delay ripples is small and the effective ratin dispersion is equal to the mean ratin dispersion [2]. We, therefore, nelect the effect of ripples in the followin analysis. The filter transfer function is modeled as F (ω) =exp(iω 2 /2), (2) where is the averae lumped dispersion of the ratin. Equations (1) and (2) can be solved by the variational method. We choose a Gaussian ansatz q(z, t) =x 1 exp { (t x 2) 2 /x 2 3 + i [ x 4 (t x 2) 2 /2 +x (t x 2)+x 6]}, (3) where x 1, x 2, x 3, x 4, x,andx 6 depend on z and correspond to the amplitude, temporal position, width, quadratic phase chirp, center frequency and phase, respectively, of the pulse. The evolution of the pulse width and chirp parameters in the optical fibers is iven by the followin coupled equations, dx 3 dz dx 4 dz = βx 3x 4, (4) = ( β x 2 4 4 ) 2γE, x 4 3 x 3 3 () where E = x 2 1x 3 is a constant proportional to the enery of the Gaussian pulse. We launch the pulse at the mid-point of the anomalous dispersion fiber into the DM systems. Durin the pulse propaation, the DM soliton reaches the minimum
pulse width, x 3, at the mid-point of the anomalous dispersion fiber. When DM soliton reaches the end of the fiber sement, the pulse width breathes to the maximum (x 3m) and the pulse will enter a CFG for dispersion compensation. To model the effect of the CFG, we let the lenth of the ratin shrinks to zero while holdin the lumped dispersion of the ratin to a constant. Since the nonlinear Kerr effect induced by ratins is proportional to the ratin lenth, we solve Eqs. (4) and () by settin the nonlinear coefficient γ = in the chirped fiber ratin, we have the effect of the CFG is determined as x 2 3out = x 2 3inH, x 4out = [x 4in (x 2 4in +4/x 4 3in)]/H, (6) H = 2 (x 2 4in +4/x 4 3in) 2x 4in +1, where y in and y out represent the values of the parameter y at the input and the output of the ratin. The DM soliton breathes to the maximum pulse width x 3m, at the input of the ratin, i.e., x 3in = x 3m. We found that the width of DM soliton enters the ratin is the same as that output of the ratin, i.e. x 3in = x 3out. From Eqs. (6), the ratin will reverse the chirp of the soliton without chanin its width if H = 1. Then, x 4in can be expressed as ( ) x 4in± = 1 x 4 3m 4 2 1 ±, (7) x 2 3m where the subscript ± represents the two possible values of the chirp for any iven lumped ratin dispersion value and maximum pulse width x 3m. But for the complete DM system comprisin of fiber and ratin, we find only one value of the chirp (either x 4in+ or x 4in ) isusefulfortheperiodic evolution of the pulse parameters. We can determine which value of x 4in should be used for a iven minimum pulse width of the DM systems, x 3. When x 3m = 2, we can obtain the correspondin minimum pulse width called T min and both values of x 4in+ and x 4in are equal to 1/. This value of T min is used to choose whether x 4in+ or x 4in should be used. For any iven DM systems, if the minimum pulse width in the system, x 3, is less than the value of T min, wehavetouse x 4in+ in Eq. (7), otherwise we use x 4in. Thus the value of chirp enters the ratin can be calculated. After determinin the choice of x 4in, we will find the pulse enery by usin the coupled equations which overns the pulse evolution in optical fibers. Takin the derivative of Eq. (4) with respect to z and then substitutin Eq. (), the resultin equation is a function of x 3. Interatin the equation with respect to x 3,weet ( ) 2 dx3 = 4β2 2 2βγE +2c, (8) dz x 2 3 x 3 where c is the constant of interation. At the middle of the anomalous dispersion fiber, the pulse width reaches its minimum, i.e., dx 3/dz =. The value of interation constant is c =2β 2 /x 2 3 + 2βγE /x 3. (9) Then, substitutin Eqs. (4) and (9) into Eq. (8) by puttin the parameters of the pulse reaches the ratin, i.e., x 3 = x 3m and x 4 = x 4in. Wehave E = x2 4inβx 3 3mx 3 2 2γ(x 3m x 3 ) 2β(x3m + x 3 ) γx 3mx 3. () Interatin Eq. (8) with respect to z, we find the lenth of the fiber to be L =2G γβe ln(4cx3 2 2γβE ) c, (11) c where G = R + γβe 2c 2c ln(2 2cR +4cx c 3m 2 2γβE ), R =2cx 2 3m 2 2βγE x 3m 4β 2. III. Lossless ratin compensated DM systems We have to find the value of T min for any iven DM system with values of L, β, γ and in order to determine the use of x 4in+ and x 4in in the calculation. Substitutin the condition on choosin x 4in, i.e., x 3m = 2 and x 4in =1/, in Eqs. () and (11), the resultin equation becomes a transcendental equation with the variable x 3 which can be numerically solved to obtain the value of T min. Then, we compare the value of T min and our required pulse width x 3 for the DM systems, the choice of x 4in in Eq. (7) is determined. The pulse enery can be calculated from Eq. (). We launch the initial pulse at the mid-point of fiber sements in DM systems and the initial pulse width is the minimum. Given the input Gaussian pulse width x 3, we know the choice of x 4in. The values of parameter β and γ are iven by the DM systems. Therefore, we have to find x 3m in order to obtain the pulse enery for the required pulse width of DM solitons. Usin Eqs. (7), () and (11), we et a transcendental equation for x 3m which can be solved by numerical iterations. Then, the Gaussian pulse enery can be calculated by Eq. (). Therefore, the initial pulse parameters of stable soliton solution for any iven DM system compensated by CFGs is found. The results obtained by variational analysis are well-known to be qualitatively correct. To illustrate the effectiveness of our method for lossless DM systems, we use direct numerical simulations to find the periodic DM soliton solutions and compare the simulated results with the analytical results. We consider a DM system consists of fiber dispersion coefficient of 1 ps/km/nm with lenth of km and we compared three DM systems that respectively used three values for lumped ratin dispersion as 39.73 ps/nm, 43. ps/nm and 46.8 ps/nm for the same fiber dispersion and the lenth of dispersion map. So, the averae dispersion in three systems is different:.21 ps/km/nm,.14 ps/km/nm and.7 ps/km/nm. The values of T min that determine the choice of x 4in in these DM systems are.82 ps, 6.69 ps and 7.38 ps, respectively. Fiures 1 (a), (b) and (c), respectively show the input full width at half maximum intensity (FWHM) of three different DM systems versus the input pulse enery E. We use the results of our analytical method as the initial parameters in the numerical simulation. The solid and dashed curves respectively represent the results obtained from our method and the numerical simulation. Results presented in Fis. 1 show that our procedure is very useful in findin the input parameters of soliton solutions of any iven DM system. IV. Lossy ratin compensated DM systems For findin the fixed point parameters of the DM line with loss and ain, we adopt the followin simple procedure. Let
..1..2 (b)..1..2 (c) (a)..1..2 E (pj) Fiure 1: The initial pulse width as a function of pulse enery E in three ratin dispersions: (a) 39.73 ps/nm, (b) 43. ps/nm and (c) 46.8 ps/nm. us consider DM systems with anomalous averae dispersion which are useful for communications []. In the absence of optical losses the presence of the ratin at the mid-point of the fiber is ood as the input enery is available throuhout the dispersion map. The fiber lenths in the first section (i.e., the section of the fiber before the ratin) and the second section (i.e., the section of the fiber after the ratin) in each dispersion map will be the same and equal to L/2. Due to the losses the input enery will not be available throuhout the dispersion map. As the enery will be exponentially decreasin, we like to shift the location of the ratin in such a way that the averae dispersion of the second section of the dispersion map is decreased with respect to the first section similar to the decrease in enery. So after shiftin the location of the ratin, the modified dispersion map will have lenth of fiber in the first section and L 2 lenth of fiber in the second section. But the total fiber lenth of each dispersion map + L 2 = L will not be altered and hence the averae dispersion of the DM system. The new fiber lenths at each section can be calculated by adjustin the respective averae dispersion. If E is the enery available at the output of the ratin then the averae enery in the second section of the dispersion map will be E a = E [1 exp( αl 2)], (12) αl 2 where α is the loss coefficient of the fiber. Now we decrease the averae dispersion of the second section of the map with the same proportion as the enery: β a2 = βa [1 exp( αl 2)], (13) αl 2 where the averae dispersion of the second-half of the dispersion map with the assumption L 2 = L/2 isβ a =( + Lβ)/L (note that the ratin dispersion value is equally divided as /2 for calculatin the averae dispersion of the individual section of the dispersion map). We can also express the adjusted averae dispersion of the second section as β a2 = 1 L 2 ( 2 + βl2 ). (14) Equatin Eqs. (13) and (14) we derive the transcendental equation with L 2 as 2 βa + βl2 = [1 exp( αl2)], () α which can be solved iteratively by assumin the initial value of L 2 = L/2. But for most practical amplification lenths ( 7 km), we find that the difference between the first iteration value of L 2 resultin from assumption that L 2 = L/2 andthe exact solution of the transcendental equation () is less than 1%. Hence with a valid assumption that L 2 = L/2 in Eq. (13) and equatin it to Eq. (14), we can calculate the new fiber lenth L 2 as αl L 2 = 4β a[1 exp( αl/2)] 2αβL. (16) Hence the decrease in the fiber lenth in the second section of the dispersion map will be L a = L/2 L 2. Then the new lenth of the first section of the fiber has to be = L/2+L a for maintainin the same total lenth and averae dispersion of the dispersion map. In total we are systematically shiftin the location of the ratin in the dispersion map from the knowlede of the averae enery available to the DM system due to losses. To calculate the input parameters of the fixed point we construct a virtual lossless dispersion map with averae dispersion hiher than that of the first section of the lossy dispersion map. In other words, we can say that the lossy DM system after shiftin the ratin location can be mapped to a lossless DM system with hiher averae dispersion than the lossy system. For that we consider the lossless DM system with same lumped ratin dispersion and fiber GVD parameter β but with lenth 2. We use the above method for findin the input pulse parameters (x 3 and E ) of the virtual lossless dispersion map. We find that the pulse width x 3 and the enery E (calculated for the virtual lossless dispersion map) work very well as the input pulse width and averae enery available for the first-half of the lossy DM system. That is after findin x 3 and E for the virtual lossless map the input enery for the lossy system can be calculated as E α E in = 1 exp( αl. (17) 1) To summarize, for lossy DM line we need to calculate the fiber lenth L a of the desired dispersion map as L a = L 2 { 1 } α. (18) 2β a[1 exp( αl/2)] βαl Then modify the dispersion map as L/2+L a lenth of anomalous dispersion fiber followed by ratin and then L/2 L a lenth of anomalous dispersion fiber. The input pulse width (x 3 ) of the modified lossy DM system will be the same as the virtual lossless DM system with fiber lenth L +2L a.butthe input enery (E in) of the modified lossy DM system has to be calculated usin Eq. (17), from the knowlede of the input enery (E ) of the virtual lossless DM system. Fiures 2 (a),
(a) Lossy Dispersion Map L / 2 (first section) (b) Modified Lossy Dispersion Map = L / 2 + L a (first section) L / 2 (second section) (c) Virtual Lossless Dispersion Map L 2 = L / 2 - L a (second section) Fiure 2: Schematic diaram showin the modelin of the lossy DM system..1.2.3 (b).1.2.3 (c) (a).1.2.3 E (pj) in Fiure 3: Plot showin the input pulse width at the beinnin of the first section of the dispersion map of the lossy DM system with lumped ratin dispersion values (a) -39.73 ps/nm, (b) -43. ps/nm and (c) -46.8 ps/nm. Solid and dashed curves correspond to the results of our method and averain method respectively. (b) and (c) respectively show the schematic of the iven dispersion map, modified dispersion map and the virtual lossless dispersion map. To show the effectiveness of our method for lossy DM system, we consider the same above lossless cases considered in Fi. 1 but now with loss coefficient α =.2 db/km. Forthese DM systems, usin Eq. () we have numerically calculated the L 2 values to be 22.76 km, 23.4 km and 24.29 km respectively. Also, usin Eq. (16) we have analytically calculated the L 2 values to be 22.62 km, 23.49 km and 24.28 km respectively. From these L 2 values one can see that the differences are less than 1%. As these L 2 values are very close it is not makin any difference when we use any of these L 2 value in the averain method. Hence one can use Eq. (16) to analytically calculate the value of L 2. Fiures 3 (a), (b) and (c), respectively show the input pulse widths of three different DM systems with losses and ain versus the input enery E in. The solid and dashed curves respectively represent the results obtained from our method and Nijhof et al averain method [3]. From our method we consider the input pulse is a chirp-free Gaussian pulse. But when we use our results as the input for the averain method, the resultin numerical fixed point has some small initial chirp. Fiure 4 shows the value of the initial chirp calculated from averain method versus the input enery E in. The solid, dashed and dot-dashed curves represents the chirp values from DM system with lumped ratin dispersion values -39.73 ps/nm, -43. ps/nm and -46.8 ps/nm, respectively. This shows that our method is also very effective for the lossy DM system. V. Conclusion In conclusion we have presented an easy method for findin the input parameters of all the possible fixed points for Input chirp (THz / ps).1.2.3.4..6.7.1.2.3 E (pj) in Fiure 4: Plot showin the input chirp at the beinnin of the first section of the dispersion map of the lossy DM system with lumped ratin dispersion values -39.73 ps/nm (solid curve), -43. ps/nm (dashed curve) and -46.8 ps/nm (dot-dashed curve).
any iven DM system. Here we have considered the case of a CFG as a lumped dispersion compensator. But our method can be used not only in the case of CFG dispersion compensator but also with any lumped dispersion compensator which can be approximated as a point function without any loss or nonlinearity. Althouh the input parameters that can be obtained from our method deviate slihtly from those obtained from the averain method particularly in the lossy case where there is some small chirp, in practical situation our method canbeusedasanhandytoolformodelinanydmsystem compensated by CFGs. Acknowledments The authors acknowlede the support of the Research Grants Council of the Hon Kon Special Administrative Reion, China (Project No. PolyU96/98E). References [1] S. K. Turitsyn and V. K. Mezentsev, Chirped solitons with stron confinement in transmission links with in-line fiber Bra ratins, Opt. Lett., vol. 23, pp. 6 62, 1998. [2] Y. H. C. Kwan, P. K. A. Wai and H. Y. Tam, Effect of roupdelay ripples on dispersion-manaed soliton communication systems with chirped fiber ratins, Opt. Lett., vol. 26, pp. 99 961, 1. [3] J.H.B.Nijhof,W.ForysiakandN.J.Doran, Theaverain method for findin exactly periodic dispersion-manaed solitons, IEEE J. Sel. Topics Quantum Electron., vol. 6, pp. 33 336,. [4] K.Nakkeeran,Y.H.C.KwanandP.K.A.Wai, Methodto find the stationary solution parameters of chirped fiber ratin compensated dispersion-manaed fiber systems, Opt. Comm., vol. 2, pp. 3 321, 3. [] M. Nakazawa, A. Sahara and H. Kubota, Propaation of a solitonlike nonlinear pulse in averae normal roup-velocity dispersion and its unsuitability for hih-speed, lon-distance optical transmission, J. Opt. Soc. Am. B, vol. 18, pp. 49 418, 1.