Lie-transform averaging in nonlinear optical transmission systems with strong and rapid periodic dispersion variations

Using Lie-transform techniques, we derive higher-order corrections to the path-averaged model governing evolution of Ž . dispersion-managed solitons in the spectral domain. The result holds in the case of arbitrary including moderate and strong dispersion. The general theory is illustrated by deriving the exact formulas q for a specific symmetric dispersion map. q 2000 Published by Elsevier Science B.V. All rights reserved. Development of ultrafast high-bit-rate optical communication lines is in the focus of extensive research because of the present growing demands on the capacity of transmission systems. High capacity Ž . communication system design has two objectives: i long-haul transmission systems using low dispersion Ž . Ž . fibers dispersion shifted fibers and ii upgrading Ž existing fiber links based on highly dispersive in the main fiber transparency window at 1.55 mm wave. length standard telecommunication fibers. The main factor that limits the bit-rate is pulsebroadening due to the chromatic dispersion of the optical fiber. This broadening is characterized by the Ž Ž .2 .y1 dispersion length Z ; d= BR . Here, d is dis the fiber chromatic dispersion and BR is the bitrate. The dispersion length Z is the distance at which dis ) Corresponding author. Fax: q1-505-665-5757; e-mail: [email protected] the pulse-width approximately doubles due to dispersive broadening. This distance decreases as inverse square of the bit-rate. Another important factor which limits capacity of the fiber links is the nonlinearity of Ž . the fiber refractive index Kerr nonlinearity nsn0 Ž qa I where n is linear part of the refractive index, 0 I is pulse intensity, and a is coefficient of the Kerr . nonlinearity . The spectrum of an optical pulse with characteristic power P will experience noticeable 0 nonlinear distortion at distances greater than the Ž .y1 characteristic nonlinear length, Z s aP . nl 0 In traditional long-haul systems using low dispersion fibers the distance between optical amplifiers required for compensating fiber losses is considerably shorter than that of both the characteristic dispersion length Z and the characteristic nonlinear dis length Z . In other words, both dispersion and nonnl linearity can be treated as perturbations on the scale of the distance between amplifiers and, to first order, only the fiber losses and periodic amplification are 0375-9601r00r$ see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. Ž . PII: S0375-9601 99 00901-9 ( ) I. GabitoÕ et al.rPhysics Letters A 265 2000 274–281 275 significant factors. These factors cause amplitude oscillations, but the shape of the pulse remains approximately unchanged. At the large-scales, pulse propagation in these communication systems is deŽ scribed by the well-established guiding-center path. w x average theory 1–4 . The nonlinear Schrodinger ̈ Ž . equation NLSE is the basis for path-averaged propagation model. For the well-known case of upgrading existing optical links, based on standard mono-mode fiber Ž . SMF , different physical scenarios and modeling Ž equations are valid. SMF has rather high approxi. mately ;17 psrnmPkm dispersion in the 1.55 mm window of optical transparency. As a comparison, the dispersion value for dispersion shifted fiber used in long haul transmission links is ;1 psrnmPkm. The negative impact of fiber chromatic dispersion on the data stream for the same value of bitrate is differs by more greater than order of magnitude for these two values of dispersion. For increasing bitrates system performance degrades as quadratic function of the bitrate value. For multigigabit transmission at 1.55 mm, the corresponding dispersion length in SMF is approximately equal to the amplification distance in the existing networks. Consequently traŽ . ditional guiding-center path-averaged soliton theory can not be applied. The limitations caused by fiber chromatic dispersion can be minimized by dispersion compensation, using pieces of fiber with high dispersion of the opposite sign to the dispersion w x of the original transmission fiber 5 . Dispersion compensation is an attractive technique to enhance the transmission capacity of fiber communication lines. The master equation for dispersion-managed systems is the perturbed NLSE with fast periodic dispersion variations. Stable propagation of disperŽ . sion-managed DM soliton is possible in such sysw x tems 6–10 . The true DM soliton is a periodic solution of the master equation that is entirely recovw x ered after each compensation period 8–10 . In the case of weak dispersion, the Lie transform is a powerful method which can be used to average the w x basic equation directly 11 . For strong dispersion, however, direct averaging is not possible due to the large variations of the coefficient of dispersion. Different theoretical approaches have been developed to describe path-averaged propagation of DM soliton: w x multi-scale analysis 12,13 ; different averaging w x methods 11,14,15 , including averaging in the specw x tral domain 9,10,16,17 and an expansion of DM soliton in the basis of the chirped Gauss–Hermite w x functions 15,18 . Due to the practical importance of the problem, it is very useful to develop different analytical methods to describe the properties of DM soliton. A variety of complementary mathematical approaches can be advantageously exploited to find an optimal and economical description of any specific practical application. Note that the problem treated is not only of a high practical interest, but it is also an interesting fundamental mathematical problem. To determine the limitations of the averaged models it is important to further develop and extend averaging methods to derive first-order corrections to the averaged equations. The averaging approaches considered to date have much in common. To make the averaging procedure possible, one should first eliminate the large variations of the coefficient of the dispersion. In other terms, it is necessary to rewrite the master equation in a different form before applying Lie techniques. This can be done in two ways. One possibility is to make the lens transform accounting for the dynamics of self-similar pulse core. The lens transformation was introduced for the first time for the description of light focusing w x in cubic media 19 . For optical telecommunication w x models it was independently applied in 20 . This approach is appropriate for describing the single w x optical pulse dynamics. A second method 9,10 is to apply a Fourier transform in order to remove rapid variations from the basic equation, thus preparing the equation for averaging. This approach, in addition to single pulse dynamics, can be applied for the analysis of pulse interactions for wavelength division mulŽ . tiplexing techniques WDM , when several data streams with different carrier frequencies propagate through the same fiber. Presently, the WDM technique is the main approach for optical fiber communications. In this Letter, we extend the analysis of Ref. w x 9,10 and using Lie-transforms, we derive higherorder corrections for the path-averaged model in the w x frequency domain 21 . We start our analysis from the master equation 1 2 ) iq q d z q qq q sRq . 1 Ž . Ž . z t t 2 ( ) I. GabitoÕ et al.rPhysics Letters A 265 2000 274–281 276 Ž . Here, d z measures dispersion variations, z is a distance normalized with respect to the soliton period found for the equivalent uniform-dispersion line with the same average dispersion as in the system under consideration. The right-hand-term, Rq, models fast pulse amplitude variations, due to the lossesrgain, that could be present in addition to the dispersion variations, we assume that z <1 is a a fast scale for both dispersion and lossrgain variations. When dispersion variations are moderate in size, za dz d z <1 , 2 Ž . Ž . H 0 w x then, the averaging method 11 , based on the Lie Ž . transform, can be applied to 1 and the resulting ‘averaged’ system represents an almost ‘pure’ NLSE Ž . with small second order in z correction, provided a resonances are absent. These results confirm the stability of solitons under small periodic amplitude perturbations. When dispersion variations are strong, za dz d z G1 , 3 Ž . Ž . H 0 w x then the averaging method 11 can not be applied Ž . directly to 1 . Nevertheless, by applying simpler w x means, an averaged system was derived 9,10 for Ž . the case 3 . This system, which, in general, is no w x longer a NLSE, was obtained 9,10 in the zeroth Ž . order in z only, because the calculation of impora Ž . tant higher order corrections in z could not be a done by simple means. We present a generalization of the averaging method, based on the Lie transform, can be used for Ž . the general case 3 as well. Our method makes it possible to derive the averaged system to any order. The problem of pulse propagation in an optical transmission line composed of optical fibers with Ž alternating dispersion characteristics dispersion . management is one important example of a system Ž . modeled by 1 . Here Rqs ig qq iG z q , 4 Ž . Ž . where g-0 is the damping constant. The amplification necessary to compensate losses, is defined by Ž . G z N G z sG d zynz , 5 Ž . Ž . Ž . Ý 0 a ns0 i.e. a periodic sequence of d-functions. The small parameter z is the amplifier spacing. For many a practical cases, damping term and dispersion terms are not small at all, while the nonlinear term, qq , can be treated as a small perturbation. Before applying the averaging method one needs to prepare the Ž . system 1 by reducing it to the standard form with a w x small right hand side 22 dxrdZse f x ,Z 6 Ž . Ž . in two steps. First, by performing the transformation q t , z sa z w t , z , idardzsRa , 7 Ž . Ž . Ž . Ž . the large damping coefficient g and the d-function Ž . amplification term from the system 1 are removed. We then obtain a system with a variable nonlinearity 2Ž . coefficient a z 1 2 2 ) iw q d z w qa z w w s0 . 8 Ž . Ž . Ž . z t t 2 Ž . 2Ž . The rapidly varying coefficients d z and a z can be split into two parts ̃ 2 ̃ 2 : 2 : d z s d qd , a z s I z s I q I . Ž . Ž . Ž . 9 Ž . 2 : 2 : Here, the mean values d , I and the periodic ̃ ̃ function I are of order 1, while the variable part d of the dispersion can be much greater than 1 for strong dispersion variations. To eliminate this large coeffiŽ . cient from the model system 8 , we need to perform the second step of applying a Fourier transform – based reformulation: ` 1 w t , z s dv u v , z Ž . Ž . H v 2p y` = ̃ 2 : i Id1 2 2 ̃ exp yiv ty v d q iv , 1 2 : 2 2 I