Novel Approaches to Numerical Modeling of Periodic Dispersion-Managed Fiber Communication Systems

IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 6, NO. 2, MARCH/APRIL 2000 263 Novel Approaches to Numerical Modeling of Periodic Dispersion-Managed Fiber Communication Systems Sergei K. Turitsyn, Michail P. Fedoruk, Elena G. Shapiro, Vladimir K. Mezentsev, and Elena G. Turitsyna Abstract We present two approaches to numerical modeling of periodic dispersion-managed (DM) fiberoptic communication systems. The first approach is a path-average mapping method giving analytical expression for the transfer function (over a system period) for arbitrary cascaded DM system with different periods of the amplification and dispersion compensation. The second method is an expansion of the signal in periodic DM transmission lines in the complete basis of Gauss Hermite functions with the leading Gaussian zero mode. Theoretical results are verified by direct numerical simulations. Index Terms Author, please supply index terms. E-mail keywords@ieee.org for info. I. INTRODUCTION OPTIMIZATION of optical fiber communication systems is the focus of intensive research nowadays, because of its great practical importance (see, e.g., [1] [48]). Optimal design of high bit-rates multichannel transmission lines effectively using optical fiber bandwidth is a rather complex problem typically requiring massive time-consuming numerical simulations. Though a rather wide range of different data format are investigated, the most studies are concentrated on two principal variants: nonreturn-to-zero (NRZ) and return-to-zero (RZ). Recently, many studies have been focused on the dispersion-managed (DM)-chirped RZ pulses transmission. In the ideal system, this format can be called DM soliton (true periodic signal recovering at the ends of the compensation cells), but a more general case of chirped RZ pulse also demonstrates its great potential for transmission [7]. The energy and chirp of periodic (that are stably reproduced at the ends of the compensation sections) or quasi-periodic (that are reproduced approximately from cell to cell) RZ signal depend on many system parameters (dispersion map, amplification distance, dispersion compensation period), making numerical optimization of the system difficult. Nonoptimized arbitrary RZ pulse launched into the system will emit radiation whose level can be rather high, leading to significant degradation of the system performance. The input Manuscript received July 1, 1999; revised February 4, 2000. This work was supported by the Russian Foundation for Basic Research, under Grant RFBR-99-02-16688, Volkswagen Stiftung, under Grant I/74 686, and INTAS, under Grant 96-0413. S. K. Turitsyn and E. G. Turitsyna are with the Photonics Research Group, School of Engineering and Applied Science, Aston University, B4 7ET Birmingham, U.K. E. G. Shapiro and V. K. Mezentsev are with the Institute of Automation and Electrometry, 630090 Novosibirsk, Russia. M. P. Fedoruk is with the Institute of Computational Technologies, Russian Academy of Science, 630090 Novosibirsk, Russia. Publisher Item Identifier S 1077-260X(00)03858-2. pulse chirp and energy should fit, respectively, the chirp and energy of the periodic (quasi-periodic) DM pulse for a specific dispersion map to diminish the shedding of radiation from the pulse into a dispersive pedestal [26] [48]. Optimization of the input signal parameters in DM fibers links is, therefore, a complicated problem requiring intensive numerical simulations. Though many theoretical approaches have already been developed in the last few years, still many practical problems are open. Obviously, better understanding of the fundamental features of the information carriers in DM lines will lead to further progress in the improvement of the system performance. Saving of the computational time is of crucial importance in the modeling of dense wave-division multiplexing (WDM) systems. It is of evident interest to develop novel advanced theoretical and numerical methods adapted to the massive numerical simulations. Recently, efficient numerical methods have been suggested allowing us to reduce substantially simulation time in long-term numerical optimization of optical fiber links [38] [40]. Further investigations in this direction seem to be an interesting area of research. In this article, we present two useful approaches to the design of periodic DM fiber transmission systems: path-average signal mapping and an expansion of DM signal in the basis of the periodic chirped Gauss Hermite functions. The path-average approach makes it possible to save on the computational time performing some analytical calculations to describe signal transformation over one system period. This method is especially useful when the length of the dispersion compensation section is much larger than the amplification distance and the low power signal is transmitted. In this case, an analytical formula for the signal transfer over compensation section allows us to save significantly on computational time. As it is well known, three major factors cause optical signal degradation and distortion in long-haul high bit-rates fiber communication systems: fiber loss, group-velocity dispersion (GVD), and nonlinearity. Signal power attenuation can be compensated using the optical fiber amplifiers (though recovering is not complete, because amplified spontaneous emission noise is added to the signal, degrading signal-to-noise ratio). Linear signal distortion caused by the GVD in fiber transmission systems can be almost suppressed by the dispersion compensation (mapping) technique. However, the nonlinear effects can still be the primary reason for signal degradation, especially in long-haul transmission systems. In a general situation, signal transformation along the fiber line is caused by a combined action of three mentioned effects and cannot be described in a simple way. The evolution of optical signal can only be determined by numerical simulations of the nonlinear Schrödinger equation (NLSE). Ideal information carrier pulse should recover its parameters (temporal width, 1077 260X/00$10.00 2000 IEEE

264 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 6, NO. 2, MARCH/APRIL 2000 peak power, chirp, spectral width) after passing any single transmission section in the cascaded (periodic) system. However, in practical, complex optical communication systems using WDM this ideal situation can hardly be realized. A more practical goal would be to diminish signal degradation caused by GVD and nonlinear effects. An important information would be how the optical signal phase and form are changed (transferred) from section to section. Transmission of optical signal through the cascaded communication system can be viewed as a mapping problem: action of a single transmission cell can be considered as a transformation of any input field into the signal at the output of the cell. Output of the th section plays a role of the input for the th section and so on. Action of the th section on the signal can be formally written as, which is a clear prescription that straightforwardly can be used for the mapping along a transmission line. Such a transfer function presents the combined effects of attenuation, nonlinearity, and GVD averaged (integrated) over the system period. The transfer function can be easily found for the linear systems, but in general, is nonlinear and nonlocal (integral) operator. In most practical systems of interest, nonlinear effects are relatively small over one compensation period and only accumulation of these effects with propagation affects the signal transmission. Therefore, one can assume that in the leading approximation the signal power and phase evolve in a quasi-linear way, which means in particular, that in the spectral domain, power is changed only because of loss and amplification and the GVD affects only signal phase. Therefore, variations of signal parameters over one period can be averaged out, and accumulating changes are then given by the path-averaged model. The path-averaged propagation equation can be both used for identification of the true periodic carrier characteristics (shape, energy, and chirp for a given pulsewidth) and for evaluation of the slow (accumulated) distortion of any input signal. The second approach to the numerical modeling of optical signal transmission in periodic DM systems that we consider in this paper is an expansion of the transmitting optical signal in the basis of the chirped Gauss Hermite functions. Using expansion of DM pulse in the basis of the chirped Gauss Hermite functions, we will demonstrate that already a few first terms in this expansion give an excellent approximation of DM soliton dynamics, making this basis useful for practical numerical simulations of periodic transmission lines. Concluding the introduction, we present path-average mapping and Gauss Hermite expansion approaches to the design of periodic DM fiberoptic communication systems. We derive a transfer function over one period of the cascaded DM system in the general case of different periods of the amplification and the dispersion compensation. Using the path-average model, we obtain conditions on the stable carrier signal transmission. As a particular result, we prove that the so-called lossless model (see, e.g., [15]) can be applied to practical long-haul fiber transmission systems. As an example of the application of the Gauss Hermite expansion, we investigate DM signal propagation in periodic optical links based on SMF. More specifically, we consider a postcompensation transmission system [23], which uses erbium-doped fiber amplifiers to compensate the effect of fiber losses and units with dispersion-compensating fibers (DCF) to compensate for the chromatic dispersion of the SMF. We demonstrate that instead of solving a partial differential equation, we can solve a few ordinary differential equations to approximate signal transmission with a high accuracy. II. BASIC MODEL The optical pulse propagation in a cascaded transmission system with varying dispersion is governed by Here, is the propagation distance in [km], is the retarded time in [ps], is the optical power in [W], and is the first-order group velocity dispersion measured in [ps /km]. We write,, and as functions of to account for the change of these parameters from fiber to fiber. It is customary to express the coefficient by the associated dispersion parameter by, where is the speed of light and is measured in ps/(nm km). We denote the nonlinear coefficient by, where is the nonlinear refractive index, m is the carrier wavelength, is the effective fiber area, and are the amplifier locations. For simplicity, we consider below a periodic amplification with the period.if is constant between two consecutive amplifiers, is an amplification coefficient after the fiber span between the th and th amplifier. The loss coefficient accounts for the fiber attenuation along a fiber span before the th amplifier, where is given in db/km. High local dispersion significantly changes pulse dynamics in comparison to systems with a constant group velocity dispersion, even if the path-average dispersions are identical. A slow (average) dynamics on the large scales is determined by the effects of nonlinearity, residual (path-averaged) dispersion, and average effects of the fast dynamics. It is customary to make the following transformation:. The evolution of the optical signal envelope along the cascaded fiber transmission system is then given by the NLS equation, with periodic coefficients that can be written in the following form: (2) Here, the field is related to the envelope of the electrical field ; in other words, using instead of a customary, we accounted already for the power variations caused by loss and amplification and moved this dependence into periodic, the dispersion. The periodic function ( ) describes periodic compensation of dispersion with the period and average dispersion. The periodic function describes power decay caused by fiber loss and lumped action of the amplifiers that is accounted through transformation of the pulse power at junctions corresponding to the locations of the optical amplifiers. (1)

TURITSYN et al.: PERIODIC DM FIBER COMMUNICATION SYSTEMS 265 Thus, a power variation between two consequetive amplifiers is given by for. We assume that amplifiers are spaced periodically with the period, so that. We consider a general case when and are rational commensurable, namely, with integer and. Averaging throughout the paper is over a minimal common period. III. METHOD I: PATH-AVERAGE MODEL IN THE SPECTRAL DOMAIN To derive an analytical expression for the transfer function over one section, let us apply following [16], [17] Fourier-like transformation from to a slowly varying field here,. Physical interpretation of this transformation is rather transparent [16]. In the linear propagation regime, the spectral bandwidth is not changed by a rapidly varying dispersion. Only a pulse phase follows rapid oscillations of the local dispersion. Therefore, effects of nonlinearity and small residual dispersion can be accounted as a slow evolution of the envelope ( ) of the quasilinear solution. Propagation equation for the field reads Here (3) (4) corrections caused by average dispersion in the nonlinear term and the kernel function (we will use this expression for in the remaining part of the paper). Requirement yields a condition on the shape of the true periodic carrier signal (DM soliton). For an arbitrary input signal, like, for instance, chirped RZ-formatted data, we can, instead, require minimization of the signal distortion over the single period. Of course, higher order schemes can be easily introduced to make simulations of the one-period transform more accurate. In the numerical simulation presented below, we have used the following procedure: 1) neglecting nonlinearity, a field at a half of the period is formally calculated; 2) using this intermediate step, a field at the end of the section is found; 3) we improve approximation for the field at the intermediate half a period step taking ; and, finally, 4) using this field at the half period, we calculate the output field It is important to note that can be calculated analytically for any particular DM system. To be specific, let us consider as an example, a two-step dispersion map with the amplification distance and dispersion compensation period ( ). Dispersion if and if. The mean-free function defined above can be found as if and if. After some calculations, assuming that the average dispersion is much smaller than the local dispersions, it can be found that the kernel of the transfer function in such a system is and is a delta-function. We can see that in the right-hand-side of this equation we have small parameters, either the average dispersion or the nonlinear term, which means that the evolution of the field is indeed slow ( experiences only small changes over a single period). Integrating over one period [from to ] and assuming that effect of the nonlinearity on the pulse evolution over one period is small, we get the transfer operator with (6) and. It is seen that if nonlinearity and residual (average) dispersion are neglected, we have complete recovering of the pulse at the end of the section. In the practically important limit of a small average dispersion,, here is a characteristic signal bandwidth, we can neglect (5) Here, gain. It is interesting to look at some particular limits in this general formula. First, if (uniform dispersion along the system), we reproduce the result of Mollenauer et al.:, and because is a constant, the path-averaged model is the integrable NLSE. The second limit is the so-called lossless model [15] ( ). In this case,. Note also that the obtained formula can be used to describe four-wave mixing between different channels in WDM systems. The path-averaged signal evolution in the spectral domain is governed by the equation The steady-state solution of this equation describes DM soliton [periodic solution of (1)]. The advantage of the developed path-averaged approach is that it can be applied to an arbitrary input signal and, therefore, can be used for design of DM soliton systems [24] as well as fiber links using more general chirped RZ signals [7]. (7) (8)

266 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 6, NO. 2, MARCH/APRIL 2000 Fig. 1. Transfer function T(X) (in the systems with L =2MZ Z )is plotted (solid line) for different M: bottom M =1, middle M =3, and top M =10. Dotted lines show corresponding transfer functions for the lossless model. Here, amplification distance Z =50km, X =(!0! )(!0! )Z d. A. Comparison of Direct Numerical Simulations and the Path-Average Approach First, we justify now the use of the lossless system [15] for modeling of the practical (with fiber loss) fiber transmission system. The developed theory above confirms that the periodic amplification and dispersion compensation can be handled as separate problems, provided that amplification distance is substantially different from the period of dispersion map. This is illustrated by Fig. 1, where transfer function is shown by a solid line for different (here, amplification distance km and db/km). It is seen that for function is indeed getting closer and closer to that one (shown by dotted line) for the lossless model multiplied by the path-averaged factor. This proves that the power budget and the dispersion mapping, effectively, can be handled separately in long-haul transoceanic optical communication systems, where amplification distance is typically much shorter than the dispersion compensation period. In Fig. 2, the shape of the DM soliton for a lossless system found using the path-average mapping condition (filled squares) by numerical simulations of the full model (solid line) is plotted [arbitrarily normalized solution of (2) with and is shown]. In Figs. 3 5, we compare evolution of the general RZ input signal having Gaussian shape in the full model (1) and in the above-described path-averaged model. Fig. 3 shows evolution of the signal bandwidth over a distance of 8000 km. The solid line corresponds to direct numerical simulations of the full model, and rhombs are for the simulations based on the path-average mapping at the ends of the compensation sections. The input Gaussian signal with the peak power 1 mw and width ps propagates in the system composed from fibers with ps/(nm km) and ps/(nm km) and the length 200 km each. Amplification distance is 50 km (. We can see a good agreement during a rather long propagation distance. In Fig. 4, the evolution of the maximum of the spectral power (relative its value at ) is shown for the same signal as in Fig. 3, calculated using the path-average mapping (rhombs) and direct numerical simulations of the full model (solid line). In Fig. 5, signal spectral shape after transmission over 8000 km is plotted. Comparison is shown of the spectral power distributions after 8000 km found by path-average mapping (dashed line) and by direct numerical simulations of the full model (solid line). The same input signal and system parameters is in Fig. 3. Again, agreement is rather good. This comparison indicates that the suggested method can be useful in long-term numerical simulations. The presented method works well when nonlinear effects should be taken into account, but powers are still not too high and SPM effects over one compensation period do not significantly affect signal form. B. Path-Average Theory of Intrachannel Four-Wave Mixing In this section, applying the path-average approach presented above, we develop a simple theory of the so-called intrachannel four-wave mixing (ICFWM) [42], [43] that is one of the major limiting factors for DM bit-overlapped RZ transmission. Bit overlapping caused by periodic large pulse spreading during propagation occurs in the systems with strong DM, for instance, using standard fibers at bit rates of 40 Gbit/s and more. Intrachannel FWM manifests itself by generation of shadow ( ghost ) pulses in the middle of the zero time slots caused by FWM between spectral components within a single channel. In this section, applying the path-average theory of ICFWM developed above, we describe initial growth, saturation, and slow (path-averaged) oscillations of FWM components. Note that even such an approximate theory is capable of identifying regimes of DM solitons transmission with suppressed ICFWM. As a particular application of the general theory, we examine here the line studied in [42] with the amplification distance equal to the compensation period, located at the end of the dispersion map., and the amplifier

TURITSYN et al.: PERIODIC DM FIBER COMMUNICATION SYSTEMS 267 Fig. 2. Shape of the DM soliton for the lossless system found using the path-average mapping condition q(!; (k +1)Z )=e q(!; kz ) (filled squares), the true periodic solution (DM soliton) found numerically solving the full model (solid line). Arbitrarily normalized solution of (2) with d(z) =65 +0:15 and c = 1is shown. As shown in the previous sections, to describe signal waveform transform over one compensation period, it is convenient to apply a Fourier-like transformation from an original optical field to a slowly varying envelope (9) Here, the periodic part of the accumulative dispersion with and the dispersion if and if,,. Indexes (1, 2) correspond to SMF and DCF, respectively. The path-averaged propagation equation on the slow-varying envelope is given by (8), with,. Here, the power variation function Fig. 3. Evolution of the signal bandwidth over 8000 km. Comparison of the path-average mapping (rhombs) and direct numerical simulations of the full model (solid line). Input Gaussian signal with the peak power 1 mw and width T =33ps propagates in the system composed from fibers with D = 2:6 ps/(nm1km) and D = 02:2 ps/(nm1km) and the length 200 km each. Amplification distance is 50 km (M =0:5 L=Z =4). (10),. The matrix element can be easily calculated for the considered system Here,,. Function has the meaning of the spectral density (efficiency) of the FWM. To describe generation of the FWM term at some, we decompose the total path-average wave field into four parts (in accordance with the process ):, with. Assuming

268 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 6, NO. 2, MARCH/APRIL 2000 that the FWM term is small compared with the other terms, we can linearize the path-average equation. Slow (path-averaged) evolution of the FWM components at is governed by (11) This equation describes generation and evolution of FWM components along a cascaded DM fiber line in a general situation. Now, we apply the path-average model to describe the intrachannel FWM process. We would like to point out that we present here a simplified model of the process. For instance, we will limit consideration by the main ICFWM process considering central frequency and two side bands only (this process gives the main contribution into the growth of shadow pulses). More general variants of a path-average theory can be easily formulated. The aim of this section is to demonstrate that even such a simple model of the effect allows us to describe basic features of intrachannel FWM and to describe system design advantageous for suppression of the ghost pulses. To be specific, consider, for instance, 40-Gbit/s transmission with a corresponding ps time slot. As mentioned in [42], spectral power distribution for a long-enough bit pattern consists from few narrow peaks around the central carrier frequency. The main side peaks are spectral components of input signal (in a single channel) at separated by from the central carrier frequency. Amplitudes of the side peaks decay very fast; therefore, we predict that the main contribution to ICFWM at the carrier frequency will be generated by the process. Spectral widths of the central peak (at ) and spectral components at ( ) are determined by statistical properties and length of the pattern and tend to zero when the length of the transmitted word tends to infinity. Then, formally, we can approximate these peaks by delta-functions. To calculate generation of ICFWM term at, we decompose the total path-average wave field as, with. Assuming that at the beginning the ghost pulses are absent ( ), we get for a slow (path-averaged) initial linear growth (small ) of the ghost pulses amplitude at caused by ICFWM process : ; here,. The power of the shadow waves generated by the process caused by ICFWM growths (for small ) as. Spectral power of FWM terms is proportional to and, therefore, points of minima correspond to operation regimes with suppressed FWM [44]. For a given choice of system parameters, as a function of only slightly varies at large amplification lengths. However, for shorter amplification distances, FWM can be enhanced operating in the region of local maxima. The derived formula for can be used for suppression of the FWM effects by system design. To describe saturation of the FWM and oscillations of the ghost pulses after the initial growth stage is completed, let us ad- Fig. 4. The same input signal and system as in Fig. 3. Evolution of the maximum of the spectral power (relative its value at z =0). Comparison of the path-average mapping (rhombs), and direct numerical simulations of the full model (solid line). Fig. 5. Comparison of the spectral power distributions after transmission over 8000 km found by path-average mapping (dashed line) and by direct numerical simulations of the full model (solid line). The same input signal and system parameters as in Fig. 3. ditionally assume that the spectral power distribution propagates in the leading order without significant distortion, similar to a single path-averaged DM soliton evolution. Even though in the typical case the spectral power distribution changes (slightly) during the path-average evolution, the main peaks are still at,, and. The phase of the whole signal pattern (in the spectral domain) approximately grows linearly with distance similar to a single path-averaged DM soliton. Again, for simplicity, we calculate only the main contribution to FWM from the signal spectral components at,, and. In other words, we approximate spectral power distribution for long random input pattern as. Here, is a corresponding quasi-momentum of the central peak. Evolution of the ICFWM term at the carrier frequency is given by

TURITSYN et al.: PERIODIC DM FIBER COMMUNICATION SYSTEMS 269 (12) Assuming, we can solve this equation and get for the power of ICFWM term (13) Here,. Fig. 6 illustrates saturation of initial growth and slow (path-average) oscillations of the ICFWM components for the following pattern considered in numerical simulations: 01100110010011101000110101001100. Note that in the numerical simulations, we have launched input Gaussian signal with optimized parameters rather than a perfect DM soliton, and this is one of the reasons why ICFWM components oscillations in a practical system deviate from the proper sinusoidal form given by (13). It should be pointed out, however, that the period of ICFWM components oscillations given by (13) is in pretty good agreement with numerical results shown in Fig. 6. As it is seen from Fig. 6 and (13), we can suppress ICFWM either by decreasing the slope of the initial growth (proportional to ) or (less reliable, but, in principle, also possible approach) by adjusting the period of ICFWM oscillations to the desirable transmission distance. Fig. 6. Peak power evolution (stroboscopically shown at the ends of the compensation sections) of the ghost pulses that emerge in the middle of the time slots occupied by zeros. Here, solid line bit number 4, long-dashed line bit #8, dashed line bit #12, dotted line bit #16, dash-dotted line bit #23, dash-doubledotted line bit #28. TABLE I PARAMETERS OF THE SYSTEM IV. METHOD II: DM PULSE PROPAGATION PRESENTED IN THE COMPLETE BASIS OF THE CHIRPED GAUSS HERMITE FUNCTIONS The second method we discuss in this paper is an expansion of the propagating signal in the complete basis of the chirped Gauss Hermite functions. To be specific (but without loss of generality), we analyze a postcompensation transmission system [23], which uses erbium-doped fiber amplifiers to compensate the effect of fiber losses and units of DCF to compensate for the chromatic dispersion of the SMF. The system setup is as follows: the transmission line consists of postcompensating sections composed of pieces of 102-km SMF and 17.245 km. The amplifier gain equalizes the loss between two consecutive amplifiers, both for SMF and DCF pieces. The parameters of the system are presented in Table I. A. Pulse Expansion and the Path-Average Model in the Time Domain As shown in [45] (see also [46] [48]), the envelope of the the electrical field of DM pulse in the general case can be advantageously expanded using a complete basis of chirped Gauss Hermite functions here, is the envelope of the electrical field ( is measured in W, length is measured in km, and time in ps, respectively). and are the periodic solutions of the equations where (15) and,if. Here again the upper indexes (1, 2) correspond to SMF and DCF, respectively. Equation (15) should be solved with periodic boundary conditions: and. is a power constant to be determined from a requirement that and are periodic solutions to (15). Varying, we obtain for periodic solutions of (15) curves in the planes and. These equations are basic for the design of the DM fiber links. The normalized orthogonal Gauss Hermite functions are (14) (16)

270 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 6, NO. 2, MARCH/APRIL 2000 Here, is the th-order Hermite polynomial; therefore, a zero term in the above expansion is a Gaussian pulse. Finally, and is found from. Fig. 7 shows the dispersion map (solid line) and periodic solutions of (15) describing fast evolution of the pulsewidth and chirp over one compensation period: (solid line with filled square), (long-dashed line), and (short-dashed line). Coefficients in this expansion are determined from a system of the ordinary differential equations (see for details [48]) Here, we introduce notation and (17) Fig. 7. The dispersion map (solid line) and periodic solutions of (15) describing fast evolution of the pulsewidth and chirp over one compensation period: T (solid line with filled square), M (long-dashed line), and R (short-dashed line). (18) Because integrals of the form can be calculated analytically, it is possible to determine any and that are some fixed numbers. Symmetrical integrals are,. The other are zero if. Note that for known coefficients can be determined as of the DM soliton is engaged in the zero self-similar mode, it is enough to consider only few-mode approximation to describe the most important properties of DM soliton. Solving equations for a few coupled, we can predict dynamics of an arbitrary initial chirped RZ signal. Additionally, in the case of a low power signal, we again can develop path-average theory. The path-average procedure in time domain can be described as a splitting of into slow ( ) and fast ( ) varying parts ( under the assumption that the rapidly varying part is small compared with the slow-varying one. Averaging over one period, in the leading order, we obtain for the following system of equations: (19) and, thus, arbitrary initial field distribution can be expanded in the complete set of chirped Gauss Hermite functions. We would like to point out that this method allows us to describe not only true periodic carrier signal, but also any arbitrary RZ signal transmission along DM lines. Practical guidelines of exploiting this expansion are as follows: 1) to determine and for given dispersion map using (15) (in the case of in-line filtering with the boundary conditions accounting for the filtering); 2) to calculate defined as ; 3) to present input signal as, where is a launch point; 4) (20) Considering the solution of the form,we obtain the expansion of the DM soliton in the complete basis of the chirped Gauss Hermite functions. The shape of any DM soliton can be found from a solution of the equation (21) 5) to calculate evolution of using as initial conditions in (17). To find true periodic soliton, we should solve (19) for with the periodic boundary conditions. Even though this procedure does not look simple at all in comparison with direct numerics, it has an important advantage. When almost all of the energy True DM soliton, then, can be found solving this set of the algebraic equations. It should be pointed out that the path-average description given by (20) formally works only for a DM soliton with low power. Previous results (before averaging) are very general and applied in a wide range of parameters, including rather powerful signals. A more simple and direct path-average description (20) is valid only when nonlinearity is relatively

TURITSYN et al.: PERIODIC DM FIBER COMMUNICATION SYSTEMS 271 small (characteristic nonlinear distance is larger than the compensation period). This description is very similar to the pathaveraged condition for the conventional soliton in systems with uniform dispersion [1]. Nevertheless, in some way, the path-average (20) can be used even in the case of a large amplitude soliton. Now, we consider how few-mode solutions of the algebraic equations (21) can be exploited so that formally the path-average concept is not valid. Formally, the expansion (14) has one arbitrary parameter. This freedom can be exploited to approximate a DM soliton even if formally the path-average theory is not valid. The use of nonlinear equations (15) to describe rapid evolution of the signal over one period leads to the substantial reduction of the second mode ( ) in the expansion. In this approach, a nonlinear effect related to the interaction of the zero and the second modes is effectively included into the fast dynamics of the signal over one period. Therefore, parameter in (21) can be approximately found from the condition. Zero-mode Gaussian approximation with such gives a good approximation of an optimal input signal with reduced secondary oscillations during propagation. Theoretical results of this section will be verified below by intensive numerical simulations. B. Comparison of Direct Numerical Simulations and Signal Presentation in the Basis of the Gauss Hermite Functions An expansion of the propagating signal in the complete basis of the chirped Gauss Hermite functions gives an advantage of reducing original partial differential equations to the set of ordinary differential equations. This procedure can be considered as a variant of the Galerkin approximation for this nonlinear evolution problem. Now, we demonstrate that already few modes are enough to describe a carrier signal with good enough accuracy. Parameters of the signal are shown in the figures, and parameters of the system in the following figures are described in Table I. In Fig. 8, a comparison is shown of the two-mode approximation ( ; dashed line), and three-mode approximation ( ; short-dashed line), with the power distribution of true DM soliton (solid line) at the beginning of the section ( ): a) logarithmic scale and b) conventional scale. Fig. 9 shows (in the logarithmic scale) dependence of the coefficients (compared with the zero mode )in the expansion of DM soliton on taken at. It is seen that effectively in the core DM soliton is close to Gaussian pulse. Dynamics over one period of the first four nonzero coefficients scaled by the zero (Gaussian) mode, (long-dashed line), (dashed-dotted line), and (short-dashed line), is presented in Fig. 10. Approximation of the signal evolution over one period by few modes is presented in Figs. 11 and 12. Fig. 11 shows peak power evolution over one period of the DM soliton (solid line), two-mode approximation ( modes dashed line), and four-mode approximation ( modes short-dashed line). Evolution of the pulsewidth (FWHM) over one period is shown in Fig. 12: true DM soliton (solid line), two-mode approximation ( modes dashed line), and four-mode approximation ( modes short-dashed line). These results, indeed, prove that the dynamics over one compensation period could be well ap- Fig. 8. Comparison of the two-mode approximation (n =0; 4; dashed line) and three-mode approximation (n =0; 4; 6; short-dashed line) with the power distribution of true DM soliton (solid line) at the beginning of the section (z = 0): (a) logarithmic scale and (b) conventional scale. Fig. 9. Dependence of the coefficients jb (z)j (compared with the zero mode jb j ) in the expansion of DM soliton on n taken at z =0is shown in the logarithmic scale. It is seen that effectively in the core DM soliton is close to Gaussian pulse. proximated by the few modes in the expansion. Next, we should prove that this feature holds in the long-term evolution over many periods. Fig. 13 shows a slow evolution (shown strobo-

272 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 6, NO. 2, MARCH/APRIL 2000 Fig. 10. Dynamics over one period of the first four nonzero coefficients jb j scaled by the zero (Gaussian) mode: n =2(long-dashed line), n =4 (dashed-dotted line), and n =6(short-dashed line). Fig. 12. Evolution of the pulsewidth (FWHM) over one period is shown for true DM soliton (solid line), two-mode approximation (0 +4modes dashed line), and four-mode approximation (0+2+4+6modes short-dashed line). Fig. 11. Peak power evolution over one period is shown for true DM soliton (solid line), two-mode approximation (0 + 4 modes dashed line), and four-mode approximation (0 +2+4+6modes short-dashed line). scopically at the ends of sections) of the input signal having the form of the two-mode (dashed line), four-mode (short-dashed line), and nine-mode (dotted line cannot be distinguished from the solid line) approximations and true DM soliton (solid line). To verify the approach based on the path-average (20), we perform long-term simulations of the evolution of input signals given by (14) with different initial parameters. It is well known that an arbitrary input pulse launched in the DM system, in addition to the rapid periodic oscillations, experiences slow quasi-periodic variations of the parameters, as shown in Fig. 14. To describe quantitatively these secondary oscillations, we introduce the following characteristic parameter : (22) The smaller, the more the signal is close to the periodic one. A perfect DM soliton has. In Figs. 15 and 16, we present results of these massive numerical simulations. Each point in Fig. 13. Slow dynamics (shown stroboscopically at the ends of sections) of the input signal having form of the two-mode (dashed line), four-mode (short-dashed line), and nine-mode (dotted line cannot be distinguished from the solid line) approximations and true DM soliton (solid line). Fig. 15 corresponds to the numerical simulations of a propagation of some input signal given by (14) over typical total distance of 8000 km. Different curves here correspond to input pulses given by (14) with different. Coefficients are approximated by (note that, formally, the path-average theory should not work in this range of parameters) found from (21) for different initial parameters. An important observation from Fig. 15 is that almost always minimal secondary oscillations (minimal ) correspond to the condition. Fig. 16 shows counterplot of the isolines with different as functions of and. The bold line corresponds to the condition. Again, we can see that the minimum of secondary oscillations corresponds to the requirement. These results indicate that even simple path-average theory can be applied in a rather wide range of parameters to minimize secondary oscillations during signal transmission. Note also that Gaussian approximation used in the previously developed variational approach and in the more advanced root-mean-square momentum method is nothing

TURITSYN et al.: PERIODIC DM FIBER COMMUNICATION SYSTEMS 273 more than the zero mode in the expansion of the true DM soliton in a complete set of chirped Gauss Hermite functions. The higher order modes in the above expansion present an inherent part of DM soliton, even though the zero mode (Gaussian) can hold most of the pulse energy. As proved above (see also [45] [48]), already a few modes in this expansion can provide an excellent description of DM pulse propagation, making this method useful for practical numerical simulations of periodic transmission lines. V. CONCLUSIONS Fig. 14. Scheme showing slow evolution (stroboscopically at the ends of sections) of an arbitrary RZ input signal. Parameter s = (P 0 P )=(P + P ) characterizes the amplitude of the secondary oscillations. Fig. 15. Minimization of the secondary oscillations in the slow evolution. Different curves correspond to input pulses given by (14) with different k. Coefficients B are approximated by F found from (21) for different initial parameters T (0). Minimal secondary oscillations (minimal s) correspond to the condition B =0. In conclusion, we have presented two novel approaches to the design of DM fiberoptic communication systems. The first approach, the path-average mapping method, can be useful to investigate low power signal propagation in periodic transmission systems, with the dispersion compensation period much larger than the amplification distance. We derived path-averaged transfer function (over system period) of the cascaded DM system in the general case of different lengths of the amplification and the dispersion compensation. Using the path-average model, we obtain conditions on the stable carrier signal transmission. As a particular result, we have justified the application of the so-called lossless model [15] to practical long-haul fiber transmission systems. We have also presented a path-average theory of intrachannel FWM. The second method discussed in this paper is the expansion of an arbitrary RZ signal in the basis of the chirped Gauss Hermite functions. As an example of the application of such an expansion, we investigate DM RZ signal transmission in periodic optical links based on SMF. Instead of solving the basic partial differential equation, we can solve few ordinary differential equations to approximate signal transmission with good accuracy. This procedure can be considered as a variant of the Galerkin approximation for this particular nonlinear evolution problem. These relatively simple and computer time-saving approaches can be useful in massive numerical simulations requested for optimization of DM fiber links using DM soliton, chirped RZ, or any other signal formats. REFERENCES Fig. 16. Counterplot of the isolines with different s in the plane (k, T (0)). Bold line corresponds to the condition B =0. [1] L. F. Mollenauer, J. P. Gordon, and P. V. Mamyshev, Solitons in high bitrate, long-distance transmission, in Optical Fiber Telecommunications, I. P. Kaminow and T. L. Koch, Eds. New York: Academic, 1997, vol. IIIA. [2] L. F. Mollenauer, P. V. Mamyshev, and M. J. Neubelt, Demonstration of soliton WDM transmission at up to 8 2 10Gbit/s, error-free over transoceanic distances, in OFC 96, San Jose, CA, Post Deadline Presentation, PD22-1. [3] J. P. Gordon and H. A. Haus, Random walk of coherently amplified solitons in optical fiber transmission, Opt. Lett., vol. 11, pp. 665 667, 1986. [4] R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, Four-photon mixing and high-speed WDM systems, J. Lightwave Technol., vol. 13, pp. 841 849, 1995. [5] C. Lin, H. Kogelnik, and L. G. Cohen, Optical-pulse equalization of low-dispersion transmission in single-mode fibers in the 1.3 1.7 m spectral region, Opt. Lett., vol. 5, pp. 476 478, 1980.

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