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Such systems with constant AD can be broadly divided into two main categories, namely conventional disper- sion-managed (DM) systems, where the amplifier ...
IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 14, NO. 9, SEPTEMBER 2002

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Average Dispersion Decreasing Densely Dispersion-Managed Fiber Transmission Systems A. B. Moubissi, K. Nakkeeran, P. Tchofo Dinda, and S. Wabnitz

Abstract—The authors propose an average dispersion decreasing densely dispersion-managed (A4DM) fiber line, which can substantially improve the performance of high-speed optical transmission systems. They show that the A4DM fiber lines have many advantages over the densely dispersion-managed fiber lines. Index Terms—Dispersion management, optical fiber communication, ultrashort pulse propagation. Fig. 1. Schematic diagram of one amplification span of (a) DDM and (b) A4DM transmission lines.

I. INTRODUCTION

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ECENT studies have shown that a periodic dispersion compensation with a constant average dispersion (AD) seems to be an effective way for improving optical transmission systems [1]–[5]. Such systems with constant AD can be broadly divided into two main categories, namely conventional disperis sion-managed (DM) systems, where the amplifier spacing equal to or smaller than the dispersion period and densely dispersion-managed (DDM) systems, where [1]. Enhanced pulse power in DM systems increases the signal-to-noise ratio (SNR), reduces Gordon–Haus jitter, and thus improves transmission system performance. However, in conventional DM systems, this advantage can become a serious drawback in the case of a transmission with high bit rates of 40 Gb/s or more per channel. Indeed, the energy of the DM soliton increases with a decrease of the pulse width. As a result, for short pulses the required soliton power can become too high to be realized in practice [2]. An additional detrimental factor lies in pulse interactions, which also becomes an important issue with an increased signal power. These interactions limit the transmission speed of the pulse. This letter proposes an average dispersion decreasing densely dispersion-managed (A4DM) fiber lines which can be easily designed analytically for any desired pulse (energy and width) and fiber (dispersion, nonlinearity, and losses) parameters. More importantly, we demonstrate that a chirp-free Gaussian pulse can propagate extremely well in A4DM lines as a fixed point, with the same input parameters for any amplification distance and with minimal intrachannel interaction. Manuscript received February 19, 2002; revised April 30, 2002. This work was supported in part by the Ministère de l’Education Nationale de la Recherche et de la Technologie (Contract ACI Jeunes 2015). This work was carried out under the Contract URP/4.00 between the University of Burgundy and the Alcatel Research Corporation (URP/C/01/0145). The work of K. Nakkeeran was supported by the Centre National de la Recherche Scientifique (CNRS) under a Research Associate fellowship. A. B. Moubissi, K. Nakkeeran and P. Tchofo Dinda are with the Laboratoire de Physique de l’Université de Bourgogne, 21078 Dijon Cédex, France. S. Wabnitz was with Alcatel Research and Innovation, 91460 Marcoussis, France. He is now with Xtera Communications, Allen, TX 75013 USA. Publisher Item Identifier 10.1109/LPT.2002.801068.

II. THEORY AND DESIGN OF A4DM LINES It is a known fact that the AD of a DM fiber system is directly related to the energy required for the periodic evolution of the DM solitons [3]. Before the novel idea of dispersion compensation, researchers were trying to design a dispersion decreasing fiber system to counterbalance the decreasing nonlinearity (due to optical losses) [6]. But the difficulty of fabricating such types of fiber has not made this concept a reality. Now, in DM fiber lines, one can easily vary the AD of the different dispersion maps by simply varying the lengths of the normal and anomalous fiber sections. Because of the optical losses, energy will be decreasing between two consecutive amplifiers, thus causing a decrease of the nonlinearity. One can similarly decrease the AD in such a way as to balance the local nonlinearity. To achieve this balance, we design a new type of DDM fiber system that we call “A4DM system,” in which the AD decreases from one dispersion map to another within the amplifier span in such a way to follow the decrease of energy due to losses. To this end, we can make an approximation that the energy within one dispersion . So we can consider map is not varying so much as that in each map, the energy is almost constant and is balanced by the respective AD. Thus, in our A4DM system, the AD will decrease exactly like the energy but in a discretized fashion from one map to another within the amplification span, as schematically represented in Fig. 1, where the horizontal dotted lines indicate the AD. Fig. 1(a) and (b) represents a conventional DDM system and our A4DM system, respectively. For designing the A4DM system we adopt the following simple procedure. We use two types of fiber for all the maps, with positive and negative group-velocity dispersions denoted and , respectively. In each amplifier span, the maps as are numbered from 1 to , and the length of the normal (anomalous) fiber section of the map number is denoted as ( ). We impose the same total length (fixed by the design of the first map) for all the maps: , for all . Assuming a desired input energy and pulse and width, we analytically calculate the fiber lengths of the first dispersion map of the amplifier span as a lossless case [7]. Hence, the input pulse (fixed point) will

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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 14, NO. 9, SEPTEMBER 2002

TABLE I (a) A4DM SYSTEM

(b) DDM SYSTEM

be the chirp-free Gaussian pulse (assumed for analytically designing the first map). The AD of this map is denoted as . If the input energy of the soliton is , then the energy at the beginning of the second dispersion , where map will be are the loss parameters for the two types of fiber. As the energy decreases exponentially, we therefore exponentially decrease the AD to keep a constant balance between the dispersion and the nonlinearity. But we make the approximation that the energy remains constant throughout the first dispersion map . Then, the energy remains to and decreases abruptly to throughout the second map before decreasing abruptly to as the pulse enters the third map, and so on. Following this energy variation, we decrease the AD from one map to the next map in the same proportion as that of the energy decrease. Hence, we take the AD of the second . Then from map to be and the total length the knowledge of of the map, we can calculate the fiber lengths of the second and , by means of the formulas dispersion map: and . We follow the same procedure for obtaining the lengths of the fiber sections for all the remaining dispersion maps in the amplification period. So, in general, for an A4DM system with dispersion maps in each amplification period, the anomalous and normal fiber lengths from the second up to the th map can be calculated from

(1) . For A4DM fiber lines, we have noticed where the following very interesting features. 1) The fixed point for the lossless line corresponding to the first dispersion map of the A4DM system works extremely well as the stationary solution for entire A4DM system with losses and gain.

Fig. 2. Transmission performance of the A4DM system. (a1), (b1), (c1): Amplitude and Timing Q factor versus propagation distance . (a2), (b2), (c2): Eye patterns. Hereafter, the factor is given in linear units. 6 corresponds to a bit-error ratio of 10 .

Q

z

Q=

2) As a result, any change in the total number of maps in the amplification span of the A4DM line will not change the fixed point of the entire system. Actually, the desired . number of maps fixes the amplifier spacing 3) The magnitude of the slow dynamics of the fixed point reduces as increases. 4) The interaction length increases with an increased number of dispersion maps in the amplification span. This feature is simply due to the fact that for a given input pulse’s energy, as increases the span AD (which is the average of all the AD of individual dispersion maps) of the entire system decreases, and also the A4DM soliton propagates for a longer distance in the low energy regime within each amplification span. However, this advantage of reducing the A4DM soliton interaction with an increased number of dispersion maps is limited by the SNR of the system. Hence, a tradeoff between the SNR and the A4DM soliton interaction will finalize the number of dispersion maps in the amplifier span of the desired transmission system. III. NUMERICAL SIMULATION To illustrate the effectiveness of our A4DM system we have considered the pulse propagation in a single channel line (with perfect cable) operating at 160 Gb/s. To this end, we have designed three A4DM systems corresponding to amplification spans of 25, 50, and 75 km, with the same following parameters: ps/nm/km; third-order second-order dispersion ps/nm /km; self-phase modulation dispersion

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of the A4DM line [see Fig. 1(b)]. Using the Nijhof et al. [8] averaging method, we have obtained the parameters of the DDM fixed points having the same spectral bandwidth as for the associated A4DM systems. The DDM input pulse parameters are given in Table I(b). By comparing Figs. 2 and 3, one can clearly observe as general features that the A4DM solitons (Fig. 2) lead to a dramatic improvement in transmission performance when compared to their DDM counterparts (Fig. 3), for the same spectral efficiency. We attribute this reduction (by a factor larger than two) in the transmission distance of DDM solitons to their much higher energies than that of their A4DM counterparts, resulting from a relatively high span average dispersion of the DDM line, as can be seen in Table I(b). The very high power level in the DDM system favors the SNR and thus reduces the amplitude jitter, hence a high amplitude factor is visible in Fig. 3(a1) and (b1), for small and moderate amplification spans, respectively. Meanwhile, it also favors the pulse interactions, which therefore become the main detrimental factor in the system performance. It then comes out that conventional DDM systems lead to power levels for stationary pulses that are much higher than the really needed power for transmitting pulses having the same spectral efficiency as for the A4DM solitons. Thus, the results in Fig. 2 demonstrate the effectiveness of A4DM solitons in high-speed long-distance transmission lines. Fig. 3. Transmission performance of the DDM system. (a1), (b1), (c1): Amplitude and timing factor versus propagation distance . (a2), (b2), (c2): Eye patterns.

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W ; dB/km; Raman gain of ; amplifier noise figure of dB, Gaussian THz (placed in-line at each amplifier filter of bandwidth site) to reduce the timing jitter, and a map strength of 1.6 for minimal pulse–pulse interaction [4], [5]. Map strength value 1.6 is not a constraint on A4DM system. One can design the A4DM system for any desired map strength and can always find better performance than the DDM system. The pulse’s (at chirp free full width at half maximum (FWHM) point), pulse’s energy , map length , and span AD ( ) for all the three A4DM systems are given in Table I(a). For the simulations, we have solved the nonlinear Schrödinger equation including the above-mentioned higher order effects. The transmission performance is evaluated by means of the and , respectively. amplitude and timing Q factors, Fig. 2, which shows the transmission performance of A4DM , demonstrates a high solitons bit pattern stability of A4DM solitons for small, moderate, and very long amplification spans, with excellent performances over several thousands of kilometers. The best performance is obtained for km), with a worst the smallest amplification span ( factor higher than 6 over more than 7.5 Mm [Fig. 2(a1) and (a2)]. For very long amplification spans, the transmission performance is limited by the SNR. On the other hand, we can also clearly illustrate the advantage of A4DM solitons over conventional DDM solitons with the help of Fig. 3, which shows the transmission performance . The DDM of DDM solitons bit pattern map [which is represented by the dashed box in Fig. 1(a)] corresponds precisely to the first map of the amplification period

IV. CONCLUSION In this letter, we have proposed a new kind of DDM line called the A4DM line. It has many advantages over the conventional DDM line such as the analytical way of designing the system, an increased amplification span fixed only by the SNR of the transmission system which will not change the fixed point parameters of the A4DM system, and finally, the robustness of the A4DM system to input chirp-free Gaussian pulses. Without being too speculative, the authors suspect that the A4DM soliton, which can propagate with a reduced energy compared to that of its counterpart DDM soliton, may reduce the impact of detrimental nonlinear effects. REFERENCES [1] A. H. Liang, H. Toda, and A. Hasegawa, “High speed soliton transmission in dense periodic fibers,” Opt. Lett., vol. 24, pp. 799–801, 1999. [2] D. S. Govan, W. Forysiak, and N. J. Doran, “Long-distance 40-gbit/s soliton transmission over standard fiber by use of dispersion management,” Opt. Lett., vol. 23, pp. 1523–1525, 1998. [3] A. Berntson, N. J. Doran, and J. H. B. Nijhof, “Power dependence of dispersion-managed solitons for anomalous, zero, and normal path-average dispersion,” Opt. Lett., vol. 23, pp. 900–902, 1998. [4] T. Yu, E. A. Golovchenko, A. N. Pilipetskii, and C. R. Menyuk, “Dispersion-managed soliton interactions in optical fibers,” Opt. Lett., vol. 22, pp. 793–795, 1997. [5] J. Martensson and A. Berntson, “Dispersion-managed solitons for 160-Gb/s data transmission,” IEEE Photon. Technol. Lett., vol. 13, pp. 666–668, 2001. [6] A. J. Stentz, R. W. Boyd, and A. F. Evans, “Dramatically improved transmission of ultrashort solitons through 40 km of dispersion-decreasing fiber,” Opt. Lett., vol. 20, pp. 1170–1172, 1995. [7] K. Nakkeeran, A. B. Moubissi, P. Tchofo Dinda, and S. Wabnitz, “Analytical method for designing dispersion-managed fiber systems,” Opt. Lett., vol. 26, pp. 1544–1546, 2001. [8] J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable soliton-like propagation in dispersion managed systems with net anomalous, zero and normal dispersion,” Electron. Lett., vol. 33, pp. 1726–1727, 1997.