Massive WDM and TDM soliton transmission systems

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MASSIVE WDM AND TDM SOLITON TRANSMISSION SYSTEMS

SOLID-STATE SCIENCE AND TECHNOLOGY LIBRARY VOLUME 6 Editorial Advisory Board L. R. Carley, Carnegie Mellon University, Pittsburgh, USA G. Declerck, IMEC, Leuven, Belgium F. M. Klaassen, University of Technology, Eindhoven, The Netherlands

Aims and Scope of the Series The aim of this series is to present monographs on semiconductor materials processing and device technology, discussing theory formation and experimental characterization of solid-state devices in relation to their application in electronic systems, their manufacturing, their reliability, and their limitations (fundamental or technology dependent). This area is highly interdisciplinary and embraces the cross-section of physics of condensed matter, materials science and electrical engineering.

Undisputedly during the second half of this century world society is rapidly changing owing to the revolutionary impact of new solid-state based concepts. Underlying this spectacular product development is a steady progress in solid-state electronics, an area of applied physics exploiting basic physical concepts established during the first half of this century. Since their invention, transistors of various types and their corresponding integrated circuits (ICs) have been widely exploited covering progress in such areas as microminiaturization, megabit complexity, gigabit speed, accurate data conversion and/or high power applications. In addition, a growing number of devices are being developed exploiting the interaction between electrons and radiation, heat, pressure, etc., preferably by merging with ICs. Possible themes are (sub)micron structures and nanostructures (applying thin layers, multi’layers and multi-dimensional configurations); micro-optic and micro-(electro)mechanical devices; hightemperature superconducting devices; high-speed and high-frequency electronic devices; sensors and actuators; and integrated opto-electronic devices (glass-fibre communications, optical recording and storage, flat-panel displays). The texts will be of a level suitable for graduate students, researchers in the above fields, practitione rs engineers, consultants, etc., with an emphasis on readability, clarity, relevance and applicability.

The titles published in this series are listed at the end of this volume.

Massive WDM and TDM Soliton Transmission Systems A ROSC Symposium

Edited by

Akira Hasegawa Research Professor,

Kochi University of Technology and Consultant, NTT Science and Core Technology Laboratory Group,

Japan

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

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TABLE OF CONTENTS PREFACE.............................................................................................. xi ORGANIZER AND PROGRAM COMMITTEE...................................... xiii RECENT PROGRESS OF OPTICAL UNDERSEA CABLE SYSTEMS . . . . . 1 M. Suzuki, N. Edagawa, I. Morita, N. Takeda, K. Imai, K. Tanaka and T. Tsuritani KDD R&D Laboratories Inc., Japan

80 GBIT/S MULTI-CHANNEL SOLITON TRANSMISSION OVER TRANSOCEANIC DISTANCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 M. Nakazawa, H. Kubota, K. Suzuki, E. Yamada and A. Sahara NTT Network Innovation Laboratories, Japan

MULTI-SOLITON TRANSMISSION AND PULSE SHEPHERDING IN BIT-PARALLEL WDM OPTICAL FIBER SYSTEMS . . . . . . . . . . . . . . . . . 41 Yu. S. Kivshar and E. A. Ostrovskaya Australian Photonics Cooperative Research Centre, Optical Sciences Centre, Research School of Physical Sciences and Engineering, The Australian National University, Australia

OPTICAL MODULATION AND DISPERSION COMPENSATION TECHNIQUES FOR ULTRA-HIGH-CAPACITY TDM/WDM TRANSMISSION SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 G. Ishikawa, H. Ooi, Y. Akiyama and T. Chikama Fujitsu Laboratories Ltd., Japan

ON THE EVOLUTION AND INTERACTION OF DISPERSION-MANAGED SOLITONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 M. J. Ablowitz, G. Biondini and E. S. Olson Department of Applied Mathematics, University of Colorado, USA.

vi EXPERIMENTAL DEMONSTRATION OF MASSIVE WDM OVER TRANSOCEANIC DISTANCES USING DISPERSION MANAGED SOLITONS . . . . . . . . . . . . . . . . . . . . . . . . 115

L. F. Mollenauer, P. V. Mamyshev, J. Gripp, M. J. Neubelt and N. Mamysheva, Bell Labs–Lucent Technologies, USA. Lars Grüner-Nielsen and Torben Veng, Lucent Denmark, Denmark

ON THE DISPERSION MANAGED SOLITON – The Guiding-center Theory Revisited – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Y. Kodama, Department of Mathematics Ohio State University, USA.

TDM AND WDM WITH CHIRPED SOLITONS IN OPTICAL TRANSMISSION SYSTEMS WITH DISTRIBUTED AMPLIFICATION . . 139

K. Hizanidis, N. Efremidis and A. Stavdas Department of Electrical and Computer Engineering, National Technical University of Athens, Greece D. J. Frantzeskakis and H. E. Nistazakis Department of Physics, University of Athens, Greece B. A. Malomed, Department of Interdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Israel

LONG-HAUL DISPERSION MANAGED SOLITON WDM SYSTEMS TOWARDS TERABIT CAPACITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

K. Fukuchi, T. Ito, Y. Inada and T. Suzaki C&C Media Research Laboratories, NEC Corporation, Japan

SPECTRAL EFFICIENCY IN WDM SOLITON TRANSMISSIONS . . . . . . . 173 S. Wabnitz, B. Biotteau, P. Brindel, B. Dany, O. Leclerc, P. Le Lourec, F. Neddam, D. Rouvillain and J. L. Beylat, Alcatel CRC, France E. Pincemin, France Telecom CNET/DTD/RTO, France

vii ANALYSIS AND DESIGN OF WAVELENGTH-DIVISION MULTIPLEXED DISPERSION-MANAGED SOLITON TRANSMISSION AT 40 G B I T / S / C H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

M. Matsumoto, Graduate School of Engineering, Osaka University, Japan A. Hasegawa, Kochi University of Technology and NTT Science and Core Technology Laboratory Group, Japan OPTIMIZATION OF DISPERSION COMPENSATION FOR LONG DISTANCE 40 Gbit/s SOLITON TRANSMISSION LINES BY THE Q-MAP M E T H O D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

K. Shimoura, I. Yamashita and S. Seikai, Technical Research Center, The Kansai Electric Power Co., Inc., Japan LONG DISTANCE TRANSMISSION OF FILTERED DISPERSION-MANAGED SOLITONS AT 40 GB/S BIT R A T E . . . . . . . . . 225

V. S. Grigoryan, P. Sinha, C. R. Menyuk and G. M. Carter Department of Computer Science and Electrical Engineering, University of Maryland Baltimore County, USA. OPTICAL COMMUNICATION SYSTEMS WITH SCHORT-SCALE DISPERSION MANAGEMENT............... 235

S. K. Turitsyn, N. J. Doran and E. G. Turistyna Photonics Research Group, School of Engineering and Applied Science, Aston University, UK. E. G. Shapiro, Institute of Automation and Electrometry, Russia M. P. Fedoruk and S. B. Medvedev, Institute of Computational Technologies, Russia REAL TIME PMD COMPENSATION FOR RZ TRANSMISSION S Y S T E M S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

M. Romagnoli, P. Franco, R. Corsini and A. Schiffini Pirelli Cavi e Sistemi, s. p. a., Italy M. Midrio, Istituto Nazionale per la Fisica della Materia, Dipartimento di Ingegneria Elettrica Gestionale e Meccanica, Università degli Studi di Udine, Italy

viii PROPAGATION OF 3-PS DISPERSION-MANAGED SOLITON PULSE

UNDER THE INFLUENCE OF THIRD-ORDER DISPERISON . . . . . . . . . . 265 Y. Takushima, X. Wang and K. Kikuchi Research Center for Advanced Science and Technology, University of Tokyo, Japan

TOLERANCE OF SCALAR AND VECTOR SOLITONS TO RANDOM VARIATIONS OF MAP PARAMETERS IN DISPERSION MANAGED OPTICAL FIBER LINKS . . . . . . . . . . . . . . . . . 277

F. Kh. Abdullaev, B. B. Baizakov, B. A. Umarov and D. V. Navotny Physical-Technical Institute of the Uzbek Academy of Sciences, Uzbekistan M. R. B. Wahiddin, Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Malaysia

QUANTUM CORRELATIONS OF COLLIDING SOLITONS . . . . . . . . . . . . . 289

A. Sizmann, F. König, M. Zielonka, R. Steidl and T. Rechtenwald Lehrstuhl für Optik, Physikalisches Institut der Universität Erlangen-Nürnberg, Germany

SYMMETRY-BREAKING AND BISTABILITY FOR

DISPERSION-MANAGED S O L I T O N S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 J. H. B. Nijhof and N. J. Doran Photonics Research Group, Aston University, UK.

40 GBIT/S MULTIPLE DISPERSION MANAGED SOLITON TRANSMISSION OVER 2700 KM . . . . . . . . . . . . . . . . . . . . . . . . . . 309

A. R. Pratt, H. Murai and Y. Ozeki Network Systems Development Center, Network Systems Business Group, Oki Electric Industry Co., Ltd., Japan

ix ENABLING FIBER TECHNOLOGIES FOR MASSIVE WDM AND TDM SOLITON TRANSMISSION S Y S T E M S . . . . . . . . . . . . . 327

S. Namiki, Opto-technology Lab, Furukawa Electric Co., Ltd., Japan

COLLISION-INDUCED IMPAIRMENTS IN DISPERSION MANAGED FIBER S Y S T E M S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

S. Kumar and A. F. Evans, Corning Incorporated Fiber Communications, USA.

ULTRA LOW NONLINEARITY PURE SILICA CORE FIBER AND ITS APPLICATION TO HYBRID TRANSMISSION L I N E S . . . . . . . . 365

T. Kato, M. Tsukitani, M. Hirano, E. Yanada, M. Onishi, M. Nakamura, Y. Ohga, E. Sasaoka, Y. Makio and M. Nishimura, Yokohama Research Laboratories, Sumitomo Electric Industries, Ltd., Japan

FIBER DESIGN FOR DISPERSION MANAGED SOLITON SYSTEMS : THE C H A L L E N G E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

W. A. Reed, Bell Labs–Lucent Technologies, USA.

40 GBIT/S RECIRCULATING LOOP EXPERIMENTS ON DISPERSION MANAGED STANDARD F I B R E . . . . . . . . . . . . . . . . . . . . . 387

P. Harper, S. B. Alleston, D. S. Govan, W. Forysiak, I. Bennion and N. J. Doran, Photonics Research Group, School of Engineering, Aston University, UK.

HANDLING NOISE IN SUPERCONTINUUM GENERATION FOR WDM APPLICATION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

H. Kubota, K. R. Tamura and M. Nakazawa NTT Network Innovation Laboratories, Japan

x DENSE-WDM SOLITON SYSTEMS USING CHANNELISOLATING NOTCH FILTERS (“SOLITON RAIL ” ) . . . . . . . . . . . . . . . . . 411 B. A. Malomed, Department of Interdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Israel A. Docherty, P. L. Chu and G. D. Peng

Optical Communications Group, School of Electrical Engineering, University of New South Wales, Australia INDEX................................................................................................. 425

LIST OF CONTRIBUTORS (SPEAKERS).......................................... 429

Preface This book summarizes the proceedings of the invited talks presented at the “International Symposium on Massive TDM and WDM Optical Soliton Transmission Systems” held in Kyoto during November 9–12, 1999. The symposium is the third of the series organized by Research Group for Optical Soliton Communications (ROSC) chaired by Akira Hasegawa. The research group, ROSC, was established in Japan in April 1995 with a support of the Japanese Ministry of Post and Telecommunications to promote collaboration and information exchange among communication service companies, communication industries and

academic circles in the theory and application of optical solitons. The symposium attracted enthusiastic response from worldwide researchers in the field of soliton based communications and intensive discussions were made. In the symposium held in 1997, new concept of soliton transmission based on dispersion management of optical fibers were presented. This new soliton is now called the dispersion managed soliton. The present symposium mainly focuses the theoretical and experimental developments of dispersion managed solitons. It is remarkable that the concept of the dispersion managed soliton, which was just born two years ago when the naming was not even given yet, has become the center of soliton research in two years. The dispersion managed

soliton has an enhanced power in maintaining reasonable signal to noise ratio, yet has reduced Gordon-Haus timing jitter by reduced average dispersion. The dispersion managed soliton also has demonstrated its power in soliton based WDM transmissions. During two years’ time since previous symposium, the linear transmission groups around the world have come to the conclusion that the most efficient linear format called NRZ modulation faces significant degradation of signal quality due to nonlinear effects in fibers and have practically discarded this format. Instead, they presently promote the use of the RZ modulation format, somewhat

similar to the dispersion managed soliton. They also developed technique of dispersion management in order to avoid cross talks due to four wave mixing

in WDM transmissions. Meanwhile, optical soliton group, by recognizing the effectiveness of dispersion managed soliton in reducing the Gordon–Haus jitter, have also discarded the use of ideal soliton and accepted stationary nonlinear

pulse which has chirp and pulse width oscillation. In this way both linear and soliton communities have approached each other to the common area in the mode of transmission. However, there remains a clear difference in attitude between these two groups. The linear transmission people allows the RZ pulse to overlap significantly during the transmission. Pulse is restored its original shape by dispersion compensation either periodically or at the terminal. While the soliton group have designed pulses not to overlap or minime the overlapping during transmission to avoid collision induced timing jitter. In addition

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linear group attempted to minimize the power level in order to avoid nonlinear effects both in self phase modulation and in cross phase modulation therefore forward error correctors (FEC) are usually needed for long distance transmissions. Soliton group allows a large power level thereby large signal to noise ratio is maintained and avoids the use of the forward error correctors. With FEC, linear group have achieved 1 Tbit/s transmission over trans-oceanic distances. However, the distance of transmission without FEC is limited to the same level as the soliton transmission which has been achieved recently by NEC group in Japan. In addition, the NEC result which is presented in the proceeding uses 20 Gbit/s transmission per channel while the linear transmission rate per channel has been limited to 10 Gbit/s in trans-oceanic distance experiments. The well separated pulses at any point in transmission line and unnecessity of the use of FEC in dispersion managed soliton systems render great advantage in building optical networkings. Thus, the future of dispersion managed soliton transmission will depends on how much of the transmission rate per channel can be increased keeping significant number of WDM channels and the transmission distance. The proceeding summarizes some of the most recent theoretical and experiment results for the dispersion managed soliton systems. The reader would find remarkable progress in this field made in last two years. In addition to the soliton related works, the present symposium invited presentation by fiber

manufacturers in order to learn the up-to-date technologies in products suitable for dispersion managed soliton systems. The reader will also find remarkable progress in this area. In the preparation of the proceeding, Messrs. H. Takehara and S. Hidaka have devoted a significant amount of their time to make the manuscript consistent and well ordered. On behalf of ROSC, the editor would like express his appreciation for those efforts without which the publication of the book would have faced significant delay. The editor would also like to thank the financial support given by the Support Center for Advanced Telecommunications Technology Research, Foundation (SCAT) in which Mr. Takehara serves as the executive director. Kyoto, January 2000

Akira HASEGAWA

ORGANIZER AND PROGRAM COMMITTEE

Organizer Research Group foe Optical Soliton Communications (ROSC) Chairman: Akira HASEGAWA, Kohchi University of Technology

Program Committee Nick, J. DORAN, Aston University Katsumi IWATSUKI, Nippon Telegraph and Telephone Corp. Masayuki MATSUMOTO, Osaka University Kiyoshi NAKAGAWA, Yamagata University Kazuo SAKAI, KDD Laboratory, Co., Ltd.

Secretariat

Hiroshi TAKEHARA and Sumiyasu HIDAKA Support Center for Advanced Telecommunications Technology Research, Foundation (SCAT) Tel.: +81–3–3351–0540, Fax.: +81–3–3351-1624 E-mail : [email protected]

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RECENT PROGRESS OF OPTICAL UNDERSEA CABLE SYSTEMS

M. SUZUKI, N. EDAGAWA, I. MORITA, N. TAKEDA, K. IMAI, K. TANAKA AND T. TSURITANI KDD Laboratories Inc. 2-1-15 Ohara, Kamifukuoka-shi, Saitama, 356-8502, Japan

Abstract. Transmission capacity for optical undersea cable systems is growing remarkably and a more than 500-fold increase has been achieved for commercial systems over the past 10 years. This paper reviews the key technologies to support next generation 160 Gbit/s undersea cable systems; 10 Gbit/s-based dense wavelength division multiplexing (WDM) technology, low nonlinear fibers, and wideband amplifiers. Recent progress towards the 1 Tera-bit/s system and dispersion managed soliton transmission for

future higher bit rate WDM are also discussed.

1. Introduction Transmission capacity for optical undersea cable systems is remarkably growing. The capacity in TPC-3, the first optical fiber cable in Pacific Ocean installed in 1989, was 280 Mbit/s per fiber pair. The emergence of Erbium-doped fiber amplifier (EDFA) paved the way for drastic increase in capacity for optical undersea cables, and large capacity optical amplifier undersea cable systems with 5 Gbit/s per fiber pair, such as TPC-5CN and APCN, were constructed in Asia-Pacific region in 1995–1996. Recent 10 Gbit/s-based WDM (Wavelength Division Multiplexing) technologies together with new fibers and new amplifiers enable us to further increase in capacity up to 160 Gbit/s [1] and 160 Gbit/s WDM systems will be installed both in Pacific Ocean (Japan–US and PC–1) and Atlantic Ocean (TAT–14) in 2000. This represents a more than 500-fold increase in capacity for commercial undersea cable systems over the past 10 years. Research interest is now being directed towards the development of undersea cable systems with a transmission capacity of 1 Tera-bit/s or more. 1 A. Hasegawa (ed.), Massive WDM and TDM Soliton Transmission Systems, 1–15. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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In this paper, the key technologies for next generation 160 Gbit/s optical undersea cable systems and recent progress towards Tera-bit/s systems are reviewed. 2. Technologies for 160 Gbit/s WDM Systems

2. 1. OPTICAL FIBER

In 10 Gbit/s based long-haul WDM transmission, nonlinear waveform distortion caused by the interaction between fiber nonlinear effect and large accumulated chromatic dispersion during transmission is significantly limits the transmission performance [1]. To reduce the nonlinear effect and dispersion slope of the transmission line, a hybrid fiber span of large core fiber (LCF) and non-zero dispersion shifted fiber (NZ-DSF) has been proposed [1, 2]. Figure 1 shows an example of the hybrid fiber span. The LCF with large effective area is placed at the high signal power portion after EDFA to reduce the nonlinear effect. NZ–DSF with low dispersion slope and moderate is placed after LCF where signal power is lower to keep the average dispersion slope low. Table 1 lists typical optical parameters of LCF and NZ-DSF. With this hybrid span configuration, both nonlinear effect and dispersion slope can be effectively reduced and we can take full advantage of LCF characteristics. 2. 2. OPTICAL AMPLIFIER

Low-noise and wide-band width EDFAs have been developed for new generation systems. 980 nm pumping allow us to reduce the noise figure of EDFAs significantly compared with conventional 1480 nm pumping. Typical noise figure for 980 nm pumped EDFA is 4. 0 dB. Highly reliable 980nm-pump laser diode modules have been already developed for 160 Gbit/s undersea

RECENT PROGRESS OF OPTICAL UNDERSEA CABLE SYSTEMS 3

cable systems [3]. Broad and flat gain bandwidth of optical amplifiers is one of the key issues to support massive WDM signals. By using one kind of gain equalizer (Fabry-Perot filter) in every repeater, flat gain shape over

12 nm was achieved throughout over 18000 km [4]. Figure 2 shows spectra for forty WDM signals before and after 18414 km transmission with 50 km amplifier spacing. Even without pre-emphasis, the peak power variation among 40 signals was only less than 5 dB. This homogenous gain equalization scheme is effective to minimize peak power variation and excess nonlinear effect during transmission compared with block gain equalization scheme.

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2.3. PULSE FORMAT

Channel bit rate of 10 Gbit/s is used in the 160 Gbit/s systems to meet SDH (Synchronous Digital Hierarchy) STM–64 networks. To mitigate the nonlinear waveform distortion, which becomes severe with increasing channel bit rate, RZ signal format with pre-chirping has been introduced [1, 2].

Since all the pulses are isolated each other and have the same waveform independently of the data pattern in RZ format, the pattern dependence of SPM-induced waveform distortion observed in conventional NRZ format can be minimized. RZ format is more robust against nonlinear effect of the fiber than NRZ format, since uniform nonlinear chirp during transmission can be partially managed by linear chirp at the transmitter. 2.4. 160 GBIT/S TRANSMISSION EXPERIMENT

Transmission performance was measured with 3600 km long recirculating loop [1]. There were 70 amplifier/fiber spans in the amplifier chain with an average span length of 51.7 km. The fiber in the spans was part LCF/NZDSF hybrid fiber with a dispersion of – 2 ps/nm/km, and part standard SMF with a dispersion of +18 ps/nm/km. The ratio of hybrid fiber span and SMF span is 9:1 so that the dispersion at a wavelength near the center of WDM channels periodically returned to zero. 16-channel, 10.7 Gbit/s transmitter with uniform channel spacing of 0.7 nm was used and chirped RZ signals with PRBS were generated. Figure 3 shows transmission performance after 10850 km for repeater output power level of 11 dBm. Average Q factor was 15.5 dB, i.e., BER The performance degradation for the channels far from zero dispersion wavelength was observed in this experiment. This demonstrates that transmission performance of these channels is limited by the interaction of their accumulated chromatic dispersion and the nonlinear effect of the fiber. However, the performance

can be improved by reducing the nonlinear effect with optimization of span length and repeater power level. WDM transoceanic systems based on these enabling technologies will be installed in the Pacific (Japan US, and PC–1) and the Atlantic (TAT–14) as shown in Fig. 4 by the end of 20th century. 3. Capacity Expansion with LCF/NZ-DSF Based Dispersion Management 3.1. 500 GBIT/S TRANSMISSION OVER 4000 KM

With improvement and refinement of 160 Gbit/s technology, some capacity expansion is possible. Gbit/s over 4036 km transmission with 0.4 nm channel spacing has been demonstrated by 400 km long recirculating

RECENT PROGRESS OF OPTICAL UNDERSEA CABLE SYSTEMS

5

loop experiments, where modified EDFA with wider gain bandwidth over 20 nm and LCF/NZ–DSF hybrid fiber spans of 40 km were used [5]. Figure 5(a)

shows the optical spectrum after 4036 km. Gain bandwidth was doubled compared with the 160 Gbit/s system by using a two-stage long period fiber grating as a gain equalizer in each amplifier. Figure 5(b) shows obtained Qfactor for WDM transmission. Thanks to the low average dispersion slope of error-free (bit error ratio ) transmission has been achieved for all channels. Average Q factor for 50 channels was 16. 2 dB.

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3.2. EXPECTED CAPACITY For transatlantic distances, error-free transmission (BER over 6050 km with 0.6 nm channel spacing [6] and error free transmission over 5160 km with 0.4 nm channel spacing [7] have been demonstrated. For relatively shorter distances, 800 Gbit/s over 3400 km has been also demonstrated [8]. The expected capacity from this kind of DWDM and conventional dispersion management based on LCF will be about 2–2.7 Pbit/s-km depending on the system length. 4. Advanced dispersion management using SMF and slope compensating DCF

For further capacity expansion, performance degradation due to interaction between fiber nonlinear effect and large accumulated dispersion for the channels far from zero-dispersion wavelength should be overcome. For this purpose, some dispersion slope compensation scheme is needed. In particular, the method of constructing dispersion flattened transmission spans with standard SMF and slope compensating dispersion compensation fiber (SCDCF) [9]-[l l ] is quite attractive, because of its simple configuration. Furthermore, the high local dispersion and large effective area of SMF can minimize the nonlinear effects such as four-wave mixing (FWM) and crossphase modulation (XPM). Figure 6 shows a comparison of the accumulated dispersion with transmission distance for conventional dispersion map using NZ–DSF and advanced dispersion map using standard SMF and slope compensating DCF.

RECENT PROGRESS OF OPTICAL UNDERSEA CABLE SYSTEMS

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With this advanced dispersion map, the dispersion flattened fiber link can be realized, and performance improvement for edge channels is expected. The effectiveness of low-slope dispersion management using standard SMF and slope-compensating DCF has been experimentally confirmed by DWDM transmission with channel spacing of 0.5 nm over 7280 km [11]. Performance was measured using 280 km long recirculating loop consists of seven EDFAs and six spans of 47 km fiber. Each span is a hybrid of standard SMF(40 km) and SC–DCF(7 km). Figure 7 shows the transmission performance after 7280 km. As shown in Fig. 7, the wavelength dependency of the transmission performance is almost eliminated due to the low dispersion slope of which is less than a tenth the value for conventional dispersion map. Average Q-factor of 15.9 dB was achieved. 5. Tera-bit/s Transmission over Transoceanic Distances

To achieve much higher capacity such as 1 Tera-bit/s, we need many breakthroughs to overcome inherent technological challenges such as lack of bandwidth and degradations due to fiber nonlinearity. In this section we show the recently developed key technology to achieve 1 Tbit/s transmission over transoceanic distances and introduce experimental results on 1 Tbit/s

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0. 3 nm spacing) 6200 km transmission [12]. 5. 1. CHANNEL SPACING

Since 10 Gbit/s seems to be the most promising channel bit rate for nearterm undersea cable applications, we need to transmit 100 channels to achieve 1 Tbit/s capacity. Considering the fact that 30 nm is the practical bandwidth-limit for silica-based single-band EDFAs, the channel spacing should be less than 0. 3 nm to accommodate 100 channels in the 30 nm bandwidth. 5. 2. OPTICAL REPEATER

For 30 nm bandwidth, either C-band or L-band EDFA is basically applicable. However, considering the necessary repeater output power to carry Tera-bit/s signals, efficiency is a crucial point to alleviate severe pump power requirement of much more than 100 mW for undersea cable applications. From this perspective, C-band EDFA is the right choice although this requires rather complicated gain-equalising filter-shape.

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5. 3. TRANSMISSION FIBER

30 nm-bandwith requires to use slope-compensating hybrid fiber configuration, such as SMF+SCDCF [11] or SMF+RDF [10]. To alleviate the increasing impact of fiber nonlinearity due to narrower channel spacing and larger channel count, the fiber placed at the output of the repeater should have a larger and moderately large dispersion compared with the conventional SMF whose is around 5. 4. EXPERIMENTAL SET-UP

Figure 8 shows a schematic diagram of the experimental set-up. We used 100 CW LDs as the signal light sources, whose wavelengths were equally spaced by 0. 3 nm ranging from 1533. 7 nm to 1563. 4 nm. Each set of odd and even channels was combined separately and then fed into two Mach-Zehnder modulators in cascade for data-coding at 10. 66 Gbit/s with a pseudo-random binary sequence and RZ-formatting to produce 40 ps signal pulses, respectively. In the following section, slight phase modulation was applied to improve transmission performance. At the last stage, the even and odd channels were combined using a polarising beam splitter and launched into the transmission line in orthogonal state of polarisation with each other to reduce FWM during the fiber transmission [2]. The 280 km recirculating loop consisted of six 47 km-long spans of transmission fiber and seven optical repeaters based on single-stage bidirectionally-pumped Cband EDFAs. The pumping wavelength was 980 nm for forward pumping and 1480 nm for backward pumping. The average repeater outputpower was about 17 dBm. Each repeater was gain-equalised with a 3-stage longperiod fiber grating to achieve a gain-flatness of less than over 30 nm. To flatten the gain shape better after 6200 km transmission, we used a Mach-Zehnder filter with an FSR of 100 nm in the loop as shown in Fig. 8. We used two kinds of fibers to configure all the fiber spans. The first segment at the output of the EDFAs is an -enlarged positive dispersion fiber (EE–PDF) based on step-index core-profile and the second segment is an SCDCF, which has a negative dispersion and negative dispersion slope to form dispersion flattened transmission spans. The typical parameters of these fibers are listed in Table 2. To compensate for the residual dispersion and dispersion slope of the loop, SCDCF with –715 ps/nm/km was inserted at the beginning of the loop. The average dispersion slope of the loop was and the average dispersion at channel-50 was –0. 3 ps/nm/km. The loss of transmission spans including splicing loss was about 11. 8 dB on average. In the receiver, the desired channel was selected with a cascade of dielec-

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tric optical bandpass filters. The accumulated dispersion for WDM channels was compensated with the dispersion compensation fiber (DCF) after transmission.

5.5. RESULTS AND DISCUSSIONS

Figure 9(a) shows the optical spectrum after 6200 km transmission. The signal spectrum was well equalised over 30 nm even after a concatenation of 154 amplifiers. The average optical SNR obtained was 14.4 dB/0.1 nm. Figure 9(b) shows the Q-factor of 100 WDM signals. The average Q-factor obtained was 14.1 dB. On a large scale, the Q-factor increases with wavelength. We have found that this is partly because of rather large dispersion slope resulting in larger accumulated dispersion for shorter wavelengths and partly because of insufficient optical SNR management in this demonstration, and, therefore, the worst Q-factor can be much improved with some

RECENT PROGRESS OF OPTICAL UNDERSEA CABLE SYSTEMS

more refinement of the experimental set-up. As a result, the key technologies described above were found very effective to achieve Tera-bit/s-scale transmission over transoceanic distances. 6. Progress of 10 Gbit/s-based WDM Transmission

Figure 10 summarizes reported capacity-distance product for 10 Gbit/sbased WDM long distance transmission experiments. Open squares show the results achieved with Large Core Fiber-based systems. For the transpacific distances ( ), expected capacity is about 160 Gbit/s, but

the capacity will be doubled for the transatlantic distances ( km). Closed triangles represent the results using advanced dispersion maps with standard SMF and slope compensating DCF or RDF. The advantage of the advanced dispersion map with low dispersion slope becomes obvious for the longer transmission distances around 9000 km. For the transpacific systems, expected capacity is twice as large as that of the conventional dispersion maps. By reducing the nonlinear effects by employing -enlarged

fibers, 1 Tbit/s-6200 km without FEC using single-stage C-band amplifiers [12] and 1 Tbit/s-10000 km with FEC using C-band and L-band amplifiers have been achieved [13] as depicted as closed circles in Fig. 10. Note that the potential transmission performance demonstrated in these different 1 Tbit/s experiments are, in fact, effectively equivalent considering the expected BER performance improvement owing to FEC gain, but that the system using just C-band amplifiers [12] seems suitable for near future

11

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Tera-bit/s undersea cable applications because of the practical constraints for undersea cable systems, such as tight pump power requirement, repeater housing size and feeding power limitation. 7. Towards Multi-Tera Bit/s Systems

For multi-Tera bit/s transmission systems, higher bit-rate WDM transmission are attractive, since it has a potential to increase the aggregate capacity without increasing wavelength count. So far, 400 Gbit/s ( Gbit/s) over 2000 km [14], 1.02 Tbit/s ( Gbit/s) over 1000 km [9], and 1.1 Tbit/s ( Gbit/s) over 3020 km [15] transmission experiments have been reported. As for 40 Gbit/s single-channel long distance soliton transmission, over 10200 km transmission with dispersion managed DSF [16], and over 1220 km transmission with dispersion managed SMF [17] have been achieved. Figure 11 shows the BER performance and waveform for the 40 Gbit/s over 10200 km transmission experiment. These major results have been achieved by employing dispersion managed soliton transmission scheme or periodic dispersion compensation scheme

RECENT PROGRESS OF OPTICAL UNDERSEA CABLE SYSTEMS 13

[18], which can reduce Gordon-Haus timing jitter in soliton-based high bit

rate transmission. However, it seems difficult to achieve multi-Tera bit/s

long distance WDM transmission with just dispersion managed soliton. For further expansion in both transmission distance and capacity with higherbit rate soliton WDM, a breakthrough such as more advanced dispersion management which will be much more robust against performance degradation due to fiber nonlinear effect is expected.

8. Conclusion

The recent progress in undersea cable systems is reviewed. The capacity per fiber pair will reach Tera-bit/s in early stage of 21st century. The remarkable increase in capacity has been supported by many technological breakthroughs, such as wideband Er-doped fiber amplifier, high power pumping LD, nonlinear fiber technology, and 10 Gbit/s opto-electronics technologies. Towards multi-tera bit/s era, some breakthrough enabling higher-bit rate soliton WDM transmission is expected. Acknowledgements

The authors thank the experimental works of A. Agata and H. Yamauchi, and wish to thank Drs. T. Muratani, K. Suzuki, S. Akiba and Y. Matsushima of KDD R&D Laboratories Inc. for their continued encouragement.

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References 1. Suzuki, M., Kidorf, H., Edagawa, N., Taga, H., Takeda, N., Imai, K., Yamamoto, S., Ma, M., Kerfoot, F., Maybach, R., Adelman. H., Arya, V., Chen, C., Evangelides, S., Gray, D., Pedersen, B. and Puc, A. : 170 Gbit/s transmission over 10850 km using large core fiber, OFC’98, PD17, (1998). 2. Bergano, N., Davidson, C., Ma, M., Pilepetskii, A., Evangelides, S., Kidorf, H., Darvie, J., Golvchenko, E., Rottwitt, K., Corbett, P., Menges, R., Milles, M., Pedersen, B., Peckham, D., Abramov, A. and Vengsarkar, A. : 320 Gbit/s WDM transmission (64×5 Gb/s) over 7200 km using large mode fiber spans and chirped returnto-zero signals, OFC’98, PD12, (1998). 3. Usami, M., Edagawa, N., Matsushima, Y., Horie, H., Fujimori, T., Sakamoto, I. and Gotoh, H. : OFC’99, PD39, (1999). 4. Takeda, N., Taga, H., Imai, K., Horiuchi, Y., Edagawa, N., Suzuki, M. and Yamamoto, S. : 40 WDM 2.5 Gbit/s transmission over 12000 km using widely gainflattened low-noise 980 nm-pumped EDFAs, Electronics Letters, Vol.34, (1998), pp.381-382. 5. Imai, K., Tsuritani, T., Takeda, N., Tanaka, K., Edagawa, N. and Suzuki, M. : 500 Gbit/s (50×10 Gb/s) WDM transmission over 4000 km using broadband EDFAs and low dispersion slope fiber, OFC’99, PD5, (1999). 6.

7. 8. 9. 10. 11.

12.

13. 14.

Tsuritani, T., Imai K., Edagawa, N. and Suzuki, M. : 340 Gbit/s (32×10.66 Gbit/s)

WDM transmission over 6054 km using hybrid fiber spans of large core fiber and WDM fiber with low dispersion slope, Electronics Letters, Vol.35, (1999), pp.646647. Vareille, G., Pitel, F., Uhel, R., Basserier, G., Gollet, J., Bourret, G. and Marcerou, J. : 340 Gb/s ( Gb/s, 50 GHz spacing DWDM) straight line transmission over 6380 km, OFC’99, PD18, (1999). Vareille, G., Pitel, F., Hugbart, A., Uhel, R. and Marcerou, J. F. : 800 Gb/s (, Gbit/s DWDM, 28 GHz spacing) error-free transmission over 3400 km, ECOC’99, PD2-11, (1999), pp.44-45. Guen, D., Burgo, S., Moulinard, M., Grot, D., Henry, M., Favre, F. and Georges, T. : Narrow band 1.02 Tbit/s (51×20 Gbit/s) soliton DWDM transmission over 1000 km of standard fiber with 100 km amplifier spans, OFC’99, PD4, (1999). Murakami, M., Matsuda, T. and Imai, T. : Quarter Terabit (25×10 Gb/s) over 9288 km WDM transmission experiment using nonlinear supported RZ pulse in higher order fiber dispersion managed link, ECOC’98, PD, (1998), pp.79-81. Tanaka, K., Tsuritani, T., Edagawa, N. and Suzuki, M. : 320 Gbit/s (32×10.7 Gbit/s) error-free transmission over 7280 km using dispersion flattened fiber link with standard SMF and slope compensating DCF, Electronics Letters, Vol.35, (1999), pp.1860-1862. Tsuritani, T., Takeda, N., Imai, K., Tanaka, K., Agata, A., Morita, I., Yamauchi, H., Edagawa, N. and Suzuki, M. : 1 Tbit/s (100×10.7 Gbit/s) transoceanic transmission using 30 nm-wide broadband optical repeaters with -enlarged positive dispersion fibre and slope-compensating DCF, ECOC’99, PD2-8, (1999), pp.38-39. Naito, T., Shimojoh, N., Tanaka, T., Nakamoto, H., Doi, M., Ueki, T. and Suyama, M. : 1 Terabit/s WDM transmission over 10,000 km, ECOC’99, PD2-1, (1999), pp.24-25. Tanaka, K., Morita, I., Edagawa, N., Suzuki, M. and Yamamoto, S. : 400 Gbit/s (20×20 Gbit/s) dense WDM soliton-based RZ signal transmission using dispersion flattened fibre, Electronics Letters, Vol.34, (1998), pp.2257-2258.

15.

Fukuchi, K., Kakui, M., Sasaki, A., Ito, T., Inada, Y., Tsuzaki, T., Shitomi, T., Fujii, K. Shikii, S., Sugahara, H. and Hasegawa, A. : ECOC’99, PD2-10, (1999), pp.42-43. 16. Morita, I., Tanaka, K., Edagawa, N. and Suzuki, M. : 40 Gbit/s single-channel soliton transmission over 10200 km without active inline transmission control,

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15

ECOC’98, PD2, (1998), pp.47-51. Alleston, S., Harper, P., Penketh, I., Bennion, I. and Doran, N. : 1220 km propagation of 40 Gbit/s single channel RZ data over dispersion managed standard (no-dispersion shifted ) fibre, OFC’99, PD3, (1999). 18. Suzuki, M., Morita, I., Edagawa, N., Yamamoto, S., Taga, H. and Akiba, S. : Reduction of Gordon-Haus timing jitter by periodic dispersion compensation in soliton transmission, Electronics Letters, Vol.31, (1995), pp.2037-2039. 17.

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80 GBIT/S MULTI-CHANNEL SOLITON TRANSMISSION OVER TRANSOCEANIC DISTANCES

M. NAKAZAWA, H. KUBOTA, K. SUZUKI, E. YAMADA AND A. SAHARA NTT Network Innovation Laboratories 1-1 Hikarino-oka, Yokosuka-shi, Kanagawa-ken 239-0847, Japan

Abstract. The in-line modulation schemes up to 80 Gbit/s per channel and its two-channel WDM soliton transmission over 10000 km are described. The polarization mode dispersion can be also reduced by the in-line modulation.

1. Introduction

In a linear fiber transmission system, the group velocity dispersion (GVD) broadens the pulse width and fiber loss reduces the signal intensity. If one uses optical solitons as the signals, the dispersion problem is overcome and the loss is compensated for by erbium-doped fiber amplifiers (EDFAs). However the use of solitons causes new problems such as soliton-soliton interaction, Gordon-Haus jitter [1], and the accumulation of ASE noise. We proposed to overcome the Gordon-Haus limit, ASE accumulation, and soliton-soliton interaction [2]-[5]. This technique is called “soliton control or in-line modulation” and was used to achieve a one million km soliton transmission [5]. The in-line modulation scheme is very similar to active or passive modelocking technologies. In the active and passive mode-locking of lasers, synchronous modulation is used for pulse shaping and retiming and a saturable absorber is used for noise reduction and pulse shaping. In-line synchronous modulation (soliton control in the time domain) enables us to retime the position of the soliton pulse which experiences jitter as a result of amplified spontaneous emission (ASE) noise. It is also possible to remove the interaction forces which inevitably occur between closely adjacent solitons. It 17 A. Hasegawa (ed.), Massive WDM and TDM Transmission Systems, 17–39. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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has been proven theoretically that soliton data transmission over unlimited distances is possible with this technique since periodic synchronous modulation can reduce the ASE noise to a very low level [4]. A bandpass filter with a narrow bandwidth was also installed in the transmission system as a soliton control in the frequency domain to stabilize the soliton energy [3]-[5]. 2. Principle of In-line Modulation and Its Performance

Gordon-Haus jitter has a cubic dependence on the propagation distance and is a major factor in limiting soliton transmission distances [1]. The variance in the jitter is given by

where G is amplifier gain expressed by is normalized fiber is amplifier spacing, h is Planck’s constant, is the nonlinear index, D is GVD, L is transmission distance, is the full width at half maximum pulse width, and is effective area. For example, for a pulse width of 30 ps, a repeater spacing of 50 km, a loss of 0.25 dB/km, a GVD of –0.4 ps/km/nm, and a spot size of the Gordon-Haus jitter is as large as 19 ps after a propagation of ten thousand kms. The limit of the bit rate and distance product which provides a BER of is given by

loss,

where R is the bit rate of the system and

is the detector window width. For example, when and the maximum transmission distance L is only 7800 km. To ensure a smaller BER, the total transmission distance should be shortened. Figure 1 shows the principle of noise reduction through the use of synchronous modulation [3]. When modulation is applied to a soliton, it can be reshaped and retimed for transmission, but noise, including ASE noise and nonsoliton components, which has a small amplitude is modulated and disperses after a transmission of a certain distance. This is because a low amplitude signal is simply a dispersive wave. After a transmission of a certain distance, the next modulator is installed. Thus, the soliton is controlled and the noise remains at a very low level. We assume a sinusoidal modulation for the shaping function f(t) and that its extinction ratio is For example, means an extinction ratio of 20 dB. The signal pulse shape is assumed to be a hyperbolic secant squared.

80 GBIT/S MULTI-CHANNEL SOLITON TRANSMISSION

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where T is the period of the shaping function ( ). The transfer function for the noise, or noise reduction ratio, due to f(t) is per shaping if the sinusoidally modulated noise disperses during propagation and becomes cw noise again before it reaches the next modulator. The signal power transmitted through the modulator is

where and is assumed to be . It is necessary to compensate for the attenuation to maintain the soliton pulse. The required excess gain is . The next amplifier adds noise but the noise is also reduced by succeeding modulators. Thus the total amount of noise from k amplifiers is written as

20

where

M. NAKAZAWA ET AL.

is the ASE power from each amplifier. Here,

therefore, is always less than unity so that converges for any x value. For example, (20 dB modulation) and gives . The Mach-Zehnder intensity modulator has a sinusoidal transmittivity as a function of the applied voltage, and the driving voltage is sinusoidal. Therefore, the transmission function has a steeper edge than a sinusoidal modulation which is assumed here. In an actual situation, this asymmetric modulation function can remove the noise more efficiently. Another large advantage of soliton control is that the amplifier spacing La can be extended to of the order of a soliton period . Theory suggests that the pulse distortion is proportional to and dispersive rionsoliton waves are increased with increases in This result means that should be much smaller than unity in order to achieve ultralong distance soliton transmission. Higher bit rate communication requires a shorter optical pulse which shortens The ASE noise added by the EDFAs also limits the amplifier spacing. It has been stated that the EDFA gain G should be less than 10 dB because excess noise grows at a rate of These difficulties can be removed through the use of soliton control which reshapes the soliton pulse and removes dispersive nonsoliton waves as well as ASE noise [8]. The effectiveness of the in-line modulation has been proved at 10 Gbit/s using a 500 km-long optical fiber loop [9]. Fixed data patterns thus obtained at 10 Gbit/s are shown in Fig. 2(a) and (b), which correspond to transmitted signals over 50 million km and 180 million km, respectively. The data patterns in (a) and (b) were and respectively. It is important to note that the accumulation of ASE and non-soliton components at “0” signal is negligible. The BER characteristics under soliton control were also measured to determine whether or not the present soliton control technique can really preserve data without deterioration. The BER characteristics at a bit rate of 10 Gbit/s after transmissions of were the same, which means that there was no degradation at all in the data signal after transmission over one million km [2, 9]. After a one million km transmission, the eye was clearly open and this suggests that Gordon-Haus jitter has been eliminated by soliton control. These results indicate that soliton transmission control is a very

powerful technique for sending data over long distances.

80 GBIT/S MULTI-CHANNEL SOLITON TRANSMISSION

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3. High-speed Single Channel Soliton Transmission Experiments with In-line Modulation 3.1. 40 GBIT/S AND 80 GBIT/S SINGLE CHANNEL SOLITON TRANSMISSION OVER 10000 KM

The advantage of soliton transmission over a conventional WDM system is that the capacity of a single channel becomes much larger than that of a WDM system. In this section, we describe a single channel 40 Gbit/s soliton transmission over 70000 km and an 80 Gbit/s transmission over 10000 km realized by using in-line synchronous modulation [10, 11]. Both experiments were performed in a 250 km dispersion-shifted fiber loop, as shown in Fig. 3, where an 80 Gbit/s in-line modulation scheme is given in the dotted area. The optical soliton source was a 10 GHz har-

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monically mode-locked erbium fiber laser at 1550. 5 nm [12]. The pulse and spectral widths for 40 Gbit/s were 5 ps and 0. 50 nm, respectively

and those for 80 Gbit/s were 3. 0 ps and 0. 81 nm, respectively. The pulse was modulated at 10 Gbit/s with a – 1 PRBS using a lithium niobate (LN) intensity modulator. In order to obtain a 10 GHz clock signal from the transmitted 40 Gbit/s signal, 10 GHz soliton units were superimposed on each other with slightly different amplitudes. This technique is also useful for reducing soliton-soliton interaction. The amplifier spacing was 50 km and the average fiber loss for the span was 12. 5 dB. We used a four-segment dispersion-decreasing configuration to reduce the influence of the dispersive waves. For the 40 Gbit/s experiment, the four 12. 5-km long dispersion shifted fibers (DSFs) we used had GVDs of 0. 24, 0. 06, –0. 04 and –0. 10 ps/km/nm. We set the average GVD at approximately 0. 04 ps/km/nm. The average launched power was +7. 0 dBm and the peak power into the first segment was 50 mW, which corresponded to an almost soliton. For the 80 Gbit/s experiment, the average launched power was +9. 5 dBm and the peak power into the first segment was 74 mW, which corresponded to an N = 1. 1 soliton. The 80 Gbit/s transmission configuration is similar to that used for the 40 Gbit/s-70000 km transmission [10] except for the polarization multiplexing technique. To achieve soliton control

80 GBIT/S MULTI-CHANNEL SOLITON TRANSMISSION

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at 80 Gbit/s based on 40 Gbit/s electronics, we employed a polarisation multiplexing technique. That is, a 40 Gbit/s single polarization signal was polarization-multiplexed and interleaved with an 80 Gbit/s TDM signal by using a 7. 5 m-long PANDA fiber. After a 250 km transmission through the loop, we applied in-line synchronous modulation and narrowband optical filtering, in which the filter bandwidth was 0. 80 nm for 40 Gbit/s and 1. 1 nm for 80 Gbit/s. The 40 GHz clock signal was extracted from part of the transmitted soliton pulses with an ultrahigh-speed photodetector which had a 50 GHz bandwidth,

and a high Q dielectric filter. The LN modulator for soliton control was driven by the extracted 40 GHz clock. To obtain 80 Gbit/s in-line modulation, the polarization-multiplexed 80 Gbit/s signal was separated into two 40 Gbit/s signals with a polarization beam splitter (PBS) and single polarization soliton control was employed at 40 Gbit/s. This part is shown in the dotted line of Fig. 3. One clock signal was applied to two orthogonal signals. The LN modulators for soliton control were driven by the extracted 40 GHz clock, where each modulator operated at a single polarisation. Simultaneously, the amplitude level of the clock signal was fed back to a polarisation controller (PC) to obtain maximum clock power. This ensured that the orthogonal channel was also automatically optimized. After the inline modulation, two orthogonal 40 Gbit/s signals passed through a delay unit and were reconverted to an 80 Gbit/s signal with another PBS. The transmitted 40 Gbit/s soliton data signal was demultiplexed into a 10 Gbit/s signal by using an electro-absorption (EA) intensity modulator. The 10 GHz clock signal was extracted from the transmitted signal. Then, by adjusting the DC bias voltage and the amplitude of the 10 GHz clock signal for the EA modulator, the gate width of the modulator was set at 20 ps to extract a 10 Gbit/s signal from the 40 Gbit/s signal. The demultiplexed signal was detected with a 10 Gbit/s optical receiver and BER was measured. The transmitted 80 Gbit/s soliton data signal was demultiplexed into a 10 Gbit/s signal by using a PBS (80 Gbit/s to 40 Gbit/s ) and an EA modulator (40 Gbit/s to 10 Gbit/s). Figure 4(a) and (b) show the measured BERs corresponding to 40 and 80 Gbit/s. In Fig. 4(a) the open circles indicate the BER with soliton control after a transmission of 70000 km. The filled circles show the BER without soliton control after a transmission of 4500 km, and the triangles show the BER at 0 km. When soliton control was not employed, the maximum transmission distance was 4500 km. The BER beyond 4500 km increased rapidly with an increase in the transmission distance due to the accumulation of the noise component and timing jitter. In contrast, when we employed the in-line modulation scheme [2], the maximum transmission distance was greatly extended up to 70000 km. When the received

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power was greater than –31.0 dBm, no error appeared at a -order error counter setting, which indicates that the BER was less than The inset photograph shows an eye pattern after a 70000 km transmission, in which the eye is clearly open.

In Fig. 4(b), we show 8 channel 10 Gbit/s signals, four of which were are vertical and four of which were from orthogonal components. The inset

photograph shows one of the eye patterns which was demultiplexed to 10 Gbit/s and the figure shows a spectral profile of the 80 Gbit/s soliton signal after 10000 km transmission. The solid line with open diamonds indicates the BER at 0 km. The power penalty after the 10000 km transmission was

typically 2.5 dB and all data fell within a power penalty difference of 1.5 dB. When the received power was larger than –27.5 dBm, no error appeared

at a -order error counter setting, which indicates that the BER was less than . Without the in-line control, the transmission distance was approximately 1750 km, but we were able to extend it to 10000 km by using in-line modulation. These results show that an ultrahigh bit-rate well beyond a 100 Gbit/s soliton can be transmitted over 10000 km by using in-line synchronous modulation. This technique will be more effective when the single-channel

80 GBIT/S MULTI-CHANNEL SOLITON TRANSMISSION

25

bit-rate is higher. The speed of the electronics may be limited to 100 Gbit/s and this makes all optical soliton control using high-speed optical switching more important [13]. 3.2. SUPPRESSION OF POLARIZATION MODE DISPERSION USING IN-LINE SYNCHRONOUS MODULATION

The in-line modulation technique is also very useful for reducing polarization mode dispersion (PMD) [14]. When the bit rate of a system is increased, the system suffers from not only second and third order dispersion, but also PMD for which there is no effective compensation technique. However, PMD can be sufficiently reduced by using in-line modulation since a pulse that has been statistically broadened by PMD can be reshaped and shortened with an optical gating function. PMD causes random pulse broadening without any spectral change [15]. Therefore, spectral manipulation is not very effective in reducing PMD, although the center wavelength can be fixed. In contrast, in-line modulation can in principle reduce the wings of two pulses and reshape the waveform itself [2, 3], which is useful for reducing the PMD effect. The effectiveness with which PMD induced soliton broadening can be suppressed by using soliton control was recently reported [16]. However, when soliton system design is discussed via numerical simulation, it is important to evaluate not only the evolution of single pulses but also soliton-soliton interaction and the effect of ASE noise. When the bit rate becomes high, these effects play a very important role. In this section, we evaluate the transmission quality of high speed (40 and 80 Gbit/s) soliton systems using a Q-factor [17, 18], for various PMD conditions, and show that in-line synchronous modulation is especially useful for extending the transmission distance in the presence of large PMD. The effect of PMD can be simulated by considering the fiber as a cascade of short pieces with a constant fiber length and birefringence . At each connection point between segments, the polarisation state is changed randomly and this causes statistical coupling between two orthogonal modes. According to this model, the PMD, , is given by . The propagation characteristics of the two polarized modes are governed by a coupled nonlinear Schrödinger equation [15]. The conditions we adopted were as follows. The initial signal was a 32 bit pseudorandom sequence, and . The transmission distance was defined as that at which the Q factor was lower than 7. The Q factor was calculated after the signal had passed a second-order Butterworth type low-pass electrical filter with a bandwidth equal to 65 % of the bit rate. We used a random number in the polarization rotation and calculated the

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mean value of the transmission distance from 8 computer runs under the

same condition. Here we consider 40 and 80 Gbit/s return-to-zero (RZ) systems with a repeater spacing of 80 km. The loss, third order dispersion, effective mode area and non-linear Kerr coefficient of the fiber were 0.25 dB/km, 0.07 and , respectively. The EDFA noise

figure was 6 dB. We installed a Lorentz shaped optical filter with a bandwidth of 3 nm at every EDFA. The group velocity dispersion of the fiber, D, was uniform over the whole transmission line, and we calculated the transmission distance at different GVD values. The pulse was sech shaped and the FWHM was 5 ps for 40 Gbit/s and 2.5 ps for 80 Gbit/s. The peak power of the soliton pulse, P, was optimized to maximize the transmission distance for each GVD value. When we used in-line synchronous modulation, we installed an amplitude modulator every few repeaters, and changed the bandwidth of all of the optical filters from 3 to 1.5 nm. Figure 5(a-l) and (a-2) show the transmission distance in the 40 Gbit/s

RZ system as a function of GVD. In Fig. 5(a-l), the PMD was varied from without soliton control. When the PMD was

ps/km/nm,

the maximum transmission distance was 4700 km ( ). This power corresponded to that of an

soliton, which means that an optical soliton was transmitted under this condition. However, when the PMD was the transmission

distance was reduced to 1600 km ( ), which is 1/3 of the value without PMD. This clearly shows that when the PMD becomes larger, even optical solitons cannot be transmitted over long distances. When we used in-line synchronous modulation, there was a

marked increase in the transmission distance even in the presence of large PMD. In Fig. 5(a-2), an in-line synchronous modulator was installed every 320 or 160 km, and the PMD was set at The maximum

transmission distance was extended to 6800 km with a modulator spacing of 320 km ( ), and further extended to 11000 km with a modulator spacing of 160 km ( ). The best result was obtained when the optical pulse was transmitted as an optical soliton. That is, in-line synchronous modulation is very effective in extending the maximum transmission distance in the

presence of large PMD. Figure 5(b-l) and (b-2) show the transmission distance in an 80 Gbit/s RZ system as a function of GVD. In Fig. 5(b-l), we calculated the distance without soliton control and for PMDs of 0.0 to . It can be seen that the maximum transmission distance was 1300 km for a PMD of 0.0 , but this was reduced to 500 km for a PMD of ( ),

80 GBIT/S MULTI-CHANNEL SOLITON TRANSMISSION

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which is about 1/3 of the value without PMD. In Fig. 5(b-2), an amplitude modulator was installed every 160 or 80 km for a PMD of The maximum transmission distance was 1500 km for a modulator spacing of 160 km ), and was extended to 6300 km for a modulator spacing of 80 km ( dBm). It is also very important to note that the maximum transmission distance was achieved when the optical pulse was transmitted as an optical

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soliton. This result indicates that in-line synchronous modulation is very effective in extending the maximum transmission distance in the presence

of large PMD even for a bit rate of 80 Gbit/s. When we apply the in-line modulation technique, it is important to note that the transmission line has the characteristics of a soliton. Even though the line is dispersion-managed, the in-line modulation is still very effective as long as the dispersion swing is small, for example a small GVD change of ps/km/nm. However, if the dispersion swing is of the order of 10–15 ps/km/nm, the nonlinearity which supports the soliton nature becomes so weak that in-line modulation is not effective for maintaining the largely dispersion-managed soliton. That is the pulse becomes almost

linear. In such a case a soliton in-line modulator, which consists of an inline modulator, an optical filter, and nonlinear fiber to maintain the pure soliton effect, plays a very important role [19]. 4. Ultrahigh Speed WDM Soliton Transmission with In-line Modulation

4.1. Q MAP EVALUATION OF WDM SOLITON TRANSMISSION AND ITS SUPERB CHARACTERISTICS

Dispersion-managed (DM) soliton transmission is very advantageous not only for TDM but also for WDM in realizing a large-capacity long-distance system. The biggest advantage of WDM soliton transmission is that one channel can carry a much greater capacity than one NRZ channel, thus enabling us to reduce the number of channels. In addition, a combination of TDM and WDM is very flexible. It had been thought that the WDM soliton was not so advantageous because of the soliton collisions (cross phase modulation) between different wavelengths. However, dispersion management

of the transmission line works very well to reduce the soliton collision effect and even a 1 Tbit/s (20 Gbit/ channels) transmission over 1000 km with a span of 100 km has been reported, in which a dispersion-compensated single-mode fiber was used [20]. There have been several reports of WDM soliton transmission over ultralong distances such as 80 Gbit/s (10 Gbit/ channels) over 9000 km using the sliding filter method [21] and 160 Gbit/s (20 Gbit/ channels) over 10000 km with the synchronous modulation method [22]. In addition, high capacity transmission experiments have also been demonstrated without soliton control [23, 24]. The dispersion-tuned synchronous modulation technique was also reported as an alternative to delay time control [25]. When the in-line modulation method is used, it is highly advantageous to increase the bit rate as it is similar to

laser mode-locking at an ultrahigh repetition rate. By contrast, a chirped RZ transmission experiments with a total ca-

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pacity of as high as 160 Gbit/s has been successfully reported in which the single channel capacity was 5 Gbit/s [26]. Recently 1 Tbit/s (10 Gbit/ channels) transmissions over 6000–10000 km have been reported [27, 28]. In this section, we describe the condition under which the WDM–DM soliton is advantageous compared with WDM–NRZ transmission with the goal of long distance transmission. Then, we show that it is possible to send 160 Gbit/s WDM soliton data (20 Gbit/ channels) over 10000 km

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by using in-line soliton control. Figure 6 shows “Q maps” for WDM–NRZ and soliton systems [29]. In the Q map for a WDM system, the Q value is represented by that of the worst channel so that the best performance is given by the worst channel. Figure 6(a) and (b) correspond to uniform dispersion and large dispersion allocation cases, respectively. There are a total of 10 channels, and the bit rate is 10 Gbit/s (a pulse width of 25 ps). The channel separation is 1 nm. In Fig. 6(a), one span consists of an 80 km fiber with zero GVD and a 5 km fiber with an arbitrary GVD to change the average GVD. In Fig. 6(b), the 80 km fiber has a GVD of 16 ps/km/nm for the soliton and the GVD of the 5 km DCF was varied according to the average GVD. With NRZ transmission, the 80 km fiber has a normal GVD of –16 ps/km/nm. A dispersion slope compensation (DSC) fiber is also installed every 320 km. In both figures, the WDM Q map for NRZ is shown on the left and that for the soliton is shown on the right. As shown in Fig. 6(a), WDM soliton transmission has characteristics comparable to those of WDM NRZ transmission. Although severe soliton collision occurs in a fiber with a uniform GVD, it can be reduced by adopting dispersion slope compensation. That is by making the soliton power in each channel the same. Without DCF, the soliton transmission is worse than NRZ transmission. However, as shown in Fig. 6(b), the situation is completely reversed. That is, the WDM DM soliton transmission has much better Q value characteristics than the WDM NRZ transmission. This means that soliton collisions (cross phase modulation) at different wavelengths are reduced due to a process of pulse broadening in DM soliton transmission. These results indicate that the DM soliton is also applicable to WDM and provides a better result than the conventional NRZ system. Figure 7 shows the transmission distance as a function of the dispersion of the transmission fiber (first segment). One span has two segments with different dispersions. In Fig. 7(a), the dispersion of the transmission fiber (first segment) is anomalous, and in Fig. 7(b), the dispersion of the transmission fiber (first segment) is normal. We installed a DCF every 80 km (every span), and a DSCF every 320 km (every four spans). The transmitted signal power and the average dispersion were optimized for each dispersion of the transmission fiber. A very interesting feature can be found in Fig. 7. When the dispersion of the transmission fiber (first segment) is small, that is ps/km/nm (less than approximately 1 ps/km/nm in Fig. 7(a) or less than approximately –1 ps/km/nm in Fig. 7(b)), a larger dispersion allocation gives a longer transmission distance at approximately the same rate for the soliton and NRZ systems. The reason for this is that four wave mixing (FWM) is suppressed when the dispersion allocation becomes larger.

When the dispersion is further increased (more than approximately 1

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31

ps/km/nm in Fig. 7(a) or more than approximately –1 ps/km/nm in Fig. 7(b)), the transmission distance increases gradually with increasing disper-

sion allocation in the soliton system, but it becomes almost constant and independent of the dispersion-allocation in the NRZ system. This is because the interaction between the different channels and collision effects are still reduced in the soliton system when a soliton pulse with a high level power

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is broadened due to the large dispersion. However, this reduction is very small for the NRZ system. When the dispersion of the transmission fiber is as much as 16 ps/km/nm, which is the dispersion of standard single-mode fiber, the transmission distance of the soliton system is almost twice that

of the NRZ system. These results are in good agreement with the experimental results we reported in reference [30]. It is important to note that

the large GVD allows us to transmit at greater power and to reduce the FWM by broadening the pulse, hence the low peak power. Therefore, we obtain a high signal to noise ratio even in a WDM system. 4.2. 160 GBIT/S (20 GBIT/S×8) AND (80 GBIT/S×2) WDM SOLITON TRANSMISSION OVER 10000 KM

We undertook a 160 Gbit/s (20 Gbit/; channels) WDM soliton transmission experiment in a 250 km dispersion-shifted fiber loop, as shown in Fig. 8. The key to successfully increasing the system performance from 100 to 160 Gbit/s is the adoption of a polarization scrambler and a phase modu-

lator at the input [22]-[24]. The amplifier spacing was 50 km and the average fiber loss for one span was 12.5 dB. We used eight wavelength-stabilized cw LDs which were equally spaced between 1552.0 and 1562.5 nm at 1.5 nm

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intervals. We employed a four-segment dispersion-decreasing configuration. The use of the dispersion-decreasing configuration is a very powerful way of reducing collision effects and dispersive waves although it requires a rather more complicated selection procedure than when we use fibers with uniform

dispersion. We used two polarization-insensitive EA modulators to convert the cw beams into 11–13 ps optical pulses at 20 GHz. One modulator was for signals at 1552.0, 1555.0, 1558.0 and 1561.0 nm (group 1) and the other for 1553.5, 1556.5, 1559.5 and 1562.5 nm (group 2). Two LN modulators were independently used for the data coding with a – 1 pseudorandom signal at 20 Gbit/s. Data signals for group 1 were modulated with a format and those for group 2 were modulated with a format in order to obtain uncorrelated neighboring channels. After amplifying each pulse to

the corresponding soliton power level, all signals were polarization scrambled at low speed and synchronously phase-modulated at 20 GHz, and then introduced into the 250 km loop. The EDFAs were pumped by LDs and had a relatively large bandwidth for WDM use. The total transmitted power was approximately +10 dBm.

After the 250 km transmission, the eight channel soliton signals were separated into their respective wavelengths with a WDM coupler. To keep

the same GVD in each channel, we installed a dispersion compensation

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fiber (DCF) in each channel after a 250 km four-step dispersion-decreasing configuration. In this case, the average GVD for all channels was set at 0.12 ps/km/nm by connecting a different DCF to each channel. The DCFs we used have GVDs of 21.3, 47.0, 72.8, 98.8, 124.9, 151.1, 177.7 and 204.5 ps/nm corresponding to signals at 1552.0, 1553.5, 1555.0, 1556.5, 1558.0, 1559.5, 1561.0 and 1562.5 nm, respectively. Since the average GVD is kept at 0.12 ps/km/nm in each channel, the average power for each channel was kept at the same value Then we applied in-line modulation and narrowband filtering . Three modulators controlled two channels each and two other modulators controlled one channel each. The bandwidth of the optical filter was approximately 0.35–0.45 nm. To obtain polarization insensitive synchronous modulation characteristics, we connected two LN modulators orthogonally. This modulator was driven by an extracted 20 GHz clock for each channel. Then, while keeping the soliton power at an appropriate level, the pulses were multiplexed again through the WDM coupler and fed back to the input of the loop. The eight clock signals for in-line modulation were cleanly extracted. When each channel was not well controlled, the wing of the 20 GHz clock signal had a broad and high pedestal which did not appear when the solitons were under control. This means that even in the presence of strong interactions between solitons with different wavelengths, stable solitons can be transmitted over long distances. The BERs after a 10000 km transmission are shown in Fig. 9(a). When the received power was larger than –28.0 dBm, no error appeared at a -order error-counter setting, which indicates that the BER was less than . The maximum power penalty difference was 1 dB. The inset photograph shows an eye pattern after a 10000 km transmission at 1555.0 nm. When the polarization scrambler was installed the number of the channels can be increased and when the phase modulation is also applied, a more stable transmission with a lower penalty was achieved. Figure 9(b) shows how the BERs degrade as the propagation distance is increased. The BERs started to worsen after a transmission of 12000 km. This indicates that even when soliton transmission control is employed in each channel, it is not possible to send WDM soliton over unlimited distances. This is attributed to the fact that there are many collisions between different channels. However, it was possible to send a 160 Gbit/s (20 Gbit/ channels) WDM soliton at least 10000 km. These results indicate that ultra high capacity WDM transmission over 10000 km is also feasible through the use of soliton technology. Figure 10 shows our setup for 160 Git/s ( Gbit/s) WDM soliton transmission over 10000 km [31]. In Subsection 3.1, we described single channel 80 Gbit/s soliton transmission with in-line modulation. We ex-

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tended the 80 Gbit/s soliton to a WDM scheme [11]. This change was not difficult since we were able to use the same synchronous modulators for two channels. The channel wavelengths were 1550.5 and 1556.0 nm. We employed a polarization multiplexing technique to increase the bitrate from 40 to 80 Gbit/s. This enabled us to realize soliton control at 80 Gbit/s based on 40 Gbit/s electronics [31]. We undertook the experiment with a 250 km fiber loop, which consisted of 5 spans. Each span was 50 km long and the average fiber loss per span was 12.5 dB. We used a four-segment dispersion-decreasing configuration to reduce the influence of the dispersive waves. The four 12.5 km long DSFs had GVDs of 0.24, 0.06, –0.04 and –0.10 ps/km/nm at 1550.5 nm and 0.65, 0.47, 0.37 and 0.31 ps/km/nm at 1556.0 nm. We set the average GVD at approximately 0.04 ps/km/nm by using DCF. The average launched powers at 1550.5 and 1556.0 nm were +9.5 and +10.0 dBm, respectively, and the corresponding peak powers into the first segment at each wavelength were 74 and 83 mW, respectively. These value also correspond to and solitons, respectively. We used two 10 GHz harmonically mode-locked erbium fiber lasers operating at 1550.5 and 1556.0 nm. The pulse and spectral widths were 3.0 ps and 0.81 nm, respectively ( ) in both lasers. The output pulses were simultaneously modulated at 10 Gbit/s with a – 1 PRBS using

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an LN modulator. Then, a PLC was used to optically multiplex the 10 Gbit/s signals into a 40 Gbit/s data train. After that, the 40 Gbit/s single polarization signals were converted into two 80 Gbit/s TDM signals with different wavelengths by injecting the two 40 Gbit/s signals in a direction

45 degrees from the axis of the PANDA fiber and by using a group delay difference between the two axes. Since the group delay difference between the two axes of the PANDA fiber at 1550.5 nm is the same as that at 1556.0 nm, we used the same PANDA fiber that we used in a previous study. After a 250 km transmission through the loop, we applied in-line modulation to each wavelength and optical filtering. The filter bandwidth was 1.1 nm. First we installed a WDM delay unit to synchronize the timing between the two channels at each modulator. Each delay unit is capable of precisely tuning the pulse peak timing. Then we coupled two channels into a PBS. The PBS separated the polarization-multiplexed 80 Gbit/s signal into two 40 Gbit/s signals and single polarization soliton control was employed at 40 Gbit/s. As regards the synchronous modulation, a 40 GHz clock was extracted from part of the 1550.5 nm transmitted soliton pulses with a photodetector by using a 50 GHz bandwidth and a high Q dielectric filter. The clock signal was then supplied to two 40 Gbit/s orthogonal signals in both channels. Each single-polarization LN modulator for soliton control driven by the extracted 40 GHz had the same modulation performance for both 1550.5 and 1556.0 nm. Simultaneously, the amplitude level of the clock signal at 1550.5 nm was fed back to a PC to obtain maximum clock power. This ensured that the orthogonal channel was also automatically optimized. The clock signal at 1556.0 nm was also extracted and fed back to another PC. After the in-line modulation, two orthogonal 40 Gbit/s signals at 1550.5 and 1556.0 nm were reconverted to two 80 Gbit/s signals with another PBS. Figure 11 (a) and (b) show the measured BERs. We measured 8 channel 10 Gbit/s signals for each channel, four of which were vertical and four

of which were from orthogonal components. The inset photograph shows one of the eye patterns which was demultiplexed to 10 Gbit/s after the 10000 km transmission. The solid line with open diamonds indicates the BER at 0 km. The power penalty after the 10000 km transmission was typically 3.5 dB and all data fell within a power penalty difference of 2.0 dB. When the received power was larger than –27.5 dBm, no error appeared at an error counter setting of which indicates that the BER was less than . Without the in-line control, the transmission distance was approximately 1500 km, but we were able to extend it to 10000 km by using in-line modulation. This result shows that an ultrahigh bit-rate TDM/WDM soliton can be transmitted over 10000 km by using in-line synchronous modulation.

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5. Summary

We described recent progress on ultrahigh speed TDM/WDM soliton transmission with the use of in-line modulation. There are two possible ways of realizing the potential of the ultrahigh speed soliton transmission. One is

a TDM system with soliton control, which enables us to send an ultrahigh speed single channel transmission over transoceanic distances. For this purpose a single channel 80 Gbit/s transmission over 10000 km was described. If one has an ultrahigh frequency modulation scheme, one can achieve the higher bit rate than 80 Gbit/s. The other is a WDM system which has much higher speed single channel than that in a conventional WDM system. For

this purpose, we described 20 Gbit/ channel and 80 Gbit/ channels transmissions over 10000 km. These results indicates that in-line modulation is very advantageous for high speed, long distance soliton transmission. Acknowledgment

The authors would like to express their thanks to Drs. M. Kawachi and K. Sato for their fruitful comments and unceasing encouragement.

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References 1. Gordon, J. P., and Haus, H. A. : Random walk of coherently amplified solitons in optical fiber transmission, Opt. Lett., Vol.11, No.10, (1986), p.665. 2. Nakazawa, M., Yamada, E., Kubota, H. and Suzuki, K. : 10 Gbit/s soliton transmission over one million kilometers, Electron. Lett., Vol.27, No.14, (1991), pp. 12703. 4. 5.

6. 7. 8.

1271. Kubota, H. and Nakazawa, M. : Soliton transmission control in time and frequency domains, IEEE J. Quantum Electron., Vol.29, No.7, (1993), pp.2189-2197. Nakazawa, M., Kubota, H., Yamada, E. and Suzuki, K. : Infinite-distance soliton transmission with soliton controls in time and frequency domains, Electron. Lett., Vol. 28, No.12, (1991), pp.1099-1101. Mecozzi, A., Moores, J. D., Haus, H. A. and Lai, Y. : Soliton transmission control, Opt. Lett., Vol.16, No.23, (1991), p.1841. Hasegawa, A. and Kodama, Y. : Guiding-center soliton, Phys. Rev. Lett., Vol.66, No.2, (1991), p.161. Gordon, J. P. and Mollenauer, L. F. : Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission, IEEE J. Lightwave Technol., Vol.9, No.2, (1991), p.170. Kubota, H. and Nakazawa, M. : Soliton transmission with long amplifier spacing under soliton control, Electron. Lett, Vol.29, No.20, (1993), pp.1780-1782; see also

Aubin, G., Montalant, T., Moulu, J., Nortier, B., Pirio, F. and Thomine, J. B. : 9.

10. 11.

12. 13. 14. 15. 16.

17. 18.

Record amplifier span of 105 km in a soliton transmission experiment at 10 Gbit/s over 1 Mm, Electron. Lett., Vol.31, No.3, (1995), pp.217-219. Nakazawa, M., Suzuki, K., Yamada, E., Kubota, H., Kimura, Y. and Takaya, M. : Experimental demonstration of soliton data transmission over unlimited distances with soliton control in time and frequency domains, Electron. Lett., Vol.29, No.9, (1993), pp.729-731. Suzuki, K., Kubota, H., Sahara, A. and Nakazawa, M. : 40 Gbit/s single channel optical soliton transmission over 70,000 km using in-line synchronous modulation and optical filtering, Electron. Lett., Vol.34, No.l, (1998), pp.98-99. Nakazawa, M., Suzuki, K. and Kubota, H. : Single-channel 80 Gbit/s soliton transmission over 10,000 km using in-line synchronous modulation, Electron. Lett., Vol.35, No.2, (1999), pp.162-163. Nakazawa, M., Yoshida, E. and Kimura, Y. : Ultrastable harmonically and regeneratively modelocked polarization-maintaining erbium fibre ring laser, Electron. Lett., Vol.30, No.19, (1994), pp.1603-1604. Bigo, S., Leclerc, O., Brindel, P., Vendrôme, G., Desurvire, E., Doussière, P. and Ducellier, T. : 20 Gbit/s all-optical regenerator, OFC’97, PD22, (1997). Sahara, A., Kubota, H. and Nakazawa, M. : Ultra-high speed soliton transmission in the presence of polarisation mode dispersion using in-line synchronous modulation, Electron. Lett., vol.35, No.l, (1999), pp.1-2. Menyuk, C. R.: Nonlinear pulse propagation in birefringent optical fibers, IEEE. J. Quantum Electron., Vol,23, (1987), pp.174-176. Matsuiuoto, M., Akagi, Y. and Hasegawa, A. : Propagation of solitons in fibers with randomly varying birefringence: Effects of soliton transmission control, IEEE. J. Lightwave Technol., Vol.15, (1997), pp.584-589. Bergano, N. S., Kerfoot, F. W. and Davidson, C. R. : Margin measurements in optical amplifier systems, Photon. Tech. Lett., Vol.5, (1993), pp.304-306. Sahara, A., Kubota, H. and Nakazawa, M. : Q-factor contour mapping for evaluation of optical transmission systems:soliton against NRZ against RZ pulses at zero group

velocity dispersion, Electron. Lett., Vol.32, (1996), pp. 915-916. 19. Dany, B. et al.: A transoceanic Gbit/s system combining dispersion-managed soliton transmission and new black box in-line optical modulator, Electron. Lett., (1999), pp.418-419; see also ibid. : Feasibility of

Gbit/s dispersion-managed

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21.

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transoceanic systems with high spectral efficiency, ECOC’99, (1999), pp.I152-153. Le Guen, D., DelBurgo, S., Moulinard, M. L., Grot, D., Henry, M., Favre, F. and Georges, T. : Narrow band 1.02 Tbit/s Gbit/s) soliton DWDM transmission over 1000 km of standard fiber with 100 km amplifier span, OFC/IOOC’99, PD4, (1999). Mollenauer, L. F., Mamyshev- P. V. and Neubelt, M. J. : Demonstration of soliton WDM transmission at up to Gbit/s, error-free over transoceanic distances, OFC’96, PD22, (1996).

22. Nakazawa, M., Suzuki, K., Kubota, H., Sahara, A. and Yamada, E. : 160 Gbit/s WDM (20 Gbit/ channels) soliton transmission over 10000 km using in-line modulation and optical filtering, Electron. Lett., Vol.34, No.1, (1998), pp.103-104. 23. Suzuki, M., Morita, I., Edagawa, N., Yamamoto, S. and Akiba, S. : 20 Gbit/s-based soliton WDM transmission over transoceanic distances using periodic compensation

of dispersion and its slope, ECOC’96, ThB3.4., (1996). Suzuki, M., Edagawa, N., Morita, I., Yamamoto, S. and Akiba, S. : Soliton-based return-to-zero transmission over transoceanic distances by periodic dispersion compensation, J. Opt. Soc. Am., Vol.B-14, (1997), pp.2953-2959. 25. Desurvire, E., Leclerc, O. and Audouin, O. : Synchronous in-line regeneration of wavelength-division multiplexed soliton signals in optical fibers, Opt. Lett., Vol.21,

24.

No.4, (1996), pp. 1026-1028. 26. Bergano, N. S. et al. : Long-haul WDM transmission using channel modulation; A 160 Gb/s (32×5 Gb/s) 9,300 km demonstration, OFC’97, PD16, (1997). 27.

Naito, T., Shimojoh, N., Tanaka, T., Nakanmoto, H., Doi, M., Ueki, T. and Suyama, M. : 1 Terabit/s WDM transmission over 10,000 km, ECOC’99, PD2-1, (1999). 28. Tsuritani, T., Takeda, N., Imai, K., Tanaka, K., A. gata, A., Morita, L, Yamauchi, H., Edagawa, N. and Suzuki, M. : 1 Tbit/s transoceanic transmission using 30 nm-wide broadband optical repeaters with

-enlarged positive dispersion

fibre and slope-compensating DCF, ECOC’99, PD2-8, (1999). 29. Sahara, A., Kubota, H. and Nakazawa, M. : Comparison of the dispersion allocated WDM (10 Gbit/s×10 channels) optical soliton and NRZ systems using a Q map, Opt. Commun., Vol.160, (1999), pp.139-145.

30. Nakazawa, M., Yamada, E., Kubota, H., Yamamoto, T. and Sahara, A. : Numerical and experimental comparison of soliton, RZ pulse and NRZ pulses under two-step dispersion allocation, Electron. Lett., Vol.33, No.17, (1997), pp.1480-1482.

31.

Nakazawa, M., Suzuki, K. and Kubota, H. : 160 Gbit/s(80 Gbit/s×2 channels) WDM soliton transmission over 10000 km using in-line synchronous modulation, Electron. Lett., Vol.35, No.16, (1999), pp.1358-1359.

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MULTI-SOLITON TRANSMISSION AND PULSE SHEPHERDING IN BIT-PARALLEL WDM OPTICAL FIBER SYSTEMS

YU. S. KIVSHAR AND E. A. OSTROVSKAYA

Australian Photonics Cooperative Research Centre Optical Sciences Centre, Research School of Physical Sciences and Engineering, The Australian National University ACT 0200 Canberra, Australia Abstract. We study the interaction of pulses in wavelength division multiplexed (WDM) systems in hit-parallel optical fiber transmission links for

high-performance computer networks. The model is described by a system of nonlinear Schrödinger (NLS) equations for the envelopes of optical pulses propagating in a single-mode nonlinear fiber. These equations are coupled

due to the cross-phase-modulation (CPM) effects caused by the intensity dependence of the refractive index. We describe two major effects: (i) manipulation and control of pulses co-propagating on different wavelengths by a shepherd pulse, and (ii) bit-parallel WDM soliton transmission due to the nonlinear CPM effect. We also provide a brief summary of other systems where the multi-mode multi-soliton propagation occurs (such as optical fiber arrays and incoherent spatial solitons), and highlight specific features of the WDM soliton systems.

1. Introduction

As is well known, the concept of long-distance error-free optical communications based on single-mode-fiber optical solitons has failed to satisfy the current demands of technology for a number of reasons; solitons are susceptible to jitter, periodic amplification restricts their capacity to a narrow bandwidth, to name but two. Some ideas for the improvement of the performance of soliton-based communication systems, such as sliding-guiding filters, active control and phase conjugation, led to an apparently unacceptable increase in complexity of the system design. Dispersion management 41 A. Hasegawa (ed.), Massive WDM and TDM Soliton Transmission Systems, 41–61. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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is the next important step in utilizing the attractive properties of soliton pulses for practical long-distance transmission systems without incurring growth in complexity. Being applied to a multi-channel wavelength division multiplexed (WDM) system, dispersion management of soliton pulses brings a many-fold application of solitons using the return-to-zero (RZ) data format. The path pioneered by the soliton dispersion management has led to the current situation whereby almost all new systems employ an RZ format thus exploiting the soliton properties. The attractive idea of using solitons for increasing the performance of high-speed WDM optical fiber systems can be used in other types of optical fiber links and networks. As was suggested by Bergman et al. [1, 2], optical solitons can be useful in a new scheme of interconnecting and local area network transmission for computer communications, based on spectrally encoding one or more computer words into a wavelength datagram. This concept leads to a bit-parallel wavelength fiber optical link that uniquely maintains channel time alignment. It requires a special multi-wavelength technique for transmitting and maintaining time aligned pulses (in fact, solitons) as parallel bits propagating through an optical fiber. In this lecture, we briefly overview these ideas and discuss a number of the unique properties of soliton-based bit-parallel WDM fiber transmission systems. In

particular, we develop the first analytical theory of the so-called shepherding effect earlier observed in numerical simulations [2], when a strong (shepherd) pulse at a separate wavelength makes possible the manipulation and control of pulses co-propagating on different wavelengths in a WDM multichannel system. We also discuss an experimental evidence of this effect, as well as some further ideas for bit-parallel WDM pulse transmission. For comparison, we briefly overview some other models of multi-mode multisoliton systems, such as optical fiber arrays, incoherent spatial solitons, the optical conveyor belt, etc. In conclusion, we briefly mention a number of important issues in this field that still remain unsolved. 2. Bit-parallel Wavelength Optical Links and Networks

In the high-performance computing and communications, the impact of a rapid rise in microprocessor performance over the last few years has been most notable in the supercomputing industry with massively parallel processor (MPP) architectures comprised of hundreds or thousands of microprocessors. One of the most important design parameters for such MPP supercomputers is the development of an intra-processor communication network that provides high enough performance so that the communication time is much smaller than the time spent computing. For a perfect network, the distributed RAM located at each processor node in the com-

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puter should look as though it is local. It is expected that next generation systems will require at least ten times larger memory bandwidth and eventually transmit as high as 10 Gigabytes per second. Such high single-channel speeds are unlikely to come from the traditional telecommunications industry, and up until now, the inexpensive computer interconnector has been

the parallel ribbon cable. More recently, the computer industry has turned from copper wire to fiber optics ribbon cables, and now several manufactures are introducing fiber optics links as low cost computer interconnectors. Several problems in using fiber optics ribbon cable should be mentioned.

First, as the speeds rise to multiple Gigabytes per second per bit line, it becomes difficult to maintain time alignment of the parallel fiber channel bits. Secondly, the interconnectors should allow an increase in bandwidth

and ultimately, the aggregate bandwidth required to interconnect two large MPP supercomputers should compare with the bandwidth of the internal communication network of the machine. All this requires the development of parallel communication link tech-

nologies. Bit-parallel wavelength fiber optics link (see Fig. 1) for very high bandwidth computer communications offers many of the advantages of both parallel fiber ribbon cable and conventional WDM, but with few of their drawbacks. A bit-parallel wavelength (BPW) system codes signals using different wavelengths before sending them over a single-mode optical fiber (see Fig. 1). BPW differs from traditional WDM in that no parallel to serial conversion is necessary (i.e. no multiplexing is required), parallel pulses are launched simultaneously on different wavelengths. The key property of such systems is time alignment of the pulses for a given signal byte. The idea to use solitons for such BPW systems looks very attractive. First of all, a soliton maintains its shape and, unlike the case of longdistance communication networks, it does not require periodic amplifica-

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YU. S. KIVSHAR AND E. A. OSTROVSKAYA

tion over such short distances. Second, interaction of solitons that belong to different wavelengths is incoherent (no four-wave mixing effects), and therefore simple. The important issue is how to keep solitons time aligned on a

number of different wavelengths. Below, we discuss several concepts related to the use of solitons for BPW systems, including the so-called shepherding effect (see also Refs. [1,2]). 3. Theoretical Model The successful design of low-loss dispersion-shifted and dispersion-flattened optical fibers with low dispersion over a relatively large wavelength range can be used to reduce or completely eliminate the group-velocity mismatch

for the multi-channel WDM systems resulting in the desirable simultaneous arrival of time-aligned bit pulses, creating a new class of BPW links used in high speed ( Gbyte/s) single fiber computer buses [1]. As has been mentioned above, in spite of the intrinsically small value of the nonlinearity-

induced change in the refractive index of fused silica, nonlinear effects in optical fibers cannot be ignored even at relatively low powers. In particular, in WDM systems with simultaneous transmission of pulses of different wavelengths, the cross-phase modulation (CPM) effects should be taken into account. Although CPM will not cause energy to be exchanged among the

different wavelengths, it leads to pulse interaction so that the pulse shapes and locations can be altered significantly. To describe the multi-channel WDM transmission of co-propagating wave envelopes in a nonlinear optical fiber including the CPM effect, we follow the standard derivation [3] and the original work of Yeh and Bergman [2] (see also Refs. [4]-[7]) and consider N coupled nonlinear Schrödinger (NLS) equations ( )

where, for the j–th wave, is the slowly varying amplitude, is the j–th wave group velocity, is the dispersion coefficient, is the absorption coefficient, and is the nonlinear coefficient where is the carrier frequency on the j–th wave, is a fiber effective core area, and for silica. Before making any further renormalizations of the general system (1), we mention that, since the fiber lengths, L, typically involved in bit-parallel networks are only a small fraction of the absorption length (i.e. ), fiber loss can be neglected. Moreover, for CPM to take effect significantly, the group-velocity mismatch must be held close to zero. That is why we

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take (zero walk-off between the modes) and assume the effect of walk-off to be a small perturbation. Introducing the normalizing coefficients and measuring the wave envelopes, , in units of the incident optical power of the central beam, we derive the dimensionless system of coupled equations,

where time is measured in the units of the pulse width in the central channel, , the propagation distance is measured in the units of and so that and . Equations (2) are the fundamental dimensionless nonlinear model for the bit-parallel WDM transmission problem. 4. The Pulse Shepherding Effect

As was first mentioned by Yeh and Bergman [2], when two or more optical pulses co-propagate simultaneously and affect each other through the intensity dependence of the refractive index, CPM can be used to produce an interesting pulse shepherding effect. In particular, Yeh and Bergman studied numerically the evolution of two pulses whose operating wavelengths are separated by . For this case, the four-wave-mixing effect is negligible. When these pulses are initially offset by a 1/2 pulse width, the pulses tend to attract each other. As is seen in Fig. 2 (upper row), the leading pulse is pulled back while the trailing pulse is pushed forward so that these pulses tend to align with each other. However, when the two co-propagating pulses, on two separate wavelengths, are separated by a sufficiently large distance, these two pulses will not interact with each other. This is demonstrated in Fig. 3 (upper row) where the two co-propagating pulses are separated by one pulse width. Each pulse propagates independently as if without the presence of the other pulse. Two widely separated pulses can be brought significantly closer to each other by the launching of another pulse on a separate wavelength ( ) with the proper magnitude and at the proper time. According to Yeh and Bergman, this pulse is called the shepherd pulse because of its shepherding behavior on the other pulses, as is shown in both Fig. 2 (lower row) and Fig. 3 (lower row). Computer simulations [2] showed that a low-magnitude shepherd pulse does not possess sufficient attractive strength to pull the shepherded pulses together. Additionally, it was noticed that at a very large amplitude of the shepherd pulse the shepherded pulses tend to break up, and for the broadened shepherd pulse the effect is getting weaker. Therefore, these (let us

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YU. S. KIVSHAR AND E. A. OSTROVSKAYA

call than preliminary) computer simulations (see also Ref. [4] for the case suggest that there exists an optimum pulse with a certain magnitude, pulse width, and pulse shape that can provide the best alignment for these pulses. As a matter of fact, this observation means that the most effective shepherding effect should be observed for the pulses that are close to the so-called ”nonlinear modes” of the model, that is solitons of the WDM model (2). 5. Two-channel WDM System

To analyse the nonlinear modes as stationary localised states of the WDM model (2) and to get a deeper insight into the results of the numerical simulations of Yeh and Bergman described above, first we consider a twochannel WDM system described by the system of equations,

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47

We are looking for the stationary solutions of the system (3) in the form,

and therefore obtain the system of equations for the normalized mode amplitudes,

where the amplitudes and time are measured in the units of and respectively, and we define System (5), in its more general form, has been extensively studied by a number of researchers (see, e.g. Ref. [8] for an overview of the results). When Eqs. (5) possess a localised solution in the form of the fundamental optical soliton (3, 9],

with the normalised power

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YU. S. KIVSHAR AND E. A. OSTROVSKAYA

For small i.e. for the second equation of Eqs. (5) generates a familiar eigenvalue problem of linear waveguiding,

where in the form

The Eq. (7) possesses a localized solution

where , and is the corresponding eigenvalue. This solution describes a low-amplitude zeroth-order mode guided by the fundamental soliton (6).

The solution described by Eqs. (6) and (7) is valid as long as in Eqs. (5) is close to . It represents a limiting case of a more general two-mode soliton solution that has nonzcroth constituents and but one of the components is no longer much smaller than the other. This

composite soliton family can be characterised by the dependence of the total power in both the components, , on the parameter as shown in Fig. 4. On such a diagram, the -independent power of the central

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49

pulse, , is represented by a straight horizontal line (see Fig. 4). At the point (open circle in Fig. 4) the central pulse undergoes a bifurcation, and the new branch of a two-mode solution emerges. Such a solution exists for

a one-component soliton

, and it has a total power smaller than that of

. Near the bifurcation point

, the solution

consists of a large soliton pulse of the central wavelength that guides a small second component (Fig. 4C, dashed). With changing , the shapes of these solution components evolve, interchanging amplitudes (see the point D in Fig. 4), and the first component disappears near the second bifurcation point (open circle in Fig. 4), where it merges with the branch of the one-mode solitons of the second component (dashed curve with the point B in Fig. 4), i.e. the solution with and

where

that has the normalised power

System (5) has one more exact solution. Indeed, let us look for a solution on these coupled equations in the form of two pulses of equal widths, i.e.

Substituting the ansatz (9) into Eqs. (5), we obtain:

and

Therefore, for any values of and , there exists a point on the bifurcation diagram (Fig. 4) where the widths of the pulses on two different wavelengths coincide; this point corresponds to the exact solution (9) and (10) (in fact,

it is the point D at in Fig. 4). Families of the stationary solutions presented on the bifurcation diagram in Fig. 4 characterise completely the lowest-order localised solutions of the system (5): two single-mode (0) and (1) (solid and dashed, with the point B) and one mixed-mode (dashed, with the points D and C) solutions. Near the bifurcation points, one of the components of the mixed-mode soliton becomes small and the solution can be considered as a localised mode guided by an effective waveguide created by the other, larger, soliton component. Therefore, the shepherding effect discussed by Yeh and Bergman can then be understood as the excitation of such a guided mode driven and controlled by the soliton-induced waveguide created by the pulse at a different wavelength.

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YU. S. KIVSHAR AND E. A. OSTROVSKAYA

6. Generalisation to Multi-channel WDM Systems

6.1. ANALYTICAL RESULTS The analysis is more involved when the number of the channels is larger, i.e. However, even in this case some of the results can be easily obtained analytically. First of all, using a normalization similar to that for the dual WDM model and looking for stationary solutions in the form (4), we obtain the normalized equations for the N coupled envelopes [cf. Eqs. (5)],

where the amplitudes and time are measured in the units of and respectively, and First of all, the system (11) has an exact analytical solution for N coupled components, the so-called WDM soliton. Indeed, looking for solutions in the form [cf. Eqs. (9)]

we obtain

where [7], we take

and the system of N coupled algebraic equations,

. In a very special case, recently considered in Ref. and the solution of the Eqs. (13) is simple:

When all ratios are close to 1, the solution of Eqs. (13) can also be found with the help of perturbation theory, where is defined by Eq. (14), and and are given by a solution of a system of N linear equations. Similar to the dual WDM system, the analytical solutions can also be obtained near the bifurcation points when the central frequency pulse ( 0) is large. Then, linearizing Eqs. (11) for small , we obtain N–1

linear (decoupled) equations,

BIT-PARALLEL SOLITON TRANSMISSION

where

51

Localized solutions of Eq. (15) exist only at certain

values of the spectral parameter , and such solutions represent guided modes of the effective waveguide created by the soliton at the central wavelength. The first two bifurcation points,

correspond to the fundamental (zeroth-order) and the first-order modes of such a soliton-induced waveguide. Since, by definition, the parameters and

are close to 1, the soliton-induced waveguide is always single-moded,

i.e. it does not support the second mode. This is an important physical result which is a consequence of the fact that the cross-phase modulation effect is two times stronger than the self-phase modulation one [the factor “2” in Eqs. (11)]. Figure 5 shows the parameter space with two bifurcation surfaces defined by Eqs. (16), such that the lower surface corresponds to the bifurcation of the fundamental guided mode, and the upper surface

corresponds to the bifurcation of the first-order mode. It is clear that when all the parameters are close to those of the central mode, the soliton-induced waveguide remains single-moded. Existence of a solution of the N coupled NLS equations (11) in the form of a large pulse of the central wavelength (at ) that guides N – 1 lower-amplitude pulses of the neighbouring wavelengths is a manifestation

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YU. S. KIVSHAR AND E. A. OSTROVSKAYA

of the shepherding effect first observed in numerical simulations by Yeh and Bergman [2]. 6.2. PARTICULAR EXAMPLES:

AND

To confirm the general features described above for arbitrary N and to demonstrate new properties of the multi-channel WDM solitons for we first consider the case To make the bifurcation picture clearer, we select the following set of the normalized parameters: and As above, the solitary waves of this three-wave system can be found as localised solutions of the system (11) with Figure 6 presents several families of such localized solutions. As above, due to the assumed normalization, the power of the central-wavelength component does not depend on and is represented by a straight line Thin dashed and dotted bifurcating curves correspond to

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solitons of two separate dual WDM systems, (0+1) and (0+2). A small offset in those curves is a result of a small difference in the parameters and . Bold solid curves in Fig. 6 correspond to the mutually coupled localized solutions of the three-mode system. First of all, at the lowest bifurcation point(the lower open circle), a two-mode soliton (1+2) emerges from a single-soliton branch (1) (dotted branch), and it gives birth to a family of three-mode solitons (0+1+2) (branch A–B) that bifurcates from the two-mode branch at the second bifurcation point (middle open circle). Two examples of such three-wave composite solitons are shown in Fig. 6 (botom row). The solution A is an exact solution (13) at for whereas the solution B is close to the second analytical solution described by Eqs. (15) and (16). Importantly, for different values of the parameters

( ), the uppermost bifurcation point (upper open circle in Fig. 6) is not predicted by a simple linear theory and, due to the nonlinear mode coupling, it gets shifted from the branch of the central-wavelength soliton (straight line) to a two-mode branch (0+1) (dotted curve). As a result, following the lowest values in Fig. 6 from above, we pass the following sequence of the soliton branches and corresponding bifurcations:

If the modal parameters are selected closer to each other, the first link of the cascade disappears and the three-mode soliton bifurcates from the central-wavelength pulse, as predicted by the linear theory and Eqs. (16). The general picture of the bifurcation cascades is preserved for larger N. As another particular example, we consider the case with the parameters: and The corresponding results are presented in Fig. 7, together with the examples of four-mode WDM solitons. Similar to the cases and discussed above, the four-mode solitary waves emerge via bifurcation of the lower-order (this time, three-mode) soliton, and the examples of such solutions corresponding to the points A and B are shown in Fig. 7 (bottom row). It is interesting to mention that the branch of these four-wave solitons (branch A–B in Fig. 7) bifurcates from two different three-wave solutions, (0+1+2) and (1+2+3).

The bifurcation cascades observed for this model at

are again a

manifestation of a small variance in the modal parameters. When a difference between the dispersion and nonlinearity coefficients of the pulses belonging to different wavelengths vanishes, all the bifurcation points merge

in a single, multiply degenerated bifurcation point located on the branch , and a mixed-mode solution near the bifurcation point corresponds to the linear modes guided by the central-wavelength soliton (shepherd) pulse.

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7. Experimental Demonstrations

To observe the effects discussed above and, in particular, the shepherding effect and WDM multi-mode solitons, the walk-off should be mini-

mized already in the linear regime. An all-optical long-distance ( ) bit-parallel WDM single-fiber link with 12 bit-parallel channels having 1 Gbyte/s capacity was designed and reported by Bergman et al. [6]. This system functionally resembles an optical fiber ribbon cable, except that all the bits pass on one fiber-optic waveguide. The validity of this concept was demonstrated in a quasi-linear regime [6] for the case of two WDM channels at wavelengths 1530 and 1535 nm carrying 1 ns pulses on each channel and sent through a single 25. 2-km long Corning dispersion-shifted fiber. The actual walk-off was 200 ps, well within the allowed set-up. Bergman et al. [6] pointed out that this result implies that 30 bit-parallel beams spaced 1 nm apart between 1530–1560

nm, each carrying a 1 Gbytes/s signal, can be sent through a fiber carrying

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55

information at a 30 Gbytes/s rate. These data mean that the speed-distance product for this link is about 94 Gbytes/s-km, a number far beyond the best that a fiber ribbon can offer. The property of the dynamically shepherded pulses can be used in the generation of time-aligned picosecond pulses in WDM transmission. This effect was studied numerically and demonstrated experimentally by Yeh et al. [5]. The idea is similar to that of the pulse shepherding effect. A high-power, picosecond pulse (the shepherd pulse) is launched on a given wavelength. A number of low-power beams that are selected based on WDM format are launched without any signal pulses (“a noise”) into the same single-mode nonlinear fiber. All wavelengths co-propagate with the central wavelength carrying the shepherd pulse in this fiber. Due to the CPM coupling between the wavelengths, the shepherd pulse generates time-aligned pulses in these lower-power WDM channels. It is required that the walk-off among all the wavelengths be kept at a minimum acceptable value. From the viewpoint of the cascading bifurcations of the stationary solutions discussed above, this effect can be understood in a rather simple way. Indeed, when a seeded signal on a different wavelength is launched together with the shepherd pulse, the primary pulse develops instability, and it evolves to the lower branch on the power diagram (see, e. g., Figure 4), the latter corresponding to a localised solution in the form of a shepherd pulse that carries small-amplitude guided modes. Experiments verifying this concept were performed for both bright and dark shepherd pulses [5]. For the case of a bright shepherd pulse, two beams,

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YU. S. KIVSHAR AND E. A. OSTROVSKAYA

one from a ring laser, carrying the shepherd pulse at 1535 nm, and the other from a diode laser, carrying no pulse at 1557 nm, were combined and sent through the Corning single-mode dispersion-shifted fiber. The input and

output pulses on these beams are shown in Fig. 8. As predicted by the theory, due to the shepherd effect a bright pulse is generated on the diode laser beam because its operating wavelength falls in the anomalous groupvelocity dispersion range of the fiber. 8. Comparison with Other Multi-mode Multi-soliton Models

It is useful to make a comparison of the bit-parallel WDM soliton transmission model described above with some other well-studied models of the multi-mode multi-soliton propagation. This will allow one to understand a number of unique features of the WDM solitons, also providing some directions for the future research of bit-parallel fiber optics links. 8.1. NONLINEAR FIBER ARRAYS

Optical devices based on arrays of nonlinear directional couplers have been proposed as possible elements for demultiplexing and switching of information both in the continuous-wave [10] and in the pulsed [11] regimes. The fabrication of such arrays is known to be possible as well [12]. One of the practical realisations of such an array is identical fibers arranged in a line. The equations governing the dynamics of pulses in the array are:

where

is the electric-field envelope in the n–th fiber (we consider N + 1

fibers, so that neighbouring fibers,

), is the coupling coefficient between is the group-velocity dispersion, and is the

nonlinear coefficient characterising the Kerr effect. Due to a linear coupling between the neighbouring fibers, the propagation of the pulse in the central (e.g., n = 0) fiber always involves the coupling to its neighboring fibers. For example, if a soliton with the profile is launched into the central fiber with n = 0, the pulses in the neighbouring fibers can be found by solving the linearized forced equations [13], and the normalized solutions are

which become smaller for larger amplitude of the leading soliton at

This case should be compared with the shepherding effect discussed

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57

above, where a strong pulse carries weaker pulses at different wavelengths. As a result of the linear coupling in the fiber array, the amplitudes of weak pulses depend strongly on the central pulse, thus making the scheme very sensitive to the pulse in the central fiber, in a sharp contrast with the basic properties of the multi-channel WDM soliton transmission scheme based on the shepherding effect. 8. 2. INCOHERENT SPATIAL SOLITONS Spatial optical solitons appear as self-guided beams when the beam spreading due to diffraction is exactly balanced by the beam self-focusing effect caused by the nonlinear refractive index of an optical medium. A spatial optical soliton is usually viewed as a mode of the waveguide it induces [14], and until recently it was associated with the self-trapping of coherent optical beams. When the light is emitted by an incoherent source, it is hard to expect some solitonic effects. However, when the light is launched into a self-focusing nonlinear medium with a slow response, it can still propagate as a spatial soliton called an incoherent soliton; this phenomenon has been recently predicted and observed experimentally [15]. There exist, two different approaches to describe partially coherent beams and incoherent spatial solitons. The first one is based on the mutual coherence function which is a measure of correlation between amplitudes of the beam in two different points. The second approach is the description of a partially coherent soliton as a multi-mode self-induced waveguide [16]. Indeed, in a slowly-responding medium, when any two modes are mutually incoherent, the stationary structure can be considered as an effective waveguide consisting of many properly populated modes, mathematically

described by a system of incoherently coupled NLS equations. A direct analogy between the incoherently interacting beams, guided by a common effective waveguide, and shepherding of the optical pulses at different wavelengths by a leading soliton can be drawn. However, the crucial difference between incoherent solitons and the WDM solitons discussed above is the specific nature of the photorefractive (saturable) nonlinearity which supports such incoherent spatial solitons. In a medium with a slow response, the change of a refractive index for a given beam component depends on the total beam intensity, with equal contribution from the self- and crossphase modulation effects. For low intensities, this model is integrable, and it is described by the vector multi-component Manakov equations (see, e. g., references in Ref. [8]). Such a system is degenerate in a sense that only fundamental solitons of exactly the same intensity and width can co-exist in the medium. As a consequence, the trapping and guiding of the lowerpowcr pulse by a leading one, i. e. the shepherding effect, is not possible in a

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YU. S. KIVSHAR AND E. A. OSTROVSKAYA

wide region of parameters. Saturation of the nonlinearity introduces some new features into the system and, most importantly, enables a fundamental mode to trap higher-order modes [17] which usually makes the multi-mode spatial solitons unstable. 8.3. OPTICAL CONVEYOR BELT

An integrable Manakov model [8] has also been usefully employed in the case when the effective waveguide is induced by a dark soliton. Such a dark-soliton-induced waveguide is always single-moded (even in the case of saturable nonlinearity [18]), and it can guide either a small-amplitude or large-amplitude bright beam, being described by the exact solutions of the Manakov model for the case of the defocusing Kerr nonlinearity [19]. In the linear regime, the guiding properties of a dark soliton were employed in suggesting a new scheme of the data transmission with the help of an uniform periodic train of dark solitons where the data stream was encoded in linear pulses of small but orthogonal polarization [20]. This scheme, as many others, is based on a single wavelength and, therefore, the coherent phase-dependent effect of the soliton collisional interactions is crucially important for the overall robustness of the soliton train. For example, as was demonstrated in Ref. [18], the interaction of (even smallamplitude) bright beams guided by dark-soliton waveguides depends on their relative phase which strongly affects the waveguiding properties. In contrast, the nonlinear mode coupling in the WDM systems is incoherent, i.e. phase-insensitive, and is positively utilized in the scheme. Another serious drawback of the transmission scheme suggested in Ref. [20] is the use of a periodic train of dark-soliton supporting structure. Such a periodic train induces an effective periodic potential where neighboring “potential wells” are weakly coupled to each other. Using the analogy with the motion of an electron in a crystal lattice, we expect that the eigenfunctions of this system are Bloch waves, and the eigenvalues become degenerate and form a band structure. This means that any localized signal carried by a selected dark soliton should “tunnel” through the train, thus losing the initial data. 8.4. SOME OTHER RELATED CONCEPTS

An idea, quite opposite to that of the optical conveyor belt [20], was suggested by Shipulin et al. [21] (see also Ref. [22]), who proposed suppressing the soliton’s timing jitter in optical communication systems by launching, parallel to the information-carrier soliton stream, a periodic array of pulses in another mode, which may be either of an orthogonal polarisation or, more realistically, of another wavelength. This support array induces,

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through the cross-phase-modulation effect, an effective periodic potential pinning of the signal solitons. As opposed to the idea of the conveyor belt, the supporting structure can be a small-amplitude signal, so that an effective “tunneling” of the large-amplitude signal pulse is suppressed by the nonlinearity, and the signal pulse remains pinned by the supporting structure. To check this idea directly, Shipulin et al. [21] performed numerical simulations of the two coupled NLS equations for the information and support channels, including the jitter-generating random perturbations acting on both of them. These simulations demonstrated that the support structure strongly suppresses the soliton jitter, in accordance with the theoretical analysis based on the corresponding Fokker-Planck equation. The stabilization seems even better if the supporting small-amplitude signal is launched in the normal-dispersion wavelength region [22]. A similar idea has been recently suggested for the soliton stabilization by fixed- or sliding-frequency notch filters providing channel isolation in the WDM system [23]. Such a “soliton rail” has been shown (both analytically and numerically) to provide a robust pulse trapping which, at the same time, reduces the effect of soliton collisions. In fact, a notch filter creates a kind of an active waveguide that demonstrates much better performance than other types of isolating optical filters based on Fabry-Perot etalons or various fiber-grating schemes. 9. Conclusions We have presented a comprehensive overview of the basic concepts of multisoliton pulse transmission in bit-parallel WDM optical fiber systems. From

the mathematical point of view, the corresponding model is described by a nonintegrable system of (many) coupled NLS equations whose properties are not well understood yet. In particular, it has a number of similarities with other nonlinear models, e. g. those studied for the pulse propagation in fiber arrays, and the dynamics of incoherent spatial solitons, but it also has many new features. We have discussed in detail two important effects in such bit-parallel wavelength pulse transmission, i. e. the pulse shepherding effect when a large-amplitude (shepherd) pulse creates an effective waveguide for pulses of different wavelengths guiding all of them; and the structure and stability of multi-mode WDM solitons consisting of many time-aligned wavelengths. Many of the issues related to this problem still remain open. In particular, it seems natural to expect that for the model with any N wavelengths there exists a close connection between the shepherding effect described by cascading bifurcations of the isolatedwavelength pulse (shepherd pulse) and multi-component WDM solitons, as was demonstrated for the cases of Moreover, a rigorous proof

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of stability of the multi-soliton pulses as localised solutions of N coupled NLS equations is still an open problem. The influence of the walk-off on the stability of the WDM multi-component solitons and the robustness of the shepherding effect is a very important issue that should be addressed in the future analysis. Acknowledgements

One of the authors (Yu. S. Kivshar) is indebted to L. Bergman and C. Yeh for a warm hospitality during his short visit to the Jet Propulsion Laboratory in Pasadena and useful discussions. He also thanks A. Hasegawa for the invitation to report these results at the ROSC Symposium in Kyoto, and for valuable suggestions and encouragement, and S. K. Turitsyn and V. S. Grigoryan for useful comments. References 1.

Bergman, L. A., Mendez, A. J. and Lome, L. S.: SPIE Crit. Rev., CR62, (1996), p. 210. 2. Yeh, C. and Bergman, L . : J. Appl. Phys., 80, (1996), p.3174. 3. Agrawal, G. P . : Nonlinear Fiber Optics, Academic Press, New York, Sec. 7. 1, (1995).

4.

Yeh, C. and Bergman, L. : Phys. Rev., E57, (1998), p. 2398.

5.

Yeh, C., Bergman, L., Moroonkian, J. and S. Monacos, S. : Phys. Rev., E57, (1998), p.6135. Bergman, L., Moroonkian, J. and Yeh, C. : J. Lightwave Tech., 16, (1998), p.1577. Yeh, C. and Bergman, L. : Phys. Rev., E60, (1999), p.2306. Yang, J. : Physica D, 108, (1997), p.92.

6. 7. 8. 9.

Hasegawa, A. and Kodama, Y. : Solitons in Optical Communications, Clarendon Press, Oxford, (1995). 10. Christodoulides, D. N. and Joseph, R. I. : Opt. Lett., 13, (1988), p.794; SchmidtHattenberger, C., Trutschel, U. and Lederer, F. : Opt. Lett., 16, (1991), p.294; Kivshar, Yu. S. Opt. Lett., 18, (1993), p.1147.

11.

Soto-Crespo, J. M. and Wright, E. M. : J. Appl. Phys., 70, (1991), p.7240; Aceves,

18.

A. B., De Angelis, C., Luther, G. G. and Rubenchik, A. M. : Opt. Lett., 19, (1994), p.1186; Aceves, A. B., Luther, G. G., De Angelis, C., Rubenchik, A. M. and Turitsyn, S. K. : Phys. Rev. Lett., 75, (1995), p.73. Mortimore, D. B. and Arkwright, J. W. : Appl. Opt., 29, (1990), p.1814; ibid, 30, (1991), p.650. Aceves, A. B., De Angelis, C., Rubenchik, A. M. and Turitsyn, S. K. : Opt. Lett., 19, (1994), p.329. Segev, M. and Stegeman, G. : Physics Today, No.8, (1998), p.42. Mitchell, M. and Segev, M. : Nature, 387, (1997), p.880. Mitchell, M., Segev, M., Coskun, T. and Christodoulides, D. N. : Phys. Rev. Lett., 79, (1997), p.4990. Ostrovskaya, E. A., Kivshar, Yu. S., Skryabin, D. and Firth, W. : Phys. Rev. Lett., 83, (1999), p.296. Ostrovskaya, E. A. and Kivshar, Yu. S. : Opt. Lett., 23, (1998), p.1268.

19.

Sheppard, A. P. and Kivshar, Yu. S. : Phys. Rev., E55, (1997), p.4773.

12. 13.

14. 15. 16. 17.

20. Miller, P. D., Akhmediev, N. N. and Ankiewicz, A. Opt. Lett., 21, (1996), p.1132.

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Shipulin, A., Onishchukov, G. and Malomed, B. : J. Opt. Soc. Am., B14, (1997), p.3393.

22. Malomed, B. and Shipulin, A. : Opt. Comun., 162, (1999), p.140. 23. Malomed, B., Peng, G. D. and Chu, P. L. : Opt. Lett., 24, (1999), p.1100.

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OPTICAL MODULATION AND DISPERSION COMPENSATION TECHNIQUES FOR ULTRA-HIGH-CAPACITY TDM/WDM TRANSMISSION SYSTEMS

G. ISHIKAWA, H. OOI, Y. AKIYAMA AND T. CHIKAMA

Fujitsu Laboratories Ltd. 4-1-1 Kamikodanaka, Nakahara-ku, Kawasaki 211-8588, Japan

Abstract. To realize ultra-high-capacity optical transmission systems, the

combination of faster TDM and denser WDM is essential, and it is important to select an optical modulation scheme suitable to fiber type, transmission distance, and total capacity. We compared the dispersion tolerance at 40 Gbit/s for the optical duobinary, NRZ, and RZ modulation schemes. In transmission systems with a relatively small fiber launched power where the SPM GVD effect is small, the optical duobinary scheme with large dispersion tolerance has a significant advantage. On the other hand, in a transmission system with large SPM–GVD effect, precise dispersion compensation is essential for any optical modulation scheme. Using strict dispersion compensation at every in-line repeater for each channel and out-band FEC, 40 Gbit/ channel (200 GHz spacing) NRZ signals can be transmitted over 600 km NZ–DSF. We introduced our 40 Gbit/s automatic dispersion compensation experiment using a tunable laser and discussed dispersion slope compensation.

1. Introduction

The rapidly growing worldwide Internet and IP traffic is the driving force

behind the development of optical transmission systems with multi-Terabit capacity. For such systems, the combination of faster time-division-multiplexing (TDM) and denser wavelength-division-multiplexing (WDM) channels is essential for higher spectral efficiencies and lower per bit costs. We believe that the 40 Gbit/s/channel WDM system is the 63 A. Hasegawa (ed.), Massive WDM and TDM Soliton Transmission Systems, 63–74. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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G. ISHIKAWA, H. O O I , Y. AKIYAMA AND T. CHIKAMA

next evolutionary step from current 10 Gbit/s/channel WDM systems [1]-

[3]. However, at 40 Gbit/s transmission, dispersion tolerance is less than several ten ps/nm and waveform distortion due to the interaction effect between self-phase modulation (SPM) and chromatic dispersion, called the SPM–GVD effect, which severely restricts the transmission distance. The dispersion tolerance is very dependent on the optical modulation spectrum bandwidth, which is determined by bit-rate and optical modulation

schemes; e.g. non-return-to-zero (NRZ), return-to-zero (RZ) and optical duobinary formats [4]-[6]. The optical crosstalk due to Four-wave mixing (FWM) as well as the optical modulation scheme determines transmission distance. Furthermore, for optical amplifier systems with multiple in-line repeaters, we have to consider that the degradation of the optical signalto-noise ratio (OSNR) restricts the transmission distance. In recently developed submarine ultra-long-distance WDM systems, the RZ modulation scheme is applied because a higher Q-factor than that in the NRZ scheme can be obtained under the same operation conditions for the in-line repeater [7]. The forward-error-correction (FEC) technique [7, 8] and Raman amplification [2] are also very effective for overcoming OSNR degradation.

To realize long-distance 40 Gbit/s/channel WDM systems, it is important to select an optical modulation scheme suitable to fiber type, transmission distance, and total capacity. In any case, precise dispersion compensation is essential. In Section 2, we compare dispersion tolerance at 40 Gbit/s for the optical duobinary, NRZ, and RZ modulation schemes. In Section 3, we discuss system design for 40 Gbit/s/channel WDM transmission. In

Section 4, we introduce our 40 Gbit/s automatic dispersion compensation experiments and discuss dispersion slope compensation. 2. 40 Gbit/s Optical Modulation Schemes 2.1. OPTICAL DUOBINARY MODULATION

Figure 1(a) shows a schematic diagram of the 40 Gbit/s system with an optical duobinary modulation scheme. The pre-coded binary signals were converted to three-level duobinary signals with fifth-order Bessel-Thomson

low-pass-filters (LPFs). Their cut-off frequency was a quarter of the bit rate, i.e. 10 GHz for 40 Gbit/s systems. The optical duobinary signal can be generated by a dual-drive Mach-Zehnder modulator driven with three-level electrical duobinary signals. As shown in Fig. 1(b), the electrical “ – 1 ” , “0” and “+1” levels are mapped into optical “ON (phase 0)”, “OFF”

and “ON (phase )” levels, respectively. We can use a conventional NRZ receiver for detecting these optical duobinary signals. The realization of a pre-coder consisting of a 40 Gbit/s EX–OR circuit with a 1 bit delay and

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the compactness of dual-driving electronics remain for practical use. Figure 1(c) compares the optical spectra between optical duobinary and NRZ schemes. The spectrum bandwidth of the optical duobinary is almost half that of NRZ, which yields a greater dispersion tolerance, and can realize closer wavelength channel spacing for WDM systems. Furthermore, the carrier frequency component is suppressed in the optical duobinary spectrum, which leads to a high stimulated Brillouin scattering threshold

[4]. 2.2. DISPERSION TOLERANCE COMPARISON Here, we compare dispersion tolerances between (a) optical duobinary, (b) NRZ and (c) RZ ( ) modulation schemes. The ( -parameters for NRZ and RZ schemes were set into zero. In this simulation, we varied the dispersion compensation values after 50-km-long SMF (chromatic dispersion: + 18 ps/nm/km) transmission at the fiber launched power, , from 0 to +15 dBm. Figure 2(a)–(c) shows the eye-opening penalty as a function of residual dispersion, the difference in dispersion between the transmission line and the dispersion compensator for three kinds of optical modulation

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schemes. At small fiber launched power ( ), the dispersion tolerance at a 1 dB penalty for the optical duobinary scheme (440 ps/nm) was more

than twice that for the NRZ scheme (150 ps/nm). For the RZ scheme, dispersion tolerance becomes smaller as the duty ratio becomes smaller, however, at a residual dispersion of just 0 ps/nm, the highest upper-limit of was obtained because soliton-like transmission occurred. And the upper-limit of with less than 1 dB penalty at 0 ps/nm for NRZ scheme (+9 dBm) was 4 dB larger than that for the optical duobinary scheme (+5 dBm). The residual dispersion with a minimum penalty at dBm for the optical duobinary scheme far from 0 ps/nm (around at +220 ps/nm). These results suggest that in transmission systems with a relatively small where the SPM–GVD effect is small, the optical duobinary scheme with large dispersion tolerance has a significant advantage. On the other hand, in a transmission system with a large where the SPM–GVD effect is large, precise dispersion compensation is essential for any optical modulation scheme. If we can ensure a 0 ps/nm strict dispersion compensation, the RZ scheme has an advantage for long-haul transmission because span budget larger than other modulation schemes can be maintained. However, an RZ signal with a smaller duty ratio has a larger optical spectrum bandwidth, which is a disadvantage for dense-WDM systems. Therefore, it is important to optimize the duty ratio in WDM transmission with the RZ scheme. 3. 40 Gbit/s/channel WDM System Design 3.1. SYSTEM MODEL

In this section, we will discuss the system design for 40 Gbit/s/channel WDM transmission. Figure 3 shows a system model and some assumptions.

Here, we adopt the NRZ modulation scheme. As shown in Fig. 1(c), the optical spectrum bandwidth at 30 dB down of the NRZ ( ) schemes were 130 GHz. Then, with the ITU–T grid in mind, we determined the channel wavelength spacing to be 200 GHz. In this study, we used non-zero dispersion fiber (NZ–DSF) which has been developed to reduce FWM generation efficiency [9, 10]. We assumed a zerodispersion wavelength to be shorter than a 1550 nm band and allocated 16 channels on C band (1535–560 nm). The chromatic dispersion was from +3.1 to +5.2 ps/nm/km and the dispersion slope was for all channels. The in-line repeater spacing was 100 km and the noise figure

(NF) of all optical amplifiers was 6 dB. The fiber loss was 0.2 dB/km and the system gain for each in-line repeater spacing was 25 dB including a margin of 5 dB. The dispersion compensation was optimized at every in-

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line repeater for each channel. The compensation values at in-line repeaters for one channel were equal. In our system design, we first determined the upper-limit of the fiber launched power, , from the waveform distortion due to the SPM–GVD effect or FWM crosstalk. We allowed for the waveform after transmission

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G. ISHIKAWA, H. OOI, Y. AKIYAMA AND T. CHIKAMA

so as to simultaneously satisfy both conditions; (a) an eye-opening penalty of less than 1 dB and (b) a phase opening of more than 70 %. This phase opening was defined as the ratio of the phase margin to the one time slot of 25 ps at the center level in the amplitude of the back-to-back waveform.

On the other hand, we considered that FWM crosstalk should be less than –36 dB. For long-haul transmission, there is OSNR degradation and the biterror-rate (BER) is inferior because of the accumulation of the amplifiedspontaneous- emission (ASE) noise from the in-line repeaters. Therefore, the forward-error-correction (FEC) coding is a promising way of improving transmission distance while maintaining transmission quality. An out-

band FEC has already been used in the ultra-long-haul submarine 5 or 10 Gbit/s/channel WDM systems [7]. The Q-factor is determined by the OSNR. If the Q-factor is more than 11.8 dB at the receiver, we achieve a BER of with an out-band FEC. For 40 Gbit/s transmission systems having large receiver bandwidth, OSNR degradation is one of the dominant factors for transmission distance limitation. Then, in this study, we assumed the use of an out-band FEC for 40 Gbit/s/channel WDM transmission. 3.2. 40-GBIT/S/CHANNEL WDM TRANSMISSION OVER NZ-DSF Figure 4(a) shows that the upper limit of determined by waveform distortion due to the SPM–GVD effect and FWM crosstalk. Because the chromatic dispersion of NZ–DSF was optimized to be larger than that of dispersion shifted fiber (DSF) to reduce the FWM efficiency, the upper limit of was dominated by the SPM–GVD effect. Figure 4(b) shows the Q-factors as a function of transmission distance. Using an out-band FEC, the transmission distance was 600 km at 11.8 dB Q-factor. 4. Dispersion Compensation Techniques

4.1. AUTOMATIC DISPERSION COMPENSATION

For 40 Gbit/s transmission, complete dispersion compensation is essential because of very small dispersion tolerance. However, in terrestrial systems, neither repeater spacing nor chromatic dispersion are uniform, so that the optimum dispersion compensation value may be different at each repeater spacing. Additionally, chromatic dispersion in a transmission line changes over time due to environmental influences on the cable, for example, changes in temperature. When the temperature of a 500-km-long transmission line changes by 100 °C, chromatic dispersion will change by about 100 ps/nm, which is comparable to the dispersion tolerance for a 40 Gbit/s NRZ transmission. Therefore, automatic dispersion compensation

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using a variable dispersion compensator is a key issue for ultra-high-speed transmission systems at 40 Gbit/s or higher [11]-[13]. For variable dispersion compensators, an integrated-optic device based on a lattice-form programmable optical filter on a planar lightwave circuit [14] and a fiber grating with temperature [15] or stress [16] gradient have been proposed. However, for either device, the wavelength bandwidth, tunable range, and reliability are insufficient for practical use. Recently, another variable dispersion compensator using a virtually imaged phased array called VIPA has been proposed [17]. Since the VIPA uses multiple reflections of light in a glass plate, it can produce chromatic dispersion in a range from –2000 ps/nm to +2000 ps/nm and can compensate for simultaneous multiple channels over a 50 nm bandwidth. If the optical bandwidth for one channel is optimized and the set of focusing lens and the mirrors can be mechanically displaced along the lens axis, this device will be applied to 40 Gbit/s/channel WDM systems with automatic dispersion compensation. In this section, we present an automatic dispersion compensation experiment for 40 Gbit/s NRZ transmission using a tunable laser as a light source. In spite of the variable dispersion compensator, we automatically set the signal wavelength at a zero-dispersion wavelength of 100 km DSF to

reduce fiber dispersion [11]. Our automatic dispersion compensation system offers the following features: (a) It provides a simple method for monitoring dispersion in a transmission fiber by utilizing the intensity of a 40 GHz frequency component in a baseband spectrum; (b) The signal wavelength can be set at zero-dispersion wavelength after a wide-scan of the signal wavelength upon commencing system operation; (c) Signal wavelength tracking

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is applied to keep the fiber dispersion at zero by wavelength dithering during system operation.

4.1.1. Dispersion Monitor We have already confirmed that the intensity of the 40 GHz frequency component extracted from the received 40 Gbit/s NRZ baseband signals can be used for the dispersion monitor [12]. Although another monitoring using the phase- to amplitude-modulation (PM–AM) conversion effect has been proposed [13], we believe that our monitoring method will lead to a simpler system configuration. Figure 5 shows the 40 GHz component intensity as a function of the signal wavelength at –35, +5 and +65 °C in a 40 Gbit/s NRZ transmission over 100 km DSF. The 40 Gbit/s NRZ signal has no 40 GHz intensity at a zero-dispersion wavelength. The wavelengths at intensities of zero, 1550.8, 1551.8 and 1553.4 nm, indicate a zero-dispersion wavelength at each temperature. As the dispersion slope was the total dispersion of the 100 km DSF for a constant wavelength changed by 18 ps/nm when subjected to a temperature change of 100 °C.

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4.1.2. System Demonstration Figure 6(a) shows our experimental setup. A 100 km DSF transmission line was thermally controlled in a thermostat capable of changing the temperature from –35 to +65 °C. At the receiver end, a 40 GHz frequency component was extracted from the baseband signal by the bandpath filter and measured by the power meter. The data of the 40 GHz intensity was transferred to a personal computer (PC) through an A/D converter. This PC with original software carried out synchronous feedback control of the tunable external-cavity laser in the transmitter and of the tunable

optical filter with a bandwidth of 5 nm in the optical preamplifier. The back-to-back receiver sensitivity at a BER of was –27.0 dBm. Figure 6(b) shows the changes in temperature and signal wavelength as found during the demonstration. We initially set the temperature of the 100 km DSF to –35 °C. After signal wavelength scanning from 1535 to 1565 nm, the wavelength was automatically set to 1550.8 nm, where the 40 GHz intensity was zero, as shown in Fig. 5. We then gradually increased the DSF temperature to +65 °C. During system operation, we enabled signal wavelength dithering (dithering range: dithering step: 0.1 nm) to track the signal wavelength to its optimum. Figure 6(c) shows signal wavelengths and power penalties with and without feedback control during the temperature change. The temperature dependence of zero-dispersion wavelength is estimated to be 0.026 nm/°C, which agrees well with the value of 0.03 nm/°C measured by FWM efficiency [18]. During the temperature change of 100 °C, corresponding to a dispersion change of 18 ps/nm, we confirmed that the optical waveform did not change and that there was no penalty. 4.2. DISPERSION SLOPE COMPENSATION

The dispersion slope compensation is indispensable for WDM systems for receiving all channels equally well. Considering equipment size and cost, simultaneous compensation for multi-channels with a wavelength range of more than 30 nm for one device is desirable. In commercialized WDM systems, the dispersion slope of the dispersion compensating fiber (DCF) has already been optimized to cancel the dispersion slope in the transmission line [19]. Recently, both 10 Gbit/s/channel and 40 Gbit/s/channel WDM transmission experiments with new transmission-line dispersion management that represent a combination of the conventional SMF and the reverse dispersion fiber (RDF) were performed [7, 20]. RDF has a negative large dispersion value from –16 to –50 ps/nm/km [21]. In this dispersion management, FWM and cross phase modulation (XPM) were effectively suppressed because the local chromatic dispersion value is large. RDF has

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a negative dispersion slope, which results in a compensation of the dispersion slope. 5. Conclusion

To realize ultra-high-capacity optical transmission systems, a combination

of faster TDM and denser WDM is essential, and it is important to select an optical modulation scheme suitable to fiber type, transmission distance, and total capacity. We compared the dispersion tolerances for optical

duobinary, NRZ and RZ modulation schemes at 40 Gbit/s. In transmission systems with a relatively small fiber launched power where the SPM–GVD effect is small, the optical duobinary scheme with large dispersion tolerance has a significant advantage. On the other hand, in a transmission system with large SPM–GVD effect, precise dispersion compensation is essential for any optical modulation scheme. Using an out-band FEC, 40 Gbit/s

NRZ signals of 16 channels with 200 GHz spacing on C band can be transmitted over a 600 km NZ–DSF. For 40 Gbit/s/channel WDM systems, the strict dispersion compensation at every in-line repeater for each channel is

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indispensable. We conducted our 40 Gbit/s automatic dispersion compensation experiment using a tunable laser. When a variable dispersion compensator suitable to the 40 Gbit/s system is developed in the near future, the dispersion monitor and the feedback control technique can be applied as they are used in this experiment. For multi-channel WDM transmission over a wide wavelength range, dispersion slope compensation is essential. Furthermore, for higher-speed TDM transmissions, the polarization-modedispersion (PMD) is another limitation restricting transmission distance. The newly developed fiber has a very small PMD value of less than 0.01 However, for a transmission over installed old fiber with a large PMD value of more than several automatic PMD compensation is inevitable [22]-[24]. Acknowledgments

The authors would like to thank Dr. T. Hoshida and T. Terahara for their contributions to the simulations for WDM transmission.

References 1.

Ishikawa G. : 40 Gbit/s transmission using optical time-division multiplexing and demultiplexing, in A. Hasegawa (ed.), New Trends in Optical Soliton Transmission systems, (1998), pp.381-390. 2. Nielsen, T. N., Stenz, A. J., Hansen, P. B., Chen, Z. J., Vengsarkar, D. S., Strasser,

T. A., Rottwitt, K., Park, J. H., Stulz, S., Cabot, S., Feder, K. S., Westbrook, P.

3.

S. and Kosonski, S. G. : 1.6 Tbit/s ( ) transmission over nonzero-dispersion fiber using hybrid Raman/erbium-doped amplifiers, ECOC’99, PD2-2, (1999), pp.26-27. Elbers, J. P., Scheerer, C., Fäber, A., Glingener, C., Chöflin, A., Gottwald, E. and Fischer, G. : 3.2 Tbit/s (

4.

) bidirectional DWDM/ETDM transmission,

ECOC’99, PD2-5, (1999), pp.32-33. Yonenaga, K. and Kuwano, S. : Dispersion-tolerant optical transmission system

using duobinary transmitter and binary receiver, J. Lightwave Technol., Vol.15, No.8, (1998), pp.1530-1537.

5. Ono, T., Yano, Y., Fukuchi. K., Ito, T., Yamazaki, H., Yamaguchi, M. and Emura, K. : Characteristics of optical duobinary signals in Terabit/s capacity, high-spectral efficiency WDM systems, J. Lightwave Technol., Vol.16, No.5, (1998), pp.788-797. 6.

Yano, Y., Ono, T., Fukuchi, K., Ito, T., Yamazaki, H., Yamaguchi, M. and Emura, K. : 2.6 Terabit/s WDM transmission experiment using optical duobinary coding, ECOC’96, ThB3.1, (1996), 5.3-5.6. 7. Naito, T., Shimojoh N., Tanaka, T., Nakamoto, H., Doi, M., Ueda, T. and Suyama, M. : 1 Tbit/s WDM transmission over 10000 km, ECOC’99, PD2-1, (1999), pp.2425.

8. Sato, K., Shinbashi, M., Taniguchi, A. and Wakabayashi, T. : SONET/SDH optical 9.

transmission system, Fujitsu Sci. Tech. J., Vol.1, No.35, (1999),pp.l3-24. International Telecommunication Union – Telecommunication Standardization Sector (ITU-T), Recommendations G.655.

10. Onaka, H. : Fiber dispersion management in WDM systems, OECC’96, 17B2-2, (1996), pp.110-111.

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11. Ishikawa, G. and Ooi, H. : Demonstration of automatic dispersion equalization in 40-Gbit/s OTDM transmission, ECOC’98, (1998), pp.519-520. 12. Akiyama, Y., Ooi, H. and Ishikawa, G. : Automatic dispersion equalization in 40Gbit/s transmission by seamless-switching between multiple signal wavelengths, ECOC’99, TuC1.5,I, (1999), pp.150-151. 13. Tomizawa, M., Sano, A., Yamabayashi, Y. and Hagimoto, K. : Automatic dispersion equalization for installing high-speed optical transmission systems, J. Lightwave Technol., Vol.16, No.2, (1998), pp.184-191. 14. Takiguchi, K., Okamoto, K., Suzuki, S. and Ohmori, Y : Planar lightwave circuit optical dispersion equalizer, IEEE Photon. Technol. Lett., Vol.6, No.1, (1994), pp.86-88. 15. Lamming, R. I., Robinson, N., Scrivener, P. L., Zervas, M. N., Barcelos, S., Reekie, L. and Tucknott. A. : A dispersion tunable grating in a 10 Gb/s 100-220-km-step index fiber link, IEEE Photon. Technol. Lett., Vol.8, No.3, (1996), pp. 428-430. 16. Ohn, M. M., Alavie, A. T., Maaskant, R., Xu, M. G., Bilodeau, F. and Hill, K. O. : Tunable fiber grating dispersion using a piezoelectric stack, OFC’97, WJ3, (1997), pp.155-156. 17. Shirasaki, M. : Chromatic dispersion compensation using virtually imaged phased array, OAA’97, PDP-8, (1997). 18. Onaka, H., Otsuka, K., Miyata, H. and Chikama, T. : Measuring the longitudinal distribution of Four-wave mixing efficiency in dispersion-shifted fibers, IEEE

Photon. Technol. Lett., Vol.6, No.12, (1994), pp.1454-1456. Onaka, H., Miyata, H., Ishikawa, G., Otsuka, K., Ooi, H., Kai, Y., Kinoshita, S., Seino, M., Nishimoto, H. and Chikama, T. : 1.1 Tb/s WDM transmission over a 150 km 1.3 zero-dispersion single-mode fiber, OFC’96, PD19, (1996). 20. Yonenaga, K., Matsuura, A., Kuwahara, S., Yoneyama, M., Miyamoto, Y., Yamabayashi, Y., Hagimoto, K., Noguchi, K. and Miyazawa, H. : Dispersioncompensation-free 400-Gbit/s ( transmission experiment using zero-dispersion-flattened transmission line, OECC’98, PD1-3, (1998), pp.6-7. 19.

21.

Mukasa, K., Akasaka, Y., Suzuki, Y. and Kamiya, Y. : Novel network fiber to

manage dispersion at 1.55 with combination of 1.3 zero dispersion single mode fiber, ECOC’97, (1997), pp. 127-130. 22. Ishikawa, G. and Ooi, H. : Polarization-mode dispersion sensitivity and monitoring in 40-Gbit/s OTDM and 10-Ghit/s NRZ transmission experiments, OFC’98, WC5, (1998), pp.117-119. 23. Ooi, H., Akiyama, Y. and Ishikawa, G. : Automatic polarization-mode dispersion compensation in 40-Gbit/s transmission, OFC/IOOC’99, WE5, (1999). 24. Ishikawa, G., Ooi, H. and Akiyama, Y. : 40-Gbit/s transmission over high-PMD fiber with automatic PMD compensation, APCC/OECC’99, Vol.1, (1999), pp.424-427.

ON THE EVOLUTION AND INTERACTION OF DISPERSION-MANAGED SOLITONS

M. J. ABLOWITZ, G. BIONDINI AND E. S. OLSON Department of Applied Mathematics, University of Colorado

Campus Box 526, Boulder, CO 80309-0526, USA. Abstract. The dynamics of localized optical pulses in a strongly dispersionmanaged fiber-optic communication system is studied. A multiscale expansion on a perturbed nonlinear Schrödinger equation (PNLS) that includes the effects of loss, amplification and dispersion management, is employed, and the dynamics of the pulse in the Fourier domain is decomposed into a rapidly varying phase and slowly evolving amplitude. The fast component is calculated exactly, and a nonlocal equation for the slow evolution

of the amplitude is derived. The equation admits a two-parameter class of traveling wave solutions, called dispersion-managed solitons. A comparison with numerical solutions of the original PNLS equation is discussed. The evolution and interactions of these solitons are studied via numerical simulations of both the nonlocal NLS and the PNLS equations. It is found that interaction can lead to serious nonadiabatic effects and soliton collapse. Pulse interactions are found to be strongly dependent upon the values of

the system parameters.

1. Introduction

The one-dimensional nonlinear Schrödinger equation (hereafter NLS), derived from Maxwell equations [1, 2] as an asymptotic equation governing

the evolution of the slowly varying envelope of a quasi-monochromatic optical pulse, has proven to be invaluable in studying the complex dynamics of pulses in long-distance communication systems. It is well-known that the NLS has soliton solutions, which we refer to here as “classical” solitons, and that the NLS is integrable via the Inverse Scattering Transform [3, 4]. As a consequence, classical NLS solitons always interact “smoothly” and 75 A. Hasegawa (ed.), Massive WDM and TDM Soliton Transmission Systems, 75–114. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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elastically; namely, the amplitudes and velocities of the solitons after the interaction are the same as those prior to the interaction. The NLS equation is also a natural averaged equation when considering the evolution of pulses in the presence of perturbations such as damping/amplification [5, 6]

and moderate management of the fiber dispersion [7, 8, 9]. Recently, considerable work has been devoted to the study of communication systems which use strong dispersion management (see e.g. Refs [10]– [41]). The reason for this widespread interest lies in the remarkable properties of these systems (as compared to systems with a constant value of dispersion). Consequently, considerable effort has been dedicated in the last few years to understanding the remarkable properties of optical pulses in dispersion-managed systems [12]–[26] and their interactions [27]–[36]. Also, the potential of dispersion managed transmission systems has been demonstrated through numerical simulations [37, 38] and especially by a recent

series of striking experimental demonstrations of soliton transmission at ultra-high bit rate capacities [39, 40, 41]. In this paper we present a study of pulse evolution and interactions in systems with strong dispersion management. We begin with a perturbed NLS (PNLS) equation containing loss, amplification and strong dispersion management. In agreement with standard values of experimental parameters (e.g. Ref. [41]), each of these terms is taken to be rapidly varying, and, in addition, the term describing dispersion management is assumed to be large. By formulating an appropriate perturbation expansion (in Section 2), we show that the dynamics of the optical pulse can be decomposed in the Fourier domain into the product of a slowly evolving amplitude and a rapidly varying phase, which describes the (Fourier) chirp of the pulse and is determined by the large periodic variations of the dispersion. We show that the amplitude satisfies an average equation, (cf. Eq. (21)) which reduces to the usual NLS equation in the limit of weak dispersion management. We refer to this equation as the dispersion-managed nonlinear Schrödinger equation, or DMNLS equation. The equation is nonlocal in both the frequency (Fourier) and time domains. (We note that a similar equation was obtained by Hamiltonian methods in Ref. [9].) Basic properties and useful symmetries of the equation are obtained, and the physically relevant case of a two-step piecewise constant dispersion maps is analyzed, where the kernels of the DMNLS equation in the time and Fourier domain are explicitly obtained. In Section 3 we focus on traveling wave solutions, and we derive an equation that governs a one-parameter family of stationary pulses whose modulus is independent of the evolution variable, and we employ a computational algorithm to find fixed points of the associated integral operator. Since the DMNLS equation is shown to be Galileian invariant (in Section

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2), this family actually describes a two-parameter class of traveling wave solutions. We call these pulses dispersion-managed (DM) solitons. Note that in the literature the term DM soliton has sometimes been used to indicate breathing solutions of the PNLS equation with strong dispersion management. To avoid confusion, we will refer to the latter solutions as PNLS solitons. The structure of these DM solitons is characterized by a Gaussian center with exponentially decaying oscillating tails. Numerical comparisons are performed between the DMNLS and PNLS equations. The stationary solutions of the DMNLS equation are shown to be good approximations of the periodic solutions of the PNLS equation in a significant region of the relevant parameter space when the effects of loss/amplification are not included. In Section 4 we analyze the evolution and interaction properties of DM solitons. We formulate a numerical algorithm to obtain general time dependent solutions of DMNLS and show numerically that the DM soliton solutions obtained by fixed point methods are stable solutions to DMNLS over the time scales considered. Then we discuss the interaction properties of these DM solitons. It is found that, in certain parameter regimes, e.g. for small enough relative velocity, or equivalently, for small enough frequency separation, DM solitons exhibit significant nonadiabatic interaction effects and, below a certain threshold, they can annihilate one another. The latter situation is referred to as collapse. The reason for this effect is that the DMNLS equation should be considered as “nonintegrable”, and collapse is a typical nonintegrable effect (cf. Ref. [42]). In fiber-optic communications, the scenario is further complicated by the following. Even if parameter values are such that collapse does not occur, if the values are sufficiently close to the critical point these non-adiabatic effects can be so severe that even a few collisions (which in a fiber-optic communication system could occur over relatively short transmission distances) would produce unacceptably high bit error rates. Our belief is that the best possible performance is obtained when the only perturbation considered is that due to dispersion management, i.e. when one omits loss/amplification. In this case we demonstrate different parameter regimes where collapse occurs. We show that, for a parameter regime consistent with recent NEC experiments [41] serious nonadiabatic effects begin to be noticed when the frequency separation is less than approximately three spectral widths. In fact, the collision effect of DM solitons appear to be in some cases less significant than those of classical solitons. (Recall that, while classical solitons do not annihilate one another in the time domain, there can be very strong effects in the frequency domain [43].) These results are consistent with the experimental observation that massive WDM transmission can be carried out in this parameter regime.

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In the future we plan to investigate the properties in DM solitons in wider parameter regimes, as well as to analyze the effects of loss/amplification in the strong interaction regime. 2. The Dispersion-managed Nonlinear Schrödinger Equation

The starting point of our analysis is the following perturbed NLS (PNLS) equation:

where all quantities are expressed in dimensionless units; t is the retarded

time, z is the propagation distance, and u the slowly varying envelope of the optical field, rescaled to take into account damping and amplification (that is, the optical amplitude is proportional to The function represents the local value of fiber dispersion, while

de-

scribes the periodic power variation due to loss and amplification. The dimensionless variables t, z, D and u are related to the corresponding quantities in the laboratory frame by:

physical units). The time

(where E(z,t) is the field envelope in , distance , fiber dispersion and power

are normalization constants which we define as follows: Note: overstrike characters may be incorrect. (where is the full

width at half maximum of the pulse and is a constant number of order unity, defined according to the type of pulse considered), (where is the nonlinear length of the pulse [44]) and is the pulse peak power. Once these quantities are fixed, follows as a constraint. In the following sections we will often compare our theory with recent longdistance soliton transmission experiments [41], for which ps and With our choice of normalizations, these values correspond to and (for which in turn yield The quantity

appearing in Eq. (1) is the characteristic (dimension-

less) distance between amplifiers, which we assume to be small compared to the dispersion length; that is, (If amplifiers are placed 50 km apart as in Ref. [41], in the above mentioned units.) We also take the dispersion map period to coincide with the amplifier distance so that both g (·) and D (·) have period one. If we take the origin of the dispersion map to coincide with the position of the amplifiers, takes the simple form

ON THE PROPERTIES OF DISPERSION-MANAGED SOLITONS

where

79

is the dimensionless loss coefficient of the fiber. In the follow-

ing, all averages will be taken over one period of the dispersion map,

As is customary,

is normalized so that that is, The lossless case corresponds to which yields . The pure NLS case is obtained when there is no loss and no dispersion management, i.e. 2.1. THE MULTISCALE PERTURBATION EXPANSION

In order to model strong dispersion management, we split the fiber dispersion in two components: a path-average constant value and a term describing the large, rapid variations corresponding to the large local values of dispersion:

The function is taken to have average zero (i.e. ), so that the path-average dispersion reduces to . The proportionality factor in front of is chosen so that both and are quantities of order one. In terms of dimensional quantities, and Equation (1) contains both slowly and rapidly varying terms and both small and large quantities. Therefore it is not convenient for describing accurately the long-term behavior of the solutions. In order to obtain this behavior, we use as an expansion parameter, and we introduce the fast and slow z scales as and respectively. Also, we expand the

field u in powers of

Then we can break Eq. (1) in a series of equations corresponding to the different powers of In general, at order we have where

with

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and so on. In the following we solve the above perturbation expansion recursively in powers of 2.2. ORDER

LEADING ORDER PULSE DECOMPOSITION

At leading order in the expansion we need to solve

where F[u] is given in Eq. (6). Equation (8) is equivalent to saying that, to leading order, the evolution of the pulse is determined solely by the large variations of about its mean, and that nonlinearity and residual dispersion represent only a small perturbation to the linear solution (cf. Ref. [24]). We solve Eq. (8) employing Fourier transforms, which we define as

In what follows, we will often make use of the convolution theorem: if the inner product of two functions f(x), g(x) is defined as the convolution integral

the convolution theorem states that

and

. Rewriting Eq. (8) in the Fourier domain, we can separate the Fourier transform of into its slowly and rapidly varying parts and write the general solution as

where and

where is for now arbitrary. The “integration constant” represents the slowly evolving amplitude of while contains the fast, periodic oscillations due to the large local values of the dispersion. Note

ON THE PROPERTIES OF DISPERSION-MANAGED SOLITONS that, if is real, represents the chirp of the Fourier transform we have

81

. Inverting

where

(and is defined to be taken on the principal branch, with The function U(Z, t) is at this stage arbitrary, and needs to be determined at higher orders in the expansion. Note that and where is the Dirac delta. Therefore, at those points for which the solution coincides with U(Z, t ) . We also note that, since and are even in t and respectively, the parity of is determined by the parity of U(·,t). (That is, if U(Z, t) is even with respect to t, then is also even with respect to t; vice versa, if U(Z, t) is odd is also odd.) 2.3. ORDER 1: THE DMNLS EQUATION

At the next-to-leading order in the expansion we need to solve

where write

is defined in Eq. (7b). Again, using Fourier transforms we can

where needs to be determined at the next order in the perturbation expansion. In order to avoid secularities, we need the integrand in the RHS to have zero mean. Or, in other words, the solvability of the inhomogeneous problem in the space of the periodic functions requires the following Fredholm condition (which expresses the orthogonality of the forcing w.r.t. the solutions of the homogeneous problem) to be satisfied:

This condition will determine the unknown function U(Z,t).

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To get the evolution equation in the Fourier space, we write explicitly the Fredholm condition (18) in the frequency domain. Then, expressing as the Fourier transform of and substituting the for their Fourier representation, after computing the relevant integrals we get the following nonlinear evolution equation, which we refer to as the dispersion-managed nonlinear Schrödinger equation (hereafter DMNLS):

where the kernel r(x) is given by

Note that, if then , and Eq. (19) reduces to the usual NLS equation written in the Fourier domain. To write the corresponding evolution equation in the time domain, we simply take the inverse Fourier transform of (19). Then, after some algebra, we have

where

Equations (19) and (21) is the fundamental asymptotic equation that governs the evolution of an optical pulse in a strongly dispersion-managed system, written either in the time (21) or the frequency (19) domain. As opposed to Eq. (1), all fast variation and large quantities have been removed, and the equation contains only slowly varying quantities of order one. We also note that Eqs. (19,21) are not limited to the case and it applies equally well to describe pulse evolution with zero or normal values of average dispersion. Equations of nonlinear Schrödinger type with nonlocal nonlinearities arise in other physical situations as well. (In the water waves case, these equations are known to be integrable; cf. Ref. [45].)

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2.4. SYMMETRIES AND CONSERVATION LAWS

In this section we list some basic properties of the DMNLS equation. 1. If (that is, in the case where the fiber dispersion becomes a constant) r (x) reduces to and becomes a two-dimensional Dirac delta: . As a consequence, in the limit Eqs. (19) and (21) reduce to the usual NLS equation, written in the time and Fourier domain, respectively. 2. Equation (19) is invariant under the combined transformations

Therefore the constant appearing in Eq. (13) can be chosen arbitrarily, and does not affect the solution of the original problem (1), since it amounts to a combined phase transformation of and . However, as we show in the next two sections, the choice of is important when looking for special solutions of Eq. (21). 3. Like the NLS equation, the nonlocal NLS equation (21) is symmetric for That is, if U(Z,t) is a solution of Eq. (21), then

U(Z,–t) is also a solution (since

). The same can

be said for Eq. (19) in the Fourier domain: that is, the equation is symmetric

for ). Eq. (21) is also invariant under and and Eq. (19) is invariant under and 4. Like the NLS equation, Eq. (21) is invariant under Galileian “boosts”. That is, if U(Z,t) is a solution of Eq. (21), then, for any real

also satisfies Eq. (21). Similarly, if

is a solution of Eq. (19), then

also satisfies Eq. (19). As an effect of the transformation, the mean frequency of is shifted by an amount and the velocity of is shifted by an amount The Galileian invariance will play an important role when looking for traveling wave solutions and studying pulse interactions. 5. The DMNLS equation (21) admits a number of conservation laws. The first few conserved quantities are: the pulse energy

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momentum

and Hamiltonian

While the pulse energy has the same form as the energy for the perturbed

NLS equation (1), the momentum and the Hamiltonian are very different. In fact, the corresponding quantities for the perturbed NLS are

Note that the Hamiltonian (27b) is not a constant of the motion, unlike Hamiltonian (26c). 6. The existence of a Hamiltonian implies that we can also write the DMNLS in Hamiltonian form:

where {A, B} is the usual Poisson bracket between two functionals:

In the following we consider the special situation of a piecewise constant

map, where the integration kernels can be written in a simple form, and in some cases computed explicitly. 2.5. TWO-STEP DISPERSION MAPS

All the results presented so far are valid for any generic dispersion map. However, in all practical situations, dispersion management is performed

by splicing together two or more sections of fiber of given length and with different value of fiber dispersion. Therefore it is useful to consider the special case of a piecewise constant dispersion map. That is, we prescribe

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Imposing becomes

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of the interval [0,1], with and we take

then yields

The function

where and is arbitrary. In what follows, we are going to fix by requiring that In the special case of a two-step map (that is, when only two fiber segments are used) we have

Then the condition yields or ). It proves convenient to introduce the parameter ), which provides a measure of the normalized map strength. Note that the actual values of dispersion in Eq. (1) are given by with corresponding to large variations even if s is of order unity. In fact, we can combine the previous definitions for s and write

Expressing

in terms of D(·) and using we also find In particular, if we have (with ). Note also that

where is the usual definition of map strength in terms of dimensional quantities introduced in Ref. [16].

Any dispersion map can be parameterized by specifying either or In fact, we can express the local values of the dispersion in terms of s and as

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conversely, we obtain s and

from

and

as

The choice is a somewhat better parameterization of the dispersion map in that the lossless kernel r(x) does not depend on ; cf. Eq. (37) below.

For the two-step map described in Eq. (32), imposing yields and As a consequence, the points where is zero are the midpoints of each fiber segment: at these values of £ the solution coincides with U(Z, t). In light of these considerations, it proves convenient to perform a shift so that the origin coincides with one of these points, and write the dispersion map as the periodic extension of

with given by (35a). Such a map is shown in Fig. 1. With this definition (and if is chosen on the discontinuities) the dispersion map is even in , and the points where the solution coincides with

U(Z, t) are condition

yields

Correspondingly, if Eqs. (36) are used, the and

2.5.1. Lossless case Using Eq. (31), for any generic piecewise constant dispersion map we can compute the kernel r(x) defined in Eq. (20). In particular, in the lossless case (i.e. when ), the kernel r(x) for a two-step map assumes a very

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simple form. Independently of the choice of the origin for the dispersion map, after a straightforward integration the kernel r(x) is found to be

As a consequence,

can be written as

where

is the cosine integral. Note that, since the strength of the coupling between different frequencies (which is a measure of the effective nonlinearity of the system) is always less than for the ordinary NLS equation. Also, since if we obtain that is a rather unusual representation of the two-dimensional Dirac delta.

2.5.2. Lossy case In the lossy case the kernel r ( x ) depends on the relative position of the amplifier with respect to the dispersion map. For concreteness, consider the symmetric two-step map described in Eq. (36), and define to represent the position of the amplifier within the dispersion map. That

is, and means that the amplifier is positioned at the midpoint of the anomalous fiber segment. The function is then expressed as (cf. Eq. (2))

where we have introduced the quantity Note that, although both and separately depend on the unit length their product G does not, and uniquely defines the loss accumulated by the pulse over an amplifier distance. With 0.20dB/km as a typical value for the fiber loss coefficient and an amplifier spacing of 50 km we get Although the resulting formulae are more convoluted, the kernel r ( x ) in the lossy case is computed with methods similar to the ones used in the lossless case. Namely, we get r(x) from Eq. (20), with given by Eq. (13), by Eq. (36) and by Eq. (39). If (that is, if the amplifier is located in the anomalous fiber segment) the resulting expression for the kernel is

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From Eq. (40) we see that, unlike the lossless case, the kernel r(x) depends explicitly on the parameter and also on in a nontrivial way. We also note that, unlike the lossless case, no choice of parameters can insure that the kernel is real for all values of x. This fact will have important consequences when looking for special solutions of the DMNLS equation. Obviously, r(x) still goes to if also, r(x) reduces to Eq. (37) if

In what follows, we will use the following set of parameter values: (corresponding to fiber segments of equal length) (corresponding to amplifiers positioned at the midpoint of the anomalous fiber segment) and (different choices of correspond to translations of the map origin). In this case, the kernel r(x) becomes

The kernel is also considerably more complicated than in the lossless case. However, in the following sections, all the calculations will be performed with the DMNLS equation in the Fourier domain. Knowledge of the kernel r(x) alone is therefore sufficient to study the properties of the equation and its solution. Therefore we omit the description of the kernel in the time domain. 3. Special Solutions

In this section we look for traveling wave solutions of the fundamental evolution equation (21). We recall that the equation is invariant under Galileian transformations. Therefore, if the stationary solutions of Eq. (21) are known, all traveling wave solutions can then be obtained by performing a Galileian transformation on these solutions. Motivated by this observation, we look for solutions of the form

Defining to be the Fourier transform of f ( t ) , and using the evolution equation (19) in the Fourier domain, we can write the following nonlinear integral equation for

We call this equation the “time”-independent DMNLS equation. By scaling

arguments it can be easily shown that, if F1 (ω) is a solution of Eq. (43) (with

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r(x) given by either Eq. (37) or (41)) corresponding to the eigenvalue and , then is also a solution, corresponding to the eigenvalue and map strength Using this property, a whole family of solutions can be generated from the knowledge of just one member. Note also that if the inverse transforms compare as All of the solutions in the previous family correspond to the same value of Alternatively, if is a solution corresponding to average dispersion 1 and then, if is a solution corresponding to average dispersion and for any fixed value of the dispersion map strength s. To solve the integral equation (43), we must resort to numerical calculations. We first rewrite the equation as

where

Our goal is to solve Eq. (44) iteratively. Ideally, we would like to choose an initial guess and apply a simple Neumann iteration procedure for Unfortunately, this simple iteration scheme

is unstable when directly applied to Eq. (44). In order to suppress the instability, we use a method which was originally introduced [46] in order to study stationary solution of the Kadomtsev-Petviashvili equation. Namely, we introduce the constants

Then, instead of solving Eq. (44), we consider the equation

Clearly, every solution of Eq. (44) is also a solution of Eq. (47), and vice versa if . and is a solution of Eq. (44), is a solution of Eq. (47) However, the degree of homogeneity of the RHS of Eq. (44) is 3, while that of Eq. (47) is Applying the same iteration procedure described previously, we now find rapid convergence for with the best results achieved for For or no convergence is observed.

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Once the stationary solution is known, we can use Eq. (11) to reconstruct the full z dependence of the leading order solution of the NLS equation (1). Namely,

The solution in the time domain can then be obtained by simply taking an inverse Fourier transform. In what follows we restrict ourselves to the case We examine the lossless case first. 3.1. LOSSLESS CASE Noting the symmetry of Eq. (21) and the reality of the kernels and we can look for real and even solutions f(t). Correspondingly, will also be real and even. Under these conditions the solution at the midpoints of the fiber segments (where is zero) coincides with f(t) up to a constant phase. Also, since the values of the at the endpoints of the fiber are symmetrical w.r.t. zero, the values of at these points arc the complex conjugate of each other. Figure 2 (upper) shows a plot of and The solid lines represent the positive part of the dot-dashed lines the negative part. The dashed line represents a hyperbolic secant pulse with the same FWHM in the frequency domain. Figure 2 (lower) shows the corresponding plot of f(t) vs. t. Again, the solid lines represent the positive part, the dot-dashed lines the negative part (we note that, in contrast, the familiar soliton solutions of NLS are always positive both in the time and frequency domains), and the dashed line is a hyperbolic secant pulse which has the same FWHM in the temporal domain. The main features of these new pulses are a Gaussian-like center and exponentially decaying tails, oscillating with a frequency that increases with increasing distance from the center of the pulse. We note that the energy of the pulse is more concentrated in the center, when compared to NLS pulses of the same FWHM. Figure 3 illustrates the profiles of f(t) for a sequence of different values of s between 0 and 16, for a fixed value of By virtue of the scaling symmetry mentioned in the previous section, the plot also represents qualitatively a class of solutions corresponding to a fixed value of s and varying values of From Fig. 3 we also see that, keeping fixed, the width of the solution increases with s. Thus, in order to compare pulses with the same FWHM and different values of map strength s one needs to consider a different value of for each value of s. We should mention that the integral equation (43) can also be used to look for stationary solutions in the zero or normal average dispersion regimes, i.e. when In fact,

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we already obtained such solutions, and they possess the same structure described above. For moderately weak dispersion maps (and if has only support for frequencies near zero), one can expand r(x) in Taylor series around Then the double convolution integrals in Eqs. (19) and (21) yield the NLS equation with corrections. The

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corrections represent terms of shock type, which are partially responsible for the self-steepening of the solution. We note however that the first order correction alone cannot reproduce the complicated structure of the solutions (exponential oscillating tails). From the knowledge of , we can reconstruct the leading-order solutions of the PNLS equations via Eq. (48). The resulting behavior, shown in Fig. 4, is a pulse whose shape and properties change periodically inside the map. Note that the pulse width is maximum when the peak amplitude is minimum and vice versa. The breathing pulse displayed in Fig. 4 shows a remarkable correspondence with direct simulations of the perturbed NLS Eq. (1) with strong dispersion management (cf. Fig. 2 in Ref. [18]). In the next section we examine this correspondence in further detail. 3.2. COMPARISON BETWEEN THE DMNLS AND NUMERICAL SOLUTIONS OF THE FULL PNLS To test our model we compared our results direct numerical simulations of

the full model (Eq. (1)), which we integrated using either a second-order split-step method [47, 48] or an integrable discretization of the NLS equation [49]. When necessary, we supplemented our code with averaging tools [18]

ON THE PROPERTIES OF DISPERSION-MANAGED SOLITONS

in order to find the periodic solutions. Two quantities that are useful in comparing the properties of the two equations are the RMS pulse width and the average chirp, defined respectively as (for a pulse centered about

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In particular, c(z) is defined so that, if (i.e., if the pulse phase varies quadratically in t and is even in t), then (i.e., c(z) is half the coefficient of Therefore, if q(z, t) is real (or if it can be made real by multiplication with a constant or linear phase) at some specific locations in the fiber, then the chirp c(z) is zero at those points. In Fig. 5 we plot the chirp c(z) versus the peak pulse amplitude for the leading order solution of the perturbation expansion, as reconstructed from the DMNLS soliton solutions via Eq. (48), and we compare them with numerical solutions of the full PNLS equation with the same energy, for and for a few values of (with corresponding pulse energies for the solutions of PNLS) and From Fig. 5 we see that, for moderate values of the expansion parameter the leading order term in the multiscale expansion (4) provides a good representation of the full dynamics of the pulse within the dispersion map. For the DMNLS-reconstructed pulse, the peak amplitude is maximum (and, correspondingly, the pulse width is minimum) when the chirp c(z) is zero, i.e. at the midpoints of both fiber segments. For the numerical solutions of PNLS, higher order corrections break the degeneracy between the two fiber segments; as a consequence, the maximum amplitude is reached only at the midpoint of the anomalous fiber segment. In Fig. 6 we plot the norm of the difference between the stationary

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solution of DMNLS and the numerical solution of PNLS with the same

energy, sampled at the midpoint of the fiber segment in our notation). The difference is normalized to the norm of the numerical solution of PNLS. Namely, we plot

(with

for various values of

(corre-

sponding to different pulse energies for and five values of We note that, as is decreased, the breathing solutions of Eq. (1) converge to the solutions obtained from the DMNLS equation. In particular, the convergence rate is linear in which confirms that the leading order term in

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the multiscale perturbation expansion (4) approximates the solution of the original problem (1) up to . Figure 7 shows the (relative) difference between the RMS pulse width and the pulse peak amplitudes between the two models, for the same parameter values used in Fig. 6. The small discrepancies between the solutions of the two equations depend on the high order terms in the perturbation expansion, which are also responsible for the fast evolution of the pulse amplitude in the Fourier domain. We have calculated the higher order corrections to the DMNLS equation, with methods similar to the ones presented here. However, the analysis of their effect on the dispersion managed solitons is outside the scope of the present paper. We will report on the higher order corrections in a separate publication.

3.3. LOSSY CASE When the loss parameter G is not zero, we can still use Eq. (43) to look for stationary solutions of DMNLS. However, Eq. (37) does not apply, and the kernel r(x) must be modified accordingly. Here we use Eq. (41), which corresponds to equal step sizes and amplifiers positioned at the midpoint of the anomalous fiber segment. Unlike the lossless case, the kernel r(x) is not

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real; therefore we cannot ask

to be real and even. Thanks to partial symmetries still possessed by r ( x ) , the kernel in the time domain is still real; however, it is not even in t and separately anymore, but only in their product: Figure 8 displays real and imaginary part of stationary solutions of the DMNLS equation in the lossy case. Figures 8a,c corresponds to Figures 8b,d to Parts (a) and (b) show the real part of the solution, while parts (c) and (d) show the corresponding

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imaginary part. These plots should be compared with Fig. 2 (lower) in the lossless case. The imaginary part of f (t) corresponds to a non-zero chirp of the pulse. We recall that, unlike the lossless case, the kernel r(x) depends on the relative position of the amplifier and the dispersion map. Our choice of parameters (cf. Eq. (41)) is such that the plots in Fig. 7 correspond to the pulse profile at the midpoint of the anomalous fiber segment. However, it is well known that, in systems with dispersion management and damping/amplification, the location of the chirp-free points is no longer at the midpoint of the anomalous fiber segment [14]. In Fig. 9 we show the fast evolution of the chirp c(z) versus the peak amplitude inside one dispersion map. As in Fig. 5, we reconstruct the leading-order solution of the multiscale perturbation expansion using Eq. (48) and we compare it with same-energy numerical solutions of PNLS. Plots (a, c) are relative

to

for NLS), while plots (b, d) correspond to

For each model, the black dot denotes the value of pulse parameters at the amplifier (i.e. at the midpoint of the fiber segment). Plots (a-b) and (c-d) should be compared with Fig. 5b and 5d

in the lossless case, respectively. We recall that, for the DMNLS equation, the kernel r ( x ) (and therefore the stationary solutions) depends only on

the product

and not on

and

separately, while numerical

solutions of the PNLS depend explicitly on in a similar way as described in Fig. 5. Unlike the lossless case, the evolution of the pulse parameters in not symmetrical about the line Note that, although the pulses have a non-zero chirp at the midpoint of the anomalous fiber segment, they becomes real at some special point inside the map. Therefore it is conceivable that, with different choices of amplifier location and/or map origin (corresponding to different reductions of the kernel (40)), it could be possible to find purely real solutions of the DMNLS equation with loss/amplification effects included. However, further analysis will be required in order to fully understand this issue. 4. Evolution and Interaction Properties of Dispersion-managed Solitons In the previous sections we derived an averaged equation governing dispersion managed solitons, discussed its symmetry properties, described a numerical algorithm to obtain soliton solutions, and presented special solutions and their properties. In this section we analyze relevant “time”dependent properties of the DM solitons from the DMNLS equation (19) for two-step dispersion maps, where the kernel r ( x ) is given by Eq. (37).

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4.1. EVOLUTION AND STABILITY OF DISPERSION-MANAGED

SOLITONS We begin by briefly discussing the method by which we solve the DMNLS

equation (19). We solve the equation in Fourier space, which means solving a system of coupled integro-differential equations in the evolution variable Z (as opposed to a nonlocal partial differential equation in the temporal domain). Although in principle the system is infinite-dimensional, any computational scheme approximates it with a finite number of degrees of freedom. The double integral in Eq. (19) is discretized by using an extended fourth order quadrature scheme. The ODE’s are solved by an adaptive, fourth order Runge-Kutta algorithm. Typically we take 128 points in the Fourier domain. We monitor the total pulse energy; the typical energy change is less than 0.1 % (less than 1 % in all cases). Note that, if the map strength

s vanishes the DMNLS equation (19) reduces to the classical NLS equation rewritten in Fourier space. First we study the stability properties of DM solitons. We take the “pure” DM soliton solutions (obtained via Eq. (43) with Eq. (37), as explained in Section 3) as initial conditions in the time dependent DMNLS equation. We integrated the DMNLS over (in some cases) 300 nonlinear lengths, which, for a typical nonlinear length of 400 km, yields a total distance of about 120000 km. We found the DM solitons to be numerically

ON THE PROPERTIES OF DISPERSION-MANAGED SOLITONS 101 stable in all cases examined. A typical case is illustrated in Fig. 10, with and initial condition corresponding to Note also that, when the numerically integrated solution is used to reconstruct the leading order solution of the multiscale perturbation expansion via Eq. (48), the resulting behavior of the pulse is numerically equivalent to the one shown in Fig. 4. Although a mathematical proof of stability is not available at present, we must nevertheless consider the DM soliton as being stable over the length scales considered. 4.2. DISPERSION-MANAGED SOLITON INTERACTIONS In this subsection we study the interaction properties of DM solitons in the lossless case. In the future we will extend our analysis to the lossy case. Dispersion-managed solitons are in many cases approximately described by a Gaussian shape (e.g. see Ref. [24]). When comparing our theory to recent experimental results of soliton transmission, we find it convenient to approximate the stationary DM soliton solution in the Fourier domain as where

with a and b real and positive. The amplitude is determined by matching the amplitude of the exact DM soliton solution (as obtained from Eq. (43)). Similarly, the function is computed by matching the RMS pulse width in the time domain. The full width at half maximum (FWHM) of the the above Gaussian in the spectral domain is while the FWHM in the time domain is (cf. Eq. (52) below at By reconstructing the leading order solution of the multiscale expansion (Eq. (48)), the previous choice corresponds to approximating the full pulse in the time domain with the following chirped Gaussian:

(where is given by Eq. (13), with corresponding to a pulse intensity

given by Eq. (36) and

When studying the properties of DM solitons, we found that lossless DM solitons are characterized by two important dimensionless quantities. One of them is the “modified map strength”, (i.e. the ratio of map

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strength to average dispersion). The second one is the expansion factor E, defined as the ratio of the maximum and minimum full width at half maximum of the pulse. Once we fix M, each yields a value for the pulse energy and the expansion factor E via the fixed point equation (43). We calculate the values of E either directly from the DM pulse (via Eq. (48)) or (when matching to experimental data) from the Gaussian approximation (52). In particular, using Eq. (52) we have When we change s and but keep their ratio M fixed, we find that, in the parameter regimes analyzed, different values of correspond to different expansion ratios. In Figs. 11-13 we review typical interactions between classical soliton solutions of NLS with In the comoving frame of reference, NLS solitons are given by (Here we set ) Nontrivial velocities are then obtained via the Galileian transformation (24). In Figs. 11-13 (and in all subsequent figures in Section 4), part (a) represents the Fourier domain and part (b) the temporal domain. In physical space, classical NLS solitons always interact smoothly; i.e. the velocities of the solitons before/after interaction are the same. However, it is well known that, even for classical solitons, there can be significant interaction effects in Fourier space [43]. Figure 11 shows an interaction with relative velocity Note that, when where is the frequency separation between the two pulses. For classical solitons, this corresponds to 4.46 spectral widths. (The spectral width of a pulse, is defined as the full width at half maximum in the Fourier domain, For classical solitons, arc cosh We also mention that, for NLS solitons, the (temporary) frequency shift due to soliton interaction decreases as cf. Ref. [50]. In Fig. 11 we see only small interaction effects in the frequency domain. When (corresponding to 2.23 spectral widths, see Fig. 12) we see that noticeable interaction effects arise in the frequency domain. When the relative velocity is further decreased to (0.84 spectral widths) we note very significant interaction effects, as plotted in Fig. 13. Next we discuss interaction effects between DM solitons. We carried out interactions over wide parameter regimes and we did a parallel study using parameters based on recent NEC data [41]. In the former case we found that significant nonadiabatic interaction effects could occur even with relatively wide frequency separations. In Figs. 14 and 15 we illustrate the situation for two typical cases, corresponding to and Figure 14 is obtained with corresponding to a spectral width of and a velocity/frequency separation (yielding a frequency separation of 4.89 spectral widths). Here we observe serious nonadiabatic effects—namely, collapse. Figure 15 corresponds to corresponding to

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0.774) and a frequency/velocity separation (corresponding to 5.05 spectral widths). Here we find that collapse occurs even for such a wide separation in the frequency domain. Note that, in all interactions presented in this section, we take the pulses to be separated initially by five times their FWHM in the time domain, a compromise which would be hard to avoid when doing WDM experiments with pulse trains. Note

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also that collapse was previously found to occur in numerical simulations of soliton interactions at zero average dispersion [27]. From these calculations it can be concluded that, should one try to carry out WDM soliton transmission with experimental parameter values close to these dimensionless ratios, serious problems are likely to ensue. On the other hand, when we change the dimensionless parameters we

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can get into regimes which are quite favorable for soliton transmission. For example, when we used NEC data to determine the relevant parameters [41] we found extremely positive results. With our scalings, NEC data yields the dimensionless numbers and which correspond to an expansion ratio of approximately and a spectral width Figures 16–18 depict interactions at three

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values of velocity separation and corresponding frequency separation. Figure 16 describes a situation where there arc only minor effects due to the interaction. Here we use a velocity separation of which corresponds to a frequency separation of yielding a separation of 4.41 spectral widths. Figure 16 shows that the four-wave mixing contribution which one has for classical solitons disappears. This looks to be a very

ON THE PROPERTIES OF DISPERSION-MANAGED SOLITONS 107

favorable regime. Figure 17 illustrates that serious nonadiabatic effects can occur. This corresponds to decreased relative velocity which corresponds to yielding a separation of 2.21 spectral widths. Again, the interaction in the Fourier domain is better than that of classical solitons—but this is not the case in the time domain (cf. Fig. 12). Finally, Fig. 18 illustrates that if we decrease the relative velocities still further to

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(which corresponds to

we observe

more serious nonadiabatic effects. In Figs. 17 and 18 we see that nonadiabatic effects occur only at such

small relative velocities and frequency separations that they are well below the values typically used in WDM experiments. (NEC experiments used a wavelength separation of 0.8nm, which, for a Gaussian pulse with

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corresponds to 4.09 spectral widths.) Presumably, such small frequency separations would not be used in an experimental setup, and one would not carry out WDM experiments with frequency separation below approximately 3 spectral widths. We remind the reader that all the calculations presented in this section are relative to the lossless case; the situation might be somewhat different in the lossy case. However, our point

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of view is that the best possible interactions (i.e. closest to “integrable”) should corne from the lossless case. This hypothesis will need to be verified. It is clear from these results that different parameter ranges can lead to substantially different results and different implications regarding WDM transmission for DM solitons. In the future we will continue these studies in the lossy case. We hope that these type of calculations will enable us to

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provide realistic indications of favorable parameter regimes for wavelengthdivision multiplexed dispersion-managed soliton transmission experiments. 5. Conclusions

In this paper we have studied the dynamics of optical pulses in a strongly dispersion-managed system. By introducing a suitable multiscale perturbation expansion, we decompose the pulse in the Fourier domain into a slowly evolving amplitude and a rapid phase which describes the chirp of the pulse. The fast phase can be calculated explicitly, and is driven by the large variations of the dispersion about the average. The evolution of the amplitude is given by a novel nonlocal equation of nonlinear Schrödinger type, called the DMNLS equation (19). Stationary solutions of this equation describe pulses with a Gaussian center and exponential oscillating tails. In the lossless case, these solutions, as well as the dynamics of the whole pulse (as reconstructed from the perturbation expansion), agree remarkably well with numerical simulations of the full NLS equation with dispersion management. Computationally, we found that the DM solitons evolved stably over very long distances compared to length scales of transmission experiments. Two dimensionless quantities were found to be particularly important: the modified map strength

and the ex-

pansion factor E (which, once M is fixed, uniquely determines the eigenvalue

and the pulse energy). When interacting the pulses in the lossless case, we found parameter regimes where strong nonadiabatic effects and even soliton collapse could occur at relatively large values of velocity/frequency separation. On the other hand, when we used data corresponding to recent WDM dispersion-managed soliton transmission [41] as the basis of the computations, we found that the parameter regimes where strong nonadiabatic effects and soliton collapse were much more limited, and pulse interactions were very favorable to WDM soliton transmission, in the sense that strong interaction effects were only found for very small frequency separations, which are well below the typical values used in actual transmission experiments to date. Acknowledgments

A special thanks goes to A. Hasegawa for encouraging this investigation and to T. Hirooka for many valuable remarks and for providing valuable data. We also thank S. Chakravarty, W. L. Kath and A. Zozulya for fruitful discussions. This effort was partially sponsored by the National Science Foundation, under grant number ECS-9800152 and by the Air Force Office of Scientific Research, Air Force Materials Command, USAF, under grant number F49620-97-1-0017.

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Hasegawa, A. and Kodama, Y. : The guiding-center soliton in optical fibers, Opt.

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9. Gabitov, I. R. and Turitsyn, S. K. : Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation, Opt. Lett., 21, (1996), p.327. 10.

Mollenauer, L. F. and Mamyshev, P. V. : Massive Wavelength-Division Multiplexing with Solitons, J. Quantum Electron., 34, (1998), p.2089. 11. Turitsyn, S. K., Schäfer, T., Spatschek, K. H. and Mezenetsev, V. K. : Path-averaged chirped optical soliton in dispersion-managed fiber communication lines, Opt. Commun., 163, (1999), p.122. 12. Ablowitz, M. J . and Biondini, G. : Multiscale pulse dynamics in communication systems with strong dispersion management, Opt. Lett., 23, (1998), p.1668. 13. Kodama, Y., Kumar, S. and Maruta, A. : Chirped nonlinear pulse propagation in a dispersion-compensated system, Opt. Lett., 22, (1997), p. 1689.

14. Yang, T. -S., Kath, W. L. and Turitsyn, S. K. : Optimal dispersion maps for wavelength-division-multiplexed soliton transmission, Opt. Lett., 23, (1998), p.597. Chen, Y. and Haus, H. A. : Dispersion-managed solitons with net positive dispersion, Opt. Lett., 23, (1998), p.1013. 16. Smith, N. J., Doran, N. J., Knox, F. M. and Forysiak, W. : Energy scaling characteristics of solitons in strongly dispersion-managed fibers, Opt. Lett., 21, (1996), 1981. 17. Devaney, J. F. L., Forysiak, W., Niculae, A. M. and Doran, N. J. : Soliton collisions in 15.

dispersion-managed wavelength-division multiplexed systems, Opt. Lett., 22, (1997), 18.

p.1695. Nijhof, J. H. B., Doran, N. J., Forysiak, W. and Knox, F. M. : Stable soliton-like

propagation in dispersion managed systems with net anomalous, zero and normal dispersion, Electron. Lett., 33, (1997), p.1726. 19. Golovchenko, E. A., Jacob, J. M., Pilipetskii, A. N., Menyuk, C. R. and Carter, G. 20. 21.

M. : Dispersion-managed solitons in a fiber loop with in-line filtering, Opt. Lett., 22, (1997), p.289. Grigoryan, V. S., Yu, T., Golovchenko, E. A., Menyuk, C. R. and Pilipetskii, A. N. : Dispersion-managed soliton dynamics, Opt. Lett., 22, (1997), p.1609. Grigoryan, V. S. and Menyuk, C. R. : Dispersion-managed solitons at normal aver-

age dispersion, Opt. Lett., 23, (1998), p.609. 22. Kutz, N. J. and Evangelides, S. G. : Dispersion-managed breathers with average 23.

normal dispersion, Opt. Lett., 23, (1998), p.685. Kutz, .1. N., Holmes, P., Evangelides, S. G. and Gordon, J. P. : Hamiltonian dy-

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namics of dispersion-managed breathers, J. Opt. Soc. Am. B, 15, (1998), p.87. 24. Turitsyn, S. K. : Theory of pulse propagation in high-bit rate optical transmission systems with strong dispersion management, Sov. Phys. JETP Lett., 65, (1997),

p.845; Stability of an optical soliton with Gaussian tails, Phys. Rev. E, 56, (1997), R3784. 25. Gabitov, I. R., Shapiro, E. G. and Turitsyn, S. K. : Optical pulse dynamics in fiber links with dispersion compensation, Opt. Commun., 134, (1997), p.31. 26. Turitsyn, S. K. and E. G. Shapiro, E. G. : Dispersion-managed solitons in optical

amplifier transmission systems with zero average dispersion, Opt. Lett., 23, (1998), p.682. 27. Chen, Y. and Haus, H. A. : Collisions in dispersion-managed soliton propagation, Opt. Lett., 24, (1999), p.217. 28. Forysiak, W., Devaney, J. F. L., Smith, N. J. and Doran, N. J. : Dispersion management for wavelength-division multiplexed soliton transmission, Opt. Lett., 22, (1997), p.600.

29. Golovchenko, E. A., Pilipetskii, A. N. and Menyuk, C. R. : Periodic dispersion 30.

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management in soliton wavelength-division multiplexed transmission with sliding filters, Opt. Lett., 22, (1997), p.1156. Hirooka, T. and Hasegawa, A. : Chirped soliton interaction in strongly dispersionmanaged wave-length-division-multiplexed systems, Opt. Lett., 23, (1999), p.768.

Kaup, D. J., Malomed, B. A. and Yang, J. : Interchannel pulse collision in a wavelength-division multiplexed system with strong dispersion management, Opt. Lett., 23, (1998), p.1600.

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Malomed, B. A. : Suppression of jitter and interactions by means of dispersion management, Opt. Commun., 147, (1998), p.157. 33. Mamyshev, P. V. and Mollenauer, L. F. : Soliton collisions in wavelength-division multiplexed dispersion-managed systems, Opt. Lett., 24, (1999), p.448. 34.

Niculae, A. M., Forysiak, W., Gloag, A. J., Nijhof, J. H. B. and Doran, N. J. : Soli-

ton collisions with wavelength-division multiplexed systems with strong dispersion

management, Opt. Lett., 23, (1998), p.1354. 35. Sugahara, H., Kato, H. and Kodama, Y. : Maximum reductions of collision-induced 36.

frequency shift in soliton-WDM systems with dispersion compensation, Electron. Lett., 33, (1997), p.1065. Sugahara, H., Maruta, A. and Kodama, Y. : Optimal allocation of amplifiers in a

dispersion-managed line for a wavelength-division multiplexed soliton transmission 37.

system, Opt. Lett., 24, (1999), p. 145. Sahara, A., Kubota, H. and Nakazawa, M. : Comparison of the dispersion allocated

WDM (10 Gbit/s 10 channels) optical soliton and NRZ systems using a Q map, Opt. Commun., 160, (1999), p.139. 38. Pincemin, E., Leclerc, O. and Desurvivre, E. : Feasibility of 1 Tbit/s (25 40 Gbit/s) trans-oceanic optically regenerated systems, Opt. Lett., 24, (1999), p.720. 39. Alleston, S. B., Harper, P., Penketh, I. S., Bennion, I. and Doran, N. J. : 1220 km propagation of 40 Gbit/s single channel RZ data over dispersion managed standard (non-dispersion shifter) fibre, OFC’99, PD3, (1999). 40. Le Guen, D. et al. : Narrow bank 1.02 Tb/s (51 20 Gbit/s) soliton DWDM trans-

mission over 1000 km of standard fiber with 100 km amplifier spans, OFC’99, PD4, (1999). Fukuchi, K. et al. : 1.1 Tb/s (55 20 Gb/s) dense WDM soliton transmission over 3020 km widely-dispersion-managed transmission line employing 1.55-1.58 hybrid repeaters, ECOC’99, PD42, (1999). 42. Ablowitz, M. J., Kruskal, M. D. and Ladik, J. F. : Solitary Wave Collisions, Siam 41.

J. Appl. Math., 36, (1979), p.428. 43. Benner, A. F., Sauer, J. R. and Ablowitz, M. J. : Interaction effects on wavelength multiplexed soliton data packets, J. Opt. Soc. Am. B, 10, (1993), p.2331.

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a fluid of finite depth, Phys. Lett. A, 197, (1995), p.401. 46. Petviashvili, V. I. : Equation of an extraordinary soliton, Sov. J. Plasma Phys., 2, (1976), p.257. 47. Hardin, R. H. and Tappert, F. : Application of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations, SIAM Rev. 48.

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EXPERIMENTAL DEMONSTRATION OF MASSIVE WDM OVER TRANSOCEANIC DISTANCES USING DISPERSION MANAGED SOLITONS

L. F. MOLLENAUER, P. V. MAMYSHEV, J. GRIPP, M. J. NEUBELT AND N. MAMYSHEVA

Bell Labs, Lucent Technologies

Holmdel, NJ 07733, USA. AND

LARS GRÜNER-NIELSEN AND TORBEN VENG

Lucent Denmark

Priorparken 680, DK-2605 Brondby, Denmark Abstract. By combining a special dispersion map having nearly constant path-average dispersion, a hybrid amplification scheme involving backwardpumped Raman gain, and sliding-frequency guiding filters, we have demonstrated massive WDM at 10 Gbit/s per channel, error free for all channels), without the use of forward error correction, over greater

than 9000 km, using dispersion managed solitons. The number of channels (27) was limited only by a temporary lack of amplifier power and gain flatness. Terabit capacities are to be expected in the near future. In the immediate aftermath of the first experimental demonstration of dispersion managed solitons [1], much of the progress made at further understanding [2]- [5] tended to focus on issues of single channel transmission, so that contemporaneous examinations [6] of WDM were sparse. It should be noted, however, that commercial interest lies almost exclusively in massive or “dense” WDM. Furthermore, in the world of terrestrial networks, there is increasing interest (driven by the economics of dense WDM itself) in all-optical transmission over distances many times greater than the traditional 400–500 km spacing between centers of electronic regeneration and signal switching. Of pioneering importance in this regard is the work of T. Georges and colleagues from CNET, who seem to have set their sights on achievement of the highest possible spectral efficiency in WDM over inter115 A. Hasegawa (ed.), Massive WDM and TDM Soliton Transmission Systems, 115–127. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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mediate distances (1–2 thousands of km), through experiments [7] based on a kind of “pulse-overlapped” [8] transmission. In this paper we describe our own attempts to address the issues of dense WDM over ultra long distances, and describe a successful experimental test that should help to define the possibilities. It uses a special dispersion map based on some recently available Lucent fibers, a hybrid amplification scheme involving Raman gain, and sliding-frequency guiding filters. We have already argued elsewhere [9]-[11] that the filters are the necessary component of ultra long distance WDM transmission with dispersion managed solitons, so that will be taken as a given here. Ultra-long distance, massive WDM places many simultaneous and stringent demands on the design of the dispersion map and its pulse behavior. First, the map must possess a path-average dispersion that is nearly constant over the required wavelength range. But it must also provide for adequate soliton pulse energy at low exhibit negligible adjacent pulse interaction within each channel, and be compatible with the use of guiding filters. Finally, with respect to interaction between channels, four-wave mixing must be negligible and effects of the inevitable XPM (cross phase modulation) must be reduced to a minimum. Newly developed fiber types have made it possible to create such maps. We present here a particularly well-behaved one.

The map (Fig. 1) combines three fibers, a low-slope, non-zero dispersion shifted fiber (Lucent TWRS, or “TrueWave® Reduced Slope”), a new ultra

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high slope DCF (dispersion compensating fiber) from Lucent Denmark, and standard single mode fiber. Although the slope of the new DCF is not quite large enough to simultaneously compensate the slope and D of the TWRS fiber alone (see Table 1), the addition some standard fiber can allow for exact simultaneous attainment of the desired (very small) and zero slope of at any desired mean wavelength. This simultaneous control is possible because the addition of the standard fiber forces the increase of the amount DCF (and hence of its negative contribution to the average slope) much faster than it increases the average slope of the combined positive D fibers (again see Table 1). To create the spans from the real-world fibers, we first surveyed our stock with the dispersion OTDR [12, 13], an instrument which quickly (within a few seconds) yields an accurate map of D(z) for each spool. We then put that information into a simple computer program which calculated the exact lengths of each of the three types to be combined to make spans of 56.0 km length, with ps/nm-km and zero slope at 1550 nm. For each of the completed spans, was quickly and accurately determined from differential time of flight measurements on pulse trains separated in wavelength by about 2 nm. Figure 2 shows the mean behavior for the six spans of our recirculating loop, where the curve is an extremely shallow parabola with peak at 1547 nm. The curves for the individual spans were identical to one another, except that the peaks of the parabolas were scattered at random in a band nm about the mean; the scatter primarily represents small errors in the values we assumed for the dispersion slopes. The DCF was spliced into the spans using a special technique developed at Lucent Denmark, and uniformly resulted in splice losses of dB at each end of the DCF. That splice loss, plus the excess losses of the DCF itself, brought the total mean span loss to 14.0 dB, or just 2.24 dB above that for a 56 km span made completely of ordinary transmission fibers (with loss rates of 0.21 dB/km). Raman gain from backward pumping was used to compensate about 70 % of the span loss (see the intensity profile of Fig. 1). while the remainder

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was compensated by a low-gain Erbium fiber amplifier. (To compensate all of the span loss with Raman gain can lead to significant excess noise from Rayleigh double back scattering of the signals themselves.) The required Raman pump power of about 235 mW for each span was provided by a pair of laser diodes, polarization-multiplexed together, lasing simultaneously on dozens of modes in a band about 10 nm wide, and centered at 1450 nm. The hybrid amplification scheme was used to enable the 56 km spans (which in turn allow for the optimum map strength) to be used with negligible noise penalty, in contrast to the several dB penalty that would result from the use of lumped amplifiers alone [14]. Raman gain also has the great advantage for WDM, that both its excellent noise figure (3.5 dB) and gain-band shape are completely independent of pump and signal power levels, in contrast to the behavior of Erbium fiber amplifiers. Finally, the net gain profile of the hybrid scheme is conveniently almost flat over the entire Erbium C band (see Fig. 3). In order to facilitate discovery of an optimum dispersion map, the pulse behavior (pulse width, bandwidth, and chirp parameter as functions of distance) of each tentative design was first explored with a sophisticated Maple program using the efficient ODE (ordinary differential equation) method that we have described elsewhere [15]. Figure 4 shows typical behavior, as thus determined, for the map of Fig. 1, and for practical values of and of path-average pulse energy. First, note that the maximum pulse width is always less than 40 % of the bit period (for 10 Gbit/s), that it tends to occur in the lower energy regions of the map, and that it is always accompanied by a large chirp; these characteristics, taken together, make the adjacent pulse interaction truly negligible [8], as confirmed both by exact numerical

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simulations and by our real world transmission experiments. Second, note that the chirp parameter is nearly zero at the lumped amplifiers; although not absolutely necessary, this characteristic tends to be convenient. Finally,

Fig. 5 shows path-average pulse energy vs for various fixed values of the minimum (unchirped) pulse width) in the section of the map. Note that for the region of most suitable for WDM (approx. 0.15 , path-average pulse energies of (adequate for error free transmission at 10000 km, and far beyond with sliding filters)

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can be had with pulsewidths in the ps range. In all cases, the pulse energies are at least several times greater than the energies of ordinary solitons for the same combinations of pulse width and For the map of Fig. 1, the only significant penalty in WDM stems from

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the frequency shifts induced by the XPM of colliding solitons [11]. Figure 6 shows the frequency shift, as a function of distance over a complete collision, of one of a pair of colliding solitons from adjacent channels. Numerical analysis of the curve of Fig. 6 shows that the area under the complete curve is accurately zero, so there is no net time shift, just as predicted from the simple theory of guiding filters. It should also be noted that just a bit be-

yond the negative peak of the frequency shift curve, the pulses have become completely separated, so the decay of back to zero represents the pure filter effect. Thus, from that last part of the curve, we can infer a filter damping length for frequency shifts of As we have noted elsewhere [11], however, with strong dispersion management, there are many inevitable partial collisions at the beginning and end of the transmission, and, without filters, these tend to induce excessively large net frequency and time shifts. The problem is sufficiently complex that full numerical simulation of a multi-channel transmission involving long trains of random bit sequences are required to allow for a reasonably accurate estimate of the WDM penalties. Figure 7 shows the results of such a simulation, the eye diagrams for each of eight channels at 10000 km in a transmission line

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involving the map of Fig. 1. These results are encouraging, as they tend to show a largely open eye and no more than a 10–15 ps spread in pulse arrival times. On the other hand, they also illustrate the down side of filters, viz., their tendency to transform frequency deviations into amplitude jitter. The source for the WDM experiments (Fig. 8) consists of two banks of DFB lasers, the one corresponding to the even, the other to the odd channels, each multiplexed together and sent through a Mach-Zehnder modulator, sinusoidally driven at 5 GHz to carve nearly Gaussian-shaped pulses (two pulses for each cycle of the drive voltage), and at 10 GHz to provide a nearly linear chirp of adjustable magnitude. After a second modulator imposes independent random data sequences on each of the two sets of channels, they are brought together in a polarization combiner, such that adjacent channels are orthogonally polarized. Finally, the pulses are compressed and unchirped by a length of standard fiber. The recirculating loop consisted of six spans like that of Fig. 1, for a total length of 336 km; for further details, see Fig. 9. To provide a base line for the WDM experiments, we first made careful examination of a single-channel transmission. The nearly constant value at long distances of the 20 GHz component of the detected pulse train (Fig. 10) implies a very small (std. deviation Gordon-Haus timing jitter, as would be expected from the combination of large pulse energy

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low and the jitter-quenching effect of the guiding filters. The observed eye diagram at 16 Mm (Fig. 11) looks very good, where standard deviations of the ones and zeros measured by the sampling scope yield a linear which in turn implies that the BER should be better than about Nevertheless, the directly

measured BER at 16 Mm was . At present, we do not know whether this discrepancy is caused by some sort of defect in the BER measurement itself, or whether it truly corresponds to the existence of long, non-Gaussiari tails of the pertinent probability distributions. Results of the dense WDM experiments tended to be remarkably similar to those with the single channel. With careful adjustment of the loop, the eye of a typical WDM channel (Fig. 12) could be made to look almost indistinguishable from that of the single channel (Fig. 11). Here, too, the inferred linear is inconsistent with the directly measured, typical BER (Fig. 13) of a To obtain the results of Figs. 12 and 13, however, it was necessary to reduce the filter sliding rate to about 60 % of that used for the single channel transmission. This required reduction in sliding rate (to the point where exponential growth in the noise is not completely suppressed) is the most significant penalty associated with dense WDM in our system. (It tends to restrict the freedom we would otherwise have with sliding-frequency filters, to extend the error free distances almost indefinitely.) Finally, we note that we found it helpful for the WDM

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experiments to increase from By far the most difficult task in the WDM experiments is that of maintaining equal pulse energies in all channels, as exemplified (Fig. 14) by the somewhat ragged appearance of the received spectrum at 9 Mm. The difficulty stems largely from the fact that with the recirculating loop, small defects in the amplifiers and in the gain equalizer tend to be magnified by

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a factor equal to the number of round trips. Fortunately, in a real (linear)

transmission line, the defects from the various segments of the line would tend to be distributed at random, and hence tend to add to a much smaller net defect. It should also be noted that the guiding filters are less effective in regulating the pulse energies of dispersion managed solitons than they are with the energies of ordinary solitons. This is because of the fact that with dispersion management, the pulse bandwidths increase several times more slowly with increasing pulse energy than they do with ordinary solitons. An improved type of filter characteristic may help with this problem. With improvements in the amplifiers, in the gain equalization, and in the guiding filters themselves, we believe that the above results should be

greatly extendible in both distance and in system capacity. In particular, our hybrid amplification scheme should rather easily yield a dead-flat gain band of 40 nm width, for a system capacity with the present channel spac-

ing of 75 GHz (0.6 nm) of 2/3 Terabit/s. Numerical simulations show, however, that the penalties for using 62.5 GHz (0.5 nm) or even 50 GHz (0.4 nm) channel spacings, where the system capacities would be 0.8 and 1.0 Terabit/s, respectively, are not that great. Finally, it should be noted that with only minor changes in the fiber parameters, the map of Fig. 1 can be easily extended, with only very small additional noise penalty, to cover the 80 km spacing typically found between amplifier huts in the continental United States. Furthermore, one can have the effect of sliding-frequency filters with the combination of fixed- frequency filters (presumably tuned to some network standard for channel wavelengths) and acousto-optic modulators (which act as frequency shifters) located at each amplifier. Thus, this work shows that dense WDM transmission using dispersion managed solitons is not only useful for transoceanic

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transmission, but could also be the basis for complex, all-optical networks extending across a large geographic area, such as that of the North American continent. We wish to thank R. B. Kummer of Lucent Atlanta for supplying us with the TWRS fiber, S. Kosinski for assistance with the splicing, and J. Cloonan for winding the fibers. References 1. Suzuki, M., Morita, I., Edagawa, N., Yamamoto, S., Taga, H. and Akiba, S. : Re2.

3. 4. 5. 6.

duction of Gordon-Haus timing jitter by periodic dispersion compensation in soliton transmission, Electon. Lett., 31, (1975), pp.2027- 2029. Smith, N. L., Knox, F. M., Doran, N. J., Bloe, K. J. and Bennion, I. : Enhanced power solitons in optical fibers with periodic dispersion management, Electron. Lett., 32, (1996), pp.54-55. Smith, N. J., Forysiak, W. and Doran, N. J. : Reduced Gordon-Haus jitter due to enhanced power solitons in strongly dispersion managed systems, Electron. Lett., 32, (1996), pp.2085-2086. Carter, G., Jacob, J. M., Menyuk, C. R., Golovchenko, E. A. and Pilipetskii, A. N. : Timing jitter reduction for a dispersion-managed soliton system: experimental evidence, Opt. Lett., 23, (1997), pp.513-515. Turitsyn, S. K., Mezentsev, V. K. and Shapiro, E. G. : Dispersion-managed solitons and optimization of the dispersion management, Opt. Fiber Tech., 4, (1998), pp.384452. Niculae, A. M., Forysiak, W., Gloag, A. G., Nijhof, J. H. B. and Doran, N. J. : Soli-

ton collisions with wavelength-division multiplexed systems with strong dispersion management, Opt. Lett., 23, (1998), pp.1354-1356. 7. Le Guen, D., Del Burgo, S., Moulinard, M. L., Grot, D., Henry, M., Favre, F. and Georges, T. : Narrow band 1.02 Thit/s soliton DWDM transmission over 1000 km of standard fiber with 100 km amplifier spans, OFC’99, PD4, (1999). 8. Mamyshev, P. V. and Mamysheva, N. A. : Pulse-overlapped dispersion-managed data transmission and intra-channel four-wave mixing, Opt. Lett., 24, (1999), pp. 1454-1456. 9. Mollenauer, L. F., Mamyshev, P. V. and Gordon, J. P. : Effect of guiding filters on the behavior of dispersion-managed solitons, Opt. Lett., 24, (1999), pp.220-222. 10. Mollenauer, L. F., Bonney, R., Gordon, J. P. and Mamyshev, P. V. : Dispersionmanaged solitons for terrestrial applications, Opt. Lett., 24, (1999), pp.285-287. 11. Mamyshev, P. V. and Mollenauer, L. F. : Soliton collisions in wavelength-divisionmultiplexed dispersion-managed systems, Opt. Lett., 24, (1999), pp.448-450. 12. Mollenauer, L. F., Mamyshev, P. V. and Neubelt, M. J. : Method for facile and accurate measurement of optical fiber dispersion maps, Opt. Lett., 21, (1996), pp.1724-

1726. 13.

Gripp, J. and Mollenauer, L. F. : Enhanced range for OTDR-like dispersion map

14.

See Section 3.1 of Mollenauer, L. F., Gordon, J. P. and Mamyshev, P. V. : Solitons in high bit rate, long distance transmission, Chap. 12 of Optical Fiber Telecommunications III, Academic Press.

15.

Gordon, J. P. and Mollenauer, L. F. : Scheme for characterization of dispersionmanaged solitons, Opt. Lett., 24, (1999), pp.223-225.

measurements, Opt. Lett., 23, (1999), pp.1603-1605.

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ON THE DISPERSION MANAGED SOLITON The guiding-center theory revisited

Y. KODAMA Department of Mathematics Ohio State University 231 West 18th Avenue Columbus, OH 43210, USA

1. Introduction The dispersion managed (DM) soliton is a periodic pulse solution of the nonlinear Schrödinger (NLS) equation with periodically varying coefficients along the distance Z,

where q represents the complex envelope of electric field in a fiber, and T and Z are normalized time and distance. The coefficients a(Z) and d(Z) represent the periodic amplification and dispersion management. Here we assume that the periods of those functions are short as compared with the nonlinear distance which is assumed to be 1. Then the problem concerned here is to find an averaged equation for the perturbed problem (1). Since we consider the case , the simple average of this equation fails to describe the solution with certain accuracy. We here consider only the lossless case, but the extension of the present discussion is straightforward. We denote the average dispersion where represents the path-average over the period . Then we write with and

where and are the differential polynomials representing the dispersion and nonlinear terms,

129 A. Hasegawa (ed.), Massive WDM and TDM Soliton Transmission Systems, 129–138. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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In this paper, we study EqD (1) in the framework of the guiding center theory developed in references [5, 6] (see also reference [7]). The guiding center theory consists of the Lie exponential transformation and the averaging method. We here give a general formulation of the theory. The

original formulation of the theory was defined on the space of differential polynomials (see references [7, 8]). The theory can be easily formulated in the Fourier spectral domain, which leads to the Hamiltonian averaging method recently discussed in references [4, 11, 12]. Then the Lie exponential transform gives a canonical transform of the system. We also discuss the averaged equation (or normal form) in both limits of weak and strong dispersion managements. In the final section, we shoe the onintegrability of the averaged equation in an asymptotic sense. 2. Guiding-center Theory

The basic idea of the guiding-center theory is to transform a perturbed system into a simple (averaged) equation by using the Lie transformation. Here we present the theory in a general form, so that the formulation in the Fourier spectral domain becomes rather trivial: Let us denote the perturbed equation such as Eq. (1) and the transformed equation as the averaged one,

where and represent the Lie derivatives with respect to the vector fields X and Y (see reference [8] for those notations). The Lie transform is given by

where

is a vector field associated with the generating function Under the transformation (6) we have

for any (differential) polynomials F. Then from the equations (4) and (5),

we have the following equation for the generating function,

from which we obtain the equation in the form of an adjoint action,

THE DM SOLITON

131

Notice that Eq. (9) is written in terms of the transformed variable u. Equation (9) can be expressed in the expansion form,

Using the Lie algebra homomorphism,

for any vector fields X, Y, Eq. (10) can be writen on the vector fields,

where This is the equation to determine the averaged equation (5) and the generating function We also formulate the transformation for the case where the vector fields X and Y are given by hamiltonian vector fields with hamiltonians H and G, i.e.

The hamiltonian for Eq. (1) is given by

The generating function function S,

of the Lie transformation (6) is given by the

Again, using the Lie algebra homomorphism between the vetor fields and the hamiltonian functions,

where {H, G} is the Poisson bracket defined by

we can write the transformation (12) in the form,

where In the next section, we apply those formulae to the DM soliton equation (1), and find the averaged equation.

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Before closing this sction, we put a comment on these equations in the Fourier spectral domain, where the Fourier transform is given by

The vector field

is then defined as

In particular, if the vector field is given by a Hamiltonian vector field, say,

where the set of Fourier transform gives the canonical pair, and is the Hamiltonian function, then the present formulation can be easily put into a Hamiltonian framework with the Poisson bracket,

The Lie transform then gives a canonical transformation with a generating function ,

The hamiltonian form of Eq. (1) in the Fourier spectral domain is given by

with the Hamiltonian

where form,

Then writing the Hamiltonian in the

one can use the method of hamiltonian averaging which is equivalent to the guiding-center theory. (See references [3, 11, 12] for the details.) 3. The DM Soliton As an Averaged Soliton We now compute the generating function and determine the averaged equation of (1) to describe the DM soliton. In order to do so, we make a series of Lie transformations denoted by

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133

where and the averaged one may be obtained as We write the equation for as

In the first step

we have Equation (2) with

To absorb the second term, we choose the generating function where the periodic function

as

satisfies

Note here that we have the order that is, the integration of in the region leads to a smoothing of the function This is the key of the averaging method, and a successive transmation gives a natural ordering with the small parameter Note also that the Lie transformation with is just the integral transformation,

where

is the Green function given by

Thus the dispersion management induces a chirp in the solution q(T , Z), which is also called a “chirped RZ pulse”. The transformed equation Y (1) is then obtained as

The (simple) averaged equation of given by

denoted by

is then

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Y. KODAMA

If the function is of the order (i.e. a weak dispersion management), a finite truncation of the expansion form (36) may be used, and the DM soliton is just a guiding-center soliton found in reference [6] (see also reference [9] for the higher order correction of in Eq. (36)). In the case one should use the form (35) or the expansion form (36) with many terms (with a validity of convergence). Equation (35) is the averaged equation obtained in [1, 2]. We will discuss this equation in some

details in the next section. Now we proceed the second step of the transformation: Let us first write the

in the form,

Then we make a transformation to absorb for example the term in . Then is given by with

The transformed equation

then becomes

The idea of this successive transformation is that in each step the dispersion management function is smoothed down through the integration, and we obtain a natural ordering . By making successive transformations, one can also push the average free terms in (37) into higher orders. Now the averaged equation is given by

We thus see that if the is of order 1, the averaged equation of gives a good approximate equation for the DM soliton. However if is still large, one should proceed to the next transformation to push the order higher. We summarize the result as follows:

THE DM SOLITON Theorem 1 If the function

135

in (31) has the order of 1, Equation (1) can

be approximated by

where

is given by Eq. (35), and with the solution

where the generating function 4.

we have

is given by (30).

The Averaged Equation (42)

Here we discuss some properties of the averaged equation (42), and present a recent result in reference [12] for its integrability in a strong management limit. It is convinient to write the integral transform (32) with the generating function in the Fourier spectral domain,

Then the equation for

where the kernel

with

in (37) then becomes [1, 11]

is given by

Let us now consider the case of 2-step dispersion management whose is given by

where and , Then the average equation of (45) is given by replacing the rapidly oscillating kernel in (45) with its average,

where α represents the strength of the management. In the case of a weak dispersion management with the averaged equation (42) can be approximated by the integrable (unperturbed) NLS equation (just note and the DM soliton is the guiding-center soliton found in reference [6]. Recently Zakharov in reference [12] discussed the limit of strong dispersion management, and found that the amplitudes of the Fourier harmonics

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Y. KODAMA

are constant in this limit. This implies that the equation is somewhat close to a linear one. We now briefly summarize his result. The key idea of taking a strong management limit is to find an approximate function for the averaged kernel. First we note for large

from which we have

Notice that the kernel becomes so small for strong management, implying that the equation becomes almost linear. Then the equation (42) may be written in the form,

Here the integral may be taken as a principal sense, in another word, this equation is valid up to logarithmic accuracy. One of the main feature of this equation is that the four wave mixing is suppressed, and the power spectral becomes a constant in Z. In this sense, the averaged equation (42) is integrable in the strong limit. However one should also note that Eq. (42) is only valid when the strength (see the previous section). To determine the (approximate) shape of the stationary solution, one should solve the averaged equation (42) for for a real function and some constants The function then satisfies

It is interesting to compute a next correction to the above strong management limit for finding an approximate shape 5. Nonintegrability

We here discuss the nonintegrability of the averaged equation (42) in an asymptotic sense. The key to show the nonintegrability is to prove the nonexistence of an additional integral. We here define the integrability as the existence of infinitely many integrals in differential polynomial form (this is also the usual definition of the integrability). Then as was shown in [8], the existence of integrals implies that one can transform the perturbed equation into an integrable normal form in an asymptotic sense. The tranformation

THE DM SOLITON

137

can be also given by the Lie transformation (18) where we drop the Zderivative terms, i.e. We write the averaged equation (42) as a perturbed hamiltonian system,

where is the NLS hamiltonian, the average of (14), and the higher order correction is given by (see Equation (36)

where we assumed . Since

and leads to only nonlinear correction (i.e. no

linear terms) to the NLS equation, the (transformed) integrable equation

up to this order is just the NLS equation, that is, we have with and Thus we try to solve the following equation for the generating function

In order to solve this equation, we first note that the general form of differential polynomial having the same order of the hamiltonian density of is a generic element of a 6-dimensional real space,

Here we dropped the elements having i in the coefficients, which give disjoint elements in the space of (see reference [8] for the details). Following reference [8], we can assume the basis of the solution space for in Eq. (55), where the power density, mentum density,

the mothe hamiltonian density, and the density of the third NLS hierarchy. Here implies the indefinite integral, Thus the solution space for is of only 3-dimensional one, and to find the general solution of one needs 3 constraints for the space in Eq. (56). We then found that in Eq. (54) does NOT satisfy the conditions, implying that the averaged equation (53) is not integrable at this order in asymptotic sense, that is, no additional integrals at this order. We would also like to mention the work [10] which also shows the nonintegrability of the averaged equation (53) in the same spirit of the present

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discussion. The work used the method of Zakharov-Shulman [13] which is defined on the Fourier spectral domain. References 1.

Ablowitz, M. J. and Biondini, G. : Multiscale pulse dynamics in communication systems with strong dispersion management, Opt. Lett., 23, (1998), pp.1668-1670. 2. Gabitov, I. and Turitsyn, S. K. : Breathing solitons in optical fiber links, JETP Lett., 63, (1996), pp.861-866. Gabitov, I. and Marshall, I. : Removing the time dependence in a rapidly oscillating Hamiltonian, Nonlinearity, 11, (1998), pp.845-857. 4. Gabitov, I. et al. : Quasi-integrability of DM soliton equation, presented in IMACS conference at Univ. Georgia, (1999). 3.

5. Hasegawa, A. and Kodama, Y. : Guiding-center soliton in optical fibers, Opt. Lett., 6.

15, (1990), pp.1443-1445. Hasegawa, A. and Kodama, Y. : Guiding-centre soliton in fibres with periodically varying dispersion, Opt. Lett., 16, (1991), pp.1385-1387.

7.

Hasegawa, A. and Kodama, Y. : Solitons in optical communications, Oxford Univ. Press, (1995).

8.

Kodama, Y. and Mikhailov, A. V. : Obstacles to asymptotic integrability, in Alge-

braic aspects of integrable systems in memory of Irene Dorfman, Fokas, A. S. and

Gelfand, I. M.(Eds.), Birkhäuser, (1996), pp.173-204. 9.

10.

Kodama, Y. and Maruta, A. : Optimal design of dispersion management for soliton wavelength-division-multiplexed system, Opt. Lett., 22, (1997), pp.1692-1694. Lakoba, T. I. : Nonintegrability of equations governing pulse propagation in dispersion-managed fibers, Preprint of Univ. of Rochester, (1999).

11.

Medvedev, S. B. and Turitsyn, S. K. : Hamilton averaging and integrability in

12.

nonlinear systems with periodically varying dispersion, JETP Lett., 69, (1999), p.465. Zakharov, V. E. : Propagation of optical pulses in nonlinear systems with varying dispersion, in Optical solton: Theoretical challenges and industrial perspectives,

(Eds.) Zakharov, V. E. and Wabnitz, S., Springer, (1999), pp.73-89.

13.

Zakharov, V. E. and Shulman, E. I. : in What is integrability?; Zakhrov, V. E.(Ed.), Springer, (1991), pp.185-249.

TDM AND WDM WITH CHIRPED SOLITONS IN OPTICAL TRANSMISSION SYSTEMS WITH DISTRIBUTED AMPLIFICATION

K. HIZANIDIS, N. EFREMIDIS AND A. STAVDAS Department of Electrical and Computer Engineering, National Technical University of Athens,

9 Iroon Polytexniou, 157 73 Athens, Greece D. J. FRANTZESKAKIS AND H. E. NISTAZAKIS

Department of Physics, University of Athens, Panepistemiopolis, 157 84 Athens, Greece AND B. A. MALOMED Department of Interdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel Abstract. The existence, formation and stability of solitons in a system of linearly coupled complex cubic Ginzburg-Landau (CGL) equations is studied in detail taking into account linear distributed gain, dispersive losses and Kerr nonlinearity in the doped component and linear losses in the passive one in a dual-core nonlinear fiber. Exact analytical chirped soliton solutions are derived and their propagation features and stability are investigated. Interaction of the soliton solutions and applications to TDM and WDM chirped soliton transmission are studied. The collision-induced delay of the interacting solitons in the WDM case is estimated as well. There is a strong evidence that, in both TDM and WDM cases, the chirped solitons are stabilized by the presence of the passive core. A possibility of generating stable chirped solitons out of unchirped ones is also demonstrated. The lumped version of this model could be the basis for a plausible optical transmission system with periodic amplification via a dual-core EDFA confined in the amplification hub. 139 A. Hasegawa (ed.). Massive WDM and TDM Soliton Transmission Systems, 139–160. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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1. Introduction

It is commonly known that solitons may exist in optical fibers in the region of anomalous dispersion [1]. The losses, which are inevitably present in the fibers, can be usually compensated by fiber amplifiers (EDFAs) [2]. However, EDFAs also give rise to various instabilities and related detrimental effects, the most dangerous one being a random jitter of the solitons induced by their interaction with optical noise which is generated by the amplifiers [3]. Schemes providing for stabilization of the transmission of periodically amplified solitons have been successfully developed, based on various techniques such as guiding (fixed- or sliding-frequency) filters [4], dispersion management [5], or combination of both [6]. A basic model that takes into account the constant group-velocity dispersion (GVD), (dispersion-management schemes are not considered here), Kerr’s nonlinearity, amplification, and fixed-frequency guiding filters is a perturbed nonlinear Schrödinger (NLS) equation for the electric field envelope function

where and are the propagation distance and reduced time, is the dispersion coefficient [1] and are respectively, the net gain and the effective filtering strength (inversely propotional to the square of the EDFA’s gain bandwidth [2]), which are uniformly distributed along the long fiber link. This formulation also applies to systems with periodic amplification if the discrete character of the amplification and filtering can be disregarded for sufficiently broad solitons (with temporal width whose period is essentially larger than the amplification spacing [3]. Upon normalizing the quantities involved becomes,

where This is the complex cubic Ginzburg-Landau (CGL) equation, which is a known model equation for the nonlinear patternformation [7]. It is well known that the CGL equation (2) has an exact pulse solution [8], which describes a stationary bright pulse with an internal chirp. It is necessary to stress that, while the unperturbed NLS equation (i.e., Eq. (2) without the right-hand side) has bright soliton solutions only in the case of anomalous dispersion the full Eq. (2) has exact pulse solutions for both signs of the dispersion. The reason is that, while for the unperturbed NLS equation, the compensation between the nonlinearity and dispersion is only possible when the dispersion is anomalous, for

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the CGL equation the nonlinearity, dispersion and filtering may all be in balance through the internal chirp of the stationary soliton. A fundamental weakness of the exact pulse solution to Eq. (2) is that it is unstable, simply because the zero solution is unstable against small perturbations This instability reflects a fundamental problem which exists in systems in which the transmission of solitons is supported by means of a distributed linear gain. However, the rate at which the instability grows and, eventually, destroys the pulse, strongly depends on the relation between the dispersion and filtering, that is, on the value and sign of the coefficient D in Eq. (2). Recently, this issue was investigated by means of direct numerical simulations in Ref. [9]. It was demonstrated that, in the case of normal dispersion, the distance of the stable propagation

(followed by an instability-induced blowup) is much larger (by a factor of up to than in the anomalous-dispersion case. This distance reaches a well-pronounced maximum at D close to an optimum value, and then gradually decreases with the further increase of inside the normal-dispersion region. Thus, quasi-stable transmission of pulses in the normal-dispersion range of the carrier wavelengths is possible, and may be actually more stable than in the anomalous-dispersion range. Besides better stability, employing the normal-dispersion range for the soliton-based communications may have a number of other obvious advantages, such as the use of the wavelength band below Nevertheless, full stability of the pulses cannot be achieved within the framework of Eq. (2). A new approach allowing one to suppress the instability of the zero state and, thus, to facilitate the generation of absolutely stable solitons, is to linearly couple the fiber to an additional parallel lossy core [10, 11]. As far as the communication systems with periodically spaced amplifiers and filters is concerned, one can add short segments of the parallel lossy core, integrating them into one unit within the amplification hub since the coupling length between the two cores is very short (typically on the order of several centimeters). For broad solitons with a period sufficiently larger than the amplification spacing, this new ingredient can then be assumed to be uniformly distributed (as well as the gain and filtering) along the communication line. This is an issue of current and future investigation which goes beyond the aim of this work. The system with the extra lossy core is bistable: it has stable states in the form of the zero solution and nontrivial soliton with uniquely determined parameters, which are separated by an unstable soliton with a smaller amplitude and larger width. It was demonstrated [12] that the stable soliton in this dual-core model can be found in an exact form. However, the analysis in Ref. [12] was rather sketchy and comprised limited parametric regions.

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In this work, a systematic analysis (in the three-dimentional parameter space of the pulse transmission in the model stabilized by means of the parallel lossy core will be presented. The character of the instability will be studied in detail along with the bifurcating analytical solutions. It will be demonstrated that absolutely stable pulses can propagate at both normal and anomalous values of the dispersion. This opens way for a very efficient use of the fiber’s bandwidth in the WDM mode [13]. It will be also demonstrated that stable chirped solitons can be tailored out of unchirped solitons lanched in a long distributed EDFA. The paper is organized as follows: In Section 2, the distributed model is formulated and estimates for the values of the parameters involved are given. The exact solitary-pulse solutions are also described in detail. In Section 3, a systematic analysis of the stability of the zero solution is performed. This provides for a necessary basis for the direct numerical analysis of the soliton’s stability presented in Section 4. The main result of Section 4 is a three-dimensional picture that shows a full stability region in the model’s parameters space. In the first part, of Section 5, the interactions between two stable solitons separated by some temporal delay (TDM) is presented. The result is that, irrespective of the initial phase difference between the pulses, they eventually merge into a single one. In the second part of Section 5 the respective behavior for the WDM case is presented. The major result here is the robust character of the propagation as compared to the TDM case and the eventual bifurcation towards asymptotic states (one is zero) for the modes involved. Finally, Section 6 summarizes the main results of this work. 2. Formulation of the Distributed Stabilized Model In the distributed model the propagation of two parallel-coupled cores is described by a system of the following form [12] (cf. Eq. (2)):

Here,

is the normalized field amplitude in the extra (passive) core with V being the electric field’s envelope function in that core) which is characterized by a loss constant, a coupling constant, K, and a phase-velocity mismatch with respect to the the active core, (all normalized to the gain g). The nonlinearity and dispersion in the additional core may be neglected since absorption and coupling are by far the major effects. The filtering inside the passive core and a possible group-velocity

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mismatch between the two cores may also also be considered as minor effects and can therefore be neglected. The system of Eqs. (3) and (4) possesses an exact analytical solution, which follows the pattern of the original solitary-wave solution to the CGL

(perturbed NLS) equation (2) [8]: Introducing the ansatz,

with F(t) being a real function, one may easily obtain the following linear relation between the amplitudes and

and a complex ordinary differential equation for F(t),

Separating the real and imaginary parts in Eq. (7) leads to the following equations which must hold simultaneously,

The compatibility conditions for the ordinary differential equations lead, after some algebraic manipulations, to relations

where is the relative phase velocity mismatch. The equation for F(t) now becomes

This equation admits the solution

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under the conditions

which leads to the Hocking Stewartson (Pereira Stenflo) [8] type of the solutions (5),

The "chirp" parameter

is given by the expression

while the amplitude and the soliton’s inverse width the following equations

are determined by

and

which always yield physically acceptable solutions under the conditions (11). Thus, an exact analytical solution in the form of a chirped solitary pulse is available in the present model. However, we will see in the following sections that stability analysis for this solution is a fairly complicated problem, which we solve numerically. To estimate relevant values of the normalized parameters kept in the model, we recall that the gain bandwidth of the EDFA, which produces the gain, is which can easily be reduced by means of the filters down to typical value for the fiber losses is so that the filtering coefficient takes values in an interval A typical value of the physical dispersion coefficient in the telecommunication fibers is [1], so that takes values within a broad interval It should also be mentioned that the dispersion coefficient of an EDFA fiber may vary from positive to negative (its value being, at the same time, quite different from the respectly value for an undoped common dispersion-shifted fiber) in a relatively narrow band measured in nanometers. This variation is due to the contribution of the resonant dispersion associated with the dopant [2]. Therefore, one may expect the possibility of significant differences between the dispersion characteristics of the active (EDFA) and passive (common dispersion shifted and lossy fiber) core. Typical values used throughout the rest of the paper are _ (reasonable for an EDFA length of about 10 to 20 Km

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which, respectively, give the values or As far as the pulse width is concerned, one can easily obtain ps and ps for (anomalous) and (normal) EDFA dispersion respectively. The normalized amplitude in all these cases is about or 6 depending upon the case. This leads to a peak intensity requirements of the order of a few hundreds mW per 3. Stability of the Zero Solution Proceeding to the stability analysis, we first of all notice that the solitary pulse (12), (13) cannot be stable unless its background, i.e., the zero solution, is stable. Recall that it is exactly an instability of the zero solution which renders unstable all the solitary-pulse solutions to the CGL equation (1). In order to investigate the stability of the zero solution, we linearize Eqs. (3) and (4), substituting into them infinitesimal perturbations

where and are the (complex) wavenumber and (real) frequency of the perturbations, respectively. Then, the stability region in the plane of the model’s parameters is determined by the condition Im which, after some algebra, yields the inequality

Evidently, this inequality holds, provided that Therefore, we need to consider Eq. (18) only for . Additionally, it is readily seen that the condition must hold for the right-hand side of Eq. (18) to remain finite at these values of In the case can be strongly simplified in some special cases, e.g., yielding

This is a known condition [10, 11, 12] which defines an area in the

plane, limited by two curves shown in Fig. and where the trivial solution is stable in the above-mentioned special cases. In what follows below, we will, chiefly, concentrate on the most straightforward case (this case, for instance, takes place when the two cores are physically identical, all the asymmetry between them being created by the fact that only in one of them the resonant dopant is pumped by an

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external source of light), although changes brought about by will also be considered. Setting we will fix D, aiming to analyze the stability condition (18) in the plane. As for the choice of the value of D, it was mentioned above [9] that, in the case of the single CGL equation (1), the propagation distance (before the onset of the soliton’s instability) took its maximum value around hence it seems natural to dwell, first of all, on this value. Upon investigating the condition (18) numerically, we have found, varying the arbitrary realfrequency that there exist a hyperbola-like curve, which further reduces the stability region of the zero solution defined by Eq. (19) (that is, the corner region I is excluded), as is shown in Fig. 1. It is noticed that in the case this numerically found curve coincides with the curve while for large values of it asymptotically

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approaches the straight line 4. The Stability of the Solitons Proceeding from the stability conditions for the zero solutions to the full stability analysis for the solitary-pulse solutions given by Eqs. (12) and (13), one should, first of all, isolate a region in the plane where these solutions actually exist. The existence condition may be readily found upon utilizing Eq. (9), which may have one or three real solutions for k, that determines the soliton’s inverse width [see Eq. (15)]. Evidently, the condition selects physical solutions of the cubic equation (9). Following this way, we have numerically found that there is an additional hyperbolalike curve (a thick solid one in Fig. 1) in the plane, below which the existence of at least one real positive value of is guaranteed. Beyond this curve, there exists (in region IV), the solutions given by Eqs. (12) and (13)

cease to exist. Thus, we have found a finite, wing-shaped region in the plane (regions II and III, Fig. 1), where the two conditions, viz., the stability of the zero solution and the existence of the exact solitary-wave one, are satisfied. However, direct numerical simulations of the solitary-pulse’s stability

demonstrate that it is really stable only in the shaded portion (region II), in the present case with (and On the contrary, in the slim region III the solitary solutions are unstable. Inside the region II, the soliton is found to be completely stable over indefinitely long propagation distances, showing no change in its shape. In Fig. 2, we display a typical example of a robust pulse in the case and (this point is located approximately in the center of the stability region (II) in Fig. 1). Inside the region I, as we have seen, the zero solution is unstable. As is illustrated by Fig. 3 for and in this case the “laminar” evolution of the pulse is eventually followed by a blowup. Comparison with the results of Ref. [9] demonstrates that the propagation distance before the onset of the blowup in the present dual-core model is about an order of magnitude larger than in the single-core one at the same values of the common parameters of the two models [9]. The further one moves away from the region II (deeper into the region I) the wider the range of frequencies that generate the instability becomes, resulting in a decrease of the propagation distance. Inside the region III, the soliton decays to zero. In Fig. 4, we display a typical example of such a case for and the soliton

propagates, initially, with small changes in its shape, but then it rapidly decays to nothing, in compliance with the fact that the zero solution is stable in this region.

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Figure 5 summarizes the stability of the solitary-wave solutions, along with the stability of the zero background, as one varies the coupling parameter K at fixed values of the other parameters, namely

and In this figure, the peak powers of the solutions are plotted vs. K. The lightly dashed curve corresponds to a root of the cubic equation (9) which formally gives this solution is, thus, unphysical. However, there are two other roots of the cubic equation which lead to positive , which correspond to existing solutions that are either stable (the solid curve), or unstable (the dashed curve). Coming back to the four regions distinguished in Fig. 1, we conclude that, inside the region I, both the zero and solitary-wave solutions are unstable. Then, as K is increased, we cross into the region II, where there exist both stable and unstable solitary-wave solutions, the zero solution being stable. In this region, the stable soliton, along with the stable zero background, act as attractors for the unstable solitary wave solution: simulations demonstrate that, in the subregion II(a), the unstable pulse evolves into the stable trivial solution, while, in the subregion II(b), it evolves into the stable pulse. Lastly, in the region III, both solitary-wave solutions are unstable and decay into the zero solution. Note that the upward and the downward arrows in Fig. 5 indicate the direction of the attraction.

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So far, we analyzed the stability of the solitary wave for fixed values and In order to investigate the change of the stability region with varying D and we note, first, that the stability region II of the solitary wave almost covers the region of stability of the zero solution (regions II and III). Thus, it would be natural to evaluate numerically the area of the stability region for the solitary wave, upon determining the boundary curves separating the regions I, II and III, IV. The results

are shown in Fig. 6, where the stability region’s area is plotted vs. D for As it can be seen, in the case the stability area is an even, parabola-like function of D that increases with As ones increases to positive values the stability area monotonically decreases, creating an interval around where all the solutions are unstable. On the other hand, if we decrease to negative values, the stability area docs not decrease monotonically. In particular, for it

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is observed that, for the stability area increases, while for negative values it almost coincides with that corresponding to Coming back to the most essential case we can collect all the data concerning the stability of the exact solitary-pulse solutions in the form of a three-dimensional picture displayed in Fig. 7. This figure shows the stability region in the parameter space from two different directions in this space. This result, showing fairly large stability regions at positive and

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negative D (i.e., anomalous and normal group-velocity dispersion in the

main core) is the main result of this work. 5. TDM and WDM with the Chirped Solitons

Having found stable solitary pulses, it is natural to study their interactions. We do this, simulating configurations defined by the following initial

condition for Eqs. (3) and (4):

Eqs. (20) and (21) represent two pulses in each core, with a temporal separation and phase difference between them. The pulses evolve

practically without changes in their shape and separation, before their collision which almost instantly takes place, apparently, as a result of attraction between them. As it is seen in a typical example displayed in Fig. 8, the interaction results in a merger of the two initial pulses into a single one, which finally relaxes into a stationary pulse, virtually coinciding with the

TDM AND WDM WITH CHIRPED SOLITONS

exact solution given by Eqs. (12) and (13). In this example, we have

153

, while the initial pulse separation and phase difference are (i.e., about and As one can see in Fig. 8, the interaction of the pulse tails in the region between the pulses generate local peaks and dips (due to the presence of the chirp in each pulse), that tend to increase along with the propagation distance. Actually, the interaction does not seem to change significantly the initial pulse shape up to the point where the two pulses begin to really interact, and finally merge into one stable pulse at As the initial separation , is increased, the collision distance is naturally found to grow exponentially, as is shown in Fig. 9. The initial phase difference does not significantly affect the interaction, the collision distance only slightly varying with Moreover, even in the case when the usual NLS solitons are well known to repel each other [1], the chirped stable pulses existing in the present model again merge into a new single pulse. The interaction between the pulses leading into their merger is, of course, a very detrimental effect for the information transfer, but, using the data presented in Fig. 9, one can easily select parameters of the solitonbased communication system so that the collision distance will be larger than the actual distance of the solitons’ propagation, hence the merger will not take place.

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In Fig. 10 the time-domain interaction of two soliton pulses (TDM) for a case with positive dispersion Figs. 10(a)–10(b)) is compared to a case with negative dispersion , Figs. 10(c)–10(d)) for both the

single-core EDFA model (Figs. 10(b)–10(d)) and the one stabilized by the extra core (Figs. 10(a)–10(c)). The temporal separation is (i.e., about and (in Figs. 10(a)–10(c)) or (in

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Figs. 10(b)–10(d)). It should be emphasized that the propagation distances shown in the Figure (as well as in the whole of this Section) are normalized to Km. Notice that, in the normal-dispersion regime, the chirp parameter given by Eq. (14) is much higher than the respective one in the anomalous dispersion regime. It is evident that the presence of high chirp weakens the interaction between the two pulses in an average sense along while low chirp cannot prevent fast “melting” of the pulses into one.

Furthermore, the presence of coupling, which (as we have shown) quaranties the stability of the background, combined with high chirp leads to large propagation distances without distortion in the pulse shape and in their temporal separation (a common drawback in the conventional TDM soliton communication senarios). The coupling also improves the transmission in the weak chirp case (in the anomalous dispersion regime for the EDFA) since it prolongs the distance of unperturbed propagation. In the WDM case, on the other hand, two modes in the active core are ineracting non-linearly through cross-phase modulation (XPM), while they are linearly coupled to their respective excitatations in the passive core. This case is therefore modeled by the following four CGL equations,

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where

is the difference in the group velocity of the two WDM modes. Let be the exact solution to the dual-core model given by (Eqs. (12) and (13); we then approximate the “vectorial” (two-channel) solution for the WDM as:

which is an exact solution in the case of zero group velocity mismatch. On the other hand, Eqs. (22) and (23) also admit another set of approximate “scallar” (one-channel) solutions

for

which are exact solutions provided that or when the same group velocity mismatch between the two modes applies for the excitations in the passive core as well. Both above-mentioned approximate solutions can be used as initial conditions (at the launching point ). In the

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following only Eq. (24) will be used as initial condition since Eqs. (25) and (26) do not describe an inreraction between the solitons in different channels. In Fig. 11 the evolution of two solitons in different channels of the WDM system is depicted for and . There is a small shift to the right (left) for the respective mode in Eq. (22) with at the very beginning. This shift constitutes a time delay between the two modes at the end of the communication line and it is, in general, small as compared to the pulse width. The solitary pulses propagate over a long distance seemingly unperturbed and, at some point, the one with an initial shift to the left passew the other. The latter pulse dies out while the former one continues to move. That is, a single pulses survives eventually. In Fig. 12, three different cases with the same and varying are presented by a diagram which provides for a clear picture of both the maximum propagation distance and the asymptotic character of the solutions for large It should be mentioned that in all three cases the pulses have the same peak power. There is, of course a small shift we mentioned before which is, however, very decisive for the survivability of the pulses. It should also be mentioned that the asymptotic value of the peak power for the modes which survive is indeed 3 times their initial value in accord with the factor Finally, in Figs. 13(a) and (b) the free propagation distance (i.e., prior to the absorption of one pulse by the other) and the time delay (temporal separation) between two WDM modes, respectively, are presented as functions of the group-velocity mismatch between the two channels for and Lastly in Fig. 14 launcing of an unchirped soliton (given by Eq. (12) with

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and its subsequent evolution during the propagation are presented. The parameters used are the typical ones, i.e., and We observe that the pulse, after a transition distance, tends asymptotically to the exact chirped soliton solution given by Eq. (12) with the chirp given by Eq. (14). This feature establishes the capability of the system (active EDFA and parallel-running passive lossy core) to produce chirped solitons out of initially unchirped ones.

6. Conclusion In this work, we have studied in detail stability of exact chirped solitarypulse solutions in a model in which stabilization of the pulses is achieved by means of an extra lossy core which is parallel-coupled to the main one. We have demonstrated that, in the model’s three-dimensional parameter space, there is a vast region where the pulses are fully stable, for both signs of the group-velocity dispersion, normal and anomalous. These results open way to a stable transmission of optical solitons in the normal-dispersion region and, thus, to an essential expansion of the bandwidth offered by the nonlinear optical fibers for telecommunications. In the cases when the pulses are unstable, we have studied in detail the development of the instability, which may end up by either a blowup or collapse to zero. The interactions between two stable solitons separated by some temporal delay (TDM) is investigated as well. The result is that, irrespective of the initial phase difference between the pulses, they eventually merge into a

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single one. In the WDM case, however, the maximum propagation distance is controlled by the group velocity mismatch between the WDM modes. By appropriately minimizing the group velocity mismatch, greater propagation distances can then be achieved as compared to the TDM case. Another interesting result is the eventual transformation of the WDM modes towards asymptotic states in which one channel is empty while the other one carries a stable chirped pulse, with the peak power 3 times as large as its value in the same channel. Another interesting feature the distributed EDFA with dispersive losses and a parallel-running lossy core is a transformation of initially unchirped solitons. They evolve, as they propagate along the line, into the stable exact chirped-soliton solution. Finally, for communication systems with periodically spaced amplifiers and filters one can envision adding short segments of an extra lossy core which is parallel-coupled to the main one inside the amplification hub. For broad solitons with the period essentially larger than the amplification spacing, this new ingredient can then be assumed to be uniformly distributed (as well as the gain and the filtering) along the communication line. This is an issue of current and future investigation which goes beyond the aim of this work. Acknowledgements

This work has been supported by the General Secretariat of Research and Technology of the Hellenic Ministry of Development (PENED-95 Grants 1242 and 644), by the Institute of Communicaions and Computer Systems of the National Technical University of Athens (ICCS–NTUA), and by the Special Research Account of the University of Athens. B.A.M. appreciates the hospitality of the Department of Physics at the University of Athens.

References 1. Agrawal, G. P. : Nonlinear Fiber Optics, Academic Press, (1995). 2. Desurvire, E. : Erbium-Doped Fiber Amplifibers, John Wiley & Sons, (1994).

3. Hasegawa, A. and Kodama, Y. : Solitons in Optical Communications, Oxford Uni4. 5. 6.

7.

versity Press, (1995). Mecozzi, A., Moores, J. D., Haus, H. A. and Lai, Y. : Soliton transmission control, Opt. Lett., 16, (1991), pp.1841-1843; Kodama, Y. and Hasegawa, A. Opt. Lett., 17, (1992), pp.1841-1843. Hasegawa, A., Kodama, Y. and A. Maruta, A. : Recent progress in dispersionmanaged soliton transmission technologies, Opt. Fiber Technol., 3, (1997), pp.197213. Matsumoto, M. : Theory of streched-pulse Transmission Dispersion-managed fibers, Opt. Lett., 22, (1997), pp.1238-1240; Malomed, B. A. : Jitter suppression guiding filters in combination with dispersion management, Opt. Lett., 23, (1998), pp.12502152; Berntson, A. and Malomed, B. A. : Dispersion management with filtering, Opt. Lett., 24, (1999), pp.507-509. Cross, M. C. and Hohenberg, P. C. : Patern formation outside of equilibrium, Rev.

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8.

Hocking, L. M. and Stewartson, K. : On the nonlinear responce of a marginally

unstable plane parallel flow to a two-dimensional disturbance, Proc. Roy. Soc. London Ser. A, 326, (1972), pp.289-313; Pereira, N. R. Stenflo, L. : Phys. Fluids, 20, (1977), pp.1733. 9. Malomed, B. A., Frantzeskakis, D. J., Nistazakis, H. E., Tsigopoulos, A. and Hizanidis, K. : Dissipative solitons under the action of third order dispersion, Phys. Rev.

E, 60, (1999), pp.3324-3331.

10. Malomed, B. A. and Winful, H. G. : Stable solitons in two-component active systems, Phys. Rev. E, 53, (1996), pp.5365. 11.

Atai, J. and Malomed, B. A. : Stability and interactions of solitons in two-component

active systems, Phys. Rev. E, 54, (1996), pp.4371-4374. 12. 13.

Atai, J. and Malomed, B. A. : Exact stable pulses in asymmetric linearly coupled Ginzbourg-Landau equations, Phys. Lett. A, 246, (1998), pp.412-422. Mollenauer, L. F., Evangelides, S. G. and Gordon, J. P. : Wavelenght division multiplexing with solitons in ultra-long distance transmission using lamped amplifiers, J. Lightwave Technol, 9, (1991), pp.362-367; Mamyshev, P. V. and Mollenauer, L. F. : Wavelenght-division-multiplexing channel energy self-equalization in a soliton

transmission line by guidings, Opt. Lett., 21, (1996), pp.1658-1660.

14. Nakazawa, M. and Kurokawa, K. : Femtosecond soliton transmission in 18 kmloiig dispersion-shifted, distributed erbium-doped fibre amplifier, IEEE J. Quantum

Electron., 27, (1991), p.1369.

LONG-HAUL DISPERSION MANAGED SOLITON WDM SYSTEMS TOWARDS TERABIT CAPACITY

K. FUKUCHI, T. ITO, Y. INADA AND T. SUZAKI C C Media Research Laboratories, NEC Corporation 4-1-1 Miyazaki, Miyamae-ku, Kawasaki 216-8555, Japan

Abstract. Long-distance WDM soliton system transmitting terabit-persecond capacity is discussed. In order to achieve cost-effective Tb/s transmission systems, the data rate for each channel was set at 20 Gb/s and the dispersion-managed soliton transmission was employed for long-haul transmission of such high-speed signals. The dense WDM transmission with 0.8 nm channel spacing has been realized by adopting the polarization interleave multiplexing, which effectively suppressed the waveform degradation induced by the cross phase modulation. To transmit all the WDM soliton signals simultaneously, we developed a wideband and dispersion-flattened transmission line. The 44 nm transmission bandwidth was achieved by use of both 1.55 and optical repeaters, and the chromatic dispersion difference was as small as 0.1 ps/nm/km by a hybrid fiber span of SMF and a dispersion-slope-compensating negative dispersion fiber (NDF). With the developed transmission line, 20 Gb/s 55-channel-WDM dispersion-managed soliton transmission was successfully conducted over 3020 km.

1. Introduction Long-distance Tb/s transmission is now an important issue for realizing world-wide large-capacity networks, and several Tb/s transmission experiments have been reported recently [l]-[3]. For such high-capacity systems in practical application, use of high bit rate signal per wavelength channel with electrical time division multiplexing (ETDM) is preferable. It will reduce the system cost, system size and complexity because the number of wavelength channels will be reduced. In order to transmit such high bit rate signals over long distance, it is effective to employ the dispersion-managed soliton transmission technique. 161 A. Hasegawa (ed.), Massive WDM and TDM Soliton Transmission Systems, 161–172. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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The performance of the dispersion-managed soliton transmission have been demonstrated in 10000 km transmission experiments at 20 Gb/s [4] and at 40 Gb/s [5], and the tolerance of the soliton system was confirmed to be large enough for practical use by an optimum system design [6]. Furthermore, there were several successful demonstrations of dispersion-managed soliton transmission in conjunction with wavelength division multiplexing (WDM) [1, 7, 8]. Especially, the soliton transmission through a highlydispersive SMF in combination with a dispersion compensating fiber (DCF) has enabled a dense WDM system, because it can effectively suppress the four wave mixing effect even in narrow wavelength separations. With the configuration, 1000 km transmission of fifty-one 20 Gb/s soliton signals (totally 1.02 Tb/s) was reported [1]. For extending the distance of the dense WDM soliton transmission, however, the cross phase modulation (XPM) effect between wavelength channels becomes the problem. The XPM induces serious signal degradation, such as timing jitter or waveform distortion. The transmission degradation due to the XPM effect depends on the channel spacing, the launch power and the signal polarization states. Especially, the channel spacing cannot be too narrow, so that the wider transmission bandwidth will be required for long-haul Tb/s transmission. Furthermore, the average chromatic dispersion of the transmission line should be uniform to satisfy the soliton condition in the entire wavelength range. In this paper, we describe the long-distance dense WDM dispersionmanaged soliton transmission system design towards Tb/s capacity. To realize a narrow channel spacing, we experimentally investigate the optimum polarization multiplexing scheme to effectively suppress the XPM effect at 20 Gb/s-based WDM soliton systems. To realize wide wavelength window for WDM dispersion managed soliton transmission, we developed wide-band hybrid optical repeater amplifiers and a widely dispersion flattened transmission fiber which consisted of SMF and negative dispersion fiber. The performance of the developed transmission line and the polarization multiplexing scheme was confirmed its feasibility in a fifty-five channel WDM soliton transmission experiment at 20 Gb/s (totally 1.1 Tb/s capacity). 2. Investigation of polarization multiplexing scheme for XPM effect suppression

In WDM soliton transmission with narrow channel spacing, the most seri-

ous issues is the fiber nonlinear effects between wavelength channel. Among the nonlinear effects, the four wave mixing (FWM) effect can be suppressed by using the high local dispersion fiber such as the standard SMF. In such

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a case, the dominant nonlinear effect is the cross phase modulation (XPM) effect. The XPM effect induces the nonlinear phase shift on the signal wavelength, and the phase shift generates the waveform distortion or timing jitter on the signal with the fiber chromatic dispersion. In case of high bit rate such as 20 Gb/s, the timing jitter is more severe because the time slot of each bit is short. One effective scheme for XPM effect suppression is to assign the orthogonal polarization states for signals that affect each other. The optical phase shift induced by the XPM effect with orthogonal polarization states is 1/3 to that when the polarization states are parallel [9]. The reduction in nonlinear effect by orthogonal polarization states is confirmed by alternate polarization modulation, with which interaction of neighboring pulses was reduced in single channel transmission [10]. In this chapter, the transmission characteristics for various polarization multiplexing schemes are investigated by a loop transmission experiment to find the optimum scheme for dense WDM [11]. Figure 1 shows the experimental setup. For the transmission line, we used the hybrid fiber span of high-dispersion SMF and negative dispersion fiber (NDF) [12] to suppress the FWM effect. Five wavelength sources were used and the center channel, in which the wavelength was 1552.1 nm, was used for measurement. Each wavelength sources were modulated to form 20 Gb/s RZ pulse signals and then multiplexed. The transmitter output waveform is shown in the inset of Fig. 1. At the transmitter output, an NDF having dispersion was placed to add the prechirp on the signal. The loop line consisted of the 10 hybrid fiber spans, each of which consisted of 20 km SMF in first half and 20 km NDF in latter half. At

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the receiver, the measurement channel was demultiplexed by the 0.35 nm bandwidth optical band-pass filter, adopted the dispersion compensation by a SMF, and then received by the 20 Gb/s electrical receiver. Figure 2 shows three types of polarization multiplexing scheme investigated in this experiment. The polarization aligned multiplexing was used

to evaluate the worst case for XPM effect impairment. The polarization optical time division multiplexing (polarization-OTDM), which is equivalent to alternate polarization modulation scheme, is effective to suppress neighboring pulse interaction in each wavelength channel. But the polarization states of neighboring wavelength channel in the same time slot are not always orthogonal because of the walk-off effect. In the polarization interleave multiplexing, on the other hand, each wavelength channel is in a single polarization state and the polarization states between neighboring wavelength channels are orthogonal. This scheme is effective to suppress XPM effect.

Figure 3(a) shows the measured transmission distance against the wavelength spacing for three multiplexing schemes. In channel spacing more than 1.0 nm, the polarization-OTDM scheme exhibited the longest transmission distance than other two schemes. This was because the neighboring pulse interaction was a dominant degradation factor in this region and the polarization OTDM suppressed it. The advantage of the polarization OTDM scheme can be also understood from the measured transmission distance of single channel experiment, which are shown by dotted- and dashed-line in Fig. 3. In the case of less than 1.0 nm channel spacing, on the other hand, the longest transmission distance was achieved by the polarization interleave multiplexing. Figures 3(b) and 3(c) shows the signal eye-patterns at 2800 km for polarization-OTDM and the polarization interleave multiplexing with channel spacing of 0.8 nm. Although the waveform distortion between pulses was suppressed with the polarization-OTDM, relatively large

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timing jitter was observed. While with polarization interleave multiplexing, the timing jitter was small, indicating the good suppression of XPM effect

induced nonlinear phase shift. Considering these results, the XPM effect is found to be dominant for transmission distance limitation in this region, and the polarization interleave multiplexing scheme is more suitable for improve the spectral efficiency in long distance transmission system. 3.

Tb/s transmission line design

3.1. CHANNEL SPACING AND LAUNCH POWER DESIGN As indicated in previous section, the XPM effect impairment can be suppressed by polarization interleave multiplexing. However, it is still severe when the channel spacing gets narrower and the launch power gets higher.

Here, we estimated the allowable launch power and channel spacing value for 20 Gb/s-based WDM transmission system. In this estimation, a transmission distance of 3000 km was assumed as our target, which covers many

parts of undersea transmission systems. Figure 4 shows the calculated available maximum launch power to give 0.5, 1.0 and 2.0 dB eye-opening penalty against the channel spacing in a 20 Gb/s-based-WDM transmission after 3000 km. In this calculation, polarization interleaved multiplexing was assumed. The optical SNR after 3000 km is also shown in the right-hand axis against the launch power, which assumed 5.2 dB repeater noise figure and the 11 dB span loss. The dotted

line shows the allowable minimum launch power. This is estimated from the optical SNR and our receiver characteristics. From these calculations, available area for channel spacing and launch power is shown as the gray area in the figure. This result indicates that there is no available power

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window for 0.4 nm spacing to satisfy both enough optical SNR and the XPM degradation suppression. Based on this design, we determined to use a launch-power and a 0.8 nm channel spacing that results in a 40 nm optical bandwidth for 1 Tb/s capacity.

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HYBRID REPEATER

To provide the 40 nm transmission window, we have developed a wideband repeater which was a hybrid optical amplifier. Figure 5 shows a diagram of the hybrid repeater. It consisted of a band splitter at input, and EDFAs, and a band combiner

at output. Both amplifiers were designed to operate at –1 dBm/ch output power and 11.3 dB gain. To get both a low noise-figure and a high output power, both EDFAs utilized a dual-stage design, wherein the first stage was pumped at 980 nm wavelength and the second stage was bi-directionally pumped at 1480 nm. In the EDFA, the gain spectrum is flattened by employing the hybrid silica-based EDF configuration composed of P/A1 co-doped and Al co-doped EDFs with a few stages of Fabry-Perot etalons as a gain equalizer (GEQ) [13]. Although the dual-stage configuration is employed, the GEQ is inserted not in the mid-stage but at the output end of each EDFA in order to improve the noise characteristics. Employing this EDFA scheme, a flat gain spectrum with 0.6 dB variation was achieved in the range 1535 to 1560 nm, and the noise figure was 5.5 dB. While in the EDFA, the gain variation was as good as 0.4

dB in the range 1570 to 1590 nm using only Al co-doped EDFs and pump power optimization. The noise figure was 5.0 dB in the 3.3. WIDELY DISPERSION MANAGED TRANSMISSION LINE

In order to transmit the WDM soliton signals simultaneously, not only the wide bandwidth and power uniformity but also the average dispersion uniformity, or in other words flat dispersion characteristic, are required because all the channel should satisfy the soliton condition. For this purpose, we have developed a widely dispersion flattened transmission fiber. The

transmission fiber was a combination of a single mode fiber (SMF) and a negative dispersion fiber (NDF). The SMF had a pure-silica core for low loss characteristics. The NDF [12] was designed both to compensate the dispersion at and to minimize the loss in the two bands. The residual dispersion slope in the was compensated by a large positive-dispersion-slope fiber placed inside the repeater. Each span consisted of the SMF in the first half and the NDF in the second half. The span length was 45.7 km. Figure 6 shows the averaged loss and dispersion for developed 8 fiber spans. The loss difference between 1550 nm and 1590 nm was 0.02 dB/km. This loss increase resulted in an opticalSNR decrease of only 0.9 dB after 3000 km. The average dispersion was successfully flattened at in the and in the The 0.1 ps/nm/km difference in dispersion was small enough for soliton transmission at all wavelengths in

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these bands with the aid of prechirp optimization. 4. Long-distance 1.1 Tb/s transmission experiment

With the developed repeater amplifiers and the transmission fiber spans, we have conducted a 1.1 Tb/s transmission experiment. Figure 7 shows

the experimental setup. In the we used 30 DFB–LDs, which are ranging from 1536 to 1559.2 nm. In the 25 DFB–LDs were used which were ranging from 1572.4 to 1591.6 nm. The wavelength separation was 0.8 nm. For both bands, even and odd channels were multi-

plexed separately and modulated to form a 20 Gb/s RZ signal. The pulse width was set at 25 ps. To suppress the XPM effect, polarization interleave multiplexing was employed by coupling the modulated lights by polarization beam splitters in both bands. The prechirp value, added by an NDF at the end of transmitter, was optimized for each wavelength. The loop transmission line consisted of 8 spans of the dispersion-flattened fiber, and 2 extra spans having 50 km span length were placed at the loop output. Inside the loop, extra repeaters, which involved gain equalizer and dispersion slope compensation fiber, were placed after every 4 spans. At the receiver end, the measurement channel was extracted by an 0.8 nm spaced AWG demultiplexer and was received by a 20 Gb/s electrical receiver.

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Figure 8 shows the optical spectrum after 3020 km transmission. By adopting the precise gain equalization, a 24 nm bandwidth in the band and 20 nm bandwidth in the were achieved after 82

repeaters. The power difference over the 44 nm window was as small as 2.5 dB. Figure 9 shows the bit error rate measured at 3020 km. For all 55 channels, the bit error rates were less than Inside the Fig. 9, the eye

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diagrams for channel 1, 23, 37 and 54 are also shown. The eye diagrams were similar, indicating the uniformity of the soliton transmission each wavelength. The XPM-effect-induced timing jitter in the eye-diagrams were small by the polarization interleave multiplexing and the optimum setting of launch power and the channel spacing. From the results of measured bit error rate and eye diagrams, the flatness of the gain profile and dispersion in the developed transmission line was uniform enough to transmit 20 Gb/s dispersion managed soliton signals. 5. Conclusion In conclusion, a long-distance 20 Gb/s-based-WDM dispersion-managed soliton system transmitting Tb/s capacity is experimentally investigated. For the long distance Tb/s transmission, it is important not only to reduce

channel spacing, but also to extend of available optical window. For narrow channel spacing transmission, various polarization multiplexing schemes

were investigated, and the polarization interleave multiplexing was found to be the most suitable in our system. It is because the polarization in-

terleave multiplexing effectively suppressed the waveform degradation induced by the cross phase modulation, which is a dominant factor in the

dense WDM case. For extending the optical transmission bandwidth, on the other hand, we have developed the

hybrid optical re-

peaters, which exhibited the 44 nm bandwidth. We have also fabricated the dispersion flattened transmission fiber span, which consisted of puresilica-core SMF and an NDF, in order to satisfy the soliton condition for

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all WDM channels over that wide bandwidth. With the fabricated transmission line, 55-channel-WDM dispersion-managed soliton transmission at 20 Gb/s was successfully conducted over 3020 km. These results indicate the possibility of long-haul Tb/s transmission systems with high bit rate signal in each wavelength, which will lead to a cost-effective Tb/s system realization. Acknowledgements

This work is supported by Telecommunications Advancement Organization of Japan (TAO). The 1.1 Tb/s transmission experiment (in Sections

3 and 4) was a joint work of NEC Corporation, Sumitomo Electric Industries Inc.(SEI), Oki Electric Company Ltd., Osaka University and Kochi University of Technology. The author would like to express their thanks to Prof. A. Hasegawa from Kochi University of technology, Drs. S. Goto, M. Shikada and K. Emura from NEC, M. Nishimura and S. Tanaka from SEI, Y. Ozeki from Oki for their continuous encouragement and project management. They also thank to M. Kakui, T. Tsuzaki and T. Shitomi from SEI, A. Sasaki, K. Fujii, S. Shikii, A. Pratt and N. Kitahara from Oki for providing the repeater amplifiers. They would be grateful to Prof. M. Matsumoto, Drs. A. Maruta, H. Toda and H. Sugawara from Osaka University and T. Ono, Y. Yano, G. C. Gupta and M. Morie from NEC for supporting the system design and help of experiment. They also thank to Y. Aoki and T. Ogata from NEC for their support in the polarization multiplexing scheme evaluation experiments (in Section 2). References 1.

Guen, D. L., Burgo, S. D., Moulinard, M. L., Grot, D., Henry, M., Fevre, F. and Georges, T. : Narrow band 1.02 Tbit/s (51 20 Gbit/s) soliton DWDM transmission

over 1000 km of standard fiber with 100 km amplifier spacing, OFC’99 Tech. Digest, PD4, (1999). Naito, T., Shimojoh, N., Tanaka, T., Nakamoto, H., Doi, M., Ueki, T. and Suyama, M. : 1 Terabit/s WDM Transmission over 10,000 km, ECOC’99, PD2-1, (1999). 3. Tsuritani, T., Takeda, N., Imai, K., Tanaka, K., Agata, A., Morita, I., Yarnauchi, H., Edagawa, N. and Suzuki, M. : 1 Tb/s (100 10.7 Gbit/s) Transoceanic Transmission Using 30 nm-Wide Broadband Optical Repeaters with Positive Dispersion Fiber and Slope-Compensating DCF, ECOC’99, PD2-8, (1999). 4. Morita, L, Suzuki, M., Edagawa, N., Tanaka, K. and Yamamoto, S. : Performance improvement by initial phase modulation in 20 Gbit/s soliton-based RZ transmission with periodic dispersion compensation, Electron. Lett., 33, (1997), pp.1021-1022.

2.

5. Morita, I., Tanaka, K., Edagawa, N. and Suzuki, M. : 40 Gb/s single-channel soliton transmission over 10,200 km without active inline transmission control, ECOC’98, Vol.3, (1998), pp.49-51. 6. Gupta, G. C., Fukuchi, K., Inada, Y. and Suzaki, T. : Experimental demonstration

of highly tolerable dispersion-managed soliton system designed for trans-oceanic distance transmission, OFC’99 Tech. Digest, WC5, (1999).

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7. Morita, I., Suzuki, M., Edagawa, N., Tanaka, K. and Yamamoto, S. : Long-haul soliton WDM transmission with periodic dispersion compensation and dispersion slope

compensation, in A. Hasegawa (Ed.), New Trends in Optical Soliton Transmission Systems, Kluwer Academic Publishers, (1997), pp.355-365. 8. Tanaka, K., Morita, I., Suzuki, M., Edagawa, N. and Yamamoto, S. : 400 Gbit/s (20×20 Gbit/s) dense WDM transmission using soliton-based RZ signals, ECOC’98, Vol.1, (1998), pp.85-86. 9.

Agrawal, G. P. : Nonlinear Fiber Optics, Academic Press, (1995), Chap. 7.

10.

Ito, T., Inada, Y., Fukuchi, K. and Suzaki, T. : Suppression of nonlinear waveform

11.

distortion using alternate polarization modulation for long distance 10 Gb/s based WDM transmission systems, OFC’99 Tech. Digest, ThQ7, (1999). Inada, Y., Fukuchi, K., Suzaki, T. and Aoki, Y. : Suppression of XPM effect by

12. 13.

polarization interleaved multiplexing in long-distance 20 Gbit/s-based dense WDM transmission, ECOC’99, II-WeC3.5, (1999), pp.140-141. Mukasa, K., Akasaka, Y., Suzuki, Y. and Kamiya, T. : Novel network fiber to manage dispersion at with combination of zero dispersion single mode fiber, Proc. ECOC’97, Vol.1, (1997), pp.127-130. Kakui, M., Kashiwada, T., Shigematsu, M, Onishi, M. and Nishimura, M. : Gainflattened hybrid silica-based Er-doped fiber amplifiers designed for more than 25 nm optical bandwidth, IEICE Trans. Electron., Vol.E81-C, No.8, (1998), pp.12851292.

SPECTRAL EFFICIENCY

IN WDM SOLITON TRANSMISSIONS

S. WABNITZ, B. BIOTTEAU, P. BRINDEL, B. DANY, O. LECLERC, P. LE LOUREC, F. NEDDAM, D. ROUVILLAIN AND J. L. BEYLAT Alcatel CRC

Route de Nozay, 91460 Marcoussis, France AND E. PINCEMIN France Telecom CNET/DTD/RTO, 2 Av. P. Marzin, 22307 Lannion Cedex, France. Abstract. Dense wavelength division multiplexing for submarine transmission applications is subject to nonlinear limitations such as intra-channel and inter-channel cross-phase modulation and interactions. We discuss novel strategies to optimize the spectral efficiency of the transmission by properly tailoring either the dispersion map in passive systems or the controlling

filter and synchronous modulator in regenerated systems.

1. Introduction

In this work, we present recent research developments at Alcatel Corporate Research Center on submarine fiber-optic transmission systems that combine the techniques of dispersion management, wavelength multiplexing, and self-phase modulation compensation by means of the soliton signal format. As widely known, dispersion management permits to effectively suppress both the noise-induced timing jitter [1] and four wave mixing products in WDM systems [2]-[9]. Whenever the return-to-zero (RZ) or dispersion management (DM) soliton format arc used for the signal pulses, the main nonlinear limitation to the transmission capacity (besides amplifier noise accumulation) is given by cross-phase modulation (XPM) that results from the collisions of pulses in adjacent, closely spaced wavelength channels [10]. In the first part of this work, we analyse how the dispersion map may be op173 A. Hasegawa (ed.), Massive WDM and TDM Soliton Transmission Systems, 173–193. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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timised to reduce the timing jitters that are observed in a dense wavelength division multiplexed (DWDM) transmission system, mainly as a result of

partial, uncompensated collisions which occur at the beginning and at the end of the link. The second part of the paper is dedicated to describe an alternative approach to solving the nonlinear capacity limitations by means of a combination of active and passive soliton control in both frequency (filtering) and time (synchronous modulation) domain, combined with periodic de/multiplexing of the WDM channels. In fact, one may include in the regenerator either a phase modulator or a nonlinear gain/pulse reshaping element, which in combination with filtering and intensity modulation stabilizes amplitude, frequency and timing fluctuations of the DM solitons. Although soliton-based systems permit the highest single-channel capacities in optical fibers, they are prone to instabilities in WDM systems owing to the periodic amplification which leads to asymmetric collisions [7. 10, 11]. These collisions result in permanent frequency shifts and translate into potentially large XPM-induced timing jitter at the system output.

One way to understand the dynamics of a WDM soliton collision is by looking at the Fourier spectrum of the whole field as two NLS solitons cross each other. By doing that, one observes the generation of several four-wave mix-

ing sidebands at the collision midpoint. These sidebands are subsequently converted back into the individual spectra of the two solitons once the collision is completed. If the symmetry of the soliton power in the first and

second half of the collision is broken (e.g., whenever the collision is centered at an amplifier), a set of FWM sidebands results from the collision, in addition to the permanent frequency and timing shift of the main pulses. These waves may interfere with other channels or may be amplified in the presence of controlling filters, thus leading to significant signal deterioration. It is quite clear, from one hand, that dispersion management permits to effectively suppress FWM sidebands. As a result, one may expect that the effect of asymmetric collisions will be strongly reduced by using this technique. Moreover, resonances between the amplifier spacing and the col-

lision length for closely spaced channels may also be removed by the DM. Indeed, a relatively slow collision which occurs in an uniform GVD fiber link is replaced by a series of short-scale non-resonant collisions in a DM link [6, 7],[12]-[16].

On the other hand, especially in the case of strong DM, it is clear that periodic dispersion variations introduce a large periodic perturbation to the pulse propagation, which in turn may lead to the onset of novel propagation instabilities. Moreover, even if each individual fast collision is non-resonant, the accumulation of several collisions may still yield a significant residual frequency shift if a proper balance is lacking within the whole set of collisions [13]. In this work we shall discuss how to reduce the

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impact of XPM and resonant collisions in DWDM long-haul transmissions

either by a proper design of the dispersion map in passive systems, or by a proper design of the soliton control in regenerated systems. 2. Dispersion Maps for N×20 Gbit/s Transmissions The dispersion management of return-to-zero (RZ) pulses permits to compensate for self-phase modulation (SPM) through the soliton effect, notably

at higher pulse energies than in the case of uniform GVD links [17]. Indeed, recent experiments have demonstrated the potential of DWDM transmissions for Tbit/s capacities over a few thousands of km [18, 19]. Further capacity improvements with the DM technique may require, from the one hand, the availability of wider transmission bands, which implies the com-

pensation of both second and third-order dispersion through the so-called higher-order DM (henceforth HODM) technique [20]. Moreover, the spectral efficiency of the transmission may be improved either by decreasing the channel spacing or by increasing the channel bit-rate (for a given channel spacing). The single channel rate is basically limited by interactions among adjacent pulses [21], which in turn may be reduced by means of polarization optical time division multiplexing (OTDM). Moreover, as we

discussed above, in DWDM transmissions a major source of impairment is set by collision-induced timing jitter [22]. We will analyse here the impact of intra-channel interactions and of XPM-induced jitter in a HODM. The optimisation parameter, which as we shall sec permits to control the XPM, is the span-averaged dispersion (or SAD) in each basic SMF/RDF cell, which may be tuned for a given

value of the PAD. Figure 1 shows the basic loop of a periodic HODM link, which consists of five 34 km spans of single-mode fiber (SMF) and reverse-

dispersion fiber (RDF), plus a single SMF span to adjust the PAD . The

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lengths of the SMF (X km) and RDF

in each basic cell fix the

SAD. The final SMF of length Y km is set to reduce the path-averaged dispersion (PAD) to relatively low values. The above map structure allows to control the XPM by means of tuning the intrinsic delay between two neighboring WDM channels (Fig. 2). At the mean wavelength of we took the GVDs and The effective areas and the losses of the two fibers are for the SMF and RDF, respectively. The nonlinear fiber indexes have the same value of and in each case third-order dispersion was compensated for at each loop. Figure 3 displays the SAD-dependence of the temporal width of the DM soliton solution in the HODM, for different values of the signal power (at

20 Gbit/s) at the amplifier output. Here the anomalous PAD is As can be seen, the DM soliton time duration grows larger as the SAD decreases: for a power of the chirp-free pulse width grows from 30 ps to 46 ps as the SAD decreases from to On the other hand, Fig. 4 illustrates the evolution over a DM period of the DM soliton parameters (peak power, chirp, bandwidth and time width), as it is obtained from the variational approach [21, 24] for a relatively small SAD of Here and in the following,

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the average signal power at the amplifier output is

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For the sake

of clarity, we removed from the plot the periodic evolution of pulse energy

due to fiber loss. To reduce the prechirp, the input pulse was injected after a single SMF/RDF span. The corresponding evolution of the DM soliton parameters in the case of a relatively large SAD is illustrated in Fig. 5. A main source of nonlinear limitation to the maximum transmission distance is the intra-channel interaction (or collision) between adjacent DM pulses. Indeed, Fig. 6 shows the dependence of this distance for 20 Gbit/s DM solitons as a function of the SAD, for parallel and orthogonal adjacent pulses, respectively. For orthogonal pulses, the output pulse chirp is compensated for by means of a dispersion grating. As can be seen, interactions are strongly reduced by polarization OTDM: in this case, the maximum distance remains above 4 Mm for all values of the SAD. Whereas with parallel pulses the interactions reduce the transmission distance below 3 Mm as soon as the SAD decreases below The increase of interactions which is observed for large negative SAD is for the most part due to the pulse width increase of the DM solitons (see Fig. 3). Another major source of nonlinear impairment for DWDM systems is the inter-channel XPM. Indeed, collisions of pulses in adjacent channels

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lead to significant timing jitter at the system output [22]. Figure 7 illustrates the timing shift at 3.5 Mm (as seen at a chirp-free point) that a DM soliton experiences upon collision with a pulse in the adjacent channel

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(for 50 GHz channel spacing), as a function of the initial time separation, and for different SAD values. As can be seen from the dot-dashed curve in Fig. 7, for small SAD (here the peak timing shifts are obtained for a delay between the pulses of either zero or 150 ps: these values correspond to pulses which are fully overlapping either at the system input or at its end at 3.5 km (for a PAD This confirms the statement that the most detrimental effect of XPM is provided by the half collisions that occur for pulses that are in-phase in the two adjacent channels at either the input or the end of the link. Moreover, Fig. 7 shows that decreasing the SAD below displaces the positions, and nearly halves the peak value of the XPM-induced time shifts. Indeed, a SAD of introduces a delay of 55 ps between the two colliding pulses after each cell. Recall that each half collision leads to either a blue or a red shift into each of the colliding pulses [12, 13, 19]; moreover, owing to pulse attenuation due to fiber loss, the accumulated frequency shift results from the portion of the collision that is situated after the amplifier. As a consequence, whenever the intrinsic span delay is of one pulse width or longer,

two initially overlapped pulses that experience a half collision in the first cell will undergo a full collision in the next cell. Hence the colliding pulse sees subsequent frequency shifts of opposite sign, which tend to cancel out. In Fig. 7, the timing shifs were computed by using the variational approach for DM soliton collisions [21], which reproduces well the numerical results. So far we considered the interaction between two individual bits in the

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same or in different channels: in order to assess the overall impact, of XPM, we applied a statistical approach to evaluate the variance of the total timing jitter resulting from all possible collisions over the given system length L.

After evaluating the timing shift in channel A due to a collision with a pulse in channel B, as a function of their initial time separation, one should calculate the total collision-induced square timing shift by averaging over all possible sequences. The procedure employed to calculate this timing shift was recently outlined in a paper by Sugahara et al. [22]. Let us recall the

basic steps that permit one to estimate the XPM-induced timing jitter [22]. Denote with the timing shift of a soliton pulse at the system output, resulting from a single collision at the position . Since the decision at receiver is based on regenerated clock, it is the relative timing

shift between subsequent bits in a given channel (and not the absolute timing shift of an individual pulse) that determines the detection errors. The relative timing shift between pulses at slot, say, 0 and at slot k in

channel A (owing to collisions with pulses at slot 0 and slot k in channel B. respectively), reads as

As a result of all the N collisions over the link of length L, the total relative

timing shift between pulses at slot 0 and at slot k in channel A is

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The resulting variance of the timing shift reads as

One finally obtains the total average timing shift as

Figures 8 and 9 show the calculated total variance of the XPM-induced timing shift distribution as a function of distance, for a transmission of five channels with 0.4 nm spacing, different values of the SAD, and PADs of or respectively. The above figures confirm that, by decreasing the SAD below one strongly reduces the XPM jitter, owing to the progressive decorrelation of the most harmful initially in-phase pulse sequences in adjacent channels. Note that increasing the PAD above zero also permits to reduce the XPM jitter. The optimal SAD of corresponds to just above one pulse width of intrinsic delay at each SMF/RDF span between the pulses spaced 0.4 nm apart.

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On the other hand, Fig. 10 illustrates the effect of changing the signal power and PAD d on the XPM timing jitter. As can be seen, a value of

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yields a larger reduction of jitter with respect to when the power is decreased from 0 dBm down to On the other hand, for a fixed power level, the optimal PAD varies with the power. The post-compensation of the residual pulse chirp at the end of each loop may also have a different effect, depending on the sign of the pulse chirp. Indeed, Fig. 11 shows that the XPM jitter at the chirp-free point (solid curves) is increased or reduced with respect to the jitter measured at the end of span without post-compensation (dashed curves), depending on the sign of the pulse chirp. In fact, the jitter is reduced (increased) by postcompensation whenever the pulse chirp is positive (negative), respectively

(see Figs. 4 5). We validated the above results by means of full numerical simulations using the Alcatel OCEAN beam propagation code, where all nonlinear effects are simultaneously taken into account. Amplifier noise was not included in the simulations in order to single out the role of nonlinear effects. Figure 12 shows the Q factor (after electrical filtering) as a function of the SAD, as it is calculated at 2 and 3 Mm, respectively. The power in each of the five channels was fixed to whereas optimal values for the pre-chirp and post-chirp were numerically adjusted; the chirp-free input pulse width was set to 30 ps in all cases. As can be seen, the system performance is strongly improved by decreasing the SAD down to The further deterioration that is observed for smaller values of SAD is ascribed

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to intra-channel interactions and to the fact that an input pulse with a 30 ps chirp-free duration no longer matches a DM soliton. Indeed, DM soli-

ton pulses with chirp-free pulse widths above 40 ps cannot be used in a 20 Gbit/s system as their broad tails would lead to an unsustainable degradation of the extinction ratio. Finally, Fig. 13 shows the evolution with distance of the Q factor for selected channels as it was calculated for the optimal case of 140 ps/nm delay per span in a transmission; these simulations, which include the accumulated amplifier noise, show that error-free transmission distances of 3 Mm are possible with 50 GHz spacing, as long as the SMF/RDF dispersion map is properly designed.

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3. Regenerated N×40 Gbit / s Soliton Transmissions

In the previous section, we have seen how to increase the efficiency of the DWDM by means of a control of the XPM and pulse interactions through the map design. Another approach to reduce the nonlinear degradations to the pulse propagation is to include in-line control elements such as asynchronous guiding filters or nonlinear gain elements, or synchronous amplitude or phase modulators [25]-[28]. The all-optical “3R” (reamplification, reshaping and retiming) signal regeneration which is permitted by these control elements is the more essential as the hit rate per channel increases

above 20 Gbit/s, whenever error-free transoceanic transmission distances are required. In this section, we discuss some recent developments in the optimization of the in-line control of DM solitons, with the scope of achieving spectrally efficient transoceanic transmissions. The control-induced improvement to the propagation stability of DM solitons is required for the following reasons. In spite of the fact that DM solitons exhibit a power enhancement with respect to NLS solitons, for a

given average GVD or chirp-free time width, still the average dispersion in DM systems is very small, which entails that the powers involved always remains relatively low. Moreover, as we have seen, the low-power requirement is also necessary for DM solitons in order to reduce the inter-pulse nonlinear effects, namely interactions and XPM. These effects are particularly troublesome in DM systems owing to the periodic (hence strongly non-integrable) nature of the dispersion map. All in all, one may view the

propagation of a DM soliton pulse as close to that of a linear pulse, or, at least, it is in this regime that competing nonlinear effects are best suppressed and thus DM solitons work better. This entails that the control of DM solitons (e.g., by means of in-line filters) is inherently difficult, because the restoring forces that act on the pulse rest on the nonlinear or particlelike nature of the soliton, and are thus less effective in the quasi-linear limit [29]-[32]. To counter these drawbacks, the solution is inserting extra lumped non-

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linear elements such as for example a nonlinear gain element [33]-[36] or an amplifier and a nonlinear fiber, which introduce back nonlinearity in the control bit of the transmission. As a result (see Fig. 14), the most effective strategy for stabilizing DM solitons consists in decoupling the propagation from the control: while the first should be as linear as possible, the second should be as nonlinear as possible [37]-[39]. Note that the dynamics of the pulse parameters of a DM soliton within each basic cell makes the in-line control problem more complex than for the case of nonlinear Schrödinger

(NLS) solitons, where the only fast variation is the linear amplitude oscillation. For NLS solitons, the action of lumped control elements may be effectively distributed along the link, and simple averaged equations for the solitons parameters are sufficient to predict pulse stability. On the other

hand, for DM solitons the precise positioning of the control elements in the map is crucial for the stability of the control, as it has been shown to be the case for control filters and modulators [30]-[32], [40]. This sensitivity may pose problems from a practical point of view, since in a system it may be

difficult to have access to a precise point in the dispersion map for inserting the control elements. Whereas the strategy that has been pursued here is that of fixing the position of a block of control elements at the chirpfree position in the dispersion map, and to finely tune the parameters of a

proper combination of control elements and extra dispersive fibers in the block, in order to optimize the strength of the control of the DM soliton [37]-[39],[41]. Note that the choice of using chirp-free pulses at the regenerator input appears as the most natural, since the minimum pulsewidth permits the best match with the synchronous modulation and to minimize dispersive wave generation. Figure 15 shows two such examples of control blocks. On the left, a phase modulator (PM) followed by an amplifier and a highly dispersive fiber (HDF) lead to soliton re-timing [37]-[39]: indeed, the phase modulator acts as a temporal lens that focuses the pulses back to the center of the time slot (which is determined by a clock-recovery circuit). Next, the pulse amplitude and time width are controlled by a filter (F) and an intensity modulator (IM). On the right of Fig. 14, we show a simpler design of the control block where a single clock recovery circuit is present [41, 42]:

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here the amplifier and the HDF behave as a sort of fast saturable absorber (or “2R” regenerator): in combination with the bandpass filter, a nonlinear filtering action results before the modulator. In the regenerator block of Fig. 14, the amplifier is followed by a sample of nonlinear, dispersive fiber which permits to generate a NLS soliton whenever a certain critical amplitude of the DM soliton (of fixed time width). If the initial DM amplitude fluctuates about the critical value, dispersive waves are generated and subsequently cut by the modulator/filter combination, hence the overall losses may be dramatically increased. The regenerating is completed by the intensity/phase modulator for pulse re-timing. Note also that the above described device is compatible with multiple wavelength operation: as we shall see, nonlinear regeneration is effective in suppressing the frequency and time position shifts that result from XPM through incomplete collisions. To clarify the stabilizing action of the nonlinear (as opposed to linear) filter control of DM solitons, let us compare the evolution of a perturbed DM pulse through the a series of dispersion maps and regeneration cells (cf. Fig. 16). We consider here the restoring force of the combination of modulator and filters on the fluctuations of energy and arrival time of a single DM pulse. The dispersion map involves 40 km of DSF with a GVD of . and about 1 km of DCF with a GVD of the average dispersion was kept as low as at 1550 nm. In the absence of in-line control, steady propagation of a DM soliton was obtained for pulse width, pre-chirp and injected power of 7.5 ps, and respectively. For a comparison, in the simulations we inserted (at the chirp-free point in the DCF span) once every two DM spans (i.e., the regenerator spacing is 80 km) a “quasi-linear” optical regeneration block composed of a Gaussian

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filter and an ideal intensity modulator. An input time shift was imposed by applying a given input delay to the pulse with respect to the center of its bit

slot. The dashed curves in Fig. 16 show the corresponding evolution of the pulse position and energy with propagation distance: As can be seen, the pulse center and power fluctuations arc slowly damped by the quasilinear regenerator. Indeed, as well known, IM cancels the time jitter since a pulse experiences time-dependent losses at, the modulator which lead to a pullback force to the center of the bit slot. However, the very mechanism that reduces the jitter in turn introduces pulse energy fluctuations. Including the relatively weak restoring force of the filters permits a gradual energy stabilization of the pulses. Indeed, for NLS solitons, the energy stabilization by means of filters is achieved thanks to the strict relationship between pulse energy and spectral bandwidth: any increase (decrease) of the soliton energy leads to a corresponding increase (decrease) of its spectral bandwidth, which leads to larger (reduced) filtering loss, and eventually the initial energy increase (decrease) is reversed and a negative feedback action results which ensures pulse stability. Unfortunately, DM solitons exhibit such a loose; relationship between spectral width and pulse energy that very large energy variations amount to only slight changes of the pulse bandwidth, so that the restoring force of the niters is much reduced. On the other hand, the solid curves in Fig. 16 show that a comparatively hard damping of pulse timing and energy fluctuations is observed by inserting after the amplifier a 2.4 km long span of HDF (with a GVD of ). Figure 16 clearly shows that the pulse energy returns to the steady-state values much more rapidly whenever the nonlinear filter is applied.

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The performance of a nonlinear regenerator for the control of DM solitons was validated in a four-span 160 km loop experiment using one regenerator per loop (i.e., every four DM spans) [41]. We limit ourselves to consider here the case of single channel transmissions only. Recent experiments have also demonstrated the good performance of this strategy for Gbit/s transmissions over transoceanic distances [42]. As shown in Fig. 17, we compared the action of a quasi-linear regenerator (left) with that of a nonlinear regenerator (right). Pulses at 10 GHz repetition rate from gain-switched DFB laser were compressed down to 6 ps by means of a Bragg grating, coded with a PRBS at 10 Gbit/s and time multiplexed to provide the 40 Gbit/s data stream. The dispersion map comprised four spans of 40 km DSF (with a GVD of ) and three spans

of DCF with a GVD of plus a final DCF with the GVD of The regenerator was placed at a chirp-free point located within this last DCF. Additional pre and post-compensation via a SMF was added before and after the regenerator, to finely tune the pre and post-chirp of the pulses entering and leaving the regenerator, respectively. The resulting average dispersion (not including the regenerator, which is considered as a pulse reshaping element and thus in a first approximation may be decoupled from the propagation) was then adjusted to an estimated anomalous value of The noise figure of the four 980 nm pumped in-line amplifiers was of 4.7 dB, and the input power that was launched into each span was of 3.75 dBm. The nonlinear regenerator included an EDFA delivering 16.4 dBm into a 3.5 km HDF with the constant GVD of The output pulse power into the HDF corresponds to the one-soliton solution of the NLS equation for a time width of 5.5 ps. Since the amplitude and width fluctuations are strongly coupled for NLS solitons, any DM soliton amplitude fluctuation before the EDFA will translate into both energy scattering into dispersive waves and time width fluctuations (hence varying losses from the filter) at the HDF output end. The modulator was an InP polarization insensitive Mach-Zehnder interferometer, with adjustable IM/PM response. Figure 17 shows the eye diagrams (before and after the 40 Gbit/s demultiplexing to 10 Gbit/s) with quasi-linear (left) or nonlinear (right) regeneration. As can be seen, in the first case the maximum error-free distance was of just 1280 km, whereas with nonlinear filters transmissions over distances exceeding 10 Mm were readily achieved. Indeed, a steady-state evolution of the transmission was observed as early as 2.5 Mm, with an asymptotic value of the quality factor Note that the optimal filter bandwidth could be decreased from 2.2 nm down to 0.7 nm when passing from a quasilinear to a nonlinear regenerator, thus showing that nonlinear filtering permits to substantially improve the filter strength without incurring into the amplification of background

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noise. Nonlinear regeneration is compatible with DWDM, at least whenever the channels are de/multiplexed before and after each regenerator stage [39]. Figure 18 shows the results of a numerical study of the performance of a regenerated transoceanic Gbit/s transmission system, where the renegerator contains an initial phase modulator as shown in the left side of Fig. 15. The WDM channel spacing was set to 1.6 nm, and we considered a 128 bits PRBS sequence of 7 ps pulses. In the simulations, we compared the transmission performance without regeneration (dashed curve) with that including nonlinear regenerators (set of grey and black curves). On Fig. 18 we plot the evolution with transmission distance of the Q-factors for all of the 16 WDM channels. As it can be seen, with nonlinear regeneration the Q-factors reach stable asymptotic values greater than 6 at 10000 km. Clearly this is not the case without regeneration, where the maximum error-free distance is limited to less than 1000 km. The observed asymptotic stabilization of the Q-factors with distance is ascribed to the efficient control of the pulse energy and jitter which is provided by the combination of DM/NLS soliton conversion and filtering/modulation in the regeneration block. Finally, in Fig. 19 we plot the maximum transmission distance that is achievable with nonlinear regeneration as a function of the channel spacing: for spacings larger than 1.3 nm (160 GHz), error-free transmissions above 6 Mm are possible. Similar results, with an even better asymptotic decay of the performance of all channels, are obtained when simulating DWDM transmissions with a nonlinear filter regenerator as in the right-hand side

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We envisage that even higher spectral efficiencies will be possible for regenerated Gbit/s transmissions by combining with the proper nonlin-

ear regenerator design the dispersion map design optimization as discussed in the first section of this work. Further work is under way in this direction. Moreover, a breakthough could be achieved towards the practical feasibility and spectral efficiency of DM soliton regenerated systems whenever it will

be possible to control a bunch of channels by means of the same regenerator block. 4. Conclusions

In conclusion, we examined the role of pulse interactions in higher-order DM soliton-based WDM transmissions, and found that intra-channel interactions may be neglected using polarization multiplexing. Whereas interchannel XPM may be strongly reduced by properly tailoring the local time delay between adjacent channels. We have also shown that for channel bit

rates of 40 Gbit/s, error-free transoceanic distances may be bridged by controlling amplitude and timing fluctuations of the signal pulses by means of properly designed nonlinear regenerators.

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Acknowledgement

We gratefully acknowledge E. Desurvire for his support and contribution to the research activity described in this work.

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30. Matsumoto, M. : Opt. Lett., Vol.23, (1998), p.901. 31. Matsumoto, M. : J. Opt. Soc. Am. B, 15, (1998), p.2831. 32. Merlaud, F. and Georges, T. : ECOC’98, pp.497-498. 33. Kodama, Y., Romagnoli, M. and Wabnitz, S. : Electron. Lett., Vol.28, (1992), p.981. 34. Govan, D. S., Smith, N. J., Knox, F. M. and Doran, N. J. : J. Opt. Soc. Am. D, Vol.14, (1997), p.2960. 35. Hirooka, T. and S. Wabnitz, S. : Electron. Lett., Vol.35, (1999), p.55; Hirooka, T. and Wabnitz, S. : Opt. Fiber Techn., in press, (2000). 36. Harper, P., Penketh, I. S., Alleston, S. B., Bennion, I. and Doran, N. J. : Electron. Lett., Vol.34, (1998), p.1997. 37. Brindel, P., Dany, P., Leclerc, O.and Desurvire, E. : Electron. Lett., 35, (1999), p.480. 38.

39. 40. 41. 42.

43.

Dany, B., Brindel, P., Leclerc, O. and Desurvire, E. : Electron. Lett., 35, (1999),

p.418. Dany, B., Brindel, P., Leclerc, O. and Desurvire, E. : ECOC’99, Tu C1.6. Tonello, A. Capobianco, A., Wabnitz, S. and Turitsyn, S. : Optics Communications, in press, (2000). Brindel, P., Leclerc, O., Rouvillain, D., Dany. B., Desurvire, E. and Nouchi, P. : Electron. Lett., 36, (2000), p.61. Leclerc, O., Brindel, P., Rouvillain, D., Pincemin, E., Dany, B., Desurvire, E., Duchet, C., Shen, A., Blache, F., Grard, E., Coquelin, A., Goix, M., Bouchoule, S. and Nouchi, P. : Electron. Lett., 36, (2000), p.58. Dany, B., Pincemin, E., Brindel, P. and Leclerc, O. : OFC2000, in press, (2000).

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ANALYSIS AND DESIGN OF WAVELENGTH-DIVISION MULTIPLEXED DISPERSION-MANAGED SOLITON TRANSMISSION AT 40 GBIT/S/CH

M. MATSUMOTO Graduate School of Engineering, Osaka University 2-1 Yarnada-oka, Suita, Osaka 565-0871, Japan AND A. HASEGAWA Kochi University of Technology and

NTT Science and Core Technology Laboratory Group Abstract. Solitons in dispersion-managed (DM) fibers are promising candidates for a modulation format to be used in long-distance high-speed optical fiber data transmission. In this paper we perform a bit-error-rate analysis of a wavelength-division multiplexed (WDM) DM soliton system operated at 40 Gbit/s per channel. Wo consider the intra-channel pulse-topulse interaction, the noise-induced timing jitter, the collision-induced time

shifts, and the signal—-ASE (amplified spontaneous emission) and ASE—ASE beat noise as the factors limiting the system performance. Effects of filters in reducing the collision-induced timing jitter in WDM transmission are also examined. It is shown that the minimum channel spacing required for long-distance transmission can be reduced by the use of in-line filters especially when the noise amplification due to the filter excess gain is suppressed.

1. Introduction

Most of the latest large-capacity and long-distance transmission experiments use channel bit rate of 10 Gbit/s [1, 2] with a few

exceptions using 20 Gbit/s per channel [3, 4]. In view of the reduction of system complexity, ease of system management, and possible improvement in spectral efficiency, higher channel bit rate more than 20 Gbit/s is desired, 195 A. Hasegawa (ed.). Massive WDM and TDM Soliton Transmission Systems, 195–210. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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which stimulates research on long-distance wavelength-division-multiplexed (WDM) systems operated at 40 Gbit/s per channel [5]. As the channel bit rate is increased, peak power of the optical pulse must be increased in order to maintain pulse energy that is required for obtaining enough signal-to-noise ratio at the receiver. The large optical

power induces large nonlinear effects in fibers resulting in significant signal distortion limiting the transmission distance. Soliton is a natural candidate for the modulation format to be used in high-speed systems because the self phase modulation (SPM), which is one of the principal nonlinear effects in fibers, is positively used to form its pulse shape. Dispersion-managed (DM) solitons in fibers having alternating sign of group-velocity dispersion (GVD)

have further advantages. The effect of energy enhancement allows us to use lowered averaged fiber dispersion, which leads to the reduction of timing jitter caused by noise and other effects [6]. 40 Gbit/s transmission over 10200 km using a DM soliton has been demonstrated without use of soliton control techniques [7].

In WDM systems, performance of the system is degraded by the interchaunel nonlinear crosstalk caused by four-wave mixing (FWM) and crossphase modulation (XPM). While the effect of FWM can be reduced by the large local fiber dispersion in DM systems [8, 9], the effect of XPM places severe limits on the transmission distance and minimum channel separation [10]-[12]. The use of soliton transmission control techniques such as those by means of narrow-hand in-line filters and synchronous modulators is beneficial in improving the system performance [11, 13]. In this paper we analyze the performance of single- and multi-channel DM soliton systems operated at 40 Gbit/s/ch with and without using inline filters. 2. Single-Channel Transmission

Figure 1 shows a unit cell of a DM line analyzed in this paper. It consists of equal lengths of anomalous- and normal-dispersion fibers (absolute value

of GVD is assumed to be a few ps/nm/km) and an amplifier immediately before the anomalous-dispersion fiber. A narrowband filter may be inserted at the output of the amplifier to improve the system performance. Here we restrict ourselves to a study of the behavior of pulses that are periodically

stationary with the same period as the dispersion map. Figure 2 shows an example of the variation of the pulse width of DM-

solitori solutions in the absence of filters obtained by a variational approach assuming a Gaussian pulse shape. The difference of the GVD between the anomalous- and normal-dispersion fibers and the pulse energy are 4 ps/nm/km arid

(just after the amplifier)

, respectively. The

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pulse energy is close to the minimum value satisfying the signal-to-noise ratio requirement for a transmission distance of km [14]. Other numerical parameters used are span length fiber loss dB/km, amplifier noise figure and fiber nonlinearity Dispersion slope of the fiber is assumed to be zero or almost completely compensated by the use of a proper combination of fibers having positive and negative dispersion slopes. Figure 2 shows that stretching of the pulse is larger for smaller averaged fiber dispersion. This is because

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large energy enhancement, which is obtained by a large stretching ratio, is needed for smaller averaged dispersion when the pulse energy is kept constant. The transmission-distance limitation imposed by the pulse-to-pulse interaction is then examined. We first solve the nonlinear Schrödiriger equation numerically to obtain a transmitted signal waveform of a pulse train having a pseudorandom bit pattern (32 or 64 bits). We next calculate the average and variance of the receiver current including noise by using [15]

respectively, where e, hv, p(t),h(t), N, M and are the electron charge, photon energy, power waveform of the received pulse train, impulse response of the electrical filter, power spectrum density of the accumulated ASE noise at the receiver amplifier spontaneous emission factor, G: amplifier gain, K: number of amplifiers), the number of noise polarization ( for unpolarized detection), and the bandwidth of the

optical filter at the receiver, respectively. The first and the second terms in Eq. (2) represent signal-ASE and ASE-ASE beat noise, respectively. The bit error rate is then given by

where

(

period, k:integer) and L are the threshold

current, the time at which the detection is made for the k-th bit, and the sum of the number of marks and spaces in the pulse train, respectively, and are optimized for the BER to be minimized.

In Fig. 3 we plot the Q factor determined by the pulse-to-pulse interaction and amplitude noise versus the averaged fiber dispersion for two different dispersion differences The transmission distance is 4000–8000 km. The Q factor is related to the BER by erfc and 7 correspond to

and

respectively, for exam-

ple. The detection of the signal is made at 7.5 km from the amplifier in the anomalous-dispersion fiber, or equivalently, the signal is detected at the amplifier followed by dispersion compensation that is equal to the

product of the length and the GVD of the fiber. We assume

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100 GHz, and the cutoff frequency of the electrical filter (5th-order Bessel filter) 30 GHz.

Figure 3 shows that the averaged fiber dispersion should be in a range to avoid large pulse-to-pulse interaction. Too small causes large pulse-to-pulse interaction due to the large pulse stretching as shown in Fig. 2, while too large causes enhanced translation of frequency shift to temporal shift. Figure 3 also shows that the effect of interaction is more significant for because the pulse

width is larger for larger when the averaged dispersion and the pulse energy is fixed. In Fig. 4 we plot the eye patterns at points a and b in Fig. 3. In the above evaluation of Q factor we do not consider the effect of timing jitter caused by the nonlinear interaction of the signal and the amplifier noise. Although the jitter is considerably reduced owing to the energy enhancement in the DM soliton systems, it can not be neglected for high-speed transmission. We then analyze the effect of noise-induced timing jitter. When a white noise that satisfies

is added to a DM soliton having a Gaussian shape of the form

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the noise gives fluctuations to the frequency and the temporal position of the pulse, whose variance and covariance are given by [16]-[19]

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The timing jitter after passing K amplifiers is then given by

where and are the accumulated dispersion in one dispersionmap period and an additional accumulated dispersion between the K-th amplifier and the receiver. appears when the pulse has a frequency chirp at the amplifier [20].

Figure 5 shows the noise-induced timing jitter evaluated by Eq. (9). The pulse energy is constant at The Gordon-Haus timing jitter of a path-averaged standard soliton having the same energy is also plotted for comparison. It is noted that the jitter of the standard soliton does not depend on the fiber dispersion when the pulse energy is kept constant. We then calculate the bit error rate caused by the timing jitter and amplitude noise. We use the following expression:

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where and are the error rates when a space surrounded by two spaces, a space surrounded by one mark arid one space, a space surrounded by two marks, and a mark arc transmitted, respectively. Detailed expressions for the error rates are given in the Appendix A. Figure

6 shows the Q factor versus for and 6 ps/nm/km. When is small, smaller than about 0.015 ps/nm/km for 6000 km transmission, for example, the Q factor is determined not by the timing jitter but by the amplitude noise. Because the spectral width of the pulse is narrower and the timing jitter is smaller for larger for the fixed pulse energy, the Q factor is larger for larger This is opposite to the dependency of the Q factor determined by the pulse-to-pulse interaction on (Fig. 3). There exists, therefore, an optimum dispersion difference with which the longest transmission distance is obtained. For the system considered here and about 4.0 and 0.025 ps/nm/km, respectively, are considered to be optimum.

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3. Wavelength-Division Multiplexed Transmission System performance of WDM systems is degraded by inter-channel nonlinear crosstalk caused by four-wave mixing (FWMindexfour-wave mixing (FWM) and cross-phase modulation (XPM). Here we do not consider the

performance degradation due to FWM because the FWM efficiency in DM systems is significantly reduced owing to the large local fiber GVD. When two pulses in different channels collide, XPM causes time shifts to the pulses [10]-[12]. We again use the variational approach assuming a Gaussian pulse shape to evaluate the time shift induced by a collision of a pair of pulses [21, 22]. We assume that the polarization state of the pulse in each channel changes rapidly during propagation and traverses uniformly on the Poincaré sphere. We further assume that the relative state of polarization between different wavelength channels also changes rapidly and randomly. We therefore take the relative strength of XPM to SPM as 1.5.

The solid curve in Fig. 7 is the collision-induced time shift at km versus the initial time separation at when we do not use guiding filters. Channel spacing is The fiber dispersion and the pulse energy used are and Other system parameters are the same as those in the

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previous section. For these numerical parameters each pulse can interact with 14 pulses in the other 40 Gbit/s channel during propagation over 6000 km. The total time shift of a pulse is given by a sum of the individual time shift as

where are random variables that take either 0 or 1. The variance of is then given by [23]

We plot the standard deviation of the time shift versus the transmission distance without use of guiding filters in Fig. 8 by solid curves : they grow almost proportionally to the distance. In Fig. 9 we plot the standard deviation of the time shift as a function of which shows that the collision-induced timing jitter decreases rapidly as the channel spacing is increased. The curves shown in Fig. 9 can be approximated by a power-low function, which is then used to estimate the collision-induced timing jitter for

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arbitrary number of channels and different channel spacings. Solid curves in Fig. 10 are the timing jitter of the central channel, which suffers the largest jitter, for the number of channels and 15 at z = 6000 km versus the channel spacing when we do not use guiding filters. It is found that the difference between the jitters for and 15 is small, indicating that the most of the jitter comes from the collision with the immediate neighbor channels. It is also found that the timing jitter is intolerably large at z =

6000 km unless the channel spacing is larger than about 4 nm. A promising method to reduce the timing jitter is to use transmission control such as in-line filtering [11] and synchronous modulation [13]. We then examine the effectiveness of the use of in-line filters. The dashed curve in Fig. 7 is the collision-induced time shift when filters arc inserted at the exit of every amplifiers. The filters have a Gaussian frequency response with a 3 dB bandwidth of 250 GHz, which is 4.5 times the pulse spectral width. From the figure we find that 1) the filters reduce but not suppress the time shift caused by the collision near the entrance of the system (initial overlap), which occurs for

2) the filters well suppress the time shift due to complete collision, which occurs for but

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3) time shifts caused by the collision occurring near the end of the transmission are not reduced because the filters are not strong enough for the collision-induced frequency shift to be dumped quickly. The time shifts remained unsuppressed limit the effectiveness of the filters for short transmission distances as shown by dashed curves in Fig. 8. For longer distances, however, the filters have significant effect in reducing the timing jitter. Finally we estimate the maximum achievable transmission distance versus the channel spacing. The Q factor is evaluated as in the previous section but we now include the effect of collision-induced timing jitter. We use the sum of the noise-induced and the collision-induced timing variances as the variance of the probability density function f(t) appearing in Eqs. (14)(18). When the effect of filters is considered, we use the expressions given in Appendix B instead of Eq. (9) for calculating the noise-induced timing jitter. Figure 11 shows the maximum transmission distance, at which the Q factor degrades to 7, versus the channel spacing. Although the number of channels is assumed to be 15, further addition of channels has little effect on the achievable distance. The solid curve is the result without filters.

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For the filtered transmission we examine two cases: the amplification of the ASE noise by the excess gain that is needed to compensate for the filter loss is taken into account (dashed curve) or is not taken into account (dotted curve). From the figure we find: 1. Large channel spacing is needed to achieve long-distance transmission when in-line filters are not used. 2. Transoceanic-distance transmission will be possible with channel spacing nm if we use in-line filters (250 GHz bandwidth) and the amplification of ASE noise is suppressed. This will possibly be achieved by using sliding-frequency filters [11] or saturable absorbers that eliminate low-power radiation. 3. Filters accompanying the amplification of ASE noise, which corresponds to fixed-frequency filters, have moderate effect in increasing the transmission distance for The maximum distance will be limited by the growth of the noise. 4. Transoceanic-distance WDM transmission with channel spacing well below 2 nm may not be achieved by the filter control alone. Effects of sliding-frequency filters in stabilizing the pulse frequency is somewhat different from that of fixed frequency filters [24]-[26]. More de-

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tailed analysis is needed to estimate more accurately the benefit of the sliding-frequency filters in WDM DM soliton systems. 4. Conclusion

In this paper we analyzed the performance of single- and multi-channel DM soliton transmission operated at 40 Gbit/s/ch. We examined the influence of performance-limiting factors such as intra-channel pulse-to-pulse interaction, noise-induced timing jitter, and collision-induced timing jitter. Effect of in-line filters in a WDM system was also studied and it was shown that the filters, particularly when the amplification of ASE noise is suppressed by mean of frequency sliding or other methods, will make long-distance transmission possible with moderate channel spacings. In the present analysis we ignored the effects of dispersion slope and polarization-mode dispersion. More accurate modeling of the system needs inclusion of these effects. Acknowledgements

This work is partially supported by Telecommunication Advanced Organization of Japan (TAO). References 1. Suyama, M. : 1 Tbit/s WDM transmission over 10000 km, ECOC’99, PD2-1, (1999). 2. Tsuritani, T., Takeda, N., Imai, K., Tanaka, K., Agata, A., Morita, I., Yamauchi, H., Edagawa, N. and Suzuki, M. : 1 Tbit/s transoceanic transmission

using 30 nm wide broadband optical repeaters with 3. 4.

5. 6. 7.

positive dispersion

fibre and slope-compensation DCF, ECOC’99, PD2-8, (1999). Le Guen, D., Del Burgo, S., Moulinard, M. L., Grot, D., Henry, M., Favre, F. and Georges, T. : Narrow band 1.02 Tbit/s soliton DWDM transmission over 1000 km of standard fiber with 100 km amplifier span, OFC’99, PD4, (1999). Fukuchi, K., Kakui, M., Sasaki, A., Ito, T., Inada, Y., Tsuzaki, T., Shitomi, T., Fujii, K., Shikii, S., Sugahara, H. and Hasegawa, A. : 1.1-Tb/s dense WDM soliton transmission over 3,020-km widely- dispersion-managed transmission line employing 1.55/1.58-mm hybrid repeaters, ECOC’99, PD2-10, (1999). Pincemin, E., Leclerc, O. and Desurvire, E. : Feasibility of 1 Tbit/s transoceanic optically regenerated systems, Opt. Lett., 24, (1999), pp.720-722. Smith, N. J., Forysiak, W. and Doran, N. J. : Reduced Gordon-Haus jitter due to enhanced power solitons in strongly dispersion managed systems, Electron. Lett., 32, (1996), pp.2085-2086. Morita, I., Tanaka, K., Edagawa, N. and Suzuki, M. : 40 Gbit/s single-channel

soliton transmission over 10200 km without active inline transmission control, ECOC’98, PD, (1998), pp.49-51.

8.

Kurtzke, C. : Suppression of fiber nonlinearities by appropriate dispersion manage-

9.

ment, IEEE Photon. Technol. Lett., 5, (1993), pp.1250-1253. Mollenauer, L. F. and Mamyshev, P. V. : Massive wavelength-division multiplexing with solitons, IEEE J. Quantum Electron., 34, (1998), pp.2089-2102.

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Kaup, D. J., Malomed, B. A. and Yang, J. : Interchannel pulse collision in a wavelength-division-multiplexed system with strong dispersion management., Opt.

Lett., 23, (1998), pp.1600-1602. Mamyshev, P. V. and Mollenauer, L. F. : Soliton collisions in wavelength-divisionmultiplexed dispersion-managed systems, Opt. Lett., 24, (1999), pp.448-450. 12. Sugahara, H. and Maruta, A. : Timing jitter of a strongly-dispersion-managed soliton in a WDM system, Nonlinear Guided Waves and Their Applications, ThA4, (1999). 13. Nakazawa, M., Suzuki. K., Kubota, H., Sahara, A. and Yamada, E. : 160 Gbit/s WDM soliton transmission over 10000 km using in-line synchronous modulation and optical filtering, Electron. Lett., 34, (1998), pp.103104. 11.

14.

Tonguz, O. K. : Impact of spontaneous emission noise on the sensitivity of directdetection lightwave receivers using optical amplifiers, Electron. Lett., 26, (1990),

pp.1343-1344. Sano, A., Miyamoto, Y., Kataoka, T. and Hagimoto, K. : Long-span repeaterless transmission systems with optical amplifiers using pulse width management, J. Lightwave Technol., 16, (1998), pp.977-985. 16. Kumar, S. and Lederer, F. : Gordon-Haus effect in dispersion-managed soliton systems, Opt. Lett., 22, (1997), pp.1870-1872. 17. Georges, T., Favre, F. and Le Guen, D. : Theoretical and experimental study of soliton transmission in dispersion managed links, IEICE Trans. Electr., E81-C, (1998), pp.226-231. 18. Okarnawari, T., Maruta, A. and Kodama, Y. : Reduction of Gordon-Haus jitter in a dispersion compensated optical transmission system : analysis, Opt. Commun.,

15.

149, (1998), pp.261-266.

19. 20.

21. 22. 23. 24.

25. 26.

Kutz, J. N. and Wai, P. K. A. : Gordon-Haus timing jitter reduction in dispersionmanaged soliton communications, IEEE Photon. Technol. Lett., 10, (1998), pp.702704. Mu, R. -M., Grigoryan, V. S., Menyuk, C. R., Golovchenko, E. A. and Pilipetskii,

A. N. : Timing-jitter reduction in a dispersion-managed soliton system, Opt. Lett., 23, (1998), pp.930-932. Masumoto, M. : Analysis of interaction between stretched pulses propagating in dispersion-managed fibers, IEEE Photon. Technol. Lett., 10, (1998), pp.373-375. Hirooka, T. and Hasegawa, A. : Chirped soliton interaction in strongly dispersionmanaged wavelength-division-multiplexirig systems, Opt. Lett., 23, (1998), pp.768770. Meccozi, A. : Timing jitter in wavelength-division-multiplexed filtered solitou transmission, J. Opt. Soc. Am. B, 15, (1998), pp.152-161. Mecozzi, A., Midrio, M. and Romagnoli, M. : Timing jitter in soliton transmission with sliding filters, Opt. Lett., 21, (1996), pp.402-404. Matsumoto, M. : Analysis of filter control of dispersion-managed soliton transmission, J. Opt. Soc. Am. B, 15, (1998), pp.2831-2837. Merlaud, F. and Georges, T. : Influence of sliding frequency filtering on dispersionmanaged soliton, ECOC’99, (1999), TuA3.3, (1999).

Appendix A

The error rates

and

in (10) are given by

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where f(t) is the Gaussian probability density function for the pulse position whose variance is given by Eq. (9). in Eq. (13) is the ASE-ASE beat noise that is the square root of the second term in Eq. (2).

and

in Eqs. (14)–(16) are the average and variance of the receiver

current waveform when an isolated pulse is transmitted. We assume that and have their peaks at t = 0 and are even functions of and in Eq. (15) account for an additional eye closure due to the presence of marks at both neighbors and are given by

Appendix B The noise-induced timing jitter controlled by filters is given by

where the filters are assumed to have a Gaussian frequency response

and C are the width and the chirp of the pulse (see Eq. (5)) at the entrance of the filter.

OPTIMIZATION OF DISPERSION COMPENSATION FOR LONG DISTANCE 40 GBIT/S SOLITON TRANSMISSION LINES BY THE Q-MAP METHOD

K. SHIMOURA, I. YAMASHITA AND S. SEIKAI

Technical Research Center The Kansai Electric Power Co., Inc. 3-11-20 Nakoji, Amagasaki, Hyogo 661-0974, Japan Abstract. We investigate periodic dispersion compensated 40 Gbit/s soliton transmission lines by a numerical simulation and transmission exper-

iment. We developed the simulation method for the optimization of these lines using Q-maps. The optimal dispersion compensation becomes a constant value of in different compensation periods, because of the optimal strength of the dispersion management. In the experiment, we

evaluated transmission performance in the parameters of signal wavelength, signal power, and the location of dispersion compensation elements. The

location is quite important in the design of such lines. In the 640 km transmission experiment using 2-pieces of dispersion compensation fibers, we observed error-free transmission in the wavelength range of 1.2 nm in the optimally designed line. This line design is practical for the next generation

of 40 Gbit/s based communication systems.

1. Introduction

The 10 Gbit/s transmission systems using dispersion-shifted fibers are commonly used in Japan. The RZ-based 40 Gbit/s systems are expected to be the next generation medium distance high capacity communication systems. In this region, fiber dispersion effects and fiber nonlinear effects become critical. Dispersion managed (DM) soliton is an attractive solution in this area. This scheme also can be applied for ultra-long distance communication [l]-[3]. The periodic dispersion compensated lines are modeled by uniform dispersion fibers and linear dispersion compensation elements. The anoma211 A. Hasegawa (ed.), Massive WDM and TDM Soliton Transmission Systems, 211–223. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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lous dispersion fiber-based transmission line was proposed for reducing the Gordon-House jitter and soliton interactions without decreasing the signal power [4]. It was also reported that the initial frequency chirping enhances the power margin and dispersion tolerance [5]. On the other hand, normal dispersion fiber-based transmission lines also have good performances for soliton transmission [6]. We have shown numerically that two extremely stable transmission conditions exist in this scheme [7]. And at these conditions, the strength of the dispersion management becomes optimal to minimize the soliton interaction [8]. From this result, the dense (short-scale, continuous) dispersion management scheme is proposed for ultra high bit rate transmission systems [9]-[11]. In this paper, we show the numerical optimization method for the design of dispersion compensated 40 Gbit/s systems. 2. Q-map Method

In the 40 Gbit/s system, the signal power becomes large and we must consider the nonlinear effect and the dispersion effect. We must optimize many parameters simultaneously for the design of dispersion-managed lines. Q-factor contour mapping (Q-map) is a practical method for this purpose

[12]. Q-factor represents the signal-to-noise ratio at the receiver decision for the [13]. Figure 1 shows the definition of Q-factor for RZ pulses. In the definition (a), only one pulse is considered and we can easily estimate the stability of the pulse in a short calculation time. The definition (b) is the ordinary Q-factor definition applied to NRZ systems. This definition can be applied to RZ systems with some modification of the receiver bandwidth. We used 1.4 times of the base band frequency in the simulations. This definition considers soliton interaction effects and jitter effects, and a relatively short bit pattern can be used because the interaction between adjacent pulses is dominant in soliton systems. We proposed some kinds of Q-maps for the optimization of periodic dispersion compensated lines [7]. Figure 2 shows the dispersion maps of the simulation model. The dispersion-shifted fibers of dispersion D and length La are connected with optical amplifiers. A linear dispersion compensation element of Dc is installed in every Nc amplifier spans. D and Dc can be pos itive (anomalous) or negative (normal) values, and the average dispersion Dav is calculated from these values by the equation (1). circuit in voltage or current unit, and requires

The nonlinear Schrödinger equation (NLS), with a third order dispersion term of and a fiber attenuation term of 0.2 dB/km, is

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213

solved by the split step Fourier method. The factor 1/2 of the nonlinear operator B originates from the averaging of the electrical field over the cross section of the fiber [14]. The effective core area is and the Kerrcoefficient is The amplifier noise figure is 4 dB and a 6 nm width filter is used in each amplifier. We executed these calculations by the Mathematica 4.0 on the Windows NT operating system. 3. Optimization of Dispersion Compensation The chirped Gaussian pulse of 7.5 ps full-width at half-maximum (FWHM) is considered as the initial pulse and the 12-bit pattern “001011101100 ” is assumed. Figure 4 shows the example of the Q-maps on the Dav – Dc plane

for the different compensation period Nc. The amplifier spacing the initial peak power

and the transmission distance

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Two parameter zones are clearly demarcated for the stable transmission. One mode has a positive Dc value of around and the other mode has a negative value of around and these values do not strictly depend on Nc. The positive Dc requires normal dispersion fibers as the transmission line, and negative DC uses anomalous dispersion fibers. The optimal Dav is in the slightly anomalous region around in both cases. In case (a), the local dispersions of the transmission fibers D are and respectively. In case (b), D is and respectively. In the longer Nc case, the local dispersion becomes small, and nonlinear signal distortion effects become critical. The non-transmissible zone between transmissible zones is expanded and affected by the dispersion fluctuation of the transmission fiber. This effect determines the upper limit of the dispersion compensation period in higher bit-rate systems. Figure 5 also shows the Q-maps on the plane, but for different amplifier spacings. in case (a), and in case (b). The compensation period is in both cases. In these cases, the optimal Dc is also about but the optimal Dav is slightly shifted. In the longer case, the optimal Dav has a smaller value. Figure 6 shows the Q-maps on the plane for a normal dispersion case and an anomalous dispersion case The amplifier spacing and transmission distance Mm. The fiber input power Pav is calculated from Pm by considering the pulse duty ratio of 0.3 and

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the mark ratio of 0.5. The optimal Dav is about and the optimal Pav is about in both cases, but better transmission

is achieved in the normal dispersion case (a). We can easily estimate the dispersion tolerance and the power margin of the transmissible area and get,

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4. The Strength of the Dispersion Management

It is interesting that in the different Nc or La cases, the optimal dispersion compensation Dc has almost the same value. This means that the important parameter of the line design is not the local dispersion D, but the dispersion compensation Dc. These results are explained by the soliton interaction mechanism in the dispersion-managed lines. In the dispersion compensated line, the positive and the negative dispersions are almost cancelled, therefore the strength of dispersion management S is approximated

by

where for range. Ts (ps) is the full-width at half maximum of the dispersion-managed soliton at the chirp-free point, and Dc (ps/nm) is the dispersion compensation value. Figure 7 indicates the pulse widths variations for a normal dispersion case and an anomalous dispersion case by the one-pulse transmission calculation. In the Fig. 7(a), the pulse width converges to the steady state by the guiding filter mechanism [15]. In the Fig. 7(b), the pulse width becomes minimal around the center point of the dispersion compensation spans, and Ts can be estimated from this point [16]. If we input and to the Eq. (3), we get This value is almost same as the previous estimation of

OPTIMIZATION OF DISPERSION COMPENSATION

217

5. 40 Gbit/s Soliton Transmission Experiment

40 Gbit/s based systems are considered to be the next generation high capacity medium distance communication systems [17, 18]. We evaluate experimentally the transmission performance of the 640 km single-channel straight-line systems in the parameters of signal wavelength, signal power, and the location of the dispersion compensation elements. We recognized the optimization of the location expands the wavelength tolerance remarkably without any initial frequency chirping. The experimental setup is shown in the Fig. 8. The optical pulse is generated by the mode-locked laser diode (MLLD), and the pulse width is 5.7 ps. This MLLD can be tuned between 1530 nm and 1560 nm wavelength range. The MLLD is triggered by 10 GHz clock. The output pulse is modulated by the lithium niobate modulator (LN-mod). The 10 Gbit/s, data pattern is optically multiplexed to 40 Gbit/s signal by a PLC-multiplexer. The dispersion-shifted fibers (DSF) are used as the transmission line. The amplifier spacing is 80 km consisting of 4-pieces of 20 km length DSF. The zero-dispersion wavelength of the each fiber is distributed between 1535 nm and 1560 nm, and their standard deviation is 6.1 nm. The 2-pieces of dispersion compensation fibers (DCF) of and are installed in the transmission line. The averaged zero-dispersion wavelength is 1547.9 nm (without DCF) and 1549.5 nm (with DCF). The amplifier noise figure is 5 dB, and no-filter is used in each amplifier. The fiber loss rate is 0.21 dB/km, and the dispersion slope is 0.07 The average dispersion of the transmission line can be changed

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according to the laser-wavelength. In the receiver, 10 GHz clock signal is

recovered by the 40 GHz phase locked loop circuit (PLL), and the 40 Gbit/s data stream is demultiplexed to 10 Gbit/s signal using an electroabsorbtion modulator (EA-mod). 6. Optimization of the Signal Power In dispersion compensated soliton systems, the optimal average dispersion is shifted nearly to zero and the power margins are expanded compared

OPTIMIZATION OF DISPERSION COMPENSATION

21 9

to the ordinary soliton systems [2, 4]. Figure 9(a) shows the signal power and the wavelength dependence of the transmission performance. The fiber input power is changed from to The DCFs are installed in 320 km and 640 km points. The optimal amplifier output power is . and the good transmission is achieved between and . At the error free transmission is observed in the wavelength range of 0.4 nm: 1549.8 nm–1550.2 nm, where the dispersion is to Figure 9(b) shows the estimation of the transmissible area by numerical simulation. This is a Q-map on the Dav – Pav plane, and requires

for the BER less than is achieved at the Pav from to

From this figure, the good transmission to and the Dav from In this simulation, the dispersion

fluctuation of the transmission fiber is not considered, and the fiber input power is estimated from the pulse peak power without considering the amplifier spontaneous emission (ASE) power. These two reasons will explain the difference between the experimental result and the simulation result. 7. Location Effect of the Dispersion Compensation Elements

In the dispersion compensated soliton systems, the pulse width becomes minimal around the center point of the dispersion compensation spans, where the frequency chirp becomes zero. Therefore, if we put the pulse source and the receiver at the chirp-free points, any pre-chirping or chirp compensation at the receiver is not required to get the optimal DM-soliton transmission. Figure 10 shows the observed pulse widths in two types of dispersion

compensation lines by the streak camera. In the case (a) the DCFs are installed in the end point of the compensation spans, 320 km and 640 km. In the case (b) these are installed in the center point of the spans, 160 km and 480 km. The average pulse widths at the receiver for 1549.2 nm to 1550.4 nm signals are 20.2 ps in the case (a), and 8.8 ps in the case (b). The pulse broadening is suppressed and intersyrnbol interference (ISI) is reduced in the case (b). Figure 11 shows the measured bit error rate for these two

types of lines. In the case (b), the error-free transmission is observed in the wavelength range of 1.2 nm: 1549.4 nm–1550.6 nm, expanded 3 times compared to the case (a). Figure 12 shows the Q-maps on the initial frequency chirp parame-

ter C and the dispersion compensation Dc plane. Dav is fixed and Pm is In the ordinary DCF allocation case (a), (down-chirping) is required for the positive Dc lines, and

(up-

chirping) is required for the negative Dc lines for the optimal transmission

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K. SHIMOURA, I. YAMASHITA AND S. SEIKAI

[5]. These initial frequency chirps compress pulses linearly in the transmission fibers in both cases [16]. In the chirp-free DCF allocation case (b), pre-chirping is not required for the optimal transmission, and transmission performance is improved especially in the positive Dc lines [7].

OPTIMIZATION OF DISPERSION COMPENSATION

221

8. Conclusion

We numerically and experimentally analyzed the optimization method for the dispersion compensated soliton lines. From the simulation results, the optimal dispersion compensation for 40 Gbit/s systems is about

K. SHIMOURA, I. YAMASHITA AND S. SEIKAI

222

and does not strictly depend on the compensation periods. This result is explained by the interaction mechanism of the dispersion-managed soliton, and simplifies the dispersion design of the transmission lines. We can easily specify the dispersion compensation elements according to the Eq. (3). We experimentally evaluated the 40 Gbit/s, 640 km dispersion managed soliton transmission lines using 2-pieces of dispersion compensation fibers. The transmission performance is remarkably improved by the location effect of the dispersion compensation elements. We observed error-free transmission in the wavelength range of 1.2 nm in the chirp-free DCF allocation; 3-times improvement to the ordinary allocation is achieved. The numerical simulation method using Q-maps is applied to the estimation of the DM-soliton transmissible conditions. The optimal transmission condition shows good agreement with the experimental results, but wavelength margin or power margin is slightly different. By considering the dispersion fluctuation of the transmission fiber, and fiber input power estimation including ASE noise power will improve the accuracy of the

numerical simulation. References 1.

Mollenauer, L. F., Evangelides Jr., S. G. and Haus, H. A.: Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber, J. Lightwave Technol., 9, (1991), pp.194-197. 2. Nakazawa, M., Kubota, H., Suzuki, K., Yamada, E. and Sahara, A.: Superb characteristics of dispersion-allocated soliton transmission in TDM and WDM systems, in A. Hasegawa (Ed.), New trends in optical solitun transmission systems, Kluwer Academic Publishers, (1998), pp.197-224. 3. Morita, I., Tanaka, K., Edagawa, N., and Suzuki, M.: 40 Gbit/s single-channel soliton transmission over 10200 km without active inline transmission control, ECOC’98, PD, (1998), pp.49-51. 4. Suzuki, M., Morita, I., Edagawa, N., Yamamoto, S., Taga, H. and Akiba, S.: Reduction of Gordon-Haus timing jitter by periodic dispersion compensation in soliton transmission, Electron. Lett., 31, (1995), pp.2027-2029. 5. Morita, I., Suzuki, M., Edagawa, N., Tanaka, K., Yamamoto, S. and Akiba, S.: Performance improvement by initial phase modulation in 20 Gbit/s soliton-based RZ transmission with periodic dispersion compensation, Electron. Lett., 33, (1997), pp.1021-1022.

6. Jacob, J. M., Golovchenko, E. A., Pilipetskii, A. N., Carter, G. M. and Menyuk, C. R.: Experimental demonstration of soliton transmission over 28 Mm using mostly

normal dispersion fiber, IEEE Photon. Technol. Lett., 9, (1997), pp.130-132. 7. 8. 9.

Shimoura, K. and Seikai, S.: Two extremely stable conditions of optical soliton transmission in periodic dispersion compensation lines, IEEE Photon. Technol. Lett., 11, (1999), pp.200-202. Yu, T., Golovchenko, E. A., Pilipetskii, A. N. and Menyuk, C. R.: Dispersion- managed soliton interactions in optical fibers, Optics Lett., 22, (1997), pp.793-795. Liang, A., Toda, H. and Hasegawa, A.: High speed optical transmission with dense

dispersion managed soliton, ECOC’99, 1, (1999), pp.386-387. 10.

Turitsyn, S. K., Fedoruk, M. P., Doran, N. J. and Forysiak, W.: Optical soliton transmission in fiber lines with short-scale dispersion management, ECOC’99, 1,

OPTIMIZATION OF DISPERSION COMPENSATION

22 3

(1999), pp.382-383.

11. Auis, H., Berkey, G., Bordogna, G., Cavallari, M., Charboimier, B., Evans, A., Hardcastle, I., Jones, M., Pettitt, G., Shaw, B., Srikant, V. and Wakefield, J.: Continuous dispersion managed fiber for very high speed soliton systems, ECOC’99, 1, (1999),

pp.230-231. 12. Sahara, A., Kubota, H. and Nakazawa, M.: Q-factor contour mapping for evaluation of optical transmission systems: soliton against NRZ against RZ pulse at zero group

velocity dispersion, Electron. Lett., 32, (1996), pp.915-916.

13.

Bergano, N. S., Kerfoot, F. W. and Davidson, C. R.: Margin measurements in optical amplifier systems, IEEE Photon. Technol. Lett., 5, (1993), pp.304-306.

14. Hasegawa, A. and Kodama, Y.: Signal transmission by optical solitons in monomode

fiber, Proceeding of the IEEE, 69, (1981), pp.1145-1150. 15. 16.

Hasegawa, A. and Kodama, Y.: Solitons in Optical Communications, Oxford University Press, (1995), Chap.8. Agrawal, G. P.: Nonlinear Fiber Optics, Academic Press, (1995), Chap.3.

17. Sahara, A., Suzuki, K., Kubota. H., Komukai, T., Yamada, E., Imai, T., Tamura, K. and Nakazawa, M.: Single channel 40 Gbit/s soliton transmission field experiment over 1000 km in Tokyo metropolitan optical loop network using dispersion compensation, Electron. Lett., 34, (1998), pp.2154- 2155.

18. Nielsen, T. N., Stentz, A. J., Hansen, P. B., Chen, Z. J., Vengsarkar, D. S., Strasser, T. A., Rottwitt, K., Park, J. H., Stulz, S., Cabot, S., Feder, K. S., Westbrook, P. S. and Kosinski, S. G.: 1.6 Tb/s transmission over nonzerodispersion fiber using hybrid Raman/Erbium-doped inline amplifiers, ECOC’99, PD2-2, (1999), pp.26-27.

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LONG DISTANCE TRANSMISSION OF FILTERED DISPERSION-MANAGED SOLITONS AT 40 GB/S BIT RATE

V. S. GRIGORYAN, P. SINHA, C. R. MENYUK AND G. M. CARTER Department of Computer Science and Electrical Engineering University of Maryland Baltimore County 1000 Hilltop Circle Baltimore, MD 21250, USA.

Abstract. A 40 Gb/s long distance, single channel, dispersion-managed soliton system was designed and realized experimentally in a recirculating

fiber loop. Transmission of 40 Gb/s data over 6400 km at the bit error rate of was achieved.

1. Introduction To keep pace with an explosively growing demand for increased transmission capacity in long distance fiber communications, new systems with multi-terabit per second data traffic are needed. Wave division multiplexed (WDM) systems with 40 Gb/s bit rate per channel are likely to become the next generation of transmission systems as higher bit rates per channel generally imply higher spectral efficiencies and lower cost per bit. The first step to achieve this goal is to realize single-channel 40 Gb/s long distance transmission. Recently, the first experimental implementation of error free transmission of 40 Gb/s dispersion-managed soliton (DMS) data was reported over 8600 km [1] and 10200 km [2] using co- and cross-polarized adjacent pulses respectively. However, the fiber used in these experiments had a low local dispersion of 0.3 ps/nm.km, which is likely to cause problems

if one uses this system for a dense WDM transmission due to a significant four-wave mixing and cross-phase modulation even if the dispersion slope is ideally compensated. The four-wave mixing and cross-phase modulation decrease if the local dispersion increases in the dispersion-managed fibers. However, we note that fibers with a high local dispersion will require a short dispersion map period for stable DMS propagation and, consequently, 225 A. Hasegawa (ed.), Massive WDM and TDM Soliton Transmission Systems, 225–234. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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a large number of splices per unit length that will result in high loss. In the present paper we use a dispersion map consisting mostly of the dispersion

shifted fiber with dispersion, four times larger than the absolute value of the dispersion used in Refs. [1, 2]. The purpose of the present paper is to obtain insight into the key physical effects and the fundamental limitations impairing the transmission of DMS data streams at a 40 Gb/s bit rate. In the paper we carefully study both theoretically and experimentally single channel long distance transmission of DMS at 40 Gb/s in a recirculating fiber loop. We have developed and validated a model [3] that accurately describes all major mechanisms affecting a pulse propagation in a recirculating fiber loop. We, first, used a computationally fast semi-analytical approach [4] carefully validated by a complete model [3] and our experimental results to rapidly explore a wide range of the parameter space in order to find the dispersion map, location of the amplifiers and filter parameters that are optimal with respect to the timing and amplitude jitter of the signal pulses as well as to the accumulation of the amplified spontaneous emission (ASE) noise in ZEROs (spaces). Next, we used this optimum dispersion map configuration to explore in detail the dynamics of the signal-signal and signal-noise interactions using the complete model. This paper is organized as follows: The model is described in Section 1. We discuss dynamic properties of DMS in a recirculating fiber loop in Section 3. Section 4 contains our experimental results. Finally, Section 5 concludes the paper. 2. Modeling Equations In our model we use a modified nonlinear Schrödinger equation that may be written as

Here, the pulse envelope q is normalized as where E is the electric field envelope, is the Kerr coefficient, is the central frequency, is the fiber effective cross section area, and c is the speed of light. The characteristic dispersion length where is a characteristic scale time, and is a scaling dispersion. In our simulation, we chose ps which roughly equals the FWHM soliton duration at the points of maximum compression, and The distance z is normalized as where Z is the physical propagation distance in the recirculating loop. The dimensionless time t is the physical retarded time T normalized to Other quantities

LONG DISTANCE TRANSMISSION OF DM SOLITONS

are normalized as follows: the third order dispersion, and gain or loss. The net gain coefficient

22 7

is where is the power may be written as:

where is the gain coefficient of the amplifier located at the propagation distance generally depends on time t and distance z inside the amplifier due to the saturation effect, is the loss coefficient of the fiber, and is the amplifier length. The ASE noise source has the autocorrelation function

where is the Planck's constant and equals the spontaneous emission factor when and is zero elsewhere. Lumped filtering in the system was modeled as where and are the Fourier components of q at the input and the output of the filter and is the filter transmission.

In order to accurately describe the evolution of the DMS pulses in the recirculating loop, it is of critical importance to carefully calculate the effects of the amplifier saturation. The time dynamics of the gain coefficient of the amplifier spaced at the propagation distance is described by the rate equation

where The quantity is the unsaturated gain of the of the amplifier spaced at the propagation distance is the normalized relaxation time which in physical units we set equal to 1 msec, and is the saturation power which in physical units we set to 10 mW. We note that due to the slow relaxation of the amplifiers, the gain coefficient evolves on the a time scale of milliseconds while q(z, t) evolves on the time scale of picoseconds, allowing us to separate the slow evolution of from the fast evolution of q(z, t). If a recirculating loop length is about 100 km long so that one round trip in the loop is approximately 0,5 msec. As the gain coefficient does not change significantly in time during one round trip in the loop one can consider as a function of the distance z and the round trip number r. We than obtain from Eq. (4) that the gain coefficient of the amplifier located at the point averaged over

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one round trip time in the loop is

where N pulses fill the loop with the average time separation

is the

saturation energy of the amplifiers,

is the

average energy of the pulse over N pulses at fixed 2, and Having at each z determined the slow variation of the gain from Eq. (5), we then determine the fast variation of q(z, t ) from Eq. (1) by assuming that

the gain is independent of t on each round trip. We note that in steady state, Eq. (5) becomes,

where is constant as a function of r. However, to obtain accurate results for the convergence of initial pulse shapes to their final values, one must use Eq. (5) rather than Eq. (6). 3. Design of the Loop Experiment Figure 1 shows the outline of the recirculating fiber loop that we designed. In the model we used the following parameters of the loop. The loop consists

LONG DISTANCE TRANSMISSION OF DM SOLITONS

22 9

of four dispersion map period. Each dispersion map period consists of a span of the dispersion shifted fiber (DSF) with the normal dispersion of and the length of 25 km and a span of the standard fiber (SF) with the anomalous dispersion of 16.7 ps/nm-km and the length of about 1.8 km. The dispersion slope is in both DSF and SF spans. One amplifier was located per each map period. We used an optical bandpass filter (OBF) with different FWHM bandwidth from 2.8 nm to 6 nm and different shapes. Loss coefficient in the fiber varied between 0.21 dB/km and 0.288 dB/km. Losses at the OBF were set to about 2 dB and total losses at the acustooptic switch and the input-output coupler were set to 7 dB. In our calculations the spontaneous emission factor was set to 1.3. Figure 2 shows dependence of the DMS pulse duration on distance inside the loop for different location of the amplifier inside the map period. The average dispersion in the loop is 0.005 ps/nm-km. The peak power of the DMS at the midpoint of the anomalous dispersion span is about 10.5 mW. One can see that the stretching factor of the DMS changes significantly depending on the amplifier location. A large stretching factor results in a strong inter-pulse interaction and significantly limits the transmission distance. In Fig. 2 the stretching factor is a minimum when the amplifier is located at about 19.6 km from the beginning of the normal dispersion span in each dispersion map period. Hereafter the amplifiers in the loop

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will be set at these points. Figure 3(a) shows dependence of timing jitter on distance. As is seen in Fig. 3(a) the timing jitter at 20000 km remains smaller than 1.2 ps so that the contribution of the timing jitter to the bit error rate remains smaller than assuming the Gaussian distribution of the timing jitter. In Fig. 3(b) we plot the standard deviation of the DMS pulse energy normalized to the average DMS pulse energy (dots) and the average noise intensity normalized to the average intensity in the pulse train. Both Figs. 3(a) and (b) are taken at the midpoint of the anomalous span at the output of the loop and are results of the averaging over 100 realizations of the ASE noise. In Figs. 1–3 the loss in the fiber is 0.21 dB/km and OBF is a Gaussian filter with FWHM bandwidth of 5 nm. The spread of the data for the amplitude jitter is much larger than that of the average ASE noise intensity because we averaged the ASE noise intensity over a larger number of bits in each realization whereas we processed only one DMS pulse in each realization. From Ref. [5] we infer that the normalized standard deviation of the pulse energy of 0.167 corresponds to a bit error rate of Consequently, we estimate that both the timing and the amplitude jitter of the signal pulses as well as the ASE noise in the

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231

ZEROs arc within the limits of error free transmission over 20000 km. 4. Experimental Results

Figure 1 shows the setup of the experiment for the recirculating fiber loop that was designed based on the model described in Sections 2 and 3. The data source was a commercially available synchronously mode-locked fiber laser operating at 10 GHz repetition rate with spectrum limited pulses of about 3 ps pulse duration. The pulses were encoded with a pseudorandom bit pattern. The data stream was, first, divided into two streams and recombined with a relative delay of about 150 ps to produce 20 Gb/s data and then divided into two streams again and recombined again to produce 40 Gb/s data stream. The data stream was then launched in the loop from the midpoint of the anomalous dispersion span. The output data were time-division-demultiplexed by using optical gates generated by the electro-optic modulator driven by the clock-recovered sinusoidal electrical

wave in two stages: from 40 Gb/s to 20 Gb/s and from 20 Gb/s to 10 Gb/s. The demultiplexed data at 10 Gb/s were detected by the bit error rate (BER) tester. In the experiment we used an OBF with the transmission function illustrated in Fig. 4. Figure 5 shows dependence of the voltage decision levels to detect ONE's and ZERO’s at the bit error rate of One can see from Fig. 5 that the decision level in ONE’s is stable while the decision level in ZERO’s degrades significantly at 4000 km. This degradation occurs because of a stable propagation of the DMS pulses and a relatively fast noise accumulation in ZERO’s. Conversely, the phase margins shown in Fig. 6 for receiving data at the bit error rate of turn to be remarkably stable owing to a very small timing jitter. The maximum error free transmission distance with in the experiment was 6400 km. This distance is significantly smaller then

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predicted in Section 3. The reduction occurs due to the following reason. Our model shows that propagation of the DMS 40 Gb/s pulse train is sensitive to the curvature of the filter near its top even though the FWHM bandwidth of the filter remains the same. Variation of the filter transmission coefficient near its top of about 0.2 dB over the wavelength range of half of the DMS pulse bandwidth (of about 0.6 nm) results in a dramatic reduction of the soliton propagation distance. We illustrate this behavior in Fig. 7 where we plot the evolution of 8 bit DMS pulse train 01110010 in the loop with OBF shown in Fig. 4. In Fig. 7 we see a significant degradation of the pulse train after 8000 km because of a rapid accumulation of the noise in ZERO'S due to the net gain of the low frequency ASE noise components. On the other hand, if the filter is too flat then the noise accumulates in a broad bandwidth resulting in an increase of the integrated noise power. Because of the energy balance this increase can occur only in expense of the signal power. We note, however, that in a WDM system with no in-

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line filters this increase does not occur because the bandwidth per channel remains small. The loss coefficient in Fig. 7 of 0.288 dB/km corresponds to the effective fiber losses in the experiment. The extra loss in the fiber, due to a number of splices in the loop, causes an extra noise that also contributes to the reduction of the stable propagation distance. Our model shows that use of a Gaussian OBF with 5 nm bandwidth or other filter with the same curvature at the top of the filter will significantly increase

the stable propagation distance. 5. Conclusion We carefully analyzed dynamic properties of the signal pulses and the noise

in long distance single channel 40 Gb/s dispersion-managed soliton system in a recirculating fiber loop. We showed that the stretching factor of the DMS is sensitive to the amplifier location and found the optimum amplifier location inside the dispersion map. We discovered that DMS propagation is sensitive to the shape of the optical bandpass filter near its top. We implemented our system design in the experiment and observed an error free transmission of 40 Gb/s data over 6400 km. We have shown that the limi-

tation of the transmission distance is due to noise accumulation in ZEROs. The noise accumulation occurs because of strong curvature of the filter near its top. We have found the optimal curvature of the filter needed for stable data transmission over more than 10000 km. Acknowledgement

One of the authors is very grateful to Gevorg Grigoryan for his valuable help in preparing the graphic material for the paper.

V. S. GRIGORYAN ET AL.

234 References

1. Morita, I., Tanaka, K., Edagawa, N., Yamamoto, S. and Suzuki, M. : 40 Gbit/s Single-Channel Transmission Over 8600 km Using Periodic Dispersion Compensation, Electron. Lett., 34, (1998), pp.1863-1865. 2.

Morita, I., Tanaka, K., Edagawa, N. and Suzuki, M. : Impact of the Dispersion

Map on Long-Haul 40 Gbit/s Single-Channel Soliton Transmission with Periodic 3.

Dispersion Compensation, OFC'99, Technical Digest, FD1-1, (1999), pp.62-64. Grigoryan, V. S., Menyuk, C. R. and Mu R.-M. : Calculation of Timing and Amplitude Jitter in Dispersion-Managed Optical Fiber Communications Using Lineariza-

tion, J. Lightwave Technol., 17, (1999), pp.1347-1356. 4.

Mu, R.-M, Grigoryan, V. S., Menyuk, C. R., Carter, G. M. and Jacob, J. M. : Comparison of Theory and Experiment for Dispersion-Managed Solitons in a Recirculating Fiber Loop, To be published in IEEE Journal of Selected Topics in

5.

Marcuse, D. : Derivation of Analytical Expressions for the Bit-Error Probability in Lightwave Systems with Optical Amplifiers, J. Lightwave Technol., 8, (1990), pp.1816-1823.

Quantum Electronics, 5, No.6, (1999).

OPTICAL COMMUNICATION SYSTEMS WITH SCHORT-SCALE DISPERSION MANAGEMENT

S. K. TURITSYN, N. J. DORAN AND E. G. TURITSYNA Photonics Research Group

School of Engineering and Applied Science Aston University, Birmingham, B4 7ET, UK E. G. SHAPIRO Institute of Automation and Electrometry, 630090, Novosibirsk, Russia

AND

M. P. FEDORUK AND S. B. MEDVEDEV Institute of Computational Technologies, 630090, Novosibirsk, Russia Abstract. We investigate theoretically and numerically properties of disper-

sion-managed (DM) solitons in fiber lines with dispersion compensation period L much shorter than amplification distance We present the pathaveraged theory of DM transmission lines with short-scale management. Applying a quasi-identical transformation we demonstrate that the path-

averaged dynamics in such systems can be described in some limits by the integrable model. Using advantages of the integrable limit (weak map) we have demonstrated in numerical simulations DM soliton transmission at 40

Gb/s over more than 9000 km and at 80 Gb/s over 4500 km in systems with short-scale management.

1. Introduction

Ultra-high capacity wave-division-multiplexing (WDM) data transmission requires very dense packing of the carrier signal both in time and in frequency domains. Then intersymbol and interchannel interactions become the main detrimental factors limiting data transmission. The dispersion management technique allows the increase of bit-rate per channel and the 235 A. Hasegawa (ed.), Massive WDM and TDM Soliton Transmission Systems, © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

235–251.

236 S. K. TURITSYN, N. J. DORAN AND E. G. TURITSYNA ET AL.

suppression of interchannel interaction in WDM systems in comparison with the traditional soliton transmission [1]. The DM soliton is a novel type of optical information carrier with many attractive properties (see e.g. Refs. [2]- [28] and references therein) combining features of the traditional fundamental soliton and dispersion-managed non-return-to-zero (NRZ) transmission. DM soliton has enhanced power [6] in comparison with the corresponding fundamental soliton. This increases the signal-tonoise ratio (SNR), reduces Gordon-Haus jitter and, thus, improves transmission system performance. However, in systems/(transmission regimes) limited by nonlinear pulse interactions rather than by noise, the enhanced soliton power can become a less attractive feature. For instance, data transmission with high bit-rates of 40 Gb/s and more per channel requires dense pulse packing and, consequently, short soliton widths. The energy of DM soliton increases with decrease of the pulse width (or in other terms, with increase of the map strength). Average power of the traditional soliton signal increases with increase of the bit-rate (assuming soliton width to be a fraction of the time slot) as a square of the bit-rate. For DM soliton this growth is even more drastical. As a result, for short pulses the DM soliton power can become too high to be realized in practice [27]. Additionally, soliton interaction becomes an important issue with the increase of the signal power [27]. Energy control by corresponding reduction of the average dispersion is limited by fluctuations of the dispersion along the fiber and higher-order dispersive effects. Therefore, in designing soliton-based (and also general return-to-zero signal) transmission systems it is necessary to keep the soliton power large enough for the SNR requirement and suppressed jitter, and at the same time, not too large to avoid strong soliton interaction and to meet the telecommunication standards on the signal power. One way to find such an optimum for high bit-rates DM transmission is to use a chirped-return-to-zero signal with less power than DM soliton power in the corresponding system. Even though such carriers are not stable in a strict mathematical sense, and will emit radiation with propagation, they can be successfully used in practical systems. A challenge for the soliton theory, however, is to find high-bit-rate transmission regimes with a true periodic soliton-like signal propagation. Short-scale dispersion management is a means of controlling the DM soliton energy whilst keeping the average dispersion not too small and taking advantage of four-wave-mixing (FWM) suppression in WDM transmission by high local dispersion.

Traditional dispersion management for long-haul transmission typically assumes amplification distance to be much shorter than the dispersion compensation period. Existing technologies make it possible to manufacture fiber with the continuous alternation of positive and negative disper-

OPTICAL COMMUNICATION SYSTEMS WITH DM

237

sion sections of few kilometers long without any splicing [12]. This regime seems to be very promising for realization of ultra-high-bit rate transmission [12]-[17]. The fundamental properties of optical signal transmission in this regime are not yet studied. In this paper we investigate optical pulse transmission in DM fiber systems with compensation length much shorter than the amplification distance. Compared with lossless models, in systems with different periods of amplification and dispersion compensation (L) there is an important new degree of freedom - parameter Shortscale dispersion compensation leads to a reduction of the DM soliton power if we fix all system parameters and pulse width and will vary only Below we show that short-scale management can be considered as a possibility of an advantageous practical realization of the regime with weak map.

2. Basic Model Let us first recall the basic equations and the notations. The optical pulse propagation in a cascaded transmission system with varying dispersion is governed by

Here z is the propagation distance in [km], t is the retarded time in [ps], is the optical power in [W], D(z) is the group velocity dispersion measured in in ps/nm-km. We assume periodic dispersion management with the period are the amplifier locations. We consider a periodic amplification with the period If is constant between two consecutive amplifiers, then is an amplification coefficient after the fiber span between the k-th and (k – l)-th amplifier;

is the nonlinear refractive index; 0.05 In

(with

is the effective fiber area and

in dB/km) is fiber loss of the corresponding fiber,

is

speed of light, is the carrier wavelength. We consider a general case when L and are rational commensurable, namely, with integer n and m. In this paper we focus on the systems with shortscale management with It is customary to make the following transformation from original optical field E(z, t) to The evolution of the scaled envelope A is then given by the NLS equation with periodic coefficients

S. K. TURITSYN, N. J. DORAN AND E. G. TURITSYNA ET AL.

238

here

3. Path-average Model In this section we briefly recall derivation of the path-average model [7, 24]

describing change of the signal waveform over one compensation period. Equation (3) governing evolution (in z) of an optical pulse can be written in the Hamiltonian form:

with the Hamiltonian

The true breathing soliton presents a solution of Eq. (3) of the form with a periodic function

Of interest is to find a systematic way to describe a family of periodic solutions F with different quasi-momentum k. The basic idea suggested in

Ref. [7] is to use the small parameter to derive a path-averaged model that gives a regular, leading order in description of the breathing soliton. Averaging cannot be performed directly in Eq. (1) in the case of the large variations of However, a path-averaged propagation equation can be obtained in the frequency domain [7]. We will show that in some important limits an averaged equation for the periodic breathing pulse can be transformed to the integrable NLS equation. First, to eliminate the periodic dependence of the linear part we apply following [7] the so-called Floquet-Lyapunov transformation

Here is a Fourier transform of Important observation (will be used below) is that for a fixed amplitude of d amplitude of the variation of the function R decreases with the increase of It can be easily found that In the new variables the equation takes the form

OPTICAL COMMUNICATION SYSTEMS WITH DM

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here

exp is and Note that depends only on the specific combination of the frequencies given by the resonance surface Both the Fourier and the Floquet-Lyapunov transform (6) are canonical and the transformed Hamiltonian H reads

Now we apply Hamiltonian averaging. Let us make the following change of variables

In the leading order in

a path-averaged equation has the form

The Hamiltonian averaging introduced here represents a regular way to calculate next order corrections to the averaged model. Note that equation (10) possesses a remarkable property. The matrix element is a function of and on the resonant surface both and its derivative over are regular. This observation allows us to make the following quasi-identical transformation, which eliminates the variable part of the matrix element

where This transformation has no singularities. If the integral part in this transform is small compared with then in the leading order we get for

This is nothing more, but the integrable nonlinear Schrödinger equation written in the frequency domain. Obviously, this transformation is quasiidentical only if the integral in Eq. (11) is small compared with This

240 S. K. TURITSYN, N. J. DORAN AND E. G. TURITSYNA ET AL. is not true in a general case and that is why, in a general case a pathaveraged DM soliton given by solution of Eq. (10) has a form different from cosh-shaped NLSE soliton. However, if the kernel function in Eq. (11) is small

then the averaged model can be reduced to the NLSE. In other terms, this is a condition on the functions c(z) and d(z) that makes possible quasiidentical transformation. Path-averaged DM soliton propagation in systems satisfying requirement Eq. (13) is close to dynamics of the traditional soliton whilst keeping all advantages of suppression of FWM by high local dispersion. 4. Systems with Short-scale Management In this section we calculate matrix element for systems with a shortscale management and demonstrate that a path average propagation in this regime, (even with the large variations of the dispersion ) can be described by the integrable NLS equation. Matrix element T plays an important role in description of the FW mixing [25]. To be specific, let us consider a two-step dispersion map with the amplification distance and dispersion compensation period krn. Dispersion if if here The mean-free function R defined above can be found as if and After straightforward calculations, it can be shown that the matrix element in such a system is

Here The function is plotted in Fig. 1 for different m. Here dB/km, _ Real (top) and imaginary (bottom) parts of T are plotted for different (dashed line), ( solid line) and (dotted line). For the real part of T one cannot see difference between three curves on this scale. In the limit we obviously recover results of the traditional path-average (guding-center) soliton theory [9]-[11]. In Fig. 2 the function is plotted versus Y. Here and Zeroes of this function correspond to operation regimes with

OPTICAL COMMUNICATION SYSTEMS WITH DM

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suppressed FWM [25]. We estimate now the matrix element of the quasiidentical transformation

One can see that with increase of m (for the fixed other parameters) the path-averaged model (10) governing DM soliton propagation converges to the integrable NLS equation. It is seen also from Fig. 1 that with increase of m the imaginary part of T decreases, but the real part does

242 S. K. TURITSYN, N. J. DORAN AND E. G. TURITSYNA ET AL.

not change significantly. It is interesting to note that in the limit of a very short-scale management (large m) we again obtain for T the lossless model approximation (multiplied by the factor However, increase of m (decrease of L) under the fixed characteristic bandwidth of the signal makes insignificant oscillatory structure of the kernel. This means that if T(Y) is practically concentrated in some region then for large m the corresponding region in will be larger than for small m. For the pulses with the same spectral width this will mean that T is much flatter for large m and, as a matter of fact, for large m (small L) function T can be well approximated by a value T(0). As a result, NLSE model works rather well in this limit and solution (of the path-averaged model!) should be close to cosh-like soliton of the NLSE. Note that although it is known that for the lossless model in the so-called weak map limit [6, 7, 21] the DM soliton shape is close to cosh, this is not so obvious for system with loss and different periods of amplification and dispersion variations. In particular, this means that all the control techniques developed for the improvement of the traditional soliton transmission can be directly used in these systems.

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5. Single Pulse Propagation

In this section we consider results of numerical simulations for single pulse

propagation in systems with short-scale management. In contrast to the lossless model, here the evolution of soliton parameters over one period is asymmetric here due to loss. Rapid variations of the pulse width, peak power and chirp are accompanied by the exponential decay of the power due to loss. Nevertheless, numerical simulations have revealed that there exists a

true periodic solution that reproduces itself at the end of the compensation cell (in this case - at the end of the amplification period). In Fig. 3 evolutions

of the DM soliton (here strength of the map

peak power (right

top), chirp (left bottom) and full-width at half maximum (right bottom)

along one section are shown for a transmission system with the short-scale dispersion map (left top). The amplification distance is 40 km and the

dispersion compensation length is 4 km. The following parameters have been used in the simulations: dispersion in the two-step map is (see Fig. 3), nonlinear coefficient fiber loss

The observed DM soliton is very stable and propagates without radiation as it is seen in Fig. 4 (here and system parameters are the same as for Fig. 3. Figure 4 (bottom) illustrates the slow dynamics of the DM soliton in such systems. The pulse is shown stro-

244 S. K. TURITSYN, N. J. DORAN AND E. G. TURITSYNA ET AL.

boscopically (logarithmic scale) at the ends of the amplification sections. Figure 4 (top) shows the power evolution (normal scale) over one periodic

section. The DM soliton identified here has reduced power compared with the previously studied DM soliton regimes for the same width and propagating in fiber system with the same average dispersion (all the same parameters except ratio). This observation is illustrated by Fig. 5

where we present results of the modelling based on the zero-mode Gaussian approximation of DM soliton (in the expansion using complete basis of the chirped Gauss-Hermite functions, see for details Ref. [28]). Using this approach we have built up an evolution of DM soliton peak power dependence

OPTICAL COMMUNICATION SYSTEMS WITH DM

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on the pulse width, with the change of the dispersion compensation length keeping the same the average dispersion and the amplification distance. In

Fig. 5 the dependence of DM soliton peak power on the pulse width at the beginning of the compensation section is shown for different ratios of the dispersion period to the amplification distance km here): (solid line), 1 (long-dashed line), 0.5 (dashed line), 0.2 (dotted line) and 0.1 (dashed-dotted line). We show also for checking the peak power dependence for true DM soliton found numerically (in the full model) in the case (squares) and (rhombs). Note that the energy of the short-scale DM soliton is very close to that of the conventional soliton (though pulse is chirped and experiences breathing oscillations of the width and chirp during propagation). This is because the effective map strength is small here due to small L. It is seen from Fig. 5 that, indeed, short-scale dispersion management provides reduced power DM soliton for the same pulse width (and the same average dispersion and the same other parameters except ratio Taking into account a very fast growth of the soliton power with reduction

246 S. K. TURITSYN, N. J. DORAN AND E. G. TURITSYNA ET AL.

of the pulse width (after the curves in Fig. 5 pass some “critical” turn points, for instance, for such point, is around 16 ps), this effect can he very important for high-bit-rates transmission using short pulses. 6. Soliton Interaction Nonlinear pulse-to-pulse interaction is one the main limiting factors in highbit-rate optical data transmission. In this section we present results on soliton interaction in systems with short-scale management with amplification period and dispersion compensation period Numerical simulations in this section have included third-order dispersion and Raman effects. An important advantage of operating close to the integrable limit (weak maps) discussed above is that the well developed techniques to suppress soliton interaction can be applied. Figures 6 and 7 show the effect of initial phase alternation of neighboring solitons. Figure 6 shows propagation of two in phase solitons initially separated by 10 ps (100 Gb/s). The solitons collapse after approximately 300 km. In contrast, DM solitons with initial phase shift can propagate without fusion over 5000 km. Here strength of the map the peak power of the single soliton is 5.44 mW and the pulse width is 2.93 ps at the chirp-free point

OPTICAL COMMUNICATION SYSTEMS WITH DM

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(0.56 km from the end of the map). Recall that as reported in [19] interaction of DM solitons with large S does not depend on the initial phase

shift. Figures 8 and 9 show pattern transmission at 40 Gb/s (Fig. 8) without

248 S. K. TURITSYN, N. J. DORAN AND E. G. TURITSYNA ET AL.

initial phase alternation and 80 Gb/s (Fig. 9) with phase alternation. 40 Gb/s soliton signal can be transmitted over more than 9000 km (actually we have stopped the simulations at this point without obersving any significant pattern distortion). At 80 Gb/s bit rate long distance transmission was possible (in the considered regime) only with initial phase alternation. Note that solitons are very powerful and noise was not the main limiting factor in these simulations. Figure 10 shows an improvement of the system performance by using initial phase alternation. It is plotted total transmission distance versus average dispersion at 80 Gb/s. Here

strength of the map is pulse width in the minimum is 3.51 ps. The total transmission distance has been defined as a distance at which Q-factor becomes less than 6. The solid line is for initial pulses with phase alternation and dashed line is for in phase input pulses. Here dispersion (with varying average dispersion). It is seen that short-scale dispersion-managed systems are attractive candidates for transmission optical data at ultra-high-bit srates more than 100 Gb/s per channel. Optimization of such lines will lead to further improvement of the system performance.

OPTICAL COMMUNICATION SYSTEMS WITH DM

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7. Conclusions

In conclusion, we have identified a regime of stable optical pulse propagation in fiber systems with the short-scale dispersion management when the compensation period is much shorter than the amplification distance. The DM soliton in systems with short-scale management has reduced power compared to the usual DM soliton of the same width (and the same amplification distance and average dispersion). Short-scale management is a means of controlling the strength of the map (and consequently, pulse energy, interactions an so on) whilst keeping the average dispersion finite and taking advantage of FWM suppression in WDM by high local dispersion. We show that the path-averaged dynamics of chirped DM solitons in systems with short-scale management for weak maps is close to that in the integrable model. Therefore, DM solitons in such system possess the dual advantages of being chirped (that is important for suppression of the four-wave mixing in WDM systems) and of the integrable path-averaged dynamics, that allows the use of well developed mathematical tools for studying practical perturbations. Appling the initial phase alternation (the

technique originally developed for improvement of the traditional soliton systems) we demonstrate transmission at 80 Gb/s over 4300 km without slope (higher dispersion) compensation. We have demonstrated also that without phase alternation optical signal can be transmitted at 40 Gb/s over 9000 km and more in systems with short-scale management. Optimization

250 S. K. TURITSYN, N. J. DORAN AND E. G. TURITSYNA ET AL.

of the considered transmission lines will lead to further improvement of the system performance. Acknowledgements

The support of EPRSC, INTAS and RFBR is acknowledged.

References 1.

Mollenauer, L. F., Mamyshev, P. V. and Neubelt, M. J. : Demonstration of soliton WDM transmission at up to

error-free over transoceanic distances,

OFC’96, PD22-1, (1996). 2. Knox, F. M., Forysiak, W. and Doran, N. J. : 10 Gbit/s soliton communication systems over standard fibre at and the use of dispersion compensation, IEEE J. Lightwave Technol., 13, (1995), pp.1955-1963. 3. Suzuki, M., Morita, I., Edagawa, N., Yamamoto, S., Taga, H. and Akiba, S. : Reduction of Gordon-Haus timing jitter by periodic dispersion compensation in soliton transmission, Electron. Lett., 31, (1995), pp.2027-2028. 4.

Nakazawa, M. and Kubota, H. : Optical soliton communication in a positively and negatively dispertion-allocated optical fiber transmission line, Electron. Lett., 31,

(1995), pp.216-217. 5. Haus, H. A., Tamura, K., Nelson, L. E. and Ippen, E. P. : Stretched-pulse additive pulse mode-locking in fiber ring lasers: Theory and Experiment, IEEE J. Quantum 6.

7.

8. 9.

Electronics., 31, (1995), pp.591-603. Smith, N., Knox, F. M., Doran, N. J., Blow, K. J. and Bennion, I. : Enhanced power

solitons in optical fiber transmission line, Electron. Lett., 32, (1996), pp.54-55. Gabitov, I. and Turitsyn, S. K. : Averaged pulse dynamics in a cascaded transmission system with passive dispertion compensation, Opt. Lett., 21, (1996), pp.327330. Le Guen, D., Favre, F.. Moulinard, M. L., Henry, M., Devaux, F. and Georges, T. : 320 Gbit/s soliton WDM transmission over 1100 km with 100 km dispersioncompensated spans of standard fibre, ECOC’97, PD5, (1997), p.25. Mollenauer, L. F., Evangelides Jr., S. G. and Haus, H. A. : Long-distance soliton

propagation using lumped amplifiers and dispersion-shifted fiber, IEEE J. Lightwave Tech., 9, (1991), pp.194-201. 10.

Hasegawa, A. and Kodama,Y. : Guiding-center soliton in optical fibers, Opt. Lett.,

11.

15, (1990), pp.1443-1445. Blow, K. J. and Doran, N. J. : Average soliton dynamics and the operation of

12.

soliton systems with lumped amplifiers, IEEE Photon. Technol. Lett., 3, (1991), pp.369-371 . Evans, A. F. : Novel fibers for soliton communications, in Optical Fiber Commu-

nication Conference, OSA Technical Digest Series, Vol.2, (1998), p.22. 13. Liang, A. H., Hasegawa, A. and Toda, H. : High-speed soliton transmission in dense periodic fibers, Opt. Lett., 24, (1999), pp.799-801. 14. Turitsyn, S. K., Fedoruk, M. P. and Gornakova, A. : Reduced-power optical solitons in fiber lines with short-scale dispersion management, Opt. Lett., 24, (1999), pp.869-871. 15. Liang, A. H., Hasegawa, A. and Toda H. : Dense periodic fibers with ultralow fourwave mixing over a broad wavelength range, Opt. Lett., 24, (1999), pp.1094-1096. 16.

Richardson, L. J., Forysiak, W. and Doran, N. J. : Energy dependence of dispersion

17.

managed solitons in short period dispersion maps, in Quantum Electronics and Photonics Conference, QE14, (1999), p.209. Anis, H. , Berkey, G., Bordogna, G., Cavallari, M., Charbonnier, B., Evans, A.,

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Hardcastle, I., Jones, M., Pettitt, G., Shaw, B., Srikant, V. and Wakefield, J. : Continuous dispersion managed fiber for very high speed soliton system, ECOC’99,

I, (1999), p.230. 18.

Georges, T. and Charbonnier, B. : Reduction of the dispersive wave in periodically

amplified links with initially chirped solitons, IEEE Photon. Techn. Lett., 9, (1997), pp.127-128. 19.

Golovchenko, E. A., Pilipetskii, A. N. and Menyuk, C. R. : Dispersion-managed soliton interactions in optical fibers, Opt. Lett., 22, (1997), pp.793-795. 20. Breuer, D., Kueppers, F., Mattheus, A., Shapiro, E. G., Gabitov, I. and Turitsyn, S. K. : Symmetrical dispersion compensation for standard monomode-fiber-

based communication systems with large amplifying spacing, Opt. Lett., 22, (1997), pp.546-549 . 21. Yang, T.S. and Kath, W. L. : Analysis of enhanced-power solitons in dispersionmanaged optical fibers, Opt. Lett., 22, (1997), pp.985-987. 22. Kumar, S. and Hasegawa, A. : Quasi-soliton propagation in dispersion managed

optical fibers, Opt. Lett., 22, (1997), pp.372-375. 23.

Hasegawa, A., Kodama, Y. and Maruta, A. : Recent Progress in DispersionManaged Soliton Transmission Technologies, Opt. Fiber Techn., 3, (1997), pp.197213. 24. Medvedev, S. B. and Turitsyn, S. K. : Hamiltonian averaging and integrability in nonlinear systems with periodically varying dispersion, JETP Letters, 69, (1999),

pp.465-469. 25.

Burtsev, S. K. and Gabitov, I. : Four-wave mixing in fiber links with dispersion

management, in Proc. of II International Symposium on Physics and applications of Optical Solitons in Fibers, (1997), p.261. 26. Kutz, J. N., Holmes, P., Evangelides Jr., S. G. and Gordon, J. P. : Hamiltonian dynamics of dispertion managed breathers, JOSA B, Vol.15, (1998), pp.87-94. 27.

Govan, D. S., Forysiak, W. and Doran, N. J. : Long-distance 40-Gbit/s soliton transmission over standard fiber by use of dispersion management, Opt. Lett., 23,

(1998), pp.1523-1525.

28. Turitsyn, S. K., Shapiro, E. G. and Mezentsev, V. K. : Dispersion-managed solitons and optimization of the dispersion management, Opt. Fiber Techn., Invited paper, Vol.4, (1998), p.384.

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REAL TIME PMD COMPENSATION

FOR RZ TRANSMISSION SYSTEMS

M. ROMAGNOLI, P. FRANCO, R. CORSINI AND A. SCHIFFINI Pirelli Cavi e Sistemi, s.p.a., vide Sarca 222, 20126 Milano, Italy AND M. MIDRIO Istituto Nazionale per la Fisica della Materia Dipartimento di Ingegneria Elettrica Gestionale e Meccunica Università degli Studi di Udine, vide dalle Scienze 208, 33100 Udine, Itdy

1. Introduction

One of the major drawbacks in high bit-rate transmission systems is the presence of random birefringence of the fibers leading to polarization mode

dispersion ew techniques of fiber fabrication permit to achieve low values of PMD, but this does not necessarily hold for the large amount of already installed fibers. For those, the value of PMD may result increased with respect to that of new fibers either because of the phenomenon of stress relaxation in aged silica and because of the old fabrication techniques. In practice, the transferring of the lab technology to the field is often limited by this problem. Attempts to envisage PMD compensation techniques have been carried out [l]-[4]. Considering that the main difficulty arises from the time dependent stochastic nature of PMD, all the reported compensation devices are required to continuously feedback the input signal. Because of this reason these devices have a compensation rate that does not account for relatively fast fiber DGD fluctuations. Moreover second order PMD is a further issue to be taken into account in the compensation technique. This contribution becomes increasingly important in high bit-rate transmission systems [5, 6]. Means to compensate for order PMD have been reported [3], but in the practical implementation are quite cumbersome. The basic idea we propose in this work relies on a technique that independently of the 253 A. Hasegawa (ed.), Massive WDM and TDM Soliton Transmission Systems, 253–263. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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input state of polarization substantially converts temporal fluctuations into frequency fluctuations. This means, when used at the receiver, it permits to restore the signal profile in the temporal domain leading therefore to a substantial opening of the electrical eye.

2. Theory

Let us consider an optical transmission system affected by polarization mode dispersion (PMD). As it is well known, if the system is linear, the electrical field of a pulse which has travelled down the link may be written, at the output face, as

where

and

are the complex fields in the two orthogonal principal states

of polarization (PSP) at the fiber output and

the pulse shape at the fiber input, with and the percentage of the pulse power lying in the input PSP corresponding to and respectively. Finally, and are the arrival time for the two states, respectively. If the origin of the time axis is chosen in correspondence of the arrival time instant a pulse would

have in the absence of PMD, it turns out that

with a Maxwellian distributed random variable which accounts for the link differential group delay (DGD). The random variable is completely statistically described when its average value is given, and this reads as with the link length. We now want to design a device

that compensated for the link DGD. Ideally, what we are looking for is a device that, once inserted at the receiver stage, 1. may exactly compensate for the timing jitters of their values, at least within a bit time;

and

irrespectively

2. does not change the pulse shape. In the following, we show that such a device may be designed by exploiting a phase modulator, which has to be operated synchronously to the transmission bit stream, and a dispersive medium. Moreover, we will show that this device can exactly match the two above requirements for a pulse having a Gaussian shape. To this end, we first recall a general result. Consider the propagation of

the optical pulse

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in a linearly dispersive medium, with dispersion D. As it is pretty well known, if one neglects the transit time, the pulse shape at the output of

the dispersive medium may be written as

where

The 1. 2. 3.

dispersive propagation hence has a threefold effect on the pulse it introduces a phase displacement, given by the term; it modifies the pulse shape, which changes from it causes a time shift, equal to This is expected, since may be regarded as a frequency detuning with respect to a suitable frequency reference frame, and the propagation takes place in a dispersive medium. We now use these results to design the PMD compensator. Suppose that one of the components the optical pulse which has travelled in the PMDaffected link has reached the detection stage with a delay with respect to the origin of the time axis (see Fig. 1) We introduce the pulse in the device schematically depicted in Fig. 2, where we suppose that the phase modulation is given by a truncated parabolic curve, synchronously driven with respect to the pulse arrival time in the absence of PMD. By moving the time axis origin so that it coincides with the delay it turns out that signal of Fig 2 is given by

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where

is the optical pulse shape, and is related to the phase modulation depth. On the basis of the general features of dispersive propagation we mentioned above, we may conclude that when this pulse crosses the dispersive medium, it undergoes the three effects cited above. In particular, we may use the induced time shift to retime the pulse. To this end, we note that, letting be the dispersion of the dispersive medium, we need to have

Note that in this condition, which allows the first requirement for the PMD compensator to be satisfied, the only values of and are involved, being it independent on the delay In other words, the above relation between and says that it is possible to exactly compensate for an arbitrary delay by only acting on the parameters of the phase modulator

and of the dispersive medium. The only limitation is posed by the fact that the pulse has to be within the phase modulation curve, i.e. the arrival delay has to be lower than half of the bit time. Quite obviously, this nice result is related to the peculiar modulation curve we have chosen; indeed being it parabolic, induces a frequency shift which linearly scales with the delay We now turn our attention to the second requirement we want the compensator to match. As we have seen above, when the pulse propagates inside the dispersive medium it undergoes a shape change, which leads from Hence, the compensator may preserve pulse shape if the input pulse is an eigenfunction for a dispersive medium, namely a Gaussian shaped function

In addition, we note that the above expression for shows that, at the output of the phase modulator, the pulse has the quadratic chirp In order to preserve the pulse shape at the compensator output, the dispersive medium has to be such that at its output the chirp has turned to and this is obtained with an overall dispersion

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This is a second relation between the parameters of the phase modulator and of the dispersive medium, and, once combined with the first one we

have derived before, yields the design criteria for the PMD compensator. As a matter of fact, it turns out that the phase modulation depth, and the dispersion of the dispersive medium should respectively be given by

Note that the same design criteria may alternatively be obtained by noticing that the pulse shape at the compensator output is proportional to the Fourier transform of the input pulse. The reader may find details on this approach in Ref. [7]. 3. Numerical results In order to validate the above theory, we performed two set of numerical simulations by considering a system affected by PMD, and measuring the eye closure in the presence and in the absence of the PMD compensator. In the first set of simulations, that were repeated up to 3000 times for each case we considered, a single pulse is splitted into two components. These are delayed the one with the respect to the other of a quantity,

which represents the PMD effect. The signal obtained in this way is used to feed a photodiode, which is followed by a Bessel–Thompson electrical filter. The eye closure at the filter input and output are computed, either in the presence or in the absence of the PMD compensator. Note that, in these simulations, the PMD compensator was supposed to provide a truncated parabolic modulation curve. Whereas, in the second set of simulations, that we describe below, the phase modulator was supposed to be driven by a sinusoidal wave. Concerning the differential delay we modeled it as a Maxwellian distributed random variable. To construct it, it suffices to note that the Maxwellian distribution is completely specified by its mean value only. Using this property, we set the average differential group delay at the desired level, and considered three gaussian distributed random variables, say and having zero mean, unitary variance, and being independent the one of the others. The differential group delay is evaluated

as

In the simulations, the input pulse was one single Gaussian shaped pulse, with full–width at half maximum duration The bit rate of the transmission was The mean differential group delay was

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The statistical distribution of the eye closure measured at the electrical filter input in the absence of the PMD compensator is reported in Fig. 3a). The extimated eye closure is shows the eye closure in the presence of a PMD compensator having a phase modulation depth, and a dispersive device giving an overall dispersion The eye closure has been compressed down to

Analogous results are finally reported in Fig. 4, for the eye closure at the filter output. The extimated eye closure in the absence and in the presence

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of the PMD compensator is and respectively. Figure 5 refers to the second set of simulations we performed. As we mentioned above, in this set of simulations we considered a more realistic case, in which the phase modulator was driven by a sinusoidal wave. In addition, we use a split step vectorial code to simulate pulse propagation, and in this way we kept into account for all the PMD orders. The parameters of the simulations were the following. The link was constituted of six amplified spans, each 100 km long. The fiber attenuation was 0.25 dB/km. Fiber dispersion was compensated for at each amplification stage, with a compensation percentage equal to 90 %. The data stream at 10 Gbit/s, modeled by a sequence of 16 pulses, was made of 35 ps long Gaussian pulses. We set the line PMD to 1.00, 1.16 and These values corresponds to overall DGD equal to 24.5, 28.4 and 30.6 ps, respectively. Recalling the the DGD is Maxwellian distributed, this also means that the “istantaneous” DGD has probability of being larger that 73.5, 84.6 and 91.8 ps, respectively. Note that this level of probability is important in the framework of optical communication systems. As a matter of fact, it would translate in having DGD larger than the above values for 5.2 minutes per year.

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The results of the simulations are shown in Fig. 5, in which each point results from statistics over more than random realizations of birefringence of the considered link. Figure 5 illustrates the complementary probability density as a function of the eye-diagram penalty of the system for the three different values of PMD. Based on the simulation, we show that for the second largest value of istantaneous DGD (84.6 ps) there is a probability of that the eye closure will be greater than 0.8 dB. The good performance of PMD compensation achieved with sinusoidal phase modulation as an alternative to the ideal case of parabolic phase modulation is indeed

verified. 4. Experimental results

To test the theory we performed a lab-trial based on the experimental setup shown in Fig. 6 where the transmission line is simulated with a pro-

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grammable polarization controller and a variable optical vectorial delay. A 10 GHz rep-rate source was able to deliver a train of 35 ps long pulses. The generated train of pulses was sent into a Mach-Zehnder modulator driven by a pattern generator operating at 10 Gbit/s. The data stream was then sent through a polarization controller and a PMD emulator with programmable DGD. The signal affected by instanteneous DGD was then sent to the PMD compensator (PMDC), then to a amplifier-attenuator pair that was used to vary the optical signal to noise ratio, and then sent to detection unit. The detail of the PMDC is shown in Fig. 6b). The 20 km long span of step-index fiber gave a delay whereas the modulation amplitude was thus giving As an example we report in Fig. 7, a few minutes persistency eye diagram measured with PMDC turned off (upper trace), and on (lower trace) PMDC for input (from left to right). By increasing the input DGD the eye diagram shows an increasing penalty that is excellently compensated for by the PMD up to For input ps the restoration of the initial eye closure is almost good as for the other cases but we note that this is the limit of correct operation of the device. The explanation relies on the fact that in the experimental set up we used sinusoidal modulation that for instantaneous delays larger than

from the center of the time slot deviates from the parabolic

approximation we used in the theory for the phase modulation.

The performance of the device is reported in Fig. 8. The BER was mea-

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sured as a function of the signal to noise ratio at the receiver. In the figure we reported two sets of measurements concerning the case with (empty symbols) and without (filled symbols) PMDC. Each set refers respectively to (circle), 40 ps (squares) and 50 ps (triangles). It is worth noticing that for 50 ps of DGD the penalty reduction achieved with the insertion of the PMDC at the receiver was 6.1 dB at

5. Conclusion

In conclusion we have demonstrated that PMD compensation is achievable with a simple technique that exploits the property of the dispersive propagation of a phase-modulated signal. The amount of PMD that can be compensated for is expressed in terms of maximum instantaneous that we achieved from our lab trials. Present scheme of PMDC is synchronous with the modulation frequency and its restoring performance is limited to half time slot. The maximum compensation range of DGD is sufficient for most of the applications in terrestrial transmission systems. References 1.

Heismann, F., Fishman, D. A. and Wilson, D. L. : Automatic compensation of firstorder polarization mode dispersion in a 10 Gbit/s transmission system, ECOC’98, (1998), pp.529-530.

REAL TIME PMD COMPENSATION . . . 2.

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Chbat, M. W., Soigné, J. P., Fuerst, T., Anthony, J. T., Lanne, S., Fèvrier, H., Desthieux, B. M., Bush, A. B. and Pennickx, D. : Long term field demonstration of

optical PMD compensation on an installed OC–192 link, OFC’99, PD12, (1999). 3. Glingener, C., Schöpflin, A., Färbert, A., Fischer, G., Noé, R., Sandel. D., Hinz, S., Yoshida-Dierolf, M., Mirvoda, V., Feise, G.. Herrmann, H., Ricken, R., Sohler, W. and Wehrmann, F. : Polarization mode dispersion compensation at 20 Gbit/s with 4.

5. 6.

7. 8.

9.

a compact distributed equalizer in OFC’99, PD29, (1999). Bülow, H. : Limitation of optical first-order PMD compensation, OFC'99, WE1,

(1999). Ciprut, P., Gisin, B., Gisin, N., Passy, R., Von der Weid, J. P., Prieto, F. and Zimrner, C. W. : Second-order polarization mode dispersion: impact on analog and digital transmissions, IEEE J. Lightwave Technol.,Vol.l6, (1998), pp.757-771. Foschini, G. J., Jopson, R. M., Nelson, L. E. and Kogelnik, H. : The statistics

of PMD-induced chromatic fiber dispersion. IEEE J. Lightwave Technol., Vol.17, (1999), pp.1560-1565. Romagnoli, M., Franco, P., Corsini, R., Schiffini, A. and Midrio, M. : Time-domain Fourier optics for polarization-mode dispersion compensation, Opt. Lett., Vol.24, (1999), pp.1197-1199. Andresciani, D., Curti, F., Matera, F. and Daino, B. : (1987). Measurement of the group-delay difference between the principal states of polarization on a lowbirefringence terrestrial fiber cable, IEEE J. Lightwave Technol., Vol.5, (1987), pp.1618-1622. Poole, C. D. and Wagner, R. E. : Phenomenological approach to polarization dispersion in long single-mode fibers, Electron. Lett., Vol.22, (1986), pp.1029-1030.

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PROPAGATION OF 3-PS DISPERSION-MANAGED SOLITON PULSE UNDER THE INFLUENCE OF THIRD-ORDER DISPERISON

Y. TAKUSHIMA, X. WANG AND K. KIKUCHI Research Center for Advanced Science and Technology, University of Tokyo 4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan

1. Introduction

Dispersion-managed (DM) solitons are stationary pulses formed by the interaction between the periodically distributed group-velocity dispersion (GVD) and the fiber nonlinearity. A transmission system using DM solitons is an attractive candidate for future high-speed and long-distance optical communications, because the existence of an eigensolution in the DM link ensures the stability of pulse transmission [l]-[3]. In addition, the small average GVD suppresses the Gordon-Hans jitter and soliton-soliton interaction, which are obstacles inherent in soliton systems [4]-[6]. In high-speed DM soliton systems, as the pulse width becomes narrower with increase in the bit rate, perturbation of the dispersion slope (or the third-order dispersion, TOD) becomes more significant. However, most of experiments on DM soliton transmission were carried out under the condition that the TOD could be ignored, and only a few theoretical studies have been reported on the influence of TOD. In Ref. [7], it has been theoretically shown that the TOD makes the soliton's profile asymmetric under the condition of weak nonlinearity but that the impact of TOD on DM solitons is not significant when the amount of TOD is small. In Ref. [8], we have numerically shown that there exists an eigensolution to DM soliton under the influence of the TOD, which can stably travel over multiple DM periods. The aim of this paper is to demonstrate the ultrashort pulse transmission in a periodically amplified fiber link using DM solitons under the influence of the TOD. The outline of this paper is as follows: In Section 2, we numerically obtain the eigensolution under the influence of the TOD in 265 A. Hasegawa (ed.), Massive WDM and TDM Soliton Transmission Systems, 265–276

© 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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a lossless fiber link with a symmetric dispersion map, and then discuss its properties [8]. In Section 3, we calculate the eigensolution to DM solitons in a practical DM fiber link, which was actually used in the experiment. We adjust the arrangement of anomalous and normal diserpsion fibers so that the eigensolution is nearly chirp-free at the input end. By using this arrangement, it becomes much easier to prepare the input pulse close to the eigensolution. In Section 4, we demonstrate 3.0-ps pulse transmission through the DM link in which the TOD length is as long as the dispersionmanaged period [9, 10].

2. Eigensolution to DM solitons under the influence of TOD Consider a periodically dispersion-managed link with a period of which consists of alternating sections of anomalous and normal dispersion fibers. Evolution of the complex electric field is governed by the nonlinear Schrödinger equation which includes the TOD term:

where is the normalized dispersion having a value of and is the normalized third-order dispersion. We numerically find a periodic solution which satisfies where the constant is the phase shift during propagation through a segment and is the delay time in the group-velocity frame. Such a solution can travel through the dispersion-managed transmission line without distortion. When the TOD is zero, it has been known that there exists a solution for any value of and it can be calculated from a solution with a nonmoving center: On the other hand, when the TOD is not zero, the solution does not always exist for all We modify the algorithm given in Reference [1] in order to take the delay into account, and calculate the eigensolution numerically [8]. As an example, we calculate eigensolutions in a strong dispersion management map. Figures and show waveforms, instantaneous frequencies and spectra at the midpoint of the anomalous dispersion fiber for various values of Here, we assume that and the period of dispersion management (shown in the top figure in Fig. 2). The energy of the pulse, is set to 13.2. The solid curves show those of the eigensolution for As is well known, the center part of the waveform is nearly Gaussian, but there are many zero points in far-field tails. The full width at half maximum, is 1.6, and the deduced map-strength,

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The dotted and dash-dotted curves in Fig. 1 correspond to the eigenso-

lutions for and 1.2, respectively. As is clearly seen in Fig. l(a), the waveform of the eigensolutions for non-zero TOD is very similar to that for and the symmetric pulse shape is maintained. The frequency chirp of the eigensolutions is greatly modified by TOD. When the TOD is not present, the eigensolution is unchirped at the mid-

point of the anomalous dispersion fiber [1]. On the other hand, the eigensolutions for non-zero TOD have the nonlinear frequency chirp as shown in Fig. l(b). The instantaneous frequency is kept near zero in the vicinity of the center, but shifts toward lower frequences at leading and trailing

edges. When increases, as indicated by the dash-dotted curve, the nonlinear chirp becomes strong. As a result, the peak of the spectrum moves to the lower frequency region, as shown in Fig. l(c). Figure 2 illustrates variations of the pulse width and the bandwidth over two DM periods for It is observed that the pulse width and the bandwidth arc symmetrical around the center of each section. The pulse width becomes the smallest at the midpoint of the anomalous fiber section

and the time-bandwidth product of the pulse is also minimum at this point. This means that the shortest pulse is almost chirp-free. These features are the same as those of the eigensolution for

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In order to examine the stability of eigensolutions, we calculate their evolution over many DM periods. Figure 3 shows the evolution of the waveform of the eigensolution for (dotted curves in Fig. 1). To illustrate the stability, the vertical axis is shown on a log scale. Extremely high stability of the eigensolution is shown in this figure. 3. Eigensolution to DM soliton in a realistic system and the dispersion map design

The above analysis deals with an ideal dispersion-managed link where the fiber loss is neglected and the dispersion map is symmetric. In practical systems, however, the fiber loss cannot be ignored and is compensated periodically by optical amplifiers. Also the dispersion value of compensation

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fibers is usually much larger than that of transmission fibers for realistic system design. Hereafter, we consider a more practical link taking the effect of the fiber loss and the asymmetric dispersion map into account. As an example, we calculate a dispersion managed link where one period

consists of a 40-km-long dispersion-shifted fiber (DSF) and a 1.6-km-long single-mode fibers (SMF) (shown in the inset in Fig. 4(a)). GVD values of the SMF and the DSF are 15.8 and respectively. Both kinds of fiber are assumed to have the same TOD of and the fiber loss of 0.25 dB/km. These values are the same as those of the fibers actually used in the experiment. We calculate the eigensolution of DM solitons in this dispersion-managed link by the method described in Section 2. The pulse energy is set to 1.1 pJ so that the minimun pulse width of the eigensolution within one period of the DM link is about 3 ps. Figures 4(a) and (b) show the evolution of the pulse width and the bandwidth of the eigen soliton solution within one period of the DM link. The shortest pulse exists in the anomalous section (i.e. in the SMF). The variations of the pulse width and the bandwidth are not symmetric about the midpoint of each section due to the influence of fiber loss. In fact, the eigensolution has the shortest pulse width and is almost chirp-free at about 400 m ahead of the SMF end. The eigensolution is a chirped pulse at the input end of each DM period. For stable DM soliton transmission, the pulse launched into the DM fiber link should be identical to the eigensolution, and hence the simultaneous control of the waveform and the chirp is indispensable. However, it

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is rather difficult as the pulse width becomes shorter. One way to overcome this difficulty is to design the dispersion map so that the eigensolution is

chirp-free at the input end. For this purpose, we rearrange the dispersion map as shown in the inset in Fig. 5(a); one period of the dispersion map consists of a 400-m SMF, a 40-km DSF and a 1.2-km SMF in this order. We calculate eigensolutions in the modified fiber link. Figure 5 shows the evolution of the pulse width and the bandwidth of the eigensolution within one period of the modified fiber link. The eigensolution has a minimum pulse width at the input end. The waveform and the spectrum of the eigensolution at the input end are shown in Fig. 6. The full width at half maximum (FWHM) is 3.2 ps and its time-bandwidth product is 0.486. The central part of the spectrum is similar to that of the Gaussian pulse, but the siderobe has asymmetric ripples due to the TOD. With this map configuration, we confirmed by simulation that the solution shown in Fig. 6 can travel stably over multiple DM periods without any distortion, as shown in Fig. 7(a). The TOD length calculated from the

pulse width is 46 km, which is of the same order of the averaged-dispersion

length (20.8 km) and the dispersion management period (41.6 km). The nonlinear length becomes much shorter than the TOD length because the strong dispersion management enhances the soliton energy. In a conventional soliton system having the same characteristics lengths, the strong interaction between the TOD and the fiber nonlinearity causes severe distortion of both spectrum and pulse shape within a distance much shorter than the TOD length [12]. To conduct a transmission experiment, it is almost impossible to produce an initial pulse identical to the eigensolution. However, when the input

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pulse is close to the eigensolution, the pulse can travel without severe wavefrom distortion. As an example, we calculated the evolution of the waveform of a pulse, whose energy and pulse width are the same as those of the eigensolution. The result is illustrated in Fig. 7(b). Though the pulse width shows some small oscillation, it is quite stable.

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Next, we calculate the evolution of the pulse when the allocation of the dispersion is changed. The dispersion map consists of an 800-m SMF, a 40-km DSF and an 800-m SMF in this order. Figure 7(c) shows the

evolution of the pulse in this dispersion map. The pulse width and energy of the input pulse are kept the same as those used in the calculation of Fig. 7(b). Although this dispersion map has the same average dispersion and average TOD as those of the previous one, the eigensolution at the input end is not chirp-free, and its pulse width is also different from that used in the calculation of Fig. 7(b). Figure 7(c) shows that the launched

pulse can not approach to the eigensolution, and the pulse transmission is degraded severely and exhibit strong pulse width oscillation. Therefore, the proper choice of the lengths of SMF’s is important to obtain the stable pulse propagation. 4. Experimental setup and results

On the basis of the eigen soliton solution condition obtained in the previous section, we construct a transmission system shown in Fig. 8. The fiber link consists of two DM periods. Each period is composed of three fibers as indicated by the dotted line box. The average zero-dispersion wavelength

of this link is measured to be 1550.3 nm, and the average TOD is 0.073 The total length of the fiber link is 83.2 km.

In order to produce pulse trains having uniform characteristics in a wide wavelength range, we used a tunable mode-locked semiconductor laser (MLLD) with a repetition rate of 10 GHz and a pulse width of 1.9 ps in conjunction with an optical bandpass filter for the precise adjustment of

the pulse width. First, the initial chirp of the pulse from the MLLD is compensated by a 25-m SMF, and then the pulse is filtered out by the bandpass

filter with a 3-dB bandwidth of 0.9 nm. Figure 9(a) shows examples of the

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autocorrelation trace and the spectrum of the pulse. The autocorrelation trace agrees well with that of a pulse envelope. The autocorrelation width and the deduced pulse width (FWHM) are 4.7 ps and 3.0 ps, respectively. In the tuning wavelength range, the variations of both the pulse width and the bandwidth are kept within The 3.0-ps pulse train is then launched into the DM link. The output pulse was evaluated by using an optical spectrum analyzer and a Si-APD autocorrelator with small polarization dependence [11]. First, we set the center wavelength and the launched power to 1551.7 nm and 11.5 dBm, respectively, so that the 3-ps pulse becomes close to the eigensolution shown in Fig. 6. Figure 9(b) shows the autocorrelation trace and the spectrum of the output pulse after propagation in the 83.2km DM link. The original autocorrelation width of 4.7 ps is maintained even though the transmission distance is twice the TOD length (the TOD length for a 3.0-ps pulse in this link is 46 km). The small pedestal component observed in Fig. 9(b) is considered to be due to a dispersive wave, which is produced because the launched pulse is not exactly equal

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to the eigensolution. In spite of the high input power, the excess spectrum broadening due to the self-phase modulation is well suppressed, and the output spectrum is still compact after propagation. This indicates that the DM soliton is formed while emitting dispersive wave components. Next, we measured the autocorrelation width of the output pulse by changing the wavelength and the averaged power (Fig. 10). The dots in Fig. 10 show combinations of these parameters where the measurement was made. The contour map in Fig. 10 is drawn by extrapolating autocorrelation widths measured at each point.

From this figure, we find that when the average input power is low enough to make the nonlinearity insignificant, the pulse width remains short only near the zero-dispersion wavelength with its width a little more expanded by TOD. As the input power becomes higher, the short pulse region gradually moves into the anomalous dispersion area, since the zerodispersion transmission is now degraded by the interaction between fiber nonlinearity and TOD [12]-[14], while the DM soliton is formed in the anomalous dispersion region. Especially, when the input power and center

wavelength approach the eigensolution condition, undistorted pulse transmission is realized. It should be also noted that the pulse width is maintained in the wide wavelength range near the DM soliton condition. This

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means that the DM soliton can provide us with higher dispersion tolerance than the linear DM system.

To confirm the importance of the dispersion map design, we adjusted both the lengths of the SMF’s to be 800 m. This dispersion map is the same as that used in Fig. 7(c). The result is shown in Fig. 11. This contour map of autocorrelation width differs significantly from Fig. 10. Obviously, the short pulse transmission is realized only around the zero-dispersion wavelength. Stable transmission is prohibited by the interaction between the fiber nonlinearity and the TOD when the power increases [12]-[14]. In the anomalous dispersion area, no stable soliton-like transmission area exists and the pulse broadens so much after 83.2 km transmission. The reason is that the input pulse differs markedly from the eigensolution of this map. 5. Conclusions

We have shown that the eigensolution to DM soliton can exist under the influence of the TOD by numerical calculations. It was demonstrated that 3.0-ps pulse train, which was close to the eigen solution to DM soliton, could be transmitted over an 83.2-km distance under the strong influence

of TOD.

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Acknowledgement The authors wish to thank K. Inoue of NTT for his support for this study. References 1.

Nijhof, J. H. B. , Doran, N. J., Forysiak, W. and Knox, F. M. : Stable soliton-like propagation in dispersion managed systems with net anomalous, zero and normal

dispersion, Electron. Lett., 33, (1997), pp.1726-27. 2. Turitsyn, S. K. , Mezentsev, V. K. and Shapiro, E. G. : Dispersion-managed solitons and optimization of the dispersion management, Optical Fiber Technol., 4, (1998), pp.384-452. 3. Lakoba, T. I. and Kaup, D. J. : Hermite-Gaussian expansion for pulse propagation in strongly dispersion managed fibers, Phys. Rev. E, Vol.58, No.5, (1998), pp.67286741. 4. Suzuki, M. , Morita, I., Yamamoto, S., Edagawa, N., Taga, H. and Akiba, S. : Timing jitter reduction by peoriodic dispersion compensation in soliton transmission, OFC’95, PD20, (1995). 5. Smith, N. J., Forysiak, W. and Doran, N. J. : Reduction Gordon-Haus jitter due to enhanced power solitons in strongly dispersion managed systems, Electron. Lett.,

6. 7.

32, (1996), pp.2087-2088. Yu, T., Golovchenko, E. A., Pilipetskii, A. N. and Menyuk, C. R. : Dispersionmanaged soliton interactions in optical fibers, Opt. Lett., 22, (1997), pp.793-795. Lakoba, T. I. and Agrawal, G. P. : Effect of third-order dispersion on dispersionmanaged solitons, Conference on Lasers and Electro-Optics (CLEO’99), CWC5,

9.

(1999). Takushima, Y. and Kikuchi, K. : Dispersion-managed soliton under the influence of third-order dispersion, CLEO’99, CWC2, (1999). Takushima, Y., Wang, X. and Kikuchi, K. : Transmission of 3-ps dispersion man-

10.

aged soliton pulses over an 80-km distance long under the influence of third-order dispersion, Electron. Lett., 35, (1999), pp.739-740. Wang, X., Takushima, Y. and Kikuchi, K. : Influence of dispersion map on 3-ps pulse

8.

transmission in an 80-km dispersion-managed fiber link, Conference on Lasers and Electro-Optics/Pacific Rim (CLEO/Pacific Rim ’99), ThB2, (1999). 11. Kikuchi, K. : Highly sensitive interferometric autocorrelator using Si avalanche photodiode as two-photon absorber, Electron. Lett., 34, (1998), pp.123-125. 12. Wai, P. K. A., Menyuk, C. R., Chen, H. H. and Lee, Y. C. : Effect of axial inhomogeneity on solitons near the zero dispersion point, IEEE J. Quantum Electron., 24,

13. 14.

(1988), pp.373-381. Naka, A. and Saito, S. : In-line amplifier transmission distance determined by selfphase modulation and group-velocity dispersion. J. Lightwave Technol., 12, (1994), pp.280-287. Wang, X., Takushima, Y. and Kikuchi, K. : Analysis of dispersion-managed optical fiber transmission system and performance restriction from third-order dispersion, IEICE Transaction on Electron./Commun. E82-B, (1999), pp.1141-1148.

TOLERANCE OF SCALAR AND VECTOR SOLITONS TO RANDOM VARIATIONS OF MAP PARAMETERS IN DISPERSION MANAGED OPTICAL FIBER LINKS

F. KH. ABDULLAEV, B. B. BAIZAKOV, B. A. UMAROV AND D. V. NAVOTNY Physical-Technical Institute of

the Uzbek Academy of Sciences, G. Mavlyanov str. 2-b, 700084, Tashkent, Uzbekistan AND M. R. B. WAHIDDIN

Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50603, Kuala Lumpur, Malaysia

Abstract. The propagation of scalar and vector solitons in dispersion managed optical fiber links with randomly varying map parameters has been studied. Two types of randomness are considered for scalar solitons: ran-

dom dispersion magnitudes and random lengths of fiber spans. In the case of vector solitons the consequence of the random fiber birefringence has been examined. Disintegration of a scalar soliton and splitting of a vector soliton propagating in such optical communication line are shown to occur by numerical simulations. Comparison between results of numerical simulations, based on the full NLSE and corresponding variational equations reveals their good agreement. Theoretical prediction for the scalar soliton break-up distance in the averaged dynamics limit has been confirmed by numerical solution of corresponding stochastic ODE’s.

1. Introduction

Basic features of dispersion managed solitons, which make them attractive for long haul optical communications, seem to be understood in sufficient detail by now [1]. In technical realization of transmission systems for ul277 A. Hasegawa (ed.), Massive WDM and TDM Soliton Transmission Systems, 277–288. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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tralong distances, one of the important issues would be the tolerance of a currier pulse to random deviations of the dispersion map parameters from designed values. Specifically, what fluctuations of parameters of dispersion map a DM soliton can withstand, covering expected distance without loosing integrity. Since an optical transmission line based on the dispersion compensation technique represents a chain of periodically linked pieces of fibers with alternating anomalous and normal group velocity dispersion (GVD), it is reasonable to suggest, that in practical situations corresponding fiber pieces will not be identical. Most likely there will be a random distribution of fiber GVD magnitudes, span lengths, birefringence etc. over certain mean values. As a result of breaking of periodicity, DM soliton suffers random perturbations along its path, and eventually disintegrates or splits after some propagation distance. The objective of this study is the theoretical and numerical investigation of DM soliton dynamics in optical transmission lines with random dispersion map, when dispersion magnitudes of spans or span lengths are ran-

domly distributed over certain mean values. In the case of vector DM solitons, the purpose is to investigate the consenquence of the random fluctuations of the fiber birefringence. Main question to be addressed will be what

distance a scalar/vector DM soliton covers before it disintegrates/splits due to randomness of the dispersion map. The present study deals with the lossless case. 2. Disintegration of a Scalar Soliton

The propagation of a scalar DM soliton is governed by the nonlinear Schrödinger equation (NLSE) for a dimensionless envelope of the electric field

where is a stepwise function describing the dispersion map, which in its turn may be represented as consisting of periodic and random parts

In the absence of the randomness it would be a periodic function where and are the fiber segment lengths. We will be concerned with the Gaussian white noise model for namely

where a standard deviation of dispersion magnitudes and span lengths respectively.

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279

We employ the variational approach developed by Anderson [2] to reduce the underlying NLSE to a system of ODE’s for DM soliton parameters. With this purpose we will search for the solution to Eq. (1) as a localized

waveform

where

and

are the complex amplitude, width, chirp parameter

and phase respectively,

is the localized function specifying the pulse

profile. Here the conserving quantity is

Performing standard calculations [2] we arrive to the following variational equations, describing the evolution of the soliton parameters

where

For the Gaussian ansatz and for

one obtains corresponding values are

Before proceeding to the stochastic ODE simulations, it is important to examine in what extent results of variational equations agree with numerical

solution of the full NLSE Eq. (1). To this end we consider the pulse propagation along a dispersion managed line with regular map parameters. Fiber segments are supposed to have and lengths Corresponding path averaged dispersion is Taking for the pulse duration we have dispersion distance corresponding to Obviously, it is convenient to use normalized parameters and dimensionless distance normalized to Initial Gaussian pulse parameters are chosen corresponding to stationary point on the phase plane of Eqs. (6, 7). Figure 1 illustrates the extent of agreement between results of variational approach

and full NLSE’s. We performed numerical simulations of the pulse propagation in optical

fiber links with a random dispersion map using both the full NLS Eq. (1),

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and corresponding variational equations Eqs. (6, 7). Perturbations experienced by a soliton propagating along fiber links with randomly varying

dispersion magnitudes of spans or span lengths will result in its continuous broadening. Mean square of the soliton's width (at starting point of each unit cell), normalized to its initial value is shown in Fig. 2 a). It may be drawn a conclusion from this figure, that a DM soliton can propagate more than 100 unit cells without noticeable distortion even at enough big random deviations of dispersion magnitudes of spans (which amounts to This conclusion is consistent with the result of Ref. [3]. To compare relative weights of two kinds of disorder in DM systems: randomness of dispersion magnitudes of spans and span lengths in disintegration of a soliton, we calculated growth rate for the energy which is a normalized constant of motion of variational equations in the absence of randomness [4]. It should be pointed out that initial values for the soliton amplitude and chirp are taken corresponding to stationary points on the phase plane of the system Eqs. (6. 7). Figure 2 b) shows that randomness of dispersion magnitudes is more detrimental for stability of a DM soliton compared to that of randomness of span lengths at standard deviations for both cases.

It is of interest to compare robustness of DM and conventional solitons against random fluctuations of the fiber dispersion. Figure 3 a) il-

lustrating this issue suggests, that DM solitons are notably more stable.

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In these calculations, for DM case the fiber spans are supposed to have and are randomly modulated with standard deviation For the conventional case the fiber dispersion is taken randomly varying around the corresponding pathaveraged value with the same Stochastic ODE’s Eqs. (6, 7)

are averaged over 400 realizations. When the fiber span lengths are small compared to dispersion magnitudes (both in dimensionless units), the averaged dynamics approach is valid [4]. In this approach a real map with the fiber segments of alternating anomalous and normal dispersions is replaced by a uniform fiber with the path averaged dispersion In this formulation the problem can be considered as a particular case of the random Kepler problem in optical solitons context, recently reported in Ref. [5]. We note that the Kepler problem and its relation with optical soliton systems has already been explored before [6, 7]. In Ref. [5] employing the action-angle variables an explicit analytical expression has been derived for the expected distance until which a soliton propagates along the fiber

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with randomly varying dispersion, before it disintegrates

where is the action variable [5]. So, for high energy pulses the decay length is strongly reduced. Comparison of our numerical simulation results with the prediction of the above theory is shown in Fig. 3 b). The criterion (when the separatrix trajectory is reached starting from the value H < 0 corresponding to the stationary point is accepted for complete disintegration of a soliton. Stochastic equations Eqs. (6, 7) are averaged over 400 realisations. Typical behaviour of a soliton propagating along DM line with randomly varying dispersion magnitudes of spans obtained by numerical solution of the full NLS Eq. (1) is shown in Fig. 4. A good agreement with the results of variatioual equations (Fig. 3 b)), regarding the soliton break up distance may be stated from this figure. It should be pointed out that the agreement between predictions based on the variational approach and numerical solution of the full NLSE becomes poor when linear waves emission by a soliton is involved. This corresponds to near-separatrix trajectories of variational equations. 3. Splitting of a Vector Soliton Let us consider the vector optical soliton propagation in a birefringent fiber with dispersion compensation. The governing equations are the modified coupled NLSE’s for the dimensionless envelopes and of the electric fields in each polarisation [8, 9, 10]

here is a stepwize function describing the dispersion map. The parameter for the birefringent fiber [9], and when the averaging over the random linear birefringence is performed [11]. The group velocity delay is assumed to be a random function

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Here

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is a Gaussian noise with the correlation time

When

We will search for the solution as a localized waveform and apply the variational principle [2, 12]. The Lagrangian density is

and we take Gaussian trial functions

where and are the amplitude, width, chirp parameter, phase and coordinate of the center of pulses respectively. To obtain the equations for the pulse parameters we use the variational approach developed by Anderson [2]. Initially we calculate the averaged Lagrangian

Substitution of (13) into (14) gives

From the variational principle

we obtain the equations for the solitori parameters

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where and relative distance between soliton centers. This system can be rewritten in the following simplified form

is the

where In particular cases the above set of stochastic equations admits the rigorous analysis. According to the dynamical system theory [4, 13] for

small span lengths the averaged dynamics can be described by asymptotic equations. Let us suppose that the widths of solitons, observed with the period of DM, close to the steady value and their changes under the action of perturbations arc small. Under these conditions the following

system of asymptotic equations may be derived

Here the function

is replaced by its averaged value

From Eqs.

(23) and (24) we obtain the equation for small oscillations of

where

From Eq. (25) it is easy to get the mean square value of fluctuations of

For large distances of propagation we find the diffusion coefficient

Also it is easy to show that the frequency of small oscillations of

to

Using the result obtained for

we can find the mean value

is equal

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Further we have

The diffusion coefficient is then

So the DM soliton splits under the white noise perturbation. The splitting distance is order of It is interesting that the diffusion rate is defined only by as in the case of the ordinary optical solitons [14]. So we

can conclude that the DM vector solitons are similar to the ordinary vector solitons with respect to random birefringence perturbations. This prediction complies with the results of numerical simulations [15]. Taking the radiation effects into account gives the value for the critical noise intensity, which is capable to split a DM soliton. This requires the generalization of the procedure developed in Ref. [16], and will be considered separately. These expressions can only be used for rough estimations, and their validity should be verified by numerical simulations. We have simulated the governing coupled stochastic NLSE’s with different values of the noise intensity Averaging over 50 realizations has been performed for PDE’s and 200 for ODE’s. In Fig. 5 a) and 5 b) we plot the averaged evolution of chirp and intensity for Calculations are performed taking the following parameters for the dispersion map: km, and the initial pulse From these graphs we can see that the variational equations describe the true dynamics of these parameters accurately even for quite big values of the noise, and individual DM solitons are stable with respect to stochastically inhomogeneous birefringence. In Fig.5c the evolution of the mean square of the distance between the centers of solitons for is presented. The diffusion growth of is observed for all used values of the noise intensity, and this is the main contribution of random birefringence to DM vector solitons dynamics. We assume, that the splitting of a vector soliton takes place when the mean distance between centers of its and components becomes comparable with the soliton's width 4. Conclusions

In conclusion, we have performed theoretical analysis and numerical simulations of the scalar and vector soliton propagation in dispersion managed fiber links with randomly varying parameters. Disintegration of a scalar soliton and splitting of a vector soliton correspondingly due to randomness of map parameters and fiber birefringence are shown to occur by numerical simulations. Enhanced tolerance of DM solitons to random fluctuations

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of fiber GVD compared to that of conventional soli tons has been demonstarated. Theoretical prediction for the scalar soliton break-up distance in the averaged dynamics limit has been confirmed by numerical simulations of corresponding stochastic ODE’s. With the help of Lagrangian formalism the approximate system of equations has been derived for parameters of DM vector solitons. The predictions for the evolution of the vector soliton parameters from variational equations have been compared with the results of numerical modeling of governing coupled NLSE’s and that the variational approach gives reliable results. The inhomogeneous birefringence affects mainly the relative distance and frequency of solitons, while the chirp and intensity are slightly affected. Randomly inhomogeneous birefringence leads to diffusional growth of the mean square of relative distance, and may split the vector soliton to its constitutional components. References 1.

Turitsyn, S. K., Doran, N. J., Nijhof, J. H. B., Mezentsev, V. K., Schäfer, T. and Forysiak, W. : Dispersion-managed solitons, in Optical solitons: Theoretical challenges and industrial perspectives, (Eds.) Zakharov, V. E. and Wabnitz, S., Springer, (1999). 2. Anderson, D. : Variational approach to non-linear pulse propagation in optical fibers, Phys. Rev., A27, (1983), pp.3135-3145. 3. Matsumoto, M. and Haus, H. A. : Streched-pulse optical fiber communications, IEEE Photon. Technol. Lett., 9, (1997), pp.785-787. 4. Kutz, J. N., Holmes, P., Evangelidis, C. G. and Gordon, J. P. : Hamiltonian dy-

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5. 6. 7.

8.

Abdullaev, F. Kh., Bronski, J. C. and Papanicolaou, G. C. : (1999). Solitons perturbations and the random Kepler problem, Physica, D135, n.1-2, (1999), pp.303-315. Malomed, B. A., Parker, D. F. and Smith, N. F. : Resonant shape oscillations and decay of a soliton in a periodically inhomogeneous nonlinear optical fiber, Phys. Rev., E48, (1993), pp.1418-1425. Abdullaev, F. Kh. and Caputo, J. G. : Validation of the variational approach for solitons in fibers with periodic dispersion, Phys. Rev., E58, (1998), pp.6637-6647.

Georges, T. : Soliton interaction in dispersion-managed links, J. Opt. Soc.Am., B15,

(1998), pp.1553-1560. Agrawal, G. P. : Nonlinear Fiber Optics, Academic Press, (1989). Abdullaev, F. Kh., Darmanyan, S. A. and Khabibullaev, P. K. : Optical Solitons, Spririger-Verlag, (1993). 11. Wai, P. K. A. and Menyuk, C. R. : Polarization mode dispersion, decorrelation,

9. 10.

and diffusion in optical fibers with randomly varying birefringence, J.Lightwave

Technol., 14, (1996), pp.148-157. 12.

Ueda, T. and Kath, W. L. : Dynamics of coupled solitons in nonlinear optical fibers,

Phys.Rev., A42, (1990), pp.563-571. 13. Gukkenheimer, J . and Holmes, P. : Nonlinear oscillations, dynamical systems and

bifurcations of vector fields, Springer-Verlag, (1983). 14.

Kivshar, Y. S. and Konotop, V. V. : Vector solitons in optical fibers with random

15.

birefringence, Kvantovaya. Elektron., 17, (1990), pp.1599-1602. Zhang, X., Karlsson, M., Andrekson, P., Bertilsson, K. : Impact of polarization

mode dispersion in dispesrsion-managed soliton systems, Electr. Lett., 34, (1998), 16.

pp.1122-1124. Lakoba, T. I. and Kaup, D. J. : Perturbation theory for the Manakov soliton and its applications to pulse propagation in randomly birefringent fibers, Phys.Rev., E56, (1997), pp.6147-6165.

QUANTUM CORRELATIONS OF COLLIDING SOLITONS

A. SIZMANN, F. KÖNIG, M. ZIELONKA, R. STEIDL AND T. RECHTENWALD Lehrstuhl für Optik, Physikalisches Institut der Universität Erlangen-Nürnberg, Staudtstr. 7/B2, D-90158 Erlangen, Germany

Abstract. Soliton collisions in a WDM transmission system produce power and phase correlations among the interacting channels. The quantum limits of the cross-phase-modulation-induced coupling of solitons were intensively investigated in theory and in experiment, however, only in terms of numberphase correlations. A recent multi-mode analysis of propagating solitons and of the soliton collisions shows intra- and inter-pulse photon-number correlations. This analysis provides new insights into the quantum structure of solitons, into the back-action evading detection of the photon number and into a mechanism of intensity modulation cross-talk in WDM transmission systems in the presence of spectral filters.

1. Introduction

Fibre-optic solitons showed unique non-classical behaviour in a variety of experiments [1]-[3]. The quantum properties of solitons are interesting for defining fundamental limits of terabaud communication systems [4] and high-precision measurements. Quantum-noise limited measurements are particularly useful for assessing the potential for improvement of a particular technology. The quantum limits of a basic component, device or system can then serve as an absolute reference against which the performance under operating conditions can be compared and beyond which it cannot be improved. The soliton collision is a well-understood elementary process in classical fibre optics. Therefore, two important questions arise in the context of quantum measurements and WDM communication systems: which unique and novel quantum features do solitons acquire in a 289

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collision and in which way do these features affect the fundamental bounds on signal-to-noise ratios? In the long history of non-classical fibre-optic experiments, two recent developments stand out [3]: substantial photon-number noise reduction below the shot-noise limit was achieved by spectral filtering of solitons and by interference of the soliton with a dispersive pulse in a highly transmissive non-linear loop mirror. The observed non-classical phenomena are interesting and useful for several reasons. First, shot-noise limited solitons are transformed into solitons with photon-number noise far below and far above the shot-noise limit. Therefore, the ultimate noise limits as well as the actually observed noise levels of an optical communication link employing spectral filters and/or non-linear loop mirrors differs significantly from that of a linear link. The propagation and accumulation of amplifier noise must be modeled accordingly. Second, proper design of filters and interferometers allows for passive in-line reduction of classical as well as quantum noise. Based on the fast optical Kerr non-linearity of glass, ultra-broadband noise reduction is expected. Third, these methods reduce photon-number noise and are therefore directly applicable to IM/DD systems, in contrast to earlier non-classical experiments which required coherent detection of the noiseless field quadrature. Furthermore, these non-classical phenomena are related to useful non-linear optical input-output functions. For example, the non-linear optical loop mirror has the capability of simultaneously reducing photon-number noise on the signal, reducing amplified spontaneous emission noise and stabilising the output power for an extended range of input powers [5], i.e. acting as a noise “squeezer”, saturable absorber and optical limiter, respectively. Therefore, quantum noise reduction and research on novel multifunctional devices are related to some extent. Next, the observed non-classical phenomena are more complex in the case of mutually interacting solitons as inter-pulse quantum correlations add to the complex quantum structure of each soliton. Finally, there are interesting quantum phenomena beyond the fundamental limits of classical applications. The quantum nature of light gives rise to new opportunities in quantum information processing which are beyond the information processing capabilities in a classical world [6, 7]. Through a collision, optical solitons acquire a permanent timing shift and phase change. At the quantum level, correlations between the photon number and phase of the collision partners are created due to cross-phase modulation. These number-phase correlations were investigated in theory and in experiment towards the realisation of back-action evading quantum measurements [3], [8]-[11] and towards all-optical switching [12, 13]. In a WDM communication system, inter-channel photon-number modulation and timing shift is expected to be an important limiting factor in addition

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to number-phase coupling. In this paper, a recent multi-mode analysis of propagating solitons and of the soliton collisions is discussed which shows that inter-channel photonnumber correlations are expected to occur in the presence of spectral filters. First, the multi-mode quantum structure of solitons is briefly reviewed. Next, numerical results of inter-pulse photon-number correlations and their relevance to applications are discussed. Then, experimental results of signal tapping based on the frequency shift of ultrashort pulses are briefly presented. The final section summarises the paper. 2. Multi-mode photon-number correlations of solitons

2.1. SPECTRALLY RESOLVED QUANTUM MEASUREMENTS In 1995, spectral filtering of coherent solitons was unexpectedly found to reduce photon-number noise below the shot-noise limit. The experimental method used by Friberg et al. [14, 15] in the picosecond pulse regime and later by Spalter et al. [16, 17] in the femtosecond regime was fairly simple: a pulse was launched into a fibre and was spectrally band-pass filtered after emerging from the fibre end. The photocurrent noise was detected and compared with the shot-noise level by using the sum and difference photocurrents of two detectors in a balanced two-port detection scheme. A variable band-pass filter was used to optimise the upper and lower cut-off frequencies. In the first experiment 2.3 dB of photon-number noise reduction below the shot-noise limit were observed with 2.7 picosecond pulses filtered after 4.5 soliton periods [14, 15]. Later, up to 3.8 dB noise reduction below shot noise was achieved with femtosecond pulses in a 100 soliton periods long fibre [17]. A variety of fibre length, filter functions and dispersion characteristics (incl. normal dispersion [18]) were investigated. An overview over theoretical predictions and experimental results of photon-number noise “squeezing” by spectral filtering is given in reference [3]. The mechanism for photon-number noise reduction can be understood in terms of the multi-mode quantum structure of solitons [19]-[22]. Spectrally resolved quantum measurements showed that some spectral domains of the soliton are negatively correlated with others in the same pulse [21]. If a spectral filter transmits those spectral domains that are negatively correlated, then their fluctuations cancel in the total transmitted photon flux. Thus a particularly “quiet” photon flux far below shot noise is achieved, as was predicted earlier for twin-beam correlations in optical fibres [23]. In order to measure the multi-mode photon-number correlations with high resolution, a series of spectrally resolved quantum measurements on 130-fs solitons were performed [21]. Theoretical work on the multi-mode correlations showed very similar results even without including the Raman

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effect. The theoretical analysis was recently extended to include the Raman effect, higher-order bound solitons and soliton collisions. 2.2. MODEL

Soliton quantum dynamics in the anomalous dispersion regime are modeled by the quantum non-linear Schrödinger equation

where length and time variables are scaled to the dispersion length and to a characteristic pulse duration With a linearisation approximation [24] the quantum field uncertainties are represented by the small perturbation of the normalised classical envelope function . The initial condition at the fibre input is a coherent state with . Photonnumber variances and covariances are then calculated for a discrete set of spectral domains from

The multi-modo photon-number correlations of the spectral domains i and j are given by the correlation coefficient

The squeezing ratio , i.e. the ratio of the photon-number variance in the spectral interval i and the average photon number which represents the shot-noise variance in this interval, is given by the diagonal element

The correlation matrix calculated for a fundamental shot-noise limited soliton after 3 soliton periods of propagation is shown in Fig. 1. Clearly visible is the emergence of negatively and positively correlated spectral domains. The grey-scale coded plot shows that the modes in the centre of the spectrum tend cancel the photon-number fluctuations of the modes next to the centre (strong negatively correlated parts in Fig. 1). The overall pattern becomes more complex as the soliton propagates further down the fibre. A band-pass filter that removes the outlying parts of the spectrum tends to transmit the negatively correlated domains which lead to strong photocurrent noise reduction.

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3. Multi-mode photon-number correlations of a soliton collision 3.1. NUMERICAL RESULTS

The simulation of a soliton collision was based on the unbound N = 2 soliton solution of the non-linear Schrödinger equation. It was assumed that initially there were no inter-pulse photon-number correlations and that the N = 2 solution represents two shot-noise limited solitons long before the collision. Figure 2 shows the correlation matrix in the centre of the collision [25]. The range of frequencies includes both soliton spectra. The correlation matrix can be subdivided into four quadrants, two showing the intra-pulse correlation pattern of each soliton, similar to Fig. 1, and two off-diagonal quadrants representing the inter-pulse correlations. The new and unique quantum feature of the solitons is the emergence of inter-pulse photon-number correlations during pulse overlap. The inter-pulse correlation pattern of Fig. 2 is related to the collisioninduced frequency shift. During a collision, the pulses accelerate towards the collision centre. This is equivalent to a red-shift of the slow (red) soliton and a blue-shift of the fast (blue) soliton in an anomalous-dispersion fibre. The inter-pulse correlation pattern originates from the fact that the frequency shift of one pulse is proportional to the energy of the other pulse. Therefore, the inner (outer) half of one pulse should be negatively (positively) correlated with the total photon number of the collision partner. Indeed, the matrix shows that the inner (outer) spectral domains are negatively (positively) correlated with those of the other pulse. However, one has to keep in mind that this simple classical argument was not intended to explain all features of the quantum correlation pattern. There is more structure in the inter-pulse quadrants than the simple argument can account for. 3.2. RELEVANCE TO QUANTUM MEASUREMENTS AND OPTICAL COMMUNICATION The inter-pulse photon-number correlations are interesting and useful in many ways. For example, the collision-induced frequency shift may be used for noiseless optical tapping. Quantum correlations, if strong enough, may allow for sub-shot-noise detection of the photon-number of the collision partner. In fact, a back-action evading quantum measurement is conceivable that reads out the photon-number of a signal without changing the photon number. Figure 3 shows a schematic outline of such a noiseless tap. The collision partner (“probe soliton”) of the signal soliton will exhibit photon-number fluctuations that arc correlated with the total photon-number of the signal,

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if only those spectral domains of the probe are detected that contain strong inter-pulse correlations. The back action onto the signal is a permanent frequency shift, however with no change in photon number. The theoretical signal-to-noise ratios that can be achieved with this quantum optical tap will be discussed elsewhere [26].

Further consequences of the inter-pulse photon-number correlations are fundamental bounds on channel cross-talk in a WDM transmission system due to perturbed collisions. The transient features such as the frequency shift and correlation pattern (Fig. 2) become permanent when a partial non-linear collision occurs in an amplifier or at a dispersion discontinuity. Spectral filters must be employed to eliminate frequency shifts. This, however, leads to a modulation of the pulse energy in one channel proportional to that of the collision partner in the other channel. Further down the line, these collision-induced pulse energy modulations are expected to be eliminated by the repeated action of spectral filters which have been shown

to stabilize the pulse energy. Therefore, the filter-assisted mechanism of opto-optical amplitude modulation will lead to cross-talk only near the detector side, similar to what is expected for timing jitter caused by partial collisions.

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4. Experimental demonstration of all-optical tapping The transient frequency-shift and spectral-filter scheme of colliding solitons was used for optical switching and pulse detection. A pair of 170-fs pulses separated by 15 nm around a centre wavelength of 1.5 was launched into

14 m of fibre and was separated in the centre of the collision. Subsequent spectral filtering of the probe produced an intensity modulation when the signal pulse was chopped on and off. With a narrow optical band-pass filter applied to the probe beam, a visibility of 53 % of the photocurrent modulation was achieved. This scheme, based on an ultrafast non-linearity, qualifies for reading out individual bits of a terabaud TDM data stream. For this purpose the probe pulse timing should be derived from an all-optical local clock recovery circuit [27, 28]. The tap will be transparent to the signal pulse. 5. Conclusion The transient spectral correlations of solitons offer a new avenue towards

optical switching and back-action evading quantum measurements. The multi-mode analysis gives insight into photon-number correlations among solitons that experience spectral filtering after a partial collision. We identified spectral domains of positive and negative inter-pulse photon-number correlations which are useful for realising a quantum or classical tap. In addition, we demonstrated opto-optical modulation transfer with 53 % visibility in a transparent tapping scheme. Worthwhile perspectives for further developments are the optimisation and demonstration of sub-shotnoise tapping and the generation of quantum-entangled multi-mode fields for quantum information processing. Further research on the optimisation of WDM system components with respect to minimum channel coupling should take the multi-mode correlation features of the soliton collision into account. Finally, a perspective is the realisation of a high-resolution timedemultiplexing tap for terabaud communication systems, based on interpulse spectral photon-number correlations. References 1.

Drummond, P. D., Shelby, R. M., Priberg, S. R. and Yamamoto, Y. : Quantum solitons in optical fibres, Nature, 365, (1993), pp.307-313. 2. Friberg, S. R. : Quantum Mechanical Aspects of Soliton Propagation: Recent Theoretical and Experimental Studies, in Special Section on Nonlinear Theory and its Applications, IEICE Trans. Electron., (1996). 3.

Sizmann, A. and Leuchs, G. : The optical Kerr effect and quantum optics in fibers,

in Progress in Optics, XXXIX, Wolf, E.(Ed.), Elsevier, (1999), pp.373-469.

4. Corney, J. F., Drummond, P. D. and Liebman, A. : Quantum noise limits to terabaud communications, Opt. Comm., 140, (1997), pp.211-215.

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5. Smith, N. J. and Doran, N. J. : Picosecond soliton transmission using concatenated nonlinear optical loop-mirror intensity filters, J. Opt. Soc. Am., B12, (1995), pp.1117–1125. 6. Furusawa, A., Sorensen, J. L., Braunstein, S. L., Fuchs, C. A., Kimble, H. J. and Polzig, E. S. : Science, 282, (1998), pp.706-709. 7. Leuchs, G., Ralph, T. C., Silberhorn, C. and Korolkova, N. : Scheme for the generation of entangled solitons for quantum communication, J. Mod. Optics, 46, (1999), pp.1927-1939. 8. Haus, H. A., Watanabe, K. and Yamamoto, Y. : Quantum-nondemolition measurement of optical solitons, J. Opt. Soc. Am., B6, (1989), pp.1138-1148. 9. Sakai, Y., Hawkins, R. J. and Friberg, S. R. : Soliton-collision interferometer for the quantum nondemolition measurement of photon number: numerical results, Opt. Lett., 15, (1990), pp.239–241. 10. Friberg, S. R., Machida, S. and Yamamoto, Y. : Quantum-Nondemoliton Measurement of the Photon Number of an Optical Soliton, Phys. Rev. Lett., 69, (1992), pp.3165-3168. 11. Friberg, S. R., Machida, S., Imoto, N., Watanabe, K. and Mukai, T. : Quantum Nondemolition Detection via Successive Back-Action-Evasion Measurements: A Step Towards the Experimental Demonstration of Quantum State Reduction, in Quan-

tum Coherence and Decoherence, Fujikawa, K. and Ono, Y. A. (Eds.); Elsevier, (1996), pp.85-88. 12. Friberg, S. R. : Demonstration of colliding-soliton all-optical switching, Appl. Phys. Lett., 63, (1993), pp.429-431. 13. Moores, J. D., Bergman, K., Haus, H. A. and Ippen, E. P. : Demonstration of optical switching by means of solitary wave collisions on a fiber ring reflector, Opt. Lett., 16, (1991), pp.138-140 14.

15. 16. 17. 18.

Friberg, S. R., Machida, S. and Levanon, A. : CLEO/Pacific Rim’95, Invited Pa-

per TuF2, (1995). Friberg, S. R., Machida, S., Werner, M. J., Levanon, A. and Mukai, T. : Observation of Optical Soliton Photon-Number Squeezing, Phys. Rev. Lett., 77, (1996), pp.37753778. Spälter, S., Burk, M., U., Bohm, M., Sizmann, A. and Leuchs, G. : Photon number squeezing of spectrally filtered sub-picosecond optical solitons, Europhys. Lett., 38, (1997), pp.335-340. Spälter, S., Burk, M., U., Sizmann, A. and Leuchs, G. : Propagation of quantum properties of sub-picosecond solitons in a fiber, Opt. Expr., 2, (1998), pp.77-83. König, F., Spälter, S., Shumay, I., Sizmann, A., Fauster, T. and Leuchs, G. : Fibreoptic photon-number squeezing in the normal group-velocity dispersion regime, J.

Mod. Opt., 45, (1998), pp.2425-2431. 19. Werner, M. J. : Quantum statistics of fundamental and higher-order coherent quantum solitons in Raman-active waveguides, Phys. Rev., A54, (1996), pp.2567-2570. 20. Werner, M. J. and Friberg, S. R. : Phase transitions and the internal noise structure of nonlinear Schrödinger equation solitons, Phys. Rev. Lett., 79, (1997), pp.41434146.

21. Spälter, S., Korolkova, N., König, F., Sizmann, A. and Leuchs, G. : Observation of multimode quantum correlations in fiber optical solitons, Phys. Rev. Lett., 81, (1998), pp.786-789. 22.

Levandovsky, D., Vasilyev, M. V. and Kumar, P. : Perturbation theory of quantum

solitons: continuum evolution and optimum squeezing by spectral filtering, Opt. Lett., 24, (1999), pp.43-45. 23. Kennedy, T, A. B. : Quantum theory of cross-phase-modulational instability: Twinbeam correlations in a process, Phys. Rev., A44, (1991), pp.2113-2123. 24. Haus, H. A. and Lai, Y. : J. Opt. Soc. Am., B7, (1990), pp.386-392. 25. Zielonka, M. A. : Photonenzahlkorrelationen im Spektrum van Quantensolitonen,

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26.

27. 28.

Zielouka, M. A., König, F. and Sizmann, A. : Transient quantum correlations of

interacting solitons, in preparation. Lucek, J. K. and Smith, K. : All-optical signal regenerator, Opt. Lett., 18, (1993), pp.22-24. Sartorius, B., Bornholdt, C., Brox, O., Ehrke, H. J., Hoffrnann, D., Ludwig, R. and Möhrle, M. : All-optical clock recovery module based on self-pulsating DFB laser, Electron. Lett., 34, (1998), pp.1664-1665.

SYMMETRY-BREAKING AND BISTABILITY FOR DISPERSION-MANAGED SOLITONS

J. H. B. NIJHOF AND N. J. DORAN Photonics Research Group, Aston University,

Aston Triangle, Birmingham

7ET, United Kingdom

Abstract. In a dispersion map with a region of sufficiently low dispersion, there will be a periodic solution with a chirp-free point in that region. Dispersion maps with multiple regions of low local dispersion can have multiple periodic solutions, and in a symmetric map the symmetric solution

can become unstable.

1. Introduction Current research on dispersion-managed (DM) solitons seems to be focussed on two areas. One topic is the “bit-overlap solitons”, high-bit-rate systems that include long segments of standard single mode fibre, in which the pulses

undergo considerable dispersive broadening and overlap with many neighbouring pulses [1, 5]. The other topic is systems with short period maps, with dispersion map periods much smaller than the amplification period. These systems, in which one has more freedom to tailor the dispersion, should allow even higher bit-rates [2, 4, 7]. For such a short period system one can envisage two ways of fabricating the dispersion map. One possibility is to splice together short fibres. Another would be to fabricate “ready-made” fibres in which the dispersion varies continuously. The second option might appear to be better, because it would reduce splicing losses. Also, in such a system with a sinusoidallike dispersion map, dispersion-managed solitons tend to experience less dispersive broadening. Therefore one would think that they would be less sensitive to intra-channel pulse to pulse interactions. But this paper will show that in such a system, the (symmetric) DM soliton becomes unstable, and instead the only stable solutions are those that have a chirp-free point in a region of low dispersion, so that the overall 299 A. Hasegawa (ed.), Massive WDM and TDM Soliton Transmission Systems, 299–308. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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dispersive broadening is larger. And indeed it seems that even when the symmetric solution still is stable, the pulse in such a system is more sensitive to pulse-to-pulse interaction than the DM soliton for a two-stage map. 2. The system

We consider a symmetric 6-stage dispersion map, as shown in Fig. 1. Each of the six segments has the same length and their group velocity is respectively. In each segment the pulse satisfies the Nonlinear Schrödinger equation

SYMMETRY-BREAKING AND BISTABILITY FOR DM SOLITONS 301

The origin is chosen in the midpoint of the segment with dispersion With this choice satisfies the symmetry relations . The symmetry of the map and the fact that the average dispersion is zero are not essential for the argument, but they make it easier to interpret the results. The high dispersion is fixed to and the low dispersion is varied from ! to . For both boundary cases the dispersion map is effectively a two-stage map — in the one case with a map length of 2, and in the other case with a map length of 6. Most of the results below are obtained from the standard variational model, where a Gaussian pulse shape is assumed:

In terms of the pulse width T and the chirp C the evolution is given by and , where E is the (constant) energy. The diagrams below show the chirp C as defined above and the FWHM pulse width 3. Symmetry-breaking

Because of the symmetry of the map, if is a solution of Eq. (1), then so is . For the two-stage maps with the only solution is the normal DM soliton which is exactly unchirped at But when is sufficiently lower than , this symmetric solution becomes

unstable, and two asymmetric solutions appear, which are chirped at the midpoint of the dispersion map. This can be seen from the bifurcation

diagrams Figs. 2 and 3. Both figures show the pulse width and the chirp at for the stable and unstable periodic solutions. Figure 2 shows the bifurcation diagrams for two different energies, and both for the variational model, and demonstrates that the bifurcations are almost independent of the pulse energy. Figure 3 compares the bifurcation diagram for the variational model with that of the full PDE. They match very well. When is very low compared to then the high dispersion region becomes effectively linear: the pulse broadens so fast that the peak power goes to zero. This can be seen from Fig. 4, which shows the evolution of the pulse width, chirp and bandwidth over one dispersion map period for The two stable asymmetric mirror solutions are mirror images of each other. Note that each one of them only shows a change in bandwidth, i.e. any nonlinear effect, in one of the regions of low dispersion. In effect,

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the pulse only sees a two-stage dispersion map of length 2l with dispersion The rest is just linear ballast. Therefore in the limit that goes to zero, the chirp-free point is in the center of one of the regions of low dispersion, and the pulse approaches the ‘standard’ DM soliton for the low dispersion region. For instance, in this limit the pulse width in the midpoint of the low dispersion segment will be given by (or 4.8 in the variational model) [3].

Because of the linear dispersion, that means that at the chirp C then tends to , and the pulse width at is proportional to . Consequently, in Figs. 2 and 3, the pulse width of the stable branches goes to infinity as goes to zero, but the chirp is continuous. When the low dispersion increases relative to the high dispersion, the

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chirp-free point moves outward. The bifurcation occurs almost exactly when

the chirp-free point is at the boundary of the low dispersion segment. This happens at

and at

for all three cases considered.

Figure 5 shows the evolution of the pulse width and the chirp during one dispersion map for , just after the bifurcation. It shows the unstable symmetric solution (dashed) and one of the stable asymmetric solutions. There is another asymmetric solution which is its mirror image. All three periodic solutions are very close in this case.

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4. Bistability

For the variational model there is a narrow region around in which there are five periodic solutions: a symmetric stable one, an asymmetric unstable one with its mirror image, and an asymmetric stable one with its mirror image. They can be identified from the Poincaré diagram, Fig. 6. Figure 7 shows the evolution of the pulse width and the chirp for the symmetric solutions and the two solutions with positive chirp at From the evolution of the chirp one can see that for the stable asymmetric solution, the chirp is zero almost exactly at and at i.e. at boundaries between the low and the high dispersion.

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So the variational model suggests that in this case there is true bistability. But it might be an artifact of the variational model. If one compares the bifurcation for the variational model with that of the full PDE in Fig. 3, one sees that the bifurcation diagram for the full PDE does not show the five solutions. For the full PDE, the pulse width changes continuously from the symmetric stable to the asymmetric stable branch, whereas for the variational model there is a jump. 5. Discussion

In a dispersion map with a region of sufficiently low (constant) dispersion, there will be a periodic solution with a chirp-free point in that region. If the dispersion map is sinusoidal, it seems that there is no stable periodic solution anymore, but instead a quasi-stationary solution with a chirp-free point almost at the zero of the dispersion, a slowly increasing bandwidth, and an enormous amount of dispersive broadening. If a dispersion map has more than one region of low dispersion, separated by regions of high dispersion, for each of these low dispersion regions there can be a stable periodic solution with a chirp-free point in it. In effect, the high dispersion regions (and all the other low-dispersion regions)

become linear. The paper has only discussed highly symmetric lossless maps, but the conclusions are not limited to those. Loss, asymmetry, or a non-zero average dispersion do not fundamentally change the results.

SYMMETRY-BREAKING AND BISTABILITY FOR DM SOLITONS 307

6. Conclusion

In conclusion, it seems that for short period dispersion maps, the best way to fabricate the fibre is by splicing together fibres, so as to get a piecewise constant dispersion map without regions of low dispersion. Acknowledgements

This work was supported by an EPSRC grant.

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References 1.

Alleston, S. B., Harper, P., Penketh, I., Bennion, I., Doran, N. J. and Ellis, A. D. : 1000 km transmission of 40 Gbit/s single channel RZ data over dispersion managed standard (non-dispersion shifted) fibre, Electron. Lett., 35(10), (1999), pp.823-824. 2. Anis, H., Berkey, G., Bordogna, G., Cavallari, M., Charbonnier, B., Evans, A., Hardcastle, I., Jones, M., Pettitt, G., Shaw, B., Srikant, V. and Wakefield, J. : Continuous dispersion managed fiber for very high speed soliton systems, ECOC’99, I-230, (1999). 3. Berntson, A., Doran, N. J., Forysiak, W. and Nijhof, J. H. B. : The power dependence of dispersion managed solitons for anomalous, zero and normal path-average dispersion, Opt. Lett., 23(12), (1999), pp.900-902. 4. Hirooka, T., Nakada, T., Liang, A. and Hasegawa, A. : 160 Gb/s soliton transmission in a densely dispersion-managed fiber in the presence of variable dispersion and polarization-mode dispersion, in Nonlinear Waves and Their Applications (NLGW’99), ThA2, (1999). 5. Mamyshev, P. V. and Mamysheva, N. A. : Pulse-overlapped dispersion-managed data transmission and infra-channel four-wave mixing, in Nonlinear Waves and Their Applications (NLGW’99), ThA1, (1999). 6. Nijhof, J. H. B., Doran, N. J., Forysiak, W. and Knox, F. M. : Stable soliton-like propagation in dispersion managed systems with net anomalous, zero and normal

dispersion, Electron. Lett., 33(20), (1997), pp.1726-1727. 7. Turitsyn, S. K., Fedoruk, M. P. and Gornakova, A. : Reduced power optical soliton in fiber lines with short-scale dispersion management, Opt. Lett., 24, (1999), p.869.

40 GBIT/S MULTIPLE DISPERSION MANAGED SOLITON TRANSMISSION OVER 2700 KM

A. R. PRATT, H. MURAI AND Y. OZEKI Network Systems Development Center,

Network Systems Business Group, Oki Electric Industry Co., Ltd. 550-5 Higashiasaka, Hachioji, Tokyo 193-8550, Japan Abstract. Multiple dispersion managed solitons have been investigated

both theoretically and experimentally. As predicted by numerical simulations, soliton transmission at 40 Gbit/s is successfully demonstrated over 2700 km.

1. Introduction Bit rates on a single wavelength channel are soon expected to take off

beyond the commercially available 10 Gbit/s, to 40 Gbit/s or even approaching 100 Gbit/s, against the back drop of ever increasing Internet and Data-com. traffic. Optical fiber per se is no longer seen as a panacea for bandwidth exhaustion. Chromatic dispersion and the optical nonlinear Kerr effect typify the limitations of standard fiber as a transmission medium for high bit-rate signals. The optical soliton [1] however, represents an ingenious way to balance the dispersion and optical Kerr effect to alleviate the problems in high bit-rate systems. The resent discovery of dispersion managed (DM) soliton transmission [2, 3], has opened the door to the development of entirely new methods of sending high bit-rate data over various types of dispersion fiber [4, 5]. As for DM-soliton transmission at 40 Gbit/s, there have only been a few reports, one from Aston University [6] which reported 1000 km transmission over DM single mode fiber (SMF) and one from NTT [7, 8], who reported 40 Gbit/s transmission over an installed fiber network. In this paper, we discuss the issue of increasing the transmission distance for 40 Gbit/s solitons over DM-SMF by the use of multiple dispersion 309 A. Hasegawa (ed.), Massive WDM and TDM Soliton Transmission Systems, 309-326. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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management (MDM), and describe a new dispersion map, which enables longer transmission distances . The simulation results are compared with those of a re-circulating loop transmission experiment in order to verify and analyze the effectiveness of the new MDM map configuration. 2. 40 Gbit/s Soliton Transmission in Standard Fiber Systems

When a deployed fiber base of SMF is optimized to fully utilize its bandwidth, dispersion compensation is considered a pre-requisite for conventional transmission. Dispersion compensation can be achieved in SMFbased systems by simply adding an ‘anomalous’ dispersion element in series

with the transmission fiber. Dispersion-compensating fiber (DCF) is generally used to counterbalance the large local dispersion of the SMF. It is well known that soliton transmission can be supported in such a system when the average dispersion of each amplifier span satisfies the soliton condition. Using an alternating average dispersion scheme, soliton propagation over 2600 km at 20 Gbit/s has already been demonstrated in our laboratory [9]. 2.1. SHORT-PERIOD DISPERSION MANAGEMENT At 40 Gbit/s the non-linear interaction between neighboring soliton pulses increases, since the pulses spread out in regions of high local dispersion, and the conditions for obtaining stable soliton transmission become ever more stringent. Although polarization mode multiplexing (PDM) [10], can be used to reduce this interaction, at very high speeds the soliton-soliton interaction still represents the limiting factor to soliton transmission distances. At such high speeds, short-period or dense dispersion compensation has been proposed by several authors to increase the soliton transmission. By reducing the dispersion compensation spacing the soliton-soliton interaction is reduced and the soliton pulse-shape is retained after propagation. To investigate 40 Gbit/s soliton transmission over dispersion compen-

sated SMF, a perturbation method based on a Lagrangian formalization [11] of the non-linear Schödinger equation (NLSE) was used to calculate the steady state DM-solitori solutions for different dispersion periods (N). For a fixed total SMF length of 40 km, the dispersion period (N) was varied between and as shown in Fig. 1. Calculations were performed as a function of the average span dispersion, which was adjusted by changing the length of the DCF. For and the optimum peak power and chirp parameters for steady state soliton solutions arc plotted in Fig. 2 for (a) and (b) There were no steady

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state solutions for short pulse propagation in the case of large dispersion compensation spacing (i.e. N = 1). For N = 4 and a pulse width of 10 ps, a peak power as high as 130 mW was required for stable soliton propagation at an average dispersion of 0.05 ps/nm/km. At a bit rate of 40 Gbit/s

this corresponds to an average input power of 14 dBm. That is to say, 10 ps pulses at 40 Gbit/s cannot be realistically propagated unless a shorter dispersion distance is employed.

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The calculated results concur that a dispersion period greater than 4 is required for 40 Gbit/s transmission if a relatively large amplifier spacing is to be maintained. Short period DM systems with SMF lengths less than 10 km however, are not ideal for upgrading pre-installed fiber systems. In this sense, we have considered an alternative multiple dispersion scheme to increase the soliton transmission distance at 40 Gbit/s. 2.2. SOLITON-LIKE PROPAGATION BY NON-LINEAR CHIRPED PULSES In the first instance we considered the dispersion map shown in Fig. 3(a). Zero dispersion fiber (DSF), is inserted between the optical amplifier and a conventional two stage dispersion compensated transmission fiber consist-

ing of standard SMF and DCF. The dispersion of the SMF and DCF are set so that the dispersion averaged over the amplifier span (D AV ) is slightly anomalous. When a pulse is launched into the fiber, chirp is induced by self-phase modulation (SPM) in the DSF. In the transmission fiber, the SPM-induced chirp and dispersion may interact in such a way that solitonlike self-trapping can occur. That is to say, when the average dispersion of the transmission fiber is small enough, the SPM-induced chirp and the dispersion-induced chirp almost cancel each other to maintain soliton-like propagation. In this scenario, non-linear effects dominate the pulse evolution in the DSF and dispersion effects dominate the transmission in the SMF. When the length of the amplifier span (Z a ) is much less than the dis-

persion length (L d ), where (Ld) is evaluated for the average span dispersion, the pulse retains its shape after propagation. Simulations were performed to investigate the feasibility of 40 Gbit/s soliton propagation over the dispersion compensated SMF system in Fig. 3(a). The calculated results indicate that a pulse peak power of 10 mW is capable of supporting stable propagation when an unchirped 10 ps pulse is launched into the fiber. This value is much lower than the corresponding optimal power for 40 Gbit/s dense-DM soliton transmission. However, for a fixed average dispersion, the optimum launch power is sensitive to the DSF length. This dependence arises because stable pulse propagation requires a fixed degree of SPM-induced chirp to balance the dispersion of the transmission fiber. When non-linear effects in the transmission fiber are neglected, the optimum power for stable pulse propagation was found to increase as the DSF length was reduced. However in more practical terms the transmission fiber does exhibit a degree of nonlinearity. When the non-linearity of the transmission fiber is included in the simulations a slightly higher peak power is required to stabilize the pulse propagation for short DSF lengths. The presence of non-linearity in the transmission fiber also enhances soliton-soliton interactions due to the as-

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sociated pulse broadening in the transmission fiber. This effect can largely be removed by adjusting the length of the DSF. Although the dispersion map shown in Fig. 3(a) is deemed suitable for the propagation of a single wavelength channel, inclusion of DSF is not

suitable for use in future optical transmission systems based on dense wavelength division multiplexing (DWDM). Cross phase modulation (XPM) has been extensively studied [12, 13], and it is well known that XPM in DSF seriously limits WDM soliton transmission. Higher order dispersion effects,

which cannot be easily compensated, introduce pulse distortion, which also inhibit the transmission when the fiber dispersion is very small. In the next section a refinement of the dispersion map presented in Fig. 3(a) is dis-

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cussed with respect to circumventing the limitations imposed by dispersion

slope and XPM for WDM applications. 2.2.1. Improved dispersion management for more stable soliton-like propagation To improve the robustness of the soliton transmission discussed in the preceding section, the DSF shown in Fig. 3(a) can be replaced by dispersion managed fiber. A modified dispersion map is shown in Fig. 3(b). 2 km of non-zero dispersion shifted fiber (NDSF) and 8 km of DCF have been considered at the beginning of the fiber span. The dispersion averaged over this non-linear region, (NL), was set to zero, while the local dispersion was typically 16 ps/nm/km for the NDSF and – 4 ps/nm/km for the DCF. Since the local dispersion is non-zero across the entire amplifier span, the problems described in the preceding section can be effectively removed. However, in such a strongly dispersion managed system, the pulse still propagates with large pulse width fluctuations. Figure 4(a) shows the calculated pulse width variance as a function of transmission distance. The symbols in the figure correspond to the variational method, whereas the solid lines represent the NLSE numerically solved by a split-step-Fourier method [14]. The initial pulse width was varied between 7 and 12 ps, while keeping the launched input power constant. As shown in Fig. 4 (a), the amplitude of the pulse width fluctuations increase as the pulse width is reduced. Most noticeably at shorter pulse widths

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there is some discrepancy between the two calculations. This difference is attributed to dispersive waves, which modify the waveform during propagation. After transmission the pulse shape can no longer be described simply by a chirped Gaussian. which is not accounted for using the Lagrangian formalism. The split-step-Fourier method on the other hand is applicable to any waveform, and the simulated results using this approach are considered more accurate. The pulse width variance shown in Fig. 4(a) however, can be stabilized if the pulse is pre-chirped prior to propagation. Figure 4(b) shows the simulated pulse width variance for a pre-chirp value of –16 ps/nm. The results show that the pulse width fluctuations can be effectively eliminated over the, range of pulse widths considered. Pre-chirping is a well-known technique [15, 16], and is widely used in DM-soliton systems. Q-value. By solving the NLSE, we can evaluate the transmission characteristics of multiple dispersion-managed systems by simply calculating the soliton Q-value [17, 18]. The Q-value represents the signal to noise ratio of the transmitted eye pattern and is directly related to the bit-error rate (BER) of the transmission. The Q-values were extracted from the simulations by generating eye patterns for a fixed transmission distance. In Fig. 5 we plot how the Q-value change as a function of the transmission distance for the dispersion managed system shown in Fig. 3(b). The ini-

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tial pulse was taken to be 9 ps wide, with a peak power of 18.6 mW. The data bit rate was 40 Gbit/s and the pulse was pre-chirped at –15.7 ps/nm,

consistent with the data presented in Fig. 4. Amplified spontaneous emission (ASE) noise was included directly in the calculations by assuming an erbium-doped fiber amplifier (EDFA) noise figure of 6 dB. In Fig. 5 we compare the NLSE simulations with the theoretical optical signal-to-noise ratio (SNR). The plotted data points (‘x’) represent the average Q-values and account for the spread in the data across a sampled time window. For a Q-value of 6, corresponding to a BER of the maximum propagation distance approaches 2500 km. This distance is limited dominantly by soliton-soliton interactions. It is clear from Fig. 5, that the proposed multiple dispersion-managed system is capable of supporting 40 Gbit/s transmission over distances approaching 3000 km. Since the dispersion compensation is limited to only 2 short sections at the beginning and end of each amplifier span, the proposed technique is well suited to upgrading pre-installed fiber systems for 40 Gbit/s transmission. 3. 40 Gbit/s MDM-soliton transmission over 2700 km

We have confirmed the simulation results reported in the preceding sections by performing a re-circulating loop experiment over two amplifier spans. The experiment comprised of 40 Gbit/s data transmission over mul-

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tiple dispersion managed SMF. The precise dispersion map of the measured fiber is shown in Fig. 6. In order to introduce chirp in the non-linear region (without pulse compression, or broadening), we have used a three-segment configuration to keep the average dispersion of this region as close to zero

as possible. The precise fiber allocation was determined principally by fiber availability and consisted of 2 km of SMF, followed by 10 km of NDSF and 0.3 km of DCF. The average dispersion of the non-linear section (measured at 1554 nm) was –0.065 ps/nm/km for span 1 and –0.039 ps/nm/km

for span 2. The transmission fiber consisted of 31 km of SMF followed by the DCF. Two linear configurations for the DCF have been considered, which we label type I and type II, as shown in Fig. 6. The two configurations were investigated to ascertain the degree of non-linearity in the

SMF. The average span dispersion measured for span 1 and span 2 were 0.05 ps/nm/km and 0.03 ps/nm/km, respectively, with a dispersion slope of 0.02 ps/nm/nm/km.

A schematic diagram of the experimental setup is shown in Fig. 7. The fiber loop consisted of two amplifier spans, with amplifier spacing of 49 km and 48.5 km, respectively. A third EDFA was included in the loop to compensate for the loop loss, including the switching loss and band pass filter (BPF). The optical pulse stream was generated using a sinusoidal-driven

electroabsorption (EA) modulator to convert a continues wave (cw) light beam into ps pulses at 20 GHz. Data encoding at 20 Gbit/s with a

pseudo random binary sequence (PRBS) was performed using a second EA modulator. A 40 Gbit/s data train was generated by optically bitinterleaving two 20 Gbit/s pulse trains with orthogonal polarization states. The generated pulse had a Fourier-transform limited Gaussian pulse shape, with a pulse width of The pulse was pre-chirped prior to prop-

agation using DCF (–20 ps/nm) outside the transmission loop. The averaged pulse power launched into the SMF was varied in the range, 2 dBm to 12 dBm, in order to determine the optimum power for maximum transmission. After transmission, the soliton signal was optically de-multiplexed

using EA modulators into two 20 Gbit/s pulse trains, which were further down converted to 10 Gbit/s for detection. 3.1. RESULTS AND DISCUSSIONS

Figure 8 shows the BER characteristics after transmission over MDM fiber of type I, for an averaged launched power of 9 dBm. This input power corresponded to the optimum value required to achieve maximum transmission.

For 40 Gbit/s transmission a

BER was achieved at a transmission

distance of 2700 km. This result agrees well with the numerical simulations

shown in Fig. 5. Also included in Fig. 8 are the detected optical eye pat-

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terns for distances of 0 and 2752 km, which show clear eye openings. These results clearly confirm that transmission distances approaching 3000 km can he achieved in the proposed MDM scheme. To investigate the linearity of the SMF and DCF in the MDM-system discussed, we have compared the transmission characteristics of the type I and II configurations. Figure 9 shows the maximum transmission distance plotted as a function of input power for the two configurations. The transmission characteristics of the two systems show a similar overall input power dependence, confirming that the ordering of the DCF is not of critical importance. This confirms the basic assumption that the SMF / DCF combination can be considered effectively linear. The numerical simulations shown in Fig. 9(b) also confirmed this general tendency. However, when we consider both the experimental and simulated results in more detail, the

precise position of the DCF docs play a subtle role. Most noticeably, the type I configuration shows a slightly longer transmission distance and improved input power tolerance, when compared to type II. This suggests that the SMF and DCF do exhibit a certain degree of non-linearity. The degree of non-linearity depends on the launched fiber power, and is expected to increase with increasing input power. The type I configuration was more

effective in reducing soliton-soiliton interactions in the SMF, and did exhibit a slightly longer maximum transmission distance than the type II configuration. Furthermore, for a transmission distance greater than 2000

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km, the type I configuration showed a greater input power tolerance. As shown in Fig. 9(a), the measured input tolerance for 2000-km-transmission for the type I configuration was 4.5 dB ranging from 6.5 dBm to 11 dBm. Compared to the type II configuration, this improvement in input tolerance occurs mainly at higher input powers, where the non-linear effects are expected to be more pronounced. However, in spite of the superior transmission characteristics of the type I configuration, the improvements are at the expense of the maximum SMF length. Clearly a type II configuration is more suited for the purpose of up-grading a pre-installed fiber base for 40 Gbit/s transmission. The performance of the soliton transmission at 40 Gbit/s was also evaluated by contour mapping to assess the power and dispersion tolerance of the multiple dispersion managed scheme. Figure 10(a) shows a “transmitted distance contour map” of the fiber input power (EDFA output power) and average dispersion for the type I configuration. The average span dispersion was varied between 0.03 and 0.11 ps/nm/km by adjusting the signal wavelength of the input pulse. As shown in Fig. 10(a), the contour map is asymmetric about zero dispersion, supporting enhanced propagating in

the anomalous dispersion region consistent with soliton transmission. The contour map is also clearly bounded. The upper boundary is limited by soliton-soliton interactions and the lower boundary limited by the soliton

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condition. At the optimum input level (of dBm) the average dispersion tolerance for 2000 km transmission was 0.02 ps/nm/km, between 0.04 and 0.06 ps/nm/km.

We have also simulated the transmission distance contour map by solving the NLSE. The preliminary results, which do not include the effects of dispersion slope, or polarization mode dispersion (PMD), are shown in Fig. 10(b) for the type I MDM configuration. For the calculations, the Q-value was set to 6, corresponding to a BER of , On first inspection there is a good overall agreement between the simulated results and the measured data. For a simulated pulse width of 9 ps, the maximum transmission distance exceeds 3000 krn for an average dispersion of ps/nm/km, and an optimum input power of 9 dBm, where the affect of the constant EDFA output was also included in the calculations, to account for the measured SNR reduction with increasing power. Furthermore, as indicated by the contours in Fig. 10(b), the dispersion tolerance for 2000 km transmission is ps/nm/km. This value is approximately double the value observed experimentally. The discrepancy is currently under investigation, but may be related to PMD effects or the precise ASE line shape, which were not included in the preliminary simulations.

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4. WDM-MDM soliton transmission

As well as increasing the bit rate on a single wavelength channel, one of the major objectives of modern dispersion managed systems is the transmission of wavelength multiplexed signals, with little or no penalty in the transmis-

sion of each wavelength from that achieved in single channel transmission. The capability to up grade to WDM is therefore an important issue for future massive WDM and TDM applications. In this respect, we have addressed the suitability of the MDM maps shown in Fig. 6 for WDM-DM soliton propagation. A WDM-MDM environment was simulated, and in this section we compare the WDM transmission characteristics with those reported in the preceding sections for single channel MDM soliton transmis-

sion. The initial WDM calculations (using the variational method) showed no clear differences between the two dispersion maps, labeled type I and type II, and the discussion in this section is limited to a type II map. In a WDM environment, XPM and four wave mixing (FWM), as well as the soliton-soliton interaction represent the main effects, which limit the soliton transmission. FWM is considered a trivial effect in the present work since the large local dispersion leads to a phase mismatch between

the FWM components. XPM on the other hand, introduces timing jitter due to a collision induced frequency shift, due to collisions between pulses belonging to different channels. When the interaction distance between col-

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liding pulses is large, XPM is expected to be the dominant inter-channel

interaction. In the case of 40 Gbit/s WDM-DM soliton transmission, the transmission becomes increasingly more difficult as the channel spacing approaches 100 GHz. Figure 11 (a) shows the accumulated timing jitter for two wavelength channels as a function of channel spacing. The single chan-

nel data is also plotted for comparison. For a channel spacing greater than 200 GHz, the increase in timing jitter due to inter channel interactions is small. If we assume timing jitter less than 1 ps is required for error free

transmission, a propagation distance approaching 2000 km is predicted for a channel spacing of 200 GHz. At long transmission distances the timing jitter increases much more rapidly due to soliton-soliton interactions, as indicated by the timing jitter plotted for the single channel. When the channel spacing was further reduced to 100 GHz a marked increase in timing jitter was observed and the maximum transmission distance significantly reduced. Compared to the single channel MDM soliton transmission, although a serious reduction in system performance was observed for a channel spacing of 100 GHz, virtually no penalty in the transmission of each wavelength was observed for a channel spacing of 200 GHz.

Figure 11 (b) shows the increase in timing jitter as the number of wavelength channel was increased for a fixed channel spacing of 200 GHz. The

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plotted data corresponds to the worst channel in each case. Clearly the effect of adding additional channels becomes less significant as the total number of channels increases, indicating that the XPM-induccd timing jitter is dominated by nearest neighbor interactions. This is because of the larger phase mismatch between channels other than the nearest neighbors. By solving the NLSE, we have evaluated the transmission characteristics of the WDM–MDM systems in more detail for eight wavelength channels with a channel spacing of 200 GHz. In Fig. 12(a) we plot how the Q-values change as a function of the transmission distance. The simulations were based on the experimental conditions reported in Section 3. The average span loss was 20 dB, the initial pulse was taken to be 9 ps wide, and the average light power for stable pulse propagation was about 2.3 dBm at an average dispersion of 0.042 ps/nm/km. The data bit rate was 40 Gbit/s and the pulse was pre-chirped at –26.5 ps/nm. As shown in Fig. 12(a), for the pre-mentioned conditions SNR degradation limits the soliton transmission. For single channel transmission, simply increasing the launched input power can circumvent this SNR degradation. However in a WDM environment, increasing the input power only enhances XPM. When the launched power was increased to 4.5 dBm, the propagation distance of the central channels was reduced. At the higher input power the error free transmission distance of the outer most wavelength channels were increased to 2000 km and limited by SNR. However, the maximum propagation distance for the

two central channels were reduced by XPM effects to

In order to increase the SNR, while keeping the XPM effects constant, we have considered reducing the loss of the DCF in the linear region of the transmission line. In the simulations we reduced the loss of the DCF by

dB, such that the span loss was of the order of 15.5 dB. Under this assumption the optimum conditions for stable pulse propagation are almost identical to those shown in Fig. 12(a). This is true since the DCF fiber in the linear section of the transmission line does not contribute significantly to

the non-linear pulse propagation, but only modifies the linear properties of the pulse propagation (i.e. the SNR). The Q-values plotted as a function of transmission distance for a reduced span loss are shown in Fig. 12(b). For a Q-value of 6, corresponding to a BER of , the maximum propagation distance for the worse WDM channel transmission exceeds 2000 km. The maximum propagation distance for the outer WDM channel approaches 2500 km, which is consistent with the value calculated for single channel propagation. Under the assumption of reduced span loss the limiting factor to the transmission is no longer the SNR (as shown in Fig. 12(a)), but

instead, XPM for the central WDM channels and soliton-soliton interaction for the outer channels. However when compared to simple increasing the launched input power, by reducing the span loss of the transmission fiber

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the maximum propagation distance for all the multi-wavelength channels is increased. It is clear from Fig. 12 that the proposed MDM scheme is capable of supporting WDM transmission at bit rates of 40 Gbit/s/ch and over distances approaching 2000 km. Furthermore, since the number of wavelength channels in excess of 4 channels, does not significantly effect the transmission characteristics, it should be possible to multiplex a much larger number of channels than 8, will still maintaining long transmission distances. 5. Conclusions

In conclusion we have investigated soliton propagation in multiple dispersion managed SMF both numerically and experimentally. Based on numerical simulations a novel dispersion map was proposed for extending the soliton transmission distance at a bit rate of 40 Gbit/s/ch. A re-circulating transmission experiment was performed to validate the theoretical predictions, and 40 Gbit/s MDM–soliton transmission was successfully demonstrated over distances greater than 2700 km. The amplifier span length used in the experiments was ., which was achieved using relative long lengths of SMF, demonstrating the usefulness of the proposed scheme for up-grading pre-installed fiber systems.

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To demonstrate the suitability of the proposed MDM soliton transmission system for massive WDM and TDM applications, we have presented simulation results comparing 8-channel WDM transmission with the single channel transmission experiments. For 8 WDM channels and 200 GHz channel spacing, the feasibility of 40 Gbit/s/ch transmission over 2000 km was demonstrated. XPM was shown to limit the transmission of the central channels, whereas the soliton-soliton interaction limited the transmission of the outer wavelength channels, which showed almost the same transmission characteristics as the single channel transmission. In the discussion presented in Section 4. the effect of dispersion slope was ignored. When there is more than one wavelength channel present, dispersion slope may modify the soliton transmission. Dispersion slope compensation is therefore considered necessary for WDM applications. This may be achieved using several pairs of alternating SMF and DCF in the non-linear section of the transmission line, while still maintaining a long SMF length. This technique may also help to suppress XPM effects, since a periodical dispersion structure will help to reduce the accumulation of frequency shift induced by colliding pulses belonging to different channels. Investigations are currently underway to fully optimize the MDM-maps shown in Fig. 6 for WDM applications. This optimization includes the reduction of span loss in order to realize WDM -MDM soliton transmission with little or no

penalty in the transmission of each wavelength channel compared to the experimental results presented in Section 3 for a single channel. Acknowledgements

This work was supported by the soliton based, total all-optical communication network research project of the Telecommunication Advanced Organization of Japan (TAO).

References 1.

Hasegawa, A. and Tappert, F. D. : Transmission of Stationary Nonlinear Optical Pulses in Dispersive Dielectric Fibres. I. Anomalous Dispersion, Appl. Phys. Lett., 23, (197),pp.142-144. 2. Suzuki, M., Morita, I., Yamamoto, S., Edagawa, N., Taga, H. and Akiba, S. : Reduction of Gordon-Haus timing jitter by periodic dispersion compensation in soliton transmission, Electron. Lett., 31, (1995), pp.2027-2029. 3. Nakazawa, M. and Kubota, H. : Optical soliton communication in positively and negatively dispersion-allocated optical fibre transmission line, Electron. Lett., 31,

(1995), pp.216-217. Knox, F. M., Forysiak, W. and Doran, N. J.: 10-Gbit/s Soliton Communication Systems Over Standard Fiber at and the Use of Dispersion Compensation, J. Lightwave Technol., 13, (1995), pp.1955-1962. 5. Yamada, E., Kubota, H., Yamamoto, T., Sahara, A. and Nakazawa, M. : 10 Gbit/s 10,000 km Dispersion-Allocated Soliton Transmission Using Conventional 4.

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6. Alleston, S. B., Happer, P., Penketh, I. S., Bennion, I., Doran N. J. and Ellis, A. D. : 1000 km transmission of 40 Gbit/s single channel RZ data over dispersion managed 7.

standard (non-dispersion shifted) fibre, Electron. Lett., 35, (1999), pp.823-824. Sahara, A., Suzuki, K., Kubota, H., Komukai, T., Yamada, E., Imai, T., Tamura, K. and Nakazawa, M. : 40 Gbits/s Soliton Transmission Field Experiment over 1020 km and its extension to 1360 km Using in-line Synchronous Modulation, OFC’99, Tech. Dig., THI1, (1999).

8.

Suzuki, K., Kubota, H., Sahara, A. and Nakazawa, M. : 40Gbit/s single channel optical soliton transmission over 70,000km using in-line synchronous modulation and optical filtering, Electron. Lett., 34, (1998), pp.98-99. 9. Murai, H., Shikata, M. and Ozeki, Y. : 20 GBIT/S PDM Soliton Transmission Experiment in Dispersion Compensated Standard Fiber Systems, in A. Hasegawa (Ed.), New trends in Optical Soliton Transmission Systems, Kluwer Academic Publishers, (1998), pp.167-181. 10. Hasegawa, A. and Kodama, Y. : Interaction between Solitons in the Same Channel, in Solitons in Optical Communications, Clarendon press, Oxford, (1995), pp.151172. 11. Georges, T. and Favre, F. : Transmission Systems Based on Dispersion- Managed Solitons: Theory and Experiment, in A. Hasegawa (Ed.), New trends in Optical Soliton Transmission Systems, Kluwer Academic Publishers, (1998), pp.317-340.

12.

Sugawara, H., Kato H. and Kodama, Y. : Maximum reduction of collision induced frequency shift in soliton-WDM systems with dispersion compensation, Electron.

Lett., 33, (1997), pp.1065-1066. 13.

14. 15. 16.

Mollenauer, L. F., Stephen, S. G., Evangelides, G. and Gordon, J. P. : Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped

amplifier, J. Lightwave Technol., 9, (1991), pp.362-367. Agrawal, G. P. : Nonlinear fiber optics, Academic Press, (1995). pp.28-59. Georges, T. and Charbonnier, B. : Reduction of the dispersive wave in periodically amplified links with initially chirped solitons, Photon. Technol. Lett., 9, (1997), pp.127-129. Favre, F., Le Guen, D., Moulinard, M. L., Henry, M., Michaud, G., Devaux, F., Legros, E., Charbornnier, B. and Georges, T. : Demonstration of soliton transmission at 20 Gbit/s over 2200 km of standard fibre with dispersion compensation and

pre-chirping, Electron. Lett., 33, (1997), pp.511-512. 17.

Bergano, N. S., Kerfoot, F. W. and Davidson, C. R. : Margin measurements in

optical amplifier systems, IEEE Photon. Technol. Lett., 5, (1993),pp.304-306. 18. Sahara, A., Kubota, H. and Nakazawa, M. : Q-factor contour mapping for evaluation of optical transmission systems : soliton against NRZ against RZ pulses at zero group velocity dispersion, Electron. Lett., 32, (1996), pp.915-916.

ENABLING FIBER TECHNOLOGIES FOR MASSIVE WDM AND TDM SOLITON TRANSMISSION SYSTEMS

S. NAMIKI Opto-technology Lab, Furukawa Electric Co., Ltd.

6 Yawata Kaigan Dori, Ichihara 290-8555, Japan Abstract. This paper reviews recent advances of optical fibers especially for massive WDM and TDM soliton transmission systems. Our main focuses are dispersion managed fibers such as RDF (reverse dispersion fiber) and DCF (dispersion compensating fiber), broadband Raman amplification in transmission fibers, and high nonlinearity fibers for optical signal processing such as wavelength conversion and soliton pulse train compression. We also review a new method to measure nonlinear coefficient of fibers with

relatively high dispersion such as RDF.

1. Introduction : Transition Toward Dispersion Managed Soliton and Related Fiber Technologies

Demands to ultra-high-capacity optical transmissions are increasing year by year. As a powerful solution to increase the capacity of optical transmissions, the application of optical soliton has been extensively studied [1, 2]. For single channel transmissions, soliton seemed to be the ultimate solution by using sliding frequency guiding filters [3]. However, for WDM transmissions, the collisions between different channels deteriorate the transmission performances in case of so-called either ‘average soliton’ or ‘guiding-center soliton’ transmission systems due to the shedding of dispersive waves from perturbed solitons [4]. On the other hand, due to demands to overhaul and upgrade of the conventional single mode fiber with the zero dispersion wavelength at 1310 nm (SMF), a new fiber that compensates for the group velocity dispersion (GVD) of SMF at 1550 nm, called ‘dispersion compensating fiber (DCF)’ was intensively developed and commercialized [5]. The combination of SMF and DCF was demonstrated to be superior to non-zero dispersion shifted fibers (NZ-DSF) [6]. Also, in the transmis327

A. Hasegawa (ed.), Massive WDM and TDM Soliton Transmission Systems, 327–350. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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sion lines consisting of SMF and DCF, a quasi soliton (now widely called ‘dispersion managed soliton, DM soliton’) was found to exist and behave differently and even better than the conventional soliton in a uniform dispersion transmission line such as dispersion shifted fiber (DSF) [7]-[10]. A variational method with the trial function of a chirped Gaussian was applied to describe well DM soliton behavior [11]. Independently, a similar phenomenon was discovered by Tamura et al. [12] in a passively mode-locked fiber ring laser with a ‘dispersion managed’ cavity, which is composed of two fiber segments with the opposite dispersions. By Taylor-expanding up to second order the nonlinear phase shift around the pulse center only, Haus et al. have shown that the pulse tends to be a Gaussian rather than a sech shape [13] as observed experimentally [14]. This pulse is called ‘Stretched pulse’. Also, a perturbation theory of the stretched pulse was developed [15, 16], which predicts that the ASE induced timing jitter can be less than Gordon-Haus jitter of soliton [17] because the average dispersion can be made smaller, and that the pulse energy can be larger than that of soliton for the same average GVD of the cavity fiber. These predictions have also been observed for DM soliton respectively in Refs. [8] and [9]. The perturbation theory has predicted that the stretched pulse scarcely sheds the dispersive waves as opposed to soliton. These features may be related to the fact that DM soliton tends to be immune to collisions as compared with soliton. Also, it should be noted that conventional transmission methods have been improved by using chirped RZ (CRZ) format [18], in which the pulse shape becomes more like a Gaussian, or a dispersion managed nonlinear solitary wave. Likewise, nonlinear solitary waves were found to persist in dispersion managed fibers like a soliton, independently by different research groups, almost at a time.

Although qualitative behavior of stretched pulse and DM soliton is similarly described, the critical difference between the stretched pulse and WDM/TDM DM soliton is whether or not we must take into account pulseto-pulse interactions: in stretched pulse lasers there is usually only one pulse in a cavity while a WDM/TDM DM soliton suffers from massive collisions. The collisions are mainly of two kinds: one is inter-channel interactions where pulses in different WDM channels are collided and walking off of each other due to different group velocities, and the other is intra-channel interactions where pulses in the same WDM channel are overlapped and interacting each other due to the ‘stretching’ of DM solitons. Therefore, the design of massive WDM/TDM DM soliton transmission systems requires a thorough optimization to mitigate these interactions. For example, a large GVD reduces the effect of inter-channel interactions by washing out both the phase and cross-phase modulation between channels, while too large GVD may increase the intra-channel interactions as pulses spaced by

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a given duration would stretch so much as to overlap with the adjacent pulses. Thus, the parameters of the optimization at a given bit rate for transmission fibers are mainly GVD and nonlinear coefficient It should be emphasized that nonlinearity of fiber aggravates the performance of

WDM/TDM transmissions anyway, and dispersion management alleviates the effects of nonlinearity. Therefore, an ideal fiber should have low nonlinearity with moderate amount of dispersion management. However, we must not forget that there is a design rule for the fiber: the fabrication of

the fibers with opposite GVD characteristics is strongly related to other fiber transmission characteristics such as loss, nonlinearity and PMD. Each parameter is not independent of others. Therefore, we should discuss what

would be an optimal fiber from the viewpoints of both theory and practice. To aid design and fabrication efforts, one has to measure the nonlinear coefficient of the developed fibers. There have been several methods proposed for the measurement of nonlinear coefficient, such as XPM method [19], SPM method [20] and methods using four-wave mixing (FWM) [21, 22]. Especially, the nonlinear coefficient measurement of fibers with relatively large dispersion has not been well developed. In this paper, we will touch upon a new measurement method based on four-wave mixing generation that enables us to measure nonlinear coefficient of the such fibers [23, 24]. Raman amplification in transmission fibers [25] are attracting intensive attention because it is broadband by a broadband WDM pumping, and low noise due to distributed amplification [26]. The feature of low noise is essential to improvement of transmission performance, not just because it maintains better SNR but also because by larger margin for SNR, it allows signal level to be lower so as to avoid adverse nonlinear effects. These features are indeed valuable as DM fiber with normal dispersion such as DCF and RDF tends to have relatively large loss and nonlinearity. Although a DM soliton alone is in principle robust and superior by the virtue of the balance between nonlinearity and average dispersion, loss and nonlinearity still hurt the performance of high-bit-rate- and multi-channeltransmissions. Since Raman amplification is more efficient for fibers with large nonlinearity, Raman amplification in DCF or RDF is quite effective in compensating for the loss and increasing the SNR, and hence reducing the nonlinear effects as a result [27]. Recently, the maturity of 1480 nm pump laser diodes has successfully enabled us to realize field-ready ultrawideband Raman amplification in transmission fibers [28]. The achievement of 1 Tb/s WDM transmission over 10000 km has been reported to employ both DM fibers and Raman amplification in DM fibers [29]. As the optical transport capacity increases, the system will transform into more and more sophisticated networks, which are often referred to as

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photonic networks. In photonic networks, the more the network configuration becomes complex, the more wavelength conversion is necessary to save

the total number of WDM channels. Particularly, the number of WDM channels is increasing year by year, and up to date more than 100 channel WDM transmissions are reported [30]. Suppose there are more than 100 WDM channel signals being transmitted individually and routed differently through network nodes interconnected in a complicated and sophisticated manner, one should definitely need some smart way of managing the overwhelming number of channels. The direction to go in such a situation may be to build a hierarchical structure of managing channels, in which each sub-set of WDM channels is so defined as to be handled as one. In this context, multichannel wavelength conversion may play an important role. For this purpose, a fiberoptic parametric wavelength converter may be most suitable. Because of almost instantaneous response time of Kerr nonlinearity of optical fiber, the wavelength conversion process through FWM in the fiber is bit-rate transparent and of multichannel. We demonstrate that a high nonlinearity (HNL-) dispersion shifted fiber (DSF) is capable of ultrawideband wavelength conversion [31]. Another application of HNL–DSF is soliton compression. Using a comblike dispersion profiled fiber (CDPF) consisting of segments of DSF–SMF pair, a two tone beat signal can be compressed to be a soliton pulse train [32]. For efficient and broadband sideband generation in DSF segments,

it is better to use HNL–DSF rather than conventional DSF. In fact, we show that the soliton compression using HNL–DSF is more efficient than conventional approaches [33]. This technique is expected to be a robust and key component for inexpensive ultrafast TDM soliton transmitters in

future. Also, this soliton source could be used for spectrally sliced multicast transmitters [34].

The organization of this paper is as follows: in Section 2 we review dispersion managed fibers, DCF and RDF. We compare SMF-DCF with SMF–RDF transmission lines, and discuss some optimization issues of DM fibers. Section 3 refers to a new FWM-based method to measure the nonlinear coefficient of RDF which has relatively large dispersion. Because conventional FWM-based measurement method did not take effects of dispersion into account, we start with an approximate analytic solution to FWM generation in the presence of dispersion and apply it to the measurement. Ultra-broadband Raman amplification is shown in Section 4, where we describe a new technique, WDM pumping for broadband pumping, which based on planar lightwave circuit (PLC) technologies and fiber Bragg grat-

ing (FBG) stabilized 14xx nm pump laser diodes. Section 5 refers to a newly developed HNL–DSF and its applications, namely ultra-wideband wavelength conversion based on four-wave mixing and soliton compression

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based on CDPF. Section 6 is the summary with a brief mark on future directions. 2. Dispersion Managed Fibers

2.1. DESIGN FOR FIBERS WITH NORMAL DISPERSION AND NEGATIVE SLOPE

Since the material dispersion of silica has anomalous dispersion and positive dispersion slope in the transmission window, it is necessary to add a considerable waveguide dispersion in order to offset the material dispersion and realize normal dispersion and negative dispersion slope. Fortunately, the so-called W-shape refractive index profile is known to realize such waveguide dispersion. Figure 1 shows a schematic diagram of the refractive index profile of W-shape fiber. And Fig. 2 plots the material dispersion of silica versus wavelength along with an example of waveguide dispersion of Wshape fiber. Figure 2 illustrates how the waveguide dispersion modifies the material dispersion so successfully as to realize desired dispersion characteristics. The depth and/or the position of null in the waveguide dispersion curve can be adjusted by changing the structural parameters of refractive index profile of W-shape. In so doing, one can design and realize dispersion managed fibers such as DCF (dispersion compensating fiber) [35] and RDF (reverse dispersion fiber) [36].

2.2. DCF AND RDF, CHARACTERISTICS AND COMPARISON

Figure 3 plots the dispersion coefficients for SMF, DCF and RDF versus wavelength around 1550 nm. It can be seen that both DCF and RDF are

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meant to offset the dispersion of SMF over entire erbium doped fiber amplifier (EDFA) band by different lengths, respectively. Since DCF is spooled and packaged in a rack of the transmission systems, it has a several times

as large magnitude of dispersion coefficient as that of SMF so that it may occupy as small space as possible. Recently, we have developed a small diameter DCF to save the space by more than 30 % compared with conventional DCF modules. Approximately, five to seven times as short DCF as SMF would offset the dispersion of SMF in a wideband range, while RDF compensates for the dispersion of SMF by a comparable length of fiber. This is because RDF is meant to deploy as a transmission fiber in the same manner as conventional SMF or DSF. The development of RDF cable is now intensively promoted to achieve stable characteristics as ribbon-slotted-rod and loose-tube cables over operating temperatures. The Table 1 compares the transmission characteristics of DCF and RDF. of SMF is the smallest, and then RDF and DCF follows. It is anticipated that a small results in low loss, large MFD and , and hence small nonlinearity, and more importantly, small PMD. In this sense, it is advantageous to reduce the amount of or doped germanium for better performances in terms of loss, nonlinearity and PMD. Therefore, when we think of a counterpart of SMF for constructing a DM fiber line, we would go with smaller fibers such as RDF rather than DCF. Also, RDF turns out to achieve much higher yield than DCF. However, in real field systems, there are many SMF deployed. The use of DCF in The refractive index difference

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a repeater is the simplest and cost effective solution to upgrade the old systems. For future deployment, RDF may be a better solution for massive WDM/TDM transmissions. A decisive benefit of these DM fibers is the flat averaged dispersion characteristics over a broadband signal range. Figure 4 shows the averaged dispersion of SMF–RDF fiber line. It is shown that the dispersion is accurately managed over the entire operating bandwidth of EDFA, which is the critical key to realization of massive WDM/TDM transmissions. Likewise, using SMF–RDF DM line, many record breaking transmission experiments have been reported [37]-[39]. 2.3. ONGOING ISSUES, NEW DM FIBERS

In the previous subsection, we argued that SMF–RDF might be better than SMF–DCF for future massive transmission systems. However, a detailed

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analysis has yet to be conducted to scrutinize the ultimate solution. The transmission performances depend on the loss, nonlinearity and PMD as well as parameters of dispersion management. For better performances, one should pursue low loss fiber as much as possible. Namely, one should optimize the DM fiber with respect to the total span loss. The second issue is how to avoid nonlinearity as much as possible. It is well known that nonlinear phase shift is calculated by integrating over transmission distance the product of the nonlinear coefficient and signal power level as a function

of distance [40]. Because the SMF has a smaller nonlinear coefficient, it is more advantageous to use longer SMF with respect to its counterpart. Also, even lower nonlinearity fibers than SMF should be used, e.g. pure silica core SMF or large mode field diameter SMF. It should be noted that DCF is much shorter than RDF along with corresponding lengths of SMF, which results in lower nonlinearity and loss as a whole. Therefore, a new fiber with intermediate normal dispersion between DCF and RDF should perform better than RDF-based systems. Of course, we should wait to give such a conclusion until we see if the loss of RDF could be as small as SMF in future. Also, as bit rate increases, PMD will increasingly limit the transmission. Since RDF has relatively smaller PMD, it is much beneficial to use it rather than DCF. In this way, one has to optimize DM fiber in terms of loss, nonlinearity and PMD. We also importantly note that the use of Raman amplification in fibers will change the whole optimization scenario discussed above, because it compensates for the loss and saves optical SNR of the total system, and hence provides larger margins to

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lessen nonlinear effects significantly. SMF–RDF is more efficient in terms of Raman amplification. In addition to the above optimization issues, or maybe more importantly, one has to scrutinize on the parameters of dispersion management. The important parameters of dispersion management are dispersion coefficients and segment lengths. As discussed in introduction, two kinds of collisions are involved: one is inter-channel and the other is intra-channel. To avoid inter-channel collisions, the local dispersion coefficient should be as large as possible, while to avoid intra-channel collisions, the total dispersion (dispersion coefficient times distance) of each segment should not be too large as the pulses in a channel spread and overlap each other. Since SMF is shorter, SMF–RDF line has less intra-channel collisions compared with SMF–DCF line. For higher bit rate pulses, because the stretching due to dispersion is larger, either smaller dispersion coefficient or shorter segment length is required. Because the former may suffer from inter-channel collisions more, the latter may be an intriguing solution [41]. However, it could suffer from larger excess loss mainly due to splicing loss between the different fibers. One attempt is proposed to realize a seamless dispersion managed fiber with a modest dispersion coefficient and segment length [42], however, a drawback of such the fiber may be a relatively large loss and nonlinearity, and difficulty to handle in real fields. The optimization of DM fiber is still ongoing. It is also important to note that the optimization depends on the bit rate of pulses. Therefore, when the high speed optoelectronic devices or practical technologies for high bit

rate OTDM are developed, the suitable DM fibers should be designed and

developed accordingly. 3.

Nonlinear Coefficient Measurement of RDF

We study a new method to simultaneously measure fiber nonlinear coefficient and group velocity dispersion, using a technique that is based on degenerate four-wave mixing (DFWM). Generally, it is difficult to analytically solve the coupled nonlinear mode equations which express DFWM.

We introduce an approximate solution taking into account nonlinear effects, such as the SPM and XPM of pump light [43]. Under this approximation, the conversion efficiency (Gc) can be analytically expressed and depends on the pump power, the phase mismatch condition and the nonlinear coefficient Therefore, the dispersion and nonlinear coefficients at pump wavelength can be decided simultaneously by measuring the conversion efficiency (Gc) versus the input pump power, through fitting the experimental data with the analytic formula including two parameters and In order to validate the above approximation, the fiber has to be rel-

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atively short. In this experiment, we used a 830 m long RDF. The experimental set-up is shown in Fig. 5. We measured the conversion efficiency (Gc) versus the launched pump power. Keeping constant the wavelength spacing between the pump and probe waves, we measured the nonlinear and dispersion coefficients for various center wavelengths of the pump and probe waves and obtained the wavelength dependence of dispersion around 1550 nm. The results arc shown in Fig. 6. The average value of nonlinear coefficient was with the standard deviation of 5.4 %. The experimental points of dispersions are fitted to a straight line. From this fitting, the dispersion coefficient D at 1550 nm and the dispersion slope were found to be –15.77 ps/nm/km and , respectively. Table 2 compares the values from Fig. 6 with the nonlinear coefficient measured by XPM method and the dispersion coefficient by the phase shift measurement technique (HP83467A) of the same fiber. A good agreement is obtained. By this measurement, it is easy to measure nonlinear coefficients of different fibers with various dispersions in region. This method can be used for constructing dispersion managed transmission lines for WDM/TDM soliton transmissions.

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4. Broadband Raman Amplifiers 4.1. HIGH POWER PUMP LASERS AND PLC COMBINER

In order to increase pumping power of EDFA, one has to introduce a multiplexing technique of more than one pump lasers, as the output from one pump laser is limited up to 250 mW to date [44]. Recently, pump lasers with fiber Bragg grating (FBG) as external cavity have been developed [45].

This technology enables us to efficiently multiplex plurality of pump lasers with different wavelength through an integrated planar lightwave circuit (PLC) [46]. We call this combined pump light source ‘high power pump-

ing unit (HPU)’. Figure 7 shows the schematic of HPU. The PLC device comprises seven imbalanced Mach-Zehnder interferometers for multiplexing eight wavelengths. The output-light-to- current (I–L) characteristics are plotted in Fig. 8. It shows that the output power exceeds 1 W. 4.2. RAMAN AMPLIFICATION IN DCF

We have applied a HPU to Raman amplification in DCF [47]. The setup is depicted in Fig. 9. The net gain spectrum is shown in Fig. 10. By using four wavelengths, 1435, 1450, 1465 and 1480 nm, 50 nm bandwidth of transparency is obtained. Because the Raman gain is distributed over distances in DCF, it has an effect to increase the overall optical SNR. Figure 11 shows an experimental setup for comparison of the noise figure of the in-line EDFAs with DCF with and without Raman gain in DCF. DCRA in

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the figure stands for dispersion compensating Raman amplifier. We compare three cases, A, B and C. In case A, EDFA is composed of just two stages of EDFAs. Case B is added a DCF as an intermidiate stage device between the two stages. Then, we turn on the Raman gain by launching the pump lasers into DCF in case C. Figure 12 shows the resulting noise

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figure versus wavelength for each case. The results clearly indicate that the Raman gain successfully reduces the noise figure by at most 1.5 dB, as comparing type B with C.

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4.3. RAMAN AMPLIFICATION IN TRANSMISSION FIBERS The above results are analogous to how Raman gain improves the SNR of

transmission lines. If we apply Raman gain in a repeater span, the optical SNR will improve to a considerable extent [26]. As we discussed in Section 1 and Subsection 2.3, major drawbacks of DM fiber are excess loss and

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nonlinearity due to the waveguide structure to realize normal dispersion. Because Raman gain is derived from fiber nonlinearity, normal dispersion fibers in general have an efficient Raman gain cross-section. Raman amplification in DM fiber is thus effective for not only reducing excess loss and increasing optical SNR, but also reducing the nonlinearity. Also, a fair amount of normal dispersion of DM fiber helps to mitigate adverse effects of nonlinearity on signals. Astonishing results have been obtained by using Raman amplification in DM fibers [48]. Another attractive feature of Raman amplification is limitless wide bandwidth. By using twelve wavelengths of pump lasers ranging from 1405 to 1510 nm as a HPU, we have achieved a flat Raman gain over 100 nm without using any gain equalizing filters in SMF, DSF and RDF [28]. We have so adjusted each pump power at different wavelength as to equalize the Raman gain. Figure 13 shows the 100 nm flat Raman gain, and Fig. 14 plots the pump power allocation upon wavelength for equalization of Raman gain over 100 nm. Comparing a fiber with the others, RDF is most efficient in terms of gain and total amount of pump power. This reflects on the magnitude of nonlinearity; RDF has the largest nonlinearity while SMF has the smallest. The pump lasers for Raman amplifiers used in above experiments are the same kind as the pump lasers for EDFAs, that are already deployed widely with the proven reliability. Therefore, Raman amplifiers are promising not only because of its efficacy but also its robustness.

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5. s and Their Applications 5.1. DESIGN OF HNL-DSF

We developed and fabricated a HNL–DSF by the vapor-phase axial deposition method [49]. The refractive index profile of the fiber is schematically

shown in Fig. 15. The transmission characteristics of the HNL–DSF are shown in Table 3. The nonlinear coefficient of the fiber is 13.8 The value is 5.14 times larger than the value of the conventional DSF. 5.2. MULTICHANNEL WAVELENGTH CONVERTER

We used 100 m of the above ‘non-PM’ HNL–DSF to achieve more than 91 nm broadband wavelength conversion [31]. Schematic diagram of the experimental setup is shown in Fig. 16.

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The pump and the signal waves are both amplified by EDFAs then combined by a fused coupler. SOP (State Of Polarization) of the waves are coincided by using a polarizer at the input of the HNL–DSF. The pump wavelength is set at the zero-dispersion wavelength of the HNL–DSF. Launched pump and signal powers are 100 mW (20dBm) and 1 mW (0dBm)/channel, respectively. We used 26 WDM signals ranging 1570-1611 nm to demonstrate simultaneous wavelength conversion. The output spectrum from the HNL–DSF is shown in Fig. 17. Figure 17 clearly indicates that signals fully

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belonging to one EDFA band, e.g. L-band, can be simultaneously converted into those belonging to the other EDFA band, i.e. C-band, and vice versa. We achieved 91.3 nm broadband simultaneous conversion. To our knowledge, the bandwidth is the broadest result of the published fiber FWM based parametric wavelength conversion experiments. We note that a 72 nm broadband wavelength conversion was achieved using a polarization maintaining (PM) fiber [50], while in this case we used ‘non-PM’ fiber. The maximum conversion efficiency in this case was –18.9 dB. However, the pump power was well below the SBS (Stimulated Brillouin Scattering) threshold and could be increased up to more than 200 mW, in which case the maximum conversion efficiency would be more than –15 dB. 5.3. SOLITON COMPRESSOR [33]

The short optical pulse sources with high repetition rates are expected to play a key role for increasing the transmission capacity in future. Since electric methods are difficult to generate pulses at a repetition rate of higher than 100 GHz, soliton pulse compression of the beat modulation between

two optical carriers [51] is an attractive method for applications in TDM optical communications. This technique permits us to use standard tunable laser sources and to tune the repetition rate and the center wavelength of

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the pulse train simply by adjusting the frequencies of the input lasers. The techniques of beat signal conversion were previously demonstrated with dispersion-decreasing fiber (DDF) [52]. An alternative approach is to profile the fiber dispersion using segments of conventional fibers with different

dispersions. Using a comb-like dispersion profiled fiber (CDPF) consisting of alternating the segments of dispersion shifted and standard telecommunication fibers (DSF and SMF) respectively [32, 53, 54], high repetition rate sub-picosecond pulse train were generated. An advantage of CDPF is that we can control the dispersion and the nonlinear phase shifts by selecting appropriate lengths of fiber segments. Previous works using a CDPF have utilized conventional standard DSF and SMF. In this subsection, we demonstrate a short comb-like dispersion profiled fiber using the HNL–DSF to achieve 104 GHz soliton pulse train with 328 fs pulsewidth. We have constructed the comb-like dispersion profiled fiber by use of the same HNL–DSF as in the previous subsections and commercially available SMF. The fiber consisted of 5 pairs of HNL–DSF and SMF and had a total length of 693.6 m. The characteristics of fibers in CDPF are shown in Table 4. The experimental setup is shown in Fig. 18. A dual frequency beat signal was generated using two tunable-wavelength lasers (HP8368F). The wavelength space between two waves was set at 0.831 nm in order to generate pulse train at a near 100 GHz repetition rate. The center wavelength was tuned to coincide with the zero dispersion wavelength of the HNL–DSF in order to generate four-wave mixing efficiently. Two tunable laser sources were phase modulated using the built-in modulation function to suppress the Stimulated Brillouin Scattering (SBS). Two signal powers were set equal and the polarization states were aligned parallel by means of adjusting polarization controllers to minimize the insertion loss of the polarizer. Both waves are combined by a fiber fused coupler. The beat sig-

nal was amplified by an EDFA. The launched beat signal power to the first HNL–DSF of CDPF was set to +27.5 dBm that is below the limit of SBS. The SHG autocorrelation trace and optical spectrum of input beat signal and output pulses after transmission of the 5th pair-CDPF are shown in Fig. 19. For the pulse transmitted through the 5th pair-CDPF, the pulse repetition rates was 104 GHz and the measured autocorrelation full-width

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at half-maximum (FWHM) was 505 fs, which gives a pulsewidth of 328 fs, assuming a pulse-intensity profile. The duty cycle of the pulse out-

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put from the 5th pair-CDPF was 0.034 and the period-to-duration ratio (T is the period and is the FWHM of the pulse) reached 29.3, which is to best of our knowledge, the largest compression ratio using this scheme. The large compression ratio may be mainly attributed to the use

of HNL–DSF for efficient sideband generation.

6. Summary As transmission capacity and distance increase, the role of transmission fibers becomes more important because nonlinear interactions and ASE

noise accumulate through fibers. A smart way may be the use of soliton which is a steady state solution to the nonlinear Schrödinger equation. However, reality is that one ought to allow dispersion to vary over distance and to handle an enormous number of perturbed solitons colliding each other at the same time. Dispersion managed soliton seems to be an optimal answer to the challenge for massive WDM and TDM transmissions. Although it is periodically ‘breathing’, it takes advantages from the balance between

GVD and SPM like soliton, and hence maintain its pulse shape over long distances, and besides, it mitigates other undesirable nonlinear effects such as FWM and XPM, and thus relatively immune to collisions. In this regard, the proper design of DM fiber receives immense importance. This paper has compared SMF–RDF and SMF–DCF and discussed some ongoing issues on further optimizations. We also have shown that by using FWM accounting for dispersion, the dispersion and nonlinear coefficient of RDF can be measured simultaneously. This technique is of much importance as DM fibers tend to have relatively large dispersion. Raman amplification in DM fibers has been shown to be efficient and effective to reduce the system noise and hence nonlinearity by compensating for fiber loss. For future networking, high nonlinearity fibers have been demonstrated to successfully apply to broadband simultaneous wavelength conversion as well as efficient soliton compression. We believe that optical fiber technologies play a key role in enabling massive WDM/TDM soliton transmissions.

Acknowledgments

The author would like to acknowledge his collaborators regarding to this report who are, in alphabetical order, Y. Akasaka, S. Arai, O. Aso, Y. Emori, K. Mukasa, R. Sugisaki, Y. Suzuki, M. Tadakuma, K. Tanaka and T. Yagi. He also thank Mr. H. Miyazawa and Mr. T. Fukazawa for their supports.

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Hasegawa, A. and Kodama, Y. : Solitons in Optical Communications, Oxford University Press, Oxford, (1995). 2. Haus, H. A. and Wong, W. S. : Solitons in optical communications, Rev. Mod. Phys., 68, (1996), pp.423-444. 3. Mamyshev, P. V. and Mollenauer, L. F. : Stability of soliton propagation with sliding-frequency guiding filters, Opt. Lett., 19: (1994), pp.2083-2085; see also, Mamyshev, P. V. and Mollenauer, L. F. : Wavelength-division multiplexing channel energy self-equalization in a soliton transmission line by guiding filters, Opt. Lett., 21, (1996). pp.1658-1660. 4. Mamyshev, P. V. and Mollenauer, L. F. : Pseudo-phase-matched four-wave mixing in soliton wavelength-division multiplexing transmission, Opt. Lett., 21, (1996), pp.396-398.

5. See for example, Akasaka, Y., Sugisaki, R. and T. Kamiya : Dispersion compensating technique of 1300 nm zero-dispersion SM fiber to get flat dispersion at 1550 nm range, ECOC’95, Vol.2, (1995), pp.605-608. 6.

Garrett, L.D., Eiselt, M. and Tkach, R. : Experimental comparison of WDM sys-

tem capacity in conventional and non-zero dispersion shifted fiber, OAA ’98, TuB2, (1998), pp.84-87. 7.

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mun., 87, (1991), pp.15-18 8. Suzuki, M., Morita, I., Edagawa, N., Yamamoto, S., Taga, H. and Akiba, S. : Reduction of Gordon-Haus timing jitter by periodic dispersion compensation in soliton transmission, Electron. Lett, 31, (1995), pp.2027-2028. 9. Smith, N. J., Knox, F. M., Doran, N. J., Blow, K. J. and Bennion, I. : Enhanced power solitons in optical fibres with periodic dispersion management, Electron. Lett.,

32, (1996), pp.54-55. 10.

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IEEE Photon. Technol. Lett., 9, (1997), pp.785-787. Georges, T. and Favre, F. : Transmission systems based on dispersion-managed solitons: theory and experiment, in A. Hasegawa (ed.), New Trends in Optical Soliton

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Transmission Systems, Kluwer Academic Publishers, (1998), pp.317-340. Tamura, K., Haus, H. A., Ippen, E. P. and Nelson, L. E. : 77-fs pulse generation from a stretched-pulse additive pulse mode locked all-fiber ring laser, Opt. Lett., 18, (1993), pp.1080-1082. Haus, H. A., Tamura, K., Nelson, L. E. and Ippen, E. P. : Stretched-pulse additive pulse mode-locking in fiber ring lasers: theory and experiment, IEEE J. Quantum Electron., QE-31, (1995), pp.591-598. Tamura, K., Nelson, L. E., Haus, H. A. and Ippen, E. P. : Soliton versus nonsoliton operation of fiber ring lasers, Appl. Phys. Lett., 64, (1994), pp.149-151. Namiki, S. and Haus, H. A. : Noise analysis of the stretched pulse laser: Part I — Theory, IEEE J. Quantum Electron., 33, (1997), pp.649-659. Yu, C. X., Namiki, S. and Haus, H. A. : Noise analysis of the stretched pulse laser: Part II — Experiment, IEEE J. Quantum Electron., 33, (1997), pp.660-668. Gordon, J. P. and Haus, H. A. : Random walk of coherently amplified solitons in

optical fiber transmission, Opt. Lett., 11, (1986), pp.665-667. 18. Bergano, N. S. et, al. : 640 Gb/s transmission of sixty-four 10 Gb/s WDM channels over 7200 km with 0.33 (bits/s)/Hz spectral efficiency, OFC’99, PD2., (1999). 19. Wada, A., Tsun, T.-O. and Yamauchi, R. : Measurement of nonlinear-index coefficients of optical fibers through the cross-phase modulation using delayed-self-

heterodyne technique, ECOC’92, (1992), pp.45-48. 20.

Stolen, R. H . and Lin, C. : Self-phase-modulation in silica optical fibers, Phys. Rev., A 17, (1978). pp.1448-1453.

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fiber at 1.55 Opt. Lett., 21, (1996), pp.1966-1968. Bigo, S. and Chbat, M. W. : Measurement of the impact of fiber nonlinearities on high data rate, dispersion-managed WDM systems, SOFM, (1998), pp.77- 82. Tadakuma, M., Aso, O. and Namiki, S. : Nonlinear coefficient measurement of reverse dispersion fiber using four wave mixing, ECOC’99, Vol.2, (1999), pp.224-225. Tadakuma, M., Aso, O. and Namiki, S. : Nonlinear coefficient measurement of reverse dispersion fiber using four wave mixing, OFMC. (1999), pp.71-74. Stolen, R. H. and Ippen, E. P. : Raman gain in glass optical waveguides, Appl. Phys. Lett., 22, (1973), pp.276-278. Nissov, M., Davidson, C. R., Rottwitt, K., Menges, R., Corbett, P. C., Innis, D. and Bergano, N. S. : 100 Gb/s WDM transmission over 7200 km using distributed Raman amplification, ECOC’97, Vol.5, (1997), pp.9-12. Kawakami, H., Miyamoto, Y., Yonenaga, K. and Toba, H. : Highly efficient distributed Raman amplification system in a zero-dispersion-flattened transmission line, OAA ’99, ThB5, (1999). Emori, Y. and Namiki, S. : 100 nm bandwidth flat gain Raman amplifiers pumped and gain-equalized by 12-wavelength-channel WDM high power laser diodes, OFC’99, PD19, (1999). Naito, T., Shimojoh, N., Tanaka, T., Nakamoto, H., Doi, M., Ueki, T. and Suyama, M. : 1 Tbit/s WDM transmission over 10000 km, ECOC’99, PD2-1, (1999), pp.2425. See for example, Bigo, S. et al. : 1.5 Terabit/s WDM transmission of 150 channels at

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Aso, O., Tadakuma, M. and Namiki, S. : More than 91 nm broadband four-wave

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25. 26. 27. 28. 29.

10 Gbit/s over

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fibre, ECOC’99, PD2-9, (1999), pp.40-41.

mixing based fiber parametric wavelength converter, ECOC’99, PD1-10, (1999),

pp.20-21. 32. Chernikov, S. V., Taylor, J. R. and Kashyap, R. : Integrated all optical fibre source of multigigahertz soliton pulse train, Electron. Lett., 29, (1993), pp.1788-1789. 33. Tadakuma, M., Aso, O. and Namiki, S. : A 104GHz 328fs soliton pulse train generation through a comb-like dispersion profiled fiber using short high nonlinearity,

OFC 2000, ThL3, (2000). 34. Collings, B. C., Mitchell, M. L., Boivin, L. and Knox, W. H. : A 1022 channel WDM transmitter, ECOC’99, PD1-4, (1999), pp.8-9. 35. Akasaka, Y., Sugisaki, R. and Kamiya, T. : Dispersion compensating technique of 1300 nm zero-dispersion SM fiber to get flat dispersion at 1550 nm, ECOC’95, Vol.2, (1995),pp.605-608. 36. Mukasa, K., Akasaka, Y., Suzuki, Y. and Kamiya, T. : Novel network fiber to manage dispersion at 1.55 with combination of 1.3 zero dispersion single mode fiber, ECOC’97, Vol.1, (1997), pp.127-130. 37. Miyamoto, Y. et al. : 1.04-Tbit/s DWDM transmission experiment based on alternate- polarization 80-Gbit/s OTDM signals, ECOC’98, Vol.3 PD Papers, (1998), pp.53-54; see also, Murakami, M. et al. : Quarter terabit over 9288 km WDM transmission experiment using nonlinear supported RZ pulse in higher order fiber dispersion managed line, ECOC’98, Vol.3 PD Papers, (1998), pp. 77-78.

38. Yamamoto, T., Yoshida, E., Tamura, K. R. and Nakazawa, M. : Single-channel 640 39.

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Gbit/s TDM transmission over 100 km, ECOC’99, Vol.2, (1999), pp.38-39. Fukuchi, K., Kakui, M., Sasaki, A., Ito, T., Inada, Y., Tsuzaki, T., Shitomi, T., Fujii, K., Shikii, S., Sugahara, H. and Hasegawa, A. : 1.1 Tb/s dense WDM soliton transmission over 3,020-km widely- dispersion-managed transmission line employing hybrid repeaters, ECOC’99, PD2-10, (1999), pp.4243. Agrawal, G. P. : Nonlinear fiberoptics, ed., Academic Press, (1995), pp.90-92.

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Liang, A. H., Toda, H. and Hasegawa, A. : Dense periodic fibers with ultralow fourwave mixing over a broad wavelength range, Opt. Lett., 24, (1999), pp. 1094-1096. 42. Anis, H. et al. : Continuous dispersion managed fiber for very high speed soliton systems, ECOC’99, Vol.1, (1999), pp.230-231.

43.

Stolen, R. and Bjorkholm, J. E. : Parametric amplification and frequency conversion

in optical fibers, IEEE J. Quantum Electron., QE-18, (1982), pp.1062-1072. 44. Kimura, T., Tsukiji, N., Iketani, A., Kimura, N., Murata, H. and Ikegami, Y. : High temperature operation quarter watt 1480 nm pump LD module, OAA ’99, ThD12, (1999). 45. Koyanagi, S., Mugino, A., Aikiyo, T. and Ikegami, Y. : The ultra high-power 1480 nm pump laser diode module with fiber Bragg-grating, OAA’98, MC2, (1998).

46. Tanaka, K., Iwashita, K., Tashiro, Y., Namiki, S. and Ozawa, S. : Low-loss integrated Mach-Zehnder interferometer-type eight-wavelength multiplexer for 1480 nm band

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48. 49.

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pumping, OFC’99, TuH5, (1999). Emori, Y., Akasaka, Y. and Namiki, S. : Less than 4.7 dB noise figure broadband in-line EDFA with a Raman amplified - 1300 ps/nm DCF pumped by multi-channel WDM laser diodes, OAA’98, PD3, (1998). Ohhira, R. et al. : NRZ WDM transmission experiment over using distributed Raman amplification in RDF, ECOC’99, Vol.2, (1999), pp.176-177; see also Ref. 29. Aso, O., Arai, S., Yagi, T., Tadakuma, M., Suzuki, Y. and Namiki, S. : Broadband wavelength conversion using a short high-nonlinearity non-polarization-maintaining

fiber, ECOC’99, Vol.2, (1999), pp.226-227. Watanabe, S., Takeda, S. and Chikama, T. : Interband wavelength conversion of 320 Gb/s WDM signal using a polarization-insensitive fiber four-wave mixer, ECOC’98, PD3, (1998), pp.85-86. Dianov, E.M., Mamyshev, P. V., Prokhorov, A. M. and Chernikov, S. V. : Generation of a train of fundamental solitons at a high repetition rate in optical fibers,

Opt. Lett., 14, (1989), pp.1008-1010. 52. Chernikov, S.V., Taylor, J. R., Mamyshev, P. V. and Dianov, E. M. : Generation of soliton pulse train in optical fibre using two cw singlemode diode lasers, Electron. Lett., 28, (1992), pp.931-932. 53. Swanson, E.A. and Chinn, S. R. : 40-GHz pulse train generation using soliton compression of a Mach-Zehnder modulator output, IEEE Phton. Technol. Lett., 7, (1995), pp.114-116. 54. Agata, A., Kurebayashi, R., Suzuki, A., Maruta, A. and Hasegawa, A. : Optimal design of comb-like dispersion profiled fiber for picosecond optical soliton source, OECC’98, Technical Digest, (1998), pp.190-191.

COLLISION-INDUCED IMPAIRMENTS IN DISPERSION MANAGED FIBER SYSTEMS

S. KUMAR AND A. F. EVANS Corning Incorporated Fiber Communications, SP-AR-01-2, Corning, New York - 14831, USA. Abstract. Recent advances in fiber characteristics of dispersion-managed fiber is reviewed. We have studied the transmission of single channel RZ systems as well as collision-induced impairments in WDM systems in dispersionmanaged fibers. With short period dispersion managed fiber, single channel

100 Gb/s is achievable without transmission control. By emplyoing strong dispersion maps, the collision-induced impairments such as four wave mixing and timing jitter due to cross-phase modulation can be suppressed.

1. Introduction

Dispersion management techniques [1]-[11] are attractive for ultra high bit rate wavelength division multiplexed (WDM) systems owing to the fact

that (i) nonlinear impairments are reduced by the dispersion management and (ii) nearly net zero dispersion of the transmission line can be achieved. Several recent transmission experiments have demonstrated superior spectral efficiency by combining dispersion management with return-tozero modulation format (chirped RZ, dispersion-managed solitons, etc.) [10, 11]. Given the finite spectral width of the low-loss window of optical fiber, improved spectral efficiency is key to pushing fiber capacity well into the Terabit/s regime. Extending the TDM or WDM dimension also requires innovations in fiber design and understanding of impact of the nonlinear impairments of densely-packed wavelength channels. In this paper, we review recent advances in fiber design and fabrication that optimize dispersionmanagement for very high speed systems, RZ single-channel systems and study the collision-induced impairments imposed by WDM. 351 A. Hasegawa (ed.), Massive WDM and TDM Soliton Transmission Systems, 351–363. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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In the case of ideal solitons (lossless), four wave mixing(FWM) components generated during collision are re-absorbed by solitons and hence, there is negligible FWM noise in a ultra long ideal soliton WDM systems. The only contribution to the FWM arises from the half-collisions occur-

ing at the transmitters/detectors. Ideal soliton systems can be realized by means of dispersion decreasing fibers [12]-[15] or distributed amplifications

[16, 17]. In the case of conventional soliton systems with lumped amplifiers, the periodicity of amplifiers provide the required phase matching and FWM components generated during collision are not re-absorbed by solitons and hence, FWM components grow with distance [15]. In this paper, we study the growth of FWM components in a dispersion managed fiber during collision and show that FWM components are significantly suppressed by dispersion management. Example calculations show that there is more than 30 dB reduction in FWM energy by the use of dispersion management. We have studied the FWM growth as a function of pre-chirp, dispersion management period and the local dispersions of dispersion managed fiber (DMF), and results show that the FWM components are significantly suppressed by longer periods and/or larger local dispersion. Energy in a FWM sideband displays a resonant feature with pre-chirp. Additionally, Collision-induced frequency shift in dispersion managed systems has attracted a considerable attention [18]-[23]. However, the previous studies have mainly focussed on the residual frequency shift. Here, we focus on the maximum frequency shift due to half-collisions. In terrestrial systems, the typical transmission distance does not exceed 1000 km and we have found that half-collisions occuring at transmitters/detectors are the dominant noise sources as compared to the residual frequency shift due to full collisions. The frequency shift due to half-collisions increase initially for extremely short dispersion management periods and then, it decreases linearly for half-periods longer than 0.7 km. Most previous dispersion-management schemes have been span-by-span compensation within the line-in optical amplifier or two-step dispersion maps between amplifier sites. This frequency of compensation has a good dispersion map strength for 10 to 20 Gb/s transmission. However, as bandwidth demand pushes fiber capacity, network granularity on the 40 to 80 Gb/s single-channel level becomes desirable. To maintain the optimum map strength, the frequency of compensation needs to decrease inversely with the data rate:

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where is the pulse width and are the dispersions and lengths of the two sections of DMF, respectively. It has been shown that 1.65 minimizes single channel soliton interactions [24]. For transmission exceeding 10 Gb/s at terrestrial distances (500–1000 km), this is a dominant impairment [25]. Therefore, 40 to 80 Gb/s requires dispersion compensation every 1 to 10 km. Progress in fabricating such fibers either continuously (without fiber splices) or in a managed-cable approach is addressed in the first part of this paper. 2. Dispersion-Managed Fiber

Recent fiber designs not only compensate first order dispersion but also second order, dispersion slope. Slope compensation is a basic requirement of next generation transmission links that push either the spectral or temporal bandwidth — more channels or faster channels. In principle, any of these refractive index profile design solutions can be applied in a dispersionmanaged cable configuration to improve transmission greater than 10 Gb/s. However, frequent splice points degrade the link by adding additional loss. A more interesting solution is continuously fabricated fiber with built-in dispersion and slope compensation [26, 27]. The dispersion properties for such a fiber are schematically shown in Fig. 1. The OTDR trace for of a 20 km length of dispersion-managed fiber shown in Fig. 2(a) illustrates its continuous nature. The step-like changes of the OTDR signal at km intervals indicate the fiber transitional interface between dispersion sections of different sign. They are the result

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of different back-scatter capture efficiency caused by a small change in the fiber’s effective area. The effective area difference between the sections of opposite dispersion signs are larger at 1310 nm than at 1550 nm as evidenced in the figure. Figure 2(b) is the average of OTDR traces from both ends of the fiber. Bi-directional OTDR separates the effective area variations from loss variations. As can be seen, there is no excess loss in the dispersion transitions. This result should be emphasized since with an effective area of 28 that this fiber has, the fiber splices needed for dispersion-managed cable would be expected to have a 0.2 to 0.5 dB loss depending on the effective area mismatch of the fibers. One of the tradeoffs in fiber design is the desire for large effective area and low or compensating dispersion slope. Usually the effective area decreases as the dispersion slope is lowered. For this example, we have pushed slope compensation to as good as what is achievable with combining transmission fiber with discrete, dispersion-compensating fiber. The resulting relatively small effective area may not be a serious issue if the modulation format is designed to take advantage of the enhanced self-phase modulation nonlinearity. Other properties of this fiber that make it well-suited for high speed applications are a magnitude of local dispersion that can be designed from 0 to 10 ps/nm·km, near zero average dispersion, and equal and opposite dispersion slope for near zero average dispersion slope. The fiber for the recently reported 100 Gb/s transmission experiment had local dispersion, 0.05 ps/nm·km average dispersion, lo-

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cal dispersion slope and average dispersion slope [28]. A standard loop configuration consisting of three spans of dispersion-managed fiber with a total length of 101 km is used to demonstrate 100 Gb/s over 1000 km [28]. The average span dispersion at the transmitter wavelength of 1556 nm are 0.09 ps/nm·km for span 1, –0.04 ps/nm·km for span 2 and 0.04 ps/nm·km for span 3. The span lengths are 35.5 km for span 1, 33.6 km for span 2, and 31.1 km for span 3. The local dispersion alternates with a symmetric 1 km period. Additionally, the dispersion slope has an average value of 0.004, 0.009 and for the three spans. These slope values are maintained over the entire 1550 nm band (1500 to 1600 nm). The measured linear Q was greater than 6 for propagation distances up to 1000 km, equivalent to error rates below [28]. The main limitation to grearer error-free distances is the local dispersion variations in the fibers that make up the spans. It is expected that transmission control schemes such as partial regenerators would deliver significant improvement in performance. Finally, the near zero dispersion slope built-in to this fiber is necessary for consistent single channel performance across the entire erbium band, a prerequisite for extending the system capability to WDM transmission. The dispersion map period of 1 km employed is significantly lower than is typical of previous experiments, resulting in single-channel transmission at an equivalent rate of 100 Gbit/s over 1000 km. This dispersion managed fiber system not only has an average dispersion near zero (0.03 ps/nm·km) but also has near zero built-in dispersion slope compensation. The average slope of measured over the wavelength band 1500-1600 nm equals the best results achieved using discrete dispersion compensating fiber. This result shows that DMF enables very high capacity, single channel transmission. WDM will be needed for higher capacity. System simulations using similar fiber profile designs that carefully trade-off dispersion slope and effective area have shown 1 Tb/s is achievable. 3. Four Wave Mixing For WDM applications, it is necessary to understand the collision-induced impairments such as FWM and timing jitter due to cross-phase modulation. Figure 3 shows the FWM components generated by a soliton collision in a two-channel periodically-amplified soliton WDM system. The following parameters are assumed: peak power = 11 mW, pulse width = 5 ps, Bit rate = 40 Gb/s, channel separation dispersion fiber loss = 0.21 dB/km, amplifier spacing = 80 km. Initial separation between pulses of two channels are 25 ps and therefore, collision occurs at The solitons are tem-

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porally well separated after 900 km. FWM components generated during the collision is not re-absorbed by solitons due to phase matching provided by periodic amplifiers. By setting the fiber loss we have verified that FWM components generated during the collision are re-absorbed by solitons. After introducing a band pass filter around one of the FWM sidebands, FWM components are integrated in frequency domain to obtain the FWM energy. Instead of a constant dispersion fiber, dispersion managed fiber (DMF) with the following parameters is used as a transmission fiber: dispersions

Lengths

and the Gaussian pulse of width 5 ps (FWHM) is launched in both channels with a temporal separation of 25 ps between pulses. The average dispersion, pulse width, channel separation and the initial temporal separation are kept fixed for all the following simulations. The broken line in Fig. 4 shows the FWM energy as function of propagation distance in the DMF. For comparison, the FWM energy in a constant dispersion fiber (classical soliton system)

with the same average dispersion and the same peak power is shown in the solid line. As can be seen, there is a 30 dB reduction in FWM energy by this dispersion management. In both cases, FWM components remain constant after the collision is complete. In a trans-oceanic system, there are multiple collisions and FWM components grow linearly with distance as in the cw case. A pre-chirp is introduced to the DMF system by inserting a section of

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normal dispersion fiber immediately after the transmitter. Figure 5 shows the FWM energy as a function of pre-chirp for different dispersion maps. In Fig. 5, the parameter S (defined in Eq. (1)) is 1.6 for all the dispersion map. The top curve is obtained by using a dispersion map with = 8 = 2.5 km. For the given 5, lower local dispersion (and longer length) decreases the FWM efficiency. From Fig. 5, it can also be seen that there is a resonance in FWM for a pre-chirp of 4.7 ps/nm for all the dispersion maps. In Fig. 6, FWM energy is plotted as a function of the half-period for different local dispersion . As can be expected, FWM efficiency decreases as the half-period and/or dispersion increases. The strong dispersion management suppresses the FWM sidebands. 4.

Collision-induced frequency shift

In a terrestrial system, the maximum transmission distance is less than 1000 km and number of collisions with the neighboring channel is typically one

or two. Therefore, impairments due to half-collision is more dominant than that due to full collisions. We have numerically estimated the frequency shift of channel 1 during a collision. To calculate the frequency shift, we have introduced a rectangular bandpass filter for channel 1 whose bandwidth is

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same as the channel spacing. The mean frequency of the channel 1 after the filter is computed by

where is the Fourier transform of the field after filtering. A tail of the spectrum of channel 2 falling within the filter bandwidth causes an error in the computation of frequency shift of channel 1 due to collision. In our parameter space, this error does not exceed 1 GHz. In Fig. 7, the frequency shift - central frequency of channel 1) is plotted as a function of propagation distance for the ideal soliton system with constant dispersion (solid line) and for dispersion managed fiber system (broken line). The parameters are same as that used in Figs. 3 and 4. As can be seen, the maximum frequency shift due to half-collision is reduced by a factor of 3 by using the dispersion managed fiber. However, the timing shift is proportional to the area under the curve which is roughly same for both systems. The collision induced frequency shift can cause impairments in the following ways: (i) Part of the spectrum of channel i falling within the bandwidth of filter corresponding to channel increases. In Fig. 7, this effect would be most serious if the transmission distance is around 400 km

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corresponding to a half-collision. (ii) Integrated frequency shift multiplied by the average dispersion is equal to the total timing shift of the pulse which gives rise to the timing jitter. In Fig. 8, the frequency shift due to half-collision (corresponding to the maximum in Fig. 7) is plotted as a function of pre-chirp for different dispersion maps. The parameters used are same as that in Fig. 5. The parameter S is 1.6 for all the dispersion maps. Unlike the FWM energy, the frequency shift depends only on S. The frequency shift and FWM energy decrease with lower pre-chirps. But moderate amount of pre-chirp is required to form a stable pulse. In Fig. 9, the frequency shift due to half collision is plotted as a function of the half-period for different local dispersion The maximum

frequency shift increases initially for extremely short periods and then, it decreases for half-periods longer than 0.7 km. FWM efficiency and the maximum frequency shift are significantly reduced by the strong dispersion management. But the pulse breathing increases with the dispersion management strength which increases the pulse interaction with the adjacent bit [24, 29, 30]. 5. Conclusions

Dispersion and especially dispersion slope management will be increasingly important to design and control. Dispersion-managed cable and fiber ap-

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proaches offer solutions for single-channel capacity greater than 10 Gb/s.

Using the dispersion-managed fibers, 100 Gb/s transmission over 1000 km is achievable. In a classical soliton system of this bit rate using an average and 2.3 ps pulses, the amplifier spacing needs to be on the order of 10 km. It is clear that dispersion managed fiber enables greater amplifier spacing, and additionally offers performance advantages at higher bit rates resulting from energy enhancement and lower Gordon-Haus jitter and soliton-soliton interactions. It is believed that without transmission control this system performance at 100 Gbit/s is achievable only with short period dispersion managed fiber. For WDM applications, we have estimated the collision-induced impairments in a dispersion managed fiber systems. The four wave mixing gencrated during the collision has been studied for different dispersion maps. By comparing the energy in a four wave mixing sideband in a constant dispersion fiber (conventional soliton system) and a dispersion managed fiber, we have found that the FWM energy is suppressed by more than 30 dB by using the dispersion managed fibers. The dispersion managed fiber with strong dispersion map is also beneficial in reducing the collision-induced

frequency shift. When the parameter S is fixed, FWM sidebands are better suppressed by lower local dispersion (and longer period). However, the frequency shift due to half-collision depends only on S irrespective of dis-

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persions and DM periods. Acknowledgements

The authors wish to thank H. Anis, G. Bordogna, M. Cavallari, B. Charbonnier, I. Hardcastle, M. Jones, G. Pettitt, B. Shaw and J. Wakefield, their collaborators at Nortel Networks, for the transmission work, V. Srikant, G. Berkey and Chris Tennent of Corning Incorporated for fiber design and fabrication, and G. Luther of Corning Incorporated for discussion on numerical simulations. References 1. Suzuki, M., Morita, I., Edagawa, N., Yamamoto, S., Taga, H. and Akiba, S. : Reduction of Gordon-Haus jitter by periodic dispersion compensation in soliton trans-

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7. Jacob, J. M., Golovchenko, E. A., Pilipetskii, A. N., Carter, G. M. and Menyuk, C. M. : Experimental demonstration of soliton transmission over 28 Mm using mostly normal dispersion fiber, Photon. Technol. Lett, 9, (1997), pp.130-132. 8. Matsumoto, M. : Opt. Lett., 22, (1997), p.1238. 9. Liang, A., Toda, H. and Hasegawa, A. : Opt. Lett., 24, (1999), pp.1094. 10. Bergano, N. et al, OFC’99, PD-2, (1999). 11. Le Guen, D. et al, OFC’99, PD-4, (1999). 12. Hasegawa, A., Kumar, S. and Kodama, Y. : Reduction of collision-induced time jitters in dispersion-managed soliton transmission systems, Opt. Lett., 21, (1996), pp.39-41. 13.

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20. Devaney, F. L., Forysiak, W., Nicule, A. M. and Doran, N. J. : Opt. Lett., 22, (1997), pp.1695-1697. 21. Hirooka, T. and Hasegawa, A. : Opt. Lett., 23, (1998), pp.768-770. 22. Nicule, A. M., Forysiak, W., Gloag, A. J., Nijhof, J. H. B. and Doran, N. J. : Opt. Lett.,23, (1997), pp.1354-1356. 23. Sugahara, H., Maruta, A. and Kodama, Y. : Opt. Lett., 24, (1999), pp.145-147. 24. Yu, T., Golovchenko, E. A., Pilipetskii, A. N. and Menyuk, C. R. : Opt. Lett., 22, (1997), p.793. 25. Evans, A. F. and J. V. Wright, J. V. : IEEE Photonics Technology Letters, 7, (1995), p.117.

26. Bhagavatula, V. A., Berkey, G., Chowdhury, D. and Evans, A. : OECC’97, (1997), p.54. 27.

Bhagavatula, V. A., Berkey, G., Chowdhury, D., A. Evans, A. and Li, M. J. : OFC’98, TuD2, (1998), p.21.

28. Anis, H., et al, : ECOC’99, I, (1999), p.230. 29. Matsumoto, M. : IEEE Photon. Technol. Lett, 10, (1998), p.373. 30. Kumar, S., Wald, M., Lederer, F. and Hasegawa, A. : Opt. Lett., 23, (1998), p.1019.

ULTRA LOW NONLINEARITY PURE SILICA CORE FIBER AND ITS APPLICATION

TO HYBRID TRANSMISSION LINES

T. KATO, M. TSUKITANI, M. HIRANO, E. YANADA, M. ONISHI, M. NAKAMURA, Y. OHGA, E. SASAOKA, Y. MAKIO AND M. NISHIMURA

Yokohama Research Laboratories, Sumitomo Electric Industries, Ltd. 1 Taya-cho, Sakae-ku, Yokohama 244-8588, Japan

Abstract. Hybrid transmission lines consisting of dispersion unshifted single mode fiber and dispersion compensating fiber has been explored. For the dispersion unshifted fiber, new design pure silica core fiber with an effective area over and attenuation of 0.171 dB/km at has been successfully developed. It exhibits ultra low nonlinearity and excellent bending loss performance. Optimum design of the hybrid line employing the new pure silica core fiber has also been examined by tailoring the characteristics of the dispersion compensating fiber. Hybrid transmission lines actually fabricated based on the optimized design exhibit a span loss of 0.205 dB/km, a span dispersion slope of and low nonlinearity with the equivalent effective area of

1. Introduction

Optical transmission lines with low nonlinearity and low dispersion slope are now being required for long-haul large capacity WDM (Wavelength

Division Multiplexed) transmission systems. Recent investigation [1]- [3] has shown that a combination of dispersion-unshifted single mode fiber and dispersion compensating fiber is one of the promising transmission line configurations to fulfill those requirements. In this hybrid transmission line, the former fiber, which has relatively large chromatic dispersion at is effective for reducing the fiber nonlinearity effects, such as four wave mixing (FWM), and the latter compensates for the accumulated dis365 A. Hasegawa (ed.), Massive WDM and TDM Soliton Transmission Systems, 365–377. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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persion and dispersion slope. Since dispersion compensating fibers tend to have slightly higher attenuation than the standard fibers, use of low loss pure silica core fibers (PSCFs), in place of the standard single mode fibers (SMFs), should be beneficial to reduction in the overall attenuation of the transmission line [2]. It is also desirable to further reduce the nonlinearity of the first fibers in order to compensate for relatively high nonlinearity of the following dispersion compensating fibers. In addition, design optimization of the hybrid line through proper selection and adjustment of the characteristics of the dispersion compensating fibers is essential to maximize the total performance. In this paper, a new design of low loss pure silica core fiber with depressed cladding is proposed to achieve ultimately low nonlinearity with maintaining good bending loss characteristics. In addition, the optimum design of the hybrid line is explored by employing the new pure silica core fibers and newly designed dispersion compensating fibers, considering the nonlinearity, the transmission loss and the average dispersion slope. Performance of actually fabricated hybrid transmission lines based on the optimization will also be presented. These hybrid transmission lines have been successfully used in the recent 1 Tbit/s, 10000 km transmission experiment [4]. 2. Low Nonlinearity Pure Silica Core Fiber [5]

2.1. THEORETICAL CALCULATION Figure 1 shows the refractive index profile examined in this study. The core is pure silica, and the inner and the outer claddings are doped with fluorine. In general, the effective area can be enlarged by simply lowering the refractive index difference , however, it tends to degrade the bending performance. It is well known that the depressed cladding is one of the effective ways of improving the bending characteristics through confining the optical power into the core [6]. The macrobending loss of the new PSCF with the depressed cladding has been examined theoretically [7]-[10]. Figure 2 shows the relationship between the effective area and the macrobending loss for a diameter of 20 mm calculated for PSCFs, where the refractive index difference is fixed to and the ratio of 2b to 2a is fixed to 2.5, 3.0, 4.0 or 5.0. The index difference and the diameter of the inner cladding 2b are varied while keeping the effective cutoff wavelength to 1.50 The calculated macrobending loss values have been somewhat modified so as to agree with experimental results of conventional PSCFs plotted in the figure. From Fig. 2, it has been found that the macrobending loss can be reduced, while maintaining the same effective area, by properly adjusting the 2b/2a ratio or the depressed cladding width. When

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the width of the depressed cladding is too narrow, the fundamental mode becomes leaky because of the outer cladding and the macrobending loss is increased. It is clear that

can be enlarged up to

without

degrading the bending performance so severely. If the macrobending loss is tolerated to the level of the standard SMF, it is anticipated that can

be increased to ratios.

or even larger by choosing relatively lower 2b/2a

2.2. FIBER CHARACTERISTICS

Table 1 summarizes characteristics of an example of fibers fabricated with and Measurement wavelength is 1.55 except for the cutoff wavelength. Its is more than The dispersion and the dispersion slope are slightly larger than those of the conventional PSCF, which has a dispersion of around 19 ps/nm/km and a dispersion slope of around The attenuation of the new PSCF is as low as those of the conventional PSCFs and is considerably lower than those of the standard doped core fibers, which are around

We have evaluated the nonlinear coefficient of the new design PSCF by the cross phase modulation (XPM) method [11]. Measurement results are shown in Fig. 3, together with results of other commercial fibers. In addition to the large the new design PSCF exhibits an ultimately low nonlinear refractive index because of no dopant in the core. As a result, the nonlinear coefficient of the fabricated fiber is 30 %

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smaller than that of the standard SMF. Measurement results of the macrobending loss are plotted in Fig. 2, together with results of the conventional PSCFs, - enlarged PSCFs with the matched cladding and the standard SMFs. Improvement of the macrobending loss achieved by the depressed cladding design agrees well with the calculation results. The macrobending performance of the fabricated fibers is significantly better than that of the standard SMFs. We have also examined the microbending loss by the winding test [12]. The loss increase

is measured at

after winding the test fiber with a constant tension

of 100 g on a 280 mm diameter bobbin whose surface is covered with sandpaper. The averaged particle size of the sandpaper is about Figure

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4 shows the test result. Results of the conventional fibers are also plotted. The new PSCF is somewhat inferior to the conventional fibers in terms of the microbending performance, but yet it is not so severely degraded in spite of the large

3. Hybrid Transmission Line 3.1. DESIGN OPTIMIZATION [13]

The configuration of the hybrid transmission line is illustrated in Fig. 5. The first part of the line is the enlarged PSCF described in Section 2. The second part is the dispersion compensating fiber (DCF). The repeater span of the hybrid line is assumed to be 50 km and the average span dispersion is set to 0 ps/nm/km at In the design optimization of the hybrid line and the DCF, we have focused on two parameters; the span loss and the nonlinearity. The nonlinearity has been quantitatively examined by calculating the cumulative phase shift due to the self-phase-modulation (SPM) expressed by Ref. [14]

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where k is the wave number, is the nonlinear refractive index, is the effective area and is the optical power at the longitudinal position

decreases as the light propagates because of the transmission loss of fibers. In the calculation, in order to keep the signal to noise ratio the same, the signal power launched into the line is adjusted so as to achieve a constant power at the output end of the span. As for the DCF, the W-type refractive index profile has also been employed as illustrated in Fig. 5. The core is doped with germanium, the inner cladding is doped with fluorine and the outer cladding is pure silica. In this study, the refractive index difference is fixed to The refractive index difference the core diameter 2a and the inner cladding diameter 2b are adjusted to obtain certain dispersion and dispersion slope with keeping the macrobending loss of 3 dB/m for a diameter of 20 mm. By using the known dependence of the nonlinear refractive index on the concentration [15], empirical relationship between the transmission loss and the concentration and other design relationships, the length ratio and the structural parameters of the DCF have been optimized. We

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have also introduced a new parameter, the equivalent which is defined by of a single fiber forming the whole line which gives the same cumulative phase shift as the hybrid line. The single fiber is assumed to be the

conventional dispersion shifted fiber (DSF) which has attenuation of 0.21 dB/km, an effective area of

and the nonlinear refractive index of

Figure 6 shows the averaged span loss and the normalized cumulative phase shift calculated as a function of the DCF length ratio with varying the average dispersion slope. The DCF length ratio represents the percentage

of the DCF length in the span. When the DCF length ratio is reduced, the average span loss becomes high. This is because the large refractive index difference necessary to obtain large negative dispersion tends to increase the attenuation of the DCF. When the DCF length ratio is large, the equivalent becomes small because the length of the DCF having a relatively small becomes long and deteriorates the overall nonlinerity. It should also be noted that as the average dispersion slope is reduced to zero, the performance of the hybrid transmission line is degraded in terms of both the span loss and the nonlinearity, because the larger refractive index difference and the smaller are required to keep the bending performance of the DCF to be used. It has been found that the hybrid line gives the minimum value of the span loss when the DCF length ratio is about 40 % of the whole line, and that the equivalent is maximized when the DCF length ratio is approximately 30 %. From these calculations, the average span loss of 0.19 dB/km, the equivalent and the average dispersion slope of 0.01 are found achievable, if the DCF length ratio is chosen to be around 30 %.

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3.2. FIBER CHARACTERISTICS

Based on the design optimization described above, we have fabricated the enlarged PSCFs and the design-optimized DCFs corresponding to the DCF length ratio of about 35 %. Total length of the fibers manufactured is more than 1000 km. Table 2 summarizes the average, the maximum and the minimum values of characteristics of the PSCFs, the DCFs and the hybrid lines composed of these fibers with the line lengths of about 50 km. We have obtained the equivalent of the average dispersion slope of 0.020 and the average span loss of 0.205 dB/km including splice loss. The equivalent is a little smaller than the calculation results. This is because design of the fabricated fibers is slightly off the calculation, and it gives the macrobending loss considerably better than predicted. Figure 7 shows examples of spectral attenuation of the PSCF and the DCF. It should be noted that no bending-induced loss increase is observed below 1600 nm in both fibers. Figure 7 also shows spectral attenuation curves measured for the hybrid lines exhibiting the maximum span loss and the minimum span loss at The measured span loss includes splice loss between the PSCF and the DCF. The average splice loss is 0.17 dB and this has been realized by heat treatment after conventional fusion splicing. Figure 8 shows spectral dispersion curves of the hybrid lines with the maximum and the minimum span dispersion slopes. It is clear that the dispersion slopes are significantly smaller than that of the conventional NZ–DSF (plotted for comparison in the same figure).

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3.3. EFFICIENCY OF FWM GENERATION

In order to examine the effectiveness of the developed hybrid line for WDM transmission, the efficiency of FWM generation has been measured and compared with the large-core NZ–DSF. Table 3 summarizes characteristics of the fibers used in this study. The hybrid line with the smallest equivalent

was selected for the worst case evaluation.

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Figure 9 shows the measurement setup for the efficiency of the FWM generation [16]. In this evaluation, two continuous wave lights with each input power level of +7 dBm were launched into the fiber, and the wavelengths were scanned from 1540 nm to 1565 nm with keeping the constant wavelength spacing between the two light 0.8 nm (100 GHz) or 0.4 nm (50 GHz). The power level of the produced FWM light was monitored by an optical spectrum analyzer. Measurement results are shown in Fig. 10. When the wavelength spacing is 0.8 nm, the FWM product power levels observed

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in the PSCF+DCF hybrid line are approximately 10 dB lower than those in the NZ–DSF, although the equivalent of the hybrid line is smaller. This is possibly because the absolute dispersion of each fiber in the hybrid line is significantly larger. Furthermore, the FWM efficiency in the hybrid line with a 0.4 nm wavelength spacing is found nearly equal to that in the

NZ–DSF with a 0.8 nm wavelength spacing, suggesting the possibility of doubling the number of WDM channels within the same bandwidth. In addition, considering the dispersion flatness, expansion of the usable bandwidth should be possible, which leads to further increase in the number of channels.

4. Summary

We have examined the hybrid transmission line consisting of the dispersionunshifted single mode fiber and the dispersion compensating fiber. For the dispersion-unshifted fiber, new design pure silica core fiber with an effective area over and attenuation of 0.171 dB/km at has been successfully developed. It exhibits ultra low nonlinearity and excellent bending loss performance. Design optimization study has indicated that the hybrid line employing this new pure silica core fiber has a possibility of realizing dispersion flattened transmission lines with low loss and low nonlinearity by optimizing the length and characteristics of the dispersion compensating fiber. Based on the optimized design, we have successfully developed

ultimately high performance dispersion-flattened hybrid transmission lines consisting of the -enlarged pure silica core fiber and the dispersion compensating fiber. They exhibit the span loss as low as 0.205 dB/km, the span dispersion slope of and low nonlinearity with the equiv-

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alent effective area of Efficiency of the FWM generation has also been examined. Results have indicated that this transmission line should be highly effective for dense WDM transmission. References 1.

Mukasa, K., Akasaka, Y., Suzuki, Y. and Kamiya, T.: Novel network fiber to manage

dispersion at

with combination of

zero dispersion single mode fiber,

Proceedings of the ECOC’97, 1, (1997), pp.127-130. 2.

Kashiwada, T., Onishi, M., Ishikawa, S., Kato, T., Okuno, T. and Nishimura M.: Ultra-low chromatic and polarization mode dispersion hybrid fiber links for ultrahigh speed transmission systems, Technical Digest of the OECC’98, 15C1-3, (1998), pp.364-365. 3. Murakami, M., Maeda, M. and Imai, T.: Long-haul WDM transmission experiment using higher order fiber dispersion management technique, Proceedings of the ECOC’98, 1, (1998), pp.313-314. 4. Naito, T., Shimojoh, N., Tanaka, T., Nakamoto, H., Doi, M., Ueki, T. and Suyama, M.: 1 Terabit/s WDM transmission over 10,000 km, Proceedings of the ECOC’99, PD2-1, (1999), pp.24-25. 5. Kato, T., Hirano, M., Onishi, M. and Nishimura, M.: Ultra-low nonlinearity low-loss pure silica core fiber for long-haul WDM transmission, Electron. Lett., 35, (1999), pp.1615-1617. 6. Kalish, D. and Cohen, L. G.: Single-mode fiber: from research and development to manufacturing, Technical Journal, 66, issue 1, (1987), pp.19-32. 7. Marcuse, D.: Influence of curvature on the losses of doubly clad fibers, Appl. Opt., 21, (1982), pp.4208-4213.

8. Cohen, L. G., Marcuse, D. and Mammel W. L.: Radiation leaky-mode losses in single-mode lightguides with depressed-index claddings, IEEE J. Quantum Elec-

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tron., 18, (1982), pp.1467-1982. 9.

Shah, V.: Curvature dependence of the effective cutoff wavelength in single-mode

10.

fibers, J. Lightwave Technol., 5, (1987), pp.35-43. Hatton, W. H. and Nishimura, M.: Accurately predicting the cutoff wavelength of cabled single-mode fiber, J. Lightwave Technol., 8, (1990), pp.1577-1583.

11. Kato, T., Suetsugu, Y., Takagi, M., Sasaoka, E. and Nishimura, M.: Measurement of the nonlinear refractive index in optical fiber by the cross-phase- modulation method using depolarized pump light, Opt. Lett., 20, (1995), pp.988-991.

12. Katsurashima, W., Kitayama, Y., Oishi, K., Kakuta, T. and Akasaka, N.: Microbending loss of thin coating single mode fiber for ultra-high-count cable, Proceeding of the IWCS’92, (1992), pp.13-19.

13. Sasaoka, E., Tsukitani, M. and Nishimura, M.: Design optimization of SMF–DCF hybrid transmission lines for long haul large capacity WDM transmission systems, presented in the OECC’99, C2S2B, (1999).

14.

Agrawal, G. P.: Nonlinear Fiber Optics, Academic Press, (1995). 15. Kato, T., Suetsugu, Y.and Nishimura, M.: Estimation of nonlinear refractive index in various silica-based glasses for optical fibers, Opt. Lett., 20, (1995), pp.2279-2281.

16. Hirano, M., Kato, T., Ishihara, T., Nakamura, M., Yokoyama, Y., Onishi, M., Sasaoka, E., Makio, Y. and Nishimura, M.: Novel ring core-dispersion-shifted fiber

with depressed cladding and its four-wave mixing efficiency, Proceedings of the ECOC’99, 2, (1999), pp.278-279.

FIBER DESIGN FOR DISPERSION MANAGED SOLITON SYSTEMS : THE CHALLENGE

W. A. REED Bell Labs – Lucent Technologies Murray Hill, NJ, USA.

1. Introduction

Dispersion managed dense WDM soliton (DMS) systems require that along the transmission span the dispersion of the optical fiber alternates between positive and negative values. The lengths of the different fibers types composing the span are also constrained in that the pulse can not expand more

than one half of the bit period, and the length averaged dispersion at the end of the span must be near zero for all wavelengths. In this paper I will discuss a set of design goals, describe the design process, and suggest possible designs for fiber pairs suitable for dispersion managed soliton (DMS) systems. 2. General Optical Fiber Design Goals and Constraints

For DMS systems the goal for the optical fiber designer is to identify pairs of optical fiber that have dispersion over dispersion slope ratios that are the same in magnitude but opposite in sign, i.e. pairs of fibers that have

where D is the dispersion, S is the dispersion slope and L is the length of the fibers. In addition to satisfying the dispersion requirements, both fibers must have low loss, be single moded at the operating wavelengths, have acceptable macro and micro bend loss, have acceptable effective areas and be manufacturable. This last requirement means that the values of the 379

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refractive index and the dimensions of each region of the index profile must lie within the capability of the manufacturing process. More specifically, dispersion managed fibers must have non-zero dispersion over the specified transmission window (the optimal value depends upon the specific system), and a dispersion slope that is as low as possible (so that the solitons at each wavelength ‘breath’ the same amount). In most cases fiber loss is more a function of how the fiber is manufactured than a function of the design. However it has been known for some time that the loss of a fiber increases as the refractive index of the core increases (and the core diameter decreases to keep the fiber single mode)[l]. For a core index greater than about 1.5 % this loss increases with index faster than one would expect due to the increase in doping. It has also been known experimentally that some of this ‘anomalous’ loss can be recovered if the fiber core is graded instead of a step index. A plausible explanation for this anomalous loss has been presented by Lines et al [2]. The physical cause of this loss is random fluctuations of the core diameter

which occur during the drawing process. The functional dependence of this loss is:

where

Thus designs that have index profiles with small lous loss than fibers with a step index core.

will have lower anoma-

3. The Design Process

The dispersion of optical fibers has two components, material and waveguide. The material dispersion is approximately the same in all silica fibers and is essentially equal to the dispersion of silica, since the effect of germania and fluorine doping is small. Thus to design fibers with specific values of dispersion and dispersion slope, the only component available to the designer is the waveguide dispersion. The waveguide dispersion is manipulated

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by adjusting the refractive index profile as a function of fiber radius. This is shown in Fig. 1. On the top left is the profile of a standard single fiber. The material, waveguide and total dispersion for this fiber is shown on the lower left. Since there is negligible waveguide dispersion in this design the total dispersion is very similar to the material dispersion. The right hand side of Fig. 1 illustrates how the waveguide dispersion can be modified by changing the fiber’s index profile.

The dispersion and other properties of optical fibers can be modeled quite accurately by solving Maxwell’s equations, imposing the cylindrical

boundary conditions, and incorporating the effects of materials dispersion. Typically these models are applied in the ‘forward’ direction in that a refractive index profile is the input and the optical properties are calculated

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for this profile. For simple design problems this approach works quite well. However for more complicated designs and more demanding dispersion requirements this forward approach is very time consuming and frustrating. We therefore have developed a ‘reverse’ process where the desired optical properties of the fiber are the input and the model searches design space for one or more index profiles that satisfy the target properties [3, 4]. Since fiber design space has many local minima, it is also necessary to develop programs that search design space so one has confidence that the ‘best’ design has been found. 4. Fiber Designs

When considering pairs of fibers suitable for DMS systems it seems rea-

sonable to start with one fiber of the pair be a design that is currently in production. Therefore I will explore a set of designs were one fiber of the pair is standard single mode (SSM) fiber and a second set of designs were one fiber of the pair is Lucent’s True Wave Reduced Slope ® design. These two pairs should give the system designer considerable flexibility in adjusting the span lengths, etc. of DMS systems.

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4.1. STANDARD SINGLE MODE (SSM) AND SSM MINUS (SSMM) DESIGNS

Standard single mode fiber (SSM) has a dispersion of about 17 ps/km·nm at 1550 nm, a dispersion slope of about 0.055 and an effective area of about 80 It is possible to compensate this fiber at ratios of 1:1 and 8:1 over the spectral range between 1530 nm and 1600 nm. The total dispersion for SSM, SMM–1XM and SMM–8XM is shown in Fig. 2 and the compensated dispersion per km for the two fiber pairs is shown in Fig. 3. As can be seen in Fig. 3, the compensation for SSM combined with SMM–1XM is significantly better over a 100 nm wavelength range than for SSM combined with SMM–8XM. 4.2. TRUE WAVE RS ® AND TWRS MINUS Lucent’s True Wave RS fiber has a has a dispersion of about 4.5 ps/km·nm at 1550 nm, a dispersion slope of about and an effective area of about Two fiber designs have been identified which have

dispersion that can compensate True Wave RS in ratios of 1:1 and 10:1. The total dispersion for these three fibers is shown in Fig. 4 and the compensated dispersion per km for the two fiber pairs is shown in Fig. 5. As can be seen in Fig. 5, the combination of True Wave RS and TWRSM–1X is slightly better over a 100 nm wavelength range than the combination of TWRS and TWRSM–10X.

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5. Conclusion It has been shown that it possible to design pairs of optical fibers which have opposite signs of dispersion and dispersion slope. These fiber pairs can give

the designer of dispersion managed soliton systems considerable flexibility in selecting system span lengths which can optimize the application.

Acknowledgments I would like to acknowledge Lawrence Cowsar and Linda Kaufmann who developed the computer programs that perform the ‘reverse’ method of fiber design. I would also like to acknowledge Dave Peckham and Stig Knudsen for discussions on fiber design, and Linn Mollenauer for helping me begin to understand how soliton sytems work. References 1. 2.

Anslie, B. J., Beales, K. L., Cooper, D. M. and Day, C. R.: SPIE, 425, (1983), p.15. Lines, M. E., Reed, W. A., diGiovanni, D. J. and J.R. Hamblin, J. R. : Electronics Letters, 35, (1999), p.1009. 3. Nelson, B. P. : Electronics Letters, 20, (1984), p.705. 4. Cowsar, L., Kaufman, L. and Reed, W. A. : private communication.

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40 GBIT/S RECIRCULATING LOOP EXPERIMENTS ON DISPERSION MANAGED STANDARD FIBRE

P. HARPER, S. B. ALLESTON, D. S. GOVAN, W. FORYSIAK, I. BENNION AND N. J. DORAN Photonics Research Group School of Engineering, Aston University Birmingham, England B4 7ET

As demand for greater bandwidth has increased, it has become apparent that the technologies required to meet the demand are being pushed to the limit. Traditionally, the way to increase capacity has been to move to higher bit rates, and despite the success of DWDM, there has been a gradual migration to 10 Gbit/s systems from 2.5 Gbit/s and it is likely that the increase in system base rate will continue and that 40 Gb/s systems will soon be introduced. Both the modulation format and the fibre type used are extremely important considerations when designing such high capacity systems. Although there are now many types of fibre available with a wide range of second and third order properties, nearly all systems with line rates of 10 Gbit/s or greater use some form of either periodic or lumped dispersion compensation. This management of the average dispersion allows high dispersion fibre to be used beyond the dispersion limit and also reduces non-linear penalties in WDM systems caused by cross phase modulation (XPM) and four wave mixing. Whilst most terrestrial systems continue to use the non-return to zero (NRZ) format even at 10 Gbit/s, the superior properties of return to zero (RZ), and more specifically the chirped return to zero (CRZ) format, have been recognised for long haul applications, allowing both laboratory demonstrations [1] and real systems with long reaches and massive (greater than 320 Gbit/s) capacities [2]. An even more promising format is the dispersion managed soliton, where the fibre non-linearity is employed to balance a small amount of residual anomalous dispersion. This technique provides excellent system performance, as has been demonstrated in many recent experiments [3, 4, 5]. When dispersion managed solitons are used there is an evolution of the pulse width and spectrum within the dispersion map which is dependent 387 A. Hasegawa (ed.), Massive WDM and TDM Soliton Transmission Systems, 387–401. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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on the local dispersion, the span length and the pulse width [6]. Numerical simulations [6] and experiments [7] show that the stable pulses in two step symmetric dispersion managed systems have two transform limited positions within the map, one in each of the two fibres. Away from the transform limited position there is an increase in the pulse width as the pulses become chirped. The rate of increase of the pulse width away from the transform limited width is dependent on the strength of the dispersion map defined as [8]

where and are the group velocity dispersion and length of the fibres and t is the transform limited pulse width. In the majority of dispersion managed systems which have been studied to date, map strengths in the region 1–10 have been used. However,

when considering the case of dispersion compensated standard fibre systems with a 40 Gb/s bit rate, map strengths can be in excess of 100 and the dispersion lengths within the fibres are reduced to less than a kilometre.

In this regime, the data pulses broaden rapidly and neighbouring pulses can be completely overlapped for a large proportion of the transmission path. Non-linear interactions between both adjacent and well separated pulses becomes the limiting factor in this regime, leading to both intra-channel XPM and inter-symbol four wave mixing [9]. In this paper we will present recent experimental and numerical simulation results which highlight the importance of correct transmitter and detector positioning in such strongly dispersion managed systems. The rapid evolution of the pulses makes it essential that the position where the pulses are launched into the dispersion map is well matched with the sign and magnitude of the pulse chirp. Furthermore, the short dispersion length means that the system performance is extremely sensitive to the position of the detector within the map at the end of the system. We will also present experimental results which show that 40 Gb/s dispersion managed soliton transmission is possible over 1000 km in standard fibre using dispersion compensation. Our results show that there is little penalty in increasing the standard fibre span from 32 km to 75 km and indicate that installed systems can be upgraded to a 40 Gb/s data rate using this technique. This is the first time that dispersion managed solitons have been used in such strong dispersion maps and the stable pulses in our simulations and experiments do not follow the usual empirical energy enhancement equation for dispersion managed solitons. Instead the optimum pulse energy is only a few times that of the equivalent uniform dispersion average soliton. Initial experiments were conducted in a single span recirculating loop with an amplifier span of 39.1 km. The experimental set-up used is shown

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schematically in Fig. 1. The experimental set-up is fairly complicated and so each of the three separate section, the 40 Gb/s data generator, the recirculating loop and the clock recovery/demultiplexing will be described separately. Bit error ratio and eye diagram measurements were to be performed at 10 Gb/s and so it was necessary to multiplex from 10 Gb/s to the experimental data rate of 40 Gb/s. This multiplexing was done optically using a 40 Gb/s data generator which consisted of a 10 GHz short pulse source, a 10 Gb/s data modulator and a 10–40 GHz bit interleaver The pulse source used in this experiment was an external cavity mode locked semiconductor laser (ECMLL). This gave 5 ps pulses with a time bandwidth product of 0.45. The operating wavelength could be tuned using an external cavity grating and the operating wavelength was tuned to 1544 nm in this experiment. In the 40 Gb/s data generator, a 10 Gb/s PRBS data pattern derived from an electronic pattern generator was imposed onto the 10 GHz pulse stream from the pulse source using a lithium niobate amplitude modulator. The resultant 10 Gb/s RZ data pattern could then be multiplexed to 40 Gb/s using the double Mach-Zehnder bit interleaver (MUX) shown schematically in Fig. 2.

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In order to generate a 40 Gb/s data pattern, the 10 Gb/s output from the amplitude modulator was split using a 3 dB coupler. One of the outputs

from this coupler was passed through a polarisation controller (PC) whilst the other passed through a variable fibre delay line. The two outputs were then recombined using another 3 dB coupler. The length of the fibre delay line was such that there was a delay of several bit intervals between the two recombined signals and the length of the fibre delay line was fine tuned to give a total delay of +50 ps. The output signal was therefore at

20 Gb/s. The 20 Gb/s output was then used as the input to a second stage to give a multiplexed 40 Gb/s data output. The purpose of the polarisation controllers was to allow the polarisation states of the OTDM channels to be altered. In this experiment, a single polarisation output was required and this was ensured by positioning a fibre polariser (POL) and polarisation controller at the data generator output. By careful adjustment of the polarisation controllers and fibre delays in the interleaver, a good quality equal amplitude, single polarisation 40 Gb/s output with de-correlation of the individual 10 Gb/s OTDM channels was achieved at the recirculating loop input. Figure 3 shows eye diagrams, taken using a sampling oscilloscope, of the 10 Gb/s amplitude modulator output, the MUX 20 Gb/s monitor output and the fibre polariser output. After transmission the received 40 Gb/s optical data pattern had to be demultiplexed to 10 Gb/s and the 10 GHz clock had to be recovered to allow bit error rate measurements to be taken. Both clock recovery and demultiplexing were accomplished with a single electro-absorption modulator (EAM) using the set-up shown in Fig. 4. The 40 Gb/s data was used as the input to the EAM. The EAM was driven by the 10 GHz output of a voltage controlled oscillator (VCO). This generated a switching window of ps at a 10 GHz rate. This switching window demultiplexed one of the 10 Gb/s

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OTDM channels from the 40 Gb/s input and this demultiplexed channel was used for clock recovery. An electrical 10 GHz component was derived from the EAM optical output using a 15 GHz photodiode. This signal was then mixed with the VCO output and the resultant error signal was fed to a phase locked loop controller (PLL) which kept the VCO frequency locked to the optical 10 GHz signal. Thus, once locking was achieved, the EAM switching window was locked onto one of the 10 Gb/s OTDM channels. By splitting the EAM optical demultiplexed output and taking a tap of the VCO output as a clock recovered trigger signal, bit error ratio (BER) and eye diagrams of the demultiplexed signal could be taken. A disadvantage of using this clock recovery/demultiplexing scheme is that it was not possible to choose each OTDM channel in turn. Once the clock recovery was set the demultiplexer remained locked to a given channel. It was however possible to switch between channels by unlocking the PLL and then re-locking but the channel choice was random and hence the bit error rate measurements taken were an average of the performance of all 4 OTDM channels as opposed to measurements of each channel in turn. Figure 5 shows eye diagrams taken for back-to-back performance of the clock recovery/demux showing that all four channels can be demultiplexed from the 40 Gb/s input. The demultiplexed eyes clearly show that all four channels could be demultiplexed with a high extinction ratio. Typically, the Q value of the demultiplexed channel was corresponding to a BER of The transmission of the 40 Gb/s data pattern was performed in the

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recirculating loop shown in Fig. 1. An acousto optic modulator (AOM) was

used to control the gating into the loop and the pump diode of the loop EDFA was electronically controlled to clear the signal from the loop at the end of each transmission/measurement cycle. The transmission fibre in the

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loop consisted of 32.3 km of standard fibre (SF) and a 6.8 km dispersion compensating fibre (DCF) module. This fibre combination gave a average dispersion of +0.03 ps/nm·km at the operating wavelength. In addition to dispersion compensation, the DCF also gave partial dispersion slope compensation resulting in a net slope of For a 5 ps pulse width, the strength of the dispersion map was and so rapid expansion of the pulses from their transform limited width was seen. A single erbium doped fibre amplifier (EDFA) was used to compensate for the loss in the loop. The EDFA was followed by a 2.3 nm bandpass filter. This filter had a degree of polarisation dependent loss (PDL) and so a polarisation controller (PC) was required in the loop. In order to investigate the dependence of system performance on launch position the SF was split into two sections which were positioned before and after the recirculating loop input/output coupler. By changing the fraction of the total SF length in each of these sections the relative launch position could be varied. The maximum error free distance was measured for each launch position. To quantify the measurement, the launch position was defined as the fraction of the total standard fibre length which was positioned before the DCF in the dispersion map i.e. a launch position of zero corresponds to launching into the DCF and a launch position of one is equivalent to launching at the start of the SF span. Figure 6 shows the results of maximum error free distance as a function of launch position and also shows a bit error rate versus transmission distance curve

taken at the optimum launch position.

These experimental results show that the system performed best when

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the pulses were launched at a position within the standard fibre with of the fibre positioned before the DCF. The graph on the right hand side of Fig. 6 shows that for this launch position the maximum error free distance was 1220 km. Although there was a clear optimum launch position, even in the worst case scenario transmission over 500 km was possible. Eye diagrams of the demultiplexed output were taken for back-to-back performance and at a transmission distance of 1000 km. These eye diagrams, shown in Fig. 7, give a good indication of the limits in the system. It is clear that there was very little build up in the zero level, the factors causing errors were timing and amplitude jitter due to interaction between the pulses — the Gordon-Haus limit for this system was over 3500 km and the conventional soliton-soliton interaction limit was over 14000 km. In the experiments, the optimum pulse power was pJ at the SF input which is equivalent to roughly twice the equivalent uniform dispersion average soliton energy. This energy enhancement factor is less than would be expected for a conventional dispersion managed soliton [10] however in this

experiment we used a map strength far in excess of those normally considered and in such cases it is not clear how the energy enhancement factor scales. This low stable pulse power was however seen in both experiments and numerical simulations.

Figure 8 shows numerical simulation results of Q value versus propagation distance taken using the same experimental parameters and pulse energy. There was very good agreement between the experiment and simulations in terms of maximum transmission distance. A Q value of 6 is equivalent to a BER of and in the simulations a transmission distance

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of 1300 km was possible compared with the 1220 km which was achieved experimentally. The numerical simulations eye diagram taken at a distance of 1300 km also shows the same degradation as the experimental eye diagram in Fig. 7 — amplitude and timing jitter due to pulse interactions was limiting the system.

Since this experiment was performed in a recirculating loop, the launch and detection positions were at the same point in the dispersion map. In a real system this would not necessarily be the case and so it would be essential to position the detector at an appropriate position. The importance of the detector position is shown in Figs. 9 and 10 which show experimental eye diagrams and numerical simulations results for different detector positions. Figure 9 shows that at the optimum detector position a clean 40 Gb/s data pattern was detected and channels could easily be demultiplexed error free. The demultiplexed channel is shown in the right hand side of Fig. 9. With the detector offset from the optimum position by 600 m of SF, pulse broadening degraded the quality of the 40 Gb/s eye and there was a subsequent increase in the BER of the demultiplexed channels. With an increase in the detector offset to 2.2 km of SF, there was a total overlap of the pulses in the 40 Gb/s eye. In this situation clock recovery was not possible as there was no 10 GHz component to the optical signal and no channel could be demultiplexed. Using a local clock the pulse broadening of the 10 Gb/s pulses due to the detector offset was monitored and is shown in the bottom trace on the right hand side of Fig. 9. The numerical simulation results in Fig. 10 show the same effect. The left hand diagram shows the received data pattern when the detector is positioned at different points in the dispersion map. The optimum positions are at one of the transform limited points and in Fig. 10 distances are

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measured relative to the transform limited point in the standard fibre span. With the detector positioned at this optimum position (i.e. at a distance = 0 km) the launched data pattern 11001011 was recovered. However as the detector was moved away from the optimum position there was a rapid degradation in the quality of the received signal. This degradation in system performance was quantified by taking Q value measurements as a function of detector position. The right hand graph in Fig. 10 shows such measurements for two different pulse widths namely 2 ps and 7 ps. For both pulse

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widths there was a rapid degradation in Q value as the detector was moved from the optimum position. The decrease in Q value was more rapid for the 2 ps pulse width due to the larger map strength associated with these pulses. Map strength and dispersion length are inversely proportional to

the square of the pulse width and so as pulse width is decreased the pulses spread more rapidly from the minimum transform limited width. Therefore the higher the map strength, the more critical the detector position becomes. In the case of the 2 ps pulses, the Q decreased from an input value of 29 to the minimum acceptable value of 6 with the detector de-tuned by only km from the optimum position. With the 7 ps pulse width a Q value of could be achieved with km of de-tuning of the detector position. In the case of the upgrade of installed systems, the end point of the system would be at the output of a standard fibre span and so a dispersion compensating fibre (or ave. dispersion element) would be required to bring the pulses back to the transform limited width. DCF has a higher dispersion value than SF and so the dispersion length is shorter and the dependence of Q value on detector position is even more pronounced than the case illustrated in Fig. 10. For this application it would be very desirable to have a tuneable dispersion element to allow the optimum position to be found more easily than the trial and error method of using different lengths of DCF and also that the detector position can be altered to take

account of any changes to the system due to ageing, repairs or upgrades. Although the experimental results described above show that transmission over 1000 km is possible over standard fibre at 40 Gb/s, the results are not directly applicable to the problem of installed system upgrade due to the small standard fibre span of 32 km. A second experiment was therefore performed with a 74.6 km standard fibre span to consider this problem. The loop was reconfigured as shown in Fig. 11. An additional DCF was required to compensate for the increased length of standard fibre. The two DCF modules were positioned together with EDFAs at both the input and

output of the DCFs (a second EDFA was required in this experiment to compensate for the additional loss introduced by the increased span length). In this experiment a different source laser was used. Here a 10 GHz fibre laser which gave near transform limited 2 ps pulses was used. This reduction in pulse width along with the increase in span length gave an even greater dispersion map strength than in the previous experiment. Here the map strength was increased to There was therefore an enormous amount of pulse breathing in the dispersion map and the data pulses were completely overlapped for the majority of the transmission. Because of the narrower pulse width the optical bandwidth of the pulses was increased and so the 2.3 nm filter was removed from the loop and a broadband high-pass ASE suppression filter with a 1541 nm cut off wavelength was used instead.

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This filter was used to filter out the 1530 nm EDFA fluorescence peak and not as a soliton guiding filter. After this change in filters, the PC could be removed from the loop as the ASE filter had negligible PDL. At the operating wavelength of 1557 nm the average dispersion of the loop was +0.04 ps/nm·km which is slightly higher than the +0.03 ps/nm·km value of the 32 km SF span experiment. There was however a more marked difference is the dispersion slopes in the two experiments. In the 75 km SF span experiment the additional DCF module used gave a high degree of dispersion slope compensation giving a net dispersion slope for the recirculating loop of The 75 km of SF was again split into two sections to optimise the launch position. A 27 km length preceded the DCF modules and the remaining 47.6 km was positioned after the DCF. A further change to the experimental set-up was that a pair of AOMs were now used to control the loop transmission/measurement cycle.

Experimental BERs as a function of distance and eye diagrams for this experiment are shown in Fig. 12. For comparison the BER results for the 32km SF span experiment are also shown. As in the previous experiment, only a modest energy enhancement factor was observed. The pre and post DCF EDFA output powers were 1.5 dBm and 10 dBm corresponding to a pulse energy enhancement factor of From the BER measurements it is immediately clear that there was no significant difference in the maximum error free distance for the two experiments and with the 75 km SF span a total transmission distance of

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1160 km was achieved. The difference in the slopes of the two BER versus distance curves is attributable to the differences in experimental parameters such as average dispersion and pulse width and is not a consequence of the increased span length. The 75 km SF span experiment actually gave better performance at BERs greater that due to improvements in the loop configuration and clock recovery. However, it is likely that if the 32 km SF span experiment were repeated with this new set-up the performance would be improved. There is no evidence from the experimental results to suggest that the increase in map strength from with the 32 km SF span to with the 75 km span, caused an increase in the pulse interaction. It is therefore likely that that the interaction between pulses is strongest before the pulses become completely overlapped. Indeed since the interaction is a nonlinear phenomenon, the low peak powers of the temporally broadened pulses is likely to reduce the interaction strength.

The km error free transmission distance achieved with a 75 km SF span is a good indication that it should be possible to upgrade installed standard fibre systems to a 40 Gb/s data rate using dispersion compensation. The eye diagrams in Fig. 12 again show that the limiting factor was amplitude and timing jitter. Despite the increase in amplifier span (and hence span loss) the ASE noise limit was still beyond the error free distance. Measurements of the pulse spectral width were taken at the loop output as a function of transmission distance. The experimental results in Fig. 13 show this data plotted in 3D and also as a contour plot. There was no change in the spectral width of the pulses as the transmission distance was increased. There was however an increase in the noise level in the wings

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of the spectrum which was due to ASE noise increase. The spectral plot also shows that a ripple developed in the pulse spectrum as distance increased. This was simply due to a small ripple in the spectrum of the ASE suppression filter used in the loop which became more pronounced as the

number of circulations increased.

In conclusion, we have shown that in very strongly dispersion managed systems it is crucial to have the transmitter and receiver positioned at ap-

propriate points in the dispersion map. With the enormous map strengths that are encountered in 40 Gb/s dispersion compensated standard fibre experiments, the pulses very quickly expand from their transform limited pulse width and with the receiver offset from this optimum position by even a few dispersion lengths the system performance is degraded substantially. In order to optimise the transmitter position it is necessary to match the launch position into the dispersion map with the chirp of the input pulses. This can be done by either choosing the correct position for the given input

pulses or by pre-chirping the input pulses to match a given launch point. This leads to a substantial improvement in system performance — in our

experiments the error free distance could be doubled by optimising the launch position. Our experiments and simulations have shown that with both 32 km and 75 km standard fibre spans it is possible to achieve error free 40 Gb/s transmission over 1000 km by using dispersion compensating fibre positioned at the amplifier stages to reduce the average system dispersion. In these experiments extremely high (several hundreds) map strengths were encountered and the systems were limited by pulse interactions. However, increasing the map strength from 50 to 700 did not result in a significant increase in these interactions. In our very strongly disper-

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sion managed systems the stable pulses were found to have a much lower energy that anticipated. In our experiments and simulations only a modest energy enhancement factor of was observed and the energy enhancement factor was not strongly dependent on map strength-roughly the same enhancement was observed with map strengths of 50 and 700. Our results give a good indication that by using dispersion managed solitons in this new regime it should be possible to upgrade installed standard fibre systems to a 40 Gb/s data rate. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Bergano, N. S. et al. : OFC’99, PD2, (1999). Vareille, G. et al. : OFC’99, PD18, (1999). Harper, P. et al. : Opt. Lett., 24(12), (1999), pp.802-804. Le Guen, D. et al. : OFC’99, PD4, (1999). Jabob, J. M. et al. : Electron. Lett., 33(13), (1997), pp.1128-1130. Smith, N. J. et al. : Electron. Lett., 32(1), (1997), pp.54-55. Jacob, J. M. et al. : Photon. Technol. Lett., 10(4), (1998), pp.546-548. Nijhof, J. H. B. et al. : Electron. Lett., 34(5), (1997), pp.481-482. Mamyshev, P. et al. : NLGW’99, ThA1-1, (1999). Smith, N. J. et al. : Opt. Lett., 21(24), (1996), pp.1981-1982.

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HANDLING NOISE IN SUPERCONTINUUM GENERATION FOR WDM APPLICATION

H. KUBOTA, K. R. TAMURA AND M. NAKAZAWA NTT Network Innovation Laboratories, 1-1 Hikarinooka, Yokosuka,Kanagawa, 239-0847 JAPAN

Recently, the super continuum (SC) has been generating considerable interest as a multi-wavelength optical source for wavelength division multiplexing (WDM) communication [1, 2]. It has been reported that dispersion decreasing fiber (DDF) can broaden the SC spectral width much more than dispersion shifted fiber (DSF) [3, 4]. The effect of pump fluctuation in SC generation was investigated [5]. However, the extent to which timing jitter exists has not yet been fully investigated. Hence there has been no investigation into how the coherency of a pulse train is maintained during SC generation. Such an investigation is of great importance the SC source is to be used as a source for high speed communications. In this paper, we report that there is significant coherence degradation during SC generation as the result of the random excitation of high-order solitons initiated by amplified spontaneous emission (ASE) noise. These high-order solitons are excited in both DSF and dispersion flattened fiber (DFF) because of their low dispersion. However, we show that there is no such coherence degradation when DDF is used, where fundamental or low-order solitons are excited. We compared the coherence degradation during SC generation in three kinds of fiber, DDF, DSF and DFF. The DDF was 0.91 km long, and its group velocity dispersion (GVD) was 9 ps/km/nm at the input end and nearly zero at the output end. The DSF was 1 km long, and it had a zero GVD wavelength of 1539 nm. The DFF was 2 km long and it had a GVD

of 0.2 ps/km/nm at

with a dispersion slope of 0.004

The input was a 3 ps transform-limited Gaussian pulse of 1542 n m , generated by a regeneratively and harmonically mode-locked fiber laser [6]. To generate spectral broadening nonlinear effects, the pulse to few watts with a high-power erbium-doped fiber amplifier (EDFA). Figure 1(a), (b) and (c) show the structure of the SC spectra of DDF, DSF and DFF, respectively sliced every 10 nm. When the DDF is used as shown in Fig. l (a), 403 A. Hasegawa (ed.), Massive WDM and TDM Soliton Transmission Systems, 403–410. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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10 GHz longitudinal modes corresponding to the input pulse repetition are clearly seen in each spectrum. Their corresponding pulse shapes maintain clean waveforms and the pulse width was 16 ps for all spectral components.

DDF made it possible to shorten the input pulse width almost adiabatic through soliton compression effect which preserves coherence of the pulse train. The DSF or DFF is used as shown in Fig. 1(b) and (c), there is a noticeable difference from Fig. 1(a). In Fig. 1(b) and (c), the longitudinal modes have almost disappeared in each spectrum and continuous wave (CW) components cover the whole spectral region. Such a spectral profile indicates that the output waveform has become incoherent. This coherence degradation is the evolution of high-order solitons randomly initiated by the

ASE noise when the EDFA is used. It is noted here that DFF is not effective for maintaining coherence. In Fig. 1(a), it is interesting to note that there are CW components at the longer wavelength region, while gradual increase

in the amount of CW components towards the shorter wavelength region. This can be understood from the fact that the pulse always experiences anomalous GVD in the longer wavelength region even though the GVD becomes close to zero so that the waveform change is more adiabatic than in the shorter wavelength region. Figure 2 shows numerically how a high order soliton train is randomly distorted in the presence of different noise patterns. Figure 2(a) is the waveform change without noise and Fig. 2(b) and (c) are those with noise. Here, the soliton number is the peak power is 8 W, and the input pulse width is 3 ps. The added noise power level is below the soliton power within the 3 dB bandwidth. This corresponds to a 6 dB NF and 20 dB gain.

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The origin of this random evolution is modulation instability (MI) or four wave mixing (FWM) which occurs at the top of the soliton pulse. It is also interesting to note that the build-up of the soliton pulse with noise is faster than that in Fig. 2(a). The reason is as follows. We have already reported that the origin of the evolution of the high order soliton is the FWM (or MI effect) of the soliton pulse itself which occurs on the top of the pulse peak at the beginning of propagation [7]. The amplitude ripple is automatically generated on the top of the soliton through FWM (MI) without noise and is amplified with its gain. FWM or MI has an optical gain, which means that, if there is broad band noise at the beginning, this noise is amplified by the gain and the propagating high order soliton is strongly affected. Therefore, in the presence of noise, even when a neatly repetitive high order soliton pulse train is initially coupled into a fiber, each pulse of the output pulse train has an independent fluctuation with a large amplitude which eventually degrades the time and frequency coherence of the pulse train. The randomness of a high order soliton strongly depends on its power and the input noise amplitude. When there is a large amount of ASE noise and a large FWM gain, the output pulse becomes completely random, and the coherence is completely lost. In the anomalous GVD region, the broader SC is generated by a more intense pulse. Such a pulse is an even higher order soliton. However, a high order soliton has a high nonlinear gain, so the broader SC has less coherence. The coherence degradation during high-order soliton pulse compression is caused mainly by the interaction between ASE and MI gain. The discussion suggests that the pulse quality during the soliton compression may

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not be degraded if ASE component that corresponds to the MI gain region is eliminated. This idea leads to a simple method for improving pulse train quality by filtering out the ASE in the vicinity of the MI gain maxima. By use of band eliminating filters (BEFs). Here we assume a BEF that has a square-well shape and is positioned at the input of the optical fiber. Figure 3(a) and (b) shows the output pulse train of the compressed pulse without and with BEF, and Fig. 3(c) shows details of the BEF and MI gain profile. Input power is 7 W which corresponds to was set at The MI frequency was 1.1 THz so that the BEF cut the frequency region from 0.5 to 2 THz on both sides. This frequency corresponds to a wavelength of 4 nm to 16 nm. Without the BEF, as shown in Fig. 3(a), the waveform changed pulse by pulse. By contrast with the BEF, the pulse intensity was almost the same for each pulse.

Figure 4(a) and (b) show the spectrum of the output pulse train without and with the BEF, respectively. Part of the spectrum is expanded on the right-hand side of each figure. The extinction ratio of the sidebands in Fig. 4(b-2) is more than 20 dB, while that in Fig. 4(a-2) is only 10 dB or less. It

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can be clearly seen in these figures that the BEF improved the coherence of the pulse train by removing the MI from the initial ASE. In other words, the quality of the pulse train was not maintained in all wavelength regions. Therefore, when we try to extract a pulse train from this broadened spectrum for WDM applications, the allowable spectral range in Fig. 4(a) is smaller than that in Fig. 4(b) even though the spectral broadening is almost the same in the two figures. These results indicate that a simple BEF can greatly enhance the coherence of the pulse train over a wide spectral range. In practice, the BEF is replaced by a simple band pass filter because the spectral components higher than the MI gain region do not affect the pulse compression characteristics.

If intense ultra short pulses can be used to generate a broad SPM, a fiber with normal GVD is a good candidate for SC generation [8]. Since there is no phase matching condition for MI gain in the normal GVD region, the MI gain can be completely eliminated by using the normal GVD wavelength

region. Below the stimulated Raman scattering (SRS) threshold, spectral broadening is mainly caused by SPM. The SRS threshold for a 10 ps pulse in an optical fiber is of the order of 100 W [9]. One of the limitations of the bandwidth during SC generation lies in the dispersion slope of the normal

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GVD fiber. When is zero, the spectrum broadens symmetrically until the input pulse becomes rectangular shape, as is well-known [10]. When is not zero, some of the broadened spectral components encroach into the anomalous GVD region. MI then occurs because of the high input power. Figure 5 shows waveform and spectral evolution in a normal GVD fiber with dispersion slope. A 3 ps pulse with 1542 nm in wavelength is launched into 1 km DSF with zero GVD of 1544 nm. The core diameter of the DSF is and the average input power is 100 m\V. Figure 5(a) shows autocorrelation trace of the input and output pulse. A small ripple at a peak of the output pulse indicates occurrence of the MI. Corresponding output pulse spectrum is shown in Fig. 5(b), part of which is get into anomalous GVD region. Figure 5(c) shows spectrum details sliced 7 different spectrum position. 10 GHz longitudinal modes are clearly seen in normal GVD region and they disappears in anomalous GVD region.

Figure 6 shows waveform and spectral broadening in normal GVD fibers in 100 m increments. The GVD of Figure 6(a) and (b) are – 0.2 ps/km/nm and – 1.0 ps/km/nm, and dispersion slope is The input peak power is as high as 20 W and no ASE is incorporated in these calculations. In Fig. 6(a), the zero GVD wavelength is only 3 nm apart from the signal wavelength. A part of the broadened pulse spectrum easily lies on the anomalous GVD region. In Fig. 6(b), the wavelength separation is 14 nm, the MI evolution is suppressed within that wavelength region. A fiber

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with larger GVD can generate broader spectrum without MI generation although it requires higher input peak power. It is because pulse broadens so quickly due to the large GVD before generating sufficient SPM. So that a fiber with small dispersion slope, such as dispersion flattened fiber is good solution for generating broad coherent spectrum in normal GVD region.

In conclusion, we have shown that a high order soliton evolves randomly and is independently distorted in the presence of ASE noise. This random evolution in the soliton waveform is interpreted as that the random noise is amplified by MI gain. The random evolution can be suppressed by the use of a DDF, in witch low order soliton is excited, by use of BEF to eliminate

ASE noise, and by a use of normal GVD fiber with low dispersion slope. These are useful for producing a coherent SC pulse and the pulse will be applicable to an optical source for high-speed WDM communication. References 1.

Morioka, T., Kawanishi, S., Mori, K. and Saruwatari, M. : Nearly penalty-free, supercontinuum Gbit/s pulse generation over 1535-1560 nm, Electron. Lett., Vol.30,

No.10, (1994), pp.790-791. 2. 3.

Kawanishi, S. et al., Proc. of OECC’97, PDP1-1, (1997), p.14. Okuno, T., Onishi, M. and Nishimura, M. : Dispersion-flattened and decreasing fiber for ultra-broadband supercontinuum generation, Proc. of ECOC’97, PD, (1997), pp.77-80.

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4. Mori, K., Takara, H., Kawanishi, S., Saruwatari, M. and Morioka, T. : Flatly broadened supercontinuum spectrum generated in a dispersion decreasing fibre with a convex dispersion profile, Electron. Lett., Vol.33, No.21, (1997), pp.1806-1808. 5. Mori, K., Takara, H. and Kawanishi, S. : The effect of pump fluctuation in supercontinuum pulse generation, Nonlinear Guided Waves Their Applications 98, (1998), 6.

p.276. Nakazawa, M., Yoshida, E. and Kimura,Y. : Ultrastable harmonically and regeneratively modelocked polarisation-maintaining erbium fiber ring laser, Electron. Lett.,

Vol.30, (1994), pp.1603-1604. 7.

Nakazawa, M., Suzuki, K., Kubota, H. and Haus, H. A. : High-order solitons and

modulation instability, Phys. Rev., Vol.39, No.11, (1989), pp.5768-5776.

8. Fork, R. L., Shank, C. V., Hirlimann, C., Yen, R. and Tomlinson, W. J. : Femtosec9. 10.

ond white-light continuum pulses, Opt. Lett., Vol.8, No.l, (1983), pp.1-3. Agrawal, G. P. : Nonlinear Fiber Optics, Academic Press, (1989). Lassen, H. E., Mengel, F., Tromborg, B., Albertson, N. C. and Christiansen, P. L.

: Evolution of chirped pulses in nonlinear single-mode fibers, Opt. Lett., Vol.10, No.1, (1985), pp.34-36.

DENSE-WDM SOLITON SYSTEMS USING CHANNELISOLATING NOTCH FILTERS (“SOLITON RAIL”)

B.A. MALOMED

Department of Interdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel AND

A. DOCHERTY, P.L. CHU AND G.D. PENG Optical Communications Group, School of Electrical Engineering,

University of New South Wales, Sydney 2052, Australia

Abstract. We present the results of detailed analytical and numerical investigations of the soliton dynamics in single-channel and multi-channel systems supported by notch filters separating the channels in the spectral domain. In the analytical part of the work, we study properties of the dispersion-managed (Gaussian) soliton, and demonstrate that it has strong

advantages in comparison with the usual soliton in a constant-dispersion system. The numerical part is focussed on direct simulations of the most dangerous collisions between initially overlapping solitons in a realistic twochannel system with lumped finite-spectral-width notch filters. The simulations reveal unexpected features of the collisions in the presence of the notch filters, that may be quite useful for applications.

1. Introduction

Wavelength-division multiplexing (WDM) with spectral separation nm opens new perspectives for soliton-bascd optical communication systems. The most serious problem in using WDM in the soliton mode is the

crosstalk induced by collisions of pulses in different channels [l]-[4]. Recently it was demonstrated that the collision-induced jitter can be strongly reduced by means of dispersion management (DM) [2, 3, 4]. To further 411 A. Hasegawa (ed.), Massive WDM and TDM Soliton Transmission Systems, 411-424. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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reduce the interaction between different channels in the WDM system, isolating optical filters, based on the Fabry-Perot ètalons [5] or various fibergrating schemes, can be used [6]. An objective of this work is to study transmission, stability and interchannel interactions of the usual (sech) solitons and DM ones (Gaussians) in a WDM system in which the channels are separated by narrow notch filters. An idealized model of this system was recently introduced in Ref. [7]. It assumed that each (nth) channel of (the same) spectral width was supported by two infinitely narrow notches at the frequencies (a “soliton multi-track rail”). The simplest twochannel version of the model is described by two coupled NLS equations,

where are envelopes of the electromagnetic fields in the two spectrally separated channels, c is a group-velocity difference between them, and are the filtering strength and compensatory gain. The dispersion coefficient is a function of z in the case of DM, the rest of the notation being standard. Note that the model can be easily extended to the case of the frequency-sliding filters, replacing each by

being a common frequency-sliding rate. It is relevant to note that, as it was suggested in Ref. [7], a comb-like system of narrow notches with equal spectral separations between them can be based on quantum dots, i.e., potential wells of a small size a with trapped electrons. As was demonstrated in Ref. [7], the wavelength separation 1 nm between the channels, or, in terms of the frequency, GHz, can be provided by quantum dots with a size nm, which is quite realistic. In this paper, we continue the work begun in Ref. [7] in several directions. In Section 2, we develop an analytical treatment of a soliton in one channel confined by the two ideal (infinitely narrow) notches, with an emphasis on the DM (Gaussian) soliton, that was not considered at all in

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the first work [7]. Although only one channel is explicitly considered, we actually analyze a collision of the soliton with its counterpart in another channel, which is modelled by the action of an external force on a given soliton. As well as the constant-dispersion model considered in Ref. [7], its DM generalization that we deal with in Section 2 assumes distributed filtering and gain. The latter assumption may be not justified for the case when the DM period L and filtering/amplification spacing coincide [8], but it is relevant for cases when L is essentially larger than which may definitely be of practical importance (see, e.g., Ref. [9]). A general inference following from the the results displayed in Section 2 is that the DM-supported Gaussian pulses clearly provide for important advantages as compared to the traditional sech solitons, viz. more channels can be accommodated in the same bandwidth, and the gain necessary to compensate the filterinduced losses is, roughly, twenty times as small (hence much less noise will be generated by the amplifiers). In Section 3, we present results of direct numerical simulations illustrating suppression of a strong soliton perturbation (initial frequency shift) by the notch filters in one channel with constant dispersion. In this section, we are dealing with the notch filters that have a finite width in the spectral domain, but we still assume uniform distribution of the filtering and amplification along the propagation distance z. The simulations suggest an interesting feature, whose detailed analysis is postponed to another work, viz., that “regularized” notch filters, with a smooth absorption profile, may be more efficient than the simplest rectangular filters. In Section 4, we present results of direct simulations of a far more realistic version of the constant-dispersion model in its lumped form, with a finite separation between the filters. As well as in Section 3, we assume a finite width of the notches in the spectral domain. In this section, we concentrate on the most important issue of simulating collisions between the solitons in two channels. The simulations reveal a number of unexpected features, e.g., the excess gain necessary to compensate the losses inflicted on the soliton by the notch filters has a clear trend to increase with the increase of while is fixed. We propose a simple explanation to this counter-intuitive result. 2. The Single-Channel Model with Dispersion Management

In this section, we consider the model represented by Eq. (1) only, setting and dropping the superscripts pertaining to the channel’s number. If the terms on r.h.s. of Eq. (1) are treated as perturbations, and use is made of the known energy- and momentum-balance technique,

414

B.A. MALOMED, A. DOCHERTY, P.L. CHU, G.D. PENG

evolution equations for the energy and central frequency of an arbitrary pulse can be derived, as it was done in Ref. [7]:

Then, for the usual soliton, with the temporal width TO in Eq. (1)), Eqs. (4) and (5) amount to:

It is obvious that Eqs. (6) and (7) have a fixed point (FP) with corresponding to a soliton frequency-locked at the channel’s centre. The necessary compensating gain is given by Eq. (6):

where the relative width of the channel is:

Further analysis of Eqs. (6) and (7) reveals that this FP is stable provided that In the case of DM, the soliton assumes a nearly Gaussian form [10, 11],

where being the pulse’s peak power and minimum width. Substitution of Eq. (10) into Eqs. (4) and (5) yields the equations:

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whose FP needs the compensating gain (cf. Eq. (8))

This FP is stable provided that For both the sech and DM (Gaussian) pulses, the FPs found above exist for any value of the relative width W. Therefore, we seek, within the framework of the present idealized model (with the infinitely narrow in the spectral space and uniformly distributed along z notch filters) for an optimal value of W, that provides for the strongest stability of the trapped pulse. Further analysis of Eqs. (6) - (13) shows that the fastest suppression of the central-frequency's fluctuations is attained at

This yields the ratio of the optimal channel widths

i.e., the DM (Gaussian) pulses allow, roughly, 70 % more channels, in the same frequency range, than the sech solitons. The above analysis can be extended to the case of the sliding-frequency filters. With the corresponding substitution (3), we look for FP solutions by setting in Eqs. (7) and (12). The accordingly modified (sliding) FP is characterized by the bias parameter which is a function of the normalized sliding rate, The final equations relating b and for the sech and DM (Gaussian) pulses take the form, respectively,

The solution for b exists provided that the normalized sliding rate does not exceed a certain maximum value , The dependence of on W , obtained numerically from Eqs. (16) and (17), is plotted in Fig. 1. It is easy to show that in both cases (for the usual and DM solitons) (W) always vanishes exactly at the above-mentioned stability boundary for the nonsliding solution, and that the FP solutions for the sliding-frequency case remains stable as long as it exists.

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B.A. MALOMED, A. DOCHERTY, P.L. CHU, G.D. PENG

An ultimate objective of having filters of any type is to stabilize the

soliton against jitter caused by various perturbations, such as the noise produced by the amplifiers (the Gordon-Haus effect), or interactions with other solitons, in the same channel or in different ones. The suppression of the pulse’s jitter T(z), i.e., its random temporal shift from the expected position, by the filters is determined by a “friction coefficient” in an evolution equation for the deviation of the soliton’s central frequency from its unperturbed value,

where the “friction force” (the first term on the r.h.s.) is nothing else but a linearization of the expressions from Eqs. (7) or (12), and F(z) is an external disturbance, e.g., a random force corresponding to the Gordon-Haus jitter

[12], or a force of the interaction with another soliton. The latter force can be written in a relatively simple explicit form if it is produced by the interaction with a soliton in the adjacent WDM channel, provided that the inverse group-velocity difference c in Eqs. (1) and (2) is large enough, so that

the solitons pass through each other quickly. In this case, one may, in the lowest-order approximation, substitute the temporal separation between the two soliton by

In the case of two identical sech solitons, the

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corresponding force that should be inserted into Eq. (18) is

For the case of the DM solitons, the expression for the force obtained in the same approximation is given by Eq. (7) of Ref. [3].

Coming back to the friction coefficient in Eq. (18), the linearization of Eqs. (7) and (12) yields the values of a for both types of the pulses in the optimum cases defined above:

As was demonstrated in Ref. [3], the largest contribution to the crosstalk in the WDM system is produced by “incomplete collisions” between initially overlapping solitons in two adjacent channels (they are opposed to “complete collisions”, in the course of which solitons are initially far separated,

pass through each other and separate again). The incomplete collisions give rise to a frequency shift, which, in turn, results in a linearly growing relative (normalized to the soliton’s width) position shift For ps, the incomplete-collisioninduced relative shift is which implies full destruction of the signal. If the filters are present, the situation is altogether different. We call

the filters strong if the relaxation length

, see Eq. (18), is much smaller

than the collision length, (recalling that c is the inverse-group-velocity difference between the channels.) In this case, an approximate solution to Eq. (18) is yielding, instead of the frequency shift, only a

net temporal-position shift,

being the net collision-induced frequency shift in the absence of the filters . The expression (22) immediately tells us that the complete collisions, having (simply because the function F(z) is odd, see, e.g., Eq. (19), Ref. [3]), produce no net effect at all, while the incomplete collision results in a negligible relative shift that can be estimated as with

measured in ps.

As follows from Eq. (20), the underlying condition, suggests taking The excess gain necessary to compensate the loss inflicted by the filtering is

In the above-mentioned typical case

this yields

dB/km. Comparing it

418

B.A. MALOMED, A. DOCHERTY, P.L. CHU, G.D. PENG

to the basic gain necessary to compensate the fiber loss, ~ 0.2 dB/km, we conclude that the extra gain for the usual (sech) solitons, roughly, doubles the net in-line gain. For the DM Gaussian solitons, a similar analysis, using Eq. (21) and the results from Ref. [3], leads to a conclusion that the crosstalk can be completely suppressed by the notch filters, requiring the extra gain which is only ~ 5 % of the basic gain. 3. Numerical Simulations of a Single Channel with Notch Filters Extensive numerical simulations are necessary to understand the real-world properties of the soliton system with notch filters. The first objective is to simulate the dynamics of a soliton in a single channel bounded by the notches. The simulations reported in this section and in the following one were performed by means of the standard split-step scheme using the fast Fourier transform in an integration domain with periodic (in

) boundary condi-

tions. The total domain size was 2 ns. This is quite sufficient for a full

numerical investigation of the dynamics of solitons with temporal width 10 ~ 20 ps, which is the case of the direct interest for applications. At this first stage of the simulations, we still assume distributed filtering and amplification, which actually implies that all the energy is removed from the notches, that have a finite width in the spectral space, at each step of the numerical integration. A typical result of the simulations for the single channel is displayed in Fig. 2, where the soliton was initially strongly perturbed by a shift of its central frequency from its equilibrium position in the middle of the channel. The curves in Fig. 2 demonstrate a gradual recentralization of the soliton under the action of the filters. The soliton has width spectral FWHM of 19 GHz and

a soliton period of 10 km

the notches have a width

of their centers being separated by the interval GHz. For comparison, on the same plot we show the results provided by some “regularized” finite-width notch filter, which has a smooth, rather than rectangular, absorption profile, as well as similar results for a more familiar super-Gaussian bandpass (rather than notch) filter, whose pass

band is close to a rectangular with the width 55.70 GHz, centered in the middle of the channel. Parameters of the alternative filtering schemes are selected to provide a stationary soliton of the same width at the same value of distributed gain. It is not surprising that the bandpass filter provides a better stabilization (faster recentralization) in the single channel than the notch filters, as the use of the latter ones makes sense only in multichannel systems. However, it is noteworthy that the “regularized” notch filters give rise to

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a considerably better result than the simple rectangular ones. Actually, we have noted a trend that, in many cases (including two-channel systems), various types of regularized notch filters indeed operate better than the rectangular filters. However, a detailed analysis of this very technical issue is beyond the scope of this paper. 4. Suppression of the Crosstalk Effects by Notch Filters in the Two-Channel System

In this section, we concentrate on direct numerical simulations of the most troublesome effect in the WDM soliton-based system, viz., collision between the solitons in two channels. Here, we will display the results for the most dangerous case of the “incomplete collision”, when the two solitons are completely overlapped at the initial point We assume that both solitons are identical, having the initial shape cosh with the dispersion in both channels being The difference in the inverse group velocities (see the parameter c in Eqs. (1) and (2)) is which, for the solitons with the above-mentioned width, gives a collision length of 300 km. The total propagation length in the simulations was 750 km. The lumped notch filters were separated by a finite spacing Simulations were run for various values of as well as for many different values

420

B.A. MALOMED, A. DOCHERTY, P.L. CHU, G.D. PENG

of the notch width and spectral separation between the notches in one channel. However, the amplification is still taken (to simplify the simulations) in the distributed form.

The results of the simulations are summarized in the three-dimensional plots displayed in Figs. 3 and 4. In each figure, the portion (a) shows the main result of the incomplete collision, viz., the residual frequency shift of each soliton after the collision distance of 300 km, while the portion (b) displays another crucially important, characteristic, the excess gain necessary for the compensation of the losses induced by the filters. The data displayed in Fig. 3 reveal two quite noteworthy features. First, the residual frequency shift, naturally decreasing with the increase of the notch’s spectral width, i.e., absorption strength, practically does not depend at all upon the spatial separation between the filters. Second, the size of the necessary compensatory gain quite unexpectedly increases with the increase of The latter feature, nevertheless, can be seen to have a natural explana-

tion. Indeed, for the values of the parameters adopted for the simulations, the soliton’s dispersion length, that determines a characteristic propagation distance necessary for an internal rearrangement of the soliton, is 200 km. If the filters are placed with a much smaller spacing, the soliton does not have a chance to refill the spectral holes “burnt” by the notch filters, hence each new filter can only remove a relatively small portion of the energy. On the contrary to this, sparcely spaced filters remove much more energy, as the soliton, receiving the energy from the amplifiers, has enough time to “heal” the spectral holes between the passage of two adjacent filters. In any case, the results presented in Fig. 3 strongly suggest that the most efficient operation regime should be sought for with the spacing between the filters much smaller than the soliton’s dispersion length. The results displayed in Fig. 4 show a natural trend for attenuation of the stabilization effect of the notches with the increase of the spectral separation between them in one channel, It is seen that the effect of the change of notch spectral separation is as strong as the effect of change in notch width on the residual frequency shift, unlike the effect of filter spacing. It is clear, however, that notches that are too closely spaced do not allow the propagation of a stable soliton (see Section 2), so operation is not possible in that region. From Fig. 4(b) it is seen that not only does the gain required increase with decreasing as intuition would suggest, but also there definitely exists a specific value of below which the required compensating gain very steeply increases with no considerable decrease in the residual frequency shift. This value is 13 Ghz. This may be naturally compared to the optimal value predicted by Eqs. (9) and (14), that was obtained above by means of the analytical perturbation theory for the

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422

B.A. MALOMED, A. DOCHERTY, P.L. CHU, G.D. PENG

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distributed-filtering model. With the same values of parameters that were used to generate Fig. 4, Eqs. (9) and (14) yield GHz. The agreement between the numerical and analytical results for the optimum spectral separation between the notches is reasonable, taking into regard

the many assumptions adopted in the analytical consideration. Thus we have found a value of that provides for the optimal operation regime for the notch filters in terms of the efficiency of suppression of the collision-induced frequency shift versus the nessesary excess gain. Finally, we notice that, as seen in Fig. 4, is nearly independant of the notch width although the effect is most notable for large 5. Conclusion

In this work, we have presented results of detailed analytical and numerical investigation of the soliton dynamics in single-channel and multichannel systems supported by the notch filters separating the channels in the spectral domain. In the analytical part of the work, we have studied the properties of the dispersion-managed (Gaussian) soliton, and demonstrated that it has strong advantages in comparison with the usual soliton in the constantdispersion system. The numerical part concentrated on direct simulations of the most dangerous collisions between initially overlapping solitons in a realistic two-channel system with lumped finite-spectral-width notch filters. The simulations reveal unexpected features of the collisions in the presence of the notch filters, that may be quite useful for applications. References 1. Wabnitz, S. : Stabilization of Sliding-Filtered Soliton WDM Transmissions by Dispersion Compensating Fibres, Opt. Lett., 21, (1996), p.638; Kumar, S., Kodama, Y. and Hasegawa, A. : Optimal dispersion management schemes for WDM soliton

5.

systems, Electron. Lett., 33, (1997), p.459. Niculae, A. N., Forysiak, W., Gloag, A. J., Nijhof, J. H. B. and Doran, N. J. : Soliton Collisions with WDM systems with strong dispersion management, Opt. Lett., 23, (1998), p.1354. Kaup, D. J., Malomed, B. A and Yang, J. : Interchannel pulse colision in a WDM system with strong Dispersion Management, Opt. Lett., 23, (1998), p.1600. Mamyshev, P. V. and Mollenauer, L. F. : Soliton collisions in wavelength-divisionmultiplexed dispersion-managed systems, Opt. Lett., 24, (1999), p.448. Golovchenko, E. A., Pilipetskii, A. N. and Menyuk, C. R. : Minimum channel spacing

6.

in filtered soliton WDM transmission, Opt. Lett., 21, (1996), p.195. Orlov, S. S., Yariv, A. and Van Essen, S. : Coupled-mode analysis of fibre-optic add-

2. 3. 4.

7. 8.

drop filters for dense wavelength division multiplexing, Opt. Lett., 22, (1997), p.688; Gu, X. J. : WDM isolation fiber filter and light source using cascading long-period fiber gratings, Opt. Lett., 23, (1998), p.509. Malomed, B. A, Peng, G. D and Chu, P. L. : Soliton WDM System using channelisolating notch filters, Opt. Lett., 24, (1999), p.1100. Matsumoto, M. : Instability of dispersion-managed solitons in a system with filter-

424

B.A. MALOMED, A. DOCHERTY, P.L. CHU, G.D. PENG ing, Opt. Lett., 23, (1998), p.1901.

9.

Malomed, B. A., Matera, F. and Settembre, M. : Reduction of the jitter for returnto-zcro signals, Opt. Comm., 143, (1997), p.193-198. 10. Kodama, Y., Kumar, S. and Maruta, A. : Chirped nonlinear pulse propogation in a

dispersion-compensated system, Opt. Lett., 22, (1997), p.1689; Kutz, J. N., Holmes, P., Evangelides, S. J. and Gordon, J. P. : Hamiltonian dynamics of dispersionmanaged breathers, J. Opt. Soc. Am. B, 15, (1998), p.87; Berntson, A., Doran, N. J., Forysiak, W. and Nijhof, J. H. B. : Power dependence of DM solitons for anomalous, zero and normal path-average dispersion, Opt. Lett., 23, (1998), p.900902; Turitsyn, S. K. : Breathing self-similar dynamics and oscillatory tails of the chirped dispersion-managed soliton, Phys. Rev. E, 58, (1998), pp.1256-1259. 11. Lakoba, T. I., Yang, J., Kaup, D. J. and Malomed, B. A. : Conditions for stationary pulse propagation in the strong dispersion management regime, Opt. Comm., 149, (1998), p.366. 12. Mecozzi, A., Moores, J. D., Haus, H. A. and Lai, Y. : Soliton transmission con-

trol, Opt. Lett., 16, (1991), p.1841; Kodama, Y. and Hasegawa, A. : Generation of asymptotically stable optical solitons and suppression of the Gordon-Haus effect, Opt. Lett., 17, (1992), p.31.

INDEX

1.55 and 1.58 optical repeaters, 161 20 Gb/s, 161 40 Gb/s, 387–391, 395, 397, 399, 400 40 Gb/s dispersion managed soliton transmission, 388 40 Gbit/s/channel WDM system, 63 40 Gbit/s soliton, 309, 310, 211 40 Gbit/s soliton transmission experiment, 217

collision of soliton, 413 comb-like dispersion profiled fiber (CDPF), 330, 345 compensated dispersion, 383 correlation matrix, 292, 294 cross-phase modulation (XPM), 203 cross-talk, 295 cross phase modulation (XPM), 71 DCF, 331 DCRA, 337 degenerate four-wave mixing (DFWM), 335 dense wavelength division multiplexing (DWDM), 313 differential group delay (DGD), 253, 254, 257, 261 disintegration, 277, 282 disintegration of soliton, 280, 286 dispersion, 211 dispersion-decreasing fiber (DDF), 345 dispersion-flattened transmission line, 161 dispersion comparision, 383 dispersion compensating fiber, 365, 327 dispersion compensating fiber (DCF), 71 dispersion compensating Raman amplifier(DCRA), 338 dispersion compensation, 66 dispersion compensation elements, 211 dispersion compensator, 65

all-optical tapping, 296 anomalous loss, 380 automatic dispersion compensation, 68 automatic PMD compensation, 73 back-action evading quantum measurement, 290, 294

band-pas s filter, 292 bifurcation, 48, 52, 301 testability, 305 bit-error-rate (BER), 68 bit-parallel transmission, 41 bit-parallel wavelength fiber, 43 broadband Raman amplification, 327 broadband Raman amplifier, 337 channel wavelength spacing, 66 chirp, 141, 152

chirped RZ (CRZ) format, 328 chromatic dispersion, 64 collision, 411, 413 collision-induced time shift, 203 425

426

dispersion managed, 277, 279, 286, 379, 388, 380 dispersion managed (DM) soliton, 129, 278, 280, 286, 265 dispersion managed (DM) soliton transmission, 309

dispersion managed cable, 354, 360 dispersion managed fiber, 351–353, 355, 356, 359, 361 dispersion managed fibers, 331 dispersion managed single mode fiber (DM-SMF), 309 dispersion managed soliton, 115, 116, 126, 299, 300, 310, 387, 211 dispersion managed soliton system, 379, 385 dispersion management, 351, 352, 356, 357, 360, 314 dispersion management technique, 351 dispersion monitor, 70 dispersion shifted fiber (DSF), 68 dispersion slope, 365, 265 dispersion tolerance, 64 DM soliton, 286

effective area, 365 eigensolution, 265 experiment, 388, 394, 398, 211 eye patterns, 199 feedback control, 71 fiber design, 382, 383 fiber FWM based parametric wavelength conversion, 344 fiberoptic parametric wavelength converter, 330 fiber with normal dispersion and

negative slope, 331 forward-error-correction (FEC), 64 four-wave mixing (FWM), 64

four-wave mixing (FWM) crosstalk, 67 Ginzburg-Landau, 139, 140 Gordon-Haus jitter, 328 Gordon-Haus timing jitter, 201 guiding-center soliton, 135 guiding-center theory, 130

hamiltonian averaging, 132 high nonlinearity (HNL) – dispersion shifted fiber (DSF), 330, 342 high nonlinearity fiber, 327, 342 integrable normal form, 136 inter-pulse correlation pattern, 294 inter-pulse photon-number correlation, 291, 294 inter-pulse spectral photon-number correlation, 296 intra-pulse correlation pattern, 294 ispersion-slope-compensating negative dispersion fiber, 161 ITU–T grid, 66 Kerr nonlinearity, 330

Lie transformation, 130 location, 211 location effect of the dispersion compensation element, 219 massive WDM, 115, 116 material dispersion, 380, 381 Maxwell’s equation, 381 multi-mode photon-number correlation, 291 multi-mode photon-number correlations, 294 multi-mode soliton, 42, 56 multi-soliton pulse transmission, 59 multichannel wavelength conversion, 330

427

multichannel wavelength converter, 342 multiple dispersion managed (MDM), 317 multiple dispersion managed (MDM) soliton, 309, 316 multiple dispersion managed single mode fiber (MDM-SMF), 317

PMD compensation, 253, 260 polarization-mode-dispersion (PMD), 73 polarization mode dispersion (PMD), 253, 254, 257, 261 pre-coder, 64 pulse-to-pulse interaction, 198 pump laser, 337 pure silica core, 365

nlinearity, 365 noise figure (NF), 66 noise reduction, 290 non-classical phenomena, 290 non-linear loop mirror, 290

Q-factor, 64, 198

non-PM HNL-DSF, 342 non-return-to-zero (NRZ), 64 non-zero dispersion fiber (NZ–DSF), 66 nonintegrability, 136 nonlinear coefficient, 329

nonlinear coefficient measurement,

335 nonlinear coefficient of fiber, 327

optical communication, 265 optical duobinary format, 64 optical fiber, 327 optical limiter, 290 optical modulation scheme, 64 optical signal-to-noise ratio (OSNR), 64 optical signal processing, 327 optimization, 211 optimization of dispersion compensation, 213

optimization of signal power, 218

OSNR degradation, 68 passively mode-locked fiber ring laser, 328 photon-number noise reduction, 290

PLC combiner, 337

Q-map, 211, 212 Q-map Method, 212 quantum dots, 412 quantum limit, 289 quantum non-linear Schrödinger equation, 292

Raman amplification, 64, 329, 340 randomness, 277, 278, 280, 286 RDF, 331 residual dispersion, 66 return-to-zero (RZ), 64 reverse dispersion fiber (RDF), 71, 327 saturable absorber, 290 self-phase modulation (SPM), 64 shepherding effect, 42, 45, 54 signal-ASE and ASE-ASE beat noise, 198 signal wavelength dithering, 71 simulation, 211 single mode fiber (SMF), 310, 312, 318 SMF, 65 soliton, 41, 57, 139, 140, 142, 145, 147, 152, 277, 279, 281, 282, 285, 310 soliton-based communication, 141 soliton break-up distance, 287 soliton collision, 289 soliton compression, 330

428

soliton compressor, 344 soliton pulse compression, 344 soliton pulse train compression, 327 spectral filtering, 290, 296 spectrally resolved quantum measurement, 291 splitting, 286 SPM–GVD effect, 64 stability, 412 standard fibre, 388, 393, 394, 396,

397, 399 standard fibre experiment, 400 stimulated Brillouin scattering, 65 strength of dispersion management, 211, 216 stretched pulse, 328 strongly dispersion managed, 400 strong management limit, 135

Tb/s transmission, 161 TDM, 139, 152 third-order dispersion (TOD), 265 time-division-multiplexing (TDM), 63 timing jitter, 199 transmission control, 205 ultra long distance WDM transmission, 116 ultrashort pulse transmission, 265

variable dispersion compensator, 69 VIPA, 69 virtually imaged phased array, 69 waveguide dispersion, 380, 381 wavelength-division-multiplexing (WDM), 63 wavelength conversion, 327 wavelength division multiplexed (WDM), 41, 42, 44, 46, 50, 351 wavelength division multiplexed (WDM) multi-mode soliton, 54

wavelength division multiplexing (WDM) soliton, 313 wavelength division multiplexing multiple dispersion managed (WDM-MDM) soliton, 321 WDM, 152, 115 WDM experiment, 122 weak dispersion management, 134 zero-dispersion wavelength, 66

List of Contributors (Speakers)

Abdullaev, Fatkhulla Physical-Technical Institute Uzbek Academy of Sciences G. Mavlyanov str. 2-b

700084, Tashkent

Uzbekistan Phone

: +998-712-35-4338

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Ablowitz, Mark Department of Applied Mathematics

University of Colorado, Boulder Box 526, Boulder, CO 80309-0526 USA

Phone

: +1-303-492-5502

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: +1-303-492-4066

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Forysiak, Wladek Photonics Research Group Aston University Aston Triangle, Bermingham B4 7ET UK

Phone

: +44-121-359-3611

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: +44-121-359-0156

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: [email protected] 429

430

Fukuchi, Kiyoshi C&C Media Research Laboratories NEC Corporation

4-1-1 Miyazaki, Miyamae-ku Kawasaki, Kanagawa 216-8555 Japan Phone

: +81-44-856-8117

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Grigoryan, Vladimir S.

Corvis Corporation 7015 Albert Einstein Drive PO Box 9400 Columbia, MD 21046-9400 Phone Fax

: +1-443-259-4311 :

+1-443-259-4444

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Hasegawa, Akira Present Office Address #403, 19-1 Awataguchi Sanjobo-cho

Higashiyama-ku, Kyoto 605-0035 Japan Phone

: +81-75-525-0700

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431

Hizanidis, Kyriakos Department of Electrical and Computer Engineering National Technical University of Athens 9 Iroon Polytechniou, Zografos 157 73

Athens Greece

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: +30-1-772-3685 :

+30-1-772-3513

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Ishikawa, George Fujitsu Laboratories Ltd. 4-1-1 Kamikodanaka, Nakahara-ku Kawasaki 211-8588 Japan

Phone

: +81-44-754-2641

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: +81-44-754-2640

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Kato, Takatoshi

Sumitomo Electric Industries, Ltd. 1 Taya-cho Sakae-ku, Yokohama 244-8588 Japan

Phone

: +81-45-853-7171

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: +81-45-851-1557

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432

Kivshar, Yuri Optical Science Centre Australian National University

Canbera, ACT 0200 Australia

Phone Fax

: +61-26-249-3081 : +61-26-249-5184

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Kodama, Yuji Department of Mathematics

Ohio State University 231 West 18th. Ave.

Columbus, OH 43210 USA

Phone

: +1-614-292-0692

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: +1-614-292-1479

E-mail : [email protected]

Kubota, Hirokazu

NTT Network Innovation Laboratories 1-1 Hikarino-oka, Yokosuka Kanagawa 239-0847 Japan

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: +81-468-59-4609

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Kumar, Shiva Corning Inc. SP-AR-01-2, 154

Corning Inc., Corning NY14831 USA

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: +l-607-974-3751

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Malomed, Boris A. Department of Interdisciplinary Studies

Faculty of Engineering

Tel Aviv University Tel aviv 69978 Israel Phone

: +972-3-640-6413

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: +972-3-640-6399

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: [email protected]

Matsumoto, Masayuki

Department of Communications Engineering Osaka University

2-1 Yamada-oka, Suita Osaka 565-0871 Japan Phone

: +81-6-6879-7729

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: +81-6-6879-7774

E-mail : [email protected]

434

Midrio, Michele Dipartimento di Ingegneria Elettrica Gestionale e Meccanica

Universita degli Studi di Udine Viale Delle Scienze 208 33100 Udine Italy Phone

: +39-0432-558292

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: +39-0432-558251

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: [email protected]

Mollenauer, Linn

Lightwave Systems Research Bell Labs-Lucent Technologies Rm 4C-306 Holmdel, NJ 07733 USA Phone

: +1-732-949-5766

Fax

: +l-732-949-5784

E-mail

: [email protected]

Nakazawa, Masataka NTT Network Innovation Laboratories 1-1 Hikarino-oka, Yokosuka Kanagawa 239-0847

Japan Phone

: +81-468-59-4637

Fax

: +81-468-59-3396

E-mail

: [email protected]

435

Namiki, Shu

Opto-technology Lab. Furukawa Electric Co., Ltd. 6 Yawata Kaigan Dori

Ichihara 290-8555

Japan Phone

: +81-436-42-1724

Fax

: +81-436-42-9340

E-mail : [email protected]

Nijhof, Jeroen H. B. Photonics Research Group

Aston University

Aston Triangle Birmingham, B29 5DY UK Phone

: +44-121-359-3611

Fax

: +44-121-359-0156

E-mail : [email protected]

Pratt, Andrew R. Oki Electric Industry Co., Ltd. 550-5 Higashi-asakawa Hachioji, Tokyo 193-8550 Japan

Phone

: +81-426-62-6762

Fax

: +81-426-67-6581

E-mail

: [email protected]

436

Reed, William A. Bell Labs-Lucent Technologies

Rm 1D153, 600 Mountain Ave. Murray Hill NJ 07974 USA

Phone

: +1-908-582-2426

Fax

:+1-908-582-6055

E-mail

: [email protected]

Shimoura, Kazuhiro

Technical Research Center The Kansai Electric Power Co., Inc.

3-11-20 Nakoji, Amagasaki Hyogo 661-0974

Japan Phone

: +81-6-6494-9748

Fax

: +81-6-6494-9728

E-mail

: [email protected]

Sizmann, Andreas F. Lehrstuhl für Optik Physikalisches Institut der Universität –Erlangen-Nürnberg

Staudtstr. 7/B2

D-90158 Erlangen Germany

Phone

: +49-9131-85-28375

Fax

:+49-9131-13508

E-mail

: [email protected]

437

Suzuki, Masatoshi KDD R&D Laboratories 2-1-15 Ohara, Kamifukuoka

Saitama 356-8502

Japan Phone

: +81-492-78-7535

Fax

: +81-492-78-7516

E-mail

: [email protected]

Turitsyn, Sergei Photonics Research Group

Aston University Aston Triangle, Bermingham B4 7ET UK

Phone

: +44-121-359-3611

Fax

: +44-121-359-0156

E-mail

: [email protected]

Wabnitz, Stefan Undersea Transmission Group Alcatel CRC

Route de Nozay, Marcoussis 91460 France Phone

: +33-1-6963-1160

Fax

: +33-1-6963-1865

E-mail

: [email protected]

438

Wang, Xiaomin Research Center for Advanced Science and Technology

University of Tokyo 4-6-1 Komaba, Meguro-ku

Tokyo 153-8904 Japan Phone

: +81-3-3481-4438

Fax

: +81-3-3481-4576

E-mail

: [email protected]

SOLID-STATE SCIENCE AND TECHNOLOGY LIBRARY 1.

2. 3.

A.J.P. Theuwissen: Solid-State Imaging with Charge-Coupled Devices. 1995 ISBN 0-7923-3456-6 M.O. van Deventer: Fundamentals of Bidirectional Transmission over a Single Optical Fibre. 1996 ISBN 0-7923-3613-5 A. Hasegawa (ed.): Physics and Applications of Optical Solitons in Fibres ’95. 1996 ISBN 0-7923-4155-4

4. M.A. Trishenkov: Detection of Low-Level Optical Signals. Photodetectors, Focal Plane Arrays and Systems. 1997 ISBN 0-7923-4691-2 5. A. Hasegawa (ed.): New Trends in Optical Soliton Transmission Systems. Proceedings of the 6.

Symposium held in Kyoto, Japan, 1821 November 1997. 1998 ISBN 0-7923-5147-9 A. Hasegawa (ed.): Massive WDM and TDM Soliton Transmission Systems. A ROSC Symposium. 2000 ISBN 0-7923-6517-8

KLUWER ACADEMIC PUBLISHERS – DORDRECHT / BOSTON / LONDON

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