Semianalytical Models for Network Performance Evaluation

  • Ronald Holzlöhner
  • Oleg V. Sinkin
  • Vladimir S. Grigoryan
Part of the Optical Networks book series (OPNW)


We discuss the linearization and the momentum methods as two complementary approaches for analyzing signal statistics in optical communications systems governed by the nonlinear Schrödinger equation. Based on the linearization, we derive the covariance matrix method that allows us to accurately compute the bit error rates. The momentum method represents an alternative approach for computationally efficient analysis of the amplitude and timing jitter, as well as signal statistics.


Timing Jitter Optical Noise Target Channel Phase Jitter Amplify Spontaneous Emission Noise 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors gratefully acknowledge valuable contributions to this work by Prof. Curtis R. Menyuk, Prof. Gary M. Carter, and Prof. John Zweck with the University of Maryland Baltimore County (UMBC), Baltimore, MD, as well as Prof. William L. Kath, Northwestern University, Chicago, IL.


  1. 1.
    Ablowitz MJ, Ahrens C, Biondini G, Chakravarty S, Docherty A (2004) Reduction of collision-induced timing shifts in dispersion-managed quasi-linear systems with periodic-group-delay dispersion compensation. Opt Lett 29(20):2354–2356Google Scholar
  2. 2.
    Ablowitz MJ, Biondini G, Biswas A, Docherty A, Hirooka T, Chakravarty S (2002) Collision-induced timing shifts in dispersion-managed soliton systems. Opt Lett 27(5):318–320Google Scholar
  3. 3.
    Ablowitz MJ, Biondini G, Chakravarty S, Horne RL (1998) On timing jitter in wavelength-division multiplexed soliton systems. Opt Commun 150(1–6):305–318Google Scholar
  4. 4.
    Ablowitz MJ, Docherty A, Hirooka T (2003) Incomplete collisions in strongly dispersion-managed return-to-zero communication systems. Opt Lett 28(4):1191–1193Google Scholar
  5. 5.
    Ablowitz MJ, Hirooka T (2000) Resonant nonlinear intrachannel interactions in strongly dispersion-managed systems. Opt Lett 25(24):1750–1752Google Scholar
  6. 6.
    Abramowitz M, Stegun IA (1965) Handbook of mathematical functions. Dover, New YorkGoogle Scholar
  7. 7.
    Agrawal GP (1995) Nonlinear fiber optics, 2nd edn. Academic Press, LondonGoogle Scholar
  8. 8.
    Agrawal GP (1997) Fiber-optics communication systems, 2nd edn. Wiley, New YorkGoogle Scholar
  9. 9.
    Ahrens C, Ablowitz MJ, Docherty A, Sinkin OV, Zweck J, Grigoryan VS, Menyuk CR (2006) Asymptotic analysis of collision-induced timing shifts in return-to-zero quasi-linear systems with pre- and post-dispersion compensation. Opt Lett 31(1):5–7Google Scholar
  10. 10.
    Anderson CJ, Lyle JA (1994) Technique for evaluating system performance using Q in numerical simulations exhibiting intersymbol interference. Electron Lett 30(1):71–72Google Scholar
  11. 11.
    Arnold L (1992) Stochastic differential equations: theory and applications. Krieger Publishing Company, Malabar. (Reprinted from Wiley, New York, 1974)Google Scholar
  12. 12.
    Bakhshi B, Vaa M, Golovchenko EA, Patterson WW, Maybach RL, Bergano NS (2001) Comparison of CRZ, RZ, and NRZ modulation formats in a \(64\times12.3 \hbox{Gb/s}\) WDM transmission experiment over 9,000 km. In: Proceedings of IEEE/OSA optical fiber communication conference (OFC), paper WF4, Anaheim, CAGoogle Scholar
  13. 13.
    Bellotti G, Varani M, Francia C, Bononi A (1998) Intensity distortion induced by cross-phase modulation and chromatic dispersion in optical-fiber transmissions with dispersion compensation. IEEE Photon Technol Lett 10(12):1745–1747Google Scholar
  14. 14.
    Bennetin G, Galgani L, Strelcyn JM (1976) Kolgomorov entropy and numerical experiments. Phys Rev A 14(6):2338–2345Google Scholar
  15. 15.
    Berg BA (1998) Algorithmic aspects of multicanonical simulations. Nucl Phys Proc Suppl 63(1–3):982–984Google Scholar
  16. 16.
    Berg BA, Neuhaus F (1992) Multicanonical ensemble: a new approach to simulate first-order phase transitions. Phys Rev Lett 68(1):9–11Google Scholar
  17. 17.
    Bergano NS, Kerfoot FW, Davidson CR (1993) Margin measurements in optical amplifier systems. IEEE Photon Technol Lett 5(3):304–306Google Scholar
  18. 18.
    Bigo S, Bellotti G, Chbat M (1999) Investigation of cross-phase modulation limitation over various types of fiber infrastructures. IEEE Photon Technol Lett 11(5):605–607Google Scholar
  19. 19.
    Bigo S, Idler W, Antona J-C, Charlet G, Simonneau C, Gorleir M, Molina M, Borne S, de Barros C, Sillard P, Tran P, Dischler R, Poehlmann W, Nouchi P, Frignac Y (2001) Transmission of 125 WDM channels at 42.7 Gbit/s (5 Tbit/s capacity) over \(12 \times 100 \hbox{km}\) of TeraLight Ultra fibre. In: Proceedings of European conference on optical communications (ECOC) 01, postdeadline paper PDM11, Amsterdam, The NetherlandsGoogle Scholar
  20. 20.
    Biondini G, Kath WL, Menyuk CR (2002) Importance sampling for polarization-mode dispersion. IEEE Photon Technol Lett 14(3):310–312Google Scholar
  21. 21.
    Biondini G, Kath WL, Menyuk CR (2004) Importance sampling for polarization-mode dispersion: techniques and applications. IEEE/OSA J Lightwave Technol 22(4):1201–1215Google Scholar
  22. 22.
    Bosco G, Carena A, Curri V, Gaudino R, Poggiolini P, Benedetto S (2000) A novel analytical method for the BER evaluation in optical systems affected by parametric gain. IEEE Photon Technol Lett 12(2):152–154Google Scholar
  23. 23.
    Bratley BL, Fox BL, Schrage LE (1987) A guide to simulation. Springer-Verlag, New YorkGoogle Scholar
  24. 24.
    Cai J-X, Foursa DG, Liu L, Davidson CR, Cai Y, Patterson WW, Lucero AJ, Bakhshi B, Mohs G, Corbett PC, Gupta V, Anderson W, Vaa M, Domagala G, Mazurczyk M, Li H, Jiang S, Nissov M, Pilipetskii AN, Bergano NS (2005) RZ-DPSK field trial over 13,100 km of installed non-slope-matched submarine fibers. IEEE/OSA J Lightwave Technol 23(1):95–103Google Scholar
  25. 25.
    Cai J-X, Nissov M, Pilipetskii AN, Davidson CR, Mu R-M, Mills MA, Xu L, Foursa D, Menges R, Corbett PC, Sutton D, Bergano NS (2001) \(1.28 \hbox{Tb/s} (32 \times 40 \hbox{Gb/s})\) transmission over 4,500 km. In: Proceedings of European conference on optical communications (ECOC) 01, postdeadline paper PDM12, Amsterdam, The NetherlandsGoogle Scholar
  26. 26.
    Cai J-X, Nissov M, Davidson CR, Cai Y, Pilipetskii AN, Li H, Mills MA, Mu R-M, Feiste U, Xu L, Lucero AJ, Foursa DG, Bergano NS (2002) Transmission of thirty-eight 40 Gb/s channels (> 1.5 Tb/s) over transoceanic distance. In: Proceedings of IEEE/OSA optical fiber communication conference (OFC), postdeadline paper FC4, Anaheim, CAGoogle Scholar
  27. 27.
    Carena A, Curri V, Gaudino R, Poggiolini P, Benedetto S (1997) New analytical results on fiber parametric gain and its effects on ASE noise. IEEE Photon Technol Lett 9(4):535–537Google Scholar
  28. 28.
    Cartaxo A (1998) Impact of modulation frequency on cross-phase modulation effect in intensity modulation-direct detection WDM systems. IEEE Photon Technol Lett 10(9):1268–1270Google Scholar
  29. 29.
    Cartaxo A (1999) Cross-phase modulation in intensity modulation direct detection WDM systems with multiple optical amplifiers and dispersion compensators. IEEE/OSA J Lightwave Technol 17(2):178–190Google Scholar
  30. 30.
    Carter GM, Jacob JM (1998) Dynamics of solitons in filtered dispersion-managed systems. IEEE Photon Technol Lett 10(4):546–548Google Scholar
  31. 31.
    Carter GM, Jacob JM, Menyuk CR, Golovchenko EA, Pillipetskii AN (1997) Timing-jitter reduction for a dispersion-managed soliton system: experimental evidence. Opt Lett 22(8):513–515Google Scholar
  32. 32.
    Chen JC, Lu D, Sadowsky JS, Yao K (1993) On importance sampling in digital communications—part I: fundamentals. IEEE J Sel Areas Commun 11(3):289–299Google Scholar
  33. 33.
    Chen Y, Haus HA (1998) Dispersion-managed solitons with net positive dispersion. Opt Lett 23(13):1013–1015Google Scholar
  34. 34.
    Chiang TK, Kagi N, Marhic ME, Kazovsky LG (1996) Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators. IEEE/OSA J Lightwave Technol 14(3):249–260Google Scholar
  35. 35.
    Ciaramella E (2002) Nonlinear impairments in extremely dense WDM systems. IEEE Photon Technol Lett 14(6):804–806Google Scholar
  36. 36.
    Ciaramella E, Forestieri E (2005) Analytical approximation of nonlinear distortions. IEEE Photon Technol Lett 17(1):91–93Google Scholar
  37. 37.
    Coelho LD, Molle L, Gross D, Hanik N, Freund R, Caspar C, Schmidt ED, Spinnler B (2009) Modeling nonlinear phase noise in differentially phase-modulated optical communication systems. Opt Express 17(5):3226–3241Google Scholar
  38. 38.
    Cohen SD, Hindmarsh AC (1994) CVODE User Guide. Lawrence Livermore National LaboratoryGoogle Scholar
  39. 39.
    Demir A (2007) Nonlinear phase noise in optical-fiber-communication systems. IEEE/OSA J Lightwave Technol 25(8):2002–2032Google Scholar
  40. 40.
    Dieci L, Eirola T (1994) Positive definitenness in the numerical solution of Riccati differential equations. Num Math 67(3):303–313MATHMathSciNetGoogle Scholar
  41. 41.
    Doob JL (1954) The Brownian movement and stochastic equations. In: Wax N (eds) Selected papers on noise and stochastic processes. Dover, New YorkGoogle Scholar
  42. 42.
    Essiambre RJ, Foschini GJ, Kramer G, Winzer PJ (2010) Capacity limits of information transmission in optically-routed fiber networks. Bell Labs Tech J 14(4):149–162Google Scholar
  43. 43.
    Forestieri E (2000) Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre- and postdetection filtering. IEEE/OSA J Lightwave Technol 18(11):1493–1503Google Scholar
  44. 44.
    Forghieri F, Tkach RW, Chraplyvy AR (1997) Fiber nonlinearities and their impact on transmission systems. In: IP Kaminow, Koch TL (eds) Optical fiber telecommunications III A. Academic Press, LondonGoogle Scholar
  45. 45.
    Franklin J (1968) Matrix theory. Prentice-Hall, New JerseyMATHGoogle Scholar
  46. 46.
    Georges T (1995) Bit-error rate degradation of interacting solitons owing to non-Gaussian statistics. Electron Lett 31(14):1174–1175Google Scholar
  47. 47.
    Georges T (1995) Perturbation theory for the assessment of soliton transmission control. Opt Fiber Technol 1(2):97–116MathSciNetGoogle Scholar
  48. 48.
    Georges T (1996) Study of the non-Gaussian timing jitter statistics induced by soliton interaction and filtering. Opt Commum 123(4–6):617–623Google Scholar
  49. 49.
    Georges T, Favre F, Guen DL (1998) Theoretical and experimental study of soliton transmission in dispersion managed links. Inst Electron Inf Commun. Eng Trans Electron E81-C(2):226–231Google Scholar
  50. 50.
    Gnauck AH, Winzer PJ (2005) Optical phase-shift-keyed transmission. IEEE/OSA J Lightwave Technol 23(1):115–130Google Scholar
  51. 51.
    Golovchenko EA, Jacob JM, Pillipetskii AN, Menyuk CR, Carter GM (1997) Dispersion-managed solitons in a fiber loop with in-line filtering. Opt Lett 22(5):289–291Google Scholar
  52. 52.
    Golovchenko EA, Pilipetskii AN, Bergano NS (2000) Transmission properties of chirped return-to-zero pulses and nonlinear intersymbol interference in 10 GBit/s WDM transmission. In: Proceedings of IEEE/OSA optical fiber communication conference (OFC), paper FC3, Baltimore, MDGoogle Scholar
  53. 53.
    Golovchenko EA, Pilipetskii AN, Bergano NS, Davidsen CR, Khatri FI, Kimball RM, Mazurczyk VJ (2000) Modeling of transoceanic fiber-optic WDM communications systems. IEEE J Sel Topics Quant Electron 6(2):337–347Google Scholar
  54. 54.
    Gordon JP, Haus HA (1986) Random walk of coherently amplified solitons in optical fiber transmission. Opt Lett 11(10):665–667Google Scholar
  55. 55.
    Grigoryan VS, Menyuk CR (1998) Dispersion-managed solitons at normal average dispersion. Opt Lett 23(8):609–611Google Scholar
  56. 56.
    Grigoryan VS, Menyuk CR, Mu RM (1999) Calculation of timing and amplitude jitter in dispersion-managed optical fiber communications using linearization. IEEE/OSA J Lightwave Technol 17(8):1347–1356Google Scholar
  57. 57.
    Grigoryan VS, Richter A (2000) Efficient approach for modeling collision-induced timing jitter in WDM return-to-zero dispersion-managed systems. IEEE/OSA J Lightwave Technol 18(8):1148–1154Google Scholar
  58. 58.
    Grigoryan VS, Veselka J, Sardesai HP (2008) Highly efficient method for BER modeling in quasi-linear fibers and its validation in a 40 Gb/s DWDM test bed. In: Proceedings of IEEE/OSA optical fiber communication conference (OFC), paper JThA53, San Diego, CAGoogle Scholar
  59. 59.
    Hahn PM, Jeruchim MC (1987) Developments in the theory and application of importance sampling. IEEE Trans Commun 35(7):706–714Google Scholar
  60. 60.
    Hasegawa A (1995) Solitons in optical communications. Clarendon, OxfordMATHGoogle Scholar
  61. 61.
    Haus HA (1991) Quantum noise in solitonlike repeater systems. J Opt Soc Am B 8(5):1122–1126Google Scholar
  62. 62.
    Holzlöhner R (2003) A covariance matrix method for the computation of bit errors in optical transmission systems. Ph.D. thesis, University of Maryland Baltimore County. Baltimore, Maryland, USAGoogle Scholar
  63. 63.
    Holzlöhner R, Ereifej HN, Carter GM, Menyuk CR (2002) Experimental and theoretical characterization of a 40 Gb/s long-haul single-channel transmission system. IEEE/OSA J Lightwave Technol 20(7):1124–1131Google Scholar
  64. 64.
    Holzlöhner R, Grigoryan VS, Menyuk CR, Kath WL (2002) Accurate calculation of eye diagrams and bit error rates in long-haul transmission systems using linearization. IEEE/OSA J Lightwave Technol 20(3):389–400Google Scholar
  65. 65.
    Holzlöhner R, Mahadevan A, Menyuk C, Morris J, Zweck J (2005) Evaluation of the very low BER of FEC codes using dual adaptive importance sampling. IEEE Commun Lett 9(2):163–165Google Scholar
  66. 66.
    Holzlöhner R, Menyuk CR (2003) The use of multicanonical Monte Carlo simulations to obtain accurate bit error rates in optical communications systems. Opt Lett 28(20):1894–1896Google Scholar
  67. 67.
    Holzlöhner R, Menyuk CR, Kath WL VSG (2003) A covariance matrix method to compute bit error rates in a highly nonlinear dispersion-managed soliton system. IEEE Photon Technol Lett 15(5):688–690Google Scholar
  68. 68.
    Hui R, Chowdhury D, Newhouse M, O’Sullivan M, Poettcker M (1997) Nonlinear amplification of noise in fibers with dispersion and its impact in optically amplified systems. IEEE Photon Technol Lett 9(3):392–394Google Scholar
  69. 69.
    Hui R, Demarest KR, Allen CT (1999) Cross-phase modulation in multispan WDM optical fiber systems. IEEE/OSA J Lightwave Technol 17(6):1018–1026Google Scholar
  70. 70.
    Hui R, O’Sullivan M, Robinson A, Taylor M (1997) Modulation instability and its impact in multispan optical amplified IMDD systems: theory and experiments. IEEE/OSA J Lightwave Technol 15(7):1071–1081Google Scholar
  71. 71.
    Humblet PA, Azizo glu M (1991) On the bit error rate of lightwave systems with optical amplifiers. IEEE/OSA J Lightwave Technol 9(11):1576–1582Google Scholar
  72. 72.
    Iannone E, Matera F, Mecozzi A, Settembre M (1998) Nonlinear optical communication networks. Microwave and optical engineering. Wiley, New YorkGoogle Scholar
  73. 73.
    Jenkins RB, Sauer JR, Chakravarty S, Ablowitz MJ (1995) Data-dependent timing jitter in wavelength-division-multiplexing soliton systems. Opt Lett 20(19):1964–1966Google Scholar
  74. 74.
    Jiang Z, Chongcheng F (2003) A comprehensive study on XPM- and SRS-induced noise in cascaded IM-DD optical fiber transmission systems. IEEE/OSA J Lightwave Technol 21(4):953–960Google Scholar
  75. 75.
    Kaminov, IP (eds) (1997) TLK optical fiber telecommunications IIIA. Academic Press, San DiegoGoogle Scholar
  76. 76.
    Kaminov, IP (eds) (2002) Optical fiber telecommunications IVB. Academic Press, San DiegoGoogle Scholar
  77. 77.
    Kaup DJ (1990) Perturbation theory for solitons in optical fibers. Phys Rev A 42(9):5689–5694MathSciNetGoogle Scholar
  78. 78.
    Kazovsky L, Benedetto S, Willner A (1996) Optical fiber comminication systems. Artech House, BostonGoogle Scholar
  79. 79.
    Killey RI, Thiele HJ, Mikhailov V, Bayvel P (2000) Prediction of transmission penalties due to cross-phase modulation in WDM systems using a simplified technique. IEEE Photon Technol Lett 12(7):804–806Google Scholar
  80. 80.
    Kovsh DI, Liu L, Bakhshi B, Pilipetskii AN, Golovchenko EA, Bergano NS (2002) Reducing interchannel crosstalk in long-haul DWDM systems. IEEE J Sel Topics Quant Electron 8(3):597–602Google Scholar
  81. 81.
    Kumar S, Hasegawa A (1997) Quasi-soliton propagation in dispersion-managed optical fibers. Opt Lett 22(6):372–375Google Scholar
  82. 82.
    Kutz JN, Evangelides SG (1998) Dispersion-managed breathers with average normal dispersion. Opt Lett 23(9):685–687Google Scholar
  83. 83.
    Lakoba TI, Yung J, Kaup DJ, Malomed BA (1998) Conditions for stationary pulse propagation in the strong dispersion management regime. Opt Commun 149(4–6):366–375Google Scholar
  84. 84.
    Le Guen F., Del Burgo S, Moulinard ML, Grot D, Henry, M, Favre F, Georges T (1999) Narrow band 1.02 Tbit/s (51 × 20 Gbit/s) soliton DWDM transmission over 1,000 km of standard fiber with 100 km amplifier spans. In: Proceedings of IEEE/OSA optical fiber communication conference (OFC), postdeadline paper PD4, Washington, DCGoogle Scholar
  85. 85.
    Lee JS, Shim CS (1994) Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain. IEEE/OSA J Lightwave Technol 12(7):1224–1229Google Scholar
  86. 86.
    Lima AO, Lima IT, CR M (2005) Error estimation in multicanonical Monte Carlo simulations with applications to polarization-mode-dispersion emulators. IEEE/OSA J Lightwave Technol 23(11):3781–3789Google Scholar
  87. 87.
    Lima IT, Lima AO, Sun Y, Jiao H, Zweck J, CR M (2005) A receiver model for optical fiber communication systems with arbitrarily polarized noise. IEEE/OSA J Lightwave Technol 23(3):1478–1490Google Scholar
  88. 88.
    Lima IT, Lima AO, Zweck J, Menyuk CR (2003) Efficient computation of outage probabilities due to polarization effects in a WDM system using a reduced Stokes model and importance sampling. IEEE Photon Technol Lett 15(1):45–47Google Scholar
  89. 89.
    Liu X, Wei X, Mollenauer LF, McKinstrie CJ, Xie C (2003) Collision-induced time shift of a dispersion-managed soliton and its minimization in wavelength-division-multiplexed transmission. Opt Lett 28(16):1148–1150Google Scholar
  90. 90.
    DeLong KW, Trebino R, Kane DJ (1994) Comparison of ultrashort pulse frequency-resolved optical gating traces for three common beam geometries. J Opt Soc Am B 11(9):1595–1608Google Scholar
  91. 91.
    Loudon R (1983) The quantum theory of light. Clarendon, OxfordGoogle Scholar
  92. 92.
    Lu T, Yevick D (2005) Efficient multicanonical algorithms. IEEE Photon Technol Lett 17(4):861–863Google Scholar
  93. 93.
    Luis RS, Cartaxo AVT (2005) Analytical characterization of SPM impact on XPM-induced degradation in dispersion-compensated WDM systems. IEEE/OSA J Lightwave Technol 23(3):1503–1513Google Scholar
  94. 94.
    Lyubomirsky I, Qui T, Roman J, Nayfeh M, Frankel M, Taylor MG (2003) Interplay of fiber nonlinearity and optical filtering in ultradense WDM. IEEE Photon Technol Lett 15(1):147–149Google Scholar
  95. 95.
    Mamyshev PV, Mamysheva NA (1999) Pulse-overlapped dispersion-managed data transmission and intrachannel four-wave mixing. Opt Lett 24(21):1454–1456Google Scholar
  96. 96.
    Marcuse D (1990) Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers. IEEE/OSA J Lightwave Technol 8(12):1816–1823Google Scholar
  97. 97.
    Marks B (2000) Dispersion management in fiber optic communications. Dissertation, Northwestern University, ChicagoGoogle Scholar
  98. 98.
    Mecozzi A (1994) Long-distance transmission at zero dispersion: Combined effect of the Kerr nonlinearity and the noise of the in-line amplifiers. J Opt Soc Am B 11(3):462–469Google Scholar
  99. 99.
    Menyuk CR (1995) Non-Gaussian corrections to the Gordon–Haus distribution resulting from soliton interactions. Opt Lett 20(3):285–287Google Scholar
  100. 100.
    Menyuk CR (1999) Application of multiple-length-scale methods to the study of optical fiber fiber transmission. J Eng Math 36(1–2):113–136MATHMathSciNetGoogle Scholar
  101. 101.
    Metropolis N, Ulam S (1949) The Monte Carlo method. J Am Stat Assoc 44(247):335–341MATHMathSciNetGoogle Scholar
  102. 102.
    Moore RO, Biondini G, Kath WL (2003) Importance sampling for noise-induced amplitude and timing jitter in soliton transmission systems. Opt Lett 28(2):105–107Google Scholar
  103. 103.
    Mu RM, Grigoryan VS, Menyuk CR, Carter GM, Jacob JM (2000) Comparison of theory and experiment for dispersion-managed solitons in a recirculating fiber loop. IEEE J Sel Topics Quant Electron 6(2):248–257Google Scholar
  104. 104.
    Mu RM, Menyuk CR (2001) Symmetric slope compensation in a long-haul WDM system using the CRZ format. IEEE Photon Technol Lett 13(8):797–799Google Scholar
  105. 105.
    Mu RM, Menyuk CR (2002) Convergence of the chirped return-to-zero and dispersion managed soliton modulation formats in WDM systems. IEEE/OSA J Lightwave Technol 20(4):608–617Google Scholar
  106. 106.
    Mu RM, Yu T, Grigoryan VS, Menyuk CR (2000) Convergence of the CRZ and DMS formats in WDM systems using dispersion management. In: Proceedings of IEEE/OSA optical fiber communication conference (OFC), paper FC1, Baltimore, MDGoogle Scholar
  107. 107.
    Mu RM, Yu T, Grigoryan VS, Menyuk CR (2002) Dynamics of the chirped return-to-zero modulation format. IEEE/OSA J Lightwave Technol 20(1):47–57Google Scholar
  108. 108.
    Narimanov EE, Mitra P (2002) The channel capacity of a fiber optics communication system: perturbation theory. IEEE/OSA J Lightwave Technol 20(3):530–537Google Scholar
  109. 109.
    Nelson LE, Jopson RM, Gnauk AH, Chraplyvy AR (1999) Resonances in cross-phase modulation impairment in wavelength-division-multiplexed lightwave transmission. IEEE Photon Technol Lett 11(7):907–909Google Scholar
  110. 110.
    Neokosmidis I, Kamalakis T, Chipouras A, Sphicopoulos T (2005) Estimation of the four-wave mixing noise probability-density function by the multicanonical Monte Carlo method. Opt Lett 30(1):11–13Google Scholar
  111. 111.
    Nijhof JHB, Doran NJ, Forysiak W, Knox FM (1997) Stable soliton-like propagation in dispersion managed systems with net anomalous, zero, and normal dispersion. Electron Lett 33(20):1726–1727Google Scholar
  112. 112.
    Ogawa K, Tzeng LD, Park YK, Sano E (1997) Advances in high bit rate transmission systems. In: Kaminow IE, Koch TL (eds.) Optical fiber telecommunications, vol IIIA, chap 11, Academic, San Diego, pp 336–372Google Scholar
  113. 113.
    Papoulis A (1991) Probability, random variables, and stochastic processes, 3rd edn. McGraw-Hill, New YorkGoogle Scholar
  114. 114.
    Pellegrini W, Zweck J, Menyuk CR, Holzlöhner R (2005) Computation of bit error ratios for a dense WDM system using the noise covariance matrix and multicanonical Monte Carlo methods. IEEE Photon Technol Lett 17(8):1644–1646Google Scholar
  115. 115.
    Pilipetskii AN, Mazurczyk VJ, Chen CJ (1999) The effect of dispersion compensation on system performance when nonlinearities are important. IEEE Photon Technol Lett 11(2):284–286Google Scholar
  116. 116.
    Proakis JG (1995) Digital communications. McGraw-Hill, New YorkGoogle Scholar
  117. 117.
    Risken H (1989) The Fokker–Planck equation, 2nd edn. Springer, BerlinGoogle Scholar
  118. 118.
    Schwartz L (1966) Théorie des distributions I. Hermann, ParisGoogle Scholar
  119. 119.
    Secondini M (2006) Optical communication theory and techniques for high bit-rate systems. Ph.D. thesis, Scuola Superiore Sant’Anna, Pisa, ItalyGoogle Scholar
  120. 120.
    Secondini M, Forestieri E, Menyuk CR (2006) A novel perturbation method for signal–noise interaction in nonlinear dispersive fibers. In: Proceedings of IEEE/OSA optical fiber communication conference (OFC), paper OthD3, Anaheim, CAGoogle Scholar
  121. 121.
    Shapiro EG, Fedoruk MP, Turitsyn SK (2001) Numerical estimate of BER in optical systems with strong patterning effects. Electron Lett 37(19):1179–1181Google Scholar
  122. 122.
    Shtaif M, Eiselt M (1998) Analysis of intensity interference caused by cross-phase modulation in dispersive optical fibers. IEEE Photon Technol Lett 10(7):979–981Google Scholar
  123. 123.
    Sidje RB (1998) Expokit: A software package for computing matrix exponentials. ACM Trans Math Softw 24(1):130–156MATHGoogle Scholar
  124. 124.
    Sinkin OV (2006) Calculation of bit error rates in optical fiber communications systems in the presence of nonlinear distortion and noise. Ph.D. thesis, University of Maryland Baltimore County, Baltimore, Maryland, USAGoogle Scholar
  125. 125.
    Sinkin OV, Grigoryan VS, Holzlöhner R, Kalra A, Zweck J, Menyuk CR (2004) Calculation of error probability in WDM RZ systems in presence of bit-pattern-dependent nonlinearity and of noise. In: Proceedings of IEEE/OSA optical fiber communication conference (OFC), paper TuN4, Los Angeles, CAGoogle Scholar
  126. 126.
    Sinkin OV, Grigoryan VS, Menyuk CR (2007) Accurate probabilistic treatment of bit-pattern-dependent nonlinear distortions in BER calculations for WDM RZ systems. IEEE/OSA J Lightwave Technol 25(10):2959–2968Google Scholar
  127. 127.
    Sinkin OV, Grigoryan VS, Zweck J, Menyuk CR (2006) Calculation of the bit-error ratio in wavelength-division-multiplexed return-to-zero systems when the nonlinear penalty is dominated by collision-induced timing jitter. In: Proceedings of IEEE/OSA optical fiber communication conference (OFC), paper JThB3, Anaheim, CAGoogle Scholar
  128. 128.
    Sinkin OV, Grigoryan VS, Zweck J, Menyuk CR, Docherty A, Ablowitz M (2005) Calculation, characterization, and application of the time shift function in wavelength-division-multiplexed return-to-zero systems. Opt Lett 30(16):2056–2058Google Scholar
  129. 129.
    Sinkin OV, Holzlöhner R, Zweck J, Curtis CR (2003) Optimization of the split-step Fourier method in modeling optical fiber communications systems. IEEE/OSA J Lightwave Technol 21(1):61–68Google Scholar
  130. 130.
    Sinkin OV, Zweck J, Menyuk CR (2003) Effects of the nonlinearly-induced timing and amplitude jitter on the performance of different modulation formats in WDM optical fiber communications systems. In: Proceedings of IEEE/OSA optical fiber communication conference (OFC), paper TuF5, Atlanta, GAGoogle Scholar
  131. 131.
    Smith NJ, Doran NJ, Forysiak W, Knox FM (1997) Soliton transmission using periodic dispersion compensation. IEEE/OSA J Lightwave Technol 15(10):1808–1822Google Scholar
  132. 132.
    Smith NJ, Forysiak NJ, Doran W (1996) Reduced Gordon–Haus jitter due to enhanced power solitons in strongly dispersion managed systems. Electron Lett 32(22):2085–2086Google Scholar
  133. 133.
    Strang G (1980) Linear algebra and its applications. Academic Publishing, New YorkGoogle Scholar
  134. 134.
    Sugahara H, Kato H, Inoue T, Maruta A, Kodama Y (1999) Optimal dispersion management for a wavelength division multiplexed optical soliton transmission system. IEEE/OSA J Lightwave Technol 17(9):1547–1559Google Scholar
  135. 135.
    Sugahara H, Maruta A, Kodama Y (1999) Optimal allocation of amplifiers in a dispersion-managed line for a wavelength-division-multiplexed soliton transmission system. Opt Lett 24(3):145–147Google Scholar
  136. 136.
    Sun H, Wu KT, Roberts K (2008) Real-time measurements of a 40 Gb/s coherent system. Opt Express 16(2):873–879Google Scholar
  137. 137.
    Thiele HJ, Killey RI, Bayvel P (1999) Influence of transmission distance on XPM-induced intensity distortion in dispersion-managed, amplified fibre links. Electron Lett 35(5):408–409Google Scholar
  138. 138.
    Thiele HJ, Killey RI, Bayvel P (2000) Simple technique to determine cross-phase modulation induced penalties in WDM transmission. In: Proceedings of IEEE/OSA optical fiber communication conference (OFC), paper ThM2, Baltimore, MDGoogle Scholar
  139. 139.
    Tsuritani T, Agata A, Morita I, Edagawa N, Shigeyuki A (2004) Ultra-long-haul 40-Gbit/s-based DWDM transmission using optically prefiltered CS-RZ signals. IEEE J Sel Topics Quant Electron 10(2):403–411Google Scholar
  140. 140.
    Turitsyn SK (1997) Theory of average pulse propagation in high bit rate optical transmission systems with strong dispersion. Lett J Exp Theor Phys 65(11):845–850Google Scholar
  141. 141.
    Turitsyn SK, Shapiro EG (1998) Dispersion-managed solitons in optical amplifier transmission systems with zero average dispersion. Opt Lett 23(9):682–684Google Scholar
  142. 142.
    Vaa M, Bakhshi B, Golovchenko EA, Chai Y, Heismann F, Li H, Arend M, Patterson WW, Simons AL, Harvey GT, Maybach RL, Bergano N (2001) Demonstration of a \(640\,\hbox{Gbit/s} \times 7\text{,}000\, \hbox{km}\) submarine transmission system technology ready for field deployment. In: Proceedings of IEEE/OSA optical fiber communication conference (OFC), paper WF5, Anaheim, CAGoogle Scholar
  143. 143.
    Veach E (1997) Robust Monte Carlo methods for light transport simulations. Ph.D. thesis, Stanford University, Stanford, USAGoogle Scholar
  144. 144.
    Yadin Y, Shtaif M, Orenstein M (2005) Bit-error rate of optical DPSK in fiber systems by multicanonical Monte Carlo simulations. IEEE Photon Technol Lett 17(6):1355–1357Google Scholar
  145. 145.
    Yang TS, Kath WL, Evangelides Jr SG (1999) Optimal prechirping for dispersion-managed transmission of return-to-zero pulses. In: Proceedings of IEEE/OSA optical fiber communication conference (OFC), paper ThQ4, Washington, DCGoogle Scholar
  146. 146.
    Yevick D (2002) Multicanonical communication system modeling—application to PMD statistics. IEEE Photon Technol Lett 14(11):1512–1514Google Scholar
  147. 147.
    Yu T, Mu RM, Grigoryan VS, Menyuk CR (1999) Energy enhancement of dispersion-managed solitons in optical fiber transmission systems with lumped amplifiers. IEEE Photon Technol Lett 11(12):75–77Google Scholar
  148. 148.
    Yu T, Reimer WM, Grigoryan VS, Menyuk CR (2000) A mean field approach for simulating wavelength-division multiplexed systems. IEEE Photon Technol Lett 12(4):443–445Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Ronald Holzlöhner
    • 1
  • Oleg V. Sinkin
    • 2
  • Vladimir S. Grigoryan
    • 3
  1. 1.Telescope Laser Systems DepartmentEuropean Southern Observatory (ESO)Garching b. MünchenGermany
  2. 2.Tyco Electronics Subsea CommunicationsEatontownUSA
  3. 3.CienaLinthicumUSA

Personalised recommendations