1. Shedding and Interaction of solitons in imperfect medium Misha Chertkov (Theoretical Division, LANL) LANL, 02/05/03 ``Statistical Physics of Fiber Optics Communications” Zoltan Toroczkai (LANL) Pavel Lushnikov (LANL) Jamison Moeser (Brown U) Tobias Schaefer (Brown U) Avner Peleg (LANL) In collaboration with Ildar Gabitov (LANL) Igor Kolokolov (Budker Inst.) Vladimir Lebedev (Landau Inst.) Yeo-Jin Chung (LANL) Sasha Dyachenko (Landau Inst.)

2. What was the idea? Fiber Optics. Statistics. What we did first (Pinning method of pulse confinement in a fiber with fluctuating dispersion) Shedding and interaction of solitons in imperfect mediumWhat we did recently (Shedding and interaction of solitons in imperfect medium) Other activities and plans (Polarization Mode Dispersion, Wave-length Division Mulitplexing, Dispersion Management, etc) Suggestions for Extensive DNS, Experiment, Field Trials

3. Fiber Electrodynamics NLS in the envelope approximation Monomode Weak nonlinearity, slow in z rescaling averaging over amplifiers

4. Nonlinear Schrodinger Equation Soliton solution Dispersion balances nonlinearity Integrability (Zakharov & Shabat ‘72) - dispersion length - pulse width - nonlinearity length - pulse amplitude Model A

5. Dispersion management Model B Lin, Kogelnik, Cohen ‘80 dispersion compensation aims to prevent broadening of the pulse (in linear regime) four wave mixing (nonlinearity) is suppressed effect of additive noise is suppressed Breathing solution - DM soliton no exact solution nearly (but not exactly) Gaussian shape mechanism: balance of disp. and nonl. Turitsyn et al/Optics Comm 163 (1999) 122 Gabitov, Turitsyn ‘96 Smith,Knox,Doran,Blow,Binnion ‘96

6. Noise in dispersion. Statistical Description. Questions: Does an initially localized pulse survive propagation? Are probability distribution functions of various pulse parameters getting steady? DSF, Gripp, Mollenauer Opt. Lett. 23, 1603, 1998 Optical-time-domain-reflection method. Measurements from only one end of fiber by phase mismatch at the Stokes frequency Mollenauer, Mamyshev, Neubelt ‘96 Stochastic model (unrestricted noise) Noise is conservative No jitter Abdullaev and co-authors ‘96-’00

7. Question: Is there a constraint that one can impose on the random chromatic dispersion to reduce pulse broadening? Describes slow evolution of the original field if nonlinearity is weak Unrestricted noise Nonlinearity dies (as z increases) == Pulse degradation correlation length Noise is strong D >> 1

8. Pinning method Constraint prescription: the accumulated dispersion should be pinned to zero periodically or quasi-periodically The restricted model Periodic quasi-periodic Random uniformly distributed ]-.5,.5[

9. Restricted (pinned) noise Model A Weak nonlinearity at L>>l is self-averaged !!! noise average The nonlinear kernel does not decay (with z) !!! noise and pinning period average The averaged equation does have a steady (soliton like) solution in the restricted case

11. Restricted (pinned) noise. DM case. The averaged equation does have a steady solution Weak nonlinearity

13. Numerical Simulations Fourier split-step scheme Fourier modes Model A Model B

17. Moral Practical recommendations for improving fiber system performance that is limited by randomness in chromatic dispersion. The limitation originates from the accumulation of the integral dispersion. The distance between naturally occurring nearest zeros grows with fiber length. This growth causes pulse degradation. We have shown that the signal can be stabilized by periodic or quasi-periodic pinning of the accumulated dispersion. M.Chertkov,I. Gabitov,J.Moeser US patent+PNAS 98, 14211 (2001) M.C.,I.G., P. Lushnikov, Z.Toroczkai, JOSAB (2002)

18. M.Chertkov,I. Gabitov, I.Kolokolov, V.Lebedev, JETP Lett. 10/01 MC, Y. Chung, A. Dyachenko, IG,IK,VL PRE Feb 2003 Shedding and Interaction of solitons in imperfect mediumQuestions: *What statistics does describe the radiation emitted due to disorder by a single soliton, pattern of solitons? How far do the radiation wings extend from the peak of the soliton(s)? What is the structure of the wings? *How strong is the radiation mediating interaction between the solitons? How is the interaction modified if we vary the soliton positions and phases within a pattern of solitons? Method Second order adiabatic pert. theory (Kaup ’90) + Statistical averaging z>>1 Model A

19. Single Soliton Story “Asymptotic freedom”: soliton is distinguishable from the radiation at any z Self-averaging

20. Radiation tail + forerunner

21. Interaction of Shedding Solitons

22. Soliton position shift is Gaussian zero mean random variable Soliton phase mismatch

23. Multi-soliton case Infinite pattern (continuous flow of information) The z-dependence is similar to the one described by Elgin-Gordon-Haus jitter

24. Pinning Bare case Pinned case Single soliton decay Two-soliton interaction

25. Statistical Physics of Fiber Communications We planned to addressed: Single pulse dynamics Fluctuating dispersion Dispersion Management and Fluctuations Raman term +noise Polarization Mode Dispersion Additive (Elgin-Gordon-Hauss) noise optimization Joint effect of the additive and multiplicative noises Mutual equilibrium of a pulse and radiation closed in a box (wave turbulence on a top of a pulse) driven by a noise Many-pulse, -channel interaction Statistics of the noise driven by the interaction Suppression of the four-wave mixing (ghost pulses) by the pinning? Dynamics in a channel under the WDM Multi mode fibers noise induced enhancement of the information flow...