Statistical physics approach to evaluation of outage probability in optical communications Misha Chertkov (Theoretical Division, LANL) In collaboration with Vladimir Chernyak (Corning) Ildar Gabitov (LANL + Tucson) Igor Kolokolov (Landau Inst.) Vladimir Lebedev (Landau Inst.) Avner Peleg (LANL)

What is the idea: Fiber Optics + Statistics. Introduction: Material. Fiber Electro-dynamics. Noise. Disorder. Impairment Consequence Amplifier Noise jitter, degradation Birefringent disorder Polarization Mode Dispersion broadening, pulse splitting, jitter Joint effect of noise and birefringent disorder Bit-error-rate. Does it fluctuate? Bit-error-rate. Does it fluctuate? How to evaluate/calculate BER (<<1) ? Practical consequences for optical communications Theoretical interest e.g. analogy with spin-glasses

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

Linear vs Nonlinear Dispersion Management Soliton solution Dispersion balances nonlinearity Integrability (Zakharov & Shabat ‘72) Information Coding Return-to-Zero (RZ) Non-Return-to-Zero (NRZ) Differential Phase Coding RZ

Polarization complex two-component first second order BIREFRINGENCE matrixes (2*2, traceless, self-adjoint)

Additive (amplifier) noise It causes: (1) pulse jitter (walk away from the slot) (2) pulse degradation Linear: jitter and amplitude degradation are equally important Soliton: jitter essentially more important than amplitude degradation Elgin (1985) Gordon-Haus for successful fiber performance short correlated ! i.e. different for different pulses 3

Disorder in Birefringence ordered exponential Getting rid of fast polarization axis rotation PMD Pauli matrixes weak isotropic Disorder

Polarization Mode Dispersion (PMD) Linear 0 pulse splitting broadening jitter Poole, Wagner ‘86 Poole ’90;’91 Statistics of PMD vector is Gaussian. Differential group delay (DGD) Polarization (PMD) vector (of first order)

input output Eye diagram ``intensity” PDF

Bit-Error-Rate 2) Build histogram (PDF) of pulse Intensity collecting statistics over many slots (separately for initially empty and filled slots) 1) Measure intensity in each slot ! Electrical filter+sampling window function Linear operator for (a)Optical filter (b)``Compensation” tricks 3) BERdecision level

Filters and ``tricks” Electrical filter +sampling window function Optical filter ``Setting the clock” First order PMD compensation

Noise and disorder. Order of averaging. Calculate BER for given realization of disorder (averaging over noise) Does BER (as a functional of disorder) fluctuate ? Linear model

C. Xie, H. Sunnerud, M. Karlsson, P.Andrekson, ``Polarization-Mode Dispersion-Induced in soliton Transmission systems”, IEEE Photonics Techn. Lett. Vol.13,Oct Monte-Carlo numerics with fiber realizations (artificial rescaling of decision level)

Noise average in the interesting range one has to keep in only the leading in term!!

Optical filter always applies Bare case ``Setting the clock” (no chirp) First order PMD compensation

Bare case PDF of Bit-Error-Rate Saddle-point (optimal fluctuation) calculations Setting the clock First order compensation (nonzero chirp) First order compensation (zero chirp)

Grossly underestimated Gaussian expectation Long (algebraic!) tail

C. Xie, H. Sunnerud, M. Karlsson, P.Andrekson, ``Polarization-Mode Dispersion-Induced in soliton Transmission systems”, IEEE Photonics Techn. Lett. Vol.13,Oct

Example: No compensation Timing jitter First order with chirp First order no chirp

Higher-order compensation main fiber c4 c3c2c1 compensating fibers The idea: to achieve higher (p) compensating degree

Periodic c4c3c2c main fiber c4 c3c2c1 compensating fibers ``Standard” Quasi-periodic c4c3c2c11234 For Q-periodic --- Need !!!!! anti-stokes refraction measurement of birefringence (Hunter,Gisin,Gisin ’99) Q-periodic guarantees much stronger p-dependence of compensation than the ``standard” one

Linear V.Chernyak,MC,I.Kolokolov,V.Lebedev Phys.RevE to appear; Optics. Lett. 28, (2003); Optics. Express. 11, 1607 (2003); JETP Lett. 78, (2003) VC,MC,I. Gabitov,IK,VL, to appear in special issue of Journal of Lightware Technology (invited) Nonlinear( soliton transmission) VC,MC,IK, Avner Peleg submitted to Euro.Phys.Lett Bare case Soliton jitter (due to noise) is the dominant destructive factor

Analogy with Functional Order Parameter approach for glassy states in infinite-range exchange spin systems Double (super) statistics Amplifier Noise Thermal Birefringent Disorder Exchange, J Pulse intensity Glassy states overlap, q PDF BER Overlap Probability, Extended (algebraic like) tail of the double statistics !! No replicas!!! Replicas+Numerics

Conclusions Noise and disorder CAN NOT be considered separately ! Probability Distribution Function of BER is the proper method/tool of extreme outages (for PMD) and their compensation analysis No other alternative to the theory in evaluation of the extremely low valued BER