SheddingShedding and and InteractionInteraction of of solitonssolitons in in imperfectimperfect medium 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 withIn collaboration withIldar Gabitov (LANL)Igor Kolokolov (Budker Inst.)Vladimir Lebedev (Landau Inst.)Yeo-Jin Chung (LANL)Sasha Dyachenko (Landau Inst.)

•What was the idea? Fiber Optics. Statistics.

•What we did first (Pinning method of pulse confinement in a fiber with fluctuating dispersion)

•What we did recently (Shedding and interaction of solitons in Shedding and interaction of solitons in imperfect mediumimperfect medium)

•Other activities and plans (Polarization Mode Dispersion, Wave-length Division Mulitplexing, Dispersion Management, etc)

• Suggestions for Extensive DNS, Experiment, Field Trials

Fiber Electrodynamics

0222 tz di

NLS in the envelope approximation

,

,

DH

BE

t

t

,

,

0

0

PED

HB

.0

,0

D

B

0/

,1222

cEE

nll

)( 00)(),()( tziezyxFnrE

•Monomode•Weak nonlinearity,• slow in z

2|| Enl

N

kkk

ttgr

z

zzrzG

ziGdv

ii

1

22

2)(

)(2

gr

z

vztt

zGdz

0

)'('exp

rescalingaveraging over amplifiers

Nonlinear Schrodinger Equation

0d Soliton solution

dba

bt

bizdazt

22

2

,/cosh

exp)0,( Dispersion balances nonlinearity

dnl zz

Integrability (Zakharov & Shabat ‘72)

0222 tz di

bd

bzd

2 - dispersion length

- pulse widtha

aznl 2

1 - nonlinearity length

- pulse amplitude

Model A

Dispersion management Model B

Lin, Kogelnik, Cohen ‘80

DM

DM

dd

ddzd

0

0)(

•dispersion compensation aims to preventbroadening 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 ‘96Smith,Knox,Doran,Blow,Binnion ‘96

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, 1998Optical-time-domain-reflection method.Measurements from only one end of fiber by phase mismatch at the Stokes frequencyMollenauer, Mamyshev, Neubelt ‘96

Stochastic model (unrestricted noise)

2121

det

)()(

0

)()(

zzDzz

zdzd

Noise is conservative No jitter

Abdullaev and co-authors ‘96-’00

Question: Is there a constraint that one can impose on the random chromatic dispersion to reduce pulse broadening?

223

22

21

*321

0

)('

3213,2,12

0 ')'(exp2 0

0det

zdzddzi

z dzziedidi

z

)()'('exp);(0

02 zdzddztidtz

z

Describes slow evolution of the original field if nonlinearity is weak

Unrestricted noise)()( det zdzd 02)(

22 tz zdi

Nonlinearity dies (as z increases)== Pulse degradation

2

exp)'('exp2

0

zDzdzi

z

DbDz //1 42

correlation length

Noise is strong D >> 1

Pinning method

Constraint prescription: the accumulated dispersion should be pinned to zero periodically or quasi-periodically

jj llyzDyz

1

1The restricted model

11

1

lll

lll

jj

jjPeriodic

quasi-periodicRandom uniformlydistributed ]-.5,.5[

Restricted Restricted (pinned)(pinned) noise noise

223

22

21

*321

2

3213,2,12

0 2

)(exp2

zlDzdidi z

0det dd Model A

0

2

0

4

,d

bzz

D

bzl NL

Weak nonlinearityWeak nonlinearity

)'('exp

00

zdziL

dz zL

at L>>l is self-averaged !!!

noise average

The nonlinear kernel does not decay (with z) !!!

44

exp2

)(exp

2

2

22DlErfi

Dl

DlzlDz

lz

noise and pinning period average

The averaged equationThe averaged equationdoes have a steadydoes have a steady (soliton like) (soliton like)solutionsolution in the in the restrictedrestricted case case

Restricted (pinned) noise. DM case.

The averaged equationThe averaged equationdoes have a steady solutiondoes have a steady solution

0

2

0

4

,,,d

bzzz

D

bzl NLDM

Weak nonlinearityWeak nonlinearity

*

321

2)('

3213,2,12

0

2

)(exp

2

0

0det

l

zzlDe

didi

z

dzddzi

z

223

22

21

Numerical Simulations

Fourier split-step scheme Fourier modes

132

01.0

180,180

stepz

t

Model A Model B

1

0

cosh|);(|

1

tto

d

6.2/

0

2

79.0|);(|

5

15.0

t

DM

eto

d

d

10,5,1

1010

5.2,1.054

l

N

D

MoralMoral

Practical recommendations for improving fiber system performancethat is limited by randomness in chromatic dispersion.

The limitation originates from the accumulation of the integraldispersion. The distance between naturally occurring nearest zerosgrows with fiber length. This growth causes pulse degradation.

We have shown that the signal can be stabilized by periodic orquasi-periodic pinning of the accumulated dispersion.

M.Chertkov,I. Gabitov,J.MoeserUS patent+PNAS 98, 14211 (2001)M.C.,I.G., P. Lushnikov, Z.Toroczkai, JOSAB (2002)

M.Chertkov,I. Gabitov,I.Kolokolov, V.Lebedev,JETP Lett. 10/01MC, Y. Chung, A. Dyachenko, IG,IK,VL PRE Feb 2003

Shedding and Interaction of solitons in imperfect medium

1

)'()'()(

)(1)(

0222

D

zzDzz

zzd

di tz

Questions:Questions: *What statistics does describe the radiation emitted due to disorderby 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

Single Soliton StorySingle Soliton Story

v

yt

ytiidzi

z

cosh

expexp

0

2

4/115/321)( Dzz

“Asymptotic freedom”:soliton is distinguishable fromthe radiation at any z

Self-averagingSelf-averaging

222 vdt

1&,2

exp256

15

1&/,/ln128

15

1&/,/ln512

15

,,2

exp

,ln16

4

3

4/13

4/134/74/3

132

3

12

2

2

zDtzz

t

z

t

zDztDztzz

zDDzttzDz

Dtzz

t

z

Dt

Dztt

zD

v

Radiation tailRadiation tail + + forerunnerforerunner

Interaction of Shedding SolitonsInteraction of Shedding Solitons

m m

mmm vyt

ytiiiz

cosh

)(expexp

2*221

2

2*222

111

1

)(cosh

cosh

tanhRe

cosh

tanhRe

)(||4cosh

tanh

2

)(

vvx

xdxiP

vx

xdxF

vx

xdxF

vvvx

xdxF

Py

FFFzF

z

xv

vv

z

vvvz

1y 2y

14.0

)'()'()(

02

G

zzGDzFzF

F

32221

3221

1

3

)2cos1(83

4)(

0)(

GzDyy

zGDzy

zy

Soliton position shift isSoliton position shift isGaussian zero mean Gaussian zero mean random variablerandom variable

Soliton phase mismatchSoliton phase mismatch

Multi-soliton case

r

i

zDNz

zNDy

deg3/23/1

int

322

~

~

Infinite patternInfinite pattern (continuous flow of information)(continuous flow of information)

22/12/3 ~~

~

DzNDzy

zN

The z-dependence is similar The z-dependence is similar to the one described by to the one described by Elgin-Gordon-Haus jitterElgin-Gordon-Haus jitter

Pinning

jj llyzDyz

1

1 yzDl

yz ''12

2~~

8/1210 315/21)(

zDlz

03.21

)'()'()(

~

~22

1

~2

G

zGDy

zzGDzFzF

4/115/321)( Dzz

14.03

4

)'()'()(

3221

2

G

zGDy

zzGDzFzF

Bare caseBare case Pinned casePinned case

Single soliton decaySingle soliton decay

Two-soliton interactionTwo-soliton interaction

Statistical Physics of Fiber Communications

We planned to addressed: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 WDMMulti mode fibers• noise induced enhancement of the information flow• ...