Pulse confinement Pulse confinement in optical fibersin optical fibers

with random dispersionwith random dispersion

Misha Chertkov (LANL) Ildar Gabitov (LANL) Jamey Moser (Brown U.)

http:/cnls.lanl.gov/~chertkov/Fiber/

Los Alamos, 03/13/01

Thanks toI. KolokolovV. LebedevP. LushnikovZ. Torozkai

Fiber Electrodynamics, Mono-mode, NLS,

Dispersion vs Kerr nonlinearity,Dispersion management

Variations of fiber parameters, Noise in NLS,

Statistics of pulse propagation:objects, questions

Random dispersion, pinning,Random dispersion, pinning,pulse confinement.pulse confinement.

Theory and numerics.Theory and numerics.

Polarization Mode Dispersion,etc

Nonlinear Schrodinger EquationModel A

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

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

Questions: Does an initially localized pulse survive propagation? Are probability distribution functions of various pulse parameters getting steady?

Answers

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

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

Strong nonlinearityStrong nonlinearity

)'('exp00

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

Strong nonlinearityStrong nonlinearity

*

321

2)('

3213,2,12

0

2

)(exp

2

0

0det

l

zzlDe

didi

z

dzddzi

z

223

22

21

Numerical Computations

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.

Polarization Mode Dispersion (PMD)Polarization Mode Dispersion (PMD)

02)(ˆ 2

ttz dzmii

random birefringence

)'(ˆ'exp)(ˆ

0

zmdziTzWz

biravernoise z

zzW 3exp)(.

Decay of the effective nonlinearity== pulse degradation

MC,IG, I. Kolokolov, T. Schaefer

Pinning ?! of )'(ˆ'

0

zmdzz

birlzpinned

zzzW exp)(

.

Nonlinear PMD cannot be suppressed completely but an essential reduction of pulse broadening can be achieved

Statistical Physics of Fiber Communications

Problems to be addressed:Problems to be addressed:

Single pulse dynamics•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?Multi mode fibers• noise induced enhancement of the information flow• ...

Fibrulence == Fiber Turbulence