Pulse confinement in optical fibers

with random dispersion

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

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

Puebla, 12/06

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, Multimode random coupling,etc

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

Fiber parameters and jargon

200

2

2

0

0

0

2

2

2

Pd

Sn

vzt

t

Pzz

PE

eff

gr

- peak pulse - Kerr nonlinearity - nonl. retractive index - wavelength - core area - group velocity - char. pulse width - second order dispersion2

0

2

0

gr

eff

v

S

n

P

Typical values for Dispersion Shifted Fiber (DSF)

kmP

kmWt

mWtPkm

ps

ps

nm

502

10

4

2

01.7

1550

0

11

0

2

2

0

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 (analytical & numerical)

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

Unrestricted noise)()( det zdzd

Nonlinearity is weak (first diagram)

NLzDbDz //1 42

Nonlinearity dies (as z increases) == Pulse degradation

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

z

dzzi0

')'(exp

223

22

21

*321

2)('

3213,2,12

0 2exp2 0

0det

Dz

edidi

z

dzddzi

z

)()'('exp);(0

02 zdzddztidtz

z

)(z Describes slow evolution of the original field

02)(22 tz zdi

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 (pinned) noise02)(

22 tz zdi

223

22

21

*321

*2)('

3213,2,12

0 2exp2 0

0det

Dz

edidi

z

dzddzi

z

,

,1

1

1*

jj

jll

zlzzunrestricted

restricted

)()( det zdzd

The nonlinear kernel does not decay (with z) in the restricted case !!!

44

exp2

exp2

2

2

*

2DlErfi

Dl

Dlz

D

lz

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

DM case

DM

DM

zz

z

z

zdzddzi

DM

4/sin2

)'('exp0

0det

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.

Mollenauer, et al. Opt. Lett. 25, 704 (2000)Long haul transmission experiments on fibers

constrained from the spans of different types.

The periodic compensation of the overall dispersion

to the fixed residual value (achieved via insertion of an

extra span) optimizes propagation of pulses.

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