Statistical physicsStatistical physics approach approach to evaluation of to evaluation of outage probabilityoutage probability

in in optical communicationsoptical communications

Misha Chertkov (Theoretical Division, LANL)

In collaboration withIn 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 effectJoint effect of of noisenoise and and birefringent birefringent disorderdisorder

Bit-error-rate.Bit-error-rate. Does it fluctuate?Does it fluctuate?How to How to evaluateevaluate/calculate BER (<<1) ? /calculate BER (<<1) ?

Practical consequences for optical communications

Theoretical intereste.g. analogy with spin-glasses

Fiber Electrodynamics

);(2)(22 ztzdi tz

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

Linear Linear vsvs NonlinearNonlinear

);(2)(22 ztzdi tz

)(zd

z

Dispersion ManagementDispersion Management

);(222 ztdi tz

0d Soliton solution

dba

bt

bizdazt

22

2

,/cosh

exp)0,( Dispersion balances

nonlinearity

Integrability (Zakharov & Shabat ‘72)

2

2 1~a

zdbz nld

Information CodingInformation Coding

Return-to-Zero (RZ)Return-to-Zero (RZ) Non-Return-to-Zero (NRZ)Non-Return-to-Zero (NRZ) Differential Phase CodingDifferential Phase Coding

RZRZ 1 1 0 1 01 1 0 1 0

Polarization

tizikj

tizikjj eeE 00 *

21

2,1

0

ee

e

Ej

j

z

22

21

*22

21

**

4

2

||||3

4

3

26/

),()()()(

EKerr

etcKerrnonlinearzdzimzi ttz

2

1Ψ complextwo-component

)(ˆ

)(ˆ

zm

z firstsecond

order BIREFRINGENCE matrixes(2*2, traceless, self-adjoint)

Additive (amplifier) noiseAdditive (amplifier) noise );(... zti z

)()();();( 21212211 zzttDztzt

It causes: (1) pulse jitter (walk away from the slot) (2) pulse degradation

Linear:jitter and amplitude degradationare equally important

Soliton:jitter essentially more importantthan amplitude degradation Elgin (1985) Gordon-Haus

0IZD

for successful fiber performance

short correlated ! i.e. different for different pulses

3

Disorder in Birefringence

rhszimzi tz

)()(

rhszVrhs

zVzmzVzm

ztzVzt

)(ˆ

)(ˆ)(ˆ)(ˆ)(ˆ

);()(ˆ);(1

ΨΨ

z

z

zdzizV

VzVzizV

0

)'(ˆ'exp)(ˆ

1̂)0(ˆ),(ˆ)(ˆ)(ˆ

ordered exponential

Getting rid of fast polarization axis rotation

PMD),,(

ˆ)()(ˆ)3()2()1( hhhh

zhzm

')'()( )()( zzDzhzh ijm

ji

Pauli matrixes2bZDm

weakisotropic

Disorder

Polarization Mode Dispersion (PMD)

z

tz

zmdzizW

zWz

rhszimi

0

)'(ˆ'exp)(ˆ

)0()(ˆ)(

)(

ΨΨ

Linear

0 pulse splittingbroadening jitter

Poole, Wagner ‘86Poole ’90;’91

Statistics of PMD vector is Gaussian. zD

P

m2

exp4

|)|(2

2

3

3

Differential group delay (DGD)

ˆ)(ˆ)(ˆ);(ˆ 1

izWzWzJ Polarization (PMD) vector(of first order)

1 1 0 1 01 1 0 1 0

1 1 1 1 11 0 0 0 0

input

output

Eye diagram

``intensity”

PD

F

Bit-Error-Rate

2) Build histogram (PDF) of pulse Intensity collecting statistics over many slots (separately for initially empty and filled slots)

)(P

)(P

1

0

I

I

1)0(,0)0(

);()()(

10

2

II

ZttdtGZI ΨK

1) Measure intensity in each slot !

Electrical filter+samplingwindow function

Linear operator for(a) Optical filter(b) ``Compensation” tricks

3) BER

)(P

)(P

1

0

01

0

1

10

IdIB

IdIB

dec

dec

I

I

10 dec

Idecision level

Filters and ``tricks”

2

);()()( ZttdtGZI ΨK

Electrical filter+sampling window function

|)|()( tTtG

)'('

)(0

'tte

dttK

tt

opt

Optical filter

)()( 0tttKclock ``Setting the clock”

Z

tzmdzK0

1 )'(ˆ'expFirst order PMD compensation

Noise and disorder. Order of averaging.

ˆ)()(ˆ

')'()(

)()();();(

);()()(

)()(

21212211

2

zhzm

zzDzhzh

zzttDztzt

ztzdzimi

ijmdisorder

ji

noise

ttz

Calculate BER for given realization of disorder (averaging over noise)

)}({ zhB Does BER (as a functional

of disorder) fluctuate ?

Linear model

BER10log

C. Xie, H. Sunnerud,M. Karlsson, P.Andrekson,``Polarization-Mode Dispersion-Induced in soliton Transmission systems”,IEEE Photonics Techn. Lett.Vol.13,Oct. 2001.

Monte-Carlo numericswith 10 000 fiber realizations(artificial rescaling of decisionlevel)

)';()(ˆ)(ˆ');(

)()(ˆ);(

1

0

0

ztzWzWdzzt

tzWztz

Ψ

Ψ

z

t

z

t zmdzzddzizW00

2 )'(ˆ'exp)'('exp)(ˆ

2121 ),(),( ttzDtZtZ

Noise average

00

2

0

01

)}({exp

|)()}({

IZD

zhB

KKtdtGIdIzhB

saddle

noise

Idec

ZDIB

00 ~/1ln

10

IZD

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

)}({ zh

)(zh

)'('

)(0

'tte

dttK

tt

opt

Optical filter always applies

)'('

ln

0

00

zhdzH

IBZDZ

Bare case

)( 231 HObH

``Setting the clock” (no chirp)

)( 3222

212 HObHH

First order PMD compensation

)()]''()'(

)''()'(['''

3212

'

0

21

0

'2

HObzhzh

zhzhdzdzzZ

Bare case

B

dB

B

B

ID

ZbD

dBBS

m

0

2210

22

ln2

exp

)(

||2

]2[1*

)(

'2

2

0

2

10

mDI

bD

B

dBBdBBS

3/2

03/2

3

23/20 ln2.4ln

B

B

ZD

bIZDS

m

PDF of Bit-Error-Rate

Saddle-point(optimal fluctuation) calculations

2bZDm

Setting the clock

First order compensation (nonzero chirp)

First order compensation (zero chirp)

Grossly underestimatedGaussian expectation

Long (algebraic!) tail

0ln B

B

)(ln0BS

S

BER10log

C. Xie, H. Sunnerud,M. Karlsson, P.Andrekson,``Polarization-Mode Dispersion-Induced in soliton Transmission systems”,IEEE Photonics Techn. Lett.Vol.13,Oct. 2001.

Example:

35.0

15.0

12.0

06.0

06.0

3

'2

2

1

0

013.0

25

/2.012

500,2

460

1010

2

0

10*

120

bZD

psb

kmpsDk

kmZ

ZDI

BB

m

m

1

*

)(B

dBBPutage

13

4

10*2

10*4

04.0

35.0

Outage

No compensation

Timing jitter

First order with chirp

First order no chirp

Higher-order compensation

main fiber c4 c3 c2 c1

compensating fibers

)( 1 pppp HObH

The idea: to achieve higher (p) compensating degree

ZDIBBZDbOutage mp

p2

00*2 ln~ln

Periodicc4c3c2c11 2 3 4

main fiber c4 c3 c2 c1

compensating fibers

``Standard”

ZDIBBZDbOutage mp

p2

00*2 ln~ln

Quasi-periodicc4 c3 c2 c11 2 3 4

nl

ln tjjn

nl

ln tjjn

NNc

zdzhK

zdzhTU

tUKUKK

)1(1

)1(

01111

ˆ)(exp

ˆ)(expˆ

)(ˆˆ

For Q-periodic --- Need !!!!!anti-stokes refractionmeasurement of birefringence(Hunter,Gisin,Gisin ’99)

0*

0'2

2

ln2

~ln BBID

bDpOutage

m

Q-periodic guarantees much stronger p-dependence of compensation than the ``standard” one

LinearV.Chernyak,MC,I.Kolokolov,V.LebedevPhys.RevE to appear; Optics. Lett. 28, (2003); Optics. Express. 11, 1607 (2003);JETP Lett. 78, 198-201 (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

0

2

0

22

0

2

0

252

1ln

ln~ln~)(ln

B

BB

ZD

yB

BID

yZDBS

m

d

m

dBare case

Soliton jitter (due to noise) is the dominant destructive factor

Analogy withAnalogy with Functional Order Parameter Functional Order Parameter approachapproach

for for glassyglassy states in infinite-range exchange states in infinite-range exchange spin systemsspin systems

Double Double (super)(super) statistics 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

)'('max

max2

qPdqQ J

q

qJ

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