Ph ddefence

University of ZagrebFaculty of Electrical Engineering and Computing

Analysis of Signal Propagation in Optical Fiber Based on

Finite - Difference Method

Sonja Zentner Pilinsky

Doctoral Thesis, Zagreb, 2003

22

Contents:

1. Introduction

2. Pulse propagation in optical fiber

3. Numerical model and its accuracy

4. Selected simulation results

5. Conclusions

33

1. Introduction

Motivation and goal

What we model

How we model

Additional devices needed

44

Motivation

- need for accurate program with all effects included:

new sophisticated optical links

upgrade of existing fiber links

- expensive experiments

why to model optical link

- linear fiber communications at the edge (bit rates, capacity)- optical transmission very sensitive to:

dispersion (cromatic, polarization mode)lossnonlinear effectsnoise

55

Goal - to model modern optical links

dispersion map

66

What we model

Nonlinear Schrödinger equation (NLSE)

0

2

2 2

d

d

nonlinear coeff.describing SPM

fiber loss in dB/km

time

1

1

gv

distance along fiber

optical pulsecomplex envelope

2 0

eff

n

cA

77

How we model

- FDM Cranck - Nicholson

- pseudospectral SSFM

- testing accuracy on canonical problems

- comparison with OptiSystem 2.0.

Models for additional devices

- EDFA model (G, ASE noise)

- Optical filter model (transfer function)

88

2. Pulse propagation in optical fiber

Propagation equation

Fiber loss

Group velocity dispersion

Self - phase modulation

Polarization dispersion

99

Maxwell equations

0f

t

t

f

BE

DH J

D

B

0

0

D E P

B H M

Optical fiber:-no sources Jf ,f = 0 -nonmag.mat M = 0

10, ,L t t t t dt

P r E r

30 1 2 3 1 2 3 1 2 3, , , , , ,NL ijklt t t t t t t t t t dt dt dt

P r E r E r E r

1010

Assumption and approximation:

2/ E E E

2 2 E E E E

WGA (ncore – ncladding)/ ncore << 1

- the EM field maintains its polarization along the fiber

- Weakly guiding approximation

1111

- PNL is treated as a small perturbation to PL

- nonlinear effects: Kerr and Raman (neglected for T0 > 1ps)

instantaneousresponse

3 31 2 3 1 2 3, ,ijkl t t t t t t t t t t t t

- SVEA - slowly varying envelope approximation

- envelope is slowly varying in z and t - removes backscattered part of the envelope

0 , 1 L NLkc

230

3, , , ,

4NL NL NLP t E t E t r r r

1212

22

32 0 0

4

,2 3

, 8

,

I

xxxx effeff eff

F x y dxdyn k Z

AA n cA

F x y dxdy

Propagation equation for pulse complex envelope:

SVEA assumption: 00 0, , , e j zE F x y A z r

HE11 mode

0

0

,

,, a

J a

F x y aJ a e a

1313

- [dB/km]=-10log-[1/km]

- absorption (intrinsic and extrinsic)- scattering - linear: Rayleigh and Mie

- nonlinear: Raman and Brillouin

Fiber loss:

1414

- caused by material and waveguide dispersion- mathematically described by

2 3

0 1 0 2 0 3 0

1 1..

2 3!.

n

c

1

1 ps

kmg

g

n

c v

0 0

2 23 20

2 2 2 2

ps

2

d n d n

c d c d km

0 0

2 3 3

3 02 3

1 ps3

km

d n d n

c d d

20

2D

TL

30

3D

TL

Group velocity dispersion:

1515

2

2

2,2

I En E n n

Z Kerr effect

SPMXPMFWM

2 0 2 0 1

Wkm

I I

eff eff

n k n

A cA

SPM without dispersion:

2

00

,

1,

z

NL

NL

U je U U

z L

AU L

PP

Self Phase Modulation:

0 2

2 2 In z n I t z

,

2

, 0,

1, 0, ,

NLi z T

zeff

NL effNL

U z T U T e

z ez T U T z

L

1616

Polarization mode dispersion

- caused by circular asymmetries in the fiber

- locally birefringence

2x y

x y

nc cn n

1 1

2B

x y

Ln

- measure of pulse splitting in biref. fiber - DGD

g

L d n d nL L

v d c c d

1717

222

2 2

22 2

2 2

2

2 2 2 3

2

2 2 2 3

x x xx x y x

y y yy y x y

A A AjA j A A A

z t t

A A AjA j A A A

z t t

PMD (cont.)

- alternative method for linear optical element

,

out in

a b

b a

J A J A

0

00

2 20 0 0

1

x

y

jxx

jy y

x y

a eE

EE a e

E E E

J

DGD 2 22 a b a’() [a(+) – a()] /

- globally - birefringence combined with random polarization mode coupling:

1818

3. Numerical model and its accuracy

FDM or SSFM ?

Accuracy check and comparison with

OptiSystem 2.0

EDFA model and filter model

1919

- FDMs: Crank - Nicholson scheme

- pseudospectral method: SSFM

Nonlinear PDE modeling:

Criterion for selected FDM model:

- accuracy- stability

2020

- solving numerical scheme to prescribed initial values and boundary conditions

- errors: modeling, truncation, round-off

FDM steps

- dividing solution region into a grid of nodes

- PDE finite difference equivalent (numerical stability!!)

2121

D e r i v a t i v e F i n i t e d i f f e r e n c e a p p r o x i m a t i o n T y p e E r r o r

1i i

t

F D O ( t )

1i i

t

B D O ( t )

1 1

2i i

t

C D O ( t 2 )

2 14 3

2i i i

t

F D O ( t 2 )

1 23 4

2i i i

t

B D O ( t 2 )

t

2 1 1 28 8

1 2i i i i

t

C D O ( t 4 )

2 1

2

2i i i

t

F D O ( t 2 )

1 2

2

2i i i

t

B D O ( t 2 )

2 1

2

2i i i

t

C D O ( t 2 )

t t

2 1 1 2

2

1 6 3 0 1 6i i i i i

t

C D O ( t 4 )

Accuracy

2222

2

2.

A Aconst

z t

First order (Euler)

1 1

1

2

2. i i

n n nn nii i const

z t

- one step, explicit, unstable

11 1 11 1

1 1 11

2 2

2 2.

2i i i i

n n n n n nn ni ii i const

z t t

Crank-Nicholson

- one step, implicit, accurate (1 in z, 2 in t), uncond. stable

1 1

1 1

2

2.

2i i

n n nn nii i const

z t

Leapfrog

- two step, explicit, accurate (2 in z, 2 in t), always unstable

Dufort-Frankel

1 1

1 11 1

2.

2i i

n n n nn ni ii i const

z t

- two step, explicit, accurate (2 in z, 2 in t), uncond. stable

Various FDM schemes for eq.

2323

Accuracy

1. comparison with analytic solutions for simple problems2. Comparison with simulations obtained by OptiSystem 2.0

1

1 1

NMNM exi i

i

NM NMNM NM ex exi i i i

i i

AKC a jb

1

1 NMAXex ZMAXi i

i

ERNMAX

2

1

1 NMex NMi i

i

SERNM

Mean error Mean square error

Correlation coefficient

2 2

arg

AKC a b

bAKC arctg

a

Measure of accuracy:

2424

M EAN TIM E ER R O R = 2.718702574396359E-005 SQ U AR E M TE = 3.188251689591302E-009 AU TO C O R R ELATIO N = 0.999997613973628 - j 1 .334916945068728E-005| AKC | = 0.999997614062729 arg(AKC ) = -7.648528944430967E-004

Gaussian pulse

2

202

0 0,T

TA T A e

2

0

20 220

0 20 2

,

T

T j zTA z T A e

T j z

Analytic solution:

Input pulse:

FDM:ER = 1.77E-004SER = 1.36E-0071-|AKC| = 3.8E-005arg (AKC)= 7E-004

SSFM:ER = 1.88E-004SER = 1.56E-0071-|AKC| = 4.4E-005arg (AKC)= 2.75E-003

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

1 26 51 76 101

126

151

176

201

226

251

276

301

326

351

376

401

426

451

476

501

number of points in time window

pu

lse

po

wer

[W

]

analytic solutionOptiSystemFiberProp

Inputpulse

fiber

A0 = 0.01 W1/2 = 0

T0 = 40 ps D = 16 ps/kmnm

0 = 1550 nm = 0

2525

Hyperbolic secant pulse:2

2

2 22 2

A j AA j A A

z T

22 0 0

0 20

, , , D

D NL

P TLA z TU N

L T LP

222

2 2sgn

2

U j UjN U U

0, sechu N 2, sechj

u e

input pulse analytic solutioninput pulse fiberP0 = 22.6 mW = 0T0 = 2.7 ps

2 = -0.243 ps2/km

0 = 1552 nm = 1.475 W-1km-1

-0.0004

-0.0003

-0.0002

-0.0001

0

0.0001

0.0002

0.0003

0.0004

0.0005

0 100 200 300 400 500


anal

ytic

val

ue

- co

mp

ute

r si

mu

l.

FiberPropOptiSystem

normalization:

2626

-3.00E-03

-2.00E-03

-1.00E-03

0.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

0 100 200 300 400 500 600


an

aly

tic

so

luti

on

- p

rog

r.s

imu

lati

on

OptiSystem

FiberProp

Second order soliton pulse: input pulse

analytic solution

0, sechu N

4

22cosh 3 6cosh

, 2cosh 4 4cosh 2 3cos 4

jj T T e

U T eT T

input pulse fiberP0 = 90.4 mW = 0T0 = 2.7 ps

2 = -0.243 ps2/km

0 = 1552 nm = 1.475 W-1km-1

20

022 2D

Tz L

L = 2z0 = 94.25 km

2727

-8.00E-03

-6.00E-03

-4.00E-03

-2.00E-03

0.00E+00

2.00E-03

4.00E-03

6.00E-03

8.00E-03

0 100 200 300 400 500


an

aly

tic

al s

olu

tio

n -

pro

gr.

sim

ula

tio

n

OptiSystem

FiberProp

Third order soliton pulse: input pulse

analyticsolution

0, sechu N

L = 5z0 = 235.67 km

4

2

12 8 8 16

2cosh 8 32cosh 2 36cosh 4 16cosh 6, 3

cosh 9 9cosh 7 64cosh 3 36cosh

20cosh 4 80cosh 2 5 45 20

36cosh 5 cos 4 20cosh 3 cos 12 90cosh cos 8

jj

j j j j

T T T T eU T e

T T T T

T T e e e e

T T T

N = 3P0 = 203.4 mW

2828

4th order soliton pulse - NO analytic solution:

N = 4P0 = 361.56 mWL = 2z0 = 94.25 km

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

1 33 65 97 129 161 193 225 257 289 321 353 385 417 449 481


pu

lse

po

we

r [W

]

FiberProp

OptiSystem

2929

EDFA model

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

1 26 51 76 101

126

151

176

201

226

251

276

301

326

351

376

401

426

451

476

501


op

tica

l po

wer

[W

]

FiberPropOptiSystem

FIBER: = 0.1 dB/km2 = -0.243 ps2/km = 1.475 1/Wkm

INPUT PULSE:N = 1P0 = 32.54 mWT0 = 2.7 ps0 = 1552 nm

EDFA: = 30 nm 0 = 1552 nm G = 4.98 dB nsp = 1.23

3030

Filter model

2

0

1 21

2f f

B

0

11 cos

2f f

B

2

0

2exp ln 2 f f

B

2

0

1

21 f f

B

Parabolic – shape characteristic

Cosine – shape characteristic

Lorentzian – shape characteristic

Gaussian – shape characteristic

Fabry-Perotfilter !!!

3131

B

B S

freq u e n cy

filte r tran sfe r fu n c tio n

so lito n sp e c tru m

Why are filters used in nonlinear optical links?

compensation of Gordon-Haus effect

filtering at the receivers end

Filter model (cont.)

3232

4. Selected simulation results

FiberProp and its abilities

High bit rates soliton systems

Gordon-Haus effect and its compensation

Dispersion-compensated and

dispersion-managed systems

Polarization dispersion

Dispersion compensated system

3333

Simple FiberProp scheme:

3434

fiber param. EDFA param. input pulse = 0.1 dB/km = 30 nm P0 = 31.48 mW

2 = -0.243 ps2/km G = 2.497 dB T0 = 1.543 ps = 3.28 W-1km-1 nsp = 1.5

0 = 1550 nm

40 Gb/s soliton transmission

PRBS01111010

La = 25 kmL = 500 km

3535

80 Gb/s soliton transmission

fiber param. EDFA param. input pulse = 0 = 30 nm P0 = 38 mW

2 = -0.243 ps2/km G = 2.497 dB T0 = 1.543 ps = 3.28 W-1km-1 nsp = 1.5

0 = 1550 nm

PRBS0011110011100110

La = 25 kmL = 350 km

3636

0

0.005

0.01

0.015

0.02

0.025

0.03

1 20

39

58

77

96

115

13

4

15

3

17

2

19

1

21

0

22

9

24

8

26

7

28

6

30

5

32

4

34

3

36

2

38

1

40

0

41

9

43

8

45

7

47

6

49

5

number of grid points

po

wer

[W

]

fiber param. EDFA param. input pulse = 0.1 dB/km = 30 nm P0 = 14.83 mW

2 = -0.243 ps2/km 0 = 1550 nm T0 = 5 ps

= 1.475 W-1km-1 G = 5 dB N = 1.5La = 50 km nsp = 2.2

0 = 1552 nm

L = 2500 km

tW = 1.5 T0

max 2972.59 kmTL

Gordon-Haus transmission distance limitation:

1 a

aL

LQ

e

22

2 cD

3737

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

1 20

39

58

77

96

115

13

4

15

3

17

2

19

1

21

0

22

9

24

8

26

7

28

6

30

5

32

4

34

3

36

2

38

1

40

0

41

9

43

8

45

7

47

6

49

5


po

wer

[W

]

Inserted Fabry-Perot filterbefore receiver:

2

0

1

21 f f

B

f0 = 193.414 THz (0 = 1550 nm)B = 100 GHz

Gordon-Haus transmission distance limitation (cont.):

3838

0

0.005

0.01

0.015

0.02

0.025

0.03

1 23 45 67 89 111

133

155

177

199

221

243

265

287

309

331

353

375

397

419

441

463

485

507


pow

er [W

]

L = 4000 km

tW = 3 T0 Ltmax = 4718.7 km 2

3max

2

0,13721

FWHM w eff aT

sp

T t A L QL

n n Dh G

Gordon-Haus transmission distance limitation (cont.):

39390

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

1 20

39

58

77

96

115

13

4

15

3

17

2

19

1

21

0

22

9

24

8

26

7

28

6

30

5

32

4

34

3

36

2

38

1

40

0

41

9

43

8

45

7

47

6

49

5


pow

er [W

]

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

1 20

39

58

77

96

115

13

4

15

3

17

2

19

1

21

0

22

9

24

8

26

7

28

6

30

5

32

4

34

3

36

2

38

1

40

0

41

9

43

8

45

7

47

6

49

5


po

wer

[W

]

Fabry - Perot filter after each EDFA:

f0 = 193.414 THzB = 100 GHz

f0 = 193.414 THzB = 360 GHz

EDFA: G = 5.1081 dB

Inserted Fabry-Perot filter before receiver:

4040

Fiber parametersTwo level disp. map Four level disp.map

[dB/km] 0.078 0.15

1 [ps2/km] 0.396 0.1275

2 [ps2/km] -0.294 0.051

3 [ps2/km] - -0.0765

4 [ps2/km] - -0.306

ave [ps2/km] -0.064 -0.051

L1 [km] 30.00 12.5L2 [km] 60.00 12.5L3 [km] - 12.5L4 [km - 12.5 [W-1km-1] 2.65 2.65

Input pulse shapesech sech

T0 [ps] 8.56 2.7N 1.4 1.4q0 2.92 4.63

Amplifier parametersLa [km] 90 50G [dB] 7 7.383

0 [nm] 1550 1550 [nm] 30 30nsp 1 1

DM soliton system:

20 Gb/s

40 Gb/s PRBS01101110

LL

L2L1

4141

Polarization mode dispersion:

0.E+00

5.E-03

1.E-02

2.E-02

2.E-02

3.E-02

3.E-02

4.E-02

4.E-02

1

22

43

64

85

10

6

12

7

14

8

16

9

19

0

21

1

23

2

25

3

27

4

29

5

31

6

33

7

35

8

37

9

40

0

42

1

44

2

46

3

48

4

50

5


po

wer

[W

]

Fiber param.:, , , = 0n = 4.210-5

h = 4.16 mL = 64h = 266.24 m

input pulse: GaussT0 = 3 psP0 = 40 mW0 = 1550 nm

4242

N=0.9

N=1.0

N=1.1

N=1.2-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

0 1 2 3 4 5 6 7

soliton periods z/z0

disl

ocat

ion

of p

ulse

s t/T

0

Fiber parameters: = 0 = -20 ps2/km = 1.475 1/Wkmn = 4.210-5

input pulse:lin. pol. at45 at fiber axesT0 = 5 ps0 = 1550 nm

Polarization mode dispersion:

soliton prop. in birefringent fiber:

20

02

7 7 72 2D

TL z L

43430 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

45

50

Polarization mode dispersion -interaction compensation:

0.00 10.00 20.00 30.00 40.00

tim e [ps]

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

dist

ance

[km

]

L = 85.76 km

Fiber: = 02 = -20 ps2/km = 1.3 1/Wkm

Input pulse:N = 1P0 = 32.54 mWT0 = 2 ps0 = 1552 nmq 0 = 4

L = 51.46 km

4444

Dispersion comp. system:

0.00 200.00 400.00 600.000.00

500.00

1000.00

1500.00

2000.00

2

0

1

20 0,

mT

TA T A e

Super Gauss (m=3)

T0 = 50 psP0 = 0.66 mW0 = 1557.6 nm

FIBER PARAMETERS:L1 = 89.8 km L2 = 16.2 kmL1 = 23(L1 + L2) = 2438 km1 = 0.05231 1/km 2 = 0.13 1/km21 = -22.23 ps/km 22 = 119.48 ps/km1 = 2 = 1.35 1/Wkm

EDFA: = 10 nm0 = 1557.6 nmG = 9.8 dBnsp = 2.78

FABRY-PEROT FILTER:f0 = 192.6 THzB = 370 GHz

4545

Conclusions:

The derivation of optical pulse propagation equation is given in details. All important effects influencing pulse propagation in optical fiber are analyzed: fiber loss, cromatic dispersion, polarization mode dispersion, nonlinear effects (especially self-phase modulation)

Several numerical models are analyzed and the most accurate one chosen for propagation equation modeling. The accuracy is tested on simple canonical problems and later on compared with commercially available software.

EDFA model strictly in time domain is developed, with special attention given to ASE noise model. EDFA model and optical filter model are included in computer program FiberProp

4646

The new approach to Gordon-Haus limitation derivation is given. Timing jitter due to Gordon-Haus effect and its suppression was analyzed with the FiberProp computer program.

Numerous examples of soliton and dispersion-managed soliton transmission systems are analyzed and guidelines for their design are given.

Conclusions (cont.):

Technology

Ph ddefence