eec239a 15 notes 03 nonlinearfibers

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EEC 239A Optical Communication Systems and NetworkingProf. S. J. B. Yoo, UC Davis Copyright©

Chapter II Propagation of Signals in Optical Fiber (cont’d)

Fiber Nonlinearities

(Chapter 2.4 and Section 5.8 and Appendix E.2 of text)

Notes_3_2EEC 239A Optical Communication Systems and NetworkingProf. S. J. B. Yoo, UC Davis Copyright©

Fiber Nonlinearities

• Optical Fibers are Far ‘Better’ than Copper Wires but Not Ideal; in addition to dispersion, there are nonlinearities

• Many Nonlinear Mechanisms

Self Phase Modulation

Cross Phase Modulation

Four-Wave Mixing

Stimulated Raman Scattering

Stimulated Brillouin Scattering

NonlinearIndex Effect

StimulatedProcess

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SELF PHASE MODULATION

• The material index , and the group velocity is Intensity Dependent!!

t'/

t/Propagation

13d)-(5 ~ where ,y Equvilentl 200

0 InnInInc


Pictorial-Self Phase Modulation

Propagation direction

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CROSS-PHASE MODULATION (XPM)

• Essentially the same mechanism (optical Kerr effect) as SPM causes XPM, and SPM is a special case of XPM

1

2

• Presence of 2 pulse affects the propagation or the phase of 1 pulse• Occurs only when they overlap in time


Pictorial-Cross Phase Modulation

s

due to reduced interaction length

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Pictorial-Stimulated Processes (SBS)

• Lightwave (photon) and Acousticwave (phonon) interaction causes SBS, SRS, etc.

optical wave in

acoustic wave

optical wave trans

SBS (doppler shifted)


Pictorial-Stimulated Processes (SRS)

• Lightwave (photon) and Acousticwave (phonon) interaction causes SBS, SRS, etc.

optical wave in

acoustic wave

optical wave transp p

s

s

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Fiber Nonlinearities-2

• All Nonlinear Optical Mechanisms, by definition are ‘optical power dependent’. Typically, they are due to ‘third order nonlinearity’ whose magnitudes are often depicted by (3) and the nonlinear polarization is PNL ~ 0 (3) E3

• This nonlinear polarization is responsible for the nonlinear optical effects and will scale nonlinearly with Intensity.

• For a given power, Large effective area ( Aeff) fiber provides lower intensity and lower nonlinear optical effects

• Nonlinear optical effects will be dominant before absorption reduces the power level, hence effective length of interacting region Leff~1/bs


Effective Length

Effective Length

P

Leff

P

L L

Equivalent for nonlinear effects

For long lengths /1 L

eff eL

/1L

kmLkmdB eff 20,/22.0

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Effective Area

Effective Area

I

I

r r 2/1/effA

Equivalent for nonlinear effects

,

,2

2

rIrdrd

rIrdrdAeff

Typical value 250 m


Linear and Nonlinear Optics-1

Polarization in linear world was:

In the real world(including nonlinearity)

Again,

,~

,~,~ 1

0 rErrPL

,~

,~

,~

,~,~

0 rPrPrErrP NLLT

,~

,~,~ 1

0 rErrPL

,~

,~

,~

:,~,~

,~

:,~,~ 3

02

0 rErErErrErErrPNL

Optical fiber has no Because it is centrosymmetric

,~ 2 r

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• See appendix F:

These are frequency domain responses.in the time domain (for linear):

t

L trEttrtrP ''10 ,,,

t t t

NL dtdtdttrEtrEtrEttttttrtrP 3213213213

0 ,,,,,,,



• For isotropic and time independent

trEtrPNL ,, 330

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SELF PHASE MODULATION-1

• The material index , and the group velocity is Intensity Dependent!!

• n= n0 + n2 I

t/

Propagation

t'/

low-high-low index

SPM induces chirp! (effectively negative chirp parameter or >0)



• Instead of linear case

‘Optical Kerr Effect’ gives

anc

12500

or

bc

1251 100

aEC

1354

31 2310

0 or

bEnC

1354

31 23

20

0 or approximately

cEn

nC

1358

3 2300

Equivalently, 13d-5 ~ , 20

0 InnInwhereInC o

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• Nonlinear Length is defined as:

• Consider initially unchirped Gaussian pulse with envelope:

P0 is the peak power of the pulse

• If the link length is comparable to or greater than the LNL then the pulse acquires a distance-dependent chirp.

2

2

,0

eU

022 Pn

AL e

NL



• Read through Appendix E.2, and use equation E.18 which shows that the SPM induced phase change is:

• Then the instantaneous frequency and the chirp factor are:

2

e

L

L

NL

2

0

212

2

2

2

eL

L

eL

L

NLspm

NL

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• For L > Le , replace L with Le

• For = 0.22 dB/km, Le =20 km


1

2

CROSS-PHASE MODULATION (XPM)-1

• Essentially the same mechanism causes XPM, and SPM is a special case of XPM

• Presence of 2 pulse affects the propagation or the phase of 1 pulse

t/

t/

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CROSS-PHASE MODULATION (XPM)-2

• XPM Occurs only when pulses overlap in time– Propagation constant seen by 1 pulse in presence of 2 pulse is

cEEEn

nc

bEEEn

nc

aEEEc

14528

3

elyapproximator ,14524

31

or ,14524

31

212

130

0

212

10

0

212

1310

0


NONLINEAR WAVE EQUATIONSPM induced Chirp-1

• The text book Appendix covers nonliear Schrodinger Equation which we will skip here and use the result

• Nonlinear Length defines as the following compares the length of fiber to the possible nonlinear impairments

• SPM induces chirp, and the pulse width after propagating a distance L is given as

pulse theofpower peak a is P and 8

3 where

1552

023

0

En

n

Pn

AL eff

NL

17-5 33

4121

2

2

2

2

0 DNL

eff

DNL

effL

L

L

L

L

L

L

L

L

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NONLINEAR WAVE EQUATIONSPM induced Chirp-2

Which includes Dispersion (LD), Loss (Leff), and SPM(LNL)induced Chirp

see Fig 5.31 of text

1mW

10 mW

100 mWL1

0t

tL


FOUR WAVE MIXING-1

• INTERMIXING GENERATES NEW FREQUENCY TERMS

before

after

• In general, it is a mixing between three waves generating a fourth wave at a new frequency

ztEE

EP

iii

NL

cos where

23

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FOUR WAVE MIXING-2

• The power Pijk of the mixing term is related through

where Pi, Pj, Pk, are power level of each component and dijk is a degeneracy factor. ijk is a phasematching term given as

17-5 8

2

23

effkjieffeff

ijkijkijkijk LPPP

cnA

dP

19-5

where18-5exp1

2/sinexp41

i

2

22

22

2

ijkkj

abs

abs

abs

absijk

L

LL


FOUR WAVE MIXING-3

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Stimulated Brillouin ScatteringStimulated Raman Scattering

Simulated scattering gain

Wcm

Wcm

ePP

raman

Brillouin

APLinout

/105.3

/10212

9

/


Stimulated Brillouin Scattering-1

Stimulated Brillouin Scattering

•Lowest threshold nonlinearly-Without amplifiers-With N amplifiers

mWPth 12NPP th

Nth /1

But

•Backward interaction

-Isolators can eliminate the effect-Bidirectional transmission impossible

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From reference

Brillouin

laserL

effth v

v

e

AP

1

21

modulation external 1Gb/s with

case canonicalour for 12mW

area) core(smaller sholdslower thre have fibers shifted Dispersion

55.1@20

mMHzvBrillouin


Stimulated Brillouin Scattering-3• From textbook (similar definition): SBS threshold becomes

• gB : Brillouin Gain coefficient, ~ 4 E –11 m/W

• b: Normalized propagation constant of the wavegude (between 0 and 1)

• f: Brillouin bandwidth ~ 20 MHz• Pth= 1.3 mW for b=1, Aeff=50 m2, Leff= 20 km, fsource <<20

Mhz

• Pth= 14.4 mW for b=1, Aeff=50 m2, Leff= 20 km fsource =200 Mhz

B

source

effB

effth f

f

Lg

bAP 1

21

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Stimulated Raman Scattering-1



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Opticalfrequency

Opticalfrequency

See 5.8.3 of text



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Stimulated Brillouin ScatteringStimulated Raman Scattering

• SBS -- Backward Light Propagation if narrow linewidth

• SRS -- Power transfer from short to long wavelength

• Solitons– Nonlinearity balances Dispersion to achieve eigen mode at a given power level– Since Kerr coefficient (n2) is positive in most materials including fibers Solitons can be seen

in positive dispersion fiber– analogy to Dark solitons and Spatial Solitons

Dispersion

Nonlinearity

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Solitons

• Solitons– Nonlinear Schrodinger Equation– Nonlinearity balances Dispersion to achieve eigen

mode at a given power level– Since Kerr coefficient (n2) is positive in most

materials including fibers Solitons can be seen in positive dispersion fiber

– analogy to Dark solitons and Spatial Solitons

Dispersion

Nonlinearity

eec239a 15 notes 03 nonlinearfibers

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