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Laser Research Institute

University of StellenboschWWW.LASER-RESEARCH.CO.ZA

Simulation of Nonlinear Effects in Optical Fibres

F.H. Mountfort, Dr J.P. Burger, Dr J-N. Maran

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Outline• Introduction• The nonlinear Schrödinger equation• Terms of the nonlinear Schrödinger equation• Numerical method of simulation• Results

– Group Velocity Dispersion (GVD)– Self-Phase Modulation (SPM)– Combined GVD & SPM

• The Ginzburg-Landau equation• Conclusion

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Introduction

• Potentially disruptive nonlinear behaviour• Accurately simulate all nonlinear effects• Design for high energy, ultrashort pulse fibre

amplifiers• Simulation involves numerically solving the

nonlinear Schrödinger equation by means of the finite difference method

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The Nonlinear Schrödinger Equation (NLSE)

• Use Maxwell’s equations to obtain:

• But

• Hence

• Use slowly varying envelope approximation– Factor out rapidly varying time dependence

2 2

02 2 2

1

c t t

E PE

NLL PPP

2

2

02

2

02

2

22 1

tttcNLL

PPE

E

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• Wave equation for slowly varying amplitude:– Work in Fourier domain → Factor out rapidly

varying spatial dependence– Ansatz:

• where– F(x,y) = modal distribution– = slowly varying function– = spatial dependence

• Use retarded time:

ziezyxFrE 000 ,~,,

~

0,~ z

gT t z v

zie 0

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• Finally NLSE:

• In summary:– Maxwell equations NLSE:

• with a description of the the field in terms of a slowly varying amplitude

2

2

2

22

1

2

T

i

zi

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Terms of the Equation

• Absorption : ; α = absorption coefficient

• Group velocity dispersion: ; β2 = GVD parameter • Self-phase modulation:

Where

area = effective area of the core n2 = nonlinear index coefficient k = wave number

2

i

2

2 2

1

2 T

2||

area

kn2

22

2 2

1

2 2

ii

z T

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Numerical Method of Simulation

• Propagation of a pulse in time with propagation distance

• Moving time frame traveling with pulse– Enables pulse to stay within computational

window• Finite difference method employed• Difference equation used to approximate a

derivative

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FIBRE

TIME TIME

Illustration of Task

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Group Velocity Dispersion (GVD)• Neglecting SPM and absorption:

• Different frequency components of the pulse travel at different speeds

• Two different dispersion regimes:

– Normal dispersion regime: β2 > 0

– Anomalous dispersion regime: β2 < 0

– Normal regime: red travels faster blue– Anomalous regime: blue travels faster

2

2 2

1

2i

z T

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Illustration Of Traveling Frequency Components

Propagation

Initial pulse Final pulse

•Time delay in the arrival of different frequency components is called a chirp

TIME TIME

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GVD Cont.• For significant GVD: with

• Initial unchirped, Gaussian pulse:

• Amplitude at any distance z:

dL L

2

20

0, exp2

TT

T

20

1 22 0 22

0 2

, exp2

T Tz T

T i zT i z

20

2d

TL

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1.5 2 2.5 3 3.5

x 10-11

0

0.2

0.4

0.6

0.8

Dispersion of the pulse with propagation distance

Time Span [s]

Nor

mal

ised

Inte

nsity

Pulse @ 0Ld

Pulse @ 2Ld

Pulse @ 4Ld

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-5 0 5 10

x 1011

0

0.5

1

1.5

2

2.5

x 10-12 Numerical spectra of the pulse

Frequency [Hz]

Spe

ctra

l Den

sity

Spectrum @ 0Ld

Spectrum @ 2Ld

Spectrum @ 4Ld

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Self-Phase Modulation (SPM)• Neglecting GVD:

• For SPM: with

• Amplitude at z:

• Where

• New frequency components continuously generated

• Spectral broadening occurs• Pulse does not change

2

2

ii

z

nlL L

, 0, exp ,NLz T T i z T

2, 0, eff

NLNL

zz T T L

1

0nlL P

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2 2.2 2.4 2.6 2.8 3

x 10-11

0

0.2

0.4

0.6

0.8

1

Time [s]

Nor

mal

ised

Pow

erNumerical Powers of the Pulse

Pulse @ 0Lnl

Pulse @ 2Lnl

Pulse @ 4Lnl

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-5 0 5

x 1012

0

0.5

1

1.5

2

2.5

x 10-12

Frequency [Hz]

Spe

ctra

l Den

sity

Numerical Spectra

Spectrum @ 0Lnl

Spectrum @ 2Lnl

Spectrum @ 4Lnl

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SPM Cont.• The following should hold:

– Max Phase shift ≈ (No. of peaks – 1/2)π– Govind P. Agrawal. Nonlinear Fibre Optics. Academic Press, 2nd edition.

-5 0 5

x 1012

0

0.5

1

1.5

2

2.5

x 10-12

Frequency [Hz]

Spectr

al D

ensity

Numerical Spectra

Spectrum @ 0Lnl

Spectrum @ 2Lnl

Spectrum @ 4Lnl

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Combined GVD and SPM

• For both GVD and SPM:

• Normal dispersion regime:– Pulse broadens more rapidly than normal– Spectral broadening less prevalent

• Anomalous dispersion regime:– Pulse broadens less rapidly than normal– Spectrum narrows

d nlL L

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FWHM with propagation distance for different dispersion regimes

0 200 400 600 800 1000 12000

500

1000

1500

2000

2500

Distance [m]

FW

HM

[s]

Normal regimeAnomalous regimeNormal regime no SPM

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Spectra in Normal Regime Spectra in Anomalous Regime

Frequency [Hz]

-5 0 5

x 1011

0

1

2

3

4

5

6

x 10-24

Inte

nsity

-2 0 2 4

x 1011

0

2

4

6

8

x 10-24

@ 0Ld

@ 1Ld

@ 2Ld

@ 3Ld

@ 4Ld

@ 5Ld

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The Ginzburg-Landau Equation (GLE)

• This takes dopant into account

• Results do not agree exactly with published results of Agrawal– Govind P. Agrawal. “Optical Pulse Propagation in Doped Fibre

Amplifiers”. Physical Review A, 44(11):7493-7501, December 1991.

2 220 2 0

2 20 0

1

2 2

ig T gii

z n T n

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My results

Agrawal’s results

-50

5

x 10-12

0

0

50

100

150

Time Span [s]

No

rma

lise

d P

ow

er

Ld

2Ld

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Conclusion

• NLSE resuts in good agreement with previously published results

• Discrepancy exits with published results for GLE

• Future work:– Use C– Improve time and spacial resolutions– Collaborate with ENNSAT, France

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Many THANKSTo

Dr J.P. Burger&

Dr J-N. Maran&