1. Simulation of Nonlinear Effects in Optical Fibres
Laser Research Institute
University of Stellenbosch
Simulation of Nonlinear Effects in Optical Fibres
F.H. Mountfort, Dr J.P. Burger, Dr J-N. Maran

2. 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

3. 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

4. The Nonlinear Schrödinger Equation (NLSE)
Use Maxwell’s equations to obtain:
But
Hence
Use slowly varying envelope approximation
Factor out rapidly varying time dependence

5. 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:

6. Finally NLSE: In summary: Maxwell equations NLSE:
with a description of the the field in terms of a slowly varying amplitude

7. 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

8. 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

9. Illustration of Task
FIBRE
TIME
TIME

10. 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

11. Illustration Of Traveling Frequency Components
Initial pulse
Final pulse
Propagation
TIME
TIME
Time delay in the arrival of different frequency components is called a chirp

12. GVD Cont. For significant GVD: with Initial unchirped, Gaussian pulse:
Amplitude at any distance z:

15. 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

18. 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.

19. 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

20. FWHM with propagation distance for different dispersion regimes

21. Spectra in Normal Regime Spectra in Anomalous Regime
Frequency [Hz]

22. 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): , December 1991.

23. My results
Agrawal’s results

24. 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

25. Many THANKS To
Dr J.P. Burger
&
Dr J-N. Maran