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Laser Research Institute University of Stellenbosch WWW.LASER-RESEARCH.CO.ZA

Simulation of Nonlinear Effects in Optical Fibres F.H. Mountfort , Dr J.P. Burger, Dr J-N. Maran. Laser Research Institute University of Stellenbosch WWW.LASER-RESEARCH.CO.ZA. Outline. Introduction The nonlinear Schr ö dinger equation Terms of the nonlinear Schr ö dinger equation

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Laser Research Institute University of Stellenbosch WWW.LASER-RESEARCH.CO.ZA

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

  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:

  13. 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

  14. 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.

  15. 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

  16. FWHM with propagation distance for different dispersion regimes

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

  18. 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.

  19. My results Agrawal’s results

  20. 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

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

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