Simulation of Nonlinear Effects in Optical Fibres 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

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

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

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

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:

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

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

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

Illustration of Task FIBRE TIME TIME

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

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

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

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

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.

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

FWHM with propagation distance for different dispersion regimes

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

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.

My results Agrawal’s results

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

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