Ultrafast Nonlinear Fibre Optics and Supercontinuum Generation

Abstract

This chapter presents a concise overview of nonlinear fibre optics, focusing particularly on the physics of supercontinuum generation in optical fibre for different regimes of pulse duration and for propagation in dispersion varying fibre. Soliton dynamics and their influence are emphasized, and basic numerical code is provided allowing higher-order soliton effects to be studied within a nonlinear Schrodinger equation model.

Appendix

This MATLAB code using the split-step Fourier technique is a very simple implementation to show the correspondence between the description in the text and Fig. 8.1 concerning the basic soliton solutions of the NLSE.

% - - - - - - - - - - - - - - - - - - - - WARNING- - - - - - - - - - - - - - - - - - - - - - - - -

% This code is written in simple dimensional form with no

% effort to be particularly robust numerically.

% Higher order effects are neglected with this implementation

clear,close all

lambda0 = 850e-9;     % input wavelength [m]

c = 299792458;        % speed of light [m/s]

F0 = c/lambda0;

% Time and Frequency Arrays

Npts = 256;

Tmax = 0.5e-12;

dT = 2*Tmax/(Npts-1);

TT = [−Npts/2:(Npts/2)-1]*dT;       % [ps]

FF = [−Npts/2:(Npts/2)-1] ./(2*Tmax);   % [THz]

WW = 2*pi*FF;

% Fibre parameters

beta2 = −1.275e-026;

gamma = 0.10;           %nonlinear length

% Soliton order and other parameters

N = 1;

T0 = 30e-15;             %nonlinear length

P0 = N^2*abs(beta2)/(T0^2*gamma);   %nonlinear length

% Length scales

L_D = T0^2/abs(beta2);  %dispersion length

L_NL = 1/(gamma*P0);    %nonlinear length

L_sol = pi/2*L_D;

% Input pulse

A0 = sqrt(P0)*sech(TT/T0);

% - - - - - - - - - - - - - - - - - - - - - Propagation - - - - - - - - - - - - - - - - - - - - - - -

Nz = 1,000;    % number of steps

Lz = 2*L_sol;   % propagation length

dz = Lz/Nz             % steps length

Nplots = 11;   % number of plots

% Operators

D_op = beta2/2*WW.^2*dz;

N_op = gamma*dz;

sel = round(Nz/(Nplots-1));

pik = linspace(0,Lz,Nplots)$\hbox{quotesingle}$;

Ip = zeros(Nplots,Npts);

Ip_TF = zeros(Nplots,Npts);

Ip(1,:) = abs(A0).^2;

Ip_TF(1,:) = abs(fftshift(ifft(A0))).^2;

disp([$\hbox{quotesingle}$Courbe No. 1 de $\hbox{quotesingle}$ num2str(Nplots)])

A1 = A0;

for ii = 2:Nz

• A_TF = fftshift(ifft(fftshift(A1))).*exp(i*D_op);

  •  A1 = fftshift(fft(fftshift(A_TF)));

  •  A1 = A1.*exp(i*N_op*abs(A1).^2);

•  if (ii/sel) == round(ii/sel)

  •  disp([$\hbox{quotesingle}$Courbe No. $\hbox{quotesingle}$ num2str(1 + ii/sel) $\hbox{quotesingle}$ de $\hbox{quotesingle}$ num2str(Nplots)])

  •  Ip(1 + (ii/sel),:) = abs(A1).^2;

  •  Ip_TF(1 + (ii/sel),:) = abs(A_TF).^2;

•  end

end

% - - - - - - - - - - - - - - - - - - - - - - Display - - - - - - - - - - - - - - - - - - - - - - - - -

figure(1)

a = waterfall(1e-12*FF,pik/L_sol,Ip_TF);

xl = xlabel($\hbox{quotesingle}$Frequency(THz)$\hbox{quotesingle}$),set(xl,$\hbox{quotesingle}$Rotation$\hbox{quotesingle}$,32,$\hbox{quotesingle}$fontsize$\hbox{quotesingle}$,16)

yl = ylabel($\hbox{quotesingle}$Distance z/L_{sol}$\hbox{quotesingle}$),set(yl,$\hbox{quotesingle}$Rotation$\hbox{quotesingle}$,-30,$\hbox{quotesingle}$fontsize$\hbox{quotesingle}$,16)

zl = zlabel($\hbox{quotesingle}$Spectrum (arb.)$\hbox{quotesingle}$,$\hbox{quotesingle}$fontsize$\hbox{quotesingle}$,16)

set(gcf,$\hbox{quotesingle}$colormap$\hbox{quotesingle}$,[0 0 0])

set(gca,$\hbox{quotesingle}$Xgrid$\hbox{quotesingle}$,$\hbox{quotesingle}$off$\hbox{quotesingle}$,$\hbox{quotesingle}$Ygrid$\hbox{quotesingle}$,$\hbox{quotesingle}$off$\hbox{quotesingle}$,$\hbox{quotesingle}$Zgrid$\hbox{quotesingle}$,$\hbox{quotesingle}$off$\hbox{quotesingle}$,$\hbox{quotesingle}$fontsize$\hbox{quotesingle}$,16, …

$\hbox{quotesingle}$Linewidth$\hbox{quotesingle}$,1)

view(135,50)

axis([−150 150 0 2 0 max(max(Ip_TF))])

figure(2)

b = waterfall(1e12*TT,pik/L_sol,Ip);

xl = xlabel($\hbox{quotesingle}$Time (ps)$\hbox{quotesingle}$),set(xl,$\hbox{quotesingle}$Rotation$\hbox{quotesingle}$,32,$\hbox{quotesingle}$fontsize$\hbox{quotesingle}$,16)

yl = ylabel($\hbox{quotesingle}$Distance z/L_{sol}$\hbox{quotesingle}$),set(yl,$\hbox{quotesingle}$Rotation$\hbox{quotesingle}$,-30,$\hbox{quotesingle}$fontsize$\hbox{quotesingle}$,16)

zl = zlabel($\hbox{quotesingle}$Intensity (W)$\hbox{quotesingle}$,$\hbox{quotesingle}$fontsize$\hbox{quotesingle}$,16)

set(gcf,$\hbox{quotesingle}$colormap$\hbox{quotesingle}$,[0 0 0])

set(gca,$\hbox{quotesingle}$Xgrid$\hbox{quotesingle}$,$\hbox{quotesingle}$off$\hbox{quotesingle}$,$\hbox{quotesingle}$Ygrid$\hbox{quotesingle}$,$\hbox{quotesingle}$off$\hbox{quotesingle}$,$\hbox{quotesingle}$Zgrid$\hbox{quotesingle}$,$\hbox{quotesingle}$off$\hbox{quotesingle}$,$\hbox{quotesingle}$fontsize$\hbox{quotesingle}$,16, …

$\hbox{quotesingle}$Linewidth$\hbox{quotesingle}$,1)

view(135,50)

axis([−0.5 0.5 0 2 0 max(max(Ip))])