Write different super gaussian function with same fwhm but different order

Hi,

I consider a gaussian pulse : how do I define in order to have . I did solve : leading to .

But when I plot and measure my fwhm I get varying values of the fwhm meanwhile the value at MAXpower/3 does not change... Where did I go wrong?

Here is my code :

%%Definition des parametres d'entree de la source%%%%%%%%%%%%%%%%%%%%%%%%%%
freq_pulse=400e3;
T_pulse=1/freq_pulse;
Pmoy=40e-6;
tau0=55e-12;
E0=Pmoy/freq_pulse;
pulse_form=0;%               ->0 : gaussien 1: sech
Chirp=0;
lambda0=1030e-9;

ResolutionSSF=100;
Nsamples=2^16;
TimeWindow=15;
betterfactor=10;

Timewin=tau0*TimeWindow;    %sqrt(Nsamples)*
dtau=Timewin/Nsamples;              %time sep samples
Fs=1/dtau;                            %frequence d'echantillonage
dF=1/Timewin;                         %sep freq d'echantillonage
domega=2*pi*dF;                     %angular freq

%%Definition de l'espace temporel
tau=(-Nsamples/2:(Nsamples/2)-1)*dtau*1e12;

%%Definition de l'espace frequentiel
Dfreq=(-Nsamples/2:1:(Nsamples/2)-1)*dF;

%%Definition de l'espace spectral
omega0=2*pi*3e8/lambda0;
dL=2*pi*3e8/omega0^2*domega*1e9;
Domega=(-Nsamples/2:Nsamples/2-1)*domega;
Lambda=((-Nsamples/2:Nsamples/2-1)*dL+lambda0*1e9);
%tau0=tau0;

Pmoy=200e-6;
for degGauss=3
    tausupG=tau0/(2^(3/2)*log(2)^(1/(2*degGauss)));
    
    %tau0=tau0/(2*sqrt(2*log(2)));
    %u_gauss=exp(-log(2)*(4*(tau*1e-12).^2./(tau0.^2)).^degGauss);
    u_gauss=exp(-((tau*1e-12).^2./(2*tau0.^2)).^degGauss);
end

Thanks

Answer 1

[David Goodmanson]

Hi Hugo,

There seems to be some confusion as to whether the constant that is used as the reference constant between various values of m is fwhm or t0. If it's t0 (fwhm still being constant) then:

tau_m  = tau1*log(2)^((m-1)/(2*m))  % I renamed tau0 to tau1 since it corresponds to m = 1.

Adding the following to the end of your code

Pmoy=200e-6;


tau1 = tau0;
u_gauss = [];
for m = 1:3
  taum = tau1*log(2)^((m-1)/(2*m));
  u_gaussm = exp(-(((tau*1e-12).^2./(2*taum.^2))).^m);
  u_gauss = [u_gauss;u_gaussm];
end
figure(1)
plot(tau,u_gauss)
grid on

results in

(saving by concatenation is certainly not recommended for variables with a large number of elements (it can be slow) and is a bad habit to get into in general but in a case like this it's fine).

[Hugo]

Thank you very much!