Problem with Inverse Fourier Transform

I was trying to generate a time domain pulse from its frequency domain spectrum; I am unable to produce it correctly: the time domain oscillations are not "smooth" and it is much narrower than expected from the Gaussian time-bandwidth product. Here it is:

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w = 2*pi*(0.3300 : 3e-05 : 0.4200); %frequency range

w0 = 2*pi*0.375; %central frequency

dw = 2*pi*0.009./(2*sqrt(2*log(2))); %frequency width (standard deviation)

%---------------------

Nw = 1./(dw*sqrt(2*pi));

zw = (w-w0)/dw;

Sw = Nw.*exp(-(1/2)*(zw.^2)); %intensity in frequency domain (spectrum)

%---------------------

Ew0 = sqrt(Sw); %spectral amplitude

phiw = 0; %spectral phase

Ew = Ew0.*exp(i*phiw); %pulse in frequency domain

%---------------------

t = (-1500:1:1500)./(3001*2*pi*3e-05); %time range

Et = ifftshift(ifft(Ew)); %pulse in time domain

St = (abs(Et)).^2; %intensity in time domain

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subplot (2,2,1)

plot (w,real(Ew))

subplot (2,2,2)

plot (w,Sw)

subplot (2,2,3)

plot (t, real(Et))

xlim ([-100 100])

subplot (2,2,4)

plot (t, St)

xlim ([-100 100])

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