MATLAB: Going from FFT to IFFT, shouldn’t I get the same graph I initially started with

fftifft

Hi, I have the code below where I am converting electric field in the frequency domain to the time domain and back to the frequency domain. I don't know why I am not getting the same graph I started with after my FFT and IFFT. Any help is much appreciated.
 % Constants
  c = 299792458; % speed of light in a vacuum
  FWHM = 30e-15; % pulse duration 
  T = FWHM/(2*(sqrt(log(2)))); % (T=tau)
  lambda0 = 800e-9; % central wavelngth 
  w0 = (2*pi*c)/lambda0; % central angular frequency
  phi_t = 0; % phase (in time or freqency)
  eta = 0; % chirp (2nd derivitive of phase)
  lambda = (740e-9:20e-9:860e-9); % wavelength interval
  w = (2.*pi.*c)./lambda; % angular frequency conversion
% electric field in the time domain
E_t = exp((-t.^2/(2*T.^2)) + (i*w0*t-(1/2)*i*eta*t.^2)); %*phi_t)); 
% FFT
dt = t(2)-t(1); % difference in time (T)
E_w = (fftshift(fft(E_t)))*dt; % fft 
Fs = 1/dt; % sampling frequency
df = Fs/length(E_t); % difference in sampling frequency
freq = -Fs/2+df:df:Fs/2; % frequency
lambda = c./freq; % wavelength
I_w = abs(E_w).^2; % intensity 
% IFFT
dw = w(2)-w(1); % difference in frequency
ifftE_t =(ifft(E_w))*dw; % ifft
Ts = 1/dw; % sampling time
dt = Ts/length(E_w); % difference in sampling time
time = -Ts/2+dt:dt:Ts/2; % time
% PLOT
subplot(3, 1, 1);
plot(t, real(E_t));
title('Gaussian Pulse Signal'); 
xlabel('time (s)'); 
ylabel('E_t'); 
subplot(3, 1, 2);
plot(lambda,I_w);
set(gca,'xlim',[0 9e-7]);
title('FFT')
xlabel('\lambda');
ylabel('I_\lambda');
subplot(3, 1, 3)
plot(time, real(ifftE_t))
title('IFFT')
xlabel('time (s)')
ylabel('E_t')

Best Answer

You used fftshift() on the results of the fft(), but you did not fftshift() back before doing the ifft()
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