Question

I am using the FFT function in Matlab in an attempt to analyze the output of a Travelling Wave Laser Model.

The of the model is in the time domain in the form (real, imaginary), with the idea being to apply the FFT to the complex output, to obtain phase and amplitude information in the frequency domain:

%load time_domain field data
data = load('fft_data.asc');

% Calc total energy in the time domain
N = size(data,1);
dt = data(2,1) - data (1,1);
field_td = complex (data(:,4), data(:,5));


wavelength = 1550e-9;
df = 1/N/dt;
frequency = (1:N)*df;
dl = wavelength^2/3e8/N/dt;
lambda = -(1:N)*dl +wavelength + N*dl/2;

%Calc FFT
FT = fft(field_td);
FT = fftshift(FT);
counter=1;
phase=angle(FT);
amptry=abs(FT);
unwraptry=unwrap(phase);

Following the unwrapping, a best fit was applied to the phase in the region of interest, and then subtracted from the phase itself in an attempt to remove wavelength dependence of phase in the region of interest.

for i=1:N % correct phase and produce new IFFT input
    bestfit(i)=1.679*(10^10)*lambda(i)-26160;
    correctedphase(i)=unwraptry(i)-bestfit(i);
    ReverseFFTinput(i)= complex(amptry(i)*cos(correctedphase(i)),amptry(i)*sin(correctedphase(i)));
end

Having performed the best fit manually, I now have the Inverse FFT input as shown above.

pleasework=ifft(ReverseFFTinput);

from which I can now extract the phase and amplitude information in the time domain:

newphasetime=angle(pleasework);
newamplitude=abs(pleasework);

However, although the output for the phase is greatly different compared to the input in the time domain

the amplitude of the corrected data seems to have varied little (if at all!),

despite the scaling of the phase. Physically speaking this does not seem correct, as my understanding is that removing wavelength dependence of phase should ‘compress’ the pulsed input i.e shorten pulse width but heighten peak.

My main question is whether I have failed to use the inverse FFT correctly, or the forward FFT or both, or is this something like a windowing or normalization issue?

Sorry for the long winded question! And thanks in advance.

Answer 1

You’re actually seeing two effects.

First the expected one goes. You’re talking about “removing wavelength dependence of phase”. If you did exactly that – zeroed out the phase completely – you would actually get a slightly compressed peak.
What you actually do is that you add a linear function to the phase. This does not compress anything; it is a well-known transformation that is equivalent to shifting the peaks in time domain. Just a textbook property of the Fourier transform.

Then goes the unintended one. You convert the spectrum obtained with fft with fftshift for better display. Thus before using ifft to convert it back you need to apply ifftshift first. As you don’t, the spectrum is effectively shifted in frequency domain. This results in your time domain phase being added a linear function of time, so the difference between the adjacent points which used to be near zero is now about pi.