Plot the Intensity and phase of variation of electric field.

Hello everyone.

I have a set of data : "y_spectrum" shows the electric field amplitude and "t_spectrum" shows the time.

Is there anyway to plot the Intensity of this variation and its phase?

Also, after doing Fourier transform , is it possible to obtain its distribution and its phase?

Thank you very much.

Answer 1

[Star Strider]

The Fourier transform is easy enough to calculate and plot:

t = dlmread('t_spectrum_11480.txt','',1,0);
E = dlmread('y_spectrum_11480.txt','',1,0);

Fs = 1/mean(diff(t));                                       % Sampling Frequency
[Er,tr] = resample(E,t,Fs);                                 % Resample To Uniform Sampling Interval
L = numel(t);                                               % Length Of Signal
Fn = Fs/2;                                                  % Nyquist Frequency
FT_E = fft(E)/L;                                            % Fourier Transform
Fv = linspace(0,1,fix(L/2)+1)*Fn;                           % Frequency Vector
Iv = 1:numel(Fv);                                           % Index Vector (One-Sided Spectrum)

figure
plot(tr, Er)
grid
title('Time Dommain')

figure
subplot(2,1,1)
plot(Fv, abs(FT_E(Iv))*2)
grid
ylabel('Amplitude Units')
title('Amplitude')
subplot(2,1,2)
plot(Fv, angle(FT_E(Iv)))
grid
xlabel('Frequency (Hz?)')
ylabel('Radians')
title('Phase')

figure
subplot(2,1,1)
plot(Fv, abs(FT_E(Iv))*2)
grid
ylabel('Amplitude Units')
title('Amplitude')
xlim([0 1E+15])
subplot(2,1,2)
plot(Fv, angle(FT_E(Iv)))
grid
xlabel('Frequency (Hz?)')
ylabel('Radians')
title('Phase')
xlim([0 1E+15])

The first plot is the time domain of the resampled signal, the second is the full Fourier spectrum, and the third is the ‘zoomed’ Fourier spectrum.

[dang khoa tran]

Thank you

From this distribution, I tried to do the inverse FFT to get the variation in time domain from the Fourier spectrum. And then to identify the spectrum and its phase but I failed to do so. I use the following :

So could you please recommend any idea?

fs=1E-15;
A=ifftshift(ifft(ifftshift(FT_E)));
I_t_reconstruct=abs(A.^ 2);
I_t_max=max(I_t_reconstruct);
I_t_reconstruct=I_t_reconstruct./I_t_max;
figure;
subplot(2,1,1),plot(tr/fs,abs(ifftshift((I_t_reconstruct))));
xlabel('time (fs)');ylabel('E(t) (a.u.)');
title('the reconstructed E(t) field');
subplot(2,1,2),plot(tr/fs,angle(I_t_reconstruct));axis([-inf, inf, 0 inf]);
xlabel('time (fs)');ylabel('phase of E(t) (rad)');
title('the phase of E(t) field');

[Star Strider]

As always, my pleasure!

I do not understand the reason you are using the inverse Fourier transform to reconstruct the original time-domain function. My code creates time-domain data with uniform sampling intervals (necessary in order to use the fft function and all other signal processing techniques), then calculates the full Fourier transform, and plots the one-sided Fourier transform.

In any event, do not use fftshift. Use the original unshifted Fourier transform to calculate the ifft, if you must.

[dang khoa tran]

So to obtain the phase distribution of the time-domain function, I tried :

But the phase appears weird when it is similar in the whole process although the shape is different in some moment.

So do you have any idea on this?

figure
subplot(2,1,1)
plot(tr, Er)
grid
title('Time Dommain')
subplot(2,1,2)
plot(tr,unwrap(angle(Er)))
title('Phase')

[Star Strider]

The plot is the time domain of the signal, with the phase, although I do not know how the phase of that time domain signal was calculated. (It appears to go almost randoimly from 0 to π.) The plotted phase is just what the unwrapped phase of the Fourier transform returns. I do not know the reason the unwrapped phase appears as it does.

[dang khoa tran]

Ok. Anyway, from the full Fourier transform , is there any ways to shift it so that the central frequency will appear in the center of the scale as shown in the figure?

[Star Strider]

Yes.

Use the fftshift function to do that. (It is best to do that on the complex fft result, then take the absolute values (magnitudes) and phases (angles) of the shifted version.)

However in MATLAB, it is necessary to use the unshifted Fourier transform to calculate the ifft.

[dang khoa tran]

In fact, I have tried this before but when using fftshift , it does not appear in the center of scale. Instead, it shifts to a different frequency. So do you have another approach?

[Star Strider]

It does actually center the scale. It is necessary for you to create a new frequency vector to use the shifted Fourier transform, for example to create frequency vector ‘Fv’ for the shifted Fourier transform:

Fv = linspace(-Fn, Fn, numel(FT));

where ‘Fn’ is the Nyquist frequency (one-half the sampling frequency) and ‘FT’ is the original complex Fourier transform (here assuming a vector, use the appropriate size function arguments, instead of numel, for an array).

[dang khoa tran]

Ok.

Thank you very much for your time and your suggestion.

[Star Strider]

As always, my pleasure!