FFT nomalization problem ?

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Regis Robles
Regis Robles 2016-3-14
评论： Star Strider 2016-3-16
Values of my intensity profile are much bigger after I used fftshift(fft2()) and values should be equal. My question is : is there a normalization problem when I do the fft() of my field U2 and what to do to have the right result ?
Here is my code so far :
clear all;
close all;
clc;
%%constants declaration and parameter calculation.
E0 = 10; %maximum field amplitude.
z = 6.8e-3; %distance between fiber and lens in m.
w0 = 15e-6; %waist of beamlets in m.
lambda = 0.98e-6; %wavelength in m.
Nx = 3; %number of fiber over x axis.
Ny = 3; %number of fiber over y axis.
dx = .4e-3; %pitche over x axis in m.
dy = .4e-3; %pitche over y axis in m.
zr = pi*w0^2/lambda; %Rayleigh range in m.
k = 2*pi/lambda; %wavenumber in m-1.
wz = w0*sqrt(1+(z^2)/(zr^2)); %w(z) in m.
Rz = z*(1+(zr^2)/(z^2)); %R(z) in m.
phiz = atan(z/zr); %Gouy phase.
R = 1.4*wz; %lens radius in m.
%%[X,Y] domain creation.
Npts = 2000;
Nspots = 9;
res1 = 1e-6;
[X1,Y1] = meshgrid((-Npts/2:Npts/2-1).*res1,(-Npts/2:Npts/2-1).*res1);
res2 = 50e-6;
[X2,Y2] = meshgrid((-Npts/2:Npts/2-1).*res2,(-Npts/2:Npts/2-1).*res2);
%%aperture center vectors.
X0 = [-dx 0 dx -dx 0 dx -dx 0 dx];
Y0 = [-dy -dy -dy 0 0 0 dy dy dy];
%%field/intensity calculation and plot.
U1 = zeros(Npts,Npts,Nspots);
U2 = zeros(Npts,Npts,Nspots);
tmp = E0*(w0/wz);
for j = 1:length(X0)
U1(:,:,j) = tmp*exp(-((X1-X0(j)).^2+(Y1-Y0(j)).^2)/(wz.^2));
U2(:,:,j) = tmp*exp(-((X2-X0(j)).^2+(Y2-Y0(j)).^2)/(wz.^2));
end
U1 = sum(U1,3);
U2 = sum(U2,3);
UCC = conj(U1);
I = U1.*UCC;
figure(1);
imagesc(X1(1,:),Y1(:,1),I);
title('Real intensity profile');
xlabel('x in m');
ylabel('y in m');
%%FFT calculation and plot.
Uf = fftshift(fft2(U2));
UfCC = conj(Uf);
If = Uf.*UfCC;
figure(2);
imagesc(X2(1,:),Y2(:,1),If);
title('FFT intensity profile');
xlabel('x in m');
ylabel('y in m');

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Star Strider
Star Strider 2016-3-14
I rarely use fft2, but I know that with fft it is necessary to normalise it by dividing it by the length of the time-domain signal.
See if:
Uf = fftshift(fft2(U2))/numel(U2);
does what you want.
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Regis Robles
Regis Robles 2016-3-16
编辑：Regis Robles 2016-3-16
You thought well. I am having problem with the magnitude because a should have the same scaling before and after I used fftshift(fft2(U2));. I think I am doing wrong with fft thus I will check how FT really works. Thanks for your help.

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