Diffraction using fourier transform in Matlab - matlab

Question

I am very new to Matlab and am trying to implement the following fresnel diffraction using the fourier transform:
This is taken from the following wikipedia page: http://en.wikipedia.org/wiki/Fresnel_diffraction
I am trying a square aperture of width 5 cm.
clc;clear;
lambda=1*10^-6;
w=.05;
z=2.0;
k=(2*pi)/lambda;
x1=linspace(-0.2,0.2,2048);
y1=linspace(-0.2,0.2,2048);
U1=((abs(x1)<=(w)/2))&((abs(y1)<=(w)/2));
u1=double(U1);
figure(1)
plot(x1,u1)
g=u1'*exp(1i*(pi/(lambda*z))*(x1.^2+y1.^2));
G=fftshift(fft2(g));
h=(exp(1i*k*z)/(1i*lambda*z))*exp(1i*(pi/(lambda*z))*(x1.^2+y1.^2));
H=fftshift(fft2(h));
u2=H*G;
figure(2)
plot(x1,abs(u2));
When I plot the field u2, for any of the distances, z, that I try, it just shows up like some random pattern and not the expected diffraction pattern for a square aperture.
Like I said, I am very new to MATLAB and find it difficult to understand. I think I am making this more complicated than it needs to be, and am implementing the integrals incorrectly.
Any pointers or advice? I am quite stuck...
Thanks!

Answer 1

Looks like you want to simulate Fresnel diffraction with this code.
With a few changes you can view the actual image. Shown here is the amplitude. The intensity would be u2.*conj(u2).
Note however that critical sampling will occur much further away than 2 meters, more like 100 meters, for the physical parameters you are using.
Also there needs to be some additional re-scaling on the output plane, not done here. Otherwise the diffraction pattern will get very small for low Fresnel numbers.
clc;clear;
N=1048;
L=0.4;
lambda=1*10^-6;
w=.05;
z=500.0;
k=(2*pi)/lambda;
src_delta = L/N;
critical_sampling_z = N*src_delta^2/lambda;
[x1 y1] = meshgrid((-N/2 : N/2-1) * src_delta);
X = double(abs(x1)<w/2); Y = double(abs(y1)<w/2);
u1 = X.*Y;
figure(1);
colormap(gray); imagesc(mat2gray(u1));
g=u1'*exp(1i*(pi/(lambda*z))*(x1.^2+y1.^2));
G=fftshift(fft2(g));
h=(exp(1i*k*z)/(1i*lambda*z))*exp(1i*(pi/(lambda*z))*(x1.^2+y1.^2));
H=fftshift(fft2(h));
u2=H*G;
figure(2);
colormap(gray); imagesc(mat2gray(abs(u2)));