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I'm trying to simulate the Fresnel diffraction by using this expersion :

$$ A(x')=\frac{1}{j\lambda z}e^{jk(z+\frac{x'^2}{2z})}F(A^{trans}(x)e^{jk\frac{x^2}{2z}})_{u=\frac{x'}{\lambda z}} $$

So when I use this formula my problem is that I don't know how to take the good frequency, here ## \frac{x'}{\lambda z}##

and in my simulation, the result is that the light is not diverging after the rectangular aperture. I don't know how to fix this problem, because I wan't to plot the (x,z) plan so z is not constant. When I plot only one column, which corresponf to a z fix, I've the good profile of intensity but it's steel not diverging when I choose different z value. So here is my Matlab (R2017b) simulation code :

I used python color to make it easier to read, but the % are not seen as comments sorry. And here you have the results :Code (Python):%%%%%% Simulation of the Fresnel Diffraction, Matlab %%%%%%%

%% Initialization %%

clear all

close all

%% Grating %%

for i_grating = 1:250 % creation of a rectangular aperture

A_t(i_grating) = 0;

end

for i_grating = 251:350;

A_t(i_grating) = 1;

end

for i_grating = 351:601

A_t(i_grating) = 0;

end

%% Parameters %%

Lambda = 795e-9; % wavelength

k = 2*pi/Lambda; % the wave vector

z = [0.06 :0.01: 6]; % distance from the grating

dx = 20e-6; % size of a pixel in micrometer

x =[-6000e-6 :dx: 6000e-6]; % coordinate on the grating plane

x2 =[-6000e-6 :dx: 6000e-6]; % coordinate on the screen plane

plot(A_t) % show the amplitude just after the grating(z=0+)

%% Simulation %%

for i_z = 1:length(z) % first loop on the distance to the grating

for i_x = 1:length(x) % loop on all point of the grating

f(i_x)=A_t(i_x)*exp(1i*k*((x(i_x))^2)/(2*z(i_z)));

end

A = fftshift(fft(f));

for i_x2 = 1:length(x2) % we do a loop for all point in the observing plane at z = cst

C =(1/(1i*Lambda*z(i_z)))*exp(1i*k*(z(i_z)+(x2(i_x2))^2)/(2*z(i_z)));

A_p(i_x2,i_z) = abs(C*A(i_x2));

end

end

%% 2D Map %%

figure;

imagesc(A_p);

colormap jet; % color representing the light intensity

On the right, what we see is the light intensity, the x axis correspond to the vertical axis and the z axis to the horizontal axis.

thanking you ,

CijiG

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# MATLAB FFT problem for Fresnel simulation

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