FFT problem for Fresnel simulation

[CijiG]

Hi all,

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 :

%%%%%%    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

I used python color to make it easier to read, but the % are not seen as comments sorry. And here you have the results :

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

 

[mmwave]

Dear CijiG,

Thank you for sharing your work on simulating Fresnel diffraction. It seems like you are using the correct expression for the Fresnel diffraction, but you are having trouble with choosing the right frequency for your simulation.

The frequency, in this case, is related to the wavelength and the distance from the grating. So, for each point on the observing plane (z=constant), you need to calculate the corresponding frequency using the expression u = (x'/lambda*z). This will give you the correct frequency for each point on the observing plane.

As for the issue of the light not diverging after the rectangular aperture, it could be due to the size of your aperture or the distance between the grating and the observing plane. You may need to adjust these parameters to see the desired diverging effect.

I hope this helps and good luck with your simulation!