Fraunhofer diffraction of circular aperture

Hello,

I am trying to see how a speckle pattern changes with distance after it scatters from a rough surface.

I am using Fraunhofer in its Fourier form, but I do not understand where distance comes into play. For example, the code below

Lambda=.633;        %Wavelength of laser (micron)
D=50;               %Diameter of aperture (micron)
Z_Meters=.1;        %Screen distance in meters
Z=Z_Meters*10^6;    %Screen distance in microns

MeshSpacing=1;      %Sampling across aperture (micron)
MeshSize=200;       %Size of Screen (micron)

% Calculate and show the Amplitude across the aperture.
[XGrid,YGrid]=meshgrid((-MeshSize/2:MeshSpacing:MeshSize/2),(-MeshSize/2:MeshSpacing:MeshSize/2));
R=sqrt(XGrid.^2+YGrid.^2);
A=R<=D/2;  %The Amplitude across aperture (plane wave has constant amplitude, phase)
figure;imshow(A);title('Amplitude Across Aperture')

% Do the the 2D fft and show the result.
U=fftshift(fft2(A));
I=abs(U).^2;
figure;imshow(I,[]);title('Intensity at Screen')

I am only taking the FT of the circular aperture, how would my output change if instead of 0.1 meters from the screen, I was at 100 meters? I do not see how I can include the distance anywhere. Does someone know the answer?

Cheers.

Answer 1

[Image Analyst]

The Fraunhofer diffraction pattern is the Fourier Transform of the aperture times the illumination pattern at the aperture, which is what you get at a distance of infinity from the aperture. The Fresnel pattern is what you get at non-infinite distances. I don't know the formula for the Fresnel equation off the top of my head (despite having a Ph.D. in optics), but you can look it up, like maybe here in WIkipedia

[Lucrezia Cester]

Isn't Fresnel only for near field? What if you have a diverging solution? Although your answer makes a lot of sense. Thank you so much!

[Image Analyst]

Well, correct, sort of. Fresnel diffraction is valid everywhere from the aperture itself, out to an infinite distance beyond it. In the special case of an infinite distance, the Fresnel pattern is the same as the Fraunhofer pattern. Right next to the aperture, the pattern is essentially the illumination pattern masked by the aperture. As your observation plane moves farther and farther away from the aperture, it starts to look less like the aperture and more like the Fraunhofer pattern.

[Lucrezia Cester]

Can I ask you one more question? Does the distance at which one reaches the Fraunhofer regime changes in different situations? It seems to me that it changes from the outputs of my code.

[Image Analyst]

The larger the aperture is with respect to the wavelength of light, the closer the Fraunhofer zone will be. Imagine if the aperture was gigantic - the transmitted light pattern would very quickly be just essentially the shadow of the hole, no matter how far you were from the hole (for colimated light of course).

[Lucrezia Cester]

Thank you so so so much!

[Lucrezia Cester]

Would you have a good book to recommend that talks about this specificalities?

[Image Analyst]

I don't know of any off the top of my head.