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I'm trying to figure out how I can possibly know the real equation of the intensity of the diffraction pattern of a double slit (I don't care about interference for now). Here are my thoughts:

Textbook say the intensity is $D = (\sin(\alpha)/\alpha)^2$ with $\alpha = kd\sin x$ for both single and double slit. (for double slit, you just multiply with the interference pattern, but the envelope stays the same).

Now, if we put both slits far apart, we should see two bumps in the diffraction pattern but I can't get those bumps with that equation. What I did is change the $x$ in the diffraction for $x + \Delta$ and then add the two patterns with the correct spacing. The problem with that method is if I bring the two slits back close together, I don't get the same result as before which proves me wrong.

I understand that the equation I use have some constraint and I might not fit in, but any ideas on how I can solve the problem? I have done the experiment and I can really see a shift in the pattern if I close on of the two slit, but can't figure out how the get the theory about it.

Here is an example of the 2 bumps I'm talking about: In orange and green are the two pattern. Red is the sum of both with the two bums and blue is the theory equation I gave earlier.

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After reducing the distance (and dividing by two the intensity) I get:

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  • $\begingroup$ I think your approach is correct. Let us say that there is no interference, e.g. there is red light passing through one slit and blue light through the other, so that the intensities simply sum up. You have the single slit diffraction pattern D(x). A slit displaced by d along x axis gives D(x-d) and the other, displaced by -d, gives D(x+d). Whatever is the distance, since there is no interference, you simply get I(x)=D(x-d)+D(x+d). If they are not displaced at all, you get I(x)=2D(x). Please clarify if you need something more. $\endgroup$ Commented May 8 at 15:38
  • $\begingroup$ The slits are not far enough apart? Related What happens if separation between slits is greater than separation between slits and screen in YDSE experiment? $\endgroup$
    – Farcher
    Commented May 8 at 15:39

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Your equation is obtained from the standard Fraunhofer diffraction conditions which require the assumption that the size of the aperture is much smaller than the distance from the aperture to the screen. In this regime, the diffracted amplitude is proportional to the Fourier transform of the aperture and moving the aperture does not change the intensity, only the phase of the diffracted amplitude. So the pattern does not move. In order for the pattern to move upon moving the slit, you have to use a different approximation such as Fresnel diffraction.

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