I'm trying to calculate the fraunhofer diffraction pattern from a aperture composed of multiple simple shapes, but I've ran into some trouble when trying to simulate the results and I really dont know if my derivation of the pattern is wrong or if my code is simply acting up.
The aperture is simply a rectangle with some arbitrary width W1 and height H1, centered in the aperture plane, and two more rectangles with some arbitrary width W2 and height H2 on each side of the central rectangle. So the composite shape could be an H, or a plus sign, or some other shape, it doesnt really matter. As far as I can tell, the fraunhofer diffraction patter should simply be proportional to the Fourier transform of the aperture function, evaluated at $(k_x,k_y) = (\frac{x}{\lambda z},\frac{y}{\lambda z})$. In this case then, the aperture function $f(x,y)$ is "simply" $$f(x,y) = rect(\frac{x}{W1})rect(\frac{y}{H1})+rect(\frac{x-\frac{W1+W2}{2}}{W2})rect(\frac{y}{H2})+rect(\frac{x+\frac{W1+W2}{2}}{W2})rect(\frac{y}{H2})$$
and the Fourier transform is
$$F(k_x,k_y) = W1 \cdot H1 \cdot sinc(W1 k_x)sinc(H1 k_y) + \exp(-i k_x \frac{W1+W2}{W1})\cdot W1 \cdot H1 \cdot sinc(W1 k_x)sinc(H1 k_y) + \exp(i k_x \frac{W1+W2}{W2}) \cdot W2 \cdot H2 \cdot sinc(W2 k_x)sinc(H2 k_y) $$
My question is, is this allowed? To define the aperture piecewise in the way above, and then use the fourier shift theorem to obtain the diffraction pattern as the sum of phase shifted sinc functions. Im asking because when I implement it in python, the pattern differs from the pattern I obtain using a DFT of the same aperture. Appreciate any input, I'm at my wits end.
