Scaling diffraction patterns (using FFT)

Question

I am trying to write a program to compute the FFT of 1D complex aperture functions to determine diffraction patterns. Supposing I can produce the diffraction patterns successfully, I am unsure as to how to 'scale' the patterns so that the intensity on the screen is given as a function of position y. i.e. I need to convert the pixels in the Fourier Transform to positions on the screen (y).

$$\psi(y) \propto \Delta \sum_{j=0}^{N-1}A(x_j)exp(\frac{-ikx_j y}{D})$$ $$x_j = (j - (N/2))\Delta$$ where $\Delta$ is the spacing between aperture samples of the aperture function, N is the number of samples, $k=\frac{2\pi}{\lambda}$ and the other variables are apparent from the diagram below.

The FFT is of the form below, where $h_m$ represents samples of the aperture function? $$H_j = \sum_{m=0}^{N-1}h_m e^{2\pi imj/N}$$ Q: How do I go from the FFT of the aperture function so that it is scaled to be a function of screen position?

Answer 1

$k$ (better: $k_x$) is the $x$ component of the diffracted wave vector. You can construct the angle associated with the wave vector from the knowledge of the magnitude of the wave vector $|k| = \frac{2\pi}{\lambda}$ and $k_x$. Then figure out where that wave vector will hit the screen.