Scaling diffraction patterns (using FFT)

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