I want to model the action of a lens on radiation from a point dipole. I have found examples but am struggling to interrupt them. My knowledge of Fourier optics is that from the Helmholtz equation, $E_{out} = \mathcal{F}(E_{in})$, where $E$ is the electric field strength in the imaging (output) and focal (input) plane of a lens.
In the examples I have found, the radiation pattern of the dipole is represented with the function $f(\theta)$, and emission is considered into plane wave modes with wavevector $k$, and the dipole position labelled $r$;
$s(\theta)=\sqrt{f(\theta)} \exp(i k.r)$
To model the effect of a lens they consider the dipole position in one dimension $x$. The next step is to say that this distribution of radiation into these modes, is equivalent to radiation into a sum of modes given by the Fourier coefficients,
$c_{\alpha} = \int_{-1}^{1}d(\cos\theta)\bigg[\sqrt{f(\theta)} \exp(i k x \cos\theta)\bigg]\exp(I \pi \alpha \cos\theta)$
For $f(\theta)=constant$, the magnitude of the coefficients follows a $\text{sinc}(z-\lambda\alpha)$ function. This is the sort of distribution I would expect to see for the electric field in the image plane. The coefficients are discrete though, and we did not start with the field defined in the focal plane. I'm lacking some understanding of this procedure?
One example of this derivation starts on page 18 of this paper, in the subsection "Imaging": https://arxiv.org/pdf/quant-ph/0611067.pdf
