Revisiting PSF models: theory and implementation

We present a unifying framework for point-spread function (PSF) models in high-numerical-aperture optical microscopy. Starting from the Richards-Wolf integral, we demonstrate the mathematical equivalence between Fourier-based and Bessel-based approaches as alternative parameterizations of the same propagation integral. This framework enables us to generalize correction factors such as Gibson-Lanni aberrations, apodisation, and arbitrary phase masks to both scalar and vectorial models. We provide an open-source PyTorch implementation that achieves high computational performance on both CPU and GPU. Our results show that Bessel propagators excel for axisymmetric pupils while Fourier propagators handle general cases. The library integrates seamlessly with deep learning workflows through automatic differentiation, facilitating applications in deconvolution, PSF engineering, and computational microscopy.