Calculating Point Spread Functions: Methods, Pitfalls and Solutions

We discuss advantages and disadvantages of various ways of calculating an optical Point Spread Function (PSF) and present novel Fourier-based techniques for computing vector PSF. The knowledge of the exact structure of the PSF of a given optical system is of interest in fluorescence microscopy to be able to perform high-quality image reconstructions. Even, if we know how an aberrant optical path deviates from the original design, the corresponding PSF is often hard to calculate, as the phase and amplitude modifications need to be modelled in detail. Accurate PSF models need to account for the vector nature of the electric fields in particular for high numerical apertures. Compared to the computation of a commonly used scalar PSF model, the vectorial model is computationally more expensive, yet more accurate. State-of-the-art scalar and vector PSF models exist, but they all have their pros and cons. Many real-space-based models fall into the sampling pitfall near the centre of the image, yielding integrated plane intensities which are not constant near the nominal focus position, violating energy conservation. This and other problems which typically arise when calculating PSFs are discussed and their shortfalls are quantitatively compared. A highly oversampled Richards and Wolf model is chosen as the gold standard for our quantitative comparison due its ability to represent the ideal field accurately, albeit being practically very slow in the calculation. Fourier-based methods are shown to be computationally very efficient and radial symmetry assumption are not needed making it easy to include non-centro-symmetric aberrations. For this reason newly presented methods such as the SincR and the Fourier-Shell method are essentially based on multidimensional Fourier-transformations.