A new approach for the numerical calculation of diffraction patterns using starshades

Context. We studied the imaging of exoplanetary systems using starshades, which are externally occulted coronagraphs in space. Aims. We provide a new method for precisely evaluating the stray light due to the star and a rapid calculation of the point spread functions in the presence of vignetting effects from the external occulter. Our study used shaped occulter configurations published in the literature, in particular, the SISTER NI2 and NW2 systems. Methods. The wavefront at the telescope aperture was computed using the classic Fresnel filtering method. The Fourier transform of the occulter was obtained with the highest possible precision using an approach initially developed for radio antennas, known as the polygonal shape factor. Results. We show that the Fresnel diffraction for a finite spatial field operates at very low frequencies only, and that it is sufficient to calculate the Fourier transforms there. Diffraction patterns computed numerically fully agree with theoretical predictions. The central parts of diffractions of petal and apodized occulters are identical over a large central area that increases in size with the number of petals. These diffraction patterns are used to compute the point spread functions. We computed the stray light for a non-point source star; this shows that starshades are not sensitive to star leakage, with a star diameter limit for a given configuration. We also computed signal-to-noise ratios for a perfect experiment limited by photon noise.