{'fraction_non_alphanumeric': 0.024146054181389872, 'fraction_numerical': 0.011778563015312132, 'mean_word_length': 4.480645161290322, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Influence of surface errors on the design of PIAA mirrors using numerical and semi-analytical propagation models

Since the radius of curvature of a mirror cannot be zero, the apodization that is created by a phase-induced amplitude apodizer (PIAA) formed by a pair of mirrors cannot be zero at the edge of the pupil. If contrasts lower than 10-10 must be obtained, then an additional apodizer must be used with the PIAA mirrors. This has a consequence on the throughput of the system, as well as on its inner working angle (IWA). The intensity distribution in the final pupil plane computed in the ray-optics approximation is misleading, and diffraction must be taken into account to evaluate the true performance of the system. We compute the propagated electric field using two different tools: the semi-analytical model developed by Pueyo and a purely numerical model based on the Huygens integral. It is shown that for higher Fresnel numbers, the agreement between the beams computed using both propagators is stronger, and that for too low Fresnel numbers, the contrast computed using the semi-analytical model can be 2 orders of magnitude higher than the one computed by a numerical evaluation of the Huygens integral. We then study the impact of surface aberrations introduced on the mirrors of the PIAA. The surface quality of the mirrors limits the performance of the system, and the IWA increases linearly with the root-mean-square (RMS) of the aberrations. For a typical set of mirrors, errors of 10nm RMS can increase the IWA by 0.5 to 1λ/D for a contrast of 10-10, and, in the case of a contrast of 10-8, the IWA is maintained to 2 λ/D as long as the errors are smaller than 20nm RMS.