Coronagraph instruments on future space telescopes will enable the direct detection and characterization of Earth-like exoplanets around Sun-like stars for the first time. The quest for the optimal optical coronagraph designs has made rapid progress in recent years thanks to the Segmented Coronagraph Design and Analysis (SCDA) initiative led by the Exoplanet Exploration Program at NASAs Jet Propulsion Laboratory. As a result, several types of high-performance designs have emerged that make use of dual deformable mirrors to (1) correct for optical aberrations and (2) suppress diffracted starlight from obstructions and discontinuities in the telescope pupil. However, the algorithms used to compute the optimal deformable mirror surface tend to be computationally intensive, prohibiting large scale design surveys. Here, we utilize the Fast Linearized Coronagraph Optimizer (FALCO), a tool that allows for rapid optimization of deformable mirror shapes, to explore trade-offs in coronagraph designs for obstructed and segmented space telescopes. We compare designs for representative shaped pupil Lyot and vortex coronagraphs, two of the most promising concepts for the LUVOIR space mission concept. We analyze the optical performance of each design, including their throughput and ability to passively suppress light from partially resolved stars in the presence of low-order aberrations. Our main result is that deformable mirror based apodization can sufficiently suppress diffraction from support struts and inter-segment gaps whose widths are on the order of $sim$0.1% of the primary mirror diameter to detect Earth-sized planets within a few tens of milliarcseconds from the star.
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