In order to maximize the potential of the next-generation of large space-based observatory to detect and characterize Earth-like exoplanets, coronagraphs must be designed that can obtain high (~10-10) contrasts in the presence of realistic low-order aberrations and finite stellar diameters. Unfortunately, for telescopes with central obstructions, maintaining aberration robustness for most coronagraph designs entails significant losses in either throughput or inner working angle. This has resulted in stringent limitations for exo-Earth yields on planned future on-axis telescopes, such as LUVOIR-A. We address this limitation with modified versions of apodized charge 6 and charge 8 vortex coronagraphs which use multiple stages of focal plane mask. These multi-stage apodized vortex coronagraphs (MSAVCs) produce dark holes with contrast <10-10 and mitigate the flux due to tip/tilt offsets as large as 0.05 λ/D while obtaining core throughputs that are a factor of ~2 higher than similarly constrained single-stage apodized vortex coronagraphs. The MSAVCs we present are robust to several low-order aberrations, and we discuss the possibility of explicitly constraining low-order aberrations further. Furthermore, we demonstrate mitigating flux due to misalignment between focal plane masks, thus overcoming a significant hurdle in implementing multi-stage vortex designs. By using a parametric expression to estimate the yield of a charge 6 MSAVC for a 10% central obstruction relative to an off-axis charge 6 vortex coronagraph on an 8-m telescope, we estimate it may be possible to retain ~67% of the off-axis yield
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