The Apodized Pupil Lyot Coronagraph (APLC) is a diffraction suppression system installed in the recently deployed instruments Palomar/P1640, Gemini/GPI, and VLT/SPHERE to allow direct imaging and spectroscopy of circumstellar environments. Using a prolate apodization, the current implementations offer raw contrasts down to $10^{-7}$ at 0.2 arcsec from a star over a wide bandpass (20%), in the presence of central obstruction and struts, enabling the study of young or massive gaseous planets. Observations of older or lighter companions at smaller separations would require improvements in terms of inner working angle (IWA) and contrast, but the methods originally used for these designs were not able to fully explore the parameter space. We here propose a novel approach to improve the APLC performance. Our method relies on the linear properties of the coronagraphic electric field with the apodization at any wavelength to develop numerical solutions producing coronagraphic star images with high-contrast region in broadband light. We explore the parameter space by considering different aperture geometries, contrast levels, dark-zone sizes, bandpasses, and focal plane mask sizes. We present an application of these solutions to the case of Gemini/GPI with a design delivering a $10^{-8}$ raw contrast at 0.19 arcsec and offering a significantly reduced sensitivity to low-order aberrations compared to the current implementation. Optimal solutions have also been found to reach $10^{-10}$ contrast in broadband light regardless of the telescope aperture shape (in particular the central obstruction size), with effective IWA in the $2-3.5lambda/D$ range, therefore making the APLC a suitable option for the future exoplanet direct imagers on the ground or in space.
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