Exoplanet imaging and spectroscopy are now routinely achieved by dedicated instruments on large ground-based observatories (e.g. Gemini/GPI, VLT/SPHERE, or Subaru/SCExAO). In addition to extreme adaptive optics (ExAO) and post-processing methods, these facilities make use of the most advanced coronagraphs to suppress light of an observed star and enable the observation of circumstellar environments. The Apodized Pupil Lyot Coronagraph (APLC) is one of the leading coronagraphic baseline in the current generation of instruments. This concept combines a pupil apodization, an opaque focal plane mask (FPM), and a Lyot stop. APLC can be optimized for a range of applications and designs exist for on-axis segmented aperture telescopes at $10^{10}$ contrast in broadband light. In this communication, we propose novel designs to push the limits of this concept further by modifying the nature of the FPM from its standard opaque mask to a smaller size occulting spot surrounded by circular phase shifting zones. We present the formalism of this new concept which solutions find two possible applications: 1) upgrades for the current generation of ExAO coronagraphs since these solutions remain compatible with the existing designs and will provide better inner working angle, contrast and throughput, and 2) coronagraphy at $10^{10}$ contrast for future flagship missions such as LUVOIR, with the goal to increase the throughput of the existing designs for the observation of Earth-like planets around nearby stars.
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