SPHERE (which stands for Spectro-Polarimetric High-contrast Exoplanet REsearch) is a second-generation Very Large Telescope (VLT) instrument dedicated to high-contrast direct imaging of exoplanets which firstlight is scheduled for 2011. Within this complex instrument one of the central components is the apodized Lyot coronagraph (ALC). The present paper reports on the most interesting aspects and results of the whole numerical study made during the design of the ALC for SPHERE/VLT. The method followed for this study is purely numerical, but with an end-to-end approach which is largely fed by a number of instrumental feedbacks. The results obtained and presented in this paper firstly permit to finalize the optical design before Exp Astron (2011) 30:39-58 laboratory performance testing of the ALC being built for SPHERE/VLT (see paper II "Laboratory tests and performances"), but will also hopefully help conceiving future other instruments alike, for example within the very promising extremely large telescope perspective.
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