Coronagraphs of the apodized pupil and shaped pupil varieties use the Fraunhofer diffraction properties of amplitude masks to create regions of high contrast in the vicinity of a target star. Here we present a hybrid coronagraph architecture in which a binary, hard-edged shaped pupil mask replaces the gray, smooth apodizer of the apodized pupil Lyot coronagraph (APLC). For any contrast and bandwidth goal in this configuration, as long as the prescribed region of contrast is restricted to a finite area in the image, a shaped pupil is the apodizer with the highest transmission. We relate the starlight cancellation mechanism to that of the conventional APLC. We introduce a new class of solutions in which the amplitude profile of the Lyot stop, instead of being fixed as a padded replica of the telescope aperture, is jointly optimized with the apodizer. Finally, we describe shaped pupil Lyot coronagraph (SPLC) designs for the baseline architecture of the WFIRST-AFTA coronagraph. These SPLCs help to enable two scientific objectives of the WFIRST-AFTA mission: (i) broadband spectroscopy to characterize exoplanet atmospheres in reflected starlight and (ii) debris disk imaging. and second a "Lyot stop" to block the outer edge of the re-collimated on-axis beam before it is reimaged. 12 To take advantage of the diffraction-limited imaging capabilities of high-order adaptive optics (AO) systems, beginning in the 1990s classical Lyot designs were revised for high-contrast stellar coronagraphy. [13][14][15][16][17][18] Through Fourier optical analysis and modeling, researchers soon discovered the remarkable performance benefits of apodizing the entrance pupil of a coronagraph. [19][20][21] Since then, the transmission profile of this apodizer mask has been a topic of vigorous study. [22][23][24][25][26][27] One resulting family of designs, the apodized pupil Lyot coronagraph (APLC), has been successfully integrated with several AO-fed cameras to facilitate deep observations of young exoplanetary systems at near-infrared wavelengths.As an alternative to a coronagraph with two or more mask planes, pupil apodization by itself is perhaps the simplest and oldest way to reject unwanted starlight from a telescope image. 28,29 Fraunhofer diffraction theory dictates the way any change in the shape or transmission profile of the entrance pupil redistributes a star's energy in the image plane. This relationship can be used to design an apodizer whose point spread function (PSF) has a zone of high contrast near the star, without additional coronagraph masks. This is the shaped pupil approach developed by N. J. Kasdin and collaborators, who pioneered the optimization of apodizers with binary-valued transmission. [30][31][32][33] In recent years, shaped pupil solutions have evolved to work around arbitrary two-dimensional telescope apertures, in parallel with similar breakthroughs in APLC design. [34][35][36][37][38][39] The relative simplicity of a single mask, however, comes with a sacrifice in how close the dark search region can be push...