  We present a study of Lyot style (i.e., classical, band-limited, and Gaussian
occulter) coronagraphy on extremely large, highly-segmented telescopes. We show
that while increased telescope diameter is always an advantage for high dynamic
range science (assuming wavefront errors have been corrected sufficiently
well), segmentation itself sets a limit on the performance of Lyot
coronagraphs. Diffraction from inter-segment gaps sets a floor to the
achievable extinction of on-axis starlight with Lyot coronagraphy. We derive an
analytical expression for the manner in which coronagraphic suppression of an
on-axis source decreases with increasing gap size when the segments are placed
in a spatially periodic array over the telescope aperture, regardless of the
details of the arrangement. A simple Lyot stop masking out pupil edges produces
good extinction of the central peak in the point-spread function (PSF), but
leaves satellite images caused by inter-segment gaps essentially unaffected.
Masking out the bright segment gaps in the Lyot plane with a reticulated mask
reduces the satellite images'intensity to a contrast of 5x10^{-9} on a 30 m
telescope with 10 mm gaps, at the expense of an increase in the brightness of
the central peak. The morphology of interesting targets will dictate which Lyot
stop geometry is preferable: the reticulated Lyot stop produces a conveniently
uni-modal PSF, whereas a simple Lyot stop produces an extended array of
satellite spots. A cryogenic reticulate Lyot stop will also benefit both direct
and coronagraphic mid-IR imaging.
