  We study the way Lyot coronagraphs with unapodized entrance pupils respond to
small, low order phase aberrations. This study is applicable to ground-based
adaptive optics coronagraphs operating at 90% and higher Strehl ratios, as well
as to some space-based coronagraphs with intrinsically higher Strehl ratio
imaging. We utilize a second order expansion of the monochromatic point-spread
function (written as a power spectrum of a power series in the phase aberration
over clear aperture) to derive analytical expressions for the response of a
`band-limited' Lyot coronagraph (BLC) to small, low order, phase aberrations.
The BLC possesses a focal plane mask with an occulting spot whose opacity
profile is a spatially band-limited function rather than a hard-edged, opaque
disk. The BLC is, to first order, insensitive to tilt and astigmatism.
Undersizing the stop in the re-imaged pupil plane (the Lyot plane) following
the focal plane mask can alleviate second order effects of astigmatism, at the
expense of system throughput and angular resolution. The optimal degree of such
undersizing depends on individual instrument designs and goals. Our analytical
work engenders physical insight, and complements existing numerical work on
this subject. Our methods can be extended to treat the passage of higher order
aberrations through band-limited Lyot coronagraphs, by using our polynomial
decomposition or an analogous Fourier approach.
