  We describe the symmetries present in the point-spread function (PSF) of an
optical system either located in space or corrected by an adaptive o to Strehl
ratios of about 70% and higher. We present a formalism for expanding the PSF to
arbitrary order in terms of powers of the Fourier transform of the residual
phase error, over an arbitrarily shaped and apodized entrance aperture. For
traditional unapodized apertures at high Strehl ratios, bright speckles pinned
to the bright Airy rings are part of an antisymmetric perturbation of the
perfect PSF, arising from the term that is first order in the residual phase
error. There are two symmetric second degree terms. One is negative at the
center, and, like the first order term, is modulated by the perfect image's
field strength -- it reduces to the Marechal approximation at the center of the
PSF. The other is non-negative everywhere, zero at the image center, and can be
responsible for an extended halo -- which limits the dynamic range of faint
companion detection in the darkest portions of the image. In regimes where one
or the other term dominates the speckles in an image, the symmetry of the
dominant term can be exploited to reduce the effect of those speckles,
potentially by an order of magnitude or more. We demonstrate the effects of
both secondary obscuration and pupil apodization on the structure of residual
speckles, and discuss how these symmetries can be exploited by appropriate
telescope and instrument design, observing strategies, and filter bandwidths to
improve the dynamic range of high dynamic range AO and space-based
observations. Finally, we show that our analysis is relevant to high dynamic
range coronagraphy.
