Static point-spread function correction dominating higher-order speckle terms at high adaptive correction

At high adaptive correction, the randomly shifting speckles familiar in conventional astronomical imaging become organized into patterns with distinct regularities that may permit partial suppression of the image noise they produce. Mathematically, the phase exponential in the Fourier-optical imaging expression may be expanded in a Taylor series in remnant phase $\varphi$, which is small at very high correction, leading to a perturbed point-spread function (PSF) that is a sum of algebraic terms, each of distinct spatial symmetry. At sufficiently high correction, one need deal with only a few of the lowest-order terms. A first-order expansion gives an ideal PSF plus two terms, linear and quadratic, describing the two brightest, physically most relevant kinds of speckle. A second-order expansion gives three new terms, the brightest of which is primarily a static correction to the PSF, with a much smaller true speckle component. When the correction is great enough to isolate individual speckle terms, the two terms from the first-order expansion alone determine the essential physics. A general observational strategy is outlined for reducing speckle noise in highly corrected companion searches, dominated by a few speckle terms of definite spatial symmetry.