High-contrast imaging from space must overcome two major noise sources to successfully detect a terrestrial planet angularly close to its parent star: photon noise from diffracted starlight and speckle noise from starlight scattered by instrumentally generated wave front perturbation. Coronagraphs tackle only the photon noise contribution by reducing diffracted starlight at the location of a planet. Speckle noise should be addressed with adaptive optics systems. Following the tracks of Malbet, Yu, and Shao, we develop in this paper two analytical methods for wave front sensing and control that aims at creating "dark holes," i.e., areas of the image plane cleared of speckles, assuming an ideal coronagraph and small aberrations. The first method, "speckle field nulling," is a fast FFT-based algorithm that requires the deformable-mirror influence functions to have identical shapes. The second method, "speckle energy minimization," is more general and provides the optimal deformable mirror shape via matrix inversion. With an N × N deformable mirror, the size of the matrix to be inverted is either N2 × N2 in the general case or only N × N if the influence functions can be written as the tensor product of two one-dimensional functions. Moreover, speckle energy minimization makes it possible to trade off some of the dark hole area against an improved contrast. For both methods, complex wave front aberrations (amplitude and phase) are measured using just three images taken with the science camera (no dedicated wave front sensing channel is used); therefore, there are no noncommon path errors. We assess the theoretical performance of both methods with numerical simulations including realistic speckle noise and experimental influence functions. We find that these speckle-nulling techniques should be able to improve the contrast by several orders of magnitude.
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