8 September 1998 Nature of the branch point problem in adaptive optics
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For this discussion of the nature of the problem presented to an adaptive optics system by branch points I shall assume that the adaptive optics system uses a unit shear shearing interferometer to make the basic measurements, and uses a standard least mean square reconstructor algorithm to convert the measurement values to commands that are sent to a deformable mirror. The shearing interferometer senses the fact that optical field at adjacent positions in the aperture are not perfectly coherent with respect to each other.* The processing of the data from the shearing interferometer should result in commands being sent to the deformable mirror which will cause the optical field to be adjusted in such a way the (nearly) perfect coherence will exist between all pairs of adjacent positions in the aperture—which means that there will be coherence between all of the optical field in the aperture. I shall show that under conditions of optical field distortion that are commonly characterized by the statement that there are branch points present, the adaptive optics system fails to utilize all of data gathered by the wave front sensor and that as a consequence there is some degree of incoherence remaining in the optical field after it has been reflected from the deformable mirror, and will show how to quantitatively treat this residual error.
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David L. Fried, David L. Fried, } "Nature of the branch point problem in adaptive optics", Proc. SPIE 3381, Airborne Laser Advanced Technology, (8 September 1998); doi: 10.1117/12.323953; https://doi.org/10.1117/12.323953

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