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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 67, Iss. 3 — Mar. 1, 1977
  • pp: 370–375

Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements

David L. Fried  »View Author Affiliations

JOSA, Vol. 67, Issue 3, pp. 370-375 (1977)

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The problem of fitting a wave-front distortion estimate to a (single - instant) set of phase-difference measurements has been formulated as an unweighted least-square problem. The least-square equations have been developed as a set of simultaneous equations for a square array of phase-difference sensors, with phase estimates at the corner of each measurement element. (This corresponds to the standard Hartmann configuration and to one version of a shearing interferometer of a predetection compensation wave-front sensor.) The noise dependence in the solution of the simultaneous equations is found to be expressible in terms of the solution to a particular version of the measurement inputs to the simultaneous equation, a sort of "Green’s-function" solution. The noise version of the simultaneous equations is solved using relaxation techniques for array sizes from 4 × 4 to 40 × 40 phase estimation points, and the mean-square wave-front error calculated as a function of the mean-square phase-difference measurement error. It is found that the results can be approximated within a fraction of a percent accuracy by 〈(δΦ)2〉 = 0.6558[1 + 0.24441n(N22pd, where 〈(δΦ)2〉 is the mean-square error (rad2) in the estimation of the wave-front distortion, for a square array consisting of N2 square subaperture elements over which two phase-difference measurements are made—one phase difference across the x dimension and the other difference across the y dimension. Here σ2pd is the mean-square error (rad2) in each phase-difference measurement.

© 1977 the Optical Society of America

David L. Fried , "Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements," J. Opt. Soc. Am. 67, 370-375 (1977)

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  1. It should be noted that while there has been an extensive body of work, and some previously published results pertaining to adaptive optics design and wave-front distortion sensing, a few examples of which we list in the references below, almost none of this work considered the approach of measuring an array of phase differences and thus had to consider the data fitting problem that we consider here.
  2. T. E. O'Meara, U. S. Patent 3,764,213, "Return Wave Phase Control Adaptive Array," (9 October 1973).
  3. J. W. Hardy, J. Feinleib, and J. Wyant, "Real Time Phase Correction of Optical Imaging Systems," in the Digest of Technical Papers of the OSA Topical Meeting on Optical Propagation Through Turbulence, 9–11 July 1974, Univ. of Colorado, Boulder, Colo. (This paper considers use of a shearing interferometer to measure phase differences, but there is no discussion of the wave-front fitting problem.)
  4. W. T. Cathey, C. L. Hayes, W. C. Davis, and V. F. Pizzurro, "Compensation for Atmospheric Phase Effects At 10.6 µ," Appl. Opt. 9, 701 (1970).
  5. R. A. Muller and A. Buffington, "Real-time correction of atmospherically degraded telescope images through image sharpening," J. Opt. Soc. Am. 64, 1200–1210 (1974).
  6. C. A. Primmerman and D. G. Fouche, "Thermal-Blooming Compensation: Experimental Observation Using a Deformable Mirror System," Appl. Opt. 15, 900 (1976).
  7. W. B. Bridges, P. T. Brunner, S. P. Lazzara, T. A. Nussmeier, T. R. O'Meara, J. A. Sanguinet, and W. P. Brown, Jr., "Coherent Optical Adaptive Techniques," Appl. Opt. 13, 291 (1974).
  8. J. E. Pearson, "Atmospheric Turbulence Compensation Using Coherent Optical Adaptive Techniques," Appl. Opt. 15, 662 (1976).

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