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Applied Optics

Applied Optics


  • Vol. 39, Iss. 2 — Jan. 10, 2000
  • pp: 250–268

Generalized wave-front reconstruction algorithm applied in a Shack–Hartmann test

Weiyao Zou and Zhenchao Zhang  »View Author Affiliations

Applied Optics, Vol. 39, Issue 2, pp. 250-268 (2000)

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A generalized numerical wave-front reconstruction method is proposed that is suitable for diversified irregular pupil shapes of optical systems to be measured. That is, to make a generalized and regular normal equation set, the test domain is extended to a regular square shape. The compatibility of this method is discussed in detail, and efficient algorithms (such as the Cholesky method) for solving this normal equation set are given. In addition, the authors give strict analyses of not only the error propagation in the wave-front estimate but also of the discretization errors of this domain extension algorithm. Finally, some application examples are given to demonstrate this algorithm.

© 2000 Optical Society of America

OCIS Codes
(010.7350) Atmospheric and oceanic optics : Wave-front sensing
(100.5070) Image processing : Phase retrieval
(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure
(220.4840) Optical design and fabrication : Testing

Original Manuscript: March 3, 1999
Revised Manuscript: July 14, 1999
Published: January 10, 2000

Weiyao Zou and Zhenchao Zhang, "Generalized wave-front reconstruction algorithm applied in a Shack–Hartmann test," Appl. Opt. 39, 250-268 (2000)

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  1. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980). [CrossRef]
  2. K. R. Freischlad, C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. 3, 1852–1861 (1986). [CrossRef]
  3. J. W. Hardy, J. E. Lefebvre, C. L. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977). [CrossRef]
  4. D. L. Fried, “Least-squares fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977). [CrossRef]
  5. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977). [CrossRef]
  6. R. J. Noll, “Phase estimates from slope-type wavefront sensors,” J. Opt. Soc. Am. 68, 139–140 (1978). [CrossRef]
  7. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase difference,” J. Opt. Soc. Am. 69, 393–399 (1979). [CrossRef]
  8. J. Hermann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980). [CrossRef]
  9. F. Roddier, C. Roddier, “Wavefront reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327 (1991). [CrossRef] [PubMed]
  10. R. G. Lane, M. Tallon, “Wave-front reconstruction using a Shack–Hartmann sensor,” Appl. Opt. 31, 6902–6908 (1992). [CrossRef] [PubMed]
  11. I. Ghozeil, “Hartmann and other screen tests,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978).
  12. E. T. Pearson, “Hartmann test data reduction,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. SPIE1236, 628–632 (1990). [CrossRef]
  13. K. Freischlad, “Wavefront integration from difference data,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 212–218 (1992).
  14. D. Su, S. Jiang, L. Shao, “A sort of algorithm of wavefront reconstruction for Shack–Hartmann test,” in Proceedings of the European Southern Observatory Conference on Progress in Telescope and Instrumentation Technologies, M. H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 289–292.
  15. D. Su, S. Jiang, W. Zou, S. Yang, S. Yang, H. Zhang, Q. Zhu, “Experiment system of thin-mirror active optics,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 609–621 (1994). [CrossRef]
  16. C. Chen, Equations of Mathematical Physics (China High Educational Press, Beijing, 1992).
  17. J. Stoer, R. Bulirsch, Introduction to Numerical Analysis (Springer-Verlag, New York, 1994) (Chinese edition).
  18. R. J. Zielinski, B. M. Levine, B. McNeil, “Hartmann sensors for optical testing,” in Optical Manufacturing and Testing II, H. Stahl, ed., Proc. SPIE3134, 398–406 (1997). [CrossRef]
  19. W. Zou, “Figure control of large segmented mirror telescope,” Master of Science thesis (Nanjing Astronomical Instruments Research Center, Chinese Academy of Sciences, Nanjing, China, 1996), p. 7.
  20. B. Jian, “New method for shearing wavefront reconstruction and its applications in wave aberration evaluation,” Ph.D. dissertation (Zhejiang University, Hangzhou, China, 1995), p. 6.
  21. M. Zhang, Department of Applied Mathematics, Southeast University, Nanjing 210096, China (personal communication, 1999).

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