OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 13, Iss. 8 — Aug. 1, 1996
  • pp: 1670–1682

Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination

T. E. Gureyev and K. A. Nugent  »View Author Affiliations


JOSA A, Vol. 13, Issue 8, pp. 1670-1682 (1996)
http://dx.doi.org/10.1364/JOSAA.13.001670


View Full Text Article

Enhanced HTML    Acrobat PDF (345 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

In a previous paper [ J. Opt. Soc. Am. A 12, 1932 ( 1995)] we presented a method for phase recovery with the transport-of-intensity equation by use of a series expansion. Here we develop a different method for the solution of this equation, which allows recovery of the phase in the case of nonuniform illumination. Though also based on the orthogonal series expansion, the new method does not require any separate boundary conditions and can be more easily adjusted for apertures of various shapes. The discussion is primarily for the case of a circular aperture and Zernike polynomials, but we also outline the solution for a rectangular aperture and Fourier harmonics. The latter example may have some substantial advantages, given the availability of the fast Fourier transform.

© 1996 Optical Society of America

History
Original Manuscript: June 21, 1995
Revised Manuscript: November 28, 1995
Manuscript Accepted: February 7, 1996
Published: August 1, 1996

Citation
T. E. Gureyev and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination," J. Opt. Soc. Am. A 13, 1670-1682 (1996)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-13-8-1670


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Sec. 9.10.
  2. R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990). [CrossRef]
  3. T. C. Wedberg, J. J. Stamnes, “Comparison of phase retrieval methods for optical diffraction tomography,” Pure Appl. Opt. 4, 39–54 (1995). [CrossRef]
  4. R. K. Tyson, Principles of Adaptive Optics (Academic, San Diego, Calif., 1991).
  5. J. Susini, R. Baker, M. Krumrey, W. Schwegle, Å. Kvick, “Adaptive x-ray mirror prototype: first results,” Rev. Sci. Instrum. 66, 2048–2052 (1995). [CrossRef]
  6. A. W. Dreher, J. F. Bille, R. N. Weinreb, “Active optical depth resolution improvement of the laser tomographic scanner,” Appl. Opt. 28, 804–808 (1989). [CrossRef] [PubMed]
  7. M. V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Probl. 11, 1–28 (1995). [CrossRef]
  8. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,”J. Opt. Soc. Am. 73, 1434–1441 (1983). [CrossRef]
  9. M. R. Teague, “Irradiance moments: their propagation and use for unique retrieval of phase,”J. Opt. Soc. Am. 72, 1199–1209 (1982). [CrossRef]
  10. F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225 (1988). [CrossRef] [PubMed]
  11. F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt. 29, 1402–1403 (1990). [CrossRef] [PubMed]
  12. C. Roddier, F. Roddier, “Wave-front reconstruction from defocused images and the testing of ground-based optical telescopes,” J. Opt. Soc. Am. A 10, 2277–2287 (1993). [CrossRef]
  13. T. E. Gureyev, A. Roberts, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A 12, 1932–1941 (1995). [CrossRef]
  14. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 9.2.
  15. S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).
  16. R. J. Noll, “Zernike polynomials and atmospheric turbulence,”J. Opt. Soc. Am. 66, 207–211 (1976). [CrossRef]
  17. P. Hickson, “Wave-front curvature sensing from a single defocused image,” J. Opt. Soc. Am. A 11, 1667–1673 (1994). [CrossRef]
  18. T. E. Gureyev, A. Roberts, K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995). [CrossRef]
  19. O. A. Oleinik, E. V. Radkevič, Second Order Equations with Non-negative Characteristic Form (Plenum, New York, 1973). [CrossRef]
  20. L. V. Kantorovich, V. I. Krylov, Approximate Methods of Higher Analysis (Wiley, New York, 1964).
  21. A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).
  22. P. C. Sabatier, ed., Inverse Methods in Action (Springer, Berlin, 1990). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited