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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • 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)

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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

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

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)

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  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]

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