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Applied Optics

Applied Optics


  • Vol. 43, Iss. 12 — Apr. 20, 2004
  • pp: 2418–2430

Optical Phase Retrieval by Use of First Born- and Rytov-Type Approximations

Timur E. Gureyev, Timothy J. Davis, Andrew Pogany, Sheridan C. Mayo, and Stephen W. Wilkins  »View Author Affiliations

Applied Optics, Vol. 43, Issue 12, pp. 2418-2430 (2004)

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The first Born and Rytov approximations of scattering theory are introduced in their less familiar near-field versions. Two algorithms for phase retrieval based on these approximations are then described. It is shown theoretically and by numerical simulations that, despite the differences in their formulation, the two algorithms deliver fairly similar results when used for optical phase retrieval in the near and intermediate fields. The algorithms are applied to derive explicit solutions to four phase-retrieval problems of practical relevance to quantitative phase-contrast imaging and tomography. An example of successful phase reconstruction by use of the Born-type algorithm with an experimental x-ray image is presented.

© 2004 Optical Society of America

OCIS Codes
(090.0090) Holography : Holography
(100.3190) Image processing : Inverse problems
(100.5070) Image processing : Phase retrieval
(110.7440) Imaging systems : X-ray imaging
(120.5050) Instrumentation, measurement, and metrology : Phase measurement
(340.7440) X-ray optics : X-ray imaging

Timur E. Gureyev, Timothy J. Davis, Andrew Pogany, Sheridan C. Mayo, and Stephen W. Wilkins, "Optical Phase Retrieval by Use of First Born- and Rytov-Type Approximations," Appl. Opt. 43, 2418-2430 (2004)

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  1. K. A. Nugent, D. Paganin, and T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27–32 (2001).
  2. D. R. Luke, J. V. Burke, and R. G. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM (Soc. Ind. Appl. Math.) Rev. 44, 169–224 (2002).
  3. R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
  4. M. Born and E. Wolf, Principles of Optics, 7th ed.(Cambridge U. Press, Cambridge, UK, 1999).
  5. F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9, 686–698 (1942).
  6. F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt. 29, 1402–1403 (1990).
  7. D. T. Miller, “Retinal imaging and vision at the frontiers of adaptive optics,” Phys. Today 53, 31–36 (2000).
  8. J. M. Cowley, Diffraction Physics, 3rd ed.(Elsevier, Amsterdam, 1995).
  9. D. Van Dyck and W. Coene, “A new procedure for wave function restoration in high resolution electron microscopy,” Optik (Stuttgart) 77, 125–128 (1987).
  10. A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
  11. S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
  12. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
  13. T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Physics D 32, 563–567 (1999).
  14. P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
  15. J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
  16. M. H. Maleki and A. J. Devaney, “Phase-retrieval and intensity-only reconstruction algorithms for optical diffraction tomography,” J. Opt. Soc. Am. A 10, 1086–1092 (1993).
  17. M. H. Maleki and A. J. Devaney, “Noniterative reconstruction of complex-valued objects from two intensity measurements,” Opt. Eng. 33, 3243–3253 (1994).
  18. T. C. Wedberg and J. J. Stamnes, “Comparison of phase-retrieval methods for optical diffraction tomography,” Pure Appl. Opt. 4, 39–54 (1995).
  19. J. C. H. Spence, M. Howells, L. D. Marks, and J. Miao, “Lensless imaging: a workshop on ‘new approaches to the phase problem for non-periodic objects, ’” Ultramicroscopy 90, 1–6 (2001).
  20. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  21. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
  22. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
  23. J. C. Spence, U. Weierstall, and M. Howells, “Phase recovery and lensless imaging by iterative methods in optical, x-ray and electron diffraction,” Philos. Trans. R. Soc. London Ser. A 360, 875–895 (2002).
  24. I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (2001).
  25. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
  26. S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, and A. W. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2002).
  27. A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
  28. J. P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik (Stuttgart) 49, 121–125 (1977).
  29. M. Op de Beeck, D. Van Dyck, and W. Coene, “Wave function reconstruction in HRTEM: the parabola method,” Ultramicroscopy 64, 167–183 (1996).
  30. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991).
  31. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (Dover, New York, 1967).
  32. J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, and M. A. O’Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
  33. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
  34. D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
  35. T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multi-energy x-rays,” Phys. Rev. Lett. 86, 5827–5830 (2001).
  36. A. N. Tichonov and V. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).
  37. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Bristol, UK, 1998).
  38. T. J. Davis, “A unified treatment of small-angle x-ray scattering, x-ray refraction and absorption using the Rytov approximation,” Acta Crystallogr. Sect. A 50, 686–690 (1994).

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