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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 21, Iss. 8 — Aug. 30, 2004
  • pp: 1407–1416

Reconstruction of spatial, phase, and coherence properties of light

Miroslav Ježek and Zdeněk Hradil  »View Author Affiliations

JOSA A, Vol. 21, Issue 8, pp. 1407-1416 (2004)

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Image reconstruction of partially coherent light is interpreted as quantum-state reconstruction. An efficient method based on the maximum-likelihood estimation is proposed for acquiring information from blurred intensity measurements affected by noise. Connections with incoherent-image restoration are pointed out. The feasibility of the method is demonstrated numerically. Spatial and correlation details significantly below the diffraction limit are revealed in the reconstructed pattern.

© 2004 Optical Society of America

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(100.3010) Image processing : Image reconstruction techniques
(100.5070) Image processing : Phase retrieval
(100.6640) Image processing : Superresolution
(100.6950) Image processing : Tomographic image processing
(110.4980) Imaging systems : Partial coherence in imaging

Original Manuscript: October 9, 2003
Revised Manuscript: February 5, 2004
Manuscript Accepted: February 5, 2004
Published: August 1, 2004

Miroslav Ježek and Zdeněk Hradil, "Reconstruction of spatial, phase, and coherence properties of light," J. Opt. Soc. Am. A 21, 1407-1416 (2004)

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  1. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. X.
  2. J. Peřina, Coherence of Light, 2nd ed. (Reidel, Dordrecht, The Netherlands, 1985).
  3. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  4. M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, London, 1998).
  5. P. Jacquinot, B. Roizen-Dossier, “Apodisation,” in Progress in Optics, Vol. III, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Chap. 2, pp. 29–186.
  6. M. Dyba, S. W. Hell, “Focal spots of size λ/23 open up far-field fluorescence microscopy at 33 nm axial resolution,” Phys. Rev. Lett. 88, 163901 (2002). [CrossRef]
  7. E. H. K. Stelzer, “Beyond the diffraction limit?” Nature (London) 417, 806–807 (2002). [CrossRef]
  8. M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994). [CrossRef] [PubMed]
  9. W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55–59 (1972). [CrossRef]
  10. A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1970).
  11. P. A. M. Dirac, The Principles of Quantum Mechanics, 3rd ed. (Clarendon, Oxford, UK, 1958).
  12. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932). [CrossRef]
  13. J. Ville, “Theorie et applications de la notion de signal analytique,” Cables Transm. 2A, 61–74 (1948).
  14. A. S. Holevo, “Statistical decision theory for quantum systems,” J. Multivar. Anal. 3, 337–394 (1973). [CrossRef]
  15. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, New York, 1976).
  16. D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962). [CrossRef]
  17. S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. Assoc. Comput. Mach. 10, 97–101 (1963). [CrossRef]
  18. K. Miller, “Least squares methods for ill-posed problems with a prescribed bound,” SIAM J. Math. Anal. 1, 52–74 (1970). [CrossRef]
  19. A. N. Tikhonov, V. Y. Arsenin, Solution of Ill-Posed Problems (Wiley, New York, 1977).
  20. G. E. Backus, F. Gilbert, “The resolving power of growth earth data,” Geophys. J. R. Astron. Soc. 16, 169–205 (1968). [CrossRef]
  21. G. E. Backus, F. Gilbert, “Uniqueness in the inversion of inaccurate gross earth data,” Philos. Trans. R. Soc. London Ser. A 266, 123–192 (1970). [CrossRef]
  22. R. W. Gerchberg, “Superresolution through error energy reduction,” Opt. Acta 21, 709–720 (1974). [CrossRef]
  23. B. R. Frieden, “Band-unlimited reconstruction of optical objects and spectra,” J. Opt. Soc. Am. 57, 1013–1019 (1967). [CrossRef]
  24. B. R. Frieden, “On arbitrarily perfect imagery with a finite aperture,” Opt. Acta 16, 795–807 (1969). [CrossRef]
  25. G. Toraldo di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am. 59, 799–804 (1969). [CrossRef] [PubMed]
  26. B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions,” in Progress in Optics, Vol. IX, E. Wolf, ed. (North-Holland, Amsterdam, 1971), Chap. 8, pp. 311–407.
  27. J. Peřina, V. Peřinová, “Optical imaging with partially coherent non-thermal light. II. Reconstruction of object from its image and similarity between object and its image,” Opt. Acta 16, 309–320 (1969). [CrossRef]
  28. J. Peřina, V. Peřinová, Z. Braunerová, “Super-resolution in linear systems with noise,” Opt. Appl. VII, 79–83 (1977).
  29. R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The application of dispersion relations (Hilbert transforms) to phase retrieval,” J. Phys. D Appl. Phys. 7, L65–L68 (1974). [CrossRef]
  30. R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976). [CrossRef]
  31. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  32. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]
  33. D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part I—theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982). [CrossRef]
  34. W. Kim, “Two-dimensional phase retrieval using a window function,” Opt. Lett. 26, 134–136 (2001). [CrossRef]
  35. R. W. Gerchberg, “A new approach to phase retrieval of a wave front,” J. Mod. Opt. 49, 1185–1196 (2002). [CrossRef]
  36. H. H. Bauschke, P. L. Combettes, D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002). [CrossRef]
  37. M. R. Teague, “Irradiance moments: their propagation and use for unique retrieval of phases,” J. Opt. Soc. Am. 72, 1199–1209 (1982). [CrossRef]
  38. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983). [CrossRef]
  39. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays, Phys. Rev. Lett. 77, 2961–2964 (1996). [CrossRef] [PubMed]
  40. D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998). [CrossRef]
  41. L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001). [CrossRef]
  42. M. J. Bastiaans, K. B. Wolf, “Phase reconstruction from intensity measurements in linear systems,” J. Opt. Soc. Am. A 20, 1046–1049 (2003). [CrossRef]
  43. D. F. V. James, G. S. Agarwal, “Generalized Radon transform for tomographic measurement of short pulses,” J. Opt. Soc. Am. B 12, 704–708 (1995). [CrossRef]
  44. X. Liu, K.-H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19–30 (2003). [CrossRef]
  45. J. Bertrand, P. Bertrand, “A tomographic approach to Wigner’s function,” Found. Phys. 17, 397–405 (1987). [CrossRef]
  46. K. Vogel, H. Risken, “Determination of quasiprobability distribution in terms of probability distribution for the rotated quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989). [CrossRef] [PubMed]
  47. D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993). [CrossRef] [PubMed]
  48. C. Kurtsiefer, T. Pfau, J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms,” Nature (London) 386, 150–153 (1997). [CrossRef]
  49. G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, New York, 1980).
  50. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988).
  51. F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986).
  52. S. Quabis, R. Dorn, M. Eberler, O. Göckl, G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001). [CrossRef]
  53. G. M. D’Ariano, C. Macchiavello, M. G. A. Paris, “Detection of the density matrix through optical homodyne tomography without filtered back projection,” Phys. Rev. A 50, 4298–4302 (1994). [CrossRef] [PubMed]
  54. G. M. D’Ariano, C. Macchiavello, M. G. A. Paris, “A fictitious photons method for tomographic imaging,” Opt. Commun. 129, 6–12 (1996). [CrossRef]
  55. S. Schiller, G. Breitenbach, S. F. Pereira, T. Muller, J. Mlynek, “Quantum statistics of the squeezed vacuum by measurement of the density matrix in the number state representation,” Phys. Rev. Lett. 77, 2933–2936 (1996). [CrossRef] [PubMed]
  56. G. Breitenbach, S. Schiller, J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature (London) 387, 471–475 (1997). [CrossRef]
  57. A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402–1-4 (2001). [CrossRef] [PubMed]
  58. S. Mancini, V. I. Man’ko, P. Tombesi, “Wigner function and probability distribution for shifted and squeezed quadratures,” Quantum Semiclassic. Opt. 7, 615–623 (1995). [CrossRef]
  59. M. A. Man’ko, “Electromagnetic signal processing and noncommutative tomography,” J. Russ. Laser Res. 23, 433–448 (2002). [CrossRef]
  60. C. W. Helstrom, “Image restoration by the method of least squares,” J. Opt. Soc. Am. 57, 297–303 (1967). [CrossRef]
  61. Y. Biraud, “A new approach for increasing the resolving power by data processing,” Astron. Astrophys. 1, 124–127 (1969).
  62. T. Opatrný, D. G. Welsch, W. Vogel, “Least-squares inversion for density-matrix reconstruction,” Phys. Rev. A 56, 1788–1799 (1997). [CrossRef]
  63. A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximumlikelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B 39, 1–38 (1977).
  64. R. A. Fisher, “On the mathematical foundations of theoretical statistics,” Philos. Trans. R. Soc. London Ser. A 222, 309–368 (1922). [CrossRef]
  65. R. A. Fisher, “Theory of statistical estimation,” Proc. Cambridge Philos. Soc. 22, 700–725 (1925). [CrossRef]
  66. A. Rockmore, A. Macovski, “A maximum likelihood approach to emission image reconstruction from projections,” IEEE Trans. Nucl. Sci. 23, 1428–1432 (1976). [CrossRef]
  67. L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982). [CrossRef] [PubMed]
  68. D. L. Snyder, M. I. Miller, J. L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imaging 6, 228–238 (1987). [CrossRef] [PubMed]
  69. A. Rockmore, A. Macovski, “A maximum likelihood approach to transmission image reconstruction from projections,” IEEE Trans. Nucl. Sci. 24, 1929–1935 (1977). [CrossRef]
  70. J. Řeháček, Z. Hradil, M. Zawisky, W. Treimer, M. Strobl, “Maximum likelihood absorption tomography,” Europhys. Lett. 59, 694–700 (2002). [CrossRef]
  71. Y. Vardi, D. Lee, “From image deblurring to optimal investments: Maximum likelihood solutions for positive linear inverse problems,” J. R. Statist. Soc. B 55, 569–612 (1993).
  72. B. R. Frieden, “Applications to optics and wave mechanics of the criterion of maximum Cramer–Rao bound,” J. Mod. Opt. 35, 1297–1316 (1988). [CrossRef]
  73. B. R. Frieden, Physics From Fisher Information. A Unification (Cambridge U. Press, Cambridge, UK, 1998; reprinted 1999).
  74. J. Řeháček, Z. Hradil, “Invariant information and quantum state estimation,” Phys. Rev. Lett. 88, 130401–1-4 (2002). [CrossRef] [PubMed]
  75. C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948). [CrossRef]
  76. E. T. Jaynes, “Information theory and statistical mechanics,” Phys. Rev. 106, 620–630; Phys. Rev. 108, 171–190 (1957).
  77. B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Am. 62, 511–518 (1972). [CrossRef] [PubMed]
  78. S. F. Gull, G. J. Daniell, “Image reconstruction from incomplete and noisy data,” Nature (London) 272, 686–690 (1978). [CrossRef]
  79. B. R. Frieden, D. J. Graser, “Closed-form maximum entropy image restoration,” Opt. Commun. 146, 79–84 (1998). [CrossRef]
  80. V. Bužek, G. Adam, G. Drobný, “Quantum state reconstruction and detection of quantum coherences on different observation levels,” Phys. Rev. A 54, 804–820 (1996). [CrossRef] [PubMed]
  81. Z. Hradil, “Quantum-state estimation,” Phys. Rev. A 55, R1561–R1564 (1997). [CrossRef]
  82. Z. Hradil, J. Summhammer, H. Rauch, “Quantum tomography as normalization of incompatible observation,” Phys. Lett. A 261, 20–24 (1999). [CrossRef]
  83. Z. Hradil, J. Summhammer, “Quantum theory of incompatible observations,” J. Phys. A Math. Gen. 33, 7607–7612 (2000). [CrossRef]
  84. J. Řeháček, Z. Hradil, M. Ježek, “Iterative algorithm for reconstruction of entangled states,” Phys. Rev. A 63, 040303–1-4 (2001). [CrossRef]
  85. M. Ježek, J. Fiurášek, Z. Hradil, “Quantum inference of states and processes,” Phys. Rev. A 68, 012305–1-7 (2003). [CrossRef]
  86. K. Banaszek, G. M. D’Ariano, M. G. A. Paris, M. F. Sacchi, “Maximum-likelihood estimation of the density matrix,” Phys. Rev. A 61, 010304–1-4 (2000). [CrossRef]
  87. Z. Hradil, J. Summhammer, G. Badurek, H. Rauch, “Reconstruction of the spin state,” Phys. Rev. A 62, 014101 (2000). [CrossRef]
  88. S. A. Babichev, B. Brezger, A. I. Lvovsky, “Remote preparation of a single-mode photonic qubit by measuring field quadrature noise,” Phys. Rev. Lett. 92, 047903–1-4 (2003). [CrossRef]
  89. A. I. Lvovsky, “Iterative maximum-likelihood reconstruction in quantum homodyne tomography” (2003); arXiv:quant-ph/0311097.

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