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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 8 — Apr. 9, 2012
  • pp: 8296–8308

Experimental compressive phase space tomography

Lei Tian, Justin Lee, Se Baek Oh, and George Barbastathis  »View Author Affiliations


Optics Express, Vol. 20, Issue 8, pp. 8296-8308 (2012)
http://dx.doi.org/10.1364/OE.20.008296


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Abstract

Phase space tomography estimates correlation functions entirely from snapshots in the evolution of the wave function along a time or space variable. In contrast, traditional interferometric methods require measurement of multiple two–point correlations. However, as in every tomographic formulation, undersampling poses a severe limitation. Here we present the first, to our knowledge, experimental demonstration of compressive reconstruction of the classical optical correlation function, i.e. the mutual intensity function. Our compressive algorithm makes explicit use of the physically justifiable assumption of a low–entropy source (or state.) Since the source was directly accessible in our classical experiment, we were able to compare the compressive estimate of the mutual intensity to an independent ground–truth estimate from the van Cittert–Zernike theorem and verify substantial quantitative improvements in the reconstruction.

© 2012 OSA

OCIS Codes
(030.0030) Coherence and statistical optics : Coherence and statistical optics
(070.0070) Fourier optics and signal processing : Fourier optics and signal processing
(100.6950) Image processing : Tomographic image processing
(050.5082) Diffraction and gratings : Phase space in wave options

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: January 13, 2012
Revised Manuscript: February 24, 2012
Manuscript Accepted: March 20, 2012
Published: March 26, 2012

Virtual Issues
April 9, 2012 Spotlight on Optics

Citation
Lei Tian, Justin Lee, Se Baek Oh, and George Barbastathis, "Experimental compressive phase space tomography," Opt. Express 20, 8296-8308 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-8-8296


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References

  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  2. K. Blum, Density Matrix Theory and Applications (Plenum Press, 1981).
  3. K. Itoh and Y. Ohtsuka, “Fourier-transform spectral imaging: retrieval of source information from three-dimensional spatial coherence,” J. Opt. Soc. Am. A3, 94–100 (1986). [CrossRef]
  4. D. L. Marks, R. A. Stack, and D. J. Brady, “Three-dimensional coherence imaging in the Fresnel domain,” Appl. Opt.38, 1332–1342 (1999). [CrossRef]
  5. J. W. Goodman, Statistical Optics (Wiley-Interscience, 2000).
  6. K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett.68, 2261–2264 (1992). [CrossRef] [PubMed]
  7. M. G. Raymer, M. Beck, and D. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett.72, 1137–1140 (1994). [CrossRef] [PubMed]
  8. C. Q. Tran, A. G. Peele, A. Roberts, K. A. Nugent, D. Paterson, and I. McNulty, “X-ray imaging: a generalized approach using phase-space tomography,” J. Opt. Soc. Am. A22, 1691–1700 (2005). [CrossRef]
  9. M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett.18, 2041–2043 (1993). [CrossRef] [PubMed]
  10. K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A40, 2847–2849 (1989). [CrossRef] [PubMed]
  11. D. T. Smithey, M. Beck, M. G. Raymer, and 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]
  12. U. Leonhardt, “Quantum–state tomography and discrete Wigner function,” Phys. Rev. Lett.74, 4101–4105 (1995). [CrossRef] [PubMed]
  13. C. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of Helium atoms,” Nature (London)386, 150–153 (1997). [CrossRef]
  14. E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory52, 489–509 (2006). [CrossRef]
  15. E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Comm. Pure Appl. Math.59, 1207–1223 (2006). [CrossRef]
  16. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inform. Theory52, 1289–1306 (2006). [CrossRef]
  17. E. J. Candès and B. Recht, “Exact matrix completion via convex optimization,” Found. Comput. Math.9, 717–772 (2009). [CrossRef]
  18. E. J. Candès and T. Tao, “The power of convex relaxation: near-optimal matrix completion,” IEEE Trans. Inform. Theory56, 2053–2080 (2010). [CrossRef]
  19. D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett.105, 150401 (2010). [CrossRef]
  20. D. Gross, “Recovering low-rank matrices from few coefficients in any basis,” IEEE Trans. Inf. Theory57, 1548–1566 (2011). [CrossRef]
  21. E. J. Candès, T. Strohmer, and V. Voroninski, “Phaselift: exact and stable signal recovery from magnitude measurements via convex programming,” ArXiv: 1109.4499v1 (2011).
  22. E. J. Candès, Y. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” ArXiv: 1109.0573 (2011).
  23. Y. Shechtman, Y. C. Eldar, A. Szameit, and M. Segev, “Sparsity based sub-wavelength imaging with partially incoherent light via quadratic compressed sensing,” Opt. Express19, 14807–14822 (2011). [CrossRef] [PubMed]
  24. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am.72, 343–351 (1982). [CrossRef]
  25. D. Pelliccia, A. Y. Nikulin, H. O. Moser, and K. A. Nugent, “Experimental characterization of the coherence properties of hard x-ray sources,” Opt. Express19, 8073–8078 (2011). [CrossRef] [PubMed]
  26. K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun.44, 323–326 (1983). [CrossRef]
  27. K.-H. Brenner and J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta.31, 213–223 (1984). [CrossRef]
  28. J. Tu, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E55, 1946–1949 (1997). [CrossRef]
  29. E. J. Candès and Y. Plan, “Matrix completion with noise,” ArXiv: 0903.3131 (2009).
  30. J.-F. Cai, E. J. Candès, and Z. Shen, “A singular value thresholding algorithm for matrix completion,” ArXiv: 0810.3286 (2008).
  31. A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am.72, 923–928 (1982). [CrossRef]
  32. A. C. Kak and M. Slaney, Principle of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 2001). [CrossRef]
  33. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun.25, 26–30 (1978). [CrossRef]
  34. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A3, 1227–1238 (1986). [CrossRef]
  35. A. Starikov, “Effective number of degrees of freedom of partially coherent sources,” J. Opt. Soc. Am.72, 1538–1544 (1982). [CrossRef]
  36. M. J. Bastiaans, “New class of uncertainty relations for partially coherent light,” J. Opt. Soc. Am. A1, 711–715 (1984). [CrossRef]

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