Abstract

We report on the application of a simple propagation-based phase-space tomographic technique to the determination of characteristic projections through the mutual optical intensity and the generalized radiance of a scalar, quasi-monochromatic partially coherent wave field. This method is applied to the reconstruction of the coherence functions of an initially spatially coherent optical wave field that has propagated through a suspension of polystyrene microspheres. As anticipated, we see that the field separates into a ballistic, or unscattered, component and a scattered component with a much shorter coherence length. Good agreement is obtained between experimental results and the results of a model based on a wave-transport equation.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. John, G. Pang, and Y. Yang, "Optical coherence propagation and imaging in a multiple scattering medium," J. Biomed. Opt. 1, 180-191 (1996).
    [CrossRef]
  2. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
  3. A. Wax and J. E. Thomas, "Measurement of smoothed Wigner phase-space distribution for small-angle scattering in a turbid medium," J. Opt. Soc. Am. A 15, 1896-1908 (1998).
    [CrossRef]
  4. C.-C. Cheng and M. G. Raymer, "Propagation of transverse optical coherence in random multiple scattering media," Phys. Rev. A 62, 023811 (2000).
    [CrossRef]
  5. T. Anhut, B. Karamata, T. Lasser, M. G. Raymer, and L. Wenke, "Measurement of scattered light Wigner functions by phase space tomography and implications for parallel OCT," Proc. SPIE 4956, 120-125 (2003).
    [CrossRef]
  6. R. S. Bennick, S. J. Bentley, and R. W. Boyd, " 'Two-photon' coincidence imaging with a classical source," Phys. Rev. Lett. 89, 113601 (2002).
    [CrossRef]
  7. A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost imaging with thermal light: comparing entanglement and classical correlation," Phys. Rev. Lett. 93, 093602 (2004).
    [CrossRef] [PubMed]
  8. Y. Takayama, R. Z. Tai, T. Hatano, T. Miyahara, W. Okamoto, and Y. Kagoshima, "Measurement of the coherence of synchrotron radiation," J. Synchrotron Radiat. 5, 456-458 (1998).
    [CrossRef]
  9. J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, and K. A. Nugent, "Measurement of the spatial coherence function of undulator radiation using a phase mask," Phys. Rev. Lett. 90, 074801 (2003).
    [CrossRef] [PubMed]
  10. C. Iaconis and I. A. Walmsley, "Direct measurement of the two-point field correlation function," Opt. Lett. 21, 1783-1785 (1996).
    [CrossRef] [PubMed]
  11. C. K. Hitzenberger, "Measurement of the spatial coherence of superluminescent diodes," J. Mod. Opt. 46, 1763-1774 (1999).
  12. C.-C. Cheng and M. G. Raymer, "A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light," J. Mod. Opt. 47, 1237-1246 (2000).
    [CrossRef]
  13. A. Wax, S. Bali, G. A. Alphonse, and J. E. Thomas, "Characterizing the coherence of broadband sources using phase space contours," J. Biomed. Opt. 4, 482-489 (1999).
    [CrossRef]
  14. K. A. Nugent, "Wave field determination using three-dimensional intensity information," Phys. Rev. Lett. 68, 2261-2264 (1992).
    [CrossRef] [PubMed]
  15. M. G. Raymer, M. Beck, and D. F. McAlister, "Complex wave-field reconstruction using phase-space tomography," Phys. Rev. Lett. 72, 1137-1140 (1994).
    [CrossRef] [PubMed]
  16. 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. A 22, 1691-1700 (2005).
    [CrossRef]
  17. C. Q. Tran, G. J. Williams, A. Roberts, S. Flewett, A. G. Peele, D. Paterson, M. D. de Jonge, and K. A. Nugent, "Experimental measurement of the four-dimensional coherence function for an undulator x ray source," Phys. Rev. Lett. 98, 224801 (2007).
    [CrossRef] [PubMed]
  18. M. J. Bastiaans, "Application of the Wigner distribution function to partially coherent light," J. Opt. Soc. Am. A 3, 1227-1238 (1986).
    [CrossRef]
  19. G. Hazak, "Comment on 'Wave field determination using three-dimensional intensity information'," Phys. Rev. Lett. 69, 2874 (1992).
    [CrossRef] [PubMed]

2007 (1)

C. Q. Tran, G. J. Williams, A. Roberts, S. Flewett, A. G. Peele, D. Paterson, M. D. de Jonge, and K. A. Nugent, "Experimental measurement of the four-dimensional coherence function for an undulator x ray source," Phys. Rev. Lett. 98, 224801 (2007).
[CrossRef] [PubMed]

2005 (1)

2004 (1)

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost imaging with thermal light: comparing entanglement and classical correlation," Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef] [PubMed]

2003 (2)

T. Anhut, B. Karamata, T. Lasser, M. G. Raymer, and L. Wenke, "Measurement of scattered light Wigner functions by phase space tomography and implications for parallel OCT," Proc. SPIE 4956, 120-125 (2003).
[CrossRef]

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, and K. A. Nugent, "Measurement of the spatial coherence function of undulator radiation using a phase mask," Phys. Rev. Lett. 90, 074801 (2003).
[CrossRef] [PubMed]

2002 (1)

R. S. Bennick, S. J. Bentley, and R. W. Boyd, " 'Two-photon' coincidence imaging with a classical source," Phys. Rev. Lett. 89, 113601 (2002).
[CrossRef]

2000 (2)

C.-C. Cheng and M. G. Raymer, "A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light," J. Mod. Opt. 47, 1237-1246 (2000).
[CrossRef]

C.-C. Cheng and M. G. Raymer, "Propagation of transverse optical coherence in random multiple scattering media," Phys. Rev. A 62, 023811 (2000).
[CrossRef]

1999 (2)

A. Wax, S. Bali, G. A. Alphonse, and J. E. Thomas, "Characterizing the coherence of broadband sources using phase space contours," J. Biomed. Opt. 4, 482-489 (1999).
[CrossRef]

C. K. Hitzenberger, "Measurement of the spatial coherence of superluminescent diodes," J. Mod. Opt. 46, 1763-1774 (1999).

1998 (2)

Y. Takayama, R. Z. Tai, T. Hatano, T. Miyahara, W. Okamoto, and Y. Kagoshima, "Measurement of the coherence of synchrotron radiation," J. Synchrotron Radiat. 5, 456-458 (1998).
[CrossRef]

A. Wax and J. E. Thomas, "Measurement of smoothed Wigner phase-space distribution for small-angle scattering in a turbid medium," J. Opt. Soc. Am. A 15, 1896-1908 (1998).
[CrossRef]

1996 (2)

S. John, G. Pang, and Y. Yang, "Optical coherence propagation and imaging in a multiple scattering medium," J. Biomed. Opt. 1, 180-191 (1996).
[CrossRef]

C. Iaconis and I. A. Walmsley, "Direct measurement of the two-point field correlation function," Opt. Lett. 21, 1783-1785 (1996).
[CrossRef] [PubMed]

1994 (1)

M. G. Raymer, M. Beck, and D. F. McAlister, "Complex wave-field reconstruction using phase-space tomography," Phys. Rev. Lett. 72, 1137-1140 (1994).
[CrossRef] [PubMed]

1992 (2)

G. Hazak, "Comment on 'Wave field determination using three-dimensional intensity information'," Phys. Rev. Lett. 69, 2874 (1992).
[CrossRef] [PubMed]

K. A. Nugent, "Wave field determination using three-dimensional intensity information," Phys. Rev. Lett. 68, 2261-2264 (1992).
[CrossRef] [PubMed]

1986 (1)

Alphonse, G. A.

A. Wax, S. Bali, G. A. Alphonse, and J. E. Thomas, "Characterizing the coherence of broadband sources using phase space contours," J. Biomed. Opt. 4, 482-489 (1999).
[CrossRef]

Anhut, T.

T. Anhut, B. Karamata, T. Lasser, M. G. Raymer, and L. Wenke, "Measurement of scattered light Wigner functions by phase space tomography and implications for parallel OCT," Proc. SPIE 4956, 120-125 (2003).
[CrossRef]

Bache, M.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost imaging with thermal light: comparing entanglement and classical correlation," Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef] [PubMed]

Bali, S.

A. Wax, S. Bali, G. A. Alphonse, and J. E. Thomas, "Characterizing the coherence of broadband sources using phase space contours," J. Biomed. Opt. 4, 482-489 (1999).
[CrossRef]

Bastiaans, M. J.

Beck, M.

M. G. Raymer, M. Beck, and D. F. McAlister, "Complex wave-field reconstruction using phase-space tomography," Phys. Rev. Lett. 72, 1137-1140 (1994).
[CrossRef] [PubMed]

Bennick, R. S.

R. S. Bennick, S. J. Bentley, and R. W. Boyd, " 'Two-photon' coincidence imaging with a classical source," Phys. Rev. Lett. 89, 113601 (2002).
[CrossRef]

Bentley, S. J.

R. S. Bennick, S. J. Bentley, and R. W. Boyd, " 'Two-photon' coincidence imaging with a classical source," Phys. Rev. Lett. 89, 113601 (2002).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Boyd, R. W.

R. S. Bennick, S. J. Bentley, and R. W. Boyd, " 'Two-photon' coincidence imaging with a classical source," Phys. Rev. Lett. 89, 113601 (2002).
[CrossRef]

Brambilla, E.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost imaging with thermal light: comparing entanglement and classical correlation," Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef] [PubMed]

Chantler, C. T.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, and K. A. Nugent, "Measurement of the spatial coherence function of undulator radiation using a phase mask," Phys. Rev. Lett. 90, 074801 (2003).
[CrossRef] [PubMed]

Cheng, C.-C.

C.-C. Cheng and M. G. Raymer, "Propagation of transverse optical coherence in random multiple scattering media," Phys. Rev. A 62, 023811 (2000).
[CrossRef]

C.-C. Cheng and M. G. Raymer, "A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light," J. Mod. Opt. 47, 1237-1246 (2000).
[CrossRef]

de Jonge, M. D.

C. Q. Tran, G. J. Williams, A. Roberts, S. Flewett, A. G. Peele, D. Paterson, M. D. de Jonge, and K. A. Nugent, "Experimental measurement of the four-dimensional coherence function for an undulator x ray source," Phys. Rev. Lett. 98, 224801 (2007).
[CrossRef] [PubMed]

Flewett, S.

C. Q. Tran, G. J. Williams, A. Roberts, S. Flewett, A. G. Peele, D. Paterson, M. D. de Jonge, and K. A. Nugent, "Experimental measurement of the four-dimensional coherence function for an undulator x ray source," Phys. Rev. Lett. 98, 224801 (2007).
[CrossRef] [PubMed]

Gatti, A.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost imaging with thermal light: comparing entanglement and classical correlation," Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef] [PubMed]

Hatano, T.

Y. Takayama, R. Z. Tai, T. Hatano, T. Miyahara, W. Okamoto, and Y. Kagoshima, "Measurement of the coherence of synchrotron radiation," J. Synchrotron Radiat. 5, 456-458 (1998).
[CrossRef]

Hazak, G.

G. Hazak, "Comment on 'Wave field determination using three-dimensional intensity information'," Phys. Rev. Lett. 69, 2874 (1992).
[CrossRef] [PubMed]

Hitzenberger, C. K.

C. K. Hitzenberger, "Measurement of the spatial coherence of superluminescent diodes," J. Mod. Opt. 46, 1763-1774 (1999).

Iaconis, C.

John, S.

S. John, G. Pang, and Y. Yang, "Optical coherence propagation and imaging in a multiple scattering medium," J. Biomed. Opt. 1, 180-191 (1996).
[CrossRef]

Kagoshima, Y.

Y. Takayama, R. Z. Tai, T. Hatano, T. Miyahara, W. Okamoto, and Y. Kagoshima, "Measurement of the coherence of synchrotron radiation," J. Synchrotron Radiat. 5, 456-458 (1998).
[CrossRef]

Karamata, B.

T. Anhut, B. Karamata, T. Lasser, M. G. Raymer, and L. Wenke, "Measurement of scattered light Wigner functions by phase space tomography and implications for parallel OCT," Proc. SPIE 4956, 120-125 (2003).
[CrossRef]

Lasser, T.

T. Anhut, B. Karamata, T. Lasser, M. G. Raymer, and L. Wenke, "Measurement of scattered light Wigner functions by phase space tomography and implications for parallel OCT," Proc. SPIE 4956, 120-125 (2003).
[CrossRef]

Lin, J. J. A.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, and K. A. Nugent, "Measurement of the spatial coherence function of undulator radiation using a phase mask," Phys. Rev. Lett. 90, 074801 (2003).
[CrossRef] [PubMed]

Lugiato, L. A.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost imaging with thermal light: comparing entanglement and classical correlation," Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef] [PubMed]

McAlister, D. F.

M. G. Raymer, M. Beck, and D. F. McAlister, "Complex wave-field reconstruction using phase-space tomography," Phys. Rev. Lett. 72, 1137-1140 (1994).
[CrossRef] [PubMed]

McMahon, P. J.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, and K. A. Nugent, "Measurement of the spatial coherence function of undulator radiation using a phase mask," Phys. Rev. Lett. 90, 074801 (2003).
[CrossRef] [PubMed]

McNulty, I.

Miyahara, T.

Y. Takayama, R. Z. Tai, T. Hatano, T. Miyahara, W. Okamoto, and Y. Kagoshima, "Measurement of the coherence of synchrotron radiation," J. Synchrotron Radiat. 5, 456-458 (1998).
[CrossRef]

Nugent, K. A.

C. Q. Tran, G. J. Williams, A. Roberts, S. Flewett, A. G. Peele, D. Paterson, M. D. de Jonge, and K. A. Nugent, "Experimental measurement of the four-dimensional coherence function for an undulator x ray source," Phys. Rev. Lett. 98, 224801 (2007).
[CrossRef] [PubMed]

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. A 22, 1691-1700 (2005).
[CrossRef]

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, and K. A. Nugent, "Measurement of the spatial coherence function of undulator radiation using a phase mask," Phys. Rev. Lett. 90, 074801 (2003).
[CrossRef] [PubMed]

K. A. Nugent, "Wave field determination using three-dimensional intensity information," Phys. Rev. Lett. 68, 2261-2264 (1992).
[CrossRef] [PubMed]

Okamoto, W.

Y. Takayama, R. Z. Tai, T. Hatano, T. Miyahara, W. Okamoto, and Y. Kagoshima, "Measurement of the coherence of synchrotron radiation," J. Synchrotron Radiat. 5, 456-458 (1998).
[CrossRef]

Pang, G.

S. John, G. Pang, and Y. Yang, "Optical coherence propagation and imaging in a multiple scattering medium," J. Biomed. Opt. 1, 180-191 (1996).
[CrossRef]

Paterson, D.

C. Q. Tran, G. J. Williams, A. Roberts, S. Flewett, A. G. Peele, D. Paterson, M. D. de Jonge, and K. A. Nugent, "Experimental measurement of the four-dimensional coherence function for an undulator x ray source," Phys. Rev. Lett. 98, 224801 (2007).
[CrossRef] [PubMed]

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. A 22, 1691-1700 (2005).
[CrossRef]

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, and K. A. Nugent, "Measurement of the spatial coherence function of undulator radiation using a phase mask," Phys. Rev. Lett. 90, 074801 (2003).
[CrossRef] [PubMed]

Peele, A. G.

C. Q. Tran, G. J. Williams, A. Roberts, S. Flewett, A. G. Peele, D. Paterson, M. D. de Jonge, and K. A. Nugent, "Experimental measurement of the four-dimensional coherence function for an undulator x ray source," Phys. Rev. Lett. 98, 224801 (2007).
[CrossRef] [PubMed]

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. A 22, 1691-1700 (2005).
[CrossRef]

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, and K. A. Nugent, "Measurement of the spatial coherence function of undulator radiation using a phase mask," Phys. Rev. Lett. 90, 074801 (2003).
[CrossRef] [PubMed]

Raymer, M. G.

T. Anhut, B. Karamata, T. Lasser, M. G. Raymer, and L. Wenke, "Measurement of scattered light Wigner functions by phase space tomography and implications for parallel OCT," Proc. SPIE 4956, 120-125 (2003).
[CrossRef]

C.-C. Cheng and M. G. Raymer, "A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light," J. Mod. Opt. 47, 1237-1246 (2000).
[CrossRef]

C.-C. Cheng and M. G. Raymer, "Propagation of transverse optical coherence in random multiple scattering media," Phys. Rev. A 62, 023811 (2000).
[CrossRef]

M. G. Raymer, M. Beck, and D. F. McAlister, "Complex wave-field reconstruction using phase-space tomography," Phys. Rev. Lett. 72, 1137-1140 (1994).
[CrossRef] [PubMed]

Roberts, A.

C. Q. Tran, G. J. Williams, A. Roberts, S. Flewett, A. G. Peele, D. Paterson, M. D. de Jonge, and K. A. Nugent, "Experimental measurement of the four-dimensional coherence function for an undulator x ray source," Phys. Rev. Lett. 98, 224801 (2007).
[CrossRef] [PubMed]

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. A 22, 1691-1700 (2005).
[CrossRef]

Tai, R. Z.

Y. Takayama, R. Z. Tai, T. Hatano, T. Miyahara, W. Okamoto, and Y. Kagoshima, "Measurement of the coherence of synchrotron radiation," J. Synchrotron Radiat. 5, 456-458 (1998).
[CrossRef]

Takayama, Y.

Y. Takayama, R. Z. Tai, T. Hatano, T. Miyahara, W. Okamoto, and Y. Kagoshima, "Measurement of the coherence of synchrotron radiation," J. Synchrotron Radiat. 5, 456-458 (1998).
[CrossRef]

Thomas, J. E.

A. Wax, S. Bali, G. A. Alphonse, and J. E. Thomas, "Characterizing the coherence of broadband sources using phase space contours," J. Biomed. Opt. 4, 482-489 (1999).
[CrossRef]

A. Wax and J. E. Thomas, "Measurement of smoothed Wigner phase-space distribution for small-angle scattering in a turbid medium," J. Opt. Soc. Am. A 15, 1896-1908 (1998).
[CrossRef]

Tran, C. Q.

C. Q. Tran, G. J. Williams, A. Roberts, S. Flewett, A. G. Peele, D. Paterson, M. D. de Jonge, and K. A. Nugent, "Experimental measurement of the four-dimensional coherence function for an undulator x ray source," Phys. Rev. Lett. 98, 224801 (2007).
[CrossRef] [PubMed]

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. A 22, 1691-1700 (2005).
[CrossRef]

Walmsley, I. A.

Wax, A.

A. Wax, S. Bali, G. A. Alphonse, and J. E. Thomas, "Characterizing the coherence of broadband sources using phase space contours," J. Biomed. Opt. 4, 482-489 (1999).
[CrossRef]

A. Wax and J. E. Thomas, "Measurement of smoothed Wigner phase-space distribution for small-angle scattering in a turbid medium," J. Opt. Soc. Am. A 15, 1896-1908 (1998).
[CrossRef]

Wenke, L.

T. Anhut, B. Karamata, T. Lasser, M. G. Raymer, and L. Wenke, "Measurement of scattered light Wigner functions by phase space tomography and implications for parallel OCT," Proc. SPIE 4956, 120-125 (2003).
[CrossRef]

Williams, G. J.

C. Q. Tran, G. J. Williams, A. Roberts, S. Flewett, A. G. Peele, D. Paterson, M. D. de Jonge, and K. A. Nugent, "Experimental measurement of the four-dimensional coherence function for an undulator x ray source," Phys. Rev. Lett. 98, 224801 (2007).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Yang, Y.

S. John, G. Pang, and Y. Yang, "Optical coherence propagation and imaging in a multiple scattering medium," J. Biomed. Opt. 1, 180-191 (1996).
[CrossRef]

J. Biomed. Opt. (2)

A. Wax, S. Bali, G. A. Alphonse, and J. E. Thomas, "Characterizing the coherence of broadband sources using phase space contours," J. Biomed. Opt. 4, 482-489 (1999).
[CrossRef]

S. John, G. Pang, and Y. Yang, "Optical coherence propagation and imaging in a multiple scattering medium," J. Biomed. Opt. 1, 180-191 (1996).
[CrossRef]

J. Mod. Opt. (2)

C. K. Hitzenberger, "Measurement of the spatial coherence of superluminescent diodes," J. Mod. Opt. 46, 1763-1774 (1999).

C.-C. Cheng and M. G. Raymer, "A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light," J. Mod. Opt. 47, 1237-1246 (2000).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Synchrotron Radiat. (1)

Y. Takayama, R. Z. Tai, T. Hatano, T. Miyahara, W. Okamoto, and Y. Kagoshima, "Measurement of the coherence of synchrotron radiation," J. Synchrotron Radiat. 5, 456-458 (1998).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

C.-C. Cheng and M. G. Raymer, "Propagation of transverse optical coherence in random multiple scattering media," Phys. Rev. A 62, 023811 (2000).
[CrossRef]

Phys. Rev. Lett. (7)

R. S. Bennick, S. J. Bentley, and R. W. Boyd, " 'Two-photon' coincidence imaging with a classical source," Phys. Rev. Lett. 89, 113601 (2002).
[CrossRef]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost imaging with thermal light: comparing entanglement and classical correlation," Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef] [PubMed]

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, and K. A. Nugent, "Measurement of the spatial coherence function of undulator radiation using a phase mask," Phys. Rev. Lett. 90, 074801 (2003).
[CrossRef] [PubMed]

C. Q. Tran, G. J. Williams, A. Roberts, S. Flewett, A. G. Peele, D. Paterson, M. D. de Jonge, and K. A. Nugent, "Experimental measurement of the four-dimensional coherence function for an undulator x ray source," Phys. Rev. Lett. 98, 224801 (2007).
[CrossRef] [PubMed]

G. Hazak, "Comment on 'Wave field determination using three-dimensional intensity information'," Phys. Rev. Lett. 69, 2874 (1992).
[CrossRef] [PubMed]

K. A. Nugent, "Wave field determination using three-dimensional intensity information," Phys. Rev. Lett. 68, 2261-2264 (1992).
[CrossRef] [PubMed]

M. G. Raymer, M. Beck, and D. F. McAlister, "Complex wave-field reconstruction using phase-space tomography," Phys. Rev. Lett. 72, 1137-1140 (1994).
[CrossRef] [PubMed]

Proc. SPIE (1)

T. Anhut, B. Karamata, T. Lasser, M. G. Raymer, and L. Wenke, "Measurement of scattered light Wigner functions by phase space tomography and implications for parallel OCT," Proc. SPIE 4956, 120-125 (2003).
[CrossRef]

Other (1)

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Schematic showing the principle of noninterferometric phase-space tomography for wave field recovery.

Fig. 2
Fig. 2

Surface plot showing the recovered spatial coherence function, (a) modeled and determined experimentally along (b) x and (c) y directions for a medium consisting of 19 μ m diameter spheres with a density of 4 × 10 11 m 3 in a water/glycerol mixture.

Fig. 3
Fig. 3

The surface plot of the generalized radiance, (a) modeled and determined experimentally along (b) x and (c) y directions for a medium consisting of 19 μ m diameter spheres with a density of 4 × 10 11 m 3 in a water/glycerol mixture.

Fig. 4
Fig. 4

Profile along the separation coordinate at x = 0 of the projection of the spatial coherence function calculated using the model (solid curve) and determined experimentally (dashed curve) for various number densities ( N ) of polystyrene microspheres of diameter 19 μ m . The path length of the turbid medium was 1 cm . The incident field was coherent with a central wavelength of 681 nm and a beam radius of 388.9 ± 0.3 μ m . (a) N = 2 × 10 11 m 3 , (b) N = 4 × 10 11 m 3 , (c) N = 6 × 10 11 m 3 , (d) N = 9 × 10 11 m 3 , and (e) N = 15 × 10 11 m 3 .

Fig. 5
Fig. 5

Profile along the momentum coordinate at x = 0 of the projection across the y coordinate calculated using the model (solid curve) and determined experimentally (dashed curve). Other data as per Fig. 4.

Equations (31)

Equations on this page are rendered with MathJax. Learn more.

G ( r 1 , r 2 ; z ) = E ( r 1 ; z ) E * ( r 2 ; z ) ,
B ( x , u ; z ) = J ( x , s ; z ) e i s u d s ,
I ( x ; z ) = B ( x , u ; z ) d u .
B ( x , u ; z ) = B ( x z u k , u ; 0 ) ,
B ̃ ( ρ , z k ρ ̃ ; 0 ) = I ̃ ( ρ , z ) ,
J ( 1 ) ( x , s x , s y ; z ) = J ( x , y , s x , s y ; z ) d y ,
B ( 1 ) ( x , u x ; z ) = B ( x , y , u x , u y ; z ) d y d u y .
G ( x , y ; z ) = G x ( x 1 , x 2 ; z ) G y ( y 1 , y 2 ; z ) ,
J N ( x , s ; z ) = d q e i q x e I 1 ( s , q , z ) I 0 ( s , q , z ) ,
I 0 ( s , q ; z ) = 1 ( 2 π ) 2 d x e i q x J ( x , s z q k ; 0 ) ,
I 1 ( s , q ; z ) = μ T z N σ N π θ 0 q exp [ k 2 θ 0 2 4 ( s 2 q s 2 q 2 ) ] , × { erf ( k θ 0 2 q q s ) erf [ k θ 0 2 q ( q s z q k ) ] }
J ( x , s ; 0 ) = 1 π a 2 exp [ ( x 2 + 1 4 s 2 ) a 2 ] ,
I 0 ( s , q ; z ) = 1 ( 2 π ) 2 exp [ ( s z q k a 2 + a 2 q 2 4 ) ] .
J ( 1 ) ( x , s x , s y ; z ) = 1 ( 2 π ) 2 ( k z ) 2 J ( 1 ) ( x , s x , s y ; 0 ) exp [ i k z ( s x s x ) ( x x ) ] d x d s x ,
J ( 1 ) ( x , s x , s y ; z ) = 2 π d q x e i q x x e I 1 ( s x , s y , q x , 0 ; z ) I 0 ( s x , s y , q x , 0 ; z ) ,
I 0 ( s x , s y , q x , 0 ; z ) = 1 ( 2 π ) 2 d x e i q x x J ( 1 ) ( x , s x q x z k , s y ; 0 ) ,
I 1 ( s x , s y , q x , 0 ; z ) = μ T z N σ N π θ 0 q x exp ( k 2 θ 0 2 s y 2 4 ) { erf [ k θ 0 s x 2 ] erf [ k θ 0 2 ( s x z q x k ) ] } .
B ̃ ( p , s ) = B ̃ ( p , s ) ,
J ( x , s ; z ) = 1 ( 2 π ) 2 B ( x , u ; z ) e i u s d u .
B ̃ ( p , q ; z ) = B ̃ ( x , u ; z ) e i ( x p + u q ) d x d u ,
B ( x , u ; z ) = B ̃ ( p , q ; z ) e i ( x p + u q ) d q d p ,
J ( x , s ; z ) = B ̃ ( p , s ; z ) e i x p d p .
J ( x , y , s x , s y ; z ) = B ̃ ( p x , p y , s x , s y ; z ) e i ( x p x + y p y ) d p x d p y .
J ( 1 ) ( x , s x , s y ; z ) = J ( x , y , s x , s y ; z ) d y .
J ( 1 ) ( x , s x , s y ; z ) = 2 π B ̃ ( p x , 0 , s x , s y ; z ) e i x p x d p x .
J ( 1 ) ( x , s x , 0 ; z ) = 2 π B ̃ ( p x , 0 , s x , 0 ; z ) e i x p x d p x .
I ̃ ( p ; z ) = I ( x , y ; z ) e i ( x p x + y p y ) d x d y ,
I ̃ ( p x , 0 ; z ) = I ( 1 ) ( x , z ) e i x p x d x ,
B ( x , y , u x , u y ; 0 ) d y d u y = 2 π J ( 1 ) ( x , s x , 0 ; 0 ) e i u x s x d s x .
J ( r , s ; z ) = J x ( x , s x ; z ) J y ( y , s y ; z ) ,
J ( 1 ) ( x , s x , 0 ; z ) = J x ( x , s x ; z ) I y ( y ; z ) d y = K y J x ( x , s x ; z ) ,

Metrics