Abstract

We present a new approach for constructing optical phase-space-time-frequency tomography (OPSTFT) of an optical wave field. This tomography can be measured by using a novel four-window optical imaging system based on two local oscillator fields balanced heterodyne detection. The OPSTFT is a Wigner distribution function of two independent Fourier Transform pairs, i.e., phase-space and time-frequency. From its theoretical and experimental aspects, it can provide information of position, momentum, time and frequency of a spatial light field with precision beyond the uncertainty principle. Besides the distributions of xp and tω, the OPSTFT can provide four other distributions such as xt, pt, xω and pω. We simulate the OPSTFT for a light field obscured by a wire and a single-line absorption filter. We believe that the four-window system can provide spatial and temporal properties of a wave field for quantum image processing and biophotonics.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. F. Lee, F. Reil, S. Bali, A. Wax, and J. E. Thomas, “Heterodyne measurement of Wigner distributions for classical optical fields,” Opt. Lett. 24, 1370–1372 (1999).
    [CrossRef]
  2. V. Bollen, Y. M. Sua, and K. F. Lee, “Direct measurement of the Kirkwood-Rihaczek distribution for the spatial properties of a coherent light beam,” Phys. Rev. A 81, 063826 (2010).
    [CrossRef]
  3. F. Reil and J. E. Thomas, “Observation of phase conjugation of light arising from enhanced backscattering in a random medium,” Phys. Rev. Lett. 95, 143903 (2005).
    [CrossRef] [PubMed]
  4. E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  5. M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
    [CrossRef]
  6. Ch. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of Helium atoms,” Nature 386, 150–153 (1997).
    [CrossRef]
  7. D. Leibfried, T. Pfau, and C. Monroe, “Shadows and mirrors: reconstructing quantum states of atom motion,” Phys. Today 51(4), 22–28 (April1998).
    [CrossRef]
  8. K. Wodkiewicz and G. H. Herling, “Classical and non-classical interference,” Phys. Rev. A 57, 815–821 (1998).
  9. K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
    [CrossRef] [PubMed]
  10. J. Bertrand and P. Bertrand, “A tomographic approach to Wigner function,” Found. Phys. 17, 397–405 (1987).
    [CrossRef]
  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. A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
    [CrossRef]
  13. 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]
  14. D. F. McAlister, M. Beck, L. Clarke, A. Meyer, and M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
    [CrossRef] [PubMed]
  15. 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]
  16. B. J. Smith, B. Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30, 3365–3367 (2005).
    [CrossRef]
  17. K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. F. Morizur, P. K. Lam, and H. A. Bachor, “Entangling the Spatial Properties of Laser Beams,” Science 321, 541–543 (2008).
    [CrossRef] [PubMed]
  18. M. I. Kolobiv, Quantum Imaging , 1st ed. (Springer, 2006).
  19. V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled Images from Four-Wave Mixing,” Science 321, 544–547 (2008).
    [CrossRef] [PubMed]
  20. M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1539–1589 (1999).
    [CrossRef]
  21. W. H. Zurek, “Sub-Planck structure in phase space and its relevance for quantum decoherence,” Nature 412, 712–717 (2001).
    [CrossRef] [PubMed]
  22. F. Toscano, D. A. R. Dalvit, L. Davidovich, and W. H. Zurek, “Sub-Planck phase-space structures and Heisenberg-limited measurements,” Phys. Rev. A 73, 023803 (2006).
    [CrossRef]
  23. 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]
  24. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vols. I and II.

2010 (1)

V. Bollen, Y. M. Sua, and K. F. Lee, “Direct measurement of the Kirkwood-Rihaczek distribution for the spatial properties of a coherent light beam,” Phys. Rev. A 81, 063826 (2010).
[CrossRef]

2009 (1)

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[CrossRef]

2008 (2)

K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. F. Morizur, P. K. Lam, and H. A. Bachor, “Entangling the Spatial Properties of Laser Beams,” Science 321, 541–543 (2008).
[CrossRef] [PubMed]

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled Images from Four-Wave Mixing,” Science 321, 544–547 (2008).
[CrossRef] [PubMed]

2006 (2)

F. Toscano, D. A. R. Dalvit, L. Davidovich, and W. H. Zurek, “Sub-Planck phase-space structures and Heisenberg-limited measurements,” Phys. Rev. A 73, 023803 (2006).
[CrossRef]

M. I. Kolobiv, Quantum Imaging , 1st ed. (Springer, 2006).

2005 (2)

F. Reil and J. E. Thomas, “Observation of phase conjugation of light arising from enhanced backscattering in a random medium,” Phys. Rev. Lett. 95, 143903 (2005).
[CrossRef] [PubMed]

B. J. Smith, B. Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30, 3365–3367 (2005).
[CrossRef]

2001 (1)

W. H. Zurek, “Sub-Planck structure in phase space and its relevance for quantum decoherence,” Nature 412, 712–717 (2001).
[CrossRef] [PubMed]

1999 (2)

1998 (2)

D. Leibfried, T. Pfau, and C. Monroe, “Shadows and mirrors: reconstructing quantum states of atom motion,” Phys. Today 51(4), 22–28 (April1998).
[CrossRef]

K. Wodkiewicz and G. H. Herling, “Classical and non-classical interference,” Phys. Rev. A 57, 815–821 (1998).

1997 (1)

Ch. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of Helium atoms,” Nature 386, 150–153 (1997).
[CrossRef]

1996 (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]

1995 (1)

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]

1993 (2)

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]

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]

1989 (1)

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef] [PubMed]

1987 (1)

J. Bertrand and P. Bertrand, “A tomographic approach to Wigner function,” Found. Phys. 17, 397–405 (1987).
[CrossRef]

1984 (1)

M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

1978 (1)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vols. I and II.

1932 (1)

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Bachor, H. A.

K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. F. Morizur, P. K. Lam, and H. A. Bachor, “Entangling the Spatial Properties of Laser Beams,” Science 321, 541–543 (2008).
[CrossRef] [PubMed]

Bali, S.

Banaszek, K.

Beck, M.

D. F. McAlister, M. Beck, L. Clarke, A. Meyer, and M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
[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]

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]

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]

Bertrand, J.

J. Bertrand and P. Bertrand, “A tomographic approach to Wigner function,” Found. Phys. 17, 397–405 (1987).
[CrossRef]

Bertrand, P.

J. Bertrand and P. Bertrand, “A tomographic approach to Wigner function,” Found. Phys. 17, 397–405 (1987).
[CrossRef]

Bollen, V.

V. Bollen, Y. M. Sua, and K. F. Lee, “Direct measurement of the Kirkwood-Rihaczek distribution for the spatial properties of a coherent light beam,” Phys. Rev. A 81, 063826 (2010).
[CrossRef]

Boyer, V.

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled Images from Four-Wave Mixing,” Science 321, 544–547 (2008).
[CrossRef] [PubMed]

Clarke, L.

Dalvit, D. A. R.

F. Toscano, D. A. R. Dalvit, L. Davidovich, and W. H. Zurek, “Sub-Planck phase-space structures and Heisenberg-limited measurements,” Phys. Rev. A 73, 023803 (2006).
[CrossRef]

Davidovich, L.

F. Toscano, D. A. R. Dalvit, L. Davidovich, and W. H. Zurek, “Sub-Planck phase-space structures and Heisenberg-limited measurements,” Phys. Rev. A 73, 023803 (2006).
[CrossRef]

Delaubert, V.

K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. F. Morizur, P. K. Lam, and H. A. Bachor, “Entangling the Spatial Properties of Laser Beams,” Science 321, 541–543 (2008).
[CrossRef] [PubMed]

Faridani, A.

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]

Harb, C.

K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. F. Morizur, P. K. Lam, and H. A. Bachor, “Entangling the Spatial Properties of Laser Beams,” Science 321, 541–543 (2008).
[CrossRef] [PubMed]

Herling, G. H.

K. Wodkiewicz and G. H. Herling, “Classical and non-classical interference,” Phys. Rev. A 57, 815–821 (1998).

Hillery, M.

M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vols. I and II.

Janousek, J.

K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. F. Morizur, P. K. Lam, and H. A. Bachor, “Entangling the Spatial Properties of Laser Beams,” Science 321, 541–543 (2008).
[CrossRef] [PubMed]

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]

Killett, B.

Kolobiv, M. I.

M. I. Kolobiv, Quantum Imaging , 1st ed. (Springer, 2006).

Kolobov, M. I.

M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1539–1589 (1999).
[CrossRef]

Kurtsiefer, Ch.

Ch. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of Helium atoms,” Nature 386, 150–153 (1997).
[CrossRef]

Lam, P. K.

K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. F. Morizur, P. K. Lam, and H. A. Bachor, “Entangling the Spatial Properties of Laser Beams,” Science 321, 541–543 (2008).
[CrossRef] [PubMed]

Lee, K. F.

V. Bollen, Y. M. Sua, and K. F. Lee, “Direct measurement of the Kirkwood-Rihaczek distribution for the spatial properties of a coherent light beam,” Phys. Rev. A 81, 063826 (2010).
[CrossRef]

K. F. Lee, F. Reil, S. Bali, A. Wax, and J. E. Thomas, “Heterodyne measurement of Wigner distributions for classical optical fields,” Opt. Lett. 24, 1370–1372 (1999).
[CrossRef]

Leibfried, D.

D. Leibfried, T. Pfau, and C. Monroe, “Shadows and mirrors: reconstructing quantum states of atom motion,” Phys. Today 51(4), 22–28 (April1998).
[CrossRef]

Lett, P. D.

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled Images from Four-Wave Mixing,” Science 321, 544–547 (2008).
[CrossRef] [PubMed]

Lvovsky, A. I.

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[CrossRef]

Marino, A. M.

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled Images from Four-Wave Mixing,” Science 321, 544–547 (2008).
[CrossRef] [PubMed]

McAlister, D. F.

Meyer, A.

Mlynek, J.

Ch. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of Helium atoms,” Nature 386, 150–153 (1997).
[CrossRef]

Monroe, C.

D. Leibfried, T. Pfau, and C. Monroe, “Shadows and mirrors: reconstructing quantum states of atom motion,” Phys. Today 51(4), 22–28 (April1998).
[CrossRef]

Morizur, J. F.

K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. F. Morizur, P. K. Lam, and H. A. Bachor, “Entangling the Spatial Properties of Laser Beams,” Science 321, 541–543 (2008).
[CrossRef] [PubMed]

O’Connell, R. F.

M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[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]

Pfau, T.

D. Leibfried, T. Pfau, and C. Monroe, “Shadows and mirrors: reconstructing quantum states of atom motion,” Phys. Today 51(4), 22–28 (April1998).
[CrossRef]

Ch. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of Helium atoms,” Nature 386, 150–153 (1997).
[CrossRef]

Pooser, R. C.

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled Images from Four-Wave Mixing,” Science 321, 544–547 (2008).
[CrossRef] [PubMed]

Raymer, M. G.

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[CrossRef]

B. J. Smith, B. Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30, 3365–3367 (2005).
[CrossRef]

D. F. McAlister, M. Beck, L. Clarke, A. Meyer, and M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
[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]

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]

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]

Reil, F.

F. Reil and J. E. Thomas, “Observation of phase conjugation of light arising from enhanced backscattering in a random medium,” Phys. Rev. Lett. 95, 143903 (2005).
[CrossRef] [PubMed]

K. F. Lee, F. Reil, S. Bali, A. Wax, and J. E. Thomas, “Heterodyne measurement of Wigner distributions for classical optical fields,” Opt. Lett. 24, 1370–1372 (1999).
[CrossRef]

Risken, H.

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef] [PubMed]

Scully, M. O.

M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Smith, B. J.

Smithey, D. T.

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]

Sua, Y. M.

V. Bollen, Y. M. Sua, and K. F. Lee, “Direct measurement of the Kirkwood-Rihaczek distribution for the spatial properties of a coherent light beam,” Phys. Rev. A 81, 063826 (2010).
[CrossRef]

Thomas, J. E.

F. Reil and J. E. Thomas, “Observation of phase conjugation of light arising from enhanced backscattering in a random medium,” Phys. Rev. Lett. 95, 143903 (2005).
[CrossRef] [PubMed]

K. F. Lee, F. Reil, S. Bali, A. Wax, and J. E. Thomas, “Heterodyne measurement of Wigner distributions for classical optical fields,” Opt. Lett. 24, 1370–1372 (1999).
[CrossRef]

Toscano, F.

F. Toscano, D. A. R. Dalvit, L. Davidovich, and W. H. Zurek, “Sub-Planck phase-space structures and Heisenberg-limited measurements,” Phys. Rev. A 73, 023803 (2006).
[CrossRef]

Treps, N.

K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. F. Morizur, P. K. Lam, and H. A. Bachor, “Entangling the Spatial Properties of Laser Beams,” Science 321, 541–543 (2008).
[CrossRef] [PubMed]

Vogel, K.

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef] [PubMed]

Wagner, K.

K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. F. Morizur, P. K. Lam, and H. A. Bachor, “Entangling the Spatial Properties of Laser Beams,” Science 321, 541–543 (2008).
[CrossRef] [PubMed]

Walmsley, I. A.

Wax, A.

Wigner, E. P.

M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Wodkiewicz, K.

K. Wodkiewicz and G. H. Herling, “Classical and non-classical interference,” Phys. Rev. A 57, 815–821 (1998).

Wong, V.

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]

Zou, H.

K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. F. Morizur, P. K. Lam, and H. A. Bachor, “Entangling the Spatial Properties of Laser Beams,” Science 321, 541–543 (2008).
[CrossRef] [PubMed]

Zurek, W. H.

F. Toscano, D. A. R. Dalvit, L. Davidovich, and W. H. Zurek, “Sub-Planck phase-space structures and Heisenberg-limited measurements,” Phys. Rev. A 73, 023803 (2006).
[CrossRef]

W. H. Zurek, “Sub-Planck structure in phase space and its relevance for quantum decoherence,” Nature 412, 712–717 (2001).
[CrossRef] [PubMed]

Found. Phys. (1)

J. Bertrand and P. Bertrand, “A tomographic approach to Wigner function,” Found. Phys. 17, 397–405 (1987).
[CrossRef]

J. Biomed. Opt. (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]

Nature (2)

W. H. Zurek, “Sub-Planck structure in phase space and its relevance for quantum decoherence,” Nature 412, 712–717 (2001).
[CrossRef] [PubMed]

Ch. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of Helium atoms,” Nature 386, 150–153 (1997).
[CrossRef]

Opt. Lett. (4)

Phys. Rep. (1)

M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Phys. Rev. (1)

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Phys. Rev. A (4)

V. Bollen, Y. M. Sua, and K. F. Lee, “Direct measurement of the Kirkwood-Rihaczek distribution for the spatial properties of a coherent light beam,” Phys. Rev. A 81, 063826 (2010).
[CrossRef]

K. Wodkiewicz and G. H. Herling, “Classical and non-classical interference,” Phys. Rev. A 57, 815–821 (1998).

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef] [PubMed]

F. Toscano, D. A. R. Dalvit, L. Davidovich, and W. H. Zurek, “Sub-Planck phase-space structures and Heisenberg-limited measurements,” Phys. Rev. A 73, 023803 (2006).
[CrossRef]

Phys. Rev. Lett. (3)

F. Reil and J. E. Thomas, “Observation of phase conjugation of light arising from enhanced backscattering in a random medium,” Phys. Rev. Lett. 95, 143903 (2005).
[CrossRef] [PubMed]

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]

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]

Phys. Today (1)

D. Leibfried, T. Pfau, and C. Monroe, “Shadows and mirrors: reconstructing quantum states of atom motion,” Phys. Today 51(4), 22–28 (April1998).
[CrossRef]

Rev. Mod. Phys. (2)

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[CrossRef]

M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1539–1589 (1999).
[CrossRef]

Science (2)

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled Images from Four-Wave Mixing,” Science 321, 544–547 (2008).
[CrossRef] [PubMed]

K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. F. Morizur, P. K. Lam, and H. A. Bachor, “Entangling the Spatial Properties of Laser Beams,” Science 321, 541–543 (2008).
[CrossRef] [PubMed]

Other (2)

M. I. Kolobiv, Quantum Imaging , 1st ed. (Springer, 2006).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vols. I and II.

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 (6)

Fig. 1
Fig. 1

(a) the x-p and (b)ω-t representations of the LO fields. The shaped area is position-momentum and time-frequency resolutions showing beyond the uncertainty limits.

Fig. 2
Fig. 2

Four-window balanced heterodyne detection for measuring OPSTFD.

Fig. 3
Fig. 3

The Kirkwood-Rihaczek (KR) distribution for the product field of Eq. (16) and a wire function. (a) real and (b) imaginary parts of �� (x, p, 0, 0). (c) real and (d) imaginary parts of �� (0.4, 0, ω, t).

Fig. 4
Fig. 4

The optical phase-space-time-frequency tomography (OPSTFT) for the product field of Eq. (16) and a wire function. (a)�� (x, p, 0, 0), (b)�� (0, 0, ω, t), (c)�� (0, 2, ω, t), (d)�� (0, p, 0, t), (e)�� (0, p, ω, 0), (f) �� (x, 0, 0, t), and (g) �� (x, 0, ω, 0).

Fig. 5
Fig. 5

The Kirkwood-Rihaczek (KR) distribution for the product field of Eq. (17) and a filter function. (a) real and (b) imaginary parts of �� (0, 0, ω, t). (c) real and (d) imaginary parts of �� (x, p, 0.2, 0).

Fig. 6
Fig. 6

The optical phase-space-time-frequency tomography (OPSTFT) for the product field of Eq. (17) and a filter function. (a)�� (0, 0, ω, t), (b)�� (x, p, 0, 0), (c)�� (x, p, 0, 3), (d)�� (0, p, 0, t), (e)�� (0, p, ω, 0), (f) �� (x,0, 0, t), and (g) �� (x, 0, ω, 0).

Equations (17)

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

𝒲 ( x , p , ω , t ) = d ε 2 π e i ε p d Ω 2 π e i Ω t * ( x + ε 2 , ω + Ω 2 ) ( x ε 2 , ω Ω 2 )
V B = dx d ω E LO * ( x , ω , z D ) E S ( x , ω , z D )
V B ( d x , d ω ) = dx d ω E LO * ( x d x , ω d ω , z D ) E S ( x , ω , z D ) .
E LO ( x d x , ω d ω , z D ) = k i 2 π f dx e i k ( x x ) 2 2 f e i kx 2 2 f E LO ( x d x , ω d ω , z = 0 ) ,
E S ( x , ω , z D ) = k i 2 π f dx e i k ( x x ) 2 2 f e i k ( x d p ) 2 2 f e i ω τ E S ( x , ω , z = 0 ) .
| V B ( d x , d ω ) | 2 = d ω d x E LO * ( x d x , ω d ω ) E S ( x , ω ) e i kd p x f e i ω τ × d ω dx E LO ( x d x , ω d ω ) E S * ( x , ω ) e i kd p x f e i ω τ
| V B | 2 = d ω o d η ω d x o d η E LO * ( x o d x + η 2 , ω o d o + η ω 2 ) E LO ( x o d x η 2 , ω o d ω η ω 2 ) × E S * ( x o η 2 , ω o η ω 2 ) E S ( x o + η 2 , ω o + η ω 2 ) e i kd p η f e i η ω τ .
E S * ( x o + η 2 , ω o + η ω 2 ) E S ( x o η 2 , ω o η ω 2 ) = d p d t e i η p e i η ω t 𝒲 S ( x o , p , ω o , t ) .
| V B | 2 = d ω o d x o d p d t d η ω d η E LO * ( x o d x + η 2 , ω o d ω + η ω 2 ) × E LO ( x o d x η 2 , ω o d ω η ω 2 ) × 𝒲 ( x o , p , ω o , t ) e i η ( k d p f + p ) e i η ω ( τ + t ) .
𝒲 LO ( x o d x , p + k d p f , ω o d ω , t τ ) = d η d η ω e i η ( p + k d p f ) e i η ω ( t τ ) × E LO * ( x o d x , p + k d p f , ω o d ω + η 2 ) × E LO ( x o d x η 2 ω o d ω η ω 2 ) .
| V B | 2 = d x d ω d p d t 𝒲 LO ( x d x , p + k d p f , ω d ω , t τ ) 𝒲 S ( x , p , ω , t ) )
E LO ( x , ω ) = E o [ exp ( x 2 2 a 2 ) exp ( ω 2 2 α 2 ) ) + γ exp ( x 2 2 A 2 ) exp ( ω 2 2 β 2 ) e i ϕ ]
𝒲 LO ( x , p , ω , t ) exp [ 2 x 2 A 2 2 a 2 p 2 + 2 ω 2 α 2 2 β 2 t 2 ] cos ( 2 x p + 2 ω t + ϕ ) cos ( 2 x p + 2 ω t + ϕ )
| V B | 2 𝒦 ( x o , p o , ω o , t o ) dx d p d ω d t e [ 2 i ( x x 0 ) ( p p o ) + 2 i ( ω ω o ) ( t t o ) ] 𝒲 S ( x , p , ω , t ) E S * ( x o , ω o ) E S ( p o , t o ) exp ( i x o p o + i ω o t o ) S R + i S I
𝒲 ( x , p , ω , t ) d x o d p o d ω o d t o cos [ 2 ( x x o ) ( p p o ) + 2 ( ω ω o ) ( t t o ) ] S R ( x o , p o , ω o , t o ) + d x o d p o d ω o d t o sin [ 2 ( x x o ) ( p p o ) + 2 ( ω ω o ) ( t t o ) ] S I ( x o , p o , ω o , t o )
( x , t ) exp [ x 2 2 σ x 2 ] exp [ i kx 2 2 R ] exp [ t 2 2 σ t 2 ] exp [ i ω c t ] ,
( x , ω ) exp [ x 2 2 σ x 2 ] exp [ i kx 2 2 R ] exp [ ( ω ω c ) 2 2 σ ω 2 ] ,

Metrics