Abstract

We demonstrate a two-window heterodyne method for measuring the xp cross correlation, E*xEp, of an optical field E for transverse position x and transverse momentum p. This scheme permits independent control of the x and p resolution. A simple linear transform of the xp correlation function yields the Wigner phase-space distribution. This technique is useful for both coherent and low-coherence light sources and may permit new biological imaging techniques based on transverse coherence measurement with time gating. We point out an interesting analogy between xp correlation measurements for classical-wave and quantum fields.

© 1999 Optical Society of America

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  1. M. Hillery, R. F. O’Connel, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
    [CrossRef]
  2. L. Cohen, Time–Frequency Analysis (Prentice-Hall, Englewood Hills, N.J., 1995).
  3. P. J. Loughlin, J. W. Pitton, and L. E. Atlas, IEEE Trans. Signal Process. 42, 2696 (1994).
    [CrossRef]
  4. A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, Science 282, 706 (1998).
    [CrossRef] [PubMed]
  5. A. Wax and J. E. Thomas, Opt. Lett. 21, 1427 (1996).
    [CrossRef] [PubMed]
  6. N. D. Cartwright, Physica A 83, 210 (1976).
    [CrossRef]
  7. A. Wax and J. E. Thomas, J. Opt. Soc. Am. A 15, 1896 (1998).
    [CrossRef]
  8. A. Wax, S. Bali, and J. E. Thomas, Opt. Lett. 24, 1188 (1999).
    [CrossRef]
  9. A. Wax and J. E. Thomas, Proc. SPIE 3598, 2 (1999).
    [CrossRef]
  10. A. Wax and J. E. Thomas, Proc. SPIE 3726, 494 (1999).
    [CrossRef]
  11. A. Wax, “Optical phase space distributions for coherence tomography,” Ph.D. dissertation (Duke University, Durham, N.C., 1999).
  12. C. Iaconis and I. A. Walmsley, Opt. Lett. 21, 1783 (1996).
    [CrossRef] [PubMed]
  13. C.-C. Cheng and M. G. Raymer, Phys. Rev. Lett. 82, 4807 (1999).
    [CrossRef]
  14. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1976), Chap. 3, pp. 29–57.
  15. K. Wódkiewicz and G. H. Herling, Phys. Rev. A 57, 815 (1998); N. J. Cerf, C. Adami, and P. G. Kwiat, Phys. Rev. A 53, R1477 (1998).
    [CrossRef]

1999

A. Wax, S. Bali, and J. E. Thomas, Opt. Lett. 24, 1188 (1999).
[CrossRef]

A. Wax and J. E. Thomas, Proc. SPIE 3598, 2 (1999).
[CrossRef]

A. Wax and J. E. Thomas, Proc. SPIE 3726, 494 (1999).
[CrossRef]

C.-C. Cheng and M. G. Raymer, Phys. Rev. Lett. 82, 4807 (1999).
[CrossRef]

1998

K. Wódkiewicz and G. H. Herling, Phys. Rev. A 57, 815 (1998); N. J. Cerf, C. Adami, and P. G. Kwiat, Phys. Rev. A 53, R1477 (1998).
[CrossRef]

A. Wax and J. E. Thomas, J. Opt. Soc. Am. A 15, 1896 (1998).
[CrossRef]

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, Science 282, 706 (1998).
[CrossRef] [PubMed]

1996

1994

P. J. Loughlin, J. W. Pitton, and L. E. Atlas, IEEE Trans. Signal Process. 42, 2696 (1994).
[CrossRef]

1984

M. Hillery, R. F. O’Connel, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
[CrossRef]

1976

N. D. Cartwright, Physica A 83, 210 (1976).
[CrossRef]

Atlas, L. E.

P. J. Loughlin, J. W. Pitton, and L. E. Atlas, IEEE Trans. Signal Process. 42, 2696 (1994).
[CrossRef]

Bali, S.

Braunstein, S. L.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, Science 282, 706 (1998).
[CrossRef] [PubMed]

Cartwright, N. D.

N. D. Cartwright, Physica A 83, 210 (1976).
[CrossRef]

Cheng, C.-C.

C.-C. Cheng and M. G. Raymer, Phys. Rev. Lett. 82, 4807 (1999).
[CrossRef]

Cohen, L.

L. Cohen, Time–Frequency Analysis (Prentice-Hall, Englewood Hills, N.J., 1995).

Fuchs, C. A.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, Science 282, 706 (1998).
[CrossRef] [PubMed]

Furusawa, A.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, Science 282, 706 (1998).
[CrossRef] [PubMed]

Herling, G. H.

K. Wódkiewicz and G. H. Herling, Phys. Rev. A 57, 815 (1998); N. J. Cerf, C. Adami, and P. G. Kwiat, Phys. Rev. A 53, R1477 (1998).
[CrossRef]

Hillery, M.

M. Hillery, R. F. O’Connel, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
[CrossRef]

Iaconis, C.

Kimble, H. J.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, Science 282, 706 (1998).
[CrossRef] [PubMed]

Loughlin, P. J.

P. J. Loughlin, J. W. Pitton, and L. E. Atlas, IEEE Trans. Signal Process. 42, 2696 (1994).
[CrossRef]

O’Connel, R. F.

M. Hillery, R. F. O’Connel, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
[CrossRef]

Pitton, J. W.

P. J. Loughlin, J. W. Pitton, and L. E. Atlas, IEEE Trans. Signal Process. 42, 2696 (1994).
[CrossRef]

Polzik, E. S.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, Science 282, 706 (1998).
[CrossRef] [PubMed]

Raymer, M. G.

C.-C. Cheng and M. G. Raymer, Phys. Rev. Lett. 82, 4807 (1999).
[CrossRef]

Scully, M. O.

M. Hillery, R. F. O’Connel, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
[CrossRef]

Sørensen, J. L.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, Science 282, 706 (1998).
[CrossRef] [PubMed]

Thomas, J. E.

Walmsley, I. A.

Wax, A.

A. Wax and J. E. Thomas, Proc. SPIE 3726, 494 (1999).
[CrossRef]

A. Wax and J. E. Thomas, Proc. SPIE 3598, 2 (1999).
[CrossRef]

A. Wax, S. Bali, and J. E. Thomas, Opt. Lett. 24, 1188 (1999).
[CrossRef]

A. Wax and J. E. Thomas, J. Opt. Soc. Am. A 15, 1896 (1998).
[CrossRef]

A. Wax and J. E. Thomas, Opt. Lett. 21, 1427 (1996).
[CrossRef] [PubMed]

A. Wax, “Optical phase space distributions for coherence tomography,” Ph.D. dissertation (Duke University, Durham, N.C., 1999).

Wigner, E. P.

M. Hillery, R. F. O’Connel, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
[CrossRef]

Wódkiewicz, K.

K. Wódkiewicz and G. H. Herling, Phys. Rev. A 57, 815 (1998); N. J. Cerf, C. Adami, and P. G. Kwiat, Phys. Rev. A 53, R1477 (1998).
[CrossRef]

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1976), Chap. 3, pp. 29–57.

IEEE Trans. Signal Process.

P. J. Loughlin, J. W. Pitton, and L. E. Atlas, IEEE Trans. Signal Process. 42, 2696 (1994).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Phys. Rep.

M. Hillery, R. F. O’Connel, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
[CrossRef]

Phys. Rev. A

K. Wódkiewicz and G. H. Herling, Phys. Rev. A 57, 815 (1998); N. J. Cerf, C. Adami, and P. G. Kwiat, Phys. Rev. A 53, R1477 (1998).
[CrossRef]

Phys. Rev. Lett.

C.-C. Cheng and M. G. Raymer, Phys. Rev. Lett. 82, 4807 (1999).
[CrossRef]

Physica A

N. D. Cartwright, Physica A 83, 210 (1976).
[CrossRef]

Proc. SPIE

A. Wax and J. E. Thomas, Proc. SPIE 3598, 2 (1999).
[CrossRef]

A. Wax and J. E. Thomas, Proc. SPIE 3726, 494 (1999).
[CrossRef]

Science

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, Science 282, 706 (1998).
[CrossRef] [PubMed]

Other

L. Cohen, Time–Frequency Analysis (Prentice-Hall, Englewood Hills, N.J., 1995).

A. Wax, “Optical phase space distributions for coherence tomography,” Ph.D. dissertation (Duke University, Durham, N.C., 1999).

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1976), Chap. 3, pp. 29–57.

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Figures (2)

Fig. 1
Fig. 1

Scheme for two-window heterodyne measurement of Wigner distributions. See text for definitions.

Fig. 2
Fig. 2

Measured Wigner distribution for a Gaussian signal beam obscured by a wire: (a) in-phase lock-in signal SRx,p, (b) out-of-phase lock-in signal SIx,p, (c) recovered Wigner distribution showing negative regions. Note that the position distribution dpWx,p=0 for x near the wire, as it should. (d) Three-dimensional view of recovered Wigner distribution. Top, data; bottom, theoretical prediction.

Equations (6)

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

Wx,p=d2πexpipE*x+/2Ex-/2.
VBdx,dp2dxdpWLOx-dx,p+kdp/fWSx,p.
ELOx=E0exp-x22a2+β exp-x22A2expiθ,
WLOx,pexp-2x2A2-2a2p2cos2xp+θcos2xp+θ,
Sx0,p0=dxdpπexp2ix-x0×p-p0WSx,p=E*x0Ep0expix0p0.
WSx,p=dx0dp0πcos2x-x0p-p0×SRx0,p0+dx0dp0πsin2x-x0p-p0×SIx0,p0,

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