Abstract

We demonstrate that optical heterodyne imaging directly measures smoothed Wigner phase space distributions. This method may be broadly applicable to fundamental studies of light propagation and tomographic imaging. Basic physical properties of Wigner distributions are illustrated by experimental measurements.

© 1996 Optical Society of America

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References

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  1. E. P. Wigner, Phys. Rev. Lett. 40, 749 (1932).
  2. M. Hillery, R. F. O'Connel, M. O. Scully, E. P. Wigner, Phys. Rep. 106, 121 (1984).
    [CrossRef]
  3. D. F. McAlister, M. Beck, L. Clarke, A. Mayer, M. G. Raymer, Opt. Lett. 20, 1181 (1995).
    [CrossRef] [PubMed]
  4. M. G. Raymer, C. Cheng, D. M. Toloudis, M. Anderson, M. Beck, in Advances in Optical Imaging and Photon Migration (Optical Society of America, Washington, D.C., 1996), pp. 236–238.
  5. S. John, G. Pang, Y. Yang, Proc. SPIE 2389, 64 (1995).
    [CrossRef]
  6. See, for example, V. J. Corcoran, J. Appl. Phys. 36, 1819 (1965); A. E. Siegman, Appl. Opt. 5, 1588 (1966); S. Cohen, Appl. Opt. 14, 1953 (1975); A. Migdall, B. Roop, Y. C. Zheng, J. E. Hardis, G. J. Xia, Appl. Opt. 29, 5136 (1990).
    [CrossRef] [PubMed]
  7. Recent heterodyne studies in turbid media include K. P. Chan, M. Yamada, B. Devaraj, H. Inaba, Opt. Lett. 20, 492 (1995); M. Toida, M. Kondo, T. Ichimura, H. Inaba, Appl. Phys. B 52, 391 (1991).
    [CrossRef] [PubMed]
  8. The mean-square beat is positive definite and takes the form of a smoothed Wigner distribution. See N. D. Cartwright, Physica 83A, 210 (1976).
  9. H. P. Yuen, V. W. S. Chan, Opt. Lett. 8, 177 (1983).
    [CrossRef] [PubMed]
  10. This method has been used in light beating spectroscopy; see H. Z. Cummins, H. L. Swinney, in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1970), Vol. VIII, Chap. 3, pp. 133–200.
    [CrossRef]
  11. This method has been used by G. L. Abbas, V. W. S. Chan, T. K. Yee, IEEE J. Lightwave Technol. 3, 1110 (1985).
    [CrossRef]
  12. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, & Winston, New York, 1976), Chap. 3, p. 35.
  13. A. Wax, J. E. Thomas, in Advances in Optical Imaging and Photon Migration (Optical Society of America, Washington, D.C., 1996), pp. 238–242.
  14. The magnitude of the mean beat amplitude (rather than the mean square) is usually measured in this case. See, for example, J. A. Izatt, H. R. Hee, G. M. Owen, E. A. Swanson, J. G. Fujimoto, Opt. Lett. 19, 590 (1994).
    [CrossRef] [PubMed]

1995

1994

1985

This method has been used by G. L. Abbas, V. W. S. Chan, T. K. Yee, IEEE J. Lightwave Technol. 3, 1110 (1985).
[CrossRef]

1984

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

1983

1976

The mean-square beat is positive definite and takes the form of a smoothed Wigner distribution. See N. D. Cartwright, Physica 83A, 210 (1976).

1965

See, for example, V. J. Corcoran, J. Appl. Phys. 36, 1819 (1965); A. E. Siegman, Appl. Opt. 5, 1588 (1966); S. Cohen, Appl. Opt. 14, 1953 (1975); A. Migdall, B. Roop, Y. C. Zheng, J. E. Hardis, G. J. Xia, Appl. Opt. 29, 5136 (1990).
[CrossRef] [PubMed]

1932

E. P. Wigner, Phys. Rev. Lett. 40, 749 (1932).

Abbas, G. L.

This method has been used by G. L. Abbas, V. W. S. Chan, T. K. Yee, IEEE J. Lightwave Technol. 3, 1110 (1985).
[CrossRef]

Anderson, M.

M. G. Raymer, C. Cheng, D. M. Toloudis, M. Anderson, M. Beck, in Advances in Optical Imaging and Photon Migration (Optical Society of America, Washington, D.C., 1996), pp. 236–238.

Beck, M.

D. F. McAlister, M. Beck, L. Clarke, A. Mayer, M. G. Raymer, Opt. Lett. 20, 1181 (1995).
[CrossRef] [PubMed]

M. G. Raymer, C. Cheng, D. M. Toloudis, M. Anderson, M. Beck, in Advances in Optical Imaging and Photon Migration (Optical Society of America, Washington, D.C., 1996), pp. 236–238.

Cartwright, N. D.

The mean-square beat is positive definite and takes the form of a smoothed Wigner distribution. See N. D. Cartwright, Physica 83A, 210 (1976).

Chan, K. P.

Chan, V. W. S.

This method has been used by G. L. Abbas, V. W. S. Chan, T. K. Yee, IEEE J. Lightwave Technol. 3, 1110 (1985).
[CrossRef]

H. P. Yuen, V. W. S. Chan, Opt. Lett. 8, 177 (1983).
[CrossRef] [PubMed]

Cheng, C.

M. G. Raymer, C. Cheng, D. M. Toloudis, M. Anderson, M. Beck, in Advances in Optical Imaging and Photon Migration (Optical Society of America, Washington, D.C., 1996), pp. 236–238.

Clarke, L.

Corcoran, V. J.

See, for example, V. J. Corcoran, J. Appl. Phys. 36, 1819 (1965); A. E. Siegman, Appl. Opt. 5, 1588 (1966); S. Cohen, Appl. Opt. 14, 1953 (1975); A. Migdall, B. Roop, Y. C. Zheng, J. E. Hardis, G. J. Xia, Appl. Opt. 29, 5136 (1990).
[CrossRef] [PubMed]

Cummins, H. Z.

This method has been used in light beating spectroscopy; see H. Z. Cummins, H. L. Swinney, in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1970), Vol. VIII, Chap. 3, pp. 133–200.
[CrossRef]

Devaraj, B.

Fujimoto, J. G.

Hee, H. R.

Hillery, M.

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

Inaba, H.

Izatt, J. A.

John, S.

S. John, G. Pang, Y. Yang, Proc. SPIE 2389, 64 (1995).
[CrossRef]

Mayer, A.

McAlister, D. F.

O'Connel, R. F.

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

Owen, G. M.

Pang, G.

S. John, G. Pang, Y. Yang, Proc. SPIE 2389, 64 (1995).
[CrossRef]

Raymer, M. G.

D. F. McAlister, M. Beck, L. Clarke, A. Mayer, M. G. Raymer, Opt. Lett. 20, 1181 (1995).
[CrossRef] [PubMed]

M. G. Raymer, C. Cheng, D. M. Toloudis, M. Anderson, M. Beck, in Advances in Optical Imaging and Photon Migration (Optical Society of America, Washington, D.C., 1996), pp. 236–238.

Scully, M. O.

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

Swanson, E. A.

Swinney, H. L.

This method has been used in light beating spectroscopy; see H. Z. Cummins, H. L. Swinney, in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1970), Vol. VIII, Chap. 3, pp. 133–200.
[CrossRef]

Thomas, J. E.

A. Wax, J. E. Thomas, in Advances in Optical Imaging and Photon Migration (Optical Society of America, Washington, D.C., 1996), pp. 238–242.

Toloudis, D. M.

M. G. Raymer, C. Cheng, D. M. Toloudis, M. Anderson, M. Beck, in Advances in Optical Imaging and Photon Migration (Optical Society of America, Washington, D.C., 1996), pp. 236–238.

Wax, A.

A. Wax, J. E. Thomas, in Advances in Optical Imaging and Photon Migration (Optical Society of America, Washington, D.C., 1996), pp. 238–242.

Wigner, E. P.

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

E. P. Wigner, Phys. Rev. Lett. 40, 749 (1932).

Yamada, M.

Yang, Y.

S. John, G. Pang, Y. Yang, Proc. SPIE 2389, 64 (1995).
[CrossRef]

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, & Winston, New York, 1976), Chap. 3, p. 35.

Yee, T. K.

This method has been used by G. L. Abbas, V. W. S. Chan, T. K. Yee, IEEE J. Lightwave Technol. 3, 1110 (1985).
[CrossRef]

Yuen, H. P.

IEEE J. Lightwave Technol.

This method has been used by G. L. Abbas, V. W. S. Chan, T. K. Yee, IEEE J. Lightwave Technol. 3, 1110 (1985).
[CrossRef]

J. Appl. Phys.

See, for example, V. J. Corcoran, J. Appl. Phys. 36, 1819 (1965); A. E. Siegman, Appl. Opt. 5, 1588 (1966); S. Cohen, Appl. Opt. 14, 1953 (1975); A. Migdall, B. Roop, Y. C. Zheng, J. E. Hardis, G. J. Xia, Appl. Opt. 29, 5136 (1990).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rep.

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

Phys. Rev. Lett.

E. P. Wigner, Phys. Rev. Lett. 40, 749 (1932).

Physica

The mean-square beat is positive definite and takes the form of a smoothed Wigner distribution. See N. D. Cartwright, Physica 83A, 210 (1976).

Proc. SPIE

S. John, G. Pang, Y. Yang, Proc. SPIE 2389, 64 (1995).
[CrossRef]

Other

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, & Winston, New York, 1976), Chap. 3, p. 35.

A. Wax, J. E. Thomas, in Advances in Optical Imaging and Photon Migration (Optical Society of America, Washington, D.C., 1996), pp. 238–242.

This method has been used in light beating spectroscopy; see H. Z. Cummins, H. L. Swinney, in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1970), Vol. VIII, Chap. 3, pp. 133–200.
[CrossRef]

M. G. Raymer, C. Cheng, D. M. Toloudis, M. Anderson, M. Beck, in Advances in Optical Imaging and Photon Migration (Optical Society of America, Washington, D.C., 1996), pp. 236–238.

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Figures (3)

Fig. 1
Fig. 1

Scheme for heterodyne measurement of Wigner phase space distributions. The displacement dx of mirror M1 determines the position x, and the displacement dp of lens L2 determines the momentum p. A/O's, acousto-optic modulators.

Fig. 2
Fig. 2

Measured Wigner phase space contours for Gaussian signal beams: (a) beam waist (flat wave front), (b) diverging (positive wave-front curvature), (c) converging (negative wave-front curvature).

Fig. 3
Fig. 3

Measured Wigner phase space contours for two spatially separated, mutually coherent beams: (a) Phase space contour, (b) position profile for momentum p = 0, (c) momentum profile at position x = 0. Dotted curves, data; solid curve, theory.

Equations (5)

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W ( x , p ) = d 2 π exp ( i p ) E * ( x + / 2 ) E ( x / 2 ) ,
V B 2 | d x E LO * ( x , z D ) E S ( x , z D ) | 2 = | d x E LO * ( x d x , z = 0 ) E S ( x , z = 0 ) × exp ( i k d p f x ) | 2 .
V B ( d x , d p ) 2 d x d p     W LO ( x d x , p + k d p / f ) W S ( x , p ) .
W ( x , p ) = ( 1 / π ) exp ( x 2 / w 2 ) × exp [ w 2 ( p k x / R ) 2 ] .
W S ( x , p ) = W G ( x d / 2 , p ) + W G ( x + d / 2 , p ) + 2 W G ( x , p ) cos ( d p + φ ) ,

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