Abstract

Phase-space tomography is experimentally demonstrated for the determination of the spatially varying amplitude and phase of a quasi-monochromatic optical field by measurements of intensity only. Both fully and partially coherent sources are characterized. The method, which makes use of the fractional-order Fourier transform, also yields the Wigner distribution of the field and works in one or two dimensions.

© 1995 Optical Society of America

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References

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  1. J. R. Fienup, Appl. Opt. 21, 2758 (1982).
    [CrossRef] [PubMed]
  2. K. Creath, in Progress in Optics XXVI, E. Wolf, ed. (Elsevier, Amsterdam, 1988), p. 349.
    [CrossRef]
  3. M. G. Raymer, M. Beck, D. F. McAlister, Phys. Rev. Lett. 72, 1137 (1994)Quantum Optics VI, D. F. Walls, J. D. Harvey, eds. (Springer-Verlag, Berlin, 1994), p. 245.
    [CrossRef] [PubMed]
  4. M. Beck, M. G. Raymer, I. A. Walmsley, V. Wong, Opt. Lett. 18, 2041 (1993)D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani, Phys. Rev. Lett. 70, 1244 (1993)
    [CrossRef] [PubMed]
  5. N. Streibl, Opt. Commun. 49, 6 (1984); K. Ichikawa, A. W. Lohmann, M. Takeda, Appl. Opt. 27, 3433 (1988).
    [CrossRef] [PubMed]
  6. D. Kohler, L. Mandel, J. Opt. Soc. Am. 63, 126 (1972).
    [CrossRef]
  7. K.-H. Brenner, A. W. Lohmann, Opt. Commun. 42, 310 (1982).
    [CrossRef]
  8. D. F. McAlister, M. Beck, M. G. Raymer, presented at the Annual Meeting of the Optical Society of America, Dallas, Tex., October 2–7, 1994.
  9. M. Hillery, R. F. O’Connel, M. O. Scully, E. P. Wigner, Phys. Rep. 106, 121 (1984)M. J. Bastiaans, J. Opt. Soc. Am. A 3, 1227 (1986).
    [CrossRef]
  10. V. Namias, J. Inst. Math. Appl. 25, 241 (1980); H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, J. Opt. Soc. Am. A 11, 547 (1994); A. W. Lohmann, J. Opt. Soc. Am. A 10, 2181 (1993).
    [CrossRef]
  11. J. Bertrand, P. Bertrand, Found. Phys. 17, 397 (1987)P. Bertrand, J. Bertrand, Reck. Aerosp. 5, 277 (1985)
    [CrossRef]
  12. G. T. Herman, Image Reconstruction From Projections: The Fundamentals of Computerized Tomography (Academic, New York, 1980), p. 279.
  13. B. Eppich, N. Reng, presented at the Laser Beam Control, Diagnostics, and Standards Conference, San Jose, Calif., February 6–7, 1995.

1994 (1)

M. G. Raymer, M. Beck, D. F. McAlister, Phys. Rev. Lett. 72, 1137 (1994)Quantum Optics VI, D. F. Walls, J. D. Harvey, eds. (Springer-Verlag, Berlin, 1994), p. 245.
[CrossRef] [PubMed]

1993 (1)

1987 (1)

J. Bertrand, P. Bertrand, Found. Phys. 17, 397 (1987)P. Bertrand, J. Bertrand, Reck. Aerosp. 5, 277 (1985)
[CrossRef]

1984 (2)

N. Streibl, Opt. Commun. 49, 6 (1984); K. Ichikawa, A. W. Lohmann, M. Takeda, Appl. Opt. 27, 3433 (1988).
[CrossRef] [PubMed]

M. Hillery, R. F. O’Connel, M. O. Scully, E. P. Wigner, Phys. Rep. 106, 121 (1984)M. J. Bastiaans, J. Opt. Soc. Am. A 3, 1227 (1986).
[CrossRef]

1982 (2)

K.-H. Brenner, A. W. Lohmann, Opt. Commun. 42, 310 (1982).
[CrossRef]

J. R. Fienup, Appl. Opt. 21, 2758 (1982).
[CrossRef] [PubMed]

1980 (1)

V. Namias, J. Inst. Math. Appl. 25, 241 (1980); H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, J. Opt. Soc. Am. A 11, 547 (1994); A. W. Lohmann, J. Opt. Soc. Am. A 10, 2181 (1993).
[CrossRef]

1972 (1)

Beck, M.

M. G. Raymer, M. Beck, D. F. McAlister, Phys. Rev. Lett. 72, 1137 (1994)Quantum Optics VI, D. F. Walls, J. D. Harvey, eds. (Springer-Verlag, Berlin, 1994), p. 245.
[CrossRef] [PubMed]

M. Beck, M. G. Raymer, I. A. Walmsley, V. Wong, Opt. Lett. 18, 2041 (1993)D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani, Phys. Rev. Lett. 70, 1244 (1993)
[CrossRef] [PubMed]

D. F. McAlister, M. Beck, M. G. Raymer, presented at the Annual Meeting of the Optical Society of America, Dallas, Tex., October 2–7, 1994.

Bertrand, J.

J. Bertrand, P. Bertrand, Found. Phys. 17, 397 (1987)P. Bertrand, J. Bertrand, Reck. Aerosp. 5, 277 (1985)
[CrossRef]

Bertrand, P.

J. Bertrand, P. Bertrand, Found. Phys. 17, 397 (1987)P. Bertrand, J. Bertrand, Reck. Aerosp. 5, 277 (1985)
[CrossRef]

Brenner, K.-H.

K.-H. Brenner, A. W. Lohmann, Opt. Commun. 42, 310 (1982).
[CrossRef]

Creath, K.

K. Creath, in Progress in Optics XXVI, E. Wolf, ed. (Elsevier, Amsterdam, 1988), p. 349.
[CrossRef]

Eppich, B.

B. Eppich, N. Reng, presented at the Laser Beam Control, Diagnostics, and Standards Conference, San Jose, Calif., February 6–7, 1995.

Fienup, J. R.

Herman, G. T.

G. T. Herman, Image Reconstruction From Projections: The Fundamentals of Computerized Tomography (Academic, New York, 1980), p. 279.

Hillery, M.

M. Hillery, R. F. O’Connel, M. O. Scully, E. P. Wigner, Phys. Rep. 106, 121 (1984)M. J. Bastiaans, J. Opt. Soc. Am. A 3, 1227 (1986).
[CrossRef]

Kohler, D.

Lohmann, A. W.

K.-H. Brenner, A. W. Lohmann, Opt. Commun. 42, 310 (1982).
[CrossRef]

Mandel, L.

McAlister, D. F.

M. G. Raymer, M. Beck, D. F. McAlister, Phys. Rev. Lett. 72, 1137 (1994)Quantum Optics VI, D. F. Walls, J. D. Harvey, eds. (Springer-Verlag, Berlin, 1994), p. 245.
[CrossRef] [PubMed]

D. F. McAlister, M. Beck, M. G. Raymer, presented at the Annual Meeting of the Optical Society of America, Dallas, Tex., October 2–7, 1994.

Namias, V.

V. Namias, J. Inst. Math. Appl. 25, 241 (1980); H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, J. Opt. Soc. Am. A 11, 547 (1994); A. W. Lohmann, J. Opt. Soc. Am. A 10, 2181 (1993).
[CrossRef]

O’Connel, R. F.

M. Hillery, R. F. O’Connel, M. O. Scully, E. P. Wigner, Phys. Rep. 106, 121 (1984)M. J. Bastiaans, J. Opt. Soc. Am. A 3, 1227 (1986).
[CrossRef]

Raymer, M. G.

M. G. Raymer, M. Beck, D. F. McAlister, Phys. Rev. Lett. 72, 1137 (1994)Quantum Optics VI, D. F. Walls, J. D. Harvey, eds. (Springer-Verlag, Berlin, 1994), p. 245.
[CrossRef] [PubMed]

M. Beck, M. G. Raymer, I. A. Walmsley, V. Wong, Opt. Lett. 18, 2041 (1993)D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani, Phys. Rev. Lett. 70, 1244 (1993)
[CrossRef] [PubMed]

D. F. McAlister, M. Beck, M. G. Raymer, presented at the Annual Meeting of the Optical Society of America, Dallas, Tex., October 2–7, 1994.

Reng, N.

B. Eppich, N. Reng, presented at the Laser Beam Control, Diagnostics, and Standards Conference, San Jose, Calif., February 6–7, 1995.

Scully, M. O.

M. Hillery, R. F. O’Connel, M. O. Scully, E. P. Wigner, Phys. Rep. 106, 121 (1984)M. J. Bastiaans, J. Opt. Soc. Am. A 3, 1227 (1986).
[CrossRef]

Streibl, N.

N. Streibl, Opt. Commun. 49, 6 (1984); K. Ichikawa, A. W. Lohmann, M. Takeda, Appl. Opt. 27, 3433 (1988).
[CrossRef] [PubMed]

Walmsley, I. A.

Wigner, E. P.

M. Hillery, R. F. O’Connel, M. O. Scully, E. P. Wigner, Phys. Rep. 106, 121 (1984)M. J. Bastiaans, J. Opt. Soc. Am. A 3, 1227 (1986).
[CrossRef]

Wong, V.

Appl. Opt. (1)

Found. Phys. (1)

J. Bertrand, P. Bertrand, Found. Phys. 17, 397 (1987)P. Bertrand, J. Bertrand, Reck. Aerosp. 5, 277 (1985)
[CrossRef]

J. Inst. Math. Appl. (1)

V. Namias, J. Inst. Math. Appl. 25, 241 (1980); H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, J. Opt. Soc. Am. A 11, 547 (1994); A. W. Lohmann, J. Opt. Soc. Am. A 10, 2181 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

N. Streibl, Opt. Commun. 49, 6 (1984); K. Ichikawa, A. W. Lohmann, M. Takeda, Appl. Opt. 27, 3433 (1988).
[CrossRef] [PubMed]

K.-H. Brenner, A. W. Lohmann, Opt. Commun. 42, 310 (1982).
[CrossRef]

Opt. Lett. (1)

Phys. Rep. (1)

M. Hillery, R. F. O’Connel, M. O. Scully, E. P. Wigner, Phys. Rep. 106, 121 (1984)M. J. Bastiaans, J. Opt. Soc. Am. A 3, 1227 (1986).
[CrossRef]

Phys. Rev. Lett. (1)

M. G. Raymer, M. Beck, D. F. McAlister, Phys. Rev. Lett. 72, 1137 (1994)Quantum Optics VI, D. F. Walls, J. D. Harvey, eds. (Springer-Verlag, Berlin, 1994), p. 245.
[CrossRef] [PubMed]

Other (4)

K. Creath, in Progress in Optics XXVI, E. Wolf, ed. (Elsevier, Amsterdam, 1988), p. 349.
[CrossRef]

D. F. McAlister, M. Beck, M. G. Raymer, presented at the Annual Meeting of the Optical Society of America, Dallas, Tex., October 2–7, 1994.

G. T. Herman, Image Reconstruction From Projections: The Fundamentals of Computerized Tomography (Academic, New York, 1980), p. 279.

B. Eppich, N. Reng, presented at the Laser Beam Control, Diagnostics, and Standards Conference, San Jose, Calif., February 6–7, 1995.

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Figures (3)

Fig. 1
Fig. 1

Experimental setup.

Fig. 2
Fig. 2

Measured scaled intensity distributions for (a) the fully coherent field and (b) the partially coherent field.

Fig. 3
Fig. 3

Equal separation contours showing the magnitude squared of the reconstructed field correlation function, |Γ(χ,χ′)|2, for (a) the fully coherent field and (b) the partially coherent field. (c) The intensity profile (solid curve) and the phase structure (dashed curves) of the reconstructed complex wave field obtained in the fully coherent case. Where the intensity is near zero the phase is undefined and is not plotted.

Equations (4)

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Γ ( x , x ) = E ( x ) E * ( x ) ,
W ( χ , η ) = 1 π - E ( χ + χ ) E * ( χ - χ ) × exp ( - i 2 χ η ) d χ .
P θ ( χ θ ) = E θ ( χ θ ) 2 ,
P θ ( χ θ ) = I [ - χ θ ( L csc θ / k x 0 ) , D , d , f ] .

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