Abstract

We propose a setup that can generate the sectional ambiguity function of a two-dimensional real light source. The setup is easy to implement; the theoretical analysis and experimental results are given.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. A. C. M. Claasen, W. F. G. Meklenbräuker, “The Wigner distribution—a tool for time–frequency signal analysis. III. Relations with other time-frequency transformations,” Philips J. Res. 35, 372–389 (1980).
  2. L. Cohen, “Time–frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
    [CrossRef]
  3. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  4. D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1997), Vol. 37, pp. 1–56.
    [CrossRef]
  5. P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, London, 1953).
  6. K. H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
    [CrossRef]
  7. J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
    [CrossRef]
  8. R. L. Easton, A. J. Ticknor, H. H. Bartelt, “Application of the Radon transform to optical production of the Wigner distribution function,” Opt. Eng. 23, 738–744 (1984).
    [CrossRef]
  9. Y. Li, G. Eichmann, M. Conner, “Optical Wigner distribution and ambiguity function for complex signals and images,” Opt. Commun. 67, 177–179 (1988).
    [CrossRef]
  10. K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
    [CrossRef]
  11. H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
    [CrossRef]
  12. M. J. Bastiaans, “Wigner distribution function display: a supplement to ambiguity function display using a single 1-D input,” Appl. Opt. 19, 192–195 (1980).
    [CrossRef] [PubMed]
  13. G. Shabtay, D. Mendlovic, Z. Zalevsky, “Proposal for optical implementation of the Wigner distribution function,” Appl. Opt. 37, 2142–2144 (1998).
    [CrossRef]
  14. T. Iwai, A. K. Gupta, T. Asakura, “Simultaneous optical production of the sectional Wigner distribution function for a two-dimensional object,” Opt. Commun. 58, 15–19 (1986).
    [CrossRef]
  15. M. Conner, Y. Li, “Optical generation of the Wigner distribution of 2-D real signals,” Appl. Opt. 24, 3825–3829 (1985).
    [CrossRef] [PubMed]
  16. R. Bamler, H. Glünder, “The Wigner distribution function of two-dimensional signals coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
    [CrossRef]

1998

1997

J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

1989

L. Cohen, “Time–frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

1988

Y. Li, G. Eichmann, M. Conner, “Optical Wigner distribution and ambiguity function for complex signals and images,” Opt. Commun. 67, 177–179 (1988).
[CrossRef]

1986

T. Iwai, A. K. Gupta, T. Asakura, “Simultaneous optical production of the sectional Wigner distribution function for a two-dimensional object,” Opt. Commun. 58, 15–19 (1986).
[CrossRef]

1985

1984

R. L. Easton, A. J. Ticknor, H. H. Bartelt, “Application of the Radon transform to optical production of the Wigner distribution function,” Opt. Eng. 23, 738–744 (1984).
[CrossRef]

K. H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

1983

R. Bamler, H. Glünder, “The Wigner distribution function of two-dimensional signals coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
[CrossRef]

1982

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

1980

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

M. J. Bastiaans, “Wigner distribution function display: a supplement to ambiguity function display using a single 1-D input,” Appl. Opt. 19, 192–195 (1980).
[CrossRef] [PubMed]

T. A. C. M. Claasen, W. F. G. Meklenbräuker, “The Wigner distribution—a tool for time–frequency signal analysis. III. Relations with other time-frequency transformations,” Philips J. Res. 35, 372–389 (1980).

1932

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

Asakura, T.

T. Iwai, A. K. Gupta, T. Asakura, “Simultaneous optical production of the sectional Wigner distribution function for a two-dimensional object,” Opt. Commun. 58, 15–19 (1986).
[CrossRef]

Bamler, R.

R. Bamler, H. Glünder, “The Wigner distribution function of two-dimensional signals coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
[CrossRef]

Bartelt, H. H.

R. L. Easton, A. J. Ticknor, H. H. Bartelt, “Application of the Radon transform to optical production of the Wigner distribution function,” Opt. Eng. 23, 738–744 (1984).
[CrossRef]

Bartelt, H. O.

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Bastiaans, M. J.

Brenner, K. H.

K. H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Claasen, T. A. C. M.

T. A. C. M. Claasen, W. F. G. Meklenbräuker, “The Wigner distribution—a tool for time–frequency signal analysis. III. Relations with other time-frequency transformations,” Philips J. Res. 35, 372–389 (1980).

Cohen, L.

L. Cohen, “Time–frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

Conner, M.

Y. Li, G. Eichmann, M. Conner, “Optical Wigner distribution and ambiguity function for complex signals and images,” Opt. Commun. 67, 177–179 (1988).
[CrossRef]

M. Conner, Y. Li, “Optical generation of the Wigner distribution of 2-D real signals,” Appl. Opt. 24, 3825–3829 (1985).
[CrossRef] [PubMed]

Dragoman, D.

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1997), Vol. 37, pp. 1–56.
[CrossRef]

Easton, R. L.

R. L. Easton, A. J. Ticknor, H. H. Bartelt, “Application of the Radon transform to optical production of the Wigner distribution function,” Opt. Eng. 23, 738–744 (1984).
[CrossRef]

Eichmann, G.

Y. Li, G. Eichmann, M. Conner, “Optical Wigner distribution and ambiguity function for complex signals and images,” Opt. Commun. 67, 177–179 (1988).
[CrossRef]

Glünder, H.

R. Bamler, H. Glünder, “The Wigner distribution function of two-dimensional signals coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
[CrossRef]

Gupta, A. K.

T. Iwai, A. K. Gupta, T. Asakura, “Simultaneous optical production of the sectional Wigner distribution function for a two-dimensional object,” Opt. Commun. 58, 15–19 (1986).
[CrossRef]

Iwai, T.

T. Iwai, A. K. Gupta, T. Asakura, “Simultaneous optical production of the sectional Wigner distribution function for a two-dimensional object,” Opt. Commun. 58, 15–19 (1986).
[CrossRef]

Li, Y.

Y. Li, G. Eichmann, M. Conner, “Optical Wigner distribution and ambiguity function for complex signals and images,” Opt. Commun. 67, 177–179 (1988).
[CrossRef]

M. Conner, Y. Li, “Optical generation of the Wigner distribution of 2-D real signals,” Appl. Opt. 24, 3825–3829 (1985).
[CrossRef] [PubMed]

Lohmann, A. W.

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Meklenbräuker, W. F. G.

T. A. C. M. Claasen, W. F. G. Meklenbräuker, “The Wigner distribution—a tool for time–frequency signal analysis. III. Relations with other time-frequency transformations,” Philips J. Res. 35, 372–389 (1980).

Mendlovic, D.

Ojeda-Castaneda, J.

K. H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

Shabtay, G.

Tamura, S.

J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

Ticknor, A. J.

R. L. Easton, A. J. Ticknor, H. H. Bartelt, “Application of the Radon transform to optical production of the Wigner distribution function,” Opt. Eng. 23, 738–744 (1984).
[CrossRef]

Tu, J.

J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

Wigner, E.

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

Woodward, P. M.

P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, London, 1953).

Zalevsky, Z.

Appl. Opt.

Opt. Acta

R. Bamler, H. Glünder, “The Wigner distribution function of two-dimensional signals coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
[CrossRef]

K. H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

Opt. Commun.

Y. Li, G. Eichmann, M. Conner, “Optical Wigner distribution and ambiguity function for complex signals and images,” Opt. Commun. 67, 177–179 (1988).
[CrossRef]

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

T. Iwai, A. K. Gupta, T. Asakura, “Simultaneous optical production of the sectional Wigner distribution function for a two-dimensional object,” Opt. Commun. 58, 15–19 (1986).
[CrossRef]

Opt. Eng.

R. L. Easton, A. J. Ticknor, H. H. Bartelt, “Application of the Radon transform to optical production of the Wigner distribution function,” Opt. Eng. 23, 738–744 (1984).
[CrossRef]

Philips J. Res.

T. A. C. M. Claasen, W. F. G. Meklenbräuker, “The Wigner distribution—a tool for time–frequency signal analysis. III. Relations with other time-frequency transformations,” Philips J. Res. 35, 372–389 (1980).

Phys. Rev.

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

Phys. Rev. E

J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

Proc. IEEE

L. Cohen, “Time–frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

Other

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1997), Vol. 37, pp. 1–56.
[CrossRef]

P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, London, 1953).

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

Result of numerically Fourier transforming |g| 2 for an elliptical Gaussian light beam tilted at 45°. (a) All three terms of F(p, q), (b) magnified picture of the contour plot of Ax, Δy, p - θ x , q - θ y ). See text for the parameters used in the simulation.

Fig. 2
Fig. 2

(a) Principle of generating the second, virtual light source by refraction inside a prism, (b) setup used for generating the sectional AF’s.

Fig. 3
Fig. 3

Negative of the picture of the CCD camera showing the two spatially separated light sources after passing through the prism. Note the difference in the intensities of the two sources. The tilted line indicates the separation between the centers of the two light distributions.

Fig. 4
Fig. 4

Modified setup with two beam splitters for the generation of the sectional AF. The images of the tilted Gaussian slit after the first and the second beam splitter are shown in the insets.

Fig. 5
Fig. 5

Spatially separated pair of images of the original light source with no phase difference. (a) Δy ≅ 1 mm, (b) Δy ≅ 0.33 mm, (c) Δy ≅ 0 mm. Note the equality of the intensities of the two sources.

Fig. 6
Fig. 6

Recovered contour plot of A(0, Δy, 0, q - θ y ) for θ y ≅ 15°.

Equations (9)

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

fr, r=ϕr+r2ϕ*r-r2
Wr, ν= fr, rexpikνrdr,
Ar, ν= fr, rexpikνrdr,
Wr, ν= Ar, νexpikνr-νrdνdr,
Ar, ν= Wr, νexpikνr-νrdνdr.
gx, y=ϕx+Δx2, y+Δy2expik2xθx+yθy+ϕx-Δx2, y-Δy2exp-ik2xθx+yθy,
Fp, q= |gx, y|2 expikpx+qydxdy=2A0, 0, p, qcospΔx+qΔy+A-Δx, -Δy, p+θx, q+θy+AΔx, Δy, p-θx, q-θy.
θx=θ2-θ1=arcsinn sin3α+δ-arcsinn sinα+δ,
Δx=l cos θ1cosθ2-θ1/2cos δcosα+δcosα+δcos3α+δ-1.

Metrics