Abstract

New techniques as well as extensions to existing techniques for the optical production of 2-D Wigner distributions are presented. Our investigation starts with a straightforward technique involving a pair of relay lenses and a Fourier transform lens. Then we explore a second technique requiring only one lens and a rotational capability. We also present a technique for display of a single frequency variable vs a single spatial variable. Finally, we discuss a dual-channel display technique in which two different sections of the Wigner distribution are displayed simultaneously.

© 1985 Optical Society of America

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References

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  1. E. Wigner, “On the Quantum Correction for Thermodynamic Equilibrium,” Phys. Rev. 40, 749 (1932).
    [CrossRef]
  2. M. J. Bastiaans, “The Wigner Distribution Function Applied to Optical Signals and Systems,” Opt. Commun. 25, 26 (1978).
    [CrossRef]
  3. M. J. Bastiaans, “The Wigner Distribution Function and Hamilton’s Characteristics of a Geometrical Optical System,” Opt. Commun. 30, 321 (1979).
    [CrossRef]
  4. J. Z. Jiao, B. Wang, H. Liu, “Wigner Distribution Function and Optical Geometrical Transformation,” Appl. Opt. 23, 1249 (1984).
    [CrossRef] [PubMed]
  5. K. H. Brenner, J. O. Castaneda, “Ambiguity Function and Wigner Distribution Function Applied to Partially Coherent Imagery,” Opt. Acta 31, 213 (1984).
    [CrossRef]
  6. M. I. Skolnik, Radar Handbook (McGraw-Hill, New York, 1970).
  7. L. Jacobson, H. Wechsler, “The Wigner Distribution As a Tool for Deriving an Invariant Representation of 2-D Images,” in Proceedings, PRIP82IEEE Computer Society Conference on Pattern Recognition and Image Processing, Las Vegas, Nev.14–17 June, 218 (1982).
  8. K. H. Brenner, H. O. Bartelt, A. W. Lohmann, “The Wigner Distribution Function and its Optical Production,” Opt. Commun. 42, 32 (1982).
  9. M. J. Bastiaans, “Wigner Distribution Function Display: A Supplement to Ambiguity Function Display Using a Single 1-D Input,” Appl. Opt. 19, 192 (1980).
    [CrossRef] [PubMed]
  10. R. J. Marks, J. F. Walkup, T. F. Krile, “Ambiguity Function Display: An Improved Coherent Processor,” Appl. Opt. 16, 746 (1977).
    [CrossRef]
  11. G. Eichmann, B. Z. Dong, “Two-Dimensional Optical Filtering of 1-D Signals,” Appl. Opt. 21, 3152 (1982).
    [CrossRef] [PubMed]
  12. R. Bamler, H. Glunder, “Coherent-Optical Generation of the Wigner Distribution Function of Real-Valued 2D Signals,” in Proceedings, International Conference on Optical Computing, Cambridge, Mass. (1983).
  13. T. A. C. M. Claasen, W. F. G. Mechlenbrauker, “The Wigner Distribution—A Tool for Time-Frequency Signal Analysis, Part I: Continuous Time Signals,” Philips J. Res. 35, 217 (1980).
  14. H. H. Szu, J. A. Blodgett, “Wigner Distribution and Ambiguity Function,” in Optics in Four Dimensions-1980, AIP Conf. Proc. 65, 292 (1981).
  15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  16. L. P. Boivin, “Multiple Imaging Using Various Types of Simple Phase Gratings,” Appl. Opt. 11, 1782 (1972).
    [CrossRef] [PubMed]
  17. F. T. S. Yu, M. S. Dymek, “Optical Information Parallel Processing: A Technique,” Appl. Opt. 20, 1450 (1981).
    [CrossRef] [PubMed]
  18. A. M. Tai, F. T. S. Yu, “Synchronous Dual-Channel Optical Spectrum Analyzer,” Appl. Opt. 18, 1297 (1979).
    [CrossRef] [PubMed]

1984 (2)

J. Z. Jiao, B. Wang, H. Liu, “Wigner Distribution Function and Optical Geometrical Transformation,” Appl. Opt. 23, 1249 (1984).
[CrossRef] [PubMed]

K. H. Brenner, J. O. Castaneda, “Ambiguity Function and Wigner Distribution Function Applied to Partially Coherent Imagery,” Opt. Acta 31, 213 (1984).
[CrossRef]

1982 (2)

K. H. Brenner, H. O. Bartelt, A. W. Lohmann, “The Wigner Distribution Function and its Optical Production,” Opt. Commun. 42, 32 (1982).

G. Eichmann, B. Z. Dong, “Two-Dimensional Optical Filtering of 1-D Signals,” Appl. Opt. 21, 3152 (1982).
[CrossRef] [PubMed]

1981 (2)

H. H. Szu, J. A. Blodgett, “Wigner Distribution and Ambiguity Function,” in Optics in Four Dimensions-1980, AIP Conf. Proc. 65, 292 (1981).

F. T. S. Yu, M. S. Dymek, “Optical Information Parallel Processing: A Technique,” Appl. Opt. 20, 1450 (1981).
[CrossRef] [PubMed]

1980 (2)

T. A. C. M. Claasen, W. F. G. Mechlenbrauker, “The Wigner Distribution—A Tool for Time-Frequency Signal Analysis, Part I: Continuous Time Signals,” Philips J. Res. 35, 217 (1980).

M. J. Bastiaans, “Wigner Distribution Function Display: A Supplement to Ambiguity Function Display Using a Single 1-D Input,” Appl. Opt. 19, 192 (1980).
[CrossRef] [PubMed]

1979 (2)

A. M. Tai, F. T. S. Yu, “Synchronous Dual-Channel Optical Spectrum Analyzer,” Appl. Opt. 18, 1297 (1979).
[CrossRef] [PubMed]

M. J. Bastiaans, “The Wigner Distribution Function and Hamilton’s Characteristics of a Geometrical Optical System,” Opt. Commun. 30, 321 (1979).
[CrossRef]

1978 (1)

M. J. Bastiaans, “The Wigner Distribution Function Applied to Optical Signals and Systems,” Opt. Commun. 25, 26 (1978).
[CrossRef]

1977 (1)

1972 (1)

1932 (1)

E. Wigner, “On the Quantum Correction for Thermodynamic Equilibrium,” Phys. Rev. 40, 749 (1932).
[CrossRef]

Bamler, R.

R. Bamler, H. Glunder, “Coherent-Optical Generation of the Wigner Distribution Function of Real-Valued 2D Signals,” in Proceedings, International Conference on Optical Computing, Cambridge, Mass. (1983).

Bartelt, H. O.

K. H. Brenner, H. O. Bartelt, A. W. Lohmann, “The Wigner Distribution Function and its Optical Production,” Opt. Commun. 42, 32 (1982).

Bastiaans, M. J.

M. J. Bastiaans, “Wigner Distribution Function Display: A Supplement to Ambiguity Function Display Using a Single 1-D Input,” Appl. Opt. 19, 192 (1980).
[CrossRef] [PubMed]

M. J. Bastiaans, “The Wigner Distribution Function and Hamilton’s Characteristics of a Geometrical Optical System,” Opt. Commun. 30, 321 (1979).
[CrossRef]

M. J. Bastiaans, “The Wigner Distribution Function Applied to Optical Signals and Systems,” Opt. Commun. 25, 26 (1978).
[CrossRef]

Blodgett, J. A.

H. H. Szu, J. A. Blodgett, “Wigner Distribution and Ambiguity Function,” in Optics in Four Dimensions-1980, AIP Conf. Proc. 65, 292 (1981).

Boivin, L. P.

Brenner, K. H.

K. H. Brenner, J. O. Castaneda, “Ambiguity Function and Wigner Distribution Function Applied to Partially Coherent Imagery,” Opt. Acta 31, 213 (1984).
[CrossRef]

K. H. Brenner, H. O. Bartelt, A. W. Lohmann, “The Wigner Distribution Function and its Optical Production,” Opt. Commun. 42, 32 (1982).

Castaneda, J. O.

K. H. Brenner, J. O. Castaneda, “Ambiguity Function and Wigner Distribution Function Applied to Partially Coherent Imagery,” Opt. Acta 31, 213 (1984).
[CrossRef]

Claasen, T. A. C. M.

T. A. C. M. Claasen, W. F. G. Mechlenbrauker, “The Wigner Distribution—A Tool for Time-Frequency Signal Analysis, Part I: Continuous Time Signals,” Philips J. Res. 35, 217 (1980).

Dong, B. Z.

Dymek, M. S.

Eichmann, G.

Glunder, H.

R. Bamler, H. Glunder, “Coherent-Optical Generation of the Wigner Distribution Function of Real-Valued 2D Signals,” in Proceedings, International Conference on Optical Computing, Cambridge, Mass. (1983).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Jacobson, L.

L. Jacobson, H. Wechsler, “The Wigner Distribution As a Tool for Deriving an Invariant Representation of 2-D Images,” in Proceedings, PRIP82IEEE Computer Society Conference on Pattern Recognition and Image Processing, Las Vegas, Nev.14–17 June, 218 (1982).

Jiao, J. Z.

Krile, T. F.

Liu, H.

Lohmann, A. W.

K. H. Brenner, H. O. Bartelt, A. W. Lohmann, “The Wigner Distribution Function and its Optical Production,” Opt. Commun. 42, 32 (1982).

Marks, R. J.

Mechlenbrauker, W. F. G.

T. A. C. M. Claasen, W. F. G. Mechlenbrauker, “The Wigner Distribution—A Tool for Time-Frequency Signal Analysis, Part I: Continuous Time Signals,” Philips J. Res. 35, 217 (1980).

Skolnik, M. I.

M. I. Skolnik, Radar Handbook (McGraw-Hill, New York, 1970).

Szu, H. H.

H. H. Szu, J. A. Blodgett, “Wigner Distribution and Ambiguity Function,” in Optics in Four Dimensions-1980, AIP Conf. Proc. 65, 292 (1981).

Tai, A. M.

Walkup, J. F.

Wang, B.

Wechsler, H.

L. Jacobson, H. Wechsler, “The Wigner Distribution As a Tool for Deriving an Invariant Representation of 2-D Images,” in Proceedings, PRIP82IEEE Computer Society Conference on Pattern Recognition and Image Processing, Las Vegas, Nev.14–17 June, 218 (1982).

Wigner, E.

E. Wigner, “On the Quantum Correction for Thermodynamic Equilibrium,” Phys. Rev. 40, 749 (1932).
[CrossRef]

Yu, F. T. S.

Appl. Opt. (7)

Opt. Acta (1)

K. H. Brenner, J. O. Castaneda, “Ambiguity Function and Wigner Distribution Function Applied to Partially Coherent Imagery,” Opt. Acta 31, 213 (1984).
[CrossRef]

Opt. Commun. (3)

M. J. Bastiaans, “The Wigner Distribution Function Applied to Optical Signals and Systems,” Opt. Commun. 25, 26 (1978).
[CrossRef]

M. J. Bastiaans, “The Wigner Distribution Function and Hamilton’s Characteristics of a Geometrical Optical System,” Opt. Commun. 30, 321 (1979).
[CrossRef]

K. H. Brenner, H. O. Bartelt, A. W. Lohmann, “The Wigner Distribution Function and its Optical Production,” Opt. Commun. 42, 32 (1982).

Optics in Four Dimensions-1980 (1)

H. H. Szu, J. A. Blodgett, “Wigner Distribution and Ambiguity Function,” in Optics in Four Dimensions-1980, AIP Conf. Proc. 65, 292 (1981).

Philips J. Res. (1)

T. A. C. M. Claasen, W. F. G. Mechlenbrauker, “The Wigner Distribution—A Tool for Time-Frequency Signal Analysis, Part I: Continuous Time Signals,” Philips J. Res. 35, 217 (1980).

Phys. Rev. (1)

E. Wigner, “On the Quantum Correction for Thermodynamic Equilibrium,” Phys. Rev. 40, 749 (1932).
[CrossRef]

Other (4)

M. I. Skolnik, Radar Handbook (McGraw-Hill, New York, 1970).

L. Jacobson, H. Wechsler, “The Wigner Distribution As a Tool for Deriving an Invariant Representation of 2-D Images,” in Proceedings, PRIP82IEEE Computer Society Conference on Pattern Recognition and Image Processing, Las Vegas, Nev.14–17 June, 218 (1982).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

R. Bamler, H. Glunder, “Coherent-Optical Generation of the Wigner Distribution Function of Real-Valued 2D Signals,” in Proceedings, International Conference on Optical Computing, Cambridge, Mass. (1983).

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Figures (7)

Fig. 1
Fig. 1

Three lens setup to generate the Wigner distribution function Wf(x0y0,u,υ): P1,P3, planes inserted with input signals; L1,L2, relay lenses; L3, Fourier transform lens; P4, output plane.

Fig. 2
Fig. 2

One lens setup to generate the Wigner distribution function Wf(x0y0u,υ): P1,P2, planes inserted with input signals; L, Fourier transform lens; P3, output plane.

Fig. 3
Fig. 3

One lens setup to generate the Wigner distribution function Wf(x0yu0υ): T1,T2, signal transparencies moving apart as arrow indicates; L, Fourier transform lens; S, 1-D narrow slit; R, recording plate moving horizontally as arrow indicates.

Fig. 4
Fig. 4

One lens setup for dual-channel generation of Wigner distribution functions: H, λ/2 plate; C1,C2, two channels inserted with two pairs of different signals; L, Fourier transform lens; P, linear polarizer; R, recording material.

Fig. 5
Fig. 5

Results of Fig. 1: (a) input signal; (b) multiplication of the signal; (c) corresponding Wigner distribution.

Fig. 6
Fig. 6

Results of Fig. 2: (a) multiplication of the signal when input is Fig. 5(a); (b) corresponding WD.

Fig. 7
Fig. 7

Results of Fig. 4: (a) two signals O and K; (b) output for signal O when filtered out the channel for K; (c) output for signal K when filtered out the channel for O; (d) results when two channels have cross talk.

Equations (14)

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W f ( x , y ; u , υ ) = f ( x + 1 2 x , y + 1 2 y ) f * ( x 1 2 x , y 1 2 y ) · exp [ i 2 π ( u x + υ y ) ] d x d y ,
W f ( x , y ; u , υ ) = F ( u + 1 2 u , υ + 1 2 υ ) F * ( u 1 2 u , υ 1 2 υ ) · exp [ i 2 π ( x u + y υ ) ] d u d υ ,
A f ( x , y ; u , υ ) = F ( x + 1 2 x , y + 1 2 y ) f * ( x 1 2 x , y 1 2 y ) · exp [ i 2 π ( u x + υ y ) ] d x d y ,
A f ( x , y ; u , υ ) = F ( u + 1 2 u , υ + 1 2 υ ) F * ( u 1 2 u , υ 1 2 υ ) · exp [ i 2 π ( x u + y υ ) ] d u d υ .
W f ( x , y ; u , υ ) = A f ( x , y ; u υ ) exp [ i 2 π ( u x + u y u x υ y ) ] d x d y d u d υ .
W f ( x , y ; u , υ ) = g ( x , y , u , υ , x , y ) f * ( x 1 2 x , y 1 2 y ) d x d y ,
g ( x , y , u , υ , x , y ) = f ( x + 1 2 x , y + 1 2 y ) · exp [ i 2 π ( u x + υ y ) ] .
W f ( x , y ; u , υ ) = h ( x , y , x , y ) exp [ i 2 π ( u x + υ y ) ] d x d y ,
h ( x , y , x , y ) = f ( x + 1 2 x , y + 1 2 y ) f * ( x 1 2 x , y 1 2 y ) .
W f ˜ f ( x , y ; u , υ ) = f ˜ ( x + 1 2 x , y + 1 2 y ) f * ( x 1 2 x , y 1 2 y ) · exp [ i 2 π ( u x + υ y ) ] d x d y ,
H ( f x , f y ) = e j k d e j π λ d ( f x 2 + f y 2 ) ,
λ z f x ˜ = 1 50 .
PC = i k d 3 16 f 2 ( cos θ + a d sin θ ) ( sin θ a d cos θ ) 2 ,
W f ( x 0 , y ; u 0 , υ ) = f ( x 0 + 1 2 x , y + 1 2 y ) × f * ( x 0 1 2 x , y 1 2 y ) · exp [ i 2 π ( u 0 x + υ y ) ] d x d y ,

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