Abstract

Shows that the amplitude and phase of an optical field can be recovered from only a section of phase-space distribution functions, the information contained in other sections being redundant. Experimental implications of this result are discussed, and a physical interpretation of it is offered. In particular the sampling problem in tomography is solved for any field distribution.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
    [CrossRef]
  2. L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
    [CrossRef]
  3. K.-H. Brenner, J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
    [CrossRef]
  4. A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974).
    [CrossRef]
  5. M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
    [CrossRef] [PubMed]
  6. T. Alieva, M. J. Bastiaans, “Phase-space distributions in quasi-polar coordinates and the fractional Fourier transform,” J. Opt. Soc. Am. A 17, 2324–2329 (2000).
    [CrossRef]
  7. J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
    [CrossRef]
  8. D. Dragoman, M. Dragoman, K.-H. Brenner, “Amplitude and phase recovery of rotationally symmetric beams,” Appl. Opt. 41, 5512–5518 (2002).
    [CrossRef] [PubMed]
  9. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  10. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  11. T. E. Gureyev, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
    [CrossRef]
  12. N. Nakajima, “Phase retrieval from Fresnel zone intensity measurements by use of Gaussian filtering,” Appl. Opt. 37, 6219–6226 (1998).
    [CrossRef]
  13. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968) Chap. 4.
  14. K.-H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
    [CrossRef]
  15. G. Shabtay, D. Mendlovic, Z. Zalevsky, “Proposal for optical implementation of the Wigner distribution function,” Appl. Opt. 37, 2142–2144 (1998).
    [CrossRef]
  16. 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]
  17. Y. Li, G. Eichmann, M. Conner, “Optical Wigner distribution and ambiguity function for complex signals and images,” Opt. Commun. 67, 177–179 (1988).
    [CrossRef]
  18. M. Conner, Y. Li, “Optical generation of the Wigner distribution of 2-D real signals,” Appl. Opt. 24, 3825–3829 (1985).
    [CrossRef] [PubMed]
  19. R. Bamler, H. Glünder, “The Wigner distribution function of two-dimensional signals. Coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
    [CrossRef]
  20. W. D. Furlan, D. Zalvidea, G. Saavedra, “Synthesis of filters for specified axial irradiance by use of phase-space tomography,” Opt. Commun. 189, 15–19 (2001).
    [CrossRef]

2002 (1)

2001 (1)

W. D. Furlan, D. Zalvidea, G. Saavedra, “Synthesis of filters for specified axial irradiance by use of phase-space tomography,” Opt. Commun. 189, 15–19 (2001).
[CrossRef]

2000 (1)

1998 (2)

1997 (2)

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

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
[CrossRef]

1996 (1)

1994 (1)

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

1989 (1)

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

1988 (1)

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 (1)

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 (1)

1984 (1)

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

1983 (1)

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 (2)

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

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

1974 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Alieva, T.

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]

Bastiaans, M. J.

Beck, M.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Brenner, K.-H.

D. Dragoman, M. Dragoman, K.-H. Brenner, “Amplitude and phase recovery of rotationally symmetric beams,” Appl. Opt. 41, 5512–5518 (2002).
[CrossRef] [PubMed]

K.-H. Brenner, J. Ojeda-Castañeda, “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]

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.

Dragoman, M.

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]

Fienup, J. R.

Furlan, W. D.

W. D. Furlan, D. Zalvidea, G. Saavedra, “Synthesis of filters for specified axial irradiance by use of phase-space tomography,” Opt. Commun. 189, 15–19 (2001).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

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]

Gureyev, T. E.

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]

McAlister, D. F.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Mendlovic, D.

Nakajima, N.

Nugent, K. A.

Ojeda-Castañeda, J.

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

Papoulis, A.

A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974).
[CrossRef]

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968) Chap. 4.

Raymer, M. G.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Saavedra, G.

W. D. Furlan, D. Zalvidea, G. Saavedra, “Synthesis of filters for specified axial irradiance by use of phase-space tomography,” Opt. Commun. 189, 15–19 (2001).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

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]

Tu, J.

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

Zalevsky, Z.

Zalvidea, D.

W. D. Furlan, D. Zalvidea, G. Saavedra, “Synthesis of filters for specified axial irradiance by use of phase-space tomography,” Opt. Commun. 189, 15–19 (2001).
[CrossRef]

Appl. Opt. (5)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Acta (2)

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

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

Opt. Commun. (4)

W. D. Furlan, D. Zalvidea, G. Saavedra, “Synthesis of filters for specified axial irradiance by use of phase-space tomography,” Opt. Commun. 189, 15–19 (2001).
[CrossRef]

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

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

Optik (Stuttgart) (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Phys. Rev. E (1)

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

Phys. Rev. Lett. (1)

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Proc. IEEE (1)

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

Prog. Opt. (1)

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
[CrossRef]

Other (1)

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968) Chap. 4.

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 (1)

Fig. 1
Fig. 1

Illustration of the sampling rate in AF determination from tomographic measurements.

Equations (18)

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

AFr, p=12π ψr+r2ψ * r-r2expirpdr,
Ar+r02Ar-r02=12πAFr0, p×exp-irpdp =Fr, r0,
exp2iφr+r02-2iφr-r02 =AFr0, pexp-irpdpAF * r0, qexpirqdq =Gr, r0,
Ar=expgr=expn=1N anϕnr, φr=n=1N bnϕnr.
n=1N anϕnr+r02+ϕnr-r02=ln Fr, r0,
n=1N bnϕnr+r02-ϕnr-r02=-i2ln Gr, r0.
Fr, r0=2π-1nAFr0, pnexp-irpnΔpn,
Gr, r0=nAFr0, pnexp-irpnΔpnnAF * r0, pnexpirpnΔpn.
A2r=2π-1limr00AFr0, p×exp-irpdp,
exp2ir0φr=AFr0, pexp-irpdpAF * r0, pexpirpdp.
Wr, p=12π ψr+r2ψ * r-r2expirpdr.
Ar0+r2Ar0-r2=12π Wr0, p×exp-irpdp,
exp2iφr0+r2-2iφr0-r2 = Wr0, pexp-irpdp Wr0, pexpirpdp.
Ar2A-r2=2π-1limr00 Wr0, p×exp-irpdp,
exp2iφr2-φ-r2+2ir0φr2-φ-r2= Wr0, pexp-irpdp Wr0, pexpirpdp.
AFx0, p=123ax0+1w2exp-x024w2-2iax03-p243iax0+1w2.
AFx0, y0, px, py=w02w22w4-w04exp-x02+y024w02-2x0y04w2exp-w02w22w2+w02×a0+ax0+y0-px+py22-w02w22w2-w02a0-ax0-y0-px-py22.
ψrψ * 0=2π-1AFr, pexp-irp2dp.

Metrics