Abstract

Two methods are presented for the amplitude and phase recovery of optical beams with rotational symmetry. These are the tomographic method based on the ambiguity function and the one-step wavefront recovery based on the measurement of a phase-space distribution closely related to the Wigner distribution function. The results obtained from these two methods are compared, and the appropriateness of using either one of them for specific situations is discussed.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H.-F. Shih, T.-P. Yang, M. O. Freeman, J.-K. Wang, H.-F. Yau, D.-R. Hua, “Holographic laser-module with dual wavelength for digital versatile disc optical head,” Jpn. J. Appl. Phys. 38, 1750–1754 (1999).
    [CrossRef]
  2. T. E. Gureyev, A. Roberts, K. A. Nuget, “Partially coherent fields, the transport of the intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
    [CrossRef]
  3. C. Iaconis, I. A. Walmsley, “Direct measurement of the two-point field correlation function,” Opt. Lett. 21, 1783–1785 (1996).
    [CrossRef] [PubMed]
  4. D. Dragoman, M. Dragoman, “Characterization of wavefront of light beams by use of tunneling cantilevers,” Appl. Opt. 40, 678–682 (2001).
    [CrossRef]
  5. D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
    [CrossRef]
  6. G. T. Hermann, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic Press, New York, 1980).
  7. D. F. McAlister, M. Beck, L. Clarke, A. Mayer, M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
    [CrossRef] [PubMed]
  8. 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]
  9. M. Beck, M. G. Raymer, I. A. Walmsley, V. Wong, “Cronocyclic tomography for measuring the amplitude and phase structure of the pulses,” Opt. Lett. 18, 2041–2043 (1993).
    [CrossRef]
  10. J. Paye, “The cronocyclic representation of ultrashort light pulses,” IEEE J. Quantum Electron. 28, 2262–2272 (1992).
    [CrossRef]
  11. D. Dragoman, M. Dragoman, K.-H. Brenner, “Variant fractional Fourier transformer for optical pulses,” Opt. Lett. 24, 933–935 (1999).
    [CrossRef]
  12. R. Trebino, D. J. Kane, “The dilemma of ultra-short-laser-pulse intensity-and-phase measurement and applications,” IEEE J. Quantum Electron. 35, 416–420 (1999).
  13. J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
    [CrossRef]
  14. D. Dragoman, “Can Wigner transform of a two-dimensional rotationally symmetric beam be fully recovered from the Wigner transform of its one-dimensional approximation?” Opt. Lett. 25, 281–283 (2000).
    [CrossRef]
  15. K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “The ambiguity function as polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
    [CrossRef]
  16. R. H. Clarke, “Ambiguity function for time-varying random diffracted fields,” J. Opt. Soc. A 3, 1224–1226 (1986).
    [CrossRef]
  17. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Ambiguity function as a design tool for high focal depth,” Appl. Opt. 4, 790–795 (1998).
  18. D. Dragoman, M. Dragoman, K.-H. Brenner, “Optical realization of the ambiguity function of real two-dimensional light sources,” Appl. Opt. 39, 2912–2917 (2000).
    [CrossRef]
  19. K.-H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
    [CrossRef]
  20. K. F. Lee, F. Reil, S. Bali, A. Wax, J. E. Thomas, “Heterodyne measurement of Wigner distributions for classical optical fields,” Opt. Lett. 19, 1370–1372 (1999).
    [CrossRef]

2001 (1)

2000 (2)

1999 (4)

D. Dragoman, M. Dragoman, K.-H. Brenner, “Variant fractional Fourier transformer for optical pulses,” Opt. Lett. 24, 933–935 (1999).
[CrossRef]

K. F. Lee, F. Reil, S. Bali, A. Wax, J. E. Thomas, “Heterodyne measurement of Wigner distributions for classical optical fields,” Opt. Lett. 19, 1370–1372 (1999).
[CrossRef]

H.-F. Shih, T.-P. Yang, M. O. Freeman, J.-K. Wang, H.-F. Yau, D.-R. Hua, “Holographic laser-module with dual wavelength for digital versatile disc optical head,” Jpn. J. Appl. Phys. 38, 1750–1754 (1999).
[CrossRef]

R. Trebino, D. J. Kane, “The dilemma of ultra-short-laser-pulse intensity-and-phase measurement and applications,” IEEE J. Quantum Electron. 35, 416–420 (1999).

1998 (1)

J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Ambiguity function as a design tool for high focal depth,” Appl. Opt. 4, 790–795 (1998).

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)

1995 (2)

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]

1993 (1)

1992 (1)

J. Paye, “The cronocyclic representation of ultrashort light pulses,” IEEE J. Quantum Electron. 28, 2262–2272 (1992).
[CrossRef]

1986 (1)

R. H. Clarke, “Ambiguity function for time-varying random diffracted fields,” J. Opt. Soc. A 3, 1224–1226 (1986).
[CrossRef]

1983 (1)

K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “The ambiguity function as polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

1982 (1)

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

Bali, S.

Beck, M.

Berriel-Valdos, L. R.

J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Ambiguity function as a design tool for high focal depth,” Appl. Opt. 4, 790–795 (1998).

Brenner, K.-H.

D. Dragoman, M. Dragoman, K.-H. Brenner, “Optical realization of the ambiguity function of real two-dimensional light sources,” Appl. Opt. 39, 2912–2917 (2000).
[CrossRef]

D. Dragoman, M. Dragoman, K.-H. Brenner, “Variant fractional Fourier transformer for optical pulses,” Opt. Lett. 24, 933–935 (1999).
[CrossRef]

K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “The ambiguity function as polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

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

Clarke, L.

Clarke, R. H.

R. H. Clarke, “Ambiguity function for time-varying random diffracted fields,” J. Opt. Soc. A 3, 1224–1226 (1986).
[CrossRef]

Dragoman, D.

Dragoman, M.

Freeman, M. O.

H.-F. Shih, T.-P. Yang, M. O. Freeman, J.-K. Wang, H.-F. Yau, D.-R. Hua, “Holographic laser-module with dual wavelength for digital versatile disc optical head,” Jpn. J. Appl. Phys. 38, 1750–1754 (1999).
[CrossRef]

Gureyev, T. E.

Hermann, G. T.

G. T. Hermann, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic Press, New York, 1980).

Hua, D.-R.

H.-F. Shih, T.-P. Yang, M. O. Freeman, J.-K. Wang, H.-F. Yau, D.-R. Hua, “Holographic laser-module with dual wavelength for digital versatile disc optical head,” Jpn. J. Appl. Phys. 38, 1750–1754 (1999).
[CrossRef]

Iaconis, C.

Kane, D. J.

R. Trebino, D. J. Kane, “The dilemma of ultra-short-laser-pulse intensity-and-phase measurement and applications,” IEEE J. Quantum Electron. 35, 416–420 (1999).

Lee, K. F.

Lohmann, A. W.

K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “The ambiguity function as polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

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

Mayer, A.

McAlister, D. F.

Montes, E.

J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Ambiguity function as a design tool for high focal depth,” Appl. Opt. 4, 790–795 (1998).

Nuget, K. A.

Ojeda-Castaneda, J.

J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Ambiguity function as a design tool for high focal depth,” Appl. Opt. 4, 790–795 (1998).

K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “The ambiguity function as polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Paye, J.

J. Paye, “The cronocyclic representation of ultrashort light pulses,” IEEE J. Quantum Electron. 28, 2262–2272 (1992).
[CrossRef]

Raymer, M. G.

Reil, F.

Roberts, A.

Shih, H.-F.

H.-F. Shih, T.-P. Yang, M. O. Freeman, J.-K. Wang, H.-F. Yau, D.-R. Hua, “Holographic laser-module with dual wavelength for digital versatile disc optical head,” Jpn. J. Appl. Phys. 38, 1750–1754 (1999).
[CrossRef]

Tamura, S.

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

Thomas, J. E.

Trebino, R.

R. Trebino, D. J. Kane, “The dilemma of ultra-short-laser-pulse intensity-and-phase measurement and applications,” IEEE J. Quantum Electron. 35, 416–420 (1999).

Tu, J.

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

Walmsley, I. A.

Wang, J.-K.

H.-F. Shih, T.-P. Yang, M. O. Freeman, J.-K. Wang, H.-F. Yau, D.-R. Hua, “Holographic laser-module with dual wavelength for digital versatile disc optical head,” Jpn. J. Appl. Phys. 38, 1750–1754 (1999).
[CrossRef]

Wax, A.

Wong, V.

Yang, T.-P.

H.-F. Shih, T.-P. Yang, M. O. Freeman, J.-K. Wang, H.-F. Yau, D.-R. Hua, “Holographic laser-module with dual wavelength for digital versatile disc optical head,” Jpn. J. Appl. Phys. 38, 1750–1754 (1999).
[CrossRef]

Yau, H.-F.

H.-F. Shih, T.-P. Yang, M. O. Freeman, J.-K. Wang, H.-F. Yau, D.-R. Hua, “Holographic laser-module with dual wavelength for digital versatile disc optical head,” Jpn. J. Appl. Phys. 38, 1750–1754 (1999).
[CrossRef]

Appl. Opt. (3)

IEEE J. Quantum Electron. (2)

J. Paye, “The cronocyclic representation of ultrashort light pulses,” IEEE J. Quantum Electron. 28, 2262–2272 (1992).
[CrossRef]

R. Trebino, D. J. Kane, “The dilemma of ultra-short-laser-pulse intensity-and-phase measurement and applications,” IEEE J. Quantum Electron. 35, 416–420 (1999).

J. Opt. Soc. A (1)

R. H. Clarke, “Ambiguity function for time-varying random diffracted fields,” J. Opt. Soc. A 3, 1224–1226 (1986).
[CrossRef]

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys. (1)

H.-F. Shih, T.-P. Yang, M. O. Freeman, J.-K. Wang, H.-F. Yau, D.-R. Hua, “Holographic laser-module with dual wavelength for digital versatile disc optical head,” Jpn. J. Appl. Phys. 38, 1750–1754 (1999).
[CrossRef]

Opt. Commun. (2)

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

K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “The ambiguity function as polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Opt. Lett. (6)

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]

Prog. Opt. (1)

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

Other (1)

G. T. Hermann, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic Press, New York, 1980).

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 (5)

Fig. 1
Fig. 1

Real part of the ambiguity function calculated from intensity samples measured at different propagation planes.

Fig. 2
Fig. 2

(a) Recovered normalized amplitude profile A (solid line) and the experimental normalized amplitude distribution in the reference plane (dotted line). (b) Recovered phase profile by use of the tomographic method and the best-fit parabolic profile.

Fig. 3
Fig. 3

Set-up for measuring the Wigner-like distribution of rotationally symmetric beams.

Fig. 5
Fig. 5

(a) Recovered normalized amplitude A (solid line) and the experimental normalized amplitude distribution in the reference plane (dotted line). (b) Recovered phase profiles before and after subtracting the contribution of the cylindrical lens in Fig. 3 and its best-fit parabolic approximations.

Fig. 4
Fig. 4

(a) Intensity pattern measured by the CCD camera of Fig. 3 for a typical He-Ne laser, and (b) its computed Fourier transform along the y direction.

Equations (9)

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

AFx, q; z= φx+x2; zφ*x-x2; z×exp-ixqdx,
AFDx-Bq, -Cx+Aq; z=AFx, q; 0,
xqz=ABCDxq0.
 Ix, zexp-i xqAdx=AFx=- BqA, q; 0,
φxφ*0=12π  AFx, qexpi xq2dq.
Wx, p= φx+x/2φ*x-x/2expixpdx.
Ψx, y=φx cos α+y sin α×expiky cos α-x sin α2/2f×expiγy+φx cos α-y sin α×expiky cos α+x sin α2/2f×exp-iγy.
W(x, p)= φ(x+y/2)φ*(x-y/2)×exp(ikxy/f)exp(ipy)dy
φxφ*0expi kx22f=12π- Wx2, p×exp-ipxdp.

Metrics