Abstract

A coherent rotationally symmetric two-dimensional beam is essentially one-dimensional in content: It is fully determined by the one-dimensional sample along a diagonal of the circularly symmetric field distribution in a transverse plane. The linear transform that reconstructs the four-dimensional Wigner distribution of the full two-dimensional beam from the two-dimensional Wigner distribution of the one-dimensional sample is presented.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. P. Wigner, Phys. Rev. 40, 749 (1932).
    [CrossRef]
  2. M. J. Bastiaans, Opt. Commun. 25, 26 (1978); J. Opt. Soc. Am. 69, 1710 (1979); D. Dragoman, Prog. Opt. 37, 1 (1997).
    [CrossRef]
  3. R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 29, 3273 (1984); Phys. Rev. A 31, 2419 (1985).
    [CrossRef]
  4. R. Simon and N. Mukunda, J. Opt. Soc. Am. A 10, 95 (1993).
    [CrossRef]
  5. D. Dragoman, Opt. Lett. 25, 281 (2000).
    [CrossRef]
  6. Strictly speaking, we should add that we lose an unimportant overall phase in going to the Wigner description.
  7. R. J. Marks, J. F. Walkup, and T. F. Krile, Appl. Opt. 16, 746 (1977); H. O. Bartelt, K. H. Brenner, and A. W. Lohmann, Opt. Commun. 32, 32 (1980); K. H. Brenner and A. W. Lohmann, Opt. Commun. 42, 310 (1982); G. E. Eichmann and B. Z. Dong, Appl. Opt. 21, 3152 (1982); G. Shabtay, D. Mendlovic, and Z. Zalevsky, Appl. Opt. 37, 2142 (1998).
    [CrossRef] [PubMed]
  8. R. Balmer and H. Glünder, Opt. Acta 30, 1789 (1983); M. Conner and Y. Li, Appl. Opt. 24, 3825 (1985); T. Iwai, A. K. Gupta, and T. Asakura, Opt. Commun. 58, 15 (1986).
    [CrossRef]
  9. D. Gloge and D. Marcuse, J. Opt. Soc. Am. 59, 1629 (1969).
  10. See, for instance, E. D. Rainville, Special Functions (Macmillan, New York, 1963), p. 216.
  11. F. Gori, Opt. Commun. 34, 301 (1980); E. Wolf, J. Opt. Soc. Am. 72, 343 (1982).
    [CrossRef]

2000 (1)

1993 (1)

1984 (1)

R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 29, 3273 (1984); Phys. Rev. A 31, 2419 (1985).
[CrossRef]

1983 (1)

R. Balmer and H. Glünder, Opt. Acta 30, 1789 (1983); M. Conner and Y. Li, Appl. Opt. 24, 3825 (1985); T. Iwai, A. K. Gupta, and T. Asakura, Opt. Commun. 58, 15 (1986).
[CrossRef]

1980 (1)

F. Gori, Opt. Commun. 34, 301 (1980); E. Wolf, J. Opt. Soc. Am. 72, 343 (1982).
[CrossRef]

1978 (1)

M. J. Bastiaans, Opt. Commun. 25, 26 (1978); J. Opt. Soc. Am. 69, 1710 (1979); D. Dragoman, Prog. Opt. 37, 1 (1997).
[CrossRef]

1977 (1)

1969 (1)

1932 (1)

E. P. Wigner, Phys. Rev. 40, 749 (1932).
[CrossRef]

Balmer, R.

R. Balmer and H. Glünder, Opt. Acta 30, 1789 (1983); M. Conner and Y. Li, Appl. Opt. 24, 3825 (1985); T. Iwai, A. K. Gupta, and T. Asakura, Opt. Commun. 58, 15 (1986).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, Opt. Commun. 25, 26 (1978); J. Opt. Soc. Am. 69, 1710 (1979); D. Dragoman, Prog. Opt. 37, 1 (1997).
[CrossRef]

Dragoman, D.

Gloge, D.

Glünder, H.

R. Balmer and H. Glünder, Opt. Acta 30, 1789 (1983); M. Conner and Y. Li, Appl. Opt. 24, 3825 (1985); T. Iwai, A. K. Gupta, and T. Asakura, Opt. Commun. 58, 15 (1986).
[CrossRef]

Gori, F.

F. Gori, Opt. Commun. 34, 301 (1980); E. Wolf, J. Opt. Soc. Am. 72, 343 (1982).
[CrossRef]

Krile, T. F.

Marcuse, D.

Marks, R. J.

Mukunda, N.

R. Simon and N. Mukunda, J. Opt. Soc. Am. A 10, 95 (1993).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 29, 3273 (1984); Phys. Rev. A 31, 2419 (1985).
[CrossRef]

Rainville, E. D.

See, for instance, E. D. Rainville, Special Functions (Macmillan, New York, 1963), p. 216.

Simon, R.

R. Simon and N. Mukunda, J. Opt. Soc. Am. A 10, 95 (1993).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 29, 3273 (1984); Phys. Rev. A 31, 2419 (1985).
[CrossRef]

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 29, 3273 (1984); Phys. Rev. A 31, 2419 (1985).
[CrossRef]

Walkup, J. F.

Wigner, E. P.

E. P. Wigner, Phys. Rev. 40, 749 (1932).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

Opt. Acta (1)

R. Balmer and H. Glünder, Opt. Acta 30, 1789 (1983); M. Conner and Y. Li, Appl. Opt. 24, 3825 (1985); T. Iwai, A. K. Gupta, and T. Asakura, Opt. Commun. 58, 15 (1986).
[CrossRef]

Opt. Commun. (2)

M. J. Bastiaans, Opt. Commun. 25, 26 (1978); J. Opt. Soc. Am. 69, 1710 (1979); D. Dragoman, Prog. Opt. 37, 1 (1997).
[CrossRef]

F. Gori, Opt. Commun. 34, 301 (1980); E. Wolf, J. Opt. Soc. Am. 72, 343 (1982).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

E. P. Wigner, Phys. Rev. 40, 749 (1932).
[CrossRef]

Phys. Rev. A (1)

R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 29, 3273 (1984); Phys. Rev. A 31, 2419 (1985).
[CrossRef]

Other (2)

Strictly speaking, we should add that we lose an unimportant overall phase in going to the Wigner description.

See, for instance, E. D. Rainville, Special Functions (Macmillan, New York, 1963), p. 216.

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

Arrangement for producing a one-dimensional sample of a rotationally invariant two-dimensional field.

Equations (15)

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

Ψρ=cδyΨx,0=cδyψx,
Ψρ=cψ-x=cψx.
WΨΨ*2ξ=d2ξKξ;ξWψψ*1ξ.
WΨΦ*2ξ=12πƛ2d2Δρexp-ip·Δρ×Ψρ+1/2ΔρΦ*ρ-1/2Δρ.
d2ρd2ρΨρΦ*ρΨρΦ*ρ*=2πƛ2d4ξWΨΦ*2ξWΨΦ*2ξ*.
Hψϕ*=Hψ1Hψ1*, HΨΦ*2=HΨ2HΨ2*.
ψnx=NnH2n2x/wexp-x2/w2, d2xψmxψn*x=δmn, m,n=0,1,
H2nx=-1n22nn!Ln-1/2x2.
Φnρ=NnLn-1/22ρ2/w2exp-ρ2/w2.
WΨΨ*2ξ=d2ξKξ,ξWψψ*1ξ, Kξ;ξ=2πƛ2m,nW˜mn2ξWmn1ξ*.
Ψρ=ncnΦnρ.
WΨΨ*2ξ=m,ncmcn*W˜mn2ξ.
ψx=ncnψnx.
Wψψ*1ξ=m,ncmcn*Wmn1ξ.
cmcn*=2πƛ2d2ξWmn1ξ*Wψψ*1ξ.

Metrics