Abstract

A phase-space description of the Laguerre–Gaussian modes is presented that points out the connection between these modes and the Hermite–Gaussian modes. We exploit the underlying phase-space symmetries to derive a closed-form expression for the Wigner distribution function of Laguerre–Gaussian modes. This expression is as compact as the one for the Hermite–Gaussian modes.

© 2000 Optical Society of America

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References

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  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); R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 29, 3273 (1984); Phys. Rev. A 31, 2419 (1985). For a review, see D. Dragoman, Prog. Opt. 37, 1 (1997).
    [CrossRef]
  3. A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), Chap. 19.
  4. H. J. Groenewold, Physica 12, 405 (1946).
    [CrossRef]
  5. R. Gase, IEEE J. Quantum Electron. 31, 1811 (1995).
    [CrossRef]
  6. D. Gloge and D. Marcuse, J. Opt. Soc. Am. 59, 1629 (1969).
  7. R. Simon, K. Sundar, and N. Mukunda, J. Opt. Soc. Am. A 10, 2008 (1993).
    [CrossRef]
  8. G. Indebetouw, J. Mod. Opt. 40, 73 (1993); M. W. Beigersbergen, L. Allen, H.E.L.O. Van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993); S. J. Van Enk, Opt. Commun. 102, 59 (1993); H. He, M. E. J. Friese, N. R. Heckenberg, and Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995); J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998); G. S. Agarwal, R. R. Puri, and R. P. Singh, Phys. Rev. A 56, 4207 (1997); R. Piestein and J. Shamir, J. Opt. Soc. Am. A 15, 3039 (1998); F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, J. Mod. Opt. 45, 539 (1998); L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. 39, 291 (1999); S. Ramee and R. Simon, J. Opt. Soc. Am. A 17, 84 (2000).
    [CrossRef] [PubMed]
  9. D. Danakas and P. K. Aravind, Phys. Rev. A 45, 1973 (1992).
    [CrossRef] [PubMed]
  10. K. Sundar, N. Mukunda, and R. Simon, J. Opt. Soc. Am. A 12, 560 (1995).
    [CrossRef]
  11. R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 37, 2028 (1988).
    [CrossRef]

1995 (2)

1993 (2)

G. Indebetouw, J. Mod. Opt. 40, 73 (1993); M. W. Beigersbergen, L. Allen, H.E.L.O. Van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993); S. J. Van Enk, Opt. Commun. 102, 59 (1993); H. He, M. E. J. Friese, N. R. Heckenberg, and Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995); J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998); G. S. Agarwal, R. R. Puri, and R. P. Singh, Phys. Rev. A 56, 4207 (1997); R. Piestein and J. Shamir, J. Opt. Soc. Am. A 15, 3039 (1998); F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, J. Mod. Opt. 45, 539 (1998); L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. 39, 291 (1999); S. Ramee and R. Simon, J. Opt. Soc. Am. A 17, 84 (2000).
[CrossRef] [PubMed]

R. Simon, K. Sundar, and N. Mukunda, J. Opt. Soc. Am. A 10, 2008 (1993).
[CrossRef]

1992 (1)

D. Danakas and P. K. Aravind, Phys. Rev. A 45, 1973 (1992).
[CrossRef] [PubMed]

1988 (1)

R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 37, 2028 (1988).
[CrossRef]

1978 (1)

M. J. Bastiaans, Opt. Commun. 25, 26 (1978); J. Opt. Soc. Am. 69, 1710 (1979); R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 29, 3273 (1984); Phys. Rev. A 31, 2419 (1985). For a review, see D. Dragoman, Prog. Opt. 37, 1 (1997).
[CrossRef]

1969 (1)

1946 (1)

H. J. Groenewold, Physica 12, 405 (1946).
[CrossRef]

1932 (1)

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

Aravind, P. K.

D. Danakas and P. K. Aravind, Phys. Rev. A 45, 1973 (1992).
[CrossRef] [PubMed]

Bastiaans, M. J.

M. J. Bastiaans, Opt. Commun. 25, 26 (1978); J. Opt. Soc. Am. 69, 1710 (1979); R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 29, 3273 (1984); Phys. Rev. A 31, 2419 (1985). For a review, see D. Dragoman, Prog. Opt. 37, 1 (1997).
[CrossRef]

Danakas, D.

D. Danakas and P. K. Aravind, Phys. Rev. A 45, 1973 (1992).
[CrossRef] [PubMed]

Gase, R.

R. Gase, IEEE J. Quantum Electron. 31, 1811 (1995).
[CrossRef]

Gloge, D.

Groenewold, H. J.

H. J. Groenewold, Physica 12, 405 (1946).
[CrossRef]

Indebetouw, G.

G. Indebetouw, J. Mod. Opt. 40, 73 (1993); M. W. Beigersbergen, L. Allen, H.E.L.O. Van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993); S. J. Van Enk, Opt. Commun. 102, 59 (1993); H. He, M. E. J. Friese, N. R. Heckenberg, and Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995); J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998); G. S. Agarwal, R. R. Puri, and R. P. Singh, Phys. Rev. A 56, 4207 (1997); R. Piestein and J. Shamir, J. Opt. Soc. Am. A 15, 3039 (1998); F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, J. Mod. Opt. 45, 539 (1998); L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. 39, 291 (1999); S. Ramee and R. Simon, J. Opt. Soc. Am. A 17, 84 (2000).
[CrossRef] [PubMed]

Marcuse, D.

Mukunda, N.

Siegman, A. E.

A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), Chap. 19.

Simon, R.

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 37, 2028 (1988).
[CrossRef]

Sundar, K.

Wigner, E. P.

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

IEEE J. Quantum Electron. (1)

R. Gase, IEEE J. Quantum Electron. 31, 1811 (1995).
[CrossRef]

J. Mod. Opt. (1)

G. Indebetouw, J. Mod. Opt. 40, 73 (1993); M. W. Beigersbergen, L. Allen, H.E.L.O. Van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993); S. J. Van Enk, Opt. Commun. 102, 59 (1993); H. He, M. E. J. Friese, N. R. Heckenberg, and Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995); J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998); G. S. Agarwal, R. R. Puri, and R. P. Singh, Phys. Rev. A 56, 4207 (1997); R. Piestein and J. Shamir, J. Opt. Soc. Am. A 15, 3039 (1998); F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, J. Mod. Opt. 45, 539 (1998); L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. 39, 291 (1999); S. Ramee and R. Simon, J. Opt. Soc. Am. A 17, 84 (2000).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

M. J. Bastiaans, Opt. Commun. 25, 26 (1978); J. Opt. Soc. Am. 69, 1710 (1979); R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 29, 3273 (1984); Phys. Rev. A 31, 2419 (1985). For a review, see D. Dragoman, Prog. Opt. 37, 1 (1997).
[CrossRef]

Phys. Rev. (1)

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

Phys. Rev. A (2)

D. Danakas and P. K. Aravind, Phys. Rev. A 45, 1973 (1992).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 37, 2028 (1988).
[CrossRef]

Physica (1)

H. J. Groenewold, Physica 12, 405 (1946).
[CrossRef]

Other (1)

A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), Chap. 19.

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Equations (20)

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Wρ,p=1/2πƛ2d2Δρ exp-ip·Δρ/ƛ×ψρ+½Δρψ*ρ-½Δρ.
ψnx=2π2nwn!1/2Hn2x/wexp-x2/w2,
Wnx,px=-1nπƛLn4x2w2+w2px24ƛ2×exp-2x2w2+w2px24ƛ2.
Ψn1,n2HGρψn1xψn2y, n1,n2=0,1,,
Wn1,n2HGρ,p=Wn1x,pxWn2y,py=-1n1+n2πƛLn14x2w2+w2px24ƛ2×Ln24y2w2+w2py24ƛ2×exp-2x2+y2w2+w24ƛ2px2+py2.
Φj,mLGρ=2πw2j-m!j+m!1/22ρw2m×expi2mθLj-m2m2ρ2w2exp-ρ2w2,
Φj,mLGρ=n1,n2cj,m;n1,n2Ψn1,n2HGρ,
xˆ:ψρxψρ, pˆx:ψρƛixψρ, yˆ:ψρyψρ, pˆy:ψρƛiyψρ.
S:WξWξ=WS-1ξ.
Tˆ0=α-1xˆ2+yˆ2+αpˆx2+pˆy2/4ƛ-12, Tˆ3=α-1xˆ2-yˆ2+αpˆx2-pˆy2/4ƛ, Tˆ1=α-1xˆyˆ+αpˆxpˆy/2ƛ, Tˆ2=xˆpˆy-yˆpˆx/2ƛ.
Tˆa,Tˆb=iabcTˆc, Tˆa,Tˆ0=0
Tˆ0Ψn1,n2HGρ=n1+n22Ψn1,n2HGρ, Tˆ3Ψn1,n2HGρ=n1-n22Ψn1,n2HGρ.
Tˆ0Φj,mLGρ=jΦj,mLGρ, Tˆ2Φj,mLGρ=mΦj,mLGρ,
ξˆexpi2ϕTˆ1ξˆexp-i2ϕTˆ1=S1ϕξˆ, S1ϕ=c00αs0cαs00-α-1sc0-α-1s00c,
expiπ2Tˆ1Tˆ0 exp-iπ2Tˆ1=Tˆ0, expiπ2Tˆ1Tˆ3 exp-iπ2Tˆ1=Tˆ2.
Φj,mLGρ=expiπ2Tˆ1Ψn1,n2HGρ, n1=j+m, n2=j-m.
Wj,mLGξ=Wn1,n2HGS1-π/4-1ξ, n1=j+m, n2=j-m.
Q012x2+y2w2+w24ƛ2px2+py2, Q2xpy-ypx/2ƛ, Q312x2-y2w2+w24ƛ2px2-py2.
Wn1,n2HGξ=-1n1+n2πƛ-1Ln14Q0+Q3×Ln24Q0-Q3exp-4Q0.
Wj,mLGξ=-12jπƛ-1Lj+m4Q0+Q2×Lj-m4Q0-Q2exp-4Q0.

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