Abstract

Ladder operators for the radial index of the paraxial optical modes in the cylindrical coordinates are calculated. The operators obey the su(1,1) algebra commutation relations. Based on this Lie algebra, we found that coherent modes constructed as eigenstates of the destruction operator or resulting from the action of the displacement operator on the fundamental mode are different. Some properties of these two kinds of radial coherent modes are studied in detail.

© 2012 Optical Society of America

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References

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  1. S. Franke-Arnold, L. Allen, and M. J. Padgett, Laser Photon. Rev. 2, 299 (2008).
    [CrossRef]
  2. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pasko, S. M. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
    [CrossRef]
  3. A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, Nature 412, 313 (2001).
    [CrossRef]
  4. E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, Nat. Photon. 3, 720 (2009).
    [CrossRef]
  5. E. Karimi, S. Sluserenko, B. Piccirillo, L. Marrucci, and E. Santamato, Phys. Rev. A 81, 053813 (2010).
    [CrossRef]
  6. M. Padgett, J. Courtial, and L. Allen, Phys. Today 57(5), 35 (2004).
    [CrossRef]
  7. M. V. Berry, J. Opt. A 6, 259 (2004).
    [CrossRef]
  8. A. Ya. Bekshaev and A. I. Karamoch, Opt. Commun. 281, 1366 (2008).
    [CrossRef]
  9. E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, Opt. Lett. 32, 3053 (2007).
    [CrossRef]
  10. E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, Opt. Lett. 34, 1225 (2009).
    [CrossRef]
  11. E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, Appl. Phys. Lett. 94, 231124 (2009).
    [CrossRef]
  12. G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, J. Eur. Opt. Soc. Rapid Pub. 2, 07014(2007).
    [CrossRef]
  13. A. Perelomov, Generalized Coherent States and Their Applications (Springer, 1986).
  14. E. Schrödinger, Naturwissenschaften 14, 664 (1926).
    [CrossRef]
  15. A. O. Barut and L. Girardello, Commun. Math. Phys. 21, 41 (1971).
    [CrossRef]
  16. The following dimensionless cylindrical and Cartesian coordinates have been used: (ρ=r/w0, ϕ, ζ=z/zR) and (u=x/w0, v=y/w0, ζ=z/zR), respectively, where w0 is the beam waist and zR=πw02/λ is the beam Rayleigh range.
  17. A. E. Siegman, Lasers (University Science Books, 1986).

2010 (1)

E. Karimi, S. Sluserenko, B. Piccirillo, L. Marrucci, and E. Santamato, Phys. Rev. A 81, 053813 (2010).
[CrossRef]

2009 (3)

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, Nat. Photon. 3, 720 (2009).
[CrossRef]

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, Appl. Phys. Lett. 94, 231124 (2009).
[CrossRef]

E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, Opt. Lett. 34, 1225 (2009).
[CrossRef]

2008 (2)

S. Franke-Arnold, L. Allen, and M. J. Padgett, Laser Photon. Rev. 2, 299 (2008).
[CrossRef]

A. Ya. Bekshaev and A. I. Karamoch, Opt. Commun. 281, 1366 (2008).
[CrossRef]

2007 (2)

G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, J. Eur. Opt. Soc. Rapid Pub. 2, 07014(2007).
[CrossRef]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, Opt. Lett. 32, 3053 (2007).
[CrossRef]

2004 (3)

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pasko, S. M. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
[CrossRef]

M. Padgett, J. Courtial, and L. Allen, Phys. Today 57(5), 35 (2004).
[CrossRef]

M. V. Berry, J. Opt. A 6, 259 (2004).
[CrossRef]

2001 (1)

A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

1971 (1)

A. O. Barut and L. Girardello, Commun. Math. Phys. 21, 41 (1971).
[CrossRef]

1926 (1)

E. Schrödinger, Naturwissenschaften 14, 664 (1926).
[CrossRef]

Allen, L.

S. Franke-Arnold, L. Allen, and M. J. Padgett, Laser Photon. Rev. 2, 299 (2008).
[CrossRef]

M. Padgett, J. Courtial, and L. Allen, Phys. Today 57(5), 35 (2004).
[CrossRef]

Barnett, S. M.

Barut, A. O.

A. O. Barut and L. Girardello, Commun. Math. Phys. 21, 41 (1971).
[CrossRef]

Bekshaev, A. Ya.

A. Ya. Bekshaev and A. I. Karamoch, Opt. Commun. 281, 1366 (2008).
[CrossRef]

Berry, M. V.

M. V. Berry, J. Opt. A 6, 259 (2004).
[CrossRef]

Carrasco, S.

G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, J. Eur. Opt. Soc. Rapid Pub. 2, 07014(2007).
[CrossRef]

Courtial, J.

De Martini, F.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, Nat. Photon. 3, 720 (2009).
[CrossRef]

Franke-Arnold, S.

Gibson, G.

Girardello, L.

A. O. Barut and L. Girardello, Commun. Math. Phys. 21, 41 (1971).
[CrossRef]

Karamoch, A. I.

A. Ya. Bekshaev and A. I. Karamoch, Opt. Commun. 281, 1366 (2008).
[CrossRef]

Karimi, E.

E. Karimi, S. Sluserenko, B. Piccirillo, L. Marrucci, and E. Santamato, Phys. Rev. A 81, 053813 (2010).
[CrossRef]

E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, Opt. Lett. 34, 1225 (2009).
[CrossRef]

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, Appl. Phys. Lett. 94, 231124 (2009).
[CrossRef]

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, Nat. Photon. 3, 720 (2009).
[CrossRef]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, Opt. Lett. 32, 3053 (2007).
[CrossRef]

Mair, A.

A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Marrucci, L.

E. Karimi, S. Sluserenko, B. Piccirillo, L. Marrucci, and E. Santamato, Phys. Rev. A 81, 053813 (2010).
[CrossRef]

E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, Opt. Lett. 34, 1225 (2009).
[CrossRef]

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, Nat. Photon. 3, 720 (2009).
[CrossRef]

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, Appl. Phys. Lett. 94, 231124 (2009).
[CrossRef]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, Opt. Lett. 32, 3053 (2007).
[CrossRef]

Molina-Terriza, G.

G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, J. Eur. Opt. Soc. Rapid Pub. 2, 07014(2007).
[CrossRef]

Nagali, E.

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, Appl. Phys. Lett. 94, 231124 (2009).
[CrossRef]

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, Nat. Photon. 3, 720 (2009).
[CrossRef]

Padgett, M.

M. Padgett, J. Courtial, and L. Allen, Phys. Today 57(5), 35 (2004).
[CrossRef]

Padgett, M. J.

Pasko, V.

Perelomov, A.

A. Perelomov, Generalized Coherent States and Their Applications (Springer, 1986).

Piccirillo, B.

E. Karimi, S. Sluserenko, B. Piccirillo, L. Marrucci, and E. Santamato, Phys. Rev. A 81, 053813 (2010).
[CrossRef]

E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, Opt. Lett. 34, 1225 (2009).
[CrossRef]

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, Nat. Photon. 3, 720 (2009).
[CrossRef]

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, Appl. Phys. Lett. 94, 231124 (2009).
[CrossRef]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, Opt. Lett. 32, 3053 (2007).
[CrossRef]

Rebane, L.

G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, J. Eur. Opt. Soc. Rapid Pub. 2, 07014(2007).
[CrossRef]

Sansoni, L.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, Nat. Photon. 3, 720 (2009).
[CrossRef]

Santamato, E.

E. Karimi, S. Sluserenko, B. Piccirillo, L. Marrucci, and E. Santamato, Phys. Rev. A 81, 053813 (2010).
[CrossRef]

E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, Opt. Lett. 34, 1225 (2009).
[CrossRef]

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, Nat. Photon. 3, 720 (2009).
[CrossRef]

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, Appl. Phys. Lett. 94, 231124 (2009).
[CrossRef]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, Opt. Lett. 32, 3053 (2007).
[CrossRef]

Schrödinger, E.

E. Schrödinger, Naturwissenschaften 14, 664 (1926).
[CrossRef]

Sciarrino, F.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, Nat. Photon. 3, 720 (2009).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Sluserenko, S.

E. Karimi, S. Sluserenko, B. Piccirillo, L. Marrucci, and E. Santamato, Phys. Rev. A 81, 053813 (2010).
[CrossRef]

Torner, L.

G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, J. Eur. Opt. Soc. Rapid Pub. 2, 07014(2007).
[CrossRef]

Torres, J. P.

G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, J. Eur. Opt. Soc. Rapid Pub. 2, 07014(2007).
[CrossRef]

Vasnetsov, M.

Vaziri, A.

A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Welhs, G.

A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Zeilinger, A.

A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Zito, G.

Appl. Phys. Lett. (1)

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, Appl. Phys. Lett. 94, 231124 (2009).
[CrossRef]

Commun. Math. Phys. (1)

A. O. Barut and L. Girardello, Commun. Math. Phys. 21, 41 (1971).
[CrossRef]

J. Eur. Opt. Soc. Rapid Pub. (1)

G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, J. Eur. Opt. Soc. Rapid Pub. 2, 07014(2007).
[CrossRef]

J. Opt. A (1)

M. V. Berry, J. Opt. A 6, 259 (2004).
[CrossRef]

Laser Photon. Rev. (1)

S. Franke-Arnold, L. Allen, and M. J. Padgett, Laser Photon. Rev. 2, 299 (2008).
[CrossRef]

Nat. Photon. (1)

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, Nat. Photon. 3, 720 (2009).
[CrossRef]

Nature (1)

A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Naturwissenschaften (1)

E. Schrödinger, Naturwissenschaften 14, 664 (1926).
[CrossRef]

Opt. Commun. (1)

A. Ya. Bekshaev and A. I. Karamoch, Opt. Commun. 281, 1366 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (1)

E. Karimi, S. Sluserenko, B. Piccirillo, L. Marrucci, and E. Santamato, Phys. Rev. A 81, 053813 (2010).
[CrossRef]

Phys. Today (1)

M. Padgett, J. Courtial, and L. Allen, Phys. Today 57(5), 35 (2004).
[CrossRef]

Other (3)

A. Perelomov, Generalized Coherent States and Their Applications (Springer, 1986).

The following dimensionless cylindrical and Cartesian coordinates have been used: (ρ=r/w0, ϕ, ζ=z/zR) and (u=x/w0, v=y/w0, ζ=z/zR), respectively, where w0 is the beam waist and zR=πw02/λ is the beam Rayleigh range.

A. E. Siegman, Lasers (University Science Books, 1986).

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Figures (3)

Fig. 1.
Fig. 1.

Simulated spectra of different RCSs in the basis of LG modes. (a) m=1, |α|=0.2; (b) m=3, |α|=0.2; (c) m=3, |α|=0.5; and (d) m=3, |α|=0.9.

Fig. 2.
Fig. 2.

Simulated propagation of two different RCSs. (a) m=1 and (b) m=2 with α=0.6.

Fig. 3.
Fig. 3.

Simulated propagation of two different RISs: (a) m=1 with η=1 and (b) m=2 with η=10. The later case has several rings at the pupils, which during propagation collapses to a doughnut shape.

Equations (13)

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

(2+4iζ)ψ(ρ,ϕ,ζ)=0,
LGp,m(ρ,ϕ,0)=2|m|+1p!π(p+|m|)!ρ|m|eρ2Lp|m|(2ρ2),
P^+|p,m=P+|p+1,mP^|p,m=P|p1,m.
[P^+,P^]=2P^0[P^0,P^±]=±P^±,
p^=18ρρ(ρρ)+(|m|28ρ2+ρ22|m|+12).
|αm=e(αP^+αP^)|0,m,
|αm=(1|α|2)(|m|+12)p=0(p+|m|)!|m|!p!αp|p,m,
|m|πd2α(1|α|2)2|αmα|m=I^,
RCSα,m(ρ,ϕ,0)=2|m|+1π|m|!(1α*1α)|m|+12eimϕ(1+α)1αρ2ρ|m|.
P^|ηm=η|ηm.
|ηm=η|m|2I|m|(2|η|)p=0ηpp!(p+|m|)!|p,m,
2πd2ηKm(2|η|)Im(2|η|)|ηmη|m=I^,
RISη,m(ρ,ϕ,0)=2πI|m|(2|η|)eimϕ+ηρ2J|m|(22ηρ).

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