Abstract

We propose a method to manipulate the intensity and phase distributions of a beam with non-zero orbital angular momentum (OAM). We investigate the superposition of coherent consecutive OAM modes, with concordant topological charges values, showing that it is possible to predict and control the phase and the radial and angular dimension of the resulting beam by acting on the number of superposed modes (N) and on their minimum value of the OAM (mmin). A general analysis from the Wigner function formalism is adopted for the geometric characterization of the beam.

© 2014 Optical Society of America

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  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  25. D. Judge, “On the uncertainty relation forLzϕ, ” Phys. Lett. 5(3), 189 (1963).
    [CrossRef]

2012

2011

2010

E. Serabyn, D. Mawet, and R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature 464(7291), 1018–1020 (2010).
[CrossRef] [PubMed]

T. Ando, N. Matsumoto, Y. Ohtake, Y. Takiguchi, and T. Inoue, “Structure of optic singularities in coaxial superposition of Laguerre-Gaussian modes,” J. Opt. Soc. Am. A 27(12), 2602–2612 (2010).
[CrossRef]

2009

2004

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
[CrossRef] [PubMed]

2003

2002

A. Vaziri, G. Weihs, and A. Zeilinger, “Superpositions of the orbital angular momentum for applications in quantum experiments,” J. Opt. B Quantum Semiclassical Opt. 4(2), S47–S51 (2002).
[CrossRef]

2000

1999

A. Y. Bekshaev, “Intensity moments of a laser beam formed by superposition of Hermite-Gaussian modes,” Fotoelektronika 8, 2–13 (1999).

1997

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

1995

1994

Y. A. Anan’ev and A. Y. Bekshaev, “Theory of intensity moments for arbitrary light beams,” Opt. Spectrosc. 76, 558–568 (1994).

1993

1992

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

1987

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36(8), 3868–3880 (1987).
[CrossRef] [PubMed]

1983

1980

1963

D. Judge, “On the uncertainty relation forLzϕ, ” Phys. Lett. 5(3), 189 (1963).
[CrossRef]

Agarwal, G. S.

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Alonso, M. A.

Anan’ev, Y. A.

Y. A. Anan’ev and A. Y. Bekshaev, “Theory of intensity moments for arbitrary light beams,” Opt. Spectrosc. 76, 558–568 (1994).

Ando, T.

Andrews, L. C.

Barbieri, C.

Baumann, S. M.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Bekshaev, A. Y.

A. Y. Bekshaev, “Intensity moments of a laser beam formed by superposition of Hermite-Gaussian modes,” Fotoelektronika 8, 2–13 (1999).

Y. A. Anan’ev and A. Y. Bekshaev, “Theory of intensity moments for arbitrary light beams,” Opt. Spectrosc. 76, 558–568 (1994).

Bianchini, A.

Burruss, R.

E. Serabyn, D. Mawet, and R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature 464(7291), 1018–1020 (2010).
[CrossRef] [PubMed]

Carter, W. H.

Dragoman, D.

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

Galvez, E. J.

Garoli, D.

Inoue, T.

Judge, D.

D. Judge, “On the uncertainty relation forLzϕ, ” Phys. Lett. 5(3), 189 (1963).
[CrossRef]

Kalb, D. M.

MacMillan, L. H.

Maleev, I. D.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
[CrossRef] [PubMed]

I. D. Maleev and G. A. Swartzlander., “Composite optical vortices,” J. Opt. Soc. Am. B 20(6), 1169–1176 (2003).
[CrossRef]

Marathay, A. S.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
[CrossRef] [PubMed]

Mari, E.

Matsumoto, N.

Mawet, D.

E. Serabyn, D. Mawet, and R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature 464(7291), 1018–1020 (2010).
[CrossRef] [PubMed]

Mukunda, N.

Ohtake, Y.

Ongarello, T.

Palacios, D. M.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
[CrossRef] [PubMed]

Parisi, G.

Phillips, R. L.

Romanato, F.

Serabyn, E.

E. Serabyn, D. Mawet, and R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature 464(7291), 1018–1020 (2010).
[CrossRef] [PubMed]

Simon, R.

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36(8), 3868–3880 (1987).
[CrossRef] [PubMed]

Sundar, K.

Swartzlander, G. A.

Takiguchi, Y.

Tamburini, F.

Thidé, B.

Vaziri, A.

A. Vaziri, G. Weihs, and A. Zeilinger, “Superpositions of the orbital angular momentum for applications in quantum experiments,” J. Opt. B Quantum Semiclassical Opt. 4(2), S47–S51 (2002).
[CrossRef]

Weihs, G.

A. Vaziri, G. Weihs, and A. Zeilinger, “Superpositions of the orbital angular momentum for applications in quantum experiments,” J. Opt. B Quantum Semiclassical Opt. 4(2), S47–S51 (2002).
[CrossRef]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Zeilinger, A.

A. Vaziri, G. Weihs, and A. Zeilinger, “Superpositions of the orbital angular momentum for applications in quantum experiments,” J. Opt. B Quantum Semiclassical Opt. 4(2), S47–S51 (2002).
[CrossRef]

Zilio, P.

Adv. Opt. Photon.

Appl. Opt.

Fotoelektronika

A. Y. Bekshaev, “Intensity moments of a laser beam formed by superposition of Hermite-Gaussian modes,” Fotoelektronika 8, 2–13 (1999).

J. Opt. B Quantum Semiclassical Opt.

A. Vaziri, G. Weihs, and A. Zeilinger, “Superpositions of the orbital angular momentum for applications in quantum experiments,” J. Opt. B Quantum Semiclassical Opt. 4(2), S47–S51 (2002).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Nature

E. Serabyn, D. Mawet, and R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature 464(7291), 1018–1020 (2010).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Opt. Spectrosc.

Y. A. Anan’ev and A. Y. Bekshaev, “Theory of intensity moments for arbitrary light beams,” Opt. Spectrosc. 76, 558–568 (1994).

Phys. Lett.

D. Judge, “On the uncertainty relation forLzϕ, ” Phys. Lett. 5(3), 189 (1963).
[CrossRef]

Phys. Rev. A

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36(8), 3868–3880 (1987).
[CrossRef] [PubMed]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
[CrossRef] [PubMed]

Prog. Optics

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

Other

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, 2007).

A. M. C. Aguirre and T. Alieva, “Orbital angular momentum Density of beam given as superposition of Hermite-Laguerre-Guass function,” in PIERS Proceedings, (Marrakesh, Marocco, 2011), pp. 250–254.

T. Alieva, C. Alejandro, and M. J. Bastiaans, “Mathematical formalism in wave optics,” in Mathematical Optics, V. Lakshminarayanan, M. L. Calvo, T. Alieva ed. (CRC Press, 2013).

A. Camara and T. Alieva, in PIERS Proceedings, (Cambrige, MA, 2010), pp. 505–508.

J. Alda, “ Laser and Gaussian Beam Propagation and Transformation,” in Encyclopedia of Optical Engineering, R. G. Driggers ed. (CRC Press, 2003).

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Figures (6)

Fig. 1
Fig. 1

Plots of the superposition of concordant, consecutive and coherent OAM modes. In the left column are reported the plots with A m = 2 / ( π | m | ! ) . Right column: A m = 1 . Normalized intensity of scalar field for N = 16, m min = 1 a), b) and for N = 7, m min = 1 c), d).

Fig. 2
Fig. 2

Schematic representation of angular, ϑ f a r and radial, s(z), position of the resulting beam.

Fig. 3
Fig. 3

Examples of s(z) at different distances for different number of superimposed OAM modes. All the OAM modes have the same beam waist and their m-values are concordant and consecutive.

Fig. 4
Fig. 4

Graphic representation of the relationship between the angular aperture and the number of superimposed OAM modes at several values of m min . Inset: exemplified plot for Θand s(z).

Fig. 5
Fig. 5

Phase (a) and intensity (b) plot for N = 5 and m min = 1 . Phase (c) and Intensity (d) plot for N = 5 and m min = 3 .

Fig. 6
Fig. 6

Simulation of the phase variation embedded in the angular regionΘmeasured at distance s from the center for three different values of m min .(a) N = 4 and (b) N = 8.

Tables (1)

Tables Icon

Table 1 Resuming table

Equations (13)

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u p,m (r,ϕ,z)=A 1 w(z) ( r 2 w(z) ) |m| L p |m| ( 2 r 2 w (z) 2 ) e r 2 /w (z) 2 e ik r 2 2R(z) e i( 2p+| m |+1 )ζ(z) e imϕ ,
u tot (r,ϕ,z)= m u p,m (r,ϕ,z) e i δ m ,
u tot (r,ϕ,z)= m min m max u m = 1 w(z) e r 2 /w (z) 2 e ik r 2 2R(z) m= m min m max A m α m e i( 2| m |+1 )ζ(z) e imϕ ,
u tot (r,ϕ,0)= 1 w 0 e r 2 / w 0 2 α (0) m min e i m min ϕ α (0) 2N 2α (0) N cos(Nϕ)+1 α (0) 2 2α(0)cosϕ+1 .
u t o t (r,ϕ,0)= 1 w 0 e r 2 / w 0 2 m= m min 0 m max 2 π| m |! ( r 2 w 0 ) |m| e imϕ .
w ( z ) = w 0 [ 1 + ( λ z π w 0 2 ) 2 ] 1 / 2
W L G T O T ( r , p ; 0 ) = m = m min m max n m m max W L G ( m , n ) ( r , p ; 0 ) ,
W L G ( m , n ) ( r , p ; 0 ) = { 2 | m n | 2 ( 1 ) m 4 λ 2 m ! n ! [ x i y w 0 + s i g n ( m n ) π w 0 λ ( p y i p x ) ] | m n | L m | m n | [ 2 ( x 2 + y 2 w 0 2 + w 0 2 π 2 λ 2 ( p x 2 + p y 2 ) ) 2 π λ ( x p y y p x ) ] exp [ 2 x 2 + y 2 w 0 2 2 w 0 2 π 2 λ 2 ( p x 2 + p y 2 ) ] } ,
I = 4 m = m min m max W L G ( m , m ) ( r , p ; 0 ) d r d p = N .
s ( z ) = 2 r 2 I = ( 1 I 0 π π r 2 m = m m i n m max W L G ( m , m ) ( r , p ; z ) d r 2 d ϕ r d p ) 1 2 = w ( z ) 2 ( N + 2 m min + 1 2 ) ,
ϑ f a r = 2 p r 2 I = ( 1 I 0 π π p 2 m = m m i n m max W L G ( m , m ) ( r , p ; 0 ) d p 2 d ϕ p d r ) 1 2 = λ 2 π 2 w 0 2 ( N + 2 m min + 1 2 ) .
Θ = 2 ϕ 2  = 2 ( 1 I 0 π π ϕ 2 n = m m i n m max m = m m i n m max W L G ( m , n ) ( r , p ; 0 ) r d r d ϕ d p ) 1 2 = 2 π 2 3 + 1 N m = m min m max n > m m max 4 ( 1 ) n m Γ ( n + m 2 + 1 2 ) ( n m ) 2 m ! n ! ,
atan( ( E tot (r,ϕ)) ( E tot (r,ϕ)) )=0,

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