Abstract

We propose a reconstruction of vortex beams based on implementation of quadratic transformations in the orbital angular momentum. The information is encoded in a superposition of Bessel-like nondiffracting beams. The measurement of the angular probability distribution at different positions allows for the reconstruction of the Wigner function.

© 2010 Optical Society of America

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References

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  1. M. Soskin and M. V. Vasnetsov, Singular Optics, Progress in Optics (Elsevier, 2001), Vol. 41.
  2. M. R. Dennis, K. O’Holleran, and M. J. Padgett, Singular Optics: Optical Vortices and Polarization Singularities, Progress in Optics (Elsevier, 2009), Vol. 53.
  3. S. Franke-Arnold, L. Allen, and M. Padgett, Laser Photon. Rev. 2, 299 (2008).
    [CrossRef]
  4. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
    [CrossRef] [PubMed]
  5. G. Molina-Terriza, A. Vaziri, J. Řeháček, Z. Hradil, and A. Zeilinger, Phys. Rev. Lett. 92, 167903 (2004).
    [CrossRef] [PubMed]
  6. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
    [CrossRef] [PubMed]
  7. M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys. Rev. A 49, 3119 (1994).
    [CrossRef] [PubMed]
  8. G. C. G. Berkhout and M. W. Beijersbergen, Phys. Rev. Lett. 101, 100801 (2008).
    [CrossRef] [PubMed]
  9. M.G. A.Paris and J.Řeháček, eds., Quantum State Estimation, Lecture Notes in Physics (Springer, 2004), Vol. 649.
  10. A. I. Lvovsky and M. G. Raymer, Rev. Mod. Phys. 81, 299 (2009).
    [CrossRef]
  11. J. Řeháček, Z. Bouchal, R. Čelechovský, Z. Hradil, and L. L. Sánchez-Soto, Phys. Rev. A 77, 032110 (2008).
    [CrossRef]
  12. Z. Hradil, J. Řeháček, Z. Bouchal, R. Čelechovský, and L. L. Sánchez-Soto, Phys. Rev. Lett. 97, 243601 (2006).
    [CrossRef]
  13. M. Asorey, P. Facchi, V. I. Man’ko, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, Phys. Rev. A 76, 012117 (2007).
    [CrossRef]
  14. I. Rigas, L. L. Sánchez-Soto, A. B. Klimov, J. Řeháček, and Z. Hradil, Phys. Rev. A 78, 060101(R) (2008).
    [CrossRef]

2009 (1)

A. I. Lvovsky and M. G. Raymer, Rev. Mod. Phys. 81, 299 (2009).
[CrossRef]

2008 (4)

J. Řeháček, Z. Bouchal, R. Čelechovský, Z. Hradil, and L. L. Sánchez-Soto, Phys. Rev. A 77, 032110 (2008).
[CrossRef]

G. C. G. Berkhout and M. W. Beijersbergen, Phys. Rev. Lett. 101, 100801 (2008).
[CrossRef] [PubMed]

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

I. Rigas, L. L. Sánchez-Soto, A. B. Klimov, J. Řeháček, and Z. Hradil, Phys. Rev. A 78, 060101(R) (2008).
[CrossRef]

2007 (1)

M. Asorey, P. Facchi, V. I. Man’ko, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, Phys. Rev. A 76, 012117 (2007).
[CrossRef]

2006 (1)

Z. Hradil, J. Řeháček, Z. Bouchal, R. Čelechovský, and L. L. Sánchez-Soto, Phys. Rev. Lett. 97, 243601 (2006).
[CrossRef]

2004 (1)

G. Molina-Terriza, A. Vaziri, J. Řeháček, Z. Hradil, and A. Zeilinger, Phys. Rev. Lett. 92, 167903 (2004).
[CrossRef] [PubMed]

2002 (1)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

2001 (1)

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

1994 (1)

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys. Rev. A 49, 3119 (1994).
[CrossRef] [PubMed]

Allen, L.

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

Asorey, M.

M. Asorey, P. Facchi, V. I. Man’ko, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, Phys. Rev. A 76, 012117 (2007).
[CrossRef]

Barnett, S. M.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Beijersbergen, M. W.

G. C. G. Berkhout and M. W. Beijersbergen, Phys. Rev. Lett. 101, 100801 (2008).
[CrossRef] [PubMed]

Berkhout, G. C. G.

G. C. G. Berkhout and M. W. Beijersbergen, Phys. Rev. Lett. 101, 100801 (2008).
[CrossRef] [PubMed]

Bouchal, Z.

J. Řeháček, Z. Bouchal, R. Čelechovský, Z. Hradil, and L. L. Sánchez-Soto, Phys. Rev. A 77, 032110 (2008).
[CrossRef]

Z. Hradil, J. Řeháček, Z. Bouchal, R. Čelechovský, and L. L. Sánchez-Soto, Phys. Rev. Lett. 97, 243601 (2006).
[CrossRef]

Celechovský, R.

J. Řeháček, Z. Bouchal, R. Čelechovský, Z. Hradil, and L. L. Sánchez-Soto, Phys. Rev. A 77, 032110 (2008).
[CrossRef]

Z. Hradil, J. Řeháček, Z. Bouchal, R. Čelechovský, and L. L. Sánchez-Soto, Phys. Rev. Lett. 97, 243601 (2006).
[CrossRef]

Courtial, J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Singular Optics: Optical Vortices and Polarization Singularities, Progress in Optics (Elsevier, 2009), Vol. 53.

Facchi, P.

M. Asorey, P. Facchi, V. I. Man’ko, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, Phys. Rev. A 76, 012117 (2007).
[CrossRef]

Franke-Arnold, S.

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

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Harris, M.

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys. Rev. A 49, 3119 (1994).
[CrossRef] [PubMed]

Hill, C. A.

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys. Rev. A 49, 3119 (1994).
[CrossRef] [PubMed]

Hradil, Z.

J. Řeháček, Z. Bouchal, R. Čelechovský, Z. Hradil, and L. L. Sánchez-Soto, Phys. Rev. A 77, 032110 (2008).
[CrossRef]

I. Rigas, L. L. Sánchez-Soto, A. B. Klimov, J. Řeháček, and Z. Hradil, Phys. Rev. A 78, 060101(R) (2008).
[CrossRef]

Z. Hradil, J. Řeháček, Z. Bouchal, R. Čelechovský, and L. L. Sánchez-Soto, Phys. Rev. Lett. 97, 243601 (2006).
[CrossRef]

G. Molina-Terriza, A. Vaziri, J. Řeháček, Z. Hradil, and A. Zeilinger, Phys. Rev. Lett. 92, 167903 (2004).
[CrossRef] [PubMed]

Klimov, A. B.

I. Rigas, L. L. Sánchez-Soto, A. B. Klimov, J. Řeháček, and Z. Hradil, Phys. Rev. A 78, 060101(R) (2008).
[CrossRef]

Leach, J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Lvovsky, A. I.

A. I. Lvovsky and M. G. Raymer, Rev. Mod. Phys. 81, 299 (2009).
[CrossRef]

Mair, A.

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

Man’ko, V. I.

M. Asorey, P. Facchi, V. I. Man’ko, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, Phys. Rev. A 76, 012117 (2007).
[CrossRef]

Marmo, G.

M. Asorey, P. Facchi, V. I. Man’ko, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, Phys. Rev. A 76, 012117 (2007).
[CrossRef]

Molina-Terriza, G.

G. Molina-Terriza, A. Vaziri, J. Řeháček, Z. Hradil, and A. Zeilinger, Phys. Rev. Lett. 92, 167903 (2004).
[CrossRef] [PubMed]

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Singular Optics: Optical Vortices and Polarization Singularities, Progress in Optics (Elsevier, 2009), Vol. 53.

Padgett, M.

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

Padgett, M. J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Singular Optics: Optical Vortices and Polarization Singularities, Progress in Optics (Elsevier, 2009), Vol. 53.

Pascazio, S.

M. Asorey, P. Facchi, V. I. Man’ko, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, Phys. Rev. A 76, 012117 (2007).
[CrossRef]

Raymer, M. G.

A. I. Lvovsky and M. G. Raymer, Rev. Mod. Phys. 81, 299 (2009).
[CrossRef]

Rehácek, J.

J. Řeháček, Z. Bouchal, R. Čelechovský, Z. Hradil, and L. L. Sánchez-Soto, Phys. Rev. A 77, 032110 (2008).
[CrossRef]

I. Rigas, L. L. Sánchez-Soto, A. B. Klimov, J. Řeháček, and Z. Hradil, Phys. Rev. A 78, 060101(R) (2008).
[CrossRef]

Z. Hradil, J. Řeháček, Z. Bouchal, R. Čelechovský, and L. L. Sánchez-Soto, Phys. Rev. Lett. 97, 243601 (2006).
[CrossRef]

G. Molina-Terriza, A. Vaziri, J. Řeháček, Z. Hradil, and A. Zeilinger, Phys. Rev. Lett. 92, 167903 (2004).
[CrossRef] [PubMed]

Rigas, I.

I. Rigas, L. L. Sánchez-Soto, A. B. Klimov, J. Řeháček, and Z. Hradil, Phys. Rev. A 78, 060101(R) (2008).
[CrossRef]

Sánchez-Soto, L. L.

I. Rigas, L. L. Sánchez-Soto, A. B. Klimov, J. Řeháček, and Z. Hradil, Phys. Rev. A 78, 060101(R) (2008).
[CrossRef]

J. Řeháček, Z. Bouchal, R. Čelechovský, Z. Hradil, and L. L. Sánchez-Soto, Phys. Rev. A 77, 032110 (2008).
[CrossRef]

Z. Hradil, J. Řeháček, Z. Bouchal, R. Čelechovský, and L. L. Sánchez-Soto, Phys. Rev. Lett. 97, 243601 (2006).
[CrossRef]

Soskin, M.

M. Soskin and M. V. Vasnetsov, Singular Optics, Progress in Optics (Elsevier, 2001), Vol. 41.

Sudarshan, E. C. G.

M. Asorey, P. Facchi, V. I. Man’ko, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, Phys. Rev. A 76, 012117 (2007).
[CrossRef]

Tapster, P. R.

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys. Rev. A 49, 3119 (1994).
[CrossRef] [PubMed]

Vasnetsov, M. V.

M. Soskin and M. V. Vasnetsov, Singular Optics, Progress in Optics (Elsevier, 2001), Vol. 41.

Vaughan, J. M.

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys. Rev. A 49, 3119 (1994).
[CrossRef] [PubMed]

Vaziri, A.

G. Molina-Terriza, A. Vaziri, J. Řeháček, Z. Hradil, and A. Zeilinger, Phys. Rev. Lett. 92, 167903 (2004).
[CrossRef] [PubMed]

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

Weihs, G.

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

Zeilinger, A.

G. Molina-Terriza, A. Vaziri, J. Řeháček, Z. Hradil, and A. Zeilinger, Phys. Rev. Lett. 92, 167903 (2004).
[CrossRef] [PubMed]

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

Laser Photon. Rev. (1)

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

Nature (1)

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

Phys. Rev. A (4)

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys. Rev. A 49, 3119 (1994).
[CrossRef] [PubMed]

J. Řeháček, Z. Bouchal, R. Čelechovský, Z. Hradil, and L. L. Sánchez-Soto, Phys. Rev. A 77, 032110 (2008).
[CrossRef]

M. Asorey, P. Facchi, V. I. Man’ko, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, Phys. Rev. A 76, 012117 (2007).
[CrossRef]

I. Rigas, L. L. Sánchez-Soto, A. B. Klimov, J. Řeháček, and Z. Hradil, Phys. Rev. A 78, 060101(R) (2008).
[CrossRef]

Phys. Rev. Lett. (4)

Z. Hradil, J. Řeháček, Z. Bouchal, R. Čelechovský, and L. L. Sánchez-Soto, Phys. Rev. Lett. 97, 243601 (2006).
[CrossRef]

G. C. G. Berkhout and M. W. Beijersbergen, Phys. Rev. Lett. 101, 100801 (2008).
[CrossRef] [PubMed]

G. Molina-Terriza, A. Vaziri, J. Řeháček, Z. Hradil, and A. Zeilinger, Phys. Rev. Lett. 92, 167903 (2004).
[CrossRef] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

A. I. Lvovsky and M. G. Raymer, Rev. Mod. Phys. 81, 299 (2009).
[CrossRef]

Other (3)

M.G. A.Paris and J.Řeháček, eds., Quantum State Estimation, Lecture Notes in Physics (Springer, 2004), Vol. 649.

M. Soskin and M. V. Vasnetsov, Singular Optics, Progress in Optics (Elsevier, 2001), Vol. 41.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Singular Optics: Optical Vortices and Polarization Singularities, Progress in Optics (Elsevier, 2009), Vol. 53.

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Figures (2)

Fig. 1
Fig. 1

Calculated (left) and experimentally generated (right) transverse intensity distribution for a normalized superposition of the states ψ 1 ( r , ϕ ) and ψ 2 ( r , ϕ ) .

Fig. 2
Fig. 2

Reconstructed Wigner function [Eq. (12)] corresponding to the same state as in Fig. 1. The left panel shows this function on the unit cylinder (whose vertical axis corresponds to the OAM), while the right panel shows the unwrapped plot in the Cartesian axis.

Equations (12)

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

W ( , ϕ ) = 1 2 π π π d ϕ e i ϕ ϕ ϕ / 2 | ϱ | ϕ + ϕ / 2 ,
W ( , ϕ ) = 1 2 π Z π π d ϕ e i ( ϕ ϕ ) ϱ ( , ϕ ) ,
ω ( ϕ , ζ ) = ϕ | exp ( i ζ L 2 / 2 ) ϱ exp ( i ζ L 2 / 2 ) | ϕ .
ϱ ( , ϕ ) = 1 2 π d ϕ e i ϕ ω ( ϕ , ϕ / ) ,
ψ ( r , φ ) = r , φ | = A ( r ) exp ( i φ ) ,
ψ ( r , φ , z ) = e i a 2 z ψ ( r , φ ) .
r 2 d 2 A ( r ) d r 2 + r d A ( r ) d r + ( 2 r 2 2 a k 2 ) A ( r ) = 0 ,
ψ ( r , φ , z ) = N J ( 2 a k r ) e i a 2 z e i φ .
ω ( ϕ , ζ ) = 1 2 π | Z c e i ϕ e i ζ 2 / 2 | 2 .
ω ( ϕ , 0 ) = P ( ϕ ) = | d φ d r r ψ ϕ * ( r , φ ) ψ ( r , φ ) | 2 ,
ψ ϕ ( r , φ ) = Z e i ( φ ϕ ) J ( 2 a k r ) .
W ( , ϕ ) = 1 4 π ( δ , 1 + δ , 2 ) + ( 1 ) + 1 π 2 ( 3 2 ) cos ϕ .

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