Abstract

Orbital angular momentum (OAM) of laser beams has potential application in free space optical communication, but it is sensitive against pointing instabilities of the beam, i.e. shift (lateral displacement) and tilt (deflection of the beam). This work proposes a method to correct the distorted OAM spectrum by using the mean square value of the orbital angular momentum as an indicator. Qualitative analysis is given, and the numerical simulation is carried out for demonstration. The results show that the mean square value can be used to determine the beam axis of the superimposed helical beams. The initial OAM spectrum can be recovered.

© 2008 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, 8185-8190 (1992).
    [CrossRef] [PubMed]
  2. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
    [CrossRef] [PubMed]
  3. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas'ko, S. M. Barnett, and S. Franke-Arnold, "Free-space information transfer using light beams carrying orbital angular momentum," Opt. Express 12, 5448-5456 (2004).
    [CrossRef] [PubMed]
  4. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313-316 (2001).
    [CrossRef] [PubMed]
  5. G. Molina-Terriza, J. P. Torres, and L. Torner, "Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum," Phys. Rev. Lett. 88, 013601/1-4 (2002).
    [CrossRef] [PubMed]
  6. L. Torner, J. P. Torres, and S. Carrasco, "Digital spiral imaging," Opt Express 13, 873-881 (2005).
    [CrossRef] [PubMed]
  7. Y.-D. Liu, C.-Q. Gao, M.-W. Gao, and F. Li, "Realizing high density optical data storage by using orbital angular momentum of light beam," Acta Phys. Sin. 56, 854-858 (2007) (in Chinese).
  8. M. V. Vasnetsov, V. A. Pas'ko, and M. S. Soskin, "Analysis of orbital angular momentum of a misaligned optical beam," New J. Phys. 7, 46/1-17 (2005).
    [CrossRef]
  9. S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, "Uncertainty principle for angular position and angular momentum," New J. Phys. 6, 103/1-7 (2004).
    [CrossRef]
  10. H. Weber, "Propagation of higher-order intensity moments in quadratic-index media," Opt. Quantum Electron. 24, 1027-1049 (1992).
    [CrossRef]
  11. R. Zambrini and S. M. Barnett, "Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum," Phys. Rev. Lett. 96, 113901/1-4 (2006).
    [CrossRef] [PubMed]
  12. A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam," Phys. Rev. Lett. 88, 053601/1-4 (2002).
    [CrossRef] [PubMed]
  13. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the Orbital Angular Momentum of a Single Photon," Phys. Rev. Lett. 88, 257901/1-4 (2002).
    [CrossRef] [PubMed]
  14. G. Nienhuis, "Doppler effect induced by rotating lenses," Opt. Commun. 132, 8-14 (1996).
    [CrossRef]
  15. J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828-4830 (1998).
    [CrossRef]

2007 (1)

Y.-D. Liu, C.-Q. Gao, M.-W. Gao, and F. Li, "Realizing high density optical data storage by using orbital angular momentum of light beam," Acta Phys. Sin. 56, 854-858 (2007) (in Chinese).

2005 (1)

L. Torner, J. P. Torres, and S. Carrasco, "Digital spiral imaging," Opt Express 13, 873-881 (2005).
[CrossRef] [PubMed]

2004 (1)

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313-316 (2001).
[CrossRef] [PubMed]

1998 (1)

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828-4830 (1998).
[CrossRef]

1996 (1)

G. Nienhuis, "Doppler effect induced by rotating lenses," Opt. Commun. 132, 8-14 (1996).
[CrossRef]

1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

1992 (2)

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, 8185-8190 (1992).
[CrossRef] [PubMed]

H. Weber, "Propagation of higher-order intensity moments in quadratic-index media," Opt. Quantum Electron. 24, 1027-1049 (1992).
[CrossRef]

Allen, L.

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828-4830 (1998).
[CrossRef]

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, 8185-8190 (1992).
[CrossRef] [PubMed]

Barnett, 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, 8185-8190 (1992).
[CrossRef] [PubMed]

Carrasco, S.

L. Torner, J. P. Torres, and S. Carrasco, "Digital spiral imaging," Opt Express 13, 873-881 (2005).
[CrossRef] [PubMed]

Courtial, J.

Dholakia, K.

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828-4830 (1998).
[CrossRef]

Franke-Arnold, S.

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Gao, C.-Q.

Y.-D. Liu, C.-Q. Gao, M.-W. Gao, and F. Li, "Realizing high density optical data storage by using orbital angular momentum of light beam," Acta Phys. Sin. 56, 854-858 (2007) (in Chinese).

Gao, M.-W.

Y.-D. Liu, C.-Q. Gao, M.-W. Gao, and F. Li, "Realizing high density optical data storage by using orbital angular momentum of light beam," Acta Phys. Sin. 56, 854-858 (2007) (in Chinese).

Gibson, G.

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Li, F.

Y.-D. Liu, C.-Q. Gao, M.-W. Gao, and F. Li, "Realizing high density optical data storage by using orbital angular momentum of light beam," Acta Phys. Sin. 56, 854-858 (2007) (in Chinese).

Liu, Y.-D.

Y.-D. Liu, C.-Q. Gao, M.-W. Gao, and F. Li, "Realizing high density optical data storage by using orbital angular momentum of light beam," Acta Phys. Sin. 56, 854-858 (2007) (in Chinese).

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313-316 (2001).
[CrossRef] [PubMed]

Nienhuis, G.

G. Nienhuis, "Doppler effect induced by rotating lenses," Opt. Commun. 132, 8-14 (1996).
[CrossRef]

Padgett, M. J.

Pas'ko, V.

Robertson, D. A.

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828-4830 (1998).
[CrossRef]

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

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, 8185-8190 (1992).
[CrossRef] [PubMed]

Torner, L.

L. Torner, J. P. Torres, and S. Carrasco, "Digital spiral imaging," Opt Express 13, 873-881 (2005).
[CrossRef] [PubMed]

Torres, J. P.

L. Torner, J. P. Torres, and S. Carrasco, "Digital spiral imaging," Opt Express 13, 873-881 (2005).
[CrossRef] [PubMed]

Vasnetsov, M.

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313-316 (2001).
[CrossRef] [PubMed]

Weber, H.

H. Weber, "Propagation of higher-order intensity moments in quadratic-index media," Opt. Quantum Electron. 24, 1027-1049 (1992).
[CrossRef]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313-316 (2001).
[CrossRef] [PubMed]

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, 8185-8190 (1992).
[CrossRef] [PubMed]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313-316 (2001).
[CrossRef] [PubMed]

Acta Phys. Sin. (1)

Y.-D. Liu, C.-Q. Gao, M.-W. Gao, and F. Li, "Realizing high density optical data storage by using orbital angular momentum of light beam," Acta Phys. Sin. 56, 854-858 (2007) (in Chinese).

Nature (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313-316 (2001).
[CrossRef] [PubMed]

Opt Express (1)

L. Torner, J. P. Torres, and S. Carrasco, "Digital spiral imaging," Opt Express 13, 873-881 (2005).
[CrossRef] [PubMed]

Opt. Commun. (1)

G. Nienhuis, "Doppler effect induced by rotating lenses," Opt. Commun. 132, 8-14 (1996).
[CrossRef]

Opt. Express (1)

Opt. Quantum Electron. (1)

H. Weber, "Propagation of higher-order intensity moments in quadratic-index media," Opt. Quantum Electron. 24, 1027-1049 (1992).
[CrossRef]

Phys. Rev. A (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, 8185-8190 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828-4830 (1998).
[CrossRef]

Other (6)

R. Zambrini and S. M. Barnett, "Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum," Phys. Rev. Lett. 96, 113901/1-4 (2006).
[CrossRef] [PubMed]

A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam," Phys. Rev. Lett. 88, 053601/1-4 (2002).
[CrossRef] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the Orbital Angular Momentum of a Single Photon," Phys. Rev. Lett. 88, 257901/1-4 (2002).
[CrossRef] [PubMed]

G. Molina-Terriza, J. P. Torres, and L. Torner, "Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum," Phys. Rev. Lett. 88, 013601/1-4 (2002).
[CrossRef] [PubMed]

M. V. Vasnetsov, V. A. Pas'ko, and M. S. Soskin, "Analysis of orbital angular momentum of a misaligned optical beam," New J. Phys. 7, 46/1-17 (2005).
[CrossRef]

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, "Uncertainty principle for angular position and angular momentum," New J. Phys. 6, 103/1-7 (2004).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

(Color online) Misalignment, (a) shift (lateral displacement) and (b) tilt (direction deflection).

Fig. 2.
Fig. 2.

(Color online) An example of the beam center defined by its intensity moment in the near-field (dark cross) which is not collinear with the real beam axis (white cross) The beam is the superposition of two Laguerre-Gaussian beams with radial and azimuthal indices (p, l)=(0,0) and (0,1), respectively.

Fig. 3.
Fig. 3.

OAM spectrum of the incident beam.

Fig. 4.
Fig. 4.

(Color online) Amplitude (a) distribution and angular (b) distribution.

Fig. 5.
Fig. 5.

(Color online) The V value varies with the shift, (a) contour plot, (b) mesh plot, and (c) x-direction and y-direction through the center.

Fig. 6.
Fig. 6.

(Color online) The information entropy I varying with the shift, (a) contour plot, (b) mesh plot, and (c) x, y-directions through the center.

Fig. 7.
Fig. 7.

Schematic routine for orbital angular momentum spectrum correction.

Tables (1)

Tables Icon

Table 1. Expansion coefficients of the initial normalized beam on Laguerre-Gaussian beams

Equations (35)

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M ( θ ) = ψ exp ( i θ l ̂ z ) ψ ,
P l = 1 2 π 0 2 π d θ M ( θ ) exp ( i l θ ) .
pl p , l p , l = 1 ,
l ̂ z p , l = l p , l .
M ( θ ) = ψ exp ( i θ l ̂ z ) ψ
= pl p l ψ p , l p , l exp ( i θ l ̂ z ) p , l p , l ψ .
= pl exp ( i l θ ) ψ p , l p , l ψ
P l = p ψ p , l p , l ψ .
ψ = T Φ ,
T LD ( r 0 , η ) u ( x , y ) = u ( x r 0 cos η , y r 0 sin η ) ,
T DD ( η , γ ) u ( x , y ) = exp [ ikr sin γ cos ( φ η ) ] u ( x , y ) ,
P l = p p 0 l 0 p l T p 0 , l 0 p 0 l 0 Φ 2 .
M pl , p 0 l 0 LD = u pl * ( x , y ) u p 0 l 0 ( x r 0 cos η , y + r 0 sin η ) d x d y
= 1 4 π 2 F 1 { F 2 [ u pl * ( x , y ) ]
F [ u p 0 l 0 ( x r 0 cos η , y + r 0 sin η ) ] d x d y } .
F [ u pl ( x , y ) w 0 ] = i 2 p + l 2 π u pl ( κ x , κ y ) 2 w 0 ,
F [ u pl ( x x 0 , y y 0 ) w 0 ] = i 2 p + l 2 π u pl ( κ x , κ y ) 2 w 0 exp ( i κ x x 0 i κ y y 0 ) .
M pl , p 0 l 0 LD = ( 1 ) p + p 0 + l i l + l 0 u pl * ( κ x , κ y ) 2 w 0 u p 0 l 0 ( κ x , κ y ) 2 w 0
× exp ( i κ x r 0 cos η i κ y r 0 sin η ) d κ x d κ y .
M pl , p 0 l 0 LD = ( 1 ) p + p 0 + l i l + l 0 u pl * ( x , y ) w 0 u p 0 l 0 ( x , y ) w 0
× exp ( i 2 r r 0 w 0 2 cos ( φ η ) ) d x d y ,
M pl , p 0 l 0 LD = ( 1 ) p + p 0 + l i l + l 0 p l exp ( i 2 r r 0 w 0 2 cos ( φ η ) ) p 0 l 0 .
exp ( i z sin θ ) = n = + J n ( z ) exp ( i n θ )
exp [ i ( l l 0 + n ) φ ] d φ = 2 π δ ( n ( l l 0 ) ) ,
M pl , p 0 l 0 LD = ( 1 ) p + p 0 + l i l + l 0 exp [ i ( l l 0 ) ( η + π 2 ) ]
× p l J l l 0 ( 2 r r 0 w 0 2 ) exp [ i ( l l 0 ) φ ] p 0 l 0 .
M pl , p 0 l 0 DD = exp [ i ( l l 0 ) ( π 2 η ) ]
× p , l J l l 0 ( k r sin γ ) exp [ i ( l l 0 ) φ ] p 0 , l 0 .
T = T DD T LD = T LD T DD .
M pl , p 0 l 0 pl T DD T LD p 0 l 0
= p l pl T DD p l p l T LD p 0 l 0
= p l M pl , p l DD M p l , p 0 l 0 LD
V = ψ I ̂ z 2 ψ ψ ψ ψ I ̂ z ψ 2 ψ ψ = l P l ( l l ¯ ) 2 = l P l l 2 ( l P l l ) 2 .
V = r 0 2 k 2 z R ( 2 p 0 + l 0 + 1 ) ( r 0 transverse shift of the axis ) ,
I = l P l ln P l .

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