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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References

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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).

2006 (1)

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]

2005 (2)

L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt Express 13, 873–881 (2005).
[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]

2004 (2)

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]

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]

2002 (3)

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]

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]

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.

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. 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.

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]

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]

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]

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]

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.

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]

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]

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]

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]

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.

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]

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]

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]

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]

Leach, J.

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]

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]

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).

MacVicar, I.

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]

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]

Molina-Terriza, G.

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]

Nienhuis, G.

G. Nienhuis, “Doppler effect induced by rotating lenses,” Opt. Commun. 132, 8–14 (1996).
[Crossref]

O’Neil, A. T.

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]

Padgett, M.

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]

Padgett, M. J.

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]

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]

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]

Pas’ko, V.

Pas’ko, V. A.

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]

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]

Soskin, M. S.

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]

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]

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]

Torres, J. P.

L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt Express 13, 873–881 (2005).
[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]

Vasnetsov, M.

Vasnetsov, M. V.

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]

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]

Yao, E.

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]

Zambrini, R.

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]

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]

New J. Phys. (2)

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]

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. (6)

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]

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]

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]

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]

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Figures (7)
Fig. 1.
Fig. 1. (Color online) Misalignment, (a) shift (lateral displacement) and (b) tilt (direction deflection).

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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.

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Fig. 3.
Fig. 3. OAM spectrum of the incident beam.

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Fig. 4.
Fig. 4. (Color online) Amplitude (a) distribution and angular (b) distribution.

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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.

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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.

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Fig. 7.
Fig. 7. Schematic routine for orbital angular momentum spectrum correction.

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Tables (1)

Tables Icon

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

View Table

Equations (35)

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

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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