Abstract

We present an analysis of the combined effects of tilt and lateral displacement on the orbital angular momentum spectrum of a laser beam. Our theory explains the symmetries and properties of the spectrum under the influence of misalignments. We apply the theory to establish a reliable and efficient method for determining and subsequently eliminating tilt and lateral displacement. An improved technique for obtaining the orbital angular momentum spectrum employing Laguerre–Gaussian modes is proposed. Finally, a numerical experiment is carried out to verify the method.

© 2010 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, 8185–8189 (1992).
    [CrossRef] [PubMed]
  2. R. A. Beth, “Mechanical detection and measurement of the angular momentum,” Phys. Rev. 50, 115–125 (1936).
    [CrossRef]
  3. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Taylor & Francis, 2003).
    [CrossRef]
  4. 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]
  5. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
    [CrossRef] [PubMed]
  6. K. Ladavac and D. G. Grier, “Assembly of 3-dimensional structures using programmable holographic optical tweezers,” Opt. Express 12, 1144–1149 (2004).
    [CrossRef] [PubMed]
  7. G. Foo, D. M. David, and G. A. Swartzlander, “Optical vortex coronagraph,” Opt. Lett. 30, 3308–3310 (2005).
    [CrossRef]
  8. B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. 99, 087701 (2007).
    [CrossRef] [PubMed]
  9. G. Anzolin, F. Tamburini, A. Bianchini, G. Umbriaco, and C. Barbieri, “Optical vortices with starlight,” Astron. Astrophys. 488, 1159–1165 (2008).
    [CrossRef]
  10. D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer, 2000).
  11. 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]
  12. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
    [CrossRef]
  13. 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]
  14. 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 (2005).
    [CrossRef]
  15. Y. Liu, C. Gao, X. Qi, and H. Weber, “Orbital angular momentum (OAM) spectrum correction in free space optical communication,” Opt. Express 16, 7091–7101 (2008).
    [CrossRef] [PubMed]
  16. 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 (2001).
    [CrossRef]
  17. S. M. Barnett and R. Zambrini, “Resolution in rotation measurements,” J. Mod. Opt. 53, 613–625 (2006).
    [CrossRef]
  18. S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92, 217901 (2004).
    [CrossRef] [PubMed]
  19. M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Opt. Lett. 28, 2285–2287 (2003).
    [CrossRef] [PubMed]
  20. 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 (2002).
    [CrossRef] [PubMed]
  21. C. Scott, Introduction to Optics and Optical Imaging (IEEE, 1998).
  22. V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Light field decomposition in angular harmonics by means of diffractive optics,” J. Mod. Opt. 45, 1495–1506 (1998).
    [CrossRef]
  23. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  24. G. Anzolin, F. Tamburini, A. Bianchini, and C. Barbieri, “Method to measure off-axis displacements based on the analysis of the intensity distribution of a vortex beam,” Phys. Rev. A 79, 033845 (2009).
    [CrossRef]
  25. S. Cui and Y. C. Soh, “Improved measurement accuracy of the quadrant detector through improvement of linearity index,” Appl. Phys. Lett. 96, 081102 (2010).
    [CrossRef]

2010 (1)

S. Cui and Y. C. Soh, “Improved measurement accuracy of the quadrant detector through improvement of linearity index,” Appl. Phys. Lett. 96, 081102 (2010).
[CrossRef]

2009 (1)

G. Anzolin, F. Tamburini, A. Bianchini, and C. Barbieri, “Method to measure off-axis displacements based on the analysis of the intensity distribution of a vortex beam,” Phys. Rev. A 79, 033845 (2009).
[CrossRef]

2008 (2)

G. Anzolin, F. Tamburini, A. Bianchini, G. Umbriaco, and C. Barbieri, “Optical vortices with starlight,” Astron. Astrophys. 488, 1159–1165 (2008).
[CrossRef]

Y. Liu, C. Gao, X. Qi, and H. Weber, “Orbital angular momentum (OAM) spectrum correction in free space optical communication,” Opt. Express 16, 7091–7101 (2008).
[CrossRef] [PubMed]

2007 (2)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. 99, 087701 (2007).
[CrossRef] [PubMed]

2006 (1)

S. M. Barnett and R. Zambrini, “Resolution in rotation measurements,” J. Mod. Opt. 53, 613–625 (2006).
[CrossRef]

2005 (2)

G. Foo, D. M. David, and G. A. Swartzlander, “Optical vortex coronagraph,” Opt. Lett. 30, 3308–3310 (2005).
[CrossRef]

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 (2005).
[CrossRef]

2004 (3)

2003 (3)

M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Opt. Lett. 28, 2285–2287 (2003).
[CrossRef] [PubMed]

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef] [PubMed]

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Taylor & Francis, 2003).
[CrossRef]

2002 (1)

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 (2002).
[CrossRef] [PubMed]

2001 (2)

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]

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 (2001).
[CrossRef]

2000 (1)

D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer, 2000).

1998 (2)

C. Scott, Introduction to Optics and Optical Imaging (IEEE, 1998).

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Light field decomposition in angular harmonics by means of diffractive optics,” J. Mod. Opt. 45, 1495–1506 (1998).
[CrossRef]

1996 (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

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 (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, 8185–8189 (1992).
[CrossRef] [PubMed]

1936 (1)

R. A. Beth, “Mechanical detection and measurement of the angular momentum,” Phys. Rev. 50, 115–125 (1936).
[CrossRef]

Aiello, A.

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92, 217901 (2004).
[CrossRef] [PubMed]

Allen, L.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Taylor & Francis, 2003).
[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, 8185–8189 (1992).
[CrossRef] [PubMed]

Anzolin, G.

G. Anzolin, F. Tamburini, A. Bianchini, and C. Barbieri, “Method to measure off-axis displacements based on the analysis of the intensity distribution of a vortex beam,” Phys. Rev. A 79, 033845 (2009).
[CrossRef]

G. Anzolin, F. Tamburini, A. Bianchini, G. Umbriaco, and C. Barbieri, “Optical vortices with starlight,” Astron. Astrophys. 488, 1159–1165 (2008).
[CrossRef]

Barbieri, C.

G. Anzolin, F. Tamburini, A. Bianchini, and C. Barbieri, “Method to measure off-axis displacements based on the analysis of the intensity distribution of a vortex beam,” Phys. Rev. A 79, 033845 (2009).
[CrossRef]

G. Anzolin, F. Tamburini, A. Bianchini, G. Umbriaco, and C. Barbieri, “Optical vortices with starlight,” Astron. Astrophys. 488, 1159–1165 (2008).
[CrossRef]

Barnett, S. M.

S. M. Barnett and R. Zambrini, “Resolution in rotation measurements,” J. Mod. Opt. 53, 613–625 (2006).
[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]

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Taylor & Francis, 2003).
[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 (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, 8185–8189 (1992).
[CrossRef] [PubMed]

Bergman, J.

B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. 99, 087701 (2007).
[CrossRef] [PubMed]

Beth, R. A.

R. A. Beth, “Mechanical detection and measurement of the angular momentum,” Phys. Rev. 50, 115–125 (1936).
[CrossRef]

Bianchini, A.

G. Anzolin, F. Tamburini, A. Bianchini, and C. Barbieri, “Method to measure off-axis displacements based on the analysis of the intensity distribution of a vortex beam,” Phys. Rev. A 79, 033845 (2009).
[CrossRef]

G. Anzolin, F. Tamburini, A. Bianchini, G. Umbriaco, and C. Barbieri, “Optical vortices with starlight,” Astron. Astrophys. 488, 1159–1165 (2008).
[CrossRef]

Bouwmeester, D.

D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer, 2000).

Carozzi, T. D.

B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. 99, 087701 (2007).
[CrossRef] [PubMed]

Courtial, J.

Cui, S.

S. Cui and Y. C. Soh, “Improved measurement accuracy of the quadrant detector through improvement of linearity index,” Appl. Phys. Lett. 96, 081102 (2010).
[CrossRef]

David, D. M.

Ekert, A.

D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer, 2000).

Eliel, E. R.

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92, 217901 (2004).
[CrossRef] [PubMed]

Foo, G.

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.

Gibson, G.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Grier, D. 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]

Ibragimov, N. H.

B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. 99, 087701 (2007).
[CrossRef] [PubMed]

Istomin, Y. N.

B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. 99, 087701 (2007).
[CrossRef] [PubMed]

Khamitova, R.

B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. 99, 087701 (2007).
[CrossRef] [PubMed]

Khonina, S. N.

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Light field decomposition in angular harmonics by means of diffractive optics,” J. Mod. Opt. 45, 1495–1506 (1998).
[CrossRef]

Kotlyar, V. V.

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Light field decomposition in angular harmonics by means of diffractive optics,” J. Mod. Opt. 45, 1495–1506 (1998).
[CrossRef]

Ladavac, K.

Leach, J.

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 (2002).
[CrossRef] [PubMed]

Liu, Y.

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, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

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 (2001).
[CrossRef]

Nienhuis, G.

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92, 217901 (2004).
[CrossRef] [PubMed]

Oemrawsingh, S. S. R.

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92, 217901 (2004).
[CrossRef] [PubMed]

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]

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Taylor & Francis, 2003).
[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 (2002).
[CrossRef] [PubMed]

Palmer, K.

B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. 99, 087701 (2007).
[CrossRef] [PubMed]

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 (2005).
[CrossRef]

Petrov, D. V.

Qi, X.

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]

Scott, C.

C. Scott, Introduction to Optics and Optical Imaging (IEEE, 1998).

Sjöholm, J.

B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. 99, 087701 (2007).
[CrossRef] [PubMed]

Soh, Y. C.

S. Cui and Y. C. Soh, “Improved measurement accuracy of the quadrant detector through improvement of linearity index,” Appl. Phys. Lett. 96, 081102 (2010).
[CrossRef]

Soifer, V. A.

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Light field decomposition in angular harmonics by means of diffractive optics,” J. Mod. Opt. 45, 1495–1506 (1998).
[CrossRef]

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 (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, 8185–8189 (1992).
[CrossRef] [PubMed]

Swartzlander, G. A.

Tamburini, F.

G. Anzolin, F. Tamburini, A. Bianchini, and C. Barbieri, “Method to measure off-axis displacements based on the analysis of the intensity distribution of a vortex beam,” Phys. Rev. A 79, 033845 (2009).
[CrossRef]

G. Anzolin, F. Tamburini, A. Bianchini, G. Umbriaco, and C. Barbieri, “Optical vortices with starlight,” Astron. Astrophys. 488, 1159–1165 (2008).
[CrossRef]

Then, H.

B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. 99, 087701 (2007).
[CrossRef] [PubMed]

Thidé, B.

B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. 99, 087701 (2007).
[CrossRef] [PubMed]

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Opt. Lett. 28, 2285–2287 (2003).
[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 (2001).
[CrossRef]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Opt. Lett. 28, 2285–2287 (2003).
[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 (2001).
[CrossRef]

Umbriaco, G.

G. Anzolin, F. Tamburini, A. Bianchini, G. Umbriaco, and C. Barbieri, “Optical vortices with starlight,” Astron. Astrophys. 488, 1159–1165 (2008).
[CrossRef]

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 (2005).
[CrossRef]

M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Opt. Lett. 28, 2285–2287 (2003).
[CrossRef] [PubMed]

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.

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.

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92, 217901 (2004).
[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, 8185–8189 (1992).
[CrossRef] [PubMed]

Zambrini, R.

S. M. Barnett and R. Zambrini, “Resolution in rotation measurements,” J. Mod. Opt. 53, 613–625 (2006).
[CrossRef]

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]

D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer, 2000).

Appl. Phys. Lett. (1)

S. Cui and Y. C. Soh, “Improved measurement accuracy of the quadrant detector through improvement of linearity index,” Appl. Phys. Lett. 96, 081102 (2010).
[CrossRef]

Astron. Astrophys. (1)

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Figures (5)

Fig. 1
Fig. 1

Schematic illustration of a misaligned Gaussian beam with lateral displacement and tilt. The dashed line depicts the propagation direction of the beam. z-axis is taken as the reference axis.

Fig. 2
Fig. 2

Weight C + 1 of the OAM state | + 1 as a function of normalized lateral displacement and tilt ( u 0 = 2 r 0 / w 0 and v 0 = k w 0   sin   α ) evolves with the relative azimuthal angle β.

Fig. 3
Fig. 3

OAM spectra subject to various misalignments. (a) Tilt only ( u 0 = 0 , v 0 = 1 , and β = 0.5 ); (b) lateral-displacement only ( u 0 = 1.52 , v 0 = 0 , and β = 0.9 ); (c) combination of tilt and lateral displacement ( u 0 = 1.52 , v 0 = 1 , and β = 0.9 ).

Fig. 4
Fig. 4

Experiment schematic to measure the weight of OAM states. The point detectors are used to measure the strength of the cross-correlation intensity at various locations, which is equivalent to lateral shift of the input.

Fig. 5
Fig. 5

Numerical simulation of the proposed experiment. (a) and (b) are the amplitude and phase distribution in the reference plane, respectively (the origin of the coordinates is denoted with “+”); (c) and (d) are the two-dimensional amplitude and phase function of the matched filter designed to measure the weight of LG 0 , 1 mode, respectively. The other filter for the measurement of LG 0 , + 1 has exactly the same amplitude function but a conjugate phase function. (e) and (f) are the intensity distributions of the output planes corresponding to the different matched filters for the measurement of LG 0 , 1 and LG 0 , + 1 modes, respectively (the locations of the detectors are denoted with “+” and “ ,” while the cross also indicates the origin of the coordinates in the output plane).

Equations (18)

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E ( r , θ ; r 0 , θ 0 , α , η ) = 2 π 1 w 0 exp ( r 2 + r 0 2 w 0 2 ) exp ( 2 r 0 r   cos ( θ θ 0 ) w 0 2 ) exp [ i k r   sin   α   cos ( θ η ) ] ,
C n ( r 0 , θ 0 , α , η ) = 0 | 0 2 π E ( r , θ ; r 0 , θ 0 , α , η ) exp ( i n θ ) d θ | 2 r d r 2 π 0 0 2 π | E ( r , θ ; r 0 , θ 0 , α , η ) | 2 r d θ d r .
C n ( u 0 , v 0 , β ) = exp ( u 0 2 + v 0 2 4 ) ( u 0 4 + v 0 4 + 2 u 0 2 v 0 2   cos ( 2 β ) u 0 2 + v 0 2 ± 2 u 0 v 0   sin   β ) | n | I | n | ( u 0 4 + v 0 4 + 2 u 0 2 v 0 2   cos ( 2 β ) 4 ) ,
C n = 2 | n | | n | ! exp ( u 0 2 2 ) u 0 2 | n | .
C n = exp ( ξ 2 4 ) I | n | ( ξ 2 4 ) ,
γ ( u 0 , v 0 , β ) = C 1 C + 1 = u 0 2 + v 0 2 + 2 u 0 v 0   sin   β u 0 2 + v 0 2 2 u 0 v 0   sin   β .
( 1 γ ) ( x 0 2 + y 0 2 + k 2 w 0 4 sin 2 α 4 ) + ( 1 + γ ) k w 0 2   sin   α ( y 0   cos   η x 0   sin   η ) = 0.
( 1 γ ) [ ( x 0 + δ ) 2 + y 0 2 + k 2 w 0 4 sin 2 α 4 ] + ( 1 + γ ) k w 0 2   sin   α [ y 0   cos   η ( x 0 + δ ) sin   η ] = 0.
2 ( 1 γ ) δ x 0 + ( 1 γ ) z 1 + ( 1 + γ ) z 2 ( 1 + γ ) δ z 3 + ( 1 γ ) δ 2 = 0.
[ 2 ( 1 γ 0 ) δ 0 1 γ 0 1 + γ 0 ( 1 + γ 0 ) δ 0 2 ( 1 γ 1 ) δ 1 1 γ 1 1 + γ 1 ( 1 + γ 1 ) δ 1 2 ( 1 γ 2 ) δ 2 1 γ 2 1 + γ 2 ( 1 + γ 2 ) δ 2 2 ( 1 γ 3 ) δ 3 1 γ 3 1 + γ 3 ( 1 + γ 3 ) δ 3 ] [ x 0 z 1 z 2 z 3 ] = [ ( 1 γ 0 ) δ 0 2 ( 1 γ 1 ) δ 1 2 ( 1 γ 2 ) δ 2 2 ( 1 γ 3 ) δ 3 2 ] ,
y 0 2 + k 2 w 0 4 sin 2 α / 4 = z 1 x 0 2 ,
k w 0 2   sin   α y 0   cos   η = x 0 z 3 + z 2 ,
k w 0 2   sin   α   sin   η = z 3 .
v ( x , y ) = g ( ξ , σ ) h ( ξ + x , σ + y ) d ξ d σ ,
E p , n LG ( r , θ ) = 2 p ! π ( | n | + p ) ! 1 w 0 ( 2 r w 0 ) | n | L p | n | ( 2 r 2 w 0 2 ) exp ( r 2 w 0 2 ) exp ( i n θ ) ,
C p , n LG = 0 0 2 π E ( r , θ ; u 0 , v 0 , β ) E p , n LG ¯ r d θ d r = 1 p ! ( | n | + p ) ! ( 1 2 ) 6 p + 3 | n | exp ( u 0 2 + v 0 2 4 ) [ u 0 4 + v 0 4 + 2 u 0 2 v 0 2   cos ( 2 β ) ] p + | n | ( u 0 2 + v 0 2 ± 2 u 0 v 0   sin   β ) | n | ,
γ LG = C p , 1 LG C p , + 1 LG = u 0 2 + v 0 2 + 2 u 0 v 0   sin   β u 0 2 + v 0 2 2 u 0 v 0   sin   β ,     p = 0 , 1 , 2 , ,
g ( ξ x 0 , σ y 0 ) h ( ξ + x , σ + y ) d ξ d σ = v ( x + x 0 , y + y 0 ) .

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