Abstract

With the current and upcoming applications of beams carrying orbital angular momentum (OAM), there will be the need to generate beams and measure their OAM spectrum with high accuracy. The instrumental OAM spectrum distortion is connected to the effect of its optical aberrations on the OAM content of the beams that the instrument creates or measures. Until now, the effect of the well-known Zernike aberrations has been studied partially, assuming vortex beams with trivial radial phase components. However, the traditional Zernike polynomials are not best suitable when dealing with vortex beams, as their OAM spectrum is highly sensitive to some Zernike terms, and completely insensitive to others. We propose the use of a new basis, the OAM-Zernike basis, which consists of the radial aberrations as described by radial Zernike polynomials and of the azimuthal aberrations described in the OAM basis. The traditional tools for the characterization of aberrations of optical instruments can be used, and the results translated to the new basis. This permits the straightforward calculation of the effect of any optical system, such as an OAM detection stage, on the OAM spectrum of an incoming beam. This knowledge permits to correct, a posteriori, the effect of instrumental OAM spectrum distortion on the measured spectra. We also found that the knowledge of the radial aberrations is important, as they affect the efficiency of the detection, and in some cases its accuracy. In this new framework, we study the effect of aberrations in common OAM detection methods, and encourage the characterization of those systems using this approach.

© 2010 Optical Society of America

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References

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  1. 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 (2002).
    [CrossRef] [PubMed]
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    [CrossRef]
  3. J. Lin, X.-C. Yuan, S. H. Tao, and R. E. Burge, "Multiplexing free-space optical signals using superimposed collinear orbital angular momentum states," Appl. Opt. 46, 4680-4685 (2007).
    [CrossRef] [PubMed]
  4. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, "Free-space information transfer using light beams carrying orbital angular momentum," Opt. Express 12, 5448-5456 (2004).
    [CrossRef] [PubMed]
  5. 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]
  6. S. Franke-Arnold, L. Allen, and M. Padgett, "Advances in optical angular momentum," Laser Photonics Rev. 2, 299-313 (2008).
    [CrossRef]
  7. G. A. Swartzlander, Jr., "Peering into darkness with a vortex spatial filter," Opt. Lett. 26, 497-499 (2001).
    [CrossRef]
  8. A. Jesacher, A. Schwaighofer, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, "Wavefront correction of spatial light modulators using an optical vortex image," Opt. Express 15, 5801-5808 (2007).
    [CrossRef] [PubMed]
  9. G. A. Tyler, and R. W. Boyd, "Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum," Opt. Lett. 34, 142-144 (2009).
    [CrossRef] [PubMed]
  10. G. Gbur, and R. K. Tyson, "Vortex beam propagation through atmospheric turbulence and topological charge conservation," J. Opt. Soc. Am. A 25, 225-230 (2008).
    [CrossRef]
  11. C. Paterson, "Atmospheric Turbulence and Orbital Angular Momentum of Single Photons for Optical Communication," Phys. Rev. Lett. 94, 153901 (2005).
    [CrossRef] [PubMed]
  12. Z. Yi-Xin and C. Ji, "Effects of turbulent aberrations on probability distribution of orbital angular momentum for optical communication," Chin. Phys. Lett. 26, 074220 (4pp) (2009).
    [CrossRef]
  13. C. Jenkins, "Optical vortex coronagraphs on ground-based telescopes," Mon. Not. R. Astron. Soc. 384, 515-524 (2008).
    [CrossRef]
  14. B. R. Boruah, and M. A. Neil, "Susceptibility to and correction of azimuthal aberrations in singular light beams," Opt. Express 14, 10377-10385 (2006).
    [CrossRef] [PubMed]
  15. E. Compain, S. Poirier, and B. Drevillon, "General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers," Appl. Opt. 38, 3490-3502 (1999).
    [CrossRef]
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    [CrossRef]
  17. N. Uribe-Patarroyo, A. Alvarez-Herrero, and T. Belenguer, "Measurement of the quantum superposition state of an imaging ensemble of photons prepared in orbital angular momentum states using a phase-diversity method," Phys. Rev. A 81, 053822 (2010).
    [CrossRef]
  18. J. C. Wyant, and K. Creath, "Basic Wavefront Aberration Theory for Optical Metrology," in "Applied Optics and Optical Engineering, Volume XI,", vol. 11, R. R. Shannon & J. C. Wyant, ed. (1992), vol. 11, pp. 27-39.
  19. M. Born, and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Pergamon Press, New York, 1959).
    [PubMed]
  20. W. Swantner, and W. W. Chow, "Gram-schmidt orthonormalization of Zernike polynomials for general aperture shapes," Appl. Opt. 33, 1832-1837 (1994).
    [CrossRef] [PubMed]
  21. N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, "Laser beams with phase singularities," Opt. Quantum Electron. 24, S951-S962 (1992).
    [CrossRef]
  22. S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, "Uncertainty principle for angular position and angular momentum," N. J. Phys. 6, 103 (2004).
    [CrossRef]
  23. J. W. Goodman, Introduction to Fourier optics (Roberts and Co. Publishers, Englewood, CO, 2005), 3rd ed.
  24. 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]
  25. G. C. G. Berkhout, and M. W. Beijersbergen, "Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics," J. Opt. A, Pure Appl. Opt. 11, 094021 (2009).
    [CrossRef]
  26. N. Uribe-Patarroyo, A. Alvarez-Herrero, A. López Ariste, A. Asensio Ramos, T. Belenguer, R. Manso Sainz, C. LeMen, and B. Gelly, "Detecting photons with orbital angular momentum in extended astronomical objects: application to solar observations," (unpublished).

2010 (1)

N. Uribe-Patarroyo, A. Alvarez-Herrero, and T. Belenguer, "Measurement of the quantum superposition state of an imaging ensemble of photons prepared in orbital angular momentum states using a phase-diversity method," Phys. Rev. A 81, 053822 (2010).
[CrossRef]

2009 (3)

G. C. G. Berkhout, and M. W. Beijersbergen, "Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics," J. Opt. A, Pure Appl. Opt. 11, 094021 (2009).
[CrossRef]

Z. Yi-Xin and C. Ji, "Effects of turbulent aberrations on probability distribution of orbital angular momentum for optical communication," Chin. Phys. Lett. 26, 074220 (4pp) (2009).
[CrossRef]

G. A. Tyler, and R. W. Boyd, "Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum," Opt. Lett. 34, 142-144 (2009).
[CrossRef] [PubMed]

2008 (3)

G. Gbur, and R. K. Tyson, "Vortex beam propagation through atmospheric turbulence and topological charge conservation," J. Opt. Soc. Am. A 25, 225-230 (2008).
[CrossRef]

C. Jenkins, "Optical vortex coronagraphs on ground-based telescopes," Mon. Not. R. Astron. Soc. 384, 515-524 (2008).
[CrossRef]

S. Franke-Arnold, L. Allen, and M. Padgett, "Advances in optical angular momentum," Laser Photonics Rev. 2, 299-313 (2008).
[CrossRef]

2007 (2)

2006 (1)

2005 (1)

C. Paterson, "Atmospheric Turbulence and Orbital Angular Momentum of Single Photons for Optical Communication," Phys. Rev. Lett. 94, 153901 (2005).
[CrossRef] [PubMed]

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," N. J. Phys. 6, 103 (2004).
[CrossRef]

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. 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 (2)

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]

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 (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]

G. A. Swartzlander, Jr., "Peering into darkness with a vortex spatial filter," Opt. Lett. 26, 497-499 (2001).
[CrossRef]

1999 (1)

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]

1994 (1)

1992 (1)

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, "Laser beams with phase singularities," Opt. Quantum Electron. 24, S951-S962 (1992).
[CrossRef]

Allen, L.

S. Franke-Arnold, L. Allen, and M. Padgett, "Advances in optical angular momentum," Laser Photonics Rev. 2, 299-313 (2008).
[CrossRef]

Alvarez-Herrero, A.

N. Uribe-Patarroyo, A. Alvarez-Herrero, and T. Belenguer, "Measurement of the quantum superposition state of an imaging ensemble of photons prepared in orbital angular momentum states using a phase-diversity method," Phys. Rev. A 81, 053822 (2010).
[CrossRef]

Barnett, S.

Barnett, S. M.

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, "Uncertainty principle for angular position and angular momentum," N. J. Phys. 6, 103 (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 (2002).
[CrossRef] [PubMed]

Beijersbergen, M. W.

G. C. G. Berkhout, and M. W. Beijersbergen, "Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics," J. Opt. A, Pure Appl. Opt. 11, 094021 (2009).
[CrossRef]

Belenguer, T.

N. Uribe-Patarroyo, A. Alvarez-Herrero, and T. Belenguer, "Measurement of the quantum superposition state of an imaging ensemble of photons prepared in orbital angular momentum states using a phase-diversity method," Phys. Rev. A 81, 053822 (2010).
[CrossRef]

Berkhout, G. C. G.

G. C. G. Berkhout, and M. W. Beijersbergen, "Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics," J. Opt. A, Pure Appl. Opt. 11, 094021 (2009).
[CrossRef]

Bernet, S.

Boruah, B. R.

Boyd, R. W.

Burge, R. E.

Chow, W. W.

Compain, E.

Courtial, J.

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

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, "Free-space information transfer using light beams carrying orbital angular momentum," Opt. Express 12, 5448-5456 (2004).
[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 (2002).
[CrossRef] [PubMed]

Drevillon, B.

Franke-Arnold, S.

S. Franke-Arnold, L. Allen, and M. Padgett, "Advances in optical angular momentum," Laser Photonics Rev. 2, 299-313 (2008).
[CrossRef]

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

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, "Free-space information transfer using light beams carrying orbital angular momentum," Opt. Express 12, 5448-5456 (2004).
[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 (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]

Fürhapter, S.

Gbur, G.

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]

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, "Laser beams with phase singularities," Opt. Quantum Electron. 24, S951-S962 (1992).
[CrossRef]

Jenkins, C.

C. Jenkins, "Optical vortex coronagraphs on ground-based telescopes," Mon. Not. R. Astron. Soc. 384, 515-524 (2008).
[CrossRef]

Jesacher, A.

Ji, C.

Z. Yi-Xin and C. Ji, "Effects of turbulent aberrations on probability distribution of orbital angular momentum for optical communication," Chin. Phys. Lett. 26, 074220 (4pp) (2009).
[CrossRef]

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," N. J. Phys. 6, 103 (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 (2002).
[CrossRef] [PubMed]

Lin, J.

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]

Maurer, C.

McDuff, R.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, "Laser beams with phase singularities," Opt. Quantum Electron. 24, S951-S962 (1992).
[CrossRef]

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 (2002).
[CrossRef] [PubMed]

Neil, M. A.

Padgett, M.

S. Franke-Arnold, L. Allen, and M. Padgett, "Advances in optical angular momentum," Laser Photonics Rev. 2, 299-313 (2008).
[CrossRef]

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

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, "Free-space information transfer using light beams carrying orbital angular momentum," Opt. Express 12, 5448-5456 (2004).
[CrossRef] [PubMed]

Padgett, M. 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]

Pas’ko, V.

Paterson, C.

C. Paterson, "Atmospheric Turbulence and Orbital Angular Momentum of Single Photons for Optical Communication," Phys. Rev. Lett. 94, 153901 (2005).
[CrossRef] [PubMed]

Poirier, S.

Ritsch-Marte, M.

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]

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, "Laser beams with phase singularities," Opt. Quantum Electron. 24, S951-S962 (1992).
[CrossRef]

Schwaighofer, A.

Smith, C. P.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, "Laser beams with phase singularities," Opt. Quantum Electron. 24, S951-S962 (1992).
[CrossRef]

Swantner, W.

Swartzlander, G. A.

Tao, S. H.

Torner, L.

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 (2002).
[CrossRef] [PubMed]

Torres, J. P.

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 (2002).
[CrossRef] [PubMed]

Tyler, G. A.

Tyson, R. K.

Uribe-Patarroyo, N.

N. Uribe-Patarroyo, A. Alvarez-Herrero, and T. Belenguer, "Measurement of the quantum superposition state of an imaging ensemble of photons prepared in orbital angular momentum states using a phase-diversity method," Phys. Rev. A 81, 053822 (2010).
[CrossRef]

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]

Wegener, M. J.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, "Laser beams with phase singularities," Opt. Quantum Electron. 24, S951-S962 (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]

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," N. J. Phys. 6, 103 (2004).
[CrossRef]

Yi-Xin, Z.

Z. Yi-Xin and C. Ji, "Effects of turbulent aberrations on probability distribution of orbital angular momentum for optical communication," Chin. Phys. Lett. 26, 074220 (4pp) (2009).
[CrossRef]

Yuan, X.-C.

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]

Appl. Opt. (3)

Chin. Phys. Lett. (1)

Z. Yi-Xin and C. Ji, "Effects of turbulent aberrations on probability distribution of orbital angular momentum for optical communication," Chin. Phys. Lett. 26, 074220 (4pp) (2009).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

G. C. G. Berkhout, and M. W. Beijersbergen, "Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics," J. Opt. A, Pure Appl. Opt. 11, 094021 (2009).
[CrossRef]

J. Opt. Soc. Am. A (1)

Laser Photonics Rev. (1)

S. Franke-Arnold, L. Allen, and M. Padgett, "Advances in optical angular momentum," Laser Photonics Rev. 2, 299-313 (2008).
[CrossRef]

Mon. Not. R. Astron. Soc. (1)

C. Jenkins, "Optical vortex coronagraphs on ground-based telescopes," Mon. Not. R. Astron. Soc. 384, 515-524 (2008).
[CrossRef]

N. J. Phys. (1)

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

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]

Opt. Express (3)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, "Laser beams with phase singularities," Opt. Quantum Electron. 24, S951-S962 (1992).
[CrossRef]

Phys. Rev. A (1)

N. Uribe-Patarroyo, A. Alvarez-Herrero, and T. Belenguer, "Measurement of the quantum superposition state of an imaging ensemble of photons prepared in orbital angular momentum states using a phase-diversity method," Phys. Rev. A 81, 053822 (2010).
[CrossRef]

Phys. Rev. Lett. (4)

C. Paterson, "Atmospheric Turbulence and Orbital Angular Momentum of Single Photons for Optical Communication," Phys. Rev. Lett. 94, 153901 (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 (2002).
[CrossRef] [PubMed]

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. 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]

Other (5)

J. W. Goodman, Introduction to Fourier optics (Roberts and Co. Publishers, Englewood, CO, 2005), 3rd ed.

N. Uribe-Patarroyo, A. Alvarez-Herrero, A. López Ariste, A. Asensio Ramos, T. Belenguer, R. Manso Sainz, C. LeMen, and B. Gelly, "Detecting photons with orbital angular momentum in extended astronomical objects: application to solar observations," (unpublished).

J. C. Wyant, and K. Creath, "Basic Wavefront Aberration Theory for Optical Metrology," in "Applied Optics and Optical Engineering, Volume XI,", vol. 11, R. R. Shannon & J. C. Wyant, ed. (1992), vol. 11, pp. 27-39.

M. Born, and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Pergamon Press, New York, 1959).
[PubMed]

H. G. Tompkins, and E. A. Irene, eds., Handbook of Ellipsometry (Springer, Berlin, 2005).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Zernike projection of vortex +1. The piston term 0, not shown completely, is equal to π 3/2.

Fig. 2
Fig. 2

Reconstructed vortex +1 from 527 Zernike polynomials. In white, lines of equal phase from 0 to 2π rad each 0.5π rad.

Fig. 3
Fig. 3

The effect of non-radial aberrations. PSF for plane wave (a) and for +1 vortex (b) with no aberrations. PSF for spherical and coma-aberrated ( 0.5 Z 2 0 + 0.25 Z 3 1 ) plane wave (c) and +1 vortex (d).

Fig. 4
Fig. 4

The effect of radial aberrations. PSF for vortex +1 (a) with defocus and spherical aberration ( 0.4 Z 2 0 + 0.07 Z 4 0 ) and PSF for non aberrated vortex +3 (b). Peak intensity in (a) is 0.43 times of that in (b).

Fig. 5
Fig. 5

PSF profile for no aberration, 0.25 Z 2 0 and 0.5 Z 2 0 defocus. The vertical black bars correspond to pinhole radius for SNR 100 and 10, and the dashed vertical bar indicates Airy radius. At 0.5 Z 2 0 defocus, the fiber or pinhole detector would confuse l = 1 for l = 0.

Equations (14)

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E ( ρ , ϕ ) e c 1 x 2 ρ l e i l ϕ L p | l | ( c 2 ρ 2 ) ,
P n ± m ( l ) = a n ± m 0 2 π d ϕ 0 1 d ρ ρ arg ( e i l ϕ ) Z n ± m ( ρ , ϕ ) ,
a n ± m = { 0 2 π d ϕ 0 1 d ρ ρ [ Z n ± m ( ρ , ϕ ) ] 2 } 1 / 2 = { κ π 2 n + 2 } 1 / 2 a n ,
P n ± m ( l ) = a n 0 1 d ρ ρ R n m ( ρ ) × t = 1 | l | 0 2 π / | l | l ϕ cs [ m ( ϕ + ( t 1 ) 2 π | l | ) ] d ϕ = a n K n m I ± m , l ,
I + m , 1 = 0 for m 0 , I 0 , 1 = 2 π 2 ,
I m , 1 = 2 π m for m 0 ,
K n m = s = 0 ( n m ) / 2 ( 1 ) s ( n s ) ! ( n + 2 2 s ) s ! ( n + m 2 s ) ! ( n m 2 s ) ! ,
K n m = 0 1 d ρ ρ R 0 0 ( ρ ) R n 0 ( ρ ) = a 0 δ 0 n = 1 2 δ 0 n ,
Ψ ( ρ , ϕ ) = e i n b n R n ( ρ ) l , p c l , p r ( ρ ) e i l ϕ ,
Ψ i ( ρ , ϕ ) = l , p c l , p r i ( ρ ) e i l ϕ .
Ψ o ( ρ , ϕ ) = e i n b n R n ( ρ ) l , p d l , p r o ( ρ ) e i l ϕ ,
G P ( ρ , ϕ ) = e i n b n R n ( ρ ) l d l e i l ϕ ,
ψ ( ρ , ϕ ) = e i m ϕ G P ( ρ , ϕ ) = e i n b n R n ( ρ ) l d l e i ( m + l ) ϕ ,
SNR ( m ) = 0 2 π d ϕ 0 r p d ρ ρ A m l m A l 0 2 π d ϕ 0 r p d ρ ρ 2 | l m | + 1 2 A m ( A m 1 + A m + 1 ) r p 2 r p 2 ,

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