Abstract

The overall spatial structure of a general partially coherent field is shown to be connected with the cross-correlation between the so-called spiral modes, understood as the terms of the spiral-harmonics series expansion of the field. The formalism based on the beam irradiance-moments is used, and the light field is globally described by the beam width, the far-field divergence, the beam quality factor, the orientation of the beam profile and the angular orbital momentum, given as the sum of its asymmetrical and vortex parts. This overall spatial description is expressed in terms of the intermodal coherence features (cross-correlation between spiral modes). The above analytical results are also illustrated by means of an example.

© 2009 OSA

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  1. Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and ornital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281, 1968–1975 (2008).
  2. R. Martínez-Herrero and A. Manjavacas, “Overall second-order parametric characterization of light beams propagating through spiral phase elements,” Opt. Commun. 282(4), 473–477 (2009).
    [CrossRef]
  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, 0136011–0136014 (2002).
  4. 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(23), 2285–2287 (2003).
    [CrossRef] [PubMed]
  5. L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005).
    [CrossRef] [PubMed]
  6. 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(11), 8185–8189 (1992).
    [CrossRef] [PubMed]
  7. L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
    [CrossRef]
  8. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
    [CrossRef]
  9. M. V. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
    [CrossRef]
  10. H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28(12), 968–970 (2003).
    [CrossRef] [PubMed]
  11. T. Alieva and M. J. Bastiaans, “Evolution of the vortex and the asymmetrical parts of orbital angular momentum in separable first-order optical systems,” Opt. Lett. 29(14), 1587–1589 (2004).
    [CrossRef] [PubMed]
  12. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
    [CrossRef] [PubMed]
  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(22), 5448–5456 (2004).
    [CrossRef] [PubMed]
  14. R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65(5), 322–328 (1988).
    [CrossRef]
  15. S. Lavi, R. Prochaska, and E. Keren, “Generalized beam parameters and transformation law for partially coherent light,” Appl. Opt. 27(17), 3696–3703 (1988).
    [CrossRef] [PubMed]
  16. M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first- order optical systems,” Optik (Stuttg.) 82, 173–181 (1989).
  17. A. E. Siegman, “New developments in laser resonators” in Laser Resonators,” Proc. SPIE 1224, 2–14 (1990).
    [CrossRef]
  18. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), S1027–1049 (1992).
    [CrossRef]
  19. R. Martínez-Herrero and P. M. Mejías, “Expansion of the cross-spectral density function of general fields and its application to beam characterization,” Opt. Commun. 94(4), 197–202 (1992).
    [CrossRef]
  20. R. Martínez-Herrero, P. M. Mejías, G. Piquero, and J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarized laser beams,” Prog. Quantum Electron. 26(2), 65–130 (2002).
    [CrossRef]
  21. A. Ya. Bekshaev, M. V. Vasnetsov, V. G. Denisenko, and M. S. Soskin, “Transformation of the orbital angular momentum of a beam with optical vortex in an astigmatic optical system,” JETP Lett. 75(3), 127–130 (2002).
    [CrossRef]
  22. A. Ya Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. A 20(8), 1635–1643 (2003).
    [CrossRef]
  23. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25(1), 26–30 (1978).
    [CrossRef]
  24. R. Martínez-Herrero and P. M. Mejías, “On the control of the spatial orientation of the transverse profile of a light beam,” Opt. Express 14(3), 1086–1093 (2006).
    [CrossRef] [PubMed]
  25. R. Martínez-Herrero and P. M. Mejías, “On the spatial orientation of the transverse irradiance profile of partially coherent beams,” Opt. Express 14(8), 3294–3303 (2006).
    [CrossRef] [PubMed]
  26. R. Simon and N. Mukunda, “Twisted Gaussian-Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993).
    [CrossRef]
  27. F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini, and G. Schirripa Spagnolo, “Coherent and partially coherent twisting beams,” Opt. Rev. 1, 143–145 (1994).
  28. G. Nemes and A. E. Siegman, “Measurement of all ten second-order moments of an astigmatic beam by use of rotating simple astigmatic (anamorphic) optics,” J. Opt. Soc. Am. A 11(8), 2257–2264 (1994).
    [CrossRef]

2009 (1)

R. Martínez-Herrero and A. Manjavacas, “Overall second-order parametric characterization of light beams propagating through spiral phase elements,” Opt. Commun. 282(4), 473–477 (2009).
[CrossRef]

2008 (1)

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and ornital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281, 1968–1975 (2008).

2006 (2)

2005 (1)

2004 (2)

2003 (3)

2002 (3)

R. Martínez-Herrero, P. M. Mejías, G. Piquero, and J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarized laser beams,” Prog. Quantum Electron. 26(2), 65–130 (2002).
[CrossRef]

A. Ya. Bekshaev, M. V. Vasnetsov, V. G. Denisenko, and M. S. Soskin, “Transformation of the orbital angular momentum of a beam with optical vortex in an astigmatic optical system,” JETP Lett. 75(3), 127–130 (2002).
[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, 0136011–0136014 (2002).

2001 (2)

M. V. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[CrossRef] [PubMed]

1999 (1)

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[CrossRef]

1997 (1)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
[CrossRef]

1994 (2)

F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini, and G. Schirripa Spagnolo, “Coherent and partially coherent twisting beams,” Opt. Rev. 1, 143–145 (1994).

G. Nemes and A. E. Siegman, “Measurement of all ten second-order moments of an astigmatic beam by use of rotating simple astigmatic (anamorphic) optics,” J. Opt. Soc. Am. A 11(8), 2257–2264 (1994).
[CrossRef]

1993 (1)

1992 (3)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), S1027–1049 (1992).
[CrossRef]

R. Martínez-Herrero and P. M. Mejías, “Expansion of the cross-spectral density function of general fields and its application to beam characterization,” Opt. Commun. 94(4), 197–202 (1992).
[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(11), 8185–8189 (1992).
[CrossRef] [PubMed]

1990 (1)

A. E. Siegman, “New developments in laser resonators” in Laser Resonators,” Proc. SPIE 1224, 2–14 (1990).
[CrossRef]

1989 (1)

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first- order optical systems,” Optik (Stuttg.) 82, 173–181 (1989).

1988 (2)

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65(5), 322–328 (1988).
[CrossRef]

S. Lavi, R. Prochaska, and E. Keren, “Generalized beam parameters and transformation law for partially coherent light,” Appl. Opt. 27(17), 3696–3703 (1988).
[CrossRef] [PubMed]

1978 (1)

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25(1), 26–30 (1978).
[CrossRef]

Alieva, T.

Allen, L.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[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(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Babiker, M.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[CrossRef]

Bagini, V.

F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini, and G. Schirripa Spagnolo, “Coherent and partially coherent twisting beams,” Opt. Rev. 1, 143–145 (1994).

Barnett, S. M.

Bastiaans, M. J.

T. Alieva and M. J. Bastiaans, “Evolution of the vortex and the asymmetrical parts of orbital angular momentum in separable first-order optical systems,” Opt. Lett. 29(14), 1587–1589 (2004).
[CrossRef] [PubMed]

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first- order optical systems,” Optik (Stuttg.) 82, 173–181 (1989).

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25(1), 26–30 (1978).
[CrossRef]

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(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Bekshaev, A. Ya.

A. Ya. Bekshaev, M. V. Vasnetsov, V. G. Denisenko, and M. S. Soskin, “Transformation of the orbital angular momentum of a beam with optical vortex in an astigmatic optical system,” JETP Lett. 75(3), 127–130 (2002).
[CrossRef]

Carrasco, S.

Courtial, J.

Denisenko, V. G.

A. Ya. Bekshaev, M. V. Vasnetsov, V. G. Denisenko, and M. S. Soskin, “Transformation of the orbital angular momentum of a beam with optical vortex in an astigmatic optical system,” JETP Lett. 75(3), 127–130 (2002).
[CrossRef]

Franke-Arnold, S.

Frezza, F.

F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini, and G. Schirripa Spagnolo, “Coherent and partially coherent twisting beams,” Opt. Rev. 1, 143–145 (1994).

Gao, C.

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and ornital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281, 1968–1975 (2008).

Gao, M.

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and ornital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281, 1968–1975 (2008).

Gbur, G.

Gibson, G.

Gori, F.

F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini, and G. Schirripa Spagnolo, “Coherent and partially coherent twisting beams,” Opt. Rev. 1, 143–145 (1994).

Gorshkov, V. N.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
[CrossRef]

Heckenberg, N. R.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
[CrossRef]

Keren, E.

Lavi, S.

Li, F.

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and ornital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281, 1968–1975 (2008).

Liu, Y.

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and ornital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281, 1968–1975 (2008).

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[CrossRef] [PubMed]

Malos, J. T.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
[CrossRef]

Manjavacas, A.

R. Martínez-Herrero and A. Manjavacas, “Overall second-order parametric characterization of light beams propagating through spiral phase elements,” Opt. Commun. 282(4), 473–477 (2009).
[CrossRef]

Martínez-Herrero, R.

R. Martínez-Herrero and A. Manjavacas, “Overall second-order parametric characterization of light beams propagating through spiral phase elements,” Opt. Commun. 282(4), 473–477 (2009).
[CrossRef]

R. Martínez-Herrero and P. M. Mejías, “On the control of the spatial orientation of the transverse profile of a light beam,” Opt. Express 14(3), 1086–1093 (2006).
[CrossRef] [PubMed]

R. Martínez-Herrero and P. M. Mejías, “On the spatial orientation of the transverse irradiance profile of partially coherent beams,” Opt. Express 14(8), 3294–3303 (2006).
[CrossRef] [PubMed]

R. Martínez-Herrero, P. M. Mejías, G. Piquero, and J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarized laser beams,” Prog. Quantum Electron. 26(2), 65–130 (2002).
[CrossRef]

R. Martínez-Herrero and P. M. Mejías, “Expansion of the cross-spectral density function of general fields and its application to beam characterization,” Opt. Commun. 94(4), 197–202 (1992).
[CrossRef]

Mejías, P. M.

R. Martínez-Herrero and P. M. Mejías, “On the spatial orientation of the transverse irradiance profile of partially coherent beams,” Opt. Express 14(8), 3294–3303 (2006).
[CrossRef] [PubMed]

R. Martínez-Herrero and P. M. Mejías, “On the control of the spatial orientation of the transverse profile of a light beam,” Opt. Express 14(3), 1086–1093 (2006).
[CrossRef] [PubMed]

R. Martínez-Herrero, P. M. Mejías, G. Piquero, and J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarized laser beams,” Prog. Quantum Electron. 26(2), 65–130 (2002).
[CrossRef]

R. Martínez-Herrero and P. M. Mejías, “Expansion of the cross-spectral density function of general fields and its application to beam characterization,” Opt. Commun. 94(4), 197–202 (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, 0136011–0136014 (2002).

Movilla, J. M.

R. Martínez-Herrero, P. M. Mejías, G. Piquero, and J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarized laser beams,” Prog. Quantum Electron. 26(2), 65–130 (2002).
[CrossRef]

Mukunda, N.

R. Simon and N. Mukunda, “Twisted Gaussian-Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993).
[CrossRef]

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65(5), 322–328 (1988).
[CrossRef]

Nemes, G.

Padgett, M. J.

Pas’ko, V.

Petrov, D. V.

Piquero, G.

R. Martínez-Herrero, P. M. Mejías, G. Piquero, and J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarized laser beams,” Prog. Quantum Electron. 26(2), 65–130 (2002).
[CrossRef]

Prochaska, R.

Santarsiero, M.

F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini, and G. Schirripa Spagnolo, “Coherent and partially coherent twisting beams,” Opt. Rev. 1, 143–145 (1994).

Schettini, G.

F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini, and G. Schirripa Spagnolo, “Coherent and partially coherent twisting beams,” Opt. Rev. 1, 143–145 (1994).

Schirripa Spagnolo, G.

F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini, and G. Schirripa Spagnolo, “Coherent and partially coherent twisting beams,” Opt. Rev. 1, 143–145 (1994).

Schouten, H. F.

Siegman, A. E.

Simon, R.

R. Simon and N. Mukunda, “Twisted Gaussian-Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993).
[CrossRef]

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65(5), 322–328 (1988).
[CrossRef]

Soskin, M. S.

A. Ya Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. A 20(8), 1635–1643 (2003).
[CrossRef]

A. Ya. Bekshaev, M. V. Vasnetsov, V. G. Denisenko, and M. S. Soskin, “Transformation of the orbital angular momentum of a beam with optical vortex in an astigmatic optical system,” JETP Lett. 75(3), 127–130 (2002).
[CrossRef]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
[CrossRef]

Soskin, M. V.

M. V. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[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(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Sudarshan, E. C. G.

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65(5), 322–328 (1988).
[CrossRef]

Torner, L.

L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005).
[CrossRef] [PubMed]

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(23), 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, 0136011–0136014 (2002).

Torres, J. P.

L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005).
[CrossRef] [PubMed]

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(23), 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, 0136011–0136014 (2002).

Vasnetsov, M.

Vasnetsov, M. V.

A. Ya Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. A 20(8), 1635–1643 (2003).
[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(23), 2285–2287 (2003).
[CrossRef] [PubMed]

A. Ya. Bekshaev, M. V. Vasnetsov, V. G. Denisenko, and M. S. Soskin, “Transformation of the orbital angular momentum of a beam with optical vortex in an astigmatic optical system,” JETP Lett. 75(3), 127–130 (2002).
[CrossRef]

M. V. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
[CrossRef]

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[CrossRef] [PubMed]

Visser, T. D.

Weber, H.

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), S1027–1049 (1992).
[CrossRef]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 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(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Wolf, E.

Ya Bekshaev, A.

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[CrossRef] [PubMed]

Appl. Opt. (1)

J. Opt. Soc. Am. A (3)

JETP Lett. (1)

A. Ya. Bekshaev, M. V. Vasnetsov, V. G. Denisenko, and M. S. Soskin, “Transformation of the orbital angular momentum of a beam with optical vortex in an astigmatic optical system,” JETP Lett. 75(3), 127–130 (2002).
[CrossRef]

Nature (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[CrossRef] [PubMed]

Opt. Commun. (5)

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65(5), 322–328 (1988).
[CrossRef]

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and ornital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281, 1968–1975 (2008).

R. Martínez-Herrero and A. Manjavacas, “Overall second-order parametric characterization of light beams propagating through spiral phase elements,” Opt. Commun. 282(4), 473–477 (2009).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25(1), 26–30 (1978).
[CrossRef]

R. Martínez-Herrero and P. M. Mejías, “Expansion of the cross-spectral density function of general fields and its application to beam characterization,” Opt. Commun. 94(4), 197–202 (1992).
[CrossRef]

Opt. Express (4)

Opt. Lett. (3)

Opt. Quantum Electron. (1)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), S1027–1049 (1992).
[CrossRef]

Opt. Rev. (1)

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Figures (4)

Fig. 1
Fig. 1

Transverse distribution (pseudo-coloured) proportional to the irradiance profile of the field considered in the example analysed in Section 4 (cf. Equations (17) and (24)), for the values w = 100λ, λ = 0.5 μm, σΨ = 0. As usual, the horizontal and vertical directions correspond to the x- and y-axis, respectively. The length of the side of each square is 4w.

Fig. 3
Fig. 3

The same as in Fig. 1 but now with σΨ = 1.

Fig. 4
Fig. 4

Rotation angle α of the beam profile upon free propagation, for the example considered in Section 4.

Fig. 2
Fig. 2

The same as in Fig. 1 but now with σΨ = 0.5.

Equations (50)

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E(r,θ)=nEn(r)exp(inθ),
En(r)=12π02πE(r,θ)exp(inθ)dθ.
<xmynupvq>1Ixmynupvq h(r,η,z)drdη,
h(r,η,z)=W(r,s)exp(ikηs)ds,
tan2α(z)=2<xy>+2z(<xv>+<yu>)+2z2<uv><x2><y2>+2z(<xu><yv>)+z2(<u2><v2>),
Jz=Ic(<xvyu>),
Jz(a)=Ic[1<x2+y2>(<x2y2><xv+yu>+2<xy><yvxu>)],
Jz(v)=JzJz(a).
M^(0)=(<x2+y2><xu+yv><xu+yv><u2+v2>),
M^(1)=(<x2y2><xuyv><xuyv><u2v2>),
M^(2)=(2<xy><xv+yu><xv+yu>2<uv>),
M^(3)=(0<xvyu><yuxv>0).
(M^(i))output=S^(M^(i))inputS^t,
W(r1,θ1,r2,θ2)=n,mEn*(r1)Em(r2)¯exp(inθ1)exp(imθ2),
W(r1,θ1,r2,θ2)=n,mWnm(r1,θ1,r2,θ2),
Wnm=En*(r1)Em(r2)¯exp(inθ1)exp(imθ2).
M^(0)=nM^nn(0),
M^(1)=nM^n1n+1(1),
M^(2)=nM^n1n+1(2),
M^(3)=nM^nn(3),
M^nn(0)]11=2πI0r2|En|2¯rdr,
M^nn(0)]22=2πk2I0(|En'|2¯+n2r2|En|2¯)rdr,
M^nn(0)]12=M^nn(0)]21=2πkI0rIm{En'En*¯}rdr,
M^n1n+1(1)]11=2πI0r2Re{En+1*En1¯}rdr,
M^n1n+1(1)]22==2πk2I0Re{(En+1')*En1'¯n21r2En+1*En1¯n1r(En+1')*En1¯+n+1r(En1')*En+1¯}rdr,
M^n1n+1(1)]12=M^n1n+1(1)]21=2πkI0rIm{En+1'En1*¯+n+1rEn+1En1*¯}rdr,
M^n1n+1(2)]11=2πI0r2Im{En+1*En1¯}rdr,
M^n1n+1(2)]22==2πk2I0Im{(En+1')*En1'¯n21r2En+1*En1¯n1r(En+1')*En1¯n+1r(En1')*En+1¯}rdr,
M^n1n+1(2)]12=M^n1n+1(2)]21=2πkI0rRe{(En+1')*En1¯+n+1rEn+1*En1¯}rdr,
M^nn(3)]11=M^nn(3)]22=0,
M^nn(3)]12=M^nn(3)]21=2πkI0rRe{nr|En|2¯}rdr.
E(r,θ)=f(r)exp(inθ)[αexp(iθ)+βexp(iθ)],
I(r,θ)=f2(r)[α2¯+β2¯+2Re{αβ*¯exp(2iθ}],
I(r,θ)=f2(r)(α2¯+β2¯)(1+σΨcos2θ),
σ=αβ*¯(α2¯β2¯)1/2,
Ψ=2(α2¯ β2¯)1/2α2¯+β2¯.
0|σ|1.
0Ψ1.
M^(0)=(<r2>00ξ+(1+n2+2ncos2γ)ρ),
M^(1)=σΨ2(<r2>00ξ+(1n2)ρ),
M^(2)=σΨ2(0nknk0),
M^(3)=(01k(n+cos2γ)1k(n+cos2γ)0),
<r2>=0r2f2(r)rdr0f2(r)rdr,
ξ=1k20[f'(r)]2rdr0f2(r)rdr,
cos2γ=β2¯α2¯α2¯+β2¯,
ρ=1k20f2(r)r2rdr0f2(r)rdr.
Jz(a)=Ic(nkσΨ2),
Jz=Ic1k(n+cos2γ).
f(r)=(rw)n+1exp(r22w2).
tan2α(z)=2nzkw2(n+2)+z2kw2(2n).

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