Abstract

We introduce a new class of partially coherent vortex beams in which the angular momentum of the beam is provided from two different sources: the underlying vortex of the random beam and the “twist” given to the ensemble of beams. The statistical and propagation properties of such beams are investigated, and their orbital angular momentum properties are analyzed. The combination of distinct orbital angular momentum sources allows unusual behaviors that were previously unobserved.

© 2018 Optical Society of America

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Partially coherent vortex beams of arbitrary order

C. S. D. Stahl and G. Gbur
J. Opt. Soc. Am. A 34(10) 1793-1799 (2017)

References

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  1. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 219–276.
  2. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: Optical vortices and polarization singularities,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2009), Vol. 53, pp. 293–363.
  3. G. J. Gbur, Singular Optics (CRC Press, 2017).
  4. J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander, “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97, 053901 (2006).
    [Crossref]
  5. D. Palacios, D. Rozas, and G. A. Swartzlander, “Observed scattering into a dark optical vortex core,” Phys. Rev. Lett. 88, 103902 (2002).
    [Crossref]
  6. A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
    [Crossref]
  7. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997).
    [Crossref]
  8. K. Ladavac and D. G. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express 12, 1144–1149 (2004).
    [Crossref]
  9. 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]
  10. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
    [Crossref]
  11. 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]
  12. J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. 47, 2414–2429 (2008).
    [Crossref]
  13. B. Rodenburg, M. P. J. Lavery, M. Malik, M. N. O’sullivan, M. Mirhosseini, D. J. Robertson, M. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. 37, 3735–3737 (2012).
    [Crossref]
  14. G. Gbur, “Partially coherent beam propagation in atmospheric turbulence [invited],” J. Opt. Soc. Am. A 31, 2038–2045 (2014).
    [Crossref]
  15. R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [Crossref]
  16. C. S. D. Stahl and G. Gbur, “Partially coherent vortex beams of arbitrary order,” J. Opt. Soc. Am. A 34, 1793–1799 (2017).
    [Crossref]
  17. G. Gbur, “Partially coherent vortex beams,” Proc. SPIE 10549, 1054903 (2018).
    [Crossref]
  18. D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
    [Crossref]
  19. E. Wolf, “New theory of partial coherence in the space-frequency domain. part 1: spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [Crossref]
  20. G. Gbur, T. D. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent fields,” J. Opt. A 6, S239–S242 (2004).
    [Crossref]
  21. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
    [Crossref]
  22. D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
    [Crossref]
  23. S. M. Kim and G. Gbur, “Angular momentum conservation in partially coherent wave fields,” Phys. Rev. A 86, 043814 (2012).
    [Crossref]
  24. 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 (2002).
    [Crossref]
  25. M. Berry, “Paraxial beams of spinning light,” Proc. SPIE 3487, 6–11 (1998).
    [Crossref]
  26. P. Galajdá and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. 78, 249–251 (2001).
    [Crossref]
  27. M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, “Optically driven micromachine elements,” Appl. Phys. Lett. 78, 547–549 (2001).
    [Crossref]

2018 (1)

G. Gbur, “Partially coherent vortex beams,” Proc. SPIE 10549, 1054903 (2018).
[Crossref]

2017 (1)

2014 (1)

2012 (3)

B. Rodenburg, M. P. J. Lavery, M. Malik, M. N. O’sullivan, M. Mirhosseini, D. J. Robertson, M. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. 37, 3735–3737 (2012).
[Crossref]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

S. M. Kim and G. Gbur, “Angular momentum conservation in partially coherent wave fields,” Phys. Rev. A 86, 043814 (2012).
[Crossref]

2008 (2)

2007 (1)

2006 (1)

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander, “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97, 053901 (2006).
[Crossref]

2005 (1)

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[Crossref]

2004 (4)

K. Ladavac and D. G. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express 12, 1144–1149 (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]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[Crossref]

G. Gbur, T. D. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent fields,” J. Opt. A 6, S239–S242 (2004).
[Crossref]

2002 (2)

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 (2002).
[Crossref]

D. Palacios, D. Rozas, and G. A. Swartzlander, “Observed scattering into a dark optical vortex core,” Phys. Rev. Lett. 88, 103902 (2002).
[Crossref]

2001 (2)

P. Galajdá and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. 78, 249–251 (2001).
[Crossref]

M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, “Optically driven micromachine elements,” Appl. Phys. Lett. 78, 547–549 (2001).
[Crossref]

1998 (1)

M. Berry, “Paraxial beams of spinning light,” Proc. SPIE 3487, 6–11 (1998).
[Crossref]

1997 (1)

1994 (1)

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

1993 (1)

1982 (1)

Ahmed, N.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[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 (2002).
[Crossref]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997).
[Crossref]

Ambrosini, D.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

Anguita, J. A.

Bagini, V.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

Barnett, S. M.

Bernet, S.

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[Crossref]

Berry, M.

M. Berry, “Paraxial beams of spinning light,” Proc. SPIE 3487, 6–11 (1998).
[Crossref]

Boyd, R. W.

Courtial, J.

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: Optical vortices and polarization singularities,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2009), Vol. 53, pp. 293–363.

Dholakia, K.

Dolinar, S.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Fazal, I. M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Foo, G.

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander, “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97, 053901 (2006).
[Crossref]

Franke-Arnold, S.

Friese, M. E. J.

M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, “Optically driven micromachine elements,” Appl. Phys. Lett. 78, 547–549 (2001).
[Crossref]

Fürhapter, S.

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[Crossref]

Galajdá, P.

P. Galajdá and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. 78, 249–251 (2001).
[Crossref]

Gbur, G.

G. Gbur, “Partially coherent vortex beams,” Proc. SPIE 10549, 1054903 (2018).
[Crossref]

C. S. D. Stahl and G. Gbur, “Partially coherent vortex beams of arbitrary order,” J. Opt. Soc. Am. A 34, 1793–1799 (2017).
[Crossref]

G. Gbur, “Partially coherent beam propagation in atmospheric turbulence [invited],” J. Opt. Soc. Am. A 31, 2038–2045 (2014).
[Crossref]

S. M. Kim and G. Gbur, “Angular momentum conservation in partially coherent wave fields,” Phys. Rev. A 86, 043814 (2012).
[Crossref]

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]

G. Gbur, T. D. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent fields,” J. Opt. A 6, S239–S242 (2004).
[Crossref]

Gbur, G. J.

G. J. Gbur, Singular Optics (CRC Press, 2017).

Gibson, G.

Gold, J.

M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, “Optically driven micromachine elements,” Appl. Phys. Lett. 78, 547–549 (2001).
[Crossref]

Gori, F.

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
[Crossref]

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

Grier, D. G.

Hagberg, P.

M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, “Optically driven micromachine elements,” Appl. Phys. Lett. 78, 547–549 (2001).
[Crossref]

Hanstorp, D.

M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, “Optically driven micromachine elements,” Appl. Phys. Lett. 78, 547–549 (2001).
[Crossref]

Huang, H.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Jesacher, A.

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[Crossref]

Johnson, E. G.

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander, “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97, 053901 (2006).
[Crossref]

Kim, S. M.

S. M. Kim and G. Gbur, “Angular momentum conservation in partially coherent wave fields,” Phys. Rev. A 86, 043814 (2012).
[Crossref]

Ladavac, K.

Lavery, M. P. J.

Lee, J. H.

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander, “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97, 053901 (2006).
[Crossref]

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 (2002).
[Crossref]

Maleev, I. D.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[Crossref]

Malik, M.

Marathay, A. S.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[Crossref]

Mirhosseini, M.

Mukunda, N.

Neifeld, M. A.

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: Optical vortices and polarization singularities,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2009), Vol. 53, pp. 293–363.

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 (2002).
[Crossref]

O’sullivan, M. N.

Ormos, P.

P. Galajdá and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. 78, 249–251 (2001).
[Crossref]

Padgett, M.

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]

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 (2002).
[Crossref]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997).
[Crossref]

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: Optical vortices and polarization singularities,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2009), Vol. 53, pp. 293–363.

Palacios, D.

D. Palacios, D. Rozas, and G. A. Swartzlander, “Observed scattering into a dark optical vortex core,” Phys. Rev. Lett. 88, 103902 (2002).
[Crossref]

Palacios, D. M.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[Crossref]

Pas’ko, V.

Ren, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Ritsch-Marte, M.

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[Crossref]

Robertson, D. J.

Rodenburg, B.

Rozas, D.

D. Palacios, D. Rozas, and G. A. Swartzlander, “Observed scattering into a dark optical vortex core,” Phys. Rev. Lett. 88, 103902 (2002).
[Crossref]

Rubinsztein-Dunlop, H.

M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, “Optically driven micromachine elements,” Appl. Phys. Lett. 78, 547–549 (2001).
[Crossref]

Santarsiero, M.

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
[Crossref]

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

Simon, R.

Simpson, N. B.

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 219–276.

Stahl, C. S. D.

Swartzlander, G. A.

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander, “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97, 053901 (2006).
[Crossref]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[Crossref]

D. Palacios, D. Rozas, and G. A. Swartzlander, “Observed scattering into a dark optical vortex core,” Phys. Rev. Lett. 88, 103902 (2002).
[Crossref]

Tur, M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Tyson, R. K.

Vasic, B. V.

Vasnetsov, M.

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 219–276.

Visser, T. D.

G. Gbur, T. D. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent fields,” J. Opt. A 6, S239–S242 (2004).
[Crossref]

Wang, J.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Willner, A. E.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Wolf, E.

G. Gbur, T. D. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent fields,” J. Opt. A 6, S239–S242 (2004).
[Crossref]

E. Wolf, “New theory of partial coherence in the space-frequency domain. part 1: spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[Crossref]

Yan, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Yang, J.-Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Yue, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

P. Galajdá and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. 78, 249–251 (2001).
[Crossref]

M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, “Optically driven micromachine elements,” Appl. Phys. Lett. 78, 547–549 (2001).
[Crossref]

J. Mod. Opt. (1)

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

J. Opt. A (1)

G. Gbur, T. D. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent fields,” J. Opt. A 6, S239–S242 (2004).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Nat. Photonics (1)

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. A (1)

S. M. Kim and G. Gbur, “Angular momentum conservation in partially coherent wave fields,” Phys. Rev. A 86, 043814 (2012).
[Crossref]

Phys. Rev. Lett. (5)

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 (2002).
[Crossref]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[Crossref]

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander, “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97, 053901 (2006).
[Crossref]

D. Palacios, D. Rozas, and G. A. Swartzlander, “Observed scattering into a dark optical vortex core,” Phys. Rev. Lett. 88, 103902 (2002).
[Crossref]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[Crossref]

Proc. SPIE (2)

G. Gbur, “Partially coherent vortex beams,” Proc. SPIE 10549, 1054903 (2018).
[Crossref]

M. Berry, “Paraxial beams of spinning light,” Proc. SPIE 3487, 6–11 (1998).
[Crossref]

Other (3)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 219–276.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: Optical vortices and polarization singularities,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2009), Vol. 53, pp. 293–363.

G. J. Gbur, Singular Optics (CRC Press, 2017).

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Figures (6)

Fig. 1.
Fig. 1. Intensity of the field in the source plane for (a) different states of coherence and (b) different mode orders. In (a), we have σ = 0.01    cm (solid), σ = 1    cm (dashed), and σ = 2    cm (dotted), with m = 1 . In (b), we have m = 1 (solid), m = 2 (dashed), and m = 4 (dotted), with σ = 1    cm .
Fig. 2.
Fig. 2. Intensity of the field on propagation for (a)  α = 0 and (b)  α = 5 × 10 6 α max . The solid lines are z = 0 , the dashed lines are z = 2    km , and the dotted lines are z = 5    km . In all cases, σ = 1    cm and m = 1 .
Fig. 3.
Fig. 3. Phase of E ( r ) , for (a)  m = 1 , (b)  m = 2 , and (c)  m = 5 . Here we have taken σ = 1    cm , α = 5 × 10 6 α max , and z = 2000    m . Black indicates a phase of 0, white indicates a phase of π .
Fig. 4.
Fig. 4. Radius of the zero ring for a tvGSM of order m = 1 , with σ = 2    cm , z = 2    km . The range of α is 5 × 10 5 α max < α < 5 × 10 5 α max .
Fig. 5.
Fig. 5. Plots related to a zero net OAM beam. (a) The total OAM as a function of α , with l z = 0 highlighted with a dot. (b) The OAM density m z as a function of r , showing the positively rotating core and negatively rotating outskirts. Here m = 1 , σ = 1    cm , and α = 809    m 2 .
Fig. 6.
Fig. 6. OAM density for a beam with suppressed quadratic dependence, leaving a “dead zone” in its core. Here m = 1 , σ = 1    cm , and α = 2.381 × 10 3    m 2 .

Equations (60)

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W ( r 1 , r 2 , ω ) = U ˜ ( r 1 , ω ) U ( r 2 , ω ) ω ,
W ( r 1 , r 2 ) = S ( r 1 ) S ( r 2 ) μ ( r 1 , r 2 ) ,
W ( r 1 , r 2 ) = P ( r 0 ) U ˜ ( r 1 , r 0 ) U ( r 2 , r 0 ) d 2 r 0 ,
P ( r 0 ) = P 0 exp [ ( x 0 2 + y 0 2 ) / 2 σ 2 ] ,
U ( r , r 0 ) = U 0 Δ m exp [ [ ( x x 0 ) 2 + ( y y 0 ) 2 ] / 2 Δ 2 ] [ ( x x 0 ) + i ( y y 0 ) ] m ,
U ( r , r 0 ) = U 0 exp [ [ ( x x 0 ) 2 + ( y y 0 ) 2 ] / 2 Δ 2 ] exp [ 2 π i α ( x 0 y y 0 x ) ] ,
U ( r , r 0 ) = U 0 Δ m exp [ [ ( x x 0 ) 2 + ( y y 0 ) 2 ] / 2 Δ 2 ] × exp [ 2 π i α ( x 0 y y 0 x ) ] [ ( x x 0 ) + i ( y y 0 ) ] m ,
U ( r , r 0 , z ) = U 0 β exp [ i k 0 z ] [ ( x + i y ) ( 1 + i γ ) ( x 0 + i y 0 ) ] m ( Δ β ) m × exp [ | r r 0 | 2 / 2 Δ 2 β ] exp [ i π γ α β | r 0 | 2 ] × exp [ 2 π i α ( y x 0 x y 0 ) / β ] ,
γ 2 π α z k 0 ,
β 1 z i k 0 Δ 2 ,
W ( r 1 , r 2 , z ) = C ( z ) F ( r 1 , r 2 , z ) k = 0 m a k ( D 1 D 2 ) m k .
a k = ( m k ) 2 Γ ( k + 1 ) A 4 m 2 k + 2 ,
C ( z ) = ( 1 + γ 2 ) m π P 0 | U 0 | 2 | β | 2 1 | Δ β | 2 m ,
A 2 = 1 Δ 2 | β | 2 + 4 π 2 α 2 z 2 k 0 2 Δ 2 | β | 2 + 1 2 σ 2 ,
D 1 [ A 2 1 i γ 1 2 Δ 2 β ˜ π α β ˜ ] ( x 1 i y 1 ) [ 1 2 Δ 2 β π α β ] ( x 2 i y 2 ) ,
D 2 [ A 2 1 + i γ 1 2 Δ 2 β π α β ] ( x 2 + i y 2 ) [ 1 2 Δ 2 β ˜ π α β ˜ ] ( x 1 + i y 1 ) .
F ( r 1 , r 2 , z ) = exp [ N 2 r 1 2 ] exp [ N ˜ 2 r 2 2 ] exp [ M 2 | r 2 r 1 | 2 ] exp [ 2 π i α A 2 | β | 2 Δ 2 r 1 r 2 ] ,
N 2 β 4 σ 2 Δ 2 A 2 | β | 2 2 π 2 α 2 z A 2 | β | 2 i k 0 Δ 2 ,
M 2 1 2 A 2 | β | 2 [ 1 2 Δ 4 + 2 π 2 α 2 ] ,
α max = k 0 2 π σ .
σ S ( z ) = 2 σ A | β | Δ = 2 σ 2 + Δ 2 + ( 8 π 2 α 2 σ 2 k 0 2 + 1 k 0 2 Δ 2 ) z 2 .
S ( r , z ) = C ( z ) exp [ r 2 / σ S 2 ] k = 0 m a k | Q | 2 ( m k ) r 2 ( m k ) ,
Q 1 | β | 2 [ A 2 | β | 2 1 i γ 1 Δ 2 2 π i α z k 0 Δ 2 ] = 1 1 i γ 1 2 σ 2 .
σ μ = ( 1 Δ 2 + 1 2 σ 2 ) + ( 4 π 2 α 2 k 0 2 Δ 2 + 1 k 0 2 Δ 4 ) z 2 1 4 Δ 4 + π 2 α 2 .
( x + i y ) = ( 1 + i γ ) ( x 0 + i y 0 ) ,
r = 1 + γ 2 r 0 .
t ¯ = m r 0 < b / 1 + γ 2 P ( r 0 ) d 2 r 0 .
t ¯ = m { 1 exp [ b 2 / 2 σ 2 ( 1 + γ 2 ) ] } .
W ^ ( r 1 , r 2 ) = k = 0 m a k [ G ˜ η ˜ 1 H η ˜ 2 ] m k [ G η 2 H ˜ η 1 ] m k ,
G A 2 1 + i γ 1 2 Δ 2 β π α β ,
H 1 2 Δ 2 β π α β .
E ( r ) = W ^ ( r , r ) .
E ( r ) = 1 A 2 m + 2 k = 0 m ( m k ) 2 Γ ( k + 1 ) [ | J | 2 r 2 A 2 ] m k ,
J A 2 1 i γ i z k 0 Δ 4 | β | 2 π α | β | 2 .
L z ( r , z ) = ε 0 k 0 Im { ϕ 2 W ( r 1 , r 2 ) } r 1 = r 2 = r .
L z ( r , z ) = ε 0 k C ( z ) exp [ r 2 / σ S 2 ] { 2 π α A 2 | β | 2 Δ 2 k = 0 m a k | Q | 2 ( m k ) r 2 ( m k + 1 ) + R k = 0 m a k ( m k ) | Q | 2 ( m k 1 ) r 2 ( m k ) } ,
l z = ω L z ( r , z ) d 2 r S z ( r , z ) d 2 r ,
S z ( r , ω ) = k μ 0 ω C ( z ) exp [ r 2 / σ S 2 ] k = 0 m a k | Q | 2 ( m k ) r 2 ( m k ) .
l z = { m + 4 π α σ 2 [ 1 + m Δ 2 2 σ 2 1 + Δ 2 2 σ 2 ] } .
m z = ω L z ( r , z ) S z ( r , z ) .
m z { 2 π α A 2 Δ 2 r 2 + a m 1 a m R r 2 } .
U ( r , z ) = i k 0 2 π z U 0 ( r ) exp [ i k 0 2 z | r r | 2 ] d 2 r ,
U 0 ( r ) = U 0 Δ m exp [ ( x 2 + y 2 ) / 2 Δ 2 ] exp [ 2 π i α ( x 0 y y 0 x ) ] [ x + i y ] m .
U ( r , z ) = S ( x , y , z ) exp [ p ( x 2 + y 2 ) ] exp [ i ( q x x + q y y ) ] [ x + i y ] m d x d y ,
p 1 2 Δ 2 i k 0 2 z ,
q x 2 π α y 0 k 0 z x ,
q y 2 π α x 0 k 0 z y ,
S ( x , y , z ) i k 0 2 π z U 0 Δ m exp [ i k 0 z ] exp [ i k 0 2 z ( x 2 + y 2 ) ] .
U ( r , z ) = 2 π i m S ( x , y , z ) ( q x + i q y ) m ( 2 p ) m + 1 exp [ q x 2 + q y 2 4 p ] .
U ( r , z ) = U 0 β e i k 0 z [ η ( 1 + i γ ) η 0 ] m ( Δ β ) m exp [ | η η 0 | 2 / 2 Δ 2 β ] × exp [ i 2 π 2 α 2 z β k 0 | η 0 | 2 ] exp [ π α ( η η ˜ 0 η ˜ η 0 ) / β ] .
W ( r 1 , r 2 , z ) = P 0 | U 0 | 2 | β | 2 1 | Δ β | 2 m exp [ | η 1 | 2 / 2 Δ 2 β ˜ ] exp [ | η 2 | 2 / 2 Δ 2 β ] exp [ A 2 | η 0 | 2 ] exp [ A B ˜ η 0 + A B η ˜ 0 ] [ η ˜ 1 ( 1 i γ ) η ˜ 0 ] m [ η 2 ( 1 + i γ ) η 0 ] m d 2 r 0 ,
A B = η 1 2 Δ 2 β ˜ + η 2 2 Δ 2 β π α η 1 β ˜ + π α η 2 β ,
A B ˜ = η ˜ 1 2 Δ 2 β ˜ + η ˜ 2 2 Δ 2 β + π α η ˜ 1 β ˜ π α η ˜ 2 β .
W ( r 1 , r 2 , z ) = P 0 | U 0 | 2 | β | 2 1 | Δ β | 2 m exp [ | η 1 | 2 / 2 Δ 2 β ˜ ] exp [ | η 2 | 2 / 2 Δ 2 β ] exp [ B B ˜ ] × exp [ ( A η 0 B ) ( A η ˜ 0 B ˜ ) ] × [ η ˜ 1 ( 1 i γ ) η ˜ 0 ] m [ η 2 ( 1 + i γ ) η 0 ] m d 2 r 0 .
η 0 η 0 + B / A , η ˜ 0 η ˜ 0 + B ˜ / A .
W ( r 1 , r 2 , z ) = 1 π ( 1 + γ 2 ) m C ( z ) F ( r 1 , r 2 , z ) A 4 m exp [ A 2 | η 0 | 2 ] × [ η ˜ 1 ( 1 i γ ) ( η ˜ 0 + B ˜ / A ) ] m [ η 2 ( 1 + i γ ) ( η 0 + B / A ) ] m d 2 r 0 .
W ( r 1 , r 2 , z ) = 1 π C ( z ) F ( r 1 , r 2 , z ) exp [ A 2 | η 0 | 2 ] [ D 1 A 2 η ˜ 0 ] m [ D 2 A 2 η 0 ] m d 2 r 0 .
[ D 1 A 2 η ˜ 0 ] m = l = 0 m ( m l ) ( D 1 ) m l ( A 2 η ˜ 0 ) l .
W ( r 1 , r 2 , z ) = 2 C ( z ) F ( r 1 , r 2 , z ) k = 0 m ( m k ) 2 ( D 1 D 2 ) m k A 4 m × 0 exp [ A 2 ρ 0 2 ] ( A 4 ρ 0 2 ) k ρ 0 d ρ 0 .
0 exp [ u 2 ] u 2 k + 1 d u = Γ ( k + 1 ) 2 .