Abstract

Singular beams have circulating energy components. When such beams are focused by low numerical aperture systems suffering from astigmatic aberration, these circulating energy components get modified. The phase gradient introduced by this type of aberration splits the higher charge vortices. The dependence of the charge, the aberration coefficient, and the size of the aperture on the nature of the splitting process are reported in this paper. The transverse components of the Poynting vector fields that can be derived from the phase gradient vector field distributions are further decomposed into solenoidal and irrotational components using the Helmholtz–Hodge decomposition method. The solenoidal components relate to the orbital angular momentum of the beams, and the irrotational components are useful in the transport of intensity equations for phase retrieval.

© 2014 Optical Society of America

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  1. V. V. Kotlyar, A. A. Kovalev, R. V. Skidanov, O. Y. Moiseev, and V. A. Soifer, “Diffraction of a finite-radius plane wave and a Gaussian beam by a helical axicon and a spiral phase plate,” J. Opt. Soc. Am. A 24, 1955–1964 (2007).
    [CrossRef]
  2. V. V. Kotlyar, A. A. Kovalev, S. N. Khonina, R. V. Skidanov, V. A. Soifer, H. Elfstrom, N. Tossavainen, and J. Turunen, “Diffraction of conic and Gaussian beams by a spiral phase plate,” J. Opt. A Pure Appl. Opt. 45, 2656–2665 (2006).
  3. R. K. Singh, P. Senthilkumaran, and K. Singh, “Influence of astigmatism and defocusing on the focusing of a singular beam,” Opt. Commun. 270, 128–138 (2007).
    [CrossRef]
  4. R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of astigmatism on the diffraction of a vortex-carrying beam with Gaussian background,” J. Opt. A Pure Appl. Opt. 9, 543–554 (2007).
    [CrossRef]
  5. R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of a vortex carrying beam with Gaussian background by an apertured system in presence of coma,” Opt. Commun. 281, 923–934 (2008).
    [CrossRef]
  6. R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of coma on the focusing of an apertured singular beam,” Opt. Lasers Eng. 45, 488–494 (2007).
    [CrossRef]
  7. A. Y. Bekshaev and M. S. Soskin, “Transverse energy flows in vectorial fields of paraxial beams with singularities,” Opt. Commun. 271, 332–348 (2007).
    [CrossRef]
  8. A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13, 053001 (2011).
    [CrossRef]
  9. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
    [CrossRef]
  10. A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004).
    [CrossRef]
  11. A. Y. Bekshaev and A. I. Karamoch, “Astigmatic telescopic transformation of a high-order optical vortex,” Opt. Commun. 281, 5687–5696 (2008).
    [CrossRef]
  12. E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
    [CrossRef]
  13. S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, P. Paakkonen, and J. Turunen, “Astigmatic Bessel laser beams,” J. Mod. Opt. 51, 677–686 (2004).
    [CrossRef]
  14. V. V. Kotlyar, S. N. Khonina, A. A. Almazov, V. A. Soifer, K. Jefimovs, and J. Turunen, “Elliptic Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 23, 43–56 (2006).
    [CrossRef]
  15. P. Vaity, J. Banerji, and R. P. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013).
    [CrossRef]
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    [CrossRef]
  17. A. M. Stewart, “Angular momentum of light,” J. Mod. Opt. 52, 1145–1154 (2005).
    [CrossRef]
  18. S. F. Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
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  22. D. J. Griffiths, Introduction to Electrodynamics (Prentice-Hall, 1999).
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  24. K. Polthier and E. Preuss, “Identifying vector field singularities using a discrete Hodge decomposition,” in Visualization and Mathematics III, H. C. Hege and K. Polthier, eds. (Springer-Verlag, 2002), pp. 113–134.
  25. F. M. Denaro, “On the application of the Helmholtz–Hodge decomposition in projection methods for incompressible flows with general boundary conditions,” Int. J. Numer. Methods Fluids 43, 43–69 (2003).
    [CrossRef]
  26. H. Bhatia, G. Norgard, V. Pascucci, and P. T. Bremer, “The Helmholtz–Hodge decomposition: a survey,” IEEE Trans. Vis. Comput. Graph. 19, 1386–1404 (2013).
  27. M. Bahl and P. Senthilkumaran, “Helmholtz Hodge decomposition of scalar optical fields,” J. Opt. Soc. Am. A 29, 2421–2427 (2012).
    [CrossRef]
  28. P. Senthilkumaran, F. Wyrowski, and H. Schimmel, “Vortex stagnation problem in iterative Fourier transform algorithms,” Opt. Lasers Eng. 43, 43–56 (2005).
    [CrossRef]
  29. P. Senthilkumaran and F. Wyrowski, “Phase synthesis in wave-optical engineering: mapping and diffuser-type approaches,” J. Mod. Opt. 49, 1831–1850 (2002).
    [CrossRef]
  30. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).
  31. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
    [CrossRef]
  32. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
    [CrossRef]
  33. J. O. Castaneda and G. Ramirez, “Zone plates for zero axial irradiance,” Opt. Lett. 18, 87–89 (1993).
    [CrossRef]
  34. L. E. Helsenth, “Smallest focal hole,” Opt. Commun. 257, 1–8 (2006).
    [CrossRef]
  35. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
    [CrossRef]

2013

P. Vaity, J. Banerji, and R. P. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013).
[CrossRef]

J. Petruccelli, L. Tian, and G. Barbastathis, “The transport of intensity equation for optical path length recovery using partially coherent illumination,” Opt. Express 21, 14430–14441 (2013).
[CrossRef]

H. Bhatia, G. Norgard, V. Pascucci, and P. T. Bremer, “The Helmholtz–Hodge decomposition: a survey,” IEEE Trans. Vis. Comput. Graph. 19, 1386–1404 (2013).

2012

2011

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).
[CrossRef]

A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13, 053001 (2011).
[CrossRef]

2008

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of a vortex carrying beam with Gaussian background by an apertured system in presence of coma,” Opt. Commun. 281, 923–934 (2008).
[CrossRef]

A. Y. Bekshaev and A. I. Karamoch, “Astigmatic telescopic transformation of a high-order optical vortex,” Opt. Commun. 281, 5687–5696 (2008).
[CrossRef]

S. F. Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
[CrossRef]

2007

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

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of coma on the focusing of an apertured singular beam,” Opt. Lasers Eng. 45, 488–494 (2007).
[CrossRef]

A. Y. Bekshaev and M. S. Soskin, “Transverse energy flows in vectorial fields of paraxial beams with singularities,” Opt. Commun. 271, 332–348 (2007).
[CrossRef]

V. V. Kotlyar, A. A. Kovalev, R. V. Skidanov, O. Y. Moiseev, and V. A. Soifer, “Diffraction of a finite-radius plane wave and a Gaussian beam by a helical axicon and a spiral phase plate,” J. Opt. Soc. Am. A 24, 1955–1964 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Influence of astigmatism and defocusing on the focusing of a singular beam,” Opt. Commun. 270, 128–138 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of astigmatism on the diffraction of a vortex-carrying beam with Gaussian background,” J. Opt. A Pure Appl. Opt. 9, 543–554 (2007).
[CrossRef]

2006

V. V. Kotlyar, A. A. Kovalev, S. N. Khonina, R. V. Skidanov, V. A. Soifer, H. Elfstrom, N. Tossavainen, and J. Turunen, “Diffraction of conic and Gaussian beams by a spiral phase plate,” J. Opt. A Pure Appl. Opt. 45, 2656–2665 (2006).

V. V. Kotlyar, S. N. Khonina, A. A. Almazov, V. A. Soifer, K. Jefimovs, and J. Turunen, “Elliptic Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 23, 43–56 (2006).
[CrossRef]

L. E. Helsenth, “Smallest focal hole,” Opt. Commun. 257, 1–8 (2006).
[CrossRef]

2005

P. Senthilkumaran, F. Wyrowski, and H. Schimmel, “Vortex stagnation problem in iterative Fourier transform algorithms,” Opt. Lasers Eng. 43, 43–56 (2005).
[CrossRef]

A. M. Stewart, “Angular momentum of light,” J. Mod. Opt. 52, 1145–1154 (2005).
[CrossRef]

2004

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, P. Paakkonen, and J. Turunen, “Astigmatic Bessel laser beams,” J. Mod. Opt. 51, 677–686 (2004).
[CrossRef]

2003

F. M. Denaro, “On the application of the Helmholtz–Hodge decomposition in projection methods for incompressible flows with general boundary conditions,” Int. J. Numer. Methods Fluids 43, 43–69 (2003).
[CrossRef]

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

2002

P. Senthilkumaran and F. Wyrowski, “Phase synthesis in wave-optical engineering: mapping and diffuser-type approaches,” J. Mod. Opt. 49, 1831–1850 (2002).
[CrossRef]

1997

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

1993

J. O. Castaneda and G. Ramirez, “Zone plates for zero axial irradiance,” Opt. Lett. 18, 87–89 (1993).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

1992

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]

1991

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

1983

Abramochkin, E.

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

Allen, L.

S. F. Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[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]

Almazov, A. A.

Arfken, G.

H. Weber and G. Arfken, Essential Mathematical Methods for Physicists (Academic, 2003).

Arnold, S. F.

S. F. Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
[CrossRef]

Bahl, M.

Banerji, J.

P. Vaity, J. Banerji, and R. P. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013).
[CrossRef]

Barbastathis, G.

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[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]

Bekshaev, A.

A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13, 053001 (2011).
[CrossRef]

Bekshaev, A. Y.

A. Y. Bekshaev and A. I. Karamoch, “Astigmatic telescopic transformation of a high-order optical vortex,” Opt. Commun. 281, 5687–5696 (2008).
[CrossRef]

A. Y. Bekshaev and M. S. Soskin, “Transverse energy flows in vectorial fields of paraxial beams with singularities,” Opt. Commun. 271, 332–348 (2007).
[CrossRef]

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004).
[CrossRef]

Bhatia, H.

H. Bhatia, G. Norgard, V. Pascucci, and P. T. Bremer, “The Helmholtz–Hodge decomposition: a survey,” IEEE Trans. Vis. Comput. Graph. 19, 1386–1404 (2013).

Bliokh, K.

A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13, 053001 (2011).
[CrossRef]

Bremer, P. T.

H. Bhatia, G. Norgard, V. Pascucci, and P. T. Bremer, “The Helmholtz–Hodge decomposition: a survey,” IEEE Trans. Vis. Comput. Graph. 19, 1386–1404 (2013).

Castaneda, J. O.

Denaro, F. M.

F. M. Denaro, “On the application of the Helmholtz–Hodge decomposition in projection methods for incompressible flows with general boundary conditions,” Int. J. Numer. Methods Fluids 43, 43–69 (2003).
[CrossRef]

Elfstrom, H.

V. V. Kotlyar, A. A. Kovalev, S. N. Khonina, R. V. Skidanov, V. A. Soifer, H. Elfstrom, N. Tossavainen, and J. Turunen, “Diffraction of conic and Gaussian beams by a spiral phase plate,” J. Opt. A Pure Appl. Opt. 45, 2656–2665 (2006).

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Grier, D. G.

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

Griffiths, D. J.

D. J. Griffiths, Introduction to Electrodynamics (Prentice-Hall, 1999).

Helsenth, L. E.

L. E. Helsenth, “Smallest focal hole,” Opt. Commun. 257, 1–8 (2006).
[CrossRef]

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Jefimovs, K.

V. V. Kotlyar, S. N. Khonina, A. A. Almazov, V. A. Soifer, K. Jefimovs, and J. Turunen, “Elliptic Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 23, 43–56 (2006).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, P. Paakkonen, and J. Turunen, “Astigmatic Bessel laser beams,” J. Mod. Opt. 51, 677–686 (2004).
[CrossRef]

Karamoch, A. I.

A. Y. Bekshaev and A. I. Karamoch, “Astigmatic telescopic transformation of a high-order optical vortex,” Opt. Commun. 281, 5687–5696 (2008).
[CrossRef]

Khonina, S. N.

V. V. Kotlyar, A. A. Kovalev, S. N. Khonina, R. V. Skidanov, V. A. Soifer, H. Elfstrom, N. Tossavainen, and J. Turunen, “Diffraction of conic and Gaussian beams by a spiral phase plate,” J. Opt. A Pure Appl. Opt. 45, 2656–2665 (2006).

V. V. Kotlyar, S. N. Khonina, A. A. Almazov, V. A. Soifer, K. Jefimovs, and J. Turunen, “Elliptic Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 23, 43–56 (2006).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, P. Paakkonen, and J. Turunen, “Astigmatic Bessel laser beams,” J. Mod. Opt. 51, 677–686 (2004).
[CrossRef]

Kotlyar, V. V.

V. V. Kotlyar, A. A. Kovalev, R. V. Skidanov, O. Y. Moiseev, and V. A. Soifer, “Diffraction of a finite-radius plane wave and a Gaussian beam by a helical axicon and a spiral phase plate,” J. Opt. Soc. Am. A 24, 1955–1964 (2007).
[CrossRef]

V. V. Kotlyar, A. A. Kovalev, S. N. Khonina, R. V. Skidanov, V. A. Soifer, H. Elfstrom, N. Tossavainen, and J. Turunen, “Diffraction of conic and Gaussian beams by a spiral phase plate,” J. Opt. A Pure Appl. Opt. 45, 2656–2665 (2006).

V. V. Kotlyar, S. N. Khonina, A. A. Almazov, V. A. Soifer, K. Jefimovs, and J. Turunen, “Elliptic Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 23, 43–56 (2006).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, P. Paakkonen, and J. Turunen, “Astigmatic Bessel laser beams,” J. Mod. Opt. 51, 677–686 (2004).
[CrossRef]

Kovalev, A. A.

V. V. Kotlyar, A. A. Kovalev, R. V. Skidanov, O. Y. Moiseev, and V. A. Soifer, “Diffraction of a finite-radius plane wave and a Gaussian beam by a helical axicon and a spiral phase plate,” J. Opt. Soc. Am. A 24, 1955–1964 (2007).
[CrossRef]

V. V. Kotlyar, A. A. Kovalev, S. N. Khonina, R. V. Skidanov, V. A. Soifer, H. Elfstrom, N. Tossavainen, and J. Turunen, “Diffraction of conic and Gaussian beams by a spiral phase plate,” J. Opt. A Pure Appl. Opt. 45, 2656–2665 (2006).

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Moiseev, O. Y.

Molina-Terriza, G.

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

Norgard, G.

H. Bhatia, G. Norgard, V. Pascucci, and P. T. Bremer, “The Helmholtz–Hodge decomposition: a survey,” IEEE Trans. Vis. Comput. Graph. 19, 1386–1404 (2013).

Paakkonen, P.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, P. Paakkonen, and J. Turunen, “Astigmatic Bessel laser beams,” J. Mod. Opt. 51, 677–686 (2004).
[CrossRef]

Padgett, M.

S. F. Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
[CrossRef]

Padgett, M. J.

Pascucci, V.

H. Bhatia, G. Norgard, V. Pascucci, and P. T. Bremer, “The Helmholtz–Hodge decomposition: a survey,” IEEE Trans. Vis. Comput. Graph. 19, 1386–1404 (2013).

Petruccelli, J.

Polthier, K.

K. Polthier and E. Preuss, “Identifying vector field singularities using a discrete Hodge decomposition,” in Visualization and Mathematics III, H. C. Hege and K. Polthier, eds. (Springer-Verlag, 2002), pp. 113–134.

Preuss, E.

K. Polthier and E. Preuss, “Identifying vector field singularities using a discrete Hodge decomposition,” in Visualization and Mathematics III, H. C. Hege and K. Polthier, eds. (Springer-Verlag, 2002), pp. 113–134.

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Ramirez, G.

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Schimmel, H.

P. Senthilkumaran, F. Wyrowski, and H. Schimmel, “Vortex stagnation problem in iterative Fourier transform algorithms,” Opt. Lasers Eng. 43, 43–56 (2005).
[CrossRef]

Senthilkumaran, P.

M. Bahl and P. Senthilkumaran, “Helmholtz Hodge decomposition of scalar optical fields,” J. Opt. Soc. Am. A 29, 2421–2427 (2012).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of a vortex carrying beam with Gaussian background by an apertured system in presence of coma,” Opt. Commun. 281, 923–934 (2008).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of astigmatism on the diffraction of a vortex-carrying beam with Gaussian background,” J. Opt. A Pure Appl. Opt. 9, 543–554 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Influence of astigmatism and defocusing on the focusing of a singular beam,” Opt. Commun. 270, 128–138 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of coma on the focusing of an apertured singular beam,” Opt. Lasers Eng. 45, 488–494 (2007).
[CrossRef]

P. Senthilkumaran, F. Wyrowski, and H. Schimmel, “Vortex stagnation problem in iterative Fourier transform algorithms,” Opt. Lasers Eng. 43, 43–56 (2005).
[CrossRef]

P. Senthilkumaran and F. Wyrowski, “Phase synthesis in wave-optical engineering: mapping and diffuser-type approaches,” J. Mod. Opt. 49, 1831–1850 (2002).
[CrossRef]

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Singh, K.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of a vortex carrying beam with Gaussian background by an apertured system in presence of coma,” Opt. Commun. 281, 923–934 (2008).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of astigmatism on the diffraction of a vortex-carrying beam with Gaussian background,” J. Opt. A Pure Appl. Opt. 9, 543–554 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Influence of astigmatism and defocusing on the focusing of a singular beam,” Opt. Commun. 270, 128–138 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of coma on the focusing of an apertured singular beam,” Opt. Lasers Eng. 45, 488–494 (2007).
[CrossRef]

Singh, R. K.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of a vortex carrying beam with Gaussian background by an apertured system in presence of coma,” Opt. Commun. 281, 923–934 (2008).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of coma on the focusing of an apertured singular beam,” Opt. Lasers Eng. 45, 488–494 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of astigmatism on the diffraction of a vortex-carrying beam with Gaussian background,” J. Opt. A Pure Appl. Opt. 9, 543–554 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Influence of astigmatism and defocusing on the focusing of a singular beam,” Opt. Commun. 270, 128–138 (2007).
[CrossRef]

Singh, R. P.

P. Vaity, J. Banerji, and R. P. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013).
[CrossRef]

Skidanov, R. V.

V. V. Kotlyar, A. A. Kovalev, R. V. Skidanov, O. Y. Moiseev, and V. A. Soifer, “Diffraction of a finite-radius plane wave and a Gaussian beam by a helical axicon and a spiral phase plate,” J. Opt. Soc. Am. A 24, 1955–1964 (2007).
[CrossRef]

V. V. Kotlyar, A. A. Kovalev, S. N. Khonina, R. V. Skidanov, V. A. Soifer, H. Elfstrom, N. Tossavainen, and J. Turunen, “Diffraction of conic and Gaussian beams by a spiral phase plate,” J. Opt. A Pure Appl. Opt. 45, 2656–2665 (2006).

Soifer, V. A.

V. V. Kotlyar, A. A. Kovalev, R. V. Skidanov, O. Y. Moiseev, and V. A. Soifer, “Diffraction of a finite-radius plane wave and a Gaussian beam by a helical axicon and a spiral phase plate,” J. Opt. Soc. Am. A 24, 1955–1964 (2007).
[CrossRef]

V. V. Kotlyar, A. A. Kovalev, S. N. Khonina, R. V. Skidanov, V. A. Soifer, H. Elfstrom, N. Tossavainen, and J. Turunen, “Diffraction of conic and Gaussian beams by a spiral phase plate,” J. Opt. A Pure Appl. Opt. 45, 2656–2665 (2006).

V. V. Kotlyar, S. N. Khonina, A. A. Almazov, V. A. Soifer, K. Jefimovs, and J. Turunen, “Elliptic Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 23, 43–56 (2006).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, P. Paakkonen, and J. Turunen, “Astigmatic Bessel laser beams,” J. Mod. Opt. 51, 677–686 (2004).
[CrossRef]

Soskin, M.

A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13, 053001 (2011).
[CrossRef]

Soskin, M. S.

A. Y. Bekshaev and M. S. Soskin, “Transverse energy flows in vectorial fields of paraxial beams with singularities,” Opt. Commun. 271, 332–348 (2007).
[CrossRef]

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004).
[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]

Stewart, A. M.

A. M. Stewart, “Angular momentum of light,” J. Mod. Opt. 52, 1145–1154 (2005).
[CrossRef]

Teague, M. R.

Tian, L.

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Torner, L.

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

Torres, J. P.

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

Tossavainen, N.

V. V. Kotlyar, A. A. Kovalev, S. N. Khonina, R. V. Skidanov, V. A. Soifer, H. Elfstrom, N. Tossavainen, and J. Turunen, “Diffraction of conic and Gaussian beams by a spiral phase plate,” J. Opt. A Pure Appl. Opt. 45, 2656–2665 (2006).

Turunen, J.

V. V. Kotlyar, A. A. Kovalev, S. N. Khonina, R. V. Skidanov, V. A. Soifer, H. Elfstrom, N. Tossavainen, and J. Turunen, “Diffraction of conic and Gaussian beams by a spiral phase plate,” J. Opt. A Pure Appl. Opt. 45, 2656–2665 (2006).

V. V. Kotlyar, S. N. Khonina, A. A. Almazov, V. A. Soifer, K. Jefimovs, and J. Turunen, “Elliptic Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 23, 43–56 (2006).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, P. Paakkonen, and J. Turunen, “Astigmatic Bessel laser beams,” J. Mod. Opt. 51, 677–686 (2004).
[CrossRef]

Vaity, P.

P. Vaity, J. Banerji, and R. P. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013).
[CrossRef]

van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Vasnetsov, M. V.

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004).
[CrossRef]

Volostnikov, V.

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

Weber, H.

H. Weber and G. Arfken, Essential Mathematical Methods for Physicists (Academic, 2003).

Woerdman, J. P.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[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]

Wyrowski, F.

P. Senthilkumaran, F. Wyrowski, and H. Schimmel, “Vortex stagnation problem in iterative Fourier transform algorithms,” Opt. Lasers Eng. 43, 43–56 (2005).
[CrossRef]

P. Senthilkumaran and F. Wyrowski, “Phase synthesis in wave-optical engineering: mapping and diffuser-type approaches,” J. Mod. Opt. 49, 1831–1850 (2002).
[CrossRef]

Yao, A. M.

Adv. Opt. Photon.

IEEE Trans. Vis. Comput. Graph.

H. Bhatia, G. Norgard, V. Pascucci, and P. T. Bremer, “The Helmholtz–Hodge decomposition: a survey,” IEEE Trans. Vis. Comput. Graph. 19, 1386–1404 (2013).

Int. J. Numer. Methods Fluids

F. M. Denaro, “On the application of the Helmholtz–Hodge decomposition in projection methods for incompressible flows with general boundary conditions,” Int. J. Numer. Methods Fluids 43, 43–69 (2003).
[CrossRef]

J. Mod. Opt.

P. Senthilkumaran and F. Wyrowski, “Phase synthesis in wave-optical engineering: mapping and diffuser-type approaches,” J. Mod. Opt. 49, 1831–1850 (2002).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, P. Paakkonen, and J. Turunen, “Astigmatic Bessel laser beams,” J. Mod. Opt. 51, 677–686 (2004).
[CrossRef]

A. M. Stewart, “Angular momentum of light,” J. Mod. Opt. 52, 1145–1154 (2005).
[CrossRef]

J. Opt.

A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13, 053001 (2011).
[CrossRef]

J. Opt. A Pure Appl. Opt.

V. V. Kotlyar, A. A. Kovalev, S. N. Khonina, R. V. Skidanov, V. A. Soifer, H. Elfstrom, N. Tossavainen, and J. Turunen, “Diffraction of conic and Gaussian beams by a spiral phase plate,” J. Opt. A Pure Appl. Opt. 45, 2656–2665 (2006).

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of astigmatism on the diffraction of a vortex-carrying beam with Gaussian background,” J. Opt. A Pure Appl. Opt. 9, 543–554 (2007).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Laser Photon. Rev.

S. F. Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
[CrossRef]

Nat. Phys.

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

Nature

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

Opt. Commun.

L. E. Helsenth, “Smallest focal hole,” Opt. Commun. 257, 1–8 (2006).
[CrossRef]

A. Y. Bekshaev and M. S. Soskin, “Transverse energy flows in vectorial fields of paraxial beams with singularities,” Opt. Commun. 271, 332–348 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of a vortex carrying beam with Gaussian background by an apertured system in presence of coma,” Opt. Commun. 281, 923–934 (2008).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Influence of astigmatism and defocusing on the focusing of a singular beam,” Opt. Commun. 270, 128–138 (2007).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004).
[CrossRef]

A. Y. Bekshaev and A. I. Karamoch, “Astigmatic telescopic transformation of a high-order optical vortex,” Opt. Commun. 281, 5687–5696 (2008).
[CrossRef]

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

P. Senthilkumaran, F. Wyrowski, and H. Schimmel, “Vortex stagnation problem in iterative Fourier transform algorithms,” Opt. Lasers Eng. 43, 43–56 (2005).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of coma on the focusing of an apertured singular beam,” Opt. Lasers Eng. 45, 488–494 (2007).
[CrossRef]

Opt. Lett.

Phys. Lett. A

P. Vaity, J. Banerji, and R. P. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013).
[CrossRef]

Phys. Rev. A

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]

Phys. Rev. Lett.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Other

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

D. J. Griffiths, Introduction to Electrodynamics (Prentice-Hall, 1999).

H. Weber and G. Arfken, Essential Mathematical Methods for Physicists (Academic, 2003).

K. Polthier and E. Preuss, “Identifying vector field singularities using a discrete Hodge decomposition,” in Visualization and Mathematics III, H. C. Hege and K. Polthier, eds. (Springer-Verlag, 2002), pp. 113–134.

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Figures (11)

Fig. 1.
Fig. 1.

3D phase profile of (a) vortex field (m=3) and (b) astigmatic aberration field (Aa=1).

Fig. 2.
Fig. 2.

(a) Phase profile and (b)–(e) phase gradient field streamlines for a circular aperture limited vortex beam when astigmatism is added: (b) Aa=0; (c) Aa=0.5; (d) Aa=1.0; and (e) Aa=1.5.

Fig. 3.
Fig. 3.

Coordinate system used in evaluating Fresnel–Kirchoff diffraction integral.

Fig. 4.
Fig. 4.

(a) Transverse phase profile. (b) Intensity distribution at the focal plane for the astigmatically aberrated vortex field.

Fig. 5.
Fig. 5.

Helmholtz–-Hodge decomposition of focused astigmatically aberrated vortex field (m=3). (a) Solenoidal component and (b) irrotational component.

Fig. 6.
Fig. 6.

Dependence of the nature of splitting on the charge of the input beam. (a) Phase profile of a vortex beam at the exit pupil of a lens. (b) Intensity profile at the focal plane evaluated using the Fresnel–Kirchoff diffraction integral. (Inset) Phase profile at the focal plane. (c) Hodge solenoidal component that shows optical currents swirling counterclockwise about the singular points, and (d) irrotational component, for (I) m=4 and (II) m=5 astigmatically aberrated vortex beams.

Fig. 7.
Fig. 7.

Dependence of the nature of splitting on the polarity of the input beam. An m=3 astigmatically aberrated vortex beam. (a) Phase profile of the beam at the exit pupil of a lens. (b) Intensity profile at the focal plane evaluated using the Fresnel–Kirchoff diffraction integral. (Inset) Phase profile at the focal plane. (c) Hodge solenoidal component that shows optical currents swirling counterclockwise about three singular points. (d) Irrotational component.

Fig. 8.
Fig. 8.

Dependence of the nature of splitting on the magnitude of the aberration coefficient for an astigmatically aberrated vortex beam, m=3. (a) Phase profile of a vortex beam at the exit pupil of a lens. (b) Intensity profile at the focal plane evaluated using the Fresnel–Kirchoff diffraction integral. (Inset) Phase profile at the focal plane. (c) Hodge solenoidal component that shows optical currents swirling clockwise about singular points, and (d) irrotational component for (I) Aa=0.025, (II) Aa=0.5, (III) Aa=1.0, and (IV) Aa=1.5.

Fig. 9.
Fig. 9.

Dependence of the nature of splitting on the polarity of the aberration coefficient. (a) Phase profile of a vortex beam at the exit pupil of a lens. (b) Intensity profile at the focal plane evaluated using the Fresnel–Kirchoff diffraction integral. (Inset) Phase profile at the focal plane. (c) Hodge solenoidal component that shows optical currents swirling clockwise about three singular points, and (d) irrotational component, for m=3 astigmatically aberrated vortex beam with (I) Aa=0.5 and (II) Aa=0.5.

Fig. 10.
Fig. 10.

Dependence of the nature of splitting on the size of the aperture. (a) Phase profile of a vortex beam at the exit pupil of a lens. (b) Intensity profile at the focal plane evaluated using the Fresnel–Kirchoff diffraction integral. (Inset) Phase profile at the focal plane. (c) Hodge solenoidal component that shows optical currents swirling clockwise about three singular points, and (d) irrotational component for an m=3 astigmatically aberrated vortex beam with Ac=0.5 and aperture size as (I) 0.14, (II) 0.27, (III) 0.78, and (IV) 1.

Fig. 11.
Fig. 11.

Dependence of the nature of splitting on the size of the aperture. (a) Phase profile of a vortex beam at the exit pupil of a lens. (b) Intensity profile at the focal plane evaluated using the Fresnel–Kirchoff diffraction integral. (Inset) Phase profile at the focal plane. (c) Hodge solenoidal component that shows optical currents swirling clockwise about singular points, and (d) irrotational component for an m=4 astigmatically aberrated vortex beam with Ac=0.5 and aperture size (I) 0.14 and (II) 0.78.

Equations (6)

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

U(r,ϕ,z)=0ρ002πAexp(imθ)exp(i(2π/λ)×Aaρ2cos2(θ))×exp(i2πrρcos(θϕ))ρdρdθ.
F⃗=f⃗1+f⃗2∇⃗φ+∇⃗×A⃗,
φ(r)=14πVb(r)rdτ,
A⃗(r)=14πVc⃗(r)rdτ.
·(I∇⃗Tψ)=kzI,
·(S⃗T)=kzI.

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