Abstract

It is shown that the vector field decomposition method, namely, the Helmholtz Hodge decomposition, can also be applied to analyze scalar optical fields that are ubiquitously present in interference and diffraction optics. A phase gradient field that depicts the propagation and Poynting vector directions can hence be separated into solenoidal and irrotational components.

© 2012 Optical Society of America

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References

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  1. A. Globus, C. Levit, and T. Lasinki, “A tool for visualizing the topology of three-dimensional vector fields,” in Proceedings of IEEE on Visualization (IEEE, 1991), pp. 33–40.
  2. 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.
  3. Y. Tong, S. Lombeyda, A. N. Hirani, and M. Desbrun, “Discrete multiscale vector field decomposition,” ACM Trans. Graph. 22, 445–452 (2003).
    [CrossRef]
  4. F. Petronetto, A. Paiva, M. Lage, G. Tavares, H. Lopes, and T. Lewiner, “Meshless Helmholtz-Hodge decomposition,” IEEE Trans. Vis. Comput. Graphics 16, 338–342 (2010).
    [CrossRef]
  5. 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]
  6. I. Kaya, A. P. Santhanam, C. Imielinska, and J. Rolland, “Modeling air-flow in the tracheobronchial tree using computational fluid dynamics,” in Proceedings of 2007 MICCAI Workshop on Computational Biomechanics (2007), pp. 142–151.
  7. R. Fedkiw, J. Stam, and H. W. Jensen, “Visual simulation of smoke,” in Proceedings of ACM SIGGRAPH 2001, Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, Eugene Fiume, ed. (Association for Computing Machinery, 2001), pp. 15–22.
  8. A. M. Stewart, “Angular momentum of light,” J. Mod. Opt. 52, 1145–1154 (2005).
    [CrossRef]
  9. M. Hattori and S. Komatsu, “An exact formulation of a filter for rotation in phase gradients and its applications to wavefront reconstruction problems,” J. Mod. Opt. 50, 1705–1723(2003).
    [CrossRef]
  10. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
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  11. M. Berry, “Optical currents,” J. Opt. A 11, 1464–1475 (2009).
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  13. L. Allen and M. J. Padgett, “The Poynting vector in Laguerre Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
    [CrossRef]
  14. 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]
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    [CrossRef]
  19. I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
    [CrossRef]
  20. I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
    [CrossRef]
  21. P. Senthilkumaran, F. Wyrowski, and H. Schimmel, “Vortex stagnation problem in iterative Fourier transform algorithms,” Opt. Lasers Eng. 43, 43–56 (2005).
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  22. S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt. 46, 2893–2898 (2007).
    [CrossRef]
  23. J. Xavier, S. Vyas, P. Senthilkumaran, and J. Joseph, “Tailored complex 3D vortex lattice structures by perturbed multiples of three-plane waves,” Appl. Opt. 51, 1872–1878 (2012).
    [CrossRef]
  24. D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, “Vortex lattice generation using interferometric techniques based on lateral shearing,” Opt. Commun. 282, 2692–2698 (2009).
    [CrossRef]
  25. J. P. Prenel and D. Ambrosini, “Flow visualization and beyond,” Opt. Lasers Eng. 50, 1–7 (2012).
    [CrossRef]
  26. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Elsevier, 2005).
  27. T. McGraw, T. Kawai, I. Yassine, and L. Zhu, “Visualizing high-order symmetric tensor field structure with differential operators,” J. Appl. Math. 2011, 1–27 (2011).
    [CrossRef]
  28. Finite difference method, http://en.wikipedia.org/wiki/Finite_difference_method .

2012 (2)

2011 (2)

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

T. McGraw, T. Kawai, I. Yassine, and L. Zhu, “Visualizing high-order symmetric tensor field structure with differential operators,” J. Appl. Math. 2011, 1–27 (2011).
[CrossRef]

2010 (1)

F. Petronetto, A. Paiva, M. Lage, G. Tavares, H. Lopes, and T. Lewiner, “Meshless Helmholtz-Hodge decomposition,” IEEE Trans. Vis. Comput. Graphics 16, 338–342 (2010).
[CrossRef]

2009 (2)

M. Berry, “Optical currents,” J. Opt. A 11, 1464–1475 (2009).
[CrossRef]

D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, “Vortex lattice generation using interferometric techniques based on lateral shearing,” Opt. Commun. 282, 2692–2698 (2009).
[CrossRef]

2008 (1)

S. F. Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–315(2008).
[CrossRef]

2007 (2)

S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt. 46, 2893–2898 (2007).
[CrossRef]

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

2005 (2)

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

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

2003 (3)

M. Hattori and S. Komatsu, “An exact formulation of a filter for rotation in phase gradients and its applications to wavefront reconstruction problems,” J. Mod. Opt. 50, 1705–1723(2003).
[CrossRef]

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]

Y. Tong, S. Lombeyda, A. N. Hirani, and M. Desbrun, “Discrete multiscale vector field decomposition,” ACM Trans. Graph. 22, 445–452 (2003).
[CrossRef]

2001 (1)

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

2000 (1)

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
[CrossRef]

1997 (1)

M. Padgett and L. Allen, “Optical tweezers and spanners,” Phys. World 10, 35–38 (1997).

1995 (1)

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

1993 (1)

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef]

Allen, L.

S. F. Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–315(2008).
[CrossRef]

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
[CrossRef]

M. Padgett and L. Allen, “Optical tweezers and spanners,” Phys. World 10, 35–38 (1997).

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]

Ambrosini, D.

J. P. Prenel and D. Ambrosini, “Flow visualization and beyond,” Opt. Lasers Eng. 50, 1–7 (2012).
[CrossRef]

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Elsevier, 2005).

Arnold, S. F.

S. F. Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–315(2008).
[CrossRef]

Basistiy, I. V.

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Bazhenov, V. Y.

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[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, 8185–8189 (1992).
[CrossRef]

Berry, M.

M. Berry, “Optical currents,” J. Opt. A 11, 1464–1475 (2009).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 2002).

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]

Desbrun, M.

Y. Tong, S. Lombeyda, A. N. Hirani, and M. Desbrun, “Discrete multiscale vector field decomposition,” ACM Trans. Graph. 22, 445–452 (2003).
[CrossRef]

Fedkiw, R.

R. Fedkiw, J. Stam, and H. W. Jensen, “Visual simulation of smoke,” in Proceedings of ACM SIGGRAPH 2001, Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, Eugene Fiume, ed. (Association for Computing Machinery, 2001), pp. 15–22.

Ghai, D. P.

D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, “Vortex lattice generation using interferometric techniques based on lateral shearing,” Opt. Commun. 282, 2692–2698 (2009).
[CrossRef]

Globus, A.

A. Globus, C. Levit, and T. Lasinki, “A tool for visualizing the topology of three-dimensional vector fields,” in Proceedings of IEEE on Visualization (IEEE, 1991), pp. 33–40.

Hattori, M.

M. Hattori and S. Komatsu, “An exact formulation of a filter for rotation in phase gradients and its applications to wavefront reconstruction problems,” J. Mod. Opt. 50, 1705–1723(2003).
[CrossRef]

Hirani, A. N.

Y. Tong, S. Lombeyda, A. N. Hirani, and M. Desbrun, “Discrete multiscale vector field decomposition,” ACM Trans. Graph. 22, 445–452 (2003).
[CrossRef]

Imielinska, C.

I. Kaya, A. P. Santhanam, C. Imielinska, and J. Rolland, “Modeling air-flow in the tracheobronchial tree using computational fluid dynamics,” in Proceedings of 2007 MICCAI Workshop on Computational Biomechanics (2007), pp. 142–151.

Jensen, H. W.

R. Fedkiw, J. Stam, and H. W. Jensen, “Visual simulation of smoke,” in Proceedings of ACM SIGGRAPH 2001, Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, Eugene Fiume, ed. (Association for Computing Machinery, 2001), pp. 15–22.

Joseph, J.

Kawai, T.

T. McGraw, T. Kawai, I. Yassine, and L. Zhu, “Visualizing high-order symmetric tensor field structure with differential operators,” J. Appl. Math. 2011, 1–27 (2011).
[CrossRef]

Kaya, I.

I. Kaya, A. P. Santhanam, C. Imielinska, and J. Rolland, “Modeling air-flow in the tracheobronchial tree using computational fluid dynamics,” in Proceedings of 2007 MICCAI Workshop on Computational Biomechanics (2007), pp. 142–151.

Komatsu, S.

M. Hattori and S. Komatsu, “An exact formulation of a filter for rotation in phase gradients and its applications to wavefront reconstruction problems,” J. Mod. Opt. 50, 1705–1723(2003).
[CrossRef]

Lage, M.

F. Petronetto, A. Paiva, M. Lage, G. Tavares, H. Lopes, and T. Lewiner, “Meshless Helmholtz-Hodge decomposition,” IEEE Trans. Vis. Comput. Graphics 16, 338–342 (2010).
[CrossRef]

Lasinki, T.

A. Globus, C. Levit, and T. Lasinki, “A tool for visualizing the topology of three-dimensional vector fields,” in Proceedings of IEEE on Visualization (IEEE, 1991), pp. 33–40.

Levit, C.

A. Globus, C. Levit, and T. Lasinki, “A tool for visualizing the topology of three-dimensional vector fields,” in Proceedings of IEEE on Visualization (IEEE, 1991), pp. 33–40.

Lewiner, T.

F. Petronetto, A. Paiva, M. Lage, G. Tavares, H. Lopes, and T. Lewiner, “Meshless Helmholtz-Hodge decomposition,” IEEE Trans. Vis. Comput. Graphics 16, 338–342 (2010).
[CrossRef]

Lombeyda, S.

Y. Tong, S. Lombeyda, A. N. Hirani, and M. Desbrun, “Discrete multiscale vector field decomposition,” ACM Trans. Graph. 22, 445–452 (2003).
[CrossRef]

Lopes, H.

F. Petronetto, A. Paiva, M. Lage, G. Tavares, H. Lopes, and T. Lewiner, “Meshless Helmholtz-Hodge decomposition,” IEEE Trans. Vis. Comput. Graphics 16, 338–342 (2010).
[CrossRef]

McGraw, T.

T. McGraw, T. Kawai, I. Yassine, and L. Zhu, “Visualizing high-order symmetric tensor field structure with differential operators,” J. Appl. Math. 2011, 1–27 (2011).
[CrossRef]

Molina-Terriza, G.

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

Padgett, M.

S. F. Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–315(2008).
[CrossRef]

M. Padgett and L. Allen, “Optical tweezers and spanners,” Phys. World 10, 35–38 (1997).

Padgett, M. J.

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

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
[CrossRef]

Paiva, A.

F. Petronetto, A. Paiva, M. Lage, G. Tavares, H. Lopes, and T. Lewiner, “Meshless Helmholtz-Hodge decomposition,” IEEE Trans. Vis. Comput. Graphics 16, 338–342 (2010).
[CrossRef]

Petronetto, F.

F. Petronetto, A. Paiva, M. Lage, G. Tavares, H. Lopes, and T. Lewiner, “Meshless Helmholtz-Hodge decomposition,” IEEE Trans. Vis. Comput. Graphics 16, 338–342 (2010).
[CrossRef]

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.

Prenel, J. P.

J. P. Prenel and D. Ambrosini, “Flow visualization and beyond,” Opt. Lasers Eng. 50, 1–7 (2012).
[CrossRef]

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.

Rolland, J.

I. Kaya, A. P. Santhanam, C. Imielinska, and J. Rolland, “Modeling air-flow in the tracheobronchial tree using computational fluid dynamics,” in Proceedings of 2007 MICCAI Workshop on Computational Biomechanics (2007), pp. 142–151.

Santhanam, A. P.

I. Kaya, A. P. Santhanam, C. Imielinska, and J. Rolland, “Modeling air-flow in the tracheobronchial tree using computational fluid dynamics,” in Proceedings of 2007 MICCAI Workshop on Computational Biomechanics (2007), pp. 142–151.

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.

J. Xavier, S. Vyas, P. Senthilkumaran, and J. Joseph, “Tailored complex 3D vortex lattice structures by perturbed multiples of three-plane waves,” Appl. Opt. 51, 1872–1878 (2012).
[CrossRef]

D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, “Vortex lattice generation using interferometric techniques based on lateral shearing,” Opt. Commun. 282, 2692–2698 (2009).
[CrossRef]

S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt. 46, 2893–2898 (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]

Sirohi, R. S.

D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, “Vortex lattice generation using interferometric techniques based on lateral shearing,” Opt. Commun. 282, 2692–2698 (2009).
[CrossRef]

Soskin, M. S.

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

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[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]

Stam, J.

R. Fedkiw, J. Stam, and H. W. Jensen, “Visual simulation of smoke,” in Proceedings of ACM SIGGRAPH 2001, Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, Eugene Fiume, ed. (Association for Computing Machinery, 2001), pp. 15–22.

Stewart, A. M.

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

Tavares, G.

F. Petronetto, A. Paiva, M. Lage, G. Tavares, H. Lopes, and T. Lewiner, “Meshless Helmholtz-Hodge decomposition,” IEEE Trans. Vis. Comput. Graphics 16, 338–342 (2010).
[CrossRef]

Tong, Y.

Y. Tong, S. Lombeyda, A. N. Hirani, and M. Desbrun, “Discrete multiscale vector field decomposition,” ACM Trans. Graph. 22, 445–452 (2003).
[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]

Vasnetsov, M. V.

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

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Vyas, S.

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Elsevier, 2005).

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, 8185–8189 (1992).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 2002).

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]

Xavier, J.

Yao, A. M.

Yassine, I.

T. McGraw, T. Kawai, I. Yassine, and L. Zhu, “Visualizing high-order symmetric tensor field structure with differential operators,” J. Appl. Math. 2011, 1–27 (2011).
[CrossRef]

Zhu, L.

T. McGraw, T. Kawai, I. Yassine, and L. Zhu, “Visualizing high-order symmetric tensor field structure with differential operators,” J. Appl. Math. 2011, 1–27 (2011).
[CrossRef]

ACM Trans. Graph. (1)

Y. Tong, S. Lombeyda, A. N. Hirani, and M. Desbrun, “Discrete multiscale vector field decomposition,” ACM Trans. Graph. 22, 445–452 (2003).
[CrossRef]

Adv. Opt. Photon. (1)

Appl. Opt. (2)

IEEE Trans. Vis. Comput. Graphics (1)

F. Petronetto, A. Paiva, M. Lage, G. Tavares, H. Lopes, and T. Lewiner, “Meshless Helmholtz-Hodge decomposition,” IEEE Trans. Vis. Comput. Graphics 16, 338–342 (2010).
[CrossRef]

Int. J. Numer. Methods Fluids (1)

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. Appl. Math. (1)

T. McGraw, T. Kawai, I. Yassine, and L. Zhu, “Visualizing high-order symmetric tensor field structure with differential operators,” J. Appl. Math. 2011, 1–27 (2011).
[CrossRef]

J. Mod. Opt. (2)

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

M. Hattori and S. Komatsu, “An exact formulation of a filter for rotation in phase gradients and its applications to wavefront reconstruction problems,” J. Mod. Opt. 50, 1705–1723(2003).
[CrossRef]

J. Opt. A (1)

M. Berry, “Optical currents,” J. Opt. A 11, 1464–1475 (2009).
[CrossRef]

Laser Photon. Rev. (1)

S. F. Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–315(2008).
[CrossRef]

Nat. Phys. (1)

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

Opt. Commun. (4)

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
[CrossRef]

D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, “Vortex lattice generation using interferometric techniques based on lateral shearing,” Opt. Commun. 282, 2692–2698 (2009).
[CrossRef]

Opt. Lasers Eng. (2)

J. P. Prenel and D. Ambrosini, “Flow visualization and beyond,” Opt. Lasers Eng. 50, 1–7 (2012).
[CrossRef]

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

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef]

Phys. World (1)

M. Padgett and L. Allen, “Optical tweezers and spanners,” Phys. World 10, 35–38 (1997).

Prog. Opt. (1)

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

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M. Born and E. Wolf, Principles of Optics (Cambridge University, 2002).

A. Globus, C. Levit, and T. Lasinki, “A tool for visualizing the topology of three-dimensional vector fields,” in Proceedings of IEEE on Visualization (IEEE, 1991), pp. 33–40.

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.

I. Kaya, A. P. Santhanam, C. Imielinska, and J. Rolland, “Modeling air-flow in the tracheobronchial tree using computational fluid dynamics,” in Proceedings of 2007 MICCAI Workshop on Computational Biomechanics (2007), pp. 142–151.

R. Fedkiw, J. Stam, and H. W. Jensen, “Visual simulation of smoke,” in Proceedings of ACM SIGGRAPH 2001, Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, Eugene Fiume, ed. (Association for Computing Machinery, 2001), pp. 15–22.

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Finite difference method, http://en.wikipedia.org/wiki/Finite_difference_method .

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Figures (5)

Fig. 1.
Fig. 1.

HHD of a spherical beam with positive divergence. (a) Transverse (x-y plane) phase profile for a spherical beam with a positive curvature. The smooth variation of color from green to white depicts the shape of the wavefront. The phase distribution has maxima at the center and decreases toward the boundaries, rendering it a spherical wavefront with a positive divergence. (b) Phase gradient field lines of the beam superimposed on the phase profile. (c) Flow lines of the solenoidal component of the Hodge decomposed field. Inset, magnified view of the beam near the center. (d) Irrotational component with field lines diverging from the center.

Fig. 2.
Fig. 2.

HHD of a spherical beam with negative divergence. (a) Transverse (x-y plane) phase profile for a spherical beam with a negative curvature. The smooth variation of color from white to green depicts the shape of the wavefront. The phase distribution has minima at the center and increases toward the boundaries, thus rendering it a spherical shape with a negative divergence. (b) Phase gradient field lines of the beam superimposed on the phase profile. (c) Flow lines of the solenoidal component of the Hodge decomposed field. Inset, magnified view of the beam near the center. (d) Irrotational component with field lines converging toward the centre.

Fig. 3.
Fig. 3.

HHD of a random vortex field added to a positively diverging spherical beam. (a) Transverse phase profile of a random vortex field added to a spherical beam with a positive curvature. The smooth variation of color from green to white in the background shows that the phase distribution has maxima at the center and reduces toward the boundaries exhibiting a positive divergence. (b) Phase gradient field lines of the beam superimposed on the phase profile. (c) Flow lines of the solenoidal component of the Hodge decomposed field. The field lines circulate about the vortex centers. (d) Irrotational component with diverging field lines. This is the vortex free field.

Fig. 4.
Fig. 4.

HHD of a random vortex field added to a negatively diverging spherical beam. (a) Transverse phase profile of a random vortex field added to a spherical beam with a negative curvature. The smooth variation of color from white to green in the background shows that the phase distribution has minima at the center and increases toward the boundaries exhibiting a negative divergence. (b) Phase gradient field lines of the beam superimposed on the phase profile. (c), (d) Represent the Hodge decomposed field. (c) Flow lines of the solenoidal component of (b). The field lines circulate about the vortex centers. (d) Irrotational component with converging field lines. This is the vortex-free field.

Fig. 5.
Fig. 5.

HHD for a three beam interference pattern. (a) Transverse phase profile of the original field. Inset, cross-sectional view of the intensity profile in the x-y plane. (b) Phase gradient field lines of the corresponding pattern. (c), (d) Represent the Hodge decomposed vortex lattice field. (c) Phase gradient field lines of the extracted solenoidal component of (b). (d) Irrotational component with flow lines forming a web-like pattern.

Equations (14)

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ϕ=Im[ψ*ψ]I,
F=f1+f2φ+×A,
φ(r)=14πVb(r)rdv,
A(r)=14πVc(r)rdv.
df⃗=fxxx^+fyyy^+fzzz^.
=12[0100101010010010].
x,3D=ImImxx,
y,3D=ImyImy,
z,3D=zImImz.
×f=[x^y^z^xyzfxfyfz]=[0zyz0xyx0][fxfyfz].
·f=fxx+fyy+fzz=[xyz][fxfyfz].
[0zy000z0x000yx0000000xyzI00I000I00I000I00I][f1xf1yf1zf2xf2yf2z]=[0000fxfyfz].
[0zy000z0x000yx0000000xyzI00I000I00I000I00I],while[f1xf1yf1zf2xf2yf2z]and[0000fxfyfz]
W=diag([αααβγγγ]).

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