Abstract

The generation of multiple vortex cones using an annular aperture array and a spatial light modulator (SLM) is studied. The direct downscale imaging of an SLM on the surface of an annular aperture enables the direct-phase modulation of the annular aperture. It is experimentally demonstrated that the direct-phase modulation of an annular aperture array can control both the topological charge and the horizontal positions of multiple vortex cones simultaneously.

© 2012 Optical Society of America

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References

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  1. Z. Jiang, Q. Lu, and Z. Liu, “Propagation of apertured Bessel beams,” Appl. Opt. 34, 7183–7185 (1995).
    [CrossRef]
  2. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327(1994).
    [CrossRef]
  3. Z. Bouchal, “Vortex array carried by a pseudo-nondiffracting beam,” J. Opt. Soc. Am. A 21, 1694–1702 (2004).
    [CrossRef]
  4. J. Courtial, R. Zambrini, M. R. Dennis, and M. Vasnetsov, “Angular momentum of optical vortex arrays,” Opt. Express 14, 938–949 (2006).
    [CrossRef]
  5. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207–211 (1998).
    [CrossRef]
  6. C.-Y. Hwang, D. Choi, K.-Y. Kim, and B. Lee, “Dual Airy beam,” Opt. Express 18, 23504–23516 (2010).
    [CrossRef]
  7. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
    [CrossRef]
  8. S. Kim, Y. Lim, H. Kim, J. Park, and B. Lee, “Optical beam focusing by a single subwavelength metal slit surrounded by chirped dielectric surface gratings,” Appl. Phys. Lett. 92, 013103 (2008).
    [CrossRef]
  9. H. He, M. Friese, N. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
    [CrossRef]
  10. 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]
  11. J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. 28, 872–874 (2003).
    [CrossRef]
  12. J. Curtis and D. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
    [CrossRef]
  13. C.-S. Guo, Y.-N. Yu, and Z. Hong, “Optical sorting using an array of optical vortices with fractional topological charge,” Opt. Commun. 283, 1889–1893 (2010).
    [CrossRef]
  14. H. Kim, J. Cho, J. Park, S. Han, and S. Seo, “Generation of mid-field concentrated beam arrays using periodic metal annular apertures,” Appl. Opt. 51, 1076–1085 (2012).
    [CrossRef]
  15. C.-S. Guo, X. Liu, J.-L. He, and H.-T. Wang, “Optimal annulus structures of optical vortices,” Opt. Express 12, 4625–4634 (2004).
    [CrossRef]
  16. T. Cizmar and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express 17, 15558–15570 (2009).
    [CrossRef]
  17. W. M. Lee, X. C. Yuan, and W. C. Cheong, “Optical vortex beam shaping by use of highly efficient irregular spiral phase plates for optical micromanipulation,” Opt. Lett. 29, 1796–1798 (2004).
    [CrossRef]
  18. Y. Izdebskaya, V. Shvedov, and A. Volyar, “Generation of higher-order optical vortices by a dielectric wedge,” Opt. Lett. 30, 2472–2474 (2005).
    [CrossRef]
  19. J. E. Morris, T. Cizmar, H. I. C. Dalgarno, R. F. Marchington, F. J. Gunn-Moore, and K. Dholakia, “Realization of curved Bessel beams: propagation around obstructions,” J. Opt. 12, 124002 (2010).
    [CrossRef]
  20. V. V. Kotlyar, A. A. Almazov, S. N. Khonina, and V. A. Soifer, “Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate,” J. Opt. Soc. Am. A 22, 849–861 (2005).
    [CrossRef]
  21. J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2004).

2012 (1)

2010 (3)

C.-Y. Hwang, D. Choi, K.-Y. Kim, and B. Lee, “Dual Airy beam,” Opt. Express 18, 23504–23516 (2010).
[CrossRef]

C.-S. Guo, Y.-N. Yu, and Z. Hong, “Optical sorting using an array of optical vortices with fractional topological charge,” Opt. Commun. 283, 1889–1893 (2010).
[CrossRef]

J. E. Morris, T. Cizmar, H. I. C. Dalgarno, R. F. Marchington, F. J. Gunn-Moore, and K. Dholakia, “Realization of curved Bessel beams: propagation around obstructions,” J. Opt. 12, 124002 (2010).
[CrossRef]

2009 (1)

2008 (1)

S. Kim, Y. Lim, H. Kim, J. Park, and B. Lee, “Optical beam focusing by a single subwavelength metal slit surrounded by chirped dielectric surface gratings,” Appl. Phys. Lett. 92, 013103 (2008).
[CrossRef]

2006 (1)

2005 (2)

2004 (3)

2003 (2)

J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. 28, 872–874 (2003).
[CrossRef]

J. Curtis and D. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
[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]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef]

1998 (1)

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207–211 (1998).
[CrossRef]

1995 (2)

Z. Jiang, Q. Lu, and Z. Liu, “Propagation of apertured Bessel beams,” Appl. Opt. 34, 7183–7185 (1995).
[CrossRef]

H. He, M. Friese, N. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef]

1994 (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327(1994).
[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]

Almazov, A. A.

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327(1994).
[CrossRef]

Bouchal, Z.

Z. Bouchal, “Vortex array carried by a pseudo-nondiffracting beam,” J. Opt. Soc. Am. A 21, 1694–1702 (2004).
[CrossRef]

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207–211 (1998).
[CrossRef]

Cheong, W. C.

Chlup, M.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207–211 (1998).
[CrossRef]

Cho, J.

Choi, D.

Cizmar, T.

J. E. Morris, T. Cizmar, H. I. C. Dalgarno, R. F. Marchington, F. J. Gunn-Moore, and K. Dholakia, “Realization of curved Bessel beams: propagation around obstructions,” J. Opt. 12, 124002 (2010).
[CrossRef]

T. Cizmar and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express 17, 15558–15570 (2009).
[CrossRef]

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327(1994).
[CrossRef]

Courtial, J.

Curtis, J.

J. Curtis and D. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef]

Curtis, J. E.

Dalgarno, H. I. C.

J. E. Morris, T. Cizmar, H. I. C. Dalgarno, R. F. Marchington, F. J. Gunn-Moore, and K. Dholakia, “Realization of curved Bessel beams: propagation around obstructions,” J. Opt. 12, 124002 (2010).
[CrossRef]

Degiron, A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef]

Dennis, M. R.

Devaux, E.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef]

Dholakia, K.

J. E. Morris, T. Cizmar, H. I. C. Dalgarno, R. F. Marchington, F. J. Gunn-Moore, and K. Dholakia, “Realization of curved Bessel beams: propagation around obstructions,” J. Opt. 12, 124002 (2010).
[CrossRef]

T. Cizmar and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express 17, 15558–15570 (2009).
[CrossRef]

Ebbesen, T. W.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef]

Friese, M.

H. He, M. Friese, N. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef]

Garcia-Vidal, F. J.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef]

Goodman, J.

J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2004).

Grier, D.

J. Curtis and D. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef]

Grier, D. G.

Gunn-Moore, F. J.

J. E. Morris, T. Cizmar, H. I. C. Dalgarno, R. F. Marchington, F. J. Gunn-Moore, and K. Dholakia, “Realization of curved Bessel beams: propagation around obstructions,” J. Opt. 12, 124002 (2010).
[CrossRef]

Guo, C.-S.

C.-S. Guo, Y.-N. Yu, and Z. Hong, “Optical sorting using an array of optical vortices with fractional topological charge,” Opt. Commun. 283, 1889–1893 (2010).
[CrossRef]

C.-S. Guo, X. Liu, J.-L. He, and H.-T. Wang, “Optimal annulus structures of optical vortices,” Opt. Express 12, 4625–4634 (2004).
[CrossRef]

Han, S.

He, H.

H. He, M. Friese, N. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef]

He, J.-L.

Heckenberg, N.

H. He, M. Friese, N. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef]

Hong, Z.

C.-S. Guo, Y.-N. Yu, and Z. Hong, “Optical sorting using an array of optical vortices with fractional topological charge,” Opt. Commun. 283, 1889–1893 (2010).
[CrossRef]

Hwang, C.-Y.

Izdebskaya, Y.

Jiang, Z.

Khonina, S. N.

Kim, H.

H. Kim, J. Cho, J. Park, S. Han, and S. Seo, “Generation of mid-field concentrated beam arrays using periodic metal annular apertures,” Appl. Opt. 51, 1076–1085 (2012).
[CrossRef]

S. Kim, Y. Lim, H. Kim, J. Park, and B. Lee, “Optical beam focusing by a single subwavelength metal slit surrounded by chirped dielectric surface gratings,” Appl. Phys. Lett. 92, 013103 (2008).
[CrossRef]

Kim, K.-Y.

Kim, S.

S. Kim, Y. Lim, H. Kim, J. Park, and B. Lee, “Optical beam focusing by a single subwavelength metal slit surrounded by chirped dielectric surface gratings,” Appl. Phys. Lett. 92, 013103 (2008).
[CrossRef]

Kotlyar, V. V.

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327(1994).
[CrossRef]

Lee, B.

C.-Y. Hwang, D. Choi, K.-Y. Kim, and B. Lee, “Dual Airy beam,” Opt. Express 18, 23504–23516 (2010).
[CrossRef]

S. Kim, Y. Lim, H. Kim, J. Park, and B. Lee, “Optical beam focusing by a single subwavelength metal slit surrounded by chirped dielectric surface gratings,” Appl. Phys. Lett. 92, 013103 (2008).
[CrossRef]

Lee, W. M.

Lezec, H. J.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef]

Lim, Y.

S. Kim, Y. Lim, H. Kim, J. Park, and B. Lee, “Optical beam focusing by a single subwavelength metal slit surrounded by chirped dielectric surface gratings,” Appl. Phys. Lett. 92, 013103 (2008).
[CrossRef]

Linke, R. A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef]

Liu, X.

Liu, Z.

Lu, Q.

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]

Marchington, R. F.

J. E. Morris, T. Cizmar, H. I. C. Dalgarno, R. F. Marchington, F. J. Gunn-Moore, and K. Dholakia, “Realization of curved Bessel beams: propagation around obstructions,” J. Opt. 12, 124002 (2010).
[CrossRef]

Martin-Moreno, L.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef]

Morris, J. E.

J. E. Morris, T. Cizmar, H. I. C. Dalgarno, R. F. Marchington, F. J. Gunn-Moore, and K. Dholakia, “Realization of curved Bessel beams: propagation around obstructions,” J. Opt. 12, 124002 (2010).
[CrossRef]

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]

Padgett, M. J.

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]

Park, J.

H. Kim, J. Cho, J. Park, S. Han, and S. Seo, “Generation of mid-field concentrated beam arrays using periodic metal annular apertures,” Appl. Opt. 51, 1076–1085 (2012).
[CrossRef]

S. Kim, Y. Lim, H. Kim, J. Park, and B. Lee, “Optical beam focusing by a single subwavelength metal slit surrounded by chirped dielectric surface gratings,” Appl. Phys. Lett. 92, 013103 (2008).
[CrossRef]

Rubinsztein-Dunlop, H.

H. He, M. Friese, N. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef]

Seo, S.

Shvedov, V.

Soifer, V. A.

Vasnetsov, M.

Volyar, A.

Wagner, J.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207–211 (1998).
[CrossRef]

Wang, H.-T.

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327(1994).
[CrossRef]

Yu, Y.-N.

C.-S. Guo, Y.-N. Yu, and Z. Hong, “Optical sorting using an array of optical vortices with fractional topological charge,” Opt. Commun. 283, 1889–1893 (2010).
[CrossRef]

Yuan, X. C.

Zambrini, R.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

S. Kim, Y. Lim, H. Kim, J. Park, and B. Lee, “Optical beam focusing by a single subwavelength metal slit surrounded by chirped dielectric surface gratings,” Appl. Phys. Lett. 92, 013103 (2008).
[CrossRef]

J. Opt. (1)

J. E. Morris, T. Cizmar, H. I. C. Dalgarno, R. F. Marchington, F. J. Gunn-Moore, and K. Dholakia, “Realization of curved Bessel beams: propagation around obstructions,” J. Opt. 12, 124002 (2010).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (3)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327(1994).
[CrossRef]

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207–211 (1998).
[CrossRef]

C.-S. Guo, Y.-N. Yu, and Z. Hong, “Optical sorting using an array of optical vortices with fractional topological charge,” Opt. Commun. 283, 1889–1893 (2010).
[CrossRef]

Opt. Express (4)

Opt. Lett. (3)

Phys. Rev. Lett. (3)

J. Curtis and D. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef]

H. He, M. Friese, N. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef]

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]

Science (1)

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef]

Other (1)

J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2004).

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Figures (7)

Fig. 1.
Fig. 1.

Phase profiles of (a) the circular aperture and (b) the annular aperture are shown, where the phase profiles are given by exp ( i 3 φ ) , where φ as given by φ = tan 1 ( y / x ) varies from 0 to 2 π . The x - z plane cross-sections of (c) the circular aperture and (d) the annular aperture show a slightly diverging profile of the vortex cone. The x - y plane cross-sections ( z = 300 μm ) of (e) the circular aperture and (f) the annular aperture show vortex profiles with bright vortex ring.

Fig. 2.
Fig. 2.

Effect of the pixelated spiral phase modulation of the circular aperture on the formation of midfield vortex field formation: (a) phase profile of the circular aperture with a topological charge of 20 and (b) the resulting optical intensity profile at z = 10 μm . (c) The pixelated phase profile with the pixel size 1.24 μm × 1.24 μm and the same topological charge and (d) the resulting optical intensity profile at z = 10 μm .

Fig. 3.
Fig. 3.

Effect of the pixelated spiral phase modulation of the annular aperture on the formation of midfield vortex field formation: (a) phase profile of the annular aperture with a topological charge of 20 and (b) the resulting optical intensity profile at z = 10 μm . (c) The pixelated phase profile with the pixel size 1.24 μm × 1.24 μm and the same topological charge and (d) the resulting optical intensity profile at z = 10 μm .

Fig. 4.
Fig. 4.

Phase modulation profiles of the zeroth to the fourth topological charges to be displayed on the SLM and the x y cross-sections of the corresponding optical field profiles at z = 0.3 mm and 0.6 mm.

Fig. 5.
Fig. 5.

(a) SEM image of the annular aperture inscribed on the silver surface. (b) The overall experimental setup is shown, as are (c) optical intensity profiles experimentally measured at the planes ( z = 0.3 mm and z = 0.6 mm ) above the aperture surface.

Fig. 6.
Fig. 6.

(a) Phase modulation profiles and (b) a CCD image of annular aperture array. (c) Simulation and (d) experimental results of multiple vortex cones. The right upper vortex cone denotes the zeroth-order topological charge (no vortex) and the left upper, left lower, and right lower vortex cones denote the third-order, fifth-order, and fourth-order topological charges, respectively.

Fig. 7.
Fig. 7.

Phase profiles to induce [(a)–(c)] the spatial shifting of the vortex cones, [(d)–(f)] the corresponding simulations, and [(g)–(i)] the experiment results. In this experiment, the vortex cones in the first and third quadrant regions are shifted. The dotted reference lines are drawn for a comparison of the spatial positions of the vortex cones.

Equations (2)

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F ( x , y , z ) = A ˜ ( α , β ) exp ( j 2 π ( α x + β y + γ z ) ) d α d β ,
A ˜ ( α , β ) = A ( x , y ) exp ( j m φ ( x , y ) ) exp ( j 2 π ( α x + β y ) ) d α d β .

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