Abstract

In this work we use a phase-only spatial light modulator (SLM) to mimic a ring-slit aperture, containing multiple azimuthally varying phases at different radial positions. The optical Fourier transform of such an aperture is currently known and its intensity profile has been shown to rotate along its propagation axis. Here we investigate the near-field of the ring-slit aperture and show, both experimentally and theoretically, that although the near-field possesses similar attributes to its Fourier transform, its intensity profile exhibits no rotation as it propagates.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and the transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
    [CrossRef]
  2. J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301(2000).
    [CrossRef]
  3. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16, 9411–9416(2008).
    [CrossRef]
  4. 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]
  5. H. He, M. E. J. Friese, N. R. 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]
  6. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
    [CrossRef]
  7. A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental Two-Photon Three-Dimensional Quantum Entanglement,” Phys. Rev. Lett. 89, 240401–4 (2002).
    [CrossRef]
  8. J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).
  9. R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher-order Bessel beams,” Opt. Express 17, 23389–23395 (2009).
    [CrossRef]
  10. R. Rop, A. Dudley, C. López-Mariscal, and A. Forbes, “Measuring the rotation rates of superpositions of higher-order Bessel beams,” J. Mod. Opt. 59, 259–267 (2011) http://dx.doi.org/10.1080/09500340.2011.631714 .
  11. V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
    [CrossRef]
  12. J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef]
  13. D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. 28, 657–659 (2003).
    [CrossRef]
  14. R. H. Jordan and D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution,” Opt. Lett. 19, 427–429 (1994).
    [CrossRef]
  15. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  16. D. W. K. Wong and G. Chen, “Redistribution of the zero order by the use of a phase checkerboard pattern in computer generated holograms,” Appl. Opt. 47, 602–610 (2008).
    [CrossRef]
  17. A. Dudley, R. Vasilyeu, V. Belyi, N. Khilo, P. Ropot, and A. Forbes, “Controlling the evolution of nondiffracting speckle by complex amplitude modulation on a phase-only spatial light modulator,” Opt. Commun. 285, 5–12 (2012).
    [CrossRef]
  18. T. A. King, W. Hogervorst, N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Formation of higher-order Bessel light beams in biaxial crystals,” Opt. Commun. 187, 407–414 (2001).
    [CrossRef]

2012

A. Dudley, R. Vasilyeu, V. Belyi, N. Khilo, P. Ropot, and A. Forbes, “Controlling the evolution of nondiffracting speckle by complex amplitude modulation on a phase-only spatial light modulator,” Opt. Commun. 285, 5–12 (2012).
[CrossRef]

2009

2008

2007

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

2003

2002

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental Two-Photon Three-Dimensional Quantum Entanglement,” Phys. Rev. Lett. 89, 240401–4 (2002).
[CrossRef]

2001

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

T. A. King, W. Hogervorst, N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Formation of higher-order Bessel light beams in biaxial crystals,” Opt. Commun. 187, 407–414 (2001).
[CrossRef]

2000

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301(2000).
[CrossRef]

1995

H. He, M. E. J. Friese, N. R. 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

1993

M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and the 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]

1987

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Alfano, R. R.

Allen, L.

M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and the 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]

Arlt, J.

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301(2000).
[CrossRef]

Barreiro, J. T.

J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and the 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]

Belyi, V.

A. Dudley, R. Vasilyeu, V. Belyi, N. Khilo, P. Ropot, and A. Forbes, “Controlling the evolution of nondiffracting speckle by complex amplitude modulation on a phase-only spatial light modulator,” Opt. Commun. 285, 5–12 (2012).
[CrossRef]

Chen, G.

Dholakia, K.

D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. 28, 657–659 (2003).
[CrossRef]

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301(2000).
[CrossRef]

Dudley, A.

A. Dudley, R. Vasilyeu, V. Belyi, N. Khilo, P. Ropot, and A. Forbes, “Controlling the evolution of nondiffracting speckle by complex amplitude modulation on a phase-only spatial light modulator,” Opt. Commun. 285, 5–12 (2012).
[CrossRef]

R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher-order Bessel beams,” Opt. Express 17, 23389–23395 (2009).
[CrossRef]

Durnin, J.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Eberly, J. H.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Forbes, A.

A. Dudley, R. Vasilyeu, V. Belyi, N. Khilo, P. Ropot, and A. Forbes, “Controlling the evolution of nondiffracting speckle by complex amplitude modulation on a phase-only spatial light modulator,” Opt. Commun. 285, 5–12 (2012).
[CrossRef]

R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher-order Bessel beams,” Opt. Express 17, 23389–23395 (2009).
[CrossRef]

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. 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]

Garcés-Chávez, V.

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Hall, D. G.

He, H.

H. He, M. E. J. Friese, N. R. 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]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. 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]

Hogervorst, W.

T. A. King, W. Hogervorst, N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Formation of higher-order Bessel light beams in biaxial crystals,” Opt. Commun. 187, 407–414 (2001).
[CrossRef]

Jordan, R. H.

Kazak, N. S.

T. A. King, W. Hogervorst, N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Formation of higher-order Bessel light beams in biaxial crystals,” Opt. Commun. 187, 407–414 (2001).
[CrossRef]

Khilo, N.

A. Dudley, R. Vasilyeu, V. Belyi, N. Khilo, P. Ropot, and A. Forbes, “Controlling the evolution of nondiffracting speckle by complex amplitude modulation on a phase-only spatial light modulator,” Opt. Commun. 285, 5–12 (2012).
[CrossRef]

R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher-order Bessel beams,” Opt. Express 17, 23389–23395 (2009).
[CrossRef]

Khilo, N. A.

T. A. King, W. Hogervorst, N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Formation of higher-order Bessel light beams in biaxial crystals,” Opt. Commun. 187, 407–414 (2001).
[CrossRef]

Khonina, S. N.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

King, T. A.

T. A. King, W. Hogervorst, N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Formation of higher-order Bessel light beams in biaxial crystals,” Opt. Commun. 187, 407–414 (2001).
[CrossRef]

Kotlyar, V. V.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

Kwiat, P. G.

J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).

Mair, A.

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

McGloin, D.

Miceli, J.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Ropot, P.

A. Dudley, R. Vasilyeu, V. Belyi, N. Khilo, P. Ropot, and A. Forbes, “Controlling the evolution of nondiffracting speckle by complex amplitude modulation on a phase-only spatial light modulator,” Opt. Commun. 285, 5–12 (2012).
[CrossRef]

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. 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]

Ryzhevich, A. A.

T. A. King, W. Hogervorst, N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Formation of higher-order Bessel light beams in biaxial crystals,” Opt. Commun. 187, 407–414 (2001).
[CrossRef]

Skidanov, R. V.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

Soifer, V. A.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[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]

Sztul, H. I.

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 the transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Vasilyeu, R.

A. Dudley, R. Vasilyeu, V. Belyi, N. Khilo, P. Ropot, and A. Forbes, “Controlling the evolution of nondiffracting speckle by complex amplitude modulation on a phase-only spatial light modulator,” Opt. Commun. 285, 5–12 (2012).
[CrossRef]

R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher-order Bessel beams,” Opt. Express 17, 23389–23395 (2009).
[CrossRef]

Vaziri, A.

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental Two-Photon Three-Dimensional Quantum Entanglement,” Phys. Rev. Lett. 89, 240401–4 (2002).
[CrossRef]

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

Wei, T.-C.

J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).

Weihs, G.

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental Two-Photon Three-Dimensional Quantum Entanglement,” Phys. Rev. Lett. 89, 240401–4 (2002).
[CrossRef]

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

Woerdman, J. P.

M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and the 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]

Wong, D. W. K.

Zeilinger, A.

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental Two-Photon Three-Dimensional Quantum Entanglement,” Phys. Rev. Lett. 89, 240401–4 (2002).
[CrossRef]

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

Appl. Opt.

Nat. Phys.

J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).

Nature

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

Opt. Commun.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

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

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301(2000).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

A. Dudley, R. Vasilyeu, V. Belyi, N. Khilo, P. Ropot, and A. Forbes, “Controlling the evolution of nondiffracting speckle by complex amplitude modulation on a phase-only spatial light modulator,” Opt. Commun. 285, 5–12 (2012).
[CrossRef]

T. A. King, W. Hogervorst, N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Formation of higher-order Bessel light beams in biaxial crystals,” Opt. Commun. 187, 407–414 (2001).
[CrossRef]

Opt. Express

Opt. Lett.

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.

H. He, M. E. J. Friese, N. R. 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. Vaziri, G. Weihs, and A. Zeilinger, “Experimental Two-Photon Three-Dimensional Quantum Entanglement,” Phys. Rev. Lett. 89, 240401–4 (2002).
[CrossRef]

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Other

R. Rop, A. Dudley, C. López-Mariscal, and A. Forbes, “Measuring the rotation rates of superpositions of higher-order Bessel beams,” J. Mod. Opt. 59, 259–267 (2011) http://dx.doi.org/10.1080/09500340.2011.631714 .

Supplementary Material (1)

» Media 1: AVI (620 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1.
Fig. 1.

Schematic of the generation of the annular field, propagating from near- (P1) to far-field (P3). The green (red) rays denote the rays originating from the outer (inner) ring-slit. The three rings (at the bottom of the schematic) aid the illustration, as to how the annular fields overlap and become completely indistinguishable as the propagation from the aperture increases.

Fig. 2.
Fig. 2.

Density plot of the transmission function described in Eq. (1) for an azimuthal mode index of l=3.

Fig. 3.
Fig. 3.

(a) Density plot and (b) cross-sectional plot of the ring-slit aperture described in Eq. (3) for the following parameters: ring-slit radius, R0=2, ring-slit width, Δ=0.5, and gradient, n=10 (the units are arbitrary).

Fig. 4.
Fig. 4.

Schematic of the experimental setup for investigating the field formed by a ring-slit hologram, as well as the propagation and Fourier transform of such a field. L: lens (f1=25mm; f2=75mm; f3=100mm; and f4=100mm); M: mirror; LCD: liquid crystal display; D: diaphragm; CCD: CCD camera. The planes of interest are marked P1, P2, and P3. P3 is the Fourier plane of the ring-slit hologram; P1 is the relayed-field (in both phase and amplitude) at the ring-slit hologram; and P2 occurs a distance of 2f after L4.

Fig. 5.
Fig. 5.

First row: ring-slit holograms addressed to LCD1. A zoomed-in section of three and four ring-slits are depicted as inserts (1g) and (1h). Second and third rows: experimentally produced and theoretically calculated fields produced in the Fourier plane (i.e., plane P3), respectively. Fourth and fifth rows: experimental and theoretical fields, respectively, produced at plane P1 (i.e., the “singularity”-fields). The white “X” marks the singularities. Sixth and seventh rows: experimental and theoretical fields, respectively, produced at plane P2 (the “spiral”-fields).

Fig. 6.
Fig. 6.

(a) Experimentally recorded field at plane P1 for a ring-slit consisting of the following azimuthal phases: linner=+3 and louter=3. Theoretical prediction is given as an insert. The red, dashed ring marks the line for which the intensity profile is plotted. (b) The solid black curve is the experimental intensity profile and the red dashed curve is the theoretical intensity profile, cos(2lϕ).

Fig. 7.
Fig. 7.

Experimental intensity profiles of the field captured at evenly spaced intervals from plane P1 to plane P2. The distances from plane P1 are given as (a) 0 mm, (b) 10 mm, (c) 20 mm, (d) 30 mm, (e) 40 mm, (f) 50 mm, (g) 60 mm, and (h) 70 mm. The white arrows illustrate the movement of a selected singularity. Inserts are given for the theoretical predictions.

Fig. 8.
Fig. 8.

First column: ring-slit hologram applied to LCD1. Second column: corresponding optical fields for the ring-slits. The white arrows mark the locations of the singularities. Third column: Fourier transform of the ring-slit hologram. Fourth column: corresponding “spiral”-field, produced at plane P2. white arrow marks the handedness of the “spokes.” Theoretical predictions are accompanied as inserts.

Fig. 9.
Fig. 9.

Video clip containing experimental images for the field occurring before and after the Fourier plane for an incoming “spiral”-field produced at plane P2 (Media 1).

Equations (16)

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

t(r,ϕ)={exp(ilϕ)R1Δ2rR1+Δ2exp(ilϕ)R2Δ2rR2+Δ20elsewhere,
AP1(r,ϕ,z0)={exp(ilϕ)exp(ik1zz0)R1Δ2rR1+Δ2exp(ilϕ)exp(ik2zz0)R2Δ2rR2+Δ2.0elsewhere
ARing(r)=exp(((rR0)Δ/2)n)
AP1(r,ϕ,z0)=ARing1(r)exp(ilϕ)exp(ik1zz0)+ARing2(r)exp(ilϕ)exp(ik2zz0)=exp(((rR1)Δ/2)n)exp(ilϕ)exp(ik1zz0)+exp(((rR2)Δ/2)n)exp(ilϕ)exp(ik2zz0).
AP1(r,ϕ,z0)=exp(((rR0)Δ/2)n)(exp(ilϕ)exp(ik1zz0)+exp(ilϕ)exp(ik2zz0)).
IP1(r,ϕ,z0)=4cos2(k1zz0k2zz0+2lϕ2)(cosh(2(rR0)Δ)sinh(2(rR0)Δ))2n.
dϕdz0=k2zk1z2l.
dϕdz0=0.
AP2(r,ϕ,z)=eikziλz02πR1Δ2R2+Δ2AP1(r,ϕ,z)exp(ik2z(r12+r22rr1cos(ϕ1ϕ))r1dr1dϕ1
AP2(r,ϕ,z)=ABG1(r,ϕ,z)+ABG2(r,ϕ,z)=(11+(zzr)2)(exp[i(kzk1r2z2kΦ(z))(1ω2(z)ik2R(z))(r2+(k1rzk)2)+ik1zz+ilϕ]Jl(k1rr1+i(z/zr))+exp[i(kzk2r2z2kΦ(z))(1ω2(z)ik2R(z))(r2+(k2rzk)2)+ik2zzilϕ]Jl(k2rr1+i(z/zr))).
dϕdzk2zk1z2l.
AP3(r,ϕ,z)=iλz02πR1Δ2R2+Δ2t(r,ϕ1)exp[ik2f(1zf)r12]exp[ikrr1fcos(ϕ1ϕ)]r1dr1dϕ1.
AP3(r,ϕ,z)=Jl(k1rr)exp(ilϕ)exp(ik1zz)+Jl(k2rr)exp(ilϕ)exp(ik2zz).
IP3(r,ϕ,z)Jl2(k1rr)+Jl2(k2rr)+2Jl(k1rr)Jl(k2rr)cos(k1zzk2zz+2lϕ).
IP3(r,ϕ,z)2Jl2(krr)((1)l+1+2(1)lcos(k1zzk2zz+2lϕ)).
dϕdz=k2zk1z2l.

Metrics