Abstract

The study and application of optical vortices have gained significant prominence over the last two decades. An interesting challenge remains the determination of the azimuthal index (topological charge) of an optical vortex beam for a range of applications. We explore the diffraction of such beams from a triangular aperture and observe that the form of the resultant diffraction pattern is dependent upon both the magnitude and sign of the azimuthal index and this is valid for both monochromatic and broadband light fields. For the first time we demonstrate that this behavior is related not only to the azimuthal index but crucially the Gouy phase component of the incident beam. In particular, we explore the far field diffraction pattern for incident fields incident upon a triangular aperture possessing non-integer values of the azimuthal index . Such fields have a complex vortex structure. We are able to infer the birth of a vortex which occurs at half-integer values of and explore its evolution by observations of the diffraction pattern. These results demonstrate the extended versatility of a triangular aperture for the study of optical vortices.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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] [PubMed]
  2. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336, 165–190 (1974).
    [CrossRef]
  3. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinszstein-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] [PubMed]
  4. K. Dholakia and W. M. Lee, “Optical trapping takes shape: the use of structured light fields,” Adv. At. Mol. Phys. 56, 261–337 (2008).
  5. V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
    [CrossRef] [PubMed]
  6. 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]
  7. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2001).
    [CrossRef]
  8. H. I. Sztul and R. R. Alfano, “Double-slit interference with Laguerre-Gaussian beams,” Opt. Lett. 31, 999–1001 (2006).
    [CrossRef] [PubMed]
  9. C. S. Guo, L. L. Lu, and H. T. Wang, “Characterizing topological charge of optical vortices by using an annular aperture,” Opt. Lett. 34, 3686–3688 (2009).
    [CrossRef] [PubMed]
  10. J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
    [CrossRef] [PubMed]
  11. S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
    [CrossRef] [PubMed]
  12. I. V. Basistiy, M. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
    [CrossRef]
  13. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A, Pure Appl. Opt. 6, 259–268 (2004).
    [CrossRef]
  14. W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239, 129–135 (2004).
    [CrossRef]
  15. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” N. J. Phys. 6, 71 (2004).
    [CrossRef]
  16. R. C. Smith and J. S. Marsh, “Diffraction patterns of simple apertures,” J. Opt. Soc. Am. 64, 798–803 (1974).
    [CrossRef]
  17. See, for example,A. E. Siegman, Lasers (University Science, 2004), Chap. 16.4.
  18. J. Arlt, “Handedness and azimuthal energy flow of optical vortex beams,” J. Mod. Opt. 50, 1573–1580 (2003).

2010 (1)

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef] [PubMed]

2009 (1)

2008 (1)

K. Dholakia and W. M. Lee, “Optical trapping takes shape: the use of structured light fields,” Adv. At. Mol. Phys. 56, 261–337 (2008).

2006 (1)

2005 (1)

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef] [PubMed]

2004 (3)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A, Pure Appl. Opt. 6, 259–268 (2004).
[CrossRef]

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239, 129–135 (2004).
[CrossRef]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” N. J. Phys. 6, 71 (2004).
[CrossRef]

2003 (2)

J. Arlt, “Handedness and azimuthal energy flow of optical vortex beams,” J. Mod. Opt. 50, 1573–1580 (2003).

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef] [PubMed]

2002 (1)

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]

2001 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2001).
[CrossRef]

1995 (2)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinszstein-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] [PubMed]

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

1974 (2)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336, 165–190 (1974).
[CrossRef]

R. C. Smith and J. S. Marsh, “Diffraction patterns of simple apertures,” J. Opt. Soc. Am. 64, 798–803 (1974).
[CrossRef]

’t Hooft, G. W.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef] [PubMed]

Aiello, A.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef] [PubMed]

Alfano, R. R.

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]

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] [PubMed]

Arlt, J.

J. Arlt, “Handedness and azimuthal energy flow of optical vortex beams,” J. Mod. Opt. 50, 1573–1580 (2003).

Basistiy, I. V.

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

Berry, M. V.

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A, Pure Appl. Opt. 6, 259–268 (2004).
[CrossRef]

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336, 165–190 (1974).
[CrossRef]

Chavez-Cerda, S.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef] [PubMed]

Dholakia, K.

K. Dholakia and W. M. Lee, “Optical trapping takes shape: the use of structured light fields,” Adv. At. Mol. Phys. 56, 261–337 (2008).

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239, 129–135 (2004).
[CrossRef]

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef] [PubMed]

Dultz, W.

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef] [PubMed]

Eliel, E. R.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef] [PubMed]

Fonseca, E. J. S.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef] [PubMed]

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinszstein-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] [PubMed]

Garces-Chavez, V.

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef] [PubMed]

Guo, C. S.

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinszstein-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] [PubMed]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinszstein-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] [PubMed]

Hickmann, J. M.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef] [PubMed]

Leach, J.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” N. J. Phys. 6, 71 (2004).
[CrossRef]

Lee, W. M.

K. Dholakia and W. M. Lee, “Optical trapping takes shape: the use of structured light fields,” Adv. At. Mol. Phys. 56, 261–337 (2008).

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239, 129–135 (2004).
[CrossRef]

Lu, L. L.

Ma, X.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef] [PubMed]

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]

Marsh, J. S.

McGloin, D.

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef] [PubMed]

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2001).
[CrossRef]

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336, 165–190 (1974).
[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]

Oemrawsingh, S. S. R.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef] [PubMed]

Padgett, M. J.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” N. J. Phys. 6, 71 (2004).
[CrossRef]

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef] [PubMed]

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]

Rubinszstein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinszstein-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] [PubMed]

Schmitzer, H.

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef] [PubMed]

Smith, R. C.

Soares, W. C.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef] [PubMed]

Soskin, M.

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

Sztul, H. I.

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2001).
[CrossRef]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2001).
[CrossRef]

Vasnetsov, M. V.

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

Voigt, D.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef] [PubMed]

Wang, H. T.

Woerdman, J. P.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef] [PubMed]

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] [PubMed]

Yao, E.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” N. J. Phys. 6, 71 (2004).
[CrossRef]

Yuan, X. C.

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239, 129–135 (2004).
[CrossRef]

Adv. At. Mol. Phys. (1)

K. Dholakia and W. M. Lee, “Optical trapping takes shape: the use of structured light fields,” Adv. At. Mol. Phys. 56, 261–337 (2008).

J. Mod. Opt. (1)

J. Arlt, “Handedness and azimuthal energy flow of optical vortex beams,” J. Mod. Opt. 50, 1573–1580 (2003).

J. Opt. A, Pure Appl. Opt. (1)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A, Pure Appl. Opt. 6, 259–268 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

N. J. Phys. (1)

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” N. J. Phys. 6, 71 (2004).
[CrossRef]

Opt. Commun. (2)

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239, 129–135 (2004).
[CrossRef]

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

Opt. Lett. (2)

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] [PubMed]

Phys. Rev. Lett. (6)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinszstein-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] [PubMed]

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef] [PubMed]

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef] [PubMed]

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef] [PubMed]

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]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2001).
[CrossRef]

Proc. R. Soc. Lond. A Math. Phys. Sci. (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336, 165–190 (1974).
[CrossRef]

Other (1)

See, for example,A. E. Siegman, Lasers (University Science, 2004), Chap. 16.4.

Supplementary Material (1)

» Media 1: MOV (661 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 (8)

Fig. 1
Fig. 1

Numerical simulations of the far-field intensity profile for azimuthal indices = 2, 3, 4 in the upper row, and = −2, −3, −4 in the lower row. These intensity patterns reveal the truncated optical lattice generated by the triangular aperture for incident optical vortices with integer azimuthal indices, and the fact that the intensity patterns are rotated by 180° under reversal of the sign of the azimuthal index.

Fig. 2
Fig. 2

Intensity patterns obtained for = 2 → 3 in steps of = 0.1. These intensity patterns illustrate the birth of an optical vortex (region indicated by dashed ellipse) as the azimuthal index passes through a half-integer value.

Fig. 3
Fig. 3

( Media 1) The video includes an animation in which the intensity pattern is shown as the azimuthal index is varied from = 2 → 5, so that several examples can be viewed. To aid viewing the animation pauses at integer values of so that the reader can view the different examples involving variation between integer values of azimuthal index.

Fig. 4
Fig. 4

Experimental setup for monochromatic studies: L = lens, SLM = spatial light modulator, M = mirror, CCD = Charge Coupled Device camera. Focal widths of lenses: f1 = 50 mm, f2 = 250 mm, f3 = 750 mm, f4 = 250 mm, f5 = 100 cm. For the broadband studies, a prism in the back focal plane of L4 compensated for dispersion mediated by the first SLM, and a static triangular aperture was used instead of the second SLM to avoid issues related to dispersion.

Fig. 5
Fig. 5

Measured monochromatic far-field intensity profile for azimuthal indices = 2,3,4 in the upper row, and = −2, −3, −4 in the lower row. This should be compared to the theoretical prediction seen in Fig. 1.

Fig. 6
Fig. 6

Measured monochromatic far-field intensity patterns obtained for incident light fields with = 2 → 3 in steps of 0.1. The dashed ellipse indicates the region where the birth of the vortex is observed. Compare to theoretical prediction in Fig. 2.

Fig. 7
Fig. 7

Measured broadband intensity patterns obtained for = 2 → 3 in steps of 0.1. The dashed ellipse indicates the region where the birth of the vortex is observed. The discrepancy between the broadband and monochromatic data in terms of pattern orientation and region of vortex birth is explained in the final paragraph of this section 3.

Fig. 8
Fig. 8

Spatial dependency of the birth of the vortex on the orientation of the triangular aperture. The figure shows measured monochromatic intensity patterns obtained for = 2.6 and different orientations of the triangular aperture as indicated on the right-bottom corner of the patterns. The dashed rectangles indicate where the birth of the vortex should be observed according to the rotation of the diffraction pattern. The dashed ellipse shows the spatial location where the birth of the vortex is actually observed.

Equations (13)

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

E ( ) ( x , y , θ ) = A ( x + i sgn ( ) y ) | | E ( ) ( ρ , θ , 0 ) = A ρ | | e i θ
E ( ) ( X , Y , z ) d x d y t ( x , y ) ( x + i sgn ( ) y ) | | e i ( X x + Y y ) ,
E ( ) ( ρ , θ , z = 0 ) = t ( x , y ) ( x + i sgn ( ) y ) | | = t ( ρ , θ ) ρ | | e i θ .
U p , m ( ρ , θ , z ) = u p , | m | ( ρ , z ) e i k ρ 2 / 2 R ( z ) e i m θ e i ( 2 p + | m | + 1 ) Φ ( z ) ,
u p , | m | ( ρ , z ) = ( 2 p ! ( 1 + δ 0 , | m | ) π ( | m | + p ) ! ) 1 w ( z ) ( 2 ρ w ( z ) ) | m | L p | m | ( 2 ρ 2 w 2 ( z ) ) e ρ 2 / w 2 ( z ) ,
E ( ) ( ρ , θ , z ) = p = 0 m = C p , m ( ) u p , | m | ( ρ , z ) e i k ρ 2 / 2 R ( z ) e i m θ e i ( 2 p + | m | + 1 ) Φ ( z ) ,
C p , m ( ) = 0 ρ d ρ 0 2 π d θ E ( ) ( ρ , θ , 0 ) U p , m * ( ρ , θ , 0 ) = 0 ρ d ρ 0 2 π d θ t ( ρ , θ ) ρ | | e i θ ( m ) u p , | m | ( ρ , 0 ) .
C p , m ( ) = ( C p , m ( ) ) * ,
E ( ) ( ρ , θ , z ) = e i k ρ 2 / 2 R ( z ) p = 0 i ( 1 ) p m = C p , m ( ) u p , | m | ( ρ , z ) e i ( m θ + | m | π / 2 ) = e i k ρ 2 / 2 R ( z ) F ( ) ( ρ , θ , z ) ,
F ( ) ( ρ , θ , z ) = p = 0 i ( 1 ) p m = C p , m ( ) u p , | m | ( ρ , z ) e i ( m θ + | m | π / 2 ) .
I ( ) ( ρ , θ , z ) = | E ( ) ( ρ , θ , z ) | 2 = | F ( ) ( ρ , θ , z ) | 2 .
E ( ) ( ρ , θ , z ) = e i k ρ 2 / 2 R ( z ) p = 0 i ( 1 ) p m = C p , m ( ) u p , | m | ( ρ , z ) e i ( m θ + | m | π / 2 ) = e i k ρ 2 / 2 R ( z ) p = 0 i ( 1 ) p m = C p , m ( ) u p , | m | ( ρ , z ) e i ( m θ + | m | π / 2 ) = e i k ρ 2 / 2 R ( z ) p = 0 i ( 1 ) p m = ( C p , m ( ) ) * u p , | m | ( ρ , z ) e i ( m θ + | m | π / 2 ) e i m π = e i k ρ 2 / 2 R ( z ) p = 0 i ( 1 ) p m = ( C p , m ( ) ) * u p , | m | ( ρ , z ) e i ( m ( θ π ) + | m | π / 2 ) = e i k ρ 2 / 2 R ( z ) [ F ( m ) ( ρ , θ π , z ) ] * ,
I ( ) ( ρ , θ , z ) = | F ( ) ( ρ , θ π , z ) | 2 = I ( ) ( ρ , θ π , z ) .

Metrics