Abstract

A simple technique for determining the topological charge of supercontinuum optical vortices is presented. The spatial dispersion inherent to generating broadband vortices with a single forked grating is compensated with a double-pass arrangement from a single spatial light modulator. The vortex charge is determined by inspecting the diffraction pattern through a triangular aperture. It is shown that the topological charge is constant, and can be consistently measured, across a wide range of colors.

© 2012 Optical Society of America

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References

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  1. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. A 336, 165–190 (1974).
    [CrossRef]
  2. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transfomation of Laguerre-Guassian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
    [CrossRef]
  3. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Progress in Optics (Elsevier, 2009), Vol. 53, pp. 293–363.
    [CrossRef]
  4. L. Allen and M. Padgett, “The orbital angular momentum of light: An introduction,” in Twisted Photons: Applications of Light with Orbital Angular Momentum1st ed., J. P. Torres and L. Torner, eds. (Wiley-VCH, 2011).
  5. K. Bezuhanov, A. Dreischuh, G. G. Paulus, M. G. Schätzel, and H. Walther, “Vortices in femtosecond laser fields,” Opt. Lett. 29, 1942–1944 (2004).
    [CrossRef]
  6. I. G. Mariyenko, J. Strohaber, and C. J. G. J. Uiterwaal, “Creation of optical vortices in femtosecond pulses,” Opt. Express 13, 7599–7608 (2005).
    [CrossRef]
  7. K. Bezuhanov, A. Dreischuh, G. G. Paulus, M. G. Schätzel, H. Walther, D. Neshev, W. Królikowski, and Y. Kivshar, “Spatial phase dislocations in femtosecond laser pulses,” J. Opt. Soc. Am. B 23, 26–35 (2006).
    [CrossRef]
  8. I. Zeylikovich, H. I. Sztul, V. Kartazaev, T. Le, and R. R. Alfano, “Ultrashort Laguerre-Gaussian pulses with angular and group velocity dispersion compensation,” Opt. Lett. 32, 2025–2027 (2007).
    [CrossRef]
  9. A. Schwarz and W. Rudolph, “Dispersion-compensating beam shaper for femtosecond optical vortex beams,” Opt. Lett. 33, 2970–2972 (2008).
    [CrossRef]
  10. Ó. Martínez-Matos, José A. Rodrigo, M. P. Hernández-Garay, J. G. Izquierdo, R. Weigand, M. L. Calvo, P. Cheben, P. Vaveliuk, and L. Bañares, “Generation of femtosecond paraxial beams with arbitrary spatial distribution,” Opt. Lett. 35, 652–654 (2010).
    [CrossRef]
  11. A. J. Wright, J. M. Girkin, G. M. Gibson, J. Leach, and M. J. Padgett, “Transfer of orbital angular momentum from a super-continuum, white-light beam,” Opt. Express 16, 9495–9500 (2008).
    [CrossRef]
  12. M. V. Berry, “Coloured phase singularities,” New J. Phys. 4, 66.1–66.14 (2002).
    [CrossRef]
  13. J. Leach and M. J. Padgett, “Observation of chromatic effects near a white-light vortex,” New J. Phys. 5, 154 (2003).
    [CrossRef]
  14. J. Leach, G. M. Gibson, M. J. Padgett, E. Esposito, G. McConnell, A. J. Wright, and J. M. Girkin, “Generation of achromatic Bessel beams using a compensated spatial light modulator,” Opt. Express 14, 5581–5587 (2006).
    [CrossRef]
  15. Y. Tokizane, K. Oka, and R. Morita, “Supercontinuum optical vortex pulse generation without spatial or topological-charge dispersion,” Opt. Express 17, 14517–14525 (2009).
    [CrossRef]
  16. I. Moreno, J. A. Davis, B. M. L. Pascoguin, M. J. Mitry, and D. M. Cottrell, “Vortex sensing diffraction gratings,” Opt. Lett. 34, 2927–2929 (2009).
    [CrossRef]
  17. G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
    [CrossRef]
  18. M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt. 13, 064006 (2011).
    [CrossRef]
  19. X. C. Yuan, N. Zhang, and R. E. Burge, “Extending the detection range of optical vortices by Dammann vortex gratings,” Opt. Lett. 35, 3495–3497 (2010).
    [CrossRef]
  20. V. Denisenko, V. Shvedov, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, A. Volyar, M. Soskin, and Y. S. Kivshar, “Determination of topological charges of polychromatic optical vortices,” Opt. Express 17, 23374–23379 (2009).
    [CrossRef]
  21. V. G. Shvedov, C. Hnatovsky, W. Krolikowski, and A. V. Rode, “Efficient beam converter for the generation of high-power femtosecond vortices,” Opt. Lett. 35, 2660–2662 (2010).
    [CrossRef]
  22. 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 light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
    [CrossRef]
  23. L. E. E. de Araujo and M. E. Anderson, “Measuring vortex charge with a triangular aperture,” Opt. Lett. 36, 787–789 (2011).
    [CrossRef]
  24. A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express 19, 5760–5771 (2011).
    [CrossRef]
  25. M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999).
    [CrossRef]
  26. D. Ganic, X-S. Gan, M. Gu, M. Hain, S. Somalingam, S. Stankovic, and T. Tschudi, “Generation of doughnut laser beams by use of a liquid-crystal cell with a conversion efficiency near 100%,” Opt. Lett. 27, 1351–1353 (2002).
    [CrossRef]
  27. K. Crabtree, J. A. Davis, and I. Moreno, “Optical processing with vortex-producing lenses,” Appl. Opt. 43, 1360–1367 (2004).
    [CrossRef]
  28. G. F. Brand, “The forked blazed grating as a quasioptical element For gyrotron applications,” Internation. J. Infrared Mill. Waves 20, 1207–1219 (1999).
    [CrossRef]
  29. A. Ya Bekshaev, and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281, 1366–1374(2008).
    [CrossRef]
  30. R. C. Smith and J. S. Marsh, “Diffraction patterns of simple apertures,” J. Opt. Soc. Am. 64, 798–803 (1974).
    [CrossRef]
  31. J. Albero, I. Moreno, and J. Campos, “Enhancement of the broadband modulation diffraction efficiency of liquid-crystal displays,” Opt. Lett. 37, 52–54 (2012).
    [CrossRef]
  32. A. Jesacher, A. Schwaighofer, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Wavefront correction of spatial light modulators using an optical vortex image,” Opt. Express 15, 5801–5808 (2007).
    [CrossRef]
  33. E. M. Kosik, A. Radunsky, I. A. Walmsley, and C. Dorrer, “Interferometric technique for measuring broadband ultrashort pulses at the sampling limit,” Opt. Lett. 30, 326–328 (2005).
    [CrossRef]

2012 (1)

2011 (3)

2010 (5)

2009 (3)

2008 (3)

2007 (2)

2006 (2)

2005 (2)

2004 (2)

2003 (1)

J. Leach and M. J. Padgett, “Observation of chromatic effects near a white-light vortex,” New J. Phys. 5, 154 (2003).
[CrossRef]

2002 (2)

1999 (2)

M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999).
[CrossRef]

G. F. Brand, “The forked blazed grating as a quasioptical element For gyrotron applications,” Internation. J. Infrared Mill. Waves 20, 1207–1219 (1999).
[CrossRef]

1992 (1)

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

1974 (2)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. A 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]

Albero, J.

Alfano, R. R.

Allen, L.

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

L. Allen and M. Padgett, “The orbital angular momentum of light: An introduction,” in Twisted Photons: Applications of Light with Orbital Angular Momentum1st ed., J. P. Torres and L. Torner, eds. (Wiley-VCH, 2011).

Anderson, M. E.

Bañares, L.

Baumgartl, J.

Beijersbergen, M. W.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

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

Bekshaev, A. Ya

A. Ya Bekshaev, and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281, 1366–1374(2008).
[CrossRef]

Berkhout, G. C. G.

M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt. 13, 064006 (2011).
[CrossRef]

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

Bernet, S.

Berry, M. V.

M. V. Berry, “Coloured phase singularities,” New J. Phys. 4, 66.1–66.14 (2002).
[CrossRef]

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

Bezuhanov, K.

Brand, G. F.

G. F. Brand, “The forked blazed grating as a quasioptical element For gyrotron applications,” Internation. J. Infrared Mill. Waves 20, 1207–1219 (1999).
[CrossRef]

Burge, R. E.

Calvo, M. L.

Campos, J.

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 light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef]

Cheben, P.

Cottrell, D. M.

Courtial, J.

M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt. 13, 064006 (2011).
[CrossRef]

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

Crabtree, K.

Davis, J. A.

de Araujo, L. E. E.

Denisenko, V.

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Progress in Optics (Elsevier, 2009), Vol. 53, pp. 293–363.
[CrossRef]

Desyatnikov, A. S.

Dholakia, K.

Dorrer, C.

Dreischuh, A.

Esposito, E.

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 light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef]

Fürhapter, S.

Gan, X-S.

Ganic, D.

Gibson, G. M.

Girkin, J. M.

Gu, M.

Hain, M.

Haist, T.

Hernández-Garay, M. P.

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 light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef]

Hnatovsky, C.

Izquierdo, J. G.

Jesacher, A.

Karamoch, A. I.

A. Ya Bekshaev, and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281, 1366–1374(2008).
[CrossRef]

Kartazaev, V.

Kivshar, Y.

Kivshar, Y. S.

Kosik, E. M.

Krolikowski, W.

Królikowski, W.

Lavery, M. P. J.

M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt. 13, 064006 (2011).
[CrossRef]

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

Le, T.

Leach, J.

Mariyenko, I. G.

Marsh, J. S.

Martínez-Matos, Ó.

Maurer, C.

McConnell, G.

Mitry, M. J.

Moreno, I.

Morita, R.

Mourka, A.

Neshev, D.

Neshev, D. N.

Nye, J. F.

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

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Progress in Optics (Elsevier, 2009), Vol. 53, pp. 293–363.
[CrossRef]

Oka, K.

Padgett, M.

L. Allen and M. Padgett, “The orbital angular momentum of light: An introduction,” in Twisted Photons: Applications of Light with Orbital Angular Momentum1st ed., J. P. Torres and L. Torner, eds. (Wiley-VCH, 2011).

Padgett, M. J.

M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt. 13, 064006 (2011).
[CrossRef]

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

A. J. Wright, J. M. Girkin, G. M. Gibson, J. Leach, and M. J. Padgett, “Transfer of orbital angular momentum from a super-continuum, white-light beam,” Opt. Express 16, 9495–9500 (2008).
[CrossRef]

J. Leach, G. M. Gibson, M. J. Padgett, E. Esposito, G. McConnell, A. J. Wright, and J. M. Girkin, “Generation of achromatic Bessel beams using a compensated spatial light modulator,” Opt. Express 14, 5581–5587 (2006).
[CrossRef]

J. Leach and M. J. Padgett, “Observation of chromatic effects near a white-light vortex,” New J. Phys. 5, 154 (2003).
[CrossRef]

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Progress in Optics (Elsevier, 2009), Vol. 53, pp. 293–363.
[CrossRef]

Pascoguin, B. M. L.

Paulus, G. G.

Radunsky, A.

Reicherter, M.

Ritsch-Marte, M.

Rode, A. V.

Rodrigo, José A.

Rudolph, W.

Schätzel, M. G.

Schwaighofer, A.

Schwarz, A.

Shanor, C.

Shvedov, V.

Shvedov, V. G.

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 light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef]

Somalingam, S.

Soskin, M.

Spreeuw, R. J. C.

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

Stankovic, S.

Strohaber, J.

Sztul, H. I.

Tiziani, H. J.

Tokizane, Y.

Tschudi, T.

Uiterwaal, C. J. G. J.

Vaveliuk, P.

Volyar, A.

Wagemann, E. U.

Walmsley, I. A.

Walther, H.

Weigand, R.

Woerdman, J. P.

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

Wright, A. J.

Wright, E. M.

Yuan, X. C.

Zeylikovich, I.

Zhang, N.

Appl. Opt. (1)

Internation. J. Infrared Mill. Waves (1)

G. F. Brand, “The forked blazed grating as a quasioptical element For gyrotron applications,” Internation. J. Infrared Mill. Waves 20, 1207–1219 (1999).
[CrossRef]

J. Opt. (1)

M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt. 13, 064006 (2011).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

New J. Phys. (2)

M. V. Berry, “Coloured phase singularities,” New J. Phys. 4, 66.1–66.14 (2002).
[CrossRef]

J. Leach and M. J. Padgett, “Observation of chromatic effects near a white-light vortex,” New J. Phys. 5, 154 (2003).
[CrossRef]

Opt. Commun. (1)

A. Ya Bekshaev, and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281, 1366–1374(2008).
[CrossRef]

Opt. Express (7)

Opt. Lett. (12)

J. Albero, I. Moreno, and J. Campos, “Enhancement of the broadband modulation diffraction efficiency of liquid-crystal displays,” Opt. Lett. 37, 52–54 (2012).
[CrossRef]

I. Moreno, J. A. Davis, B. M. L. Pascoguin, M. J. Mitry, and D. M. Cottrell, “Vortex sensing diffraction gratings,” Opt. Lett. 34, 2927–2929 (2009).
[CrossRef]

Ó. Martínez-Matos, José A. Rodrigo, M. P. Hernández-Garay, J. G. Izquierdo, R. Weigand, M. L. Calvo, P. Cheben, P. Vaveliuk, and L. Bañares, “Generation of femtosecond paraxial beams with arbitrary spatial distribution,” Opt. Lett. 35, 652–654 (2010).
[CrossRef]

V. G. Shvedov, C. Hnatovsky, W. Krolikowski, and A. V. Rode, “Efficient beam converter for the generation of high-power femtosecond vortices,” Opt. Lett. 35, 2660–2662 (2010).
[CrossRef]

X. C. Yuan, N. Zhang, and R. E. Burge, “Extending the detection range of optical vortices by Dammann vortex gratings,” Opt. Lett. 35, 3495–3497 (2010).
[CrossRef]

L. E. E. de Araujo and M. E. Anderson, “Measuring vortex charge with a triangular aperture,” Opt. Lett. 36, 787–789 (2011).
[CrossRef]

A. Schwarz and W. Rudolph, “Dispersion-compensating beam shaper for femtosecond optical vortex beams,” Opt. Lett. 33, 2970–2972 (2008).
[CrossRef]

I. Zeylikovich, H. I. Sztul, V. Kartazaev, T. Le, and R. R. Alfano, “Ultrashort Laguerre-Gaussian pulses with angular and group velocity dispersion compensation,” Opt. Lett. 32, 2025–2027 (2007).
[CrossRef]

D. Ganic, X-S. Gan, M. Gu, M. Hain, S. Somalingam, S. Stankovic, and T. Tschudi, “Generation of doughnut laser beams by use of a liquid-crystal cell with a conversion efficiency near 100%,” Opt. Lett. 27, 1351–1353 (2002).
[CrossRef]

K. Bezuhanov, A. Dreischuh, G. G. Paulus, M. G. Schätzel, and H. Walther, “Vortices in femtosecond laser fields,” Opt. Lett. 29, 1942–1944 (2004).
[CrossRef]

E. M. Kosik, A. Radunsky, I. A. Walmsley, and C. Dorrer, “Interferometric technique for measuring broadband ultrashort pulses at the sampling limit,” Opt. Lett. 30, 326–328 (2005).
[CrossRef]

M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999).
[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 transfomation of Laguerre-Guassian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef]

Phys. Rev. Lett. (2)

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

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 light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef]

Proc. R. Soc. A (1)

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

Other (2)

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Progress in Optics (Elsevier, 2009), Vol. 53, pp. 293–363.
[CrossRef]

L. Allen and M. Padgett, “The orbital angular momentum of light: An introduction,” in Twisted Photons: Applications of Light with Orbital Angular Momentum1st ed., J. P. Torres and L. Torner, eds. (Wiley-VCH, 2011).

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Figures (9)

Fig. 1.
Fig. 1.

Spiral phase combined with a phase tilt results in the familiar forked phase pattern. (a) A spiral phase ranging from 010π is (b) displayed modulo 2π, resulting in a phase plate with five 02π segments. Likewise, (c) a phase tilt is (d) displayed modulo 2π, resulting in a blazed grating. The addition of these two patterns results in (e) a forked phase pattern.

Fig. 2.
Fig. 2.

Unfolded, 6f optical system used for spatial chirp correction. From left to right, the field amplitude at each plane is indicated by U(x,y). A grating is placed at plane 0 (the x0, y0 plane, which is the first reflection from the SLM), nothing is placed at plane 2 (reflection from our mirror), and a forked grating is placed at plane 4 (second pass off the SLM). The outgoing field is at plane 6. Lenses L1–L3 are all identical.

Fig. 3.
Fig. 3.

Calculated cross sections of optical vortices (charge m=3) emerging from the (a) doubly chirped and (b) zero-chirp optical system. The three vortices shown in both cases correspond to the wavelengths 550 nm (dashed green line), 580 nm (solid orange line), and 619 nm (dotted red line). For the calculations, we used f=70cm, w0=173μm, and grating period d=325μm. These parameters were chosen to approximate the conditions of our experiment.

Fig. 4.
Fig. 4.

Experimental apparatus for generation of dispersion-free supercontinuum vortices. The items in the dashed box (triangular aperture directly in front of lens) are added to measure the topological charge. SCG, supercontinuum generator; M1, M2, mirrors; L1, 70 cm focal length achromatic lens, 2” diameter; L2=10cm and L3=15cm, focal lengths, 1” diameter; SLM, spatial light modulator. Beam path is approximate, as beams are large, filling each half of the SLM. Path lengths are as follows: SCG to L1=70cm, L1 to SLM=70cm, L1 to Fourier plane=70cm, Fourier plane to L220cm, L2 to L320cm, L3 to CCD=15cm.

Fig. 5.
Fig. 5.

Screenshot of the grayscale pattern displayed on the SLM. The left half of the SLM displays an apodized forked grating for generating a vortex beam. The right half displays only a blazed grating.

Fig. 6.
Fig. 6.

Charge 3 vortex is generated in the doubly dispersive case for (a) continuous wave (CW), (b) femtosecond (fs), and (c) supercontinuum (NIR SC). The dispersion-corrected case is shown for (d) CW, (e) fs, and (f) NIR SC.

Fig. 7.
Fig. 7.

Color vortices are recorded with a color DSLR CCD array. The left column shows the doubly dispersive results at wavelengths (a) 550 nm, (b) 580 nm, (c) 619 nm, and (d) full spectrum. The right column shows the dispersion-corrected results for (e) 550 nm, (f) 580 nm, (g) 619 nm, and (h) full spectrum.

Fig. 8.
Fig. 8.

False-color diffraction patterns in the near infrared from a 0.73 mm triangular aperture of charge 3 vortices. The left column shows the doubly dispersive case for (a) cw, (b) femtosecond, and (c) supercontinuum modes. The right column shows the dispersion-corrected case for (d) cw, (e) femtosecond, and (f) supercontinuum modes.

Fig. 9.
Fig. 9.

True-color diffraction patterns in the visible for a charge 3 supercontinuum vortex from a 0.73 mm triangular aperture: (a) 523 nm, (b) 580 nm, (c) 620 nm, and (d) full spectrum.

Equations (10)

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u0(x,y)exp{x2+y2w02},
uG(x0,y0)exp{x02+y02w02}exp{in2πdx0},
u2f(x2,y2)exp{x02+y02w02}exp{in2πdx0}×exp{ikf(x0x2+y0y2)}dx0dy0.
ua(x2,y2)exp{(x2λf/d)2+y22(λf/πw0)2}.
u4f(x4,y4)++exp{(x2λf/d)2+y22(λf/πw0)2}×exp{ikf(x2x4+y2y4)}dx2dy2.
u4f(x4,y4)exp{x42+y42w02}exp{i2πdx4}.
uFG(x4,y4)exp{x42+y42w02}exp[imtan1(y4/x4)]×exp{i2πdx4(1+N)},
uFG(r4,φ4)exp{r42w02}exp{imφ4}×exp[i2πd(1+N)r4cosφ4].
E(r6,φ6)(pq)2w02π3/24(i)mexp{(pq)2w028}×[Ym12((pq)2w028)Ym+12((pq)2w028)],
E(kx,ky)δ(kx23m/a),

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