Abstract

Optical vortices can be easily generated using a multilevel spiral phase plate (SPP). However the quality of the generated vortex beam depends on the number of phase segments. We review the theory for this multilevel SPP using a Fourier expansion of integer topological charged vortices. We then experimentally examine the validity of this expansion using a fabricated Dammann vortex diffraction grating spectrum analyzer. The Dammann vortex diffraction grating is fabricated using SU-8 photoresist on glass substrate and yields uniform diffraction efficiency across a desired number of diffracted orders. Experimental results show the extra angular harmonics of a multilevel SPP and agree with the theory.

© 2010 OSA

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  1. 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(5), 826–829 (1995).
    [CrossRef] [PubMed]
  2. J. A. Davis, D. E. McNamara, D. M. Cottrell, and J. Campos, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25(2), 99–101 (2000).
    [CrossRef]
  3. C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230(1), 134–142 (2008).
    [CrossRef] [PubMed]
  4. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
    [CrossRef] [PubMed]
  5. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
    [CrossRef] [PubMed]
  6. S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
    [CrossRef]
  7. 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(5–6), 321–327 (1994).
    [CrossRef]
  8. W. C. Cheong, W. M. Lee, X. C. Yuan, L. S. Zhang, K. Dholakia, and H. Wang, “Direct electron-beam writing of continuous spiral phase plates in negative resist with high power efficiency for optical manipulation,” Appl. Phys. Lett. 85(23), 5784–5786 (2004).
    [CrossRef]
  9. C.-S. Guo, D.-M. Xue, Y.-J. Han, and J. Ding, “Optimal phase steps of multi-level spiral phase plates,” Opt. Commun. 268(2), 235–239 (2006).
    [CrossRef]
  10. V. V. Kotlyar and A. A. Kovalev, “Fraunhofer diffraction of the plane wave by a multilevel (quantized) spiral phase plate,” Opt. Lett. 33(2), 189–191 (2008).
    [CrossRef] [PubMed]
  11. I. Moreno, J. A. Davis, B. M. L. Pascoguin, M. J. Mitry, and D. M. Cottrell, “Vortex sensing diffraction gratings,” Opt. Lett. 34(19), 2927–2929 (2009).
    [CrossRef] [PubMed]
  12. I. Moreno, J. A. Davis, I. Ruiz, and D. M. Cottrell, “Decomposition of radially and azimuthally polarized beams using a circular-polarization and vortex-sensing diffraction grating,” Opt. Express 18(7), 7173–7183 (2010).
    [CrossRef] [PubMed]
  13. N. Zhang, J. A. Davis, I Moreno, J Lin, K.-J Moh, D. M Cottrell, and X Yuan, “Analysis of fractional vortex beams using a vortex grating spectrum analyzer,” Appl. Opt. 49(13), 2456–2462 (2010).
    [CrossRef]
  14. S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Pääkkönen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48(10), 1543–1557 (2001).
  15. G. J. Swanson and W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng . 28(Compendex), 605–608 (1989).
  16. J. Jahns and S. J. Walker, “Two-dimensional array of diffractive microlenses fabricated by thin film deposition,” Appl. Opt. 29(7), 931–936 (1990).
    [CrossRef] [PubMed]
  17. H. Dammann and K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3(5), 312–315 (1971).
    [CrossRef]
  18. H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta (Lond.) 24(4), 505–515 (1977).
  19. U. Krackhardt and N. Streibl, “Design of dammann-gratings for array generation,” Opt. Commun. 74(1–2), 31–36 (1989).
    [CrossRef]
  20. I. Moreno, J. A. Davis, D. M. Cottrell, N. Zhang, and X. C. Yuan, “Encoding generalized phase functions on Dammann gratings,” Opt. Lett. 35(10), 1536–1538 (2010).
    [CrossRef] [PubMed]

2010 (3)

I. Moreno, J. A. Davis, I. Ruiz, and D. M. Cottrell, “Decomposition of radially and azimuthally polarized beams using a circular-polarization and vortex-sensing diffraction grating,” Opt. Express 18(7), 7173–7183 (2010).
[CrossRef] [PubMed]

N. Zhang, J. A. Davis, I Moreno, J Lin, K.-J Moh, D. M Cottrell, and X Yuan, “Analysis of fractional vortex beams using a vortex grating spectrum analyzer,” Appl. Opt. 49(13), 2456–2462 (2010).
[CrossRef]

I. Moreno, J. A. Davis, D. M. Cottrell, N. Zhang, and X. C. Yuan, “Encoding generalized phase functions on Dammann gratings,” Opt. Lett. 35(10), 1536–1538 (2010).
[CrossRef] [PubMed]

2009 (1)

I. Moreno, J. A. Davis, B. M. L. Pascoguin, M. J. Mitry, and D. M. Cottrell, “Vortex sensing diffraction gratings,” Opt. Lett. 34(19), 2927–2929 (2009).
[CrossRef] [PubMed]

2008 (2)

V. V. Kotlyar and A. A. Kovalev, “Fraunhofer diffraction of the plane wave by a multilevel (quantized) spiral phase plate,” Opt. Lett. 33(2), 189–191 (2008).
[CrossRef] [PubMed]

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230(1), 134–142 (2008).
[CrossRef] [PubMed]

2006 (1)

C.-S. Guo, D.-M. Xue, Y.-J. Han, and J. Ding, “Optimal phase steps of multi-level spiral phase plates,” Opt. Commun. 268(2), 235–239 (2006).
[CrossRef]

2004 (2)

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
[CrossRef] [PubMed]

W. C. Cheong, W. M. Lee, X. C. Yuan, L. S. Zhang, K. Dholakia, and H. Wang, “Direct electron-beam writing of continuous spiral phase plates in negative resist with high power efficiency for optical manipulation,” Appl. Phys. Lett. 85(23), 5784–5786 (2004).
[CrossRef]

2001 (1)

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Pääkkönen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48(10), 1543–1557 (2001).

2000 (1)

J. A. Davis, D. E. McNamara, D. M. Cottrell, and J. Campos, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25(2), 99–101 (2000).
[CrossRef]

1995 (1)

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(5), 826–829 (1995).
[CrossRef] [PubMed]

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(5–6), 321–327 (1994).
[CrossRef]

1992 (2)

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
[CrossRef] [PubMed]

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[CrossRef]

1990 (1)

J. Jahns and S. J. Walker, “Two-dimensional array of diffractive microlenses fabricated by thin film deposition,” Appl. Opt. 29(7), 931–936 (1990).
[CrossRef] [PubMed]

1989 (2)

G. J. Swanson and W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng . 28(Compendex), 605–608 (1989).

U. Krackhardt and N. Streibl, “Design of dammann-gratings for array generation,” Opt. Commun. 74(1–2), 31–36 (1989).
[CrossRef]

1977 (1)

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta (Lond.) 24(4), 505–515 (1977).

1971 (1)

H. Dammann and K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3(5), 312–315 (1971).
[CrossRef]

Barnett, S.

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
[CrossRef] [PubMed]

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(5–6), 321–327 (1994).
[CrossRef]

Bernet, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230(1), 134–142 (2008).
[CrossRef] [PubMed]

Campos, J.

J. A. Davis, D. E. McNamara, D. M. Cottrell, and J. Campos, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25(2), 99–101 (2000).
[CrossRef]

Cheong, W. C.

W. C. Cheong, W. M. Lee, X. C. Yuan, L. S. Zhang, K. Dholakia, and H. Wang, “Direct electron-beam writing of continuous spiral phase plates in negative resist with high power efficiency for optical manipulation,” Appl. Phys. Lett. 85(23), 5784–5786 (2004).
[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(5–6), 321–327 (1994).
[CrossRef]

Cottrell, D. M

N. Zhang, J. A. Davis, I Moreno, J Lin, K.-J Moh, D. M Cottrell, and X Yuan, “Analysis of fractional vortex beams using a vortex grating spectrum analyzer,” Appl. Opt. 49(13), 2456–2462 (2010).
[CrossRef]

Cottrell, D. M.

I. Moreno, J. A. Davis, I. Ruiz, and D. M. Cottrell, “Decomposition of radially and azimuthally polarized beams using a circular-polarization and vortex-sensing diffraction grating,” Opt. Express 18(7), 7173–7183 (2010).
[CrossRef] [PubMed]

I. Moreno, J. A. Davis, D. M. Cottrell, N. Zhang, and X. C. Yuan, “Encoding generalized phase functions on Dammann gratings,” Opt. Lett. 35(10), 1536–1538 (2010).
[CrossRef] [PubMed]

I. Moreno, J. A. Davis, B. M. L. Pascoguin, M. J. Mitry, and D. M. Cottrell, “Vortex sensing diffraction gratings,” Opt. Lett. 34(19), 2927–2929 (2009).
[CrossRef] [PubMed]

J. A. Davis, D. E. McNamara, D. M. Cottrell, and J. Campos, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25(2), 99–101 (2000).
[CrossRef]

Courtial, J.

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
[CrossRef] [PubMed]

Dammann, H.

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta (Lond.) 24(4), 505–515 (1977).

H. Dammann and K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3(5), 312–315 (1971).
[CrossRef]

Davis, J. A.

I. Moreno, J. A. Davis, D. M. Cottrell, N. Zhang, and X. C. Yuan, “Encoding generalized phase functions on Dammann gratings,” Opt. Lett. 35(10), 1536–1538 (2010).
[CrossRef] [PubMed]

I. Moreno, J. A. Davis, I. Ruiz, and D. M. Cottrell, “Decomposition of radially and azimuthally polarized beams using a circular-polarization and vortex-sensing diffraction grating,” Opt. Express 18(7), 7173–7183 (2010).
[CrossRef] [PubMed]

N. Zhang, J. A. Davis, I Moreno, J Lin, K.-J Moh, D. M Cottrell, and X Yuan, “Analysis of fractional vortex beams using a vortex grating spectrum analyzer,” Appl. Opt. 49(13), 2456–2462 (2010).
[CrossRef]

I. Moreno, J. A. Davis, B. M. L. Pascoguin, M. J. Mitry, and D. M. Cottrell, “Vortex sensing diffraction gratings,” Opt. Lett. 34(19), 2927–2929 (2009).
[CrossRef] [PubMed]

J. A. Davis, D. E. McNamara, D. M. Cottrell, and J. Campos, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25(2), 99–101 (2000).
[CrossRef]

Dholakia, K.

W. C. Cheong, W. M. Lee, X. C. Yuan, L. S. Zhang, K. Dholakia, and H. Wang, “Direct electron-beam writing of continuous spiral phase plates in negative resist with high power efficiency for optical manipulation,” Appl. Phys. Lett. 85(23), 5784–5786 (2004).
[CrossRef]

Ding, J.

C.-S. Guo, D.-M. Xue, Y.-J. Han, and J. Ding, “Optimal phase steps of multi-level spiral phase plates,” Opt. Commun. 268(2), 235–239 (2006).
[CrossRef]

Franke-Arnold, S.

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
[CrossRef] [PubMed]

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(5), 826–829 (1995).
[CrossRef] [PubMed]

Fürhapter, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230(1), 134–142 (2008).
[CrossRef] [PubMed]

Gibson, G.

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
[CrossRef] [PubMed]

Görtler, K.

H. Dammann and K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3(5), 312–315 (1971).
[CrossRef]

Guo, C.-S.

C.-S. Guo, D.-M. Xue, Y.-J. Han, and J. Ding, “Optimal phase steps of multi-level spiral phase plates,” Opt. Commun. 268(2), 235–239 (2006).
[CrossRef]

Han, Y.-J.

C.-S. Guo, D.-M. Xue, Y.-J. Han, and J. Ding, “Optimal phase steps of multi-level spiral phase plates,” Opt. Commun. 268(2), 235–239 (2006).
[CrossRef]

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(5), 826–829 (1995).
[CrossRef] [PubMed]

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(5), 826–829 (1995).
[CrossRef] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
[CrossRef] [PubMed]

Jahns, J.

J. Jahns and S. J. Walker, “Two-dimensional array of diffractive microlenses fabricated by thin film deposition,” Appl. Opt. 29(7), 931–936 (1990).
[CrossRef] [PubMed]

Jesacher, A.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230(1), 134–142 (2008).
[CrossRef] [PubMed]

Khonina, S. N.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Pääkkönen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48(10), 1543–1557 (2001).

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[CrossRef]

Klotz, E.

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta (Lond.) 24(4), 505–515 (1977).

Kotlyar, V. V.

V. V. Kotlyar and A. A. Kovalev, “Fraunhofer diffraction of the plane wave by a multilevel (quantized) spiral phase plate,” Opt. Lett. 33(2), 189–191 (2008).
[CrossRef] [PubMed]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Pääkkönen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48(10), 1543–1557 (2001).

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[CrossRef]

Kovalev, A. A.

V. V. Kotlyar and A. A. Kovalev, “Fraunhofer diffraction of the plane wave by a multilevel (quantized) spiral phase plate,” Opt. Lett. 33(2), 189–191 (2008).
[CrossRef] [PubMed]

Krackhardt, U.

U. Krackhardt and N. Streibl, “Design of dammann-gratings for array generation,” Opt. Commun. 74(1–2), 31–36 (1989).
[CrossRef]

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(5–6), 321–327 (1994).
[CrossRef]

Lee, W. M.

W. C. Cheong, W. M. Lee, X. C. Yuan, L. S. Zhang, K. Dholakia, and H. Wang, “Direct electron-beam writing of continuous spiral phase plates in negative resist with high power efficiency for optical manipulation,” Appl. Phys. Lett. 85(23), 5784–5786 (2004).
[CrossRef]

Lin, J

N. Zhang, J. A. Davis, I Moreno, J Lin, K.-J Moh, D. M Cottrell, and X Yuan, “Analysis of fractional vortex beams using a vortex grating spectrum analyzer,” Appl. Opt. 49(13), 2456–2462 (2010).
[CrossRef]

Maurer, C.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230(1), 134–142 (2008).
[CrossRef] [PubMed]

McDuff, R.

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
[CrossRef] [PubMed]

McNamara, D. E.

J. A. Davis, D. E. McNamara, D. M. Cottrell, and J. Campos, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25(2), 99–101 (2000).
[CrossRef]

Mitry, M. J.

I. Moreno, J. A. Davis, B. M. L. Pascoguin, M. J. Mitry, and D. M. Cottrell, “Vortex sensing diffraction gratings,” Opt. Lett. 34(19), 2927–2929 (2009).
[CrossRef] [PubMed]

Moh, K.-J

N. Zhang, J. A. Davis, I Moreno, J Lin, K.-J Moh, D. M Cottrell, and X Yuan, “Analysis of fractional vortex beams using a vortex grating spectrum analyzer,” Appl. Opt. 49(13), 2456–2462 (2010).
[CrossRef]

Moreno, I

N. Zhang, J. A. Davis, I Moreno, J Lin, K.-J Moh, D. M Cottrell, and X Yuan, “Analysis of fractional vortex beams using a vortex grating spectrum analyzer,” Appl. Opt. 49(13), 2456–2462 (2010).
[CrossRef]

Moreno, I.

I. Moreno, J. A. Davis, I. Ruiz, and D. M. Cottrell, “Decomposition of radially and azimuthally polarized beams using a circular-polarization and vortex-sensing diffraction grating,” Opt. Express 18(7), 7173–7183 (2010).
[CrossRef] [PubMed]

I. Moreno, J. A. Davis, D. M. Cottrell, N. Zhang, and X. C. Yuan, “Encoding generalized phase functions on Dammann gratings,” Opt. Lett. 35(10), 1536–1538 (2010).
[CrossRef] [PubMed]

I. Moreno, J. A. Davis, B. M. L. Pascoguin, M. J. Mitry, and D. M. Cottrell, “Vortex sensing diffraction gratings,” Opt. Lett. 34(19), 2927–2929 (2009).
[CrossRef] [PubMed]

Pääkkönen, P.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Pääkkönen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48(10), 1543–1557 (2001).

Padgett, M.

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
[CrossRef] [PubMed]

Pas’ko, V.

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
[CrossRef] [PubMed]

Pascoguin, B. M. L.

I. Moreno, J. A. Davis, B. M. L. Pascoguin, M. J. Mitry, and D. M. Cottrell, “Vortex sensing diffraction gratings,” Opt. Lett. 34(19), 2927–2929 (2009).
[CrossRef] [PubMed]

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230(1), 134–142 (2008).
[CrossRef] [PubMed]

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(5), 826–829 (1995).
[CrossRef] [PubMed]

Ruiz, I.

I. Moreno, J. A. Davis, I. Ruiz, and D. M. Cottrell, “Decomposition of radially and azimuthally polarized beams using a circular-polarization and vortex-sensing diffraction grating,” Opt. Express 18(7), 7173–7183 (2010).
[CrossRef] [PubMed]

Shinkaryev, M. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[CrossRef]

Simonen, J.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Pääkkönen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48(10), 1543–1557 (2001).

Smith, C. P.

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
[CrossRef] [PubMed]

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Pääkkönen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48(10), 1543–1557 (2001).

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[CrossRef]

Streibl, N.

U. Krackhardt and N. Streibl, “Design of dammann-gratings for array generation,” Opt. Commun. 74(1–2), 31–36 (1989).
[CrossRef]

Swanson, G. J.

G. J. Swanson and W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng . 28(Compendex), 605–608 (1989).

Turunen, J.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Pääkkönen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48(10), 1543–1557 (2001).

Uspleniev, G. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[CrossRef]

Vasnetsov, M.

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
[CrossRef] [PubMed]

Veldkamp, W. B.

G. J. Swanson and W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng . 28(Compendex), 605–608 (1989).

Walker, S. J.

J. Jahns and S. J. Walker, “Two-dimensional array of diffractive microlenses fabricated by thin film deposition,” Appl. Opt. 29(7), 931–936 (1990).
[CrossRef] [PubMed]

Wang, H.

W. C. Cheong, W. M. Lee, X. C. Yuan, L. S. Zhang, K. Dholakia, and H. Wang, “Direct electron-beam writing of continuous spiral phase plates in negative resist with high power efficiency for optical manipulation,” Appl. Phys. Lett. 85(23), 5784–5786 (2004).
[CrossRef]

White, A. G.

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
[CrossRef] [PubMed]

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(5–6), 321–327 (1994).
[CrossRef]

Xue, D.-M.

C.-S. Guo, D.-M. Xue, Y.-J. Han, and J. Ding, “Optimal phase steps of multi-level spiral phase plates,” Opt. Commun. 268(2), 235–239 (2006).
[CrossRef]

Yuan, X

N. Zhang, J. A. Davis, I Moreno, J Lin, K.-J Moh, D. M Cottrell, and X Yuan, “Analysis of fractional vortex beams using a vortex grating spectrum analyzer,” Appl. Opt. 49(13), 2456–2462 (2010).
[CrossRef]

Yuan, X. C.

I. Moreno, J. A. Davis, D. M. Cottrell, N. Zhang, and X. C. Yuan, “Encoding generalized phase functions on Dammann gratings,” Opt. Lett. 35(10), 1536–1538 (2010).
[CrossRef] [PubMed]

W. C. Cheong, W. M. Lee, X. C. Yuan, L. S. Zhang, K. Dholakia, and H. Wang, “Direct electron-beam writing of continuous spiral phase plates in negative resist with high power efficiency for optical manipulation,” Appl. Phys. Lett. 85(23), 5784–5786 (2004).
[CrossRef]

Zhang, L. S.

W. C. Cheong, W. M. Lee, X. C. Yuan, L. S. Zhang, K. Dholakia, and H. Wang, “Direct electron-beam writing of continuous spiral phase plates in negative resist with high power efficiency for optical manipulation,” Appl. Phys. Lett. 85(23), 5784–5786 (2004).
[CrossRef]

Zhang, N.

N. Zhang, J. A. Davis, I Moreno, J Lin, K.-J Moh, D. M Cottrell, and X Yuan, “Analysis of fractional vortex beams using a vortex grating spectrum analyzer,” Appl. Opt. 49(13), 2456–2462 (2010).
[CrossRef]

I. Moreno, J. A. Davis, D. M. Cottrell, N. Zhang, and X. C. Yuan, “Encoding generalized phase functions on Dammann gratings,” Opt. Lett. 35(10), 1536–1538 (2010).
[CrossRef] [PubMed]

Appl. Opt. (2)

N. Zhang, J. A. Davis, I Moreno, J Lin, K.-J Moh, D. M Cottrell, and X Yuan, “Analysis of fractional vortex beams using a vortex grating spectrum analyzer,” Appl. Opt. 49(13), 2456–2462 (2010).
[CrossRef]

J. Jahns and S. J. Walker, “Two-dimensional array of diffractive microlenses fabricated by thin film deposition,” Appl. Opt. 29(7), 931–936 (1990).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

W. C. Cheong, W. M. Lee, X. C. Yuan, L. S. Zhang, K. Dholakia, and H. Wang, “Direct electron-beam writing of continuous spiral phase plates in negative resist with high power efficiency for optical manipulation,” Appl. Phys. Lett. 85(23), 5784–5786 (2004).
[CrossRef]

J. Microsc. (1)

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230(1), 134–142 (2008).
[CrossRef] [PubMed]

J. Mod. Opt. (2)

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Pääkkönen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48(10), 1543–1557 (2001).

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[CrossRef]

Opt. Acta (Lond.) (1)

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta (Lond.) 24(4), 505–515 (1977).

Opt. Commun. (4)

U. Krackhardt and N. Streibl, “Design of dammann-gratings for array generation,” Opt. Commun. 74(1–2), 31–36 (1989).
[CrossRef]

H. Dammann and K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3(5), 312–315 (1971).
[CrossRef]

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(5–6), 321–327 (1994).
[CrossRef]

C.-S. Guo, D.-M. Xue, Y.-J. Han, and J. Ding, “Optimal phase steps of multi-level spiral phase plates,” Opt. Commun. 268(2), 235–239 (2006).
[CrossRef]

Opt. Eng (1)

G. J. Swanson and W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng . 28(Compendex), 605–608 (1989).

Opt. Express (2)

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
[CrossRef] [PubMed]

I. Moreno, J. A. Davis, I. Ruiz, and D. M. Cottrell, “Decomposition of radially and azimuthally polarized beams using a circular-polarization and vortex-sensing diffraction grating,” Opt. Express 18(7), 7173–7183 (2010).
[CrossRef] [PubMed]

Opt. Lett. (5)

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
[CrossRef] [PubMed]

J. A. Davis, D. E. McNamara, D. M. Cottrell, and J. Campos, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25(2), 99–101 (2000).
[CrossRef]

V. V. Kotlyar and A. A. Kovalev, “Fraunhofer diffraction of the plane wave by a multilevel (quantized) spiral phase plate,” Opt. Lett. 33(2), 189–191 (2008).
[CrossRef] [PubMed]

I. Moreno, J. A. Davis, B. M. L. Pascoguin, M. J. Mitry, and D. M. Cottrell, “Vortex sensing diffraction gratings,” Opt. Lett. 34(19), 2927–2929 (2009).
[CrossRef] [PubMed]

I. Moreno, J. A. Davis, D. M. Cottrell, N. Zhang, and X. C. Yuan, “Encoding generalized phase functions on Dammann gratings,” Opt. Lett. 35(10), 1536–1538 (2010).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

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(5), 826–829 (1995).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

Sketches of quantized SPPs (first row) and the respective optical vortices generated experimentally (second row), with topological charge L = 1 and step number from M = 2 to M = 5, the grayscale represents 0 (black) to 2π (white) phase values.

Fig. 2
Fig. 2

(a) - (d): The intensity spectrum of topological charge components of an L = 1 quantized SPP with step number M = 2 to M = 5. (e): The intensity of topological charge 1 component Im = 1,L = 1 versus number of steps M of a multilevel SPP.

Fig. 3
Fig. 3

The Fourier spectra of SPPs with (a) M = 3, L = 1; (b) M = 6, L = 2; and (c) M = 9, L = 3. The diffraction efficiencies are the same when the M/L value is constant (M/L = 3).

Fig. 5
Fig. 5

Experimental detection results using the Dammann vortex sensing grating. The input vortex beam has topological charge L = 1. The number of steps of the quantized SPP are listed on the right, and the number of diffracted order are shown at the bottom. The arrows indicate the detected vortex components.

Fig. 4
Fig. 4

(a) 1-D Dammann grating and (b) Dammann vortex grating with embedded topological charge 1, with phase values shown as 0 (black) and π (gray). (c) and (d) are their diffraction patterns respectively.

Equations (5)

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u s ( r , θ ) = exp [ i Δ Φ f l o o r ( M θ 2 π ) ] ,
u s ( r , θ ) = + C m exp ( i m θ ) .
C m = 1 2 π 0 2 π u s ( r , θ ) exp ( i m θ ) d θ .
I m = | C m | 2 = { sinc 2 ( m M ) , m L M = 0 , ± 1 , ± 2 , ... 0 , o t h e r w i s e .
g ( x ) = n = + c n exp [ i ( n q θ + n γ x ) ] ,

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