Abstract

We derive what we believe to be new analytical relations to describe the Fraunhofer diffraction of the finite-radius plane wave by a helical axicon (HA) and a spiral phase plate (SPP). The solutions are deduced in the form of a series of the Bessel functions for the HA and a finite sum of the Bessel functions for the SPP. The solution for the HA changes to that for the SPP if the axicon parameter is set equal to zero. We also derive what we believe to be new analytical relations to describe the Fresnel and Fraunhofer diffraction of the Gaussian beam by a HA are derived. The solutions are deduced in the form of a series of the hypergeometric functions. We have fabricated by photolithography a binary diffractive optical element (a HA with number n=10) able to produce in the focal plane of a spherical lens an optical vortex, which was then used to perform rotation of several polystyrene beads of diameter 5μm.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. V. Shinkaryev, and G. V. Uspleniev, "Trochoson," Opt. Commun. 91, 158-162 (1992).
    [CrossRef]
  2. S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, "The phase rotor filter," J. Mod. Opt. 39, 1147-1154 (1992).
    [CrossRef]
  3. J. A. Davis, E. McNamara, D. M. Cottrell, and J. Campos, "Image processing with the radial Hilbert transform: theory and experiments," Opt. Lett. 25, 99-101 (2000).
    [CrossRef]
  4. C. Guo, Y. Han, J. Xu, and J. Ding, "Radial Hilbert transform with Laguerre-Gaussian spatial filters," Opt. Lett. 31, 1394-1396 (2006).
    [CrossRef] [PubMed]
  5. S. Furhapter, A. Jesacher, S. Beraet, and M. Ritsch-Marte, "Spiral interferometry," Opt. Lett. 30, 1953-1955 (2005).
    [CrossRef] [PubMed]
  6. S. Bernet, A. Jesacher, S. Furhapter, C. Maurer, and M. Ritsch-Marte, "Quantitative imaging of complex samples by spiral phase contrast microscopy," Opt. Express 14, 3792-3805 (2006).
    [CrossRef] [PubMed]
  7. I. G. Foo, D. M. Palecios, and G. A. Swartzlander, "Optical vortex coronograph," Opt. Lett. 30, 3308-3310 (2005).
    [CrossRef]
  8. J. E. Curtis and D. G. Grier, "Structure of optical vortices," Phys. Rev. Lett. 90, 1.33901-1-3 (2003).
    [CrossRef]
  9. J. Lin, X. Yuan, S. H. Tao, and R. E. Burge, "Synthesis of multiple collinear helical modes generated by a phase-only element," J. Opt. Soc. Am. A 23, 1214-1218 (2006).
    [CrossRef]
  10. J. A. Davis, J. Guertin, and D. M. Cottrell, "Diffraction-free beam generated with programmable spatial light modulators," Appl. Opt. 32, 6368-6370 (1993).
    [CrossRef] [PubMed]
  11. J. A. Davis, E. Carcole, and D. M. Cottrell, "Intensity and phase measurements of nondiffracting beams generated with the magneto-optic spatial light modulator," Appl. Opt. 35, 593-598 (1996).
    [CrossRef] [PubMed]
  12. J. A. Davis, E. Carcole, and D. M. Cottrell, "Nondiffracting interference patterns generated with programmable spatial light modulators," Appl. Opt. 35, 599-602 (1996).
    [CrossRef] [PubMed]
  13. N. Chattrapiban, E. A. Rogers, D. Cofield, W. T. Hill, and R. Roy, "Generation of nondiffracting Bessel beams by use of a spatial light modulator," Opt. Lett. 28, 2183-2185 (2003).
    [CrossRef] [PubMed]
  14. A. Hakola, A. Shevchenko, S. C. Buchter, M. Kaivola, and N. V. Tabiryan, "Creation of a narrow Bessel-like laser beam using a nematic liquid crystal," J. Opt. Soc. Am. B 23, 637-641 (2006).
    [CrossRef]
  15. R. Chakraborty and A. Ghosh, "Generation of an elliptic Bessel beam," Opt. Lett. 31, 38-40 (2006).
    [CrossRef] [PubMed]
  16. J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, "Generation of helical Ince-Gaussian beams with a liquid-crystal display," Opt. Lett. 31, 649-651 (2006).
    [CrossRef] [PubMed]
  17. F. K. Fatemi and M. Bashkansky, "Generation of hollow beams by using a binary spatial light modulator," Opt. Lett. 31, 864-866 (2006).
    [CrossRef] [PubMed]
  18. V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Soifer, H. Elfstrom, J. Turunen, "Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate," J. Opt. Soc. Am. A 22, 849-861 (2005).
    [CrossRef]
  19. V. V. Kotlyar, S. N. Khonina, A. A. Kovalev, V. A. Soifer, H. Elfstrom, and J. Turunen, "Diffraction of a plane, finite-radius wave by a spiral phase plate," Opt. Lett. 31, 1597-1599 (2006).
    [CrossRef] [PubMed]
  20. V. V. Kotlyar, A. A. Kovalev, S. N. Khonina, R. V. Skidanov, V. A. Soifer, H. Elfstrom, N. Tossavainen, and J. Turunen, "Diffraction of conic and Gaussian beams by a spiral phase plate," Appl. Opt. 45, 2656-2665 (2006).
    [CrossRef] [PubMed]
  21. B. P. S. Ahluwalia, W. C. Cheong, X.-C. Yuan, L.-S. Zhang, S.-H. Tao, J. Bu, and H. Wang, "Design and fabrication of a double-axicon for generation of tailorable self-imaged three-dimensional intensity voids," Opt. Lett. 31, 987-989 (2006).
    [CrossRef] [PubMed]
  22. D. Rozas, C. T. Law, and G. A. Swartzlander, "Propagation dynamics of optical vortices," J. Opt. Soc. Am. B 14, 3054-3065 (1997).
    [CrossRef]
  23. M. R. Dennis, "Rows of optical vortices from elliptically perturbing a high-order beam," Opt. Lett. 31, 1325-1327 (2006).
    [CrossRef] [PubMed]
  24. D. Ling, J. Li, and J. Chen, "Analysis of eigenfields in the axicon-based Bessel-Gauss resonator by the transfer-matrix method," J. Opt. Soc. Am. A 23, 912-918 (2006).
    [CrossRef]
  25. 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, 5784-5786 (2004).
    [CrossRef]
  26. C. Guo, X. Liu, X. Ren, and H. Wang, "Optimal annular computer-generated holograms for the generation of optical vortices," J. Opt. Soc. Am. A 22, 385-390 (2005).
    [CrossRef]
  27. J. Lin, X. Yuan, S. H. Tao, and R. E. Burge, "Variable-radius focused optical vortex with suppressed sidelobes," Opt. Lett. 31, 1600-1602 (2006).
    [CrossRef] [PubMed]
  28. J. Courtial, G. Whyte, Z. Bouchel, and J. Wagner, "Iterative algorithm for holographic shaping of non-diffracting and self-imaging light beams," Opt. Express 14, 2108-2116 (2006).
    [CrossRef] [PubMed]
  29. G. Whyte and J. Courtial, "Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg-Saxton algorithm," New J. Phys. 7, 1-12 (2005).
    [CrossRef]
  30. Q. Wang, X. W. Sun, P. Shum, and X. J. Yin, "Dynamic switching of optical vortices with dynamic gamma-correction liquid-crystal spiral phase plate," Opt. Express 13, 10285-10291 (2005).
    [CrossRef] [PubMed]
  31. J. Lin, X. Yuan, S. H. Tao, X. Peng, and H. B. Nin, "Deterministic approach to the generation of modified helical beams for optical manipulation," Opt. Express 13, 3862-3867 (2005).
    [CrossRef] [PubMed]
  32. J. Hahn, H. Kim, K. Choi, and B. Lee, "Real-time digital holographic beam-shaping system with a genetic feedback tuning loop," Appl. Opt. 45, 915-924 (2006).
    [CrossRef] [PubMed]
  33. D. Cojoc, E. Di Fabrizio, L. Businaro, S. Carbini, F. Romanato, L. Vaccari, and M. Altissimo, "Design and fabrication of diffractive optical elements for optical tweezer arrays by means of e-beam lithography," Microelectron. Eng. 61-62, 963-969 (2002).
    [CrossRef]
  34. G. A. Swartzlander, "Broadband nulling of a vortex phase mask," Opt. Lett. 30, 2876-2878 (2005).
    [CrossRef] [PubMed]
  35. A. P. Prudnikov, Y. A. Brichkov, and O. I. Marichev, Integrals and Series. Special Functions (Nauka, 1983).
  36. V.A.Soifer, ed., Method for Computer Design of Diffractive Optical Elements (Wiley, 2002).

2006 (15)

C. Guo, Y. Han, J. Xu, and J. Ding, "Radial Hilbert transform with Laguerre-Gaussian spatial filters," Opt. Lett. 31, 1394-1396 (2006).
[CrossRef] [PubMed]

S. Bernet, A. Jesacher, S. Furhapter, C. Maurer, and M. Ritsch-Marte, "Quantitative imaging of complex samples by spiral phase contrast microscopy," Opt. Express 14, 3792-3805 (2006).
[CrossRef] [PubMed]

J. Lin, X. Yuan, S. H. Tao, and R. E. Burge, "Synthesis of multiple collinear helical modes generated by a phase-only element," J. Opt. Soc. Am. A 23, 1214-1218 (2006).
[CrossRef]

A. Hakola, A. Shevchenko, S. C. Buchter, M. Kaivola, and N. V. Tabiryan, "Creation of a narrow Bessel-like laser beam using a nematic liquid crystal," J. Opt. Soc. Am. B 23, 637-641 (2006).
[CrossRef]

R. Chakraborty and A. Ghosh, "Generation of an elliptic Bessel beam," Opt. Lett. 31, 38-40 (2006).
[CrossRef] [PubMed]

J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, "Generation of helical Ince-Gaussian beams with a liquid-crystal display," Opt. Lett. 31, 649-651 (2006).
[CrossRef] [PubMed]

F. K. Fatemi and M. Bashkansky, "Generation of hollow beams by using a binary spatial light modulator," Opt. Lett. 31, 864-866 (2006).
[CrossRef] [PubMed]

V. V. Kotlyar, S. N. Khonina, A. A. Kovalev, V. A. Soifer, H. Elfstrom, and J. Turunen, "Diffraction of a plane, finite-radius wave by a spiral phase plate," Opt. Lett. 31, 1597-1599 (2006).
[CrossRef] [PubMed]

V. V. Kotlyar, A. A. Kovalev, S. N. Khonina, R. V. Skidanov, V. A. Soifer, H. Elfstrom, N. Tossavainen, and J. Turunen, "Diffraction of conic and Gaussian beams by a spiral phase plate," Appl. Opt. 45, 2656-2665 (2006).
[CrossRef] [PubMed]

B. P. S. Ahluwalia, W. C. Cheong, X.-C. Yuan, L.-S. Zhang, S.-H. Tao, J. Bu, and H. Wang, "Design and fabrication of a double-axicon for generation of tailorable self-imaged three-dimensional intensity voids," Opt. Lett. 31, 987-989 (2006).
[CrossRef] [PubMed]

J. Lin, X. Yuan, S. H. Tao, and R. E. Burge, "Variable-radius focused optical vortex with suppressed sidelobes," Opt. Lett. 31, 1600-1602 (2006).
[CrossRef] [PubMed]

J. Courtial, G. Whyte, Z. Bouchel, and J. Wagner, "Iterative algorithm for holographic shaping of non-diffracting and self-imaging light beams," Opt. Express 14, 2108-2116 (2006).
[CrossRef] [PubMed]

M. R. Dennis, "Rows of optical vortices from elliptically perturbing a high-order beam," Opt. Lett. 31, 1325-1327 (2006).
[CrossRef] [PubMed]

D. Ling, J. Li, and J. Chen, "Analysis of eigenfields in the axicon-based Bessel-Gauss resonator by the transfer-matrix method," J. Opt. Soc. Am. A 23, 912-918 (2006).
[CrossRef]

J. Hahn, H. Kim, K. Choi, and B. Lee, "Real-time digital holographic beam-shaping system with a genetic feedback tuning loop," Appl. Opt. 45, 915-924 (2006).
[CrossRef] [PubMed]

2005 (8)

2004 (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, 5784-5786 (2004).
[CrossRef]

2003 (2)

2002 (2)

D. Cojoc, E. Di Fabrizio, L. Businaro, S. Carbini, F. Romanato, L. Vaccari, and M. Altissimo, "Design and fabrication of diffractive optical elements for optical tweezer arrays by means of e-beam lithography," Microelectron. Eng. 61-62, 963-969 (2002).
[CrossRef]

V.A.Soifer, ed., Method for Computer Design of Diffractive Optical Elements (Wiley, 2002).

2000 (1)

1997 (1)

1996 (2)

1993 (1)

1992 (2)

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. V. Shinkaryev, and G. V. Uspleniev, "Trochoson," Opt. Commun. 91, 158-162 (1992).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, "The phase rotor filter," J. Mod. Opt. 39, 1147-1154 (1992).
[CrossRef]

1983 (1)

A. P. Prudnikov, Y. A. Brichkov, and O. I. Marichev, Integrals and Series. Special Functions (Nauka, 1983).

Ahluwalia, B. P. S.

Almazov, A. A.

Altissimo, M.

D. Cojoc, E. Di Fabrizio, L. Businaro, S. Carbini, F. Romanato, L. Vaccari, and M. Altissimo, "Design and fabrication of diffractive optical elements for optical tweezer arrays by means of e-beam lithography," Microelectron. Eng. 61-62, 963-969 (2002).
[CrossRef]

Bandres, M. A.

Bashkansky, M.

Bentley, J. B.

Beraet, S.

Bernet, S.

Bouchel, Z.

Brichkov, Y. A.

A. P. Prudnikov, Y. A. Brichkov, and O. I. Marichev, Integrals and Series. Special Functions (Nauka, 1983).

Bu, J.

Buchter, S. C.

Burge, R. E.

Businaro, L.

D. Cojoc, E. Di Fabrizio, L. Businaro, S. Carbini, F. Romanato, L. Vaccari, and M. Altissimo, "Design and fabrication of diffractive optical elements for optical tweezer arrays by means of e-beam lithography," Microelectron. Eng. 61-62, 963-969 (2002).
[CrossRef]

Campos, J.

Carbini, S.

D. Cojoc, E. Di Fabrizio, L. Businaro, S. Carbini, F. Romanato, L. Vaccari, and M. Altissimo, "Design and fabrication of diffractive optical elements for optical tweezer arrays by means of e-beam lithography," Microelectron. Eng. 61-62, 963-969 (2002).
[CrossRef]

Carcole, E.

Chakraborty, R.

Chattrapiban, N.

Chen, J.

Cheong, W. C.

B. P. S. Ahluwalia, W. C. Cheong, X.-C. Yuan, L.-S. Zhang, S.-H. Tao, J. Bu, and H. Wang, "Design and fabrication of a double-axicon for generation of tailorable self-imaged three-dimensional intensity voids," Opt. Lett. 31, 987-989 (2006).
[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, 5784-5786 (2004).
[CrossRef]

Choi, K.

Cofield, D.

Cojoc, D.

D. Cojoc, E. Di Fabrizio, L. Businaro, S. Carbini, F. Romanato, L. Vaccari, and M. Altissimo, "Design and fabrication of diffractive optical elements for optical tweezer arrays by means of e-beam lithography," Microelectron. Eng. 61-62, 963-969 (2002).
[CrossRef]

Cottrell, D. M.

Courtial, J.

J. Courtial, G. Whyte, Z. Bouchel, and J. Wagner, "Iterative algorithm for holographic shaping of non-diffracting and self-imaging light beams," Opt. Express 14, 2108-2116 (2006).
[CrossRef] [PubMed]

G. Whyte and J. Courtial, "Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg-Saxton algorithm," New J. Phys. 7, 1-12 (2005).
[CrossRef]

Curtis, J. E.

J. E. Curtis and D. G. Grier, "Structure of optical vortices," Phys. Rev. Lett. 90, 1.33901-1-3 (2003).
[CrossRef]

Davis, J. A.

Dennis, M. R.

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, 5784-5786 (2004).
[CrossRef]

Di, E.

D. Cojoc, E. Di Fabrizio, L. Businaro, S. Carbini, F. Romanato, L. Vaccari, and M. Altissimo, "Design and fabrication of diffractive optical elements for optical tweezer arrays by means of e-beam lithography," Microelectron. Eng. 61-62, 963-969 (2002).
[CrossRef]

Ding, J.

Elfstrom, H.

Fatemi, F. K.

Foo, I. G.

Furhapter, S.

Ghosh, A.

Grier, D. G.

J. E. Curtis and D. G. Grier, "Structure of optical vortices," Phys. Rev. Lett. 90, 1.33901-1-3 (2003).
[CrossRef]

Guertin, J.

Guo, C.

Gutiérrez-Vega, J. C.

Hahn, J.

Hakola, A.

Han, Y.

Hill, W. T.

Jesacher, A.

Kaivola, M.

Khonina, S. N.

Kim, H.

Kotlyar, V. V.

Kovalev, A. A.

Law, C. T.

Lee, B.

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, 5784-5786 (2004).
[CrossRef]

Li, J.

Lin, J.

Ling, D.

Liu, X.

Marichev, O. I.

A. P. Prudnikov, Y. A. Brichkov, and O. I. Marichev, Integrals and Series. Special Functions (Nauka, 1983).

Maurer, C.

McNamara, E.

Nin, H. B.

Palecios, D. M.

Peng, X.

Prudnikov, A. P.

A. P. Prudnikov, Y. A. Brichkov, and O. I. Marichev, Integrals and Series. Special Functions (Nauka, 1983).

Ren, X.

Ritsch-Marte, M.

Rogers, E. A.

Romanato, F.

D. Cojoc, E. Di Fabrizio, L. Businaro, S. Carbini, F. Romanato, L. Vaccari, and M. Altissimo, "Design and fabrication of diffractive optical elements for optical tweezer arrays by means of e-beam lithography," Microelectron. Eng. 61-62, 963-969 (2002).
[CrossRef]

Roy, R.

Rozas, D.

Shevchenko, A.

Shinkarev, M. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, "The phase rotor filter," J. Mod. Opt. 39, 1147-1154 (1992).
[CrossRef]

Shinkaryev, M. V.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. V. Shinkaryev, and G. V. Uspleniev, "Trochoson," Opt. Commun. 91, 158-162 (1992).
[CrossRef]

Shum, P.

Skidanov, R. V.

Soifer, V. A.

Sun, X. W.

Swartzlander, G. A.

Tabiryan, N. V.

Tao, S. H.

Tao, S.-H.

Tossavainen, N.

Turunen, J.

Uspleniev, G. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, "The phase rotor filter," J. Mod. Opt. 39, 1147-1154 (1992).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. V. Shinkaryev, and G. V. Uspleniev, "Trochoson," Opt. Commun. 91, 158-162 (1992).
[CrossRef]

Vaccari, L.

D. Cojoc, E. Di Fabrizio, L. Businaro, S. Carbini, F. Romanato, L. Vaccari, and M. Altissimo, "Design and fabrication of diffractive optical elements for optical tweezer arrays by means of e-beam lithography," Microelectron. Eng. 61-62, 963-969 (2002).
[CrossRef]

Wagner, J.

Wang, H.

Wang, Q.

Whyte, G.

J. Courtial, G. Whyte, Z. Bouchel, and J. Wagner, "Iterative algorithm for holographic shaping of non-diffracting and self-imaging light beams," Opt. Express 14, 2108-2116 (2006).
[CrossRef] [PubMed]

G. Whyte and J. Courtial, "Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg-Saxton algorithm," New J. Phys. 7, 1-12 (2005).
[CrossRef]

Xu, J.

Yin, X. J.

Yuan, X.

Yuan, X.-C.

B. P. S. Ahluwalia, W. C. Cheong, X.-C. Yuan, L.-S. Zhang, S.-H. Tao, J. Bu, and H. Wang, "Design and fabrication of a double-axicon for generation of tailorable self-imaged three-dimensional intensity voids," Opt. Lett. 31, 987-989 (2006).
[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, 5784-5786 (2004).
[CrossRef]

Zhang, L.-S.

B. P. S. Ahluwalia, W. C. Cheong, X.-C. Yuan, L.-S. Zhang, S.-H. Tao, J. Bu, and H. Wang, "Design and fabrication of a double-axicon for generation of tailorable self-imaged three-dimensional intensity voids," Opt. Lett. 31, 987-989 (2006).
[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, 5784-5786 (2004).
[CrossRef]

Appl. Opt. (5)

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, 5784-5786 (2004).
[CrossRef]

J. Mod. Opt. (1)

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, "The phase rotor filter," J. Mod. Opt. 39, 1147-1154 (1992).
[CrossRef]

J. Opt. Soc. Am. A (4)

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

Microelectron. Eng. (1)

D. Cojoc, E. Di Fabrizio, L. Businaro, S. Carbini, F. Romanato, L. Vaccari, and M. Altissimo, "Design and fabrication of diffractive optical elements for optical tweezer arrays by means of e-beam lithography," Microelectron. Eng. 61-62, 963-969 (2002).
[CrossRef]

New J. Phys. (1)

G. Whyte and J. Courtial, "Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg-Saxton algorithm," New J. Phys. 7, 1-12 (2005).
[CrossRef]

Opt. Commun. (1)

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. V. Shinkaryev, and G. V. Uspleniev, "Trochoson," Opt. Commun. 91, 158-162 (1992).
[CrossRef]

Opt. Express (4)

Opt. Lett. (13)

G. A. Swartzlander, "Broadband nulling of a vortex phase mask," Opt. Lett. 30, 2876-2878 (2005).
[CrossRef] [PubMed]

M. R. Dennis, "Rows of optical vortices from elliptically perturbing a high-order beam," Opt. Lett. 31, 1325-1327 (2006).
[CrossRef] [PubMed]

J. Lin, X. Yuan, S. H. Tao, and R. E. Burge, "Variable-radius focused optical vortex with suppressed sidelobes," Opt. Lett. 31, 1600-1602 (2006).
[CrossRef] [PubMed]

V. V. Kotlyar, S. N. Khonina, A. A. Kovalev, V. A. Soifer, H. Elfstrom, and J. Turunen, "Diffraction of a plane, finite-radius wave by a spiral phase plate," Opt. Lett. 31, 1597-1599 (2006).
[CrossRef] [PubMed]

B. P. S. Ahluwalia, W. C. Cheong, X.-C. Yuan, L.-S. Zhang, S.-H. Tao, J. Bu, and H. Wang, "Design and fabrication of a double-axicon for generation of tailorable self-imaged three-dimensional intensity voids," Opt. Lett. 31, 987-989 (2006).
[CrossRef] [PubMed]

I. G. Foo, D. M. Palecios, and G. A. Swartzlander, "Optical vortex coronograph," Opt. Lett. 30, 3308-3310 (2005).
[CrossRef]

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

C. Guo, Y. Han, J. Xu, and J. Ding, "Radial Hilbert transform with Laguerre-Gaussian spatial filters," Opt. Lett. 31, 1394-1396 (2006).
[CrossRef] [PubMed]

S. Furhapter, A. Jesacher, S. Beraet, and M. Ritsch-Marte, "Spiral interferometry," Opt. Lett. 30, 1953-1955 (2005).
[CrossRef] [PubMed]

R. Chakraborty and A. Ghosh, "Generation of an elliptic Bessel beam," Opt. Lett. 31, 38-40 (2006).
[CrossRef] [PubMed]

J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, "Generation of helical Ince-Gaussian beams with a liquid-crystal display," Opt. Lett. 31, 649-651 (2006).
[CrossRef] [PubMed]

F. K. Fatemi and M. Bashkansky, "Generation of hollow beams by using a binary spatial light modulator," Opt. Lett. 31, 864-866 (2006).
[CrossRef] [PubMed]

N. Chattrapiban, E. A. Rogers, D. Cofield, W. T. Hill, and R. Roy, "Generation of nondiffracting Bessel beams by use of a spatial light modulator," Opt. Lett. 28, 2183-2185 (2003).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

J. E. Curtis and D. G. Grier, "Structure of optical vortices," Phys. Rev. Lett. 90, 1.33901-1-3 (2003).
[CrossRef]

Other (2)

A. P. Prudnikov, Y. A. Brichkov, and O. I. Marichev, Integrals and Series. Special Functions (Nauka, 1983).

V.A.Soifer, ed., Method for Computer Design of Diffractive Optical Elements (Wiley, 2002).

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 (10)

Fig. 1
Fig. 1

Radial profile of the Fraunhofer diffraction pattern (amplitude E n ( ρ , θ ) ) for a plane, finite-radius wave diffracted by an HA.

Fig. 2
Fig. 2

Calculated complex amplitude modules E n ( ρ , θ ) vs the radial coordinate for different numbers of the SPP: (a) n = 2 , (b) n = 16 , and (c) n = 50 .

Fig. 3
Fig. 3

Plots of the functions from the left and right parts in Eq. (25): the bold curve is a plot of E n ( ρ , θ ) function, and the thin curve is a plot of the function ( 2 R ρ ) ( n + 1 ) ( n + 2 ) J n + 1 ( k R ρ f ) . Calculation parameters: R = 1 mm , f = 100 mm , λ = 633 nm , (a) n = 2 , and (b) n = 4 .

Fig. 4
Fig. 4

Plots of the functions from the left and right parts in Eq. (26): the bold curve is a plot of E n ( ρ , θ ) function, and the thin curve is a plot of the function ( k R 2 ) ( 2 f ) J n ( k R ρ f ) .

Fig. 5
Fig. 5

Radial profile of the Fresnel diffraction pattern (amplitude F n ( ρ , θ ) at a distance z = 200 mm ) for the Gaussian beam ( λ = 633 nm , w = 1 mm ) diffracted by the HA ( n = 8 ) : (a) α = 0 , (b) α = 20 mm 1 , and (c) α = 50 mm 1 .

Fig. 6
Fig. 6

Radial profile of the Fresnel diffraction pattern (amplitude F n ( ρ , θ ) ) for the Gaussian beam ( λ = 633 nm , w = 1 mm ) diffracted by the HA ( n = 8 , α = 20 mm 1 ) : (a) z = 400 mm and (b) z = 500 mm .

Fig. 7
Fig. 7

Radial profile of the Fresnel diffraction pattern (amplitude F n ( ρ , θ ) at the distance z = 200 mm ) for the Gaussian beam ( λ = 633 nm , w = 1 mm ) diffracted by the HA ( α = 20 mm 1 ) : (a) n = 20 and (b) n = 40 .

Fig. 8
Fig. 8

(a) Phase of a binary HA of 10th order, (b) calculated diffraction pattern of a plane wave by the DOE of Fig. 8a.

Fig. 9
Fig. 9

(a) Relief of the binary HA, obtained by interferometry (the central fragment of size 260 μ m × 350 μ m ) and (b) experimental radial profile of the light ring intensity generated in the lens focal plane ( f = 460 mm ) .

Fig. 10
Fig. 10

Ten polystyrene beads of diameter 5 μ m are moving along the bright ring of radius 37.5 μ m produced by the HA with number n = 10 , the average velocity being 4 μ m s . Frames (a) and (b) are taken at a 3 s interval.

Tables (1)

Tables Icon

Table 1 Approximate Values of the Radius ρ v of the Maximal Amplitude E n ( ρ , θ )

Equations (45)

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

E 0 ( r ) = circl ( r R ) = { 1 , r R 0 , r > R ,
τ n ( r , φ ) = exp ( i α r + i n φ ) ,
E n ( ρ , θ ) = i k 2 π f 0 R 0 2 π exp [ i α r + i n φ i k f r ρ cos ( φ θ ) ] r d r d φ = ( i ) n + 1 k f exp ( i n θ ) m = 0 ( i α ) m m ! 0 R r m + 1 J n ( k f ρ r ) d r ,
0 R x λ J v ( a x ) d x = ( a R ) v + λ + 1 a λ + 1 2 v ( v + λ + 1 ) v ! F 2 1 [ v + λ + 1 2 , v + λ + 3 2 , v + 1 , ( a R 2 ) 2 ] ,
E n ( ρ , θ ) = ( i ) n + 1 exp ( i n θ ) n ! ( k R 2 f ) ( k R ρ 2 f ) n × m = 0 ( i α R ) m ( m + n + 2 ) m ! F 2 1 [ m + n + 2 2 , m + n + 4 2 , n + 1 , ( k R ρ 2 f ) 2 ] ,
F 2 1 ( a , b , c , x ) = m = 0 ( a ) m x m ( b ) m ( c ) m m ! ,
ρ l = λ f γ l π R ,
E n ( ρ , θ ) = ( i ) n + 1 exp ( i n θ ) ( n + 2 ) n ! ( k R 2 f ) ( k R ρ 2 f ) n F 2 1 [ n + 2 2 , n + 4 2 , n + 1 , ( k R ρ 2 f ) 2 ] .
E n ( ρ , θ , z ) = 2 exp ( i z 0 ρ ¯ 2 z + i n θ ) ρ ¯ n n ! ( i z 0 z ) n + 1 × m = 0 ( i z 0 z ) m ( 2 m + n + 2 ) m ! F 2 1 [ 2 m + n + 2 2 , 2 m + n + 4 2 , n + 1 , ( z 0 ρ ¯ z ) 2 ] ,
E n ( ρ , θ ) = ( i ) n + 1 k exp ( i n θ ) 2 π f [ ( i ) n I 1 n α exp ( i α R ) m = + i m ( i R I 1 m + I 1 m α ) J m + n ( R ρ ¯ ) ] ,
I 1 n = 0 2 π exp ( i n φ ) d φ α + ρ ¯ cos φ ,
I 1 n = { 2 π sgn α α 2 ρ ¯ 2 χ n , 0 < ρ ¯ < α π i ( β * n β n ) ρ ¯ 2 α 2 , ρ ¯ > α ) ,
{ β = α + i ρ ¯ 2 α 2 ρ ¯ χ = α + sgn α α 2 ρ ¯ 2 ρ ¯ ,
sgn α = { 1 , α > 0 , 1 , α < 0 ,
I 1 n α
= { 2 π χ n α sgn α + n α 2 ρ ¯ 2 ( α 2 ρ ¯ 2 ) 3 2 , 0 < ρ ¯ < α π i [ α ( β * n β n ) ( ρ ¯ 2 α 2 ) 3 2 i n ( β * n + β n ) ρ ¯ 2 α 2 ] , ρ ¯ > α ] .
I 1 n = { 0 , n = 2 m 2 π i n + 1 ρ ¯ , n = 2 m + 1 ,
I 1 n α = { 2 π i n n ρ ¯ 2 , n = 2 m 0 , n = 2 m + 1 .
E n ( ρ , θ ) = ( i ) n + 1 k exp ( i n θ ) f ρ ¯ 2 { n [ 1 J 0 ( y ) 2 m = 1 ( n 2 ) 2 J 2 m ( y ) ] y J n 1 ( y ) , n = 2 m n [ 0 y J 0 ( t ) d t 2 m = 1 ( n 1 ) 2 J 2 m 1 ( y ) ] y J n 1 ( y ) , n = 2 m + 1 ] ,
0 y J 0 ( t ) d t = y 2 { π J 1 ( y ) H 0 ( y ) + J 0 ( y ) [ 2 π H 1 ( y ) ] } ,
E 0 ( ρ , θ ) = i k R 2 f J 1 ( y ) y ,
E 1 ( ρ , θ ) = k exp ( i θ ) f ρ ¯ 2 [ 0 y J 0 ( t ) d t y J 0 ( y ) ] ,
E 2 ( ρ , θ ) = i k exp ( i 2 θ ) f ρ ¯ 2 [ 2 2 J 0 ( y ) y J 1 ( y ) ] .
E n ( ρ , θ ) k [ n y J n ( y ) ] f ρ ¯ 2 k f ρ ¯ 2 [ n 2 y π sin ( y n + 1 2 π ) ] .
ρ 0 λ f n 2 4 R .
E n + 2 ( ρ , θ ) = n + 2 n exp ( i 2 θ ) E n ( ρ , θ ) + 2 ( i ) n + 1 R ρ ( n + 1 n ) exp [ i ( n + 2 ) θ ] J n + 1 ( k f R ρ ) .
E n ( ρ m , θ ) = 2 R ρ m ( n + 1 n + 2 ) J n + 1 ( k R ρ m f ) .
E n ( ρ s , θ ) = ( k R 2 2 f ) J n ( k R ρ s f ) .
E n ( ρ s , θ ) 2 ( k R 2 2 f ) 2 J 0 n ,
f γ n , 1 k R < ρ v < f γ n , 1 k R ,
ρ v γ ¯ n f k R , γ ¯ n = γ n , 1 + γ n , 1 2
ρ v = γ n 1 , 1 f k R ,
E 0 ( r ) = exp ( r 2 w 2 )
F n ( ρ , θ , z ) = i k 2 π z exp ( i k z + i k ρ 2 2 z ) 0 R 0 2 π exp [ r 2 w 2 + i α r + i n φ + i k r 2 2 z i k z ρ r cos ( φ θ ) ] r d r d φ .
0 x λ + 1 exp ( p x 2 ) J v ( c x ) d x = c v p ( v + λ + 2 ) 2 2 v + 1 v ! Γ ( v + λ + 2 2 ) F 1 1 [ v + λ + 2 2 , v + 1 , ( c 2 p ) 2 ] ,
F n ( ρ , θ , z ) = ( i ) n + 1 k z exp [ i n θ + i k z + i k ρ 2 2 z ] ( k ρ 2 z ) n γ ( n + 2 ) 2 2 n + 1 n ! × m = 0 ( i α ) m γ m 2 m ! Γ ( m + n + 2 2 ) F 1 1 [ m + n + 2 2 , n + 1 , ( k ρ 2 z γ ) 2 ] ,
F 1 1 ( a , b , x ) = m = 0 ( a ) m x m ( b ) m m ! .
ρ l = w z a l z 0 ( 1 + z 0 2 z 2 ) 1 4 ,
F n ( ρ , θ , z , α = 0 ) = ( i ) n + 1 k z exp [ i ( n θ + k z ) + i k ρ 2 2 z ] ( k ρ 2 z ) n × γ ( n + 2 ) 2 2 n + 1 n ! Γ ( n + 2 2 ) F 1 1 [ n + 2 2 , n + 1 , ( k ρ 2 z γ ) 2 ] .
J ( n 1 ) 2 ( x ) = ( x 2 ) ( n 1 ) 2 exp ( i x ) Γ ( n 1 2 ) F 1 1 ( n 2 , n ; 2 i x ) ,
F 1 1 ( n 2 , n + 1 ; 2 i x ) = ( i d d x + 2 ) F 1 1 ( n 2 , n ; 2 i x ) ,
F n ( ρ , θ , z , α = 0 ) = ( i ) n + 1 π 2 ( z 0 z ) 2 ( ρ w ) [ 1 + ( z 0 z ) 2 ] 3 4 × exp [ i 3 2 tan 1 ( z 0 z ) i k ρ 2 2 R 0 ( z ) + i k ρ 2 2 z ρ 2 w 2 ( z ) + i n θ + i k z ] × { I n 1 2 [ ρ 2 ( 1 w 2 ( z ) + i k 2 R 0 ( z ) ) ] I n + 1 2 [ ρ 2 ( 1 w 2 ( z ) + i k 2 R 0 ( z ) ) ] } ,
F n ( ρ , θ , z , ) = ( i ) n + 1 z 0 2 n n ! z exp ( i n θ + i k z ) ( z 0 ρ z w ) n × m = 0 ( i α w ) m m ! Γ ( m + n + 2 2 ) F 1 1 [ m + n + 2 2 , n + 1 , ( z 0 ρ z w ) 2 ] .
F n ( ρ , θ , z , α = 0 ) = ( i ) n + 1 z 0 2 n n ! z exp ( i n θ + i k z ) ( z 0 ρ z w ) n × Γ ( n + 2 2 ) F 1 1 [ n + 2 2 , n + 1 , ( z 0 ρ z w ) 2 ] .
τ n β ( r , φ ) = sign [ cos ( α r + n φ + β r cos φ ) ] rect ( r R ) ,

Metrics