Abstract

This paper discusses paraxial diffraction at a spiral phase plate of a limited light field whose phase is constant (a plane wave) while its amplitude varies as a power function with integer exponent n (positive or negative). It is shown that the Fraunhofer diffraction in this case is described by a Bessel function of the first kind of (n+1)st or (n−1)st order. Light fields that form diffraction patterns described by Bessel functions in the far zone are called simple optical vortices. Expressions are obtained for the amplitude of the Fraunhofer diffraction pattern of pure optical vortices and hypergeometric modes. The experimental part of the paper describes the formation of a vortex beam with an intensity distribution in the form of a double ring of arbitrary radius. Diffraction optical elements were created in this case by three methods: by electron lithography, by photolithography, and by means of a liquid-crystal display. Experiments are presented on the rotation of microparticles in a double ring.

© 2007 Optical Society of America

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  1. K. J. Moh, X. C. Yuan, W. C. Cheong, L. S. Zhang, J. Lin, B. P. S. Ahluwalia, and H. Wang, "High-power efficient multiple optical vortices in a single beam generated by a kinoform-type spiral phase plate," Appl. Opt. 45, 1153 (2006).
    [CrossRef]
  2. J. W. Sung, H. Hockel, J. D. Brown, and E. G. Johnson, "Development of a two-dimensional phase-grating mask for fabrication of an analog-resist profile," Appl. Opt. 45, 33 (2006).
    [CrossRef]
  3. P. Kurzynowski, W. A. Wozniak, and E. Fraçzek, "Optical vortices generation using the Wollaston prism," Appl. Opt. 45, 7898 (2006).
    [CrossRef]
  4. G. A. Swartzlander, "Achromatic optical vortex lens," Opt. Lett. 31, 2042 (2006).
    [CrossRef]
  5. N. Chattrapiban, E. A. Rogers, I. V. Arakelyan, R. Roy, and W. T. Hill, "Laser beams with embedded vortices: tools for atom optics," J. Opt. Soc. Am. B 23, 94 (2006).
    [CrossRef]
  6. J. Lin, X. C. Yuan, S. H. Tao, and R. E. Burge, "Variable-radius focused optical vortex with suppressed sidelobes," Opt. Lett. 31, 1600 (2006).
    [CrossRef]
  7. 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 (2006).
    [CrossRef]
  8. J. B. Bentley, J. A. Davis, J. Albero, and I. Moreno, "Self-interferometric technique for visualization of phase patterns encoded onto a liquid-crystal display," Appl. Opt. 45, 7791 (2006).
    [CrossRef]
  9. A. Jesacher, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, "Reverse orbiting of microparticles in optical vortices," Opt. Lett. 31, 2824 (2006).
  10. A. Bekshaev and M. Soskin, "Rotational transformations and transverse energy flow in paraxial light beams: linear azimuthons," Opt. Lett. 31, 2199 (2006).
    [CrossRef]
  11. M. R. Dennis, "Rows of optical vortices from elliptically perturbing a high-order beam," Opt. Lett. 31, 1325 (2006).
    [CrossRef]
  12. 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 (2006).
    [CrossRef]
  13. 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 (2006).
    [CrossRef]
  14. V. V. Kotlyar, S. N. Khonina, A. A. Almazov, V. A. Soifer, K. Jefimovs, and J. Turunen, "Elliptic Laguerre-Gaussian beams," J. Opt. Soc. Am. A 23, 43 (2006).
  15. R. Chakraborty and A. Ghosh, "Generation of an elliptic hollow beam using Mathieu and Bessel functions," J. Opt. Soc. Am. A 23, 2278 (2006).
  16. Y. Cai and S. He, "Propagation of hollow Gaussian beams through apertured paraxial optical systems," J. Opt. Soc. Am. A 23, 1410 (2006).
  17. Z. Mei and D. Zhao, "Controllable elliptical dark-hollow beams," J. Opt. Soc. Am. A 23, 919 (2006).
  18. Q. Zhan, "Properties of circularly polarized vortex beams," Opt. Lett. 31, 867 (2006).
    [CrossRef]
  19. Y. Izdebskaya, T. Fadeyeva, V. Shvedov, and A. Volyar, "Vortex-bearing array of singular beams with very high orbital angular momentum," Opt. Lett. 31, 2523 (2006).
  20. R. Menon and H. I. Smith, "Absorbance-modulation optical lithography," J. Opt. Soc. Am. A 23, 2290 (2006).
    [CrossRef]
  21. A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, "Spiral interferogram analysis," J. Opt. Soc. Am. A 23, 1400 (2006).
  22. D. M. Palacios and S. L. Hunyadi, "Low-order aberration sensitivity of an optical vortex coronagraph," Opt. Lett. 31, 2981 (2006).
  23. C. Guo, Y. Han, J. Xu, and J. Ding, "Radial Hilbert transform with Laguerre-Gaussian spatial filters," Opt. Lett. 31, 1394 (2006).
    [CrossRef]
  24. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series (Gordon and Breach, New York, 1986; Nauka, Moscow, 1983).
  25. V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Soifer, H. Elfstrom, and 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 (2005).
    [CrossRef]
  26. W. Miller, Symmetry and separation of variables (Addison-Wesley, Reading, Mass., 1977).
  27. V. V. Kotlyar, S. N. Khonina, A. A. Almazov, and V. A. Soĭfer, "Optical pure vortices and hypergeometric modes," Komp. Optika 27, 21 (2005).
  28. D. Rozas, C. T. Law, and G. A. Swartzlander, "Propagation dynamics of optical vortices," J. Opt. Soc. Am. B 14, 3054 (1997).
  29. J. Arlt and M. J. Padgett, "Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam," Opt. Lett. 25, 191 (2000).
    [CrossRef]
  30. 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 (2006).
    [CrossRef]

2006

K. J. Moh, X. C. Yuan, W. C. Cheong, L. S. Zhang, J. Lin, B. P. S. Ahluwalia, and H. Wang, "High-power efficient multiple optical vortices in a single beam generated by a kinoform-type spiral phase plate," Appl. Opt. 45, 1153 (2006).
[CrossRef]

J. W. Sung, H. Hockel, J. D. Brown, and E. G. Johnson, "Development of a two-dimensional phase-grating mask for fabrication of an analog-resist profile," Appl. Opt. 45, 33 (2006).
[CrossRef]

P. Kurzynowski, W. A. Wozniak, and E. Fraçzek, "Optical vortices generation using the Wollaston prism," Appl. Opt. 45, 7898 (2006).
[CrossRef]

G. A. Swartzlander, "Achromatic optical vortex lens," Opt. Lett. 31, 2042 (2006).
[CrossRef]

N. Chattrapiban, E. A. Rogers, I. V. Arakelyan, R. Roy, and W. T. Hill, "Laser beams with embedded vortices: tools for atom optics," J. Opt. Soc. Am. B 23, 94 (2006).
[CrossRef]

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

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 (2006).
[CrossRef]

J. B. Bentley, J. A. Davis, J. Albero, and I. Moreno, "Self-interferometric technique for visualization of phase patterns encoded onto a liquid-crystal display," Appl. Opt. 45, 7791 (2006).
[CrossRef]

A. Jesacher, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, "Reverse orbiting of microparticles in optical vortices," Opt. Lett. 31, 2824 (2006).

A. Bekshaev and M. Soskin, "Rotational transformations and transverse energy flow in paraxial light beams: linear azimuthons," Opt. Lett. 31, 2199 (2006).
[CrossRef]

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

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 (2006).
[CrossRef]

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 (2006).
[CrossRef]

V. V. Kotlyar, S. N. Khonina, A. A. Almazov, V. A. Soifer, K. Jefimovs, and J. Turunen, "Elliptic Laguerre-Gaussian beams," J. Opt. Soc. Am. A 23, 43 (2006).

R. Chakraborty and A. Ghosh, "Generation of an elliptic hollow beam using Mathieu and Bessel functions," J. Opt. Soc. Am. A 23, 2278 (2006).

Y. Cai and S. He, "Propagation of hollow Gaussian beams through apertured paraxial optical systems," J. Opt. Soc. Am. A 23, 1410 (2006).

Z. Mei and D. Zhao, "Controllable elliptical dark-hollow beams," J. Opt. Soc. Am. A 23, 919 (2006).

Q. Zhan, "Properties of circularly polarized vortex beams," Opt. Lett. 31, 867 (2006).
[CrossRef]

Y. Izdebskaya, T. Fadeyeva, V. Shvedov, and A. Volyar, "Vortex-bearing array of singular beams with very high orbital angular momentum," Opt. Lett. 31, 2523 (2006).

R. Menon and H. I. Smith, "Absorbance-modulation optical lithography," J. Opt. Soc. Am. A 23, 2290 (2006).
[CrossRef]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, "Spiral interferogram analysis," J. Opt. Soc. Am. A 23, 1400 (2006).

D. M. Palacios and S. L. Hunyadi, "Low-order aberration sensitivity of an optical vortex coronagraph," Opt. Lett. 31, 2981 (2006).

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

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 (2006).
[CrossRef]

2005

2000

1997

Appl. Opt.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Komp. Optika

V. V. Kotlyar, S. N. Khonina, A. A. Almazov, and V. A. Soĭfer, "Optical pure vortices and hypergeometric modes," Komp. Optika 27, 21 (2005).

Opt. Lett.

J. Arlt and M. J. Padgett, "Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam," Opt. Lett. 25, 191 (2000).
[CrossRef]

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 (2006).
[CrossRef]

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

D. M. Palacios and S. L. Hunyadi, "Low-order aberration sensitivity of an optical vortex coronagraph," Opt. Lett. 31, 2981 (2006).

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

Q. Zhan, "Properties of circularly polarized vortex beams," Opt. Lett. 31, 867 (2006).
[CrossRef]

Y. Izdebskaya, T. Fadeyeva, V. Shvedov, and A. Volyar, "Vortex-bearing array of singular beams with very high orbital angular momentum," Opt. Lett. 31, 2523 (2006).

A. Jesacher, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, "Reverse orbiting of microparticles in optical vortices," Opt. Lett. 31, 2824 (2006).

A. Bekshaev and M. Soskin, "Rotational transformations and transverse energy flow in paraxial light beams: linear azimuthons," Opt. Lett. 31, 2199 (2006).
[CrossRef]

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

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 (2006).
[CrossRef]

G. A. Swartzlander, "Achromatic optical vortex lens," Opt. Lett. 31, 2042 (2006).
[CrossRef]

Other

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series (Gordon and Breach, New York, 1986; Nauka, Moscow, 1983).

W. Miller, Symmetry and separation of variables (Addison-Wesley, Reading, Mass., 1977).

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