Abstract

An analytical expression for the spatial spectrum of the conic wave diffracted by a spiral phase plate (SPP) with arbitrary integer singularity of order n is obtained. Conic wave diffraction by the SPP is equivalent to plane-wave diffraction by a helical axicon. A comparison of the conic wave and Gaussian beam diffraction on a SPP is made. It is shown that in both cases a light ring is formed, with the intensity function growing in proportion to ρ2n at small values of radial variable ρ and decreasing as n2ρ4 at large ρ. By use of direct e-beam writing on the resist, a 32 level SPP of the 2nd order and diameter 5  mm is manufactured. By use of this SPP, a He–Ne laser beam is transformed into a beam with phase singularity and ringlike intensity distribution. A four-order binary diffractive optical element (DOE) with its transmittance proportional to a linear superposition of four angular harmonics is also manufactured. With this DOE, simultaneous optical trapping of several polystyrene beads of diameter 5   μm is performed.

© 2006 Optical Society of America

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References

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  1. S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, " The rotor phase filter," J. Mod. Opt. 39, 1147- 1154 ( 1992).
    [CrossRef]
  2. S. S. R. Oemrawsingh, J. A. W. van Houwelinger, E. R. Eliel, J. R. Woerdman, E. J. K. Vestegen, J. G. Kloosterboer, and G. W. Hooft, " Production and characterization of spiral phase plates for optical wavelengths," Appl. Opt. 43, 688- 694 ( 2004).
    [CrossRef] [PubMed]
  3. W. G. 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]
  4. W. M. Lee, B. P. S. Ahluwalia, X. -C. Yuan, W. C. Cheong, and K. Dholakia, " Optical steering of high and low index microparticles by manipulating an off-axis optical vortex," J. Opt. A Pure Appl. Opt. 7, 1- 6 ( 2005).
    [CrossRef]
  5. K. Sueda, G. Miyaji, N. Miyanaga, and M. Nakatsura, " Laguerre-Gaussian beam generated with a multilevel spiral phase plate for high intensity laser pulses," Opt. Express 12, 3548- 3553 ( 2004).
    [CrossRef] [PubMed]
  6. 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- 861 ( 2005).
    [CrossRef]
  7. S. Sundbeck, I. Gruzberg, and D. G. Grier, " Structure and scaling of helical modes of light," Opt. Lett. 30, 1- 13 ( 2005).
    [CrossRef]
  8. S. Furhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, " Spiral phase contrast imaging in microscopy," Opt. Express 13, 689- 694 ( 2005).
    [CrossRef] [PubMed]
  9. D. Ganic, X. Gan, and M. Gu, " Focusing of doughnut laser beams by a high numerical-aperture objective in free space," Opt. Express 11, 2747- 2752 ( 2003).
    [CrossRef] [PubMed]
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    [CrossRef]
  11. C. Paterson and R. Smith, " Higher-order Bessel waves produced by axicon-type computer-generated holograms," Opt. Commun. 124, 123- 130 ( 1996).
    [CrossRef]
  12. S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, " Rotation of microparticles with Bessel beams generated by diffractive elements," J. Mod. Opt. 51, 2167- 2184 ( 2004).
    [CrossRef]
  13. C. A. Alonzo, P. J. Rodrigo, and J. Gluckstad, " Helico-conical optical beams: a product of helical and conical phase fronts," Opt. Express 13, 1749- 1760 ( 2005).
    [CrossRef] [PubMed]
  14. A. Niv, G. Biener, V. Kleiner, and E. Hasman, " Spiral phase elements obtained by use of discrete space-variant subwavelength gratings," Opt. Commun. 251, 306- 314 ( 2005).
    [CrossRef]
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    [CrossRef]
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2005

W. M. Lee, B. P. S. Ahluwalia, X. -C. Yuan, W. C. Cheong, and K. Dholakia, " Optical steering of high and low index microparticles by manipulating an off-axis optical vortex," J. Opt. A Pure Appl. Opt. 7, 1- 6 ( 2005).
[CrossRef]

S. Sundbeck, I. Gruzberg, and D. G. Grier, " Structure and scaling of helical modes of light," Opt. Lett. 30, 1- 13 ( 2005).
[CrossRef]

A. Niv, G. Biener, V. Kleiner, and E. Hasman, " Spiral phase elements obtained by use of discrete space-variant subwavelength gratings," Opt. Commun. 251, 306- 314 ( 2005).
[CrossRef]

S. Furhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, " Spiral phase contrast imaging in microscopy," Opt. Express 13, 689- 694 ( 2005).
[CrossRef] [PubMed]

C. A. Alonzo, P. J. Rodrigo, and J. Gluckstad, " Helico-conical optical beams: a product of helical and conical phase fronts," Opt. Express 13, 1749- 1760 ( 2005).
[CrossRef] [PubMed]

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- 861 ( 2005).
[CrossRef]

2004

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, " Rotation of microparticles with Bessel beams generated by diffractive elements," J. Mod. Opt. 51, 2167- 2184 ( 2004).
[CrossRef]

S. S. R. Oemrawsingh, J. A. W. van Houwelinger, E. R. Eliel, J. R. Woerdman, E. J. K. Vestegen, J. G. Kloosterboer, and G. W. Hooft, " Production and characterization of spiral phase plates for optical wavelengths," Appl. Opt. 43, 688- 694 ( 2004).
[CrossRef] [PubMed]

K. Sueda, G. Miyaji, N. Miyanaga, and M. Nakatsura, " Laguerre-Gaussian beam generated with a multilevel spiral phase plate for high intensity laser pulses," Opt. Express 12, 3548- 3553 ( 2004).
[CrossRef] [PubMed]

W. G. 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

1998

1996

C. Paterson and R. Smith, " Higher-order Bessel waves produced by axicon-type computer-generated holograms," Opt. Commun. 124, 123- 130 ( 1996).
[CrossRef]

1992

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

Ahluwalia, B. P. S.

W. M. Lee, B. P. S. Ahluwalia, X. -C. Yuan, W. C. Cheong, and K. Dholakia, " Optical steering of high and low index microparticles by manipulating an off-axis optical vortex," J. Opt. A Pure Appl. Opt. 7, 1- 6 ( 2005).
[CrossRef]

Almazov, A. A.

Alonzo, C. A.

Bernet, S.

Biener, G.

A. Niv, G. Biener, V. Kleiner, and E. Hasman, " Spiral phase elements obtained by use of discrete space-variant subwavelength gratings," Opt. Commun. 251, 306- 314 ( 2005).
[CrossRef]

Brichkov, Y. A.

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

Cheong, W. C.

W. M. Lee, B. P. S. Ahluwalia, X. -C. Yuan, W. C. Cheong, and K. Dholakia, " Optical steering of high and low index microparticles by manipulating an off-axis optical vortex," J. Opt. A Pure Appl. Opt. 7, 1- 6 ( 2005).
[CrossRef]

Cheong, W. G.

W. G. 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]

Dholakia, K.

W. M. Lee, B. P. S. Ahluwalia, X. -C. Yuan, W. C. Cheong, and K. Dholakia, " Optical steering of high and low index microparticles by manipulating an off-axis optical vortex," J. Opt. A Pure Appl. Opt. 7, 1- 6 ( 2005).
[CrossRef]

W. G. 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]

Elfstrom, H.

Eliel, E. R.

Furhapter, S.

Gan, X.

Ganic, D.

Gluckstad, J.

Grier, D. G.

S. Sundbeck, I. Gruzberg, and D. G. Grier, " Structure and scaling of helical modes of light," Opt. Lett. 30, 1- 13 ( 2005).
[CrossRef]

Gruzberg, I.

S. Sundbeck, I. Gruzberg, and D. G. Grier, " Structure and scaling of helical modes of light," Opt. Lett. 30, 1- 13 ( 2005).
[CrossRef]

Gu, M.

Hasman, E.

A. Niv, G. Biener, V. Kleiner, and E. Hasman, " Spiral phase elements obtained by use of discrete space-variant subwavelength gratings," Opt. Commun. 251, 306- 314 ( 2005).
[CrossRef]

Hooft, G. W.

Jefimovs, K.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, " Rotation of microparticles with Bessel beams generated by diffractive elements," J. Mod. Opt. 51, 2167- 2184 ( 2004).
[CrossRef]

Jesacher, A.

Khonina, S. N.

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- 861 ( 2005).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, " Rotation of microparticles with Bessel beams generated by diffractive elements," J. Mod. Opt. 51, 2167- 2184 ( 2004).
[CrossRef]

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

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

Kleiner, V.

A. Niv, G. Biener, V. Kleiner, and E. Hasman, " Spiral phase elements obtained by use of discrete space-variant subwavelength gratings," Opt. Commun. 251, 306- 314 ( 2005).
[CrossRef]

Kloosterboer, J. G.

Kotlyar, V. V.

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- 861 ( 2005).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, " Rotation of microparticles with Bessel beams generated by diffractive elements," J. Mod. Opt. 51, 2167- 2184 ( 2004).
[CrossRef]

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

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

Lee, W. M.

W. M. Lee, B. P. S. Ahluwalia, X. -C. Yuan, W. C. Cheong, and K. Dholakia, " Optical steering of high and low index microparticles by manipulating an off-axis optical vortex," J. Opt. A Pure Appl. Opt. 7, 1- 6 ( 2005).
[CrossRef]

W. G. 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]

Marichev, O. I.

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

Miyaji, G.

Miyanaga, N.

Nakatsura, M.

Niv, A.

A. Niv, G. Biener, V. Kleiner, and E. Hasman, " Spiral phase elements obtained by use of discrete space-variant subwavelength gratings," Opt. Commun. 251, 306- 314 ( 2005).
[CrossRef]

Oemrawsingh, S. S. R.

Paterson, C.

C. Paterson and R. Smith, " Higher-order Bessel waves produced by axicon-type computer-generated holograms," Opt. Commun. 124, 123- 130 ( 1996).
[CrossRef]

Prudnikov, A. P.

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

Ritsch-Marte, M.

Rodrigo, P. J.

Rozes, D.

Saks, Z. S.

Shinkarev, M. V.

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

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

Simonen, J.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, " Rotation of microparticles with Bessel beams generated by diffractive elements," J. Mod. Opt. 51, 2167- 2184 ( 2004).
[CrossRef]

Skidanov, R. V.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, " Rotation of microparticles with Bessel beams generated by diffractive elements," J. Mod. Opt. 51, 2167- 2184 ( 2004).
[CrossRef]

Smith, R.

C. Paterson and R. Smith, " Higher-order Bessel waves produced by axicon-type computer-generated holograms," Opt. Commun. 124, 123- 130 ( 1996).
[CrossRef]

Soifer, V. A.

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- 861 ( 2005).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, " Rotation of microparticles with Bessel beams generated by diffractive elements," J. Mod. Opt. 51, 2167- 2184 ( 2004).
[CrossRef]

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

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

Sueda, K.

Sundbeck, S.

S. Sundbeck, I. Gruzberg, and D. G. Grier, " Structure and scaling of helical modes of light," Opt. Lett. 30, 1- 13 ( 2005).
[CrossRef]

Swatzlander, G. A.

Turunen, J.

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- 861 ( 2005).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, " Rotation of microparticles with Bessel beams generated by diffractive elements," J. Mod. Opt. 51, 2167- 2184 ( 2004).
[CrossRef]

Uspleniev, G. V.

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

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

van Houwelinger, J. A. W.

Vestegen, E. J. K.

Wang, H.

W. G. 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]

Woerdman, J. R.

Yuan, X. -C.

W. M. Lee, B. P. S. Ahluwalia, X. -C. Yuan, W. C. Cheong, and K. Dholakia, " Optical steering of high and low index microparticles by manipulating an off-axis optical vortex," J. Opt. A Pure Appl. Opt. 7, 1- 6 ( 2005).
[CrossRef]

W. G. 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.

W. G. 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.

Appl. Phys. Lett.

W. G. 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.

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

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, " Rotation of microparticles with Bessel beams generated by diffractive elements," J. Mod. Opt. 51, 2167- 2184 ( 2004).
[CrossRef]

J. Opt. A Pure Appl. Opt.

W. M. Lee, B. P. S. Ahluwalia, X. -C. Yuan, W. C. Cheong, and K. Dholakia, " Optical steering of high and low index microparticles by manipulating an off-axis optical vortex," J. Opt. A Pure Appl. Opt. 7, 1- 6 ( 2005).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

A. Niv, G. Biener, V. Kleiner, and E. Hasman, " Spiral phase elements obtained by use of discrete space-variant subwavelength gratings," Opt. Commun. 251, 306- 314 ( 2005).
[CrossRef]

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

C. Paterson and R. Smith, " Higher-order Bessel waves produced by axicon-type computer-generated holograms," Opt. Commun. 124, 123- 130 ( 1996).
[CrossRef]

Opt. Express

Opt. Lett.

S. Sundbeck, I. Gruzberg, and D. G. Grier, " Structure and scaling of helical modes of light," Opt. Lett. 30, 1- 13 ( 2005).
[CrossRef]

Other

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

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Figures (7)

Fig. 1
Fig. 1

Radial intensity distribution in the spatial spectrum plane when a conic wave is diffracted by a SPP (k∕f = 1 mm−2). (a) Infinite aperture, different orders of singularities n: 0 (curve A), 3 (curve B), and 5 (curve C). (b) Finite aperture, (R = 35 mm), different orders of singularities n: 0, 3, 5, 10 (curves A, B, C, and D respectively). (c) Singularity order n = 3, different apertures: infinite (curve A) and R = 50, 35, 20 mm (curves B, C, and D, respectively).

Fig. 2
Fig. 2

Generation of a laser field with second-order phase singularity: (a) desired phase distribution, (b) central part of the SPP microrelief. Shown also are the field intensity distributions registered by a CCD camera at several distances from the SPP: (c) z = 10 mm, (d) z = 40 mm, (e) z = 80 mm, (f) at the focal plane of the lens.

Fig. 3
Fig. 3

Fraunhofer diffraction patterns for (a) a conic wave (diffraction axicon's parameter, α = 44.5 mm−1) and (b) a Gaussian beam (waist radius, ω = 0.8 mm) diffracted by a SPP with numbers (a) n = 5 and (b) n = 2.

Fig. 4
Fig. 4

Diffraction of (a) a conic wave and (b) a Gaussian beam by a SPP: solid curves are experimental and solid curves with circles are theoretical intensity distributions.

Fig. 5
Fig. 5

Four-order DOE to generate laser beams with singularities of the ±3rd and ±7th orders.

Fig. 6
Fig. 6

Microparticle motion in a laser beam with the 7th order angular harmonic.

Fig. 7
Fig. 7

Trapping of a group of particles in the 3rd order angular harmonic.

Equations (185)

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

ρ 2 n
n 2 ρ 4
5   mm
5   μm
exp ( i n φ )
n = 3
5   μm
831   nm
3%
n = 1
( 500   μm
1 .04   μm
633 nm
10%
3   μm
n = 1.59
789   nm
SiO 2
100   mm
928   nm
2 .5   mm
1082   nm
514   nm
n = 80
n 1
5   μm
6   mm
10 .6   μm
r 2 n
n 2 r 4
( w r ) 2 n
n 2 r 4
n = 2
n = 5
n = 2
± 3
± 7
f n ( r , θ ) = exp ( i α r + i n θ ) , n = 0 , ± 1 , ± 2 , ,
n
α = 2 π ( n 1 ) λ tan ( ν / 2 ) .
2 β
g n ( r , θ ) = exp ( i n θ ) , n = 0 , ± 1 , ± 2 , .
α = 2 π sin β λ .
F n ( ρ , φ ) = i k 2 π f 0 0 2 π exp [ i α r + i n θ i k f r ρ × cos ( θ φ ) ] r d r d θ ,
k = 2 π / λ
F n ( ρ , φ ) = ( i ) n + 2 k f × exp ( i n φ ) d d α 0 exp ( i α r ) J n ( k f r ρ ) d r ,
J n ( x )
0 exp ( i b x ) J n ( c x ) d x =
{ exp [ i n arcsin ( b / c ) ] ( c 2 b 2 ) 1 / 2 b < c c n i n + 1 [ b + ( b 2 c 2 ) 1 / 2 ] n ( b 2 c 2 ) 1 / 2 b > c ,
F n ( ρ , φ ) = k f exp ( i n φ ) × { ρ ¯ n ( α + i n ρ ¯ 2 α 2 ) ( ρ ¯ 2 α 2 ) 3 / 2 ( α + i ρ ¯ 2 α 2 ) n α < ρ ¯ i ρ ¯ n ( α + n α 2 ρ ¯ 2 ) ( α 2 ρ ¯ 2 ) 3 / 2 ( α + α 2 ρ ¯ 2 ) n α > ρ ¯ ,
ρ ¯ = k ρ / f
n = 0
F n ( ρ ) = k f { α ( ρ ¯ 2 α 2 ) 3 / 2     α < ρ ¯ i α ( α 2 ρ ¯ 2 ) 3 / 2 α > ρ ¯ .
F n ( ρ , φ )
α = ρ ¯
ρ 0 = α f / k
I ¯ 0
I ¯ n ( ρ ) = | F n ( ρ , φ ) | 2 = ( k f ) 2 { α 2 + n 2 ( ρ ¯ 2 α 2 ) ( ρ ¯ 2 α 2 ) 3 α < ρ ¯ , ρ ¯ 2 n ( α + n α 2 ρ 2 ) 2 ( α 2 ρ ¯ 2 ) 3 ( α + α 2 ρ ¯ 2 ) 2 n α > ρ ¯ .
I ¯ 0
n = 0
ρ 1   and   ρ 2
ρ 1 , 2 = f k [ α 2 ± ( α k I ¯ 0 f ) 2 / 3 ] 1 / 2 .
I ¯ 0 ( ρ ) = I ¯ 0
ρ 1 + ρ 2
α < ( k / f ) 1 / 2 I ¯ 0 - 1 / 4
ρ = 0
I ¯ 0 [ I ¯ 0 ( ρ = 0 ) > I ¯ 0 ]
ρ = 0
I ¯ n ( ρ = 0 ) = { 0 n 0 ( k f α 2 ) 2 n = 0 .
( n 0 )
I ¯ n ( 0 ) = 0
( ρ = 0 )
α ρ ¯
I ¯ n ( ρ ) = ( k f ) 2 ( ρ ¯ α ) 2 n ( n + 1 ) 2 4 n α 4 .
ρ ¯
α / ρ ¯
I ¯ n ( ρ ¯ 0 , α 0 ) α 4 .
ρ ¯ = 0
α 0
( ρ = 0 )
I ¯ n ( ρ ¯ = 0 ) = 0
I ¯ n ( ρ ¯ ) ( ρ ¯ α ) 2 n , ρ ¯ α .
α ρ ¯
I ¯ n ( ρ ) ( f n k ρ 2 ) 2 .
α = 0
I ¯ n ( ρ ¯ ) = { 0       ρ ¯ = 0 ( k n f ρ ¯ 2 ) 2       ρ ¯ 0 .
( k / f = 1 mm 2 )
ρ ¯
ρ ¯ = 2 mm 1
ρ ¯
7 %
ρ ¯
ρ ¯ = 2 mm −1
α = ρ ¯
α > ρ ¯ , F 0 ( ρ )
α < ρ ¯
F 0 ( ρ )
α = ρ ¯
F 0 ( ρ )
π / 2   rad
α = ρ ¯
F 0 ( ρ )
F 0 ( ρ )
n = 0
α = 0
F 0 ( ρ ) = i k f 0 J 0 ( r ρ ¯ ) r d r = i k δ ( ρ ¯ ) f ρ ¯ .
δ ( x ) d x = 1 ,
0 F 0 ( ρ ) ρ d ρ = i f 2 k .
F 0 ( ρ )
α = ρ ¯
0 exp [ ( i α + β ) r ] J 0 ( ρ ¯ r ) r d r = i α + β ( ρ ¯ 2 α 2 + β 2 + 2 i α β ) 3 / 2 .
β = 0
α   and   ρ ¯
F 0 ( ρ , β ) = α k f ( ρ ¯ 2 α 2 + 2 i α β ) 3 / 2 .
β = 0
F 0 ( ρ , β = 0 )
0 F 0 ( ρ , β ) ρ d ρ = i f 2 k ( 1 2 i β ) 1 / 2 .
β = 0
α = 0
β = 0
α = ρ ¯
I ¯ 0 ( ρ , β ) = | F 0 ( ρ , β ) | 2 = ( α k f ) 2 [ ( ρ ¯ 2 α 2 ) 2 + 4 α 2 β 2 ] 3 / 2 .
0 I ¯ 0 ( ρ , β ) ρ d ρ = 1 8 β 2 [ 1 + ( 1 + 4 β 2 α 2 ) 1 / 2 ] .
β 0
f n ( r , θ ) = exp ( r 2 w 2 + i n θ ) ,
F n ( ρ , φ ) = ( i ) n + 1 k f exp ( i n φ ) 0 exp ( r 2 w 2 ) J n ( k f r ρ ) r d r .
0 exp ( p x 2 ) J n ( c x ) x d x = c π 8 p 3 / 2 exp ( c 2 8 p ) [ I ( n 1 ) / 2 ( c 2 8 p ) I ( n + 1 ) / 2 ( c 2 8 p ) ] .
I ν ( x )
F n ( ρ , φ ) = ( i ) n + 1 exp ( i n φ ) ( k w 2 4 f ) 2 π x     exp ( x ) [ I ( n 1 ) / 2 ( x ) I ( n + 1 ) / 2 ( x ) ] ,
x = 1 2 ( k w ρ 2 f ) 2 .
I ¯ n ( ρ ) = | F n ( ρ , φ ) | 2 = 2 π ( k w 2 4 f ) 2 x exp ( 2 x ) [ I ( n 1 ) / 2 ( x ) I ( n + 1 ) / 2 ( x ) ] 2 .
x = 0
( if   n 0 )
I ¯ n ( 0 ) = 0
x exp ( 2 x )
( n 4 x ) I ( n 1 ) / 2 ( x ) + ( n + 4 x ) I ( n + 1 ) / 2 ( x ) = 0.
ρ ( or   x )
I ν ( x ) exp ( x ) 2 π x ( 1 4 ν 2 1 8 x ) , x 1 .
x
I ¯ n ( ρ ) ( n f k ρ 2 ) 2 .
ρ
( ω )
I ν ( x ) ( x 2 ) ν Γ 1 ( ν + 1 ) , x 1 ,
Γ ( x )
ρ 0
I ¯ n ( ρ ) π Γ 2 ( n + 1 2 ) ( k w 2 f ) ( k w ρ 4 f ) 2 n .
I ¯ n ( ρ ) ( w ρ ) 2 n , ρ 1 .
1 / α
I ¯ n ( ρ 0 , w ) w 2 , ρ w = const .
ρ = 0
I ¯ n ( ρ = 0 ) = 0
500   μm
2 .5   mm × 2 .5   mm
5   μm × 5   μm
λ =633   nm
1320   nm
1341   nm
n r = 1.457
1.5 %
514   nm
543   nm
633   nm
2 π ( 1−1 / N )
( 200 ×
f = 135   mm
f = 135   mm
R = 2.5   mm
α =44 .5   mm 1
n = 5
λ = 0.633   μm
1   mW
2 π m
ω = 0.8   mm
n = 2
ρ 0 = α f / k = 605   μm
Δ ρ = 2 λ f / R 68   μm
ρ 2 = 0.46 λ f / σ 45   μm
± 3 rd   and   ± 7 th
ρ 2 n
n 2 ρ 4
5   mm
n = 2
n = 5
n = 2
( 5   μm

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