Abstract

We derive analytical expressions containing a hypergeometric function to describe the Fresnel and Fraunhofer diffraction of a plane wave of circular and ringlike cross section by a spiral phase plate (SPP) of an arbitrary integer order. Experimental diffraction patterns generated by an SPP fabricated in resist through direct e-beam writing are in good agreement with the theoretical intensity distribution.

© 2006 Optical Society of America

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  1. S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, J. Mod. Opt. 39, 1147 (1992).
    [CrossRef]
  2. M. V. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, Opt. Commun. 112, 321 (1994).
    [CrossRef]
  3. J. E. Curtis and D. G. Grier, Phys. Rev. Lett. 90, 133901 (2003).
    [CrossRef] [PubMed]
  4. 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, Appl. Opt. 43, 688 (2004).
    [CrossRef] [PubMed]
  5. W. G. Cheong, W. M. Lee, X.-C. Yuan, L.-S. Zhang, K. Dholakia, and H. Wang, Appl. Phys. Lett. 85, 5784 (2004).
    [CrossRef]
  6. W. M. Lee, B. P. S. Ahluwalia, X.-C. Yuan, W. C. Cheong, and K. Dholakia, J. Opt. A 7, 1 (2005).
    [CrossRef]
  7. K. Sueda, G. Miyaji, N. Miyanaga, and M. Nakatsura, Opt. Express 12, 3548 (2004).
    [CrossRef] [PubMed]
  8. S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, J. Mod. Opt. 51, 2167 (2004).
    [CrossRef]
  9. S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, Phys. Rev. Lett. 92, 217901 (2004).
    [CrossRef] [PubMed]
  10. V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Soifer, H. Elfstrom, and J. Turunen, J. Opt. Soc. Am. A 22, 849 (2005).
    [CrossRef]
  11. M. V. Berry, J. Opt. A 6, 259 (2004).
    [CrossRef]
  12. Z. S. Sacks, D. Rozas, G. A. Swartzlander Jr., J. Opt. Soc. Am. B 15, 2226 (1998).
    [CrossRef]
  13. C. Guo, X. Liu, J. He, and H. Wang, Opt. Express 12, 4625 (2004).
    [CrossRef] [PubMed]
  14. S. Sundbeck, I. Gruzberg, and D. G. Grier, Opt. Lett. 30, 477 (2005).
    [CrossRef] [PubMed]
  15. A. P. Prudnikov, Y. A. Brichkov, and O. I. Marichev, Integrals and Series: Special Functions (Nauka, 1983).

2005 (3)

2004 (7)

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, Appl. Opt. 43, 688 (2004).
[CrossRef] [PubMed]

K. Sueda, G. Miyaji, N. Miyanaga, and M. Nakatsura, Opt. Express 12, 3548 (2004).
[CrossRef] [PubMed]

C. Guo, X. Liu, J. He, and H. Wang, Opt. Express 12, 4625 (2004).
[CrossRef] [PubMed]

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, J. Mod. Opt. 51, 2167 (2004).
[CrossRef]

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, Phys. Rev. Lett. 92, 217901 (2004).
[CrossRef] [PubMed]

M. V. Berry, J. Opt. A 6, 259 (2004).
[CrossRef]

W. G. Cheong, W. M. Lee, X.-C. Yuan, L.-S. Zhang, K. Dholakia, and H. Wang, Appl. Phys. Lett. 85, 5784 (2004).
[CrossRef]

2003 (1)

J. E. Curtis and D. G. Grier, Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef] [PubMed]

1998 (1)

1994 (1)

M. V. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, Opt. Commun. 112, 321 (1994).
[CrossRef]

1992 (1)

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, J. Mod. Opt. 39, 1147 (1992).
[CrossRef]

Ahluwalia, B. P. S.

W. M. Lee, B. P. S. Ahluwalia, X.-C. Yuan, W. C. Cheong, and K. Dholakia, J. Opt. A 7, 1 (2005).
[CrossRef]

Aiello, A.

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, Phys. Rev. Lett. 92, 217901 (2004).
[CrossRef] [PubMed]

Almazov, A. A.

Beijersbergen, M. V.

M. V. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, Opt. Commun. 112, 321 (1994).
[CrossRef]

Berry, M. V.

M. V. Berry, J. Opt. A 6, 259 (2004).
[CrossRef]

Brichkov, Y. A.

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

Cheong, W. C.

W. M. Lee, B. P. S. Ahluwalia, X.-C. Yuan, W. C. Cheong, and K. Dholakia, J. Opt. A 7, 1 (2005).
[CrossRef]

Cheong, W. G.

W. G. Cheong, W. M. Lee, X.-C. Yuan, L.-S. Zhang, K. Dholakia, and H. Wang, Appl. Phys. Lett. 85, 5784 (2004).
[CrossRef]

Coerwinkel, R. P. C.

M. V. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, Opt. Commun. 112, 321 (1994).
[CrossRef]

Curtis, J. E.

J. E. Curtis and D. G. Grier, Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef] [PubMed]

Dholakia, K.

W. M. Lee, B. P. S. Ahluwalia, X.-C. Yuan, W. C. Cheong, and K. Dholakia, J. Opt. A 7, 1 (2005).
[CrossRef]

W. G. Cheong, W. M. Lee, X.-C. Yuan, L.-S. Zhang, K. Dholakia, and H. Wang, Appl. Phys. Lett. 85, 5784 (2004).
[CrossRef]

Elfstrom, H.

Eliel, E. R.

Grier, D. G.

Gruzberg, I.

Guo, C.

He, J.

Hooft, G. W.

Jefimovs, K.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, J. Mod. Opt. 51, 2167 (2004).
[CrossRef]

Khonina, S. N.

V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Soifer, H. Elfstrom, and J. Turunen, J. Opt. Soc. Am. A 22, 849 (2005).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, J. Mod. Opt. 51, 2167 (2004).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, J. Mod. Opt. 39, 1147 (1992).
[CrossRef]

Kloosterboer, J. G.

Kotlyar, V. V.

V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Soifer, H. Elfstrom, and J. Turunen, J. Opt. Soc. Am. A 22, 849 (2005).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, J. Mod. Opt. 51, 2167 (2004).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, J. Mod. Opt. 39, 1147 (1992).
[CrossRef]

Kristensen, M.

M. V. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, Opt. Commun. 112, 321 (1994).
[CrossRef]

Lee, W. M.

W. M. Lee, B. P. S. Ahluwalia, X.-C. Yuan, W. C. Cheong, and K. Dholakia, J. Opt. A 7, 1 (2005).
[CrossRef]

W. G. Cheong, W. M. Lee, X.-C. Yuan, L.-S. Zhang, K. Dholakia, and H. Wang, Appl. Phys. Lett. 85, 5784 (2004).
[CrossRef]

Liu, X.

Marichev, O. I.

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

Miyaji, G.

Miyanaga, N.

Nakatsura, M.

Nienhuis, G.

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, Phys. Rev. Lett. 92, 217901 (2004).
[CrossRef] [PubMed]

Oemrawsingh, S. S. R.

Prudnikov, A. P.

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

Rozas, D.

Sacks, Z. S.

Shinkarev, M. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, J. Mod. Opt. 39, 1147 (1992).
[CrossRef]

Simonen, J.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, J. Mod. Opt. 51, 2167 (2004).
[CrossRef]

Skidanov, R. V.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, J. Mod. Opt. 51, 2167 (2004).
[CrossRef]

Soifer, V. A.

V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Soifer, H. Elfstrom, and J. Turunen, J. Opt. Soc. Am. A 22, 849 (2005).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, J. Mod. Opt. 51, 2167 (2004).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, J. Mod. Opt. 39, 1147 (1992).
[CrossRef]

Sueda, K.

Sundbeck, S.

Swartzlander, G. A.

Turunen, J.

V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Soifer, H. Elfstrom, and J. Turunen, J. Opt. Soc. Am. A 22, 849 (2005).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, J. Mod. Opt. 51, 2167 (2004).
[CrossRef]

Uspleniev, G. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, J. Mod. Opt. 39, 1147 (1992).
[CrossRef]

van Houwelinger, J. A. W.

Vestegen, E. J. K.

Wang, H.

C. Guo, X. Liu, J. He, and H. Wang, Opt. Express 12, 4625 (2004).
[CrossRef] [PubMed]

W. G. Cheong, W. M. Lee, X.-C. Yuan, L.-S. Zhang, K. Dholakia, and H. Wang, Appl. Phys. Lett. 85, 5784 (2004).
[CrossRef]

Woerdman, J. P.

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, Phys. Rev. Lett. 92, 217901 (2004).
[CrossRef] [PubMed]

M. V. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, Opt. Commun. 112, 321 (1994).
[CrossRef]

Woerdman, J. R.

Yuan, X.-C.

W. M. Lee, B. P. S. Ahluwalia, X.-C. Yuan, W. C. Cheong, and K. Dholakia, J. Opt. A 7, 1 (2005).
[CrossRef]

W. G. Cheong, W. M. Lee, X.-C. Yuan, L.-S. Zhang, K. Dholakia, and H. Wang, Appl. Phys. Lett. 85, 5784 (2004).
[CrossRef]

Zhang, L.-S.

W. G. Cheong, W. M. Lee, X.-C. Yuan, L.-S. Zhang, K. Dholakia, and H. Wang, Appl. Phys. Lett. 85, 5784 (2004).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

W. G. Cheong, W. M. Lee, X.-C. Yuan, L.-S. Zhang, K. Dholakia, and H. Wang, Appl. Phys. Lett. 85, 5784 (2004).
[CrossRef]

J. Mod. Opt. (2)

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, J. Mod. Opt. 39, 1147 (1992).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, J. Mod. Opt. 51, 2167 (2004).
[CrossRef]

J. Opt. A (2)

M. V. Berry, J. Opt. A 6, 259 (2004).
[CrossRef]

W. M. Lee, B. P. S. Ahluwalia, X.-C. Yuan, W. C. Cheong, and K. Dholakia, J. Opt. A 7, 1 (2005).
[CrossRef]

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

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

Opt. Commun. (1)

M. V. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, Opt. Commun. 112, 321 (1994).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. Lett. (2)

J. E. Curtis and D. G. Grier, Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef] [PubMed]

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, Phys. Rev. Lett. 92, 217901 (2004).
[CrossRef] [PubMed]

Other (1)

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

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Figures (4)

Fig. 1
Fig. 1

Radial intensity distributions of the Fraunhofer diffraction pattern for a plane wave of radius R = 1 mm diffracted by an SPP with n = 1 (curve 1), n = 2 (curve 2), and n = 3 (curve 3).

Fig. 2
Fig. 2

Radial intensity distribution of the Fraunhofer diffraction pattern of a plane wave diffracted by an SPP with n = 2 at R = 3 mm (curve 1), R = 2.5 mm (curve 2), and R = 2 mm (curve 3).

Fig. 3
Fig. 3

(a) The SPP surface profile ( n = 3 ) , the Fresnel diffraction pattern for a plane wave of radius R = 1.25 mm and wavelength λ = 0.633 μ m at distance z = 80 mm from the SPP. (b) Experiment and theory: (c) without and (d) with displacement (0.1 R) between the centers of the beam and the SPP.

Fig. 4
Fig. 4

Fraunhofer diffraction of a plane wave of radius R = 1.25 mm and wavelength λ = 0.633 μ m by an SPP with n = 3 , generated at the focal plane of a Fourier lens of focus 150 mm . (a) Intensity distribution and intensity cross sections: (b) horizontal and (c) vertical (A, theory, B, experiment).

Equations (8)

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

E 0 ( r ) = circl ( r R ) = { 1 r R 0 r > R } .
τ ( φ ) = exp ( i n φ ) , n = 0 , ± 1 , ± 2 , ,
E n R ( ρ , θ ) = ( i ) n + 1 k f exp ( i n θ ) 0 R J n ( k f r ρ ) r d r = ( 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 ] ,
I n ( ξ ) R 4 ξ 2 n F 2 1 [ n + 2 2 , n + 4 2 , n + 1 ; ( k ξ 2 f ) 2 ] 2 ,
I n ( ρ n ) = ( k R 2 2 f ) 2 J n 2 ( k R ρ n f ) ,
ρ n γ n 1 , 1 λ f 2 π R ,
E ( ρ , θ ) = E n R ( ρ , θ ) E n R 1 ( ρ , θ ) ,
E n ( ρ , θ , z ) = 2 exp ( i z 0 ρ ¯ 2 z + i n θ ) ( i z 0 z ) n + 1 ρ ¯ n n ! × 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 ] ,

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