Abstract

The Fraunhofer diffraction pattern from a partially blocked spiral phase plate (SPP) produces a partial vortex output pattern that is rotated by 90 degrees compared with the input. The rotation direction depends on whether the angular phase pattern increases in the clockwise or counterclockwise direction. In this work, we present an explanation of this effect based on careful examination of classical diffraction theory and show new experimental results. This approach is very convenient for easily determining the sign of the vortex charge.

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References

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  1. P. A. M. Dirac, “Quantized singularities in the electromagnetic field,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character 133(821), 60–72 (1931).
    [CrossRef]
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    [CrossRef]
  3. 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]
  4. 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]
  5. 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]
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    [CrossRef]
  14. 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(5), 849–861 (2005).
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    [CrossRef]
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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]

2005 (3)

2000 (1)

1996 (1)

1995 (1)

H. He, M. E. 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]

1993 (1)

Z. Jaroszewicz and A. Kolodziejczyk, “Zone plates performing generalized Hankel transforms and their metrological applications,” Opt. Commun. 102(5-6), 391–396 (1993).
[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]

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[CrossRef]

1931 (1)

P. A. M. Dirac, “Quantized singularities in the electromagnetic field,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character 133(821), 60–72 (1931).
[CrossRef]

Almazov, A. A.

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]

Bentley, J. B.

Berry, M. V.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[CrossRef]

Campos, J.

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.

Davis, J. A.

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]

Dirac, P. A. M.

P. A. M. Dirac, “Quantized singularities in the electromagnetic field,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character 133(821), 60–72 (1931).
[CrossRef]

Elfstrom, H.

Friese, M. E.

H. He, M. E. 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]

Gahagan, K. T.

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. 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. 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]

Jaroszewicz, Z.

Z. Jaroszewicz and A. Kolodziejczyk, “Zone plates performing generalized Hankel transforms and their metrological applications,” Opt. Commun. 102(5-6), 391–396 (1993).
[CrossRef]

Khonina, S. N.

Kolodziejczyk, A.

Z. Jaroszewicz and A. Kolodziejczyk, “Zone plates performing generalized Hankel transforms and their metrological applications,” Opt. Commun. 102(5-6), 391–396 (1993).
[CrossRef]

Kotlyar, V. V.

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]

McDuff, R.

McNamara, D. E.

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[CrossRef]

Rubinsztein-Dunlop, H.

H. He, M. E. 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]

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]

Shum, P.

Smith, C. P.

Soifer, V. A.

Sun, X. W.

Swartzlander, G. A.

Turunen, J.

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]

Wang, Q.

White, A. G.

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]

Yin, X. J.

J. Mod. Opt. (1)

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]

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

Opt. Commun. (3)

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]

Z. Jaroszewicz and A. Kolodziejczyk, “Zone plates performing generalized Hankel transforms and their metrological applications,” Opt. Commun. 102(5-6), 391–396 (1993).
[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. Express (1)

Opt. Lett. (4)

Phys. Rev. Lett. (1)

H. He, M. E. 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]

Proc. R. Soc. Lond. A Math. Phys. Sci. (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[CrossRef]

Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character (1)

P. A. M. Dirac, “Quantized singularities in the electromagnetic field,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character 133(821), 60–72 (1931).
[CrossRef]

Other (3)

M. Born and E. Wolf, Principles of Optics (Macmillan, 1964), p. 395.

E. Hecht, Optics (Addison Wesley, 2002), p. 467.
[PubMed]

J. Matthews and R. L. Walker, Mathematical Methods of Physics (Benjamin, 1964), p. 177.

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Figures (3)

Fig. 1
Fig. 1

(a) SPP with CCW ( m = 30 ) angular phase, (b) partially blocked SPP with CCW ( m = 30 ) angular phase, (c) partially blocked SPP with CW ( m = + 30 ) angular phase, Figs. 1(d-f) show the output diffraction patterns formed from the patterns in Figs. 1(a-c).

Fig. 2
Fig. 2

Partially blocked SPP with CCW ( m = 30 ) angular phase with transmissive regions having (a) 135 degrees, (b) 90 degrees, (c) 45 degrees. Figures 2(d-f) show the output diffraction patterns formed from the patterns in Figs. 2(a-c).

Fig. 3
Fig. 3

Multiplexed blocked SPP patterns with (a) combination of CCW and CW SPP patterns with m = ± 30 , (b) combination of CCW SPP pattern with m = 50 and CW SPP pattern with m = + 30 , (c) combination with the left portion having a CW SPP pattern with m = + 30 and the right portion having a CCW SPP pattern with m = 30 . Figures 3(d-f) show the output diffraction patterns formed from the patterns in Figs. 3(a-c).

Equations (9)

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V m ( ρ , ϕ ) = e i m ϕ .
E ( r , Φ ) = C 0 R ρ d ρ 0 2 π d ϕ e i m ϕ e i ( k r ρ / f ) cos ( ϕ Φ ) .
E ( r , Φ ) = e i m ( Φ π / 2 ) 0 R ρ d ρ 0 2 π d x e i m x e i ( k r ρ / f ) sin x .
J m ( z ) = 1 2 π 0 2 π e i m τ e i z sin τ d τ .
E ( r , Φ ) = e i m ( Φ π / 2 ) C 0 R J m ( k r ρ / f ) ρ d ρ
J m ( x ) = ( 1 ) m J m ( x ) = e i m π J m ( x ) .
E ( r , Φ ) = e i m ( Φ + π / 2 ) C 0 R J m ( k r ρ / f ) ρ d ρ .
V m ( ρ , ϕ ) Rect { ϕ / ϕ 0 } = e i m ϕ Rect { ϕ / ϕ 0 } = j = a j e i ( m + j ) ϕ .
E ( r , Φ ) = C j = a j e i ( m + j ) ( Φ π / 2 ) 0 R J m + j ( k r ρ / f ) ρ d ρ .

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