Abstract

We carry out an experimental and theoretical study of the focusing properties of a Gaussian Schell-model (GSM) vortex beam. It is found that we can shape the beam profile of the focused GSM vortex beam by varying its initial spatial coherence width. Focused dark hollow, flat-topped, and Gaussian beam spots can be obtained in our experiment, which will be useful for trapping particles. The experimental results agree well with the theoretical results.

© 2011 Optical Society of America

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References

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    [CrossRef]

2011 (1)

2010 (1)

2009 (3)

2008 (3)

2006 (3)

2004 (1)

2003 (2)

J. Yin, W. Gao, and Y. Zhu, in Progress in Optics, E.Wolf, ed. (North-Holland, 2003), Vol.  44, p. 119.
[CrossRef]

G. Gbur and T. D. Visser, Opt. Commun. 222, 117 (2003).
[CrossRef]

2002 (2)

2001 (1)

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics (Elsevier, 2001), Vol.  42, p. 219.
[CrossRef]

2000 (1)

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, and M. Nakatsuka, Opt. Rev. 7, 216 (2000).
[CrossRef]

Baykal, Y.

Cai, Y.

Dan, Y.

Exter, M. P.

Eyyuboglu, H. T.

Gao, W.

J. Yin, W. Gao, and Y. Zhu, in Progress in Optics, E.Wolf, ed. (North-Holland, 2003), Vol.  44, p. 119.
[CrossRef]

Gbur, G.

G. Gbur and T. D. Visser, Opt. Commun. 222, 117 (2003).
[CrossRef]

He, S.

Jitsuno, T.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, and M. Nakatsuka, Opt. Rev. 7, 216 (2000).
[CrossRef]

Korotkova, O.

Leger, J. R.

Li, J.

J. Li and B. Lu, J. Opt. A 11, 045710 (2009).
[CrossRef]

Lin, Q.

Lu, B.

J. Li and B. Lu, J. Opt. A 11, 045710 (2009).
[CrossRef]

Lu, X.

Matsuoka, S.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, and M. Nakatsuka, Opt. Rev. 7, 216 (2000).
[CrossRef]

Miyanaga, N.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, and M. Nakatsuka, Opt. Rev. 7, 216 (2000).
[CrossRef]

Nakatsuka, M.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, and M. Nakatsuka, Opt. Rev. 7, 216 (2000).
[CrossRef]

Nishi, N.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, and M. Nakatsuka, Opt. Rev. 7, 216 (2000).
[CrossRef]

Pires, H. D. L.

Qu, J.

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics (Elsevier, 2001), Vol.  42, p. 219.
[CrossRef]

Takeda, M.

W. Wang and M. Takeda, Phys. Rev. Lett. 96, 223904 (2006).
[CrossRef] [PubMed]

Tsubakimoto, K.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, and M. Nakatsuka, Opt. Rev. 7, 216 (2000).
[CrossRef]

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics (Elsevier, 2001), Vol.  42, p. 219.
[CrossRef]

Visser, T. D.

G. Gbur and T. D. Visser, Opt. Commun. 222, 117 (2003).
[CrossRef]

Wang, F.

Wang, W.

W. Wang and M. Takeda, Phys. Rev. Lett. 96, 223904 (2006).
[CrossRef] [PubMed]

Wang, Y.

Woudenberg, J.

Yin, J.

J. Yin, W. Gao, and Y. Zhu, in Progress in Optics, E.Wolf, ed. (North-Holland, 2003), Vol.  44, p. 119.
[CrossRef]

Yuan, Y.

Zhan, Q.

Zhang, B.

Zhao, C.

Zhu, Y.

J. Yin, W. Gao, and Y. Zhu, in Progress in Optics, E.Wolf, ed. (North-Holland, 2003), Vol.  44, p. 119.
[CrossRef]

Appl. Opt. (1)

J. Opt. A (1)

J. Li and B. Lu, J. Opt. A 11, 045710 (2009).
[CrossRef]

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

Opt. Commun. (1)

G. Gbur and T. D. Visser, Opt. Commun. 222, 117 (2003).
[CrossRef]

Opt. Express (6)

Opt. Lett. (4)

Opt. Rev. (1)

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, and M. Nakatsuka, Opt. Rev. 7, 216 (2000).
[CrossRef]

Phys. Rev. Lett. (1)

W. Wang and M. Takeda, Phys. Rev. Lett. 96, 223904 (2006).
[CrossRef] [PubMed]

Other (2)

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics (Elsevier, 2001), Vol.  42, p. 219.
[CrossRef]

J. Yin, W. Gao, and Y. Zhu, in Progress in Optics, E.Wolf, ed. (North-Holland, 2003), Vol.  44, p. 119.
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Experimental setup for generating a GSM vortex beam and measuring its focused intensity. L 1 , L 2 , L 3 , thin lenses; M, mirror; RGGP, rotating ground-glass plate; GAF, Gaussian amplitude filter; SPP, spiral phase plate; BPA, beam profile analyzer; PC, personal computer.

Fig. 2
Fig. 2

Experimental setup for measuring the transverse coherence width σ g of the generated GSM beam. BS, 50 50 beam splitter; D 1 , D 2 single photon detectors.

Fig. 3
Fig. 3

Experimental results (dotted curves) of the normalized FOCFs and corresponding Gaussian fit (solid curves) of the experimental results for two different focused beam spot sizes ( 0.08 mm and 0.15 mm ) on the RGGP.

Fig. 4
Fig. 4

Experimental results of the focused intensity distribution and the corresponding cross line (dotted curve) of the generated GSM vortex beam for three different σ g . The solid curves are calculated by theoretical formulae.

Equations (15)

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W ( r 1 , r 2 , 0 ) = exp [ r 1 2 + r 2 2 4 σ 0 2 ( r 1 r 2 ) 2 2 σ g 2 ] exp [ i l ( φ 1 φ 2 ) ] ,
I ( ρ , z ) = 1 ( λ B ) 2 W ( r 1 , r 2 , 0 ) d 2 r 1 d 2 r 2 × exp [ i k A 2 B ( r 1 2 r 2 2 ) i k B ( r 1 · ρ r 2 · ρ ) ] ,
r s = ( r 1 + r 2 ) / 2 , Δ r = r 1 r 2 .
I ( ρ , z ) = 1 ( λ B ) 2 exp [ ( Δ r ) 2 2 σ g 2 + i k B Δ r · ρ ] × P * ( r s + Δ r / 2 ) P ( r s Δ r / 2 ) d 2 r s d 2 Δ r ,
P * ( r s + Δ r / 2 ) = exp [ ( 1 4 σ 0 2 i k A 2 B ) ( r s + Δ r / 2 ) 2 ] exp ( i l φ + ) ,
P ( r s Δ r / 2 ) = exp [ ( 1 4 σ 0 2 + i k A 2 B ) ( r s Δ r / 2 ) 2 ] exp ( i l φ ) ,
P ( r s Δ r / 2 ) = 1 λ 2 B 2 P ˜ ( u λ B ) exp [ i k B ( r s Δ r / 2 ) · u ] d 2 u ,
P * ( r s + Δ r / 2 ) = 1 λ 2 B 2 P ˜ * ( u 1 λ B ) exp [ i k B ( r s + Δ r / 2 ) · u 1 ] d 2 u 1 .
I ( ρ , z ) = 1 ( λ B ) 4 | P ˜ ( u λ B ) | 2 g ˜ ( u + ρ λ B ) d 2 u ,
g ˜ ( u + ρ λ B ) = 2 π σ g 2 exp [ 2 π 2 σ g 2 ( u + ρ ) 2 λ 2 B 2 ] .
| P ˜ ( u / λ B ) | 2 = | π 2.5 M 3 u 2 λ B exp ( π 2 M 2 u 2 2 λ 2 B 2 ) | 2 × | [ I 0 ( π 2 M 2 u 2 2 λ 2 B 2 ) I 1 ( π 2 M 2 u 2 2 λ 2 B 2 ) ] | 2 ,
I ( ρ , z ) = π 7 σ g 2 | M | 6 λ 6 B 6 0 exp [ 2 π 2 σ g 2 ( u 2 + ρ 2 ) λ 2 B 2 ] I 0 ( 4 π 2 σ g 2 u ρ λ 2 B 2 ) × | exp ( π 2 u 2 M 2 2 λ 2 B 2 ) [ I 0 ( π 2 u 2 M 2 2 λ 2 B 2 ) I 1 ( π 2 u 2 M 2 2 λ 2 B 2 ) ] | 2 u 3 d u .
( A B C D ) = ( 1 f 0 1 ) ( 1 0 1 / f 1 ) ( 1 f 0 1 ) = ( 0 f 1 / f 0 ) .
g ( 2 ) ( u 1 u 2 , τ ) = I ( u 1 , t ) I ( u 2 , t + τ ) I ( u 1 , t ) I ( u 2 , t + τ ) ,
g x x ( 2 ) ( u 1 u 2 , τ = 0 ) = 1 + exp [ ( u 1 u 2 ) 2 / σ g 2 ] .

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