Abstract

We demonstrate that hollow Gaussian beams can be obtained from Fourier transform of the differentials of a Gaussian beam, and thus they can be generated by spatial filtering in the Fourier domain with spatial filters that consist of binomial combinations of even-order Hermite polynomials. A typical 4f optical system and a Michelson interferometer type system are proposed to implement the proposed scheme. Numerical results have proved the validity and effectiveness of this method. Furthermore, other polynomial Gaussian beams can also be generated by using this scheme. This approach is simple and may find significant applications in generating the dark hollow beams for nanophotonic technology.

© 2007 Optical Society of America

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References

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2006

2005

Y. K. Wu, J. Li, and J. Wu, Phys. Rev. Lett. 94, 134802 (2005).
[CrossRef] [PubMed]

2004

2003

2000

1998

J. Yin, Y. Zhu, W. Jhe, and Y. Wang, Phys. Rev. A 58, 509 (1998).
[CrossRef]

1997

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, Phys. Rev. Lett. 78, 4714 (1997).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Experimental illustration of HGB generation with a 4 f system. The polynomial filter P n ( u , v ) is displayed on the SLM.

Fig. 2
Fig. 2

Experimental setup of HGB generation with a Michelson interferometer-like 4 f system, where filter P n ( u , v ) is divided into positive and negative parts. The λ 4 wave plate implements a π phase delay.

Fig. 3
Fig. 3

Numerical results of HGB generation by the configuration shown in Figs. 1, 2 with different orders, n = 1 , 3, and 6.

Equations (15)

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H g n ( x , y ) = G 0 ( x 2 + y 2 ω 0 2 ) n exp ( x 2 + y 2 ω 0 2 ) ,
E ( x , y , z ) = G ( z ) exp [ x 2 + y 2 ω 2 ( z ) ] × exp [ i k x 2 + y 2 2 R ( z ) ] ,
E ( x , y ) = G 0 exp ( x 2 + y 2 ω 0 2 ) .
E ̃ ( u , v ) = E ( x , y ) exp [ 2 i π ( x u + y v ) ] d x d y = G 0 π ω 0 2 exp [ π 2 ω 0 2 ( u 2 + v 2 ) ] = G 0 ( 1 π ω ̃ 0 2 ) exp ( u 2 + v 2 ω ̃ 0 2 ) ,
F { n f ( x , y ) x n } = ( 2 i π ) n u n F { f ( x , y ) } ( u , v ) .
F { ( 1 ) n 2 n E ( x , y ) x 2 n } ( u , v ) = ( 2 π ) 2 n u 2 n E ̃ ( u , v ) .
H n ( x ) = ( 1 ) n exp ( x 2 ) d n d x n exp ( x 2 ) .
F { ( 1 4 ) n H 2 n ( x ω 0 ) E ( x , y ) } = ( u 2 ω ̃ 0 2 ) n E ̃ ( u , v ) .
F { ( 1 4 ) n k = 0 n C n k H 2 k ( x ω 0 ) H 2 n 2 k ( y ω 0 ) E ( x , y ) } = ( u 2 ω ̃ 0 2 + v 2 ω ̃ 0 2 ) n E ̃ ( u , v ) ,
P n ( u , v ) = ( 1 4 ) n k = 0 n C n k H 2 k ( u ω ̃ 0 ) H 2 n 2 k ( v ω ̃ 0 ) .
H g n = ( x 2 ω 0 2 + y 2 ω 0 2 ) n E ( x , y ) .
P n , ( u , v ) = { 0 , if P n ( u , v ) 0 , P n ( u , v ) , otherwise
P n , + ( u , v ) = { 0 , if P n ( u , v ) 0 P n ( u , v ) , otherwise .
k = 0 m j = 0 n W k , j ( x ω 0 ) k ( y ω 0 ) j exp ( x 2 + y 2 ω 0 2 ) .
F { k = 0 m j = 0 n ( i 2 ) k + j W k , j H k ( π ω 0 u ) H j ( π ω 0 v ) E ̃ ( u , v ) } = k = 0 m j = 0 n W k , j ( x ω 0 ) k ( y ω 0 ) j exp ( x 2 + y 2 ω 0 2 ) .

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