Abstract

A theoretical model is proposed to approximately describe an anomalous hollow beam of elliptical symmetry with an elliptical solid core, which was observed in experiment recently [Phys. Rev. Lett. 94, 134802 (2005) ]. Expressions for the propagation factor and effective beam spot size for the anomalous hollow beam are derived. Based on the Collins integral formula, an analytical propagation formula for the anomalous hollow beam passing through a paraxial ABCD optical system is derived. The propagation properties in free space are studied graphically.

© 2007 Optical Society of America

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References

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2007 (1)

2006 (1)

2005 (1)

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

2004 (2)

Y. Cai and Q. Lin, J. Opt. Soc. Am. A 21, 1058 (2004).
[CrossRef]

F. E. S. Vetelino and L. C. Andrews, Proc. SPIE 5160, 86 (2004).
[CrossRef]

2003 (2)

J. Yin, W. Gao, and Y. Zhu, in Progress in Optics, E.Wolf, ed. (North-Holland, 2003), Vol. 44, pp. 119-204.
[CrossRef]

Y. Cai, X. Lu, and Q. Lin, Opt. Lett. 28, 1084 (2003).
[CrossRef] [PubMed]

2000 (2)

1997 (1)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

1996 (1)

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef] [PubMed]

1993 (1)

1992 (1)

1990 (1)

A. E. Siegman, Proc. SPIE 1224, 2 (1990).
[CrossRef]

1987 (1)

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[CrossRef]

1980 (1)

1970 (1)

1969 (1)

Ahmad, M. A.

Andrews, L. C.

F. E. S. Vetelino and L. C. Andrews, Proc. SPIE 5160, 86 (2004).
[CrossRef]

Arlt, J.

J. Arlt and K. Dholakia, Opt. Commun. 177, 297 (2000).
[CrossRef]

Arnaud, J. A.

Cai, Y.

Carter, W. H.

Chavez-Cerda, S.

Collins, S. A.

Dholakia, K.

J. Arlt and K. Dholakia, Opt. Commun. 177, 297 (2000).
[CrossRef]

Gao, W.

J. Yin, W. Gao, and Y. Zhu, in Progress in Optics, E.Wolf, ed. (North-Holland, 2003), Vol. 44, pp. 119-204.
[CrossRef]

Gori, F.

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[CrossRef]

Gutierrez-Vega, J. C.

He, S.

Heckenberg, N. R.

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Ito, H.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef] [PubMed]

Iturbe-Castillo, M. D.

Jhe, W.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef] [PubMed]

Kogelink, H.

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Lee, K. I.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef] [PubMed]

Li, J.

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

Lin, J.

Lin, Q.

Littman, M. G.

Liu, J.

Liu, S.

Liu, Z.

Lu, X.

McDuff, R.

Nakata, T.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef] [PubMed]

Ohtsu, M.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef] [PubMed]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[CrossRef]

Sakaki, K.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef] [PubMed]

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Proc. SPIE 1224, 2 (1990).
[CrossRef]

Smith, C. P.

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Vetelino, F. E. S.

F. E. S. Vetelino and L. C. Andrews, Proc. SPIE 5160, 86 (2004).
[CrossRef]

Wang, X.

White, A. G.

Wu, J.

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

Wu, Y. K.

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

Yin, J.

J. Yin, W. Gao, and Y. Zhu, in Progress in Optics, E.Wolf, ed. (North-Holland, 2003), Vol. 44, pp. 119-204.
[CrossRef]

Zhao, H.

Zhu, Y.

J. Yin, W. Gao, and Y. Zhu, in Progress in Optics, E.Wolf, ed. (North-Holland, 2003), Vol. 44, pp. 119-204.
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

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

Opt. Commun. (2)

J. Arlt and K. Dholakia, Opt. Commun. 177, 297 (2000).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[CrossRef]

Opt. Express (1)

Opt. Lett. (5)

Phys. Rev. Lett. (3)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef] [PubMed]

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

Proc. SPIE (2)

A. E. Siegman, Proc. SPIE 1224, 2 (1990).
[CrossRef]

F. E. S. Vetelino and L. C. Andrews, Proc. SPIE 5160, 86 (2004).
[CrossRef]

Other (1)

J. Yin, W. Gao, and Y. Zhu, in Progress in Optics, E.Wolf, ed. (North-Holland, 2003), Vol. 44, pp. 119-204.
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Contour graph for the normalized irradiance of an anomalous hollow beam for four different sets of w 0 x , w 0 y , and α (a) w 0 x = w 0 y = 1 mm , α = 0 ; (b) w 0 x = 2 mm , w 0 y = 1 mm , α = 0 ; (c) w 0 x = 2 mm , w 0 y = 1 mm , α = π 4 ; (d) w 0 x = 2 mm , w 0 y = 1 mm , α = 3 π 4 .

Fig. 2
Fig. 2

Normalized 3D-irradiance distribution of an anomalous hollow beam and cross line ( y = 0 ) in free space at several different propagation distances (a) z = 0.5 m , (b) z = 3 m , (c) z = 10 m , (d) z = 50 m .

Equations (12)

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E ( x , y , 0 ) = exp ( x 2 w 0 x 2 y 2 w 0 y 2 ) H 2 ( 2 x 2 w 0 x 2 + 2 y 2 w 0 y 2 ) ,
H 2 ( x ) = 2 + 4 x 2 ,
E ( x , y , 0 ) = ( 2 + 8 x 2 w 0 x 2 + 8 y 2 w 0 y 2 ) exp ( x 2 w 0 x 2 y 2 w 0 y 2 ) .
W s z = 2 s 2 E ( x , y , z ) 2 d x d y E ( x , y , z ) 2 d x d y , ( s = x , y ) ,
M x 2 = 4 π Δ x Δ p x , M y 2 = 4 π Δ y Δ p y ,
Δ s = 1 I ( s s ¯ ) 2 E ( x , y , 0 ) 2 d x d y ,
Δ p s = 1 I ( p s p ¯ s ) 2 E ̃ ( p x , p y , 0 ) 2 d p x d p y , ( s = x , y ) ,
s ¯ = 1 I s E ( x , y , 0 ) 2 d x d y , p ¯ s = 1 I p s E ̃ ( p x , p y , 0 ) 2 d p x d p y ,
I = E ( x , y , 0 ) 2 d x d y = E ̃ ( p x , p y , 0 ) 2 d p x d p y
E ( x 1 , y 1 , z ) = i k 2 π B exp ( i k z ) exp [ i k D 2 B ( x 1 2 + y 1 2 ) ] × E ( x , y , 0 ) exp [ i k A 2 B ( x 2 + y 2 ) + i k B ( x 1 x + y 1 y ) ] d x d y ,
E ( x 1 , y 1 , z ) = 2 i k B exp ( i k z ) exp [ i k D 2 B ( x 1 2 + y 1 2 ) ] 1 ( k A w 0 x 2 2 i B ) 2 ( i k A w 0 y 2 + 2 B ) 2 ( i k A B + 2 w 0 x 2 ) ( i k A B + 2 w 0 y 2 ) ( α + β x 1 2 + γ y 1 2 ) exp [ 1 2 k 2 ( w 0 x 2 2 B 2 + i A B k w 0 x 2 x 1 2 + w 0 y 2 2 B 2 + i A B k w 0 y 2 y 1 2 ) ] ,
W s z = 36 B 2 + 17 A 2 k 2 w 0 s 4 10 k 2 w 0 s 2 , ( s = x , y ) .

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