Abstract

Based on the vector angular spectrum representation and the method of stationary phase, the analytical vectorial structure of the cylindrically polarized Laguerre–Gaussian beam diffracted at a circular aperture is derived in the far field, which provides an approach to further comprehend the vectorial properties of the apertured cylindrically polarized beams. The radially polarized, azimuthally polarized, and unapertured cases can be viewed as the special cases of our general result. The analyses show that the far-field energy flux distributions of the entire beam, the TE term, and the TM term depend on the beam order, the ratio of the waist width to the wavelength, the truncation parameter, and the angle between the electric field vector and the radial direction.

© 2011 Optical Society of America

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2006

M. Meier, V. Romano, and T. Feurer, Appl. Phys. A 86, 329 (2006).

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2004

2001

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K. S. Youngworth and T. G. Brown, Proc. SPIE 3919, 75 (2000).

1999

V. G. Niziev and A. V. Nesterov, J. Phys. D 32, 1455(1999).

1998

Bosch, S.

Brown, T. G.

K. S. Youngworth and T. G. Brown, Proc. SPIE 3919, 75 (2000).

Bu, J.

Burge, R. E.

Carnicer, A.

Chang, R. S.

Choi, H.

W. Kim, N. Park, Y. Yoon, H. Choi, and Y. Park, Opt. Rev. 14, 236 (2007).

Chu, X.

G. Zhou, X. Chu, and J. Zheng, Opt. Commun. 281, 1929 (2008).

Deng, D.

Feurer, T.

M. Meier, V. Romano, and T. Feurer, Appl. Phys. A 86, 329 (2006).

Gao, B. Z.

Gori, F.

Guo, Q.

Gupta, D. N.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, Phys. Lett. A 368, 402 (2007).

Jackel, S.

Kant, N.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, Phys. Lett. A 368, 402 (2007).

Kim, D. E.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, Phys. Lett. A 368, 402 (2007).

Kim, W.

W. Kim, N. Park, Y. Yoon, H. Choi, and Y. Park, Opt. Rev. 14, 236 (2007).

Lumer, Y.

Machavariani, G.

Martínez-Herrero, R.

Meier, M.

M. Meier, V. Romano, and T. Feurer, Appl. Phys. A 86, 329 (2006).

Meir, A.

Mejías, P. M.

Moh, K. J.

Moshe, I.

Nesterov, A. V.

V. G. Niziev, R. S. Chang, and A. V. Nesterov, Appl. Opt. 45, 8393 (2006).
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V. G. Niziev and A. V. Nesterov, J. Phys. D 32, 1455(1999).

Niziev, V. G.

V. G. Niziev, R. S. Chang, and A. V. Nesterov, Appl. Opt. 45, 8393 (2006).
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V. G. Niziev and A. V. Nesterov, J. Phys. D 32, 1455(1999).

Park, N.

W. Kim, N. Park, Y. Yoon, H. Choi, and Y. Park, Opt. Rev. 14, 236 (2007).

Park, Y.

W. Kim, N. Park, Y. Yoon, H. Choi, and Y. Park, Opt. Rev. 14, 236 (2007).

Romano, V.

M. Meier, V. Romano, and T. Feurer, Appl. Phys. A 86, 329 (2006).

Suk, H.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, Phys. Lett. A 368, 402 (2007).

Tovar, A. A.

Wu, L.

Yan, S.

Yang, X.

Yao, B.

Yoon, Y.

W. Kim, N. Park, Y. Yoon, H. Choi, and Y. Park, Opt. Rev. 14, 236 (2007).

Youngworth, K. S.

K. S. Youngworth and T. G. Brown, Proc. SPIE 3919, 75 (2000).

Yuan, X. C.

Zhan, Q.

Zheng, J.

G. Zhou, X. Chu, and J. Zheng, Opt. Commun. 281, 1929 (2008).

Zhou, G.

G. Zhou, J. Opt. Soc. Am. A 27, 1750 (2010).

G. Zhou, J. Opt. Soc. Am. A 26, 1654 (2009).

G. Zhou, X. Chu, and J. Zheng, Opt. Commun. 281, 1929 (2008).

Appl. Opt.

Appl. Phys. A

M. Meier, V. Romano, and T. Feurer, Appl. Phys. A 86, 329 (2006).

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. D

V. G. Niziev and A. V. Nesterov, J. Phys. D 32, 1455(1999).

Opt. Commun.

G. Zhou, X. Chu, and J. Zheng, Opt. Commun. 281, 1929 (2008).

Opt. Express

Opt. Lett.

Opt. Rev.

W. Kim, N. Park, Y. Yoon, H. Choi, and Y. Park, Opt. Rev. 14, 236 (2007).

Phys. Lett. A

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, Phys. Lett. A 368, 402 (2007).

Proc. SPIE

K. S. Youngworth and T. G. Brown, Proc. SPIE 3919, 75 (2000).

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Figures (3)

Fig. 1
Fig. 1

Cylindrically polarized vector beam with ϕ rotation from the radial direction.

Fig. 2
Fig. 2

Normalized cross-section energy flux distributions of S z ( x , 0 , z ) (solid curve), S z ( x , 0 , z ) TE (dashed curve), and S z ( x , 0 , z ) TM (dotted curve) at the plane z / λ = 100 and ϕ = π / 4 . (a) n = 0 , ω 0 / λ = 2 , δ = 2 ; (b) n = 0 , ω 0 / λ = 1 , δ = 2 ; (c) n = 0 , ω 0 / λ = 1 , δ = 0.5 ; (d) n = 2 , ω 0 / λ = 2 , δ = 2 ; (e) n = 2 , ω 0 / λ = 1 , δ = 2 ; (f) n = 2 , ω 0 / λ = 1 , δ = 0.5 .

Fig. 3
Fig. 3

Normalized cross-section energy flux distributions of S z ( x , 0 , z ) at the plane z / λ = 100 for the parameters of ϕ = 0 (solid curve), π / 4 (dashed curve), π / 2 (dotted curve), n = 2 , ω 0 / λ = 1 , δ = 1 .

Equations (27)

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E n 1 ( x 0 , y 0 , 0 ) = E n 1 x ( x 0 , y 0 , 0 ) e x + E n 1 y ( x 0 , y 0 , 0 ) e y ,
E n 1 x ( x 0 , y 0 , 0 ) = A x L n 1 ( 2 ρ 0 2 ω 0 2 ) exp ( ρ 0 2 ω 0 2 ) t ( x 0 , y 0 ) ,
E n 1 y ( x 0 , y 0 , 0 ) = A y L n 1 ( 2 ρ 0 2 ω 0 2 ) exp ( ρ 0 2 ω 0 2 ) t ( x 0 , y 0 ) ,
t ( x 0 , y 0 ) = { 1 x 0 2 + y 0 2 a 2 0 otherwise ,
E n 1 ( r ) = + [ A n 1 x ( p , q ) e x + A n 1 y ( p , q ) e y e z / A n 1 z ( p , q ) ] exp ( i k v ) d p d q ,
A n 1 x ( p , q ) = 1 λ 2 + E n 1 x ( x 0 , y 0 , 0 ) exp ( i k v 0 ) d x 0 d y 0 ,
A n 1 y ( p , q ) = 1 λ 2 + E n 1 y ( x 0 , y 0 , 0 ) exp ( i k v 0 ) d x 0 d y 0 ,
E n 1 ( r ) = E TE ( r ) + E TM ( r ) ,
E TE ( r ) = + 1 b 2 [ q A n 1 x ( p , q ) p A n 1 y ( p , q ) ] ( q e x p e y ) × exp ( i k v ) d p d q ,
E TM ( r ) = + 1 b 2 [ p A n 1 x ( p , q ) + q A n 1 y ( p , q ) ] × [ p e x + q e y ( b 2 / m ) e z ] exp ( i k v ) d p d q ,
H n 1 ( r ) = H TE ( r ) + H TM ( r ) ,
H TE ( r ) = ε μ + 1 b 2 [ q A n 1 x ( p , q ) p A n 1 y ( p , q ) ] ( p m e x + q m e y b 2 e z ) exp ( i k v ) d p d q ,
H TM ( r ) = ε μ + 1 b 2 m [ p A n 1 x ( p , q ) + q A n 1 y ( p , q ) ] ( q e x + p e y ) exp ( i k v ) d p d q ,
E TE ( r ) = 2 E 0 k z sin ϕ ω 0 ρ r 2 exp ( i k r ) T n ( r ) s 1 ,
E TM ( r ) = 2 E 0 k z cos ϕ ω 0 ρ r 2 exp ( i k r ) T n ( r ) s 2 ,
H TE ( r ) = ε μ 2 E 0 k z 2 sin ϕ ω 0 ρ r 3 exp ( i k r ) T n ( r ) s 2 ,
H TM ( r ) = ε μ 2 E 0 k cos ϕ ω 0 ρ r exp ( i k r ) T n ( r ) s 1 ,
T n ( r ) = 0 a ρ 0 2 L n 1 ( 2 ρ 0 2 ω 0 2 ) exp ( ρ 0 2 ω 0 2 ) J 1 ( k ρ ρ 0 r ) d ρ 0 ,
E TE ( r ) = 2 E 0 k z sin ϕ ρ r 2 exp ( i k r ) I n ( r ) s 1 ,
E TM ( r ) = 2 E 0 k z cos ϕ ρ r 2 exp ( i k r ) I n ( r ) s 2 ,
H TE ( r ) = ε μ 2 E 0 k z 2 sin ϕ ρ r 3 exp ( i k r ) I n ( r ) s 2 ,
H TM ( r ) = ε μ 2 E 0 k cos ϕ ρ r exp ( i k r ) I n ( r ) s 1 ,
I n ( r ) = l = 0 n u = 0 ( 1 ) l + u ( n + 1 ) ! 2 l 1 ( l + 1 ) ! l ! ( n l ) ! u ! ( u + 1 ) ! ( k ρ 2 r ) 2 u + 1 × ω 0 2 u + 3 [ Γ ( l + u + 2 , δ 2 ) ( l + u + 1 ) ! ] ,
E TE ( r ) = ( 1 ) n 2 E 0 k 2 z ω 0 3 sin ϕ 4 r 3 I n ( r ) s 1 ,
E TM ( r ) = ( 1 ) n + 1 2 E 0 k 2 z ω 0 3 cos ϕ 4 r 3 I n ( r ) s 2 ,
H TM ( r ) = ε μ ( 1 ) n 2 E 0 k 2 z 2 ω 0 3 sin ϕ 4 r 4 I n ( r ) s 2 ,
H TM ( r ) = ε μ ( 1 ) n 2 E 0 k 2 ω 0 3 cos ϕ 4 r 2 I n ( r ) s 1 ,

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