Abstract

On the basis of the vectorial Rayleigh diffraction integral, the analytical expressions for the electromagnetic fields of the radially polarized beams diffracted at a circular aperture are derived, which helps us investigate the propagation properties of the apertured radially polarized beams in the nonparaxial and paraxial regimes. The unapertured and paraxial cases can be viewed as the special cases of the general result obtained in this paper. The analyses indicate that the nonparaxiality of the apertured radially polarized beams depends on the ratio of the waist width to the wavelength and the truncation parameter. In addition, the truncation parameter and the beam order have a great impact on the beam diffraction effect and the beam evolution behavior.

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References

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2008

G. MacHavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarization beams,” Opt. Commun. 281(4), 732–738 (2008).
[CrossRef]

S. Yan and B. Yao, “Accurate description of a radially polarized Gaussian beam,” Phys. Rev. A 77(2), 023827 (2008).
[CrossRef]

2007

2006

2005

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80(6), 707–713 (2005).
[CrossRef]

2004

2003

2002

Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88(9), 095005 (2002).
[CrossRef] [PubMed]

2001

2000

K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE 3919, 75–85 (2000).
[CrossRef]

1999

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32(13), 1455–1461 (1999).
[CrossRef]

1998

Ahmed, M. A.

M. A. Ahmed, J. Schulz, A. Voss, O. Parriaux, J. C. Pommier, and T. Graf, “Radially polarized 3kW beam from a CO2 laser with an intracavity resonant grating mirror,” Opt. Lett. 32(13), 1824–1826 (2007).
[CrossRef] [PubMed]

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80(6), 707–713 (2005).
[CrossRef]

Brown, T. G.

K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE 3919, 75–85 (2000).
[CrossRef]

Bu, J.

Burge, R. E.

Deng, D.

Duan, K.

Feurer, T.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys., A Mater. Sci. Process. 86(3), 329–334 (2007).
[CrossRef]

Gao, B. Z.

Glur, H.

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80(6), 707–713 (2005).
[CrossRef]

Gori, F.

Graf, T.

M. A. Ahmed, J. Schulz, A. Voss, O. Parriaux, J. C. Pommier, and T. Graf, “Radially polarized 3kW beam from a CO2 laser with an intracavity resonant grating mirror,” Opt. Lett. 32(13), 1824–1826 (2007).
[CrossRef] [PubMed]

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80(6), 707–713 (2005).
[CrossRef]

Guo, Q.

Jackel, S.

G. MacHavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarization beams,” Opt. Commun. 281(4), 732–738 (2008).
[CrossRef]

I. Moshe, S. Jackel, and A. Meir, “Production of radially or azimuthally polarized beams in solid-state lasers and the elimination of thermally induced birefringence effects,” Opt. Lett. 28(10), 807–809 (2003).
[CrossRef] [PubMed]

Keitel, C. H.

Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88(9), 095005 (2002).
[CrossRef] [PubMed]

Lü, B.

Lumer, Y.

G. MacHavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarization beams,” Opt. Commun. 281(4), 732–738 (2008).
[CrossRef]

MacHavariani, G.

G. MacHavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarization beams,” Opt. Commun. 281(4), 732–738 (2008).
[CrossRef]

Meier, M.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys., A Mater. Sci. Process. 86(3), 329–334 (2007).
[CrossRef]

Meir, A.

G. MacHavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarization beams,” Opt. Commun. 281(4), 732–738 (2008).
[CrossRef]

I. Moshe, S. Jackel, and A. Meir, “Production of radially or azimuthally polarized beams in solid-state lasers and the elimination of thermally induced birefringence effects,” Opt. Lett. 28(10), 807–809 (2003).
[CrossRef] [PubMed]

Moh, K. J.

Moser, T.

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80(6), 707–713 (2005).
[CrossRef]

Moshe, I.

G. MacHavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarization beams,” Opt. Commun. 281(4), 732–738 (2008).
[CrossRef]

I. Moshe, S. Jackel, and A. Meir, “Production of radially or azimuthally polarized beams in solid-state lasers and the elimination of thermally induced birefringence effects,” Opt. Lett. 28(10), 807–809 (2003).
[CrossRef] [PubMed]

Nesterov, A. V.

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32(13), 1455–1461 (1999).
[CrossRef]

Niziev, V. G.

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32(13), 1455–1461 (1999).
[CrossRef]

Parriaux, O.

M. A. Ahmed, J. Schulz, A. Voss, O. Parriaux, J. C. Pommier, and T. Graf, “Radially polarized 3kW beam from a CO2 laser with an intracavity resonant grating mirror,” Opt. Lett. 32(13), 1824–1826 (2007).
[CrossRef] [PubMed]

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80(6), 707–713 (2005).
[CrossRef]

Pigeon, F.

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80(6), 707–713 (2005).
[CrossRef]

Pommier, J. C.

Romano, V.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys., A Mater. Sci. Process. 86(3), 329–334 (2007).
[CrossRef]

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80(6), 707–713 (2005).
[CrossRef]

Salamin, Y. I.

Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88(9), 095005 (2002).
[CrossRef] [PubMed]

Schulz, J.

Tovar, A. A.

Voss, A.

Wu, L.

Yan, S.

Yang, X.

Yao, B.

Youngworth, K. S.

K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE 3919, 75–85 (2000).
[CrossRef]

Yuan, X. C.

Zhan, Q.

Appl. Opt.

Appl. Phys. B

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80(6), 707–713 (2005).
[CrossRef]

Appl. Phys., A Mater. Sci. Process.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys., A Mater. Sci. Process. 86(3), 329–334 (2007).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. D

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32(13), 1455–1461 (1999).
[CrossRef]

Opt. Commun.

G. MacHavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarization beams,” Opt. Commun. 281(4), 732–738 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

S. Yan and B. Yao, “Accurate description of a radially polarized Gaussian beam,” Phys. Rev. A 77(2), 023827 (2008).
[CrossRef]

Phys. Rev. Lett.

Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88(9), 095005 (2002).
[CrossRef] [PubMed]

Proc. SPIE

K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE 3919, 75–85 (2000).
[CrossRef]

Other

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1966).

I. S. Gradshteyn, and I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic Press, New York, 1994).

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Figures (7)

Fig. 1
Fig. 1

Nonparaxial (a) and paraxial (b) energy flux distributions of the apertured radially polarized beams versus x / λ and y / λ at the plane z = 10 z R for n = 0 , ω 0 / λ = 2 , δ = 2 . Solid and dashed curves in the panel (c) denote the corresponding cross-section energy flux distributions at y = 0 for the nonparaxial and paraxial cases, respectively.

Fig. 2
Fig. 2

The same legend as in Fig. 1 except that the parameters n = 0 , ω 0 / λ = 1 , δ = 2 .

Fig. 3
Fig. 3

The same legend as in Fig. 1 except that the parameters n = 0 , ω 0 / λ = 1 , δ = 0.5 .

Fig. 4
Fig. 4

The same legend as in Fig. 1 except that the parameters n = 2 , ω 0 / λ = 2 , δ = 2 .

Fig. 5
Fig. 5

The same legend as in Fig. 1 except that the parameters n = 2 , ω 0 / λ = 1 , δ = 2 .

Fig. 6
Fig. 6

The same legend as in Fig. 1 except that the parameters n = 2 , ω 0 / λ = 1 , δ = 0.5 .

Fig. 7
Fig. 7

Normalized maximum intensity of the longitudinal component of the electric field I z , max ( x , 0 , 10 z R ) / I x , max ( x , 0 , 10 z R ) versus ω 0 / λ in the cases of (a) δ = 0.5 (solid curve), 1 (dashed curve), 3 (dotted curve), n = 0 and (b) n = 0 (solid curve), 1 (dashed curve), 2 (dotted curve), δ = 1.

Equations (20)

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E n 1 0 ( x 0 , y 0 , 0 ) = E n 1 x 0 ( x 0 , y 0 , 0 ) e x + E n 1 y 0 ( x 0 , y 0 , 0 ) e y
E n 1 x 0 ( x 0 , y 0 , 0 ) = 2 E 0 x 0 ω 0 L n 1 ( 2 ρ 0 2 ω 0 2 ) exp ( ρ 0 2 ω 0 2 )
E n 1 y 0 ( x 0 , y 0 , 0 ) = 2 E 0 y 0 ω 0 L n 1 ( 2 ρ 0 2 ω 0 2 ) exp ( ρ 0 2 ω 0 2 )
E n 1 x ( x 0 , y 0 , 0 ) = 2 E 0 x 0 ω 0 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 ) = 2 E 0 y 0 ω 0 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 o t h e r w i s e
E n 1 x ( r ) = 1 2 π + E n 1 x ( x 0 , y 0 , 0 ) G ( r , r 0 ) z d x 0 d y 0
E n 1 y ( r ) = 1 2 π + E n 1 y ( x 0 , y 0 , 0 ) G ( r , r 0 ) z d x 0 d y 0
E n 1 z ( r ) = 1 2 π + [ E n 1 x ( x 0 , y 0 , 0 ) G ( r , r 0 ) x + E n 1 y ( x 0 , y 0 , 0 ) G ( r , r 0 ) y ] d x 0 d y 0
G ( r , r 0 ) = exp ( i k | r r 0 | ) | r r 0 |
| r r 0 | r + x 0 2 + y 0 2 2 x x 0 2 y y 0 2 r
E n 1 x ( r ) = 2 E 0 k z x ρ exp ( i k r ) r 2 ω 0 0 a ρ 0 2 L n 1 ( 2 ρ 0 2 ω 0 2 ) exp ( g ρ 0 2 ) J 1 ( k ρ ρ 0 r ) d ρ 0
E n 1 y ( r ) = 2 E 0 k z y ρ exp ( i k r ) r 2 ω 0 0 a ρ 0 2 L n 1 ( 2 ρ 0 2 ω 0 2 ) exp ( g ρ 0 2 ) J 1 ( k ρ ρ 0 r ) d ρ 0
E n 1 z ( r ) = 2 E 0 k exp ( i k r ) r 2 ω 0 [ ρ 0 a ρ 0 2 L n 1 ( 2 ρ 0 2 ω 0 2 ) exp ( g ρ 0 2 ) J 1 ( k ρ ρ 0 r ) d ρ 0 i 0 a ρ 0 3 L n 1 ( 2 ρ 0 2 ω 0 2 ) exp ( g ρ 0 2 ) J 0 ( k ρ ρ 0 r ) d ρ 0 ]
Γ ( α , x ) = x e t t α 1 d t
E n 1 ( r ) = 2 E 0 k exp ( i k r ) r 2 l = 0 n u = 0 ( n + 1 ) ! 2 l 1 ( l + 1 ) ! l ! ( n l ) ! ω 0 2 l + 1 { z ρ ( x e x + y e y ) 1 u ! ( u + 1 ) ! ( k ρ 2 r ) 2 u + 1 + e z [ ρ u ! ( u + 1 ) ! ( k ρ 2 r ) 2 u + 1 + i u ! u ! ( k ρ 2 r ) 2 u ] } [ Γ ( l + u + 2 , g a 2 ) ( l + u + 1 ) ! ] g l + u + 2
H n 1 ( r ) = i 2 ε μ E 0 exp ( i k r ) ( y e x x e y ) l = 0 n u = 0 ( n + 1 ) ! 2 l 1 ( l + 1 ) ! l ! ( n l ) ! ω 0 2 l + 1 × { ( i k r 2 ) r 4 [ ρ u ! ( u + 1 ) ! ( k ρ 2 r ) 2 u + 1 i u ! u ! ( k ρ 2 r ) 2 u ] [ Γ ( l + u + 2 , g a 2 ) ( l + u + 1 ) ! ] g l + u + 2 + [ 2 + ( i k r 2 ) z 2 r 2 ] 1 ρ r 2 u ! ( u + 1 ) ! ( k ρ 2 r ) 2 u + 1 [ Γ ( l + u + 2 , g a 2 ) ( l + u + 1 ) ! ] g l + u + 2 + k 2 r 5 [ ( i ( 3 ρ 2 r 2 ρ ) 1 u ! ( u + 1 ) ! ( k ρ 2 r ) 2 u + 1 + z 2 u ! ( u + 2 ) ! ( k ρ 2 r ) 2 u + 2 + 1 u ! u ! ( k ρ 2 r ) 2 u ) × [ Γ ( l + u + 3 , g a 2 ) ( l + u + 2 ) ! ] g l + u + 3 + z 2 u ! u ! ( k ρ 2 r ) 2 u [ Γ ( l + u + 2 , g a 2 ) ( l + u + 1 ) ! ] g l + u + 2 ] k z 2 2 r 5 [ ( i ρ u ! ( u + 1 ) ! ( k ρ 2 r ) 2 u + 1 + 1 u ! ( u + 2 ) ! ( k ρ 2 r ) 2 u + 2 ) [ Γ ( l + u + 3 , g a 2 ) ( l + u + 2 ) ! ] g l + u + 3 + 1 u ! u ! ( k ρ 2 r ) 2 u [ Γ ( l + u + 2 , g a 2 ) ( l + u + 1 ) ! ] g l + u + 2 ] }
r z + x 2 + y 2 2 z
E n 1 p ( r ) = 2 E 0 k exp [ i k ( z + ρ 2 2 z ) ] l = 0 n u = 0 ( n + 1 ) ! 2 l 1 ( l + 1 ) ! l ! ( n l ) ! ω 0 2 l + 1 { 1 ρ z ( x e x + y e y ) × 1 u ! ( u + 1 ) ! ( k ρ 2 z ) 2 u + 1 + e z 1 z 2 [ ρ u ! ( u + 1 ) ! ( k ρ 2 z ) 2 u + 1 + i u ! u ! ( k ρ 2 z ) 2 u ] } × [ Γ ( l + u + 2 , t a 2 ) ( l + u + 1 ) ! ] t l + u + 2
H n 1 p ( r ) = i 2 ε μ E 0 exp [ i k ( z + ρ 2 2 z ) ] ( y e x x e y ) l = 0 n u = 0 ( n + 1 ) ! 2 l 1 ( l + 1 ) ! l ! ( n l ) ! ω 0 2 l + 1 × { ( i k z 2 ) z 4 [ ρ u ! ( u + 1 ) ! ( k ρ 2 z ) 2 u + 1 i u ! u ! ( k ρ 2 z ) 2 u ] [ Γ ( l + u + 2 , t a 2 ) ( l + u + 1 ) ! ] t l + u + 2 + i k ρ z u ! ( u + 1 ) ! ( k ρ 2 z ) 2 u + 1 [ Γ ( l + u + 2 , t a 2 ) ( l + u + 1 ) ! ] t l + u + 2 + k 2 z 5 [ ( i ( 3 ρ 2 z 2 ρ ) 1 u ! ( u + 1 ) ! ( k ρ 2 z ) 2 u + 1 + z 2 u ! ( u + 2 ) ! ( k ρ 2 z ) 2 u + 2 + 1 u ! u ! ( k ρ 2 z ) 2 u ) × [ Γ ( l + u + 3 , t a 2 ) ( l + u + 2 ) ! ] t l + u + 3 + z 2 u ! u ! ( k ρ 2 z ) 2 u [ Γ ( l + u + 2 , t a 2 ) ( l + u + 1 ) ! ] t l + u + 2 ] k 2 z 3 [ ( i ρ u ! ( u + 1 ) ! ( k ρ 2 z ) 2 u + 1 + 1 u ! ( u + 2 ) ! ( k ρ 2 z ) 2 u + 2 ) [ Γ ( l + u + 3 , t a 2 ) ( l + u + 2 ) ! ] t l + u + 3 + 1 u ! u ! ( k ρ 2 z ) 2 u [ Γ ( l + u + 2 , t a 2 ) ( l + u + 1 ) ! ] t l + u + 2 ] }

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