Abstract

We study the nonparaxial diffraction of a Gaussian vortex beam with initial radial polarization and an arbitrary integer topological charge n. Analytical relationships for the radial, azimuthal, and longitudinal components of the E-vector are deduced. At n=0, the azimuthal component of the field equals zero, with the radial and axial components becoming coincident with the relationships reported in [J. Opt. Soc. Am. A 26, 1366 (2009) ]. At any n>1, the vortex beam intensity on the optical axis equals zero, whereas at n=1 (1) an intensity peak is found in the focus. Explicit analytical relationships for a Gaussian vortex beam with initial elliptical polarization are also derived. Relationships that describe the nonparaxial radially polarized Gaussian beam are deduced as a linear combination of the Gaussian vortex beams with n=1 (1) and left- and right-hand circular polarization.

© 2010 Optical Society of America

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References

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  26. W. H. Carter, “Focal shift and concept of effective Fresnel number for a Gaussian laser beam,” Appl. Opt. 21, 1989-1994 (1982).
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    [CrossRef]
  28. Q. Zhu, “Description of the propagation of a radially polarized beam with the scalar Kirchhoff diffraction,” J. Mod. Opt. 56(14), 1621-1625 (2009).
    [CrossRef]

2009

V. V. Kotlyar and A. A. Kovalev, “Nonparaxial hypergeometric beams,” J. Opt. A, Pure Appl. Opt. 11, 045711 (2009).
[CrossRef]

M. Rashid, O. M. Marago, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 11, 065204 (2009).
[CrossRef]

Q. Zhu, “Description of the propagation of a radially polarized beam with the scalar Kirchhoff diffraction,” J. Mod. Opt. 56(14), 1621-1625 (2009).
[CrossRef]

B. Chen and J. Pu, “Tight focusing of elliptically polarized vortex beams,” Appl. Opt. 48, 1288-1294 (2009).
[CrossRef]

P. Banerjee, G. Cook, and D. Evans, “A q-parameter approach to analysis of propagation, focusing, and waveguiding of radially polarized Gaussian beams,” J. Opt. Soc. Am. A 26, 1366-1374 (2009).
[CrossRef]

2008

2007

2006

2005

2004

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers by Laguerre-Gaussian beams,” Opt. Commun. 237, 89-95 (2004).
[CrossRef]

2003

2001

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space variant subwavelength metal strip grating,” Appl. Phys. Lett. 79, 1587-1589 (2001).
[CrossRef]

1999

1990

1982

1981

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211-215 (1981).
[CrossRef]

Abramovitz, M.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Applied Math. Series, 1965).

Aït-Ameur, K.

Almazov, A.

Armstrong, D. J.

Banerjee, P.

Biener, G.

Bomzon, Z.

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space variant subwavelength metal strip grating,” Appl. Phys. Lett. 79, 1587-1589 (2001).
[CrossRef]

Carter, W. H.

Chen, B.

Cook, G.

Deng, D.

Denis, R. de S.

Dong, J.

Elfstrom, H.

Evans, D.

Ford, D. H.

Gahagan, K. T.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier, 2007).

Hasman, E.

Hierle, R.

Jones, P. H.

M. Rashid, O. M. Marago, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 11, 065204 (2009).
[CrossRef]

Khonina, S.

Khonina, S. N.

Kimura, W. D.

Kleiner, V.

Kotlyar, V.

Kotlyar, V. V.

Kovalev, A. A.

Li, Y.

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211-215 (1981).
[CrossRef]

Lou, Q.

Luneburg, R. K.

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

Marago, O. M.

M. Rashid, O. M. Marago, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 11, 065204 (2009).
[CrossRef]

Mei, Z.

Moiseev, O. Yu.

Nesterov, A. V.

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

Niv, A.

Niziev, V. G.

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

Passilly, N.

Petrov, D.

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers by Laguerre-Gaussian beams,” Opt. Commun. 237, 89-95 (2004).
[CrossRef]

Philips, M. C.

Pu, J.

Rashid, M.

M. Rashid, O. M. Marago, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 11, 065204 (2009).
[CrossRef]

Roch, J.-F.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier, 2007).

Salamin, Y. I.

Y. I. Salamin, “Electron acceleration from rest in vacuum by an axicon Gaussian laser beam,” Phys. Rev. A 73, 043402 (2006).
[CrossRef]

Skidanov, R. V.

Smith, A. V.

Soifer, V.

Soifer, V. A.

Stegun, I. A.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Applied Math. Series, 1965).

Swartzlander, G. A.

Tidwell, S. C.

Tossavainen, N.

Treussart, F.

Turunen, J.

Volpe, G.

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers by Laguerre-Gaussian beams,” Opt. Commun. 237, 89-95 (2004).
[CrossRef]

Wei, Y.

Wolf, E.

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211-215 (1981).
[CrossRef]

Wu, G.

Yirmiyahu, Y.

Zhao, D.

Zhou, J.

Zhu, Q.

Q. Zhu, “Description of the propagation of a radially polarized beam with the scalar Kirchhoff diffraction,” J. Mod. Opt. 56(14), 1621-1625 (2009).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space variant subwavelength metal strip grating,” Appl. Phys. Lett. 79, 1587-1589 (2001).
[CrossRef]

Computer Optics

A. A. Kovalev and V. V. Kotlyar, “Nonparaxial vector diffraction of a Gaussian beam by a spiral phase plate,” Computer Optics 31, 19-22 (2007) (in Russian).

J. Mod. Opt.

Q. Zhu, “Description of the propagation of a radially polarized beam with the scalar Kirchhoff diffraction,” J. Mod. Opt. 56(14), 1621-1625 (2009).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

V. V. Kotlyar and A. A. Kovalev, “Nonparaxial hypergeometric beams,” J. Opt. A, Pure Appl. Opt. 11, 045711 (2009).
[CrossRef]

M. Rashid, O. M. Marago, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 11, 065204 (2009).
[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, 1455-1461 (1999).
[CrossRef]

Opt. Commun.

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers by Laguerre-Gaussian beams,” Opt. Commun. 237, 89-95 (2004).
[CrossRef]

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211-215 (1981).
[CrossRef]

Opt. Express

Phys. Rev. A

Y. I. Salamin, “Electron acceleration from rest in vacuum by an axicon Gaussian laser beam,” Phys. Rev. A 73, 043402 (2006).
[CrossRef]

Other

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

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier, 2007).

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Applied Math. Series, 1965).

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Figures (4)

Fig. 1
Fig. 1

Amplitude (real part ) of a radially polarized Gaussian beam ( n = 0 ) in the plane of the lens’ geometrical focus: (a) radial component E r ; (b) longitudinal component E z ; (c ) intensity I .

Fig. 2
Fig. 2

Amplitude (real part ) of a Gaussian vortex beam ( n = 1 ) with initial radial polarization in the plane of the lens’ geometrical focus: (a) radial component E r ; (b) azimuthal component E ϕ ; (c) longitudinal component E z ; (d) intensity I.

Fig. 3
Fig. 3

Modeling a non-vortex Gaussian beam ( n = 0 ) with the radial polarization in the initial plane ( z = 0 ) using the FDTD method: (a) instantaneous amplitude E x in the X Z plane (horizontal dashed line is position of focus, dotted lines are borders of simulation area); (b) time-averaged diffraction pattern in the plane z = 4 μ m ; (c) corresponding horizontal cross-section.

Fig. 4
Fig. 4

Modeling a vortex Gaussian beam ( n = 1 ) with the radial polarization in the initial plane ( z = 0 ) using the FDTD method: (a) instantaneous amplitude E x in the X Z plane (horizontal dashed line is position of focus, dotted lines are borders of simulation area); (b) time-averaged diffraction pattern in the plane z = 4 μ m ; (c) horizontal cross-section .

Equations (46)

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{ E x ( u , v , z ) = 1 2 π R 2 E x ( x , y , 0 ) z [ exp ( i k R ) R ] d x d y , E y ( u , v , z ) = 1 2 π R 2 E y ( x , y , 0 ) z [ exp ( i k R ) R ] d x d y , E z ( u , v , z ) = 1 2 π R 2 { E x ( x , y , 0 ) u [ exp ( i k R ) R ] } { + E y ( u , v , 0 ) v [ exp ( i k R ) R ] } d x d y ,
[ E r E ϕ ] = [ cos ϕ sin ϕ sin ϕ cos ϕ ] [ E x E y ] ,
[ E x E y ] = [ cos ϕ sin ϕ sin ϕ cos ϕ ] [ E r E ϕ ] ,
cos ϕ = x x 2 + y 2 , sin ϕ = y x 2 + y 2 .
E r ( ρ , θ , z ) = 1 2 π R 2 [ E r ( r , ϕ , 0 ) cos ( ϕ θ ) E ϕ ( r , ϕ , 0 ) sin ( ϕ θ ) ] z [ exp ( i k R ) R ] r d r d ϕ ,
E ϕ ( ρ , θ , z ) = 1 2 π R 2 [ E ϕ ( r , ϕ , 0 ) cos ( ϕ θ ) + E r ( r , ϕ , 0 ) sin ( ϕ θ ) ] z [ exp ( i k R ) R ] r d r d ϕ .
0 2 π sin ( ϕ θ ) z [ exp ( i k R ) R ] d ϕ = 0 2 π z [ exp ( i k R ) R ] d cos ( ϕ θ ) = | χ [ cos ( ϕ θ ) ] | ϕ = 0 2 π = 0 ,
z [ exp ( i k z 2 + ρ 2 + r 2 2 ρ r ξ ) z 2 + ρ 2 + r 2 2 ρ r ξ ] .
z [ exp ( i k R ) R ] i k z z 2 + ρ 2 exp ( i k z 2 + ρ 2 ) exp ( i k r 2 2 z 2 + ρ 2 ) exp [ i k ρ r cos ( ϕ θ ) z 2 + ρ 2 ] .
E r ( ρ , θ , z ) = i k z 2 π ( z 2 + ρ 2 ) exp ( i k z 2 + ρ 2 ) R 2 [ E r ( r , ϕ + θ , 0 ) cos ϕ E ϕ ( r , ϕ + θ , 0 ) sin ϕ ] exp ( i k r 2 2 z 2 + ρ 2 ) exp ( i k ρ r cos ϕ z 2 + ρ 2 ) r d r d ϕ ,
E ϕ ( ρ , θ , z ) = i k z 2 π ( z 2 + ρ 2 ) exp ( i k z 2 + ρ 2 ) R 2 [ E ϕ ( r , ϕ + θ , 0 ) cos ϕ + E r ( r , ϕ + θ , 0 ) sin ϕ ] exp ( i k r 2 2 z 2 + ρ 2 ) exp ( i k ρ r cos ϕ z 2 + ρ 2 ) r d r d ϕ .
{ E r ( r , ϕ , 0 ) = A r ( r ) exp ( i n ϕ ) , E ϕ ( r , ϕ , 0 ) = 0 , }
E r ( ρ , θ , z ) = ( i ) n k z 2 ( z 2 + ρ 2 ) exp ( i k z 2 + ρ 2 + i n θ ) × 0 A r ( r ) exp ( i k r 2 2 z 2 + ρ 2 ) [ J n 1 ( k ρ r z 2 + ρ 2 ) J n + 1 ( k ρ r z 2 + ρ 2 ) ] r d r ,
{ E x ( x , y , 0 ) = E 0 ( 2 ω 0 ) x exp [ α ( x 2 + y 2 ) ] , E y ( x , y , 0 ) = E 0 ( 2 ω 0 ) y exp [ α ( x 2 + y 2 ) ] , }
E r ( r , ϕ , 0 ) = E 0 ( 2 ω 0 ) r exp ( α r 2 ) .
E r ( ρ , θ , z ) = ( i ) n k z E 0 2 ω 0 ( z 2 + ρ 2 ) exp ( i k z 2 + ρ 2 + i n θ ) 0 r 2 exp ( β r 2 ) [ J n 1 ( γ r ) J n + 1 ( γ r ) ] d r ,
γ = k ρ z 2 + ρ 2 , β = α i k 2 z 2 + ρ 2 = α q z 2 + ρ 2 ,
q = z 2 + ρ 2 i k 2 α .
0 exp ( α x 2 ) J v ( β x ) d x = π 2 α exp ( y ) I v 2 ( y ) ,
0 x 2 exp ( α x 2 ) J ν ( β x ) d x = π 4 α α exp ( y ) [ ( ν 1 + 2 y ) I ν 2 ( y ) 2 y I ( ν 2 ) / 2 ( y ) ] .
E r ( ρ , θ , z ) = ( i ) n π k z E 0 4 2 ω 0 β β ( z 2 + ρ 2 ) [ ( n 4 x ) I ( n 1 ) / 2 ( x ) + ( n + 4 x ) I ( n + 1 ) / 2 ( x ) ] exp ( i k z 2 + ρ 2 x + i n θ ) ,
E ϕ ( ρ , θ , z ) = ( i ) n + 1 n π k z E 0 4 2 ω 0 β β ( z 2 + ρ 2 ) [ I ( n 1 ) / 2 ( x ) + I ( n + 1 ) / 2 ( x ) ] exp ( i k z 2 + ρ 2 x + i n θ ) .
E r ( ρ , θ , z ) = k 2 z E 0 ρ 2 2 ω 0 α 2 q 2 z 2 + ρ 2 exp ( i k z 2 + ρ 2 k 2 ρ 2 4 α q z 2 + ρ 2 ) .
E r ( ρ , θ , z ) = k 2 E 0 ρ 2 2 q 2 α 2 ω 0 exp ( i k z ) exp ( i k ρ 2 2 q ) .
E z ( ρ , θ , z ) = 1 2 π R 2 { E r ( r , ϕ , 0 ) [ r ρ cos ( ϕ θ ) ] + E ϕ ( r , ϕ , 0 ) ρ sin ( ϕ θ ) } 1 R R [ exp ( i k R ) R ] r d r d ϕ .
E z ( ρ , θ , z ) i k 2 π ( z 2 + ρ 2 ) exp ( i k z 2 + ρ 2 ) R 2 E r ( r , ϕ , 0 ) [ r ρ cos ( ϕ θ ) ] exp ( i k r 2 2 z 2 + ρ 2 ) exp [ i k ρ r cos ( ϕ θ ) z 2 + ρ 2 ] r d r d ϕ .
E z ( ρ , θ , z ) = ( i ) n i k 2 ( z 2 + ρ 2 ) exp ( i k z 2 + ρ 2 + i n θ ) { 2 0 r 2 A r ( r ) exp ( i k r 2 2 z 2 + ρ 2 ) J n ( k ρ r z 2 + ρ 2 ) d r + i ρ 0 r A r ( r ) exp ( i k r 2 2 z 2 + ρ 2 ) [ J n + 1 ( k ρ r z 2 + ρ 2 ) J n 1 ( k ρ r z 2 + ρ 2 ) ] d r } .
E z ( ρ , θ , z ) = ( i ) n i k E 0 2 ω 0 ( z 2 + ρ 2 ) exp ( i k z 2 + ρ 2 + i n θ ) { 2 0 r 3 exp ( β r 2 ) J n ( γ r ) d r + i ρ 0 r 2 exp ( β r 2 ) [ J n + 1 ( γ r ) J n 1 ( γ r ) ] d r } .
β = α i k 2 z 2 + ρ 2 , γ = k ρ z 2 + ρ 2 .
0 x exp ( α x 2 ) J ν ( β x ) d x = π β 8 α 3 2 exp ( β 2 8 α ) [ I ( ν 1 ) / 2 ( β 2 8 α ) I ( ν + 1 ) / 2 ( β 2 8 α ) ] ,
0 x 3 exp ( β x 2 ) J ν ( γ x ) d x = π γ 8 β 2 β exp ( t ) { 3 2 [ I ( ν 1 ) / 2 ( t ) I ( ν + 1 ) / 2 ( t ) ] + t 2 [ I ( ν 3 ) / 2 ( t ) 3 I ( ν 1 ) / 2 ( t ) + 3 I ( ν + 1 ) / 2 ( t ) I ( ν + 3 ) / 2 ( t ) ] } ,
E z ( ρ , θ , z ) = ( i ) n π i k E 0 4 2 β 2 β ω 0 ( z 2 + ρ 2 ) exp ( i k z 2 + ρ 2 t + i n θ ) { 3 γ 2 [ I ( n 1 ) / 2 ( t ) I ( n + 1 ) / 2 ( t ) ] + γ t 2 [ I ( n 3 ) / 2 ( t ) 3 I ( n 1 ) / 2 ( t ) + 3 I ( n + 1 ) / 2 ( t ) I ( n + 3 ) / 2 ( t ) ] i β ρ [ ( n 4 t ) I ( n 1 ) / 2 ( t ) + ( n + 4 t ) I ( n + 1 ) / 2 ( t ) ] } .
E z ( ρ , θ , z ) = ( i ) n π i k E 0 4 2 β 2 β ω 0 ( z 2 + ρ 2 ) exp ( i k z 2 + ρ 2 t + i n θ ) × { [ γ ( n + 2 ) 2 2 γ t i β ρ ( n 4 t ) ] I ( n 1 ) 2 ( t ) + [ γ ( n 2 ) 2 + 2 γ t i β ρ ( n + 4 t ) ] I ( n + 1 ) 2 ( t ) } .
E z ( ρ , θ , z ) = i k E 0 2 q 2 α 2 ω 0 ( 1 + i k ρ 2 2 q ) exp ( i k z 2 + ρ 2 k 2 ρ 2 4 q α z 2 + ρ 2 ) .
i k z 2 + ρ 2 k 2 ρ 2 4 q α z 2 + ρ 2 i k z + i k ρ 2 2 z k 2 ρ 2 4 q α z = i k z + i k ρ 2 2 q .
E z ( ρ , θ , z ) i k E 0 2 q 2 α 2 ω 0 ( 1 + i k ρ 2 2 q ) exp ( i k z + i k ρ 2 2 q ) .
{ E x ( r , ϕ , 0 ) = B x exp ( r 2 ω 2 ) exp ( i n ϕ ) , E y ( r , ϕ , 0 ) = B y exp ( r 2 ω 2 ) exp ( i n ϕ ) , }
{ E x , y ( ρ , θ , z ) = ( i ) n + 1 B x , y k z exp ( i n θ + i k ρ 2 + z 2 ) ρ 2 + z 2 c π 8 p 3 2 exp ( y ) [ I ( n 1 ) 2 ( y ) I ( n + 1 ) 2 ( y ) ] , E z ( ρ , θ , z ) = ( i ) n k ρ 2 + z 2 exp ( i k ρ 2 + z 2 + i n θ ) π 8 p 3 2 × exp ( y ) ( B x i B y 2 exp ( i θ ) { ( n + 3 3 y ) [ I ( n + 1 ) 2 ( y ) I ( n + 3 ) 2 ( y ) ] + y [ I ( n 1 ) 2 ( y ) I ( n + 5 ) 2 ( y ) ] } ) B x + i B y 2 exp ( i θ ) { ( n + 1 3 y ) [ I ( n 1 ) 2 ( y ) I ( n + 1 ) 2 ( y ) ] + y [ I ( n 3 ) 2 ( y ) I ( n + 3 ) 2 ( y ) ] } ( i ( B x cos θ + B y sin θ ) c ρ [ I ( n 1 ) 2 ( y ) I ( n + 1 ) 2 ( y ) ] ) , }
p = 1 ω 2 i k 2 ρ 2 + z 2 , c = k ρ ρ 2 + z 2 , y = c 2 8 p ,
{ E r ( ρ , θ , z ) = B x k z c π 4 ( ρ 2 + z 2 ) p 3 2 × exp ( i k ρ 2 + z 2 y ) [ I 0 ( y ) I 1 ( y ) ] , E ϕ ( ρ , θ , z ) = 0 , E z ( ρ , θ , z ) = i B x k π 4 ( ρ 2 + z 2 ) p 3 2 exp ( i k ρ 2 + z 2 y ) × { ( 2 3 y + i c ρ ) [ I 0 ( y ) I 1 ( y ) ] } { + y [ I 1 ( y ) I 2 ( y ) ] } .
E 1 ( x , y , z = 0 ) = exp ( r 2 ω 2 ) r [ cos ϕ sin ϕ ] ,
E 2 ( x , y , z = 0 ) = exp ( r 2 ω 2 ) [ cos ϕ sin ϕ ] ,
{ E r ( ρ , θ , z ) = 0 , E ϕ ( ρ , θ , z ) = i B x k z c π 4 ( ρ 2 + z 2 ) p 3 2 × exp ( i k ρ 2 + z 2 y ) [ I 0 ( y ) I 1 ( y ) ] , E z ( ρ , θ , z ) = 0 .
E z ( ρ , θ , z ) = i k E 0 exp ( i k z ) 2 α 2 q 2 ω 0 ,
z = f 1 + ( f z 0 ) 2 ,
z f f = 1 1 + ( z 0 f ) 2 .

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