Abstract

Based on the vectorial Rayleigh-Sommerfeld formulas under the weak nonparaxial approximation, we investigate the propagation behavior of a lowest-order Laguerre-Gaussian beam with azimuthal-variant states of polarization. We present the analytical expressions for the radial, azimuthal, and longitudinal components of the electric field with an arbitrary integer topological charge m focused by a nonaperturing thin lens. We illustrate the three-dimensional optical intensities, energy flux distributions, beam waists, and focal shifts of the focused azimuthal-variant vector beams under the nonparaxial and paraxial approximations.

© 2012 OSA

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
  4. X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett.105, 253602 (2010).
    [CrossRef]
  5. V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A27, 372–380 (2010).
    [CrossRef]
  6. X. Jia and Y. Wang “Vectorial structure of far field of cylindrically polarized beams diffracted at a circular aperture,” Opt. Lett.36, 295–297 (2011).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]

2012 (1)

2011 (4)

X. Jia and Y. Wang “Vectorial structure of far field of cylindrically polarized beams diffracted at a circular aperture,” Opt. Lett.36, 295–297 (2011).
[CrossRef] [PubMed]

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B102, 205–213 (2011).
[CrossRef]

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett.106, 123901 (2011).
[CrossRef] [PubMed]

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Lensless focusing of hypergeometric laser beams,” J. Opt.13, 075703 (2011).
[CrossRef]

2010 (3)

2009 (5)

2008 (1)

2007 (3)

2006 (2)

A. F. Abouraddy and K. C. Toussaint, “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett.96, 153901 (2006).
[CrossRef] [PubMed]

D. M. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B23, 1228–1234 (2006).
[CrossRef]

2005 (1)

2004 (2)

2003 (2)

B. Lü and K. Duan, “Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture,” Opt. Lett.28, 2440–2442 (2003).
[CrossRef] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91, 233901 (2003).
[CrossRef] [PubMed]

2002 (1)

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun.202, 17–20 (2002).
[CrossRef]

2001 (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett.86, 5251–5254 (2001).
[CrossRef] [PubMed]

2000 (1)

1999 (1)

1998 (1)

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A253, 358–379 (1959).
[CrossRef]

Abouraddy, A. F.

A. F. Abouraddy and K. C. Toussaint, “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett.96, 153901 (2006).
[CrossRef] [PubMed]

Ahn, J. S.

Banerjee, P. P.

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett.86, 5251–5254 (2001).
[CrossRef] [PubMed]

Borghi, R.

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett.86, 5251–5254 (2001).
[CrossRef] [PubMed]

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7, 77–87 (2000).
[CrossRef] [PubMed]

Cai, Y.

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B102, 205–213 (2011).
[CrossRef]

Chen, J.

B. Gu, F. Ye, K. Lou, Y. Li, J. Chen, and H. T. Wang, “Vectorial self-diffraction effect in optically Kerr medium,” Opt. Express20, 149–157 (2012).
[CrossRef] [PubMed]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett.105, 253602 (2010).
[CrossRef]

Ciattoni, A.

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun.202, 17–20 (2002).
[CrossRef]

Cook, G.

Crosignani, B.

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun.202, 17–20 (2002).
[CrossRef]

Deng, D.

Deng, D. M.

Ding, J. P.

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett.105, 253602 (2010).
[CrossRef]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91, 233901 (2003).
[CrossRef] [PubMed]

Duan, K.

Evans, D. R.

Gillen, G. D.

Greene, P. L.

Gu, B.

Guha, S.

Guo, C. S.

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett.105, 253602 (2010).
[CrossRef]

Guo, Q.

Hall, D. G.

Hnatovsky, C.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett.106, 123901 (2011).
[CrossRef] [PubMed]

Jia, X.

Jones, P. H.

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

Kihm, H. W.

Kihm, J. E.

Kim, D. S.

Kotlyar, V. V.

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Lensless focusing of hypergeometric laser beams,” J. Opt.13, 075703 (2011).
[CrossRef]

V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A27, 372–380 (2010).
[CrossRef]

Kovalev, A. A.

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Lensless focusing of hypergeometric laser beams,” J. Opt.13, 075703 (2011).
[CrossRef]

V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A27, 372–380 (2010).
[CrossRef]

Krolikowski, W.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett.106, 123901 (2011).
[CrossRef] [PubMed]

Lan, S.

Lee, K. G.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91, 233901 (2003).
[CrossRef] [PubMed]

Li, B.

Li, X.

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B102, 205–213 (2011).
[CrossRef]

Li, Y.

Li, Y. N.

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett.105, 253602 (2010).
[CrossRef]

Lou, K.

Lü, B.

Maragò, O. M.

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

Mei, Z.

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett.86, 5251–5254 (2001).
[CrossRef] [PubMed]

Porto, P. D.

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun.202, 17–20 (2002).
[CrossRef]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91, 233901 (2003).
[CrossRef] [PubMed]

Rashid, M.

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

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A253, 358–379 (1959).
[CrossRef]

Rode, A.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett.106, 123901 (2011).
[CrossRef] [PubMed]

Santarsiero, M.

Shvedov, V.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett.106, 123901 (2011).
[CrossRef] [PubMed]

Soifer, V. A.

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Lensless focusing of hypergeometric laser beams,” J. Opt.13, 075703 (2011).
[CrossRef]

Toussaint, K. C.

A. F. Abouraddy and K. C. Toussaint, “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett.96, 153901 (2006).
[CrossRef] [PubMed]

Tovar, A. A.

Wang, H. T.

B. Gu, F. Ye, K. Lou, Y. Li, J. Chen, and H. T. Wang, “Vectorial self-diffraction effect in optically Kerr medium,” Opt. Express20, 149–157 (2012).
[CrossRef] [PubMed]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett.105, 253602 (2010).
[CrossRef]

Wang, X. L.

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett.105, 253602 (2010).
[CrossRef]

Wang, Y.

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A253, 358–379 (1959).
[CrossRef]

Wu, L. J.

Yang, X. B.

Ye, F.

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett.86, 5251–5254 (2001).
[CrossRef] [PubMed]

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7, 77–87 (2000).
[CrossRef] [PubMed]

Zhao, D.

Zhou, G.

Appl. Phys. B (1)

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B102, 205–213 (2011).
[CrossRef]

J. Opt. (1)

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Lensless focusing of hypergeometric laser beams,” J. Opt.13, 075703 (2011).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

M. Rashid, O. M. Maragò, 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 (7)

J. Opt. Soc. Am. B (4)

Opt. Commun. (1)

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun.202, 17–20 (2002).
[CrossRef]

Opt. Express (7)

Opt. Lett. (2)

Phys. Rev. Lett. (5)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91, 233901 (2003).
[CrossRef] [PubMed]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett.86, 5251–5254 (2001).
[CrossRef] [PubMed]

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett.106, 123901 (2011).
[CrossRef] [PubMed]

A. F. Abouraddy and K. C. Toussaint, “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett.96, 153901 (2006).
[CrossRef] [PubMed]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett.105, 253602 (2010).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A253, 358–379 (1959).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Nonparaxial intensity patterns (top row) and the corresponding cross-section intensity profiles when y = 0 (middle row) of a vector beam with m = 1 for φ0 = π/4 at the plane of the lens’ geometrical focus by taking λ = 633 nm, E0 = 1 (a.u.), ω0 = 1 μm, and f = 4 μm. The intensity patterns are normalized by I G Max ( x , y , f ). Circles in (e)–(h) give the corresponding paraxial intensity profiles. The bottom row gives the nonparaxial intensity patterns through the focus, normalized by I G Max ( x , 0 , z ). Dotted and dashed lines in (l) are positions of the true focus and the lens’ geometrical focus, respectively.

Fig. 2
Fig. 2

Beam waists ρ0 of nonparaxial (paraxial) focused vector beam with m = 1 and φ0 = π/4 through focus for λ = 633 nm, E0 = 1 (a.u.), ω0 = 1 μm, and f = 4 μm.

Fig. 3
Fig. 3

Nonparaxial energy flux patterns of a vector beam with m = 1 at the planes of (a) the lens’ geometrical focus and (b) the true focus. Solid lines (circles) in (c) and (d) denote the corresponding nonparaxial (paraxial) energy flux profiles for y = 0 at z = f and z = 0.55 f, respectively. Numerical parameters: φ0 = π/4, λ = 633 nm, E0 = 1 (a.u.), ω0 = 1 μm, and f = 4 μm.

Fig. 4
Fig. 4

Normalized paraxial intensity patterns of vector beams with different topological charges m at focus (top row) and through focus (lower row), by taking φ0 = 0, λ = 532 nm, E0 = 1 (a.u.), ω0 = 2.5 mm, and f = 8 mm.

Fig. 5
Fig. 5

Beam waists ρ0 of the paraxial focused vector beams with different topological charges through focus, by taking φ0 = 0, λ = 532 nm, E0 = 1 (a.u.), ω0 = 2.5 mm, and f = 8 mm.

Fig. 6
Fig. 6

Paraxial cross-section energy flux profiles of the vector beams with (a) different m for φ0 = 0 and (b) m = 3 for different φ0 at the plane of z = f, by taking λ = 532 nm, E0 = 1 (a.u.), ω0 = 2.5 mm, f = 8 mm, and y = 0.

Equations (34)

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E ( r , ϕ , 0 ) = E r ( r , ϕ , 0 ) e ^ r + E ϕ ( r , ϕ , 0 ) e ^ ϕ ,
E r ( r , ϕ , 0 ) = A ( r ) cos ( m ϕ ϕ + φ 0 ) ,
E ϕ ( r , ϕ , 0 ) = A ( r ) sin ( m ϕ ϕ + φ 0 ) .
E r ( ρ , θ , z ) = i k z 2 π ξ 2 exp ( i k ξ ) × 0 0 2 π [ E r ( r , ϕ + θ , 0 ) cos ϕ E ϕ ( r , ϕ + θ , 0 ) sin ϕ ] × exp ( i k r 2 2 ξ ) exp ( i γ r cos ϕ ) r d r d ϕ ,
E ϕ ( ρ , θ , z ) = i k z 2 π ξ 2 exp ( i k ξ ) × 0 0 2 π [ E r ( r , ϕ + θ , 0 ) sin ϕ + E ϕ ( r , ϕ + θ , 0 ) cos ϕ ] × exp ( i k r 2 2 ξ ) exp ( i γ r cos ϕ ) r d r d ϕ ,
E z ( ρ , θ , z ) = i k 2 π ξ 2 exp ( i k ξ ) × 0 0 2 π { E r ( r , ϕ + θ , 0 ) [ r ρ cos ϕ ] + E ϕ ( r , ϕ + θ , 0 ) ρ sin ϕ } × exp ( i k r 2 2 ξ ) exp ( i γ r cos ϕ ) r d r d ϕ ,
0 2 π cos ( m ϕ + φ 0 ) exp [ i x cos ( ϕ θ ) ] d ϕ 2 π ( i ) m J m ( x ) cos ( m θ + φ 0 ) ,
0 2 π sin ( m ϕ + φ 0 ) exp [ i x cos ( ϕ θ ) ] d ϕ 2 π ( i ) m J m ( x ) sin ( m θ + φ 0 ) ,
E r ( ρ , θ , z ) = ( i ) m + 1 k z ξ 2 exp ( i k ξ ) cos Ψ 0 A ( r ) exp ( i k r 2 2 ξ ) J m ( γ r ) r d r ,
E ϕ ( ρ , θ , z ) = ( i ) m + 1 k z ξ 2 exp ( i k ξ ) sin Ψ 0 A ( r ) exp ( i k r 2 2 ξ ) J m ( γ r ) r d r ,
E z ( ρ , θ , z ) = ( i ) m + 1 k ξ 2 exp ( i k ξ ) cos Ψ 0 A ( r ) [ ρ J m ( γ r ) i r J m 1 ( γ r ) ] exp ( i k r 2 2 ξ ) r d r ,
A ( r ) = E 0 ( 2 ω 0 ) r exp ( α r 2 ) ,
0 r 2 e β r 2 J m ( γ r ) d r = π 4 β 3 / 2 e t [ 2 t I ( m 2 ) / 2 ( t ) ( m 1 + 2 t ) I m / 2 ( t ) ] ,
0 r 3 e β r 2 J m 1 ( γ r ) d r = π γ 16 β 5 / 2 e t [ ( m + 1 4 t ) I ( m 2 ) / 2 ( t ) + ( m 3 + 4 t ) I m / 2 ( t ) ] ,
E r ( ρ , θ , z ) = ( i ) m + 1 π E 0 k z 2 2 β 3 / 2 ω 0 ξ 2 exp ( i k ξ t ) cos Ψ × [ 2 t I ( m 2 ) / 2 ( t ) ( m 1 + 2 t ) I m / 2 ( t ) ] ,
E ϕ ( ρ , θ , z ) = ( i ) m + 1 π E 0 k z 2 2 β 3 / 2 ω 0 ξ 2 exp ( i k ξ t ) sin Ψ × [ 2 t I ( m 2 ) / 2 ( t ) ( m 1 + 2 t ) I m / 2 ( t ) ] ,
E z ( ρ , θ , z ) = ( i ) m + 1 π E 0 k 2 2 β 3 / 2 ω 0 ξ 2 exp ( i k ξ t ) cos Ψ × { [ i γ 4 β ( m + 1 4 t ) 2 ρ t ] I ( m 2 ) / 2 ( t ) + [ i γ 4 β ( m 3 + 4 t ) + ρ ( m 1 + 2 t ) ] I m / 2 ( t ) } ,
E r ( ρ , θ , z ) = E 0 k 2 ρ z 2 2 ω 0 α 2 q 2 ξ exp ( i k ξ k 2 ρ 2 4 α q ξ ) ,
E ϕ ( ρ , θ , z ) = 0 ,
E z ( ρ , θ , z ) = i E 0 k 2 ω 0 α 2 q 2 ( 1 + i k ρ 2 2 q ) exp ( i k ξ k 2 ρ 2 4 α q ξ ) ,
E r ( ρ , θ , z ) = 0 ,
E ϕ ( ρ , θ , z ) = E 0 k 2 ρ z 2 2 ω 0 α 2 q 2 ξ exp ( i k ξ k 2 ρ 2 4 α q ξ ) ,
E z ( ρ , θ , z ) = 0 .
E r p ( ρ , θ , z ) = ( i ) m + 1 π E 0 k 2 2 β 3 / 2 ω 0 z exp ( i k z t ) cos Ψ × [ 2 t I ( m 2 ) / 2 ( t ) ( m 1 + 2 t ) I m / 2 ( t ) ] ,
E ϕ p ( ρ , θ , z ) = ( i ) m + 1 π E 0 k 2 2 β 3 / 2 ω 0 z exp ( i k z t ) sin Ψ × [ 2 t I ( m 2 ) / 2 ( t ) ( m 1 + 2 t ) I ( m / 2 ) ( t ) ] ,
E z p ( ρ , θ , z ) = ( i ) m + 1 π E 0 k 2 2 β 3 / 2 ω 0 z 2 exp ( i k z t ) cos Ψ × { [ i γ 4 β ( m + 1 4 t ) 2 ρ t ] I ( m 2 ) / 2 ( t ) + [ i γ 4 β ( m 3 + 4 t ) + ρ ( m 1 + 2 t ) ] I m / 2 ( t ) } ,
E r p ( ρ , θ , z ) = E 0 k 2 ρ 2 2 ω 0 α 2 q 2 exp ( i k z + i k ρ 2 2 q ) ,
E ϕ p ( ρ , θ , z ) = 0 ,
E z p ( ρ , θ , z ) = i E 0 k 2 ω 0 α 2 q 2 ( 1 + i k ρ 2 2 q ) exp ( i k z + i k ρ 2 2 q ) ,
E r p ( ρ , θ , z ) = 0 ,
E ϕ p ( ρ , θ , z ) = E 0 k 2 ρ 2 2 ω 0 α 2 q 2 exp ( i k z + i k ρ 2 2 q ) ,
E z p ( ρ , θ , z ) = 0 .
0 2 π 0 ρ 0 I G ( ρ , θ , z ) ρ d ρ d θ 0 2 π 0 I G ( ρ , θ , z ) ρ d ρ d θ = 0.8 ,
S z = 1 2 Re [ E ( ρ , θ , z ) × H * ( ρ , θ , z ) ] z .

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