Abstract

The evanescent wave of the cylindrical vector field is analyzed using the vector angular spectrum of the electromagnetic beam. Comparison between the contributions of the TE and TM terms of both the propagating and the evanescent waves associated with the cylindrical vector field in free space is demonstrated. The physical pictures of the evanescent wave and the propagating wave are well illustrated from the vectorial structure, which provides a new approach to manipulating laser beams by choosing the states of polarization in the cross-section of the field.

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References

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    [Crossref]
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    [Crossref] [PubMed]
  3. B. Gu, F. Ye, K. Lou, Y. Li, J. Chen, and H. T. Wang, “Vectorial self-diffraction effect in optically Kerr medium,” Opt. Express 20(1), 149–157 (2012).
    [Crossref] [PubMed]
  4. Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
    [Crossref] [PubMed]
  5. R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
    [Crossref] [PubMed]
  6. X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105(25), 253602 (2010).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  16. M. F. Imani and A. Grbic, “Tailoring near-field patterns with concentrically corrugated plates,” Appl. Phys. Lett. 95(11), 111107 (2009).
    [Crossref]
  17. R. Merlin, “Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing,” Science 317(5840), 927–929 (2007).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  20. R. Martínez-Herrero, P. M. Mejías, I. Juvells, and A. Carnicer, “Transverse and longitudinal components of the propagating and evanescent waves associated to radially polarized nonparaxial fields,” Appl. Phys. B 106(1), 151–159 (2012).
    [Crossref]
  21. R. Martínez-Herrero, P. M. Mejías, and A. Carnicer, “Evanescent field of vectorial highly non-paraxial beams,” Opt. Express 16(5), 2845–2858 (2008).
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    [Crossref]
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    [Crossref] [PubMed]

2013 (1)

R. P. Chen and K. H. Chew, “Far field properties of a vortex Airy beam,” Laser and Particle Beams 31(01), 9–15 (2013).
[Crossref]

2012 (4)

R. Martínez-Herrero, P. M. Mejías, I. Juvells, and A. Carnicer, “Transverse and longitudinal components of the propagating and evanescent waves associated to radially polarized nonparaxial fields,” Appl. Phys. B 106(1), 151–159 (2012).
[Crossref]

B. Gu and Y. Cui, “Nonparaxial and paraxial focusing of azimuthal-variant vector beams,” Opt. Express 20(16), 17684–17694 (2012).
[Crossref] [PubMed]

B. Gu, F. Ye, K. Lou, Y. Li, J. Chen, and H. T. Wang, “Vectorial self-diffraction effect in optically Kerr medium,” Opt. Express 20(1), 149–157 (2012).
[Crossref] [PubMed]

M. F. Imani and A. Grbic, “Generating evanescent Bessel beams using near-field plates,” IEEE Trans. Antenn. Propag. 60(7), 3155–3164 (2012).
[Crossref]

2011 (1)

A. Grbic, R. Merlin, E. M. Thomas, and M. F. Imani, “Near-field plates: metamaterial surfaces/arrays for subwavelength focusing and probing,” Proc. IEEE 99(10), 1806–1815 (2011).
[Crossref]

2010 (2)

X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18(10), 10786–10795 (2010).
[Crossref] [PubMed]

X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105(25), 253602 (2010).
[Crossref] [PubMed]

2009 (2)

Q. W. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 (2009).
[Crossref]

M. F. Imani and A. Grbic, “Tailoring near-field patterns with concentrically corrugated plates,” Appl. Phys. Lett. 95(11), 111107 (2009).
[Crossref]

2008 (4)

A. Grbic, L. Jiang, and R. Merlin, “Near-field plates: subdiffraction focusing with patterned surfaces,” Science 320(5875), 511–513 (2008).
[Crossref] [PubMed]

L. E. Helseth, “Radiationless electromagnetic interference shaping of evanescent cylindrical vector waves,” Phys. Rev. A 78(1), 013819 (2008).
[Crossref]

R. Martínez-Herrero, P. M. Mejías, and A. Carnicer, “Evanescent field of vectorial highly non-paraxial beams,” Opt. Express 16(5), 2845–2858 (2008).
[Crossref] [PubMed]

G. Zhou, “The analytical vectorial structure of a nonparaxial Gaussian beam close to the source,” Opt. Express 16(6), 3504–3514 (2008).
[Crossref] [PubMed]

2007 (1)

R. Merlin, “Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing,” Science 317(5840), 927–929 (2007).
[Crossref] [PubMed]

2006 (2)

2005 (1)

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

2002 (1)

2001 (1)

1999 (2)

1997 (1)

A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. 136(1-2), 114–124 (1997).
[Crossref]

1996 (1)

1983 (1)

G. P. Agrawal and M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27(3), 1693–1695 (1983).
[Crossref]

1976 (1)

G. C. Sherman, J. J. Stamnes, and E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17(5), 760–776 (1976).
[Crossref]

Agrawal, G. P.

G. P. Agrawal and M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27(3), 1693–1695 (1983).
[Crossref]

Bosch, S.

Carney, P. S.

Carnicer, A.

Carnicer,, A.

R. Martínez-Herrero, P. M. Mejías, I. Juvells, and A. Carnicer, “Transverse and longitudinal components of the propagating and evanescent waves associated to radially polarized nonparaxial fields,” Appl. Phys. B 106(1), 151–159 (2012).
[Crossref]

Chen, J.

Chen, R. P.

R. P. Chen and K. H. Chew, “Far field properties of a vortex Airy beam,” Laser and Particle Beams 31(01), 9–15 (2013).
[Crossref]

Chew, K. H.

R. P. Chen and K. H. Chew, “Far field properties of a vortex Airy beam,” Laser and Particle Beams 31(01), 9–15 (2013).
[Crossref]

Cui, Y.

Ding, J.

X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105(25), 253602 (2010).
[Crossref] [PubMed]

Ding, J. P.

Doicu, A.

A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. 136(1-2), 114–124 (1997).
[Crossref]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Duan, K.

Grbic, A.

M. F. Imani and A. Grbic, “Generating evanescent Bessel beams using near-field plates,” IEEE Trans. Antenn. Propag. 60(7), 3155–3164 (2012).
[Crossref]

A. Grbic, R. Merlin, E. M. Thomas, and M. F. Imani, “Near-field plates: metamaterial surfaces/arrays for subwavelength focusing and probing,” Proc. IEEE 99(10), 1806–1815 (2011).
[Crossref]

M. F. Imani and A. Grbic, “Tailoring near-field patterns with concentrically corrugated plates,” Appl. Phys. Lett. 95(11), 111107 (2009).
[Crossref]

A. Grbic, L. Jiang, and R. Merlin, “Near-field plates: subdiffraction focusing with patterned surfaces,” Science 320(5875), 511–513 (2008).
[Crossref] [PubMed]

Gu, B.

Guo, C. S.

X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18(10), 10786–10795 (2010).
[Crossref] [PubMed]

X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105(25), 253602 (2010).
[Crossref] [PubMed]

Guo, H.

Hall, D. G.

Helseth, L. E.

L. E. Helseth, “Radiationless electromagnetic interference shaping of evanescent cylindrical vector waves,” Phys. Rev. A 78(1), 013819 (2008).
[Crossref]

Imani, M. F.

M. F. Imani and A. Grbic, “Generating evanescent Bessel beams using near-field plates,” IEEE Trans. Antenn. Propag. 60(7), 3155–3164 (2012).
[Crossref]

A. Grbic, R. Merlin, E. M. Thomas, and M. F. Imani, “Near-field plates: metamaterial surfaces/arrays for subwavelength focusing and probing,” Proc. IEEE 99(10), 1806–1815 (2011).
[Crossref]

M. F. Imani and A. Grbic, “Tailoring near-field patterns with concentrically corrugated plates,” Appl. Phys. Lett. 95(11), 111107 (2009).
[Crossref]

Jiang, L.

A. Grbic, L. Jiang, and R. Merlin, “Near-field plates: subdiffraction focusing with patterned surfaces,” Science 320(5875), 511–513 (2008).
[Crossref] [PubMed]

Juvells, I.

R. Martínez-Herrero, P. M. Mejías, I. Juvells, and A. Carnicer, “Transverse and longitudinal components of the propagating and evanescent waves associated to radially polarized nonparaxial fields,” Appl. Phys. B 106(1), 151–159 (2012).
[Crossref]

Lalor, E.

G. C. Sherman, J. J. Stamnes, and E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17(5), 760–776 (1976).
[Crossref]

Lax, M.

G. P. Agrawal and M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27(3), 1693–1695 (1983).
[Crossref]

Leger, J.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Li, Y.

B. Gu, F. Ye, K. Lou, Y. Li, J. Chen, and H. T. Wang, “Vectorial self-diffraction effect in optically Kerr medium,” Opt. Express 20(1), 149–157 (2012).
[Crossref] [PubMed]

X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105(25), 253602 (2010).
[Crossref] [PubMed]

Li, Y. N.

Lou, K.

Lü, B.

Martínez-Herrero, R.

Mejías, P. M.

Merlin, R.

A. Grbic, R. Merlin, E. M. Thomas, and M. F. Imani, “Near-field plates: metamaterial surfaces/arrays for subwavelength focusing and probing,” Proc. IEEE 99(10), 1806–1815 (2011).
[Crossref]

A. Grbic, L. Jiang, and R. Merlin, “Near-field plates: subdiffraction focusing with patterned surfaces,” Science 320(5875), 511–513 (2008).
[Crossref] [PubMed]

R. Merlin, “Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing,” Science 317(5840), 927–929 (2007).
[Crossref] [PubMed]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Saghafi, S.

Shchegrov, A. V.

Sheppard, C. J. R.

Sherman, G. C.

G. C. Sherman, J. J. Stamnes, and E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17(5), 760–776 (1976).
[Crossref]

Stamnes, J. J.

G. C. Sherman, J. J. Stamnes, and E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17(5), 760–776 (1976).
[Crossref]

Thomas, E. M.

A. Grbic, R. Merlin, E. M. Thomas, and M. F. Imani, “Near-field plates: metamaterial surfaces/arrays for subwavelength focusing and probing,” Proc. IEEE 99(10), 1806–1815 (2011).
[Crossref]

Wang, H. T.

Wang, X. L.

X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105(25), 253602 (2010).
[Crossref] [PubMed]

X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18(10), 10786–10795 (2010).
[Crossref] [PubMed]

Wriedt, T.

A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. 136(1-2), 114–124 (1997).
[Crossref]

Ye, F.

Zhan, Q.

Zhan, Q. W.

Zhou, G.

Zhuang, S.

Adv. Opt. Photon. (1)

Appl. Phys. B (1)

R. Martínez-Herrero, P. M. Mejías, I. Juvells, and A. Carnicer, “Transverse and longitudinal components of the propagating and evanescent waves associated to radially polarized nonparaxial fields,” Appl. Phys. B 106(1), 151–159 (2012).
[Crossref]

Appl. Phys. Lett. (1)

M. F. Imani and A. Grbic, “Tailoring near-field patterns with concentrically corrugated plates,” Appl. Phys. Lett. 95(11), 111107 (2009).
[Crossref]

IEEE Trans. Antenn. Propag. (1)

M. F. Imani and A. Grbic, “Generating evanescent Bessel beams using near-field plates,” IEEE Trans. Antenn. Propag. 60(7), 3155–3164 (2012).
[Crossref]

J. Math. Phys. (1)

G. C. Sherman, J. J. Stamnes, and E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17(5), 760–776 (1976).
[Crossref]

J. Opt. Soc. Am. A (3)

Laser and Particle Beams (1)

R. P. Chen and K. H. Chew, “Far field properties of a vortex Airy beam,” Laser and Particle Beams 31(01), 9–15 (2013).
[Crossref]

Opt. Commun. (1)

A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. 136(1-2), 114–124 (1997).
[Crossref]

Opt. Express (7)

Opt. Lett. (3)

Phys. Rev. A (2)

G. P. Agrawal and M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27(3), 1693–1695 (1983).
[Crossref]

L. E. Helseth, “Radiationless electromagnetic interference shaping of evanescent cylindrical vector waves,” Phys. Rev. A 78(1), 013819 (2008).
[Crossref]

Phys. Rev. Lett. (2)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105(25), 253602 (2010).
[Crossref] [PubMed]

Proc. IEEE (1)

A. Grbic, R. Merlin, E. M. Thomas, and M. F. Imani, “Near-field plates: metamaterial surfaces/arrays for subwavelength focusing and probing,” Proc. IEEE 99(10), 1806–1815 (2011).
[Crossref]

Science (2)

R. Merlin, “Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing,” Science 317(5840), 927–929 (2007).
[Crossref] [PubMed]

A. Grbic, L. Jiang, and R. Merlin, “Near-field plates: subdiffraction focusing with patterned surfaces,” Science 320(5875), 511–513 (2008).
[Crossref] [PubMed]

Other (3)

J. W. Goodman, Introduction to Fourier Optics. (Greenwood Village: Roberts and Company, 2004).

K. E. Okan, Diffraction, Fourier Optics and Imaging. (Wiley & Sons, 2007).

F. De Fornel, Evanescent Waves: From Newtonian Optics to Atomic Optics (Springer. 2001).

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Figures (5)

Fig. 1
Fig. 1

Ratio of the propagating and evanescent components of the field as a function of the propagation distances with (a) w = 0.1λ; (b) w = 0.5λ.

Fig. 2
Fig. 2

Intensity distribution of the propagating and evanescent components of the field as a function of the propagation distances with m = 2, θ0 = π/4, w = 0.1λ, z = 0.2λ. (a) TE term of the propagating component. The scale is in the order of 10−3; (b) TM term of the propagating wave; (c) propagating wave; (d) TE term of evanescent wave; (e) TM term of evanescent wave; (f) evanescent wave; (g) total field (both propagating and evanescent waves).

Fig. 3
Fig. 3

Intensity distribution of the propagating and evanescent components of the field as a function of the propagation distances with m = 1, θ0 = π/4, w = 0.1λ, z = 0.2λ. (a) TE term of the propagating component. The scale is in the order of 10−3; (b) TM term of the propagating wave; (c) propagating wave; (d) TE term of the evanescent wave; (e) TM term of the evanescent wave; (f) evanescent wave; (g) total field (both propagating and evanescent waves).

Fig. 4
Fig. 4

Intensity distribution of the propagating and evanescent components of the field as a function of the propagation distances with m = 0, θ0 = π/4, w = 0.1λ, z = 0.2λ. (a) TE term of the propagating component; (b) TM term of the propagating wave; (c) propagating wave; (d) TE term of the evanescent wave; (e) TM term of the evanescent wave; (f) evanescent wave; (g) total field (both propagating and evanescent waves).

Fig. 5
Fig. 5

Intensity distribution of the propagating and evanescent components of the field with w = 0.1λ, z = 0.2λ as a function of the propagation distance for radial polarization: (a) propagating wave; (b) evanescent wave; (c) total field (both propagating and evanescent wave); and for azimuthal polarization: (d) propagating wave; (e) evanescent wave; (f) total field (both propagating and evanescent waves).

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

E(r,θ)=A(r)P(θ)=A(r)[cos(mθ+ θ 0 ) e ^ x +sin(mθ+ θ 0 ) e ^ y ],
( A x (ρcosϕ,ρsinϕ) A y (ρcosϕ,ρsinϕ) )= ( k 2π ) 2 ( 0 0 2π A(r)cos(mθ+ θ 0 )exp[ikrρcos(θϕ)ikξz]rdθdr 0 0 2π A(r)sin(mθ+ θ 0 )exp[ikrρcos(θϕ)ikξz]rdθdr ),
u=ρcosϕ, v=ρsinϕ, ξ= 1 ρ 2 .
A(ρcosϕ,ρsinϕ)= ( k 2π ) 2 π k w 3 ρ 8 exp( k 2 w 2 ρ 2 /8) [ I (m1)/2 ( k 2 w 2 ρ 2 /8) I (m+1)/2 ( k 2 w 2 ρ 2 /8) ] ×[cos(mϕ+ θ 0 ) e ^ x +sin(mϕ+ θ 0 ) e ^ y cos(mϕϕ+ θ 0 )ρ/ξ e ^ z ].
E(r)= A(ρcosϕ,ρsinϕ)exp[ ik(ux+vy+ξz) ]dudv = i 3m k 3 w 3 16 π 0 e k 2 w 2 ρ 2 8 J m (krρ)[ I (m1)/2 ( k 2 w 2 ρ 2 8 ) I (m+1)/2 ( k 2 w 2 ρ 2 8 ) ] [ cos(mθ+ θ 0 ) e ^ x +sin(mθ+ θ 0 ) e ^ y cos(mθθ+ θ 0 )ρ/ξ e ^ z ]exp( ikξz ) ρ 2 dρ.
E ( r )= E TE ( r )+ E TM ( r ).
E TE (r)= 1 2 ( k 2π ) 2 0 π k w 3 ρ 8 exp( k 2 w 2 ρ 2 /8) [ I (m1)/2 ( k 2 w 2 ρ 2 /8) I (m+1)/2 ( k 2 w 2 ρ 2 /8) ] exp( ikξz )ρdρ { [ 2π i m J m (krρ)cos(mθ+ θ 0 )+2π i m2 J m2 (krρ)cos((m2)θ+ θ 0 ) ] e ^ x + [ 2π i m J m (krρ)sin(mθ+ θ 0 )2π i m J m2 (krρ)sin((m2)θ+ θ 0 ) ] e ^ y }
E TM (r)= 1 2 ( k 2π ) 2 0 π k w 3 ρ 8 exp( k 2 w 2 ρ 2 /8) [ I (m1)/2 ( k 2 w 2 ρ 2 /8) I (m+1)/2 ( k 2 w 2 ρ 2 /8) ] exp( ikξz )ρdρ { [ 2π i m J m (krρ)cos(mθ+ θ 0 )2π i m J m2 (krρ)cos((m2)θ+ θ 0 ) ] e ^ x +[ 2π i m J m (krρ)sin(mθ+ θ 0 )+2π i m J m2 (krρ)sin((m2)θ+ θ 0 ) ] e ^ y y 4π i m1 J m1 (krρ)cos((m1)θ+ θ 0 )ρ/ξ e ^ z z }.
E TE ev (r)= 1 2 ( k 2π ) 2 1 π k w 3 ρ 8 exp( k 2 w 2 ρ 2 /8) [ I (m1)/2 ( k 2 w 2 ρ 2 /8) I (m+1)/2 ( k 2 w 2 ρ 2 /8) ] exp( ikζz )ρdρ { [ 2π i m J m (krρ)cos(mθ+ θ 0 )+2π i m2 J m2 (krρ)cos((m2)θ+ θ 0 ) ] e ^ x + [ 2π i m J m (krρ)sin(mθ+ θ 0 )2π i m J m2 (krρ)sin((m2)θ+ θ 0 ) ] e ^ y }.
E TM ev (r)= 1 2 ( k 2π ) 2 1 π k w 3 ρ 8 exp( k 2 w 2 ρ 2 /8)[ I (m1)/2 ( k 2 w 2 ρ 2 /8) I (m+1)/2 ( k 2 w 2 ρ 2 /8) ] exp( ikξz )ρdρ { 1 2 ρ 2 1 [ 2π i m J m (krρ)cos(mθ+ θ 0 )2π i m J m2 (krρ)cos((m2)θ+ θ 0 ) ] e ^ x + 1 2 ρ 2 1 [ 2π i m J m (krρ)sin(mθ+ θ 0 )+2π i m J m2 (krρ)sin((m2)θ+ θ 0 ) ] e ^ y iρ 2 ρ 2 1 ρ 2 1 4π i m1 J m1 (krρ)cos((m1)θ+ θ 0 ) e ^ z },
I pr = | E pr | 2 dxdy= 0 1 0 2π [ | a | 2 + | b | 2 ]ρdρdϕ ,
I ev = | E ev | 2 dxdy= 1 0 2π [ | a | 2 + | b ev | 2 ]exp(2kz ρ 2 1 )ρdρdϕ ,
a= ( k 2π ) 2 π k w 3 ρexp( k 2 w 2 ρ 2 /8) 8 [ I (m1)/2 ( k 2 w 2 ρ 2 /8) I (m+1)/2 ( k 2 w 2 ρ 2 /8) ]sin((1m)ϕ θ 0 ),
b= ( k 2π ) 2 π k w 3 ρexp( k 2 w 2 ρ 2 /8) 8 1 ρ 2 [ I (m1)/2 ( k 2 w 2 ρ 2 /8) I (m+1)/2 ( k 2 w 2 ρ 2 /8) ]cos((1m)ϕ θ 0 ),
b ev =i ( k 2π ) 2 π k w 3 ρexp( k 2 w 2 ρ 2 /8) 8 2 ρ 2 1 ρ 2 1 [ I (m1)/2 ( k 2 w 2 ρ 2 /8) I (m+1)/2 ( k 2 w 2 ρ 2 /8) ]cos((1m)ϕ θ 0 ).

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