Abstract

In terms of the angular spectrum representation, general expressions are given to describe the free-space propagation of electromagnetic fields with radial or azimuthal polarization structure at a transverse plane. The transverse distributions of the radial, azimuthal and longitudinal components of these fields are also analysed. In particular, the on-axis behavior upon free propagation is studied. Furthermore, the special but important case of those fields that retain their polarization character (radial or azimuthal) under propagation is also considered. The analytical results are illustrated by application to some examples.

© 2008 Optical Society of America

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  1. A. A. Tovar, "Production and propagation of cylindrical polarized Laguerre-Gaussian beams," J. Opt. Soc. Am. A 15, 2705-2711 (1998).
    [CrossRef]
  2. A. V. Nesterov and V. G. Niziev, "Laser beams with axially symmetric polarization," J. Phys. D 33, 1817-1822 (2000).
    [CrossRef]
  3. R. Oron, S. Blit, N. Davidson, and A. A. Fiesem, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
    [CrossRef]
  4. J. Tervo, P. Vahimaa, and J. Turunen, "On the propagation-invariant and self-imaging intensity distributions of electromagnetic fields," J. Mod. Opt. 49, 1537-1543 (2002).
    [CrossRef]
  5. P. Pääkkönen, J. Tervo, P. Vahimaa, J. Turunen, and F. Gori, "General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields," Opt. Express 10, 949-959 (2002).
    [PubMed]
  6. D. J. Armstrong, M. C. Philips, and A. V. Smith, "Generation of radially polarized beams with an image rotating resonator," Appl. Opt. 42, 3550-3554 (2003).
    [CrossRef] [PubMed]
  7. J. Tervo, "Azimuthal polarization and partial coherence," J. Opt. Soc. Am. A,  20, 1974-1980 (2003).
    [CrossRef]
  8. N. Pasilly, R. de S. Denis, K. Aït-Ameur, F. Treussart, R. Hierle, and J-F Roch, "Simple interferometric technique for generation of a radially polarized light beam," J. Opt. Soc. Am. A 22, 984-991 (2005).
    [CrossRef]
  9. M. S. Roth, E. W. Wyss, H. Glur, and H. P. Weber, "Generation of radially polarized beams in a Nd:YAG laser with self-adaptive overcompensation of the thermal lens," Opt. Lett. 30, 1665-1667 (2005).
    [CrossRef] [PubMed]
  10. D. M. Deng, "Nonparaxial propagation of radially polarized light beams," J. Opt. Soc. Am. B 23, 1228-1234 (2006).
    [CrossRef]
  11. D. Deng, Q. Guo, L. Wu, and X. Yang, "Propagation of radially polarized elegant light beams," J. Opt. Soc. Am. B 24, 636-643 (2007).
    [CrossRef]
  12. D. Deng and Q. Guo, "Analytical vectorial structure of radially polarized light beams," Opt. Lett. 32, 2711-2713 (2007).
    [CrossRef] [PubMed]
  13. P. Varga and P. Török, "Exact and approximate solutions of Maxwell???s equations for a confocal cavity," Opt. Lett. 21, 1523-1525 (1996).
    [CrossRef] [PubMed]
  14. P. Varga and P. Török, "The Gaussian wave solution of Maxvell???s equations and the validity of the scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
    [CrossRef]
  15. C. J. R. Sheppard and S. Saghafi, "Electromagnetic Gaussian beams beyond the paraxial approximation," J. Opt. Soc. Am. A 16, 1381-1386 (1999).
    [CrossRef]
  16. C. J. R. Sheppard, "Polarization of almost-planes waves," J. Opt. Soc. Am. A 17, 335-341 (2000).
    [CrossRef]
  17. R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, "Vectorial structure of nonparaxial electromagnetic beams," J. Opt. Soc. Am. A 18, 1678-1680 (2001).
    [CrossRef]
  18. P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarizad laser beams," Prog. Quantum Electron. 26, 65-130 (2002).
    [CrossRef]
  19. G. Zhou, X. Chu, and L. Zhao, "Propagation characteristics of TM Gaussian beam," Opt. Laser Technol. 37, 470-474 (2005).
    [CrossRef]
  20. R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, "Structure of the transverse profile of Gaussian-model non-paraxial electromagnetic beams," J. Opt. A: Pure Appl. Opt. 8, 524-530 (2006).
    [CrossRef]
  21. P. C. Chaumet, "Fully vectorial highly nonparaxial beam close to the waist," J. Opt. Soc. Am. A 23, 3197-3202 (2006).
    [CrossRef]
  22. R. Martínez-Herrero, P. M. Mejías, and A. Carnicer, "Evanescent field of vectorial highly non-paraxial beams," Opt. Express 16, 2845-2858 (2008).
    [CrossRef] [PubMed]

2008 (1)

2007 (2)

2006 (3)

D. M. Deng, "Nonparaxial propagation of radially polarized light beams," J. Opt. Soc. Am. B 23, 1228-1234 (2006).
[CrossRef]

P. C. Chaumet, "Fully vectorial highly nonparaxial beam close to the waist," J. Opt. Soc. Am. A 23, 3197-3202 (2006).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, "Structure of the transverse profile of Gaussian-model non-paraxial electromagnetic beams," J. Opt. A: Pure Appl. Opt. 8, 524-530 (2006).
[CrossRef]

2005 (3)

2003 (2)

2002 (3)

P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarizad laser beams," Prog. Quantum Electron. 26, 65-130 (2002).
[CrossRef]

P. Pääkkönen, J. Tervo, P. Vahimaa, J. Turunen, and F. Gori, "General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields," Opt. Express 10, 949-959 (2002).
[PubMed]

J. Tervo, P. Vahimaa, and J. Turunen, "On the propagation-invariant and self-imaging intensity distributions of electromagnetic fields," J. Mod. Opt. 49, 1537-1543 (2002).
[CrossRef]

2001 (1)

2000 (3)

A. V. Nesterov and V. G. Niziev, "Laser beams with axially symmetric polarization," J. Phys. D 33, 1817-1822 (2000).
[CrossRef]

R. Oron, S. Blit, N. Davidson, and A. A. Fiesem, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

C. J. R. Sheppard, "Polarization of almost-planes waves," J. Opt. Soc. Am. A 17, 335-341 (2000).
[CrossRef]

1999 (1)

1998 (2)

A. A. Tovar, "Production and propagation of cylindrical polarized Laguerre-Gaussian beams," J. Opt. Soc. Am. A 15, 2705-2711 (1998).
[CrossRef]

P. Varga and P. Török, "The Gaussian wave solution of Maxvell???s equations and the validity of the scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
[CrossRef]

1996 (1)

Armstrong, D. J.

Blit, S.

R. Oron, S. Blit, N. Davidson, and A. A. Fiesem, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Bosch, S.

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, "Structure of the transverse profile of Gaussian-model non-paraxial electromagnetic beams," J. Opt. A: Pure Appl. Opt. 8, 524-530 (2006).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, "Vectorial structure of nonparaxial electromagnetic beams," J. Opt. Soc. Am. A 18, 1678-1680 (2001).
[CrossRef]

Carnicer, A.

Chaumet, P. C.

Chu, X.

G. Zhou, X. Chu, and L. Zhao, "Propagation characteristics of TM Gaussian beam," Opt. Laser Technol. 37, 470-474 (2005).
[CrossRef]

Davidson, N.

R. Oron, S. Blit, N. Davidson, and A. A. Fiesem, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Deng, D.

Deng, D. M.

Fiesem, A. A.

R. Oron, S. Blit, N. Davidson, and A. A. Fiesem, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Glur, H.

Gori, F.

Guo, Q.

Martínez-Herrero, R.

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

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, "Structure of the transverse profile of Gaussian-model non-paraxial electromagnetic beams," J. Opt. A: Pure Appl. Opt. 8, 524-530 (2006).
[CrossRef]

P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarizad laser beams," Prog. Quantum Electron. 26, 65-130 (2002).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, "Vectorial structure of nonparaxial electromagnetic beams," J. Opt. Soc. Am. A 18, 1678-1680 (2001).
[CrossRef]

Mejías, P. M.

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

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, "Structure of the transverse profile of Gaussian-model non-paraxial electromagnetic beams," J. Opt. A: Pure Appl. Opt. 8, 524-530 (2006).
[CrossRef]

P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarizad laser beams," Prog. Quantum Electron. 26, 65-130 (2002).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, "Vectorial structure of nonparaxial electromagnetic beams," J. Opt. Soc. Am. A 18, 1678-1680 (2001).
[CrossRef]

Movilla, J. M.

P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarizad laser beams," Prog. Quantum Electron. 26, 65-130 (2002).
[CrossRef]

Nesterov, A. V.

A. V. Nesterov and V. G. Niziev, "Laser beams with axially symmetric polarization," J. Phys. D 33, 1817-1822 (2000).
[CrossRef]

Niziev, V. G.

A. V. Nesterov and V. G. Niziev, "Laser beams with axially symmetric polarization," J. Phys. D 33, 1817-1822 (2000).
[CrossRef]

Oron, R.

R. Oron, S. Blit, N. Davidson, and A. A. Fiesem, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Pääkkönen, P.

Pasilly, N.

Philips, M. C.

Piquero, G.

P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarizad laser beams," Prog. Quantum Electron. 26, 65-130 (2002).
[CrossRef]

Roth, M. S.

Saghafi, S.

Sheppard, C. J. R.

Smith, A. V.

Tervo, J.

Török, P.

P. Varga and P. Török, "The Gaussian wave solution of Maxvell???s equations and the validity of the scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
[CrossRef]

P. Varga and P. Török, "Exact and approximate solutions of Maxwell???s equations for a confocal cavity," Opt. Lett. 21, 1523-1525 (1996).
[CrossRef] [PubMed]

Tovar, A. A.

Turunen, J.

J. Tervo, P. Vahimaa, and J. Turunen, "On the propagation-invariant and self-imaging intensity distributions of electromagnetic fields," J. Mod. Opt. 49, 1537-1543 (2002).
[CrossRef]

P. Pääkkönen, J. Tervo, P. Vahimaa, J. Turunen, and F. Gori, "General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields," Opt. Express 10, 949-959 (2002).
[PubMed]

Vahimaa, P.

P. Pääkkönen, J. Tervo, P. Vahimaa, J. Turunen, and F. Gori, "General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields," Opt. Express 10, 949-959 (2002).
[PubMed]

J. Tervo, P. Vahimaa, and J. Turunen, "On the propagation-invariant and self-imaging intensity distributions of electromagnetic fields," J. Mod. Opt. 49, 1537-1543 (2002).
[CrossRef]

Varga, P.

P. Varga and P. Török, "The Gaussian wave solution of Maxvell???s equations and the validity of the scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
[CrossRef]

P. Varga and P. Török, "Exact and approximate solutions of Maxwell???s equations for a confocal cavity," Opt. Lett. 21, 1523-1525 (1996).
[CrossRef] [PubMed]

Weber, H. P.

Wu, L.

Wyss, E. W.

Yang, X.

Zhao, L.

G. Zhou, X. Chu, and L. Zhao, "Propagation characteristics of TM Gaussian beam," Opt. Laser Technol. 37, 470-474 (2005).
[CrossRef]

Zhou, G.

G. Zhou, X. Chu, and L. Zhao, "Propagation characteristics of TM Gaussian beam," Opt. Laser Technol. 37, 470-474 (2005).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

R. Oron, S. Blit, N. Davidson, and A. A. Fiesem, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

J. Mod. Opt. (1)

J. Tervo, P. Vahimaa, and J. Turunen, "On the propagation-invariant and self-imaging intensity distributions of electromagnetic fields," J. Mod. Opt. 49, 1537-1543 (2002).
[CrossRef]

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

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, "Structure of the transverse profile of Gaussian-model non-paraxial electromagnetic beams," J. Opt. A: Pure Appl. Opt. 8, 524-530 (2006).
[CrossRef]

J. Opt. Soc. Am. A (7)

J. Opt. Soc. Am. B (2)

J. Phys. D (1)

A. V. Nesterov and V. G. Niziev, "Laser beams with axially symmetric polarization," J. Phys. D 33, 1817-1822 (2000).
[CrossRef]

Opt. Commun. (1)

P. Varga and P. Török, "The Gaussian wave solution of Maxvell???s equations and the validity of the scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
[CrossRef]

Opt. Express (2)

Opt. Laser Technol. (1)

G. Zhou, X. Chu, and L. Zhao, "Propagation characteristics of TM Gaussian beam," Opt. Laser Technol. 37, 470-474 (2005).
[CrossRef]

Opt. Lett. (3)

Prog. Quantum Electron. (1)

P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarizad laser beams," Prog. Quantum Electron. 26, 65-130 (2002).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

Squared modulus of the fields defined by Eqs. (32) and (33) in terms of at the initial plane z=0. Continuous line: p=1; dotted line: p=2; dashed line: p=3. Ordinates are given in arbitrary units. Note that, at plane z=0, we have E θ=0.

Fig. 2.
Fig. 2.

Squared modulus of the radial field component, |ER |2 (dotted line), and azimuthal component, |Eθ |2 (continuous line) in terms of , at the transverse plane z=λ. The curves are plotted for p=1. Remember that, on the propagation axis, the field is circularly polarized.

Fig. 3.
Fig. 3.

The same as in Fig. 2, but now calculated at the transverse plane z=20 λ.

Fig. 4.
Fig. 4.

Ordinates give the ratio q (defined by Eq. (35)) in terms of in the limit z→∞. The curves correspond to the cases p=1 (continuous line); p=2 (dotted line); and p=3 (dashed line).

Fig. 5.
Fig. 5.

The same as in Fig. 2 but now with p=2. On the axis (=0) the transverse field components vanish, and the field only exhibits a longitudinal component.

Fig. 6.
Fig. 6.

The same as in Fig. 5 but now calculated at the transverse plane z=20 λ. Again, the transverse components of the field vanish on the axis.

Equations (64)

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E ( x , y , z ) = E ˜ ( u , v , z ) exp [ i k ( xu + yv ) ] dudv ,
E ˜ ( ρ , ϕ , z ) = E 0 ˜ ( ρ , ϕ ) exp ( ikz 1 ρ 2 ) ,
H ˜ ( ρ , ϕ , z ) = s ( ρ , ϕ ) × E ˜ ( ρ , ϕ , z ) ,
E ( R , θ , z ) = 0 1 0 2 π [ a ( ρ , ϕ ) e 1 ( ϕ ) + b ( ρ , ϕ ) e 2 ( ρ , ϕ ) ] ×
exp [ ik R ρ cos ( ϕ θ ) ] exp ikz 1 ρ 2 ρ d ρ d ϕ
E R ( R , θ , z ) = cos θ E x ( R , θ , z ) + sin θ E y ( R , θ , z ) ,
E θ ( R , θ , z ) = sin θ E x ( R , θ , z ) + cos θ E y ( R , θ , z ) ,
E R ( R , θ , z ) = 0 1 0 2 π [ a ( ρ , ϕ ) sin ( θ ϕ ) + b ˜ ( ρ , ϕ ) cos ( θ ϕ ) ] ×
        exp [ i k ρ R cos ( θ ϕ ) ] exp ( i kz 1 ρ 2 ) ρ d ρ d ϕ ,
E θ ( R , θ , z ) = 0 1 0 2 π [ a ( ρ , ϕ ) cos ( θ ϕ ) + b ˜ ( ρ , ϕ ) sin ( θ ϕ ) ] ×
exp [ i k ρ R cos ( θ ϕ ) ] exp ( i kz 1 ρ 2 ) ρ d ρ d ϕ ,
0 1 0 2 π [ a ( ρ , ϕ ) cos ( θ ϕ ) + b ˜ ( ρ , ϕ ) sin ( θ ϕ ) ] exp [ i k ρ R cos ( θ ϕ ) ] ρ d ρ d ϕ = 0 .
0 2 π d ϕ 0 1 a ( ρ , ϕ ) ρ { exp [ i k R ρ cos ( θ ϕ ) ] } ρ d ρ +
0 1 d ρ 0 2 π b ˜ ( ρ , ϕ ) ϕ { exp [ i k R ρ cos ( θ ϕ ) ] } d ϕ = 0 .
I 1 = 0 2 π d ϕ { [ ρ a ( ρ , ϕ ) exp [ i k R ρ cos ( θ ϕ ) ] ] } ρ = 0 ρ = 1
0 2 π 0 1 [ ρ a ρ ] exp [ i k R ρ cos ( θ ϕ ) ] d ρ d ϕ
I 1 = 0 2 π 0 1 [ ρ a ρ ] exp [ ikRρ cos ( θ ϕ ) ] d ρd ϕ .
I 2 = - 0 2 π 0 1 [ b ~ ϕ ] exp [ ikRρ cos ( θ ϕ ) ] d ρd ϕ ,
0 2 π 0 1 [ ρa ρ + b ~ ϕ ] exp [ ikRρ cos ( θ ϕ ) ] d ρd ϕ = 0 .
ρ a ρ + b ~ ϕ = 0 .
a ( ρ , ϕ ) = 1 ρ F ( ρ , ϕ ) ϕ ,
b ~ ( ρ , ϕ ) = F ( ρ , ϕ ) ρ ,
E rad ( R , θ , z ) = 0 1 0 2 π [ 1 ρ F ( ρ , ϕ ) ϕ e 1 ( ϕ ) + 1 1 ρ 2 F ( ρ , ϕ ) ρ e 2 ( ρ , θ ) ] ×
exp [ ikRρ cos ( θ ϕ ) ] exp ( ikz 1 ρ 2 ) ρd ρd ϕ .
F ( ρ , ϕ ) = m exp ( i m ϕ ) f m ( ρ ) , m = 0 , ± 1 , ± 2 ,
f m ( ρ ) = 1 2 π 0 2 π F ( ρ , ϕ ) exp ( i m ϕ ) d ϕ ,
E r a d ( R , θ , z ) = [ m i exp ( i m θ ) A m ( R , z ) ] u R ( θ ) + [ m exp ( i m θ ) B m ( R , z ) ] u θ ( θ ) +
+ [ m exp ( i m θ ) C m ( R , z ) ] u z
A m ( R , z ) = π ( i ) m 0 1 exp ( i k z 1 ρ 2 ) ×
{ m f m ( ρ ) [ J m 1 ( k R ρ ) + J m + 1 ( k R ρ ) ] + ρ f m ( ρ ) [ J m + 1 ( k R ρ ) - J m 1 ( k R ρ ) ] } d ρ ,
B m ( R , z ) = π ( i ) m 0 1 exp ( i k z 1 ρ 2 ) ×
{ m f m ( ρ ) [ J m 1 ( k R ρ ) J m + 1 ( k R ρ ) ] + ρ f m ( ρ ) [ J m + 1 ( k R ρ ) + J m 1 ( K R ρ ) ] } d ρ ,
C m ( R , z ) = 2 π ( i ) m 0 1 ρ 2 f m ( ρ ) 1 ρ 2 J m ( k R ρ ) exp ( i k z 1 ρ 2 ) d ρ ,
u R = ( cos θ , sin θ , 0 ) ,
u θ = ( sin θ , cos θ , 0 ) ,
u z = ( 0 , 0 , 1 ) .
( E r a d ) x ( 0 , z ) = i A 1 ( 0 , z ) + i A 1 ( 0 , z ) ,
( E r a d ) y ( 0 , z ) = A 1 ( 0 , z ) + A 1 ( 0 , z ) ,
( E r a d ) z ( 0 , z ) = 2 π 0 1 f m = 0 ( ρ ) 1 ρ 2 ρ 2 J 0 ( k R ρ ) exp ( i k z 1 ρ 2 ) d ρ ,
0 1 0 2 π [ a ( ρ , ϕ ) sin ( θ ϕ ) + b ~ ( ρ , ϕ ) cos ( θ ϕ ) ] exp [ i k ρ R cos ( θ ϕ ) ] ρ d ρ d ϕ = 0 .
a ϕ ρ b ~ ρ = 0 .
a ( ρ , ϕ ) = G ρ ,
b ~ ( ρ , ϕ ) = 1 ρ G ϕ ,
E azim ( R , θ , z ) = 0 1 0 2 π [ G ρ e 1 ( ϕ ) + 1 ρ 1 1 ρ 2 G ϕ e 2 ( ρ , θ ) ] ×
exp [ i k R ρ cos ( θ ϕ ) ] exp ( i k z 1 ρ 2 ) ρ d ρ d ϕ .
G ( ρ , ϕ ) = m exp ( i m ϕ ) g m ( ρ ) ,
E azim ( R , θ , z ) = [ m exp ( i m θ ) U m ( R , z ) ] u R ( θ ) + [ m i exp ( i m θ ) V m ( R , z ) ] u θ ( θ ) +
+ [ m exp ( i m θ ) W m ( R , z ) ] u z ,
U m ( R , z ) = π ( i ) m 0 1 exp ( i k z 1 ρ 2 ) ×
{ ρ g m ( ρ ) [ J m 1 ( k R ρ ) + J m + 1 ( k R ρ ) ] + m g m ( ρ ) [ J m 1 ( k R ρ ) J m + 1 ( k R ρ ) ] } d ρ ,
V m ( R , z ) = π ( i ) m 0 1 exp ( i k z 1 ρ 2 ) ×
{ ρ g m ( ρ ) [ J m + 1 ( k R ρ ) J m 1 ( k R ρ ) ] m g m ( ρ ) [ J m 1 ( k R ρ ) + J m + 1 ( k R ρ ) ] } d ρ ,
W m ( R , z ) = 2 m π ( i ) m 1 0 1 ρ g m ( ρ ) 1 ρ 2 J m ( k R ρ ) exp ( i k z 1 ρ 2 ) d ρ ,
( E azim ) x ( 0 , z ) = U 1 ( 0 , z ) + U 1 ( 0 , z ) ,
( E azim ) y ( 0 , z ) = i U 1 ( 0 , z ) i U 1 ( 0 , z ) ,
( E azim ) z ( 0 , z ) = 0 .
E azim ( R , θ , z ) = H ( R , z ) u θ ( θ ) ,
H ( R , z ) = 2 π i 0 1 d G ( ρ ) d ρ J 1 ( kRρ ) exp ( ikz 1 ρ 2 ) ρ d ρ .
F ( ρ , ϕ ) = δ ( ρ a ) exp ( ip ϕ ) , a ( 0 , 1 ) ,
E R [ R ˜ J p ( R ˜ ) + 2 i z ˜ dJ p d R ˜ ] exp ( ikz 1 a 2 ) exp ( ip θ ) ,
E θ 2 p z ˜ J p ( R ˜ ) R ˜ exp ( ikz 1 a 2 ) exp ( ip θ ) ,
R ˜ = kaR ,
z ˜ = ka 2 1 a 2 z ,
q = E θ 2 E R 2 + E θ 2 ,

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