Abstract

We analyze the phase behavior of strongly focused, radially polarized electromagnetic fields. It is shown that, under certain circumstances, the spacing between successive wavefronts can be either greater or smaller than that of a plane wave of the same frequency. Also, this spacing can be significantly larger than that which is predicted for a linearly polarized field that is focused by the same system.

© 2005 Optical Society of America

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  3. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
    [CrossRef]
  4. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
    [CrossRef]
  5. R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef] [PubMed]
  6. L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
    [CrossRef] [PubMed]
  7. C. J.R. Sheppard, A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. 43, 4322–4327 (2004).
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  8. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004). http://www.opticsexpress.org.
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  9. E. H. Linfoot, E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. London, Sect. B, 69, 823–832 (1956).
    [CrossRef]
  10. J. T. Foley, E. Wolf, “Wave-front spacing in the focal region of high-numerical-aperture systems,” Opt. Lett. 30, 1312–1314 (2005).
    [CrossRef] [PubMed]
  11. K. S. Youngworth, T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” in Proc. SPIE 3919, 75–85 (2000).
    [CrossRef]
  12. R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
    [CrossRef]
  13. P. W. Milonni, J. H. Eberly, Lasers (Wiley, 1988). See especially Section 14.8.
  14. J. J. Stamnes, Waves in Focal Region (Hilger, 1986). See especially Chap. 16.
  15. B. Richards, E. Wolf, “The Airy pattern in systems of high angular aperture,” Proc. Phys. Soc. London, Sect. B 69, 854–856 (1956).
    [CrossRef]
  16. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London, Ser. A 253, 349–357 (1959).
    [CrossRef]
  17. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
    [CrossRef]
  18. A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561–B1565 (1965).
    [CrossRef]
  19. A. Boivin, J. Dow, E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967).
    [CrossRef]
  20. K. S. Youngworth, T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000). http://www.opticsexpress.org.
    [CrossRef] [PubMed]
  21. M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999). See especially Sec. 4.5.1.
    [CrossRef]

2005

2004

2003

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

2001

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

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

2000

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

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

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

K. S. Youngworth, T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000). http://www.opticsexpress.org.
[CrossRef] [PubMed]

1996

1994

1967

1965

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561–B1565 (1965).
[CrossRef]

1959

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London, Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

1956

B. Richards, E. Wolf, “The Airy pattern in systems of high angular aperture,” Proc. Phys. Soc. London, Sect. B 69, 854–856 (1956).
[CrossRef]

E. H. Linfoot, E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. London, Sect. B, 69, 823–832 (1956).
[CrossRef]

Beversluis, M. R.

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

Blit, S.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Boivin, A.

A. Boivin, J. Dow, E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967).
[CrossRef]

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561–B1565 (1965).
[CrossRef]

Bomzon, Z.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999). See especially Sec. 4.5.1.
[CrossRef]

Brown, T. G.

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

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

K. S. Youngworth, T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000). http://www.opticsexpress.org.
[CrossRef] [PubMed]

Choudhury, A.

Davidson, N.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Dorn, R.

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

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Dow, J.

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Eberly, J. H.

P. W. Milonni, J. H. Eberly, Lasers (Wiley, 1988). See especially Section 14.8.

Foley, J. T.

Friesem, A. A.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Hall, D. G.

Hasman, E.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Jordan, R. H.

Leuchs, G.

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

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Linfoot, E. H.

E. H. Linfoot, E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. London, Sect. B, 69, 823–832 (1956).
[CrossRef]

Milonni, P. W.

P. W. Milonni, J. H. Eberly, Lasers (Wiley, 1988). See especially Section 14.8.

Novotny, L.

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

Oron, R.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Quabis, S.

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

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

B. Richards, E. Wolf, “The Airy pattern in systems of high angular aperture,” Proc. Phys. Soc. London, Sect. B 69, 854–856 (1956).
[CrossRef]

Sheppard, C. J.R.

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Region (Hilger, 1986). See especially Chap. 16.

Wolf, E.

J. T. Foley, E. Wolf, “Wave-front spacing in the focal region of high-numerical-aperture systems,” Opt. Lett. 30, 1312–1314 (2005).
[CrossRef] [PubMed]

A. Boivin, J. Dow, E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967).
[CrossRef]

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561–B1565 (1965).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London, Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

E. H. Linfoot, E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. London, Sect. B, 69, 823–832 (1956).
[CrossRef]

B. Richards, E. Wolf, “The Airy pattern in systems of high angular aperture,” Proc. Phys. Soc. London, Sect. B 69, 854–856 (1956).
[CrossRef]

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999). See especially Sec. 4.5.1.
[CrossRef]

Youngworth, K. S.

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

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

K. S. Youngworth, T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000). http://www.opticsexpress.org.
[CrossRef] [PubMed]

Zhan, Q.

Appl. Opt.

Appl. Phys. B

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

Appl. Phys. Lett.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev.

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561–B1565 (1965).
[CrossRef]

Phys. Rev. Lett.

R. Dorn, S. Quabis, 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, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

Proc. Phys. Soc. London, Sect. B

E. H. Linfoot, E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. London, Sect. B, 69, 823–832 (1956).
[CrossRef]

B. Richards, E. Wolf, “The Airy pattern in systems of high angular aperture,” Proc. Phys. Soc. London, Sect. B 69, 854–856 (1956).
[CrossRef]

Proc. R. Soc. London, Ser. A

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London, Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

Proc. SPIE

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

Other

P. W. Milonni, J. H. Eberly, Lasers (Wiley, 1988). See especially Section 14.8.

J. J. Stamnes, Waves in Focal Region (Hilger, 1986). See especially Chap. 16.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999). See especially Sec. 4.5.1.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Illustration of a high-NA focusing system. The incident beam propagates along the z axis.

Fig. 2
Fig. 2

Example of the angular amplitude function l ( θ ) for two different values of the beam-spot size w 0 . For the upper curve w 0 = 0.02 m , for the lower curve w 0 = 0.01 m . In both cases f = 0.01 m .

Fig. 3
Fig. 3

Longitudinal electric field component E z ( 0 , 0 , z ) along the z axis. In this example, NA = 0.75 , and β = f w 0 = 2 .

Fig. 4
Fig. 4

Longitudinal electric field component E z ( 0 , 0 , z ) along the z axis. In this example, NA = 0.75 , and β = f w 0 = 1 .

Fig. 5
Fig. 5

Longitudinal electric field component E z ( 0 , 0 , z ) along the z axis. In this example NA = 0.3 , and β = f w 0 = 2 .

Tables (1)

Tables Icon

Table 1 Position and Spacings (Both Expressed in Wavelengths) of the First Five Axial Zeros of the Longitudinal Electric Field Component E z in the Focal Region of a High-NA System a

Equations (29)

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

E ̃ 10 ( x , y , z ) = 2 3 2 A w 0 w 2 ( z ) x x ̂ exp { i [ k z 2 tan 1 ( z z 0 ) ] } × exp [ i k ( x 2 + y 2 ) 2 R ( z ) ] exp [ ( x 2 + y 2 ) w 2 ( z ) ] ,
E ̃ 01 ( x , y , z ) = 2 3 2 A w 0 w 2 ( z ) y y ̂ exp { i [ k z 2 tan 1 ( z z 0 ) ] } × exp [ i k ( x 2 + y 2 ) 2 R ( z ) ] exp [ ( x 2 + y 2 ) w 2 ( z ) ] .
R ( z ) = z + z 0 2 z ,
w ( z ) = w 0 ( 1 + z 2 z 0 2 ) 1 2 ,
z 0 = π w 0 2 λ .
E ( x , y , z , t ) = [ E ̃ 10 ( x , y , z ) + E ̃ 01 ( x , y , z ) ] exp ( i ω t ) ,
E ( ρ , z , t ) = 2 3 2 A w 0 w 2 ( z ) ρ ̂ l 0 ( ρ ) exp { i [ k z 2 tan 1 ( z z 0 ) ] } exp ( i ω t ) ,
l 0 ( ρ ) = ρ exp [ i k ρ 2 2 R ( z ) ] exp [ ρ 2 w 2 ( z ) ] .
E ( r , t ) = Re [ e ( r ) exp ( i ω t ) ] ,
H ( r , t ) = Re [ h ( r ) exp ( i ω t ) ] ,
e ( r ) = i k 2 π Ω a ( s x , s y ) s z exp [ i k ( s x x + s y y + s z z ) ] d s x d s y ,
h ( r ) = i k 2 π Ω b ( s x , s y ) s z exp [ i k ( s x x + s y y + s z z ) ] d s x d s y .
s = ( sin θ cos ϕ , sin θ sin ϕ , cos θ ) .
g 0 = ( cos ϕ , sin ϕ , 0 ) ,
g 1 = ( cos θ cos ϕ , cos θ sin ϕ , sin θ ) .
e ( 0 ) = l 0 ( ρ ) [ e r ( 0 ) g 0 + e ϕ ( 0 ) ( g 0 × k ) ] ,
a = f l ( θ ) cos 1 2 θ [ e r ( 0 ) g 1 + e ϕ ( 0 ) ( g 1 × s ) ] .
l ( θ ) = f sin θ exp [ i k f 2 sin 2 θ 2 R ( z ) ] exp [ f 2 sin 2 θ w 2 ( z ) ] .
l ( θ ) = f sin θ exp [ f 2 sin 2 θ w 0 2 ] , ( z = 0 ) ,
s r = ρ P sin θ cos ( ϕ P ϕ ) + z P cos θ .
e ( P ) = i k f 2 π 0 α 0 2 π l ( θ ) sin θ cos 1 2 θ × exp { i k [ ρ P sin θ cos ( ϕ P ϕ ) + z P cos θ ] } × ( cos θ cos ϕ , cos θ sin ϕ , sin θ ) d θ d ϕ ,
g P = ( cos ϕ P , sin ϕ P , 0 ) .
e ( P ) = e ρ ( P ) g P + e z ( P ) k ,
e z ( P ) = i k f 2 π 0 α 0 2 π l ( θ ) sin 2 θ cos 1 2 θ × exp { i k [ ρ P sin θ cos ( ϕ P ϕ ) + z P cos θ ] } d θ d ϕ ,
e ρ ( P ) = i k f 2 π 0 α 0 2 π l ( θ ) sin θ cos 3 2 θ cos ( ϕ P ϕ ) × exp { i k [ ρ P sin θ cos ( ϕ P ϕ ) + z P cos θ ] } d θ d ϕ .
e z ( P ) = i k f 0 α l ( θ ) sin 2 θ cos 1 2 θ × exp ( i k z P cos θ ) J 0 ( k ρ P sin θ ) d θ ,
e ρ ( P ) = k f 0 α l ( θ ) sin θ cos 3 2 θ × exp ( i k z P cos θ ) J 1 ( k ρ P sin θ ) d θ ,
E z ( 0 , 0 , z ) = k f 0 α l ( θ ) sin 2 θ cos 1 2 θ sin ( k z cos θ ) d θ ,
= k f 2 0 α sin 3 θ cos 1 2 θ sin ( k z cos θ ) × exp ( β 2 sin 2 θ ) d θ ,

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