Abstract

The paraxial Bessel beam is obtained by applying an approximation in the wavenumbers. The scattering of the beams by a circular aperture in an absorbing screen is investigated. The scattered fields are expressed in terms of the Fresnel integrals by evaluating the Kirchhoff diffraction integral in the paraxial approximation. The results are examined numerically.

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  1. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651-654 (1987).
    [CrossRef]
  2. J. Durnin, J. J. Miceli Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987).
    [CrossRef] [PubMed]
  3. D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46, 15-28 (2005).
    [CrossRef]
  4. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207-211 (1998).
    [CrossRef]
  5. S. H. Tao and X. Yuan, “Self-reconstruction property of fractional Bessel beams,” J. Opt. Soc. Am. A 21, 1192-1197 (2004).
    [CrossRef]
  6. M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo and S. Chavez-Cerda, “Conical dynamics of Bessel beams,” Opt. Eng. 46, 078001 (2007).
    [CrossRef]
  7. J. Turunen, A. Vasara, and A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959-3962 (1988).
    [CrossRef] [PubMed]
  8. Y. Lin, W. Seka, J. H. Eberly, H. Huang, and D. L. Brown, “Experimental investigation of Bessel beam characteristics,” Appl. Opt. 31, 2708-2713 (1992).
    [CrossRef] [PubMed]
  9. P. Muys and E. Vandamme, “Direct generation of Bessel beams,” Appl. Opt. 41, 6375-6379 (2002).
    [CrossRef] [PubMed]
  10. J. Durnin, J. J. Miceli Jr., and J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79-80 (1988).
    [CrossRef] [PubMed]
  11. P. L. Overfelt and C. S. Kenney, “Comparison of the propagation characteristics of Bessel, Bessel-Gauss and Gaussian beams diffracted by a circular aperture,” J. Opt. Soc. Am. A 8, 732-745 (1991).
    [CrossRef]
  12. Z. Jiang, Q. Lu, and Z. Liu, “Propagation of apertured Bessel beams,” Appl. Opt. 34, 7183-7185 (1995).
    [CrossRef] [PubMed]
  13. R. Borghi, M. Santarsiero, and F. Gori, “Axial intensity of apertured Bessel beams,” J. Opt. Soc. Am. A 14, 23-26 (1997).
    [CrossRef]
  14. D. Ding and X. Liu, “Approximate description for Bessel, Bessel-Gauss and Gaussian beams with finite aperture,” J. Opt. Soc. Am. A 16, 1286-1293 (1999).
    [CrossRef]
  15. R. Piestun, Y. Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” J. Opt. Soc. Am. A 17, 294-303 (2000).
    [CrossRef]
  16. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684-685 (1971).
    [CrossRef]
  17. Y. Z. Umul, “Modified theory of physical optics,” Opt. Express 12, 4959-4972 (2004).
    [CrossRef] [PubMed]
  18. Y. Z. Umul, “Modified diffraction theory of Kirchhoff,” J. Opt. Soc. Am. A 25, 1850-1860 (2008).
    [CrossRef]
  19. Y. Z. Umul, “Improved equivalent source theory,” J. Opt. Soc. Am. A 26, 1799-1805 (2009).
    [CrossRef]
  20. Y. Z. Umul, “General formulation of the edge-diffracted paraxial waves,” Opt. Laser Technol. 41, 778-782 (2009).
    [CrossRef]
  21. Y. Z. Umul, “Scattering of a line source by a cylindrical parabolic impedance surface,” J. Opt. Soc. Am. A 25, 1652-1659 (2008).
    [CrossRef]
  22. Y. Z. Umul, “Uniform theory for the diffraction of evanescent plane waves,” J. Opt. Soc. Am. A 24, 2426-2430 (2007).
    [CrossRef]
  23. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116-130 (1962).
    [CrossRef] [PubMed]
  24. R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting screen,” Proc. IEEE 62, 1448-1461 (1974).
    [CrossRef]
  25. Y. Z. Umul, “Modified theory of physical optics solution of the impedance half plane problem,” IEEE Trans. Antennas Propag. 54, 2048-2053 (2006).
    [CrossRef]
  26. T. Young, “The Bakerian lecture: On the theory of light and colors,” Philos. Trans. R. Soc. London, Ser. A 92, 12-48 (1802).
    [CrossRef]
  27. A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature (London) 180, 160-162 (1957).
    [CrossRef]
  28. Y. Z. Umul, “Alternative interpretation of the edge-diffraction phenomenon,” J. Opt. Soc. Am. A 25, 582-587 (2008).
    [CrossRef]
  29. M. Moshinsky, “Diffraction in time,” Phys. Rev. 88, 625-631 (1952).
    [CrossRef]
  30. M. Moshinsky, “Diffraction in time and the time-energy uncertainty relation,” Am. J. Phys. 44, 1037-1042 (1976).
    [CrossRef]

2009 (2)

Y. Z. Umul, “Improved equivalent source theory,” J. Opt. Soc. Am. A 26, 1799-1805 (2009).
[CrossRef]

Y. Z. Umul, “General formulation of the edge-diffracted paraxial waves,” Opt. Laser Technol. 41, 778-782 (2009).
[CrossRef]

2008 (3)

2007 (2)

Y. Z. Umul, “Uniform theory for the diffraction of evanescent plane waves,” J. Opt. Soc. Am. A 24, 2426-2430 (2007).
[CrossRef]

M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo and S. Chavez-Cerda, “Conical dynamics of Bessel beams,” Opt. Eng. 46, 078001 (2007).
[CrossRef]

2006 (1)

Y. Z. Umul, “Modified theory of physical optics solution of the impedance half plane problem,” IEEE Trans. Antennas Propag. 54, 2048-2053 (2006).
[CrossRef]

2005 (1)

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

2004 (2)

2002 (1)

2000 (1)

1999 (1)

1998 (1)

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207-211 (1998).
[CrossRef]

1997 (1)

1995 (1)

1992 (1)

1991 (1)

1988 (2)

1987 (2)

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

1976 (1)

M. Moshinsky, “Diffraction in time and the time-energy uncertainty relation,” Am. J. Phys. 44, 1037-1042 (1976).
[CrossRef]

1974 (1)

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting screen,” Proc. IEEE 62, 1448-1461 (1974).
[CrossRef]

1971 (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684-685 (1971).
[CrossRef]

1962 (1)

1957 (1)

A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature (London) 180, 160-162 (1957).
[CrossRef]

1952 (1)

M. Moshinsky, “Diffraction in time,” Phys. Rev. 88, 625-631 (1952).
[CrossRef]

1802 (1)

T. Young, “The Bakerian lecture: On the theory of light and colors,” Philos. Trans. R. Soc. London, Ser. A 92, 12-48 (1802).
[CrossRef]

Anguiano-Morales, M.

M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo and S. Chavez-Cerda, “Conical dynamics of Bessel beams,” Opt. Eng. 46, 078001 (2007).
[CrossRef]

Borghi, R.

Bouchal, Z.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207-211 (1998).
[CrossRef]

Brown, D. L.

Chavez-Cerda, S.

M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo and S. Chavez-Cerda, “Conical dynamics of Bessel beams,” Opt. Eng. 46, 078001 (2007).
[CrossRef]

Chlup, M.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207-211 (1998).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684-685 (1971).
[CrossRef]

Dholakia, K.

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

Ding, D.

Durnin, J.

Eberly, J. H.

Friberg, A. T.

Gori, F.

Huang, H.

Iturbe-Castillo, M. D.

M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo and S. Chavez-Cerda, “Conical dynamics of Bessel beams,” Opt. Eng. 46, 078001 (2007).
[CrossRef]

Jiang, Z.

Keller, J. B.

Kenney, C. S.

Kouyoumjian, R. G.

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting screen,” Proc. IEEE 62, 1448-1461 (1974).
[CrossRef]

Lin, Y.

Liu, X.

Liu, Z.

Lu, Q.

McGloin, D.

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

Méndez-Otero, M. M.

M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo and S. Chavez-Cerda, “Conical dynamics of Bessel beams,” Opt. Eng. 46, 078001 (2007).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli Jr., and J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79-80 (1988).
[CrossRef] [PubMed]

J. Durnin, J. J. Miceli Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Moshinsky, M.

M. Moshinsky, “Diffraction in time and the time-energy uncertainty relation,” Am. J. Phys. 44, 1037-1042 (1976).
[CrossRef]

M. Moshinsky, “Diffraction in time,” Phys. Rev. 88, 625-631 (1952).
[CrossRef]

Muys, P.

Overfelt, P. L.

Pathak, P. H.

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting screen,” Proc. IEEE 62, 1448-1461 (1974).
[CrossRef]

Piestun, R.

Rubinowicz, A.

A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature (London) 180, 160-162 (1957).
[CrossRef]

Santarsiero, M.

Schechner, Y. Y.

Seka, W.

Shamir, J.

Tao, S. H.

Turunen, J.

Umul, Y. Z.

Vandamme, E.

Vasara, A.

Wagner, J.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207-211 (1998).
[CrossRef]

Young, T.

T. Young, “The Bakerian lecture: On the theory of light and colors,” Philos. Trans. R. Soc. London, Ser. A 92, 12-48 (1802).
[CrossRef]

Yuan, X.

Am. J. Phys. (1)

M. Moshinsky, “Diffraction in time and the time-energy uncertainty relation,” Am. J. Phys. 44, 1037-1042 (1976).
[CrossRef]

Appl. Opt. (4)

Contemp. Phys. (1)

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

Electron. Lett. (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684-685 (1971).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

Y. Z. Umul, “Modified theory of physical optics solution of the impedance half plane problem,” IEEE Trans. Antennas Propag. 54, 2048-2053 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (11)

Y. Z. Umul, “Modified diffraction theory of Kirchhoff,” J. Opt. Soc. Am. A 25, 1850-1860 (2008).
[CrossRef]

Y. Z. Umul, “Improved equivalent source theory,” J. Opt. Soc. Am. A 26, 1799-1805 (2009).
[CrossRef]

Y. Z. Umul, “Scattering of a line source by a cylindrical parabolic impedance surface,” J. Opt. Soc. Am. A 25, 1652-1659 (2008).
[CrossRef]

Y. Z. Umul, “Uniform theory for the diffraction of evanescent plane waves,” J. Opt. Soc. Am. A 24, 2426-2430 (2007).
[CrossRef]

P. L. Overfelt and C. S. Kenney, “Comparison of the propagation characteristics of Bessel, Bessel-Gauss and Gaussian beams diffracted by a circular aperture,” J. Opt. Soc. Am. A 8, 732-745 (1991).
[CrossRef]

Y. Z. Umul, “Alternative interpretation of the edge-diffraction phenomenon,” J. Opt. Soc. Am. A 25, 582-587 (2008).
[CrossRef]

S. H. Tao and X. Yuan, “Self-reconstruction property of fractional Bessel beams,” J. Opt. Soc. Am. A 21, 1192-1197 (2004).
[CrossRef]

R. Borghi, M. Santarsiero, and F. Gori, “Axial intensity of apertured Bessel beams,” J. Opt. Soc. Am. A 14, 23-26 (1997).
[CrossRef]

D. Ding and X. Liu, “Approximate description for Bessel, Bessel-Gauss and Gaussian beams with finite aperture,” J. Opt. Soc. Am. A 16, 1286-1293 (1999).
[CrossRef]

R. Piestun, Y. Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” J. Opt. Soc. Am. A 17, 294-303 (2000).
[CrossRef]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

Nature (London) (1)

A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature (London) 180, 160-162 (1957).
[CrossRef]

Opt. Commun. (1)

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207-211 (1998).
[CrossRef]

Opt. Eng. (1)

M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo and S. Chavez-Cerda, “Conical dynamics of Bessel beams,” Opt. Eng. 46, 078001 (2007).
[CrossRef]

Opt. Express (1)

Opt. Laser Technol. (1)

Y. Z. Umul, “General formulation of the edge-diffracted paraxial waves,” Opt. Laser Technol. 41, 778-782 (2009).
[CrossRef]

Opt. Lett. (1)

Philos. Trans. R. Soc. London, Ser. A (1)

T. Young, “The Bakerian lecture: On the theory of light and colors,” Philos. Trans. R. Soc. London, Ser. A 92, 12-48 (1802).
[CrossRef]

Phys. Rev. (1)

M. Moshinsky, “Diffraction in time,” Phys. Rev. 88, 625-631 (1952).
[CrossRef]

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Proc. IEEE (1)

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting screen,” Proc. IEEE 62, 1448-1461 (1974).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Geometry of the scattering problem.

Fig. 2
Fig. 2

Propagation of a paraxial Bessel beam in free space.

Fig. 3
Fig. 3

Scattered ray in a two-dimensional plane.

Fig. 4
Fig. 4

Comparison of the scattered waves.

Fig. 5
Fig. 5

Axial diffraction pattern.

Fig. 6
Fig. 6

Variation of the scattered wave according to η.

Fig. 7
Fig. 7

Axial scattered fields for different values of a.

Equations (52)

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

u ( P ) = u 0 J 0 ( α ρ ) exp ( j β z ) ,
β = k 2 α 2
u ( P ) = u 0 J 0 ( α ρ ) exp [ j ( β k ) z ] exp ( j k z ) .
β = k 1 α 2 k 2 ,
β k α 2 2 k
u ( P ) = u 0 J 0 ( α ρ ) exp ( j α 2 2 k z ) exp ( j k z ) ,
α = k cos η
θ = t g 1 ρ z
u s ( P ) = j k 2 π ρ = 0 a ϕ = 0 2 π u i ( Q ) sin χ η 2 exp ( j k R 1 ) R 1 ρ d ρ d ϕ
R 1 z + ρ 2 + ρ 2 2 ρ ρ cos ( ϕ ϕ ) 2 z
R 1 z
u s ( P ) = j k 2 π exp ( j k z ) z exp ( j k ρ 2 2 z ) ρ = 0 a ϕ = 0 2 π u i ( Q ) exp [ j k ρ 2 2 ρ ρ cos ( ϕ ϕ ) 2 z ] ρ d ρ d ϕ
I = ϕ = 0 2 π exp [ j k ρ ρ cos ( ϕ ϕ ) z ] d ϕ
J 0 ( x ) = 1 2 π 0 2 π exp ( j x cos τ ) d τ .
I = 2 π J 0 ( k ρ ρ z ) .
u s ( P ) = j k exp ( j k z ) z exp ( j k ρ 2 2 z ) ρ = 0 a u i ( Q ) J 0 ( k ρ ρ z ) exp ( j k ρ 2 2 z ) ρ d ρ
u s ( P ) = j k exp ( j k z ) z exp ( j k ρ 2 2 z ) ρ = 0 a J 0 ( α ρ ) J 0 ( k ρ ρ z ) exp ( j k ρ 2 2 z ) ρ d ρ ,
R = ( ρ ρ ) 2 + z 2 ,
R z + ( ρ ρ ) 2 2 z
exp ( j k R ) exp [ j k ( z + ρ 2 2 z + ρ 2 2 z ρ ρ z ) ]
exp [ j k ( z + ρ 2 2 z + ρ 2 2 z ) ] .
J 0 ( k ρ ρ z ) exp ( j π 4 ) z 2 π k ρ ρ exp ( j k ρ ρ z )
u s ( P ) = k 2 π ρ z exp ( j π 4 ) exp ( j k z ) exp ( j k ρ 2 2 z ) ρ = 0 a J 0 ( α ρ ) exp [ j k ( ρ 2 2 z ρ ρ z ) ] ρ d ρ .
u s ( P ) = 1 2 π k α ρ z exp ( j π 4 ) exp [ j k ( z + ρ 2 2 z ) ] [ exp ( j π 4 ) I 1 + exp ( j π 4 ) I 2 ] ,
I 1 = ρ = 0 a exp [ j k ( ρ 2 2 z ρ z ρ α k ρ ) ] d ρ
I 2 = ρ = 0 a exp [ j k ( ρ 2 2 z ρ z ρ + α k ρ ) ] d ρ ,
ψ 1 = ρ 2 2 z ρ z ρ α k ρ ,
ψ 1 ρ = k ( ρ ρ ) α z k z .
ρ s = ρ + α k z
2 ψ 1 ρ 2 = 1 z .
ψ ρ 2 2 z α 2 2 k 2 z α k ρ + 1 2 z ( ρ ρ s ) 2
I 1 = exp ( j k ρ 2 2 z ) exp ( j α 2 2 k z ) exp ( j α ρ ) exp [ j k 2 z ( ρ ρ s ) 2 ] d ρ
I 1 = 2 π z k exp ( j π 4 ) exp ( j k ρ 2 2 z ) exp ( j α 2 2 k z ) exp ( j α ρ )
I 2 = 2 π z k exp ( j π 4 ) exp ( j k ρ 2 2 z ) exp ( j α 2 2 k z ) exp ( j α ρ ) .
u GO = exp ( j k z ) exp ( j α 2 2 k z ) J 0 ( α ρ )
0 a f ( x ) exp [ j k g ( x ) ] d x 1 j k f ( a ) g ( a ) exp [ j k g ( a ) ]
I e 1 = 1 j k k z k ( a ρ ) α z exp [ j k ( a 2 2 z ρ a z α a k ) ]
I e 2 = 1 j k k z k ( a ρ ) + α z exp [ j k ( a 2 2 z ρ a z + α a k ) ] ,
ξ 1 = k ( a ρ ) α z 2 k z
ξ 2 = k ( a ρ ) + α z 2 k z
I e 1 = 2 π z k exp [ j ( k ρ 2 2 z + α 2 2 k z + α ρ π 4 ) ] u de 1
I e 2 = 2 π z k exp [ j ( k ρ 2 2 z + α 2 2 k z α ρ π 4 ) ] u de 2
u de 1 = exp ( j π 4 ) 2 π exp ( j k ξ 1 2 ) ξ 1
u de 2 = exp ( j π 4 ) 2 π exp ( j k ξ 2 2 ) ξ 2 ,
u d = 1 2 exp [ j ( k z α 2 2 k z ) ] [ H 0 ( 1 ) ( α ρ ) u de 1 + H 0 ( 2 ) ( α ρ ) u de 2 ]
ξ 1 , 2 = 0 .
sign ( x ) F [ | x | ] exp ( j π 4 ) 2 π exp ( j x 2 ) x ,
F [ x ] = exp ( j π 4 ) π x exp ( j t 2 ) d t .
u d = 1 2 exp [ j ( k z α 2 2 k z ) ] { H 0 ( 1 ) ( α ρ ) sign ( ξ 1 ) F [ | ξ 1 | ] + H 0 ( 2 ) ( α ρ ) sign ( ξ 2 ) F [ | ξ 2 | ] }
u GO = 1 2 exp [ j ( k z α 2 2 k z ) ] [ H 0 ( 1 ) ( α ρ ) U ( ξ 1 ) + H 0 ( 2 ) ( α ρ ) U ( ξ 2 ) ] ,
u s = 1 2 exp [ j ( k z α 2 2 k z ) ] [ H 0 ( 1 ) ( α ρ ) F ( ξ 1 ) + H 0 ( 2 ) ( α ρ ) F ( ξ 2 ) ]
u s ( P ) = j k exp ( j k z ) z exp ( j k ρ 2 2 z ) ρ = 0 a J 0 ( α ρ ) J 0 ( k ρ ρ z ) exp ( j k ρ 2 2 z ) ρ d ρ ,

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