Abstract

Based on quadratic approximation and δ expansion in the first order, the analytical expressions for multi-Gaussian beams in a turbulent atmosphere have been derived. By comparing the two approaches with numerical calculations, the relative errors of average intensity are investigated. As special cases, the relative errors for Gaussian beams, flattened Gaussian beams, and annular beams are investigated. The investigation shows that the method of δ expansion in the first order agrees well with numerical calculations, no matter what the effect of turbulence. If the effect of turbulence is large enough, the relative error of on-axis intensity trends to a constant. The maximum of relative error is about 2.8%. However, quadratic approximation does not give satisfying results under some circumstance even when the effect of turbulence is small. The relative error of on-axis intensity reaches 9.2% when the effect of turbulence is large enough.

© 2010 Optical Society of America

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  1. H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical system,” J. Opt. Soc. Am. A 4, 1931-1948 (1987).
    [CrossRef]
  2. Y. Cai and S. He, “Propagation of various dark hollow beams in turbulent atmosphere,” Opt. Express 14, 1353-1367 (2006).
    [CrossRef] [PubMed]
  3. H. T. Eyyuboğlu and Y. Baykal, “Reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Express 12, 4659-4674 (2004).
    [CrossRef] [PubMed]
  4. H. T. Eyyuboğlu, C. Arpali, and Y. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express 14, 4196-4207 (2006).
    [CrossRef] [PubMed]
  5. Y. Baykal and H. T. Eyyuboğlu, “Scintillations of incoherent flat-topped Gaussian source field in turbulence,” Appl. Opt. 46, 5044 (2007).
    [CrossRef] [PubMed]
  6. Y. Baykal and H. T. Eyyuboğlu, “Scintillation index of flat-topped Gaussian beams,” Appl. Opt. 45, 3793-3797 (2006).
    [CrossRef] [PubMed]
  7. Y. Zhang and G. Wang, “Slant path average intensity of finite optical beam propagating in turbulent atmosphere,” Chin. Opt. Lett. 10, 559-562 (2006).
  8. X. Chu, Y. Ni, and G. Zhou, “Propagation analysis of flattened circular Gaussian beams with a circular aperture in turbulent atmosphere,” Opt. Commun. 274, 274-280 (2007).
    [CrossRef]
  9. X. Chu, Z. Liu, and Y. Wu, “Propagation of a general multi-Gaussian beam in turbulent atmosphere in a slant path,” J. Opt. Soc. Am. A 25, 74-79 (2008).
    [CrossRef]
  10. X. Chu, “Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere,” Opt. Express 15, 17613 (2007).
    [CrossRef] [PubMed]
  11. Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16, 18437 (2008).
    [CrossRef] [PubMed]
  12. X. Du and D. Zhao, “Polarization modulation of stochastic electromagnetic beams on propagation through the turbulent atmosphere,” Opt. Express 17, 4257-4262 (2009).
    [CrossRef] [PubMed]
  13. J. Pu, “Invariance of spectrum and polarization of electromagnetic Gaussian Schell-model beams propagating in free space,” Chin. Phys. Lett. 4, 196-198 (2006).
  14. X. Ji, E. Zhang, and B. Lu, “Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 259, 1-6 (2006).
    [CrossRef]
  15. R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424-1443 (1980).
    [CrossRef]
  16. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207-211 (1976).
    [CrossRef]
  17. L. Andrews and R. Phillips, Laser Beam Propagation in Random Media (SPIE, 1998), Chap. 6.
  18. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Chap. 20.
  19. J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37, 671-684 (1990).
    [CrossRef]
  20. Y. Dan, B. Zhang, and P. Pan, “Propagation of partially coherent flat-topped beams through a turbulent atmosphere,” J. Opt. Soc. Am. A 25, 2223-2231 (2008).
    [CrossRef]
  21. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005), Chap. 7.
    [CrossRef]
  22. R. F. Lutomirski and H. T. Yura, “Wave structure function and mutual coherence function of an optical wave in a turbulent atmosphere,” J. Opt. Soc. Am. 61, 482-487 (1971).
    [CrossRef]
  23. H. T. Eyyuboğlu, S. Altay, and Y. Baykal, “Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,” Opt. Commun. 264, 25-34 (2006).
    [CrossRef]
  24. S. M. Wandzura, “Meaning of quadratic structure functions,” J. Opt. Soc. Am. 70, 745 (1980).
    [CrossRef]
  25. S. M. Wandzura, “Systematic corrections to quadratic approximations for power-law structure functions: the delta expansion,” J. Opt. Soc. Am. 71, 321-326 (1981).
    [CrossRef]
  26. M. S. Belen'kii and V. L. Mironov, “Coherence of a laser beam field in a turbulent atmosphere,” Sov. J. Quantum Electron. 10, 1042-1047 (1980).
    [CrossRef]
  27. M. S. Belen'kii, V. M. Buldakov, and V. L. Mironov, “Spectrum of the spatial coherence function of a laser beam field in a turbulent atmosphere,” Radiophys. Quantum Electron. 23, 851-854 (1980).
    [CrossRef]
  28. M. A. Plonus and S. J. Wang, “Quadratic structure functions and scintillation,” Appl. Opt. 24, 570-571 (1985).
    [CrossRef] [PubMed]
  29. Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. 27, 1007-1009 (2002).
    [CrossRef]
  30. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69, 1297-1304 (1979).
    [CrossRef]
  31. I. S. Gradysteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1980).

2009 (1)

2008 (3)

2007 (3)

2006 (7)

J. Pu, “Invariance of spectrum and polarization of electromagnetic Gaussian Schell-model beams propagating in free space,” Chin. Phys. Lett. 4, 196-198 (2006).

X. Ji, E. Zhang, and B. Lu, “Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 259, 1-6 (2006).
[CrossRef]

Y. Zhang and G. Wang, “Slant path average intensity of finite optical beam propagating in turbulent atmosphere,” Chin. Opt. Lett. 10, 559-562 (2006).

H. T. Eyyuboğlu, S. Altay, and Y. Baykal, “Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,” Opt. Commun. 264, 25-34 (2006).
[CrossRef]

Y. Cai and S. He, “Propagation of various dark hollow beams in turbulent atmosphere,” Opt. Express 14, 1353-1367 (2006).
[CrossRef] [PubMed]

H. T. Eyyuboğlu, C. Arpali, and Y. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express 14, 4196-4207 (2006).
[CrossRef] [PubMed]

Y. Baykal and H. T. Eyyuboğlu, “Scintillation index of flat-topped Gaussian beams,” Appl. Opt. 45, 3793-3797 (2006).
[CrossRef] [PubMed]

2005 (1)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005), Chap. 7.
[CrossRef]

2004 (1)

2002 (1)

1998 (1)

L. Andrews and R. Phillips, Laser Beam Propagation in Random Media (SPIE, 1998), Chap. 6.

1990 (1)

J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

1987 (1)

1985 (1)

1981 (1)

1980 (5)

S. M. Wandzura, “Meaning of quadratic structure functions,” J. Opt. Soc. Am. 70, 745 (1980).
[CrossRef]

M. S. Belen'kii and V. L. Mironov, “Coherence of a laser beam field in a turbulent atmosphere,” Sov. J. Quantum Electron. 10, 1042-1047 (1980).
[CrossRef]

M. S. Belen'kii, V. M. Buldakov, and V. L. Mironov, “Spectrum of the spatial coherence function of a laser beam field in a turbulent atmosphere,” Radiophys. Quantum Electron. 23, 851-854 (1980).
[CrossRef]

I. S. Gradysteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1980).

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424-1443 (1980).
[CrossRef]

1979 (1)

1978 (1)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Chap. 20.

1976 (1)

1971 (1)

Altay, S.

H. T. Eyyuboğlu, S. Altay, and Y. Baykal, “Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,” Opt. Commun. 264, 25-34 (2006).
[CrossRef]

Andrews, L.

L. Andrews and R. Phillips, Laser Beam Propagation in Random Media (SPIE, 1998), Chap. 6.

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005), Chap. 7.
[CrossRef]

Arpali, C.

Baykal, Y.

Belen'kii, M. S.

M. S. Belen'kii, V. M. Buldakov, and V. L. Mironov, “Spectrum of the spatial coherence function of a laser beam field in a turbulent atmosphere,” Radiophys. Quantum Electron. 23, 851-854 (1980).
[CrossRef]

M. S. Belen'kii and V. L. Mironov, “Coherence of a laser beam field in a turbulent atmosphere,” Sov. J. Quantum Electron. 10, 1042-1047 (1980).
[CrossRef]

Buldakov, V. M.

M. S. Belen'kii, V. M. Buldakov, and V. L. Mironov, “Spectrum of the spatial coherence function of a laser beam field in a turbulent atmosphere,” Radiophys. Quantum Electron. 23, 851-854 (1980).
[CrossRef]

Cai, Y.

Chu, X.

Dan, Y.

Du, X.

Eyyuboglu, H. T.

Fante, R. L.

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424-1443 (1980).
[CrossRef]

Gradysteyn, I. S.

I. S. Gradysteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1980).

Hanson, S. G.

He, S.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Chap. 20.

Ji, X.

X. Ji, E. Zhang, and B. Lu, “Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 259, 1-6 (2006).
[CrossRef]

Li, Y.

Liu, Z.

Lu, B.

X. Ji, E. Zhang, and B. Lu, “Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 259, 1-6 (2006).
[CrossRef]

Lutomirski, R. F.

Mironov, V. L.

M. S. Belen'kii and V. L. Mironov, “Coherence of a laser beam field in a turbulent atmosphere,” Sov. J. Quantum Electron. 10, 1042-1047 (1980).
[CrossRef]

M. S. Belen'kii, V. M. Buldakov, and V. L. Mironov, “Spectrum of the spatial coherence function of a laser beam field in a turbulent atmosphere,” Radiophys. Quantum Electron. 23, 851-854 (1980).
[CrossRef]

Ni, Y.

X. Chu, Y. Ni, and G. Zhou, “Propagation analysis of flattened circular Gaussian beams with a circular aperture in turbulent atmosphere,” Opt. Commun. 274, 274-280 (2007).
[CrossRef]

Noll, R. J.

Pan, P.

Phillips, R.

L. Andrews and R. Phillips, Laser Beam Propagation in Random Media (SPIE, 1998), Chap. 6.

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005), Chap. 7.
[CrossRef]

Plonus, M. A.

Pu, J.

J. Pu, “Invariance of spectrum and polarization of electromagnetic Gaussian Schell-model beams propagating in free space,” Chin. Phys. Lett. 4, 196-198 (2006).

Ryzhik, I. M.

I. S. Gradysteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1980).

Wandzura, S. M.

Wang, G.

Y. Zhang and G. Wang, “Slant path average intensity of finite optical beam propagating in turbulent atmosphere,” Chin. Opt. Lett. 10, 559-562 (2006).

Wang, S. C. H.

Wang, S. J.

Wu, J.

J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

Wu, Y.

Yura, H. T.

Zhang, B.

Zhang, E.

X. Ji, E. Zhang, and B. Lu, “Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 259, 1-6 (2006).
[CrossRef]

Zhang, Y.

Y. Zhang and G. Wang, “Slant path average intensity of finite optical beam propagating in turbulent atmosphere,” Chin. Opt. Lett. 10, 559-562 (2006).

Zhao, D.

Zhou, G.

X. Chu, Y. Ni, and G. Zhou, “Propagation analysis of flattened circular Gaussian beams with a circular aperture in turbulent atmosphere,” Opt. Commun. 274, 274-280 (2007).
[CrossRef]

Zhu, Y.

Appl. Opt. (3)

Chin. Opt. Lett. (1)

Y. Zhang and G. Wang, “Slant path average intensity of finite optical beam propagating in turbulent atmosphere,” Chin. Opt. Lett. 10, 559-562 (2006).

Chin. Phys. Lett. (1)

J. Pu, “Invariance of spectrum and polarization of electromagnetic Gaussian Schell-model beams propagating in free space,” Chin. Phys. Lett. 4, 196-198 (2006).

J. Mod. Opt. (1)

J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

J. Opt. Soc. Am. (5)

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

Opt. Commun. (3)

H. T. Eyyuboğlu, S. Altay, and Y. Baykal, “Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,” Opt. Commun. 264, 25-34 (2006).
[CrossRef]

X. Chu, Y. Ni, and G. Zhou, “Propagation analysis of flattened circular Gaussian beams with a circular aperture in turbulent atmosphere,” Opt. Commun. 274, 274-280 (2007).
[CrossRef]

X. Ji, E. Zhang, and B. Lu, “Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 259, 1-6 (2006).
[CrossRef]

Opt. Express (6)

Opt. Lett. (1)

Proc. IEEE (1)

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424-1443 (1980).
[CrossRef]

Radiophys. Quantum Electron. (1)

M. S. Belen'kii, V. M. Buldakov, and V. L. Mironov, “Spectrum of the spatial coherence function of a laser beam field in a turbulent atmosphere,” Radiophys. Quantum Electron. 23, 851-854 (1980).
[CrossRef]

Sov. J. Quantum Electron. (1)

M. S. Belen'kii and V. L. Mironov, “Coherence of a laser beam field in a turbulent atmosphere,” Sov. J. Quantum Electron. 10, 1042-1047 (1980).
[CrossRef]

Other (4)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005), Chap. 7.
[CrossRef]

I. S. Gradysteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1980).

L. Andrews and R. Phillips, Laser Beam Propagation in Random Media (SPIE, 1998), Chap. 6.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Chap. 20.

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Figures (7)

Fig. 1
Fig. 1

Normalized intensity distribution of the MGB with different parameters.

Fig. 2
Fig. 2

Mutual coherence function in different orders of the expansion.

Fig. 3
Fig. 3

Variations of relative error of on-axis intensity with different approaches.

Fig. 4
Fig. 4

Relative error of off-axis intensity with different s: (a)  s = 0.1 , (b)  s = 1 , and (c)  s = 5 .

Fig. 5
Fig. 5

Relations between the relative errors and s with different N, τ 1 , and τ 2 .

Fig. 6
Fig. 6

Intensity distribution of annular beams with different parameters and approaches.

Fig. 7
Fig. 7

Variation of relative error of on-axis intensity with different parameters.

Equations (29)

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u 0 ( r 0 , 0 ) = n = 1 N A n exp [ ( 1 w n 2 + i k 2 R ) r 0 2 + i φ n ] ,
I N ( x 0 , 0 , z ) = I ( x 0 , 0 , z ) max [ I ( x 0 , 0 , z ) ] ,
I ( r , z ) = k 2 / ( 2 π z ) 2 d 2 r 01 d 2 r 02 u 0 ( r 01 ) u 0 * ( r 02 ) exp { i k [ ( r r 01 ) 2 ( r r 02 ) 2 ] / ( 2 z ) } × exp [ ψ ( r 01 , r ) + ψ * ( r 02 , r ) ]
M ( r 01 r 02 ) = exp [ ψ ( r 01 , r ) + ψ * ( r 02 , r ) ] = exp [ 0.5 D w ( r 01 r 02 ) ]
I ( r , z ) = k 2 2 π z 2 0 H ( q ) M ( q ) J 0 ( k z r q ) q d q ,
H ( q ) = u 0 ( p + q 2 ) u 0 * ( p q 2 ) exp ( i k z p · q ) d 2 p .
H ( q ) = m = 1 N n = 1 N A m A n exp [ i ( φ m φ n ) ] π w m 2 w n 2 w m 2 + w n 2 exp { [ k w m 2 ( R z ) + 2 i R z ] [ k w n 2 ( R z ) 2 i R z ] 4 R 2 z 2 ( w m 2 + w n 2 ) q 2 } .
I q ( r , z ) = m = 1 N n = 1 N A m A n exp [ i ( φ m φ n ) ] 2 w m 2 w n 2 w 2 ( w m 2 + w n 2 ) exp ( 2 r 2 w 2 ) ,
w 2 = 8 R 2 z 2 ( w m 2 + w n 2 ) + 2 ρ 0 2 [ k w m 2 ( R z ) + 2 i R z ] [ k w n 2 ( R z ) 2 i R z ] k 2 R 2 ρ 0 2 ( w m 2 + w n 2 ) .
exp ( χ 2 δ ) = [ 1 + χ 2 δ ln ( χ ) 1 2 χ 2 ln 2 ( χ ) ( 1 χ 2 ) δ 2 ] exp ( χ 2 ) + O ( δ 3 ) ,
I δ ( r , z ) = m = 1 N n = 1 N A m A n exp [ i ( φ m φ n ) ] k 2 w m 2 w n 2 ρ 0 2 2 z 2 ( w m 2 + w n 2 ) 0 exp ( ρ 0 2 + a 2 a 2 χ 2 ) [ 1 + χ 2 δ ln ( χ ) ] J 0 ( k ρ 0 z r χ ) χ d χ ,
a 2 = 4 R 2 z 2 ( w m 2 + w n 2 ) [ k w m 2 ( R z ) + 2 i R z ] [ k w n 2 ( R z ) 2 i R z ] .
I δ ( r , z ) = I q ( r , z ) + Δ I ( r , z ) .
J 0 ( k ρ 0 z r x ) = j = 0 ( 1 ) j ( j ! ) 2 ( k ρ 0 2 z r x ) 2 j .
Δ I ( r , z ) = m = 1 N n = 1 N j = 0 ( 2 ) j ( j + 1 ) j ! A m A n exp [ i ( φ m φ n ) ] w m 2 w n 2 δ t 3 2 ( w m 2 + w n 2 ) t 4 [ ln ( t 3 2 t 2 ) + ψ ( j + 2 ) ] ( r t ) 2 j ,
Δ I ( r , z ) = m = 1 N n = 1 N A m A n exp [ i ( φ m φ n ) ] t 3 2 w m 2 w n 2 t 4 ( w m 2 + w n 2 ) exp ( 2 t 2 r 2 ) δ × { [ 2 exp ( 2 t 2 r 2 ) ] + ( 2 t 2 r 2 1 ) [ chi ( 2 t 2 r 2 ) + shi ( 2 t 2 r 2 ) ln ( 2 t 3 2 t 4 r 2 ) ] } .
Δ I ( 0 , z ) = m = 1 N n = 1 N A m A n exp [ i ( φ m φ n ) ] t 3 2 w m 2 w n 2 w m 2 + w n 2 [ 1 γ + ln ( t 3 2 t 2 ) ] δ ,
I q ( r , z ) = 1 τ 2 exp ( 2 r 2 τ 2 ) .
I δ ( r , z ) = I q ( r , z ) ( 1 + δ τ 3 2 2 τ 2 { [ 2 exp ( 2 τ 2 r 2 ) ] + ( 2 τ 2 r 2 1 ) [ chi ( 2 τ 2 r 2 ) + shi ( 2 τ 2 r 2 ) ln ( 2 τ 3 2 τ 4 r 2 ) ] } ) .
I δ ( 0 , z ) = 1 τ 12 2 ( 1 + s 2 ) { 1 + δ s 2 2 ( 1 + s 2 ) [ 1 γ + ln ( s 2 1 + s 2 ) ] } .
I ( r , z ) = 1 τ 12 2 0 exp [ 1 2 q 2 ( q s 2 ) 5 / 3 ] J 0 ( 2 r q τ 12 ) q d q ,
σ ν = 1 I ν ( r , z ) / I ( r , z ) ( ν = q , δ ) .
I ( 0 , z ) = m = 1 N n = 1 N ( 1 ) m + n ( N m ) ( N n ) 2 m + n 0 exp [ ( τ 1 + i m τ 2 ) ( τ 1 i n τ 2 ) ( m + n ) q 2 ( τ 3 q 2 ) 5 / 3 ] q d q ,
I q ( 0 , z ) = m = 1 N n = 1 N ( 1 ) m + n ( N m ) ( N n ) 2 ( m + n ) τ 2 ,
I δ ( 0 , z ) = I q ( 0 , z ) + m = 1 N n = 1 N ( 1 ) m + n ( N m ) ( N n ) δ τ 3 2 ( m + n ) τ 4 [ 1 γ + ln ( τ 3 2 τ 4 ) ] ,
τ = t / w 0 = 2 τ 1 2 + 2 m n τ 2 2 + ( m + n ) τ 3 2 + 2 i ( m n ) τ 1 τ 2 m + n .
I ( r , z ) = n = 1 N n = 1 N i m n 2 m + n 0 exp [ ( τ 1 + i m τ 2 ) ( τ 1 i n τ 2 ) ( m + n ) q 2 ( 2 2 τ 3 q ) 5 / 3 ] J 0 ( 2 r q ) q d q ,
I q ( r , z ) = n = 1 N n = 1 N i m n 2 ( m + n ) τ 2 exp ( 2 r 2 τ 2 ) ,
I δ ( r , z ) = n = 1 N n = 1 N i m n 2 ( m + n ) τ 2 exp ( 2 r 2 τ 2 ) ( 1 + τ 3 2 δ 2 τ 2 ) × { 2 exp ( 2 r 2 τ 2 ) + ( 2 r 2 τ 2 1 ) [ chi ( 2 r 2 τ 2 ) + shi ( 2 r 2 τ 2 ) ln ( 2 τ 3 2 τ 4 r 2 ) ] } .

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