Abstract

Analytical formulas are derived for the average intensity, the root-mean-square (rms) angular width, and the M2-factor of Laguerre-Gaussian correlated Schell-model (LGCSM) beam propagating in non-Kolmogorov turbulence. The influence of the beam and turbulence parameters on the LGCSM beam is numerically calculated. It is shown that the quality of the LGCSM beam can be improved by choosing appropriate beam or turbulence parameter values. It is also found that the LGCSM beam has advantage over the Gaussian Schell-model (GSM) beam for reducing the turbulence-induced degradation. Our results will have some theoretical reference value for optical communications.

© 2016 Optical Society of America

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References

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  1. M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
    [Crossref]
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    [Crossref]
  3. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 122 (2008).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  7. L. Guo, Y. Chen, L. Liu, and Y. Cai, “Propagation of a Laguerre–Gaussian correlated Schell-model beam beyond the paraxial approximation,” Opt. Commun. 352, 127–134 (2015).
    [Crossref]
  8. J. Cang, P. Xiu, and X. Liu, “Propagation of Laguerre–Gaussian and Bessel–Gaussian Schell-model beams through paraxial optical systems in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  20. Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008).
    [Crossref] [PubMed]
  21. Z. Song, Z. Liu, K. Zhou, Q. Sun, and S. Liu, “Propagation factors of multi-sinc Schell-model beams in non-Kolmogorov turbulence,” Opt. Express 24(2), 1804–1813 (2016).
    [Crossref] [PubMed]

2016 (1)

2015 (2)

L. Guo, Y. Chen, L. Liu, and Y. Cai, “Propagation of a Laguerre–Gaussian correlated Schell-model beam beyond the paraxial approximation,” Opt. Commun. 352, 127–134 (2015).
[Crossref]

Z. Zhu, L. Liu, F. Wang, and Y. Cai, “Evolution properties of a Laguerre-Gaussian correlated Schell-model beam propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 32(3), 374–380 (2015).
[Crossref] [PubMed]

2014 (4)

2013 (4)

2010 (1)

2008 (2)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 122 (2008).

Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008).
[Crossref] [PubMed]

2005 (1)

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE 5987(598702), 598702 (2005).
[Crossref]

2003 (2)

1999 (1)

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

1993 (1)

I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[Crossref]

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 122 (2008).

Barchers, J. D.

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

Belen’kii, M. S.

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

Brown, J. M.

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

Cai, Y.

Cang, J.

J. Cang, P. Xiu, and X. Liu, “Propagation of Laguerre–Gaussian and Bessel–Gaussian Schell-model beams through paraxial optical systems in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013).
[Crossref]

Chen, R.

Chen, Y.

Dan, Y.

Dogariu, A.

Elias, L. R.

I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[Crossref]

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 122 (2008).

Fugate, R. Q.

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

Golbraikh, E.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE 5987(598702), 598702 (2005).
[Crossref]

Guo, L.

L. Guo, Y. Chen, L. Liu, and Y. Cai, “Propagation of a Laguerre–Gaussian correlated Schell-model beam beyond the paraxial approximation,” Opt. Commun. 352, 127–134 (2015).
[Crossref]

Karis, S. J.

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

Kimel, I.

I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[Crossref]

Kopeika, N. S.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE 5987(598702), 598702 (2005).
[Crossref]

Korotkova, O.

Kumar, A.

Liu, L.

Liu, S.

Liu, X.

J. Cang, P. Xiu, and X. Liu, “Propagation of Laguerre–Gaussian and Bessel–Gaussian Schell-model beams through paraxial optical systems in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013).
[Crossref]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

Liu, Z.

Mei, Z.

Osmon, C. L.

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 122 (2008).

Prabhakar, S.

Reddy, S. G.

Shchepakina, E.

Shirai, T.

Singh, R. P.

Song, Z.

Sun, Q.

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 122 (2008).

Wang, F.

Wolf, E.

Wu, G.

Xiu, P.

J. Cang, P. Xiu, and X. Liu, “Propagation of Laguerre–Gaussian and Bessel–Gaussian Schell-model beams through paraxial optical systems in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013).
[Crossref]

Yuan, Y.

Zhang, B.

Zhao, C.

Zhou, K.

Zhu, S.

Zhu, Z.

Zilberman, A.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE 5987(598702), 598702 (2005).
[Crossref]

IEEE J. Quantum Electron. (1)

I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[Crossref]

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

Opt. Commun. (1)

L. Guo, Y. Chen, L. Liu, and Y. Cai, “Propagation of a Laguerre–Gaussian correlated Schell-model beam beyond the paraxial approximation,” Opt. Commun. 352, 127–134 (2015).
[Crossref]

Opt. Eng. (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 122 (2008).

Opt. Express (6)

Opt. Laser Technol. (1)

J. Cang, P. Xiu, and X. Liu, “Propagation of Laguerre–Gaussian and Bessel–Gaussian Schell-model beams through paraxial optical systems in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013).
[Crossref]

Opt. Lett. (5)

Proc. SPIE (2)

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE 5987(598702), 598702 (2005).
[Crossref]

Other (2)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, 1994).

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Figures (8)

Fig. 1
Fig. 1 The normalized intensity of LGCSM beam with different mode order n and transverse coherence width δ in non-Kolmogorov turbulence at several propagation distances. The calculation parameters are: C ˜ n 2 = 10 14 m 3α , λ = 632.8nm, σ = 1mm, κ = 10, α = 3.8, L0 = 1m, and l0 = 0.01m.
Fig. 2
Fig. 2 The normalized intensity of LGCSM beam in non-Kolmogorov turbulence at propagation distance z = 6km for different power-law exponent α, outer scale L0, inner scale l0, and structure constant C ˜ n 2 . The calculation parameters are: n = 1, and δ = 1.5mm. (a) C ˜ n 2 = 10 14 m 3α , L0 = 1m, and l0 = 0.01m. (b) C ˜ n 2 = 10 14 m 3α , α = 3.8, and l0 = 0.01m. (c) C ˜ n 2 = 10 14 m 3α , α = 3.8, and L0 = 1m. (d) α = 3.8, L0 = 1m, and l0 = 0.01m.
Fig. 3
Fig. 3 The normalized rms angular width of LGCSM beams propagating in non-Kolmogorov turbulence with different mode order n. The calculation parameters are: C ˜ n 2 = 10 15 m 3α , λ = 632.8nm, σ = 0.01m, δ = 0.005m, κ = 10, α = 3.8, L0 = 1m, and l0 = 0.01m.
Fig. 4
Fig. 4 The normalized M2-factor of LGCSM beam at propagation distance z = 6km in non-Kolmogorov turbulence as a function of (a) wavelength λ, (b) beam width σ, and (c) transverse coherence width δ.
Fig. 5
Fig. 5 The normalized M2-factor of LGCSM beam at propagation distance z = 6km in non-Kolmogorov turbulence as a function of α.
Fig. 6
Fig. 6 The normalized M2-factor of LGCSM beam at propagation distance z = 6km in non-Kolmogorov turbulence as a function of C ˜ n 2 .
Fig. 7
Fig. 7 The normalized M2-factor of LGCSM beam with mode order n = 1 propagating in non-Kolmogorov turbulence for different power-law exponent α, outer scale L0 and inner scale l0.
Fig. 8
Fig. 8 The normalized M2-factor of LGCSM beams with different mode order propagating in non-Kolmogorov turbulence.

Equations (33)

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W( ρ 1 ' , ρ 2 ' ,0 )=exp( | ρ 1 ' | 2 + | ρ 2 ' | 2 4 σ 2 | ρ 1 ' ρ 2 ' | 2 2 δ 2 ) L n ( | ρ 1 ' ρ 2 ' | 2 2 δ 2 ),
W( ρ 1 , ρ 2 ,z )= k 2 4 π 2 z 2 - - - - W( ρ 1 ' , ρ 2 ' ,0 )exp[ ik 2z ( ρ 1 ' ρ 1 ) 2 + ik 2z ( ρ 2 ' ρ 2 ) 2 ] ×exp Ψ( ρ 1 ' , ρ 1 )+ Ψ * ( ρ 2 ' , ρ 2 ) d 2 ρ 1 ' d 2 ρ 2 ' ,
exp Ψ( ρ 1 ' ,ρ )+ Ψ * ( ρ 2 ' ,ρ ) =exp{ 4 π 2 k 2 z 0 1 0 κ Φ n ( κ,α )[ 1 J 0 ( κξ| ρ 1 ' ρ 2 ' | ) ]dκdξ },
J 0 ( κξ| ρ 1 ' ρ 2 ' | )~1 1 4 ( κξ| ρ 1 ' ρ 2 ' | ) 2 ,
Φ n ( κ,α )=A( α ) C ˜ n 2 exp[ ( κ 2 / κ m 2 ) ] κ 2 + κ 0 2 ,0κ<,3<α<4.
c( α )= [ Γ( 5α/2 )A( α ) 2π /3 ] [ 1/ ( α5 ) ] ,
A( α )=Γ( α1 ) cos( απ /2 ) / 4 π 2 ,
exp Ψ( ρ 1 ' ,ρ )+ Ψ * ( ρ 2 ' ,ρ ) =exp( 1 3 π 2 k 2 z | ρ 1 ' ρ 2 ' | 2 T ),
T= 0 κ 3 Φ n ( κ,α )dκ = A( α ) C ˜ n 2 2 κ m 2α βexp( κ 0 2 κ m 2 )Γ( 2 α 2 , κ 0 2 κ m 2 )2 κ 0 4α α2 ,
I( x,y,z )=W( ρ,ρ,z ) = k 2 4 z 2 g=0 n h=0 2g j=0 2n2g m=0 2gh p=0 2n2gj s=0 [ h 2 ] t=0 [ m 2 ] u=0 [ j 2 ] v=0 [ p 2 ] ( n g )( 2g h )( 2n2g j )( 2gh m )( 2n2gj p ) × ( 1 ) hs+t+ju+v+n 2 h2n+j 2 3n i h+2sm+2tj+2up+2v δ h+2sj+2u f 3 m2t+p2v × f 4 h+2sm+2tj+2up+2v2 2 1 f 1 ( 1 1 f 1 δ 2 ) 2gh+2n2gj 2 1 n! h! s!( h2s )! m! t!( m2t )! j! u!( j2u )! × p! v!( p2v )! exp[ ( k 2 4 f 1 z 2 + f 5 2 4 f 4 )( x 2 + y 2 ) ] H h2s+m2t ( i f 5 x 2 f 4 ) H j2u+p2v ( i f 5 y 2 f 4 ) × H 2ghm ( ikx z 2 f 1 2 δ 2 f 1 ) H 2n2gjp ( iky z 2 f 1 2 δ 2 f 1 ),
f 1 = 1 4 σ 2 + 1 2 δ 2 + ik 2z + π 2 k 2 zT 3 , f 2 = 1 δ 2 + 2 3 π 2 k 2 zT, f 3 = f 2 2 f 1 2 δ 2 f 1 , f 4 = 1 4 σ 2 + 1 2 δ 2 ik 2z + π 2 k 2 zT 3 f 2 2 4 f 1 , f 5 = ik f 2 2 f 1 z ik z .
L n m ( ρ 2 )= ( 1 ) n 2 2n n! r=0 n ( n r ) H 2r ( x ) H 2( nr ) ( y ),
H n ( x+y )= 2 n 2 k=0 n ( n k ) H nk ( 2 x ) H k ( 2 y ) ,
exp[ ( xy ) 2 ] H n ( αx )dx= π ( 1 α 2 ) n 2 H n ( αy 1 α 2 ),
H n ( x )= k=0 [ n 2 ] ( 1 ) k n! k!( n2k )! ( 2x ) n2k ,
x n exp[ ( xb ) 2 ]dx = ( 2i ) n π H n ( ib ).
ρ ' = ρ 1 ' + ρ 2 ' 2 , ρ d ' = ρ 1 ' ρ 2 ' ,ρ= ρ 1 + ρ 2 2 , ρ d = ρ 1 ρ 2 .
W( ρ, ρ d ,z )= ( k 2πz ) 2 W( ρ ' , ρ d ' ,0 )exp{ ik z [ ( ρ ρ ' )( ρ d ρ d ' ) ]H( ρ d , ρ d ' ,z ) } d 2 ρ ' d 2 ρ d ' ,
W( ρ, ρ d ,z )= ( 1 2π ) 2 W( ρ '' , ρ d + z k κ d ,0 ) d 2 ρ '' d 2 κ d ×exp[ iρ κ d +i ρ '' κ d π 2 k 2 z 3 ( z 2 k 2 κ d 2 +3 z k κ d ρ d +3 ρ d 2 )T ],
W( ρ '' , ρ d + z k κ d ,0 )=exp[ 1 2 σ 2 ρ ''2 ( 1 8 σ 2 + 1 2 δ 2 ) ( ρ d + z k κ d ) 2 ] L n [ 1 2 δ 2 ( ρ d + z k κ d ) 2 ].
h( ρ,θ,z )= ( k 2π ) 2 W( ρ, ρ d ,z )exp( ikθ ρ d ) d 2 ρ d ,
h( ρ,θ,z )= k 2 16 π 4 2π σ 2 L n [ 1 2 δ 2 ( ρ d + z k κ d ) 2 ] ×exp( a ρ d 2 b κ d 2 c ρ d κ d ikθ ρ d iρ κ d ) d 2 κ d d 2 ρ d ,
a= 1 8 σ 2 + 1 2 δ 2 + π 2 k 2 zT,b= z 2 8 k 2 σ 2 + z 2 2 k 2 δ 2 + σ 2 2 + π 2 z 3 T 3 ,c= z 4k σ 2 + z k δ 2 + π 2 k z 2 T.
x n 1 y n 2 θ x m 1 θ y m 2 = 1 P x n 1 y n 2 θ x m 1 θ y m 2 h( ρ,θ,z ) d 2 ρ d 2 θ ,
P= h( ρ,θ,z ) d 2 ρ d 2 θ .
ρ 2 = z 2 k 2 [ 1 2 σ 2 + 2 δ 2 ( 1+n ) ]+2 σ 2 + 4 π 2 z 3 T 3 ,
θ 2 = 1 k 2 [ 1 2 σ 2 + 2 δ 2 ( 1+n ) ]+4 π 2 zT,
ρθ = z k 2 [ 1 2 σ 2 + 2 δ 2 ( 1+n ) ]2 π 2 z 2 T.
θ N ( z ) ( | θ θ | 2 ) 1 2 = ( θ 2 ) 1 2 = { 1 k 2 [ 1 2 σ 2 + 2 δ 2 ( 1+n ) ]+4 π 2 zT } 1 2 ,
M 2 ( z )=k ( ρ 2 θ 2 ρθ 2 ) 1 2 =k{ [ z 2 2 k 2 σ 2 + 2 z 2 k 2 δ 2 ( 1+n )+2 σ 2 + 4 π 2 z 3 T 3 ][ 1 2 k 2 σ 2 + 2 k 2 δ 2 ( 1+n )+4 π 2 zT ] [ z 2 k 2 σ 2 + 2z k 2 δ 2 ( 1+n )+2 π 2 z 2 T ] 2 } 1 2 .
I( x,y,z )=W( ρ,ρ,z ) = k 2 4 π 2 z 2 exp[ ( x 1 '2 + y 1 '2 )+( x 2 '2 + y 2 '2 ) 4 σ 2 ( x 1 ' x 2 ' ) 2 + ( y 1 ' y 2 ' ) 2 2 δ 2 ] ×exp{ ik 2z [ ( x 1 ' x ) 2 + ( y 1 ' y ) 2 ]+ ik 2z [ ( x 2 ' x ) 2 + ( y 2 ' y ) 2 ] } L n ( ( x 1 ' x 2 ' ) 2 + ( y 1 ' y 2 ' ) 2 2 δ 2 ) ×exp{ 1 3 π 2 k 2 zT[ ( x 1 ' x 2 ' ) 2 + ( y 1 ' y 2 ' ) 2 ] }d x 1 ' d x 2 ' d y 1 ' d y 2 ' .
I( x,y,z )=W( ρ,ρ,z ) = k 2 4 π 2 z 2 d x 2 ' d y 2 ' g=0 n h=0 2g j=0 2n2g ( n g )( 2g h )( 2n2g j ) ( 1 ) n 2 3n n! H h ( x 2 ' δ ) H j ( y 2 ' δ ) ×exp( x 2 '2 + y 2 '2 4 σ 2 + x 2 '2 y 2 '2 2 δ 2 )exp[ ik 2z ( x 2 '2 2 x 2 ' x+ y 2 ' 2 y 2 ' y ) ]exp[ 1 3 π 2 k 2 zT( x 2 '2 y 2 '2 ) ] × H 2gh ( x 1 ' δ )exp[ f 1 x 1 '2 +( f 2 x 2 ' + ikx z ) x 1 ' ]d x 1 ' × H 2n2gj ( y 1 ' δ )exp[ f 1 y 1 '2 +( f 2 y 2 ' + iky z ) y 1 ' ]d y 1 ' .
I( x,y,z )=W( ρ,ρ,z ) = k 2 4 π 2 z 2 g=0 n h=0 2g j=0 2n2g m=0 2gh p=0 2n2gj ( n g )( 2g h )( 2n2g j )( 2gh m )( 2n2gj p ) 2 h+2nj 2 3n × 1 f 1 ( 1 ) n n! ( 1 1 f 1 δ 2 ) h+2nj 2 exp[ k 2 4 f 1 z 2 ( x 2 + y 2 ) ] H 2ghm ( ikx z 2 f 1 2 δ 2 f 1 ) × H 2n2gjp ( iky z 2 f 1 2 δ 2 f 1 ) H h ( x 2 ' δ ) H m ( f 3 x 2 ' )exp[ f 4 x 2 '2 + f 5 x x 2 ' ]d x 2 ' × H h ( y 2 ' δ ) H p ( f 3 y 2 ' )exp[ f 4 y 2 '2 + f 5 y y 2 ' ]d y 2 ' .

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