Abstract

This paper studies the propagation characteristics related to higher-order moments of decentered annular beams through non-Kolmogorov turbulence. The analytical expressions for the mean-squared beam width w, the skewness parameter A, and the kurtosis parameter K are derived. The analytical expression for the non-Kolmogorov turbulence parameter T is also derived, and the differences between two non-Kolmogorov turbulence parameters T and T are examined. It is shown that K depends on both T and T, but w and A only depend on T. K decreases monotonically as the spectral power law exponent α increases, but there exist a maximum of w and a minimum of A when α=3.112. When propagation distance z is long enough, A reaches zero, i.e., the intensity distribution is perfectly symmetric about the centroid position axis. In free space, both A>0 and A<0 may appear on propagation. However, it is always A>0 or A<0 on propagation when turbulence is not weak. On the other hand, in turbulence, the maximum of K increases as the decentered parameter increases and the obscure ratio decreases. In particular, when z is long enough, the beam spot is elliptical in free space, but it becomes circular in turbulence.

© 2013 Optical Society of America

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References

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2013 (3)

X. Q. Li and X. L. Ji, “Propagation of higher-order intensity moments through an optical system in atmospheric turbulence,” Opt. Commun. 298–299, 1–7 (2013).
[CrossRef]

Z. Mei, E. Schchepakina, and O. Korotkova, “Propagation of cosine-Gaussian correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21, 17512–17519 (2013).
[CrossRef]

J. P. Deng, X. L. Ji, Z. C. Pu, X. Q. Li, and X. H. Jia, “Influence of polychroism and decentration on spreading of laser beams propagating in non-Kolmogorov turbulence,” Opt. Commun. 301–302, 19–28 (2013).
[CrossRef]

2012 (2)

2011 (3)

2010 (3)

2009 (1)

2008 (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, 026003 (2008).
[CrossRef]

2007 (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

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 (2005).
[CrossRef]

2002 (1)

2000 (1)

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

1995 (1)

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[CrossRef]

1994 (1)

1993 (1)

P. P. Martínez-Herrero, P. M. Mejías, and H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993).
[CrossRef]

1992 (1)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
[CrossRef]

1990 (1)

M. Sanchez, J. L. H. Neira, J. Delgado, and G. Calvo, “Free propagation of high order moments of laser beam intensity distribution,” Proc. SPIE 1397, 635–638 (1990).

1941 (1)

A. N. Kolmogorov, “The local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers,” Dokl. Akad. Nauk SSSR 30, 301–305 (1941).

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, 026003 (2008).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

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

Bandres, M. A.

D. Lopez-Mago, M. A. Bandres, and J. C. Gutiérrez-Vega, “Propagation characteristics of Cartesian parabolic-Gaussian beams,” Proc. SPIE 7789, 77890Q (2010).
[CrossRef]

Baykal, Y.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge University, 1997).

Calvo, G.

M. Sanchez, J. L. H. Neira, J. Delgado, and G. Calvo, “Free propagation of high order moments of laser beam intensity distribution,” Proc. SPIE 1397, 635–638 (1990).

Chu, X. X.

Dan, Y.

Delgado, J.

M. Sanchez, J. L. H. Neira, J. Delgado, and G. Calvo, “Free propagation of high order moments of laser beam intensity distribution,” Proc. SPIE 1397, 635–638 (1990).

Deng, J. P.

J. P. Deng, X. L. Ji, Z. C. Pu, X. Q. Li, and X. H. Jia, “Influence of polychroism and decentration on spreading of laser beams propagating in non-Kolmogorov turbulence,” Opt. Commun. 301–302, 19–28 (2013).
[CrossRef]

Dou, L. Y.

Dragoman, D.

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, 026003 (2008).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

Gerçekcioglu, H.

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 (2005).
[CrossRef]

Guo, H.

Gutiérrez-Vega, J. C.

D. Lopez-Mago, M. A. Bandres, and J. C. Gutiérrez-Vega, “Propagation characteristics of Cartesian parabolic-Gaussian beams,” Proc. SPIE 7789, 77890Q (2010).
[CrossRef]

Ji, G. M.

X. L. Ji, X. Q. Li, and G. M. Ji, “Propagation of second-order moments of general truncated beams in atmospheric turbulence,” New J. Phys. 13, 103006 (2011).
[CrossRef]

Ji, X. L.

X. Q. Li and X. L. Ji, “Propagation of higher-order intensity moments through an optical system in atmospheric turbulence,” Opt. Commun. 298–299, 1–7 (2013).
[CrossRef]

J. P. Deng, X. L. Ji, Z. C. Pu, X. Q. Li, and X. H. Jia, “Influence of polychroism and decentration on spreading of laser beams propagating in non-Kolmogorov turbulence,” Opt. Commun. 301–302, 19–28 (2013).
[CrossRef]

L. Y. Dou, X. L. Ji, and P. Y. Li, “Propagation of partially coherent annular beams with decentered field in turbulence along a slant path,” Opt. Express 20, 8417–8430 (2012).
[CrossRef]

X. L. Ji, X. Q. Li, and G. M. Ji, “Propagation of second-order moments of general truncated beams in atmospheric turbulence,” New J. Phys. 13, 103006 (2011).
[CrossRef]

Jia, X. H.

J. P. Deng, X. L. Ji, Z. C. Pu, X. Q. Li, and X. H. Jia, “Influence of polychroism and decentration on spreading of laser beams propagating in non-Kolmogorov turbulence,” Opt. Commun. 301–302, 19–28 (2013).
[CrossRef]

Jiang, W.

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

Kolmogorov, A. N.

A. N. Kolmogorov, “The local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers,” Dokl. Akad. Nauk SSSR 30, 301–305 (1941).

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 (2005).
[CrossRef]

Korotkova, O.

Kutay, M. A.

H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2000).

Li, P. Y.

Li, X. Q.

J. P. Deng, X. L. Ji, Z. C. Pu, X. Q. Li, and X. H. Jia, “Influence of polychroism and decentration on spreading of laser beams propagating in non-Kolmogorov turbulence,” Opt. Commun. 301–302, 19–28 (2013).
[CrossRef]

X. Q. Li and X. L. Ji, “Propagation of higher-order intensity moments through an optical system in atmospheric turbulence,” Opt. Commun. 298–299, 1–7 (2013).
[CrossRef]

X. L. Ji, X. Q. Li, and G. M. Ji, “Propagation of second-order moments of general truncated beams in atmospheric turbulence,” New J. Phys. 13, 103006 (2011).
[CrossRef]

Li, Y.

Ling, N.

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

Liu, Z.

Lopez-Mago, D.

D. Lopez-Mago, M. A. Bandres, and J. C. Gutiérrez-Vega, “Propagation characteristics of Cartesian parabolic-Gaussian beams,” Proc. SPIE 7789, 77890Q (2010).
[CrossRef]

Luo, B.

Ma, Y.

Martínez-Herrero, P. P.

P. P. Martínez-Herrero, P. M. Mejías, and H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993).
[CrossRef]

Martínez-Herrero, R.

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[CrossRef]

Mei, Z.

Mejías, P. M.

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[CrossRef]

P. P. Martínez-Herrero, P. M. Mejías, and H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993).
[CrossRef]

Neira, J. L. H.

M. Sanchez, J. L. H. Neira, J. Delgado, and G. Calvo, “Free propagation of high order moments of laser beam intensity distribution,” Proc. SPIE 1397, 635–638 (1990).

Ozaktas, H. M.

H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2000).

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, 026003 (2008).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

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

Piquero, G.

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[CrossRef]

Pu, Z. C.

J. P. Deng, X. L. Ji, Z. C. Pu, X. Q. Li, and X. H. Jia, “Influence of polychroism and decentration on spreading of laser beams propagating in non-Kolmogorov turbulence,” Opt. Commun. 301–302, 19–28 (2013).
[CrossRef]

Rao, C.

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

Ruppert, D.

D. Ruppert, Statistics and Data Analysis for Financial Engineering (Springer, 2011).

Sanchez, M.

M. Sanchez, J. L. H. Neira, J. Delgado, and G. Calvo, “Free propagation of high order moments of laser beam intensity distribution,” Proc. SPIE 1397, 635–638 (1990).

Schchepakina, E.

Shchepakina, E.

Si, L.

Siegman, A. E.

A. E. Siegman, Laser (University Science, 1986).

Tao, R.

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, 026003 (2008).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

Weber, H.

P. P. Martínez-Herrero, P. M. Mejías, and H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993).
[CrossRef]

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge University, 1997).

Wu, G.

Yu, S.

Zalevsky, Z.

H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2000).

Zhang, B.

Zhou, P.

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 (2005).
[CrossRef]

Appl. Opt. (1)

Dokl. Akad. Nauk SSSR (1)

A. N. Kolmogorov, “The local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers,” Dokl. Akad. Nauk SSSR 30, 301–305 (1941).

J. Mod. Opt. (1)

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

J. Opt. Soc. Am. A (1)

New J. Phys. (1)

X. L. Ji, X. Q. Li, and G. M. Ji, “Propagation of second-order moments of general truncated beams in atmospheric turbulence,” New J. Phys. 13, 103006 (2011).
[CrossRef]

Opt. Commun. (3)

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[CrossRef]

X. Q. Li and X. L. Ji, “Propagation of higher-order intensity moments through an optical system in atmospheric turbulence,” Opt. Commun. 298–299, 1–7 (2013).
[CrossRef]

J. P. Deng, X. L. Ji, Z. C. Pu, X. Q. Li, and X. H. Jia, “Influence of polychroism and decentration on spreading of laser beams propagating in non-Kolmogorov turbulence,” Opt. Commun. 301–302, 19–28 (2013).
[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, 026003 (2008).
[CrossRef]

Opt. Express (3)

Opt. Lett. (5)

Opt. Quantum Electron. (2)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
[CrossRef]

P. P. Martínez-Herrero, P. M. Mejías, and H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993).
[CrossRef]

Proc. SPIE (4)

M. Sanchez, J. L. H. Neira, J. Delgado, and G. Calvo, “Free propagation of high order moments of laser beam intensity distribution,” Proc. SPIE 1397, 635–638 (1990).

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

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

D. Lopez-Mago, M. A. Bandres, and J. C. Gutiérrez-Vega, “Propagation characteristics of Cartesian parabolic-Gaussian beams,” Proc. SPIE 7789, 77890Q (2010).
[CrossRef]

Other (5)

D. Ruppert, Statistics and Data Analysis for Financial Engineering (Springer, 2011).

H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2000).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge University, 1997).

A. E. Siegman, Laser (University Science, 1986).

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

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Figures (14)

Fig. 1.
Fig. 1.

Changes of turbulence parameters T and T versus the generalized exponent α.

Fig. 2.
Fig. 2.

Changes of (a) the mean-squared beam width w, (b) the skewness parameter A, and (c) the kurtosis parameter K versus the generalized exponent α.

Fig. 3.
Fig. 3.

Changes of the mean-squared beam width w for different β versus the propagation distance z (ε=0.3), (a) in free space and (b) in turbulence (α=3.2, C˜n2=1014m3α).

Fig. 4.
Fig. 4.

Changes of the mean-squared beam width w for different ε versus the propagation distance z (β=8), (a) in free space and (b) in turbulence (α=3.2, C˜n2=1014m3α).

Fig. 5.
Fig. 5.

Changes of the skewness parameter A for different β versus the propagation distance z (ε=0.3), (a) in free space and (b) in turbulence (α=3.2, C˜n2=1014m3α).

Fig. 6.
Fig. 6.

Changes of the skewness parameter A for different ε versus the propagation distance z (β=8), (a) in free space and (b) in turbulence (α=3.2, C˜n2=1014m3α).

Fig. 7.
Fig. 7.

Changes of the skewness parameter A for different C˜n2 versus the propagation distance z (ε=0.3, β=8).

Fig. 8.
Fig. 8.

Changes of the kurtosis parameter K for different β versus the propagation distance z (ε=0.3), (a) in free space and (b) in turbulence (α=3.2, C˜n2=1014m3α).

Fig. 9.
Fig. 9.

Changes of the kurtosis parameter K for different ε versus the propagation distance z (β=8), (a) in free space and (b) in turbulence (α=3.2, C˜n2=1014m3α).

Fig. 10.
Fig. 10.

Changes of the kurtosis parameter K for different C˜n2 versus the propagation distance z (ε=0.3, β=8).

Fig. 11.
Fig. 11.

3D average intensity distributions I(x,y,z) and the contour lines at different propagation distances in free space (ε=0.3, β=8).

Fig. 12.
Fig. 12.

3D average intensity distributions I(x,y,z) and the contour lines at different propagation distances in turbulence (ε=0.3, β=8, α=3.2, C˜n2=1014m3α).

Fig. 13.
Fig. 13.

2D normalized average intensity distributions I(x,0,z)/Imax at different z in free space (ε=0.3, β=8).

Fig. 14.
Fig. 14.

2D normalized average intensity distributions I(x,0,z)/Imax at different z in turbulence (ε=0.3, β=8, α=3.2, C˜n2=1014m3α).

Equations (51)

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

W(ρ,ρd,z)=(k2πz)2W(ρ,ρd,0)exp{ikz[(ρρ)·(ρdρd)]H(ρd,ρd,z)}d2ρd2ρd,
H(ρd,ρd,z)=4π2k2z010κΦn(κ,α){1J0[κ|ξρd+(1ξ)ρd|]}dκdξ,
Φn(κ,α)=A(α)C˜n2exp[(κ2/κm2)](κ2+κ02)α/2,0κ<,3<α<4,
A(α)=14π2Γ(α1)cos(απ2),c(α)=[Γ(5α2)A(α)23π]1/(α5),
H(xd,yd,xd,yd,z)=4π2k2zs=1(1)s+1(s!)222s×010κ2s+1Φn(κ,α){[ξxd+(1ξ)xd]2+[ξyd+(1ξ)yd]2}sdκdξ.
h(ρ,θ,z)=(k2π)2W(ρ,ρd,z)exp(ikθ·ρd)d2ρd,
h(ρ,θ,z)=(k2π)41z2W(ρ,ρd,0)d2ρd2ρdd2ρd×exp{ikz[(ρρ)·(ρdρd)]ikθ·ρdH(ρd,ρd,z)}.
xn1yn2θxm1θym2=1Pxn1yn2θxm1θym2h(ρ,θ,z)d2ρd2θ,
δ(ρdρd)=(k2π)2exp[±ikθ(ρdρd)]d2θ,
f(x)δ(n)(x)dx=(1)nf(n)(0),(n=1,2,3),
xn1yn2θxm1θym2=1PG(ρ,θ,z)h(ρ,θ,0)d2ρd2θ,
G(ρ,θ,z)=(k2π)41z2d2ρd2θd2ρdd2ρdxn1yn2θxm1θym2×exp{ikz[(ρρ)·(ρdρd)]+ikθ·ρdikθ·ρdH(ρd,ρd,z)}.
G(ρ,θ,z)=(i)n1+n2+m1+m2zn1+n2kn1+n2+m1+m2d2ρdd2ρd×exp{ikzρ(ρdρd)+ikθ·ρdH(xd,yd;xd,yd,z)}×δ(n1)(xdxd)δ(n2)(ydyd)δ(m1)(xd)δ(m2)(yd).
x=x0+θx0z,
x2=(x0+θx0z)2+23z3T,
x3=(x0+θx0z)3+2(x0+θx0z)z3T,
x4=(x0+θx0z)4+4(x0+θx0z)2z3T+43z6T2+310k2z5T,
T=π20κ3Φn(κ,α)dκ,
T=π20κ5Φn(κ,α)dκ,
uxν1eμxdx=μνΓ(ν,μu),(Reμ>0,Reν>0),
Γ(α+1,x)=αΓ(α,x)+xαex,
T=π2A(α)C˜n22(α2)[Bκm2αexp(κ02κm2)Γ(2α2,κ02κm2)2κ04α],
T=π2A(α)C˜n24(α2)[Bκm2αexp(κ02κm2)Γ(2α2,κ02κm2)+2(α2)κ04ακm2+κ06α],
B=2κ022κm2+ακm2,
B=(4α)(α2)κm44κ044(α2)κ02κm2.
E(ρ1,0)=(1βx)[t=1Nαtexp(tρ12w02)t=1Nαtexp(tρ12w02)],
W(ρ1,ρ2,0)=E(ρ1,0)E*(ρ2,0).
h(ρ,θ,0)=k2πt=1Nr=1Ns=14αtαr(1)s+1ps[12βx+β2x2β22psβ2ps2(qsx+ikθx)2]×exp[psρ2+1ps(qsρ+ikθ)2],
p1=tw02+rw02,p2=tw02+rw02,p3=tw02+rw02,p4=tw02+rw02,
x0=βπPt=1Nr=1Ns=14αtαr(1)s+1ps2,
θx0=iβπkPt=1Nr=1Ns=14αtαrqs(1)s+1ps2,
x02=πPt=1Nr=1Ns=14(1)s+1αtαr12ps2(1+3β22ps),
θ02=πk2Pt=1Nr=1Ns=14(1)s+1αtαr[Gs2(1+3β22ps)],
x0θx0=iπkPt=1Nr=1Ns=14(1)s+1αtαrqs2ps2(1+3β22ps),
x03=πPt=1Nr=1Ns=14(1)s+1αtαr3β2ps3,
θ03=iπk3Pt=1Nr=1Ns=14(1)s+1αtαr3βqsGs2ps,
x02θx0=iπkPt=1Nr=1Ns=14(1)s+1αtαr3qsβ2ps3,
x0θx02=πk2Pt=1Nr=1Ns=14(1)s+1αtαrβ2ps(13qs2ps2).
x04=πPt=1Nr=1Ns=14(1)s+1αtαr34ps3(1+5β22ps),
θx04=πk4Pt=1Nr=1Ns=14(1)s+1αtαr3psGs24(1+5β22ps),
x03θx0=iπkPt=1Nr=1Ns=14(1)s+1αtαr3qs4ps3(1+5β22ps),
x0θx03=iπk3Pt=1Nr=1Ns=14(1)s+1αtαr3qsGs4ps(1+5β22ps),
x02θx02=πk2Pt=1Nr=1Ns=14(1)s+1αtαr14ps(13qs2ps2)(1+5β22ps).
w=2(xxc)21/2=2(x2xc2)1/2,
A=(xxc)3((xxc)2)3/2=x33xcx2+2xc3(x2xc2)3/2,
K=(xxc)4((xxc)2)2=x44xcx3+6xc2x23xc4(x2xc2)2.
I(x,y,z)=k24z2t=1Nr=1Ns=14(1)s+1αtαrCs2Ds2exp[k2ps4Cs2Ds2z2(x2+y2)]×{1βFx2Ds2+β2Cs2[2βDs2+βF2x22Ds4ρ02(iβkxz+2ρ02)Fx2Ds2+ikxz]},
C1=C2=tw02+1ρ02ik2z,
D1=rw02+1ρ02+ik2z1C12ρ04,
F=ikz(11Cs2ρ02),
ρ0=(k2zT/3)1/2,

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