Abstract

The propagation of an elegant Hermite-Gaussian beam (EHGB) in turbulent atmosphere is investigated. Analytical propagation formulae for the average intensity and effective beam size of an EHGB in turbulent atmosphere are derived based on the extended Huygens-Fresnel integral. The corresponding results of a standard Hermite-Gaussian beam (SHGB) in turbulent atmosphere are also derived for the convenience of comparison. The intensity and spreading properties of EHGBs and SHGBs in turbulent atmosphere are studied numerically and comparatively. It is found that the propagation properties of EHGBs and SHGBs are much different from their properties in free space, and the EHGB and SHGB with higher orders are less affected by the turbulence. What’s more, the SHGB spreads more rapidly than the EHGB in turbulent atmosphere under the same conditions. Our results will be useful in long-distance free-space optical communications.

© 2009 OSA

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References

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2008 (4)

2007 (6)

2006 (2)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[CrossRef]

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
[CrossRef] [PubMed]

2005 (3)

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005).
[CrossRef]

H. T. Eyyuboğlu, “Hermite-cosine-Gaussian laser beam and its propagation characteristics in turbulent atmosphere,” J. Opt. Soc. Am. A 22(8), 1527–1535 (2005).
[CrossRef]

Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. 245(1-6), 21–26 (2005).
[CrossRef]

2004 (1)

2003 (1)

2002 (2)

Y. Cai and Q. Lin, “The elliptical Hermite-Gaussian beam and its propagation through paraxial systems,” Opt. Commun. 207, 139–147 (2002).

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. 41(5), 1097–1103 (2002).
[CrossRef]

2001 (2)

O. Mata-Mendez and F. Chavez-Rivas, “Diffraction of Gaussian and Hermite-Gaussia beams by finite gratings,” J. Opt. Soc. Am. A 18(3), 537–545 (2001).
[CrossRef]

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun. 191(3-6), 173–179 (2001).
[CrossRef]

2000 (1)

B. Lu and H. Ma, “A comparative study of elegant and standard Hermite-Gaussian beams,” Opt. Commun. 174(1-4), 99–104 (2000).
[CrossRef]

1999 (1)

H. Laabs, C. Gao, and H. Weber, “Twisting of three-dimensional Hermite-Gaussian beams,” J. Mod. Opt. 46, 709–719 (1999).

1998 (2)

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).
[CrossRef]

H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998).
[CrossRef]

1997 (2)

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997).
[CrossRef]

A. Belmonte, A. Comerón, J. A. Rubio, J. Bará, and E. Fernández, “Atmospheric-turbulence-induced power-fade statistics for a multiaperture optical receiver,” Appl. Opt. 36(33), 8632–8638 (1997).
[CrossRef] [PubMed]

1990 (1)

1986 (1)

E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. 3(4), 465–469 (1986).
[CrossRef]

1985 (1)

1980 (1)

1979 (1)

1977 (1)

1973 (1)

1967 (1)

Z. I. Feizulin and Y. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10(1), 33–35 (1967).
[CrossRef]

Alavinejad, M.

M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).
[CrossRef]

Bará, J.

Baykal, Y.

Belmonte, A.

Cai, Y.

Carter, W. H.

Chavez-Rivas, F.

Chen, C.

Chen, X.

Chen, Y.

Chen, Y. F.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997).
[CrossRef]

Chen, Z.

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9(12), 1123–1130 (2007).
[CrossRef]

Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. 245(1-6), 21–26 (2005).
[CrossRef]

Chu, X.

Comerón, A.

Dogariu, A.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005).
[CrossRef]

Eyyuboglu, H. T.

Feizulin, Z. I.

Z. I. Feizulin and Y. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10(1), 33–35 (1967).
[CrossRef]

Felsen, L. B.

Fernández, E.

Gao, C.

H. Laabs, C. Gao, and H. Weber, “Twisting of three-dimensional Hermite-Gaussian beams,” J. Mod. Opt. 46, 709–719 (1999).

Ghafary, B.

M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).
[CrossRef]

Gilchrest, Y. V.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. 41(5), 1097–1103 (2002).
[CrossRef]

Guo, H.

Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. 245(1-6), 21–26 (2005).
[CrossRef]

Gutierrez-Vega, J. C.

He, S.

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
[CrossRef] [PubMed]

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[CrossRef]

Huang, T. M.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997).
[CrossRef]

Ji, X.

Kao, C. F.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997).
[CrossRef]

Kashani, F. D.

M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).
[CrossRef]

Kojima, T.

Korotkova, O.

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
[CrossRef]

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005).
[CrossRef]

Kravtsov, Y. A.

Z. I. Feizulin and Y. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10(1), 33–35 (1967).
[CrossRef]

Laabs, H.

H. Laabs, C. Gao, and H. Weber, “Twisting of three-dimensional Hermite-Gaussian beams,” J. Mod. Opt. 46, 709–719 (1999).

H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998).
[CrossRef]

Li, X.

Lin, Q.

Y. Cai and Q. Lin, “Decentered elliptical Hermite–Gaussian beam,” J. Opt. Soc. Am. A 20(6), 1111–1119 (2003).
[CrossRef]

Y. Cai and Q. Lin, “The elliptical Hermite-Gaussian beam and its propagation through paraxial systems,” Opt. Commun. 207, 139–147 (2002).

Lu, B.

Luk, K. M.

Ma, H.

B. Lu and H. Ma, “A comparative study of elegant and standard Hermite-Gaussian beams,” Opt. Commun. 174(1-4), 99–104 (2000).
[CrossRef]

Macon, B. R.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. 41(5), 1097–1103 (2002).
[CrossRef]

Mata-Mendez, O.

Noriega-Manez, R. J.

Piper, J. A.

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun. 191(3-6), 173–179 (2001).
[CrossRef]

Plonus, M. A.

Pu, J.

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9(12), 1123–1130 (2007).
[CrossRef]

Qiu, Y.

Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. 245(1-6), 21–26 (2005).
[CrossRef]

Rubio, J. A.

Saghafi, S.

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun. 191(3-6), 173–179 (2001).
[CrossRef]

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).
[CrossRef]

Salem, M.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005).
[CrossRef]

Sheppard, C. J. R.

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun. 191(3-6), 173–179 (2001).
[CrossRef]

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).
[CrossRef]

Shin, S. Y.

Siegman, A. E.

Tang, H.

Wang, C. L.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997).
[CrossRef]

Wang, S. C.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997).
[CrossRef]

Wang, S. C. H.

Weber, H.

H. Laabs, C. Gao, and H. Weber, “Twisting of three-dimensional Hermite-Gaussian beams,” J. Mod. Opt. 46, 709–719 (1999).

Wolf, E.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005).
[CrossRef]

Young, C. Y.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. 41(5), 1097–1103 (2002).
[CrossRef]

Yu, P. K.

Zauderer, E.

E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. 3(4), 465–469 (1986).
[CrossRef]

Zheng, X.

Zhou, G.

Zhu, K.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[CrossRef]

IEEE J. Quantum Electron. (1)

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997).
[CrossRef]

J. Mod. Opt. (1)

H. Laabs, C. Gao, and H. Weber, “Twisting of three-dimensional Hermite-Gaussian beams,” J. Mod. Opt. 46, 709–719 (1999).

J. Opt. A, Pure Appl. Opt. (1)

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9(12), 1123–1130 (2007).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (8)

Opt. Commun. (6)

Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. 245(1-6), 21–26 (2005).
[CrossRef]

H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998).
[CrossRef]

Y. Cai and Q. Lin, “The elliptical Hermite-Gaussian beam and its propagation through paraxial systems,” Opt. Commun. 207, 139–147 (2002).

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).
[CrossRef]

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun. 191(3-6), 173–179 (2001).
[CrossRef]

B. Lu and H. Ma, “A comparative study of elegant and standard Hermite-Gaussian beams,” Opt. Commun. 174(1-4), 99–104 (2000).
[CrossRef]

Opt. Eng. (1)

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. 41(5), 1097–1103 (2002).
[CrossRef]

Opt. Express (5)

Opt. Lasers Eng. (1)

M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).
[CrossRef]

Opt. Lett. (2)

Radiophys. Quantum Electron. (1)

Z. I. Feizulin and Y. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10(1), 33–35 (1967).
[CrossRef]

Waves Random Complex Media (1)

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005).
[CrossRef]

Other (4)

I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple apertures,” in Free-Space Laser Communication Technologies IX, G. Mecherle, ed., Proc. SPIE 2990, 102–113 (1997).

Y. E. Yenice, and B. G. Evans, “Adaptive beam-size control for ground-to-space laser communications,” in Free-Space Laser Communication Technologies X, G. Mecherle, ed., Proc. SPIE 3266, 221–230 (1998).

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

A. E. Siegman, LASERS, University Science Books, Mill Valley, 1986.

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Figures (6)

Fig. 1
Fig. 1

Normalized intensity distribution of an EHGB for different values of m with m = n at z = 0 (a) m = 1, (b) m = 2, (c) m = 3.

Fig. 2
Fig. 2

Normalized intensity distribution of a SHGB for different values of m with m = n and w0=0.02m at z = 0 (a) m = 1, (b) m = 2, (c) m = 3.

Fig. 3
Fig. 3

Normalized intensity distribution of an EHGB with m = n = 2 at several different propagation distances in free space (a) z = 1km, (b) z = 3km, (c) z = 5km, (d) z = 10km.

Fig. 4
Fig. 4

Normalized intensity distribution of an EHGB with m = n = 2 at several different propagation distances in turbulent atmosphere with Cn2=1014m2/3 (a) z = 1km, (b) z = 1.5km, (c) z = 2km, (d) z = 3km.

Fig. 5
Fig. 5

Normalized intensity distribution of an EHGB for different values of m with m = n at z = 3km in turbulent atmosphere with Cn2=1014m2/3 (a) m = 1, (b) m = 3, (c) m = 5, (d) m = 8.

Fig. 6
Fig. 6

Effective beam size of an EHGB (solid lines) and an SHGB (dotted line) with m = n = 2 and λ=632.8nmversus the propagation distance z in turbulent atmosphere for different values of the structure constant (a) Cn2=1013m2/3 (b) Cn2=1014m2/3 (c) Cn2=1015m2/3 (d) Cn2=0 (free space).

Equations (19)

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

E(x1,y1,0)=Hm(x1w0)Hn(y1w0)exp(x12w02y12w02), 
E(x1,y1,0)=Hm(2x1w0)Hn(2y1w0)exp(x12w02y12w02). 
I(px,py,z)=k24π2z2E(x1,y1,0)E*(x2,y2,0)exp[ik2z(x1px)2ik2z(y1py)2]                        ×exp[ik2z(x2px)2+ik2z(y2py)21ρ02(x1x2)21ρ02(y1y2)2]dx1dx2dy1dy2,
<I(px,py,z)>=k24π2z2w02πe1(11e1)m/2exp(k2w02px24e1z2)j1=0[m2](1)j1m!j1!(m2j1)!l1=0[m2](1)l1m!l1!(m2l1)!t1=0m2l1(m2l1t1)                         ×(2)2m2j12l1(g1)t1(h1x)m2l1t1(m+t12j1)!exp(f1x2e2)(f1xe2)m+t12j1πe2q1=0[m+t12j12]1q1!(m+t12j12q1)!                       ×(e24f1x2)q1w02πe1(11e1)n/2exp(k2w02py24e1z2)j'1=0[n2](1)j'1n!j'1!(n2j'1)!l'1=0[n2](1)l'1n!l'1!(n2l'1)!t'1=0n2l'1(n2l'1t'1)(2)2n2j'12l'1                       ×(g1y)t'1(h1y)n2l'1t'1(n+t'12j'1)!exp(f1y2e2)(f1ye2)n+t'12j'1πe2q'1=0[n+t'12j'12]1q'1!(n+t'12j'12q'1)!(e24f1y2)q'1,
e1=1ikw022z+w02ρ02,  g1=(1e12e1)1/2w02ρ02, h1x=ikw0px2z(1e12e1)1/2, h1y=ikw0py2z(1e12e1)1/2e2=1+ik2zw02+w02ρ02w04e1ρ04,   f1x=ikw0px2zw03ikpx2ze1ρ02,   f1y=ikw0py2zw03ikpy2ze1ρ02.       
Hn(l)=k=0[n2](1)kn!k!(n2k)!(2l)n2k,                 
exp[(xy)22u]Hn(x)dx=2πu(12u)n/2Hn[y(12u)1/2],  
+xnexp(px2+2qx)dx=n!exp(q2p)(qp)nπpk=0[n2]1k!(n2k)!(p4q2)k.  
Wxz(z)=2x2<I(x,y,z)>dxdy<I(x,y,z)>dxdy.   
Wxz(z)=A1A2,
A1=k24π2z2w02πe1(11e1)m/2j1=0[m2](1)j1m!j1!(m2j1)!l1=0[m2](1)l1m!l1!(m2l1)!t1=0m2l1(m2l1t1)       ×(2)2m2j12l1(g1)t1(m+t12j1)!πe2q1=0[m+t12j12]1q1!(m+t12j12q1)!(1e2)m+t12j1       ×(e24)q1(f2)m+t12j12q1(h2)m2l1t1×Γ(2m2j12q12l1+32)(k2w024e1z2f22e2)2m2j12q12l1+3,              
A2=k24π2z2w02πe1(11e1)m/2j1=0[m2](1)j1m!j1!(m2j1)!l1=0[m2](1)l1m!l1!(m2l1)!t1=0m2l1(m2l1t1)      ×(2)2m2j12l1(g1)t1(m+t12j1)!πe2q1=0[m+t12j12]1q1!(m+t12j12q1)!(1e2)m+t12j1     ×(e24)q1(f2)m+t12j12q1(h2)m2l1t1×Γ(2m2j12q12l1+12)2(k2w024e1z2f22e2)2m2j12q12l1+1,                 
f2=ikw02zw03ik2ze1ρ02,h2=ikw02z(1e12e1)1/2. 
<I(px,py,z)>=k24π2z2w022πes1(11es1)m/2exp(k2w02px28es1z2)j1=0[m2](1)j1m!j1!(m2j1)!l1=0[m2](1)l1m!l1!(m2l1)!t1=0m2l1(m2l1t1)                          ×(2)2m2j12l1(gs1)t1(hs1x)m2l1t1(m+t12j1)!exp(f3x2es2)(f3xes2)m+t12j1                          ×πes2q1=0[m+t12j12]1q1!(m+t12j12q1)!(es24f3x2)q1w022πes1(11es1)n/2                          ×exp(k2w02py28es1z2)j'1=0[n2](1)j'1n!j'1!(n2j'1)!l'1=0[n2](1)l'1n!l'1!(n2l'1)!t'1=0n2l'1(n2l'1t'1)(2)2n2j'12l'1(gs1)t'1                          ×(hs1y)n2l'1t'1(n+t'12j'1)!exp(f3y2es2)(f3yes2)n+t'12j'1πes2                          ×q'1=0[n+t'12j'12]1q'1!(n+t'12j'12q'1)!(es24f3y2)q'1
es1=12ikw024z+w022ρ02, gs1=(1es12es1)1/2w022ρ02, hs1x=ikw0px22z(1es12es1)1/2, hs1y=ikw0py22z(1es12es1)1/2es2=12+ikw024z+w022ρ02w044es1ρ04,  f3x=ikw0px22zw03ikpx42zes1ρ02, f3y=ikw0py22zw03ikpy42zes1ρ02. 
Wsxz(z)=As1As2, 
As1=k24π2z2w022πes1(11es1)m/2j1=0[m2](1)j1m!j1!(m2j1)!l1=0[m2](1)l1m!l1!(m2l1)!t1=0m2l1(m2l1t1)         ×(2)2m2j12l1(gs1)t1(m+t12j1)!πes2q1=0[m+t12j12]1q1!(m+t12j12q1)!         ×(1es2)m+t12j1(es24)q1(f4)m+t12j12q1(hs2)m2l1t1×Γ(2m2j12q12l1+32)(k2w028es1z2f42es2)2m2j12q12l1+3, 
As2=k24π2z2w022πes1(11es1)m/2j1=0[m2](1)j1m!j1!(m2j1)!l1=0[m2](1)l1m!l1!(m2l1)!t1=0m2l1(m2l1t1)       ×(2)2m2j12l1(gs1)t1(m+t12j1)!πes2q1=0[m+t12j12]1q1!(m+t12j12q1)!       ×(1es2)m+t12j1(es24)q1(f4)m+t12j12q1(hs2)m2l1t1×Γ(2m2j12q12l1+12)2(k2w028es1z2f42es2)2m2j12q12l1+1,
f4=ikw022zw03ik42zes1ρ02,hs2=ikw022z(1es12es1)1/2.

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