Abstract

To study the performance of atmospheric optical links by using Hermite–sinusoidal-Gaussian laser beam sources, we derive the log-amplitude and the phase correlation and structure functions of such beams in a turbulent atmosphere. Our formulations correctly reduce to the known higher-order mode correlation and structure functions, which in turn reduce to the fundamental-mode (TEM00-mode) results. Several special cases of our formulation are presented, among which the case involving Hermite–cosh-Gaussian dependence is especially noted, since this case is of interest to us owing to the nature of cosh dependence exhibiting the concentration of the energy in the outer lobes of the beam.

© 2004 Optical Society of America

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References

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  1. C. C. Davis, I. I. Smolyaninov, S. D. Milner, “Flexible optical wireless links and networks,” IEEE Commun. Mag. 41, 51–57 (2003).
    [CrossRef]
  2. H. Izadpanah, T. Elbatt, V. Kukshya, F. Dolezal, B. K. Ryu, “High-availability free space optical and RF hybrid wireless networks,” IEEE Wireless Commun. 10, 45–55 (2003).
    [CrossRef]
  3. H. A. Willebrand, B. S. Ghuman, “Fiber optics without fiber,” IEEE Spectrum 38, 40–45 (2001).
    [CrossRef]
  4. X. Zhu, J. M. Kahn, J. Wang, “Mitigation of turbulence-induced scintillation noise in free-space optical links using temporal-domain detection techniques,” IEEE Photon. Technol. Lett. 15, 623–625 (2003).
    [CrossRef]
  5. X. Zhu, J. M. Kahn, “Performance bounds for coded free-space optical communications through atmospheric turbulence channels,” IEEE Trans. Commun. 51, 1233–1239 (2003).
    [CrossRef]
  6. L. W. Casperson, A. A. Tovar, “Hermite–sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 15, 954–961 (1998).
    [CrossRef]
  7. A. A. Tovar, L. W. Casperson, “Production and propagation of Hermite–sinusoidal-Gaussian laser beams,” J. Opt. Soc. Am. A 15, 2425–2432 (1998).
    [CrossRef]
  8. L. W. Casperson, D. G. Hall, A. A. Tovar, “Sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 14, 3341–3348 (1997).
    [CrossRef]
  9. B. Lü, H. Ma, B. Zhang, “Propagation properties of cosh-Gaussian beams,” Opt. Commun. 164, 165–170 (1999).
    [CrossRef]
  10. B. Lü, S. Luo, “Beam propagation factor of hard-edge diffracted cosh-Gaussian beams,” Opt. Commun. 178, 275–281 (2000).
    [CrossRef]
  11. S. Yu, H. Guo, X. Fu, W. Hu, “Propagation properties of elegant Hermite–cosh-Gaussian laser beams,” Opt. Commun. 204, 59–66 (2002).
    [CrossRef]
  12. D. Zhao, H. Mao, H. Liu, “Propagation of off-axial Hermite–cosh-Gaussian laser beams,” J. Opt. A Pure Appl. Opt. 6, 77–83 (2004).
    [CrossRef]
  13. Y. Baykal, “Correlation and structure functions for multimode-laser-beam incidence in atmospheric turbulence,” J. Opt. Soc. Am. A 4, 817–819 (1987).
    [CrossRef]
  14. A. Ishimaru, “Fluctuations in the parameters of spherical waves propagating in a turbulent atmosphere,” Radio Sci. 4, 295–305 (1969).
    [CrossRef]
  15. I. S. Gradysteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980).

2004

D. Zhao, H. Mao, H. Liu, “Propagation of off-axial Hermite–cosh-Gaussian laser beams,” J. Opt. A Pure Appl. Opt. 6, 77–83 (2004).
[CrossRef]

2003

X. Zhu, J. M. Kahn, J. Wang, “Mitigation of turbulence-induced scintillation noise in free-space optical links using temporal-domain detection techniques,” IEEE Photon. Technol. Lett. 15, 623–625 (2003).
[CrossRef]

X. Zhu, J. M. Kahn, “Performance bounds for coded free-space optical communications through atmospheric turbulence channels,” IEEE Trans. Commun. 51, 1233–1239 (2003).
[CrossRef]

C. C. Davis, I. I. Smolyaninov, S. D. Milner, “Flexible optical wireless links and networks,” IEEE Commun. Mag. 41, 51–57 (2003).
[CrossRef]

H. Izadpanah, T. Elbatt, V. Kukshya, F. Dolezal, B. K. Ryu, “High-availability free space optical and RF hybrid wireless networks,” IEEE Wireless Commun. 10, 45–55 (2003).
[CrossRef]

2002

S. Yu, H. Guo, X. Fu, W. Hu, “Propagation properties of elegant Hermite–cosh-Gaussian laser beams,” Opt. Commun. 204, 59–66 (2002).
[CrossRef]

2001

H. A. Willebrand, B. S. Ghuman, “Fiber optics without fiber,” IEEE Spectrum 38, 40–45 (2001).
[CrossRef]

2000

B. Lü, S. Luo, “Beam propagation factor of hard-edge diffracted cosh-Gaussian beams,” Opt. Commun. 178, 275–281 (2000).
[CrossRef]

1999

B. Lü, H. Ma, B. Zhang, “Propagation properties of cosh-Gaussian beams,” Opt. Commun. 164, 165–170 (1999).
[CrossRef]

1998

1997

1987

1969

A. Ishimaru, “Fluctuations in the parameters of spherical waves propagating in a turbulent atmosphere,” Radio Sci. 4, 295–305 (1969).
[CrossRef]

Baykal, Y.

Casperson, L. W.

Davis, C. C.

C. C. Davis, I. I. Smolyaninov, S. D. Milner, “Flexible optical wireless links and networks,” IEEE Commun. Mag. 41, 51–57 (2003).
[CrossRef]

Dolezal, F.

H. Izadpanah, T. Elbatt, V. Kukshya, F. Dolezal, B. K. Ryu, “High-availability free space optical and RF hybrid wireless networks,” IEEE Wireless Commun. 10, 45–55 (2003).
[CrossRef]

Elbatt, T.

H. Izadpanah, T. Elbatt, V. Kukshya, F. Dolezal, B. K. Ryu, “High-availability free space optical and RF hybrid wireless networks,” IEEE Wireless Commun. 10, 45–55 (2003).
[CrossRef]

Fu, X.

S. Yu, H. Guo, X. Fu, W. Hu, “Propagation properties of elegant Hermite–cosh-Gaussian laser beams,” Opt. Commun. 204, 59–66 (2002).
[CrossRef]

Ghuman, B. S.

H. A. Willebrand, B. S. Ghuman, “Fiber optics without fiber,” IEEE Spectrum 38, 40–45 (2001).
[CrossRef]

Gradysteyn, I. S.

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

Guo, H.

S. Yu, H. Guo, X. Fu, W. Hu, “Propagation properties of elegant Hermite–cosh-Gaussian laser beams,” Opt. Commun. 204, 59–66 (2002).
[CrossRef]

Hall, D. G.

Hu, W.

S. Yu, H. Guo, X. Fu, W. Hu, “Propagation properties of elegant Hermite–cosh-Gaussian laser beams,” Opt. Commun. 204, 59–66 (2002).
[CrossRef]

Ishimaru, A.

A. Ishimaru, “Fluctuations in the parameters of spherical waves propagating in a turbulent atmosphere,” Radio Sci. 4, 295–305 (1969).
[CrossRef]

Izadpanah, H.

H. Izadpanah, T. Elbatt, V. Kukshya, F. Dolezal, B. K. Ryu, “High-availability free space optical and RF hybrid wireless networks,” IEEE Wireless Commun. 10, 45–55 (2003).
[CrossRef]

Kahn, J. M.

X. Zhu, J. M. Kahn, J. Wang, “Mitigation of turbulence-induced scintillation noise in free-space optical links using temporal-domain detection techniques,” IEEE Photon. Technol. Lett. 15, 623–625 (2003).
[CrossRef]

X. Zhu, J. M. Kahn, “Performance bounds for coded free-space optical communications through atmospheric turbulence channels,” IEEE Trans. Commun. 51, 1233–1239 (2003).
[CrossRef]

Kukshya, V.

H. Izadpanah, T. Elbatt, V. Kukshya, F. Dolezal, B. K. Ryu, “High-availability free space optical and RF hybrid wireless networks,” IEEE Wireless Commun. 10, 45–55 (2003).
[CrossRef]

Liu, H.

D. Zhao, H. Mao, H. Liu, “Propagation of off-axial Hermite–cosh-Gaussian laser beams,” J. Opt. A Pure Appl. Opt. 6, 77–83 (2004).
[CrossRef]

Lü, B.

B. Lü, S. Luo, “Beam propagation factor of hard-edge diffracted cosh-Gaussian beams,” Opt. Commun. 178, 275–281 (2000).
[CrossRef]

B. Lü, H. Ma, B. Zhang, “Propagation properties of cosh-Gaussian beams,” Opt. Commun. 164, 165–170 (1999).
[CrossRef]

Luo, S.

B. Lü, S. Luo, “Beam propagation factor of hard-edge diffracted cosh-Gaussian beams,” Opt. Commun. 178, 275–281 (2000).
[CrossRef]

Ma, H.

B. Lü, H. Ma, B. Zhang, “Propagation properties of cosh-Gaussian beams,” Opt. Commun. 164, 165–170 (1999).
[CrossRef]

Mao, H.

D. Zhao, H. Mao, H. Liu, “Propagation of off-axial Hermite–cosh-Gaussian laser beams,” J. Opt. A Pure Appl. Opt. 6, 77–83 (2004).
[CrossRef]

Milner, S. D.

C. C. Davis, I. I. Smolyaninov, S. D. Milner, “Flexible optical wireless links and networks,” IEEE Commun. Mag. 41, 51–57 (2003).
[CrossRef]

Ryu, B. K.

H. Izadpanah, T. Elbatt, V. Kukshya, F. Dolezal, B. K. Ryu, “High-availability free space optical and RF hybrid wireless networks,” IEEE Wireless Commun. 10, 45–55 (2003).
[CrossRef]

Ryzhik, I. M.

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

Smolyaninov, I. I.

C. C. Davis, I. I. Smolyaninov, S. D. Milner, “Flexible optical wireless links and networks,” IEEE Commun. Mag. 41, 51–57 (2003).
[CrossRef]

Tovar, A. A.

Wang, J.

X. Zhu, J. M. Kahn, J. Wang, “Mitigation of turbulence-induced scintillation noise in free-space optical links using temporal-domain detection techniques,” IEEE Photon. Technol. Lett. 15, 623–625 (2003).
[CrossRef]

Willebrand, H. A.

H. A. Willebrand, B. S. Ghuman, “Fiber optics without fiber,” IEEE Spectrum 38, 40–45 (2001).
[CrossRef]

Yu, S.

S. Yu, H. Guo, X. Fu, W. Hu, “Propagation properties of elegant Hermite–cosh-Gaussian laser beams,” Opt. Commun. 204, 59–66 (2002).
[CrossRef]

Zhang, B.

B. Lü, H. Ma, B. Zhang, “Propagation properties of cosh-Gaussian beams,” Opt. Commun. 164, 165–170 (1999).
[CrossRef]

Zhao, D.

D. Zhao, H. Mao, H. Liu, “Propagation of off-axial Hermite–cosh-Gaussian laser beams,” J. Opt. A Pure Appl. Opt. 6, 77–83 (2004).
[CrossRef]

Zhu, X.

X. Zhu, J. M. Kahn, “Performance bounds for coded free-space optical communications through atmospheric turbulence channels,” IEEE Trans. Commun. 51, 1233–1239 (2003).
[CrossRef]

X. Zhu, J. M. Kahn, J. Wang, “Mitigation of turbulence-induced scintillation noise in free-space optical links using temporal-domain detection techniques,” IEEE Photon. Technol. Lett. 15, 623–625 (2003).
[CrossRef]

IEEE Commun. Mag.

C. C. Davis, I. I. Smolyaninov, S. D. Milner, “Flexible optical wireless links and networks,” IEEE Commun. Mag. 41, 51–57 (2003).
[CrossRef]

IEEE Photon. Technol. Lett.

X. Zhu, J. M. Kahn, J. Wang, “Mitigation of turbulence-induced scintillation noise in free-space optical links using temporal-domain detection techniques,” IEEE Photon. Technol. Lett. 15, 623–625 (2003).
[CrossRef]

IEEE Spectrum

H. A. Willebrand, B. S. Ghuman, “Fiber optics without fiber,” IEEE Spectrum 38, 40–45 (2001).
[CrossRef]

IEEE Trans. Commun.

X. Zhu, J. M. Kahn, “Performance bounds for coded free-space optical communications through atmospheric turbulence channels,” IEEE Trans. Commun. 51, 1233–1239 (2003).
[CrossRef]

IEEE Wireless Commun.

H. Izadpanah, T. Elbatt, V. Kukshya, F. Dolezal, B. K. Ryu, “High-availability free space optical and RF hybrid wireless networks,” IEEE Wireless Commun. 10, 45–55 (2003).
[CrossRef]

J. Opt. A Pure Appl. Opt.

D. Zhao, H. Mao, H. Liu, “Propagation of off-axial Hermite–cosh-Gaussian laser beams,” J. Opt. A Pure Appl. Opt. 6, 77–83 (2004).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

B. Lü, H. Ma, B. Zhang, “Propagation properties of cosh-Gaussian beams,” Opt. Commun. 164, 165–170 (1999).
[CrossRef]

B. Lü, S. Luo, “Beam propagation factor of hard-edge diffracted cosh-Gaussian beams,” Opt. Commun. 178, 275–281 (2000).
[CrossRef]

S. Yu, H. Guo, X. Fu, W. Hu, “Propagation properties of elegant Hermite–cosh-Gaussian laser beams,” Opt. Commun. 204, 59–66 (2002).
[CrossRef]

Radio Sci.

A. Ishimaru, “Fluctuations in the parameters of spherical waves propagating in a turbulent atmosphere,” Radio Sci. 4, 295–305 (1969).
[CrossRef]

Other

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

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Figures (2)

Fig. 1
Fig. 1

Propagation geometry.

Fig. 2
Fig. 2

Variance of the log-amplitude fluctuations Bχ(0, 0, L)=σχ2 for the plane wave obtained as the limiting case (when l1=1 and l2=0, Vx=Vy=0, n=m=0, bx=by=0, αs=, ax=ay=0, and Fx=Fy=) of Eq. (26) versus the length L of a horizontal link in weak atmospheric turbulence.

Equations (101)

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un,mOA(sx, sy, z=0)
=An,mHn(axsx+bx)Hm(aysy+by)
×exp[-12k(αxsx2+αysy2)]×exp[-i(Vxsx+Vysy)]×exp(-iϕ),
αx=1/(kαsx2)+i/Fx,
αy=1/(kαsy2)+i/Fy,
un,mFS(p, z)=k exp(ikz)/(2πiz)×-dsx-dsyun,m(sx, sy, z=0)×exp{[ik/(2z)]×[(sx-px)2+(sy-py)2]},
un,mFS,OA(p, z)
=An,mexp(ikz)exp(-iϕ)(1-2iax2zAx/k)n/2×(1-2iay2zAy/k)m/2Ax1/2Ay1/2×exp[-iVx2zAx/(2k)]×exp[-iVy2zAy/(2k)]×exp(-kαxAxpx2/2)exp(-kαyAypy2/2)exp(-iVxAxpx)exp(-iVyAypy)×Hn(β2xpx+β1x)Hm(β2ypy+β1y),
Ax=1/(1+iαxz),
Ay=1/(1+iαyz),
β2x=axAx1/2/{1-iz[(2ax2/k)-αx]}1/2,
β1x=β2x[zaxVx+kbx(1+iαxz)]/(kax).
Bχ(p1, p2, L)=π Re0Ldη0κdκ02πdθ×[G1(p1, p2, η, κ, θ, L)+G2(p1, p2, η, κ, θ, L)]Φn(κ),
G1(p1, p2, η, κ, θ, L)
={-k2/[E(p1, 0)E(p2, 0)]}
×exp(b3xκ2cos2 θ+b3yκ2sin2 θ)
×exp[iγx(px1-px2)κ cos θ
+iγy(py1-py2)κ sin θ]
×E(p1, κ)E(p2, -κ),
G2(p1, p2, η, κ, θ, L)
={k2/[E(p1, 0)E*(p2, 0)]}
×exp(ζ3xκ2cos2 θ
+ζ3yκ2sin2 θ)exp
×[i(γxpx1-γx*px2)κ cos θ
+i(γypy1-γy*py2)κ sin θ]
×exp(ζ2xκ cos θ
+ζ2yκ sin θ)E(p1, κ)E*(p2, κ),
E(p1, κ)=Hn(g4xpx1+g5xκ cos θ+g7x)×Hm(g4ypy1+g5yκ sin θ+g7y),
E(p2, κ)=Hn(g4xpx2+g5xκ cos θ+g7x)×Hm(g4ypy2+g5yκ sin θ+g7y),
γx=(1+iαxη)/(1+iαxL),
b3x=iγx(η-L)/k,
g4x=ax/((1+iαxL)×{1-iL[(2ax2/k)-αx]})1/2,
g5x=g4x(η-L)/k,
g7x=g4x{(LVx/k)+[bx(1+iαxL)/ax]},
ζ2x=[2L(L-η)Vx]/×(k2αsx2{1-(2L/Fx)+(L2/Fx2)+[L2/(k2αsx4)]}),
ζ3x=[Im(γx)](L-η)/k,
Dχ(p1, p2, L)
=π Re0Ldη0κdκ02πdθ[G1(p1, p1, η, κ, θ, L)+G2(p1, p1, η, κ, θ, L)+G1(p2, p2, η, κ, θ, L)+G2(p2, p2, η, κ, θ, L)-2G1(p1, p2, η, κ, θ, L)-2G2(p1, p2, η, κ, θ, L)]Φn(κ).
BS(p1, p2, L)=-π Re0Ldη0κdκ02πdθ×[G1(p1, p2, η, κ, θ, L)-G2(p1, p2, η, κ, θ, L)]Φn(κ).
DS(p1, p2, L)
=-π Re0Ldη0κdκ02πdθ×[G1(p1, p1, η, κ, θ, L)-G2(p1, p1, η, κ, θ, L)+G1(p2, p2, η, κ, θ, L)-G2(p2, p2, η, κ, θ, L)-2G1(p1, p2, η, κ, θ, L)+2G2(p1, p2, η, κ, θ, L)]Φn(κ).
un,mHSG(sx, sy, z=0)
=l1An,mHn(axsx+bx)Hm(aysy+by)×exp[-12k(αxsx2+αysy2)]×exp[-i(Vxsx+Vysy)]exp(-iϕ)+l2An,m×Hn(axsx+bx)Hm(aysy+by)×exp[-12k(αxsx2+αysy2)]×exp[-i(Yxsx+Yysy)]exp(-iϕ)=l1un,mOA(sx, sy, z=0)|Vx,Vy+l2un,mOA(sx, sy, z=0)|Yx,Yy,
un,mFS,HSG(p, z)=l1un,mFS,OA(p, z)|Vx,Vy+l2un,mFS,OA(p, z)|Yx,Yy,
un,mFS,OA(p, z)|Yx,Yy
HN(px, py, L, κx, κy, z)
=HN1(px, py, L, κx, κy, , z)+HN2(px, py, L, κx, κy, z),
HN1(p)=l1fV(p)fVY(p)HV(p),
HN2(p)=l2fY(p)fVY(p)HY(p),
fVY(p)=1/[l1fV(p)+l2fY(p)],
fV(p)=exp{-iVx2L/[2k(1+iαxL)]}×exp{-iVy2L/[2k(1+iαyL)]}×exp{-iVxpx/[(1+iαxL)]}×exp{-iVypy/[(1+iαyL)]}Hn(g4xpx+g7x)Hm×(g4ypy+g7y).
Bχ(p1, p2, L)=π Re0Ldη0κdκ02πdθ×[G1N(p1, p2, η, κ, θ, L)+G2N(p1, p2, η, κ, θ, L)]Φn(κ),
G1N(p1, p2)
=fVY(p1)fVY(p2)[l12fV(p1)fV(p2)G1V+l1l2fV(p1)fY(p2)G1VY+l2l1fY(p1)fV(p2)G1YV+l22fY(p1)fY(p2)G1Y],
G2N(p1, p2)
=fVY(p1)[fVY(p2)]*{|l1|2fV(p1)[fV(p2)]*G2V+l1(l2)*fV(p1)[fY(p2)]*G2VY+l2(l1)*fY(p1)[fV(p2)]*G2YV+|l2|2fY(p1)×[fY(p2)]*G2Y},
G1VY={-k2/[E(p1, 0)EY(p2, 0)]}exp(b3xκ2cos2 θ+b3yκ2sin2 θ)exp{[iγx(px1-px2)+g1xVY]κ cos θ+[iγy(py1-py2)+g1yVY]κ sin θ}E(p1, κ)EY(p2, -κ),
G1YV={-k2/[EY(p1, 0)E(p2, 0)]}exp(b3xκ2cos2 θ+b3yκ2sin2 θ)exp{[iγx(px1-px2)-g1xVY]κ cos θ+[iγy(py1-py2)-g1yVY]κ sin θ}EY(p1, κ)E(p2, -κ),
G2VY=(k2/{E(p1, 0)[EY(p2, 0)]*})exp(ζ3xκ2cos2 θ+ξ3yκ2sin2 θ)exp{[i(γxpx1-γx*px2)+g2xVY]κ cos θ+[i(γypy1-γy*py2)+g2yVY]κ sin θ}E(p1, κ)[EY(p2, κ)]*,
G2YV=(k2/{EY(p1, 0)[E(p2, 0)]*})exp(ζ3xκ2cos2 θ+ζ3yκ2sin2 θ)exp{[i(γxpx1-γx*px2)+(g2xVY)*]κ cos θ+[i(γypy1-γy*py2)+(g2yVY)*]κ sin θ}EY(p1, κ)[E(p2, κ)]*,
g1xVY=i(L-η)(Vx-Yx)/[k(1+iαxL)],
g1yVY=i(L-η)(Vy-Yy)/[k(1+iαyL)],
g2xVY={i(L-η)Vx/[k(1+iαxL)]}+{i(η-L)(Yx)*/[k(1-iαx*L)]},
g2yVY={i(L-η)Vy/[k(1+iαyL)]}+{i(η-L)(Yy)*/[k(1-iαy*L)]},
Dχ(p1, p2, L)
=π Re0Ldη0κdκ02πdθ×[G1N(p1, p1, η, κ, θ, L)+G2N(p1, p1, η, κ, θ, L)+G1N(p2, p2, η, κ, θ, L)+G2N(p2, p2, η, κ, θ, L)-2G1N(p1, p2, η, κ, θ, L)-2G2N(p1, p2, η, κ, θ, L)]Φn(κ)
BS(p1, p2, L)=-π Re0Ldη0κdκ02πdθ×[G1N(p1, p2, η, κ, θ, L)-G2N(p1, p2, η, κ, θ, L)]Φn(κ).
DS(p1, p2, L)
=-π Re0Ldη0κdκ02πdθ×[G1N(p1, p1, η, κ, θ, L)-G2N(p1, p1, η, κ, θ, L)+G1N(p2, p2, η, κ, θ, L)-G2N(p2, p2, η, κ, θ, L)-2G1N(p1, p2, η, κ, θ, L)+2G2N  (p1, p2, η, κ, θ, L)]Φn(κ).
u(p, z)=un,mFS(p, z)exp[ψ(p, z)],
ψ(p, z)=k2/[2πun,mFS(p, z)]Vd3rn1(r)un,mFS(r)×exp(ik|r-r|)/|r-r|
n1(px, py, z)=--exp(iκxpx+iκypy)dZn(κx, κy, z),
exp(ik|r-r|)/|r-r|
exp(ik{(z-z)+[(px-px)2+(py-py)2]/[2(z-z)]})/(z-z).
Ψ(p, L)=0Ldz--H(px, py, L, κx, κy, z)×dZn(κx, κy, z),
H(px, py, L, κx, κy, z)
=k2/[2κ(L-z)un,mFS(p, L)]-dpx-dpy×exp(iκxpx+iκypy)un,mFS(p, z)exp(ik{(z-z)+[(px-px)2+(py-py)2]/[2(z-z)]})/(z-z),
Ψ(p, L)=χ(p, L)+iS(p, L),
χ(p, L)=12[ψ(p, L)+ψ*(p, L)],
S(p, L)=[1/(2i)][ψ(p, L)-ψ*(p, L)].
χ(p, L)=0Ldz--T1(px, py, L, κx, κy, z)dZn×(κx, κy, z),
T1(px, py, L, κx, κy, z)
=12[H(px, py, L, κx, κy, z)+H*(px, py, L, -κx, -κy, z)],
Bχ(p1, p2, L)=χ(p1, L)χ(p2, L),
Bχ(p1, p2, L)
=2π0Ldη-dκx-dκyT1×(px1, py1, L, κx, κy, η)×T1(px2, py2, L, -κx, -κy, η)Φn(κ),
dZn*(κx, κy, z)
=dZn(-κx, -κy, z),
dZn(κx1, κy1, z1)dZn(κx2, κy2, z2)
=δ(κx1+κx2)δ(κy1+κy2)×Fn(κx1, κy1, z1-z2)dκx1dκy1dκx2dκy2,
0Ldz10Ldz2f(η, zd)
=0Ldzdzd/2L-zd/2dη[f(η, zd)-f(η, -zd)]20Ldη0dzdf(η, zd).
0dzdFn(κx1, κy1, zd)=πΦn(κ).
BS(p1, p2, L)=S(p1, L)S(p2, L),
S(p, L)=0Ldz--T2(px, py, L, κx, κy, z)×dZn(κx, κy, z),
T2(px, py, L, κx, κy, z)
=[1/(2i)][H(px, py, L, κx, κy, z)-H*(px, py, L, -κx, -κy, z)].
BS(p1, p2, L)=2π0Ldη-dκx-dκy×T2(px1, py1, L, κx, κy, η)×T2(px2, py2, L, -κx, -κy, η)Φn(κ).
Dχ(p1, p2, L)=[χ(p1, L)-χ(p2, L)]2=Bχ(p1, p1, L)+Bχ(p2, p2, L)-2Bχ(p1, p2, L),
DS(p1, p2, L)=[S(p1, L)-S(p2, L)]2=BS(p1, p1, L)+BS(p2, p2, L)-2BS(p1, p2, L),
Dψ=Dχ+DS.

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