Abstract

This review paper addresses typical mistakes and omissions that involve theoretical research and modeling of optical propagation through atmospheric turbulence. We discuss the disregard of some general properties of narrow-angle propagation in refractive random media, the careless use of simplified models of turbulence, and omissions in the calculations of the second moment of the propagating wave. We also review some misconceptions regarding short-exposure imaging, propagation of polarized waves, and calculations of the scintillation index of the beam waves.

© 2012 Optical Society of America

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  1. S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1989).
  2. M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, Vol. XXXII, E. Wolf, eds. (North-Holland, 1993), pp. 205–268.
  3. V. I. Gelfgat, “Reflection in a scattering medium,” Sov. Phys. Acoust. 22, 65–66 (1976).
  4. V. P. Lukin and M. I. Charnotskii, “The reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12, 602–605 (1982).
    [CrossRef]
  5. M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
    [CrossRef]
  6. V. P. Lukin and M. I. Charnotskii, “Reverse wave propagation in a randomly-inhomogeneous medium,” Russ. Phys. J. 28, 894–904 (1985).
  7. V. U. Zavorotnyi, “Origin of intensity fluctuations in the image of an incoherent object observed through a turbulent medium,” Opt. Spectrosc. 65, 575–576 (1988).
  8. M. I. Charnotskii, “Turbulence effects on the imaging of an object with a sharp edge: asymptotic technique and aperture-plane statistics,” J. Opt. Soc. Am. A 13, 1094–1105 (1996).
    [CrossRef]
  9. M. I. Charnotskii, “Asymptotic analysis of flux fluctuation averaging and finite-size source scintillations in random media,” Waves Random Media 1, 223–243 (1991).
    [CrossRef]
  10. M. I. Charnotskii, “Coupling turbulence-distorted wave front to fiber: Wave propagation theory perspective,” Proc. SPIE 7814, 78140I1 (2010).
  11. L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271–291 (2001).
  12. M. I. Charnotskii, “Coherence of beam arrays propagating in the turbulent atmosphere,” Proc. SPIE 7685, 76850Q (2010).
    [CrossRef]
  13. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  14. S. M. Wandzura, “Meaning of quadratic structure functions,” J. Opt. Soc. Am. 70, 745–747 (1980).
    [CrossRef]
  15. G. J. Baker, “Gaussian beam weak scintillation: low-order turbulence effects and applicability of the Rytov method,” J. Opt. Soc. Am. A 23, 395–417 (2006).
    [CrossRef]
  16. M. I. Charnotskii, “Laser beam propagation in the low-order turbulence: Exact solution,” Proc. SPIE 7324, 734203(2009).
  17. M. I. Charnotskii, “Weak and strong off-axis beam scintillations and beam wander for propagation in turbulence,” Proc. SPIE 7865, 786502 (2010).
  18. M. I. Charnotskii, “Coherence of beam arrays propagating in the turbulent atmosphere,” Proc. SPIE 7685, 76850Q(2010).
    [CrossRef]
  19. M. I. Charnotskii, “Beam scintillation for the ground-to-space propagation,” J. Opt. Soc. Am. A 27, 2169–2187(2010).
    [CrossRef]
  20. M. I. Charnotskii, “Statistics of the point spread function for imaging through turbulence,” Proc. SPIE 8014, 80140W(2011).
    [CrossRef]
  21. M. I. Charnotskii and G. J. Baker, “Practical calculation of the beam scintillation index based on the rigorous asymptotic propagation theory,” Proc. SPIE 8038, 803804 (2011).
    [CrossRef]
  22. M. I. Charnotskii, “Superresolution in dewarped anisoplanatic images,” Appl. Opt. 47, 5110–5116 (2008).
    [CrossRef]
  23. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  24. D. H. Tofsted, “Reanalysis of turbulence effects on short-exposure passive imaging,” Opt. Eng. 50, 016001 (2011).
    [CrossRef]
  25. M. I. Charnotskii, “Anisoplanatic short-exposure imaging in turbulence,” J. Opt. Soc. Am. A 10, 492–501 (1993).
    [CrossRef]

2011 (3)

M. I. Charnotskii, “Statistics of the point spread function for imaging through turbulence,” Proc. SPIE 8014, 80140W(2011).
[CrossRef]

M. I. Charnotskii and G. J. Baker, “Practical calculation of the beam scintillation index based on the rigorous asymptotic propagation theory,” Proc. SPIE 8038, 803804 (2011).
[CrossRef]

D. H. Tofsted, “Reanalysis of turbulence effects on short-exposure passive imaging,” Opt. Eng. 50, 016001 (2011).
[CrossRef]

2010 (5)

M. I. Charnotskii, “Coherence of beam arrays propagating in the turbulent atmosphere,” Proc. SPIE 7685, 76850Q (2010).
[CrossRef]

M. I. Charnotskii, “Weak and strong off-axis beam scintillations and beam wander for propagation in turbulence,” Proc. SPIE 7865, 786502 (2010).

M. I. Charnotskii, “Coherence of beam arrays propagating in the turbulent atmosphere,” Proc. SPIE 7685, 76850Q(2010).
[CrossRef]

M. I. Charnotskii, “Coupling turbulence-distorted wave front to fiber: Wave propagation theory perspective,” Proc. SPIE 7814, 78140I1 (2010).

M. I. Charnotskii, “Beam scintillation for the ground-to-space propagation,” J. Opt. Soc. Am. A 27, 2169–2187(2010).
[CrossRef]

2009 (1)

M. I. Charnotskii, “Laser beam propagation in the low-order turbulence: Exact solution,” Proc. SPIE 7324, 734203(2009).

2008 (1)

2006 (1)

2001 (1)

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271–291 (2001).

1996 (1)

1994 (1)

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
[CrossRef]

1993 (1)

1991 (1)

M. I. Charnotskii, “Asymptotic analysis of flux fluctuation averaging and finite-size source scintillations in random media,” Waves Random Media 1, 223–243 (1991).
[CrossRef]

1988 (1)

V. U. Zavorotnyi, “Origin of intensity fluctuations in the image of an incoherent object observed through a turbulent medium,” Opt. Spectrosc. 65, 575–576 (1988).

1985 (1)

V. P. Lukin and M. I. Charnotskii, “Reverse wave propagation in a randomly-inhomogeneous medium,” Russ. Phys. J. 28, 894–904 (1985).

1982 (1)

V. P. Lukin and M. I. Charnotskii, “The reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12, 602–605 (1982).
[CrossRef]

1980 (1)

1976 (1)

V. I. Gelfgat, “Reflection in a scattering medium,” Sov. Phys. Acoust. 22, 65–66 (1976).

1966 (1)

Al-Habash, M. A.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271–291 (2001).

Andrews, L. C.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271–291 (2001).

Baker, G. J.

M. I. Charnotskii and G. J. Baker, “Practical calculation of the beam scintillation index based on the rigorous asymptotic propagation theory,” Proc. SPIE 8038, 803804 (2011).
[CrossRef]

G. J. Baker, “Gaussian beam weak scintillation: low-order turbulence effects and applicability of the Rytov method,” J. Opt. Soc. Am. A 23, 395–417 (2006).
[CrossRef]

Charnotskii, M. I.

M. I. Charnotskii and G. J. Baker, “Practical calculation of the beam scintillation index based on the rigorous asymptotic propagation theory,” Proc. SPIE 8038, 803804 (2011).
[CrossRef]

M. I. Charnotskii, “Statistics of the point spread function for imaging through turbulence,” Proc. SPIE 8014, 80140W(2011).
[CrossRef]

M. I. Charnotskii, “Coherence of beam arrays propagating in the turbulent atmosphere,” Proc. SPIE 7685, 76850Q(2010).
[CrossRef]

M. I. Charnotskii, “Coupling turbulence-distorted wave front to fiber: Wave propagation theory perspective,” Proc. SPIE 7814, 78140I1 (2010).

M. I. Charnotskii, “Beam scintillation for the ground-to-space propagation,” J. Opt. Soc. Am. A 27, 2169–2187(2010).
[CrossRef]

M. I. Charnotskii, “Weak and strong off-axis beam scintillations and beam wander for propagation in turbulence,” Proc. SPIE 7865, 786502 (2010).

M. I. Charnotskii, “Coherence of beam arrays propagating in the turbulent atmosphere,” Proc. SPIE 7685, 76850Q (2010).
[CrossRef]

M. I. Charnotskii, “Laser beam propagation in the low-order turbulence: Exact solution,” Proc. SPIE 7324, 734203(2009).

M. I. Charnotskii, “Superresolution in dewarped anisoplanatic images,” Appl. Opt. 47, 5110–5116 (2008).
[CrossRef]

M. I. Charnotskii, “Turbulence effects on the imaging of an object with a sharp edge: asymptotic technique and aperture-plane statistics,” J. Opt. Soc. Am. A 13, 1094–1105 (1996).
[CrossRef]

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
[CrossRef]

M. I. Charnotskii, “Anisoplanatic short-exposure imaging in turbulence,” J. Opt. Soc. Am. A 10, 492–501 (1993).
[CrossRef]

M. I. Charnotskii, “Asymptotic analysis of flux fluctuation averaging and finite-size source scintillations in random media,” Waves Random Media 1, 223–243 (1991).
[CrossRef]

V. P. Lukin and M. I. Charnotskii, “Reverse wave propagation in a randomly-inhomogeneous medium,” Russ. Phys. J. 28, 894–904 (1985).

V. P. Lukin and M. I. Charnotskii, “The reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12, 602–605 (1982).
[CrossRef]

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, Vol. XXXII, E. Wolf, eds. (North-Holland, 1993), pp. 205–268.

Fried, D. L.

Gelfgat, V. I.

V. I. Gelfgat, “Reflection in a scattering medium,” Sov. Phys. Acoust. 22, 65–66 (1976).

Gozani, J.

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, Vol. XXXII, E. Wolf, eds. (North-Holland, 1993), pp. 205–268.

Hopen, C. Y.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271–291 (2001).

Kravtsov, Yu. A.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1989).

Lukin, V. P.

V. P. Lukin and M. I. Charnotskii, “Reverse wave propagation in a randomly-inhomogeneous medium,” Russ. Phys. J. 28, 894–904 (1985).

V. P. Lukin and M. I. Charnotskii, “The reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12, 602–605 (1982).
[CrossRef]

Phillips, R. L.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271–291 (2001).

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1989).

Tatarskii, V. I.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1989).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, Vol. XXXII, E. Wolf, eds. (North-Holland, 1993), pp. 205–268.

Tofsted, D. H.

D. H. Tofsted, “Reanalysis of turbulence effects on short-exposure passive imaging,” Opt. Eng. 50, 016001 (2011).
[CrossRef]

Wandzura, S. M.

Zavorotny, V. U.

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, Vol. XXXII, E. Wolf, eds. (North-Holland, 1993), pp. 205–268.

Zavorotnyi, V. U.

V. U. Zavorotnyi, “Origin of intensity fluctuations in the image of an incoherent object observed through a turbulent medium,” Opt. Spectrosc. 65, 575–576 (1988).

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Opt. Eng. (1)

D. H. Tofsted, “Reanalysis of turbulence effects on short-exposure passive imaging,” Opt. Eng. 50, 016001 (2011).
[CrossRef]

Opt. Spectrosc. (1)

V. U. Zavorotnyi, “Origin of intensity fluctuations in the image of an incoherent object observed through a turbulent medium,” Opt. Spectrosc. 65, 575–576 (1988).

Proc. SPIE (7)

M. I. Charnotskii, “Coupling turbulence-distorted wave front to fiber: Wave propagation theory perspective,” Proc. SPIE 7814, 78140I1 (2010).

M. I. Charnotskii, “Coherence of beam arrays propagating in the turbulent atmosphere,” Proc. SPIE 7685, 76850Q (2010).
[CrossRef]

M. I. Charnotskii, “Laser beam propagation in the low-order turbulence: Exact solution,” Proc. SPIE 7324, 734203(2009).

M. I. Charnotskii, “Weak and strong off-axis beam scintillations and beam wander for propagation in turbulence,” Proc. SPIE 7865, 786502 (2010).

M. I. Charnotskii, “Coherence of beam arrays propagating in the turbulent atmosphere,” Proc. SPIE 7685, 76850Q(2010).
[CrossRef]

M. I. Charnotskii, “Statistics of the point spread function for imaging through turbulence,” Proc. SPIE 8014, 80140W(2011).
[CrossRef]

M. I. Charnotskii and G. J. Baker, “Practical calculation of the beam scintillation index based on the rigorous asymptotic propagation theory,” Proc. SPIE 8038, 803804 (2011).
[CrossRef]

Russ. Phys. J. (1)

V. P. Lukin and M. I. Charnotskii, “Reverse wave propagation in a randomly-inhomogeneous medium,” Russ. Phys. J. 28, 894–904 (1985).

Sov. J. Quantum Electron. (1)

V. P. Lukin and M. I. Charnotskii, “The reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12, 602–605 (1982).
[CrossRef]

Sov. Phys. Acoust. (1)

V. I. Gelfgat, “Reflection in a scattering medium,” Sov. Phys. Acoust. 22, 65–66 (1976).

Waves Random Media (3)

M. I. Charnotskii, “Asymptotic analysis of flux fluctuation averaging and finite-size source scintillations in random media,” Waves Random Media 1, 223–243 (1991).
[CrossRef]

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
[CrossRef]

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271–291 (2001).

Other (3)

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1989).

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, Vol. XXXII, E. Wolf, eds. (North-Holland, 1993), pp. 205–268.

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Figures (9)

Fig. 1.
Fig. 1.

Dependence of the normalized extrapolation error on the point position for different exponents α of the power-law structure function (33).

Fig. 2.
Fig. 2.

Dependence of the spherical wave SI for the wavelength 0.63 μm and path length 1 km on the structure constant. Heavy solid line, α=5/3; short dashed line, α=1.4; and long dashed line, α=1.9.

Fig. 3.
Fig. 3.

Dependence of the spherical wave SI for the wavelength 0.63 μm, and path length 1 km on the coherence radius parameter q and “Rytov variance”. Heavy solid line, α=5/3; short dashed line, α=1.4; and long dashed line, α=1.9.

Fig. 4.
Fig. 4.

Turbulent PSF for Kolmogorov spectrum. Dashed line, r11/3 dependence.

Fig. 5.
Fig. 5.

Turbulent PSF for Kolmogorov spectrum (18) and QSF (33).

Fig. 6.
Fig. 6.

Domains of the weak and strong beam spread at the (N,q) plane. Focusing parameter d=1L/F.

Fig. 7.
Fig. 7.

Domains of the weak and strong beam scintillation at the (N,q) plane for various focusing conditions. Focusing parameter d=1L/F.

Fig. 8.
Fig. 8.

Domains of the weak and strong beam scintillation at the (N,q) plane. (a) Vicinity of focus for ground to space path. Parameter θ1 is the ratio of the path length L to the effective thickness of the turbulent atmosphere. (b) Aperture-averaged scintillation from the point source.

Fig. 9.
Fig. 9.

Dependence of normalized structure functions on the separation and Fresnel number N. Solid lines, SE structure functions of [25]. Dashed lines, Fried’s [23] model [Eq. (57)]. Dotted line, LT structure function. Crosses, N=0.1; squares, N=0.3; diamonds, N=1; and unmarked line, N>10.

Equations (59)

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±2ikzu(r,z)+Δu(r,z)+2k2n˜(r,z)u(r,z)=0.
u(r2,z2)=d2r1u(r1,z1)G(r1,z1r2,z2),
G(r1,z1r2,z2)=k2πi|z1z2|exp(ik(r1r2)22|z1z2|).
G(r1,z1r2,z2)=G(r2,z2r1,z1).
P=d2RO(R)I(R)=d2RO(R)|G(0,0R,L)|2.
I(0,0)=d2RO(R)|G(R,L0,0)|2.
u(0,L)=d2rA(r)exp(ikLr2)G(r,00,L).
u(0,L)=Cd2rG(0,lr,0)×A(r)exp[ik(1l+1L)r2]G(r,00,L).
G(r1,z1r3,z3)=d2r2G(r1,z1r2,z2)×G(r2,z2r3,z3).
±ikzγ(R,ρ,z)+R·ργ(R,ρ,z)+k2[n˜(R+ρ2,z)n˜(Rρ2,z)]γ(R,ρ,z)=0.
P(z)=d2Rγ(R,ρ,z)|ρ=0.
zP(z)=0.
d2Ru(R,z1)u*(R,z1)=d2r11d2r2u(r1,z1)u*(r2,z1)d2RG(r1,0R,z2)G*(r2,0R,z2).
d2RG(r1,0R,z2)G*(r2,0R,z2)=δ(r1r2).
d2Rd2ρbI(R,ρ)=0.
d2ρbI(ρ)=0.
G(r1,z1r2,z2)=d2r2G(r1,z1r3,z3)×G*(r3,z3r2,z2).
Φn(κ,z)=0.033Cn2(z)κ11/3.
Φn(κ)=σn2ln38ππexp(κ2ln24).
u(r,z=0)=exp(r22a2)
σI2=8πk20Ldzd2κΦn(κ,z)exp[κ2a2(1+N2)(1zL)2]sin2[κ2a2N2(1+N2)(1zL)2+κ2z2k(1zL)].
N=ka2L.
σI28πk20Ldzd2κΦn(κ,z)sin2[κ2z2k(1zL)]=2.25k7/6L11/601dtCn2(Lt)t5/6(1t)5/6=O(q5/6),
σI28πk20Ldzd2κΦn(κ,z)sin2[κ2L2k(1zL)]=2.25k7/6L11/601dtCn2(Lt)(1t)5/6=O(q5/6),
q=krC2L,rC=[1.46k2L01Cn2(Lt)(1t)5/3dt]3/5
σI28πk2ln3L2a201dtσn2(Lt)D(t)C(t)[C2(t)+D2(t)],C(t)MN+(1t)21+N2,D(t)t(1t)N+N(1t)21+N2.
σI2=π2(kln)301dtσn2(Lt)t2(1t)21M3,M>1>N,σI2=π2(kln)301dtσn2(Lt)(1t)21M3,M>1>1/N,σI2=π2(kln)301dtσn2(Lt)1M,(1>M>1/N)(1>M>N),σI2=ππ4(kln)3σn2(L)NM,1<N<1/M,σI2=ππ4(kln)3σn2(L)1MN,M<N<1.
±2ikzΓ+(Δ1Δ2)R·ρΓ+iπk32H(ρ,z)Γ(R,ρ,z)=0,
H(ρ,z)=4d2κΦn(κ,z)[1cos(κ·ρ)].
Γ(R,ρ,L)=k24π2L2d2R0d2ρ0Γ0(R0,ρ)exp[ikL(RR0)(ρρ0)12D(ρ,ρ0)],
D(ρ,ρ0)=πk220LdzH[ρ0(1zL)+ρzL,z]
D(ρ0)=2(ρ0rC)5/3,
D(ρ0)2(ρ0rC)2,
S(r+ρ)S(r)=(Dxx(ρ)Dxy(ρ)Dxy(ρ)Dyy(ρ)).
S˜(x)=aS(x1)+bS(x2),
S˜(x)=12[(1D(xx1)D(xx2)D(x1x2))S(x1)+(1D(xx2)D(xx1)D(x1x2))S(x2)].
ΔS2=14D(x1x2)[2D(x1x2)(D(x1)D(x2))(D(x1)D(x2))2D2(x1x2)].
D(x)(|x|rC)α,
ΔS2=D(x1)4|1t|α[2|1t|α+2|t|α|1t|α(1|t|α)2|1t|2α],t=x1x2.
12D(ρ)=ρ2rC2=πk20Ldzd2κΦn(κ,z)[1cos(κ·ρ(1zL))].
Φn(κ,z)=A(z)Δδ(κ),0LdzA(z)(1zL)2=1πk2rC2,
2ikzΓ4+21·2Γ4+iπk32[2H(r1,z)+2H(r2,z)H(r1+r2,z)H(r1r2,z)]Γ4(r1,r2,z)=0,
Γ4(r1,r2,z)=u(r1+r22,z)u(r1+r22,z)u*(r1r22,z)u*(r1r22,z).
Φn(κ,z)=Γ(α+1)4π2sin(π2(α1))Cn2(z)κ2α,1<α<2.
I(R)=k24π2L2d2R0d2ρ0Γ0(R0,ρ0)exp[ik2L(R0R)·ρ0+iψ(R0+ρ02,R)iψ(R0ρ02,R)],
I(R)=k24π2L2d2Rd2ρΓ0(R,ρ)exp[ik2L(RR)·ρ12D(ρ)],
I^(κ)=14π2d2RI(R)exp(iκ·R)=I^FS(κ)exp[12D(κLk)],
I^FS(κ)=14π2d2R0Γ0(R0,κLk)exp(iκ·R0)
I(R)=d2R1IFS(R1)P(RR1),
P(R)=14π2d2κexp(12D(κLk)iR·κ).
aRMS2=R2d2RI(R)d2RI(R).
R2d2RI(R)=L24π2k2Δρ[I^FS(kLρ)exp(12D(ρ))]ρ=0.
P(R)(2π)3/2Γ(196)(Lkrc)5/3R11/3.
aST2=aFS2IFS(0)I(0)
2iku1(r,z)z+Δru1+2k2n˜(r,z)u1=2x(u2n˜y),2iku2(r,z)z+Δru2+2k2n˜(r,z)u2=2y(u1n˜x),r=xx^+yy^.
σI(1)2=2π0Ldzz2(1zL)2κ2d2κΦn(κ,z)exp(κ2a2(1zL)2).
σI(1)2=0.60a7/30LdzCn2(z)z2(1zL)2=O(q5/6N7/6).
σI(2)2=0.33k4a10/3[0LdzCn2(z)(1zL)5/3]2=O(q5/3N5/3)
PSFSEFr(r)=14π2d2ρexp[12DSEFr(ρ)+ikLρ·r],DSEFr(ρ)=2rC5/3[ρ5/3Vρ2],

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