Abstract

An analytical expression for the log-amplitude correlation function based on the Rytov approximation is derived for spherical wave propagation through an anisotropic non-Kolmogorov refractive turbulent atmosphere. The expression reduces correctly to the previously published analytic expressions for the case of spherical wave propagation through isotropic Kolmogorov turbulence. These results agree well with a wave-optics simulation based on the more general Fresnel approximation, as well as with numerical evaluations, for low-to-moderate strengths of turbulence. These results are useful for understanding the potential impact of deviations from the standard isotropic Kolmogorov spectrum.

© 2013 Optical Society of America

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References

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  2. A. D. Wheelon, Electromagnetic Scintillation (Cambridge University, 2001), Vol. 1, Chaps. 2 and 4.
  3. A. D. Wheelon, Electromagnetic Scintillation (Cambridge University, 2001), Vol. 2, Chap. 3.
  4. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
  5. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6457, 64570T (2007).
  6. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
  7. V. S. Rao Gudimetla, R. B. Holmes, and J. F. Riker, “Analytical expressions for the log-amplitude correlation function for plane wave through anisotropic non-Kolmogorov refractive turbulence,” J. Opt. Soc. Am. A 29, 2622–2627 (2012).
    [CrossRef]
  8. V. S. Rao Gudimetla, R. B. Holmes, C. Smith, and G. Needham, “Analytical expressions for the log-amplitude correlation function of a plane wave through anisotropic atmospheric refractive turbulence,” J. Opt. Soc. Am. A 29, 832–841 (2012).
    [CrossRef]
  9. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer, 1994).
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2012 (2)

2007 (2)

W. M. Hughes and R. B. Holmes, “Pupil-plane imager for scintillometry over long horizontal paths,” Appl. Opt. 46, 7099–7109 (2007).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6457, 64570T (2007).

1995 (1)

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).

1993 (1)

1970 (1)

V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 1, 199–201 (1970).

1967 (1)

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,” Proc. SPIE 6457, 64570T (2007).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

Beresnev, L. A.

M. A. Vorontsov, G. W. Carhart, V. S. R. Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 AMOS Technologies Conference, S. Ryan, ed. (The Maui Economic Development Board, 2010), p. E18.

Carhart, G. W.

M. A. Vorontsov, G. W. Carhart, V. S. R. Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 AMOS Technologies Conference, S. Ryan, ed. (The Maui Economic Development Board, 2010), p. E18.

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,” Proc. SPIE 6457, 64570T (2007).

Flatte, S. M.

Fried, D. L.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1965), p. 973.

Gudimetla, V. S. R.

M. A. Vorontsov, G. W. Carhart, V. S. R. Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 AMOS Technologies Conference, S. Ryan, ed. (The Maui Economic Development Board, 2010), p. E18.

Holmes, R. B.

Hughes, W. M.

Lachinova, S. L.

M. A. Vorontsov, G. W. Carhart, V. S. R. Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 AMOS Technologies Conference, S. Ryan, ed. (The Maui Economic Development Board, 2010), p. E18.

Liu, J.

M. A. Vorontsov, G. W. Carhart, V. S. R. Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 AMOS Technologies Conference, S. Ryan, ed. (The Maui Economic Development Board, 2010), p. E18.

Martin, J.

Needham, G.

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,” Proc. SPIE 6457, 64570T (2007).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

Rao Gudimetla, V. S.

Rehder, K.

M. A. Vorontsov, G. W. Carhart, V. S. R. Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 AMOS Technologies Conference, S. Ryan, ed. (The Maui Economic Development Board, 2010), p. E18.

Riker, J. F.

V. S. Rao Gudimetla, R. B. Holmes, and J. F. Riker, “Analytical expressions for the log-amplitude correlation function for plane wave through anisotropic non-Kolmogorov refractive turbulence,” J. Opt. Soc. Am. A 29, 2622–2627 (2012).
[CrossRef]

M. A. Vorontsov, G. W. Carhart, V. S. R. Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 AMOS Technologies Conference, S. Ryan, ed. (The Maui Economic Development Board, 2010), p. E18.

Roggemann, M. C.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1965), p. 973.

Sasiela, R. J.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer, 1994).

Smith, C.

Stevenson, E.

M. A. Vorontsov, G. W. Carhart, V. S. R. Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 AMOS Technologies Conference, S. Ryan, ed. (The Maui Economic Development Board, 2010), p. E18.

Stribling, B. E.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).

Talanov, V. I.

V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 1, 199–201 (1970).

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,” Proc. SPIE 6457, 64570T (2007).

Vorontsov, M. A.

M. A. Vorontsov, G. W. Carhart, V. S. R. Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 AMOS Technologies Conference, S. Ryan, ed. (The Maui Economic Development Board, 2010), p. E18.

Wang, G.-Y.

Welsh, B. M.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).

Weyrauch, T.

M. A. Vorontsov, G. W. Carhart, V. S. R. Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 AMOS Technologies Conference, S. Ryan, ed. (The Maui Economic Development Board, 2010), p. E18.

Wheelon, A. D.

A. D. Wheelon, Electromagnetic Scintillation (Cambridge University, 2001), Vol. 1, Chaps. 2 and 4.

A. D. Wheelon, Electromagnetic Scintillation (Cambridge University, 2001), Vol. 2, Chap. 3.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

JETP Lett. (1)

V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 1, 199–201 (1970).

Proc. SPIE (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,” Proc. SPIE 6457, 64570T (2007).

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).

Other (6)

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer, 1994).

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1965), p. 973.

M. A. Vorontsov, G. W. Carhart, V. S. R. Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 AMOS Technologies Conference, S. Ryan, ed. (The Maui Economic Development Board, 2010), p. E18.

A. D. Wheelon, Electromagnetic Scintillation (Cambridge University, 2001), Vol. 1, Chaps. 2 and 4.

A. D. Wheelon, Electromagnetic Scintillation (Cambridge University, 2001), Vol. 2, Chap. 3.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

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Figures (4)

Fig. 1.
Fig. 1.

Comparison of the analytical log-amplitude correlation function with simulation results for propagation length=30km, generalized Rytov variance=0.1, anisotropy=1, and varying Kolmogorov exponent. Left: correlation along strong-turbulence axis; right: weak-turbulence axis. Analytic results: squares, exponent=19/6; circles, exponent=22/6; triangles, exponent=23/6. Simulation results: solid line, exponent=19/6; dashed line, exponent=22/6; dotted line, exponent=23/6.

Fig. 2.
Fig. 2.

Comparison of the analytical log-amplitude correlation function with simulation results for propagation length=100km, generalized Rytov variance=0.1, anisotropy=1, and varying Kolmogorov exponent. Left: correlation along strong-turbulence axis; right: weak-turbulence axis. Analytic results: squares, exponent=19/6; circles, exponent=22/6; triangles, exponent=23/6. Simulation results: solid line, exponent=19/6; dashed line, exponent=22/6; dotted line, exponent=23/6.

Fig. 3.
Fig. 3.

Comparison of the analytical log-amplitude correlation function with simulation results for propagation length=30km, generalized Rytov variance=0.1, Kolmogorov exponent=19/6, and varying anisotropy. Left: correlation along strong-turbulence axis; right: weak-turbulence axis. Analytic results: squares, anisotropy (ε)=0; circles, anisotropy=1; triangles, anisotropy=2. Simulation results: solid line, anisotropy=0; dashed line, anisotropy=1; dotted line, anisotropy=2.

Fig. 4.
Fig. 4.

Comparison of the analytical log-amplitude correlation function with simulation results for propagation length=30km, generalized Rytov variance=0.3, anisotropy=1, and varying Kolmogorov exponent. Left: correlation along strong-turbulence axis; right: weak-turbulence axis. Analytic results: squares, exponent=19/6; circles, exponent=22/6; triangles, exponent=23/6. Simulation results: solid line, exponent=19/6; dashed line, exponent=22/6; dotted line, exponent=23/6.

Equations (21)

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Bχ(ρ,θρ)=πk2L01duKdK02πdθρeiKuρcos(θρ){1cos[K2Lu(1u)k]}φn(K),
φ(K)=A(α)β(a2Kx2+b2Ky2+Kz2)α/2.
φ(K)=A(α)β(ab)α/2(a2Kx2+b2Ky2)α/2.
φn(K)=A(α)βKα(1+ε2)α/4(1+ε2sin2(θK))α/2.
Bχ(ρ,θρ)=A(α)πk2Lβ(1+ε2)α/401du0dKKα+1{1cos[K2Lu(1u)k]}02πdθKeiKρucos(θKθρ)1(1+ε2sin2(θK))α/2.
Bχ(ρ,θρ)=12A(α)πkα/2+3Lα/2β(1+ε2)α/4F(0,ε2)01du(u(1u))α/210dqqα/2[1cos(q)]J0([ku(1u)ρ2]1/2q1/2)+A(α)πkα/2+3Lα/2β(1+ε2)α/4m=1(1)mcos(2mθρ)F(m,ε2)01du(u(1u))α/210dqqα/2[1cos(q)]J2m([ku(1u)ρ2]1/2q1/2),
F(m,ε2)=π+πdθKcos(2mθK)(1+ε2sin2(θK))α/2.
σχ2=12A(α)πkα/2+3Lα/2β2π01du(u(1u))α/210dqqα/2[1cos(q)]=A(α)kα/2+3Lα/2βΓ(α/2)Γ(α)π32cos(πα/4).
Qm=01du(u(1u))α/210dqqα/2{1cos(q)}J2m((kuL(1u)ρ2)1/2q1/2).
Qm1=01du(u(1u))α/210dqqα/2J2m((kuL(1u)ρ2)1/2q1/2),
Qm2=01du(u(1u))α/210dqqα/2cos(q)J2m((kuL(1u)ρ2)1/2q1/2).
Qm1=01duuα22(kLρ2)1+α/20dxxα+1J2m(x)=2α/2α1(k8Lρ2)1+α/2Γ[m/2+1/2α/4m/2+1α/4m/2+α/4m/2+α/4+1/2].
Qm2=01du(u(1u))α/212α/2π2πi0ds(ku2L(1u)ρ2)sΓ[s/2α/4+1/2s+ms/2+α/4s+m+1].
Qm2=23+α/22πiπ1Γ(α)0ds(k8Lρ2)2sΓ[s+α/4s+α/4+1/2sα/4+1/2s+α/4+1/2s+m/2s+m/2+1/2s+m/2+1/2s+m/2+1].
s=n+m/2,n=0,1,2,s=n+m/2+1/2,n=0,1,2,3,s=α/41/2n,n=0s=n+α/4+1/2,n=0,1,2,3.
Bχ(ρ,θρ)=[12A(α)πkα/2+3Lα/2β(1+ε2)α/4F(0,ε2)(Q01Q02)+A(α)πkα/2+3Lα/2β(1+ε2)α/4m=1(1)mcos(2mθρ)F(m,ε2)(Qm1Qm2)],whereQm2=Qm2a+Qmb2+Qm2c.
Qm2a=23+α/22πiπ1Γ(α)n=0,1,2,3,(1)nn!(k8Lρ2)2n+α/2+1Γ[n+α/2+1/2n+α/2+1n+1n+α/4+m/2+1n+α/4+m/2+3/2]Γ[nα/41/2+m/2nα/4+m/2].
Qm2a=α23α/2(k8Lρ2)α/2+1Γ[m/2α/4m/2α/41/2α/4+m/2+1α/4+m/2+3]{F34(A;B,(k8Lρ2)2)},withA=[α/2+1/2,α/2+1,1],B=[α/4+m/2+1,α/4+m/2+3/2,α/4m/2+3/2,α/4m/2+1].
Qm2b=23+α/22πiπ1Γ(α)0ds(k8Lρ2)2sΓ[s+α/4s+α/4+1/2sα/4+1/2s+m/2+1/2s+m/2+1]Γ[s+α/4+1/2s+m/2s+m/2+1/2]Qm2b=23+α/2π1Γ(α)(k8Lρ2)mΓ[m/2+α/4m/2+α/4+1/2m/2α/4+1/2m/2α/4+1/21/2m+1/2m+1]{F23(A;B,(k8Lρ2)2)}withA=[m/2+α/4,m/2+α/4+1/2]andB=[m+1/2,m+1,1/2].
Qm2c=23+α/22πiπ1Γ(α)n=1,2,3,4,(1)nn!(k8Lρ2)2n+m+1Γ[n+m/2+1/2+α/4n+m/2+α/4+1n+m/2α/4+1n+m+1n+m/2+3/]Γ[nm+α/4n1/2].
Qm2c=23+α/2πΓ(α)(k8Lρ2)m+1Γ[m/2+1/2+α/4m/2+α/4+1m/2α/4+1m/2+α/41/2m+1m/2+3/2]{F33(A;B;(k8Lρ2)2)},withA=[m/2+1/2+α/4,m/2+α/4+1]andB=[1/2m+1m/2+3/2].

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