Abstract

Considerable amount of data has been collected in the past asserting that atmospheric turbulence has regions where it exhibits anisotropic statistics. For instance, it is known that the fluctuations in the refractive index within the first meter above the ground are typically stronger in the vertical direction compared with those in the horizontal directions. We have investigated the second-order statistical properties of a Gaussian Schell-model (GSM) beam traversing anisotropic atmospheric turbulence along a horizontal path. Analytical expression is rigorously derived for the cross-spectral density function of a GSM beam. It is shown that the spread of the beam and its coherence properties become different in two transverse directions due to anisotropy. In the limiting case when the source coherence width becomes infinite our results reduce to those for Gaussian beam propagation. Source partial coherence is shown to mitigate anisotropy at sub-kilometer distances.

© 2016 Optical Society of America

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References

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2016 (2)

2015 (3)

2014 (3)

M. Yao, I. Toselli, and O. Korotkova, “Propagation of electromagnetic stochastic beams in anisotropic turbulence,” Opt. Express 22(26), 31608–31619 (2014).
[Crossref] [PubMed]

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

L. Y. Cui, B. D. Xue, X. G. Cao, and F. G. Zhou, “Atmospheric turbulence MTF for optical waves’ propagation through anisotropic non-Kolmogorov atmospheric turbulence,” Opt. Laser Technol. 63, 70–75 (2014).
[Crossref]

2013 (2)

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

A. Attili and F. Bisetti, “Fluctuations of a passive scalar in a turbulent mixing layer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 88(3), 033013 (2013).
[Crossref] [PubMed]

2011 (1)

2005 (2)

L. Biferale and I. Procaccia, “Anisotropy in turbulent flows and in turbulent transport,” Phys. Rep. 414(2–3), 43–164 (2005).
[Crossref]

J. D. Phillips, M. E. Goda, and J. Schmidt, “Atmospheric turbulence simulation using liquid crystal light modulators,” Proc. SPIE 5894, 589406 (2005).
[Crossref]

2003 (3)

A. S. Gurvich and V. Kan, “Structure of Air Density Irregularities in the Stratosphere from Spacecraft Observations of Stellar Scintillation: 1. Three-Dimensional Spectrum Model and Recovery of Its Parameters,” Izv., Atmos. Ocean. Phys. 39, 300–310 (2003).

A. S. Gurvich and V. Kan, “Structure of Air Density Irregularities in the Stratosphere from Spacecraft Observations of Stellar Scintillation: 2. Characteristic Scales, Structure Characteristics, and Kinetic Energy Dissipation,” Izv., Atmos. Ocean. Phys. 39, 311–321 (2003).

T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20(6), 1094–1102 (2003).
[Crossref] [PubMed]

2002 (1)

2001 (1)

L. Biferale and M. Vergassola, “Isotropy versus anisotropy in small-scale turbulence,” Phys. Fluids 13(8), 2139–2141 (2001).
[Crossref]

1996 (1)

1994 (3)

F. Dalaudier, C. Sidi, M. Crochet, and J. Vernin, “Direct evidence of ‘Sheets’ in the atmospheric temperature field,” J. Atmos. Sci. 51(2), 237–248 (1994).
[Crossref]

V. A. Banakh and I. N. Smalikho, “Propagation of a laser beam through the stratosphere,” Atmos. Oceanic Opt. 7, 736–743 (1994).

A. I. Kon, “Qualitative theory of amplitude and phase fluctuations in a medium with anisotropic turbulent irregularity,” Waves Random Complex Media 4(3), 297–306 (1994).
[Crossref]

1992 (1)

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

1972 (1)

1970 (1)

Agrawal, B.

Andrews, L. C.

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

Attili, A.

A. Attili and F. Bisetti, “Fluctuations of a passive scalar in a turbulent mixing layer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 88(3), 033013 (2013).
[Crossref] [PubMed]

Banakh, V. A.

V. A. Banakh and I. N. Smalikho, “Propagation of a laser beam through the stratosphere,” Atmos. Oceanic Opt. 7, 736–743 (1994).

Biferale, L.

L. Biferale and I. Procaccia, “Anisotropy in turbulent flows and in turbulent transport,” Phys. Rep. 414(2–3), 43–164 (2005).
[Crossref]

L. Biferale and M. Vergassola, “Isotropy versus anisotropy in small-scale turbulence,” Phys. Fluids 13(8), 2139–2141 (2001).
[Crossref]

Bisetti, F.

A. Attili and F. Bisetti, “Fluctuations of a passive scalar in a turbulent mixing layer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 88(3), 033013 (2013).
[Crossref] [PubMed]

Bowersox, R. D.

Cai, Y.

Cao, X. G.

L. Y. Cui, B. D. Xue, X. G. Cao, and F. G. Zhou, “Atmospheric turbulence MTF for optical waves’ propagation through anisotropic non-Kolmogorov atmospheric turbulence,” Opt. Laser Technol. 63, 70–75 (2014).
[Crossref]

Chen, C.

Consortini, A.

Crabbs, R.

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

Crochet, M.

F. Dalaudier, C. Sidi, M. Crochet, and J. Vernin, “Direct evidence of ‘Sheets’ in the atmospheric temperature field,” J. Atmos. Sci. 51(2), 237–248 (1994).
[Crossref]

Cui, L. Y.

L. Y. Cui, B. D. Xue, X. G. Cao, and F. G. Zhou, “Atmospheric turbulence MTF for optical waves’ propagation through anisotropic non-Kolmogorov atmospheric turbulence,” Opt. Laser Technol. 63, 70–75 (2014).
[Crossref]

Dalaudier, F.

F. Dalaudier, C. Sidi, M. Crochet, and J. Vernin, “Direct evidence of ‘Sheets’ in the atmospheric temperature field,” J. Atmos. Sci. 51(2), 237–248 (1994).
[Crossref]

Dogariu, A.

Gardner, P. J.

Gbur, G.

Goda, M. E.

J. D. Phillips, M. E. Goda, and J. Schmidt, “Atmospheric turbulence simulation using liquid crystal light modulators,” Proc. SPIE 5894, 589406 (2005).
[Crossref]

Grechko, G. M.

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Gurvich, A. S.

A. S. Gurvich and V. Kan, “Structure of Air Density Irregularities in the Stratosphere from Spacecraft Observations of Stellar Scintillation: 1. Three-Dimensional Spectrum Model and Recovery of Its Parameters,” Izv., Atmos. Ocean. Phys. 39, 300–310 (2003).

A. S. Gurvich and V. Kan, “Structure of Air Density Irregularities in the Stratosphere from Spacecraft Observations of Stellar Scintillation: 2. Characteristic Scales, Structure Characteristics, and Kinetic Energy Dissipation,” Izv., Atmos. Ocean. Phys. 39, 311–321 (2003).

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Kan, V.

A. S. Gurvich and V. Kan, “Structure of Air Density Irregularities in the Stratosphere from Spacecraft Observations of Stellar Scintillation: 2. Characteristic Scales, Structure Characteristics, and Kinetic Energy Dissipation,” Izv., Atmos. Ocean. Phys. 39, 311–321 (2003).

A. S. Gurvich and V. Kan, “Structure of Air Density Irregularities in the Stratosphere from Spacecraft Observations of Stellar Scintillation: 1. Three-Dimensional Spectrum Model and Recovery of Its Parameters,” Izv., Atmos. Ocean. Phys. 39, 300–310 (2003).

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Kireev, S. V.

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Kon, A. I.

A. I. Kon, “Qualitative theory of amplitude and phase fluctuations in a medium with anisotropic turbulent irregularity,” Waves Random Complex Media 4(3), 297–306 (1994).
[Crossref]

Korotkova, O.

Leclerc, T.

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

Li, Y.

Li, Z.

Liu, X.

Luke, T. E.

Phillips, J. D.

J. D. Phillips, M. E. Goda, and J. Schmidt, “Atmospheric turbulence simulation using liquid crystal light modulators,” Proc. SPIE 5894, 589406 (2005).
[Crossref]

Phillips, R. L.

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

Procaccia, I.

L. Biferale and I. Procaccia, “Anisotropy in turbulent flows and in turbulent transport,” Phys. Rep. 414(2–3), 43–164 (2005).
[Crossref]

Ren, B.

Restaino, S.

Roggemann, M. C.

Ronchi, L.

Savchenko, S. A.

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Schmidt, J.

J. D. Phillips, M. E. Goda, and J. Schmidt, “Atmospheric turbulence simulation using liquid crystal light modulators,” Proc. SPIE 5894, 589406 (2005).
[Crossref]

Shirai, T.

Sidi, C.

F. Dalaudier, C. Sidi, M. Crochet, and J. Vernin, “Direct evidence of ‘Sheets’ in the atmospheric temperature field,” J. Atmos. Sci. 51(2), 237–248 (1994).
[Crossref]

Smalikho, I. N.

V. A. Banakh and I. N. Smalikho, “Propagation of a laser beam through the stratosphere,” Atmos. Oceanic Opt. 7, 736–743 (1994).

Stefanutti, L.

Tong, S.

Toselli, I.

Vergassola, M.

L. Biferale and M. Vergassola, “Isotropy versus anisotropy in small-scale turbulence,” Phys. Fluids 13(8), 2139–2141 (2001).
[Crossref]

Vernin, J.

F. Dalaudier, C. Sidi, M. Crochet, and J. Vernin, “Direct evidence of ‘Sheets’ in the atmospheric temperature field,” J. Atmos. Sci. 51(2), 237–248 (1994).
[Crossref]

Voelz, D.

X. Xiao and D. Voelz, “Wave optics simulation of anisotropic turbulence,” in IEEE Aerospace Conference, Big Sky, Montana, 2015.

Voelz, D. G.

Wang, H.

Wang, J.

Welsh, B. M.

Wolf, E.

Xiao, X.

Xue, B. D.

L. Y. Cui, B. D. Xue, X. G. Cao, and F. G. Zhou, “Atmospheric turbulence MTF for optical waves’ propagation through anisotropic non-Kolmogorov atmospheric turbulence,” Opt. Laser Technol. 63, 70–75 (2014).
[Crossref]

Yang, H.

Yao, M.

Yura, H. T.

Zhao, D.

Zhou, F. G.

L. Y. Cui, B. D. Xue, X. G. Cao, and F. G. Zhou, “Atmospheric turbulence MTF for optical waves’ propagation through anisotropic non-Kolmogorov atmospheric turbulence,” Opt. Laser Technol. 63, 70–75 (2014).
[Crossref]

Zhu, S.

Adv. Space Res. (1)

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Appl. Opt. (5)

Atmos. Oceanic Opt. (1)

V. A. Banakh and I. N. Smalikho, “Propagation of a laser beam through the stratosphere,” Atmos. Oceanic Opt. 7, 736–743 (1994).

Izv., Atmos. Ocean. Phys. (2)

A. S. Gurvich and V. Kan, “Structure of Air Density Irregularities in the Stratosphere from Spacecraft Observations of Stellar Scintillation: 1. Three-Dimensional Spectrum Model and Recovery of Its Parameters,” Izv., Atmos. Ocean. Phys. 39, 300–310 (2003).

A. S. Gurvich and V. Kan, “Structure of Air Density Irregularities in the Stratosphere from Spacecraft Observations of Stellar Scintillation: 2. Characteristic Scales, Structure Characteristics, and Kinetic Energy Dissipation,” Izv., Atmos. Ocean. Phys. 39, 311–321 (2003).

J. Atmos. Sci. (1)

F. Dalaudier, C. Sidi, M. Crochet, and J. Vernin, “Direct evidence of ‘Sheets’ in the atmospheric temperature field,” J. Atmos. Sci. 51(2), 237–248 (1994).
[Crossref]

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

Opt. Express (4)

Opt. Laser Technol. (1)

L. Y. Cui, B. D. Xue, X. G. Cao, and F. G. Zhou, “Atmospheric turbulence MTF for optical waves’ propagation through anisotropic non-Kolmogorov atmospheric turbulence,” Opt. Laser Technol. 63, 70–75 (2014).
[Crossref]

Phys. Fluids (1)

L. Biferale and M. Vergassola, “Isotropy versus anisotropy in small-scale turbulence,” Phys. Fluids 13(8), 2139–2141 (2001).
[Crossref]

Phys. Rep. (1)

L. Biferale and I. Procaccia, “Anisotropy in turbulent flows and in turbulent transport,” Phys. Rep. 414(2–3), 43–164 (2005).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

A. Attili and F. Bisetti, “Fluctuations of a passive scalar in a turbulent mixing layer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 88(3), 033013 (2013).
[Crossref] [PubMed]

Proc. SPIE (3)

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

J. D. Phillips, M. E. Goda, and J. Schmidt, “Atmospheric turbulence simulation using liquid crystal light modulators,” Proc. SPIE 5894, 589406 (2005).
[Crossref]

Waves Random Complex Media (1)

A. I. Kon, “Qualitative theory of amplitude and phase fluctuations in a medium with anisotropic turbulent irregularity,” Waves Random Complex Media 4(3), 297–306 (1994).
[Crossref]

Other (3)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

X. Xiao and D. Voelz, “Wave optics simulation of anisotropic turbulence,” in IEEE Aerospace Conference, Big Sky, Montana, 2015.

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

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Figures (8)

Fig. 1
Fig. 1 Illustrating beam propagation in anisotropic turbulence along horizontal paths.
Fig. 2
Fig. 2 Density plots (false-color rendering) of SD of GSM beams at several propagation distances for three values of source coherence width.
Fig. 3
Fig. 3 Density plots (false-color rendering): the modulus of DOC of GSM beams at several propagation distances for three values of initial coherence width.
Fig. 4
Fig. 4 Variation of (a) second-fourth terms in Eq. (19), (b)-(c) first-third terms in Eq. (20) with the propagation distance. The parameters in the calculation are same with those in Fig. 2 or Fig. 3.
Fig. 5
Fig. 5 R.m.s. beam widths in x and y directions as a function of the propagation distance [(a)-(c)] and Rytov variance [(d)-(f)] for three values of C ˜ n 2 .
Fig. 6
Fig. 6 R.m.s. coherence widths in x and y directions as a function of propagation distance [(a)-(c)] and Rytov variance [(d-(f))] for three values of C ˜ n 2 .
Fig. 7
Fig. 7 Changes of the ellipticity of the SD with (a) propagation distance, (b) anisotropic factor μx, (c) power law α and (d) turbulence structure parameter.
Fig. 8
Fig. 8 Changes of the ellipticity of the DOC distribution with (a) propagation distance, (b) anisotropic factor μx, (c) power law α and (d) turbulence structure parameter.

Equations (21)

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W (0) ( r 1 , r 2 ,ω)=exp( r 1 2 + r 2 2 4 σ 0 2 )exp[ | r 2 r 1 | 2 2 δ 0 2 ],
W( ρ 1 , ρ 2 ,z)= 1 λ 2 z 2 W (0) ( r 1 , r 2 )exp[ ik 2z ( ρ 1 r 1 ) 2 + ik 2z ( ρ 2 r 2 ) 2 ] × exp[ ψ*( r 1 , ρ 1 ,z)+ψ( r 2 , ρ 2 ,z) ] d 2 r 1 d 2 r 2 ,
exp( ψ*( r 1 , ρ 1 ,z)( r 2 , ρ 2 ,z) ) =exp{ 2π k 2 z 0 1 dt d 2 κ Φ n (κ)[ 1exp( t ρ d ·κ+(1t) r d ·κ ) ] },
exp( ψ*( r 1 , ρ 1 ,z)( r 2 , ρ 2 ,z) ) =exp{ 4 π 2 k 2 z 0 1 dt κdκ Φ n (κ)[ 1 J 0 ( κ| t ρ d +(1t) r d | ) ] },
Φ n ( κ x , κ y ,0 )= μ x μ y μ z A(α) C ˜ n 2 exp( κ x 2 / κ mx 2 κ y 2 / κ my 2 ) ( μ x 2 κ x 2 + μ y 2 κ y 2 + κ 0 2 ) α/2 , 3<α<4,
A(α)= 1 4 π 2 Γ(α1)cos( απ 2 ),
c(α)= [ 2πΓ(5α/2)A(α) 3 ] 1/(α5) .
exp( ψ*( r 1 , ρ 1 ,z)( r 2 , ρ 2 ,z) ) =exp{ 4 π 2 k 2 z μ x μ y 0 1 dt d κ x ' d κ y ' Φ ' n ( κ ' )[ 1exp( t ρ d ' · κ ' +(1t) r d ' · κ ' ) ] }
Φ ' n (κ')= μ x μ y μ z A(α) C ˜ n 2 ( | κ' | 2 + κ 0 2 ) α/2 exp( | κ' | 2 / κ m 2 ),
exp( ψ*( r 1 , ρ 1 ,z)( r 2 , ρ 2 ,z) ) =exp{ 4 π 2 k 2 z μ x μ y 0 1 dt 0 κ'dκ'Φ ' n (κ')[ 1 J 0 ( κ'| tρ ' d +(1t)r ' d | ) ] },
exp( ψ*( r 1 , ρ 1 ,z)( r 2 , ρ 2 ,z) ) =exp[ π 2 k 2 zT( ξ d 2 + ξ d x d + x d 2 ) x 2 ]exp[ π 2 k 2 zT( η d 2 + η d y d + y d 2 ) y 2 ],
T= 0 dκ'κ ' 3 Φ ' n (κ')/ μ x μ y = μ z A(α) 2(α2) C ˜ n 2 [ β κ m 2α exp( κ 0 2 / κ m 2 ) Γ 1 (2α/2, κ 0 2 / κ m 2 )2 κ 0 4α ],
W( ρ 1 , ρ 2 ,z)= W x ( ξ 1 , ξ 2 ) W y ( η 1 , η 2 ),
W x ( ξ 1 , ξ 2 )= 1 Δ x (z) exp( ξ 1 2 + ξ 2 2 4 σ 0 2 Δ x (z) )exp( ik( ξ 1 2 ξ 2 2 ) 2 R x (z) ) ×exp[ ( 1 2 δ 0 2 Δ x (z) + π 2 k 2 Tz 3 μ x 2 ( 1+ 2 Δ x (z) ) π 4 k 2 T 2 z 4 18 μ x 4 Δ x (z) σ 0 2 ) ( ξ 1 ξ 2 ) 2 ].
Δ x (z)=1+[ 1 4 k 2 σ 0 4 + 1 k 2 σ 0 2 ( 1 δ 0 2 + 2 π 2 k 2 Tz 3 μ x 2 ) ] z 2 ,
R x (z)=z+ σ 0 2 z π 2 T z 4 /3 μ x 2 ( Δ x (z)1 ) σ 0 2 + π 2 T z 3 /3 μ x 2 .
S(ρ,z)=W(ρ,ρ,z),
μ( ρ 1 , ρ 2 ,z)= W( ρ 1 , ρ 2 ,z) S( ρ 1 ,z)S( ρ 2 ,z) .
σ i (z)= σ 0 Δ i (z) = σ 0 2 + z 2 4 k 2 σ 0 2 + z 2 k 2 δ 0 2 + 2 π 2 T z 3 3 μ i 2 ,(i=x,y),
δ i (z)= [ 1 δ 0 2 Δ i (z) + 2 π 2 k 2 Tz 3 μ i 2 + 2 π 2 k 2 Tz 3 μ i 2 Δ i (z) ( 2 π 2 T z 3 6 μ i 2 σ 0 2 ) ] 1/2 ,(i=x,y)
f(z)= P(z)Q(z) P(z) ,

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