Abstract

The physical model of strong optical scintillation of plane waves is extended to the case of spherical-wave propagation. The spherical-wave amplitude filter function is derived and compared to the corresponding plane-wave result. Specifically, we calculate the spherical- to plane-wave log-amplitude-variance ratio. For weak fluctuations, our results are identical to the results of perturbation theory, with which the spherical-wave log-amplitude variance is less than the corresponding plane-wave result. On the other hand, for strong fluctuations (i.e., in the saturation regime), the plane- and spherical-wave log-amplitude variances saturate to the same constant value. Expressions are presented, in the saturation regime, for the plane- and spherical-wave log-amplitude covariance function and compared with the corresponding results in the regime of weak fluctuations.

© 1974 Optical Society of America

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References

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  1. H. T. Yura, J. Opt. Soc. Am. 64, 59 (1974).
    [Crossref]
  2. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, U. S. Dept. of Commerce, Springfield, Va., 1971).
  3. The limits on the integrals appearing in Eq. (19) are understood to be contained implicitly in Φn(Q).
  4. Indeed, Eq. (24) suggests that the amplitude filter function in the saturation regime is independent of the initial beam geometry, and is equal to Q2(Q2+ Q02)−1. This implies that the log-amplitude variance for an arbitrary beam wave saturates to the same constant, independent of the initial beam geometry (Ref. 5).
  5. M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, Akad. Nauk. SSSR, Otdelenie Oceanologi, Fiziki, Atmosfery i. Geografii, pp. 1–39, report of work prior to publication, Moscow (1973). English translation available from The Aerospace Corp. Library Services, Literature Research Group, P. O. Box 02057, Los Angeles, Calif. 90009. Translation No. LRG-73-T-28.
  6. D. A. de Wolf, J. Opt. Soc. Am. 64, 360 (1974).
    [Crossref]
  7. H. T. Yura, J. Opt. Soc. Am. 64, 1211 (1974).
    [Crossref]
  8. J. R. Dunphy and J. R. Kerr, J. Opt. Soc. Am. 63, 981 (1973).
    [Crossref]
  9. D. L. Fried, J. Opt. Soc. Am. 57, 169 (1967).
    [Crossref]
  10. H. T. Yura, J. Opt. Soc. Am. 64, 357 (1974).
    [Crossref]

1974 (4)

1973 (1)

1967 (1)

de Wolf, D. A.

Dunphy, J. R.

Fried, D. L.

Gracheva, M. E.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, Akad. Nauk. SSSR, Otdelenie Oceanologi, Fiziki, Atmosfery i. Geografii, pp. 1–39, report of work prior to publication, Moscow (1973). English translation available from The Aerospace Corp. Library Services, Literature Research Group, P. O. Box 02057, Los Angeles, Calif. 90009. Translation No. LRG-73-T-28.

Gurvich, A. S.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, Akad. Nauk. SSSR, Otdelenie Oceanologi, Fiziki, Atmosfery i. Geografii, pp. 1–39, report of work prior to publication, Moscow (1973). English translation available from The Aerospace Corp. Library Services, Literature Research Group, P. O. Box 02057, Los Angeles, Calif. 90009. Translation No. LRG-73-T-28.

Kashkarov, S. S.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, Akad. Nauk. SSSR, Otdelenie Oceanologi, Fiziki, Atmosfery i. Geografii, pp. 1–39, report of work prior to publication, Moscow (1973). English translation available from The Aerospace Corp. Library Services, Literature Research Group, P. O. Box 02057, Los Angeles, Calif. 90009. Translation No. LRG-73-T-28.

Kerr, J. R.

Pokasov, V. V.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, Akad. Nauk. SSSR, Otdelenie Oceanologi, Fiziki, Atmosfery i. Geografii, pp. 1–39, report of work prior to publication, Moscow (1973). English translation available from The Aerospace Corp. Library Services, Literature Research Group, P. O. Box 02057, Los Angeles, Calif. 90009. Translation No. LRG-73-T-28.

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, U. S. Dept. of Commerce, Springfield, Va., 1971).

Yura, H. T.

J. Opt. Soc. Am. (6)

Other (4)

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, U. S. Dept. of Commerce, Springfield, Va., 1971).

The limits on the integrals appearing in Eq. (19) are understood to be contained implicitly in Φn(Q).

Indeed, Eq. (24) suggests that the amplitude filter function in the saturation regime is independent of the initial beam geometry, and is equal to Q2(Q2+ Q02)−1. This implies that the log-amplitude variance for an arbitrary beam wave saturates to the same constant, independent of the initial beam geometry (Ref. 5).

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, Akad. Nauk. SSSR, Otdelenie Oceanologi, Fiziki, Atmosfery i. Geografii, pp. 1–39, report of work prior to publication, Moscow (1973). English translation available from The Aerospace Corp. Library Services, Literature Research Group, P. O. Box 02057, Los Angeles, Calif. 90009. Translation No. LRG-73-T-28.

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Figures (4)

Fig. 1
Fig. 1

Plane-wave to spherical-wave log-amplitude ratio as a function of L/zc for various α(= zc/ka2).

Fig. 2
Fig. 2

Plane- to spherical-wave log-amplitude variance ratio as a function of σT2 for the Kolmogorov spectrum in the wave-optics regime (Lkl02).

Fig. 3
Fig. 3

Normalized log-amplitude covariance function of plane and spherical waves as a function of ρ/ρ0 for the Kolmogorov spectrum in the saturation regime (σT2 ≫ 1).

Fig. 4
Fig. 4

Normalized temporal-frequency spectrum of a plane wave for the Kolmogorov spectrum in the saturation regime. For σT2 ≫ 1, the normalized temporal spectrum is a universal function of ν/νs, where νs = vn/2πρ0 (vn is normal component of the wind velocity to the optic axis). This result is in contrast to the regime of weak fluctuations (σT2 ≪ 1) where the normalized temporal-frequency spectrum is a universal function of ν/ν0, where ν 0 = v n / 2 π [ L / k ] 1 2.

Equations (31)

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σ a 2 ~ n 1 2 0 L d z a ( z ) 2 ( z / L ) 2 [ ( z / k a ) 2 ( z / L ) 2 + a 2 ] .
σ a 2 ~ n 1 2 a 3 0 L d z ( L - z ) 2 ( z / L ) 2 [ ( z / k a a ¯ ) 2 ( z / L ) 2 + 1 ] ,
1 a ¯ 2 1 a 2 + 1 ρ 0 2 ( z ) + 1 ρ 0 2 ( L - z ) ,
σ a 2 = L / z c 1 + L / z c [ 1 - 4 Re { Δ - 1 2 tan - 1 ( i Q a 2 Δ 1 2 ) } ] ,
z c ( k 2 n 1 2 a ) - 1 ,
Q a 2 = L k a 2 ( 1 + L z c ) 1 2 ,
Δ = Q a 4 + 4 i Q a 2 ,
χ s 2 = 2 π k 2 L 0 d K K Φ n ( K ) f A s ( K ) ,
f A s ( K ) = Q 4 0 1 d t t 2 ( 1 - t ) 2 1 + t 2 ( 1 - t ) 2 Q 2 [ Q 2 + Q 0 2 ( t ) ] .
Q 2 = L K 2 k
Q 0 2 ( t ) = L k [ 1 ρ 0 2 ( L t ) + 1 ρ 0 2 [ L ( 1 - t ) ] ] .
Φ n ( K ) = 0.033 C n 2 K 11 / 3 ,             L 0 - 1 K l 0 - 1 ,
f A s ( Q ) Q 4 0 1 d t t 2 ( 1 - t ) 2 1 + t 2 ( 1 - t ) 2 Q 2 ( Q 2 + Q 0 2 ) ,
Q 0 2 = L k ρ 0 2 ( L )
~ ( σ T 2 ) 6 / 5
σ T 2 = C n 2 k 7 / 6 L 11 / 6 .
f A p ( Q ) Q 4 0 1 d t ( 1 - t ) 2 1 + ( 1 - t ) 2 Q 2 ( Q 2 + Q 0 2 )
= ( Q 2 Q 2 + Q 0 2 ) [ 1 - tan - 1 { Q ( Q 2 + Q 0 2 ) } 1 2 Q ( Q 2 + Q 0 2 ) 1 2 ] ,
R = χ p 2 χ s 2 ,
R = f A p ( Q ) Φ n ( Q ) Q d Q f A s ( Q ) Φ n ( Q ) Q d Q ,
R = Q 1 Q 2 f A p ( Q ) Q - 8 / 3 d Q Q 1 Q 2 f A s ( Q ) Q - 8 / 3 d Q ,
Q 1 = ( L k L 0 2 ) 1 2
Q 2 = ( L k l 0 2 ) 1 2 .
R = 0 1 d t ( 1 - t ) 2 0 1 d t t 2 ( 1 - t ) 2 = 10 ,
R 0 d Q Q 4 3 0 1 d t t 2 [ 1 + t 2 Q 4 ] - 1 0 d Q Q 4 3 0 1 d t t 2 ( 1 - t ) 2 [ 1 + t 2 ( 1 - t ) 2 Q 4 ] - 1 .
f A s ( Q ) = f A p ( Q ) = Q 2 Q 2 + Q 0 2 .
R = χ p 2 χ s 2 = 1.
b A ( ρ ) B A ( ρ ) B A ( 0 ) = 1 π 0 1 d t 0 J 0 ( x ρ t / ρ 0 ) 1 + x 2 x - 2 3 d x             ( σ T 2 1 ) ,
lim ρ 0 b A ( ρ ) { 1 - 3 4 π ( ρ / ρ 0 ) 5 / 3 + ( plane wave ) 1 - 9 32 π ( ρ / ρ 0 ) 5 / 3 + ( spherical wave )
lim ρ b A ( ρ ) { 3 π ( ρ 0 / ρ ) 1 / 3 + ( plane wave ) 9 2 π ( ρ 0 / ρ ) 1 / 3 + ( spherical wave ) .
lim ρ 0 b A ( ρ ) 1 - O ( k ρ 2 L ) 5 / 6 + and lim ρ b A ( ρ ) O ( L k ρ 2 ) 7 / 6 + ,