Abstract

There is strong evidence that the amplitude of a light wave propagating through turbulence becomes Rayleigh distributed (i.e., the irradiance becomes exponentially distributed) in the limit of strong turbulence, which implies that the log-amplitude variance tends to π2/24. We find that the theory by Clifford et al. [ J. Opt. Soc. Am. 64, 148– 154 ( 1974)] for saturation of scintillation by strong refractive turbulence can be made to obey this limit for power-law refractive-index spectra. However, for a nonzero inner scale of turbulence (no matter how small), the theory predicts that log-amplitude variance tends to zero in the limit of strong turbulence. A generalization of the theory is derived that obeys the π2/24 limit for arbitrary refractive-index spectra, a nonzero inner scale being a particular case. The new theory has no arbitrary parameters. Both old and new modulation transfer functions have different behavior for nonzero inner scale at both very large and very small spatial wave numbers when compared with the case of zero inner scale. This differing behavior affects the log-amplitude variance even if the Fresnel-zone size is much greater than the inner scale, provided that the lateral coherence length of phase is less than the inner scale. This differing behavior also applies at all spatial wave numbers if the Rytov variance is strongly affected by the inner scale. For strong (but finite) turbulence strength, the predicted log-amplitude variance is larger for a smaller ratio of Fresnel-zone size to inner scale, which is in quantitative agreement with observations.

© 1981 Optical Society of America

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References

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  1. S. F. Clifford, G. R. Ochs, and R. W. Lawrence, “Saturation of optical scintillation by strong turbulence,” J. Opt. Soc. Am. 64, 148–154 (1974).
    [Crossref]
  2. R. J. Hill and S. F. Clifford, “Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation,” J. Opt. Soc. Am. 68, 892–899 (1978).
    [Crossref]
  3. R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–562 (1978).
    [Crossref]
  4. S. F. Clifford and H. T. Yura, “Equivalence of two theories of strong optical scintillation,” J. Opt. Soc. Am. 64, 1641–1644 (1974).
    [Crossref]
  5. R. J. Hill and M. H. Ackley, “Function routines for integrals involving the Bessel functions J0 and J1,” (U.S. Government Printing Office, Washington, D.C., 1980).
  6. R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
    [Crossref]
  7. S. F. Clifford and R. J. Hill, “Relation between irradiance and log-amplitude variance for optical scintillation described by the K distribution,” J. Opt. Soc. Am. 71, 112–114 (1981).
    [Crossref]
  8. R. J. Hill and S. F. Clifford, “The bump in the variance of log-intensity,” in Technical Digest of Topical Meeting on Optical Propagation through Turbulence, Rain, and Fog (Optical Society of America, Washington, D.C., 1977).
  9. G. R. Ochs, “Measurements of 0.63 μ m laser-beam scintillation in strong atmospheric turbulence,” (U.S. Government Printing Office, Washington, D.C., 1969).

1981 (1)

1978 (2)

1975 (1)

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[Crossref]

1974 (2)

Ackley, M. H.

R. J. Hill and M. H. Ackley, “Function routines for integrals involving the Bessel functions J0 and J1,” (U.S. Government Printing Office, Washington, D.C., 1980).

Clifford, S. F.

Fante, R. L.

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[Crossref]

Hill, R. J.

S. F. Clifford and R. J. Hill, “Relation between irradiance and log-amplitude variance for optical scintillation described by the K distribution,” J. Opt. Soc. Am. 71, 112–114 (1981).
[Crossref]

R. J. Hill and S. F. Clifford, “Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation,” J. Opt. Soc. Am. 68, 892–899 (1978).
[Crossref]

R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–562 (1978).
[Crossref]

R. J. Hill and S. F. Clifford, “The bump in the variance of log-intensity,” in Technical Digest of Topical Meeting on Optical Propagation through Turbulence, Rain, and Fog (Optical Society of America, Washington, D.C., 1977).

R. J. Hill and M. H. Ackley, “Function routines for integrals involving the Bessel functions J0 and J1,” (U.S. Government Printing Office, Washington, D.C., 1980).

Lawrence, R. W.

Ochs, G. R.

S. F. Clifford, G. R. Ochs, and R. W. Lawrence, “Saturation of optical scintillation by strong turbulence,” J. Opt. Soc. Am. 64, 148–154 (1974).
[Crossref]

G. R. Ochs, “Measurements of 0.63 μ m laser-beam scintillation in strong atmospheric turbulence,” (U.S. Government Printing Office, Washington, D.C., 1969).

Yura, H. T.

J. Fluid Mech. (1)

R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–562 (1978).
[Crossref]

J. Opt. Soc. Am. (4)

Proc. IEEE (1)

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[Crossref]

Other (3)

R. J. Hill and S. F. Clifford, “The bump in the variance of log-intensity,” in Technical Digest of Topical Meeting on Optical Propagation through Turbulence, Rain, and Fog (Optical Society of America, Washington, D.C., 1977).

G. R. Ochs, “Measurements of 0.63 μ m laser-beam scintillation in strong atmospheric turbulence,” (U.S. Government Printing Office, Washington, D.C., 1969).

R. J. Hill and M. H. Ackley, “Function routines for integrals involving the Bessel functions J0 and J1,” (U.S. Government Printing Office, Washington, D.C., 1980).

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Figures (11)

Fig. 1
Fig. 1

The spectra Φn (solid curve) and Ψ (dashed curve) versus Kolmogorov scaled spatial wave number κη. The spectra are scaled so that they tend to unity in the inertial range.

Fig. 2
Fig. 2

f(y) from Eqs. (9) and (11) versus y = κ2Lu(1 = u)/2k. The short-dashed curve, which is hidden beneath the uppermost solid curve on the left side, is from Φnκ−11/3 for all κ. The solid curves (from numerical integration) and long-dashed curves (from function routines of Ref. 5) use the accurate Φn in Fig. 1. Equation (14) defines V.

Fig. 3
Fig. 3

Logarithm of MST from Eqs. (7) and (11) is scaled by turbulence strength parameter Yη and plotted versus Kolmogorov scaled spatial wavenumber. Meaning of short-dashed, solid, and long-dashed curves is same as in Fig. 2. Equation (14) defines V.

Fig. 4
Fig. 4

Variance of log amplitude computed from the original theory [Eqs. (1) and (2)] for several values of λ L / l 0. Dashed curve is l0 = 0 case, i.e., Φnκ11/3 for all κ. Dots are data of Ref. 1. A tick at right side marks π2/24.

Fig. 5
Fig. 5

F(y) versus y = κ2Lu(1 − u)/2k. The short-dashed curve, which is hidden beneath solid curves near the center, is from Eq. (56), that is, Φnκ−11/3 for all κ. The solid curves (from numerical integration) and long-dashed curves (from function routines of Ref. 5) from Eq. (57) use the accurate Φn in Fig. 1. Equation (14) defines V.

Fig. 6
Fig. 6

Logarithm of MST scaled by turbulence-strength parameter Yη is plotted versus Kolmogorov scaled spatial wave number. Short-dashed curves are from Eq. (7) by replacing f(y) by the F(y) in Eq. (56), that is, l0 = 0, so Φnκ11/3 for all κ. Solid and long-dashed curves as in Fig. 5. Equation (14) defines V.

Fig. 7
Fig. 7

Log-amplitude variance computed from the new theory [Eqs. (1) and (55)] for several values of λ L / l 0. Dashed curve is l0 = 0 case, i.e., Φnκ11/3 for all κ. Dots are data Ref. 1. For reasons independent of the agreement of the curve λ L / l 0 = 7 with the data, it is most appropriate to compare the bulk of the data with the curve for λ L / l 0 = 7. A tick at the right side marks π2/24.

Fig. 8
Fig. 8

Computed log-amplitude variance for several λ L / l 0 is plotted versus Rytov variance when the latter includes the inner-scale effect. Solid line is Rytov asymptote. Short-dashed curve is case l0 = 0, i.e., Φnκ−11/3 for all κ. Long-dashed curve is upper limit for small λ L / l 0.

Fig. 9
Fig. 9

Inner-scale effect on σχ2 for several values of strength-of-turbulence parameter σT2 = 0.124k7/6L11/6Cn2. Solid curve is Rytov case, i.e., σT2 → 0. Dashed line is Rytov asymptote for λ L / l 0 1.

Fig. 10
Fig. 10

Measured (dots) and predicted (symbols) log-amplitude variance for several wind speeds. Symbols correspond to the following values of λ L / l 0: squares, 4; diamonds, 7; triangles, 20. Inner scale l0 decreases as wind speed increases according to Table 1.

Fig. 11
Fig. 11

Phase-coherence length for spherical waves using Φn in Fig. 1. Coherence length is scaled by both ρ1 and ρ2 and shown for both an infinite and a finite outer scale of turbulence.

Tables (1)

Tables Icon

Table 1 Inner Scale at 2 m Above Ground Deduced from Mean Wind Speed for Table Mountain, Boulder, Colorado.

Equations (93)

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σ χ 2 = 4 π 2 k 2 L × 0 1 d u 0 d κ κ Φ n ( κ ) sin 2 [ κ 2 L u ( 1 - u ) 2 k ] M S T ,
M S T = exp ( - 4 π 2 k 2 L 0 1 d u u - 2 γ κ d κ κ Φ n ( κ / u ) × { 1 - J 0 [ κ κ L u ( 1 - u ) k ] } ) ,
M S T = exp ( - 4 π 2 k 2 L γ κ d κ Ψ ( κ ) × { 1 - J 0 [ κ κ L u ( 1 - u ) k ] } ) ,
Ψ ( κ ) = κ Φ n ( κ ) d κ .
Φ n ( κ ) = 0.033 C n 2 κ - 11 / 3
σ χ 2 = 2.95 σ T 2 0 1 d u [ u ( 1 - u ) ] 5 / 6 × 0 d y y - 11 / 6 sin 2 ( y ) M S T 0 ,
M S T 0 = exp { - σ T 2 [ u ( 1 - u ) ] 5 / 6 f ( y ) } ,
σ T 2 = 0.124 k 7 / 6 L 11 / 6 C n 2 ,
f ( y ) = 7.02 y 5 / 6 α y d ξ ξ - 8 / 3 [ 1 - J 0 ( ξ ) ] , y = κ 2 L u ( 1 - u ) / 2 k , α = 2 γ .
σ χ 2 = 1.30 Y η 0 1 d u 0 d κ ˆ κ ˆ Φ ˜ n ( κ ˆ ) sin 2 ( κ ˆ 2 V / 2 ) M S T ,
M S T = exp { - 1.30 Y η γ κ ˆ d κ ˜ Ψ ˜ ( κ ˜ ) [ 1 - J 0 ( κ ˜ κ ˆ V ) ] } ,
Ψ ˜ ( κ ˜ ) = κ ˜ Φ ˜ n ( κ ˜ ) d κ ˜ ,
Y η = k 2 L C n 2 η 5 / 3 = σ T 2 ( η / L / k ) 5 / 3 / 0.124 ,
V = u ( 1 - u ) ( L / k / η ) 2 ,
κ ˆ = κ η ,
κ ˜ = κ η ,
Φ ˜ n = Φ n / 0.033 C n 2 η 11 / 3 .
f ( y ) 7.85 y 5 / 6 ,             y 1 ,
f ( y ) 4.21 α - 5 / 3 y - 5 / 6 ,             y 1.
M S T 0 exp { - 0.546 k 1 / 3 L 8 / 3 C n 2 [ u ( 1 - u ) ] 5 / 3 κ 5 / 3 } ,             y 1.
M S T 0 exp ( - 0.930 α - 5 / 3 k 2 L C n 2 κ - 5 / 3 ) ,             y 1.
γ κ ˆ d κ ˜ Ψ ˜ ( κ ˜ ) [ 1 - J 0 ( κ ˜ κ ˆ V ) ] ( κ ˆ V ) 5 / 3 × 0 d ξ ξ - 8 / 3 [ 1 - J 0 ( ξ ) ] ,
M S T exp { - σ η 2 [ u ( 1 - u ) ] 2 κ ˆ 2 } ,
σ η 2 = 1.30 12 L 3 C n 2 η - 7 / 3 0 d κ ˜ κ ˜ 3 Φ ˜ n ( κ ˜ ) ;
σ χ 2 ( low ) 1.30 2 Y η k - 2 L 2 η - 4 0 0.5 d u [ u ( 1 - u ) ] 2 × 0 0.5 d κ ˆ κ ˆ 4 / 3 exp { - σ η 2 [ u ( 1 - u ) ] 2 κ ˆ 2 } .
σ χ 2 ( low ) 2 A ( σ η 2 ) - 1 / 6 0 0.5 d u [ u ( 1 - u ) ] - 1 / 3 × 0 ξ d z z 1 / 6 e - z ,
ξ = σ η 2 [ u ( 1 - u ) ] 2 / 4 ,
A = ³ / [ 0 d κ ˜ κ ˜ 3 Φ ˜ n ( κ ˜ ) ] - 1 .
σ η 2 ( l o w ) A ( σ η 2 ) - 1 / 6 [ Γ ( 2 / 3 ) ] 2 Γ ( 7 / 6 ) / Γ ( 4 / 3 ) ,
σ χ 2 ( h i g h ) 0.35 α 5 / 3 × 0 1 d u β ( u ) 1 d y y - 11 / 6 e - y - 5 / 6 β ( u ) ,
β ( u ) = 4.2 σ T 2 α - 5 / 3 [ u ( 1 - u ) ] 5 / 6 .
σ χ 2 ( high ) 0.420 α 5 / 3 .
π 2 / 24 = 0.420 α 5 / 3 ,
α = 0.987 ,             γ = α / 2 = 0.494 ;
σ χ 2 ( high ) 0.35 α 5 / 3 β × 0 1 d u [ u ( 1 - u ) ] - 1 d z z - 11 / 6 exp ( - β z - 5 / 6 ) ,
β = 4.2 σ T 2 α - 5 / 3 .
σ χ 2 ( high ) 0.7 α 5 / 3 β 0 d K K - 8 / 3 e - β K - 5 / 3 .
σ χ 2 ( high ) 1.30 2 Y η 0.5 d κ ˆ κ ˆ Φ ˜ n ( κ ˆ ) × exp [ - 1.30 Y η γ κ ˆ d κ ˜ κ ˜ Φ ˜ n ( κ ˜ ) d κ ˜ ] .
Φ ˜ n = e - a κ ˆ ,
σ χ 2 ( high ) ( B / 2 ) - d x x exp ( - x - B e - γ x ) ,
x = a κ ˆ ,
B = 1.30 Y η a - 2 .
σ χ 2 ( high ) - 2 B 0 d z z ln ( z ) e - B z
= 2 [ ln ( B ) - 0.423 ] / B .
σ χ 2 ( high ) 0 ,             for l 0 0 and σ T 2 .
M S T = exp ( - 4 π 2 k 2 L 0 1 d u u - 2 × 0 d κ κ Φ n ( κ / u ) h ( κ / κ ) { 1 - J 0 [ κ κ L u ( 1 - u ) k ] } ) .
σ χ 2 ( high ) 2 π 2 k 2 L 0 d κ κ Φ n ( κ ) exp [ - 4 π 2 k 2 L 0 1 u × 0 d K K Φ n ( K ) h ( K u , κ ) ] .
d κ κ Φ n ( κ ) 0 1 d u 0 d K K Φ n ( K ) h ( K u , κ ) κ d κ ,
0 1 d u h ( K u , κ ) κ δ ( κ - K ) ,
0 K d ξ h ( ξ , κ ) κ K δ ( κ - K ) .
h ( K , κ ) = A K [ K θ ( K - κ ) ] ,
σ χ 2 ( high ) 0.5 0 d q e A q = 0.5 / A ,
q = - 4 π 2 k 2 L κ d K K Φ n ( K ) .
A = 12 / π 2 .
h ( κ , κ ) = ( 12 / π 2 ) [ θ ( κ - κ ) + κ δ ( κ - κ ) ] .
M S T = exp ( - 48 k 2 L { κ Ψ ( κ ) [ 1 - J 0 ( κ 2 U ) ] + κ d κ Ψ ( κ ) [ 1 - J 0 ( κ κ U ) ] } ) ,
U = L u ( 1 - u ) / k .
F ( y ) = 8.54 { 0.315 y - 5 / 6 [ 1 - J 0 ( 2 y ) ] + y 5 / 6 × 2 y d ξ ξ - 8 / 3 [ 1 - J 0 ( ξ ) ] } .
M S T = exp ( - 1.584 Y η { κ ˆ Ψ ˜ ( κ ˆ ) [ 1 - J 0 ( κ ˆ 2 V ) ] + κ ˆ d κ ˜ Ψ ˜ ( κ ˜ ) [ 1 - J 0 ( κ ˜ κ ˆ V ) ] } ) .
F ( y ) 9.45 y 5 / 6 ,             y 1 ,
F ( y ) 4.30 y - 5 / 6 ,             y 1.
σ χ 2 ( high ) = I 1 + I 2 ,
I 1 1.30 2 Y η ( 2 / V ) 0.5 0.2 d κ ˆ κ ˆ - 8 / 3 exp ( - 0.950 Y η κ ˆ - 5 / 3 ) ,
I 2 1.30 2 Y η 0.2 d κ ˆ κ ˆ Φ ˜ n ( κ ˆ ) exp [ - 1.584 Y η κ ˆ d κ ˜ κ ˜ Φ ˜ n ( κ ˜ ) ] ,
I 1 π 2 24 e - 13.89 Y η ,
q = - 1.584 Y η κ ˆ d κ ˆ κ ˆ Φ ˜ n ( κ ˜ )
I 2 π 2 24 { 1 - exp [ - 1.584 Y η 0.2 d κ ˜ κ ˜ Φ ˜ n ( κ ˜ ) ] } .
0.2 d κ ˜ κ ˜ Φ ˜ n ( κ ˜ ) = 9.5.
e - 13.89 Y p = 1 - e - 15.05 Y p .
κ ˆ = [ 1.584 12 ( 0.664 ) I Φ ] - 3 V - 1 = 5 / V ,
I Φ 0 d κ ˜ κ ˜ 3 Φ ˜ n ( κ ˜ ) = 2.94.
σ χ 2 ( low ) = 0 1 ( L 1 + L 2 ) d u ,
L 1 1.3 4 Y η V 2 5 / V ( 2 / V ) 0.5 d κ ˆ κ ˆ 4 / 3 exp ( - 0.664 Y η V 5 / 3 κ ˆ 5 / 3 ) ,
L 2 1.3 4 Y η V 2 0 5 / V d κ ˆ κ ˆ 4 / 3 exp ( - 1.584 12 I Φ Y η V 2 κ ˆ 2 ) .
L 1 0.35 Y η - 2 / 5 V - 1 / 3 9.72 Y η 1.18 Y η V 5 / 6 d z z 2 / 5 e - z ,
L 2 0.49 Y η - 1 / 6 V - 1 / 3 0 9.72 Y η d ξ ξ 1 / 6 e - ξ ,
1.417 Y P 7 / 30 0 9.72 Y P d z z 1 / 6 e - z = 9.72 Y P d ξ ξ 2 / 5 e - ξ ,
0 1 d u V - 1 / 3 = 2 ( L / k η ) - 2 / 3 .
Y l 0 k 2 L C n 2 l 0 5 / 3 = 28.1 Y η ,
ρ 0 l 0 implies Y l 0 1.34.
σ χ 2 1.30 4 L 3 C n 2 η - 7 / 3 0 1 d u [ u ( 1 - u ) ] 2 × 0 d κ ˆ κ ˆ 5 Φ ˜ n ( κ ˆ ) M S T ,
M S T exp { - 1.584 4 L 3 C n 2 η - 7 / 3 [ u ( 1 - u ) ] 2 [ κ ˆ 5 Ψ ˜ ( κ ˆ ) + κ ˆ 2 κ ˆ d κ ˜ κ ˜ 2 Ψ ˜ ( κ ˜ ) ] } ,
u * = K V Ū / ln ( h / h r ) ,
= u * 3 / K V h .
l 0 = 7.4 ( ν 3 / ) 1 / 4 ,
1 = 4 π 2 k 2 L 0 1 0 [ 1 - J 0 ( κ u ρ 0 ) ] Φ n ( κ ) κ d κ d u .
1 = 4 π 2 k 2 L 0 [ 1 - J 0 ( κ ρ 0 ) ] Ψ ( κ ) d κ .
1 = 4 π 2 0.033 Y η 0 [ 1 - J 0 ( κ ˜ ρ 0 / η ) ] Ψ ˜ ( κ ˜ ) d κ ˜ .
ρ 1 ( k 2 L C n 2 ) - 3 / 5 ,
ρ 2 ( k 2 L C n 2 l 0 - 1 / 3 ) - 1 / 2 .
ρ 0 1.44 ( k 2 L C n 2 ) - 3 / 5 = 1.44 l 0 Y l 0 - 3 / 5             for             Y l 0 0.1 ,
ρ 0 1.27 ( k 2 L C n 2 l 0 - 1 / 3 ) - 1 / 2 = 1.27 l 0 Y l 0 - 1 / 2             for             Y l 0 4.0.
ρ 0 l 0 implies Y l 0 1.34.