Abstract

We have investigated five types of atmospheric optical-turbulence inner scales for their effects on normalized laser irradiance variance in the Rytov and early saturation regimes: (1) zero inner scale, (2) Gaussian inner scale, (3) Hill’s viscous-convective enhancement inner scale, (4) Frehlich’s parameterization of the viscous-convective enhancement, and (5) turbulence spectrum truncation because of the discrete grid representation. Wave-optics computer simulations yielded normalized irradiance variances within 2% of the results from numerical integrations of the Rytov–Tatarskii predictions. In the Rytov regime a Gaussian inner scale reduces the normalized irradiance variance compared with the zero-inner-scale case, and the viscous-convective inner scale first raises, then lowers the irradiance variance as the inner-scale size increases. In the saturation regime all inner-scale models increase the intensity variance for a spherical wave.

© 1994 Optical Society of America

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References

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  1. V. I. Tatarskii, Wave Propagation in a Turbulent Medium, R. A. Silverman, trans. (McGraw-Hill, New York, 1961).
  2. J. M. Martin, S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27, 2111–2126 (1988).
    [CrossRef] [PubMed]
  3. J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 805–814 (1986).
    [CrossRef]
  4. J. M. Martin, S. M. Flatté, “Simulation of point-source scintillation through three-dimensional random media,” J. Opt. Soc. Am. A 7, 838–847 (1990).
    [CrossRef]
  5. S. M. Flatté, G. Wang, J. Martin, “Irradiance variance of optical waves through atmospheric turbulence by numerical simulation and comparison with experiment,” J. Opt. Soc. Am. A 10, 2363–2370 (1993).
    [CrossRef]
  6. R. J. Hill, S. F. Clifford, “Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation,” J. Opt. Soc. Am. 68, 892–899 (1978).
    [CrossRef]
  7. R. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
    [CrossRef]
  8. B. Ellerbroek, Star Field Optical Range, U.S. Air Force Phillips Laboratory, Kirtland Air Force Base, N.M. 87117 (personal communication, March1993).
  9. P. H. Roberts, “A wave optics propagation code,” Rep. TR-760 (Optical Sciences Company, Placentia, Calif., 1986).
  10. D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
    [CrossRef]

1993 (1)

1992 (1)

R. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
[CrossRef]

1990 (1)

1988 (1)

1986 (1)

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 805–814 (1986).
[CrossRef]

1983 (1)

D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
[CrossRef]

1978 (1)

Clifford, S. F.

Codona, J. L.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 805–814 (1986).
[CrossRef]

Creamer, D. B.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 805–814 (1986).
[CrossRef]

Ellerbroek, B.

B. Ellerbroek, Star Field Optical Range, U.S. Air Force Phillips Laboratory, Kirtland Air Force Base, N.M. 87117 (personal communication, March1993).

Flatté, S. M.

Frehlich, R.

R. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
[CrossRef]

Frehlich, R. G.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 805–814 (1986).
[CrossRef]

Henyey, F. S.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 805–814 (1986).
[CrossRef]

Hill, R. J.

Knepp, D. L.

D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
[CrossRef]

Martin, J.

Martin, J. M.

Roberts, P. H.

P. H. Roberts, “A wave optics propagation code,” Rep. TR-760 (Optical Sciences Company, Placentia, Calif., 1986).

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium, R. A. Silverman, trans. (McGraw-Hill, New York, 1961).

Wang, G.

Appl. Opt. (1)

J. Atmos. Sci. (1)

R. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Proc. IEEE (1)

D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
[CrossRef]

Radio Sci. (1)

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 805–814 (1986).
[CrossRef]

Other (3)

V. I. Tatarskii, Wave Propagation in a Turbulent Medium, R. A. Silverman, trans. (McGraw-Hill, New York, 1961).

B. Ellerbroek, Star Field Optical Range, U.S. Air Force Phillips Laboratory, Kirtland Air Force Base, N.M. 87117 (personal communication, March1993).

P. H. Roberts, “A wave optics propagation code,” Rep. TR-760 (Optical Sciences Company, Placentia, Calif., 1986).

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Figures (7)

Fig. 1
Fig. 1

Inner scales for the three-dimensional spectrum of the refractive-index fluctuations: solid curves, Hill viscous-convective enhancement inner scales; dashed curves, Frehlich viscous-convective enhancement inner scales; dotted curves, Gaussian inner scales; box, numerical-grid cutoff for a 1024 × 1024 mesh. The inner scale values of 2, 3, 4, 5, 10, and 15 cm relate to stratospheric propagation over 200 km, with λ = 5 × 10−7 m.

Fig. 2
Fig. 2

Normalized irradiance variance in the Rytov regime for Gaussian inner scales: thick dashed line, Rytov–Tatarskii theory with a zero inner scale; dotted lines, numerical integration of Eq. (7) with Gaussian inner scales; solid lines, computer simulation with Gaussian inner scales.

Fig. 3
Fig. 3

Normalized irradiance variance for β0 2 = 5 × 10−4 and Gaussian inner scales for numerical-grid cutoff and 5, 10, and 15 cm: dotted curve, numerical integration of Eq. (7); solid curve computer-simulation results. The normalized irradiance variation decreased almost linearly with increasing inner-scale size.

Fig. 4
Fig. 4

Normalized irradiance variance in the Rytov regime for the Hill viscous-convective enhancement inner scale: thick dashed line, Rytov–Tatarskii theory with a zero inner scale; dotted lines, numerical integration of the Rytov–Tatarskii results, Eq. (7), with Hill inner scales; solid lines, computer simulation with Hill inner scales.

Fig. 5
Fig. 5

Normalized irradiance variance for β0 2 = 5 × 10−4 and viscous-convective enhancement inner scales of numerical-grid cutoff and 2, 3, 4, 5, 6, 7, 10, and 15 cm: dotted curve, numerical integration of Eq. (7) with the Hill inner scale; solid curve, computer simulation with the Hill inner scale; dashed curve, computer simulation with the Frehlich inner scale. The normalized irradiance variance reached a maximum for l 0 ≈ 4 cm. The Hill and Frehlich versions agreed within 3%.

Fig. 6
Fig. 6

Normalized irradiance variances from computer simulation for a spherical wave with zero inner scale (solid curve) and Gaussian inner scales of 5 cm (dashed curve), 10 cm (dashed–dotted curve), and 15 cm (dotted curve). In the Rytov regime the normalized irradiance variance monotonically decreased with increasing inner-scale size. In the saturation regime the normalized irradiance variance monotonically increased with increasing inner-scale size. A transition in the behavior occurred around β0 2 ≈ 1–3.

Fig. 7
Fig. 7

Normalized irradiance variances from computer simulation for zero-inner-scale (solid curve) and Hill viscous-convective inner scales of 5 cm (dashed curve), 10 cm (dashed-dotted curve), and 15 cm (dotted curve).

Equations (18)

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2 u + k 2 n 2 ( r ) u = 0.
n ( r ) = 1 + n 1 ( r ) ,
2 Ψ 1 + 2 Ψ 0 Ψ 1 + 2 k 2 n 1 ( r ) = 0.
Ψ 1 ( r ) = k 2 2 π u 0 ( r ) v n 1 ( r ) u 0 ( r ) exp ( i k r - r ) r - r d V .
u 0 ( r ) = Q exp ( i k r ) r ,             Q = constant ,
Ψ 1 ( r ) k 2 z 2 π v n 1 ( r ) z ( z - z ) exp [ i k z 2 ( x 2 + y 2 ) + z 2 ( x 2 + y 2 ) - 2 z z ( x x + y y ) 2 z z ( z - z ) ] d V .
X 2 ¯ = 4 π 2 k 2 0 d K K Φ n ( K ) 0 L d z sin 2 [ K 2 z ( L - z ) 2 k L ] ,
σ I 2 I ¯ 2 = exp ( 4 X 2 ¯ ) - 1.
X 2 ¯ = 0.124 C n 2 k 7 / 6 L 11 / 6 .
σ I 2 I ¯ 2 = exp ( 4 X 2 ¯ ) - 1 = exp ( 0.497 C n 2 k 7 / 6 L 11 / 6 ) - 1.
σ I 2 I ¯ 2 0.497 C n 2 k 7 / 6 L 11 / 6 β 0 2 ,
Φ n ( K ) = 0.033 C n 2 K - 11 / 3 F ( K l 0 ) ,
F ( K l 0 ) = 1 ,             0 < K < .
F ( K l 0 ) = exp [ - ( K l 0 5.9 ) 2 ]
F ( K l 0 ) = { 1 , 0 < K < K max 0 , K > K max .
u ( r , z ) = - i λ z d ρ u ( ρ , 0 ) exp [ - 2 π i λ ( z 2 + r - ρ 2 ) 1 / 2 ] .
u ( r , z ) = 1 2 π d K exp ( i K · r ) exp ( - i π λ z K 2 ) × d ρ u ( ρ , 0 ) exp ( - i K · ρ ) .
ϕ = Re ( 0.0984 k ( C n 2 Δ z ) 1 / 2 ( N δ x ) 5 / 6 × F T { ( n x 2 + n y 2 ) - 11 / 12 [ F ( K l 0 ) ] 1 / 2 G ( n x , n y ) } ) ,

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