Abstract

The calculation results obtained with different outer-scale dependent atmospheric turbulence models are presented. Three outer-scale models are considered: the exponential model, the von Kármán model, and the Greenwood model. A generalization of the outer-scale models is suggested. The calculation results of the refractive-index structure functions and of the structure functions of phase and spatial–temporal Zernike coefficient characteristics are compared for the cases of different outer-scale models. In the case of Kolmogorov turbulence the analytical expressions for the temporal correlation functions and spectra of the Zernike coefficients are obtained. It is shown that all considered models give nearly the same results.

© 1995 Optical Society of America

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References

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  1. E. I. Gelfer, A. I. Kon, A. N. Cheremuhin, “The shifts correlation of the gravity centers of the focused light beam in the turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16, 245–253 (1973) (in Russian).
  2. G. W. Reinhardt, S. A. Collins, “Outer-scale effects in turbulence-degraded light-beam spectra,” J. Opt. Soc. Am. 62, 1526–1528 (1972).
    [CrossRef]
  3. D. P. Greenwood, D. O. Tarazano, “A proposed form for the atmospheric microtemperature spatial spectrum in the input range,” Rep. RADC-TR-74-19 (ADA 776294/1GI) (Rome Air Development Center, Hanscom Air Force Base, Mass., 1974).
  4. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  5. R. J. Sasiela, J. D. Shelton, “Transverse spectral filtering and Mellin transform techniques applied to the effect of outer scale on tilt and tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646–659 (1993).
    [CrossRef]
  6. V. V. Voitsekhovich, “The temporal characteristics of an adaptive astronomical system,” Prepr. IKI AN SSSR, 873, 1–24 (1984) (in Russian).
  7. V. V. Voitsekhovich, “The outer scale of the turbulence influence on the wave-front distortions,” Izv. AN SSSR Fiz. Atmosferi Okeana 22, 427–429 (1986) (in Russian).
  8. V. V. Voitsekhovich, V. B. Gubin, A. V. Mikulich, “The estimation of the adaptive astronomical system parameters on the basis of the experimental dates,” Opt. Atmosferi 1, 66–70 (1988) (in Russian).
  9. G. N. Watson, Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1948).
  10. O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions (Ellis Horwood, New York, 1983).
  11. Y. L. Luke, Mathematical Functions and Their Approximations (Academic, New York, 1975).
  12. J. Y. Wang, J. K. Markey, “Modal compensation of atmospheric turbulence phase distortion,” J. Opt. Soc. Am. 68, 78–87 (1978).
    [CrossRef]
  13. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  14. F. Roddier, M. J. Northcott, J. E. Graves, D. L. McKenna, D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations: time-delay and isoplanicity error in partial adaptive compensation,” J. Opt. Soc. Am. A 10, 957–965 (1993).
    [CrossRef]
  15. C. B. Hogge, R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. AP-24, 144–154 (1976).
    [CrossRef]

1993

1988

V. V. Voitsekhovich, V. B. Gubin, A. V. Mikulich, “The estimation of the adaptive astronomical system parameters on the basis of the experimental dates,” Opt. Atmosferi 1, 66–70 (1988) (in Russian).

1986

V. V. Voitsekhovich, “The outer scale of the turbulence influence on the wave-front distortions,” Izv. AN SSSR Fiz. Atmosferi Okeana 22, 427–429 (1986) (in Russian).

1984

V. V. Voitsekhovich, “The temporal characteristics of an adaptive astronomical system,” Prepr. IKI AN SSSR, 873, 1–24 (1984) (in Russian).

1978

1976

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
[CrossRef]

C. B. Hogge, R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. AP-24, 144–154 (1976).
[CrossRef]

1973

E. I. Gelfer, A. I. Kon, A. N. Cheremuhin, “The shifts correlation of the gravity centers of the focused light beam in the turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16, 245–253 (1973) (in Russian).

1972

Butts, R. R.

C. B. Hogge, R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. AP-24, 144–154 (1976).
[CrossRef]

Cheremuhin, A. N.

E. I. Gelfer, A. I. Kon, A. N. Cheremuhin, “The shifts correlation of the gravity centers of the focused light beam in the turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16, 245–253 (1973) (in Russian).

Collins, S. A.

Gelfer, E. I.

E. I. Gelfer, A. I. Kon, A. N. Cheremuhin, “The shifts correlation of the gravity centers of the focused light beam in the turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16, 245–253 (1973) (in Russian).

Graves, J. E.

Greenwood, D. P.

D. P. Greenwood, D. O. Tarazano, “A proposed form for the atmospheric microtemperature spatial spectrum in the input range,” Rep. RADC-TR-74-19 (ADA 776294/1GI) (Rome Air Development Center, Hanscom Air Force Base, Mass., 1974).

Gubin, V. B.

V. V. Voitsekhovich, V. B. Gubin, A. V. Mikulich, “The estimation of the adaptive astronomical system parameters on the basis of the experimental dates,” Opt. Atmosferi 1, 66–70 (1988) (in Russian).

Hogge, C. B.

C. B. Hogge, R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. AP-24, 144–154 (1976).
[CrossRef]

Kon, A. I.

E. I. Gelfer, A. I. Kon, A. N. Cheremuhin, “The shifts correlation of the gravity centers of the focused light beam in the turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16, 245–253 (1973) (in Russian).

Luke, Y. L.

Y. L. Luke, Mathematical Functions and Their Approximations (Academic, New York, 1975).

Marichev, O. I.

O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions (Ellis Horwood, New York, 1983).

Markey, J. K.

McKenna, D. L.

Mikulich, A. V.

V. V. Voitsekhovich, V. B. Gubin, A. V. Mikulich, “The estimation of the adaptive astronomical system parameters on the basis of the experimental dates,” Opt. Atmosferi 1, 66–70 (1988) (in Russian).

Noll, R. J.

Northcott, M. J.

Reinhardt, G. W.

Roddier, D.

Roddier, F.

Sasiela, R. J.

Shelton, J. D.

Tarazano, D. O.

D. P. Greenwood, D. O. Tarazano, “A proposed form for the atmospheric microtemperature spatial spectrum in the input range,” Rep. RADC-TR-74-19 (ADA 776294/1GI) (Rome Air Development Center, Hanscom Air Force Base, Mass., 1974).

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

Voitsekhovich, V. V.

V. V. Voitsekhovich, V. B. Gubin, A. V. Mikulich, “The estimation of the adaptive astronomical system parameters on the basis of the experimental dates,” Opt. Atmosferi 1, 66–70 (1988) (in Russian).

V. V. Voitsekhovich, “The outer scale of the turbulence influence on the wave-front distortions,” Izv. AN SSSR Fiz. Atmosferi Okeana 22, 427–429 (1986) (in Russian).

V. V. Voitsekhovich, “The temporal characteristics of an adaptive astronomical system,” Prepr. IKI AN SSSR, 873, 1–24 (1984) (in Russian).

Wang, J. Y.

Watson, G. N.

G. N. Watson, Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1948).

IEEE Trans. Antennas Propag.

C. B. Hogge, R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. AP-24, 144–154 (1976).
[CrossRef]

Izv. AN SSSR Fiz. Atmosferi Okeana

V. V. Voitsekhovich, “The outer scale of the turbulence influence on the wave-front distortions,” Izv. AN SSSR Fiz. Atmosferi Okeana 22, 427–429 (1986) (in Russian).

Izv. Vyssh. Uchebn. Zaved. Radiofiz.

E. I. Gelfer, A. I. Kon, A. N. Cheremuhin, “The shifts correlation of the gravity centers of the focused light beam in the turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16, 245–253 (1973) (in Russian).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Atmosferi

V. V. Voitsekhovich, V. B. Gubin, A. V. Mikulich, “The estimation of the adaptive astronomical system parameters on the basis of the experimental dates,” Opt. Atmosferi 1, 66–70 (1988) (in Russian).

Prepr. IKI AN SSSR

V. V. Voitsekhovich, “The temporal characteristics of an adaptive astronomical system,” Prepr. IKI AN SSSR, 873, 1–24 (1984) (in Russian).

Other

G. N. Watson, Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1948).

O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions (Ellis Horwood, New York, 1983).

Y. L. Luke, Mathematical Functions and Their Approximations (Academic, New York, 1975).

D. P. Greenwood, D. O. Tarazano, “A proposed form for the atmospheric microtemperature spatial spectrum in the input range,” Rep. RADC-TR-74-19 (ADA 776294/1GI) (Rome Air Development Center, Hanscom Air Force Base, Mass., 1974).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

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Figures (7)

Fig. 1
Fig. 1

Refractive-index structure functions: initial parts.

Fig. 2
Fig. 2

Refractive-index structure functions: middle and asymptotical parts.

Fig. 3
Fig. 3

Structure functions of phase. Initial parts.

Fig. 4
Fig. 4

Structure functions of phase. Middle and asymptotical parts.

Fig. 5
Fig. 5

Mean square of the tilt: dependence on the outer scale.

Fig. 6
Fig. 6

Temporal correlation functions of the tilt.

Fig. 7
Fig. 7

Temporal spectra of the tilt.

Equations (43)

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D n ( r ) = 8 π × 0.033 C n 2 0 d x x 2 [ 1 - sin ( x r ) x r ] Φ n ( x , L 0 ) ,
lim L 0 Φ n ( x , L 0 ) = x - 11 / 3 ,
i r d x [ x - 11 / 3 - Φ n ( x , L 0 ) ] 2 = min ,
0 d x x 2 Φ n ( x , L 0 )
0 d x x Φ n ( x , L 0 )
Φ n ( x , L 0 ) = x - 11 / 3 [ 1 - f ( x / x 0 ) ] .
( * ) lim L 0 f ( x / x 0 ) = 0 , ( * * ) f ( x / x 0 ) x - 0 = 1 - O ( x α ) , α > 5 / 3 , ( * * * ) f ( x / x 0 ) x - < A x - δ , δ > 2 / 3.
Φ n ( x , L 0 ) = x - β ( x γ + x 0 b γ ) - η .
β + γ η = 11 / 3 , β > 0 , γ > 0 , η > 0 , β < 1 / γ η + 2.
Φ n E ( x , L 0 ) = x - 11 / 3 [ 1 - exp ( - x 2 / x 01 2 ) ] .
Φ n K ( x , L 0 ) = ( x 2 + x 02 2 ) - 11 / 6 .
Φ n G ( x , L 0 ) = ( x 2 + x x 03 ) - 11 / 6 .
x 01 = k 01 / L 0 , x 02 = k 02 / L 0 , x 03 = k 03 / L 0 ,
f = exp ( - x 2 / x 0 d 2 ) ,             x 0 d = x 01 .
β = 0 , γ = 2 , η = 11 / 6 , x 0 b = x 02 , β = 11 / 6 , γ = 1 , η = 11 / 6 , x 0 b = x 03 ,
x 01 = 2.04 x 02 = 1.88 x 03 .
x 01 = 2 π / L 0 ,             x 02 = 3.075 / L 0 ,             x 01 = 3.349 / L 0 .
D n E ( r ) = 6.88 ( r r n ) 2 / 3 + 11.59 ( x 01 r n ) - 2 / 3 × [ 1 - 1 F 1 ( - 1 3 ; 3 2 ; - r 2 x 01 2 4 ) ] ,
D n K ( r ) = 6.88 ( r r n ) 2 / 3 F 0 1 ( ; 4 3 ; r 2 x 02 2 4 ) + 7.20 ( x 02 r n ) - 2 / 3 [ 1 - F 0 1 ( ; 2 3 ; r 2 x 02 2 4 ) ] ,
D n G ( r ) = 6.88 ( r r n ) 2 / 3 F 2 3 ( 11 12 , 17 12 ; 4 3 , 11 6 , 1 2 ; - r 2 x 03 2 4 ) + 7.62 ( x 03 r n ) - 2 / 3 × [ 1 - F 2 3 ( 7 12 , 13 12 ; 2 3 , 1 6 , 3 2 ; - r 2 x 03 2 4 ) ] - 8.18 r x 03 ( r r n ) 2 / 3 × F 2 3 ( 17 12 , 23 12 ; 7 3 , 11 6 , 3 2 ; - r 2 x 03 2 4 ) ,
D n ( r ) = 6.88 ( r / r n ) 2 / 3 .
D n K ( r ) = 7.20 ( x 02 r n ) - 2 / 3 - 4.27 ( r / r n ) 1 / 3 ( x 02 r n ) - 1 / 3 K 1 / 3 ( r x 02 ) .
D n E ( r ) ~ 11.59 ( x 01 r n ) - 2 / 3 [ 1 - 0.66 ( r x 01 ) - 4 / 3 ] ,             r x 01 ,
D n K ( r ) ~ 7.20 ( x 02 r n ) - 2 / 3 [ 1 - 0.74 ( r x 02 ) - 1 / 6 exp ( - r x 02 ) ] ,             r x 02 ,
D n G ( r ) ~ 7.62 ( x 03 r n ) - 2 / 3 [ 1 - 1.08 ( r x 03 ) - 7 / 6 ] ,             r x 03 .
D S ( r ) = 6.16 r 0 - 5 / 3 0 d x x [ 1 - J 0 ( x r ) ] Φ n ( x , L 0 ) ,
D S E ( r ) = 6.88 ( r r 0 ) 5 / 3 + 20.56 ( x 01 r 0 ) - 5 / 3 × [ 1 - F 1 1 ( - 5 6 ; 1 ; - r 2 x 01 2 4 ) ] ,
D S K ( r ) = 6.88 ( r r 0 ) 5 / 3 F 0 1 ( ; 11 6 ; r 2 x 02 2 4 ) + 3.69 ( x 02 r 0 ) - 5 / 3 [ 1 - F 0 1 ( ; 1 6 ; r 2 x 02 2 4 ) ] ,
D S G ( r ) = 6.88 ( r r 0 ) 5 / 3 F 2 3 ( 11 12 , 17 12 ; 11 6 , 11 6 , 1 2 ; - r 2 x 03 2 4 ) + 32.89 ( x 03 r 0 ) - 5 / 3 × [ 1 - F 2 3 ( 1 12 , 17 12 ; 1 6 , - 1 3 , 1 ; - r 2 x 03 2 4 ) ] - 2.27 r x 03 ( r r 0 ) 5 / 3 × F 2 3 ( 17 12 , 23 12 ; 7 3 , 7 3 , 3 2 ; - r 2 x 03 2 4 ) .
D n K ( r ) = 3.694 [ ( x 02 r 0 ) - 5 / 3 - ( r / r 0 ) 5 / 6 ( x 02 r 0 ) - 5 / 6 K - 5 / 6 ( r x 02 ) ] .
D S E ( r ) ~ 20.56 ( x 01 r 0 ) - 5 / 3 [ 1 - 0.93 ( r x 01 ) - 1 / 3 ] ,             r x 01 ,
D S K ( r ) ~ 3.69 ( x 02 r 0 ) - 5 / 3 [ 1 - 1.247 ( r x 02 ) 1 / 3 exp ( - r x 02 ) ] ,             r x 02 ,
D S G ( r ) ~ 32.89 ( x 03 r 0 ) - 5 / 3 [ 1 - 1.85 ( r x 03 ) - 1 / 6 ] ,             r x 03 .
B l ( τ ) = 12.32 ( n + 1 ) D - 2 r 0 - 5 / 3 0 d x x - 1 J n + 1 2 ( x ) × Φ n ( 2 D x , L 0 ) [ J 0 ( 2 v τ D x ) + ( - 1 ) l + m ( 1 - δ 0 m ) × J 2 m ( 2 v τ D x ) cos ( 2 m θ ) ] ,
a l 2 = B l ( 0 ) = 12.32 ( n + 1 ) D - 2 r 0 - 5 / 3 × 0 d x x - 1 J n + 1 2 ( x ) Φ n ( 2 D x , L 0 ) .
W l ( ν ) = 2 0 d τ B l ( τ ) cos ( 2 π ν τ ) .
b l ( τ ) = B l ( τ ) / B l ( 0 ) , w l ( τ ) = W l ( ν ) / 0 d ν W l ( ν ) .
w l ( ν ) = 4 0 d τ b l ( τ ) cos ( 2 π ν τ ) .
ν w l ( ν ) = 2 π 1 d x x - 1 J n + 1 2 ( ν p x ) ( x 2 - 1 ) - 1 / 2 Φ n ( 2 D ν p x , L 0 ) × [ 1 + ( - 1 ) l + m ( 1 - δ 0 m ) cos ( 2 m arcsin 1 x ) × cos ( 2 m θ ) ] / 0 d x x - 1 J n + 1 2 ( x ) Φ n ( 2 D x , L 0 ) ,
B l ( τ ) = 0.97 ( n + 1 ) ( D / r 0 ) 5 / 3 [ I b ( n , 0 , τ ) + I b ( n , m , τ ) ( - 1 ) l + m ( 1 - δ 0 m ) cos ( 2 m θ ) ] ,
I b ( n , m , τ ) = ( 2 τ p ) 2 m × 2 - 14 / 3 Γ ( ¹⁴ / - 2 m ) Γ ( n + m - / ) Γ ( n - m + ²³ / ) Γ 2 ( ¹⁷ / - m ) Γ ( 2 m + 1 ) × F 3 2 ( n + m - / , m - n - ¹⁷ / , m - ¹¹ / ; m - / , 2 m + 1 ; τ p 2 ) + ( 2 τ p ) 14 / 3 2 - 2 m Γ ( 2 m - ¹⁴ / ) Γ ( m + ¹⁰ / ) Γ ( m - ¹¹ / ) Γ ( ½ ) × F 3 2 ( n + ³ / , - n - ½ , ½ ; ¹⁰ / - m , ¹⁰ / + m ; τ p 2 )             if τ p 1 ; I b ( n , m , τ ) = ( τ p ) 5 / 3 - 2 n 2 - 3 Γ ( n + m - / ) Γ ( m - n + ¹¹ / ) Γ 2 ( n + 2 ) × F 3 2 ( n + ³ / , n + m - / , n - m - / ; n + 2 , 2 n + 3 ; 1 / τ p 2 )             if τ p 1 , τ p = v τ / D .
ν w l ( ν ) = 1 I 0 [ I w ( n , 0 , ν p ) + I w ( n , m , ν p ) ( - 1 ) l + m ( 1 - δ 0 m ) cos ( 2 m θ ) ] ,
I w ( n , m , ν p ) = ( ν p 2 ) 2 n - 5 / 3 × π - 1 / 2 2 - 11 / 3 cos ( m π ) Γ ( / - n ) Γ ( ¹¹ / - n ) Γ ( ¹¹ / - n - m ) Γ ( ¹¹ / - n + m ) Γ 2 ( n + 2 ) × F 3 4 ( n + ³ / , n + m - / , n - m - / ; n - , n - / , n + 2 , 2 n + 3 ; - ν p 2 ) + ν p π - 1 2 - 14 / 3 Γ ( n - / ) Γ ( ¹⁷ / ) Γ ( n + ¹³ / ) Γ 2 ( ¹⁰ / ) × F 3 4 ( ¹⁷ / , ½ + m , ½ - m ; / - n , n + ¹³ / , ½ ; ¹⁰ / ; - ν p 2 ) , ν p = π ν D / v , I 0 = 2 - 14 / 3 Γ ( ¹⁴ / ) Γ ( n - / ) Γ ( n + ²³ / ) Γ 2 ( ¹⁷ / ) .

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