Abstract

The outer-scale influence on spatial and temporal characteristics of turbulence-induced wave-front distortions is discussed. The calculation results for the image centroid and the Zernike modes are presented. Two methods of the outer-scale estimation based on tilt and defocus and on image centroid measurements are suggested. The application of the results obtained to adaptive optics problems is considered. Finally, a possibility of prediction of the wave-front statistical characteristics is discussed.

© 1995 Optical Society of America

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  1. J. M. Marlotti, G. P. Di Benedetto, “Pathlength stability of synthetic aperture telescopes: the case of a 25 cm Cerga interferometer,” in Proc. I.A.U. Colloq. 79, 257–265 (1984).
  2. C. E. Coulman, J. Vernin, Y. Coqueugniot, J. L. Caccia, “Outer scale of turbulence appropriate to modeling refractive-index structure profiles,” Appl. Opt. 27, 155–160 (1988).
    [CrossRef] [PubMed]
  3. V. V. Voitsekhovich, V. B. Gubin, A. V. Mikulich, “The estimation of parameters of an adaptive astronomical system on the basis of experimental data,” Opt. Atmosferi. 1, 66–70 (1988) (in Russian).
  4. M. Bester, W. C. Duchi, C. G. Degiacomi, L. J. Greenbill, C. H. Townes, “Atmospheric fluctuations: empirical structure function and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
    [CrossRef]
  5. N. Takato, M. Iye, I. Yamaguchi, “Atmospheric turbulence of small outer scale,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (European Southern Observatory, Garching, Germany, 1993), pp. 15–20.
  6. D. F. Buscher, J. T. Armstrong, C. A. Hummel, A. Quirrenbach, D. Mozurkewich, K. J. Johnston, C. S. Denison, M. M. Colavita, M. Shao, “Interferometric seeing measurements on Mt. Wilson: power spectra and outer scales,” Appl. Opt. 34, 1081–1096 (1995).
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  10. A. Consortini, L. Ronchi, E. Moroder, “Role of the outer scale of turbulence in atmospheric degradation of optical images,” J. Opt. Soc. Am. 63, 1246–1248 (1973).
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  11. 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, 646–660 (1993).
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  12. D. M. Winker, “Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence,” J. Opt. Soc. Am. A 8, 1568–1573 (1991).
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  13. H. T. Yura, M. T. Tavis, “Centroid anisoplanatism,” J. Opt. Soc. Am. A 2, 765–773 (1985).
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  15. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
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  17. 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).
  18. 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, Hanscon Air Force Base, Mass., 1974).
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  20. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
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  21. V. V. Voitsekhovich, “Temporal characteristics of an adaptive astronomical system,” Prepr. IKI AN SSSR 873, 1–24 (1984) (in Russian).
  22. M. Sarazin, F. Roddier, “The ESO differential image motion monitor,” Astron. Astrophys. 227, 294–300 (1990).
  23. F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt. 29, 1402–1403 (1990).
    [CrossRef] [PubMed]

1995 (1)

1993 (2)

1992 (1)

M. Bester, W. C. Duchi, C. G. Degiacomi, L. J. Greenbill, C. H. Townes, “Atmospheric fluctuations: empirical structure function and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

1991 (1)

1990 (2)

M. Sarazin, F. Roddier, “The ESO differential image motion monitor,” Astron. Astrophys. 227, 294–300 (1990).

F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt. 29, 1402–1403 (1990).
[CrossRef] [PubMed]

1988 (2)

C. E. Coulman, J. Vernin, Y. Coqueugniot, J. L. Caccia, “Outer scale of turbulence appropriate to modeling refractive-index structure profiles,” Appl. Opt. 27, 155–160 (1988).
[CrossRef] [PubMed]

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

1985 (1)

1984 (2)

J. M. Marlotti, G. P. Di Benedetto, “Pathlength stability of synthetic aperture telescopes: the case of a 25 cm Cerga interferometer,” in Proc. I.A.U. Colloq. 79, 257–265 (1984).

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

1982 (1)

1979 (1)

1978 (1)

1976 (1)

1973 (2)

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).

A. Consortini, L. Ronchi, E. Moroder, “Role of the outer scale of turbulence in atmospheric degradation of optical images,” J. Opt. Soc. Am. 63, 1246–1248 (1973).
[CrossRef]

1972 (1)

1966 (1)

Armstrong, J. T.

Bester, M.

M. Bester, W. C. Duchi, C. G. Degiacomi, L. J. Greenbill, C. H. Townes, “Atmospheric fluctuations: empirical structure function and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Buscher, D. F.

Caccia, J. L.

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).

Colavita, M. M.

Collins, S. A.

Consortini, A.

Coqueugniot, Y.

Coulman, C. E.

Degiacomi, C. G.

M. Bester, W. C. Duchi, C. G. Degiacomi, L. J. Greenbill, C. H. Townes, “Atmospheric fluctuations: empirical structure function and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Denison, C. S.

Di Benedetto, G. P.

J. M. Marlotti, G. P. Di Benedetto, “Pathlength stability of synthetic aperture telescopes: the case of a 25 cm Cerga interferometer,” in Proc. I.A.U. Colloq. 79, 257–265 (1984).

Duchi, W. C.

M. Bester, W. C. Duchi, C. G. Degiacomi, L. J. Greenbill, C. H. Townes, “Atmospheric fluctuations: empirical structure function and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Fried, D. L.

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.

Greenbill, L. J.

M. Bester, W. C. Duchi, C. G. Degiacomi, L. J. Greenbill, C. H. Townes, “Atmospheric fluctuations: empirical structure function and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

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, Hanscon Air Force Base, Mass., 1974).

Gubin, V. B.

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

Hummel, C. A.

Iye, M.

N. Takato, M. Iye, I. Yamaguchi, “Atmospheric turbulence of small outer scale,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (European Southern Observatory, Garching, Germany, 1993), pp. 15–20.

Johnston, K. J.

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).

Markey, J. K.

Marlotti, J. M.

J. M. Marlotti, G. P. Di Benedetto, “Pathlength stability of synthetic aperture telescopes: the case of a 25 cm Cerga interferometer,” in Proc. I.A.U. Colloq. 79, 257–265 (1984).

McKenna, D. L.

Mikulich, A. V.

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

Moroder, E.

Mozurkewich, D.

Noll, R. J.

Northcott, M. J.

Quirrenbach, A.

Reinhardt, G. W.

Roddier, D.

Roddier, F.

Ronchi, L.

Sarazin, M.

M. Sarazin, F. Roddier, “The ESO differential image motion monitor,” Astron. Astrophys. 227, 294–300 (1990).

Sasiela, R. J.

Shao, M.

Shelton, J. D.

Takato, N.

N. Takato, M. Iye, I. Yamaguchi, “Atmospheric turbulence of small outer scale,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (European Southern Observatory, Garching, Germany, 1993), pp. 15–20.

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, Hanscon Air Force Base, Mass., 1974).

Tatarski, V. I.

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

Tavis, M. T.

Teague, M. R.

Townes, C. H.

M. Bester, W. C. Duchi, C. G. Degiacomi, L. J. Greenbill, C. H. Townes, “Atmospheric fluctuations: empirical structure function and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Vernin, J.

Voitsekhovich, V. V.

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

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

Wang, J. V.

Wang, J. Y.

Winker, D. M.

Yamaguchi, I.

N. Takato, M. Iye, I. Yamaguchi, “Atmospheric turbulence of small outer scale,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (European Southern Observatory, Garching, Germany, 1993), pp. 15–20.

Yura, H. T.

Appl. Opt. (3)

Astron. Astrophys. (1)

M. Sarazin, F. Roddier, “The ESO differential image motion monitor,” Astron. Astrophys. 227, 294–300 (1990).

Astrophys. J. (1)

M. Bester, W. C. Duchi, C. G. Degiacomi, L. J. Greenbill, C. H. Townes, “Atmospheric fluctuations: empirical structure function and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (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).

J. Opt. Soc. Am. (7)

J. Opt. Soc. Am. A (4)

Opt. Atmosferi. (1)

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

Prepr. IKI AN SSSR (1)

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

Proc. I.A.U. Colloq. (1)

J. M. Marlotti, G. P. Di Benedetto, “Pathlength stability of synthetic aperture telescopes: the case of a 25 cm Cerga interferometer,” in Proc. I.A.U. Colloq. 79, 257–265 (1984).

Other (3)

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

N. Takato, M. Iye, I. Yamaguchi, “Atmospheric turbulence of small outer scale,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (European Southern Observatory, Garching, Germany, 1993), pp. 15–20.

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, Hanscon Air Force Base, Mass., 1974).

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Figures (10)

Fig. 1
Fig. 1

Temporal correlation functions of the IC (solid curves) and tilt (dashed curves) for the case of wind direction θ = π/4. The outer scale strongly affects these functions.

Fig. 2
Fig. 2

Temporal correlation functions of the IC for different wind directions θ. The whole range of variation of these functions with θ is shown.

Fig. 3
Fig. 3

Temporal correlation functions of the first four Zernike modes. The wind direction θ is chosen from the equation cos 2 = 0. The correlation time decreases with an increase in the order of aberration. Def., defocus; ast., astigmatism.

Fig. 4
Fig. 4

Temporal correlation functions of the defocus (def.) and coma. The coma’s functions are shown for the case of wind direction θ = π/4. The outer scale affects these functions weakly. The outer-scale influence falls off with an increase in the order of aberration.

Fig. 5
Fig. 5

Mean squares of the Zernike coefficients. Normalized values are shown. The effect of the outer scale is significant for the tilt in the case of a small value of L0/D. Def., defocus; ast., astigmatism.

Fig. 6
Fig. 6

Tilt contribution to the total wave-front distortion and the residual error after tip-tilt correction. The effect of the outer scale is significant in the case of a small outer-scale value or a large aperture size.

Fig. 7
Fig. 7

Mechanism of the outer-scale influence. This plot illustrates how the energy of distortion transmits from the tilts into higher-order aberrations with a decrease in the outer scale.

Fig. 8
Fig. 8

Functions (a) ξ0, ξ01, and (b) ξ02. These functions depend only on the outer scale. They can be obtained from experiment, and each can be used for an estimation of the outer-scale value.

Fig. 9
Fig. 9

Functions (a) ξ0, ξ01, and (b) ξ02 for the case of a small aperture. The curves show that these functions allow us to estimate the outer-scale value using a small aperture size.

Fig. 10
Fig. 10

Error of the outer-scale estimation associated with a finite integration range. The curves are of help in choosing the needed integration time T. They can also be used to determine the error of the outer-scale estimation for given T.

Equations (36)

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ρ c ( t ) = d 2 r r I ( r , t ) / d 2 r I ( r , t ) ,
ρ c ( t ) = f k R d 2 ρ S ( ρ , t ) ,
B ρ ( τ ) = { x c ( t ) x c ( t + τ ) , y c ( t ) y c ( t + τ ) } ,
B ρ ( τ ) = f 2 k 2 R d 2 ρ 1 d 2 ρ 2 ρ 1 , ρ 2 B s ( ρ 1 , ρ 2 , τ ) ,
B s ( ρ 1 , ρ 2 , τ ) = 0.21 k 2 0 H d h C n 2 ( h ) d 2 χ Φ n ( χ , L 0 ) × exp { i χ [ ρ 1 ρ 2 υ ( h ) τ ] } ,
B s ( ρ 1 , ρ 2 , τ ) = 0.49 r 0 5 / 3 d 2 χ Φ n ( χ , L 0 ) × exp { i χ ( ρ 1 ρ 2 υ τ ) } ,
r 0 = 1.68 [ k 2 0 H d h C n 2 ( h ) ] 3 / 5 .
B ρ ( τ ) = 4.78 ( D / r 0 ) 5 / 3 ( f D k ) 2 0 d χ χ Φ n ( χ , L 0 / D ) J 1 2 ( χ ) × [ J 0 ( 2 υ | τ | D χ ) ± J 2 ( 2 υ | τ | D χ ) cos ( 2 θ ) ] ,
Φ n ( χ , L 0 / D ) = χ 11 / 3 { 1 exp [ ( χ L 0 / π D ) 2 ] } .
Z l even ( 2 ρ / D ) = 8 ( n + 1 ) R n m ( 2 ρ / D ) cos ( m φ ) , m 0 , Z l odd ( 2 ρ / D ) = 8 ( n + 1 ) R n m ( 2 ρ / D ) sin ( m φ ) , m 0 , Z l ( 2 ρ / D ) = 4 ( n + 1 ) R n 0 ( 2 ρ / D ) , m = 0 ,
S ( ρ , t ) = l = 1 a l ( t ) Z l ( 2 ρ / D ) .
a l ( t ) = 1 π D 2 R d 2 ρ S ( ρ , t ) Z l ( 2 ρ / D ) .
B l ( τ ) = a l ( t ) a l ( t + τ ) .
B l ( τ ) = ( 1 π D 2 ) 2 R d 2 ρ 1 d 2 ρ 2 B s ( ρ 1 , ρ 2 , τ ) × Z l ( 2 ρ 1 / D ) Z l ( 2 ρ 2 / D ) .
B l ( τ ) = 0.97 ( n + 1 ) ( D / r 0 ) 5 / 3 0 d χ χ 1 Φ n ( χ , L 0 / D ) × J n + 1 2 ( χ ) [ J 0 ( 2 υ | τ | D χ ) + ( 1 ) l + m ( 1 δ 0 m ) × J 2 m ( 2 υ | τ | D χ ) cos ( 2 m θ ) ] ,
B 2 , 3 ( τ ) = 1.94 ( D / r 0 ) 5 / 3 0 d χ χ 1 Φ n ( χ , L 0 / D ) J 2 2 ( χ ) × [ J 0 ( 2 υ | τ | D χ ) J 2 ( 2 υ | τ | D χ ) cos ( 2 θ ) ] ,
B 4 ( τ ) = 3.88 ( D / r 0 ) 5 / 3 0 d χ χ 1 Φ n ( χ , L 0 / D ) × J 3 2 ( χ ) J 0 ( 2 υ | τ | D χ ) .
B t + ( τ ) = B 2 ( τ ) + B 3 ( τ ) ; b t + ( τ ) = B t + ( τ ) / B t + ( 0 ) ; B t ( τ ) = B 2 ( τ ) B 3 ( τ ) ; b t ( τ ) = B t ( τ ) / B t ( 0 ) ; b d ( τ ) = B 4 ( τ ) / B 4 ( 0 ) .
τ k = 0 d τ τ k B ( τ ) .
τ 0 t + = D 2 υ Ω ( 2 , 2 ) / Ω ( 1 , 2 ) , τ 0 t = D 2 υ cos ( 2 θ ) Ω ( 2 , 2 ) / Ω ( 1 , 2 ) , τ 0 d = D 2 υ Ω ( 2 , 3 ) / Ω ( 1 , 3 ) ,
Ω ( α , l ) = 0 d χ χ α Φ n ( χ , L 0 / D ) J l 2 ( χ ) .
ξ 0 ( L 0 / D ) = τ 0 t + / τ 0 d .
B x , y ( τ ) = 4.78 ( D / r 0 ) 5 / 3 ( f D k ) 2 0 d χ χ Φ n ( χ , L 0 / D ) J 1 2 ( χ ) × [ J 0 ( 2 υ | τ | D χ ) J 2 ( 2 υ | τ | D χ ) cos ( 2 θ ) ] ,
B + ( τ ) = B x ( τ ) + B y ( τ ) ; b + ( τ ) = B + ( τ ) / B + ( 0 ) ; B ( τ ) = B x ( τ ) B y ( τ ) ; b ( τ ) = B ( τ ) / B ( 0 ) ;
τ 0 + = D 2 υ Ω ( 0 , 1 ) / Ω ( 1 , 1 ) , τ 0 = D 2 υ cos ( 2 θ ) Ω ( 0 , 1 ) / Ω ( 1 , 1 ) .
w f ( ν ) = d τ b f ( τ ) exp ( i ν τ ) .
ν k = 0 d ν ν k w f ( ν ) .
ν 1 + = 0 d ν ν w + ( ν ) = 4 υ D Ω ( 2 , 1 ) / Ω ( 1 , 1 ) ,
ν 2 + = 0 d ν ν 2 w + ( ν ) = 2 π ( υ D ) 2 Ω ( 3 , 1 ) / Ω ( 1 , 1 ) ,
w + ( ν ) = d τ b + ( τ ) exp ( i ν τ ) .
d τ J 0 ( a | τ | ) exp ( i ν τ ) = rect ( ν / a ) / a 2 ν 2 .
ξ 01 ( L 0 / D ) = τ 0 + ν 1 + , ξ 02 ( L 0 / D ) = ( τ 0 + ) 2 ν 2 + .
ξ 0 * ( T , L 0 / D ) = 0 d χ χ 1 Φ n ( χ , L 0 / D ) J 2 2 ( χ ) 1 F 2 ( a χ ) Ω ( 1 , 3 ) 0 d χ χ 1 Φ n ( χ , L 0 / D ) J 3 2 ( χ ) 1 F 2 ( a χ ) Ω ( 1 , 2 ) , ξ 0 k * ( T , L 0 / D ) = [ 0 d χ χ Φ n ( χ , L 0 / D ) J 1 2 ( χ ) F 1 2 ( a χ ) Ω ( 1 , 1 ) ] k × [ ξ 0 k ( L 0 / D ) ( 2 a ) 1 k + 1 a / 2 d χ Φ n ( χ , L 0 / D ) × J 1 2 ( χ ) F 1 2 ( a / χ ) / Ω ( 1 , 1 ) ] , a = υ T / D , F 1 2 ( a χ ) = F 1 2 ( 1 / 2 ; 1 , 3 / 2 ; a 2 χ 2 ) , F 2 1 ( a χ ) = F 2 1 ( 1 / 2 , k / 2 + 1 / 2 ; k / 2 + 3 / 2 ; a 2 / 4 χ 2 ) ,
σ = ( L 0 L 0 * ) / L 0 ,
ξ ( L 0 / D ) = ξ * ( T , L 0 * / D ) .
τ 0 t + = 0 d h [ C n 2 ( h ) / υ ( h ) ] / 0 d h C n 2 ( h ) × [ Ω ( 2 , 2 ) / Ω ( 1 , 2 ) ] , τ 0 d = 0 d h [ C n 2 ( h ) / υ ( h ) ] / 0 d h C n 2 ( h ) × [ Ω ( 2 , 3 ) / Ω ( 1 , 3 ) ] ,

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