Abstract

A photoelectric apparatus enabled us to obtain new estimates of the longitudinal and transverse correlation functions for the angle-of-arrival fluctuations on the entrance pupil of a solar telescope. The inertial model of turbulence was verified for spatial shifts up to 35 cm. Measurements of the turbulent-layers velocity were taken. The values found for Fried’s parameter r0, the integral C2Ndh of the structure constant for the variations of the refractive index of the air over the thickness of the lower atmosphere, and the root-mean-square fluctuation of the angles of arrival appear to agree with independent estimates.

© 1978 Optical Society of America

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References

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  1. R. E. Hufnagel, Proceedings of AGARD Conference (Ankara) (1967).
  2. N. V. Bystrova and A. N. Demidova, Solnecnye Dannye 3, 73 (1961).
  3. J. Borgnino and F. Martin, J. Opt. (Paris) 8, 319–326 (1977).
    [Crossref]
  4. J. Borgnino and F. Martin, J. Opt. Soc. Am. 67, 1065 (1977).
    [Crossref]
  5. F. Martin and J. Borgnino, “Statistical analysis of wavefront random deformations produced by atmospheric turbulence near the ground.” 2nd part, “Correlation function estimation by numerical processing”, J. Opt. (Paris) (to be published).
  6. D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
    [Crossref]
  7. C. Roddier, J. Opt. Soc. Am. 66, 478 (1976).
    [Crossref]
  8. A. Rocca, F. Roddier, and J. Vernin, J. Opt. Soc. Am. 64, 1000 (1974).
    [Crossref]
  9. C. E. Coulman, Sol. Phys. 7, 122 (1969).
    [Crossref]
  10. M. A. Kallistratova, Radiofizika 9, 50 (1966).
  11. P. N. Brandt, Sol. Phys. 7, 187 (1969).
    [Crossref]
  12. G. Ricort, Thesis, 3rd cycle, (Nice University, France, 1973).
  13. C. Aime, Thesis (Nice University, France, 1977).
  14. V. I. Tatarski, “The effects of the turbulent atmosphere on wave propagation” (1971) translated by Israel Program for Scientific Translations.

1977 (2)

J. Borgnino and F. Martin, J. Opt. (Paris) 8, 319–326 (1977).
[Crossref]

J. Borgnino and F. Martin, J. Opt. Soc. Am. 67, 1065 (1977).
[Crossref]

1976 (1)

1974 (1)

1969 (2)

C. E. Coulman, Sol. Phys. 7, 122 (1969).
[Crossref]

P. N. Brandt, Sol. Phys. 7, 187 (1969).
[Crossref]

1967 (1)

R. E. Hufnagel, Proceedings of AGARD Conference (Ankara) (1967).

1966 (2)

D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
[Crossref]

M. A. Kallistratova, Radiofizika 9, 50 (1966).

1961 (1)

N. V. Bystrova and A. N. Demidova, Solnecnye Dannye 3, 73 (1961).

Aime, C.

C. Aime, Thesis (Nice University, France, 1977).

Borgnino, J.

J. Borgnino and F. Martin, J. Opt. (Paris) 8, 319–326 (1977).
[Crossref]

J. Borgnino and F. Martin, J. Opt. Soc. Am. 67, 1065 (1977).
[Crossref]

F. Martin and J. Borgnino, “Statistical analysis of wavefront random deformations produced by atmospheric turbulence near the ground.” 2nd part, “Correlation function estimation by numerical processing”, J. Opt. (Paris) (to be published).

Brandt, P. N.

P. N. Brandt, Sol. Phys. 7, 187 (1969).
[Crossref]

Bystrova, N. V.

N. V. Bystrova and A. N. Demidova, Solnecnye Dannye 3, 73 (1961).

Coulman, C. E.

C. E. Coulman, Sol. Phys. 7, 122 (1969).
[Crossref]

Demidova, A. N.

N. V. Bystrova and A. N. Demidova, Solnecnye Dannye 3, 73 (1961).

Fried, D. L.

Hufnagel, R. E.

R. E. Hufnagel, Proceedings of AGARD Conference (Ankara) (1967).

Kallistratova, M. A.

M. A. Kallistratova, Radiofizika 9, 50 (1966).

Martin, F.

J. Borgnino and F. Martin, J. Opt. (Paris) 8, 319–326 (1977).
[Crossref]

J. Borgnino and F. Martin, J. Opt. Soc. Am. 67, 1065 (1977).
[Crossref]

F. Martin and J. Borgnino, “Statistical analysis of wavefront random deformations produced by atmospheric turbulence near the ground.” 2nd part, “Correlation function estimation by numerical processing”, J. Opt. (Paris) (to be published).

Ricort, G.

G. Ricort, Thesis, 3rd cycle, (Nice University, France, 1973).

Rocca, A.

Roddier, C.

Roddier, F.

Tatarski, V. I.

V. I. Tatarski, “The effects of the turbulent atmosphere on wave propagation” (1971) translated by Israel Program for Scientific Translations.

Vernin, J.

J. Opt. (Paris) (1)

J. Borgnino and F. Martin, J. Opt. (Paris) 8, 319–326 (1977).
[Crossref]

J. Opt. Soc. Am. (4)

Proceedings of AGARD Conference (Ankara) (1)

R. E. Hufnagel, Proceedings of AGARD Conference (Ankara) (1967).

Radiofizika (1)

M. A. Kallistratova, Radiofizika 9, 50 (1966).

Sol. Phys. (2)

P. N. Brandt, Sol. Phys. 7, 187 (1969).
[Crossref]

C. E. Coulman, Sol. Phys. 7, 122 (1969).
[Crossref]

Solnecnye Dannye (1)

N. V. Bystrova and A. N. Demidova, Solnecnye Dannye 3, 73 (1961).

Other (4)

F. Martin and J. Borgnino, “Statistical analysis of wavefront random deformations produced by atmospheric turbulence near the ground.” 2nd part, “Correlation function estimation by numerical processing”, J. Opt. (Paris) (to be published).

G. Ricort, Thesis, 3rd cycle, (Nice University, France, 1973).

C. Aime, Thesis (Nice University, France, 1977).

V. I. Tatarski, “The effects of the turbulent atmosphere on wave propagation” (1971) translated by Israel Program for Scientific Translations.

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Figures (10)

FIG. 1
FIG. 1

Arrangement of the photodiodes in the plane of the pupil image. d is the distance between the two nearest photodiodes.

FIG. 2
FIG. 2

Electronical apparatus. (a) Recording. The signal given by each photodiode is amplified a first time (× gi), then after subtraction of a constant value (− Ai) amplified again (× Gi) and finally recorded. (b) Processing. The autocorrelations of the signals 1, 2, 3, and 4 (x = 0) are computed successively, then the cross correlation between the signals 1 and 2 (x = d), 3 and 4 (x = 2d), 2 and 3 (x = 3d), 1 and 3 (x = 4d), 2 and 4 (x = 5d), 1 and 4 (x = 6d).

FIG. 3
FIG. 3

Transverse autocorrelation of the angle-of-arrival fluctuations. Date, 09/05/77; local time, 10H52′. Correlation coefficient r = 0.99 − (▲, experimental points). Slope of the straight line is −0.37. σφ is the root-mean-square fluctuation of the angles of arrival. The unit indicated on the left scale is a voltage from a correlator.

FIG. 4
FIG. 4

Longitudinal autocorrelation of the angle-of-arrival fluctuations. Date, 09/05/77; local time, 11H15′. Correlation coefficient r = 0.99 − (▲, experimental points). Slope of the straight line is −0.31. σφ is the root-mean-square fluctuation of the angle of arrival; v is the velocity of the turbulent layer. The unit indicated on the left scale is a voltage from a correlator.

FIG. 5
FIG. 5

Longitudinal autocorrelation of the angle-of-arrival fluctuations. Date, 09/05/77; local time, 12H17′. Correlation coefficient r = 0.998 − (▲, experimental points). Slope of the straight line is −0.35; σφ is the root-mean-square fluctuation of the angles of arrival; v is the velocity of the turbulent layer. The unit indicated on the left scale is a voltage from a correlator.

FIG. 6
FIG. 6

Transverse autocorrelation of the angle-of-arrival fluctuations. Date, 09/05/77; local time, 12H28′. Correlation coefficient r = 0.993 − (▲, experimental points). Slope of the straight-line is −0.36. σφ is the root-mean-square fluctuation of the angles of arrival. The unit indicated on the left scale is a voltage from a correlator.

FIG. 7
FIG. 7

Transverse autocorrelation of the angle-of-arrival fluctuations. Date, 09/05/77; local time, 13H45′. Correlation coefficient r = 0.98 − (▲, experimental points). Slope of the straight-line is −0.336. σφ is the root-mean-square fluctuation of the angles of arrival. The unit indicated of the left scale is a voltage from a correlator.

FIG. 8
FIG. 8

Longitudinal autocorrelation of the angle-of-arrival fluctuations. Date, 09/05/77; local time, 13H54′. Correlation coefficient r = 0.99 − (▲, experimental points). Slope of the straight-line is −0.327; σφ is the root-mean-square fluctuation of the angles of arrival; v is the velocity of the turbulent layer. The unit indicated on the left scale is a voltage from a correlator.

FIG. 9
FIG. 9

Measurement of the turbulent-layer velocity. (a) Principle. The alignment of the photodiodes is placed in the direction of propagation of the turbulent layer (TL) which moves with a velocity v. ρ is the distance between the photodiodes numbers i(Phi) and j(Phj). A phase deformation detected by the photodiode number i is detected by the photodiode number j after a time delay τ for which the spatio-temporal correlation function shows a maximum. The velocity of the turbulent layer is given by v = ρ/τ. CR is the correlator. (b) Example of spatiotemporal correlation function. Between two consecutive points, we have Δτ = 3.33 ms. The distance between the photodiodes, considered on the entrance pupil of the telescope is ρ = 0.058 m. In this example v ≃ 5.8 m/s.

FIG. 10
FIG. 10

Longitudinal autocorrelation of the angle-of-arrival fluctuations. Date, 09/05/77; local time, 12H05′. ▲, experimental points. The slope of the straight line (1) is −0.422 (correlation coefficient r = 0.998). The values of CEL(x) are the raw results. The slope of straight line (2) is −0.33 (correlation coefficient r = 0.99). In this case, the values of CEL(x) are obtained by adding a posteriori to the raw results a constant measuring the loss of energy due to the low-frequency filtering. The unit indicated on left scale is a voltage from a correlator.

Equations (19)

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Δ E ( x , y ) = k φ ( x , y ) / y ,
C AAL ( x , 0 ) = 5.73 r 0 - 5 / 3 x - 1 / 3 ,
C AAT ( 0 , y ) = ( 2 / 3 ) 5.73 r 0 - 5 / 3 y - 1 / 3 ,
C EL ( x , 0 ) = 5.73 k 2 r 0 - 5 / 3 x - 1 / 3
C ET ( 0 , y ) = ( 2 / 3 ) 5.73 k 2 r 0 - 5 / 3 y - 1 / 3 .
C N 2 d h = 0.062 λ 2 cos θ r 0 - 5 / 3
S i = S i + s i ,
s i = k i φ / y ,
Δ V i = k i φ y | max .
φ y | max = 2 π a λ f ,
k i = Δ V i λ f / 2 π a .
C i j ( x ) = k i k j C φ ( x )
C i j ( x ) = ( λ 2 f 2 / 4 π 2 a 2 ) Δ V i Δ V j C φ ( x ) .
Δ V i = Δ V 1 [ C i i ( 0 ) / C 11 ( 0 ) ] 1 / 2
C i j ( x ) = ( λ 2 f 2 / 4 π 2 a 2 ) Δ V 1 2 × { [ C i i ( 0 ) C j j ( 0 ) ] 1 / 2 / C 11 ( 0 ) } C φ ( x ) .
C EL ( x ) = C i j ( x ) C 11 ( 0 ) / [ C i i ( 0 ) C j j ( 0 ) ] 1 / 2 = 5.73 ( λ 2 f 2 / 4 π 2 a 2 ) Δ V 1 2 r 0 - 5 / 3 x - 1 / 3
C ET ( y ) = C i j ( y ) C 11 ( 0 ) / [ C i i ( 0 ) C j j ( 0 ) ] 1 / 2 = ( 2 / 3 ) 5.73 ( λ 2 f 2 / 4 π 2 a 2 ) Δ V 1 2 r 0 - 5 / 3 y - 1 / 3 .
L = v / ν c .
C N 80 × 10 - 6 p C T / T 2 ,