Abstract

Two He–Ne lasers operating at 6328 Å have been utilized to make simultaneous measurements of the effects of scintillation over homogeneous optical paths of 650 and 1300 m to study the transfer of laser radiation through a turbulent medium. At the path terminus, multiple sampling of each laser beam was effected by use of a photo-optical technique that records a 61-cm cross section of an optical beam. Concurrent with the optical data, wind-speed and direction recordings were made at multiple points along the optical path in order to estimate the homogeneity of meteorological conditions. Near the path terminus, measurements of wind shear and temperature lapse were taken. In addition, high-speed-thermometry techniques were utilized to compute one-dimensional temperature spectra as well as the thermal structure coefficient CT. Data were gathered during temperature-lapse, neutral, and inversion conditions. Log-irradiance scans derived from the optical data were used to compute log-irradiance power spectra, variance, and other statistical quantities. From these optical and meteorological data, optical-filter functions were calculated for spatial frequencies above 87 cycles/m and are used for comparison with current theories. The saturation of the log-irradiance data is again observed, and the isotropy of the irradiance fluctuations is examined.

© 1970 Optical Society of America

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References

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  1. J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
    [CrossRef]
  2. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill, New York, 1960).
  3. A. Ishimaru, Radio Sci. 4 (New Series), 295 (1969).
    [CrossRef]
  4. F. P. Carlson, J. Opt. Soc. Am. 59, 1343 (1969).
    [CrossRef]
  5. V. I. Tatarski and A. I. Kon, Izv. Vuzov, Radiofizika 8, 870 (1965). Here the authors consider only the perturbation in the phase structure function and, hence, the angle-of-arrival fluctuations. However, the beam profile is formulated in terms of a gaussian amplitude rather than a gaussian irradiance and is therefore not a solution for the TEM00 mode beam.
  6. M. E. Gracheva and A. S. Gurvich, Izv. Vuzov, Radiofizika 8, 717 (1965).
  7. P. H. Deitz and N. J. Wright, J. Opt. Soc. Am. 59, 527 (1969).
    [CrossRef]
  8. V. I. Tatarski, Sov. Phys. JETP 22, 1083 (1966).
  9. V. I. Tatarski, Sov. Phys. JETP 19, 946 (1964).
  10. V. I. Tatarski and M. E. Gertsenshtein, Sov. Phys. JETP 17, 458 (1963).
  11. D. A. deWolf, J. Opt. Soc. Am. 58, 461 (1968).
    [CrossRef]
  12. Reference 2, p. 136.
  13. J. M. Caborn, Brit. Forestry Comm. Bull. 29, 1950.
  14. Reference 2, p. 40.
  15. L. R. Tsvang, Izv. ANSSSR, Geophys. Ser. 8, 1252 (1960).
  16. Private correspondence, through contract, with D. J. Portman and co-workers, Dept. of Meteorology and Oceanography, University of Michigan, Ann Arbor, Mich.
  17. P. H. Deitz, in Modern Optics, edited by J. Fox (Polytechnic Press, Brooklyn, N. Y., 1967), pp. 766 ff.
  18. Reference 2, p. 215.
  19. Private communication with M. A. Martin.
  20. Photometrics, Inc., independently scanned the films and determined the power spectra of the identical exposures. Their results are in direct agreement with those reported here.
  21. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Bateman Manuscript Project, California Institute of Technology (McGraw–Hill, New York, 1953), Vol. I, p. 15.

1969 (3)

1968 (2)

D. A. deWolf, J. Opt. Soc. Am. 58, 461 (1968).
[CrossRef]

J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
[CrossRef]

1966 (1)

V. I. Tatarski, Sov. Phys. JETP 22, 1083 (1966).

1965 (2)

V. I. Tatarski and A. I. Kon, Izv. Vuzov, Radiofizika 8, 870 (1965). Here the authors consider only the perturbation in the phase structure function and, hence, the angle-of-arrival fluctuations. However, the beam profile is formulated in terms of a gaussian amplitude rather than a gaussian irradiance and is therefore not a solution for the TEM00 mode beam.

M. E. Gracheva and A. S. Gurvich, Izv. Vuzov, Radiofizika 8, 717 (1965).

1964 (1)

V. I. Tatarski, Sov. Phys. JETP 19, 946 (1964).

1963 (1)

V. I. Tatarski and M. E. Gertsenshtein, Sov. Phys. JETP 17, 458 (1963).

1960 (1)

L. R. Tsvang, Izv. ANSSSR, Geophys. Ser. 8, 1252 (1960).

1950 (1)

J. M. Caborn, Brit. Forestry Comm. Bull. 29, 1950.

Caborn, J. M.

J. M. Caborn, Brit. Forestry Comm. Bull. 29, 1950.

Carlson, F. P.

Deitz, P. H.

P. H. Deitz and N. J. Wright, J. Opt. Soc. Am. 59, 527 (1969).
[CrossRef]

P. H. Deitz, in Modern Optics, edited by J. Fox (Polytechnic Press, Brooklyn, N. Y., 1967), pp. 766 ff.

deWolf, D. A.

Erdelyi, A.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Bateman Manuscript Project, California Institute of Technology (McGraw–Hill, New York, 1953), Vol. I, p. 15.

Gertsenshtein, M. E.

V. I. Tatarski and M. E. Gertsenshtein, Sov. Phys. JETP 17, 458 (1963).

Gracheva, M. E.

M. E. Gracheva and A. S. Gurvich, Izv. Vuzov, Radiofizika 8, 717 (1965).

Gurvich, A. S.

M. E. Gracheva and A. S. Gurvich, Izv. Vuzov, Radiofizika 8, 717 (1965).

Ishimaru, A.

A. Ishimaru, Radio Sci. 4 (New Series), 295 (1969).
[CrossRef]

Kon, A. I.

V. I. Tatarski and A. I. Kon, Izv. Vuzov, Radiofizika 8, 870 (1965). Here the authors consider only the perturbation in the phase structure function and, hence, the angle-of-arrival fluctuations. However, the beam profile is formulated in terms of a gaussian amplitude rather than a gaussian irradiance and is therefore not a solution for the TEM00 mode beam.

Magnus, W.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Bateman Manuscript Project, California Institute of Technology (McGraw–Hill, New York, 1953), Vol. I, p. 15.

Martin, M. A.

Private communication with M. A. Martin.

Oberhettinger, F.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Bateman Manuscript Project, California Institute of Technology (McGraw–Hill, New York, 1953), Vol. I, p. 15.

Portman, D. J.

Private correspondence, through contract, with D. J. Portman and co-workers, Dept. of Meteorology and Oceanography, University of Michigan, Ann Arbor, Mich.

Strohbehn, J. W.

J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
[CrossRef]

Tatarski, V. I.

V. I. Tatarski, Sov. Phys. JETP 22, 1083 (1966).

V. I. Tatarski and A. I. Kon, Izv. Vuzov, Radiofizika 8, 870 (1965). Here the authors consider only the perturbation in the phase structure function and, hence, the angle-of-arrival fluctuations. However, the beam profile is formulated in terms of a gaussian amplitude rather than a gaussian irradiance and is therefore not a solution for the TEM00 mode beam.

V. I. Tatarski, Sov. Phys. JETP 19, 946 (1964).

V. I. Tatarski and M. E. Gertsenshtein, Sov. Phys. JETP 17, 458 (1963).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill, New York, 1960).

Tricomi, F. G.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Bateman Manuscript Project, California Institute of Technology (McGraw–Hill, New York, 1953), Vol. I, p. 15.

Tsvang, L. R.

L. R. Tsvang, Izv. ANSSSR, Geophys. Ser. 8, 1252 (1960).

Wright, N. J.

Brit. Forestry Comm. Bull. (1)

J. M. Caborn, Brit. Forestry Comm. Bull. 29, 1950.

Izv. ANSSSR, Geophys. Ser. (1)

L. R. Tsvang, Izv. ANSSSR, Geophys. Ser. 8, 1252 (1960).

Izv. Vuzov, Radiofizika (2)

V. I. Tatarski and A. I. Kon, Izv. Vuzov, Radiofizika 8, 870 (1965). Here the authors consider only the perturbation in the phase structure function and, hence, the angle-of-arrival fluctuations. However, the beam profile is formulated in terms of a gaussian amplitude rather than a gaussian irradiance and is therefore not a solution for the TEM00 mode beam.

M. E. Gracheva and A. S. Gurvich, Izv. Vuzov, Radiofizika 8, 717 (1965).

J. Opt. Soc. Am. (3)

Proc. IEEE (1)

J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
[CrossRef]

Radio Sci. (1)

A. Ishimaru, Radio Sci. 4 (New Series), 295 (1969).
[CrossRef]

Sov. Phys. JETP (3)

V. I. Tatarski, Sov. Phys. JETP 22, 1083 (1966).

V. I. Tatarski, Sov. Phys. JETP 19, 946 (1964).

V. I. Tatarski and M. E. Gertsenshtein, Sov. Phys. JETP 17, 458 (1963).

Other (9)

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill, New York, 1960).

Reference 2, p. 136.

Reference 2, p. 40.

Private correspondence, through contract, with D. J. Portman and co-workers, Dept. of Meteorology and Oceanography, University of Michigan, Ann Arbor, Mich.

P. H. Deitz, in Modern Optics, edited by J. Fox (Polytechnic Press, Brooklyn, N. Y., 1967), pp. 766 ff.

Reference 2, p. 215.

Private communication with M. A. Martin.

Photometrics, Inc., independently scanned the films and determined the power spectra of the identical exposures. Their results are in direct agreement with those reported here.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Bateman Manuscript Project, California Institute of Technology (McGraw–Hill, New York, 1953), Vol. I, p. 15.

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Figures (14)

Fig. 1
Fig. 1

The optical-filter function, θ(κ/2π), as a function of spatial frequency. The predictions of Tatarski (PST) and Carlson and Ishimaru (BST) are shown by solid lines, and experimental results for several times are shown by broken lines for a range of 650 m.

Fig. 2
Fig. 2

The optical-filter function, θ(κ/2π), as a function of spatial frequency. The predictions of Tatarski (PST) and Carlson and Ishimaru (BST) are shown by solid lines, and experimental results for several times are shown by broken lines for a 1300-m range.

Fig. 3
Fig. 3

Standard errors of log irradiance as a function of the plane-wave Tatarski prediction. (A) PST; (B) PMST (Tatarski); (C) BST; (D) PMST (DeWolf). The associated times of day are ⋯ 1130h; ——1430h;—·—1900h; ——2100h; — — —2200 h.

Fig. 4
Fig. 4

The index-of-refraction spectra, Φn(1)(κ/2π), as a function of spatial frequency. The associated times of day and Richardson number are ▲, ●1130 h, −0.159; ■1430 h, −0.217.

Fig. 5
Fig. 5

The index-of-refraction spectra, Φn(1)(κ/2π), as a function of spatial frequency. The associated times of day and Richardson number are ▲1900 h, 0.146; ●2100 h, 0.162; ■2200 h, 0.482.

Fig. 6
Fig. 6

Ratio of vertical-to-horizontal-scan power spectra as a function of spatial frequency for 1400 h. The ranges are —— 650 m and - - - - - 1300 m.

Fig. 7
Fig. 7

Vertical (0)-and-horizontal (X)-scan power spectra for a 30-nsec-pulsed ruby laser.

Fig. 8
Fig. 8

Average-scan power spectra for 1130 h and 650-m range.

Fig. 9
Fig. 9

Typical photograph for 1130 h and 650-m range.

Fig. 10
Fig. 10

Average-scan power spectra for 1130 h and 1300-m range.

Fig. 11
Fig. 11

Average-scan power spectra for 1900 h and 650-m range.

Fig. 12
Fig. 12

Average-scan power spectra for 1900 h and 1300-m range.

Fig. 13
Fig. 13

Average-scan power spectra for 2200 h and 650-m range.

Fig. 14
Fig. 14

Average-scan power spectra for 2200 h and 1300-m range.

Tables (2)

Tables Icon

Table I Meteorological data.

Tables Icon

Table II Statistical averages.

Equations (60)

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Φ χ ( 2 ) ( κ , 0 ) = π k 2 L θ ( κ ) Φ n ( 3 ) ( κ )
χ 2 = 2 π 0 Φ χ ( 2 ) ( κ , 0 ) κ d κ .
θ ( κ ) = 1 - ( k / κ 2 L ) sin ( κ 2 L / k ) ,
χ T 2 = 0.307 C n 2 k 7 / 6 L 11 / 6 ,
θ ( κ ) = 1 2 ( π q b ) 1 2 erf [ ( q b ) 1 2 ] - Re { 1 2 ( π q b 1 ) 1 2 exp ( - q / 4 b 1 ) × [ erf ( - i ( q ) 1 2 2 ( b 1 ) 1 2 + ( q b 1 ) 1 2 ) - erf ( - i ( q ) 1 2 2 ( b 1 ) 1 2 ) ] } ,
q = κ 2 L / k , b = α L / ( 1 + α 2 L ) , and n 1 = α L / ( 1 - i α L ) .
χ 2 = π 2 0.033 C n 2 Γ ( - 5 6 ) k 7 / 6 × L 11 / 6 ( α L 1 + α 2 L 2 ) 5 / 6 [ 3 8 - g ( α L ) ] ,
g ( α L ) = Re [ 6 11 ( 1 + i α L i α L ) 5 / 6 F 2 1 ( - 5 6 ; 1 ; 17 / 6 ; i α L ) ] .
χ 2 = 1 4 ln [ 2 - exp ( - 4 χ T 2 ) ] ,
R i = g ( T ¯ 1 - T ¯ 2 ) ( z 1 z 2 ) 1 2 log ( z 1 / z 2 ) T ambient ( V ¯ 1 - V ¯ 2 ) 2 ,
σ I 2 ( I - I ) 2 ,
σ 2 χ 2 ( 2 χ ) 2 = ln ( 1 + σ I 2 / I 2 ) .
Φ ( 1 ) ( κ ) = Φ ( 1 ) ( κ 0 ) exp { A ln ( κ / κ 0 ) + B [ ln ( κ / κ 0 ) ] 2 } ,
Φ ( 3 ) ( κ ) = ( - 1 / 2 π κ ) ( d / d κ ) Φ ( 1 ) ( κ ) .
Φ ( 2 ) ( κ ) = - d γ Φ ( 3 ) [ ( γ 2 + κ 2 ) 1 2 ] = - 1 2 π - d γ 1 ( γ 2 + κ 2 ) 1 2 d d ( γ 2 + κ 2 ) 1 2 × Φ ( 1 ) [ ( γ 2 + κ 2 ) 1 2 ] .
Φ ( 2 ) ( κ ) = - Φ ( 1 ) ( κ ) 2 π d d κ e a ( θ ) k = 0 1 ( 4 B ) k κ 2 k k ! d 2 k d κ 2 k × β ( A - 1 + 2 θ B ; 1 2 ) ,
θ = ln ( κ / κ 0 ) ,             a ( θ ) = A θ + E θ 2 ,
x = κ - 1 ( γ 2 + κ 2 ) 1 2 .
Φ ( 2 ) ( κ ) = Φ ( 1 ) ( κ 0 ) κ π 1 d x ( x 2 - 1 ) 1 2 d d x × [ exp A ( θ + x ˜ ) + B ( θ + x ˜ ) 2 ] ,
x ˜ = ln x             and             θ = ln ( κ / κ 0 ) .
d d x exp [ A ( θ + x ˜ ) + B ( θ + x ˜ ) 2 ] = x - 1 d d x ˜ exp [ A ( θ + x ˜ ) + B ( θ + x ˜ ) 2 ] .
à = A - 1.
Φ ( 2 ) ( κ ) = - Φ ( 1 ) ( κ 0 ) π d d κ exp ( A θ + B θ 2 ) × 1 d x ( x 2 - 1 ) 1 2 exp ( A x ˜ + 2 B x ˜ θ + B x ˜ 2 ) .
exp ( Ã x ˜ + 2 B x ˜ θ + B x ˜ 2 ) = exp [ ( Ã + 2 B θ ) x ˜ ] k = 0 ( B x ˜ 2 ) k k ! = k = 0 1 ( 4 B ) k κ 2 k k ! d 2 k d κ 2 k exp [ ( Ã + 2 B θ ) x ˜ ] .
Φ ( 2 ) ( κ ) = - Φ ( 1 ) ( κ 0 ) π d d κ × [ e a ( θ ) k = 0 ( 1 4 B ) k κ 2 k k ! d 2 k d κ 2 k G ( θ ) ] ,
G ( θ ) = 1 d x ( x 2 - 1 ) 1 2 x ( Ã + 2 B θ )
a ( θ ) = A θ + B θ 2 .
x 2 = 1 / r ,
G ( θ ) = 1 2 0 1 ( 1 - r ) - 1 2 r - 1 2 ( Ã + 2 B θ ) - 1 d r .
G ( θ ) = 1 2 β ( 1 2 Ã + 2 B θ ; 1 2 ) .
( Ã + 2 B θ ) < 0.
z = ( B ) 1 2 x ˜ ,
N ( θ ) = 1 ( B ) 1 2 × 0 exp { [ M ( θ ) z / ( B ) 1 2 ] + ( sgn B ) z 2 } d z { 1 - exp [ - 2 z / ( B ) 1 2 ] } 1 2 ,
M ( θ ) = Ã + 2 B θ .
N ( θ ) = 1 2 0 1 x - M ( θ ) / 2 - 1 ( 1 - x ) - 1 2 d x + 1 2 m = 1 B m 2 2 m m ! × 0 exp [ 2 m ln y - M ( θ ) 2 y - 1 2 ln ( 1 - e - y ) ] d y .
f ( y ) = 2 m / y - M ( θ ) / 2 - 1 / 2 ( 1 - e - y ) = 0.
y 0 2 ( 2 m - 1 2 ) / M ( θ ) ,
1 - e - y ~ y .
y 0 4 m / M ( θ ) + 1.
f ( y ) ( 2 m - 1 2 ) ln y 0 - M ( θ ) / 2 y 0 .
0 d y e f ( y ) e f ( y 0 ) 0 exp [ - 1 2 f ( y 0 ) ( y - y 0 ) 2 d y ] e f ( y 0 ) ( π 2 f ( y 0 ) ) 1 2 ( 1 + erf { [ f ( y 0 ) y 0 / 2 ] 1 2 } ) .
f ( y 0 ) = M ( θ ) 2 / 4 ( 2 m - 1 2 ) .
N ( θ ) 1 2 β ( M ( θ ) / 2 ; 1 2 ) + 1 2 h = 1 w B n n ! 2 Γ ( 2 n + 1 2 ) M ( θ ) n + 1 2 × { 1 + erf [ ( m - 1 4 ) 1 2 ] } ,
w = 4 M ( θ ) 2 / B .
Φ ( 2 ) ( κ ) = - Φ ( 1 ) ( κ 0 ) 2 π κ 0 [ A + 2 B ln ( κ / κ 0 ) ] Q × ( κ κ 0 ) [ A - 1 + B ln ( κ / κ 0 ) ]
Q = β [ M ( θ ) / 2 ; 1 2 ] [ 1 + B M ( θ ) M ( θ ) [ 1 + M ( θ ) ] q ( M ( θ ) 2 ) ] + 1 2 n = 1 w [ C n + 2 B M ( θ ) ( θ 1 + M ( θ ) ) D n ]
C n = B n Γ ( 2 n + 1 2 ) Γ ( n + 1 ) M ( θ ) 2 n + 1 2 { 1 + erf [ ( n - 1 4 ) 1 2 ] }
D n = - B n Γ ( 2 n + 3 2 ) Γ ( n + 1 ) M ( θ ) 2 n + 3 2 { 1 + erf [ ( n - 1 4 ) 1 2 ] } ,
q ( M ( θ ) 2 ) = ψ ( M ( θ ) 2 ) - ψ { 1 2 [ 1 + M ( θ ) ] } .
Q = β [ M ( θ ) / 2 ; 1 2 ]
Φ ( 2 ) ( κ ) = - Φ ( 1 ) ( κ 0 ) 2 π κ 0 ( κ κ 0 ) A - 1 A β ( A - 1 / 2 ; 1 2 ) .
D f ( x - x , y - y ) = 2 - d κ 1 d κ 2 { 1 - cos [ κ 1 ( x - x ) + κ 2 ( y - y ) ] } Φ f ( 2 ) ( κ 1 , κ 2 , 0 ) .
x = x - v x t ; x = x - v x t .
D f [ x - x - v x ( t - t ) , y - y ] 2 = - d κ 1 d κ 2 { 1 - cos [ κ 1 ( x - x ) + κ 2 ( y - y ) - v x κ 1 ( t - t ) ] } Φ f ( 2 ) ( κ 1 , κ 2 , 0 ) .
D f ( x - x , y - y ) av = 1 τ 2 0 τ d t d t × D f [ x - x - v ( t - t ) , y - y ] ,
D f ( x - x , y - y ) av = 2 d κ 1 d κ 2 { 1 - 1 τ 2 0 τ d t d t cos [ κ 1 ( x - x ) + κ 2 ( y - y ) - v x κ 1 ( t - t ) ] } Φ f ( 2 ) ( κ 1 , κ 2 , 0 ) .
D f ( x - x , y - y ) av = 2 - { 1 - cos [ κ ˜ 1 ( x - x ) + κ ˜ 2 ( y - y ) ] } × Φ f ( 2 ) av ( κ ˜ 1 , κ ˜ 2 , 0 ) d κ ˜ 1 d κ ˜ 2 .
Φ f ( 2 ) av ( κ , 0 ) = 2 ( κ · v ) 2 τ 2 [ 1 - cos ( κ · v τ ) ] Φ f ( 2 ) ( κ , 0 ) .
T ( κ , 0 ) = 2 ( κ · v ) 2 τ 2 [ 1 - cos ( κ · v τ ) ] ,
f = ( 2 × 10 - 3 V ) - 1 cycles / m