Abstract

The dependence of the complex optical transfer function of an image-forming system on range through the atmosphere has been investigated experimentally. The simple case of a horizontal line of sight over flat terrain has been studied and empirical equations have been obtained which relate the transfer function to range and to the atmospheric temperature structure function. These equations agree with a general theory of wave propagation in a turbulent medium and are therefore applicable to a wide range of situations, provided that the distribution of the temperature structure function along the line of sight can be obtained. The experimental results also furnish evidence that the Kolmogorov expression for the temperature structure function (“the two-thirds law”) is applicable to scale sizes of about a centimeter or so.

© 1966 Optical Society of America

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References

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  1. L. A. Chernov, Wave Propagation in a Random Medium, translated by R. A. Silverman (McGraw-Hill Book Co., Inc., New York, 1960).
  2. V. I. Tatarski, Wave Propagation in a Turbulent Medium, translated by R. A. Silverman (McGraw-Hill Book Co., Inc., Inc., New York, 1961).
  3. J. R. Meyer-Arendt and C. B. Emmanuel, Optical Scintillation: a Survey of the Literature (National Bureau of Standards Technical Note 225, April1965).
    [Crossref]
  4. E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co. Inc., Reading, Massachusetts, 1963).
  5. R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).
    [Crossref]
  6. C. E. Coulman, J. opt. Soc. Am. 55, 806 (1965).
    [Crossref]
  7. J. L. Lumley and H. A. Panofsky, The Structure of Atmos-bhieric Turbulence (Interscience Publishers, New York, 1964).
  8. L. R. Tswang, Izv. Akad. Nauk SSSR, Ser. Geofiz. No. 8, 1252 (1960).
  9. V. I. Tatarski, Izv. Akad. Nauk SSSR, Ser. Geofiz. No. 6689 (1956).
  10. E. K. Webb, Appl. Opt. 3, 1329 (1964).
    [Crossref]
  11. R. T. Leslie, D. Shaw, and C. E. Coulman, J. Opt. Soc. Am. 56, 1261 (1966).
    [Crossref]
  12. O. G. Sutton, Atmospheric Turbulence (Methuen and Co., London, 1955).
  13. W. C. Swinbank, Quart. J. Roy. Meteorol. Soc. 90, 119 (1964).
    [Crossref]
  14. V. I. Tatarski, Ref. 2, Chap. 8.
  15. S. M. Rytov, Izv. Akad. Nauk, SSSR, Ser. Fiz. No. 2,  223 (1937).
  16. J. L. Lumley and H. A. Panofsky, Ref. 7, Chap. 5.
  17. V. I. Tatarski, Ref. 2, Chap. 9 and Note 9(e), p. 273.
  18. E. Djurle and A. Bäck, J. Opt. Soc. Am. 51, 1029 (1961).
    [Crossref]
  19. C. B. Rogers, J. Opt. Soc. Am. 55, 1151 (1965).
    [Crossref]

1966 (1)

1965 (3)

J. R. Meyer-Arendt and C. B. Emmanuel, Optical Scintillation: a Survey of the Literature (National Bureau of Standards Technical Note 225, April1965).
[Crossref]

C. E. Coulman, J. opt. Soc. Am. 55, 806 (1965).
[Crossref]

C. B. Rogers, J. Opt. Soc. Am. 55, 1151 (1965).
[Crossref]

1964 (3)

1961 (1)

1960 (1)

L. R. Tswang, Izv. Akad. Nauk SSSR, Ser. Geofiz. No. 8, 1252 (1960).

1956 (1)

V. I. Tatarski, Izv. Akad. Nauk SSSR, Ser. Geofiz. No. 6689 (1956).

1937 (1)

S. M. Rytov, Izv. Akad. Nauk, SSSR, Ser. Fiz. No. 2,  223 (1937).

Bäck, A.

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium, translated by R. A. Silverman (McGraw-Hill Book Co., Inc., New York, 1960).

Coulman, C. E.

Djurle, E.

Emmanuel, C. B.

J. R. Meyer-Arendt and C. B. Emmanuel, Optical Scintillation: a Survey of the Literature (National Bureau of Standards Technical Note 225, April1965).
[Crossref]

Hufnagel, R. E.

Leslie, R. T.

Lumley, J. L.

J. L. Lumley and H. A. Panofsky, Ref. 7, Chap. 5.

J. L. Lumley and H. A. Panofsky, The Structure of Atmos-bhieric Turbulence (Interscience Publishers, New York, 1964).

Meyer-Arendt, J. R.

J. R. Meyer-Arendt and C. B. Emmanuel, Optical Scintillation: a Survey of the Literature (National Bureau of Standards Technical Note 225, April1965).
[Crossref]

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co. Inc., Reading, Massachusetts, 1963).

Panofsky, H. A.

J. L. Lumley and H. A. Panofsky, The Structure of Atmos-bhieric Turbulence (Interscience Publishers, New York, 1964).

J. L. Lumley and H. A. Panofsky, Ref. 7, Chap. 5.

Rogers, C. B.

Rytov, S. M.

S. M. Rytov, Izv. Akad. Nauk, SSSR, Ser. Fiz. No. 2,  223 (1937).

Shaw, D.

Stanley, N. R.

Sutton, O. G.

O. G. Sutton, Atmospheric Turbulence (Methuen and Co., London, 1955).

Swinbank, W. C.

W. C. Swinbank, Quart. J. Roy. Meteorol. Soc. 90, 119 (1964).
[Crossref]

Tatarski, V. I.

V. I. Tatarski, Izv. Akad. Nauk SSSR, Ser. Geofiz. No. 6689 (1956).

V. I. Tatarski, Wave Propagation in a Turbulent Medium, translated by R. A. Silverman (McGraw-Hill Book Co., Inc., Inc., New York, 1961).

V. I. Tatarski, Ref. 2, Chap. 8.

V. I. Tatarski, Ref. 2, Chap. 9 and Note 9(e), p. 273.

Tswang, L. R.

L. R. Tswang, Izv. Akad. Nauk SSSR, Ser. Geofiz. No. 8, 1252 (1960).

Webb, E. K.

Appl. Opt. (1)

Izv. Akad. Nauk SSSR (2)

L. R. Tswang, Izv. Akad. Nauk SSSR, Ser. Geofiz. No. 8, 1252 (1960).

V. I. Tatarski, Izv. Akad. Nauk SSSR, Ser. Geofiz. No. 6689 (1956).

Izv. Akad. Nauk, SSSR (1)

S. M. Rytov, Izv. Akad. Nauk, SSSR, Ser. Fiz. No. 2,  223 (1937).

J. Opt. Soc. Am. (4)

Optical Scintillation: a Survey of the Literature (1)

J. R. Meyer-Arendt and C. B. Emmanuel, Optical Scintillation: a Survey of the Literature (National Bureau of Standards Technical Note 225, April1965).
[Crossref]

Quart. J. Roy. Meteorol. Soc. (1)

W. C. Swinbank, Quart. J. Roy. Meteorol. Soc. 90, 119 (1964).
[Crossref]

Other (8)

V. I. Tatarski, Ref. 2, Chap. 8.

O. G. Sutton, Atmospheric Turbulence (Methuen and Co., London, 1955).

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co. Inc., Reading, Massachusetts, 1963).

J. L. Lumley and H. A. Panofsky, The Structure of Atmos-bhieric Turbulence (Interscience Publishers, New York, 1964).

L. A. Chernov, Wave Propagation in a Random Medium, translated by R. A. Silverman (McGraw-Hill Book Co., Inc., New York, 1960).

V. I. Tatarski, Wave Propagation in a Turbulent Medium, translated by R. A. Silverman (McGraw-Hill Book Co., Inc., Inc., New York, 1961).

J. L. Lumley and H. A. Panofsky, Ref. 7, Chap. 5.

V. I. Tatarski, Ref. 2, Chap. 9 and Note 9(e), p. 273.

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Figures (8)

F. 1
F. 1

Records of modulus and argument of optical transfer function: (a) under temperature-lapse conditions C T 0.1 ° C cm 1 3. (b) under near-neutral conditions, C T 0.03 ° C cm 1 3. Normalized spatial frequency f = 0.10, range X = 180 m.

F. 2
F. 2

Logarithmic modulation transfer function is approximately a linear function of CT2. The slope depends on the range X. Data points · refer to, X = 100 m, ✕ refer to X = 180 m, △ refer to X = 260 m, ○ refer to X = 400 m, □ refer to X = 800 m, and ▲ refer to X = 260 m at the Hay test site. Normalized spatial frequency f = 0.10.

F. 3
F. 3

As for Fig. 2 but with f = 0.15.

F. 4
F. 4

As for Fig. 2 but with f = 0.23.

F. 5
F. 5

Relationship between logarithmic modulation transfer function and CT2 for various values of averaging time τ. X = 260 m, f = 0.10. (a) Theoretical result for τ → ∞, (b) experimental result for τ≃0.1 sec, and (c) experimental result for τ≃0.04 sec. For τ≃0.02 sec the result is not significantly different from (c).

F. 6
F. 6

rms angular fluctuation of image position ( Δ α ) 2 1 2 is approximately a linear function of CT0.8. The slope depends on the range. (a) X = 100 m, (b) X = 180 m, (c) X = 260 m, (d) X = 400 m, and (e) X = 800 m. Data points ✕ refer to measurements made at Hay.

F. 7
F. 7

Values of the Obukhov parameter L estimated by the approximate method agree closely with values calculated from precise vertical wind- and temperature-profile measurements.

F. 8
F. 8

Values of Richardson’s number Ri estimated by the approximate method agree closely with values calculated from precise wind- and temperature-profile measurements.

Tables (2)

Tables Icon

Table I Three sets of values of the parameters in Eq. (7). Each set corresponds to a local minimum of the residual sum of squared deviations from the model but these minima are not significantly different from each other.

Tables Icon

Table II Comparison of predicted and measured results for modulation transfer function for C T = 0.031 ° C cm 1 3, 1520-m range, and three values of spatial frequency f.

Equations (21)

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D T ( r ) = [ T ( ζ ) T ( ζ + r ] 2 ,
D T ( r ) = C T 2 r 2 3 ,
C T 2 = a 2 T * 2 / ( k z ) 3 2 ,
C T 2 = a 2 T * 16 / 9 u * 4 / 9 ( 0.03 θ / g ) 2 / 9 ( k z ) 8 / 9 ( 1 R i ) 1 3 ,
L = u * 3 c p ρ θ / kgH ,
ln M ( f ) τ = β C T 2 + constant ,
ln M ( f ) τ = γ 1 f γ 2 C T 2 X γ 3 + A ( f ) .
ln M ( f ) τ = A ( f ) .
( Δ α ) 2 1 2 = γ 4 C T γ 5 X γ 6 ,
γ 4 = 1.016
γ 5 = 0.800 s . e . = 0.06
γ 6 = 0.433 s . e . = 0.06 .
( Hay , 260 - m range ) γ 5 = 0.90 ( s . e . = 0.16 )
( Airfield , 260 - m range ) γ 5 = 0.76 ( s . e . = 0.06 )
( All ranges ) γ 5 = 0.80 ( s . e . = 0.06 ) .
D n ( r ) = C n 2 r 2 3 ,
D s ( r ) = 2.91 ( 2 π λ ) 2 r 5 / 3 0 X 1 C n 2 ( X ) d X ,
D s ( r ) = 2.91 ( 2 π / λ ) 2 r 5 / 3 C n 2 X 1 .
( Δ α ) 2 = γ [ r 1 1 3 C T 2 ] ,
M ( λ F f ) = exp { 1 2 [ D s ( r ) ] } ,
r / λ F = f ,