Abstract

The concept of the quantity ro as a length that measures the effect of atmospheric turbulence on optical systems that are wavefront distortion sensitive is briefly reviewed. It is pointed out that no precise set of data on ro for propagation down through the atmosphere has been published. Using astronomical seeing data obtained by Hoag and by Meinel it is shown how statistics for ro can be derived, at least for observatory quality sites. We find that the two sets of data are in very good agreement and conclude that ro is variable from night to night and distributed according to a log-normal distribution. For 0.55-μm light and zenith propagation, the median value of ro is 0.114 m and changes by a factor of 1.36 for occurrences one standard deviation from the median.

© 1974 Optical Society of America

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Corrections

D. L. Fried, "Evaluation of ro for propagation down through the atmosphere: correction," Appl. Opt. 14, 2567-2567 (1975)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-14-11-2567

References

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  1. D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965).
    [CrossRef]
  2. D. L. Fried, “The Effect of Wavefront Distortion on the Performance of an Ideal Optical Heterodyne Receiver and an Ideal Camera,” in Proceedings of the Conference on Atmospheric Limitations to Optical Propagation, sponsored by Central Radio Propagation Laboratory and National Center for Atmospheric Research, Boulder, Colorado (1965), p. 192.
  3. I. Goldstein, P. A. Miles, A. Chabot, Proc. IEEE 53, 1172 (1965). Here deff is used in place of ro, but the quantities are identical.
    [CrossRef]
  4. D. M. Chase, J. Opt. Soc. Am. 56, 33 (1966). Here ae is used in place of ro, but the quantities are identical.
    [CrossRef]
  5. D. L. Fried, IEEE J. Quantum Electron. QE-3, 213 (1967).
    [CrossRef]
  6. D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
    [CrossRef]
  7. R. F. Lutomirski, H. T. Yura, Appl. Opt. 10, 1652 (1971). Here ρo is used in place of ro. These quantities differ by a factor of (3.44)−3/5 = 0.477, so that ρo = 0.477 ro. (The results cited refer to the case where the outer scale of turbulence is much larger than ro.)
    [CrossRef] [PubMed]
  8. D. Korff, J. Opt. Soc. Am. 63, 971 (1973).
    [CrossRef]
  9. D. Kelsall, J. Opt. Soc. Am. 63, 1472 (1973).
    [CrossRef]
  10. T. J. Gilmartin, J. Z. Hultz, “Atmospheric Optical Coherence Measurements at 10.6 μm and 0.63 μm,” in Optics Research2, 43 (1972), published by MIT Lincoln Laboratory.
  11. J. L. Bufton, Appl. Opt. 12, 1785 (1973).
    [CrossRef] [PubMed]
  12. A. A. Hoag, Bull. Astron. 24, Part 2, 269 (1964).
  13. A. B. Meinel, “Final Report on the Site Selection Survey for the National Astronomical Observatory,” Contributions from the Kitt Peak National Observatory, No. 45 (Oct.1963).
  14. R. E. Hufnagel, N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).
    [CrossRef]
  15. C. B. Johnson, Appl. Opt. 12, 1031 (1973).
    [CrossRef] [PubMed]
  16. A factor of (2π)−1 is apparently missing from the right-hand side of Eq. (5) of Ref. 15. That this is actually the case can be confirmed by noting that according to Eq. (7) of our work, Fe(x) for x = +∞ will not equal unity, as implied by the data in Table V of Ref. 15 unless the factor (2π)−1 is introduced.
  17. A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Triconi, Table of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1, p. 117, Eq. 3.1(1).
  18. W. E. Milne, Numerical Calculus (Princeton U.P., Princeton, 1949), p. 83, Eq. (1).

1973 (4)

1971 (1)

1967 (1)

D. L. Fried, IEEE J. Quantum Electron. QE-3, 213 (1967).
[CrossRef]

1966 (2)

1965 (2)

I. Goldstein, P. A. Miles, A. Chabot, Proc. IEEE 53, 1172 (1965). Here deff is used in place of ro, but the quantities are identical.
[CrossRef]

D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965).
[CrossRef]

1964 (2)

A. A. Hoag, Bull. Astron. 24, Part 2, 269 (1964).

R. E. Hufnagel, N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).
[CrossRef]

Bufton, J. L.

Chabot, A.

I. Goldstein, P. A. Miles, A. Chabot, Proc. IEEE 53, 1172 (1965). Here deff is used in place of ro, but the quantities are identical.
[CrossRef]

Chase, D. M.

Erdélyi, A.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Triconi, Table of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1, p. 117, Eq. 3.1(1).

Fried, D. L.

D. L. Fried, IEEE J. Quantum Electron. QE-3, 213 (1967).
[CrossRef]

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

D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965).
[CrossRef]

D. L. Fried, “The Effect of Wavefront Distortion on the Performance of an Ideal Optical Heterodyne Receiver and an Ideal Camera,” in Proceedings of the Conference on Atmospheric Limitations to Optical Propagation, sponsored by Central Radio Propagation Laboratory and National Center for Atmospheric Research, Boulder, Colorado (1965), p. 192.

Gilmartin, T. J.

T. J. Gilmartin, J. Z. Hultz, “Atmospheric Optical Coherence Measurements at 10.6 μm and 0.63 μm,” in Optics Research2, 43 (1972), published by MIT Lincoln Laboratory.

Goldstein, I.

I. Goldstein, P. A. Miles, A. Chabot, Proc. IEEE 53, 1172 (1965). Here deff is used in place of ro, but the quantities are identical.
[CrossRef]

Hoag, A. A.

A. A. Hoag, Bull. Astron. 24, Part 2, 269 (1964).

Hufnagel, R. E.

Hultz, J. Z.

T. J. Gilmartin, J. Z. Hultz, “Atmospheric Optical Coherence Measurements at 10.6 μm and 0.63 μm,” in Optics Research2, 43 (1972), published by MIT Lincoln Laboratory.

Johnson, C. B.

Kelsall, D.

Korff, D.

Lutomirski, R. F.

Magnus, W.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Triconi, Table of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1, p. 117, Eq. 3.1(1).

Meinel, A. B.

A. B. Meinel, “Final Report on the Site Selection Survey for the National Astronomical Observatory,” Contributions from the Kitt Peak National Observatory, No. 45 (Oct.1963).

Miles, P. A.

I. Goldstein, P. A. Miles, A. Chabot, Proc. IEEE 53, 1172 (1965). Here deff is used in place of ro, but the quantities are identical.
[CrossRef]

Milne, W. E.

W. E. Milne, Numerical Calculus (Princeton U.P., Princeton, 1949), p. 83, Eq. (1).

Oberhettinger, F.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Triconi, Table of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1, p. 117, Eq. 3.1(1).

Stanley, N. R.

Triconi, F. G.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Triconi, Table of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1, p. 117, Eq. 3.1(1).

Yura, H. T.

Appl. Opt. (3)

Bull. Astron. (1)

A. A. Hoag, Bull. Astron. 24, Part 2, 269 (1964).

IEEE J. Quantum Electron. (1)

D. L. Fried, IEEE J. Quantum Electron. QE-3, 213 (1967).
[CrossRef]

J. Opt. Soc. Am. (6)

Proc. IEEE (1)

I. Goldstein, P. A. Miles, A. Chabot, Proc. IEEE 53, 1172 (1965). Here deff is used in place of ro, but the quantities are identical.
[CrossRef]

Other (6)

D. L. Fried, “The Effect of Wavefront Distortion on the Performance of an Ideal Optical Heterodyne Receiver and an Ideal Camera,” in Proceedings of the Conference on Atmospheric Limitations to Optical Propagation, sponsored by Central Radio Propagation Laboratory and National Center for Atmospheric Research, Boulder, Colorado (1965), p. 192.

T. J. Gilmartin, J. Z. Hultz, “Atmospheric Optical Coherence Measurements at 10.6 μm and 0.63 μm,” in Optics Research2, 43 (1972), published by MIT Lincoln Laboratory.

A. B. Meinel, “Final Report on the Site Selection Survey for the National Astronomical Observatory,” Contributions from the Kitt Peak National Observatory, No. 45 (Oct.1963).

A factor of (2π)−1 is apparently missing from the right-hand side of Eq. (5) of Ref. 15. That this is actually the case can be confirmed by noting that according to Eq. (7) of our work, Fe(x) for x = +∞ will not equal unity, as implied by the data in Table V of Ref. 15 unless the factor (2π)−1 is introduced.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Triconi, Table of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1, p. 117, Eq. 3.1(1).

W. E. Milne, Numerical Calculus (Princeton U.P., Princeton, 1949), p. 83, Eq. (1).

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Figures (2)

Fig. 1
Fig. 1

Cumulative probability distribution for slit width. The scales are chosen so that the straight line approximation to the data represents a log-normal distribution. ro values calculated from the slit width for 5500 Å light are shown on the right-hand side. The experimental data are from Hoag.12

Fig. 2
Fig. 2

Cumulative probability distribution for slit width. The same comments apply as for Fig. 1, except that here the experimental data are from Meinel.13

Tables (1)

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Table I Edge Response Function Fe(x)

Equations (12)

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r o = ( 6.88 / a ) 3 / 5 ,
a = 2.91 ( 2 π / λ ) 2 path d s C n 2 ( s ) Q ( s ) .
τ ( f ) = exp [ 6.88 ( λ f / r o ) 5 / 3 ] .
F e ( x ) = ( 2 π ) 1 x d ξ + d ω exp [ ( ω / ω o ) n ] × exp ( i ω ξ ) .
p ( ξ , η ) = ( 2 π ) 2 + d ω x + d ω y exp [ 6.88 × ( λ 2 π r o ) 5 / 3 ( ω x 2 + ω y 2 ) 5 / 6 ] exp [ i ( ω x ξ + ω y η ) ] .
κ ( x ) = x d ξ + d η p ( ξ , η ) .
f ( w ) = ( 2 π ) 1 + d u + d υ f ( υ ) exp [ i u ( w υ ) ] ,
κ ( x ) = ( 2 π ) 1 x d ξ + d ω x exp [ 6.88 ( λ ω x 2 π r o ) 5 / 3 ] exp ( i ω x ξ ) ,
ω o = ( 2 π r o / λ ) ( 6.88 ) 3 / 5 = 1.975 r o / λ ,
[ n = 5 / 3 , F e ( x ) = 0.16 ] x = 1.33 / ω o
r o = 1.35 λ / | 2 x | .
( r o ) median = 0.114 ( λ / 5.5 × 10 7 ) 3 / 5 ( sec θ ) 3 / 5 ,

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