Abstract

The aim of this paper is to determine exactly the optical transfer function corresponding to the time-averaged image-degrading effects of atmospheric turbulence. First the average transfer function is shown to be related to the spatial coherence function for the light entering the imaging system. Next an exact closed solution is found for the coherence propagation equation. This yields the desired coherence function in terms of the statistics of the random fluctuations of the atmospheric index of refraction. Published meteorological data are analyzed to determine empirical values for the required index statistics. Of particular interest is the variation of turbulent index fluctuations with altitude. Finally, quantitative predictions of image degradation are made and shown to agree with observed data.

PDF Article

References

  • View by:
  • |
  • |

  1. S. Q. Duntley, J. Soc. Motion Picture Television Engs. 67, 231 (1958).
  2. R. E. Hufnagel and N. R. Stanley, "The Propagation of Average Mutual Coherence from a Point Source in a Random Medium" (to be published).
  3. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), pp. 479–484.
  4. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, Inc., New York, 1961), pp. 93–97.
  5. For clarity of presentation, it is implied throughout this paper that the light is monochromatic, but this restriction does not appear to be essential to the basic argument or (for quasimono-chromatic light) to the final results. To demonstrate this it is sufficient to see that Eq. (4.4) results from the propagation equations for mutual coherence6 when the light is quasimonochromatic and techniques are applied which are analogous to those used here.
  6. G. B. Parrent, Jr., R. A. Shore, and T. J. Skinner, J. Math. Phys. 3, 678 (1962).
  7. For the remainder of this paper, the explicit dependence of A and N on time t is usually not written.
  8. G. Keller, Astron. J. 58, 113 (1953).
  9. S. H. Reiger, "Atmospheric Turbulence and the Scintillation of Starlight," The Rand Corporation Report No. R-406-PR (1962).
  10. C. G. Little, Monthly Notices Roy. Astron. Soc. (London) 111, 289 (1951).
  11. D. J. Portman, F. C. Elder, E. Ryznar, and V. E. Noble, J. Geophys. RCs. 67, 3223 (1962).
  12. V. I. Tatarski, Ref. 4, p. 210.
  13. L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill Book Company, Inc., New York, 1960), pp. 28–34.
  14. V. I. Tatarski, Ref. 4, pp. 122–128.
  15. L. A. Chernov, Ref. 13, pp. 58–67.
  16. V. I. Tatarski, Ref. 4, p. 238.
  17. H. Scheffler, Astron. Nachr. 285, 21 (1958).
  18. To illustrate this, let B = x+iθand k= 105cm-1. An rms image shimmer of 0.4 arc second = 2 ×10-6 radian corresponds to ∇θ≈2× 10-6k= 0.2 cm-1. Since image shimmer decreases rapidly for apertures over 10 cm in diameter, it follows that ∇θ must change appreciably in the same distance. Hence ∇2θ≈0.02 cm-2. Since the variations (scintillation shadow bands) appear visually19 as "blobs" of diameter ≈ 10 cm, it follows, for x≈ 1, that ∇x≈ 0.1 cm-1 and ∇2x≈0.0l cm-2.
  19. M. H. Wimbush, "Optical Astronomical Seeing: a Review," U. S. Air Force Cambridge Research Laboratory, Report No. 697 (1961), pp. 6–13.
  20. H. Cramer, Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1954), p. 213.
  21. V. I. Tatarski, Ref. 4, pp. 8–12.
  22. U. S. Air Force, Handbook of Geophysics (The Macmillan Company, New York, 1960), pp. 13–2.
  23. E. E. Gossard, IRE Trans. Antennas Propagation 10, 186 (1960).
  24. J. O. Hinze, Turbulence (McGraw-Hill Book Company, Inc., New York, 1959), pp. 182–196.
  25. V. I. Tatarski, Ref. 4, pp. 40–58 and 189–197.
  26. R. Bolgiano, J. Geophys. Res. 64, 2226 (1959).
  27. Equation (6.3) is valid only over the range ri <<r0, where ri≲3cm is called the inner scale of the turbulence and r0≲ 103cm is called the outer scale of the turbulence. For the purposes of this paper this restriction is ignored, since it often has a negligible effect on the final result. A future paper is planned which will treat this matter more exactly.41
  28. L. R. Zwang, Bull. Acad. Sci. USSR, Geophys. Ser. No. 8 (Engl. Transl.) 1960, 1117.
  29. H. H. Lettau, J. Meteorol. 18, 125 (1961).
  30. F. A. Gifford, J. Atmospheric Sci. 19, 205 (1962).
  31. U. S. Air Force, Ref. 22, pp. 5–12.
  32. U. O. Lappe and B. Davidson, "The Power Spectral Analysis of Concurrent Airplane and Tower Measurements of Atmospheric Turbulence," New York University College of Engineering Research Division (1960).
  33. H. A. Panofsky, J. Geophys. Res. 67, 3161 (1962).
  34. U. S. Air Force, Ref. 22, pp. 1–10.
  35. J. R. Scoggins, Proceedings of National Symposium on Winds for Aerospace Vehicle Design (U. S. Air Force Cambridge Research Laboratory, 1962), Vol. II, pp. 85–88.
  36. N. A. Engler and J. B. Wright, Ref. 35, pp. 127–149.
  37. U. S. Air Force, Ref. 22, pp. 1–13.
  38. C. M. Crain, as quoted by S. H. Reiger on p. 29 of Ref. 9.
  39. R. Hosfeld, "Measurements of the Size of Stellar Images," AFCRC-TN-55-873 Joint Scientific Report 2 (June 1955).
  40. E. Djurle and A. Bäck, J. Opt. Soc. Am. 51, 1029 (1961).
  41. R. E. Hufnagel, "The Effect of the Inner Scale of Atmospheric Turbulence on the Propagation of Mutual Coherence" (to be published).
  42. 42 M. Loève, Probability Theory (D. Van Nostrand Company, Inc., New York, 1960), p. 470.
  43. M. Loève, Ref. 42, p. 471.

Bäck, A.

E. Djurle and A. Bäck, J. Opt. Soc. Am. 51, 1029 (1961).

Bolgiano, R.

R. Bolgiano, J. Geophys. Res. 64, 2226 (1959).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), pp. 479–484.

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill Book Company, Inc., New York, 1960), pp. 28–34.

L. A. Chernov, Ref. 13, pp. 58–67.

Crain, C. M.

C. M. Crain, as quoted by S. H. Reiger on p. 29 of Ref. 9.

Cramer, H.

H. Cramer, Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1954), p. 213.

Davidson, B.

U. O. Lappe and B. Davidson, "The Power Spectral Analysis of Concurrent Airplane and Tower Measurements of Atmospheric Turbulence," New York University College of Engineering Research Division (1960).

Djurle, E.

E. Djurle and A. Bäck, J. Opt. Soc. Am. 51, 1029 (1961).

Duntley, S. Q.

S. Q. Duntley, J. Soc. Motion Picture Television Engs. 67, 231 (1958).

Elder, F. C.

D. J. Portman, F. C. Elder, E. Ryznar, and V. E. Noble, J. Geophys. RCs. 67, 3223 (1962).

Engler, N. A.

N. A. Engler and J. B. Wright, Ref. 35, pp. 127–149.

Gifford, F. A.

F. A. Gifford, J. Atmospheric Sci. 19, 205 (1962).

Gossard, E. E.

E. E. Gossard, IRE Trans. Antennas Propagation 10, 186 (1960).

Hinze, J. O.

J. O. Hinze, Turbulence (McGraw-Hill Book Company, Inc., New York, 1959), pp. 182–196.

Hosfeld, R.

R. Hosfeld, "Measurements of the Size of Stellar Images," AFCRC-TN-55-873 Joint Scientific Report 2 (June 1955).

Hufnagel, R. E.

R. E. Hufnagel, "The Effect of the Inner Scale of Atmospheric Turbulence on the Propagation of Mutual Coherence" (to be published).

R. E. Hufnagel and N. R. Stanley, "The Propagation of Average Mutual Coherence from a Point Source in a Random Medium" (to be published).

Keller, G.

G. Keller, Astron. J. 58, 113 (1953).

Lappe, U. O.

U. O. Lappe and B. Davidson, "The Power Spectral Analysis of Concurrent Airplane and Tower Measurements of Atmospheric Turbulence," New York University College of Engineering Research Division (1960).

Lettau, H. H.

H. H. Lettau, J. Meteorol. 18, 125 (1961).

Little, C. G.

C. G. Little, Monthly Notices Roy. Astron. Soc. (London) 111, 289 (1951).

Loève, M.

42 M. Loève, Probability Theory (D. Van Nostrand Company, Inc., New York, 1960), p. 470.

M. Loève, Ref. 42, p. 471.

Noble, V. E.

D. J. Portman, F. C. Elder, E. Ryznar, and V. E. Noble, J. Geophys. RCs. 67, 3223 (1962).

Panofsky, H. A.

H. A. Panofsky, J. Geophys. Res. 67, 3161 (1962).

Parrent, Jr., G. B.

G. B. Parrent, Jr., R. A. Shore, and T. J. Skinner, J. Math. Phys. 3, 678 (1962).

Portman, D. J.

D. J. Portman, F. C. Elder, E. Ryznar, and V. E. Noble, J. Geophys. RCs. 67, 3223 (1962).

Reiger, S. H.

S. H. Reiger, "Atmospheric Turbulence and the Scintillation of Starlight," The Rand Corporation Report No. R-406-PR (1962).

Ryznar, E.

D. J. Portman, F. C. Elder, E. Ryznar, and V. E. Noble, J. Geophys. RCs. 67, 3223 (1962).

Scheffler, H.

H. Scheffler, Astron. Nachr. 285, 21 (1958).

Scoggins, J. R.

J. R. Scoggins, Proceedings of National Symposium on Winds for Aerospace Vehicle Design (U. S. Air Force Cambridge Research Laboratory, 1962), Vol. II, pp. 85–88.

Shore, R. A.

G. B. Parrent, Jr., R. A. Shore, and T. J. Skinner, J. Math. Phys. 3, 678 (1962).

Skinner, T. J.

G. B. Parrent, Jr., R. A. Shore, and T. J. Skinner, J. Math. Phys. 3, 678 (1962).

Stanley, N. R.

R. E. Hufnagel and N. R. Stanley, "The Propagation of Average Mutual Coherence from a Point Source in a Random Medium" (to be published).

Tatarski, V. I.

V. I. Tatarski, Ref. 4, pp. 8–12.

V. I. Tatarski, Ref. 4, p. 238.

V. I. Tatarski, Ref. 4, pp. 122–128.

V. I. Tatarski, Ref. 4, p. 210.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, Inc., New York, 1961), pp. 93–97.

V. I. Tatarski, Ref. 4, pp. 40–58 and 189–197.

Wimbush, M. H.

M. H. Wimbush, "Optical Astronomical Seeing: a Review," U. S. Air Force Cambridge Research Laboratory, Report No. 697 (1961), pp. 6–13.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), pp. 479–484.

Wright, J. B.

N. A. Engler and J. B. Wright, Ref. 35, pp. 127–149.

Zwang, L. R.

L. R. Zwang, Bull. Acad. Sci. USSR, Geophys. Ser. No. 8 (Engl. Transl.) 1960, 1117.

Other

S. Q. Duntley, J. Soc. Motion Picture Television Engs. 67, 231 (1958).

R. E. Hufnagel and N. R. Stanley, "The Propagation of Average Mutual Coherence from a Point Source in a Random Medium" (to be published).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), pp. 479–484.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, Inc., New York, 1961), pp. 93–97.

For clarity of presentation, it is implied throughout this paper that the light is monochromatic, but this restriction does not appear to be essential to the basic argument or (for quasimono-chromatic light) to the final results. To demonstrate this it is sufficient to see that Eq. (4.4) results from the propagation equations for mutual coherence6 when the light is quasimonochromatic and techniques are applied which are analogous to those used here.

G. B. Parrent, Jr., R. A. Shore, and T. J. Skinner, J. Math. Phys. 3, 678 (1962).

For the remainder of this paper, the explicit dependence of A and N on time t is usually not written.

G. Keller, Astron. J. 58, 113 (1953).

S. H. Reiger, "Atmospheric Turbulence and the Scintillation of Starlight," The Rand Corporation Report No. R-406-PR (1962).

C. G. Little, Monthly Notices Roy. Astron. Soc. (London) 111, 289 (1951).

D. J. Portman, F. C. Elder, E. Ryznar, and V. E. Noble, J. Geophys. RCs. 67, 3223 (1962).

V. I. Tatarski, Ref. 4, p. 210.

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill Book Company, Inc., New York, 1960), pp. 28–34.

V. I. Tatarski, Ref. 4, pp. 122–128.

L. A. Chernov, Ref. 13, pp. 58–67.

V. I. Tatarski, Ref. 4, p. 238.

H. Scheffler, Astron. Nachr. 285, 21 (1958).

To illustrate this, let B = x+iθand k= 105cm-1. An rms image shimmer of 0.4 arc second = 2 ×10-6 radian corresponds to ∇θ≈2× 10-6k= 0.2 cm-1. Since image shimmer decreases rapidly for apertures over 10 cm in diameter, it follows that ∇θ must change appreciably in the same distance. Hence ∇2θ≈0.02 cm-2. Since the variations (scintillation shadow bands) appear visually19 as "blobs" of diameter ≈ 10 cm, it follows, for x≈ 1, that ∇x≈ 0.1 cm-1 and ∇2x≈0.0l cm-2.

M. H. Wimbush, "Optical Astronomical Seeing: a Review," U. S. Air Force Cambridge Research Laboratory, Report No. 697 (1961), pp. 6–13.

H. Cramer, Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1954), p. 213.

V. I. Tatarski, Ref. 4, pp. 8–12.

U. S. Air Force, Handbook of Geophysics (The Macmillan Company, New York, 1960), pp. 13–2.

E. E. Gossard, IRE Trans. Antennas Propagation 10, 186 (1960).

J. O. Hinze, Turbulence (McGraw-Hill Book Company, Inc., New York, 1959), pp. 182–196.

V. I. Tatarski, Ref. 4, pp. 40–58 and 189–197.

R. Bolgiano, J. Geophys. Res. 64, 2226 (1959).

Equation (6.3) is valid only over the range ri <<r0, where ri≲3cm is called the inner scale of the turbulence and r0≲ 103cm is called the outer scale of the turbulence. For the purposes of this paper this restriction is ignored, since it often has a negligible effect on the final result. A future paper is planned which will treat this matter more exactly.41

L. R. Zwang, Bull. Acad. Sci. USSR, Geophys. Ser. No. 8 (Engl. Transl.) 1960, 1117.

H. H. Lettau, J. Meteorol. 18, 125 (1961).

F. A. Gifford, J. Atmospheric Sci. 19, 205 (1962).

U. S. Air Force, Ref. 22, pp. 5–12.

U. O. Lappe and B. Davidson, "The Power Spectral Analysis of Concurrent Airplane and Tower Measurements of Atmospheric Turbulence," New York University College of Engineering Research Division (1960).

H. A. Panofsky, J. Geophys. Res. 67, 3161 (1962).

U. S. Air Force, Ref. 22, pp. 1–10.

J. R. Scoggins, Proceedings of National Symposium on Winds for Aerospace Vehicle Design (U. S. Air Force Cambridge Research Laboratory, 1962), Vol. II, pp. 85–88.

N. A. Engler and J. B. Wright, Ref. 35, pp. 127–149.

U. S. Air Force, Ref. 22, pp. 1–13.

C. M. Crain, as quoted by S. H. Reiger on p. 29 of Ref. 9.

R. Hosfeld, "Measurements of the Size of Stellar Images," AFCRC-TN-55-873 Joint Scientific Report 2 (June 1955).

E. Djurle and A. Bäck, J. Opt. Soc. Am. 51, 1029 (1961).

R. E. Hufnagel, "The Effect of the Inner Scale of Atmospheric Turbulence on the Propagation of Mutual Coherence" (to be published).

42 M. Loève, Probability Theory (D. Van Nostrand Company, Inc., New York, 1960), p. 470.

M. Loève, Ref. 42, p. 471.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.