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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 54, Iss. 1 — Jan. 1, 1964
  • pp: 52–60

Modulation Transfer Function Associated with Image Transmission through Turbulent Media

R. E. HUFNAGEL and N. R. STANLEY  »View Author Affiliations


JOSA, Vol. 54, Issue 1, pp. 52-60 (1964)
http://dx.doi.org/10.1364/JOSA.54.000052


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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.

Citation
R. E. HUFNAGEL and N. R. STANLEY, "Modulation Transfer Function Associated with Image Transmission through Turbulent Media," J. Opt. Soc. Am. 54, 52-60 (1964)
http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-54-1-52


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References

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

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