Abstract

<p>The resolution of images formed from turbulence-degraded lensless Fourier-transform holograms is analyzed for gaussian and exponential refractive-index structure function models, first by a geometrical-optics method and then by Schmeltzer’s series-expansion method, under the assumption that the random logamplitude and phase perturbations across the entrance pupil of the recording apparatus are locally stationary processes with gaussian statistics. Both long and short exposures are treated.</p><p>The resolution of turbulence-degraded holographic images is governed primarily by a function <i>m</i><sup>2</sup>(<i>r</i><sub>1</sub>), for which analytical expressions are derived and results of experimental measurements are given. The analytical results obtained for <i>m</i><sup>2</sup>(<i>r</i><sub>1</sub>) by the geometrical-optics method are identical with those obtained by Schmeltzer’s series-expansion method. The measurements were made by an interferometric technique for horizontal atmospheric propagation paths of 86 and 542 m. In addition, holographic images of an extended object were obtained for the 86-m path.</p>

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  1. J. D. Gaskill, J. Opt. Soc. Am. 58, 600 (1968).
  2. D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
  3. Reference 1, Eq. (33).
  4. R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1967).
  5. D. L. Fried, J. Opt. Soc. Am. 57, 268 (1967). In footnote 9 of this, Fried states that Schmeltzer's method is as accurate, or inaccurate, as the Rytov method. It follows that the conditions for validity are the same for the two methods.
  6. D. L. Fried, J. Opt. Soc. Am. 56, 1380 (1966).
  7. D. L. Fried, J. Opt. Soc. Am. 57, 175 (1967).
  8. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Co., New York, 1961).
  9. D. L. Fried and J. D. Cloud, J. Opt. Soc. Am. 56, 1667 (1966).
  10. Reference 8, pp. 32–34, 58, I50.
  11. J. W. Strohbehn, J. Geophys. Res. 71, 5793 (1966).
  12. Reference 1, Eq. (19).
  13. Reference 7, Fig. 1. It may be seen that the log-amplitude covariance function is essentially zero for r2>1.88(λz)1/2.
  14. J. W. Strohbehn, J. Opt. Soc. Am. 58, 139 (1968).
  15. The quantity R0 may also be thought of as the correlation length of the refractive-index perturbations. We make the distinction between R0 and L0 because L0 is not a well-defined quantity, while R0, as used here, has a more definite meaning. We retain the symbol CN2 for the structure constant, even though this notation is usually associated with the r2/3 structure function. As a result, our CN2 may have a slightly different value than that of the constant normally referred to.
  16. Reference 8, p. 34.
  17. Reference 11.
  18. Reference 4.
  19. Reference 6, Eq. (2.15).
  20. J. D. Gaskill, "Holographic Imaging Through a Randomly Inhomogeneous Medium" (Ph.D. dissertation, Stanford University, 1968), Appendix C. (Available from University Microfilms, Inc., Ann Arbor, Mich.)
  21. Reference 7, Eq. (3.3)
  22. These spectra have been calculated by Strohbehn (Ref. 11), but were recalculated here because of the different constants in the exponents of the structure functions.
  23. Reference 6, Eq. (2.21).
  24. Reference 1, Eq. (52).
  25. Reference 2, p. 1375.
  26. Reference 20, pp. 70–72.
  27. The resolution limit referred to here is that obtained by using the Rayleigh criterion, which is usually associated with incoherent illumination. For coherent imaging, as in the present case, the resolution of a system may be considerably poorer due to the speckle effect.
  28. Reference 8, p. 120.
  29. Reference 8, p. 128.
  30. Reference 14.
  31. W. P. Brown, Jr., J. Opt. Soc. Am. 57, 1539 (1967).
  32. L. S. Taylor, J. Opt. Soc. Am. 58, 57 (1968).
  33. L. S. Taylor, J. Opt. Soc. Am. 58, 705 (1968).
  34. Reference 6, Eq. (2.25).
  35. Reference 8, Eq. (8.20).
  36. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill Book Company, New York, 1965), p. 264, Table 12.9.
  37. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press Inc., New York, 1966), 4th ed., 2nd Corrected Printing, Eq. 3.365 (1).
  38. Reference 37, Eq. 8.407 (1).
  39. Reference 37, 6.561 (4).
  40. Reference 37, Eq. 8.550 (2).

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill Book Company, New York, 1965), p. 264, Table 12.9.

Brown, Jr., W. P.

W. P. Brown, Jr., J. Opt. Soc. Am. 57, 1539 (1967).

Cloud, J. D.

D. L. Fried and J. D. Cloud, J. Opt. Soc. Am. 56, 1667 (1966).

Fried, D. L.

D. L. Fried, J. Opt. Soc. Am. 57, 175 (1967).

D. L. Fried and J. D. Cloud, J. Opt. Soc. Am. 56, 1667 (1966).

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

D. L. Fried, J. Opt. Soc. Am. 56, 1380 (1966).

D. L. Fried, J. Opt. Soc. Am. 57, 268 (1967). In footnote 9 of this, Fried states that Schmeltzer's method is as accurate, or inaccurate, as the Rytov method. It follows that the conditions for validity are the same for the two methods.

Gaskill, J. D.

J. D. Gaskill, J. Opt. Soc. Am. 58, 600 (1968).

J. D. Gaskill, "Holographic Imaging Through a Randomly Inhomogeneous Medium" (Ph.D. dissertation, Stanford University, 1968), Appendix C. (Available from University Microfilms, Inc., Ann Arbor, Mich.)

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press Inc., New York, 1966), 4th ed., 2nd Corrected Printing, Eq. 3.365 (1).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press Inc., New York, 1966), 4th ed., 2nd Corrected Printing, Eq. 3.365 (1).

Schmeltzer, R. A.

R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1967).

Strohbehn, J. W.

J. W. Strohbehn, J. Geophys. Res. 71, 5793 (1966).

J. W. Strohbehn, J. Opt. Soc. Am. 58, 139 (1968).

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Co., New York, 1961).

Taylor, L. S.

L. S. Taylor, J. Opt. Soc. Am. 58, 705 (1968).

L. S. Taylor, J. Opt. Soc. Am. 58, 57 (1968).

Other (40)

J. D. Gaskill, J. Opt. Soc. Am. 58, 600 (1968).

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

Reference 1, Eq. (33).

R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1967).

D. L. Fried, J. Opt. Soc. Am. 57, 268 (1967). In footnote 9 of this, Fried states that Schmeltzer's method is as accurate, or inaccurate, as the Rytov method. It follows that the conditions for validity are the same for the two methods.

D. L. Fried, J. Opt. Soc. Am. 56, 1380 (1966).

D. L. Fried, J. Opt. Soc. Am. 57, 175 (1967).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Co., New York, 1961).

D. L. Fried and J. D. Cloud, J. Opt. Soc. Am. 56, 1667 (1966).

Reference 8, pp. 32–34, 58, I50.

J. W. Strohbehn, J. Geophys. Res. 71, 5793 (1966).

Reference 1, Eq. (19).

Reference 7, Fig. 1. It may be seen that the log-amplitude covariance function is essentially zero for r2>1.88(λz)1/2.

J. W. Strohbehn, J. Opt. Soc. Am. 58, 139 (1968).

The quantity R0 may also be thought of as the correlation length of the refractive-index perturbations. We make the distinction between R0 and L0 because L0 is not a well-defined quantity, while R0, as used here, has a more definite meaning. We retain the symbol CN2 for the structure constant, even though this notation is usually associated with the r2/3 structure function. As a result, our CN2 may have a slightly different value than that of the constant normally referred to.

Reference 8, p. 34.

Reference 11.

Reference 4.

Reference 6, Eq. (2.15).

J. D. Gaskill, "Holographic Imaging Through a Randomly Inhomogeneous Medium" (Ph.D. dissertation, Stanford University, 1968), Appendix C. (Available from University Microfilms, Inc., Ann Arbor, Mich.)

Reference 7, Eq. (3.3)

These spectra have been calculated by Strohbehn (Ref. 11), but were recalculated here because of the different constants in the exponents of the structure functions.

Reference 6, Eq. (2.21).

Reference 1, Eq. (52).

Reference 2, p. 1375.

Reference 20, pp. 70–72.

The resolution limit referred to here is that obtained by using the Rayleigh criterion, which is usually associated with incoherent illumination. For coherent imaging, as in the present case, the resolution of a system may be considerably poorer due to the speckle effect.

Reference 8, p. 120.

Reference 8, p. 128.

Reference 14.

W. P. Brown, Jr., J. Opt. Soc. Am. 57, 1539 (1967).

L. S. Taylor, J. Opt. Soc. Am. 58, 57 (1968).

L. S. Taylor, J. Opt. Soc. Am. 58, 705 (1968).

Reference 6, Eq. (2.25).

Reference 8, Eq. (8.20).

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill Book Company, New York, 1965), p. 264, Table 12.9.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press Inc., New York, 1966), 4th ed., 2nd Corrected Printing, Eq. 3.365 (1).

Reference 37, Eq. 8.407 (1).

Reference 37, 6.561 (4).

Reference 37, Eq. 8.550 (2).

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