Abstract

The applicability of the Rytov approximation to the calculation of the characteristics of optical propagation in a weakly inhomogeneous random medium is investigated. The condition that the mean square value of the second term in the associated perturbation expansion be smaller than that of the first is adopted as a criterion for the validity of the Rytov approximation. It is shown that there is a very severe range limitation on the validity of the Rytov approximation for optical propagation in the lower portions of the earth’s atmosphere. Furthermore, comparison of the limiting form for large <i>x</i>/<i>kl</i><sup>2</sup> of the validity condition derived in this paper with the condition on the validity of the Born approximation obtained independently by Mintzer, and by Kay and Silverman reveals that the Rytov and Born approximations have the same domain of validity in this limiting case. The equivalence of the validity conditions for the Rytov and Born approximations contradicts the statements of Tatarski and Chernov who contend that the Rytov approximation is superior to the Born approximation. It is conjectured that the Rytov and Born approximations have the same domain of validity for all <i>x</i>/<i>kl</i><sup>2</sup>.

PDF Article

References

  • View by:
  • |
  • |

  1. R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).
  2. A. M. Obukhov, Izvestiya Akad. Nauk., Geophys. Ser. 2, 155 (1953) [translated by W. C. Hoffman in Project RAND Report T-47 (1955)].
  3. L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill Book Co., Inc., New York, 1960).
  4. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Co., Inc., New York, 1961).
  5. W. C. Hoffman, IRE Trans. Antennas Propagation Special Suppl., S301 (1959).
  6. A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions, R. A. Silverman, Ed. (Prentice Hall, Englewood Cliffs, N. J., 1962).
  7. V. I. Tatarski, Ref. 4, pp. 128–130.
  8. See Ref. 4, pp. 130–137.
  9. Actually, the assumption of gaussian correlation implicit in the use of (26) is not required in the evaluation of (32). It is possible to proceed in the manner of Tatarski, i.e., let ∫0Fv(k1,u)du=πϕ(k,0), where ϕ is the three-dimensional spectral density of the refractive index fluctuations. In this case ϕ(K,0) need not correspond to a gaussian correlation function. Equation (26) is employed in the calculation of L, however, to conform with the approach required in the evaluation of the integrals involving M2.
  10. W. Magnus and F. Oberhettinger, Formulas and Theorems for the Functions of Mathematical Physics (Chelsea Publishing Co., New York, 1954), p. 137.
  11. A. Erdélyi, Asymptotic Expansions (Dover Publications, Inc., New York, 1956), pp. 8–13.
  12. D. Mintzer, J. Acoust. Soc. Am. 25, 1107 (1953).
  13. I. Kay and R. A. Silverman, Nuovo Cimento Suppl. 9 (Ser. 10) 626 (1958).
  14. V. I. Tatarski, Ref. 4, pp. 122–126.
  15. A. M. Obukhov, Ref. 2, pp. 58–66.
  16. D. A. de Wolf, J. Opt. Soc. Am. 55, 812 (1965).
  17. V. I. Tatarski, Ref. 4, p. 148.
  18. V. I. Tatarski, Ref. 4, pp. 120–121.
  19. R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 57 (1964).
  20. Generally, the strength of the turbulence decreases as the altitude above the earth is increased. The data tabulated by Hufnagel and Stanley1 indicate that the magnitude of the fluctuation is decreased by a factor of 10 at 3 km.
  21. F. E. Goodwin, paper presented at the conference on Atmospheric Limitations on Optical Propagation, Boulder, Colorado (1965).

Chernov, L. A.

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

de Wolf, D. A.

D. A. de Wolf, J. Opt. Soc. Am. 55, 812 (1965).

Erdélyi, A.

A. Erdélyi, Asymptotic Expansions (Dover Publications, Inc., New York, 1956), pp. 8–13.

Goodwin, F. E.

F. E. Goodwin, paper presented at the conference on Atmospheric Limitations on Optical Propagation, Boulder, Colorado (1965).

Hoffman, W. C.

W. C. Hoffman, IRE Trans. Antennas Propagation Special Suppl., S301 (1959).

Hufnagel, R. E.

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

R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 57 (1964).

Kay, I.

I. Kay and R. A. Silverman, Nuovo Cimento Suppl. 9 (Ser. 10) 626 (1958).

Magnus, W.

W. Magnus and F. Oberhettinger, Formulas and Theorems for the Functions of Mathematical Physics (Chelsea Publishing Co., New York, 1954), p. 137.

Mintzer, D.

D. Mintzer, J. Acoust. Soc. Am. 25, 1107 (1953).

Oberhettinger, F.

W. Magnus and F. Oberhettinger, Formulas and Theorems for the Functions of Mathematical Physics (Chelsea Publishing Co., New York, 1954), p. 137.

Obukhov, A. M.

A. M. Obukhov, Izvestiya Akad. Nauk., Geophys. Ser. 2, 155 (1953) [translated by W. C. Hoffman in Project RAND Report T-47 (1955)].

A. M. Obukhov, Ref. 2, pp. 58–66.

Silverman, R. A.

I. Kay and R. A. Silverman, Nuovo Cimento Suppl. 9 (Ser. 10) 626 (1958).

Stanley, N. R.

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

R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 57 (1964).

Tatarski, V. I.

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

V. I. Tatarski, Ref. 4, pp. 128–130.

V. I. Tatarski, Ref. 4, pp. 122–126.

V. I. Tatarski, Ref. 4, p. 148.

V. I. Tatarski, Ref. 4, pp. 120–121.

Yaglom, A. M.

A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions, R. A. Silverman, Ed. (Prentice Hall, Englewood Cliffs, N. J., 1962).

Other (21)

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

A. M. Obukhov, Izvestiya Akad. Nauk., Geophys. Ser. 2, 155 (1953) [translated by W. C. Hoffman in Project RAND Report T-47 (1955)].

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

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

W. C. Hoffman, IRE Trans. Antennas Propagation Special Suppl., S301 (1959).

A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions, R. A. Silverman, Ed. (Prentice Hall, Englewood Cliffs, N. J., 1962).

V. I. Tatarski, Ref. 4, pp. 128–130.

See Ref. 4, pp. 130–137.

Actually, the assumption of gaussian correlation implicit in the use of (26) is not required in the evaluation of (32). It is possible to proceed in the manner of Tatarski, i.e., let ∫0Fv(k1,u)du=πϕ(k,0), where ϕ is the three-dimensional spectral density of the refractive index fluctuations. In this case ϕ(K,0) need not correspond to a gaussian correlation function. Equation (26) is employed in the calculation of L, however, to conform with the approach required in the evaluation of the integrals involving M2.

W. Magnus and F. Oberhettinger, Formulas and Theorems for the Functions of Mathematical Physics (Chelsea Publishing Co., New York, 1954), p. 137.

A. Erdélyi, Asymptotic Expansions (Dover Publications, Inc., New York, 1956), pp. 8–13.

D. Mintzer, J. Acoust. Soc. Am. 25, 1107 (1953).

I. Kay and R. A. Silverman, Nuovo Cimento Suppl. 9 (Ser. 10) 626 (1958).

V. I. Tatarski, Ref. 4, pp. 122–126.

A. M. Obukhov, Ref. 2, pp. 58–66.

D. A. de Wolf, J. Opt. Soc. Am. 55, 812 (1965).

V. I. Tatarski, Ref. 4, p. 148.

V. I. Tatarski, Ref. 4, pp. 120–121.

R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 57 (1964).

Generally, the strength of the turbulence decreases as the altitude above the earth is increased. The data tabulated by Hufnagel and Stanley1 indicate that the magnitude of the fluctuation is decreased by a factor of 10 at 3 km.

F. E. Goodwin, paper presented at the conference on Atmospheric Limitations on Optical Propagation, Boulder, Colorado (1965).

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.