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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 61, Iss. 10 — Oct. 1, 1971
  • pp: 1285–1292

Temporal-Frequency Spectra for a Spherical Wave Propagating through Atmospheric Turbulence

S. F. CLIFFORD  »View Author Affiliations

JOSA, Vol. 61, Issue 10, pp. 1285-1292 (1971)

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Tatarski has found the frequency spectra for the amplitude, phase, and phase-difference fluctuations of an infinite plane wave propagating through turbulence. Many practical optical beams, used in atmospheric studies, closely resemble point sources, for which the spherical-wave theory is more applicable. The same spectra, calculated for spherical waves, reveal contributions at higher frequencies for amplitude scintillations, nearly identical phase results, and a phase-difference spectrum with no nulls, in contrast with the plane-wave results. Comparison with recent data is shown.

S. F. CLIFFORD, "Temporal-Frequency Spectra for a Spherical Wave Propagating through Atmospheric Turbulence," J. Opt. Soc. Am. 61, 1285-1292 (1971)

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  1. R. S. Lawrence and J. W. Strohbeln, Proc. IEEE 58, 1523 (1970).
  2. Formulas derived here in the context of optical propagation, where n is function of the random temperature field, extend directly to microwave or acoustic propagation with n interpreted as a function of the humidity fields or temperature and wind fields, respectively (see Refs. 3–5).
  3. R. W. Lee and J. C. Harp, Proc. IEEE 57, 375 (1969).
  4. S. F. Clifford and J. W. Strohbehn, Proc. IEEE AP-18, 264 (1970).
  5. S. F. Clifford and E. H. Brown, J. Acoust. Soc. Am. 48, 1123 (1970).
  6. V. I. Tatarski, Wave Propagation in a Turbulent Atmosphere (Nauka, Moscow, 1967) (in Russian).
  7. The phase fluctuations, because they are most sensitive to the largest turbulent eddy scales, have an undefined spatial covariance function for a Kolmogorov spectrum of turbulence. Therefore, the phase power spectrum should be defined in terms of structure functions (Ref. 1) that are less sensitive to large scale variations. The two definitions give the same spectrum when the covariance function exists.
  8. Tables of Integral Transforms, edited by A. Erdelyi (McGraw-Hill, New York, 1959), Vol. 1, p. 43, No. 1.
  9. Reference 8, Vol. 1, p. 311, No. 30.
  10. Reference 8, Vol. 2., p. 234, No. 12.
  11. The integral in Ref. 9 converges for λ-ρ≥2 if Re(a)=0, a condition satisfied by Eq. (A1).
  12. Handboak of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (U. S. Govt. Printing Office, Washington, 1964; Dover, New York, 1965).
  13. Reference 8, Vol. 2, p. 200, No. 95.
  14. Reference 12, p. 363, No. 9.1.75.
  15. Reference 8, Vol. 1, p. 43, No. 2.
  16. Reference 12, p. 361, No. 9.1.44.

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