Abstract

An extended, Rayleigh-Sommerfeld integral method is used to derive expressions for the mutual coherence function and radiation intensity derived from a planar, partially coherent source propagating through the atmosphere. The derived results reduce to previous results for (i) coherent radiation propagation in the atmosphere and (ii) the relations relating the far-field intensity angular distribution and the source coherence for a partially coherent source <i>in vacuo</i>. A mathematical description of the predicted results in terms of the vacuum distribution and scattering functions (related to the Fourier-transformed two-source mutual coherence function) is permitted by this development. Analytical results are calculated for a homogeneous atmosphere and a source coherence that simulates a laser-illuminated rough surface. The effective far-field range is determined by the source size, wavelength, and source coherence length. The phase of the calculated mutual coherence function is determined by the field-point separation for off-axial propagation directions. Numerical results for the amplitude and phase coherence lengths are calculated and illustrated as a function of the source size, source coherence length, propagation angle, range, and refractive-index structure constant.

© 1978 Optical Society of America

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  1. M. J. Beran, "Propagations of a finite beam in a random medium," J. Opt. Soc. Am. 60, 518–521 (1970).
  2. R. L. Fante, "Mutual coherence function and frequency spectrum of a laser beam propagating through atmospheric turbulence," J. Opt. Soc. Am. 64, 592–598 (1974).
  3. V. I. Tatarskii, "The effects of the turbulent atmosphere on wave propagation," translated from the Russian TT-68-50464, 472 pp., National Technical Information Service, U.S. Dept. of Commerce, Springfield, Va. 22151.
  4. W. P. Brown, "Second moment of a wave propagating in a random medium," J. Opt. Soc. Am. 61, 1051–1059 (1971).
  5. H. T. Yura, "Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium," Appl. Opt. 11, 1399–1406 (1972).
  6. M. J. Beran, "Coherence equations governing propagation through random media," Radio. Sci. 10, 15–21 (1975).
  7. M. H. Lee, J. F. Holmes, and J. R. Kerr, "Statistics of speckle propagation through the turbulent atmosphere," J. Opt. Soc. Am. 66, 1164–1172 (1976).
  8. R. F. Lutomirski and H. T. Yura, "Propagation of a finite optical beam in an inhomgeneous medium," Appl. Opt. 10, 1652–1658 (1971).
  9. E. Wolf and W. H. Carter, "Angular distribution of radiant intensity from sources of different degrees of spatial coherence," Opt. Commun. 13, 205–209 (1975).
  10. R. L. Fante and J. L. Poirier, "Mutual coherence function of a finite optical beam in a turbulent medium," Appl. Opt. 12, 2247–2248 (1973).
  11. E. W. Marchand and E. Wolf, "Angular correlation and the far-zone behavior of partially coherent fields," J. Opt. Soc. Am. 62, 379–385 (1972).
  12. The vector problem can be treated using the technique described in this paper together with the formalism described in J. C. Leader, "The generalized partial coherence of a radiation source and its far-field" Optica Acta (to be published).
  13. A. Sommerfeld, "Optics," Lectures on Theoretical Physics (Academic, New York, 1954), Vol IV, pp. 197–201.
  14. R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1964), pp. 311–319.
  15. M. Born and E. Wolf, Principzes of Optics, 3rd (revised) edition (Pergamon, New York, 1965), pp. 512–513.
  16. Use has been made of the fact that R- = (|p|2 + z2)1/2 - (|p′|2 + z2)1/2 ≅ (1/2z) × (|p|2 - |p′|2) = P+ · p-/z.
  17. M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964), Eqs. 4–25.
  18. D. L. Fried, "Optical resolution through a randomly inhomogeneous medium for very long and very short exposures," J. Opt. Soc. Am. 56, 1372–1384 (1966).
  19. A. Kon and V. Feizulin, "Fluctuations in the parameters of spherical waves propagating in a turbulant atmosphere," Radiophys. Quantum Electron. (USSR) 13, 51–53 (1970).
  20. R. L. Fante, "Two-source spherical wave structure functions in atmospheric turbulence," J. Opt. Soc. Am. 66, 74 (1976).
  21. A. V. Artem'ev and A. S. Gurvich, "Experimental study of coherence function spectra," Radiophys. Quantum Electron. (USSR) 14, 580–583 (1971).
  22. D. E. Barrick, "Rough surface scattering based on the specular point theory," IEEE Trans. Antennas Propag. AP-16, 449–454 (1968).
  23. J. R. Dunphy and J. R. Kerr, "Turbulence effects on target illumination by laser sources: phenomonological analysis and experimental results," Appl. Opt. 16, 1345–1358 (1977).
  24. Ref 17, pp. 50–52.
  25. J. C. Leader, "An analysis of the spatial coherence of laser light scattered from a surface with two scales of roughness," J. Opt. Soc. Am. 66, 536–546 (1976).
  26. J. C. Leader, "Incoherent backscatter from rough surfaces: the two scale model re-examined," Radio. Sci. (to be published June/July 1978).

1977

1976

1975

M. J. Beran, "Coherence equations governing propagation through random media," Radio. Sci. 10, 15–21 (1975).

E. Wolf and W. H. Carter, "Angular distribution of radiant intensity from sources of different degrees of spatial coherence," Opt. Commun. 13, 205–209 (1975).

1974

1973

1972

1971

1970

M. J. Beran, "Propagations of a finite beam in a random medium," J. Opt. Soc. Am. 60, 518–521 (1970).

A. Kon and V. Feizulin, "Fluctuations in the parameters of spherical waves propagating in a turbulant atmosphere," Radiophys. Quantum Electron. (USSR) 13, 51–53 (1970).

1968

D. E. Barrick, "Rough surface scattering based on the specular point theory," IEEE Trans. Antennas Propag. AP-16, 449–454 (1968).

1966

1954

A. Sommerfeld, "Optics," Lectures on Theoretical Physics (Academic, New York, 1954), Vol IV, pp. 197–201.

Artem’ev, A. V.

A. V. Artem'ev and A. S. Gurvich, "Experimental study of coherence function spectra," Radiophys. Quantum Electron. (USSR) 14, 580–583 (1971).

Barrick, D. E.

D. E. Barrick, "Rough surface scattering based on the specular point theory," IEEE Trans. Antennas Propag. AP-16, 449–454 (1968).

Beran, M. J.

M. J. Beran, "Coherence equations governing propagation through random media," Radio. Sci. 10, 15–21 (1975).

M. J. Beran, "Propagations of a finite beam in a random medium," J. Opt. Soc. Am. 60, 518–521 (1970).

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964), Eqs. 4–25.

Born, M.

M. Born and E. Wolf, Principzes of Optics, 3rd (revised) edition (Pergamon, New York, 1965), pp. 512–513.

Brown, W. P.

Carter, W. H.

E. Wolf and W. H. Carter, "Angular distribution of radiant intensity from sources of different degrees of spatial coherence," Opt. Commun. 13, 205–209 (1975).

Dunphy, J. R.

Fante, R. L.

Feizulin, V.

A. Kon and V. Feizulin, "Fluctuations in the parameters of spherical waves propagating in a turbulant atmosphere," Radiophys. Quantum Electron. (USSR) 13, 51–53 (1970).

Fried, D. L.

Gurvich, A. S.

A. V. Artem'ev and A. S. Gurvich, "Experimental study of coherence function spectra," Radiophys. Quantum Electron. (USSR) 14, 580–583 (1971).

Holmes, J. F.

Kerr, J. R.

Kon, A.

A. Kon and V. Feizulin, "Fluctuations in the parameters of spherical waves propagating in a turbulant atmosphere," Radiophys. Quantum Electron. (USSR) 13, 51–53 (1970).

Leader, J. C.

J. C. Leader, "Incoherent backscatter from rough surfaces: the two scale model re-examined," Radio. Sci. (to be published June/July 1978).

J. C. Leader, "An analysis of the spatial coherence of laser light scattered from a surface with two scales of roughness," J. Opt. Soc. Am. 66, 536–546 (1976).

The vector problem can be treated using the technique described in this paper together with the formalism described in J. C. Leader, "The generalized partial coherence of a radiation source and its far-field" Optica Acta (to be published).

Lee, M. H.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1964), pp. 311–319.

Lutomirski, R. F.

Marchand, E. W.

Parrent, G. B.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964), Eqs. 4–25.

Poirier, J. L.

Sommerfeld, A.

A. Sommerfeld, "Optics," Lectures on Theoretical Physics (Academic, New York, 1954), Vol IV, pp. 197–201.

Tatarskii, V. I.

V. I. Tatarskii, "The effects of the turbulent atmosphere on wave propagation," translated from the Russian TT-68-50464, 472 pp., National Technical Information Service, U.S. Dept. of Commerce, Springfield, Va. 22151.

Wolf, E.

E. Wolf and W. H. Carter, "Angular distribution of radiant intensity from sources of different degrees of spatial coherence," Opt. Commun. 13, 205–209 (1975).

E. W. Marchand and E. Wolf, "Angular correlation and the far-zone behavior of partially coherent fields," J. Opt. Soc. Am. 62, 379–385 (1972).

M. Born and E. Wolf, Principzes of Optics, 3rd (revised) edition (Pergamon, New York, 1965), pp. 512–513.

Yura, H. T.

Appl. Opt.

IEEE Trans. Antennas Propag.

D. E. Barrick, "Rough surface scattering based on the specular point theory," IEEE Trans. Antennas Propag. AP-16, 449–454 (1968).

J. Opt. Soc. Am.

Opt. Commun.

E. Wolf and W. H. Carter, "Angular distribution of radiant intensity from sources of different degrees of spatial coherence," Opt. Commun. 13, 205–209 (1975).

Radio. Sci.

M. J. Beran, "Coherence equations governing propagation through random media," Radio. Sci. 10, 15–21 (1975).

Radiophys. Quantum Electron.

A. Kon and V. Feizulin, "Fluctuations in the parameters of spherical waves propagating in a turbulant atmosphere," Radiophys. Quantum Electron. (USSR) 13, 51–53 (1970).

A. V. Artem'ev and A. S. Gurvich, "Experimental study of coherence function spectra," Radiophys. Quantum Electron. (USSR) 14, 580–583 (1971).

Other

V. I. Tatarskii, "The effects of the turbulent atmosphere on wave propagation," translated from the Russian TT-68-50464, 472 pp., National Technical Information Service, U.S. Dept. of Commerce, Springfield, Va. 22151.

The vector problem can be treated using the technique described in this paper together with the formalism described in J. C. Leader, "The generalized partial coherence of a radiation source and its far-field" Optica Acta (to be published).

A. Sommerfeld, "Optics," Lectures on Theoretical Physics (Academic, New York, 1954), Vol IV, pp. 197–201.

R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1964), pp. 311–319.

M. Born and E. Wolf, Principzes of Optics, 3rd (revised) edition (Pergamon, New York, 1965), pp. 512–513.

Use has been made of the fact that R- = (|p|2 + z2)1/2 - (|p′|2 + z2)1/2 ≅ (1/2z) × (|p|2 - |p′|2) = P+ · p-/z.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964), Eqs. 4–25.

Ref 17, pp. 50–52.

J. C. Leader, "Incoherent backscatter from rough surfaces: the two scale model re-examined," Radio. Sci. (to be published June/July 1978).

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