Abstract

The extended Huygens-Fresnel principle is used to formulate general expressions of the mutual intensity function for a finite optical source with partial spatial coherence propagating in the weakly turbulent atmosphere. Formulations are developed for both the focused and collimated Gaussian beam. Generalized criterion for the effective far-field range is defined in terms of the source aperture, optical wave number, source coherence, and characteristic length associated with the atmospheric turbulence. Beam spread and lateral coherence length in the near and far field are investigated for a combination of parameter variations and the physical implications discussed. Finally, analytic results are calculated and plotted to illustrate the functional behavior of relevant physical parameters.

© 1979 Optical Society of America

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  1. In the literature the term "mutual coherence function" is sometimes used synonymously with "mutual intensity function." We note that the latter is a special case of the former, i.e., τ = 0 in Eq.(3).
  2. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, VA, 1971).
  3. H. T. Yura, "Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium," Appl. Opt. 11, 1399–1406 (1972).
  4. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2 (Academic, New York, 1978), p. 422.
  5. 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).
  6. A. I. Kon and V. I. Tatarskii, "On the theory of the propagation of partially coherent light beams in a turbulent atmosphere," Radiophys. Quantum Electron. 15, 1187–1192 (1972).
  7. W. H. Carter and E. Wolf, "Coherence and radiometry with quasihomogeneous planar sources," J. Opt. Soc. Am. 67, 785–796 (1977).
  8. A. M. Whitman and M. J. Beran, "Beam spread of laser light propagation in a random medium," J. Opt. Soc. Am. 60, 1595–1602 (1970).
  9. J. C. Leader, "Atmospheric propagation of partially coherent radiation," J. Opt. Soc. Am. 68, 175–185 (1978).
  10. Z. I. Feizulin and Y. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. Quantum Electron. 10, 33–35 (1967).
  11. We note that, for a statistically homogeneous source distribution,the relation of the form Γs2(r1, r2) ≡ 〈us(r1)u*s(r2)〉s= F(r1 − r2) should hold for all position vectors r1 and r2 in the source plane z = 0, implying a source of infinite dimension. Obviously, this requirement is never satisfied in practice. In this paper we assume "statistical homogeneity" for a finite source in the approximate sense that the above relation holds whenever the point r1 and r2 both lie within the source aperture.
  12. S. C. H. Wang, M. A. Plonus, and C. F. Ouyang, "Irradiance scintillation of a partially coherent source in extremely strong turbulence," Appl. Opt. 18, 1133–1135 (1979).
  13. A. I. Kon and Z. I. Feizulin, "Fluctuations in the parameters of spherical waves propagating in a turbulent atmosphere," Radiophys. Quantum Electron. 13, 51–53 (1970).
  14. R. L. Fante, "Two-source spherical wave structure functions in atmospheric turbulence," J. Opt. Soc. Am. 66, 74 (1976).
  15. For a demonstration of the accuracy of this approximation, let us consider a coherent plane wave in a random medium. It is well known (Ref. 4, p. 445) that the plane-wave coherence length derived from the five-thirds law is ρc = (8/3)−3/5ρ00 being the spherical wave coherence length). The result in section V, using the quadratic approximation, shows that ρc = ρ0/√3, which is within a 4% agreement. One should notice, however, that although this approximation is reasonable for the second moment, it results in a significant difference for the higher-order moments.
  16. In Young's experiment, this corresponds to calculating the visibility from the interference pattern on the optical axis.
  17. M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice Hall, Englewood Cliff, N.J., 1964).
  18. H. Fujii, T. Asakura, and Y. Shindo, "Measurements of surface roughness properties by means of laser speckle techniques," Opt. Commun. 16, 68–72 (1976).
  19. A. C. Shell, "The multiple plate antenna," Doctoral Dissertation, Massachusetts Institute of Technology, 1961 (unpublished).
  20. M. Born and E. Wolf, Principles of Optics, 5th Ed. (Pergamon, Oxford, 1975).
  21. Special note should be taken in interpreting the results of Lee et al., about the dominant scale size for the focused case in very weak turbulence. The "coherent" TEM00 source with radius αT is focused on the diffuse target on which the spot size reduces to the order αF = λF/παT (assuming negligible turbulence effect). The scattered beam, having suffered a random phase delay from point to point over the target, then acts as an incoherent planar source propagating to the receiver. Since the receiver falls definitely into the far field of the new (secondary) source as in Eq. (21), the coherence length is ~2L/kαF = αT, as predicted in Ref. 5.

1979 (1)

1978 (1)

1977 (1)

1976 (3)

1972 (2)

H. T. Yura, "Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium," Appl. Opt. 11, 1399–1406 (1972).

A. I. Kon and V. I. Tatarskii, "On the theory of the propagation of partially coherent light beams in a turbulent atmosphere," Radiophys. Quantum Electron. 15, 1187–1192 (1972).

1970 (2)

A. M. Whitman and M. J. Beran, "Beam spread of laser light propagation in a random medium," J. Opt. Soc. Am. 60, 1595–1602 (1970).

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

1967 (1)

Z. I. Feizulin and Y. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. Quantum Electron. 10, 33–35 (1967).

Asakura, T.

H. Fujii, T. Asakura, and Y. Shindo, "Measurements of surface roughness properties by means of laser speckle techniques," Opt. Commun. 16, 68–72 (1976).

Beran, M. J.

A. M. Whitman and M. J. Beran, "Beam spread of laser light propagation in a random medium," J. Opt. Soc. Am. 60, 1595–1602 (1970).

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice Hall, Englewood Cliff, N.J., 1964).

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th Ed. (Pergamon, Oxford, 1975).

Carter, W. H.

Fante, R. L.

Feizulin, Z. I.

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

Z. I. Feizulin and Y. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. Quantum Electron. 10, 33–35 (1967).

Fujii, H.

H. Fujii, T. Asakura, and Y. Shindo, "Measurements of surface roughness properties by means of laser speckle techniques," Opt. Commun. 16, 68–72 (1976).

Holmes, J. F.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2 (Academic, New York, 1978), p. 422.

Kerr, J. R.

Kon, A. I.

A. I. Kon and V. I. Tatarskii, "On the theory of the propagation of partially coherent light beams in a turbulent atmosphere," Radiophys. Quantum Electron. 15, 1187–1192 (1972).

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

Kravtsov, Y.

Z. I. Feizulin and Y. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. Quantum Electron. 10, 33–35 (1967).

Leader, J. C.

Lee, M. H.

Ouyang, C. F.

Parrent, G. B.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice Hall, Englewood Cliff, N.J., 1964).

Plonus, M. A.

Shell, A. C.

A. C. Shell, "The multiple plate antenna," Doctoral Dissertation, Massachusetts Institute of Technology, 1961 (unpublished).

Shindo, Y.

H. Fujii, T. Asakura, and Y. Shindo, "Measurements of surface roughness properties by means of laser speckle techniques," Opt. Commun. 16, 68–72 (1976).

Tatarskii, V. I.

A. I. Kon and V. I. Tatarskii, "On the theory of the propagation of partially coherent light beams in a turbulent atmosphere," Radiophys. Quantum Electron. 15, 1187–1192 (1972).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, VA, 1971).

Wang, S. C. H.

Whitman, A. M.

Wolf, E.

Yura, H. T.

Appl. Opt. (2)

J. Opt. Soc. Am. (5)

Opt. Commun. (1)

H. Fujii, T. Asakura, and Y. Shindo, "Measurements of surface roughness properties by means of laser speckle techniques," Opt. Commun. 16, 68–72 (1976).

Radiophys. Quantum Electron. (3)

A. I. Kon and V. I. Tatarskii, "On the theory of the propagation of partially coherent light beams in a turbulent atmosphere," Radiophys. Quantum Electron. 15, 1187–1192 (1972).

Z. I. Feizulin and Y. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. Quantum Electron. 10, 33–35 (1967).

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

Other (10)

For a demonstration of the accuracy of this approximation, let us consider a coherent plane wave in a random medium. It is well known (Ref. 4, p. 445) that the plane-wave coherence length derived from the five-thirds law is ρc = (8/3)−3/5ρ00 being the spherical wave coherence length). The result in section V, using the quadratic approximation, shows that ρc = ρ0/√3, which is within a 4% agreement. One should notice, however, that although this approximation is reasonable for the second moment, it results in a significant difference for the higher-order moments.

In Young's experiment, this corresponds to calculating the visibility from the interference pattern on the optical axis.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice Hall, Englewood Cliff, N.J., 1964).

We note that, for a statistically homogeneous source distribution,the relation of the form Γs2(r1, r2) ≡ 〈us(r1)u*s(r2)〉s= F(r1 − r2) should hold for all position vectors r1 and r2 in the source plane z = 0, implying a source of infinite dimension. Obviously, this requirement is never satisfied in practice. In this paper we assume "statistical homogeneity" for a finite source in the approximate sense that the above relation holds whenever the point r1 and r2 both lie within the source aperture.

In the literature the term "mutual coherence function" is sometimes used synonymously with "mutual intensity function." We note that the latter is a special case of the former, i.e., τ = 0 in Eq.(3).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, VA, 1971).

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2 (Academic, New York, 1978), p. 422.

A. C. Shell, "The multiple plate antenna," Doctoral Dissertation, Massachusetts Institute of Technology, 1961 (unpublished).

M. Born and E. Wolf, Principles of Optics, 5th Ed. (Pergamon, Oxford, 1975).

Special note should be taken in interpreting the results of Lee et al., about the dominant scale size for the focused case in very weak turbulence. The "coherent" TEM00 source with radius αT is focused on the diffuse target on which the spot size reduces to the order αF = λF/παT (assuming negligible turbulence effect). The scattered beam, having suffered a random phase delay from point to point over the target, then acts as an incoherent planar source propagating to the receiver. Since the receiver falls definitely into the far field of the new (secondary) source as in Eq. (21), the coherence length is ~2L/kαF = αT, as predicted in Ref. 5.

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