Abstract

A source having a deterministic beam-wave amplitude distribution with a spatially random phase variation is assumed. The source distibution simulates laser reflectance from (or transmission through) a rough surface with arbitrary height deviation and a correlation yielding a Gaussian intensity covariance. The intensity covariance resulting from the assumed source propagating through atmospheric turbulence is calculated using a formalism developed previously. The resultant eight-fold intergral is evaluated in closed form retaining all phase, log-amplitude, and cross phase-log-amplitude structure functions by employing the quadratic approximation for the complex phase. Limiting case conditions of (i) a field from a partially coherent source propagating in vacuo (speckle) and (ii) a coherent beam-wave propagating through turbulence are examined. Speckle contrast calculations replicate published data using less restrictive assumptions than formerly employed, while turbulent atmosphere beam-wave calculations appear more physically reasonable than results of Ishimaru. General-case calculations show that the normalized intensity variance (contrast or fluctuation parameter) increases less rapidly with increasing turbulence as the phase variance of the source increases. A saturation phenomenon is observed at high turbulence levels as the coherence decreases, The inability to sustain high spatial frequency speckle in turbulence is reflected in the calculated intensity covariance function.

© 1979 Optical Society of America

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  1. R. L. Fante, "Electromagnetic beam propagation in turbulent media," IEEE Proc. 63,1669–1692 (1975).
  2. A. S. Gurvich and V. I. Tatarskii, "Coherence and intensity fluctu- ations of light in the turbulent atmosphere," Radio Sci. 10, 3–14 (1975).
  3. V. A. Banakh and V. L. Mironov, "Phase approximation of the Huygens- Kirchhoff method in problems of laser-beam propagation in the turbulent atmosphere," Opt. Lett. 1, 172–174 (1977).
  4. J. C. Leader, "Spatial coherence measurements of Rayleigh-scattered light," J. Opt. Soc. Am. 65, 740–741 (1975).
  5. J. C. Leader, "The generalized partial coherence of a radiation source and its far-field," Optica Acta 25, 395–413 (1978).
  6. 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).
  7. J. C. Leader, "Atmospheric propagation of partially coherent radiation," J. Opt. Soc. Am. 68, 175–185 (1978).
  8. M. J. Beran and G. B. Parrent, Jr., Theory of Partial Coherence (Prentice Hall, Englewood Cliffs, NJ, 1964), pp 7–11, 27–35.
  9. J. C. Leader, "An analysis of the frequency spectrum of laser light scattered from moving rough objects," J. Opt. Soc. Am. 67, 1091–1098 (1977).
  10. A. Sommerfeld, "Optics", Lectures on Theoretical Physics (Academic, New York, 1954) Vol. IV, pp 197–201.
  11. R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1964), pp 311–319.
  12. Z. 1. Feizulin and Yu. A. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. and Quantum Electron. (USSR) 10, 68–73 (1967).
  13. R. L. Fante, "Two-source spherical wave structure functions in atmospheric turbulence," J. Opt. Soc. Am. 66,74 (1976).
  14. A. V. Artem'ev and A. S. Gurvich, "Experimental study of coherence function spectra," Radiophys. Quantum Electron. (USSR) 14, 580–583 (1971).
  15. H. M. Pedersen, "Theory of speckle dependence on surface roughness," J. Opt. Soc. Am. 66, 1204–1210 (1976).
  16. J. Ohtsubo and T. Asakura, "Statistical properties of laser speckle produced in the diffraction field," Appl. Opt. 16, 1742–1753 (1977).
  17. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, (Pergamon, New York, 1963).
  18. The unit vectors are evaluated using the formulas in Appendix C of Ref. 7.
  19. J. C. Dainty, Laser Speckle (Springer-Verlag, New York, 1975).
  20. H. Fujii, T. Asakura, and Y. Shindo, "Measurements of surface roughness properties by means of laser speckle techniques," Opt. Comm. 16, 68–72 (1976).
  21. The phase deviation for transmission experiments is one half the value predicted by Eq. (21). The experiments reported in Ref. 16 were transmission experiments.
  22. A. Ishimaru, "Fluctuations of a beam wave propagating through a locally homogeneous medium," Radio Sci. 4, 295–305 (1969).
  23. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1968), pp 556–566.
  24. Ibid., pp 504–535.
  25. R. S. Lawrence and J. W. Strohbehn, "A survey of clear-air propagation effects relevant to optical communications," IEEE Proc. 58, 1523–1545 (1970).
  26. M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pakasov, "Similarity correlations and their experimental verification in the case of strong intensity fluctuations of the laser radiation," Akademiia Nauk SSSR. Otdelenie Okeanologii, Fiziki, Atmosfery; Geografii. (Preprint, Moscow, (1973). Aerospace Corp., translation No. LRG-73-T-28).
  27. Note added in review.
  28. P A. Pincus, M. E. Fossey, J. F. Holmes, and J. R. Kerr, "Speckle propagation through turbulence: Experimental," 3. Opt. Soc. Am. 68, 760–762 (1978).
  29. A phase deviation of σø = 5.0 and coherence length of ρu = 5.0 μm was assumed commensurate with the diffuse nature of the scattering target (Scotchlite).
  30. M. H. Lee, 3. F. Holmes, and 3. R. Kerr, "Statistics of speckle propagation through the turbulent atmosphere," J. Opt. Soc. Am. 66, 1164–1172 (1976).

1978

J. C. Leader, "The generalized partial coherence of a radiation source and its far-field," Optica Acta 25, 395–413 (1978).

J. C. Leader, "Atmospheric propagation of partially coherent radiation," J. Opt. Soc. Am. 68, 175–185 (1978).

P A. Pincus, M. E. Fossey, J. F. Holmes, and J. R. Kerr, "Speckle propagation through turbulence: Experimental," 3. Opt. Soc. Am. 68, 760–762 (1978).

1977

1976

1975

J. C. Leader, "Spatial coherence measurements of Rayleigh-scattered light," J. Opt. Soc. Am. 65, 740–741 (1975).

R. L. Fante, "Electromagnetic beam propagation in turbulent media," IEEE Proc. 63,1669–1692 (1975).

A. S. Gurvich and V. I. Tatarskii, "Coherence and intensity fluctu- ations of light in the turbulent atmosphere," Radio Sci. 10, 3–14 (1975).

1971

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

1970

R. S. Lawrence and J. W. Strohbehn, "A survey of clear-air propagation effects relevant to optical communications," IEEE Proc. 58, 1523–1545 (1970).

1969

A. Ishimaru, "Fluctuations of a beam wave propagating through a locally homogeneous medium," Radio Sci. 4, 295–305 (1969).

1967

Z. 1. Feizulin and Yu. A. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. and Quantum Electron. (USSR) 10, 68–73 (1967).

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1968), pp 556–566.

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).

Asakura, T.

J. Ohtsubo and T. Asakura, "Statistical properties of laser speckle produced in the diffraction field," Appl. Opt. 16, 1742–1753 (1977).

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

Banakh, V. A.

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, (Pergamon, New York, 1963).

Beran, M. J.

M. J. Beran and G. B. Parrent, Jr., Theory of Partial Coherence (Prentice Hall, Englewood Cliffs, NJ, 1964), pp 7–11, 27–35.

Dainty, J. C.

J. C. Dainty, Laser Speckle (Springer-Verlag, New York, 1975).

Fante, R. L.

R. L. Fante, "Two-source spherical wave structure functions in atmospheric turbulence," J. Opt. Soc. Am. 66,74 (1976).

R. L. Fante, "Electromagnetic beam propagation in turbulent media," IEEE Proc. 63,1669–1692 (1975).

Feizulin, Z. 1.

Z. 1. Feizulin and Yu. A. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. and Quantum Electron. (USSR) 10, 68–73 (1967).

Fossey, M. E.

P A. Pincus, M. E. Fossey, J. F. Holmes, and J. R. Kerr, "Speckle propagation through turbulence: Experimental," 3. Opt. Soc. Am. 68, 760–762 (1978).

Fujii, H.

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

Gracheva, M. E.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pakasov, "Similarity correlations and their experimental verification in the case of strong intensity fluctuations of the laser radiation," Akademiia Nauk SSSR. Otdelenie Okeanologii, Fiziki, Atmosfery; Geografii. (Preprint, Moscow, (1973). Aerospace Corp., translation No. LRG-73-T-28).

Gurvich, A. S.

A. S. Gurvich and V. I. Tatarskii, "Coherence and intensity fluctu- ations of light in the turbulent atmosphere," Radio Sci. 10, 3–14 (1975).

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

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pakasov, "Similarity correlations and their experimental verification in the case of strong intensity fluctuations of the laser radiation," Akademiia Nauk SSSR. Otdelenie Okeanologii, Fiziki, Atmosfery; Geografii. (Preprint, Moscow, (1973). Aerospace Corp., translation No. LRG-73-T-28).

Holmes, J. F.

P A. Pincus, M. E. Fossey, J. F. Holmes, and J. R. Kerr, "Speckle propagation through turbulence: Experimental," 3. Opt. Soc. Am. 68, 760–762 (1978).

M. H. Lee, 3. F. Holmes, and 3. R. Kerr, "Statistics of speckle propagation through the turbulent atmosphere," J. Opt. Soc. Am. 66, 1164–1172 (1976).

Ishimaru, A.

A. Ishimaru, "Fluctuations of a beam wave propagating through a locally homogeneous medium," Radio Sci. 4, 295–305 (1969).

Kashkarov, S. S.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pakasov, "Similarity correlations and their experimental verification in the case of strong intensity fluctuations of the laser radiation," Akademiia Nauk SSSR. Otdelenie Okeanologii, Fiziki, Atmosfery; Geografii. (Preprint, Moscow, (1973). Aerospace Corp., translation No. LRG-73-T-28).

Kerr, J. R.

P A. Pincus, M. E. Fossey, J. F. Holmes, and J. R. Kerr, "Speckle propagation through turbulence: Experimental," 3. Opt. Soc. Am. 68, 760–762 (1978).

M. H. Lee, 3. F. Holmes, and 3. R. Kerr, "Statistics of speckle propagation through the turbulent atmosphere," J. Opt. Soc. Am. 66, 1164–1172 (1976).

Kravtsov, Yu. A.

Z. 1. Feizulin and Yu. A. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. and Quantum Electron. (USSR) 10, 68–73 (1967).

Lawrence, R. S.

R. S. Lawrence and J. W. Strohbehn, "A survey of clear-air propagation effects relevant to optical communications," IEEE Proc. 58, 1523–1545 (1970).

Leader, J. C.

Lee, M. H.

Luneburg, R. K.

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

Mironov, V. L.

Ohtsubo, J.

Pakasov, V. V.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pakasov, "Similarity correlations and their experimental verification in the case of strong intensity fluctuations of the laser radiation," Akademiia Nauk SSSR. Otdelenie Okeanologii, Fiziki, Atmosfery; Geografii. (Preprint, Moscow, (1973). Aerospace Corp., translation No. LRG-73-T-28).

Parrent, Jr., G. B.

M. J. Beran and G. B. Parrent, Jr., Theory of Partial Coherence (Prentice Hall, Englewood Cliffs, NJ, 1964), pp 7–11, 27–35.

Pedersen, H. M.

Pincus, P A.

P A. Pincus, M. E. Fossey, J. F. Holmes, and J. R. Kerr, "Speckle propagation through turbulence: Experimental," 3. Opt. Soc. Am. 68, 760–762 (1978).

Shindo, Y.

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

Sommerfeld, A.

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

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, (Pergamon, New York, 1963).

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1968), pp 556–566.

Strohbehn, J. W.

R. S. Lawrence and J. W. Strohbehn, "A survey of clear-air propagation effects relevant to optical communications," IEEE Proc. 58, 1523–1545 (1970).

Tatarskii, V. I.

A. S. Gurvich and V. I. Tatarskii, "Coherence and intensity fluctu- ations of light in the turbulent atmosphere," Radio Sci. 10, 3–14 (1975).

3. Opt. Soc. Am.

P A. Pincus, M. E. Fossey, J. F. Holmes, and J. R. Kerr, "Speckle propagation through turbulence: Experimental," 3. Opt. Soc. Am. 68, 760–762 (1978).

Appl. Opt.

IEEE Proc.

R. L. Fante, "Electromagnetic beam propagation in turbulent media," IEEE Proc. 63,1669–1692 (1975).

R. S. Lawrence and J. W. Strohbehn, "A survey of clear-air propagation effects relevant to optical communications," IEEE Proc. 58, 1523–1545 (1970).

J. Opt. Soc. Am.

Opt. Comm.

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

Opt. Lett.

Optica Acta

J. C. Leader, "The generalized partial coherence of a radiation source and its far-field," Optica Acta 25, 395–413 (1978).

Radio Sci.

A. S. Gurvich and V. I. Tatarskii, "Coherence and intensity fluctu- ations of light in the turbulent atmosphere," Radio Sci. 10, 3–14 (1975).

A. Ishimaru, "Fluctuations of a beam wave propagating through a locally homogeneous medium," Radio Sci. 4, 295–305 (1969).

Radiophys. and Quantum Electron. USSR

Z. 1. Feizulin and Yu. A. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. and Quantum Electron. (USSR) 10, 68–73 (1967).

Radiophys. Quantum Electron. USSR

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

Other

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, (Pergamon, New York, 1963).

The unit vectors are evaluated using the formulas in Appendix C of Ref. 7.

J. C. Dainty, Laser Speckle (Springer-Verlag, New York, 1975).

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. J. Beran and G. B. Parrent, Jr., Theory of Partial Coherence (Prentice Hall, Englewood Cliffs, NJ, 1964), pp 7–11, 27–35.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1968), pp 556–566.

Ibid., pp 504–535.

The phase deviation for transmission experiments is one half the value predicted by Eq. (21). The experiments reported in Ref. 16 were transmission experiments.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pakasov, "Similarity correlations and their experimental verification in the case of strong intensity fluctuations of the laser radiation," Akademiia Nauk SSSR. Otdelenie Okeanologii, Fiziki, Atmosfery; Geografii. (Preprint, Moscow, (1973). Aerospace Corp., translation No. LRG-73-T-28).

Note added in review.

A phase deviation of σø = 5.0 and coherence length of ρu = 5.0 μm was assumed commensurate with the diffuse nature of the scattering target (Scotchlite).

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