Abstract

Consideration of the density of rays emanating from an infinite, corrugated Gaussian surface with fractal slope reveals a hitherto unexplored shortwave scattering regime that is uncomplicated by the presence of caustics in the scattered wave field. The statistical and coherence properties of the ray-density fluctuations in this regime are calculated as a function of fractal dimension <i>D</i>, and it is shown that in the Brownian case (<i>D</i> = 1.5) the problem can be solved exactly. The properties of the intensity pattern in a coherent scattering configuration are also investigated. The contrast of the pattern is computed as a function of propagation distance, and the asymptotic behavior in the strong scattering limit is again found to be exactly solvable when <i>D</i> = 1.5. It is shown that the intensity fluctuations are <i>K</i> distributed in this case. The effects of a finite outer-scale size are evaluated and discussed.

© 1982 Optical Society of America

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  1. J. C. Dainty, ed., Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975).
  2. E. Jakeman, "Speckle statistics with a small number of scatterers," Proc. Soc. Photo-Opt. Instrum. Eng. 243, 9–19 (1980).
  3. M. Nakagami, "The m-distribution—a general formula of intensity distribution of rapid fading," in Statistical Methods in Radio Wave Propagation (Pergamon, New York, 1960), pp. 3–36.
  4. M. M. Siddiqui and G. H. Weiss, "Families of distributions for hourly median power and instantaneous power of received radio signals," J. Res. Nat. Bur. Stand. 67D, 753–762(1963).
  5. P. Beckmann, Probability in Communication Engineering (Harcourt Brace and World, New York, 1967), p. 167.
  6. E. Jakeman and P. N. Pusey, "A model for non-Rayleigh microwave sea echo," IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
  7. G. Parry, P. N. Pusey, E. Jakeman, and J. G. McWhirter, "The statistical and correlation properties of light scattered by a random phase screen," in Coherence and Quantum Optics IV, L. Mandel and E. Wolf, eds. (Plenum, New York, 1978), pp. 351–361.
  8. G. Parry and P. N. Pusey, "K-distributions in atmospheric propagation of laser light," J. Opt. Soc. Am. 69, 796–798 (1979).
  9. E. Jakeman and P. N. Pusey, "Photon counting statistics of optical scintillation," Inverse Scattering Problems in Optics, Vol. 20 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 73-116.
  10. G. Parry, "Measurements of atmospheric turbulence induced intensity fluctuations in a laser beam," Opt. Acta 28, 715–728 (1981).
  11. S. F. Clifford and R. J. Hill, "Relation between irradiance and log-amplitude variance for optical scintillation described by the K distribution," J. Opt. Soc. Am. 71, 112–114 (1981).
  12. R. L. Phillips and L. C. Andrews, University of Central Florida Research Rep. No. N61339-79-D-0105 (University of Central Florida, Orlando, Fla., 1980).
  13. K. D. Ward, "Compound representation of high resolution sea clutter," Electron. Lett. 17, 561–653 (1981).
  14. B. J. Hoenders, E. Jakeman, H. P. Baltes, and B. Steinle, "K correlations and facet models in diffuse scattering," Opt. Acta 26, 1307–1319 (1979).
  15. A. Zardecki, "Statistical features of phase screens from scattering data," Inverse Source Problems in Optics, Vol. 9 of Topics in Current Physics (Springer-Verlag, Berlin, 1978), pp. 155–189.
  16. M. V. Berry, "Diffractals," J. Phys. A 12, 781–797 (1979).
  17. B. B. Mandelbrot, Fractals (Freeman, San Francisco, 1977).
  18. M. V. Berry, "Focussing and twinkling: critical exponents from catastrophes in non-Gaussian random shortwaves," J. Phys. A 10, 2061–2081 (1977).
  19. M. V. Berry and C. Upstill, "Catastrophe optics: morphologies of caustics and their diffraction patterns," in Progress in Optics XIII (North-Holland, Amsterdam, 1980), pp. 258–346.
  20. V. H. Rumsey, "Scintillations due to a concentrated layer with a power law turbulence spectrum," Radio Sci. 10, 107–114 (1975).
  21. C. L. Rino, "A power law phase screen model for ionospheric scintillation," Radio Sci. 14, 1135–1155 (1979).
  22. B. J. Uscinski, H. G. Booker, and M. Marians, "Intensity fluctuations due to a deep phase screen with power-law spectrum," Proc. Roy. Soc. Lond. A 374, 503–530 (1981).
  23. E. Jakeman, "On the statistics of K-distributed noise," J. Phys. A 13, 31–48 (1980).
  24. W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958).
  25. E. Jakeman, "Scattering by a corrugated random surface with fractal slope," J. Phys. A Math. Nucl. Gen. 15, L55–L59 (1982).
  26. R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766–2788 (1963).
  27. E. Jakeman and J. G. McWhirter, "Correlation function dependence of scintillation behind a random phase screen," J. Phys. A 10, 1599–1643 (1977).
  28. L. S. Taylor and C. I. Infosino, "Diffraction theory of optical scintillation due to turbulent layers," J. Opt. Soc. Am. 65,78–84 (1975).
  29. E. Jakeman and P. N. Pusey, "Significance of K distributions in scattering experiments," Phys. Rev. Lett. 40, 546–550 (1978).
  30. Lord Rayleigh, "On the problem of random vibrations and of random flights in one, two or three dimensions," Philos. Mag. 37, 321–347 (1919).
  31. N. A. J. Hastings and J. B. Peacock, Statistical Distributions (Butterworths, London, 1974).
  32. A. Hewish, "The diffraction of radio waves in passing through a phase changing ionosphere," Proc. Roy. Soc. Lond. A 209, 81 (1951).
  33. V. I. Shishov, "Dependence of the form of the scintillation spectrum on the form of the spectrum of refractive-index inhomogeneities," Radiophys. Quantum Electron. 17, 1287–1292 (1974).
  34. V. I. Shishov, "Scintillation spectrum as a function of the spectrum of refractive index inhomogeneities," Radiosphys. Quantum Electron. 19, 62–68 (1976).

1982

E. Jakeman, "Scattering by a corrugated random surface with fractal slope," J. Phys. A Math. Nucl. Gen. 15, L55–L59 (1982).

1981

B. J. Uscinski, H. G. Booker, and M. Marians, "Intensity fluctuations due to a deep phase screen with power-law spectrum," Proc. Roy. Soc. Lond. A 374, 503–530 (1981).

K. D. Ward, "Compound representation of high resolution sea clutter," Electron. Lett. 17, 561–653 (1981).

G. Parry, "Measurements of atmospheric turbulence induced intensity fluctuations in a laser beam," Opt. Acta 28, 715–728 (1981).

S. F. Clifford and R. J. Hill, "Relation between irradiance and log-amplitude variance for optical scintillation described by the K distribution," J. Opt. Soc. Am. 71, 112–114 (1981).

1980

E. Jakeman, "On the statistics of K-distributed noise," J. Phys. A 13, 31–48 (1980).

E. Jakeman, "Speckle statistics with a small number of scatterers," Proc. Soc. Photo-Opt. Instrum. Eng. 243, 9–19 (1980).

1979

C. L. Rino, "A power law phase screen model for ionospheric scintillation," Radio Sci. 14, 1135–1155 (1979).

B. J. Hoenders, E. Jakeman, H. P. Baltes, and B. Steinle, "K correlations and facet models in diffuse scattering," Opt. Acta 26, 1307–1319 (1979).

M. V. Berry, "Diffractals," J. Phys. A 12, 781–797 (1979).

G. Parry and P. N. Pusey, "K-distributions in atmospheric propagation of laser light," J. Opt. Soc. Am. 69, 796–798 (1979).

1978

E. Jakeman and P. N. Pusey, "Significance of K distributions in scattering experiments," Phys. Rev. Lett. 40, 546–550 (1978).

1977

E. Jakeman and J. G. McWhirter, "Correlation function dependence of scintillation behind a random phase screen," J. Phys. A 10, 1599–1643 (1977).

M. V. Berry, "Focussing and twinkling: critical exponents from catastrophes in non-Gaussian random shortwaves," J. Phys. A 10, 2061–2081 (1977).

1976

E. Jakeman and P. N. Pusey, "A model for non-Rayleigh microwave sea echo," IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).

V. I. Shishov, "Scintillation spectrum as a function of the spectrum of refractive index inhomogeneities," Radiosphys. Quantum Electron. 19, 62–68 (1976).

1975

L. S. Taylor and C. I. Infosino, "Diffraction theory of optical scintillation due to turbulent layers," J. Opt. Soc. Am. 65,78–84 (1975).

V. H. Rumsey, "Scintillations due to a concentrated layer with a power law turbulence spectrum," Radio Sci. 10, 107–114 (1975).

1974

V. I. Shishov, "Dependence of the form of the scintillation spectrum on the form of the spectrum of refractive-index inhomogeneities," Radiophys. Quantum Electron. 17, 1287–1292 (1974).

1963

R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766–2788 (1963).

M. M. Siddiqui and G. H. Weiss, "Families of distributions for hourly median power and instantaneous power of received radio signals," J. Res. Nat. Bur. Stand. 67D, 753–762(1963).

1951

A. Hewish, "The diffraction of radio waves in passing through a phase changing ionosphere," Proc. Roy. Soc. Lond. A 209, 81 (1951).

1919

Lord Rayleigh, "On the problem of random vibrations and of random flights in one, two or three dimensions," Philos. Mag. 37, 321–347 (1919).

Andrews, L. C.

R. L. Phillips and L. C. Andrews, University of Central Florida Research Rep. No. N61339-79-D-0105 (University of Central Florida, Orlando, Fla., 1980).

Baltes, H. P.

B. J. Hoenders, E. Jakeman, H. P. Baltes, and B. Steinle, "K correlations and facet models in diffuse scattering," Opt. Acta 26, 1307–1319 (1979).

Beckmann, P.

P. Beckmann, Probability in Communication Engineering (Harcourt Brace and World, New York, 1967), p. 167.

Berry, M. V.

M. V. Berry, "Diffractals," J. Phys. A 12, 781–797 (1979).

M. V. Berry, "Focussing and twinkling: critical exponents from catastrophes in non-Gaussian random shortwaves," J. Phys. A 10, 2061–2081 (1977).

M. V. Berry and C. Upstill, "Catastrophe optics: morphologies of caustics and their diffraction patterns," in Progress in Optics XIII (North-Holland, Amsterdam, 1980), pp. 258–346.

Booker, H. G.

B. J. Uscinski, H. G. Booker, and M. Marians, "Intensity fluctuations due to a deep phase screen with power-law spectrum," Proc. Roy. Soc. Lond. A 374, 503–530 (1981).

Clifford, S. F.

Davenport, W. B.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958).

Glauber, R. J.

R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766–2788 (1963).

Hastings, N. A. J.

N. A. J. Hastings and J. B. Peacock, Statistical Distributions (Butterworths, London, 1974).

Hewish, A.

A. Hewish, "The diffraction of radio waves in passing through a phase changing ionosphere," Proc. Roy. Soc. Lond. A 209, 81 (1951).

Hill, R. J.

Hoenders, B. J.

B. J. Hoenders, E. Jakeman, H. P. Baltes, and B. Steinle, "K correlations and facet models in diffuse scattering," Opt. Acta 26, 1307–1319 (1979).

Infosino, C. I.

Jakeman, E.

E. Jakeman, "Scattering by a corrugated random surface with fractal slope," J. Phys. A Math. Nucl. Gen. 15, L55–L59 (1982).

E. Jakeman, "On the statistics of K-distributed noise," J. Phys. A 13, 31–48 (1980).

E. Jakeman, "Speckle statistics with a small number of scatterers," Proc. Soc. Photo-Opt. Instrum. Eng. 243, 9–19 (1980).

B. J. Hoenders, E. Jakeman, H. P. Baltes, and B. Steinle, "K correlations and facet models in diffuse scattering," Opt. Acta 26, 1307–1319 (1979).

E. Jakeman and P. N. Pusey, "Significance of K distributions in scattering experiments," Phys. Rev. Lett. 40, 546–550 (1978).

E. Jakeman and J. G. McWhirter, "Correlation function dependence of scintillation behind a random phase screen," J. Phys. A 10, 1599–1643 (1977).

E. Jakeman and P. N. Pusey, "A model for non-Rayleigh microwave sea echo," IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).

E. Jakeman and P. N. Pusey, "Photon counting statistics of optical scintillation," Inverse Scattering Problems in Optics, Vol. 20 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 73-116.

G. Parry, P. N. Pusey, E. Jakeman, and J. G. McWhirter, "The statistical and correlation properties of light scattered by a random phase screen," in Coherence and Quantum Optics IV, L. Mandel and E. Wolf, eds. (Plenum, New York, 1978), pp. 351–361.

Mandelbrot, B. B.

B. B. Mandelbrot, Fractals (Freeman, San Francisco, 1977).

Marians, M.

B. J. Uscinski, H. G. Booker, and M. Marians, "Intensity fluctuations due to a deep phase screen with power-law spectrum," Proc. Roy. Soc. Lond. A 374, 503–530 (1981).

McWhirter, J. G.

E. Jakeman and J. G. McWhirter, "Correlation function dependence of scintillation behind a random phase screen," J. Phys. A 10, 1599–1643 (1977).

G. Parry, P. N. Pusey, E. Jakeman, and J. G. McWhirter, "The statistical and correlation properties of light scattered by a random phase screen," in Coherence and Quantum Optics IV, L. Mandel and E. Wolf, eds. (Plenum, New York, 1978), pp. 351–361.

Nakagami, M.

M. Nakagami, "The m-distribution—a general formula of intensity distribution of rapid fading," in Statistical Methods in Radio Wave Propagation (Pergamon, New York, 1960), pp. 3–36.

Parry, G.

G. Parry, "Measurements of atmospheric turbulence induced intensity fluctuations in a laser beam," Opt. Acta 28, 715–728 (1981).

G. Parry and P. N. Pusey, "K-distributions in atmospheric propagation of laser light," J. Opt. Soc. Am. 69, 796–798 (1979).

G. Parry, P. N. Pusey, E. Jakeman, and J. G. McWhirter, "The statistical and correlation properties of light scattered by a random phase screen," in Coherence and Quantum Optics IV, L. Mandel and E. Wolf, eds. (Plenum, New York, 1978), pp. 351–361.

Peacock, J. B.

N. A. J. Hastings and J. B. Peacock, Statistical Distributions (Butterworths, London, 1974).

Phillips, R. L.

R. L. Phillips and L. C. Andrews, University of Central Florida Research Rep. No. N61339-79-D-0105 (University of Central Florida, Orlando, Fla., 1980).

Pusey, P. N.

G. Parry and P. N. Pusey, "K-distributions in atmospheric propagation of laser light," J. Opt. Soc. Am. 69, 796–798 (1979).

E. Jakeman and P. N. Pusey, "Significance of K distributions in scattering experiments," Phys. Rev. Lett. 40, 546–550 (1978).

E. Jakeman and P. N. Pusey, "A model for non-Rayleigh microwave sea echo," IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).

E. Jakeman and P. N. Pusey, "Photon counting statistics of optical scintillation," Inverse Scattering Problems in Optics, Vol. 20 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 73-116.

G. Parry, P. N. Pusey, E. Jakeman, and J. G. McWhirter, "The statistical and correlation properties of light scattered by a random phase screen," in Coherence and Quantum Optics IV, L. Mandel and E. Wolf, eds. (Plenum, New York, 1978), pp. 351–361.

Rayleigh, Lord

Lord Rayleigh, "On the problem of random vibrations and of random flights in one, two or three dimensions," Philos. Mag. 37, 321–347 (1919).

Rino, C. L.

C. L. Rino, "A power law phase screen model for ionospheric scintillation," Radio Sci. 14, 1135–1155 (1979).

Root, W. L.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958).

Rumsey, V. H.

V. H. Rumsey, "Scintillations due to a concentrated layer with a power law turbulence spectrum," Radio Sci. 10, 107–114 (1975).

Shishov, V. I.

V. I. Shishov, "Scintillation spectrum as a function of the spectrum of refractive index inhomogeneities," Radiosphys. Quantum Electron. 19, 62–68 (1976).

V. I. Shishov, "Dependence of the form of the scintillation spectrum on the form of the spectrum of refractive-index inhomogeneities," Radiophys. Quantum Electron. 17, 1287–1292 (1974).

Siddiqui, M. M.

M. M. Siddiqui and G. H. Weiss, "Families of distributions for hourly median power and instantaneous power of received radio signals," J. Res. Nat. Bur. Stand. 67D, 753–762(1963).

Steinle, B.

B. J. Hoenders, E. Jakeman, H. P. Baltes, and B. Steinle, "K correlations and facet models in diffuse scattering," Opt. Acta 26, 1307–1319 (1979).

Taylor, L. S.

Upstill, C.

M. V. Berry and C. Upstill, "Catastrophe optics: morphologies of caustics and their diffraction patterns," in Progress in Optics XIII (North-Holland, Amsterdam, 1980), pp. 258–346.

Uscinski, B. J.

B. J. Uscinski, H. G. Booker, and M. Marians, "Intensity fluctuations due to a deep phase screen with power-law spectrum," Proc. Roy. Soc. Lond. A 374, 503–530 (1981).

Ward, K. D.

K. D. Ward, "Compound representation of high resolution sea clutter," Electron. Lett. 17, 561–653 (1981).

Weiss, G. H.

M. M. Siddiqui and G. H. Weiss, "Families of distributions for hourly median power and instantaneous power of received radio signals," J. Res. Nat. Bur. Stand. 67D, 753–762(1963).

Zardecki, A.

A. Zardecki, "Statistical features of phase screens from scattering data," Inverse Source Problems in Optics, Vol. 9 of Topics in Current Physics (Springer-Verlag, Berlin, 1978), pp. 155–189.

Electron. Lett.

K. D. Ward, "Compound representation of high resolution sea clutter," Electron. Lett. 17, 561–653 (1981).

IEEE Trans. Antennas Propag.

E. Jakeman and P. N. Pusey, "A model for non-Rayleigh microwave sea echo," IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).

J. Opt. Soc. Am.

J. Phys. A

M. V. Berry, "Diffractals," J. Phys. A 12, 781–797 (1979).

M. V. Berry, "Focussing and twinkling: critical exponents from catastrophes in non-Gaussian random shortwaves," J. Phys. A 10, 2061–2081 (1977).

E. Jakeman, "On the statistics of K-distributed noise," J. Phys. A 13, 31–48 (1980).

E. Jakeman and J. G. McWhirter, "Correlation function dependence of scintillation behind a random phase screen," J. Phys. A 10, 1599–1643 (1977).

J. Phys. A Math. Nucl. Gen.

E. Jakeman, "Scattering by a corrugated random surface with fractal slope," J. Phys. A Math. Nucl. Gen. 15, L55–L59 (1982).

J. Res. Nat. Bur. Stand.

M. M. Siddiqui and G. H. Weiss, "Families of distributions for hourly median power and instantaneous power of received radio signals," J. Res. Nat. Bur. Stand. 67D, 753–762(1963).

Opt. Acta

G. Parry, "Measurements of atmospheric turbulence induced intensity fluctuations in a laser beam," Opt. Acta 28, 715–728 (1981).

B. J. Hoenders, E. Jakeman, H. P. Baltes, and B. Steinle, "K correlations and facet models in diffuse scattering," Opt. Acta 26, 1307–1319 (1979).

Philos. Mag.

Lord Rayleigh, "On the problem of random vibrations and of random flights in one, two or three dimensions," Philos. Mag. 37, 321–347 (1919).

Phys. Rev.

R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766–2788 (1963).

Phys. Rev. Lett.

E. Jakeman and P. N. Pusey, "Significance of K distributions in scattering experiments," Phys. Rev. Lett. 40, 546–550 (1978).

Proc. Roy. Soc. Lond. A

A. Hewish, "The diffraction of radio waves in passing through a phase changing ionosphere," Proc. Roy. Soc. Lond. A 209, 81 (1951).

B. J. Uscinski, H. G. Booker, and M. Marians, "Intensity fluctuations due to a deep phase screen with power-law spectrum," Proc. Roy. Soc. Lond. A 374, 503–530 (1981).

Proc. Soc. Photo-Opt. Instrum. Eng.

E. Jakeman, "Speckle statistics with a small number of scatterers," Proc. Soc. Photo-Opt. Instrum. Eng. 243, 9–19 (1980).

Radio Sci.

V. H. Rumsey, "Scintillations due to a concentrated layer with a power law turbulence spectrum," Radio Sci. 10, 107–114 (1975).

C. L. Rino, "A power law phase screen model for ionospheric scintillation," Radio Sci. 14, 1135–1155 (1979).

Radiophys. Quantum Electron.

V. I. Shishov, "Dependence of the form of the scintillation spectrum on the form of the spectrum of refractive-index inhomogeneities," Radiophys. Quantum Electron. 17, 1287–1292 (1974).

Radiosphys. Quantum Electron.

V. I. Shishov, "Scintillation spectrum as a function of the spectrum of refractive index inhomogeneities," Radiosphys. Quantum Electron. 19, 62–68 (1976).

Other

E. Jakeman and P. N. Pusey, "Photon counting statistics of optical scintillation," Inverse Scattering Problems in Optics, Vol. 20 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 73-116.

M. Nakagami, "The m-distribution—a general formula of intensity distribution of rapid fading," in Statistical Methods in Radio Wave Propagation (Pergamon, New York, 1960), pp. 3–36.

P. Beckmann, Probability in Communication Engineering (Harcourt Brace and World, New York, 1967), p. 167.

N. A. J. Hastings and J. B. Peacock, Statistical Distributions (Butterworths, London, 1974).

G. Parry, P. N. Pusey, E. Jakeman, and J. G. McWhirter, "The statistical and correlation properties of light scattered by a random phase screen," in Coherence and Quantum Optics IV, L. Mandel and E. Wolf, eds. (Plenum, New York, 1978), pp. 351–361.

M. V. Berry and C. Upstill, "Catastrophe optics: morphologies of caustics and their diffraction patterns," in Progress in Optics XIII (North-Holland, Amsterdam, 1980), pp. 258–346.

A. Zardecki, "Statistical features of phase screens from scattering data," Inverse Source Problems in Optics, Vol. 9 of Topics in Current Physics (Springer-Verlag, Berlin, 1978), pp. 155–189.

B. B. Mandelbrot, Fractals (Freeman, San Francisco, 1977).

R. L. Phillips and L. C. Andrews, University of Central Florida Research Rep. No. N61339-79-D-0105 (University of Central Florida, Orlando, Fla., 1980).

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958).

J. C. Dainty, ed., Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975).

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