Abstract

The statistical properties of a filtered random phasor are calculated. This problem arises in remote-sensing techniques such as coherent spectroscopy and non-Gaussian scattering and is relevant more generally to the field of nonlinear signal processing. Results are obtained analytically and by numerical simulation for a variety of phase fluctuation models. The problem is found to be analogous to scattering of light from a random-phase-changing screen. This provides a simple interpretation of the results and suggests new ways for characterizing both phase variable and filter.

© 1998 Optical Society of America

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References

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  1. M. Harris, G. N. Pearson, J. M. Vaughan, “Electrical method of spectral filtering within the Schawlow–Townes laser linewidth,” Electron. Lett. 30, 1678–1679 (1994).
    [CrossRef]
  2. T. R. Watts, K. I. Hopcraft, T. R. Falkner, “Single measurements on probability density functions and their use in non-Gaussian light scattering,” J. Phys. A 29, 7501–7517 (1996).
    [CrossRef]
  3. H. Taub, D. L. Schilling, Principles of Communication Systems (McGraw-Hill, New York, 1986).
  4. E. Jakeman, J. G. McWhirter, “Correlation dependence of the scintillation behind a deep random phase screen,” J. Phys. A 10, 1599–1643 (1977).
    [CrossRef]
  5. H. G. Booker, “Application of refractive scintillation theory to radiotransmission through the ionosphere and the solar wind and to reflection from a rough ocean,” J. Atmos. Terr. Phys. 43, 1215–1233 (1981).
    [CrossRef]
  6. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  7. J. C. Dainty, ed., Laser Speckle and Related Phenomena (Plenum, New York, 1975).
  8. R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. A58, 382–400 (1962).
  9. E. Jakeman, P. N. Pusey, “The statistics of light scattered by a random phase screen,” J. Phys. A 6, L88–L92 (1973).
    [CrossRef]
  10. E. Renshaw, “The higher order covariance structure of the telegraph wave,” J. Appl. Prob. 25, 744–751 (1988).
    [CrossRef]
  11. M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 (1979).
    [CrossRef]
  12. R. Centeno Neelen, D. M. Boersma, M. P. Van Exter, G. Nienhuis, J. P. Woerdman, “Spectral filtering within the Schawlow–Townes linewidth of a semiconductor laser,” Phys. Rev. Lett. 69, 593–596 (1992).
    [CrossRef] [PubMed]
  13. R. Buckley, “A Fokker–Planck equation for the probability distribution of the field scattered by a random phase screen,” in Wave Propagation and Scattering, B. J. Uscinsky, ed. (Clarendon, Oxford, 1986), pp. 113–127.
  14. E. Jakeman, J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B: 26, 125–131 (1981).
    [CrossRef]
  15. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  16. D. L. Jordan, R. C. Hollins, E. Jakeman, “Experimental measurements of non-Gaussian scattering by a fractal diffuser,” Appl. Phys. B: Photophys. Laser Chem. 31, 179–186 (1983).
    [CrossRef]
  17. E. Jakeman, B. J. Hoenders, “Scattering by a surface of rectangular grooves,” Opt. Acta 29, 1587–1598 (1982).
    [CrossRef]
  18. E. Jakeman, P. N. Pusey, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen. I. Theory,” J. Phys. A 8, 369–391 (1975).
    [CrossRef]
  19. E. Jakeman, E. Renshaw, “Correlated random walk model for scattering,” J. Opt. Soc. Am. A 4, 1206–1212 (1987).
    [CrossRef]
  20. E. Jakeman, “Optical scattering experiments,” in Wave Propagation and Scattering, B. J. Uscinsky, ed. (Clarendon, Oxford, 1986), pp. 241–259.
  21. A. D. Bruce, “Probability density functions for collective coordinates in Ising-like systems,” J. Phys. C 14, 3667–3688 (1981).
    [CrossRef]
  22. J. G. Walker, E. Jakeman, “Observation of sub-fractal behavior in a light scattering system,” Opt. Acta 31, 1185–1196 (1984).
    [CrossRef]
  23. B. J. Hoenders, E. Jakeman, H. P. Baltes, B. Steinle, “K correlations and facet models in diffuse scattering,” Opt. Acta 26, 1307–1319 (1979).
    [CrossRef]
  24. A. S. Glass, H. P. Baltes, “The significance of far-zone coherence for sources or scatterers with hidden periodicity,” Opt. Acta 29, 169–185 (1982).
    [CrossRef]
  25. H. Z. Cummins, E. R. Pike, eds., Photon Correlation Spectroscopy and Velocimetry (Plenum, New York, 1977).
  26. H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. London, Ser. A 242, 579–609 (1950).
    [CrossRef]

1996

T. R. Watts, K. I. Hopcraft, T. R. Falkner, “Single measurements on probability density functions and their use in non-Gaussian light scattering,” J. Phys. A 29, 7501–7517 (1996).
[CrossRef]

1994

M. Harris, G. N. Pearson, J. M. Vaughan, “Electrical method of spectral filtering within the Schawlow–Townes laser linewidth,” Electron. Lett. 30, 1678–1679 (1994).
[CrossRef]

1992

R. Centeno Neelen, D. M. Boersma, M. P. Van Exter, G. Nienhuis, J. P. Woerdman, “Spectral filtering within the Schawlow–Townes linewidth of a semiconductor laser,” Phys. Rev. Lett. 69, 593–596 (1992).
[CrossRef] [PubMed]

1988

E. Renshaw, “The higher order covariance structure of the telegraph wave,” J. Appl. Prob. 25, 744–751 (1988).
[CrossRef]

1987

1984

J. G. Walker, E. Jakeman, “Observation of sub-fractal behavior in a light scattering system,” Opt. Acta 31, 1185–1196 (1984).
[CrossRef]

1983

D. L. Jordan, R. C. Hollins, E. Jakeman, “Experimental measurements of non-Gaussian scattering by a fractal diffuser,” Appl. Phys. B: Photophys. Laser Chem. 31, 179–186 (1983).
[CrossRef]

1982

E. Jakeman, B. J. Hoenders, “Scattering by a surface of rectangular grooves,” Opt. Acta 29, 1587–1598 (1982).
[CrossRef]

A. S. Glass, H. P. Baltes, “The significance of far-zone coherence for sources or scatterers with hidden periodicity,” Opt. Acta 29, 169–185 (1982).
[CrossRef]

1981

A. D. Bruce, “Probability density functions for collective coordinates in Ising-like systems,” J. Phys. C 14, 3667–3688 (1981).
[CrossRef]

E. Jakeman, J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B: 26, 125–131 (1981).
[CrossRef]

H. G. Booker, “Application of refractive scintillation theory to radiotransmission through the ionosphere and the solar wind and to reflection from a rough ocean,” J. Atmos. Terr. Phys. 43, 1215–1233 (1981).
[CrossRef]

1979

M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 (1979).
[CrossRef]

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

1977

E. Jakeman, J. G. McWhirter, “Correlation dependence of the scintillation behind a deep random phase screen,” J. Phys. A 10, 1599–1643 (1977).
[CrossRef]

1975

E. Jakeman, P. N. Pusey, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen. I. Theory,” J. Phys. A 8, 369–391 (1975).
[CrossRef]

1973

E. Jakeman, P. N. Pusey, “The statistics of light scattered by a random phase screen,” J. Phys. A 6, L88–L92 (1973).
[CrossRef]

1950

H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. London, Ser. A 242, 579–609 (1950).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Baltes, H. P.

A. S. Glass, H. P. Baltes, “The significance of far-zone coherence for sources or scatterers with hidden periodicity,” Opt. Acta 29, 169–185 (1982).
[CrossRef]

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

Berry, M. V.

M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 (1979).
[CrossRef]

Boersma, D. M.

R. Centeno Neelen, D. M. Boersma, M. P. Van Exter, G. Nienhuis, J. P. Woerdman, “Spectral filtering within the Schawlow–Townes linewidth of a semiconductor laser,” Phys. Rev. Lett. 69, 593–596 (1992).
[CrossRef] [PubMed]

Booker, H. G.

H. G. Booker, “Application of refractive scintillation theory to radiotransmission through the ionosphere and the solar wind and to reflection from a rough ocean,” J. Atmos. Terr. Phys. 43, 1215–1233 (1981).
[CrossRef]

H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. London, Ser. A 242, 579–609 (1950).
[CrossRef]

Bruce, A. D.

A. D. Bruce, “Probability density functions for collective coordinates in Ising-like systems,” J. Phys. C 14, 3667–3688 (1981).
[CrossRef]

Buckley, R.

R. Buckley, “A Fokker–Planck equation for the probability distribution of the field scattered by a random phase screen,” in Wave Propagation and Scattering, B. J. Uscinsky, ed. (Clarendon, Oxford, 1986), pp. 113–127.

Centeno Neelen, R.

R. Centeno Neelen, D. M. Boersma, M. P. Van Exter, G. Nienhuis, J. P. Woerdman, “Spectral filtering within the Schawlow–Townes linewidth of a semiconductor laser,” Phys. Rev. Lett. 69, 593–596 (1992).
[CrossRef] [PubMed]

Falkner, T. R.

T. R. Watts, K. I. Hopcraft, T. R. Falkner, “Single measurements on probability density functions and their use in non-Gaussian light scattering,” J. Phys. A 29, 7501–7517 (1996).
[CrossRef]

Glass, A. S.

A. S. Glass, H. P. Baltes, “The significance of far-zone coherence for sources or scatterers with hidden periodicity,” Opt. Acta 29, 169–185 (1982).
[CrossRef]

Harris, M.

M. Harris, G. N. Pearson, J. M. Vaughan, “Electrical method of spectral filtering within the Schawlow–Townes laser linewidth,” Electron. Lett. 30, 1678–1679 (1994).
[CrossRef]

Hoenders, B. J.

E. Jakeman, B. J. Hoenders, “Scattering by a surface of rectangular grooves,” Opt. Acta 29, 1587–1598 (1982).
[CrossRef]

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

Hollins, R. C.

D. L. Jordan, R. C. Hollins, E. Jakeman, “Experimental measurements of non-Gaussian scattering by a fractal diffuser,” Appl. Phys. B: Photophys. Laser Chem. 31, 179–186 (1983).
[CrossRef]

Hopcraft, K. I.

T. R. Watts, K. I. Hopcraft, T. R. Falkner, “Single measurements on probability density functions and their use in non-Gaussian light scattering,” J. Phys. A 29, 7501–7517 (1996).
[CrossRef]

Ishimaru, A.

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

Jakeman, E.

E. Jakeman, E. Renshaw, “Correlated random walk model for scattering,” J. Opt. Soc. Am. A 4, 1206–1212 (1987).
[CrossRef]

J. G. Walker, E. Jakeman, “Observation of sub-fractal behavior in a light scattering system,” Opt. Acta 31, 1185–1196 (1984).
[CrossRef]

D. L. Jordan, R. C. Hollins, E. Jakeman, “Experimental measurements of non-Gaussian scattering by a fractal diffuser,” Appl. Phys. B: Photophys. Laser Chem. 31, 179–186 (1983).
[CrossRef]

E. Jakeman, B. J. Hoenders, “Scattering by a surface of rectangular grooves,” Opt. Acta 29, 1587–1598 (1982).
[CrossRef]

E. Jakeman, J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B: 26, 125–131 (1981).
[CrossRef]

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

E. Jakeman, J. G. McWhirter, “Correlation dependence of the scintillation behind a deep random phase screen,” J. Phys. A 10, 1599–1643 (1977).
[CrossRef]

E. Jakeman, P. N. Pusey, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen. I. Theory,” J. Phys. A 8, 369–391 (1975).
[CrossRef]

E. Jakeman, P. N. Pusey, “The statistics of light scattered by a random phase screen,” J. Phys. A 6, L88–L92 (1973).
[CrossRef]

E. Jakeman, “Optical scattering experiments,” in Wave Propagation and Scattering, B. J. Uscinsky, ed. (Clarendon, Oxford, 1986), pp. 241–259.

Jordan, D. L.

D. L. Jordan, R. C. Hollins, E. Jakeman, “Experimental measurements of non-Gaussian scattering by a fractal diffuser,” Appl. Phys. B: Photophys. Laser Chem. 31, 179–186 (1983).
[CrossRef]

McWhirter, J. G.

E. Jakeman, J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B: 26, 125–131 (1981).
[CrossRef]

E. Jakeman, J. G. McWhirter, “Correlation dependence of the scintillation behind a deep random phase screen,” J. Phys. A 10, 1599–1643 (1977).
[CrossRef]

Mercier, R. P.

R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. A58, 382–400 (1962).

Nienhuis, G.

R. Centeno Neelen, D. M. Boersma, M. P. Van Exter, G. Nienhuis, J. P. Woerdman, “Spectral filtering within the Schawlow–Townes linewidth of a semiconductor laser,” Phys. Rev. Lett. 69, 593–596 (1992).
[CrossRef] [PubMed]

Pearson, G. N.

M. Harris, G. N. Pearson, J. M. Vaughan, “Electrical method of spectral filtering within the Schawlow–Townes laser linewidth,” Electron. Lett. 30, 1678–1679 (1994).
[CrossRef]

Pusey, P. N.

E. Jakeman, P. N. Pusey, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen. I. Theory,” J. Phys. A 8, 369–391 (1975).
[CrossRef]

E. Jakeman, P. N. Pusey, “The statistics of light scattered by a random phase screen,” J. Phys. A 6, L88–L92 (1973).
[CrossRef]

Ratcliffe, J. A.

H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. London, Ser. A 242, 579–609 (1950).
[CrossRef]

Renshaw, E.

E. Renshaw, “The higher order covariance structure of the telegraph wave,” J. Appl. Prob. 25, 744–751 (1988).
[CrossRef]

E. Jakeman, E. Renshaw, “Correlated random walk model for scattering,” J. Opt. Soc. Am. A 4, 1206–1212 (1987).
[CrossRef]

Schilling, D. L.

H. Taub, D. L. Schilling, Principles of Communication Systems (McGraw-Hill, New York, 1986).

Shinn, D. H.

H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. London, Ser. A 242, 579–609 (1950).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Steinle, B.

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

Taub, H.

H. Taub, D. L. Schilling, Principles of Communication Systems (McGraw-Hill, New York, 1986).

Van Exter, M. P.

R. Centeno Neelen, D. M. Boersma, M. P. Van Exter, G. Nienhuis, J. P. Woerdman, “Spectral filtering within the Schawlow–Townes linewidth of a semiconductor laser,” Phys. Rev. Lett. 69, 593–596 (1992).
[CrossRef] [PubMed]

Vaughan, J. M.

M. Harris, G. N. Pearson, J. M. Vaughan, “Electrical method of spectral filtering within the Schawlow–Townes laser linewidth,” Electron. Lett. 30, 1678–1679 (1994).
[CrossRef]

Walker, J. G.

J. G. Walker, E. Jakeman, “Observation of sub-fractal behavior in a light scattering system,” Opt. Acta 31, 1185–1196 (1984).
[CrossRef]

Watts, T. R.

T. R. Watts, K. I. Hopcraft, T. R. Falkner, “Single measurements on probability density functions and their use in non-Gaussian light scattering,” J. Phys. A 29, 7501–7517 (1996).
[CrossRef]

Woerdman, J. P.

R. Centeno Neelen, D. M. Boersma, M. P. Van Exter, G. Nienhuis, J. P. Woerdman, “Spectral filtering within the Schawlow–Townes linewidth of a semiconductor laser,” Phys. Rev. Lett. 69, 593–596 (1992).
[CrossRef] [PubMed]

Appl. Phys. B:

E. Jakeman, J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B: 26, 125–131 (1981).
[CrossRef]

Appl. Phys. B: Photophys. Laser Chem.

D. L. Jordan, R. C. Hollins, E. Jakeman, “Experimental measurements of non-Gaussian scattering by a fractal diffuser,” Appl. Phys. B: Photophys. Laser Chem. 31, 179–186 (1983).
[CrossRef]

Electron. Lett.

M. Harris, G. N. Pearson, J. M. Vaughan, “Electrical method of spectral filtering within the Schawlow–Townes laser linewidth,” Electron. Lett. 30, 1678–1679 (1994).
[CrossRef]

J. Appl. Prob.

E. Renshaw, “The higher order covariance structure of the telegraph wave,” J. Appl. Prob. 25, 744–751 (1988).
[CrossRef]

J. Atmos. Terr. Phys.

H. G. Booker, “Application of refractive scintillation theory to radiotransmission through the ionosphere and the solar wind and to reflection from a rough ocean,” J. Atmos. Terr. Phys. 43, 1215–1233 (1981).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. A

E. Jakeman, P. N. Pusey, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen. I. Theory,” J. Phys. A 8, 369–391 (1975).
[CrossRef]

T. R. Watts, K. I. Hopcraft, T. R. Falkner, “Single measurements on probability density functions and their use in non-Gaussian light scattering,” J. Phys. A 29, 7501–7517 (1996).
[CrossRef]

E. Jakeman, J. G. McWhirter, “Correlation dependence of the scintillation behind a deep random phase screen,” J. Phys. A 10, 1599–1643 (1977).
[CrossRef]

E. Jakeman, P. N. Pusey, “The statistics of light scattered by a random phase screen,” J. Phys. A 6, L88–L92 (1973).
[CrossRef]

M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 (1979).
[CrossRef]

J. Phys. C

A. D. Bruce, “Probability density functions for collective coordinates in Ising-like systems,” J. Phys. C 14, 3667–3688 (1981).
[CrossRef]

Opt. Acta

J. G. Walker, E. Jakeman, “Observation of sub-fractal behavior in a light scattering system,” Opt. Acta 31, 1185–1196 (1984).
[CrossRef]

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

A. S. Glass, H. P. Baltes, “The significance of far-zone coherence for sources or scatterers with hidden periodicity,” Opt. Acta 29, 169–185 (1982).
[CrossRef]

E. Jakeman, B. J. Hoenders, “Scattering by a surface of rectangular grooves,” Opt. Acta 29, 1587–1598 (1982).
[CrossRef]

Philos. Trans. R. Soc. London, Ser. A

H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. London, Ser. A 242, 579–609 (1950).
[CrossRef]

Phys. Rev. Lett.

R. Centeno Neelen, D. M. Boersma, M. P. Van Exter, G. Nienhuis, J. P. Woerdman, “Spectral filtering within the Schawlow–Townes linewidth of a semiconductor laser,” Phys. Rev. Lett. 69, 593–596 (1992).
[CrossRef] [PubMed]

Other

R. Buckley, “A Fokker–Planck equation for the probability distribution of the field scattered by a random phase screen,” in Wave Propagation and Scattering, B. J. Uscinsky, ed. (Clarendon, Oxford, 1986), pp. 113–127.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

H. Taub, D. L. Schilling, Principles of Communication Systems (McGraw-Hill, New York, 1986).

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

J. C. Dainty, ed., Laser Speckle and Related Phenomena (Plenum, New York, 1975).

R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. A58, 382–400 (1962).

E. Jakeman, “Optical scattering experiments,” in Wave Propagation and Scattering, B. J. Uscinsky, ed. (Clarendon, Oxford, 1986), pp. 241–259.

H. Z. Cummins, E. R. Pike, eds., Photon Correlation Spectroscopy and Velocimetry (Plenum, New York, 1977).

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Figures (13)

Fig. 1
Fig. 1

Second moment of the intensity for the telegraph wave, plotted as a function of (a) the ratio Λ of filter width to spectral linewidth for different values of the normalized frequency offset Ω and (b) the normalized frequency offset Ω for different values of Λ.

Fig. 2
Fig. 2

Telegraph wave probability density function plotted for three values of the ratio Λ of filter width to spectral linewidth.

Fig. 3
Fig. 3

Same as Fig. 1, but for second moment of the intensity for the Brownian fractal.

Fig. 4
Fig. 4

Second moment of the intensity for the smoothly varying model, plotted as a function of the parameter q for three values of f. The points are the results of numerical simulations (circles for f=6 and squares for f=4). The solid curves are from the approximate formula of Eq. (19).

Fig. 5
Fig. 5

Numerical simulations of the telegraph wave for the broadband filter (Λ=2): (a) output time series (the corresponding input series is shown at the top of the plot, not to scale), (b) probability density function. The solid curve in (b) is from Eq. (11).

Fig. 6
Fig. 6

Same as Fig. 5, but for numerical simulations of the telegraph wave for the narrow-band filter (Λ=0.1).

Fig. 7
Fig. 7

Numerical simulations of the Brownian fractal for the broadband filter (Λ=3.2): (a) input phase and filtered intensity time series, (b) probability density function.

Fig. 8
Fig. 8

Same as Fig. 7, but for numerical simulations of the Brownian fractal for the narrow-band filter (Λ=0.16).

Fig. 9
Fig. 9

Second moment of the intensity for the Brownian fractal. The numerical results are the circles, and the solid curve is from Eq. (17).

Fig. 10
Fig. 10

Filtered time series for the smoothly varying model: (a) phase; intensity with (b) λ=0.05, (c) λ=0.01, (d) λ=0.02.

Fig. 11
Fig. 11

Computed probability densities for the smoothly varying model: (a) λ=0.05, (b) λ=0.01, (c) λ=0.02.

Fig. 12
Fig. 12

Probability density of Fig. 11(b), replotted on a log scale and compared with the linear result for a negative exponential distribution with the same mean.

Fig. 13
Fig. 13

Geometry of phase screen scattering.

Equations (30)

Equations on this page are rendered with MathJax. Learn more.

S(t)=λ-tdt exp[iϕ(t)+λ(t-t)+iω(t-t)],
ϕ(t)=hT(t),whereT(t)=±1.
ϕ(t)ϕ(t+τ)=h2 exp(-γ|τ|),
i=1nT(ti)
=exp-γi=1n/2(t2i-t2j-1)forevenn0foroddn.
exp(ihT)=cos(h)+iT sin(h),
S(t)=λ-tdt T(t)exp[(λ+iω)(t-t)].
|S|2=Λ(Λ+1)/[(Λ+1)2+Ω2],
|S|4|S|22=2(1+3Λ)[(1+Λ)2+Ω2](1+Λ)[(1+3Λ)2+Ω2]+Λ[(1+Λ)2+Ω2][Λ(1+Λ)(1+3Λ)-Ω2(2+5Λ)]2(Λ2+Ω2)(1+Λ)2[(1+3Λ)2+Ω2],
SN=λN-tdt1-tdt2-tdtN ×n=1N exp[λ(tn-t)]T(tn)=N!λN-tdt1-t1dt2-tN-1dtN ×n=1N exp[λ(tn-t)]T(tn).
S2N=(2N)!λ2N-tdt1-t1dt2-t2N-1dt2N ×expn=12N[λ(tn-t)+γ(t2n-t2n-1)]=(2N)!λ2N0dt10dt20dt2N ×exp-n=12N[λ(2N-n+1)+γt2n]=(2N)!Γ(12(1+Λ-1))22NN!Γ(N+12(1+Λ-1)),
P(S)
=Γ(12(1+Λ-1))(1-S2)-1+1/2ΛπΓ(12Λ-1),0<|S|<10,otherwise.
[ϕ(t)-ϕ(t+τ)]2=2γ|τ|.
exp{i[ϕ(0)-ϕ(τ)]}=exp(-γ|τ|),
|S|2=Λ(1+Λ)/[(1+Λ)2+Ω2].
exp[i(ϕ1-ϕ2+ϕ3-ϕ4)
=exp[-12(D12+D14+D23+D34-D13-D24)]
|S|4=λ40dτ-τdt3-τdt2-τdt1  ×exp[-γ(|t1-t2|+|t1|+|t2-t3|+|t3|-|t1-t3|-|t2|)]  ×exp[-λ(t1+t2+t3+4τ)+iω(t1-t2+t3)].
|S|4|S|22=2(1+3Λ)[(1+Λ)2+Ω2](1+Λ)[(1+3Λ)2+Ω2]+Λ[(1+Λ)2+Ω2](1+Λ)2  ×(1+Λ)(1+3Λ)(2+Λ)-Ω2(4+5Λ)[(1+3Λ)2+Ω2][(2+Λ)2+Ω2].
ϕ(0)ϕ(t)=ϕ02 exp(-t2/τ2),
I2I2=2 exp[(q2/4f2)-qf]erfc(q/f2)erfc2(q/2 f2)+qf2 exp(-q2/8f2)erfc(q/2 f2)2q2π[1-(1+q)  ×exp(-q)]+E1(q)-E1(qf),
Sn+1=exp(iϕn)δt+exp[-(λ+iω)δt]Snδt.
S(t)=z exp[-(λ+iω)t]±11+iΩ{1-exp[-(λ+iω)t]},
E-dx A(x)exp[iϕ(x)+ikx sin θ],
4λ30dy-yydxxdt exp{-ϕ02[2-ρ(t+x)-ρ(t-x)
-ρ(t+y)-ρ(t-y)+ρ(x+y)+ρ(x-y)]
-λ[2t+2y-2x]},
ϕ022-2ρ(x)-2ρ(y)+ρ(x+y)+ρ(x-y)+2 t2τ2ρ(x)1-2 x2τ2+ρ(y)1-2 y2τ2.
2 ϕ02τ2(x2+t2),y>τ/3(region1),2 ϕ02τ2(3x2y2+2t2),yτ/3(region2).

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