Abstract

We discuss the properties of enhanced non-Gaussian fluctuations that occur in radiation scattered by a moving deep random phase screen (e.g., a perfectly conducting, very rough surface) when the scattering configuration is such that the electric field at the point of observation is composed of relatively few independent randomly phased contributions. Fraunhofer and Fresnel region effects are considered both from the information-content point of view and in the context of unwanted noise in measuring and detection systems.

© 1976 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. D. Rigden and E. I. Gordon, Proc. Inst. Radio Eng. 50, 2367–2368 (1962).
  2. B. M. Oliver, Proc. IEEE 51, 220–221 (1963).
    [Crossref]
  3. “Light Beating and Photon-Correlation Spectroscopy” (Proceedings NATO Advanced Study Institute), edited H. Z. Cummins and E. R. Pike (Plenum, New York, 1974).
  4. “Laser Speckle and Related Phenomena,” Topics in Applied Physics Vol. 9, edited by J. C. Dainty (Springer, Berlin, 1975).
  5. G. Parry in Ref. 4.
  6. K. Nagata, T. Umehara, and J. Nishiwaki, Jpn. J. Appl. Phys. 12, 1693–1698 (1973).
    [Crossref]
  7. E. Jakeman and P. N. Pusey, J. Phys. A: Math. Nucl. Gen. 8, 369–391 (1975).
    [Crossref]
  8. P. N. Pusey and E. Jakeman, J. Phys. A: Math. Nucl. Gen. 8, 392–410 (1975).
    [Crossref]
  9. “Engineering Uses of Coherent Light,” in Proceedings of 1975 Strathclyde Conference, edited by J. Robertson (Academic, London, 1976).
  10. L. H. Tanner, Appl. Opt. 13, 2026–2034 (1974).
    [Crossref] [PubMed]
  11. L. Ronchi and A. Fontana, Opt. Acta 22, 243–246 (1975).
    [Crossref]
  12. E. Jakeman, J. Phys. A: Math. Nucl. Gen. 8, L23–28 (1975).
    [Crossref]
  13. T. S. Mckechnie in Ref. 4.
  14. E. Jakeman and J. G. McWhirter, J. Phys. A: Math. Gen. 9, 785–797 (1976).
    [Crossref]
  15. P. N. Pusey, J. Phys. D: Appl. Phys. 9, 1399–1409 (1976).
    [Crossref]
  16. L. T. Little and A. Hewish, Mon. Not. R. Astron. Soc. 134, 221–237 (1966).
  17. D. G. Singleton, J. Atom. Terr. Phys. 32, 187–208 (1970).
    [Crossref]
  18. R. P. Mercier, Proc. Cambridge Philos. Soc. A58, 382–400 (1962).
    [Crossref]
  19. E. N. Bramley and M. Young, Proc. IEEE 114, 553–556 (1967).
  20. E. Jakeman and P. N. Pusey, (Royal Radar Establishment, Malvern, Worcs., England, May1974).
  21. E. E. Salpeter, Astrophys. J. 147, 433–448 (1967).
    [Crossref]
  22. D. W. Schaefer and B. J. Berne, Phys. Rev. Lett. 28, 475 (1972).
    [Crossref]
  23. D. W. Schaefer and P. N. Pusey, Phys. Rev. Lett. 29, 843 (1972).
    [Crossref]
  24. D. W. Schaefer and B. J. Berne, Biophys. J. 15, 785–94 (1975).
  25. V. Bluemel, L. M. Narducci, and R. A. Tuft, J. Opt. Soc. Am. 62, 1309 (1972).
    [Crossref]
  26. F. Scudieri, M. Bertolotti, and R. Bartolino, Appl. Opt. 13, 181–185 (1974).
    [Crossref] [PubMed]
  27. M. Bertolotti in Ref. 3.
  28. J. W. Wright, IEEE Trans. Antennas Propag. AP-16, 217–223 (1968).
    [Crossref]
  29. E. Jakeman, E. R. Pike, and P. N. Pusey, Nature 263, 215–217 (1976).
    [Crossref]
  30. J. J. Burke, J. Opt. Soc. Am. 60, 1262–1264 (1970).
    [Crossref]
  31. A. T. Young, Appl. Opt. 8, 869–885 (1969).
    [Crossref] [PubMed]
  32. J. Vernin and F. Roddier, J. Opt. Soc. Am. 63, 270–273 (1973).
    [Crossref]
  33. A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, Proc. IEEE 63, 790–811 (1975).
    [Crossref]
  34. M. I. Skolnik, (Naval Research Laboratory, Washington, D. C., July1969).
  35. H. Goldstein in Propagation of Short Radio Waves, edited by D. E. KerrMcGraw-Hill, New York, 1951).
  36. K. R. Schmidt, P. M. Thieband, and J. S. Waller, (Naval Research Laboratory, Washington, D. C., July1972).
  37. G. Bishop in Determination and Use of Radar Scattering Characteristics, AGARD Lecture Series No. 59 (1973).
  38. Rayleigh (J. W. Strutt), Philos. Mag. 6, 321–347 (1919).
  39. E. Jakeman and P. N. Pusey, IEEE Trans. Antennas Propag. (to be published).

1976 (3)

E. Jakeman and J. G. McWhirter, J. Phys. A: Math. Gen. 9, 785–797 (1976).
[Crossref]

P. N. Pusey, J. Phys. D: Appl. Phys. 9, 1399–1409 (1976).
[Crossref]

E. Jakeman, E. R. Pike, and P. N. Pusey, Nature 263, 215–217 (1976).
[Crossref]

1975 (6)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, Proc. IEEE 63, 790–811 (1975).
[Crossref]

L. Ronchi and A. Fontana, Opt. Acta 22, 243–246 (1975).
[Crossref]

E. Jakeman, J. Phys. A: Math. Nucl. Gen. 8, L23–28 (1975).
[Crossref]

E. Jakeman and P. N. Pusey, J. Phys. A: Math. Nucl. Gen. 8, 369–391 (1975).
[Crossref]

P. N. Pusey and E. Jakeman, J. Phys. A: Math. Nucl. Gen. 8, 392–410 (1975).
[Crossref]

D. W. Schaefer and B. J. Berne, Biophys. J. 15, 785–94 (1975).

1974 (2)

1973 (2)

K. Nagata, T. Umehara, and J. Nishiwaki, Jpn. J. Appl. Phys. 12, 1693–1698 (1973).
[Crossref]

J. Vernin and F. Roddier, J. Opt. Soc. Am. 63, 270–273 (1973).
[Crossref]

1972 (3)

V. Bluemel, L. M. Narducci, and R. A. Tuft, J. Opt. Soc. Am. 62, 1309 (1972).
[Crossref]

D. W. Schaefer and B. J. Berne, Phys. Rev. Lett. 28, 475 (1972).
[Crossref]

D. W. Schaefer and P. N. Pusey, Phys. Rev. Lett. 29, 843 (1972).
[Crossref]

1970 (2)

D. G. Singleton, J. Atom. Terr. Phys. 32, 187–208 (1970).
[Crossref]

J. J. Burke, J. Opt. Soc. Am. 60, 1262–1264 (1970).
[Crossref]

1969 (1)

1968 (1)

J. W. Wright, IEEE Trans. Antennas Propag. AP-16, 217–223 (1968).
[Crossref]

1967 (2)

E. N. Bramley and M. Young, Proc. IEEE 114, 553–556 (1967).

E. E. Salpeter, Astrophys. J. 147, 433–448 (1967).
[Crossref]

1966 (1)

L. T. Little and A. Hewish, Mon. Not. R. Astron. Soc. 134, 221–237 (1966).

1963 (1)

B. M. Oliver, Proc. IEEE 51, 220–221 (1963).
[Crossref]

1962 (2)

J. D. Rigden and E. I. Gordon, Proc. Inst. Radio Eng. 50, 2367–2368 (1962).

R. P. Mercier, Proc. Cambridge Philos. Soc. A58, 382–400 (1962).
[Crossref]

1919 (1)

Rayleigh (J. W. Strutt), Philos. Mag. 6, 321–347 (1919).

Bartolino, R.

Berne, B. J.

D. W. Schaefer and B. J. Berne, Biophys. J. 15, 785–94 (1975).

D. W. Schaefer and B. J. Berne, Phys. Rev. Lett. 28, 475 (1972).
[Crossref]

Bertolotti, M.

Bishop, G.

G. Bishop in Determination and Use of Radar Scattering Characteristics, AGARD Lecture Series No. 59 (1973).

Bluemel, V.

Bramley, E. N.

E. N. Bramley and M. Young, Proc. IEEE 114, 553–556 (1967).

Bunkin, F. V.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, Proc. IEEE 63, 790–811 (1975).
[Crossref]

Burke, J. J.

Fontana, A.

L. Ronchi and A. Fontana, Opt. Acta 22, 243–246 (1975).
[Crossref]

Gochelashvily, K. S.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, Proc. IEEE 63, 790–811 (1975).
[Crossref]

Goldstein, H.

H. Goldstein in Propagation of Short Radio Waves, edited by D. E. KerrMcGraw-Hill, New York, 1951).

Gordon, E. I.

J. D. Rigden and E. I. Gordon, Proc. Inst. Radio Eng. 50, 2367–2368 (1962).

Hewish, A.

L. T. Little and A. Hewish, Mon. Not. R. Astron. Soc. 134, 221–237 (1966).

Jakeman, E.

E. Jakeman and J. G. McWhirter, J. Phys. A: Math. Gen. 9, 785–797 (1976).
[Crossref]

E. Jakeman, E. R. Pike, and P. N. Pusey, Nature 263, 215–217 (1976).
[Crossref]

E. Jakeman and P. N. Pusey, J. Phys. A: Math. Nucl. Gen. 8, 369–391 (1975).
[Crossref]

E. Jakeman, J. Phys. A: Math. Nucl. Gen. 8, L23–28 (1975).
[Crossref]

P. N. Pusey and E. Jakeman, J. Phys. A: Math. Nucl. Gen. 8, 392–410 (1975).
[Crossref]

E. Jakeman and P. N. Pusey, IEEE Trans. Antennas Propag. (to be published).

E. Jakeman and P. N. Pusey, (Royal Radar Establishment, Malvern, Worcs., England, May1974).

Little, L. T.

L. T. Little and A. Hewish, Mon. Not. R. Astron. Soc. 134, 221–237 (1966).

Mckechnie, T. S.

T. S. Mckechnie in Ref. 4.

McWhirter, J. G.

E. Jakeman and J. G. McWhirter, J. Phys. A: Math. Gen. 9, 785–797 (1976).
[Crossref]

Mercier, R. P.

R. P. Mercier, Proc. Cambridge Philos. Soc. A58, 382–400 (1962).
[Crossref]

Nagata, K.

K. Nagata, T. Umehara, and J. Nishiwaki, Jpn. J. Appl. Phys. 12, 1693–1698 (1973).
[Crossref]

Narducci, L. M.

Nishiwaki, J.

K. Nagata, T. Umehara, and J. Nishiwaki, Jpn. J. Appl. Phys. 12, 1693–1698 (1973).
[Crossref]

Oliver, B. M.

B. M. Oliver, Proc. IEEE 51, 220–221 (1963).
[Crossref]

Parry, G.

G. Parry in Ref. 4.

Pike, E. R.

E. Jakeman, E. R. Pike, and P. N. Pusey, Nature 263, 215–217 (1976).
[Crossref]

Prokhorov, A. M.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, Proc. IEEE 63, 790–811 (1975).
[Crossref]

Pusey, P. N.

E. Jakeman, E. R. Pike, and P. N. Pusey, Nature 263, 215–217 (1976).
[Crossref]

P. N. Pusey, J. Phys. D: Appl. Phys. 9, 1399–1409 (1976).
[Crossref]

E. Jakeman and P. N. Pusey, J. Phys. A: Math. Nucl. Gen. 8, 369–391 (1975).
[Crossref]

P. N. Pusey and E. Jakeman, J. Phys. A: Math. Nucl. Gen. 8, 392–410 (1975).
[Crossref]

D. W. Schaefer and P. N. Pusey, Phys. Rev. Lett. 29, 843 (1972).
[Crossref]

E. Jakeman and P. N. Pusey, (Royal Radar Establishment, Malvern, Worcs., England, May1974).

E. Jakeman and P. N. Pusey, IEEE Trans. Antennas Propag. (to be published).

Rayleigh,

Rayleigh (J. W. Strutt), Philos. Mag. 6, 321–347 (1919).

Rigden, J. D.

J. D. Rigden and E. I. Gordon, Proc. Inst. Radio Eng. 50, 2367–2368 (1962).

Roddier, F.

Ronchi, L.

L. Ronchi and A. Fontana, Opt. Acta 22, 243–246 (1975).
[Crossref]

Salpeter, E. E.

E. E. Salpeter, Astrophys. J. 147, 433–448 (1967).
[Crossref]

Schaefer, D. W.

D. W. Schaefer and B. J. Berne, Biophys. J. 15, 785–94 (1975).

D. W. Schaefer and B. J. Berne, Phys. Rev. Lett. 28, 475 (1972).
[Crossref]

D. W. Schaefer and P. N. Pusey, Phys. Rev. Lett. 29, 843 (1972).
[Crossref]

Schmidt, K. R.

K. R. Schmidt, P. M. Thieband, and J. S. Waller, (Naval Research Laboratory, Washington, D. C., July1972).

Scudieri, F.

Shishov, V. I.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, Proc. IEEE 63, 790–811 (1975).
[Crossref]

Singleton, D. G.

D. G. Singleton, J. Atom. Terr. Phys. 32, 187–208 (1970).
[Crossref]

Skolnik, M. I.

M. I. Skolnik, (Naval Research Laboratory, Washington, D. C., July1969).

Tanner, L. H.

Thieband, P. M.

K. R. Schmidt, P. M. Thieband, and J. S. Waller, (Naval Research Laboratory, Washington, D. C., July1972).

Tuft, R. A.

Umehara, T.

K. Nagata, T. Umehara, and J. Nishiwaki, Jpn. J. Appl. Phys. 12, 1693–1698 (1973).
[Crossref]

Vernin, J.

Waller, J. S.

K. R. Schmidt, P. M. Thieband, and J. S. Waller, (Naval Research Laboratory, Washington, D. C., July1972).

Wright, J. W.

J. W. Wright, IEEE Trans. Antennas Propag. AP-16, 217–223 (1968).
[Crossref]

Young, A. T.

Young, M.

E. N. Bramley and M. Young, Proc. IEEE 114, 553–556 (1967).

Appl. Opt. (3)

Astrophys. J. (1)

E. E. Salpeter, Astrophys. J. 147, 433–448 (1967).
[Crossref]

Biophys. J. (1)

D. W. Schaefer and B. J. Berne, Biophys. J. 15, 785–94 (1975).

IEEE Trans. Antennas Propag. (1)

J. W. Wright, IEEE Trans. Antennas Propag. AP-16, 217–223 (1968).
[Crossref]

J. Atom. Terr. Phys. (1)

D. G. Singleton, J. Atom. Terr. Phys. 32, 187–208 (1970).
[Crossref]

J. Opt. Soc. Am. (3)

J. Phys. A: Math. Gen. (1)

E. Jakeman and J. G. McWhirter, J. Phys. A: Math. Gen. 9, 785–797 (1976).
[Crossref]

J. Phys. A: Math. Nucl. Gen. (3)

E. Jakeman, J. Phys. A: Math. Nucl. Gen. 8, L23–28 (1975).
[Crossref]

E. Jakeman and P. N. Pusey, J. Phys. A: Math. Nucl. Gen. 8, 369–391 (1975).
[Crossref]

P. N. Pusey and E. Jakeman, J. Phys. A: Math. Nucl. Gen. 8, 392–410 (1975).
[Crossref]

J. Phys. D: Appl. Phys. (1)

P. N. Pusey, J. Phys. D: Appl. Phys. 9, 1399–1409 (1976).
[Crossref]

Jpn. J. Appl. Phys. (1)

K. Nagata, T. Umehara, and J. Nishiwaki, Jpn. J. Appl. Phys. 12, 1693–1698 (1973).
[Crossref]

Mon. Not. R. Astron. Soc. (1)

L. T. Little and A. Hewish, Mon. Not. R. Astron. Soc. 134, 221–237 (1966).

Nature (1)

E. Jakeman, E. R. Pike, and P. N. Pusey, Nature 263, 215–217 (1976).
[Crossref]

Opt. Acta (1)

L. Ronchi and A. Fontana, Opt. Acta 22, 243–246 (1975).
[Crossref]

Philos. Mag. (1)

Rayleigh (J. W. Strutt), Philos. Mag. 6, 321–347 (1919).

Phys. Rev. Lett. (2)

D. W. Schaefer and B. J. Berne, Phys. Rev. Lett. 28, 475 (1972).
[Crossref]

D. W. Schaefer and P. N. Pusey, Phys. Rev. Lett. 29, 843 (1972).
[Crossref]

Proc. Cambridge Philos. Soc. (1)

R. P. Mercier, Proc. Cambridge Philos. Soc. A58, 382–400 (1962).
[Crossref]

Proc. IEEE (3)

E. N. Bramley and M. Young, Proc. IEEE 114, 553–556 (1967).

B. M. Oliver, Proc. IEEE 51, 220–221 (1963).
[Crossref]

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, Proc. IEEE 63, 790–811 (1975).
[Crossref]

Proc. Inst. Radio Eng. (1)

J. D. Rigden and E. I. Gordon, Proc. Inst. Radio Eng. 50, 2367–2368 (1962).

Other (12)

“Engineering Uses of Coherent Light,” in Proceedings of 1975 Strathclyde Conference, edited by J. Robertson (Academic, London, 1976).

“Light Beating and Photon-Correlation Spectroscopy” (Proceedings NATO Advanced Study Institute), edited H. Z. Cummins and E. R. Pike (Plenum, New York, 1974).

“Laser Speckle and Related Phenomena,” Topics in Applied Physics Vol. 9, edited by J. C. Dainty (Springer, Berlin, 1975).

G. Parry in Ref. 4.

E. Jakeman and P. N. Pusey, (Royal Radar Establishment, Malvern, Worcs., England, May1974).

T. S. Mckechnie in Ref. 4.

M. I. Skolnik, (Naval Research Laboratory, Washington, D. C., July1969).

H. Goldstein in Propagation of Short Radio Waves, edited by D. E. KerrMcGraw-Hill, New York, 1951).

K. R. Schmidt, P. M. Thieband, and J. S. Waller, (Naval Research Laboratory, Washington, D. C., July1972).

G. Bishop in Determination and Use of Radar Scattering Characteristics, AGARD Lecture Series No. 59 (1973).

E. Jakeman and P. N. Pusey, IEEE Trans. Antennas Propag. (to be published).

M. Bertolotti in Ref. 3.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

FIG. 1
FIG. 1

Dependence of normalized second intensity moment (n[2] = 1 + σ2) on illuminated area for light dynamically scattered (scattering angle ~ 22°) by a 50 μm layer of liquid crystal (Ref. 8).

FIG. 2
FIG. 2

Photograph of instantaneous non-Gaussian speckle pattern obtained when a dynamic scattering liquid crystal is illuminated by a focused laser beam.

FIG. 3
FIG. 3

Spatial correlation function of light dynamically scattered by a liquid-crystal sample. One photomultipler detector was situated at scattering angle θ ~ 29° and the other was varied in angular position. The narrow central peak is due to the Gaussian speckle. (Points indicate experimental values, continuous line indicates simple theory).

FIG. 4
FIG. 4

Temporal correlation function of light dynamically scattered by a liquid-crystal sample. The upper trace is the correlation function with beam radius 12.5 μm showing both Gaussian and non-Gaussian contributions. The third trace is for beam radius ≳ 500 μm, i.e., N ≫ 1, showing only the Gaussian contribution. (See Ref. 8 for a full explanation of this diagram.)

FIG. 5
FIG. 5

Space-time correlation function for light scattered by a moving rough surface in the presence of wave-front curvature as a function of detector separation d ≡ |rr′|: (a) d ~ 0.06 cm, (b) d = 0.51 cm, (c) d = 0.89 cm. (See Ref. 15 for further details.)

FIG. 6
FIG. 6

Normalized mean-square intensity (1 + σ2) in the Fresnal region of a phase screen with ϕ 2 ¯ = 10: (a) Numerically computed result (Ref. 19), (b) behavior predicted by Eq. (21).

Equations (28)

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

E ( r , t ) = e i ω t i = 1 N a i ( r , t ) e i ϕ i ( r , t ) .
I ( r ) = N a 2 ( r ) ,
σ 2 ( r ) = I 2 ( r ) - I ( r ) 2 I ( r ) 2 = 1 - 2 N + a 4 ( r ) N a 2 ( r ) 2 .
a 2 n = A n Ω Q ( ω = r ˆ ) ,
I ( r ) = N A Ω Q ( r ˆ ) ,
σ 2 ( r ) = 1 - 2 N + 1 N Ω Q ( r ˆ ) .
I ( r ) I ( r ) I ( r ) I ( r ) = ( 1 - 1 N ) ( 1 + E * ( r ) E ( r ) 2 I ( r ) I ( r ) ) + a 2 ( r ) a 2 ( r ) N a 2 ( r ) a 2 ( r ) ,
E * ( r ) E ( r ) = N a ( r ) a ( r ) e i [ ϕ ( r ) - ϕ ( r ) ] .
I ( t ) I ( t ) I 2 = ( 1 - 1 N ) ( 1 + E * ( t ) E ( t ) 2 I 2 ) + 1 N a 2 ( t ) a 2 ( t ) a 2 2 ,
E * ( t ) E ( t ) = N exp i ω ( t - t ) a ( t ) a ( t ) exp i [ ϕ ( t ) - ϕ ( t ) ] .
I ( r , t ) I ( r , t + τ ) I ( r ) I ( r ) = 1 + exp ( - ( τ - τ d ) 2 τ c 2 ) × exp [ ( τ d τ c ) 2 - k 2 ( r - r ) 2 W e 2 4 Z 2 ] ,
W e = W .
τ d = ( k 2 W 4 / 2 Z ν 2 ) ν · ( r - r ) 1 + ( k 2 W 4 / 4 Z 2 )
τ c = W ν ( 1 + k 2 W 4 4 Z 2 ) - 1 / 2 ,
E + ( r , t ) = i E 0 ( 1 + cos θ ) 2 λ r exp [ i ( k r - ω t ) ] × - + d r exp ( i k r · ( r - 2 r ) 2 r ) exp ( i ϕ ( r , t ) - r 2 W 2 ) ,
ϕ ( r , t ) ϕ ( r , t ) = ϕ 2 ¯ [ 1 - r - r 2 / ξ 2 + ]
τ c = ( W 2 / ν W e ) / [ ( 1 + ( k 2 W 4 / 4 Z 2 ) ] 1 / 2
W e - 2 = W - 2 + k 2 ξ 2 / 8 ϕ 2 ¯ Z 2 .
σ 2 ( r ) = 2 exp ( - ξ 2 W e 2 ) - 1 + 2 ϕ 2 ¯ W 6 ξ 2 W e 2 ( W 2 + W 0 2 ) exp [ k 2 ξ 2 W e 4 sin 2 θ / 2 ϕ 2 ¯ W 2 ( W 2 + W e 2 ) ] ( 1 + k 2 W 4 / 4 Z 2 ) × { [ cos x - π ϕ 2 ¯ x exp ( - ξ 2 2 W 2 ) cosech π x ϕ 2 ¯ ] 2 + [ sin x - π ξ 2 2 W 2 ϕ ¯ 2 exp ( - ξ 2 2 W 2 ) ( cosech π x ϕ 2 ¯ - ϕ ¯ 2 π x ) ] 2 } ,
σ 2 ( r ) 1 - 2 ξ 2 W 2 + ϕ 2 ¯ ξ 2 4 W 2 exp k 2 ξ 2 sin 2 θ 4 ϕ 2 ¯ ,
σ 2 ( r ) = 2 exp ( - 2 x 2 ϕ 2 ¯ ) - 1 + ( cos x - π x ϕ 2 ¯ cosech π x ϕ 2 ¯ ) 2 + sin 2 x ,
σ 2 ( r ) = 1 + 1 / N .
σ 2 ( r ) = 1 - 2 ξ 2 W 2 + k 2 ξ 2 h 2 ¯ cos 4 θ / 2 W 2 × exp ξ 2 tan 2 θ / 2 4 h 2 ¯ ,
P N ( I ; r ) = 1 2 0 u d u J 0 ( u I ) J 0 ( u a ( r ) ) N ,
J 0 ( u a ( r ) ) = 0 J 0 ( a u ) p ( a ; r ) d a ,
p ( a ; r ) = 2 b Γ ( 1 + ν ) ( b a 2 ) ν + 1 K ν ( b a ) ,             ν > - 1
P N ( I ; r ) = b / I Γ ( M ) ( b I 2 ) M K M - 1 ( b I ) ,
I n = ( 2 b ) 2 n n ! Γ ( n + M ) Γ ( M ) .