Abstract

First-order statistical properties of the speckle field and its intensity in the Fraunhofer diffraction region that is produced by random Koch fractals are investigated by means of computer simulations in comparison with the ordinary fully developed speckle.

© 1993 Optical Society of America

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References

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  1. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1982), Chap. 6, pp. 37–57, andB. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1982), Chap. 39, pp. 362–365.
  2. M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 (1979).
    [Crossref]
  3. E. Jakeman, “Scattering by a corrugated random surface with fractal slope,” J. Phys. A 15, L55–L59 (1982).
    [Crossref]
  4. E. Jakeman, “Fresnel scattering by a corrugated random surface with fractal slope,” J. Opt. Soc. Am. 72, 1034–1041 (1982).
    [Crossref]
  5. D. L. Jaggard, Y. Kim, “Diffraction by band-limited fractal screens,” J. Opt. Soc. Am. A 4, 1055–1062 (1987).
    [Crossref]
  6. D. L. Jaggard, X. Sun, “Scattering from fractally corrugated surfaces,” J. Opt. Soc. Am. A 7, 1131–1139 (1990).
    [Crossref]
  7. C. Allain, M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B 33, 3566–3569 (1986).
    [Crossref]
  8. J. Uozumi, H. Kimura, T. Asakura, “Fraunhofer diffraction by Koch fractals,” J. Mod. Opt. 37, 1011–1031 (1990).
    [Crossref]
  9. J. Uozumi, H. Kimura, T. Asakura, “Fraunhofer diffraction by Koch fractals: the dimensionality,” J. Mod. Opt. 38, 1335–1347 (1991).
    [Crossref]
  10. Y. Sakurada, J. Uozumi, T. Asakura, “Fresnel diffraction by one-dimensional regular fractals,” Pure Appl. Opt. 1, 29–40 (1992).
    [Crossref]
  11. J. Uozumi, H. Kimura, T. Asakura, “Laser diffraction by randomized Koch fractals,” Waves Random Media 1, 73–80 (1991).
    [Crossref]
  12. S. K. Sinha, “Scattering from fractal structures,” Physica D 38, 310–314 (1989).
    [Crossref]
  13. T. Vicsek, Fractal Growth Phenomena (World Scientific, Singapore, 1989), Chap. 4, pp. 71–79, andT. Vicsek, Fractal Growth Phenomena (World Scientific, Singapore, 1989) Chap. 8, pp. 246–257.
  14. A. K. Dogariu, J. Uozumi, T. Asakura, “Enhancement of the backscattered intensity from fractal aggregates,” Waves Random Media (to be published).
  15. J. F. Muzy, B. Pouligny, E. Freysz, F. Argoul, A. Arneodo, “Optical-diffraction measurement of fractal dimensions and f(α) spectrum,” Phys. Rev. A 45, 8961–8964 (1992).
    [Crossref] [PubMed]
  16. J. W. Goodman, “Statistical properties of laser speckle patterns,”in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 9–75.
    [Crossref]
  17. E. Jakeman, P. N. Pusey, “Photon-counting statistics of optical scintillation,”in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, New York, 1980), pp. 73–116.
    [Crossref]

1992 (2)

Y. Sakurada, J. Uozumi, T. Asakura, “Fresnel diffraction by one-dimensional regular fractals,” Pure Appl. Opt. 1, 29–40 (1992).
[Crossref]

J. F. Muzy, B. Pouligny, E. Freysz, F. Argoul, A. Arneodo, “Optical-diffraction measurement of fractal dimensions and f(α) spectrum,” Phys. Rev. A 45, 8961–8964 (1992).
[Crossref] [PubMed]

1991 (2)

J. Uozumi, H. Kimura, T. Asakura, “Laser diffraction by randomized Koch fractals,” Waves Random Media 1, 73–80 (1991).
[Crossref]

J. Uozumi, H. Kimura, T. Asakura, “Fraunhofer diffraction by Koch fractals: the dimensionality,” J. Mod. Opt. 38, 1335–1347 (1991).
[Crossref]

1990 (2)

J. Uozumi, H. Kimura, T. Asakura, “Fraunhofer diffraction by Koch fractals,” J. Mod. Opt. 37, 1011–1031 (1990).
[Crossref]

D. L. Jaggard, X. Sun, “Scattering from fractally corrugated surfaces,” J. Opt. Soc. Am. A 7, 1131–1139 (1990).
[Crossref]

1989 (1)

S. K. Sinha, “Scattering from fractal structures,” Physica D 38, 310–314 (1989).
[Crossref]

1987 (1)

1986 (1)

C. Allain, M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B 33, 3566–3569 (1986).
[Crossref]

1982 (2)

E. Jakeman, “Scattering by a corrugated random surface with fractal slope,” J. Phys. A 15, L55–L59 (1982).
[Crossref]

E. Jakeman, “Fresnel scattering by a corrugated random surface with fractal slope,” J. Opt. Soc. Am. 72, 1034–1041 (1982).
[Crossref]

1979 (1)

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

Allain, C.

C. Allain, M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B 33, 3566–3569 (1986).
[Crossref]

Argoul, F.

J. F. Muzy, B. Pouligny, E. Freysz, F. Argoul, A. Arneodo, “Optical-diffraction measurement of fractal dimensions and f(α) spectrum,” Phys. Rev. A 45, 8961–8964 (1992).
[Crossref] [PubMed]

Arneodo, A.

J. F. Muzy, B. Pouligny, E. Freysz, F. Argoul, A. Arneodo, “Optical-diffraction measurement of fractal dimensions and f(α) spectrum,” Phys. Rev. A 45, 8961–8964 (1992).
[Crossref] [PubMed]

Asakura, T.

Y. Sakurada, J. Uozumi, T. Asakura, “Fresnel diffraction by one-dimensional regular fractals,” Pure Appl. Opt. 1, 29–40 (1992).
[Crossref]

J. Uozumi, H. Kimura, T. Asakura, “Laser diffraction by randomized Koch fractals,” Waves Random Media 1, 73–80 (1991).
[Crossref]

J. Uozumi, H. Kimura, T. Asakura, “Fraunhofer diffraction by Koch fractals: the dimensionality,” J. Mod. Opt. 38, 1335–1347 (1991).
[Crossref]

J. Uozumi, H. Kimura, T. Asakura, “Fraunhofer diffraction by Koch fractals,” J. Mod. Opt. 37, 1011–1031 (1990).
[Crossref]

A. K. Dogariu, J. Uozumi, T. Asakura, “Enhancement of the backscattered intensity from fractal aggregates,” Waves Random Media (to be published).

Berry, M. V.

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

Cloitre, M.

C. Allain, M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B 33, 3566–3569 (1986).
[Crossref]

Dogariu, A. K.

A. K. Dogariu, J. Uozumi, T. Asakura, “Enhancement of the backscattered intensity from fractal aggregates,” Waves Random Media (to be published).

Freysz, E.

J. F. Muzy, B. Pouligny, E. Freysz, F. Argoul, A. Arneodo, “Optical-diffraction measurement of fractal dimensions and f(α) spectrum,” Phys. Rev. A 45, 8961–8964 (1992).
[Crossref] [PubMed]

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,”in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 9–75.
[Crossref]

Jaggard, D. L.

Jakeman, E.

E. Jakeman, “Fresnel scattering by a corrugated random surface with fractal slope,” J. Opt. Soc. Am. 72, 1034–1041 (1982).
[Crossref]

E. Jakeman, “Scattering by a corrugated random surface with fractal slope,” J. Phys. A 15, L55–L59 (1982).
[Crossref]

E. Jakeman, P. N. Pusey, “Photon-counting statistics of optical scintillation,”in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, New York, 1980), pp. 73–116.
[Crossref]

Kim, Y.

Kimura, H.

J. Uozumi, H. Kimura, T. Asakura, “Laser diffraction by randomized Koch fractals,” Waves Random Media 1, 73–80 (1991).
[Crossref]

J. Uozumi, H. Kimura, T. Asakura, “Fraunhofer diffraction by Koch fractals: the dimensionality,” J. Mod. Opt. 38, 1335–1347 (1991).
[Crossref]

J. Uozumi, H. Kimura, T. Asakura, “Fraunhofer diffraction by Koch fractals,” J. Mod. Opt. 37, 1011–1031 (1990).
[Crossref]

Mandelbrot, B. B.

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1982), Chap. 6, pp. 37–57, andB. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1982), Chap. 39, pp. 362–365.

Muzy, J. F.

J. F. Muzy, B. Pouligny, E. Freysz, F. Argoul, A. Arneodo, “Optical-diffraction measurement of fractal dimensions and f(α) spectrum,” Phys. Rev. A 45, 8961–8964 (1992).
[Crossref] [PubMed]

Pouligny, B.

J. F. Muzy, B. Pouligny, E. Freysz, F. Argoul, A. Arneodo, “Optical-diffraction measurement of fractal dimensions and f(α) spectrum,” Phys. Rev. A 45, 8961–8964 (1992).
[Crossref] [PubMed]

Pusey, P. N.

E. Jakeman, P. N. Pusey, “Photon-counting statistics of optical scintillation,”in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, New York, 1980), pp. 73–116.
[Crossref]

Sakurada, Y.

Y. Sakurada, J. Uozumi, T. Asakura, “Fresnel diffraction by one-dimensional regular fractals,” Pure Appl. Opt. 1, 29–40 (1992).
[Crossref]

Sinha, S. K.

S. K. Sinha, “Scattering from fractal structures,” Physica D 38, 310–314 (1989).
[Crossref]

Sun, X.

Uozumi, J.

Y. Sakurada, J. Uozumi, T. Asakura, “Fresnel diffraction by one-dimensional regular fractals,” Pure Appl. Opt. 1, 29–40 (1992).
[Crossref]

J. Uozumi, H. Kimura, T. Asakura, “Fraunhofer diffraction by Koch fractals: the dimensionality,” J. Mod. Opt. 38, 1335–1347 (1991).
[Crossref]

J. Uozumi, H. Kimura, T. Asakura, “Laser diffraction by randomized Koch fractals,” Waves Random Media 1, 73–80 (1991).
[Crossref]

J. Uozumi, H. Kimura, T. Asakura, “Fraunhofer diffraction by Koch fractals,” J. Mod. Opt. 37, 1011–1031 (1990).
[Crossref]

A. K. Dogariu, J. Uozumi, T. Asakura, “Enhancement of the backscattered intensity from fractal aggregates,” Waves Random Media (to be published).

Vicsek, T.

T. Vicsek, Fractal Growth Phenomena (World Scientific, Singapore, 1989), Chap. 4, pp. 71–79, andT. Vicsek, Fractal Growth Phenomena (World Scientific, Singapore, 1989) Chap. 8, pp. 246–257.

J. Mod. Opt. (2)

J. Uozumi, H. Kimura, T. Asakura, “Fraunhofer diffraction by Koch fractals,” J. Mod. Opt. 37, 1011–1031 (1990).
[Crossref]

J. Uozumi, H. Kimura, T. Asakura, “Fraunhofer diffraction by Koch fractals: the dimensionality,” J. Mod. Opt. 38, 1335–1347 (1991).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Phys. A (2)

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

E. Jakeman, “Scattering by a corrugated random surface with fractal slope,” J. Phys. A 15, L55–L59 (1982).
[Crossref]

Phys. Rev. A (1)

J. F. Muzy, B. Pouligny, E. Freysz, F. Argoul, A. Arneodo, “Optical-diffraction measurement of fractal dimensions and f(α) spectrum,” Phys. Rev. A 45, 8961–8964 (1992).
[Crossref] [PubMed]

Phys. Rev. B (1)

C. Allain, M. Cloitre, “Optical diffraction on fractals,” Phys. Rev. B 33, 3566–3569 (1986).
[Crossref]

Physica D (1)

S. K. Sinha, “Scattering from fractal structures,” Physica D 38, 310–314 (1989).
[Crossref]

Pure Appl. Opt. (1)

Y. Sakurada, J. Uozumi, T. Asakura, “Fresnel diffraction by one-dimensional regular fractals,” Pure Appl. Opt. 1, 29–40 (1992).
[Crossref]

Waves Random Media (1)

J. Uozumi, H. Kimura, T. Asakura, “Laser diffraction by randomized Koch fractals,” Waves Random Media 1, 73–80 (1991).
[Crossref]

Other (5)

T. Vicsek, Fractal Growth Phenomena (World Scientific, Singapore, 1989), Chap. 4, pp. 71–79, andT. Vicsek, Fractal Growth Phenomena (World Scientific, Singapore, 1989) Chap. 8, pp. 246–257.

A. K. Dogariu, J. Uozumi, T. Asakura, “Enhancement of the backscattered intensity from fractal aggregates,” Waves Random Media (to be published).

J. W. Goodman, “Statistical properties of laser speckle patterns,”in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 9–75.
[Crossref]

E. Jakeman, P. N. Pusey, “Photon-counting statistics of optical scintillation,”in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, New York, 1980), pp. 73–116.
[Crossref]

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1982), Chap. 6, pp. 37–57, andB. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1982), Chap. 39, pp. 362–365.

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Figures (6)

Fig. 1
Fig. 1

Three different generators with the same parameter m = 4 and different parameters of μ: (a) 3.5263, (b) 3, (c) 2.3784.

Fig. 2
Fig. 2

Objects and their Fraunhofer patterns of Koch fractals. The dimension D of the objects is (a) 1.1, (b) 1.2618, and (c) 1.6. Each result is arrayed in such a way that randomness σ increases from top to bottom. All figures of the Fraunhofer patterns are shown for a square area defined by diagonal points (−384,0) and (382.5, 382.5).

Fig. 3
Fig. 3

Probability density functions of intensity distributions of diffraction patterns produced by Koch fractals. The layout of the parameters is the same as Fig. 2.

Fig. 4
Fig. 4

PDF's of phase distributions of diffraction patterns produced by Koch fractals. Results are arrayed so that randomness σ increases from top to bottom and dimension D increases from left to right.

Fig. 5
Fig. 5

PDF's of complex amplitudes in diffraction fields that are produced by Koch fractals. The fractal dimension is fixed to D = 1.2618, while the randomness σ is set to 0, 1, and 15 deg.

Fig. 6
Fig. 6

Root-sum-square value e(σ) of the difference between the joint PDF of the complex amplitude by Koch fractals and the associated joint Gaussian density fit.

Equations (10)

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

K N ( p ) = Π i = 1 N { M [ 1 / μ ] k i = 1 m T [ d ( k i ) ] R [ θ ( k i ) ] } K 0 ( p ) ,
K 0 ( p ) = { δ ( y ) , | x | L / 2 0 , otherwise .
D = log m log μ .
K N ( q ) = μ N L k 1 = 1 m k 2 = 1 m k N = 1 m exp { 2 π i Lq × r = 1 N μ r ( k r ) cos [ ϕ α ( k r ) Θ r 1 ] } × sinc [ μ N Lq cos ( ϕ Θ N ) ] ,
Θ υ = s = 1 υ θ ( k s ) .
e ( σ ) = ( k = 1 M l = 1 N { p [ U r ( k ) , U i ( l ) ] N [ U r ( k ) , U i ( l ) ] } 2 ) 1 / 2 ,
N ( U r , U i ) = 1 2 π σ r σ i ( 1 ρ 2 ) 1 / 2 × exp { 1 2 ( 1 ρ 2 ) [ ( U r η r ) 2 σ r 2 2 ρ ( U r η r ) ( U i η i ) σ r σ i + ( U i η i ) 2 σ i 2 ] } ,
C = σ I I = ( 4 I s σ r 2 + 2 σ r 4 + 2 σ i 4 ) 1 / 2 I s + σ r 2 + σ i 2 ,
C = [ 2 ( σ r 4 + σ i 4 ) ] 1 / 2 σ r 2 + σ i 2 ,
1 C 2 .

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