Abstract

The multifractal description of rough surfaces is discussed and the mechanisms for generation of fractal and multifractal height distributions of inhomogeneities for rough surfaces are simulated. The original technique for estimating the spectrum of singularities is proposed for the study of these distributions.

© 2002 Optical Society of America

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References

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  1. M. Francon, Laser Speckle and Applications in Optics (Academic, New York, 1979).
  2. J. Uozumi, N. Asakura, Fractal Optics (Hokkaido University Press, Sapporo, Japan, 1995).
  3. M. V. Berry, J. H. Hannay, “Topography of random surfaces,” Nature 273, 573–576 (1978).
    [CrossRef]
  4. M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Bahan, ed. (North-Holland, Amsterdam, 1981).
  5. J. F. Nye, M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
    [CrossRef]
  6. M. S. Soskin, M. V. Vasnetsov, “Nonlinear singular optics,” Pure Appl. Opt. 7, 301–311 (1998).
    [CrossRef]
  7. O. V. Angelsky, S. G. Hanson, P. P. Maksimyak, Use of Optical Correlation Techniques for Characterizing Scattering Objects and Media, Vol. PM71 of SPIE Press Monograph Series (SPIE Press, Bellingham, Wash., 1999).
  8. A. S. Toporets, Optics of Rough Surfaces (Mashinostroenie, Leningrad, 1988), in Russian.
  9. J. M. Bennett, L. Mattson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).
  10. P. Beckmann, A. Spizzichino, Reflection of Electromagnetic Waves from Rough Surfaces (Basingstoke, Macmillan, London, 1963).
  11. A. Arneodo, G. Grasseau, M. Holschneider, “Wavelet transform of multifractals,” Phys. Rev. Lett. 61, 2281–2284 (1988).
    [CrossRef] [PubMed]
  12. M. Berry, I. Marzoli, W. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39–44 (2001).
  13. P. Hall, S. Davies, “On direction invariance of fractal dimension on a surface,” Appl. Phys. A 60, 271–274 (1995).
    [CrossRef]
  14. M. Hasegawa, J. Liu, Y. Konishi, “Characterization of engineering surfaces by fractal analysis,” Int. J. Jpn. Soc. Prec. Eng. 27, 192–196 (1993).
  15. E. L. Church, “Fractal surface finish,” Appl. Opt. 27, 1518–1526 (1988).
    [CrossRef] [PubMed]
  16. J. Kertész, T. Vicsek, “Self-affine surfaces,” in Fractals in Science, A. Bunde, S. Havlin, eds. (Springer-Verlag, Berlin, 1994).
    [CrossRef]
  17. T. Vicsek, “Dynamics of growing self-affine surfaces,” in From Statistical Physics to Chaos, G. Györgyi, I. Kondor, L. Sasvári, T. Tél, eds. (World Scientific, Singapore, 1992).
  18. A. Dogariu, J. Uozumi, T. Asakura, “Sources of error in optical measurements of fractal dimension,” Pure Appl. Opt. 2, 339–350 (1993).
    [CrossRef]
  19. A. Dogariu, J. Uozumi, T. Asakura, “Angular power spectra of fractal structures,” J. Mod. Opt. 41, 729–738 (1994).
    [CrossRef]
  20. Y.-P. Zhao, C.-F. Cheng, G.-C. Wang, T.-M. Lu, “Power law behavior in diffraction from fractal surfaces,” Surf. Sci. Lett. 409, L703–L708 (1998).
    [CrossRef]
  21. O. V. Angelsky, P. P. Maksimyak, V. V. Ryukhtin, S. G. Hanson, “New feasibilities for characterizing rough surfaces by optical-correlation techniques,” Appl. Opt. 40, 5693–5707 (2001).
    [CrossRef]
  22. V. G. Zakharov, “Elaboration and application of wavelet-analysis to nonlinear hydrodynamical systems,” Ph.D. dissertation (Institute of Mechanics of Continua, Ural Division, Russian Academy of Sciences, Perm, Russia, 1997), in Russian.
  23. B. B. Mandelbrot, Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1984).
  24. N. M. Astafyeva, “Wavelet-analysis: theoretical principles and application examples,” Usp. Phys. Nauk 166, 1145–1170 (1996), in Russian.
    [CrossRef]
  25. A. Arneodo, “Wavelet analysis of fractals: from the mathematical concept to experimental reality,” in Wavelets: Theory and Application, M. Y. Hussaini, ed. (Oxford University, New York, 1996), pp. 352–497.

2001

1998

Y.-P. Zhao, C.-F. Cheng, G.-C. Wang, T.-M. Lu, “Power law behavior in diffraction from fractal surfaces,” Surf. Sci. Lett. 409, L703–L708 (1998).
[CrossRef]

M. S. Soskin, M. V. Vasnetsov, “Nonlinear singular optics,” Pure Appl. Opt. 7, 301–311 (1998).
[CrossRef]

1996

N. M. Astafyeva, “Wavelet-analysis: theoretical principles and application examples,” Usp. Phys. Nauk 166, 1145–1170 (1996), in Russian.
[CrossRef]

1995

P. Hall, S. Davies, “On direction invariance of fractal dimension on a surface,” Appl. Phys. A 60, 271–274 (1995).
[CrossRef]

1994

A. Dogariu, J. Uozumi, T. Asakura, “Angular power spectra of fractal structures,” J. Mod. Opt. 41, 729–738 (1994).
[CrossRef]

1993

A. Dogariu, J. Uozumi, T. Asakura, “Sources of error in optical measurements of fractal dimension,” Pure Appl. Opt. 2, 339–350 (1993).
[CrossRef]

M. Hasegawa, J. Liu, Y. Konishi, “Characterization of engineering surfaces by fractal analysis,” Int. J. Jpn. Soc. Prec. Eng. 27, 192–196 (1993).

1988

E. L. Church, “Fractal surface finish,” Appl. Opt. 27, 1518–1526 (1988).
[CrossRef] [PubMed]

A. Arneodo, G. Grasseau, M. Holschneider, “Wavelet transform of multifractals,” Phys. Rev. Lett. 61, 2281–2284 (1988).
[CrossRef] [PubMed]

1978

M. V. Berry, J. H. Hannay, “Topography of random surfaces,” Nature 273, 573–576 (1978).
[CrossRef]

1974

J. F. Nye, M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Angelsky, O. V.

O. V. Angelsky, P. P. Maksimyak, V. V. Ryukhtin, S. G. Hanson, “New feasibilities for characterizing rough surfaces by optical-correlation techniques,” Appl. Opt. 40, 5693–5707 (2001).
[CrossRef]

O. V. Angelsky, S. G. Hanson, P. P. Maksimyak, Use of Optical Correlation Techniques for Characterizing Scattering Objects and Media, Vol. PM71 of SPIE Press Monograph Series (SPIE Press, Bellingham, Wash., 1999).

Arneodo, A.

A. Arneodo, G. Grasseau, M. Holschneider, “Wavelet transform of multifractals,” Phys. Rev. Lett. 61, 2281–2284 (1988).
[CrossRef] [PubMed]

A. Arneodo, “Wavelet analysis of fractals: from the mathematical concept to experimental reality,” in Wavelets: Theory and Application, M. Y. Hussaini, ed. (Oxford University, New York, 1996), pp. 352–497.

Asakura, N.

J. Uozumi, N. Asakura, Fractal Optics (Hokkaido University Press, Sapporo, Japan, 1995).

Asakura, T.

A. Dogariu, J. Uozumi, T. Asakura, “Angular power spectra of fractal structures,” J. Mod. Opt. 41, 729–738 (1994).
[CrossRef]

A. Dogariu, J. Uozumi, T. Asakura, “Sources of error in optical measurements of fractal dimension,” Pure Appl. Opt. 2, 339–350 (1993).
[CrossRef]

Astafyeva, N. M.

N. M. Astafyeva, “Wavelet-analysis: theoretical principles and application examples,” Usp. Phys. Nauk 166, 1145–1170 (1996), in Russian.
[CrossRef]

Beckmann, P.

P. Beckmann, A. Spizzichino, Reflection of Electromagnetic Waves from Rough Surfaces (Basingstoke, Macmillan, London, 1963).

Bennett, J. M.

J. M. Bennett, L. Mattson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

Berry, M.

M. Berry, I. Marzoli, W. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39–44 (2001).

J. F. Nye, M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Bahan, ed. (North-Holland, Amsterdam, 1981).

Berry, M. V.

M. V. Berry, J. H. Hannay, “Topography of random surfaces,” Nature 273, 573–576 (1978).
[CrossRef]

Cheng, C.-F.

Y.-P. Zhao, C.-F. Cheng, G.-C. Wang, T.-M. Lu, “Power law behavior in diffraction from fractal surfaces,” Surf. Sci. Lett. 409, L703–L708 (1998).
[CrossRef]

Church, E. L.

Davies, S.

P. Hall, S. Davies, “On direction invariance of fractal dimension on a surface,” Appl. Phys. A 60, 271–274 (1995).
[CrossRef]

Dogariu, A.

A. Dogariu, J. Uozumi, T. Asakura, “Angular power spectra of fractal structures,” J. Mod. Opt. 41, 729–738 (1994).
[CrossRef]

A. Dogariu, J. Uozumi, T. Asakura, “Sources of error in optical measurements of fractal dimension,” Pure Appl. Opt. 2, 339–350 (1993).
[CrossRef]

Francon, M.

M. Francon, Laser Speckle and Applications in Optics (Academic, New York, 1979).

Grasseau, G.

A. Arneodo, G. Grasseau, M. Holschneider, “Wavelet transform of multifractals,” Phys. Rev. Lett. 61, 2281–2284 (1988).
[CrossRef] [PubMed]

Hall, P.

P. Hall, S. Davies, “On direction invariance of fractal dimension on a surface,” Appl. Phys. A 60, 271–274 (1995).
[CrossRef]

Hannay, J. H.

M. V. Berry, J. H. Hannay, “Topography of random surfaces,” Nature 273, 573–576 (1978).
[CrossRef]

Hanson, S. G.

O. V. Angelsky, P. P. Maksimyak, V. V. Ryukhtin, S. G. Hanson, “New feasibilities for characterizing rough surfaces by optical-correlation techniques,” Appl. Opt. 40, 5693–5707 (2001).
[CrossRef]

O. V. Angelsky, S. G. Hanson, P. P. Maksimyak, Use of Optical Correlation Techniques for Characterizing Scattering Objects and Media, Vol. PM71 of SPIE Press Monograph Series (SPIE Press, Bellingham, Wash., 1999).

Hasegawa, M.

M. Hasegawa, J. Liu, Y. Konishi, “Characterization of engineering surfaces by fractal analysis,” Int. J. Jpn. Soc. Prec. Eng. 27, 192–196 (1993).

Holschneider, M.

A. Arneodo, G. Grasseau, M. Holschneider, “Wavelet transform of multifractals,” Phys. Rev. Lett. 61, 2281–2284 (1988).
[CrossRef] [PubMed]

Kertész, J.

J. Kertész, T. Vicsek, “Self-affine surfaces,” in Fractals in Science, A. Bunde, S. Havlin, eds. (Springer-Verlag, Berlin, 1994).
[CrossRef]

Konishi, Y.

M. Hasegawa, J. Liu, Y. Konishi, “Characterization of engineering surfaces by fractal analysis,” Int. J. Jpn. Soc. Prec. Eng. 27, 192–196 (1993).

Liu, J.

M. Hasegawa, J. Liu, Y. Konishi, “Characterization of engineering surfaces by fractal analysis,” Int. J. Jpn. Soc. Prec. Eng. 27, 192–196 (1993).

Lu, T.-M.

Y.-P. Zhao, C.-F. Cheng, G.-C. Wang, T.-M. Lu, “Power law behavior in diffraction from fractal surfaces,” Surf. Sci. Lett. 409, L703–L708 (1998).
[CrossRef]

Maksimyak, P. P.

O. V. Angelsky, P. P. Maksimyak, V. V. Ryukhtin, S. G. Hanson, “New feasibilities for characterizing rough surfaces by optical-correlation techniques,” Appl. Opt. 40, 5693–5707 (2001).
[CrossRef]

O. V. Angelsky, S. G. Hanson, P. P. Maksimyak, Use of Optical Correlation Techniques for Characterizing Scattering Objects and Media, Vol. PM71 of SPIE Press Monograph Series (SPIE Press, Bellingham, Wash., 1999).

Mandelbrot, B. B.

B. B. Mandelbrot, Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1984).

Marzoli, I.

M. Berry, I. Marzoli, W. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39–44 (2001).

Mattson, L.

J. M. Bennett, L. Mattson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

Nye, J. F.

J. F. Nye, M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Ryukhtin, V. V.

Schleich, W.

M. Berry, I. Marzoli, W. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39–44 (2001).

Soskin, M. S.

M. S. Soskin, M. V. Vasnetsov, “Nonlinear singular optics,” Pure Appl. Opt. 7, 301–311 (1998).
[CrossRef]

Spizzichino, A.

P. Beckmann, A. Spizzichino, Reflection of Electromagnetic Waves from Rough Surfaces (Basingstoke, Macmillan, London, 1963).

Toporets, A. S.

A. S. Toporets, Optics of Rough Surfaces (Mashinostroenie, Leningrad, 1988), in Russian.

Uozumi, J.

A. Dogariu, J. Uozumi, T. Asakura, “Angular power spectra of fractal structures,” J. Mod. Opt. 41, 729–738 (1994).
[CrossRef]

A. Dogariu, J. Uozumi, T. Asakura, “Sources of error in optical measurements of fractal dimension,” Pure Appl. Opt. 2, 339–350 (1993).
[CrossRef]

J. Uozumi, N. Asakura, Fractal Optics (Hokkaido University Press, Sapporo, Japan, 1995).

Vasnetsov, M. V.

M. S. Soskin, M. V. Vasnetsov, “Nonlinear singular optics,” Pure Appl. Opt. 7, 301–311 (1998).
[CrossRef]

Vicsek, T.

J. Kertész, T. Vicsek, “Self-affine surfaces,” in Fractals in Science, A. Bunde, S. Havlin, eds. (Springer-Verlag, Berlin, 1994).
[CrossRef]

T. Vicsek, “Dynamics of growing self-affine surfaces,” in From Statistical Physics to Chaos, G. Györgyi, I. Kondor, L. Sasvári, T. Tél, eds. (World Scientific, Singapore, 1992).

Wang, G.-C.

Y.-P. Zhao, C.-F. Cheng, G.-C. Wang, T.-M. Lu, “Power law behavior in diffraction from fractal surfaces,” Surf. Sci. Lett. 409, L703–L708 (1998).
[CrossRef]

Zakharov, V. G.

V. G. Zakharov, “Elaboration and application of wavelet-analysis to nonlinear hydrodynamical systems,” Ph.D. dissertation (Institute of Mechanics of Continua, Ural Division, Russian Academy of Sciences, Perm, Russia, 1997), in Russian.

Zhao, Y.-P.

Y.-P. Zhao, C.-F. Cheng, G.-C. Wang, T.-M. Lu, “Power law behavior in diffraction from fractal surfaces,” Surf. Sci. Lett. 409, L703–L708 (1998).
[CrossRef]

Appl. Opt.

Appl. Phys. A

P. Hall, S. Davies, “On direction invariance of fractal dimension on a surface,” Appl. Phys. A 60, 271–274 (1995).
[CrossRef]

Int. J. Jpn. Soc. Prec. Eng.

M. Hasegawa, J. Liu, Y. Konishi, “Characterization of engineering surfaces by fractal analysis,” Int. J. Jpn. Soc. Prec. Eng. 27, 192–196 (1993).

J. Mod. Opt.

A. Dogariu, J. Uozumi, T. Asakura, “Angular power spectra of fractal structures,” J. Mod. Opt. 41, 729–738 (1994).
[CrossRef]

Nature

M. V. Berry, J. H. Hannay, “Topography of random surfaces,” Nature 273, 573–576 (1978).
[CrossRef]

Phys. Rev. Lett.

A. Arneodo, G. Grasseau, M. Holschneider, “Wavelet transform of multifractals,” Phys. Rev. Lett. 61, 2281–2284 (1988).
[CrossRef] [PubMed]

Phys. World

M. Berry, I. Marzoli, W. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39–44 (2001).

Proc. R. Soc. London Ser. A

J. F. Nye, M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Pure Appl. Opt.

M. S. Soskin, M. V. Vasnetsov, “Nonlinear singular optics,” Pure Appl. Opt. 7, 301–311 (1998).
[CrossRef]

A. Dogariu, J. Uozumi, T. Asakura, “Sources of error in optical measurements of fractal dimension,” Pure Appl. Opt. 2, 339–350 (1993).
[CrossRef]

Surf. Sci. Lett.

Y.-P. Zhao, C.-F. Cheng, G.-C. Wang, T.-M. Lu, “Power law behavior in diffraction from fractal surfaces,” Surf. Sci. Lett. 409, L703–L708 (1998).
[CrossRef]

Usp. Phys. Nauk

N. M. Astafyeva, “Wavelet-analysis: theoretical principles and application examples,” Usp. Phys. Nauk 166, 1145–1170 (1996), in Russian.
[CrossRef]

Other

A. Arneodo, “Wavelet analysis of fractals: from the mathematical concept to experimental reality,” in Wavelets: Theory and Application, M. Y. Hussaini, ed. (Oxford University, New York, 1996), pp. 352–497.

V. G. Zakharov, “Elaboration and application of wavelet-analysis to nonlinear hydrodynamical systems,” Ph.D. dissertation (Institute of Mechanics of Continua, Ural Division, Russian Academy of Sciences, Perm, Russia, 1997), in Russian.

B. B. Mandelbrot, Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1984).

J. Kertész, T. Vicsek, “Self-affine surfaces,” in Fractals in Science, A. Bunde, S. Havlin, eds. (Springer-Verlag, Berlin, 1994).
[CrossRef]

T. Vicsek, “Dynamics of growing self-affine surfaces,” in From Statistical Physics to Chaos, G. Györgyi, I. Kondor, L. Sasvári, T. Tél, eds. (World Scientific, Singapore, 1992).

O. V. Angelsky, S. G. Hanson, P. P. Maksimyak, Use of Optical Correlation Techniques for Characterizing Scattering Objects and Media, Vol. PM71 of SPIE Press Monograph Series (SPIE Press, Bellingham, Wash., 1999).

A. S. Toporets, Optics of Rough Surfaces (Mashinostroenie, Leningrad, 1988), in Russian.

J. M. Bennett, L. Mattson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

P. Beckmann, A. Spizzichino, Reflection of Electromagnetic Waves from Rough Surfaces (Basingstoke, Macmillan, London, 1963).

M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Bahan, ed. (North-Holland, Amsterdam, 1981).

M. Francon, Laser Speckle and Applications in Optics (Academic, New York, 1979).

J. Uozumi, N. Asakura, Fractal Optics (Hokkaido University Press, Sapporo, Japan, 1995).

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Figures (17)

Fig. 1
Fig. 1

Rough surface samples of glass obtained with crown glass No. 8 when the surface was polished with an abrasive powder: upper row, first sample has an average grain size of 28 µm; bottom row, second sample has an average grain size of 100 µm; the samples were magnified 100, 300, 1000, and 3000 times.

Fig. 2
Fig. 2

Spatial distributions and power spectra of discrete [(e), (f), (g), (h)] and continuous [(a), (b), (c), (d)] signals used in computer simulation.

Fig. 3
Fig. 3

(a) Spatial distribution and (b) power spectrum of the signal that results from the addition of the initial signals shown in Figs. 2(e) and 2(g).

Fig. 4
Fig. 4

(a) Spatial distribution and (b) power spectrum of the signal that results from the addition of the initial signals shown in Figs. 2(a) and 2(c). The dashed curves show the slopes of power spectra of the initial signals.

Fig. 5
Fig. 5

(a) Spatial distribution and (b) power spectrum of the signal that results from the addition of the initial signals shown in Figs. 2(a) and 2(e).

Fig. 6
Fig. 6

(a) Spatial distribution and (b) power spectrum of the signal that resulted when we multiplied the initial signals shown in Figs. 2(a) and 2(e). The dashed lines represent two slopes that we observed in the power spectrum of the resulting signal on a log-log representation.

Fig. 7
Fig. 7

(a) Spatial distribution and (b) power spectrum of the signal that resulted when we multiplied the initial signals shown in Figs. 2(a) and 2(c). The dashed lines represent two slopes that we observed in the power spectrum of the resulting signal on a log-log representation.

Fig. 8
Fig. 8

Choice of the height magnitudes of the analyzed distribution h(x) spaced by b along the x axis (upper graph), and approximation of the initial distribution h(x) by the broken curve h b (x) on the b scale (lower graph).

Fig. 9
Fig. 9

Straight-line segment b lying within the scale interval that ranges from minimal detail a to the total distribution width R.

Fig. 10
Fig. 10

Density distribution function of the trial distribution P(N n b ) versus the approximating link lengths and scale b (see Fig. 2).

Fig. 11
Fig. 11

Dependence of the average relative length N 0 b of the analyzed distribution versus the scale of interest b.

Fig. 12
Fig. 12

Test signals and the corresponding spectra of singularities: (a) Koch fractal (D = 1.2619), (b) harmonic curve (D = 1), (c) random distribution (D ≤ 2). The solid curve represents the envelope. As the sampling rate increases, the dimension spectrum narrows.

Fig. 13
Fig. 13

Explicit form of the signal shown in Fig. 2(a) with (a) dimension D = 1.6 and its spectra of singularities estimated over (b) 256, (c) 1024, (d) 4096 sampling points.

Fig. 14
Fig. 14

Explicit form of the signal shown in Fig. 2(c) with (a) dimension D = 1.8 and its spectra of singularities estimated over (b) 256, (c) 1024, (d) 4096 sampling points.

Fig. 15
Fig. 15

Explicit form of the signal obtained by (a) summation of the signals shown in Figs. 2(a) and 2(c) and its spectra of singularities estimated over (b) 256, (c) 1024, (d) 4096 sampling points.

Fig. 16
Fig. 16

Explicit form of the signal obtained by (a) multiplication of the signals shown in Figs. 2(a) and 2(c) and its spectra of singularities estimated over (b) 256, (c) 1024, (d) 4096 sampling points.

Fig. 17
Fig. 17

Explicit form of the signal obtained from (a) a profilogram of a real rough surface (No. 2) and its spectra of singularities estimated over (b) 256, (c) 1024, (d) 4096 sampling points.

Equations (2)

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Nnb=m Lmb/Lnb,
DNbmax=D0N0bmax Nbmax

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