Abstract

This paper reports experimental studies on speckles produced by the rough silver films. The speckles on the rough glass/silver surfaces are measured with a microscopic imaging system. The structures of speckle patterns have the characteristics of fractals and multi-scaled sizes. We find that with the increase of the silver film thickness, the contrast of the speckles increases, and the intensity probability density functions gradually transit to exponential decay. We calculate the global and the local correlation functions of the speckle patterns, and find that both the fractal exponent and correlation length of the small-sized speckles decrease with the thickness of the silver films. We use the mechanisms of rough dielectric interface scattering and random surface plasmon waves to give the preliminary explanations for the evolutions of the speckles.

© 2013 OSA

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  6. R. Cerbino, “Correlations of light in the deep Fresnel region: An extended Van Cittert and Zernike theorem,” Phys. Rev. A75(5), 053815 (2007).
    [CrossRef]
  7. M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation function g(r),” Phys. Rev. Lett.85(7), 1416–1419 (2000).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  19. S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B Condens. Matter38(4), 2297–2311 (1988).
    [CrossRef] [PubMed]
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2012 (1)

2010 (3)

S. E. Skipetrov, J. Peuser, R. Cerbino, P. Zakharov, B. Weber, and F. Scheffold, “Noise in laser speckle correlation and imaging techniques,” Opt. Express18(14), 14519–14534 (2010).
[CrossRef] [PubMed]

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9(3), 193–204 (2010).
[CrossRef] [PubMed]

Z. Y. Fan and A. O. Govorov, “Plasmonic circular dichroism of chiral metal nanoparticle assemblies,” Nano Lett.10(7), 2580–2587 (2010).
[CrossRef] [PubMed]

2009 (2)

Z. W. Liu, Y. Wang, J. Yao, H. Lee, W. Srituravanich, and X. Zhang, “Broad band two-dimensional manipulation of surface plasmons,” Nano Lett.9(1), 462–466 (2009).
[CrossRef] [PubMed]

J. Sorrentini, M. Zerrad, and C. Amra, “Statistical signatures of random media and their correlation to polarization properties,” Opt. Lett.34(16), 2429–2431 (2009).
[CrossRef] [PubMed]

2008 (5)

J. W. Goodman, “Speckle with a finite number of steps,” Appl. Opt.47(4), A111–A118 (2008).
[CrossRef] [PubMed]

D. Han, M. Wang, and J. Zhou, “Fractal analysis of self-mixing speckle signal in velocity sensing,” Opt. Express16(5), 3204–3211 (2008).
[CrossRef] [PubMed]

A. S. Ulyanov, “Application of laser speckles for identification of tissues with pathological changes,” Quant. Elec.38(6), 557–562 (2008).
[CrossRef]

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: Theory,” Phys. Rev. A78(6), 063806 (2008).
[CrossRef]

J. Laverdant, S. Buil, B. Berini, and X. Quelin, “Polarization dependent near-field speckle of random gold films,” Phys. Rev. B77(16), 165406 (2008).
[CrossRef]

2007 (1)

R. Cerbino, “Correlations of light in the deep Fresnel region: An extended Van Cittert and Zernike theorem,” Phys. Rev. A75(5), 053815 (2007).
[CrossRef]

2004 (1)

C. F. Cheng, M. Liu, N. Y. Zhang, S. Y. Teng, H. S. Song, and Z. Z. Xu, “M. Liu1, N. Y. Zhang, S. Y. Teng, H. S. Song and Z. Z. Xu, “Speckle intensity correlation in the diffraction region near rough surfaces and simulational experiments for extraction of surface parameters,” Europhys. Lett.65(6), 779–784 (2004).
[CrossRef]

2002 (1)

Z. Q. Tian, B. Ren, and D. Y. Wu, “Surface-enhanced Raman scattering: from noble to transition metals and from rough surfaces to ordered nanostructures,” J. Phys. Chem. B106(37), 9463–9483 (2002).
[CrossRef]

2001 (1)

2000 (2)

D. P. Qi, D. L. Liu, S. Y. Teng, N. Y. Zhang, and C. F. Cheng, “Morphology analysis atomic force microscope and light scattering study for random scattering screens,” Acta Phys. Sin.49(7), 1260–1266 (2000) (in Chinese).

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation function g(r),” Phys. Rev. Lett.85(7), 1416–1419 (2000).
[CrossRef] [PubMed]

1992 (1)

J. A. Sánchez-Gil and M. Nieto-Vesperinas, “Resonance effects in multiple light scattering from statistically rough metallic surfaces,” Phys. Rev. B Condens. Mater.45(15), 8623–8633 (1992).
[CrossRef] [PubMed]

1988 (1)

S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B Condens. Matter38(4), 2297–2311 (1988).
[CrossRef] [PubMed]

1984 (1)

E. Jakeman, “Speckle statistics with a small number of scatters,” Opt. Eng.23(4), 234453 (1984).
[CrossRef]

Amra, C.

Angelsky, O. V.

Barnard, E. S.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9(3), 193–204 (2010).
[CrossRef] [PubMed]

Berini, B.

J. Laverdant, S. Buil, B. Berini, and X. Quelin, “Polarization dependent near-field speckle of random gold films,” Phys. Rev. B77(16), 165406 (2008).
[CrossRef]

Brongersma, M. L.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9(3), 193–204 (2010).
[CrossRef] [PubMed]

Buil, S.

J. Laverdant, S. Buil, B. Berini, and X. Quelin, “Polarization dependent near-field speckle of random gold films,” Phys. Rev. B77(16), 165406 (2008).
[CrossRef]

Cai, W.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9(3), 193–204 (2010).
[CrossRef] [PubMed]

Carpineti, M.

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation function g(r),” Phys. Rev. Lett.85(7), 1416–1419 (2000).
[CrossRef] [PubMed]

Cerbino, R.

S. E. Skipetrov, J. Peuser, R. Cerbino, P. Zakharov, B. Weber, and F. Scheffold, “Noise in laser speckle correlation and imaging techniques,” Opt. Express18(14), 14519–14534 (2010).
[CrossRef] [PubMed]

R. Cerbino, “Correlations of light in the deep Fresnel region: An extended Van Cittert and Zernike theorem,” Phys. Rev. A75(5), 053815 (2007).
[CrossRef]

Cheng, C. F.

C. F. Cheng, M. Liu, N. Y. Zhang, S. Y. Teng, H. S. Song, and Z. Z. Xu, “M. Liu1, N. Y. Zhang, S. Y. Teng, H. S. Song and Z. Z. Xu, “Speckle intensity correlation in the diffraction region near rough surfaces and simulational experiments for extraction of surface parameters,” Europhys. Lett.65(6), 779–784 (2004).
[CrossRef]

D. P. Qi, D. L. Liu, S. Y. Teng, N. Y. Zhang, and C. F. Cheng, “Morphology analysis atomic force microscope and light scattering study for random scattering screens,” Acta Phys. Sin.49(7), 1260–1266 (2000) (in Chinese).

Fan, Z. Y.

Z. Y. Fan and A. O. Govorov, “Plasmonic circular dichroism of chiral metal nanoparticle assemblies,” Nano Lett.10(7), 2580–2587 (2010).
[CrossRef] [PubMed]

Ferri, F.

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: Theory,” Phys. Rev. A78(6), 063806 (2008).
[CrossRef]

Garoff, S.

S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B Condens. Matter38(4), 2297–2311 (1988).
[CrossRef] [PubMed]

Gatti, A.

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: Theory,” Phys. Rev. A78(6), 063806 (2008).
[CrossRef]

Giglio, M.

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation function g(r),” Phys. Rev. Lett.85(7), 1416–1419 (2000).
[CrossRef] [PubMed]

Goodman, J. W.

Govorov, A. O.

Z. Y. Fan and A. O. Govorov, “Plasmonic circular dichroism of chiral metal nanoparticle assemblies,” Nano Lett.10(7), 2580–2587 (2010).
[CrossRef] [PubMed]

Han, D.

Hanson, S. G.

Häusler, G.

Hybl, O.

Jakeman, E.

E. Jakeman, “Speckle statistics with a small number of scatters,” Opt. Eng.23(4), 234453 (1984).
[CrossRef]

Jun, Y. C.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9(3), 193–204 (2010).
[CrossRef] [PubMed]

Laverdant, J.

J. Laverdant, S. Buil, B. Berini, and X. Quelin, “Polarization dependent near-field speckle of random gold films,” Phys. Rev. B77(16), 165406 (2008).
[CrossRef]

Lee, H.

Z. W. Liu, Y. Wang, J. Yao, H. Lee, W. Srituravanich, and X. Zhang, “Broad band two-dimensional manipulation of surface plasmons,” Nano Lett.9(1), 462–466 (2009).
[CrossRef] [PubMed]

Liu, D. L.

D. P. Qi, D. L. Liu, S. Y. Teng, N. Y. Zhang, and C. F. Cheng, “Morphology analysis atomic force microscope and light scattering study for random scattering screens,” Acta Phys. Sin.49(7), 1260–1266 (2000) (in Chinese).

Liu, M.

C. F. Cheng, M. Liu, N. Y. Zhang, S. Y. Teng, H. S. Song, and Z. Z. Xu, “M. Liu1, N. Y. Zhang, S. Y. Teng, H. S. Song and Z. Z. Xu, “Speckle intensity correlation in the diffraction region near rough surfaces and simulational experiments for extraction of surface parameters,” Europhys. Lett.65(6), 779–784 (2004).
[CrossRef]

Liu, Z. W.

Z. W. Liu, Y. Wang, J. Yao, H. Lee, W. Srituravanich, and X. Zhang, “Broad band two-dimensional manipulation of surface plasmons,” Nano Lett.9(1), 462–466 (2009).
[CrossRef] [PubMed]

Magatti, D.

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: Theory,” Phys. Rev. A78(6), 063806 (2008).
[CrossRef]

Maksimyak, P. P.

Nieto-Vesperinas, M.

J. A. Sánchez-Gil and M. Nieto-Vesperinas, “Resonance effects in multiple light scattering from statistically rough metallic surfaces,” Phys. Rev. B Condens. Mater.45(15), 8623–8633 (1992).
[CrossRef] [PubMed]

Peuser, J.

Qi, D. P.

D. P. Qi, D. L. Liu, S. Y. Teng, N. Y. Zhang, and C. F. Cheng, “Morphology analysis atomic force microscope and light scattering study for random scattering screens,” Acta Phys. Sin.49(7), 1260–1266 (2000) (in Chinese).

Quelin, X.

J. Laverdant, S. Buil, B. Berini, and X. Quelin, “Polarization dependent near-field speckle of random gold films,” Phys. Rev. B77(16), 165406 (2008).
[CrossRef]

Ren, B.

Z. Q. Tian, B. Ren, and D. Y. Wu, “Surface-enhanced Raman scattering: from noble to transition metals and from rough surfaces to ordered nanostructures,” J. Phys. Chem. B106(37), 9463–9483 (2002).
[CrossRef]

Ryukhtin, V. V.

Sánchez-Gil, J. A.

J. A. Sánchez-Gil and M. Nieto-Vesperinas, “Resonance effects in multiple light scattering from statistically rough metallic surfaces,” Phys. Rev. B Condens. Mater.45(15), 8623–8633 (1992).
[CrossRef] [PubMed]

Scheffold, F.

Schuller, J. A.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9(3), 193–204 (2010).
[CrossRef] [PubMed]

Sinha, S. K.

S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B Condens. Matter38(4), 2297–2311 (1988).
[CrossRef] [PubMed]

Sirota, E. B.

S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B Condens. Matter38(4), 2297–2311 (1988).
[CrossRef] [PubMed]

Skipetrov, S. E.

Song, H. S.

C. F. Cheng, M. Liu, N. Y. Zhang, S. Y. Teng, H. S. Song, and Z. Z. Xu, “M. Liu1, N. Y. Zhang, S. Y. Teng, H. S. Song and Z. Z. Xu, “Speckle intensity correlation in the diffraction region near rough surfaces and simulational experiments for extraction of surface parameters,” Europhys. Lett.65(6), 779–784 (2004).
[CrossRef]

Sorrentini, J.

Srituravanich, W.

Z. W. Liu, Y. Wang, J. Yao, H. Lee, W. Srituravanich, and X. Zhang, “Broad band two-dimensional manipulation of surface plasmons,” Nano Lett.9(1), 462–466 (2009).
[CrossRef] [PubMed]

Stanley, H. B.

S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B Condens. Matter38(4), 2297–2311 (1988).
[CrossRef] [PubMed]

Teng, S. Y.

C. F. Cheng, M. Liu, N. Y. Zhang, S. Y. Teng, H. S. Song, and Z. Z. Xu, “M. Liu1, N. Y. Zhang, S. Y. Teng, H. S. Song and Z. Z. Xu, “Speckle intensity correlation in the diffraction region near rough surfaces and simulational experiments for extraction of surface parameters,” Europhys. Lett.65(6), 779–784 (2004).
[CrossRef]

D. P. Qi, D. L. Liu, S. Y. Teng, N. Y. Zhang, and C. F. Cheng, “Morphology analysis atomic force microscope and light scattering study for random scattering screens,” Acta Phys. Sin.49(7), 1260–1266 (2000) (in Chinese).

Tian, Z. Q.

Z. Q. Tian, B. Ren, and D. Y. Wu, “Surface-enhanced Raman scattering: from noble to transition metals and from rough surfaces to ordered nanostructures,” J. Phys. Chem. B106(37), 9463–9483 (2002).
[CrossRef]

Ulyanov, A. S.

A. S. Ulyanov, “Application of laser speckles for identification of tissues with pathological changes,” Quant. Elec.38(6), 557–562 (2008).
[CrossRef]

Vailati, A.

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation function g(r),” Phys. Rev. Lett.85(7), 1416–1419 (2000).
[CrossRef] [PubMed]

Wang, M.

Wang, Y.

Z. W. Liu, Y. Wang, J. Yao, H. Lee, W. Srituravanich, and X. Zhang, “Broad band two-dimensional manipulation of surface plasmons,” Nano Lett.9(1), 462–466 (2009).
[CrossRef] [PubMed]

Weber, B.

White, J. S.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9(3), 193–204 (2010).
[CrossRef] [PubMed]

Wiesner, B.

Wu, D. Y.

Z. Q. Tian, B. Ren, and D. Y. Wu, “Surface-enhanced Raman scattering: from noble to transition metals and from rough surfaces to ordered nanostructures,” J. Phys. Chem. B106(37), 9463–9483 (2002).
[CrossRef]

Xu, Z. Z.

C. F. Cheng, M. Liu, N. Y. Zhang, S. Y. Teng, H. S. Song, and Z. Z. Xu, “M. Liu1, N. Y. Zhang, S. Y. Teng, H. S. Song and Z. Z. Xu, “Speckle intensity correlation in the diffraction region near rough surfaces and simulational experiments for extraction of surface parameters,” Europhys. Lett.65(6), 779–784 (2004).
[CrossRef]

Yao, J.

Z. W. Liu, Y. Wang, J. Yao, H. Lee, W. Srituravanich, and X. Zhang, “Broad band two-dimensional manipulation of surface plasmons,” Nano Lett.9(1), 462–466 (2009).
[CrossRef] [PubMed]

Zakharov, P.

Zerrad, M.

Zhang, N. Y.

C. F. Cheng, M. Liu, N. Y. Zhang, S. Y. Teng, H. S. Song, and Z. Z. Xu, “M. Liu1, N. Y. Zhang, S. Y. Teng, H. S. Song and Z. Z. Xu, “Speckle intensity correlation in the diffraction region near rough surfaces and simulational experiments for extraction of surface parameters,” Europhys. Lett.65(6), 779–784 (2004).
[CrossRef]

D. P. Qi, D. L. Liu, S. Y. Teng, N. Y. Zhang, and C. F. Cheng, “Morphology analysis atomic force microscope and light scattering study for random scattering screens,” Acta Phys. Sin.49(7), 1260–1266 (2000) (in Chinese).

Zhang, X.

Z. W. Liu, Y. Wang, J. Yao, H. Lee, W. Srituravanich, and X. Zhang, “Broad band two-dimensional manipulation of surface plasmons,” Nano Lett.9(1), 462–466 (2009).
[CrossRef] [PubMed]

Zhou, J.

Acta Phys. Sin. (1)

D. P. Qi, D. L. Liu, S. Y. Teng, N. Y. Zhang, and C. F. Cheng, “Morphology analysis atomic force microscope and light scattering study for random scattering screens,” Acta Phys. Sin.49(7), 1260–1266 (2000) (in Chinese).

Appl. Opt. (3)

Europhys. Lett. (1)

C. F. Cheng, M. Liu, N. Y. Zhang, S. Y. Teng, H. S. Song, and Z. Z. Xu, “M. Liu1, N. Y. Zhang, S. Y. Teng, H. S. Song and Z. Z. Xu, “Speckle intensity correlation in the diffraction region near rough surfaces and simulational experiments for extraction of surface parameters,” Europhys. Lett.65(6), 779–784 (2004).
[CrossRef]

J. Phys. Chem. B (1)

Z. Q. Tian, B. Ren, and D. Y. Wu, “Surface-enhanced Raman scattering: from noble to transition metals and from rough surfaces to ordered nanostructures,” J. Phys. Chem. B106(37), 9463–9483 (2002).
[CrossRef]

Nano Lett. (2)

Z. W. Liu, Y. Wang, J. Yao, H. Lee, W. Srituravanich, and X. Zhang, “Broad band two-dimensional manipulation of surface plasmons,” Nano Lett.9(1), 462–466 (2009).
[CrossRef] [PubMed]

Z. Y. Fan and A. O. Govorov, “Plasmonic circular dichroism of chiral metal nanoparticle assemblies,” Nano Lett.10(7), 2580–2587 (2010).
[CrossRef] [PubMed]

Nat. Mater. (1)

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9(3), 193–204 (2010).
[CrossRef] [PubMed]

Opt. Eng. (1)

E. Jakeman, “Speckle statistics with a small number of scatters,” Opt. Eng.23(4), 234453 (1984).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (2)

R. Cerbino, “Correlations of light in the deep Fresnel region: An extended Van Cittert and Zernike theorem,” Phys. Rev. A75(5), 053815 (2007).
[CrossRef]

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: Theory,” Phys. Rev. A78(6), 063806 (2008).
[CrossRef]

Phys. Rev. B (1)

J. Laverdant, S. Buil, B. Berini, and X. Quelin, “Polarization dependent near-field speckle of random gold films,” Phys. Rev. B77(16), 165406 (2008).
[CrossRef]

Phys. Rev. B Condens. Mater. (1)

J. A. Sánchez-Gil and M. Nieto-Vesperinas, “Resonance effects in multiple light scattering from statistically rough metallic surfaces,” Phys. Rev. B Condens. Mater.45(15), 8623–8633 (1992).
[CrossRef] [PubMed]

Phys. Rev. B Condens. Matter (1)

S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B Condens. Matter38(4), 2297–2311 (1988).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation function g(r),” Phys. Rev. Lett.85(7), 1416–1419 (2000).
[CrossRef] [PubMed]

Quant. Elec. (1)

A. S. Ulyanov, “Application of laser speckles for identification of tissues with pathological changes,” Quant. Elec.38(6), 557–562 (2008).
[CrossRef]

Other (3)

J. W. Goodman, Speckle Phenonmena in Optics: Theory and Applications (Ben Robert & Company, 2007).

Y. P. Zhao, G. C. Wang, and T. M. Lu, Characterization of Amorphous and Crystalline Rough Surface: Principles and Applications, (Academic Press, 2001).

H. Raether, Surface Plasmons on Smooth and Rrough Surfaces and on Gratings (Springer, 1988).

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Figures (7)

Fig. 1
Fig. 1

The experiment system of obtaining the speckle patterns.

Fig. 2
Fig. 2

(a) (Right) The AFM image of the ground glass sample; (b) (Left) The simulated speckle pattern using the AMF image data in (a).

Fig. 3
Fig. 3

Speckle patterns with smaller magnification for the samples No.1-No.6, respectively.

Fig. 4
Fig. 4

Speckle patterns with large magnification for the samples No. 1-No. 6, respectively.

Fig. 5
Fig. 5

Diagrammatic sketches for the speckle formation (a) (Right) on the rough dielectric surface, (b) (Left) on the rough dielectric/metal interface.

Fig. 6
Fig. 6

(a) (Right) Curves of speckle contrast and r. The blue is obtained by calculation on speckle pattern intensity; the green is obtained by fitting the p(I) in Eq. (7) to experimental data; the red is r obtained by the fitting; (b) (Left) Probability density function curves of speckles for all the samples.

Fig. 7
Fig. 7

(a) (Right) The autocorrelation function curves of the global speckles. The inset is the curve and the fit curve for the 100nm sample. (b) (Left) The autocorrelation function curves of the local speckles. The inset is the curve and the fit curve for the 60nm sample.

Tables (2)

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Table 1 The global correlation length δ and fractal exponent α I

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Table 2 The correlation length δ s of the small-scale speckles.

Equations (15)

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H(ρ)=H(ρ)=< [h( r 0 +ρ)h( r 0 )] 2 >=2 w 2 [1exp[ (ρ/ξ) 2α ]
H(ρ)={ 2 w 2 (ρ/ξ) 2α            for  ρ<<ξ, 2 w 2                            for ρ>>ξ 
U(x,y)= 1 4π 1 A exp{jk [r+(m1)h]}×{jk[M/ r 2 +(m1)B/r]+M/ r 3 }d x 1 d y 1
C= [(< I 2 ><I > 2 )/<I > 2 ] 1/ 2 ,
p(I)  d(I/<I>)=1,
p(I)= 1 <I> exp( I <I> )
p(I)= 1 < I s > exp((I+ I c )/< I s >) I 0 (2 I I c /< I s >)
p(I)= 1 < I s > exp[(I/< I s >+r)] I 0 (2 Ir/< I s > )
C= 1+2r /(r+1),
R I (ρ;ρ+d)=<I(ρ)I(ρ+d)>,
γ(d)=<I(ρ)I(ρ+d)>/< I 2 >
γ(d)=(1β)exp[ (d/δ) 2 α I ]+β,
C= 1/(1β)1 ,
γ loc (n1,n2) ( d i )= 1 2m< I 2 > loc j2=1 m j1=1 mi 1 (mi) I( x j1 y j2 )I( x j1 +iΔ y j2 ) + 1 2m< I 2 > loc j1=1 m j2=1 mi 1 (mi) I( x j1 y j2 )I( x j1 y j2 +iΔ),
γ loc ( d i )= 1 M 2 n2=1 M n1=1 M γ loc (n1,n2) ( d i )

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