Abstract

In this work, a linear grating is used to project a periodic light intensity distribution on a rough interface, and the near field transmitted light scattering is studied. It is shown theoretically that the intensity in the Fresnel regime depends on statistical properties of the rough interface and the light intensity period. The self-image contrast exponentially depends on the interface height–height correlation function. The correlation is obtained in terms of multiplication of the self-image number and the period of the light intensity distribution. Therefore, the roughness and the correlation length of the interface can be obtained by determining the contrast of the self-images when the light intensity period is smaller than the interface correlation length. For periods longer than twice the correlation length, the contrast measurements only provide the interface roughness. In experimental studies, the roughness of interfaces is determined by square gratings with periods much longer than the correlation lengths. The rough interfaces are prepared by roughening sheet glass by powders of different grit numbers. The results for different gratings and light wavelengths are quite consistent.

© 2013 Optical Society of America

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2011

2009

2008

2007

2004

T. M. Elfouhaily and C. A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
[CrossRef]

2003

2001

2000

1997

D. J. Whitehouse, “Review article: surface metrology,” Meas. Sci. Technol. 8, 955–972 (1997).
[CrossRef]

1996

1993

1992

Y. Oh, K. Sarabandi, and F. T. Ulaby, “An empirical model and an inversion technique for radar scattering from bare soil surfaces,” IEEE Trans. Geosci. Remote Sens. 30, 370–381 (1992).
[CrossRef]

1991

1990

1987

1984

1982

1977

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269–287 (1977).
[CrossRef]

1976

1961

Amra, C.

Angelsky, O. V.

Beckman, P.

P. Beckman and A. Spizzochino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, 1963).

Bennett, H. E.

Bennett, J. M.

Boreman, G. D.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 2002), pp. 421–430.

Burkovets, D. N.

Chandley, P. J.

P. J. Chandley, “Surface roughness measurements from coherent light scattering,” Opt. Quantum Electron. 8, 323–327 (1976).
[CrossRef]

Chew, W. C.

K. F. Warnick and W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media 11, R1–R30 (2001).
[CrossRef]

Choi, N.

Dainty, J. C.

Dashtdar, M.

M. Dashtdar and M. T. Tavassoly, “Roughness measurement using threshold angle of image formation,” Opt. Eng. 50, 123601 (2011).
[CrossRef]

M. Dashtdar and M. T. Tavassoly, “Redshift and blueshift in the spectra of lights coherently and diffusely scattered from random rough interfaces,” J. Opt. Soc. Am. A 26, 2134–2138 (2009).
[CrossRef]

M. Dashtdar and M. T. Tavassoly, “Determination of height distribution on a rough interface by measuring the coherently transmitted or reflected light intensity,” J. Opt. Soc. Am. A 25, 2509–2517 (2008).
[CrossRef]

M. T. Tavassoly and M. Dashtdar, “Height distribution on a rough plane and specularly diffracted light amplitude are Fourier transform pair,” Opt. Commun. 281, 2397–2405 (2008).
[CrossRef]

Dewees, R. V.

Dogariu, A.

Elfouhaily, T. M.

T. M. Elfouhaily and C. A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
[CrossRef]

Fainman, Y.

Gonzalez-Rodriguez, P.

Guenther, K. H.

Guerin, C. A.

T. M. Elfouhaily and C. A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
[CrossRef]

Hanson, S. G.

Harvey, J. E.

Johnson, L. F.

Kim, A. D.

Krywonos, A.

Lenz, E.

Lu, T.-M.

Y.-P. Zhao, G.-C. Wang, and T.-M. Lu, Characterization of Amorphous and Crystalline Rough Surface—Principles and Applications Vol. 37 of Experimental Methods in the Physical Sciences (Academic, 2001), pp. 133–156.

Maksimyak, P. P.

Marx, E.

Mendez, E. R.

Moran, M. B.

Nee, S.-M. F.

Nee, T.-W.

Nieto-Vesperinas, M.

O’Donnell, K. A.

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Adam Hilger, 1991).

Oh, Y.

Y. Oh, K. Sarabandi, and F. T. Ulaby, “An empirical model and an inversion technique for radar scattering from bare soil surfaces,” IEEE Trans. Geosci. Remote Sens. 30, 370–381 (1992).
[CrossRef]

Porteus, J. O.

Ryukhtin, V. V.

Sanchez-Gil, J. A.

Sant, A. J.

Sarabandi, K.

Y. Oh, K. Sarabandi, and F. T. Ulaby, “An empirical model and an inversion technique for radar scattering from bare soil surfaces,” IEEE Trans. Geosci. Remote Sens. 30, 370–381 (1992).
[CrossRef]

Sassi, I.

Shamir, J.

Sifaoui, M. S.

Spizzochino, A.

P. Beckman and A. Spizzochino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, 1963).

Tavassoly, M. T.

M. Dashtdar and M. T. Tavassoly, “Roughness measurement using threshold angle of image formation,” Opt. Eng. 50, 123601 (2011).
[CrossRef]

M. Dashtdar and M. T. Tavassoly, “Redshift and blueshift in the spectra of lights coherently and diffusely scattered from random rough interfaces,” J. Opt. Soc. Am. A 26, 2134–2138 (2009).
[CrossRef]

M. Dashtdar and M. T. Tavassoly, “Determination of height distribution on a rough interface by measuring the coherently transmitted or reflected light intensity,” J. Opt. Soc. Am. A 25, 2509–2517 (2008).
[CrossRef]

M. T. Tavassoly and M. Dashtdar, “Height distribution on a rough plane and specularly diffracted light amplitude are Fourier transform pair,” Opt. Commun. 281, 2397–2405 (2008).
[CrossRef]

Ulaby, F. T.

Y. Oh, K. Sarabandi, and F. T. Ulaby, “An empirical model and an inversion technique for radar scattering from bare soil surfaces,” IEEE Trans. Geosci. Remote Sens. 30, 370–381 (1992).
[CrossRef]

Vorburger, T. V.

Wang, G.-C.

Y.-P. Zhao, G.-C. Wang, and T.-M. Lu, Characterization of Amorphous and Crystalline Rough Surface—Principles and Applications Vol. 37 of Experimental Methods in the Physical Sciences (Academic, 2001), pp. 133–156.

Warnick, K. F.

K. F. Warnick and W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media 11, R1–R30 (2001).
[CrossRef]

Welford, W. T.

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269–287 (1977).
[CrossRef]

Whitehouse, D. J.

D. J. Whitehouse, “Review article: surface metrology,” Meas. Sci. Technol. 8, 955–972 (1997).
[CrossRef]

Wierer, P. G.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 2002), pp. 421–430.

Zhao, Y.-P.

Y.-P. Zhao, G.-C. Wang, and T.-M. Lu, Characterization of Amorphous and Crystalline Rough Surface—Principles and Applications Vol. 37 of Experimental Methods in the Physical Sciences (Academic, 2001), pp. 133–156.

Appl. Opt.

J. M. Bennett, “Measurement of the rms roughness, autocovariance and other statistical properties of optical surfaces using a FECO scanning interferometer,” Appl. Opt. 15, 2705–2721 (1976).
[CrossRef]

Y. Fainman, E. Lenz, and J. Shamir, “Optical profilometer: a new method for high sensitivity and wide dynamic range,” Appl. Opt. 21, 3200–3208 (1982).
[CrossRef]

K. H. Guenther, P. G. Wierer, and J. M. Bennett, “Surface roughness measurements of low-scatter mirrors and roughness standards,” Appl. Opt. 23, 3820–3836 (1984).
[CrossRef]

O. V. Angelsky and P. P. Maksimyak, “Optical diagnostics of random phase objects,” Appl. Opt. 29, 2894–2898 (1990).
[CrossRef]

E. Marx and T. V. Vorburger, “Direct and inverse problems for light scattered by rough surfaces,” Appl. Opt. 29, 3613–3626 (1990).
[CrossRef]

K. A. O’Donnell, “Effect of finite stylus width in surface contact profilomery,” Appl. Opt. 32, 4922–4928 (1993).
[CrossRef]

C. Amra, “From light scattering to the microstructure of thin-film multilayers,” Appl. Opt. 32, 5481–5491 (1993).
[CrossRef]

S.-M. F. Nee, R. V. Dewees, T.-W. Nee, L. F. Johnson, and M. B. Moran, “Slope distribution of a rough surface measured by transmission scattering and polarization,” Appl. Opt. 39, 1561–1569 (2000).
[CrossRef]

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

O. V. Angelsky, D. N. Burkovets, P. P. Maksimyak, and S. G. Hanson, “Applicability of the singular-optics concept for diagnostics of random and fractal rough surfaces,” Appl. Opt. 42, 4529–4540 (2003).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

Y. Oh, K. Sarabandi, and F. T. Ulaby, “An empirical model and an inversion technique for radar scattering from bare soil surfaces,” IEEE Trans. Geosci. Remote Sens. 30, 370–381 (1992).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Meas. Sci. Technol.

D. J. Whitehouse, “Review article: surface metrology,” Meas. Sci. Technol. 8, 955–972 (1997).
[CrossRef]

Opt. Commun.

M. T. Tavassoly and M. Dashtdar, “Height distribution on a rough plane and specularly diffracted light amplitude are Fourier transform pair,” Opt. Commun. 281, 2397–2405 (2008).
[CrossRef]

Opt. Eng.

M. Dashtdar and M. T. Tavassoly, “Roughness measurement using threshold angle of image formation,” Opt. Eng. 50, 123601 (2011).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

P. J. Chandley, “Surface roughness measurements from coherent light scattering,” Opt. Quantum Electron. 8, 323–327 (1976).
[CrossRef]

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269–287 (1977).
[CrossRef]

Waves Random Media

T. M. Elfouhaily and C. A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
[CrossRef]

K. F. Warnick and W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media 11, R1–R30 (2001).
[CrossRef]

Other

P. Beckman and A. Spizzochino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, 1963).

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Adam Hilger, 1991).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 2002), pp. 421–430.

Y.-P. Zhao, G.-C. Wang, and T.-M. Lu, Characterization of Amorphous and Crystalline Rough Surface—Principles and Applications Vol. 37 of Experimental Methods in the Physical Sciences (Academic, 2001), pp. 133–156.

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Figures (6)

Fig. 1.
Fig. 1.

Rough interface and a grating with its direction parallel to the η axis located in the ξη plane. The observation plane is located at distance z from the ξη plane.

Fig. 2.
Fig. 2.

(a) Fringes in terms of distances between a grating with a period of 20 μm (50lines/mm) and the observation plane for λ=633nm and I0=255. The grating is located on the rough interfaces with σ=0.2μm and the Gaussian correlation function with (b) λ0=2d, (c) λ0=d, and (d) λ0=0.5d.

Fig. 3.
Fig. 3.

Normalized intensity (I(x,y)/I0) for x=0 corresponding to Fig. 2.

Fig. 4.
Fig. 4.

Contrast of the Talbot images versus the number of self-images for the gratings with five different periods on the rough interface with σ=0.2μm and λ0=40μm. The incident light wavelength is 633 nm.

Fig. 5.
Fig. 5.

Experimental setup. The sample rough interface is located at the Talbot distance of a square grating. A CCD records the intensity distribution at the Talbot distances of the periodic light distribution on the rough interface.

Fig. 6.
Fig. 6.

Intensity distribution at the first Talbot distance, (a) for reference and for the samples with roughnesses specified by (b) 3000, (c) 1500, and (d) 1000 grit numbers; (a)–(d) the corresponding average intensity distributions along the grating lines.

Tables (3)

Tables Icon

Table 1. Sample Roughnesses, σ, and Correlation Lengths, λ0, Obtained by AFM

Tables Icon

Table 2. Samples Roughnesses, σ, Obtained by Gratings of Different Periods, d, for Incident Light of Wavelength 633 nm

Tables Icon

Table 3. Roughnesses, σ, of the Sample Corresponding to a 3000 Grit Number Obtained by Gratings of Different Periods, d, for Incident Light of Wavelength 532 nm

Equations (16)

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t(ξ,η)=aexp[ik(n1)h],
g(ξ,η)=12[1+v0cos(2πξd)],
ψ(ξ,η)=t(ξ,η)g(ξ,η).
ψ(x,y)=eikziλzt(ξ,η)g(ξ,η)exp{ik2z[(xξ)2+(yη)2]}dξdη,
I(x,y)=ψ*(x,y)ψ(x,y)=a24λ2z2exp[ik(n1)(h(ξ,η)h(ξ,η))]×[1+v02cos(2πξd)cos(2πξd)+v0cos(2πξd)+v0cos(2πξd)]×exp{+ik2z[(xξ)2+(yη)2]}×exp{ik2z[(xξ)2+(yη)2]}dξdηdξdη.
I(x,y)=a216λ2z2χ(X,X){1+v02cos[π(X+Y)d]cos[π(XY)d]+v0cos[π(X+Y)d]+v0cos[π(XY)d]}×exp[+ik2z(XY+XY2Xx2Xy)]dXdYdXdY.
I(x,y)=I04{χ(0,0)+2v0χ(λzd,0)cos(2πxd)cos(πλzd2)+v022[χ(0,0)+χ(2λzd,0)cos(4πxd)]},
I(x,y)=I04{1+2v0χ(λzd)cos(2πxd)cos(πλzd2)+v022[1+χ(2λzd)cos(4πxd)]}.
χ(X)=exp[12k2(n1)2H(X)],
H(X)=2σ2(1C(X)),
I(x,y)=I04{1±2v0χ(Nd)cos(2πxd)+v022[1+χ(2Nd)cos(4πxd)]}.
V=4v0exp[12k2(n1)2H(Nd)]2+v02+v02exp[12k2(n1)2H(2Nd)].
V=4v0exp[k2(n1)2σ2]2+v02+v02exp[k2(n1)2σ2].
g(ξ,η)=12[1+v0n=1cncos(2nπξd)],
I(x,y)=I04{1+2v0ncnχ(nλzd)cos(2nπxd)cos(n2πλzd2)+v022nmcncmcos(πλz(m2n2)d2)×[χ(λz(mn)d)cos(2πx(mn)d)+χ(λz(m+n)d)cos(2πx(m+n)d)]}.
V=4v0exp[k2(n1)2σ2]2+2v02.

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