Abstract

The scattering of polarized light from a dielectric film sandwiched between two different semi-infinite dielectric media is studied experimentally and theoretically. The illuminated interface is planar, while the back interface is a two-dimensional randomly rough interface. We consider here only the case in which the medium of incidence is optically more dense than the substrate, in which case effects due to the presence of a critical angle for total internal reflection occur. A reduced Rayleigh equation for the scattering amplitudes is solved by a rigorous, purely numerical, nonperturbative approach. The solutions are used to calculate the reflectivity of the structure and the mean differential reflection coefficient. Optical analogues of Yoneda peaks are present in the results obtained. The computational results are compared with experimental data for the in-plane mean differential reflection coefficient, and good agreement between theory and experiment is found.

© 2016 Optical Society of America

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References

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  1. J. Nakayama, H. Ogura, and B. Matsumoto, “A probabilistic theory of scattering from a random rough surface,” Radio Sci. 15, 1049–1057 (1980).
    [Crossref]
  2. T. Kawanishi, H. Ogura, and Z. L. Wang, “Scattering of an electromagnetic wave from a slightly random dielectric surface: Yoneda peak and Brewster angle in incoherent scattering,” Wave. Random Media 7, 351–384 (1997).
    [Crossref]
  3. Y. Yoneda, “Anomalous Surface Reflection of X Rays,” Phys. Rev. 131, 2010–2013 (1963).
    [Crossref]
  4. A. Soubret, G. Berginc, and C. Bourrely, “Application of reduced Rayleigh equations to electromagnetic wave scattering by two-dimensional randomly rough surfaces,” Phys. Rev. B 63, 245411 (2001).
    [Crossref]
  5. T. Nordam, P. A. Letnes, and I. Simonsen, “Numerical simulations of scattering of light from two-dimensional rough surfaces using the reduced Rayleigh equation,” Front. Phys. 1, 8 (2013).
    [Crossref]
  6. Ø. S. Hetland, A. A. Maradudin, T. Nordam, and I. Simonsen, “Numerical studies of the scattering of light from a two-dimensional randomly rough interface between two dielectric media,” Phys. Rev. A 93, 053819 (2016).
    [Crossref]
  7. E. R. Méndez, G. D. Jiménez, and A. A. Maradudin, “A simple model of a one-dimensional, randomly rough, non-Gaussian surface,” Proc. SPIE 9961, 99610D (2016).
  8. P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
    [Crossref]
  9. B. E. Warren and J. S. Clarke, “Interpretation of the Anomalous Surface Reflection of X Rays,” J. Appl. Phys. 36, 324–325 (1965).
    [Crossref]
  10. G. H. Vineyard, “Grazing-incidence diffraction and the distorted-wave approximation for the study of surfaces,” Phys. Rev. B 26, 4146–4159 (1982).
    [Crossref]
  11. S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
    [Crossref]
  12. For reasons of convenience, a negative value of the polar angle of scattering, θs, is used to denote the case q̂‖ = − k̂‖.

2016 (2)

Ø. S. Hetland, A. A. Maradudin, T. Nordam, and I. Simonsen, “Numerical studies of the scattering of light from a two-dimensional randomly rough interface between two dielectric media,” Phys. Rev. A 93, 053819 (2016).
[Crossref]

E. R. Méndez, G. D. Jiménez, and A. A. Maradudin, “A simple model of a one-dimensional, randomly rough, non-Gaussian surface,” Proc. SPIE 9961, 99610D (2016).

2013 (1)

T. Nordam, P. A. Letnes, and I. Simonsen, “Numerical simulations of scattering of light from two-dimensional rough surfaces using the reduced Rayleigh equation,” Front. Phys. 1, 8 (2013).
[Crossref]

2001 (1)

A. Soubret, G. Berginc, and C. Bourrely, “Application of reduced Rayleigh equations to electromagnetic wave scattering by two-dimensional randomly rough surfaces,” Phys. Rev. B 63, 245411 (2001).
[Crossref]

1997 (1)

T. Kawanishi, H. Ogura, and Z. L. Wang, “Scattering of an electromagnetic wave from a slightly random dielectric surface: Yoneda peak and Brewster angle in incoherent scattering,” Wave. Random Media 7, 351–384 (1997).
[Crossref]

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 38, 2297–2311 (1988).
[Crossref]

1982 (1)

G. H. Vineyard, “Grazing-incidence diffraction and the distorted-wave approximation for the study of surfaces,” Phys. Rev. B 26, 4146–4159 (1982).
[Crossref]

1980 (1)

J. Nakayama, H. Ogura, and B. Matsumoto, “A probabilistic theory of scattering from a random rough surface,” Radio Sci. 15, 1049–1057 (1980).
[Crossref]

1978 (1)

P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
[Crossref]

1965 (1)

B. E. Warren and J. S. Clarke, “Interpretation of the Anomalous Surface Reflection of X Rays,” J. Appl. Phys. 36, 324–325 (1965).
[Crossref]

1963 (1)

Y. Yoneda, “Anomalous Surface Reflection of X Rays,” Phys. Rev. 131, 2010–2013 (1963).
[Crossref]

Berginc, G.

A. Soubret, G. Berginc, and C. Bourrely, “Application of reduced Rayleigh equations to electromagnetic wave scattering by two-dimensional randomly rough surfaces,” Phys. Rev. B 63, 245411 (2001).
[Crossref]

Bourrely, C.

A. Soubret, G. Berginc, and C. Bourrely, “Application of reduced Rayleigh equations to electromagnetic wave scattering by two-dimensional randomly rough surfaces,” Phys. Rev. B 63, 245411 (2001).
[Crossref]

Clarke, J. S.

B. E. Warren and J. S. Clarke, “Interpretation of the Anomalous Surface Reflection of X Rays,” J. Appl. Phys. 36, 324–325 (1965).
[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 38, 2297–2311 (1988).
[Crossref]

Gray, P. F.

P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
[Crossref]

Hetland, Ø. S.

Ø. S. Hetland, A. A. Maradudin, T. Nordam, and I. Simonsen, “Numerical studies of the scattering of light from a two-dimensional randomly rough interface between two dielectric media,” Phys. Rev. A 93, 053819 (2016).
[Crossref]

Jiménez, G. D.

E. R. Méndez, G. D. Jiménez, and A. A. Maradudin, “A simple model of a one-dimensional, randomly rough, non-Gaussian surface,” Proc. SPIE 9961, 99610D (2016).

Kawanishi, T.

T. Kawanishi, H. Ogura, and Z. L. Wang, “Scattering of an electromagnetic wave from a slightly random dielectric surface: Yoneda peak and Brewster angle in incoherent scattering,” Wave. Random Media 7, 351–384 (1997).
[Crossref]

Letnes, P. A.

T. Nordam, P. A. Letnes, and I. Simonsen, “Numerical simulations of scattering of light from two-dimensional rough surfaces using the reduced Rayleigh equation,” Front. Phys. 1, 8 (2013).
[Crossref]

Maradudin, A. A.

Ø. S. Hetland, A. A. Maradudin, T. Nordam, and I. Simonsen, “Numerical studies of the scattering of light from a two-dimensional randomly rough interface between two dielectric media,” Phys. Rev. A 93, 053819 (2016).
[Crossref]

E. R. Méndez, G. D. Jiménez, and A. A. Maradudin, “A simple model of a one-dimensional, randomly rough, non-Gaussian surface,” Proc. SPIE 9961, 99610D (2016).

Matsumoto, B.

J. Nakayama, H. Ogura, and B. Matsumoto, “A probabilistic theory of scattering from a random rough surface,” Radio Sci. 15, 1049–1057 (1980).
[Crossref]

Méndez, E. R.

E. R. Méndez, G. D. Jiménez, and A. A. Maradudin, “A simple model of a one-dimensional, randomly rough, non-Gaussian surface,” Proc. SPIE 9961, 99610D (2016).

Nakayama, J.

J. Nakayama, H. Ogura, and B. Matsumoto, “A probabilistic theory of scattering from a random rough surface,” Radio Sci. 15, 1049–1057 (1980).
[Crossref]

Nordam, T.

Ø. S. Hetland, A. A. Maradudin, T. Nordam, and I. Simonsen, “Numerical studies of the scattering of light from a two-dimensional randomly rough interface between two dielectric media,” Phys. Rev. A 93, 053819 (2016).
[Crossref]

T. Nordam, P. A. Letnes, and I. Simonsen, “Numerical simulations of scattering of light from two-dimensional rough surfaces using the reduced Rayleigh equation,” Front. Phys. 1, 8 (2013).
[Crossref]

Ogura, H.

T. Kawanishi, H. Ogura, and Z. L. Wang, “Scattering of an electromagnetic wave from a slightly random dielectric surface: Yoneda peak and Brewster angle in incoherent scattering,” Wave. Random Media 7, 351–384 (1997).
[Crossref]

J. Nakayama, H. Ogura, and B. Matsumoto, “A probabilistic theory of scattering from a random rough surface,” Radio Sci. 15, 1049–1057 (1980).
[Crossref]

Simonsen, I.

Ø. S. Hetland, A. A. Maradudin, T. Nordam, and I. Simonsen, “Numerical studies of the scattering of light from a two-dimensional randomly rough interface between two dielectric media,” Phys. Rev. A 93, 053819 (2016).
[Crossref]

T. Nordam, P. A. Letnes, and I. Simonsen, “Numerical simulations of scattering of light from two-dimensional rough surfaces using the reduced Rayleigh equation,” Front. Phys. 1, 8 (2013).
[Crossref]

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 38, 2297–2311 (1988).
[Crossref]

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 38, 2297–2311 (1988).
[Crossref]

Soubret, A.

A. Soubret, G. Berginc, and C. Bourrely, “Application of reduced Rayleigh equations to electromagnetic wave scattering by two-dimensional randomly rough surfaces,” Phys. Rev. B 63, 245411 (2001).
[Crossref]

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 38, 2297–2311 (1988).
[Crossref]

Vineyard, G. H.

G. H. Vineyard, “Grazing-incidence diffraction and the distorted-wave approximation for the study of surfaces,” Phys. Rev. B 26, 4146–4159 (1982).
[Crossref]

Wang, Z. L.

T. Kawanishi, H. Ogura, and Z. L. Wang, “Scattering of an electromagnetic wave from a slightly random dielectric surface: Yoneda peak and Brewster angle in incoherent scattering,” Wave. Random Media 7, 351–384 (1997).
[Crossref]

Warren, B. E.

B. E. Warren and J. S. Clarke, “Interpretation of the Anomalous Surface Reflection of X Rays,” J. Appl. Phys. 36, 324–325 (1965).
[Crossref]

Yoneda, Y.

Y. Yoneda, “Anomalous Surface Reflection of X Rays,” Phys. Rev. 131, 2010–2013 (1963).
[Crossref]

Front. Phys. (1)

T. Nordam, P. A. Letnes, and I. Simonsen, “Numerical simulations of scattering of light from two-dimensional rough surfaces using the reduced Rayleigh equation,” Front. Phys. 1, 8 (2013).
[Crossref]

J. Appl. Phys. (1)

B. E. Warren and J. S. Clarke, “Interpretation of the Anomalous Surface Reflection of X Rays,” J. Appl. Phys. 36, 324–325 (1965).
[Crossref]

Opt. Acta (1)

P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
[Crossref]

Phys. Rev. (1)

Y. Yoneda, “Anomalous Surface Reflection of X Rays,” Phys. Rev. 131, 2010–2013 (1963).
[Crossref]

Phys. Rev. A (1)

Ø. S. Hetland, A. A. Maradudin, T. Nordam, and I. Simonsen, “Numerical studies of the scattering of light from a two-dimensional randomly rough interface between two dielectric media,” Phys. Rev. A 93, 053819 (2016).
[Crossref]

Phys. Rev. B (3)

A. Soubret, G. Berginc, and C. Bourrely, “Application of reduced Rayleigh equations to electromagnetic wave scattering by two-dimensional randomly rough surfaces,” Phys. Rev. B 63, 245411 (2001).
[Crossref]

G. H. Vineyard, “Grazing-incidence diffraction and the distorted-wave approximation for the study of surfaces,” Phys. Rev. B 26, 4146–4159 (1982).
[Crossref]

S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[Crossref]

Proc. SPIE (1)

E. R. Méndez, G. D. Jiménez, and A. A. Maradudin, “A simple model of a one-dimensional, randomly rough, non-Gaussian surface,” Proc. SPIE 9961, 99610D (2016).

Radio Sci. (1)

J. Nakayama, H. Ogura, and B. Matsumoto, “A probabilistic theory of scattering from a random rough surface,” Radio Sci. 15, 1049–1057 (1980).
[Crossref]

Wave. Random Media (1)

T. Kawanishi, H. Ogura, and Z. L. Wang, “Scattering of an electromagnetic wave from a slightly random dielectric surface: Yoneda peak and Brewster angle in incoherent scattering,” Wave. Random Media 7, 351–384 (1997).
[Crossref]

Other (1)

For reasons of convenience, a negative value of the polar angle of scattering, θs, is used to denote the case q̂‖ = − k̂‖.

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Figures (7)

Fig. 1
Fig. 1 Schematic diagram of the sample geometry.
Fig. 2
Fig. 2 The in-plane angular dependence of 〈∂Rαα/∂Ωsincoh for a set of θ0 obtained by the numerical approach outlined in the text. The open symbols represents the data points obtained in the simulations, while the lines connecting them, are only included as a guide to the eye. Notice how the amplitudes of the data vary from panel to panel.
Fig. 3
Fig. 3 Same as Fig. 2 but now presenting experimental measurements. The data sets in each panel were scaled by a common factor so that a unit amplitude corresponds to the maximum value of the corresponding numerical simulation results presented in Fig. 2.
Fig. 4
Fig. 4 Contour maps of the in-plane dependence of the raw simulation data for the co-polarized mean DRC as functions of θs and θ0. Figure 4(a) represents the contour map of 〈∂Rpp/∂Ωsincoh and its inset details the behavior around (θs, θ0) = (θc, θc). In Fig. 4(b) the logarithm of the same data, log〈∂Rpp/∂Ωsincoh, are presented but over the full range of polar scattering angles −90° < θs < 90°. The vertical dashed center line (black) indicates θs = 0°.
Fig. 5
Fig. 5 Photographs showing the spatial intensity distributions formed on the rough aluminum screen [see Fig. 6] for a set of polar angles of incidence, θ0 = θs, as indicated in the figure.
Fig. 6
Fig. 6 Schematics of the geometry used to take the photographs presented in Fig. 5.
Fig. 7
Fig. 7 The reflectivity, α (θ0), of the scattering system studied. The dashed lines denotes the Fresnel reflection coefficient, i.e. the reflectivity of a corresponding film with planar surfaces.

Equations (18)

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E ( x | ω ) inc = [ e ^ p ( i ) ( k ) E 0 p ( k ) + e ^ s ( i ) ( k ) E 0 s ( k ) ] exp [ i k x i α 1 ( k ) x 3 ]
E ( x | ω ) sc = d 2 q ( 2 π ) 2 [ e ^ p ( s ) ( q ) A p ( q ) + e ^ s ( s ) ( q ) A s ( q ) ] exp [ i q x + i α 1 ( q ) x 3 ] ,
e ^ p ( i ) ( k ) = k ^ α 1 ( k ) + x ^ 3 k ε 1 ω c
e ^ s ( i ) ( k ) = k ^ × x ^ 3
e ^ p ( s ) ( q ) = q ^ α 1 ( q ) + x ^ 3 q ε 1 ω c
e ^ s ( s ) ( q ) = q ^ × x ^ 3 .
A α ( q ) = β R α β ( q | k ) E 0 β ( k ) .
d 2 q ( 2 π ) 2 M + ( p | q ) R ( q | k ) = M ( p | k ) ,
R ( q | k ) = ( R p p ( q | k ) R p s ( q | k ) R s p ( q | k ) R s s ( q | k ) )
M ± ( p | q ) = ( M p p ± ( p | q ) M p s ± ( p | q ) M s p ± ( p | q ) M s s ± ( p | q ) ) ,
M p p ± ( p | q ) = 1 α 2 ( q ) { I ( p | q ) [ p q + α 3 ( p ) p ^ q ^ α 2 ( q ) ] [ ε 1 α 2 ( q ) ± ε 2 α 1 ( q ) ] + I + ( p | q ) [ p q α 3 ( p ) p ^ q ^ α 2 ( q ) ] [ ε 1 α 2 ( q ) ε 2 α 1 ( q ) ] }
M s p ± ( p | q ) = ε 3 ω c [ p ^ × q ^ ] 3 × { I ( p | q ) [ ε 1 α 2 ( q ) ± ε 2 α 1 ( q ) ] I + ( p | q ) [ ε 1 α 2 ( q ) ε 2 α 1 ( q ) ] }
M p s ± ( p | q ) = ε 1 ε 2 ω c [ p ^ × q ^ ] 3 α 3 ( p ) α 2 ( q ) × { I ( p | q ) [ α 2 ( q ) ± α 1 ( q ) ] + I + ( p | q ) [ α 2 ( q ) α 1 ( q ) ] }
M p s ± ( p | q ) = ε 1 ε 2 ε 3 ( ω c ) 2 p ^ q ^ 1 α 2 ( q ) × { I ( p | q ) [ α 2 ( q ) ± α 1 ( q ) ] + I + ( p | q ) [ α 2 ( q ) α 1 ( q ) ] } .
I ± ( p | q ) = exp { i [ α 3 ( p ) ± α 2 ( q ) ] d } I ( α 3 ( p ) ± α 2 ( q ) | p q | ) α 3 ( p ) ± α 2 ( q ) ,
I ( γ | Q ) = d 2 x exp [ i γ ζ ( x ) ] exp ( i Q x ) .
R α β ( q | k ) Ω s incoh = ε 1 S ( ω 2 π c ) 2 cos 2 θ s cos θ 0 [ | R α β ( q | k ) | 2 | R α β ( q | k ) | 2 ] .
α ( θ 0 ) = | R α ( k ) | 2 = | R α ( ε 1 ( ω / c ) sin θ 0 ) | 2 ,

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