Abstract

The light scattering of rough metallic surfaces with roughness levels ranging from a few to several hundred nanometers is modeled and compared to experimental data. Different modeling approaches such as the classical Rayleigh-Rice vector perturbation theory and the new Generalized Harvey-Shack theory are used and critically assessed with respect to ranges of validity, accuracy, and practicability. Based on theoretical calculations and comparisons with Rigorous Coupled Wave Analysis for sinusoidal phase gratings, it is demonstrated that the approximate scatter models yield surprisingly accurate results and at the same time provide insight into light scattering phenomena. For stochastically rough metal surfaces, the predicted angles resolved scattering is compared to experimental results at 325 nm, 532 nm, and 1064 nm. In addition, the possibilities of retrieving roughness information from measured scattering data for different roughness regimes are discussed.

© 2011 OSA

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References

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  1. E. L. Church and P. Z. Takacs, “Specification of surface figure and finish in terms of system performance,” Appl. Opt. 32(19), 3344–3353 (1993).
    [CrossRef] [PubMed]
  2. J. E. Harvey, K. L. Lewotsky, and A. Kotha, “Effects of surface scatter on the optical performance of x-ray synchrotron beam-line mirrors,” Appl. Opt. 34(16), 3024–3032 (1995).
    [CrossRef] [PubMed]
  3. A. Duparré, “Scattering from surfaces and thin films,” in Encyclopedia of Modern Optics, B. D. Guenther, D. G. Steel, and L. Bayvel, eds. (Elsevier, 2004).
  4. C. Rockstuhl, S. Fahr, K. Bittkau, T. Beckers, R. Carius, F.-J. Haug, T. Söderström, C. Ballif, and F. Lederer, “Comparison and optimization of randomly textured surfaces in thin-film solar cells,” Opt. Express 18(S3Suppl 3), A335–A341 (2010).
    [CrossRef] [PubMed]
  5. S. Schröder, A. Duparré, K. Füchsel, N. Kaiser, A. Tünnermann, and J. E. Harvey, “Scattering of Roughened TCO Films - Modeling and Measurement,” in Optical Interference Coatings, OSA Technical Digest (Optical Society of America, 2010), paper ThD3.
  6. M. Flemming, L. Coriand, and A. Duparré, “Ultra-hydrophobicity through stochastic surface roughness,” J. Adhes. Sci. Technol. 23(3), 381–400 (2009).
    [CrossRef]
  7. S. Schröder, A. Duparré, “Finish assessment of complex surfaces by advanced light scattering techniques,” Proc. SPIE 7102, paper 0F (2008).
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    [CrossRef]
  9. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, (Pergamon Press, 1963).
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    [CrossRef]
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  12. J. M. Elson and J. M. Bennett, “Vector scattering theory,” Opt. Eng. 18, 116–124 (1979).
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    [CrossRef]
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  20. A. Duparré, J. Ferre-Borrull, S. Gliech, G. Notni, J. Steinert, and J. M. Bennett, “Surface characterization techniques for determining the root-mean-square roughness and power spectral densities of optical components,” Appl. Opt. 41(1), 154–171 (2002).
    [CrossRef] [PubMed]
  21. “Optics and optical instruments-test methods for radiation scattered by optical components,” ISO 13696:2002 (International Organization for Standardization, 2002).
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    [CrossRef]
  23. M. Trost, S. Schröder, T. Feigl, A. Duparré, and A. Tünnermann, “Influence of the substrate finish and thin film roughness on the optical performance of Mo/Si multilayers,” Appl. Opt. 50(9), C148–C153 (2011).
    [CrossRef] [PubMed]
  24. A. Krywonos, “Predicting surface scatter using a linear systems formulation of non-paraxial scalar diffraction,” Ph.D. Dissertation, College of Optics and Photonics, University of Central Florida (2006).
  25. J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough surfaces,” Proc. SPIE 7426, 74260I, 74260I-9 (2009).
    [CrossRef]
  26. A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” Accepted for publication in J. Opt. Soc. Am. A (March 25, 2011)
  27. J. C. Stover, “Experimental confirmation of the Rayleigh-Rice obliquity factor,” Proc. SPIE 7792, 77920J, 77920J-5 (2010).
    [CrossRef]
  28. S. Schröder, T. Herffurth, H. Blaschke, and A. Duparré, “Angle-resolved scattering: an effective method for characterizing thin-film coatings,” Appl. Opt. 50(9), C164–C171 (2011).
    [CrossRef] [PubMed]
  29. A. von Finck, M. Hauptvogel, and A. Duparré, “Instrument for close-to-process light scatter measurements of thin film coatings and substrates,” in Optical Interference Coatings, OSA Technical Digest (Optical Society of America, 2010), paper ThD4.
  30. J. E. Harvey, A. Krywonos, and D. Bogunovic, “Nonparaxial scalar treatment of sinusoidal phase gratings,” J. Opt. Soc. Am. A 23(4), 858–865 (2006).
    [CrossRef]

2011 (3)

2010 (3)

C. Rockstuhl, S. Fahr, K. Bittkau, T. Beckers, R. Carius, F.-J. Haug, T. Söderström, C. Ballif, and F. Lederer, “Comparison and optimization of randomly textured surfaces in thin-film solar cells,” Opt. Express 18(S3Suppl 3), A335–A341 (2010).
[CrossRef] [PubMed]

J. C. Stover, “Experimental confirmation of the Rayleigh-Rice obliquity factor,” Proc. SPIE 7792, 77920J, 77920J-5 (2010).
[CrossRef]

J. E. Harvey, N. Choi, and A. Krywonos, “Scattering from moderately rough interfaces between two arbitrary media,” Proc. SPIE 7794, 77940V, 77940V-11 (2010).
[CrossRef]

2009 (2)

J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough surfaces,” Proc. SPIE 7426, 74260I, 74260I-9 (2009).
[CrossRef]

M. Flemming, L. Coriand, and A. Duparré, “Ultra-hydrophobicity through stochastic surface roughness,” J. Adhes. Sci. Technol. 23(3), 381–400 (2009).
[CrossRef]

2007 (1)

J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattering angles,” Opt. Eng. 46(7), 078002 (2007).
[CrossRef]

2006 (1)

2004 (1)

T. M. Elfouhaily and C. A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14(4), R1–R40 (2004).
[CrossRef]

2002 (1)

1995 (1)

1993 (2)

1984 (1)

E. L. Church, “Statistical effects in the measurement and characterization of smooth scattering surfaces,” Proc. SPIE 511, 18 (1984).

1983 (1)

1979 (2)

E. L. Church, H. A. Jenkinson, and J. M. Zavada, “Relationship #res,” Opt. Eng. 18, 125 (1979).

J. M. Elson and J. M. Bennett, “Vector scattering theory,” Opt. Eng. 18, 116–124 (1979).

1951 (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4(2-3), 351–378 (1951).
[CrossRef]

Ballif, C.

Beckers, T.

Bennett, J. M.

Bittkau, K.

Blaschke, H.

Bogunovic, D.

Carius, R.

Choi, N.

A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” Accepted for publication in J. Opt. Soc. Am. A (March 25, 2011)

J. E. Harvey, N. Choi, and A. Krywonos, “Scattering from moderately rough interfaces between two arbitrary media,” Proc. SPIE 7794, 77940V, 77940V-11 (2010).
[CrossRef]

J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough surfaces,” Proc. SPIE 7426, 74260I, 74260I-9 (2009).
[CrossRef]

Church, E. L.

E. L. Church and P. Z. Takacs, “Specification of surface figure and finish in terms of system performance,” Appl. Opt. 32(19), 3344–3353 (1993).
[CrossRef] [PubMed]

E. L. Church, “Statistical effects in the measurement and characterization of smooth scattering surfaces,” Proc. SPIE 511, 18 (1984).

E. L. Church, H. A. Jenkinson, and J. M. Zavada, “Relationship #res,” Opt. Eng. 18, 125 (1979).

Coriand, L.

M. Flemming, L. Coriand, and A. Duparré, “Ultra-hydrophobicity through stochastic surface roughness,” J. Adhes. Sci. Technol. 23(3), 381–400 (2009).
[CrossRef]

Duparré, 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(4), R1–R40 (2004).
[CrossRef]

Elson, J. M.

J. M. Elson and J. M. Bennett, “Vector scattering theory,” Opt. Eng. 18, 116–124 (1979).

Fahr, S.

Feigl, T.

Ferre-Borrull, J.

Flemming, M.

M. Flemming, L. Coriand, and A. Duparré, “Ultra-hydrophobicity through stochastic surface roughness,” J. Adhes. Sci. Technol. 23(3), 381–400 (2009).
[CrossRef]

Gaylord, T. K.

Gliech, S.

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(4), R1–R40 (2004).
[CrossRef]

Harvey, J. E.

A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” Accepted for publication in J. Opt. Soc. Am. A (March 25, 2011)

J. E. Harvey, N. Choi, and A. Krywonos, “Scattering from moderately rough interfaces between two arbitrary media,” Proc. SPIE 7794, 77940V, 77940V-11 (2010).
[CrossRef]

J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough surfaces,” Proc. SPIE 7426, 74260I, 74260I-9 (2009).
[CrossRef]

J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattering angles,” Opt. Eng. 46(7), 078002 (2007).
[CrossRef]

J. E. Harvey, A. Krywonos, and D. Bogunovic, “Nonparaxial scalar treatment of sinusoidal phase gratings,” J. Opt. Soc. Am. A 23(4), 858–865 (2006).
[CrossRef]

J. E. Harvey, K. L. Lewotsky, and A. Kotha, “Effects of surface scatter on the optical performance of x-ray synchrotron beam-line mirrors,” Appl. Opt. 34(16), 3024–3032 (1995).
[CrossRef] [PubMed]

Haug, F.-J.

Herffurth, T.

Jenkinson, H. A.

E. L. Church, H. A. Jenkinson, and J. M. Zavada, “Relationship #res,” Opt. Eng. 18, 125 (1979).

Kotha, A.

Krywonos, A.

A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” Accepted for publication in J. Opt. Soc. Am. A (March 25, 2011)

J. E. Harvey, N. Choi, and A. Krywonos, “Scattering from moderately rough interfaces between two arbitrary media,” Proc. SPIE 7794, 77940V, 77940V-11 (2010).
[CrossRef]

J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough surfaces,” Proc. SPIE 7426, 74260I, 74260I-9 (2009).
[CrossRef]

J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattering angles,” Opt. Eng. 46(7), 078002 (2007).
[CrossRef]

J. E. Harvey, A. Krywonos, and D. Bogunovic, “Nonparaxial scalar treatment of sinusoidal phase gratings,” J. Opt. Soc. Am. A 23(4), 858–865 (2006).
[CrossRef]

Lederer, F.

Lettieri, T. R.

Lewotsky, K. L.

Marcen, J.

J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough surfaces,” Proc. SPIE 7426, 74260I, 74260I-9 (2009).
[CrossRef]

Marx, E.

Moharam, M. G.

Notni, G.

Rice, S. O.

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4(2-3), 351–378 (1951).
[CrossRef]

Rockstuhl, C.

Schröder, S.

Söderström, T.

Steinert, J.

Stover, J. C.

J. C. Stover, “Experimental confirmation of the Rayleigh-Rice obliquity factor,” Proc. SPIE 7792, 77920J, 77920J-5 (2010).
[CrossRef]

Takacs, P. Z.

Trost, M.

Tünnermann, A.

Vernold, C. L.

J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattering angles,” Opt. Eng. 46(7), 078002 (2007).
[CrossRef]

Vorburger, T. V.

Zavada, J. M.

E. L. Church, H. A. Jenkinson, and J. M. Zavada, “Relationship #res,” Opt. Eng. 18, 125 (1979).

Appl. Opt. (6)

Commun. Pure Appl. Math. (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4(2-3), 351–378 (1951).
[CrossRef]

J. Adhes. Sci. Technol. (1)

M. Flemming, L. Coriand, and A. Duparré, “Ultra-hydrophobicity through stochastic surface roughness,” J. Adhes. Sci. Technol. 23(3), 381–400 (2009).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. E. Harvey, A. Krywonos, and D. Bogunovic, “Nonparaxial scalar treatment of sinusoidal phase gratings,” J. Opt. Soc. Am. A 23(4), 858–865 (2006).
[CrossRef]

A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” Accepted for publication in J. Opt. Soc. Am. A (March 25, 2011)

Opt. Eng. (3)

E. L. Church, H. A. Jenkinson, and J. M. Zavada, “Relationship #res,” Opt. Eng. 18, 125 (1979).

J. M. Elson and J. M. Bennett, “Vector scattering theory,” Opt. Eng. 18, 116–124 (1979).

J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattering angles,” Opt. Eng. 46(7), 078002 (2007).
[CrossRef]

Opt. Express (1)

Proc. SPIE (4)

J. C. Stover, “Experimental confirmation of the Rayleigh-Rice obliquity factor,” Proc. SPIE 7792, 77920J, 77920J-5 (2010).
[CrossRef]

J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough surfaces,” Proc. SPIE 7426, 74260I, 74260I-9 (2009).
[CrossRef]

J. E. Harvey, N. Choi, and A. Krywonos, “Scattering from moderately rough interfaces between two arbitrary media,” Proc. SPIE 7794, 77940V, 77940V-11 (2010).
[CrossRef]

E. L. Church, “Statistical effects in the measurement and characterization of smooth scattering surfaces,” Proc. SPIE 511, 18 (1984).

Waves Random Media (1)

T. M. Elfouhaily and C. A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14(4), R1–R40 (2004).
[CrossRef]

Other (10)

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, (Pergamon Press, 1963).

S. Schröder, A. Duparré, “Finish assessment of complex surfaces by advanced light scattering techniques,” Proc. SPIE 7102, paper 0F (2008).

J. C. Stover, Optical Scattering, Measurement and Analysis, 2nd ed. (SPIE Press, 1995).

J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D. Dissertation, University of Arizona (1976).

A. Duparré, “Scattering from surfaces and thin films,” in Encyclopedia of Modern Optics, B. D. Guenther, D. G. Steel, and L. Bayvel, eds. (Elsevier, 2004).

A. Krywonos, “Predicting surface scatter using a linear systems formulation of non-paraxial scalar diffraction,” Ph.D. Dissertation, College of Optics and Photonics, University of Central Florida (2006).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

S. Schröder, A. Duparré, K. Füchsel, N. Kaiser, A. Tünnermann, and J. E. Harvey, “Scattering of Roughened TCO Films - Modeling and Measurement,” in Optical Interference Coatings, OSA Technical Digest (Optical Society of America, 2010), paper ThD3.

“Optics and optical instruments-test methods for radiation scattered by optical components,” ISO 13696:2002 (International Organization for Standardization, 2002).

A. von Finck, M. Hauptvogel, and A. Duparré, “Instrument for close-to-process light scatter measurements of thin film coatings and substrates,” in Optical Interference Coatings, OSA Technical Digest (Optical Society of America, 2010), paper ThD4.

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Figures (4)

Fig. 1
Fig. 1

First order scattering efficiency of sinusoidal phase gratings with different periods p and amplitudes a at a wavelength of 532 nm. Diffraction angles between 0 degrees and 90 degrees were achieved by adjusting the angle of incidence.

Fig. 2
Fig. 2

Master-PSDs of steel surfaces. The arrows indicate the upper bandwidth limits of the relevant spatial frequency range for three illumination wavelengths and assuming normal incidence.

Fig. 3
Fig. 3

Results of ARS measurements at 325 nm of steel surfaces with different surface structures represented as different roughness values. The heights of the specular peaks are indicated by horizontal bars.

Fig. 4
Fig. 4

ARS of surfaces with different roughness levels. Measurement results at 325 nm, 532 nm, and 1064 nm and modelling using the Rayleigh-Rice theory (RR) and the Generalized Harvey-Shack theory (GHS). (a) sample A, (b) sample B, and (c) sample C at 325 nm, 532 nm, and 1064 nm.

Tables (1)

Tables Icon

Table 1 Total rms Roughness and Ratio of Relevant Roughness to Wavelength for Samples A, B, and C

Equations (10)

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

P S D 2 ( f x , f y ) = lim L 1 L 2 | 0 L 0 L z ( x , y ) e 2 π i ( f x x + f y y ) d x d y | 2 .
P S D ( f ) = 1 2 π 0 2 π P S D ( f , ψ ) d ψ .
σ = [ 2 π f min f max P S D ( f ) f   d f ] 1 2
A R S ( θ s ) = Δ P s ( θ s ) Δ Ω s P i .
T S b = P s P i = 2 π 2 ° 85 ° A R S ( θ s ) sin θ s d θ s .
A R S ( θ s ) = 16 π 2 λ 4 γ i γ s 2 θ s Q     PSD ( f ) .
H s ( x ^ , y ^ ; γ i , γ s ) = exp { [ 2 π σ ^ r e l ( γ i + γ s ) ] 2 [ 1 A C V ( x ^ , y ^ ) σ t o t a l 2 ] } ,
A R S ( θ s ) = Q γ s F { H s ( x ^ , y ^ ; γ i , γ s ) } .
P 1 / P i = ( 2 π a λ ) 2 Q cos θ s cos θ i .
P m / P i = Q J m 2 ( 2 π a λ ( γ i + γ s ) 2 ) m = min max J m 2 ( 2 π a λ ( γ i + γ s ) 2 ) .

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