Abstract

A 2-D symmetric model is incorporated into the calculation of the ellipsometric parameters Ψ and Δ for surface roughness and texture characterization based on the effective medium theory. The least-squares fits of the experimental data at a 5-μm IR wavelength for rough fused silica samples at multiple angles of incidence give the standard deviations of Ψ and Δ of about twice the instrumental errors. The effective thickness and the depolarization factor obtained by ellipsometry agree with the roughness and average height-to-halfwidth ratio of voids obtained by stylus profilometry. The surface texture can be characterized by the fit depolarization factors set. The excellent agreement between theory and experiments indicates that ellipsometry can be a promising nondestructive technique for rough-surface evaluation.

© 1988 Optical Society of America

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References

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  1. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  2. C. F. Fenstermaker, F. L. McCrackin, “Errors Arising from Surface Roughness in Ellipsometric Measurement of the Refractive Index of a Surface,” Surf. Sci. 16, 85 (1969).
    [Crossref]
  3. D. E. Aspnes, “Optical Properties of Thin Films,” Thin Solid Films 89, 249 (1982).
    [Crossref]
  4. D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of Effective Medium Models of Microscopic Surface Roughness by Spectroscopic Ellipsometry,” Phys. Rev. B 20, 3292 (1979); D. E. Aspnes, “Studies of Surface Thin Film and Interface Properties by Automatic Spectroscopic Ellipsometry,” J. Vac. Sci. Technol. 18, 289 (1981).
    [Crossref]
  5. J. P. Marton, E. C. Chang, “Surface Roughness Interpretation of Ellipsometer Measurements Using the Maxwell-Garnett Theory,” J. Appl. Phys. 45, 5008 (1974).
    [Crossref]
  6. I. Ohlidal, F. Lukes, “Ellipsometric Parameters of Rough Surfaces and of a System Substrate—Thin Film with Rough Boundaries,” Opt. Acta 19, 817 (1972); I. Ohlidal, F. Lukes, K. Navratil, “Rough Silicon Surfaces Studied by Optical Methods,” Surf. Sci. 45, 91 (1974).
    [Crossref]
  7. E. L. Church, J. Zavada, “Residual Surface Roughness of Diamond Turned Optics,” Appl. Opt. 14, 1788 (1975).
    [Crossref] [PubMed]
  8. T. V. Vorburger, K. C. Ludema, “Ellipsometry of Rough Surfaces,” Appl. Opt. 19, 561 (1980).
    [Crossref] [PubMed]
  9. J. R. Blanco, P. J. McMarr, K. Vedam, “Roughness Measurements by Spectroscopic Ellipsometry,” Appl. Opt. 24, 3773 (1985).
    [Crossref] [PubMed]
  10. T. Smith, “Effect of Surface Roughness on Ellipsometry of Aluminium,” Surf. Sci. 56, 252 (1976).
    [Crossref]
  11. S. F. Nee, H. E. Bennett, “Nondestructive Evaluation of Surface Roughness in the 0.01- to 1.0-μm Range Using Infrared Ellipsometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 675, 260 (1986).
  12. I. J. M. M. Raayjmakers, M. J. Verkerk, “Characterization of the Topography of Vacuum-Deposited Films. 2: Ellipsometry,” Appl. Opt. 25, 3610 (1986).
    [Crossref]
  13. G. H. Bu-Abbud et al., “Roughness Studies of Ion Beam Processed Molybdenum Surfaces,” J. Appl. Phys. 59, 257 (1986).
    [Crossref]
  14. D. K. Burge, H. E. Bennett, “Effect of a Thin Surface Film on the Ellipsometric Determination of Optical Constants,” J. Opt. Soc. Am. 54, 1428 (1964).
    [Crossref]
  15. K. Vedam, “Characterization of Defects in Real Surfaces by Ellipsometry,” Surf. Sci. 56, 221 (1976).
    [Crossref]
  16. K. Riedling, “Error Effects in the Ellipsometric Investigation of Thin Films,” Thin Solid Films 75, 355 (1981).
    [Crossref]
  17. M. E. Pedinoff, O. M. Stafsudd, “Multiple Angle Ellipsometric Analysis of Surface Layers and Surface Layer Contaminants,” Appl. Opt. 21, 518 (1982).
    [Crossref] [PubMed]
  18. S. F. Nee, H. E. Bennett, “Quantitative Characterization of Rough SiO2 Surfaces by Infrared Ellipsometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 818, 34 (1987).
  19. G. M. Nomarski, “Differential Microinterferometer with Polarized Waves,” J. Phys. Rad. 16, 9 (1955).
  20. T. A. Leonard, J. Loomis, K. G. Haarding, M. Scott, “Design and Construction of Three Infrared Ellipsometers for Thin Film Research,” Opt. Eng. 21, 971 (1982); also UDRI-TR-84-128.
    [Crossref]
  21. E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985), p. 761.
  22. J. M. Bennett, J. H. Dancy, “Stylus Profiling Instrument for Measuring Statistical Properties of Smooth Optical Surfaces,” Appl. Opt. 20, 1785 (1981).
    [Crossref] [PubMed]
  23. J. C. Maxwell-Garnett, “Colours in Metal Glasses, in Metallic Films and in Solutions,” Philos. Trans. R. Soc. London 205, 237 (1906).
    [Crossref]
  24. D. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitatskonstanten und Leitfahigkeiten der Mischkorper aus isotropen Substanzen,” Ann. Phys. Leipzig 24, 636 (1935).
    [Crossref]
  25. C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1967), p. 378.
  26. L. Davis, J. L. Greenstein, “The Polarization of Starlight by Aligned Dust Grains,” Astrophys. J. 114, 220 (1951).
    [Crossref]
  27. D. den Engelsen, “Ellipsometry of Anisotropic Films,” J. Opt. Soc. Am. 61, 1460 (1971).
    [Crossref]
  28. H. Osterberg, Optical Design, Sec. 21, MIL-HDBK-141 (1962).
  29. L. G. Parratt, Probability and Experimental Errors in Science (Wiley, New York, 1961), p. 140.

1987 (1)

S. F. Nee, H. E. Bennett, “Quantitative Characterization of Rough SiO2 Surfaces by Infrared Ellipsometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 818, 34 (1987).

1986 (3)

S. F. Nee, H. E. Bennett, “Nondestructive Evaluation of Surface Roughness in the 0.01- to 1.0-μm Range Using Infrared Ellipsometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 675, 260 (1986).

I. J. M. M. Raayjmakers, M. J. Verkerk, “Characterization of the Topography of Vacuum-Deposited Films. 2: Ellipsometry,” Appl. Opt. 25, 3610 (1986).
[Crossref]

G. H. Bu-Abbud et al., “Roughness Studies of Ion Beam Processed Molybdenum Surfaces,” J. Appl. Phys. 59, 257 (1986).
[Crossref]

1985 (1)

1982 (3)

D. E. Aspnes, “Optical Properties of Thin Films,” Thin Solid Films 89, 249 (1982).
[Crossref]

M. E. Pedinoff, O. M. Stafsudd, “Multiple Angle Ellipsometric Analysis of Surface Layers and Surface Layer Contaminants,” Appl. Opt. 21, 518 (1982).
[Crossref] [PubMed]

T. A. Leonard, J. Loomis, K. G. Haarding, M. Scott, “Design and Construction of Three Infrared Ellipsometers for Thin Film Research,” Opt. Eng. 21, 971 (1982); also UDRI-TR-84-128.
[Crossref]

1981 (2)

1980 (1)

1979 (1)

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of Effective Medium Models of Microscopic Surface Roughness by Spectroscopic Ellipsometry,” Phys. Rev. B 20, 3292 (1979); D. E. Aspnes, “Studies of Surface Thin Film and Interface Properties by Automatic Spectroscopic Ellipsometry,” J. Vac. Sci. Technol. 18, 289 (1981).
[Crossref]

1976 (2)

T. Smith, “Effect of Surface Roughness on Ellipsometry of Aluminium,” Surf. Sci. 56, 252 (1976).
[Crossref]

K. Vedam, “Characterization of Defects in Real Surfaces by Ellipsometry,” Surf. Sci. 56, 221 (1976).
[Crossref]

1975 (1)

1974 (1)

J. P. Marton, E. C. Chang, “Surface Roughness Interpretation of Ellipsometer Measurements Using the Maxwell-Garnett Theory,” J. Appl. Phys. 45, 5008 (1974).
[Crossref]

1972 (1)

I. Ohlidal, F. Lukes, “Ellipsometric Parameters of Rough Surfaces and of a System Substrate—Thin Film with Rough Boundaries,” Opt. Acta 19, 817 (1972); I. Ohlidal, F. Lukes, K. Navratil, “Rough Silicon Surfaces Studied by Optical Methods,” Surf. Sci. 45, 91 (1974).
[Crossref]

1971 (1)

1969 (1)

C. F. Fenstermaker, F. L. McCrackin, “Errors Arising from Surface Roughness in Ellipsometric Measurement of the Refractive Index of a Surface,” Surf. Sci. 16, 85 (1969).
[Crossref]

1964 (1)

1955 (1)

G. M. Nomarski, “Differential Microinterferometer with Polarized Waves,” J. Phys. Rad. 16, 9 (1955).

1951 (1)

L. Davis, J. L. Greenstein, “The Polarization of Starlight by Aligned Dust Grains,” Astrophys. J. 114, 220 (1951).
[Crossref]

1935 (1)

D. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitatskonstanten und Leitfahigkeiten der Mischkorper aus isotropen Substanzen,” Ann. Phys. Leipzig 24, 636 (1935).
[Crossref]

1906 (1)

J. C. Maxwell-Garnett, “Colours in Metal Glasses, in Metallic Films and in Solutions,” Philos. Trans. R. Soc. London 205, 237 (1906).
[Crossref]

Aspnes, D. E.

D. E. Aspnes, “Optical Properties of Thin Films,” Thin Solid Films 89, 249 (1982).
[Crossref]

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of Effective Medium Models of Microscopic Surface Roughness by Spectroscopic Ellipsometry,” Phys. Rev. B 20, 3292 (1979); D. E. Aspnes, “Studies of Surface Thin Film and Interface Properties by Automatic Spectroscopic Ellipsometry,” J. Vac. Sci. Technol. 18, 289 (1981).
[Crossref]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bennett, H. E.

S. F. Nee, H. E. Bennett, “Quantitative Characterization of Rough SiO2 Surfaces by Infrared Ellipsometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 818, 34 (1987).

S. F. Nee, H. E. Bennett, “Nondestructive Evaluation of Surface Roughness in the 0.01- to 1.0-μm Range Using Infrared Ellipsometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 675, 260 (1986).

D. K. Burge, H. E. Bennett, “Effect of a Thin Surface Film on the Ellipsometric Determination of Optical Constants,” J. Opt. Soc. Am. 54, 1428 (1964).
[Crossref]

Bennett, J. M.

Blanco, J. R.

Bruggeman, D. A. G.

D. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitatskonstanten und Leitfahigkeiten der Mischkorper aus isotropen Substanzen,” Ann. Phys. Leipzig 24, 636 (1935).
[Crossref]

Bu-Abbud, G. H.

G. H. Bu-Abbud et al., “Roughness Studies of Ion Beam Processed Molybdenum Surfaces,” J. Appl. Phys. 59, 257 (1986).
[Crossref]

Burge, D. K.

Chang, E. C.

J. P. Marton, E. C. Chang, “Surface Roughness Interpretation of Ellipsometer Measurements Using the Maxwell-Garnett Theory,” J. Appl. Phys. 45, 5008 (1974).
[Crossref]

Church, E. L.

Dancy, J. H.

Davis, L.

L. Davis, J. L. Greenstein, “The Polarization of Starlight by Aligned Dust Grains,” Astrophys. J. 114, 220 (1951).
[Crossref]

den Engelsen, D.

Fenstermaker, C. F.

C. F. Fenstermaker, F. L. McCrackin, “Errors Arising from Surface Roughness in Ellipsometric Measurement of the Refractive Index of a Surface,” Surf. Sci. 16, 85 (1969).
[Crossref]

Greenstein, J. L.

L. Davis, J. L. Greenstein, “The Polarization of Starlight by Aligned Dust Grains,” Astrophys. J. 114, 220 (1951).
[Crossref]

Haarding, K. G.

T. A. Leonard, J. Loomis, K. G. Haarding, M. Scott, “Design and Construction of Three Infrared Ellipsometers for Thin Film Research,” Opt. Eng. 21, 971 (1982); also UDRI-TR-84-128.
[Crossref]

Hottier, F.

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of Effective Medium Models of Microscopic Surface Roughness by Spectroscopic Ellipsometry,” Phys. Rev. B 20, 3292 (1979); D. E. Aspnes, “Studies of Surface Thin Film and Interface Properties by Automatic Spectroscopic Ellipsometry,” J. Vac. Sci. Technol. 18, 289 (1981).
[Crossref]

Kittel, C.

C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1967), p. 378.

Leonard, T. A.

T. A. Leonard, J. Loomis, K. G. Haarding, M. Scott, “Design and Construction of Three Infrared Ellipsometers for Thin Film Research,” Opt. Eng. 21, 971 (1982); also UDRI-TR-84-128.
[Crossref]

Loomis, J.

T. A. Leonard, J. Loomis, K. G. Haarding, M. Scott, “Design and Construction of Three Infrared Ellipsometers for Thin Film Research,” Opt. Eng. 21, 971 (1982); also UDRI-TR-84-128.
[Crossref]

Ludema, K. C.

Lukes, F.

I. Ohlidal, F. Lukes, “Ellipsometric Parameters of Rough Surfaces and of a System Substrate—Thin Film with Rough Boundaries,” Opt. Acta 19, 817 (1972); I. Ohlidal, F. Lukes, K. Navratil, “Rough Silicon Surfaces Studied by Optical Methods,” Surf. Sci. 45, 91 (1974).
[Crossref]

Marton, J. P.

J. P. Marton, E. C. Chang, “Surface Roughness Interpretation of Ellipsometer Measurements Using the Maxwell-Garnett Theory,” J. Appl. Phys. 45, 5008 (1974).
[Crossref]

Maxwell-Garnett, J. C.

J. C. Maxwell-Garnett, “Colours in Metal Glasses, in Metallic Films and in Solutions,” Philos. Trans. R. Soc. London 205, 237 (1906).
[Crossref]

McCrackin, F. L.

C. F. Fenstermaker, F. L. McCrackin, “Errors Arising from Surface Roughness in Ellipsometric Measurement of the Refractive Index of a Surface,” Surf. Sci. 16, 85 (1969).
[Crossref]

McMarr, P. J.

Nee, S. F.

S. F. Nee, H. E. Bennett, “Quantitative Characterization of Rough SiO2 Surfaces by Infrared Ellipsometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 818, 34 (1987).

S. F. Nee, H. E. Bennett, “Nondestructive Evaluation of Surface Roughness in the 0.01- to 1.0-μm Range Using Infrared Ellipsometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 675, 260 (1986).

Nomarski, G. M.

G. M. Nomarski, “Differential Microinterferometer with Polarized Waves,” J. Phys. Rad. 16, 9 (1955).

Ohlidal, I.

I. Ohlidal, F. Lukes, “Ellipsometric Parameters of Rough Surfaces and of a System Substrate—Thin Film with Rough Boundaries,” Opt. Acta 19, 817 (1972); I. Ohlidal, F. Lukes, K. Navratil, “Rough Silicon Surfaces Studied by Optical Methods,” Surf. Sci. 45, 91 (1974).
[Crossref]

Osterberg, H.

H. Osterberg, Optical Design, Sec. 21, MIL-HDBK-141 (1962).

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985), p. 761.

Parratt, L. G.

L. G. Parratt, Probability and Experimental Errors in Science (Wiley, New York, 1961), p. 140.

Pedinoff, M. E.

Raayjmakers, I. J. M. M.

Riedling, K.

K. Riedling, “Error Effects in the Ellipsometric Investigation of Thin Films,” Thin Solid Films 75, 355 (1981).
[Crossref]

Scott, M.

T. A. Leonard, J. Loomis, K. G. Haarding, M. Scott, “Design and Construction of Three Infrared Ellipsometers for Thin Film Research,” Opt. Eng. 21, 971 (1982); also UDRI-TR-84-128.
[Crossref]

Smith, T.

T. Smith, “Effect of Surface Roughness on Ellipsometry of Aluminium,” Surf. Sci. 56, 252 (1976).
[Crossref]

Stafsudd, O. M.

Theeten, J. B.

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of Effective Medium Models of Microscopic Surface Roughness by Spectroscopic Ellipsometry,” Phys. Rev. B 20, 3292 (1979); D. E. Aspnes, “Studies of Surface Thin Film and Interface Properties by Automatic Spectroscopic Ellipsometry,” J. Vac. Sci. Technol. 18, 289 (1981).
[Crossref]

Vedam, K.

J. R. Blanco, P. J. McMarr, K. Vedam, “Roughness Measurements by Spectroscopic Ellipsometry,” Appl. Opt. 24, 3773 (1985).
[Crossref] [PubMed]

K. Vedam, “Characterization of Defects in Real Surfaces by Ellipsometry,” Surf. Sci. 56, 221 (1976).
[Crossref]

Verkerk, M. J.

Vorburger, T. V.

Zavada, J.

Ann. Phys. Leipzig (1)

D. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitatskonstanten und Leitfahigkeiten der Mischkorper aus isotropen Substanzen,” Ann. Phys. Leipzig 24, 636 (1935).
[Crossref]

Appl. Opt. (6)

Astrophys. J. (1)

L. Davis, J. L. Greenstein, “The Polarization of Starlight by Aligned Dust Grains,” Astrophys. J. 114, 220 (1951).
[Crossref]

J. Appl. Phys. (2)

J. P. Marton, E. C. Chang, “Surface Roughness Interpretation of Ellipsometer Measurements Using the Maxwell-Garnett Theory,” J. Appl. Phys. 45, 5008 (1974).
[Crossref]

G. H. Bu-Abbud et al., “Roughness Studies of Ion Beam Processed Molybdenum Surfaces,” J. Appl. Phys. 59, 257 (1986).
[Crossref]

J. Opt. Soc. Am. (2)

J. Phys. Rad. (1)

G. M. Nomarski, “Differential Microinterferometer with Polarized Waves,” J. Phys. Rad. 16, 9 (1955).

Opt. Acta (1)

I. Ohlidal, F. Lukes, “Ellipsometric Parameters of Rough Surfaces and of a System Substrate—Thin Film with Rough Boundaries,” Opt. Acta 19, 817 (1972); I. Ohlidal, F. Lukes, K. Navratil, “Rough Silicon Surfaces Studied by Optical Methods,” Surf. Sci. 45, 91 (1974).
[Crossref]

Opt. Eng. (1)

T. A. Leonard, J. Loomis, K. G. Haarding, M. Scott, “Design and Construction of Three Infrared Ellipsometers for Thin Film Research,” Opt. Eng. 21, 971 (1982); also UDRI-TR-84-128.
[Crossref]

Philos. Trans. R. Soc. London (1)

J. C. Maxwell-Garnett, “Colours in Metal Glasses, in Metallic Films and in Solutions,” Philos. Trans. R. Soc. London 205, 237 (1906).
[Crossref]

Phys. Rev. B (1)

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of Effective Medium Models of Microscopic Surface Roughness by Spectroscopic Ellipsometry,” Phys. Rev. B 20, 3292 (1979); D. E. Aspnes, “Studies of Surface Thin Film and Interface Properties by Automatic Spectroscopic Ellipsometry,” J. Vac. Sci. Technol. 18, 289 (1981).
[Crossref]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

S. F. Nee, H. E. Bennett, “Quantitative Characterization of Rough SiO2 Surfaces by Infrared Ellipsometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 818, 34 (1987).

S. F. Nee, H. E. Bennett, “Nondestructive Evaluation of Surface Roughness in the 0.01- to 1.0-μm Range Using Infrared Ellipsometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 675, 260 (1986).

Surf. Sci. (3)

K. Vedam, “Characterization of Defects in Real Surfaces by Ellipsometry,” Surf. Sci. 56, 221 (1976).
[Crossref]

C. F. Fenstermaker, F. L. McCrackin, “Errors Arising from Surface Roughness in Ellipsometric Measurement of the Refractive Index of a Surface,” Surf. Sci. 16, 85 (1969).
[Crossref]

T. Smith, “Effect of Surface Roughness on Ellipsometry of Aluminium,” Surf. Sci. 56, 252 (1976).
[Crossref]

Thin Solid Films (2)

D. E. Aspnes, “Optical Properties of Thin Films,” Thin Solid Films 89, 249 (1982).
[Crossref]

K. Riedling, “Error Effects in the Ellipsometric Investigation of Thin Films,” Thin Solid Films 75, 355 (1981).
[Crossref]

Other (5)

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1967), p. 378.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985), p. 761.

H. Osterberg, Optical Design, Sec. 21, MIL-HDBK-141 (1962).

L. G. Parratt, Probability and Experimental Errors in Science (Wiley, New York, 1961), p. 140.

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Figures (17)

Fig. 1
Fig. 1

Nomarski micrographs with 90× magnification for rough fused-silica samples: (a) 4, (b) 32, (c) 43. Each micrograph corresponds to an area of 1 × 1.25 mm on the sample.

Fig. 2
Fig. 2

Major components of the Naval Weapons Center’s ellipsometer as configured for automated null operation. The compensator in conjunction with two polarizers and sample can be aligned to produce a null signal (after Ref. 20).

Fig. 3
Fig. 3

Talystep profiles for rough fused-silica sample 4.

Fig. 4
Fig. 4

Rough surface as an effective surface layer. The rough surface is equivalent to an anisotropic surface layer with an effective thickness d and effective dielectric constants of || and z for fields parallel and perpendicular to the surface.

Fig. 5
Fig. 5

Talystep profiles for rough fused-silica sample 32.

Fig. 6
Fig. 6

Talystep profiles for rough fused-silica sample 43.

Fig. 7
Fig. 7

Ψ as a function of incident angle θ for rough fused-silica sample 4 at 5.0-μm wavelength. Triangles are the experimental points, and the solid line is the best-fit theoretical curve for the 2-D symmetric void model.

Fig. 8
Fig. 8

Δ as a function of incident angle θ for rough fused-silica sample 4 at 5.0-μm wavelength. Again, triangles are the experimental points, and the solid line is the best-fit theoretical curve.

Fig. 9
Fig. 9

Relative Ψ with respect to an ideal flat as a function of incident angle θ at 5-μm wavelength for samples 4, 32, and 43. Sample 4 has the largest deviation but is not the roughest sample.

Fig. 10
Fig. 10

Relative Δ with respect to an ideal flat as a function of incident angle θ at 5-μm wavelength for samples 4,32, and 43. Some data of sample 4 near the Brewster region are out of scale and not included in the figure. DΔ is better noticeable than DΨ for sample 43.

Fig. 11
Fig. 11

Least deviation σΨ,Δ for sample 4 for different assumed values of parallel depolarization factor q||. The best-fit q|| is 0.15, which gives a minimum deviation of 0.0298°.

Fig. 12
Fig. 12

Relative Ψ for sample 4 with respect to an ideal flat as a function of incident angle θ at 5-μm wavelength. Triangles represent experimental data, and the solid line is the best-fit theoretical curve.

Fig. 13
Fig. 13

Relative Δ for sample 4 with respect to an ideal flat as a function of incident angle θ at a 5-μm wavelength. Input data are the same as Fig. 12.

Fig. 14
Fig. 14

Least deviation σΨ,Δ for sample 32 for different assumed values of q||. The best-fit q|| is 0.039, which gives a minimum deviation of 0.0243°.

Fig. 15
Fig. 15

Least deviation σΨ,Δ for sample 43 for different assumed values of q||. The best-fit q|| is 0.007, which gives a minimum deviation of 0.0283°.

Fig. 16
Fig. 16

Best-fit effective thickness d vs the best-fit void fraction f for sample 4 for different values of q|| · q|| is 1/3 at S, 1/2 at C, and 0.1 at A. Point B represents the best-fit d and f at q|| = 0.15.

Fig. 17
Fig. 17

Best-fit effective thickness d vs the best-fit void fraction f for samples 4,32, and 43. All curves of d(f) show quadratic behavior. The best-fit points shown as empty diamonds fit nicelyon a straight line.

Tables (3)

Tables Icon

Table I Talystep Profilometric Results for Rough Fused-Silica Samples

Tables Icon

Table II Ellipsometric Data for Rough Fused-Silica Sample 4 and Errors from a Model Calculation Based on Aspnes’s General EMT with Bruggeman Approximation for a 2-D Symmetric Rough Layer with q|| = 0.15, f = 0.1418, and d = 0.9858 μm; Units are Given by Degrees.

Tables Icon

Table III Ellipsometric Best-Fit Results for Rough Fused-Silica Samples Based on Aspnes’s Generalized EMT; MG Denotes Maxwell-Garnett EMT, and BM Denotes Bruggeman Approximation; Measurements are Made at λ = 5 μm.

Equations (32)

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

j = q j a b + ( 1 - q j ) h ( f a a + f b b ) ( 1 - q j ) h + q j ( f a b + f b a ) , n j = j ,
q x + q y + q z = 1.
q = q x = q y = ( 1 - q z ) / 2.
q z = 1 1 - x 2 ( 1 - x cos - 1 x 1 - x 2 )             for x 1 ,
= 1 x 2 - 1 ( x cosh - 1 x x 2 - 1 - 1 )             for x 1.
ρ = r p r s = tan Ψ exp ( i Δ ) ,
r s = r 01 s + r 12 s exp ( i β s ) 1 + r 01 s r 12 s exp ( i β s ) .
β s = 2 k d n ˜ 1
β p = 2 k d n 1 n ˜ 1 z / n 1 z ,
n ˜ j = j - 0 sin 2 θ             j = 0 , 1 , 2.
n ˜ j z = j z - 0 z sin 2 θ
r 01 s = n ˜ 0 - n ˜ 1 n ˜ 0 + n ˜ 1 ,
r 01 p = n ˜ 0 z / n 0 n 0 z - n ˜ 1 z / n 1 n 1 z n ˜ 0 z / n 0 n 0 z + n ˜ 1 z / n 1 n 1 z .
σ Ψ 2 = 1 N - 1 i = 1 N [ Ψ - Ψ ( θ i ) ] 2 ,
σ Δ 2 = 1 N - 1 i = 1 N [ Δ i - Δ ( θ i ) ] 2 ,
σ Ψ , Δ 2 = σ Ψ 2 + σ Δ 2 / 36.
d 2 2 σ z 2 2 σ z .
z ( x , y ) = i j A i j cos ( k x i x + a i j ) cos ( k y j y + b i j )
z ( x , y * ) = i j A i j cos ( k x i x + a i j )
= i B i cos ( k x i x + c i )             for 0 x L ,
A i j = A i j cos ( k y j y * + b i j ) ,
B i cos ( k x i x + c i ) = j A i j cos ( k x i x + a i j ) .
σ z 2 = δ z 2 = 1 i L i i = 1 N 0 L i [ z i ( x ) - z ¯ ] 2 d x ,
σ z 2 = 1 N L i = 1 N { 0 L [ z i ( x ) - z ¯ i ] 2 d x + L ( z ¯ i - z ¯ ) 2 }
= 1 N i = 1 N σ z i 2 + 1 N i = 1 N ( z ¯ i - z ¯ ) 2 ,
σ z 2 = 1 N i = 1 N σ z i 2 σ z 2 .
σ z 2 = 1 L 2 0 L d x 0 L d y [ z ( x , y ) - z ¯ ] 2 .
δ σ z 2 = 1 N - 1 i = 1 N ( σ z i - σ z ) 2 .
rms ( σ z ) = δ σ z 2 .
ρ Ψ = 1 - i δ Ψ i 2 / i D Ψ i 2 ,
ρ Δ = 1 - i δ Δ i 2 / i D Δ i 2 .
d 2 = 2 2 σ z ,

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