Abstract

Polarization of specular reflection and near-specular scattering (NSS) by a randomly rough surface is investigated by the use of a Mueller matrix formulation. The collective effect by a rough surface on the average specular field results in reflectance loss and polarization, which can be explained by an effective medium theory. Effects of random NSS can be represented by a scattering matrix that is partially coherent and polarized. The incoherent and unpolarized part of scattering causes depolarization, and the coherent and polarized parts of scattering change the apparent polarization properties of specular reflection. Results of a simulation and least-squares fit of ellipsometric data to the models including the NSS effect, for a black anodized aluminum sample, are presented. Simultaneous least-squares fits for both ellipsometric data and reflectance data at multiple angles of incidence at three different wavelengths gave approximately the same rms roughness, which agrees with the profilometric values reported previously.

© 1996 Optical Society of America

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  7. S. F. Nee, H. E. Bennett, D. L. Decker, S. D. Greene, A. A. Ogloza, “Analysis of diamond-turned optics using ellipsometry,” in Mirrors and Windows for High Power/High Energy Laser Systems, C. A. Klein, ed., Proc. SPIE1047, 226–237 (1989).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  38. R. Simon, “Mueller matrices and depolarization criteria,” J. Mod. Opt. 34, 569–575 (1987).
    [CrossRef]
  39. A. B. Kostinski, “Depolarization criterion for incoherent scattering,” Appl. Opt. 31, 3506–3508 (1992).
    [CrossRef] [PubMed]

1994

1993

1992

1991

S. F. Nee, “Error reduction for a serious compensator imperfection for null ellipsometry,” J. Opt. Soc. Am. A 8, 314–321 (1991).
[CrossRef]

K. A. O'Donnell, M. E. Knotts, “Polarization dependence of scattering from one-dimensional rough surfaces,” J. Opt. Soc. Am. A 8, 1127–1131 (1991).

1988

1987

R. Simon, “Mueller matrices and depolarization criteria,” J. Mod. Opt. 34, 569–575 (1987).
[CrossRef]

V. J. Iafelice, W. S. Bickel, “Polarized light-scattering matrix elements for select perfect and perturbed optical surfaces,” Appl. Opt. 26, 2410–2415 (1987).
[CrossRef]

W. S. Bickel, R. R. Zito, V. Iafelice, “Polarized light scattering from metal surfaces,” J. Appl. Phys. 61, 5392–5398 (1987).
[CrossRef]

K. Kim, L. Mandel, E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433–437 (1987).
[CrossRef]

1986

G. H. Bu-Abbud, D. L. Mathine, P. Snyder, J. A. Woollam, D. Poker, J. Bennett, D. Ingram, P. P. Pronko, “Roughness studies of ion beam processed molybdenum surfaces,” J. Appl. Phys. 59, 257–262 (1986).
[CrossRef]

I. J. M. M. Raayjmakers, M. J. Verkerk, “Characterization of the topography of vacuum-deposited films. 2: Ellipsometry,” Appl. Opt. 25, 3610–3615 (1986).
[CrossRef]

1985

J. R. Blanco, P. J. McMarr, K. Vedam, “Roughness measurements by spectroscopic ellipsometry,” Appl. Opt. 24, 3773–3779 (1985).
[CrossRef] [PubMed]

J. J. Gil, E. Bernabeu, “A depolarization criterion in Müller matrices,” Opt. Acta 32, 259–261 (1985).
[CrossRef]

1982

D. E. Aspnes, “Optical properties of thin films,” Thin Solid Films 89, 249–262 (1982).
[CrossRef]

1981

1980

1979

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

1976

T. Smith, “Effect of surface roughness on ellipsometry of aluminium,” Surf. Sci. 56, 252–272 (1976).
[CrossRef]

1974

I. Ohlidal, F. Lukes, K. Navratil, “Rough silicon surfaces studies by optical methods,” Surf. Sci. 45, 91–116 (1974).
[CrossRef]

J. P. Marton, E. C. Chang, “Surface roughness interpretation of ellipsometer measurements using the Maxwell-Garnett theory,” J. Appl. Phys. 45, 5008–5014 (1974).
[CrossRef]

1972

I. Ohlidal, F. Lukes, “Ellipsometric parameters of rough surfaces and of a system substrate—thin film with rough boundaries,” Opt. Acta 19, 817–943 (1972).
[CrossRef]

1971

1969

C. F. Fenstermaker, F. L. McCrackin, “Errors arising from surface roughness in ellipsometric measurement of the refractive index of a surface,” Surf. Sci. 16, 85–96 (1969).
[CrossRef]

1959

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
[CrossRef]

1954

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[CrossRef]

1935

D. A. G. Bruggeman, “Berechnung verschiedener physika-lischer Konstanten von heterogenen Substanzen. I. Dielek-trizitaätskonstanten und Leitfaähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. Leipzig 24, 636 (1935).
[CrossRef]

1906

J. C. Maxwell-Garnett, “Colours in metal glasses in metallic films and in solutions,” Philos. Trans. R. Soc. London 205, 237 (1906).
[CrossRef]

Archibald, P. C.

S. F. Nee, J. M. Bennett, P. C. Archibald, “Reflection, scattering and polarization from very rough surfaces,” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 202–212 (1993).

Aspnes, D. E.

D. E. Aspnes, “Optical properties of thin films,” Thin Solid Films 89, 249–262 (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, 3292–3302 (1979).
[CrossRef]

Azzam, R. M. A.

Bashara, N. M.

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, Part I (Pergamon, New York, 1963).

Bennett, H. E.

S. F. Nee, H. E. Bennett, “Nondestructive evaluation of surface roughness in the 0.01 to 1.0 μm range using infrared ellipsometry,” in Stray Radiation V, R. P. Breault, ed., Proc. SPIE675, 260–269 (1986).

S. F. Nee, H. E. Bennett, D. L. Decker, S. D. Greene, A. A. Ogloza, “Analysis of diamond-turned optics using ellipsometry,” in Mirrors and Windows for High Power/High Energy Laser Systems, C. A. Klein, ed., Proc. SPIE1047, 226–237 (1989).

S. F. Nee, H. E. Bennett, “Quantitative characterization of rough SiO2 surfaces by infrared ellipsometry,” in Current Developments in Optical Engineering II, R. E. Fischer, W. J. Smith, eds., Proc. SPIE818, 34–45 (1987).

S. F. Nee, H. E. Bennett, “Characterization of optical blacks by infrared ellipsometry and reflectometry,” in Stray Radiation in Optical Systems, R. P. Breault, ed., Proc. SPIE1331, 249–260 (1990).
[CrossRef]

Bennett, J.

G. H. Bu-Abbud, D. L. Mathine, P. Snyder, J. A. Woollam, D. Poker, J. Bennett, D. Ingram, P. P. Pronko, “Roughness studies of ion beam processed molybdenum surfaces,” J. Appl. Phys. 59, 257–262 (1986).
[CrossRef]

Bennett, J. M.

S. F. Nee, J. M. Bennett, P. C. Archibald, “Reflection, scattering and polarization from very rough surfaces,” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 202–212 (1993).

Bernabeu, E.

J. J. Gil, E. Bernabeu, “A depolarization criterion in Müller matrices,” Opt. Acta 32, 259–261 (1985).
[CrossRef]

Bickel, W. S.

V. J. Iafelice, W. S. Bickel, “Polarized light-scattering matrix elements for select perfect and perturbed optical surfaces,” Appl. Opt. 26, 2410–2415 (1987).
[CrossRef]

W. S. Bickel, R. R. Zito, V. Iafelice, “Polarized light scattering from metal surfaces,” J. Appl. Phys. 61, 5392–5398 (1987).
[CrossRef]

Blanco, J. R.

Bruggeman, D. A. G.

D. A. G. Bruggeman, “Berechnung verschiedener physika-lischer Konstanten von heterogenen Substanzen. I. Dielek-trizitaätskonstanten und Leitfaähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. Leipzig 24, 636 (1935).
[CrossRef]

Bu-Abbud, G. H.

G. H. Bu-Abbud, D. L. Mathine, P. Snyder, J. A. Woollam, D. Poker, J. Bennett, D. Ingram, P. P. Pronko, “Roughness studies of ion beam processed molybdenum surfaces,” J. Appl. Phys. 59, 257–262 (1986).
[CrossRef]

Chang, E. C.

J. P. Marton, E. C. Chang, “Surface roughness interpretation of ellipsometer measurements using the Maxwell-Garnett theory,” J. Appl. Phys. 45, 5008–5014 (1974).
[CrossRef]

Decker, D. L.

S. F. Nee, H. E. Bennett, D. L. Decker, S. D. Greene, A. A. Ogloza, “Analysis of diamond-turned optics using ellipsometry,” in Mirrors and Windows for High Power/High Energy Laser Systems, C. A. Klein, ed., Proc. SPIE1047, 226–237 (1989).

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, 85–96 (1969).
[CrossRef]

Fry, E. S.

Gil, J. J.

J. J. Gil, E. Bernabeu, “A depolarization criterion in Müller matrices,” Opt. Acta 32, 259–261 (1985).
[CrossRef]

Greene, S. D.

S. F. Nee, H. E. Bennett, D. L. Decker, S. D. Greene, A. A. Ogloza, “Analysis of diamond-turned optics using ellipsometry,” in Mirrors and Windows for High Power/High Energy Laser Systems, C. A. Klein, ed., Proc. SPIE1047, 226–237 (1989).

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, 3292–3302 (1979).
[CrossRef]

Iafelice, V.

W. S. Bickel, R. R. Zito, V. Iafelice, “Polarized light scattering from metal surfaces,” J. Appl. Phys. 61, 5392–5398 (1987).
[CrossRef]

Iafelice, V. J.

Ingram, D.

G. H. Bu-Abbud, D. L. Mathine, P. Snyder, J. A. Woollam, D. Poker, J. Bennett, D. Ingram, P. P. Pronko, “Roughness studies of ion beam processed molybdenum surfaces,” J. Appl. Phys. 59, 257–262 (1986).
[CrossRef]

Kattawar, G. W.

Kim, K.

Knotts, M. E.

Kostinski, A. B.

Ludema, K. C.

Lukes, F.

I. Ohlidal, F. Lukes, K. Navratil, “Rough silicon surfaces studies by optical methods,” Surf. Sci. 45, 91–116 (1974).
[CrossRef]

I. Ohlidal, F. Lukes, “Ellipsometric parameters of rough surfaces and of a system substrate—thin film with rough boundaries,” Opt. Acta 19, 817–943 (1972).
[CrossRef]

Mandel, L.

Marton, J. P.

J. P. Marton, E. C. Chang, “Surface roughness interpretation of ellipsometer measurements using the Maxwell-Garnett theory,” J. Appl. Phys. 45, 5008–5014 (1974).
[CrossRef]

Mathine, D. L.

G. H. Bu-Abbud, D. L. Mathine, P. Snyder, J. A. Woollam, D. Poker, J. Bennett, D. Ingram, P. P. Pronko, “Roughness studies of ion beam processed molybdenum surfaces,” J. Appl. Phys. 59, 257–262 (1986).
[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, 85–96 (1969).
[CrossRef]

McMarr, P. J.

Michel, T. R.

Navratil, K.

I. Ohlidal, F. Lukes, K. Navratil, “Rough silicon surfaces studies by optical methods,” Surf. Sci. 45, 91–116 (1974).
[CrossRef]

Nee, S. F.

S. F. Nee, “Ellipsometric view on reflection and scattering from optical blacks,” Appl. Opt. 31, 1549–1556 (1992).
[CrossRef] [PubMed]

S. F. Nee, “Error reduction for a serious compensator imperfection for null ellipsometry,” J. Opt. Soc. Am. A 8, 314–321 (1991).
[CrossRef]

S. F. Nee, “Ellipsometric analysis for surface roughness and texture,” Appl. Opt. 27, 2819–2831 (1988).
[CrossRef] [PubMed]

S. F. Nee, J. M. Bennett, P. C. Archibald, “Reflection, scattering and polarization from very rough surfaces,” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 202–212 (1993).

S. F. Nee, H. E. Bennett, “Nondestructive evaluation of surface roughness in the 0.01 to 1.0 μm range using infrared ellipsometry,” in Stray Radiation V, R. P. Breault, ed., Proc. SPIE675, 260–269 (1986).

S. F. Nee, “Effects of near-specular scattering on polarimetry,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 304–313 (1994).
[CrossRef]

S. F. Nee, “The effects of incoherent scattering on ellipsometry,” in Polarization Analysis and Measurement, D. H. Gold-stein, R. A. Chipman, eds., Proc. SPIE1746, 119–127 (1992).
[CrossRef]

S. F. Nee, H. E. Bennett, “Quantitative characterization of rough SiO2 surfaces by infrared ellipsometry,” in Current Developments in Optical Engineering II, R. E. Fischer, W. J. Smith, eds., Proc. SPIE818, 34–45 (1987).

S. F. Nee, H. E. Bennett, “Characterization of optical blacks by infrared ellipsometry and reflectometry,” in Stray Radiation in Optical Systems, R. P. Breault, ed., Proc. SPIE1331, 249–260 (1990).
[CrossRef]

S. F. Nee, H. E. Bennett, D. L. Decker, S. D. Greene, A. A. Ogloza, “Analysis of diamond-turned optics using ellipsometry,” in Mirrors and Windows for High Power/High Energy Laser Systems, C. A. Klein, ed., Proc. SPIE1047, 226–237 (1989).

O'Donnell, K. A.

Ogloza, A. A.

S. F. Nee, H. E. Bennett, D. L. Decker, S. D. Greene, A. A. Ogloza, “Analysis of diamond-turned optics using ellipsometry,” in Mirrors and Windows for High Power/High Energy Laser Systems, C. A. Klein, ed., Proc. SPIE1047, 226–237 (1989).

Ohlidal, I.

I. Ohlidal, F. Lukes, K. Navratil, “Rough silicon surfaces studies by optical methods,” Surf. Sci. 45, 91–116 (1974).
[CrossRef]

I. Ohlidal, F. Lukes, “Ellipsometric parameters of rough surfaces and of a system substrate—thin film with rough boundaries,” Opt. Acta 19, 817–943 (1972).
[CrossRef]

Poker, D.

G. H. Bu-Abbud, D. L. Mathine, P. Snyder, J. A. Woollam, D. Poker, J. Bennett, D. Ingram, P. P. Pronko, “Roughness studies of ion beam processed molybdenum surfaces,” J. Appl. Phys. 59, 257–262 (1986).
[CrossRef]

Pronko, P. P.

G. H. Bu-Abbud, D. L. Mathine, P. Snyder, J. A. Woollam, D. Poker, J. Bennett, D. Ingram, P. P. Pronko, “Roughness studies of ion beam processed molybdenum surfaces,” J. Appl. Phys. 59, 257–262 (1986).
[CrossRef]

Raayjmakers, I. J. M. M.

Simon, R.

R. Simon, “Mueller matrices and depolarization criteria,” J. Mod. Opt. 34, 569–575 (1987).
[CrossRef]

Smith, T.

T. Smith, “Effect of surface roughness on ellipsometry of aluminium,” Surf. Sci. 56, 252–272 (1976).
[CrossRef]

Snyder, P.

G. H. Bu-Abbud, D. L. Mathine, P. Snyder, J. A. Woollam, D. Poker, J. Bennett, D. Ingram, P. P. Pronko, “Roughness studies of ion beam processed molybdenum surfaces,” J. Appl. Phys. 59, 257–262 (1986).
[CrossRef]

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, Part I (Pergamon, New York, 1963).

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, 3292–3302 (1979).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 5.

Vedam, K.

Verkerk, M. J.

Vorburger, T. V.

Wolf, E.

K. Kim, L. Mandel, E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433–437 (1987).
[CrossRef]

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
[CrossRef]

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[CrossRef]

Woollam, J. A.

G. H. Bu-Abbud, D. L. Mathine, P. Snyder, J. A. Woollam, D. Poker, J. Bennett, D. Ingram, P. P. Pronko, “Roughness studies of ion beam processed molybdenum surfaces,” J. Appl. Phys. 59, 257–262 (1986).
[CrossRef]

Zito, R. R.

W. S. Bickel, R. R. Zito, V. Iafelice, “Polarized light scattering from metal surfaces,” J. Appl. Phys. 61, 5392–5398 (1987).
[CrossRef]

Ann. Phys. Leipzig

D. A. G. Bruggeman, “Berechnung verschiedener physika-lischer Konstanten von heterogenen Substanzen. I. Dielek-trizitaätskonstanten und Leitfaähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. Leipzig 24, 636 (1935).
[CrossRef]

Appl. Opt.

J. Appl. Phys.

J. P. Marton, E. C. Chang, “Surface roughness interpretation of ellipsometer measurements using the Maxwell-Garnett theory,” J. Appl. Phys. 45, 5008–5014 (1974).
[CrossRef]

G. H. Bu-Abbud, D. L. Mathine, P. Snyder, J. A. Woollam, D. Poker, J. Bennett, D. Ingram, P. P. Pronko, “Roughness studies of ion beam processed molybdenum surfaces,” J. Appl. Phys. 59, 257–262 (1986).
[CrossRef]

W. S. Bickel, R. R. Zito, V. Iafelice, “Polarized light scattering from metal surfaces,” J. Appl. Phys. 61, 5392–5398 (1987).
[CrossRef]

J. Mod. Opt.

R. Simon, “Mueller matrices and depolarization criteria,” J. Mod. Opt. 34, 569–575 (1987).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nuovo Cimento

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[CrossRef]

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
[CrossRef]

Opt. Acta

J. J. Gil, E. Bernabeu, “A depolarization criterion in Müller matrices,” Opt. Acta 32, 259–261 (1985).
[CrossRef]

I. Ohlidal, F. Lukes, “Ellipsometric parameters of rough surfaces and of a system substrate—thin film with rough boundaries,” Opt. Acta 19, 817–943 (1972).
[CrossRef]

Philos. Trans. R. Soc. London

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

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

Surf. Sci.

T. Smith, “Effect of surface roughness on ellipsometry of aluminium,” Surf. Sci. 56, 252–272 (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, 85–96 (1969).
[CrossRef]

I. Ohlidal, F. Lukes, K. Navratil, “Rough silicon surfaces studies by optical methods,” Surf. Sci. 45, 91–116 (1974).
[CrossRef]

Thin Solid Films

D. E. Aspnes, “Optical properties of thin films,” Thin Solid Films 89, 249–262 (1982).
[CrossRef]

Other

S. F. Nee, H. E. Bennett, “Quantitative characterization of rough SiO2 surfaces by infrared ellipsometry,” in Current Developments in Optical Engineering II, R. E. Fischer, W. J. Smith, eds., Proc. SPIE818, 34–45 (1987).

S. F. Nee, H. E. Bennett, “Nondestructive evaluation of surface roughness in the 0.01 to 1.0 μm range using infrared ellipsometry,” in Stray Radiation V, R. P. Breault, ed., Proc. SPIE675, 260–269 (1986).

S. F. Nee, H. E. Bennett, D. L. Decker, S. D. Greene, A. A. Ogloza, “Analysis of diamond-turned optics using ellipsometry,” in Mirrors and Windows for High Power/High Energy Laser Systems, C. A. Klein, ed., Proc. SPIE1047, 226–237 (1989).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 5.

S. F. Nee, J. M. Bennett, P. C. Archibald, “Reflection, scattering and polarization from very rough surfaces,” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 202–212 (1993).

S. F. Nee, H. E. Bennett, “Characterization of optical blacks by infrared ellipsometry and reflectometry,” in Stray Radiation in Optical Systems, R. P. Breault, ed., Proc. SPIE1331, 249–260 (1990).
[CrossRef]

S. F. Nee, “The effects of incoherent scattering on ellipsometry,” in Polarization Analysis and Measurement, D. H. Gold-stein, R. A. Chipman, eds., Proc. SPIE1746, 119–127 (1992).
[CrossRef]

S. F. Nee, “Effects of near-specular scattering on polarimetry,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 304–313 (1994).
[CrossRef]

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

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Figures (7)

Fig. 1
Fig. 1

Simulated curves for δψ(θ) at λ = 3.39 μm for different effects.

Fig. 2
Fig. 2

Plot similar to that of Fig. 2 for λ = 5.0 μm.

Fig. 3
Fig. 3

Simulated data of Δ for different effects at λ = 3.39 μm.

Fig. 4
Fig. 4

Simulated data of δΔ for different effects at λ = 3.39 μm.

Fig. 5
Fig. 5

Residual errors σψ,Δ for Model A (B′ = C = 0) and Model B (B′C ≠ 0), using different numbers (N) of ellipsometric data; crosses and triangles adjacent to σψ,Δ of a specific model represent σψ and σΔ for each model.

Fig. 6
Fig. 6

Deviations of the measured ψ from the simulated values for different models.

Fig. 7
Fig. 7

Deviations of the measured Δ from the simulated values for different models.

Tables (5)

Tables Icon

Table 1 Best-Fit Values of n, k, and d for the Ellipsometric Data at Multiple Angles of Incidence for a Black Anodized Aluminum Sample a

Tables Icon

Table 2 Ellipsometric Data for a Black Anodized Aluminum Sample at Multiple Angles of Incidence at a 3.39-mm Wavelength

Tables Icon

Table 3 Best-Fit Results for the Ellipsometric Data for a Black Anodized Aluminum Sample at 3.39 μm

Tables Icon

Table 4 Best-Fit Results for Simultaneous Regression of Both Ellipsometric and Reflectance Data for a Black Anodized Aluminum Sample at 5.0, 3.39 μm, and 0.633 μm

Tables Icon

Table 5 Conditions of Measurements for Both Ellipsometric and Reflectance Data for a Black Anodized Aluminum Sample at 5.0, 3.39, and 0.633 μm

Equations (46)

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E x = ( r x x + s x x ) E x o + s x y E y o + b x , E y = ( r y y + s y y ) E y o + s y x E x o + b y .
s i j = 0 = b i , i , j = 1 , 2 .
E x = r x x E x o , E y = r y y E y o .
r x x / r y y = tan ψ exp ( i Δ ) .
r i i o = r i i o exp ( g / 2 ) , g = ( 4 π σ z cos θ / λ ) 2 .
r i i = r i i o + r i i , i = 1 , 2 .
s i j r k k * = 0 = b k r i i * = b k s i j * , i , j , k = 1 , 2 .
E i E k * = j ; m = 1 2 [ ( r i i δ i j + s i j ) E j o + b i ] × [ ( r k k * δ k m + s k m * ) E m o * + b k * ] , = r i i r k k * E i o E k o * + j ; m = 1 2 s i j s k m * × E j o E m o * + b i b k * .
X = M X o .
M = R + S .
R = R [ 1 cos 2 ψ 0 0 cos 2 ψ 1 0 0 0 0 sin 2 ψ cos Δ sin 2 ψ sin Δ 0 0 sin 2 ψ sin Δ sin 2 ψ cos Δ ] ,
R = ( r x x r x x * + r y y r y y * ) / 2 .
i ; j = 1 4 M i j 2 = 4 M 11 2 .
b i b j * = b i b i * δ i j .
X b = ( I b , 0 , 0 , 0 ) , I b = b x b x * + b y b y * .
S 11 = ( s x x s x x * + s y y s y y * + 2 s x y s x y * ) / 2 + I b / I o , S 22 = ( s x x s x x * + s y y s y y * 2 s x y s x y * ) / 2 , S 12 = S 21 = ( s x x s x x * s y y s y y * ) / 2 , S 33 + i S 34 = S 44 i S 43 = s x x s y y * .
S = [ U + V + W U cos 2 ψ s 0 0 U cos 2 ψ s U V 0 0 0 0 γ U cos Δ s γ U sin Δ s 0 0 γ U sin Δ s γ U cos Δ s ] ,
U = ( s x x s x x * + s y y s y y * ) / 2 , V = ( s x y s x y * + s y x s y x * ) / 2 , W = I b / I o , tan 2 ψ s = s x x s x x * / s y y s y y * , tan Δ s = Im s x x s y y * / Re s x x s y y * , γ = | s x x s y y * | / U .
S u = [ U + V + W 0 0 0 0 U V 0 0 0 0 0 0 0 0 0 0 ] .
i ; j = 1 4 M i j 2 4 M 11 2 .
M 11 = R + U + V + W = R + S 11 , M 22 = R + U V , M 12 = M 21 = R cos 2 ψ cos U cos 2 ψ s , M 33 = M 44 = R sin 2 ψ cos Δ + γ U cos Δ s , M 34 = M 43 = R sin 2 ψ sin Δ + γ U sin Δ s .
M = R [ 1 + u + υ + w cos 2 ψ 0 0 cos 2 ψ 1 + u υ 0 0 0 0 sin 2 ψ cos Δ sin 2 ψ sin Δ 0 0 sin 2 ψ sin Δ sin 2 ψ cos Δ ] .
M = R [ 1 P cos 2 ψ 0 0 P cos 2 ψ 1 2 υ 0 0 0 0 P sin 2 ψ cos Δ P sin 2 ψ sin Δ 0 0 P sin 2 ψ sin Δ P sin 2 ψ cos Δ ] .
tan Δ = R sin 2 ψ sin Δ + γ U sin Δ s R sin 2 ψ cos Δ + γ U cos Δ s ,
tan 2 2 ψ = R 2 sin 2 2 ψ + 2 U R γ sin 2 ψ cos ( Δ Δ s ) + U 2 γ 2 ( R cos 2 ψ + U cos 2 ψ s ) 2 ,
R = R + S 11 , υ = ( V + W / 2 ) / R ,
P = { R 2 + 2 U R [ cos 2 ψ cos 2 ψ s + γ sin 2 ψ cos ( Δ Δ s ) ] + U 2 ( γ 2 + cos 2 2 ψ s ) } 1 / 2 / R .
R * = exp ( g ) [ R e + B F ( θ ) R o ξ ( g ) ] ,
B = π T 2 Δ Ω / λ 2 , ξ ( g ) = m = 1 g m / m ! m .
R x x = 2 ( R sin 2 ψ + U sin 2 ψ s ) + W , R y y = 2 ( R cos 2 ψ + U cos 2 ψ s ) + W .
R * = ( R x x + R y y ) / 2 = R + U + W .
R = R e exp ( g ) , U = B R o F ( θ ) ξ ( g ) exp ( g ) ,
d = 2 2 σ z .
tan δ Δ = U γ sin ( Δ s Δ ) / [ R sin 2 ψ + U γ cos ( Δ s Δ ) ] .
tan 2 δ ψ = U γ cos 2 ψ U cos 2 ψ s sin 2 ψ R + U cos 2 ψ s cos 2 ψ + U γ sin 2 ψ .
σ ψ , Δ 2 = σ ψ 2 + σ Δ 2 / 4 ,
σ ψ , Δ , R 2 = σ ψ 2 + σ Δ 2 / 4 + σ ( ln R s ) 2 / w 2 ,
M = [ ( E 1 + E 2 + E 3 + E 4 ) / 2 ( E 1 E 2 E 3 + E 4 ) / 2 F 13 + F 42 G 13 G 42 ( E 1 E 2 + E 3 E 4 ) / 2 ( E 1 + E 2 E 3 E 4 ) / 2 F 13 F 42 G 13 + G 42 F 14 + F 32 F 14 F 32 F 12 + F 34 G 12 + G 34 G 14 + G 32 G 14 G 32 G 12 + G 34 F 12 F 34 ] ,
E i = T i T i * = | T i | 2 , i = 1 , 2 , 3 , 4
F i j = F j i = Re T i T j * = Re T j T i * , i , j = 1 , 2 , 3 , 4 , i j
G i j = G j i = Im T i T j * = lm T j T i * , i , j = 1 , 2 , 3 , 4.
S 12 = S 21 = ( s x x s x x * s y y s y y * ) / 2 ,
S 34 = S 43 = G 12 = Im s x x s y y * .
S 33 = S 44 = F 12 = Re s x x s y y * .
S 11 = ( s x x s x x * + s y y s y y * + 2 s x y s x y * ) / 2 + I b / I o ,
S 22 = ( s x x s x x * + s y y s y y * 2 s x y s x y * ) / 2 .

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