Abstract

The reflectance R(θ) and ellipsometric parameters for a black anodized aluminum sample were measured at wavelengths of 3.39 and 5.0 μm and at multiple incident angles. A three-phase model in which the dielectric constant for the rough layer is computed from Bruggeman’s effective medium theory is used to reduce the ellipsometric data and to calculate the reference specular reflectance. Beckmann’s scattering theory, modified for polarization effect, was used to reduce the reflectance data. A self-consistent solution was found in which the ellipsometric regressed effective thickness of the rough layer agrees with the reflectometric estimated roughness, and in which the measured reflectance data agree with Beckmann’s scattering theory.

© 1992 Optical Society of America

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References

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  1. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waυes from Rough Surfaces, Part I (Pergamon, New York, 1963).
  2. H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. Inst. Electr. Eng. Part 3 101, 209–214 (1954).
  3. W. S. Ament, “Toward a theory of reflection by a rough surface,” Proc. IRE 41, 142–146 (1953).
    [CrossRef]
  4. H. E. Bennett, J. O. Porteus, “Relation between surface roughness and specular reflectance at normal incidence,” J. Opt. Soc. Am. 51, 123–129 (1961).
    [CrossRef]
  5. J. O. Porteus, “Relation between the height distribution of a rough surface and the reflectance at normal incidence,” J. Opt. Soc. Am. 53, 1394–1402 (1963).
    [CrossRef]
  6. 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]
  7. T. Smith, “Effect of surface roughness on ellipsometry of aluminium,” Surf. Sci. 56, 252–271 (1976).
    [CrossRef]
  8. J. P. Marton, E. C. Chan, “Surface roughness interpretation of ellipsometer measurements using the Maxwell-Garnett theory,” J. Appl. Phys. 45, 5008–5014 (1974).
    [CrossRef]
  9. 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); D. E. Aspnes, “Studies of surface thin film and interface properties by automatic spectroscopic ellipsometry,” J. Vac. Sci. Technol. 18, 289–295 (1981).
    [CrossRef]
  10. S. F. Nee, “Ellipsometric analysis for surface roughness and texture,” Appl. Opt. 27, 2819–2831 (1988).
    [CrossRef] [PubMed]
  11. D. E. Aspnes, “Optical properties of thin films,” Thin Solid Films 89, 249–262 (1982).
    [CrossRef]
  12. D. A. G. Bruggeman, “Berechung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. Leipzig 24, 636 (1935).
    [CrossRef]
  13. J. C. Maxwell-Garnett, “Colours in metal glasses in metallic films and in solutions,” Philos. Trans. R. Soc. London 205, 237 (1906).
    [CrossRef]
  14. 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]
  15. 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, 257–261 (1986).
    [CrossRef]
  16. S. F. Nee, H. E. Bennett, D. L. Decker, F. 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. Soc. Photo-Opt. Instrum. Eng.1047, 226–237 (1989).
  17. Soe-Mie F. Nee, “Error reductions for a serious compensator imperfection for null ellipsometry,” J. Opt. Soc. Am. A 8, 314–321 (1991).
    [CrossRef]
  18. 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. Soc. Photo-Opt. Instrum. Eng.1331, 249–260 (1990).
  19. 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); Tech. Rep. UDRI-TR-84-128 (University of Dayton Research Institute, Dayton, Ohio, 1984).
    [CrossRef]

1991 (1)

1988 (1)

1986 (2)

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]

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, 257–261 (1986).
[CrossRef]

1982 (2)

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); Tech. Rep. UDRI-TR-84-128 (University of Dayton Research Institute, Dayton, Ohio, 1984).
[CrossRef]

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

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, 3292–3302 (1979); D. E. Aspnes, “Studies of surface thin film and interface properties by automatic spectroscopic ellipsometry,” J. Vac. Sci. Technol. 18, 289–295 (1981).
[CrossRef]

1976 (1)

T. Smith, “Effect of surface roughness on ellipsometry of aluminium,” Surf. Sci. 56, 252–271 (1976).
[CrossRef]

1974 (1)

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

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, 85–96 (1969).
[CrossRef]

1963 (1)

1961 (1)

1954 (1)

H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. Inst. Electr. Eng. Part 3 101, 209–214 (1954).

1953 (1)

W. S. Ament, “Toward a theory of reflection by a rough surface,” Proc. IRE 41, 142–146 (1953).
[CrossRef]

1935 (1)

D. A. G. Bruggeman, “Berechung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper 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]

Ament, W. S.

W. S. Ament, “Toward a theory of reflection by a rough surface,” Proc. IRE 41, 142–146 (1953).
[CrossRef]

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); D. E. Aspnes, “Studies of surface thin film and interface properties by automatic spectroscopic ellipsometry,” J. Vac. Sci. Technol. 18, 289–295 (1981).
[CrossRef]

Beckmann, P.

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

Bennett, H. E.

H. E. Bennett, J. O. Porteus, “Relation between surface roughness and specular reflectance at normal incidence,” J. Opt. Soc. Am. 51, 123–129 (1961).
[CrossRef]

S. F. Nee, H. E. Bennett, D. L. Decker, F. 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. Soc. Photo-Opt. Instrum. Eng.1047, 226–237 (1989).

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. Soc. Photo-Opt. Instrum. Eng.1331, 249–260 (1990).

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, 257–261 (1986).
[CrossRef]

Bruggeman, D. A. G.

D. A. G. Bruggeman, “Berechung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten 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, 257–261 (1986).
[CrossRef]

Chan, E. C.

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

Davies, H.

H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. Inst. Electr. Eng. Part 3 101, 209–214 (1954).

Decker, D. L.

S. F. Nee, H. E. Bennett, D. L. Decker, F. 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. Soc. Photo-Opt. Instrum. Eng.1047, 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]

Greene, F. D.

S. F. Nee, H. E. Bennett, D. L. Decker, F. 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. Soc. Photo-Opt. Instrum. Eng.1047, 226–237 (1989).

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); Tech. Rep. UDRI-TR-84-128 (University of Dayton Research Institute, Dayton, Ohio, 1984).
[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, 3292–3302 (1979); D. E. Aspnes, “Studies of surface thin film and interface properties by automatic spectroscopic ellipsometry,” J. Vac. Sci. Technol. 18, 289–295 (1981).
[CrossRef]

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, 257–261 (1986).
[CrossRef]

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); Tech. Rep. UDRI-TR-84-128 (University of Dayton Research Institute, Dayton, Ohio, 1984).
[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); Tech. Rep. UDRI-TR-84-128 (University of Dayton Research Institute, Dayton, Ohio, 1984).
[CrossRef]

Marton, J. P.

J. P. Marton, E. C. Chan, “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, 257–261 (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]

Nee, S. F.

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

S. F. Nee, H. E. Bennett, D. L. Decker, F. 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. Soc. Photo-Opt. Instrum. Eng.1047, 226–237 (1989).

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. Soc. Photo-Opt. Instrum. Eng.1331, 249–260 (1990).

Nee, Soe-Mie F.

Ogloza, A. A.

S. F. Nee, H. E. Bennett, D. L. Decker, F. 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. Soc. Photo-Opt. Instrum. Eng.1047, 226–237 (1989).

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, 257–261 (1986).
[CrossRef]

Porteus, J. O.

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, 257–261 (1986).
[CrossRef]

Raayjmakers, I. J. M. M.

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); Tech. Rep. UDRI-TR-84-128 (University of Dayton Research Institute, Dayton, Ohio, 1984).
[CrossRef]

Smith, T.

T. Smith, “Effect of surface roughness on ellipsometry of aluminium,” Surf. Sci. 56, 252–271 (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, 257–261 (1986).
[CrossRef]

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waυes 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); D. E. Aspnes, “Studies of surface thin film and interface properties by automatic spectroscopic ellipsometry,” J. Vac. Sci. Technol. 18, 289–295 (1981).
[CrossRef]

Verkerk, M. J.

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, 257–261 (1986).
[CrossRef]

Ann. Phys. Leipzig (1)

D. A. G. Bruggeman, “Berechung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. Leipzig 24, 636 (1935).
[CrossRef]

Appl. Opt. (2)

J. Appl. Phys. (2)

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, 257–261 (1986).
[CrossRef]

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

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

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); Tech. Rep. UDRI-TR-84-128 (University of Dayton Research Institute, Dayton, Ohio, 1984).
[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, 3292–3302 (1979); D. E. Aspnes, “Studies of surface thin film and interface properties by automatic spectroscopic ellipsometry,” J. Vac. Sci. Technol. 18, 289–295 (1981).
[CrossRef]

Proc. Inst. Electr. Eng. Part 3 (1)

H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. Inst. Electr. Eng. Part 3 101, 209–214 (1954).

Proc. IRE (1)

W. S. Ament, “Toward a theory of reflection by a rough surface,” Proc. IRE 41, 142–146 (1953).
[CrossRef]

Surf. Sci. (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, 85–96 (1969).
[CrossRef]

T. Smith, “Effect of surface roughness on ellipsometry of aluminium,” Surf. Sci. 56, 252–271 (1976).
[CrossRef]

Thin Solid Films (1)

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

Other (3)

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. Soc. Photo-Opt. Instrum. Eng.1331, 249–260 (1990).

S. F. Nee, H. E. Bennett, D. L. Decker, F. 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. Soc. Photo-Opt. Instrum. Eng.1047, 226–237 (1989).

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

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Figures (11)

Fig. 1
Fig. 1

Functions of ln(R/Ro) versus g for reflection from a random rough surface for B given as 0.01 and 0.04. The slope s for small g for real data can be used to estimate go and σz.

Fig. 2
Fig. 2

Characteristic plot of ln(Rsc/Ro) versus ln(g) for B = 1. For ln(g) > 2, the slope sc can be approximated by −1 − 1/g. This characteristic is a good check on experimental results for g ≫ 1, where the near-angle scattering dominates.

Fig. 3
Fig. 3

Measured ψ at multiple incident angles θi at 3.39- and 5.0-μm wavelengths for a black anodized aluminum sample.

Fig. 4
Fig. 4

Measured Δ (θi) at 3.39- and 5.0-μm wavelengths for the anodized aluminum sample.

Fig. 5
Fig. 5

Rs(θ) and Rp(θ) measured at λ = 3.39 μm for the anodized aluminum sample.

Fig. 6
Fig. 6

Rs(θ) and Rp(θ) measured at λ = 5.0 μm for the anodized aluminum sample.

Fig. 7
Fig. 7

Plot of ln(Rs*/Rs′) versus cos2θ for reflectance data for the black anodized aluminum sample at λ = 3.39 μm with Rs′ calculated from the best-fit set of solutions in Table I. The curve looks like the characteristic curves in Fig. 1.

Fig. 8
Fig. 8

Plots of ln(Rs*/Rs′) for small cos2 θ and their best-fit Unes for various sets of solutions for the black anodized aluminum sample at λ = 3.39 μm. The slopes for these best-fit lines give raw estimates of go and roughness.

Fig. 9
Fig. 9

Bλ2 at λ = 3.39 μm versus Bλ2 at λ = 5 μm for various sets of solutions given in Tables III and IV. The set of solutions closest to the diagonal line is the best-fit one.

Fig. 10
Fig. 10

Plot of ln(Rs) versus cos2 θ for the black anodized aluminum sample at λ = 3.39 μm. The solid line is calculated from Eq. (6) from the best-fit set of solutions.

Fig. 11
Fig. 11

Plot similar to that of Fig. 10 for λ = 5.0 μm for the best-fit set of solutions.

Tables (4)

Tables Icon

Table I Best-Fit Values of d, n, and k at λ = 3.39 μm for a Black Anodized Aluminum Sample Regressed from Ellipsometric Data at 10 Different θ Using Two- and Three-Phase Modelsa

Tables Icon

Table II Best-Fit Values of d, n, and k at λ = 5.0 μm for a Black Anodized Aluminum Sample Regressed from Ellipsometric Data at Seven Different θ Using Two- and Three-Phase Models

Tables Icon

Table III Best-Fit rms Roughness σz and B to the Reflectance Data for Various Models and for Different Ellipsometric Solutions at λ = 3.39 μm for the Black Anodized Aluminum Sample

Tables Icon

Table IV Best-Fit σz and B to the Reflectance Data for Various Models and for Different Ellipsometric Solutions at λ = 5.0 μm for the Black Anodized Aluminum Sample

Equations (22)

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R = R o exp ( g ) [ 1 + B ξ ( g ) ] ,
g = g o cos 2 θ, g o = ( 4 πσ z / λ ) 2 ,
B = π T 2 Δ Ω / λ 2 ,
ξ ( g ) = m = 1 g m / m ! m .
ρ = r p / r s = tan ψ exp ( i Δ ) ,
R = exp ( g ) [ R + B R o ξ ( g ) ] .
d 2 ( 2 ) 1 / 2 σ z .
Y = ln ( R / R ) = g o x + ln [ 1 + B ξ ( g ) R o / R ] ,
Y sc = ln ( R sc / R o ) = g + ln [ B ξ ( g ) ] ,
ln ( B / g o ) ln ( x ) for g 1 .
s c = d Y sc / d u = g + [ exp ( g ) 1 ] / ξ ( g ) .
s c 1 1 / g , for g 1 .
δψ i = ψ ( n , k , d , θ i ) ψ i .
δψ i = ψ i n δ n + ψ i k δ k + ψ i d δ d .
σ ψ 2 = 1 N 1 i = 1 N ( δψ i ) 2 ,
σ ψ, Δ 2 = σ ψ 2 + σ Δ 2 / w 2 ,
δ n 1 N i ( δψ i ψ i / n + δ Δ i Δ i / n ) .
δ Y i = δ R s i / R s i = ln ( R s i * ) ln [ R ( σ z , B , R , R s o , θ i ) ] .
x 1 x 2 = τ sec θ 1 cos φ, y 1 y 2 = τ sin φ,
d x d y = τ sec θ 1 d τ d φ .
D { ρ } = 2 π F 3 2 A cos θ 1 0 J o ( υ x y τ ) [ χ 2 ( υ z , υ z ) χ ( υ z ) χ * ( υ z ) ] τ d τ ,
υ x y = ( υ x 2 sec 2 θ 1 + υ y 2 ) 1 / 2 .

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