Abstract

The effects of surface roughness on the determination of the optical properties of materials by the reflection method were investigated. We showed that the refractive indices of a material with a moderately rough surface can be obtained by the reflection method, provided that the radio of the angular reflectances at both states of polarization is known. Specifically, we showed that for a nonspecular surface with specularity index as low as 0.07 the real and imaginary parts of the complex refractive index can be determined with an average deviation from its actual value of 2.9% and 5.0%, respectively.

© 1991 Optical Society of America

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References

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  1. J. R. Beattie, G. K. T. Conn, “Optical constants of metals in the infrared—principles of measurements,” Philos. Mag. 46, 222–234 (1955).
  2. J. R. Beattie, “Optical constants of metals in the infrared– experimental methods,” Philos. Mag. 46, 235–245 (1955).
  3. J. T. McCartney, S. Ergun, “Optical properties of graphite and coal,” Fuel 37, 272–282 (1958).
  4. L. A. Gilbert, “Refractive indices and absorption coefficients of coal in bulk measured in the range 6,000 to 24,000 Å by a polarized light technique,” Fuel 41, 351–358 (1962).
  5. P. J. Foster, C. R. Howarth, “Optical constants of carbons and coals in the infrared,” Carbon 6, 719–729 (1963).
    [Crossref]
  6. W. H. Dalzell, A. F. Sarofim, “Optical constants of soot and their application to heat-flux calculations,” Trans. ASME 21, 103–104 (1969).
  7. E. A. Taft, H. R. Philipp, “Optical properties of graphite,” Phys. Rev. A 138, 197–202 (1965).
  8. M. W. Williams, E. T. Arackawa, “Optical properties of glassy carbon from 0 to 82 eV,” J. Appl. Phys. 43, 3460–3463 (1972).
    [Crossref]
  9. V. P. Tomaselli, R. Rivera, D. C. Edewaard, K. D. Möller, “Infrared optical constants of black powders determined from reflection measurements,” Appl. Opt. 20, 3961–3967 (1981).
    [Crossref] [PubMed]
  10. J. D. Felske, T. T. Charalampopoulos, H. Hura, “Determination of the refractive indices of soot particles from the reflectivities of compressed soot pellets,” Combust. Sci. Technol. 37, 263–284 (1984).
    [Crossref]
  11. D. G. Goodwin, M. Mitchner, “Measurements of the near infrared optical properties of coal ash,” presented at the American Society of Mechanical Engineers Heat Transfer Conference, 1984.
  12. C. E. Batten, “Spectral optical constants of soot from polarized angular reflectance measurements,” Appl. Opt. 24, 1193–1199 (1985).
    [Crossref] [PubMed]
  13. I. Simon, H. O. McMahon, “Study of the structure of quartz cristobalite and vitreous silica by reflection in the infrared,” J. Chem. Phys. 21, 23–30 (1953).
    [Crossref]
  14. W. R. Hunter, “Errors in using the reflectance vs. angle of incidence method for measuring optical constants,” J. Opt. Soc. Am. 55, 1197–1204 (1965).
    [Crossref]
  15. D. G. Avery, “An improved method for measurements of optical constants by reflection,” Proc. Phys. Soc. B 65, 425–428 (1952).
    [Crossref]
  16. S. P. F. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis, and proposal for new methods based on the Brewster angle,” Proc. Phys. Soc. (London) 77, 949–957 (1961).
    [Crossref]
  17. R. F. Miller, A. J. Taylor, L. S. Julien, “The optimum angle of incidence for determining optical constants from reflection measurements,” J. Phys. D 3, 1957–1961 (1970).
    [Crossref]
  18. J. Janzen, “The refractive index of colloidal carbon,” J. Colloid Interface Sci. 69, 436–447 (1979).
    [Crossref]
  19. B. J. Stagg, T. T. Charalampopoulos, “Method to minimize the effects of polarizer leakage on reflectivity measurements,” Appl. Opt. 29, 4638–4645 (1990).
    [Crossref] [PubMed]
  20. H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. Inst. Electr. Eng. 101, 209–214 (1954).
  21. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1987).
  22. A. F. Houchens, R. G. Hering, “Bidirectional reflectance of rough metal surfaces,” in AIAA Thermophysics Specialist Conference (Academic, New York, 1967), paper 67-319.
  23. M. E. Bennett, J. O. Porteus, “Relation between the surface roughness and specular reflectance at normal incidence,” J. Opt. Soc. Am. 51, 123–129 (1961).
    [Crossref]
  24. H. E. Bennett, “Specular reflectance of aluminized ground glass and the height distribution of surface irregularities,” J. Opt. Soc. Am. 53, 1389–1394 (1963).
    [Crossref]
  25. I. Ohlidal, F. Lukes, K. Navratil, “Rough silicon surfaces studied by optical methods,” Surf. Sci. 45, 91–116 (1974).
    [Crossref]
  26. D. H. Hensler, “Light scattering from fused polycrystalline aluminum oxide surfaces,” Appl. Opt. 11, 2522–2528 (1972).
    [Crossref] [PubMed]

1990 (1)

1985 (1)

1984 (1)

J. D. Felske, T. T. Charalampopoulos, H. Hura, “Determination of the refractive indices of soot particles from the reflectivities of compressed soot pellets,” Combust. Sci. Technol. 37, 263–284 (1984).
[Crossref]

1981 (1)

1979 (1)

J. Janzen, “The refractive index of colloidal carbon,” J. Colloid Interface Sci. 69, 436–447 (1979).
[Crossref]

1974 (1)

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

1972 (2)

D. H. Hensler, “Light scattering from fused polycrystalline aluminum oxide surfaces,” Appl. Opt. 11, 2522–2528 (1972).
[Crossref] [PubMed]

M. W. Williams, E. T. Arackawa, “Optical properties of glassy carbon from 0 to 82 eV,” J. Appl. Phys. 43, 3460–3463 (1972).
[Crossref]

1970 (1)

R. F. Miller, A. J. Taylor, L. S. Julien, “The optimum angle of incidence for determining optical constants from reflection measurements,” J. Phys. D 3, 1957–1961 (1970).
[Crossref]

1969 (1)

W. H. Dalzell, A. F. Sarofim, “Optical constants of soot and their application to heat-flux calculations,” Trans. ASME 21, 103–104 (1969).

1965 (2)

1963 (2)

1962 (1)

L. A. Gilbert, “Refractive indices and absorption coefficients of coal in bulk measured in the range 6,000 to 24,000 Å by a polarized light technique,” Fuel 41, 351–358 (1962).

1961 (2)

S. P. F. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis, and proposal for new methods based on the Brewster angle,” Proc. Phys. Soc. (London) 77, 949–957 (1961).
[Crossref]

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

1958 (1)

J. T. McCartney, S. Ergun, “Optical properties of graphite and coal,” Fuel 37, 272–282 (1958).

1955 (2)

J. R. Beattie, G. K. T. Conn, “Optical constants of metals in the infrared—principles of measurements,” Philos. Mag. 46, 222–234 (1955).

J. R. Beattie, “Optical constants of metals in the infrared– experimental methods,” Philos. Mag. 46, 235–245 (1955).

1954 (1)

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

1953 (1)

I. Simon, H. O. McMahon, “Study of the structure of quartz cristobalite and vitreous silica by reflection in the infrared,” J. Chem. Phys. 21, 23–30 (1953).
[Crossref]

1952 (1)

D. G. Avery, “An improved method for measurements of optical constants by reflection,” Proc. Phys. Soc. B 65, 425–428 (1952).
[Crossref]

Arackawa, E. T.

M. W. Williams, E. T. Arackawa, “Optical properties of glassy carbon from 0 to 82 eV,” J. Appl. Phys. 43, 3460–3463 (1972).
[Crossref]

Avery, D. G.

D. G. Avery, “An improved method for measurements of optical constants by reflection,” Proc. Phys. Soc. B 65, 425–428 (1952).
[Crossref]

Batten, C. E.

Beattie, J. R.

J. R. Beattie, G. K. T. Conn, “Optical constants of metals in the infrared—principles of measurements,” Philos. Mag. 46, 222–234 (1955).

J. R. Beattie, “Optical constants of metals in the infrared– experimental methods,” Philos. Mag. 46, 235–245 (1955).

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1987).

Bennett, H. E.

Bennett, M. E.

Charalampopoulos, T. T.

B. J. Stagg, T. T. Charalampopoulos, “Method to minimize the effects of polarizer leakage on reflectivity measurements,” Appl. Opt. 29, 4638–4645 (1990).
[Crossref] [PubMed]

J. D. Felske, T. T. Charalampopoulos, H. Hura, “Determination of the refractive indices of soot particles from the reflectivities of compressed soot pellets,” Combust. Sci. Technol. 37, 263–284 (1984).
[Crossref]

Conn, G. K. T.

J. R. Beattie, G. K. T. Conn, “Optical constants of metals in the infrared—principles of measurements,” Philos. Mag. 46, 222–234 (1955).

Dalzell, W. H.

W. H. Dalzell, A. F. Sarofim, “Optical constants of soot and their application to heat-flux calculations,” Trans. ASME 21, 103–104 (1969).

Davies, H.

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

Edewaard, D. C.

Ergun, S.

J. T. McCartney, S. Ergun, “Optical properties of graphite and coal,” Fuel 37, 272–282 (1958).

Felske, J. D.

J. D. Felske, T. T. Charalampopoulos, H. Hura, “Determination of the refractive indices of soot particles from the reflectivities of compressed soot pellets,” Combust. Sci. Technol. 37, 263–284 (1984).
[Crossref]

Foster, P. J.

P. J. Foster, C. R. Howarth, “Optical constants of carbons and coals in the infrared,” Carbon 6, 719–729 (1963).
[Crossref]

Gilbert, L. A.

L. A. Gilbert, “Refractive indices and absorption coefficients of coal in bulk measured in the range 6,000 to 24,000 Å by a polarized light technique,” Fuel 41, 351–358 (1962).

Goodwin, D. G.

D. G. Goodwin, M. Mitchner, “Measurements of the near infrared optical properties of coal ash,” presented at the American Society of Mechanical Engineers Heat Transfer Conference, 1984.

Hensler, D. H.

Hering, R. G.

A. F. Houchens, R. G. Hering, “Bidirectional reflectance of rough metal surfaces,” in AIAA Thermophysics Specialist Conference (Academic, New York, 1967), paper 67-319.

Houchens, A. F.

A. F. Houchens, R. G. Hering, “Bidirectional reflectance of rough metal surfaces,” in AIAA Thermophysics Specialist Conference (Academic, New York, 1967), paper 67-319.

Howarth, C. R.

P. J. Foster, C. R. Howarth, “Optical constants of carbons and coals in the infrared,” Carbon 6, 719–729 (1963).
[Crossref]

Humphreys-Owen, S. P. F.

S. P. F. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis, and proposal for new methods based on the Brewster angle,” Proc. Phys. Soc. (London) 77, 949–957 (1961).
[Crossref]

Hunter, W. R.

Hura, H.

J. D. Felske, T. T. Charalampopoulos, H. Hura, “Determination of the refractive indices of soot particles from the reflectivities of compressed soot pellets,” Combust. Sci. Technol. 37, 263–284 (1984).
[Crossref]

Janzen, J.

J. Janzen, “The refractive index of colloidal carbon,” J. Colloid Interface Sci. 69, 436–447 (1979).
[Crossref]

Julien, L. S.

R. F. Miller, A. J. Taylor, L. S. Julien, “The optimum angle of incidence for determining optical constants from reflection measurements,” J. Phys. D 3, 1957–1961 (1970).
[Crossref]

Lukes, F.

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

McCartney, J. T.

J. T. McCartney, S. Ergun, “Optical properties of graphite and coal,” Fuel 37, 272–282 (1958).

McMahon, H. O.

I. Simon, H. O. McMahon, “Study of the structure of quartz cristobalite and vitreous silica by reflection in the infrared,” J. Chem. Phys. 21, 23–30 (1953).
[Crossref]

Miller, R. F.

R. F. Miller, A. J. Taylor, L. S. Julien, “The optimum angle of incidence for determining optical constants from reflection measurements,” J. Phys. D 3, 1957–1961 (1970).
[Crossref]

Mitchner, M.

D. G. Goodwin, M. Mitchner, “Measurements of the near infrared optical properties of coal ash,” presented at the American Society of Mechanical Engineers Heat Transfer Conference, 1984.

Möller, K. D.

Navratil, K.

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

Ohlidal, I.

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

Philipp, H. R.

E. A. Taft, H. R. Philipp, “Optical properties of graphite,” Phys. Rev. A 138, 197–202 (1965).

Porteus, J. O.

Rivera, R.

Sarofim, A. F.

W. H. Dalzell, A. F. Sarofim, “Optical constants of soot and their application to heat-flux calculations,” Trans. ASME 21, 103–104 (1969).

Simon, I.

I. Simon, H. O. McMahon, “Study of the structure of quartz cristobalite and vitreous silica by reflection in the infrared,” J. Chem. Phys. 21, 23–30 (1953).
[Crossref]

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1987).

Stagg, B. J.

Taft, E. A.

E. A. Taft, H. R. Philipp, “Optical properties of graphite,” Phys. Rev. A 138, 197–202 (1965).

Taylor, A. J.

R. F. Miller, A. J. Taylor, L. S. Julien, “The optimum angle of incidence for determining optical constants from reflection measurements,” J. Phys. D 3, 1957–1961 (1970).
[Crossref]

Tomaselli, V. P.

Williams, M. W.

M. W. Williams, E. T. Arackawa, “Optical properties of glassy carbon from 0 to 82 eV,” J. Appl. Phys. 43, 3460–3463 (1972).
[Crossref]

Appl. Opt. (4)

Carbon (1)

P. J. Foster, C. R. Howarth, “Optical constants of carbons and coals in the infrared,” Carbon 6, 719–729 (1963).
[Crossref]

Combust. Sci. Technol. (1)

J. D. Felske, T. T. Charalampopoulos, H. Hura, “Determination of the refractive indices of soot particles from the reflectivities of compressed soot pellets,” Combust. Sci. Technol. 37, 263–284 (1984).
[Crossref]

Fuel (2)

J. T. McCartney, S. Ergun, “Optical properties of graphite and coal,” Fuel 37, 272–282 (1958).

L. A. Gilbert, “Refractive indices and absorption coefficients of coal in bulk measured in the range 6,000 to 24,000 Å by a polarized light technique,” Fuel 41, 351–358 (1962).

J. Appl. Phys. (1)

M. W. Williams, E. T. Arackawa, “Optical properties of glassy carbon from 0 to 82 eV,” J. Appl. Phys. 43, 3460–3463 (1972).
[Crossref]

J. Chem. Phys. (1)

I. Simon, H. O. McMahon, “Study of the structure of quartz cristobalite and vitreous silica by reflection in the infrared,” J. Chem. Phys. 21, 23–30 (1953).
[Crossref]

J. Colloid Interface Sci. (1)

J. Janzen, “The refractive index of colloidal carbon,” J. Colloid Interface Sci. 69, 436–447 (1979).
[Crossref]

J. Opt. Soc. Am. (3)

J. Phys. D (1)

R. F. Miller, A. J. Taylor, L. S. Julien, “The optimum angle of incidence for determining optical constants from reflection measurements,” J. Phys. D 3, 1957–1961 (1970).
[Crossref]

Philos. Mag. (2)

J. R. Beattie, G. K. T. Conn, “Optical constants of metals in the infrared—principles of measurements,” Philos. Mag. 46, 222–234 (1955).

J. R. Beattie, “Optical constants of metals in the infrared– experimental methods,” Philos. Mag. 46, 235–245 (1955).

Phys. Rev. A (1)

E. A. Taft, H. R. Philipp, “Optical properties of graphite,” Phys. Rev. A 138, 197–202 (1965).

Proc. Inst. Electr. Eng. (1)

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

Proc. Phys. Soc. (London) (1)

S. P. F. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis, and proposal for new methods based on the Brewster angle,” Proc. Phys. Soc. (London) 77, 949–957 (1961).
[Crossref]

Proc. Phys. Soc. B (1)

D. G. Avery, “An improved method for measurements of optical constants by reflection,” Proc. Phys. Soc. B 65, 425–428 (1952).
[Crossref]

Surf. Sci. (1)

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

Trans. ASME (1)

W. H. Dalzell, A. F. Sarofim, “Optical constants of soot and their application to heat-flux calculations,” Trans. ASME 21, 103–104 (1969).

Other (3)

D. G. Goodwin, M. Mitchner, “Measurements of the near infrared optical properties of coal ash,” presented at the American Society of Mechanical Engineers Heat Transfer Conference, 1984.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1987).

A. F. Houchens, R. G. Hering, “Bidirectional reflectance of rough metal surfaces,” in AIAA Thermophysics Specialist Conference (Academic, New York, 1967), paper 67-319.

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Figures (5)

Fig. 1
Fig. 1

Reflectometer–monochromator–detection system: LS, light source; RCM, rotatable concave mirror with a 63.5-mm focal length; I, iris; LC, light chopper; P1, 50-mm-diameter plane mirror; C1, 50-mm-diameter spherical mirror with a 150-mm focal length; SH, sample holder; P2, 50-mm-diamter plane mirror; C2, 50-mm-diameter spherical mirror with a 200-mm focal length; RB, rotatable base; P, polarizer; MS's, monochromator slits; M, grating monochromator; F, cut on the wavelength filter; D, detector.

Fig. 2
Fig. 2

(a) Parallel (R) and perpendicular (R) reflectivities of the carbon rod surface at the wavelength of λ = 3.5 mm and a specularity index S.I. = 0.076 as a function of the angle of incidence. Diamonds are the experimental data, and solid curves correspond to the predictions of the Fresnel equations. The inferred refractive index m ¯ is 3.708–1.598i. (b) Reflectance ratio and state of polarization of the carbon rod surface at a wavelength of λ = 3.5 μm and a specularity index of S.I. = 0.076 as a function of the angle of incidence. Diamonds are the data, and solid curves correspond to the predictions of the Fresnel equations. The inferred refractive index m ¯ is 3.708–1.598i.

Fig. 3
Fig. 3

(a) Parallel (R) and perpendicular (R) reflectivities of the carbon rod surface at a wavelength of λ = 3.5 μm and a specularity index S.I. = 0.99 as a function of the angle of incidence. Diamonds are the experimental data, and solid curves correspond to the predictions of the Fresnel equations. The inferred refractive index m ¯ is 3.875–1.534i. (b) Reflectance ratio and state of polarization of the carbon rod surface at a wavelength of λ = 3.5 μm and a specularity index S.I. = 0.99 as a function of the angle of incidence. Diamonds are the data, and solid curves correspond to the predictions of the Fresnel equations. The inferred refractive index m ¯ is 3.875–1.534i.

Fig. 4
Fig. 4

Real part of the refractive index as a function of the specularity index of the carbon rod surface inferred from the inversion of the reflectance ratio (◊) and from the parallel component of the reflectivity (■).

Fig. 5
Fig. 5

Imaginary part of the refractive index as a function of the specularity index of the carbon rod surface inferred from the inversion of the reflectance ratio (◊) and from the parallel component of the reflectivity (■).

Tables (1)

Tables Icon

Table I Complex Refractive Indices at Various Values of Specularity Index Compared with the Refractive Index at S.I. = 0.99

Equations (11)

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

ρ D = e f + f π ( α / λ ) 2 cos ( θ ) Δ ω ,
f = [ 4 π cos ( θ ) σ λ ] 2 .
ρ B S = e f [ 1 + π ( α λ ) 2 cos ( θ ) Δ ω m = 1 f m m m ! ] .
ρ s = exp { [ 4 π cos ( θ ) σ λ ] 2 } .
R ( m ¯ , θ , σ λ ) = ρ ( θ , σ λ ) R 0 ( m ¯ , θ ) ,
R = ρ R 0 , ,
R = ρ R 0 , .
R R = R 0 , R 0 , .
R 0 , 2 R 0 , θ = 45 = 1.0.
F = i = 1 N ( G th , i G exp , i ) 2 ,
G = R R .

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