Abstract

The purpose of this study is to develop a technique that will provide the maximum sensitivity of the reflection technique for the determination of the optical properties of materials. Analytical relations are presented that yield the maximum sensitivity of the technique for a given range of angles of incidence and refractive indices. The methods of the reflectance ratio at one and two angles of incidence are compared. The angular range of 75° to 80° was found to be suitable for the maximum sensitivity for the reflectance ratio at one angle of incidence, whereas for the two angles of incidence the ranges 45° to 60° and 75° to 85° are suitable. Furthermore, angular-reflectance measurements on a specular carbon-rod surface at the wavelength of 3.5 μm were used to demonstrate the advantages of inverting full-angular-range data as opposed to one or two angle measurements. The limitations of the measurements at normal incidence and 45° are also assessed.

© 1992 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 measurement,” Philos. Mag. 46, 222–234 (1955).
  2. J. R. Beattie, “Optical constants of metals in the infrared—experimental methods,” Philos. Mag. 46, 235–244 (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 6000 to 24000 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 constant of soot and their application to heat-flux calculations,” Trans. ASME 91, 100–104 (1969).
    [CrossRef]
  7. E. A. Taft, H. R. Phillip, “Optical properties of graphite,” Phys. Rev. 138, A197–A202 (1965).
    [CrossRef]
  8. M. W. Williams, E. T. Arakawa, “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, 263–283 (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.
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    [CrossRef] [PubMed]
  13. I. Simon, H. O. McMahon, “Study of the structure of quartz, cristobalite, and vitreous silica by reflection in infrared,” J. Chem. Phys. 21, 23–30 (1953).
    [CrossRef]
  14. T. S. Robinson, W. C. Price, “The determination of the infrared absorption spectra from reflection measurements,” Proc. Phys. Soc. London Ser. B 66, 969–974 (1953).
    [CrossRef]
  15. 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]
  16. D. G. Avery, “An improved method for measurements of optical constants by reflection,” Proc. Phys. Soc. London Ser. B 65, 425–428 (1952).
    [CrossRef]
  17. 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 Ser. B 77, 949–957 (1961).
  18. R. F. Miller, A. J. Taylor, L. S. Julien, “The optimum angle of incidence for determining optical constants from reflectance measurements,” J. Phys. D 3, 1957–1961 (1970).
    [CrossRef]
  19. B. J. Stagg, T. T. Charalampopoulos, “Surface-roughness effects on the determination of optical properties of materials by the reflection method,” Appl. Opt. 30, 4113–4118 (1991).
    [CrossRef] [PubMed]
  20. 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]
  21. B. J. Stagg, T. T. Charalampopoulos, “Method for azimuthal alignment in fixed angle ellipsommetry,” Appl. Opt. 31, 479–484 (1992).
    [CrossRef] [PubMed]
  22. D. M. Kolb, “Determination of the optical constants of solids by reflectance—ratio measurements at non-normal incidence,” J. Opt. Soc. Am. 62, 599–600 (1972).
    [CrossRef]

1992

1991

1990

1985

1984

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, 263–283 (1984).
[CrossRef]

1981

1972

D. M. Kolb, “Determination of the optical constants of solids by reflectance—ratio measurements at non-normal incidence,” J. Opt. Soc. Am. 62, 599–600 (1972).
[CrossRef]

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

1970

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

1969

W. H. Dalzell, A. F. Sarofim, “Optical constant of soot and their application to heat-flux calculations,” Trans. ASME 91, 100–104 (1969).
[CrossRef]

1965

1963

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

1962

L. A. Gilbert, “Refractive indices and absorption coefficients of coal in bulk measured in the range 6000 to 24000 Å by a polarized light technique,” Fuel 41, 351–358 (1962).

1961

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 Ser. B 77, 949–957 (1961).

1958

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

1955

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

J. R. Beattie, “Optical constants of metals in the infrared—experimental methods,” Philos. Mag. 46, 235–244 (1955).

1953

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

T. S. Robinson, W. C. Price, “The determination of the infrared absorption spectra from reflection measurements,” Proc. Phys. Soc. London Ser. B 66, 969–974 (1953).
[CrossRef]

1952

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

Arakawa, E. T.

M. W. Williams, E. T. Arakawa, “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. London Ser. 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 measurement,” Philos. Mag. 46, 222–234 (1955).

J. R. Beattie, “Optical constants of metals in the infrared—experimental methods,” Philos. Mag. 46, 235–244 (1955).

Charalampopoulos, T. T.

Conn, G. K. T.

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

Dalzell, W. H.

W. H. Dalzell, A. F. Sarofim, “Optical constant of soot and their application to heat-flux calculations,” Trans. ASME 91, 100–104 (1969).
[CrossRef]

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, 263–283 (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 6000 to 24000 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.

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 Ser. B 77, 949–957 (1961).

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, 263–283 (1984).
[CrossRef]

Julien, L. S.

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

Kolb, D. M.

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 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 reflectance 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.

Phillip, H. R.

E. A. Taft, H. R. Phillip, “Optical properties of graphite,” Phys. Rev. 138, A197–A202 (1965).
[CrossRef]

Price, W. C.

T. S. Robinson, W. C. Price, “The determination of the infrared absorption spectra from reflection measurements,” Proc. Phys. Soc. London Ser. B 66, 969–974 (1953).
[CrossRef]

Rivera, R.

Robinson, T. S.

T. S. Robinson, W. C. Price, “The determination of the infrared absorption spectra from reflection measurements,” Proc. Phys. Soc. London Ser. B 66, 969–974 (1953).
[CrossRef]

Sarofim, A. F.

W. H. Dalzell, A. F. Sarofim, “Optical constant of soot and their application to heat-flux calculations,” Trans. ASME 91, 100–104 (1969).
[CrossRef]

Simon, I.

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

Stagg, B. J.

Taft, E. A.

E. A. Taft, H. R. Phillip, “Optical properties of graphite,” Phys. Rev. 138, A197–A202 (1965).
[CrossRef]

Taylor, A. J.

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

Tomaselli, V. P.

Williams, M. W.

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

Appl. Opt.

Carbon

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

Combust. Sci. Technol.

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, 263–283 (1984).
[CrossRef]

Fuel

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 6000 to 24000 Å by a polarized light technique,” Fuel 41, 351–358 (1962).

J. Appl. Phys.

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

J. Chem. Phys.

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

J. Opt. Soc. Am.

J. Phys. D

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

Philos. Mag.

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

J. R. Beattie, “Optical constants of metals in the infrared—experimental methods,” Philos. Mag. 46, 235–244 (1955).

Phys. Rev.

E. A. Taft, H. R. Phillip, “Optical properties of graphite,” Phys. Rev. 138, A197–A202 (1965).
[CrossRef]

Proc. Phys. Soc. London Ser. B

T. S. Robinson, W. C. Price, “The determination of the infrared absorption spectra from reflection measurements,” Proc. Phys. Soc. London Ser. B 66, 969–974 (1953).
[CrossRef]

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

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 Ser. B 77, 949–957 (1961).

Trans. ASME

W. H. Dalzell, A. F. Sarofim, “Optical constant of soot and their application to heat-flux calculations,” Trans. ASME 91, 100–104 (1969).
[CrossRef]

Other

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.

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Figures (12)

Fig. 1
Fig. 1

Sensitivity chart for R||/R at the angles of incidence of 60° and 80°.

Fig. 2
Fig. 2

Sensitivity chart for R||/R at the angles of incidence of 45° and 75°.

Fig. 3
Fig. 3

Sensitivity chart for R||/R at the angles of incidence of 25° and 45°.

Fig. 4
Fig. 4

Sensitivity chart for reflectivities in both planes of polarization (R|| and R) at the angle of incidence of 16°.

Fig. 5
Fig. 5

Sensitivity chart for reflectivities in both planes of polarization (R|| and R) at the angle of incidence of 75°.

Fig. 6
Fig. 6

Expanded view of one grid block area in the (x, y) plane.

Fig. 7
Fig. 7

Optimum angles of incidence for method I.

Fig. 8
Fig. 8

Normalized sensitivity for method I as function of the real (n) and imaginary (k) parts of the refractive index evaluated at the optimum angles of incidence.

Fig. 9
Fig. 9

Optimum angle of incidence for method II.

Fig. 10
Fig. 10

Normalized sensitivity for method II as function of the real (n) and imaginary (k) parts of the refractive index evaluated at the optimum angle of incidence.

Fig. 11
Fig. 11

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

Fig. 12
Fig. 12

Parallel (R||) and perpendicular (R) reflectivities of the carbon-rob surface at the wavelength λ = 3.5 μm and specularity index S.I. = 0.99 as function of the angle of incidence. Open diamonds are the experimental data and solid curves correspond to the predictions of the Fresnel equations. The inferred refractive index is m ¯ = 3.875 − 1.534i.

Tables (2)

Tables Icon

Table 1 Real (n) and Imaginary (k) Parts of the Refractive Index Inferred from the Inversion of Theoretical Valuesa Perturbed by a −5% Error

Tables Icon

Table 2 Refractive Indices of an Amorphous Rod at a 3.5-μm Wavelength Inferred from the Inversion of Data in the Angular Range and in Two Different Sets of Angles

Equations (15)

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

R = I , r I , i ,
R = I , r I , i ,
R = a 2 + b 2 - 2 a cos θ + cos 2 θ a 2 + b 2 + 2 a cos θ + cos 2 θ ,
R = a 2 + b 2 - 2 a sin θ tan θ + sin 2 θ tan 2 θ a 2 + b 2 + 2 a sin θ tan θ + sin 2 θ tan 2 θ R ,
2 a 2 = [ ( n 2 - k 2 - sin 2 θ ) 2 + 4 n 2 k 2 ] 1 / 2 + ( n 2 - k 2 - sin 2 θ ) , 2 b 2 = [ ( n 2 - k 2 - sin 2 θ ) 2 + 4 n 2 k 2 ] 1 / 2 - ( n 2 - k 2 - sin 2 θ ) .
R 2 R | θ = 45 ° = 1.0 ,
A = a b sin ( γ ) ,
a = { [ x ( n + Δ n , k ) - x ( n , k ) ] 2 + [ y ( n + Δ n , k ) - y ( n , k ) ] 2 } 1 / 2 Δ n [ ( x n ) 2 + ( y n ) 2 ] 1 / 2 , b Δ k [ ( x k ) 2 + ( y k ) 2 ] 1 / 2 .
α = tan - 1 [ ( y / n ) ( x / n ) ] , β = tan - 1 [ ( y / k ) ( x / k ) ] .
sin ( γ ) = { ± [ 1 + tan 2 ( α ) ] - 1 / 2 } × { ± [ 1 + tan 2 ( β ) ] - 1 / 2 } { tan ( β ) - tan ( α ) } .
A ( Δ n ) ( Δ k ) | ( x n ) ( y k ) - ( x k ) ( y n ) | .
A N = A ( Δ n ) ( Δ k ) .
F = A N - 2 .
F = i = 1 N p G th , i - G exp , i ] 2 ,
G = R R .

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