Abstract

Separation of the scattering contribution from the total optical attenuation is necessary to determine the absorption portion of nonhomogeneous media such as naturally occurring minerals. In order to investigate experimentally the applicability of a previously developed two-flux radiative transfer model that takes into account surface reflections and collimated incidence, we have measured pulverized and sintered scattering standards prepared from a glass of known absorption coefficient variation. The new model produces an accuracy improvement up to a factor of 2.5 over the Kubelka-Munk theory. Off-axis scattering measurements were made with improved instrumentation between 0.33 μm and 2.7 μm. The model was then applied to the mineral rhodochrosite in this range to obtain accurate values of scattering and absorption.

© 1973 Optical Society of America

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References

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  1. W. W. Wendlandt, H. G. Hecht, Reflectance Spectroscopy (Wiley, New York, 1966), pp. 27–33.
  2. P. Kubelka, F. Munk, Z. Tech. Phys. 12, 593 (1931).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  5. W. G. Egan, T. Hilgeman, Appl. Opt. 10, 2132 (1971).
    [CrossRef] [PubMed]
  6. J. Reichman, Appl. Opt. 12, this issue 0000 (1973).
    [CrossRef] [PubMed]
  7. S. E. Orchard, J. Opt. Soc. Am. 59, 1584 (1969).
    [CrossRef]
  8. Corning Color Filter Glasses, Technical Optics, Optical Products Department, Corning Glass Works, Corning, N.Y. (1970).
  9. Handbook of Chemistry and Physics, C. D. Hodgman, Ed. (Chemical Rubber Publishing Co., Cleveland, Ohio, 1962), 44th ed., p. 3091.
  10. W. G. Egan, J. F. Becker, Appl. Opt. 8, 720 (1969).
    [CrossRef] [PubMed]
  11. W. G. Egan, Icarus 10, 223 (1969).
    [CrossRef]

1973 (1)

J. Reichman, Appl. Opt. 12, this issue 0000 (1973).
[CrossRef] [PubMed]

1971 (1)

1969 (3)

1968 (1)

1948 (1)

1931 (1)

P. Kubelka, F. Munk, Z. Tech. Phys. 12, 593 (1931).

Becker, J. F.

Egan, W. G.

Hecht, H. G.

W. W. Wendlandt, H. G. Hecht, Reflectance Spectroscopy (Wiley, New York, 1966), pp. 27–33.

Hilgeman, T.

Hunt, G. R.

Kubelka, P.

P. Kubelka, J. Opt. Soc. Am. 38, 448 (1948).
[CrossRef] [PubMed]

P. Kubelka, F. Munk, Z. Tech. Phys. 12, 593 (1931).

Munk, F.

P. Kubelka, F. Munk, Z. Tech. Phys. 12, 593 (1931).

Orchard, S. E.

Reichman, J.

J. Reichman, Appl. Opt. 12, this issue 0000 (1973).
[CrossRef] [PubMed]

Vincent, R. K.

Wendlandt, W. W.

W. W. Wendlandt, H. G. Hecht, Reflectance Spectroscopy (Wiley, New York, 1966), pp. 27–33.

Appl. Opt. (4)

Icarus (1)

W. G. Egan, Icarus 10, 223 (1969).
[CrossRef]

J. Opt. Soc. Am. (2)

Z. Tech. Phys. (1)

P. Kubelka, F. Munk, Z. Tech. Phys. 12, 593 (1931).

Other (3)

W. W. Wendlandt, H. G. Hecht, Reflectance Spectroscopy (Wiley, New York, 1966), pp. 27–33.

Corning Color Filter Glasses, Technical Optics, Optical Products Department, Corning Glass Works, Corning, N.Y. (1970).

Handbook of Chemistry and Physics, C. D. Hodgman, Ed. (Chemical Rubber Publishing Co., Cleveland, Ohio, 1962), 44th ed., p. 3091.

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Figures (10)

Fig. 1
Fig. 1

Solutions for β and τ in terms of transmittance (T) and reflectance (R) for n = 1.5.

Fig. 2
Fig. 2

Microphotograph of sintered CS1-64 glass containing air bubbles.

Fig. 3
Fig. 3

Distribution of pore diameters in sintered CS1-64 bubble glass shown in Fig. 2.

Fig. 4
Fig. 4

Layout of optical system of Perkin-Elmer type 13U spectrograph for diffuse reflectance and transmission measurements.

Fig. 5
Fig. 5

Weighted diffuse scattering curves for MgCO3 and CS1-64 powder standard at 0.400 μm; T = 0.65% and R = 91.3% relative to MgCO3 (see text).

Fig. 6
Fig. 6

Absolute 2π reflectance MgCO3 between 0.25 μm and 2.3 μm as determined on a Gier-Dunkle absolute integrating sphere; calibration beyond 2.3 μ based on our own goniometer measurements.

Fig. 7
Fig. 7

Absolute diffuse transmission and reflection measurements on 0.0432-cm thin section of rhodochrosite.

Fig. 8
Fig. 8

Plot of nk between 0.33 μm and 2.7 μm for rhodochrosite.

Fig. 9
Fig. 9

Plot of s between 0.33 μm and 2.7 μm for rhodochrosite.

Fig. 10
Fig. 10

Reflectance of rhodochrosite measured under various experimental arrangements. - - - - Polished bulk rhodochrosite (diffuse); — -rhodochrosite powder <1 μ (diffuse relative to MgO);— — polished bulk rhodochrosite (specular at 60° incidence); — – — polished bulk rhodochrosite (specular at 8° incidence; expanded scale at right).

Tables (3)

Tables Icon

Table I Index to Curves on Fig. 1

Tables Icon

Table II Corning CS1-64 Glass: Measured Absolute Percent Reflectance (R) and Transmission (T)

Tables Icon

Table III Corning CS1-64 Glass: Data Analysis

Equations (15)

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T = exp ( - α t ) ,
N = n - i n k ,
α = 4 π n k / λ ,
R D = R [ 1 - exp ( - 2 τ ) ] / [ 1 - R 2 exp ( - 2 τ ) ] ,
T D = ( 1 - R 2 ) exp ( - τ ) / [ 1 - R 2 exp ( - 2 τ ) ] ,
R = ( 1 - β ) / ( 1 + β ) ,
β = [ α / ( α + s ) ] 1 / 2 ,
τ = 2 [ α ( α + s ) ] 1 / 2 d .
T = ( { [ T c + ρ 0 R c exp ( - Γ ) ] ( 1 - ρ i R D ) + ρ i T D [ R c + ρ 0 T c exp ( - Γ ) ] } ( 1 - ρ 0 ) ( 1 - ρ i ) / [ 1 - ρ 0 2 exp ( - 2 Γ ) ] [ ( 1 - ρ i R D ) 2 - ρ i 2 T D 2 ] ) + ( 1 - ρ 0 ) 2 exp ( - Γ ) / [ 1 - ρ 0 2 exp ( - 2 Γ ) ] ,
R = { [ R c + ρ 0 T c exp ( - Γ ) ] ( 1 - ρ i R D ) + ρ i T c T D + ρ i ρ 0 T D R c exp ( - Γ ) } ( 1 - ρ i ) ( 1 - ρ 0 ) / [ 1 - ρ 0 2 exp ( - 2 Γ ) ] [ ( 1 - ρ i R D ) 2 - ρ i 2 T D 2 ] + { ( 1 - ρ 0 ) 2 ρ 0 exp ( - 2 Γ ) / [ 1 - ρ 0 2 exp ( - 2 Γ ) ] } + ρ 0 ,
ρ 0 = normal incidence surface reflection = [ ( n - 1 ) / ( n + 1 ) ] 2 ,
ρ i = internal diffuse surface reflectance = 1 - [ ( 1 - ρ e ) ] / [ n 2 ]
ρ e = external diffuse surface reflectance , - 0.4399 + 0.7099 n - 0.3319 n 2 + 0.0636 n 3 ( curve fit totabular data of Ref . 7 ) ,
T c = [ ( 1 - β 2 ) / ( 8 β 2 - 2 ) ] ( { ( R - 3 R 2 ) exp [ - ( 2 τ + Γ ) ] + ( 3 - R ) exp ( - Γ ) + 3 ( R 2 - 1 ) exp ( - τ ) } / [ 1 - R 2 exp ( - 2 τ ) ] ) ,
R c = [ ( 1 - β 2 ) / ( 8 β 2 - 2 ) ] ( { ( R 2 - 1 ) exp [ 1 - ( τ + Γ ) ] + exp ( - 2 τ ) ( 3 R - R 2 ) + 1 - 3 R } / [ 1 - R 2 exp ( - 2 τ ) ] ) .

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