Determination of Kubelka–Munk scattering and absorption coefficients by diffuse illumination

Robert Molenaar; Jaap J. ten Bosch; Jaap R. Zijp

doi:10.1364/AO.38.002068

Applied Optics
Vol. 38,
Issue 10,
pp. 2068-2077
(1999)
•https://doi.org/10.1364/AO.38.002068

Determination of Kubelka–Munk scattering and absorption coefficients by diffuse illumination

Robert Molenaar, Jaap J. ten Bosch, and Jaap R. Zijp

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Robert Molenaar, Jaap J. ten Bosch, and Jaap R. Zijp, "Determination of Kubelka–Munk scattering and absorption coefficients by diffuse illumination," Appl. Opt. 38, 2068-2077 (1999)

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Abstract

The Kubelka–Munk theory, although it provides an equation that relates the reflection of a sample under diffuse illumination to certain of its properties, does not take boundary reflectance into account. Boundary reflection is always present because there is always a difference between the refractive indices of the sample and of the surrounding medium. We describe how a half-sphere is used to achieve diffuse illumination, and we present and exemplify equations that correct for boundary reflection with measurements of four composite restorative dental materials. The refractive index of the sample is measured with a matching technique that employs a glycerol–water mixture. Edge loss errors are estimated.

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Assumed True Values of Hypothetical Sample				Calculated True Reflectances with Two Backgrounds		Values Calculated from True Reflectances, Assuming no Specular Reflectance [Eq. (5)]
d	n	S	K	Black, $R_{0}^{'}$	White, R ^′	S	Error (%)	K	Error (%)
1	1.4	1.0	0.010	0.592	0.962	1.16	+16	0.020	+100
1	1.5	1.0	0.020	0.525	0.918	1.19	+19	0.045	+125
2	1.4	1.5	0.010	0.666	0.929	1.09	-27	0.019	+90

Material	Thickness (mm)	Area (mm²)
Kerr prodigy A2
Thick	1.71 ± 0.02	294
Thin	0.412 ± 0.004	265
Enamel plus Dentina A2
Thick	1.79 ± 0.03	224
Thin	0.36 ± 0.01	94
Enamel plus Smalto White 1
Thick	1.81 ± 0.02	273
Thin	0.38 ± 0.02	213
Enamel plus Smalto Clear 1
Thick	1.85 ± 0.01	259
Thin	0.436 ± 0.003	189

Sample	Wavelength
Sample	n (λ = 633 nm)	n (λ = 594 nm)	n (λ = 543 nm)
Kerr Prodigy A2	1.4035 ± 0.0008	1.4095 ± 0.0006	1.4141 ± 0.0004
Enamel Dentina A2	1.4052 ± 0.0008	1.4080 ± 0.0004	1.4094 ± 0.0002
Enamel Smalto White I	1.4031 ± 0.0023	1.4098 ± 0.0005	1.4142 ± 0.0005
Enamel Smalto Clear I	1.4037 ± 0.0007	1.4102 ± 0.0007	1.4149 ± 0.0005

Data	Wavelength (nm)
Data	543.5	594.1	632.8
Thick sample
$R_{0}^{'}$	0.492	0.535	0.534
T ^′	0.277	0.325	0.341
$R_{above}^{'}$	0.417	0.422	0.409
R ₀	0.570	0.594	0.583
T	0.297	0.329	0.349
r _i	0.539	0.536	0.532
S (mm^-1)	1.030	1.008	0.943
K (mm^-1)	0.088	0.048	0.042
Thin sample
$R_{0}^{'}$	0.355	0.352	0.342
T ^′	0.575	0.609	0.623
$R_{above}^{'}$	0.220	0.210	0.202
R ₀	0.205	0.189	0.176
T	0.741	0.781	0.796
r _i	0.298	0.283	0.269

Data	Wavelength (nm)
Data	543.5	594.1	632.8
Thick sample
$R_{0}^{'}$	0.559	0.596	0.591
T ^′	0.224	0.270	0.288
$R_{above}^{'}$	0.511	0.516	0.500
R ₀	0.665	0.684	0.672
T	0.210	0.242	0.262
r _i	0.536	0.535	0.533
S (mm^-1)	1.548	1.471	1.351
K (mm^-1)	0.079	0.044	0.039
Thin sample
$R_{0}^{'}$	0.389	0.381	0.369
T ^′	0.576	0.607	0.623
$R_{above}^{'}$	0.224	0.210	0.200
R ₀	0.225	0.200	0.183
T	0.750	0.791	0.796
r _i	0.352	0.332	0.315

Data	Wavelength (nm)
Data	543.5	594.1	632.8
Thick sample
$R_{0}^{'}$	0.283	0.280	0.275
T ^′	0.614	0.632	0.642
$R_{above}^{'}$	0.159	0.156	0.153
R ₀	0.112	0.107	0.103
T	0.801	0.816	0.824
r _i	0.244	0.229	0.222
Thin sample
$R_{0}^{'}$	0.191	0.189	0.187
T ^′	0.788	0.797	0.801
$R_{above}^{'}$	0.119	0.117	0.116
R ₀	0.090	0.046	0.046
T	0.887	0.940	0.942
r _i	0.544	0.096	0.094

Data	Wavelength (nm)
Data	543.5	594.1	632.8
Thick sample
$R_{0}^{'}$	0.310	0.306	0.301
T ^′	0.574	0.593	0.605
$R_{above}^{'}$	0.185	0.180	0.177
R ₀	0.152	0.144	0.139
T	0.753	0.773	0.782
r _i	0.277	0.264	0.252
Thin sample
$R_{0}^{'}$	0.194	0.192	0.190
T ^′	0.784	0.797	0.798
$R_{above}^{'}$	0.118	0.117	0.115
R ₀	0.047	0.046	0.045
T	0.932	0.940	0.943
r _i	0.104	0.100	0.099

Data	Wavelength (nm)
Data	543.5	594.1	632.8
Thick sample
$R_{0}^{'}$	0.283	0.280	0.275
T ^′	0.614	0.632	0.642
$R_{above}^{'}$	0.159	0.156	0.153
R ₀	0.112	0.107	0.103
T	0.801	0.816	0.824
r _i	0.244	0.229	0.222
Thin sample
$R_{0}^{'}$	0.191	0.189	0.187
T ^′	0.788	0.797	0.801
$R_{above}^{'}$	0.119	0.117	0.116
R ₀	0.090	0.046	0.046
T	0.887	0.940	0.942
r _i	0.544	0.096	0.094

Data	Wavelength (nm)
Data	543.5	594.1	632.8
Thick sample
$R_{0}^{'}$	0.310	0.306	0.301
T ^′	0.574	0.593	0.605
$R_{above}^{'}$	0.185	0.180	0.177
R ₀	0.152	0.144	0.139
T	0.753	0.773	0.782
r _i	0.277	0.264	0.252
Thin sample
$R_{0}^{'}$	0.194	0.192	0.190
T ^′	0.784	0.797	0.798
$R_{above}^{'}$	0.118	0.117	0.115
R ₀	0.047	0.046	0.045
T	0.932	0.940	0.943
r _i	0.104	0.100	0.099

Abstract

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Tables (7)

Equations (31)

Applied Optics