Total Reflectances of Opaque Diffusers

W. R. Blevin; W. J. Brown

doi:10.1364/JOSA.52.001250

Journal of the Optical Society of America
Vol. 52,
Issue 11,
pp. 1250-1255
(1962)
•https://doi.org/10.1364/JOSA.52.001250

Total Reflectances of Opaque Diffusers

W. R. Blevin and W. J. Brown

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W. R. Blevin and W. J. Brown, "Total Reflectances of Opaque Diffusers," J. Opt. Soc. Am. 52, 1250-1255 (1962)

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Abstract

The change in reflectance of a diffuser, due to a known change in the ratio of its absorption and scattering coefficients, can be predicted from the simple Kubelka–Munk equations or from Giovanelli’s more detailed theory. These predictions may differ by up to about 0.01 in reflectance. Differences of this magnitude have also been found between reflectance changes predicted by Giovanelli’s theory and measured changes. There is evidence that diffuser theory cannot give greater accuracy unless based on a scattering distribution which is more realistic than isotropic scattering. In the absence of such a theory, the Kubelka–Munk theory is as reliable as any for practical reflectance problems with opaque diffusers.

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		Reflectance

${\tilde{ω}}_{0}$	α/σ	n=1	n=1.333	n=1.46	n=1.50
1.00000	0.00000	1.0000	1.0000	1.0000	1.0000

0.99999	0.00001	0.991	0.986	0.983	0.982
0.99997	0.00003	0.984	0.975	0.971	0.970
0.99990	0.00010	0.972	0.956	0.948	0.945
0.99970	0.00030	0.951	0.925	0.913	0.909
0.99900	0.00100	0.9128	0.868	0.848	0.8414

0.9975	0.00251	0.866	0.802	0.774	0.765
0.9950	0.00503	0.8170	0.736	0.701	0.6910
0.9900	0.01010	0.7528	0.6520	0.613	0.6020
0.9750	0.02564	0.6409	0.520	0.479	0.4685
0.9500	0.05263	0.5356	0.4073	0.370	0.3612

0.9	0.1111	0.4150	0.295	0.266	0.2600
0.8	0.2500	0.2853	0.1898	0.174	0.1720
0.7	0.4286	0.2087	0.137	0.129	0.1289
0.6	0.6667	0.1554	0.1030	0.101	0.1024
0.5	1.0000	0.1152	0.079	0.082	0.0841

0.4	1.500	0.0834	0.061	0.068	0.0708
0.3	2.333	0.0572	0.047	0.057	0.0605
0.2	4.000	0.0352	0.036	0.048	0.0523
0.1	9.000	0.0164	0.028	0.041	0.0456
0.0	∞	0.0000	0.0204	0.0350	0.0400

	Measured reflectance				(Measured reflectance–predicted reflectance)
(i) Mixture of aqueous suspensions of barium sulfate and carbon.
Volume of carbon suspension (ml)	546 mμ				546 mμ
0.0000	0.889				p
0.0014	0.884				−0.001
0.0044	0.876				−0.002
0.0144	0.852				−0.001
0.0444	0.801				−0.005
0.1444	0.703				−0.008
0.4444	0.565				−0.008
1.444	0.389				−0.005
4.444	0.234				p
14.44	0.108				−0.007
86.44	0.036				−0.002
(ii) Mixture of aqueous suspensions of zinc oxide and carbon.
	400 mμ	500 mμ	600 mμ	700 mμ	400 mμ	500 mμ	600 mμ	700 mμ

0.00000	0.808	0.949	0.938	0.913	p	p	p	p
0.00014	0.806	0.944	0.935	0.910	−0.001	+0.001	+0.003	+0.001
0.00044	0.804	0.933	0.923	0.899	0.000	+0.002	+0.001	−0.002
0.00144	0.795	0.905	0.895	0.877	−0.001	+0.002	0.000	−0.002
0.00444	0.780	0.856	0.846	0.832	+0.003	+0.007	+0.003	+0.001
0.01444	0.727	0.765	0.757	0.744	+0.003	+0.005	+0.003	−0.001
0.04444	0.638	0.641	0.630	0.618	+0.016	+0.012	+0.009	+0.006
0.1444	0.492	0.469	0.458	0.445	+0.023	+0.014	+0.011	+0.006
0.4444	0.320	0.286	0.279	0.271	+0.012	p	p	p
1.444	0.159	0.137	0.131	0.126	p	−0.009	−0.011	−0.011
4.444	0.068	0.058	0.057	0.055	−0.011	−0.013	−0.011	−0.012
14.44	0.033	0.030	0.031	0.031	−0.006	−0.006	−0.004	−0.004
(iii) Mixture of aqueous suspensions of titanium dioxide (rutile) and carbon.
	400 mμ	500 mμ	600 mμ	700 mμ	400 mμ	500 mμ	600 mμ	700 mμ

0.00000	0.383	0.926	0.932	0.910	p	p	p	p
0.00003	0.383	0.928	0.928	0.909	0.000	+0.003	−0.003	0.000
0.00013	0.384	0.920	0.925	0.905	+0.001	0.000	−0.002	−0.001
0.00043	0.386	0.909	0.913	0.897	+0.003	0.000	−0.002	0.000
0.00143	0.385	0.876	0.880	0.871	+0.004	−0.004	−0.006	−0.003
0.00443	0.378	0.822	0.828	0.819	0.000	−0.001	−0.002	−0.006
0.01443	0.366	0.718	0.731	0.729	0.000	−0.006	−0.003	−0.004
0.04443	0.335	0.579	0.596	0.597	−0.003	−0.004	+0.001	−0.001
0.1444	0.268	0.400	0.420	0.423	0.000	−0.002	+0.005	+0.004
0.4444	0.178	0.240	0.252	0.254	+0.003	p	p	p
1.444	0.093	0.113	0.118	0.118	p	−0.006	−0.008	−0.010
4.444	0.049	0.056	0.055	0.056	+0.001	−0.002	−0.006	−0.006
14.44	0.032	0.035	0.034	0.032	+0.002	+0.003	+0.001	−0.001
(iv) Commercially supplied aqueous suspension of yellow pigment mixed with an aqueous suspension of carbon.
	500 mμ	525 mμ	550 mμ	700 mμ	500 mμ	525 mμ	550 mμ	700 mμ

0.00	0.160	0.566	0.758	0.885	p	p	p	p
0.01	0.159	0.567	0.754	0.862	−0.001	+0.002	0.000	−0.004
0.04	0.158	0.562	0.745	0.817	−0.002	0.000	0.000	−0.009
0.14	0.158	0.539	0.712	0.730	−0.002	−0.016	−0.007	−0.014
0.44	0.160	0.525	0.646	0.603	+0.001	−0.010	−0.011	−0.013
1.44	0.156	0.469	0.529	0.436	−0.001	−0.010	−0.011	−0.010
4.44	0.151	0.374	0.381	0.278	−0.001	−0.002	−0.002	p
14.44	0.135	0.240	0.223	0.145	−0.002	0.000	p	+0.003
44.44	0.103	0.130	0.115	0.074	−0.004	p	0.000	+0.005
144.4	0.067	0.065	0.058	0.051	p	+0.003	+0.004	+0.016
(v) White enamel paint greatly diluted in mineral turpentine, mixed with black enamel paint.
Volume of black paint (ml)	400 mμ	500 mμ	600 mμ	700 mμ	400 mμ	500 mμ	600 mμ	700 mμ
Volume of black paint (ml)
0.000	0.350	0.873	0.913	0.927	p	p	p	p
0.025	0.349	0.848	0.877	0.885	0.000	−0.001	−0.001	0.000
0.10	0.347	0.797	0.818	0.818	0.000	−0.001	0.000	0.000
0.35	0.334	0.701	0.713	0.712	−0.002	+0.006	+0.002	+0.007
1.1	0.311	0.562	0.575	0.573	−0.002	+0.004	+0.005	+0.011
3.6	0.258	0.390	0.399	0.398	−0.001	+0.008	+0.006	+0.015
13.6	0.164	0.209	0.218	0.209	+0.001	p	p	p
43.6	0.094	0.107	0.105	0.102	p	−0.002	−0.009	−0.007
143.6	0.057	0.058	0.057	0.054	0.000	−0.004	−0.006	−0.008
(vi) Red enamel paint greatly diluted in mineral turpentine, mixed with black enamel paint.
	620 mμ	660 mμ		700 mμ	620 mμ	660 mμ		700 mμ

0.00	0.476	0.742		0.779	p	p		p
0.01	0.472	0.731		0.760	−0.001	+0.001		0.000
0.04	0.463	0.708		0.716	−0.003	+0.012		+0.004
0.14	0.445	0.621		0.622	0.000	+0.003		+0.005
0.44	0.398	0.499		0.484	+0.005	+0.006		+0.005
1.44	0.304	0.334		0.314	+0.005	+0.003		+0.002
4.44	0.200	0.197		0.179	+0.008	p		p
14.44	0.104	0.096		0.078	p	−0.005		−0.015
44.44	0.059	0.056		0.053	−0.003	−0.005		−0.004
130.4	0.046	0.044		0.042	+0.002	0.000		0.000

		Reflectance

${\tilde{ω}}_{0}$	α/σ	n=1	n=1.333	n=1.46	n=1.50
1.00000	0.00000	1.0000	1.0000	1.0000	1.0000

0.99999	0.00001	0.991	0.986	0.983	0.982
0.99997	0.00003	0.984	0.975	0.971	0.970
0.99990	0.00010	0.972	0.956	0.948	0.945
0.99970	0.00030	0.951	0.925	0.913	0.909
0.99900	0.00100	0.9128	0.868	0.848	0.8414

0.9975	0.00251	0.866	0.802	0.774	0.765
0.9950	0.00503	0.8170	0.736	0.701	0.6910
0.9900	0.01010	0.7528	0.6520	0.613	0.6020
0.9750	0.02564	0.6409	0.520	0.479	0.4685
0.9500	0.05263	0.5356	0.4073	0.370	0.3612

0.9	0.1111	0.4150	0.295	0.266	0.2600
0.8	0.2500	0.2853	0.1898	0.174	0.1720
0.7	0.4286	0.2087	0.137	0.129	0.1289
0.6	0.6667	0.1554	0.1030	0.101	0.1024
0.5	1.0000	0.1152	0.079	0.082	0.0841

0.4	1.500	0.0834	0.061	0.068	0.0708
0.3	2.333	0.0572	0.047	0.057	0.0605
0.2	4.000	0.0352	0.036	0.048	0.0523
0.1	9.000	0.0164	0.028	0.041	0.0456
0.0	∞	0.0000	0.0204	0.0350	0.0400

	Measured reflectance				(Measured reflectance–predicted reflectance)
(i) Mixture of aqueous suspensions of barium sulfate and carbon.
Volume of carbon suspension (ml)	546 mμ				546 mμ
0.0000	0.889				p
0.0014	0.884				−0.001
0.0044	0.876				−0.002
0.0144	0.852				−0.001
0.0444	0.801				−0.005
0.1444	0.703				−0.008
0.4444	0.565				−0.008
1.444	0.389				−0.005
4.444	0.234				p
14.44	0.108				−0.007
86.44	0.036				−0.002
(ii) Mixture of aqueous suspensions of zinc oxide and carbon.
	400 mμ	500 mμ	600 mμ	700 mμ	400 mμ	500 mμ	600 mμ	700 mμ

0.00000	0.808	0.949	0.938	0.913	p	p	p	p
0.00014	0.806	0.944	0.935	0.910	−0.001	+0.001	+0.003	+0.001
0.00044	0.804	0.933	0.923	0.899	0.000	+0.002	+0.001	−0.002
0.00144	0.795	0.905	0.895	0.877	−0.001	+0.002	0.000	−0.002
0.00444	0.780	0.856	0.846	0.832	+0.003	+0.007	+0.003	+0.001
0.01444	0.727	0.765	0.757	0.744	+0.003	+0.005	+0.003	−0.001
0.04444	0.638	0.641	0.630	0.618	+0.016	+0.012	+0.009	+0.006
0.1444	0.492	0.469	0.458	0.445	+0.023	+0.014	+0.011	+0.006
0.4444	0.320	0.286	0.279	0.271	+0.012	p	p	p
1.444	0.159	0.137	0.131	0.126	p	−0.009	−0.011	−0.011
4.444	0.068	0.058	0.057	0.055	−0.011	−0.013	−0.011	−0.012
14.44	0.033	0.030	0.031	0.031	−0.006	−0.006	−0.004	−0.004
(iii) Mixture of aqueous suspensions of titanium dioxide (rutile) and carbon.
	400 mμ	500 mμ	600 mμ	700 mμ	400 mμ	500 mμ	600 mμ	700 mμ

0.00000	0.383	0.926	0.932	0.910	p	p	p	p
0.00003	0.383	0.928	0.928	0.909	0.000	+0.003	−0.003	0.000
0.00013	0.384	0.920	0.925	0.905	+0.001	0.000	−0.002	−0.001
0.00043	0.386	0.909	0.913	0.897	+0.003	0.000	−0.002	0.000
0.00143	0.385	0.876	0.880	0.871	+0.004	−0.004	−0.006	−0.003
0.00443	0.378	0.822	0.828	0.819	0.000	−0.001	−0.002	−0.006
0.01443	0.366	0.718	0.731	0.729	0.000	−0.006	−0.003	−0.004
0.04443	0.335	0.579	0.596	0.597	−0.003	−0.004	+0.001	−0.001
0.1444	0.268	0.400	0.420	0.423	0.000	−0.002	+0.005	+0.004
0.4444	0.178	0.240	0.252	0.254	+0.003	p	p	p
1.444	0.093	0.113	0.118	0.118	p	−0.006	−0.008	−0.010
4.444	0.049	0.056	0.055	0.056	+0.001	−0.002	−0.006	−0.006
14.44	0.032	0.035	0.034	0.032	+0.002	+0.003	+0.001	−0.001
(iv) Commercially supplied aqueous suspension of yellow pigment mixed with an aqueous suspension of carbon.
	500 mμ	525 mμ	550 mμ	700 mμ	500 mμ	525 mμ	550 mμ	700 mμ

0.00	0.160	0.566	0.758	0.885	p	p	p	p
0.01	0.159	0.567	0.754	0.862	−0.001	+0.002	0.000	−0.004
0.04	0.158	0.562	0.745	0.817	−0.002	0.000	0.000	−0.009
0.14	0.158	0.539	0.712	0.730	−0.002	−0.016	−0.007	−0.014
0.44	0.160	0.525	0.646	0.603	+0.001	−0.010	−0.011	−0.013
1.44	0.156	0.469	0.529	0.436	−0.001	−0.010	−0.011	−0.010
4.44	0.151	0.374	0.381	0.278	−0.001	−0.002	−0.002	p
14.44	0.135	0.240	0.223	0.145	−0.002	0.000	p	+0.003
44.44	0.103	0.130	0.115	0.074	−0.004	p	0.000	+0.005
144.4	0.067	0.065	0.058	0.051	p	+0.003	+0.004	+0.016
(v) White enamel paint greatly diluted in mineral turpentine, mixed with black enamel paint.
Volume of black paint (ml)	400 mμ	500 mμ	600 mμ	700 mμ	400 mμ	500 mμ	600 mμ	700 mμ
Volume of black paint (ml)
0.000	0.350	0.873	0.913	0.927	p	p	p	p
0.025	0.349	0.848	0.877	0.885	0.000	−0.001	−0.001	0.000
0.10	0.347	0.797	0.818	0.818	0.000	−0.001	0.000	0.000
0.35	0.334	0.701	0.713	0.712	−0.002	+0.006	+0.002	+0.007
1.1	0.311	0.562	0.575	0.573	−0.002	+0.004	+0.005	+0.011
3.6	0.258	0.390	0.399	0.398	−0.001	+0.008	+0.006	+0.015
13.6	0.164	0.209	0.218	0.209	+0.001	p	p	p
43.6	0.094	0.107	0.105	0.102	p	−0.002	−0.009	−0.007
143.6	0.057	0.058	0.057	0.054	0.000	−0.004	−0.006	−0.008
(vi) Red enamel paint greatly diluted in mineral turpentine, mixed with black enamel paint.
	620 mμ	660 mμ		700 mμ	620 mμ	660 mμ		700 mμ

0.00	0.476	0.742		0.779	p	p		p
0.01	0.472	0.731		0.760	−0.001	+0.001		0.000
0.04	0.463	0.708		0.716	−0.003	+0.012		+0.004
0.14	0.445	0.621		0.622	0.000	+0.003		+0.005
0.44	0.398	0.499		0.484	+0.005	+0.006		+0.005
1.44	0.304	0.334		0.314	+0.005	+0.003		+0.002
4.44	0.200	0.197		0.179	+0.008	p		p
14.44	0.104	0.096		0.078	p	−0.005		−0.015
44.44	0.059	0.056		0.053	−0.003	−0.005		−0.004
130.4	0.046	0.044		0.042	+0.002	0.000		0.000

Abstract

Cited By

Figures (2)

Tables (2)

Equations (3)

Journal of the Optical Society of America