Reflectance of Perfect Diffuse and Specular Samples in the Integrating Sphere

Bjarne J. Hisdal

doi:10.1364/JOSA.55.001122

Journal of the Optical Society of America
Vol. 55,
Issue 9,
pp. 1122-1125
(1965)
•https://doi.org/10.1364/JOSA.55.001122

Reflectance of Perfect Diffuse and Specular Samples in the Integrating Sphere

Bjarne J. Hisdal

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Bjarne J. Hisdal, "Reflectance of Perfect Diffuse and Specular Samples in the Integrating Sphere," J. Opt. Soc. Am. 55, 1122-1125 (1965)

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Abstract

Expressions for the reflectance of finite, perfect diffuse, and perfect specular samples measured in an integrating sphere wlth finite ports and a uniform, perfectly diffuse reflecting wall, are derived by solving a set of linear equations. The derivation is easy compared with the long computations necessary when using an integral equation approach.

Expressions are given both for the comparison and the substitution method of measurement, with curved or flat sample and standard. For finite, flat, diffuse, and specular samples, computed examples indicate the range of sample sizes within which previous approximate expressions may be used, without introducing significant errors.

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Sample and standard	Comparison method [Fig.1(a)]	Substitution method [Fig. 1(b)]
Curved sample and standard	(Ia) $\begin{array}{l} \frac{L_{1 s}}{L_{1 st}} & = \frac{r_{2} [r_{3} F_{32} F_{13} - F_{12} (r_{3} F_{33} - 1)]}{r_{3} [r_{2} F_{12} F_{23} - F_{13} (r_{2} F_{22} - 1)]} = \frac{r_{2}}{r_{3}} \\ r_{2} & = r_{3} \cdot \frac{L_{1 s}}{L_{1 st}} \end{array}$	(Ie) $\frac{r_{2} [(1 - r_{3} F_{33}) (1 - r_{1} F_{11}) - r_{1} r_{3} F_{31} \cdot F_{13}]}{r_{3} [(1 - r_{2} F_{22}) (1 - r_{1} F_{11}) - r_{1} r_{2} F_{21} \cdot F_{12}]} = \frac{r_{2} (1 - r_{3} \frac{A_{2}}{A} - r_{1} \frac{A_{1}}{A})}{r_{3} (1 - r_{2} \frac{A_{2}}{A} - r_{1} \frac{A_{1}}{A})} \frac{r_{3} \cdot \frac{L_{1 s}}{L_{1 st}} (1 - r_{1} \frac{A_{1}}{A})}{1 - r_{1} \frac{A_{1}}{A} - r_{3} \frac{A_{2}}{A} (1 - \frac{L_{1 s}}{L_{1 st}})}$
Flat sample (small-area approx.), curved standard with r₃ = r₁	(Ib) $\begin{array}{l} \frac{L_{1 s}}{L_{1 st}} & = \frac{r_{2}}{r_{1} (1 + r_{2} \frac{A_{2}}{A})} \\ r_{2} & = \frac{r_{1} \cdot \frac{L_{1 s}}{L_{1 st}}}{1 - \frac{L_{1 s}}{L_{1 st}} \cdot r_{1} \cdot \frac{A_{2}}{A}} \end{array}$	(If) $\begin{matrix} \frac{r_{2} [1 - r_{1} \frac{A_{1}}{A} - r_{1} \frac{A_{2}}{A}]}{r_{2} [1 - r_{1} \frac{A_{1}}{A} - r_{1} r_{2} \frac{A_{2}}{A} \cdot \frac{A_{1}}{A}]} \\ \frac{r_{1} \frac{L_{1 s}}{L_{1 st}} (1 - r_{1} \frac{A_{1}}{A})}{1 - r_{1} \frac{A_{1}}{A} - r_{1} \frac{A_{2}}{A} (1 - r_{1} \frac{A_{1}}{A} \cdot \frac{L_{1 s}}{L_{1 st}})} \end{matrix}$
Flat sample and standard (small area approx.)	(Ic) $\begin{array}{l} \frac{L_{1 s}}{L_{1 st}} & = \frac{r_{2} (1 + r_{3} \cdot \frac{A_{2}}{A})}{r_{3} (1 + r_{2} \cdot \frac{A_{2}}{A})} \\ r_{2} & = \frac{r_{3} \cdot \frac{L_{1 s}}{L_{1 st}}}{1 + r_{3} \cdot \frac{A_{2}}{A} (1 - \frac{L_{1 s}}{L_{1 st}}) \cdot} \end{array}$	(Ig) $\begin{matrix} \frac{r_{2} [1 - r_{1} \frac{A_{1}}{A} - r_{1} r_{3} \frac{A_{1}}{A} \cdot \frac{A_{2}}{A}]}{r_{3} [1 - r_{1} \frac{A_{1}}{A} - r_{1} r_{2} \frac{A_{1}}{A} \cdot \frac{A_{2}}{A}]} \\ \frac{r_{3} \frac{L_{1 s}}{L_{1 st}} (1 - r_{1} \frac{A_{1}}{A})}{1 - r_{1} \frac{A_{1}}{A} - r_{1} r_{3} \frac{A_{1}}{A} \cdot \frac{A_{2}}{A} (1 - \frac{L_{1 s}}{L_{1 st}})} \end{matrix}$
Flat sample and standard of finite area	(Id) $\begin{array}{l} \frac{L_{1 s}}{L_{1 st}} & = \frac{r_{2} (1 + r_{3} \frac{A_{2 c}}{A - A_{2 c}})}{r_{3} (1 + r_{2} \frac{A_{2 c}}{A - A_{2 c}})} \\ r_{2} & = \frac{r_{3} \frac{L_{1 s}}{L_{1 st}}}{1 + r_{3} \frac{A_{2 c}}{A - A_{2 c}} (1 - \frac{L_{1 s}}{L_{1 st}})} \end{array}$	(Ih) $\begin{matrix} \frac{r_{2} [1 - r_{1} \frac{A_{1}}{A} - r_{1} r_{3} \frac{A_{2 c}}{A} \cdot \frac{A_{1}}{A - A_{2 c}}]}{r_{3} [1 - r_{1} \frac{A_{1}}{A} - r_{1} r_{2} \frac{A_{2 c}}{A} \cdot \frac{A_{1}}{A - A_{2 c}}]} \\ \frac{r_{3} \frac{L_{1 s}}{L_{1 st}} (1 - r_{1} \frac{A_{1}}{A})}{1 - r_{1} \frac{A_{1}}{A} - r_{1} r_{3} \frac{A_{1}}{A} \frac{A_{2 c}}{A - A_{2 c}} (1 - \frac{L_{1 s}}{L_{1 st}})} \end{matrix}$

Method	$\frac{A_{1}}{A}$	$\frac{A_{2}}{A}$	r₂ from (Ia)	r₂ from (Ib)	$\frac{A_{1}}{A}$	$\frac{A_{2 c}}{A - A_{2 c}}$	r₂ from (Id)		$\frac{A_{1}}{A}$	$\frac{A_{2}}{A}$	r₂ from (Ie)	r₂ from (Ig)	$\frac{A_{1}}{A}$	$\frac{A_{2 c}}{A - A_{2 c}}$	r₂ from (Ih)
Comparison	0.96	0.01	0.4850	0.4827	0.96	0.01010	0.4826	Substitution	0.97	0.01	0.5280	0.5256	0.97	0.01010	0.5260
	0.92	0.03	0.4850	0.4780	0.92	0.03093	0.4778		0.95	0.03	0.5965	0.5849	0.95	0.03093	0.5887
	0.88	0.05	0.4850	0.4735	0.88	0.05263	0.4729		0.93	0.05	0.6445	0.6246	0.93	0.05263	0.6342
	0.78	0.10	0.4850	0.4626	0.78	0.11111	0.4602		0.88	0.10	0.7285	0.6762	0.88	0.11111	0.7072

Comparison method			Substitution method
r₂ from (5.7)	$\frac{A_{2 c}}{A - A_{2 c}}$	r₂ from (5.6a)	$\frac{A_{1}}{A}$	$\frac{A_{2 c}}{A - A_{2 c}}$	r₂ from (5.6b)
0.5000	0.01010	0.4976	0.97	0.01010	0.5409
0.5000	0.03093	0.4928	0.95	0.03093	0.6029
0.5000	0.05263	0.4879	0.93	0.05263	0.6476
0.5000	0.11111	0.4752	0.88	0.11111	0.7189

Sample and standard	Comparison method [Fig.1(a)]	Substitution method [Fig. 1(b)]
Curved sample and standard	(Ia) $\begin{array}{l} \frac{L_{1 s}}{L_{1 st}} & = \frac{r_{2} [r_{3} F_{32} F_{13} - F_{12} (r_{3} F_{33} - 1)]}{r_{3} [r_{2} F_{12} F_{23} - F_{13} (r_{2} F_{22} - 1)]} = \frac{r_{2}}{r_{3}} \\ r_{2} & = r_{3} \cdot \frac{L_{1 s}}{L_{1 st}} \end{array}$	(Ie) $\frac{r_{2} [(1 - r_{3} F_{33}) (1 - r_{1} F_{11}) - r_{1} r_{3} F_{31} \cdot F_{13}]}{r_{3} [(1 - r_{2} F_{22}) (1 - r_{1} F_{11}) - r_{1} r_{2} F_{21} \cdot F_{12}]} = \frac{r_{2} (1 - r_{3} \frac{A_{2}}{A} - r_{1} \frac{A_{1}}{A})}{r_{3} (1 - r_{2} \frac{A_{2}}{A} - r_{1} \frac{A_{1}}{A})} \frac{r_{3} \cdot \frac{L_{1 s}}{L_{1 st}} (1 - r_{1} \frac{A_{1}}{A})}{1 - r_{1} \frac{A_{1}}{A} - r_{3} \frac{A_{2}}{A} (1 - \frac{L_{1 s}}{L_{1 st}})}$
Flat sample (small-area approx.), curved standard with r₃ = r₁	(Ib) $\begin{array}{l} \frac{L_{1 s}}{L_{1 st}} & = \frac{r_{2}}{r_{1} (1 + r_{2} \frac{A_{2}}{A})} \\ r_{2} & = \frac{r_{1} \cdot \frac{L_{1 s}}{L_{1 st}}}{1 - \frac{L_{1 s}}{L_{1 st}} \cdot r_{1} \cdot \frac{A_{2}}{A}} \end{array}$	(If) $\begin{matrix} \frac{r_{2} [1 - r_{1} \frac{A_{1}}{A} - r_{1} \frac{A_{2}}{A}]}{r_{2} [1 - r_{1} \frac{A_{1}}{A} - r_{1} r_{2} \frac{A_{2}}{A} \cdot \frac{A_{1}}{A}]} \\ \frac{r_{1} \frac{L_{1 s}}{L_{1 st}} (1 - r_{1} \frac{A_{1}}{A})}{1 - r_{1} \frac{A_{1}}{A} - r_{1} \frac{A_{2}}{A} (1 - r_{1} \frac{A_{1}}{A} \cdot \frac{L_{1 s}}{L_{1 st}})} \end{matrix}$
Flat sample and standard (small area approx.)	(Ic) $\begin{array}{l} \frac{L_{1 s}}{L_{1 st}} & = \frac{r_{2} (1 + r_{3} \cdot \frac{A_{2}}{A})}{r_{3} (1 + r_{2} \cdot \frac{A_{2}}{A})} \\ r_{2} & = \frac{r_{3} \cdot \frac{L_{1 s}}{L_{1 st}}}{1 + r_{3} \cdot \frac{A_{2}}{A} (1 - \frac{L_{1 s}}{L_{1 st}}) \cdot} \end{array}$	(Ig) $\begin{matrix} \frac{r_{2} [1 - r_{1} \frac{A_{1}}{A} - r_{1} r_{3} \frac{A_{1}}{A} \cdot \frac{A_{2}}{A}]}{r_{3} [1 - r_{1} \frac{A_{1}}{A} - r_{1} r_{2} \frac{A_{1}}{A} \cdot \frac{A_{2}}{A}]} \\ \frac{r_{3} \frac{L_{1 s}}{L_{1 st}} (1 - r_{1} \frac{A_{1}}{A})}{1 - r_{1} \frac{A_{1}}{A} - r_{1} r_{3} \frac{A_{1}}{A} \cdot \frac{A_{2}}{A} (1 - \frac{L_{1 s}}{L_{1 st}})} \end{matrix}$
Flat sample and standard of finite area	(Id) $\begin{array}{l} \frac{L_{1 s}}{L_{1 st}} & = \frac{r_{2} (1 + r_{3} \frac{A_{2 c}}{A - A_{2 c}})}{r_{3} (1 + r_{2} \frac{A_{2 c}}{A - A_{2 c}})} \\ r_{2} & = \frac{r_{3} \frac{L_{1 s}}{L_{1 st}}}{1 + r_{3} \frac{A_{2 c}}{A - A_{2 c}} (1 - \frac{L_{1 s}}{L_{1 st}})} \end{array}$	(Ih) $\begin{matrix} \frac{r_{2} [1 - r_{1} \frac{A_{1}}{A} - r_{1} r_{3} \frac{A_{2 c}}{A} \cdot \frac{A_{1}}{A - A_{2 c}}]}{r_{3} [1 - r_{1} \frac{A_{1}}{A} - r_{1} r_{2} \frac{A_{2 c}}{A} \cdot \frac{A_{1}}{A - A_{2 c}}]} \\ \frac{r_{3} \frac{L_{1 s}}{L_{1 st}} (1 - r_{1} \frac{A_{1}}{A})}{1 - r_{1} \frac{A_{1}}{A} - r_{1} r_{3} \frac{A_{1}}{A} \frac{A_{2 c}}{A - A_{2 c}} (1 - \frac{L_{1 s}}{L_{1 st}})} \end{matrix}$

Method	$\frac{A_{1}}{A}$	$\frac{A_{2}}{A}$	r₂ from (Ia)	r₂ from (Ib)	$\frac{A_{1}}{A}$	$\frac{A_{2 c}}{A - A_{2 c}}$	r₂ from (Id)		$\frac{A_{1}}{A}$	$\frac{A_{2}}{A}$	r₂ from (Ie)	r₂ from (Ig)	$\frac{A_{1}}{A}$	$\frac{A_{2 c}}{A - A_{2 c}}$	r₂ from (Ih)
Comparison	0.96	0.01	0.4850	0.4827	0.96	0.01010	0.4826	Substitution	0.97	0.01	0.5280	0.5256	0.97	0.01010	0.5260
	0.92	0.03	0.4850	0.4780	0.92	0.03093	0.4778		0.95	0.03	0.5965	0.5849	0.95	0.03093	0.5887
	0.88	0.05	0.4850	0.4735	0.88	0.05263	0.4729		0.93	0.05	0.6445	0.6246	0.93	0.05263	0.6342
	0.78	0.10	0.4850	0.4626	0.78	0.11111	0.4602		0.88	0.10	0.7285	0.6762	0.88	0.11111	0.7072

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Equations (40)

Journal of the Optical Society of America