## Abstract

An analytical treatment is given which shows that the peak luminance gain and lobe width of a back or front projection screen material can be used as coordinates on a graph to determine the transmission or reflection efficiency. The analysis assumes that the angular distribution of the gain can be expressed as a power of a cosine function. Also, a screen photometer is described which is based upon the adaptation of a spectrometer to measure screens having retroreflective characteristics. The photometer is capable of measuring luminance on the retroreflective axis and out to 15° on either side of the incident beam. Screen samples, 2.54 cm sq, can be oriented through angles of ±30° to the incident beam and luminances from 1 ft-L to 10,000 ft-L (from about 3.5 cd/m^{2} to 35,000 cd/m^{2}) can be handled. The instrument has three modes of operation: visual, photoelectric, and photographic. A graph is presented illustrating the application of the analytical technique to provide a rapid performance evaluation and comparison of screen efficiencies.

© 1970 Optical Society of America

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### Equations (27)

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(1)
$$L\left(\theta \right)=\left(E/\pi \right)G\left(\theta \right).$$
(2)
$$F\left(\beta \right)={\displaystyle {\int}_{0}^{A}{\displaystyle {\int}_{0}^{\beta}L\left(\theta \right)\phantom{\rule{0.2em}{0ex}}\left(\text{cos}\theta dA\right)\phantom{\rule{0.2em}{0ex}}\left(2\pi \phantom{\rule{0.2em}{0ex}}\text{sin}\theta \phantom{\rule{0.2em}{0ex}}d\theta \right)}}.$$
(3)
$$F\left(\beta \right)=2\phantom{\rule{0.2em}{0ex}}{\displaystyle {\int}_{0}^{A}EdA}\phantom{\rule{0.2em}{0ex}}{\displaystyle {\int}_{\beta}^{0}G\left(\theta \right)\phantom{\rule{0.2em}{0ex}}\text{cos}\theta \phantom{\rule{0.2em}{0ex}}d\left(\text{cos}\theta \right)}.$$
(4)
$$G\left(\theta \right)={G}_{0}{\left(\text{cos}\theta \right)}^{p},$$
(6)
$$-\pi /2\le \theta \le \pi /2.$$
(7)
$$F\left(\beta \right)=2{G}_{0}\phantom{\rule{0.2em}{0ex}}{\displaystyle {\int}_{0}^{A}EdA}\phantom{\rule{0.2em}{0ex}}{\displaystyle {\int}_{\beta}^{0}{\left(\text{cos}\theta \right)}^{p+1}\phantom{\rule{0.2em}{0ex}}d\left(\text{cos}\theta \right)}.$$
(8)
$$F\left(\beta \right)=2{G}_{0}\phantom{\rule{0.2em}{0ex}}\left[\frac{1-{\left(\text{cos}\beta \right)}^{p+2}}{p+2}\right]\phantom{\rule{0.2em}{0ex}}{\displaystyle {\int}_{0}^{A}EdA}.$$
(9)
$$F\phantom{\rule{0.2em}{0ex}}\left(\frac{\pi}{2}\right)=\frac{2{G}_{0}}{p+2}\phantom{\rule{0.2em}{0ex}}{\displaystyle {\int}_{0}^{A}EdA}.$$
(10)
$$f\left(\beta \right)=\frac{F\left(\beta \right)}{F\left(\pi /2\right)}=1-{\left(\text{cos}\beta \right)}^{p}{\left(\text{cos}\beta \right)}^{2}.$$
(11)
$$\begin{array}{c}{\left(\text{cos}\beta \right)}^{p}=G\left(\beta \right)/{G}_{0},\\ \therefore \phantom{\rule{0.2em}{0ex}}f\left(\beta \right)=1-\left[G\left(\beta \right)/{G}_{0}\right]\phantom{\rule{0.2em}{0ex}}{\left(\text{cos}\beta \right)}^{2}.\end{array}$$
(12)
$$G\left(\gamma \right)/{G}_{0}=\mathrm{0.1.}$$
(13)
$$f\left(\gamma \right)=1-0.1{\left(\text{cos}\gamma \right)}^{2}.$$
(14)
$$m=\frac{\text{Total}\phantom{\rule{0.2em}{0ex}}\text{light}\phantom{\rule{0.2em}{0ex}}\text{flux}\phantom{\rule{0.2em}{0ex}}\text{reflected}\phantom{\rule{0.2em}{0ex}}\text{in}\phantom{\rule{0.2em}{0ex}}\text{lobe}}{\text{Total}\phantom{\rule{0.2em}{0ex}}\text{incident}\phantom{\rule{0.2em}{0ex}}\text{light}\phantom{\rule{0.2em}{0ex}}\text{flux}}=F\phantom{\rule{0.2em}{0ex}}\left(\frac{\pi}{2}\right)/{\displaystyle {\int}_{0}^{A}EdA}.$$
(15)
$$\begin{array}{cc}m=2{G}_{0}/\left(p+2\right)& m\le 1.\end{array}$$
(16)
$$p=\text{ln}\left(G\left(\beta \right)/{G}_{0}\right)/\text{ln}\left(\text{cos}\beta \right).$$
(17)
$$\frac{{G}_{0}}{m}+\frac{1/\left[2\phantom{\rule{0.2em}{0ex}}\text{ln}\left(\text{cos}\beta \right)\right]}{1/\text{ln}\left[{G}_{0}/G\left(\beta \right)\right]}=1,$$
(18)
$$\begin{array}{cc}\beta =\gamma & G\left(\gamma \right)/{G}_{0}=0.1,\end{array}$$
(19)
$$\frac{{G}_{0}}{m}+\frac{1/\left[2\phantom{\rule{0.2em}{0ex}}\text{ln}\left(\text{cos}\gamma \right)\right]}{1/\text{ln}\left(10\right)}=1.$$
(20)
$$\begin{array}{c}\varphi \left(\gamma \right)=1/\left[2\phantom{\rule{0.2em}{0ex}}\text{ln}\left(\text{cos}\gamma \right)\right],\\ {G}_{0}/m+\varphi \left(\gamma \right)/0.4343=1.\end{array}$$
(21)
$$\frac{{G}_{0}}{m}-\frac{1/{\gamma}^{2}}{0.4343}=1.$$
(23)
$${G}_{0}/m=1/0.4343{\gamma}^{2}.$$
(24)
$$\text{log}\phantom{\rule{0.2em}{0ex}}{G}_{0}=-2\phantom{\rule{0.2em}{0ex}}\text{log}{\gamma}^{\prime}+\text{log}\left(75.6{m}^{\prime}\right),$$
(25)
$$\begin{array}{ll}E={\left(18.5/R\right)}^{1.155}\hfill & 0.02<E<10\hfill \end{array},$$
(26)
$$G\left(\theta \right)=L\left(\theta \right)/{L}_{S}=0.99{\left[{R}_{S}/R\left(\theta \right)\right]}^{1.155},$$
(27)
$$F\left(\beta \right)={\displaystyle {\int}_{0}^{A}EdA}\phantom{\rule{0.2em}{0ex}}{\displaystyle {\int}_{0}^{\beta}G\left(\theta \right)d\left({\text{sin}}^{2}\theta \right)}.$$