## Abstract

An analysis of the various factors involved in several typical three-layered reflection-reducing coatings has been made from both an experimental and a theoretical standpoint. By taking into account the approximate dispersions of the materials, reasonable agreement between the experimental and calculated reflection curves has been obtained.

Interesting results were obtained on electron diffraction studies of the various coating materials. Thin films of titanium dioxide, the high index material used, were found to have crystalline patterns that varied with the coated material and that, in general, were not recognizable as titanium dioxide patterns. Thus it was considered impossible to assign any definite values to the refractive indices of such thin films without specifying the substrate.

© 1947 Optical Society of America

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### Equations (11)

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(1)
$$R={|(X-1)/(X+1)|}^{2},$$
(2)
$$\begin{array}{ll}X\hfill & =[({n}_{2}{n}_{3}{n}_{4}-{{n}_{2}}^{2}{n}_{4}tan{\omega}_{2}tan{\omega}_{3}\hfill \\ \hfill & -{{n}_{2}}^{2}{n}_{3}tan{\omega}_{2}tan{\omega}_{4}-{n}_{2}{{n}_{3}}^{2}tan{\omega}_{3}tan{\omega}_{4})\cdots \hfill \\ \hfill & +i({n}_{1}{n}_{3}{n}_{4}tan{\omega}_{2}+{n}_{1}{n}_{2}{n}_{4}tan{\omega}_{3}\hfill \\ \hfill & +{n}_{1}{n}_{2}{n}_{3}tan{\omega}_{4}-{n}_{1}{{n}_{3}}^{2}tan{\omega}_{2}tan{\omega}_{3}tan{\omega}_{4})]/\hfill \\ \hfill & \cdots [({n}_{1}{n}_{2}{n}_{3}{n}_{4}-{n}_{1}{{n}_{3}}^{2}{n}_{4}tan{\omega}_{2}tan{\omega}_{3}\hfill \\ \hfill & -{n}_{1}{n}_{3}{{n}_{4}}^{2}tan{\omega}_{2}tan{\omega}_{4}-{n}_{1}{n}_{2}{{n}_{4}}^{2}tan{\omega}_{3}tan{\omega}_{4})\hfill \\ \hfill & \cdots +i({{n}_{2}}^{2}{n}_{3}{n}_{4}tan{\omega}_{2}+{n}_{2}{{n}_{3}}^{2}{n}_{4}tan{\omega}_{3}\hfill \\ \hfill & +{n}_{2}{n}_{3}{{n}_{4}}^{2}tan{\omega}_{4}-{{n}_{2}}^{2}{{n}_{4}}^{2}tan{\omega}_{2}tan{\omega}_{3}tan{\omega}_{4})],\hfill \end{array}$$
(3)
$$tan{\omega}_{2}=tan{\omega}_{3}=tan{\omega}_{4}=\infty ,$$
(4)
$$X={n}_{1}{{n}_{3}}^{2}/{{n}_{2}}^{2}{{n}_{4}}^{2}$$
(5)
$$R={[({n}_{1}{{n}_{3}}^{2}-{{n}_{2}}^{2}{{n}_{4}}^{2})/({n}_{1}{{n}_{3}}^{2}+{{n}_{2}}^{2}{{n}_{4}}^{2})]}^{2}.$$
(6)
$${n}_{3}={n}_{2}{n}_{4}/{{n}_{1}}^{\frac{1}{2}}.$$
(7)
$$X={{n}_{2}}^{2}/{n}_{1}{{n}_{4}}^{2}$$
(8)
$$R={[({{n}_{2}}^{2}-{n}_{1}{{n}_{4}}^{2})/({{n}_{2}}^{2}+{n}_{1}{{n}_{4}}^{2})]}^{2}.$$
(9)
$${{n}_{2}}^{2}={n}_{1}{{n}_{4}}^{2}.$$
(10)
$${n}_{f}={[{n}_{g}(1+{R}^{\frac{1}{2}})/(1-{R}^{\frac{1}{2}})]}^{\frac{1}{2}},$$
(11)
$$\begin{array}{ll}{n}_{j}\hfill & =\text{index of the film},\hfill \\ {n}_{g}\hfill & =\text{index of the glass},\hfill \\ R\hfill & =\text{reflection}.\hfill \end{array}$$