## Abstract

An approximate expression is derived for the length of a stigmatic final image in multiple-traverse cells of the John U. White type. The length of the final tangential image is, for a point source,

$$\mathrm{\Delta}{L}_{T}\approx \frac{h{b}^{2}}{12{R}^{2}}\left(N-\frac{4}{N}\right),$$ where *h* is the height of the “rear” mirrors illuminated, *b* is the separation of the entrance and exit images, *R* is the common radius of curvature of the three spherical mirrors, and *N* is the number of traversals. Direct experimental measurements of image length for two sets of mirrors are in agreement with the calculated values. The results are discussed in terms of design parameters, and two new cells constructed in accordance with these principles are described.

© 1961 Optical Society of America

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### Equations (8)

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(1)
$$\mathrm{\Delta}{L}_{T}\approx \frac{h{b}^{2}}{12{R}^{2}}\left(N-\frac{4}{N}\right),$$
(2)
$$\frac{1}{{{S}^{\prime}}_{T}}+\frac{1}{S}=\frac{2}{R\hspace{0.17em}\text{cos}\varphi}$$
(3)
$$\frac{1}{{{S}^{\prime}}_{S}}+\frac{1}{S}=\frac{2\hspace{0.17em}\text{cos}\varphi}{R},$$
(4)
$$\mathrm{\Delta}{L}_{T}=\frac{2h}{{R}^{2}}\sum _{i}{{d}_{i}}^{2}.$$
(5)
$$\mathrm{\Delta}{L}_{T}=\frac{2h}{{R}^{2}}\left[{\left(\frac{b}{4}\right)}^{2}+{\left(\frac{b}{4}\right)}^{2}\right]=\frac{h{b}^{2}}{4{R}^{2}},$$
(6)
$$\mathrm{\Delta}{L}_{T}=\frac{4h{b}^{2}}{{R}^{2}}\left[{\left(\frac{\frac{N}{2}-1}{N}\right)}^{2}+\cdots +{\left(\frac{3}{N}\right)}^{2}+{\left(\frac{1}{N}\right)}^{2}\right]=\frac{h{b}^{2}}{12{R}^{2}}\left(N-\frac{4}{N}\right)=\frac{{b}^{2}}{12R{f}_{V}}\left(N-\frac{4}{N}\right).$$
(7)
$$\left(\frac{N}{4}-1\right){w}_{s}.$$
(8)
$$\mathrm{\Delta}{L}_{T}\approx \frac{N}{R}=\frac{NR}{{R}^{2}}\approx \frac{NR}{Vo{l}^{{\scriptstyle \frac{2}{3}}}},$$