## Abstract

The Ebert spectrograph has recently been used extensively as a monochromator. In view of the spherical mirror being used as a collimator-camera mirror and the plane grating as the dispersing element, the instrument suffers mainly from spherical aberration if the *f*-number of the system is smaller than about 9 for a 1-m focal length. Also, if many exit slits are needed over the entire spectrum of interest, the coma cannot be neglected. In order to obtain a fast spectrometer in which many exit slits could be used, the two surfaces in the system must be aspherized. The theory of such a system is presented in this paper. Specifically, a numerical example is worked out for an *f*/5 system with 40-in. focal length and 1200 lines/mm rulings on the grating. The spectrum of interest is 1000 to 4000 Å in first order.

© 1962 Optical Society of America

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### Equations (23)

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(1)
$$\Delta =\left({y}^{4}/2{r}^{3}\right)-\left(2{y}^{2}\delta /{r}^{2}\right).$$
(2)
$$\delta =\left({{y}^{2}}_{\text{max}}/4r\right),$$
(3)
$${\Delta}_{\text{max}}={{y}^{4}}_{\text{max}}/8{r}^{3}.$$
(4)
$$f-\text{number}\ge \frac{1}{4}{\left(f/\mathrm{\lambda}\right)}^{\frac{1}{4}}.$$
(5)
$$\Delta {h}_{s}=\left({y}^{2}/4f\right)\phantom{\rule{0.2em}{0ex}}\left(h/f\right).$$
(6)
$$\Delta {h}_{T}=3{y}^{2}h/4{f}^{2}.$$
(7)
$${\text{coma}}_{T}=\left({y}^{3}h/4{f}^{3}\right)=2{y}^{3}h/{r}^{3},$$
(8)
$${\text{coma}}_{T}=2{y}^{3}{y}_{p}/{r}^{3}.$$
(9)
$${\text{coma}}_{T}=8A{y}^{3}{y}_{p}.$$
(11)
$${A}^{\prime}{y}^{4}=\left[\left(1/4{r}^{3}\right)-\left(1/8{r}^{3}\right)\right]{y}^{4}=\left(1/8{r}^{3}\right){y}^{4}.$$
(12)
$$Z=\frac{c\left({x}^{2}+{y}^{2}\right)}{1+{\left[1-{c}^{2}\left({x}^{2}+{y}^{2}\right)\right]}^{\frac{1}{2}}}-\frac{{\left({x}^{2}+{y}^{2}\right)}^{2}}{8{r}^{3}},$$
(13)
$$z={\left({x}^{2}+{y}^{2}\right)}^{2}/2{r}^{3}.$$
(14)
$$h={y}_{p}\left[r/\left(r-2a\right)\right].$$
(15)
$$z=\frac{c\left({x}^{2}+{y}^{2}\right)}{1+{\left[1-{c}^{2}\left({x}^{2}+{y}^{2}\right)\right]}^{\frac{1}{2}}}-\frac{k{\left({x}^{2}+{y}^{2}\right)}^{2}}{8{r}^{3}},$$
(16)
$$z=\frac{c\left({x}^{2}+{y}^{2}\right)}{1+{\left[1-{c}^{2}\left(1+k\right)\phantom{\rule{0.2em}{0ex}}\left({x}^{2}+{y}^{2}\right)\right]}^{\frac{1}{2}}}.$$
(17)
$$z=\left[\left(1+k\right)/4{r}^{3}\right]{\left({x}^{2}+{y}^{2}\right)}^{2}.$$
(18)
$$2\phantom{\rule{0.2em}{0ex}}\text{sin}\theta =\mathrm{\lambda}/\sigma ,$$
(19)
$$z=\left[\left(1+k\right)/4{r}^{3}\right]{\left({x}^{2}\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{2}\theta +{y}^{2}\right)}^{2}\left(1/\text{cos}\theta \right).$$
(20)
$$z=\left[\left(1+k\right)/4{r}^{3}\right]{\text{cos}}^{3}\theta {\left({x}^{2}+{y}^{2}\right)}^{2}.$$
(21)
$$z=-0.10486549\times {10}^{-5}{\left({x}^{2}+{y}^{2}\right)}^{2},$$
(22)
$$z=-0.10567\times {10}^{-5}{\left({x}^{2}+{y}^{2}\right)}^{2}.$$
(23)
$$z=-0.10923\times {10}^{-5}{\left({x}^{2}+{y}^{2}\right)}^{2},$$