## Abstract

The analytic theory of aberrations has been used to derive an expression for the magnitude of the coma width of the image in the meridional plane in a Czerny–Turner spectrograph with unequal mirror radii. The calculated properties of a 4-m spectrograph with equal radii and a recently constructed 3.34-m spectrograph with unequal radii are compared with the results obtained by tracing individual rays. The agreement is excellent, in contrast to the results of Chandler [
J. Opt. Soc. Am. **58**,
895 (
1968)]. The lateral position of the grating for complete elimination of coma found experimentally with the 3.34-m instrument is in fair agreement with the theory. A correction to the
$\sqrt{3}$ longitudinal grating position is given for a Czerny–Turner spectrograph which results in a flatter focal surface.

© 1969 Optical Society of America

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### Equations (24)

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(1)
$$\begin{array}{l}F=r+{r}^{\prime}-w(\text{sin}\alpha +\text{sin}{\alpha}^{\prime})\\ +\frac{{w}^{2}}{2}\left[\left(\frac{{\text{cos}}^{2}\alpha}{r}-\frac{\text{cos}\alpha}{{R}_{1}}\right)+\left(\frac{{\text{cos}}^{2}{\alpha}^{\prime}}{{r}^{\prime}}-\frac{\text{cos}{\alpha}^{\prime}}{{R}_{1}}\right)\right]\\ +\hspace{0.17em}{\scriptstyle \frac{1}{2}}{w}^{3}\left[\frac{\text{sin}\alpha}{r}\left(\frac{{\text{cos}}^{2}\alpha}{r}-\frac{\text{cos}\alpha}{{R}_{1}}\right)+\frac{\text{sin}{\alpha}^{\prime}}{{r}^{\prime}}\left(\frac{{\text{cos}}^{2}{\alpha}^{\prime}}{{r}^{\prime}}-\frac{\text{cos}{\alpha}^{\prime}}{{R}_{1}}\right)\right].\end{array}$$
(2)
$$\frac{\partial F}{\partial {w}^{\prime}}(\text{for}\hspace{0.17em}{r}^{\prime}=\infty )=-\frac{1}{\text{cos}{\alpha}^{\prime}}(\text{sin}\alpha +\text{sin}{\alpha}^{\prime})+\frac{{w}^{\prime}}{{\text{cos}}^{2}{\alpha}^{\prime}}\left[\frac{{\text{cos}}^{2}\alpha}{r}-\frac{\text{cos}\alpha}{{R}_{1}}-\frac{\text{cos}{\alpha}^{\prime}}{{R}_{1}}\right]+\frac{3{{w}^{\prime}}^{2}}{2\hspace{0.17em}{\text{cos}}^{3}{\alpha}^{\prime}}\left[\frac{\text{sin}\alpha}{r}\left(\frac{{\text{cos}}^{2}\alpha}{r}-\frac{\text{cos}\alpha}{{R}_{1}}\right)\right]\stackrel{?}{=}0.$$
(3)
$$(\partial F/\partial {w}^{\prime})\hspace{0.17em}(\text{for}\hspace{0.17em}{r}^{\prime}=\infty ,{\alpha}^{\prime}=-\alpha ,r={\scriptstyle \frac{1}{2}}{R}_{1}\hspace{0.17em}\text{cos}\alpha ,{w}^{\prime}={\scriptstyle \frac{1}{2}}W\hspace{0.17em}\text{cos}{\alpha}_{g})=\frac{3{W}^{2}\hspace{0.17em}{\text{cos}}^{2}{\alpha}_{g}\hspace{0.17em}\text{sin}\alpha}{4{{R}_{1}}^{2}\hspace{0.17em}{\text{cos}}^{3}\alpha}.$$
(4)
$$\delta {\beta}_{g}=-\frac{\text{cos}{\alpha}_{g}}{\text{cos}{\beta}_{g}}\delta {\alpha}_{g}=-\frac{3{W}^{2}\hspace{0.17em}\text{sin}\alpha \hspace{0.17em}{\text{cos}}^{3}{\alpha}_{g}}{4{{R}_{1}}^{2}\hspace{0.17em}{\text{cos}}^{3}\alpha \hspace{0.17em}\text{cos}{\beta}_{g}}.$$
(5)
$${\mathrm{\Delta}}_{\alpha}={\scriptstyle \frac{1}{2}}{R}_{2}\hspace{0.17em}\text{cos}\beta \delta {\beta}_{g}=-\frac{3{W}^{2}{R}_{2}\hspace{0.17em}\text{sin}\alpha \hspace{0.17em}\text{cos}\beta \hspace{0.17em}{\text{cos}}^{3}{\alpha}_{g}}{8{{R}_{1}}^{2}\hspace{0.17em}{\text{cos}}^{3}\alpha \hspace{0.17em}\text{cos}{\beta}_{g}},$$
(6)
$$(\partial F/\partial {w}^{\prime})\hspace{0.17em}(\text{for}\hspace{0.17em}r=\infty ,{\beta}^{\prime}=-\beta ,{r}^{\prime}={\scriptstyle \frac{1}{2}}{R}_{2}\hspace{0.17em}\text{cos}\beta ,{w}^{\prime}={\scriptstyle \frac{1}{2}}W\hspace{0.17em}\text{cos}{\beta}_{g})=\frac{3{W}^{2}\hspace{0.17em}{\text{cos}}^{2}{\beta}_{g}\hspace{0.17em}\text{sin}\beta}{4{{R}_{2}}^{2}\hspace{0.17em}{\text{cos}}^{3}\beta}.$$
(7)
$${\mathrm{\Delta}}_{\beta}={\scriptstyle \frac{1}{2}}{R}_{2}\hspace{0.17em}\text{cos}\beta \frac{\partial F}{\partial {w}^{\prime}}=\frac{3{W}^{2}\hspace{0.17em}\text{cos}{\beta}_{g}\hspace{0.17em}\text{sin}\beta}{8{R}_{2}\hspace{0.17em}{\text{cos}}^{2}\beta}.$$
(8)
$$\mathrm{\Delta}={\mathrm{\Delta}}_{\alpha}+{\mathrm{\Delta}}_{\beta}=\frac{3{W}^{2}{R}_{2}\hspace{0.17em}{\text{cos}}^{2}{\alpha}_{g}\hspace{0.17em}\text{cos}\beta}{8\hspace{0.17em}{\text{cos}}^{3}\alpha}\times \left[\frac{\text{sin}\beta}{{{R}_{2}}^{2}}\xb7\frac{{\text{cos}}^{2}{\beta}_{g}\hspace{0.17em}{\text{cos}}^{3}\alpha}{{\text{cos}}^{2}{\alpha}_{g}\hspace{0.17em}{\text{cos}}^{3}\beta}-\frac{\text{sin}\alpha \hspace{0.17em}\text{cos}{\alpha}_{g}}{{{R}_{1}}^{2}\hspace{0.17em}\text{cos}{\beta}_{g}}\right].$$
(9)
$$\frac{\text{sin}\beta}{\text{sin}\alpha}=\frac{{{R}_{2}}^{2}}{{{R}_{1}}^{2}}\frac{{\text{cos}}^{3}\beta}{{\text{cos}}^{3}\alpha}\frac{{\text{cos}}^{3}{\alpha}_{g}}{{\text{cos}}^{3}{\beta}_{g}},$$
(10)
$$\beta /\alpha ={\text{cos}}^{3}{\alpha}_{g}/{\text{cos}}^{3}{\beta}_{g},$$
(11)
$$\frac{x}{{R}_{2}}=-\frac{1}{4}\left[1-3{\left(\frac{m}{{R}_{2}}\right)}^{2}\right]{\theta}^{2}+\frac{1}{48}\left[1-30{\left(\frac{m}{{R}_{2}}\right)}^{2}+48{\left(\frac{m}{{R}_{2}}\right)}^{3}-27{\left(\frac{m}{{R}_{2}}\right)}^{4}\right]{\theta}^{4}.$$
(12)
$$x(\u220a,{\theta}_{\text{min}})=x(\u220a,{\theta}_{\text{max}})$$
(13)
$$2x(\u220a,{\theta}_{\text{center}})=x(\u220a,{\theta}_{\text{min}})+x(\u220a,{\theta}_{\text{max}})$$
(14)
$$\begin{array}{ll}\hfill {\theta}_{\text{min}}& =\sqrt{3}\hspace{0.17em}{\beta}_{\text{center}}-\sqrt{3}\hspace{0.17em}L/2{R}_{2},\\ \hfill {\theta}_{\text{center}}& =\sqrt{3}\hspace{0.17em}{\beta}_{\text{center}},\end{array}$$
(15)
$${\theta}_{\text{max}}=\sqrt{3}\hspace{0.17em}{\beta}_{\text{center}}+\sqrt{3}\hspace{0.17em}L/2{R}_{2},$$
(16)
$$\alpha ={z}_{g}/2{x}_{g}.$$
(17)
$$\beta =({E}_{1}+h\mathrm{\Delta}{\beta}_{g}-{z}_{g})/2l+({E}_{2}+{\scriptstyle \frac{1}{2}}{R}_{2}\mathrm{\Delta}{\beta}_{g}-{E}_{1}-h\mathrm{\Delta}{\beta}_{g})/{R}_{2},$$
(18)
$$\begin{array}{ll}\hfill \mathrm{\Delta}{\beta}_{g}& ={\beta}_{g}-{\beta}_{g}\hspace{0.17em}(\text{center}\hspace{0.17em}\text{of}\hspace{0.17em}\text{plate})\\ \hfill l& ={[{h}^{2}-{({E}_{1}-{z}_{g})}^{2}]}^{{\scriptstyle \frac{1}{2}}}.\end{array}$$
(19)
$$\begin{array}{l}{\gamma}_{1}={z}_{g}{R}_{1}/2{x}_{g}\\ {\gamma}_{2}={E}_{2}-{R}_{2}({E}_{1}-{z}_{g})/2l.\end{array}$$
(20)
$$\begin{array}{l}{\delta}_{1}=[{{R}_{1}}^{2}-{{\gamma}_{1}}^{2}]-{R}_{1}\\ {\delta}_{2}={x}_{g}-l-{R}_{2}+{[{{R}_{2}}^{2}-{({E}_{1}-{\gamma}_{2})}^{2}]}^{{\scriptstyle \frac{1}{2}}}.\end{array}$$
(21)
$$m={\{{(l-{[{{R}_{2}}^{2}-{({E}_{1}-{\gamma}_{2})}^{2}]}^{{\scriptstyle \frac{1}{2}}})}^{2}+{({z}_{g}-{\gamma}_{2})}^{2}\}}^{{\scriptstyle \frac{1}{2}}}.$$
(22)
$$\theta =\text{arcsin}[({R}_{2}/m)\hspace{0.17em}\text{sin}\beta ].$$
(23)
$$\text{Slope}=\frac{{z}_{g}-{\gamma}_{2}}{{[{{R}_{2}}^{2}-{({E}_{1}-{\gamma}_{2})}^{2}]}^{{\scriptstyle \frac{1}{2}}}-l}.$$
(24)
$${\beta}_{g}\hspace{0.17em}(\text{center}\hspace{0.17em}\text{of}\hspace{0.17em}\text{plate})-{\alpha}_{g}=({E}_{1}-{z}_{g})/l+{z}_{g}/{x}_{g}.$$