## Abstract

The problem of mounting a twenty-one foot concave grating and slit at oblique incidence, so that the focus will lie on a previously established track having less than one inch adjustment range, is discussed. A simple technique, employing only a good transit, is described. The development of this method depends on a precise knowledge of the nature of the general focal curve. This information is found by a brief examination of grating theory. A table of permissible departures of the slit from the Rowland circle is given, such that the error in path length to the sharpest image is less than λ/4. Application of classical grating theory to the case of oblique incidence and diffraction results in an expression for maximum useful grating width. This expression is shown to differ only by a factor of 1.06 from the expression derived in a recent article by Mack, Stehn and Edlén.

© 1933 Optical Society of America

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### Equations (16)

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(1)
$${a}^{2}+{b}^{2}={r}^{2},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}2\rho x={x}^{2}+{y}^{2},$$
(2)
$${u}^{2}={(x-a)}^{2}+{(y-b)}^{2}={r}^{2}-2by+{y}^{2}+{x}^{2}-2ax={(r-by/r)}^{2}+(a/{r}^{2}-1/\rho )a{y}^{2}+(1-a/\rho ){x}^{2}.$$
(3)
$$u=(r-by/r)+{\scriptstyle \frac{1}{2}}\frac{(a/{r}^{2}-1/\rho )a{y}^{2}}{(r-by/r)}+\cdots =(r-by/r)+\frac{1}{2r}(a/{r}^{2}-1/\rho )a{y}^{2}(1+by/{r}^{2}+\cdots )+\cdots .$$
(4)
$$v=({r}^{\prime}-{b}^{\prime}y/{r}^{\prime})+\frac{1}{2{r}^{\prime}}({a}^{\prime}/{{r}^{\prime}}^{2}-1/\rho ){a}^{\prime}{y}^{2}(1+{b}^{\prime}y/{{r}^{\prime}}^{2}+\cdots )+\cdots ;$$
(5)
$$u+v=(r+{r}^{\prime})-\left(\frac{b}{r}+\frac{{b}^{\prime}}{{r}^{\prime}}\right)y+\left[\frac{a}{r}\left(\frac{a}{{r}^{2}}-\frac{1}{\rho}\right)+\frac{{a}^{\prime}}{{r}^{\prime}}\left(\frac{{a}^{\prime}}{{{r}^{\prime}}^{2}}-\frac{1}{\rho}\right)\right]\frac{{y}^{2}}{2}+\left[\frac{ab}{{r}^{3}}\left(\frac{a}{{r}^{2}}-\frac{1}{\rho}\right)+\frac{{a}^{\prime}{b}^{\prime}}{{{r}^{\prime}}^{3}}\left(\frac{{a}^{\prime}}{{{r}^{\prime}}^{2}}-\frac{1}{\rho}\right)\right]\frac{{y}^{3}}{2}+\cdots .$$
(6)
$$\text{sin}\hspace{0.17em}\theta +\text{sin}\hspace{0.17em}{\theta}^{\prime}=\pm m\mathrm{\lambda}/\sigma .$$
(7)
$$(a/r)(a/{r}^{2}-1/\rho )+({a}^{\prime}/{r}^{\prime})({a}^{\prime}/{{r}^{\prime}}^{2}-1/\rho )=0.$$
(8)
$${\text{cos}}^{2}\hspace{0.17em}\theta /r+{\text{cos}}^{2}\hspace{0.17em}{\theta}^{\prime}/{r}^{\prime}=(\text{cos}\hspace{0.17em}\theta +\text{cos}\hspace{0.17em}{\theta}^{\prime})/\rho .$$
(9)
$$({\text{cos}}^{2}\hspace{0.17em}\theta /{r}^{2})\mathrm{\Delta}r+({\text{cos}}^{2}\hspace{0.17em}{\theta}^{\prime}/{r}^{\prime})\mathrm{\Delta}{{r}^{\prime}}^{2}=0.$$
(10)
$${\text{cos}}^{2}\hspace{0.17em}\theta /{r}^{2}={\text{cos}}^{2}\hspace{0.17em}{\theta}^{\prime}/{{r}^{\prime}}^{2}=1/{\rho}^{2}.$$
(11)
$$\begin{array}{ll}\hfill a/{r}^{2}-1/\rho =& (\rho \hspace{0.17em}\text{cos}\hspace{0.17em}\theta -r)/r\rho =\mathrm{\Delta}r/r\rho ,\\ \hfill {a}^{\prime}/{{r}^{\prime}}^{2}-1/\rho =& \mathrm{\Delta}{r}^{\prime}/{r}^{\prime}\rho =-\mathrm{\Delta}r/{r}^{\prime}\rho ,\end{array}$$
(12)
$$\begin{array}{l}{\mathrm{\Delta}}_{3}(u+v)=\left[\frac{ab}{{r}^{3}}(a/{r}^{2}-1/\rho )+\frac{{a}^{\prime}{b}^{\prime}}{{{r}^{\prime}}^{3}}({a}^{\prime}/{{r}^{\prime}}^{2}-1/\rho )\right]\frac{{y}^{3}}{2}\\ =\left(\frac{ab}{{r}^{4}\rho}-\frac{{a}^{\prime}{b}^{\prime}}{{{r}^{\prime}}^{4}\rho}\right)\mathrm{\Delta}r\frac{{y}^{3}}{2}\\ =\left(\frac{\text{sin}\hspace{0.17em}\theta \hspace{0.17em}\text{cos}\hspace{0.17em}\theta}{{r}^{2}\rho}-\frac{\text{sin}\hspace{0.17em}{\theta}^{\prime}\hspace{0.17em}\text{cos}\hspace{0.17em}{\theta}^{\prime}}{{{r}^{\prime}}^{2}\rho}\right)\mathrm{\Delta}r\frac{{y}^{3}}{2}\\ =(\text{tan}\hspace{0.17em}\theta -\text{tan}\hspace{0.17em}{\theta}^{\prime})\frac{\mathrm{\Delta}r}{2}{\left(\frac{y}{\rho}\right)}^{3}.\end{array}$$
(13)
$$\begin{array}{ll}\hfill u=& r-by/r+({x}^{2}/2r)(1-a/\rho ),\\ \hfill u+v=& r+{r}^{\prime}-(\text{sin}\hspace{0.17em}\theta +\text{sin}\hspace{0.17em}{\theta}^{\prime})y+({x}^{2}/2)[(1-{\text{cos}}^{2}\hspace{0.17em}\theta )/r+(1-{\text{cos}}^{2}\hspace{0.17em}{\theta}^{\prime}){r}^{\prime}].\end{array}$$
(14)
$${\mathrm{\Delta}}_{4}(u+v)=\frac{{y}^{4}}{8{\rho}^{3}}\left[\frac{{\text{sin}}^{2}\hspace{0.17em}\theta}{\text{cos}\hspace{0.17em}\theta}+\frac{{\text{sin}}^{2}\hspace{0.17em}{\theta}^{\prime}}{\text{cos}\hspace{0.17em}{\theta}^{\prime}}\right]=\frac{{y}^{4}}{8{\rho}^{3}}[\text{tan}\hspace{0.17em}\theta \hspace{0.17em}\text{sin}\hspace{0.17em}\theta +\text{tan}\hspace{0.17em}{\theta}^{\prime}\hspace{0.17em}\text{sin}\hspace{0.17em}{\theta}^{\prime}].$$
(15)
$$2{Y}_{m}=2\hspace{0.17em}{\left[\frac{2\mathrm{\lambda}{\rho}^{3}}{\text{tan}\hspace{0.17em}\theta \hspace{0.17em}\text{sin}\hspace{0.17em}\theta +\text{tan}\hspace{0.17em}{\theta}^{\prime}\hspace{0.17em}\text{sin}\hspace{0.17em}{\theta}^{\prime}}\right]}^{{\scriptstyle \frac{1}{4}}}.$$
(16)
$$2{Y}_{\text{opt}.}=2.36\beta =2.36\hspace{0.17em}{\left[\frac{2\mathrm{\lambda}{\rho}^{3}(2/\pi )}{\text{tan}\hspace{0.17em}\theta \hspace{0.17em}\text{sin}\hspace{0.17em}\theta +\text{tan}\hspace{0.17em}{\theta}^{\prime}\hspace{0.17em}\text{sin}\hspace{0.17em}{\theta}^{\prime}}\right]}^{{\scriptstyle \frac{1}{4}}}=1.06(2{Y}_{m}).$$