## Abstract

Any image-forming optical system is usually tested for aberrations by looking at the image plane. A point source is placed at some distance from the system (this may be a very large or small distance depending on the particular system) and its image by the optical system is studied by any of many methods available. The present paper presents the theory of a method in which a grating containing a series of concentric circles is placed at the image of the point source. Characteristic patterns are derived for the usual aberrations of optical systems and some typical photographs of these patterns are presented. Finally, this method is compared to the Ronchi test in which a series of straight lines are used for the grating.

© 1966 Optical Society of America

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### Equations (13)

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(1)
$${TA}_{x}=\overline{x}=R\frac{\partial W}{\partial x};\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{TA}_{y}=\overline{y}=R\frac{\partial W}{\partial y}.$$
(2)
$${(\overline{x}-\xi )}^{2}+{(\overline{y}-\eta )}^{2}={n}^{2}{{\rho}_{1}}^{2}$$
(3)
$${(\overline{x}-\xi )}^{2}+{(\overline{y}-\eta )}^{2}=n{{\rho}_{1}}^{2}$$
(4)
$${\left(R\frac{\partial W}{\partial x}-\xi \right)}^{2}+{\left(R\frac{\partial W}{\partial y}-\eta \right)}^{2}={n}^{2}{{\rho}_{1}}^{2}.$$
(5)
$$W(x,y)=B{({x}^{2}+{y}^{2})}^{2}+\frac{\overline{z}}{2{R}^{2}}({x}^{2}+{y}^{2}),$$
(6)
$$\begin{array}{l}R\frac{\partial W}{\partial x}=4BRx\hspace{0.17em}({x}^{2}+{y}^{2})+\frac{\overline{z}x}{R}\hfill \\ R\frac{\partial W}{\partial y}=4BRy\hspace{0.17em}({x}^{2}+{y}^{2})+\frac{\overline{z}y}{R}\hfill \end{array}\}.$$
(7)
$$\begin{array}{l}R\frac{\partial W}{\partial x}=8{B}_{0}F\#u\hspace{0.17em}({u}^{2}+{v}^{2})+\frac{\overline{z}}{2F\#}u\hfill \\ R\frac{\partial W}{\partial y}=8{B}_{0}F\#v\hspace{0.17em}({u}^{2}+{v}^{2})+\frac{\overline{z}}{2F\#}v\hfill \end{array}\},$$
(8)
$$\begin{array}{l}\hfill {(8{B}_{0}F\#)}^{2}{({u}^{2}+{v}^{2})}^{3}+8{B}_{0}\overline{z}{({u}^{2}+{v}^{2})}^{2}\hfill \\ \hfill -\hspace{0.17em}16\hspace{0.17em}{B}_{0}F\#({u}^{2}+{v}^{2})(\xi u+\eta v)+{\left(\frac{\overline{z}}{2F\#}\right)}^{2}({u}^{2}+{v}^{2})\hfill \\ \hfill -\hspace{0.17em}\frac{\overline{z}}{F\#}(\xi u+\eta v)+({\xi}^{2}+{\eta}^{2})={n}^{2}{{\rho}_{1}}^{2}\hfill \end{array}\}.$$
(9)
$$W(x,y)=Fy({x}^{2}+{y}^{2})+\frac{\overline{z}}{2{R}^{2}}({x}^{2}+{y}^{2}).$$
(10)
$$\begin{array}{l}{(2{F}_{0}F\#)}^{2}({u}^{4}+10{u}^{2}{v}^{2}+9{v}^{4})+6{F}_{0}\overline{z}({u}^{2}+{v}^{2})v\hfill \\ -\hspace{0.17em}4{F}_{0}F\#\{2\xi uv+\eta \hspace{0.17em}({u}^{2}+3{v}^{2})\}+{\left(\frac{\overline{z}}{2F\#}\right)}^{2}({u}^{2}+{v}^{2})\hfill \\ -\hspace{0.17em}\frac{\overline{z}}{F\#}(\xi u+\eta v)+({\xi}^{2}+{\eta}^{2})={n}^{2}{{\rho}_{1}}^{2}\hfill \end{array}\},$$
(11)
$$W(x,y)=C({x}^{2}-{y}^{2})+\frac{\overline{z}}{2{R}^{2}}({x}^{2}+{y}^{2}).$$
(12)
$$\begin{array}{l}\left\{{(4{C}_{0}F\#)}^{2}+{\left(\frac{\overline{z}}{2F\#}\right)}^{2}\right\}({u}^{2}+{v}^{2})+4{C}_{0}\overline{z}\hspace{0.17em}({u}^{2}-{v}^{2})\hfill \\ -\hspace{0.17em}8{C}_{0}F\#(\xi u-\eta v)-\frac{\overline{z}}{F\#}(\xi u+\eta v)\hfill \\ +({\xi}^{2}+{\eta}^{2})={n}^{2}{{\rho}_{1}}^{2}\hfill \end{array}\},$$
(13)
$$R\frac{\partial W}{\partial x}\text{cos}\alpha +R\frac{\partial W}{\partial y}\text{sin}\alpha =nd,$$