## Abstract

Hopkins’s treatment of tolerance theory shows that, in designing quality optical systems, we should aim at minimizing the variance *K* of a wave-aberration difference function. This suggests that the value of *K* may serve as a diffraction-based criterion of image quality in the fine-correction stage of automatic optical design. For this purpose, it would be desirable to know the range of states of poorer correction over which *K* may still be regarded as a useful criterion. On the basis of the present investigation, it appears that a necessary condition to be satisfied is that the relative modulation of the system should exceed the range 0.60–0.69, depending on the type of wave aberration and the azimuth of the grating test object considered.

© 1969 Optical Society of America

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### Equations (18)

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(1)
$$M(s,\psi )\text{exp}[i\theta (s,\psi )]=D(s,\psi )/{D}_{0}(s,\psi ).$$
(2)
$$D(s,\psi )=\frac{1}{A}\underset{S}{\int \int}\text{exp}[iksV(x,y;s,\psi )]dxdy,$$
(3)
$$V(x,y;s,\psi )=V=(1/s)\{W(x+s/2,y)-W(x-s/2,y)\}$$
(4)
$$M(s,\psi )\text{exp}[i\theta (s,\psi )]=\frac{\text{exp}[iks\overline{V}]}{S}\underset{S}{\int \int}\text{exp}[iks(V-\overline{V})]dxdy.$$
(5)
$$M(s,\psi )=1-(2{\pi}^{2}{s}^{2}/{\mathrm{\lambda}}^{2})K(s,\psi ),$$
(6)
$$K(s,\psi )={\u3008{V}^{2}\u3009}_{\text{av}}-{\{\overline{V}\}}^{2}$$
(7)
$$\partial K/\partial {W}_{20}=0,$$
(8)
$$W(x,y)={W}_{20}({x}^{2}+{y}^{2})+{W}_{40}{({x}^{2}+{y}^{2})}^{2}+{W}_{60}{({x}^{2}+{y}^{2})}^{3},$$
(9)
$$\begin{array}{l}K(s,0)={K}_{\text{even}}\\ =\underset{mn}{\sum \sum}{A}_{m,n}(s){W}_{m,0}{W}_{n,0},\end{array}$$
(10)
$$\begin{array}{ll}\hfill {W}_{60}*& =(0.58/s+1.5)\mathrm{\lambda}\\ \hfill {B}_{46}*& =-(1.8-1.5s+1.5{s}^{2})\\ \hfill {B}_{26}*& =(0.9-1.4s+2.1{s}^{2}).\end{array}$$
(11)
$$W=f{W}_{60}[{B}_{26}({x}^{2}+{y}^{2})+{B}_{46}{({x}^{2}+{y}^{2})}^{2}+{({x}^{2}+{y}^{2})}^{3}],$$
(12)
$$N(s,0)=1-\frac{2{\pi}^{2}{s}^{2}{f}^{2}}{{\mathrm{\lambda}}^{2}}\{\underset{mn}{\sum \sum}{A}_{m,n}(s){W}_{m,0}{W}_{n,0}\},$$
(13)
$$W(x,y)={W}_{31}({x}^{2}+{y}^{2})y+{W}_{51}{({x}^{2}+{y}^{2})}^{2}y,$$
(14)
$$\begin{array}{l}K(s,\psi )={K}_{\text{odd}}\\ =\underset{mn}{\sum \sum}{C}_{m,n}(s,\psi ){W}_{m,1}{W}_{n,1},\end{array}$$
(15)
$$\begin{array}{ll}\hfill {W}_{51}*& =(0.37/s+0.83)\mathrm{\lambda}\\ \hfill {B}_{35}*& =-(1.49-1.26s+0.85{s}^{2})\end{array}\}\psi =90\xb0$$
(16)
$$\begin{array}{ll}\hfill {W}_{51}*& =(0.64/s+0.96)\mathrm{\lambda}\\ \hfill {B}_{35}*& =-(1.49-0.94s+0.41{s}^{2})\end{array}\}\psi =0.$$
(17)
$$W=f{W}_{51}[{B}_{35}({x}^{2}+{y}^{2})y+{({x}^{2}+{y}^{2})}^{2}y],$$
(18)
$$N(s,\psi )=1-\frac{2{\pi}^{2}{s}^{2}{f}^{2}}{{\mathrm{\lambda}}^{2}}\{\underset{mn}{\sum \sum}{C}_{mn}(s,\psi ){W}_{m1}{W}_{n1}\}$$