## Abstract

Most optical systems have rotational symmetry. For such systems, we establish a method of finding (a) the maximum attainable modulation transfer function (MTF) at arbitrary frequency *ω*_{0}; and (b) the required pupil function *U*(*ω*_{0}; *ρ*). Physically, the latter are absorbing films in the pupil of diffraction-limited optics. The method of solution is numerical and iterative, based upon the Newton–Raphson algorithm. Solutions (a) and (b) are established at frequencies *ω*_{0} = 0.1, 0.2, ⋯, 0.9 (× optical cutoff). The computed (a) are correct to ±0.0001 over all *ω*_{0} indicated. Quantities (b) have an average error over each pupil of ±0.002 for frequencies *ω*_{0} ≤ 0.5. With 0.6 ≤ *ω*_{0} ≤ 0.9, the error is ±0.01. The curve of maximum MTF(*ω*_{0}) seems sufficiently smooth to allow for accurate interpolation. Solutions (a) and (b) were also found over the finer subdivision *ω*_{0} = 0.05, 0.1, 0.15, ⋯, 0.8 with slightly less accuracy than above, in order to allow for interpolation of pupils *U*(*ω*_{0}; *ρ*) over values *ω*_{0}. This seems possible for 0.05 ≤ *ω*_{0} ≤ 0.40. The maximum MTF(*ω*_{0}) shows appreciable gain (e.g., 8% at *ω*_{0} = 0.2) over the MTF for uncoated, diffraction-limited optics at all *ω*_{0} except in the intermediate region 0.4 ≤ *ω*_{0} ≤ 0.6. However, in the high-frequency band 0.5 ≤ *ω*_{0} ≤ 1.0, the maximum MTF(*ω*_{0}) shows little gain over the MTF due to an uncoated, diffraction-limited pupil with the proper central obstruction. The light loss due to each *U*(*ω*_{0}; *ρ*) may be measured by the total energy transmittance and the Strehl flux ratio. These are plotted against *ω*_{0}, and indicate moderate light loss.

© 1969 Optical Society of America

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### Equations (13)

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(1)
$$U({\omega}_{0};\rho )=0\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{for}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\rho <{\omega}_{0}-{\rho}_{0},$$
(2)
$${\omega}_{0}>{\rho}_{0}.$$
(3)
$${\rho}_{1}=\{\begin{array}{ccc}{\omega}_{0}-{\rho}_{0}& \text{if}& {\omega}_{0}>{\rho}_{0}\\ 0& \text{if}& {\omega}_{0}\le {\rho}_{0}.\end{array}$$
(4)
$$U({\omega}_{0};\rho )=\{\begin{array}{ll}\sum _{m=0}^{M-1}{b}_{m}{J}_{0}({\mu}_{m}\rho ),\hfill & \text{for}\hspace{0.17em}{\rho}_{1}\le \rho \le {\rho}_{0},\hfill \\ \hfill 0\hfill & \text{for}\hspace{0.17em}\rho <{\rho}_{1}\hspace{0.17em}\text{and}\hspace{0.17em}\rho >{\rho}_{0}.\hfill \end{array}$$
(5)
$${J}_{1}({\mu}_{m}{\rho}_{0})=0,\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}m=1,2,\cdots ,M-1$$
(6)
$${b}_{0},{b}_{1},\cdots ,{b}_{M-1}\hspace{0.17em}\text{by}\hspace{0.17em}\mathbf{b},$$
(7)
$$\partial \hspace{0.17em}\text{MTF}({\omega}_{0})/\partial {b}_{m}{\mid}_{\mathbf{b}=\mathbf{B}}=0,\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}m=0,1,\cdots ,M-1.$$
(8)
$$W=W(\mathbf{b})\equiv \sum _{m=0}^{M-1}{[\partial \hspace{0.17em}\text{MTF}({\omega}_{0})/\partial {b}_{m}]}^{2}.$$
(9)
$$\begin{array}{c}\mathrm{\Delta}{b}_{m}\equiv -W\xb7\partial W/\partial {b}_{m}/\sum _{n=0}^{M}{(\partial W/\partial {b}_{n})}^{2},\\ m=0,1,\cdots ,M-1.\end{array}$$
(10)
$${\int}_{0}^{\theta}d\varphi U({\omega}_{0};{\rho}^{\prime})=\pi \hspace{0.17em}\text{MTF}({\omega}_{0})U({\omega}_{0};\rho ),$$
(11)
$${\rho}^{\prime}={({\rho}^{2}+{{\omega}_{0}}^{2}-2\rho {\omega}_{0}\hspace{0.17em}\text{cos}\varphi )}^{{\scriptstyle \frac{1}{2}}}$$
(12)
$$\theta =\{\begin{array}{cc}{\text{cos}}^{-1}[{(2\rho {\omega}_{0})}^{-1}({\rho}^{2}+{{\omega}_{0}}^{2}-{{\rho}_{0}}^{2})]& \text{for}\hspace{0.17em}\rho \ge {\rho}_{0}-{\omega}_{0}\hfill \\ \pi & \text{for}\hspace{0.17em}\rho \le {\rho}_{0}-{\omega}_{0}.\hfill \end{array}$$
(13)
$$2\pi {\int}_{{\rho}_{1}}^{{\rho}_{0}}d\rho \rho .$$