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<sub>o</sub>; and (b) the required pupil function <i>U</i>(ω<sub>0</sub>; 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<sub>0</sub>=0.1, 0.2, ..., 0.9 (× optical cutoff). The computed (a) are correct to ±0.0001 over all ω<sub>0</sub> indicated. Quantities (b) have an average error over each pupil of ±0.002 for frequencies ω<sub>0</sub>≤0.5. With 0.6≤ω<sub>0</sub>≤0.9, the error is ±0.01. The curve of maximum MTF(ω<sub>0</sub>) seems sufficiently smooth to allow for accurate interpolation. Solutions (a) and (b) were also found over the finer subdivision ω<sub>0</sub>=0.05, 0.1, 0.15, …, 0.8 with slightly less accuracy than above, in order to allow for interpolation of pupils <i>U</i>(ω<sub>0</sub>; ρ) over values ω<sub>0</sub>. This seems possible for 0.05≤ω<sub>0</sub>≤0.40. The maximum MTF(ω<sub>0</sub>) shows appreciable gain (e.g., 8% at ω<sub>0</sub>=0.2) over the MTF for uncoated, diffraction-limited optics at all ω<sub>0</sub> except in the intermediate region 0.4≤ω<sub>0</sub>≤0.6. However, in the high-frequency band 0.5≤ω<sub>0</sub>≤1.0, the maximum MTF(ω<sub>0</sub>) shows little gain over the MTF due to an uncoated, diffraction-limited pupil with the proper central obstruction. The light loss due to each <i>U</i>(ω<sub>0</sub>; ρ) may be measured by the total energy transmittance and the Strehl flux ratio. These are plotted against ω<sub>0</sub>, and indicate moderate light loss.
B. ROY FRIEDEN, "Maximum Attainable MTF for Rotationally Symmetric Lens Systems," J. Opt. Soc. Am. 59, 402-404 (1969)