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 ω0; and (b) the required pupil function U(ω0; 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 ω0 = 0.1, 0.2, ⋯, 0.9 (× optical cutoff). The computed (a) are correct to ±0.0001 over all ω0 indicated. Quantities (b) have an average error over each pupil of ±0.002 for frequencies ω0 ≤ 0.5. With 0.6 ≤ ω0 ≤ 0.9, the error is ±0.01. The curve of maximum MTF(ω0) seems sufficiently smooth to allow for accurate interpolation. Solutions (a) and (b) were also found over the finer subdivision ω0 = 0.05, 0.1, 0.15, ⋯, 0.8 with slightly less accuracy than above, in order to allow for interpolation of pupils U(ω0; ρ) over values ω0. This seems possible for 0.05 ≤ ω0 ≤ 0.40. The maximum MTF(ω0) shows appreciable gain (e.g., 8% at ω0 = 0.2) over the MTF for uncoated, diffraction-limited optics at all ω0 except in the intermediate region 0.4 ≤ ω0 ≤ 0.6. However, in the high-frequency band 0.5 ≤ ω0 ≤ 1.0, the maximum MTF(ω0) 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(ω0; ρ) may be measured by the total energy transmittance and the Strehl flux ratio. These are plotted against ω0, and indicate moderate light loss.
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