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 ωo; 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.
B. ROY FRIEDEN, "Maximum Attainable MTF for Rotationally Symmetric Lens Systems," J. Opt. Soc. Am. 59, 402-404 (1969)