While this formalism should accurately describe contrast transfer between zero and cutoff frequency for annular aperture, it is not appropriate for determining the cutoff frequency. The reason is that annular pupil produces diffraction pattern reduced in size with respect to that of clear aperture of the same diameter - in effect, a pattern nearly identical to one produced by a larger clear aperture with spherical aberration, in which the level of aberration does not cause change in cutoff frequency. In other words, it is physically impossible that a near-identical impulse response (PSF) produces two different MTF cutoff frequencies.
This conclusion is consistent with the relation between PSF and OTF, which are a Fourier pair, implying that any given PSF will produce a single corresponding OTF. Since the annular aperture PSF is narrower than that of a clear aperture of the same diameter, its OTF will be wider, i.e. will expand toward higher frequencies. Plots below illustrate the Fourier transform relationship between angular size of the PSF (first minima 1.22λ/D radians) and the extent of OTF frequency spectrum (linear cutoff period λF and frequency 1/λF, angular cutoff period λ/D period and angular cutoff frequency D/λ) for circular pupil of diameter D (top) and for half as large aperture (bottom) of identical focal ratio.
Properties of the ring structure depend on pupil transmission properties but OTF width depends only on the first minima radius. In general, reduction of transmission in the central pupil area increases the magnitude and/or size of (primarily) the 1st bright ring and reduces the central maxima radius, while reduction in the outer area suppresses the rings but increases the maxima radius; aberrations effectively alter transmission over pupil as well, so they also affect the central maxima radius and the rings' radii and magnitude.
Follows an example of the obstructed aperture PSF and its properties, indicating the same conclusion, i.e. that annular aberration-free aperture must have higher MTF cutoff frequency than circular aperture of the same diameter. Raytrace was done with D=152mm f/8.18 mirror in polychromatic light (25 wavelengths from 440nm to 680nm, weighted for photopic eye sensitivity). Base (obstructed) aperture diameter was 152mm, and the aberrated by 10% larger.
of a clear aberration-free aperture of diameter D with 0.33D
central obstruction (red) nearly coincides with the PSF of the 10% larger
(linearly, 1.1D diameter) clear aperture with λ/4 wave P-V of primary spherical (blue).
Central maxima of the obstructed aperture - reduced by about 10% in
diameter due to
the effect of
central obstruction -
is nearly identical to that of the larger aberrated clear aperture
in monochromatic light; however, raytrace in polychromatic light
indicates it is still some 7% smaller (despite the pSF maxima of the
aberrated aperture being also slightly smaller, due to the
aberration). Since the focal length is constant, the PSF maxima -
including FWHM - in the smaller obstructed aperture is actually
smaller angularly than in the larger aperture. Yet, according to the
formalism, the smaller obstructed aperture will have 10% lower cutoff frequency, given as
D/λ (cycles per radian; related
to the linear resolution as D/λ�=1/λF
lines per mm, �
being the focal length and F the focal ratio),
i.e. as much worse limiting MTF resolution. Evidently, there is no
basis for such a difference in their respective PSFs.
In fact, the plane physical parameters directly imply that the
smaller obstructed aperture will actually have higher cutoff
frequency than the larger one.
In effect, the obstructed aperture's PSF becomes comparable to that of a clear aperture larger by a factor 1/(1-ο2) having its Strehl reduced to (1-ο2)2 by spherical aberration. Since the central maxima also shrinks with the increase in primary spherical aberration, although significantly less than with c. obstruction - nearly 3% with λ/4 wavefront error, and nearly 9% with λ/2 error - it will partly offset the larger rate of obstructed maxima diameter reduction in polychromatic light.
On the graph, we see that even with the standard MTF formalism the 1-o "rule" is, for this obstruction size - placing its contrast level at about 0.68D aperture level - valid only for the lower third of the frequency range. Contrast rapidly recovers toward higher frequencies: it is at the level of 0.75D already at the 0.4 frequency, and at about 0.9D level at the bright low-contrast cutoff at ~0.55 frequency. For bright low contrast details, contrast dropping to 0.68D level would cause resolution threshold to also drop by a third (5 vs. 0), but the threshold is even in the standard MTF model better, about 27% lower than for perfect base aperture. If going by the contrast level projected from the likely obstructed plot at about 0.8D, the threshold is worsened by little over 20% vs. base unobstructed aperture (4 vs. 0). But at the actual point of intersection for the likely obstructed aperture plot, the threshold is 4-5% above the base aperture (2 vs. 0). It is about 7% smaller than the threshold for 10% larger aperture (3) when aberration-free, and a few percent larger when it has 1/4 wave of primary spherical aberration.
Raytrace gives imilar results for other central obstruction sizes. For instance, the PSF of 20% larger linear aperture aberrated by λ/2 wave P-V wavefront error of spherical aberration is nearly identical to the PSF of the smaller aperture with 0.6D CO (the only difference being in mainly offsetting energy content in the 2nd and 3rd bright ring). Here (1-ο2)2 gives as the PSF-equilizing Strehl value of 0.41, with the actual Strehl for this amount of spherical aberration being 0.38. And, yet again, despite the two nearly identical PSFs, the standard MTF formalism tells us that the larger aberrated aperture has 20% higher cutoff frequency than the smaller obstructed one.
It should be noted that FWHM shrinks due to CO at somewhat slower rate than the 1st minima radius for obstructions ~D/4 and larger. Specifically, the FWHM-deduced aperture enlargement is smaller from about 30% at 0.3D to about 40% at 0.5D CO. FWHM is more relevant to diffraction resolution than the 1st minima radius, but in polychromatic light it would be mainly offset by the larger rate of obstruction-induced central minima reduction, implied by raytrace.
Better contrast transfer of obstructed apertures than what the standard MTF formalism indicates would also help explain why the popular 1-o "rule" - based on that formalism - seems to be at least roughly working, even if it completely neglects significant differences in the total aberration level between smaller unobstructed and larger obstructed apertures (seeing, thermals, inherent aberrations, alignment, surface/coating quality, etc.).