**According to the theory, annular aperture has MTF cutoff
frequency identical to that of a circular aperture of the same
diameter. Since central obstruction (CO) results in smaller central
maxima, the cutoff frequency (i.e. limiting MTF resolution) should
be proportionally smaller. **

The usual way of obtaining OTF/MTF is from
the normalized exit pupil
autocorrelation, where the integration area for the contrast
transfer for aberration-free aperture is graphically presented as
the overlap - normalized to the circle area - of two identical circles of
the diameter equaling **λ***f*νc,
where **λ** is the wavelength,
*f*
the focal length and
**ν**c
the cutoff frequency. With
**ν**c=1/λF,
**F** being the focal ratio, the center separation of these two
circles at the cutoff frequency equals the circle (i.e. pupil)
diameter. The same formalism is used to obtain OTF/MTF (for
aberration-free aperture the phase OTF component is unity, and OTF
reduces to MTF) of annular aperture, except that the overlap area
excludes the area of central obstruction, and overlap is normalized
to the annulus area (FIG. 104).
Hence, the formalism results in the cutoff frequency for obstructed
aperture independent of the size of obstruction, and equal to that
of clear circular aperture, 1/λF.

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.

LEFT:
PSF
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
standard MTF
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.

MTF
cutoff frequency does not change for primary spherical aberration up to 1/2 wave P-V,
and somewhat larger, which implies that more energy outside the Airy
disc shouldn't affect cutoff frequency for
larger CO as well. As mentioned above, the main determinant is the
diameter of central
maxima. Since the reduction of the PSF angular size in the obstructed aperture is
well approximated by 1-*o*2
(somewhat optimistic for obstructions significantly larger than D/3), angular size of its central maxima - and
its MTF cutoff frequency - are better approximated as those corresponding
to an aperture larger by a factor of

D/(1-*o*2). Consequently, contrast transfer at any
normalized spatial frequency **ν**
of unobstructed aperture nearly corresponds to (1-*o*2)ν
frequency of the obstructed aperture.

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.).