7.1. Central obstruction effect

FIGURE 103: Relative central obstruction diameter ο expressed in units of the aperture diameter D.  The effects of obstruction are: (1) reduction in light transmission by a factor of (1-ο2), resulting from pupil obscuration, and (2) transfer of energy out of the Airy disc - mostly to the first bright ring. As a rule of thumb, relative loss of energy from the Airy disc is well approximated with double the relative obstruction area in the pupil. Consequences of the latter in regard to intensity distribution within diffraction pattern differ somewhat for near-perfect wavefront on one side, and aberrated wavefronts on the other. Depending on the type and size of wavefront deformation, presence of central obstruction may improve, worsen, or have no appreciable effect on wavefront quality within the annulus, compared to the quality of the entire wavefront. This effect is small to negligible for the usual range of central obstruction sizes.

INCONSISTENCIES

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

LEFT: Linear PSF of a clear aperture D with l/4 wave of primary spherical (blue), and 10% smaller aberration-free aperture with D/3 central obstruction (red). Central maxima of the obstructed aperture - widened by 10% due to aperture reduction, but reduced about as much due the effect of central obstruction - remains unchanged, i.e. practically equal to that of 10% larger clear aperture. Since the focal length is constant, angular central maxima size of the obstructed aperture is also equal to that of linearly 10% smaller clear aperture (makes no difference to scale the entire system down, which would give to the obstructed aperture 10% smaller linear PSF, but also as much smaller focal length, with the angular PSF unchanged), Yet, according to the MTF theory, the larger aberrated aperture will have 10% higher cutoff frequency, given as D/λ (cycles per radian; related to the linear resolution as D/λƒ=1/λF lines per mm, ƒ being the focal length and F the focal ratio), i.e. as much better limiting MTF resolution. Evidently, there is no basis for such difference in their respective PSFs. If the two PSFs are nearly identical, as they are, their actual MTF resolution limits can only be nearly identical as well.

 

RIGHT: 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. Since the reduction of the PSF angular size in the obstructed aperture is closely approximated by (1-o2) for obstructions smaller than ~D/3, and by 1-o2+o4 for larger obstructions, angular size of its central maxima - and its MTF cutoff frequency - should correspond to that of aperture larger by a factor of D/(1-o2)  and  D/(1-o2+o4), respectively. Consequently, contrast transfer at any normalized spatial frequency ν of unobstructed aperture corresponds to (1-o2)ν frequency of the obstructed aperture (in other words, by showing plots for both over identical normalized frequency range, the plot for obstructed aperture is effectively compressed horizontally by the 1-o2 factor for CO up to ~D/3, and slightly more for larger CO). When plotted over the actual resolution range for the obstructed aperture, the MTF of an aperture with 0.325D C.O. vs. its standard clear aberration-free MTF is shown at right.

    In effect, the obstructed PSF becomes similar to that of a clear aperture larger by a factor 1/(1-ο2) having its Strehl reduced to (1-ο2)2 by spherical aberration (note that 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 should be noted that FWHM shrinks due to CO at somewhat slower rate than the 1st minima radius for obstructions ~D/4 and larger. Since the FWHM is more relevant to diffraction resolution than the radius, the 1/(1-ο2) effective aperture enlargement ratio is somewhat too optimistic for obstructions larger than D/4 (specifically, the FWHM-based aperture enlargement is smaller from about 30% at 0.3D to about 40% at 0.5D CO).

    This results in a considerably different contrast transfer effect in obstructed aperture than what the usual presentations with cutoff frequency normalized to 1 indicate. While the obstructed aperture is better by relatively insignificant amount on the left side of MTF graph, it is significantly better - even better than clear aberration-free aperture - in the standard presentation on the right side. The reason why this contrast advantage might not be apparent in use is that this affects only details smaller than Airy disc diameter - a small fraction of the total of observable details. Also, in comparison of usually larger obstructed and smaller unobstructed aperture, this contrast transfer advantage of the obstructed aperture is lost due to its generally higher level of aberrations (seeing, thermals, inherent aberrations, alignment, surface/coating quality, etc.).

2. Theory models CO effect on near-monochromatic, practically coherent point-source. The light we observe is polychromatic, thus closer to incoherent. Since coherent and incoherent light have different dependencies on the changes in pupil transmission, CO effect will also be different.
While PSF has identical form for coherent and incoherent light, the actual energy and the way it is generated differ between the two. Light gathering power of astronomical telescopes, given transmission, changes with the area of aperture. So, doubling the aperture quadruples its area, hence also light gathering power and the actual peak diffraction intensity (in terms of magnitudes, every doubling in linear aperture gains 1.5 magnitude). This is because incoherent waves add up their energy individually, as a sum of their individual squared amplitudes. This is what makes incoherent light linear in intensity. On the other hand, coherent waves, due to their strong phase correlation, generate energy more efficiently: their individual amplitudes add up to a complex amplitude, which is then squared for energy. That makes them linear in amplitude, and quadratic in energy. In other words, quadrupling pupil area quadruples the complex amplitude, and increases the energy
16-fold.

These different dependencies of the energy on the pupil area imply that any given central obstruction - or any obstruction, for that matter - will have different effect in coherent vs. incoherent light. In the standard theory of diffraction in annular aperture, diffraction peak (normalized to 1 for clear aperture) is given by the relative annulus area squared - i.e. by (1-o2)2, the limit of I=[2J1(x)/x-o22J1(ox)/ox]2 for x=πr=0 (see Eq. c). This is consistent with coherent, but not with incoherent light. For the latter, the appropriate general expression would be I=[2J1(x)/x]2-[o22J1(ox)/ox]2, with the energy at every image point being a sum of the squared individual wave amplitudes, less those from the obstructed area. In terms of amplitudes, it is given as I'(r)=ΣAt2-ΣAot2. Hence, the normalized peak intensity here is reduced in proportion to 1-o4.

Since deducting the CO area amplitude contribution and squaring the difference (in coherent light) produces different result than deducting energy from this area directly (in incoherent light), the entire annular aperture PSF will differ for the two, not only its central intensity. Considering that PSF-defined intensity distribution for point-source image determines contrast and resolution of extended objects as well, any significant difference in the annular PSF will imply significant difference in the effect of CO on overall performance.

Applying the same method of deducting the contribution of the obstructed area from the contribution of the entire aperture area at every image point for coherent and incoherent light is shown below (top). The annulus area PSF (red) at right is shown negative because it is for an inverse aperture, but the actual energy (illuminance, or intensity) is always positive (constructive/destructive interference in incoherent light takes place at the level of energy, i.e. individual wave amplitude squared, where sign of amplitude, i.e. phase, determines whether the interference is constructive or destructive). Evaluating both, coherent (left side) and incoherent (right side) modality for selected values of central obstruction (o) shows significant differences between their respective annular PSFs (bottom).

It is obvious that the effect is much smaller in incoherent light (so small that the 11% energy PSF from D/3 CO area top right had to be magnified for clarity, and the resultant PSF after deducting this energy is not shown, since it would nearly coincide with the clear aperture PSF at this image scale), and here's some more details.

In general, CO effect modeled for incoherent light shows much less depressed central maxima, and nearly identical reduction in its width. In incoherent light, only two CO values are evaluated, because the effect is still negligible at 0.3D CO, and because it is somewhat informal graphic routine meant only illustrate magnitude of the effect, not to give exact values. If the CO energy energy in incoherent light is directly deducted, the effect is even smaller. But it is assumed as more likely that energy interferes as the amplitudes do, so that the energy in the first bright ring, for example, is of opposite phase to that in the central maxima, and that combines constructively (i.e. adds up) with the (missing) energy from the CO area corresponding to this section of pattern radius. Central maxima values in the table are accurate for the peak, radius and FWHM (as well as for the 2nd maxima relative peak vs. that of the central maxima), and only roughly estimated for the encircled energy, since no raytracing software uses the incoherent light modality in any form.

For the actual brightness of the 1st bright ring is not relevant its height in obstructed vs. clear aperture, shown on the graph. It is its height relative to the peak of central maxima, given in the last column at right. It shows much less of brightening in incoherent light: at D/2 CO, the 1st bright ring is as bright as at D/5 CO in coherent light. This, however, is only one side. It is more important how much energy the ring contains, i.e. its brightness combined with its width. In this respect, D/2 CO in incoherent light is closer to 0.3D CO in coherent light. Still significantly less of an effect, although graphs indicate that larger obstructions in incoherent light might have relatively more energy transferred beyond the first few bright rings.

The difference in FWHM for the range of obstructions is small in coherent, and negligible in incoherent light, indicating no appreciable difference in the resolution limit for near-equally bright stars. The energy ratio for the central maxima indicates the amount of energy lost to the rings, and can be looked at as a Strehl-like number for the left half of MTF graph, which covers most of the observable details, including resolvable low-contrast planetary detail. Here too the effect of CO is significantly smaller in incoherent light, with as large as D/2 CO losing only about 20% of the energy from the central maxima (hence qualifying as "diffraction limited").

The actual light from astronomical objects is neither incoherent, nor coherent. It is generally closer to incoherent (partially incoherent), so the actual CO effect would be, correspondingly, closer to that indicated by the incoherent model. While this modeling for the effect of central obstruction in incoherent light is, as already mentioned, informal and approximate, it can be assumed that it is significantly smaller than in coherent light, simply because taking out energy originating from the obstruction area directly has significantly less of an effect than taking out the corresponding wave amplitude in coherent light.