Any obstruction placed in the light path of an imaging system prevents waves from a portion of the wavefront to reach the focal zone. The consequence is change in wave contribution at every point of the diffraction pattern. The effect is similar to that of wavefront aberrations in that it changes pattern's intensity distribution, the specifics of it depending on the form and size of obstruction. Those common to amateur telescopes are: (1) central obstruction, usually caused by the secondary mirror, (2) secondary holder vanes and (3) apodizing mask. The light loss is significant only with apodizing mask. Real concern is the change in point-object image intensity distribution and its effect on image quality.
 

7.1. Central obstruction effect

Much has been said about the effect of central obstruction in the amateurs circles, most of it being speculation. The common notion is that it reduces effective linear aperture for low-contrast details by as much as obstruction diameter. Informal attempts have been made to find a theoretical basis for this empirical "rule". Not a few amateurs "tested" it and often concluded that it "works". What tends to be neglected is a pretty obvious fact that in any such comparison there is more than just a single factor - central obstruction - affecting low-contrast performance. Most of these factors - seeing error, overall optical quality, sensitivity to miscollimation and thermal errors, light scatter, baffling - favor smaller unobstructed aperture, usually a high-quality apochromatic refractor, over the larger reflector or catadioptric. Consequently, if the rule "works" empirically, it inadvertently proves it incorrect, as long as it insists on the difference in performance coming from the effect of central obstruction alone.

FIGURE 65: 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.

Central obstruction (CO) effect on the intensity distribution I'(r) of the diffraction pattern - or PSF of the obstructed aperture - normalized to 1 for the peak intensity of clear circular aperture, can be expressed as I'(r)=(At-Aotο2)2, with At being the normalized amplitude sum at a point of radius r in the image plane for unobstructed aberration-free aperture - given by the sum in brackets in Eq. (c) - and Aot the normalized amplitude sum for the obstruction area, obtained using the same relation form, only with t replaced by οt.

In effect, the sum of wave amplitudes originating from the obstructed area is deducted  from the sum of amplitudes of the aperture without obstruction, with the ο2 factor giving to the amplitude sum of the obstruction, normalized to 1, a proper value corresponding to its area vs. area of the whole aperture. Since the normalized sum of amplitudes for both approaches 1 as r approaches zero, peak diffraction intensity of an obstructed aperture is

I'(0) = (1-ο2)2          (60)

which is, in effect, the relative annulus area (in units of the clear aperture area) squared. This implies that the size of CO corresponding to a given peak diffraction intensity I is ο=(1-√I)1/2. PSF of an obstructed aperture is often normalized to unit based on its relative peak intensity, smaller by a factor of (1-ο2)2 than peak intensity of the corresponding unobstructed aperture, thus given as I'(r)/(1-ο2)2.

Following table presents intensity distribution within the fourth minima for unobstructed and selected obstructed apertures. Pattern radius r is in units of λF, and intensity I(r) is normalized to peak diffraction intensity, to reflect more clearly relative changes in intensity distribution; EE' is encircled energy. Analogously to central intensity, the actual relative encircled energy of obstructed aperture is smaller by a factor (1-ο2)2, or EE=(1-ο2)2EE'. For ο=0.30, energy encircled within first minima is 0.565, with 0.263 in the rings. If the photon flux in the annulus would equal that in the unobstructed aperture, both figures would be larger by a factor of 1/(1-ο2)2, or 0.682 and 0.318, respectively, and the central intensity would be identical to that of the clear aperture.
 

EFFECT OF CENTRAL OBSTRUCTION ON NORMALIZED PSF INTENSITY DISTRIBUTION (Source: Aberration Theory Made Simple, Mahajan)
Parameter		r						I'(r)						EE'
o g		0	0.10	0.20	0.30	0.40	0.50	0	0.10	0.20	0.30	0.40	0.50	0	0.10	0.20	0.30	0.40	0.50
1ST	Max.	0	0	0	0	0	0	1	1	1	1	1	1	0	0	0	0	0	0
	Min.	1.22	1.21	1.17	1.11	1.06	1	0	0	0	0	0	0	0.838	0.818	0.764	0.682	0.584	0.479
2ND	Max.	1.63	1.63	1.63	1.61	1.58	1.54	0.0175	0.0206	0.0304	0.0475	0.0707	0.0963	0.867	0.853	0.818	0.766	0.702	0.618
	Min.	2.23	2.27	2.36	2.42	2.39	2.29	0	0	0	0	0	0	0.910	0.906	0.900	0.899	0.885	0.829
3RD	Max.	2.68	2.68	2.69	2.73	2.77	2.76	0.0042	0.0031	0.0015	0.0011	0.0033	0.0124	0.922	0.914	0.904	0.902	0.893	0.859
	Min.	3.24	3.18	3.09	2.10	3.30	3.49	0	0	0	0	0	0	0.938	0.925	0.908	0.904	0.903	0.901
4TH	Max.	3.70	3.70	3.68	3.64	3.66	3.78	0.0016	0.0024	0.0037	0.0028	0.0007	0.0004	0.944	0.936	0.926	0.916	0.905	0.902
	Min.	4.24	4.32	4.37	4.22	4.04	4.12	0	0	0	0	0	0	0.952	0.949	0.947	0.929	0.907	0.903

Nominal change in the normalized peak intensity, given by ΔI = 1-I, closely approximates the relative amount of energy transferred from the disc to the rings area. For instance, 30% CO (ο=0.3) will lower normalized central diffraction intensity produced by a perfect wavefront from 1 to 0.828; at the same time, relative transfer of energy from the Airy disc to the rings area is ~0.17, or 17%. A simple rule of thumb is that the relative loss of energy up to ο~0.4 is closely approximated - as a ratio number - by 2ο2, and for ο~0.5 and larger by 1.9ο2.

The similarity extends to the contrast drop-off for the range of spatial frequencies bellow ~0.5 (approximately, left side of the MTF graph), which is the range of resolvable low-contrast details. In other words, for this range of spatial frequencies, the peak intensity I resulting from CO is comparable to the Strehl ratio for wavefront aberrations with respect to the effect on contrast and resolution. They both indicate relative amount of energy transferred to the rings area, the main factor determining contrast level at low- to mid-frequencies of the MTF.

Thus, with the RMS wavefront error ω in terms of the Strehl ratio S being given by ω=0.24√-logS, direct relation can be established between the relative linear size ο of CO and similar in effect RMS wavefront error ώ (in units of the wavelength) with respect to low-contrast detail effect as:

ώ~0.24[-log(1-ο2)2]1/2         (61)

(it can be simplified to an empirical approximation, ώ~0.24ο for ο<0.3 and ώ~0.25ο for 0.15<ο<0.3).

For ο=0.325, this gives ώ~0.075, practically equal to 1/4 wave P-V of spherical aberration level. Comparison with the effect of spherical aberration is most appropriate, due to both CO and spherical aberration causing radially symmetric intensity distribution, with the predominant pattern change being brightening and widening of the first bright ring.

However, the actual MTF graph (FIG. 66) indicates that the above formula is somewhat pessimistic in regard to the effect of CO. Obstructed aperture has significantly better contrast transfer - even from that of a perfect aperture - in the right half of MTF frequency range (i.e. for details smaller than 2λF linear, or 2λ/D in radians. Even in the left half of the graph (range of resolvable low contrast details), obstructed aperture has an edge.

The reason is the effect unique to CO (at least in its extent), namely, the reduction in size of the Airy disc caused by it. The linear disc reduction is closely approximated by a factor (1-ο2) for obstructions of ~D/3 and smaller, and by a factor (1-ο2+ο4) for larger obstructions, up to ~0.7D. Good approximation for the 1st minima reduction ratio for any obstruction size is 1-οn, with n=2+[ο2/(1-ο)] or, alternately, n=a+(1/a), with a=1-ο. Apparently, the overall smaller diffraction pattern and brighter central disc give to the obstructed aperture an edge in contrast transfer efficiency with respect to spherical aberration error of near identical nominal energy loss from the Airy disc.

*There is a small difference between polychromatic and monochromatic MTF plot even for non-refractive optics, showing as lower contrast transfer in the highest ~40% frequency range, due to longer wavelengths forming larger Airy disc. However, it doesn't change appreciably plot appearance, due to a perfect aperture being affected in the same manner.

This effect is present at all obstruction sizes. As a result, size of CO causing similar contrast drop to that caused by the amount of low spherical aberration indicated by Eq. 60 is, for the range of resolvable low contrast details (left half of the MTF graph), nearly 10% larger, linearly. Thus better approximation for the RMS error of spherical aberration similar in effect to that of the central obstruction ο for this frequency range is given by:

ώ~0.22[-log(1-ο2)2]1/2         (61.1)

(it can also be simplified to an empirical approximation, ώ~0.21ο; it is within a couple of percentage points from the true value for ο~0.4 and smaller).

For the entire range of frequencies, the MTF degradation factor - truly comparable to a Strehl number assigned to wavefront aberrations - is given by:

So = 1-ο2           (61.2)

According to it, what we can call central obstruction Strehl equivalent for 0.32D CO is 0.898, significantly better than that for 1/4 wave P-V of lower order spherical aberration (0.80). This is routinely neglected when assessing CO effect, but it is the only proper way of comparison: for the entire range of resolvable frequencies.

Hence, substituting (1-ο2) for (1-ο2)2 in Eq. 61 gives a true measure of the RMS wavefront error of lower-order spherical aberration comparable to given relative obstruction ο as:

ώt~0.24[-log(1-ο2)]1/2         (61.3)

Note that the MTF output varies somewhat from one program to another. For instance, Aberrator and ATMOS show more of a contrast drop resulting from CO. It still has an edge over spherical aberration, but the numerical constant in the above approximation is between 0.22 and 0.23.

Analogous to the contrast transfer of a perfect aperture (FIG. 45, top right), that of an obstructed aperture is given by the overlapping area of two unit-diameter circles, but this time for the two annuli, normalized to the annulus area.

Since the negative effect of CO is so similar to that of wavefront aberrations, the question of what is its maximum acceptable size can be answered in terms of the conventional aberration limit of 0.80 Strehl. The following answers this question approximately for the range of resolvable low-contrast MTF frequencies (approximately the left half of MTF graph), usually one that is of greatest interest.

Setting I=0.80 puts the maximum acceptable CO size at ~0.32D according to Eq. 61, and at ~0.35D according to Eq. 61.1. However, it assumes perfect optics. For an actual optical set of the Strehl ratio S smaller than 1, the minimum acceptable obstruction size for the combined ~0.80 Strehl level for mid MTF frequencies would be obtained from SI=0.80, with the peak intensity factor for the obstruction, as mentioned, I=(1-ο2)2. In this concept, linear obstruction has to be  smaller than 0.35D, so the ο4 factor can be neglected, and the maximum acceptable CO size for the combined ~0.80 Strehl level is:

οmax~[0.5-(0.4/S)]1/2  or, adjusted for better contrast transfer,   
οmax~[0.6-(0.48/S)]1/2                                          (62)

with  οmax being, as before, the relative obstruction diameter in units of the aperture. More exact calculation would take into account that the RMS wavefront error - hence the resulting Strehl ratio - would likely change due to the presence of obstruction. The RMS wavefront error change can be for the better, or for the worse, depending on the contribution of the obscured central part of the wavefront to its average deviation. Specifically, the RMS wavefront error in an obstructed aperture diminishes for spherical aberration and defocus, and increases for coma and astigmatism, as given by the following factors for best focus location:

- spherical aberration: (1-ο2)2  
- defocus: (1-ο2)
     - coma: (1-ο2)(1+4ο2+ο4)1/2/(1+ο2)1/2
     - astigmatism: (1+ο2+ο4)1/2

Note that these RMS values are with respect to new reference sphere, best fitted to the portion of wavefront within annulus area (the P-V error also changes, but that is not directly related to image quality). The combined peak diffraction intensity in the presence of aberrations is given by a product of the peak diffraction intensity of aberration-free obstructed aperture, and that corresponding to the RMS wavefront error ω over annulus area, in units of the wavelength, or:

I = (1-o2)2e-(2πω)2       (63)

e being, as before, the natural logarithm base e~2.72. Thus, for instance, a system with 0.37D c. obstruction and 0.074 waves RMS of spherical aberration (0.25 wave P-V) over full surface area of its optics, has the RMS error within annulus reduced to 0.055 waves RMS, resulting in the combined peak intensity of 0.66. That is noticeably better than 0.60 peak intensity that would result from using the unadjusted wavefront RMS.

In effect, in the presence of spherical aberration, CO partly compensates for its damaging effect by reducing the wavefront error. When both CO and inherent wavefront error are large enough, obstructed system can even perform better. For instance, a system with 1/2 wave P-V of lower-order spherical aberration performs better with 50% obstruction than without it (peak diffraction intensity 0.426 vs. 0.395, respectively).

Evidently, these factors may have importance with larger obstruction sizes and wavefront error levels at which the relative change induced by obstruction has appreciable effect (i.e. not too small, and not to large aberration). The effect on defocus error can be significant, the consequence of axial elongation of the central maxima in the presence of obstruction. It makes an aberration-free obstructed telescope less sensitive to defocus by a (1-ο2) factor; hence, from Eq. 25, defocus error in an obstructed telescope, given as P-V wavefront error at best focus, becomes Wd = -(1-ο2)Ld/8F2, with Ld being the longitudinal defocus and F the telescope focal ratio.

The question of the CO size at which its effect becomes insignificant can be answered in a similar manner. For perfect optics, with S=1, it is determined by any chosen Strehl figure S* considered to be the level of negligible image deterioration. Since here SI=S*=I=(1-ο2)2,

οmax~(0.5-0.5S*)1/2     or     οmax~(0.6-0.6S*)1/2        (64)

the latter adjusted for the better contrast transfer efficiency.

For imperfect optics, with the Strehl S<1, but presumably better than S*, it would be determined from

οmax~[0.6-(0.6S*/S)]1/2            (64.1)

also adjusted for better contrast transfer efficiency.

And for an aberrated optics set with S<S*, valid criterion would be how much of an additional contrast loss of extended details τE, expressed as a ratio number, is found to be either negligible or acceptable. According to it,

 
οmax~(0.6TE)1/2            (64.2)

Taking 5% additional average contrast loss (τ=0.05) on low-contrast details as a reasonable level of hard to notice contrast change, we arrive at the size of obstruction likely to produce negligible effect for most people as ο~0.17 of the aperture diameter. Of course, this applies as well to aberration-free apertures.

As  mentioned, the above consideration is for the left side of the MTF graph, i.e. resolvable low-contrast details. For the entire range of MTF frequencies, the tolerable size of CO is significantly larger, as obtained by replacing (1-ο2)2 factor by (1-ο2). In terms of the additional general contrast loss τG, over the entire range of MTF frequencies, the corresponding relative obstruction size is given by:

οmax~ TG1/2            (64.3)

Thus, while the CO size producing ~20% contrast loss for extended details (τE) is 0.35D, it is as much as 0.45D for the identical drop in general contrast level TG. However, practical importance of the right half of MTF graph for general observing is considerably less than 50%; it mainly limits to splitting near-equal in brightness double stars, and resolving high-contrast line-like features near or beyond diffraction limit (Cassini division, Moon rills). Consequently, extended-detail contrast transfer τE is more relevant indicator of the overall performance level of an obstructed aperture.

More detailed insight into the change of intensity distribution and contrast loss caused by CO is given by the PSF and MTF, respectively (FIG. 67).