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▪ CONTENTS
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7.1.1. Inconsistencies in the
theoretical concept?
▐
7.2. Spider obstruction
► 7.1.2. TELESCOPE CENTRAL OBSTRUCTION: SIZE CRITERIASince 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 lowcontrast 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 higher than 0.80, the minimum acceptable obstruction size for the combined ~0.80 Strehl level for lowtomid MTF frequencies would be obtained from SI=0.80, with the peak intensity factor for the obstruction, as mentioned, I=(1ο2)2=12ο2+ο4. 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 due to the smaller pattern, οmax~[0.6(0.48/S)]1/2 (62) with ο being, as before, the relative obstruction diameter in units of the aperture.
For the entire range of MTF
frequencies, οmax~[1(0.80/S)]1/2. Following table gives
the corresponding c.obstruction sizes for selected optics Strehl
values (in units of the aperture diameter).
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. The problem is that often it is not known which specific aberrations are inherent to the optics, or it is a mixture of multiple aberrations, including various forms of random surface deformations. In general, central obstruction will reduce aberrations causing significant wavefront deformation over the inner pupil portion. And vice versa, it will worsen those causing only insignificant wavefront deviations over the inner pupil area. Inset below addresses effect of central obstruction on wavefront aberrations in more detail. RMS wavefront error at the best focus location is affected by central obstruction as given by these relations:
 primary spherical aberration (best
focus):
ωo = ω(1ο2)2 where ωo and ω are the RMS wavefront error in obstructed and clear aperture, respectively. The corresponding graph, at left (for unit aberration in clear aperture), shows that central obstruction consistently (i.e. for any obstruction size) reduces primary spherical aberration and defocus. It slightly worsens primary coma up to about 0.4D CO, but it is quickly reversed to the reduction in aberration for larger obstructions. In general, the effect increases progressively with the obstruction size.
The only primary aberration worsened by any obstruction size is
astigmatism. However, the effect may become significant only at
obstruction sizes larger than 0.5D. The benefit of reducing defocus and
spherical aberration is more than offsetting slight worsening in coma
and astigmatism. This is particularly the case with spherical
aberration, which can be significantly reduced already at the obstruction
sizes of ~D/3. Plots at left show how the shape of wavefront deformation
at the best focus spherical aberration change with the size of central
obstruction. With the defocus to location of minimum wavefront deviation
in the presence of central obstruction larger by a factor of (1+o2)
than in circular aperture, wavefront profile at this location is given by ρ4(1+o2)ρ2,
where ρ is the height in pupil normalized to 1 for pupil radius.
Setting first derivative of it, 4ρ32(1+o2)ρ,
to zero, and solving for ρ, gives the zonal height of the
deflection zone ρd
where the PV error reaches its maximum. While it gives larger PV
wavefront error for obstructed apertures when measured from its
imaginary center coinciding with the pupil center, the error  given as
a differential between the function values for ρ=o and ρ=ρd
 diminishes with the increase in obstruction for the actual wavefront
in the annulus.
Although both, the relation and graph indicate that astigmatism becomes progressively larger with increase in central obstruction, becoming larger by a factor 31/2 even for obstruction covering the entire pupil, the latter, of course, is not so. Rather, due to its deviation increasing toward the edge in opposite directions along two perpendicular axes, the RMS error for the annulus area progressively increases, approaching 31/2 factor as the obstruction ratio approaches 1. Note that these RMS values are with respect to a new reference sphere, best fitted to the portion of wavefront within annulus area. This reference sphere is of slightly shorter radius with undercorrection, opposite with overcorrection (the effective PV error also changes, but it is the RMS error change that affects image quality). The combined peak diffraction intensity in the presence of aberrations is given by a product of the peak diffraction intensity of aberrationfree obstructed aperture, and that corresponding to the RMS wavefront error ω over annulus area, in units of the wavelength, or: (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 PV) 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 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 the inherent wavefront error are large enough, obstructed system can even perform better. For instance, a system with 1/2 wave PV of lowerorder spherical aberration performs slightly 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).
In addition to spherical aberration, the effect on defocus
error also can be significant, a consequence
of the axial elongation of the central maxima in the presence of obstruction. It makes an
aberrationfree obstructed telescope
less sensitive to defocus by a
(1ο2)
factor; hence, from
Eq. 25,
defocus error in an obstructed telescope, given as PV wavefront
error at the best focus, becomes
With aberration significantly exceeding 1/2 wave PV at the best focus, there is a shift of diffraction focus (i.e. PSF maxima) away from the point of minimum wavefront deviation. It is a consequence of the RMS wavefront error and Strehl ratio being not directly related with large, as they are for small aberrations. Since the above considerations assume focus location with the minimum wavefront error (mid focus), they are valid only for the aberration levels not exceeding 0.15 wave RMS. Plots at left represent change in the longitudinal intensity peak intensity for a system with 50% linear obstruction (o=0.5) and increasing level of primary spherical aberration. The PSF is normalized to unity for zero spherical aberration.
The question of the CO size at which its effect becomes insignificant can be answered in a similar manner as for its maximum tolerable size. For perfect optics, with S=1, it is determined by any chosen Strehl figure SN considered to be the level of negligible image deterioration. Since here SI=SN=I=(1ο2)2, οmax~(0.50.5SN)1/2 or οmax~(0.60.6SN)1/2 (64) the latter adjusted for the better contrast transfer efficiency. So, if the desired effective Strehl for resolvable lowcontrast details is SN=0.9, the corresponding maximum c. obstruction size (adjusted for better contrast transfer) with aberrationfree aperture is οmax=0.24. For imperfect optics, with the Strehl S<1, but presumably better than S*, it would be determined from οmax~[0.6(0.6SN/S)]1/2 (64.1) also adjusted for better contrast transfer efficiency. If, for instance, the optics Strehl is S=0.95, and the desired Strehl level for resolvable lowcontrast details is SN=0.9, the corresponding c. obstruction size is οmax=0.18 (also adjusted for better contrast transfer due to its relatively brighter central maxima vs. that in aberrated aperture). And for an aberrated optics set with S<SN, 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,
Taking 5% additional average contrast loss (τE=0.05) on lowcontrast 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 aberrationfree apertures. As mentioned, the above consideration is for the left side of the MTF graph, i.e. resolvable lowcontrast details. For the entire range of MTF frequencies, the tolerable size of CO is significantly larger, as obtained by replacing (1ο2)2 factor by (1ο2). 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 nearequal in brightness double stars, and resolving highcontrast linelike features near or beyond diffraction limit (Cassini division, Moon rills). Consequently, extendeddetail 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. 106).
FIGURE 106:
LEFT/MIDDLE: Change in
the central
and 1st bright ring intensity of the PSF, Airy disc size, and MTF image contrast, as central obstruction increases from 0.16D to 0.32D (slightly better
than 0.80 Strehl performance level in the low to midfrequency MTF
range) and to 0.4D,
(comparable to 1/3.4 wave PV of spherical aberration in that same
range).
Contrast recovery in the last ~40% of the MTF frequency
range is mainly result of the reduction in size of the Airy disc caused by
central obstruction. RIGHT:
Contrast transfer with 32% obstruction (ο=0.32)
nearly coincides with that of (1ο)D
aperture (dashed red line) approximately in the 0<ν<1/3
frequency range (blue area). This accounts for most extended details, from
about 1.4 times the Airy disc diameter up. However, due to contrast
recovery at higher frequencies, the
resolution limit for bright
lowcontrast details (BLC) is less
than 10% lower than in a perfect aperture. There is no simple
relation expressing
these two MTF parameters for the range of
ο
values, but MTF data can be used to interpolate the corresponding
graphs. Note that seeing error, whose averaged magnitude is in proportion to (D1/D2)5/6 will worsen actual field performance of the larger (obstructed) relative to that of the smaller aperture. Obviously, in the actual field conditions, that will lower somewhat the overall contrast level in the larger (obstructed) aperture, widening its limiting resolution gap vs. perfect aperture. However, since smaller aperture also suffers from seeing error, although smaller in magnitude, there is no significant change in their relative contrast transfers (FIG. 107, left). The effect of c. obstruction on contrast and resolution also vary somewhat with the aperture size (FIG. 107, right).
Note that nearly identical
contrast/resolution level in the larger vs. smaller aperture for the
averaged seeing error does not imply that the two will offer similar
level of performance. Seeing error constantly varies around its average
value and, in general, error reduction by any given ratio  it is
commonly up to 50%, sometimes more  benefits larger aperture more. On
the other hand, larger aperture has generally more significant other
error sources (thermals, collimation, optical quality), so the actual
score is determined as a breakdown between the magnitude of the
residual advantage of the larger aperture, when it is optically perfect,
and the level of its optical errors not related to seeing. ◄ 7.1.1. Inconsistencies in the theoretical concept? ▐ 7.2. Spider obstruction ►
