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7.1.1. Inconsistencies in the theoretical concept?      7.2. Spider obstruction
 

7.1.2. TELESCOPE CENTRAL OBSTRUCTION: SIZE CRITERIA

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 higher than 0.80, the minimum acceptable obstruction size for the combined ~0.80 Strehl level for low-to-mid MTF frequencies would be obtained from SI=0.80, with the peak intensity factor for the obstruction, as mentioned, I=(1-ο2)2=1-2ο2+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).
 

OPTICS STREHL

1

0.95

0.90

0.85

0.80

MAX. C.OBSTRUCTION (ο) FOR
PRESERVING 0.80 STREHL

mid-to-low MTF frequencies

unadjusted

0.32

0.28

0.24

0.17

0

adjusted

0.35

0.31

0.26

0.19

0

entire range of MTF frequencies

0.45

0.40

0.33

0.24

0

    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(1-ο2)2
- defocus: ωo = ω(1-ο2)
    
- primary coma (best focus): ωo = ω(1-ο2)[(1+4ο2+ο4)/(1+ο2)]1/2
    
- primary astigmatism (best focus): ωo = ω(1+ο2+ο4)1/2

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+o22, where ρ is the height in pupil normalized to 1 for pupil radius. Setting first derivative of it, 4ρ3-2(1+o2)ρ, to zero, and solving for ρ, gives the zonal height of the deflection zone ρd where the P-V error reaches its maximum. While it gives larger P-V 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.
 

C.O.
(o)
RATIO TO THE RMS WAVEFRONT ERROR OF THE ABERRATION INHERENT TO OPTICS
SPHERICAL ABERRATION DEFOCUS COMA ASTIGMATISM
0.0 1 1 1 1
0.1 0.98 0.99 1.005 1.005
0.2 0.96 0.98 1.015 1.0205
0.3 0.83 0.91 1.019 1.048
0.4 0.71 0.84 1.007 1.089
0.5 0.56 0.75 0.963 1.146

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 under-correction, opposite with over-correction (the effective P-V 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 aberration-free 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 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 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 P-V of lower-order 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 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 the best focus, becomes
Wd = -(1-ο2)Ld/8F2,
with
Ld being the longitudinal defocus and F the telescope focal ratio.

With aberration significantly exceeding 1/2 wave P-V 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 peak aberration coefficient S equals the P-V  wavefront error at the paraxial focus. For S~3 and smaller, peak diffraction intensity is at the location of mid focus, where the RMS/P-V error is the lowest, one fourth of the error at either paraxial or marginal focus. For S~4 (1 wave P-V at the mid focus), and larger, the PSF peak shifts to two locations equally separated from marginal - located at (1+o
2)S waves of defocus from the paraxial focus) and paraxial focus (location of zero defocus). Compared to the axial intensity distribution without obstruction (FIG. 36B), obstructed aperture - and particularly those with large obstruction - have generally better wavefront quality for a given level of spherical aberration, hence higher PSF peak (normalized to the PSF peak of obstructed aperture), and somewhat delayed shift of the PSF peak away from the mid focus.

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=(1-ο2)2,

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

the latter adjusted for the better contrast transfer efficiency. So, if the desired effective Strehl for resolvable low-contrast details is SN=0.9, the corresponding maximum c. obstruction size (adjusted for better contrast transfer) with aberration-free 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 low-contrast 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,

 
οmax~(0.6TE)1/2            (64.2)

Taking 5% additional average contrast loss (τE=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. 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 mid-frequency MTF range) and to 0.4D, (comparable to 1/3.4 wave P-V 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 (1-ο)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 the contrast recovery at higher frequencies, the resolution limit for bright low-contrast 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.
 BOTTOM: In general, BLC resolution threshold is affected significantly less by central obstruction than the BLC contrast level. While the contrast level for most resolvable detail sizes drops nearly linearly with the increase in linear relative central obstruction, slightly less than what the D-d rule implies (dashed line depicts perhaps more likely shape of this graph, with contrast dropping at somewhat slower rate up to about 0.15D obstruction, and after 0.35D), the resolution threshold  - reflecting contrast improvement close to the limit of resolution due to the smaller diffraction pattern - worsens much more slowly up to approximately 0.25D, then somewhat faster, but still much slower than the overall contrast up to about 0.4D - remaining still high above the relative contrast loss - has a dramatic drop from 0.4D to 0.5D, and then returns to a slow deterioration rate for larger obstructions.
Simulation of contrast transfer to images of planetary surface below (Aberrator, Cor Berrevoets) seem to support indications that the empirical D-d "rule" is too pessimistic. While such simulations have serious limitations, the obstructed aperture retained small but consistent advantage in image definition. The apparent size of Jupiter is about 3
when looked at from 15-inch distance, corresponding to the telescope view with magnification of about 350 (based on the average angular diameter of 40 arc seconds).
 

Note that seeing error, whose averaged magnitude is in proportion to (D1/D2)5/6 will worsen the actual field performance of a 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).


FIGURE 107: LEFT: Contrast transfer in 0.33D obstructed aperture vs. 0.67D unobstructed aperture. When the smaller aperture is placed in moderate-to-good seeing, simulated with 0.26 wave P-V (0.075 wave RMS, the diffraction-limited level) of defocus, with the larger aperture having correspondingly larger seeing error (0.36 wave P-V), larger aperture retains its bright low-contrast (BLC) detail resolution advantage (3 vs. 5), but to a significantly smaller extent than without seeing error (2 vs. 4). The obstructed aperture also loses its edge on contrast transfer toward resolving limit. This implies that the popular concept postulating that the effective contrast/resolution of an obstructed aperture compares to that of unobstructed smaller by the size of linear c, obstruction, expresses - at least at this obstruction level - combined effect of aperture size, obstruction and seeing. Resolution advantage in the larger aperture for the averaged seeing error, becomes more pronounced during the moments of better seeing, but diminishes when seeing worsens.  MIDDLE/RIGHT: The effect of c. obstruction in different aperture sizes. At left, smaller aperture with and without 0.26 wave P-V of defocus to simulate the averaged seeing error, and further deterioration of contrast transfer with the addition of 0.25D c.obstruction. To the right, 2.3 times larger (by diameter) aperture with the same relative obstruction size and doubled seeing error. The effect of c. obstruction on contrast transfer is greater in the smaller aperture (A vs. B), which is to expect considering that the energy transfer induced by obstruction is relatively larger vs. its seeing error level. Interpolating perfect MTF curves to coincide with the lower-to-md frequency contrast transfer sections of unobstructed and obstructed MTF plots indicates that the addition of 0.25D c.obstruction lowers contrast level for extended details over most of the range of resolution from about 0.75D to 0.61D (19% linear aperture reduction) and from 0.43D to 0.37D level (14% linear aperture reduction) in the smaller and larger aperture, respectively. Higher frequency contrast transfer in both apertures is nearly entirely determined by seeing, with little effect due to added obstruction. Loss in the limiting BLC resolution in the averaged seeing is, however, quite small, and slightly higher in the larger aperture (2 vs. 3, approx. 10% vs. 5%), which has already lost significantly more of its theoretical potential due to its larger seeing error. Seeing combined with the obstruction in the larger aperture decrease its extended object contrast level and BLC resolution to that of approximately three times smaller perfect clear aperture. Smaller aperture is only degraded to the level of about twice smaller perfect unobstructed aperture. Hence, they now compare as 2.3D/3 and D/2, or 0.75D vs. 0.5D, with the larger aperture keeping its relative contrast/resolution advantage. This, however, will diminish with the increase in seeing error.

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, within short periods of time - 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 break-down 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

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