6.2. GENERAL EFFECT OF ABERRATIONS ON IMAGE QUALITY
What is common to all wavefront aberrations is that they result in less efficient energy concentration into a point-image. Consequently, image contrast and resolution suffer. How much of image deterioration is acceptable? Optical theory has developed methods of measuring the size of various aberrations, as well as their effect on contrast and resolution. These methods are based on complex diffraction calculations, but the final results can be expressed quite simply, giving the amateurs tools needed to understand and measure the effects of wavefront aberrations.
Image contrast is defined by the relative intensities of its components. According to this simple formula, contrast of two adjacent surfaces is given by
where 1 is the normalized intensity of the brighter surface, and i the relative to it intensity of the dimmer surface. Thus, nominal contrast is independent of the brightness level (unlike detail resolution and detection, for which the absolute brightness, together with a contrast level, are determining factors). By spreading the energy out wider, wavefront aberrations lower contrast, and with it resolution and detection level.
For instance, if unaberrated image consists of two areas with relative intensities of 1 and 0.5, its contrast is 0.33. Unless the image is angularly much larger than the resolution limit of a telescope, this contrast level is appreciably lower than that inherent to the object, due to the energy spread caused by diffraction. If now, as a result of wavefront aberrations, additional 10% of the energy from either surface spreads onto the other, it changes the relative intensities to 0.95 and 0.55 or, after being normalized to the higher intensity, 1 and 0.58. This in turn lowers the contrast by 20%, to 0.266. Details of low inherent contrast, if small enough, will be lost, and the resolution capability of a telescope will be reduced in this respect. As the amount of energy spread out by wavefront aberration increases, it will lower the contrast more, beginning to affect resolution/detection of larger, more contrasty details.
The simplest indicator of the effect of wavefront aberrations and non-wavefront related diffraction factors, such as pupil obstruction of any form, is a drop in the peak intensity of the diffraction pattern (PSF) produced. It is expressed as a single number, the ratio of actual peak diffraction intensity (PDI) vs. that of a perfect aperture. Thus, this relative number has values between 0 and 1. When the PDI drop is result of wavefront deviations from perfect sphere, it is called the Strehl ratio. For factors unrelated to the wavefront it is simply a central intensity ratio.
The primary effect of wavefront aberrations and pupil obstructions - with rare exceptions - is change in the intensity distribution within diffraction pattern forming point-object image: specifically, brightening of the rings area, at the expense of the central disc's brightness. Change in the intensity distribution within the pattern is more significant with respect to its consequences to image quality than usually insignificant change in the size of the disc vs. ring area, if present. For aperture obstructions in general, the appropriate peak diffraction intensity (normalized to 1 for unobstructed perfect aperture) in incoherent light equals the relative unobstructed (the encircled energy for circular obstruction, however, due to the reduction in central maxima, is proportional to the area squared). As mentioned, it also determines the relative energy loss from the Airy disc and the associated average contrast loss.
For instance, central obstruction of D/3 that covers (1/3)2=0.11 of the pupil area, will reduce peak diffraction intensity from 1 (normalized) to (1-0.11)=0.89, but the energy encircled within the central maxima is (1-0.11)2=0.79. Likewise, spider vanes obstructing 2% of the area of this annulus will further reduce the peak intensity to 0.79(0.98)=0.774. The corresponding average contrast loss over the range of MTF frequencies is ~0.226, or 22.6%.
Similarly, peak diffraction intensity in the presence of wavefront aberrations is closely approximated by an empirical expression for the Strehl ratio. Despite both, obstructions and aberrations having similar effect in that they transfer energy out of the Airy disc, they can't be directly compared in the most important case, that of circular central obstruction, because obstruction significantly changes (reduces) the central maxima.
Somewhat similarly, effect of aberrations cannot be directly compared with the reduction in the aperture size, even if they both affect diffraction pattern. The two have similar effect of reducing image contrast and resolution, but the mechanism is different. While the energy transfer in aberrated aperture mainly enhances the ring area, with usually no significant change in the size of central maxima (for errors not larger than ~0.15 wave RMS), the effect of aperture reduction is the enlargement of the entire diffraction pattern. The relative intensity distribution within the pattern doesn't change with respect to a perfect aperture, but the enlarged angular pattern size of a smaller aperture effectively does change intensity distribution relative to the smaller pattern of larger aperture. After factoring out reduction in light gathering power - which, unlike the intensity distribution, doesn't affect nominal contrast - and resolution, both directly determined by the ratio of reduction (i.e. ratio squared and ratio, respectively), nominal contrast levels in smaller vs. larger aperture also can be directly compared only within the resolving range of the smaller aperture.
The loss in limiting stellar resolution resulting from aperture reduction can be directly quantified, but the important question is how is the contrast transfer of extended object affected in smaller vs. larger aperture. Relating integrated contrast transfers of the two in their entirety would give general information on contrast transfer and resolution of bright details with high inherent contrast. For more specific information on the relative contrast transfers of other detail types it is necessary to find out contrast transfer characteristics within resolving range of the smaller aperture vs. larger one. According to FIG. 100, such two sub-ranges within the range of resolved details in a smaller aperture are, approximately, normalized spatial frequencies 0<ν<0.11 and 0<ν<0.53, indicating the contrast level for dim and bright low contrast objects, respectively.
The overall contrast transfer being proportional to the volume of the solid figure formed by 360º rotation of the MTF curve (for aberration-free pupil, or for rotationally symmetrical aberrations) - the 3-D MTF. But it is merely a sum of the transfers for all orientations of the PSF vs. MTF bars. For aberration-free pupil, or for rotationally symmetrical aberrations the overall contrast is proportional to the area under 1-D MTF curve, and for asymmetrical aberrations, where the transfer varies with the PSF orientation, to the area under azimuthally averaged MTF curve. Thus, the comparison of contrast transfer in a smaller vs. larger aperture can be reduced to the comparison of the areas under their respective curves. Graph below illustrates contrast level difference in larger vs. smaller aperture.
At left is the standard MTF graph for four aperture sizes, with added plot for 1/4 wave p-v wavefront error of spherical aberration for the unit aperture. It shows that contrast transfer in unit aperture drops to the level of twice smaller aperture for about first 1/8 of the frequencies (details larger than three Airy disc diameters), increasing to the 25% smaller aperture level at the 0.4 frequency (Airy disc size), and to the 10% smaller aperture level at the 0.53 frequency, remaining close to the unit aperture level for the rest of frequencies. Graph on the right shows contrast transfer of the smaller apertures for unit contrast transfer of the unit aperture. Within resolving range of the respective aperture, contrast drops to 80% at 0.63, 0.41 and 0.27 frequency for the 10%, 25% and 50% smaller aperture, respectively. On the other hand, contrast drop toward the cutoff frequency is the fastest in the aperture with the smallest reduction factor of 10%.
If we simply mask the outer 10% of the aperture radius, the resulting peak intensity would drop to 0.81, but so it would across the entire pattern; in other words, relative intensity distribution wouldn't change, only the pattern gets enlarged by 11% linearly. With respect to the Airy disc of unmasked aperture, encircled energy is still as high as 96.8%, which represents the amount of energy contained within 89% radius of the Airy disc. That would be commensurate to 0.968 Strehl, but the average contrast drop vs. unmasked aperture is obviously larger than that. The reason is that the higher 60% of the MTF range depends primarily on the size of the central maxima - as does the cutoff frequency - and only the first 40%, or nearly so, on the amount of energy outside of it. With increase of the outer radius portion masked, encircled energy drops faster, to 0.908, 0.868 and 0.549 for 20%, 25% and 50% masked, respectively, but the discrepancy between encircled energy and contrast transfer remains, since it depends on both, encircled energy and pattern size. It makes direct comparison between the effect of aberrations and aperture reduction impossible, at least in the context of these performance indicators.
This consideration is unavoidably partial, relating only to the contrast level. In addition to its effect on contrast, linear aperture reduction is causing both, commensurate loss in stellar resolution and exponential loss in the light gathering power, which is not the case with either wavefront aberrations of this magnitude or common-size central obstructions.
Also, the above consideration, assuming identical level of optical quality - zero aberration - in both smaller and larger aperture, does not have much in common with the real world. While larger apertures tend to have greater error contributions from multiple sources, including optics quality, seeing alone ensures that the larger aperture at every moment suffer more image degradation - the larger it is, the more so. Factoring in only the two most significant error sources, seeing and internal thermal currents will, in general, take away most of the contrast/resolution advantage of the larger aperture. However, specifics will vary significantly with the magnitude of these errors, as well as other error sources - including optics quality - which can appreciably affect performance level as well. Still, not only the most significant factor, but also the one which in amateur telescopes cannot be influenced or minimized on any given site is the seeing error. MTF plots below illustrate the effect of seeing on contrast transfer in three different aperture size (FIG. 92).
As already mentioned, other error sources can significantly add to the seeing error. Internal thermal currents are generally more of a problem with larger apertures, but so is the seeing error. Approximating, very loosely, thermally induced RMS wavefront error by ωt~D(mm)/10,000 for every 1°C of thermal differential, gives 0.06, 0.03 and 0.015 wave RMS for 600, 300 and 150mm aperture, respectively. It indicates that even a couple degrees of thermal differential between the telescope components and air could significantly add to the seeing error (which is, for comparison, in 1 arcsec seeing roughly five times smaller than the seeing error for all three apertures).
Another factor influencing image quality in large vs. small telescopes are eye aberrations. Since eye aberrations (as the RMS wavefront error) change approximately with the square of pupil diameter, halving the aperture lowers eye aberrations at any given nominal magnification by a factor of four (assuming eye pupil not smaller than the exit pupil of a telescope).
An interesting implication by the MTF graphs is that large apertures in poor seeing can have moments of a partial black-out, or significantly lower contrast level than smaller aperture in a relatively narrow lower-frequency range, but still with better contrast toward higher frequencies.
All this suggests that, in the average field conditions, the gain in light gathering power and resolution with the increase in aperture size is more significant than small-to-non-existent (depending on the magnitude of other errors) gain in contrast/resolution. In better than average seeing large apertures will have more substantial advantage. On the other hand, considering that large telescopes tend to have greater error contributions from other sources as well (gravitational deformations, central obstruction almost invariably part of larger apertures, overall design correction level, misalignment, and so forth), their image contrast and even resolution can be actually inferior compared to smaller apertures, the worse the seeing conditions, the more so.
While MTF gives a general
indication of the effect of aberrations on image quality, for a more
complete picture of their effect it may be necessary to consider them
from different angles, related to the specific object properties, such
as object type, shape, size on the retina or brightness.