◄ 5.2. Misalignment and forced surface deviations   ▐    6.2. Aberrations and image properties ►
 

Effects of wavefront aberrations on image quality is fairly complex subject. Somewhat informally, it will be tackled along the following lines:

(1) general effects on image contrast due to a worsening of diffraction effect, with comparison to the effects of central obstruction and and aperture reduction,
(2) effect of wavefront aberrations as a function of object/image properties: magnification, telescopic brightness and inherent contrast level,
(3) effect of aberrations on diffraction pattern,
(4) Strehl ratio as an indicator of the effect of aberrations, and
(5) MTF as an indicator of the effect of wavefront aberrations.
 

6.1. Telescope aberrations: general effects

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 relative intensities of its components. According to a simple formula, contrast of two adjacent surfaces is given by c=(1-i)/(1+i), where 1 presents the normalized intensity of the brighter surface, and i the relative 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 dispersing the energy, wavefront aberrations lower contrast, and with it resolution and detection level.

For instance, if unaberrated image consists of two surfaces with relative intensities of 1 and 0.5, its contrast is 0.33. Unless the image is very large angularly, 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 into 1.05 and 0.6 or, after being normalized to the higher intensity, 1 and  0.57. This in turn lowers the contrast by 18%, to 0.27. Since the contrast level is still relatively high, it will only cause loss of sharpness, not in limiting diffraction (i.e. point source) resolution. However, 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. At some point, the point-source resolution will begin to suffer as well.

This is how wavefront aberrations affect image quality, in principle. Specific values require much more complex calculations. Luckily, as already mentioned, determining the effect of wavefront aberrations - as well as other diffraction-related effects - can be greatly simplified, still preserving sufficient accuracy for amateurs' needs.

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 peak intensity of the diffraction pattern (PSF) produced by an optical system. 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, it is called the Strehl ratio. For the factors unrelated to wavefront it is simply a central intensity ratio. One important property of this number is that it indicates quite closely the amount of energy lost from the Airy disc. For instance, 0.92 Strehl, or PDI, indicates that about 8% of the energy has been transferred out of the Airy disc. This, in turn, directly determines level of contrast achievable with the aperture.

An optical quality indicator normally presented in the form of a graph, called Modulation Transfer Function (MTF), or Contrast Transfer Function, shows how contrast in an optical system changes with the detail size. What is less known is that the Strehl ratio - and somewhat conditionally[1] the peak diffraction intensity ratio in general - also indicates the average contrast loss over the entire MTF range. Hence, 0.92 Strehl, or PDI ratio, also indicates that the average contrast loss for details of all sizes down to the limit of resolution is also 8%. This implies that the average contrast loss and loss of the energy from the Airy disc are directly related.

What ultimately simplifies understanding and practical use of these basic indicators of optical quality for the amateurs, is that they can be closely approximated in terms of the aperture size. The relation is not direct, since diffraction effects caused by wavefront aberrations, as well as obstructions, are different than those resulting from variations in aperture size, in that the former mainly affect energy distribution, while the latter changes the size of diffraction pattern as a whole, and light gathering power. Nevertheless, it is possible to establish an approximate relation between them, based on the peak diffraction intensity.

The primary effect of wavefront aberrations and pupil obstructions - with rare exceptions - is brightening of the rings area, at the expense of the central disc's brightness. The change in relative brightness is more significant with respect to its consequences to image quality than change in size of the disc vs. ring area. For aperture obstructions in general, the appropriate peak diffraction intensity (normalized to 1 for unobstructed perfect aperture), and with it the energy lost from the Airy disc and the associated average contrast loss, are all given by the relative unobstructed pupil area squared.

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)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)2=0.76.

Similarly, peak diffraction intensity in the presence of wavefront aberrations is closely approximated by the expression for the Strehl ratio. The fact that both, obstructions and aberrations have similar effect in that they transfer energy out of the Airy disc, allows for their comparison based on the peak diffraction intensity value.

On the other hand, reduction in the aperture size has quite similar effect of reducing image contrast and resolution, but the mechanism is different. Here, the effect is the enlargement of the entire diffraction pattern. Hence, it affects the pattern size, rather than intensity distribution within it. As a result, its effect on contrast depend on the aperture size, but in this case on aperture diameter, not the area.

While the exact values for the contrast degradation caused by reduction could be obtained from the relatively simple expression for contrast transfer, it would still require more than quite simple computation. A simple, yet sufficiently accurate approximation for amateurs' needs can be obtained from the MTF graph geometry, as shown on the sketch to the left (FIG. 38).

It puts the average contrast level of a smaller aperture a, expressed in units of the larger aperture, as follows:

A - for the range of details resolvable with smaller aperture, the mid-range contrast relative to that in the larger aperture is approximated by: Cs~0.875/(2-1.125a)

B - for the range of extended objects (approximately the left side of the MTF graph) of the smaller aperture, the mid-range contrast in the smaller relative to larger aperture is approximated by Ce~2.75/(4-1.25a), and

C - for the range of details resolvable in larger aperture (thus including zero contrast transfer frequencies in the smaller aperture), the mid-range contrast in the smaller aperture is lower approximately by a factor Ct~a/(2-a)

This gives that, for instance, 30% smaller aperture (a=0.7), will have the mid-range contrast lower by about 28% within its range of resolution, by 12% in the range of extended details, and by about 46% in the extended range of the larger aperture. Average contrast ratio for planetary details in smaller aperture could be - tentatively - approximated by Cs (red line approximates minimum contrast level needed for planetary details, with PTS marking approximate threshold of resolution for planetary details; PTL is the threshold for larger aperture).

Since the average contrast figure is tightly related to the peak diffraction intensity and Strehl ratio, it is possible now to compare either obstruction effect, or that of wavefront aberrations, with the effect of reduction in the aperture size. The comparison is necessarily rough, mainly due to aperture reduction causing both, loss in stellar resolution and light gathering power, which is not normally the case with the former two. In general, similar values of the PDI - either resulting from obstruction or wavefront aberrations - and Ct, Cs or Ce, will result in a roughly comparable effect (i.e. quality level) for general observing. However, the last three have different implications.

The total contrast indicator Ct includes not only loss of contrast caused by the aperture reduction within the range of detail resolvable with the reduced aperture, but also the loss in limiting resolution. So, looking for the comparable in effect aperture reduction to 0.80 Strehl or PDI on this basis, we find a~0.89, from a=2Ct/(Ct+1), obtained from the above expression for Ct, with Ct=0.8.

And in the range of detail resolvable in smaller aperture - or only range of extended details - the approximate aperture reduction comparable to the effect of aberrations, or obstruction, will be somewhat different. For instance, 0.80 Strehl or PDI would be comparable in the effect to general image quality to the linear reduction in aperture size a~0.8 (from a~1.78-0.78/Cs, putting Cs=0.8) for its range of resolvable details, and to the linear reduction in size of a~0.45 (from a~3.2-2.2/Ce, putting Ce=0.8 ) for its range of extended objects. The latter appears to be unrealistically low, but it agrees well with the MTF graph (keep in mind that this is well into the range of deep-sky objects, thus loss of light in the smaller aperture can be a significant minus).

Again, for range of planetary details, comparable aperture reduction would be approximated by the value deducted from the Cs relation, or a~0.8. Following table gives an overview of the relation between the RMS wavefront error, peak diffraction intensity (Strehl), size of central obstruction and aperture reduction.

TABLE 2: EFFECT OF WAVEFRONT ERROR IN TERMS OF PEAK DIFFRACTION INTENSITY (PDI),
COMPARABLE CENTRAL OBSTRUCTION SIZE AND APERTURE REDUCTION

There is no value for the Cs based aperture reduction comparable to 0.41 peak diffraction intensity (PDI), because no reduction in aperture (as long as it has finite size) can result in the average contrast within its resolving range lower than 50% of the average contrast of larger aperture in that same resolving range.

One has to keep in mind that other factors are always involved with a significant change in aperture size. One is the average seeing error, which changes with D5/6, and the other are eye aberrations, which also change exponentially with the pupil size. Thus reduction in aperture of, say, 50%, also reduces the average seeing error by 44%. Also, eye aberrations (as the RMS wavefront error) change approximately with the square of pupil diameter, which means that 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).

The effect of chromatism can also be approximated in terms of the PDI, thus compared to the effect of aberrations, obstructions, or aperture reduction.

[1]Central obstruction causes distinctly different contrast transfer for details close to the limit of resolution, due to it diminishing the size of central diffraction disc. But this area of contrast transfer is, in general, little affected by the loss of energy from the Airy disc for relatively low aberration level normally found in amateur telescopes. For extended detail contrast transfer (approximately the left side of an MTF graph), and resulting performance level, any given PDI and energy loss it indicates will have similar effect, regardless of the origin of diffraction disturbance (aberrations, obstruction, chromatism).
 

◄ 5.2. Misalignment and forced surface deviations   ▐    6.2. Aberrations and image properties ►
 

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