◄ 6. EFFECTS OF WAVEFRONT ABERRATIONS   ▐    6.3. Aberrations and image properties ►

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

c=(1-i)/(1+i),

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.

An optical quality indicator normally presented in the form of a graph, called Modulation Transfer Function (MTF), shows how contrast level in an optical system of known optical quality changes with the detail size. What is less known is that the Strehl ratio - and somewhat conditionally the peak diffraction intensity ratio, such as  the peak diffraction intensity resulting from central obstruction - also indicate average contrast loss over the entire MTF range of frequencies (Strehl) or, in the case of the obstructed PSF, significant portion of this 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 average contrast loss and drop in central diffraction intensity are directly related.

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) equals the relative unobstructed pupil 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)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. The corresponding average contrast loss over the range of MTF frequencies is ~0.24, or 24%. Should be noted, however, that this standard theoretical approach assumes near monochromatic point-source, which is effectively a coherent source. In the real world, light from astronomical objects, including stars, is polychromatic, thus partially (mainly) incoherent, and its effect on intensity distribution is significantly smaller (more details).

Similarly, peak diffraction intensity in the presence of wavefront aberrations is closely approximated by an empirical 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. But this comparison is limited to the main reference point, which is 0.80 PDI, the conventional "diffraction-limited" level, regarded as a relatively loose line separating acceptable or better optical quality from unacceptable. In other words, nominal DPI comparison doesn't immediately specify optical quality level in the qualitative sense.

The simplest way to relate nominal PDI values to their qualitative implications on optical quality is to have them expressed in terms of reduction in the aperture size. 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. This also allows comparison of the contrast effect of aperture reduction to that of aberrations, central obstruction and chromatism.

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. For circular aberration-free aperture this area can be approximated by a right triangle, with unit height and the base proportional to the aperture diameter; for square aperture, whose MTF plot is flat, the area is triangular. Hence the triangular area is exact for the square-pupil MTF and close approximation for the circular. Since the former is much simpler for expressing the relative area of smaller vs. larger aperture, it will be used here for that purpose.

As the triangular MTF geometry at left illustrates, halving the aperture results in 50% loss in contrast transfer over the full range of frequencies before reduction (frequency ν from 0 to 1), which is comparable to the effect of 0.43 wave P-V of primary spherical aberration (note that the specific contrast transfer distribution at this large error level is likely to vary considerably for different error types; in this particular case, contrast transfer for twice larger aperture with 0.43 wave of spherical aberration (green) is significantly lower within the resolving range of twice smaller aperture, but it is compensated for by its contrast transfer in higher frequencies). Over resolving range of the reduced, twice smaller aperture (0 to 0.5), the loss of contrast vs. that in twice as large aperture for this range of frequencies is 33%. The bright low contrast (BLC) and dim low contrast (DLC) thresholds for square aperture are slightly higher, at about 0.13 and 0.6 frequency, respectively.

As graph indicates, the differential in contrast within the range of resolvable dim low contrast details is not significant, unless the aperture ratio goes well over 2 to 1. Since this type of objects are much more affected by the loss in light gathering power and resolution than contrast, they won't be included in further consideration.

The smaller-to-larger aperture diameter ratio a equals the relative contrast transfer in a smaller vs. larger aperture. The needed height c is simply c=(1-a). It determines the area of a rectangle (dashed) whose base is a and side (1-a), as well as the two isosceles right triangles covering rest of the MTF area: one, on top of the rectangle, with sides equaling a, and the other with sides equaling (1-a). The former, when added to the rectangle, gives the MTF area over resolvable range of smaller aperture, and the latter is merely the remaining area of the larger aperture.

To assess relative contrast transfer within the BLC subrange of frequencies for smaller aperture, the area up to its BLC cutoff frequency is needed, and it is obtained in the same manner using the contrast transfer height for that frequency, cBL=0.6a.

In all, the three basic areas are those representing: (1) the total contrast transfer of larger aperture (the entire blue triangle), (2) contrast transfer of a larger aperture over the resolving range of smaller aperture (sum of the area of rectangle with the base a and side c and with the triangle on top of it), and (3) contrast transfer of a smaller aperture (the smaller right triangle with a bold hypotenuses) over its whole range. In addition, it is interesting to find out how the contrast transfer within the BLC range of smaller aperture compares to that in a larger one. In that order, in units of the MTF area of larger aperture over its entire range, they are given as:

A1=1
A2=(2-a)a
A3=a, with
A4S=0.84a   and   A4L=(1.2-0.36a)a

where the last two areas are those for the smaller and larger aperture over the range of resolvable BLC details for the former. All area indicators are relative to the unit area A1 (blue shaded at the plot above).

A graph of these contrast ratios shown below (FIG. 92A, left) indicates that the amount of contrast transfer of larger aperture over the whole range of MTF frequencies in a smaller one (A2) decreases very slowly with aperture reduction, and may be considered negligible for the aperture ratios larger than 0.9. On the other hand, the amount of contrast transfer of the smaller aperture over its range (A3) decreases linearly with the aperture, with the ratio A3/A2 decreasing as 1/(2-a), from 1 at a=1 to 0.5 at a=0, indicating that the smaller aperture ratio, the greater contrast transfer advantage of the larger aperture over the smaller aperture's resolving range.

It is similar with respect to contrast transfer over the smaller aperture's BLC range, with the ratio larger(A4L)-to-smaller(A4S) aperture MTF transfer changing with 1/(1.43-0.43a), from 1 at a=1 to 0.7 at a=0.

    The final step is obtaining the corresponding RMS wavefront error from this Strehl-like number from the general relation ω=0.24(-logS)0.5, also shown at right (dashed), with S1=1/(2-a), S2=1/(1.43-0.43a) and S3=a. The top Strehl-like plot shows degradation due to aperture reduction over the BLC range of smaller aperture, the middle plot the degradation over its entire range, and the diagonal plot the degradation over the entire range of larger aperture. The corresponding RMS wavefront errors in the common aperture ratio range of 0.5 to 0.25 are little under 0.1 for the BLC MTF range, little over it for contrast degradation over the smaller aperture entire range, and between 0.14 and 0.2 for contrast degradation over the MTF range of larger aperture.

    This aperture reduction to Strehl/RMS relation is unavoidably very 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 is can be actually inferior compared to smaller apertures, the worse the seeing conditions, the more so.

For telescopes capable of efficiently correcting for the seeing error, or operating outside an atmosphere, following table gives an overview of the relation between the RMS wavefront error, peak diffraction intensity (Strehl), size of central obstruction, chromatic aberration and aperture reduction, based on relating MTF contrast drop in smaller apertures to the appropriate drop in central diffraction intensity.

There is no values for aperture reduction causing 0.41 drop in contrast transfer over the full resolving range of smaller aperture, or 0.67 and 0.41 drop in the contrast transfer over the low-contrast detail resolving range of smaller aperture, because no aperture reduction can result in that magnitude of contrast loss in this range.

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, size on the retina or brightness.
 

◄ 6. EFFECTS OF WAVEFRONT ABERRATIONS   ▐    6.3. Aberrations and image properties ►
 

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