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6.2.General effects of aberrations   ▐    6.3.2. Aberrations and extended objects

6.3. Effect of telescope aberrations on optical image; point-image

Point image visibility   • Point source resolution

For any given nominal aberration, the effect on image quality perceived by the eye will depend on several factors. The most important are:

(1) object/detail size,
(2) detail type (point-like vs. extended,),
(3) size on the retina (magnification),
(4) telescopic brightness and
(5) inherent contrast.

It will be helpful to clarify some basic terminology. While the effect of diffraction is often treated as something different than the effect of aberrations, it is in fact one same phenomenon. What is usually called "diffraction" is the diffraction in aberration-free aperture. However, aberrated apertures merely modify this pattern - and so do obstructed apertures - into another one with different properties. In general, we can say that optical aberrations worsen diffraction effect, and that it is at its minimum in an aberration-free (still not "perfect") aperture. In other words, aberrated diffraction patterns are diffraction patterns resulting from non-spherical wavefronts.

Effect of any given magnitude of aberration changes in inverse proportion to object size . It is because any given diffraction pattern becomes effectively smaller as object size increases, which is no different from the pattern - i.e. aberration - becoming smaller with given object size (this effect is familiar from the MTF graph, where any level of aberration has less effect on contrast transfer as the frequency nears zero, i.e. an infinitely wide line pair, where image contrast reaches that of the object). Image below shows effect of aberration-free diffraction pattern, where size of object - in this case square wave patterns, generated by SYNOPSYS - diminishes or, alternately, diffraction pattern becomes larger from top to bottom, doubling at each next image (left).

The top pattern can be regarded as the object, with no appreciable diffraction effect. Adding any aberration at a level where the effect of diffraction alone is not negligible (shown diffraction pattern and the square wave are on the same scale) causes transfer of energy away from the center of diffraction pattern which, in general, further worsens image contrast. This is noticeable in the pattern image even if the added aberratios are at the "diffraction limited" level (right). Since the line separation is about eight Airy disc diameters, or 20λF wide, it corresponds to ~0.05 normalized frequency. Plot below shows contrast transfer for unaberrated aperture and aberrated with nearly 1/4 wave P-V of primary spherical aberration. At 0.05 frequency nominal contrast drop is still small; at 0.2 frequency the line pair is about two Airy disc diameters wide, half as much at 0.4, and about λF at the cutoff. Near the cutoff there is contrast reversal, when transfer becomes negative (magnified in the box at right).

It will take roughly twice larger object - i.e. twice wider line pair - to equilize contrast of 1/4 wave P-V aberrated images vs. unaberrated. Even at this, still low aberration level, differences in effect of these aberrations are noticeable, with spherical aberration spreading energy more than the others; this affects images of relatively large objects - approximately one to four Airy disc diameters - more than with other aberrations, but less with images of small object, closer to the limit of resolution (it should be noticeable if aberrations are added to the bottom pattern in the first image).

How much optical aberrations affect image quality is directly dependant on image magnification. Large aberrations may be entirely invisible at low magnifications and, vice versa, quite small aberrations may noticeably impair image quality at high magnifications. In order to establish some general guidelines in respect to the effect of aberrations, it is necessary to start with the visibility of diffraction effects in a perfect aperture.

The two main object types to consider are: (1) point-like (stellar), and (2) extended. Along with their angular size on the retina, important properties of either type are their brightness and inherent contrast, both having great effect on image appearance in either perfect or aberrated aperture (since the diffraction minimum itself acts as an aberration even in a perfect aperture). In general, the brighter image, the more of imperfections (aberrations) it will show, while higher inherent contrast makes the effect of aberrations less detrimental.

However, the very basic consideration starts with the determinants set by the eye itself.

For the effect of diffraction to be noticed, retinal image has to be large enough for the brain to create an image of finite dimensions. This requires retinal image extending over at least three cone cells (FIG. 18). With the angular size of the smallest cones (in the middle of the retina) being ~1/2 arc minute, it is approximately 1.5 arc minutes in diameter. Angular size of the retinal image needed for the average eye to start recognizing its shape is about twice as large, or ~3 arc minutes for bright, contrasty details. For low-contrast details, it is roughly double, or ~6 arc minutes, on average.

Anything smaller than ~1 arc minute appears point-like to the eye, hence neither diffraction minimum, nor whatever amount of aberration in might be containing in addition, is not visible. The larger retinal image, the more apparent will be the effect of aberration on image quality.

6.3.1. Aberrations and point-like objects

Angular size of a point-like object, such as star, in a perfect aperture depends on its brightness on the retina. It is determined by star's apparent brightness and, given light transmission coefficient, aperture size. In brightest stars, the first bright diffraction ring will appear nearly as bright as central disc of the diffraction pattern, with the disc itself being nearly as large as the Airy disc. Since the diameter of the first bright ring is about 1.8 times the Airy disc diameter, or about 8/D in arc minutes for telescope aperture D in mm (1/3D for D in inches), diffraction pattern of a bright star starts appearing non-stellar at a relative magnification M~0.2D for D in mm, or M~5D for D in inches (which is the same as 5x per inch of aperture). The form starts emerging with magnification about doubled, at ~0.4D for D in mm, and ~10D for D in inches. Its form becomes clearly defined with magnification increased by another factor of ~2.

Stars of average brightness, somewhere between the brightest and faintest visible will have significantly smaller visible central disc - approximately half the Airy disc diameter - with no noticeable ring structure. Since the ring, due to its faintness, is not visible before the disc itself becomes non-stellar, respective magnifications needed to visually recognize such a star as non-point-like object are nearly four times higher than for a bright star. And stars close to the limit of detection will remain tiny patches of light within the range of usable magnification.

This is how the eye perceives diffraction minimum (i.e. diffraction in aberration-free aperture) for point-like objects. Hence, we may conclude that point-object images in an aberration-free aperture do not appreciably differ from perfect as long as magnification remains below 0.2D for the aperture D in mm, or 5D for D in inches. This limit magnification grows exponentially with the apparent (telescopic) star brightness.

Introduction of wavefront aberrations in the optical system causes transfer of energy from the disc to the ring area. In order to affect appearance of point-object image as seen in aberration-free aperture, this transfer of energy needs to be sufficiently significant in its extent and intensity to produce an enlarged bright portion of the diffraction pattern. Very approximately, this begins to take place at the aberration level of ~0.15 wave RMS, and grows roughly in proportion to the error. While the central diffraction disc does not appreciably change in size at ~0.15 wave RMS error level, the surrounding energy pattern becomes brighter and generally somewhat larger. This, in effect, makes image of a point-object that was near-perfect in aberration-free aperture, now appear slightly more blurry; in other words, magnification that will not show effects of aberration is lower than those for aberration-free aperture. However, this will affect mostly brighter stars; those of lower brightness will not change appreciably, because the energy expanded out of the central disc is still too faint to have appreciable visual effect.

This makes larger apertures in general more sensitive to the effect of wavefront errors on star images, due to their higher visual telescopic brightness.

Since the level of aberrations inherent to amateur telescopes is generally well below 0.15 wave RMS, the effect of aberrations on visual quality of star images vs. that in aberration-free aperture may be noticeable, but not significant. However, there are exceptions to this. One is the size of aberrations farther off-axis, which is often significantly higher than 0.15 wave RMS. Another is wavefront error caused by seeing, which can be higher even in medium-size apertures.

Off-axis aberrations are usually coma and astigmatism, with the latter being especially large in conventional eyepieces. Coma is present in all Newtonian reflectors, with the angular size of sagittal coma in the eyepiece, from Eq.17, given by CSa=CS/fe=215tanε/F2, in arc minutes (for small ε simply by CSa=3.75ε/F2), ε being the apparent field radius in a zero-distortion eyepiece in degrees (for the field edge, ε=AFOV/2, with AFOV being the apparent field of view diameter in a distortion-free eyepiece) and fe the eyepiece focal length in mm. Equating this expression with 3, for the apparent size at which sagittal coma just begins to appear not point-like, gives tanε~3F2/215, for the approximate radius of the "coma-free" apparent eyepiece field, defined as the field within which the geometric sagittal coma is not recognizable as not being point-like. For an f/5 mirror, this gives 19° coma-free field radius, regardless of the eyepiece focal length. It is important to note that this relates to star images; extended objects, in particular brightly illuminated low-contrast type (planets), have more stringent coma-free criterion and, consequently, smaller coma-free field.

As a consequence of the magnification factor, i.e. non-point images appearing point-like if magnified to less than ~3 arc minute, visual coma-free field in the eyepiece is larger than "diffraction-limited" field in the image formed by the objective. Substituting h=tanεfe in Eq.17, sagittal coma in terms of eyepiece apparent field and focal length is CS=tanεfe/16F2 in mm, and with the P-V wavefront error smaller than sagittal coma by a factor of 3F, i.e. W=CS/3F=tanεfe/48F3, the P-V wavefront error at the boundary of visually coma-free field in a 20mm eyepiece in an f/5 Newtonian is about 2 wave P-V for 550nm wavelength. It is larger than the "diffraction-limited" coma field (0.42 wave P-V) by a factor of 5, which means that the linear coma free field here is also as much larger than the formal "diffraction-limited" field.

This, of course, is very much dependant on the telescopic star brightness: fainter star will not show deformation farther off, while on the bright ones will be apparent somewhat closer to field center.

With 5mm eyepiece and identical AFOV, coma wavefront error at the edge of coma-free field is four times smaller than in a 20mm eyepiece, or ~1/2 wave P-V, just above the "diffraction-limited" 0.42 wave. Sagittal coma is still ~3 arc minutes, but now, due to higher eyepiece magnification, it is appreciably smaller than the Airy disc diameter (given by 4.6F/fe, in arc minutes, for 550nm wavelength). The eyepiece focal length at which the two are about identical is found from tanεfe/48F3=0.00023 (i.e. for coma at the field radius ε being at "diffraction-limited" level), giving fe~F3/90.6tanε. For an f/5 mirror, it comes to fe~4mm, and for shorter focal length eyepieces the "coma-free" field defined by the angular size of sagittal coma below 3 arc minutes becomes smaller than diffraction-limited field. However, the actual coma-free field, as seen in the eyepiece, is always larger, since it requires sagittal coma appreciably larger than the Airy disc to produce noticeable deformation of the central maxima. Approximately, it takes place with sagittal coma larger than the Airy disc by a factor of 1.5, which corresponds to about 1.2 wave P-V coma wavefront error.

Again, these criteria are valid for the stars not too bright nor too faint. Changes in diffraction pattern due to coma error are less noticeable as telescopic star brightness subsides, and the visual "coma-free" field becomes, in effect, larger.

Also, this consideration only includes mirror coma, while for the actual field quality in the Newtonian, eyepiece astigmatism is significant or dominant factor. Finally, the visual coma-free field, in the sense of no star deformation visible still has large enough coma in its outer portion to significantly lower its performance level with respect to extended objects like Moon, planets and deep sky.

It is similar, in general, with off-axis astigmatism, only the numbers are somewhat different. As mentioned, it is eyepiece astigmatism that usually dominates off axis in visual observing. A typical conventional eyepiece will have roughly between feε2/60F2 and feε2/30F2 P-V waves of Seidel astigmatism, ε being the eyepiece field angle in degrees. A 20mm eyepiece at f/5 will, therefore, have about 1.6 wave P-V of astigmatism at the approximate 10° boundary of the coma-free visual field radius. This already exceeds coma, which is only ~1 wave P-V this far off. Considering that the smallest geometric astigmatic blur is about 15% larger than geometric sagittal coma for given P-V wavefront error (both, geometric and diffraction, for P-V errors larger than ~, and that the two are roughly proportional to the the actual diffraction blur for errors greater than ~1/2 wave, the coma-free field boundary would have visible blur nearly doubled due to the eyepiece astigmatism. In other words, the actual aberration-free visual field would be somewhat smaller.

Also, since the astigmatism RMS wavefront error for given P-V error is nearly 14% larger than that for coma, the combined loss of energy is likely beginning to noticeably degrade contrast of extended objects before the aberration shows visible deformation of the diffraction pattern of brighter stars. This means that the actual aberration-free field, as determined by these two aberrations alone, is yet smaller.

Point source resolution

The view that aberrations quickly impair resolving power of a telescope is not uncommon among the amateurs , but it is only partially true. Resolution of low contrast details is, indeed, can be very sensitive even to small aberrations, but high-contrast details - including near equal in brightness double stars - may not be. Even aberrations as large that would entirely wash out low contrast detail resolution, will not necessarily reduce the resolution of high-contrast detail. The two main deciding factors are type of aberration and relative orientation of the detail. In the context of point source resolution, of primary interest is double star resolution. Simulations at left (generated with Aberrator, Cor Berrevoets) illustrate the effect of three classical aberrations - spherical, coma and astigmatism - on the resolution of a pair of equal-brightness star separated at the Rayleigh limit. Each aberration is given at its diffraction limited level (0.80 Strehl, double column at left), as well as twice larger. For each aberration and aberration level shown are an average star without clearly pronounced diffraction rings (left), and a bright star. Finally, it is all given for unobstructed and 0.3D obstructed aperture.

What simulations suggest is that spherical aberration has little effect on this type of resolution at its "diffraction-limited" level (0.25λ), and even with the error doubled. With a bright star, the brightness of the first bright ring can impair resolution in field conditions (due to seeing error and other induced errors), but the stars are still clearly resolved.

Not so with the coma. While an average star will remain clearly resolved at the "diffraction-limited" level (0.42λ), the bright star resolution is impaired, generally more than with 0.5λ of spherical aberration. At double the error, stars are not resolved if their common axis nearly coincides with the axis of aberration - especially bright pairs - while might be only partly resolvable for other orientations (shown 45° and 90° orientations relative to the common axis). 

Similarly, the "diffraction-limited" level of astigmatism (0.37λ) will little affect resolution of a pair of average brightness if its common axis nearly coincides with the axis of astigmatic spike. If the spike is at 45 degrees, however, resolution will be impaired even at this aberration level for a pair of average brightness, and will vanish with bright stars. At double the astigmatism, a partial resolution with the common axis coinciding with the spike might be possible due to the pattern elongation, but with the spike at 45° a pair of stars at the Rayleigh limit is unresolvable regardless of the pair brightness.

As for the effect of 0.3D obstruction, it doesn't seem that it significantly influences resolution of equal doubles. In some instances, it appears that its 10% smaller central maxima does produce a slightly better resolved pair, but that is probably offset by the larger, more intense first bright ring, which not only lowers the contrast against the immediate surroundings, but also would throw more energy over the double in the presence of random movements of the portions of the pattern due to seeing.

In general, aberrations affect much more quickly the resolution of unequal doubles. Simulations at left show a pair of stars at the Rayleigh limit separation, with the primary star four magnitudes brighter. As in previous example, shown are an average, and a bright star for aberration free aperture, clear as well as 0.3D (30% linearly) obstructed. However, since the fainter companion is already invisible with the former (the first bright ring in aberration-free aperture is nearly as bright as a four magnitudes fainter star), the remaining simulations only show a bright star.

At the "diffraction-limited" level of spherical aberration the faint companion is barely detectable in the top of the first bright ring. It completely disappears with the aberration doubled.

Coma is shown only at its "diffraction-limited" level, because the faint companion is already undetectable, even with the most favorable orientation when the comatic rings spread to the opposite side from where it is located. This is caused by the pattern of the faint companion being just as much spread by coma, with its central maxima merging with the extended maxima of the bright companion.

With astigmatism at its "diffraction-limited" level, the faint companion completely disappears in the spike, if the two coincide. When the spikes is at 45°, a slight brightening between the spikes is detectable, which would quickly disappear with further error increase and the associated pattern enlargement.

The obstruction here has only a slight negative effect due to the larger and brighter first bright ring; the smaller central maxima is also likely to be slightly detrimental to resolution of unequal doubles in most situations.

Obviously, near equally bright stars of at least average telescopic brightness and those with a faint companion are two different contexts: high vs. low contrast detail. The aberration tolerance is much tighter for the latter which, in general, applies to the extended details as well.

In principle, aberrations affect extended detail images through the same basic mechanism, by expanding the image of a point source, but the consequences, as well as detail types, are somewhat different. More on next page.

6.2.General effects of aberrations   ▐    6.3.2. Aberrations and extended objects

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