◄ 6. EFFECTS OF WAVEFRONT ABERRATIONS   ▐    6.2. Aberrations and object type 2 ►
 

6.2. Aberrations and object type

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

(1) detail type (point-like vs. extended,),
(2) size on the retina (magnification),
(3) telescopic brightness and
(4) 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: diffraction of light. What is usually called "diffraction" is diffraction in a perfect aperture. However, aberrated apertures also produce diffraction pattern - and so do obstructed apertures - it only has different properties. In general, we can say that diffraction acts as an optical aberration, and that it is at its minimum in aberration-free (still not "perfect") aperture.

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 are their brightness and inherent contrast, both having profound effect on image appearance in either perfect or aberrated aperture (since diffraction minimum itself acts as an aberration even in a perfect aperture). 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. With the angular size of the smallest cones on the retina being ~1/2 arc minute, it is approximately 1.5 arc minutes in diameter.  Angular size of an image on the retina 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.

Point-like objects

Angular size of a point-like object, such as star, in a perfect aperture depends on its brightness on the retina, determined by star's apparent brightness and, given transmission coefficient, aperture size. In brightest stars, first bright diffraction ring will appear nearly as bright as central disc of the diffraction pattern, with the disc itself being close to the Airy disc diameter. 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 becomes 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. It 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 noticeably fainter first bright ring. Since the ring, due to its faintness, is not visible before the disc itself becomes non-stellar, respective magnifications for such a star 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 for point-like objects. Hence, we may conclude that visual point-object images in an aberration-free aperture do not appreciably differ from perfect as long as magnification remains bellow 0.2D in 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 causes transfer of energy from the disc to the ring area of diffraction pattern. 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 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 doesn't appreciably change in size at ~0.15 wave RMS error, the surrounding area 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 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 a visual effect.

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

Since the level of aberrations present in amateur telescopes is generally bellow 0.15 wave RMS, effect of aberrations on the 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 approximated by 2AFOV/F2, in arc minutes, AFOV being the apparent field of view of the eyepiece in degrees. Equating this expression with 1.5, for the level at which the coma just begins to show, gives AFOV~0.75F2, for the approximate diameter of the coma-free field. For an f/5 mirror, this gives nearly 20° coma-free AFOV, 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, visual coma-free field is larger than "diffraction-limited" field in the image formed by the objective. Since the sagittal coma in this context is approximated by feAFOV/1800F2, with fe being the eyepiece focal length, and the P-V wavefront error is smaller than sagittal coma by a factor of 3F, the P-V wavefront error at the boundary of coma-free field in a 20mm eyepiece in an f/5 Newtonian is approximated by W~feAFOV/5400F3, giving about 1 wave P-V of coma. This is larger than "diffraction-limited" coma level (0.42 wave) by a factor of 2.4, which means that the linear coma free field here is nearly 2.5 times greater than "diffraction-limited" field.

With 5mm eyepiece and identical AFOV, coma wavefront error at the edge of coma-free field is four times smaller, or ~0.25 wave P-V, well bellow "diffraction-limited". Sagittal coma is still ~1.5 arc minutes, but now, due to higher eyepiece magnification, it is smaller than the angular Airy disc size (given by 4.6F/fe, in arc minutes, for the 550nm wavelength). In a way, "coma-free field" still has some meaning, but in a different sense. This high magnification will show relatively subtle pattern changes due to coma not visible at lower magnifications. Since it is not to be called "coma-free" if the effect of coma can be seen, "diffraction-limited" field is here less representative of the actual performance than the "coma-free field", even if the latter has not been strictly derived to be this kind of indicator.

Again, these criteria are valid for brighter stars. Changes in diffraction pattern due to coma error will be less noticeable as the telescopic star brightness subsides, and the appropriate visual "coma-free" field will be becoming larger.

It is similar with off-axis astigmatism, only the numbers are somewhat different. As mentioned, it is eyepiece astigmatism that usually dominates here.

◄ 6. EFFECTS OF WAVEFRONT ABERRATIONS   ▐    6.2. Aberrations and object type 2 ►
 

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