6. TELESCOPE ABERRATIONS: Effects on image quality

Effects of wavefront aberrations on image quality is a fairly complex subject. Any deviation in the wavefront form away from spherical causes deterioration in the quality of point images, thus also quality of the image as a whole. The hard part is to define the specifics of this general fact. Assessing the relation between optical aberrations and image quality requires not only knowledge of the mechanism by which the aberrations influence the determinants of optical quality, but also establishing the universal criteria applicable to any and all optical aberrations. Such criteria need to define specific aberration tolerances as a reference point in fabrication, evaluation and use of astronomical optics.

 In addition, it is just as practical to be able to compare effects of aberrations to those of other common factors of image degradation in telescopes, specifically central obstruction and, at least approximately, reduction in aperture size.

Criteria for assessing the effect of aberrations on image quality were evolving together with the optical theory. The first, and probably best known, is Lord Rayleigh's 1/4 wave of optical path difference criterion for monochromatic aberrations, expanded for the application to optics fabrication by the Danjon-Couder conditions. This relatively crude concept based mainly on geometric optics was further refined by the introduction of "diffraction-limited" criterion which, as the term implies, uses diffraction calculation. Another tolerance criterion, used by optical designers, is the geometric ray spot size vs. Airy disc criterion.

Conrady and Sidgwick each laid down somewhat different criteria for chromatic aberration - specifically for secondary spectrum. 

Finally, the two indicators of the effect of aberrations on optical quality based on diffraction calculation - the Strehl ratio and contrast transfer function (MTF) - are the latest addition to this set of tools for defining optical quality as a function of the level of aberrations.

The primary factor determining image quality is the quality of the wavefront. It can be expressed in several ways: as optical path difference, or wavefront error (either P-V or RMS), peak diffraction intensity (Strehl), or geometric blur size. Table bellow summarizes general aberrations-related optical quality criteria known and used in amateur astronomy.

Following text expands on each of these concepts.

In 1878, after investigating effect of various monochromatic aberrations on the characteristics of point-image, Lord Rayleigh concluded that performance of an optical system will not be significantly impaired as long as the maximum optical path difference (OPD) resulting from aberrations does not exceed 1/4 of the wavelength. Or, as illustrated by Sidgwick (Amateur Astronomer's Handbook, p66), if the wavefront is contained between two concentric spheres separated by 1/4 wave. This is known as Rayleigh criterion (not to be confused with Rayleigh limit for point-image resolution), or tolerance. Rayleigh criterion is not strictly accurate, since different forms of nominally (i.e. in terms of the maximum OPD) identical wavefront deformation have appreciably different effect on image quality. For instance, 1/4 wave OPD of coma lowers MTF image contrast less than half as much as identical wavefront error of spherical aberration (8% vs. 20%, with 0.92 and 0.80 Strehl, respectively).

Despite its approximate nature, Rayleigh's 1/4 wave criterion was the corner stone in: (1) determining the effect of aberrations on image quality, and (2) defining the appropriate tolerances. Pretty much neglected for decades, it was given due attention in the early 20th century, particularly in the work of Joseph Conrady (Applied Optics and Optical Design, 1929).

Such a general nature of the Rayleigh's criterion did not sufficiently address specific problems resulting from smaller-scale/higher-slope surface deformations in optical fabrication, as well as the extent of the wavefront area with deformations near the maximum tolerable error (1/4 wave). To compensate for this, Danjon and Couder expanded it to the following two conditions:

(1) wavefront needs to have mild slope over most of the aperture,
with the smallest geometric blur not exceeding the Airy disc, and

(2) no portion of the wavefront can have more than 1/4 wave deviation,
and it should be significantly less than that over most of its area.

The first condition addresses the need for smoothness of optical surface (since smaller zonal and/or local errors can divert rays far from the Airy disc even when well within the 1/4 wave tolerance), and the second the fact that the more of wavefront area has deviation close to 1/4 wave, the worse its optical performance will be, including the possibility of unacceptable quality. If applied to the actual ray spot at the best focus, the first criterion is extraordinarily demanding, at the level of highest optical quality achievable in optics fabrication (one exception is astigmatism, which is significant at geometric blur sizes well bellow the Airy disc diameter).

A variant of the first Danjon-Couder condition is used in mirror making, specifically as the RTA criterion in the Foucault test. Graphic application of the principle of limiting ray scatter to a certain area around focal point (standard criterion is a circle equal to the Airy disc) is known as Millies-LaCroix envelope (or tolerance).

Similar criterion, based on the size of aberrated geometric blur vs. Airy disc diameter, is still commonly used to assess imaging quality of optical designs. While useful for quick evaluations, it is also arbitrary and inaccurate in its basic assumption that an optical system is well corrected - or "diffraction-limited" - if its geometric blur is not larger than the Airy disc. Simply put, there is no useful causal relationship between distribution of rays and diffraction image for different aberrations. For illustration, the level of balancd (best focus) primary astigmatism producing ray spot equaling the Airy disc lowers the overall (MTF) image contrast by 46% (0.54 Strehl), while spherical aberration of equal geometric blur size at the best focus location only lowers it by 2% (0.98 Strehl). It is illustrated below, with ray spot plots for common aberration forms (PSF value is the corresponding Strehl number).

Hence, in order to use this criterion productively, one needs to be aware of its specific limitations (note that this is much less of a problem when using Airy disc criterion in the Foucault test, since it is usually limited to assessing axial, rotationally symmetrical wavefront/surface forms).

The conventional "diffraction-limited" criterion is a comparatively recent concept, with origins in the work of Andre Marechal, published in 1947. Use of diffraction calculation allowed Marechal to apply broader approach, that is, to track down specific effect of both, isolated and combined monochromatic aberrations on the intensity distribution within diffraction image (note that diffraction-limited criterion is a measure of the overall imaging quality, and shouldn't be confused with diffraction resolution limit, which applies only to point-source resolution and requires generally lower minimum optical quality). Based on this, he proposed a practical definition of well corrected system as one with phase variance averaged over the pupil (φ2) not greater than 0.2 (Marechal criterion). Since φ=2πφ, where φ, in radians, is the phase analog of the RMS wavefront error, assuming φ, in units of 2π radians (full phase of a wave) nearly identical to the RMS wavefront error, in units of the wavelength (which generally holds for aberrations smaller than ~λ/2 P-V, smoothly affecting all or most of wavefront area), it implies the RMS wavefront error not larger than √0.2/2π, or 1/14 wave. According to Marechal's Strehl approximation, given by S~(1-0.5φ2)2, this corresponds to the central diffraction intensity (Strehl ratio) not lower than 0.81 (the actual Strehl value is closer to 0.82). Subsequently, the criterion is only slightly modified to 0.80 Strehl (corresponding to 1/4 wave P-V of lower-order spherical aberration - proposed by Conrady nearly two decades prior to Marechal's work - or 1/13.4 wave RMS), or better, and was associated with the "diffraction-limited" attribute (term independently coined by Ernst Abbe in his work on microscope resolution).

Although the term implies that the 0.80 Strehl level is still "diffraction-limited" optics, practically indistinguishable from the aberration-free, that is not so in most demanding applications (in addition, the term itself is ambiguous, since even heavily aberrated telescopes are still limited by diffraction, which is merely worsened by the presence of aberrations). As an illustration of that point, diffraction simulations below show the effect of diffraction on an object image, as well as the effect of adding 0.80 Strehl worth of some common aberration forms.

At left, Airy disc size changes vs. constant image size, which in effect reflects the corresponding change in aperture size (it is just the same if the image size would change relative to the Airy disc, showing how contrast and resolution diminishes as details get smaller; number at left of a slash shows the change in Airy disc size, and number at right the the corresponding change in image size with respect to the starting pattern, second from bottom). At right, to the starting pattern, already heavily affected by zero-aberration diffraction, is added 0.80 Strehl worth of specified aberrations. Even with already compromised image, additional loss in definition, as well as energy scatter from introducing what is regarded to be a boundary of the "diffraction limited" level of aberrations is fairly easily detectable. Defocus and astigmatism scatter less than spherical aberration and coma, but their more concentrated energy spread is actually more detrimental to the small detail resolution.

It is important to understand, though, that the criterion is properly applied only to the aggregate system error, not the objective system (i.e. optical surfaces) alone, as it is often done. For instance, 0.95 system combined with 0.84 Strehl degradation ratio of induced errors (seeing, thermals, miscollimation, etc.) is still within diffraction-limited range, while the same induced error pushes 0.80 Strehl system down to 0.67 Strehl level. The latter will often be assumed to be representative of 0.80 Strehl level, which is not correct.

This provided satisfactory tolerance criterion for optical quality - as coarse as such a general criterion inevitably has to be - for monochromatic aberrations. However, tolerance for chromatic aberration - specifically secondary spectrum - was still relying on the two criteria based on Rayleigh's 1/4 wave standard. The older one, defined by Conrady (Applied Optics I, p201), assumes that the maximum acceptable separation - i.e. defocus error - between the combined C/F focus and best (green) focus is, due to the lower eye sensitivity to those wavelengths, two and a half times the Rayleigh tolerance, or 0.625 waves. Applying this defocus value on the minimum secondary spectrum possible with standard achromat (f/2,000, f being the focal length), Conrady determines the corresponding focal length as fmin=D2/10,000λ, or the corresponding focal ratio Fmin=D/10,000λ for the aperture D and wavelength λ (both in the same units). Using λ=0.00002 inch, Conrady arrived at Fmin=5D, for D in inches.

 A quarter century later, Sidgwick proposes a more forgiving criterion, assuming that the maximum acceptable aberration is with the C/F focus separated by four times the Rayleigh tolerance, or 1 wave, from the brightest green focus. That leads to the corresponding minimum focal length fmin=D2/16,000λ, or Fmin=D/16,000λ (Amateur Astronomer's Handbook, p92). Taking λ=0.0000217 inch (5500 Angstrom), Sidgwick obtained Fmin=2.88D, which he rounded off to Fmin=3D, also for D in inches.

More recent alternative criterion, assuming the maximum acceptable chromatism with the defocused C/F blurs reaching 3 times the Airy disc diameter (Rutten/Venrooij, Telescope Optics p55), implies Fmin=D/14,640λ. With λ=0.00055mm (550nm), it gives Fmin=D/8, for D in mm.

In addition to being fairly arbitrary, these criteria do not address other forms of chromatism (chromatic spherical aberration and lateral color). More accurate tolerance is, analogously to the "diffraction-limited" standard for monochromatic aberrations, polychromatic Strehl of 0.80 or better. According to it, the minimum tolerable F-ratio of a standard doublet achromat at the optimized focus location (usually e-line) is about D/13,000λ (for instance, Fmin=14 for D=100mm and λ=0.00055mm), midway  between Conrady's and Sidgwick's estimate. Shifting from the e-line focus to the best polychromatic focus, with somewhat higher peak diffraction intensity, allows further loosening of the tolerance to Fmin~D/17,000λ, or about f/11 for 100mm aperture. And when achromat's improved contrast transfer efficiency over most of the MTF range, due to slightly enlarged central maxima and the resulting higher encircled energy relative to the Strehl value, is included, diffraction-limited threshold for an achromat with near-zero spherical aberration is approximately Fmin~D/20,000λ (about f/9 for 100mm aperture).

Note that this is valid for the standard e-line correction mode in achromats and photopic eye sensitivity. In night-time observing, eye sensitivity shifts toward blue, with polychromatic Strehl declining due to disproportionately larger defocus error toward this end of the spectrum. For the conventional mesopic eye sensitivity (i.e. peak sensitivity at about 530nm wavelength, relatively higher sensitivity in the blue/violet and lower in red), polychromatic Strehl of an e-line optimized achromat (i.e. with C-F correction) drops by over 10%. Consequently, the minimum F-ratio for 0.80 polychromatic Strehl at the best focus location is approximated by Fmin~D/7,000λ, for λ=0.00053mm. For 100mm aperture, that would give Fmin~27. Raytracing comes up with a bit faster system (below). Strehl (PSF) values are given for the e-line focus (top) and best diffraction focus, with "Z" denoting shift in mm from e-line focus. Note that the sensitivity figures used are approximate and based on the experimental values for mesopic sensitivity.

This consideration, however, assumes no change in eye acuity. In reality, both resolution and contrast sensitivity are significantly lower in the mesopic mode. As a result, the lowest acceptable level of optical quality is also lower. Since specifics of mesopic vision are very approximate, the minimum focal ratio cannot be determined nearly as accurately as for photopic vision. Since the increased sensitivity in both, cones and rods, comes at a price of lowered acuity, larger nominal error is probably mainly offset by the lowered detection thresholds of the eye. If so, quality level determined for the eye in photopic conditions should be good indicator of that in mesopic conditions as well.

Also, it should be pointed out that the above criteria assume negligible monochromatic aberrations. In achromats of moderate or, especially, faster focal ratio, however, it is difficult to achieve near-zero spherical aberration; and even moderate level of error will appreciably affect polychromatic Strehl, with the resulting Strehl degradation factor approximately equaling Strehl value corresponding to the spherical aberration error. Thus, for practical reasons, the tolerance for secondary spectrum should be somewhat smaller, approximately at Fmin~D/15,000λ. For instance, a well-made 100mm achromat, with about 1/8 wave P-V of spherical aberration, will be at the minimum diffraction-limited level (~0.80 polychromatic Strehl) for extended object contrast at about f/12; for the diffraction limited overall contrast, which includes details smaller than the Airy disc size, it is back at Fmin~D/13,000λ or about f/14 for 100mm aperture.

The effect of aberrations on image quality is best assessed with the Strehl ratio and MTF of the optical system. For that reason, these two quality indicators are addressed in more details. Also, more specific topics on this broad subject are added:

(1) general effects of the aberrated wavefront on image contrast, with its intensified diffraction-caused spread of energy in the point-image, including comparison to the effects of central obstruction and aperture reduction,
(2) effects of wavefront aberrations as a function of object/image properties: magnification, telescopic brightness and inherent contrast level,
(3) effect of aberrations on the appearance of diffraction pattern