◄ 2.3. Telescope magnification   ▐    3.2. Ray (geometric) aberrations ►

The former are known as wavefront aberrations; the later as ray, or geometric (ray) aberrations.

Either aberration form has its purpose. While the wavefront aberration form is more directly related to the physical fundamentals determining image quality, ray aberration form offers more convenient graphical interface for the initial evaluation of the quality level of optical systems. Since the wavefront and the rays emerging from it are directly related, there is a constant relationship between the size of wavefront aberration, and that of corresponding transverse ray aberration relative to the size of the Airy disc. This is true for any given relative aperture; obviously, change in relative aperture for any given size of the transverse aberration relative to the Airy disc (i.e. for any given wavefront error) requires the relative longitudinal aberration to change inversely.

Unrelated to the form of presentation of aberrations, it is useful to make a distinction between aberrations that are intrinsic to optical surfaces in their proper alignment, and those induced by external factors.

Intrinsic telescope aberrations are those inherent to conical surfaces, to glass medium, and those resulting from fabrication errors.

As described in previous chapters, imaging quality of a telescope rely on optical surfaces capable of producing spherical wavefronts for the image formed by objective, then transformed into flat wavefronts by the eyepiece. The final wavefront is formed by the eye, ideally of spherical shape. Spherical wavefront ensures tightest possible energy concentration in the image of a point-source and, consequently, highest contrast and resolution. In other words, the effect of diffraction, which causes the point-object image to form as a bright central disc surrounded by a number of fainter concentric rings of rapidly decreasing intensity, is at its minimum for perfectly spherical (aberration-free) wavefront.

Thus perfect telescope is the one that produces flat wavefronts exiting the eyepiece. While any combination of aberrated wavefronts at the objective and eyepiece that cancel each other out will do the trick, it is preferred to have the objective producing a near perfect spherical wavefront, and the eyepiece turning it into near perfectly flat. After that, it is up to the eye how accurate will be the final wavefront: the closer to spherical, the better.

For most people, the wavefront formed by the eye becomes nearly spherical at ~2mm pupil diameter, and practically spherical at ~1mm pupil. The larger eye pupil, the greater wavefront deviations from spherical, due to eye's optical imperfections. This, in general, has less of an effect, with larger exit pupil sizes being associated with low-power observing, when wavefront imperfections are in general more forgiving. Wavefront quality is critical for high-magnification observing at small pupil sizes, when the eye, as mentioned, produces near-spherical wavefronts, provided it is supplied with near-perfect flat wavefront by the telescope.

Any significant deviation from spherical in the shape of the wavefront formed by telescope's objective results in lower quality of its image. Assuming no aberration contribution from the eyepiece, this wavefront deformation will be transferred to the eye as an imperfectly flat wavefront coming out of the eyepiece, passing the deviation to the wavefront formed by the eye. Since the path length of a wave from any deviant, or aberrated point on the wavefront differs from the wavefront radius' length, it arrives at the focal point out of phase with the waves coming from the spherical portion of the wavefront. The greater wave path difference, the greater its phase difference, and the lower wave energy contribution at the focal point (up to 1/2 wave path difference; from 1/2 wave to 1 wave the contribution is increasing, as shown on the graph on the page bottom). The more such points on the wavefront, the more energy transferred to the outer portion of diffraction pattern, and the lower image quality.

Picture at left illustrates ray path through a lens. Rays - by definition normal to the wavefront - from faraway object enter the lens parallel with lens' optical axis. However, due to lens' curved shape, the central ray, or chief ray, enters the glass sooner than a random off-axis ray. As the random ray touches the glass in point 1, chief ray is already some distance inside the glass, at its point 1 (light inside is moving slower, hence travels shorter distance in a given amount of time). After the random ray reaches point 2, with the chief ray at its own point 2, light again starts moving faster, and by the time the chief ray hits the air at poin 3, the random ray is at its point 3. What determines the angle of refraction at the points 1 and 2 is the direction in which the waves traveling along rays over a small section (or a patch in 3-D) around the random ray are realigned into a straight line, since due to the inclination of the glass they are not in phase traveling straight anymore (waves don't refract, but since every wavefront point radiates waves in all direction, every ray drawn from wavefront point will have a wave traveling along it). In order for a lens, i.e. objective to bring all rays to the same point, it has to form spherical wavefront behind it. In other words, all ray paths through the air and glass have to have equal travel time - i.e. optical path length - from their respective points 0 on the incident wavefront, to the focus point.

The existence of path difference at the focal point implies that there is a point - or points - farther off in the image space for which the wave path difference from aberrated points at the wavefront is now smaller, and constructive energy interference greater, than in a perfect system. In other words, that the energy lost from the proximity of the focal point due to wavefront aberrations will be effectively transferred toward outer area of the diffraction pattern.

Deviation of any single point on the wavefront will not cause measurable effect on image quality, regardless of its optical path difference; however, if an area of the wavefront deviates from spherical, it will negatively affect image quality, the larger area, the more so. Energy concentration at the center of diffraction pattern becomes noticeably lower relative to energy contained in the diffraction rings, blurring the point image. The image quality of point-source deteriorates, and with it image quality of extended objects (the latter being merely point-image conglomerates). In terms of loss of resolution, expectedly, low-contrast details are affected more than those of high inherent contrast.

Wavefront deviation is commonly presented with the reference sphere as a straight line. This wavefront deviation form shouldn't be confused with the wavefront itself. Picture at left illustrates the relation between the wavefront and wavefront deviation for three common aberrations, defocus, primary spherical and coma.  The actual wavefront converging toward focal point is always for all practical purposes spherical, with the deviations from spherical usually being a small fraction of micron, with the wavefront deviation form always greatly exaggerated.

 The values of maximum positive and negative wavefront deviation from the reference sphere combined determine peak-to-valley (P-V) wavefront error. This figure alone is meaningless with respect to the damage it causes to image quality, unless related to a known form of wavefront deformation. In other words, unless both maximum wavefront deviation from spherical, as well as the wavefront form are known (FIG. 21). An example of such forms of wavefront deformations are those characteristic of the typical optical surface, conic of revolution - spherical aberration, coma, astigmatism, field curvature and distortion. The only useful input from the P-V figure alone is that it approximates the worst case scenario; that is, if the specified P-V error affects most or all of the wavefront area, it cannot be significantly worse than a wavefront with this level of P-V error of spherical aberration (assuming that larger-scale surface roughness is not significant). So 1/10 wave P-V mirror with reasonably smooth surface cannot be significantly below the quality level of 1/10 wave P-V of spherical aberration; on the other hand, it is possible that 1/4 wave P-V mirror, with the deviation limited to a small wavefront area, performs as well, or better.

Potentially significant indicator of wavefront quality is its slope, i.e. dynamics of change in the wavefront error over its radius. Wavefront slope is generally expressed as the change in optical path difference vs. change in radius value, i.e. ∆OPD/∆d, which for wavefront aberration expressed as f(x), x being the pupil zonal radius, determines its first derivative f'(x). It can be given for a section of wavefront radius, expressed either in linear or angular units. For any point at the wavefront, the slope is determined by the tangent to the wavefront for that point, as shown on the graphs below.

At left are shown wavefront deviations for primary spherical aberration at the best (left) and paraxial focus (middle), as well as for balanced secondary spherical (right), all for unit P-V error. Wavefront slope is measured in the units of wavefront error (WFE), as a projection of the tangent section over the radius onto the vertical (wavefront error) scale. The corresponding value for any point on the wavefront is given by the first derivative (y', blue plot) of the aberration function (y, red plot). For the edge point, slope for primary spherical at the best focus is 8 times the P-V error, while only 4 times at the paraxial focus. For balanced secondary, the slope is 12 times the P-V error. For the first two, the slope difference is not affecting optical quality, since the P-V to RMS ratio for either wavefront form is identical. For the balanced secondary spherical, the ratio is 1.6 times higher (which means that at the comparable RMS error level it would have as much higher edge slope, i.e. over 19), a consequence of the real steep portion at the edge covering less than 10% of the radius, and the secondary P-V maxima being smaller and at the lower, ~0.5 zone. It would, however, change for the worse, if wavefront slope would fluctuate in this same range over the radius, i.e. with the presence of a large-scale wavefront roughness. And even if balanced secondary spherical has more favorable P-V to RMS wavefront error ratio, it lags a bit behind with respect to encircled energy efficiency, having its bright rings energy spread out wider.

Simplifying the Mean Value Theorem by limiting the aberration function to (0,1) interval, gives the average slope of the wavefront as equal to its first derivative value for x=1, in this case y=f'(1)=2. Given that, mean slope value is determined by the value of x for which f(x) first derivative f'(x) equals f(1), which is here also equal to the average slope value. This value of x is found by equating the second derivative of the aberration function to 2 (since that means that the slope of the tangent to f'(x) at this point is parallel to the average slope). An example is given for the best focus primary spherical.