3. TELESCOPE ABERRATIONS: Types and causes

Any deviation of the wavefront formed by a telescope from perfect spherical - for wavefronts formed by the objective - or from perfect flat for wavefronts formed by the eyepiece, results in an optical aberration. Aberrations disturb optimum convergence of the energy, with the result being degradation of image quality. The two main forms of expressing the aberration are:

(1) at the wavefront itself, as a deviation from perfect reference sphere, and
(2) at the focal point, as a linear or angular deviation of rays projected from the wavefront.

FIGURE 11: Ray and wavefront geometry in a perfect (top) and aberrated (bottom) telescope. Ray aberrations - longitudinal, transverse and angular - result from wavefront deformation, but their numerical values have no inherent relation to the determinant of energy (re)distribution: optical path difference (OPD), creating phase differential between waves interfering in the image space. The efficacy of wave interference varies with cos2(OPDπ/λ), with the peaks at (OPD/λ)=0,1,2... and zeros at (OPD/λ)=0.5, 1.5, 2.5... This means that, for instance, 1 wave OPD at the marginal ray with respect to paraxial rays will result in corresponding waves interfering 100% constructively; however, all other zonal emitters will be out of phase, causing central intensity of point-image to plummet - if affected with spherical aberration, to less than 1/10 of its possible maximum. For OPD=λ/4, normalized combined amplitude is cos2(π/4), or 0.5.

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 aberrations 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.

Externally induced telescope aberrations are caused by: (1) alignment errors, (2) forced surface deformations caused by thermal variations, gravity and improper mounting, and (3) air currents/turbulence.

3.1. WAVEFRONT ABERRATIONS

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 perfect spherical wavefront, and the eyepiece turning it into a perfect 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 due to larger pupil sizes being associated with low-power observing of objects of lower brightness, 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 a telescope 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, and proceeding to pass 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.

On the other hand, 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 the deviant points of the wavefront is now smaller, and constructive energy interference greater, than in a perfect system. In other words, that the energy lost at the focal point due to wavefront aberrations will be effectively transferred toward the outer portions of the intensity 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 less efficient, and more efficient in the area of rings. In effect, wavefront deformations cause energy transfer from the central spot to the ring area, blurring the point image. This negatively affects both, image quality of point- and extended objects, the latter being merely a dense point-image conglomerates. In terms of loss of resolution, expectedly, low-contrast details are affected more than those of high inherent contrast.

The point of maximum wavefront deviation from a perfect sphere determines peak-to-valley (P-V) wavefront error. This figure is meaningless in regard 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 its form and areal extent of its deformation are known (FIG. 12). An example of such forms of wavefront deformations are those characteristic of typical optical surfaces, conics of revolution - spherical aberration, coma, astigmatism, field curvature and distortion.

The sign of the P-V wavefront aberration is determined by the optical path length: if it is larger than a perfect reference path (i.e. if the wave has to travel an extra length to reach the focus), the P-V error is positive, and vice versa. The term "optical path length" refers to a path length that the light wave travels in a given time; therefore, it is directly dependant on the speed of light through optical media and may differ from the geometric path length. This is why the error on, say, mirror surface (the medium is air), results in different optical path length - and error magnitude - than nominally identical error on the lens surface (the medium is glass).

FIGURE 12: Two nominally identical peak-to-valley (P-V) wavefront errors with dramatically different effect on image quality (dashed arc is a perfect reference sphere for the wavefront). In the first case (a), deviation affects most of the wavefront area, causing much larger overall error than in the second case (b), where nominally identical P-V error affects small fraction of the wavefront area, having little effect on image quality. This illustrates why it is impossible to tell what is the specific effect of a nominal P-V wavefront error alone. It only has meaning when associated with a known form of wavefront deformation, in which case an actual deviation averaged over the wavefront can be calculated, and used to assess the effect on image quality. Note that the P-V wavefront error in both cases is positive, since the optical path length for the aberrated points is greater than the reference sphere radius.

The extent of image deterioration caused by wavefront deformations is determined by its deviation from spherical, averaged over the entire wavefront. It is the so called root-mean-square (RMS) wavefront error, usually expressed in units of the wavelength of light. It is a square root of the difference between the average of squared wavefront deviations minus the square of average wavefront deviation, or RMS=(áW2ñ-áWñ2)1/2, with the áñ brackets indicating average value. For instance, if we measure wavefront deviations at three points (for simplicity) as 0.5, 0.2 and 0.1, the average of their squared values áW2ñ=0.1, while the square of their average value áWñ2=0.071. The RMS error would be given as RMS=√0.1-0.071=0.17. This amounts to a standard, or statistical deviation from a perfect reference sphere over the entire wavefront. To be meaningful, the RMS wavefront error has to be calculated for a large number of points on the wavefront (or optical surface, for the surface RMS).

By being an indicator of the average optical path deviation over the entire wavefront, RMS wavefront error is directly related to the cumulative phase loss at the center of diffraction pattern and, hence, to its peak intensity. For relatively small aberration levels, the relative drop in peak intensity of diffraction pattern caused by wavefront errors is accompanied by a transfer of nearly identical relative amount of the total energy to the area of rings. Type of aberration is irrelevant, as long as it is sufficiently low - roughly less than 1/6 wave RMS. This establishes direct relationship between the RMS wavefront error and the quality of point-source image.

Since the phase-related effect of wavefront deviations from spherical doesn't change with the sign of P-V wavefront error, the RMS error is independent of the sign of P-V wavefront deviations, thus always given as an absolute (positive) number.

Discrepancies between the RMS and actual wavefront quality for larger errors are mainly related to those affecting limited wavefront area. For instance, turned edge will increasingly contribute to the quality deterioration only up to a certain level, after which a further increase in the nominal RMS wavefront error will have little or no effect on image quality. For mirror edge, and 0.95% TE, this level is at about 1 wave P-V, or 0.13 wave RMS. At this point, central diffraction intensity is reduced to ~0.92. Increasing the error to 2 waves P-V, or 0.26 wave RMS, causes additional drop in the central intensity of only 0.01, down to ~0.91. Similar effect will be observed with narrow zones (which can't cause more of diffraction disturbance than a matching ring-like obstruction, no matter how many waves RMS deep), or any other type of local wavefront deviation.

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

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