Reflectors versus Refractors Optical Principles

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Even a cursory reading of the literature of visual double-star astronomy is sufficient to show that the field has long been heavily dominated by the refractor, which remains the instrument of choice for many visual observers. It is not, indeed, hard to find statements backed by the highest authority alleging that for this type of observation a reflector must be of substantially larger aperture to match the performance of a refractor of given size. For instance, van den Bos stated that a reflector must have a linear aperture 50% greater than that of an equivalent refractor. There is, however, no basis whatever in optical theory for such claims, nor, as will shortly be seen, do actual results at the eyepiece sustain this perception of the reflecting telescope as second-class citizen. This chapter will demonstrate that, and how, a reflector of good optical quality, maintained in proper adjustment, can be fully the equal aperture-for-aperture of the best refractor, matching the latter's resolution to the uttermost limits of visual double-star astronomy, at least on fairly equal pairs. It is not amiss to recall at this point that the study of binary stars was founded by Herschel with reflecting telescopes and that its current limits have largely been set by recent observations with reflecting systems, both in terrestrial speckle interferometry and in the Hipparcos orbital observatory.

Present purposes would not be served by entering into the minutiae of the apparently interminable debate over the relative merits of the two classes of instrument, but there are important differences between their respective imaging properties, and handling characteristics in real observing conditions, which must be recognised by any observer who aims to push telescopic performance to its limits. There are, accordingly, a few fundamental optical principles which must be borne in mind as the essential context for what is said later in this chapter specifically about reflecting telescopes. In particular, given the myths, misconceptions and dubious anecdotal evidence common in the "refractor versus reflector" debate, it seems appropriate to begin by stating clearly what are not the reasons for significant differences between the two types-not, at least, so far as double stars are concerned.

One such notion holds that residual chromatic aberration is a serious limitation to the defining power of refractors with simple doublet objectives, and that the reflector therefore has a marked superiority in this sense. That there is, in fact, no theoretical justification for this view in the case of any refractor of sufficiently long focus to be used for high-resolution imaging (say f/10, at least, for smaller apertures, rising to f/18 or so for large instruments) has been known at least since the work of Conrady.1 It was shown there that moderate levels of defocussing such as may be induced by the secondary spectrum in such a refractor, that is up to one quarter or even one half of a wavelength phase-lag, does not significantly alter the diameter of the Airy disk formed by the telescope, despite its intensity declining noticeably. Effectively, the chromatic dispersion of focus is lost within the depth of focus naturally allowed by the wave theory; this is the reason why image definition is so good in refractors despite secondary spectrum. The result is that resolution on high-contrast targets such as double stars is fully maintained, even if some low-contrast fine detail may be lost in planetary images. That this conclusion is fully borne out by practical experience is convincingly demonstrated by the magnificent achievements in high-resolution double-star astronomy of the best visual observers using the big refractors: one only need think of the Lick 36-inch regularly reaching 0.1'' in the hands of Burnham, Aitken and Hussey. Indeed, one of the greatest of recent observers of visual binaries, Paul Couteau, seems from the remarks in his well-known book2 to consider the secondary spectrum of refractors to be a positive advantage. Clearly, three-colour or apochromatic correction, whatever its benefits for the use of relatively short-focus instruments in planetary imaging, is for the double-star observer an expensive and dispensable luxury - the classical long-focus doublet objective is more than equal to the task required.

The effects of central obstructions, often alleged to degrade imaging quality of reflectors quite seriously compared with that of refractors, can similarly be dismissed. By blocking a small central patch of the incident wavefront, the secondary mirror of a reflector removes a minor portion of the light from that process of mutual interference at focus, which otherwise produces a standard Airy diffraction pattern. The result is that an equal amount of light which would previously have interfered, constructively or destructively, with this obstructed portion in the process of image formation must now be redistributed in the Airy pattern. It follows on simple grounds of energy conservation that the amount and location of this redistribution of light in the image is essentially identical with the intensity distribution in the image which would be formed alone by just the light that has actually been blocked - a statement familiar to all students of diffraction theory as Babinet's principle (the Complementary Apertures theorem).3 One can immediately see from this that, for the fairly small central obstructions of most reflectors, the amount of light redistributed in the image must be very small and, as the point-spread function of the obstructed central zone is very much wider than that of the full aperture (in inverse ratio to their diameters), this small amount of light is deflected from the Airy disk into the surrounding rings. It is, therefore, quite impossible for a secondary mirror blocking, say 5% of the incident light, to cause a redistribution of 20% of what remains from diffraction disk to rings, a change which would itself be near the limits of visual perception even on planetary images. This is the case of a "22.4% central obstruction" in the linear measure usually applied to discussions of this issue, and even this is decidedly on the large side for most Newtonians, at least, of f/6 and longer.

Central obstructions are not in fact the only possible cause of excess brightness in the diffraction rings nor, probably, indeed the most important single cause in the vast majority of reflecting telescopes. The effect of deviation of light from the Airy disk into the rings is quantified by the Strehl ratio, a parameter commonly used as a measure of imaging quality and as a basis of optical tolerance criteria, which is the peak central intensity of the diffraction pattern actually formed by an instrument, expressed as a fraction of that of the ideal Airy pattern appropriate to the case. The essential point here is that any small deformations, W, of the wavefront converging to focus, whether arising in the telescope from surface errors of the optics or from aberrations, will reduce the Strehl ratio and so cause the kind of effect commonly attributed to "central obstructions". According to Marechal's theorem, this deviation of light from disk to rings is proportional to the statistical variance (mean square) of the wavefront deformations, W, thus:

Strehl ratio = 1-

This approximation holds for W values up to about the Rayleigh "quarter-wave" tolerance limit and in that range is independent of the nature of the wavefront deformations. More than half a century after Marechal's discovery it is extraordinary how little-known this fundamental result4 appears to remain in the practical world of telescope users and makers.

In particular, it turns out that spherical aberration (SA) in small doses mimics the diffraction effects of central obstructions particularly closely, putting extra light into the rings, while leaving the size of the Airy disk unaltered. With SA just at the Rayleigh limit, Marechal's theorem shows that the Strehl ratio will already have dropped to 0.8, an effect fully as large as that of a 30% central obstruction. The conclusion is that, unless a reflector is of very high optical quality and very precisely corrected, or has an exceptionally large secondary mirror (or both), any effect of the central obstruction will be swamped by that of SA, to say nothing of other aberrations and optical errors. This is particularly significant in view of the prevalence of residual SA in reflecting telescopes: plate-glass mirrors tend to go overcorrected in typical night time falling temperatures, so older optics even from profes-

sional makers are often undercorrected, deliberately; the absence of a simple null-test for paraboloids, and the acquired skill necessary to interpret accurately the results of the Foucault test at the centre of curvature, mean that amateur-made mirrors are often only very approximately corrected; and Cassegrain systems, such as the ubiquitous SCT compacts, which focus by moving one of the main optical elements, necessarily introduce correction errors for all settings except that in which the principal focus of the primary mirror coincides exactly with the conjugate focus of the secondary. For a very interesting field survey of the effects of residual correction errors on performance of reflectors see reference 5. A further point here is that SA is proportional to (aperture)2/focal length, so the claim that the "cleaning up" of the image in a typical reflector by use of an off-axis unobstructed aperture proves that the secondary mirror is responsible for the less-than-ideal image at full aperture is obviously a misinterpretation of the evidence: simply by stopping down, both SA and "seeing" effects are drastically reduced, naturally giving rise to the observed changes in image quality.

These conclusions are entirely vindicated by practical experience. In the 12.5-inch (0.32 m) f/7 Newtonian with whose star images this author has been intimately familiar since the 1960s, increase of the normal 16% central obstruction to 32% has no perceptible effect on the diffraction image of a first magnitude star, although the brightening of the rings has become very obvious at 60% obstruction. Again, a deliberate trial of this question was made by side-by-side star tests, on the same bright star, of a 4-inch refractor and a 6-inch Newtonian having 37% central obstruction. With both instruments showing a beautifully defined Airy pattern at x200, the greater relative intensity of the rings in the reflector was so small as to be barely detectable even after many rapidly alternated comparisons. It should be noted that even this rather large obstruction only stops about 1/7 of the incident light.

In short, the unavoidable presence of a central obstruction in most reflectors does not limit their resolution, or make it inferior to that of refractors of equal aperture. On the contrary, by stopping out the centre of the mirror, the mean separation of the points on the incident wavefront is increased, thereby decreasing the size of the Airy disk which arises from their mutual interference, so the resolving power of a reflector on fairly equal double stars is actually greater than that of a refractor of the same aperture, other things being equal. In truth, this last effect is almost negligible for central obstruction much below 50% but it may surprise some readers to learn that for the highest resolution on equal pairs this author deliberately stops out the central 72% of the telescope's aperture - a 9-inch central obstruction on a 12.5-inch reflector! (None of the double-star results given later were dependent on this trick, however.) Of course, such doubles are extreme high-contrast targets and therefore react quite differently to such treatment, compared with planets or even unequal double stars, whose resolution would be seriously impaired by this tactic.

To bring this discussion to its conclusion, the real differences between refractors and reflectors which are important for high-resolution imaging of double stars are very simple and very fundamental: refractors refract, while reflectors reflect and refractors do this at four (or more) curved optical surfaces as against only one in a Newtonian. These two facts are so obvious that they are often ignored but they are, far more than any other factors, truly the crux of the matter in comparing the optical performance of the two main classes of instrument.

That image-formation is, in the one case, by refraction, and, in the other, by reflection has radical implications for the relative immunity of the refractor from image degradation due to surface errors of the optics, whether arising from inaccuracy of figuring, thermal expansion or mechanical flexure. Thinking in wave terms, one can say that the function of a telescope's optics in forming a good image of a distant star is simply to cause rays from all points of the plane wave-front incident on the aperture to travel exactly the same number of wavelengths (optical path-length) in arriving at the focus, so that they may interfere constructively there and form a bright point of light. That is all there is to image formation in the wave theory, whether by refraction or reflection (and this is precisely why results like the Airy pattern and Marechal's theorem arise) - arrival in phase of all rays at focus. The refractor achieves the necessary phase delay of the near-axial rays, relative to the peripheral rays which must follow a longer route to focus, by intercepting them with a greater thickness of dense optical medium to equalise axial and peripheral optical path lengths. That is to say, the telescope uses a convex lens. The reflector attains exactly the same result by bouncing the axial rays back up to focus from further down the tube than the peripheral rays, that is, it uses a concave mirror.

It immediately follows that this differential phase-delay, and hence quality of image, is dependent on the thickness of the objective at any point relative to that at its edge, in a refractor, but on actual longitudinal position of the mirror surface relative to the edge, in a reflector. Further, errors of glass thickness in the first case only cause optical path-length errors (^ - 1) times, or approximately half as great, while errors of surface in the second case are doubled on the reflected wave-front, as such errors are added to both the to and fro path length. Consequently, to achieve any particular level of wavefront accuracy var (W), and thus image quality (cf. Maréchal's theorem, above) in a reflector requires optical work roughly four times more accurate than in the case of a refractor and, for exactly the same reason, the latter is about four times less sensitive, optically, to uneven thermal expansion of its objective. Lastly, because mechanical flexure does not alter thickness of an objective in first approximation, while it has an immediate and direct effect on the local position of surface elements of a mirror, refractors are hugely more resistant to the optical effects of flexure.6'7

That refractors share the work of focussing light between at least four curved surfaces, compared to only one in a Newtonian, is equally fundamental and takes us to something which will be the central theme of the next few pages: optical aberrations and their avoidance or management. The requirement that a curved mirror surface return all rays incident parallel to the optical axis to focus with equal optical path lengths, so forming a fully corrected image there as discussed above, is alone sufficient to determine uniquely the form of that surface. A very simple geometrical construction shows that the mirror must be a paraboloid of revolution. In other words, the requirement that axial aberrations, specifically SA, be zero defines the optical configuration uniquely and leaves no adjustable parameters free for reducing or eliminating off-axis aberrations (apart, trivially, from the focal length). The result is that all reflectors, Newtonian, Herschelian, or prime focus, having only one curved optical surface, necessarily suffer from both coma and astigmatism. Unless other adjustable optical surfaces are introduced into the system, nothing can be done to mitigate the full force of these off-axis aberrations and, as will be seen in the next section, coma severely limits the usable field of view of all paraboloid reflectors and makes them hypersensitive to misalignment of the optics (collimation errors). A refractor objective, by contrast, possesses at least four independently adjustable curvatures and opticians have known since the time of Fraunhofer how to use this freedom to eliminate both the axial aberrations and coma, in the so-called aplanatic objective. (The need for multiple-surface adjustability to minimise aberrations is, of course, the reason why all short-focus wide-field imaging units such as camera lenses and wide-field eyepieces must have four or more components.) Most quality refractor objectives are nearly or quite apla-natic, leaving only astigmatism as the factor limiting field of view, a very much less serious constraint which leaves most refractors with a far larger field of critical definition and far less sensitivity to collimation errors than all Newtonians, at least. Compound reflectors such as Cassegrains or catadioptrics represent a halfway stage in this sense between Newtonians and aplanatic refractors but most of these pay the price of decreased (rarely eliminated) coma in increased trouble from SA. Coma arising from miscollimation in reflectors is perhaps the most obnoxious of all aberrations to the double-star observer, as it rapidly destroys the symmetry and definition of the star image: even a quarter wave of coma, that is just at the Rayleigh tolerance, is quite sufficient to make the diffraction rings contract into short, bright arcs all on one side, an image distortion quite unacceptable for critical doublestar observation - see Figure 11.1.8

What all of this amounts to in practice is that a reasonably well-made Fraunhofer achromat is a hugely more robust instrument than a typical reflector in the face of the thermal variations, mechanical flexure and shifting collimation which commonly arise in real observing conditions, and so can be relied upon far more than the comparatively delicate, fickle reflector to deliver critical definition at a moment's notice with minimal cosseting and adjustment. It is also more likely to meet the optical tolerances necessary for such diffraction-limited performance. These are the reasons why the refractor has so often been the first choice for observers of close visual binaries.

However, as will be seen shortly, none of this implies an inevitable inferiority of the reflector in this field of

astronomy, for good optics and proper management of the instrument will easily hold in check all those adverse factors to which the reflector is more sensitive, to an extent quite sufficient to deliver star images equal to any seen in a refractor. (With the possible exception of some enhancement of the diffraction rings in reflectors exhibiting residual SA. If this is the only fault, the telescope will perform just as well on equal double stars but faint companions may be swamped. For this reason, a good refractor will often outperform a reflector on contrasted pairs even when the two instruments are absolutely matched on equal doubles.) All the supposed optical defects of the reflector are removable or fictitious and, of course, a good 0.3-m reflecting telescope is a far less expensive item than an equally good 0.3-m refractor! For reaching the observational limits, however, the unrelenting emphasis must be on quality optics and their proper management, in particular to maintain accurate collimation so that all highpower images may be examined truly on axis, free of the dreaded coma. This is the subject of the next few sections. What follows is largely based on experience with a Newtonian reflector, with which this author has done most of his double-star astronomy, but results comparable with those reported here are probably within reach of good longish-focus reflectors of virtually any type, given the same aperture.

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