An Analysis of the Impact of Focal Ratio on Off-axis Images
I have been asked with increasing frequency about the performance of short focal ratio mirrors with respect to lunar/planetary observing. Essentially, the question is will an f/4.5 or f/5 mirror give truly sharp images for planetary observing. This often even extends to questions concerning f/4 mirrors. I have generally been strongly in favor of mirrors having a focal ratios of f/6 and above; and I still believe that such mirrors, on an all-around basis, offer the most reliably high performing telescope. But this does not entirely exclude mirrors of smaller focal ratio from performing excellently. Also, the shorter ratios place great demands on the ability the user to properly collimate such a mirror so that its true optical center resides in the center of the field of view. As a response to the various in queries which I have been receiving I decided to prepare an analysis for those individuals contemplating the purchase of a short ratio mirror. I hope that the following will clearly define the problems involved. The conclusions I have reached indicate that mirrors having focal ratios a short as f/4 can be made to perform within the diffraction limit and that an individual purchasing a high-quality mirror at that focal ratio or above can, if he is scrupulous about collimation, achieve excellent results.
The primary off-axis aberration formed by a paraboloidal mirror is coma. Conversely, the primary aberration formed by most eyepieces is astigmatism. By far, the astigmatism formed by a eyepieces is usually the most noticeable and annoying aberration seen while observing. The problem with astigmatism is that it is a more concentrated aberration, usually taking the form of an elongated stellar image to the extent that it almost forms what appears to be a bar shape at the edges of the field. This phenomenon is most noticeable in wide field eyepieces when observing at low to medium powers. Coma, on the other hand, is an aberration wherein the characteristic fan shape has most of its light concentrated at the apex with the intensity dropping off as one works out toward the outer regions of the fan. In other words, much of the comatic aberration is unseen. The general effect is to produce an apparent image somewhat less imperfect when casually observed than is actually the case. This is not to suggest that the aberration is inconsequential. Deleterious effects can manifest themselves in the form of blurring of fine planetary detail. One might suggest quite reasonably that if the planet is kept in the center of the field of view this is not a problem, but the question then arises as to exactly how precisely centered within the field of the do the planet must be kept before the impact of aberrations begins to noticeably impair the image. Any perfect paraboloidal mirror will focus its axial rays precisely - absolutely precisely. Its those off-axis rays that cause the trouble. Furthermore, inasmuch as we do not live in a perfect world, and opticians like me have to actually make a mirror to high accuracy (approaching perfection) questions arise as to how perfect a mirror has to be in order to form a sensibly perfect image for any given focal ratio. When making mirrors of around f/8 and above, sensible perfection is easier to achieve than when making mirrors at f/4. Also, one should note that a 6 inch mirror at f/5 is much easier to make than a 10 inch mirror at f/5. The six inch mirror need not be as accurate as the 10 inch mirror to form a sensibly perfect image.
Before I continue with the subject of mirrors it is obvious that I have suggested that eyepieces form their own aberrations separate and apart from mirrors. Therefore, I can really only address half the problem as a maker of mirrors. One infallible rule is that all optics perform better off-axially when they are of larger focal ratio. But with regard to refracting optics that perfection naturally occurring with the paraboloidal mirror does not exist. Refracting optics simply have a greater problem performing accurately at smaller focal ratios, even for on axis imagery. You may have noticed that reflecting telescopes frequently come in sizes a short as f/4, while refractors touted as planetary telescopes really are no shorter than f/6 and many are longer than that; even the apochromatic varieties. This is the result of struggling to get lenses to perform well with tubes as short as possible. If refractor makers could make high-quality planetary instruments that f/4, they would. Binocular objectives are frequently at about f/5, as are certain varieties of deep sky refractors, but these instruments are specifically designed for low-power wide field use. The refractor can be made to correct for off-axis aberrations better than a paraboloidal mirror of equal focal ratio, but the trade-off (aside from chromatic aberration problems) has to do with the correction of spherical aberration by the use of spherical curves only. Eyepiece designers have said that they could make eyepieces perform much better if they could make at least one surface aspheric, but manufacturing costs and the difficulty of making small aspheric surfaces apparently prohibits the manufacture of such eyepieces. Optics is infinitely complex and the opportunity for juggling things around infinitely tempting and intriguing.
The following spot diagrams were made showing the effects of focal ratio on paraboloidal mirrors having the following focal ratios: f/4, f/4.5, f/5, f/6 and f/8 and represent the those ratios most commonly encountered. Further explanation is required as to why I chose spots at certain off-axis positions. Inasmuch as most of my customers are interested in lunar/planetary viewing, mostly still from a visual perspective, I have attempted analyzed the practical reality of visual observing in terms of what we actually look at, how much of the sky we look at and where we look at the object within the field of view of the eyepiece; we don't always look at the center of the field, even if we have a telescope that is mechanically driven.
Let's begin with a typical object that can be generally imagined by most people, Jupiter. Jupiter subtends an angular diameter across the sky of approximately 45 arc seconds. A comfortable field of view is one having a diameter of perhaps 6 to 8 Jupiters and that within this field a central optimum viewing area of 4 to 5 Jupiters would be acceptable. The size of an optimum viewing area would be approximately 3 to 3.75, say 4, arc minutes or, 0.05 to 0.67 degrees. To my mind this represents an area that should contain an image of high enough quality to show the planet as well as it can be seen without concern over degradation of the image. With respect to lunar observing, to form another basis of comparison, 4 arc minutes is approximately 240 miles across the lunar surface. (This is not exact but close enough for conceptual purposes.) Considering that the crater Copernicus is approximately 60 miles across this would allow an observer to see approximately four Copernicuses within the field of view. One would like to feel that having Copernicus centered anywhere within that field of view would present a diffraction limited image of the object, a fairly comfortable observing situation. Accepting this, we have established that we would like a field of view, for comfortable and reasonable observing, of approximately 4 arc minutes, or roughly 1/16 of a degree. I have purposely kept all of this to very rough measurements since over-precision would be pointless and detract from the main purpose of this exercise. The whole business of what we see and how we see it, in terms of where limit lay, is imprecise and observer opinions very greatly.
But there is more to consider. If we establish a minimum field of 1/16 of a degree we have to allow for some error in collimation of the telescope. To expect people to collimate their telescopes to perfection is not realistic. So, a fair compromise might be to allow for an error of twice that amount, or a practical working field of approximately 1/8 of a degree as a basis for aberration calculation. Added to that I have also made spot diagrams at twice that distance from the center of the field or 1/4 degree. These will form boundaries from which one can analyze the realistic impact of off-axis aberrations.
Inasmuch as we're dealing with coma as the primary aberration it is interesting to note that the impact of coma increases linearly with respect to distance from the center of the field and cubically with respect to focal ratio. Simply stated, a telescope having a focal ratio of 4 will have eight times the coma of a telescope having a focal ratio of 8.
A further complication, which will cause distress to those having very large aperture telescopes, is that for any given focal ratio and field area of the sky, coma increases respective to the diameter of the mirror. In other words, a 10 inch f/5 will have less coma at 1/8 of a degree than a 20 inch f/5. Much of the complaining connected with 20 and 30 inch Dobsonian reflectors of f/4 has to do with the fact that the mere size of the mirror is radically increasing apparent coma. For any given f ratio and area of the sky to be covered, the actual physical diameter of the diffraction limited area (defined area of aberration) at the focal plane remains the same, even as the telescope grows larger.
The following spot diagrams were generated for a 10 inch paraboloidal mirror having focal ratios of f/4, f/4.5, f/5, f/6 and f/8. The indicated off-axis amount in degrees shown above each spot is the half angle, so one should multiply times 2 to get the total field. As an aid in analyzing the spot diagrams, one should understand that the ideal is that all the light be contained within the Airy disk, or as it is sometimes known, the spurious disk. This is the area where, in a perfect optical system, approximately 82 percent of all the light should be concentrated. The remaining light is distributed around the disk in the form of concentric rings, the so-called diffraction circles. (On the very best of nights these rings can be observed with a fine optical system.) Practically speaking, the quarter wave Rayleigh criterion can be achieved with a blur spot approximately twice the size of the Airy disk. After this, the image will begin to appear degraded. Readers will also note that the spot diagram representing the on-axes image is actually blurred somewhat beyond a perfect fine point. This is the result of the computer program used attempting to optimize to the extent possible across the 1/4 degree field of view.
One can see that even a 10 inch telescope working at f/4 will deliver acceptable images and benefit from a high quality mirror as long as collimation is accomplished well enough. An f/4.5 is noticeably better and things become much more comfortable at f/5. At f/6 and above performance improves considerably.
The following spot diagrams have been made for f/5 mirrors of 10", 12" and 16".
Now that we analyzed what happens to a perfect mirror when off-axis images are viewed, the next question that arises is how good the mirror has to be to achieves such results. As an optician, I state that I am able to produce mirrors having a Strehl ratio of 0.96 or better. This roughly translates into approximately Lambda/9. Whether I'm making a mirror at f/8 or a mirror at f/5 or a mirror at f/4 my job is still the same, to make a mirror of a given a wavefront accuracy. The problem is, making a mirror at f/4 with a 0.96 Strehl is much more difficult than making one at f/6 or even f/5. It has been said that a 6 inch f/8 mirror need only be made to 50 percent full parabolization from a sphere to be diffraction limited, and this is true. A 6 inch f/4 mirror needs to be enormously more accurate. At least half of the problem involved in making mirrors to high accuracy is the ability to test to the accuracy required. The other half is to be able to actually make it. But the testing component is extremely critical, and it is for that reason that I use autocollimation testing. Without such tests I find making mirrors of small focal ratio to be a grueling exercise in eye strain and shadow guessing, a haphazard operation at best. While this limits my mirror sizes to 16 inches I feel confident that while confined to those sizes I can produce mirrors to the accuracy required to achieve the results shown in the above spot diagrams.
I hope this helps clarify some of the issues concerning small focal ratio mirrors. In general, I would say that if you must go below f/5 be prepared to learn to collimate your telescope very precisely and possess a telescope that will maintain its collimation in use. While there are a wide variety of collimators on the market I cannot emphasize too strongly that one should do the final tweaking with that ultimate collimator in the sky, the defocused stellar image. It is by that alone that one can truly ascertain the quality of the image he is getting.
R. F. Royce