1.1 Telescope Evaluation
I'm going to tell you a little-known fact. Telescopes are easy to test. All that is required is a good high-magnification eyepiece, a conveniently placed star or illuminated pinhole, and some experience. In fact, telescopes are so easy to test that I recommend you check the optical train of your telescope every time you use it, as a monitor of alignment, proper optical support, and atmospheric conditions.
You will notice that the methods proposed in this book are a great deal different from the methods telescope makers use. Telescope fabrication, as contrasted with telescope evaluation, requires that corrective measures be indicated. Hence, telescope makers favor methods that lead to profiles of the optical surfaces, or at least the errors in the slope of those surfaces. They can use such information to decide where they must remove slight amounts of glass during the next figuring step.
Such methods demand specialized equipment, and more importantly they require skills best learned at the elbow of an expert. Optical shop tests tend to require delicate visual interpretations, and someone usually has to physically demonstrate the test to a person unfamiliar with it.
Telescope evaluation, on the other hand, is a test of a finished product. It requires little or no specialized equipment. The evaluator cares little how an optical piece can be improved. The test is purely a yes-no decision: Is the optical train good enough to transmit the image or is it not? The evaluation encompasses the entire optical path, even those elements not customarily included in the telescope, from the top of the atmosphere to the eye of the observer.
The star test evaluates telescopes in their final configuration, doing precisely what they were intended to do. The star test is not easily reduced to numbers, but it is very sensitive. A telescope that "passes" the star test need not be evaluated with a bench test. It has already met the most stringent criterion necessary to deliver beautiful images.
The star test is an examination of the image of a point source, most commonly a star, both in focus and on both sides of focus. The power of the star test is contained in this simple motion of the eyepiece to examine the expanded diffraction disk on both sides of focus. Not only is the expanded spot bigger and hence easier to see (the focused diffraction disks of most astronomical telescopes are tiny), it "unfolds" into a unique representation of the aberrations that caused it. In particular, an out-of-focus circular image appears similar at equal small distances inside and outside the focal point but only if the optics are excellent.1
The expanded disks will appear different if any aberration is degrading the system. The sensitivity of this test is phenomenal. My first mirror was a 200 mm f/6 reflector that regularly shows excellent images of the planets, yet still just barely passes the star test. It shows some inclination towards overcorrection, roughness, and a turned edge.
The telescope is automatically prepared to do the star test and the test is conducted with all optical elements in place. Many of the optical defects discussed in this book have nothing to do with errors on the glass, and would not even have been detected in the shop tests during fabrication. For example, the 200 mm reflector mentioned above is carefully held in a 9-point mirror support, and still shows some slight evidence that the mirror sags in its cell.
You will grow to rely on the star test as a check on alignment. A quick turn on the focuser is all that's needed to verify that misalignment is not disturbing your images. You'll come to depend on the appearance of outside-focus images to see what is happening in the upper atmosphere. When good seeing comes and goes, you can shift that evening's observing schedule to take advantage of superior tranquility.
You may rightly suspect that telescopes are not easy to make. The objective, or main optical element, of an astronomical telescope contains the most accurate macroscopic solid surfaces yet shaped by humans. Typical
1At this point, terminology should be defined to avoid subsequent confusion. In all further discussion, an image is said to be "out of focus" if it is defocused in either direction. "Inside focus" means that the eyepiece's focal plane is placed between the main optical element and focus; "outside focus" indicates that the eyepiece is withdrawn beyond focus. "In focus" and "focused" are used as synonyms, meaning that a pointlike object is focused to its minimum size.
tolerances are a thousand times smaller than the usual accuracies of a metal-cutting lathe.
As a means of comparison, let us imagine the surface of a common 200-mm (8-inch) telescope mirror expanded to 1 mile (1.6 km). If this mirror had the usual thickness ratio of small mirrors, it would be 880 feet thick (268 m). In common metal-shop practice, it is normal to machine such an 8-inch surface to a thousandth of an inch, or to a scaled accuracy of 8 inches. A wavelength of light expands from 0.000022 inches to 0.17 inches (4.4 mm) at this scale. The maximum optical error tolerable on such a surface would be only 0.55 mm, or about 2/i00 inches. Premium optics would be made to an accuracy of less than 0.01 inches (0.25 mm)—a playing-card thickness error on a disk a mile across and 300 yards high.
Clearly, testing such surface accuracy is not trivial. Common calipers, a measuring device of machinists, have a maximum accuracy of only about 10 micrometers (pm), or about 20 wavelengths of green light. Even if sufficiently accurate calipers were possible, one would have the additional problem of repeatedly placing them on the curved surface. The spread of measurements would greatly exceed the inherent accuracy of the measuring tool.
Obviously, some device or technique which can sense micro-deformations on the surface is needed. Light itself is the most appropriate tool. Less certain is the precise manner to use light to bring out these defects without demanding the same stringent measurements as gauging the surface profile by physical contact.
For example, it is not difficult to come up with completely useless ways to measure surfaces. One could place a point source of light (say, a pinhole) to one side of a mirror (see Fig. 1-1) and reflect it to a spot on a screen. By moving the mask around over the objective, we could see how the spot was forced to shift. In principle, this method would contain all the error information, but it is not the easiest way of proceeding. The spot is fuzzy, the system is difficult to align and focus, and the measurements are difficult to reduce because they are taken far from the optical axis. The position of the spot is dominated by the overall curvature of the surfaces and the motions of the mask, rather than the interesting deviations from the curvature. The way to perform an accurate and simply interpreted test eliminates first-order difficulties such as the innate curvature of the surface. It is best done along the optical axis either near the focus or the center of curvature of the lens or mirror.
Accurate tests are possible when the strengths of the testing geometry are exploited. The useless test in Fig. 1-1 becomes the very accurate caustic test when the image spot is moved near the center of curvature and the
sensor is fitted with a knife edge or wire.
Accuracy of surface shape is a function of the stiffness of the material and the definition of just what surface accuracy means. For example, most machinist's gauge blocks and flat surface plates, the absolute standards by which they check their measuring tools, are rectangular chunks of steel. Steel is a fairly inflexible material, and machinists think of steel as capable of holding its shape. These blocks are more than accurate enough for the kind of precision demanded in machining as long as the temperature of the room does not vary too much. For optical use, however, the scale of precision is much smaller.
Microscopically, a block of steel expands under heating. Its linear dimensions depend on the ambient temperature. This change wouldn't ordinarily affect optical performance—after all, a sphere is still a sphere even if it has a slightly different radius. But as temperature varies, portions of the metal change faster and internal stresses build up. The surface slightly buckles as the temperature of the piece non-uniformly follows rapidly changing outside temperature.
Likewise, glass deforms under heating. Although it swells less than metal, the thermal conductivity of glass is lower and it has a problem getting rid of heat. Mirrors are typically coated on one side with metal, which complicates the way they radiate energy. The temperature of glass is a complex function of thermal radiation, convective air cooling, and conduction through the relatively few points at which the optical disk is supported.
Glass not only changes shape under heating, it deforms under pressure. A helpful way of viewing glass disks at the wavelength scale is to think of them as sheets of rubber. If you push on the top, the surface goes down. If you incorrectly support the bottom, the whole disk sags and the upper surface will deform. Thin pieces are harder to hold flat than thick pieces, large pieces more difficult than small ones. The cells that hold optics must not pinch or warp them.
No process is as stressful to optics as fabrication. Simple grinding even has a few pitfalls. Say that a mirror disk has a slight amount of cylindrical curvature on its rear side—it rocks on the backing instead of sitting flat. Under the pressure of grinding, it will deform away from its "backbone" and suffer astigmatic curvature when the pressure is relieved.
The fabrication stage at which most errors originate, however, is polishing. It takes place on polishing pitch, a material that no one fully understands. Some experienced opticians have learned the limits and general behavior of pitch, but even after years of experience, they are often surprised by its unstable nature.
Pitch is a highly viscous fluid used in a thin layer (3 to 6 mm) covering a disk used as a tool. This lapping surface (or "lap") is usually crosshatched with grooves that allow the fluid to spread and conform more readily to the polished surface. Pitch will behave more-or-less as a solid at fast speeds and as a liquid at slow speeds. For example, if you hit it with a hammer, it shatters. Lay a stick of pure pitch across the lip of a bowl, however, and eventually you'll find it has run to the bottom.
During polishing, powdered abrasive grains sink into the pitch surface where they are held as microscopic scrapers. Certain polishing agents are more effective than others, and cause more heat to be generated in the lap. The resistance to deformation in pitch varies strongly with temperature. Its characteristics on the outer portion of the disk will vary markedly from those of the inside because wet pitch on the periphery is exposed to the air and cools more rapidly by evaporation. If polishing is stretched too long, the pitch tool dries out, becomes overheated, and loses its shape. Bad polishing habits will result in excessive wear at the edge of the optical disk, giving it a run-down appearance under sensitive testing. Not varying the stroke when using a machine may cut shallow circular channels in the optics. In the case of fast aspherical optics, more polishing must be applied on the center of the disk. Clearly, many opportunities exist for errors to find their way to the optical surface.
One way of gauging optical quality is to measure the peak-to-valley wavefront error. A wavefront is a line traced along the crest or trough of a wave. In regions far from focus, a wavefront is perpendicular to the direction of wave motion. Using a convenient example, a wavefront is the crest of a surfer's wave, parallel to the beach. The wave is moving toward the beach, at right angles to the crest.
A perfect converging wavefront is part of a sphere with its center at the focus. The light from a point source converges to the minimum spot size, called the "diffraction disk." After passing through an optical system with errors, a wavefront departs from the spherical shape, and the image spot will be larger and less intense.
Imagine two spheres with a common center at focus, somewhat like the layers of an onion. The outer layer touches the point that lags furthest behind the real wavefront and the inner one touches the point closest to the sphere's center. The different radii of these spheres defines the total wavefront error. (See Fig. 1-2.)
J.W. Strutt (Lord Rayleigh) stated an often quoted rule: If the total wavefront error—peak-to-valley—exceeds '/4 wavelength of yellow-green light (550 nm), then the optics begin to noticeably degrade. The reason that the image begins to fall apart is simple—a significant portion of the converging wavefront now has a phase mildly "disagreeing" with the majority. Rayleigh's rule is not a hard limit. Some people don't easily perceive diminished quality until total wavefront error exceeds '/.3 wavelength. Others are more discriminating, detecting degradation at '/8 wavelength and below. Much of the sensitivity to optical faults depends on the type of observing, the type of error, and the sophistication of the observer (Ceravolo et al. 1992; Texereau 1984).
Below are some variations of the ways that different optical shops state the same 1/4 -wavelength quality:
• " 1/8-wavelength surface. " Say the primary mirror has a VS-wavelength bump on it. The incident wavefront reflects from the peak of the bump while the adjacent portion of the wavefront is forced to travel to the base of the error. This section of the wavefront moves 1/8 wavelength going in and 1/8 wavelength coming out, leaving it 1/4 wavelength behind the leading edge of the wave.
• "±1/16-wave surface." If the same VS-wavelength hill is not measured base-to-crest but measured from its average position, the error
is spuriously divided by two yet again.
• "1/27-wave RMS on the surface." This widely-used measure in the optics industry is known as Maréchal's criterion (Born and Wolf 1980, p. 469). A 1/4 wavelength of spherical aberration (a large-scale departure from a sphere) approximately translates to a 1/14-wavelength RMS Maréchal criterion (Maréchal 1947). Measuring it on the surface cuts it in half again.
• 1/31-wave RMS He-Ne laser light surface accuracy. " The red light of a helium-neon laser has a longer wavelength and the same error appears smaller. The easily calculated transition to visual wavelength has not been made.
Nearly every telescope user has a hazy memory of reading about the 1/4-wavelength Rayleigh tolerance. The accuracies above apparently exceed the 1/4-wavelength limit with room to spare, but they are different descriptions of the same 1/4-wavelength tolerance.
These differing claims of surface accuracy are not inherently dishonest, as long as they are given in sufficient detail that one can pick apart their meanings. In fact, the 1/14-wavelength RMS Maréchal tolerance is superior to the Rayleigh limit because it quantifies the fraction of the wavefront that is bent away from a perfect sphere. But these accuracies are seldom written with decoding instructions, and consumers are left to wonder what the claims mean, even if they know the distinctions.
Advertising claims by commercial firms have therefore been in a confusing state for some time. In recent years, a certain fraction of consumer telescope makers have sensibly avoided the whole question of assigning numbers to their optics. They merely state that their optics are diffraction-limited and let it go at that. Such a designation is better than the artificially inflated claims above. "Diffraction-limited" conventionally means that the V14-wavelength RMS Maréchal limit has been met (Schroeder 1987).
Another factor often neglected in statements of optical quality is the slope of the error. If sharp channels, turned edges, or roughness appears on the optics, the overall wavefront error can often be contained within the expansive Rayleigh tolerance. The anomalous slope does not persist over the whole aperture. The sharply sloped fault still diverts light out of the central spot to pollute the rest of the image, but the optics are still "officially" perfect.
A. Danjon and A. Couder addressed this topic in their book Lunettes et Telescopes (Danjon and Couder 1935, pp. 518-522). They noticed that some instruments could slip by Rayleigh's limit yet possess enough surface roughness to scatter a hazy glow through lunar and planetary images. They stated that optics could not be judged as "good" until two conditions were simultaneously met:
1. Over the greatest part of the aperture, the wavefront has a mild slope and does not divert light rays outside the diffraction disk.
2. The Rayleigh 1/4-wavelength tolerance is everywhere obeyed, and over most of the aperture, deviations should be appreciably less.
Condition #2 is just the Rayleigh limit, with a verbal warning having the same goal as limiting the RMS deviation. After stating these two conditions, Danjon and Couder point out that condition #1 on the slope of the mirror is typically more difficult to meet than the 1/4-wavelength condition. Even though both wavefronts in Fig. 1-2 are within the Rayleigh tolerance, the wavefront in Fig. l-2b would do the best imaging because it is more gently sloped.
Incidentally, for aberrations that smoothly change over the whole aperture (such as the error sketched in Fig. l-2b), the maximum wavefront departure that leads to condition #1 is closer to 1/7 wavelength. Thus, optics that truly satisfy both conditions are not only good, but excellent.
Another number, commonly used as a criterion of optical quality, is the Strehl ratio of the aperture (Born and Wolf 1980, p. 462; Mahajan 1982). The Strehl ratio is defined as the intensity of the image spot at its central brightest point divided by the same image intensity without aberration. The 1/4-wavelength Rayleigh tolerance on spherical aberration causes a drop of the Strehl ratio to the value 0.8. The Strehl ratio is 1.00 with perfect optics. Marechal's criterion on the RMS aberration came from noticing that it leads to the same decrease in the Strehl ratio.
The most complete, though expansive, way of indicating optical quality is to present the detailed modulation transfer function (or MTF), which is the ability of an optical system to preserve the contrast of bar patterns of various spacings. It will be the method used in this book. No optical difficulty can escape the MTF. Dusty optics, pits on the optical surface, spider diffraction, telescope vibration, microripple, aberrations, and obstructions of any sort reveal themselves in a lowered transfer function. MTF charts have the advantage of giving the spacing of detail that the optical problem attacks.
Other equally valid measures of optical quality could easily be defined. They will be discussed in more detail in Chapter 10.
1.3 The Star Test—A Brief Overview
Observers rightly regard an out-of-focus instrument as nothing more than a problem to be cured. A telescope is either in focus or is almost useless—at least for the job it was meant to perform. When used properly, a telescope must be focused as accurately as possible.
Implicit in the customary use of a telescope is the fixed objective assumption, which regards the image produced by the objective as the whole purpose of the telescope. The eyepiece is relegated to the secondary status of a mere accessory, a magnifier. It slides along the optical axis and has only one correct setting.
The star test uses the telescope in a new way. We must assume the eyepiece has a fixed position. From this viewpoint the eyepiece's field plane is seen as the whole purpose of the exercise. The eyepiece is regarded as always being in correct focus and examines whatever occupies its field plane. The field plane is usually constricted by a sharp-edged mask called a field stop. If you invert an eyepiece and look in the bottom, this stop is usually visible as a ring inside the base. The field stop is the crisp edge you see in an eyepiece. This edge has nothing to do—as it first seems—with the boundary of the objective.
Figure 1-3 shows an idealized eyepiece. We regard the eyepiece as fixed and the objective as mobile.
If, as in the top of Fig. 1-3, the objective is located at precisely the right distance to put an image of a star on the eyepiece's field plane, the instrument is said to be in focus. The light coming from a point source leaves the rear of the eyepiece in a parallel bundle. Fig. 1-3, bottom, shows an out-of-focus instrument. Here the path of the rays depicted by solid lines exits the tube in a converging bundle that is not focused properly on the retina unless the eye's powers of accommodation are very large.
The "looking path" of two points on the out-of-focus disk is denoted by dotted lines. One imagines that the bundle of light is sliced neatly at the focal plane of the eyepiece and that the eyepiece images this slice perfectly. If the eyepiece is moved to and fro across the position of focus, each slice can be examined in turn, and the memory of all such slices forms a collective record of the behavior of light near the focus. We are allowed to use this viewpoint because the in-focus location of the eyepiece is no more special than an out-of-focus location.
Almost everywhere, the complicated situation of a converging wavefront can be approximated by replacing the wavefront with little "arrows" moving perpendicular to it. These arrows are called light rays and the intensity of such a beam can be calculated as the cross-sectional area of the ray bundle. However, elementary geometry used on a converging light cone leads to an important breakdown in the ray approximation. If a certain amount of power is incident on an area of aperture, the intensity can be calculated as this power divided by the area.2 Halfway to focus, the area of the cone has shrunk to 74 of its value right against the aperture, but it still contains the same power—so the intensity has increased 4 times. Move halfway again, and intensity further increases 4 times to a factor of 16 greater than it was at the aperture. You can double it again and again until you get to focus. What happens there?
The ray description has the area of the cone go to zero as the light approaches focus. This area must be multiplied by the intensity to make the power the same as it was all along the path. Because the speed of light through air is uniform, the energy content of the beam neither increases nor decreases. The ray approximation says that if the optics are perfect, the intensity of an image point is infinite. Needless to say, infinite intensity is impossible.
During the two hundred years between the invention of the telescope and the final acceptance of the wave theory of light, people actually believed there was no limit on optical quality. If optics were made of exquisite quality, the central spot would shrink in size—or so opticians thought. They must have agonized when their optical masterpieces, on which they had worked so diligently, still showed that disk surrounded by a system of rings.
We now know that there is a fundamental limit to imaging. Diffraction softens the image in the region of focus. For a given telescope focal length, the central spot (called the Airy disk) decreases linearly in diameter for larger apertures. The formula for the radius of the Airy disk with aperture diameter D and focal length f is
Thus, the diameter of the diffraction disk is 11/im (0.0004 inches) for a 150mm (6-inch) f/8 and 5.5/^m for a 300-mm (12-inch) f/4. Since 4 times the light was intercepted by the larger telescope and it was squeezed inside 74 the area, the larger telescope has a central image intensity 16 times as bright. A focused diffraction image appears on the left side of Fig. 1-4—the whole square is (201/)/D across.
To those accustomed to purely symbolic equations, the factor 1.22 in the expression for the Airy disk seems messy and inexact, but it is unavoidable. Its source is the circularity of the aperture. If we were to make a square objective of distance x on a side, the brightest portion of the diffraction spot would be a little square with side dimension (2%/)/x. Similarly, apertures
2 Radiometrically, this quantity is not the "intensity" at all, but should be called the "radiant flux density." However, this term is conventional among physicists.
with aberrations and obstructions have their own unique diffraction spot sizes. The Airy disk is nothing special, save that it is the perfect diffraction disk of the circular window that so many real optical devices resemble.
Diffraction is an angular effect. The angular blurring of the image is not lessened by increasing the focal length of the telescope, or equivalently, the magnification. If one doubles the focal length, the Airy disk also doubles in size. Until this independence of fundamental image blurring on focal length was appreciated, telescopes were commonly specified by their focal length instead of their apertures. Nowadays, such terminology appears quaint.
The central spot is not the whole story. Thin, ghostly rings encircle the bright spot. With large instruments under ideal conditions, they are observable out to 3 or 4 rings, but stars in small telescopes show only one ring readily.
The out-of-focus pattern of Fig. 1-4 is channeled by some circular furrows. These are diffraction rings, too, although their location and magnitude are not a simple matter to calculate. The first conjecture would be that the dark lines were the previously hidden structure of the diffraction image that is now exposed because the expanding disk filled it with light. However, this reasonable guess is wrong. The images don't behave that way at all.
As the focuser is racked in or out, the grooves continue to appear at the center and move outward, like ripples spreading from a pebble dropped in a puddle. That central point successively dims to blackness and then lightens to become the brightest part of the disk. It does so each time it creates
a new ring. The one seemingly unchanging feature of the expanding spot is the broad outer ring. It seems narrower as the focuser is turned, but it never goes away.
Notice another feature. The expanded disk is channeled, but its average brightness is more-or-less constant. The outer ring is somewhat brighter to average out that dark ring just inside of it. Still, except for the slightly brighter outer ring, the disk is remarkably flat. This principle is even more true for white-light diffraction images. Each contributing color exhibits a different number of rings in its expanded disk. The minima of one color sit on top of maxima of another color, and the net effect is largely to wash out any variation of the interior of the disk. Far out of focus, unobstructed telescopes display a flat disk with a clearly defined outer ring separated from the interior by a dark ring. Only a hint of grooved structuring exists inside.
Finally, one last characteristic is fundamental to the star test. The inside-focus disk is approximately identical to the outside-focus disk. No circularly symmetric aberration can appear the same on both sides of focus. If the patterns are similar, and if they are circular, the optics are nearly perfect.
Fi gure 1-5 shows an actual photograph of the defocused situation calculated in Fig. 1-4. The contrast of the pattern has been increased by using one very pure red light from a helium-neon laser reflected in a small reflective sphere. A hole punched in metal and placed over a small refractor created the aperture. The theoretical pattern reproduces the real behavior, even showing the terracing of the outer parts of the disk.
The appearance of a defocused image seen in Fig. 1-5 is rare. Aberrations or other optical difficulties conspire to destroy this perfection. By moving the eyepiece inside and outside focus, you can detect and possibly identify the problems that disturb your telescope. Comparing images seen at the same distances inside and outside focus is particularly powerful. The differences between these patterns will betray one of the most common optical errors, spherical aberration.
You will seldom see any one aberration unadorned by a mixture of other optical effects. The diffraction pattern is hard to diagnose using the star test without external information. Diagnosis is not the point, however. You may decide after inspection that one of the difficulties described here is dominating your system. You should regard identification of the problem only as an interesting fact if the mistake is ground into the glass or use it as a guide for telescope or site modification if the errors are correctable.
As you read this book, you will learn the appearance of the best image possible, both in and out of focus. That appearance does not vary. It may be modified by obstructions, but the effects of a secondary are predictable.
A good stellar image has a short list of identifying characteristics:
1. The in-focus stellar image is circularly symmetric; it has a dim ring hugging the outer perimeter of the diffraction disk, and rings beyond that are vanishingly dark. (Bad optics have rings too, but they are bright and too many of them can be counted. They are often asymmetric.)
2. The out-of-focus image is circularly symmetric; it is identical for all equal distances on either side of focus. (The images of aberrated apertures can be one or the other, but they are not both identical and round.)
3. The out-of-focus image has a fairly flat distribution of intensity along the radial direction, except for a slightly brighter outer ring. It is divided by diffraction grooves, but they are of very low contrast. They are mostly washed out in white light (except for the inside of the outer ring and the outside of the secondary shadow, if any).
4. If a central obstruction is used, its shadow reappears during defocus at equal distances on either side of focus.
You will also learn a systematic way of identifying optical errors and see image intensity patterns calculated for known amounts of aberration. Using this information, you may be able to estimate the size of your own telescope's aberrations and act on them if they are severe.
It is also useful to change viewpoints. Many useful concepts and procedures in modern physical optics can be used by observers to more fully understand their instruments. The first and greatest of these concepts is to view the telescope as a generalized filter. We can then use ideas developed for the electronics industry, with a few modifications changing the terminology to optics. The second concept is the idea that light is a wave. Diffraction causes fundamental limits to image quality. Using the viewpoints of wave optics and filtering theory,3 we are forced to rely less on the questionable store of folk wisdom, mythology, and belief that has accumulated around telescope use.
3 This way of looking at the process of imaging is called Fourier optics.
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