M

Fig. 2-1. Central Obstruction. Images from three perfect telescopes are shown from inside focus (left) to outside focus (right). Typical behaviors of (top) refractors, (middle) long-focus Newtonians, (bottom) commercial Schmidt-Cassegrains.

First of all, it doesn't really cause that big a difference. Those dips are extremely delicate and they are destroyed by nearly anything (as in mixing together colors in white light). Except for the minor brightenings and the shadow at the center, the disk is still fairly uniform, and it is still identical on opposite sides of focus.

For the purpose of diffraction calculations, we are allowed to pretend that the shadow is a miniature aperture, only out of phase with the larger opening (called Babinet's principle—Hecht 1987, p. 458). This tiny blockage has a much larger diffraction pattern, just as if the telescope were stopped down to the secondary's size. But remember, it is imbedded as a tiny correction to the larger pattern surrounding it. Thus, the small shadow's diffraction pattern is not visible as a distinct entity, but it causes enough coarse changes to destroy the lace-like variations seen in the out-of-focus disk of the unobstructed aperture (see Fig. 1-4 or 1-5).

The other feature of note in Fig. 2-1 is the little bright spot at the center of the out-of-focus shadows. Interestingly enough, the successful observation of such spots is part of the experimental confirmation of the wave theory of light. When Fresnel first presented his paper on the wave theory to the French Academy in 1818, one of the listeners was S.D. Poisson, a vehement opponent of the light-as-waves description (he was a propo unobstructed unobstructed

Fig. 2-1. Central Obstruction. Images from three perfect telescopes are shown from inside focus (left) to outside focus (right). Typical behaviors of (top) refractors, (middle) long-focus Newtonians, (bottom) commercial Schmidt-Cassegrains.

nent of the light-as-particles ray theory). He used Fresnel's new theory of diffraction to show that a bright spot should appear at some locations in the shadow of a circular obstacle. Poisson thought that such a ridiculous conclusion would settle the nonsense about light waves once and for all. Imagine his chagrin when this spot was found soon after. In fact, it had been discovered long before, and the report had escaped notice by optical theorists. This little bright patch is still called Poisson's spot.

The spot appearing here is a composite of the central spot expected from the full aperture and the inner negative aperture. However, a Poisson spot is observable without even using an outer aperture just by carefully arranging a circular shadow in a beam of light diverging from a pinhole.

Fig. 2-2. Misalignment. A severely misaligned reflector with a secondary obstruction of 30%. On the left is a focused image and on the right is the same image defocused. The magnification of the focused image is 6 times that of the defocused one.

2.2.2 Misalignment

Misalignment can be exposed by star testing. In fact, before testing can be continued much further, the optics must be properly collimated. This procedure is described in more detail later, but Fig. 2-2 depicts a typical misaligned Newtonian telescope. The region of good imaging may be outside a high-magnification eyepiece's field stop in a misaligned telescope. Misalignment aberrations can be straightforwardly corrected, so you should not greet their appearance with a feeling of dread. Assuming nothing is wrong with the glass, such aberrations will vanish in the center of the field with proper alignment.

The focused image has a small amount of astigmatism but mostly shows the effects of coma. The word coma comes from Latin, meaning "hair" and is also the root word of the familiar astronomical term comet. Coma, when severe, stretches the image out into two wing structures.

2.2.3 Atmospheric Motion and Turbulence

Another troubling image aberration is caused by wavefront passage through the many atmospheric turbulence cells occupying the long cylinder of air in front of the instrument. This tube of air is always part of an earth-bound optical system. Refraction by air is very small, but it does exist. It changes with temperature and pressure.

Hot air is lighter than cold air, and the ability of air to refract light increases with density. Density differences also result in a fundamental instability of the atmosphere. Sunlight chiefly heats the ground, leaving little energy in any particular volume of air in its passage through the atmosphere. The hot ground warms the air above it, and the warmer air becomes less dense. The cool air above it falls, and the warm air rises to occupy the same volume as the cool air, making a temporary vortex. At some point, the decrease in pressure cools the rising warm air, and the falling cool air warms for the opposite reason. The two volumes of air have just changed places. The ground is now slightly cooler, and both volumes of air are slightly warmer. Perhaps the upper volume of air will become unstable with respect to a still higher volume of air.

Inefficiencies of the process make the difference in temperatures of the two layers less profound, and without further heating of the ground, the process will wind down like an old clock. Such motion is an invisible cyclical heat engine, transporting energy from the hot ground to the cool upper atmosphere. The energy of sunlight percolates upward by convection through a cauldron of air.

During the daytime, this heating continues. Convection cells persist and sometimes grow. Leavened by the efficient heat transport properties of water and taking advantage of the shade provided by clouds, convection structures can expand to colossal dimensions—beyond little wisps and dust devils, beyond even thermal currents exploited by soaring birds. They can grow to summer afternoon thunderstorms. At the margins of the major convective structures are fine-scale cells, and at the edges of those little cells, even finer scale cells, until the swirls disappear into seething turbulence at microscopic scales.

It is no surprise that solar observation is usually best conducted in the morning, before the ground has heated much. A daytime test using the image of the Sun in a reflective sphere is best done in the very early morning. Also, turbulence is strong anywhere near clouds, since clouds are a flag that indicates one of these gigantic heat engines is operating.

It is nighttime that mostly concerns us, however. Without sunlight to drive the process, convection must rely on residual ground warmth or currents in the atmosphere caused masses of air at different temperatures. At some time during a clear night, the ground cools by heat radiation to have a lower temperature than the air above it. Because cool air is denser than warm air, this result is more stable. The convective cells on clear nights are less spectacular. Through near-pinholes, such as the human eye, this effect is seen as a slight spreading or bunching of light—stellar "twinkling." Through larger apertures, such as a telescope, the effect seldom appears as changes in brightness. In small telescopes, the image jumps around; in larger instruments the image is fixed but blurred.

In the outside-focus star test image, such cells look like nothing so much as the dappling of sunlight on the bottom of a swimming pool. Driven by high altitude winds, they wash across the aperture.

"focused out-of-focus out-of-foe us (later)

"focused out-of-focus out-of-foe us (later)

Fig. 2-3. Air Turbulence. A frozen moment of a turbulent image is shown in a focused pattern (left) and a defocused pattern (center). A delay of only a moment can show the example pattern at right. The focused image is magnified 5 times that of the defocused images.

Figure 2-3 shows a snapshot calculation of such a roughened diffraction disk. Look particularly at the focused image. The diffraction pattern varies so rapidly that such appearances aren't often directly observable. At least you can follow the changes in the outside-focus pattern as the cells sweep rapidly across the front of the telescope. The most visible change in the focused pattern is the angle at which splinters of light appear.

2.2.4 Tube Currents

Yet another atmospheric effect concerns the telescope user. Small telescopes are often carried from a warm house to the cold outdoors. These days, they are frequently transported to dark-sky sites in a warm auto mobile for some distance before reassembly. Even permanently mounted observatory instruments are seldom maintained at precisely the outside temperature at which they will be used. As a consequence, portions of the mounting—and most notably the mirror itself—must cool.

Glass has a high heat capacity and the accompanying low heat conductivity. In other words, glass optics have both a lot of heat to dump and the inclination to hold on to it a long time. Large, full-thickness mirrors take hours to cool.

Air convection is responsible for a large fraction of the energy transport. In the presence of gravity, the air warmed by higher temperature surfaces in the tube is displaced by falling cool air. In an open-tubed reflector that is tilted at an angle, the result is a tube current, as seen in Fig. 2-4. Other cooling effects are visible in windowed reflectors and even refractors, but they may not take precisely this form.

inside focus focused outside focus inside focus focused outside focus

Fig. 2-4. Tube Current. A common tube current appearance is the squeezed or herniated lobe on one side of the disk, and a flattened look on the other. Magnification of the focused image has been increased 6 times.

The heated air exits the telescope as it would a chimney. It generally hugs the upper side of the tube as it is ducted out. As light passes more quickly through the less-dense warm air, the wavefront there curls up like a page being turned. At certain angles, the in-focus image resembles the Moon setting over a quiet sea. After the telescope cools, these effects go away.

2.2.5 Pinched or Deformed Optics

Particularly common with overly-tight reflector cells or thin mirrors bending under their own weight is the aberration that has the appearance of Fig. 2-5. Details of the deformation will change the precise out-of-focus pattern. This deformation depends on which clip is tight, how many support points hold up the mirror, and whether the optics are supported on the edge or on the bottom. It will change with different telescope elevations.

In focus, the particular deformation modeled here results in a three-sided spike pattern.

2.2.6 Spherical Aberration

If abrasive is placed between two disks of glass and they are rubbed together with the orientation and stroke length completely random, what can be expected? At the end of thousands of such rubbing motions, part of two spheres must result—one convex and the other concave.

Thus, spheres (or more accurately, spherical bowls) are easy to make, at least compared with other three-dimensional surfaces. Unfortunately, spheres do not image properly. If a hemispherical reflector or a spherical lens is imagined, it is not difficult to see why.

Let's say light rays are incident on such surfaces as in Fig. 2-6. As the impact point deviates from a center-on-center direct hit, the focus wanders from a single point. Clearly, this aberration is an ever-present danger.

The mechanisms that evolved to correct this problem are fascinating because they point out the essential differences between refractive and reflective astronomical optics. They are almost two separate lines of development.

In the case of refractors, the spherical lens was retained and refined. In refractors, optical surfaces can be placed close together without getting in each other's way. More elaborate optical systems can be designed by invoking trade-offs between the curvature and separation of closely-spaced elements. The first optical designer to attack aberrations systematically was Joseph Fraunhofer. His masterpiece instrument, the great Dorpat achromatic refractor, was corrected not only for spherical and chromatic aberration, but off-axis coma besides. This instrument became the pattern for telescopes of the 1800s in much the same manner as Hadley's telescope had been the fundamental design of the 1700s. Fraunhofer accomplished this task with spherical surfaces (with a small amount of retouching) and clever design. Variations of Fraunhofer's airspaced 2-element doublet refractors are sold today and still give outstanding images.

Reflecting telescopes took another path. If you place a lens very near a mirror, light traverses it twice, which may or may not be useful. Rear-surface mirror telescopes have been designed (indeed, one was suggested by Newton himself in Opticks, p. 105), but they are usually specialty instruments that are difficult to construct. All-spherical reflectors are not impossible, but they had to wait for sophisticated mixed lens/mirror systems (such as the Maksutov telescope). Makers of reflectors turned to aspherical optics quite early.

focused

Fig. 2-5. Deformed Optics. The three-lobed pattern that results from too-tight mirror clips or a thin mirror that is inadequately supported. Left: focused pattern. Right: one appearance of the slightly defocused disk. Focused pattern is expanded 2.5 times.

Fig. 2-6. Spherical aberration in reflectors (a) and refractors (b).

The aspherical surface for a one-mirror astronomical telescope is a parabola that is spun on its axis like a top. This three-dimensional surface is called a "paraboloid." Unfortunately, the paraboloidal surface does not happen randomly. The maker must take the more-or-less statistical process that forms a sphere and control it (with pressure, special strokes, or small tools) to deform the surface to a paraboloid. Two-mirror systems have been devised that demand everything from hyperboloids to oblate spheroids.

Aspherizing takes good bench testing and understanding of materials and methods. As might be suspected, some telescope makers are more conscientious with such operations than others. Pitch, which is used for polishing and shaping the surface, is one of the most cantankerous materials used in any process. Fabrication can go badly when pitch is used without respect.

In my experience of testing approximately 100 nominal paraboloids, about half of the commercial mirrors have been marginally undercorrected. They have a surface somewhere between a sphere and paraboloid, barely within tolerance or slightly outside of it. A quarter of the mirrors have been severely undercorrected, and about a quarter were figured within acceptable limits. Most of the undercorrected mirrors were of the short-focus variety, between f/4 and f/6. Nearly all of the f/6 to f/8 paraboloids were adequately figured. I have seen few commercial Newtonian mirrors that were overcorrected—although many amateur-made mirrors seem to be.

For experienced opticians, paraboloidal mirrors are not all that difficult to make. Because Newtonian telescope mirrors tend to be undercorrected, the makers must be spending as little time as possible on each mirror. This practice can perhaps be explained (if not justified) by the current low prices of consumer mirrors. The makers are attempting to figure mirrors to the near edge of the tolerance, minimizing time and costs. The inevitable statistical spread means that many such mirrors will be unacceptable.

Undercorrection is shown in Fig. 2-7. Inside focus, much of the light is bunched up into the outside ring. Beyond focus, it has been pushed to a fuzzy patch in the center or to the outside of the secondary shadow. This is the signature of undercorrection. Memorize these patterns. If you have an opportunity to test Newtonians, you will see undercorrection repeatedly, and it is common enough in other types of telescopes. Spherical aberration is perhaps the only unadulterated glass error that you will ever see. The rest are usually mixed together.

Some people may misread the overall tone of these comments as a condemnation of Newtonian reflectors. Nothing of the sort is intended. The two best telescopes I have ever seen—and that includes refractors—have been exquisite Newtonians. The crisp star images of a well-made and well-aligned paraboloid are a beautiful sight. It is a shame that more of them do not perform as well as they can.

2.2.7 Rough Surfaces

Another common aberration afflicting telescopes is surface roughness. It results from using rapid polishing materials and maintaining insuffic-

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1/8 wavelength

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b) 33% obstructed b) 33% obstructed

Fig. 2-7. Spherical Aberration. Under correction appearing in a) unobstructed apertures, and b) 33% obstructed apertures. Inside focus (toward left): much of the light is collected in a strong outer ring, leaving a dim center. Middle: The focused image steadily worsens as aberration increases. Outside focus (toward right): the outer side of the image fades away and the missing light is found near the center. (Focused-image magnification is 4 times that of the defocused frames.)

ient contact between the pitch polishing lap and the optical surface being worked. The polishing operation is a strange and delicate process. Transient conditions on the surface of the lap can modify the contact between the tool and the optics. If the surface is polished by hand, one can feel the tool grab and kick almost like a living thing. Manual workers know immediately that something is wrong, and they can rewet the lap with polishing compound and press until contact is once again established.

As a simple matter of economics, optics aren't worked much by hand. A $300 telescope mirror probably requires between 2 and 4 hours of the optician's time. Such work must be prosecuted rapidly with machines or the cost of telescope mirrors would quickly escalate out of the reach of consumers. When a lap begins to seize, the machine neither notices nor cares. It has power enough to roll over the lap's squeaking complaints. If the optician has not worked out a standing procedure to avoid such difficulties, one possible consequence could be rough surfaces.

1o c used out-of-focus smooth

1o c used out-of-focus smooth

Fig. 2-8. Roughness. A small amount of surface roughness. Left: the focused image. Middle: defocused rough surface image. Right: a smooth defocused image for comparison.

Figure 2-8 is an example of the behavior of light in the diffraction pattern from an excessively rough surface. Of course, only one such appearance is illustrated. The pattern depends on such details as the scale of the roughness as well as its graininess and periodicity. These patterns, if the page is placed at sufficient distance, exhibit a weak spiky effect. These spikes can be distinguished from the roughness induced in the atmosphere by their motionless aspect. Atmospheric turbulence shifts and changes. As a result, spikes shoot out from one side of the image and then another. Rough surfaces, however, are coldly rigid. Nevertheless, the sky isn't usually steady enough to test for this problem on an actual star; most often this is an error best left to an earthbound artificial source.

2.2.8 Zonal Aberrations

Part of what makes a successful optical surface is the effect of statistical averaging. One of the most paradoxical features of optical work is that the best surfaces are the result of superficially sloppy practice. Behind the variations, however, are carefully delineated boundaries.

Machines are less random. The operator must make adjustments to add a pseudo-random component to the stroke. If insufficient artificial variations are imposed on the machine, it will tend to dig circular furrows or wavy deformations, called zonal defects, in the optical surface. Zones can also be the result of employing small polishers on a larger optical surface. Use of small polishers without sufficient blending can cause zones in both handmade and machine-fabricated optics. See Fig. 2-9.

inside focus outside focus normal inside focus outside focus normal

Fig. 2-9. Zonal Defect. Zonal aberration on an unobstructed aperture caused by a trench at 60% of the radius of the disk. Far inside of focus, the zone appears as a bright ring on the uniform disk, and outside it becomes a dark ring. On the right is an unaberratedpattern.

Zonal defects are common in small optics at the center. They appear as an indentation or a bump (the photograph in Fig. A-3 of Appendix A is an example). Zones at the center are less harmful than zones appearing at other radii. In obstructed reflectors they are confined largely to the shadow of the diagonal and rendered harmless. Even if they show, defects at the center occupy only a small part of the surface area.

2.2.9 Turned Edges

Turned-down edge is a defect where the edge does not end abruptly but curls over gradually. This special case of zonal aberration results in a surprisingly large amount of damage to the image because the edge of the aperture has a large fraction of the total surface area. It deflects more light.

Turned edge comes either from polishing pitch that is too soft or from applying incorrect pressure when the optical surface is extended over the edge of the polishing tool (Texereau 1984). It is extremely difficult to remove once generated, and the fear of it causes telescope makers to over-compensate with an extremely hard grade of pitch. Some add adulterants that change the pitch's readiness to flow. Stiff or waxy pitch often worsens the problems with rough surfaces, though, so the net effect is a trade between two noxious aberrations.

In an otherwise perfect reflector, turned-down edge appears as a softening of the ring structure inside focus and a corresponding hardening of the ring structure outside of focus. To avoid confusion, look for this effect through a strongly colored filter. Figure 2-10 shows two 25% apertures, one normal and the other with a turned edge. The light misdirected from a turned-down edge appears as a hazy glow in the vicinity of the image on the inside of focus (remove the filter if you look for this halo).

On both sides of focus, a narrow turned-down edge displays a fairly flat distribution of light in the disk.

TE inside focus

TE outside focus

TE outside focus

Fig. 2-10. Turned-Down Edge. Turned edge in a 25% obstructed aperture. A normal 25% aperture appears at the left. The inside-focus disk shows loss of contrast and a diffuse glow surrounding it. The outside-focus disk seems less affected in terms of light distribution, but the contrast is increased in the rings.

2.2.10 Astigmatism

Pure astigmatism can occur even in perfect telescopes (particularly refractors) if the system is not aligned properly. The cure is simple, and the aberration disappears quickly upon the telescope's collimation. It also occurs in Newtonian telescopes that have curved secondary mirrors. Because of the 45° tilt of the supposedly flat mirror, a bulging or concave diagonal will be expressed as astigmatism in the image.

Astigmatism in the glass itself is caused by three fabrication errors. It can result from pressing the disk against a cylindrical surface—for example, the rear surface of the disk may not be flat. Another cause is too rapid cooling of the glass disk when it was poured, freezing unrelieved stresses

2.3. Concluding Remarks

into the disk. This error is scarcely ever seen in deliberately made optical disks, but is common enough in portholes or other undocumented glass. And last, failing to rotate the optical disk with respect to the tool grinds a cylinder into the disk directly.

In all cases, out-of-focus astigmatic images will show two oval patterns at 90° to one another (see Fig. 2-11). If astigmatism is severe, certain focus positions will give stellar images that look like straight lines.

inside focus focused outside focus inside focus focused outside focus

Fig. 2-11. Astigmatism. Appearance of astigmatism immediately on either side of focus. Middle: at best focus, the pattern is a cross. Out of focus, the profile is stretched into an oval, with the direction of stretch changing a quarter turn on opposite sides of focus.

At best focus, astigmatism will show the indicated crossed pattern. Small amounts display a thickening of the first diffraction ring along the cross directions.

2.3 Concluding Remarks

This chapter is meant only as a brief tour of the enormous landscape that can be found in the star test. Do not confuse this nodding acquaintance with real expertise. The star test's vastness and complexity are astounding. You will continue learning new features of the star test many years into the future.

The size of the preceding sections is one measure of the importance given to them as optical problems. If one topic can be recommended for study, it is spherical aberration—i.e., simple correction error. Become an expert in its detection. You will see this error again and again.

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