## Wastig Pe ATgp2 cos2 e 2 141

Here is the coefficient giving the amplitude of the astigmatism, p is the distance from the axis, and 8 is the angle from the axis of astigmatism. The constant XA subtracted from the cos2 8 term is the needed focus shift to produce a Zernike best-focus aberration. Below, the astigmatism axis is conveniently placed at 0°, but it can occur at any angle. Wastig is graphed in Fig. 14-2.

For example, if AaUg is % wavelength, then W . goes from +78 wavelength to — 78 wavelength. This definition is slightly different from the usual one for primary astigmatism. Coefficient A 2^tlg is the total peak-to-valley aberration and is the number that compares most closely with the Rayleigh tolerance.

3 If disk itself was at fault, Russell Porter gave a succinct, though final, solution: "Seek out a good hard, solid hydrant. Hurl the mirror as fiercely as possible at said hydrant. Walk home." (Ingalls 1976).

Fig. 14-2. The saddle-shaped aberration function of astigmatism just after it has passed through the aperture.

Eq. 14.1 has two interesting features. The first is the way the astigmatism scales with distance from the axis. It has the same power, p2, as defocusing aberration, so one may interpret astigmatism as a pathological form of defocusing. Also, at a defocusing aberration of A = ±(1/2) A0!"g , the aberration becomes flat along one line, with the line for the "+" value oriented at right angles to the line for the "—" value. This fact demonstrates that an optimum focal point exists for each axis of the mirror and that these optimum foci are on either side of conventional focus. The actual best focus position is a compromise between these axes.

The division of focal regions is best seen in Fig. 14-3, where the same longitudinal slice through focus is viewed first from the side and then from the top. The views are mirror opposites of one another. The approximate lines of brightest focus are the three-lobed wing structures, only one of which appears in each diagram. The little bright lozenge that appears fore or aft of the wing is just the other wing viewed end-on. If these slices could be perceived in three dimensions, one would see a pair of illuminated regions vaguely resembling boomerangs at right angles to each other, overlapping nose-to-nose.

As the astigmatism gets worse, the bright region looks less like a wing and more like a long bar. The two bars separate and move apart. In the diagram, the bright regions separate and become more like lines.

The best focus diffraction pattern pinches the rings into two bars for small aberrations (see Fig. 14-4). For larger amounts (many wavelengths) of astigmatism, best focus (if any focus could be called "best") resembles a square-woven basket. At 1.5 wavelengths of aberration, an on-axis dip in intensity or a nodal minimum exists. By defocusing to ±0.75 wavelengths, we see the wings of Fig. 14-3 as they would appear face-on in a) side view

15 b) top view

Fig. 14-3. Two slice patterns depicting 1.5 wavelengths of astigmatism. The up-down dimension of each frame is 15 angle units (1.22 = Airy radius). Defocus is between -3 and 3 wavelengths. Best focus is at the center.

Fig. 14-4. Strong astigmatism (1.5 wavelengths total aberration): inside focus, best focus, outside focus.

the eyepiece.

Another interesting feature of Eq. 14.1 is that the toroidal shape is approximated by a parabolic radial dependence. Like spherical aberration, there are higher order terms to this expansion as well, but they are customarily ignored.

14.4 Filtering of Astigmatism

Since astigmatism is not circularly symmetric, the transfer function depends on how the MTF bar target is oriented. Hence, Fig. 14-5 shows each

Fraction of maximum spatial frequency

Fraction of maximum spatial frequency t5 0.6 c

I tu

Filtration caused by astigmatism

Fig. 14-5. Modulation transfer functions for astigmatism at best focus. Three amounts are shown: 3/8 wavelength, 3/4 wavelength, 1.5 wavelengths. An aberration of 3/8 wavelength is just outside the Maréchal limit (0.8 Strehl ratio). In each pair, the lower curve is for MTF bars oriented along the axes of astigmatism. The higher curve is for bars at 45°.

aberration amount as two lines, with all intermediate MTF curves approximately between them. Clearly, astigmatism is profoundly harmful at 1.5 wavelengths. The 0.8 Strehl ratio occurs at a little less than 3/8 wavelength peak-to-valley.

Because it affects the aperture on the same scale as defocusing, astigmatism is primarily an intermediate-to-high spatial frequency error. It affects the transfer of contrast only mildly at low spatial frequencies. The transfer function of astigmatism doesn't fall as quickly with increased spatial frequency as the MTF for primary spherical aberration or obstruction. Look at its performance at middle spatial frequency, however. The lowest curve even for the marginally acceptable 3/8-wavelength range degrades the MTF to an average of 70% the perfect value.

The degradation bottoms out on the lower curve of the 3/4-wavelength MTF at about 0.5 of the maximum spatial frequency. For a 200-mm aperture, details with spacing of about 1.2 arcseconds and less would be the most severely damaged. Because it attacks high spatial frequencies, astigmatism reserves its worst behavior for lunar-planetary observation. This aberration may partially account for the poor reputation of thin-mirror Newtonians for high-magnification observation.

### 14.5 Star-Test Patterns

Star-test focus runs appear in the next two figures. The aberration amount in Fig. 14-6 is 3/4 wavelength. It is shown as it would appear with 20% obstruction. Figure 14-7 shows effects of 3/8 wavelength of unobstructed astigmatism. The signature of astigmatism is the oppositely directed oval appearance on either side of focus. Recall that these patterns are calculated with an astigmatism axis aligned with the square. In real observing, astigmatism is not required to sit upright. It can be seen at any angle. As defocus is increased, the pattern becomes less elliptical. Defocusing is kept small in these figures because astigmatism most severely affects high spatial frequencies and is most readily detected close to focus.

In fact, the most useful method to detect astigmatism is to focus a dimmer star and then rock the eyepiece back and forth across focus. The astigmatism stretches the image first one way and then the other, and the aberration is immediately apparent. A focused image is supposed to appear as a cross, but you may have trouble seeing the diffraction disk at all in the telescopes for which astigmatism is likely. More common are apertures that just show a hint of the pattern, as in Fig. 14-7.

If ellipticity is seen, turn to Table 5-1 and determine how far you must move the eyepiece to defocus 4 wavelengths. For a focal ratio of f/6, the amount is only 0.025 inches or 0.63 mm. Defocus this far and carefully inspect the image of a dim star. If the pattern is distinctly elliptical, the telescope has too much astigmatism. Ideally, astigmatism is difficult to detect at 2 wavelengths defocus.

However, do not be surprised if most telescopes suffer from a trace amount of this aberration, if only because of the ever-present force of gravity. Few instruments have none.

### 14.6 Identification in Newtonian Reflectors

Just because the instrument suffers from astigmatism doesn't necessarily mean that the problem is ground into the glass. Finding astigmatic error is easy; identifying its source is more difficult.

First of all, determine if your own eye is causing the astigmatism. Simply using your other eye to look through the telescope often does not help (see my prescription above), so that's not the way to tell. You can try rotating your head with respect to the eyepiece and see if the astigmatic axis follows, but changes caused by such slight rotations are difficult to perceive.

ASTIG=0.75 OB=20% 10 normal OB=20% 10

ASTIG=0.75 OB=20% 10 normal OB=20% 10

Fig. 14-6. Focus run of %-wavelength astigmatism. The normal aperture is in the right column. Obstruction is 20%. Ellipticity is still clear at 8-wavelengths defocus.

Fig. 14-6. Focus run of %-wavelength astigmatism. The normal aperture is in the right column. Obstruction is 20%. Ellipticity is still clear at 8-wavelengths defocus.

AST = 0.37 10 perfect 10

AST = 0.37 10 perfect 10

Fig. 14-7. Unobstructed aperture with 3/8-wavelength total astigmatism. The right column depicts a perfect aperture. Stretching is visible at 4 wavelengths defocus, but is hard to see at 8 wavelengths.

Fig. 14-7. Unobstructed aperture with 3/8-wavelength total astigmatism. The right column depicts a perfect aperture. Stretching is visible at 4 wavelengths defocus, but is hard to see at 8 wavelengths.

The best way to determine if the problem is in your eye is to increase the magnification. You will make the astigmatism easier to see if it is contained in the telescope, and diminish its strength if it is contained in your eye. Few eyes are so bad that they will show strong astigmatism if the exit pupil of the telescope is set below 1 mm. The bundle of light exiting the telescope decreases in cross-sectional area with higher power, and the illuminated area of misshapen cornea is reduced significantly. As the exit pupil approaches a pinhole, your eyes perform better.

At this point, spin the eyepiece. The astigmatism axis should remain fixed. If it follows the rotation, the eyepiece is astigmatic.

The next thing to investigate is the angle along which astigmatism seems to compress or stretch the image. In refractors or Schmidt-Cassegrains, the angle is obvious. For Newtonians, you can perform the same trick used in the alignment chapter. By defocusing a long way and poking your hand halfway into the optical path, you can determine the direction of stretching. Be suspicious of the main mirror cell if the astigmatism axis is oriented along the horizon or elevation direction (that is, in the gravity axis). Suspect the diagonal cell or the diagonal itself if the astigmatism seems to be aligned with respect to the tube (in the diagonal-tilt axis).

Unfortunately, these two situations sometimes occur together, especially for large altazimuth Newtonians with a level eyepiece. Therefore, we must change the deformation for one mirror or the other. The first thing to try (especially if you're doing a level-telescope artificial source test) is to point the telescope at a star near zenith and see if the astigmatism is still visible. If the main mirror is riding on a single edge point, this step will redistribute the weight of the mirror and hence its compression. Perhaps the pattern will still be bad, but at least it will be different. It might, for example, transform into a three-sided pattern. If the image is unaffected, shake the telescope slightly and try again.

If no change is seen, main mirror warping may yet be the problem. In the case of a mirror glued into its cell, one adhesive pad may be unduly straining the mirror, and the tension might have little to do with the mirror's weight.

The worst possibility is that the mirror actually has an anomalous curvature in the glass. To determine if that problem afflicts your telescope, try a 120° rotation of the mirror. Realign the telescope and test again on a star near zenith. The error should also rotate if it is contained in the mirror.

The image rotates. Ask yourself if the mirror performed well in the past, and only recently has performed poorly. If so, the weight of evidence seems to indicate that the mounting has failed or the mirror has become chipped or cracked.

Disassemble the whole mirror cell, inspect the mirror, and carefully reassemble the cell. Make certain that the mirror is adequately held but is nowhere pinched. For mirrors glued to a flat metal plate, make certain that you do not overly tighten this plate into place.

If the telescope is still astigmatic after these corrective measures or if it has always performed badly, then consider the possibility that the aberration is either ground into the glass or the mirror substrate material was poorly annealed.

The image doesn't rotate. When a 120° rotation does not affect the axis of astigmatism, then your attention must turn to the diagonal. Again, ask yourself if it performed well up to a recent date, then suddenly turned bad. Or have images always been marginal, and you are just now investigating with the star-test techniques presented here?

If the loss of quality has been abrupt, then the diagonal mounting is probably at fault. Take it out and look it over. Have you recently taken it apart for cleaning? Have you recently readjusted it?

Many Newtonian diagonal holders use cotton wadding behind the diagonal. Often, too much cotton stuffs the holder and causes tension on the mirror. Use only enough cotton behind the diagonal to hold it into place.

Some diagonal holders have a split-cylinder construction. During reassembly, carefully widen the slit enough to give the diagonal mirror room. Make sure that a screw is not protruding from the base to strike the mirror in its back side, and be certain that the tip of the diagonal does not touch the base.

If all these mounting changes do not help, perhaps the diagonal has a spherical curvature polished into its surface. Because this sphere is at a 45° angle, the effect on the image turns to astigmatism.

You can check this poor performance by removing the diagonal holder and rigging it on a separate support (a photographic tripod is ideal). First, perform a straight star test on an inexpensive 50 or 60 mm refractor using an artificial source. Verify that the image is nicely circular. These department-store refractors are often optically good, and one should be kept for this purpose even if it is never used for observing. Then, do the star test again, looking at approximately 45° through the diagonal.4 If the diagonal has curvature, you will see that the refractor has developed a sudden case of astigmatism.

You can also check the operation of star diagonals or right-angle prisms using this auxiliary telescope technique. You may rely on any mirror satisfying such a check because the test demands equal path length over the whole surface of the right-angle bend. This condition is probably harsher

Pointing this combination can be frustrating, but you will eventually succeed.

Fig. 14-8. A sensitive test of diagonal mirrors and star diagonals.

than actual operation demands. In normal use the converging light cone leaves much of the surface dark, but during the test the whole path is equally lit.

### 14.7 Refractors or Schmidt-Cassegrains

Except for instructions concerning the diagonal, the comments and instructions dealing with Newtonian reflectors above are still useful. First, remove the right-angle prism or mirror if you are using one. Many (if not most) such devices have poor optical quality, and the problem may be cured by this simple expedient.

If astigmatism is seen in a refractor, one possible cause is poor alignment. Enough surfaces are present in such telescopes that designers, if they so choose, can strip coma away from off-axis images. Only astigmatism is left. Try to realign using instructions in Chapter 6.

The secondary is not strongly tilted in Cassegrain-style reflectors, as it was for the Newtonians described above. Thus, a spherical error in the secondary will not show itself as astigmatism. Astigmatism must come from misshapen or severely strained optics. One likely source is the secondary holder itself. Some secondary mirrors aren't sufficiently isolated from the adjustment plate at their bases. Distortions in that plate can be transmitted to the mirror and show up as astigmatism (or worse) in the image.

The secondary mirror of a Schmidt-Cassegrain adjusts by the turning of set screws with a hex-head wrench. Because of the mechanical advantage

Chapter 14. Astigmatism of such a tool, you can unknowingly put huge tensions (500 pounds or more) on the secondary cell. The secondary is the only mirror to adjust in Schmidt-Cassegrains, so if you can't eliminate astigmatism by reducing forces in the secondary holder, you must return the instrument to the maker for servicing.

### 14.8 Remedies

Nothing can be done about astigmatism in the glass except to refigure or replace the optics. Perhaps the maker will agree to refigure them for you, since the optics either left the factory with astigmatism or developed it later because of improperly annealed glass.

Luckily, astigmatism is most often the result of improperly assembled mirror cells and secondary holders. Increasingly, mirrors are held in cells by the technique of gluing them to a plate instead of the more benign method of holding them loosely by clips. The thin plates buckle and transmit their warped shape to the optics through forces applied by the adhesive pad.

We tend to judge everything by familiar uses, and in this case what we know is tightening bolts in, for example, an automobile. Everyday vibrations would quickly shake an automobile to pieces if the screws weren't tight. However, telescopes are delicate. The worst vibrations they are likely to encounter is a gentle, infrequent jiggling during transport. They don't turn at 2000 RPM like an internal combustion engine but shift position perhaps 40 times in a night. They are scientific instruments. Tighten screws firmly but gently.