Astigmatism is the tendency of an objective to focus at two distances. Each focal position is stretched, one at a right angle to the other. An average focus is found between them, with the image forming a crosslike pattern. Because of the stretching at the angles of the cross, as well as the inability to locate a unique focus, contrast of high-spatial frequency information is sharply lessened. This chapter will discuss a number of points:

1. Astigmatism has many separate causes, and most can be cured at moderate or no expense.

2. Unambiguously locating the source of astigmatism takes careful logic (and to be honest, a small amount of luck).

3. Small amounts of astigmatism are detected by gently rocking the focus about the position of best focus.

4. The way to determine whether the telescope has too much astigmatism is to try to detect non-circularity of the image at a calibrated focus setting.

14.1 Astigmatism in Eyes and Telescope Optics

Astigmatism means not stigmatic, or not focusing to a spot. In the widest possible definition, all of the nonsymmetric aberrations discussed in this book are astigmatic. The symmetric aberrations may produce fuzzy spots, but at least they are round.

This type of aberration was originally defined by Ludwig von Seidel and modified by Fritz Zernike (Born and Wolf 1980, p. 470). Astigmatism divides focus into two stubby lines separated by a short distance along the direction of eyepiece travel. The lines are at right angles to one another. A sort of "best focus" is seen at the halfway point between them, where the illumination pattern takes the shape of a cross.

Astigmatism is frequently corrected in human vision.1 The cylinder numbers in the eyeglass prescription are adjustments for astigmatism. This aberration is usually caused by a deformation of the cornea, or the transparent outer covering of the eye. Astigmatism could result from a tighter "stretch" of the cornea in one direction than in the other, as well as a misshapen eyeball. The aberration can be temporarily corrected in some cases by pulling one corner of the eyelid.

For example, my right eye has a cylinder power of 2 diopters2 at an axis of 90° and my left eye has a power of 2.5 diopters at 88°. Because I also have significant focusing correction, I see point sources stretched up in one eye and flattened in the other. The approximate 90° axis is probably no accident; the stretch direction may be related to the muscle attachment points of my eye or to the shape of the orbital cavities in my head.

The simplest example of an astigmatic lens is a cylinder with no curvature in one direction and significant curvature in the other. Truncated chords of cylinders are often used in reading magnifiers. By putting the magnifier right against the page, one can suppress the poor focusing properties of the lens and stretch tiny letters in one direction.

Truly cylindrical lenses or mirrors are almost never used on purpose and never happen accidentally. The most common form is a hybrid of spherical and cylindrical shapes. In fact, one can subtract the spherical focusing power of a lens entirely and be left with a purely astigmatic form.

A cylindrical lens that won't focus in any axis is modeled as part of a torus. A torus is shaped like a doughnut. The particular form that is most interesting here has a hole in it with exactly the same diameter as the torus' cross section. A drawing of such a doughnut appears in Fig. 14-1, with half of the torus chopped away to reveal the interesting region. Consider the tiny circle drawn on the inside of the hole. When lifted away from the rest of the torus, that circular piece has the same surface shape of an astigmatic lens with no focusing power (the other side of the lens is flat). The curve goes into the paper along the up-down direction and out of the paper to the left or right. Imagine looking down on a saddle shape,

1 A more common problem fixed in eyeglasses are the spherical corrections, which are modifications of the focusing power of the eyes. "Spherical" to an optometrist refers to the Aip2 focusing term of the expansion of Eq. 10.2. The words "spherical aberration correction" used by telescope makers designates the Atp4 term of that expansion.

2 A diopter is the reciprocal of the focal length in meters. The typical 2-meter focal length of a small telescope produces a focusing power of only 0.5 diopters.

Fig. 14-1. A tiny circle on the inside of a torus is the purest form of the astigmatic deformation.

Fig. 14-1. A tiny circle on the inside of a torus is the purest form of the astigmatic deformation.

with the left-right directions toward the pommel and seat, and up-down toward either stirrup.

Astigmatic surfaces can be modeled as combinations of purely spherical (i.e., focusing) surfaces and nonfocusing toroidal surfaces. If we move the tiny circle of Fig. 14-1 to the outside of the doughnut, its surface no longer resembles a saddle. Instead, it looks like one of these hybrid surfaces. It has lower astigmatic power than before, with some spherical curvature mixed in. In the astigmatism of a real lens, the converging wavefront doesn't truly have a saddle shape. The shape is produced only after the spherical focusing component is subtracted away.

14.2 Causes of Astigmatism

Astigmatism in telescopes is a symptom of several optical problems, some of which are listed here:

1. Misalignment (often mixed with coma).

2. Poor support of the optics' weight, causing pressure directed along an axis (the "potato chip" sag).

3. A supposedly flat diagonal or right-angle prism that is actually slightly spherical (a surprisingly common problem).

4. A slight cylindrical deformation ground or polished into the glass (a very tiny amount is found in most mirrors and lenses).

5. Poorly annealed glass (often appears in "porthole" mirrors or other thick pieces that were made with another application in mind).

6. Uncompensated astigmatism in the observer's eye.

The most disastrous of these causes are number 4, true astigmatism ground or polished into the surface, and number 5, unrelieved stresses contained in the glass itself. Saving money by using undocumented glass for the mirror substrate is a risk that many makers are willing to accept, and they often get away with it. The astigmatism that is derived from the other sources, however, is no less objectionable. Luckily, these causes are easily repaired.

In the days when all mirrors were small and thick compared with their diameter, astigmatism was comparatively rare. Even halfhearted adherence to the optical shop practice of rotating the tool with respect to the mirror usually kept astigmatism dormant. Only when stresses were frozen into the mirror disk itself (cause #5) did astigmatism appear.3

Cause #4 is more common now that large, thin mirrors are being figured. Even careful opticians can polish cylindrical curves into such a flexible surface. If conditions that result in astigmatism appear, they are usually a function of how the mirror was supported during grinding or polishing and are therefore persistent. Astigmatism, if it appears at all in such mirrors, is usually severe.

Discovering the cause of observed astigmatism to determine possible strategies for its elimination takes methodical but straightforward detective work. The process will be described in section 14.5.

14.3 Aberration Function of Astigmatism

The term added to the wavefront aberration function to account for astigmatism alone is (Born and Wolf 1980, p. 470)

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