## Spherical Aberration

The most common error on the glass is called spherical aberration. It resides to some degree on all surfaces and need not become debilitating, though that depends on its severity.

The star test for spherical aberration is surprisingly sensitive and easy to interpret. This chapter will make five points:

1. Wavefronts deformed by simple spherical aberration (fourth-order curves) are recognizable by noticing that light approximately follows the behavior of the ray-optics caustic. On one side of focus, light is taken from the outer parts of the out-of-focus disk and deposited at its center. On the other side of focus, light taken from the center brightens the outermost rings.

2. The strength of lower-order spherical aberration can be roughly esti mated by utilizing a central obstruction and comparing the breakout points of the central shadow on both sides. Correction error must be the only contributing aberration for this estimation technique to be valid.

3. Higher-order spherical aberrations, although noticeable in star tests of certain designs, are practically never seen in pure form.

4. By calculating the energy deposited within an angular radius of 1/D (near the edge of the diffraction disk), the telescope user can usefully add together the combined effects of spherical aberration and central obstruction. This encircled energy ratio yields a number similar to the Strehl ratio, but it includes the degradation of obstruction in a single standard.

5. A quarter wavelength of correction error delivers images of extended objects that are noticeably soft, but barely acceptable.

10.1 What Is Spherical Aberration?

Because they seem to have the form of shallow bowls or protrusions, it is natural to assume that perfect mirrors and lenses are sections of spheres. However, spheres are not the best geometrical form to produce an image. The following thought experiment makes this point obvious. Imagine the inner surface of a hemisphere as a mirror and light as incident from infinitely far away on the right, as in Fig. 10-1. Light striking the edges is hardly deflected; it kisses the interior and bounces around the half dome. Light incident more toward the center is deflected towards a sort of focus, but the light striking near the axis is directed the farthest away. The focus region is expanded in Fig. 10-2 to demonstrate that the focus is bent from a geometric point to a horn-shaped caustic envelope.

Fig. 10-1. Half-dome spherical mirror. The focus is more poorly denned as the incoming ray approaches the edge.

Fig. 10-1. Half-dome spherical mirror. The focus is more poorly denned as the incoming ray approaches the edge.

The original meaning of caustic was "burning," so an optical caustic is at or near focus. In ray tracing, the term acquired a slightly different meaning. A caustic is a curve or surface along which rays seem to pile up.

It refers to locations where the geometric spreading or convergence of a ray bundle does not give the correct value of the intensity. Caustics are places where one must resort to diffraction theory.

The type of spherical aberration shown in Fig. 10-1 is called spherical undercorrection. With overcorrection, the order is reversed. Centrally incident light crosses the axis too near the objective, and light incident at the edge crosses the axis at the most distant point. In overcorrection, the horn points the other way.

The best surface configuration to image light varies from telescope to telescope. Some objectives, such as compound refractor lenses, correct for spherical aberration by bending or separating spherical surfaces. Here the designer uses a trick to retain the easily made spherical shape. A similar bending occurs in the Maksutov design. The meniscus shell doesn't really possess much focusing power because the rear surface has about the same curvature as the front. The real purpose is aberration modification. Other telescopes share the burden of correcting the main focusing element's spherical aberration with an oddly deformed corrector plate, as in the common Schmidt-Cassegrain telescope.

For telescopes with only one focusing mirror (i.e., Newtonian reflectors), the proper geometric form is a paraboloid. The flat secondary diverts the beam so the observer's head doesn't get in the way, but it doesn't actively participate in forming the image. The paraboloid is one special case of a family of surfaces called conic sections of revolution.

Classical Cassegrains combine a paraboloidal primary with a hyperboloidal secondary to achieve the requisite spherical correction. One can take out the secondary mirror of a classical Cassegrain and install a diagonal to use it as a Newtonian. The secondary mirror must correct its inherent spherical aberration independently. One might reasonably guess that the secondary would have to be a convex paraboloid, but the secondary doesn't do the same job as the primary, so it must be curved differently.1

Other Cassegrain-style instruments, such as the Dall-Kirkham type, only correct part of the spherical aberration of the whole system in each mirror. The convex secondary remains spherical. This small mirror adds a component of spherical aberration of opposite sign to a concave spherical primary mirror, but the amount is not sufficient to correct the system completely. As a consequence, the properly made primary mirror of a Dall-Kirkham is deformed to a prolate spheroid (between a sphere and a paraboloid). Dall-Kirkhams are popular with telescope makers because the spherical secondary is easy to make. However, they suffer severe off-axis coma.

1 One peculiar Cassegrain-style design, the afocal Mersenne, has a paraboloidal secondary mirror. This curious telescope doesn't need an eyepiece (King 1955, pp. 49-50.)

The Ritchey-Chretien design goes the other way. By putting stronger hyperboloidal curves on both the primary and secondary mirrors, the designer can achieve a degree of coma correction superior to the classical Cassegrain. However, these telescopes are hard to make, and are usually of interest only to professional observatories. The Hubble Space Telescope was designed as a Ritchey-Chretien.

### 10.2 The Hubble Space Telescope

The summer of 1990 featured an event making the previously esoteric topic of spherical aberration front-page news. The Hubble Space Telescope (HST) was revealed to be improperly manufactured. Newspaper reports said that it suffered from approximately 1/2 wavelength of spherical aberration and that the edge of the mirror was nearly 2 microns (or 2 fim) off.

At first, these statements were confusing. A 2 im surface error would result in a wavefront improperly curved by about 4 ^rn, or about 7.25 wavelengths of yellow-green light. If focus is adjusted to the minimum RMS deviation rather than the focus of the center zone, that value is reduced by a factor of 4 to 1.8 wavelengths.

This number was clarified when an article in Sky & Telescope mentioned that the 1/2-wavelength aberration had been measured as a root-mean-square (RMS) deviation (Sinnott 1990a). To derive the peak-to-valley value from the RMS value, we have to multiply by a factor of approximately 13.4/4, because for correction errors

V wavelength (peak-to-valley)= 134 wavelength (RMS)

so Vi wavelength RMS x( 4 J=1.68 wavelength.

The comparison that is best known to amateursâ€”with Rayleigh's '/4-wavelength toleranceâ€”in this calculation comes to 1.7 wavelengths for the Hubble Space Telescope, assuming the 1/2-wavelength RMS error was exact.

The root cause of the error was an improperly assembled device called a null tester used in the manufacture of the primary mirror. This null tester was supposed to generate a wavefront with precisely the reverse correction as the main mirror. Thus, a mirror that undid that correction would be exactly right. Unfortunately, the null tester was spaced incorrectly and presented the wrong reverse correction (S&T 1990; Capers et al. 1991).

### 10.3 Generalized Spherical Aberration

From the point of view of the star test, you don't have to think about the form of the optical surfaces, or even remember their long names. Consider only the shape of the final wavefront as subtracted from a perfect sphere. This difference can be expanded in the form of a simple polynomial function:

where p is the radial coordinate with range 0 to 1. The symbol W(p) stands for the total distortion of that wavefront away from a sphere with a center at the focus. If W(p) is zero, then the curves are the same. Let's look at each of these terms and the coefficients in front of them (the "A's") and discuss what they mean.

The first is a constant, A0 that only advances or delays the wavefront. We can think of this number as the "time" or "phase" constant, and it should be chosen so that the comparison sphere is not too far removed from the wavefront. This constant represents propagation, with different values of A0 representing snapshots taken at different times. Usually, the constant is set to zero just as the wave passes through the aperture or else adjusted for convenience.

The term A2p2 is a smooth bending, either pushing the wavefront in a small amount or pulling it out somewhat. If our reference sphere has its center placed at the wrong focal point, this term bears the brunt of the increase. Thus, A2p2 is called the "defocusing aberration" here and is the same one appearing in Fig. 4-15.

The A2 defocusing term can be conceptually included in the spherical aberration expansion, and from the point of view of star-testing, defocusing should be considered as just another aberration. However, defocusing is not a feature of the glass, so it isn't customary to refer to it as a spherical aberration term. Defocusing is so important that it is set aside and considered separately.

The fourth-order term, A4p4, is what is usually thought of as spherical aberration. Another name for this term is primary spherical aberration. Errors here are said to be "correction" errors, such as undercorrection or overcorrection.

The terms A6p6... are usually small but can become important with unusual optical systems. You can see by the ellipsis that this expansion goes on forever, but each factor An is usually much smaller than the previous coefficient. We can regard the A6 coefficient as the last important term in this chapter.

Spherical aberration can be expressed by a number of equally valid conventions. Another way of referring to the fourth- and sixth-order wavefront terms is to call them third- and fifth-order spherical aberration. These names are derived from the slope of the wavefront, not the wavefront itself. and how far the light is shifted sideways from the center of the diffraction spot. Some authors find it more convenient to consider the residual changes in focal distance with p, or "longitudinal aberration" (Kingslake 1978, p. 114). In this way of looking at errors, primary aberration is the coefficient of the p2 term. Thus, we can find the same primary spherical aberration expressed as fourth-, third-, or second-order coefficients, depending on whether the polynomial expansion refers to the wavefront, to the residual slope, or to the longitudinal aberration respectively. In this book, we will always refer to the wavefront.

### 10.4 The Aberration Functions

The aberrations are usually measured at the position of best focus2 because the spurious disk is smallest there. It's what we think when we say "the telescope is focused." When A2 is zero, Eq. 10.2 has the position of focus arbitrarily set at what is called "paraxial" focus, or the focus of the center of the mirror or lens (the narrow end of the caustic horn above). Although a convenient place for the mathematics of wavefront shape, it has nothing to do with the location where we visually perceive the tightest disk.

If we make the constant A4 non-zero, for example, we find that some defocus must be added to chase best focus as it scoots away. If we change A4 a second time, we must move the focus again. The term A6 creates even more complications. If we wish to consider pure higher-order spherical aberrations at the best focus position, we must subtract out just the right amounts of lower orders. Removing these terms can be tiresome although the process is computationally straightforward.

It is much more convenient to encapsulate just enough of the lower-order aberrations to automatically cancel them out of each term as it is increased. This step was taken, in a considerably more complicated form, by Fritz Zernike, and the resultant terms are called orthogonal Zernike polynomials (Born and Wolf 1980). The interesting terms are limited here to 4th- and 6th-order, but they exist for higher orders as well:

The primes are placed on the coefficients to indicate that they are not

2also called "diffraction focus"

the same size as in Eq. 10.2. These complicated-looking equations simplify somewhat when displayed as aberration functions (see Figs. 10-3 and 10-4). All focused patterns appearing below are referenced near the focus positions implied in these functions.

The functional form of the fourth-order Zernike polynomial has just one doughnut-shaped ring, whereas the higher order has an elevated center. The 8th-, 10th-, and higher-order polynomials add one extra curl each. In fact, as we go higher, the pattern appears increasingly corrugated. But remember, the sum of these polynomials represents smooth surfaces. The signs and amplitudes are chosen so that the washboard appearance goes away. Only for the zonal defects of Chapter 11 do anomalous contributions of high-order polynomials add up to give a non-zero result.

Although it is not plotted, the 6th-order function has a more involuted caustic shape. Imagine if a very powerful individual reached into the bell of the musical horn of Fig. 10-2 and pulled the mouthpiece halfway back through it. The resulting damage would look much like that caustic (Cagnet et al 1962).

Another minor point is that the position of diffraction focus moves slightly if the telescope is obstructed. Most of the patterns appearing in this chapter take into account this change. The longitudinal slice patterns are an exception.

10.5 Correction Error (Lower-Order Spherical Aberration)

### 10.5.1 Filtering of Spherical Aberration

The modulation transfer function (MTF) is depicted in Fig. 10-5. The drooping lines show the way contrast is reduced as spherical aberration is steadily worsened. One interesting feature is the mild increase that occurs for small aberrations near a spatial frequency 0.5 to 0.6 of maximum. This recovery corresponds to a frequency where the separation of the target bars is about the distance to the first diffraction ring. It sags on either side.

Spherical aberration (unobstructed)

Spherical aberration (unobstructed)

Fraction of maximum spatial frequency

Fraction of maximum spatial frequency

### Fig. 10-5. MTF characteristic of correction error.

The MTF worsens significantly when the aperture pupil goes from 1/8 to 1/4 wavelength of total aberration, emphasizing that optical quality begins to fail at around 1/4 wavelength of correction error.

Spherical aberration shifts light from the central disk to the outer parts of the diffraction pattern. A peculiar feature of spherical aberration is that it leaves the central core of the image alone (until the aberration is quite strong) other than to sap it of its intensity. Spherical aberration drains energy from the central Airy disk and feeds it to the rings. In Fig. 10-6, the encircled energy ratio of increasing low-order spherical aberration is shown plotted against reduced angle. Note the flat or near-flat appearance of the encircled energy ratio out to the radius of the Airy disk. This signature identifies spherical aberration of any order (even mild defocusing displays this peculiar behavior). Energy has been removed from the central disk of the aberrated pupil, but it holds the unaberrated shape fairly well until out beyond the first ring. Also in Fig. 10-6, the encircled energy ratio for 1/4 wavelength of spherical aberration intercepts zero radius at 0.8, just where the Strehl ratio says it should.

Angle (Airy disk edge at 1.22)

Fig. 10-6. Encircled energy of unobstructed apertures suffering from spherical aberration divided by the encircled energy of a perfect circular aperture.

Angle (Airy disk edge at 1.22)

Fig. 10-6. Encircled energy of unobstructed apertures suffering from spherical aberration divided by the encircled energy of a perfect circular aperture.

The effect of draining the energy in the core image is to maintain a more or less equivalent (but dimmer) Airy disk surrounded by a blurring that lessens with distance. Planetary detail suffers considerably, because low contrast dark markings are often very close to bright areas that bleed or wash over. The MTF chart indicates that the worst effects of correction error begin to occur for surface markings having separations of about 1/3 to % the resolution limit of the instrument. Thus, an aberrated telescope capable of resolving stars separated by 1 arcsecond shows planetary detail

(such as banding separated by less than about 2 or 3 arcseconds) with markedly reduced contrast.

### 10.5.2 Star-Test Patterns of Correction Error

When star testing, you may wish to view the image through a color filter to avoid color mixing effects. Because the maximum sensitivity of the human eye is around yellow or green, you probably can derive the most information from a filter centered on these colors. If you use an artificial source (such as a flashlight) some serendipitous filtering is caused by the lower color temperature of the filament. You may discover for yourself the helpful effects of filtration when star testing a telescope on Arcturus during a hazy night. The star is naturally yellow and so heavily colored by passage through the haze that it becomes orange. Few color filters cut out all of the lights of other bands, however. You shouldn't expect the monochromatic diffraction patterns calculated here to be precisely reproduced.

Of course, you are welcome to try other filters, but be aware that as you go from blue to red the wavelength error lessens. Red filters may help subtract small amplitude trash from a diffraction-pattern image when you are trying to see broad deformations of the wavefront.

The behavior of the in-focus image is depicted in Fig. 10-7, where spherical aberration runs from 0 to 1.7 wavelengths. All correction errors are different from the perfect aperture, but the first ring brightens badly for aberrations greater than '/4 wave. Since the use of a secondary is so common, a similar comparison is made for a 33% obstructed aperture in Fig. 10-8.

Out-of-focus behavior appears in Figs. 10-9 and 10-10 with each successive pair having slightly worse undercorrection. Each of these patterns is calculated for 10 wavelengths defocusing aberration. The severe 1.7 wavelength case is calculated for overcorrection (note the reversed appearance of the patterns). Here the strong outer ring appears outside of focus. We are viewing slices of the horn-shaped caustic in these figures. On one side of focus, the horn is sliced near the flaring bell. As a consequence, most of the energy is confined to a thin outer ring. On the other side of focus, the horn is sliced near the mouthpiece, so much of the energy is concentrated near the center. Still, a lot of energy spills out into the surrounding area to make the disk appear blurred.

Again, the out-of-focus behavior is depicted for a 33% obstruction in Figs. 10-11 and 10-12. The 1.7-wavelength error doesn't precisely reproduce the pattern expected of the Hubble Space Telescope before the repair mission, but this behavior is a good approximation.

With lesser amounts of spherical aberration, you need to defocus less to show the patterns well. Figure 10-13 shows the appearance when defocusing aberration is only 5 wavelengths.

### 10.5.3 Estimation of the Severity of the Problem

Correction error creates a marked contrast between the inside-focus and outside-focus star-test patterns. An experienced observer under excellent conditions can certainly detect errors smaller than 1/10 wavelength and possibly 1/20 wavelength (Welford 1960). Ironically, the star test for spherical aberration is almost too sensitive. It is so revealing that nearly any telescope fails casual inspection.

A high-resolution light detection system would allow measurements over the expanded stellar disk and determination of exactly how the aberration affects the telescope. Because the eye is a terrible radiometer, it cannot be trusted to measure brightness. People who use the eye to determine the magnitude of variable stars are only successful if they follow a careful procedure using similar comparison stars. Estimating brightness of extended objects (like defocused star disks) is hopeless. Meticulously calibrated light sensors have been used to perform this job, but such a solution requires precise knowledge of the defocus distance. It is not practical for those wishing to do a fast test. (For an example of these difficult measurements, see Burch 1985.)

A method must be developed that uses the strengths of vision instead of its weaknesses, some sort of tool that does not rely on the eye's absolute ability to determine brightness. A hint of the method appears in The Amateur's Telescope, by Rev. William F.A. Ellison, which was reprinted in Amateur Telescope Making Book One: (Ingalls 1976)

It is easy enough to see, by the out-of-focus images of a star, what is the state of correction of the mirror. A truly corrected mirror, out-of-focus, will give an expanded disk, uniformly illuminated except for faint traces of diffraction rings, having a clean, sharply defined edge, and a round black spot in the center. This black spot is the shadow of the fiat, and it should be the same size at equal distances inside and outside focus. If it is larger inside focus, the mirror is under-corrected. If it is larger outside, it is over-corrected. And many a time on a night when temperature was variable, the writer has watched a mirror change through all these phases within not very many minutes, the changes of the black spot answering faithfully to those of the thermometer... [italics in original].

Chapter 10. Spherical Aberration b) SA = -1/8 7

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