Fig. 10-7. Focused patterns for 0, 1/8, 1/4,1/3,1/2, and 1.7 wavelengths of lower-order spherical aberration. The aperture is unobstructed.

Fig. 10-9. Undercorrected apertures inside focus (left) and outside focus (right). Defocusing aberration ±10 wavelengths, a) 0, b) '/8, c) 1/4, wavelength lower-order spherical aberration. Aperture is unobstructed.

Fig. 10-11. Undercorrected apertures inside (left) and outside focus (right). Defocusing aberration ±10 wavelengths, a) 0, b) 1/8, c) '/4 wavelength lower-order spherical aberration. Aperture is 33% obstructed.

Fig. 10-11. Undercorrected apertures inside (left) and outside focus (right). Defocusing aberration ±10 wavelengths, a) 0, b) 1/8, c) '/4 wavelength lower-order spherical aberration. Aperture is 33% obstructed.

Fig. 10-12. Severe spherical aberration inside (left) and outside focus (right). Defocusing aberration is ±10 wavelengths, a) 1/3 wavelength under corrected, b) 1/2 wavelength under corrected, c) 1.7 wavelengths over corrected. Aperture is 33% obstructed.

Fig. 10-12. Severe spherical aberration inside (left) and outside focus (right). Defocusing aberration is ±10 wavelengths, a) 1/3 wavelength under corrected, b) 1/2 wavelength under corrected, c) 1.7 wavelengths over corrected. Aperture is 33% obstructed.

Fig. 10-13: Under corrected apertures inside (left) and outside focus (right). De-focusing aberrations ±5 wavelengths, a) 1/8, b) '/4, and c) 1/2 wavelength under-corrected. The aperture is unobstructed.

Fig. 10-13: Under corrected apertures inside (left) and outside focus (right). De-focusing aberrations ±5 wavelengths, a) 1/8, b) '/4, and c) 1/2 wavelength under-corrected. The aperture is unobstructed.

As it turned out, these comments were imbedded in an argument that seemed critical of star testing. Perhaps many of Ellison's readers were confused by this discussion into thinking that the star test was inadequate. Ellison's point, however, was valid. The plate glass mirrors common at that time were untestable in an environment that was rapidly varying in temperature. Any test would have failed in this situation.

Modern materials used in mirror substrates are much less prone to deform with temperature changes. As long as the telescope is close to ambient temperature, the optics are reasonably well-behaved. The test is reliable for slowly changing exterior temperatures.

In any case, the shadow spot gives testers a way to estimate the aberration. Figure 10-14a shows a longitudinal slice through a perfect aperture's focus point. The objective lens or mirror is to the left, the outside-focus direction is to the right (for an explanation of labeling, see Appendix D). Except for the spot activity along the axis, looking like beads on a string, the out-of-focus profile is almost smooth and uninteresting. In 10-14b, we are looking at the same situation with a 33% obstruction. The aperture is otherwise perfect, and this situation is symmetric.

In dark cones emerging from the center, the shadow of the diagonal seems to break out of the defocused image at a finite distance close to either side of focus. Since the image is quite small, the bright spot at the center delays the appearance of the central obstruction until the defocusing aberration is close to 2 wavelengths on either side. (See Chapter 5 for the conversion of defocusing aberration to focuser motion.) The eyepiece must be moved a little more until the spot is clearly defined. Nevertheless, notice that the breakout points for a perfect mirror are balanced; they are the same distance on either side of focus.

What happens when we add some undercorrection to the obstructed aperture? The answer is shown by Fig. 10-15.

The first point of interest is that the best focus point slides a little forward with progressively worse undercorrection. The aberration was entered as a Zernike polynomial, but those functions have a slight focus shift for obstructed apertures.

The next peculiarity is the small size of the disk inside focus compared with that of the outside. This condition is caused partly by the obstruction focus shift, but it is noticeable in Fig. 10-12 above, which has been corrected for this shift. No unique focal point exists for the aberrated wavefront. Approaching focus, the wavefront must buckle and change shape, manifesting itself in these different sizes.

Energy conservation also plays a role. In the longitudinal slice figures, a vertical line cannot be drawn that doesn't intercept an illuminated region. The intensity is never allowed to turn itself off everywhere in a sliced plane.

a) perfect unobstructed slice pattern

Spherical Aberration 32

Fig. 10-14. A longitudinal slice through the focus of a) a circular unobstructed aperture, b) a 33% obstructed aperture. Neither pattern has any aberration associated with it. The slice is taken from defocusing aberration of —8 wavelengths to +8 wavelengths. The corner angle of 32Xf/D corresponds to the ray-tracing edge of geometric shadow ±8 wavelengths defocusing. Thus, the picture has been squeezed until it resembles the cone of an f/1 system.

Fig. 10-14. A longitudinal slice through the focus of a) a circular unobstructed aperture, b) a 33% obstructed aperture. Neither pattern has any aberration associated with it. The slice is taken from defocusing aberration of —8 wavelengths to +8 wavelengths. The corner angle of 32Xf/D corresponds to the ray-tracing edge of geometric shadow ±8 wavelengths defocusing. Thus, the picture has been squeezed until it resembles the cone of an f/1 system.

In fact, if we very carefully keep track of the total energy at any value of defocus, we discover it to be the same as the total energy that passed through the aperture. The gnarls and knots are just an arrangement. An appearance of a bright ring is counterbalanced by a dark ring showing up elsewhere in the sliced plane.

The dark cones of the secondary shadow are no longer at equal offsets in the presence of correction error. This is made clearer in the stick-figured drawing of Fig. 1016. In the '/4-wavelength diagram, the central obstruction does not show itself until it is about twice as far from best focus.

Two effects are conspiring to offset the secondary-shadow breakout point. One is the clustering of energy around the rim of the horn-shaped b) SA = -1/4 OB = 33%

Fig. 10-15: 33% obstructed apertures show the differing distances of the emergence of the secondary shadow from the center of the diffraction disk. Correction error is a) -1/8 wavelength, b)-1/4 wavelength, and c) -1/4 wavelength (all undercorrected).

Fig. 10-15: 33% obstructed apertures show the differing distances of the emergence of the secondary shadow from the center of the diffraction disk. Correction error is a) -1/8 wavelength, b)-1/4 wavelength, and c) -1/4 wavelength (all undercorrected).

caustic on one side of focus. This fierce excavation of energy from the center allows the secondary shadow to burst forth more quickly. The other effect is the pile-up of energy toward the mouthpiece of the horn on the other side of focus. This intensity fills in the secondary shadow and retards its reappearance. Not until the eyepiece has been moved well past the caustic region is the secondary shadow allowed to poke free.

A criterion can be defined for low-order spherical aberration. We will demand that the ratio of the breakout distances be no more than 2:1 or 3:1. Of course, the evaluation must be performed on a 33% obstructed aperture. This test is different than the one described by Ellison. He defocused equal distances and compared the size of the shadows. Here, we will estimate the relative distances on either side of focus that the shadow firmly shows itself.

Such a criterion would be exceedingly weak if it were based only on a single theoretical plot. It comes from long experience with star-testing telescopes for which other tests had also been performed. This tolerance is not absolute by any means. The reappearance of the shadow can depend on the brightness of the star, the seeing, and the admixture of other aberrations. The tester must take into account the general performance behavior of the telescope before rejecting it for failure of the "2:1 test" alone. Nevertheless, I have seen no mirror having '/4-wavelength correction error (as determined by the zonal Foucault test) give a lower ratio.

This 2:1 ratio test is useful for other telescopes besides reflectors. The telescope should be obstructed even it has no natural secondary. Refractors can be artificially obstructed by centering a piece of paper over the telescope opening.

Reading the offset, however, is by no means a well-controlled process. If the obstruction is 25%, the cutoff drifts up to about 3:1. Since most Newtonian reflectors have obstructions of less than 33%, this test can be standardized by making a larger mask to attach to the back of the spider. Of course, the native obstruction of most Schmidt-Cassegrains is very close to 33% already.

Also, finding the breakout points of obstruction shadow is a much more straightforward process on medium to high focal-ratio telescopes. It is much easier to read on telescopes with focal ratios above f/8. The difficulty of making this estimate on faster instruments is exacerbated by rack and pinion focusers and the tiny depth of focus.

This procedure seems to work well in white light, because having a multiplicity of colors tends to wash out the minima in surrounding diffraction rings, or at least make them less distinctive. The secondary appears in all colors, but details in the disk depend on the color. Unless you're testing a refractor, try removing the color filter to check for spot size.

The source should not be too bright. Since you're inspecting the image close to focus, examining a bright star could overpower the eye and make seeing details difficult at the center of the image. If you are doing the test with an artificial source, you may want to put the illuminator at a greater distance, use a smaller reflector, or use a neutral-density eyepiece filter. If the telescope seems to have an unusually high offset, try again with a dimmer star.

The behavior of one color is shown in Fig. 10-17. The diagrams go from slightly inside focus at top left to somewhat farther outside focus at lower right. Keep in mind that this diagram is reproduced here at too great a magnification. If you have difficulty seeing where the shadow permanently and strongly reappears in Fig. 10-17, place the page at some distance. You'll find that the shadow isn't really apparent until well beyond focus.

The shadow appears to have equivalent central depressions at about -1.5 and 3.75 defocusing aberration.

3.75 compared to 1.5 seems to be a bit more than the estimate of 2:1. but recall that this transition point is a rough one. Many compromises have been made in generating these image patterns on paper. The most important approximation is that the figures are not self-luminous. Absolute image brightness and contrast have also been allowed to slide so that they could be printed on a medium with limited dynamic range.

An additional effect of defocused spherical aberration is demonstrated in Fig. 10-17. As you defocus toward the bell of the caustic (inside focus for undercorrection), the shadow breaks out abruptly and cleanly. On the other side, the shadow first appears as a soft central depression or a navel. The secondary shadow uncurls or blooms as it appears. The point of undoubted appearance is less crisp but is still appreciably different than the other side. Because of this uncertainty, the method is not suggested as a measurement technique. It is only a way of detecting unusual amounts of correction error which could cripple your telescope. The point at which you should become concerned about the correction of your telescope is when the ratio exceeds 3:1, but you cannot use this ratio method to measure spherical aberration precisely.

If an otherwise good telescope is failing this test, you could have an interfering aberration of a different type. If you suspect that the location of the shadow breakout is giving you the wrong answer, shift to a fixed-distance pattern comparison (as in Fig. 10-13) by looking up the value of the defocus in Table 5-1. Be careful of Ellison's warning. Let the optics cool down completely. Pyrex is a better material than plate glass, but its shape is not completely independent of temperature change.

Sometimes the A'6 coefficient is neglected or uncorrected. In most telescopes, this aberration makes little difference, but it can be a problem for some unusual instruments.

For example, the shape of a Schmidt corrector plate is similar to the fourth-order curve in Eq. 10.3 with a different amount of p2 cleverly chosen to minimize potential chromatic aberration. The fast spherical primary produces an aberration function with many terms in the expansion in Eq. 10.2, but the corrector plate is capable of easily correcting these terms out only to fourth order. A small value of sixth-order aberration may remain uncorrected. Various lens designs can also add trifling amounts of "secondary" spherical aberration of sixth order on the wavefront (Kingslake

Fig. 10-17. Defocused stellar images of a 33% obstructed aperture having % wavelength under correct ion. Each frame is magnified so that the perfect geometric profile is the same size as the labeled frame. Thus, the edge is at 10 units of angle for 2 wavelengths of defocusing, 20 units for 4 wavelengths, etc. The bottom of each box is marked with the defocusing aberration in wavelengths.

Fig. 10-17. Defocused stellar images of a 33% obstructed aperture having % wavelength under correct ion. Each frame is magnified so that the perfect geometric profile is the same size as the labeled frame. Thus, the edge is at 10 units of angle for 2 wavelengths of defocusing, 20 units for 4 wavelengths, etc. The bottom of each box is marked with the defocusing aberration in wavelengths.

Higher-order spherical aberration can be safely neglected in most instruments. Nevertheless, in certain ultra-fast catadioptrics or complicated refractor designs, you should not be surprised to see small amounts of the aberration described in the next section.

10.7.1 Star-Test Patterns of Higher-Order Spherical Aberration

An A'6 coefficient in Eq. 10.3 yields the patterns of Fig. 10-18. Like a fourth-order correction error, an A'6 with the opposite sign results in these patterns being reversed in focus direction.

The error looks worse in the star test than it behaves in the image. To reduce the Strehl ratio to the same 0.8 value that it possesses for 1/4 wavelength of lower-order spherical aberration, A'6 would have to be increased to about 0.4 wavelength.

The description of the higher-order caustic as a horn pulled back halfway through itself helps explain these complicated patterns. The correction error star-test patterns that appeared earlier in the chapter went from a bright outer ring on one side of focus to a fuzzy bright core on the other side. Here the fuzzy bright core appears on the same side of focus as a hole in the center (Fig. 10-18, —3 wavelengths defocused).

For that reason, high-order spherical aberration might be called the "ring aberration." It bears more than a passing resemblance to the zonal defects appearing in Chapter 11. Indeed, this aberration can be viewed as the broadest of the zonal aberrations.

Of course, higher-order spherical aberration is rarely seen in clearly identifiable form. As a residual aberration in a normal telescope, its amplitude is very low. This aberration is usually swamped by other effects. I have seen a slight amount only in one Schmidt-Cassegrain, where a dark secondary shadow was coupled with diminishing brightness toward the edge of the out-of-focus diffraction disk. The other side of focus revealed the opposite behavior, with a light secondary shadow and a strong outer ring showing simultaneously. The only reason I was capable of unambiguously seeing this small aberration was the almost complete absence of simple correction error in an exceptionally good instrument.

10.7.2 Filtering of Higher-Order Spherical Aberration

The filter graph for higher-order spherical aberration appears in Fig. 10-19. Clearly, 1/4 wavelength of aberration does not seriously affect the optics. Not until the aberration has been increased to 0.4 wavelengths does the damage become considerable. The worst part of the decrease occurs at

10 10 10

Fig. 10-18. Star-test patterns for 1/5 wavelength of higher-order spherical aberration at best focus. Obstruction is 33%.

10 10 10

Fig. 10-18. Star-test patterns for 1/5 wavelength of higher-order spherical aberration at best focus. Obstruction is 33%.

a lower spatial frequency than it did in Fig. 10-5 for low-order spherical aberration. In this case the fall occurs at about 20% of the maximum spatial frequency instead of 35%. Recalling that the maximum resolution of a 200-mm aperture is about 0.6 cycles/arcsecond, this aberrated system transfers surface details separated by less than 3 arcseconds with reduced contrast.

When the "elbow" of the MTF curve appears farther to the left, it is a sign of a more corrugated appearance of the aberration function. The surface error becomes more localized. As the optical error becomes smaller and goes through more wiggles, the corresponding MTF exhibits a sharper decline at lower spatial frequencies. As the optical errors become more localized, the MTFs at higher spatial frequencies are also reduced but don't oscillate much. The damage is already done at lower spatial frequencies. However, it should be emphasized that pure higher-order spherical aberration of this magnitude is unlikely to trouble ordinary instruments. If the optics are fabricated poorly, the bulk of the aberration is usually expressed in simple fourth-order correction error.

10.8 A Compact, Uniform Standard for Optical Quality

Consumer telescope makers and observers alike tend to divide aberration and obstruction into separate compartments, treating the two as incomparable phenomena. However, a single standard can easily be defined to cover them both. It is based on the encircled energy ratio (or EER(8)). The encircled energy ratio gives a way of comparing these two degradations on an equal footing.

Here's the way such ratios are calculated: First, we find what fraction of the energy from a point source is focused by the imperfect telescope on a tiny circle of specified angular radius at the focal plane. This number is then divided by the same fraction for a perfect, unblocked aperture of the same diameter. For example, a moderately obstructed telescope that also has a trifling amount of spherical aberration encircles 72% of its energy at a certain angle and a perfect aperture encloses 84% at the same angular radius. The encircled energy ratio would then be 72/84 = 0.86 at that angle.

Please note that all these ratios go to unity as radius goes to infinity. They are corrected for any simple dimming of the aperture by a secondary mirror or apodization. The degradation of the image caused by diffraction is more pertinent than a simple transmission loss (remember the extinction paradox of van de Hulst). About half of the lessening of the central intensity is caused by simple dimming (light that hits the rear of the secondary), and half is caused by the damaging effects of diffraction. Punishing a telescope for an effect that does not increase the point spreading is unfair, so it is normalized out. Besides, if we are going to talk of the absolute transmission, we also must know details of the coatings and internal reflections. These are unknown in the general case, so accounting for dimming caused by the secondary is an incomplete treatment.

The encircled energy ratios appearing in Figs. 9-1 and 10-6 are complete curves. For a single number that represents a quality criterion, one needs to take the EER(8) value at a specific value of 8. The question arises: what angle is best?

Unfortunately, no one angle is the last word on optical quality. We could choose an angle (or circle) very near the center of the image, or EER(8^0). This number is close to the normalized brightness ratio at the center of the diffraction disk. In fact, it is identical with the Strehl ratio in unobstructed apertures. EER taken near the center of the image, however, seems excessively tolerant of obstruction, as Fig. 9-1 demonstrates. EER(8^0) does not dip below 0.8—the cutoff point of good optical quality in the Strehl ratio—until obstruction is above 45%. One could also define the quality factor as EER within a circle of radius 8 = 1.221/D (Eq. 1.1), or the edge of the Airy diffraction disk. Fig. 9-1 shows that the values of EER(1.22) sag considerably, and have even started to rise again.

Somewhat arbitrarily, this book will use an angular radius of 8 = UD, or the angular spacing where the MTF always goes to zero. This angle has the practical advantage of catching obstructed apertures at their low points in Fig. 9-1 and has the philosophical advantage of always being related to the maximum spatial frequency of the MTF chart. This ratio will be called EER(l). In Fig. D-2, you can see the edge of the frame is at an angle of 1.22X/D. Thus a circle drawn here would be sitting in the darkness between the rings. The integrated area of EER(l) is slightly inside the bright edge of the disk.

EER(l) of apertures mixing the two optical problems of obstruction and lower-order (Zernike) spherical aberration have been collected together in Table 10-1. We see a very similar behavior to the Strehl ratio in the unobstructed top row. A quarter wavelength of spherical aberration still results in a degradation of EER(l) to 0.8. It is the second axis that is most interesting, however. It is possible to compare the loss of encircled energy ratio of obstructed, but otherwise perfect apertures. EER(l) = 0.8 for obstructions slightly less than 33% of the full diameter.

Notice that obstruction does not always diminish quality. The case of /-wavelength correction error shows a curious inversion, with increasing obstruction serving to cover up the poor figuring.

Table 10-1 EER(l) for apertures with lower-order spherical aberration.

Wavefront is refocused.

Obstruction is fraction of diameter covered.

Peak-to-valley correction error on unobstructed aperture

Table 10-1 EER(l) for apertures with lower-order spherical aberration.

Wavefront is refocused.

Obstruction is fraction of diameter covered.

Peak-to-valley correction error on unobstructed aperture

0 |
1/8* |
1/6* |
1/5*. |
1/4* |
1/3* |
1/2* | |

Obstruction | |||||||

0.00 |
1.00 |
0.95 |
0.91 |
0.87 |
0.80 |
0.67 |
0.39 |

0.15 |
0.95 |
0.91 |
0.87 |
0.84 |
0.78 |
0.66 |
0.41 |

0.20 |
0.92 |
0.88 |
0.84 |
0.81 |
0.76 |
0.65 |
0.42 |

0.25 |
0.88 |
0.84 |
0.81 |
0.78 |
0.74 |
0.64 |
0.43 |

0.30 |
0.82 |
0.79 |
0.77 |
0.75 |
0.71 |
0.63 |
0.44 |

0.33 |
0.79 |
0.76 |
0.74 |
0.72 |
0.69 |
0.61 |
0.44 |

0.40 |
0.71 |
0.69 |
0.68 |
0.66 |
0.63 |
0.58 |
0.45 |

0.50 |
0.58 |
0.57 |
0.56 |
0.56 |
0.54 |
0.51 |
0.44 |

My personal experience with a large number of telescopes having various amounts of correction error suggests the following empirical ratings. These cutoffs are necessarily hazy, and the "good" point is deliberately chosen to match the Strehl ratio (i.e., 0.8), where optics are conventionally called "diffraction-limited."

1. 0.88 1.00 excellent to perfect

2. 0.80-0.88 good to excellent

The only acceptable instruments with EER(l) below 0.70 arc specialpurpose telescopes, such as astrocameras or richest-field telescopes. No instrument having 1/3 wavelength of correction error, even if unobstructed, reaches up to this minimum standard. No aperture with an obstruction slightly larger than 40%, even if figured perfectly, ever meets it.

All telescopes are made with some spherical aberration. The perfect Newtonian paraboloid, for example, is an unattainable goal between an infinite number of prolate spheroids and hyperboloids. The question is whether the telescope suffers under the load. Once EER(l) is over 0.88 or so, spherical aberration is gratifyingly small and the optics could justifiably be called "perfect."

We saw in Chapter 3 how modulation transfer functions stacked individually. Most obstructed telescopes are teetering on the brink already. It takes very little to push them over. By such logic, we should be intolerant of any correction error, but that attitude is unrealistic.

Commercial telescope optics have always been corrected to a tolerance of about 1/4 wavelength. The way that accuracy is stated has changed, but commercial telescope makers still fabricate the same V4-wavelength optics they always did.

Let's recognize a simple fact. Making objectives to higher accuracy than '/4-wavelength is expensive. The scaling of price with quality is similar to the scaling of price with diameter. Incremental improvements in surface accuracy cost much more because we are paying not for glass but for an optician's valuable time. For better or worse (usually worse), buyers use price as a strong deciding factor.

Is more accuracy really needed? In informal tests, a telescope with a 1/4-wavelength correction error has been found difficult to distinguish from a very good telescope unless seeing is excellent and the observer is skillful (Ceravolo et dl. 1992; also see Chapter 15). For most people who observe under average skies, a V4-wavelength correction error represents an acceptable compromise between quality and the price of optics.

In the previous section, we defined those apertures with encircled energy ratios greater than 0.88 as excellent. We see this designation only applies to the upper left corner of Table 10-1, i.e., to obstructions less than about 25% or correction errors less than 1/5 wavelength. Notice that a 25% obstructed aperture with only 1/6 wavelength of correction error is still "good" at 0.81, but that a 15% obstructed aperture with a V4-wavelength error is below the cutoff at 0.78. The lesson is clear. Accurate figuring allows the telescope to get away with other difficulties.

Personally, I find the images of optics that are nudged against the Rayleigh limit a bit too soft. However, of the telescopes I've tested, most of

Chapter 10. Spherical Aberration them that obviously didn't perform well on the sky have been much worse than Rayleigh's limit. A quarter wavelength of correction error is barely acceptable if it is the only significant problem. With a reasonable 25% obstruction, such an aperture has EER(l) = 0.74, and has a transfer function better than a perfect, unobstructed aperture 1/2 to 1/3 of its size. Even with optical problems of this magnitude, a 6-inch f/8 reflector is at least as good as a perfect 3 to 4-inch apochromatic refractor. At some spatial frequencies, it is better.

Was this article helpful?

Through this ebook, you are going to learn what you will need to know all about the telescopes that can provide a fun and rewarding hobby for you and your family!

## Post a comment