Obstruction and Shading

Modifying the transmission of the aperture pupil causes changes to the diffraction pattern. Obstruction and shaded transmission are not deformations of the wavefront in the same sense as aberrations. They can occur in officially "perfect" apertures. Nevertheless, they can affect the perceived image quality in a very similar manner.

Five main points are made in this chapter:

1 Central obstructions below 20% of the aperture are indistinguishable in practice from an unobstructed aperture, and for obstruction under 25%, performance can be very good.

2. Reckless efforts to reduce central obstruction can lead to even worse images than those resulting from obstruction.

3. A spider in front of an aperture hurts the image only for dim objects next to bright sources of interference or for low contrast objects imbedded in an extended field. For most dark-field observing, the spider's effect is only cosmetic.

4. Darkening the outside portion of the aperture results in contrast improvements at low spatial frequencies, but only at the expense of high spatial frequencies.

5. Dust and scratches are cosmetic errors, similar to spider diffraction except for certain types of observing.

9.1 Central Obstruction

The most obvious and potentially the most damaging kind of transmission change is caused by the centrally placed diagonal or secondary mirror.

Some observers are almost fanatical about central obstruction. In various astronomical publications, they have made blanket statements such as "obstruction reduces contrast" without giving the spatial frequencies at which that reduction occurs. They imply that obstruction so severely damages the image that no amount can be tolerated.

However, the negative consequences of central obstruction can be readily and precisely calculated. We will see that they worsen considerably beyond a fractional obstruction of 20 to 25% of the aperture. As long as the obstruction is kept inside that fraction, the image closely approximates that of an unobstructed telescope.

tj 1

LU HI

00 0

Fig. 9-1. The transmitted encircled energies of centrally obstructed apertures divided by the encircled energy of an unobstructed aperture (later called the encircled energy ratio, or EER). The performance drops off sharply at 20-25% obstruction.

Figure 9-1 does not illustrate contrast but the closely related topic of encircled energy. In Fig. 9-1, the normalized encircled energies of the obstructed apertures are divided by the normalized encircled energy of a perfect circular aperture. Thus, the obstructed aperture ratios deviate from an ideal value of unity.

As the circle approaches the radius of the unobstructed Airy disk at 1.22, these ratios fall. The spot size is smaller in obstructed instruments, and the energy robbed from the core diffraction spot is mostly deposited in the first one or two diffraction rings. While the obstructed pattern is crossing the minimum between the central disk and the first ring, it encloses little additional energy and the unobstructed pattern gets ahead of it. Not

Fractional transmitted encircled energy compared to perfect aperture

Fractional transmitted encircled energy compared to perfect aperture

tj 1

LU HI

00 0

.-T—._______

_ 25% obstructed

15% obstructed

50% obstructed

i i ... .

Angle (Airy disk edge at 1.22)

Angle (Airy disk edge at 1.22)

Fig. 9-2. The in-focus diffraction patterns resulting from central obstruction.

until the circle encloses the first rings of both patterns does the ratio begin to recover.

In the focused diffraction patterns of Fig. 9-2, the intensities of the rings swell as obstruction is increased. By the time a 75% obstruction is reached, all pretense of optical quality is lost. Paradoxically, the diffraction disk is smaller. All of these images are calculated at the same scale and printed with the same central intensity, so this shrinkage cannot be explained as an artifact of the reproduction. This phenomenon is real. In fact, the narrowest central disk is found in an aperture that is almost entirely obstructed, but the powerful rings caused by such an aperture render it useless for fine imaging.

Filtering appears in Fig. 9-3. Again, the bottom does not drop out at middle frequencies until the obstruction is beyond 25%. An obstruction diameter under 20% of the aperture can be viewed as acceptably small. The narrower spot size shows itself as an increase in the MTF at high spatial frequencies; contrast here exceeds even the value for a perfect aperture.

Central obstruction filtration

Central obstruction filtration

o.o

Fraction of maximum spatial frequency

Fig. 9-3. MTF curves of simple central obstructions.

Fraction of maximum spatial frequency

Fig. 9-3. MTF curves of simple central obstructions.

These curves demonstrate that a little breathing room is available between 20% and 25% of the aperture. The negative effects of central obstruction have begun to show themselves, but they are saving their full fury for obstructions beyond 30%. Any aperture that is 25% obstructed can be very good, and telescopes that block 20% of the diameter can be excellent. Instruments that have been modified to achieve less obstruction than 20% are obtaining very little contrast gain and are risking other optical problems. (See the end of Chapter 10 for more information on the degradation caused by obstruction in the presence of spherical aberration.)

In the absence of hard knowledge about the tolerable obstruction, telescope makers often cure this difficulty by immoderate measures. Commonly, they build Newtonian telescopes with overly squat focusers and minimum-size diagonals.

Unfortunately, reducing diagonal size often forces a compromise with the operation of other useful features of the telescope. Perhaps it even cuts into an unforeseen safety margin. For example, a long focuser tube baffles external light. Some telescopes are designed with eyepieces set so low that light can enter them directly, a problem especially common in open-tube telescopes. Another difficulty arises when a Barlow lens must be used. The focus should be far enough away from the optical path that the Barlow won't jut out in front of the mirror.

The diagonal is also prone to curvature near its edge, so telescopes that require every bit of the diagonal for on-axis imaging often have reduced quality. Certainly, small diagonals cause vignetting, and observers must be careful to make sure that the outer parts of the field of view are adequately illuminated for their favorite objects.

Thus, a well-meaning effort to improve contrast by reducing the diagonal size might have the opposite effect. The instrument may be so malformed by such efforts that contrast is much worse than it would have been with a slightly larger obstruction.1

9.2 Spider Diffraction

The support hardware that holds a secondary mirror causes some light diffraction. For linear spider vanes, the pattern takes the form of two or more radial spikes away from a point image. Spider diffraction takes a bright, extended image and smears it to either side.

Contrast is reduced, but by how much? Clearly, if the area of the vanes is not large when projected against the mirror, the spider cannot be scattering much light. That light looks brighter because it is blurred in only a few directions.

The MTF of a spider with vanes of thickness 7128 the diameter of the mirror appears in Fig. 9-4 (this MTF is compounded with the degradation of a 20% obstruction). The thickness is 716 inch (nearly 2 mm) for an 8-inch (200 mm) mirror. Usually, small telescopes have vanes less than half that thick. Even so, the contrast is degraded only about 1.6% at first. The area of the vanes is also 1.6% of the aperture's area. Thus, a simple relationship exists between the area of small-scale obscuration and the amount of abrupt degradation in the MTF.

Spider diffraction causes the quick drop of the MTF, and thereafter the degradation is about a constant fraction of the normal behavior.2 One can see from the very slight decrease that spider diffraction is a cosmetic defect for most dark-field observation. Only in exceptional situations does it significantly affect image contrast. If, for example, a dim star resides in the spider diffraction spike of a bright companion star, this light could become a problem. The spider diffraction does not cause difficulty if the dim star is alone or at a different angle to the spike, but because the nearby star is so bright, the spike may illuminate the whole area around the dim star. Fortunately, many telescopes allow rotation of the spider or tube.

1 These examples are not meant to imply that a small secondary mirror is not a laudable goal. If telescope designers are aware of the risks, they can avoid them. The key is not to reduce the secondary size to the exclusion of every other optical consideration.

2 For modulation patterns at 45°, the MTF will recover at 71% the maximum spatial frequency and even slightly exceed the value expected in its absence. MTFs for horizontal or vertical bar targets display less violently-changing structure. See Zmek 1993.

Accuracy Ratio

Fraction of maximum spatial frequency

Fig. 9-4. Modulation transfer function for spider diffraction. The degradation is slight compared to a spiderless 20% obstructed aperture. The lowest curve is for anMTFpattern at 45° to the vanes.

Fraction of maximum spatial frequency

Fig. 9-4. Modulation transfer function for spider diffraction. The degradation is slight compared to a spiderless 20% obstructed aperture. The lowest curve is for anMTFpattern at 45° to the vanes.

On extended bright objects such as planets, spider diffraction causes a much subtler degradation. Each point on the object has its own spikes. Low contrast details can be washed out by weak light diffracted from the spider. A realistic thin-vaned spider has a surface area of about 0.5% the area of the aperture. Thus, the signal-to-noise ratio could be as low as 23 dB. Luckily, much of the stray light is beyond the edge of planets. If the vanes are as narrow as 0.5 mm, the width of the diffraction spikes is about 100 arcseconds. Spider-vane diffraction would greatly exceed the size of even a large planetary image, such as the 50 arcsecond angle subtended by Jupiter during oppositions. SNR exceeds 30-40 dB for planetary observation. Light diffracted from spider vanes becomes troublesome chiefly for lunar and solar observations, where the extent of the image is larger than the angle of diffraction.

Despite the conclusions from Fig. 9-4, spider diffraction can be an important design consideration. If the diagonal support vanes are much too thick, the diffraction spikes become brighter. More area of the mirror is intercepted, and more light is diverted. Thicker vanes also have shorter diffraction spikes. Since the light diffracts into a smaller area, it is relatively brighter.

You can see the effect of a wide spider vane by stretching a strip of black electrical tape across the front of your instrument (this experiment even works for refractors and Schmidt-Cassegrains—just keep the tape away from the lens). Direct the telescope towards a bright star and use a medium-

spider width 1/128 no spider 20

spider width 1/128 no spider 20

Fig. 9-5. An "overexposed" monochromatic spider diffraction pattern compared with the same pattern calculated with no spider.

Fig. 9-5. An "overexposed" monochromatic spider diffraction pattern compared with the same pattern calculated with no spider.

to-high magnification eyepiece. You should see a bright spike of light at right angles to the tape. This spike glistens with sparkles of color. If the star is bright enough, the spike fades out with increasing distance from the star and then brightens again, perhaps doing so many times. You're observing the side peaks of spider diffraction, similar to the rings of circular aperture diffraction.

Another design mistake is the use of extremely thick, curved-vane spiders or a diffraction-spike suppression mask. Two such masks were suggested by A. Couder using curved-edge vane covers (Ingalls, Book Two 1978, p. 620). The images don't suffer the diffraction spikes caused by conventional spiders, but an increased level of scattered light is still there—just spread around in angle. In imaging of extended objects (such as planets), it doesn't matter whether the light is confined to a diffraction spike or has been averaged over various angles. Contrast is lowered the same amount. The most important factor is the fraction of the mirror such vanes cover. The only acceptable change to spider vanes is the use of thin curved-vane spiders, or eliminating them entirely by supporting the secondary on an optical window.

A focused spider image is shown in Fig. 9-5. This rendering is not a very realistic picture because it is calculated for monochromatic light, such as a laser would produce. The dips in the spikes will be filled in with other colors in white-light images, giving only a hint of structure.

While spider diffraction is only a minor annoyance for small telescopes, it can become a major object of concern in the design of large instruments. Because mechanical structure does not scale linearly, the support requirements of large reflector secondaries can be very great indeed. Heavy Cassegrain secondaries must be supported rigidly enough that alignment does not suffer. Thick vanes are necessary, and diffraction from the vanes becomes considerable (Beyer and Clune 1988).

One should make certain at the end of coarse alignment that the secondary support vanes present the minimum interception area to the beam of incoming light. This maintenance costs nothing, and can improve the image substantially.

9.3 Shading or Apodization

Diffraction rings are among the most objectionable features of the perfect circular diffraction image. This problem is perhaps most observable at the boundary between a bright area and a dark sky background. If seeing is sufficiently good, this boundary does not appear sharp, but shows "echo" images, one or more thin lines at the edge. Observers must be aware of this phenomenon, or else they will report spurious detail in the image. It is particularly prominent in images that are strongly colored or viewed through color filters. Planetary disks display limb-darkening, so this effect is difficult to notice at the edge of a planet. However, the problem is observable elsewhere and is always a source of interference with any clustering of spotty detail. Several diffraction rings may add to create another spot where one didn't exist before.

Changing the transmission characteristics of an aperture is called apodization. "Apod-" literally means "without feet" and refers to a shading of the entrance pupil that results in lowered diffraction rings. Apodization existed informally before it was named. Jacquinot and Couder made a one-dimensional apodizer in the 1930s to suppress the dim side lines that appeared next to bright lines on spectrographic plates (Jacquinot 1958).

R.K. Luneburg widened the definition of apodization by suggesting a set of generalized problems that don't necessarily lead to smaller rings but induce modifications of the diffraction pattern to optimize any given characteristic.3 Based on Luneburg's investigations, it can be stated with some confidence that the best pupil shape in most cases is an unobstructed one (Luneburg 1964, pp. 344-359). One can set any sort of condition on the diffraction image, however, as long as some other parameter is allowed to swing freely.

For example, the diameter of the central spot can be minimized if the height of the diffraction rings does not matter. In fact, G. Toraldo di Francia designed a complex pupil shading that results in arbitrarily fine resolution and suppression of diffraction rings out to a specified radius (di Francia 1952). Unfortunately, such a shaded pupil is fantastically inefficient for all

3 For this reason, I personally prefer the term shading, which comes from antenna and acoustic array theory.

realistic apertures, diverting most of the energy of the beam to a bright ring beyond the specified radius. The field stop must be inside of this radius to prevent the bright ring from dazzling the super-resolved star.

Similarly, C.L. Dolph (1946) derived a shading technique for linear-array radar antennae that balances resolution and diffraction. It also works for slit apertures. Dolph's technique features arbitrary suppression of side lobes to a specified level surrounding the central peak. This method is also inefficient but helps most for strong sources where interference from multiple directions is troublesome.

An early systematic investigation into partially transparent coatings of lenses to achieve enhanced resolution was made by Osterberg and Wilkins in 1949. They were able to theoretically achieve a central spot diameter only 77% that of the usual Airy disk. The Strehl ratio of such an aperture is 0.21, so this resolution comes at considerable optical costs. The first diffraction ring is about 710 as high as the central peak. This behavior is similar to that of the obstructed apertures mentioned earlier, but it is carefully optimized to do the least damage and at the same time achieve the highest resolution.

All of these advanced solutions have the same general features: highresolution pupils look like soft-edged obstructed apertures, while low diffraction-ring pupils become darker toward the outside in a slow taper (Barakat 1962; Jacquinot and Roizen-Dossier 1964).

Also, these advanced techniques generally don't feature simple shading, such as would be provided by a variable strength neutral-density filter (particularly for resolution enhancement). They shift the transmission back and forth between positive and negative. Admittedly, the notion of negative transmission sounds odd. One imagines light springing from the eye and going back through the telescope in a reversed direction. The truth makes more sense—negative transmission refers to places on the aperture where the phase is reversed, or areas that have a uniform aberration of / wavelength. Of course, jerking the aperture to both sides of the phase severely reduces central spot intensity. Such filters are also difficult to make.

The only pupil shading considered here is a truncated Gaussian function. The Gaussian is the familiar bell-shaped curve representing statistical deviations in measurements. It is similar to the curve teachers sometimes consult when they assign students' grades. A Gaussian aperture starts with full transmission at the center and gradually tapers off until the edge is reached. The transmission coefficient of an aperture pupil is modeled by

where p is the radial coordinate and w is related to the width of the Gaussian. Figure 9-6 shows this transmission pattern. The word "truncated"

Normalized radial coordinate

Fig. 9-6. The transmission coefficient of the Gaussian pupil as it varies with radius. The outside of the aperture is at a radius of 1.0.

Normalized radial coordinate

Fig. 9-6. The transmission coefficient of the Gaussian pupil as it varies with radius. The outside of the aperture is at a radius of 1.0.

refers to the sharp drop at the outside edge of the aperture. As the width is decreased, this drop is less important, but the clear area in the center of the aperture decreases with smaller widths. If the transmission at the edge of the aperture is small, the usable window is also smaller.

The Gaussian function has a unique mathematical property. When one calculates the diffraction pattern of a perfect circular aperture, the result is a complicated expression that goes through many oscillations—the cause of diffraction rings. When the same calculation is made for an untruncated Gaussian-shaded pupil, the result is another Gaussian. Once a Gaussian function dies away, it does not rise again. Therefore, the diffraction pattern has no rings around it.

If the Gaussian function has a small drop-off at the edge (as it does for the truncated examples in Fig. 9-6), the rings reappear, but they are strongly suppressed. Figure 9-7 shows the focused appearance of a truncated Gaussian with w = 0.75p. First, note that the pattern is definitely a bit larger. Second, muted rings remain around the central spot. If we turn the pattern sideways, the longitudinal slice of the diffraction pattern from this truncated Gaussian pupil is seen in Fig. 9-8. We still see nodes, but this behavior is severely diminished. Another interesting characteristic of this diagram is the boxy appearance of the central lozenge. Compared to the regular aperture, the image seems to display a cohesiveness more tolerant of defocusing.

normal unshaded 8 truncated Gaussian normal unshaded 8 truncated Gaussian

Fig. 9-7. In-focus pattern of perfect unshaded and truncated Gaussian pupils.

normal unshaded 8 truncated Gaussian 8

normal unshaded 8 truncated Gaussian 8

Fig. 9-8. Longitudinally sliced image pattern of a normal and Gaussian pupil.

Fig. 9-8. Longitudinally sliced image pattern of a normal and Gaussian pupil.

What sort of changes to the image take place? The encircled energy of a w = 0.75p truncated Gaussian appears in Fig. 9-9. Remember, this diagram is corrected for the simple darkening of the aperture. Thus, the ratio of the encircled energies goes to 1 as the circle becomes very large. Amazingly, the ratio for the Gaussian-apodized aperture sweeps upward to enclose more of the transmitted energy than the normal aperture over most of the range.

This strange behavior is related to the suppression of diffraction rings. For a Gaussian pupil, all of the stray energy that normally is in distant portions of the image has already been gathered. A normal pupil encloses only 83.8% of the energy in the Airy disk, 91% inside the edge of the first ring, 93.8% inside the edge of the second ring, 95.2%, 96.1%, and so forth. Gaussian pupil transmission is the same as taking a broom and traveling around the image, sweeping all of the remaining intensity toward the center. This sweeping is not as tidy as one would like, so the energy raked inward is piled up at the edge of an enlarged diffraction disk.

Encircled energy of truncated Gaussian-shaded aperture

Fig. 9-9. Encircled energy transmitted by a truncated Gaussian pupil compared to a normal unobscured pupil.

ratio of ervcirdad energies

\ Encircled energy of normal aperture Encircled energy of Gaussian-shaded aperture

Angle (1.22 is Airy radius of unshaded pupil)

Fig. 9-9. Encircled energy transmitted by a truncated Gaussian pupil compared to a normal unobscured pupil.

Figure 9-10 shows the effect on filtering. Gaussian transmission enhances low spatial frequencies at the expense of high ones. The lowered response at high spatial frequency is understandable if one remembers that the central spot is bulkier. The enhancement at low frequencies is related to the gathering of energy from distant portions of the image. Low frequency modulation targets have wide bars. If the energy is piled close to the center of the diffraction disk rather than spread out, less light leaks from the light areas into the dark ones.

Thus, Gaussian apodizers will not show detail very near the resolution limit of the telescope with higher contrast, even though the diffraction rings are suppressed. On the other hand, much of an image's content is at low spatial frequency. The Gaussian filter will help the contrast of features that don't require every scrap of the resolving power. Another useful benefit of lowered diffraction rings is resolution of unequal double stars just outside the resolution limit, where the bright ring is inconveniently coincident with the dim star. For general extended images, however, making the rings less apparent does not help high resolution.

Apodization came to amateur astronomy with a series of letters and articles in the "Amateur Astronomer" column of Scientific American in the early 1950s. These articles culminated with a suggestion for pupil shading using layers of periodic screening (Leonard 1954). This suggestion was a clever and practical way of achieving a peaked pupil shape.

In spite of their mixed performance on the MTF chart, apodizers have been used in planetary viewing for many years. They have been vigorously

Fraction of maximum spatial frequency

Fig. 9-10. The MTF of a w = 0.75p truncated Gaussian pupil. The normal MTF is plotted also for comparison. The Gaussian shifts the frequency response from high spatial frequencies to low ones.

Fraction of maximum spatial frequency

Fig. 9-10. The MTF of a w = 0.75p truncated Gaussian pupil. The normal MTF is plotted also for comparison. The Gaussian shifts the frequency response from high spatial frequencies to low ones.

promoted by several amateurs, who claim that they are useful as "seeing" filters, although their explanations of the cause differ (Van Nuland 1983; Gordon 1984). This popularity, even though it has been confined to a small number of observers, is difficult to dismiss.

Is there a mechanism by which these filters can act to steady an image? Edberg (1984) suggested some indirect benefits, including stopping down the aperture, lessening the dazzling brightness of the planets, and covering poor optical fabrication. Gordon (1984) suggested that the moving rings associated with turbulence blur over into the Airy disk. If no rings are present, this blurring is reduced.

Another article suggested that a Gaussian filter could shift contrast rendition from high spatial frequencies—where turbulence was destroying the image anyway—to low spatial frequencies. Thus, coarser details that were still visible through the turbulent atmosphere were being rendered with higher contrast (Suiter 1986b). The MTF of a circular aperture troubled by turbulence provides evidence for that claim. Figure 9-11 illustrates the Gaussian pupil transmission MTF for a single moment of turbulence.

This figure shows how contrast is being shifted. Below a spatial frequency of about 0.4 maximum, the Gaussian-filtered system performs better than the open telescope. Below 0.1, the Gaussian-filtered instrument transfers contrast better than a perfect unaberrated system.

For example, a perfect 10-inch (250-mm) telescope can resolve MTF targets with bars separated by about XA arcsecond. However, when troubled

Gaussian "seeing" filter

Gaussian "seeing" filter

0.0

Fraction of maximum spatial frequency

Fig. 9-11. Improvement of the contrast transfer of low spatial frequencies for a w = 0.75p Gaussian-filter pupil suffering from 0.15 wavelengths RMS of turbulence aberration. The same aperture when unshaded is also shown. A single MTF orientation angle during one snapshot is depicted. At higher spatial frequencies, the curve is unstable.

Fraction of maximum spatial frequency

Fig. 9-11. Improvement of the contrast transfer of low spatial frequencies for a w = 0.75p Gaussian-filter pupil suffering from 0.15 wavelengths RMS of turbulence aberration. The same aperture when unshaded is also shown. A single MTF orientation angle during one snapshot is depicted. At higher spatial frequencies, the curve is unstable.

by 0.15 wavelengths RMS turbulence, the MTF curve fluctuates wildly for bars separated by about 0.8 arcseconds and below. We may then choose to throw away contrast at high resolutions—where the behavior is unreliable anyway—and shift it down to spatial frequencies where it can be profitably used. With Gaussian filtration, bar separations of about 1.5 arcseconds and higher are imaged with higher contrast than the unapodized aperture. Above 4 arcseconds, the apodized aperture is behaving better than a perfect circular aperture on a steady night.

Thus, a Gaussian apodizing filter does seem to help during bouts of poor seeing, but it does so only in a backhanded manner. The seeing filter is perhaps useful in order to transform bad seeing to passable, but for critical resolution nothing less than steady skies and full aperture are necessary.

9.4 Dust and Scratches on the Optics

We can formally use the admittedly esoteric topic of superposition of apertures to explain the effect of central obstruction. The central obstruction can be mathematically replaced by a negative oscillating aperture that just cancels the positively oscillating area of the full aperture on top of it— Babinet's principle. This procedure works the same for lesser obstructions such as dust particles, only in this case, the dust grains or scratches act like a myriad of pinhole or short slit apertures.

Tiny pinholes and slits do not have good resolution. They emit a diffuse glow that is spread throughout the field of view. This behavior is observable in certain benchtop experiments. If a bright image is barely hidden from view by a straightedge (as in a Foucault test), dust flaws can actually be seen shining on the darkened aperture.

We can predict the curve of the modulation transfer function without calculating it. It is very similar to the MTF calculated for spider vanes. If an outrageous fraction of the mirror or lens were covered with dust or scratches (say, 1%) the contrast would lurch down suddenly by 1%. After that, the degradation of the MTF would remain fairly constant. The reason for the sudden drop is simple. The MTF must always start at 1, but the diffuse glow affects narrow bars and wide ones equally. Not until the bars are very wide does the leakage from a bright bar not extend over the dark region.

Apertures with 1% of their area covered with dust are very grimy optics indeed. Nearly everyone keeps lenses and mirrors cleaner. Imagine how spotted the optics would look if, for each square centimeter, 1 square millimeter (about the area of the head of a pin) was blocked. A 200-mm aperture has an area of over 314 cm2. Sprinkling salt on the mirror or lens could scarcely produce 1% obstruction.

Dust and scratches, like spider diffraction, are mostly cosmetic errors except for unusual situations that feature bright, non-interesting objects close to the object to be observed. If a dim deep-sky object were observed quite near a dazzling star (NGC 404 behind Beta Andromedae or NGC 2024 near Zeta Orionis, for example), then the diffuse glow surrounding the bright star would invade the image of the dim object. Observing tricks, such as obscuring the star behind a field stop, can diminish scattering in the eyepiece and the eye, but if the main lens or mirror is dusty, the damage is already done. The stray light will peek out beyond the edge of the stop anyway. Excessive dust on the optics can also damage the detection of low contrast details on bright extended objects, such as planets, because the scattered light exists in a general haze.

The maximum amount of dirt the observer should tolerate on the optics is about 71000 of the surface area. We have already seen in Chapter 3 how this can lead to signal-to-noise ratios as low as 30 dB.4 Luckily, most of the

4 Conventionally, noise varies with time and is random in location. Thus, my calling scattered light "noise" is incorrect in the sense that light scattering appears motionless and is completely determined by details at the aperture. Nevertheless, I use the concept of the signal-to-noise ratio because scattered light similarly degrades the quality of the image. SNR is used throughout this book to compare a scattered or diffracted haze of light with the underlying "true signal," giving relative weights to widely different phenomena. Please remember, however, that such a usage should be viewed only as an analogy or beneficial fiction, and it should not be pushed too far.

halo extends beyond the edge of a planet. Hence, much of the scattering is ugly but not harmful. Like spider diffraction, the worst SNRs are reserved for very large bright-field objects, such as the Sun and Moon. It is difficult to estimate the fraction of the optics covered by dirt, but 71000 of the area is the size of a single obstruction about 730 of the diameter. On a 200 mm mirror, the accumulated grime would cover a spot 7 mm across—slightly smaller than the size of your little fingernail. Even a telescope that is that dirty will not contribute much additional scattering to a reflector with spider vanes, and the tolerance for dirt could perhaps be relaxed somewhat for such instruments.

The observer should heed one additional warning. Some telescope owners, after reading the above comments, might be tempted to clean their mirrors or lenses too frequently. Optics possess delicate coatings for which the safest prescription is to leave them alone. Overuse of even the most gentle of cleaning materials leaves a myriad of tiny scratches in coatings. Don't decide to clean mirrors on the basis of shining a light down the tube at night. All mirrors fail such a harsh inspection.

The best procedure for clean optics is not washing but prevention. Keep them covered and dry. Clean them only when you believe they have been chemically attacked or when the dust begins to visibly affect the images. Of course, specialized observers might need cleaner surfaces at all times, but no doubt these people have already learned of the considerable risks and expenses involved (more frequent aluminization, etc.). By following good maintenance procedures, you should not have to clean optics very often.5

5 Frequently, enthusiastic mirror-makers will aluminize a mirror before it is truly polished. They are either covered with a haze of pits or have a fuzzy ring out toward the edge. Unfortunately, such mirrors cannot be "cleaned" at all and are usually unsalvageable. They need to be returned to polishing.

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