IS e2naRMS 2132

It gives a more accurate number than Eq. 13.1 at large aberration amplitudes. We can invert Eq. 13.2 for a iS = 0.8 to define a "Mahajan tolerance" of about 713 wavelength RMS. For small roughnesses, however, the difference between these approximations is negligible.

13.2 The Terminology of Roughness

We must distinguish the nomenclature from the reality of surface error. Roughness is typically modeled either as a continuous spectrum from large scale to small scale or as a discontinuous composite scale spectrum. The "spectrum" in this case is not composed of light intensities plotted versus colors, but magnitude of roughness versus width scales. A composite scale is something like huge swells of water on which are superimposed tiny windblown capillary waves—smooth undulations with wrinkles. The terminology of calling lap-sized roughness "primary ripple" and small roughness "microripple" originates from assuming that two separate causes generate composite-scale roughness.

I have seen mirrors that obviously obeyed this composite-scale model very well. They were covered with smooth, wavy roughness that displayed little of the smaller-scale roughness in between the primary ripple and microripple. More often, though, mirrors appear less and less rough at more diminishing scales, but there is no one scale at which the tester can say the roughness stops (an example appears in Fig. 13-1). Such names as "primary ripple" mean less on such surfaces because there is always a scale just below it. We might call it a "not-so-large ripple" scale, followed closely by an "even smaller ripple" scale. The model described below follows this continuous spectrum behavior instead of the composite-scale that the terminology is based on. The words will continue to be used because they are convenient and firmly lodged in the literature.

Another reason for using the artificial divisions of roughness into medium-scale versus small-scale is that they are typically found in different tests. The Foucault test is good down to a small-scale of roughness intermediate between primary and microripple, but then its sensitivity fails. The investigation of microripple requires a phase-interference test that deliberately suppresses sensitivity to large-scale error.

13.3 Medium-Scale Roughness, or Primary Ripple

The appearance of a mirror suffering from primary ripple is shown in the Foucault test photograph of Fig. 13-1. The roughness is only apparent in the center of the mirror at this knife setting. Ripple extends into the bright and dark regions, though it is less visible in these areas. The roughness is dimly seen as a random structuring with a set of superimposed grooves beginning in the center and extending outward.

Fig. 13-1. Roughness visible in the central zone of a mirror in the Foucault test. At right: contrast is enhanced by subtracting an unsharp-mask image.

This 6-inch (150-mm) f/5 mirror was produced during the manufacturing boom at the last return of periodic comet Halley and is typical of the sloppy fabrication practices of the time. As bad as it looks, the roughness is estimated at only somewhere between 710 or 720 wavelength RMS (peak-to-valley roughness between '/3 and '/5-wavelength). Spherical aberration amounting to 1 wavelength overcorrection was also found during this bench test. Obviously, the spherical aberration was the worst failing.

Your eye also suffers from medium-scale roughness. Take aluminum foil and perforate it with a pin. Hold the foil about 8 to 15 cm in front of your eye and look through the pinhole at a frosted incandescent light bulb. Try to focus your eye on the lamp, not the pinhole, and cover the other eye. If you have punched the right size hole in the foil, you should see a mottled disk that roughly approximates the out-of-focus patterns seen in this book. The outside ring is perhaps the only one clearly delineated. The appearance may be cleared up slightly by placing a colored filter between the lamp and the pinhole.

As you blink, horizontal lines appear briefly on the defocused disk. You can see that some of the details change with every blink. They are probably caused by variations of the moisture thickness on the cornea. Depending on how bright the light is, you may also see some dim radial spikes outside the disk. These spikes may be caused by diffraction from the non-circular iris opening or streaks in the roughness.

The roughness is visible as coarseness in the expanded disk. This coarseness does not vary from blink to blink. The aberration is many wavelengths high, so the appearance of individual rings is obscured and confused. When I was 19, I had a fleck of metal removed from my cornea, and the evidence of the trauma can still be seen in the out-of-focus image. The roughness can originate from corneal defects, surface roughness in the eye lens, and non-uniformities in the refractive index of the eye lens.

The human eye is not even close to diffraction-limited. An eye with a 3-mm iris opening (typical during daylight) can theoretically resolve lines separated by 0.6 arcminutes, but a person who resolves lines only 1 arcminute apart is deemed to have excellent vision. How can we test telescopes to the diffraction limit through such an imperfect aperture?

In fact, the answer to this seeming paradox is quite simple. Angular errors in the instrument are magnified until they are bigger than errors in the eye. Once the separation between the finest possible details have been magnified beyond 5 arcminutes (i.e., 76 the diameter of the Moon), aberrations in the telescope begin to dominate aberrations in the eye. A 1-ineh aperture should resolve lines separated by 0.092 arcminutes, so sufficient image size is reached by 5/0.092 = 50 power/inch (20 power/cm). Somewhere beyond this magnification, even perfect telescopic images begin to become fuzzy.

Ironically, some people boast about telescopes that can "withstand more than 100 power/inch" (40 per cm). What they don't realize is that they're not bragging about the telescopes. They are inadvertently admitting the poor quality of their own visual acuities. When using extremely high mag nifications beyond 100/inch, the diffraction disk appears bigger than two-thirds the angular diameter of the full Moon.

13.3.1 The Aberration Function of Medium-Scale Roughness

To generate rough wavefronts, the fractal model described in the chapter on turbulence is again used, with certain modifications.

The first change is to suppress midpoint deviations for two iterations. This step ensures that the surfaces so generated will be uncorrelated at distances greater than 1/8 to 1/4 of the aperture. One doesn't expect that medium-scale roughness will persist over long distances, and by not allowing the surface to deviate until it is divided into a grid of 16 squares, this correlation scale is achieved. Only 16 points out of almost 13,000 are artificially clamped to zero, but the entire character of the surface is changed.

The other modification is to avoid quenching the roughness. Figure 13-1 shows fine detail over scales smaller than the tool spacing (presumably about :/8 of the diameter). In the case of turbulence it was desirable to suppress the deviation at small scale because no mechanism existed to produce it. Turbulence cells have a quasi-period of about 10 cm. Roughness in the glass, because it originates from many causes at a number of scales, will be modeled here to behave as a self-similar fractal.

Fig. 13-2. The aberration function of medium-scale roughness, also called "primary ripple" or "dog biscuit."

Figure 13-2 shows an example aberration function of medium-scale roughness. The streaks visible in Fig. 13-1 are not represented in this algorithm. One expects such grooves to diffract light into low-contrast spikes at right angles to their extent (as a spider vane does). We must not overemphasize their importance, however. The visual perception system tries to create order in what we see. It is especially fond of straight lines and often creates a line where only a hint of one is actually present. Orion's Belt, for example, is curved a great deal. The eye imposes linearity because it prefers linearity.

13.3.2 Filtering Effects of Medium-Scale Roughness

Because the scale of roughness is much smaller than the whole aperture, one expects a brisk drop at low spatial frequencies, a condition similar to turned edge. The MTF thereafter should remain a fairly fixed fraction of the perfect MTF. Thus, the average degradation drops from unity to a constant at about the correlation length (Schroeder 1987, p. 208). We see in Fig. 13-3 that the previous guess of a correlation length of 74 to 78 of the aperture is a good one. The sagging of the curves seems to have reached a steady fraction of the perfect aperture's MTF at about that range.

Fig. 13-3. Filtration caused by primary ripple. Three amounts are shown: 0.1, 0.05, and 0.025 wavelengths RMS wavefront deviation.

Roughness aberration is nonsymmetric, so three curves are plotted for target patterns with bars oriented up-down, left-right, and at a 45° angle. Because these curves represent a single realization of the rough surface rather than an average over many such surfaces, the MTF wiggles somewhat. These curves are examples of the variations that can be expected from changes in the MTF-target orientation or from slightly different surfaces.

The degradation is severe for 0.1 wavelength RMS wavefronts, but it improves rapidly for smaller roughnesses. The quality is acceptable for wavefront roughnesses less than 0.05 wavelength RMS. Some manufacturers guarantee optics that smooth, but they always give the specification on the surface rather than the wavefront. Read claims carefully.

Also shown is the smooth wavefront with RMS deviation of only 1/40 wavelength (1/80 wavelength on a mirror surface). If any reasonable care is taken, all astronomical optics can be made this smooth. Global wave-front aberrations are difficult to reduce below 1/28 wavelength RMS (1/8 wavelength peak-to-valley), but optics can easily be smoothed until the wavefront roughness is less than 1/40 wavelength RMS deviation.

13.3.3 Star Test on Medium-Scale Roughness

Two focus runs appear in Fig. 13-4 and Fig. 13-5. The first is an image sequence of an otherwise perfect 1/40-wavelength RMS roughness wavefront. Even though the wavefront is very good, the roughness is detectable in the out-of-focus images. Figure 13-5 doubles the aberration and uses a different fractally-derived wavefront. This 1/20-wavelength RMS aperture is acceptable in the MTF chart, yet it seriously distorts out-of-focus images. Fortunately, it seems to tuck away the messiness visible out-of-focus to yield a fairly crisp pattern while in focus. If this pattern were turbulence, it would be at least a 9 on the 1-10 Pickering seeing scale.

We go approximately 8 wavelengths on either side of focus (Table 5-1). If the wavefront has primary ripple close to 1/40 wavelength RMS, the effects of roughness are very delicate and hard to detect. At 1/20 wavelength RMS (about the limit of what you should tolerate), you will see it all too plainly.

Often, you must test for roughness alongside some spherical aberration. Roughness is easier to see on the soft-edged side of focus. The dim outer portions of the disk flare into a twisted, asterisk-like pattern. Don't concentrate on the roughness until you have determined that spherical aberration is acceptable. Spherical correction errors are much more damaging to high resolution images than roughness errors because their width scale is so large.

13.3.4 Roughness and Turbulence

Turbulence closely resembles roughness, so it interferes strongly with the star test for that aberration. Thus, roughness is nearly impossible to check under real skies using an actual star. Nights where the air is absolutely still are so rare that they will never coincide with a deliberately-planned star test. Besides, star tests are the last thing the observer wants to do on nights of exceptional steadiness.

An artificial source is often crucial to test for roughness. Also, testers cannot check for primary ripple anytime and anywhere. They must try for a serendipitous combination of time and place that results in a tranquil test-

Rough=1/40 RMS 10 normal OB=25% 10

Rough=1/40 RMS 10 normal OB=25% 10

Fig. 13-4. Medium-scale roughness of 1/40 wavelength RMS with defocusing aberration from -8 to +8 wavelengths. Obstruction is 25% and the perfect image is seen in the right column.

Fig. 13-4. Medium-scale roughness of 1/40 wavelength RMS with defocusing aberration from -8 to +8 wavelengths. Obstruction is 25% and the perfect image is seen in the right column.

ing path. Maybe the necessary conditions will occur during a night when the temperature is not dropping too rapidly. The likeliest good tests are conducted on a windless evening or a very early morning over grass. The artificial source test goes best when the flashlight and sphere are set up in the bright period after sunset but before twilight has ended. Use the Sun to illuminate the sphere in the early morning. If these times are difficult to arrange, test for roughness with the artificial source at night. (See Hufnagel

ROUGH=1/2Q OB=25% 10 normal OB=25% 10

ROUGH=1/2Q OB=25% 10 normal OB=25% 10

Fig. 13-5. Another primary ripple wave front, this time having a statistical deviation of 1/20 wavelength RMS, is focused from —8 to +8 wavelengths. The smooth wavefront is to the right.

Fig. 13-5. Another primary ripple wave front, this time having a statistical deviation of 1/20 wavelength RMS, is focused from —8 to +8 wavelengths. The smooth wavefront is to the right.

Also, you must be realistic in your expectations. Roughness is only important if it is a reasonable fraction of the similar turbulence aberration. Thus, roughness is judged to be objectionable only if it appears during the nights that turbulence affects the telescope least. If the image is flickering a small amount when one sets up the test, the result is not useless.

See if the fixed roughness pattern is visible even against the relatively light turbulence in front of an artificial source. If you cannot discern a fixed roughness pattern under these excellent conditions, then you may be assured that during actual use the roughness is affecting the image little.

The scale must also slide a little to accommodate local conditions. If seeing is abysmal during 99% of the nights at your location, perhaps roughness is less important. In locations with good seeing, roughness standards must be tighter.

If roughness is still grossly objectionable even after taking these mitigating conditions into account, the optics must be refigured.

13.4 Small-Scale Roughness, or Microripple

The original concern over microripple stems from efforts early in the 20th century to observe the solar atmosphere all of the time, not just during solar eclipses. The solar corona is a thin, high-temperature gas that extends out several solar diameters. Observations during eclipses were excellent, but resembled infrequent snapshots. Scientists wanted a method to view the inner corona daily and to monitor its changes. The corona is brighter than the full Moon, but it can be lost next to the hellish intensity of the Sun.

To make a telescope capable of blocking out parasitic scattered light requires more advanced methods than the empirical baffling recipes generally used by astronomers. André Couder stated that the intensity of the corona 5 arcminutes from an occluded solar image is only about 1 millionth as strong as the light intensity streaming from the unblocked Sun. Even nonuniformities in the glass of his coronagraph lens scattered light only a little less intense. He estimated the scattering from the atmosphere, even on a clear mountaintop, was about half as intense as the corona, and the scattering from the most carefully cleaned lens was equally bright (quoted by Twyman 1988, p. 585). Clearly, contrast was already suffering badly from unavoidable effects, and little room was left for scattering by microripple. The severe 1/100 wavelength RMS microripple example mentioned above by Texereau would be unacceptable. It scatters 0.4% of the energy striking the aperture, more than a thousand times too bright.

Texereau also describes a finely polished surface with microripple approximately 0.05 nm RMS deviation, or about 1/11,000 wavelength. This value inserted into Eq. 13.1 results in an intensity reduced (2n/11,000)2 from 1, or only 3 x 10-7. This tiny amount of scattered light would be even less intense by the time it was considered at an angle of 5 arcminutes. Such a surface is sufficiently smooth to be used in a coronagraph (Texereau 1984, p. 88).

Optics exhibiting the primary ripple of Fig. 13-1 demand little or no attention to the relatively subtle effect of microripple. In fact, the frac-

tal model of medium-scale roughness automatically includes a moderate amount of microripple (about 1/1000 wavelength RMS), but the presence of coarser-scale roughness dominates the lesser scale.

Nevertheless, we want to investigate what happens when primary ripple is stripped away and all that is left is small-scale roughness. Microripple is often blamed for low contrast. Can this mysterious roughness scale be responsible for so many optical worries?

13.4.1 The Aberration Function of Small-Scale Roughness

The fractal algorithm was not used in the microripple model, because we want to remove medium-scale features and concentrate on small-scale effects alone. A pseudo-random number generator was used to assign heights to the 128x128 pupil grid. Because asymmetry was expected, tending toward flat tops with sharper grooves, a square root was taken of this starting surface, and the result was normalized. The surface is shown in Fig. 13-6.

Fig. 13-6. A modeled microripple surface with amplitude expanded greatly.

13.4.2 Filtering of Small-Scale Roughness

Consider what was said above concerning correlation length. Because the microripple has a very tiny correlation length (less than a millimeter or so), we should expect a precipitous drop in the modulation transfer function, followed by a constant degradation. These effects are illustrated in Fig. 13-7. MTF drops quickly and then remains a more or less constant fraction of the perfect value.

In fact, microripple as large as 710 wavelength RMS is an unlikely event, even though it appears on the graph. It is shown only to make the fast initial drop more apparent. Texereau gave a worst-case amount of only 7100 wavelength. We can see by the behavior of the MTF graphs that

Filtering of very small scale roughness (i.e., microrippie)

Filtering of very small scale roughness (i.e., microrippie)

Fraction of maximum spatial frequency

Fig. 13-7. Various MTF curves characteristic of microrippie. The aberrations appearing here are exaggerated to show the shape.

Fraction of maximum spatial frequency

Fig. 13-7. Various MTF curves characteristic of microrippie. The aberrations appearing here are exaggerated to show the shape.

7l00 wavelength of microrippie would be indistinguishable from a perfect aperture in most dark-field observing situations. Calculated images differed little from perfection, so no star test diagrams of microrippie appear here.

13.4.3 The Great Unknown

So why do we hear all sorts of warnings about the debilitating effects of microrippie? Microrippie is one of those myths of telescope making that feeds on folk wisdom and hearsay. Part of the trouble is the difficulty of measuring it. Texereau describes a technique (originated by Lyot) that requires an attenuating phase plate, a specialized device that delays and weakens the propagation of the unscattered wavefront. Since the strength of microrippie cannot be measured easily, it gets the blame for any unidentified optical difficulty.

Also, published descriptions of the construction of unusual instruments such as coronagraphs (where microrippie does matter) tend to frighten readers. People believe that microrippie may affect more prosaic forms of ordinary observing. The light diffracted from a slightly turned far edge is vastly stronger than that coming from microrippie. Nearly every telescope with an unmasked edge has much more diffuse light than the scattering caused by the microrippie roughness.

Extended objects suffer far greater degradation from obstructions like spider vanes, mirror clips, and tiny screws projecting from the side of the secondary holder than from microrippie. For example, spiders deflect more light than any likely case of small-scale roughness. A spider with 4 vanes 0.5 mm thick on a 200-mm aperture diffracts 0.5% of the incident energy, slightly more than Texereau's 7100-wavelength worst case microripple.

A root-mean-square microripple error of 0.5 nm (about 71000-wavelength) results in a Strehl-ratio decrease of 0.00004. If this missing light is distributed over the field of interest, it results in a signal-to-noise ratio on extended objects that can be as bad as 44 dB. This value is between the 30 dB maximum defined in Chapter 9 for dirty optics and the 55 dB SNR of good magnetic tape.

For most dark-field observing, however, microripple is not very harmful. For example, 1 mm width-scale roughness scatters light into a halo about 100 arcseconds wide. Maybe 10% of that energy will cover a 20-arcsecond image of Mars. The rest of the stray light is beyond the planet's limb where it doesn't contaminate the image. Thus, the SNR improves to 54 dB, a very large value indeed. An analogy to a 50 dB signal-to-noise ratio is the darkening of the image of the Sun as viewed through a safe solar filter. If the signal were the unfiltered solar image, the noise caused by 71000-wavelength microripple would be about as strong as the filtered Sun. If you require an instrument capable of discerning dim details next to a bright source of interference, you may well need to worry about microripple. Small-scale, low-amplitude roughness is not a threat for most observers.

Fig. 13-8. A testing technique originated by Lyot reveals microripple and veins of differing hardness on the glass. (From How to Make a Telescope by Jean Texereau, Copyright ©1984 by Willmann-Bell, Inc. and used with permission.)

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