This chapter discusses the diffraction effects caused by quasi-random or asymmetric errors polished into the surface of the glass. The circular rings break up into tiny speckles in focused images, and the out-of-focus disk shows non-circular detail. The reader should take four important points from this chapter:

1. Wavefront roughness usually follows a continuous spectrum at increasingly finer scales, which telescope makers arbitrarily divide into categories such as "dog-biscuit" or "microripple." However, such distinctions are more matters of nomenclature than descriptions of real phenomena.

2. Medium-scale roughness (primary ripple or dog-biscuit) errors are the ones which most severely damage the image because they don't divert light far from the core of the image. The scattered light is hence more condensed and brighter.

3. Roughness errors are difficult to distinguish from turbulence, and careful star-testing is required to avoid unfairly judging an instrument. Tolerance for roughness errors must be viewed in the context of likely turbulence errors.

4. Roughness at the scale of microripple is of interest only to makers of specialized instruments who have already reduced other forms of diffracted light to the vanishing point. Microripple of small amplitude has little relevance to general-purpose instruments.

13.1 Roughness Scales and Effects

Harsh methods of polishing and use of fast polishing compounds can lead to random or nonperiodic error that is not a circle of revolution. This so-called "surface roughness" is not generally viewed as a problem on the global scale, the way figuring errors are perceived, but it can be harmful to the image in its own way.

We must carefully define what is meant by surface scale before introducing the deformations of that surface. Think of the aperture as being expanded to the size of the United States. The spherical curvature of the Earth is analogous to the focusing curvature of the wavefront. In the aberration function plots, this curvature has been removed as a noninstructive universal constant. The map has been flattened on the average.

Figuring errors are large-scale deformations. An example is spherical overcorrection, which starts fairly level at the center of the aperture and rises to a peak at the 70% zone. The aberration function then falls rapidly until it reaches the edge. Similarly, the center of the U.S. starts fairly flat in the plains states, rises toward the Appalachian or Rocky Mountain ranges, and then falls rapidly toward the oceans.

No serious mapmaker would suggest that the U.S. topography could be completely represented by a simple model of two ridges with a flat area between. Nevertheless, we may usefully describe the coarsest features of the landscape with broad-brush concepts like continental divide and leave the narrower details for later. Describing the figuring errors of the optical surface as "spherical aberration" or "zonal defects" is that sort of large-scale reference. Here "scale" refers not to how high the aberration is but how wide it is, or rather how persistent the aberration is over long distances.

To improve the map, the landscape is refined by adding rivers or watersheds. Many of these features extend over areas the size of an entire state. On the mirror, we can decide to measure roughness with ruler divisions of around Vl0 or V20 pf the aperture. These are "medium-scale" roughness errors. We could also map individual mountains or county-sized variations in the terrain and tile analogous "small-scale" roughness errors. With sufficient magnification, the map could chart the position of boulders, plowed fields and ditches. likewise, if we examine the optics on the molecular scale, we see a convoluted surface, but such errors are so much smaller than the wavelength of visual light that they cannot be sensed by ordinary means. The wavefront remains flat after encountering molecule-sized roughness.

Medium-scale roughness errors go by the colorful name "dog biscuit" and the less-colorful name "primary ripple." Their greatest width scale approximately matches the spaces between grooves on the polishing tool. These channels are always cut or cast into the polisher to provide space for the pitch to spread with pressure and to supply reserves of the finely-suspended polishing abrasive. The grooves are a necessary evil. If they are not cut into the lap, large-scale shape errors are generated that are even worse than moderate amounts of roughness.

Small-scale roughness errors, called "microripple," have spacings of about 1 to 2 mm. The cause of these errors is less obvious than primary ripple, but their origin is probably found in the choice and use of polishing materials. Cerium-oxide polishing compound seems to give rougher results than rouge. Waxy laps yield more rippled surfaces than pure pitch, and paper laps are worse than wax. At the basic level, however, non-uniformities in the glass itself seem to limit the smoothness. Texereau claims that the lap is able to attack the surface of the mirror through a combination of physical and chemical means, and that once begun, such errors are self-sustaining (Twyman 1988, pp. 578-584; Texereau 1984, pp. 88-91).

Certain fast-acting laps deliver a roughness with characteristic dimensions intermediate in scale between classic primary ripple and microripple, descriptively called "lemon-peel" surfaces. This appearance is rare in instruments intended for astronomical use, however. Usually, telescopes are polished on gentler materials.

We might expect the diffraction image from the roughness facets characteristic of primary ripple to be 5 to 20 times larger than the unaberrated image, but that simplified logic does not take into account the accidental correlations that occur when nearby scattering facets act in phase with one another. Antinodal bright areas and nodal dark regions will form. The net effect of mild primary ripple is to blow the scattered light into a knobby glow surrounding the image, which has its greatest brightness within a radius less than 5 times the Airy disk. Such scattered light can be a bad problem because it is condensed enough to easily see.

Let's compare that defect with the likely behavior of microripple. Texereau states that microripple is occasionally as bad as 6 nm on the wavefront and has an average spacing as small as 1 mm (Twyman 1988, p. 580). However, we are fortunate that the slope of each 1 mm facet is seldom correlated with the slopes from nearby facets, so the effective apertures of the facets only statistically combine. This 6 nm case is also the worst one; most wave-fronts have microripple below 1 nm (< V500 wavelength). Because of the small size and lack of correlation of the scattering surfaces, the diffraction pattern of scattered light from microripple is a shattered dim glow, quite similar to the aura that occurs with a turned-down edge. Microripple of small amplitude is hard to see with the Foucault knife-edge test. It requires specialized equipment to detect unambiguously.

When roughness is small, as it is for optics, it only affects the diffraction shape of the image in a minor way. It removes light from the focused image and shoves it out into a blotchy halo of small diameter for primary ripple and large diameter for microripple. The missing energy is calculated by seeing how much the central intensity is lessened.

The Strehl ratio of roughness can be calculated from an approximation (Born and Wolf 1980, p. 464). Here is represents the Strehl ratio at best focus and ctrms is the root-mean-square deviation of the wavefront (in wavelengths) as measured from the reference sphere centered on best focus:

For example, a 1/14.05-wavelength RMS deviation yields a Strehl ratio of 0.8 (the Maréchal tolerance). A 720-wavelength error typical of noticeable primary ripple gives a ratio of 0.9. A severe case of microripple might have a deviation as large as 7100 wavelength, so the intensity is reduced only to 0.996. Clearly, microripple has a very different character than primary ripple.

Another approximation to the Strehl ratio has been given by Mahajan (1982):

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