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4.5. Fabrication errors

Fabrication errors are deviations of an actual optical surface from the perfect reference surface as a result of fabrication process. As such, they come in various forms. Wavefronts produced by imperfect optical surfaces are also imperfect, suffering from aberrations.

Surface errors covering more than ~D/10 (linearly) are usually called figure errors. They include zonal errors, turned edge, and asymmetric surface deviations. Full-figure surface errors - with the error smoothly distributed over the entire surface area - will result in a form of primary wavefront aberration, such as spherical or astigmatism (the latter may be either polished into a surface - common with thin mirrors and lenses - or resulting from inner glass tensions, frequent when it is not annealed).

Random surface deviations smaller than ~D/10 are termed roughness. They are randomly scattered over the surface. Depending on their relative size, they are classified as large-, medium- and micro-ripple. Large-scale roughness is normally produced by poorly made tool and/or poor technique (uneven pressure/movement). Medium-scale roughness (also dog biscuit, or primary ripple) is usually a random pattern  resulting from the existence of empty interspaces on the polishing tool. And microripple are caused by the abrasive action of polishing material.

As the linear ripple diameter diminishes, so does its maximum P-V/RMS error, and its adverse effect on wavefront quality. In general, large and medium surface ripple double in the wavefront for reflecting surfaces, while for a lens they are reduced in the wavefront by a factor of n/(n-1)√2 (statistical probability, assuming similar degree of roughness on both lens surfaces). Given nominal wavefront P-V error caused by large or medium surface ripple is roughly as bad as a similar P-V error of spherical aberration.

As for micro-ripple, usually spanning over 1mm-2mm, their nominal RMS wavefront error is very small for optical surfaces made according to accepted proper procedures - no more than about 5nm, and usually ~1nm (~1/500 wave for the visual peak) and smaller. Hence, the effect on central intensity is negligible for general observing. Due to this small amount of energy being widely scattered, the relative contrast drop caused by micro-ripple is nearly identical at all frequencies, except for the very narrow lowest MTF frequency range, in which it drops from 1 at the zero spatial frequency (ν=0) to the relative contrast level slightly bellow that given by the Strehl ratio.

Range width of this initial drop depends on the relative average size ℓ of micro-ripple in units of the aperture radius. Roughly, spatial frequency at which this level is reached is given by ν~ℓ, ν being the spatial frequency. So, for, say, ℓ~1/50 average relative size of 1/300 wave RMS micro-ripple, contrast would drop to slightly bellow 0.99956 of the perfect aperture contrast level at ν~1/50, and remain near to that contrast ratio for the remaining ~98% o the spatial frequency range. This also indicates that the energy spread caused by the micro-ripple reaches beyond ν~1/50 frequency radius, or over 40 Airy disc radii far from diffraction peak (the cutoff frequency ν=1 is at 1/2.44 Airy disc diameters).

With lenses, glass homogeneity and optical properties are also a factor.

While the specifics of, among others, fabrication errors' effect on imaging is addressed in 4. The effects of aberrations, it may be interesting to consider the relation between surface fabrication errors and wavefront deviations. The common notion among amateurs is that the reflecting surface error doubles in the wavefront, while the refracting surface error halves in the wavefront error. While it may be so, neither is generally correct. For light reflected in nearly opposite direction (that is, for a surface nearly orthogonal to the optical axis), reflecting surface error doubles in the wavefront centered at the Gaussian (paraxial) focus, regardless of the nature of surface error.

However, errors smoothly distributed over the surface will result in a smoothly distributed wavefront deviation as well. The result is that the actual wavefront, while with doubled maximum deviation with respect to the perfect reference sphere centered at the paraxial focus may, and usually does have smaller deviation in respect to a reference sphere focusing at some other point in the proximity of paraxial focus. This point becomes the point of most efficient energy concentration, so-called best, or diffraction focus. This is the case with spherical aberration, where the wavefront error at the location of best focus is smaller that that at the paraxial focus by a factor of four (Fig. 16). In effect, with spherical aberration, the surface error halves in the wavefront. With astigmatism, the best focus wavefront error is smaller by a factor of 2/√6 vs. paraxial focus error, making it effectively 1.63 times greater than a surface error.

The "double error rule" also doesn't apply to full-figure surface errors of reflecting surfaces in multi-surface systems, such as diagonal flat or curved secondary mirrors. With a diagonal flat, the wavefront error is determined not only by the surface error, but also by the shape of surface deformation. While certain toroidal form of the diagonal flat will not induce significant aberration at best focus even with the surface error exceeding 1 wave P-V, or more, diagonal's local errors and surface roughness will be ~1.4 times greater in the wavefront. Curved secondary mirrors in two-mirror telescopes can have their surface error very much diminished in the "best-fit" wavefront, especially for deviations resulting from errors in the surface radius of curvature.

For surface errors affecting relatively small area, however, the wavefront deformation is also local, and no better reference sphere is available. Thus, this kind of surface errors effectively doubles in the wavefront, but only when the reflected light moves to a nearly opposite direction vs. incident light.

Similarly to reflecting elements, refracting surface error will follow its "rule" (i.e. surface error halving in the wavefront) rather exceptionally. Full-figure errors, like conic error or, especially, radius of curvature error, deviations from perfect result in a significantly smaller effective (best focus) wavefront error. Local surface errors do approximately halve in the wavefront, although it directly applies to a lens, not the lens surface. Refractive surface error multiplies in the wavefront only by a factor of (n-1)/n with air-to-glass surface, while by a (n-1) factor with glass-to-air surface. However, the former increases at the rear (glass-to-air) surface by a factor of n, so that the local error on lens surface multiplies in the wavefront by a factor of (n-1), regardless of the location (FIG. 25).

Most often, optical surface imperfections are of random nature, which makes them largely unrelated from one surface to another in regard to their effect on the wavefront. Hence, if the surface RMS error is known, it is possible to calculate - from the square root of the sum of their individual RMS errors squared - what would be the probable cumulative error for two or more such surfaces. Or, in reverse, a limit to the individual surface error can be set so that the cumulative system error for all surfaces combined doesn't exceed the desired maximum error level.

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