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4.7.3. Measuring chromatic error
▐
4.8.1. Testing optical
quality
► 4.8. Fabrication errors
PAGE HIGHLIGHTS Fabrication errors are deviations of the actual optical surface from perfect due fabrication process. As such, they come in various forms. Wavefronts produced by imperfect optical surfaces are also imperfect, suffering from aberrations. For reflecting optics, primary concern is the accuracy of the overall figure, followed by smoothness of the figure and surface smoothness. For refracting optics, in addition to these, errors in the wavefront can be caused by wedge, decenter, deviations in glass thickness and homogeneity. Let's start with some general remarks with respect to the relation between surface and wavefront errors. 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 of nearnormal incidence reflected back (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, socalled 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. 33). In effect, with spherical aberration, the surface error halves in the wavefront, measured vs. reference sphere centered at paraxial focus. With astigmatism, the best focus wavefront PV error has double the surface error, but the corresponding best focus RMS error is smaller by a factor of 2/√6 than the doubled surface RMS error, making it effectively 1.63 times  not twice  the surface RMS error. The "double error rule" also doesn't apply to fullfigure surface errors of reflecting surfaces in multisurface 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 the best focus even with the surface error exceeding 1 wave PV, or more, diagonal's local errors and surface roughness will be ~1.4 times greater in the wavefront. Curved secondary mirrors in twomirror telescopes can have their surface error very much diminished in the "bestfit" wavefront, especially for deviations resulting from errors in their 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 does double in
the wavefront, but only when the reflected light moves in nearly
opposite direction to that of the incident light.
Hence, instead of the commonly cited
λ/8
maximum tolerable error for a single reflecting surface, the tolerable
error vary widely with the type of reflecting surface and error type.
Since the tolerable nominal surface PV error generally increases as the
areal extent of deviation diminishes, it is more appropriate to express
the tolerance in terms of the maximum tolerable RMS wavefront error, as
outlined in table below.
Similarly to reflecting elements, refracting surface error will follow its "rule" (i.e. surface error halving in the wavefront) rather exceptionally. With fullfigure errors, like conic error or, especially, radius of curvature error, deviations from perfect result in a significantly smaller effective (best focus) wavefront error than with reflecting surface. 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 (n1)/n with airtoglass surface  n being the glass index of refraction  and by (n1) factor with glasstoair surface. However, the former is enlarged at the rear (glasstoair) surface by a factor of n, so that the local error on lens surface multiplies in the wavefront by a factor of (n1), regardless of its location (i.e. front or rear surface, FIG. 49).
FIGURE 49: Wavefront delay caused by a local error on airtoglass (a) and glasstoair (b) lens surface. As a consequence of glass media of the refractive index n lowering the speed of light by a factor of 1/n, surface deviation t on the front (airtoglass) surface generates local wavefront PV error of δ'= (n1)t/n. At the rear (glasstoair) surface, the advance further increases by a factor of n, resulting in the final PV error on the wavefront exiting the lens given by δ=(n1)t. Identical surface error δ on the rear lens surface would also result in the local wavefront advance δ=(n1)t. If combined, the two surface errors on each, front and rear surface, would produce wave retardation (PV wavefront error) given by δ=2(n1)t.
Unlike mirror surface, lens needs to satisfy a number of requirements
related to the accuracy of lens form (e.g. wedge, thickness, centering)
and glass homogeneity. While the actual tolerances can vary widely with
the lens specs and its function, some general criteria do exist. These
are cited in a University of Arizona article (J.H. Burge):
In conclusion, general tolerances for random local surface errors for given PV wavefront error W limit are δS=W/2 for mirror surface and δS=W/√2(n1) for lens surface (for statistically unrelated errors on both lens surfaces). For figure errors, δS=2W for the conic error (rotationally symmetrical surface with the error resulting from incorrect conic value) for a single mirror; single spherical lens cannot be free from spherical aberration, thus the tolerance is determined by the specific surface radii and spacing of a lens pair or a multilens assembly (which may make possible looser practical standards, due to the opportunity to vary the two factors for postfabrication corrections). For a pair of mirror surfaces, the random local error tolerance is smaller than for a single surface, δS=W/√2, as given by the square root of the errors squared, and δS=W/2(n1) for a lens doublet. Similarly to a lens pair, the tolerance for figure error for a pair of mirror surfaces can be determined after tolerances for the radii and spacing errors are set, according to Eq. 80. Most often, optical surface imperfections are of random nature, which makes them largely unrelated from one surface to another with respect 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, the other way around, 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. Surface roughness Surface errors about ~D/10 (linearly, D=aperture diameter) in extent and larger are usually called figure errors. They include zonal errors, turned edge, and asymmetric local surface deviations. Fullfigure 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 glass is not annealed). Pattern of surface deviations smaller than ~D/10 is termed roughness. It us usually randomly scattered over larger of smaller portion of the surface. Depending on their size, they are classified as large, medium and microripple (also low, medium and highfrequency ripple, respectively). Largescale roughness is not a usual term, and here will partly overlap with figure error. We'll define it as a pattern of random surface deviations with the dominant structure averaging, approximately, D/10 to D/5 in diameter. It is not a frequent form of optical surface roughness  although isolated local surface errors of that size are rather common  but it is certainly possible with poor fabrication methods. Mediumscale roughness (also dog biscuit, or primary ripple) is usually a random pattern resulting from the existence of empty interspaces on the polishing tool. Its dominant structure average is between D/10 and D/20 in diameter. And smallscale roughness, or microripple (also, secondary ripple), is caused by the abrasive action of polishing material. Average size of surface irregularity here is approximately 1mm in diameter. For special applications, such as laser optics, significantly smaller structures can be important, but for general amateur astronomy their effect is entirely negligible. As the linear diameter of ripple diminishes, so does its maximum PV/RMS error, as well as its adverse effect on wavefront quality. The sizemagnitude dependence of surface deviations is usually expressed through power density spectrum (PDS; also, power spectral density, PSD), which is the mean surface deviation squared as a function of spatial frequency (the inverse of spatial period, i.e. trace length on the surface). In other words, it shows how the amount of error varies with the size of surface trace length, which is the length, or area on the surface for which the measurements are taken. For the PDS to be meaningful, measurements must be taken for a sufficiently large number of such surface units, for every frequency, with sufficient number of sampling points. The measured surface errors are averaged over frequency; actually, over its inverse, the trace length. In the context of optics, the term power density is not really appropriate for the roughness of an optical surface, where it should signify (surface) error magnitude spectrum (spectrum is generally the change of a given parameter as a function of frequency). Once measurements are taken, the distribution of error over the frequency range becomes known. A specific "synthetic" PDS function then can be fit into the data for analytical purposes. Unlike synthetic PDS, which on a loglog plot declines from low to high frequencies in a straight line  slope of which is given by the function's exponent  error measurement of an actual surface will plot unsteady declining line, often spiking or curving away from an interpolated straight line. While some of it can be "noise" due to finite sampling, a significant spike up at any given frequency ν, or frequency interval, indicates the presence of periodic roughness structure over the surface, with the corresponding spatial period given by 1/ν. The range of frequencies is conventionally divided into low, mid and high order frequencies, with the low order generally covering figure errors (here from 1/D to ~1/0.1D), described by Siedel and Zernike aberrations, highorder frequencies causing wide scatter of light, and midorder frequencies in between the two. For an actual surface, PDS can be derived by way of Fourier transform, i.e. constructed as a cumulative averaged Fourier amplitude generated from surface phase map, or direct measurement, for every frequency in the range. In general, PDS has an exponential downward trend toward higher frequencies, i.e. smaller trace lengths. Main focus of the PDS are not figure errors; a generaltype PDS, whose purpose is illustrating a typical error distribution for certain type of purposes, assumes some (average) level of figure error, as well as some average level of smallerscale periodic surface roughness that steadily diminishes in error magnitude as its characteristic (average) length becomes smaller. One such PDS for meterclass optics, published (probably) by J. H. Burge (University of Arizona) puts the surface error level (RMS surface, doubles in the wavefront for reflecting, and about halves in refracting surface), at ~40nm over lowfrequency range (~1.7 to 0.17m spatial frequency), ~10nm over midfrequency range (~0.17m to 1mm), and ~1nm for roughness structures smaller than ~1mm (highfrequency range). But the main purpose of having PDS of an actual surface is quantifying small and microscale structures; any periodic error of this type significantly exceeding error magnitude at neighboring frequencies will generate greater total amplitude and will show as spike up in the PDS plot at that frequency. It is also possible that the rate of change in error magnitude within a relatively wider subrange of frequencies increases or decreases with respect to the rest of frequency range, in which case such subrange forms a weaker or stronger PDS slope, respectively, i.e. cumulates larger or smaller error within the subrange. PDS allows extracting the RMS surface error for the entire range of frequencies, as well as for any frequency subrange. And while the conventional RMS error for the entire surface allows us to calculate the amount of energy lost from the Airy disc, having the surface error structural profile defined by PDS makes possible to determine where that energy goes, i.e. surface scatter properties. As such, it is the main factor in scatter analysis with bidrectional scattering distribution function (BSDF). While various aspects of scatter off small and micro roughness structures can be significant for special instruments (e.g. coronograph) and for laser optics, it is much less important for generalpurpose amateur instruments. All that may be important here is the size (length) and magnitude of a periodic roughness structure showing in the PDS, which can be easily converted into both, the corresponding loss of energy from the diffraction maxima and the approximate scatter radius. The simplest power density spectrum function  and the one that fits well mirror surface in general  is PDS=C/νs, where C is the numerical constant, ν the spatial frequency (1/p, p being the spatial period, i.e. trace length) and s the exponent determining PDS slope on a loglog plot. This function produces a straight line plot over the entire range of frequencies. Somewhat more complex form, called ABC model, often used for fitting PSD into measurement data, is PDS=A/[1+(Bν)2)C, with A, B and C being the adjustable parameters (in the notation used here, A=C=numerical constant, C=s=power exponent determining the loglog plot slope toward higher frequencies, and B determines at which low frequency  unlike the previous PDS form  the plot changes its slope toward horizontal and flattens out). A variant of the ABC model, PDS=CνDs/(ν+νD)s, using the minimum spatial frequency νD=1/D parameter (corresponding to the maximum spatial period equal to aperture diameter D), along with the simplest PDS form, is used to illustrate their description of surface roughness on the graph below. The more complex form can be used to limit error in the low frequency range (figure errors) for given required roughness standard in the higher range. PDS can be either onedimensional (1D), describing surface roughness spectrum along any single line across the surface phase map (i.e. based on the corresponding single surface cross section), or twodimensional (2D), extending to the area (volume) of the surface around straight line, i.e. providing areal (vs. linear with 1D PDS) coverage of every spatial period p in the frequency range. Onedimensional PDS is sufficient to describe radially symmetrical surface, but nonsymmetrical surfaces require obtaining either multiple 1D, or a 2D PDS. PDS not only indicates the presence of periodic roughness structures on an optical surface, but also allows for a direct assessment of its RMS value. Since 1D PDS is the cumulative squared mean error for a frequency (i.e. for the corresponding spatial period), or frequency interval (when generated by Fourier transform, it is the squared discrete Fourier amplitude of surface errors at a given frequency, or frequency interval), multiplying it with the frequency interval (i.e. dividing it with the corresponding spatial period) produces the squared mean error  or RMS  for that interval. Hence the RMS surface error corresponding to such interval is the square root of the area under the curve APDS, or RMS=APDS0.5, with APDS= ΔνΣ[PDS(νi)] = ∫(νminmax)[PDS(ν)dν] = f'(νmax)f'(νmin), with Δν being the frequency interval, νmax, νmin the maximum and minimum frequency value for the interval, and f' the antiderivative of the PDS function. Antiderivative for PDS=C/νs is f'=Cν(1s)/(1s), and for CνDs/(ν+νD)s it is f'=CνDs(ν+νD)(1s)/(1s). Twodimensional PDS is not directly related to the RMS error. Taking the simpler PDS function, PDS=C/νs, and approximating the RMS function as RMS~[Cν/(1.5ν)s]0.5 with the frequency ν effectively becoming frequency interval Δν which, multiplied with the PDS for 50% higher frequency (i.e. with approximately mean PDS for the interval) gives the approximate area under PDS plot for this interval, the RMS changes in proportion to 1/(1.5sνs1)0.5=1/1.5s/2ν0.5(s1). Taking s~2 as typical for polished optical glass gives that the surface roughness RMS changes as 1/√ν which, with 1/ν equaling spatial period p, implies that the RMS roughness error changes approximately with √p, i.e. with the square root of the size of roughness structure. As illustrated below, there is no significant difference in this rate of change with either different exponent value, or the other PDS function form, except for the very low end of the frequency range.
Roughness standards for optical surfaces are not clearly defined. It is common to see the roughness figure stated without specifying for what frequency range, which makes it meaningless, knowing that surface roughness magnitude is a function of the characteristic length (size) of roughness structure. Unless we know it conforms to MILSTD10A standard, which in such case assumes it applies to roughness scale length of less than 0.03 inches. The surface roughness figures quoted are usually for a submillimeter roughness length, i.e. highfrequency domain (usually smaller than ~0.1mm) which in the context of amateur telescopes belongs to microroughness. For the conventionally polished optical surface, error induced by this roughness scale is negligible. While sometimes microroughness is specified in the addition to the figure accuracy, the roughness scale commonly left out of the specs is the midscale roughness, potentially much more harmful than microroughness.
Table below illustrates, quite
approximately, the relation between ripple diameter and the magnitude of
its deviation. For large and medium scale ripple it is assumed that the
error magnitude changes nearly in proportion to their characteristic
length, but the real surface patterns can deviate from it locally,
possibly significantly (sporadic local deviations, even if significant
nominally, do not significantly worsen the effect of prevailing
pattern). This is not in agreement with the rate of change in the PDS
context  which is
approximately in proportion to the square root of the roughness scale 
but PDS, in addition to be of very general character, does not cover
accurately what is here termed large and mediumscale roughness. While
the assumption of the error magnitude changing in proportion to the
roughness scale is not very accurate either, it should be more
appropriate for this size of roughness structure, with the ranges given
creating in effect a wider spectrum of possible rates of change in error
magnitude with the structure size.
In general, surface roughness error doubles in the wavefront for reflecting surfaces, while for a lens they are reduced in the wavefront by a factor of n/(n1)√2 (statistical probability, assuming similar degree of roughness on both lens surfaces). Assuming nearly uniform roughness structure  or nearly uniform dominant structure  over the entire surface, the corresponding surface roughness PV wavefront error is about 3 the roughness RMS error (for instance, for a singleplane sinusoidal deformation pattern the RMS is PV/2.8, for sawtooth pattern  isosceles triangle  with the same base and height as the sinusoid it is PV/3.5, for a isosceles trapezoid of the same base and height with the top half as long as the base it is PV/2.5, and so on). This indicates that the roughness PV error extending over most or all of the surface causes more damage than nominally identical PV error of conic aberrations (the PV to RMS ratio is 3.354, 5.657 and 4.899 for primary spherical aberration, coma and astigmatism, respectively). However, only largescale roughness can reach the magnitude needed to reduce the Strehl below diffraction limited 0.80. Also, the smaller relative area affected by roughness, the smaller damage; for instance, if only half of the area is affected, its effect is comparable to that of twice smaller nominal roughness error over the entire surface area (with respect to the portion of energy transferred outside the central maxima; the wider characteristic roughness period, the smaller radius of energy spread by it).
As for microripple, 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. Suiter cites Texereau
stating that the typical RMS wavefront error due to microripple is
typically a small fraction of 1/100 wave. Using other sources provides
a more detailed picture. For instance, polishing experiment at the
University of Arizona (Supersmooth optical fabrication controlling
high spatial frequency surface irregularity, Del Hoyo, Kim, Burge,
2013),
reports these numbers for the polish quality of 10inch Zerodur
substrates with varying amounts of polishing time:
Experiment with more conventional glass types, investigating the effect
of combining different polishing agents, glasses and slurry pH (Slurry
Particle Size Evolution During the Polishing of Optical Glass), reports
these numbers (1Å=0.1nm):
The experiment used continuous polishing machine and 40mm diameter glass discs, 4hour polishing sessions and 4mm measurement scan length (5 sites per disc, average). Thus it is for nearly identical roughness scale as the previous experiment, with comparable roughness figures  and with important exceptions. Low pH polishing slurry resulted in the much larger roughness error with fused silica and aluminum oxide, while medium pH slurry and aluminum oxide produced it with all three glass types (although in significantly different proportions). The cause, according to study authors, is the difference in electrical charge between the polishing agent and glass surface, resulting in agglomeration of the polishing agent (for a smooth polish, both agent and glass need to have negative charge; the latter is always negative, and the former changes from positive at low, to negative at high pH level). Obviously, the excessive roughness obtained with some pH/glass combinations are unacceptable. While the size of roughness structure in both experiments (4 and 5mm spatial period) is somewhat between the midscale roughness and microripple, what matters is its magnitude, not the formal classification. The second experiment indicates that it is possible that unacceptable level of roughness remain on a polished glass surface even after polishing session of proper duration and agent applied. It is likely that there are other scenarios where that can happen  and, on the other hand, polishing procedures that would neutralize such potentially harmful combinations  but it all boils down to knowing and following proper fabrication procedures, which includes standard quality control. Hence, assuming standard fabrication, microripple does not exceed a few nm RMS surface error, in which case its effect on central intensity is negligible for general observing. The RMS error tells us that the amount of energy scattered by microripple is less than 1% (0.99+ Strehl degradation factor). If surface roughness is modeled statistically, as a nearuniform pattern of the dominant roughness structure  any roughness structure, from large to micro scale  with the characteristic length (diameter, width) ℓ extending over the entire wavefront, its effect on contrast transfer can be expressed with a (contrast) degradation function (FIG. 50, top). This particular function is given in Schroeder's Astronomical Optics with reference to O'Neill. The product of this function with the system's MTF w/o roughness is its actual contrast transfer function (FIG. 50, bottom).
FIGURE 50: Graphs of the roughness degradation function alone (top) and its effect on MTF (transfer T and normalized spatial frequency ν) of otherwise aberrationfree aperture (bottom), for selected values of the RMS wavefront error ω and the diameter of typical roughness deviation ℓ. Assumed is that nearuniform pattern of roughness extends over the entire wavefront. Evidently, by far the most important factor is the RMS error, which tends to scale with the roughness diameter ℓ, the latter merely affecting the width of energy spread  the smaller roughness diameter, the wider energy spread, and the lower amount of energy transferred outside the central maxima. The worstcase scenario for microripple (blue) still has entirely negligible effect on contrast transfer in general observing. The near worstcase scenario for mediumscale roughness (red) does inflict noticeable contrast loss, although it is still below 10%. And near worstscenario largescale roughness (pink), with λ/10 wave RMS (~λ/3.3 wave PV) wavefront error (WFE), lowers contrast transfer very noticeably. Since largescale roughness is likely to be accompanied with mediumscale roughness, their combined effect (gray), given as a product of the two respective degradation functions, is even larger. The form of roughness degradation function plot shows that it lowers contrast uniformly over most  about 90%, or more  of MTF frequency range. Since it is the factor at left in the exponent, (2πω)2, that determines this degradation constant (factor at right shapes up the plot between zero frequency and the constant level), using this factor alone gives a good approximation of the corresponding Strehl (only slightly pessimistic). Thus, the Strehl values for these selected roughness examples are about 0.91, 0.67 and 0.62 for the red, pink and gray MTF plots, respectively. Unlike most common aberrations, roughness lowers contrast nearly evenly over the range of MTF frequencies. The width between the origin (top left) and the constant contrast drop off range  easier to determine on the degradation function plots on top  indicates the radius of energy spread. For the mediumscale roughness plots energy spreads beyond 0.08 frequency with ℓ=0.1, and beyond 0.04 with ℓ=0.05 (i.e. beyond 5 and 10 times the Airy disc radius, respectively). Due to the wide radius of scattered energy, the relative contrast drop caused by microripple  as well as any periodic surface roughness in general  is nearly identical at all MTF frequencies, except for the very narrow stretch at the low end of MTF frequency range, in which it drops from 1 at the zero spatial frequency (ν=0) to the relative contrast level slightly below that given by the Strehl ratio. The width of this initial drop depends on the relative average size ℓ of microripple in units of the aperture radius (assuming relatively limited deviations of the true roughness structure from ℓ). 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 microripple, contrast would drop to slightly below 0.99956 of the perfect aperture contrast level (Eq. 56) 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 microripple reaches beyond ν~1/50 frequency radius, or over 40 Airy disc radii far from diffraction peak (the cutoff frequency ν=1 corresponds to 1/2.44 Airy disc diameters). Alternately, the radius of light scattered by a roughness structure is given angularly as θ=sin1 (λ/ℓ), where λ is the wavelength of light, and ℓ the characteristic roughness length (i.e. average size). So, for instance, with λ=0.00055mm and ℓ=1mm, the angle of scatter is 0.032 degrees (note that sin1 is not the inverse of the sine value, but the angle corresponding to it). With lenses, glass homogeneity and optical properties are also a factor.
Overall, it can be concluded that image degradation from microripple is
negligible even in the worstcase scenario, that from mediumscale
roughness ranges from about 0.95 to 0.9 Strehl degradation factor, and
for largescale roughness from 0.9 to 0.65. For combined large/medium
scale roughness from 0.85 to 0.6. These are only approximate ranges and,
again, assume that the entire wavefront area is affected with a
consistent roughness structure. For medium and largescale roughness,
it is seldom the case, so these ranges outline the worstcase scenario
not likely to be encountered in practice. With a more realistic assumption
that only about half of the wavefront area is affected, the damage
diminishes to 0.990.96, 0.960.9 and 0.950.85 for medium, large and
combined medium and largescale roughness, respectively. ◄ 4.7.3. Measuring chromatic error ▐ 4.8.1. Testing optical quality ►
