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4.7.2. Lateral color error   ▐    4.8. Fabrication errors

4.7.3. Measuring chromatic error in an achromat: polychromatic PSF

Summing it up, an achromat optimized for a particular wavelength, will have spherical aberration canceled for that wavelength, and chromatic aberration nearly cancelled laterally, while reduced to nearly /2000 of the F/C secondary spectrum. The error at the optimum focus results from other wavelengths being: (1) defocused, and (2) affected by spherical aberration, with the latter being comparatively low or negligible. The main error component, that of chromatic defocus, can be expressed as a P-V wavefront error:

W = Pρ2        (51)

with P=-Δ/8F2 being the peak aberration coefficient, equal to the P-V wavefront error, and ρ the pupil ray height in units of the pupil radius. This error combines with the error of spherochromatism for that particular wavelength, and the combined error is finally "measured up" by the sensitivity factor of the eye.

With Δ being, for the typical achromats, ~/2000 at best, the P-V wavefront error of chromatic defocus at its red and blue foci can be written as:

W ~ D/16,000F        (51.1)

   For D=100mm and F=10, this gives 0.000625mm, or 1.29 wave P-V of defocus for the blue F-line (λ=0.000486mm), and 0.95 wave P-V of defocus for the red C-line (λ=0.000656mm).

   For film/CCD applications, defocused wavelengths are more to much more detrimental, depending on both, characteristics of the chromatic defocus and spectral sensitivity of the detector. For instance, most achromats have defocus error significantly greater toward the blue/violet end, which would seriously impair performance with a detector with high sensitivity for that range. However, if the detector is relatively insensitive in the blue/violet, a decent to good results can be achieved even with relatively fast achromats, with significant gross chromatic errors.

Contrary to the common misconception, nominal chromatic defocus in the red and blue is not an accurate indicator of the level of chromatism in an achromat, in the sense that the two change at a different rate with the change in either aperture D or focal ratio F. For instance, while the defocus error at any wavelength other than the optimized changes either in proportion to the change in aperture D, or in inverse proportion to the change in focal ratio F, resulting chromatism - measured as a change in defocus error corresponding to achromat's polychromatic peak diffraction intensity (PPDI), or polychromatic Strehl (SP) - changes at a significantly slower rate. For instance, an /10 achromat has only half the secondary spectrum of an /5, but its polychromatic Strehl is only smaller by a factor of 21/3 (0.79 vs. 0.63 for 100mm aperture diameter, which corresponds to the RMS wavefront error smaller by a factor of 0.51/2).  It further slows down for the longer-focus systems, with the Strehl for a 100mm /20 vs. /10 achromat increasing by a factor of 21/5. (0.90 vs. 0.79, corresponding to the RMS wavefront error reduction by a factor of 0.50.6).

Since the RMS wavefront error is proportional to the P-V error, the actual wavefront error for an achromat of given aperture changes approximately in inverse proportion to the square root of its focal ratio. Actual chromatic error in an /10 achromat is only 0.71 of that in an /5, but in the latter it is also only 1.4 times larger than in the former. This is what the advanced optical design software programs, using diffraction calculation, imply (note that the value of PPDI in the visual range doesn't change with scaling doublet achromat while keeping the focal-ratio-to-aperture ratio F/D constant: 100mm /12 has identical PPDI as 200mm /24).

The reason for this "strange" behavior is that much of defocused light of the farther-off-optimal wavelengths is already out of the Airy disc, and merely gets spread out wider (for instance, F and C line in the 4" /10 above have only a few percent of the energy left within the Airy disc). The spectral range relatively close to the optimized wavelength does not contribute significant "new" lost energy, since it is relatively little affected. It is only a relatively narrow spectral segment on either side of the optimized wavelength, which previously had small but appreciable error that adds significant new energy to that already transferred outside of the Airy disc.

A parallel can be drawn between any far-from-focus wavelength, or a narrow spectral range, and a surface error limited to a relatively small area. Such surface error keeps draining more energy from the Airy disc with the increase in the nominal error only up to the point when practically all the energy available from that area is lost. After that, there is no appreciable effect from the further error increase. This is why turned edge behaves as it does, or any wavefront error limited to a relatively small area. For instance, a zone 1/10 of pupil radius wide, at half the radius, going from 1 to 2 waves P-V, won't change neither peak intensity value (0.96) nor encircled energy (0.95), despite the consequent doubling of the nominal (and at 0.22 and 0.44, respectively rather substantial) RMS error. Heavily defocused far wavelengths in an achromat are alike those relatively small in area, but nominally large and effect-wise mainly drained out wavefront errors.

This helps explain surprisingly good performance of fast achromats in general - and particularly large fast achromats - which, according to their nominal secondary spectrum, should be hardly usable at all.

As it often goes in life, there is the bad side to it as well: it is that the polychromatic Strehl also improves at a slower rate with the decrease in nominal chromatic defocus; in other word, halving the focal ratio doesn't halve the chromatism. The good news is that the discrepancy between decrease in nominal defocus error and the actual chromatic error is significantly smaller here.

While polychromatic Strehl is a reliable general indicator of the effect of aberration over the range of resolvable frequencies, it gives no information about more specific effects on contrast transfer within sub-ranges of frequencies that could be of interest. In particular, how the effect of chromatism compares to the spherical aberration effect at mid-to-low MTF frequencies (details approaching Airy disc diameter, and larger; the range of planetary and deep-sky observing) and near the stellar resolution threshold. Differences in this respect can be expected based on distinctly different form of energy distribution caused by chromatic error. Due to the increasing defocus error for non-optimized wavelengths, diffraction pattern has a form similar to one caused by monochromatic defocus: the first dark ring is filled with energy, with its contrast deep dramatically reduced, or non-existent.

As a result, at any given nominal peak intensity (within the range commonly encountered with amateur telescopes), diffraction pattern affected with secondary spectrum has higher encircled energy fraction within the central maxima than what is indicated by the peak value (Strehl). For instance, while the polychromatic e-line Strehl of a 100mm /12 doublet achromat is 0.77, and its best focus Strehl 0.81, the encircled energy within the Airy disc is 0.83. It is generally higher than the encircled energy fraction with spherical aberration at the same nominal peak intensity, which is nearly identical to the encircled energy fraction. This effectively increases the relative energy contained in central maxima vs. energy transferred to the rings for given nominal Strehl, enhancing contrast transfer for extended objects. On the flip side, the slight enlargement of central maxima due to chromatic defocus does have negative effect on the efficiency of contrast transfer details in that size range. The combined effect of these two factors, as the MTF plots below illustrate, is better contrast transfer at mid-to-low frequencies, and lower in the 0.5-0.8 frequency range (approximately).

Consequently, assessment of the optical quality of an achromat varies somewhat with the focus location: peak diffraction intensity (SP) is not at the optimized (e-line) focus (SPe), but closer to d-line focus. And for any given Strehl, an achromat will have better contrast transfer in the range of extended details than an aperture with equal Strehl resulting from spherical aberration (and most others), thus can be assigned a better effective Strehl (Sext).

Graphs below (FIG. 73) and the accompanying text describe more specifically how the effect of secondary spectrum in an achromat with standard glasses depends on its aperture and focal ratio. Since its magnitude, with the given glasses, is determined by these two parameters alone, it is not surprising that it is a function of their combined form, F/D. As will be explained in more details below, peak diffraction intensity in an achromat is not at coinciding with its optimized wavelength's focus. Rather, it is shifted somewhat toward the rest of wavelengths focusing behind it. Gain in the Strehl and contrast transfer at the best focus location is relatively small, but not negligible.

Both, Strehl values and MTF are calculated by OSLO, based on 25 wavelengths from 440nm to 680nm (10nm increment), weighted for the photopic eye sensitivity, for the standard C-e-F Fraunhofer doublet achromat (BK7/F2, with secondary spectrum Δ~/2000 with respect to d-line, and Δ~/1800 with respect to e-line). Note that the same polychromatic input is used for calculating comparative effects of spherical aberration.   

FIGURE 73: Effect of secondary spectrum on image quality in terms of standard indicators of optical quality - central diffraction intensity and contrast transfer efficacy - allows for its direct comparison with other forms of aberrations, as well as a qualitative assessment of its performance level. While the nominal chromatic defocus error for given aperture D scales inversely with the focal ratio F, and with the aperture for given focal ratio, the effective error - the one that corresponds to the polychromatic Strehl, either at the best diffraction focus, e-line focus, or for the "effective" Strehl - changes approximately with the cube root of the square of either parameter. For instance, chromatic defocus error for given aperture halves nominally going from /10 to /20, but the effective error, corresponding to the best focus Strehl, only reduces by 35%, from 0.275 to 0.18 wave P-V of spherical aberration (it is nearly identical to wavefront error of defocus, with the latter being larger by 3.3% for given RMS). An /34 has the nominal chromatic defocus smaller by a factor of 0.3, but the effective wavefront error (0.12) only by a factor of 0.44, and so on.

For short-focus achromats, additional limiting factors is the emergence of higher order spherical aberration, due to strongly curved inner surfaces. This affects all the wavelengths, and quickly brings down (the remaining) optical quality with further focal ratio reduction. Small apertures are more limited by it in terms of F/D ratio: at the ratio value of 1, for D in inches, the focal ratio is still /8 for an 8-incher, while only /4 for 4-inch aperture.

There is no simple accurate expression for achromat's polychromatic Strehl, but it can be well approximated with simple empirical relations. Following text addresses more specifically the three points of assessment of achromat's optical quality:
(1) polychromatic Strehl at the optimized wavelength, (2) polychromatic Strehl at the best focus location, and (3) effective polychromatic Strehl for extended objects.

(1): Polychromatic Strehl at the optimized line focus (SPe)

It is the peak diffraction intensity measured at the optimized line (assumed e-line) focus in the standard Fraunhofer-type doublet achromat. as a function of the aperture size D and relative focal ratio F. Weighted for photopic eye sensitivity in 430-680nm range, in the standard achromat with the d-line secondary spectrum Δ~/2000, it is approximated by:

  SPe ~ 1.3(F/Dmm)1/4           (a)

For D in inches, SP~0.58(F/D")1/4. It stays close to the actual value for F/Dmm values smaller than ~0.25, which covers most practical instruments. For larger values of F/Dmm it becomes too optimistic (it gives SPe=1 for F/Dmm=0.35, which corresponds to 100mm /35, or 200mm /70).

For (F/Dmm) values greater than 0.2, up to ~0.8 (i.e. for very long focus achromats), better empirical approximation for polychromatic Strehl in an achromat is SPe~(F/Dmm)0.08, or SPe~0.8(F/D")0.07 for D in inches.

For F/Dmm values of ~0.05 and smaller, the approximated polychromatic Strehl gradually becomes too optimistic, mostly due to the overall deterioration caused by increasing spherochromatism. Smaller apertures are more affected, due to higher-order spherical aberration generated by their steep inner radii. For instance, a 150mm /4.5 has polychromatic Strehl of 0.50, while 100mm /3 drops to 0.38 Strehl, despite both having F/Dmm=0.03, for which the above approximation gives 0.54 value.

The corresponding comparable RMS error of monochromatic aberration, obtained from Eq. 56, is RMS=0.24(-logSPe)0.5.

(2): Peak polychromatic Strehl (SP)

However, polychromatic diffraction calculation also reveals a little known - yet rather obvious - fact, that best polychromatic focus in an achromat does not coincide with the focus of optimized wavelength (usually around e-line). It is shifted somewhat toward the red/blue focus, closer to the d-line focus. It is the consequence of all other wavelengths focusing farther away than the optimized wavelength, including those to which the eye is still highly sensitive. Up to a point, reduction in defocus error in all the wavelengths away from e-focus by shifting focus location toward red/blue focus overcompensates for the increase in e-line (and the immediate wavelengths) defocus. As a result, diffraction maxima generates more energy.

I am not aware that this fact, tumbled upon accidentally while going through matters related to Neil English's book about refractors, was known at all; there is no mention of it by Sidgwick or Conrady (understandably, since they did not have the benefit of accessible diffraction calculation), nor more contemporary texts on aberrations, including my cited references).

Graph bellow gives specifics of this effect for a standard 100mm /10 achromat.

This implies that best polychromatic ratio in an achromat is higher than what is indicated by (a), which approximates Strehl value at the focus of optimized wavelength, usually around green e-line. As the inset "F-Ratio Dependence" shows, the gain in polychromatic Strehl by refocusing from e-line to best polychromatic focus - which is fair to assume that the eye leads us to do - is slightly decreasing from fast toward slow systems. Again, based on OSLO output, the highest polychromatic Strehl, as specified earlier, is higher approximately by a factor of (F+7)/(F+6), F being, as before, the focal ratio number. Thus, peak polychromatic Strehl of an achromat is approximated by:

SP~ (F+7)Spe/(F+6)      (b)

with Spe being the polychromatic Strehl at the location of e-line focus, as given by (a). Note that (a) applies only to F/D values ~0.2 and smaller; for F/D values larger than ~0.2, e-line polychromatic Strehl is approximated by Spe~(F/D)0.08 for D in mm, and by Spe~0.8(F/D)0.07 for D in inches. For F/Dmm values of ~0.25 and higher, polychromatic Strehl at the best focus is closely approximated by (F/D)0.06.

This strictly applies only to achromats with negligible spherical aberration. That is fair assumption with long-focus objectives, where fabrication tolerances are very wide, but with mid and short focus achromats it is realistic to expect some degree of correction error. In general, spherical aberration will reduce the gain of refocusing. For instance, λ/4 P-V wavefront error of overcorrection will roughly cut it in half, also reducing the extent of refocusing to a similar degree. On the other hand, λ/4 of undercorrection while reducing the gain similarly, will have little effect on the extent of refocusing.

(2): The effective Strehl for extended details (Sext)

Another interesting property of the achromat revealed by diffraction calculation shows that the central maxima consistently encircles more energy than the same nominal Strehl of spherical aberration. The significance of it is that the latter is the only other "default" axial aberration, and even more that it is the dominant aberration form in apochromats. Thus comparison by the nominal polychromatic Strehl may not tell the whole story.

The effects is illustrated below with the polychromatic diffraction PSF, encircled energy and MTF plots of a 100mm f/6 achromat, whose 0.67 Strehl at the best polychromatic focus nominally equals that of 1/3 wave P-V of primary spherical aberration. While both have 0.67 nominal Strehl, the encircled energy within first maxima is 0.63 for the achromat, and 0.58 for spherical aberration. The energy nearly equalizes after the first bright ring, but energy distribution up to that point favors the achromat, and it shows as better MTF contrast transfer in the 0.15-0.5 frequency range. It is compensated for with lower contrast transfer in the 0.5-0.7 frequency range, but the lower range - approximately details 1-2 times the Airy disc size - is more important for general observing, affecting the bright low-contrast detail resolution threshold. For spherical aberration to reach that level in this frequency sub-range, it would need to be at about 0.28 wave P-V (0.76 Strehl), or to be "switched" to about 10% larger aperture.

Expectedly, the difference diminishes with the magnitude of aberration. Very  approximately, the ratio of encircled energy for secondary spectrum vs. spherical aberration changes in proportion to (F+7)/(F+6), F being, as before, the focal ratio. The distribution profile is not the same, however, with the long focus achromats apparently having more encircled energy not only within the first dark ring area, but over a significantly wider radius as well. 

 There is no indications that either brightening of the first dark ring caused by secondary spectrum, or intensifying and enlarging the 1st bright ring by spherical aberration at these levels of aberrations, lowers limiting stellar resolution - inasmuch as MTF graph can show this resolution aspect. Considering that spherical aberration and central diffraction obstruction generate similar diffraction effect, it is fair to assume that secondary spectrum would have similar advantage over central obstruction causing identical drop in the central diffraction intensity.

    Another question that can be answered using diffraction calculation is what is the difference in optical quality between long-focus achromats and apochromatic refractor. It is known that "true" apos have only about 1/10 of the secondary spectrum of a comparable achromat, or less, but this fact alone can't be used as the basis for a direct comparison. The reason is that the primary source of chromatic error in a typical apochromat is not secondary spectrum, but sphero-chromatism, which is in turn entirely negligible in long-focus achromats. Long-focus achromat aficionados tend to place it very close to the "true apo" level optically, but the MTF confirms that the latter does have noticeable advantage.

: Polychromatic (photopic eye sensitivity) MTF plots for selected 4 and 6-inch refracting objectives. A 6-inch triplet apo is well into the "true apo" zone (above 0.95 polychromatic Strehl) at f/8, and a tad short of it at f/6.5 with the best FPL53 match, Schott's ZKN7. An f/8 doublet with these glasses is nearly as good as the f/6.5 triplet and, for all practical purposes, identical to a second-class f/8 triplet with FPL51 and one of its best matches, K7. An f/8 doublet with these glasses, however, drops below 0.9 Strehl, positioned closer to long-focus achromats than the true apos. With a 4-inch aperture, a first-class doublet at f/9 comes as close to perfection as it may matter, and the second-class triplet is nearly as good
at f/7. A second-class doublet can still be a true apo at f/9, but not the first-class doublet at f/7. It is only slightly better - most likely below the threshold of perception - than a second-class doublet at f/9, in this particular case the old Meade f/9 apo. An f/15 achromat is about as much behind them as they are behind near-perfect aperture. About as much behind the f/15 is f/10 achromat, and the f/6 is nearly as much behind the f/10, as the latter is behind the true apo.
A few notes: (1) all doublets are air-spaced, and all triplets are oil-type, i.e. with lenses in contact and R2=R3 and R4=R5
(2) MTF plots are based on the design optimum, which somewhat favors the apos, which generally have much higher sensitivity to misalignment, (3) photopic Strehl is different than mesopic Strehl, which is lower and generally applies to night time observing, and (4) the plots do not include seeing error, which means that the gap between the 6-inch and 4-inch apertures is - all else unchanged - somewhat smaller under the real sky.

    Above plots are polychromatic MTFs (440-680nm) for the standard Fraunhofer doublet achromats, and several apochromats, for 4 (blue) and 6-inch apertures (pink). The difference in contrast level and resolution between long focus achromats and apochromats is smaller for the 4-inchers, as expected, considering that larger achromats have more chromatism for given focal ratio. Long-focus 4-inch achromat is close to the faster 4-inch apos, but an /10 apo has near zero chromatism. Polychromatic Strehl values given for achromats are those at the best (diffraction) focus (SP), and their MTF plots are also for this focus location.

    Note that these plots reflect only the level of chromatic correction. Longer-focus systems have the additional advantage of more relaxed fabrication and alignment tolerances, and can be expected to have generally somewhat better correction level than what the chromatic correction alone implies. Smaller apertures, of course, are less subjected to seeing error, hence the difference in contrast transfer is also somewhat less than what above plots imply.

The overall perception is probably that long-focus refractors should have somewhat better optical rating than what approximation in (a) indicates. Part of this notion results from the general tendency of assigning to the high-quality performers better optical quality than they really have. In other words, empirical criteria is based on the performance relative to other instruments; a telescope operating at a 0.9 Strehl overall optical quality, or even less, will be perceived as being close to perfection if other telescopes are operating at lower to significantly lower levels of optical quality - which is commonly the case.

In this context, the perception of higher than actual optical quality level springs from a number of error sources that are invariably present being neglected or downplayed. For instance, a superb 6" Maksutov-Cassegrain telescope that goes with 0.96 Strehl optics will be operating far below that level in the field. In the average 2 arc seconds seeing, average seeing-induced error is around 0.1 wave RMS, enough to keep it below 0.7 Strehl level half of the time, or so. It is not much better inside the tube: typical ~0.35D central obstruction alone lowers the 0.96 optics Strehl by a 0.77 degradation factor, to 0.74. Thermally induced errors are all but likely to push it further down, below 0.7 Strehl level. Combine it with the seeing error, and you have an instrument performing not much above 0.5 Strehl most of the time (misalignment error is not as much significant with Maksutov-type telescopes, in general, as it can be with some others). Yet it is regarded as a very good overall performer. Its actual optical quality level? According to the generally accepted scale shown on Fig. 48 (top), it belongs to "poor" optics.

Considering this, the 0.74 peak diffraction intensity resulting from secondary spectrum in a 6" /15 achromat - or an effective ~0.77 for the range of resolvable extended details - doesn't look bad anymore. While it still suffers the same from seeing, it is significantly less affected by thermal errors, and - has no aperture obstruction. Since its chromatic error nearly offsets the effect of Mak's central obstruction, it is likely to perform better in the field.

Combined with the approximate ratio of the increase in "effective Strehl" for the left side of MTF graph, due to the better contrast transfer resulting from higher relative encircled energy within first maxima, given by (F+10)/(F+9), best polychromatic Strehl of an achromat is approximated by Sext~(F+10)(F+7)Spe/(F+9)(F+6), or

Sext~ (F+10)Sp/(F+9)         (c)

This "effective Strehl" value better reflects achromat's performance level for details about Airy disc in size, and larger, including nearly all resolvable planetary details.

In conclusion, in order to assess level of performance of an achromat, and to compare the effect of secondary spectrum to that of central obstruction and spherical aberration, we have to depart from nominal secondary spectrum, and turn to diffraction calculation. In addition, the effects specified above - higher relative energy within 1st maxima, and shift to best polychromatic focus - need to be taken into account.

Based on these factors, following table shows the peak polychromatic Strehl SP (for the entire range of MTF frequencies) and the maximum effective polychromatic Strehl Sext for the left side of MTF graph (extended details) of an achromat with near-zero aberration in the optimized wavelength, for selected systems, with the corresponding value of primary spherical aberration, as well as central obstruction size (not adjusted for the effect of brighter central disk). The achromats are standard,  BK7/F2 , with the wavefront error in F and C nearly equalized (this secondary spectrum mode gives somewhat higher polychromatic Strehl than the canonical mode with F and C focus nearly coinciding). The apo is an FPL52/ZKN7 doublet.




spherical aberration

central obstruction

SP Sext SP Sext

4" /5 achromat







4" /10 achromat







4" /15 achromat







4" /20 achromat







6" /15 achromat







8" /10 achromat







4" f/10 apochromat







36" /10.8 (Lick refractor)







TABLE 7: Approximate comparative effects of secondary spectrum (according to OSLO output) vs. spherical aberration and central obstruction. Encircled energy (EE) is within the Airy disc radius. Comparable mid-to-low frequency P-V error of spherical aberration is obtained substituting the EE value for the peak intensity value (Sext)) in W~0.8-logS, to better reflect the effect of the out-of-disc energy, which is generally lower for achromats than what the peak diffraction intensity indicates. The comparable relative central obstruction size ο is obtained from Sext=(1-ο2)2., thus ο=[1-Sext0.5]0.5.

Considering common perceptions among amateurs, the numbers for achromats may look a bit optimistic, but that is what the raytracing results imply. One likely exception is the Lick refractor which, due to its enormous size, cannot have overall optical quality comparable to that of small amateur instruments.

Also, a first-class 4" apo can easily have better polychromatic Strehl than 0.96, up to 0.99 for a 100mm /10 system - at least in theory (in fact, a slight weakening of its positive element takes this particular 4" to 0.99 polychromatic Strehl, with the upscaled 6" improving to 0.96). However, common doublet designs are extremely sensitive to even slight deviations from design parameters; induced spherical aberration, even when only moderate, can significantly lower their polychromatic Strehl. It may not be readily apparent, since the chromatism in an apochromat is visually less noticeable as a color error, for given error level (i.e. polychromatic Strehl), due to non-optimized colors being tighter together. But it does lower contrast just the same. For that reason, assigning to the apo 0.96 SP value is more realistic (being over 0.95 Strehl, it is already within "sensibly perfect" range, so it does not matter in practical terms).

Note that triplet apos are much more forgiving in that respect than doublets, and that is their main practical advantage.

It is important to emphasize that all the above figures are valid for photopic eye sensitivity and the standard e-line optimized achromat (C-F correction). As we all know, most observing is done during night time, when eye sensitivity shifts from photopic (bright light conditions) toward partial dark adaptation (mesopic mode). It is the most complex eye mode, with both cone and rod photoreceptors simultaneously active. Cone sensitivity in the mesopic mode increases for all wavelengths, but more in the blue and red than in green/yellow. However, this expanded cone sensitivity is followed with the decrease in their acuity. Even more so, this is the case with the rods, which are much more sensitive in the blue/violet than the cones, but also having much lower resolution and contrast sensitivity. At present, there is probably not enough specific knowledge about eye's mesopic function to assess level of optical quality in this mode. In general, it is to expect that the increased sensitivity beyond photopic range, particularly toward blue/violet, effectively increases polychromatic error in both achromats and apochromats. This increase, however, is probably mainly offset by the lowered eye acuity.

Due to chromatic defocus in achromats being significantly larger toward blue/violet end of the spectrum, they may be somewhat more affected than apochromats, but it remains speculative until mesopic error level - i.e. nominal contrast and resolution limits imposed by chromatism - can be evaluated versus specific contrast/resolution abilities of the mesopic eye.

4.7.2. Lateral color error   ▐    4.8. Fabrication errors

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