4.8.2. 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 ƒ/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= Pr2 (51)

with P=-Dƒ/8F2 being the peak aberration coefficient, equal to the P-V wavefront error, and r 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 Dƒ being, for typical achromats, ~ƒ/2000 at best, the P-V wavefront error of chromatic defocus for an achromat can be written as W~D/16,000F at its red and blue foci. 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 is not an accurate indicator of the level of chromatism, 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 doubles with either doubling D or halving F, resulting chromatism - measured by the drop in polychromatic peak diffraction intensity, or polychromatic Strehl (SP) - will increase at a significantly slower rate. This is what advanced optical design software programs, using diffraction calculation, imply.

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" ƒ/10 above have only a few percent of the energy left within the Airy disc). The spectral range relatively close to the optimized wavelength also doesn't contribute "new" lost energy, since it is not affected. It is only a relatively narrow spectral segment on either side of the optimized wavelength, which previously had relatively small amount of energy outside the Airy disc, 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 a 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 pretty much like an aerially small, nominally large, and effect-wise mainly drained out wavefront error.

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

As it often goes in life, there is the bad side to it as well: it is that the SP 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 the SP 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 at within sub-ranges of frequencies that could be of interest. In particular, how the effect of chromatism compares to 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 suspected based on distinctly different form of energy distribution caused by chromatic error. Due to 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 significantly reduced.

As a result, at any given nominal peak intensity (within the range commonly encountered with amateur telescopes), diffraction pattern affected with chromatism has higher encircled energy fraction within the Airy disc than what is indicated by the peak value. For instance, while the SP of a 100mm f/12 doublet achromat is 0.77, 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). On the flip side, brightening of the first dark ring caused by chromatic defocus does have negative effect on the efficiency of contrast transfer. The combined effect of these two opposing factors, as the MTF plots bellow illustrate, is better contrast transfer at mid-to-low frequencies, and lower in the 0.5-0.8 frequency range (approximately).

The graphs bellow (FIG. 48) illustrate:

(1) how the peak of polychromatic PSF of a standard doublet achromat - and with it the effect of secondary spectrum on image quality - changes with the change in F and D,

(2) the effect of the form of intensity distribution caused by secondary spectrum on contrast transfer, with comparison to the effect of spherical aberration, and

(3) SP and contrast transfer in an achromat compared to an apochromatic refractor.

Both, SP 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 Dƒ~ƒ/2000 with respect to d-line, and Dƒ~ƒ/1800 with respect to e-line). Note that the same polychromatic input is used for calculating comparative effects of spherical aberration.

FIGURE 48: Effect of chromatism 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.
(1): Polychromatic peak diffraction intensity (PPDI=SP) in a standard Fraunhofer-type doublet achromat as a function of aperture size and relative aperture has significantly different rate than change in the nominal chromatic defocus (secondary spectrum). While there is no simple accurate expression for it, the change in PPSF can be fairly well approximated with simple relations fitting into actual values. In any instance, the detriment of chromatism with the increase in aperture size - or reduction in relative aperture - is much smaller than corresponding nominal increase in the defocus error (secondary spectrum) caused by either of the two. Note that PPDI doesn't change with scaling doublet achromat while keeping the aperture to focal length ratio constant; a 100mm ƒ/12 has identical PPDI as 200mm ƒ/24. For a given aperture D in mm, the peak diffraction intensity (PPDI, or polychromatic Strehl) weighed for photopic eye sensitivity in 440-680nm range, in function of the focal ratio F of the achromat with d-line secondary spectrum Dƒ=ƒ/2000 is approximated by:

SP ~ 1.3(F/Dmm)1/4

For D in inches, SP~0.58(F/D")1/4. It stays close to the actual value for F/Dmm of ~0.25 and smaller, which covers most practical instruments. For larger values of F/Dmm it becomes too optimistic (it gives SP=1 for F/Dmm=0.35, which corresponds to 100mm ƒ/35, or 200mm ƒ/70). For (F/Dmm)>0.25, up to (F/Dmm)~0.8, better empirical approximation is SP~(F/Dmm)0.08, or SP~0.8(F/D")0.07 for D in inches.

The corresponding comparable RMS error of monochromatic aberration, obtained from Eq. 56, is RMS=0.24(-logSP)1/2.

However, diffraction calculation shows that, due to the central maxima slightly enlarged by chromatic defocus, energy encircled within it is consistently higher than what peak diffraction intensity indicates (i.e. EE~SP, the relationship between encircled energy and peak intensity valid for most aberrations), with approximate EE/SP ratio of (F+3)/(F+2). This implies as much less energy in the ring area, and better contrast transfer for extended details.

(2) Relatively more energy kept within the central maxima, and fainter first bright ring, result in the chromatic error for given peak diffraction intensity to have somewhat better contrast transfer than spherical aberration error of identical nominal diffraction peak, in the mid-to-low MTF frequencies. The low-contrast resolution thresholds are for typical planetary and deep-sky details (Telescope Optics, Rutten/Venrooij, p215).

For the left side of MTF graph (extended object resolution) an effective contrast transfer is closer to that appropriate to peak intensity equaling EE, or higher nearly by (F+3)/(F+2) factor than its actual SP value.

On the other hand, there is no indications that brightening of the first dark ring caused by secondary spectrum, even in an achromat as fast as 100mm f/6, lowers limiting stellar resolution - inasmuch as the MTF graph can show this resolution aspect. In all, the particular form of intensity distribution within diffraction pattern in achromats results in more efficient contrast transfer within the range of resolvable extended details than with spherical aberration causing nominally identical drop in peak diffraction intensity (or central diffraction obstruction, considering its close similarity with spherical aberration in regard to intensity distribution for given relative peak intensity).

(3) 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 an apo optically, but the MTF confirms that the latter does have noticeable advantage. Plots to the left are polychromatic MTF (440-680nm) for the standard f/15 Fraunhofer doublet, an f/10 air-spaced apo doublet with O'Hara FPL52 ED glass in front, and ZKN7 crown rear lens (from ATMOS designs, the 6" is upscaled from the original 4" aperture size), and an 6" ED apo triplet with FPL51 O'Hara ED glass between K5 (front) and BK7 (rear) crowns. The difference in contrast level and resolution is relatively small, but noticeable in average field conditions; it is greater for larger aperture, as expected. If the peak diffraction values, multiplied by (F+3)/(F+2) factor to account for the higher EE effect on contrast transfer in mid-to-low-frequencies, are translated into corresponding RMS wavefront errors, it gives 0.051, 0.032, 0.079, 0.047 and 0.028 (equivalent of 1/5.8, 1/9.3, 1/3.8, 1/6.4 and 1/10.6 wave P-V of spherical aberration) for the 4" achromat, 4" ED apo, 6" achromat. 6" apo dublet and 6" apo triplet, respectively.

The overall perception is probably that long-focus refractors should have somewhat better optical rating. Most of this notion is the result of a general tendency of assigning to 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.

A number of error sources that are invariably present are neglected or downplayed. For instance, a superb 6" Maksutov-Cassegrain telescope that goes with 0.96 Strehl optics will be operating far bellow 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 bellow 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 0.77 degradation factor, to 0.74. Thermally induced errors are all but likely to push it further down, bellow 0.7 Strehl level. Combine it with the seeing error, and you have an instrument performing merely above 0.5 Strehl - at best - most of the time. Yet it is regarded as a very good overall performer (misalignment error is not as much significant with Maksutov-type telescopes, as it can be with some others).

Compared to it, 0.74 peak diffraction intensity of a 6" f/15 achromat - or an effective ~0.77 for the range of resolvable extended details - doesn't look that much off. While it still suffers the same from seeing, it is significantly less affected by thermal errors. Since its chromatic error nearly offsets with the effect of Mak's central obstruction, it is likely to perform better in the field.

Following table shows nominal polychromatic PSF peak. It includes comparable amount of lower spherical aberration, and comparable central obstruction size (not adjusted for the effect of brighter central disk), with respect to relative intensity distribution within diffraction pattern and the resulting contrast level.

Refractor	SP	EE	SP-comparable sph. aberration (P-V)		Comparable central obstruction.
Refractor	SP	EE	entire MTF range	mid-to-low frequencies	entire MTF range	mid-to-low frequencies
4" ƒ/6 achromat	0.64	0.72	1/2.8	1/3.3	0.60D	0.39D
4" ƒ/10 achromat	0.74	0.80	1/3.4	1/4	0.51D	0.33D
4" ƒ/12 achromat	0.77	0.83	1/3.7	1/4.4	0.48D	0.30D
4" ƒ/24 achromat	0.88	0.93	1/5.5	1/7	0.33D	0.19D
4" ƒ/48 achromat	0.95	0.975	1/8.3	1/11.8	0.22D	0.11D
4" ƒ/15 achromat	0.81	0.87	1/4.2	1/5.1	0.44D	0.26D
6" ƒ/15 achromat	0.71	0.79	1/3.4	1/3.9	0.52D	0.33D
8" ƒ/12 achromat	0.63	0.69	1/2.8	1/3.1	0.62D	0.41D
8" ƒ/15 achromat	0.67	0.73	1/3	1/3.4	0.57D	0.38D
4" f/10 apochromat	0.96	0.97	1/9.3	1/10.8	0.20D	0.12D
36" ƒ/10.8 (Lick refractor)	0.34	0.365	1/1.82	1/1.88	0.81D	0.63D

TABLE 4: 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 (S=SP, the raytrace value) 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. Comparable central obstruction size is obtained from SP=(1-ο2) and EE=(1-ο2)2, for the entire MTF range and mid-to-low frequencies, respectively.

◄ 4.8.1. Secondary spectrum and spherochromatism ▐ 5. INDUCED ABERRATIONS ►

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