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▪ CONTENTS
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. |
|
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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