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6.4.1.
Star testing telescopes
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6.6. Effects of aberrations: MTF
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#
**6.5. Strehl ratio**
One of the most frequently used optical
terms in both, professional and amateur circles is the *Strehl ratio*.
It is the simplest meaningful way of expressing the effect of wavefront
aberrations on image quality. By definition, Strehl ratio -
introduced by the German physicist, mathematician and astronomer Dr. Karl Strehl at the end of 19th
century - is the ratio of peak diffraction intensities of an aberrated
vs. perfect wavefront.
The ratio indicates the level of image quality in
the presence of wavefront aberrations; often times, it is used to define the
maximum acceptable level of wavefront aberration for general observing -
so-called *diffraction-limited* level - conventionally set at 0.80
Strehl.
Similar type of indicator is the *Struve ratio*, which expresses
peak diffraction intensity of aberrated vs, aberration-free line spread
function (LSF). It requires slightly tighter 0.80 ratio level
requirement for primary coma (0.58 vs. 0.63 wave P-V), more relaxed for
spherical aberration (0.27 vs. 0.25) and defocus (0.29 vs. 0.26) than
the Strehl ratio (the tolerance for astigmatism is nearly identical).
However, it has far less universal appeal than the Strehl ratio,
expressing vital property of the single most important optical
indicator, the PSF, building stone of nearly all intensity distribution
forms, including the LSF.
Wavefront deviations from perfect
spherical are directly related to the size of phase errors at all points
of wave interference that form diffraction pattern. In other words, it
is a nominal wavefront
deviation from spherical that determines the change in pattern's intensity
distribution. However, it is not the peak-to-valley nominal
aberration, which only specifies the peak of deviation, and tells
nothing about its extent over the wavefront area. It is the
root-mean-square, or RMS wavefront error,
which expresses the deviation averaged over the entire wavefront. This
average wavefront deviation determines the peak intensity of diffraction
pattern and, hence, numerical value of the Strehl ratio (note that the
RMS error itself is accurately representing the magnitude of wavefront
deviation only when it is affecting relatively large wavefront area,
which is generally the case with the
conic surface aberrations).
For relatively small errors - roughly 0.15
wave RMS, and smaller - the RMS wavefront
error, and the resulting Strehl ratio, accurately reflect the
effect of overall
change in energy distribution, regardless of the type of aberration.
With larger errors, the correlation between the RMS error and the Strehl
vanishes: larger RMS error can produce higher PSF peak intensity, and
better image quality than the lower errors.
Mid astigmatic focus, for
instance, has identical PSF peak intensity at 2 and 3 waves P-V
wavefront error, despite the latter having 50% higher RMS/P-V. Similar
RMS-to-Strehl inconsistency above 0.15 wave RMS exist for
spherical aberration,
and aberrations in general.
As a general rule for aberrations below
0.15 wave RMS, the relative drop in peak diffraction intensity
indicates how much of the energy is lost, relatively, from the Airy disc.
For instance, 0.90 Strehl indicates about 10% lower energy within
the Airy disc. But the exceptions are possible, and generally larger in
magnitude with larger error levels.
For instance, the drop in peak diffraction intensity
is nearly identical at 0.0745 wave RMS and 0.15 RMS wavefront
error - 20% and 59% respectively - for all three, spherical aberration,
coma and astigmatism. At the same time, the accompanying drop in the energy encircled within
the Airy disc is 20% and 11% at 0.745 wave RMS, and 61%, 56% and 38% at 0.15
wave RMS, for spherical aberration and coma vs. astigmatism,
respectively. However both, nominal Strehl and overall contrast level
remain nearly identical for all, due to the energy transferred by
astigmatism effectively transforming central disc into a larger,
cross-like form, reducing contrast level
over the higher range of MTF frequencies more, and less than the other
two in the lower
frequency range.
While the actual Strehl calculation requires
complex math, simple empirical expression by Mahajan gives a very close
approximation of the Strehl ratio in
terms of the RMS wavefront error:
(56)
where
e
is the natural logarithm base (2.72, rounded to two
decimals), and
**ω**
is usually the RMS wavefront error in units of the wavelength. Note that
use of the RMS wavefront error can yield inaccurate result; the actual
Strehl value - and the original form of approximation - are phase dependant, thus determined by phase
variance **φ**2
and, more directly, by the phase analog to the OPD-based RMS wavefront error,
** φ**,
with φ2=(2πφ)2.
The approximation is accurate to a
couple of percent for RMS errors of ~1/10 wave, with the difference
diminishing for smaller errors. The difference vs. exact Strehl value gradually increases with the RMS
error, but even at S~0.3 it still does not exceed 10%. It overestimates
true Strehl for balanced primary aberrations, and underestimates it for
classical aberrations.
This
approximation is also known as "extended Maréchal's approximation", as
opposed to the original Maréchal's approximation, S~(1-0.5φ2)2~[1-0.5(2πφ)2]2
which, for φ~ω,
can be written in terms of the RMS wavefront error as S~[1-2(πω)2]2.
For small RMS
errors (~1/15 wave or
less), a simpler approximation, given by **S~1-(2****πω)2**,
or **S~1-39.5ω****2**,
is also accurate; however, it becomes increasingly inaccurate with
larger RMS errors - at 1/10 wave it already underestimates the true
Strehl by more than 10%, and drops to zero at ~0.16 wave RMS **(FIG.
97)**.
For errors larger than ~1/15 wave RMS, and smaller than 1/5 wave RMS,
a simple empirical approximation **S~1-10****ω1.5**
gives slightly less accurate result than Mahajan's approximation for
RMS<0.2 (within 2%), but has better overall accuracy than the two
alternative approximations.
**FIGURE
97**:
Strehl ratio as a function of RMS wavefront error.
LEFT: Plots for three ratio approximations and the
true Strehl value for primary spherical aberration at the best focus
(balanced spherical; identical to the Siedel - i.e. Gaussian focus' -
spherical aberration) in unobstructed aperture. Strehl ratio
approximations, from the top down, Mahajan's (also known
as "extended Maréchal approximation"), Maréchal's, and simplified Maréchal's, the latter with the 4th power term in the expansion neglected. The lower two approximations are accurate for RMS errors smaller than
~0.07, while Mahajan's remains reasonably close to the true ratio value
even for RMS errors in excess of ~0.2, for classical and balanced (best
focus) aberrations in general. It remains close
to the true Strehl for spherical aberration for wavefront errors in excess of 0.25 wave RMS.
For larger RMS
errors (~0.1 wave RMS and larger) the true
Strehl ratio for best focus coma (not shown) is slightly higher than the
ratio for spherical aberration, and the ratio for best focus astigmatism
(not shown) is slightly higher then that for best focus coma, with the
latter only slightly lower than Mahajan's
approximation. Both, coma and astigmatism Strehl ratios become slightly
lower than the ratio for spherical aberration as the RMS error exceeds
~0.2. In general, for RMS wavefront errors over ~0.1 wave, Strehl value
for given large RMS error varies slightly with the aberration type. This
variation becomes more pronounced expanding to other wavefront forms;
for instance, true Strehl for Siedel coma of 0.25 wave RMS is over 60%
higher than for the three balanced forms (spherical, coma, astigmatism),
and true Strehl for Siedel astigmatism more than doubled (that despite
the RMS wavefront error keeping the same lower ratio for the balanced
aberration forms; similar
shift of the PSF peak away from the focus point with minimum wavefront
deviation occurs with spherical aberration as well).
RIGHT: Change in the Strehl ratio due to central
obstruction, for balanced primary spherical, defocus, coma and astigmatism,
all at the best focus, for
selected central obstruction sizes. Horizontal scale shows the RMS error
for zero obstruction. Change of the Strehl is the consequence of the
change in the RMS
wavefront error due to obstruction of a portion of the wavefront,
with the RMS of the annulus varying somewhat with the aberration type.
With balanced spherical and defocus, Strehl ratio of the annulus
continually increases with the size of central obstruction in a similar
way, only the magnitude of increase is larger for the
former. With coma, the change is negligible except for obstructions
nearing 0.5D and larger, for which the Strehl increases by nearly 10%, or
more (due to the shape of the comatic wavefront, with nearly flat
central area and deformation increasing toward the edge, smaller
obstructions - up to about 0.3D - cause relatively small increase in the
annulus RMS error, but as the obstruction becomes large enough to
significantly block out the deformed areas, the RMS and Strehl quickly
start improving, becoming better than for the unobstructed wavefront). With
astigmatism, the Strehl continually decreases, falling over 50% at
0.5D obstruction size.
The Strehl and RMS error are, of course, only one
side of the story. In addition to the effect of optical path difference
due to wavefront deviation from perfect sphere, wave interference at the
focus is also affected by missing wave contributions due to wavefront
obstruction. The PSF peak here is a product of the Strehl and PSF peak
degradation due to central obstruction, given by
(1-ο2)2,
where **o** is the relative linear obstruction diameter in units of aperture
diameter.
For errors larger than about 0.15 wave RMS, the correlation between the
RMS wavefront error and Strehl ratio becomes more loose, and the PSF peak shifts away
from the point of minimum wavefront deviation. Since the plots are based
on the RMS error at this point, they are accurate only up to about that
error level. Thus, part of the plot beyond 0.15 wave RMS is only approximation of the actual change in the Strehl ratio.
Conventional "diffraction-limited" aberration level is
set at the Strehl
ratio of 0.80 or, in terms of the RMS wavefront error, 0.0745 (or 1/**√**180), regardless of the type of aberration.
This only concerns wavefront quality; presence of other factors
negatively affecting image quality, such as aperture
obstruction, or chromatism, would result in further deterioration in
quality of the diffraction image. Thus achieving "diffraction-limited"
level in such circumstances requires higher wavefront quality,
according to the magnitude of additional error.
The RMS wavefront error in terms of Strehl
ratio is, from **Eq. 56**, closely approximated as ω~0.24**√**-logS. For the range of aberration mentioned, drop in
the peak intensity expressed by the Strehl ratio
also indicates the relative amount of energy transferred from the central disc to
the ring area of the diffraction pattern, given as (1-S). Moreover, this relative number
also indicates the average contrast loss over the range of resolvable
frequencies. Regardless
of the aberration type, these three basic properties of an aberrated
pattern - the relative drop in central intensity, relative amount of
energy transferred to the rings area, and averaged relative contrast loss -
are practically identical for a given RMS wavefront error.
While the Strehl ratio
furnishes very useful quantitative information about the effect of an
aberrated wavefront, it is of general nature. It doesn't give specific
indications on how the contrast varies for details of different angular
size, nor how it affects the resolution limit. Also, there are factors
affecting intensity distribution within diffraction pattern - such is
pupil
obstruction or apodization - not originating from wavefront
aberrations. Hence, the Strehl figure doesn't include such effects. The
effects of change in the pupil transmission factor due
to obstructions of various forms still can be expressed through the
PSF,
as a single number comparable to the Strehl ratio.
Strehl
and encircled/ensquared energy Potentially more versatile indicator of the effect
of aberrations is the amount of point image energy contained in a circle
of given radius, or a square of given side (**encircled**
and **ensquared energy**, respectively). It shows what portion of the energy is contained within
a circle of given radius, centered at the intensity peak of the
diffraction pattern. If specified for more
than a single radius, it gives more detailed picture of intensity
distribution.
Illustration
at left shows Point Spread Function (**PSF**)
- with its peak intensity determining the value of Strehl ratio - and encircled energy (**EE**) of a
perfect (aberration-free) and aberrated aperture (0.25 wave P-V of
primary spherical aberration), as a function of diffraction pattern radius,
given in units of **λF**. In the presence of
aberrations, the energy is spread wider, thus the energy
encircled within a given pattern radius diminishes. Encircled energy
figure can be given not only for the Airy disc, but also for any radius
of the diffraction pattern. It can indicate possible change in size of
the central disc, or furnish some other information of particular
interest. An additional **EE** value for, say, 2.5λF radius, would
indicate how much of the energy lost from the disc ended up in the first
bright ring. It gets more complicated with asymmetrical
aberrations, since the amount of energy at any radius can vary
significantly with the pupil angle. Showing this aspect of energy
distribution would require several **EE** figures for each of various radial
angles, or some kind of a graphical (contour) **EE** presentation -
far from the clear simplicity of the Strehl.
For pixel-based detectors like CCD, more relevant is the information
on ensquared energy, although the difference between the two is
generally small. The amount of energy contained within a square of given
side vary with the form of aberration, and can be significant even if
the Strehl number remains similar (**FIG.
98**).
**FIGURE 98**: Change in ensquared energy with
increase in primary spherical aberration at best focus from zero to λ/4,
λ/2 and
λ wave P-V, in comparison to four other aberration forms of comparable
RMS wavefront errors (the RMS-to-PV ratio is **√**11.25,
**√**28,
**√**12,
**√**32
and **√**24
for primary spherical,
balanced secondary spherical,
defocus, primary
coma and astigmatism, respectively). Included is also the
effect of
central obstruction for 0, 0.3, 0.4 and 0.5 obstruction ratios.
Since the RMS wavefront errors are identical, the corresponding Strehl
values - 0.80, 0.39 and 0.14 - are either practically identical, for the
RMS errors of about 0.15 and smaller, but are somewhat different for the 0.3 wave RMS
level (for large errors, focus location with the lowest RMS does not
coincide with the maximum Strehl focus location, and the
difference vary somewhat with the form of aberration). Yet the
corresponding ensquared energies may be significantly different, and
particularly for larger error levels. It is the consequence of a
different pattern of energy transfer out of the Airy disc: while for
given Strehl all aberrations have similar amounts of energy lost from
the central maxima, those with more of the transferred energy ending up
closer to the central maxima - like astigmatism and defocus - have more ensquared energy at all significant aberration levels, and more so the
larger the aberration magnitude.
In general, balanced secondary
spherical spreads the energy most extensively, while astigmatism and
defocus do the least amount of damage (encircled energy for coma is
somewhat better if its centroid, not Gaussian focus, coincides with the
square center). As a measure of the energy spread here is used the
energy content in a square with the side equaling Airy disc diameter
(87% of the total energy, marked by the **horizontal dashed line**).
Aberrations at the level of λ/4 of primary spherical are already not
negligible in this respect, with the 87% square side increasing from
about 40% (astigmatism) to 150% (balanced secondary spherical),
increasing for larger aberration levels roughly in proportion to this
initial level. The effect of central obstruction in its usual range of
sizes in imaging systems is comparatively small, roughly at the level of
λ/4 wave P-V of primary spherical up to about 0.4D obstruction size, and
somewhat worse for larger obstructions.
The blue FWHM (Full-Width-at-Half-Maximum) squares mark the diameter of
the corresponding PSF FWHM, and also show how much energy is contained
within it. The importance of the FWHM is in it being considered the
determinant of point-source resolution, hence it complements the
information on EE with this important aspect. There is little change in
the size of FWHM in this aberration magnitude range for primary and
secondary spherical and defocus, but its energy content drops
significantly with the increase in aberration. At the 0.3 wave RMS level
of defocus, there is no FWHM in terms of central maxima since the
intensity drops to zero in the center, gradually increasing away from it
(FIG. 47, 1λ P-V defocus). Half
maximum of the decreasing intensity, outlining the ring-like central dot
is quite wide, about 5.4λF in diameter. Similarly, balanced secondary
spherical at the minimum RMS focus has central maxima of lower intensity
than the surrounding rings (this inverse intensity pattern in the PSF
center is indicated by the EE plot laying near flat close to the
origin). However, the same aberration at the maximum Strehl focus
(0.24mm away axially at ƒ/8.15)
does form the central maxima, resulting in more ensquared energy closer
to pattern center, but less going farther out (that would result in
better contrast transfer in the high MTF frequencies, and worse in the
lower ones). Unlike it, primary spherical forms central maxima at both,
minimum RMS and maximum Strehl foci (separated by about 0.25mm axially),
only the former has somewhat more energy in the first few rings.
With coma, central maxima becomes elongated already at the 0.075 wave
RMS level, with the cross section of the maxima with the pattern in
sagittal (vertical) orientation being slightly narrower than that in
aberration-free aperture, but more than twice wider in the tangential
(horizontal) pattern orientation. With further increase in magnitude,
central maxima becomes larger and more elongated, also shifting away
from the point of Gaussian focus, as shown on the
coma peak intensity shift graph.
Astigmatism doesn't change the FWHM appreciably up to about 0.15 wave
RMS level, but quickly widens it after that, also developing central
depression at 0.3 wave RMS at its minimum RMS focus, although not as
deep as with 0.3 wave RMS of defocus. Its maximum Strehl focus for this
aberration level is nearly coinciding with either tangential or sagittal
focus, with a large discrepancy in the FWHM diameter along the two
perpendicular axes.
Of course, both EE and FWHM are significantly affected (enlarged) due to
the aberrations induced to the optical train, such as the seeing error,
thermal effects on the optics and air within the instrument,
miscollimation, and others. It is difficult to measure all those sources
of error separately, and it is usually the easiest approach to model
performance level with MTF and empirically
measured FWHM.
Note that the corresponding square side in microns is given by a product
of the wavelength **λ** in microns and the focal ratio **F**.
Still, encircled/ensquared energy remains
quantitative indicator of image quality. For more
specific information on the effect of wavefront aberrations on image
quality, as well as the effect of other factors affecting wave interference in the focal zone,
the calculation has to expand from the characteristics of a single point-image (PSF), to
those of the images of standardized extended objects, covering the
entire range of resolution. The needed tool is found in the **
optical transfer function** (OTF), a
Fourier transform
of the PSF.
Strehl
and MTF, Hopkins ratio
Being based on the system's PSF, Strehl ratio is directly related to its
MTF, with the PSF being the inverse Fourier transform of the MTF. In
effect, the Strehl represents the MTF averaged over all frequencies - in other words, it
represents the averaged MTF contrast transfer. Thus the quantity 1-S
represents the averaged MTF contrast loss due to the aberrations.
General consensus for general observing is that contrast loss of up to
5% is inconsequential, and that loss of up to 20% does not significantly
degrade performance.
The problem with such generalization is that: (1) contrast loss for most
aberrations is not uniform over the range of MTF frequencies, and (2)
the effect of contrast loss depend primarily on the inherent object
contrast, and it varies widely from one object type to another. Hence
20% loss may not significantly degrade performance with some objects and
details - possibly majority of them - but it will with some others,
generally those with the lowest inherent contrast. That puts the
acceptable contrast loss - depending of the object of observation -
anywhere between 20%, or somewhat more, to 5%, or somewhat less.
As for the contrast loss
variation over MTF frequencies for a given Strehl (i.e. aberration
level), it is evident on the typical MTF. Even at relatively low
aberration level, resulting in 0.80 Strehl, it can cause potentially
noticeable differences in performance with specific object types. For
clarity, it is presented as contrast transfer vs. that in a perfect
aperture normalized to 1 for every frequency, i.e. as the
MTF relative contrast (**FIG. 99**; plots
generated by *Aperture*, R. Suiter).
**FIGURE 99**: MTF contrast
variation for 0.80 Strehl. Contrast is normalized to 1 for contrast
transfer in a perfect aperture at every frequency (i.e. the contrast
transfer of a perfect aperture coincides with the top horizontal
scale). All four wavefront deformations result in 0.80 Strehl, but
the differences in their contrast transfer over local frequencies -
with the Strehl representing the average contrast over all
frequencies, the local contrast transfer is effectively a local
Strehl - can be very significant. At the resolution limit for
planetary details, for example, where less than 5% of contrast
differential can produce detectable difference in performance, the
"local Strehl" for the four 0.80 Strehl deformations ranges from
0.71 for defocus, to 0.82 for spherical aberration and turned edge.
Even with the all four
aberrations being at the "diffraction limited" level, the differences in
the contrast transfer are not negligible, and can be substantial. The
worst effect has turned edge, which underperforms at both ends of the
frequency range. At the low-frequency end, for details of about 10 Airy
disc diameters, and larger (since the cutoff frequency is 2.5 times
smaller than Airy disc diameter, frequency equaling the Airy disc
diameter is 0.4, and 0.04 is ten times larger), it quickly loses nearly
10% of the contrast. While it is still a relatively small loss,
generally speaking, it indicates wide spread of energy that can brighten
background, and soften - even wash off entirely - faint objects in
proximity of bright objects. On the high-frequency end, contrast with
turned edge begins its dive to zero as the detail size goes under half
the Airy disc diameter, hitting zero at some 96% of the resolution
limit. Needles to say, it will noticeably affect not only performance in
splitting unequal doubles, of resolving critical lunar details, but also
the resolution of near-equal doubles.
Glance at this relative contrast transfer variation over the range of
MTF frequencies indicates that the contrast drop tends to be smaller
toward either low or high frequency end, and larger over mid
frequencies. When that is the case, the aberration tolerance for such
sub-range widens. Hopkins found specific aberration tolerances producing
0.8 Hopkins ratio - the contrast drop of 20%, analogous to the Strehl
ratio - or better, for MTF frequencies equal to, or lower than 0.1. Shown at left
are peak aberration coefficients as a function of spatial frequency **
ν** in this
frequency sub-range. The
tolerances are significantly larger than in the conventional treatment
of aberrations, placing the lower limit at 0.80 Strehl; consequently, their
corresponding conventional Strehl values are significantly below 0.80,
for which the coefficient value is S=1, C=0.63, A=0.37 and P=0.26 for primary
spherical aberration, coma, astigmatism and defocus, respectively (coma
with θ=0 is for the blur orientation same as that of MTF bars).
Note that at this large aberration levels the coefficient equals the
actual P-V wavefront error only for defocus. For primary spherical
aberration, the aberration minimum for Hopkins ratio is at a point
defocused by PS=-(1.33-2.2ν+2.8ν2)S,
generally more defocused than for the point of minimum wavefront
deviation (PS=-S),
where the P-V wavefront error equals S/4.
This is the consequence of the
shift of the PSF peak
away from the minimum deviation focus as the P-V wavefront error exceeds
0.6 wave, i.e. for the peak aberration coefficient values of 2.5 and
larger. With the Strehl at these aberration levels being up to several
times higher for the PSF peak than for the Gaussian (paraxial) focus, the actual error is also significantly smaller,
corresponding to roughly 2-3 times smaller P-V wavefront error.
Similarly, for large errors of
astigmatism (about 1 wave P-V, which is nominally equal the
coefficient, and
larger), the PSF peak also shifts away from the point of minimum
wavefront deviation (mid focus, i.e. defocused from either tangential or
sagittal focus by DA=A/2)
toward sagittal and tangential focus (double peak), with the PSF at
these peaks for larger aberration being
up to several times higher than at the mid focus.
For coma, the P-V wavefront error at the tilt-corrected focus, the one
with the minimum RMS wavefront deviation, is 2/3 of
the aberration coefficient, with the actual effect on MTF contrast
ranging from the maximum for θ=0 (the blur length perpendicular to MTF
bars), to the minimum for θ=π/2 (blur length
parallel to the bars).
The shift of the PSF peak away from the
focus of minimum wavefront deviation begins with the P-V error nearing 1λ
(i.e. peak aberration coefficient 1.5).
Graphs below show MTF for the aberration level resulting in 0.80 Hopkins
ratio at 0.1 frequency (ν=0.1,
line pair width little over four Airy disc diameters), with the center
of the small square marking the 0.8 contrast drop point for this
frequency. Expectedly, all plots are at or near this point at this
frequency. Small deviations might be result of the specific MTF
algorithms applied by OSLO. For coma and astigmatism MTF shows sagittal
(**x**) and tangential (**+**), i.e. vertical and horizontal,
respectively - blur orientation.
The most convenient general indicator of the magnitude of tolerance
change with the frequency in the conventional P-V wavefront error
context is defocus, which does not have
neither axial nor tilt correction aspect. It shows the tolerance
increasing inversely to the spatial frequency, from 1/2 wave
P-V at ν=0.1, to 1 wave at ν=0.05, 2 waves at ν=0.025, and so on.
Hopkins ratio confirms the practical experience finding that observation
of dim, low contrast details - whose resolving range (inset
A, right) does not extend to frequencies significantly higher than
0.1 - has lower requirement with respect to optical quality. But it also
shows that the aberration tolerance for this type of objects varies
significantly with the type of aberration. Of the four aberrations here,
the tolerance is the stringest for primary spherical aberration (0.34λ
P-V, or 0.64 Strehl), somewhat more forgiving for defocus and
significantly more forgiving for coma and astigmatism. With coma blur
aligned with MTF bars (the case to which applies the relation for
Hopkins ratio given above), nearly 1 wave P-V of coma is needed to cause
20% contrast loss. For the perpendicular blur orientation, not more than
0.6 wave P-V, and for the average contrast drop midway between the two
orientations about 0.7 wave P-V.
The contrast is even more forgiving to astigmatism, with 1 wave P-V
needed to cause a 20% drop with the astigmatic cross aligned with MTF
bars, and 0.9 wave P-V with it rotated 45° (not shown). The top Strehl
given for astigmatism is for the minimum RMS focus, and the bottom
is the peak Strehl focus, defocused 0.14mm (at
ƒ/8.2) from the former; astigmatism is the only aberration here
for which the difference in the Strehl at the point of minimum RMS
focus vs. peak Strehl focus is significant.
Similar results, only in the direction of tightening the tolerance, can
be expected for the frequencies toward the high end (certain exception
being turned edge, which causes unacceptable contrast drop in this
sub-range at the 0.80 conventional Strehl already). On the other hand,
the conventional Strehl that would secure no more than 20% contrast drop
in the mid range, where are the resolution threshold for bright
low-contrast details, probably wouldn't be significantly below 0.90. For
ensuring no more than 5% contrast drop at mid frequencies, the
conventional Strehl would need to be above 0.97 (equivalent of 1/11 wave
P-V of spherical aberration, or better). That, however, would strictly
apply only to very small apertures with near-perfect correction and
negligible induced errors (seeing, thermals, miscollimation...). At the
relatively large error levels, the finest details are washed out, and
those more coarse that remain are generally less affected by any given
contrast drop.
◄
6.4.1.
Star testing telescopes
▐
6.6. Effects of aberrations: MTF
►
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