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▪ ** **CONTENTS
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6.6. Effects of aberrations: MTF
▐
6.7. Coherent transfer function, Fourier
transform
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#
**
6.6.1. ****
Aberration compounding, CCD contrast transfer, MTF limitations**
Expressing **combined effect**
of two or more aberrations through the MTF can be done, similarly as for
the wavefront aberrations, either precisely, through a complex
calculation, or as a single number extracted from statistically averaged probability (approximation). As the above graphs show, difference in
the effect on performance of various aberrations with a similar RMS wavefront error level becomes more
significant as the nominal error increases. For
two or more aberrations combined, the effect
is, on average, greater than either single aberration components alone.
However, they do not add up arithmetically. For relatively small, unrelated
or
random wavefront errors, the combined effect is given by the *square root of the
individual errors squared*, or
**
ωc**=
(**Σωi**2)**1/2**
(58)
with
**ωi**
being the individual RMS wavefront error. Similarly, the combined effect
on peak intensity of the PSF (Strehl) and, thus, average MTF contrast
loss can be, for mostly unrelated smaller aberrations, closely
approximated as the product of individual Strehl ratios for any number
**i** of different aberrations, Strehl being, in fact, an MTF (contrast) degradation
factor:
S**c**=
S**1**S**2
**... S**i**
(58.1)
The product **
Sc**
is a combined Strehl (the peak diffraction intensity ratio) for the errors
included; it also expresses the average MTF contrast level in units of the
average contrast level of a perfect aperture.
The conventional minimum acceptable level of
optical quality - so called *diffraction-limited* standard - is 1/4 wave P-V wavefront error of spherical aberration
or, in more universal form, the associated 1/13.4 wave RMS (0.80 Strehl). That
corresponds to 0.42 wave P-V of coma, 0.37 wave P-V of astigmatism, and
so on. This criterion is not strict - it is a somewhat lose dividing line between good and bad
optics. In
fact, since it is, in the commercial environment, habitually applied to the quality level inherent to
optical surfaces alone, it is safe to assume that the actual
quality minimum is lower, perhaps significantly. In addition to
induced aberrations considered earlier, as well as chromatism, one of the factors lowering
optical quality in addition to that determined by the wavefront
aberration level alone is the effect of central obstruction.
Beside wavefront aberrations and pupil obstructions, contrast
degradation can be caused by detector properties. Specifically, contrast
transfer of a **CCD chip** depends on
its pixel size, relative to the system Airy disc diameter. Contrast
transfer in the final image is a product of contrast transfers of a
telescope and that of the CCD chip. According to Schroeder (*Astronomical Optics*,
p309), for square pixel with the side *p* in units of
λF, contrast transfer (MTF) is given by:
with ** c**t
being the telescope MTF, the sinc term the MTF of CCD chip (the angle *
p*νπ is in radians,
converting to degrees in the numerator) and **
ν** the normalized spatial frequency
(for ν=0 **c**f
is undefined, but it approaches 1 as **c**f
approaches 0, thus represents
its limit value).
Sinc values for selected pixel sizes are
plotted to the left. For *p*=2, which for an ƒ/5 system and λ=0.5μ
corresponds to 5μ pixel size, contrast transfer drops to zero at ν=0.5
(from ct=sinπ/π=sin180°/π),
effectively halving the cutoff frequency of a telescope before
contrast
reversal (negative modulation values). In addition,
contrast level over all frequencies before reversal is degraded to the
level of a twice smaller perfect aperture, with the contrast after
reversal being a small fraction of the (original) contrast in a perfect
aperture (**FIG. 102**). Half as large pixel
preserves more of contrast transfer efficiency, but in order for it
to approach that of a telescope - assuming its optical quality is not
significantly below the usual standards - pixel size needs to be reduced to 0.5λF,
or 1/2 of the system's FWHM. According to the Nyquist criterion, this is
also the minimum needed for clearly resolved point sources at the
diffraction limit ~λ/F.
Since the first zero for the sinc factor falls
at *p*ν=1, first zero of the
combined MTF is at the spatial frequency
ν=1/*p*, regardless of the MTF of
a telescope, as plots for p=3 and p=4 show. Hence, preserving the limiting MTF resolution of a
telescope requires *p*=1, or smaller. Some positive contrast
recovery at high frequencies does occur for p>2, but it is generally
insignificant vs. the overall contrast/resolution loss.
**FIGURE 102**: Additional contrast degradation in
42% obstructed system resulting from relative pixel size **p**, in units of system's
aberration-free FWHM (~λF).
The combined system+detector MTF is a product of the system's and
pixel's MTF (i.e. product of their respective
contrast transfers at each frequency). For *p*=2, contrast
drops to zero at spatial frequency
ν=0.5; it still retains
low negative, or reverse contrast (the bright lines become dark, and vice versa).
For an f/8 imaging system and no significant aberrations present, this
level of contrast loss would correspond to 8.8μ pixel size, for
λ=0.55m.
Smaller pixels deliver better contrast transfer, and are not prone to
the occurrences of the "negative" effect in the high-frequency range.
To nearly preserve contrast transfer of the system in this case, pixel
size shouldn't exceed 2.2μ. This requirement is significantly more
relaxed in
the average field conditions, mainly due to the actual FWHM at
the detector being larger up to several times (depending on the aperture
size and length of exposure) than the aberration-free FWHM's, as a result of blurring caused by
atmospheric turbulence.
At this pixel size, the sinc factor for
ν=0.5 (approximate MTF cutoff frequency
for extended bright low-contrast details for aberration-free aperture),
is sin45/(π/4)=0.9,
degrading telescope's contrast transfer at this frequency by only 10%.
Relative contrast degradation at this pixel size does increase for higher frequencies,
which will also slightly reduce the point-source cutoff frequency.
Obviously, aberration level substantially degrading actual contrast
transfer of a telescope would correspondingly ease the limit to the
maximum pixel size needed to nearly maintain this contrast transfer
level. By significantly enlarging telescope's FWHM at the detector, the
ever present seeing error puts the actual pixel size limit significantly
above the formal Nyquist criterion level, for all but very small apertures. For
short- and long-exposure imaging, in order to determine the approximate limit to the pixel size
which nearly preserves contrast transfer efficiency of a telescope,
the λF unit in the Nyquist
criterion needs to be replaced with the respective
seeing FWHM.
Of course, other factors influencing CCD contrast transfer efficiency
also need to be considered, particularly with respect to
**signal-to-noise** (SNR) ratio, which
can significantly affect contrast transfer and overall detection
capability of imaging system. In general, the SNR ratio can be expressed
as
SNR=SQt/**√**SQt+N,
with **S** being the signal flux from the object, **Q** the
detector's quantum efficiency (defined as the ratio of detected vs.
incident photons), **t** the exposure time and **N** the sum of
all noise contributions, N=(BQ+D)t+R2,
where **B** is the background (sky) emission, **D** the detector's
dark count (thermally produced electrons within CCD chip itself) and **
R** the readout noise.
Needless to say, presence of wavefront aberrations, either inherent to
imaging system or induced, further reduces CCD contrast transfer
efficiency and limits to resolution. If chromatic errors are present,
they will affect CCD imaging efficiency significantly more than visual
image, due to the significantly higher
sensitivity of CCD chip to wavelengths toward the ends of the visual
spectrum.
A few words about
**limitations of the standardized MTF**
in projecting its output to the actual observing. One comes from the
standardized MTF being strictly exact only for the object used:
parallel, contrasty, brightly illuminated dark and bright lines with
sinusoidal intensity distribution. For
different types of objects, both contrast transfer and cutoff frequency
(limiting resolution) may be different, possibly significantly. Also, the MTF assumes continuous
spread of energy; in other words, every line interacts with all the
energy reaching it from both, left and right. Actual observing objects
are finite, and the absence of continuous spread results in the outer
areas being subjected to less of energy overlapping than the MTF
assumes. It is particularly pronounced with small, or narrow objects (**FIG.
103**).
**
**
FIGURE 103:
LEFT: Typical continuous
MTF pattern exposes every single point to all the energy from all
surrounding points that
extends to it (gray circles). All bright lines between **A** and **B** will
contribute to the energy spread out by diffraction and aberrations to the mid
point marked by the cross.
MIDDLE: With relatively small ( with regard to the diffracted/aberrated energy spread
radius) object image, significant portion of the energy lost to the Airy disc
will spread out of object area (**C**),
with comparatively less energy affecting the image itself. It would
result in better contrast transfer than what MTF indicates
for given aberration level.
RIGHT: Example of the
effect of pattern extent on contrast transfer. Image of a 3-bar square
wave pattern has higher contrast transfer at all frequencies than
continuous pattern, as well as somewhat better limiting resolution
(cutoff frequency); the effect is similar with 3-bar sinusoidal pattern
(standard MTF).
The consequence is better contrast transfer and
resolution than what the MTF (thus, also PSF and the Strehl) would
indicate, even for an object identical to the standardized MTF pattern.
Specifics of this effect vary with the object type, as well as the type and
level of aberrations present.
Finally, a limitation of the MTF - but not
OTF - is, as mentioned on the previous page, that it does not include
information on the phase shift and its possible effect on the image. Phase shift itself merely shifts
the MTF diffraction image laterally
with respect to the Gaussian (geometric) image, and shift of half the
phase (**π**
radians) will result in *contrast reversal*, i.e. switch of light
areas into dark, and vice versa. While neither is affecting the
definition of MTF pattern, which is evaluated at each isolated frequency
alone, an actual image will often contain multiple frequencies, either
isolated or in continuums. In such case, intensity distribution shifts
caused by phase shift can result in deformation of image vs. object,
which could also impair detail detection or resolution. Some examples of
this effect are given below. One of the objects used is the continuum of
frequencies contained in the MTF (aberration-free) resolving range, and
the other a radial test pattern (simulations generated by Aberrator;
aberrated images are made more contrasty than the original output, to
make the effects on patterns more obvious).
Aberration-free and aberrated PSFs are shown top left
(the plots are not proportional to the simulations scale-wise, but
either are among themselves, double PSF/MTF plots for asymmetrical
aberrations are for the best and worst PSF orientation). Magnitude of aberrations are chosen to
roughly produce the contrast transfer level corresponding to 1λ
wave P-V of defocus (with this large aberrations even the RMS error is
not a reliable indicator of error magnitude, and so is not the PSF
central intensity). Asymmetric aberrations, coma and astigmatism are
given at a rotation angle midway between their best and worst
orientation (indicated as two colored plots on their PSF and MTF graph)
- 45 and 22.5 degrees respectively (counterclockwise) - for which the
effect should be at least roughly averaged up.
Defocus causes contrast reversal, changing the
appearance of both images. Coma causes obvious asymmetry. Astigmatism
causes an interesting effect that appears as contrast reversal in the
star pattern, but only as intensity shift in the frequency continuum
pattern (the difference in appearance for the lower and upper left
portion of the image is caused by the diffraction pattern, rotated 22.5
counterclockwise, being nearly aligned with its "best" axis with the
former, and with its "worst" axis with the latter). Primary spherical
has lower contrast transfer in low and lower-mid frequencies,
compensated by higher transfer in the high-mid frequencies, but the only
change in image appearance is due to the insufficient contrast at some
frequencies. Balanced secondary spherical, even if generally similar to
the primary spherical, and at a roughly similar magnitude level, causes
noticeably different effects. And the effect of seeing is in some ways
similar to coma having, due to the extent of PSF energy spread, lower an
early contrast drop and, due to its asymmetry, detail deformations.
However, unlike coma it causes quite pronounced contrast reversals.
None of these effects that can significantly lower
image quality and detection level is indicated by the MTF. exception
being, of course, low contrast level in the case of spherical aberration
(spherical aberration also causes contrast reversal at the roughly
doubled error magnitude). While this may not be significant in general
terms, since this error magnitude is significantly larger than the
average found near axis in amateur telescopes, farther off axis this
level can be easily met and exceeded, thus effects of this type are the
reality. Also, larger amateur telescopes routinely operate at these and
larger error levels due to seeing, hence these kind of effects are no
stranger there, even on axis.
◄
6.6. Effects of aberrations: MTF
▐
6.7. Coherent transfer function, Fourier
transform
►
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