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6.5. Effects of aberrations: MTF 2
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7. OBSTRUCTION EFFECTS
► 6.5.1. MTF 3: Aberration compounding, CCD contrast transfer, MTF limitationsExpressing 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=
(Sωi2)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:
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 CCD chip. As given by Schroeder (Astronomical Optics,
p309), for square pixel with the side p in units of
λF, contrast transfer (MTF) is given by:
with ct
being the telescope MTF, the sinc term the MTF of CCD chip (the angle is in radians) and
ν the normalized spatial frequency. For p=2, which for an ƒ/5
imaging system and λ=0.5μ
determines the pixel size as 5μ, contrast transfer drops to zero at ν=0.5
(from ct=sinπ/π=sin180°/π),
effectively halving the cutoff frequency of a telescope. In addition,
contrast level within resolvable range of frequencies is degraded to the
level of twice smaller aperture (FIG. 63). Half as large pixel
significantly improves contrast transfer efficiency, but in order for it
to approach that of a telescope - assuming its optical quality is not
significantly bellow the usual standards - pixel size needs to be reduced to 0.5λF,
or one half of the system's aberration-free FWHM
(Nyquist criterion).
FIGURE 63: Additional contrast degradation in
42% obstructed system resulting from a relative pixel size p, in units of system's
aberration-free FWHM (~λF).
The combined system+detector MTF is a product of 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), with the final loss of contrast at
ν~0.9.
For an f/8 imaging system and no significant aberrations present, this
loss of contrast 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
average field conditions, mainly due to the actual FWHM radius 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 a
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
for nearly preserving 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 the level of its internal
(electron) noise, generally lower with larger pixels.
Finally, a word 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. For
different types of objects, both contrast transfer and cutoff frequency
(limiting resolution) will deviate. 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.
64).
FIGURE 64:
LEFT: Typical continuous
MTF pattern exposes every single point to all the energy from all
surrounding points that
extends to it (gray circles). All lines between A and B will
contribute to the energy spread out by diffraction and aberrations to the mid
point marked by the cross. RIGHT: 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.
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 type and
level of aberrations present.
◄
6.5. Effects of aberrations: MTF 2
▐
7. OBSTRUCTION EFFECTS
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