## 7.3. APODIZING MASK
PAGE HIGHLIGHTS Apodizing, or change of aperture's transmission properties, also affects intensity distribution within the diffraction pattern it generates. The effect of apodization was mainly subject of speculations among amateur astronomers, until Harold R. Suiter published his book and articles, during the 1990s and at the beginning of this century, using diffraction calculation to find specific answers. This text uses his analysis and his diffraction program "Aperture", with somewhat expanded coverage of the effect of Gaussian apodization and potentially useful types of apodizing masks.
There is a number of possible
patterns of altering pupil light transmission. The starting point here
is so called
A) in the pupil
according to the Gaussian
function A=exp-(r/w)2, where
exp is the natural logarithm base (Euler number, ) 2.718 under
the negative exponent, in which er is the height in the pupil
normalized to 1 (with 1 corresponding to the aperture radius, D/2) and
w the Gaussian radius, at which transmission (t) drops to
1/e. Thus, Gaussian radius indicates the relative width of the bell-shaped
amplitude curve centered around the axis (it is also related to the
Gaussian's volume, which is equal to that of a cylinder of identical
height with the base radius equal to w).
Gaussian amplitude profile over
the pupil has full amplitude (A=1) in the center, decreasing toward the
edge. The smaller Gaussian width With incoherent light, total energy in the pupil - or transmission - is proportional to the amplitude squared. Squaring, in effect, produces a narrower Gaussian, whose volume vs. volume of the cylinder equals the transmission coefficient (top right). For w=∞, 1.5, 1, 0.7 and 0.4 total transmission is 1, 0.663, 0.432, 0.241 and 0.080, respectively.
The two main opposing effects of this type of pupil apodizing are:
The encircled energy plots show that the relative
energy within 1.22λF radius steadily increases from 84% for uniform
pupil to 97% for w=0.7 Gaussian, which has less than 1% of energy
outside its central maxima. Again, further reduction in the value of
MTF plots (bottom) illustrate the gain, and
the loss due to pupil apodization. Vanishing rings energy results in
better contrast over low-to-mid
MTF
frequencies. Expanding central maxima, however, lowers it over mid-to-high frequency range.
At w~1, the gain and loss are visually near equal; loss is already
significantly larger nominally but, on the other hand, the mid-to-low
frequency range covers much wider variety of details of about Airy disc
in size and larger. This implies that somewhat greater reduction in
So
far, it was assumed that the optical system with Gaussian pupil
transmission is aberration-free. In the presence of aberrations,
Gaussian transmission gradient over the pupil does not influence shape
of the wavefront, but does influence the magnitude of wave contributions
from different portions of the pupil. In general, contribution
suppression increases toward the edge. If those portions of the
wavefront are aberrated, it will reduce their destructive contribution.
Graph at left shows how the four principal aberrations - spherical,
coma, astigmatism and defocus - given in unit value for even pupil
transmission, decrease with the Gaussian width
The contrast transfer consideration in Independently of its effect on contrast transfer, the effect of Gaussian apodization on suppressing diffraction rings - and primarily the first bright ring - is beneficial in resolving, or detecting faint companions positioned in the first bright ring of the principal star. Another benefit from Gaussian apodization is that it generally increases the tolerance for inherent system aberrations. In other words, it does reduce the negative effect of any given level of conic surface aberrations - spherical, coma, astigmatism - as well as the effect of defocus. In the systems with relatively significant inherent aberrations - particularly astigmatism - it can produce noticeable improvement in the performance level based on this factor alone. Apodization is even more effective in suppressing the effects of local/zonal surface errors affecting the extreme outer area of the wavefront , such as turned edge. And, it is always important to consider the actual conditions of use. As Suiter points out, seeing error generally worsens contrast and resolution in the high-frequency MTF range. In larger amateurs apertures, details in this range are observable only in a fraction of the time, at best (FIG. 92B). In such case, there is little or no harm from the nominal contrast drop in this range, but the advantage of improved contrast in the mid-to-low frequency range remains.
Practical problem with Gaussian pupils is that fabricating substrate
producing such an accurate fall off in pupil amplitude/transmission requires complex
technology, generally unavailable to the amateurs. Suiter solves that
problem by designing step-apodizers made of a widely available material,
an ordinary window screen (
T is given for the mask alone, to
make them directly comparable in that respect; actual transmission is
(T-o2),
o being the relative diameter of c. obstruction in units of
aperture diameter. The mask for unobstructed aperture (TOP)
produces effect comparable to the Gaussian pupil with w~0.75. PSF shows
significantly reduced 1st bright ring intensity, which is now fainter
than the 2nd bright ring in unpodized clear aperture, with the 2nd
bright ring practically disappearing (apodized PSF - black - compares to
the dotted plot, which is the clear unapodized pupil's PSF normalized to
the peak intensity of apodized pupil). MTF shows relatively small
contrast improvement from ~0.3 frequency down, but also that the
enlargement of central maxima is lowering contrast transfer in mid
frequencies, enough to significantly reduce limit to resolution for
bright low-contrast details. With 0.2D central obstruction (MIDDLE),
central maxima is noticeably smaller, although still larger than in
clear aperture. First bright ring is nearly as bright as in clear
unapodized aperture, with 2nd bright ring collapsing into a minima
between the 1st and 3rd bright ring. There is only a small gain from
somewhat dimmer 1st bright ring, but at the price of enlarged central
maxima (PSF plot for clear unapodized pupil - orange - compares to the
unapodized obstructed pupil - blue - while the apodized obstructed pupil
- black - compares to the clear unapodized pupil PSF normalized to the
apodizet pupil's central intensity - dotted). Not surprisingly, MTF plot
shows apodized obstructed aperture having only slightly better contrast
transfer than that with apodizer in the low, but inferior to it in the
high-frequency range. Apodizing 0.3D obstructed aperture (BOTTOM) results in the
1st bright ring as bright as w/o apodization, only expanding to engulf
the 2nd bright ring, with the 3rd ring nearly as bright as in a clear
unapodized aperture. Consequently, contrast transfer over mid-to-low
frequencies is nearly identical to that with obstruction alone; with the
central maxima enlarged to somewhat over its size in clear unapodized
aperture, the only significant effect is a drop of contrast transfer in
the high-frequency range.In general, this step apodizer will produce similar effect to that of Gaussian pupil transmission it is comparable to with respect to the average drop off toward the pupil edge. There is an important difference between the two, and it is that step apodizer does not affect amplitude of individual waves - it selectively blocks them, or transmits. In coherent light, it would produce reduction in the complex amplitude (i.e. the sum of individual wave amplitudes) proportional to the pupil area blocked in the total area, with the energy (light transmission) proportional to this relative area squared. For instance, a mask blocking half of the pupil area would reduce the energy fourfold. In incoherent light - which is closer to the kind of light coming from astronomical objects - the energy is a sum of squared individual wave amplitudes. Hence energy transmission is proportional to the clear pupil area: blocking half of it will have it reduced twofold. More important, however, is that this different dependence of the energy on the aperture area would cause differences in point-source PSF, hence in the corresponding MTF as well. Similarly to case of the effect of central obstruction, the effect of apodizing mask should be significantly smaller in incoherent light, where the blocked out energy is deducted directly at the level of energy, not at the level of amplitude. The diffraction relation used by Suiter is from the domain of conventional treatment of diffraction effects, assuming near-monochromatic - in effect coherent - point source. And so is, according to its output, the underlying algorithm used by OSLO. If so, effects of pupil apodizing in the actual use by amateurs are significantly weaker than these shown, although they should be generally similar in their form. In other words, it would require significantly stronger apodizing in incoherent light to achieve effects shown here in incoherent light.
With this in mind, follows an overview of some basic principles in
designing apodizing mask, and the effect of varying parameters of its
simplest configurations.
Starting with the Gaussian transmission curves above, a simpler, nearly as
effective apodizers can be put together as well (
Unfortunately, but expected, the beneficial effect of these masks also diminishes with the increase in central obstruction. Some simple narrow-purpose mask profiles would include those with a single-layer screen zone featuring a concentric ring with further reduced transmission. In general, a darkened, or opaque ring in the outer pupil area tends to reduce brightness of the first 2-3 bright rings, but at the price of shifting significant portion of energy to the rings farther out. Multiple rings have similar effect. If properly designed, such mask can shift the energy far from the central maxima, with little or no ill effect on contrast in the actual observing, while benefiting from the darkened close rings. Unfortunately, that also comes at the price of enlarged central maxima. In all, apodizing mask can produce effects beneficial for observing extended details larger than roughly twice the stellar resolution limit. It is not dramatic: generally less than the benefit of correcting 1/4 wave P-V of spherical aberration, but it is easy to employ and deploy, and every little bit can count. Apodizer can also be helpful in resolving close unequal doubles, with the faint companion situated in the first bright ring of the principal star. These beneficial effects, however, come at a price of contrast loss with details close and at the limit of resolution. In general, benefits of apodizing are greater for larger apertures, which are due to the larger seeing error typically able to resolve only details in their mid-to-low frequency range. Finally, apodizing mask will have little effect in apertures with central obstruction significantly larger than 0.2D. |