|
telescopeѲptics.net ▪ ▪ ▪ ▪ ▪▪▪▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ CONTENTS
7.2. Spider diffraction and apodizationBeside central obstruction by a smaller mirror in most reflecting telescopes, other common forms of pupil obstruction are spider vanes, supporting smaller mirror mounted inside the telescope tube and, occasionally, apodizing masks, used to modify diffraction pattern to some extent. Their effect is generally small, but it can be significant. That makes them worth of a closer look. 7.2.1. Diffraction effect of spider vanes Another frequent form of pupil obstruction is the secondary mirror support structure. Unless the secondary mirror cell is supported by an optical window, the supporting vanes - so called spider vanes - are in the optical path, altering emitting area of the wavefront and, thus, creating diffraction effect. As long as the pupil area obstructed by the vanes remains relatively small, spider diffraction is more of a cosmetic damage than seriously affecting contrast level (FIG. 68). Analogously to the central obstruction effect, what can be thought of as a
Strehl ratio degradation factor caused by spider diffraction is, in effect, the ratio of the clear (annular) pupil area with and without the vanes squared, or
with N being the vane count, τ the relative vane thickness and ο the relative size of central obstruction, both in units of the aperture diameter. The negative factor equals the relative spider area in units of the clear aperture area; this is consistent with degradation factor caused by central obstruction (Eq. 60). Average spider area is somewhere between 1% and 2% of the clear aperture area. That puts an average spider vane contrast degradation factor between 0.98 and 0.96 - below the level of 1/30 wave RMS wavefront error. Analogously to the effect of central obstruction, vane obstruction reduces central intensity of the main pattern by a (1-a)2 factor, a being the relative vane area in units of the clear aperture area (for spider vanes, it is the area of annulus), by lowering constructive interference within central maxima and intensifying it in the outer potion of the pattern. For small values of a, typical for spider vanes, the Strehl degradation factor can be written as S'~(1-2a). It also closely approximates the combined Strehl degradation factor of the spider and c. obstruction in the left side of MTF graph (extended low-contrast detail resolution) if a is their combined relative area in the aperture (this approximation is also good for the PSF maxima degradation factor for central obstructions smaller than ~0.35D). This is not quite in agreement with the popular notion that the contrast effect of spider vanes is directly proportional to their area, relative to the area of aperture. The misconception probably comes from misunderstood sequence in Suiter's "Star Testing Astronomical Telescopes", where he states that the initial quick contrast drop is in proportion to the vanes area. However, looking at the MTF graph, it is easy to see that this initial drop in contrast remains nearly unchanged linearly for nearly 2/3 of the MTF range. In other words, the actual contrast loss keeps increasing as the relative contrast value decreases for smaller spatial frequencies (detail size). The average contrast loss caused by vanes is, therefore, considerably higher, as given by Eq. 65. The actual spider effect can be much smaller, due to the energy being thrown so far from the Airy disc. For instance, a spider wane D/100 thick will have its principal spike length superimposed over diffraction pattern nearly 100 Airy disc diameters long (only a portion of it visible at best, depending on its telescopic brightness). For a 10" aperture, that is nearly 1 arc minute from the disc center. That would place most of the spike energy out of a relatively small object, not influencing its contrast. For Jupiter, roughly 2/3 of the principal spike fall outside the planet's disc, with 1/3, or so, of the spikes' energy left in, lowering the contrast. Assuming 4-vane spider and 25% obstruction, it would cause little over 1% actual average contrast loss (nearly 0.99 Strehl equivalent), not 4% as indicated by Eq. 65. On the other hand, on large objects like the Moon, nearly entire spikes' energy remains within the image, and the effective contrast degradation factor is ~0.96. There are various vane configurations possible, but the only result is a different form of energy distribution - the amount of energy transferred out of the Airy disc remains unchanged for any given vanes area. Given size of central obstruction, the vane area is directly proportional to its width - the wider vanes, the more energy spread out, the higher its peak intensity, but the shorter spike length. Specifically, intensity distribution of diffraction pattern created by a slit aperture (or, inversely, by a thin vane) is described by
with I(0) being the intensity as a function of point radius, for central intensity normalized to 1 (actual intensity is proportional to the slit/vane area), β=Sπsinθ in units of the wavelength λ, S being the edge-to-edge separation (i.e. either slit/vane's width, or length) and θ being the point angle in the image plane. The minimas occur for β=aπ, with a=1,2,3,4... First maxima is for β=θ=0, and every subsequent maxima at β=tanβ (with β on the left in radians), or for β=bπ, with b=1.43, 2.46. 3.47... This gives the second maxima intensity (for β=1.43π) as 0.047 of the central intensity, the third maxima as 0.016, the fourth 0.008, and so forth. Considering relatively low intensity of the first maxima vs. that of the main aperture - proportional to their respective areas - it is only the central peak of the spider vane diffraction pattern that may be relevant for both cosmetic visual effect and the effect on contrast transfer. With the first intensity minima falling at a constant nominal value of β, its angular radius, given by sinθ=βλ/πS (for small angles sinθ=θ in radians) is inversely proportional to the edge separation S. Thus, the longer the vane, the more narrow its spike; the wider vane, the shorter its spike. A 200x1mm vane - so with S equaling 200mm lengthwise (neglecting central obstruction), and 1mm width-wise - for λ=0.00055mm will produce first maxima nearly 4 arc minutes long (for β=π and S=1) and about 1.1 arc seconds wide (β=π, S=200); a vane twice as thick will produce maxima half as long, with its width unchanged. Thicker vanes may appear to be producing less intrusive, shorter spikes, but they drain more energy from the Airy disc, causing greater negative effect on contrast level.
Spider diffraction effect is often illustrated by the effect of a
narrow slit. While even the narrowest spider vane is still much wider
than a narrow slit, whose width is, by definition, (much) smaller than the
wavelength of light, the difference between the two with respect to their diffraction
effect is insignificant.
A spider vane is, in effect, a greatly elongated rectangular aperture. In incoherent light, which is normally
applicable to astronomical telescope, spike diffraction intensity is in
proportion to the aperture area, not the square of it, as it is with a
narrow slit in coherent light.
The form of pattern change is determined by the vane profile in
the pupil, which in turn determines intensity distribution of the vane
as an aperture. Straight vane projects a spike that is centered over
diffraction pattern, as illustrated on
FIG. 69. Since "dark aperture"
created by the vane becomes a part of the wavefront, it projects a spike
centered at the chief ray (i.e. center of the diffraction pattern), extending orthogonally to
the vane orientation, regardless of its orientation in the pupil, or
length (shorter section will produce wider, fainter spike).
Since at these small angles there is practically no difference between
the angle and its tangent, linear length of the spike maxima is
approximately 2λƒ/w,
with ƒ
being the mirror focal length; substituting the vane width w in
terms of the aperture diameter as w=ςD
gives the linear spike length as ~2λF/ς,
F being the telescope focal ratio.
As the two insets on the bottom of FIG. 69 show, it is possible
to reduce diffraction effect of a spider vane by replacing a single vane
with two or more parallel vanes. Multiple vane replaces a single central
maxima of a single vane with multiple subsiding maximas covering bearly
identical width angularly. Intensity of the central maxima is
proportional to the combined vane area, thus for the reduction in energy
transferred from the Airy disc, such multiple vane would need to have
unit vanes of lesser width than a single vane it would replace.
Curved vane spider
The intense spike produced by a straight vane can be
visually eliminated by
curving the vanes. The result is a
curved vane spider. Diffraction effect of a curved vane can be
illustrated by breaking it into a number of
smaller, practically straight sections, with varying orientations (FIG.
70a). While the total
amount of energy produced by a curved vane is identical to that of a
straight vane of equal length and thickness, it is spread out wide,
making it practically invisible (it still lowers the contrast the same,
on average).
The Strehl degradation factor is somewhat different from
that for the straight vanes (Eq. 65):
with α=180/N being the vane arch angle
in degrees.
However, the result is only slightly lower for given count (N) and
relative thickness (τ) of the vanes, reflecting slightly
greater curved vane length.
Apodizing,
or change of the aperture transmission
properties, also affects intensity distribution within
diffraction pattern. Properly made apodizing mask minimizes intensity of
the rings, at the price of somewhat enlarged central disc (FIG. 71).
FIGURE 71:
Apodizing mask's effect on unobstructed (left) and 30% obstructed
aperture (right), as
presented
by H.R. Suiter. The top two patterns are for unapodized
apertures. The effect of rings suppression seems much more
obvious with unobstructed aperture, but it is evident even
at 0.3D obstruction. However, visual effect as seen on these
simulated diffraction patterns is fairly biased, since the
loss in brightness in the ring area is more noticeable than
that of the spurious disc (in other words, with the
identical relative amount of intensity reduction, the rings
will disappear sooner than the disc (such is the case with
faint stars), without actual change in the energy
distribution within the pattern. Thus, if there is an effect
of energy distribution, it is likely smaller than what these
patterns suggest - and comes at a price. Similar effect may
be possible to produce with high-density neutral filter,
without the side-effect of the enlarged Airy disc. What may be more important benefit
from the use of an apodizing mask, is reducing the effect of possible,
fairly frequent surface defects - outer zone, turned edge,
chromatism, even figure error - that may be obscured by the mask to a
significant degree.
While the apodizing won't improve general
instrument's performance level, there is a solid anecdotal evidence that
it can improve image clarity in field conditions with significant
atmospheric turbulence. Optical theory allows for the possibility, since
the suppression of rings results in actually better than perfect
aperture contrast level for low- to lower mid-frequencies,
which is the range of resolvable extended details in compromised seeing conditions.
|