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◄ 7.1. Central obstruction effect ▐ 8. REFLECTING TELESCOPES ►

7.2. Spider diffraction and apodization

Beside 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 a closer look.

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 cosmetic than seriously affecting contrast level (FIG. 51). Analogously to the central obstruction effect, what can be thought of as the Strehl

FIGURE 51: Visual appearance of a bright star without spider effect (a), with three-vane spider effect (b), and four-vane spider effect - the two most common spider forms - (c). The effect is noticeable mainly on objects of high telescopic brightness. While the spikes caused by spider vanes can be visually distracting, the amount of energy lost from the disc is usually negligible for general observing (3-vane spider spikes are usually shorter, due to the vanes being generally thicker, as it is needed for mechanical stability in that spider configuration, but they may be less intense, since their patterns don't overlap).

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

S'= [1-2Nς/(1+u)π]2 (65)

with N being the vane count, ς the relative vane thickness and u the relative size of central obstruction, both in units of the aperture diameter. Average spider area is somewhere between 1% and 2% of the clear aperture area. That puts an average spider vane contrast deterioration factor between 0.98 and 0.96 - bellow the level of 1/30 wave RMS wavefront error.

In a simpler form, spider degradation factor S' is closely approximated by by S'=1-2a, a being the ratio of vanes area vs. area of the annulus. For central obstruction smaller than ~0.35D, this same formula will also closely approximate the cumulative degradation factor of spider and c. obstruction combined, if a is their combined relative area in the aperture.

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 roughly 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. (it is also closely approximated by S'=1-2a, a being the ratio of vanes area vs. area of the annulus).

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 some 1/3 of the spike 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, the entire spike 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. Thicker vanes may appear to be causing less intrusive, shorter spikes, but their negative effect on contrast level is greater.

Spider diffraction effect is often illustrated by the effect of a narrow slit. However, even the narrowest spider vane is still much wider than narrow slit, whose width is, by definition, (much) smaller than the wavelength of light. A spider vane is, in effect, a greatly elongated rectangular aperture. The difference between such aperture and a narrow slit with respect to their diffraction pattern is fairly insignificant. With rectangular aperture in incoherent light, which is normally applicable to astronomical telescope, spike intensity is in proportion to the aperture area, not the square of it, as it is with a narrow slit in coherent light. However, the extent of the radii of central maxima (as the radius of the first minima) are approximately inversely proportional to the width of the opening for both.

By blocking wave emission from the wavefront area it obstructs, spider vane changes wave contribution to the diffraction pattern. It literally creates a dark stripe across the wavefront. Consequently, it changes intensity distribution within the pattern by effectively withdrawing its own intensity distribution as an aperture. At some points of the diffraction pattern of the main (circular) aperture, such is the central spot, this withdrawn energy will decrease existing intensity; and at others - unfortunately, outside the Airy disc - the intensity will increase; the principle of energy conservation at work. 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. 52. Since the 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).

FIGURE 52: LEFT: Spider vane diffraction effect is one of a narrow rectangular aperture. Since it is much more extended in one dimension, its diffraction pattern extends appropriately less in that direction, compared to the extension in the direction orthogonal to it. The length ratio of the two is approximately 1/ς, with ς being the width-to-length ratio of the vane. Consequently, it makes the central spider vane maxima ~1/ς times wider than the Airy disc of a telescope. Its peak intensity versus that of the Airy disc is proportional to the ratio of the vane area to the area of the clear aperture. CENTER: Actual intensity distribution* within diffraction pattern of a 2mm by 200mm opening centered over a 200mm f/10 mirror (so ς=0.01). Its central diffraction maxima is nearly 1mm long (purple), and nearly 0.012mm wide (green, magnified at the bottom), with the length-to-width ratio of ~90 (both are somewhat smaller than expected, probably due to under-sampled calculation). At 0.0134mm @ 550nm, central maxima of the circular aperture (the mirror), is only slightly wider than the central green plot, and nearly 80 times higher (blue line). Central maxima of the vane is nearly 90 Airy disc diameters long, but how much of it is visible - if any - depends on star's telescopic brightness and magnification. This intensity distribution shows that practically all visual effect of the vane results from its central maxima. With ~80 times (the reciprocal of the area ratio) lower intensity than the telescope's PSF central maxima (showing intensity distribution over the Airy disc), it is fainter here than the first bright ring. Its own second maxima is over 20 times fainter, nearly as faint as the 5th bright ring of a perfect aperture. However, similarly to the first bright ring, the central spike appears nearly half as bright as the disc at high telescopic brightness levels, due to the logarithmic intensity response of the eye (however, since its actual intensity is so much lower, it fades away much sooner with the drop in brightness, as its intensity approaches visual threshold of detection). RIGHT: High-magnification simulation of the effect of a 4-vane spider on the diffraction pattern of a bright star; the spikes are clearly visible, extending far out from the central disc.
*Polychromatic intensity distribution, produced by OSLO, with 25 wavelengths from 440nm to 670nm, weighed for the spectral response of the eye; weighing may and may not be appropriate, but the difference is negligible - slightly higher first minima, and somewhat lower second maxima.

Thus a 3-vane spider forms three spikes centered over diffraction pattern at 120° radially, while 4-vane spider forms two spikes at a 90° angle. The total energy contained in the pattern is proportional to the vane area, and so is the peak pattern intensity. The central maxima of the diffraction spike is approximately 1/ς times the Airy disc diameter in length. It is approximated by by 2.4λ/w angularly (in radians), λ being the wavelength of light and w being the vane width.

Since at these small angles there is practically no difference between the angle and its tangent, linear length of the spike maxima is approximately 2.4λƒ/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 ~2.4λF/ς, F being the telescope focal ratio number.

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. 53a). 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): S'=[1-Nςα/180(1+υ)sin(α/2)]2, with α being the vane arch angle. However, the result is only slightly lower for given count (N) and relative thickness (ς) of the vanes.

FIGURE 53: Diffraction effect of a curved spider vane: (a) Each practically straight vane section produces an effective slit at the aperture, which projects its diffraction pattern onto the main pattern (bottom). The patterns of the individual sections are wider, due to their reduced length, but their extension is unchanged (it only varies with the vane width). The orientation of each of these patterns (1,2,3 and 4) is orthogonal to that of the vane section, with the total angle of pattern spread being approximately that of the radial angle of the vane arch α (centered on its radius of curvature). Note that the actual effectively straight sections of a curved vane are shorter than those shown on the illustration, and the pattern spread is continuous within the spread angle. To spread the energy as much as possible - over 360° angle - and, at the same time, avoid overlapping, the vanes need to be appropriately curved and positioned. The simple rule is that a spider with N vanes needs to have 180/N vane arch angle (in degrees), with every next vane separated radially either by the same (180/N degrees) angle, or by (180/N)+180 degrees angle, whichever is mechanically superior (to which side the vane is curved is irrelevant). Thus, a 4-vane spider (b) requires α=45°, and either 45° or 225° between the vanes (B' and C' are alternative positions for better stability, with the D vane preferably rotated length-wise). For a 3-vane curved spider (c), the arch angle is 60°, with the second vane separated from the first one by either 60° or 240°, and the third vane by as much from the second vane, clockwise; B' is the alternative B vane position. A 2-vane curved spider (d) needs 90° arch angle, with the second vane at either 90° or 270° from the first one (B' are alternate positions for the second vane). A single curved vane (e) would require 180° vane arch; hence more stable, circular form would double the energy spread out, which still would be at the level of a 2-vane spider.

Apodizing mask

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. 54).

FIGURE 54: 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 come 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 details in compromised seeing conditions (as well as the range of resolvable low-contrast details in even perfect seeing).

◄ 7.1. Central obstruction effect ▐ 8. REFLECTING TELESCOPES ►

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