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◄ 7.1.1. Telescope central obstruction: size criteria   ▐    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 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).

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.