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 7.1.1. Telescope central obstruction: size criteria   ▐    7.3. Apodizing mask
 

7.2. Spider vanes diffraction

More often than not, central obstruction by a smaller secondary mirror is accompanied with aperture obstruction caused by its support structure - so called spider vanes. Their effect is generally small, but it can be significant. That makes them worth of a closer look.

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. 108). Analogously to the central obstruction effect, change in the


FIGURE 108
: 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).

normalized central diffraction intensity 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 (i.e. annulus) area; this is consistent with the degradation factor caused by central obstruction (Eq. 60). Average spider area is somewhere between 1% and 2% of the clear aperture area. That puts the average spider vane central intensity degradation factor between 0.98 and 0.96. As plots at left show, it decreases linearly with the increase in obstruction (quadratic plots are near flat sections of very wide parabolas).

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 this degradation factor can be written as I(0)'~(1-2a). 

As with the CO effect, this diffraction peak degradation includes the obscuration effect, which does not affect the relative energy distribution within diffraction pattern. Factoring that in, the relative degradation is given simply by (1-a). Another similarity with the CO effect is that the 4-vane spider also makes the central diffraction maxima smaller. However, while in the case of CO it remains circular in shape, the 4-vane spider makes it slightly rectangular, by slightly flattening the central maxima on four sides where the spikes come out. The reduction is roughly commensurate to the area of obscuration, thus much smaller than with the standard CO sizes. Simulation at left shows it for a 3-inch aperture and 4-vane spider with 5mm thick vanes (no central obstruction), thus creating nearly 17% obscuration - much more than the vanes usually do. The MTF is also, for clarity, shown for this exaggerated scenario. The contrast transfer improvement over the perfect clear aperture is for the orientation shown; for the pattern rotated by 45 degrees it would be smaller. Compared to the CO effect, this contrast improvement in this orientation drops off quickly after the initial gain at mid frequencies, probably due to the vanes bringing energy into the first dark ring. With the standard thickness vanes, these effects are negligible, more so considering that the actual vane area is reduced in the presence of CO. 

The relative intensity distribution is, again as with the CO effect, based on the peak intensity w/o obscuration effect, normalized to 1. Energy distribution between the central maxima and the rings depends, in addition, on the volume corresponding to the central maxima, relative to the volume of the rings area. In the absence of exact calculations, best reference point is the effect of CO with comparable relative obscuration. Taking the same 3-inch aperture, 17% obscuration area corresponds to 41% linear CO. Graph below summarizes the effects of cross-like and circular obscuration alone, for 17% areal obscuration. The corresponding PSFs show that the 1st bright ring is significantly less bright with the cross-like obscuration than with with CO, with uneven intensity (given for three orientation, zero degrees being along the horizontal, or vertical central line of the PSF pattern), but with generally more energy in the outer area. Encircled energy graphs top right show that this energy beyond the 1st bright ring is significantly higher, despite the cross-like obscuration encircling more energy in the central maxima (approximately 80% of the energy content of Airy disc, vs. nearly 70% for the CO). The more energy beyond the 1st bright ring for the cross-like obscuration is a consequence of roughly three times more energy in the 1st bright ring with the CO. Taking that the relative encircled energy in the central maxima is, as a quality indicator, comparable to the Strehl, and for relatively small disturbance levels nominally nearly identical, leads to the conclusion that a 4-vane spider Strehl-like figure is somewhat higher than for a CO of the same relative obscuration. In other words, a standard 1-2% obscuration 4-vane spider would degrade the effective Strehl by less than the corresponding 1-2% obscuration CO, i.e. less than 10-14% linear CO.

Factor that complicates such comparison are the differences in the size of central maxima. As the magnified inset top left shows, average central maxima reduction due to the cross-like obscuration is less than half of that due to the CO. So are the related effects of increased contrast on the left side of MTF graph and higher cutoff frequency. For small relative obscurations, such are those by the typical 4-vane spiders, these high-frequency gains can be safely neglected, more so considering that the spider effect considered here includes its non-existent central portion behind the CO. The effect is limited to the contrast drop in mid and low frequencies, as shown on the bottom right. With 17% relative obscuration, contrast drop at the 0.2 frequency is 12%; a thin-vane spider obscuring 1% of the area would have little over 0.5% contrast drop there, as little as 0.995 Strehl of aberration averaged evenly over all frequencies.

It should be emphasized that this theoretical approach is strictly valid only in the context of near monochromatic point source - i.e. coherent light. In the real world, light processed by the telescopes is typically polychromatic, i.e. partly incoherent, in which case the effect of light obstructions on intensity distribution within diffraction image is significantly smaller. As in the section about the effect of central obstruction, the text continues with the standard coherent light context, but keeping in mind that it is not directly applicable to the field conditions.

Depending on the object of observation, 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.

VANE AS AN INVERSE NARROW SLIT

Spider diffraction effect is often illustrated by the effect of a narrow slit. There doesn't seem to be clearly defined width above which diffraction effect becomes that of an aperture. Hecht uses narrow slit relation for a 0.5mm by 30mm slit, which is very similar to the vane configuration, even if it is, evidently very wide with respect to the wavelength. Specifically, intensity distribution within diffraction pattern created by a slit aperture placed in front of an objective with focal length ƒ is described  by:

with I(0) being the intensity as a function of point radius, for the central intensity normalized to 1 (actual intensity depends on the slit area), β=Sπsinθ in units of the wavelength λ, S being the edge-to-edge separation (i.e. either the slit width, or length) and θ=r/ƒ being the point angle in the image plane, in radians, with r the linear point height (linear radius r=β/πi in units of λƒ/S). The numerator angle is in degrees, denominator angle in radians. 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 at 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.

    With the slit height much larger than its width, diffracted energy drops to first minima much quicker in the plane perpendicular to the slit height, negligible in comparison to the energy spread in the plane perpendicular to the slit's width (for instance, with the width-to-height ratio of 1:100, sinθ in β=Sπsinθ has to be 100 times smaller in order for β to have any given value, including the first minima at β=π.

    What seems to be more appropriate reference shape for the vane in the visual wavelengths is rectangular aperture. Its width is still much smaller than its height, but it is quite large relative to the wavelength. In the case of rectangular aperture, intensity distribution within diffraction pattern is described with a double squared sinc function:

where subscripts W and H stand for aperture's width and height, respectively, along its sides, with βW=Wxπ/ƒ and βH=H/ƒ, also in units of λ, where x and y are the linear point coordinates in image plane  in the horizontal and vertical direction, respectively. Evidently, this relation gives identical distribution of minimas and maximas in the two perpendicular planes as Eq. 66 (one perpendicular to the vane width, the other to vane's height), with either x or y being the equivalent of r under Eq. 66, and either being zero along one of the two perpendicular axes in image plane. Still, this relation is more complete since directly determining intensity distribution along both perpendicular axes, and in any chosen direction in the image plane.

With the first intensity minima falling at a constant nominal value of β, its angular radius, given by θ=x/ƒ=βWλ/πW (for small angles sinθ=θ in radians) is inversely proportional to the width W. Thus, the longer the vane, the more narrow its spike; the wider vane, the shorter its spike. A 200x1mm vane - so with W=1 and H=200 (neglecting central obstruction), for λ=0.00055mm will produce first maxima nearly 4 arc minutes long (for βW=π) and about 1.1 arc seconds wide (βH=π); 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 the contrast level.

The actual spike peak diffraction intensity, similarly to circular aperture, is proportional to its area, and can be written as I=πΦW/ƒλ2, where Φ, W, ƒ are the flux (blocked by the vane), vane width and focal length. The flux as a product of vane area and flux per unit area, and can be written as Φ=WHu, u being the flux per unit area. The W/ƒ factor reflects the effect of spike size (i.e. length), with the intensity being proportional to the width for given flux, and proportional to the square of it considering that the flux also changes in proportion to the width. Obviously, if we scale a vane to a twice larger aperture, its area - and flux blocked out - increases fourfold, and with it its spike intensity. But the same happens with the aperture's diffraction pattern, whose intensity, I=πΦ/(2λF)2, also changes with the flux, and their brightness relative to each other does not change.

If, however, it is the focal length ƒ that doubles, spike intensity relative to the star diffraction pattern also doubles, since the latter is spread onto four times larger, and the former only two times larger area.

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. 109. 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).

FIGURE 109: TOP LEFT: Spider vane diffraction effect is one of a narrow, elongated rectangular aperture. Its diffraction pattern extends appropriately less in the direction of elongation, than in the direction orthogonal to it. The ratio between these two pattern extensions is 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. Peak PSF intensities of the vane vs. Airy disc are proportional to the ratio of their respective areas. CENTER: Intensity distribution* within diffraction pattern of 2x200mm opening centered over 200mm ƒ/10 mirror (so ς=0.01), calculated by OSLO. Its central diffraction maxima is nearly 1mm long (purple), and nearly 0.01mm wide (green, magnified below). At 0.0134mm and 550nm, the Airy disc is only slightly wider than the spike width, and nearly 80 times higher (blue). The central vane maxima is ~70 Airy disc diameters long, but how much of it is visible - if any - depends on star's telescopic brightness and magnification. 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 another 21 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 (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). Above, to the right, the high-magnification simulation of the effect of a 4-vane spider on the diffraction pattern of a bright star, with the spikes clearly visible, extending far out from the central disc. Note that, all else equal, the spike width is greater the greater central obstruction, due to the inverse slit aperture formed by the vane having its length reduced by a factor (1-o)/2, o being the relative linear obstruction size (for instance, spike maxima in the presence of D/3 central obstruction will be three times wider than that of a vane spanning the entire aperture; with the  combined intensity of its two sections lower by 1/3). BOTTOM LEFT: PSF of a double slit/vane has double the central intensity of a single slit/vane of the same width, with the central maxima of a single vane effectively fractionalized into 2s-1 maximas (including central maxima), s being the slit/vane separation in units of their (equal) width W. BOTTOM RIGHT: PSF of a multiple slit/vane has the central intensity higher by a factor of N than a single slit, N being the slit/vane number. Within the width of diffraction envelope of a single slit/vane central maxima, it forms central maxima narrower approximately by a factor 1/5(N-1)s, with (N-2) subsidiary maximas between subsequent principal maximas (s-th principal maxima is suppressed by a diffraction envelope null).
*
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. Dimensions of the maxima are somewhat smaller than what monochromatic formula implies (for λ=550nm) due to polychromatic effect and, perhaps, under-sampling.

 
     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 2λ/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 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. 109 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 nearly 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 on the bright stars and planets 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. 110a). 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 peak diffraction intensity 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 the slightly greater curved vane length.


FIGURE 110: 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 alternative 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.

Using the same concept of inverse aperture, the actual energy distribution of a curved vane can be simulated. It is shown, for comparison, with those for the straight vane, and for a significantly wider vane with one straight and one curved side (note that the patterns are in their actual orientation vs. aperures-vanes). These energy distribution patterns are deducted from the diffraction distribution of the clear aperture, changing its appearance. The full arc of the curved vane-aperture is 90 degrees, but the angle of energy spread is only about 60 degrees, or little more. Three vanes of this shape, properly positioned, would spread their energu in a nearly continuous radial pattern, eliminating spikes (even a single curved vane would probably make it invisible). This vane still produces an elongated central maxima, but some 15 times shorter than a straight vane of the same thickness, thus much less pronounced.

The middle vane shape illustrates that curving and "styling" the vane shape has little influence on their diffraction effect; in general, the inverse energy they are throwing onto the main diffraction pattern will be determined by their relative size in its quantity, and by their shape - approximately - in its shape. A vane like this (it would make little difference if both sides were curved) would produce significantly shorter spike than a thin straight vane, similar to the straight vane of similar thickness. It is still well over twice longer than the curved vane spike, and with much more energy - proportional to its area - transferred out of the Airy disc. There is no shape that will magically take vane diffraction away.

                                                 
7.1.1. Telescope central obstruction: size criteria   ▐    7.3. Apodizing mask

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