Sensitivity analysis of a 16" 1.6" thick mirror

Fig.1 perfect piano wire support.

Here is a study of a 16" mirror, 1.6" thick, with two-point lateral support (+/- 45°). The first picture is the perfect case (support at CoG, no friction). The colour scale is magnified 4 times compared to the case of the 30" 2" thick mirror below. Nevertheless the surface deformations, with piston, tilt and defocus removed, is merely 0.85 nm rms (times two for the wavefront).
The effect of misplacement on the contact point is shown in figure 2. It is the case of the contact point standing 1 mm above the CoG. The surface error this time is 2.2 nm rms.
In plane friction is not important. As shown in figure 3 even a large amount of friction (friction coefficent 1) produces a mere 1.18 nm rms. In this case the direction of forces is 45° from the radius, i.e. exactly in the vertical direction.
The last figure depicts the most critical case: a small amount of friction, 0.01 of the radial force, is big enough to cause the largest deformation, which is 3.25 nm rms.
The conclusion is that a simple roller bearing, which cancels out-of-plane forces, and placed at the CoG is perhaps the most effective choice. The small amount of in plane friction is not critical. Conversely the simpler mechanical design maybe allows to better control the location and direction of reaction forces.
N.B. the cases of figure 2 and 4 use a simplified back restraining system (3 points not the complete 18 or 9 or 6 points) and thus the deformations are somewhat overestimated.


Fig.2 support 1 mm above centre of gravity.


Fig.3 support with in-plane friction (friction coefficient 1).


Fig.4 support with out of plane friction (friction coefficient 0.01),

Unusual ways of lateral restraining

Here are some "strange" ways to make a lateral support. Picture 1 ia a mirror (same of previous post) suspended at the sides of the horizontal diameter. In a certain sense it is the worst case of sling friction (infinite friction). The rms surface displacements (piston, tilt, tip and defocus removed) ia 6.5 nm.
Picture 2 is the same mirror, suspended with vertical wires attached near the top, at +/- 30° from the vertical diameter. The rms surface displacements is 7.7 nm.
Although the latter may seem bad, one should consider that the suspension is "self aligning" and maybe it might naturally stay close to the ideal configuration (TBC).

How good can it be?

Here is a comparison between a "perfect" sling and a "perfect" piano wire support. They are exactly at the centre of gravity, no friction, no out of plane forces of any kind. This is the maximum performance if everything is perfect. The mirror is 30" diameter, 2" thick, Pyrex.
As seen in the pictures, piston, tilt, tip and defocus are removed (this time accurately) and what remasins are the aberrations. Values represent surface displacements (times 2 to get wavefront delay).
The perfect sling has 2.7 nm rms surface displacements. The perfect piano wires 5.2 nm rms. The difference is near the contact points of the PW, and is due to local concentrated stresses.
Both systems are largely within tolerances. So, if one sees astigmatism it does not depend on the choice between sling or PW. It depends on manufacturing and or settings that happen to be far from the ideal configuration.

Misaligned sling suspension points

This is an analysis of the problem described by Nils Olof Carlin where a sling perfectly centered at the centre of gravity of the mirror is suspended to points that do not stand in the same plane. This causes the departure of the sling from the disk with a small angle that in turn produces a slight pull out of the mirror plane. The case may happen for improper manufacturing, but also as a consequence of mirror displacements during collimation.
The analysis here shown uses different assumptions than that of Nils: in this case the mirror is vertical and the departure angle is supposed to be 1/100 rad, which means a pull of 2.21 N (the load case is linear and one may proportionately evaluate the effect of smaller angles).
As shown the PTV surface error is 164 nm (1/1.67 waves on wavefront!!) mostly astigmatism (tilt and defocus are removed). This figure is somewhat exaggerated because the mirror is restrained by a simple three-point support on the back (for the marginal loads of the out-of-plane pull). A real 9 or 18 points support would produce reactions to the out-of-plane pulls somewhat better distributed and thus an overall lesser deformation. Nevertheless the conclusion is clear: not only the sling needs to be accurately placed under the centre of gravity, but also the suspension points need to be accurately on the sling plane.
If one supposes to be able to keep the departure angle within 1/1000 rad (the suspension points must lie within 0.38 mm from the plane assuming a base distance of 380 mm (the distance between point of departure and point of suspension), then the deformations would be 16.4 nm (1/17 waves).

Sling at perfect Centre of Gravity

This is the surface deformation for a sling supported mirror (30" diameter, 2" thick, F/5). The sling stands exactly at the centre of gravity of the mirror (22.5 mm from back side). The picture shows the residual abberations after defocus has been (rougly) removed. As seen the PTV surface z-displacements are less than 20 nm.
The following two pictures compare this case to the case of the sling standing 2.5 mm above the centre of gravity. The color scale, for comparison, is the same: -60nm to 30 nm. The conclusion is that a perfectly placed sling almost zero the residual aberrations, but this is the result of the cancellation of otherwise large deformations. Placing and keeping the sling exactly at the centre of gravity is critical. In practice any deformation of the mirror is likely to be due to "imperfections" in sling placement, friction, sling planarity (included the straight parts to the suspension points). In a word what matters is "robustness" of the design, or the sensistivity to undesired noise factors. More on this...