Mirror cell induced astigmatism

By: Arjan te Marvelde

This article investigates the amount of astigmatism that can be induced in a mirror, because it is glued to the mirror cell with silicone caulk. The astigmatism is hypothetically caused by the force couple that is imposed on the mirror by levers and triangles, when the mirror is facing low altitudes.

The mirror that is the subject of investigation, measures a typical 10" diameter by 3/4" thick, and is made of float glass (or 'plate'). The mirror cell is a 6-point flotation type, consisting of three levers.

Mathematical model

Consider the mirror to be a disk of floatglass:

The glass properties are:

Use the biharmonic equation for thin plate deformation:

Where U is the surface deviation and Q is the load distribution caused by mirror mass and gravity:

Cell construction

A six-point flotation cell, consisting of three cantilevers, is plenty sufficient for this mirror. PLOP calculates supports should be placed at appr. 59% radius, resulting in a maximum surface deviation after refocussing of well under 10nm.

For this purpose the 6 supports are reduced to three by means of 3 cantilever beams, of slightly under 75mm length each (Figure 1A). The distance of the fulcrum to the mirror backside is taken as approximately 10% of the beam length, i.e. 7.5mm (Figure 1B).

Figure 1A: Placement of cantilever beams
Figure 1B: Cantilever dimensions

When the mirror is facing zenith, the support forces (each carrying 1/6 of the mirror weight) are directed normal to the glass. PLOP optimizes a cell for a mirror being in this position.

When pointing the telescope anywhere else, the gravitational force has to be split in a component working normally on the mirror supports, and another component working sideways along the mirror surface. This lateral force component can be supported in two ways:

1. Let the mirror slide on the supports and support the edge of the mirror separately, for example with a sling or additional flotation points.
2. Glue the mirror to the six cell supports with silicone caulk.

Now consider the mirror facing horizon.
Then, in the first case, the deformation must be calculated as caused by the mirror standing on its edge supports or sling.

In the second case however, there will be a force couple imposed on the mirror by each cantilever having a (partially) vertical orientation (Figure 1B). The worst case is when the cantlever is precisely vertical, and this is why the cell orientation is like shown in Figure 1A..

In the example shown in the figure, the normal forces excerted on the mirror are + and - 20% of the force caused by the weight, which is a considerable amount.

Results

A model of this problem was evaluated with MSC-NASTRAN. The first figure shows the mirror surface deformation when pointing at zenith. The numbers are much higher than those obtained with PLOP, because they have not been compensated by refocusing. Uncompensated P-V surface deformation is here about 50nm, or l/10. Running PLOP with the same cell and mirror, but no optimization or obstruction, yield a maximum P-V surface deformation of approximately 70nm. These results are fairly consistent.

Figure 2: Surface deformation when facing zenith.

Figure 3 shows the deformation when the mirror in a cell as shown in Figure 1A is pointed to the horizon. In this case the P-V surface deformation is about 100nm, or l /5.. However, since the deformation is mostly astigmatic it cannot be compensated by refocusing, and the deformation may therefore be considered as very bad.

Figure 3: Surface deformation when facing horizon.

Conclusions

Although a six point cell is plenty accurate for the support of this 10" by ¾" mirror, glueing it to the cell will cause unacceptable amounts of astigmatism at low altitudes. Lateral forces, and presumably also the amount of deformation, will be reduced only by half at 60° altitude angle.

Of course the cell may be rotated by 60°, in order to have the lower lever horizontal, but the combined force of two upper levers will still cause unacceptable amounts of astigmatism.

Also, the distance of the fulcrum to the mirror back may possibly be decreased to minimize the force couple. But even when a 3mm strip of uluminium is used as a cantilever, glued to both the mirror and to the center support with 2mm of silicone caulk, the distance is still about 6mm.

I have not done simulation of edge support induced deformation, but at least the presented results discourage glueing the mirror to its cell.