There have been discussions on the net about how well a foam pad and flat plate can support a thin mirror. Some propose that it will work well if the mirror back and back plate are flat enough and the mirror is thick enough.
Supports of this type requires equal force across the back of the mirror. To accomplish that the back plate supporting the mirror must be flat to a certain tolerance and the foam pad must have a uniform thickness and spring constant. This investigates how flat the plate must be in order to support a 16 x 3/8 F3.5 plate glass mirror to meet the ¼ PV wave front error criteria.
I found the plate could not deviate by more than .007 inch from the edge to the middle of a 12 inch span in order not to introduce any astigmatism. I don’t derive any equations that will help others determine the tolerance for their mirror. This is just a study of what happens to this particular mirror. I think it would be wise for others who think about using that kind of support to consider the consequences for their implementation.
I used an interferometer to measure the effect of placing thin shims between the back support and the foam under the mirror. Relative deformations of the mirror surface were then measured and compared to the no shim state. Two shim thickness were used, a .063 inch and a .020 inch.
I only have a curved back mirror to measure but I can take the curve out of the equation by supporting it only at the edges and then changing one of the edge supports by known amounts and measure the edge surface change. I have done enough measurements on this mirror to know the figure to a very close degree. I have software that can subtract that figure from the measurements so that only the support induced errors are measured.
I supported the 3/8 thick 16 inch plate glass mirror on 4 pads and tilted the test stand back 45 deg. The pads are 3/4 inch wide, 3/8 inch thick foam weather strip with sticky back. They are 1.5 inches long and were placed ¾ inch under the mirror edge. It required a force of 44 oz to depress one 1/16 of an inch. The pads were placed at the edge of the mirror in the quadrants 45 deg from vertical. To minimize errors from lateral support, I used tape attached to the back of the mirror and the top of the test stand.
Here is an image of the mirror with the figuring errors removed and only the test stand deformation shown. The PV is 1.5 waves wave front error. I have a bug in my display software that is clipping the front bumps but I think you get the idea.
Here is a contour map. I found it close to PLOP when setting the sagitta to 0 and multiplying the density by .7 to factor the stand tilt of 45 deg. You can see that the right side bumps are somewhat lower but that is not important for what I’m about to do. The right image is generated from PLOP. I negated the colors to help match what I have red color as high in my software. Plop computes of PV of 1339 nm. The actual for the mirror is 1500 nm.
The data in the above images is my control. It proves to me that the mirror deforms as expected. When I make changes by carefully adding a pad to the mirror I can subtract this data from the new measurements to compute the change
. Next I will add a pad under the right side. Here is the image of the relative change (4 pad data subtracted.)
It is interesting to see what else changed. I have created a large amount of trefoil distortion. The important part however is that now this is my new reference. What I will do next is add tiny shims on top of the right side pad. That will cause the pad to compress and perhaps put more force on the mirror at that point. That will be the equivalent of the back board being higher or the foam having a slightly different spring constant.
First I added a 1/16 inch thick shim on top of the right pad. It looks a lot like the previous image but remember the previous data has been subtracted from this. So the right edge has risen yet again. In fact the right edge was raised 540 nm (a whole wave) on the mirror surface relative to the old surface.
Below is a profile through the horizontal diameter of the mirror.
A 1/16 inch is not a lot but perhaps people can keep a back board to a tighter tolerance. Next I tried only .020 thick shim. That is 3 times smaller than the previous. The image is similar but the magnitude is smaller. The surface raised only 270 nm about ½ a wave. The deviation is slightly less than 3 times smaller. So it looks like the linear relationship is one wave increase per 1/16 inch thickness deviation. To meet the ¼ wave front deviation requirement means the surface must deviate less than 1/8 wave. So we can see that the surface supporting this mirror cannot deviate by more than 1/8 X 1/16 or .007 inch.
Lets ignore small bumpy surface detail that size for the moment. What I believe I have just shown is that if there is a slight cylinder curve to the back board greater than .007 inch then it will create astigmatism in this mirror larger than the 1/8 wave we had hoped to avoid. The amount of deformation for any given thickness over a distance is proportional to the distance squared divided by the thickness. The distance for my test was between the upper right and lower right support points. This was about 12.5 inches. So the tolerance needs to be even tighter across the whole 16 inch mirror. I believe it is next to impossible to control a wooden surface to that tight a tolerance. Although it is not impossible it isn’t easy to do it with metal over 16 inches either.
I have been ignoring the astigmatism caused by the rise in the pad and just measuring the pad height. In reality that rise will cause large astigmatism. Below are 3d models of the surface with the astigmatism cause by the bump. These are computed from actual interferograms like the data above but have the astigmatism term enabled. The limit box is ¼ wave high.
Some ATM’s use PLOP to compute where support points should be placed on a mirror support. Next they place individual compliant supports like foam pads at those points on solid surfaces. For this arrangement to work, all support points must apply an equal force. If the mirror is thin enough this will fail to support the mirror properly when the surface of the back board has errors exceeding the assumed surface accuracy. For the mirror just measured the accuracy must be greater than about .007 inch.