Experiments and Design Techniques for Mirror Cells

This page contains some experiments and design information for common mirror cells. It is somewhat verbose, but you can quickly get to the design summary for a given type of cell by following the links:
 
  • 3 point cells    (click here for design summary)
  • 9 point cells   (click here for design summary)
  • 9 point cells with non-uniform supports   (click here for design summary)
  • 18 point cells  (click here for design summary)
  • 18 point cells with non-uniform supports (click here for design summary)
  • 27 point cells
  • 54 point cells

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    Custom designs: if you look here, and can't find what you want, or are so close to the limit that you want a detailed analysis.,e-mail me. Click here for contact info. Tell me mirror size, thickness, f-ratio, and cell geometry. If I can solve it in a reasonable time, I'll email you back the design measurements.

    3 Point Mirror Cells

    The first experiment more precisely determines the best radius for support. We use Plop to optimize the support radius for a 3 point cell. We also consider the radius of the secondary mirror. A 317.5mm (12.5 inch), f/5, 54mm (2.125 inch) thick mirror is used. The radius of the secondary is varied from 0 to .3 of the mirror radius. A plot of the best radius for the support, and resulting error, is shown below. The lower plot is the best radius for support, the upper plot is the RMS error.
    Design Summary for 3 point cells
    For a typical .2 radius secondary, the best support radius is .401. Click  here  to see the complete output. The table below gives the maximum diameter for each thickness. Since we only calculated deformation on 15mm diameter intervals, this is a bit conservative.
     
     
    thickness (inch) maximum diameter (inch)
    0.875 8.7
    1.0 9.3
    1.25 10.4
    1.5 11.0
    1.75 12.2
    2.125 12.8

    To see how big we can go, we run an experiment for glass of thickness 22.22, 25.4, 31.75, 38.1, 44.45, and 54 mm (.875, 1, 1.25, 1.5, 1.75 and 2.125 inches), varying the diameter from 100 to 400 mm, (3.9 to 15.7 inches)  keeping the support at radius .40. Here is a plot of the deformation; thinner mirrors have more deformation.


     

    For the complete output, click here.

    We would also like to find out if the f-ratio affects deformation much. We take a large mirror, 340 mm (13.4 inch), and 54 mm (2.125 inch) thick, and evaluate the error as the f-ratio is scanned from 3 to 10. The support radius is fixed at .40, and the error is plotted below. Note that you could optimize the support radius for each f-ratio, but the results aren't much different.  There is only a small change in error for short focal ratios below f/5 so we'll use f/5 for a standard.


     
     

     9 Point Cells

    We perform a similar experiment for 9 point cells. As a starting point, we use a 406mm (16 inch) diameter mirror, 54mm (2.125 inch) thick, and use the cell design given in Kriege and Berry (which, BTW, is an excellent book!) Kriege and Berry use CELL.EXE, written by David Chandler, to design their cells. CELL.EXE uses the approach of determining mirror segments with equal mass, and supporting each one of these by a cell point. This is a eminently reasonable approach, and gives good results for typical cases. Plop's advantage is that it takes into account structural properties of the material, which will especially affect mirror deformation with large, thin mirrors, or non-uniform support. Also, by calculating the deformation, Plop is able to determine the best way to refocus the mirror, which may in turn affect the best place to support the mirrorn.

    It turns out that CELL.EXE designs very good cells for this case: the error with .32 and .78 radius is 2.11e-06; Plop can find a cell with 1.99e-06 error, only 5% better. Interestingly, the best support radii are different by a large amount: about .33 and .72. The fact that the best radii are different from the starting point by a large amount, but the error is not much different is also a piece of good news; it hints that cell error is not highly sensitive to placement of the supports. We'll look at sensitivity in more detail later.

    Here is a plot showing the best radii for the supports, and the RMS error as the secondary radius is varied from 0 to 0.3 of the primary  radius.

    Click here for the complete output.

    How large can you go? This plot shows the error for supports at .33 and .72, 0.2 obstruction, as the diameter is varied from 300 mm to 600 mm. You can go up to 500 mm without exceeding 1/128 wave, just barely under 20 inches. Click here for the complete output.
     

    Sensitivity Analysis
    We would also like to see how sensitive the mirror cell design is to fabrication tolerances. We use Plop's scan feature to investigate this, by varying the inner support from .32 to .34, and the outer support from .71 to .73, and looking at the variation in error. It turns out that it is insignificant - less than 1%.  Therefore, don't kill yourself getting the cell accurate to .001 inch - even .1 inch will do fine! For the complete output, click here.
    Design Summary for 9 Point Cells
    Place the supports at .33 and .72 radius. For 54mm (2.125 inch) thick glass, you can go up to 500mm (19.7 inch). What the heck, live dangerously and go to 20 inches!  If you are at the ragged edge of the limits, look in the database here to get the details. The datbase lists best supports for 12" to 20" mirror, as f/ratio goes from 4.5 to 6.
     

    9 Point Cells with Non-uniform Support

    Now we get into a less common type of cell. Consider a 9-point cell, but the forces on the inner and outer support rings are allowed to be different. We investigate whether a better cell can be designed as a result of the freedom to vary support forces, and the answer is yes. Here is a plot of the support radii, and the ratio of force (outer/inner), and the resulting error, as the obstruction goes from 0 to 0.3. The bottom trace is inner support radius, next is outer support radius, next is ratio of outer force to inner, top one is RMS error. Note the noise in the inner radius. This should be a concern for investigation. It is due to the fact that the outer radius is close to the middle of two rings on the mesh, and the optimizer hops between them. It is also evidence of roughly how much tolerance you have. The error doesn't vary much across this range.

    This is interesting: the error is decreased by 30% with a non-uniform force on the cell. The support points move quite dramatically, to .19 and .64, with relative force of 3.3. We expect the 30% decrease in error to buy us an additional 15% or so in allowable diameter. In fact, that's roughly what happens. We can go to 570 mm without excessive error, or 22.4". Click here to see the output.

    Sensitivity becomes a concern at large diameters. Varying radii from .18 to .20, .63 to .65, and force from 3.2 to 3.4 on a 570mm mirror causes 11% increase in error. Note that .01 radius is 2.8mm, or nearly .11", so this is pretty sloppy, but some care is nevertheless required.

    For a comparison of 9-point cells with uniform and non-uniform forces click on the icon: 

    However, at large diameters, thickness of the mirror becomes a concern. We vary f/ratio on a 560mm mirror from f/3 to f/8. The cell is optimized for f/5, and there is a 20% increase in error at f/3. So, you can use the cookbook formula for any f-ratio up to about 530mm, but f/5 only up to 570mm. Beyond that, you had better consider f/ratio of the mirror.

    Design Summary for 9-Point Cells with Non-Uniform Force
    For f/5, place the supports at .19 and .64, with 3.3X as much force on the outer ones. For 2.125" glass, don't exceed 22.4" diameter.  If the mirror is faster, don't exceed 20.8" without looking in the database. (This will be produced after I reduce the numerical noise in Plop.)

    18 Point Cells

    At this point, design becomes a little more intricate for the thin glass that we assume. Using the previous technique, the best radius for support for .2 obstruction is found to be .378 and .764. However, studying the error as diameter is varied leads to a new surprise. Error grows much faster than the square of diameter, as was previously the case. You can only go to 750 mm, a bit less than 30", with this support. Here is the plot of error vs. diameter:

    As diameter doubles from 300 to 600, error goes up by a factor of 7. The problem is that at this scale, the sagitta becomes significant compared to the thickness. We need to adjust the support radii as the mirror increases in diameter. If this is done, the radii and error are given by the plot below. The bottom  plot is inner radius, next is outer radius, top is error.

    This shows that you can go beyond 900 mm, provided that you adjust the support radii.

    This  is also a strong hint that cookbook formulae, giving support radii as a constant percentage, won't be optimal  for large thin mirrors. Even for a constant f-ratio, the thickness at the center will vary, and so will the support radii. For a given diameter, the support radii may also vary with the f-ratio, as shown in the plot below. Here are the best radii  and error for a 30" mirror as the f-ratio is varied:

    For another aspect, consider a cell optimized for a 30"  f/5 mirror. The plot below shows the error as a function of the f-ratio of the mirror that is actually place on the cell. You can see that using a f/4 mirror on a f/5 cell causes a 20% increase in error.

    If your mirror has a sagitta that is over about .5", you may want to optimize the cell for that f-ratio.

    These sensitivities are also a strong hint that you'd better be careful manufacturing these cells. We performed a sensitivity analysis on a 30" mirror, varying support radii by +/- .01. This caused  up to a 40% increase in error. The good news is that even the worst case was 3.18e-06, well within the tolerance of 4.2e-06. Also, +/- .01 represents .15" error in a cell this size, so it is a fairly large amount of slop. So, even though an 18-point cell can conceivably go to 36", you'd better be careful.
     

    Design Summary for 18-Point Cells
    If the mirror is f/5 or longer, and under  30" diameter , place the supports at the locations in the above plot, and you'll be fine.  You can go up to 36", but had better be careful. To be precise, search here in this database for the nearest cells and interpolate.
     

    27 Point Mirror Cells

    27 point cells can take a long time to optimize. The cells that we have been looking at don't constrain the angles to be even divisions of the 360 degrees in a circle. Instead, we let the angles float as well. If you want to design a 27 point cell, start with example 38, which is for a 40" Cassegranian with a  10" diameter hole. This analysis took 52 hours on a 233 Mhz P2. A picture of the resulting deformation is shown below. Note that the best arrangement doesn't look anything like what the conventional cell design with evenly spaced supports would. However, this cell gives 3.89e-06 wavefront error with 27 points on this f/3.66 mirror, which has a sagitta of over .7". There are a number of symettries that can be seen, but the cell design doesn't look much like the standard style in, for example,
    Kreige and Berry. Following this analysis, I redefined the cell to describe the symmetries using variables. This produces a design that is perfectly symmetrical, and takes less than 2 hours to run.

     

    54 Point Mirror Cells

    54 point cells follow the same strategy as 27 point cells. Our example is a 38" mirror that is only 0.75" thick. We begin with 18 sets of 3 points, placing 5,6, and 7 sets of these at various radii that are crude guesses at the correct radii (well, this is actually a bit optimistic; the guesses were acutally pretty awful, but who cares since the optimizer will sort it out.) Click here to ee the .gr file example44/example44raw.gr. After running Plop for about 48 hours, it was killed, and the support pattern was examined. Here is a picture of it:
     

    At this point we can see that two of the sets of three points are located at approximately the same angle, and the remaining sets of points are approximately symettrical on either side of these two. We then construct a new .gr file based on estimates of the angles and radii of these points. We fix two sets of points at 60 degrees, and the others are constrained to be symmetrical by using variables. It takes about 12 hours to run this problem to completion. Here is a picture of the resulting supports and deformation. Here is the .gr file and here is the optimized result saved by Plop.