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
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
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
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
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
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.
||maximum diameter (inch)
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,
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
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.
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
Design Summary for 9 Point
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
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
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.
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
This shows that you can go beyond 900 mm, provided that you adjust the
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
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.