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.

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.

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.

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.

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.

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.

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.