For a very long time, amateur telescope makers did not have good designs to use for mirror cells. Cells designed on the basis of intuition or the equal area rule turn out to have been far from optimum.

In the 1990's, Toshimi Taki developed a finite element analysis program to determine better cell designs using numerical modeling of mirror flexing1,2. Taki's program, called Plate, revolutionized amateur cell design, but it was not easy to use. Many amateurs continued to use older designs because Plate was not sufficiently accessable. David Lewis considerably improved this situation by developing the program Plop3,4. (Plop stands for PLate OPtimizer.) Plop, and particularly the most recent version, Graphical Plop, which uses a graphical interface, is considerably easier to use.

Graphical Plop is still sufficiently difficult to learn that many amateurs are put off. In order to help them, I have decided to make a web page describing a number of mirror cell designs that have been optimized using Graphical Plop.

Amateurs should note that, as of 2002, almost all of the amateur telescope making books currently in print contain cell designs that predate the use of Plate and Plop. Do not use these old designs. Otherwise excellent books that include obsolete cell designs include:

Plop determines the support geometry for the back of the mirror. This is a key design feature, and the one most in need of quantitative engineering. Cells have other features, without which the telescope will not function. These include mirror side support, retaining clips, mechanical structure of the back support, collimation adjustments, integration with the rest of the telescope structure and such accessory functions as cooling fans. This web site only addresses the geometry of the mirror back support. Kriege and Berry's mechnical designs have been used in many successful telescopes and can be adapted to geometries derived from Plop. Many other realizations of mirror cells are workable. Web sites from amateurs describing their creations are fruitful sources of inspiration for the mechanical aspects of cell design.

For any proposed support geometry, Plop will optimize the placement of mirror support points and part pivot points in order to minimize surface deformation. Plop also produces a measure of mirror surface deformations produced by the support. An ideal cell would produce no surface deformations, but there is not a practical ideal cell. All practical cells produce some deformation. The vital question is: how much is too much? Lord Raleigh's famous criterion says that, to form a sensibly perfect image, light should not deviate more than 1/4 wavelength from it's ideal path. Since any deformation of a mirror surface is doubled in the light path, that indicates that mirror deformation should not exceed 1/8 wavelength. Most astronomers agree that, for fine visual observing at high magnification, Lord Raleigh's criterion is too loose by a factor of at least 2. On the other hand, it is difficult for even the most skilled amateur, and many professional, mirror makers to produce mirrors with less than about 1/20 - 1/30 wave surface imperfection. Since mirror cell design is more easily controlled than precise surface figure, it makes sense to use a cell design which contributes less deformation than the mirror figure. Using this criterion, we can conclude that a cell producing not more than 1/40 wave of surface deformation is sufficient for almost any amateur purpose. In some cases, cells producing as much as 1/20 wave surface deformation may be suitable. These would include telescopes to be used soley for photographic or ccd observations, where resolution is limited by the detector, telescopes to be used only at low magnification, such as richest field telescopes, and telescopes made with mirrors that are known not to be of the highest quality.

Below about 1/40 wave, there is little reason to concern oneself overly much with the actual deformation values. Plop dutifully reports deformation values such as 1/77 wave or 1/161 wave. I have tabulated these, but, except to say that the cell induced deformation is small, they should not be taken too seriously. Nobody can figure a mirror that well anyhow.

In the preceeding discussion of mirror deformation, I have left some things unspecified which need to be nailed down. The numbers in the previous discussion refer to Peak to Valley (P-V) deformation. That is, the highest deformation of the mirror minus the lowest deformation. There are good arguments that what is called the Root Mean Square (RMS) average deformation is a better measure of mirror deformation.

The following table shows RMS values of mirror deformation presented in various formats along with estimates of the Strehl ratio for each. The Raleigh criterion, commonly stated as 1/4 wave P-V at the wavefront corresponds roughly to 1/14 wave at the wavefront. (Jim Burrows)

At the Raleigh criterion, the Strehl ratio is about 0.81. Halving ones tolerance to 1/28 wave RMS at the wavefront yields a Strehl ratio of 0.95, a very significant improvement.

The formula for combining uncorrelated RMS values is RMStotal = SQRT(RMSmirror2 + RMScell2)

Example: A mirror with 7.3 nanometer RMS deviation from a parabola has a theoretical Strehl ratio (assuming the rest of the system is perfect) of 0.9669. ( I made a mirror with this value.) Putting this mirror on a cell with 1/120 wave RMS surface deformation will result in a total RMS of about: 8.405 nM = SQRT(7.32 + 4.16672) This will result in a Strehl ratio of about 0.956.

The same mirror on a cell with 1/240 wave surface deformation will result in a total RMS of about: 7.591 nM = SQRT(7.32 + 2.08332) This will result in a Strehl ratio of about 0.964.

Surface RMS Surface RMS Surface RMS WaveFront RMS Strehl
nM mm Waves Waves
25 2.5e-05 1/20 1/10 0.6738
17.855 1.79e-05 1/28 1/14 0.8186
12.5 1.25e-05 1/40 1/20 0.9037
8.92857 8.93e-06 1/56 1/28 0.9509
4.16667 4.17e-06 1/120 1/60 0.9891
3.90625 3.91e-06 1/128 1/56 0.9904
2.08333 2.08e-06 1/240 1/120 0.99726
1.953125 1.95e-06 1/256 1/128 0.9976
Reference wavelength = 500 nM

Approximate formula for Strehl ratio from RMS is taken from Jim Burrows web page http://home.earthlink.net/~burrjaw/atm/atm_math.lwp/atm_math.htm

Inside Plop itself, one has a choice of using RMS (recommended) or P-V (strongly not recommended) for the optimization calculations. I have always chosen RMS. This does not affect the fact that the final results are presented with both P-V and RMS deformation values. For each cell design, I have presented both P-V and RMS deformation values so that you can use whichever you prefer.

Plop presents the P-V and RMS deformation in millimeters. This is consistant with all of the other units in Plop which are in millimeters. For the purpose of comparing surface errors, amateurs, by longstanding precedent, think in terms of fractions of a wavelength of light. One needs to specify exactly which color of light one is comparing to. All of the entries in this web site refer to a reference wavelength of 500nM. This wavelength, in the green part of the spectrum is very near the peak sensitivity of dark adapted human vision. It is also an easy number to calculate with.

Although I am a strong supporter of System International (metric) units in technical work, I recognize that, in the United States where I live, mirror blanks are still specified in inches. Therefore, the cell designs presented here are mostly calculated for currently available U.S. mirror blank sizes. All of the cell dimensions however are presented as fractions of the blank radius. This makes it easy to scale the cell designs as needed. Simply pick the mirror blank in the tables nearest in diameter and thickness to yours and multiply its cell dimensions by your blank's radius (not radius of curvature).

The simplest cells described are 3-point cells. Almost any 8-inch or smaller mirror can be well supported on a 3-point cell. Some 10 inch and a few 12.5 inch mirrors can also use 3-point support. Note that the support circle radius for most Plop optimized 3-point cells is near 0.40 times the mirror radius. This is a perfect example of the difference between Plop optimized cells and older designs. Most older 3-point cells had the supports at 0.707 times the mirror radius. Texerau has them very near the mirror edge, perhaps at 0.95 radius. For a 1.1-inch thick, 8-inch diameter f/8 primary made of Pyrex, with 1.35-inch diameter secondary, Plop predicts 1/23 wave P-V deformation for the 0.707r case and 1/10 wave P-V deformation for the 0.95r case, but only 1/62 wave for the 0.399r case! Assuming we can trust the math, Plop clearly produces better cells.

Most reflecting telescopes use a secondary mirror that blocks the center of the primary mirror from receiving light. Plop takes this into account by ignoring the shadowed portion of the mirror when calculating deformations. In order to make realistic use of this feature, I have chosen a reasonably sized secondary for each cell design using the program Newt5. (The exact sizes are taken from the product catalog of a well regarded U.S. manufacturer of diagonal mirrors.) The secondaries are sized based on the presumption that the mirror will be used in a typical Newtonian configuration for visual observing. Newtonian secondary size is subject to change depending on the design goals of the builder, and on the available sizes. Small changes in secondary size should have only small effects on cell design. In most cases they can be ignored.

I have included, for each mirror diameter from 3-inch to 12.5-inch, cells calculated for an f/12 case with no secondary obstruction. These are included for the benefit of those building unobstructed telescopes such as schiefspieglers or Yolo's.

Different glass types have different mechnical properties. Plop contains property values for a few glass types. I have calculated results for Pyrex and Plate glass and some for BVC. (BVC is a product of ASM Products.) The results for Pyrex should be similar to those for other low expansion borosilicate glasses such as Duran and Borofloat.

If you wish to use Plop to calculate a cell for a BVC mirror, you will need to enter these values on the Material tab in Plop:

Modulus
7200 KgF/mm2
Poisson0.23
Density2.44E-6 Kg/mm3

Properties for Suprax 8488 are:

Modulus
6830 KgF/mm2
Poisson0.20
Density2.3E-6 Kg/mm3

(KgF = Kilogram Force = 1 Kg * Acceleration due to Earth's gravity at surface = 9.80665 Newton)


Footnotes

  1. Sky & Telescope, September 1994, pp. 84 - 87
  2. Sky & Telescope, April 1996, pp. 75 - 77
  3. Sky & Telescope, June 1999, pp. 132 - 135
  4. http://www.davidlewistoronto.com/plop/
  5. Newt for Windows by Dale A. Keller http://home.att.net/~dale.keller/atm/newtonians/newtsoft/newtsoft.htm

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Copyright 2002 Mark D. Holm