Adrien Millies-LaCroix introduced a graphical method of Foucault test data reduction in the February 1976 issue of Sky & Telescope Magazine^{(1)}. This simple method makes interpreting your test data quick, easy, and quite painless (especially if you're mathematically challenged like me!). I have created an Excel 97 spreadsheet that makes creating a M-L plot, both in standard and parabola-removed versions, even easier.

Recent advances in test data reduction have made the M-L plot somewhat useless. Jim Burrow's SIXTESTS program is far more accurate. It offers a surface analysis not a slope analysis like the M-L plot. Finally, it provides you a way of testing your mirror against the most stringent criteria, such as the Strehl ratio. Additionally, some very knowledgable ATMs feel that the M-L plot is misleading enough that it isn't even useful for interim testing (in other words, getting the mirror close with the M-L test, then using SIXTESTS).

You do as you wish. I will not remove the ML.XLS spreadsheet, as I know a couple of people use it and like it. However, I don't recommend that you use this as your only or even your primary data reduction method.

To start with, you would create a table with five columns and as many rows as you have zones in your test data set. In the first column, you would enter the radius of the zone. In the next column, you would calculate the value for the knife-edge shift of a perfect paraboloid with the formula (r^{2}/R) where r is the zonal radius and R is the radius of curvature. Next, you calculate the allowable tolerance with the formula (2Rp/r)^{(see footnote A)}, where p^{(see footnote B)} is the diameter of the Airy disk. p=1.22LF/D where L=wavelength of light being used for the test, F is the focal length, and D is the diameter of your mirror. In this case L is set to 0.00054864 mm^{(see footnote C)} representing the yellow-green light to which the eye is most sensitive. Then, add column 2 and column 3 to get the upper limit for column 4. Subtract column 3 from column 2 to get the lower limit and put that in column 5. Like this:

r r2/R 2Rp/r Upper Lower 10 0.03125 3.427024896 3.458274896 -3.395774896 20 0.125 1.713512448 1.838512448 -1.588512448 30 0.28125 1.142341632 1.423591632 -0.861091632 40 0.5 0.856756224 1.356756224 -0.356756224 50 0.78125 0.685404979 1.466654979 0.095845021 60 1.125 0.571170816 1.696170816 0.553829184 70 1.53125 0.489574985 2.020824985 1.041675015 80 2 0.428378112 2.428378112 1.571621888 90 2.53125 0.380780544 2.912030544 2.150469456 100 3.125 0.34270249 3.46770249 2.78229751(Values for an 8 inch [200 mm], f/8 mirror.)

Once you have created your table like this one, you plot it on graph paper. You will see that the upper and lower limit lines form a "tornado" shaped plot -- widely spaced at the innermost zone (the center of the mirror), shrinking considerably at the outer zone. The perfect parabola threads the middle of those two extremes. Your goal is to have your readings fall in between the upper and lower limit lines. So then, you take your zonal readings as you normally would with the Foucault test. As usual, you are interested in the offset of one reading from the next. So, subtract your reading for the center zone from the readings for the remaining zones. Finally, plot those adjusted readings on your M-L graph.

If you need to, you can add or subtract a constant value to all your zonal readings (an offset) to shift the lines up or down on the graph. Doing so has no relevance to your mirror's figure. Essentially, you are adjusting the plot in the same way that re-focusing adjusts the intercepted cone of light from your finished mirror.

**Note: **The M-L graph plots the slope of the surface at a given zone. Don't confuse the slope with the surface height -- you're not plotting a profile of your mirror's surface with the M-L graph.

In the ML.XLS spreadsheet, the Millies-LaCroix Chart (standard) plots the above described data points. This type of plot is very instructive. Try comparing the tornado for a 12 inch f/8 versus a 12 inch f/4. The tolerances are considerably different. Also note that the tolerances allowed for the center are considerably larger than those for the edge. As Robert Miller^{(2)} put it: "a radius defect at the very center throws light into the center of the diffraction spot no matter how severe, but a turned edge reflects light outside the spot." One final note, a sphere would plot as a straight horizontal line across all zones. Check out the tornado and see if a straight line would fall within it. If so, you can skip parabolizing and stick with a sphere. Of course, not all mirror geometries will accept this shortcut!

The M-L graph produced by the Tex.exe program is a parabola-removed M-L plot. The ML.XLS spreadsheet automatically calculates a parabola-removed plot in addition to the standard plot. Just select the the Millies-LaCroix Chart (parabola-removed) tab at the bottom of the chart area.

- Open the spreadsheet in Excel 97.
- Don't enter anything into blank or "greyed-out" cells as some contain hidden numbers & formulas
- To choose your units of measurement, enter "inch" or "mm" (no quotes) as appropriate into cell F1.
- Enter your mirror's diameter in corresponding units into cell B1.
- Enter your mirror's focal length in corresponding units into cell B2.
- Enter "m" or "f" (no quotes) into cell B4 to denote whether you are using a moving-source or fixed-source tester.
- Choose the number of zonal readings you will be taking into cell F2. The spreadsheet will calculate a table of values. (By the way, the graphs will look weird with less than 10 zonal readings, ignore the spurious data points.)
- Enter a set of readings into cells G8 through G17.
- (Optional) Enter a second set of readings into cells H8 through H17.
- Click on the desired chart tab at the bottom of the screen to change to Chart view. The Millies-Lacroix graph for your mirror is plotted automatically.
- If your readings all fall between the Upper and Lower limit lines, your mirror is sufficiently corrected and you can stop figuring. Optionally, you can enter an offset factor in cell I7 to shift your plotted values up (enter negative number) or down (enter positve number) to best-fit within the upper/lower limit tornado. Doing so does not affect the interpretation of your mirror's figure.

References:

^{(1)}A Graphical Approach To The Foucault Test, by Adrien Millies-LaCroix, Sky & Telescope Magazine, Feb. 1976, p 127.^{(2)}Stalking The Wild Parabaloid, by Robert P. Miller, Telescope Making magazine, issue 6 (Winter 1979/80), ISSN 0190-5570 (out of print).^{(3)}Testing Paraboloidal Mirrors, by Dick Suiter, Telescope Making magazine, issue 32

Footnotes:

^{A} - On 12/23, I updated the spreadsheet to support shapes other than paraboloids. I did this by including the coefficient of deformation, as described by Jean Texereau, How to Make a Telescope, 2nd Ed., pg 77. (The spreadsheet requires that this value be entered with an opposite sign as what Tex specifies.) This change necessitated changing the formula r^{2}/R to b(r^{2}/R + (h^{4}/2R^{3})). In most cases, the difference is negligible.

^{B} - p is supposed to be "rho" but HTML won't let me easily do greek letters.

^{C} - Value for L taken from Testing Paraboloidal Mirrors, by Dick Suiter, TM Magazine issue 32.

This document, its contents, and its Web representation are Copyright ©1999, Tim Poulsen. For complete copyright information, including allowed uses of this FAQ, please see Section 8. Initially created on Sunday, February 7, 1999 by Tim Poulsen, poulsen@netacc.net.