Starting Tex

Tex is started by typing ``tex'' at the DOS command line prompt. For those of you running UNIX, the program has been renamed texereau to avoid conflicting with the popular text formatting program called ``tex''. If you wish Tex to use a sphere instead of a paraboloid as the reference surface, you can add the ``-s'' option to the command line. To exit Tex, enter q to quit, or hit the interrupt character (such as control-C or BREAK).

Here is the screen you will see when you start running Tex. All the screen shots shown were taken of Tex running on a Linux system. All the graphics shown were produced by Aladdin Ghostscript.

Selecting Units

By default, Tex uses inches. If you wish to use millimeters or centimeters instead, now is the time to do it.Remember that the units must be set before the data is entered, and that all measurements must be in the same units. To select different units of measurement, select u from the menu.

Entering Mirror Data

File or Keyboard

Next, enter the data for your mirror, by selecting d from the main menu. You will be asked whether you wish to enter the information from the keyboard or from a file. The first time, you will enter the information from the keyboard, so select k. After you have entered the information, you can save it in a file, so you don't have to type it in again.

Entering the Numbers

The information I'm entering here is the sample mirror from the book ``How to Make a Telescope'' by jean texereau. The program is named after this author, and its workings closely follow the algorithms and examples found in this book.

The lines from the k on up are from the previous screen. Note that on each line, the default value is shown in square braces ``[ ]''. If you want to use this default value, simply hit ENTER without typing a new value.

As a comment, to identify the mirror, I have entered Texereau's Mirror.
The optical diameter is the part of the mirror that forms the image. It should be the diameter of the blank, minus the size of the bevel on each side. Texereau's sample mirror was an 8'' blank, with a 0.04'' bevel on each side, so I entered 8'' - 0.04'' - 0.04'', or 7.92''.
The radius of curvature for this mirror is 97.06''. Note that for a parabola, the radius of curvature is twice the focal length, so this mirror is about an f/6.
Next comes the number of zones used to test the mirror. If you haven't actually tested the mirror yet, and you're not sure how many zones you want to use, accept the default and move on. You can make a Couder mask with the default parameters and try it out, and change the number of zones later.
How many readings are there for each zone? Again, if you haven't tested the mirror yet, just take the default.
Does your Foucault tester have a fixed or moving light source (is the light source attached to the part of the tester that moves)?
Were all the readings taken on one diameter of the mirror, or did you (as Texereau suggests) take readings along one diameter, then rotate the mirror 90 degrees and take readings along this second diameter?

The following entries are to specify the openings in the Couder mask used to test the mirror. Currently, Tex assumes that the inner radius for each zone is the same as the outer radius for the previous zone. Tex does not currently support direct entry of data for pin sticks. If you use a pin stick instead of a Couder mask, you will have to fabricate Couder mask zone radii such that the midpoint of each mask opening (outer + inner)/2 is the position of the pin for that zone.

The inner radius of zone 1 is to allow you to omit testing the center of the mirror. Some people feel they cannot accurately test the center, so leave out a small circle at the center of the mirror, figuring it'll be in the diagonal's shadow anyway. For Cassegrain primaries, the center will be cut out anyway, so there's no point in testing it.
The default values for the outer radii are computed to divide the remaining surface of the mirror into zones of equal area. I have entered the numbers from Texereau's worksheet, which are close (but not identical) to the default values.

If you made a mistake, simply answer no to the confirmation at the bottom of the screen, and you will be given a chance to change the values you entered. After yo uenter the mirror data, you may want to save it in a file. To do this, select s from the main menu.

Generating a Couder Mask

Choose option p from the main menu to generate a PostScript file that can be used to print a Couder Mask on your computer's printer. The PostScript code to generate the mask will be saved in a file, by default, couder.ps. This file can be printed on any printer that understands Adobe PostScript. If your printer does not, you can get a free copy of Aladdin's Ghostscript program, which will take the PostScript file and produce output appropriate for whatever printer you do have. Tex asks for the outside diameter of your mirror blank. This will be used to draw the outline of the blank, so the mask can be properly centered on the mirror.

The couder screen will look something like this. Cut out the openings carefully with a razor blade, and trim the edges. For anything but a small mirror, you'll want to take cardboard, cut an opening in it large enough for all the mask openings to show through, and attach the printer paper to the front, so the carboard holds the paper flat and stiff.

The reason for using PostScript for the Couder mask, is that it can adapt the image to the printer ``on-the-fly''. The result is that the mask is printed to an accurate size and scale regardless of the printer or its resolution, and you can print Couder masks up to twice the longest dimension of the printable area of your paper. For 8.5x11'' paper, you can print masks for mirrors of up to about 21''. With U.S. Legal size paper in the printer, you can print masks for mirrors up to about 33''.

Entering Readings

This screen should be self explanatory. Note that the readings don't need to be ``normalized''.

Understanding the Results

After you enter the readings, the mirror data is automatically calculated and saved in the file mirror.txt. If you want to recalculate this data, select c from the main menu. The data shown here mimics that of Texereau's mirror worksheet.

Line 2 h(x) shows the Couder mask boundaries
Line 3 h(m) shows the midpoints of each zone, which should correspond to the pin positions, if you used a pin stick.
Lines 6 and 7 D1 and D2 are the average of all the readings for each zone on each diameter.
Line 14 is the error at the wavefront measured in microinches (assuming you used inches as the unit of measurement).
Maximum waverfront error is the wavefront error computed from the average of all sets of readings, using the wavelength of green light (21.6 microinches) as the standard wave.
If you entered more than one set of readings, Tex also prints the maximum wavefront error for each set of readings. This can help you determine how closely your readings correspond to each other.

Graphing the Results

Selecting g from the main menu produces a graph of mirror results like that shown below. Under MS-DOS, the graph appears directly on the screen. For other platforms, the graph is saved as a PostScript file, which can be viewed using Ghostview and Ghostscript, from Aladdin.

The top graphs shows the relative surface error of the wavefront in wavelengths of green light. Remember that the surface errors of the mirror are in the same direction, but half as big as the errors shown here. The zone boundaries are marked for reference. This graph is to help decide from what part of the mirror to remove glass, so you can alter your figuring accordingly.

The bottom graph shows the Millies-Lacroix tolerance (marked in cyan). The mirror's aberration is graphed by the bold black line. If the line goes outside the area marked by the cyan lines, it does not meet the requirements for a telescope mirror.

Using the Monte Carlo Algorithm

The Monte Carlo algorithm is designed to reduce the largest source of errors in making a Foucault test - the inaccuracy of the readings. It does this by looking at several sets of readings you've taken, and generating thousands of sets of readings that have the same statistical distribution as your readings, then computing the wavefront error for each of the generated sets of readings and plotting the results. The more consistent your readings are, the higher the probability that they are correct.

You can play with the deviations or number of simulations, but in general the defaults should be correct for your input readings. Since the calculations can take a while, a progress bar is displayed (the line of +). The resulting graph looks like this. It shows that with 80% confidence, the mirror is at least as good as 1/11 wave. Likely values are spread out from 1/16 to 1/19 wave, and it is very likely that the mirror is not as good as 1/23 wave (the red bar at the far right).