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FOUCAULT-PROGRAM
reed this notice

To be able to understand how a mirror is tested,
it is opportune to know before how it is made.

It is given therefore for discounted
the acquaintance of the optical geometry relative to the optical mirrors,
of the Foucault-method, of the apparatus of the same name, of the Couder-masks,
and of the techniques in general terms, as well of figuring as of testing.

I send back therefore to the
literature on the argument.

That one of Foucault is the classic-method for mirror-testing.
It consists in dividing the mirror in several concentric zones, and measuring then the radii of curvature of each zone, evidencing them with a special mask, called "Couder mask"
(whoose aspect can be seen in the program), and comparing then the values you find with the theoretical values, calculated by the program itself.

In order to carry out the measures is necessary a special device, called "apparatus of Foucault"

This consists in a special device able to move itself in longitudinal and transverse sense (like the undercarriage of a lathe).

The apparatus is equipped with a punctiform or threadlike light-source and with a cutting knife, whose purpose is to intercept the light reflected from the mirror under testing and to cut it in whose center of curvature, where the light-source and the knife are placed.

According to the way in which move the light-source and the cutting knife (together or separately), two types of apparatuses of Foucault can be constructed:

• A type in which only the knife moves, remaining the light-source fixed;

• Another in which light-source and cutting-knife move together.

The present program is previewed for this last type of apparatus, that is called "with the star loyal with the knife" (the "star" is the light-source).

Purpose:
Test of concave objective-mirrors (spherical, elliptic, parabolic and hyperbolic),
in the figuring-phase or at completed job.
Superfluous to add that with this program can be tested not only the home-made mirrors, but also those produced from the industry, that can sometimes have rather serious defects.

Display:
images, diagrams, tables and sounds.

Functions:
the program suggests to the operator, leaving him the possibility of choise:

• the more convenient zones-number of the Couder-mask (that can vary from a minimum of 3 for a small mirror, to ten (in this program) for a mirror of one meter diameter);

• the external radii of every single zone of the mirror;

and asks to insert with an input the data obtained from the reading in the apparatus, that is equipped to this need with a scale, able to measure longitudinal movements (variations of distance between the source and the mirror) of the order of cent of millimeter, with the aim to measure the exact radius of curvature of each zone (superfluous to add that, while a spherical mirror has the same radius of curvature for each zone, an aspheric mirror, as for example a parabolic, or in its figuring phase, has a different radius for each zone).

Results:
They are supplied in numerical shape (tables) with three decimals and in diagram shape.
The curve that appears in the diagram is a broken line,
formed from so many segments how many are the zones, and is compared with a reference-line (horizontal) that is no other than the bisector of a family of curves (of parabolas, in the event of a parabolic mirror).

In the ideal event of a perfect mirror, the diagram-curve would be coincide with said bisector
(a parabola with an infinite radius of curvature), but in the practical it moves always away from such curve, as more as great are the defects of the mirror.

The program indicates:

• if there are errors;

• where they are;

• and how big they are;

and gives the precision of the mirror, taking as unit of measure the wavelength of the light (λ),
both on the glass surface and on the wave-front, for λ=0.56 microns
(560 nanometers).

The diagram comprises a scale, whose unit of measure is exactly the wavelength.

From the examination of the diagram, can moreover be easy deduced the measures to be adopted in order to correct the errors and to bring the mirror to its perfection.

Suggestion:
With theoretic aim, or of simple curiosity, very instructive experiences can be made with this program, in order to realize which, neither Foucault-apparatus, nor Couder-mask, nor the same mirror are necessary !!!

For example, one can test as parabolic a virtual spherical mirror. Try with a 100 mm  f/4.
The result will indicate the presence of an error ( the maximum possible error ),
not being  necessary in this case neither the reading of the zones, nor the presence of any apparatus nor the mirror itself, since in this case you would know a priori that the reading would be always equal to zero (zero means, that there is no difference under the radii of curvature of the various zones of the hypothetical mirror and between these values and thouse you would find). That is obvious, being in the case of a spherical mirror).

This virtual experiment, among other things, allows to insert a greater number of zones (as 10, for example),
more than those one can insert in practical, obtaining so a line that, more than a polygonal line, seems a continuous line.

The program also asks, to insert with an imput a coefficient, called Y. The value of Y (1) serves in order to render more evident, in the diagram, the presence of errors, amplifying the scale of the diagram as much as necessary. In contrary case, the errors could pass unnoticed.

Try rather with a low Y-value (for example Y=5), increasing it subsequently until the presence of errors is turned out. The result is very instructive!

APPENDIX
( further details about the “Foucault” )

In order to understand correctly the images that we see by means of the Foucault tester, it is necessary to know beforehand how they are formed.

We know about the theory that states that a parabolic mirror concentrates in only one spot a light beam parallel to its optical axis, as the one that comes from a virtually infinite distance.

A spherical mirror, on the contrary, concentrates in only one spot the light coming from its centre of curvature, what is rather obvious, if we think that its surface is part of a sphere, that returns to its centre the whole light beam coming from its own centre. Since every light ray is perpendicular to the sphere, if we look at a spherical mirror from its centre of curvature we see it flat.

But, how do we see a parabolic mirror from its centre of curvature?

It is obvious that it is seen deformed. The secret is to know what it is like. It is seen deformed the same as a spherical mirror would be seen, if we could see it from an infinite distance!

But there is a slight difference, while the spherical mirror has only one centre of curvature, the parabolic mirror has infinite!

In practice, fortunately, these centers are all very close to each other, and all of them on the same optical axis, and we can look at the mirror from three different points: from the closest, from the furthest and from one that is in between.

So it is very easy to know what type of mirror it is:

• If, when seen from its centre of curvature, it appears plane, it means it is spherical.

• If it is not seen plane, it is non-spherical (but not necessarily parabolic).

Letting aside coarse mistakes, the image that we will see could be one of an elliptical mirror, of a semi-parabolic one, or even of a hyperbolic one.

But, if what we are doing is trying to make it parabolic, what we will probably see, is the image of a parabolic mirror, or in its figuring phase.

Nevertheless, the definite answer will be known only through a detailed analysis with the Foucault tester. By means of this device it is possible, for example:

• to analyze a spherical mirror as spherical, and the result will be in this case the one of  a perfect mirror.

• to analyze a spherical mirror as parabolic (what is done changing the deformation coefficient with the INPUT), and the result will be in this case a huge error (the maximum error possible).

Going back to previous words, that is to say, how to look at a mirror, the results will be the following:

•  If we look at it from its closest centre of curvature, it will appear as a dome;

• If we look at it from its furthest centre of curvature, it will appear as a crater;

• And if we look at it from an intermediate point, it will appear as it is seen on this photo
(in this particular case it is about a mirror with a central hole, for a Cassegrain configuration).

More precisely, the image on this photo is the one that will be obtained when the mirror is seen from the centre of curvature of the zone that is at 70.6/100 from the centre of the mirror itself. In a 200 mm diameter mirror, for example, this zone would then be the one of 70.6 mm of intermediate radius. It is not necessary to say that the image in itself (no matter how nice its aspect might be) there is no guarantee of a good parabola. This will only come from a precise analysis, measuring the radius of each zone with an approximation of a hundredth mm (these zones are infinite, but in practice three to ten can be enough, according to the mirror’s diameter and/or relative aperture. The zone-number is suggested by the program itself, and it is advisable to accept it).

The secret of Foucault’s method is that, in spite of the fact that the mirror under control is frontally-illuminated, the same is seen as if it were side-illuminated, highlighting the most minute defects. This method is so sensitive that, if it were necessary to use a microscope to highlight such minute errors, the same would need a magnification of no less than one million times!

But, how can we know if the centre of curvature from which we are looking at the mirror is the correct one, i.e., the centre of curvature of the zone that is at 70.6 hundredths of the radius of the glass disc?

It is very simple: by means of the Couder-Mask.  If in a small mirror, of 100 mm for example, we use a screen of four zones, that are more than enough, as it can be see in this drawing, the point "70.6" will be the intermediate point of the third zone from the centre, and we will know with certainty that we are in its centre of curvature at the moment of seeing the two windows of that zone (left and right) equally illuminated.

For a proper appreciation of that value, it is convenient to perform many readings, as 12 for example, rule out the maximum and minimum values, and average the remaining 10.

During the figuring phase it will be necessary to take the same measurements many times. If we draw the graph, we make a print, cut out the paper following on one side the segmented line and on the other the reference line, we make a symmetric copy equal to the cut out piece obtained and we cut out again other 5 pieces of paper equal to the one obtained in the last operation, we will have 6 cut out pieces of paper with which we will be able to form a six-pointed-star.
Placing that star on the pitch, and the polished mirror on the star, the pitch becomes depressed in correspondence with the paper.

Upon removal of the paper, and going on with figuring, the depressed part, for not making contact with the mirror, will not work, and the error represented by the graph should be corrected automatically!
Necessary requirements: good pitch, some experience, and of course a lot of patience.

If the result is good, you will be proud of telling your ... parab...!

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