Parabolizing

Criteria

As discussed before, for our mirror of 150 mm parabolizing only has to be done when the focal ratio is 8.2 or less.

 Diameter in mm Focus /f 75 100 125 150 175 200 250 300 472 700 975 1230 1505 1800 2400 3120 6.3 7.0 7.8 8.2 8.6 9.0 9.6 10.4

In the above given table the minimum focal lengths and focal ratios are given for various mirror diameters still allowing good observation results with a purely spherical shape. The figures have been calculated for the first time by Lord Rayleigh around 1900 for determining the accuracy of objectives. The calculations have been done for a maximum aberration of 1/4 lambda (wavelength) of the optical surface of the mirror. For focal ratios smaller than the ones given, the lightbeams are no more focussed in the same point. On the other hand, for a paraboloidal surface a beam coming from an infinite distance does focus in one point. We can imagine a paraboloid as a merged set of spheres with different radii, with the result, that the radial point (i.e. the point on the axis where an outgoing beam exactly reflects back) for the centre of a parabola is shorter than the one for an area near the rim. This is illustrated in the next figure.

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Normally a freshly polished mirror is perfectly spherical and should be of reasonable quality. However, the laws of optics require a parabolic shape. Whether this is to be realised or not is entirely the choice of the maker. If you already had to turn back to #1200 powder or coarser, because of scratches or other flaws, we can well imagine, that you had enough of it. In such a case you better take a break and do the job of parabolising at some other occasion, for instance after a few years when the time has come for aluminising the mirror anew. If, however, everything goes well and you still are in the mood for polishing further, keep reading the following pages, because the most captivating but also the most difficult aspect of polishing mirrors is now to come!

Parabolizing

Before starting the parabolizing procedure the pitch tool and the mirror should be matched well by pressing both together. The patterns on the pitch should be nicely square and eventually we improve the incisions for a last time. Some amateurs cut the squares at the border a little bit down to avoid a sagged-down edge, especially in this last stage. The parabolizing itself takes only a few minutes, but the optical measurements and the interpretation will take 'somewhat' more time.

We apply a good portion of cerium oxide slurry and polish with a stroke length of 3/4 of a diameter along a W pattern, also of 3/4 of a diameter. Instead of the usual 8 strokes we already turn the mirror and the tool after 2 strokes. In this way we deepen the centre of the mirror. We make about 30 strokes a minute. With cerium oxide about 50 strokes are necessary.

Some experienced polishers use 'rouge de Paris' instead of cerium oxide, because this results in a better and more even surface. In that case the number of strokes should be doubled, because this polishing material is less abrasive. While this procedure takes a few tenths of microns away from the centre, it almost has no effect near the rim. Now the mirror is thoroughly rinsed and installed on its cradle in the foucault testing device. We wait until the mirror has reached temperature equilibrium. If the image is as can be seen above we can be quite satisfied, since this figure is a typical example of a parabolic mirror with a small focal ratio.

In most manuals, books or periodicals, also in the past in our observatory, the centre of the mirror was taken as the starting point of the measurements. These were performed as described here, but instead of the 35% zone there was a 30 mm large hole in the centre of the zone-disc. This centre area was used for determining the radius as accurately as possible. This value was taken as a starting point for calculating and measuring the radial distance at 70% and 90%. However, for most beginning amateurs the accurate determination of the radius on an opening of only 30 mm was almost impossible. It is worthwhile to try it as an experiment: cover your mirror with a piece of cardboard with a 30 mm hole in the centre and start measuring. I wish you good luck!! Some beginners already have much trouble for determining the radius of a complete mirror, so what would it be for a small centre zone of 30 mm from the same distance! In our observatory workshop we found out a trick to elude (circumvent) this problem. In stead of taking the centrezone as a starting point, we choose the 35% zone because it can be probed more easily and more accurately. Consequently in our workshop the radius determinations are done for the 35%, the 70% and the 90% zone. For these measurements we need a 'zonedisc' as described here.

 Eerst worden de twee zonekaarten verstevigd door ze op een stuk karton te kleven. Daarna worden ze beiden met een film- of Stanleymes uitgesneden. Bij zonekaart 1 : lipje niet wegsnijden (dient om gemakkelijk rond te draaien) Vervolgens worden alle zwarte gebieden netjes uitgesneden.  Daarna worden met een papiersplitpennetje de twee zonekaarten precies in het midden met elkaar verbonden. De twee kaarten wordt voor de spiegel gezet met de uitgesneden zones van zonekaart nr 2 precies horizontaal. Klik op de kaarten om ze in Corel formaat te downloaden. (De zonekaarten worden enkel gebruikt om de spiegel te paraboliseren. Dus NIET bij sferische spiegels)

We place this double disc against the mirror with the apertures in the rear disc horizontally, the front disc being turned as to match its 35% apertures with those in the rear disc. We arrange the Foucault tester so that we can see both 35% zones. Then we adjust the foucault tester until the darkening of both zones occurs exactly simultaneously when cutting the knife into the reflected light beam. The handle for shifting the tester in the axial direction of the mirror is equipped with a vernier allowing reading the distance from an arbitrary zero point with an accuracy of 0.1 mm or better. After notifying this reading, the handle is turned away and the measurement is done once more, 5 times in total. The 5 figures are averaged. Next, we turn the zone disc as to match the 70% apertures. Once again: be careful, that those apertures are positioned precisely horizontally. While manipulating the disc the mirror should stay put. The slightest disturbance of the mirror makes all the former measurements useless. In the same way the radius of the mirror is measured 5 times with the 70% zone. The 5 thus obtained values are again averaged and we record the difference with the 35% zone measurements. Now we do the same for the 90% zones and we compare the experimental results with the calculated values.

 Hier wordt de 35% zone 'onder de loep' genomen.  We zien dat het linkergedeelte reeds volledig verduisterd is en een fracie later zal het rechtergedeelte zwart worden. Maw het mes staat achter de 35% van de parabool.  Het mes moet een beetje terug naar voor. Bij deze fraaie opname zien we dat het mes perfect in de 70 % zone staat.  Beide openingen worden op exact hetzelfde moment door het mes verduisterd. Bij de 90% staat het mes net iets te veel naar voor(naar de spiegel). We zien duidelijk dat  rechts al gedeeltelijk verduisterd is en links is nog helemaal niets te zien.  Bij een 30 cm f/4 spiegel kan de afstand tusen 70 en 90% zone al gauw oplopen tot enkele millimeter

Calculations

We can calculate the correct values for the three zones of our mirror (15cm, f/8.2) as followings:

 Correctie =  r˛:R Correctie = straal v/d spiegel˛ : straalpunt
 35% zone = 26.25˛ : 2460 = 0.28 mm 70% zone = 52.50˛ : 2460 = 1.12 mm 90% zone = 67.50˛ : 2460 = 1.85 mm
 Al deze berekeningen zijn enkel voor een stationaire lichtbron..!

Those values are calculated starting from the centre zone. Since we take the 35% zone as the starting point of our measurements we must reduce this value to zero, which means that we must subtract the 35% value from the two other ones, in order to obtain the neat difference between the 35% and the two other zones. The radius point of the 70% zone is therefor situated 0.84 mm behind the one of the 35% zone. For the 90% zone we find 1.75 mm.

The corrections

A comparison of our experimental values with the calculated ones enables us now to make a correct evaluation of how to proceed. The longer we parabolize, the nearer we come to a perfect parabola. However, when we push this correction too far, the mirror will no more have a parabolic shape, but a hyperbolic one. In this case it might be necessary to correct the other way. This is why it is better to stop in time, do some measurements and, if necessary, continue parabolizing.

The experience you acquired during polishing will be very useful for parabolizing. The amount of glass to be removed during this procedure is of the order of one wavelength, i.e. few tenths of a micron. Therefor extreme caution is required.

 Tolerantie in %
 Diameter mm 100 150 200 250 400 600 900 f/4 f/5 f/6 f/7 f/8 f/10 f/12 8.2 16 28 44 66 128 222 5.5 11 18 29 44 85 148 4.1 8.0 14 22 33 64 111 3.3 6.4 11 18 26 51 89 2.1 4.0 6.9 11 16 32 55 1.4 2.7 4.6 7.3 11 21 37 0.9 1.8 3.1 4.9 7.3 14 25

When, after a control session, we did not yet reach the figures required for a correct parabolization, we should proceed. For doing so you should apply a good amount of cerium oxide slurry and cold-press the mirror onto the pitch tool until you are sure that it makes perfect contact. The number of strokes still to be given can be estimated from the last measurements. When you are not far from the calculated figures you either carefully can give a limited number of strokes or you just could stop it. You can see in the table given below whether you are within the limits of tolerance. These limits depend on the diameter of the mirror and on the focal ratio. If you want an as good as perfect mirror you should get within these tolerance limits. For our mirror of 150 mm diameter and f/8.2, taken as an example, a quite generous tolerance of 44% is allowed. This is because we have a border case here: the difference between a sphere and a parabolic is very small, especially when the focal ratio is rather large. In our example a deviation of plus or minus 0.49 mm is allowed for the 70% zone and 0.81 mm for the 90% zone. This means that the experimental value of the 90% zone should fall between 1.04 and 2.66 mm!
I you went too far, i.e. if the experimental values are larger than the upper limit (calculated value plus tolerance) you should stop giving long strokes and instead make 1/3 strokes as was done before for obtaining are spherical shape. You also could switch the mirror and the tool, because with the tool on top more material is taken away from the rim. This procedure proceeds faster but is more risky for overcompensation.
In the figure on top of the preceding page you can see how the mirror looks in the Foucault test when the knife cuts the centre zone (see left), the 70% zone (centre) and the 90% zone (right). As you can see only the 70% view allows you to admire the parabolic shape. In all three cases the light source is at the left.

Enkele fouten

These drawings give an outline of aberrations frequently occurring during parabolization, together with some methods to correct them.

 Deze parabool heeft een opstaande rand(komt zelden voor).  Alvorens de rand weg te werken even de pek persen en met korte slagen deze fout wegwerken.  Indien de fout te groot is moeten we terug  viltjes kleven (afhankelijk v/d grootte van de rand) kleven we nr. 7 of  8.  Zeer regelmatig draaien om microrippel te voorkomen. Deze parabool is in het centrum niet diep genoeg. Om deze fout weg te werken gebruiken we de standaard slag om te paraboliseren maar zullen (met de spiegel boven) iets langer op de rand van de pek polijsten.  Zéér regelmatig meten.  Na elke meting de pek even persen. Deze parabool is in het centrum diep genoeg en de rand is ook correct.  Om deze ringvormige fout weg te werken moeten we viltjes kleven nr.10, of met een kleine pek werken en een derde mogelijkheid  (enkel voor ervaren handen) is de pek bijsnijden.
 Tijdens het paraboliseren of corrigeren NOOIT te lang na elkaar polijsten en alvorens een nieuwe sessie te beginnen even de pek persen...!