Updated December 11th 2003
If you are amongst the lucky people who find that the finished mirror is almost spherical to start with, and has no major 'zonal' ridges or depressions - well, what can I say !? "Good for you! "
If that is not the case - then see later in this section for info on correcting zonal defects, and other nasty problems!
The difference in cross-section, for the average amateur mirror of 6 or 8 inch diameter is only a few millionths of an inch when the sphere and parabola are compared at the edge. In the case of the 8" F8 mirror discussed in this page's 'mirror' section the difference at the edge of the mirror is about 15 millionths of an inch. (15 x 10^-6) If they are compared by making the centers tangent and the edges intersect, the difference can be even less (about 4 x 10^-6 .)
The difference between the two curves is that the parabola is a little 'flatter' at the edges and a little deeper in the center. So we can selectively polish these two areas to 'parabolise' the sphere by using a different polishing stroke.
It should be obvious that to determine what action is necessary, you must be able to measure the existing profile of the mirror. Then if it is close to a sphere you can apply the modified stroke ("A" in the diagram) which is normally used to make it into a parabola.
This stroke is like two or three letter W's together - WW(W) - and is wider and longer than the normal stroke, and is done with the mirror on top. Starting at one side with the mirror on top of the tool and offset perhaps 3" or more, you push the mirror about 3" or so up and down at the edge - gradually increasing the length of the stroke as you cross the center - up to perhaps a little over 6" and then reducing it again as you move to the other side. The long stroke across the center tends to deepen the center and the wide sideways motion works more on the edge of the mirror to reduce the edge.
The time for doing this should be short - only about 10 to 15 minutes for the first session, and the mirror must make an even number of turns. Then let the mirror settle for at least 8 hours (for pyrex) - (possibly 5 or 6 for plate glass) and check the result. The mirror surface is heated during even 10 or 15 minutes of polishing and will take a long time to settle properly. If you check it too soon - it may seem that nothing has changed and you will perhaps work it some more. Then when it has finally settled you will find that it was overdone. So be sure to give it lots of time to settle - possibly 6 to 8 hours after each 'figuring' session. It is better to wait longer to ensure a correct measurement than to rush this process.
Before starting, the mirror and tool should be properly pressed to ensure good contact, and the polish should be mixed a little thicker than for normal polishing. Any correction, either parabolising, or fixing some zonal defect, should be done by degrees - the action of the tool is sometimes not what you expect it to be ! So do only a few minutes at first, then if all is going the right way, you can simply do some more until a good parabola is obtained.
When doing this you should mark a point on the edge of the mirror, so that you know where you start. After about a minute - turn only about a 5th or 6th of the circumference of the mirror, so that you do five or six steps to make one complete turn. Usually it is best to make only two or three turns and take about ten or 15 minutes total time polishing, for the first attempt at parabolising. At the end of the last turn stop a little way short of where you started so that you do not overlap a section which you have already done.
Another method - which some people may find harder to do (if so, stick with the 'w' stroke) - is to use a wide "lazy-eight' stroke, which is a figure 8 stroke but lying on its side, as in "B" in the diagram. The normal 'w' type stroke shown in "A" is generally easier to use. In each case the jagged line shows the track made by the center of the mirror.
Another method is to modify the pitch lap, by cutting the outer squares to form a 'star' pattern. We have never tried this method - so I do not know how well it works. Modifying the tool is not always a good thing - the full lap is not something to be lightly altered. It can be very necessary for blending defects in the mirror - and smoothing it out again if previous actions have caused a problem. The other possible method is to make a smaller lap (using a plywood disc as the base) - making it about half the normal diameter of the mirror. This can be used selectively to deepen the center first, using a long stroke with some two or three inch sideways motion, and then again to reduce the edge with a shorter chordal stroke across the edge, again using some 'spreading motion' so as not to 'dig a channel', until a good parabola is obtained. In this case - the mirror may be on the bench - and the small lap used on top - but an even number of turns around the mirror should be made.
The use of a small lap has some dangers - it can very easily produce grooves in the mirror - especially if used too long, and without adequate sideways motion to blend into the other areas of the mirror. If you are determined to make a mirror accurate to a 1/5th wave, or better, you may have to take the risk. Frequently - after using the small lap you may need to smooth things out again overall with the full-size lap. Too much use of the small lap can produce a rough appearance to the surface, and re-polishing for an hour or more may be required afterwards.
The choice of which method to use is up to the mirror-maker. There are probably yet more ways to do the job - but the above should provide a couple of ways that you can try. The small tool is often the method of choice with larger and much shorter focus mirrors, where it can depart from a sphere by a large amount (such as the 20" F5 mirror we made for our Club, edge difference= about 150 millionths")
The process of accurately 'figuring' the mirror can be time-consuming and requires lots of patience. The final guide to accuracy will be the time spent testing and graphing the results of each step you make in getting the correct shape. If all goes well, with patience and perseverence you'll wind up with a smoothly parabolised mirror of a quality which would cost you dearly if ordered from a company who will guarantee an 'observatory' or 'research grade' mirror, which would require hand-finishing and careful measuring to ensure the best of quality.
This is probably the main reason, apart from the challenge of learning a new 'craft,' for making your own mirror. There is always also the feeling of satisfaction - especially when looking through the finished telescope, of having accomplished something (which although a 'simple' process can present quite a challenge to complete successfully) There are many finished mirrors available at reasonable cost from various suppliers today. Careful attention to the various aspects of making, testing and finishing your own mirror though is still likely to cost less - and you have first-hand assurance of its quality if you are successful.
If you are reading this because you are making your own mirror - or intend to do so - I wish you lots of success.
Oh -Oh - it didn't come out right ! So this is real life ??
Well, these things happen - but unless your mirror is so far away from being a reasonable curve, whether it is too deep, has a raised hump in the center, or a deep 'hole' in the center, a couple of concentric 'ridges' or a turned down edge, you may be able to correct it by selective polishing !
These things are of course only visible if you are looking at the mirror with some kind of tester. If the knife edge position is set at the center of curvature, a perfect sphere will darken across its whole face as the knife edge enters the reflected rays of light. If it is elliptical it will show some form of an 'upside down dish shape," and if it has zonal ridges or depressions they will show as lighter or darker 'rings' which are concentric to the circumference of the mirror disc.
Making a complete measurement, using a Couder screen and plotting the result will show you where your mirror departs from a perfect sphere. You may even get very lucky and find that it is already close to being a parabola. (If you need a 'refresher' on this - see the section on Testing, back at "home-page")
In the diagram - 'A' represents the kind of shadow you would see with the knife-edge cutting the rays from the edge of the mirror.(Assuming that the light is coming from the left side across the mirror) Underneath is a 'profile' of the defect - greatly exaggerated. In 'A' the defect is a raised central bump. In 'B' the drawing shows a mirror with a depressed channel a little over half-way out from center. This is one of the worst defects you could find ! If it is fairly shallow it is probably best to check your polishing lap first - it could have oversize squares in some area, or perhaps was not properly pressed. Another possible cause is applying (unconsciously) pressure on one side of the lap or mirror during the polishing stroke. Whatever the cause - firstly try about an hour of polishing again - check the squares of pitch on the lap for evenness and that no squares are contacting any other, that the channels between them are clear, make sure the tool is really well-pressed, and check your polishing stroke carefully. If this doesn't fix it - then your only possibility is to make a small lap ( about 4") diameter and try to polish the areas inside and around the channel. Not a very nice problem to solve. Perhaps the best answer is to build a new lap (full size) make sure it's pressed and polish for two or three hours.
Well - after that - the next problem, a raised ridge as shown at 'C' is a 'piece of cake' !! Any raised ridge can be reduced by centering the tool (lap) over the point on the radius where the ridge is, and using a short chordal stroke, work around the mirror for an even number of turns. Once again - do this only for a few minutes, and then check the effect. Make sure that you are blending the action into the rest of the mirror surface by moving the stroke a little to each side. And once again wait for a few hours before making a new measurement. Any polishing - even for a few minutes can heat the surface of the mirror where the polishing action has been applied. If you test it too soon, it may appear to have done nothing - but a few hours later a careful measurment will show some difference. If you do it too quickly you may repeat the action and then later, find that you have done too much.
Experience is the best guide to making corrections, and patience is always needed to ensure the best results. Also the action of the lap for a short period of time can be a little unpredictable. The action may vary quite a bit - it may be dependent on the way you hold the tool or mirror and how much pressure you may apply during the strokes when polishing for a few minutes, and it can vary in effect from person to person using the "same" stroke.
If you find more than one defect - for example a central hump and a raised zone further out : The best way to approach this kind of fault is to attack the center hump first. Then you can deal with the raised ridge after the central area is corrected. If you try to do both at the same time you can make things more complicated. The action of the lap when used for just a few minutes can give varied results and you could wind up with three areas to correct afterwards. It is better to correct one part of the mirror first, then move on to the next. You may also, after making a correction of a raised ridge - find that you need to use the normal polishing stroke for a maybe a half-hour to 'blend' things together and smooth the surface.
Depressed zones, provided that they are not deep (that is they are not grooves) and are fairly wide can be corrected by reducing the zones on either side of the depressed one. Then the whole surface can be 'blended' again by the use of the normal polishing stroke. Sometimes this kind of defect may require the use of a small lap - perhaps only 2 1/2" to 3" in diameter. Some defects just have to be dealt with this way. Although the action of a small lap is fairly predictable - and certainly only affects a small area of the mirror on which it is used - over-use can cause more problems than just going back to 'square one' and using the full lap to polish for a couple of hours. (making sure that it is pressed and in good contact, and that your stroke is carefully done.
Almost all the people I have talked to about mirror-making mention the 'dreaded turned-down edge' Why this is so I really don't know. I don't think it happens to everyone - I've certainly seen lots of mirrors without turned edges.
With the knife cutting the edge rays of the mirror - this shows as a dark ridge on the right side of the mirror, with the rest of the disc having the appearance of a 'satellite dish' or a hollow 'salad dish.' The best indicator is the graph of your readings from the tester. To correct this you can try polishing with the lap centered about an inch to inch and a quarter inside the edge, using the chordal stroke. This, if the turn down is not too serious, will have the effect of 'blending' the edge into the rest of the 'figure' and will change the focus very slightly. Causes of a t.d. edge can be either too soft a pitch or maybe too wide a polishing stroke, or again possibly unconscious pressure over the edge of the mirror during polishing, or even too high a temperature in the work area.
In the end - experience is the main requirement, along with lots of patience if your mirror turns out to have any serious zonal defects. The only way to get experience is to persist - and spend the time necessary to find out what works for you ! If you have defects other than zonal ridges or depressions - such as astigmatism, where defects appear at different cross-sections of the mirror, your only choice is to go back to polishing for a few hours. Check the condition of the lap and the arrangement of the pitch squares. If all else fails - build a new lap and try again.
I hope the above helps you make a really fine mirror and GOOD LUCK