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Grinding, Polishing and Figuring
Thin Telescope Mirrors
Part 2 - Polishing
Abridged from an article appearing in Telescope Making #13
Provided: Courtesy of Astronomy Magazine
Article's Author: Bob Kestner
I am going to start with some additions to topics covered in part 1, on grinding mirrors.
First a few cautionary notes on segmented tools. I Should have mentioned that a small amount of Ivory dish soap added to the fine abrasive gives you a smoother action if you have a segmented grinder. I have even heard of tile grinders sticking like solid glass tool if soap is not used. In working the mirror with the final abrasives, that is, those smaller than 9 micron, scratching can be a serious problem especially with a segmented tool. The edges of the tile seem to dig into the glass. If you run into this problem, there are several things you can try. Go on to grinding with your final abrasive right after grinding with the abrasive before -- just clean up and continue working. If the problem occurred because the tool was warping when it sat for a time, this will help.
Preventing scratches: Micro Grit abrasive is available mixed with talcum, which helps prevent scratching. It is signified with a T after the micron size number, for example 5T, for 5 micron with talcum. If you are having scratch problems in 3 micron, try finishing the mirror in 5 micron. Don't finish with anything coarser than 6 micron or 305 emery. You might also try using the abrasive in a thicker slurry, like cream, or adding more soap, or both. A last resort (and one that I have never tried, but I suspect it would be effective) is to bevel all the tile edges on your grinder. If nothing works and the scratches are not too numerous, forget them. They will only flaw the mirror cosmetically.
Holding the mirror: Don't let your fingers touch the bevel of your glass while rotating the mirror during grinding and polishing. When you rotate the mirror during it closer to the center of the edge. Touching the bevel heats the glass, causing to expand. Then you grind or polish the raised edge off, resulting in a turned edge when the mirror cools down.
Turned edge problems: These can be a problem with any type or thickness of glass. It's by no means restricted to thin mirrors - and it's just as often a problem with full-thickness pyrex mirrors. Most amateur mirrors in the 16"-plus size range from this problem - but it often goes unnoticed despite its obvious effect on the images in the telescope.
Ordering sheet glass: Sometimes a rough round blank turns out to be longer across one diameter than the others by more than 0.25". Usually they scribe a circle on the glass, then cut up to it. If they do a sloppy job, the blank could be out of round. I'm not sure to what degree it effects the mirror, but it seems likely to cause astigmatism. (The problem is similar to figuring an elliptical diagonal without blocking it with glass keepers on the sides.) Be sure to specify that you want a blank as round as they can make it, within reason.
Fine grinding: I change from my rough grinding area to my fine grinding and polishing area after 220, not before. #220 is still gritty stuff.
The "W" stroke: This is a casual way of describing any stroke that goes forward and backward while progressing left to right or right to left on the mirror. Take four or five back and forth strokes while moving the mirror from one side to the other. By "short stroke," I mean to hang over front and back left and right only 1 or 2 inches while the tool is on top. During fine grinding it is a good idea to vary the stroke length a small amount to avoid digging a zonal groove where the strokes start and stop.
Polishing begins with pitch, a viscous tar-like substance that serves as a carrier for polishing agent, usually a metal oxide in optical work. The pitch is coated onto, or formed into a tool that fits the optical surface to be polished. The type hardness, and behavior of the pitch are important in determining the action of polishing.
I won't go into methods of hardening and softening pitch because these are already well covered in many books for TMs. My own experience is that for a large lap, any hardness within reason will do the job. I started making mirrors with straight resin pitch with a very small amount of turpentine added. These laps were extremely hard. Lately I have used some very soft pitch laps. They work well too. Very hard pitch laps can be a nuisance because they tend to act up a lot and begin to cut sporadically. Soft laps cut evenly, but can lose their shape and need a lot of maintenance. Still, you're better off if your pitch is a little on the soft side.
The best pitch I know is Swiss black tar pitch, which comes in different hardnesses. Adolf Miller pitch #73 is what I recommend. Oddly enough, it is less expensive than many of the pitches available to amateurs.
A kilogram and a half (3.3 pounds)of pitch will make one 16" lap. I would order three kilograms (6.6 pounds) just in case I messed up a lap, and to have extra on hand for smaller figuring laps.
Polishing Agents
For efficient polishing, choose one of the many cerium-oxide-based polishing agents such as Vitrox C, which is good for polishing, or Rhodite 15, which I like for figuring. But don't go nuts trying to find these particular brand names. There are lots of good polishing agents available from many different suppliers sold under many different names. Most of them will be quite suitable to your needs.
However, stay away from "super-fast" polishing agents used for metal mirrors. These will cut much too fast for your needs on a glass telescope mirror. I don't think you'll have to worry about being sold any, since I don't know of any being sold outside the optical industry.
Your pitch lap requires a base, something to pour the pitch on. It can be flat, or it can have the radius of your mirror on it. The grinding tool is traditionally used by TMs. The base must be thick enough not to flex, and can be made of thick glass, aluminum - even plaster has been successful. What you choose is up to you. I have always used 16" or 18" X 1" thick portholes. They make excellent pitch laps. If you try plaster, I again recommend Kerr Plaster (mentioned in part 1) for its rigidity.
The diameter of the pitch lap, like the diameter of the grinding tool, is up to you to some extent. The closer the diameter of pitch lap is to the diameter of the mirror, the more likely it is to produce a spherical figure. However, I don't recommend everyone make a full sized lap because it's not practical. For starters, a full size base for the lap is not always available. There is also the problem of managing a big lap. Furthermore, the larger the lap, the more work is needed to push it back and forth on the mirror, and the harder it is just to lug around from one place to another.
Sub-diameter laps tend to overcorrect the figure of the mirror, tending to push a sphere toward a figure that is spherical in the center and parabolic at the edge. This is a manageable problem - even desirable at times. It may prove to be a new experience to TMs who are used to parabolizing from a spherical surface.
The smallest polishing lap (as opposed to figuring laps, which are much smaller) that I've seen used was on a 25" f/6 mirror. It was polished it with an 18.75'" lap - 75% of full diameter. The figure came out as described: with the center spherical and edges parabolic. It was corrected this quite easily using various sizes of figuring laps.
For a 16" mirror I would recommend a 16" lap unless a 16" base is not available. In a pinch, I would go to the 25% sub-diameter lap without anticipating too much added difficulty.
For mirrors up to 20", I still recommend the full size lap. Above 20", where laps usually become unmanageable for one person working by hand, I would recommend using a sub-diameter lap. But don't let this stop you, especially if you have a strong build and think you can handle a 24" lap.
Pouring a pitch lap for a big mirror can be quite a job if you've never done it before. If you are not careful, as well as a little lucky, you may end up doing it all over again. Pouring a big lap requires timing, and until you have done it, it's hard to get everything to fall into place. Once you have done it, successfully or not, you'll have some feel for it, and it'll become much easier.
Start by heating the pitch on a hot plate. A 5 pound coffee can is an excellent container to heat the pitch in, and it will hold enough pitch to make a 20" lap. Heat pitch slowly. One or two hours is not too long. If you try to heat it in 15, or even 30 minutes, it will probably overheat at the bottom of the can.
While it is heating, stir it now and again. Use the time to clean the pitch base so the pitch will stick firmly to its surface. Make a dam at least 1/2" high around the base with masking tape. Be sure that the tape is strong enough to separate from your base when you strip it. This usually is no problem, but if the tape tears when you strip it, it does create a problem.
Prepares the area, too. Place the base on something solid and level for pouring the pitch. Be sure to have a water slurry of polishing agent handy and a little Ivory dish soap or Windex ready. A propane torch and matches are useful but optional.
When the pitch melts, pour a little on something and let it cool to make sure it is as hard or soft as you want it. Before you pour, be sure the pitch is hot enough to insure it will stick to the base. Smear water and polisher with 2 or 3 drops of soap or a small squirt of Windex on the surface of the mirror. Then light the propane torch and set it safely aside. Start by pouring a layer of pitch on the base to a maximum depth of 1/4". Even if the base is convex, the pitch will flow into a more or less uniform layer.
At this point you should have about a quarter of the pitch still left in the can. Don't set the pitch back on the hot plate: let it cool a little for the next step. Just how cool to let it get is one of the tricky parts of this job. You don't want it so cool that you can't get enough out of the can, and you don't want it so runny or it will fail to serve its purpose. When I do the first pour, if the pitch is hotter than I planned, I set it on the cement floor. If the temperature was right, I set it on a wooden counter.
Right after your first pour, flash the flame of the torch over the surface of the pitch. This will pop all the bubbles on the surface and leave it smooth. Check the mirror to be sure it is still wet with soap and polishing agent.
When the pitch lap has cooled to just the right temperature (this you must learn by experience), pull the tape off the base. The pitch should just run over the edge a small amount. If it does not run at all, you are behind schedule. If the pitch starts running quickly off the base, you pulled the tape too soon. You can monitor the pitch before you pull the tape by just starting to pull the tape and see how fast the pitch wants to run out of the crack.
As soon as you pull the tape, slowly pour more of the somewhat-cooled pitch right in the middle of the lap. If everything is at the right temperature, the new pitch will form a convex crown on the lap
that will flow toward the edge. If there is time, flash this with the torch. By the time the crown reaches the edge zones of the lap, it should have cooled enough so it won't stick to the mirror. Grab the mirror, which should still be wet on the face with polisher and soap, wet its back so you can see through it, and lower it into the pitch.
Start lowering the mirror tilted slightly so that it contacts one edge of the lap first. Then as you set it on the lap, tilt it level with the pitch. It is very important not to lower it level on the lap because you may trap a large air bubble in the center. Move the mirror around on the pitch and keep moving it as the pitch cools. If you stop, it will stick. Keep working until the lap is in contact with the mirror all the way to the edges. If you have a full-size pitch base, the pitch will ooze out somewhat bigger than the mirror and the mirror may sink into the pitch. Push the pitch down and try to get the mirror moving around.
If you end up with a large space in the center, let the pitch cool, then separate the mirror, heat more pitch, and pour another crown. This is a common procedure for big lap and fast mirrors.
Finally, when the pitch is hard in the middle, you may stop and lift the mirror off. If you lift it off too soon, you will deform the lap when you separate them. Be sure the lap touches the mirror right to the edge. This whole procedure is done quite rapidly, although the timing depends on the temper and temperature of the pitch. If your lap base is glass, let it cool on its own after you separate them. Cold water may break it.
If your mirror is too heavy for you to pick up and turn over, place the pitch lap face down on the mirror instead of pressing the mirror on the lap. Again, let them touch at an angle to avoid trapping an air bubble which would deform the surface of the lap.
It is a good idea to keep a bucket of water nearby in case you get hot pitch on your hand. You can immerse your hand in it, then quickly get right back to work.
If the whole thing fails, you may be able to save it by heating up more pitch and pouring a layer right on top of what you have, then pressing the mirror on it. It is a good idea to heat the lap until it's tacky with the torch before pouring on new pitch.
To allow the pitch to flow and conform to the mirror surface during polishing, the surface of the lap must be cut into facets. One method for doing this is to incise the face of the solid lap with grooves. For pitch laps under 20", I use a groove spacing of 1.5", so I will suggest the same thing for you.
Start by marking the center of the lap by a small "X" cut with a single-sided razor blade. Arrange the squares off center with respect to the X. Take a straightedge and mark, with razor blade lines 1.5" apart, where the grooves will go. Gently cut the grooves into the pitch using single-sided blades, while running cold water on the lap. Cut down on one side of a line, then rotate the lap about 1/4" wide. This job is very messy and is best done outside, or in a large sink, or on lots of newspaper. Pitch sticks to everything, and it is very hard to get out of clothing. You have been warned.
Once the lap is grooved, you have to be sure it is "pressed" to conform precisely to the curve of the mirror. If you have a hard lap, warm it slowly in water or in the sun. Wet the mirror with polisher, then press them together. Put 40 or 50 pounds on top and leave them for a while. Keep them wet; if they dry up, they will stick together. If you have a softer lap, you can press it without heating beforehand.
When the mirror will not be polished for a considerable time, for example, after the day's polishing, keep the lap face down on the mirror with a single piece of waterproof paper covering the entire lap. Butcher paper works well for this if it is the type that is coated with plastic instead of wax. Be sure the lap is dry or it will wrinkle the paper.
Don't let the lap sit on the mirror for more than a day or it will start to press out. For periods of pressing between polishing spells in a single day, just remember to keep the mirror moist.
Polishing is the stage of mirror making that requires the most physical work, even more than rough grinding. What probably concerns you most is how much time it will take you to polish the mirror. Using the techniques described below, a 16" porthole takes about 10 hours to polish out with a full lap. A pyrex 25" finished in 3 micron abrasive takes about 12 hours on a lap 18.75" diameter.
For those unfamiliar with efficient polishing techniques, these times may seem much too short. If you’ve ever put in 60 hours to polish out a 12" mirror, you’ll suspect these mirrors must not be polished out properly, or their figure must be grossly defective due to the fast action of the lap. Neither of these is the case. I realize that many amateur mirrors never get perfectly polished out. My own first couple of mirrors suffered from this. I know how quickly a mirror can take a superficial polish that will look hazy when aluminized, but I’m talking about a real polish. These efficient polishing techniques produce an extremely even cut. Even though the friction is high, the work is done at a slow pace and produces a smooth surface texture.
Nor do I want to make these polishing procedures seem too easy. When I say 10 hours on a 16", that’s not two afternoons - that’s 10 hours actual time spent polishing. It is so physically demanding that I am only able to polish in 15 minute spells 2 or 3 times in an entire day.
You can not expect such short polishing times using a machine. Normal machine polishing techniques are not as efficient as hand polishing in these sizes. But don’t get me wrong about machine polishing: it is the only way to go. I work with them all day in my profession and precision optical production would be impossible without them. Just don’t feel that you’re at a disadvantage for making a mirror with your own hands.
To start with, mix your polishing agent with water. The amount of each to be used can vary quite a bit depending on how you like it, or on the brand of polisher you use. Start with 2 or 3 tablespoons to a cup of water. If you realize it’s too thick, add a little water later. After mixing, strain the mix through cheesecloth to remove lumps.
Use a squirt bottle to apply polisher and water. Squirt bottles come in many shapes and sizes. You can find suitable bottles in sporting goods stores - they are used by distance runners and other athletes - or make your own by putting a small hole in the top of a plastic bottle and turning the bottle over to squirt the polisher on the mirror. An additional squirt bottle filled with water is also helpful.
"Wet press" the pitch lap before starting. Squirt polishing compound on the mirror, then set the lap on the center of the mirror. Load about 40 pounds of weights on the lap and let it sit for 15 minutes. Be sure that it does not dry up. Take off the weights. Now you’re ready to start.
Begin polishing by moving the lap around to stir up the polish on the mirror. Add a little water if it needs it. Move the pitch lap slowly back and forth in your polishing stroke. I recommend the "W" stroke. Starting with a left overhang, push the lap forward over the front of the glass, then pull it back. Work slowly - each stroke should take one to two seconds. After four or five strokes, you’ll be over on the right. Continue the forward and back motion while progressing to the left side. At the same time, walk slowly around the barrel. Rotate the lap every now and then, or continuously, as you prefer.
The length of the stroke, and hence the amount of overhang, can vary from just a couple of inches to 5 or 6 depending on your circumstances. With a sub-diameter lap, avoid making the stroke too long or you may end up overcorrecting the mirror. With a full-size lap there is little danger of this, and you can use longer strokes.
How well the polishing lap works depends on several things. You can judge its performance by the "feel" of the lap. Most of all, you want the lap to be extremely hard to push. It should move only with a great deal of effort on your part, and then it should move slowly and smoothly across the mirror. This should cause blisters on your hands after a while. I recommend leather gloves, or tape on your hands in tender areas, or both. You may have trouble finding gloves that don’t slip on the back of the lap: I’ve found that snug gardening gloves work well when they’re a little wet.
It is during "tight" and "smooth" polishing that the mirror polishes quickly. The lap and the mirror are in full contact, you are pushing hard, the polishing action is generating heat, and the glass is polishing rapidly but evenly. Often amateur optical workers seem to be afraid when the lap begins to function correctly. They can tell that something is happening. So they stop and add more water, or more polisher - anything to make the lap slide easily again, This is the wrong thing to do: when it’s right, keep working.
Often the lap just slides around on the mirror with little or no drag at all. This loose and skiddy action is all wrong. Or it may slip and stick suddenly again, feeling anything but smooth. These problems often happen when the pitch is too hard, or too cold (due to your climate), or just cold from the night before. It may also be that the lap is not grooved well enough or has not been pressed in contact. Quite often laps will do this for no discernible reason.
The remedy that works almost 100% of the time is heating the lap. You can warm it by submerging it in hot water (about 43 degree celcius) for 5 minutes. Sunshine or a sunlamp also work well. Be careful no to heat it too much if you have soft pitch. You may have to heat the lap only once to cure it, and it may work well from that day on. You may have to heat up the lap every day you polish. Learn to love your pitch lap and cater to its idiosyncracies.
Another possible remedy is walking around the barrel in the opposite direction. For some reason, pitch laps seem to have a direction (either clockwise or counterclockwise) that they like to go. They also seem to change their preferred direction at any time. Experiment with the phenomenon as polishing progresses.
How much pressure? Lots: a slipping lap may result from lack of pressure on your part. Lean on it!
The opposite can happen also. The lap may be working so well that one person can’t push it. Usually reversing the direction you are rotating things will help. Walk around the barrel the other way, turn the mirror the other way, turn the lap the other way, etc. Making the polisher slurry either thicker or thinner may work. As a last resort you can lubricate it with 2 or 3 drops of Ivory dish soap in your squirt bottle. In any case, you want to get the lap working right or your mirror will take many more hours to polish.
Once you get things working right, you will have to experiment with different ways of holding the lap to push it back and forth. I always grab the lap over the far edge at 10 o’clock and rest my forearms across its back. This allows me to put my weight on the lap and also gibes me a smooth transition between forward and backward strokes. Another method is to put one hand on the far side of the lap at 12 o’clock, with that forearm resting on the back of the glass, and the palm of the other hand on the near side of the lap at 6 o’clock. It does not seem to harm anything if you have to hesitate between reversing the forward and backward stroke.
Don’t push the lap rapidly back and forth. It should take 1 or 2 seconds to complete one stroke. This may sound fast, but it is actually quite methodical.
Squirt polish or water on the mirror when it starts running dry. Be sure to rotate the mirror every time you complete on revolution around the mirror. Rotate it in the opposite direction from the direction you are walking. Vary the amount of rotation a little each time, and vary the point at which you stop to rotate the mirror.
Don’t try to do more than 15 minutes at one time or may be end up hurting yourself. I recommend you do about 15 minutes, then rest for 30 minutes, then do another 15 minutes, then quit for the day. After you get used to it, you can determine your own pace.
Don’t let all this deter you. Once you master it, you will think nothing of setting up a routine of polishing 2 or 3 times a day for 15 minutes and you’ll get a 16" mirror polished out in 2 weeks. You can’t expect to polish out your first big mirror in the minimum amount of time. Go easy, especially while you’re still somewhat tender in the muscles and tendons of the shoulders and arms. There is nothing wrong with taking twice as long on your first mirror as you would on your second.
It is not difficult to misjudge the polish on a mirror. Your first goal is to completely polish out the mirror in this stage before going on to figure it. If you make the assumption that the mirror will get polished out while figuring, more often than not, you’ll end up with a figured mirror that is not polished out.
When checking the polish, there is one thing you can count on: if it looks underpolished at all, it is probably worse than you think it is. The 25" f/6 mirror which I mentioned earlier looked very nearly polished after 7 hours of work. It took another 5 hours to polish it out completely.
The first and most widely known method of checking polish is to focus light on the surface. In the initial stages of polishing, use a pen-light that focuses a bright beam of light. Shine it on the cleaned surface of your mirror - right away you’ll see the beam on the surface. As polishing progresses, you can monitor which parts of the mirror are polishing quickly and which parts are lagging behind. Keep records, and continue to check the parts that originally lagged behind so they get fully polished.
When this gets ineffective, put black paper under the mirror to give a dark background. Focus an image of the filament of a bright desk lamp on the surface of the mirror with a positive lens such as a magnifying glass. Angle the light at approximately 45 degree so that the back of the mirror is not illuminated under the area being observed. If there is still haze on the mirror, you will be able to se the image of the filament on the surface.
When no haze is left, it will be virtually impossible to see the surface. You may find it difficult to get the surface clean enough so you are not just looking at dust or oil or leftover polishing agent. If it passes naked-eye inspection, examine the focused beam on the surface with a magnifier such as a low power eyepiece. Experiment with different angles from which you observe the surface, and do not consider polishing completed until it is complete.
If you plan to do all the testing of your mirror in the telescope itself (an option which I will discuss in part 3 of this series), you won’t need a test stand and testing tunnel. However, you will be better off to learn how to evaluate the mirror’s condition at the center of curvature, indoors under controlled conditions, even if final figuring is to be done in the telescope outside. If you plan on testing the mirror indoors, your testing environment is extremely important.
The test stand must allow the mirror to hang freely in sling while resting on a piece of shag carpet. The sling can be made of any flexible material that bends to the circumference of the mirror. Nylon or plastic straps are most common; I prefer old seat belts myself. The strap must not support over 180 degree of the mirror or the strap will squeeze the mirror. Nor should it support much less than 160 degree or the strap will over support the bottom of the mirror.
The sling should be hung in such a way that the surface of the mirror rests about ¼" forward of it. If the mirror is thicker than the strap is wide, then center it around the middle of the edge. The carpet just sits behind the mirror. I use the same piece I polish on. In the vertical position, the mirror rests lightly against the carpet, and so serves as a satisfactory support.
If you plan to do the critical testing "in the lab," you must take steps to shield the optical path from air currents. Your test area must be twice the focal length of your mirror and allow space for your test stand, the tester and you. If you are fortunate enough to have a room long enough to accommodate the tests, you may be able to cover up the cracks under the doors and close windows and air vents. However, if the test path must traverse more than a single area in the house, you will need to close all the doors and windows in the test area and possibly in the entire house. Close off any large airways that lead to other parts of the house, especially near the mirror. Large sheets of plastic work well; paint stores carry large plastic drop cloths, and I have seen rolls of plastic drop cloths, and I have seen rolls of plastic 10 and 12 feet wide sold in hardware stores.
If you are unfamiliar with mirror testing, you should learn about it. Testing is discussed in several TM books, including ATM Book 1. At this stage, you will begin to realize how crucial previous optical experience with smaller mirrors of conventional thickness is for making a large thin mirror. For this article, we assume you have some test experience or have the help of someone who has, and that you understand the basics of testing in this manner.
When the mirror looks shiny enough to test, test it, since you must verify early that is good enough to continue working on. Although 15 minutes of polishing will permit some testing, a shine suitable for evaluating the mirror will probably require an hour.
Begin by setting up to test the mirror with a 100 or 150 line Ronchi screen. This is an excellent quick test for overall quality and smoothness of figure. Excellent Ronchi screens can be purchased from Edmund Scientific Company. When you first look at your mirror with a light source at center of curvature, it may not look evenly illuminated. If is difficult to describe what degree of unevenness in the illumination at this stage is serious. If you have a dark area, usually the center or edge, examine it closely to see if it is properly ground out. If you can see many large pits, return to fine grinding and remove them.
If the area is free of pits, test the mirror with the Ronchi screen to see if there is a gross zone within the unpolished area. If there is, you may need to regrind the mirror, or you may be able to polish it out. Dark spots are caused by your tool bending in to the middle while grinding or by using too long of a stroke. The shape of the mirror then deviates so greatly from a sphere that the polishing lap cannot reach all of it.
If the area looks reasonable good, just keep track of it during polishing to see if the problem remedies itself within 4 or 5 hours of polishing. If you have been proceeding according to this article’s instructions, very few of you should have to deal with this. Normally the mirror will look more or less evenly illuminated after an hour of polishing.
Next, estimate how much the mirror departs from a sphere. If you have been using a full-sized lap, it is very likely you have almost a spherical surface. A sub-diameter lap will probably have a progressively longer focus toward the edge. If you are working plate glass and your test right after polishing, it will look overcorrected and in 3 hours or so, if the lap was warm, it will settle down and look more nearly spherical.
If the figure looks reasonably spherical, you’re on Easy Street! Just test for astigmatism and then continue polishing. If the curve is an overcorrected sphere, i.e., a parabola or hyperbola, check the uniformity of the curved Ronchi bands. Even though the Ronchi lines bow, they should bow with no gross breaks or zigzag areas indicating a gross zone. If this is the case, and several hours of continued polishing produce no appreciable improvement, you must investigate your fine grinding techniques and try again from #400 or 12 micron abrasive.
Next, you’ll want to determine approximately how overcorrected the mirror is. The difference between the focus at the edge and at the center of a parabola tested at the center of curvature with a stationary source and moving knife edge is:
r²/R
where r is the radius of the mirrors (i.e., half of the diameter), and R is the Radius of curvature, or twice the focal length. For a 16" f/5 mirror, it is:
8²/160=64/160=.400
If the mirror exceeds this by more than 10% or 20%, take some steps to reduce it early in the polishing process. Crude measurement of the correction is a simple matter especially if you are experienced with Foucault tester; just measure the difference of the focus of the edge and of the center. If your light source moves with your knife edge, divide the focus difference by 2. For a 16" f/5, it would be 0.200.
You can also check for overcorrection with a Ronchi tester by placing the nulled zone on the edge of the mirror, marking where the screen is, then nulling the center and marking the position. Even measuring the difference between the lines with a ruler will tell you if things are all right. Once again, remember to divide by 2 if the light source (whether it’s a pinhole or the other side of the screen) moves with the testing Ronchi screen. If the correction is too great, shorten your polishing stroke for a while.
Usually long "W" type strokes made with a sub diameter lap or larger will tend to overcorrect a sphere toward a parabola, while short "W" type strokes with a 1" overhang will bring the mirror much more spherical. In the case of a full-size lap, short strokes can even undercorrect a sphere. With a sub-diameter lap, you might make the center portion a slightly undercorrected sphere while the edge becomes overcorrected. A very short stroke will make the edge more difficult to polish out. You will find out, with time and experience, which stroke length or lengths are right for you.
At this stage your mirror should have no astigmatism. If it does, something is wrong. In testing for astigmatism, you will find out why it took so long for thin mirrors to become popular - and if things go well for you, you will learn why they eventually became accepted.
The trouble with testing thin mirrors for astigmatism is that when they are being tested vertical, they quite often bend. Even a good mirror may look astigmatic. This is another reason why thin mirrors are not used much in the optical industry, since most optical elements are used vertical. The problem becomes manageable in telescopes, where the mirror most often faces above the horizon in its floatation cell.
Astigmatism, or cylinder as it is often called, happens when one axis of the mirror has a shorter focus than the other. This shows up in the telescope as star images on either side of focus are elliptical in shape, with axes at right angles.
To test indoors for astigmatism at the center of curvature, examine a small illuminated pinhole with an eyepiece of approximately 9mm focal length. Make pinholes by poking the smallest hole you can in a piece of tin foil or .001 thick brass shim stock. A small sewing needle works well. You will discover all kinds of ways to make smaller and more uniform pinholes by putting the brass on different things and poking it different ways with the needle. If you get too good at it, you may have trouble getting enough light through your pinhole.
Place the pinhole in front of a light, preferably the bulb of your Foucault/Ronchi tester, and look at the image of the pinhole at the center of curvature with an eyepiece. Make the distance between the pinhole and the eyepiece as small as you can or off-axis aberrations will confuse you. Keep stray light from entering the eyepiece or the image of the pinhole will be washed out, and make sure the eyepiece is as square as you can make it to the optical axis of the mirror.
A non-astigmatic image is perfectly round on either side of focus and through focus. Don’t unfocus the image too far, because you are interested in examining the pinhole very near focus. Depending on the figure of the mirror, the inside focus may be too soft for this test. Outside of focus you’ll see a sharp bright ring on the edge of the image due to the edge focusing long. This should be round.
Don’t mistake irregularities in the roundness of the pinhole for problems in the mirror. The image will take on the shape of the pinhole when the image focuses. Be aware that air current can disturb this test.
If you don’t see a round image, make sure your eyepiece is square on, then rotate the eyepiece to make doubly sure it is not the fault of the eyepiece. Tilt your head to make sure it is not your eyes.
Make a note as to the shape and orientation of the problem and which side of focus you’re on. Then take the mirror off the sling, and without rotating it, reposition it on the sling again making sure it hangs freely and evenly against the carpet. Inspect the image again. If the problem now looks significantly different, then your mirror is not well supported on the test stand.
If the image still looks the same, rotate the mirror 45 degrees in the sling and again observe the out of focus image. If you still see astigmatism but it has not rotated with the mirror, then it is due to bending in the vertical position. If this is the case, you can probably make the astigmatism come and go depending on just how the mirror rests in the sling. Develop a technique for putting the mirror in its sling to minimize the effect - and then ignore it. Since it will probably be in the horizontal and vertical directions, it will not significantly affect your testing of the mirror.
With thin mirrors, this bending problem sometimes takes a form not quite like astigmatism. Sometimes called "potato chipping, " it means the bottom of the mirror is bent, but the top is not. You may see round images with a flat spot on the bottom outside focus image, the bottom area having more light in it.
Even if you see a round image apparently free of astigmatism, rotate the mirror anyway to make sure that astigmatism in the mirror has not canceled astigmatism due to the support system.
The bad news starts if the problem rotates reliably with the mirror. If the astigmatism is very slight or difficult to detect, these is a chance it will polish out. I do not recommend trying. Polishing out astigmatism by hand may prove fruitless and you can waste a lot of time and a great deal of energy.
Make one final check before despairing. Allow the mirror to come to thermal equilibrium, at least 3 to 4 hours, and then retest it. Some portholes become astigmatic while cooling.
If it is astigmatic, the solution is to go back to #400 or 12 micron grit, whichever you used, and grind again. Be extremely scrupulous in applying all the precautions outlined in part 1. Make sure the back is flat. If you are using a porthole and you did not grind the back flat, do so. Did you forget and let the carpet pile get smashed down? If the tool was on the thin side, maybe it was too thin. Examine your rotating techniques to insure you are rotating by random amounts, and not repeating positions.
Carefully determine and mark the axis of the astigmatism on the mirror. If it returns after grinding, you can see if it lies in the same axis of the mirror as it did the first time or not. If it does, it could indicate bad glass, or that the back is not flat. Even if the tool is too thin, the astigmatism should be less the second time.
Of course, the mirror can be tested for astigmatism in the telescope - just apply the same tests using a star. The diagonal adds another variable because it will introduce astigmatism if it is not flat. Nine and eight point flotation systems can also do strange things if not properly constructed or used.
The main disadvantage of astigmatism testing in the telescope is turbulence. Air turbulence blurs the image, hiding the problem if it is small. Also, uneven temperatures in the air in the tube can cause oblong star images. However, when the air is still, there is no more practical test for astigmatism than observing a star through a high-power eyepiece.
At this point - with the mirror polished out and free of astigmatism - you are ready to tackle figuring. Most of the physical labor is behind you and the work of the mind id before you. Even the most experienced opticians can find figuring a study in frustration. But the feeling of accomplishment when you complete something as rewarding as a large telescope mirror will make all the labor and frustration worthwhile.