The Joy of Mirror Making: Parabolizing

by Mel Bartels

What's parabolizing?

Testing The paraboloid...

The four parabolization methods...

Sub-diameter and oversized laps...

Parabolizing sessions and polishing speed...

My parabolization process...

The Ronchi and star test tandem...

The Ronchi test...

Star testing...

Fixing common maladies...

Some final words: a journey ends, another begins...

References...


What's parabolizing?

The Goal

The goal of parabolizing is to produce a mirror that focuses perfectly at the highest powers with no scattered light.

Our mirror must meet these two criteria:
1. The wavefront at the eyepiece cannot vary peak to valley more than 1/4 wave of green light (ideally closer to 1/8 wave). Halve this for the mirror surface error. This is the total deviation or Danjon-Couder condition #2.
2. The mirror's surface must be very smooth with fine scale deviation less than 1/60 wave. This is the slope criteria, Milles-Lacroix tornado or Danjon-Couder condition #1.

That calls for the mirror's surface to be accurate to about two-millionths of an inch, or one-twentieth of a micron (0.05 microns). The mirror's surface is carefully polished to achieve the required accuracy. Because math is often employed, this is also called 'figuring'.

Thanks to interference, perfectly focused light forms an Airy disk surrounded by rings of fainter brightness. If we meet these two criteria then our star image will look essentially perfect. Failure to meet the two criteria means that the surrounding rings will be too bright, ruining resolution and scattering light.

Every optical test devised rates mirrors on their ability to produce a perfectly focused mirror. The beauty of a telescope is that we can look through the eyepiece and judge for ourselves.

The principal defect of a spherical mirror is called, 'spherical aberration'. Parabolizing a mirror means removing spherical aberration. Parabolizing is an intensely satisfying intellectual endeavor, requiring some physical skill with a fair amount of patience and discipline. It is man the tool maker at his finest. With simple test equipment, the mirror maker can resolve and remove errors in the mirror's surface to a millionth of an inch [0.025 microns], creating a surface so large, smooth and precise that the light of astronomical objects from across the universe can be seen.

"O telescope, instrument of knowledge, more precious than any sceptre." - Johannes Kepler
"I have tried to improve telescopes and practiced continually to see with them. These instruments have play'd me so many tricks that I have at last found them out in many of their humours." - Sir William Herschel

Spherical Aberration

Every mirror maker should grind and polish out 4 inch [10cm] F/10 and 10 inch [25cm] F/4 mirrors. Before parabolizing, mount them in a temporary structure propped up on a chair aimed at Polaris or best possible in the southern hemisphere. Try to bring a star image to focus.

At the radius of curvature the light emanates from the tester and is focused back by a spherical mirror onto the tester - thus the straight Ronchi bands. But light coming from infinity focuses at half the radius of curvature's distance. This distance is the focal length of the mirror.

The 4 inch [[10cm] F/10 mirror focuses nicely with only the slightest hint of spherical aberration. But a 10 inch [25cm] F/4 will be a disaster. If you attempt to focus the central portion of the mirror then the edge zones throws light way out in a giant disc. If you attempt the focus the edge portion of the mirror then the center zone throws light way out in a giant disc. This is very ugly and will give you an appreciation of the importance of parabolizing particularly if the mirror is large or fast.

You will no doubt note that the center focuses outward compared to the edge. This is extreme spherical aberration. We say that the center is high and the edge is low. Here is the above graphic greatly enlarged illustrating the mirror zones' different foci. Pictured is a 10 inch [25cm] F/4 mirror where the difference between central zone focus and edge zone focus is 1/6 inch [4mm].

If the mirror's curve is deepened from a sphere to a parabola then the light focuses perfectly, limited only by diffraction. The amount of glass to be removed is a few millionths of an inch. The formula is r^4/(8R^3) (r=mirror radius, R=radius of curvature). For a 6" F/8, it is one hundred thousandths of an inch or about 1/2 wavelength of light. It is amazing that testing at the eyepiece or testers built from common inexpensive materials can test to a millionth of an inch. Want to know how much glass to remove?

Parabolizing Calculator

Mirror diameter
Focal ratio

Difference

Testing The paraboloid...

The Uphill Climb

It's easy to casually test a mirror to a half wave. It's much harder to critically test to a tenth wave. Mirror testing is one of those vast fields where the more you learn the less you realize you know. Malacara's "Optical Shop Testing" is an indispensible reference. Yet a few realities shine through the fog.

Separating reality from fog is learning what you do not know. There is more illusion in what you see and more reality in what you do not see. At first you observe that a parabolizing stroke has a certain effect. But this effect is dependent on so many hidden variables or contexts. As the stroke is repeated in different settings with different mirrors of different focal ratios with different pitch hardness with different ambient temperature and humidity and different polishing compound and different dilution with different pressure and speed and a whole host of factors, you observe a variety of effects. As you observe each of these effects, they then spin off into sub-universes of downstream variables or contexts. This is the beginning of mirror making wisdom.

Wisdom in mirror making is not learned; it is absorbed by constant observation and thought over a long period of time. Luckily for the first time mirror maker, the joy of taking the first step is as powerful as the joy in taking the next step. Above all, take the steps; walk forward - observe and learn. You are making arguably the finest surface possible by man or machine. It's an intrinsically personal journey that you will find necessarily frustrating yet ultimately deeply satisfying.

The diameter and speed of the mirror matter a great deal. A 6 inch [15cm] F/9 is very different from a 12 inch [30cm] F/5; a 24 inch [61cm] F/4 is a beast of another stripe. A test that works quite well for the former may run into trouble for the latter. Attitude matters. Brash, quick to judge, seeing what you want to see, leads to a poor mirror. Cautious, doubting, quiet, thoughtful, looking for defects, testing again and again - this leads to a quality mirror. As the mirror improves, the reality of its quality, of its profile, will shine through the noise in a shy and relentless manner.

While I did a great deal of long exposure cold camera astrophotography and eyepiece projection photography and later on was an early CCD imager, my consistent interest has been observing. I enjoy planetary viewing but most of all I cannot get enough of widest angle largest aperture deep space observing. Consequently I've tended towards large aperture, fast scopes. The tests I use and my experience with them are slanted towards large thin fast mirrors.

Foucault Test

As with most mirror makers, I started with the Foucault test. The Foucault test enabled me to achieve satisfactory to good mirrors. But I ran into limitations on larger faster mirrors. The edge and center zones are difficult to judge. I could see this instantly in the Star Test, a test I was learning from the old masters. After all, before Foucault invented his test (we use it differently today), mirror makers were using the Star Test. John Hadley two hundred years ago gained a reputation for accurately figured mirrors using the Star Test at the radius of curvature. This began my journey using the Star Test to first judge, then parabolize mirrors.

Caustic Test

I began using the Poor Man's Caustic Test in order to achieve better zonal readings. To my surprise years later it was discovered that the math to reduce the test readings was flawed when the zonal readings deviated from ideal. But how was I able to make mirrors for years? The answer lay with how I began to approach tests that measure the mirror's slope at various zones and then reconstruct the mirror's surface profile. I abandoned the concept that there is any acceptable tolerance in the zonal readings. Each zone must read perfectly. I discovered early on that any error in the Foucault/Caustic zonal readings meant a very real error at highest powers using the Star Test, an error often worse than anticipated. The only acceptable standard became that each zone had to read perfectly. But even that wasn't enough.

A mirror's surface has to be smooth. It cannot be wavy or what we call 'zonal', it can't have high zones and low zones. The surface should flow smoothly from edge to center. Far better some smooth spherical aberration than a mirror with ripples. The old masters said this. Lord Rayleigh said this. Many mirror makers say it. It cannot be overly emphasized.

Ronchi Test

But how to test for mirror smoothness? The Foucault knife-edge test can be used, but not zonal readings - they are too coarse and far apart. About this time Sherman Schultz at Macalaster College in St Paul Minnesota was having tremendous success using the Ronchi test with his many students. With the Ronchi test you can see the smoothness or lack thereof instantly. And it's a quick test to setup and execute, perfect for mirror making classes.

But could it be used to critically judge spherical aberration? I tried the Mosby Null test where a compensating set of curved bands are used, but found the registration difficult. I set up the task of comparing the curved Ronchi bands at a set of precise spacings to computer generated Ronchi bands of a perfect mirror. With a little practice and careful eye, I was able to produce mirrors that were not only smooth but also had good spherical aberration correction or parabolization. A quick Star Test confirmed the overall correction and if a touchup session was needed, the Ronchi test served to confirm the mirror's smoothness when done. This proved so successful that I've continued with this approach ever since.

The Numbers Game

I avoid the numbers game. After all, reducing zonal readings is a mathematical exercise. But what does that number, say 1/6 wavefront mean? Is it good enough? What if there is a turned edge or a high center or a noticeable zone? Is it all about the number or not? Behind every number is a subjective judgment, not only regarding the number itself but the about the errors that led up to the number. The reality that it's the entire telescope, the atmosphere and the observer's experience that combine to produce an outcome at the eyepiece that's judged. If it's subjective, at least to some degree, then maybe cutting to the chase and using subjective tests is the most direct way to an excellent mirror. After all, by using the Star Test you know exactly what you will get each and every night, in the observing conditions that you'll be using your scope under.

He Said, She Said

Why the variation in people's approaches, particularly in polishing and parabolizing? Some makers like thin hard pitch, e.g., Gugolz 73 and others like thick soft pitch, e.g., Gogolz 55. Some use full sized polishers, some use sub-diameter polishers. Some like me use oversized polishers. Mirror makers often point to the materials as the culprit or the savior. I can hear them preaching, “Hard pitch is the answer to everything; it'll cure what ails you". Yet in the end, it is the mirror maker's process, his personality that makes the greatest impact and generates consistent results.

This varies by generation too. The 'standard' today is quite different than the standard two generations ago. I suppose this is a useful reminder to judge the artifact, the finished mirror, and not the construct, or the mirror maker's process.

Find a Mentor

Which path to choose if you are beginning? My suggestion is to find a mentor that you like and follow his process. As you first copy then learn by repeating, you'll develop into your own personality, eventually striking out in your direction. Me? I like to learn by studying the reports of the old masters from the late 1800's when American mirror making first flourished. These makers encountered and overcame the seminal problems. Today, we have unprecedented access to information, each other, there are more of us, and digital computer processing of tester algorithms that take into account the effects of diffraction result in very high mirror quality. Testing early mirrors from decades, even a century ago, show rather mixed results compared to today's mirrors. Nonetheless, mirror making has drifted as twists and techniques have been modified and overlaid on top of the initial masters - sort of a random walk. The result is a certain lack of appreciation for the core problem in mirror making, namely parabolization. For instance, our testers can measure lots of zones, so we use sub-diameter laps that rough up the surface in order to attack the zones, forgetting the initial masters' admonition of the importance of a smooth overall mirror figure.

Open to Discovery

I advocate investigating accidents and happenstance. One day when pitch gradually squeezed past the edge of the lap as I was polishing I happened to stop and test the mirror's figure. I was sure I had done something terribly wrong. But to my astonishment, there was for the first time, no turned edge! I removed the pitch that had squeezed past the edge, polished more, finding that the turned edge had reappeared. I polished hard some more until the pitch once again squeezed past the lap's edge and found that the turned edge had disappeared. I asked a couple of professional opticians who told me about the value of oversized laps and that channelled laps rough the surface. Investigating the original masters I found that John Braschear advocated oversized laps along with petal laps, another area that I was sliding into.

Bath Interferometer Test

I continue to look at new tests. The Bath Interferometer is absolutely wonderful, a revolution in the making for amateur mirror makers. The SCOTS test, a slope test, is intriguing, and the Holomask Test shows promise. Check out a new test called the Slit Image Test (http://www.yubagold.com/tests/index.php) I've used the Ross Null test briefly, ending up using it more for overall smoothness than for exacting spherical aberration correction. Also check out the holographic mask test.

Beyond The Test Methods

What are the realities that shine through the fog? Be cautious and thoughtful, look for defects because they are surely there, confirm with the Star Test and become conversant with more than one test. Finally, practice, practice, practice by making mirror after mirror after mirror. Give them to friends or barter them for eyepieces or other goodies (I once made a 20 inch [50cm] mirror in exchange for a fully enclosed trailer to transport a large scope of mine).

Regardless of the tests you use, you will face high zones that need more polishing, low zones that frustrate you, turned edges that are plain annoying and astigmatism in large thin mirrors that can result in temporary insanity. Above all, it's the combination of sky, observer, telescope and optics that create the view. So let's get on with the task of creating the very best primary mirror we can.

Geometric Tests and the Diffractive Nature of Light

A perfect mirror is limited by the wave nature of light. Fraunhofer diffraction of a circular aperture, the mirror's rim, sets the limits of performance. The circular rim of the aperture diffracts light into expanding spherical waves that interfere with each other at focus, going in and out of phase repeatedly as the angular distance from the center grows. This creates a central dot, the Airy disk, and a series of rings of decreasing brightness. A perfect mirror will reflect 84% of the light into the Airy disk, 7% into the first ring, 3% into the second ring, and so forth, with a total of 16% of the light in the rings combined. [Oldham Optical, UK, http://www.oldham-optical.co.uk/Airy%20Disk.htm]

Less than perfect optics increase the brightness of the rings causing the star image to lose resolution. Our mirror should present very close to the ideal Airy disk with approximately the same brightness in the rings. Geometric based methods that calculate the path of the reflected light rays across the mirror face are popular and have a long history. These tests typically measure the longitudinal aberration, or the discrepancy between where the light ray geometrically would travel to compared to where it ought to be. However, geometric tests need to be used with the understanding that the light actually does not exactly go where the geometric ray traces say it goes, thanks to the diffraction of wave optics. [Jim Burrows, http://home.earthlink.net/~burrjaw/atm/t_verse.lwp/t_verse.htm and http://home.earthlink.net/~burrjaw/atm/odyframe.htm]

The four parabolization methods...

To remove the spherical aberration, we need to change the mirror's spherical shape to a paraboloidal shape by preferentially polishing glass. Here are four ways to parabolize a mirror. I've tried them all successfully.

1. In the first example you see a standard channelled lap with mirror on top and extreme strokes in width and length. This is the most commonly cited approach in telescope making books and is suitable for common mirror sizes and focal ratios. This method wears down the center and the edge.

2. The second example is the approach I use for very fast very large mirrors. The lap preferentially concentrates polishing in the center region tapering off towards the edge. I use short strokes with no side to side variation. Ellison in the early 1900's called this approach the standard way to parabolize a mirror. George McHardie states in his 1937 book, "Preparation of Mirrors for Astronomical Telescopes" that 'graduated facets' is the simplest method and strongly recommended by experts. Here's McHardie's drawing of a graduated lap.

3. The third example is very unusual from what I can gather. I've used it to parabolize 20 inch [50cm] F4 mirror. Short strokes with no side to side variation are called for.

4. The last example is also unusual. I've tried this too. Use the same short strokes with no side swing. Note that this is equivalent to a sub diameter star lap for the center and a feathered ring lap for the edge.

To form the shapes you can scratch out the areas that are not to be in contact, or you can use paper cut to shape pressed between the mirror and lap for a few minutes.

It can be quite confusing at times to contemplate that all these approaches parabolize a mirror, after all the second and third laps are perfect inverses of each other, and the fourth approach is a hybrid of the second and third laps. Here's another way to visualize parabolizing. Never forget that after parabolizing and testing from the radius of curvature, the mirror's center zone must always focus short and the mirror's edge zone must always focus long.

During parabolization, we have the luxury of increasing or shrinking the radius of curvature of the mirror's zones to float or change. Here's a graphic to illustrate.

In each of the three cases, the sphere and the parabola have different touch points.

And if the spherical mirror's surface is straightened into a horizontal line, the glass to remove for each of these cases is the gray colored volume:

Here is what the 7 inch [18cm] oversized parabolizing lap looks like. This is meant to be used mirror on top. Noteow the percentage of pitch in contact with the glass is high in the center and tapers off towards the edge.

Here's the 7 inch [18cm] pitch lap adjusted to remove parabolization from an overcorrected mirror. The pitch concentrates on the 70% zone, sharply tapering towards the edge and more gently tapering towards the center.

And where is what the pitch lap looks like after being prepared to remove the kink in the 70% zone (the mirror is sitting on top). Note how the pitch at the 70% zone is scratched away. Short strokes are used.

Here is what extreme chordal strokes looks like (10.5 inch [27cm mirror on a 11 inch [28cm] pitch lap).

The 70% Zone is Special.

Note that when we deepen the center it focuses shorter and when we deepen the edge it focuses longer. The key is that we do not polish the 70% zone. Inside of the 70% zone polishing tends to shorter the focal length and outside the 70% zone polishing tends to lengthen the focal length.

Now that we've seen these curves, we know how to fix overcorrection, which otherwise can be a real bear. I successfully employ the technique of preferentially polishing the 70% zone, tapering sharply towards the edge and gradually towards the center in order to reduce the height of the curve.

Sub-diameter and oversized laps...

The Problem with Sub-diameter Laps

It's easy to concentrate on 'hitting the number', forgoing mirror smoothness. For many, hitting the numbers is less difficult with sub-diameter laps. The problem with these laps is that they promote roughness. Imagine you are icing a cake or pouring a concrete pad. What happens when you use a tiny spatula or trowel? No matter how hard you try, the surface will not be as smooth from edge to edge as that gotten from a large spatula or a wide trowel. A surface worked with a sub-diameter lap needs smoothing with a full sized lap. But this changes the figure subtlety. I chose to learn to parabolize with full sized and ultimately oversized laps exclusively.

Zonal problems show up in 12 inch [30cm] and larger mirrors because these larger mirrors are often worked with sub-diameter tools. The first masters (Ritchey) used very large laps to generate smoother surfaces. We should not forget the lessons learned by these pioneers.

Finally, there's another drawback to sub-diameter laps that no one seems to notice. A parabolizing tool 1/3 the diameter of the mirror works at 1/9 the speed of a full sized tool and even slower compared to an oversized tool.

Parabolizing Fast Mirrors with Oversized Laps

Researching further, I found that Brashear mentioned oversized laps as a standard technique in the late 1800's. Oversized laps were used almost from the start of glass mirror and pitch tools. You see, during that era, there was an explosion of pamphlets and small books on how to do things. Telescope making was a 'big deal' back then. Holcombe had formed the first USA telescope company in the early 1800's (to the surprise of leading European intellectuals who maintained that Americans were not up to the task), followed by Fitz and Clark which was followed by Brashear and others. Check out http://tinyurl.com/pn3crhl, The Production of Optical Surfaces by John Brashear, Pittsburgh, Pennsylvania, 1881. Also see Strong's Procedures in Experimental Physics for a modern treatment of oversized laps.

I use slightly oversized laps to better control the edge. The lap pattern and strokes are the same as for standard sized laps. (Actually, any sized lap will control the edge - it is a matter of technique. Many amateurs have trouble with turned edge using subdiameter and full sized laps. I've had far less trouble with oversized laps.)

Parabolizing sessions and polishing speed...

Now that the post-polish stage has been completed resulting in a good edge and straight Ronchi bands indicating a spherical curve, it's time to begin parabolizing.

A parabolization session starts with analysis of the mirror's surface, forming an hypothesis of how the mirror's surface will be altered given a particular pitch lap and stroke pattern then finally testing the results. How long should the session be? It needs to be long enough to detect a sufficient change in the mirror's surface that hopefully makes progress but not so long that the session ruins the parabolization if the action proves deleterious.

Check out the following analysis that shows the number of sessions for three mirrors that I have detailed logs.

'Close'' means that the mirror forms an acceptable low power star image. 'Final' means that the mirror forms an excellent high power star image. 'Restart' means that the parabolization spun out of control and necessitated a return to a spherical mirror surface to begin the parabolization anew. The '2nd close' means that the second parabolization attempt forms an acceptable low power star image. And the '2nd final' means that the second parabolization attempt forms an excellent high power star image. I draw three conclusions.
The first that focal ratio matters more than aperture in determining the difficulty of parabolizing.
The second that getting 'close' is one-fourth to one-half of the journey, depending on the difficulty of the focal ratio (parabolizing an F/2.8 mirror is like zig-zagging about on ice).
The third that returning to spherical when parabolizing spins out of control is a viable strategy because the second try goes faster after learning from the first parabolization attempt. For the 10.5 inch [27cm] mirror, I determined to become expert at controlling parabolization, particularly overcorrection in the outer zones of the mirror. This ultimately proved successful. Subsequent mirrors will show if this advanced technique shortens the number of parabolizing sessions.

What's more difficult, a large mirror or a fast mirror?

In my experience, focal ratio is most correlated with effort and touchiness during parabolizing. An F3 is difficult at any size, F8 not nearly so much. Parabolizing accuracy in terms of smooth under and over correction depends solely on the focal ratio, not on aperture. For instance, consider the following chart. The graph is for worse case 1/4 wavefront; for the more demanding 1/8 wavefront, halve these values. While slower focal ratios have a larger allowable parabolic deviation percentage, because the paraboloidal correction is smaller, the deviation in absolute terms is also smaller. I derived this relationship by using a standard algorithm that calculates wave error given a set of zonal readings. I iteratively fed it zonal readings smoothly varying by a correction factor, deriving the maximum correction factor that fit the quarter wavefront error envelope.

The six factors of polishing or cutting speed
1. Glass is polished or cut away in proportion to the time the tool passes over the glass.
2. The faster the tool's speed, the more that's cut away (this is not strictly proportional).
3. Overhanging sections of the tool cut faster (gravity).
4. The leading edge of the facets and channels cut faster than the trailer edges (more polishing compound builds up in front of facets).
5. Polishing or cutting action is heaviest when and where pressure is heaviest.
6. Type of polishing compound (Ceium Oxide is faster than Rouge).
7. Thickness or concentration of polishing compound.

My parabolization process...

I developed a parabolizing procedure from my 13 inch [34cm] f/3.0 where I studied parabolizing the mirror using several procedures.Later when I had to return the 6 inch [15cm] to spherical after overcorrecting the outer zones, I streamlined the process, halving the number of sessions from thirteen to seven.

The parabolizing process is:
1. I start with mirror on top of an oversized lap, but only execute the extreme chordal strokes at the edge - I do not stroke through center. This roughs in parabolization into the central 60-70% of the mirror. This proceeds rapidly and does not need a lot of precision.
2. After significant parabolization appears in the central zones, I then switch to the second parabolization method to push the parabolization out to the edge. These are long strokes with no side swing over a parabolizing lap where the percent of pitch contact fades towards the edge. The result of this is a kink or low point at the 70% zone. This goes slower and requires some attention.
3. I alter the lap by scratching (pressing out is fine too) away pitch at the 70% zone; this in effect raises the 70% zone back up. I use short strokes with some almost no side to side motion. This gives a lot of control over the overall shape of the curve. As the parabolization reaches the edge, the Ronchi test should be switched from the inside of radius of curvature position where the test is most sensitive to the central zones of the mirror to the outside of radius of curvature position where the test is most sensitive to the edge zones of the mirror.
4. I now adjust the parabolization more closely by using the Ronchi test with precision offsets from the radius of curvature. I start by placing the Ronchi tester inside of the radius of curvature such that the bands match the appearance as given by the computer, move the tester outward the precise distance as indicated by the computer using an engineering ruler and inspect the outside of radius of curvature bands. I adjust the curve as needed. A 6 inch [15cm] F/9 mirror at this point could be serviceable at high powers (1/4 wave). A 12 inch [30cm] F/5 won't focus well at high power (1/2-1 wave). A 24 inch [61cm] F/4 might focus at low power (1 wave). How good is the mirror? Build a test rig and see for yourself - that's how you'll really learn.
5. Fine tune the parabolization by using the star test in conjunction with the Ronchi test.

Select the parabolization method (mirror on top with very long very wide strokes) if working a standard sized standard focal ratio mirror, or the second parabolization method (long strokes with mirror on top of a parabolizing lap that has progressively less contact towards the edge of the lap) if working a larger faster mirror, and begin 20 minute sessions. As with polishing, execute slow strokes with heavy even drag. Do not go too fast! Look for the swelling of the bands in the mirror's center when testing inside of the radius of curvature and look for the smoothness of the bands.

Once you find the sweet spot where parabolization gradually increases then take small steps, testing or measuring often. Sneak up on the final 100% parabolization curve. If you go over, then you will have to work to find a new lap configuration and stroke pattern that works. Usually this cannot be found and the worker returns to spherical to start over.

A Parabolizing Session

I start with warming the pitch lap by pouring warm water over it for a minute or two. I want the pitch warmed just enough so that it can be pressed into perfect contact. Too much heat will warm the glass causing all sorts of havoc. I press the lap for a few seconds, then rotate and reposition the lap slightly and press again. I repeat until satisfied with the contact. If necessary I warm the pitch again. After contact I renew the microfaceting using room temperature or slightly colder water to prevent the chips from flying too far and creating too big of a mess. I place the mirror back on the lap, rotating and moving every few seconds, until the glass and lap have equilibrated to the same temperature. This whole process takes 5-15 minutes and is necessary for consistent results. However long it takes though, don't settle for less than the desired contact or equilibrium.

Remember to keep slow heavy even drag. Rotate top piece methodically; walk around or rotate the bottom piece at a slower but regular pace. Start and stop in the same position. Don't be shocked if you are working a very large very fast mirror: you will have to push down harder on the mirror's back to maintain even drag. That's because the difference between sphere and parabola becomes quite severe.

Each session I begin with a test, at this stage, Ronchigrams, write out my analysis of the mirror, pick the biggest error, write out my plan of attack (strokes, deformed lap, accentuated pressure, time to execute or at least see if the proposed cure is making the mirror healthier or sicker), execute, then follow up with more tests to evaluate results. This is recorded in a log. The log will be your savior as you look back to see how you corrected issues that crop up again (hopefully more shyly). You will find that your personality coupled with the mirror tend to produce similar outcomes. If that particular outcome is not desired, then study your notes for what to do differently. Sometimes in desperation, doing the exact opposite is exactly the ticket! Then you can study why this worked, talk to other mirror makers, and ultimately gain a deeper insight into parabolization.

Remember that you only really need know the worse defect and if the mirror is getting better or worse. Don't become sidetracked into obsessively measuring the amount of deformity. It does not matter - it has to be removed. That's a beauty of the Ronchi test. You can see instantly the major defect and if its getting better or worse.

It's not only learning what to do and why it works, but it is also learning what to pay attention to and what to ignore. Watching an experienced mirror maker deftly go through the motions may leave you with the impression of casualness but believe me; it's all carefully thought through and controlled.

The Ronchi and star test tandem...

I've been grinding mirrors, making telescopes and observing with them for 40 years. I quietly star test every telescope (when I can get the owner to put in a high power eyepiece) I look through. I've noticed a trend. Mirror makers that used the star test or the interferometer test consistently make better mirrors.

I am going to show you the Ronchi test for roughing in the curve and testing for overall smoothness and the star test, both indoor and outdoor, for final adjustments. The Bath interferometer is explained in great detail by others (see the interferometry@yahoogroups.com discussion group for resources). I've made mirrors with other tests like the Foucault, Caustic, Poor Man's Caustic and the Wire test and have used various data reduction programs. You can find all the information you need on these tests elsewhere, but in my experience nothing beats the Ronchi and star tests or the Bath interferometer in consistency, ease of use and accuracy. Remember the primary goal is to produce a mirror that focuses light perfectly at the highest powers, not to argue over Strehl ratio. Spend more time grinding and less time arguing online, I say!

My experience from star testing many hundreds of telescopes over the decades is that every single mirror (except perhaps a handful) has discernible defects. The defects in the best mirrors have no detectable impact on the image, the defects in the average mirror has slight impact on the image, certainly outweighed by the myriad of issues that accompany telescope use.

Allow me to offer an unsolicited testimonial from well-known telescope maker and interferometrist Dale Eason. "A few years ago I met Mel in person for the first time at a Star Party in Wisconsin. We had communicated for year on the net. I had my 16 F/5 telescope whose mirror I made and knew very well from the interferometry data. The telescope itself was still a work in progress and I think the mirror was not yet coated. The telescope had no tracking and was very unstable. It jiggled when you toughed the eyepiece. Mel wanted to star test it so I let him. He did not know the interferometry data from it. He took about one minute and then he proceeded to describe its faults that I knew from interferometry and described their position on the mirror. That man can star test" --- Dale Eason

You too can learn to star test like this with practice, particularly if you star test your mirror as you parabolize.

Here is a Hartman test report by Jim Burrows on a 6 inch [15cm] F/4 mirror parabolized by me using my standard approach of the Ronchi test followed by final touchup using the star test. The mirror has a small turned edge that is masked off when in use and during the test. You can see that the RMS figure of 9nm is about 1/60 wave RMS and peak to valley of 1/20 wave (both on the surface). By the way, I saw the high zone is the star test but judged it extremely minor - the mirror was more than good enough, and I was able to suspect the zone in the Ronchi test with very careful inspection after the fact.

A 20.5 inch [52cm] F/5 mirror that I made in 1990 has been viewed through by many experienced observers. It gives an indistinguishable from perfect star test pattern at high power. On nights of excellent seeing I use it at powers of 800x-1200x. On one famous night of perfect seeing at the Oregon Star Party I used it at 6000x power.

The Ronchi test...

I use the Ronchi test for its speed and quickness to interpret. The tester is used in two positions: inside the radius of curvature and outside the radius of curvature. Shown here is a mirror that is parabolized. Testing at the radius of curvature shows that the mirror's center zones focus in front of the mirror's edge zones.

When inside the radius of curvature, the tester is closest to the mirror's center zones and furthest from the mirror's edge zones. When outside the radius of curvature the situation is reversed: the tester is closest to the mirror's edge zones and furthest from the mirror's center zones.

As the tester is moved close to a zone's focus, that zone's Ronchi bands expand, becoming further apart and thicker. Here's the Ronchigrams for a finished 13.2 inch [34cm] F/3.0 mirror (final parabolizing by star test at high power). The image on the left is outside of the radius of curvature and the next image on the right is inside the radius of curvature. Note in the first image that since the tester is closest to the mirror's edge, the bands at the edge spread apart and are thicker. The bands representing the mirror's center are spaced tightly together and are thinner since the mirror's center focuses further away. The reverse is true for the second image.

Here's a favorite visualization of mine. The paraboloidal mirror is concave at the radius of curvature by virtue of lowering the center (or equivalently lengthening the edge). A grating placed inside the radius of curvature will see the center closest, the edge farthest. Since the bands expand closer in and shrink further out, the bands will appear fattened in the center and tapered at the edge. A grating placed outside the radius of curvature sees the center farthest and the edge closest, resulting in the bands appearing thinner in the middle and spread out at the edge.

Ronchigrams outside and inside of radius of curvature

Go to my online Ronchi test software, http://www.bbastrodesigns.com/ronchi.html, enter your mirror's measurements and look at the left most inside of radius of curvature image. Your goal is to induce parabolization in your mirror until it looks close to this image. You will be inspecting for the character of the Ronchi bands (smoothly curved with no kinks or straight sections or sharp bends) and the overall curvature of the band. Keeping in mind that this initial stage is to get some parabolization into the mirror, if you have a smaller slower mirror then you may be within a fraction of a wavelength, if you have a large fast mirror you may be within a wave or two. Here are the desired Ronchi test patterns for the above mirror.

Using precision offsets from the radius of curvature and comparing to the computer generated Ronchigrams, I judge that this mirror is quite close, perhaps slightly overcorrected. target images outside and inside of radius of curvature.

Why are the curves shaped differently inside of focus compared to outside of focus? Look at this surprising graphic showing the same parabola but with different offsets from the radius of curvature. You can see these curves mirrored in the Ronchigrams, the lower curves seen outside the radius of curvature and the upper curves seen inside the radius of curvature.

Dr. Sherman Schultz's Ronchi Testing

One of the most successful practitioners was Dr. Sherman Schultz of Macalester College in St Paul Minnesota. He used the Ronchi test in his mirror making classes with countless students successfully completing their telescopes. For a interview with Dr Schultz, select here. See Telescope Making #9 for an article by Dr Schultz on the Ronchi test.

He lists the following advantages of the Ronchi test.

  1. Ease of interpretation. Leads to confidence which leads to a better mirror surface. Getting mired down in numbers, spreadsheets and equipment adjustments just isn't confidence inspiring.
  2. His students made 275 mirrors. Not a single student gave up or quit. Testing can be the point of failure for telescope makers. But not with the Ronchi test.
  3. The Ronchi test is sensitive enough to distinguish between excellent, good, fair and poor. Every little irregularity is seen. It doesn't matter if the error is 1/6 or 1/8 wave - you don't need to know the magnitude of the error. A drowning person doesn't need to know how deep the water is once he is in over his head. He just needs to know how to swim.
  4. You work until the curves match. The curves are not a mysterious table of numbers or a grid of coefficients. You can see the perfect simulated curve; remember it in your mind's eye. Each mirror diameter and focal length combination has its own curves (and for that matter, its own set of Foucault zonal measurements).
  5. You can easily detect, better than any other test, the most common issue, that of turned edge. And the next most common issue, zonal roughness.
  6. The student immediately recognizes the overall curve as compared to the sphere. The correct corrective action is quickly devised. There is no plotting of the mirror profile as reconstituted from zonal measurements, which by the way, misses issues between the arbitrarily designated zones.
  7. The test image is bright. The Ronchi test can be used in an ordinarily lit room in daytime. It's easy to setup, easy to see, easy to understand.

Another approach mentioned soon after the invention of the Ronchi test by Vasco Ronchi in 1923 is to measure the separation between Ronchigrams where the inner Ronchigram shows the central bands separated by a precise amount, say 1 inch or 2.5cm, and the outer Ronchigram shows the edge bands separated by the exact same amount. The separation between Ronchigrams should equal r^2/2R (mirror radius squared divided by twice the radius of curvature).This can be used to gain a sense of overall correction. This can be extended to intermediate zones.

Ronchi Test Examples

Here are two examples of mirrors that require extreme parabolization. The first is a 6 inch [15cm] F/2.8. These are the Ronchigrams taken after every parabolizing session, each session consuming 15-30 minutes of parabolizing time on the lap. I needed 13 sessions to match bands close enough (note: not perfectly) to switch to star testing for guidance in final parabolizing. After the outer zones became overcorrected and I could not fix them, I spent three hours polishing the mirror returning it to spherical. On the second parabolizing attempt I skipped the extreme chordal strokes, going directly to long strokes center over center over a parabolizing lap. This took seven parabolizing sessions to come close enough to justify star testing. Note: every parabolizing session is shown regardless of result. I keep notes so that I can understand and improve.

My 20 inch mirror log
My 6 inch mirror log
My 10.5 inch mirror log

6 inch [15cm] F/2.8

6 inch [15cm] target.

Step 1. Roughing in the curve in the middle part of the mirror. The first two sessions were extreme chordal strokes (no strokes through the center) on a normal oversized lap.

Step 2. Pushing parabolization out to the edge. The next six sessions where very long center over center strokes on a parabolizing lap.

Step 3. Smoothing the curve. The final four sessions short strokes on a lap with the 70% zone scratched away.

Step 4. Using precision offsets from the radius of curvature and comparing to the computer generated Ronchigrams, I judge that the mirror is quite close, perhaps slightly overcorrected particularly in the 80% zone.

Step 5. Fine tune the parabolization by using the star test in conjunction with the Ronchi test. Initial star test at 3mm exit pupil shows overcorrection; all zones do not quite focus simultaneously. Overcorrection perhaps slightly worse in mid-zones. I then attempted to fix the overcorrection which resulted in a star test where the light barely focused into a pinpoint now (so overall correction closer) but with heavily overcorrected mid-zones.

Further attempts to fix the overcorrected mid-zones by accentuated pressure and then by a deformed lap resulted in the following Ronchi test which shows that the parabolization is getting worse.

At this junction I judged it best to return to spherical and start the parabolization process anew. A touch of turned edge persisted after three hours of very short strokes on an oversized lap.

Back to the beginning of parabolization where I decided to start on step 2, pushing parabolization out to the edge using very long strokes with no side swing, mirror on top of oversized lap that's been microfaceted into a parabolizing shape.

At this point I switched to short strokes with the 60% zone on the pitch lap scratched away. You can see the parabolization build smoothly while the kink at the 60% zone disappears.

Here are the final results: star test looks very good: excellent overall correction with a touch of overcorrection in the 50-80% zones.

Compare to ideal.

10.5 inch [27cm] F/2.7

The second example is a 10.5 inch [27cm] F/2.7. Again, the images are taken after every parabolizing session, whether better, worse or indifferent. Here I needed 26 sessions to match bands close enough to switch to star testing for guidance in final parabolizing.

10.5 inch [27cm] target.

Step 1. Roughing in the cuve in the middle part of the mirror. The first session was extreme chordal strokes (no strokes through the center) on a normal oversized lap.

Step 2. Pushing parabolization out to the edge. The next six sessions where very long center over center strokes on a parabolizing lap

Step 3. Smoothing the curve.The next five sessions short strokes on a lap with the 70% zone scratched away to fix the kink in the 60% zone. This proceeded successfully until the extreme edge zones lost their parabolization and the kink consequently became worse in the last two sessions. At this point the lack of parabolization is the greater problem and so I concentrated on solving this problem.

Step 3b. Fixing the kink and putting more parabolization back into the mirror. The next 8 sessions I reverted back to the very long strokes with no side swing over a mildly parabolizing lap with the 50% zone on the pitch scratched away to minimize contact there. The kink gradually disappeared and the overall parabolization increased in a smooth fashion. You can see the curvature near the mirror's edge increasing each session.

Step 4. Using precision offsets from the radius of curvature and comparing to the computer generated Ronchigrams, I judged that the mirror was slightly undercorrected in the outer zones. Using very long strokes directly center over center (no side swing) on a parabolizing lap, I push in more correction. Sessions were 10 to 15 minutes long. The second session used a standard oversized lap that was not parabolizing (less pitch in contact towards the lap's edge). That resulted in a rougher surface with slightly reduced parabolization. Lesson learned! The next two sessions were executed with very long strokes, some side swing, on a parabolizing lap, resulting in more correction being added. Note also that the turned edge is disappearing. At this point the Ronchi bands are close to ideal.

Step 5. Fine tuning the parabolization by using the star test in conjunction with the Ronchi test, a star test at 3mm exit pupil which reveals that the mirror focuses to a pinpoint with slight and smooth undercorrection. This resulted in a great number of sessions where I zigzagged between overcorrected and undercorrected, eventually overcorrected the outer zones, then attempting to remove the excess parabolization resulting in undercorrecting the central and mid-zones, then finally pushing more parabolization into the mid-zones. Sessions were as short as seconds and as long as a couple of minutes. Like other mirrors, I came close early (see the 4th and 5th test images, were the 4th test image is slightly overcorrected and the 5th test image had a bit too much correction in the outer zones and not quite enough in the middle zones; a problem that got worse before it got better).

Here are the final results: the star test shows diagonal shadow breakout the same in both directions but with the above focus position showing a brighter ring around the diagonal and the below focus position showing a brighter ring on the outside meaning that the outer 15% is very slightly overcorrected and the inner 85% is very slightly undercorrected. Star test pattern improves as the mirror cools to ambient air temperature. These issues are very slight.

Compare to ideal.

Jerry Oltion's 12.5 inch [32cm] f4.5 Mirror

Here's Jerry Oltion's 12.5 inch [32cm] f4.5 mirror that's been parabolized to high quality: at 50x per inch of aperture [2x per millimeter] the mirror focuses sharply and has an essentially perfect star test with a slight brightening of the diagonal breakout ring outside of focus, indicating a broad high zone between the center and edge of the mirror. The third image is composed of the first two images laid on top of each other showing that the mirror's 40-80% zone is ever so slightly undercorrected. This gives you an idea of how carefully the Ronchigram should be judged.

Final comments on zonal irregularities: I discovered by accident after washing a mirror in warm water that a Ronchi test of a temporarily heated mirror makes minor zonal irregularities more obvious. There is a lot of shimmering but through it the zonal problems are exaggerated and easier to see. Also, it can help to move the Ronchi tester a great distance from the radius of curvature so that many bands cross the mirror's face. Zonal irregularities can be seen as discontinuities in the tightly spaced bands.

Star testing...

A Piece of Glass

He labored late into the night,
At early morn' his task resumed,
To fashion thus a disk of glass
Into a subtle curve, not deep,
but measured only by the shades of light.
From a simple pinhole made of foil,
Revealing to his practiced eye
Imperfections infinitesimal;
Until at least his skill produced
A curve so true the mind of man
Could not discern the wavering of a breath.

"Just a piece of glass," 'twas said,
But in that simple disk
The heavenly host
Of suns and stars, yea, universes,
Revealed their glory in the sky
For man to ponder - and adore.

-C.A. Olson, Westwood, N.J.

John Dobson

John Dobson wrote in the Celestial Observer, 1973, published in San Diego, California, "The bright spot ... is thrown out of focus first one way then the other by pushing the eyepiece in and out. The two resulting discs of light should be the same. If they are not the mirror needs to be dug in those areas that bundle too much light when the eyepiece is too far out."

Remember his simple words. He knows what he's talking about. I've star tested his 24 inch [61cm] f/6.5 mirror and it is very good. He gave me confidence that the star test was a serious, discerning and demanding test. So I learned the art of star testing. The quality of the view through the eyepiece is subjective. Stirring in numbers like peak to valley wavefront rating, r.m.s. wave error and Strehl ratio confuse as much as they clarify. The beauty of the star test is that you get what you see. And it is all done with a simple high power eyepiece on a night of good seeing. I try to star test every telescope I look through. The experience of seeing hundreds of mirrors and their defects is invaluable. Every mirror will show errors or deviations in the star test, some greater that are injurious to the view, some hard to see and completely inconsequential.

It is a simple rule of thumb: rack the eyepiece outward. Those areas of the mirror that appear excessively bright or have bright rings need more polishing. Rack the eyepiece inward. Those areas of the mirror that appear excessively bright or have bright rings need less polishing.

A Testimonial

Allow me to offer an unsolicited testimonial from well-known telescope maker and interferometrist Dale Eason. "A few years ago I met Mel in person for the first time at a Star Party in Wisconsin. We had communicated for year on the net. I had my 16 F/5 telescope whose mirror I made and knew very well from the interferometry data. The telescope itself was still a work in progress and I think the mirror was not yet coated. The telescope had no tracking and was very unstable. It jiggled when you toughed the eyepiece. Mel wanted to star test it so I let him. He did not know the interferometry data from it. He took about one minute and then he proceeded to describe its faults that I knew from interferometry and described their position on the mirror. That man can star test" --- Dale Eason

I've moved my star testing section to its own web page so that it's available to anyone wishing to star test their telescope.

Fixing common maladies

Outside of proper parabolization, edge problems are the most common maladies afflicting mirrors. A turned down edge, or TDE, is a narrow zone, no more than 1/8 inch [3mm], at the extreme edge that turns downward, focusing long. If the zone is wider then it is called a rolled edge. Rarely the extreme edge will turn upward, focusing short. This is called a turned up edge.

Turned Down Edge

A turned edge is readily be detected in the star test. A hairy edge inside of focus and a bright outer ring outside of focus indicates that the mirror's edge is focusing long.

TDE is also easily detected in the Ronchi test by looking for hooks at the ends of the bands, particularly noticeable when testing outside the radius of curvature.

There is a great deal of superstition on turned edges since they can appear unexpectedly and are not usually quickly fixed by amateur opticians. The earliest optics exhibit terrible edge problems, so it's a malady that's afflicted opticians from the earliest times.

While TDEs can be caused by too soft of pitch plowing into the mirror's edge as it is stroked back across the mirror's face, more usually TDE is caused by fear. The amateur optician, fearful of a TDE, consciously or unconsciously avoids polishing the edge with the same vigor as the rest of the mirror. Polishing in amateur hands typically drives the mirror to a shorter radius of curvature. Since the edge receives less polishing, it lags behind, hanging onto its longer radius of curvature.

Common thinking is that a TDE can be caused by warm fingers curled over the mirror's edge. I've not seen this when I grip the mirror's edge.

What to do about it? It's been said that professional opticians charge double for work good to the extreme edge. A simple solution is to very carefully bevel the turned edge off. Another solution is to use a retaining rim to restrain the mirror in the cell. One of the most popular lines of telescopes, Cave Astrola, used such a retaining rim. I never saw a turned edge in a Cave telescope! The amount of light lost is very small, maybe 0.02 magnitudes of light, which can be made up for with better coatings, cleaner mirrors or observing on clearer nights.

Removing a turned edge depends on the perspective taken. The TDE can be seen as zone that's low, in which case the fix is to polish the entire rest of the mirror. Planing down the entire mirror surface sans edge takes a long time. Harder pitch can help keep the radius of curvature from shortening, otherwise a new TDE will be continuously re-created. Use short center over center strokes with mirror on top

The better perspective is to imagine a line from the mirror center to the outside extreme edge. In this perspective the glass exhibits a bump just inside of the edge. The fix then is to remove the bump. There are several approaches. The first, recommended by Texereau's classic, "How to Make a Telescope", is to concentrate polishing on the edge. This is a powerful technique and done carelessly can exacerbate the TDE. The second, recommended by Waineo, is to use a rectangular lap, say 1x3 inches [25x75mm] in size, stroking it laterally to the mirror's edge. I like this approach because I have more control, though the roughed outer zones need to be smoothed with a larger lap. The final approach is to ignore the edge, particularly with fast optics, using a parabolizing technique that lowers the outer zones, for instance a small star lap stroked tangentially, or mirror on top with very long strokes. Done properly, the TDE will be swallowed up by the parabolization, magically disappearing.

Some final words: a journey ends, another begins...

Many watch, few observe. Keep a log or notes.

Enjoy each level of expertise that you climb through: apprentice, craftsman, master. No matter how much you learn, you will discover more that you do not know, and what you thought you learned needs revising. Don't rush the end, it only retreats further away. The way to learn mirror making is to waste time making mirrors. Never hide in pride or arrogance; it only makes you more afraid and angry of truth; keeping in mind that those who know are usually the quiet ones. As mirrors slide through your hands into telescopes, you will come to love glass and it will reward you beyond words.

"I have looked further into space than any human being did before me." - Sir William Herschel

"At the last dim horizon, we search among ghostly errors of observations for landmarks that are scarcely more substantial. The search will continue. The urge is older than history. It is not satisfied and it will not be oppressed." - Edwin Hubble

"I was interested in telescopes and the way they worked because I had an intense desire to see what things looked like, so I learned how to use telescopes and find things in the sky." - Clyde Tombaugh

"For my confirmation, I didn't get a watch and my first pair of long pants, like most Lutheran boys. I got a telescope. My mother thought it would make the best gift." - Wernher von Braun

Do not forget to savor nights under the stars with your wonderful mirror that you made with your own hands and brain. Just think of it, using simple testers and humanity's marvelous invention, glass, you can make the invisible and unfathomably distant Universe visible by shaping to astonishing accuracy the telescope mirror.

And attend or conduct mirror making classes and share your experiences and observations on mirror making.

My 20 inch mirror log
My 6 inch mirror log
My 10.5 inch mirror log

References...

- My mirror making articles at http://www.bbastrodesigns.com/tm.html
- Advanced mirror makers who are also experienced observers

- Jeff Baldwin's telescope making pages http://www.jeffbaldwin.org/atm.htm
- Bell's The Telescope
- Richard Berry's Build Your Own Telescope
- Richard Berry and David Kriege's The Dobsonian Telescope
- John Brashear's The Production of Optical Surfaces from Summarized Proceedings and a Directory of Members, 1871, http://tinyurl.com/pn3crhl
- Sam Brown's All About Telescopes
- William J. Cook's The Best of Amateur Telescope Making Journal
- John Dobson's How and Why to Make a User-Friendly Sidewalk Telescope
- Myron Emerson's Amateur Telescope Mirror Making
- David Harbour's Understanding Foucault
- Albert Highne's Portable Newtonian Telescopes
- Neale E. Howard's Standard Handbook for Telescope Making
- Albert G. Ingall's Amateur Telescope Making, Volumes 1-3
- H. Dennis Taylor's The Adjustment and Testing of Telescope Objectives
- Henry King's The History of the Telescope
- Karine and Jean-Marc Lecleire's A Manual for Amateur Telescope Makers
- Allyn J. Thompson's Making Your Own Telescope
- Allan Mackintosh's Advanced Telescope Making Techniques - Optics, Advanced Telescope Making Techniques - Mechanical
- Daniel Malacara's Optical Shop Testing
- George McHardie's Preparation of Mirrors for Astronomical Telescopes
- Robert Miller and Kenneth Wilson's Making and Enjoying Telescopes
- James Muirden's Beginner's Guide to Astronomical Telescope Making
- Donald Osterbrock's Ritchey, Hale, and Big American Telescopes
- Henry Paul's Telescopes for Skygazing
- Robert Piekiel's Testing and Evaluating the Optics of Schmidt-Cassegrain Telescopes, Making Schmidt-Cassegrain Telescope Optics, ATM's Guide to Setting up a Home Optics Shop, Tips for Making Optical Flats
- Norman Rember's Making a Refractor Telescope
- Sherman Shultz's The Macalaster Four-Goal System of Mirror Making and the Ronchi Test, Telescope Making #9
- John Strong's Procedures in Experimental Physics
- Scientific American's The Amateur Astronomer
- H.R.Suiter's Star Testing Astronomical Telescopes
- Telescope Making magazine (no longer published)
- Jean Texereau's How to Make a Telescope
- Bill Thomas' Split Image Test (http://www.yubagold.com/tests/index.php)
- Stephen J. Tonkin's Amateur Telescope Making
- John Walley's Your Telescope, a Construction Manual
- Wilkins and Moore's How to Make and Use a Telescope
- Stellafane Amateur Telescope Making pages http://stellafane.org/stellafane-main/tm/atm/ (comprehensive collection of links to web articles)

(end of parabolizing)

For more see
Introduction
Rough Grinding
Fine Grinding
Polishing
Star Testing