The Joy of Mirror Making: Parabolizing

by Mel Bartels

What's parabolizing?

Testing the paraboloid including methods...

Control and variation...

Sub-diameter and oversized laps...

The four parabolization methods using full sized tools...

Parabolizing sessions and polishing speed...

My parabolization process...

The Ronchi and star test tandem...

The Ronchi test...

Star testing...

Fixing common maladies...

How to know you are finished...

The retrospective...

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


What's parabolizing?

The goal

The goal of parabolizing is to produce a mirror that focuses light perfectly at the highest powers.

Our mirror must meet the following two criteria:

1. The wavefront at the eyepiece cannot vary more than 1/4 wave of green light peak to valley and should be much less than this over much of the wavefront. Don't forget to halve this when talking about the mirror's surface and adjust for any red to green light conversion. 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 wavefront. This is the slope criteria, Milles-Lacroix tornado or Danjon-Couder condition #1.

Meeting these conditions 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 or 50 nanometers). The mirror's surface is carefully polished to achieve the required accuracy. Because math is often employed to calculate the parabolic deviation, this stage of mirror making is called 'figuring'.

Thanks to interference and the round shape of our mirror, perfectly focused light forms an Airy disk surrounded by rings of ever fainter brightness. A poor quality mirror causes the surrounding rings to be too bright, ruining resolution and scattering light. The beauty of the star test 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 you the tool maker at your 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] F10 and 10 inch [25cm] F4 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] F10 mirror focuses nicely with only the slightest hint of spherical aberration. But a 10 inch [25cm] F4 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] F4 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" F8, 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



Notice how the mirror's center and edge cross at different locations along the mirror's axis (the edge zones actually come to focus a little beyond the mirror's longitudinal axis - basis for the Caustic Test). The mirror can be divided into zones such as the edge zone or the center zone or into zones expressed as percentages such as the 70% zone as measured from the mirror's center to the mirror's edge.

As measured from where the mirror's center rays cross the mirror's axis, each zone should cross at a point r^2/2R, where little r is the mirror's radius and big R is the Radius of Curvature (moving light source).

Consider the 10 inch [25cm] F5 as pictured. The zonal reading difference between the center and edge is 0.125 inches [3.2mm]. The parabolic deviation is 3.6 waves of green light. That is a magnification factor of 13,000! If we can read the zones to 0.01 inches [0.25mm] then we could figure the mirror to 1/35 wave.

Testing the paraboloid including methods...

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 many hidden variables. 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. 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] F9 is very different from a 12 inch [30cm] F5; a 24 inch [61cm] F4 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

Foucault's test (we use it differently today) revolutionized mirror testing. The Foucault test may be responsible for more quality mirrors than any other test. I recommend that every mirror maker learn this test and the importance of zonal readings.

Today we use a Couder mask to cover the mirrors such that any one mirror zone is visible. We then use a knife-edge traveling on the longitudinal axis to cut the light coming from the zones, measuring one at a time where each zone crosses the longitudinal axis. From this we obtain a series of zonal readings that are then compared to the ideal numbers (the r^2/2R mentioned earlier).

A comment on mirror tests and quality

It is easy to find people criticizing this test or that test. It can leave you wondering if your mirror is any good because it was figured with the Foucault test, or the Ronchi test, or the Bath Interferometric test. Opticians, both professional and amateur, and telescope users can have strong opinions. And I'm guilty of occasionally criticizing too. My experience is that the type of test used, the optician's figuring skills, even the particular test results, are not as important as the optician's personality and quality of the process used to produce the mirror. Is the process repeatable, is it consistent, is it precise, how does the process ensure that a bad mirror never escapes?

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, mirror makers were using the Star Test. John Hadley and James Short in the 1700's gained reputations for figuring mirrors using the out of focus illumination pattern. This began my journey using the Star Test to first judge, then parabolize mirrors. To be clear, others use the Foucault test brilliantly to produce consistently high quality optics.

The Poor Man's 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 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, the 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 'zony': it can't have high zones and low zones. The surface should flow smoothly from edge to center. Far better for the mirror to sport residual smooth spherical aberration than ripples. The old masters said this. Lord Rayleigh said this. Many mirror makers say it. It cannot be overly emphasized.

While I don't use this test anymore, I did use it for a number of years on smaller fast mirrors. I learned a lot from using the test. Since there's precious little information on the test and how to use it, I'm including this section for the interested.

Here you can see the mirror's 70% zone focusing past the mirror's longitudinal axis. Each zone focuses along a curve called the 'caustic'. Interestingly the caustic curve was first drawn by Leonardo da Vinci in the early 16th century who imagined that the clear lens with such a curve would magnify light. Da Vinci came tantalizingly close to inventing the telescope. He was defeated by the extraordinarily exactitude of the mirror's surface and by impossibly poor glass material for lenses.

The first shadow and last light do not focus at the mirror's axis; they are instead focus offset to the sides. These points have a displacement to and fro the mirror's face and also laterally, left and right. The full form of the Caustic test measures both the 'x' and the 'y' displacement, computing the location of focus. In the Poor Man's Caustic test, or 'version 2' of the test as the ATM book series article titles it, only the 'y' displacement to and fro the mirror is measured. The 'x' displacement is not measured; instead, using the Everett stick bar, the zone of first light and last shadow as seen on the mirror is noted.

Image the familiar donut shape as seen in a knife-edge test. Initially, as the knife-edge is moved from one side to the other, a shadow forms. The knife-edge is at the radius of curvature for the mirror's zone where the shadow first appears on the mirror's face. As the knife-edge completes its movement, the knife-edge is at the radius of curvature for that zone as the light shrinks to its final rays, leaving the mirror in shadow. I use a fixed slit for light source and a stage where the knife-edge is moved to and fro the mirror with measured micrometer movements; the knife-edge can be rocked in a wide arc laterally to the mirror (right to left). I place an Everett stick horizontally across the center of the mirror's face with pins representing the zones to measure.

Here we start with the sketch as pictured on the right with first shadow appearing on the mirror's right hand side's 70% zone. As the knife-edge cuts further, the zone's width and height increases, eventually leading to the familiar Foucault donut as the knife-edge crosses the mirror's axis. This is pictured in the center sketch. As the knife-edge continues its arc, the light begins to shrink to an arc on the left hand side of the mirror. Eventually the light disappears in a tiny arc centered on the 70% zone as pictured in the left most sketch.

Be careful when measuring the zone: as the first light or last shadow collapses from a broader arc to a thin line (assuming very narrow vertical slite high intensity light source), the point to measure appears to move to the outside edge of the arc. Demonstrate this to yourself by rocking back and forth with the knife-edge.

I settled upon at least three zones: the 30% zone, the 70% zone and the 93% zone. The center and extreme edge are difficult to measure and can be extrapolated subjectively using the knife-edge to look for a smooth curve across the center and for a good ring at the extreme edge. I screw the stage to and fro the mirror until the first shadow and last light form at the sticks placed across the center of the mirror's face. I measure from center out, then from edge inward, averaging the readings if necessary.

A quick derivation from the equations describing the mirror's radius of curvature shows that the readings need to be divided by 3 to match the readings from the fixed-source Foucault test. Essentially this test measures the change in slope of the mirror's surface, or the second derivative of the parabolic deviation from spherical.

Here's an example from a 6 inch [15cm] F4 in progress. Test results show that outer half needs a bit more parabolization.

Advantages are: focusing attention on one side then the other, unlike the Couder Mask where you have to watch both sides of the mask simultaneously; a poor curve with zones is readily apparent. Disadvantages: takes practice to discern exactly where the shadow first appears and light last disappears.

Using this test I could see the mirror's radius of curvature expand over time as the temperature in the test hall changed. I could also see the narrow sharp first shadow and last light, and that very small changes, thousands of an inch, changes the zone being read. Fast mirrors come with more distinct arcs but tighter tolerances down to a couple of thousandths of an inch.

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 Dr. 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 that style and what is in vogue varies.

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, mirror makers can measure lots of zones today, 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 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.

What impacts success

Is it the testing method? Is it using rouge instead of cerium oxide? It's you - your personality, your self-awareness, your patience and discipline - these are the greatest determinants for success.

Bath interferometer test

The Bath Interferometer is absolutely wonderful, a revolution in the making for amateur mirror makers. The reference and test light beams travel the same path, negating instrument errors. By determining OPD (Optical Path Difference), the mirror's surface profile can be calculated. Key is free software for amateurs such as DFT Fringe by Dale Eason.

Other tests

The SCOTS test, a slope test, is intriguing.
Check out a new test called the Slit Image Test (
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 and my online version.
Finally, Mark Cowan has been developing the Unmasked Foucault 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.

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.

Control and variation...

Stroking a full-sized polishing tool against the mirror using not too long of strokes or spinning a 5/6 sized tool on the mirror drives the mirror's surface spherical. What can the mirror maker vary to induce a paraboloidal curve on the mirror's face? We know that polishing is a result of time, pressure, speed, and polishing compound. While accentuated pressure is occasionally used to fix a zone, it isn't used to parabolize because it isn't consistent. We can't easily vary speed or polishing compound, so that leaves time.

We vary time of polishing by lengthening and widening the stroking, by using a smaller pitch lap, or by altering area in contact of the lap. While it's possible to combine these techniques, it's saner to vary just one and control for the others.

Lengthening and widening the stroking using mirror on top is arguably the most popular technique for mirrors up to 12 inches [30cm] in diameter. It's well described in popular telescope making books. For larger and faster mirrors, this approach has trouble achieving the deeper paraboloidal curves; zones also become an issue.

Mirror makers turn to sub-diameter laps for larger faster mirrors. Deep parabolas can be carved out and zonal irregularities attacked. However, surface smoothness is an issue.

A smooth mirror surface along with deep parabolas can be achieved with full-sized or over-sized pitch laps by varying the pitch lap's area of contact. Strokes are kept simple and constant. Since pitch is a source of variability due to hardness, thickness and temperature, this approach constrains all the variables into the pitch lap. I find great success with this approach, a technique not widely used today but popular in the past. Brashear writes, "After six years of labor I reluctantly gave up the pursuit in this direction [a machine constructed so as to give an intricate motion to the polisher]... I have been led to this conclusion: that, given a properly shaped polisher, surfaces of the highest excellence may be produced; either by hand or machine work, and that the simple rotary and reciprocal motions are all that are necessary to be given to the polishing tool."

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.

Teetering on the precipice

Highly aspheric mirrors, mirrors that are large and very fast face a crumbling precipice between larger laps and smoother laps. Larger laps smooth better but have trouble fixing narrow zones and cannot keep up with the changing slope of the mirror as the lap moves across its surface. On the otherhand, smaller laps handle the changing slope across the mirror's face but have trouble fixing large scale correction issues. Pitch as commonly used isn't up to the task, hence the popularity of smaller laps. I find success with thick and soft pitch in the shape of petals and stars used on a relatively thin plaster tool that's supported by a thick rubber mat.

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, 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.)

The four parabolization methods using full sized tools...

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. This approach is featured in Sky and Telescope magazine for December 1974, where Storm Dunlop writes how a 24 inch [60cm] F3.8 was successfully finished with a graduated lap using the 'Mason method' after other approaches had failed. 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. The lap can be warmed by soaking in hot water or by a heat lamp.

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:

How much glass is removed?

A 10 inch [25cm] F5 has a zonal difference between the edge and center of 0.125 inches [3.2mm]. Yet the surface change is but 3.6 waves of green light either by lowering the center or by lowering the edge. Parabolizing by lowering both center and edge removes 1.8 waves of glass. By contrast, changing the radius of curvature by an inch calls for about 60 waves of glass needs to be removed (0.0013 inches [0.3mm]).

Examples of parabolizing and other laps

Here is what the 7 inch [18cm] oversized parabolizing lap looks like. This is meant to be used mirror on top. Note how 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] F2.7 mirror on a 11 inch [28cm] pitch lap).

Pitch lap calculator

I wrote a pitch lap calculator to help me design my laps. It's proving useful parabolizing very fast mirrors.

Parabolizing is like sculpting

Once glass is polished away, it cannot be added back on. The only recourse is to remove all the rest of the glass. We are free to pick a new radius of curvature to minimize the amount of glass removed.

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 mid-point of the mirror's area is 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.

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 F2.8 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 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] f3.0 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.

My steps to parabolize:
0. The better the sphere, the easier the parabolization since errors in the sphere will otherwise need to be corrected concurrently.
1. I start with mirror on top of a full sized or 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 30% of the mirror. This proceeds rapidly and does not need a lot of precision.
2. I continue with extreme chordal strokes, mirror on top, until the hollowing out reaches the 60% zone.
3. 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. For large very fast mirrors I use 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. 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 continue with longest possible strokes, center over center, until parabolization reaches the edge.
5. For very fast mirrors, this may leave the center flat. A brief spell of mirror on top of a star lap will deepen the center without affecting the edge's parabolization.

Here is an extreme star lap being used to deepen the center of a 25.1 inch [638cm] F2.62 mirror that calls for 62 waves of correction.

I wrote a pitch lap calculator to help me pick the best shape. You can see from this screen shot how an extreme star lap focuses polishing on the mirror's center.

6. The final step is fine tuning the parabolization by using the star test in conjunction with the Ronchi test.

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.

Random or regular

Strokes and rotations should be precise with no variation. End at the same rotation point around the barrel as you started. Machines can make beautifully smooth and highly regular surfaces. Be like a machine.

Logging sessions

I find it useful to log my work. I can look back to see how I solved this problem but more than that, logging my work makes me detail the problem specifically and commit in writing to the fix, predicting the outcome. Learning comes from measuring and analyzing the difference between prediction and outcome, creating a plan for the next step. Create a four column log: first column state the mirror's state, second column the proposed action, third column the outcome and the fourth column the analysis.

The Ronchi and star test tandem...

I've been grinding mirrors, making telescopes and observing with them for nearly 50 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, to verify results. The Bath interferometer is explained in great detail by others (see the 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 producing a mirror that focuses light perfectly at the highest powers, not so much arguing over Strehl ratio.

My experience from star testing many hundreds of telescopes over the decades is that every single mirror 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 F5 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] F4 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] F5 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 mirror's profile, to see the entire surface at one time and to adjust final parabolization. The tester is used in two positions: inside 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] F3.0 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,, 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.

Using the Ronchi Test to parabolize a 10.5 inch [27cm] F2.7

Here is my 10.5 inch [27cm] F2.7 that required 21 waves of parabolization. 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.
before ... after

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

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.
before ... after

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.
before ... after

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.
before ... after

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.
before ... after

Here are the final results: the star test shows identical diagonal shadow breakout distances 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. This means 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 and can be seen in these Ronchigrams, particularly the overlays.

Here are the final inside and outside radius of curvature Ronchigrams:

Compare to ideal:

And overlaying the actual with theoretical (note the outer 15% is very slightly overcorrected, consistent with the very slight issues seen in the star test, demonstrating that the Ronchi test can detect very small errors):

What is best - fewer bands or more bands? Fewer bands brings the zone closer to the grating with more sensitivity, however, issues within the band may be shadowed. Consequently it is best to test with the Ronchi grating in several positions.

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] F6.5 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 F5 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

The incredible shrinking radius of curvature

Determining the radius of curvature is important for accurate test results. Desired accuracy especially for faster or larger mirrors is 1/16 of an inch or 1mm. Radius of curvature is commonly measured from the mirror's central zone. Using a knife-edge test or a Ronchi test, place the knife edge or grating such that the mirror's center zone is nulled. Using a quality tape measure, supporting the tape so that it doesn't sag, measure from the mirror's center. Wrap the tape's end in masking tape so that it won't scratch the mirror. Also take the time to measure the mirror's clear diameter.

As the mirror's center is deepened during parabolizing, periodically re-measure the radius of curvature. I didn't re-measure often enough on a 25.1 inch [638cm] F2.62, only to find the mirror undercorrected in the star test when the Ronchi test showed good correction - a disappointment. A 0.3 inch [8 cm] shrinkage in the radius of curvature from 131.6 inches [3343cm] to 131.3 inches [3335cm] matters with this mirror. Luckily the mirror needed but a bit more parabolizing which was largely added during the subsequent work session. The mirror began figuring with a radius of curvature of 131.8 inches [3348cm], so the total shrinkage was 0.5 inches [13mm].

The Bartels' rule of smoothness first, always

No matter what goes on, strive for a smooth curve on the mirror. Fixing a kink only to create another kink is no progress at all. Parabolizing consists of smoothly approaching full parabolization with complete control of the mirror's surface, i.e., first the mirror is 25% parabolized, then 50% parabolized, eventually 90% parabolized; finally all zones on the mirror are brought up to 100% parabolization concurrently.

Turned down edge (TDE)

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.

TDE is readily 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.

Here are computed Ronchigrams for a 10 inch F5 with 1/2 wavefront (1/4 wave on the mirror's surface) TDE.

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 TDE takes time because the mirror's entire surface has to be polished down to the level of the turned edge. Remember that at the radius of curvature the center has to focus short compared to the edge. Simply lowering the edge by attacking the high point where the turned edge begins to plunge downward is not completely sufficient because that will leave progressively higher zones towards the center, resulting in an under-parabolized surface that requires more work.

Consequently TDE should be dealt with before parabolizing. Either accentuated pressure with MOT or TOT or a rectangular tool about 1 inch x 4 inch [2.5x10cm] in size stroked over the high point just interior of the TDE will smooth out the TDE quickly.

With faster larger mirrors, an overcorrected edge can develop. This typically develops into a rolled edge that spins into a TDE. THe only recourse is to return to a sphere and begin parabolizing again.

Overcorrected Outer Zones

The fix for overcorrected outer zones is returning to a sphere. This can be done with 1/4 to 1/3 long center over center strokes. The edge can be flattened with a pitch lap configured so that contact lightens inward like this:

Turned Up Edge

A turned up edge is equivalent to a high center is equivalent to a low zone near the edge. Simply continuining on with parabolizing will work down the center and the edge, making the low zone magically disappear.

Interpreting zones with the Ronchi test

Look at a spherical mirror being tested at the radius of curvature compared to being tested at the telescope. Note that the sensitivity at the telescope is halved. Depending on where the grating is placed, zones will be closer or further away from the grating, making the bands tighter or conversely more spreadout.

Here are the computed Ronchigrams for a spherical mirror starting inside radius of curvature and ending outside radius of curvature

Compare to a parabolized mirror, where the radius of curvature Ronchi bands are curved but those at the eyepiece are straight.

Here are the computed Ronchigrams for a 10 inch [25cm] F5 mirror with the grating offset relative to the radius of curvature -0.3, -0.1, 0.086, 0.3, 0.5 inches, the middle position with the grating on the 70% zone.

Now we can extend this to zones. Let's look at the 70% zone again.

Here are the computed Ronchigrams for a 1/2 wavefront high 70% zone followed by computed Ronchigrams for a 1/2 wavefront low 70% zone.

We can see generalized rules:

At the radius of curvature, a band that is too flat means that the zone is long ('too flat = long' rule).

At the radius of curvature, a band that is too curved means that the zone is short ('too curved = short' rule).

Combining these rules at the radius of curvature, too flat of bands are long zones and too curvy bands are short zones.

At the telescope (or auto-collimation test), if the the bands bow outward when the grating is outside or above focus, the zone is long ('out-out = long' rule).

At the telescope (or auto-collimation test), if the the bands bow inward when the grating is inside or below focus, the zone is long ('in-in = long' rule).

At the telescope (or auto-collimation test), if the the bands bow outward when the grating is inside or below focus, the zone is short ('in-out = short' rule).

At the telescope (or auto-collimation test), if the the bands bow inward when the grating is outside or above focus, the zone is short ('out-in = short' rule).

Combining these rules at the telescope, when the band bends towards the grating, the grating's position with respect to focus tells us when the zone is short or long; when the band bends opposite the grating's position, the zone is opposite the grating's position vis-a-vis focus.

A long or high zone needs more parabolizing while a short or low zone needs the surrounding glass polished.

The fix for a low 70% zone is polishing the high center and high edge down to match the low 70% zone. Another approach is to polish the zone broadly, smearing out the kink, leaving an easier to fix high center and high edge. Regardless, the zonal reading difference between the various zones must match the calculated values. For example, a 10 inch [25cm] F5 has a zonal reading difference between the edge and the 70% zone of 0.06125 inches [1.56mm], and between the 70% zone and the edge, an additional 0.064 inches [1.62mm].

Figuring by comparing Ronchigrams

The actual and theoretical Ronchigrams can be compared or overlayed in order to determine the next figuring step.

Consider our 70% high zone. Comparing actual to theoretical, we see that the mirror surface, represented by the gray Ronchi band, needs to be polished a tad lower in the 70% zone area such that the black theoretical band is completely covered. The rule is that deepening a zone shortens it, moving it closer to the grating when inside of radius of curvature, causing the bands to spread out more.

Here's an actual example, a 25.1 inch [638cm] F2.62, showing that I need to deepen the 75% zone and a broader 25-50% zone area (the high 75% zone was measured at 1/4 wavefront undercorrected).

Here's another example with the same mirror, where the center is undercorrected by about one-third wavefront. The star test looks modestly undercorrected, passes the snap focus test but fails the diagonal breakout test.

A session on the 25.1 inch [638cm] F2.62

Starting from the example pictured above, I worked four one-minute sessions with a ring lap concentrating action on the 70% zone with a bit of contact in the central 50% of the mirror. Here's the Ronchigram afterwards followed by a difference Ronchigram showing the change in the bands.

The mirror's bands fit closely with the theoretical bands, with a hint that the 60-70% zone is still a touch undercorrected. Using my online Ronchi tool, I generated an image with 1/8 wave undercorrection at the 70% zone - the mirror's bands are a good fit.

Indeed, the star test at high power as focus is moved from outside-focus to at-focus shows a ring just before the light collapses to a dot. The brightening disappears when masking the 60-70% zone of the mirror, confirming the Ronchi analysis. A bit more work to do!

Connecting zones and deriving a polish plan.

Observe the zones, noting if they are low or high and where they start and finish. For example, both the star test and Ronchi overlay test show a low zone centered on the 75% zone. During the star test I use masks of various shapes lightly taped to the mirror's face. Here I determined the low zone was 2 inches wide, ranging from the 65% to the 85% zone. This can be seen in the Ronchi test too where the 75% zone curves too much (overcorrected) compared to ideal.

Next, starting from the center or the edge, draw each zone in succession, attaching the beginning edge of the next zone to the ending edge of the current zone, curving each zone based if they are high, on target, or low. In this example, both the extreme edge zone and broad central zone come to the same focus but the narrow 75% zone focuses short. I start with the edge zone, drawing an arc centered on a focal point. I then draw the low 75% zone, connecting its outer edge to the inner edge of the extreme edge zone. Because this 75% zone focuses short, it's focus is shorter and tighter, resulting in a curve that plunges downward with respect to the outermost zone. Finally I draw the center zone to the same focus of the extreme edge zone, connecting its starting edge to the inner edge of the low 75% zone.

Then find different parabolas, selecting the one that causes the most surety of results and least amount of glass removal. The above drawing shows that I need to focus dwell time on the connection point of the 75% zone and extreme edge zone and lower the central zone.

I then derive a pitch lap contact map reflecting the dwell time necessary.

Finally I inspect results to verify analysis.

Fixing zones

It's a rule of thumb in star testing driven parabolizing (using either real stars or indoor artificial stars) to 'polish the zone that focuses long'. More precisely we want to alter the slope of the zone so that it focuses at the same point as the rest of the mirror. We can see from the above example that polishing the zone where the shorter radius zone transitions to the longer radius zone is the correct play.

The moat effect

Be cognizant of the moat effect: using too long of stroke or polishing over too broad of area, causing a moat to the polished surrounding the high zone. Studying the above results shows evidence of the moat effect occuring on the inside of the zone.

Zonal Irregularities

I discovered by accident after washing a mirror in warm water that the 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. Allyn Thompson, in his 1947 book, "How to Make a Telescope", describes how heating the glass exaggerates zonal irregularities. 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. The first example shows rough zones towards the edge; the second example shows smooth bands.

The fix is to use the largest sized laps possible with feathered, scalloped or stared edges with good blending technique. Varying strokes, adding a little swide swing or 'W' motion helps as well as warmer shop temperatures or softer pitch.

How to know you are finished...

Two factors work against us when completing a mirror. The first is our very own psychology and the second is the difficulty in testing our mirror. We are goal centric beings. Once we set a goal, say of 1/4 wavefront or 1/10 wavefront, we become highly motivated to reach it. We optimistically interpret the tests when we are within reach of the goal. Professional training is no inoculation – twice I’ve looked over the shoulders of professional opticians while testing who became convinced that the mirror was good, blind to significant errors.

My defense is twofold: first, I use more than one type of test, and secondly, I deliberately look for errors. Each type of test shows the mirror from a different perspective. It’s harder to skip over a potential iffy result when the error is revealed in different ways. For instance, there might be a tiny kink in the Ronchigram’s bands, the interferometer shows a little zone too, and the star test reveals some light is focusing short. The kink can be reasoned away, perhaps that interferometer report is a test stand problem and maybe the mirror wasn’t cooled sufficiently during the star test - but all three?

All mirrors have defects. If I can find them then I know the resolution of my testing. Consider a mirror test that shows no errors. What is more likely: that the test isn’t sensitive or is the mirror is truly perfect? Only twice in half a century have I been unable to see any defect in a mirror and I’ve tested many hundreds of mirrors, professional and amateur. So I test, test, test until the same errors consistently appear. When defects become a small fraction of the wavelength of visible light, testing like this takes considerable time and mental effort.

One reason why there are so many types of mirror tests is that testing to a small fraction of the wavelength of visible light is hard: each test tends to illuminate a different type of defect and have trouble revealing other types of defects. For instance, Foucault, Caustic and other zonal reading tests show correction for spherical aberration nicely, but might miss a narrow zone, surface roughness or astigmatism. Subjective tests like the Ronchi show smoothness nicely, but gauging precise correction is hard and slight astigmatism even harder to see. Interferometry is caught up in testing technique in order to remove test stand and mirror support errors and can have trouble revealing very small scale surface roughness plus difficulty with very fast mirrors. The star test quickly reveals small errors but can be caught up in cooling night time temperatures and uncertainly whether the error is in the primary mirror or perhaps in the secondary mirror, the mirror support or a thermal issue in the optical path.

What standard do I use? When do I say my mirror is done? My standard is an 'indistinguishable from perfect' star test at high power; when I can no longer improve the mirror's figure. Besides mastering the technique of mirror making and mastering multiple mirror tests, I find myself playing 'wackamole' with the mirror's zones, particularly on my large thin mirrors; a Zen master of zones if you will. Fix one zone and another goes catawampus. Pacifying all the zones simultaneously is an advanced skill that must be honed. In this final stage I will take many minutes to hours to prepare for a figuring session that lasts 30 seconds to a minute, that is half to a single walk around the barrel. Then it is many minutes to hours of testing. Repeat until I am unable to make further progress.

Fight psychology and testing difficulty by using at least two different types of tests until the same defects emerge. Once you reach “mirror reality” then you can decide whether the defects are significant and work must continue or whether the defect is immaterial for the telescope’s intended purpose.

The retrospective

After finishing the mirror, conduct a retrospective. Write down:

  1. What went well
  2. What went poorly
  3. What was plain ugly
  4. What I ought to continue doing
  5. What I should stop doing
  6. What I am still confused over
Improving isn't just about avoiding mistakes; it is also recognizing the good points and continuing on with them. Focus on reality, on the facts. Examine your approach and personality. Stay positive, stay motivated. Confusions are gaps in knowledge: seek out a mentor and conduct experiments.

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


- My mirror making articles at
- Advanced mirror makers who are also experienced observers

- Jeff Baldwin's telescope making pages
- 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,
- 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
- GAP 47's machines summary
- 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 (
- 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 (comprehensive collection of links to web articles)

(end of parabolizing)

For more see
Rough Grinding
Fine Grinding
Star Testing