Section 3 of the ATM List FAQ
Mirror making

Document Table of Contents

3.0 Grinding mirrors

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I don't feel that I can adequately or competently treat this subject as well as the many books, periodicals, and web sites do, not to mention others on the ATM list far more experienced than me. However, what would this FAQ be without at least an overview of the mirror grinding process?

Using the most basic process, you start with two round pieces of glass. One goes on a workbench or grinding stand and is called the tool. The other will be slid back and forth over the tool and will become your mirror. In between these two, you add a small amount of grinding powder, grit, and a little water. You rub the two in certain patterns. Due to the nature of the abrasive action you create, the glass on top will become concave (that is, curved inwards) and the bottom piece will become convex (curved outwards).

You proceed through a series of grit stages, generally starting with 80 grit, then 120, 220, and finally 500 (grits get finer as the numbers go higher). Next you use a finish grinding powder, usually 9 micron aluminum oxide, sometimes followed with 5 micron (in this case, the actual size of the grit grains are being noted, so smaller numbers are finer particles). By now, your mirror will have taken on a fine spherical shape.

Next, you use a polishing compound like cerium oxide to polish the mirror. During this time, you test and "figure" your mirror, turning it from a sphere into a paraboloid. This is the most precise time of the process. You will use tests, like the Ronchi test, Foucault test, and star test, to determine when you have hit the paraboloidal shape you desire. Finally, you send your mirror to a coating lab to have a thin layer of reflective aluminum applied.

Of course, you'll still need to build a telescope into which you will install your mirror. Most first-time telescope builders choose to build a Dobsonian style telescope. Advanced telescope builders often build much more complex telescopes, like catadioptric, tilted-component, and refracting scopes. You might say the sky's the limit when it comes to building telescopes.

Now that I've introduced the basic method to you, I will refer you to the real pros. Get some books from your library, astronomy club, or favorite book store, check out some of the ATM links listed in this FAQ and on the other ATM pages. For some introductory ATM information, check out
   

Finally, hang out on the ATM list. Then, buy a kit or supplies, and start grinding!

3.1 Beveling the edge

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Before you start grinding, you should bevel the edges of both your mirror blank and your tool. This will prevent a chip from the edges from breaking off during grinding or polishing and scratching your mirror. Here are a couple of suggestions from the list for easy and inexpensive methods for beveling the edges.

Method 1
The Carbide Stone Method - You can pick up a carbide stone at any hardware store. Other list members have suggested the knives section of hunting stores or kitchen / cooking supply stores as other locations where you can purchase carbide stones. Also, check out stained-glass hobby shops or catalogs. Just be careful, one slip with a (hard) stone and you can knock a big chunk out of the edge. Also, a coarse stone makes the job a lot easier. I found the carbide stone from my wife's stained glass projects finished the beveling job in minutes whereas using a knife-sharpening stone took me the better part of an hour.

Method 2
Go to the hardware store and pick up a hard rubber sanding block and a few sheets of 80 or 100 grit carborumdum paper (the black stuff). Use this to put the bevel on, with water as a lubricant. You'll end up with a more rounded edge than with a stone - which makes it more chip-resistant in the long run - and there's no danger if you accidentally drop it on your blank. With too fine a choice in sandpaper (like 500), this method will be slower than method 1.

Method 3
To bevel the mirror edge, use a mixing bowl (glass or stoneware type) that your mirror will fit into, but not all the way to the bottom. Spread some grit and water over the sides of the bowl and place your mirror, face down, inside the bowl. Place your hands on the back of the mirror and wobble it around. After about five minutes, you should have a 1/8 to 3/16 inch bevel.

3.2 Removing pitch from a tool

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Members of the list have offered many creative suggestions for removing pitch from a tool. Here are a couple:

Method 1
To remove the pitch (from a glass tool at least,) wash the lap in clean water to remove the rouge or CeO. Place the tool in a plastic bag then into the freezer for about an hour. Remove from the freezer, after about 15 seconds the pitch will begin to make a crackling sound and will begin "jumping" off the tool. There will likely be a couple of spots that need to be worked at with a wooden stick. The tool can then be cleaned with turpentine and you are ready to grind again. The pitch can be reused if you take good care of it and don't let it get contaminated (grit).

Method 2
Put the tool with pitch under hot water until it is really soft. A soapy knife will quickly remove most of the pitch without mess. Now put the tool in the freezer for a bit. The icy pitch remnants shatter off the tool when attacked with a kitchen knife. It's really easy and really messy (which is why most of the pitch is removed first).

3.3 How to package a mirror for shipping to the coater

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This subject has been discussed a few times on the list. Howard Clausing of P.A. Clausing, Inc. provided some advice, which I have quoted here. I edited his response just a bit to make it generic (he referred to specific mirror sizes in his original note).

First ... wrap the mirror in a nice soft cloth. Use a cardboard box that will afford at least 2 inches of packing material between the mirror and each of the 6 faces of the carton. For example, for a six inch mirror, you need at least a 10" X 10" X 6" carton. I would recommend slightly larger.

Next cut layers of bead foam (Styrofoam) to fill the entire volume of the carton. 2" thick sheets can be found at most building supply stores. For the above example, cut three 10" x 10" x 2" sheets. Cut a 6" dia. disk (hole) out of one of the sheets. Use the solid layers for top & bottom and nestle the wrapped mirror in the middle fitted layer (in the hole). This method completely immobilizes & protects the mirror. The light weight of the bead foam saves on shipping costs.

Others on the list have suggested many other ways. Please check the archives for those solutions.

3.4 Testing for good contact during grinding

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In his "Standard Handbook of Telescope Making," N.E. Howard states that the only two surface shapes that can be in complete contact are a pair of plane (flat) surfaces and a pair of spherical surfaces. If you've been grinding, you are probably not going to still be at a flat. So, by determining the completeness of contact between your mirror and tool, you can check the shape of your mirror -- is it a sphere yet.

Many of the books recommend that you watch the bubbles through the top piece of glass. The bubbles should become small and uniform in size when you have good contact. On the ATM list, the most often recommended method for determining good contact is the Sharpie test. Use a thin permanent marker (like a "Sharpie") to draw a grid of lines across your mirror. A pair of lines should intersect at the center of your mirror. Grind for a few strokes and check the lines. The lines should be worn away evenly across the entire surface of your mirror. If not, you do not have good contact and should keep grinding. Re-draw the lines when you want to check again.

3.5 When to move on to the next grit

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How you determine when to move from one grit to the next depends one where you are in the process of grinding. In the initial stages of grinding, when you're hogging out the curve, your primary test for when to move on is the shape of your mirror. Later, you will check for the pits and defects left behind by one grit before moving on to the next.

In the beginning, when you are hogging out the basic curve, your goal is to produce a sphere. By now, you should have chosen the basic design parameters of your mirror, and most importantly right now, the focal length. The focal length is going to determine the shape of the sphere you are grinding. A long focal length scope will have a shallow curve whereas a short focal length scope will have a deep curve. To determine when to move on, you need to know how close the curve on your mirror matches the goal you have in mind.

The most common techniques for determining the curve are measuring the sagitta (or depth of the curve) or doing the spit test.

Measuring the sagitta

So, why does measuring the depth you have ground to tell you your radius of curvature? Well, check out this diagram of a sphere (in two dimensions, the scale of the curve is greatly exaggerated) and some basic measurements. In the diagram, r is the radius of your mirror, R is the radius of the sphere of which your curve is a part of, and S is the depth, or sagitta, of your curve.

Sagitta diagram

If you're a math wiz, you could prove that

S = R - Sqrt (R2 - r2)
or you could take my word for it. By measuring S, you can determine the radius of curvature of your mirror. (For a JavaScript-based sagitta calculator, check out the ATM page / Optical Miscellany / ATM Calculator page.)

Now, how do you measure the sagitta? Well, you could build or buy a spherometer, an optician's tool designed especially for measuring sagittal depths. Or, you could use a ruler and a set of automotive "feeler" gauges. Feeler gauges are used to set spark plug gaps and can be purchased just about anywhere you buy automotive parts and tools. These gauges are accurately produced to measure in the half-millimeter scale (typically), which is sufficient for now. Carefully lay the ruler across the diameter of your mirror. One by one, starting from the smallest gauge, slide the feeler gauges under the ruler at the center of your mirror until the gauge no longer fits. Now, you know your depth is somewhere between the last gauge that fit and the gauge that doesn't fit. Plug that number as the value for S into the formula above and you have a fairly accurate idea of the depth of your curve. For more accuracy, try the spit test.

The spit test

For this test, prop your mirror on its stand and wet it with a spray bottle of water. Hold a flashlight next to your head aiming both at the mirror. Move back and forth until you see the entire surface of your mirror grow brightly. When you are inside the radius of curvature (closer to the mirror), moving your head (and flashlight) down will make the glow on the mirror appear to move up, and vice versa. When you are outside of the radius of curvature (further from the mirror), moving your head (and flashlight) down will make the glow on the mirror appear to move down, and vice versa. At the radius of curvature, the mirror should glow brightly fairly uniformly across the entire surface.

Now, rumor has it that you are supposed to drop some spit on the floor. Then, measure from the mirror to the spit-mark to determine the radius of curvature of your mirror. If you are doing this in your kitchen, get your spouse's approval before attempting this technique! You can choose a technique of your own to measure from your eye's location to the mirror. I just use a tape measure and approximate.

Once you have hogged out your curve and have moved on to the "fine" grinding stages, you will need to use a different technique to determine when to move on from one grit to the next. At this point, measuring the curve isn't your goal, making sure the surface of your mirror is free from unnecessarily large pits and defects is your goal.

Checking for pits

The most basic technique for checking is to use a magnifying glass to inspect the surface of your mirror. Under relatively low magnification (5x or so) you should be able to easily spot all sorts of pits, divots, and spots on the surface of your mirror (make sure to inspect under bright lights). You can use a regular magnifying glass, jeweler's loupe, or even an eyepiece as a magnifier. To use an eyepiece, choose one with about a 25 mm focal length. Sight through it backwards at the mirror. It will act as a nice low-power magnifier.

The defects on your mirror should be evenly distributed across the surface of the mirror. Of course, the grinding you are doing is going to leave pits. That's the purpose of progressing from course to fine grits -- you remove the previous grit's pits before moving on to repeat the process with the next finer grade. What you are concerned with is unusually large pits remaining. Check for even size and even distribution. If you spot larger-than-average pits, keep grinding. You're not ready to move on. You would be best served to find the largest pits and mark them. Then, grind for a while longer and re-check the marked pits. Once they are gone, you can be sure the rest of them are gone and you'll be ready to move on.

Some recommend marking large pits by sighting through the back of the mirror and marking on the back. Others mark directly on the face of the mirror with a permanent marker (a "Sharpie"). The marker's ink will fill the pits in your mirror. If after a bit more grinding, you still see small spots of ink, you're not ready to move on.

3.6 How do you tell when you're done polishing?

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An often asked question is "how do I know when I can stop polishing." There have been many ideas posted to the list that might help you.

You are done polishing when you have removed all of the pits and surface defects from your mirror. At that stage, the front surface of your mirror should be so smooth that you cannot see it (we're talking about your mirror before you get it aluminized, obviously). Of course, if there is dust or film on the glass, you will be able to see that. Start by rinsing the surface of your mirror well to remove any dust. Let it dry completely. Use a small flashlight, holding it an inch or two away from the glass, shining towards the mirror at an angle towards you. Look closely for a reflection from the front surface of the glass. You should see right through the mirror's surface and not see the image of the bulb on the mirror surface. (You will be able to see reflections from the back surface of the mirror.) If you do see a reflection off the front surface, with perhaps a faint milky sheen, it means there are tiny pits absorbing the light.

Another method suggested to the list was to take your mirror outside on a sunny day (again, assuming your mirror is free of dust and film and still uncoated). Use a magnifying glass to focus sunlight onto your mirror. When fully polished, you should not be able to focus a point of light on to the front surface of your mirror.

The most likely place you will see unpolished surface is going to be the edges of your mirror. So, check those edges well.

3.7 Standard versus enhanced coatings

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Okay, you've finished grinding, polishing, and figuring your new mirror. Now, do you go for standard, semi-enhanced, enhanced, fully-enhanced, or some other esoteric coating? Each is more expensive than the last, so you need to know just how they will effect the performance of your scope. Click here for a calculator (JavaScript-based) that will help you compare the effectiveness of different coatings on your Newtonian (or other two-element) optics.

3.8 Removing the coating from your mirror

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If you want to remove the coatings from a mirror, before re-coating for example, you have three basic choices: chemically strip the coating, polish off the coating, or pay someone else to remove the coating. The opinions of experts vary, of course, as to which is the best.

In a note to the list, Mr. H.R. Suiter said: "ATMs should ALWAYS strip the coating themselves, inspect the surface, and retest. That way they'll know the condition of the glass before they ship, and will meet fewer surprises upon its return." He went on to recommend black rouge, especially when dealing with Pyrex mirrors. He says "black rouge is almost ideal for removing old aluminum coatings (I've used it since 1985). The reason: it scarcely cuts the harder Pyrex substrate beneath. Even on overcoated aluminum you have to rub and rub."

On the other hand...

You could inadvertantly scratch your mirror, adding extra work to your simple re-coating process. Very fine scratches and sleeks often don't show up until after the coating is applied. You wouldn't want to blame the coater for scratching your mirror if you did the scratching yourself. So, in response to Mr. Suiter's note, Howard Clausing of P.A. Clausing wrote "Further polishing on a finished surface is only an invitation for trouble. I maintain that chemical removal of a coating is by FAR the safest way to solve the problem."

To remove your own coating by chemical means, you can use Ferric Chloride, commonly sold as circuit board etching solution (available at Radio Shack and probably many other sources). Mr. Clausing warned that Draino or other drain cleaners can sometimes etch glass (these cleaners are sometimes recommended to remove coatings). While many list members have used it with no ill effects, maybe better safe than sorry?

I have found that ferric chloride also works on silver coatings. I had to use a piece of surgical cotton (available in rolls at pharmacies) to remove the most stubborn parts. Very lightly drag the cotton over the surface. Don't apply any pressure to the cotton -- the pressure from just the weight of the cotton wad is all you want to use. Be careful with ferric chloride, it will stain things a bright lemon yellow. So, wear gloves and old clothes; especially wear safety glasses.

Finally, the third opinion, expressed by Peter Nance of Precision Applied Products, is to let the coating firm remove your old coating. He says that his company charges a small additional fee ($5 US at the time of his note) to remove your old coating. Check with your favorite coating firm to see if they offer this service. Of course, if you choose this option, you lose the opportunity to check the mirror's figure and make any improvements before recoating.

3.9 A PostScript program for generating a Couder mask

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Michael Lindner has written a PostScript program that will generate a custom Couder mask when sent to a PostScript-based printer. (Or, if you use Ghostscript, you can send to other types of printers.). Download the file from here and save the file as couder.ps. You can also use this web page to create a customized PostScript file.

To print out the mask, you must first edit the file (with a text editor like Windows Notepad) and specify the particulars for your mirror. At the end of the file are three similar sections that look like this:

        6 inches mirror
        0.75 inches 1.80 inches zone
        1.80 inches 2.43 inches zone
        2.43 inches 2.93 inches zone
        showpage

The above example will create a mask for a mirror six inches in (real, not optical) diameter with three zone whose inner and outer radii are give. If you leave all three sections in, you'll get three copies of the mask. Change the numbers to suit your mirror and testing needs. Then print the program. For my system (running Windows NT), at a DOS prompt I type copy couder.ps lpt1 (my printer is connected to the LPT1 port). I assume that this will work for any DOS or Windows type system with a PostScript printer. Consult your system documentation for help with printing the file.

Once you have the printed mask, tape or glue it to your card stock (or whatever you're going to make the mask out of) and cut away! It should be printed full scale to the best accuracy of your printer.

For more information, you can read Michael's original posting to the ATM list from the Archives at: http://www.system.missouri.edu/ics/staff/andy/ATM/ARCHIVES/SEP97/0213.html.

3.10 Types of glass

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There are many types of glass, not all are suitable for grinding. Here's a brief discourse on the types of glass you are most likely to encounter.

In general, the term optical glass refers to glass used for lenses and prisms. We'll extend that to include glass used for astronomical mirrors. Optical glass used for lenses and prisms must be colorless and transparent and be free of physical defects, such as bubbles, inhomogeneities, striae, and strain. It must also have known optical qualities, for example how much it bends (refracts) light and how that bending changes with different colors. Optical glass used for mirrors can be less exacting, as the light doesn't pass through the glass. It primarily must be free of strain and have appropriate mechanical properties (appropriate hardness, elasticity, and resistance to weathering).

Plate glass and window glass are the same stuff. Both are basically "soda-lime" glass - a blend of silica (silicon dioxide), sodium oxide (from soda - sodium carbonate), and calcium oxide (lime). Plate glass is generally distinguishable by having a faint green color, most noticable when you look at the edge of a piece. Plate glass is perfectly suitable for mirrors, especially for average sized mirrors (under 12 inches or so). Plate glass is not suitable for lenses or prisms.

Pyrex™ and related glasses are "borosilicate" glasses, which have boron oxide in the mix. Pyrex is a preferred glass for telescope mirrors because it expands and contracts less with changes in temperature than plate glass. Pyrex is harder than plate glass, thus is more difficult to grind and requires more grit. Pyrex is generally colorless or slightly grey in color; older Pyrex can be yellow-ish (straw colored). Pyrex is not suitable for lenses and prisms.

Tempered glass is generally soda-lime glass that's been through a heat treatment to put the surface under a strong compressive strain; this makes it fracture into small, roughly square, fragments rather than long sharp needles when it breaks - great from safety standpoint - but also makes it almost impossible to grind without shattering. Don't try to grind this type of glass. Tempered glass is not suitable for mirrors, lenses, or prisms.

Laminated glass (or "safety" glass) consists of (typically) two layers of glass (often tempered) with a thin adhesive plastic layer in between, to keep the pieces from flying when it shatters under impact. Typically, this is the type of glass used in automotive windshields. This type of glass is pretty much useless for optical work.

Flint glass in general has a high index of refraction and a high dispersion, that is it bends light a lot and how much it bends it is dependent on the color. Flint is suitable for lenses and prisms. It will work for mirrors, but using it as such would probably be considered a waste of glass that could be better used in a different project.

Crown glass has a low index of refraction and a low dispersion, hence it bends light less than flint but that bending is less color dependent. Crown glass generally is generally harder than flint and also less prone to enviromental damage. This is why the crown element is generally the lens that sees the air and the flint element sees the tube. Crown glass is also generally less expensive than flint and easier to make. Crown is suitable for lenses and prisms. It too will work for mirrors, but using it as such would probably be considered a waste of glass that could be better used in a different project.

Exotic glass is glass that uses rare-earth or other elements to produce special optical qualities for lens and prism elements. For example, the addition of lanthanum, thorium, and tantalum enable glass designers to produce glass with high refraction and low dispersion at values greater than possible with "normal" flint and crown glass.

To create color-corrected lens assemblies, combinations of glass must be used so that each color is brought to the same focus point. A typical camera lens, for example, will contain four or more lens elements. Each of those elements will have different optical properties (index of refraction, dispersion, focal length, etc.). It is the combination of those elements' properties that make it possible to design lenses that are free of chromatic, spherical, and coma abberations, astrigmatism, field curvature, and distortion.

So, you've found a hunk of glass. How do you tell what type of glass it is?

You could hit it with a hammer. If it breaks into small, roughly square fragments then it's tempered; if it breaks into long sharp slivers then it isn't. Okay, so this isn't a very good idea.

When viewed edge on, is it green colored? If so, it is probably plate glass.

When viewed edge on, is it colorless, slightly gray, or even straw colored? If so, it could be pyrex.

Finally, you could check its density (mass/volume).
      Glass type Density
(g/cm3)
Linear coefficient
of expansion
(um/m°C)*
Index of refraction
(Nd)
      Plate      
      Pyrex 7740® 2.23 3.2 1.474
      BK7 2.46 7.1 1.517
      BaK4 2.85 7.0 1.569
      Fused silica 2.20 0.55 1.458
      Zerodur® 2.53 0.05 1.542
      Sitall®      
      BVC**      

* Units for coefficient of expansion or microns/m°C.
** BVC stands for "Black Vitreous Ceramic" and is sold by ASM Products.

How do you tell if it is tempered? Look at it through crossed polarizers - tempered glass usually shows a lot of strain bifringence. Get some polaroid material (cheap plastic clip-on shades will work) and view through the glass with one sheet on each side rotated 90 degrees to each other. For instance, clip on one pair to your glasses, hold the other polarizer in front of your light source and rotate it until you get maximum extinction. Now insert the glass in question between the polaroids. Tempered glass will exhibit distinct patterns of dark and light bands or other shapes. You might even get groovy colors. If you can see through the glass edge-on, then it will show a dramatic pattern of colored bands parallel to the faces. If you see no patterns, and light is evenly extinguished as you look through it, then it is not tempered and is pretty well annealed.

Finally, you can examine the glass closely for any brand name etched into it. Examine the edge for any sign of fire polishing. Examine the surface for little dimples where tongs may have indented the glass. Any of these indicate tempering, and you should not try to cut or grind this glass, or you will have a terrible mess.

Glass for mirrors is often chosen based on its coefficient of expansion. This value specifies how much the glass expands (or contracts) with changes in heat. This information is crucial in determining how much the figure of your mirror will change as it is cooling down or heating up. See the table above for some common values.

Once thermal equilibrium is reached, the figure will remain fixed no matter what the material. This means plate glass is perfectly suitable for a mirror. Of course, if you have a large thick mirror or don't store your scope outside (where it will remain close to the ambient temperature), your mirror might never cool completely. Thus, its figure will change during your entire observing session. In these applications, Pyrex, Sitall, or Zerodur might make a prudent choice.

3.11 Scratch-dig numbers

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In some optics references, for example the Edmund Scientific catalog, you will find glass rated by scratch/dig numbers. These numbers note the defect size and number in relation to the size of the part. The first number is for scratches, the second for digs, (i.e. pits).

According to the U.S. Military, who defined this spec:

"Scratches and Digs are defined by the U.S. Military Specification for the Inspection of Optical Components, MIL-O-13830A. Scratch numbers are essentially defined by the width of a scratch in 1/10,000 mm. However lengths of scratches and combinations of smaller scratches also contribute to the scratch number. Dig numbers are defined by the actual diameter of a dig in 1/100 mm. Smaller digs and irregularly shaped digs also affect the dig number. Both scratch and dig numbers are defined within certain areas of an optic and are defined differently for optical components by the MIL-Spec. The specification is denoted by the scratch number followed by the dig number."

80-50 Commonly acceptable cosmetic standard
60-40 Good commercial surface
40-20 Starting to be a true optical surface
20-10 High quality non-laser surface
10-10 Suitable for demanding laser applications.

So, in terms of microns, scratch widths are as follows:
80 scratch = 8 microns width
60 scratch = 6 microns width
40 scratch = 4 microns width
20 scratch = 2 microns width
10 scratch = 1 microns width

And digs are:
50 dig = 0.5mm diameter
40 dig = 0.45mm diameter
20 dig = 0.2mm diameter
10 dig = 0.1mm diameter

3.12 Abrasives - size and classification

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Stephen Tonkin maintains information on abrasives (grits) at http://www.aegis1.demon.co.uk/abrasiv.html. His information includes the particle size related to various classification schemes and also the elutriative time. The times given are the time to settle in a 1m column of water.

3.13 Foucault and slitless testers

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Foucault and slitless testers are variations on a theme; the principles behind both are the same. A light source is placed at the radius of curvature (twice the focal length) of the mirror being tested. With a Foucault test, the light source is covered by a very narrow slit. With a slitless tester, half of the light source is covered by a knife edge. The returning light is partially blocked by a knife edge parallel to the slit or knife covering the source.

If your mirror is a perfect sphere, as you move the knife edge into the returning cone of light, your mirror will darken uniformly across its face. If your mirror is not a sphere, for example it is a paraboloid, zones (concentric rings) on your mirror will darken before or after other zones. Remarkably, this gives the impression that you're seeing the actual shape of the mirror's surface.

Testing is then done by measuring the longitudinal shift (the distance towards and away from the mirror) inside or outside of the radius of curvature where certain zones darken. This is done with the help of a Couder mask, which exposes selected zones on the mirror (in this figure, three zones). To test, you simply note the micrometer readings where each of the three zones darken uniformly as the knife edge is inserted into the returning light. Read from the center out (center = zone 1). Enter these values into Tex or plot them on a Millies-Lacroix graph to determine the figure of your mirror.

For complete information on testing and data reduction, see one of the ATM books, such as Texereau's How To Make A Telescope.

Berthold Hamburger has a nice set of plans for a slitless tester on his web page (http://www.geocities.com/capecanaveral/9601/atm.html). His tester is very similar to the slit-type tester detailed by Texereau. One of these days, I'll snap a picture of my slitless tester and put it here, too. [I will add some additional links to plans for testers available on the Internet as I have time. - Tim]

Michael Lindner has kindly provided a couple of PostScript programs for making dials for your Tex-style micrometers. You can download the PostScript programs and modify them yourself (rule.ps is for creating an indicator that would run around the circumference of your micrometer knob (like Tex describes); dial.ps can be used to create a dial indicator like Berthold uses for his slitless tester; Each has comments (the lines that begin with %) that describe how to modify them to meet your needs) or you can use this web page to create customized PostScript files to meet your needs. Also, here's a PDF version of each for a 2 inch diameter knob rule.pdf, dial.pdf.

Thanks to Stephen Tonkin, I have a list of some books and periodicals in which slitless testers have been discussed. See section 5 for information on these publications. Don't forget to check out the archives, as slitless testers have been discussed on the list many times.

Allan Mackintosh, Advanced Telescope Making Techniques - Vol. 1, Willmann-Bell

Dick Suiter, On Slitless Testers, Telescope Making #22, pp16-19

Ralph K. Dakin, An Improved Foucault Testing Device, Sky and Telescope, Jan 1967, p45

Robert E. Cox, Further Improvements on a Slitless Foucault Tester, Sky and Telescope April 1967, p248.

You may also be interested in:
Robert T. Holleran, The Double-Pass Knife-edge Test, Sky and Telescope August 1995, p86.

I probably should have a section devoted to spherometers and using them for determining the optical shape of your mirror. Till then, let me mention Perru Keinänen's Spherometer Analyzer page (http://sirius.astro.utu.fi/formit/sfero/sfero.shtml). This page helps you analyze readings taken with a spherometer. Enter spherometer parameters and your readings (up to 20 across the diameter of your mirror) and the page calculates the surface shape of your mirror and residual errors as compared to theoretical sphere closest to your mirror's shape.

3.14 The wave nature of light

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We tend to think of light as moving from source to target in straight lines, as if it were made up of a stream of bullets fired from some luminous gun. For example, we can model reflections very well using light "rays." Simply draw straight lines following the rule that the angle of incidence equals the angle of reflection to figure out how light will reflect off a surface. Scientists use the term photon to describe a "particle" of light.

In some cases, such as calculating how good the optics of a telescope are, the particle model doesn't work so well. For those cases we need to look at light in another way. In addition to acting like particles, light acts as a wave. You could think of waves on the ocean--the surface of the water moves up and down rhythmically in an oscillating cycle. With light, however, the waves don't move in anything (though scientists searched long and hard for the æther that they thought light waves did move in).

Like waves on the ocean, you can measure the wavelength of light as the distance between either two peaks or two troughs. Light in the visible range of the spectrum has a wavelength of between 0.4 microns (blue) to 0.75 microns (red). (See http://wine1.sb.fsu.edu/chm1045/notes/Struct/Wave/Struct01.htm/ for more info.) A micron is 0.001 millimeters or 0.00004 inches. So visible light has a wavelength of between 0.0004 millimeters to 0.00075 millimeters.

If you've ever stood on a breakwall and watched the waves come in, you will have seen them curve around the end of the breakwall into the protected area of the harbor. This is an example of diffraction. Waves will propagate around an obstruction. Light does this too. But what would provide the obstruction in your telescope?

The edges of your telescope tube, the edges of the secondary mirror, and the vanes that support your secondary are all obstructions in the path of incoming light. Even with a refractor, you have to consider the telescope tube edges as a source of diffraction. Look at the following picture. It shows how light will diffract through a couple of slits -- they are analogous to the edges of your secondary mirror.

Regions where the waves from one slit (or one edge of your secondary) line up with the waves from the other slit are said to "constructively interfere." The intensity of the light at that spot is the sum of the intensity of each of the waves. Regions where the peak of a wave from one slit meets up with the trough of the wave from the other slit (or vice versa) are said to "destructively interfere."

You could think of the peaks of the waves as having positive values and the troughs as having negative values. (The "negative" value of a trough shouldn't be taken to mean that there's no light or it's somehow "negative light" there. At either a peak or trough, you would see light.) The zero value in between is the case of no light. If you add two positive peaks or two negative troughs, you get an even stronger (brighter) peak. Those are cases of constructive interference. Add a positive peak to a negative trough and you'll get something closer to a zero value. The top of a peak added to the bottom of a trough results in no light at all. That's destructive interference.

So why does all this matter?

Because of this wave nature of light, light from a star or anything else for that matter can never be focused to an exact point--no matter how good your optics are. The peaks and troughs of the light waves, caused by the diffraction of your scope's tube, secondary, and spider, will always interfere at least a little. The result is the airy disk and the surrounding diffraction pattern, shown here.


Size and brightness of rings exaggerated.

The best telescopes will create a perfect airy disk* and diffraction pattern. Less-than-perfect telescopes will not. Irregularities and imperfections in the shape of the optics will send more or less light into the central disk or surrounding rings. This effect is still caused by diffraction and interference. It's just that the shape of the waves that interact in the telescope are distorted by the imperfections in the optics. Thus, it is the goal of the mirror maker to produce a telescope that gives a perfect diffraction pattern.

In fact, the principle behind star testing a telescope is simple: compare the actual diffraction pattern with the ideal one. The differences reveal the severity and nature of the optical defects. Dick Suiter has written a very detailed book, Star Testing Astronomical Telescopes, on the subject.

*The diameter of the airy (central) disk is (1.22 * lambda * f)/D, where lamda is the wavelength of light, f is the focal length, and D is the aperture diameter.

3.15 What's "1/4 wave" mean?

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Your goal in figuring a mirror is to create a perfect paraboloidal shape. Your mirror's figure must be so precise as to differ from the ideal shape by mere ten-thousandths of a millimeter--that's a fraction of the wavelength of visible light. Thus, you'll see terms like "1/4 wave" to describe the level of perfection of an optic. The figure is describing the amount of deviation from the perfect optic.

Consider a mirror with a small depression that is 1/4 of a wave deep. A wave of light coming in to the mirror will have to travel 1/4 of a wavelength extra into the hole and another 1/4 wave out of the hole. The waves that hit the mirror at the hole will be 1/2 of a wave out of phase as compared to the waves that don't hit the hole when they reach the focal plane. The phase differences will cause destructive interference, creating dark regions in the image. The same would be true with a 1/4 wave high hill instead of a hole.

Let's consider a mirror with a small depression 1/8th of a wave deep and a small hill that's a 1/8th wave high. How good would this mirror be? Let's start with the terminology used to describe such a mirror. You could describe it as an 1/8th wave, 1/4 wave, or 1/2 wave mirror. It is a "1/8th wave on the surface" mirror, because that would describe the largest deviation of the surface of the mirror at any one point. It is a "1/4 wave peak-to-valley (PV)" mirror, because that would describe the error from the peak of the highest hill to the bottom of the lowest valley on the surface of the mirror. It is also a "1/2 wave on the wavefront" mirror, because that would describe the extremes of the phase deviation of the waves at the focal plane.

Most manufacturers of telescopes blur the difference. In most cases, they are reporting surface errors because the numbers sound better. After all, which would you rather buy, an 1/8th wave or 1/2 wave mirror?

Of course, a single wave rating number doesn't give us a complete picture. In fact, the example we used illustrates the problem--it had a single hill and single valley as its errors. What if those spots were in the shadow of the secondary. You'd have a perfect mirror in practice. A wave rating number could describe a single bump in the mirror. Or, there could be many errors, that all add up to the wave rating given. Or worst case, half of the mirror could be a fraction of a wave too low and the other a fraction too high. You could still describe such a mirror with a single wave rating number and that number wouldn't look all that bad.

The "wave" rating is rather ambiguous. So, opticians have developed many ways of being more precise when speaking of their optics. One is to calculate the root-mean-square (RMS), which is sort of an average across the surface. The Strehl ratio takes into account not only the difference in height, but how much of the mirror's area is off. It expresses the fraction of light that the mirror concentrates into the airy disk. A perfect mirror, therefore, has a Strehl ratio of 1--it reflects all of the light into the airy disk. In fact, when you do the Foucault test with a Couder mask, you look at a series of concentric ring-shaped zones and calculate the errors for those zones. You're determining more than just a single wave rating number for your mirror with this test.

For reasons that are too difficult for me to understand, let alone describe, it becomes very difficult to measure optics errors when they become very small fractions of a wavelength of light. In fact, some members of the ATM List will vehemently argue that saying a mirror is anything more precise than 1/8th of a wave (surface) is just stroking one's own ego. On the other hand, others have passionately argued that they could reliably measure errors (using just the Foucault test) to the range of 1/25th or even smaller. I'll leave it up to you to believe what you will. If you can make a mirror that tests to 1/8th wave on the surface, you have an excellent mirror. If you feel you need to claim better numbers than that, just be prepared to back up your claim with some real evidence (like repeatable Foucault readings).

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This document, its contents, and its Web representation are Copyright ©1996, Tim Poulsen. For complete copyright information, including allowed uses of this FAQ, please see Section 8. Initially created on Thursday, September 12, 1996 by Tim Poulsen, poulsen@netacc.net.