# Spherometers

by Mike Lockwood

These notes are based on my actual use of spherometers.  I find spherometers more convenient and often more accurate than the sun test, flashlight test, etc., done on a wet, oiled, or soapy mirror.

The size of a spherometer depends on the application.  If you just want to measure sagitta at the center of a mirror or flat, then you want a spherometer base that's nearly as big as the mirror, so long as you can make a base that doesn't flex appreciably.  However, using a base that is smaller than the mirror allows it to be moved around, and you can detect local high and low spots such as zones that are produced during fine grinding (and which may take significant time to polish out).

### Calibration

It is important to realize that you need a "standard" to calibrate the spherometer on, and it must be at least as large as the base you wish to calibrate.  This standard is a decent flat (1/2 wave to 1/4 wave is fine), or a mirror of known radius of curvature.  The spherometer is typically zeroed while on the standard, and all readings are taken with reference to that calibration point.

I have a few different spherometer bases from 2" to 6" diameter, meaning the radius of the three feet from the center is 1" to 3".  I use the 6" the most, and I may make an even larger base in the future.  All bases are aluminum disks (except for one made from a pulley) with three ball-bearing feet.  That covers my needs of measuring all or portions of mirrors of nearly all sizes.  All bases have glued-in ball bearings as feet, and I used a good pair of calipers to measure their separations.  This was used in the spherometer equations to develop more accurate tables of ROC as a function of measured sagitta for a chosen base.

However, a better calibration method is to solve for the effective radius of the spherometer feet by measuring the sagitta of a mirror for which the radius of curvature is known to an accuracy of 1/8" or less.  Measuring more than one mirror of known ROC will help you judge the ultimate accuracy of the spherometer.  Sagitta measurements can be done with uncoated mirrors or coated mirrors, so long as you are careful not to slide the spherometer base around.  Paraboloids of normal focal ratios (F/4 and slower) should work fine for calibration - the asphericity will not affect the measurements too much, but if you have any doubts, calculate the asphericity for the parabola you are using, especially if it's fast, and take it into account.

After you have calculated the effective radius of the spherometer feet from the center, you can simply plug it into the equation and calculate the ROC for various spherometer readings.  I wrote a computer script to calculate a table of ROCs for each base that I calibrate, for both concave and convex surfaces.  Note that the formulas for the sagitta are different for convex and concave surfaces assumign you use balls for the feet of the spherometer base!  I often measure mirrors from optical systems that I have no information about, so these tables save much time and possibly some calculation errors.  At minimum, carefully calculate the ideal spherometer reading for the radius of curvature you are trying to hit.

### Usage

Repeatability of the readings increases as your mirror becomes more spherical - that is, as your grit size decreases.  There are two reasons for this.  Many don't realize that a mirror is not a fairly good sphere until fine grinding.  You can easily convince yourself of this by using a spherometer with an indicator that reads to 0.00005".  Also, large pits will throw off the readings as you slide the spherometer around on the surface.  So, sliding the base (carefully) across the mirror from one side to the other, you can see significant changes in the sagitta that you measure due to both asphericity and a rough surface.

Don't worry about waiting for the glass to equilibrate unless it just came out of the freezer.  Unless it is large, thin, plate glass blank with strain, I doubt you will ever measure an effect of glass changing shape due to heat by using a spherometer.  I haven't ever seen it with Pyrex.

Flexure and thermal expansion of the base can be problems.  To avoid flexure, use metal at least 3/16" thick, and thicker for diameters over 4".  I like aluminum - this helps avoid making the base heavy, which will help avoid scratches.  To avoid thermal expansion issues with the base itself, don't handle the spherometer any more than necessary.  If you are zeroing it on a flat (or other reference surface) keep them side by side, so you can move the spherometer from one to the other with a minimum of handling.

Make sure any glass surfaces and the feet of the base and the tip of the dial indicator are clean before measuring.  A quick wipe with a fingertip does the trick.

Following these suggestions, I get repeatability down the the accuracy of my best indicator (0.00005") with my spherometer bases.  That's a little over two waves over the diameter of the spherometer base.

### Accuracy

I calibrate my spherometer bases on mirrors of know radius of curvature.  Using this calibration method on my largest spherometer base, and with the repeatability mentioned above, for mirrors of around 60"-80" focal length I can easily control radii to +-0.25" fairly easily, and thus focal length to within 1/8".

I find spherometers very useful for making flats, too - one can get the surface flat to a handful of waves before polishing.  It is also indispensible for precise control of radii, especially for Cassegrain secondary mirrors with high magnification factors, where slight differences can change the system focal length by inches.  Seems like an ideal instrument for controlling the curves on refractor lens elements, too.

The spherometer bases I use are shown below, first the top, then the bottom, the latter showing the bearing placement.

Sizes range from 2" to 6" diameter of the circle that the bearings are layed out in.  The ball bearing feet on the pulley are 3/8" diameter and are epoxied on.  The other disks have 1/4" bearings that are super glued on.  All bases are made of aluminum.  All have bushings of bronze-type material to hold the dial indicator.  The right-most three bases were pieces of aluminum plate turned in a lathe.

Holes for the bearings were drilled with a countersink bit.  A circle was scribed in while the disks were still mounted in the lathe, to aid in hole location for the bearings.  Dividers were then used to evenly space the holes around the circle so that the bearings would be as close to the same distance apart as we could get them.

A closeup of one of the bearings and the scribed circle on the large disk at bottom left, is shown below.  This disk was a pulley bought at a surplus store for ~\$7.  The distance between two of the bearings is written on the bottom of the disk, and can be seen in the top middle of the photo.

Finally, the three indicators that I commonly use are shown below, sitting in a holder that I made from scrap plywood.  The holder ensures that the plunger is not depressed while the indicators are in storage.  The indicator at left is a ~\$10 model, suitable for rough measurement and use on a Foucault tester.  The one in the back reads to 0.00025".  The one on the right gets the most use, and will read to 0.00005" quite accurately and repeatably.  How many waves is that?  How many microns?  See my mirror-making conversion chart to get an idea.