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A spherometer is used to measure the radius of curvature of a spherical surface. Most lenses have surfaces that are either spherical or close to it. For such surfaces, a spherometer provides a simple way to verify that the curvature is correct. The basis of the method is that the sag of a spherical cap can be measured and from this measurement the radius can be calculated. The equation relating the sag to the radius is

R = (y²+z²) / 2z

where R is the radius of curvature, y is the semidiameter of the spherical cap and z is the measure sag.

### Basic Types of Spherometers

A Google image search will yield pictures of a variety of spherometers. One common type is the “three ball spherometer.” It’s based on three bearing balls, usually steel or sapphire, equally spaced around a circle of known diameter. The diameter of a bearing ball can be very carefully controlled – 2.5 microns for a G25 or 1.25 for a G10 – so the main limitation on accuracy of a 3 ball spherometer is the placement of the balls. Because of the surface being tested contacts a place on the balls other than the bottom, the above equation must be modified:

R = z/2 + y²/2z ± r

where r is the semidiameter of the bearing balls. The plus sign is used for concave surfaces and the minus for convex.

Three ball spherometers are useful for telescope mirrors and other large lenses. Unfortunately, many of the lenses we use are less than 25 mm diameter, so we had to develop our own. Below is a picture of our spherometer. It is a “ring type spherometer,” which means that the base of the spherometer contacts the optical surface along the circumference of a circle. The MItutoyo digital indicator is accurate to 0.003 mm, so we can make very accurate measurements of the radius of curvature of a lens or mirror surface. The bases have a polished flat bottom with accurately turned and measured inner and outer diameters ranging from 6 to 22 mm. They are made of brass so they don’t mar the optical surface. The advantage of the ring type spherometer is that the diameter of the circle where the spherometer contacts the optical surface is accurately known, and the first equation can be used. You can buy a ring type spherometer from Edmund Optics, but it uses a dial indicator rather than a digital one. Unfortunately, the dial indicator has an 8 mm (Japanese standard) collar and most digital indicators in the U.S. are made with a 3/8″ collar, so it is not easy to upgrade.