I like to experiment with lenses and different optical combinations in my photography and cinematography. It's easy to build a design when you buy all of your glass from a lab supplier. They usually can supply CAD drawings and a full range of specifications and tolerances. If, like me, you pull elements out of old camera lenses or you buy surplus optics, you have to find other ways to get those critical dimensions. Most lenses are spherical, which means their surfaces represent a section of a sphere. It is fairly easy to determine the focal length of a lens element once you know the radius of its surfaces and the distance between them. All you need to derive the radius is find three points on the curve. The cool part is, anybody can build a simple tool to find those points quickly and accurately.
At its simplest, a spherometer is just a ring with a known inner diameter and outer diameter with a depth indicator in the middle. When placed on a convex or concave spherical surface, you immediately have all three points.
Step 1: Materials
You can use almost any materials, to any dimension. There are a handful of basic components that are necessary, though.
The most important part is the depth indicator. You can buy these pretty inexpensively but this determines, more than anything else, the accuracy of your spherometer. I was lucky to find mine at a thrift store for $3, despite it being an expensive Mitutoyo indicator. It's best to get one that measures in millimeters, since that is traditionally how lenses are specified.
You also want to pick gentle materials for the pieces that are going to make contact with the glass. I used a nylon ball bearing for the tip of the indicator and low-friction black Delrin (acetal plastic) for the ring at the bottom of the spherometer.
Other than that, you are free to riff on the materials, shapes, and dimensions. I used aluminum for the main body because it's easy to machine, and I had some available. I used an M2.5 socket head cap screw to extend the stem of the indicator, and designed the body to put the indicator tip protruding half the length of its total travel. That way, it should be equally able to measure concave and convex surfaces. The diameter was based on the relatively small elements I typically encounter.
In addition to these materials and components, you'll need access to a lathe, a drill or drill press, and a band saw or Dremel. A tap of your choosing is also helpful.
Step 2: Machining the Indicator Sleeve
If you do buy (or find) a nice expensive dial indicator, you probably don't want to mar it when securing the stem in the body of the spherometer. The standard method for avoiding this is to put a pinch sleeve between the indicator and a screw. I used brass because it's soft and, quite frankly, it's a very attractive material.
I turned a small piece on a lathe, making sure that the inner diameter matches the stem of the indicator as closely as possible. The most important aspect of the spherometer is the concentricity of all components. If the tip of the indicator is held off center, your readings will be completely inaccurate. You also want to keep track of the outer diameter of this sleeve, because you'll need to match it when turning the main body.
After you're pleased with the size and shape, you'll need to split the ring with a band saw or a rotary tool like a Dremel. This is necessary to give the sleeve the ability to clamp tightly around the indicator.
Step 3: Machining the Main Body
Next, I turned the main body out of 6061 aluminum round stock. First, a hole was drilled through the entire piece, to allow the stem of the indicator to pass through. This diameter isn't crucial, as long as there is enough room for the indicator to raise and lower without interference or friction.
Next, I bored a pocket to fit the brass sleeve and checked it to make sure the fit was snug. Too much play will throw the measurements off. It's also important that the face the brass will sit on be totally square, for the same accuracy concern.
I flipped the piece in the chuck and bored another, smaller pocket in the opposite end. This is going to be for the Delrin ring that will sit on the glass. This diameter is open to interpretation. I made it a nominal .250", but as long as you match the diameter in the mating piece, you'll be fine.
The only other work you have to do is drill and tap two holes to secure the brass sleeve and the Delrin ring. I tapped both holes for M3 threads, since they offer a good balance of size and holding power. I decided to use a set screw to hold the Delrin in place, and a thumbscrew for the brass to make it easy to adjust the indicator height as necessary.
Step 4: The Delrin Ring
After the indicator, this is the most critical piece. Every diameter must be totally accurate, both for concentricity and reliable, repeatable readings.
I picked Delrin because it has very low friction, it machines very easily and holds tight tolerances fairly well. Delrin is manufactured by DuPont, but generic versions are also known as acetal. You can also use nylon or acrylic, depending on what tools and materials you have available to you. The most important feature is that maintain consistent dimensions, as the inner and outer diameters are the basis for two-thirds of the raw information used to find the radius of the optical surfaces. It doesn't matter what you pick, but round metric numbers will probably be easier to remember and keep the process as straightforward as possible. I used 5mm for the ID and 15mm for the OD, though I am considering making a series of swappable Delrin rings to accommodate any lens I might encounter.
I also included a pilot diameter on the Delrin piece. This fits neatly inside the main body to ensure everything is centered, and it gives the set screw something to bite on without affecting the important surfaces.
Step 5: Assembly
There aren't many pieces, so it should go pretty smoothly. You should have a collection of parts like those pictured above. When assembled, they should form a tidy little instrument. Make sure everything looks centered on the bottom before taking your first measurements.
Step 6: Using the Pocket Spherometer
So, now that you've assembled the spherometer and verified that everything looks concentric, it's time to zero out the indicator.
First, find a perfectly flat and smooth surface. Traditionally, a precision-ground granite block is used, but since this is a pocket instrument, a glass smartphone screen is just as suitable. Place the spherometer flat against the screen so that the spring-loaded stem doesn't throw off your reading. This is going to be your zero reading, so loosen the thumbscrew securing the indicator in the sleeve, and slide it up and down until you're close to reading zero. Tighten the thumbscrew again, then rotate the number ring on the indicator dial to set it exactly to zero. Record the measurement. In my case, the indicator has a total travel of 5mm. I zeroed it at exactly 1mm, as shown by the smaller dial on the indicator face.
Next, verify the relevant diameter on your Delrin ring. I am going to measure a convex surface, so I measured the ID of the ring and got 5.00mm exactly. It's important to record if your diameter is a little off, like 4.88mm, because that will affect your math later.
Place the end of the spherometer anywhere on the lens element you want to measure. Sometimes it can good to take a handful of measurements, both to eliminate error in any one reading, and to make sure you aren't dealing with an aspherical lens. Once you're satisfied, record the new measurement and calculate the difference between that and your zero reading. You now have everything you need to find the radius of the lens.
The formula is very simple: R = (r^2 / 2h) + (h / 2), where "r" is the radius (NOT the diameter) of the Delrin ring and "h" is the difference between your zero measurement and the reading from the lens. If you're measuring a convex surface, use the radius of the hole in your Delrin ring. If you're measuring a concave surface, you'll use the outside measurement of the Delrin. It should be apparent if you use the wrong one.
I measured the front element of my 50mm f1.4 Nikon lens. It has a very pronounced curve, which I found to be 28.464mm (you can check my math in the photo above). You should be able to tell by eye that something is wrong if you get a strangely large or small number. If I had used the diameter and not the radius of the spherometer, I would have gotten 113.691mm, which is obviously too gentle a curve.
That's all you have to do. If you want to generate CAD drawings of the orphaned optics you find, you can find the radii of the surfaces and measure the edge thickness. From there, you can use the lensmaker's equation to find your focal length (https://www.boundless.com/physics/textbooks/boundless-physics-textbook/geometric-optics-24/lenses-170/the-lensmaker-s-equation-615-4333/)