I purchased a pair of projection lenses for possible use in optical experiments, such as spatial filtering and achromatic Lloyd's mirror fringes. These weighed 3 lb. each, and were in a sturdy mount 95 mm in diameter and 96 mm long. They were advertised as "triplets", but I have not disassembled them yet to see exactly what is inside. They were a good value, costing less than steak per pound. To use them, I needed to know their focal length and the location of the focal points (or principal points). What is required for this is some way to hold the lenses and the other items required so the measurements can be easily made. That is, an *optical bench* was needed.

Not having a suitable optical bench, I decided to make a serviceable substitute from easily available materials. Home Depot sells excellent white poplar planed dimensional boards. Select boards that are clear and straight; all that I saw were acceptable. The base for the bench was a 4' length of 1 x 6. The guides were 3' lengths of 1 x 2 (actually 1/2" x 1-1/2"). These were spaced so that a third 1 x 2 fit between them. They were glued to the base with Elmer's glue, clamping them with screws at each end. The screws were removed when the glue had dried. Holders for a miniature lamp source and a screen were made from a 4" lengths of 1 x 2 and 90 mm squares cut from a 1 x 4 (actually 1/2" x 3-1/2" x 3-1/2"), connected by sheet-metal brackets (available already bent and with 2 holes in each leg) and two 1/2 x 6 screws in each leg. It is easy to make holders this way, or by using a 1 x 1 square screwed to the 4" length of 1 x 2 as a basis. The screen was a 90 mm square of white paper glued to the wood. The holders fit snugly enough that there was no need to clamp them. A millimetre scale on the guides would be a useful addition, but I could not find one immediately. The heavy lenses rest stably between the guides, but are easily moved.

The source was a 16V 1.3W miniature lamp. The power source was a 120V - 12V transformer (using one-half of the centre-tapped secondary). Lacking a lamp socket, #18 AWG wire was wrapped around the screw base, while a #20 wire was soldered to the contact at the tip of the base. These wires were brought up to a terminal strip at the top and soldered there. The 12 V gave a satisfactory filament temperature, bright enough but well below brilliance. The lamp will last a lot longer at this reduced voltage.

The first step is to determine the location of the focal points relative to the lens. This is done by *autocollimation*, illustrated in the diagram. A mirror redirects rays back through the mirror, so that rays from the focal point F will be returned to F after reflection. The mirror may be a back-silvered mirror of any reasonable quality. The source is a small hole illuminated from the back, or the filament of a lamp. An aperture in the screen allows the light from the source to shine through, while the returned light falls on the screen. This screen may be arranged so that it is in exactly the same plane as the source (very easy if the source is a hole punched in the screen). The lens is moved back and forth until the image is focused on the screen. The distance d is measured relative to any fixed point on the lens system (if this point is the first vertex, it is called the *front focal length*, FFL. Of course, it is not the focal length, which is measured to the principal plane, whose location we have not yet determined. The lens system is reversed to determine the focal point F' on the other side in the same way.

The second step is to find the focal length of the lens system. Since it is in air, f = f'. The object and screen are placed a distance d apart, which should be somewhat greater than four times the expected focal length. Two positions for the lens system will be found that give sharp images of the source on the screen. In the diagram, x and x' are the two values for the object distance, measured from the focal point. The corresponding values for the image distance will be x' and x, respectively. Newton's lens formula gives f^{2} = xx'. The sum of x and x' is d - s, where s is the distance between the two focal points F and F' of the lens system. The difference x' - x is the amount that the lens is moved between the two points giving sharp images. These equations can be solved for x and x', and f will then be the square root of their product. In fact, f = [(d - s)^{2} - t^{2}]^{1/2}/2. Now the principal points H and H' can be located as a distance f from the corresponding focal points.

For my lens, the focal points were 172 mm from the ends of the mount, and the mount was 96 mm long, so the distance s between the focal points was 440 mm. The lens was symmetrical. The distance d was set at 952 mm, so d - s = 512 mm. The lens had to be moved a distance t = 182 mm between the points giving focused images. Therefore, f = 239 mm. The principal planes are located 29 mm from the ends of the mount, and the secondary principal point H' comes in front of the primary principal point H.

B. K. Johnson, *Optics and Optical Instruments* (New York: Dover,1960). Describes many other measurements on the optical bench. Focal length measurement by the method of this article is explained on p. 30 in a slightly confusing way.

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Composed by J. B. Calvert

Created 1 September 2007

Last revised