“It works” isn’t something you can measure. A spec is a requirement written as a number you can check. A good spec has three parts: the quantity, the target, and the way you will measure it. Compare:
| Vague | A real spec |
|---|---|
| “A good periscope.” | “Raises the line of sight 0.30 m; the exit ray is parallel to the entering ray within 2°.” |
| “A strong magnifier.” | “Magnification at least 4×, measured with two rulers.” |
| “A telescope that works.” | “M at least 4×, a tube of 60 cm or less, and brick mortar visible at 8 m.” |
Engineers follow the same loop every time: spec → predict → build → measure → compare. You predict first, so that when you measure, the difference between the two numbers actually tells you something. If you only measure afterward, any result looks fine. The gap between prediction and measurement is the real result, and it points you to the part of the build that needs work.
A periscope is two flat mirrors, each set at 45° to the light path. The first turns the beam by 90° and the second turns it back. The vertical distance between the mirrors is how much the line of sight is raised, so a 0.30 m raise needs mirrors 0.30 m apart.
What makes it work is that the mirrors are parallel to each other. From Activity 6-1-1, two mirrors turn a ray by a total of 2α, where α is the angle between them. For a periscope you want that to be zero, so the exiting beam travels in exactly the direction it entered:
Two more design points. First, a 45° mirror stretches a beam, so for a tube of width w, each mirror must be at least 1.4 × w long, or it will clip the picture. A 4.0 cm wide tube needs mirrors at least 5.7 cm long. Second, you can replace each mirror with a 45° right-angle prism: light hits the long face at 45°, which is more than the 42° critical angle of glass or acrylic, so it reflects totally (Activity 6-1-2), with no silver coating to wear off. Household mirrors reflect only about 85–90% of the light, so a two-mirror periscope passes about 70–80%.
A periscope keeps the image upright. Two reflections undo each other’s left-right reversal, so unlike a single mirror, what you see through it isn’t mirror-imaged.
Start with mirror 1 at 0° and mirror 2 at 1.5°. Then tilt both by the same amount and watch the direction error disappear. Then move them in opposite directions and watch it double.
A beam enters a periscope from the left, bounces off two mirrors, and heads for a wall. Each slider tilts one mirror away from a perfect 45°. (The tilts are drawn to scale in angle, but the distances are not.)
A magnifier is one converging lens with the object held at, or just inside, its focal length. What a magnifier does is make the object subtend a bigger angle at your eye, which is called angular magnification. Compared with looking at the object from the closest comfortable distance of 25 cm, a lens of focal length f gives about
A shorter focal length means a stronger lens and more magnification, but it also means you have to hold the lens closer to the object and you see a smaller patch of it. “Bigger” isn’t automatically “better.”
How to measure M. You can’t read magnification off the lens, but you can measure it with two millimeter rulers. Put one ruler under the lens, and hold the other 25 cm from your free eye. With both eyes open, your brain overlays the two views. If 1 mm of the magnified ruler spans N mm of the unaided ruler, then M = N. Repeat it a few times and average.
A simple telescope has two converging lenses. The objective, with a long focal length, makes a small real image of a faraway scene at its focal point. The eyepiece, with a short focal length, works as a magnifier on that image. The magnification is the ratio of the focal lengths:
For the eyepiece to work as a magnifier, it must sit one of its focal lengths past the objective’s image. For a faraway object that image is one objective focal length behind the objective, so the tube length is
The image you see is inverted, which doesn’t matter for this project. A wider objective collects more light, so the view is brighter, while a higher magnification gives a narrower, dimmer, shakier view.
A nearby object isn’t at infinity, so the objective’s image doesn’t form at its focal point. From the lens equation, it forms at d_i = f_o d_o ÷ (d_o − f_o), a little farther back. The eyepiece has to follow it.
How to measure M. Aim at a regular pattern such as bricks or a ruled sheet 8–10 m away. Look through the telescope with one eye while the other eye looks at the same pattern unaided. With both eyes open, count how many unaided-eye bricks fit across one brick in the telescope’s view. That count is M.
Try combinations until you find one that meets a 4× spec and a 60 cm tube. Then slide the object closer and see how much longer the tube has to get.
Choose an objective and an eyepiece, and see the magnification and the tube length you need. Then pick how far away the object is, and watch the eyepiece have to slide back for nearby things.
A good demo makes the measurement visible. In 60–90 seconds, cover these in order:
Four questions. They are not graded. Answer them to find out what still needs work.
You’ve now designed, built, and measured an optical device. Next you’ll do the design half on paper, for a real person with a real need and a real limit on size or cost.