|The lens as a prism. Prentice's rule.|
PAGE References to Optical Formulas Tutorial: (first reference is to edition 1 / second reference is to edition 2).
We have already described lenses with their curved surfaces as a series of very short straight sides each angled a little more than the last one. The whole point of the lens is that it causes no deviation at the center, a very little amount of deviation a very short distance from the center, and then progressively increasing amounts of deviation at points progressively further from the center.
In other words, a lens is simply varying amounts of prism.
Plus lenses are made of varying amounts of prism with their bases together, or with the bases toward the optical center.
Minus lenses are made of varying amounts of prism with their apices together, or with the apices toward the optical center.
Go back to the link that showed with movement for a minus lens and against movement for a plus lens. What is happening here can be explained by the amount of prism that is present at the point in the lens were the edge of the page is. The image is displaced by some amount that is related to how far the object is from the optical center.
So now we come to the second most used formula in optical theory -- Prentice's rule. This formula gives us an approximation of the amount of prism present at any given point on a lens based on the dioptric power of the lens and the distance that the point is from the optical center of the lens.
Prentice's rule is:
There are several versions of this formula. Since we use it for measurements that we take in the lab, and since we usually use mm in the lab instead of cm, there is another form of the formula that I prefer for you to use. This form of the formula does not require you to remember to convert the measurement to cm.
If you like geometry and algebra and you want to see it, here is the proof of Prentice's rule. This is not required, so you don't need to go there unless you want to.
Read through pages (77-81 / 99-102) at the top. Follow the examples, but don't do the exercises yet.
Consider a spherical lens with a power of +1.00 diopters. At the very center of this lens there is no prism. 1 mm away from the center of the lens there is (1)(1) 10 = 0.1 prism. But there are an infinite number of points on the lens that are 1 mm away from the center: you can draw a circle on the lens, 1 mm away from the center, and at every point on this circle there will be 0.1 prism present.
Now consider the amount of prism present 5 mm away from the center. At every point on a circle 5 mm from the center there will be (5)(1) 10 = 0.5 prism present. How about the circle that is 10 mm from the center? 20 mm from the center?
Now look at a spherical lens of +8.00 power. How much prism is present at every point on a circle 1 mm from the center? 5 mm from the center? 20 mm from the center?
On the circle that is 20 mm from the center of the +8.00 DS lens, where is the image of an object going to be displaced? It will be displaced away from the center -- it will appear to move toward the edge of the lens by an amount that is .16 times the distance the object is away from the lens.
How about a -8.00 D spherical lens? The only difference between the 20 mm circle on the +8.00 DS lens and on the -8.00 DS lens is that the image is displaced away from the center on the +8.00 DS lens and toward the center on the -8.00 DS lens.
(PS -- on the +8.00 sphere the prism at 1 mm is 0.8,
mm it is 4.0, at 20 mm it is 16.)
One note that many opticians are unaware of: Prentice's rule is an approximation. It is not true for very low powers -- plano, 0.25, 0.50. So, for very low powers, if you verify the results of the formula on a lens using the prism rings in your focimeter and you get a different answer, the focimeter is probably right!
Read pages (79-81 / 99-102) of the Optical Formulas Tutorial, and do the exercises. Check your answers in the back of the textbook. Then read pages 425-427 (top left) in Systems for Ophthalmic Dispensing. We will get further into this chapter in the advanced optics course.
Do the following exercises for practice. Indicate base direction with your answers. Check the answers as you go.