Physics 103

Lens Combinations: Telescopes

 Introduction

The rules of ray-tracing have a simple consequence for lens combinations: if two lenses are mounted one after the other, then the image formed by the first lens becomes the object for the second lens. Thus one can apply the thin lens equation twice to find the image formed by the system of two thin lenses. With each application you have to measure p and q from the appropriate lens and use the f for that lens. You also have to remember the sign conventions for object position p and image position q (opposite!) If necessary, REVIEW the thin lens equation. Check your understanding of these ideas by deriving the following useful fact: two lenses which are at the same place -- essentially on top of each other -- have an effective focal length f which obeys

[Note this is NOT TRUE if the lenses are at different places!]

One of the simplest and most useful lens combinations is the astronomical telescope (below). The lens at the left is called the objective, and the lens at the right is called the eyepiece (the one you would put your eye up to). The object is at infinity, and the image is also at infinity! What good is that? you may wonder. Look at the angles in the simulation below. In particular, select the "source" at the far left by clicking it, and then drag the rays to change their angle. You will see that the telescope magnifies angles -- and if you think about how we see, you will realize that this is what we mean intuitively by magnification.

Another way to think about this is to add an "eye" at the far right to look through the telescope. The eye is a third lens and a "retina". The object for this third lens is the image formed by the second lens, so if that image is at infinity, it will be focussed nicely onto the retina (the relaxed eye can easily form images of things far away). Again select the source and drag the rays around to see how angle translates into position on the retina. The eye is seeing the image of distant stars as points, and their angular separation is magnified. If the eye were looking at a fixed constellation of stars, would the constellation look inverted through the telescope or not? Explain carefully!

How far apart should the two lenses be to make a telescope? Find qo and pe (in the notation of the thin lens equation) where the subscripts o and e refer to the objective and the eyepiece respectively. The distance between the lenses is just their sum qo+pe. Show that this is fo+fe, where again the subscripts o and e refer to the objective and the eyepiece. Also, use the principal ray through the center of each lens to derive the angular magnification of the telescope: M= - fo/fe. (Use small-angle approximation. The minus sign is a sign convention like that for the image formed by a single lens, having to do with whether the image is inverted or right-side-up.)

A Galilean telescope is just like the telescope above, except that the eyepiece is a negative lens. You can drag a focal point of the eyepiece through the lens in the applet above to bring it to the wrong side, thus making a Galilean telescope. Of course you will have to position the eyepiece in the right place, analogous to focussing a real telescope. Now how long is the telescope? What is its magnification? Is the image right-side-up or inverted? How is the analysis of the Galilean telescope different from the astronomical telescope, if at all?

You might be interested in what Galileo said about his methods for making quantitative measurements with the telescope. He had no clear theory of geometrical optics, but he knew that the telescope magnified angles. Here is his description in its entirety, from his 1610 book The Starry Messenger:

"Let ABCD be the tube and E be the eye of the observer. Then if there were no lenses in the tube, the rays would reach the object FG along the straight lines ECF and EDG. But when the lenses have been inserted, the rays go along the refracted lines ECH and EDI; thus they are brought closer together, and those which were previously directed freely to the object FG now include only the portion of it HI. The ratio of the distance EH to the line HI then being found, one may by means of a table of sines determine the size of the angle formed at the eye by the object HI, which we shall find to be but a few minutes of arc. Now, if to the lens CD we fit thin plates, some pierced with larger and some with smaller apertures, putting now one plate and now another over the lens as required, we may form at pleasure different angles subtending more or fewer minutes of arc, and by this means we may easily measure the intervals between stars which are but a few minutes apart, with no greater error than one or two minutes. And for the present let it suffice that we have touched lightly on these matters and scarcely more than mentioned them, as on some other occasion we shall explain the entire theory of this instrument."

He never wrote about the theory of the telescope again.