29.4 Images of images by Benjamin Crowell, Light and Matter  licensed under the Creative Commons Attribution-ShareAlike license.

29.4 Images of images

If you are wearing glasses right now, then the light rays from the page are being manipulated first by your glasses and then by the lens of your eye. You might think that it would be extremely difficult to analyze this, but in fact it is quite easy. In any series of optical elements (mirrors or lenses or both), each element works on the rays furnished by the previous element in exactly the same manner as if the image formed by the previous element was an actual object.

Figure h shows an example involving only mirrors. The Newtonian telescope, invented by Isaac Newton, consists of a large curved mirror, plus a second, flat mirror that brings the light out of the tube. (In very large telescopes, there may be enough room to put a camera or even a person inside the tube, in which case the second mirror is not needed.) The tube of the telescope is not vital; it is mainly a structural element, although it can also be helpful for blocking out stray light. The lens has been removed from the front of the camera body, and is not needed for this setup. Note that the two sample rays have been drawn parallel, because an astronomical telescope is used for viewing objects that are extremely far away. These two “parallel” lines actually meet at a certain point, say a crater on the moon, so they can't actually be perfectly parallel, but they are parallel for all practical purposes since we would have to follow them upward for a quarter of a million miles to get to the point where they intersect.

The large curved mirror by itself would form an image `I`, but the small flat mirror creates an image of the image, `I'`. The relationship between `I` and `I'`is exactly the same as it would be if `I` was an actual object rather than an image: `I` and `I'` are at equal distances from the plane of the mirror, and the line between them is perpendicular to the plane of the mirror.

One surprising wrinkle is that whereas a flat mirror used by itself forms a virtual image of an object that is real, here the mirror is forming a real image of virtual image `I`. This shows how pointless it would be to try to memorize lists of facts about what kinds of images are formed by various optical elements under various circumstances. You are better off simply drawing a ray diagram.

Although the main point here was to give an example of an image of an image, figure i also shows an interesting case where we need to make the distinction between magnification and angular magnification. If you are looking at the moon through this telescope, then the images `I` and `I'` are much smaller than the actual moon. Otherwise, for example, image `I` would not fit inside the telescope! However, these images are very close to your eye compared to the actual moon. The small size of the image has been more than compensated for by the shorter distance. The important thing here is the amount of angle within your field of view that the image covers, and it is this angle that has been increased. The factor by which it is increased is called the angular magnification, `M_a`.

Discussion Questions

A   Locate the images of you that will be formed if you stand between two parallel mirrors.

B   Locate the images formed by two perpendicular mirrors, as in the figure. What happens if the mirrors are not perfectly perpendicular?

C   Locate the images formed by the periscope.

29.4 Images of images by Benjamin Crowell, Light and Matter  licensed under the Creative Commons Attribution-ShareAlike license.