The ray tracing rules that we have described in previous topics specify the size and position of an image in any position relative to a lens or mirror. However, these rules do not tell the whole story when the image formed by one lens is an object for a second lens. This situation arises in telescopes, compound microscopes, etc.
To illustrate the problem, consider the lens configuration of a compound microscope (see Topic 39. The Compound Microscope). The objective lens forms a real and inverted image of the object as shown in the following figure.
All of the rays from any point on the object are focused at the same point on the image. If there were a screen or some other target at the position of the image then all of these rays really would strike the target at that position, and the intensity of light striking any point would be given by the sum of the intensities of all of these rays.
However, if there is no physical target at the image plane, then while the rays still converge to a single point in the image plane, there is nothing there for them to hit and they continue traveling in straight lines.
In the figure above, the red rays, which come from the top of the first object, converge to a point in the image plane as shown. However, if this is the image formed by the objective lens of a microscope or telescope, then there is no screen there. Instead, that image is the object for the second eye-piece lens. However, after converging to the point as shown, the red rays continue to travel in straight lines and never strike the second eye-piece lens. In other words, even though the objective lens formed an image of the tip of the original object, that point is not visible through the second lens since no rays from that point can ever strike it. Points on the original object closer to the axis can be seen through the second lens, as shown by the purple rays. This point on the original object can be seen through the eye-piece, since rays from the original object reach the eye-piece.
There are obviously intermediate points on the original object between the source of the purple rays, which has all of its rays striking the eye-piece lens, and the source of the red rays, which has none of the rays striking the eye-piece lens. For these intermediate points, some of the rays strike the eye-piece lens and so these points are visible but are not very bright. The bottom line is that the brightness of the image seen through the final eye-piece lens gradually gets dimmer for object points further and further from the axis, and there are points on the object which cannot be seen at all through the eye-piece lens.
The solution to this problem is to use a field lens, which is another positive lens located between the objective and the eye-piece as shown in the following figure.
There are a number of different designs for the strength and position of the field lens, but the general idea is always the same. The field lens bends rays that would miss the eye-piece lens (or the next lens in a multiple lens system) back towards the axis, so that they pass through the next lens instead of continuing off into space. As an example, the effect of the field lens on the red rays is shown in the figure.
On page 176, the text shows the field lens located exactly where the intermediate image would be formed if it were not present, but this is not always true. (If you are interested in this, look at the Hugyens and Ramsden designs for field lenses in a microscope).