Figure 4.11 shows a simple convex lens, which should remind you of the prisms in Figure 4.10. Instead of making a diamond shape, the lens surface is spherically curved so that incoming, parallel, horizontal rays of light converge to a point on the other side of the lens. This special place of convergence is called the focal point. Its distance from the lens center is called the focal depth or focal length.
The incoming rays in Figure 4.11 are special in two ways: 1) They are parallel, thereby corresponding to a source that is infinitely far away, and 2) they are perpendicular to the plane in which the lens is centered. If the rays are parallel but not perpendicular to the lens plane, then the focal point shifts accordingly, as shown in Figure 4.12. In this case, the focal point is not on the optical axis. There are two DOFs of incoming ray directions, leading to a focal plane that contains all of the focal points. Unfortunately, this planarity is just an approximation; Section 4.3 explains what really happens. In this idealized setting, a real image is formed in the image plane, as if it were a projection screen that is showing how the world looks in front of the lens (assuming everything in the world is very far away).
If the rays are not parallel, then it may still be possible to focus them into a real image, as shown in Figure 4.13. Suppose that a lens is given that has focal length . If the light source is placed at distance from the lens, then the rays from that will be in focus if and only if the following equation is satisfied (which is derived from Snell's law):
If the light source is placed too close to the lens, then the outgoing rays might be diverging so much that the lens cannot force them to converge. If , then the outgoing rays would be parallel ( ). If , then (4.6) yields . In this case, a real image is not formed; however, something interesting happens: The phenomenon of magnification. A virtual image appears when looking into the lens, as shown in Figure 4.14. This exactly what happens in the case of the View-Master and the VR headsets that were shown in Figure 2.11. The screen is placed so that it appears magnified. To the user viewing looking through the lenses, it appears as if the screen is infinitely far away (and quite enormous!).
Steven M LaValle 2020-01-06