Simple convex lens

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.

Figure 4.12: If the rays are not perpendicular to the lens, then the focal point is shifted away from the optical axis.

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).

Figure 4.13: In the real world, an object is not infinitely far away. When placed at distance $ s_1$ from the lens, a real image forms in a focal plane at distance $ s_2 > f$ behind the lens, as calculated using (4.6).

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 $ f$. If the light source is placed at distance $ s_1$ 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):

$\displaystyle \frac{1}{s_1} + \frac{1}{s_2} = \frac{1}{f} .$ (4.6)

Figure 4.11 corresponds to the idealized case in which $ s_1 = \infty$, for which solving (4.6) yields $ s_2 = f$. What if the object being viewed is not completely flat and lying in a plane perpendicular to the lens? In this case, there does not exist a single plane behind the lens that would bring the entire object into focus. We must tolerate the fact that most of it will be approximately in focus. Unfortunately, this is the situation almost always encountered in the real world, including the focus provided by our own eyes (see Section 4.4).

Figure 4.14: If the object is very close to the lens, then the lens cannot force its outgoing light rays to converge to a focal point. In this case, however, a virtual image appears and the lens works as a magnifying glass. This is the way lenses are commonly used for VR headsets.

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 $ s_1 = f$, then the outgoing rays would be parallel ( $ s_2 = \infty$). If $ s_1 < f$, then (4.6) yields $ s_2 < 0$. 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!).

Figure 4.15: In the case of a concave lens, parallel rays are forced to diverge. The rays can be extended backward through the lens to arrive at a focal point on the left side. The usual sign convention is that $ f < 0$ for concave lenses.

Steven M LaValle 2020-01-06