What Are Lenses?
We have shown that a mirror can produce images by using the law of reflection. We have also shown how looking directly into a pool of water produces an image by using Snell’s law for those rays which enter your eyes. Now we are going to introduce lenses, which are specially shaped materials designed to produce well-defined images by using refraction.
Before getting started, it is worth pointing out something that may seem obvious: lenses are manufactured. A typical transparent object will not be the correct shape to produce a good image. For example, consider the glass blob illustrated below.
To someone standing to the right, the light rays would not appear to come from any particular location, so there would be no (clear) image of the object on the other side. We have only chosen to draw three rays above (the normals we used are the black dashed lines) but no amount of lines would appear to come from a single location. No image of the object exists.
In contrast to our typical blob which refracts light, a lens forms a sharp image. The typical way that a lens does this is by changing its curvature (and hence changing its normal) continuously to ensure that the light rays are actually focused. This may sound contrived, after all how often do we find that the curvature changes in just the right way to keep the light in a sharp focus? Well, it is contrived! However lenses are incredibly useful in correcting eyesight, magnifying objects and seeing distant stars. It is precisely because these systems are contrived that we must pay so much for glasses – the lenses must be carefully made!
Features of Lenses
The lenses we will be discussing are either converging (shown below on the left) or diverging (below on the right). The light bends as it enters the lens (i.e. as the light goes from air to the lens material) and bends again as it exits the lens (as the light goes from the lens material to air). We are going to simplify the treatment of lenses by pretending that the light bends only once, at the center of the lens (illustrated below). This is a good approximation if the lens is thin, which almost all lens are manufactured to be.
If we knew the precise shape of the lens, we could figure out the normal at every location, and use Snell’s law for ever normal, like how we described it earlier, to find how the light exits the lens, but that would be tedious. Instead, we are going to utilize that the lenses are specially manufactured to have focal points that help determine where light rays go. It is the job of the person who makes the lens to ensure that the light behaves the way that you want!
Before addressing the noticeable features of the above figures, it is convenient to make a definition. The optical axis of a lens is the line that goes through the central part of the lens, and is parallel with the normal at the center of the lens. In the above figures, all the rays on the left hand side of the lenses are parallel to each other and to the optical axes of their respective lenses.
Notice that each lens has a point where the light rays all intersect. For the converging lens the point is where all the light rays cross, while for the diverging lens this special point is where the refracted light rays appear to be diverging from. In both cases the special point is referred to as the "focal point" because it is the point towards or away from which light rays refract.
For a converging lens the light rays actually cross at this point, and in analogy with images we call the focus real. In contrast no light rays actually cross at the focal point of the diverging lens. Instead it is our perception of light as traveling in straight lines that makes it a point source behind the diverging lens appear to exist. In this case the focus is called virtual.