Thin Lenses
The optics of a thin lens are very similar to that of mirrors from last chapter. We use all the same conventions on signs - a convention that depends on where the actual light goes. Just as with mirrors, the convention is to draw the object on the left just as with mirrors. Keep in mind that this is not a universal convention, but just a majority tendency. In fact, the makers of the GeoGebra software had a ray tracing diagram for a converging lens as an example in their materials repository which follows the opposite convention. It is shown below.
Ray Tracing Diagram of Converging Lens
Just as with mirrors, which besides being planar could come in two forms with curvature (concave and convex), we have lenses that can be flat (really a window or plate glass) as well as converging and diverging lenses.
A converging lens like the one shown in the interactive diagram above is sometimes called a biconvex lens. As the name suggests, a biconvex lens is one in which both sides of the lens are convex or sticking outward. More commonly, however, it is called a converging lens because the lens is causing incoming light rays to converge toward the principal axis. So this name suggests more about the behavior of the lens rather than its geometry.
A diverging lens is one that causes incoming light rays to diverge away from the principal axis. Such a lens is sometimes called a biconcave lens. A ray diagram for such a lens is shown below. Note that the common convention of putting the object on the left side is not followed, but the concepts are still the same.
Ray Tracing Diagram for Diverging Lens
Algebraic Equation for Image Formation
The way to algebraically solve for the image location is exactly the same as it was for mirrors: Just keep in mind the sign conventions and everything will work out fine. Furthermore, the same equation for magnification applies: The interpretation is the same as it was for mirrors. When the magnitude is less than one the image is smaller than the object, when greater than one it is larger than the object. The sign of the magnification indicates an upright image when positive and an inverted image when negative.
Lens Curvature and Focal Length
It is easy to know that a lens with two convex surfaces will be a converging lens and that a lens with two concave surfaces will be a diverging lens. However, what if a lens has one of each. It turns out that the surface with the smaller radius of curvature will dominate the behavior of the lens.
Geometrically, you are always guaranteed that if the lens is thicker in the center than at its edges, it is a converging lens, and vice versa. There is one stipulation which is almost always true. The lens material must have a higher refractive index than the surrounding medium. Since the surrounding medium is almost always air, we generally expect this to be true.
Lensmaker Equation
The quantitative connection between the refractive index of the lens material , the surrounding medium and the radius of curvatures of the two lens surfaces and is called the lensmaker equation. It states:
The conventions used for the two surface radii are the exact same as the ones we've used before. The first surface, for instance in a converging (biconvex) lens has a center of curvature on the right side of the lens (where the light goes) and is therefore positive. The back surface of the same lens has a center of curvature on the same side as the object (not where the light goes) and is therefore negative.
It should be clear from the equation that given the unusual circumstance that the surrounding medium has a higher refractive index than the lens, the sign of the focal length flips as compared with ordinary conditions in which the lens has the higher refractive index.
One more thing to note is that lenses are useless if the refractive index of the lens material matches that of the surrounding medium. This fact was exploited in a James Bond movie in which James hid a diamond in a vodka martini and simply walked right past guards who were on high alert for the potential theft of the diamond. The only problem with the movie is that the refractive index of vodka is 1.36 and that of diamond is 2.42. Assuming the guards were looking in the drinks, they would have seen it whereas in the movie it literally disappeared once submerged in the beverage.