Making a Spectacle

This week I noticed something interesting about my shadow: I could see the lenses of my glasses casting a dark spot with the rest of my head. This made me curious, since I can see through my glasses, meaning light is passing through them, so why would they cast a shadow? Taking them off revealed an even more interesting effect:

There's a dark spot in the center, but a magnified brighter area around it. I decided to try some ray-tracing, a technique in optics where we consider beams of light coming from a source and follow their path through a system. Often this is done using matrix optics, in which lenses and other elements are represented by matrices which are multiplied together to create a system. Rather than deal with that directly though, we can use Python's RayTracing package to handle the math for us.

Lenses are typically defined by their focal length, which specifies the distance from the lens where parallel beams passing through it converge. However, glasses prescriptions are given in diopters, which are the inverse of the focal length. I'm nearsighted, which requires a lens around -5 diopters. The negative means that the convergence point is on the same side of the lens as the source. This is a little clearer with a diagram: Since the Sun is far away, its light is roughly parallel, so we can look at what my lenses do to a bunch of parallel beams (click to enlarge):

This shows the light being spread out by the lens – If you follow those lines to the left, they converge on the opposite side of the lens, corresponding to the negative focal length.

That accounts for the magnified bright spot we saw above, but it's only considering the light that goes into the lens – There's also light that goes past the lens on the sides. We can add together this light with what was redirected by the lens to get the total image:

The different lines show distances from the lens to the surface of the shadow. As in the picture that started this, we see a dark spot surrounded by a brighter area! What's interesting is that, because the beams from the Sun are parallel, the shadow (drop in intensity) stays the same size for all the distances, but the brighter area gets bigger or smaller. I've been wearing glasses since the 4th grade, but somehow this is the first I've wondered about their shadow – One of those blind spots that seems obvious once you see it.