Comment & Analysis
“THIS MEANS THE POT OF GOLD AT THE END OF THE RAINBOW IS UNATTAINABLE”
Helen Czerki on caustics
Proper summer has finally arrived, along with the opportunity for sitting outside and basking like a lizard through long and lazy afternoons. On a sunny day, while eating al fresco, intense sunlight glints off plates and glasses, and the most important question of the afternoon is whether clotted cream or jam should be spread on scones first. But take a moment away from these ponderings to look at the table – really look – and you’ll see that it’s covered in bright lines and odd dark patches. The surface of your tea is decorated with two sharp semicircles on the side furthest from the Sun. Your transparent glass of water casts a weird dark shadow (even though it is transparent), cut in half with a blaze of white light. Inside the rim of a plate, there are thread-like lines which roll across the surface as you tilt it. All of these features are called caustics, and a sunny day offers the perfect opportunity to admire them.
Caustics are the sharp dividing lines between bright and dark regions, and they turn up wherever parallel light rays meet curved surfaces. Because the Sun is effectively a point source a long way away, the light rays coming from it are almost perfectly parallel. When they reach the inside of a teacup, they bounce off at different angles, depending on which part of the inside of the cup they hit. But there are some regions that many different rays get directed to (those are the bright semicircles) and some regions which are effectively forbidden – they can’t be reached by any reflected ray. In your cup, the two intense semicircles meet at a point called the cusp, which makes a distinctive pattern known as a nephroid caustic, because it’s vaguely kidney-shaped.
You’ve probably also seen caustics on the bottom of a swimming pool, where waves travelling across the surface focus light into criss-crossing bands, and direct it away from the spaces in between. The surface where you see a caustic is acting like a screen, giving you a sneak peek into where light was travelling. If you move the screen, you’ll see the pattern change, but the three-dimensional pattern itself was there all along. The light field around us is lumpy and bumpy like this all the time, especially on sunny days. But we need to put an obstacle in the way to make the lumps and bumps visible.
The final stage in the beauty of caustics brings us to the rainbow. Light paths are bent as they cross from one transparent object to the next, violet bends more than red. You may well see some coloured lines in the shadow from your water glass because each colour generates its own caustic, and they don’t line up. And so we come to the search for a pot of gold. A rainbow is formed as light from the Sun reflects and refracts its way around the inside of tiny water droplets in the sky. Afterwards, the light field for each colour is lumpy, with caustics dividing the most and least intense regions. You intercept that pattern with your own eyes, and you see the rainbow – different caustics for red, orange, yellow, green, blue, indigo and violet, glittering in a giant arc across the sky. The lumpy light field exists at every point between you and those water droplets, and your eyes detect the unique pattern at your viewpoint. But this means the pot of gold at the end of the rainbow is unattainable, because all that’s there is a three-dimensional pattern. However, the dish of butter on the table is right here, so maybe I’ll admire the caustics inside the dish and make do with that instead.