Light Caustics

Snell’s law and refraction

When a ray of light crosses the boundary between two media with different refractive indices — air to water, air to glass — it changes direction. The relationship is governed by Snell’s law: n₁ sin θ₁ = n₂ sin θ₂, where n₁ and n₂ are the refractive indices and θ₁, θ₂ are the angles of incidence and refraction measured from the surface normal. A higher refractive index means light travels more slowly in that medium, bending the ray toward the normal upon entry. This bending is the entire mechanism behind caustics: a curved surface presents a different normal direction at each point, so parallel incoming rays refract at different angles and converge or diverge depending on the local curvature.

What caustics are

Mathematically, a caustic is the envelope of a family of rays — the curve (or surface, in 3D) that is tangent to every ray in the family. At points on the caustic, neighboring rays are nearly parallel and very close together, so the light intensity spikes. In catastrophe theory, the generic caustics in 2D are folds (smooth bright curves) and cusps (where two fold lines meet at a sharp point). These are structurally stable: small perturbations to the refracting surface deform them smoothly rather than destroying them. This is why caustic patterns at the bottom of a swimming pool look so characteristic — the network of bright lines with Y-shaped junctions is a universal feature of random wavefronts.

Caustics in nature

The shimmering bright lines on the floor of a swimming pool are the most familiar example. Sunlight refracts through the constantly shifting water surface, and the caustic pattern dances as the waves evolve. But caustics appear everywhere that light encounters curved transparent or reflective surfaces. The bright curve inside a coffee cup when light hits the rim is a nephroid caustic formed by reflection. Rainbows are caustics: the bright arc at 42° is the fold caustic of the family of rays refracted and internally reflected inside raindrops, and the fainter supernumerary bows just inside the main arc are interference fringes near the caustic where wave optics become important. The brilliance of a diamond comes from caustics generated by total internal reflection at carefully cut facets — the “fire” is chromatic caustics splitting white light into spectral colors. Even gravitational lensing produces caustics: when light from a distant quasar passes near a massive galaxy, the warped spacetime acts as a refracting medium, and the caustic structure determines where multiple images of the quasar appear.

The simulation

This simulation sends parallel vertical rays downward through a sinusoidal refracting surface. At each point where a ray meets the surface, the local surface normal is computed from the derivative of the wave function, and Snell’s law determines the refracted direction. The refracted rays continue to a collection surface at the bottom, where a 1D intensity histogram accumulates how many rays arrive at each horizontal position. Where many rays converge, the histogram spikes — these are the caustic bright spots. The “Animate” button introduces a slow phase shift in the wave, so the surface ripples forward in time and the caustic pattern dances, just as it does on a real pool floor. Try increasing the amplitude to see sharper, more intense caustics, or raising the refractive index to simulate denser materials like glass or diamond.

Further reading

M.V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Progress in Optics 18 (1980). J.F. Nye, Natural Focusing and Fine Structure of Light (IOP, 1999). For the connection to catastrophe theory: V.I. Arnold, Singularities of Caustics and Wave Fronts (Kluwer, 1990).