Optics 005 · Ray worlds, boundaries, and natural optics

Caustic Factory

Why does light reflected or refracted by a curved surface gather into sharp, bright curves? Neighboring rays cross each other along a curve called the envelope, and ray optics predicts infinite brightness there. Parallel light reflected inside a circular mirror draws a nephroid, the bright curve you see in a coffee cup, with its cusp at half the radius. Wave effects keep the real brightness finite.

In a circular vessel, incident, reflected, internal and transmitted ray segments are drawn separately before their computed envelope is highlighted. A radius ruler shows the boundary itself changing size. The three scenes show the nephroid made by reflection, refraction through two interfaces and the catacaustic of a point source.

Cusp-to-vessel ratio

r_{\mathrm{cusp}}/R_v

$0.50$

Caustic span

\Delta y_{\mathrm{caustic}}

$50\%$

Direction norm error

\Delta_{|\hat{\mathbf k}|}

$0.00\pi$

Model regime $\text{valid model regime}$

Physics tutorial

Audit the physics of Caustic Factory

BackgroundA caustic is the envelope of a family of rays, where the ray picture becomes singular. It forms where $\det\!\left(\frac{\partial\mathbf r}{\partial\boldsymbol\xi}\right)=0$ . Resizing the vessel changes the real radius of its circular wall; for a circle, that is also the radius of curvature. The Lab colors the incident, reflected, internal and transmitted segments differently.

Why it mattersWhy does light reflected or refracted by a curved surface gather into sharp, bright curves?

Start with the essentials

Focus question: Why does light reflected or refracted by a curved surface gather into sharp, bright curves?
One-sentence intuition: When parallel rays reflect inside a circle, the cusp sits exactly at $r_{\mathrm{cusp}}=R/2$ . The glow on screen stays finite on purpose: in any real experiment, wave effects keep the brightness finite where ray optics says it diverges.

Core mathematical model

Envelope condition

\det\!\left(\frac{\partial\mathbf r}{\partial\boldsymbol\xi}\right)=0

Each ray carries a label. At the caustic, the map from ray label to physical position loses rank.

Cusp of the reflection caustic in a circle

r_{\mathrm{cusp}}=\frac{R}{2}

This exact result holds for parallel rays reflecting off the inside of an ideal circle.

Common difficulties

Reading infinite intensity from bright pixels

Typical misconceptionThe brightest spot on the drawn caustic tells you how intense the light really is there.

Better mental modelThe ray calculation gives the ray density and the shape of the envelope. Predicting intensity near the singularity takes the wavelength, the aperture, the coherence and a wave-optics calculation.

Run the experiment

01
Scene 1: Cup nephroid
Keep the source angle fixed and change the vessel radius. Watch the wall, the radius ruler, the ray hit points and the nephroid cusp together.
What to observe: The wall and every hit point move with the vessel radius. The cusp stays at half that radius, and the whole envelope turns with the source direction.
02
Scene 2: Refractive glass caustic
In the glass-cylinder scene, follow each ray through its entry, internal, exit and outside segments.
What to observe: The ray through the center of the cylinder goes straight through. Off-axis rays refract at both surfaces and then form the envelope beyond the cylinder.
03
Scene 3: Point-source cusp editor
Rotate the point source inside the circle and compare where neighboring rays cross with the visible cusp.
What to observe: Switching from parallel light to a point source inside the circle changes the topology of the reflection caustic. The law of reflection stays the same.