---
title: "Caustic Factory Interactive Lab | Sandbox Physics"
description: "Why does light reflected or refracted by a curved surface gather into sharp, bright curves?"
canonical_url: "https://sandboxphysics.com/labs/caustic-factory/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/caustic-factory/"
last_modified: "2026-10-27"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Geometric optics predicts an infinite ray density at a caustic. In reality, the finite wavelength, aperture and coherence keep the intensity finite. These wave effects are not modeled, and brightness is not adjusted to imitate them.

Physics tutorial

## Audit the physics of Caustic Factory

**Background** A 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 matters** Why 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 misconception** The brightest spot on the drawn caustic tells you how intense the light really is there.

**Better mental model** The 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

1. 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.

2. 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.

3. 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.

Physical conclusion

Trust a drawn caustic when every point on it traces back to a complete family of real rays, and when it stays clear where wave optics has to take over.

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)