Optical Caustics

The intuition — light piling up

Stand at the edge of a sunlit pool and look at the floor: a restless net of bright lines weaves across the tiles, brighter than the daylight around it. Hold a glass of water in the sun and a sharp curve of light burns onto the table. Tilt a coffee cup and a pointed loop of light floats on the surface. These are all the same phenomenon — a caustic.

The key idea is that a curved surface does not send parallel rays to a single perfect focus. Instead, rays from different parts of the surface cross each other along a curve (in 2D) or a surface (in 3D). Where many rays cross in the same place, the light concentrates. The bright envelope they trace out is the caustic. The name comes from the Latin caustica, "burning" — a magnifying glass focuses sunlight onto its caustic point and can set paper alight.

Crucially, a caustic is not a single focal point. A perfect lens focuses parallel light to one point, but any imperfect focusing — a ripple, a wine glass, a cylindrical cup, a raindrop — smears that focus into an extended bright curve with a characteristic shape: a smooth fold line that pinches into a sharp cusp.

How caustics form — the envelope of rays

Treat light as a family of rays leaving a surface. Each ray is a straight line; the family is parameterized by where on the surface it started (call that parameter s). As you sweep s, the rays sweep through space. The caustic is the envelope of this family — the curve that every ray is tangent to.

Think of it geometrically. Two infinitesimally separated rays, from surface points s and s + ds, generally cross at some point. As you slide s along the surface, that crossing point traces a path. That path is the caustic. Where the crossing happens determines how bright a region gets, because a whole continuum of nearby rays passes through the same neighborhood.

Two stable things can happen along a caustic, and only two in everyday situations:

• Fold. A bright line where exactly two rays merge. On one side of a fold you have two rays through each point; on the other side, none. The brightness jumps sharply at the line. Most of any caustic is fold.
• Cusp. A pointed corner where two folds meet and three rays merge. Inside the cusp there are three rays through each point; outside, one. The cusp is the brightest, most pointed feature — the tip of the heart in your coffee cup.

The governing physics

Let the surface map a ray's starting parameter s to where it lands on a screen, a function x(s). The intensity on the screen counts how many rays land in a small interval and divides by its width:

I(x) ∝ Σ_branches  1 / |dx/ds|

caustic condition:   dx/ds = 0    (rays pile up — neighboring rays land together)

When dx/ds = 0, an entire range of starting parameters s maps to nearly the same landing point x. The ray density — and therefore the intensity — diverges. In 3D the same statement is that the Jacobian of the ray map vanishes:

J = ∂(x, y)/∂(s, t) = 0     (2D surface → 2D screen)

Catastrophe theory says the way x(s) flattens near a caustic has universal forms. Near a fold the optical path length behaves like a cubic, and near a cusp like a quartic:

fold:   path length  Φ(s) = s³/3 + a·s          (Airy function intensity)
cusp:   path length  Φ(s) = s⁴/4 + b·s²/2 + c·s  (Pearcey function intensity)

Ray optics predicts infinite intensity at the caustic because dx/ds = 0. Real light is a wave, and diffraction rounds the spike to a finite peak. Solving the wave equation near a fold gives the Airy function; near a cusp it gives the Pearcey function. The peak width scales with wavelength:

fold peak width  ∝ λ^(2/3)
cusp peak width  ∝ λ^(3/4)
peak intensity   ∝ λ^(-1/3)  (fold) ,  λ^(-1/2)  (cusp)

Because these exponents are fractional and universal, the faint fringes you see on the inner side of a rainbow (supernumerary bows) have spacings set entirely by the Airy function — pure wave optics overlaid on the geometric caustic.

Reflection vs refraction caustics

Caustics come in two flavors depending on whether light bounces off or passes through the curved surface.

Type	Mechanism	Classic example	Caustic name
Catacaustic (reflection)	Light reflects off a curved mirror	Coffee cup, polished ring	Cardioid / nephroid
Diacaustic (refraction)	Light bends through a curved interface	Pool floor, wine glass, lens	Fold + cusp network
Internal-reflection caustic	Refract in, reflect inside, refract out	Rainbow (raindrop)	Fold at min-deviation angle
Gravitational caustic	Spacetime curvature bends light	Lensed quasars, Einstein rings	Fold + cusp (same math)

A circular mirror illuminated by parallel light produces a nephroid (a two-cusped curve); illuminated by a point source on its rim it produces a cardioid (one cusp — the classic coffee-cup heart). A spherical refracting surface or a wavy water film produces a dense network of folds and cusps that flickers as the surface moves.

Numbers behind the shimmer

Setting	Quantity	Typical value
Pool surface ripple	Amplitude	1–10 mm; wavelength 5–50 cm
Pool caustic flicker	Frequency	0.5–5 Hz (tracks surface waves)
Caustic brightness	Peak / background ratio	3×–10× ambient on the floor
Refraction at the surface	Bending governed by Snell	n_water = 1.33, n_air = 1.00
Cusp fringe width	Wave-optics scale (λ ≈ 550 nm)	tens of µm at lab focal lengths
Primary rainbow	Caustic deviation angle	≈ 138° (bow at ≈ 42° from antisolar point)
Secondary rainbow	Second internal reflection	bow at ≈ 51°, reversed colors

Energy is conserved throughout. The light brightening a caustic line is exactly the light removed from the dim regions around it — a caustic is a redistribution of a fixed flux, squeezed where ray tubes pinch and starved where they spread.

Where caustics show up

• Swimming pools and shallow water. The textbook caustic — sunlight refracting through a wavy surface onto the floor. The same net appears on the side of a boat hull and on the seabed in clear shallows.
• Drinking glasses and cups. Reflection off a cup's inner wall (cardioid/nephroid) and refraction through a glass of water onto a table.
• Rainbows, glories, and halos. Rainbows are fold caustics from raindrops; supernumerary arcs are the Airy fringes of that caustic.
• Gravitational lensing. Massive galaxies bend light into the same fold-and-cusp caustics. Microlensing light-curve spikes happen when a star crosses a caustic — astronomers detect exoplanets this way.
• Solar concentrators and burning glasses. The "burning" origin of the word; sloppy reflectors create hot caustic lines that can damage receivers.
• Computer graphics. Rendering realistic glass, water, and gems requires simulating caustics — photon mapping and caustic maps exist specifically for this.
• Acoustics and tsunamis. The same envelope mathematics governs sound focusing and the wave-height spikes where ocean bathymetry focuses a tsunami onto a stretch of coast.

Worked example — the coffee-cup cardioid

Take a cylindrical cup of inner radius R with a small light source sitting on the rim. A ray from the source hits the inner wall at angle θ from the contact point and reflects (angle in = angle out). Tracking where successive reflected rays cross, their envelope is a cardioid whose cusp sits at distance R/3 from the cup's center, on the side away from the source:

cardioid (polar, pole at the cusp):
  r(φ) = (2R/3) · (1 + cos φ)        — the bright caustic curve

cusp located at distance  R/3  from the cup's center
(the source point on the rim is the far tip, at r = 4R/3).

For a 4 cm radius mug, the cusp sits about 1.3 cm from the center — exactly where you see the sharp point of the heart. Swap the point source for parallel sunlight and the curve becomes a nephroid with two cusps, whose on-axis focus is the mirror's paraxial focal length R/2. The cup didn't change — only the incoming ray family did, and the caustic shape followed.

Common misconceptions and edge cases

• "A caustic is just an out-of-focus image." No — a caustic is a real geometric object, the envelope of rays, that exists whether or not anything is in focus. It is the singular set of the ray map, not a blurred point.
• "Ray optics says the brightness is infinite, so the theory is broken." Ray optics genuinely predicts a divergence; that's a feature, not a bug. It correctly flags where wave optics must take over. The Airy/Pearcey functions give the finite real answer.
• "Caustics need a lens or a mirror." Any curved interface works — a single ripple, a hair-thin film, a soap bubble, even density gradients in hot air (which is why heat shimmer above a road dances with caustic-like banding).
• "There are infinitely many caustic shapes." Generically, no. Stability arguments (catastrophe theory) restrict everyday caustics to folds and cusps; only fine-tuned symmetry produces the rarer swallowtails and umbilics.
• "The pattern moves because the light moves." The Sun barely moves on these timescales. The pattern shimmers because the surface moves — its curvature changes continuously, sliding the crossing points and creating/annihilating caustic lines in pairs.
• "Color in a caustic comes from the surface." Colored edges on caustics (and on rainbows) come from dispersion — the refractive index depends on wavelength, so each color's caustic sits at a slightly different place.