Water Caustics: Snell's Law & the Physics of Focused Light

Q: What causes bright and dark patches in a caustic pattern?

Brightness on the pool floor is proportional to the inverse of the Jacobian of the mapping from the wavy surface to the bottom. Where refracted rays converge (negative surface curvature), the Jacobian shrinks toward zero and intensity spikes toward a bright caustic line; where rays diverge, the spot dims. The caustic itself is exactly where the Jacobian equals zero.

Q: What is Snell's window?

Snell's window is the roughly 96-degree-wide cone (48.6° from vertical) within which a fish looking upward from underwater can see the entire outside world compressed into a circle, because water's critical angle for total internal reflection is arcsin(1/1.333) ≈ 48.6°. Outside that cone the water surface acts as a mirror.

Q: Why does geometric optics predict infinite brightness at a caustic?

In geometric optics, ray density literally diverges at the focusing surface, since infinitely many rays compress into zero area. Wave optics resolves this: diffraction gives a finite peak intensity described by an Airy function, with a characteristic width scaling as (wavelength × distance)^(2/3) — the physically real, finite version of the caustic's bright edge.

Snell's law and Snell's window

When a ray crosses the boundary between air and water, the refraction angle follows Snell's law:

n₁ sinθ₁ = n₂ sinθ₂     (n_water ≈ 1.333, n_air = 1.000)

Critical angle: sinθ_c = n₂/n₁  →  θ_c ≈ 48.6° for water

At θ_c, total internal reflection begins — a fish looking upward sees
the entire "sky" compressed into a 48.6° cone: Snell's window.

At normal incidence, water reflects only about 2% of light (Fresnel's F₀ = ((n₁−n₂)/(n₁+n₂))² ≈ 0.020) — but as the viewing angle approaches grazing incidence, reflectance rises toward 100%, which is why a lake looks mirror-like near the horizon but transparent looking straight down.

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Caustics as the envelope of a ray family

A caustic (from the Greek for "burning") is the envelope of a family of refracted or reflected rays. Where a moving water surface has enough negative curvature at a convex section, refracted rays converge — the intensity on the bottom is proportional to the inverse of the Jacobian of the mapping from surface to bottom:

I ∝ 1 / |J|,     J ≈ 1 + d·(n−1)·∇²h

Caustic forms where J = 0  (i.e. ∇²h = −1/(d(n−1)))
— convex ripple sections focus light into bright lines

Catastrophe theory classifies these focusing surfaces into named types: fold caustics (a bright line, the most common form from ripples), cusp caustics (a sharpened tail near a pool corner), and rarer swallowtail and elliptic umbilic forms. In pure geometric optics the ray density literally diverges at the caustic — infinite brightness — but wave optics resolves this with a finite Airy-function peak of characteristic width scaling as (wavelength × distance)^(2/3).

Frequently asked questions

What causes bright and dark patches in a caustic pattern?

Brightness is proportional to the inverse of the Jacobian of the surface-to-bottom mapping. Converging rays (negative curvature) push the Jacobian toward zero — a bright caustic; diverging rays dim the spot.

What is Snell's window?

The roughly 48.6°-from-vertical cone within which a fish looking upward sees the entire outside world compressed into a circle, bounded by water's critical angle for total internal reflection.

Why does geometric optics predict infinite brightness at a caustic?

Infinitely many rays compress into zero area at the focusing surface. Wave optics resolves this with a finite Airy-function peak whose width scales as (wavelength × distance)^(2/3).

Try it live

Everything above runs in your browser — open Water Caustics and adjust wave amplitude, frequency and depth to watch sunlight refract through a wavy surface into shimmering caustic patterns.

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