Trout and Optical Catastrophes Imaged by Andrew Kirk. "I was photographing rapidly moving caustics and a dead tree on the bottom of a wide stream, when a trout floated into the frame. I then began to attempt to catch the caustics as they crossed the trout. The distortions of the trout were too rapid to see, but the camera caught them!"  ©Andrew Kirk, shown with permission.


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Refraction by the wavy water surface distorts the trout and produces bright lines on the stream bed, ‘caustics’.

Caustics are often ascribed to ‘focusing’ by the wavy surface. Partially true – but why then are they always sharp? A lens focuses rays to a point but change the lens distance and there is only a blur. Why then are caustics not blurred? And why are their dancing patterns strangely structured and persistent rather than random? Why are they so often paired? Why do they apparently like triple junctions or pout into lip shapes? Watch them, there is order there – It transpires that the order is profound.

In its simplest form a caustic is a sheet or line of light within the water and where it intersects a stream bed (or a passing trout) we see a wavy line. It is a region of space where light rays bunch together, coalesce. But it marks something more; the patterns of rays are different on each side of a caustic sheet.  Caustics denote sharp spatial discontinuities in ray behaviour.

A comparatively recent branch of pure mathematics, theatrically named ‘catastrophe theory’, has surprising applications to caustics. Catastrophe theory deals with the behaviour of critical points on a map.   The name arises because these critical points are ones of stability and transition from one stable point to another can, in the practical world, have dire consequences.   For example, engineers apply catastrophe theory to the stability of bridges.   One stable point on their map is the desired bridge structure.  Twisted girders in the river are another stable point.   The discontinuity in behaviour representing bridge collapse is a catastrophe.

The very same mathematics describes how rays from a refracting (or reflecting) surface behave because rays of light from a surface to the eye define a map.    Caustics mark discontinuities, optical catastrophes.    The theory predicts that they are stable and occur naturally.   A rippling stream or swimming pool generates their sharp and ordered forms without special preconditions – they just happen.   Compare that to the high-tech effort needed to create a telescope lens.   The lens focus is unstable in that any minor perturbation – a change in distance to the focusing screen or an error in figuring – destroys the hard won focus into a blur.   Caustics - Nature’s focusing - stay sharp.  It is stable – human focusing is not.

A deep result from catastrophe theory is that caustics are not infinite in their variety.   There are only seven stable elementary forms. The first two, cusp and fold caustics, are at right.  The severe restrictions on form give rise to their remarkable order and to a high specificity in the way they cross and interact.   Although ever changing, their shifting patterns remain sharp, familiar and ordered.

Caustics go beyond stream beds and swimming pools.    The arc of a rainbow is one. The twinkling of stars, glitter paths on the sea, curves of light in teacups and wine glasses, the sharp spidery shapes seen by spectacle wearers when raindrops get on their lenses and some gravitational lensing are all manifestations of caustics.

Catastrophe theory strictly organises caustics. Their dance is to precisely prescribed steps - a minuet in light.

Light rays crossing and bunching together form a bright caustic surface (a fold caustic in this example).   More profoundly, a caustic marks a topological discontinuity (a catastrophe) in ray behaviour.

Upper right:
Caustic sheets below a wavy surface illuminated by an overhead sun. For each wave crest there are paired sheets (fold caustics) joined at top by a cusp caustic. Where the fold caustics intercept a stream bed they give a bright line.

Background image:
An accurate computation for vertical rays refracted by a wavy water surface. Three rays from the wave crests form the cusps. From elsewhere on the wave, two rays intersect on each point of the downward going fold caustic sheets. Rays from near the wave midpoints where the surface has an inflexion point form the very distant caustics.

The waves were made high amplitude to fit the caustics onto the page. We usually see caustics from gentler surfaces and then the pairing of caustics with brighter regions between them is less obvious.