The Cook Book

Recipe for the week of September 23 - 29

Refraction a la Toffoli and Margolus

Over the past year we have discussed examples of cellular automaton modeling in fields of applied science such as ecology, microbiology, marine biology, and chemistry. This week's soup depicts a classic experiment from the most frequent area of CA applications: physics. We show the familiar refraction pattern as a traveling wave (the blue stripe) crosses a 'lens' (the red ball) with higher refractive index than its surroundings, causing slower propagation through an 'ether' of lattice gas particles.

Particle motion and interaction is modeled by means of the previously discussed HPP Gas. Wave 'solitons' within the lens simply advance at half the speed, diffusing on alternate updates. Our graphic, generated by CAM8, shows the resulting configuration after the wave has traveled almost half way across the lens (from left to right). Originally a demo for the earlier CAM6 mixmaster, details of the source code may be found on pp.182-3 of Toffoli and Margolus' Cellular Automata Machines (MIT Press, 1987). They describe the effect as follows:

The part of the wave that enters the lens first is slowed down first; this bends the wave-front and makes the wave converge. Since we are using a circular lens the converging rays show spherical aberration : instead of a sharply defined focal point they produce a caustic - the pattern one sees when light is reflected on the inside of a cup filled with milk.

Notice that there is also a weak reflected wave. This is not an artifact of the model; in a reversible medium, theory predicts reflections whenever there is a sharp discontinuity in the index of refraction, i.e., in the speed of propagation of information. If information arrives faster than it can proceed onward, some of it must be reflected since none can be lost.

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