The Catastrophe Optics of Liquid Drop Lenses

The optical caustics associated with a number of higher umbilic catastrophes have been studied experimentally by passing light through drops of water resting on a horizontal glass slide. When the perimeters of the drops were constrained to have the symmetry of a square (4mm), the caustics were organized by X_9, which is the lowest catastrophe of corank 2 to possess a modulus. Changing the shape of the perimeter of a thin drop of this symmetry that is large enough to be affected by gravity has the effect of sweeping the modulus K through its complete range -6 to +2; in particular, one can study the physical caustics as K passes through the special value -2, which is mathematically excluded from X_9. Similar variations of K are produced by varying the size of thin drops with square outlines, or simply by adding water to a drop of fixed outline so that its profile becomes highly curved. Twofold (2mm) and fourfold (4mm) symmetries are the only ones that allow X_9, with its full range of moduli, to participate. However, the range of caustics produced by drops with these symmetries cannot be understood in terms of X_9 alone; to explain the observed details it is necessary to take account of the fact that X_9 is embedded in the higher catastrophe Y^1_{2, 2}. A theory of the drop profile under the combined effects of surface tension and gravity, valid for small slopes, leads to a calculation of the caustics in the paraxial approximation. This is completely adequate to explain the observations, for thin drops, but, as a surprising example of structural stability, it also suffices when the drops are highly curved and the paraxial approximation breaks down. However, the paraxial theory does fail, as expected, when highly curved drops are inverted so that they hang from the lower surface of the glass slide. Then the X_9 patterns due to gravity are opposed to those due to non-paraxiality; for a drop of square outline, filled nearly to the point where it is about to detach itself, and illuminated from below, the two effects exactly cancel when the side is 7.3 mm. Analysis of these patterns of fourfold symmetry helps in understanding the caustic patterns produced by drops having other symmetries, or no symmetry at all.