We have made a detailed theoretical and experimental study of the three-dimensional diffraction pattern decorating the geometrical-optics caustic surface whose form is the elliptic umbilic catastrophe in Thom’s classification. This caustic has three sheets joined along three parabolic cusped edges (‘ribs’) which touch at one singular point (the ‘focus’). Experimentally, the diffraction catastrophe was studied in light refracted by a water droplet 'lens’ with triangular perimeter, and photographed in sections perpendicular to the symmetry axis of the pattern. Theoretically, the pattern was represented by a diffraction integral
E(x,y,z)
, which was studied numerically through computer simulations and analytically by the method of stationary phase. Particular attention was concentrated on the ‘dislocation lines’ where |
E
| vanishes, since these can be considered as a skeleton on which the whole diffraction pattern is built. Within the region bounded by the caustic surface the interference of four rays produces hexagonal diffraction maxima stacked in space like the atoms of a distorted crystal with space group R3m. The dislocation lines not too close to the ribs form hexagonally puckered rings. On receding from the focus and approaching the ribs, these rings approach one another and eventually join to form ‘hairpins’, each arm of which is a tightly wound sheared helix that develops asymptotically into one of the dislocations of the cusp diffraction catastrophe previously studied by Pearcey. Outside the caustic there are also helical dislocation lines, this time formed by interference involving a complex ray. There is close agreement, down to the finest details, between observation, exact computation of
E(x,y,z)
, and the four-wave stationary-phase approximation.
Spatial organization of different sigma factor activities and c-di-GMP signalling within the three-dimensional landscape of a bacterial biofilm
