Evaluation of cuspoid and umbilic diffraction catastrophes of codimension four
As any lens maker can attest, it requires precise and deliberate design to produce the multiray intersection associated with a geometrical focal point. Further, the focus is not stable with respect to perturbations of the lens—small defects result in optical aberrations and ray intersections that are no longer colocated. In contrast to a geometrical focus, there do exist ray configurations in which the pattern of ray intersections (which correspond to bright optical loci known as caustics) are stable with respect to small perturbations of the governing physical system. Catastrophe optics shows that these stable caustics can be described using a surprisingly limited taxonomy of cases. The stability of these 'catastrophes' means that they are frequently observed in natural focusing. Indeed, catastrophe optics describe phenomena such as rainbows, sunlight sparkling from the sea surface or casting patterns on the floor of a swimming pool, the twinkling of starlight after propagation through the atmosphere, and the distinctive structures observed when light passes through a glass of water.
Catastrophe optics is not restricted to the geometric short-wave limit—the theory also provides a means to evaluate the diffraction pattern in the vicinity of the caustics. However, this calculation involves a challenging integration of highly oscillatory phase functions. In previous work, Borghi has described a tractable and efficient method for evaluating the near-caustic diffraction field for the five most elementary catastrophes. His method relies on an optimized representation of the diffraction integral as a power series. In the current paper he extends this work to two additional catastrophes: the butterfly and the parabolic umbilic. These additional cases encompass systems with a greater number of contributing physical parameters, expanding the range of scenarios in which the diffraction catastrophe can be practically calculated.