Supernumerary ice-crystal halos?

Geometric-optics singularities in the intensity profiles of refraction halos formed by randomly oriented ice crystals are softened by diffraction and decorated with fine supernumerary fringes. If the crystals have a fixed symmetry axis (as in parhelia), the geometric singularity is a square-root divergence, as in the rainbow. However, the universal curve that describes diffraction is different from the rainbow's Airy function, with weak maxima (supernumerary fringes) on the geometrically dark region inside the halo (and even fainter fringes outside); these are much smaller than their counterparts on the light side of rainbows. If the crystals have no preferred orientation (as in the 22° halo), the geometric singularity is a step. In this case the universal diffraction function has no maxima, and its supernumeraries are shoulders rather than maxima. The low contrast of the fringes is probably the main reason why supernumerary halos are rarely if ever seen.