Generalized Glory Scattering from Spherical and Spheroidal Bubbles in Water: Unfolding Axial Caustics with Harmonic Angular Perturbations of Toroidal Wavefronts.

Photographs of the cross-polarized backscattering patterns (i.e. glory scattering) for freely rising spherical air bubbles in fresh water are displayed. A physical-optics model was developed for these patterns. Observations were also made of the unfolded glory patterns produced by oblate spheroidal bubbles illuminated with polarized laser light. The cross-polarized pattern was observed, and was modeled with a physical optics approximation for two different incident polarization directions relative to the axis of rotational symmetry of the spheroidal bubble. Inverse scattering results were derived and used to estimate bubble oblateness from features of the angular scattering pattern. Axisymmetric toroidal wavefronts are pertinent to glory scattering (i.e. enhanced backscattering) by spherical bubbles. This wavefront propagates to produce a structurally unstable axial line caustic. When the axisymmetric toroidal wavefront is perturbed to have 2-fold rotational and mirror symmetry, the axial caustic unfolds into a 4-cusped curve (astroid) in cross-section with cusp points that move away from the symmetry axis as the range increases. In addition to glory scattering from bubbles, theories for other unfoldings of axial caustics were explored. The caustic surfaces associated with general harmonically perturbed toroidal wavefronts are considered. Perturbed toroidal wavefronts that have p-fold rotational and mirror symmetries have caustic cross-sections of cusped stars, with p cusps when p is odd, and 2p cusps when p is even. These wavefront perturbations have application to objects that can be produced by deforming a sphere to shapes having p-fold rotational and mirror symmetry about one axis, and inversion symmetry. The p = 2 object is a spheroid for which the rising bubble is one example. The scalar (acoustic) wavefield is computed using a Fresnel approximation of the diffraction integral. The wavefield near the cusps of an astroid caustic is similar to that of a Pearcey pattern; a result which also follows from the principles of catastrophe optics. The locations and types of rays which contribute to the scattering near the caustic surface are considered for the 2-fold symmetric perturbed toroidal wavefront. A novel expression is derived in polar coordinates for the Hessian associated with propagation. Though not analyzed in detail, these results can give insight into the unfolded glories of dielectric spheroids in air and elastic objects in water.