Optical diabolos: Configurations, nucleations, transformations, and reactions

Generically, elliptically polarized optical beams contain C points, isolated points of circular polarization embedded in a field of elliptical polarization. At a C point the major axis a and minor axis b of the polarization ellipse are degenerate. As one recedes from a C point this degeneracy is lifted in an unusual manner: surfaces a and b form paired cones that touch at their apex, the C point, to form a double cone known as a diabolo. Diabolos appear in a many different areas of science ranging from the mathematics of curved surfaces (umbilic points) to the optics of biaxial crystals (conical refraction) to the degeneracies of chaotic systems (quantum billiards) to the energy surfaces of polyatomic molecules (conical intersections). The cones of a diabolo are classified as either elliptic or hyperbolic. We show that in optical fields only hyperbolics can nucleate or annihilate, and that elliptics arise from hyperbolic transformations. We also show that hyperbolics divide into two subclasses which give rise to five different diabolo types. We formulate loop rules based on topology and geometry for the various diabolos. These rules constrain the landscapes of all systems that contain diabolos. We use the loop rules to derive the generic transformations and reactions of diabolos and to enumerate the nine basic diabolo landscapes. The theoretical predictions are confirmed using computer simulations of diabolos in elliptically polarized laser beams.