Higher-order extrema in two-dimensional wave fields

Higher-order extrema with topological indices greater than unity are discussed. Explicit constructions are given for their wave functions, and simple geometric rules are presented for analysis of their topological indices. Experimental means for verifying the theory with use of Gaussian laser beams are considered, unusual properties of optical vortices constructed from this new type of critical point are described, and applications to topologically based optical arithmetic are suggested.