Vector potentials and physical optics

The general problem considered is to obtain solutions w to the vector equation v = curl(w), where v is a given divergence−free vector field with singularities. Two methods are discussed: A special method, which applies when v is of the type which occurs in the Kirchhoff theory of diffraction, and a general method, which applies to any divergence−free vector field whatever. As an example the general method is used to obtain the Maggi−Rubinowicz representations of the Kirchhoff−Helmholtz (double) integral as a line integral. The singularities of the solutions w are known to produce important optical effects, and the nature of these singularities is largely determined by topological properties of the domain on which v is regular.