We develop the theory of the Poynting singularities (critical points of the Poynting vector) extending the theory of dynamic systems to classify and analyze optical singularities. An optical dynamic system is described by the three first-order differential equations for the image point, with the tangent to the image point trajectory being the Poynting vector. Important feature of the Poynting singularities is the existence of the polarization-induced singularities (arise due to the specific field polarization) along with the field-induced ones (appear owing to the vanishing the fields). We analyze not only isolated critical points, but the manifolds of singularities forming lines and surfaces as well. We define the types of the singular points (vortex, saddle, sink, source, and focus) using the trace and determinant of the stability matrix. Such a criterion and the study of the dependence on parameter (bifurcations) are applied for a number of examples. We offer to study the chaotic dynamic of the image point in future.

• Received 15 January 2009

DOI:https://doi.org/10.1103/PhysRevA.79.033821