The Critical Points of Poynting Vector Fields

Citation

Rizvi, Syed Azhar Abbas (1988) The Critical Points of Poynting Vector Fields. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/gj18-eq28. https://resolver.caltech.edu/CaltechETD:etd-11082007-131130

Abstract

In a thought provoking paper Maxwell [The Scientific Papers of James Clerk Maxwell, ed. W. D. Niven, vol. 2, 233-240, Dover Publications, New York (1952)] studied the flow of water on the Earth's surface and how this flow is affected by the local geography. His results linking number of hills and lake bottoms to valleys are simple and the conclusions elegant. Critical points such as summits and lake bottoms play a key role in the overall organization and structuring of the flow lines. This is the spirit in which electromagnetic power flow represented by the Poynting vector field (S) is studied in this thesis. The specialized case of a planar S field which arises due to a single electromagnetic field component E^z or H^z is dealt with here in considerable detail.

In order to analyse the behaviour of the flow lines of a plane Poynting vector field in the neighbourhood of a critical point, the S field is expanded in a Taylor series. Critical points can be classified according to their order, degeneracy or structural stability. The order of a critical point refers to the degree of the leading non zero term in the Taylor series. A critical point is non degenerate if this leading term is sufficient to give a qualitative description of the flow lines in the neighbourhood. A critical point is structurally stable if the flow lines in the neighbourhood do not change drastically when there is a small perturbation of the electromagnetic field. It is found that lowest order critical points, i.e., elementary center point and elementary saddle point, are the only structurally stable critical points. These critical points are always non degenerate. All degenerate and non elementary critical points are found to be structurally unstable. A formula for the index of rotation of the S field at a critical point is derived. The behaviour of the electric or the magnetic field component which lies in the x-y plane is also studied. It is shown that structurally unstable configurations of flow lines change into structurally stable configurations under small perturbations in such a way that the index of rotation is conserved. The statements made above in connection with the behaviour of flow lines and structural stability are illustrated with the help of examples involving linearly polarized system of interfering plane and/or cylindrical waves.

The flow lines of the S field in the vicinity of a perfectly conducting surface are studied. It is found that in structurally stable situations these lines are either parallel to the surface or they form critical points of half saddle type on this surface. Two types of problems involving flow lines and conducting surfaces are identified. The interior problem deals with the situations where all the flow lines are inside a region bounded by a perfect conductor. In the exterior problems all the flow lines are outside a region bounded by a perfectly conducting surface. Conclusions regarding the existence of critical points and the behaviour of flow lines are drawn in the two above mentioned problems. These conclusions are verified by computation of flow lines in a few well known problems of scattering and diffraction.

Finally the critical points of three dimensional Poynting vector fields are considered. A complete classification of these critical points requires further study at this time. In this thesis only structurally stable critical points are classified for these S fields. An example demonstrating the existence of such critical points is given.

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