Point-matched solutions for propagating modes on arbitrarily-shaped dielectric rods

Every arbitrarily shaped rod is shown to possess a non-radiating guided wave and a new variational solution is derived. A simpler formulation based on the point-matching principle is outlined and likely sources of computational error are scrutinized. In distinguishing between static and wave-like field solutions it is shown that a convergence criterion based on eigenvalues alone can be misleading, point-matching methods being no worse than some other numerical processes in this respect. The validity of the chosen expansion is a critical factor and two new necessary conditions imposed by the boundary geometry are deduced which are of considerable help in selecting an expansion; these being angular periodicity and radial single-valuedness. No sufficient condition for the expansion validity has been given as yet. Relevant mathematical ideas and theories about the point-matching technique are examined and recommendations made on how best to apply the technique. Computations supported by experimental or other evidence are presented including computations of an arbitrarily shaped nonconvex guide and a new equilateral triangular guide. The calculations throughout are restricted mainly to the dominant HEu mode and the present computer program is capable of giving acceptable engineering results for a variety of rod shapes which are of interest.