Modal Solution of a Point Source in a Strongly Focusing Medium

The ray solution for a point source located in a medium with varying propagation constant given by k ( z ) = h 0 √sech 2 α z β 2 exhibits for β = 0 the interesting property of the focusing of all rays at periodic distances. The problem of finding the exact solution for the electromagnetic field reduces to that of finding Green's function to the scalar wave equation, ∇ G + k 2 0 (sech 2 α z − β) G = (1/2π r )δ( r )δ( z − z 0 ). The eigenfunctions for this equation consist of a mixed spectrum, i.e., a continuous set of eigenfunctions plus a finite number of discrete eigenfunctions. The mixed spectrum is normalized and shown to form a complete set, in which the Green's function is expanded. Examination ofthe discrete spectrum in the far field reveals that all the lower order modes add in phase where ray theory predicts focusing. Integral approximations for the continuous spectrum show that it represents a nonpropagating, rapidly decaying field, which is negligible compared to the discrete spectrum, except in the region “close” to the point source.