Equivalence relation between partial angular harmonic and ray-type Green´s functions for a cylindrical dielectric layer

The relation between Green´s functions appropriate to closed and open shells is explored. The problem of constructing a rigorous reference solution for multiple reflected-ray fields traversing a section of a curved layer is addressed by exploiting the connection between traveling-wave (ray-type) and standing-wave (angular harmonic) fields in the closed layer. Utilizing Poisson summation, the discrete superposition of standing wave fields is converted into a discrete superposition of spectral integrals whose asymptotic evaluation generates traveling waves. It is possible to group together the spectral integrals that correspond to the collection of ray fields with N reflections inside the cavity, and to express this sum alternatively, and exactly, in angular harmonic form. The quality of the asymptotic (i.e. ray) approximation obtained from the spectral integrals can be checked against numerical data from the harmonic series. This problem strategy is implemented for the two-dimensional canonical configuration comprising a circular cylindrical homogeneous dielectric layer excited by an axially directed electric line current source located in the interior cavity surrounded by the layer