Ray Theory of Reflection from the Open End of a Waveguide

Ray methods have been widely used to calculate high frequency scattering by objects in an unbounded medium, but they have not been used much for waveguide problems. To show that they can be used for waveguides, we treat by ray methods reflection from an open-ended parallel plane waveguide propagating several modes. The incident mode is decomposed into two plane waves whose scattering by the edges at the termination produces the reflected field. The singly diffracted cylindrical wave originating at each. edge, which is known from the asymptotic theory of diffraction by a single wedge, is represented by means of diffracted rays. Then the stem of the fields on the multiply reflected rays is converted into modal form. This yields formulas for the reflection coefficients in the various modes due to single diffraction. In addition double and multiple diffraction are also taken into account, yielding improved formulas for the reflection coefficients. The results of extensive numerical calculations for $TE$ and $TM$ modes incident in flanged and unflanged open-ended waveguides are presented, and are compared with calculations based on exact solutions when these are available. It is found that the single diffraction results account for the average behavior of the amplitude and phase of the reflection coefficients. The inclusion of multiple diffraction accounts for the finer details and yields excellent accuracy even for wave guide height-to-wavelength ratios as small as $1/3$ in the TM case. These results suggest that the ray method also provides an accurate means for analyzing other types of waveguide discontinuities.