Phase-integral formulation of the single-layer microstrip Green´s function

The three-dimensional (3-D) Green´s function for a continuous, linearly stratified planar media, backed by a PEC ground plane, can be expressed in terms of a single contour integral involving one-dimensional (1-D) green´s function. The phase-integral and WKB methods are applied here to obtain the 1-D green´s function in the direction of continuous stratification, i.e., the z-axis. It is shown that for the spatially local wavenumber, k(z) with one simple zero, the 1-D green´s function contains Airy functions and hence need careful evaluation across the Stokes lines. The phase-integral or WKB approach avoids the computational complexities associated with discretization of the continuous media into several layers with piecewise continuous electrical parameters, and hence a single 3-D Green´s function can be derived to calculate all fields for arbitrary source and receiver locations.