Efficient Computation of 1-D Periodic Layered Mixed Potentials for the Analysis of Leaky-Wave Antennas With Vertical Elements

Abstract

An efficient mixed-potential integral equation formulation is proposed for the analysis of one-dimensional (1-D) periodic leaky-wave antennas (LWAs) based on planar stratified configurations with inclusions of arbitrarily oriented metallic or dielectric perturbations. Both the transverse and vertical components of the mixed-potential Green's functions due to a 1-D phased array of dipoles in a layered medium are computed through suitable homogeneous-medium asymptotic extractions from the standard spectral series of Floquet harmonics. An original acceleration procedure is applied for the computation of the vertical potentials, whose extracted terms can be expressed as potentials from a 1-D phased array of half-line sources in a homogeneous medium. Their numerical calculation requires a suitable modification of the Ewald method, thus resulting in new modified spectral and spatial series, having Gaussian convergence even in the case of complex modes and improper harmonics. Numerical comparisons for the 1-D periodic potentials, both in the case of bounded and unbounded (e.g., leaky) harmonics, validate the efficiency and accuracy of the proposed acceleration technique. The method is illustrated and verified by determining the dispersion behavior of both bound and leaky modes for several LWA test cases.

Convergent Expressions for Periodic Potentials in Stratified Media Using Asymptotic Extractions

“…The expressions for the extracted term z P  can be found in [6], and will not repeated here for the sake of brevity. It is interesting to give the explicit expression of the closed-form solution for the spatial-domain term p, z P  : it is a sum of potentials g p,z due to an array of half-line sources, computed by integrating the expressions for dipoles sources [6]. The integration can be performed analytically, resulting in proportional to P z through a k xn or k y factor, corresponding to xand y-derivatives of (4) and (5).…”

“…2, but two periods are chosen: p = 10 mm (solid lines) and p = 4 mm (dashed lines). The Bloch wavenumber is kx0 = (0.8−j0.1)k0 (the n = 0 harmonic is radiating and is improper [6], [11]). Coordinates: z = z′ = 0 (air/substrate interface), x = y = p/2.…”

“…values of k xn and k y can be derived once we recall those of the original spectral terms of p A G (always in the on-plane case): These functions must be computed using extraction (2), since the original series (1) is no longer a valid representation. The second-order derivatives of (4) and (5) are then needed for the (the n = 0 harmonic is radiating and is proper [6], [11]). Potentials computed with (continuous lines) and without (dashed lines) acceleration at z = 0.1 mm (off plane) and with acceleration (dotted lines) at z = 0 (on plane).…”

“…The case of vertically stratified media is a good example of a canonical geometry where Green's functions can be easily defined in a dyadic form [1]. While results are available for uniform structures, only recently an effort has been devoted to periodic Green's functions [2] [6], required to study, e.g., planar stratified structures perturbed by arbitrarily shaped periodic loading (see Fig. 1).…”

“…In [5] and [6], a method has been recently proposed to transform these terms into modified periodic homogeneousmedium Green's functions, thus yielding a simple algorithm based on a modification of the Ewald method, which grants a remarkable convergence acceleration even in the case of complex modes and improper harmonics. This method is suitable for analyzing metallic objects using the EFIE.…”