Abstract
The Ewald method is applied to accelerate the evaluation of the Green's function of an infinite periodic phased array of line sources. The Ewald representation for a cylindrical wave is obtained from the known representation for the spherical wave, and a systematic general procedure is applied to extend previous results. Only a few terms are needed to evaluate Ewald sums, which are cast in terms of error functions and exponential integrals, to high accuracy. Singularities and convergence rates are analyzed, and a recipe for selecting the Ewald splitting parameter /spl epsiv/ is given to handle both low and high frequency ranges. Indeed, it is shown analytically that the choice of the standard optimal splitting parameter /spl epsiv//sub 0/ will cause overflow errors at high frequencies. Numerical examples illustrate the results and the sensitivity of the Ewald representation to the splitting parameter /spl epsiv/.
Highlights
I N APPLYING numerical full wave methods to periodic structures, fast and accurate means for evaluating the periodic Green’s function are often needed
While the paper was in print, we have found that a similar procedure was derived in [7], where performances of various Green’s function representations for 2-D periodic arrays are compared
The proposed algorithm is efficiently applied to periodic structures when the observation point is near the planar array of sources
Summary
I N APPLYING numerical full wave methods to periodic structures, fast and accurate means for evaluating the periodic Green’s function are often needed. The Ewald method is extended in [3] to 2-D problems with one-dimensional (1-D) periodicity (i.e., a planar array of line sources) for the case of coplanar source and observation points. We present here an alternative direct procedure for applying the Ewald approach to obtain the Green’s function for an array of line sources with 1-D periodicity. We present for the first time an algorithm for choosing the Ewald splitting parameter that extends the efficiency of the method when the wavelength is somewhat larger or smaller than the periodicity. The proposed algorithm is efficiently applied to periodic structures when the observation point is near the planar array of sources. The critical distance from the array plane beyond which the Ewald method is not advantageous compared to the standard spectral grating lobe series is analyzed in [9]
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