Arbitrarily Oriented Biaxial Media Branch Point Singularities Pertinent to the Solution of Integral Equations via Method of Moments

摘要

It is impossible to solve all but a select few electromagnetic problems analytically. Multiple anisotropic layers, non-planar geometries, and randomly located scatterers make the use of numerical techniques mandatory. One such method used to solve general configurations is the Method of Moments (MoM) technique. Essentially, this technique converts an integral equation into a matrix equation. The various elements of the matrix equation contain multidimensional integrals that must be evaluated in the spatial domain, the spectral domain, or some combination of the two domains. Equation (1) is representative of the typical two dimensional integral that must be evaluated. ∫ dk{sub}y (-∞^s< k < ∞) ∫ (dk{sub}x(J{sub}t){sup}*(k{sub}y)(J{sub}t){sup}*(k{sub}x)G{sub}(αβ)(k{sub}x,k{sub}y)J{sub}e(k{sub}y)J{sub}e(k{sub}x)(-∞^s< k < ∞) (1) A few examples of the numerous research efforts utilizing spectral analysis MoM techniques are detailed in [1-3]. The ability to successfully perform the integration required for the spectral domain analysis depends on the convergence of the integrand. For lossless media the singularities are on the real axis for certain ranges of the integration variables. To speed convergence analytic continuation is often employed whereby the integration over the real axis is deformed to integration over the complex plane. Historically, these techniques were applied to isotropic layered media. Due to the azimuthal isotropy of the medium, the two dimensional spectral integrals are reducable to a single dimensional integral over the radial coordinate using the so-called Sommerfeld identity [4]. However, many of the integrated circuit substrates in common use today are not isotropic. Indeed, many are anisotropic. The ability to invoke azimuthal symmetry arguments in order to reduce the integration from two spectral coordinates to one ceases if the medium is biaxial. For such anisotropic media the successful calculation of the MoM integrals requires the calculation and tracking of the integrand singularities in two spectral dimensions, such as the k{sub}x and k{sub}y dimensions shown in (1).