On the Calculation of 2D Green's Functions Via Sommerfeld Integrals and the Spatial Singularity Method

Abstract: We provide a low-level review of the computation of Sommerfeld integration theory using the singularity expansion method (SEM) to analytically estimate the short-wavelength components of the 2dimensional Green's function. The SEM is employed to replace the infinite tail of the spectral integral by a closed-form evaluation. The various steps in the SEM substitution and the calculations are elaborately presented and discussed with emphasis on giving the missing details often not included in the published literature.


I. Introduction
Sommerfeld integrals arise in the problem of calculating Green's functions of electromagnetic sources in media containing open boundaries. For a source above stratified media, the spectral-domain Green's functions can be obtained in closed form. However, in order to obtain the corresponding spatial-domain Green's functions it is required to calculate the inverse Fourier transform. This process results in what is known as Sommerfeld integrals. The calculation of these integrals is very difficult due to the oscillatory nature of the integrand and the existence of branch points and surface pole (SWP) singularities. Discrete complex image theory (DCIT) or the spatial singularity expansion method (SEM) is a fast and efficient method that can be used to treat such problems [1][2][3][4][5][6]. Applications include computing the electromagnetic fields in cylindrical nanostructures [7,8], RF antenna system analysis and design [9], and near-field nano-optics [10].

II. Formulation of the Problem
In what follows, we assume familiarity with the concept of Green's function at the level of [11]. Consider the following spatial domain 2D Green's functions calculated using the inverse Hankel transform where   Gk  is the spectral-domain function. For review of the bessel functions 0 () Jx and the derivation of (1), see [11]. If the function   Gk  is even then we can write the above integral as The propagation constant z k is given by where i k is the wavenumber in the ith medium. We now present the computation of the spectral integral above in the following hierarchical method.  The reason of including the multiplicative factor z jk in equation (6) will be apparent when we use Sommerfeld identity later. Alternatively, the equation (6) can be re-written in the following form where it is easy to show that Substituting (7) into (2) we get The following form of Sommerfeld identity will now be used Equation (9) which is the distancegenerally complex -of the nth image.

II.b 2-Level Approach:
The two-level DCIT approach consists of sampling over the following two parameterized integration paths   :0 The corresponding path in the  k complex plane is shown in Fig. 1. It is clear that a linear variation in the z k plane shown in Fig. 1(a) is translated in the  k plane to the deformed path shown in Fig. 1(b). Notice that Now, the spectral domain function in the first integral is approximated using PM or GPOF over the path

III.c 3-Level Formulation
The 3-level formulation follows the basic idea developed in the previous section. Fig. 3 illustrates the three paths , kk    . The reason of doing this is the fact that for some problems the spectral domain Green's functions may vary considerably in one region while being smooth in other regions. Since the PM or the GPOM work only for uniform sampling, they can not be utilized directly in sampling the function unless a huge number of points is used (may be thousands).
To avoid this, we divide the main region max1 0, k    into two sub-regions where higher number of samples will be used in each region. Because the sub-regions are smaller than the original interval, the "higher" number of samples here is still less what would be required if the entire region is sampled once using uniform sampling. The suggested contour path will be given in the following way     1  2  1  1  1   2  2  2  3  2  2  2   3  3  3  3  3  2   : ,0 :