Singularity Extraction for Reflected Sommerfeld Integrals over Multilayered Media

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1 Telfor Journal, Vol. 6, No., 4. 7 Singularity Extraction for Reflected Soerfeld Integrals over Multilayered Media Vladiir V. Petrovic, Senior Meber, IEEE, Aleksandra J. Krneta, and Branko M. Kolundzija, Fellow, IEEE Abstract A new analytical-nuerical ethod based on singularity extraction is presented for efficient and accurate evaluation of one of the reflected Soerfeld integrals over ultilayered edia backed by a conducting (PEC) plane. Method is based on extraction of all the poles of the integrand. The ethod is deonstrated on one eleent of the tensor ultilayered Green s function, where only the singular part of the integrand, i.e. the reflection coefficient, is taken into consideration, as the cause of the largest calculation difficulties. It was also shown that the ethod practically eliinates all the branch cut singularities except for one. Keywords Soerfeld integrals, singularity extraction, ultilayered edia C I. INTRODUCTION ALCULATION of electroagnetic (EM) fields in the presence of a half-space or ultilayered edia, whose exact analytical solution is given by the so-called Soerfeld integrals, basically established by the classical paper [], is still in the focus of research. Applications are any. Historically the first was radio counication over the earth. Soe of the odern applications are ground penetrating radars for detecting buried objects, wall penetrating radars for detecting installations, aterial defects and inner cracks, through the wall iaging for security applications and geophysical search for gas and oil by using EM waves. However, the ost iportant ones are the increasingly growing application of the ultilayered printed circuit boards Paper received April 6, 4; accepted October, 4. Date of publication Noveber 5, 4. The associate editor coordinating the review of this anuscript and approving it for publication was Prof. George Paunovic. This paper is a revised and expanded version of the paper presented at the th Telecounications Foru TELFOR. The work presented here is partially supported by Serbian Ministry of Education, Science and Technical developent under the grant TR-5. V. Petrovic is with TES Electronic Solutions, Zettachring 8, 7567 Stuttgart, Gerany (phone: ; e-ail: vladiir.petrovic@gx.de). A. Krneta is with the School of Electrical Engineering, Univ. of Belgrade, Bul. kralja Aleksandra 7, Belgrade, Serbia (phone: ; e-ail: krneta@etf.rs). B. Kolundzija is with the School of Electrical Engineering, Univ. of Belgrade, Bul. kralja Aleksandra 7, Belgrade, Serbia (phone: ; e-ail: kol@etf.rs). (PCB) on icrowave and illieter-wave frequencies, where low-frequency approxiations and corresponding design ethods are not accurate enough. Moreover, such ultilayered PCBs often include printed or other antennas placed ostly on the top layer. A long series of scientific publications on this atter can be traced. A representative list of works can be found, e.g. in []. Both exact ethods (based on the Soerfeld integrals) and various approxiate ethods were developed. The goal of ost of these ethods is to provide an efficient and accurate evaluation of the Green function for a ultilayered structure. In this paper one of the reflected ultilayered Soerfeld integrals is chosen for deonstration of the ethod, naely the zz-coponent of a tensor Green function ( G ) for the agnetic vector-potential []. A II. ANALYTICAL FORM OF THE MULTILAYERED GREEN FUNCTION Consider a ultilayered structure, consisting of M layers ( M> ), where the botto layer is a perfectly conducting plane (Fig.). The structure is considered infinite and unifor in directions perpendicular to z-axis. Dielectric layers are non-agnetic and could be lossy or lossless, represented by their equivalent coplex perittivity, ε, and thickness, h, =,, M. r The upper half-space does not have to be vacuu, thus, in general, εr. A source and field point are above the structure and their utual position is denoted by heights z s and z f and the radial distance ρ, where any of these three relative coordinates can be zero or approach zero in the liiting case. If all the three coordinates approach zero the tensor Green function becoes singular (tends to infinity). The eleent of a tensor Green function chosen for deonstration of the ethod is g, whose copact (noralized) for [] is given by the Soerfeld integral g g = F( α, Z, Ρ) R ( α), () where Z= β( zs+ zf ), Ρ= βρ and β= ω ε µ. Function F( α, Z, Ρ ) coprises the Bessel function and an exponential function and its only singularity (in the coplex α -plane), as it contains the expression

2 8 Telfor Journal, Vol. 6, No., 4. u α =, is the branch point at α=. Function R ( α ) is often called a reflection coefficient, as it can be interpreted as a reflection coefficient for the plane wave incident to the ultilayered edia. z O M M source point Fig.. Layered edia backed by the PEC plane. For li R ( α) = R significant α iproveent of integral convergence is obtained when R ( α ) is represented as R ( α) = R+ R ( α) and the first of the resulting integrals solved analytically [4], jr F( α, Z, Ρ) R= Re / R, R= Ρ + Z. We will call function R ( α ) a reduced reflection coefficient. Extraction of this ter is iportant as the function R ( α ) for large arguents behaves as α [] and alone provides the convergence of the reaining z s integral, F( α, Z, Ρ) R ( α). This integral cannot be ρ h hm field point (PEC) ε r, µ ε r, µ ε r, µ ( ) solved analytically. More asyptotic ters could be extracted and the corresponding integrals solved analytically [], [5]. This will be addressed shortly later in the text. The oscillating nature of function F and its slow decay to zero for sall or zero values of Z ake its nuerical integration difficult. These two difficulties can, however, be solved in relatively standard and reported ways. A ore serious difficulty is that R ( α ) contains singularities in the for of poles and branch cut singularities (at points where the branch cuts of the ultivalued square root function starts). Every layer introduces one pair of these two types of singularities. For lossless layers these singularities are on the α -axis. For layers with sall losses (good dielectrics) they are very close to the α -axis, what is proven to be an even bigger difficulty for integration [4]. In this paper, the second, ore deanding, case will be chosen for deonstration of the ethod. In the ost deanding case of the Green function evaluation, and that is the case when all the three coordinates, z s, z f and ρ, approach zero, and, thus both Z and Ρ in () also approach zero, the F function is for z f ε, µ r, M - x sall arguents close to one and very slowly decaying. Thus, the integral, I = R ( α), () will be considered for deonstration of the proposed ethod. Due to α behavior of R ( α ) this integral is convergent. For solving the initial integral (), the ethod that will be presented can be cobined with techniques for efficient solving the integral if only function F is considered, leading to a hybrid and adaptive overall ethod, siilar to that reported in [], [4]. Function R ( α ) is given by the recurrence forula that is presented here according to [6] and after soe eleentary transforations, k z z+ + k th( uh) R=, z= k, k + z k + z th( u H ) () + =,, M, u z k, k, u, M= M = = α εr εr =,, M. Here th( ) is a hyperbolic tangent and H= βh. In the case that the botto layer is a PEC plane, is we considered in this work, k M=. In what follows we will consider also coplex values of α and will denote such a coplex variable as z. Fro the recurrence equations it can be seen that:. R is a function of (4) α, thus, any of its series expansions contains only even powers of α,. R ( z ) in the coplex plane has branch cut singularities at points ε, =,, M, where the function is r finite, but its derivative is infinite. It is also shown (e.g. in [7], [8]) that R( z ) has a nuber of poles in the second quadrant ( π/ < arg{ z} ). We will assue that locations of those poles, z, i=,, N, and their residuus can be efficiently nuerically deterined (e.g. as explained in [7]). To derive R ( α ) we need to obtain R i = li R ( α). For this reason we first obtain α liα th( uh) = liu th( uh) =. Next, for α, z= k for every and the result is k k εr εr R= liα =. (5) k + k ε + ε r r This is exactly the sae value as for the case when there is only one layer (the first) filling the whole lower halfspace. The value of R is the constant ter in the series expansion of R ( α ) around infinity (which is equivalent to expansion of R (/ x ) around x= ). As every expansion of R ( α ) has only even powers of α, the first ter in expansion of R ( α ) around infinity is the α ter, as already entioned.

3 Petrovic et al.: Singularity Extraction for Reflected Soerfeld Integrals over Multilayered Media 9 Fig.. Reduced reflection coefficient along the real axis. III. EXTRACTION OF POLES OF THE INTEGRAND As an exaple for deonstration of the ethod, in the rest of the paper, two-dielectric layered edia backed by a PEC plane ( M = ), with paraeters ε r= j (air), ε r= 9 j., ε r= 4 j., H =, H = will be considered. Graph of the reduced reflection coefficient R ( α ), for the chosen paraeters of the layered edia is shown in Fig., and in the coplex plane, denoted R ( z ), in Fig.. Large values of R ( α ) in Fig. are not infinite, because the three poles are soewhat displaced fro the real axis ( α -axis). However, direct nuerical integration of such a function, without the use of special ethods is highly inefficient. In the coplex plane, function R ( α ) has also branch cuts, one for each of the three functions u, equation (4), naely u,,. Each branch cut doubles the ultiplicity of the coplex functions (as the square root is a two-valued function). These branch cuts can be seen in Fig.4. The ulti-valued nature of R ( α ) is reoved by the condition π/ < arg{ z} + π/, but branch cut singularities (essential singularities) reain. following. In the coplex plane, function R ( z ) should be represented as R ( z) = R f ( z z ) + f ( z), (6) si i i= where z i are locations of poles, the pole singularities are copletely included in a properly chosen function f and R si are coefficients, directly connected to residuus of R ( z ). The reaining function f ( z ) is then considered for nuerical integration, possibly also after soe transforations. Function f ust provide that f ( z zi ) dz is convergent. The siplest choice for function f sees to be function i ( z z ). However, this does not lead to the convergence of the integral, as this function behaves in infinity as α. Instead, we considered the function e f ( x) =, (7) x whose priitive function is an exponential integral function, Ei( x ). Further properties of f ( x ) iportant x here are:. its residuu in the pole x = equals one,. its series expansion around infinity is identically equal to, and. for real arguents it is an odd function. Due to the first property, R si is the residuu of R ( z ). Due to the second property, series expansion of f ( z ) at infinity is the sae as of the function R ( z ). This is a very convenient property. Naely, the ajority of techniques for speeding up the integral convergence perfors the expansion of the integrand at infinity. By applying function (7), our ethod enables application of those techniques in an unchanged for. Fig.4. Branch cuts of the reduced reflection coefficient. Fig.. Absolute value of the reduced reflection coefficient in the coplex plane. The ethod of extracting poles fro the integrand is the Function f ( z ) has no poles and is depicted graphically, along the real, α-axis in Fig.5. Although with no poles it is obvious that it is not yet suited for nuerical integration due to its shape at α= (branch cut singularity).

4 4 Telfor Journal, Vol. 6, No., 4. IV. EXTRACTION OF BRANCH CUT SINGULARITIES OF THE INTEGRAND The reaining integrand, f ( z ) has, theoretically, branch cut singularities at points ε ri, i=,,, the first being at α = and the other two very close to α = and α =. Fro Fig.5 and also after the very detailed analysis around points α and α, we did not notice any significant singular behavior of the function. Eliination of those two branch cut singularities could be attributed to the fact that the already extracted poles were located very close to those branch cut singularities and that our extraction of those poles practically eliinated those singularities to. This property, which we observed also for other arrangeents and paraeters of the layered edia as in Fig., is not yet clearly reported or stated in the published work. This eans that only the essential singularity at α = needs to be extracted. This is efficiently and copletely done by the well-known change of variables [9], α= sinθ for < α< and α= ch( t) for α >. The transfored function f is depicted in Figs. 6 and 7. The size of the interval in which this change of variables is applied is not critical. We adopted, soewhat arbitrarily, α<. subintervals of length α=, by the use of different orders of the standard Gauss polynoial integration forula. Fig.8 shows the relative error of integration for:. direct integration of the reduced reflection coefficient, R ( α ),. direct integration after extracting the poles, i.e., direct integration of f ( α ) without the change of variables around α= and. as in point, only after a change of variables around α=. Convergence for the direct integration of R ( α ) is relatively slow. For the first accurate digit, a forula of the order about ust be applied. The fastest convergence, up to errors of about 5 has been achieved by the third ethod, resulting in 5 error of less than for the forula of the 5th order. The convergence of the second ethod is in between. Fig.7. Function f after the change of variables, for α<. Fig.5. Reflection coefficient after extraction of poles (function f ), along the real axis. Relative error.e+.e+.e-.e-.e-.e-4.e-5.e-6.e-7 Integration fro to 4 Integration forula order Fig.8. Relative error of integration. Fig.6. Function f after the change of variables, for α<. V. NUMERICAL RESULTS As a nuerical deonstration of the proposed ethod, the ethod is applied for integration in the interval (,4), in whose radius around a coplex origin all the initial singularities were located. Integration is perfored on VI. CONCLUSION The proposed ethod for extraction of singularities fro the reflection coefficient of the Soerfeld integrals for the ultilayered edia efficiently extracts the coplex poles of the reflection coefficient. As a side effect of the ethod, branch cut singularities are also practically reoved, except for the singularity at point (,) in the coplex plane, which is further reoved by a siple transforation. This enables efficient nuerical integration of the reaining integrand. The ethod does not change the behavior of the reflection coefficient function for large arguents, i.e. does not alter its series expansion around infinity, which could be very useful for further application of ethods for speeding up the convergence of the integral.

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