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 This paper presents a new approach to solve two-dimensional transverse magnetic (TM) scattering problems involving homogeneous penetrable scatterers. The scatterers can be replaced by equivalent electric surface currents provided a proper surface admittance operator is introduced relating those currents to the electric fields on the scatterer's surface. The surface operator for a scatterer with rectangular cross section can be determined analytically. The paper emphasizes that the proposed technique remains valid in a complex scattering environment and in the presence of both conducting and nonconducting scatterers. The new method allows for a considerable gain in CPU time and memory requirements and remains very accurate even in the presence of very good conductors and this from the DC regime to the high-frequency skin effect regime. Scattering of a plane wave by a periodic arrangement of rectangles or crosses embedded in a dielectric slab is used to illustrate the use of the surface admittance operator in conjunction with an integral equation method.
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 Scattering of plane waves by two-dimensional (2D), perfectly conducting or penetrable objects has been treated by many authors, both for TE and TM polarization and by various numerical techniques [Tsang, 2001]. The scatterers are either placed in homogeneous space or can be embedded in an unbounded planar stratified medium. Surface integral equation techniques are ideally suited for homogeneous scatterers, either single scatterers or periodic arrangements of scatterers. Such integral equation techniques are based on the knowledge of appropriate Green's functions. In the past, Rogier et al. , Rogier and De Zutter , Olyslager et al.  and Rogier et al.  have contributed to this field of research, both for scattering problems as well as for guided wave problems. Typically, the equivalence principle is used to introduce unknown equivalent electric and magnetic surface currents on the boundary or boundaries of the scatterer or scatterers. These equivalent currents are then found by solving integral equations that enforce the continuity of the total tangential electric and magnetic field at the scatterer's surface [Poggio and Miller, 1973; Charles et al., 1991; Donepudi et al., 2003; Ylä-Oijala and Taskinen, 2005].
 In this paper it will be shown that it suffices to introduce an equivalent electric surface current Js(r, ω) on the scatterer's surface, provided a suitable surface admittance operator is introduced relating this current at each point r of the scatterer's surface to the tangential electric fields Etan(r′, ω) at every other point on the surface. This surface admittance operator allows to replace the medium of the scatterer by the medium of the surrounding background medium the scatterer is embedded in. The remaining field problem can then be solved by solely considering the interactions between the equivalent electric surface currents and the incident field in the sole presence of the background medium the scatterers were originally embedded in. It is shown that the admittance operator yields a highly accurate description of the behavior of the scatterer for a wide range of electromagnetic material properties. This is in particular the case when the scatterer becomes highly conductive and for small skin depths. In this paper, we will restrict ourselves to the time-harmonic TM case.
 The proposed surface admittance approach was originally developed to study the skin effect for high clock rate signals propagating on RF boards, packages and chips [De Zutter and Knockaert, 2005]. More in particular, in work by De Zutter and Knockaert , attention was focused on the determination of the frequency-dependent resistance and inductance matrices per unit of length of a set of parallel signal lines. From a purely theoretical point of view, the technique presented by De Zutter and Knockaert  can be expected to be applicable to scattering problems as well. However, in work by De Zutter and Knockaert  the application of the theory is restricted to the low-frequency case (i.e., the dimensions of the considered multiconductor lines remain small with respect to the free space wavelength) and to a homogeneous background medium (homogeneous free space with or without a PEC ground plane). The main purpose of this contribution is to demonstrate the validity of the surface admittance approach for 2D TM-scattering problems. It is demonstrated that the surface admittance operator can be easily incorporated into an integral equation method, including the case of a layered background medium and/or for a periodic arrangement of the scatterers.
 In section 2, the theoretical background of the surface admittance operator is briefly recapitulated and a general expression in terms of the Dirichlet eigenfunctions of the scatterer's cross section is given. For the rectangular cross section, application of the method of moments (MoM) leads to a discretized form of the surface admittance operator: the surface admittance matrix. In section 3, this surface admittance matrix for the rectangle is briefly presented, but the reader is again referred to De Zutter and Knockaert  for more details.
 From many possible examples and in order to keep this paper concise, we have selected three numerical examples which are presented in section 5. First, the simple scattering configuration of a coated rectangular cylinder in free space is considered and the obtained results are compared with data published by Jin and Liepa . In the two other examples the background medium consists of a dielectric slab in free space and the scatterers form a periodic grid. First (lossy) dielectric scatterers with rectangular cross section are considered. Next, the rectangles are replaced by nonlossy dielectric crosses. The incident wave is a plane TM wave and both reflection and transmission coefficients are calculated, as well as the total dissipated power for the case of lossy scatterers. Numerical results are compared with those obtained with a previously published technique based on a boundary integral equation approach using equivalent electric and magnetic currents [Poggio and Miller, 1973; Rogier et al., 2000]. Before presenting the examples, section 4 starts with a brief explanation on the solution of the overall scattering problem when using a surface admittance operator. Finally, section 6 gives a brief conclusion and some prospects for further research.
2. Surface Admittance Operator
 As specified before, time-harmonic (ejωt dependence) TM polarization is considered, with the z axis in the longitudinal direction. Figure 1a shows the cross-section S of an arbitrary scatterer embedded in a piecewise homogeneous background medium. The homogeneous material of the layer the scatterer is embedded in, is characterized by the constitutive parameters , μ and σ. The scatterer itself is characterized by the constitutive parameters εsc, μsc and σsc. From Maxwell's equations we readily derive that on the boundary c of S,
with the index t referring to the tangential component of the magnetic field and where stands for the limit of the normal derivative of the electric field tending from the inside of the scatterer to c. Now suppose that the constitutive parameters of the scatterer are replaced by those of the surrounding material. On the boundary c of S we now have that
where we have introduced the subscript “0” to distinguish between the two situations. If, in the configuration of Figure 1a, we now want to replace the material of the scatterer by the material of its surrounding layer, in this way undoing the discontinuity in permittivity, conductivity and permeability due to the scatterer's presence, it suffices to introduce an equivalent surface current density Jsz, related to the value of the electric field on the boundary, by means of the differential surface admittance operator �� given by
 This is depicted in Figure 1b. Note that on the boundary c and only on c, Ez0 = Ez, as the introduced surface current does not give rise to a jump in the tangential electric field. When solving the field problem of Figure 1b, the obtained result is only identical to the one for the original configuration of Figure 1a, taken outside the scatterer. Inside the scatterer a fictitious field is obtained. To obtain data such as the total Joule losses or the scattered fields, the sole knowledge of the surface current density Jsz suffices. In the sequel we will restrict ourselves to nonmagnetic materials (μsc = μ = μ0). It has been shown by De Zutter and Knockaert  that in this case a possible way to obtain the operator is to use the Dirichlet eigenfunctions of the cross-section S. The final result is
with τ = [σ − σsc + jω( − εsc)]. The symbol k represents the wave number of the material of the scatterer, i.e., k = and k0 = is the wave number of the material the scatterer is embedded in. The ξm are the Dirichlet eigenfunctions of the cross-section S with corresponding eigenvalues λm2.
 At this point, two remarks should be made. First, it is clear that the general reasoning leading to the introduction of the surface admittance operator can easily be extended to the 2D TE case and indeed to the 3D case. For the TE case the calculations leading to (4) can be repeated but one can expect that the Neumann eigenfunctions will play a role in this case. For the 3D case, simple Dirichlet or Neumann eigenfunctions of the scatterer's volume will certainly not suffice.
 As pointed out by the reviewers, the uniqueness of the admittance operator �� poses a problem when the scatterer supports an internal resonance. This can already be anticipated when writing down (1) and (2). Indeed, consider solutions to ∇2Ez + k2Ez = 0 or to ∇2Ez0 + k02Ez0 = 0 in S which also satisfy Ez = 0 or Ez0 = 0 on the boundary c. Such solutions leave the total tangential electric field on c unchanged. Hence the unique relationship between the total tangential electric and total tangential magnetic field, on which our reasoning is based, is lost. These solutions are the internal resonances of S, with metallized boundary c, i.e., the Dirichlet eigenmodes of S. This nonuniqueness also exists for the Electric Field Integral Equation (EFIE) for a perfect conductor. Consequently, it is not surprising that in (4) the surface admittance operator becomes infinite for wave numbers corresponding to one of the λm values. In the numerical examples that follow, the lossy dielectrics remain small enough such that, for the considered frequency ranges, k0 never corresponds to a resonant wave number. Furthermore, when considering lossy dielectrics, k is complex and hence, for real frequencies, k will never hit one of the real-valued resonant wave numbers. A way to circumvent an internal resonance at a particular frequency, is to subdivide the domain S into two or more subdomains, to determine the admittance operator for each subdomain and to “glue” these domains together afterward. This gluing technique is illustrated in the example of Figure 9 in section 5 to model cross-shaped scatterers.
3. Surface Admittance Matrix for a Rectangle
 As announced in section 1, we now focus attention to a scatterer with rectangular cross section. As will become clear from the dielectric cross example in section 5, this does not automatically mean that all scatterers have to be restricted to rectangles. The method can also be used to treat scatterers that can be constructed from rectangles.
 For a rectangular scatterer (0 ≤ x ≤ a and 0 ≤ y ≤ b), the Dirichlet eigenfunctions and eigenvalues are
with λmn2 = ((mπ)/a)2+ ((nπ)/b)2. In work by De Zutter and Knockaert  it is shown that an analytical expression for Jsz can be obtained by expanding Ez on each side of the rectangle in an appropriate Fourier sine series. For example, for y = 0 and 0 ≤ x ≤ a this series is
 However, when solving the overall scattering problem, Ez and Jsz will be used in a Galerkin MoM. Hence we search for a discretized form of (4) obtained by expanding Ez in a set of pulse basis functions and by projecting Jsz on the same set of pulse functions. On the side y = 0, 0 ≤ x ≤ a, for example, Ez is written as
where tj = 1 for xj−1 < x ≤ xj and zero elsewhere, with x0 = 0 and xM = a and with M the number of pulse basis functions along the considered side. A similar expression can be put forward for Jsz, replacing the coefficients Ej by Jj/(xj − xj−1) (the factor (xj − xj−1) is necessary in view of the Galerkin testing). We can now collect all the pulse basis amplitudes Ej, on all of the four sides, into a vector E and similarly all Jj's into a vector J. Tedious, but completely analytical calculations [De Zutter and Knockaert, 2005] allow us to obtain the discretized form of �� as
Ys is the M × M differential surface admittance matrix (all entries of Ys have dimension Ω−1). We again refer the reader to De Zutter and Knockaert  for the detailed analytical expressions of the elements of Ys.
4. Overall Scattering Problem
 Before turning to the examples, let us briefly summarize how the surface admittance operator (3) and more in particular its discretized counterpart (8) for the rectangle, can be used to solve the overall scattering problem. By introducing the equivalent but as yet unknown surface current Jsz on the scatterer's boundary c, the scatterer can be removed; that is, the material of the scatterer can be replaced by that of the surrounding layer. As a consequence, the remaining problem is that of a background medium, no longer disturbed by the presence of the scatterer, in which the current Jsz is radiating. As explained in section 1, when using integral equation techniques, the knowledge of the Green's function G(r, r′) (with r = (x, y) and r′ = (x′, y′)) of the background medium is essential. It is in particular this Green's function that allows us to determine the electric field (Ez)scat everywhere in the background medium due to the radiating current Jsz. We have that
 If we now consider (9) for r on the boundary c, appropriate MoM discretization of (9), again using pulse functions, leads to
where we have used the same set of M pulse functions as in (8). The symbol G stands for the M × M MoM matrix representation of the Green's function G in (9). The unknown currents J, i.e., the amplitudes of the pulses representing the originally continuous current Jsz, can now simply be determined by enforcing the following boundary condition on c:
and solving for J. The symbol Einc stands for the pulse weighted field on c due to the incident field. For an incident plane wave, for example, and a background medium consisting of a dielectric slab in free space, the scattering, penetration and transmission of this plane wave has to be taken into account to determine the correct value of Einc inside the slab.
5. Numerical Examples
 Let us now turn to the first numerical example. Consider the coated PEC cylinder the cross section of which is shown in Figure 2, together with the incident plane TM wave. All dimensions are in units of centimeter. Calculations were performed for a free space wavelength of λ = 1 m. The PEC core is covered with a nonmagnetic coating with εr = 4 − jα. This example is also treated by Jin and Liepa  with a combined finite element-boundary element method. Figure 3 shows the backscattering cross-section σback/λ for the angle of incidence ϕ varying between 0 and 90 degrees. Results are displayed for three values of α: 4, 40 and 400. This corresponds with a conductivity σ = 1/15, 10/15 and 100/15 S/m respectively and with corresponding skin depths δ = of 11.25, 3.56 and 1.125 cm. The result for a complete PEC cylinder (α = ∞) is also displayed. The result for α = 4 corresponds very well with the results displayed by Jin and Liepa . We have added extra results for α = 40 and α = 400 to show how the backscattering cross section evolves as a function of conductivity and skin depth.
 As a second numerical example, consider a periodic grid of dielectric bars (Figure 4), buried in a nonmagnetic and lossless dielectric slab with thickness t = 18 mm and a relative permittivity of εr = 3.0. The dielectric slab is placed in free space. The bars (also nonmagnetic and lossless) of size 6 mm × 3 mm have a relative permittivity εr = 6.0 and are located 5 mm under the top surface of the dielectric slab. The center-to-center spacing between the bars is chosen to be 10 mm. The structure is excited by a TM incident plane wave Ei = Eiuz, at a free-space wavelength λ = 2 cm (i.e., about 15 GHz). As explained above, we need the Green's function of the background medium. In order to account for the periodic nature of the problem, such that in the final set of equations (11) only the discretized current on a single scatterer in the unit cell of the periodic configuration is required, we need the periodic Green's function of the problem. To determine this periodic Green's function, several approaches have been put forward in literature. Here we use an efficient and recently published technique based on the use of Perfectly Matched Layers (PML) [Rogier and De Zutter, 2004]. The crux of this method is to place PMLs, backed by perfectly conducting (PEC) plates above and below the air-slab-air configuration and to use the propagation characteristics of the resulting parallel-plate waveguide to derive the Green's function. Figure 5 shows the parallel-plate waveguide corresponding with the configuration of Figure 4. The PMLs are placed at a distance dair = 5 mm from the slab and their characteristics are chosen to be dPML = 3.5 mm, κ0 = 15, = 10.
 For further verification, the results obtained with this fast periodic Green's function calculation in conjunction with (11) are compared to the results obtained with a boundary integral equation approach based on the simultaneous use of equivalent electric and magnetic surface currents [Poggio and Miller, 1973]. We will further refer to this approach as the PMCHW approach [Umashankar et al., 1986; Jung et al., 2002]. In order to restrict the analysis to a single cell of the periodic background medium, also in this case a suitable periodic Green's function is needed. Here we use the approach presented by Rogier et al.  where the free-space Green's function in combination with the Floquet-Bloch condition is used. Hence there are two major differences between the approach used in this paper and the PMCHW with periodic Green's function: (1) in PMCHW both equivalent electric and magnetic currents on the scatterer's surface are needed as compared with the single electric current introduced here and (2) for PMCHW, also the discretization of the fields at the slab-air interface, over a complete unit cell is needed and this is avoided here altogether. In Figures 6 and 7, the power reflection coefficient R and the power transmission coefficient T are shown as a function of the angle of incidence θ. The periodicity of the scatterers is such that only a plane wave in the specular direction is reflected (accompanied by its transmitted counterpart) and that higher-order propagating modes remain below cut-off. The solid line gives the result obtained with the admittance matrix approach for 6 pulse basis functions to model Jsz along the horizontal sides and 3 for the vertical sides, i.e., M = 18 in (8). In order to check the convergence of the solution, the result obtained with 6 × 3 pulse functions is compared with the solution (the crosses in Figures 6 and 7) with the double number of pulses (12 × 6). One notices that a stable solution is found. Moreover, very good agreement is seen with the results (indicated by the plus signs in Figures 6 and 7) obtained through the PMCHW formalism. Over almost the complete range of angles, the results with the plus signs coincide with the results for half the number of unknowns and the PMCHW results. This makes the plus signs and the crosses almost indistinguishable within Figures 6 and 7. Some difference can be observed in the first peak of the curve, i.e., around 10°. For a single angle of incidence, the admittance matrix implementation takes 6.6 s of CPU time on a 1.8 G Hz 64 bit AMD Opteron processor (18.3 s for the double number of unknowns), whereas the PMCHW implementation requires 1 min 52 s of CPU time. Hence a considerable gain in speed is observed.
 Next, we investigate the effect of losses by replacing the lossless dielectric scatterers with r = 3 in Figure 4 by lossy scatterers with complex permittivity c = ε0εr + σ/(jω). Figure 8 shows the power reflection coefficient R (solid line) and the total power loss L (dotted line) as a function of skin depth δ = and again for a free-space wavelength of λ = 2 cm. The total power loss L is the total power dissipated in the scatterers. We have chosen to display the results as a function of skin depth, as this leads to the best physical insight. As in the lossless case, the obtained results are confirmed by calculations based on the PMCHW formalism, the results of which are displayed in Figure 8 by stars for R and by crosses for L. Roughly speaking, three regions of interest can be distinguished: very small skin depths as compared to the cross-sectional dimensions of the scatterer, skin depths comparable to the cross-sectional dimensions and very large skin depths. Although we only work at one particular frequency, one could also talk about the high-frequency region, the mid-frequency region and the low-frequency region resp. Remark that for very small skin depths (σ > 0.5 × 106) the results based on the PMCHW approach are no longer accurate. For a conductivity of 0.5 × 106 at 15 GHz, the skin depth is 5.81 μm, i.e., already very small with respect to the dimensions of the scatterer. The electric field is confined within a few skin depths and the calculation of the MoM interaction integrals in the PMCHW approach are no longer accurate enough to capture this extreme current crowding. However, the surface admittance operator approach presented in this paper is still capable to correctly model the skin effect including the limit of very small skin depths. In that case, i.e., in the limit for σ → ∞, it has been shown by De Zutter and Knockaert  that the surface admittance matrix Ys(8) reduces to a diagonal matrix with the diagonal elements corresponding to the scalar Leontovich local surface admittance Jsz(r) = Ys(r)Ez(r) = Ez(r) for r on c. To confirm the limiting case for σ → ∞, a separate calculation was performed in which the Leontovich local boundary condition was used for all conductivity values. These results are displayed in Figure 8 by crosses for the total loss L and by squares for the power reflection R. Remark that the Leontovich results and the admittance matrix results completely coincide for very small skin depths and that the total power loss decreases linearly with decreasing skin depth, as expected. When the skin depth increases, the Leontovich results start deviating from the exact results. For skin depths of the order of the dimensions of the scatterer (a few millimeters and more), the current induced in the scatterer will be almost homogeneously distributed over the cross section, implying that the loss L becomes proportional to σ. Hence, for high values of δ, i.e., the low-frequency limit, the loss L will decrease proportional to 1/δ2, as can be clearly observed from the numerical results. Finally, the Leontovich results for the power reflection coefficient R, are also shown on Figure 8 using the squares.
 As the third example, consider the periodic grid of lossless dielectric crosses shown in Figure 9. In order to apply the surface admittance formalism, we have subdivided one cross into either three subregions, i.e., one rectangle of dimensions 6 mm × 2 mm and two rectangles of dimensions 2 mm × 2 mm (Figure 10a), or five rectangles, each of dimensions 2 mm × 2 mm (Figure 10b). In these cases unknown surface currents have to be introduced on the circumferences of all rectangles. From a theoretical point of view one could imagine these rectangles to be separated (in theory) by very small gaps. This approach was already proven to be viable to model a rectangular coaxial shield by De Zutter and Knockaert . When using three or five rectangles, the Green's function matrix G in (11) contains the interactions between all the pulse basis functions, including the interactions between the closely spaced sides (in theory separated by a very small gap) of some of the rectangles. As the singular behavior of the Green's function is logarithmic, self-patch couplings and near neighbor couplings can be numerically calculated in a very robust way, such that the small gap of the theory can simply be ignored without compromising the numerical accuracy. The surface admittance matrix Ys now collects all the data of the different rectangles. It is a block diagonal matrix with all nondiagonal blocks equal to zero and with the diagonal blocks equal to the Ys matrices of the different rectangles. In Figures 11 and 12, the power reflection coefficient R and the power transmission coefficient T are shown as a function of the angle of incidence θ. The results obtained with 3 rectangles are given by the solid lines; those with 5 rectangles by the plus symbols. These results were obtained for pulse basis functions with pulses extending over 1 mm or equivalently 32 unknowns for the 3 rectangle case and 36 unknowns for the 5 rectangle case. Finally, as for the previous examples, comparison is again made with the results obtained with the PMCHW approach. Those results are displayed by the crosses. To prove numerical convergence, results for 0.5 mm pulses (i.e., a doubling of the number of unknowns) were also calculated. They are not displayed in the figures as, on the scale of the figures, they almost completely coincide with the already displayed results. By way of example, Table 1 shows the variation on R and T for the case of 3 rectangles when doubling the number of unknowns. In the PMCHW approach the dielectric crosses are not subdivided into rectangles and the circumference of the crosses is discretized as such. However, as explained above, additional unit cell unknowns are needed on the top and bottom surface of the dielectric slab. Again, the results obtained with the surface admittance approach are clearly in good correspondence with simulation data obtained with the PMCHW formalism. However, the formalism based on the admittance operator clearly leads to an important reduction in CPU time and memory requirement, as shown in Table 2.
Table 1. Power Reflection and Transmission Coefficient When Applying the Admittance Matrix Formalism for the Periodic Grid of Dielectric Crosses Using Three Rectangles for Two Different Discretizations
Table 2. CPU Times and Memory Requirements for the Periodic Grid of Dielectric Crosses
Admittance matrix, 3 rect. - 32 unknowns
Admittance matrix, 5 rect. - 36 unknowns
Admittance matrix, 3 rect. - 64 unknowns
Admittance matrix, 5 rect. - 72 unknowns
1 min 14 s
PMCHW approach, 216 unknowns
1 min 59 s
PMCHW approach, 312 unknowns
2 min 52 s
 In this paper we have shown that a surface admittance operator can be advantageously used to solve TM-scattering problems involving homogeneous penetrable scatterers. These scatterers can be embedded in a layered background medium and can be arranged periodically, as illustrated by the numerical examples. In the context of the solution of the overall scattering problem with integral equation techniques in conjunction with the Method-of-Moments, particular attention was devoted to the surface admittance matrix description for a scatterer of rectangular cross section. The numerical examples illustrate that the new approach leads to a considerable gain in CPU time as the introduction of both an equivalent electric and an equivalent magnetic current is avoided. Moreover, very good accuracy for highly conducting scatterers in the skin effect regime is observed. It is much more difficult to achieve such accuracy with the a set of coupled integral equations as in the PMCHW approach, owing to the fact that for very large conductivities it becomes cumbersome to accurately determine the interaction integrals. Future research involves extension of the method to scatterers of arbitrary cross section, to TE problems and to problems involving magnetic contrast.
 This work was supported by Agilent Technologies. H. Rogier is a Postdoctoral Researcher of the FWO-V. His research was supported by a grant of the DWTC/SSTC, MOTION project.