A fast technique based on perfectly matched layers to model electromagnetic scattering from wires embedded in substrates

Authors


Abstract

[1] A two-dimensional integral equation formulation is used to model electromagnetic scattering from metallic structures embedded in substrates. The new method involves a fast evaluation scheme for the Green's function of the substrate by using perfectly matched layers to obtain closed-form expressions as a series of leaky and Berenger modes. Efficient summation of these series is performed by means of the Shanks transform. The modal series can be applied to calculate all interactions, except for the self-patch contribution. The latter can be approximated with good accuracy by using the closed-form quasi-static part of the Green's function. The new technique is illustrated by studying scattering from wires embedded in a microstrip substrate.

1. Introduction

[2] Method of moments (MOM) techniques relying on integral equation formulations are very popular for full-wave field analysis in CAD tools. These formalisms only require the discretization of the fields along the boundaries of the metallic structures present in the configuration, provided the characteristics of the dielectric background medium are incorporated in the kernel functions of the integrals. Although the latter problem has been studied for quite a number of years, the calculation of the Green's function of the substrate still remains an important issue. Indeed, it is well known [Michalski and Mosig, 1997; Faché et al., 1992] that the two-dimensional (2-D) Green's functions of multilayered media can be easily computed in the spectral domain; however, the inverse Fourier transform leads to Sommerfeld-type integrals, requiring complex and time-consuming integration schemes. Therefore a number of faster alternative approaches have been proposed, such as, for example, the fast Hankel transform [Hsieh and Kuo, 1998] and the complex image technique [Chow et al., 1991; Van Hese et al., 1994]. A new technique is proposed by Derudder et al. [1999, 2001] and Bienstman et al. [2001] to calculate the Green's function by using perfectly matched layers (PML) [Bérenger, 1996; Gedney, 1996; Knockaert and De Zutter, 2000] to obtain a closed waveguide configuration. The PML then mimics an open structure, while an efficient problem description in terms of a set of discrete modes of the closed waveguide containing the PML is possible.

[3] This approach is used by Derudder et al. [1999, 2001] and Bienstman et al. [2001] to construct the Green's function Gxx for a two-dimensional configuration, for the transverse electric (TE) polarization. The problem we are studying here consists of a 2-D configuration of metallic objects buried in a substrate, as defined in section 2. The main contribution of this paper then consists in relying on results obtained by Rogier and De Zutter [2002] to apply the new PML formalism within a 2-D Galerkin MOM approach with subdomain pulse basis functions in order to expand the currents on the boundaries of the objects buried in the substrate, as discussed in section 3. More specifically, we will show that the Green's function written down in section 4.1 as an expansion in leaky and Berenger modes [Rogier and De Zutter, 2001] and accelerated by the Shanks transform, discussed in section 4.2, can be used to calculate the interaction integrals between arbitrarily oriented segments placed anywhere in the substrate. This latter statement does not hold for the self-patch contribution, but there the quasi-static part suffices to calculate this interaction with sufficient accuracy. Since for all interactions, the Green's function is available under closed form, the technique allows for a drastic improvement in CPU time, as shown by the examples in section 5. Furthermore, it is shown that the PML formalism is also useful in the calculation of the fields anywhere in the substrate, once the unknown currents are solved for. Again, a different treatment is necessary for the self-patch contribution as well as for calculating the fields in the air region above the substrate.

2. Geometry of the Problem

[4] We consider a two-dimensional configuration consisting of a substrate with thickness d and permittivity ϵr and backed by a perfectly conducting ground plane, in which perfectly conducting metallic objects can be buried, as shown in Figure 1. The substrate is assumed to be open, and the structure is illuminated by an incoming TE-polarized (E field directed along the x axis and parallel to the substrate) plane wave of the form

equation image

This incoming wave is scattered by the presence of the substrate and the ground plane, and inside the substrate the incoming field is then given by

equation image

with T(ky) the transmission coefficient at the substrate-air interface and with kr,z2 = k02 ϵrky2. We will investigate the scattering characteristics of the objects in the substrate at a certain frequency f, so a ejω t (ω = 2 πf) dependence of the fields is assumed. We are interested in the currents on the boundaries of the buried objects and in the fields in the neighborhood of the objects.

Figure 1.

Metallic objects buried in a substrate.

3. Integral Equation Description

[5] The scattering problem defined in section 2 can be easily described by the following boundary integral equation:

equation image

where Gxx(y, z; y′, z′) represents the Green's function of the substrate and the incoming field Exin, sub (y, z) is given by equation (2). The integration extends over the boundaries Ci of all metallic objects present in the substrate. Let us postpone the calculation of Gxx(y, z; y′, z′) until section 4 and sketch the solution process for obtaining the currents on the objects. We choose a set of basis functions wi to expand Jx(y′, z′) = ∑i = 1NAiwi, and we apply Galerkin weighting, yielding

equation image

where the integration extends over the supports l and l′ of the respective basis functions wi(y, z) and wj(y′, z′). A good choice for wi is to use pulse basis functions. The resulting matrix system can easily be solved, for example, by means of LU decomposition. Once the currents Jx(y′,z′) on the boundaries are known, the right-hand side of equation (3) can be used to calculate the scattered field.

4. Calculation of the Green's Function

[6] The classical approach for calculating the Green's function of the substrate shown in Figure 1 involves transforming the field problem first to the spectral domain, where the problem is reduced to solving a transmission line cascade. The spatial Green's function is then obtained by inverse Fourier transformation. Since the integrand tends to be highly oscillating and exhibits a slow decay approaching infinity, this technique is quite time-consuming. An accurate implementation requires the extraction of the surface waves in the substrate

equation image

with βguided the propagation constants of the guided modes (Im (βguided) = 0), and of the quasi-static contributions:

equation image

In this paper, however, we apply a different method, making use of a PML formalism.

4.1. PML Formalism

[7] In order to speed up the calculation of the substrate's Green's function, the air region is terminated by a perfectly matched layer (PML) with thickness dPML and with material parameters κ0 and σ0 [Derudder et al., 1999], as shown in Figure 2. Derudder et al. [2001] show that by stretching the coordinates the air region can then be combined with the PML to form a single air layer with complex thickness equation image = dair + dPML0j0/(ωϵ0)]}. In this way, the Green's function of the substrate is found by a relatively simple modal analysis of the waveguide under consideration.

Figure 2.

Pertinent to the calculation of the Green's function.

[8] The Green's function Gxx(y, z; y′, z′) for a line source at (y′, z′) carrying an x-oriented current can now be expressed as a series of TE modes. For a source lying in or on top of the substrate (0 < z′ ≤ d), the Green's function in or on top of the substrate (0 < zd) can be written as [Rogier and De Zutter, 2002]

equation image

where β represents the propagation constant of either a guided mode, a leaky mode, or a Berenger mode contribution.

4.2. Series Acceleration by Shanks Transformation

[9] Rogier and De Zutter [2002] show that the convergence speed of equation (7) can be drastically increased by applying the Shanks transform [Singh and Singh, 1990]. Furthermore, Rogier and De Zutter [2002] demonstrated that each part that composes the Green's function has a distinct convergence behavior and that the acceleration technique has to be applied to each separate series of Berenger or leaky modes. This allows one to obtain accurate results for distances |yy′| up to λ/100. Whenever the |zz′| separation is zero, the classical evaluation technique has to be applied. Indeed, when the Green's function is evaluated at the substrate-air interface, both series of Berenger and leaky modes diverge, whereas only the leaky mode series diverges when the Green's function is evaluated inside the substrate. For the self-patch calculation, however, if suffices to only take the quasi-static contribution (6) into account. For nonzero |zz′| distances both series stay accurate for even smaller distances, and the PML series expansion remains valid.

5. Examples

[10] Consider a substrate with thickness d = 9 mm and ϵr = 3, in which a metallic circular wire with radius 2 mm is buried, as shown in Figure 3. The wire is illuminated by a TE-polarized plane wave under perpendicular incidence. The configuration is studied at a frequency f = 12 GHz. We will compare between results obtained by a robust implementation of the classical spectral domain approach and by the new fast PML formalism. For the latter, the substrate is covered by a PML with thickness dPML = 3.5 mm at dair = 5 mm from the substrate-air interface. A sufficiently absorbing PML is obtained for κ0 = 10 and σ0/(ωϵ0) = 8. Let us first investigate the convergence of the solution for an increasingly finer discretization of the wire. The wire is modeled as a polygon with n sides, n = 6, 12, 24, and 48, and on each side, two segments are chosen with a representation for the current by means of pulse basis functions.

Figure 3.

A metallic wire buried in a substrate.

[11] In Figures 4 and 5 the current along the wire is represented as a function of the normalized arc length k0s along the wire. The starting point for the arc length is depicted in Figure 3. Both the new method based on the Green's function obtained by the PML formalism and the classical approach based on the spectral domain technique for the Green's function are considered. In the former, the self-patch contribution is approximated by the quasi-static contribution (6). A clear convergence for the solution is seen, and a good comparison is obtained between the two techniques. In Table 1 the CPU time needed when modeling the wire as a polygon with n sides is compared for both methods. It is seen that the use of the PML formalism reduces CPU time to only a fraction of that when the classical approach is used. It should, however, be emphasized that the implementation used for the classical technique focused on accuracy and that no efforts were made to optimize this approach. Still, important savings can be expected.

Figure 4.

Real part of the current distribution on the wire.

Figure 5.

Imaginary part of the current distribution on the wire.

Table 1. CPU Time Needed When Modeling the Wire as a Polygon With n Sides
nPML FormalismClassical Technique
610 s3 min 54 s
1242 s15 min 39 s
242 min 58 s1 h 2 min 26 s
4812 min 10 s 

[12] In Figure 6 a contour plot of the total field is shown in the neighborhood of the wire, modeled as a polygon with 24 sides. Inside the substrate the PML formalism is used, except for the self-patch, where the quasi-static contribution can be applied. In the air region above the substrate the classical approach is used for small y separations |yy′|. This is because the PML series in the air region tends to diverge for |yy′| → 0. However, for |yy′| > 2l, with l the typical length of a segment of the wire, the results of the PML formalism still remain accurate, provided the fields are calculated sufficiently far away from the PML termination.

Figure 6.

Total field around the wire.

[13] Let us now consider the same substrate with two wires buried in it, as shown in Figure 7. Again, a plane wave illuminates the configuration at 12 GHz under perpendicular incidence. The wires are modeled as polygons with n sides, n = 6, 12, 24, and 48, and on each side two segments are chosen. In Figures 8 and 9 the current distribution along the first wire and in Figures 10 and 11 the current distribution along the second wire are represented as a function of the normalized arc length k0s along the wire. Convergence is clearly observed, as well as the symmetry of the current distribution along the two wires. Note the small discrepancy that exists between the results obtained with the PML formalism and with the classical approach for n = 6. This is mainly due to the use of the quasi-static part of the Green's function to approximate the self-patch contribution. This discrepancy disappears when a finer discretization is chosen.

Figure 7.

Two wires buried in a substrate.

Figure 8.

Real part of the current distribution on the first wire.

Figure 9.

Imaginary part of the current distribution on the first wire.

Figure 10.

Real part of the current distribution on the second wire.

Figure 11.

Imaginary part of the current distribution on the second wire.

[14] In Table 2 the CPU time needed when modeling the wires as polygons with n sides is compared for both methods. The decrease in CPU time is now even more important than in the one-wire case, because the PML formalism requires a lower number of modes to represent the Green's function accurately when the |yy′| distance increases. This is typically the case for the interactions between segments on wire 1 and wire 2.

Table 2. CPU Time Needed When Modeling the Two Wires as a Polygon With n Sides
nPML FormalismClassical Technique
631 s15 min 45 s
122 min 8 s1 h 3 min 23 s
248 min 39 s 

6. Conclusions

[15] A new two-dimensional integral equation was proposed involving a fast evaluation scheme for the Green's function of the background medium by using perfectly matched layers to obtain closed-form expressions as a series of leaky and Berenger modes. The technique resulted in closed-form expressions for the Green's function used to calculate all interactions: The modal series was applied for all interactions, except for the self-patch contribution, where the quasi static-part of the Green's function was used. The new technique is illustrated by studying scattering from wires embedded in a microstrip substrate. An important decrease of the CPU is observed for the same accuracy in the field solutions.

Acknowledgments

[16] The work of H. Rogier is supported by a postdoctoral grant of the Flemish Institute for the Promotion of Scientific and Technological Research in Industry (IWT).

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