Windowed Green Function MoM for Second-Kind Surface Integral Equation Formulations of Layered Media Electromagnetic Scattering Problems

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Windowed Green Function MoM for Second-Kind

Surface Integral Equation Formulations of Layered

Media Electromagnetic Scattering Problems

Rodrigo Arrieta1and Carlos Pérez-Arancibia2

1Department of Electrical Engineering, PUC Chile (riarrieta@uc.cl)

2Department of Applied Mathematics, University of Twente (c.a.perezarancibia@utwente.nl)

Abstract—This paper presents a second-kind surface integral

equation method for the numerical solution of frequency-domain

electromagnetic scattering problems by locally perturbed layered

media in three spatial dimensions. Unlike standard approaches,

the proposed methodology does not involve the use of layer Green

functions. It instead leverages an indirect Müller formulation in

terms of free-space Green functions that entails integration over

the entire unbounded penetrable boundary. The integral equation

domain is effectively reduced to a small-area surface by means

of the windowed Green function method, which exhibits high-

order convergence as the size of the truncated surface increases.

The resulting (second-kind) windowed integral equation is then

numerically solved by means of the standard Galerkin method of

moments (MoM) using RWG basis functions. The methodology

is validated by comparison with Mie-series and Sommerfeld-

integral exact solutions as well as against a layer Green function-

based MoM. Challenging examples including realistic structures

relevant to the design of plasmonic solar cells and all-dielectric

metasurfaces, demonstrate the applicability, efﬁciency, and accu-

racy of the proposed methodology.

Index Terms—layered media, layer Green function, Sommer-

feld integrals, method of moments, dielectric cavities, solar cells,

metasurfaces

I. INTRODUCTION

Problems of electromagnetic (EM) scattering and radiation

in the presence of planar layered media have played an

important role in the development of electromagnetics since

the beginning of the 20th century, when the seminal works

of Zenneck and Sommerfeld on the propagation of radio

waves over the surface of the earth appeared [1]. Their

relevance lies in that in many application areas it is crucial

to determine the scattering from localized perturbations (e.g.,

surface roughness, small-size inclusions, meta-atoms) and/or

the ﬁeld produced by localized sources (e.g., antenna feeds)

embedded within physically large structures that, away from

a certain region of interest, can be effectively assumed as

planar and inﬁnite (e.g., the surface of the earth, silicon

substrates). This is often the case in numerous problems

in radio communications [2], remote subsurface sensing [3],

microwave circuits [4], [5], nano-optical metamaterials [6],

photonics [7], and plasmonics [8].

Popular numerical approaches to layered media scattering

include differential equation-based methods, such as the ﬁnite

difference [9] and ﬁnite element methods [10], and surface in-

Ω1, ǫ1, µ1

Ω2, ǫ2, µ2

(E1,H1)

(E2,H2)

(Esrc,Hsrc)

Fig. 1: Illustration of the scattering of a plane electromag-

netic wave by a locally perturbed penetrable half-space. The

boundary Γand the planar surface Σcoincide for (x, y, 0) far

enough from the bounded local perturbation.

tegral equation (SIE) methods, such as the method of moments

(MoM) [11] (also known as the boundary element method)

and Nyström methods [12]. Unlike differential equation-based

methods, SIE methods do not suffer from dispersion errors,

and they can easily handle unbounded domains and radiation

conditions at inﬁnity without recourse to perfectly matched

layers or approximate absorbing/transparent boundary condi-

tions for truncation of the computational domain. Additionally,

SIE methods rely on discretization of the relevant physical

boundaries, and they therefore give rise to linear systems

of reduced dimensionality which, although dense, can be

efﬁciently solved by means of iterative solvers in conjunction

with fast algorithms [13].

In classical layered media SIE formulations [14]–[19], how-

ever, all these attractive features come at the price of employ-

ing the dyadic Green function for layered media, also known

as the layer Green function (LGF), which naturally enforces

the exact transmission conditions at planar unbounded physical

boundaries. The use of the LGF poses difﬁculties in view of

the LGF evaluation cost (there is vast literature on this subject,

for which we refer the reader to the review articles [20]–[22]).

Moreover, problems involving localized perturbations (e.g.,

open cavities and bumps) give rise to additional challenges

to LGF-SIE formulations, as artiﬁcial/transparent interfaces

need to be introduced in order to properly represent the ﬁelds

surrounding the perturbations [23].

arXiv:2210.02565v1 [math.NA] 15 Aug 2022

This work presents a fully 3D EM layered-media windowed

Green function (WGF) method. The WGF method, which was

originally developed for scalar layered media problems [24]–

[26] and later extended to waveguides in the frequency and

time domains [27]–[30], completely bypasses the use of Som-

merfeld integrals or other problem-speciﬁc Green functions.

This is here achieved by ﬁrst deriving an indirect Müller

SIE [31], [32] given in terms of free-space Green functions and

featuring only weakly-singular kernels, which is posed on the

entire unbounded penetrable interface (see Secs. III and IV).

The unbounded SIE domain is then effectively truncated to

a bounded surface containing the localized perturbations by

introducing (in the surface integrals) a smooth windowing

function that effectively acts like a reﬂectionless absorber

for the surface currents leaving the windowed region (see

Sec. V). As in the case of the Helmholtz SIEs [26], the

ﬁeld errors introduced by the windowing approximation decay

faster than any negative power of the diameter of the truncated

region. A straightforward (Galerkin) MoM discretization using

Rao-Wilton-Glisson (RWG) functions is used to discretize

the resulting windowed SIE (see Sec. VI), although any

other Maxwell SIE method could be employed. A limitation

of the proposed approach is that transmission conditions at

unbounded penetrable interfaces need to be enforced via

second-kind SIEs such as Müller’s. First-kind SIEs, such as

the more popular Poggio–Miller–Chang–Harrington–Wu–Tsai

(PMCHWT) [33]–[35], could be considered provided they

are converted into equivalent second-kind SIEs by means of

Calderón preconditioners [36].

Compared to LGF-SIE formulations, the WGF formulation

involves additional unknown surface currents on planar por-

tions of the unbounded dielectric interfaces that eventually lead

to larger linear systems. In many cases this additional cost is

compensated by the fact that the associated matrix coefﬁcients

involve evaluations of the inexpensive free-space Green func-

tions and that the resulting linear system can be efﬁciently

solved iteratively by means of GMRES (see Sec. VII-C).

For problems involving multiple dielectric layers and/or small

size PEC inclusions, however, LGF formulations that leverage

the discrete complex images method (DCIM) [37]–[39] for

the evaluation of the LGF, may well outperform the WGF

methodology.

The proposed approach amounts to a ﬂexible and easy-

to-implement MoM for layered media EM problems, in the

sense that only minor modiﬁcations to existing electromag-

netic SIE solvers are needed to deliver the WGF capabilities.

The method is thoroughly validated (see Sec. VII) against

the exact Mie series scattering solution for a hemispherical

bump on a perfectly electrically conducting (PEC) half-space

(using a windowed MFIE formulation), and also against

the open-source LGF code [40]. A performance compari-

son against a LGF-MoM based on the state-of-the-art C++

library Strata [41] is presented in Sec. VII-C. Finally, the

proposed methodology is showcased by means of a variety of

challenging examples including EM scattering by a large open

cavity (Sec. VII-D), an all-dielectric metasurface (Sec. VII-E),

and a three-layer plasmonic solar cell (Sec. VII-F).

II. LAYERED MEDIA SCATTERING

We consider here the problem of time-harmonic electro-

magnetic scattering of an incident ﬁeld (Einc,Hinc)by a

penetrable locally perturbed half-space Ω2, with boundary

Γ = ∂Ω2, as depicted in Fig. 1. Letting Ω1=R3\Ω2, we

express the total electromagnetic ﬁeld as

(Etot,Htot)=(Esrc,Hsrc)+(Ej,Hj)in Ωj(1)

for j= 1,2.The known auxiliary source ﬁeld (Esrc,Hsrc)

which is given in terms of the incident ﬁeld (Einc,Hinc)under

consideration, is constructed so that the ﬁelds (Ej,Hj),j=

1,2,satisfy the homogeneous Maxwell equations

∇×Ej−iωµjHj=0and ∇×Hj+iωjEj=0in Ωj(2)

for j= 1,2, where ω > 0is the angular frequency, and j

and µjare respectively the permittivity and the permeability

within the subdomain Ωj. (We have assumed here that the

time dependence of the EM ﬁelds is given by e−iωt.) For

planewave incidences, for instance, (Esrc,Hsrc)is taken as

the exact total ﬁeld solution of the problem of scattering of the

planewave by the ﬂat lower half-space with planar boundary

Σ = {(x, y, 0) ∈R3}and constants 2and µ2(see Fig. 1).

The rationale for introducing (Esrc,Hsrc)lies in ensuring that

(Ej,Hj),j= 1,2,are outgoing waveﬁelds propagating away

from the localized perturbations or, more precisely, that they

satisfy the Silver-Müller radiation condition:

lim

|r|→∞ √µjHj×r− |r|√jEj=0in Ωj, j = 1,2,(3)

uniformly in all directions r/|r|. The explicit expressions of

the source ﬁelds utilized throughout the paper are provided in

Sec. III below.

The transmission conditions at the material interfaces,

meaning that the tangential components of (Etot,Htot)are

continuous across Γ, lead to the jump conditions

n× {E2|−−E1|+}=Msrc (4a)

n× {H2|−−H1|+}=Jsrc (4b)

on Γ, with

Msrc := ˆ

n× {Esrc|+−Esrc|−}(5a)

Jsrc := ˆ

n× {Hsrc|+−Hsrc|−}(5b)

where we have adopted the notation F(r)|±= limδ→0+ F(r±

δˆ

n(r)) for r∈Γ. As usual, the unit normal vector at r∈Γis

denoted as ˆ

n(r)and is assumed directed from Ω2to Ω1(see

Fig.1). Existence and uniqueness of solutions of the resulting

EM transmission problem are established in [42].

III. INCIDENT AND SOURCE FIELDS

Two types of incident ﬁelds (Einc,Hinc)and corresponding

auxiliary source ﬁelds (Esrc,Hsrc)are considered in this

paper, namely planewaves and electric dipoles.

Upon impinging on the planar surface Σat the interface

between the half spaces D1={z > 0}and D2={z < 0}

with wavenumbers k1and k2(kj=ω√µjj, for j= 1,2),

respectively, the incident planewave

Einc(r)=(p×k)eik·rand Hinc(r) = 1

ωµ1

k×Einc(r)(6)

with p= (px, py, pz)and k= (0, k1y,−k1z)where k1z≥0

and |k|=qk2

1y+k2

1z=k1, gives rise to a reﬂected

ﬁeld (Eref ,Href )in D1and a transmitted ﬁeld (Etrs,Htrs)

in D2. The resulting x-independent total ﬁeld, given by

(Einc +Eref ,Hinc +Href )in D1and (Etrs,Htrs)in D2,

is completely determined by the transverse component of the

ﬁelds [43]:

Einc

x(r)

Hinc

x(r)=E0

H0exp(ik1yy−ik1zz)

Eref

x(r)

Href

x(r)=E0RTE

H0RTMexp(ik1yy+ik1zz)

Etrs

x(r)

Htrs

x(r)=E0TTE

H0TTMexp(ik2yy−ik2zz)

depending on the reﬂection coefﬁcients:

RTE =µ2k1z−µ1k2z

µ2k1z+µ1k2z

, RTM =2k1z−1k2z

2k1z+1k2z

the transmission coefﬁcients:

TTE =2µ2k1z

µ2k1z+µ1k2z

, T TM =22k1z

2k1z+1k2z

the amplitudes:

E0=−pzk1y−pyk1z, H0=k2

ωµ1

and the propagation constants k2y=k1yand k2z=

qk2

2−k2

2ywith the complex square root deﬁned so that

Im k2z≥0. The EM ﬁeld can be retrieved from the transverse

components via

E=Exˆ

x−1

iω

∂Hx

∂z ˆ

y+1

iω

∂Hx

∂y ˆ

H=Hxˆ

x+1

iωµ

∂Ex

∂z ˆ

y−1

iωµ

∂Ex

∂y ˆ

With these expressions at hand we deﬁne the auxiliary

planewave source ﬁeld as

(Esrc,Hsrc) = ((Einc,Hinc)+(Eref ,Href )in Ω1

(Etrs,Htrs)in Ω2.(7)

Since by construction the source ﬁeld (7) satisﬁes the exact

transmission conditions at Σ, it holds that the current sources

Msrc and Jsrc deﬁned in (5) are supported on the (bounded)

local perturbation Γ\Σ.

In the special case when Ω2is occupied by a PEC, in

which the boundary condition ˆ

n×Etot =0holds on the

interface Γ, we have that (Esrc,Hsrc)takes the form (7) with

(Etrs,Htrs)=(0,0)and (Eref ,Href )given in terms of the

reﬂection coefﬁcients RTE =−RTM =−1.

Finally, we take

Hinc =1

iωµj∇×{Gj(·,r0)p},Einc =−1

iωj∇×Hinc (8)

with

Gj(r,r0) := eikj|r−r0|

4π|r−r0|(9)

being the (Helmholtz) free-space Green function, as the in-

cident ﬁeld produced by an electric dipole at r0∈Ωj. The

corresponding source ﬁeld is thus selected as

(Esrc,Hsrc) = ((Einc,Hinc)in Ωj

(0,0)otherwise (10)

for j= 1,2.

IV. SECOND-KIND INTEGRAL EQUATION FORMULATION

In order to approximate the unknown EM ﬁelds (Ej,Hj),

j= 1,2, we resort to a second-kind indirect Müller formula-

tion. We start by introducing the off-surface integral operators

(Sjϕ)(r) := ZΓ

Gj(r,r0)ϕ(r0) ds0+

j∇ZΓ

Gj(r,r0)∇0

s·ϕ(r0) ds0

(11)

(Djϕ)(r) := ∇ × ZΓ

Gj(r,r0)ϕ(r0) ds0(12)

for r∈R3\Γ, with ϕbeing a vector ﬁeld tangential to Γ. (In

what follows the surface integrals over Γmust be interpreted as

conditionally convergent.) The unknown EM ﬁelds (Ej,Hj)

are thus sought as

Ej(r) := k2

j(Sjv)(r) + iωµj(Dju)(r)(13a)

Hj(r) := k2

j(Sju)(r)−iωj(Djv)(r)(13b)

for r∈Ωj,j= 1,2, in terms of unknown surface currents u

and vthat are to be determined by means of a SIE posed on Γ.

Clearly, the ﬁeld deﬁned in (13) satisfy Maxwell equations (2),

in view of the fact that ∇×Sj=Djand ∇×Dj=k2

jSj.

In order to derive a SIE for the currents we make use of

the well-known jump relations:

n×(Sjϕ)|±=Tjϕand ˆ

n×(Djϕ)|±=Kjϕ±ϕ

2(14)

on r∈Γ, where

(Tjϕ)(r) := ˆ

n(r)×ZΓ

Gj(r,r0)ϕ(r0) ds0+

n(r)× ∇ZΓ

Gj(r,r0)∇0

s·ϕ(r0) ds0(15)

and

(Kjϕ)(r) := ˆ

n(r)×∇×ZΓ

Gj(r,r0)ϕ(r0) ds0.(16)

Evaluating the integral representation formulae (13) on Γ

and using (14) we obtain

n×Ej|±=k2

jTjv+iωµjn±u

2+Kjuo(17a)

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摘要：

WindowedGreenFunctionMoMforSecond-KindSurfaceIntegralEquationFormulationsofLayeredMediaElectromagneticScatteringProblemsRodrigoArrieta1andCarlosPérez-Arancibia21DepartmentofElectricalEngineering,PUCChile(riarrieta@uc.cl)2DepartmentofAppliedMathematics,UniversityofTwente(c.a.perezarancibia@utwente.nl...

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Windowed Green Function MoM for Second-Kind Surface Integral Equation Formulations of Layered Media Electromagnetic Scattering Problems

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