This document discusses a numerical method for modeling electromagnetic scattering from objects with thin coatings. It presents a method that properly weights the integration of near-singular behaviors introduced by thin coatings. The weighting is applied to the Fourier coefficients of Dirac delta functions. This allows the modeling of how varying a thin coating's properties alters an object's far-field scattering pattern.

1. Electromagnetic Scattering from Objects with Thin Coatings
Luke M Underwood
Department of Mathematics and Statistics, University of New Hampshire, 33 Academic Way, Durham, New Hampshire, 03861
Abstract
Numerically solving for the electromagnetic field on the surface of a scattering object requires integration
with Green’s functions that contain singularities. Adding a thin coating to the object introduces near-
singularities that must also be treated. A method is presented that achieves a proper weighting for integration
involving this near-singular behavior. The weighting is applied to the Fourier coefficients of Dirac delta
functions. This method is then implemented to illustrate how varying the properties of the thin coating
alters the far-field pattern.
1. Background
Electromagnetic scattering is a complex topic that has important applications, whether the electromag-
netic field is analyzed in regions near or far from a scattering object. For example, understanding the
behavior of scattered signals as seen from a distance can help improve the interpretation of reflected radar
signals. This data can also be used to find ways for preventing these reflected signals. Whatever the case,
examining models using numerical data is an effective research tool.
One particular model used to analyze scattered signals is the far-field radiation pattern. The term far-
field refers to the region of an electromagnetic field that is far away from the radiating object. A model of
the far-field pattern is a display of the relative amplitude of that field as seen from different angles around the
object. This data can thus be represented well in two dimensions using a polar coordinate system. Sectors
in which the far-field has a greater relative amplitude will appear as lobes in the model.
To solve for the far-field of a scattered signal numerically, various considerations must be taken into
account to deal with singularities that arise in the calculations. These appear as a consequence of applying
numerical integration on the surface of the object, which is necessary to determine the electromagnetic field.
Once the electromagnetic field on the surface of the object is known, the far-field pattern can be found.
Singularities become more difficult to deal with when a thin film of material is coated onto a scattering
object. Before investigating the numerical construction of such a problem, we should first become familiar
with the underlying theory. To gain an understanding of this subject, we will draw upon the derivations
made in [1]. The formulas they have composed for the numerical application provide a clear framework
upon which modifications can be made.
1.1. Solving the Electromagnetic Scattering Problem in Two Dimensions
The electromagnetic scattering problem can be solved for a known incident field by using the proper
boundary conditions. This requires that surface potentials be examined. The two-dimensional direct scat-
tering problem, as described in [1], involves finding a solution to the Helmholtz equation
u + k2
u = 0 in R2
¯Ω , (1)

2. where u(x) = u0(x)+us(x) for some known incident field u0(x) = eikx·d and unknown scattered field us. The
boundary conditions presented in [2] that must be satisfied for the scattering of an electromagnetic signal
are
u0 + us = ut on ∂Ω , (2)
ν
∂
∂υ
[u0 + us] =
∂ut
∂υ
on ∂Ω ,
where ut is the signal that is transmitted through the boundary ∂Ω and υ is the outward-pointing unit normal
on ∂Ω. If the signal has transverse-electric polarization then ν = 1, and if it has transverse-magnetic
polarization then ν = k2
1/k2
0, where k0 and k1 are the wave-numbers associated with u0 and ut. The solutions
can be described as a combination of single- and double-layer potentials. In [1], the single-layer potential
is given by
u(x) =
∂Ω
ϕ(y)Φ(x, y)ds(y), x ∈ R2
∂Ω ,
and the double-layer potential is given by
v(x) =
∂Ω
ϕ(y)
∂Φ(x, y)
∂υ(y)
ds(y), x ∈ R2
∂Ω ,
where ϕ is a continuous density and
Φ(x, y) =
i
4
H(1)
0 (k|x − y|), x y ,
is the fundamental solution to the Helmholtz equation. The density ϕ is the unknown that must be deter-
mined to calculate the far-field. It has furthermore been shown that these single-layer and double-layer
potentials can be expressed in Green’s formula for a radiating solution to the Helmholtz equation:
u(x) =
∂Ω
u(y)
∂Φ(x, y)
∂υ(y)
−
∂u
∂υ
(y)Φ(x, y) ds(y), x ∈ R2
¯Ω . (3)
Green’s formula is then used in [1] to show that
u∞(ˆx) =
eiπ
4
√
8πk ∂Ω
u(y)
∂e−ik ˆx·y
∂υ(y)
−
∂u
∂υ
(y)e−ik ˆx·y
ds(y), ˆx =
x
|x|
, (4)
where u∞ is the far-field of u. The direction from which the scattered field is measured is represented by the
normalized two-dimensional vector ˆx.
The far-field pattern can be modeled by calculating and plotting u∞ in all directions. As can be seen
in (4), the far-field depends on the electromagnetic field on the surface of an object, which in turn depends
on the object’s shape and material properties. Given enough distance, any object struck by a signal will
appear as a point source, but the amplitude of the far-field will vary when observed from different angles.
1.2. Treating the Singularity
For simplicity, the formulas appearing in this section will describe the scattering of acoustic waves from
a sound-soft object using the formulations made by [1]. This will introduce the singularities that exists and
part of the approach that is required for the numerical treatment of thin layers. The combined layer potential
2

3. u(x) =
∂Ω
∂Φ(x, y)
∂υ(y)
− iηΦ(x, y) ϕ(y)ds(y), x ∈ R2
¯Ω , (5)
with the real parameter η 0, which couples the single- and double-layer surface potentials, is employed
in [1]. As formulated, the potential u solves the exterior Dirichlet problem
u + k2
u = 0 in R2
¯Ω ,
u = f on ∂Ω ,
describing acoustic signals on a sound-soft object, only if the density ϕ is a solution to the integral equation
ϕ + Vϕ − iηJϕ = 2 f , (6)
where
(Vϕ)(x) = 2
∂Ω
∂Φ(x, y)
∂υ(y)
ϕ(y)ds(y), x ∈ ∂Ω ,
(Jϕ)(x) = 2
∂Ω
Φ(x, y)ϕ(y)ds(y), x ∈ ∂Ω .
The analytic boundary curve ∂Ω can be written using the 2π-periodic parametric representation
x(t) = [x1(t), x2(t)], 0 ≤ t ≤ 2π ,
where for the derivative function, x (t), we have |x (t)|2 > 0 for all t. Following [1], the Dirichlet boundary
integral equation (6) can then be written as
ψ(t) −
2π
0
L(t, τ) + iηM(t, τ) ψ(τ)dτ = g(t), 0 ≤ t ≤ 2π , (7)
where ψ(t) = ϕ(x(t)) and g(t) = 2 f(x(t)) are vector functions. The kernels, which only hold for t τ, are
given by
L(t, τ) =
ik
2
x2(τ) [x1(τ) − x1(t)] − x1(τ) [x2(τ) − x2(t)]
H(1)
1 (k|x(t) − x(τ)|)
|x(t) − x(τ)|
,
M(t, τ) =
i
2
H(1)
0 (k|x(t) − x(τ)|)|x (τ)| .
The Hankel functions H(1)
0 and H(1)
1 have a logarithmic singularity at t = τ. To treat this singularity, the
kernels are broken up in [1] into the following:
L(t, τ) = L1(t, τ) ln 4 sin2 t − τ
2
+ L2(t, τ) ,
M(t, τ) = M1(t, τ) ln 4 sin2 t − τ
2
+ M2(t, τ) .
By separating the logarithmic part of the kernels, they are now combined
K1(t, τ) = L1(t, τ) + iηM1(t, τ) ,
3

4. K2(t, τ) = L2(t, τ) + iηM2(t, τ) ,
such that the integral equation (7) can be written as
ψ(t) −
2π
0
K(t, τ)ψ(τ)dτ = g(t), 0 ≤ t ≤ 2π , (8)
where
K(t, τ) = K1(t, τ) ln 4 sin2 t − τ
2
+ K2(t, τ) . (9)
This kernel K is constructed in [1] to isolate the singularity such that K1 and K2 are well-behaved on the
surface. The trapezoidal rule
2π
0
f(τ)dτ ≈
π
n
2n−1
j=0
f(tj), tj =
jπ
n
can then be accurately performed on K2. According to the Nystr¨om method, a quadrature rule can also
be accurately performed on the logarithmic term as long as K1 is weighted properly to account for the
logarithm not being well-behaved. The quadrature rule used in [1] for this term is
2π
0
ln 4 sin2 t − τ
2
f(τ)dτ ≈
2n−1
j=0
R(n)
j (t)f(tj), 0 ≤ t ≤ 2π , (10)
where R(n)
j represents the jth quadrature weight when there are 2n-many points used for the integration.
Note that these weights are dependent on t. They are computed using the formula
R(n)
j (t) = −
2π
n
n−1
m=1
1
m
cos(m(t − tj)) −
π
n2
cos(n(t − tj)), j = 0, ..., 2n − 1 .
The integral equation (8) is then approximated with 2n points in [1] by
ψ(n)
(t) −
2n−1
j=0
R(n)
j (t)K1(t, tj) +
π
n
K2(t, tj) ψ(n)
(tj) = g(t), 0 ≤ t ≤ 2π .
Since t will be discretized to a particular set of points ti, from which set are the points for tj, and since
cos α(ti − tj) = cos α
π
n
|i − j| ,
it follows that
R(n)
j (ti) = R(n)
|i−j|
(0) .
It can be seen here that the quadrature weights only need to be calculated once and then shifted for integra-
tion at every ti. The equation at every ith point is given in [1] by
ψ(n)
(ti) −
2n−1
j=0
R(n)
|i−j|
K1(ti, tj) +
π
n
K2(ti, tj) ψ(n)
(tj) = g(ti), i = 0, ..., 2n − 1 , (11)
4

5. where
R(n)
j = −
2π
n
n−1
m=1
1
m
cos
mjπ
n
− (−1)j π
n2
, j = 0, ..., 2n − 1 .
Now, considering a square matrix A with elements
A(n)
ij = R(n)
|i− j|
K1(ti, tj) +
π
n
K2(ti, tj), i = 0, ..., 2n − 1, j = 0, ..., 2n − 1 , (12)
the integral equation (11) can be expressed in matrix-vector notation as
(I − A)ψ = g ,
where I is the 2n-identity matrix and the arrays ψ and g contain two columns of length 2n—one for each
spatial dimension. Once ψ is solved, such that ϕ has been solved, it is presented in [1] that the far-field can
be calculated using
u∞(ˆx) =
e−iπ
4
√
8πk ∂D
kυ(y) · ˆx + η e−ik ˆx·y
ϕ(y)ds(y), |ˆx| = 1 . (13)
1.3. Numerical Construction for Electromagnetic Scattering
To focus attention on examining the singularities that arise in numerical integration of a direct scattering
problem, acoustic signals on a sound-soft object were deemed sufficient. Although electromagnetic signals
are more complex than sound waves, the overall construct of the direct scattering problem is analogous. The
coupled Green’s formula (5), as presented in [1], will give a radiating solution u to the Helmholtz equation
for a density ϕ that satisfies the integral equations
ϕ + Vϕ + J
∂ϕ
∂υ
= f , (14)
ν + 1
2
∂
∂υ
ϕ + Wϕ + F
∂ϕ
∂υ
=
∂
∂υ
f ,
where f is the total incident field on ∂Ω and
(Vϕ)(x) =
∂Ω
∂(Φ1 − Φ0)(x, y)
∂υ(y)
ϕ(y)ds(y), x ∈ ∂Ω ,
J
∂ϕ
∂υ
(x) = −
∂Ω
(νΦ1 − Φ0)(x, y)
∂ϕ(y)
∂υ(y)
ds(y), x ∈ ∂Ω ,
(Wϕ)(x) =
∂Ω
∂2(Φ1 − Φ0)(x, y)
∂υ(x)∂υ(y)
ϕ(y)ds(y), x ∈ ∂Ω ,
F
∂ϕ
∂υ
(x) = −
∂Ω
∂(νΦ1 − Φ0)(x, y)
∂υ(x)
∂ϕ(y)
∂υ(y)
ds(y), x ∈ ∂Ω .
The function Φ0 corresponds to propagation in the first medium and Φ1 corresponds to propagation in the
penetrated medium, such that the wave-numbers may be different. These equations must hold to satisfy
the boundary conditions given in (2), which depend on the polarization of the electromagnetic field. The
5

6. four kernels by which these equations are represented numerically will contain the logarithmic singularity
discussed earlier; therefore, the kernels must be split and recombined into the form presented in (9).
To combine the two equations (14) into a single representation, we define the vector notation
¯α(t) =


α
∂α
∂υ


(t) , (15)
for any function α, and the operator
A =
V J
W F
. (16)
It will be assumed that all electromagnetic waves are transverse-electric, such that ν = 1. The integral
equations (14) can then be written as
(I + A)¯ϕ = ¯f . (17)
2. Adding a Thin Layer
Electromagnetic scattering only occurs on the surface of a homogeneous material, where there is a
change in the medium of propagation. Let us consider adding another layer of material that is not a perfect
conductor. What would happen if there were a thin film of material uniformly distributed on the surface of
an object? How might the far-field pattern appear different?
2.1. The Electromagnetic Field on Two Surfaces
The potential on both surfaces must satisfy the Helmholtz equation (1) and can be calculated using
Green’s formula (3). However, not only will the field at a single point depend upon all points on that surface,
but all points on the other surface will also contribute. The boundary conditions (2) will be combined
using (15) for ν = 1. When an electromagnetic wave encounters the first surface, the boundary condition
¯u0 + ¯us1 = ¯ut1 + ¯us2 on ∂Ωext
must hold true, where us2 is the signal scattered back from the second surface on which the transmitted
signal ut1 is incident. The boundary condition for the second surface is
¯ut1 + ¯us2 = ¯ut2 on ∂Ωint .
These signals on the two-layer object are shown in Figure 1.
The boundary conditions can be written in terms of the density of the field on each surface using
¯ϕ = ¯ϕt1 + ¯ϕs2 = ¯ϕ0 + ¯ϕs1 on ∂Ωext , (18)
¯ϕ = ¯ϕt1 + ¯ϕs2 = ¯ϕt2 on ∂Ωint . (19)
The combined electromagnetic integral equations derived in (17) will be used for both surfaces:
(I + A)¯ϕ = ¯ϕ0 + ¯ϕs2 on ∂Ωext , (20)
6

7. Figure 1: Electromagnetic Signal on a Two-Layer Object
(I + B)¯ϕ = ¯ϕt1 on ∂Ωint . (21)
The operators A and B correspond to the integration of points on one surface to a point on that same surface.
The scattered and transmitted field densities are unknowns that should be removed from the equations.
Rewriting (21) with substitution of (19) gives
−B¯ϕ = ¯ϕ − ¯ϕt1 = ¯ϕs2 on ∂Ωint , (22)
while rewriting (20) with substitution of (18) gives
−A¯ϕ = ¯ϕ − ¯ϕ0 − ¯ϕs2 = ¯ϕt1 − ¯ϕ0 on ∂Ωext . (23)
These equations can be mapped from one surface to the other. From (22) we define a C such that
¯ϕs2|int = −B¯ϕ|int ⇒ ¯ϕs2|ext = −C ¯ϕ|int , (24)
and from (23) we define a D such that
¯ϕt1|ext = ¯ϕ0|ext − A¯ϕ|ext ⇒ ¯ϕt1|int = ¯ϕ0|int − D¯ϕ|ext . (25)
So C maps the field reflected from the inner surface onto the outer surface and D maps the field transmitted
from the outer surface onto the inner surface. These are in fact just evaluations of the layer potentials defined
previously. Substituting (24) and (25) into (20) and (21) yields
(I + A)¯ϕ|ext = ¯ϕ0|ext − C ¯ϕ|int , (26)
(I + B)¯ϕ|int = ¯ϕ0|int − D¯ϕ|ext .
To parameterize the system in terms of one variable, the boundary curves ∂Ωext and ∂Ωint can be repre-
sented by the periodic vector functions xext(t) and xint(t). We can then define the following:
ψext(t) = ¯ϕ(xext(t)), gext(t) = ¯ϕ0(xext(t)) ,
ψint(t) = ¯ϕ(xint(t)), gint(t) = ¯ϕ0(xint(t)) .
7

8. These functions will all contain two vectors—one for each spatial dimension—twice the length of t because
of the included derivative terms denoted by the bar. Given a set of values for t, the system of equations (26)
can be discretized and expressed in matrix-vector notation as
I + A C
D I + B
ψext
ψint
=
gext
gint
. (27)
Note that the operators, when discretized appropriately, are matrices. This system can be solved to find ψext
for calculating the far-field.
2.2. Integration Between Surfaces
The logarithmic singularity will exist when integrating over all points contained on each surface of
the object; however, this singularity will not occur when integrating over the points of one surface to a
point on the other. This is due to the condition that there will always remain some distance between the
outside surface xext and the inside surface xint. If the thickness of the added layer x is very small, such
that ||xext(t) − xint(t)|| ≈ 0, then there will be near-singular behavior when integrating at t, which behavior
must approach a logarithmic singularity as x → 0. This near-singular behavior will destroy the accuracy of
standard numerical methods if not given proper treatment, as is achieved in what follows.
The kernels Kext and Kint have the form (9) and will represent the combined kernels contained in the
matrices A and B, respectively. The kernels modified for integration between layers will have the form
˜K(t, τ) = ˜K1(t, τ) ln 4 sin2 t − τ
2
+ 2
+ ˜K2(t, τ) , (28)
where
= (t) =
x
||xint(t)||
for xint(t) ≈ xext(t). The kernels ˜Kext and ˜Kint are contained in the matrices C and D, respectively. All
of these K and ˜K kernels, which are constructed numerically from the operators presented in (16), store
four entries for every pair of values of t and τ—two corresponding to the density ϕ and two to its derivative
∂ϕ/∂υ. Notice that the logarithmic term in (28) no longer matches the form for which the quadrature weights
in (10) were constructed. A new weighting formula for this term must be derived to continue integration
using a quadrature rule.
2.3. Treating the Near-Singularity
Consider the integral
2π
0
ln

4 sin2


(t − τ)2 + 2
2



eiqτ
dτ , (29)
where the distance due to is considered when the points t and τ are near to each other. Using properties
of exponents, the expression can be manipulated to remove t from appearing explicitly inside the integral.
This will put the integral in terms of q and . To simplify, we introduce the function
r (τ) = ln

4 sin2


√
τ2 + 2
2



 (30)
and substitute into (29) to get
8

9. 2π
0
r (t − τ)eiqτ
dτ .
By declaring a new variable z = t − τ, the function r can be written without explicit dependence on t.
Applying this change of variable results in
−
t−2π
t
r (z)eiq(t−z)
dz . (31)
Note that if the integrand is 2π-periodic in z—as it always will be for closed surface scattering in two
dimensions—then shifting the bounds of the integral by t will not alter the solution. Therefore, t can be
eliminated from the limits of integration. Using the first term in the Taylor series expansion of sin θ about
θ = 0, it is determined that
sin2


√
z2 + 2
2

 ≈


√
z2 + 2
2


2
=
z2
4
+
2
4
≈ sin2 z
2
+
2
4
. (32)
Applying (32) to (30) gives the approximation
˜r (z) = ln 4 sin2 z
2
+ 2
≈ r (z) ,
which only needs to be valid near the singularity, since the well-behaved sub-kernels contained in (28) are
dependent on one another— ˜K2 is determined by evaluating ˜K and ˜K1. The kernels are constructed to capture
the near-singularity, but they will also compensate for each other at the other points. Now, the function ˜r
will be used instead of r so that t − 2π may be subtracted from the limits of integration in (31). From here,
the explicit dependence on t and z can be separated into a product of two functions by setting
2π
0
˜r (z)eiq(t−z)
dz = D ,qeiqt
, (33)
where
D ,q =
2π
0
ln 4 sin2 z
2
+ 2
e−iqz
dz . (34)
These D ,q values can be numerically solved for any given and set of values q. When is constant with
respect to t, they are a set of constant coefficients.
Let us now consider the Fourier series expanded function
f(t) =
∞
q=−∞
dqeiqt
,
and rewrite the following integral using (33) with z = t − τ:
9

10. 2π
0
ln 4 sin2 t − τ
2
+ 2
f(τ)dτ (35)
=
2π
0
˜r (t − τ)f(τ)dτ
=
2π
0


∞
q=−∞
dq˜r (t − τ)eiqτ

 dτ
=
∞
q=−∞
dq
2π
0
˜r (t − τ)eiqτ
dτ
=
∞
q=−∞
dqD ,qeiqt
.
This gives an efficient way of calculating the weights for the expansion of f(t) under the integral. Further-
more, if f is real, then d−q = d∗
q. And by inspection of (34), it can be seen that D ,−q = D∗
,q. This allows for
the given integral to be arranged in the following way:
2π
0
ln 4 sin2 t − τ
2
+ 2
f(τ)dτ
=
∞
q=−∞
dqD ,qeiqt
= d0D ,0 +
∞
q=1
d−qD ,−qe−iqt
+ dqD ,qeiqt
= d0D ,0 +
∞
q=1
d∗
qD∗
,qe−iqt
+ dqD ,qeiqt
= d0D ,0 +
∞
q=1
(d∗
qD∗
,q + dqD ,q) cos(qt) − i(d∗
qD∗
,q − dqD ,q) sin(qt)
= d0D ,0 + 2
∞
q=1
Re{dqD ,q} cos(qt) − Im{dqD ,q} sin(qt) .
Notice that only the non-negatively indexed weights need to be solved with a numerical quadrature and then
stored when employing this representation.
Let us now consider how these weights D ,q depend on . If were constant, then only one set in q
would need to be calculated to perform the necessary integration at all points t along a given contour. This,
10

11. however, describes a special case since is scaled according to the curvature of the contour. Only if the
contour were a perfect circle would , in the parameter space t, be the same at each point. Calculating a
new set of values {D ,q} for every value of t will require many numerical integrations, each one involving
near-singular points, which will be costly in computation time. This seems to nullify the previous work of
removing t from the integral, if in fact depends on t. Fortunately, having to perform these integrations at
each step can be avoided by calculating the necessary values for D ,q through interpolation.
2.4. Employing Interpolation
Given any set of values for , and having solved D ,q numerically at each , a polynomial representation
can be derived for D ,q. The unknown set of polynomial coefficients {aq,k} are determined by constructing
and solving a linear system. Let us choose the Chebyshev polynomial of the first kind
Tk(x) = cos(k arccos(x)), k = 0, 1, 2, ... ,
which gives the recurrence relation
T0(x) = 1
T1(x) = x
Tk+1(x) = 2xTk(x) − Tk−1(x) .
Every D ,q can be expressed as a linear combination of these functions Tk using
aq,0T0( ) + aq,1T1( ) + ... + aq,N−1TN−1( ) + aq,NTN( ) = D ,q . (36)
The set of values chosen for must be unevenly-spaced and should be spaced in the same manner as the
Chebyshev nodes to minimize Runge’s phenomenon, as determined in [3]. The Chebyshev nodes over an
interval [a, b] are given by
xk =
1
2
(a + b) +
1
2
(b − a) cos
2k − 1
2N
π , k = 1, 2, ..., N .
The number of points N that are used for will determine the order of the polynomial and the number
coefficients that must be solved, lest there be an over- or under-determined system of equations.
Once the set of coefficients {aq,k} have been solved for all q, they can be saved to a file for future use.
Note, however, that this data will only be valid when every point used for integration has a corresponding
set of coefficients. As long as there already exists an array of coefficients whose dimension for q matches
the number of points being used along a contour, numerical integration can be replaced by interpolation
to solve for the set of values {D ,q}. This is a tremendous benefit considering that these weights can be
calculated at every point along the contour relatively quickly.
2.5. Using the Dirac Delta Function
The set of weights {D ,q} provides an effective way to solve the integral presented in (35) as long as the
Fourier series expansion for the function f is known. As is shown in equation (28), there is a kernel ˜K1 that
is being multiplied by the logarithm. Rather than finding the Fourier series expansion of this kernel, the
Dirac delta function can be used as the function undergoing integration and the value of the kernel can be
treated as another weight. This requires that a new variable be introduced to shift the delta function along
the domain, which is given as τ in the formula
11

12. f(τ ) =
2π
0
δ(τ − τ) f(τ)dτ .
If the Fourier series expansion of the Dirac delta function is given by
δ(τ − τ) =
∞
q=−∞
dτ,qeiqτ
,
then from (35) it follows that
2π
0
ln 4 sin2 t − τ
2
+ 2
f(τ )dτ ≈
2π
0
f(τ)
∞
q=−∞
dτ,qD ,qeiqt
dτ . (37)
This formula can be used to numerically solve the integral for any analytic function f without knowing its
Fourier series expansion. Recall that there exists a density function ¯ϕ that is acted on by the operators C
and D containing the near-singular logarithm, as seen in (26). The product of this density function with the
proper kernel is what defines the function f. Also, recall that ¯ϕ evaluated on the exterior and interior surface
composes the unknown arrays ψext and ψint, which must be solved using the linear system of equations (27).
It is therefore necessary to integrate using the Dirac delta function, since ¯ϕ will remain intact.
3. Scattering from a Perfect Circle with a Thin Coating
3.1. Implementing the Method Numerically
With a method now derived for solving the near-singular integral (35), let us model electromagnetic
scattering from a perfect circle in two-dimensions when covered by a thin layer of material. In this case,
with a uniform thickness in the outer layer, remains constant. The total number of points that will be used
along each contour is 2n. As stated earlier, this will determine the number of weights D ,q, which will now
be referred to as Dq, required for the integration. The larger the absolute value of q 0, the smaller the
weight is for Dq. These weights all decrease as is made larger, as seen in Figure 2.
The set {Dq} will be calculated with N points unevenly spaced from 0 to κ using
p = cos
pπ
N
+ 1
κ
2
, p = 0, ..., N − 1 .
The interval extending to κ = 2 is chosen to give accuracy to the polynomials that correspond to a smaller
magnitude q, since those values have greater weight. Note that although the index q will continue to be
used, the set is no longer infinite. Following (36), the polynomial for Dq is given by
aq,0T0( p) + aq,1T1( p) + ... + aq,N−1TN−1( p) = Dq( p), q = 0, ..., 2n − 1 .
The coefficients are found by solving the linear system of N equations. In the case of a perfect circle, the
set {Dq} will only need to be calculated once from the polynomial whenever changes.
To test the accuracy of the method given in (35), let us compare the results for calculating the integral
using these Dq values with the results from a numerical quadrature. The set {Dq} is solved using N = 30
coefficients. For the integration, we set n = 64 and f(t) = ecos(t−1.5). Various values are used for , as seen
in Figure 3. The integral is evaluated at every point t along the contour. The relative error is then calculated
using the two-norm. As can be seen in the table, this method works well for small , which indicates a thin
coating applied to an object.
12

13. Figure 2: Dq vs
The accuracy in using the Dirac delta function in (37) is also determined with comparison to a numerical
quadrature. Again, the parameters are set to N = 30 and n = 64, except now the function f(t) = 1 is chosen
so that the weight of the function remains constant at every point along the contour. The relative error
is displayed in Figure 4. At some point, as is decreased, accuracy is lost. In the range of values for
displayed in the table, the error is sufficient for employing the Dirac delta function.
The approximation (37) is in discrete form given by
2π
0
ln 4 sin2 t − τ
2
+ 2
f(τ )dτ ≈
π
n
n
j=−n+1
f(tj)
n
q=−n+1
dj,qDqeiqti
, i = 0, ..., 2n − 1 ,
where dj,q is the qth Fourier coefficient for δ(t − tj). We must set
f(tj) = ˜Kext1(ti, tj)ψint(tj)
for constructing matrix C in the linear system (27) and
f(tj) = ˜Kint1(ti, tj)ψext(tj)
for constructing matrix D. Recall that C and D are maps between surfaces that are close to each other and
therefore require this near-singular treatment. In either case, the function f stores two entries for every
tj—one corresponding to the density ϕ and another to its derivative ∂ϕ/∂υ. This leads to the following
numerical formulas:
13

14. Figure 3: Accuracy of Method with f(t) = ecos(t−1.5)
Figure 4: Accuracy of Method on Dirac Delta Functions
C(n)
ij =
π
n
n
q=−n+1
dj,qDqeiqti ˜Kext1(ti, tj) +
π
n
˜Kext2(ti, tj),
D(n)
ij =
π
n
n
q=−n+1
dj,qDqeiqti ˜Kint1(ti, tj) +
π
n
˜Kint2(ti, tj),
i = 0, ..., 2n − 1, j = 0, ..., 2n − 1 .
The matrices A and B are constructed using the form (12) that was established for a single-layer object:
A(n)
ij = R(n)
|i−j|
Kext1(ti, tj) +
π
n
Kext2(ti, tj),
B(n)
ij = R(n)
|i− j|
Kint1(ti, tj) +
π
n
Kint2(ti, tj),
i = 0, ..., 2n − 1, j = 0, ..., 2n − 1 .
14

15. The combined matrix constructed in (27) will have dimensions of length 8n. The density on the outer surface
ϕext can be found by solving for ψext, which is sufficient for calculating the far-field (13). The formulas for
the linear system are given numerically by
ψ(n)
ext(ti) +
2n−1
j=0
R(n)
|i−j|
Kext1(ti, tj) +
π
n
Kext2(ti, tj) ψ(n)
ext(tj)
+
π
n
2n−1
j=0


n
q=−n+1
dj,qDqeiqti ˜Kext1(ti, tj) + ˜Kext2(ti, tj)

 ψ(n)
int (tj) = g(ti),
ψ(n)
int (ti) +
2n−1
j=0
R(n)
|i−j|
Kint1(ti, tj) +
π
n
Kint2(ti, tj) ψ(n)
int (tj)
+
π
n
2n−1
j=0


n
q=−n+1
dj,qDqeiqti ˜Kint1(ti, tj) + ˜Kint2(ti, tj)

 ψ(n)
ext(tj) = g(ti),
i = 0, ..., 2n − 1 .
Note that there are near-singularities that can also arise within the kernels ˜Kext1, ˜Kext2, ˜Kint1, and ˜Kint2.
These near-singularities are effectively treated using the same interpolation strategy that was employed for
the logarithmic singularity. See for instance [4].
3.2. Examples of the Far-Field Pattern
Let us examine the far-field pattern of a transverse-electric wave scattered from a circular object that has
a thin coating of material. The pattern will depend on the thickness of the coating, denoted by x, and the
properties of the material. These properties will be accounted for by assignment of the wave-number k1 that
corresponds to propagation through the material. The wave-number of the incident signal is set k0 = 1 and
the wave-number through the object is set k2 = 3. The direction from which the incident wave approaches
will in every case be 270◦. Furthermore, the radius of the object is chosen to be 5 since the far-field pattern
contains several smooth lobes when x = 0, as seen in Figure 5. Notice that before a coating is introduced,
these lobes extend to about a relative measure of 10.
Before observing the effect that altering the thin coating has on the far-field pattern, a comparison is
made between the modified and unmodified code. The modified code includes the method required for
treating near-singularities, whereas the unmodified code can only determine the far-field for thick outer
layers. Figure 6 displays the far-field pattern produced by both when x = 0.001 and k1 = 10. There were
150 points (n = 75) used for the integration. The pattern produced by the modified code (a) is almost
identical to the pattern in Figure 5 when x = 0, but the pattern produced by the unmodified code (b) is
not at all similar. Indeed, with that small of a thickness and with the chosen parameters, there should be
no appreciable difference in the far-field patterns. This illustrates the need for treating the near-singularity
when the two layers are close together. The far-field patterns presented hereafter are determined using the
modified code.
15

16. Figure 5: Far-Field Pattern from a Perfect Circle of Radius 5
(a) Modified Code (b) Unmodified Code
Figure 6: Far-Field Pattern with x = 0.001 and k1 = 10
The accuracy of the electromagnetic field calculated on the object depends on the density of points
used for the integration. Increasing the number of points should reduce the error in the far-field calculation
in such a way that demonstrates convergence to the actual solution. If there is not a proper decay in the
magnitude of the error, then the number of points is insufficient. Figure 7 illustrates the convergence in
error relative to n = 239 when x = 0.001 and k1 = 10. Recall that there are 2n-many points around
each contour. To have confidence that the far-field patterns produced are accurate, n should be made large
16

17. enough so that small additions do not cause a noticeable change in size or shape. Since the density of
points required for accuracy depends on the shape of the actual pattern, this analysis on convergence must
be performed anytime the signal or object properties are altered. This convergence was confirmed for all
patterns presented below, which were solved using n = 201.
Figure 7: Relative Error of the Far-Field
We will fix x = 0.050, which results in ≈ 0.010 in equation (34). Figure 8 displays the far-field
pattern for four different choices made for k1. When k1 = 0.1, the wave-number for the thin coating is made
less than that of the surrounding atmosphere. When k1 = 2, the wave-number is instead made larger, but
it is still less than that of the object. It appears that this choice does not significantly alter the shape of the
far-field. When k1 = 5 or 10, the wave-number for the coating is made larger than that of the object. It is
seen here that increasing k1 can result in far-field patterns that are very different from one another.
If the goal is to minimize scattered signals, then k1 should be chosen to reduce the width and magnitude
of the lobes in the far-field pattern. The more narrow the lobes, the smaller the window in which a scattered
signal can be received. The direction in which the far-field should be reduced may also be of importance.
For example, in regard to stealth, radar transmitted from enemy aircraft will be received from the same
direction. Item (b) in Figure 8 is shown to have reduced the amplitude scattered back towards the incident
direction of 270◦. Now consider Figure 9, which shows the far-field pattern when k1 = 25. The magnitude
of the overall pattern has been reduced, yet the magnitude at 270◦ is larger than in Figure 8(b).
17

18. (a) k1 = 0.1 (b) k1 = 2
(c) k1 = 5 (d) k1 = 10
Figure 8: Far-Field Pattern with x = 0.050
The amplitude of the far-field may be further reduced by adding absorption to the thin coating. This is
modeled by adding an imaginary component to the wave-number. In Figure 10, absorption is added to the
coating from which Figure 8(c) is modeled. As the absorption is increased, the far-field pattern becomes
smaller. The wave-number in Figure 10(b) is chosen such that the pattern almost achieves a scaled down
version of that which is shown in Figure 5.
18

19. Figure 9: Far-Field Pattern with x = 0.050 and k1 = 25
(a) k1 = 5 + 0.3i (b) k1 = 5 + 2i
Figure 10: Far-Field Pattern with Absorption when x = 0.050
Figure 11 displays the far-field pattern for k1 = 25 + i at decreasing thicknesses for x—0.050, 0.025,
0.010, and 0.002. Notice that in item (a) the far-field at 90◦ has a relative amplitude larger than at any angle
in Figure 9. Therefore, adding some absorption can actually increase the amplitude of the far-field in certain
directions. Furthermore, notice that the amplitude at 270◦ has been scaled down from a relative measure
of 5 to 2 by adding absorption. The scattered field has in a sense been diverted away from the incident
direction.
19

20. Comparing the plots in Figure 11 makes it clear that varying the thickness of the thin coating may
considerably alter the shape of the far-field. Figure 12 also displays a pattern at decreasing thicknesses,
where k1 = 18. These plots further reveal that varying the thickness does not correspond to a predictable
change in scale. However, as the thickness approaches zero, the far-field pattern must eventually converge
to the pattern in Figure 5. Strict adherence to the exact thickness of coatings chosen for an object may be
crucial. This thickness must be chosen carefully to help achieve the desired pattern with whatever properties
belong to the material used in the thin layer.
(a) x = 0.050 (b) x = 0.025
(c) x = 0.010 (d) x = 0.002
Figure 11: Far-Field Pattern with k1 = 25 + i
20

21. (a) x = 0.050 (b) x = 0.025
(c) x = 0.010 (d) x = 0.002
Figure 12: Far-Field Pattern with k1 = 18
4. Final Comments
The ability to model the far-field pattern scattered from an object is no doubt beneficial to the studies of
radar and electromagnetic interference. To properly model this pattern from objects that have special thin
coatings requires additional numerical treatment. The electromagnetic field on the surface of each layer
of the object is dependent on the field radiated from the others. Calculating these electromagnetic fields
involves both singularities and near-singularities that arise in the integration. The near-singular behavior
cannot be disregarded without destroying numerical accuracy; it must be isolated and used to properly
weight the integration of a well-behaved function.
21

22. An approximated weighting can be applied to the spectral components of the well-behaved function
through a fast Fourier transform. These weights, which remain as a near-singular integral, can be solved
using a numerical quadrature and then stored; however, they do not only depend on the number of spec-
tral components being used, but on the scaled thickness of the coating as well. Calculating these weights
at every point where this scaled thickness changes—which occurs from changes in the curvature of ob-
ject’s surface—is costly in computation time. To avoid performing many numerical integrations, a set of
scaled thicknesses can be chosen to solve and store an array by which the weights for all other contained
thicknesses may be interpolated.
5. Acknowledgments
This paper was completed under the guidance of Mark Lyon, associate professor at the University of New
Hampshire.
6. References
[1] David Colton and Rainer Kress. Inverse Acoustic and Electromagnetic Scattering Theory. Springer Science+Business Media
LLC, third edition, 2013.
[2] Notes from Catalin Turc. New Jersey Institute of Technology.
[3] John H. Mathews and Kurtis D. Fink. Numerical Methods Using MATLAB. Prentice-Hall Inc, fourth edition, 2004.
[4] C. P´erez-Arancibia and O. P. Bruno. High-order integral equation methods for problems of scattering by bumps and cavities
on half-planes. Journal of the Optical Society of America A, 31(8):17381746, August 2014.
22