dcorreiaPhD

A HIGHER-ORDER PERFECTLY MATCHED LAYER FOR
OPEN-REGION, WAVEGUIDE, AND PERIODIC
ELECTROMAGNETIC PROBLEMS
BY
DAVI CORREIA
B.E., University of Brasilia, 1999
M.S., State University of Campinas, 2002
DISSERTATION
Submitted in partial fulﬁllment of the requirements
for the degree of Doctor of Philosophy in Electrical and Computer Engineering
in the Graduate College of the
University of Illinois at Urbana-Champaign, 2006
Urbana, Illinois

ABSTRACT
In this dissertation the idea of a higher-order perfectly matched layer (PML) is pro-
posed. Using a metric coefficient that includes both the regular and the complex fre-
quency shifted (CFS) PMLs, it is possible to obtain a PML that includes the advantages
of both of them. The second-order PML is applied to waveguide, periodic, and open-
region electromagnetic problems and its performance is compared to the regular’s and the
CFS-PML’s performance. The second-order PML outperforms both and its performance
is proved to be independent of the formulation or the simulation technique, since the
same behavior was observed using a stretched-coordinate approach in finite-difference
time-domain and uniaxial PML in time-domain finite-element method.
iii

To the most important women in my life:
my daugther, my mother, and my sister.
iv

ACKNOWLEDGMENTS
First, I would like to thank the Brazilian people who, through the agency CAPES
that supported me in USA, made this work possible. I express my great thanks to my
adviser, Professor Jianming Jin. With his patient and warm-hearted instruction, I had an
opportunity to walk step-by-step into the magniﬁcent palace of science and engineering.
I am also grateful for the contributions and guidance provided by my thesis committee
consisting of Professors Weng Cho Chew, Erhan Kudeki, and Andreas Cangellaris. Also,
to Professors Stephen Gedney and Jean-Pierre Berenger for some fruitful discussions in
conferences and by e-mail about PML. I should also acknowledge Dr. Zheng Lou for the
patience and time he spent working with me and for his key contributions to Chapter 7.
Finally, to my family and friends. Without their support, even when they did not
notice it, I would not have ﬁnished this dissertation. Thank you all.
v

TABLE OF CONTENTS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 1
CHAPTER 2 HIGHER-ORDER PERFECTLY MATCHED LAYERS . . . . . . 4
CHAPTER 3 APPLICATION TO OPEN-REGION PROBLEMS . . . . . . . . 9
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Numerical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4.1 Finite sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4.2 Finite plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.6 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
CHAPTER 4 APPLICATION TO WAVEGUIDE PROBLEMS . . . . . . . . . 23
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Parallel-Plate Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3 Rectangular Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3.1 Empty waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3.2 Inhomogeneous waveguide . . . . . . . . . . . . . . . . . . . . . . 27
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.5 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
CHAPTER 5 APPLICATION TO PERIODIC PROBLEMS . . . . . . . . . . . 36
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.2 Numerical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.2.1 Formulation for oblique incidence- 2D Case . . . . . . . . . . . . . 37
5.2.2 Formulation for oblique incidence- 3D case . . . . . . . . . . . . . 39
5.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.4.1 Two-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . 46
vi

5.4.2 Three-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . 48
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.6 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
CHAPTER 6 APPLICATION TO ARBITRARY MATERIALS . . . . . . . . . 58
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.2 Analysis of Left-Handed Metamaterials . . . . . . . . . . . . . . . . . . . 58
6.3 Formulation for Left-Handed Metamaterials . . . . . . . . . . . . . . . . 60
6.3.1 PML in a plasma medium . . . . . . . . . . . . . . . . . . . . . . 62
6.4 Results for Left-Handed Metamaterials . . . . . . . . . . . . . . . . . . . 64
6.4.1 Comparison with 2D results . . . . . . . . . . . . . . . . . . . . . 65
6.4.2 Three-dimensional simulation of a λ/2 dipole . . . . . . . . . . . . 66
6.5 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.6 Formulation for Gyrotropic Media . . . . . . . . . . . . . . . . . . . . . . 67
6.6.1 Uniaxial dispersive material . . . . . . . . . . . . . . . . . . . . . 68
6.6.2 FDTD-PML formulation for gyromagnetic media . . . . . . . . . 70
6.7 Results for Gyrotropic Media . . . . . . . . . . . . . . . . . . . . . . . . 72
6.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.9 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
CHAPTER 7 APPLICATION TO TIME-DOMAIN FINITE-ELEMENT
METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.5 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
CHAPTER 8 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
AUTHOR’S BIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
vii

LIST OF FIGURES
Figure Page
3.1 Computational domain. The cell size is ∆x = ∆y = 1 mm. The error is
measured at point A, three cells away from each PML interface. The line
source is located at the center. . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Relative error for the three different PMLs as a function of time step for
the 2D case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Maximum error in dB for the CFS PML as a function of κ and σ. . . . . 20
3.4 Maximum relative error in dB for the second-order PML as a function of
σ1 and σ2. The other parameters are kept constant and their value are the
optimum ones stated in the text. . . . . . . . . . . . . . . . . . . . . . . 21
3.5 Maximum relative error in dB for the second-order PML as a function of
κ2 and σ2. The other parameters are kept constant and their value are the
optimum ones stated in the text. . . . . . . . . . . . . . . . . . . . . . . 21
3.6 Relative error for the three different PMLs as a function of time step for
the 3D case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.1 Reflection error as a function of frequency for the TEM mode in a parallel-
plate waveguide using a 10-layer regular PML, CFS-PML and second-order
PML. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Reflection error for a 10-layer regular PML as a function of frequency for
the TM1 mode in a parallel-plate waveguide for κmax = 1 (solid line) and
κmax = 5 (dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 Reflection error for a 10 layer CFS-PML as a function of frequency for the
TM1 mode in a parallel-plate waveguide for different values of κ. . . . . . 30
4.4 Reflection error for a 10-layer regular, CFS and second-order PMLs as a
function of frequency for the TM1 mode in a parallel-plate waveguide. . . 31
4.5 Reflection error for a 10-layer regular PML as a function of frequency for
the TEM and TM1 modes in a parallel-plate waveguide for different values
of κmax. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.6 Reflection error for 10-layer CFS-PML as a function of frequency for the
TEM and TM1 modes in a parallel-plate waveguide for constant a (solid
line) and a linear decay in a (dashed line). . . . . . . . . . . . . . . . . . 32
viii

4.7 Reflection error for a 10-layer regular, CFS and second-order PMLs as
a function of frequency for the TEM and TM1 modes in a parallel-plate
waveguide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.8 Reflection error for a 10-layer regular, CFS and second-order PMLs as
a function of frequency for the TE10 and TE20 modes in a rectangular
waveguide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.9 Computational domain for the inhomogeneous waveguide problem excited
by a line source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.10 Spectrum of the incident field 10 cells away from the source for the inho-
mogeneous waveguide problem. . . . . . . . . . . . . . . . . . . . . . . . 34
function of frequency for the inhomogeneous waveguide problem excited
by a line source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.12 Computational domain for the microstrip problem. . . . . . . . . . . . . 35
function of frequency for the microstrip problem. . . . . . . . . . . . . . 35
5.1 Computational domain. The cell size is ∆x = ∆y = 0.5 mm. . . . . . . . 50
5.2 Results for a modulated periodic slab. The regular PML has a poor ab-
sorption around the frequency of the first Floquet mode (17.6 GHz). . . . 51
5.3 Results for a modulated periodic slab. The CFS-PML has a poor absorp-
tion for low-frequency propagating waves. . . . . . . . . . . . . . . . . . . 51
5.4 Results for a modulated periodic slab. The second-order PML incorporates
the advantages of both regular and CFS PMLs. . . . . . . . . . . . . . . 52
5.5 Geometry for the first FSS. The metal plate is immersed in a dielectric
with permittivity r = 2 and the magnetic field is polarized along the
smaller side of the rectangular PEC patch. . . . . . . . . . . . . . . . . . 52
5.6 Specular reflection coefficient for the geometry shown in Fig. 5.5 calculated
using FE-BI method and MoM. . . . . . . . . . . . . . . . . . . . . . . . 53
5.7 Geometry for the second FSS. Two metal grids are immersed in a dielectric
with permittivity r = 3 and separated by a dielectric with r = 1.01. . . 53
5.8 Results for the vertical polarization at 20◦
incidence for FDTD and FE-BI
techniques for the geometry shown in Fig. 5.7 without one of the metal
strips. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
strips. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
strips. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.11 Results for the horizontal polarization at 20◦
incidence for FDTD and FE-
BI techniques for the geometry shown in Fig. 5.7 without one of the metal
strips. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
ix

strips. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
strips. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.14 Result for the horizontal polarization at 20◦
incidence using FDTD for the
geometry shown in Fig. 5.7. . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.15 Result for the vertical polarization at 40◦
geometry shown in Fig. 5.7. . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.1 Ez-field intensity distribution at t = 1250∆t for ωp = 266.6 × 109
rad/s. . 75
6.2 Ez-field intensity distribution at t = 1250∆t for ωp = 500 × 109
rad/s. . . 75
6.3 Geometry for the 3D problem. . . . . . . . . . . . . . . . . . . . . . . . . 76
6.4 Field distribution for ωp = 266 × 109
rad/s: (a) in the xy-plane, (b) in the
zx-plane, (c) in the yz-plane. . . . . . . . . . . . . . . . . . . . . . . . . 77
6.5 Field distribution for ωp = 500 × 109
rad/s: (a) in the xy-plane, (b) in the
zx-plane, (c) in the yz-plane. . . . . . . . . . . . . . . . . . . . . . . . . 78
6.6 Spectrum of the source for a differentiated Gaussian (solid line) and for a
modulated Gaussian (dashed line). . . . . . . . . . . . . . . . . . . . . . 79
6.7 Field amplitude for the differentiated Gaussian excitation. . . . . . . . . 79
6.8 Field amplitude for the modulated Gaussian excitation. . . . . . . . . . . 80
6.9 Error introduced by a six-cells thick PML, Ex component in free-space, at
the PML interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.10 Error introduced by a six-cells thick PML, Ey component in free-space, at
6.11 Error introduced by a six-cells thick PML, Ez component in free-space, at
6.12 Error introduced by a six-cells thick PML, Ex component inside the gy-
rotropic medium, at the PML interface. . . . . . . . . . . . . . . . . . . . 82
6.13 Error introduced by a six-cells thick PML, Ey component inside the gy-
6.14 Error introduced by a six-cells thick PML, Ez component inside the gy-
7.1 Reflection errors in an empty rectangular waveguide. . . . . . . . . . . . 94
7.2 Cross section of the stripline waveguide. . . . . . . . . . . . . . . . . . . 95
7.3 Results for the stripline waveguide. . . . . . . . . . . . . . . . . . . . . . 95
7.4 Geometry of the monopole: a = 1 mm, b = 2.3 mm, h = 32.8 mm. . . . . 96
7.5 Time-domain reflected voltage calculated at the coaxial port of the mono-
pole antenna (τa = h/c). . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.6 Frequency-domain input impedance of the monopole antenna. . . . . . . 97
x

CHAPTER 1
INTRODUCTION
A basic consideration when dealing with differential-based methods in electromag-
netics, either finite-difference time-domain (FDTD) or finite-element method (FEM), is
that many problems are defined as open region problems, where the spatial domain is
unbounded in one or more directions. For obvious reasons, no computer can store an
unlimited amount of data and some type of absorbing boundary condition (ABC) must
be used to truncate the computational domain when simulating open-region problems.
Among the ABCs used in such problems, the perfectly matched layer (PML) concept is
one of the most popular.
The basic concept of the PML was first introduced by Berenger [1] as a material-
based ABC for the FDTD simulation of open-region electromagnetic problems. The
PML enjoyed a great success because of its great capability of absorbing propagating
waves regardless of their frequency. In his pioneering work, Berenger derived a split-field
formulation of Maxwell’s equations where each vector field component is split into two
orthogonal components.
After Berenger’s seminal paper, Chew and Weedon developed a more compact form for
the PML. The split-field equations were reposed in a nonsplit form that maps Maxwell’s
equations into a stretched coordinate space [2]. However, neither formulation had a
physical meaning. Gedney [3] and Sacks et al. [4] developed the uniaxial PML (UPML).
1

Here, the PML is seen as a physical medium, contrary to the split-field formulation
or the stretched-coordinate formulation. One of the main advantages of the stretched-
coordinate formulation is that it allows straightforward mathematical manipulation of the
PML equations. It also can be shown that if the metric coefficients are independent of the
material parameters, the PML matches waves propagating in a general host medium [5, 6].
In [7] the relation between the three formulation is derived in detail. In this work we use
the stretched-coordinate formulation for the FDTD simulations and uniaxial formulation
for the time-domain finite element (TDFE) simulations.
Even though the PML enjoyed a great success for absorption of propagating waves, it
is, in its original form, incapable of absorbing evanescent waves. Various efforts have been
attempted to overcome this limitation [8, 9, 10, 11]. Among these, the complex frequency
shifted (CFS)-PML [11] is particularly effective, especially after the development of a
simple convolution-based implementation [12]. The CFS-PML has since been applied to
several different problems [12, 13, 14, 15].
While the CFS-PML is highly effective in absorbing evanescent waves, it compromises
the capability of the original PML in the absorption of low-frequency propagating waves.
Therefore, for a general problem where both evanescent and low-frequency propagating
waves exist, both the CFS and the regular PMLs fail to provide a highly accurate solution.
After the CFS-PML was proposed, some authors started to realize that the best
way to improve the PML performance was by working on the metric coefficients rather
than trying to optimize the profiles for the conductivity in the metric coefficients [16,
17]. Among those attempts, the higher-order proposed by Chevalier and Inan [17] was
promising, but it did not include the CFS-PML in its formulation.
To overcome the limitations of the CFS and the regular PMLs, we proposed the
higher-order PML [18], which could retain the advantages of both the CFS and regular
PMLs. All the metric coefficients proposed so far, namely the original PML, the CFS-
2

PML, and Chevalier’s higher-order PML, are special cases of our higher-order PML. We
have shown that our second-order PML is highly effective in absorbing both evanescent
and low-frequency propagating waves in open-region periodic and nonperiodic problems
[18] and in waveguide problems [19] for FDTD simulations and open-region and waveguide
problems for TDFE simulation [20].
This work is divided as follows. In Chapter 2, we show the limitations and advan-
tages of each metric coefficient. In Chapter 3, we derive the complete formulation and
implementation of the second-order PML for open region problems. We show 2D and
3D examples to verify the performance of the second-order PML. In Chapter 4, we com-
pare the performance of the second-order PML to the CFS-PML and the regular PML
when applied to waveguide problems. In Chapter 5, we derive our own formulation for
the periodic problem and compare the performance of the second-order PML with the
CFS-PML and the regular PML. In Chapter 6 we show one potential application of an
accurate analysis of periodic structures, the left-handed metamaterials, which is drawing
increasing attention is the last years. We also shown that the second-order PML can be
implemented to arbitrary materials as long as either µ or is constant. In Chapter 7 we
present the formulation and implementation of the second-order PML to time-domain
finite element method. Finally, in Chapter 8 we present the conclusions as well as future
research.
3

CHAPTER 2
HIGHER-ORDER PERFECTLY MATCHED
LAYERS
Following the approach of stretched coordinates [2], the z-projection of Maxwell-
Ampere’s law in free-space for the modified Maxwell’s equations is shown to be
jω Ez = −
1
sy
∂
∂y
Hx +
1
sx
∂
∂x
Hy (2.1)
where sx and sy are the stretched-coordinate metric coefficients. For the regular PML,
they are given by
si = κi +
σi
jω
, i = x or y (2.2)
in which σi ≥ 0 is the conductivity profile different from zero only in the PML region
to provide attenuation for propagating waves and κi ≥ 1 is different from 1 only in the
PML region to attenuate evanescent waves. The subscript i will not be used from now
on but it should be understood that the analysis is the same for sx, sy or sz.
Several authors have explained why the regular PML has a poor absorption of evanes-
cent waves [10, 8]. Kuzuoglu and Mittra [11] proposed a different metric coefficient, called
complex-frequency shifted PML (CFS-PML), which is given by
s = κ +
σ
a + jω
. (2.3)
Here, a ≥ 0 is introduced to better absorb the evanescent waves with a = 0 outside
the PML region. Several authors have successfully applied this metric coefficient to the
4

absorption of evanescent waves [12, 13, 14]. Unfortunately, a value of a = 0 compromises
the absorption of low-frequency propagating waves. We will generalize this metric coef-
ficient for the case where more than one pole is present, hence the name of higher-order
PML.
It has been shown [21] that the reflection from the PML in the continuous space for
propagating waves is given by the imaginary part of the metric s. For the regular PML
it is shown to be
R(θ) = e−2σηd cos θ
, (2.4)
where η is the free-space impedance, d is the PML thickness and θ is the angle of incidence
at the PML interface. For simplicity, we assume here that σ is a constant across the PML.
By using the metric coefficient (2.3) for the CFS-PML, the reflection coefficient is given
by
R(θ) = e
−2σηd cos θ ω2 2
a2+ω2 2
. (2.5)
As shown in [15] and [22], the CFS-PML would have a poor absorption of low-frequency
propagating waves. In the limit when ω → 0, the CFS-PML would be completely in-
effective whereas the regular PML would still be able to absorb those waves. On the
other hand, the regular PML would not have a good performance when compared to the
CFS-PML for the absorption of evanescent waves [12].
The idea of a higher-order scheme for the metric coefficient was first proposed in [17],
where a convolution was used in its implementation. We will show that a careful selection
of the parameters will provide a metric coefficient that can keep the advantages of both
the regular and the CFS-PMLs. For a second-order PML, we have
s = κ1 +
σ1
a1 + jω
κ2 +
σ2
a2 + jω
, (2.6)
where κ1,2, σ1,2 and a1,2 are the parameters that can be chosen to control the PML
performance. Note that the second-order PML is reduced to the regular PML when
5

a1 = 0, κ2 = 1, and σ2 = 0, and to the CFS-PML when a1 = 0, κ2 = 1, and σ2 = 0. We
also note that when a1 = a2 = 0, our second-order PML is reduced to the one proposed
in [17]. In all cases, σ1,2 ≥ 0 and κ1,2 ≥ 1 only inside the PML region.
Even though it is not required theoretically that the real part of the metric coefficient
(rpml) be one at the air/PML interface, in practice setting rpml = 1 at the interface is
desirable as it reduces grid reflections. This requirement can easily be satisfied for the
regular or the CFS-PML by setting κ = 1 at the interface. However, it would require a
special treatment if the second-order PML is used because the real part of the second-
order PML is given by
rpml = κ1κ2 +
κ1a2σ2
a2
2 + (ω )2
+
κ2a1σ1
a2
1 + (ω )2
+
σ1σ2[a1a2 − (ω )2
]
[a2
1 + (ω )2][a2
2 + (ω )2]
. (2.7)
This real part reduces to 1 at the interface when σ1,2 = 0 and κ1,2 = 1 at the interface.
The imaginary part of the second-order PML, responsible for the attenuation of prop-
agating waves, is given by
ipml = −
κ1σ2ω
a2
2 + (ω )2
−
κ2σ1ω
a2
1 + (ω )2
−
σ1σ2ω (a1 + a2)
[a2
1 + (ω )2][a2
2 + (ω )2]
. (2.8)
Here, it is important to realize that, if both a1 = 0 and a2 = 0, the second-order PML will
have the same problem as the CFS-PML at low frequencies, since the reflection coefficient
for the PML, assuming for simplicity that all the variables are constant, is given by
R(θ) = e−2dηω cos θipml
(2.9)
Clearly the reflection coefficient for the PML is 1 when ω → 0. A simple way of overcom-
ing this limitation is to set either a1 or a2 to be zero. If a1 = 0, the second term in (2.8)
will be frequency independent and will resemble the absorption of the regular PML.
Even though the second-order PML adds more degrees of freedom that might help to
achieve lower reflections, the trade-off between the gain and the computational cost and
6

the time required to adjust all the variables has to be taken into account. If the problem
does not require absorption of evanescent waves, the cost both in memory requirement
and in time spent adjusting the parameters for both the CFS-PML and the second-order
PML will not pay off. Similarly, if no low-frequency propagating waves are present, the
CFS-PML would be sufficient. For general problems, where both evanescent and low-
frequency propagating waves are present, the rule of thumb is to use the best CFS-PML
available as the second factor in (2.6) and to set a1 = 0 in the second term. With this
setup, the second-order PML can perform better than both the regular and CFS-PMLs
in many different problems. We select a coefficient that would reproduce both regular
and CFS-PMLs. It is given by
s = 1 +
σ1
jω
κ2 +
σ2
a2 + jω
(2.10)
We should point out that we set κ1 = 1 and a1 = 0 in (2.10) because, for the problems
we considered, absorption of low-frequency evanescent waves was not critical. If the
absorption of those waves is critical, one should not set a1 = 0 so that the second-order
PML can recover the CFS-PML at low frequencies.
The imaginary part of the coefficient we choose in (2.10) is given by
ipml =
κ2σ1
ω
+
ω σ2
a2
2 + ω2 2
+
σ1σ2a2
ω (a2
2 + ω2 2)
(2.11)
and its reflection coefficient is
R(θ) = e
−2ηd cos θ κ2σ1+
σ2ω2 2
a2
2+ω2 2 +
a2σ1σ2
a2
2+ω2 2
. (2.12)
The first term will provide the frequency independent attenuation, just like the regular
PML. The second one is the same as the CFS-PML and will go to zero as ω → 0. Finally,
the third term will provide zero absorption when ω increases and becomes a constant
when ω → 0. Since κ2 ≥ 1 and all the terms are greater than zero, the absorption of
7

propagating waves for the higher-order PML is always greater than both the regular and
the CFS PMLs.
Now consider the absorption of evanescent waves. For this case, the real part of the
metric coefficient will play an important role. For the regular PML, the best one can do
is a profile for κ such that it is one at the interface so that grid reflections are reduced
and with a growth inside the PML so that the evanescent waves are absorbed. By adding
the term a in the metric for the CFS-PML what one is actually doing is adding an extra
term for the real part. This happens because one could interpret the CFS-PML metric
in (2.3) as
s = κ +
σa
a2 + ω2 2
−
jω
a2 + ω2 2
. (2.13)
The real part for the CFS-PML will then be
rpml = κ +
σa
a2 + ω2 2
. (2.14)
A careful choice of a will provide a very good absorption of evanescent waves [12],
[22]. Berenger also showed that an optimal choice for a = a0 could be made [22]. Unfor-
tunately, this choice would still provide a poor absorption of low-frequency propagating
waves.
By using the higher-order PML, such a problem would not appear due to the number
of degrees of freedom available. For the metric coefficient in (2.10) the real part is given
by
rpml = κ2 +
σ2a2 − σ2σ1
a2
2 + ω2 2
. (2.15)
One can simply choose a2 = σ1 + a0, with a0 as defined in [22], and σ2 = σcfs, with
σcfs being the one used for the CFS-PML and the behavior for the CFS-PML in (2.14)
for evanescent waves is recovered without having to compromise the absorption of low-
frequency propagating waves, as happened in [22]. Hence, one can have the advantages
of both regular and CFS-PMLs when using the higher-order PML.
8

CHAPTER 3
APPLICATION TO OPEN-REGION PROBLEMS
3.1 Introduction
Open-region problems are the most common problems to be solved for antennas de-
sign. Even though the use of FDTD is limited by its poor capability of handling arbitrary
geometries, its simplicity and low-cost make it the method of choice if one is solving sim-
ple problems. Since the PML was introduced by Berenger [1] to handle open-region
problems, FDTD became even more attractive. Due to the PML poor absorption of
evanescent waves, however, it is required that the PML be placed away from the scat-
terer. We will show that with the second-order PML the computational domain can be
reduced while still keeping a good performance. In this chapter we will apply the second-
order PML to open-region problems, both in 2D and 3D. The derivation for the numerical
formulation is shown in Section 3.2. The implementation is shown in Section 3.3. The
results for the 2D and 3D cases are presented in Section 3.4. Finally, the conclusions are
presented in Section 3.5.
3.2 Numerical Formulation
The entire derivation and implementation shown here assume a second-order PML,
where two complex poles are present. It should be noted, however, that the same ap-
proach could be applied for an nth
-order PML.
9

Following the stretched-coordinate approach, we can write Maxwell’s equation as
s × E = −jωµH.
Since
s × E =
1
sx
∂
∂x
ˆx × E +
1
sy
∂
∂y
ˆy × E +
1
sz
∂
∂z
ˆz × E
we have
−jωµHsx =
1
sx
∂
∂x
ˆx × E
−jωµHsy =
1
sy
∂
∂y
ˆy × E
−jωµHsz =
1
sz
∂
∂z
ˆz × E. (3.1)
In a similar fashion, since
s × H = jω E,
we have
jω Esx =
1
sx
∂
∂x
ˆx × H
jω Esy =
1
sy
∂
∂y
ˆy × H
jω Esz =
1
sz
∂
∂z
ˆz × H. (3.2)
where H = Hsx + Hsy + Hsz and E = Esx + Esy + Esz. This deﬁnition for H and E is
nonphysical, but it greatly simpliﬁes the mathematical manipulation.
Let
sx = κ1x +
σ1x
a1x + jω
κ2x +
σ2x
a2x + jω
sy = κ1y +
σ1y
a1y + jω
κ2y +
σ2y
a2y + jω
sz = κ1z +
σ1z
a1z + jω
κ2z +
σ2z
a2z + jω
. (3.3)
10

Labeling the ﬁrst factors in the right-hand side of the equations above as s1x, s1y, and
s1z and the second factors as s2x, s2y, and s2z, we can rewrite (3.1) as
−s1xjωµHsx =
1
s2x
∂
∂x
ˆx × E
−s1yjωµHsy =
1
s2y
∂
∂y
ˆy × E
−s1zjωµHsz =
1
s2z
∂
∂z
ˆz × E (3.4)
and (3.2) as
s1xjω Esx =
1
s2x
∂
∂x
ˆx × H
s1yjω Esy =
1
s2y
∂
∂y
ˆy × H
s1zjω Esz =
1
s2z
∂
∂z
ˆz × H. (3.5)
Upon carrying on the multiplication in the left-hand side of (3.4) and (3.5), we have
−jωµκ1xHsx −
jωµσ1x
a1x + jω
Hsx =
1
s2x
∂
∂x
ˆx × E
−jωµκ1yHsy −
jωµσ1y
a1y + jω
Hsy =
1
s2y
∂
∂y
ˆy × E
−jωµκ1zHsz −
jωµσ1z
a1z + jω
Hsz =
1
s2z
∂
∂z
ˆz × E (3.6)
and
jω κ1xEsx +
jω σ1x
a1x + jω
Esx =
1
s2x
∂
∂x
ˆx × H
jω κ1yEsy +
jω σ1y
a1y + jω
Esy =
1
s2y
∂
∂y
ˆy × H
jω κ1zEsz +
jω σ1z
a1z + jω
Esz =
1
s2z
∂
∂z
ˆz × H. (3.7)
We deﬁne the second terms in the left-hand side of (3.6) and (3.7) as new variables
ΨHx1, ΨHy1, ΨHz1, ΨEx1, ΨEy1, and ΨEz1, such that
11

ΨHx1 =
jωµσ1x
a1x + jω
Hsx
ΨHy1 =
jωµσ1y
a1y + jω
Hsy
ΨHz1 =
jωµσ1z
a1z + jω
Hsz
ΨEx1 =
jω σ1x
a1x + jω
Esx
ΨEy1 =
jω σ1y
a1y + jω
Esy
ΨEz1 =
jω σ1z
a1z + jω
Esz. (3.8)
If s2x = s2y = s2z = 1, this is simply the auxiliary diﬀerential equation (ADE) method
described in [23]. To generalize the ADE method, we move s2x, s2y, and s2z to the
left-hand side to obtain
−jωµκ1xκ2xHsx − κ2xΨHx1 −
jωµκ1xσ2xHsx + σ2xΨHx1
a2x + jω
=
∂
∂x
ˆx × E
−jωµκ1yκ2yHsy − κ2yΨHy1 −
jωµκ1yσ2xHsy + σ2yΨHy1
a2y + jω
=
∂
∂y
ˆy × E
−jωµκ1zκ2zHsz − κ2zΨHz1 −
jωµκ1zσ2zHsz + σ2zΨHz1
a2z + jω
=
∂
∂z
ˆz × E (3.9)
and
jω κ1xκ2xEsx + κ2xΨEx1 +
jω κ1xσ2xEsx + σ2xΨEx1
a2x + jω
=
∂
∂x
ˆx × H
jω κ1yκ2yEsy + κ2yΨEy1 +
jω κ1yσ2xEsy + σ2yΨEy1
a2y + jω
=
∂
∂y
ˆy × H
jω κ1zκ2zEsz + κ2zΨEz1 +
jω κ1zσ2zEsz + σ2zΨEz1
a2z + jω
=
∂
∂z
ˆz × H. (3.10)
We deﬁne the third terms in each equation in (3.9) and (3.10) as new variables ΨHx2,
ΨHy2, ΨHz2, ΨEx2, ΨEy2, and ΨEz2, such that
12

ΨHx2 =
jωµκ1xσ2xHsx + σ1xΨHx1
a2x + jω
ΨHy2 =
jωµκ1yσ2yHsy + σ1yΨHy1
a2y + jω
ΨHz2 =
jωµκ1zσ2zHsz + σ1zΨHz1
a2z + jω
ΨEx2 =
jω κ1xσ2xEsx + σ1xΨEx1
a2x + jω
ΨEy2 =
jω κ1yσ2yEsy + σ1yΨEy1
a2y + jω
ΨEz2 =
jω κ1zσ2zEsz + σ1zΨEz1
a2z + jω
. (3.11)
Rearranging the new variables Ψ1 and Ψ2 and their correspondent field components,
the final set of equations is given by (3.12). Here it is important to notice that ΨHx2,
ΨHy2, ΨHz2, ΨEx2, ΨEy2, and ΨEz2 depend only on their respective field component and
should be updated first. Variables ΨHx1, ΨHy1, ΨHz1, ΨEx1, ΨEy1, and ΨEz1 depend both
on their respective field components as well as on ΨHx2, ΨHy2, ΨHz2, ΨEx2, ΨEy2, and
ΨEz2, respectively, and should be updated next. Finally, Hsx, Hsy, Hsz Esx, Esy, and
Esz depend on both ΨHx2, ΨHy2, ΨHz2, ΨEx2, ΨEy2, and ΨEz2 and on ΨHx1, ΨHy1, ΨHz1,
ΨEx1, ΨEy1, and ΨEz1, respectively, and should be the last to be updated.
jωµΨHx2 + a2x +
σ2x
κ2x
ΨHx2 =
σ2x
κ2x
∂
∂x
ˆx × E
jωµΨHx1 + a1x +
σ1x
κ1x
ΨHx1 =
σ1x
κ1xκ2x
∂
∂x
ˆx × E − ΨHx2
jωµκ1xκ2xHsx =
∂
∂x
ˆx × E − κ2xΨHx1 − ΨHx2
jωµΨHy2 + a2y +
σ2y
κ2y
ΨHy2 =
σ2y
κ2y
∂
∂y
ˆy × E
jωµΨHy1 + a1y +
σ1y
κ1y
ΨHy1 =
σ1y
κ1yκ2y
∂
∂y
ˆy × E − ΨHy2
jωµκ1yκ2yHsy =
∂
∂y
ˆy × E − κ2yΨHy1 − ΨHy2
13

jωµΨHz2 + a2z +
σ2z
κ2z
ΨHz2 =
σ2z
κ2z
∂
∂z
ˆz × E
jωµΨHz1 + a1z +
σ1z
κ1z
ΨHz1 =
σ1z
κ1zκ2z
∂
∂z
ˆz × E − ΨHz2
jωµκ1zκ2zHsz =
∂
∂z
ˆz × E − κ2zΨHz1 − ΨHz2
jω ΨEx2 + a2x +
σ2x
κ2x
ΨEx2 =
σ2x
κ2x
∂
∂x
ˆx × H
jω ΨEx1 + a1x +
σ1x
κ1x
ΨEx1 =
σ1x
κ1xκ2x
∂
∂x
ˆx × H − ΨEx2
jω κ1xκ2xEsx =
∂
∂x
ˆx × H − κ2xΨEx1 − ΨEx2
jω ΨEy2 + a2y +
σ2y
κ2y
ΨEy2 =
σ2y
κ2y
∂
∂y
ˆy × H
jω ΨEy1 + a1y +
σ1y
κ1y
ΨEy1 =
σ1y
κ1yκ2y
∂
∂y
ˆy × H − ΨEy2
jω κ1yκ2yEsy =
∂
∂y
ˆy × H − κ2yΨEy1 − ΨEy2
jω ΨEz2 + a2z +
σ2z
κ2z
ΨEz2 =
σ2z
κ2z
∂
∂z
ˆz × H
jω ΨEz1 + a1z +
σ1z
κ1z
ΨEz1 =
σ1z
κ1zκ2z
∂
∂z
ˆz × H − ΨEz2
jω κ1zκ2zEsz =
∂
∂z
ˆz × H − κ2zΨEz1 − ΨEz2 . (3.12)
3.3 Implementation
The update FDTD-equations using the central diﬀerence method in time for (3.12)
is given by
Ψ
n+1/2
Hx2 = mxΨ
n−1/2
Hx2 + nx
∂
∂x
ˆx × En
Ψ
n+1/2
Hx1 = pxΨ
n−1/2
Hx1 + qx
∂
∂x
ˆx × En
− Ψ
n+1/2
Hx2
Hn+1/2
sx = Hn−1/2
sx +
∆t
µκ1xκ2x
∂
∂x
ˆx × En
− Ψ
n+1/2
Hx2 − κ2xΨ
n+1/2
Hx1
14

Ψ
n+1/2
Hy2 = myΨ
n−1/2
Hy2 + ny
∂
∂y
ˆy × En
Ψ
n+1/2
Hy1 = pyΨ
n−1/2
Hy1 + qy
∂
∂y
ˆy × En
− Ψ
n+1/2
Hy2
Hn+1/2
sy = Hn−1/2
sy +
∆t
µκ1yκ2y
∂
∂y
ˆy × En
− Ψ
n+1/2
Hy2 − κ2yΨ
n+1/2
Hy1
Ψ
n+1/2
Hz2 = mzΨ
n−1/2
Hz2 + nz
∂
∂z
ˆz × En
Ψ
n+1/2
Hz1 = pzΨ
n−1/2
Hz1 + qz
∂
∂z
ˆz × En
− Ψ
n+1/2
Hz2
Hn+1/2
sz = Hn−1/2
sz +
∆t
µκ1zκ2z
∂
∂z
ˆz × En
− Ψ
n+1/2
Hz2 − κ2zΨ
n+1/2
Hz1
Ψn+1
Ex2 = mxΨn
Ex2 + nx
∂
∂x
ˆx × Hn+1/2
Ψn+1
Ex1 = pxΨn
Ex1 + qx
∂
∂x
ˆx × Hn+1/2
− Ψn+1
Ex2
En+1
sx = En
sx +
∆t
κ1xκ2x
∂
∂x
ˆx × Hn+1/2
− Ψn+1
Ex2 − κ2xΨn+1
Ex1
Ψn+1
Ey2 = myΨn
Ey2 + ny
∂
∂y
ˆy × Hn+1/2
Ψn+1
Ey1 = pyΨn
Ey1 + qy
∂
∂y
ˆy × Hn+1/2
− Ψn+1
Ey2
En+1
sy = En
sy +
∆t
κ1yκ2y
∂
∂y
ˆy × Hn+1/2
− Ψn+1
Ey2 − κ2yΨn+1
Ey1
Ψn+1
Ez2 = mzΨn
Ez2 + nz
∂
∂z
ˆz × Hn+1/2
Ψn+1
Ez1 = pzΨn
Ez1 + qz
∂
∂z
ˆz × Hn+1/2
− Ψn+1
Ez2
En+1
sz = En
sz +
∆t
κ1zκ2z
∂
∂z
ˆz × Hn+1/2
− Ψn+1
Ez2 − κ2zΨn+1
Ez1 , (3.13)
where the spacial derivatives are calculated in the standard FDTD fashion, and the
constants m, n, p, and q can be calculated outside the main loop to save time. They are
given by
mi =
2 − (a2i + σ2i/κ2i)∆t
2 + (a2i + σ2i/κ2i)∆t
ni =
2σ2i∆t
κ2i[2 + (a2i + σ2i/κ2i)∆t]
15

pi =
2 − (a1i + σ1i/κ1i)∆t
2 + (a1i + σ1i/κ1i)∆t
qi =
2σ1i∆t
κ1iκ2i[2 + (a1i + σ1i/κ1i)∆t]
(3.14)
where i = x, y or z.
Even though the equations can be solved in the format above, the variables Hsx, Hsy,
Hsz, Esx, Esy, and Esz have no physical meaning. One has to split each one of them into
the two derivatives that will arise from each of the terms ∂
∂x
ˆx × H, ∂
∂y
ˆy × H, ∂
∂z
ˆz × H,
∂
∂x
ˆx × E, ∂
∂y
ˆy × E, and ∂
∂z
ˆz × E such that
−jωµHsxz =
1
sx
∂
∂x
Ey − jωµHsxz =
1
sx
∂
∂x
Ez − jωµHsyx =
1
sy
∂
∂y
Ez
−jωµHsyz =
1
sy
∂
∂y
Ex − jωµHszy =
1
sz
∂
∂z
Ex − jωµHszx =
1
sz
∂
∂z
Ey
jω Esxz =
1
sx
∂
∂x
Hy jω Esxy =
1
sx
∂
∂x
Hz jω Esyx =
1
sy
∂
∂y
Hz
jω Esyz =
1
sy
∂
∂y
Hx jω Eszy =
1
sz
∂
∂z
Hx jω Eszx =
1
sz
∂
∂z
Hy.
By setting Hx = Hszx + Hsyx, Hy = Hszy + Hsxy, Hz = Hsxz + Hsyz, Ex = Eszx + Esyx,
Ey = Eszy + Esxy, and Ez = Esxz + Esyz the field components are fully recovered.
3.4 Numerical Results
In this section we present the results for two different problems where the absorption
of propagating waves for all frequencies as well as evanescent waves is critical. In both
cases strong evanescent waves are present.
3.4.1 Finite sheet
The first problem consists of a 2D TE wave with an infinitely long, perfectly electric
conductor (PEC) sheet with a finite width. Figure 3.1 shows the computational domain
16

used in this simulation. The line source, infinitely long in the z-direction, is given by
Jy = −2[(t − t0)/tw]e−[(t−t0)/tw]2
, with tw = 26.53 ps and t0 = 4tw and we use ∆t = 0.5×
Courant number [21] with a cell size ∆x = 1 mm.
The problem consists of a 100-cell wide sheet surrounded by free space. We also use a
10-cell thick PML to truncate the domain. A y-polarized electric current is placed at the
center and the y-component of the electric field is measured at the end. We expect very
strong evanescent waves to appear at that point. The reference solution was obtained
by using a large 1200 × 1200 cells grid so that no field would be reflected back by the
artificial truncation during the time period of the simulation.
The error relative to the reference solution was computed as a function of time using
Error = 20 log10
|Ey(t) − Eref
y (t)|
|Eref
ymax|
(3.15)
where Ey(t) represents the field calculated with the PML at point A in Fig. 3.1, three
cells away from the PML, Eref
y (t) is the field obtained with the large domain 1200 × 1200
and Eref
ymax its maximum value over the entire simulation.
For this problem, we use the same values as in [12] for the regular PML. We use a
polynomial scaling for σ with m = 4 being its order and σmax = 0.7σopt, where
σopt =
m + 1
150π∆x
.
We also use a scaling for κ with the same value for m and κmax = 11. For the CFS-PML,
we use a = 0.05, σmax = 1.1σopt, and κmax = 1. For the higher-order PML, we set κ1 = 1,
a1 = 0, a2 = 0.09 + σ1 and
σ1 = σ1optξ7
σ2 = σ2optξ3
κ2 = 1 + κ2optξ3
,
17

where ξ is zero at the interface of the PML and 1 at the end and σ1opt = 0.175/(150π
∆x), σ2opt = 2.5/(150π∆x) and κ2opt = 7.0. With these values, the error in the electric
field measured at point A on the sheet is shown in Fig. 3.2, which clearly demonstrates
the advantage of the second-order PML over the regular and the CFS PMLs.
In order to obtain the best possible profile for the CFS PML for this problem we
performed a search for κ and σ, as shown in Fig. 3.3. It confirmed the improvement of
almost 20 dB over the regular PML stated in [12].
Based on the our experience with the second-order PML, the parameter σ2 is the most
important one to be adjusted. It is instructive to observe how the maximum reflection
error behaves as a function of σ1, σ2, and κ2. Figures 3.4 and 3.5 illustrate how the
maximum error changes for this problem.
Finally, we reduce the number of layers from 10 cells to 6 cells while keeping the same
distance from the scatterer. The higher-order PML outperforms the CFS PML by over
20 dB.
3.4.2 Finite plate
As for a 3D example, we simulate a plate immersed in a lossless free space. The finite
PEC plate has a dimension of 25 mm × 100 mm with a point source located at one corner
and the observation point at the opposite corner. Our reference solution was obtained in
a similar fashion as in [12], by placing the PML 75 cells away from the plate. We use the
second-order PML to truncate the domain for the reference solution since for the 2D case
it gave the best result. Figure 3.6 shows the result for this example, which again shows
that the second-order PML performs significantly better than both the regular and the
CFS PMLs.
For this problem, we made some small changes on the parameters both for the CFS
PML and the second-order PML. The best result for the CFS PML was obtained with
18

κmax = 7.0 rather than 1 and for the second-order PML we used σ1opt = 0.275/150π∆x
and σ2opt = 2.75/150π∆x. The other parameters remained the same.
An important advantage of the CFS-PML over the regular PML is its long-time decay
when a = 0 [15]. The higher-order PML retains this characteristic of the CFS-PML, as
can be seen both in Figs. 3.2 and 3.6. We simulated both problems for 150000 time steps
and the error remained around -100 dB level showing no linear increase in the ﬁeld.
3.5 Conclusion
In this Chapter we compared the performance of the second-order PML to the perfor-
mance of the regular PML and the CFS-PML for open-region problems. The second-order
PML outperforms the CFS-PML in both examples shown here by at least 20 dB for the
maximum reﬂection error. It also maintain the long time behavior of the CFS-PML, one
of its great advantage over the regular PML.
3.6 Figures
Figure 3.1 Computational domain. The cell size is ∆x = ∆y = 1 mm. The error
is measured at point A, three cells away from each PML interface. The line source is
located at the center.
19

0 500 1000 1500
−140
−120
−100
−80
−60
−40
−20
Time steps
Error(dB)
CFS PML
Regular PML
2nd−Order PML
Figure 3.2 Relative error for the three diﬀerent PMLs as a function of time step for the
2D case.
5 7.5 10 12.5 15 17.5
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
σ
max
κ
max
−60
−56
−56
−52
−52
−52
−52
Figure 3.3 Maximum error in dB for the CFS PML as a function of κ and σ.
20

2.5 5 7.5 10 12.5 15
0.25
0.5
0.75
1
1.25
1.5
1.75
2
σ
2
σ1
−86
−86
−82
−82
−82
−74
−74
−74
−74
−74
−60
−60
−60
−60
−60
−60
Figure 3.4 Maximum relative error in dB for the second-order PML as a function of σ1
and σ2. The other parameters are kept constant and their value are the optimum ones
stated in the text.
2.5 5 7.5 10 12.5 15
2.5
5
7.5
10
12.5
15
17.5
20
σ
2 max
κ2max
−86
−82
−82
−74
−74
−74
−74
−74
−74
−60
−60
−60
−60
−60
−60
Figure 3.5 Maximum relative error in dB for the second-order PML as a function of κ2
and σ2. The other parameters are kept constant and their value are the optimum ones
stated in the text.
21

400 600 800 1000 1200 1400 1600 1800 2000
−140
−130
−120
−110
−100
−90
−80
−70
−60
−50
−40
Time steps
Error(dB)
CFS PML
Regular PML
2nd−Order PML
Figure 3.6 Relative error for the three diﬀerent PMLs as a function of time step for the
3D case.
22

CHAPTER 4
APPLICATION TO WAVEGUIDE PROBLEMS
4.1 Introduction
We begin our analysis of waveguide problems with the simplest 2D case. An empty
parallel-plate waveguide will be simulated in Section 4.2. This example will show the
difference in the performance of different PMLs. Its is suitable for that purpose because
such waveguide can support both TEM and higher-order modes. Then we apply the
technique to different types of 3D waveguide problems in Section 4.3. An empty rec-
tangular waveguide and an inhomogeneous dielectric waveguide will be considered. In
both cases no TEM waves are supported, so the CFS-PML will have a good performance.
We finally consider a microstrip problem where the propagating spectrum includes both
low-frequency propagating waves, for which the CFS-PML does not perform well, and
higher-order modes for which the regular PML will have a poor performance. In all cases,
the reflection coefficient is calculated at the PML/air interface 10 cells away from the
source. A reference solution is obtained by simulating a waveguide long enough so no
reflection occurs during the simulation period.
4.2 Parallel-Plate Waveguide
In the past, several authors have tried to address the problem of absorbing evanescent
waves in a waveguide [9, 13, 24, 25]. In all cases, even though the absorption of those
23

waves indeed became better, the PMLs employed were completely ineffective at the cutoff
frequencies for the waveguide modes.
In this section we will optimize the performance of the regular, the CFS, and the
second-order PMLs for each different problem. The idea is to demonstrate the limitations
of each metric coefficient and their advantages. The parallel-plate waveguide is the best
choice for this study since we can excite the TEM mode individually, the TM1 mode
individually, or a combination of those two modes at the same time and compare the
performance of each metric coefficient.
Our first case of study is a simple parallel-plate waveguide excited with a TEM mode
using the same waveguide as in [9, 13, 25]. The waveguide is filled with air and has
PEC walls separated by 40 mm. Since the cutoff frequency for the TEM mode is 0Hz,
the propagating spectrum will include frequencies in the entire band. For that reason,
the regular PML will perform very well in the entire band, as can be seen in Fig. 4.1.
However, the CFS-PML eventually breaks down at low frequencies. The only way to
make the CFS-PML work for low-frequency propagating waves is to reduce the value of
a, which effectively reduces the CFS-PML to the regular PML. The performance of the
second-order PML is as good as that of the regular PML. The parameters used here for
the regular PML are κ = 1 and σ = 5ξ3
, where ξ is zero at the air/PML interface and 1
at the end. For the CFS-PML, we used the same profile with two different values for a.
Finally, for the second-order PML we used σ1 = 4.5ξ3
, σ2 = 2.9ξ3
, a1 = σ2, κ1 = κ2 = 1,
and a2 = 0.
The second case is the same parallel-plate waveguide but excited with the TM1 mode,
as was done in [9] and [13]. In this case, the regular PML is totally ineffective at the cutoff
frequency of the TM1 mode. It is possible to improve its performance for evanescent waves
at the low-frequency end of the spectrum by setting κ = 1. However, this compromises the
absorption of propagating waves at the high-frequency end. Fang and Wu [9] developed
24

a modified PML that can absorb evanescent waves without such a problem. However,
the absorption at cutoff is still problematic. Figure 4.2 shows the performance of the
regular PML for this problem with σ having the same profile as in the previous example.
As shown in [13], the CFS-PML is very effective for the absorption of the evanescent
waves below the cutoff frequency for this problem. Indeed, the performance of the CFS-
PML is better than the regular PML even if one takes into account the increase in
memory requirement from the regular PML to the CFS-PML. Although the simulations
in [13] showed large reflection at the cutoff, this problem can be alleviate by setting
κ = 1. Figure 4.3 shows the results of the CFS-PML using two different values for κmax.
As is the case with the regular PML, the increase in κ compromises the absorption of
high-frequency propagating waves. The profile used for the CFS-PML where κ = 1 is
σ = 35ξ3
, a = 0.1, and κ = 1 + 4ξ2
, with ξ zero at the air/PML interface and one at the
end. For κ = 1, we used a = 0.1 and σ = 7ξ3
.
The performance of the second-order PML is shown in Fig. 4.4, which is just slightly
better than the performance of the CFS-PML while both significantly outperform the
regular PML. It should be noted that the regular PML performs better at high frequencies
because we were mainly concerned about the maximum reflection coefficient in the entire
frequency band. Both the CFS-PML and the second-order PML can be made to better
absorb high-frequency propagating waves, but at the expense of a higher reflection around
the cutoff frequency. The profile for the second-order PML is σ1 = 12ξ2
, σ2 = 12ξ4
,
κ1 = κ2 = 1 + ξ2
, a1 = 0.1 + 0.65σ2, and a2 = 0.1.
The last case of study for the parallel-plate waveguide has a simultaneous excitation
of both the TEM and the TM1 modes. In this case, neither the regular PML nor the
CFS-PML are expected to have a good performance at low frequencies. The regular
PML will be limited by its poor absorption around the cutoff frequency and its poor
absorption of evanescent waves. The CFS-PML will be limited by its poor performance
25

for low-frequency propagating waves. Figure 4.5 shows the results of the regular PML
using different values for κmax, all having a quadratic profile. Next, we analyze the CFS-
PML performance. Figure 4.6 shows the results using of the CFS-PML with a constant
a = 0.05 and a linear decay for a = 0.05(1 − ξ), both with σ = 17ξ3
and κ = 1 + 4ξ2
. As
can be expected, the CFS-PML fails to absorb low-frequency waves, even when a has a
linear decay.
Finally, Fig. 4.7 shows the performance of the second-order PML as compared to
those of the regular and the CFS-PMLs. For the regular PML we have σ = 5ξ3
and
κ = 1. For the CFS-PML we have σ = 17ξ3
, a = 0.075, and κ = 1 + ξ2
. Finally, for the
second-order PML we have σ1 = 17ξ3
, κ1 = 1 + ξ2
, σ2 = 0.05ξ3
, a1 = 0.075 + σ2, κ2 = 1,
and a2 = 0. By doing so, we recover in the second-order PML the same behavior as the
CFS-PML for absorbing evanescent waves while achieving a much better absorption at
low frequencies.
4.3 Rectangular Waveguide
In the previous section we had different profiles for the regular, the CFS, and the
second-order PMLs, depending on the type of problem we were solving. The idea was to
optimize the three different metric coefficients for each problem. This approach, however,
is not recommended for practical purposes. It is always desirable to have a PML robust
enough to absorb waves in a general problem. Since the most general case was the one
with both the TEM and TM1 modes, we use that profiles for all the 3D problems shown
here. Hence, in all cases here we have, for the regular PML, σ = 5ξ3
and κ = 1. For the
CFS-PML, we have σ = 17ξ3
, a = 0.075, and κ = 1 + ξ2
. Finally, for the second-order
PML, we have σ1 = 17ξ3
, κ1 = 1 + ξ2
, σ2 = 0.05ξ3
, a1 = 0.075 + σ2, κ2 = 1, a2 = 0,
where ξ varies from 0 at the air/PML interface to 1 at the end.
26

4.3.1 Empty waveguide
Our first 3D problem is an empty rectangular waveguide having a cross section of
40 mm × 10 mm. We excite this waveguide with TE10 and TE20 modes. As stated in
[25] the higher-order evanescent waves attenuate faster than the lower-order ones so one
needs to set the best parameters for the first higher-order mode. But contrary to [25],
here we launch both TE10 and TE20 modes simultaneously to check the performance of
the PML when two modes are presented. The cutoff frequencies for the two modes are
3.75 and 7.5 GHz, respectively. For this problem, no TEM wave is supported. We expect
the result to be similar for the TM1 mode in the parallel-plate waveguide problem. The
CFS-PML is expected to perform better than the regular PML and the second-order
PML to be similar to the CFS-PML. This is indeed the case as shown in Fig. 4.8.
4.3.2 Inhomogeneous waveguide
The next two examples are made of a partially filled waveguide. In both cases the
upper half of the waveguide is filled with air and the lower half with a dielectric having
r = 4.
To simulate this problem without having to address the inhomogeneity inside the
PML, we used the duality principle [26]. By working with the PMC instead of PEC and
using a µr = 4, the metric coefficients become totally transparent to the medium and
we avoid the special treatment required in [21]. We computed the reflection error for
different values of σ to confirm the approach. As expected, the minimum error occurs
for the same values of σ as in the previous examples.
In the first example, we used a line source to excite multiple modes, as shown in
Fig. 4.9. The spectrum of the excited incident field is plotted in Fig. 4.10.
The performance of the regular, CFS, and second-order PMLs is exhibited in Fig. 4.11.
We can clearly see that the regular PML fails to absorb the waves around the cutoff
27

frequencies. By using the same CFS-PML as in the example for the empty waveguide
the reflection error drops by 40 dB throughout the frequency band with its maximum
value being around −30 dB compared to 0 dB for the regular PML. The second-order
PML has a similar performance to that of the CFS-PML.
In the last example, we simulate a microstrip line inside the waveguide. In this case the
structure supports a quasi-TEM mode. Hence, low-frequency propagating waves can be
excited. Similarly to the previous problem, we use a line source located in the dielectric
to excite both the quasi-TEM and waveguide modes, as shown in Fig. 4.12. Figure
4.13 shows the reflection error for this problem. Similar to the 2D problem with low-
frequency evanescent and propagating waves, the regular and the CFS-PMLs have a poor
performance at low frequencies. However, the second-order PML performs significantly
better.
4.4 Conclusion
In this chapter we studied the performance of three different types of metric coeffi-
cients for the PML. Since here it is possible to excite the propagating modes indepen-
dently from each other, the advantages of each metric coefficient is more clear. The
regular PML is the best choice when only propagating waves are present. The CFS-PML
is the best choice if strong evanescent waves are present but low-frequency propagating
waves are absent. In more general cases where both low-frequency propagating waves
and strong evanescent waves are present, the second-order PML is the best choice. Both
the regular and the CFS-PMLs are special cases of the second-order PML; hence, an
implementation of the second-order PML includes both of them at the expense of some
memory and computation time increase.
28

4.5 Figures
2 4 6 8 10 12 14 16 18 20
−110
−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Frequency (GHz)
ReflectionError(dB)
CFS−PML (a=0.1)
CFS−PML (a=0.01)
Regular PML
Second−Order PML
Figure 4.1 Reﬂection error as a function of frequency for the TEM mode in a parallel-
plate waveguide using a 10-layer regular PML, CFS-PML and second-order PML.
29

2 4 6 8 10 12 14 16 18 20
−110
−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Frequency (GHz)
ReflectionError(dB)
Regular PML (κ=1)
Regular PML (κ≠1)
Figure 4.2 Reflection error for a 10-layer regular PML as a function of frequency for the
TM1 mode in a parallel-plate waveguide for κmax = 1 (solid line) and κmax = 5 (dotted
line).
2 4 6 8 10 12 14 16 18 20
−110
−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Frequency (GHz)
ReflectionError(dB)
CFS−PML (κ=1 a=0.1)
CFS−PML (κ≠1 a=0.1)
Figure 4.3 Reflection error for a 10 layer CFS-PML as a function of frequency for the
TM1 mode in a parallel-plate waveguide for different values of κ.
30

2 4 6 8 10 12 14 16 18 20
−110
−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Frequency (GHz)
ReflectionError(dB)
Regular PML
CFS−PML
Second−Order PML
Figure 4.4 Reflection error for a 10-layer regular, CFS and second-order PMLs as a
function of frequency for the TM1 mode in a parallel-plate waveguide.
2 4 6 8 10 12 14 16 18 20
−110
−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Frequency (GHz)
ReflectionError(dB)
Regular PML (κ=1)
Regular PML (κ
max
=5)
Regular PML (κmax
=10)
Figure 4.5 Reflection error for a 10-layer regular PML as a function of frequency for
the TEM and TM1 modes in a parallel-plate waveguide for different values of κmax.
31

2 4 6 8 10 12 14 16 18 20
−110
−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Frequency (GHz)
ReflectionError(dB)
CFS−PML (a=0.05)
CFS−PML (a
max
=0.05)
Figure 4.6 Reﬂection error for 10-layer CFS-PML as a function of frequency for the
TEM and TM1 modes in a parallel-plate waveguide for constant a (solid line) and a
linear decay in a (dashed line).
2 4 6 8 10 12 14 16 18 20
−110
−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Frequency (GHz)
ReflectionError(dB)
Regular PML
CFS−PML
Second−Order
function of frequency for the TEM and TM1 modes in a parallel-plate waveguide.
32

2 4 6 8 10 12 14 16 18 20
−110
−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Frequency (GHz)
ReflectionError(dB)
Regular PML
CFS−PML
Second−Order PML
function of frequency for the TE10 and TE20 modes in a rectangular waveguide.
10 mm
40 mm
Line Source
r=4
Figure 4.9 Computational domain for the inhomogeneous waveguide problem excited
by a line source.
33

2 4 6 8 10 12 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (GHz)
AmplitudeoftheFrequencyComponents
Figure 4.10 Spectrum of the incident ﬁeld 10 cells away from the source for the inho-
mogeneous waveguide problem.
2 4 6 8 10 12 14 16 18 20
−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Frequency (GHz)
ReflectionError(dB)
Regular PML
CFS−PML
Second−Order PML
function of frequency for the inhomogeneous waveguide problem excited by a line source.
34

10 mm
40 mm
r=4
Figure 4.12 Computational domain for the microstrip problem.
2 4 6 8 10 12 14 16 18 20
−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Frequency (GHz)
ReflectionError(dB)
Regular PML
CFS−PML
Second−Order PML
function of frequency for the microstrip problem.
35

CHAPTER 5
APPLICATION TO PERIODIC PROBLEMS
5.1 Introduction
Many structures of electromagnetic interest have a periodicity in one or two dimen-
sions. The frequency selective surfaces (FSS)[27], the photonic bandgap (PBG) struc-
ture [28, 29], the antenna array, [30] and more recently, the left-handed metamaterials
[31, 32, 33] are some examples. All those problems have in common the periodicity in
one or two directions, that we will assume to be y− and z-directions. Since for periodic
problems the details of the structure are not as important as the periodicity or the general
shape of the structure, the FDTD method is a powerful choice to solve electromagnetic
problems involving periodicity.
This chapter starts with the numerical formulation of the the FDTD method for
analysis of periodic structures in Section 5.2. Section 5.3 address some details of the
implementation procedure. Numerical examples, both for 2D and 3D cases, are shown
in Section 5.4. Finally, Section 5.5 concludes the chapter.
5.2 Numerical Formulation
Our formulation for the FDTD analysis of periodic structures is equivalent to the
method proposed in [34] and [35]. Here we use the split-ﬁeld formulation to account for
36

both PML and periodicity so that the effects of attenuation in the PML region and the
delay in the grid due to oblique incidence are more obvious.
5.2.1 Formulation for oblique incidence- 2D Case
Starting from the modified Maxwell’s equation for the TM case and using the split-
field technique, we have
jωµHy =
1
sx
∂
∂x
(Esxz + Eszy) (5.1)
jωµHx = −
∂
∂y
(Esxz + Esyz) (5.2)
jω Esxz =
1
sx
∂
∂x
Hy (5.3)
jω Esyz = −
∂
∂y
Hx (5.4)
where Ez = Eszx + EsyzFor simplicity, in the 2D case we will assume µr = 1. By defining
auxiliary functions P = Eejk sin θy
/η0 and Q = Hejk sin θy
, where θ is the propagation
direction, the equations in (5.1)-(5.4) can be rewritten as
jωQy =
c0
sx
∂
∂x
(Psxz + Psyz) (5.5)
jωQx = −c0
∂
∂y
(Psxz + Psyz) + jω sin θ(Psxz + Psyz) (5.6)
jω rPsxz =
c0
sx
∂
∂x
Qy (5.7)
jω rPsyz = −c0
∂
∂y
Qx + jω sin θQx. (5.8)
The equations above cannot be implemented, yet since the term jω appears in both
sides of the equations for Qx and Psyz. The other two, Qy and Psxz, responsible for the
field attenuation in the PML, are not affected by the delay in the grid. They can be
calculated using the technique described in Chapter 3.
37

To facilitate the implementation of (5.6) and (5.8), we split the fields there into two
components such that
jωQa
x = −c0
∂
∂y
(Psxz + Pa
syz + Pb
syz) (5.9)
jωQb
x = jω sin θ(Psxz + Pa
syz + Pb
syz) (5.10)
jω rPa
syz = −c0
∂
∂y
(Qa
x + Qb
x) (5.11)
jω rPb
syz = jω sin θ(Qa
x + Qb
x). (5.12)
Equations (5.10) and (5.12) can further be written as
Pb
syz =
1
( r − sin2
θ)
[sin θQa
x + sin2
θ(Psxz + Pa
syz)] (5.13)
Qb
x = sin θ(Psxz + Pa
syz + Pb
syz). (5.14)
Equations (5.9), (5.11), (5.13), and (5.14) can readily be implemented using a FDTD
scheme.
To implement the PML we just need to work with Psxz and Qy since they are the only
variables directly affected by it. The others are already in their final form. Substituting
the metric coefficient (2.3) in the equation for Psxz and Qy and proceeding the same way
was done in Chapter 3, we have
jω rΨP2 + a2x +
σ2x
κ2x
ΨP2 =
σ2xc0
κ2x
∂
∂x
Qy
jω rΨP1 + a1x +
σ1x
κ1x
ΨP1 =
σ1x
κ1xκ2x
c0
∂
∂x
Qy − ΨP2
jω rκ1xκ2xPsxz = c0
∂
∂x
Qy − κ2xΨP1 − ΨP2 (5.15)
and
jω rΨQ2 + a2x +
σ2x
κ2x
ΨQ2 =
σ2xc0
κ2x
∂
∂x
Pz
jω rΨQ1 + a1x +
σ1x
κ1x
ΨQ1 =
σ1x
κ1xκ2x
c0
∂
∂x
Pz − ΨQ2
jω rκ1xκ2xQy = − c0
∂
∂x
Pz − κ2xΨQ1 − ΨQ2 . (5.16)
38

The ﬁnal set of equations for the 2D-case is given by
jω rΨQ2 + a2x +
σ2x
κ2x
ΨQ2 =
σ2xc0
κ2x
∂
∂x
(Psxz + Pa
syz + Pb
syz)
jω rΨQ1 + a1x +
σ1x
κ1x
ΨQ1 =
σ1x
κ1xκ2x
c0
∂
∂x
(Psxz + Pa
syz + Pb
syz) − ΨQ2
jω rκ1xκ2xQy = − c0
∂
∂x
(Psxz + Pa
syz + Pb
syz) − κ2xΨQ1 − ΨQ2
jωQa
x = −c0
∂
∂y
(Psxz + Pa
syz + Pb
syz)
jω rΨP2 + a2x +
σ2x
κ2x
ΨP2 =
σ2xc0
κ2x
∂
∂x
Qy
jω rΨP1 + a1x +
σ1x
κ1x
ΨP1 =
σ1x
κ1xκ2x
c0
∂
∂x
Qy − ΨP2
∂
∂x
Qy − κ2xΨP1 − ΨP2
jω rPa
syz = −c0
∂
∂y
(Qa
x + Qb
x)
Pb
syz =
1
( r − sin2
θ)
[sin θQa
x + sin2
θ(Psxz + Pa
syz)]
Qb
x = sin θ(Psxz + Pa
syz + Pb
syz). (5.17)
5.2.2 Formulation for oblique incidence- 3D case
Starting from the modiﬁed Maxwell’s equations
−jωµHx =
∂
∂y
Ez −
∂
∂z
Ey
−jωµHz =
1
sx
∂
∂x
Ey −
∂
∂y
Ex
−jωµHy =
∂
∂z
Ex −
1
sx
∂
∂x
Ez
jω Ex =
∂
∂y
Hz −
∂
∂z
Hy
jω Ez =
∂
∂y
Hx −
1
sx
∂
∂x
Hy
jω Ey =
1
sx
∂
∂x
Hz −
∂
∂z
Hx,
39

the first step is to separate the derivative in the x-direction, where the PML will act, to
the derivatives in the y- and z-directions, where the delay in the grid will act. That is
the main difference from my formulation to [34]. In my formulation, the effects of the
delay in the grid and of the PML are more clear, since the variables are separated [36]:
−jωµHxz =
1
sx
∂
∂x
Ey
−jωµHyz = −
∂
∂y
Ex
−jωµHxy = −
1
sx
∂
∂x
Ez
−jωµHzy =
∂
∂z
Ex
jω Exz =
1
sx
∂
∂x
Hy
jω Eyz = −
∂
∂y
Hx
jω Exy =
1
sx
∂
∂x
Hz
jω Ezy =
∂
∂z
Hx,
(5.18)
where the field components are given by Hz = Hxz+Hyz, Hy = Hxy+Hzy, Ez = Exz+Eyz,
and Ey = Exy + Ezy, with Hx and Ex remaining the same.
The equations for Hxy, Hxz, Exy, and Exz can be derived by the same technique as
the one used in Chapter 3. They will have the same format as the equations for Qy
and Pszx in the 2D-case once the P and Q are introduced, with new variables Ψ being
introduced for each component.
The remaining variables, however, cannot be directly implemented since, due to the
periodic boundary condition, a delay in the grid is required in y- and z-directions. In
40

order to implement them, we introduce two new variables P = Eejk(ˆkyy+ˆkzz)
/η0 and
Q = Hejk(ˆkyy+ˆkzz)
, where ˆkz = sin(θ) cos(φ) and ˆky = sin(θ) sin(φ), such that
−jωµrQx = c0
∂
∂y
Pz −
∂
∂z
Py + jωˆkyPz − jωˆkzPy
−jωµrQxz =
c0
sx
∂
∂x
Py
−jωµrQyz = −c0
∂
∂y
Px − jωˆkyPx
−jωµrQxy = −
c0
sx
∂
∂x
Pz +
∂
∂z
Px
−jωµrQzy = c0
∂
∂z
Px + jωˆkzPx
jω rPx = c0
∂
∂y
Qz −
∂
∂z
Qy − jωˆkyQz + jωˆkzQy
jω rPxz =
c0
sx
∂
∂x
Qy
jω rPyz = −c0
∂
∂y
Qx − jωˆkyQx
jω rPxy =
c0
sx
∂
∂x
Qz
jω rPzy = c0
∂
∂z
Qx + jωˆkzQx. (5.19)
Next, we split each of the variables Px, Pzy, Pyz, Qx, Qzy, and Qyz into two components,
a and b, such that
−jωµrQa
x = c0
∂
∂y
Pz −
∂
∂z
Py
−jωµrQa
yz = −c0
∂
∂y
Px
−jωµrQa
zy = c0
∂
∂z
Px
41

jω rPa
x = c0
∂
∂y
Qz −
∂
∂z
Qy
jω rPa
yz = −c0
∂
∂y
Qx
jω rPa
zy = c0
∂
∂z
Qx
−µrQb
x = ˆkyPz − ˆkzPy
rPb
x = ˆkyQz − ˆkzQy
µrQb
yz = ˆkyPx
−µrQb
zy = ˆkzPx
rPb
yz = −ˆkyQx
rPb
zy = ˆkzQx.
(5.20)
The variables Qb
x and Pb
x cannot be implemented yet since they depend on Pb
yz and Pb
zy
and on Qb
yz and Qb
zy, respectively, and they are not yet available. After some algebraic
manipulation, they can be rewritten as
Qb
x =
r
rµr − ˆk2
z − ˆk2
y
[ˆky(Pxz + Pa
yz +
ˆky
r
Qa
x) − ˆkz(Pxy + Pa
zy +
ˆkz
r
Qa
x)]
Pb
x =
µr
rµr − ˆk2
z − ˆk2
y
[−ˆky(Qxz + Qa
yz +
ˆky
µr
Pa
x ) + ˆkz(Qxy + Qa
zy +
ˆkz
µr
Pa
x )].
(5.21)
As in the 2D-case, the final sequence of update equation should start with the components
directly affected by the PML, then the a components and finally the b components. The
42

ﬁnal set of equations is given by
jω rΨQxy2 + a2x +
σ2x
κ2x
ΨQxy2 =
σ2xc0
κ2x
∂
∂x
(Psxz + Pa
syz + Pb
syz)
jω rΨQxy1 + a1x +
σ1x
κ1x
ΨQxy1 =
σ1x
κ1xκ2x
c0
∂
∂x
(Psxz + Pa
syz + Pb
syz) − ΨQxy2
jω rκ1xκ2xQxy = − c0
∂
∂x
(Psxz + Pa
syz + Pb
syz) − κ2xΨQxy1 − ΨQxy2
jω rΨQxz2 + a2x +
σ2x
κ2x
ΨQxz2 =
σ2xc0
κ2x
∂
∂x
(Psxy + Pa
szy + Pb
szy)
jω rΨQxz1 + a1x +
σ1x
κ1x
ΨQxz1 =
σ1x
κ1xκ2x
c0
∂
∂x
(Psxy + Pa
szy + Pb
szy) − ΨQxy2
jωµrκ1xκ2xQxz = c0
∂
∂x
(Psxy + Pa
szy + Pb
szy) − κ2xΨQxy1 − ΨQxy2
−jωµrQa
x = c0
∂
∂y
Pz −
∂
∂z
Py
−jωµrQa
yz = −c0
∂
∂y
Px
−jωµrQa
zy = c0
∂
∂z
Px
jω rΨPxy2 + a2x +
σ2x
κ2x
ΨPxy2 =
σ2xc0
κ2x
∂
∂x
(Qsxz + Qa
syz + Qb
syz)
jω rΨPxy1 + a1x +
σ1x
κ1x
ΨPxy1 =
σ1x
κ1xκ2x
c0
∂
∂x
(Qsxz + Qa
syz + Qb
syz) − ΨPxy2
jω rκ1xκ2xPsxy = c0
∂
∂x
(Qsxz + Qa
syz + Qb
syz) − κ2xΨPxy1 − ΨPxy2
jω rΨPxz2 + a2x +
σ2x
κ2x
ΨPxz2 =
σ2xc0
κ2x
∂
∂x
(Qsxy + Qa
szy + Qb
szy)
jω rΨPxz1 + a1x +
σ1x
κ1x
ΨPxz1 =
σ1x
κ1xκ2x
c0
∂
∂x
(Qsxy + Qa
szy + Qb
szy) − ΨPxz2
∂
∂x
(Qsxy + Qa
szy + Qb
szy) − κ2xΨPxz1 − ΨPxz2
jω rPa
x = c0
∂
∂y
Qz −
∂
∂z
Qy
jω rPa
yz = −c0
∂
∂y
Qx
jω rPa
zy = c0
∂
∂z
Qx
43

Qb
x =
r
rµr − ˆk2
z − ˆk2
y
[ˆky(Pxz + Pa
yz +
ˆky
r
Qa
x) −
ˆkz(Pxy + Pa
zy +
ˆkz
r
Qa
x)]
Pb
x =
µr
µr r − ˆk2
z − ˆk2
y
[−ˆky(Qxz + Qa
yz +
ˆky
µr
Pa
x ) +
ˆkz(Qxy + Qa
zy +
ˆkz
µr
Pa
x )]
µrQb
yz = ˆkyPx
−µrQb
zy = ˆkzPx
rPb
yz = −ˆkyQx
rPb
zy = ˆkzQx. (5.22)
5.3 Implementation
In order to implement (5.22), a dual grid in time has to be used, since the b components
will not have a time derivative and will require only the ﬁelds at the same time, whereas
the other components require their previous values as well. Another important point is
that the equations for Pb
x, Pb
zy, and Pb
yz involve the values of Qz and Qx. Since in the
regular FDTD simulation the E and H ﬁelds are calculated half cell away from each
other, and hence the same holds true for P and Q, an averaging in space is required.
Similarly for Qb
x, Qb
zy, and Qb
yz. Using the same constants as in (3.14) and calculating
the spacial derivative in the standard FDTD fashion, we have
ΨQxy2|n+1/2
= mxΨQxy2|n−1/2
+ nxc0
∂
∂x
(Psxz|n
+ Pa
syz|n
+ Pb
syz|n
)
ΨQxy1|n+1/2
= pxΨQxy1|n−1/2
+ qx c0
∂
∂x
(Psxz|n
+ Pa
syz|n
+ Pb
syz|n
) − ΨQxy2|n+1/2
Qxy|n+1/2
= Qxy|n−1/2
+
∆t
µrκ1xκ2x
c0
∂
∂x
(Psxz|n
+ Pa
syz|n
+ Pb
syz|n
) −
κ2xΨQxy1|n+1/2
− ΨQxy2|n+1/2
44

µrQb
x|n+1/2
=
r
µr r − ˆk2
z − ˆk2
y
[ˆky(Pxz|n+1/2
+ Pa
yz|n+1/2
+
ˆky
r
Qa
x)|n+1/2
−
ˆkz(Pxy|n+1/2
+ Pa
zy|n+1/2
+
ˆkz
r
Qa
x)|n+1/2
]
rPb
x|n+1/2
=
µr
µr r − ˆk2
z − ˆk2
y
[−ˆky(Qxz|n+1/2
+ Qa
yz|n+1/2
+
ˆky
µr
Pa
x ) +
ˆkz(Qxy|n+1/2
+ Qa
zy|n+1/2
+
ˆkz
µr
Pa
x )]
µrQb
yz|n+1/2
= ˆkyPx|n+1/2
−µrQb
zy|n+1/2
= ˆkzPx|n+1/2
rPb
yz|n+1/2
= −ˆkyQx|n+1/2
rPb
zy|n+1/2
= ˆkzQx|n+1/2
. (5.23)
5.4 Numerical Examples
5.4.1 Two-dimensional case
In this subsection, we simulate a periodic structure in a frequency range that includes
the ﬁrst higher-order Floquet mode [37, 38]. The reason is that around the frequency
where this mode starts to form and propagate there are strong evanescent waves [14].
The CFS PML has proved to be able to absorb those waves better than the regular PML
[14]. Unfortunately, if the problem requires absorption of both high-frequency evanescent
waves and low-frequency propagating waves, the CFS PML might not be the best choice
due to its poor absorption of low-frequency propagating waves. Since the regular PML
has a poor performance for evanescent waves, the use of a higher-order PML is again the
best choice.
The problem consists of a slab of two diﬀerent dielectrics periodically repeated in the
y-direction. One dielectric has r = 1.44 and the second one has r = 2.56. The structure
is illuminated by a plane wave 45◦
from the normal. The structure is 2.0 cm wide and
46

each dielectric is 0.5 cm thick. Figure 5.1 shows the computational domain. For this
case, we use the same differentiated Gaussian as the source.
Since the plane wave will be hitting the PML at 45◦
we expect that a different profile
for σ1, σ2, and κ2 would give the best results. For the regular PML, the best is a quadratic
profile with σmax = 11.1 and κ = 1 + 11ξ2
. For the CFS PML, we found that the best
values are σmax = 13.8, κ = 1 + 11ξ2
, and a = 0.5. Finally, for the second-order PML,
we used a2 = 0.6 and
σ1 =
0.175
150π∆x
ξ1.5
σ2 =
2.5
150π∆x
ξ1.5
κ2 = 1 + 7ξ3
.
We should point out that the values we use might not be the best ones since no systematic
optimization procedure was carried out.
The reference solution for this problem was obtained using a modal analysis method
[39]. The first result shows the performance of the regular PML in Fig. 5.2. Around the
frequency (17.6 GHz) where the first Floquet mode starts to form and propagate [14]
strong evanescent waves are present and the regular PML has a poor performance, as the
predicted reflection coefficient (1.2) is much larger than the true solution, which is 1.0.
We have used CFS-PML to overcome this limitation [14]. Figure 5.3 shows its result.
As expected, the CFS-PML works better than the regular PML for the evanescent waves
around 17.6 GHz but now has a poor absorption at low frequency.
The only way to overcome this limitation is by reducing a. By doing so, the absorp-
tion of propagating waves at low frequencies improves but the absorption of evanescent
waves is compromised. In the limit when a → 0, the regular PML is recovered with a
good absorption of low-frequency propagating waves but with a poor performance for
frequencies around the first Floquet mode.
47

The second-order PML works at both ends. Figure 5.4 shows its performance. The
error in the high-frequency part occurs for all the truncation techniques and is due to
the dispersion error in the FDTD simulation. For the 45◦
incidence, its maximum value
is 6o
/λ [35].
5.4.2 Three-dimensional case
In this subsection we simulate a frequency selective surface to compare the FDTD
method with the finite-element boundary-integral (FE-BI) method [40] and the method
of moments (MoM) [41] for periodic structures. The problem consists of a PEC patch
immersed in a dielectric with permittivity r = 2. Figure 5.5 shows the geometry of this
problem.
We compare the FDTD method with FE-BI and MoM for normal incidence case
(θ = 0). Figure 5.6 shows the result obtained by the three methods. Clearly, the FDTD
method agrees very well with both MoM and FE-BI.
In the next example, we simulate the FSS shown in Fig. 5.7. Notice that for the
frequency of 12 GHz, the unit cell for the metal grid is λ × λ, while it is slightly large in
length. On the other hand, it requires a very fine discretization since the region where
r = 3 is only λ/50 thick. That makes this problem not suitable for FE-BI simulation
due to the memory requirement. Since the memory requirement for FDTD simulation
is much smaller and a regular grid is enough to simulate the structure, it is strongly
recommended for this type of problem.
In order to compare our simulation to the FE-BI technique [40], we first simulate the
geometry shown in Fig. 5.7 without one of the metal strips. By doing so, the computa-
tional domain can be reduced in such a way that it is possible to be simulated by the
FE-BI code. We launch a plane wave with the electric field polarized along the metal
48

strip and compare the results for both vertical and horizontal polarizations. The results
are shown in Figs. 5.8-5.13.
Finally, we simulate the full structure shown in Fig. 5.7. For the horizontal polar-
ization, the result is very close to the result without the metal strip, as can be seen in
Fig. 5.14. This should be compared with the result from Fig. 5.11. Clearly, they are very
close, which means that the strip perpendicular to the electric field does not affect this
polarization.
The vertical polarization, on the other hand, does not show the same agreement. The
extra metal strip seems to have an effect on this case. This can be seen in Fig. 5.15.
This result should be compared to the one in Fig. 5.9. Clearly, the full structure have a
good agreement with the single metal strip case for frequencies beyond 10 GHz, but have
different behavior at lower frequencies. The explanation is that for vertical polarization,
the perpendicular metal strip excites different Floquet modes at lower frequencies that
would not be present with a single metal strip and that are not excited by the horizontal
polarization.
5.5 Conclusion
In this chapter we presented the formulation for the FDTD method for periodic
structures. The performance of the second-order PML, the CFS-PML and the regular
PML were shown. The idea of the second-order PML came when I was solving this
problem, since the CFS-PML and the regular PML would not perform well in different
frequency ranges for different reasons. A metric coefficient that would mimic both of
them would solve the problem. For the periodic problem, however, one cannot separate
the fundamental mode from the higher-order Floquet modes since the excitation is a
plane wave. Hence, contrary to the waveguide problems where for specific problems the
49

performance of the regular PML or the CFS-PML could match the performance of the
second-order PML, here the second-order PML is strongly recommended. The second-
order PML was formulated and implemented for 3D periodic structures.
5.6 Figures
Figure 5.1 Computational domain. The cell size is ∆x = ∆y = 0.5 mm.
50

2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Frequency (GHz)
SpecularReflectionCoefficient
Regular PML
Modal Analysis
Figure 5.2 Results for a modulated periodic slab. The regular PML has a poor absorp-
tion around the frequency of the ﬁrst Floquet mode (17.6 GHz).
2 4 6 8 10 12 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (GHz)
CFS PML
Modal Analysis
Figure 5.3 Results for a modulated periodic slab. The CFS-PML has a poor absorption
for low-frequency propagating waves.
51

2 4 6 8 10 12 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (GHz)
2nd−Order PML
Modal Analysis
Figure 5.4 Results for a modulated periodic slab. The second-order PML incorporates
the advantages of both regular and CFS PMLs.
10 mm
10 mm
2.5 mm
5 mm2 mm
k
θ
E
Side View Top View
Figure 5.5 Geometry for the ﬁrst FSS. The metal plate is immersed in a dielectric with
permittivity r = 2 and the magnetic ﬁeld is polarized along the smaller side of the
rectangular PEC patch.
52

8 10 12 14 16 18 20 22 24
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (GHz)
MoM
FE−BI
FDTD
Figure 5.6 Specular reﬂection coeﬃcient for the geometry shown in Fig. 5.5 calculated
using FE-BI method and MoM.
2.5 cm
2.5 cm
2.5 cm, ε =1.01
0.05 cm, ε =3.0
0.05 cm, ε =3.0
0.05 cm, ε =3.0
0.05 cm, ε =3.0
Front view
Incident plane wave
Top view
Figure 5.7 Geometry for the second FSS. Two metal grids are immersed in a dielectric
with permittivity r = 3 and separated by a dielectric with r = 1.01.
53

2 4 6 8 10 12 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (GHz)
FDTD
FE−BI
Figure 5.8 Results for the vertical polarization at 20◦
techniques for the geometry shown in Fig. 5.7 without one of the metal strips.
2 4 6 8 10 12 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (GHz)
FDTD
FE−BI
54

2 4 6 8 10 12 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (GHz)
FDTD
FE−BI
2 4 6 8 10 12 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (GHz)
FDTD
FE−BI
Figure 5.11 Results for the horizontal polarization at 20◦
incidence for FDTD and
FE-BI techniques for the geometry shown in Fig. 5.7 without one of the metal strips.
55

2 4 6 8 10 12 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (GHz)
FDTD
FE−BI
2 4 6 8 10 12 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (GHz)
FDTD
FE−BI
56

2 4 6 8 10 12 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (GHz)
Figure 5.14 Result for the horizontal polarization at 20◦
geometry shown in Fig. 5.7.
2 4 6 8 10 12 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (GHz)
Figure 5.15 Result for the vertical polarization at 40◦
geometry shown in Fig. 5.7.
57

CHAPTER 6
APPLICATION TO ARBITRARY MATERIALS
6.1 Introduction
Left-handed metamaterials (LHM) form a group of man-made materials that posses
non-naturally occurring behavior. Their main characteristic is having both permeability
and permittivity negative (µ < 0, < 0). Remarkably, the direction of wave propagation
k and the direction of the Poynting vector S are antiparallel in this case. One of the
main limitation of the PML approach to truncate a domain is its dependence on the
material permittivity ( ), since it appears on the metric coefficients si. For different
types of materials, a different has to be used. For dispersive and nonlinear materials,
that limitation can lead to a cumbersome formulation for the PML truncation. In this
chapter we describe a very simple way of using exactly the same higher-order PML
developed previously, as long as the material has constant either its permittivity or its
permeability (µ).
6.2 Analysis of Left-Handed Metamaterials
The theoretical behavior of wave propagation was first studied by Veselago [42]. The
approach is quite simple. Starting from Maxwell’s equations
× E = −jωµH
× H = jω E
58

one can easily conclude that
k × E = ωµH
k × H = −ω E (6.1)
assuming wave propagation in the form e−jk·r
, where k is the propagation direction.
From (6.1) we can see that the triad formed by E, H, and k can have different behavior
depending on the sign of and µ. If > 0 and µ > 0, then the direction of S = 1
2
E × H
and k would be the same and hence the phase velocity and Poynting vector would point
in the same direction. On the other hand, if < 0 and µ < 0, they would have opposite
sign. Such a medium would, among other things, have a negative index of refraction,
accordingly to Veselago.
Thirty years after Veselago’s work, Pendry et al. suggested separately two types of
artificial materials with < 0 [43] and µ < 0 [44]. Even though Pendry’s idea was,
at first, to use the new proposed materials for nonlinear phenomena, Shelby et al. [33]
used his results to obtain what Veselago had predicted. They claimed to have obtained
experimentally the negative index of refraction. This result was doubted by Valanju et
al. [45] who stated that the angular intensity profile interpretation in [33] is a near-field
effect. Valanju et al. also doubted, together with several other authors [46], the perfect
lens suggested by Pendry [32], where he stated that perfect lens could be obtained using
a metamaterial slab.
Ziolkowski and Heyman [47] has enlightened the discussion by numerical simulation
means. He chose FDTD to solve Maxwell’s equations directly, without any assumption
on the sign of the refractive index or direction of wave propagation inside the LHM
slab. Our goal is to increase the analysis capability by adding the PML formulation
to Ziolkowski’s approach, reducing the requirement for the computational domain, and
simulating 3D problems enabling the analysis of wave propagation in space.
59

6.3 Formulation for Left-Handed Metamaterials
The formulation used to simulate wave propagation inside a LHM followed the one
proposed by Ziolkowski et al. [47, 48]. There, the authors developed the idea of the
plasma medium used to simulate the negative µ and . Note that simply imposing µ and
negative inside the slab would produce an unstable simulation, since the ﬁeld at the
interface, for a matched case, would blow up. The plasma medium approach is hence
more general and can be applied to both matched and unmatched interfaces.
To show how the instability would show up, we take Maxwell’s equations for the 2D
TM case, for example,
∂Ez
∂t
=
∂Hy
∂x
−
∂Hx
∂y
− Jz
µ
∂Hx
∂t
= −
∂Ez
∂y
µ
∂Hy
∂t
=
∂Ez
∂x
(6.2)
one can easily see that if Hx is at the interface, for example, to update its value one
would have to take the average of µ from each side of the interface, as shown below:
Hl+1
x(i,j+1) = Hl
x(i,j+1) −
∆t
∆y × µav
[E
l+1/2
z(i,j+1) − E
l+1/2
z(i,j) ]
where the superscript l refers to the time step, and the subscripts i and j refer to the x and
y directions, respectively. For the case with a matched interface (µ1 = −µ2, 1 = − 2),
the average of the values would yield to µav = 0 which would result in an unstable
situation. The same problem occurs with Ez and Hy at the interface.
To avoid such instability, we set the negative values of µ and indirectly, using the
plasma model [47]. First, let µ and be modeled by the expressions
= o 1 −
ω2
p
ω2
µ = µo 1 −
ω2
p
ω2
(6.3)
60

with ωp being the material’s resonant frequency. Note that these expressions are similar
to the ones previously used to characterize LHM [32, 33, 43]. If we want to simulate
r = µr = −1, for example, we simply choose ωp =
√
2ω.
Substituting those expressions into Maxwell’s equations in the frequency domain, we
obtain
µojωHx + Kx =
∂Ey
∂z
−
∂Ez
∂y
µojωHy + Ky =
∂Ez
∂x
−
∂Ex
∂z
µojωHz + Kz =
∂Ex
∂y
−
∂Ey
∂x
ojωEx + Jx =
∂Hz
∂y
−
∂Hy
∂z
ojωEy + Jy =
∂Hx
∂z
−
∂Hz
∂x
ojωEz + Jz =
∂Hy
∂x
−
∂Hx
∂y
with
Kx = −jµo
ω2
p
ω
Hx
Ky = −jµo
ω2
p
ω
Hy
Kz = −jµo
ω2
p
ω
Hz
Jx = −j o
ω2
p
ω
Ex
Jy = −j o
ω2
p
ω
Ey
Jz = −j o
ω2
p
ω
Ez
61

In the time domain, we finally have
µo
∂Hx
∂t
+ Kx =
∂Ey
∂z
−
∂Ez
∂y
µo
∂Hy
∂t
+ Ky =
∂Ez
∂x
−
∂Ex
∂z
µo
∂Hz
∂t
+ Kz =
∂Ex
∂y
−
∂Ey
∂x
o
∂Ex
∂t
+ Jx =
∂Hz
∂y
−
∂Hy
∂z
o
∂Ey
∂t
+ Jy =
∂Hx
∂z
−
∂Hz
∂x
o
∂Ez
∂t
+ Jz =
∂Hy
∂x
−
∂Hx
∂y
∂Kx
∂t
= µoω2
pHx
∂Ky
∂t
= µoω2
pHy
∂Kz
∂t
= µoω2
pHz
∂Jx
∂t
= oω2
pEx
∂Jy
∂t
= oω2
pEy
∂Jz
∂t
= oω2
pEz
Note that with this new approach, we avoid the instability issue present when setting
directly µ < 0 and < 0. The application of Yee’s algorithm is then straightforward,
with Jx, Kx, Jy, Ky, Jz, and Kz being updated in each time step, as the E and H fields.
6.3.1 PML in a plasma medium
Contrary to the two-time-derivative Lorentz material absorbing boundary condition
employed to terminate the grid in [47], we decided to use the split-PML proposed by
Berenger [1]. To use Berenger’s PML, we split the each field component into two com-
ponents. To illustrate the procedure, we show how it is done for the first equation. The
62

remaining equations follow the same process. To use Belenger’s PML, we start with the
modiﬁed Maxwell’s equations and assuming the regular PML, that is,
si = 1 +
σi
jω
, i = x, y, or z (6.4)
we have
1 −
ω2
p
ω2
µ0jωHsx + σx
µ
Hsx = −
∂
∂x
ˆx × E (6.5)
1 −
ω2
p
ω2
µ0jωHsy + σy
µ
Hsy = −
∂
∂y
ˆy × E
1 −
ω2
p
ω2
µ0jωHsz + σz
µ
Hsz = −
∂
∂z
ˆz × E
1 −
ω2
p
ω2 0jωEsx + σxEsx =
∂
∂x
ˆx × H
1 −
ω2
p
ω2 0jωEsy + σyEsy =
∂
∂y
ˆy × H
1 −
ω2
p
ω2 0jωEsz + σzEsz =
∂
∂z
ˆz × H
Equation (6.5) is ﬁrst written as
µ0jωHsx + σx
µ
Hsx = −
∂
∂x
[ˆzEy − ˆyEz] − Ksx
Ksx = −
ω2
p
ω
µ0jHsx
When these two equations are splitted into two components, we have
µ0jωHsxz + σx
µ
Hsxz = −
∂Ey
∂x
− Ksxz
µ0jωHsxy + σx
µ
Hsxy =
∂Ez
∂x
− Ksxy
jωKsxz = ω2
pµ0Hsxz
jωKsxy = ω2
pµ0Hsxy
63

In the time domain, these become
µ0
∂Hsxz
∂t
+ σx
µ
Hsxz = −
∂Ey
∂x
− Ksxz
µ0
∂Hsxy
∂t
+ σx
µ
Hsxy =
∂Ez
∂x
− Ksxy
∂Ksxz
∂t
= ω2
pµ0Hsxz
∂Ksxy
∂t
= ω2
pµ0Hsxy
The same approach is done for the five remaining equations. We are now in condition
to apply Yee’s algorithm to the equations above, with the PML and the plasma model
working together in our 3D code.
6.4 Results for Left-Handed Metamaterials
The 3D-FDTD-PML simulator uses the standard leapfrog in time and a regular rec-
tangular grid in space. The magnetic field H as well as its auxiliar current K are located
at the cell center while electric field E and electric current J are located at the cell edge.
It requires averaging µ, and σ for E and J. The cell sizes in the simulations below are
∆x = ∆y = ∆z = 0.025 cm, corresponding to λo/40 (fo = 30 GHz). The time step is set
to be ∆t = 0.95∆x/(
√
3c) = 0.447 ps. For the PML, we used 8 layers with σmax = 10.
The analytical expression for σ is given by
σu = σmax
u
L
2
(6.6)
where u is either x, y or z and L is the PML’s thickness. The expression for the input
signal, the same as in [47], is given by
f(t) =



gon(t) sin(ωot) for 0 ≤ t < mTp
sin(ωot) for mTp ≤ t < (m + n)Tp
goff(t) sin(ωot) for (m + n)Tp ≤ t < (m + n + m)Tp
0 for (m + n + m)Tp ≤ t
(6.7)
64

where Tp = 1/fo is the period of one single cycle and the three-derivative smooth window
functions are given by
gon(t) = 10x3
on − 15x4
on + 6x5
on
goff(t) = 1 − [10x3
off − 15x4
off + 6x5
off]
with xon = 1 − (mTp − t)/mTp and xoff = [t − (m + n)]/mTp.
6.4.1 Comparison with 2D results
To test our code, we simulate the same problem as [47], used there to check focus
properties of LHM. The problem consists of an infinite long line source located λo/2 above
a LHM slab with thickness d = λo/2, which is surrounded by free space. The LHM slab
was located exactly in the middle of our domain, occupying x =[0 120], y =[40 80], and
z =[0 30] cells, including the PML region. The line source was located at (60,100,[0 30]),
20 cells away from the slab and the resonant frequency used for expression (6.3) was set to
be ωp = 266.6 × 109
rad/s, what gives r = µr −1. The computational domain for this
example is 120×120×30 grid points, in the x, y and z directions, respectively. With the
PML approach, we were able to reproduce the results in [47] in a much smaller domain.
Figure 6.1 shows the Ez-field distribution corresponding to the time t = 1250∆t. The
field is shown down to 40 dB below its maximum. In order to compare our simulation
to [47], we did not place a PML layer on the front and back faces, perpendicular to the
z-axis, in the 2D simulations. It should be notice that the field distribution is the same
as in Fig. 7 in [47], even with a different precision due to larger time steps and larger cell
sizes in our simulation.
Another simulation was done to compare with Fig. 8 in [47]. In this case, the slab
is located in the region ([0 120],[36 84],[0 30]), the line source at (60,88,[0 30]), and
ωp = 500 × 109
rad/s. The field is plotted down to 40 dB below its maximum and at a
65

time t = 1250∆t. As shown in Fig. 6.2, paraxial foci are located at the center of the slab
and at the opposite side from the source.
6.4.2 Three-dimensional simulation of a λ/2 dipole
Finally, we apply our code to 3D problems. The problem is to simulate a λo/2 dipole
λo/2 above a LHM slab with thickness d = λo/2. We reproduced similar geometry to
the 2D problems, but with a finite line source. The computational domain for all 3D
examples has 120 × 120 × 120 grid points. Figure 6.3 illustrates the problem’s geometry,
with the xy-plane cutting at the middle of our domain and the yz- and zx-planes cutting
along the dipole. All the fields are plotted down to 60 dB below their maxima. For the
first example, the dipole was located at (60,100,[-10 10]) with the source at its center, 20
cells away from the slab placed at ([0 120],[40 80],[0 120]) and ωp = 266.6 × 109
rad/s.
The field distribution in each plane is shown in Fig. 6.4.
For the second example, the dipole was located at (60,88,[-10 10]) with the source at
its center, 4 cells away from the slab placed at ([0 120],[36 84],[0 120]) and ωp = 500×1011
rad/s. The field distribution in each plane is shown in Fig. 6.5.
Unfortunately, the use of the second-order PML is not recommended in the simulation
of left-handed metamaterials. The reason is the presence of the permittivity of the
material in the expressions for the metric coefficients. Since the metric coefficients for
the second-order are more involved, the use of complicated expressions for both and
the metric coefficients si would lead to cumbersome expressions and would require and
amount of memory that would not justify the use of the second-order PML.
However, one can still use the second-order PML with complicated expressions for ,
with both frequency and space dependence, if the permeability µ is constant, which was
not the case for the plasma model of the left-handed metamaterials. It can be done by
making use of one of the most fundamental priciples in electromagnetics, duality, and
66

solving the problem for µ and H instead of and E. The next section explain how it is
done.
6.5 Duality
One of the basic electromagnetic principles is the duality principle [26, 49]. In order to
make Maxwell’s equations symmetric, one has to add a magnetic charge density, present
only in the mathematical formulation but absent in practice. However, since the problems
we solve here are source-free, the equations are already symmetric. They have the format
× E = −jωµH − M
× H = jω E + J
· B = 0
· D = 0,
with B = µH and D = E. The duality principle states the once one knows the solution
for a set (H, E, µ and ), another solution can be obtained by simply replacing E → −H,
H → −E, µ → and → µ.
Since the metric coeﬃcients for the PML are deﬁned in such a way that they depend
on , for problems where is not constant we will solve the dual problem, solving for µ
and H.
6.6 Formulation for Gyrotropic Media
For problems involving magnetic materials (µ not constant) with constant value for ,
the formulation is the same as the one developed in the previous chapters. For dielectric
materials with µ constant and as a function of space or frequency or both, we use the
same formulation and solve the dual problem. For problems involving variations both in
67

µ and , the use of the second-order PML becomes less attractive since it is not possible
to use duality and the metric coefficients will have to include the expressions for .
6.6.1 Uniaxial dispersive material
In this section we solve the problem of matching the second-order PML to a uniaxial
dielectric material with permittivity given by
=






1 0 0
0 1 0
0 0 z






with the constitutive parameters defined by
1 = 0 1 −
ω2
p
(ω2 − ω2
c )
z = 0 1 −
ω2
p
ω2
. (6.8)
In order to apply the second-order PML, we make use of duality, and rather than
solving directly for , which would lead to a complicated expression, we set µ = and
solve for H field instead of solving for E.
The expression for 1, as well as for z, can lead to a frequency range where there
is a change in the sign for the relative permittivity. Even though that may be possible
theoretically, that would lead to an instability inside the PML. However, for the frequency
range of interest, ωp ω and ωc ω, leading to positive values for both and z.
The formulation uses the same ADE technique developed for left-handed metamate-
rials and for the higher-order PML. To implement the PML with a uniaxial medium as
described in (6.8), we first use duality principle to solve the problem for µ and H. By
setting µ = and = 0 and, using the same PML metric coefficients as defined before
68

we have
jωµH = × E
−jω E = × H.
The equations for the E-ﬁeld are the same as the ones for free-space and can be
derived using the stretched-coordinate technique directly. For the H-ﬁeld, we have
−jωµ1Hx =
1
sy
∂
∂y
Ez −
1
sz
∂
∂z
Ey
−jωµ1Hy =
1
sz
∂
∂z
Ex −
1
sx
∂
∂x
Ez
−jωµzHz =
1
sx
∂
∂x
Ey −
1
sy
∂
∂y
Ex.
(6.9)
The next step is to split each equation into its x, y and z derivative components and
apply the ADE technique described in Chapter 3.
Another important topic in electromagnetics is the gyrotropic materials, present both
in the format of gyroelectric material, where the permittivity has the tensor form given
by
=






1 j g 0
−j g 1 0
0 0 z






or gyromagnetic material, where permeability µ is given by
µ =






µ1 jµg 0
−jµg µ1 0
0 0 µ0






.
The expressions for 1, g, z, µ1, µg, and µz depend on the material properties.
69

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