This document describes time-domain modeling of electromagnetic wave propagation in complex materials using the Transmission-Line Modeling (TLM) method. The formulation is derived from Maxwell's equations and constitutive relations using bilinear Z-transform methods. This approach can model anisotropic, bianisotropic, and frequency-dependent linear materials. Two examples are presented to validate the approach: plane wave reflection/transmission in an isotropic chiral slab, and reflectivity of a uniaxial chiral material. Close agreement with frequency-domain analyses demonstrates the method can accurately model problems without analytic solutions.

1. TIME-DOMAIN MODELING OF
ELECTROMAGNETIC WAVE PROPAGATION
IN COMPLEX MATERIALS
J. Paul, C. Christopoulos and D. W. P. Thomas
School of Electrical and Electronic Engineering
University of Nottingham
Nottingham, NG7 2RD, United Kingdom
ABSTRACT
In this study, the Transmission-Line Modeling (TLM) method is ex-
tended and applied to the time-domain simulation of electromagnetic
wave propagation in materials displaying magnetoelectric coupling. The
formulation is derived from Maxwell’s equations and the constitutive
relations using bilinear Z-transform methods leading to a general Pad´e
system. This approach is applicable to all frequency-dependent lin-
ear materials including anisotropic and bianisotropic media. The close
agreements between the results obtained from the time-domain simula-
tions and analyses for examples involving isotropic and uniaxial chiral
materials indicates that the numerical approach can be applied with
confidence to problems having no analytic solution.
1. INTRODUCTION
Both the Transmission-Line Modeling (TLM) method [1] and the Finite-
Difference Time-Domain (FDTD) method [2] are differential techniques
useful for the time-domain solution of electromagnetic problems. Previ-
ous researchers have used FDTD for the simulation of electromagnetic
wave propagation in anisotropic materials. As an example, in [3], a
model of an anisotropic material with constant parameters was devel-
oped. In [4], FDTD was extended to include the frequency-dependent
anisotropic material properties of a magnetized plasma and to a mag-
1

2. netized ferrite material in [5]. However, because of the offsets between
the electric and magnetic fields of half a space-step and half a time-step
in a FDTD grid, as noted in [3, 4], the update scheme for 3-D prob-
lems requires spatial and temporal averaging. This leads to difficulties
in the description of material discontinuities, boundary conditions and
materials exhibiting magnetoelectric coupling.
One of the main differences between FDTD and TLM is that in TLM
the electric and magnetic fields are solved at the same point in space-
time. This leads to the proposition that TLM may be easier to apply
than FDTD for the simulation of electromagnetic wave propagation in
anisotropic materials. Also the condensed nature of the TLM space-
time grid offers the possibility of describing bianisotropic materials.
Previous investigators have developed TLM procedures for anisotropic
materials, in [6] an anisotropic medium with constant parameters was
studied and in [7, 8], magnetized plasmas and ferrites were examined.
The model detailed in this article is an extension of the isotropic
formulation of Debye (first-order) frequency-dependent materials [9] to
include frequency-dependent bianisotropic materials [10]. In this ap-
proach, the model is developed throughout by the systematic applica-
tion of bilinear Z-transform techniques [11]. This leads to a general
formulation as a Pad´e system which is applicable to the time-domain
modeling of all linear frequency-dependent complex materials.
The approach is validated using two straightforward examples in
which the plane wave reflection and transmission of an isotropic second-
order chiral slab having a second-order frequency dependence [10] and
the reflectivity of a uniaxial second-order chiral material [12] are stud-
ied. The results show excellent agreement between the time-domain
models and frequency-domain analyses. Also presented are the time-
domain results of pulse propagation in these materials.
2. FORMULATION OF THE TIME-DOMAIN MODEL
Equation (1) expresses Maxwell’s curl equations in compact form using
the notation for the fields, current and flux densities of [13].
∇ × H
−∇ × E
=
Je
Jm
+
∂
∂t
D
B
(1)
Equation (2) expresses the constitutive relations for the electric and
2

3. magnetic flux densities D and B.
D
B
=
ε0 E
µ0 H
+
ε0χe ξr/c
ζr/c µ0χm
∗
E
H
(2)
In (2), χe and χm are the electric and magnetic susceptibility tensors
and the dimensionless tensors describing the magnetoelectric coupling
are ξr and ζr. Also in (2), ε0 and µ0 are the free-space permittivity
and permeability, c is the speed of light in free-space and ∗ denotes
a time-domain convolution. The constitutive relations for the electric
and magnetic current densities Je and Jm are given in (3).
Je
Jm
=
Jef
Jmf
+
σe σem
σme σm
∗
E
H
(3)
In (3), Jef and Jmf are the free electric and magnetic current densities,
σe and σm are the electric and magnetic conductivity tensors and σem
and σme are the magnetoelectric conductivity tensors. Although all
conductive materials can be described in (2), for time-domain modeling
it is useful to express conduction separately as (3). For example, a
frequency-dependent σe is used for describing plasmas [9] and σe and
σm are used in absorbing boundaries.
In the general case, the tensors of the constitutive relations of (2)
and (3) contain elements describing causal time functions. Substitution
of (2) and (3) into (1) yields
∇×H − Jef
−∇×E −Jmf
−
∂
∂t
ε0 E
µ0 H
=
σe σem
σme σm
∗
E
H
+
∂
∂t
ε0χe ξr/c
ζr/c µ0χm
∗
E
H
(4)
The model detailed in this paper is a discrete time solution of (4) in a
Cartesian grid, solving for the fields E and H at each time-step. The
possibility of modeling magnetoelectric coupling by TLM is allowed for
by the normalization of E and H so that the circuit representations of
these quantities V and i both have the dimensions of volts using
E = −V/∆ℓ , H = −i/(∆ℓ η0) (5)
In (5), ∆ℓ is the space step and η0 is the intrinsic impedance of free-
space. Similarly the free current densities are normalized to quantities
if and Vf both with the dimensions of volts using
Jef = −if /(∆ℓ2
η0) , Jmf = −Vf /(∆ℓ2
) (6)
3

4. The conductivity tensors are normalized so that their circuit represen-
tations ge, gem, gme and rm are dimensionless, i.e.
σe = ge/(∆ℓ η0), σem = gem/∆ℓ, σme = gme/∆ℓ, σm = rm η0/∆ℓ (7)
Also, the time and spatial derivatives are normalized using
∂
∂t
=
1
∆t
∂
∂¯t
, (∇ × . . .) =
1
∆ℓ
(¯∇ × . . .) (8)
In (8), ∆t is the time-step of the time-domain simulation and ¯t is
the normalized time. In 1-D models, ∆ℓ/∆t = c and in 3-D models
∆ℓ/∆t = 2c [1, 14].
Using (5), (6), (7) and (8) in (4) for a 3-D model leads to
¯∇ × i − if
−¯∇ × V − Vf
−2
∂
∂¯t
V
i
=
ge gem
gme rm
∗
V
i
+ 2
∂
∂¯t
χe ξr
ζr χm
∗
V
i
(9)
The left-hand side of (9) involves the curl operations, the free sources
and the time derivative of the fields. The solution of the left-hand side
of (9) is detailed for the 1-D and 3-D cases in Appendix A, leading to
the TLM spatial transform
¯∇ × i − if
−¯∇ × V − Vf
− 2
∂
∂¯t
V
i
TLM
−→ 2 Fr
− 4 F where F =
V
i
(10)
On the right-hand side of (10), the excitation vector Fr
is a function of
the incident voltages and any free sources. The definition of Fr
is given
in Appendix A. For the modeling of free-space, the right-hand side of
(9) is 0, i.e. the null vector. However for the description of materials,
using the complex variable z to represent the time-shift operator, the
right-hand side of (9) is transformed to the Z-domain using the bilinear
transform [11], i.e. ∂/∂¯t → 2(1 − z−1
)/(1 + z−1
)
ge gem
gme rm
∗
V
i
+2
∂
∂¯t
χe ξr
ζr χm
∗
V
i
Z
−→ σ(z)·F+4
1−z−1
1+z−1
M(z)·F(11)
The matrices on the right-hand side of (11) are
σ(z) =
ge gem
gme rm
and M(z) =
χe ξr
ζr χm
(12)
The conductivity matrix σ(z) and the material matrix M(z) may con-
tain frequency-dependent elements and this is indicated by explicitly
4

5. writing their arguments (z). Combining the right-hand sides of (10)
and (11) using (9) yields
2(1+z−1
)Fr
= (1+z−1
)4 · F + (1+z−1
)σ(z) · F + (1−z−1
)4M(z) · F(13)
where using 1 to represent the identity matrix, 4 = 4 1. In the model-
ing of matrices σ(z) or M(z) consisting of causal frequency-dependent
elements, the overall strategy is shift the frequency-dependence back to
the previous time-step by taking the partial fraction expansions of
(1+z−1
)σ(z) = σ0 + z−1
[σ1 + ¯σ(z)] (14)
(1−z−1
)M(z) = M0 − z−1
[M1 + ¯M(z)] (15)
In (14) and (15), depending on the type of material, matrices σ0, σ1,
M0 and M1 contain constant (possibly zero) elements and matrices ¯σ(z)
and ¯M(z) contain zero or frequency-dependent elements. Substituting
(14) and (15) into (13) gives
F = T · [2Fr
+ z−1
S] (16)
where the forward gain matrix T = [4 + σ0 + 4M0]−1
. The main accu-
mulator vector S is calculated using
S = 2Fr
+ κ · F − ¯σ(z) · F + 4 ¯M(z) · F (17)
where the feedback matrix κ = −[4 + σ1 − 4M1]. In the present model,
(16) and (17) are used at each time-step to obtain the vector of total
fields F from the excitation Fr
. The process is summarized in the signal
flow diagram of Fig. 1. To generate a compact notation to describe
general material functions, (16) and (17) are combined as
F = t(z) · Fr
(18)
In (18), t(z) is a 6×6 matrix of transfer functions.
3. MODELING SECOND-ORDER CHIRAL MEDIA
In this section, the formulation of section 2 is used to develop time-
domain models for the description of electromagnetic propagation in
isotropic and uniaxial chiral materials displaying a second-order fre-
quency response.
5

6. z-1
2
+
+
S
+
z-1
S
F
r
F
+
_
__κ σ(z)_ _
__ M(z)4____
T__
+
Figure 1: Field update system for a general material
3.1. Isotropic Chiral Medium
In an isotropic chiral material [10], the susceptibility and magnetoelec-
tric coupling tensors of (2) have the form:
χe = χe1 , χm = χm1 , ξr = ξr1 , ζr = −ξr1 (19)
One example is constructed by a dispersal of wire helices with ran-
dom orientations in a low-loss isotropic dielectric background having a
constant electric susceptibility χeb.
In references [10, 15], a single helix was modeled as an intercon-
nected dipole and loop and the polarizabilities were quantified using
antenna theory. From this investigation, to a first approximation the
effective material properties are second-order in form. In the Laplace
domain, the electric susceptibility due to the helices is
χec(s) =
χe0 ω2
0
(s + δ)2 + β2
(20)
In (20), s is the complex frequency [11], χe0 is the dc electric suscepti-
bility, δ is the damping frequency, β is the natural frequency and the
resonant frequency ω0 =
√
δ2 + β2. The magnetic susceptibility is
χmc(s) =
−χm∞ s2
(s + δ)2 + β2
(21)
where χm∞ is the magnetic susceptibility at high frequencies. In the
analysis of [10, 15], the higher-order modes of the equivalent loop were
neglected and above ω0, (21) goes negative. To obtain physical solu-
tions, it is necessary to introduce a background magnetic susceptibility
χmb ≥ χm∞ to account for high frequency effects not described in (21).
6

7. Note that another form of the magnetic susceptibility exists in the liter-
ature: In references [16, 17], the frequency-dependence of the magnetic
susceptibility is proposed to have the same form as (20).
The frequency dependence of the magnetoelectric parameters obey
Condon’s model [10, 18],
ξr(s) =
s τ ω2
0
(s + δ)2 + β2
(22)
where τ is the chirality time constant. Because this medium is based on
the wire and loop model, the chirality is not independent of the electric
and magnetic susceptibilities.
The time-domain model is developed by identifying the matrices of
(12). For this material, σ = 0 where 0 is the null matrix and
M =
χeb1 0
0 χmb1
+
χec1 ξr1
−ξr1 χmc1
= Mb + Mc (23)
In (23), Mb is the matrix of background susceptibilities and Mc is the
frequency-dependent matrix describing the coupled part of isotropic
chiral material response. To simplify the notation, in the development
of the time-domain model of M, the 1’s are suppressed. Using (20),
(21) and (22) in (23) the frequency-dependent material matrix in the
Laplace domain is
Mc(s) =
χec ξr
−ξr χmc
=
1
(s + δ)2 + β2
χe0 ω2
0 s τ ω2
0
−s τ ω2
0 −χm∞ s2 (24)
Equation (24) is converted to the Z-domain using the bilinear trans-
form as in (11), i.e. s → (2/∆t)(1 − z−1
)/(1 + z−1
), yielding
Mc(z) =
b0 + z−1
b1 + z−2
b2
1 − z−1a1 − z−2a2
(25)
where using D = [(2 + ∆t δ)2
+ β2
∆t2
], the feedback gains are
a1 = 2 D−1
[(2 + ∆t δ)(2 − ∆t δ) − β2
∆t2
]
a2 = −D−1
[(2 − ∆t δ)2
+ β2
∆t2
]
and the forward gain matrices are
b0 = D−1 χe0 ω2
0 ∆t2
2 ∆t τ ω2
0
−2 ∆t τ ω2
0 −4 χm∞
, b1 =2 D−1 χe0 ω2
0 ∆t2
0
0 4 χm∞
b2 = D−1 χe0 ω2
0 ∆t2
−2 ∆t τ ω2
0
2 ∆t τ ω2
0 −4 χm∞
7

8. Equation (25) is in the standard Pad´e form of a transfer function. The
model follows by substituting (25) and (23) into (15) leading to
(1 − z−1
)M(z) = M0 − z−1
[ M1 + ¯M(z)]
= Mb +b0 − z−1

Mb +
b′
0/4 +z−1
b′
1/4+z−2
b′
2/4
1 − z−1a1 − z−2a2

 (26)
The matrices in the numerator of (26) are
b′
0/4 = b0 − b1 − a1b0 , b′
1/4 = b1 − b2 − a2b0 , b′
2/4 = b2 (27)
From (26), the matrices of (15) are
M0 = Mb + b0 , M1 = Mb , 4 ¯M(z) =
b′
0 + z−1
b′
1 + z−2
b′
2
1 − z−1a1 − z−2a2
(28)
The total fields are obtained as in (16) using
F = T · [2Fr
+ z−1
S] (29)
where T = [4 + 4 Mb + 4 b0]−1
. The main accumulator vector S is
calculated using (17) in the form
S = 2Fr
+ κ · F + S1 (30)
where κ = −(4 − 4 Mb). In (30), the material accumulator vector S1 is
evaluated using
S1 = 4 ¯M(z) · F =


b′
0 + z−1
b′
1 + z−2
b′
2
1 − z−1a1 − z−2a2

 · F (31)
An efficient technique for the solution of (31) is the phase-variable state-
space method [11]. Defining state vectors X1 and X2, the discrete
state-space and output equations are
X1
X2
= z−1 a1 a2
1 0
·
X1
X2
+
1
0
F (32)
S1 = b′
0 (b′
1 + z−1
b′
2) ·
X1
X2
(33)
The algorithm requires 4 backstores per field component. The system
described by (32) and (33) is illustrated in Fig. 2.
8

9. z-1
z-1
X1
X2
X2
z-1
b0
==
+
+
S
1
+
b1
==
b2
==
++
+F
a2
a1
Figure 2: Phase-variable system for a second-order chiral medium
Although in this section only a second-order system was consid-
ered, because (31) is in a standard Pad´e form, it is straightforward to
extend this formulation for the description of higher-order frequency-
dependent material functions.
3.2. Uniaxial Chiral Medium
In a uniaxial chiral material [10, 12], the helices are randomly dis-
persed in the background material, but are aligned in a particular direc-
tion. For example, consider 1-D propagation in ˆx with E and H trans-
verse to ˆx (Appendix A.1), with the helices aligned in the ˆz-direction.
Assuming the helices are embedded in an isotropic background hav-
ing a frequency-independent electric susceptibility of χeb and have the
frequency-dependent properties of (20), (21) and (22), the reduced ma-
terial matrix of (12) is
M =





χyy
e χyz
e ξyy
r ξyz
r
χzy
e χzz
e ξzy
r ξzz
r
−ξyy
r −ξyz
r χyy
m χyz
m
−ξzy
r −ξzz
r χzy
m χzz
m





=





χeb 0 0 0
0 χeb 0 0
0 0 0 0
0 0 0 χmb





+





0 0 0 0
0 χec 0 ξr
0 0 0 0
0 −ξr 0 χmc





(34)
As in the isotropic case of (23), (34) is written as
M = Mb + Mc (35)
where Mb and Mc in (35) apply to the uniaxial case. The development
of the discrete-time model follows a similar approach to that detailed
in equations (24) through to (33).
9

10. 4. RESULTS
In this section, the time-domain models developed in sections 2 and 3
are validated against frequency-domain analysis for isotropic and uniax-
ial chiral materials having a second-order frequency dependence. Also,
time-domain results are presented for pulse propagation in these ma-
terials. Although time-domain results for propagation in chiral media
were presented in [16, 17], direct comparison with our technique would
not be straightforward as these authors used a Beltrami description of
the fields.
4.1. Isotropic Chiral Medium
As in previous developments of novel formulations in the time-domain
of frequency-dependent materials [2, 4, 5, 6], to ensure the model is
giving the correct results and thus offer a degree of validation, two
basic problems involving an isotropic chiral medium with analytic solu-
tions are investigated: Using both TLM and frequency-domain analysis
[10], the reflection and transmission coefficients of an isotropic chiral
slab having a second-order frequency dependence between two isotropic
free-spaces and the reflection coefficient of the slab with a metal back-
ing were obtained and compared.
4.1.1. Isotropic Chiral Slab—Frequency-domain Results
The thickness of the slab was selected as 200mm with properties χe0 =
χm∞ = ω0τ = 0.5, ω0 = 2π ×1000×106
and δ = 2π ×100×106
. The
background electric and magnetic susceptibilities were selected as χeb =
χmb = 1. Thus in the frequency-domain model, the overall relative
permittivity and permeability were εr =1+χeb+χec and µr =1+χmb+χmc.
The characteristic impedance was η=η0 µr/εr and the refractive index
for circularly polarized waves was n± =
√
µrεr ±jξr.
The time-domain model had a space-step of ∆ℓ = 1mm and was
excited with a ˆz-polarized pulse of electric field traveling in the +ˆx-
direction having unit magnitude. During the simulation, the time-
domain co-polarized and cross-polarized reflected and transmitted elec-
tric fields were saved for transformation to the frequency-domain for
direct comparison with the analytic results. The transmission and re-
flection coefficients obtained for circularly polarized (CP) waves [4, 5]
10

11. are compared in Fig. 3. In this diagram, for example: Tlcp is the
left-hand CP transmission coefficient and Rrcp is the right-hand CP
reflection coefficient.
-50
-40
-30
-20
-10
0
0 0.5 1 1.5 2 2.5
Reflection/transmissioncoefficientmagnitudes(dB)
Frequency (GHz)
Trcp (analytic)
Trcp (TLM)
Tlcp (analytic)
Tlcp (TLM)
Rrcp/Rlcp (analytic)
Rrcp/Rlcp (TLM)
Figure 3: Transmission and reflection coefficients of a chiral slab
4.1.2. Isotropic Chiral Slab—Time-domain Results
The time-domain response of the slab for a pulse plane wave excitation
is shown in Fig. 4. The initial pulse was ˆz-polarized, traveling in the
+ˆx direction, with a Gaussian profile, having a maximum amplitude
of 1V/m and a spatial width of 128 cells between the 0.001 amplitude
truncation points [5].
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
z-polarizedelecricfield(V/m)
Distance (m)
Free-space Free-space
Isotropic
chiral
Initial condition
834ps
1668ps
2502ps
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
y-polarizedelectricfield(V/m)
Distance (m)
Free-space Free-space
Isotropic
chiral
834ps
1668ps
2502ps
Figure 4: Time-domain response of an isotropic chiral slab
In Fig. 4 the distribution of the ˆz- and ˆy-polarized electric fields
at various times are indicated. As the incident wave enters the chiral
slab, the coupling from the ˆz-polarization to the ˆy-polarization is seen.
Also, in agreement with the frequency-domain analysis, no ˆy-polarized
reflected electric field is observed.
11

12. 4.1.3. Metal-backed Isotropic Chiral Slab
In this example, to examine the effect of the spatial discretization,
the slab of the previous section was terminated with a metal back-
ing and the simulation repeated using space-steps of 1mm, 5mm and
20mm. The co-polarized return loss magnitudes obtained using TLM
and analysis are shown on the left-hand side of Fig. 5.
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
0 0.5 1 1.5 2 2.5
Co-polarizedreturnloss(dB)
Frequency (GHz)
Analytic
1mm
5mm
20mm
-4
-2
0
2
4
0 0.5 1 1.5 2 2.5
Error(dB)
Frequency (GHz)
1mm
5mm
20mm
Figure 5: Co-polarized return loss and error of an isotropic chiral slab
with a metal backing
The error between the time-domain model and analysis is shown on
the right-hand side of Fig. 5. For the models using 1mm and 5mm cells,
the worst case errors are ∼0.1dB and ∼1.5dB respectively, both at the
chosen material resonant frequency (1GHz). The error in all cases is
mainly due to the frequency error in the bilinear transform: This error
can be reduced by applying the standard technique of prewarping the
critical frequencies prior to bilinear transformation [11].
4.2. Uniaxial Chiral Half-Space
The formulation for uniaxial chiral materials developed in section 3.2
was validated by the simulation of the reflectivity of an infinite half-
space of this medium. The results obtained using TLM are compared
with frequency-domain analysis [12] in Fig. 6. The material properties
of the uniaxial chiral material were identical to that given in section
4.1.1 for the isotopic chiral medium, except that the helices were aligned
with the ˆz direction. In Fig. 6, using the notation of [12], Ryz is the
ˆy-polarized reflection coefficient magnitude for a ˆz-polarized excita-
tion and Rzz is the ˆz-polarized reflection coefficient magnitude for a
ˆz-polarized excitation.
12

13. -40
-35
-30
-25
-20
-15
-10
-5
0
0 0.5 1 1.5 2 2.5
Reflectioncoefficientmagnitude(dB)
Frequency (GHz)
Rzz (analytic)
Rzz (TLM)
Ryz/Rzy (analytic)
Ryz/Rzy (TLM)
Ryy (analytic)
Ryy (TLM)
Figure 6: Reflection coefficients of a uniaxial chiral half-space
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
z-polarizedelectricfield(V/m)
Distance (m)
Free-space Uniaxial chiral
Initial condition
834ps
1668ps
2502ps
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
z-polarizedelectricfield(V/m)
Distance (m)
Free-space Uniaxial chiral
Initial condition
834ps
1668ps
2502ps
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
y-polarizedelectricfield(V/m)
Distance (m)
Free-space Uniaxial chiral
834ps
1668ps
2502ps
Figure 7: Time-domain response of a uniaxial chiral half-space for an
incident wave polarized in ˆz
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
y-polarizedelectricfield(V/m)
Distance (m)
Free-space Uniaxial chiral
Initial condition
834ps
1668ps
2502ps
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
z-polarizedelectricfield(V/m)
Distance (m)
Free-space Uniaxial chiral
834ps
1668ps
2502ps
Figure 8: As Fig. 7, but using a ˆy-polarized incident wave
13

14. The time-domain response of the uniaxial chiral medium is shown
in Figs. 7 and 8 for both ˆz- and ˆy-polarized traveling pulse excitations.
These figures give an insight into the dispersive behavior of the medium
in the time-domain: The ˆz-polarized transmitted wave for a ˆz-polarized
excitation is dispersed more heavily than the ˆy-polarized transmitted
wave for a ˆy-polarized excitation. Also, the cross-polarized transmitted
waves in both cases are of a similar magnitude.
5. CONCLUSIONS
In this paper, discrete time-domain models of electromagnetic wave
propagation in materials exhibiting magnetoelectric coupling have been
developed and validated. The approach adopted here was based on
bilinear Z-transform methods and the final system was expressed in a
standard Pad´e form. The essential details of the solution of the spatial
network in 1-D and 3-D problems are detailed in an Appendix.
Examples involving frequency-dependent chiral media have been
studied and the close agreement between the analytic and modeled
results demonstrate that the time-domain method may be applied to
problems having no analytic solution. Also, time-domain pulse prop-
agation in isotropic and uniaxial chiral materials were shown, giving
a further insight into the properties of electromagnetic waves in these
complex materials.
The method described has all the advantages of a full 3-D numeri-
cal model and can thus be used to study practical configurations with
complex geometrical shapes and material properties. As the technique
is implemented in the time-domain, it can also incorporate nonlineari-
ties and hence is applicable to the most general electromagnetic field-
material interactions.
APPENDIX A: BACKGROUND
To complete the formulation of section 2, it is necessary to solve for
the curl operations on the left-hand side of (10). This involves the
derivation of the TLM algorithm for the time-domain solution of the
spatial network. In this appendix, the computational algorithm for
the 1-D case is derived and the 3-D method is shown to follow as an
extension of the 1-D case.
14

15. A.1. 1-D TLM
To illustrate the general principles involved, initially, a 1-D model is
studied: For propagation in the ˆx-direction, with E and H transverse
to ˆx, where ˆx is the unit vector pointing in x,
E · ˆx = 0 , H · ˆx = 0 (36)
The left-hand side of Fig. 9 shows the 1-D spatial network applicable
to the modeling of this situation. The node has four ports (V4, V5,
V10 and V11) and the four transverse field quantities (Ey, Ez, Hy and
Hz) are indicated at the center of the cell. The curl operations are
solved using Stokes’ theorem along the integration contours Cy and Cz.
For consistency with a 3-D development in Appendix A.2, the port
numbers used in the 1-D case are taken from the 3-D node shown on
the right-hand side of Fig. 9.
V5
V11
V4
V10
Cz
Cy
Ez
Ey
Hz
Hy
x
y
z
Cz
Cy
V6
V0
V5
V11
V2
V8
V4
V10
V9
V3
V7
V1
Cx
Figure 9: 1-D and 3-D TLM spatial networks
Reduction of (4) to the 1-D case described above, gives





(∇×H)y
(∇×H)z
−(∇×E)y
−(∇×E)z





−





Jefy
Jefz
Jmfy
Jmfz





−
∂
∂t





ε0Ey
ε0Ez
µ0Hy
µ0Hz





=
σe σem
σme σm
∗





Ey
Ez
Hy
Hz





+
∂
∂t
ε0χe ξr/c
ζr/c µ0χm
∗





Ey
Ez
Hy
Hz





(37)
For modeling this particular 1-D case, the 3-D tensors have the following
form:
σe =



σxx
e 0 0
0 σyy
e σyz
e
0 σzy
e σzz
e


 (38)
15

16. Thus in (37), we may write the tensors as 2×2, for example
σe =
σyy
e σyz
e
σzy
e σzz
e
(39)
and (∇ × H)u = (∇ × H) · ˆu, where ˆu∈{ˆy,ˆz}. Using the field-circuit
equivalences of section 2 to transform (37) to a normalized form yields





V4 + V5
V10 + V11
V11 − V10
V4 − V5





−





ify
ifz
Vfy
Vfz





−
∂
∂¯t





Vy
Vz
iy
iz





=
ge gem
gme rm
∗





Vy
Vz
iy
iz





+
∂
∂¯t
χe ξr
ζr χm
∗





Vy
Vz
iy
iz





(40)
Converting (40) to the traveling wave format [9], using superscript i to
denote incident wave quantities and the notation of (12) gives
2





V4 + V5
V10 + V11
V11 − V10
V4 − V5





i
−





ify
ifz
Vfy
Vfz





−2





Vy
Vz
iy
iz





= σ(t)∗





Vy
Vz
iy
iz





+
∂
∂¯t
M(t)∗





Vy
Vz
iy
iz





(41)
The first two terms on the left-hand side of (41) are defined as the
external excitation,
2





V4 + V5
V10 + V11
V11 − V10
V4 − V5





i
−





ify
ifz
Vfy
Vfz





= 2





Vy
Vz
−iy
−iz





r
= 2 Fr
(42)
where the superscript r denotes reflected wave quantities. By defining
a matrix RT
1
, the vector of free-sources Ff and the vector of incident
voltages,
Vi T
= V4 V5 V10 V11
i
(43)
where the superscript T denotes a transposed vector, (42) can be writ-
ten compactly as
Fr
= RT
1
· Vi
− 0.5 Ff (44)
Defining the vector of total fields
F = Vy Vz iy iz
T
(45)
16

17. and by substituting (42) into (41) and transforming to the Laplace
domain using ∂/∂¯t → ¯s = s ∆t yields
2 Fr
= (2 + σ + ¯sM) · F (46)
As in (18), by defining a matrix of frequency-dependent transfer func-
tions, t = 2(2 + σ + ¯sM)−1
and transforming to the Z-domain, (46)
may be written as
F = t(z) · Fr
(47)
In order to time-step the process, we require the reflected voltages on
the transmission-lines. These are obtained using





V4
V5
V10
V11





r
=





Vy − iz
Vy + iz
Vz + iy
Vz − iy





−





V5
V4
V11
V10





i
(48)
Defining the vector of reflected voltages Vr
which is of the same form
as Vi
and the matrices R and P, (48) in concise form is
Vr
= R · F − P · Vi
= R · F − ˜Vi
(49)
In (49), ˜Vi
is the vector of voltages incident on the lines opposite those
used to obtain Vi
. In the final step of the algorithm, the connection
process, the reflected voltages are swapped between neighboring nodes
and become the incident voltages of the next time-step.
In summary, the 1-D method consists of the 3 steps of (44), (47) and
(49) as illustrated in Fig. 10. As discussed in section 2, for the mod-
eling of general material responses, only the block t(z) of this diagram
requires further development.
t(z)__ R__
Ff
i
V
~
r
r
VR
T
__V
i
__P
0.5
_
+ +
_
1
F F
Figure 10: Signal flow graph of the general TLM process
17

18. A.2. 3-D TLM
The spatial network used for 3-D problems is illustrated on the right-
hand side of Fig. 9 [14]. This node has 12 ports, (V0 . . . V11) and six
total field quantities (Ex, Ey, Ez, Hx, Hy and Hz) at the center of the
cell. Extending the development of (42) from (37) to the 3-D case, the
external excitation is:





Vx
Vy
Vz
−ix
−iy
−iz





r
=





( V0 + V1 + V2 + V3 )
( V4 + V5 + V6 + V7 )
( V8 + V9 + V10 + V11 )
− ( V6 − V7 − V8 + V9 )
− ( V10 − V11 − V0 + V1 )
− ( V2 − V3 − V4 + V5 )





i
−
1
2





ifx
ify
ifz
Vfx
Vfy
Vfz





→ Fr
= RT
1
· Vi
− 0.5 Ff (50)
As in the 1-D development in (44), the right-hand side of (50) is ob-
tained by defining a matrix RT
1
, the free-source vector Ff and the vector
of incident voltages,
Vi T
= V0 V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11
i
(51)
Defining the vector of total fields
F = Vx Vy Vz ix iy iz
T
(52)
and using (44) and (52) in (41) written for the 3-D case and transform-
ing to the Laplace domain using ∂/∂¯t → ¯s yields
2 Fr
= (4 + σ + 2¯sM) · F → F = t(z) · Fr
(53)
The right-hand side of (53) follows by defining a matrix of transfer
functions, t = 2(4 + σ + 2¯sM)−1
and transforming to the Z-domain.
The reflected voltages are obtained by extension of (48),















V0
V1
V2
V3
V4
V5
V6
V7
V8
V9
V10
V11















r
=















Vx − iy
Vx + iy
Vx + iz
Vx − iz
Vy − iz
Vy + iz
Vy + ix
Vy − ix
Vz − ix
Vz + ix
Vz + iy
Vz − iy















−















V1
V0
V3
V2
V5
V4
V7
V6
V9
V8
V11
V10















i
→ Vr
= R · F − ˜Vi
(54)
The right-hand side of (54) is developed by defining the vector of re-
flected voltages Vr
of the same form as Vi
, the matrix R and the
reordered incident vector ˜Vi
.
Equations (50), (53) and (54) show that solution of the 3-D spatial
network is simply an extended form of the 1-D case shown in Fig. 10.
18

19. REFERENCES
[1] C. Christopoulos, The Transmission-Line Modeling Method: TLM
(Piscataway, NJ, USA: IEEE Press, 1995).
[2] A. Taflove, Computational Electrodynamics: The Finite-Difference
Time-Domain Method (Norwood, MA, USA: Artech House, 1995).
[3] J. Schneider, and S. Hudson, “The Finite-Difference Time-Domain
Method Applied to Anisotropic Material”, IEEE Trans. Antennas
Propagat. 41, 994–999 (1993).
[4] F. Hunsberger, R. Luebbers, and K. Kunz. “Finite-Difference
Time-Domain Analysis of Gyrotropic Media—I: Magnetized
Plasma”, IEEE Trans. Antennas Propagat. 40, 1489–1495 (1992).
[5] K. S. Kunz and R. J. Luebbers, The Finite Difference Time Do-
main Method for Electromagnetics. (Boca Raton, FL, USA: CRC
press, 1993).
[6] L. de Menezes and W. J. R. Hoefer, “Modeling of General Consti-
tutive Relationships using SCN TLM”, IEEE Trans. Microwave
Theory Tech. 44, 854–861 (1996).
[7] S. Hein, “Synthesis of TLM Algorithms in the Propagator Inte-
gral Framework”, In Second International Workshop on Transmis-
sion Line Matrix (TLM) Modeling Theory and Applications 1–11
(1997).
[8] S. Hein, “TLM Numerical Solution of Bloch’s Equations for Mag-
netized Gyrotropic Media”, Appl. Math. Modeling 21, 221–229
(1997).
[9] J. Paul, C. Christopoulos and D. W. P. Thomas, “Modelling of
Debye material in 1-D and 2-D TLM schemes”, In International
Symposium on Electromagnetic Theory, URSI—Commission B,
Thessaloniki, Greece (1998).
[10] I. V. Lindell, A. H. Sihvola, S. A. Tretyakov and A. J. Viitanen,
Electromagnetic Waves in Chiral and Bi-Isotropic Media (Nor-
wood, MA, USA: Artech House, 1994).
19

20. [11] R. T. Stefani, C. J. Savant, B. Shahian and G. H. Hostetter, Design
of Feedback Control Systems (Philadelphia, PA, USA: Saunders
College Publishing, 1994).
[12] I. V. Lindell, and A. H. Sihvola, “Plane-Wave Reflection from Uni-
axial Chiral Interface and Its Application to Polarization Transfor-
mation”, IEEE Trans. Antennas Propagat. 43, 1397–1404 (1995).
[13] J. A. Kong, Electromagnetic Wave Theory (New York, NY, USA:
Wiley, 1986).
[14] P. B. Johns, “A Symmetrical Condensed Node for the TLM
method”, IEEE Trans. Microwave Theory Tech. 35, 370–377
(1987).
[15] S. A. Tretyakov, F. Mariotte, C. R. Simovski, T. G. Kharina
and J-P Heliot, “Analytical Antenna Model for Chiral Scatter-
ers: Comparison with Numerical and Experimental Data”, IEEE
Trans. Antennas Propagat. 44, 1006–1014 (1996).
[16] P. G. Zablocky and N. Engheta, “Transients in chiral media with
single-resonance dispersion” J. Opt. Soc. Am. A 10, 740–758
(1993).
[17] S. A. Maksimenko, G. Y. Slepyan and A. Lakhtakia, “Gaus-
sian pulse propagation in a linear, lossy chiral medium”,
J. Opt. Soc. Am. A 14, 894–900 (1997).
[18] A. H. Sihvola, “Temporal Dispersion in Chiral Composite Ma-
terials: A Theoretical Study”, J. Electromagn. Waves Appl. 6,
1177–1196 (1993).
20