1. Time-Domain Simulation of Electromagnetic Wave Propagation
in Two-Level Dielectrics
John Paul∗ Christos Christopoulos∗ and David W. P. Thomas∗
∗George Green Institute for Electromagnetics Research
School of Electrical and Electronic Engineering,
University Park, Nottingham NG7 2RD, United Kingdom.
Email: john.paul@nottingham.ac.uk
Abstract— A method for the time-domain simulation of
electromagnetic wave propagation in two-level dielectric me-
dia is developed. The technique is based on incorporating
into Transmission-Line Modelling (TLM) a semi-classical
model to simulate the quantum properties of a two-level
medium. The approach is accurate and free from numerical
instabilities. The approach was validated for the small-signal
case by a demonstration of the correspondence principle
with the classical Lorentz oscillator. Results obtained for
propagation in the two-level medium are in agreement with
previous studies in the literature. In addition, results are
shown demonstrating self-induced transparency (SIT) and
laser oscillations.
Index Terms— Time-domain electromagnetics, Classical
Lorentz oscillator, two-level quantum medium, Bloch equa-
tions, Nonlinear dielectrics.
I. INTRODUCTION
In this paper an approach for the simulation of pulse
propagation in two-level dielectric media is described.
The method is based on the Transmission-Line Modelling
(TLM) method. Solutions are obtained by a direct consid-
eration of the interaction of the classical electromagnetic
field with a two-level quantum medium. Thus the results
are free from the usual approximations of rotating waves
and slowly varying envelopes. The only approximation
introduced is the discretization of space and time.
Related work in the Finite-Difference Time-Domain
(FDTD) method is in [1], where a scheme for the sim-
ulation of pulse interactions in two-level medium is devel-
oped. This formulation was employed in [2], [3] for the
simulation of diffraction gratings. Other related work in
FDTD includes the two-level model of Nagra and York [4]
and the simulation of inhomogeneously broadened media
by Schlottau et al. [5].
As noted in [5], the approach in [1], [2], [3] suffers
from late-time inaccuracy because the numerical imple-
mentation has been destiffened. This refers to the artificial
damping of the solution at late times because of the time
integration of a constant term. In this formulation, the
constant term is removed from the differential equations
prior to discretization leading to a stable solution which is
late-time accurate.
The paper is organized as follows: In section II, the for-
mulation of the approach is described: In section II-A, the
classical Lorentz oscillator is reviewed and the polarization
is split into in-phase and in-quadrature components to
allow a straightforward extension of the classical Lorentz
oscillator to a two-level dielectric. In section II-B, the
Bloch equations describing a two-level dielectric medium
are reviewed [6] and in section II-C, it is shown that for the
small-signal excitation of the ground state, the two-level
Bloch model corresponds to the classical Lorentz oscilla-
tor. In section II-D, an outline of the one-dimensional TLM
method [7] is given and in section II-E, the discretization
of the Bloch equations for a two-level dielectric material
is shown. In section III, the quantum model is validated
for small signals by comparison with the classical Lorentz
oscillator. In addition, results are shown demonstrating
absorption, amplification, self-induced transparency (SIT)
and lasing. The paper is concluded in section IV.
II. FORMULATION
A. Classical Lorentz Oscillator
The differential equation for the polarization of a clas-
sical Lorentz oscillator is
∂2
Py
∂t2
+ 2δ0
∂Py
∂t
+ ω2
0Py =
Nae2
m
Ey (1)
where δ0 is the damping frequency, ω0 is the resonant
frequency, Na is the density of oscillators, e is the electron
charge and m is the electron mass. In terms of macroscopic
Lorentz dielectrics, the coefficient on the right-hand side
(RHS) of (1) is usually written as Nae2
/m = ε0ω2
0∆χe
where the susceptibility strength is ∆χe. Equation (1) may
be solved using a system of two first-order differential
equations, i.e.,
∂
∂t
Py1
Py2
=
−δ0 β0
−β0 −δ0
·
Py1
Py2
+
0
Nae2
β0m Ey
(2)
where w.r.t. the electric field, Py1 ≡ Py is the in-
phase component of the polarization and Py2 is the in-
quadrature component of the polarization. The damped
natural frequency is β0 = ω2
0 − δ2
0.

2. B. Two-level Dielectric
The Bloch equations for a two-level dielectric medium
are [6],
∂
∂t


ρ1
ρ2
ρ3

 =


0 β0 0
−β0 0 −2ωR
0 2ωR 0

·


ρ1
ρ2
ρ3


−


1/T2 0 0
0 1/T2 0
0 0 1/T1

·


ρ1
ρ2
ρ3 −ρ30

 (3)
In (3), [ρ1 ρ2 ρ3]T
is the state density vector, with ρ1 (ρ2)
proportional to the in-phase (in-quadrature) polarization
and ρ3 proportional to the inversion population. Also in
(3), T2 is the dephasing time, T1 is the inversion decay
time, ωR is the Rabi frequency and ρ30 is the initial
inversion state density. By specifying ρ30 = −1(1) means
that the initial condition of the medium is the ground
(excited) state. The Rabi frequency is proportional to the
electric field, i.e.,
ωR = (ea0/¯h)Ey (4)
where ea0 is the dipole moment.
Recognizing that the constant term ρ30 will cause insta-
bility in a time-domain solution, we define
∆ρ3 = ρ3 − ρ30 (5)
Substituting (5) into (3) gives
∂
∂t


ρ1
ρ2
∆ρ3

 =


−1/T2 β0 0
−β0 −1/T2 −2ωR
0 2ωR −1/T1

·


ρ1
ρ2
∆ρ3


+


0
−2ωRρ30
0

 (6)
Equations (3) can be converted to polarizations using the
relations Pyi = −Nae a0 ρi where i ∈ {1, 2, 30} and
Py3 = −Nae a0 ∆ρ3 leading to
∂
∂t


Py1
Py2
Py3

 =


−1/T2 β0 0
−β0 −1/T2 −2ωR
0 2ωR −1/T1

·


Py1
Py2
Py3


+


0
−2ωRPy30
0

 (7)
C. Correspondence for Small-Signals
To show that (7) reverts to the classical Lorentz oscil-
lator (2), assume small signals so that ∂Py3/∂t ∼ 0 and
that the medium is in the ground state so ρ30 = −1, i.e.,
Py30 = Nae a0. Using these assumptions and (4) in (7)
yields
∂
∂t
Py1
Py2
=
−δ0 β0
−β0 −δ0
·
Py1
Py2
+
0
2Nae2
a2
0
¯h Ey
(8)
where δ0 = 1/T2. Comparison of (8) with (2) shows that
when the medium is in the ground state and excited by
small signals, the two-level Bloch model has reduced to
the classical Lorentz oscillator. The only difference is in
the electric field interaction coefficients: In the classical
model (2), it is Nae2
/(β0m), whereas in the Bloch model
(8), it is 2Nae2
a2
0/¯h. Thus for correspondence between the
models for small-signals, the two interaction coefficients
may be made equal by choosing a0 = ¯h/(2β0m).
D. Outline of One-Dimensional TLM
The one-dimensional (1-D) model was developed in [7],
thus only an outline is given here. For 1-D propagation in
x, polarized in y in a non-magnetic dielectric material,
−
∂Hz
∂x
= ε0
∂Ey
∂t
+ ε0χeb
∂Ey
∂t
+
∂Py
∂t
(9)
In (9), Py represents the polarization due to the electrons
and ε0χebEy is the background polarization. The normal-
izations are
Ey = −Vy/∆ℓ , Hz = −iz/(∆ℓ η0) , Py = −py ε0/∆ℓ
(10)
where η0 is the intrinsic impedance of free-space. The time
and space derivatives in (9) are normalized using
∂
∂t
=
1
∆t
∂
∂T
,
∂
∂x
=
1
∆ℓ
∂
∂X
(11)
where ∆t is the time-step and ∆ℓ is the space-step. In
normalized form, (9) is
2(V i
4 + V i
5 ) = 2Vy + χeb
∂Vy
∂T
+
∂py
∂T
(12)
where V i
4 and V i
5 are the incident voltages on a
transmission-line segment. The discrete model of (12) is
obtained from application of the bilinear Z-transform,
∂
∂T
= 2
1 − z−1
1 + z−1
(13)
where z−1
is the time-shift operator. This leads to an
expression for the total voltage Vy.
Vy + 2Teypy = Tey(2V r
y + z−1
Sey) = Uey (14)
where the coefficient Tey = (2 + 2χeb)−1
and V r
y = V i
4 +
V i
5 . The accumulator Sey in (14) is obtained from
Sey = 2V r
y + κeyVy + 2py (15)
where κey = −(2+2χeb). Also in (14), Uey is the electric
field forcing function.

3. E. Time-domain Model of the Two-level Dielectric
The normalized form of (7) is
∂
∂T


py1
py2
py3

 =


−Ω2 B0 0
−B0 −Ω2 −2ΩR
0 2ΩR −Ω1

·


py1
py2
py3


+


0
−2ΩRpy30
0

 (16)
where B0 = β0∆t, ΩR = ωR∆t, Ω2 = ∆t/T2 and Ω1 =
∆t/T1. The normalized Rabi frequency is
ΩR = ΓVy/2 (17)
where Γ = 2ea0/(¯hc). Using (17) in (16) yields:
∂
∂T


py1
py2
py3

 =


−Ω2 B0 0
−B0 −Ω2 −ΓVy
0 ΓVy −Ω1

·


py1
py2
py3


+


0
−ΓVypy30
0

 (18)
Application of the bilinear transform (13) to (18) and
rearranging gives:
Vy = −2Teypy1 + Uey (19)
py1 = Te2[B0py2 + z−1
Upy1] (20)
py2 = Te2[−B0py1 −ΓVy(py3 +py30)+z−1
Upy2](21)
py3 = Te1[ΓVypy2 + z−1
Up3] (22)
where Te2 = 1/(2 + Ω2) and Te1 = 1/(2 + Ω1). The
forcing functions in (19–22) are found using:
Uey = Tey(2V r
y + z−1
Sey) (23)
Upy1 = κ2py1 + B0py2 (24)
Upy2 = −B0py1 + κ2py2 − ΓVy(py3 + py30) (25)
Up3 = ΓVypy2 + κ1py3 (26)
where κ2 = 2 − Ω2 and κ1 = 2 − Ω1 and the accumulator
Sey follows from (15). It is not possible to obtain an
explicit solution for equations (19–22). This system is
solved using a Gauss-Siedel iteration, i.e.,
Vy,k+1 = −2Teypy1,k + Uey (27)
py1,k+1 = Te2[B0py2,k + z−1
Upy1] (28)
py2,k+1 = Te2[−B0py1,k+1
− ΓVy,k+1(py3,k +py30)+z−1
Upy2] (29)
py3,k+1 = Te1[ΓVy,k+1py2,k+1 +z−1
Up3] (30)
where k is the iteration number. The starting values are the
those saved from the previous time-step and the iteration
is performed until the values have converged to within a
suitable tolerance parameter.
-1
-0.5
0
0.5
1
70 70.5 71 71.5 72 72.5 73 73.5 74 74.5 75
Electricfield(V/m)
Distance (um)
Linear Lorentz (lossy)
Two-level (lossy)
Linear Lorentz (gain)
Two-level (gain)
Fig. 1. Classical Lorentz oscillator compared with the two-level medium
for small-signals
III. RESULTS
A. Small-Signal Plane-Wave Propagation
The small-signal responses of the classical Lorentz os-
cillator and the two-level material are compared in Fig. 1.
The properties of the Lorentz medium were:
∆χe = ±170.445 × 10−6
, χeb = 0
β0 = 2π × 200 × 1012
s−1
, δ0 = 1010
s−1
(31)
The positive (negative) value of ∆χe gave the result for
the lossy (gain) Lorentz material. The properties of the
two-level medium were [1]:
Na = 1024
m−3
, e a0 = 10−29
C m
β0 = 2π × 200 × 1012
s−1
, T2 = T1 = 10−10
s
ρ30 = ±1 , χeb = 0 (32)
The positive (negative) value of ρ30 gave the initial in-
version population of the gain (lossy) two-level medium.
The space-step was ∆ℓ = 75nm and the problem spaces
consisted of 2000 cells with 100 cells modelling free-space
at both ends. The excitation was a derivative Gaussian
function and the simulations were stepped for 1000 time-
steps, with the time-step set to ∆t = ∆ℓ/c. Fig. 1
shows that the classical Lorentz oscillator and the two-
level medium yield almost identical results for small-signal
loss and gain cases in agreement with the discussion in
section II-C.
B. Loss and Gain of Small-Signals in a Slab
Fig. 2 shows the results for loss and gain. For these
simulations, the space-step was ∆ℓ = 9.375nm and the
problem space consisted of 1920 cells where the 960
cells in the center of the domain modelled the two-level
medium. Sinusoidal excitations of amplitudes 1V m−1
and
angular frequency 2π × 200 × 1012
s−1
were used. The
material properties were the same as those in (32), except
that T2 was modified to 10−12
s.

4. -4
-2
0
2
4
6
0 2 4 6 8 10 12 14 16 18
Electricfield(V/m)
Distance (um)
Gain
Loss
Fig. 2. Gain and loss in the two-level medium
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
58 60 62 64 66 68 70 72 74
Normalizedelectricfield/populationinversion
Distance (um)
Ey (SIT)
Inversion (SIT)
Ey (small-sig.)
Fig. 3. Self-induced transparency compared with small-signal pulse
C. Self Induced Transparency
Fig. 3 shows the response to an intense pulse. The ex-
citation pulse was a modulated hyperbolic secant function
with a peak of 5.607GV m−1
. The result after 1000 time-
steps is shown in Fig. 3. This intense pulse produces self-
induced transparency (SIT), i.e. the pulse retains its shape.
This pulse has the property of inverting the medium and
then returning it back to the ground state. Also shown in
Fig. 3 is the dispersion of a small-signal pulse with peak
1 V m−1
.
D. Laser Action
In Fig. 4, the time-domain response of the peak electric
field output into the free-space region on the RHS of the
medium is shown. The problem space was identical to the
example in Fig. 2. The material properties were as in (32),
except that T2 = 10−12
s and χeb = 11.89. The presence of
the background dielectric susceptibility causes reflections
at the ends of the layer leading to a build up of oscillations.
The problem was excited with a derivative Gaussian pulse
and was stepped for 80000 time-steps. Fig. 4 shows the
build up and decay of oscillations in the slab for a fully
inverted initial population. The spikes in this figure are
due to the excitation pulse. Note that at about 1.3ps, the
energy output has depleted the population inversion and
the oscillations begin to decay. Finally, Fig. 5 shows the
electric field distribution at the end of the simulation.
0
20
40
60
80
100
0 0.5 1 1.5 2 2.5
Peakelectricfield(MV/m)
Time (ps)
Fig. 4. Peak electric field in the free-space region on the RHS of the
laser model
-50
-40
-30
-20
-10
0
10
20
30
40
50
0 2 4 6 8 10 12 14 16 18
Electricfield(MV/m)
Distance (um)
Fig. 5. Electric field in the laser model at the end of the simulation
(∼2.5ps)
IV. CONCLUSION
In this paper we have developed a TLM approach for
the simulation of two-level dielectrics. The detail of the
numerical algorithm was given and 1-D electromagnetic
wave propagation in a two-level medium was simulated.
The TLM solutions were accurate and stable at late-
times. Results demonstrating absorption, gain, self-induced
transparency and laser action were shown.
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