Optics and Photonics Journal, 2013, 3, 15-20
doi:10.4236/opj.2013.32B004 Published Online June 2013 (http://www.scirp.org/journal/opj)
Study on the Gain Material with Four Energy Level
Model Using FDTD Method
Hui Xue1, Zhixiang Huang1, Xianliang Wu1,2
1Key Laboratory of Intelligent Computing and Signal Processing, Anhui University, Hefei, China
2Department of Physics and Electronic Engineering Hefei Normal University, Hefei, China
Received 2013
ABSTRACT
A faster numerical method based on FDTD for the four energy level atomic system is present here. The initial condi-
tions for the electrons of each level are achieving while the fields are in steady state. Polarization equation, rate equa-
tions of electronic population and Maxwell’s equations were used to describe the coupling between the atoms and elec-
tromagnetic wave. Numerical simulations, based on a finite-difference time-domain (FDTD) method, were utilized to
obtain the population inversion and lasing threshold. The validity of the model and its theory is confirmed. The time,
which we can observe the lasing phenomenon, is much shorter in our new model. Our model can be put into using in
large scale simulations in mutiph ysics to reduce the total simulated time.
Keywords: Finite-Difference Time-Domain; Gain Material; Lasing
1. Introduction
The system is always treated either semi classical or fully
quantum mechanical [2, 3] while a high-frequency light
is incident on a medium. Because of metallic nature of
metamaterials constituent metamolecules, they suffer from
high dissipative losses in the range of optical frequencies.
Losses are too large in the real applications. It is better to
incorporate the ga in media into matematerial to compen-
sate the losses. When an electromagnetic wave propa-
gates in a medium, the dipole moment in the individual
atom changes, in turn changing the total field coupling to
the medium until a steady state established. A full-vec-
torial time domain approach is utilized to do self-consistent
calculations. Fin ite-difference time-domain (FDTD) method
[4] is used as a powerful tool in modeling linear disper-
sive media [5, 6]. In an attempt to achieve more realistic
simulations two-level Maxwell-Bloch equations can be
solved using iterative predictor-corrector finite difference
time-domain FDTD methods to demonstrate saturation
and self-induced transparency [7].
To simulate lasing dynamics, we present here a faster
numerical simulation model for the four energy level
atomic system. We use the populations of each level
while the fields are achieving steady state as the initial
value, in this way, the simulation time will sharply be
reduced. The electromagnetic fields and atomic energy
level populations at any time step can be calculated in
terms of known quantities. Comparing the results of this
model with those which putting all the electrons on the
ground state level (E0). We can find that the results of
two methods are similar. So the validity of the model and
its theory is confirmed.
2. Theoretical and Numerical Model
2.1. Rate Equations
A simplified four level atomic system with energy levels
E0, E1, E2, E3 and populations upon each level N0, N1, N2,
N3 is depicted in Figure 1.
In our model, the gain atoms are embedded in the each
level of host medium aforehand. The electrons of the
ground state level are pumped to the third level by the
some external pumping mechanism (Pr). After a very
short time period

32, the electrons of the third level (E3)
fall into the second level (E2) by a non-radioactively
transition. A population accumulates in the second level
Figure 1. Schematic of the four-level atomic system model.
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opyright © 2013 SciRes. OPJ

H. XUE ET AL.
16
due to the lifetime of the laser transition is longer than
other lifetime, e.g.

32. The amount of electrons in the
second level (E2) is larger than the amount of electrons in
the first level (E1).Then a population inversion is
achieved. The lasing and optical amplification E2 will
happen at the frequency of

21 = (E2-E1)/ħ. At last, the
electrons transfer quickly and non-radiatively from the
first state level (E1) to the ground state level (E0).
In the traditional four-level atomic system model, the
populations can be modeled by the following rate equa-
tions:
33 0
32
r
NNPN
t 

 (1a)
3 22
32 21
1
a
N NN P
E
tt  

 


 (1b)
121
21 10
1
a
NN NP
E
tt  




 (1c)
121
21 10
1
a
NN NP
E
t  




t
(1d)
We set the right of the equations as zero. The physical
interpretation of this is owing to the steady state for the
populations of each level. So the corresponding homo-
geneous equations can be rewritten as follows:
3
32
NN
t
3


 (2a)
3 2
32 21
N N
t
2
N



 (2b)
12
21 10
NNN
t
1



 (2c)
0
32
NN
t
3


 (2d)
2.2. Classical Electron Oscillator Model
Using the Lorenz Model, the electric polarization in real
atomic transitions can be described by the following
equation:
22
2
00
2
()() ()() ()
r
c
dPt dPte
Pt NtEt
dt mdt

    (3)
where e is the charge of an electron. m is the mass of an
electron and N = N2  N1.

r = 1/

21 is the real decay rate
of the second level. c = 1/

21 is the classical rate.

0 is
the total energy decayrate and 

0 = 1/

21 + 2/T2. T2 is
the mean time between dephasing events.
2.3. FDTD Formulation
Presuming a two-dimensional problem of a plane wave
propagating along +y direction, using spatial and tempo-
ral interleaving of the fields and the central differencing
scheme. We can write the discretized equations for the
electric and magnetic fields as follows:
11
22
0
11
(, )(, )
2
[(, 1)(,)]
nn
xx
nn
zz
Hij Hij
tEij Eij
y 

2






(4a)
11
22
0
11
(,) (,
2
[(1,) (,)]
nn
yy
nn
zz
Hi jHi j
tEij Eij
y 

 



)
2
(4b)
1
1
11
22
11
22
(1,)(1,)
(, )(, )
1
[(,) (,
22
11
[(,) (,)
22
nn
zz
nn
zz
nn
yy
nn
xx
Ei jEi j
PijPij
tHi jHi j
x
tHij Hij
y







 








1
)]
]
(4c)
The macroscopic polarization has a nonzero compo-
nent in the x direction alone. The polarization equation
can be discretized as:
22
10
0
1 0
0
22
12
0
2
(1,) (,)
1/2
1/2
(, )
1/2
[(, )(, )](, )
1/2
nn
zz
n
z
r
nnn c
z
t
Pi jPij
t
tPij
t
et
mNij NijEij
t











 
 
 


 
(5)
Similarly, the discrete rate equations are:
132
33
32
32 0
32
2
(, )(, )
2
2(, )
2
nn
n
r
t
N
ijN ij
t
PNij
t







(6a)
11 21
021
121
22
21
1 21 33
32 21
[ (,)(,)][ (,)(,)]
(2 )
2
(, )(, )
2
[(,) (,)]
()
nnnn
zzzz
nn
nn
EijE ijPijPij
t
t
Nij Nij
t
tNijNij
t

















(6b)
11 10
010
110
11
10
1 10 22
21 10
[(, )(, )][(, )(,)]
(2 )
2
(, )(,)
2
[(,)(,)]
(2 )
nnnn
zzzz
nn
nn
EijE ijPijP ij
t
t
Nij Nij
t
tNijNij
t

















(6c)

H. XUE ET AL.
17
The initial state is t = 0. So we can embed the gain
atoms in the each level of host medium aforehand base
on above equations. There is no polarization at first.
However, the system begins to evolve while electrons are
pumped by exte rnal p umping into hig h energy level.
11
01
10
0
(, )[(, )(, )]
2
(1)( ,)
nn
n
r
t
1
n
N
ijNijN ij
tPNij





(6d)
3. Numerical Results We set parameters as fellows. The slab width is 300
nm. The gain slab width is 100nm. The discrete time and
space s teps are chosen a s ∆t = 1.667 × 10-17 s and ∆x = 1
× 10-8 m. The center frequency of the radiation 
0 is set
as 2 × 1014 Hz. ∆
ω
0 is set as 5THz. The p arameters

10,

21 and

32 are chosen as 5 × 10-14 s, 5 × 10-12 s and 5 ×
10-14 s. The total electron density (Nt) is chosen as 6 ×
1023 m-3.The total iterated time step is set as 1.5 × 106.
The frequency of the CW wave is 250 THz.
The 2D system incorporated a gain material as showing
in Figure 2.
By solving the differential equations (1a)-(1d) the
amount of electrons of each level in each time step can
be gotten as (8a)-(8b):
32
/
30
() t Nt ce


 (7a)
32 21 /
/021
21
21 32
() t
tc
Nt cee







 (7b) We first set the pump rate (Pr) as 9 × 107 1/s. A con-
tinuous wave is generated. We then let it propagate
through the gain slab. We may find from the Figures 3
(a) and (b) that there is no lasing. As we increase the
pump rate (Pr), the lasing phenomenon appear, see Fig-
ures 3(c) and (d). With the pump rate increases (Figures
3(e) and (f)), th e peak be comes larg er. The Figures 3(a),
(c) and (e) are the situations when all the electrons are
embedded on the level (E0). The Figures 3(b), (d) and (f)
are the situations when electrons are embedded on each
level base on the equations (7a)-(7d).

10 21
32
//
110
12
10 21
/
010 32
21 32 1032
() tt
t
c
Nt cee
ce












(7c)

10 21
32
//
121
032
10 21
2/
032
21 32 1032
() tt
t
c
Ntc cee
ce









 

(7d)
where

1
010 102132
()
r
cNP P 

1
r
(8a)

1
021
121102132
21 10
()
rr
c
cNP P


1


 (8b)

1
110
210102132
10 21
()
r
c
cNPP


1
r


 (8c)


010 121
32
102121 10
1
1021 32
()
r
c c
cc
NP1






 

 
(8d)
Figure 2. Schematic of the simulated structure.
(a)The situation when all the electrons are embedded on the level (E0).The pump rate is 9 × 107 1/s.

H. XUE ET AL.
18
(b) The situation when electrons are embedded on each level base on the equations (7a) - (7d). The pump rate is 9 × 107 1/s.
(c) The situation when all the electrons are embedded on the level (E0).The pump rate is 9 × 108 1/s.
(d) The situation when electrons are embedded on each level base on the equations (7a) - (7d). The pump rate is 9 × 108 1/s.

H. XUE ET AL.
19
(e)The situation when all the electrons are embedded on the level (E0).The pump rate is 9 × 109 1/s.
(f) The situation when electrons are embedded on each level base on the equations (7a) - (7d). The pump rate is 9 × 109 1/s.
Figure 3. The Electric field of the reflected wave, transmitted wave, the Fourier transform results, and the normalized dif-
ference of electron number between the upper level and lower level under different pump rate.
Populations changing can also be observed in the Fig-
ure 4. The Figure 4(a) is the situation when all the elec-
trons are embedded on the ground state level, we may
record this as case a. The Figure 4(b) is the situation
when electrons are embedded on each level base on the
equations (7a)-(7d), namely case b. Here we use the
pump rate of 9 × 108 1/s. If we increase the pump rate
further, we may see the populations begin to oscillating
strongly. It is easy to see that case b has a shorter lasing
time than the case a does. That means our model costs
Figure 5 shows the c
less time to start to lase.
omparisons of the lasing time
un
4. Conclusions
umerical model for gain materials
der two situations, case a and case b, with different
pumping rates. We may find that the case b needs less
simulated time to lase than case a. Which means our new
system takes less time to start to lase than the system
which all the electrons are embedded on the ground level.
Furthermore, when the pump rate gets higher the lasing
time which the system begins to lase is decreasing.
A modified FDTD n

H. XUE ET AL.
20
Figure 4. The two situations’ populations change in each level with different pump rate.
Figure 5. The lasing time of two situations under differen
nd numerical simulations based on the lasing dynamics
5. Acknowledgements
the National Natural Sci-
ence Foundation of Anhui Province (Nos.KJ2011A002,
[1] Anan Fang, Tstas M Soukoulis,
“Lasing in mes,” Journal of Op-
t
C
opyright © 2013 SciRes. OPJ
pump rates.
a
are presented here. The system begins to lase while the
pumping rate increased to a certain value. Our modified
model obviously reduces the lasing time (the time when
the system start to lase) and the simulation results agree
with the classic theory. Our model can be applied in lar-
ger electromagnetic and optics physics simulations to
reduce the total simulated time.
This work was supported by
ence Foundation Of China under Grant (Nos.60931002,
61101064, 51277001), Distinguished Natural Science
Foundation (No.1108085J01), Universities Natural Sci-
KJ2012A013) and 211 Project of Anhui University.
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