Faylali et al., International Journal of Advanced Engineering Technology
E-ISSN 0976-3945
Research Paper
MODELING OF OPTICAL PULSE PROPAGATION IN KERR
AND RAMAN NONLINEAR DISPERSIVE MEDIA USING
ADE-TLM METHOD
H. El Faylali, M. Iben Yaich and M. Khaladi
Address for Correspondence
Electronic & Microwaves Group Faculty of Sciences, Abdelmalek Essaadi University P. O. Box: 2121,
Tetuan 93000 Morocco
ABSTRACT
In this paper, we propose a simulation model of electromagnetic wave’s propagation, in media with different kinds of
dispersions. This approach is based on the existing Auxiliary Differential Equation (ADE) technique in the context of
the Transmission Line Matrix method with the symmetrical condensed node (SCN -TLM) and novel voltage sources.
The proposed model, named ADE-TLM, gives a full solution of Maxwell’s equations and polarisation terms which
describe the Lorentz linear dispersion, the nonlinear instantaneous Kerr and retarded Raman effects.
KEYWORDS Auxiliary Differential Equation (ADE) technique, Finite-Difference Time Domain (FDTD),
Transmission Line Matrix (TLM), dispersion, Kerr, Lorentz, Raman.
I. INTRODUCTION
Based on different techniques of discretization of
own equations, several differential numerical time
domain methods have efficiently simulated
nonlinear dispersive media. These approaches
include the linear and nonlinear properties and take
into account physical complexity of media such as
Kerr effects, Raman diffusion and/or photonic
absorption. The Finite-Difference Time-Domain
(FDTD) method and the numerical resolution of
polarization and Maxwell’s equations are often
used to characterize intrinsically nonlinear media,
and to predict the behavior of propagated
electromagnetic (EM) waves [1]-[3].
By incorporating judiciously the polarization
vector and Maxwell's equations, The FDTD
method based on the auxiliary differential equation
(ADE) technique has effectively modeled the
interaction of femtosecsond pulses in silica fibers
[4],[5]. The Z-transform technique combined with
the FDTD method, gave consistent results for the
simulation of solitons in nonlinear dispersive
media [6], [7]. In order to simulate the wave’s
propagation in dispersive media with complex
permittivity, methods based on the discrete
convolution of the dispersion relation are presented
in [8].
The nonlinear Schrödinger equation was applied to
model the propagation of short optical pulses in
Kerr media [9]. The Alternating-Direction Implicit
Finite-Difference Time-Domain (ADI-FDTD) has
been introduced as an unconditionally stable
implicit method to solve the problems of EM
radiation in nonlinear dispersive media [10].
Front of the variety of numerical techniques
employed and the rigor of the proposed approaches
and models developed by the FDTD method to
simulate EM wave propagation in nonlinear media
and predict their interactions, the TLM method
provides further opportunities for proposals and
development of new modeling tools and
investigation despite of the numerous studies
performed in this field [11]-[15].
In this paper, we increase the applicability of the
TLM numerical method to model the linear and
nonlinear dispersion properties. A novel algorithm
based on the TLM method with condensed
symmetrical node (SCN) and new voltage sources
combined with the auxiliary differential equation
ADE, is used to model the Lorentz dispersion, the
Int J Adv Engg Tech/IV/III/July-Sept.,2013/01-04
nonlinear instantaneous Kerr and the retarded
Raman effects in nanostructured nonlinear optical
media.
Numerical results that demonstrate the validity and
efficiency of the proposed algorithm TLM-ADE as
well as the nonlinear formulation of the electric
field are presented.
Excellent accordance is achieved between the
ADE-TLM calculated and exact theoretical results
for the reflection coefficient in Lorentz, Kerr and
Raman media.
II. THEORY
In our proposed model, we reformulate the
auxiliary differential equation (ADE) technique in
the context of the TLM method. This approach
allows the simulation of the propagation of optical
pulses in linear Lorentz and nonlinear Kerr and
Raman medias, by solving Maxwell's and
polarization equations describing the effects of the
media on the EM wave’s propagation. This
approach is based on the time discretization
applied to the components of the EM field and
polarization vector by using centered differencing.
A. Nonlinear Maxwell’s equations
The Maxwell-Ampere equation for an EM wave
propagating along the z-direction (i.e., E x
component) in a dispersive nonlinear medium can
be written as follows:

∂Ex (t)
1  ∂Px (t)
=
+ ∇ΛH 
−
x
∂t
ε0ε∞  ∂t

(
)
(1)
Where ε 0 is the dielectric constant of free space,
ε ∞ is the relative dielectric constant in the limit of
infinite frequency and Px (t) = PL (t) + PNL (t) is the
polarization term. The linear part PL (t) = PLorentz (t)
describes the linear Lorentz dispersive properties
of the medium and the nonlinear one
PNL (t) = PKerr (t)+ PRaman (t)
represents
the
instantaneous Kerr and retarded Raman effects.
E x and Px are respectively the components, along
the x-axis, of the electric field and polarization
vectors.
Using centered differencing scheme between time
n+1
n
steps t = (n + 1)∆l and t = n∆l , we can write
the discretized equation for the electric and
polarization
fields
in
regular
space
Faylali et al., International Journal of Advanced Engineering Technology
( ∆x = ∆y = ∆z = ∆l )
χ0(3) being the third order nonlinear susceptibility
and α , 0 ≤ α ≤ 1 , represents the relative strengths
as follows:
1
n+ 
1
∆t 
2
(Pxn+1 - Pxn )+
 ∇ΛH 
ε0ε∞
ε 0ε ∞ 
x
E nx +1 = E nx -
(2)
To develop a TLM model for the numerical
treatment of these problems we need to transform
the EM field parameters into circuit parameters
that are related to transmission lines. Substituting
in (2), the component E x by its electric equivalent
Vx
, we obtain the expression of the discretized
∆l
n+1
electric voltage component Vx along the x-axis at
∆l
ε 0ε ∞
(Pxn+1 - Pxn ) +
1
n+ 
∆t∆l 
 ∇ΛH 2 
ε 0ε ∞ 
x
(3)
The proposed ADE-TLM algorithm, unifies all
types of dispersion (of the second and the third
order) of medias expressed by the polarization
terms appearing in (3), in a single formula
expressing the component of the electric voltage
along the z-direction.
In the sequel, we give the numerical expressions of
the polarization of the second and the third order
( PL (t) and PNL (t) ) in terms of electric voltage.
B. Second-order linear dispersion
The Lorentz polarization as a function of electric
voltage is governed by the following auxiliary
differential equation:
∂2PLorentz (t)
∂P
(t)
+2δLorentz Lorentz
2
∂t
∂t
ε
ε
(
−ε ) 2
2
PLorentz (t) = 0 Lorentz ∞ ωLorentz
V(t),
+ωLorentz
∆l
(4)
frequency, δ Lorentz the damping factor and ε Lorentz
the static permittivity caused by the Lorentz
dispersion.
By using centered differencing, the polarization of
n+1
n+1
the Lorentz dispersion PLorentz at time step t
obeys to the following difference equation:
Vxn
∆l
(5)
Where
a Lorentz =
2
2-ωLorentz
∆t 2
1 + δ Lorentz ∆t
b Lorentz =c Lorentz =
1 − δ Lorentz ∆t
1 + δ Lorentz ∆t
2
ε 0 (ε Lorentz − ε ∞ )ωLorentz
∆t 2
1 + δ Lorentz ∆t
C. Third-order nonlinear dispersion
The polarization caused by the instantaneous Kerr
nonlinearity is written, as function of the electric
voltage by:
PK err (t) =
=
ε 0 χ 0( 3 )
∆l
αε0χ
∆ l3
t
V (t) ∫
-∞
(3)
0
α δ (t - t ' )
∆ l2
V 2 (t ' )dt '
|V 2 (t)|V (t)
IJAET/Vol. IV/ Issue II/April-June, 2013/
n+1
PKerr
=
αε 0 χ 0(3)
|Vxn+1 |2 Vxn+1
(7)
∆l
As a convolution, the polarization caused by the
Raman effect may be writing as:
3
ε0
PRaman (t) =
∆l
(3)
V(t)χ Raman
(t) *
V 2 (t)
∆l 2
V 2 (t)
(9)
∆l 2
Then we apply the Fourier transform to (9):
2
% ω ) = χ (3) (ω ) * ℑ  V (t) 
S(
Raman
 2 
 ∆l 
(10)
(3)
S(t) = χ Raman
(t) *
Where S and χ denote the Fourier transforms of
the corresponding time functions.
The retarded response function is:
2
(3)
(1 − α ) χ 0(3)ωRaman
χ Raman (ω )= 2
ωRaman + 2 jωδ Raman − ω 2
(11)
With
τ 12 + τ 22
τ 12τ 22
ωRaman =
Where ωLorentz is the Lorentz characteristic resonant
n+1
n
n-1
PLorentz
= a Lorentz PLorentz
+ b Lorentz PLorentz
+ c Lorentz
of the Kerr and Raman polarizations.
Analogous to the Lorentz case, the polarization of
n+1
n+1
the Kerr dispersion PKerr at time step t can be
written as follows:
(8)
To solve (8), we introduce an auxiliary variable
S(t) for the convolution:
n+1
time step t :
Vxn+1 =Vxn -
E-ISSN 0976-3945
δ Raman =
1
τ2
Next insert (11) into (10), multiply by
2
2
( ωRaman + 2 jωδ Raman − ω ) and transform to the time-
∂2
∂
and ω 2 → − 2 to
∂t
∂t
get the following auxiliary differential equation:
domain functions by jω →
2
(1- α )χ 0(3)ωRaman
∂ 2 S(t)
∂S(t)
2
δ
ω
+
2
+
S(t)
=
Vx2 (t)
Raman
Raman
∂t 2
∂t
∆l 2
(12)
The finite difference expression for the time
n+1
derivative of (12) centered at time step t
is then
given by:
c
Sn+1 = a Raman Sn +b Raman Sn-1 + Raman
Vxn 2
(13)
∆l 2
With
2
∆t 2
2-ωRaman
a Raman =
1 + δ Raman ∆t
b Raman =c Raman =
1 − δ Raman ∆t
1 + δ Raman ∆t
2
(1 − α ) χ 0(3)ωRaman
∆t 2
1 + δ Raman ∆t
n+1
Finally, the discretized Raman polarization PRaman
(6)
n+1
at time step t can be written as follows:
Faylali et al., International Journal of Advanced Engineering Technology
n+1
PRaman
=
ε0
Vxn+1Sn+1
(14)
∆l
III. ADE-TLM ALGORITHM
By substituting the terms of the polarization given
by (5), (7) and (14) in (3), we can write the
discretized nonlinear relation describing electrical
n+1
voltage at time step t as:
∆l
n+1
n
(PLorentz
-PLorentz
)
ε0ε∞
Vxn+1 =Vxn +
αχ 0(3)
∆l 2 ε ∞
(|V
| Vxn+1 -|Vxn |2 Vxn ) -
n+1 2
x
1
(Vxn Sn+1 + Vxn Sn )
ε∞
1
n+ 
∆t∆l 
 ∇ΛH 2 
ε 0ε∞ 
x
E-ISSN 0976-3945
χ (3) = 7 ×10−2 (V/m)−2 , τ1 = 12.2fs , τ 2 = 32fs
and α = 0.7 .
The considered TLM mesh dimensions are (1, 1,
10000) ∆l with space step ∆l = 8 nm. The airnonlinear dispersive medium interface is located at
z = 8∆l and excited by a unit amplitude pulse with a
sinusoidal carrier frequency fc = 0.137×1015 Hz
considering a hyperbolic secant envelope function
with a characteristic time constant of 14.6fs.
Fig. 1 shows the spatial evolution to the right of
the short pulse propagation, after 8000, 16000 and
32000 iterations for a linear dispersive medium.
Our results are in accordance with those presented
in [1], [14] and [18].
(15)
Writing the magnetic field components of (15) in
terms of incident and reflected pulses [16]-[17],
then applying the charge conservation principle for
the symmetrical condensed node SCN with a
capacitive stub of normalized admittance Yox (t)
we obtain the following expression for the
n+1
n+1
discretized electric voltage Vx at time step t :
Vxn+1 +
A 2 n+1 2 n+1
2
|Vx | Vx =
A1
4+Yox
 V1i + V2i + V9i + V12i 


+Yox V13i + 0.5Vsx 
n+1 
(16)
Where
A1 = ε ∞ + Sn+1
A2 =
αχ0(3)
Fig. 1. TLM results of the short pulse propagation
after 8000, 16000 and 32000 time steps respectively in
dispersive linear medium.
Fig. 2 shows the spatial evolution to the right of
the short pulse propagation, after 8000, 16000 and
32000 iterations for the Kerr and the Raman
medium. Our results are in accordance with those
presented in [1], [14] and [18].
∆l2
Vsx and Yox are respectively the voltage source,
and the normalized admittance terms in order to
take into account, in the TLM mesh, linear and
nonlinear dispersive properties and also Kerr
effects and Raman interactions in the nonlinear
medium. They are given by:
( ε ∞ + Sn+1 − A1 ) Vxn + 2A 2 |Vxn |2 Vxn 


Vsx = − n Vsx + 4  ∆l n+1

n
 − ε ( PLorentz -PLorentz )

 0

Yox = 4 ( ε ∞ + Sn+1 − 1)
(17)
n+1
(18)
The ADE-TLM model with voltage sources is
based on recursive calculation of normalized
admittance made in (18) and the voltage sources
expressed in (17). The obtained values are then
inserted in (16). The solution of the last equation is
then used in the calculation of reflected pulses and
in the connection process along the TLM mesh
nodes.
IV. NUMERICAL RESULTS
In order to show the efficiency of the new model
ADE-TLM with voltage sources, we present
spatial variations of electrical field propagation in
a medium having a Lorentz linear dispersion
characterized by the following [18]:
ε s = 5.25 , ε ∞ = 2.25 , ωLorentz = 0.4 ×1015 rad/s and
Fig. 2. TLM results of the short pulse propagation
after 8000, 16000 and 32000 time steps respectively in
dispersive nonlinear medium.
Next, the air-nonlinear dispersive medium
interface is exited by Gaussian unit amplitude
pulse. The damping factor is δ Lorentz = 0.1ωLorentz .
Fig. 3 compares the ADE-TLM calculated and
exact theoretical results [19] for the magnitude of
the reflection coefficient for Lorentz, Kerr and
Raman media.
δ Lorentz = 2 ×109 s−1 .
Fig. 3. Comparison of the reflection coefficient
magnitude for the Lorentz, Kerr and Raman media
as predicted by exact analytical results and computed
using ADE-TLM.
Further, the medium nonlinearity is assumed to be
characterized by [18]:
V. CONCLUSIONS
We have efficiently analyzed Maxwell's equations
Int J Adv Engg Tech/IV/III/July-Sept.,2013/01-04
Faylali et al., International Journal of Advanced Engineering Technology
and different polarization expressions for
spatiotemporal modeling propagation of optical
pulses in linear and nonlinear dispersive media
using the ADE-TLM model. In this model we
introduced a voltage sources modeling linear and
nonlinear properties and the variable admittance
concept.
The advantage of this new model is its modularity;
since it allows calculating separately each
dispersion and its accuracy was tested by he
obtained numerical results.
An excellent agreement was obtained between the
model ADE-TLM and the theoretical results.
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