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Numerical stability of solitons waves through splices in optical
fibers
Camila F. de Oliveira a,1
, Paulo L. Natti a,2,*
, Eliandro R. Cirilo a,3, Neyva M.
L. Romeiro a,4
and Érica R. Takano Nattib,5
a Universidade Estadual de Londrina, Departamento de Matemática (DMAT/UEL)
C.P. 6001, Londrina, Paraná, Brazil, 86051-990
b Pontifícia Universidade Católica do Paraná
Rua Jóquei Clube, 458, Londrina, Paraná, Brazil, 86067-000
1 E-mail: [email protected].
2 Corresponding author. E-mail: [email protected]. Web page: http://www.mat.uel.br/plnatti
Tel: 55-43-33714226. Fax: 55-43-33714216
3 E-mail: [email protected].
4 E-mail: [email protected].
5 E-mail: [email protected].
ABSTRACT. The propagation of soliton waves is simulated through splices in optical fibers, in which
fluctuations of dielectric parameters occur. The mathematical modeling of these local fluctuations of dielectric
properties of fibers was performed by Gaussian functions. By simulating soliton wave propagation in optical
fibers with Gaussian fluctuations in their dielectric properties, it was observed that the perturbed soliton
numerical solution presented higher sensitivity to fluctuations in the dielectric parameter β, a measure of the
intensity of nonlinearity in the fiber. In order to verify whether the fluctuations of β parameter in the splices of
the optical fiber generate unstable solitons, the propagation of a soliton wave, subject to this perturbation, was
simulated for large time intervals. Considering various geometric configurations and intensities of the
fluctuations of parameter β, it was found that the perturbed soliton wave stabilizes, i.e., the amplitude of the
wave oscillations decreases for increasing values of propagation distance. It is concluded that the propagation of
perturbed soliton wave presents numerical stability when subjected to local Gaussian fluctuations (perturbations)
of the dielectric parameters of the optical fiber.
Keywords: perturbed soliton propagation; optical fiber splice; numerical stability; Gaussian fluctuation; finite
difference method; Gauss-Seidel method.
PACS numbers: 02.60.Lj, 02.70.Bf, 42.65.Tg, 42.81.Dp
1. Introduction
In context of optical communication via solitons, the experiments performed in the late nineties
generated encouraging results. In 1998, Thierry Georges and his team at France Telecom, combining
optical solitons of different wavelengths, demonstrated data transmission of 1 terabit per second [1],
and in 2000, Algety Telecom, then located in Lannion, France, developed undersea telecommunication
equipment for the transmission of optical solitons. However, these promising results were not
translated into actual commercial soliton system deployments, in either terrestrial or submarine
systems, chiefly due to the Gordon–Haus (GH) jitter effect [2]. GH jitter requires a sophisticated and
expensive compensatory solution that ultimately makes the Dense Wavelength Division Multiplexing
(DWDM) soliton transmission unattractive. Consequently, in the last decade the long-haul soliton
transmission has remained as a subject of laboratory research. On the other hand, several solutions
have been proposed to minimize the jitter effect, such as Raman fiber amplifiers [3], tapering
dispersion fiber spans [4], sliding frequency guiding filters [5], in-line synchronous modulation [6],
among others. A review of the basic physics of the dynamics of solitons and futuristic applications of
solitons in optical communication can be found in [7].
In recent years, an increase in the number of theoretical and experimental works on soliton
communications, that aim to overcome the many well-known problems and improve the methods
already proposed, was published. Such studies approach themes related to the new soliton generation
processes [8,9], soliton propagation processes [10,11] and soliton stabilization processes [12-14] in
optical fibers.
This work is about propagation and stability of femtosecond optical solitons in fibers. The
propagation of these waves in optical fibers is affected by several disturbing processes. Usually, the
most important ones are group velocity dispersion (chromatic dispersion) and optical Kerr effect
(intensity dependence of the refractive index). Under certain circumstances, however, the effects of
Kerr nonlinearity and dispersion can just cancel each other, so that the temporal and spectral shapes of
the pulses is preserved even over long propagation distances. Taking only these disturbances into
account, the pulse propagation is a soliton described by a system of coupled nonlinear Schrödinger
differential equations [15-17]. The most remarkable fact about soliton waves is, actually, not the
possibility of balance of dispersion and nonlinearity, but rather that soliton solutions of nonlinear wave
equation are very stable: even for substantial deviations of the initial pulse from the exact soliton
solution, the pulse automatically finds the correct soliton shape.
On the other hand, to describe real-world fiber-optic systems, it is more realistic to include further
disturbing effects such as influence of fusion splice [18], Rayleigh scattering [19], high-order
dispersion and high-order nonlinearities [20], soliton self-steepening, Raman effect and self-frequency
shift [10], polarization-mode dispersion [13], nonlinear phase noise [21], among others [22].
It should be observed that the perturbed coupled nonlinear Schrödinger differential equations
systems, which describe wave propagation in real optical fibers, do not present analytical solution. In
the literature there are several numerical approaches whose objective is to describe the propagation of
perturbed solitons in dielectric environments, most of them using the finite difference method
[14,23,24] or the finite element method [25,26]. On the other hand, to solve numerically the resulting
system of equations, the authors use various methods like Newton's method [26], Crank-Nicolson
method [14], Runge-Kutta Method [27], among others. A review of the several numerical procedures
applied to describe the propagation of solitons in optical fibers is found in [28].
In a previous work [29], we described the propagation of soliton waves in ideal optical fibers
through a procedure based on the finite difference method and relaxation Gauss-Seidel method. By
comparing the obtained numerical results with the known analytical results, the validation of the
developed numerical procedure has been verified.
In this work, aiming to study the propagation and the stability of perturbed soliton waves, through
splices in optical fibers, the general numerical procedure developed in [29] is used. In Section 2 we
present the soliton analytical solutions of the coupled nonlinear differential equations system in case of
ideal )2( optical dielectric fibers. In Section 3, the numerical procedure to study the propagation of
the perturbed solitons in optical fibers is described. In Section 4 the propagation of perturbed soliton
waves is simulated through splices in optical fibers, where fluctuations of the dielectric parameters of
the fiber occur. The mathematical modeling of these local fluctuations of the dielectric properties was
performed by Gaussian functions with various geometric configurations and intensities. In Section 5
the main results of this work are presented.
2. Solitons in ideal dielectric fibers
This section studies the coupled non-linear complex partial differential equations (PDE) system,
obtained from Maxwell’s equations, which describe the longitudinal propagation of two coupled
electromagnetic waves (fundamental and second harmonic modes) in ideal ( )2χ dielectric optical
fibers [20]. The detailed mathematical modeling of this PDE system can be found in [16,17]. This
PDE system is given by
( ) 0-exp2
- 2
*
121
21 =+
∂
∂
∂
∂βξiaa
s
ar
ξ
ai (1)
( ) 0exp2
-- 212
22
22 =+∂
∂
∂
∂
∂
∂βξia
s
aα
s
aδi
ξ
ai ,
where 1-=i is the imaginary unit, ),(1 sξa and ),(2 sξa are complex variables that represent the
normalized amplitudes of the electrical fields of the fundamental and second harmonic waves,
respectively, with ),(*1 sξa and ),(*
2 sξa as their complex conjugates. The independent variable s has
spatial character, whereas the independent variable ξ has temporal character.
The real parameters α , β , δ and r , in (1), are related with the dielectric properties of the optical
fiber. The β quantity is a measure of the intensity of nonlinearity in the optical fiber, or a measure of
the generation rate of the second harmonic. The α quantity measures the relative dispersion of the
group velocity dispersion (GVD) of fundamental and second harmonic waves in the optical fiber. For
values 1>α , the second harmonic wave has higher dispersion than the fundamental wave and for
values 1<α , it is the fundamental wave that has higher dispersion. The r quantity is the signal of the
fundamental GVD wave. When 1r , the fundamental wave is in normal dispersion regime, but if
1-=r , the fundamental wave is in anomalous dispersion regime. Finally, parameter δ measures the
difference of group velocities of fundamental and second harmonic waves, so it accounts for the
presence of Poynting vector walk-off that occurs in birefringent media. It should be noticed that is
possible to choose the characteristics (velocity, width, amplitude, etc.) of the wave to be propagated in
the optical fiber, selecting or proposing materials with the appropriate α , ,β δ and r dielectric
properties [16,17].
The PDE system (1) presents soliton solutions [17], given by
+±= )-2
( )2-(2
3),(
2
1 βrα
δrα
rαsξa ])
-2-()
-2(
)-2(2
1[sech
22 +±× ξ
rα
δrsβ
rα
δ
αr
]} )-2
( -)-2
-)-2()-2(2
)54(( [ {exp 2
2
srα
δξ
αr
βr
αrrα
αrδri× (2)
)-2
()2-(2
3),(
2
2 βrα
δ
rα
rsξa += ])
-2-()
-2(
)-2(2
1[sech
22 +±× ξ
rα
δrsβ
rα
δ
αr
]})-2
(-)2-2
-)-2()-2(2
)54(( [2{exp 2
2
srα
δξ
β
αr
βr
αrrα
αrδri +× . (3)
In [29] a numerical procedure was developed based on the finite difference method and relaxation
Gauss-Seidel method to solve the propagation of soliton waves in optical fibers described by the PDE
system (1). By comparing the obtained numerical results with the known analytical solutions (2-3), the
validation of the developed numerical procedure was verified. This numerical development is
presented in the next section.
3. Numerical model for the propagation of solitons in optical fibers
The numerical scheme developed in [29] to solve the PDEs system (1) consists in approximating the
derivates by finite differences and resolving the algebraic system resulting from the discretization,
implicitly, by means of the relaxation Gauss-Seidel method [30,31].
The system (1) is numerically resolved in domain [ ] [ ]LLTsξ ,-,0 ×=× , where LT , . By
discretizing the variables ( )jk
asξa,111 ,
+≡ and ( )
jkasξa
,122 ,+
≡ , for max,...,1,0 kk = and njj ,...,2,1= ,
where maxk is denominated the last advance in ξ and nj the maximum number of points in s , the
propagation domain of the soliton waves is defined by a discretized computational grid of njk ×max
points, as represented in figure 1.
Fig. 1. Computational domain of the propagation of soliton waves.
Thus, by means of the method of finite differences, approaching the temporal derivates by
progressive differences, and the spatial derivates by central differences [30], the following linear
systems are generated from the differential equations (1), namely,
( )++=++++
]i-exp-[1
,,,1,11-j,1,1 2*11
11
11
111 kpoEW
p
tβaaaAaAaAA
ajkjkjkjkkjk
(4)
( )++=+++++
]iexp)(-[1 2
122
22
22
22 ,1,1,11-j,1,1 kpoEWp
tβaaAaAaAA
ajkjkjkkjk
where
( )21
ΔΔ s
r
ξ
iAp += WE AA 11
21
2 s
rAW
ξ
iApo Δ
1 =
( )22
ΔΔ s
α
ξ
iAp +=
( )22
Δ2Δ2 s
α
s
δiAE +=
( )2
22
Δ2Δ2-
s
α
s
δiAW +=
ξ
iAp Δ0
2 =
and in these approaches 1+k is the current time, k is the previous time, sΔ is the spatial quantity
discretized and ξΔ is the temporal quantity discretized, so that ξktk Δ= .
In this work, the linear system (4) is resolved by means of the relaxation Gauss-Seidel method [30-
32]. Consider this linear system for jk
a,11 +
, given explicitly by
( )++=+++
]i-exp-[1
2,2,2,3,11,12,1 2*11
11
11
111 kpoEW
p
tβaaaAaAaAA
akkkkkk
( )++=+++
]i-exp-[1
3,3,3,4,12,13,1 2*11
11
11
111 kpoEW
p
tβaaaAaAaAA
akkkkkk
( )++=+++
]i-exp-[1
4,4,4,5,13,14,1 2*11
11
11
111 kpoEW
p
tβaaaAaAaAA
akkkkkk
...
( ) ( ) ( ) .]i-exp1-,1-,-[1
2*11
11
11
111 1-,,12-j,11-,1 kpoEW
p
tβnjkanjkaaAaAaAA
anjknjknknjk
++=+++
It can be written in compact form as
,1
11
111
11,11-j,1,
,1
++=
+++
+p
EW
A
aAaABa
jkkjk
jk
where ( )kpo tβiaaaABjkjkjkjk
-exp-,,,, 2
*11
11 = with 1-,...,2 nji = .
From the initial condition j
a,01 , given by soliton solution (2), and imposing the contour conditions
01,11 =
+ka and 0
,11 =+ njk
a , for L sufficiently large, )(1 ,1
ljk
a+
is iteratively calculated by means of the
equations
,1
)1-(1
1)(1
1)(1
)(1
1,11-j,1,
,1
++=
+++
+p
lE
lW
l
l
A
aAaABa
jkkjk
jk (5)
where ...3,2,1=l is the iterative level, ( )klll
pol tβiaaaAB
jkjkjkjk-exp- )(
2)(*
1)(
11)(1
,,,,= , until the stop
criterion is fulfilled, namely,
6-)1-(
1)(
11-j2
10|-|max,1,1
<++≤≤
ll
njjkjk
aa . (6)
This method consists in determining )(1 ,1
ljk
a+
by using the already known components of
)1-(1 1,1
ljk
a++
and )(1 1-,1
ljk
a+
, with the advantage of not requiring the simultaneous storage of the two
vectors )1-(1 1,1
ljk
a++
and )(1 1-j,1
lk
a+
at each step. Likewise, )(2 ,1
ljk
a+
is resolved.
It should be noticed that in equations (5)-(6) the value 0.1=w was used for the parameter of
relaxation [32]. Such value corresponds to the optimal relaxation parameter in relation to the
variations of the dielectric parameters α , β and δ of system (1). Figure 2 presents the flowchart of
the numerical code developed for PDEs system (1).
Fig. 2. Flowchart of the numerical code developed to obtain the numerical soliton solutions.
4. Stability of perturbed soliton waves
In our previous work [29] the dielectric parameters α , β e δ in (1) were considered constant, so that
PDE system (1) presented soliton solutions. On the other hand, in real optical fibers, the dielectric
parameters α , β e δ are not constant. This section will study how localized fluctuations of the
dielectric parameters occurring in the splices of optical fibers affect the propagation of soliton waves.
The fusion splicing process of optical fibers is usually realized by means of an electric arc, but it
can be realized by laser, gas flame, or tungsten filament through which current is passed. The fusion
splicing apparatus consists of two fixtures on which the fibers are mounted. The fibers in the apparatus
are aligned and then fused. In fusion splicing, the splice loss is a direct function of the angles of
alignment and quality of the two fiber-end-faces. A splice loss under 0.1 dB is typical. Alternatives to
fusion splicing includes using optical fiber connectors or mechanical splices, both of which have
higher insertion losses, lower reliability and higher return losses than fusion splicing.
In the fusion splicing process, the local dielectric properties of the optical fiber are modified. To
analyze the stability of the perturbed soliton waves as a function of the fluctuations of the dielectric
parameters in the fiber is our aim. In this case, only numerical solutions are possible, since the
propagated waves are not solitons given by (2-3) anymore.
4.1. A localized perturbation in the optical fiber
Initially, the discretization of the computational grid will be described. For variable s , the interval
9050- << s was established, with discretization 1-100.1Δ ×=s , whereas for variable ξ , the interval
was 500 << ξ , with discretization 3-100.1Δ ×=ξ . The geometry of the computational grid was
adjusted so that the physics of the propagation of the perturbed soliton wave is within the considered
computational domain.
In the sequence, the mathematical modeling of the fluctuation of the dielectric parameters in the
splices of the optical fibers will be considered, as well as such localized fluctuation affect the stability
of the propagation of waves. In the modeling of the optical properties along the dielectric fiber used in
this work, it is supposed that, in average, the dielectric parameters take values 0.1-=r , 4/1-=α ,
2/1-=β and 4/1-=δ . It is also supposed that in the areas surrounding the splices of two optical
fibers, the optical properties α , β e δ are altered according to a Gaussian function. In the following
simulations, Gaussian fluctuations of 5% in the values of the dielectric parameters are considered.
Parameterα : Figure 3 shows the simulations of perturbations in ),(1 sξa and ),(2 sξa due to the
fluctuations in parameter α , around 4/1-=α , with maximum amplitude corresponding to 5% of the
average value of the dielectric parameter. By establishing values 0.1-=r , 2/1-=β , 4/1-=δ and
varying α by means of the Gaussian function )])10-(-exp(05.01[)4/1-( 2ξα ×+= , there is the
occurrence of small variations in the amplitudes of ),(1 sξa and ),(2 sξa , not easily visualized in
figure 3. From the Gaussian function, it is observed that the fluctuations in α occur around 10=ξ .
Fig. 3. Numerical solutions ),(1 sξa and ),(2 sξa when )])10-(-exp(05.01[)4/1-( 2ξα ×+= with
0.1-=r , 2/1-=β and 4/1-=δ .
Parameter δ : Figure 4 shows the simulations of perturbations in ),(1 sξa and ),(2 sξa due to the
fluctuations in parameter δ , around 4/1-=δ , with maximum amplitude corresponding to 5% of the
average value of the dielectric parameter. By establishing values 0.1-=r , 4/1-=α , 2/1-=β and
varying δ by means of the Gaussian function )])10-(-exp(05.01[)4/1-( 2ξδ ×+= , there is the
occurrence of small variations in the amplitudes of ),(1 sξa and ),(2 sξa , not easily visualized in
figure 4. From the Gaussian function, it is observed that the fluctuations in δ occur around 10=ξ .
Fig. 4. Numerical solution ),(1 sξa and ),(2 sξa when )])10-(-exp(05.01[)4/1-( 2ξδ ×+= with
0.1-=r , 4/1-=α and 2/1-=β .
Parameter β : Figure 5 shows the simulations of perturbations in ),(1 sξa and ),(2 sξa due to the
fluctuation in parameter β , around 2/1-=β , with maximum amplitude corresponding to 5% of the
average value of the dielectric parameter. By establishing values 0.1-=r , 4/1-=α , 4/1-=δ and
varying β by means of the Gaussian function )])10-(-exp(05.01[)2/1-( 2ξβ += , in figure 5 there is
the occurrence of visible variations in the amplitudes of ),(1 sξa and ),(2 sξa . From the Gaussian
function, it is observed that the fluctuations in β occur around 10=ξ .
Fig. 5. Numerical solution ),(1 sξa and ),(2 sξa when )])10-(-exp(05.01[)2/1-( 2ξβ ×+= with
0.1-=r , 4/1-=α and 4/1-=δ .
When figures 3, 4 and 5 are compared, it is verified that the propagation of the soliton wave was
more sensitive to the fluctuations of parameter β .
4.2. Periodic perturbations
In this subsection, the computational grid was constructed by considering, for variable s , the interval
9050- << s , with discretization 1-109.0Δ ×=s , and for variable , the interval 600 , with
discretization 3-100.1Δ ×=ξ . Again, the geometry of the computational grid was adjusted so that the
physics of the propagation of the perturbed soliton wave is completely within the considered
computational domain.
It was also considered that the fluctuations of the dielectric parameters in the splices of the optical
fibers are modeled by means of Gaussian functions. In the simulations conducted, periodic
fluctuations of 1% were considered in the values of the dielectric parameters in kξ 5= with
11,..,1=k . By establishing, for the dielectric parameters, the average values 0.1-=r , 4/1-=α ,
2/1-=β and 4/1-=δ , and considering the described periodic Gaussian perturbations, it is verified
again that ),(1 sξa and ),(2 sξa are more sensitive to the fluctuations of parameter .
Figure 6 shows the simulations of perturbations in ),(1 sξa and ),(2 sξa due to the periodic
fluctuations in parameter β , around 2/1-=β , when 0.1-=r , 4/1-=α , 4/1-=δ . In this case,
significant variations are observed in the amplitudes of ),(1 sξa and ),(2 sξa .
Fig. 6. Numerical solution ),(1 sξa and ),(2 sξa for 1% Gaussian perturbations in the β value, in
kξ 5= , for 11,..,1=k , with 0.1-=r , 4/1-=α , 4/1-=δ .
From the simulations in this subsection, it is observed that, in the context of optical communication
with localized perturbative processes, the propagation of soliton waves is more affected by the
fluctuations in parameter . Therefore, the procedures for optical fiber fusion through voltaic arcs,
for example, should be conceived so that, locally, the dielectric properties related to parameter of
the optical fiber are little affected.
4.3. Parameter and the stability of perturbed solitons
In the preceding sections, by simulating the propagation of soliton waves through localized
perturbations with various geometric configurations and intensities, it was verified that the soliton
wave is more distorted when there are fluctuation in parameter β . With the objective of verifying
whether the perturbed soliton waves achieve stability in function of the fluctuations in the dielectric
parameter β , in the sequence, the evolution of such waves with higher values of ξ will be studied.
Thus a computational grid was constructed, considering for variable s the interval 9050- << s , with
discretization 2-106.5Δ ×=s , and for variable ξ , the interval 1000 <<ξ , with discretization
3-100.1Δ ×=ξ . In all the following simulations, the values considered for the dielectric parameters
were: 0.1-=r , 4/1-=α , 4/1-=δ .
In figure 7, a Gaussian-type perturbation of 1% is considered in parameter β , around 10=ξ . It is
verified that ),(1 sξa and ),(2 sξa evolve towards a stationary situation for higher values of ξ .
Fig. 7. Numerical solution ),(1 sξa and ),(2 sξa when )])10-(-exp(01.01[)2/1-( 2ξβ ×+= with
0.1-=r , 4/1-=α and 4/1-=δ , for large values of ξ .
In figure 8, a Gaussian-type perturbation of 5% is considered in parameter β , around 10=ξ . It is
observed again that the perturbed soliton wave evolves towards a stationary situation with damped
fluctuations for increasing values of ξ .
Fig. 8. Numerical solution ),(1 sξa and ),(2 sξa when )])10-(-exp(05.01[)2/1-( 2ξβ ×+= with
0.1-=r , 4/1-=α and 4/1-=δ , for large values of ξ .
5. Conclusions
In this work, the stability of the propagation of soliton waves through optical fiber splices was studied.
In optical fiber splices made, for example, by an electric arc, it is observed that the dielectric
properties are locally altered. In order to simulate the propagation of solitons through such splices,
considered as perturbations, the fluctuations of the dielectric parameters were locally modeled by
means of Gaussian functions. By considering local and periodic configurations for the optical fiber
splices, it was verified that the perturbed soliton wave presents higher sensitivity to parameter ; in
other words, fluctuations in the dielectric parameter generate higher amplitude perturbations
(oscillations) in the soliton waves. With the objective of verifying whether such perturbations generate
unstable solitons, the evolution of the soliton wave was studied for several configurations of
perturbations along of optical fiber, for higher values of . It was verified that, after the end of the
perturbations, the perturbed soliton wave achieves stability, i.e., the amplitude of the oscillations
decrease for increasing values of . It is therefore concluded that soliton waves, subject to Gaussian
perturbations in the dielectric parameters of the optical fiber, present numerical stability.
This work is finalized by suggesting that the procedures and experimental methods utilized in the
process of optical fiber fusion are designed so that, locally, the dielectric properties related to
parameter are less altered during the fusion process. Future works intend to continue the study of
the behavior of perturbed soliton waves in non-ideal optical fibers. Such numerical studies will enable
researchers to propose appropriate materials and procedures in order to make optical communication
via solitons more viable.
Acknowledgements
We express our appreciation to the late Dr. Valdemir Garcia Ferrreira, whose contribution to this work
was of great significance.The author C.F. de Oliveira thanks the State University of Londrina for the
scholarships IC/UEL granted from August/2006 to July/2007 and from August/2008 to February/2009.
The author P.L. Natti thanks the State University of Londrina for the financial support received for this
research through FAEPE/2005 and FAEPE/2009 programs.
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