Propagation characteristics of Chirped Vector Soliton in optical fibers with variable
coefficients
Siham Aziez, Derradji Bahloul.
Département de Science de la Matière, Faculté des Sciences,
Université Hadj-Lakhdar de Batna, 1 Avenue Boukhlouf Mohamed El Hadi, Batna 05000, Algeria
aziez_siham@hotmail.fr bahloul@univ-batna.dz Abstract:
In this work we study the propagation characteristics of chirped vector solitons in optical fiber systems with variable coefficients by using the compact split step Padé scheme (CSSPS). The numerical simulations show that the chirped managed vector soliton with different perturbations maintains its characteristics during propagation along the polarization maintaining optical fiber.
Keywords: Vector soliton, Chirped soliton, Optical fibers, compact split step Padé scheme, Coupled higher-order nonlinear Schrodinger equations, temporal waveform.
1 Introduction
Vector solitons [1,2] have a rich nonlinear dynamics because of their multicomponent structure [3], and their potential applications in fiber-optic-based communication systems [4].
In general, the propagation of managed vector solitons in birefringent optical fibers is governed by the coupled NLS equations [1] with variable coefficients.
Usually, the coupled NLS equations with variable coefficients are not integrable, but may possess exact solitary-wave solutions (vector solitons) in some particular cases. So, it is necessary to use numerical methods to study the propagation of vector solitons.
The compact split step Padé scheme (CSSPS) [5] having a fourth order accuracy in space, shows a good stability and efficiency for the coupled NLS equations with variable coefficients describing vector solitons. This method shows many advantages over classical methods such as split step Fourier method or finite difference Crank Nicolson method. SSFM is time consuming especially for wavelength division multiplexing WDM systems. CSSPS method is more efficient, more rapid and well adapted for higher order time derivatives such as third order dispersion or Kerr dispersion.
2 Theoretical model
We start our discusions from the coupled nonlinear Schrödinger (CNLS) equations (describing the propagation of solitons in birefringent optical fibers) which can be written under the following vectorial form [8] :
2 0 2
2 0 2
2 2 2 1
0 0 1
v u v
i u
v u T
i v u T v
u Z
(1)
Where u(Z,T) and v(Z,T) are the slowly varying amplitudes, Z and T are normalized distance and time, δ is the difference of the group velocities for the two polarization components, Г is the linear and nonlinear gain (loss), and βis the cross-phase modulation (XPM) coefficient (For linearly birefringent fibers, β=2/3).
It is found that in real systems, the dispersion, nonlinearity, gain and loss are usually varied with the propagation distance. So, we have a new concept of managed vector solitons which has been extensively investigated. Its propagation in optical fibers is governed by the CNLS equation with variable coefficients:
v2 u2 0
2 0 2 v Z u iR
v Z u T2
2 Z 2D
i v u 1 T 0
0 1 v u Z
(2)
Where the functions D(Z) and R(z) are respectively the group velocity dispersion (GVD) and self-phase modulation (SPM).
3 Numerical model for the propagation of managed vector solitons in optical fibers
We choose the following initial conditions with linear chirp:
(3)
(4)
Where C is the linear chirp parameter and the angle α determines the relative strengths of the partial pulses in each of the two polarizations. It represents the angle of the polarization of the soliton with respect to the polarization axis of the fiber.
For our numerical simulations, we introduce the following parameters [7]:
-the varying GVD parameter
Z exp Z R Z / D
0D
(5)-the nonlinear parameter
Z R R gZ
R
0
1sin
(6)-and the gain or loss distributed parameter
2
Z
(7)Where D0 is the parameter related to the initial peak power, σ is theparameter discribed gain or loss, and R0, R1, g are the parameters described kerr nonlinearity.
For our study, we take the parameters D0=1, R0=0, R1=1 and g=1 [7].
We introduce also the pulse width [8]:
(8)
(9)
iCT T
T
u / cosh
exp 2 cos ,
0
2
2
2 2
2 2 2
, , ,
,
dT T Z u
dT T Z u T dT
T Z u
dT T Z u T w
u iCT T
T
v / cosh
exp 2 sin ,
0
2
2
2 2
2 2 2
, ,
dT T Z v T dT T Z v T wv
4 Numerical simulations
For the following cases we are going to examine the evolution of the managed vector soliton under the effect of the chirp. When α=450, the two polarization components have same strength. At the time of the propagation of the two components of an unchirped vector soliton in birefringent optical fiber with the parameters α=450, C=0, and σ=0.025, the temporal width performs a slight oscillation with the increase of the propagation distance (Figure 1-(a), 1-(b)). The variation of temporal width (FWHM) of the soliton with normalised propagation distance in this case is shown in Figure 1-(c). It can be seen from Figure 2 and figure 3 that the chirped vector soliton is periodically compressed and broadened with the increase of propagation distance in a system regardless of loss or amplification. In figure 2with the parameters α=450, C=-0.2, and σ=0.05, we can see that a negative chirp makes the soliton broadening, while; in figure 3 with the parameters α=450, C=0.1, and σ=0.025, we can see that a positive chirp leads to a soliton compression.
(a) (b)
(c) Figure 1
Evolution of the two components of an unchirped vector soliton in birefringent optical fiber (a and b), and the variation of temporal width (FWHM) of the vector soliton with the
propagation distance (c), with the parameters α=450, C=0, and σ=0.05.
0 50
100 150
200 -10
-5 0
5 10 0
0.5 1 1.5 2
distance Intensity Ix
time
|U(Z,t)|
0 50
100 150
200 -10
-5 0
5 10 0
0.5 1 1.5 2
distance Intensity Iy
time
|V(Z,t)|
0 0.5 1 1.5 2 2.5
16 18 20 22 24 26 28
z
pulse width
variation of pulse width with propagation distance
Figure 2
Evolution of the two components of an unchirped vector soliton in birefringent optical fiber with the parameters α=450, C=-0.2, and σ=0.05
Figure 3
Evolution of the two components of chirped vector soliton in birefringent optical fiber with the parameters α=450, C=0.1, and σ=0.025
Conclusion
We have studied numerically the propagation characteristics of chirped managed vector solitons in birefringent optical fibers. The propagation of vector solitons presents rich nonlinear behaviour. The temporal width (FWHM) of the two components of an unchirped vector soliton performs a slight oscillation with the increase of propagation distance. The chirped vector soliton is periodically compressed and broadened with the increase of propagation distance in a system
0 50
100 150
200 -10
-5 0
5 10 0
0.2 0.4 0.6 0.8
distance Intensity Ix
time
|U(Z,t)|
0 50
100 150
200 -10
-5 0
5 10 0
0.2 0.4 0.6 0.8
distance Intensity Iy
time
|V(Z,t)|
0 50
100 150
200 -10
-5 0
5 10 0
0.5 1 1.5 2
distance Intensity Ix
time
|U(Z,t)|
0 50
100 150
200 -10
-5 0
5 10 0
0.5 1 1.5 2
distance Intensity Iy
time
|V(Z,t)|
regardless of loss or amplification. A negative chirp makes the soliton broadening, while; a positive chirp leads to a soliton compression. The effect of chirp on the soliton temporal width of an amplification system (σ> 0) is greater than that in a loss system (σ< 0)
Acknowledgement
This work is supported by the post graduation of The Département de Science de la Matière, Faculté des Sciences of the Université de Batna, Algeria.
References
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Observation of Temporal Vector Soliton Propagation and Collision in Birefringent Fiber, DOI: 10.1103/PhysRevLett.98.053902.
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[6] Siham Aziez, Moussa Smadi, Derradji Bahloul: Semi-Implicit Operator Splitting Padé Method For Vector HNLS Solitons, AIP Conference Proceedings 09/2008; 1047(1). DOI:10.1063/1.2999940.
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