Semi-implicit operator splitting Padé method for vector HNLS solitons
Siham Aziez, Moussa Smadi and Derradji Bahloul
Département de physique, Faculté des Sciences, Université de Batna 05 Avenue Chahid Boukhlouf, 05000 Batna, Algérie
Abstract. We use in this paper the semi-implicit finite difference operator splitting Padé (OSPD) method for solving the coupled higher-order nonlinear Schrödinger equation which describes the propagation of vector solitons in optical fibers.
This method having a fourth order accuracy in space shows good stability and efficiency for the coupled HNLS equations describing vector solitons. We have tested this method for analyzing the behavior of optical pulses in birefringent fibers verifying that the third order dispersion TOD has different effects on the two polarizations and the asymmetric oscillation is significant only in one polarization.
Keywords: Birefringence, vector solitons, optical fibers, operator splitting Padé method, coupled higher-order nonlinear Schrodinger equation, Third order dispersion.
PACS: 42.81.Dp;
1-INTRODUCTION
In general, when an optical pulse is propagating in a real fiber its state of polarization is not preserved. Its two polarization states have different propagation properties through polarization mode dispersion PMD.
Although the linear effects of PMD can in many cases be neglected, when nonlinear effects (such as cross phase modulation XPM) are present the two states of polarization are coupled, an many interesting features appear such as XPM-induced phase shift. The dynamics of the two components of different polarizations in a birefringent fiber are described by a system of nonlinearly coupled nonlinear Schrödinger (CNLS) equations [1,2]. CNLS equations have attracted the attention of many authors [3-11]. It has many applications especially in optical telecommunication systems such as wavelength division multiplexing systems. These coupled equations may possess exact solitary-wave solutions in some particular cases, but it is often necessary to use numerical methods to study the propagation of vector solitons. For this purpose many numerical methods have been used such us Split Step Fourier method SSFM or finite-difference methods [1]. SSFM is commonly used for studying nonlinear phenomenon in optical fibers but it is time consuming when used for WDM systems. Several finite-difference methods have been used to solve CNLS equations. One of the commonly used is the Crank-Nicolson scheme. In this paper propose the implementation of a semi-implicit finite difference operator splitting Padé method, used
for HNLS in reference [13], in order to simulate the nonlinear propagation of vector solitons in birefringent optical fibers. The OSPD is proven to be more efficient using higher order time derivatives and shows many advantages compared to its comparable Crank Nicolson method. It is more rapid and consumes less memory.
This paper is organized as follows: In Sec. 2 we present the CNLS model that describes the propagation of vector solitons in optical fibers. Sec. 3 will be devoted to explain the numerical model. In Sec. 4 we discuss the numerical simulations for particular cases.
2-Theoretical model
In optical propagation down a birefringent fiber and in anomalous dispersion regime, we can derive the basic equations using the slowly varying envelope approximation [2]. With the consideration of the third order dispersion, the normalized coupled nonlinear Schrodinger (CNLS) equations are given by
ε
2 1
6 Γ || 2
3 ||
ε 2 1
6 Γ || 2
3 ||
We can write this equation in vector form as follows:
ε 0
0 ε
2 1
6 Γ || 2
3 || 0
0 || 2
3 ||
where √1, , !" and , !" are the slowly varying amplitudes. and ! are the space and time in the restarted frame. The real parameters #, and Γ are related to the differential group velocity of the two orthogonal components, third order dispersion, and the fiber loss.
3- Numerical model
The CNLS equations are solved using the semi- implicit operator splitting Padé method. In this method the linear and nonlinear parts of the equation are solved separately as two different subproblems. The solution of one subproblem is employed as an initial condition for the next subproblem. We solve the linear subproblem by using the semi-implicit Padé method and the midpoint rule in space.
ε 00 ε
2 1
6 Γ
The Padé scheme is given in a vector form:
1 $%"&'()*
'(+ ∆ #
2 -% 6 &- 2-%+ 1
6 $-. Γ 2
&'()* '(+
1 $%"&'()*
'(+ ∆
#
2 -% 6 &- 2-%+ 1
6 $-. Γ 2
&'()*
'(+
Where:
$'1
2 /')* ⁄ '1* ⁄ 2 $3'1
2 4')* '1*5 -' 1
∆ /')* ⁄ '1* ⁄ 2 -3' 1
2∆ 4')*'1*5 The nonlinear subproblem is given by the system
67 || 2
3 || 0
0 || 2
3 ||
is solved by an explicit fourth-order Runge–Kutta method
8|98:'()*8|98:'( 168|;18: 28|;28: 28|;38: 8|;48:"∆z The explicit fourth-order Runge–Kutta method used for the nonlinear subproblem satisfies the CFL condition is proven to be stable. The discretization of the linear part is an implicit scheme that is unconditionally stable.
4-Numerical simulations
Let us now test the OSPD method to solve the system of CNLS equations for few examples.
We start with the initial condition 0, !" 0, !" > cos B
sin B cosh1*! where > is the soliton order and the angle B 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 the following examples we are going to examine the evolution of the vector soliton under the effect of the third order dispersion near the zero dispersion when the dispersion of the second order can be neglected. We also analyze the variation of the soliton trapping threshold under the effect of the third order dispersion. For this purpose let us test OSPD method using the same parameters of the reference [3].
We consider the two following cases:
The first case corresponds to the case where the two initial amplitudes are equals B 45G". Figure 1 shows the evolution of the two components of the first vector soliton for the following parameters # 0.15, 0.3, Γ 0, > 1.
We see that the pulse width increases along the propagation at the same time there is a light shift of the center of the soliton and the two components of the vector soliton remain trapped during the propagation.
For the second case where the two initial amplitudes are unequal B 30G". Figure (2) shows the evolution of the two components of the second order vector soliton for the following parameters # 0.5, 0.6, Γ 0 N=2.
FIGURE 1. Evolution of the components of the first order soliton for the following parameters: θ JK, ε 0.15, β 0.3, Γ 0, N 1.
As in the first case we remark a similar augmentation of the pulse width and a similar shift of the center of the soliton. Where as a asymmetric oscillation on the trailing edge for the larger pulse.
. FIGURE 2. Evolution of the components of the second order soliton for the following parameters: θ JN, ε 0.5, β 0.6, Γ 0, N 2.
CONCULSION
In this work we have developed a a semi-implicit finite difference operator splitting Padé method to solve numerically the coupled nonlinear Schrödinger equation that governs the propagation of vector solitons in optical fibers. This method is proved to be stable as it satisfies the CFL condition. It has many advantages compared with split step Fourier method and finite difference Crank Nicolson methods. It is more rapid well adapted to the higher order time derivatives such as third order dispersion. We have tested this method with particular examples taken from other references and it gave good results with better CPU time. This method is readily generalized to dispersion managed fibers and nonlinear managed fibers.
ACKNOWLEDGMENTS
This work was supported by the PRIMALAB laboratory of the University of Batna. The authors thank A. Sid and Prof. A. Bouljedri for helpful discussions.
REFERENCES
1. G. P. Agrawal,”Polarization effects” in Nonlinear Fiber Optics, Academic Press, fourth edition, US, 2007, pp.
177-220.
2. A. Hasegawa and Y. Kodama,”Interaction between solitons in different channels” in Solitons in optical communications, Clarendon Press Oxford, US, 1995, pp.
183-197.
3. X. Zhang and X. Wang, Soliton propagation in birefringent optical fibers near the zero-dispersion wavelength, optik 115, 2004, pp. 36-42.
4. R. Ohhira, M. Matsumoto and A. Hasegawa, Effect of polarization orthogonalization in wavelength division multiplexing soliton transmission system, optics communications 111, 1994, pp. 39-42.
5. A. Hasegawa, Effect of polarization mode dispersion in optical soliton transmission in fibers, physica D 188, 2003, pp. 241-246.
6. C. D. Angelis, M. Santagiustina, S. Wabnitz, Stability of vector solitons in fiber laser and transmission systems, optics communications 122, 1995, pp. 23-27.
7. J. Yang, Y. Tan, Fractal dependence of vector-soliton collisions in birefringent fibers, physics letters A 280, 2001, pp. 129-138.
8. D. J. Frantzeskakis, Vector solitons supported by third- order dispersion, physics letters A 285, 2001, pp. 363- 367.
9. I. M. Uzunov and V. I. Pulov, Vector solitary waves in strongly birefringent fibers with Raman scattering, Physics Letters A,Volume 372, Issue 15, 7 April 2008, pp. 2730-2733.
10. S. Trillo and S. Wabnitz, Nonlinear modulation of coupled wave in birefringent optical fibers, Physics 11. M. F. Mahmood, W. W. Zachary, and T. L. Gill,
Nonlinear pulse propagation in elliptically birefringent optical fibers, Physica D 90, 1996, pp.
271-279.
12.S. K. Lele,, Compact finite difference schemes with spectral-like resolution, J. comput. Phys. 103, 1992, pp. 16-42.
13. Zhenli. Xu, Jingson He and Houde Han, Semi-implicit operator splitting Padé method for higher-order nonlinear Schrodinger equations, Applied Mathematics and computation , 2006, Vol 179, Issue 2, pp. 596-605
