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Dark and Bright Soliton in Fiber Optics

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Dark and Bright Soliton in Fiber Optics 2013 International Conference of Information and Communication Technology (ICoICT) Dark and Bright Soliton in Fiber Optics Subekti Ari Santoso1), Fakhrurrozi1), Octarina Nur S1) Ary Syahriar1)2) 1) Departement of Electrical Engineering, Faculty of 2)Agency for The Assessment and Application of Science and Technology, University of Al Azhar Indonesia Technology Jakarta, Indonesia Jakarta, Indonesia e-mail : [email protected] e-mail : [email protected] Abstract – Nonlinear Schrödinger equation is a general Dispersion (GVD). However, to maintain a pulse, it uses form for modeling and explaining the phenomenon of nonlinear effect of fiber-optic called Self Phase Modulation nonlinear physics system. Nonlinear Schrödinger Equation (SPM) [11]. (NSE) describes the propagation of light pulses that are Soliton propagation tends to be stable and has been stable in Kerr medium. This paper discusses the analytic developed for applications in optical communications with high formulation of nonlinear Schrödinger equation which is speed access [4]. In the field of optical communications, the influenced by Stimulated Raman Scattering and Self information signal is modulated in a pulse light and transmitted Steepening derived from Maxwell's equations. The NSE in fiber optic based on the principle of Total Internal Reflection equation is also influenced by the linear response of a [12]. dielectric material and nonlinear dielectric response. The Soliton has great potential to be applied in optical propagation profile of pulse soliton is stable and this is communications; this encourages the development of research suitable to be implemented in optical communication to to investigate the characteristic of soliton. This paper carry the information. investigates the characteristics of soliton especially for bright and dark soliton. This paper discusses the Nonlinear Keywords — Nonlinear Schrödinger Equation, Self Steepening, Stimulated Raman Scattering, Maxwell Equation, Soliton. Schrödinger equations derived from Maxwell's equation which is including the perturbation factor : Self steepening (SS), Stimulated Raman Scattering (SRS) and Thrid Order I. INTRODUCTION Dispersion (TOD). Then the soliton equation is simulated The discovery of laser technology in early 1960s stimulates numerically using MATLAB. The characteristic of soliton is many researchers to conduct the experiments especially in the investigated by changing the value of parameter β and , application of fiber optic. Fiber optic cables have the capability which are the parameter of the group velocity dispersion. The to carry data with large capacity and high speed [1]. However, simulation result is expected to describe about the fiber optic has many losses when it is transmitted for long characteristics of bright and dark soliton which can be utilized distances communication. Then the experiment is also as the reference in selecting a better pulse soliton for optical developed in investigating the light source with high intensity communications. which allows the transmission of information. Recently, several researches have been developed to apply the nonlinear waves II. NONLINEAR SCHRÖDINGER EQUATION (soliton) in optical communications to carry the information [2]. Basic equation governing the deployment of fiber optic pulse is a Nonlinear Schrödinger Equation which is derived The modern development of the soliton theory in the last from Maxwell's equations [13]. Maxwell equation is combined three decades of the 20th century has lead to a number of with the response of linear and nonlinear dielectric materials in important applications and developments in several areas of order to obtain the general equation of soliton. NSE is more contemporary physics and mathematics [3]. The soliton was appropriate to describe the propagation of picosecond pulse in first observed by Russell as surface waves in 1834 [4]. optical fiber. However for shorter pulse duration, femtosecond, Theoretical explanation of the experimental Russell is obtained NSE need a new approach to take into account other from the experimental work of Korteweg and de Vries [4] - [8], perturbation part such as the Self perturbatif steepening (SS), which found the Koteweg-de Vries equation (KdV). This is a Stimulated Raman Scatering (SRS), and Thrid Order partial differential equation whose solution describes the Dispersion (TOD) [9 ] [14] [15]. existence of soliton [8]. Soliton is the result of the removal of nonlinear effect from A. Maxwell's equations a medium with the same medium dispersion effect [9]. The Nonlinear effects in optical fiber observed in short pulsesas effect of dispersion occurs because light wave propagates at the dispersive effect can be studied by solving Maxwell's different speeds due to different frequencies [10], thus equations [16], [17]. widening the pulse wave. This effect is called Group Velocity 978-1-4673-4992-5/13/$31.00 ©2013 IEEE 344 2013 International Conference of Information and Communication Technology (ICoICT) D 0 (2.1) C. Response Nonlinear of Dielectric Materials Basic equation for wave propagation parallel to the z-axis B 0 (2.2) direction is described by the electric field E (z, t) [14] [21]: 1 B 2ED14 2 2 xE (2.3) P()NL (,) z t ct 2 2 2 2 2 z c t c t (2.10) B xH (2.4) ()NL t With P is nonlinear polarization. Where D is electric flux density (Coulombs per square In isotropic medium, where the direction of polarization P meter), B is magnetic flux density (tesla or webers per square is in the direction of the electric field β, the second order of meter), E is electric current density (ampere per square meter) susceptibility (2) is negleted, then the response of and H is magnetic field (ampere per meter). electromagnetic waves in a nonlinear medium to the electrical Maxwell equation assumes there is no charge and external medium disturbance from outside can be written as [14] [22]: electric current, then j 0 [14][18], so that: (1) (2) (3) PEEEEEEj0 jk k jkl k l jklm k l m ... DEP4 (2.5) (2.11) BHM4 (2.6) ()n is the n-th order of electric susceptibility tensor. If the With P and M is the electric dipole moment per unit volume medium is homogeneous isotropic nonmagnetic and has the and magnetic moment per unit volume respectively. For an symmetry inversion then =0, so that the refractive index isotropic dielectric material which is nonmagnetic (M=0), so of the medium depends on the intencity (Intencity Dependent of that B = H, and using vector identities then substitute D in refractive Index - IDRI) [13] [14] [22]. equation (2.5) obtain [14]: Refractive index equation is: 1422EP 2EE ( ) 0 (2.7) c c2 t 2 c t 2 n (2.12) n 0 0 0 B. Linear Response of Dielectric Material Power shifted can be described as [13] [14] [21] [22]: Material response relating the electric field (E) with the dipole moment of unity volume (P). Dipole moment per unit volume P (r, t) depends on the electric field at the point r, E DEP0 (r,t) [14] [19]. (2.13) (1 (1) (2) E 2 ...) In a linear dielectric theory, the relationship between the 0 displacement field (D) and the electric field (E) from Maxwell equation is formulated as [14] [19] [20]: (1) (2) 2 Where 0 (1 E ...) D(,)(,)(,) k w k w E k w (2.8) And E is intensity of the refractive index of the medium, formulated as [22]: Where the material dielectric tensor is [13] [14]: n n n (2.14) 02 (k , w ) 4 ( k , w ) (2.9) Where : n 1 (1) is the linear refractive index. Equation (2.9) indicates the propagation wave with 0 frequency w and wave vector k in a material. Dielectric tensor (3) non-linear refractive index. is depending on the frequency and the number of wave vector n2 [13] [14]. 978-1-4673-4992-5/13/$31.00 ©2013 IEEE 345 2013 International Conference of Information and Communication Technology (ICoICT) D. General Solution of NSE 2 E k"2 E EE In a normalized form, non-linear Schrödinger equation is (2.20) ig22 0 [13] [16] [23] [24]: 2 By replacing z and g with λ, the equation (2.20) is a general 2 iEz DE E E 0 (2.15) nonlinear Schrödinger equation [4] [22] [25] written as: Where E is a complex function that describes the 2 EE1 2 normalized electric field and z is the propagation distance, t is (2.21) i2 E E 0 the time delay, Eα is the part of temporal dispersion with z 2 coefficient D = +1 for anomalous dispersion region (GVD1 <0) and D = -1 for the normal dispersion area (GVD> 0). The If the value of λ > 0, the solution for soliton commonly value of β is the coefficient of Self-Phase Modulation [24]. known as the bright-soliton, and for the value of λ < 0, the solution is known as dark-soliton [14] [16]. Kerr effect represent the changes in refractive index of n0 to 2 ()n02 n E so that the change is obtained as III. DARK AND BRIGHT SOLITON 2 nE2 [13][14][22][21]. The wave change is influenced by A. Dark Soliton factor: NSE can be solved by including the inverse scattering in the normal dispersion [26]. However for the positive dispersion region (λ < 0) the solution in the form of hole-soliton known as 2 2n2 dark-soliton. In this area, the dark-soliton pulse cannot be nE2 (2.16) c E 2 propagated, because the solution is equal with the holes in the carrier wave of continuous light. dark-soliton will propagate in a lower rate of power and will be narrowed in a higher rate of Because of the expansion of the wave numbers power [17] [27]. (/)k n c around the center frequency, where the value of The solution for dark-soliton is: refractive index n is a function of then the decomposition of the wave modulation frequency deviates slightly from the 1 center frequency ()0 [4][25], so the equation for the wave 2 22 2v vector is: EtAmhAm(0, ) 1 sec ( ( tVz 2 )) exp( i ) 2 (3.1) " '2k k k k ()() (2.17) With: 0 02 0 mV2 tan1 tanhAm ( t 2 Vz ) Based on the Kerr effect [4], the equation (2.17) becomes: 2 1 m 2 (1mV2 )(2 ) (3.2) " A( t 2 Vz ) V ( t Vz ) ' k 2 2 k k00 k() g E (2.18) 2 2 22V 2 3 (1 m )(2 V ) A( m 3) 2 A z 22 Then by changing the value of () 0 and ()kk 0 From the formulation of dark soliton above, the pulse with and k with it / and k i / z [4][25] the equation becomes: profile can be propagated in the anomalous dispersion region.
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