Numerical simulation models of soliton systems

Abstract

This paper describes the development of numerical simulation models of soliton systems using a newly developed Fourier Series Analysis Technique (FSAT) to analyze the non-linear Schrodinger equation of soliton propagation. Generation of sub-picosecond solitons in an active mode-locked fiber ring laser with amplitude and phase modulators and soliton pulse compression mechanisms using dispersion decreasing fiber are investigated. Soliton propagation in optical fiber has been studied extensively for high data rate and ultra-long distance applications in recent years. Numerical methods to study the non-linear Schrodinger equation (NLSE) include inverse scattering method, pertubation method, and split-step Fourier method. The proposed FSAT [1] method has several advantages over the others including reduced sampling points, better computation efficiency, and ease in handling high order dispersion and nonlinear terms. In this paper, FSAT is applied to model soliton pulse generation and compression. The first numerical simulation model is to investigate the generation of sub-picosecond solitons in active mode-locked fiber ring laser. The laser model consists of a polarization preserving Er-doped single mode fiber, an amplitude modulator and a phase modulator. The model has taken into account of dispersive spreading, self-phase modulation, finite amplification bandwidth, pump depletion, and Raman self-frequency shift. The propagation of soliton pulses in the gain section of the fiber laser can be described by a simplified nonlinear equation [1]: ∂u/∂z= j ∂ 2 u/∂T 2 + j|u| 2 u + Gu + γ a G∂u/∂T - jγ R u ∂|u| 2 /∂T (1) where u is the normalized amplitude, T is the normalized time and z is the normalized distance. In (1), the first term in the right hand side describes the pulse dispersion spreading, the second term describes the Kerr effect, the third and forth terms represent the finite gain bandwidth and pump depletion of Er 3+ fiber, respectively. The last term describes the Raman self-frequency shift of the laser. Equation (1) can be solved via the FSAT method to analyze the formation of soliton pulses [2] by first expressing u(z,T) in terms of the Fourier series as shown below: N Σ n=-N ∂u n (z)/∂z exp(jneT)=- jΣ N n =-N n 2 e 2 u n (z)exp(jneT)+jΣ N n=-N Ψ n (z)exp(jneT) (2) + GΣ N n=-N u n (z)exp(jneT) + jγ a G Σ N n=-N neu n (z)exp(jneT) - jγ R Σ N n=-N Φ n (z)exp(jneT).