Agus Suryanto

A finite element scheme to study the nonlinear optical response of a finite grating without and with defect

We present a simple numerical scheme based on the finite element method (FEM) using transparent-influx boundary conditions to study the nonlinear optical response of a finite one-dimensional grating with Kerr medium. Restricting first to the linear case, we improve the standard FEM to get a fourth order accurate scheme maintaining a symmetric-tridiagonal structure of the finite element matrix. For the full nonlinear equation, we implement the improved FEM for the linear part and a standard FEM for the nonlinear part. The resulting nonlinear system of equations is solved using a weighted-averaged fixed-point iterative method combined with a continuation method. To illustrate the method, we study a periodic structure without and with defect and show that the method has no problem with large nonlinear effect. The method is also found to be able to show the optical bistability behavior of the ideal and the defect structure as a function of either the frequency or the intensity of the input light. The bistability of the ideal periodic structure can be obtained by tuning the frequency to a value close to the bottom or top linear band-edge while that of the defect structure can be produced using a frequency near the defect mode or near the bottom of the linear band-edge. The threshold value can be reduced by increasing the number of layer periods. We found that the threshold needed for the defect structure is much lower then that for a strictly periodic structure of the same length.

ON THE SWING EFFECT OF SPATIAL INHOMOGENEOUS NLS SOLITONS

In this paper, we consider the propagation of a spatial soliton in a waveguide with triangular linear refractive index profile. We propose a model that is obtained by starting with a small perturbation of the constant linear refractive index in the displacement vector of the Maxwell equation, and then deriving the NLS equation for this case. Using this model it is shown, both analytically and numerically, that the soliton beam oscillates inside the waveguide. This is as expected, but differs from the model found in the literature in which the inhomogeneity is introduced directly in the standard NLS equation. Finally, the proposed model is used to study the breakup of bound N-soliton in a triangular waveguide.

Finite element analysis of optical bistability in one-dimensional nonlinear photonic band gap structures with a defect

We present a new approach based on the recently reported finite element scheme16 to study the optical response of a finite one-dimensional nonlinear grating. Using the transmitted wave amplitude as a numerical input parameter, we are able to find all stable and unstable solutions related to a specific incident wave which build up the complete bistability curve. The method is applied to investigate the optical bistability in a nonlinear quarter-wave reflector with a defect. With a proper choice of the incident light frequency, a very low bistability threshold is predicted for an optimized defect structure.

Break up of bound-N-spatial-soliton in a ramp waveguide

We present an analytical and numerical investigation of the propagation of spatial solitons in a nonlinear waveguide with ramp linear refractive index profile (ramp waveguide). For the propagation of a single soliton beam in a ramp waveguide, the particle theory shows that the soliton beam follows a parabolic curve in the region where the linear refractive index increases and a straight line outside the waveguide. The acceleration of the soliton depends on the beam intensity: higher amplitude solitons experience higher acceleration. Numerical calculations using an implicit Crank-Nicolson scheme confirm the result of the particle theory. Combining these propagation properties with the theory about bound-N-soliton, we study the break up of such a bound-N-soliton in a ramp waveguide. In a ramp waveguide, a bound-N-soliton will always be splitted intoN independent solitons with the higher amplitude soliton emitted first. The amplitude of the separated solitons after break up are calculated using the soliton theory as if the solitons are independent. Numerical simulations show that the results agree quite well with this theoretical prediction, indicating that the interaction during break up has only little influence.

Stability preserving non-standard finite difference scheme for a harvesting Leslie–Gower predator–prey model

A non-standard finite difference scheme for a harvesting Leslie–Gower equations is constructed. It is shown that the obtained difference system has the same dynamics as the original continuous system, such as positivity of solutions, equilibria and their local stability properties, irrespective of the size of numerical time step. To illustrate the analytical results, we present some numerical simulations.

Deformation of modulated wave groups in third-order nonlinear media

We present a numerical and analytical investigation of the deformation of a modulated wave group in third-order nonlinear media. Numerical results show that an optical pulse that is initially bichromatic can deform substantially with large variations in amplitude and phase. For specific cases, the bi-chromatic pulse deforms into a train of temporal solitons. Based on the coupled phase-amplitude equation of Nonlinear Schrödinger (NLS), the initial deformation of the modulated wave-packet will be explained and an instability condition can be derived. Energy arguments are given that provide an alternative derivation of the instability condition.

Uni-directional models for narrow- and broadband pulse propagation in second order nonlinear media

We consider optical pulse propagation in one spatial direction and observe that for lossless media, the resulting Maxwell equations are of the form of an infinite dimensional Hamiltonian system evolving in the spatial direction. A simplified uni-directional model is derived for waves running mainly in one direction. For quadratic χ2-nonlinearity, this leads to variants of the Korteweg—de Vries equation (well known in fluid dynamics) with dispersion determined by the material properties. For narrow banded spectra, a corresponding envelope equation of nonlinear Schrödinger (NLS)-type, with full dispersive properties, is derived. Special attention is given to translate the NLS-solution to the physical field, which involves phase adaptations that contribute to the nonlinear dispersion relation. Then the propagation and distortion of double pumped pulses is studied by deriving uniformly valid analytic approximations. It is found, confirming but specifying previous observations, that when the quotient of amplitude and frequency difference is not small, side bands from third order effects have a contribution comparable to that of the first order terms. The unidirectional model describes the asymmetry in the distortions that are not described by the standard NLS-equation but which can be recovered when higher order dispersive effects are incorporated. The final conclusion when comparing the different models is that the uni-directional model is preferred above the NLS-model, based on its more general applicability for broad signals, its direct description of the physical fields, and the more direct analytical methods to find asymptotically valid approximations.

Weakly nonparaxial effects on the propagation of (1+1)D spatial solitons in inhomogeneous Kerr media

The widely-used approach to study the beam propagation in Kerr media is based on the slowly varying envelope approximation (SVEA) which is also known as the paraxial approximation. Within this approximation, the beam evolution is described by the nonlinear Schrodinger (NLS) equation. In this paper, we extend the NLS equation by including higher-order terms to study the effects of nonparaxiality on the soliton propagation in inhomogeneous Kerr media. The result is still a one-way wave equation which means that all back-reflections are neglected. The accuracy of this approximation exceeds the standard SVEA. By performing several numerical simulations, we show that the NLS equation produces reasonably good predictions for relatively small degrees of nonparaxiality, as expected. However, in the regions where the envelope beam is changing rapidly as in the breakup of a multisoliton bound state, the nonparaxiality plays an important role.

Stability Analysis of a Fractional Order Modified Leslie-Gower Model with Additive Allee Effect

We analyze the dynamics of a fractional order modified Leslie-Gower model with Beddington-DeAngelis functional response and additive Allee effect by means of local stability. In this respect, all possible equilibria and their existence conditions are determined and their stability properties are established. We also construct nonstandard numerical schemes based on Grunwald-Letnikov approximation. The constructed scheme is explicit and maintains the positivity of solutions. Using this scheme, we perform some numerical simulations to illustrate the dynamical behavior of the model. It is noticed that the nonstandard Grunwald-Letnikov scheme preserves the dynamical properties of the continuous model, while the classical scheme may fail to maintain those dynamical properties.

Effect of different predation rate on predator-prey model with harvesting, disease and refuge

This paper deals with predator-prey interactions with predator harvesting and prey refuge. The predator may be infective by a disease. Therefore the predator is divided into two subclasses, i.e. infective and susceptible predator. It is assumed that susceptible predator have higher predation rate than infective predator, and hence the growth rate of susceptible predator will be higher than infective predator. It is found that the model has five equilibrium points. Finally, numerical simulation are presented not only to illustrate equilibrium point but also to illustrate effect of predation rate.