1. Introduction

Solitary waves in Fiber Bragg Gratings (FBGs) exhibit a number of striking properties which makes them attractive both from the basic physics and applications point of view [1, 2]. In particular, their central frequency may lie within the photonic band gap of the corresponding linear system. Owing to this unique feature, the group velocity of these gap solitons may become arbitrarily small. Recently, different physical mechanisms were suggested that would allow to control and manipulate such localized pulses. In Refs. [3, 4] it has been demonstrated that – by means of nonresonant wave-interaction processes – one can control the group velocity and the position of gap solitons. On the other hand it has been shown that – by introducing localized defects – solitary waves may be effectively pinned at certain positions in FBGs [5]. In Ref. [6], certain aspects of soliton interaction with point defects have been considered. In the present manuscript, we report on extensive numerical studies of soliton-defect interaction processes in FBGs that provide a comprehensive picture of these systems. In particular, we have found that for certain parameter ranges, the scattering of gap solitons on point defects exhibits so-called n-bounce resonances and becomes chaotic. Similar phenomena have been discovered earlier for soliton-interaction processes in other models (see [7, 8] and references therein). Furthermore, our analysis provides a quantitative guideline for future experiments in FBGs.

2. Gap solitons and the defect modes

Nonlinear wave dynamics in FBGs with a point defect is described by the following nonlinear coupled mode equations [6]:

In these equations, E ₊ and E _- represent the envelopes of forward and backward propagating waves [1]. The point defect represents a local disturbance of the refractive index modulation in the fiber. The corresponding strength of this defect is characterized through the parameter κ. Finally, we would like to note that all variables are cast in dimensionless units and for given parameters of actual fibers, the universal results of Eq. (1) can easily be translated to physical quantities [1].

The well-known fundamental gap-soliton solutions of the unperturbed model (κ=0) can be written as [9, 10]:

Here, we have introduced the boosted spatial and temporal coordinates $X=\frac{\left(x-vt\right)}{\sqrt{1-v^2}}$ and $T=\frac{\left(t-vx\right)}{\sqrt{1-v^2}}$ and have defined y=tanh^-1(v). In addition, we have adopted the following abbreviations:

This family of soliton solutions depends on two independent parameters, ν and δ_s, where |ν|<1 and 0<δ_s<π. It can be easily seen that the soliton velocity is parametrized by ν and that for a fixed value of ν, the width, height, and spectrum of the pulse is determined by δ_s. For this reason ν and δ_s are commonly referred to as the velocity and the detuning, respectively [1, 2]. The stability analysis of these gap solitons indicates that they remain stable for detunings smaller than δ _cr≈1.011(π/2) [10, 11, 12].

Furthermore, there also exists an exact analytical solution of the perturbed equations [6]:

In these expressions, the function V is given by

and the parameter a is found to be

It is important to realize there are no defect modes for detunings δ_d<δ _min, where

Most importantly, the defect mode is stable only for δ_d≈π/2 provided that κ>0. In other words, we can deduce from Eq. (12), that for only κ>0 there always exists a stable defect mode [6] and this defect mode exhibits a rather well-defined detuning. Clearly, this defect mode lends itself to applications as an optical bit and its interaction with moving solitons provides an avenue to control and/or probe its properties.

3. Numerical simulations

In this section we present the results of extensive numerical experiments on soliton-defect interactions in FBGs that have been geared towards obtaining a comprehensive picture of the possibilities and limitations associated with this system. The actual simulations consist of a gap soliton which collides with a point defect. Initially, the defect mode is assumed to be not excited. We have obtained all of our results using a fourth-order correct implicit Runge-Kutta scheme, as proposed in Ref. [13]. Additionally, we double-checked some results using a standard split-step Fourier method. In all cases, the results agreed perfectly. The total length of the simulated fiber was taken to be l=68 with the defect in the center. This system has been discretized using 25 grid points per unit length, so that well converged results have been obtained. In order to absorb outgoing radiation, a small damping term was introduced in the outer regions of the fiber.

The free parameters of this problem comprise the detuning δ_s and velocity ν of the incoming soliton and the defect strength κ. This parameter space is too large to be thoroughly studied. However, in view of the discussion regarding the stability of defect modes [6], we can reduce the number of parameters by choosing the detuning of the colliding soliton to be δ_s=π/2 and assume κ>0. In addition, the initial separation d between the soliton center and the defect may play an important role when there is a trapped soliton at the defect. However, in our simulations the defect mode is initially not excited and we have chosen d=4.

 

Fig. 1. “Phase diagrams” of how much energy gets trapped (upper panel), transmitted (middle panel), and reflected (lower panel) when a moving soliton interacts with a point defect which initially is not excited. The color scale is normalized to the energy of the incoming soliton. The soliton detuning is chosen such that a stable nonlinear defect mode could be formed. The area marked with “A” in the upper left corner of the parameter space is shown in more detail in Fig. 3. Parameter values for typical interactions processes are marked with “1”–“4” and the corresponding simulations are displayed in Fig. 2 (see the text for more details).

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In Fig. 1, we provide numerically determined “phase diagrams” of trapped, transmitted and reflected energies of the interaction between a gap soliton and the defect. To obtain these energies, we numerically integrate along a contour in the (x,t)-plane:

For the left and right boundaries we choose -x_l=x_r=9 and the simulations are usually run for a time T _end=2(x_r-x_l)/ν that corresponds to a full round-trip thruough this interval. The region in which we consider the energy as being trapped is defined by L_d=2. As a further check, we compute I _ref+I _trap+I _trans and always find that this is equal to I _init within the numerical accuracies. The soliton velocity ν is varied between 0.05 and 0.95 in steps of 0.01 while the defect strength is varied between 0.05 and 1.0 in steps of 0.01, yielding a total of 8736 simulations. This reduced parameter space is separated into three rather distinct regions:

Most interestingly, the transition between the different regions can be either abrupt or continuous. For example, for small velocities and defect strengths, the transition between transmission and trapping regions is abrupt. For a given defect strength, one can easily determine a threshold velocity below which the soliton gets trapped and above which it passes through the defect. This behavior can be interpreted via a simple model of scattering a particle on a potential barrier [6]. However, the situation becomes more involved when the defect strength or the velocity increase. For moderate strength defects and high velocities (ν>0.5) the soliton energy is split into transmitted, trapped and reflected parts (see Fig. 2(c)). In Ref. [5], a simple criterion has been suggested that to predict whether a solitary wave will be captured or not. According to this model, the pulse can get captured if there exists a defect mode with the same frequency and less intensity than the incoming pulse. This is the so-called the resonant energy transfer criterion. It should be noted that this model is not valid for the description of processes that involve solitary wave reflection and/or transmission processes. Even for the pure trapping regime it is incomplete since for a point defect nonlinear defect modes do not exist for detunings δ_d<δ _min. In fact, our numerical simulations clearly show that there are parameter ranges where no trapping can be observed although it would be allowed by the resonant energy transfer criterion. Thus, while the potential barrier model [6] and the resonant energy transfer criterion [5] provide reasonable qualitative guidelines, the dynamics of soliton-defect interaction processes is significantly more complicated than what these simple models can describe.

 

Fig. 2. Details of typical interaction processes of a soliton with a point defect. In all panels, we denote the defect site with a line along the time axis. The individual processes and corresponding values of the parameters are given in the captions of the sub-figures. See the text and Fig. 1 for further details.

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Furthermore, a careful inspection of Fig. 1 reveals the existence of an area with a fractal pattern. This area of small soliton velocities and strong defects is marked by “A”. In Fig. 3 we provide a close-up of this area where we plot the “phase diagrams” of trapped and reflected energy for velocities ν ranging from 0.05 to 0.15 in steps of 0.002 and defect strengths κ ranging from 0.85 to 1.0 in steps of 0.002. In this parameter range, transmission plays a minor role.

We find irregularly alternating regions where either only reflection or both reflection and trapping takes place. In most cases, a considerable part (abut 50%) of the energy gets trapped and the remaining energy is reflected. This reflection happens via multiple-bounce resonances, similar to those found in solitary wave collisions [7, 8]. The resonance processes within which no energy remains trapped usually consist of only two bounces before reflection occurs. In this cases, the soliton and the defect mode appear to be in-phase, leading to complete reflection. We illustrate details of this peculiar behavior for two typical examples in Fig. 4. On the one hand, this behavior is analogous to the multiple-bounce resonances found in solitary wave collisions [7, 8]and collisions of sine-Gordon kinks with defects [14]. In Ref. [8], by means of a variational method, Goodman and Haberman give a mathematical theory of chaotic solitary wave dynamics. In this approch, solitons are treated as particles which are perturbed by the defect mode. The particles can escape to infinity, so that – according to the classical Melnikov criterion – the chaotic behavior of interacting solitary waves originates from oscillatory motion near the perturbed separatrix. There exist two- and multiple-bounce resonaces for this dynamical system. In particular, the two-bounce resonances lead to a situation when the energy of the reflected solitary wave is the same as the energy of the initial pulse. For instance, this applies to the sine-Gordon model perturbed with a defect. The dynamics of the coupled mode equations is essentially the same as that of the sine-Gordon equation with a saddle point at infinity [15]. However, the variational approach to the nonlinear coupled mode equations is extremely complex so that is becomes difficult to ascertain that the underlying approximations capture all the relevant physics. Instead, our extensive numerical simulations clearly show that the chaotic scattering of solitons is due to the multiple-bounce phenomena near the escape point [8]. Yet, this behavior is somewhat unexpected for soliton-defect interactions and certainly not anticipated from simple models such as the potential barrier model [6] or the resonant energy transfer criterion [5].

 

Fig. 3. Blow-up of the fractal part (marked by “A”) of the “Phase diagrams” of Fig. 1. The ‘Phase diagrams” describe how much energy gets trapped (left panel) and reflected (lower panel) when a moving soliton interacts with a point defect which initially is not excited. See the text for further details.

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Fig. 4. Two typical processes in the fractal parameter region “A” of the “Phase diagrams” shown in Figs. 1 and 3. In most cases, about 50% of the energy gets trapped and the remained is reflected (left panel). However, for certain parameter combinations, the soliton and the defect mode appears to be in phase and all the energy is reflected (right panel).

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4. Conclusion

We have carried out a comprehensive numerical analysis of soliton-defect interaction processes in FBGs. From these numerical experiments we can extract at least three important features. First of all, simple models such as the the potential barrier model [6] or the resonant energy transfer criterion [5] provide a rough qualitative guideline in certain cases only. The true dynamics of these system is significantly more complex than what these models suggest and detailed numerical investigations are required. Secondly, it is possible to uniquely identify a critical velocity which separates the trapping and the transmission regimes. Thirdly, and perhaps most interestingly, for slow solitons there exist resonances which lead to bouncing of the pulse at the defect before the reflection. As a result the scattering process becomes chaotic. The physical picture provided by the “phase diagrams” of Figs. 1 and 3 provides quantitative guidance for future experiments on soliton-defect interaction in FBGs and allows to discuss the realization of all-optical information processing devices based on soliton-defect interaction processes in this and similar systems such as nonlinear photonic crystals.