1. Introduction

Optical lattices have been a subject of continuously increasing research interest in the past few years. Such inhomogeneous photonic structures provide efficient mechanisms for the control of the light propagation dynamics that have no counterpart in bulk media [1–4]. The linear diffraction properties of these configurations can be managed so that the propagation characteristics of optical wave fronts can be engineered. Several linear phenomena such as optical Bloch oscillations [5, 6], Rabi oscillations [7, 8] and Anderson localization [9, 10] have been studied with some of them, although classical, being in direct analogy to quantum phenomena in solid state structures [11]. The presence of nonlinearity in these photonic structures introduces the existence of spatially localized waves due to the interplay between the linear diffraction and nonlinearity. Therefore soliton formation and dynamics can be controlled by tailoring the geometric and material properties of the configuration, resulting in interesting functionality which is promising for all-optical control concepts and applications. In parallel, Bose-Einstein condensates with either attractive or repulsive atom interactions have been successfully loaded in optical lattices and the properties of the respective lattice solitons have been investigated [12–15].

In optics, beam dynamics control includes shape, position and velocity control under propagation. In the linear regime, propagation of non-diffractive, self-bending beams has been theoretically predicted and experimentally demonstrated [16, 17] in bulk homogeneous media. These beams have been shown to propagate along parabolic trajectories without exhibiting significant diffraction for quite large propagation distances. Although, these beams have infinite power, it has been shown that, under appropriate truncation or apodization, they retain their basic properties [18–21]. Also, an analogous linear beam acceleration mechanism has been studied for the case of uniform optical lattices [22]. These linear mechanisms of beam acceleration are based on the diffraction properties of the specific beam shapes, since the homogeneous/uniform medium itself cannot, in general, modify beam velocity under propagation. Other known mechanisms for beam velocity control and acceleration include the presence of external optical potentials due to inhomogeneity of the medium. Soliton propagation in such media can be described intuitively in terms of an effective particle approach [23, 24], under which a soliton corresponds to a particle of given mass moving in an effective potential depending on both the geometry of the inhomogeneity and the soliton shape. In the absence of any dissipative (gain or loss) mechanisms the soliton (particle) mass remains constant and its motion is determined by the energy landscape ”experienced” by the soliton. For transversely inhomogeneous media we can distinguish between the cases of monotonous and periodic modulations. In the first case, we can have continuous velocity increasing or decreasing based on the slope of the respective potential [25] while in the second case the velocity oscillates periodically so that no net acceleration can take place [26]. On the other hand, a transversely periodic medium has been shown to result in average velocity variation when a longitudinal periodic modulation is introduced [4, 27–30]. This is known as a ”ratchet” effect and it is related to a symmetry breaking of soliton dynamics [31].

In this work we present a novel mechanism for soliton acceleration in transversely periodic structures with modulated linear and nonlinear refractive indices under the presence of dissipative mechanisms. The key idea is that, although the medium is longitudinally homogeneous, a soliton with varying mass, due to gain and/or loss mechanisms, ”experiences” a longitudinally varying effective potential through the dependence of the latter on the soliton mass. In these systems the soliton formation results not only from the balance between nonlinearity and diffraction but also from the competition between gain and loss mechanisms of different physical origin. In contrast to the conservative cases, the localized modes of the system have no longer neutral stability properties and become attractors of enhanced stability which can be excited by a sufficiently larger class of initial conditions [32–34].

2. Model and results

The underlying model of soliton propagation is the NonLinear Schrödinger (NLS) equation with linear and nonlinear dissipative terms and spatially modulated linear and nonlinear refractive index

 

Fig. 1 Amplitude of the effective potential versus soliton mass, as given by Eq. (5), for the cases where either the linear, the nonlinear or both refractive indices are spatially modulated with a period 2π (K = 1). α₁ = 0.05, α₂ = 0 (blue line), α₁ = 0, α₂ = 0.05 (green line), α₁ = 0.05, α₂ = −0.12 (red, thick line).

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Let us consider a soliton of mass m_in launched at a local maximum of the effective potential with zero initial velocity. Firstly we investigate the case where only the linear or the nonlinear refractive index is modulated (α₂ = 0 or α₁ = 0). In the absence of linear gain (σ₁ = 0, σ₂ > 0), the soliton will evolve with continuously decreasing mass in a potential of continuously decreasing amplitude (Fig. 1), and as a result will have a continuously increasing velocity. In that case we can only achieve acceleration of solitons having an increasing width and decreasing amplitude during propagation; such a mechanism would be useful only for a restricted propagation distance so that a specific degree of localization is retained. Under the presence of linear gain (σ₁ ≠ 0), there is fixed mass value $m_0^{\text{'}}=\sqrt{-6{\sigma }_1/{\sigma }_2}$ for which the rhs of Eq. (2) becomes zero. The gain and loss mechanisms are exactly balanced for m′₀, resulting in stable soliton propagation. A soliton with initial mass m_in > m′₀ will evolve with decreasing mass until its mass reaches the value m′₀ and then propagate with constant shape. During the transient phase, the amplitude of effective potential has been decreased and soliton has been accelerated. Subsequently, the soliton propagates with a velocity which is highly oscillating around a nonzero mean value, as shown in Fig. 2(a).

 

Fig. 2 Soliton velocity evolution versus propagation distance for the case of lattice period 2π (K = 1), under the presence of nonlinear dissipation (σ₂ = 0.005) and linear gain σ₁ = − 0.0033, for which zero dissipation mass is m′₀ = 2. (a) Velocity evolution of a soliton having m_in = 2.4 for the case where only the linear (blue line) or the nonlinear (green line) refractive index is modulated; (α₁ = 0.05, α₂ = 0) and (α₁ = 0, α₂ = 0.05), respectively. (b) Velocity evolution of a soliton having m_in = 2.4 or m_in = 1.6 for the case where both the linear and the nonlinear refractive indices are modulated (α₁ = 0.05, α₂ = −0.12). Results obtained under the effective particle approach [Eqs. (2)–(4)].

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When both linear and nonlinear refractive indices are modulated, the presence of a soliton mass value $m_0=\sqrt{-12{\alpha }_1/{\alpha }_2-K^2}$ for which the amplitude of the effective potential is zero, when −12α₁/α₂ > K², allows for two options for soliton acceleration. As it can be seen from Fig. 1, a soliton launched with an initial mass m_in > m₀ can be accelerated by reducing its mass to m₀ if the values of the gain and loss coefficients are chosen so that m′₀ = m₀. In that case, after a transient stage, the soliton eventually travels with a constant mass and as if there is no spatial modulation of the medium. In contrast to the aforementioned cases where only one of the linear or the nonlinear refractive indexes were spatially modulated, in this case soliton eventually travels with a constant, non-oscillating velocity as shown in Fig. 2(b). On the other hand, soliton acceleration with simultaneous mass increasing is possible for solitons launched with initial mass m_in < m₀ as it can be seen from the form dependence of the effective potential amplitude on the soliton mass in Fig. 1. In contrast to all previous cases, this is the only possibility for beam acceleration to a constant velocity [Fig. 2(b)] not only without degradation but also with enhancement of the beam localization, which is an important feature of the acceleration mechanism with respect to applications.

The final velocity value v_out as a function of the initial soliton mass m_in is shown in Fig. 3(a) for spatial modulations having different periods, as obtained from the effective particle Eqs. (2)–(4). Note that the initial soliton position located at the maxima of the respective effective potential differs by 2π/K for the cases m_in > m₀ and m_in < m₀, due to sign reversal of the potential. The maxima of the potential are saddle points for soliton dynamics in the absence of gain and loss. The mass dependence on z, due to the presence of gain and loss, results in an additional degree of freedom of the effective particle system, which becomes nonintegrable. Complex dependence of the soliton evolution on the initial conditions occurs in the region around a saddle point corresponding to a potential maximum, so that a stationary soliton can be accelerated to either positive or negative velocity. However, this region, corresponding to a chaotic separatrix layer is very narrow. Therefore, we can always choose an initial soliton position which is close enough to a potential maximum corresponding to high initial particle energy but slightly displaced from it in order to determine the sign of the final soliton velocity. Such a displacement from the potential maximum of 1% with respect to the lattice period has been used in all results. As it can be seen in Fig. 3(a), the modulation period K determines the value mass m₀ for which the soliton ”experiences” no effective potential as well as the maximum output velocity for given modulation amplitudes (α_i, i = 1, 2). On the other hand, the output velocity scales with the square root of α_i so that an increased modulation amplitude results in increased output velocities. In Fig. 3(a) and (b), the two cases of soliton acceleration with decreasing and increasing mass, respectively, are shown, as given by the effective particle Eqs. (2)–(4) and numerical simulations of the original NLS Eq. (1). The results are shown to be in remarkable agreement with respect to the initial stage of the acceleration process as well as to the final velocities given by the slopes of the respective lines.

 

Fig. 3 (a) Soliton final velocity versus initial mass for a lattice with α₁ = 0.05, α₂ = −0.12, σ₂ = 0.005, σ₁ = −0.0033 and K = 0.5, 1, 1.5 as obtained from the effective particle approach. (b) and (c) Transverse soliton position under propagation for the cases where m_in > m₀ (b) and m_in < m₀ (c). The lattice parameters correspond to the case m₀ = m′₀ and the initial soliton masses correspond to the points depicted in (a). Results from the effective particle approach (green line) are compared to results from numerical simulation of the original NLS Eq. (1) (blue line). A noise level of 1% with respect to soliton amplitude has been superimposed in the initial soliton profiles. The insets show the initial and final soliton shapes (amplitude and width).

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In all previous cases, relatively weakly modulated lattices and small gain/loss coefficients have been considered resulting in a qualitative and quantitative agreement between the predictions of the effective particle model and the actual soliton dynamics as described by the NLS Eq. (1), with respect to the soliton acceleration mechanism. In the following we consider a case of stronger modulation resulting in higher values of soliton output velocities. We also consider higher values of gain/loss coefficients resulting in shorter propagation lengths for the achievement of the final soliton velocities. In Fig. 4, soliton acceleration under such a configuration is depicted. It is shown that all the qualitative features of the acceleration mechanism, predicted in terms of the effective particle approach, actually persist even under the presence of a noise level superimposed to the initial soliton profile. An appropriately small value of the linear gain coefficient as well as the finite length of the propagation distance prevents the zero-solution instability to show up and more importantly to destroy the characteristic features of soliton dynamics. Therefore, soliton acceleration takes place when for both cases m_in > m₀ and m_in < m₀. In comparison to the case of weaker lattices where the soliton mass attains a fixed value after a transient stage (as predicted by the effective particle approach), the soliton mass undergoes periodic oscillations. However, the averaged value of the soliton mass is smaller or larger than m_in for the case where m_in > m₀ or m_in < m₀ in accordance to results of the effective particle approach.

 

Fig. 4 Numerical simulation of the original NLS Eq. (1) for the case of a stronger lattice having α₁ = 0.4, α₂ = −0.96, K = 0.5 and higher gain/loss coefficients σ₂ = 0.01 and σ₁ = −0.0067 (m₀ = m′₀ = 2.18) for solitons having m_in = 2.9 (a) and m_in = 1.5 (b). A noise level of 1% with respect to soliton amplitude has been superimposed in the initial soliton profiles.

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3. Discussion and conclusions

In conclusion, we have demonstrated effective mechanisms for dissipative soliton acceleration in nonlinear optical lattices under the presence of gain and loss. The specific case of linear gain and nonlinear loss corresponding to two-photon absorption has been considered. However, the soliton acceleration mechanism is directly applicable to any other combination of linear or nonlinear gain and loss. Therefore, the case of a nonlinear loss mechanism such as three photon absorption can also be considered as well as the case of a nonlinear gain compensated by either linear [37] or nonlinear loss, for which no background instability appears. The choice of the appropriate combination of the gain and loss mechanism can be based on their experimental feasibility and their dynamical controllability related to the controllability of the whole soliton acceleration mechanism.

The importance of modulating both the linear and the nonlinear refractive index has also been presented. In contrast to the case where either the linear or the nonlinear refractive index is spatially modulated, we have shown that when both indices are modulated with the same period we can have soliton acceleration and mass increasing as well as stable soliton propagation with constant non-oscillating velocity. The acceleration mechanism has been shown to persist under a wide range of lattice and gain/loss parameters while being robust under the presence of noise.