SSpatial storage of discrete dark solitons
Alejadro J. Mart´ınez and Yair Z´arate Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford,Oxford OX2 6GG, United Kingdom Nonlinear Physics Center, Research School of Physics and Engineering, Australian National University,Canberra, Australian Capital Territory 0200, Australia
The interaction between a mobile discrete dark soliton (DDS) and impurities in one-dimensionalnonlinear (Kerr) photonic lattices is studied. We found that the scattering is an inelastic processwhere the DDS can be reflected or transmitted depending on its transversal speed and the strengthof the impurities. In particular, in the reflection regime, the DDS increases its transversal speed aftereach scattering. A method for spatial storage of DDS solutions using two impurities is discussed,where the soliton can be trapped within a storage region until it reaches the critical speed neededto be transmitted. We show numerically, that this method allows the storage of multiple DDSssimultaneously.
PACS numbers: 42.65.Tg, 42.25.Fx, 42.79.Vb.
I. INTRODUCTION
Scattering of traveling waves by local impurities hasbeen a subject of extensive research, both theoreticallyand experimentally, in a broad spectrum of physics [1, 2].In the framework of optics, this interaction provides theability to control localized traveling pulses by means ofswitching mechanisms. These can depend on, for exam-ple, the power of the pulse [3], the momentum [4], or eventhe particular structure of the impurities [5, 6]. Most ofthe research in this field has focused on the scattering ofbright waves. Especially those concerning discrete opti-cal systems, where there is a high flexibility to constructimpurities by manipulating the relative refraction indexbetween the impurities and the rest of the system [7].In particular, it is possible to generate either positive ornegative impurities with different resonance response [8].Different impurities permit to control transmission, re-flection or even the localization degree of energy [9]. Itis interesting to note that these phenomena are analo-gous to the scattering of matter-waves by local impuri-ties studied in Bose-Einstein condensates [10–12]. On theother hand, less attention has been given to the scatter-ing of dark waves, namely discrete dark solitons (DDS),with local impurities. Even though there is a consider-able amount of theoretical research on the subject (seee.g. [13–15]) and DDS waves have been experimentallyobserved for more than a decade [16, 17].It is the aim of this article to study the scattering pro-cess of a mobile DDS with local linear impurities. Firstwe show that the interaction is an inelastic process, wherethe DDS increases its transversal speed after the scatter-ing. Then, following a continuous approach based on adi-abatic perturbation theory, we develop an inelastic anal-ysis for the position of the core of the DDS as functionof the propagation coordinate. This analysis allows usto obtain the critical (minimal) initial speed that a DDSrequires in order to be transmitted by a delta impurity.Applying our results to a system with two impurities, weshow that the DDS can be stored between the impurities for a certain propagation distance depending on the ini-tial speed and the strength of the impurities. Along thestorage, the DDS is successively reflected by the impuri-ties until it reaches or exceeds the critical speed to escapethe confinement. We verify numerically that the pro-posed storage mechanism applies for either self-focusingor self-defocusing systems. Where the resulting propaga-tive nonlinear waves, except for the phase, exhibit thesame behavior. Finally, the storage for multiple DDS isreviewed. For symmetrical initial conditions, relative tothe position of the impurities, the DDSs always escapein pairs.
II. MODEL
In the coupled modes framework, the propagation oflight along a nonlinear single-mode waveguide array canbe described by Ψ( x, ξ ) = (cid:80) n A n ( ξ )Φ n ( x − na ), where ξ is the propagation direction, x is the spatial transversalcoordinate and a corresponds to the separation betweenguides. The field Φ n ( x ) is the mode of the n -th waveg-uide and the complex amplitude A n is described by thediscrete nonlinear Schr¨odinger (DNLS) equation [18], − i dA n dξ = V ( A n +1 + A n − ) + (cid:32) β + M (cid:88) l =1 δ n,n l ¯ (cid:15) (cid:33) A n +¯ γ | A n | A n , (1)where β is the linear propagation constant and δ n,n l ¯ (cid:15) represents the local perturbation to the refractive index,i.e. an impurity, located in the n l -th waveguide. The pa-rameter ¯ (cid:15) accounts for the strength of the impurity andit can take positive or negative values. V is the couplingcoefficient and ¯ γ = ( ω ¯ n ) / ( cA eff ) is the nonlinear pa-rameter; with ¯ γ > γ < ω is the optical fre-quency associated with the modes, ¯ n is the Kerr coef-ficient, c is the speed of light and A eff is the effective a r X i v : . [ phy s i c s . op ti c s ] M a r a) b)c) d) FIG. 1: Examples of scattering of DDSs with an impurity in aself-defocusing nonlinear media. a) and b) show the intensitydistributions as a function of n at z = 0 (black) and z =75 (red), for a DDS with initial speed B = 0 .
2. c) andd) show the dynamic associated with two independent initialconditions in each case, with B = 0 .
5. The vertical dashedlines denote the position of the impurity. The values of (cid:15) are0 . − . area of the waveguide mode [18]. Equation (1) can berescaled in order to reduce the number of parameters byconsidering A n ( ξ ) = (cid:112) V / ¯ γ u n ( z ) e iβz/V , with ξ = V z .Thus, Eq. (1) now reads − i du n dz = ( u n +1 + u n − ) + M (cid:88) l =1 δ n,n l (cid:15)u n + γ | u n | u n , (2)with (cid:15) ≡ ¯ (cid:15)/V and γ ≡ sgn(¯ γ ), where sgn is the signumfunction. An important feature of Eq. (2) is that itremains invariant under staggered-unstaggered transfor-mation: { z, (cid:15), γ, u n } → {− z, − (cid:15), − γ, ( − n u n } . Hence,by engineering the appropriate phase, both nonlinearregimes are equivalent. In fact, the balance between non-linearity, dispersion and discreteness required by the ex-istence of DDS can be achieved in both cases [13]. Thisleads to an unstaggered (staggered) wave for the self-defocussing (self-focusing) regime. Thus, without loss ofgenerality, we will focus our study on the self-defocusingcase, with γ = −
1. It is worth mentioning that a mobileDDS propagating in a self-defocusing system without im-purities can be described by u n = A tanh( A ( n − n ))+ iB ,where ˙ n ≡ dn dz ≈ B is the transversal speed of the DDS[13]. For the sake of simplicity, we fix the backgroundfield as A + B = 1, reducing the parameter space to { B, (cid:15) } .We conducted numerical simulations of a DDS in thenonlinear waveguide array in order to characterize thescattering process due to the presence of impurities.Equation (2) was integrated with a fourth-order Runge-Kutta scheme with a fixed-step size of δz = 10 − , this isin order to conserve the total power and the energy with arelative error lower than 10 − . To consider disturbancesthat emerge naturally in an experiment, we set as a ini-tial condition, the DDS profile for the homogeneous sys- Ε B Eq. (13)Eq. (6)
Transmission Reflection
FIG. 2: Critical speed of the DDS ( B c ) as a function of thestrength of a negative impurity, (cid:15) <
0. The dots denote thecritical speed calculated numerically from Eq. (2), the dashedred line corresponds to the Hamiltonian estimation, Eq. (6),and the dashed blue line is the fit given by Eq. (13) with α = 1 . tem (without impurities) as described above. Thus, oncethe DDS begins to propagate, the energy is split amongthe DDS itself, fluctuations in the background field andlocalized impurity modes [see Figs. 1a and 1b]. The mag-nitude of the background fluctuations is typically lowerthan 10% of the amplitude of the background field, andin most of the cases it is smaller than the amplitude ofthe localized impurity mode.For a single positive impurity, i.e. (cid:15) > M = 1 inEq. (2), the DDS is always transmitted independentlyof the relation between the parameters [see Fig. 1c].Whereas, for a negative impurity, the DDS can be eitherreflected or transmitted [see Fig. 1d]. The transition de-pends on the relation between the transversal speed ofthe DDS and the strength of the impurity. The remain-der of this article will focus on the negative impuritiescase ( (cid:15) < III. ADIABATIC APPROACH
To understand the DDS dynamics in the presence ofimpurities, we perform a continuous approach of the dis-crete system. For that, we introduce a phase shift ofthe complex amplitude u n ( z ) → u ( x n , z ) e iz in Eq. (2).And considering the discrete derivative as ( u n +1 + u n − − u n ) ∝ (cid:0) ∂ u/∂x (cid:1)(cid:12)(cid:12) x = x n , we obtain − i ∂u∂z = ∂ u∂x + V ( x ) u − | u | u , (3)where a local impurity at the origin has been introducedas a delta-like potential, V ( x ) ≡ (cid:15)δ ( x ). To attain ananalytical description of the evolution of the dark soli-ton (DS), we adopt the adiabatic perturbation theorydeveloped in Ref. [12] for dark solitons interacting withimpurities in Bose-Einstein condensates. Hereby, we con-sider the complex amplitude u ( x, z ) = v ( x, z ) u b ( x ) e − iz in Eq. (3), where u b ( x ) = 1 + (cid:15) e − | x | represents thewavefunction associated with the background field in thepresence of a delta impurity. Thus, the DS profile v ( x, z )satisfies the perturbed defocusing nonlinear Schr¨odingerequation [see Ref. [12] and references therein] i ∂v∂z + ∂ v∂x − (cid:0) | v | − (cid:1) v = P ( v ) , (4)where P ( v ) = (cid:15)e − | x | [(1 − | v | v ) v − sgn( x ) ∂ x v ]. Itis known that in the unperturbed limit ( P ( v ) = 0),i.e. the defocusing NLS equation, the DS solution isgiven by v ( x, z ) = cos φ tanh ξ + i sin φ , where ξ =cos φ [ x − (sin φ ) z ] [19]. Then, following the adiabatictreatment [12], after the introduction of the impurity,the parameters become slowly varying functions of thepropagation coordinate z but the functional form re-mains unchanged. Thus, the soliton phase is promotedto φ → φ ( z ) and the soliton coordinate becomes ξ → cos φ ( z ) [ x − x ( z )]. Here, the variable x ( z ) accounts forthe core of the dark soliton. As a result, the dynamicsof the DS in the presence of an impurity is determinedby the Newtonian equation d x /dz = − dU/dx , with U ( x ) = − (cid:15)/ ( x ). This force equation is associ-ated with the Hamiltonian H = ˙ x − (cid:15) ( x ) . (5)In this case, H c = − (cid:15)/ H < H c ) or transmitted (for H > H c ) by the impurity. This defines a critical initialspeed given by B c ≈ ˙ x c = ± √− (cid:15) , (6)which corresponds to the minimum speed that a darksoliton coming from infinity must have in order to betransmitted by the impurity. Notice that here infinityis understood as an initial position at which the effectof the impurity over the DS can be neglected. Accord-ing to Eq. (6), the transition arises only when (cid:15) < x c ∈ R ), whichagrees with our previous numerical observations. Fig-ure 2 shows the comparison between the critical velocityfor the DDS ( B c ) and the Hamiltonian estimation givenabove. We highlight that the critical speed obtained bythe Hamiltonian description represents only a scaling ap-proach. This is because continuous and discrete modelsare not exactly equivalent. IV. INELASTIC DESCRIPTION
As expected, the Hamiltonian analysis [cf. Eq. (5)]leads to an elastic description of the scattering process.However, the exchange of energy involves mainly threedifferent type of waves: the DDS, which can change itswidth and speed after each scattering. The localized im-purity mode, whose characteristic size depends on (cid:15) . And a) b) c)
FIG. 3: Phase space for a system with one impurity. a) Elasticdesciption, b) inelastic description and c) numerical integra-tion of Eq. (2). finally, radiation modes that spread over the background.Such interchange between different modes produces in-elastic dynamics of scattering. In order to confirm this,we construct the numerical phase space by extractingthe speed and position of the center of the DDS from thesimulation. Since we have considered a local impurity,the speed change occurs in a region close to the impu-rity. That is, the DDS travels with a constant speeduntil the interaction with the impurity takes place. Con-secutively, the speed of the DDS is increased and then,when the DDS is far enough from the impurity, the newspeed remains constant [cf. Fig. 1]. Figure 3 shows thephase spaces for the Hamiltonian description, the numer-ical simulation, as well as, the inelastic model that wewill develop below. Numerical phase space in Fig. 3chas been made by interpolating the DDS using a piece-wise Hermite interpolation method. Then, we estimatethe speed by tracking the minimum of the interpolatedfunction as a function of z .In order to deliver a qualitative and quantitative de-scription for the inelastic behavior, we introduce a phe-nomenological correction to the Newtonian equation ob-tained after the adiabatic treatment. Based on our nu-merical observations discussed above: (i) the interactionis local and (ii) the speed of the DDS is not preserved.The position of the soliton core can thus be described by¨ x = − dU ( x ) dx (1 + ¯ α ( x ) ˙ x ) , (7)where ¯ α ( x ) ≡ α · sgn( x ). The constant parameter α accounts for the injection of energy due to the interactionwith the impurity. Note that the function sgn( x ) hasbeen introduced to preserve the invariance of the systemunder spatial inversions ( x → − x ) [cf. Eq. (3)]. Hereby,we are able now to determine the increment in the speedof a the DDS after being reflected by the impurity. Wethen perform a perturbative analysis for the final speedof the DDS at the left of the impurity approaching it withpositive speed ˙ x f, . It is important to mention that thefinal speed is evaluated at the z where the DDS returnsto its initial position, such that x i = x f . Accordingly, weintroduce the expansion ˙ x f = ˙ x f, + α ˙ x f, + α ˙ x f, + · · · in Eq. (7), obtaining¨ x f, = − dUdx (cid:12)(cid:12)(cid:12)(cid:12) x = x f , (8)¨ x f,n = dUdx (cid:12)(cid:12)(cid:12)(cid:12) x = x f ˙ x f,n − = ( − n ( n + 1)! d ˙ x n +1 f, dz . (9)The former equation is the force equation obtainedfor the unperturbed (Hamiltonian) description, Eq. (5).Thus, we set ˙ x f, = − ˙ x i = −√ (cid:112) H − U ( x i ), where ˙ x i isthe initial speed and H is the unperturbed initial energy.Equation (9) leads to an asymptotic expansion of ˙ x f inthe form˙ x f = − x i − ˙ x i α − · · · = − ˙ x i ∞ (cid:88) n =0 ˙ x ni α n ( n + 1)! . (10)This series can we written in closed form as˙ x f = (1 − e α ˙ x i ) α ≡ F ( ˙ x i ) . (11)It is worth noting that the final speed reached by the DDSafter being reflected by the impurity, only depends onthe speed that the DDS had before the interaction. Fora DDS approaching the impurity from the right (with aninitial speed − ˙ x i ) we obtain a final speed as in Eq. (10)but positive. In a compact form, it reads ˙ x f = ( e α ˙ x i − /α . Thus, a DDS colliding with the impurity from theleft or the right will experience the same increase in speedas function of its initial speed. This is consistent withnumerical simulations on the discrete system [cf. Fig. 3]. V. STORAGE MECHANISM
The capability of negative impurities to reflect DDSsallows us to design a simple, but effective, mechanismof spatial storage. This consists of a system of two im-purities ( M = 2) distant enough such that the interac-tion of the DDS with each impurity occurs independently.Therefore, a DDS initially positioned between impuritieswill remain in the storage region as long as its speed isless than the critical speed at which it is transmitted out[see Fig. 4a,b]. Within the storage region, the DDS isconsecutively reflected by the impurities delimiting thestorage region. After each collision the speed of the DDSincreases according to Eq. (11). Note that the condi-tion for the above to be valid is that the impurities aresufficiently separated. Once the DDS reaches the criti-cal speed to be transmitted by one of the the impurity,it will leave the storage region. From Eq. (11) we canbuild a recursive procedure to determine the number ofreflections that a DDS will experience before reachinga speed greater (or at least equal to) than the criticalspeed. Thus, we define the n -th critical speed B nc ≡ ln (cid:0) αB n − c (cid:1) α , (12) S t o r a g e d i s t a n ce , z s d) c) a) b)c) FIG. 4: a) and b) show the propagation of a DDS along thestorage region (hatched in green) bounded by two impurities(vertical dashed lines) for B = 0 . B = 0 .
27, respectively.The horizontal dashed lines mark the storage distance z s . c)Shows the number of reflections until the DDS is transmittedout of the storage region. White dashed lines were calculatedusing Eqs. (12) and (13) for α = 1 .
2. The circle and thestar mark the parameters used in a) and b), respectively. d)Storage distance as function of B . Red dashed and black linesare associated with the numerical simulation and Eq. (14),respectively. In a), b) and d) (cid:15) = − .
8, and the impuritiesare separated by 30 guides. as the maximum initial speed that a DDS can have suchthat there will be n reflections before escaping of the stor-age region. That is, if the initial speed of the DDS is inthe interval ( B n +1 c , B nc ), then it will be reflected n timesbefore being transmitted. Note that Eq. (12) requires oneto know the minimal initial speed at which the DDS istransmitted. A good estimation of this quantity is givenby B c = ln (cid:0) √− (cid:15) (cid:1) α . (13)Figure 4c show good agreement between the analytical n -th critical velocity, Eq. (12), and the numerical criticalspeed of the DDS ( B c ).Finally, an upper value for the storage distance z s isestimated by considering that the speed reached by theDDS after scattering with one impurity, remains constantuntil the next reflection z s = d (cid:32) − | ˙ x i | + N (cid:88) m =1 (cid:12)(cid:12) F ( m ) ( ˙ x i ) (cid:12)(cid:12) (cid:33) , if B Nc < ˙ x i < B N − c , (14)where d is the space between impurities. Fig. 4d com-pares the numerical and analytical results for a certainstorage distance. We can see that Eq. (14) provides agood estimation for few reflections, however the error in-creases according to the number of reflections. A. Storage of staggered discrete dark solitons
Throughout this article, all of our numerical simula-tions have involved the integration of Eq. (2), which in-trinsically describes a discrete system. Here, effects ofdiscreteness in the propagation of the DDS gains impor-tance mainly at very low speed, when the DDS solutionsare highly localized [13]. Faster DDSs are well describedby a continuous description. Thus, we expect that stor-age behavior arises in continuous systems as well, de-scribed by the NLS equation in the presence of narrowimpurities. However, in this case dark solitons exist onlyin the self-defocusing regime, thus storage can be possibleonly in this nonlinear regime.Otherwise, in a waveguide configuration where DNLSapplies, storage of DDS can be possible in both, self-defocusing and self-focusing nonlinear regimes, with ap-propriate phase engineering. In order to emphasize this,we compute numerical simulations in both cases andwe did not find any difference between either regime.Of course, numerically this was expected since bothregimes are mathematically equivalent because staggered-unstaggered transformation applies (see Sec. II). Figure 5shows the intensity | u n | and the real part of the opti-cal field Re { u n } associated with the dynamics of a DDSbetween two impurities. Here, we can see that the onlyone difference between both fields comes from the rela-tive phase between neighboring guides, while the inten-sity distribution is the same. VI. STORAGE OF MULTIPLE DDS
Now, an easy way to generate multiple discrete darksolitons is using as initial condition: u n (0) = 1 − C K (cid:88) k =1 sech [ D ( n − n k )] , (15)which leads to the formation of K pairs of DDSs thatpropagate with equal speed, but in opposite directions.Each pair is centered initially around n k . The speed canbe controlled by manipulating either C ∈ [0 ,
2] or
D > K = 1) andtwo ( K = 2) pairs of DDSs initially placed between twonegative impurities. Some examples are shown in fig-ure 6, where long-distance storage is observed ( z > K = 1 and the initial configuration is symmetric Staggered Unstaggered a) b)c) d)
FIG. 5: Example of propagation of staggered (left panels, γ = 1) and unstaggered (right panels, γ = −
1) discrete darksolitons in a storage regime. a) and b) represent the evolutionof the intensity, while c) and d) the real part of the opticalfield: Re { u n } . Initial speed is given by B = 0 .
3. The impuri-ties are separated by 30 guides, | (cid:15) | = 0 . with respect to the central point between the two impu-rities, the optical field will propagate symmetrically aswell and finally the pair of DDSs escape simultaneouslyafter reaching the critical speed. As in the case of oneDDS, the mechanism is the same and the DDS increasesits speed after each inelastic interaction with the impu-rities. Furthermore, the storage distance depends onlyon the initial speed and the strength of the impurities.Nevertheless, the entire dynamic along z is enriched bythe elastic collisions undergo by the DDSs.In the case of K = 2, the internal dynamic is morecomplex and either periodic or aperiodic oscillations canbe found, depending on the initial configuration. Fur-thermore, if the initial condition is symmetric, then theDDS will propagate symmetrically and DDSs will escapeby pairs as long as their speed increases enough. This isshown in figure 6, where we can see that faster DDSs es-cape, while slower DDSs are maintained within the stor-age region. VII. CONCLUSIONS
In conclusion, we have shown that the scattering of aDDS with a single impurity is an inelastic process andthe condition for reflection or transmission relies only onthe initial speed of the DDS and the strength of the im-purity. Furthermore, after the scattering, the speed ofthe DDS is increased. A phenomenological description a) b)c) d)e) f)
FIG. 6: Examples of propagation of multiple DDSs betweentwo negative impurities. Left panels are associated with onepair of DDSs ( K = 1) centered at the middle of the impurities,while right panels for two pairs of DDSs ( K = 2) separatedby n − n = 12 guides. The impurities are separated by 30guides and their positions are shown as vertical dashed lines.The parameters are: a)-b) C = 1, c)-d) C = 1 .
5, and e)-f) C = 2. In each case (cid:15) = − . B = 0 .
6. The initialintensity distribution | u n | is shown in the upper panels. for the inelastic behavior of the DDS has been proposedbased on the well known adiabatic perturbation theory,by including a non-Hamiltonian term, which accounts forthe acceleration of the DDS after each reflection with animpurity.In addition, a method of spatial storage based on twonegative impurities is proposed. We show that a DDScan be trapped within a storage region until it reaches thecritical speed needed to be transmitted. We showed thatthis method can be implemented successfully for trappingmultiple discrete dark solitons for a certain propagationdistance. Acknowledgements
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