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Discrete optical soliton scattering by local inhomogeneities
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Photonics and Nanostructures – Fundamentals and Applications 11 (2013) 95–101 www.elsevier.com/locate/photonics
Discrete optical soliton scattering by local inhomogeneities Lasha Tkeshelashvili a,b,* a
Andronikashvili Institute of Physics, Tamarashvili 6, 0177 Tbilisi, Georgia b Tbilisi State University, Chavchavadze 3, 0128 Tbilisi, Georgia
Received 27 July 2012; received in revised form 10 October 2012; accepted 30 October 2012 Available online 9 November 2012
Abstract The nonlinear wave scattering by local inhomogeneities in discrete optical systems is studied both analytically and numerically. The presented theory describes the reflection and transmission of discrete optical solitons at a point defect. In particular, the derived expressions determine the reflected and transmitted pulses from the incident one. In the range of validity, the analytical results are in excellent agreement with the numerical simulations. It is demonstrated that the point defects in structured optical materials represent effective tool for controlling and manipulation of the nonlinear light pulses. # 2012 Elsevier B.V. All rights reserved. Keywords: Discrete optical solitons; Point defects; Wave scattering
1. Introduction The studies of discrete wave dynamics in structured systems range from the realization of optical analogies of various quantum-mechanical phenomena [1,2] to practical design of functional elements for the all-optical communication networks [3,4]. In particular, the tailored light-matter interaction processes in such systems [5,6] allow to demonstrate unique effects related to the nonlinear optical pulses called solitons [7]. Solitons are localized wave packets that can propagate undistorted in homogeneous nonlinear media. That peculiarity makes nonlinear systems very attractive for applications in the field of all-optical communications [8]. However, in general, the soliton interaction with inhomogeneities is an extremely com-
* Correspondence address: Andronikashvili Institute of Physics, Tamarashvili 6, 0177 Tbilisi, Georgia. Tel.: þ995 32 239 87 83; fax: þ995 32 239 14 94. E-mail address: [email protected]. 1569-4410/$ – see front matter # 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.photonics.2012.10.001
plicated process. Indeed, the solitons represent solutions of so-called integrable nonlinear equations [9]. The inhomogeneity, at least locally, breaks the integrability of the model. In the non-integrable models the stability of nonlinear pulses is not guaranteed anymore, and the nonelastic effects such as the soliton radiative decay in the scattering processes may take place [10]. Perhaps, the effectively one-dimensional discrete structures represent the most convenient systems for study of the nonlinear wave dynamics [1,7,11]. In particular, much of the important theoretical and experimental results were obtained for arrays of optical waveguides [12], coupled nano-cavities in photonic crystals [13], metallo-dielectric systems [14,15], and the Bose–Einstein condensates in deep optical lattices [16– 18]. The universal mathematical model that governs the wave dynamics in such systems is the Discrete Nonlinear Schro¨dinger (DNLS) equation [19,20]: @c i n þ Cðcnþ1 þ cn1 Þ þ Njcn j2 cn þ en cn ¼ 0: @t (1)
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Depending on the system under consideration, t is either the temporal or spatial variable [1]. In the case when t is the temporal variable, the localized wave packets are the optical pulses. However, for the systems such as the coupled waveguide arrays, t is a spatial coordinate, and those pulses represent optical beams. The eigenmode amplitude at the site n is cn, and C gives the evanescent coupling rate between adjacent sites. N is the nonlinear coefficient. Here, the inhomogeneity is introduced through the n-dependent en. In different cases that might reflect different physical factors. For instance, in the case of arrays of optical waveguides, the variation of en from site to site may be caused by the different refractive index of the individual waveguides. It should be noted that the defect states can be introduced in the system by different means. In particular, the bond defects defined as a local variation in C were studied in [21–24]. Moreover, Ref. [25] addressed the effects caused by local inhomogeneities in the nonlinear coefficient N. The problem of soliton scattering by localized defects in binary optical lattices was considered in [26]. Below, following [27,28], it is assumed that C, as well as the nonlinear coefficient N, is constant for all sites. Thus, only the linear term associated with en defines a scatterer in the system. Nevertheless, even in such case the nonlinear effects may cause coupling to the linear defect modes [29]. Indeed, the numerical simulations reported in [27,28] show that various scattering regimes might be realized with soliton trapping, reflection, and transmission effects. In what follows, the weakly nonlinear wave scattering by point defects in the discrete optical systems is studied both analytically and numerically. In this case, the nonlinear term in Eq. (1) can be treated as a small perturbation to the linear part. That allows to derive analytical expressions for the scattered waves. Note that, only the variation in en is relevant for the wave propagation and scattering processes. Indeed, by means of cn ! expði˜etÞcn transformation, the values of en in the corresponding term of Eq. (1) can be shifted by an arbitrary constant e˜. Suppose that the point defect is located at n = 0. Then, without loss of generality, en for n 6¼ 0 can be assumed to be exactly zero, i.e. en = e d0,n. Here, d0,n is the Kronecker delta, and e determines the defect strength.
Nonlinear Schro¨dinger (NLS) equation for the carrier wave envelope [7,8]. In turn, the inverse scattering method allows to obtain the solutions of the NLS model analytically [9]. In particular, under certain conditions, the NLS equation supports the bright soliton solutions. Physically, the soliton formation is possible when the balance between the nonlinear effects and the linear dispersion (or diffraction) takes place. That happens when the wave envelope is a slowly varying function on the carrier wavelength scale. It must be stressed that the wider envelope soliton has the smaller amplitude [8,9]. Therefore, the presented theory describes the dynamics of solitons which are sufficiently broad. As a result of the localized pulse scattering process, in the final state, there exist the reflected wave propagating backwards, and the transmitted wave which tunnels to the other side of the defect. In general, the incident pulse can excite the localized defect mode as well [1,7,32]. That may cause the trapping of the incident soliton by the defect [27,28]. However, as will be shown below, such bound states are not realized for the weakly nonlinear pulses and, therefore, can be neglected. Moreover, since the point defect is assumed to be linear, in the reflection process the nonlinear frequency conversion processes do not take place. Therefore, the incident, reflected, and transmitted waves have the same wave numbers |k|.
2. Perturbation analysis
In addition, vg is the group velocity:
In the weakly nonlinear limit the problem can be simplified significantly by means of the reductive perturbation method [30,31]. In particular, the nonintegrable Eq. (1) can be reduced to the integrable
vg ¼
2.1. Transmitted wave Suppose that the incident localized pulse impinges at the point defect from the left region (n < 0). Then, according to Ref. [31], the ansatz for the transmitted wave in the right region n > 0 is: cn ¼
1 X X
mm V ðm;lÞ ðj; tÞEnðlÞ :
(2)
m¼1 lm
Here, the smallness parameter m 1 guarantees that the nonlinear term in Eq. (1) can be treated as a small perturbation. Moreover, in order to describe explicitly the wave envelope dynamics, the set of new ‘‘slow variables’’, t = m2n and j ¼ mðn vg tÞ, is introduced. ðlÞ The carrier wave is En ¼ expðil½kn vtÞ, and v is given by the dispersion relation: v ¼ 2C cosðkÞ:
dv ¼ 2C sinðkÞ: dk
(3)
(4)
The perturbation analysis based on Eqs. (1) and (2) results in the NLS equation for the slowly varying
L. Tkeshelashvili / Photonics and Nanostructures – Fundamentals and Applications 11 (2013) 95–101
envelope V(1,1)(j, t) [30,31]: iVtð1;1Þ þ
Dk ð1;1Þ V þ N k jV ð1;1Þ j2 V ð1;1Þ ¼ 0; 2 jj
(5)
with Dk ¼ cotðkÞ;
vc0 þ Cðc1 þ c1 Þ þ ec0 0: (7) c0
The slow variables in the subscripts represent the corresponding partial derivatives, i.e, f t @f/@t, etc. As can be seen from the definition of j, Eq. (5) describes the transmitted wave in the reference frame moving with the group velocity vg.
1 X X
In the left region there is only the incident pulse initially. However, the scattering process results in the reflected wave as well. Therefore, for n < 0 the ansatz for cn reads [31]: 0
0
mm U ðm;l;l Þ ðj; j; tÞEnðl;l Þ ;
(8)
(12)
m¼1 lm
However, in addition, according to Eq. (8): c0
1 X X
ðl;l0 Þ
0
mm U ðm;l;l Þ E0
:
(13)
m¼1 lþl0 m
V ð1;1Þ ¼ U ð1;1;0Þ þ U ð1;0;1Þ : Note that at n = 0 the slow variables are: t ¼ 0; j ¼ mvg t; j ¼ þmvg t:
ðl;l0 Þ
here En ¼ expðil½kn vt il0 ½kn þ vtÞ. The additional slow variable j ¼ mðn þ vg tÞ is introduced to describe the reflected wave envelope. Nevertheless, at 0 this stage of calculations, U ðm;l;l Þ envelopes are allowed to be functions of all (that is t, j and j) variables. Inserting Eq. (8) into Eq. (1) reduces the initial DNLS equation the NLS model [30,31]. In particular, the perturbation analysis shows that the slowly varying envelope of incident wave does not depend on j (i.e. U(1,1,0) = U(1,1,0)(j, t)) and: Dk ð1;1;0Þ U þ N k jU ð1;1;0Þ j2 U ð1;1;0Þ ¼ 0: 2 jj
(9)
Furthermore, the envelope of the reflected wave is independent of j (i.e. U ð1;0;1Þ ¼ U ð1;0;1Þ ðj; tÞ) and obeys: Dk ð1;0;1Þ U N k jU ð1;0;1Þ j2 U ð1;0;1Þ ¼ 0: 2 jj
(14)
Furthermore, Eq. (8) for c1 gives:
m¼1 lþl0 m
iUtð1;0;1Þ
ðlÞ
mm V ðm;lÞ E0 ;
Eq. (12) in combination with Eq. (13) results in:
2.2. Reflected wave
iUtð1;1;0Þ þ
(11)
Here, c0 is given by Eq. (2):
1 N : Nk ¼ 2 C sinðkÞ
1 X X
by the NLS equation. The equation of motion at n = 0 can be used relate the amplitudes of the transmitted and reflected waves with that of the incident wave. Indeed, since the nonlinear term is small, at the defect Eq. (1) approximately gives:
(6)
and the nonlinear coefficient reads:
cn ¼
97
(10)
Eqs. (9) and (10) are written in the reference frames moving with the group velocities vg and vg , respectively.
c1 ¼
1 X X
0
ðl;l0 Þ
mm U ðm;l;l Þ E1 ;
(15)
m¼1 lþl0 m
and Eq. (2) yields the following expression for c1: c1 ¼
1 X X
ðlÞ
mm V ðm;lÞ E1 :
(16)
m¼1 lm
At n = 1 the slow variables read: t ¼ m2 ; j ¼ mð1 vg tÞ; j ¼ mð1 þ vg tÞ: Furthermore, the slowly varying functions V(1,1), U(1,1,0), and U(1,0,1) are expanded in the Taylor series: V ð1;1Þ ðmvg t m; m2 Þ V ð1;1Þ ðmvg t; 0Þ; U ð1;1;0Þ ðmvg t m; m2 Þ U ð1;1;0Þ ðmvg t; 0Þ; U ð1;0;1Þ ðþmvg t m; m2 Þ U ð1;0;1Þ ðþmvg t; 0Þ:
2.3. Continuity conditions
Since m 1, in these expansions only the leading terms are retained. The expressions for the reflected and transmitted wave envelopes at n = 0 follow from Eqs. (11) and (14):
The above analysis shows that the nonlinear wave dynamics on both sides of the point defect is governed
U ð1;0;1Þ ðj; 0Þ ¼ R U ð1;1;0Þ ðj; 0Þ;
(17)
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with the reflection coefficient R: e ; e þ 2iC sinðkÞ
(18)
V ð1;1Þ ðj; 0Þ ¼ T U ð1;1;0Þ ðj; 0Þ;
(19)
R¼ and
with the transmission coefficient T: T¼
2iC sinðkÞ : e þ 2iC sinðkÞ
(20)
Eqs. (17) and (19) determine the reflected and transmitted waves from the incident pulse. Moreover, it is easy to see that |R|2 + |T|2 = 1. That is in full agreement with the energy conservation law since the wave trapping processes by the defect are neglected.
by the celebrated NLS equation:
2.4. Bound states It should be noted that p k p [1]. Nevertheless, according to Eqs. (17) and (19), R and T diverge for the imaginary value k = ik. Here, k obeys: sinhðkÞ ¼
e : 2C
Fig. 1. The soliton scattering process at the local inhomogeneity. Here, |cn(t)| is plotted. In this simulation the point defect is located at n = 2000 (shown by the dotted line). The values of other parameters and further details are given in the text.
(21)
As with the quantum mechanics scattering problems, the divergence of the reflection and transmission coefficients is due to the discrete energy level associated with the point defect. That level gives rise to the bound state. The frequency v = V, which corresponds to k, follows from Eqs. (3) and (21): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 2 V ¼ 2C 1 þ : (22) 2C The frequency range of the modes with real k does not overlap with V. Nevertheless, nonlinear effects may cause the strong localization of the pulse in real space. In this case, the Fourier spectrum of the wave packet may become broad enough to be in resonance with V. The weakly nonlinear pulses are not strongly localized, and therefore, the processes which lead to the bound state formation do not take place. In this context, it must be stressed that the group velocity vg is an important parameter as well [27]. 3. Results Let us consider the soliton scattering by the point defect in more details. A numerical example of such process is shown in Fig. 1. As it is demonstrated above, the dynamics of the weakly nonlinear pulses is governed
i
@F P @2 F þ þ QjFj2 F ¼ 0: @z 2 @x2
(23)
The solutions of the NLS model can be obtained by means of the inverse scattering method [8,9]. In particular, Eq. (23) supports the bright soliton solutions if PQ > 0. Therefore, since it is assumed that the bright solitons can propagate in the system, the values of k are further restricted by DkNk > 0 inequality. For instance, if C and N are simultaneously positive, |k| < p/2 as can be seen from Eqs. (6) and (7). The presented theory describes p/2 < |k| < p range too. However, for those values of k the stable optical pulses do not exist, and therefore, such cases will not be considered below. 3.1. Fundamental soliton scattering Suppose that the incident pulse represents the fundamental soliton solution of Eq. (9): rffiffiffiffiffiffi 1 Dk j Dk t sech (24) Fðj; tÞ ¼ exp i : L Nk L 2 L2 In this expression a real parameter L is the soliton width [9]. However, L is related to the soliton amplitude as well. In particular, to wider solitons correspond the smaller values of the amplitude. As was discussed above, for the given vg , the value of L must be sufficiently large to suppress the formation of the bound state. From Eqs. (17) and (19) it directly follows that the initial conditions for Eqs. (10) and (5) read: rffiffiffiffiffiffi R Dk j sech (25) FR ðj; 0Þ ¼ ; L Nk L
L. Tkeshelashvili / Photonics and Nanostructures – Fundamentals and Applications 11 (2013) 95–101
rffiffiffiffiffiffi T Dk j sech FT ðh; 0Þ ¼ ; L Nk L
(26)
Eqs. (25) and (26) define the initial value problem for the reflected and transmitted waves, respectively. The sech-type initial value problem of the NLS equation was solved in Ref. [33]. In particular, for the following initial condition: rffiffiffiffi x A P sech Fi ðx; z ¼ 0Þ ¼ (27) ; L Q L the number of generated solitons in Eq. (23) is the maximum integer M which satisfies: 1 M < jAj þ : 2
(28)
For |A| 1/2, no soliton emerges and the pulse disperses. The dispersive modes, similar to the linear waves, have vanishing amplitudes in the final state. If 1/2 |A| 3/2, Eq. (28) shows that M = 1. In this case, by sheding the dispersive radiation, the solution asymptotically relaxes to the fundamental soliton with the amplitude: rffiffiffiffi 2 P 1 ðsÞ jAj : A ¼ (29) L Q 2 In general, M > 1 and there emerges the bound state of M solitons plus the dispersive radiation [8,33]. However, such higher-order soliton solutions are not relevant for the present problem. It can be seen from Eqs. (25) and (27) that for the reflected wave A = AR, where: AR ¼ R:
(30)
Similarly, from Eqs. (26) and (27), for the transmitted wave follows A = AT: AT ¼ T:
(31)
Furthermore, Eqs. (18) and (20) give: 1 jAR j ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 þ s 1
(32)
1 jAT j ¼ pffiffiffiffiffiffiffiffiffiffiffi ; 1þs
(33)
99
According to Eqs. (28) and (32) there is no soliton in the reflected pulse for 0 < s < 1/3. Moreover, for s > 1/3, only one reflected soliton is generated with the amplitude: rffiffiffiffiffiffi 1 Dk 2 ðsÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 : AR ¼ (35) L Nk 1 þ s 1 By comparing with Eq. (24), the width of the reflected soliton LR reads: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ s 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : (36) LR ¼ L 2 1 þ s 1 For the transmitted wave Eqs. (28) and (33) show that there exists only dispersive radiation for s > 3. If 0 < s < 3 one soliton emerges in the transmitted pulse with the amplitude: rffiffiffiffiffiffi 1 Dk 2 ðsÞ pffiffiffiffiffiffiffiffiffiffiffi 1 ; AT ¼ (37) L Nk 1þs and comparison with Eq. (24) shows that the width of the transmitted soliton LT is: pffiffiffiffiffiffiffiffiffiffiffi 1þs pffiffiffiffiffiffiffiffiffiffiffi : LT ¼ L (38) 2 1þs These expressions show how the incident soliton amplitude and width transform in the scattering process. For example, for s = 1, the reflected and the transmitted solitons are approximately 2.414 times wider compared to the incident pulse. 3.2. Numerical simulations An example of the fundamental soliton scattering process at the point defect is depicted in Fig. 1. The presented result is the numerical solution of Eq. (1) with the following initial condition: rffiffiffiffiffiffi 1 Dk cn ð0Þ ¼ m sechðJÞexpðiQÞ; (39) L Nk with
where s represents the effective strength of the defect: 2 e s¼ : (34) 2C sinðkÞ Note that the lower bound for s is smin = e2/(2C)2. Eq. (34) gives the quantitative measure of the group velocity vg (see Eq. (4)) influence on the scattering process.
m ðn n0 Þ; L D k m2 Q¼ kþ ðn n0 Þ: 2 L2 J¼
(40) (41)
In this expression it is assumed that the incident soliton at t = 0 is centered around site n0 = 1500. The values of other parameters are set as follows: C = N = 1, L = 1, m = 0.05, and e = 0.5. Since both C and N are chosen to be positive, as was discussed above, k must obey
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0.06
3.5
0.05
3 0.04
μ AT
σ
(s)
2.5 2 1.5
0.03 0.02
1
0.01
0.5 0 0
0
0.5
1
1.5
0.4
0.6
0.8
1
1.2
1.4
k
k ðsÞ
Fig. 2. The dependency of s on k. For the values of the chosen parameters see the text.
|k| < p/2. In Fig. 2 the dependency of s on k is presented for the given set of parameters. For the numerical simulation shown in Fig. 1 the incident pulse has k = 0.8. The corresponding value for the effective defect strength is s = 0.122, and the amplitude of the incident soliton is 0.059. It follows from Eqs. (32) and (33) that |AR| = 0.329 and |AT| = 0.944. Then, Eq. (28) implies that the reflected pulse is the dispersive wave packet, while the transmitted pulse relaxes to the fundamental soliton in the final state. The amplitude of the generated soliton is: jcT ðt ! 1Þjmax ¼
ðsÞ mAT
¼ 0:0524;
(42)
and, according to Eq. (38), LT = 1.125. The soliton formation process exhibits a damped oscillatory behavior of the pulse amplitude and width around the analytically predicted values. For instance, the arithmetic mean of the amplitude maximum and minimum values in the first oscillation is 0.0525. That is in excellent agreement with Eq. (42). The amplitude of the reflected pulse decays monotonously with time. Similarly, for the given width of the incident soliton L = 1, the excellent agreement between the theory and numerical simulations is found for 0.4 < |k| < 1.4. For example, the corresponding analytical predictions and numerical results for the transmitted soliton amplitude are given in Fig. 3. For |k| < 0.4 the incident soliton is too narrow. Indeed, Eq. (24) shows that the incident wave packet is localized on the scale of order 2L/ m 40. In turn, for example for k = 0.3, the carrier wavelength is 2p/k 21. Therefore, the soliton envelope is not really slowly varying. In this case, wider solitons, e.g. with L = 2, must be considered. Furthermore, the incident soliton is too narrow for 1.4 < |k| < p/2 as well. Although, in this case, |k| < p/2
Fig. 3. The dependency of mAT on k. The squares show the numerical results for the arithmetic mean of the amplitude maximum and minimum in the first oscillation of the soliton parameters. The dashed line represents the theoretical prediction.
still holds for the carrier wave number, sufficiently large portion of the pulse Fourier spectrum does not fall inside the limits of this range. That causes the breakdown of the soliton. The increase of pulse width leads to the shrinkage of its Fourier spectrum, and therefore, suppresses such instabilities. That is in agreement with the numerical simulations. 4. Discussion and conclusions In certain cases the inhomogeneous medium can be treated as a collection of independent scatterers. That is often referred to as the independent scattering approach [34]. For instance, assuming that 0 < s < 3 holds, consider the soliton tunneling through a set of regular or irregular set of point defects with the equal strengths e. Moreover, suppose that the soliton width is smaller compared to the average distance lav between the neighboring defects. The presented results can be applied to this problem as well. Indeed, Eq. (38) gives the soliton width increase after each scattering event. Then, it is clear that after passing through p defects the soliton size becomes: pffiffiffiffiffiffiffiffiffiffiffi p 1þs ð pÞ pffiffiffiffiffiffiffiffiffiffiffi : LT ¼ L 2 1þs Therefore, while propagating, the soliton width increases exponentially. The independent scattering ð pÞ approach becomes invalid when LT lav . Let us assume that this is the case at p = p0. For the system sizes less than p0lav the transmitted soliton amplitude decreases exponentially with distance as (see Eq. (37)): rffiffiffiffiffiffi p 1 Dk 2 ð pÞ pffiffiffiffiffiffiffiffiffiffiffi 1 ; cT ¼ m L Nk 1þs
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where p < p0. For the larger system sizes the independent scattering approach is not valid anymore. In conclusion, in the weakly nonlinear limit, the nonintegrable DNLS equation reduces to the integrable NLS model. This allows to derive the explicit expressions for the reflected and transmitted pulses from the incident one. The analytical results suggest that local inhomogeneities can be used for the management of the soliton amplitude and width. In the range of validity, the theoretical predictions are in excellent agreement with the numerical simulations. The discussed effects allow the effective control and manipulation of slow light pulses, and so, are potentially useful for applications in the field of all-optical communications. Acknowledgements I am grateful to R. Khomeriki for useful discussions. This work is supported by Science and Technology Center in Ukraine (Grant No. 5053). References [1] F. Lederer, G.I. Stegeman, D.N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, Discrete solitons in optics, Physics Reports 463 (2008) 1–126. [2] S. Longhi, Quantum-optical analogies using photonic structures, Laser and Photonics Reviews 3 (2009) 243–261. [3] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals: Molding the Flow of Light, second ed., Princeton Univ. Press, Princeton, 2008. [4] K. Busch, G. von Freymann, S. Linden, S.F. Mingaleev, L. Tkeshelashvili, M. Wegener, Periodic nanostructures for photonics, Physics Reports 444 (2007) 101–202. [5] M. Notomi, E. Kuramochi, T. Tanabe, Large-scale arrays of ultrahigh-Q coupled nanocavities, Nature Photonics 2 (2008) 741–747. [6] T.F. Krauss, Why do we need slow light? Nature Photonics 2 (2008) 448–450. [7] Y.S. Kivshar, G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic Press, San Diego, 2003. [8] G.P. Agrawal, Nonlinear Fiber Optics, third ed., Academic Press, San Diego, 2001. [9] M.J. Ablowitz, Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons, Cambridge Univ. Press, Cambridge, 2011. [10] Y.S. Kivshar, B.A. Malomed, Dynamics of solitons in nearly integrable systems, Reviews of Modern Physics 61 (1989) 763–915. [11] D. Hennig, G.P. Tsironis, Wave transmission in nonlinear lattices, Physics Reports 307 (1999) 333–432. [12] D.N. Christodoulides, R.J. Joseph, Discrete self-focusing in nonlinear arrays of coupled waveguides, Optics Letters 13 (1988) 794–796. [13] D.N. Christodoulides, N.K. Efremidis, Discrete temporal solitons along a chain of nonlinear coupled microcavities embedded in photonic crystals, Optics Letters 27 (2002) 568–570. [14] Y. Liu, G. Bartal, D.A. Genov, X. Zhang, Subwavelength discrete solitons in nonlinear metamaterials, Physical Review Letters 99 (2007) 153901.
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[15] F. Ye, D. Mihalache, B. Hu, N.C. Panoiu, Subwavelength plasmonic lattice solitons in arrays of metallic nanowires, Physical Review Letters 104 (2010) 106802. [16] A. Trombettoni, A. Smerzi, Discrete solitons and breathers with dilute Bose–Einstein condensates, Physical Review Letters 86 (2001) 2353–2356. [17] A. Smerzi, A. Trombettoni, Nonlinear tight-binding approximation for Bose–Einstein condensates in a lattice, Physical Review A 68 (2003) 023613. [18] O. Morsch, M. Oberthaler, Dynamics of Bose–Einstein condensates in optical lattices, Reviews of Modern Physics 78 (2006) 179–215. [19] A.B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, S. Wabnitz, Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays, Physical Review E 53 (1996) 1172–1189. [20] P.G. Kevrekidis, The Discrete Nonlinear Schro¨dinger Equation: Mathematical Analysis, Numerical Computations, and Physical Perspectives, Springer, Berlin, 2009. [21] W. Kro´likowski, Y.S. Kivshar, Soliton-based optical switching in waveguide arrays, Journal of the Optical Society of America B 13 (1996) 876–887. [22] U. Peschel, R. Morandotti, J.S. Aitchison, H.S. Eisenberg, Y. Silberberg, Nonlinearly induced escape from a defect state in waveguide arrays, Applied Physics Letters 75 (1999) 1348–1350. [23] R. Morandotti, H.S. Eisenberg, D. Mandelik, Y. Silberberg, D. Modotto, M. Sorel, C.R. Stanley, J.S. Aitchison, Interactions of discrete solitons with structural defects, Optics Letters 28 (2003) 834–836. [24] M.T. Primatarowa, K.T. Stoychev, R.S. Kamburova, Interaction of solitons with bond defects in Discrete Nonlinear Schro¨dinger chains, Physical Review E 77 (2008) 066604. [25] A. Fratalocchi, G. Assanto, Governing soliton splitting in onedimensional lattices, Physical Review E 73 (2006) 046603. [26] A.A. Sukhorukov, Y.S. Kivshar, Soliton control and Bloch-wave filtering in periodic photonic lattices, Optics Letters 30 (2005) 1849–1851. [27] L. Morales-Molina, R.A. Vicencio, Trapping of discrete solitons by defects in nonlinear waveguide arrays, Optics Letters 31 (2006) 966–968. [28] F. Palmero, R. Carretero-Gonza´lez, J. Cuevas, P.G. Kevrekidis, W. Kro´likowski, Solitons in one-dimensional nonlinear Schro¨dinger lattices with a local inhomogeneity, Physical Review E 77 (2008) 036614. [29] Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, S. BarAd, Nonlinear scattering and trapping by local photonic potentials, Physical Review Letters 99 (2007) 133901. [30] T. Taniuti, Reductive perturbation method and far fields of wave equations, Progress of Theoretical Physics Supplement 55 (1974) 1–35. [31] T. Iizuka, M. Wadati, Scattering of envelope solitons by a mass impurity in nonlinear lattices, Journal of the Physical Society of Japan 61 (1992) 4344–4349. [32] H. Trompeter, U. Peschel, T. Pertsch, F. Lederer, U. Streppel, D. Michaelis, A. Bra¨uer, Tailoring guided modes in waveguide arrays, Optics Express 11 (2003) 3404–3411. [33] J. Satsuma, N. Yajima, Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media, Progress of Theoretical Physics Supplement 55 (1974) 284–306. [34] S.A. Gredeskul, Y.S. Kivshar, Propagation and scattering of nonlinear waves in disordered systems, Physics Reports 216 (1992) 1–61.
Photonics and Nanostructures – Fundamentals and Applications 11 (2013) 95–101 www.elsevier.com/locate/photonics
Discrete optical soliton scattering by local inhomogeneities Lasha Tkeshelashvili a,b,* a
Andronikashvili Institute of Physics, Tamarashvili 6, 0177 Tbilisi, Georgia b Tbilisi State University, Chavchavadze 3, 0128 Tbilisi, Georgia
Received 27 July 2012; received in revised form 10 October 2012; accepted 30 October 2012 Available online 9 November 2012
Abstract The nonlinear wave scattering by local inhomogeneities in discrete optical systems is studied both analytically and numerically. The presented theory describes the reflection and transmission of discrete optical solitons at a point defect. In particular, the derived expressions determine the reflected and transmitted pulses from the incident one. In the range of validity, the analytical results are in excellent agreement with the numerical simulations. It is demonstrated that the point defects in structured optical materials represent effective tool for controlling and manipulation of the nonlinear light pulses. # 2012 Elsevier B.V. All rights reserved. Keywords: Discrete optical solitons; Point defects; Wave scattering
1. Introduction The studies of discrete wave dynamics in structured systems range from the realization of optical analogies of various quantum-mechanical phenomena [1,2] to practical design of functional elements for the all-optical communication networks [3,4]. In particular, the tailored light-matter interaction processes in such systems [5,6] allow to demonstrate unique effects related to the nonlinear optical pulses called solitons [7]. Solitons are localized wave packets that can propagate undistorted in homogeneous nonlinear media. That peculiarity makes nonlinear systems very attractive for applications in the field of all-optical communications [8]. However, in general, the soliton interaction with inhomogeneities is an extremely com-
* Correspondence address: Andronikashvili Institute of Physics, Tamarashvili 6, 0177 Tbilisi, Georgia. Tel.: þ995 32 239 87 83; fax: þ995 32 239 14 94. E-mail address: [email protected]. 1569-4410/$ – see front matter # 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.photonics.2012.10.001
plicated process. Indeed, the solitons represent solutions of so-called integrable nonlinear equations [9]. The inhomogeneity, at least locally, breaks the integrability of the model. In the non-integrable models the stability of nonlinear pulses is not guaranteed anymore, and the nonelastic effects such as the soliton radiative decay in the scattering processes may take place [10]. Perhaps, the effectively one-dimensional discrete structures represent the most convenient systems for study of the nonlinear wave dynamics [1,7,11]. In particular, much of the important theoretical and experimental results were obtained for arrays of optical waveguides [12], coupled nano-cavities in photonic crystals [13], metallo-dielectric systems [14,15], and the Bose–Einstein condensates in deep optical lattices [16– 18]. The universal mathematical model that governs the wave dynamics in such systems is the Discrete Nonlinear Schro¨dinger (DNLS) equation [19,20]: @c i n þ Cðcnþ1 þ cn1 Þ þ Njcn j2 cn þ en cn ¼ 0: @t (1)
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Depending on the system under consideration, t is either the temporal or spatial variable [1]. In the case when t is the temporal variable, the localized wave packets are the optical pulses. However, for the systems such as the coupled waveguide arrays, t is a spatial coordinate, and those pulses represent optical beams. The eigenmode amplitude at the site n is cn, and C gives the evanescent coupling rate between adjacent sites. N is the nonlinear coefficient. Here, the inhomogeneity is introduced through the n-dependent en. In different cases that might reflect different physical factors. For instance, in the case of arrays of optical waveguides, the variation of en from site to site may be caused by the different refractive index of the individual waveguides. It should be noted that the defect states can be introduced in the system by different means. In particular, the bond defects defined as a local variation in C were studied in [21–24]. Moreover, Ref. [25] addressed the effects caused by local inhomogeneities in the nonlinear coefficient N. The problem of soliton scattering by localized defects in binary optical lattices was considered in [26]. Below, following [27,28], it is assumed that C, as well as the nonlinear coefficient N, is constant for all sites. Thus, only the linear term associated with en defines a scatterer in the system. Nevertheless, even in such case the nonlinear effects may cause coupling to the linear defect modes [29]. Indeed, the numerical simulations reported in [27,28] show that various scattering regimes might be realized with soliton trapping, reflection, and transmission effects. In what follows, the weakly nonlinear wave scattering by point defects in the discrete optical systems is studied both analytically and numerically. In this case, the nonlinear term in Eq. (1) can be treated as a small perturbation to the linear part. That allows to derive analytical expressions for the scattered waves. Note that, only the variation in en is relevant for the wave propagation and scattering processes. Indeed, by means of cn ! expði˜etÞcn transformation, the values of en in the corresponding term of Eq. (1) can be shifted by an arbitrary constant e˜. Suppose that the point defect is located at n = 0. Then, without loss of generality, en for n 6¼ 0 can be assumed to be exactly zero, i.e. en = e d0,n. Here, d0,n is the Kronecker delta, and e determines the defect strength.
Nonlinear Schro¨dinger (NLS) equation for the carrier wave envelope [7,8]. In turn, the inverse scattering method allows to obtain the solutions of the NLS model analytically [9]. In particular, under certain conditions, the NLS equation supports the bright soliton solutions. Physically, the soliton formation is possible when the balance between the nonlinear effects and the linear dispersion (or diffraction) takes place. That happens when the wave envelope is a slowly varying function on the carrier wavelength scale. It must be stressed that the wider envelope soliton has the smaller amplitude [8,9]. Therefore, the presented theory describes the dynamics of solitons which are sufficiently broad. As a result of the localized pulse scattering process, in the final state, there exist the reflected wave propagating backwards, and the transmitted wave which tunnels to the other side of the defect. In general, the incident pulse can excite the localized defect mode as well [1,7,32]. That may cause the trapping of the incident soliton by the defect [27,28]. However, as will be shown below, such bound states are not realized for the weakly nonlinear pulses and, therefore, can be neglected. Moreover, since the point defect is assumed to be linear, in the reflection process the nonlinear frequency conversion processes do not take place. Therefore, the incident, reflected, and transmitted waves have the same wave numbers |k|.
2. Perturbation analysis
In addition, vg is the group velocity:
In the weakly nonlinear limit the problem can be simplified significantly by means of the reductive perturbation method [30,31]. In particular, the nonintegrable Eq. (1) can be reduced to the integrable
vg ¼
2.1. Transmitted wave Suppose that the incident localized pulse impinges at the point defect from the left region (n < 0). Then, according to Ref. [31], the ansatz for the transmitted wave in the right region n > 0 is: cn ¼
1 X X
mm V ðm;lÞ ðj; tÞEnðlÞ :
(2)
m¼1 lm
Here, the smallness parameter m 1 guarantees that the nonlinear term in Eq. (1) can be treated as a small perturbation. Moreover, in order to describe explicitly the wave envelope dynamics, the set of new ‘‘slow variables’’, t = m2n and j ¼ mðn vg tÞ, is introduced. ðlÞ The carrier wave is En ¼ expðil½kn vtÞ, and v is given by the dispersion relation: v ¼ 2C cosðkÞ:
dv ¼ 2C sinðkÞ: dk
(3)
(4)
The perturbation analysis based on Eqs. (1) and (2) results in the NLS equation for the slowly varying
L. Tkeshelashvili / Photonics and Nanostructures – Fundamentals and Applications 11 (2013) 95–101
envelope V(1,1)(j, t) [30,31]: iVtð1;1Þ þ
Dk ð1;1Þ V þ N k jV ð1;1Þ j2 V ð1;1Þ ¼ 0; 2 jj
(5)
with Dk ¼ cotðkÞ;
vc0 þ Cðc1 þ c1 Þ þ ec0 0: (7) c0
The slow variables in the subscripts represent the corresponding partial derivatives, i.e, f t @f/@t, etc. As can be seen from the definition of j, Eq. (5) describes the transmitted wave in the reference frame moving with the group velocity vg.
1 X X
In the left region there is only the incident pulse initially. However, the scattering process results in the reflected wave as well. Therefore, for n < 0 the ansatz for cn reads [31]: 0
0
mm U ðm;l;l Þ ðj; j; tÞEnðl;l Þ ;
(8)
(12)
m¼1 lm
However, in addition, according to Eq. (8): c0
1 X X
ðl;l0 Þ
0
mm U ðm;l;l Þ E0
:
(13)
m¼1 lþl0 m
V ð1;1Þ ¼ U ð1;1;0Þ þ U ð1;0;1Þ : Note that at n = 0 the slow variables are: t ¼ 0; j ¼ mvg t; j ¼ þmvg t:
ðl;l0 Þ
here En ¼ expðil½kn vt il0 ½kn þ vtÞ. The additional slow variable j ¼ mðn þ vg tÞ is introduced to describe the reflected wave envelope. Nevertheless, at 0 this stage of calculations, U ðm;l;l Þ envelopes are allowed to be functions of all (that is t, j and j) variables. Inserting Eq. (8) into Eq. (1) reduces the initial DNLS equation the NLS model [30,31]. In particular, the perturbation analysis shows that the slowly varying envelope of incident wave does not depend on j (i.e. U(1,1,0) = U(1,1,0)(j, t)) and: Dk ð1;1;0Þ U þ N k jU ð1;1;0Þ j2 U ð1;1;0Þ ¼ 0: 2 jj
(9)
Furthermore, the envelope of the reflected wave is independent of j (i.e. U ð1;0;1Þ ¼ U ð1;0;1Þ ðj; tÞ) and obeys: Dk ð1;0;1Þ U N k jU ð1;0;1Þ j2 U ð1;0;1Þ ¼ 0: 2 jj
(14)
Furthermore, Eq. (8) for c1 gives:
m¼1 lþl0 m
iUtð1;0;1Þ
ðlÞ
mm V ðm;lÞ E0 ;
Eq. (12) in combination with Eq. (13) results in:
2.2. Reflected wave
iUtð1;1;0Þ þ
(11)
Here, c0 is given by Eq. (2):
1 N : Nk ¼ 2 C sinðkÞ
1 X X
by the NLS equation. The equation of motion at n = 0 can be used relate the amplitudes of the transmitted and reflected waves with that of the incident wave. Indeed, since the nonlinear term is small, at the defect Eq. (1) approximately gives:
(6)
and the nonlinear coefficient reads:
cn ¼
97
(10)
Eqs. (9) and (10) are written in the reference frames moving with the group velocities vg and vg , respectively.
c1 ¼
1 X X
0
ðl;l0 Þ
mm U ðm;l;l Þ E1 ;
(15)
m¼1 lþl0 m
and Eq. (2) yields the following expression for c1: c1 ¼
1 X X
ðlÞ
mm V ðm;lÞ E1 :
(16)
m¼1 lm
At n = 1 the slow variables read: t ¼ m2 ; j ¼ mð1 vg tÞ; j ¼ mð1 þ vg tÞ: Furthermore, the slowly varying functions V(1,1), U(1,1,0), and U(1,0,1) are expanded in the Taylor series: V ð1;1Þ ðmvg t m; m2 Þ V ð1;1Þ ðmvg t; 0Þ; U ð1;1;0Þ ðmvg t m; m2 Þ U ð1;1;0Þ ðmvg t; 0Þ; U ð1;0;1Þ ðþmvg t m; m2 Þ U ð1;0;1Þ ðþmvg t; 0Þ:
2.3. Continuity conditions
Since m 1, in these expansions only the leading terms are retained. The expressions for the reflected and transmitted wave envelopes at n = 0 follow from Eqs. (11) and (14):
The above analysis shows that the nonlinear wave dynamics on both sides of the point defect is governed
U ð1;0;1Þ ðj; 0Þ ¼ R U ð1;1;0Þ ðj; 0Þ;
(17)
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with the reflection coefficient R: e ; e þ 2iC sinðkÞ
(18)
V ð1;1Þ ðj; 0Þ ¼ T U ð1;1;0Þ ðj; 0Þ;
(19)
R¼ and
with the transmission coefficient T: T¼
2iC sinðkÞ : e þ 2iC sinðkÞ
(20)
Eqs. (17) and (19) determine the reflected and transmitted waves from the incident pulse. Moreover, it is easy to see that |R|2 + |T|2 = 1. That is in full agreement with the energy conservation law since the wave trapping processes by the defect are neglected.
by the celebrated NLS equation:
2.4. Bound states It should be noted that p k p [1]. Nevertheless, according to Eqs. (17) and (19), R and T diverge for the imaginary value k = ik. Here, k obeys: sinhðkÞ ¼
e : 2C
Fig. 1. The soliton scattering process at the local inhomogeneity. Here, |cn(t)| is plotted. In this simulation the point defect is located at n = 2000 (shown by the dotted line). The values of other parameters and further details are given in the text.
(21)
As with the quantum mechanics scattering problems, the divergence of the reflection and transmission coefficients is due to the discrete energy level associated with the point defect. That level gives rise to the bound state. The frequency v = V, which corresponds to k, follows from Eqs. (3) and (21): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 2 V ¼ 2C 1 þ : (22) 2C The frequency range of the modes with real k does not overlap with V. Nevertheless, nonlinear effects may cause the strong localization of the pulse in real space. In this case, the Fourier spectrum of the wave packet may become broad enough to be in resonance with V. The weakly nonlinear pulses are not strongly localized, and therefore, the processes which lead to the bound state formation do not take place. In this context, it must be stressed that the group velocity vg is an important parameter as well [27]. 3. Results Let us consider the soliton scattering by the point defect in more details. A numerical example of such process is shown in Fig. 1. As it is demonstrated above, the dynamics of the weakly nonlinear pulses is governed
i
@F P @2 F þ þ QjFj2 F ¼ 0: @z 2 @x2
(23)
The solutions of the NLS model can be obtained by means of the inverse scattering method [8,9]. In particular, Eq. (23) supports the bright soliton solutions if PQ > 0. Therefore, since it is assumed that the bright solitons can propagate in the system, the values of k are further restricted by DkNk > 0 inequality. For instance, if C and N are simultaneously positive, |k| < p/2 as can be seen from Eqs. (6) and (7). The presented theory describes p/2 < |k| < p range too. However, for those values of k the stable optical pulses do not exist, and therefore, such cases will not be considered below. 3.1. Fundamental soliton scattering Suppose that the incident pulse represents the fundamental soliton solution of Eq. (9): rffiffiffiffiffiffi 1 Dk j Dk t sech (24) Fðj; tÞ ¼ exp i : L Nk L 2 L2 In this expression a real parameter L is the soliton width [9]. However, L is related to the soliton amplitude as well. In particular, to wider solitons correspond the smaller values of the amplitude. As was discussed above, for the given vg , the value of L must be sufficiently large to suppress the formation of the bound state. From Eqs. (17) and (19) it directly follows that the initial conditions for Eqs. (10) and (5) read: rffiffiffiffiffiffi R Dk j sech (25) FR ðj; 0Þ ¼ ; L Nk L
L. Tkeshelashvili / Photonics and Nanostructures – Fundamentals and Applications 11 (2013) 95–101
rffiffiffiffiffiffi T Dk j sech FT ðh; 0Þ ¼ ; L Nk L
(26)
Eqs. (25) and (26) define the initial value problem for the reflected and transmitted waves, respectively. The sech-type initial value problem of the NLS equation was solved in Ref. [33]. In particular, for the following initial condition: rffiffiffiffi x A P sech Fi ðx; z ¼ 0Þ ¼ (27) ; L Q L the number of generated solitons in Eq. (23) is the maximum integer M which satisfies: 1 M < jAj þ : 2
(28)
For |A| 1/2, no soliton emerges and the pulse disperses. The dispersive modes, similar to the linear waves, have vanishing amplitudes in the final state. If 1/2 |A| 3/2, Eq. (28) shows that M = 1. In this case, by sheding the dispersive radiation, the solution asymptotically relaxes to the fundamental soliton with the amplitude: rffiffiffiffi 2 P 1 ðsÞ jAj : A ¼ (29) L Q 2 In general, M > 1 and there emerges the bound state of M solitons plus the dispersive radiation [8,33]. However, such higher-order soliton solutions are not relevant for the present problem. It can be seen from Eqs. (25) and (27) that for the reflected wave A = AR, where: AR ¼ R:
(30)
Similarly, from Eqs. (26) and (27), for the transmitted wave follows A = AT: AT ¼ T:
(31)
Furthermore, Eqs. (18) and (20) give: 1 jAR j ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 þ s 1
(32)
1 jAT j ¼ pffiffiffiffiffiffiffiffiffiffiffi ; 1þs
(33)
99
According to Eqs. (28) and (32) there is no soliton in the reflected pulse for 0 < s < 1/3. Moreover, for s > 1/3, only one reflected soliton is generated with the amplitude: rffiffiffiffiffiffi 1 Dk 2 ðsÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 : AR ¼ (35) L Nk 1 þ s 1 By comparing with Eq. (24), the width of the reflected soliton LR reads: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ s 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : (36) LR ¼ L 2 1 þ s 1 For the transmitted wave Eqs. (28) and (33) show that there exists only dispersive radiation for s > 3. If 0 < s < 3 one soliton emerges in the transmitted pulse with the amplitude: rffiffiffiffiffiffi 1 Dk 2 ðsÞ pffiffiffiffiffiffiffiffiffiffiffi 1 ; AT ¼ (37) L Nk 1þs and comparison with Eq. (24) shows that the width of the transmitted soliton LT is: pffiffiffiffiffiffiffiffiffiffiffi 1þs pffiffiffiffiffiffiffiffiffiffiffi : LT ¼ L (38) 2 1þs These expressions show how the incident soliton amplitude and width transform in the scattering process. For example, for s = 1, the reflected and the transmitted solitons are approximately 2.414 times wider compared to the incident pulse. 3.2. Numerical simulations An example of the fundamental soliton scattering process at the point defect is depicted in Fig. 1. The presented result is the numerical solution of Eq. (1) with the following initial condition: rffiffiffiffiffiffi 1 Dk cn ð0Þ ¼ m sechðJÞexpðiQÞ; (39) L Nk with
where s represents the effective strength of the defect: 2 e s¼ : (34) 2C sinðkÞ Note that the lower bound for s is smin = e2/(2C)2. Eq. (34) gives the quantitative measure of the group velocity vg (see Eq. (4)) influence on the scattering process.
m ðn n0 Þ; L D k m2 Q¼ kþ ðn n0 Þ: 2 L2 J¼
(40) (41)
In this expression it is assumed that the incident soliton at t = 0 is centered around site n0 = 1500. The values of other parameters are set as follows: C = N = 1, L = 1, m = 0.05, and e = 0.5. Since both C and N are chosen to be positive, as was discussed above, k must obey
100
L. Tkeshelashvili / Photonics and Nanostructures – Fundamentals and Applications 11 (2013) 95–101 4
0.06
3.5
0.05
3 0.04
μ AT
σ
(s)
2.5 2 1.5
0.03 0.02
1
0.01
0.5 0 0
0
0.5
1
1.5
0.4
0.6
0.8
1
1.2
1.4
k
k ðsÞ
Fig. 2. The dependency of s on k. For the values of the chosen parameters see the text.
|k| < p/2. In Fig. 2 the dependency of s on k is presented for the given set of parameters. For the numerical simulation shown in Fig. 1 the incident pulse has k = 0.8. The corresponding value for the effective defect strength is s = 0.122, and the amplitude of the incident soliton is 0.059. It follows from Eqs. (32) and (33) that |AR| = 0.329 and |AT| = 0.944. Then, Eq. (28) implies that the reflected pulse is the dispersive wave packet, while the transmitted pulse relaxes to the fundamental soliton in the final state. The amplitude of the generated soliton is: jcT ðt ! 1Þjmax ¼
ðsÞ mAT
¼ 0:0524;
(42)
and, according to Eq. (38), LT = 1.125. The soliton formation process exhibits a damped oscillatory behavior of the pulse amplitude and width around the analytically predicted values. For instance, the arithmetic mean of the amplitude maximum and minimum values in the first oscillation is 0.0525. That is in excellent agreement with Eq. (42). The amplitude of the reflected pulse decays monotonously with time. Similarly, for the given width of the incident soliton L = 1, the excellent agreement between the theory and numerical simulations is found for 0.4 < |k| < 1.4. For example, the corresponding analytical predictions and numerical results for the transmitted soliton amplitude are given in Fig. 3. For |k| < 0.4 the incident soliton is too narrow. Indeed, Eq. (24) shows that the incident wave packet is localized on the scale of order 2L/ m 40. In turn, for example for k = 0.3, the carrier wavelength is 2p/k 21. Therefore, the soliton envelope is not really slowly varying. In this case, wider solitons, e.g. with L = 2, must be considered. Furthermore, the incident soliton is too narrow for 1.4 < |k| < p/2 as well. Although, in this case, |k| < p/2
Fig. 3. The dependency of mAT on k. The squares show the numerical results for the arithmetic mean of the amplitude maximum and minimum in the first oscillation of the soliton parameters. The dashed line represents the theoretical prediction.
still holds for the carrier wave number, sufficiently large portion of the pulse Fourier spectrum does not fall inside the limits of this range. That causes the breakdown of the soliton. The increase of pulse width leads to the shrinkage of its Fourier spectrum, and therefore, suppresses such instabilities. That is in agreement with the numerical simulations. 4. Discussion and conclusions In certain cases the inhomogeneous medium can be treated as a collection of independent scatterers. That is often referred to as the independent scattering approach [34]. For instance, assuming that 0 < s < 3 holds, consider the soliton tunneling through a set of regular or irregular set of point defects with the equal strengths e. Moreover, suppose that the soliton width is smaller compared to the average distance lav between the neighboring defects. The presented results can be applied to this problem as well. Indeed, Eq. (38) gives the soliton width increase after each scattering event. Then, it is clear that after passing through p defects the soliton size becomes: pffiffiffiffiffiffiffiffiffiffiffi p 1þs ð pÞ pffiffiffiffiffiffiffiffiffiffiffi : LT ¼ L 2 1þs Therefore, while propagating, the soliton width increases exponentially. The independent scattering ð pÞ approach becomes invalid when LT lav . Let us assume that this is the case at p = p0. For the system sizes less than p0lav the transmitted soliton amplitude decreases exponentially with distance as (see Eq. (37)): rffiffiffiffiffiffi p 1 Dk 2 ð pÞ pffiffiffiffiffiffiffiffiffiffiffi 1 ; cT ¼ m L Nk 1þs
L. Tkeshelashvili / Photonics and Nanostructures – Fundamentals and Applications 11 (2013) 95–101
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