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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 18 — Aug. 29, 2011
  • pp: 17179–17188
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Symmetric and antisymmetric solitons in finite lattices

Shunsheng Zhong, Changming Huang, Chunyan Li, and Liangwei Dong  »View Author Affiliations


Optics Express, Vol. 19, Issue 18, pp. 17179-17188 (2011)
http://dx.doi.org/10.1364/OE.19.017179


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Abstract

We propose a simple model for the realization of symmetrically and antisymmetrically shape-preserving nonlinear waves with nonvanishing intensities at infinity. A finite lattice embedded into a defocusing saturable medium can support various families of novel solitons, including out-of-phase and in-phase solitons with symmetric and antisymmetric profiles. Although the lattice is finite, the existence and stability of solitons depend strongly on the band-gap structure of the corresponding infinite lattice. Saturable nonlinearity enhances the pedestal height and renormalized energy flow of solitons evidently. In particular, increasing the lattice site number or saturation degree of nonlinearity can considerably suppresses the instability of solitons. In addition, we find two branches of in-phase solitons in finite lattices and one branch of them can be dynamically stable. Our findings may provide a helpful hint for linking the solitons supported by infinite and finite lattices.

© 2011 OSA

Nonlinear surface waves have been investigated in the context of fluids [1

1. R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Academic Press, 1982).

], solids [2

2. G. Maugin, Nonlinear Waves in Elastic Crystals (Oxford University Press, 2000).

], plasmas [3

3. L. Stenflo, “Theory of nonlinear plasma surface waves,” Phys. Scr. T63, 59–62 (1996). [CrossRef]

] and nonlinear optics [4

4. I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, “Surface multi-gap vector solitons,” Opt. Express 14, 4780–4785 (2006). [CrossRef] [PubMed]

6

6. S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, “Observation of discrete surface solitons,” Phys. Rev. Lett. 96, 063901 (2006). [CrossRef] [PubMed]

], where optical surface waves were predicted to exist at interfaces separating periodic and homogeneous dielectric media. Recently, it was suggested theoretically and demonstrated experimentally that nonlinearity-induced self-trapping of light near the edge of a semi-infinite optical lattice can result in the formation of localized surface solitons [7

7. M. I. Molina, I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, “Discrete surface solitons in semi-infinite binary waveguide arrays,” Opt. Lett. 31, 2332–2334 (2006). [CrossRef] [PubMed]

11

11. Y. He, D. Mihalache, and B. Hu, “Soliton drift, rebound, penetration, and trapping at the interface between media with uniform and spatially modulated nonlinearities,” Opt. Lett. 35, 1716–1718 (2010). [CrossRef] [PubMed]

]. One important generalization of these ideas is the concept of surface gap solitons [12

12. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface gap solitons,” Phys. Rev. Lett. 96, 073901 (2006). [CrossRef] [PubMed]

, 13

13. C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Observation of surface gap solitons in semi-infinite waveguide arrays,” Phys. Rev. Lett. 97, 083901 (2006). [CrossRef] [PubMed]

]. Meanwhile, dark solitons in uniform or lattice modulated media with defocusing nonlinearity were investigated [14

14. Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81–197 (1998). [CrossRef]

18

18. J. Belmonte-Beitia and J. Cuevas, “Existence of dark solitons in a class of stationary nonlinear Schrödinger equations with periodically modulated nonlinearity and periodic asymptotics,” J. Math. Phys. 52, 032702 (2011). [CrossRef]

]. For a review of early works, see [19

19. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008). [CrossRef]

] and references therein.

Of particular interest is that optical pulses in the form of kink solitons may propagate stably in optical fibers [20

20. Y. S. Kivshar and B. A. Malomed, “Raman-induced optical shocks in nonlinear fibers,” Opt. Lett. 18, 485–487 (1993). [CrossRef] [PubMed]

, 21

21. S. Wabnitz, “Chiral polarization solitons in elliptically birefringent spun optical fibers,” Opt. Lett. 34, 908–910 (2009). [CrossRef] [PubMed]

]. Kink solitons are specific type of semi-localized nonlinear modes composed by a sharp transition region associating with a pedestal and many decaying oscillatory tails. In time domain, nonlinear fibers in the normal and anomalous regimes were reported to support Raman-induced optical kinks [20

20. Y. S. Kivshar and B. A. Malomed, “Raman-induced optical shocks in nonlinear fibers,” Opt. Lett. 18, 485–487 (1993). [CrossRef] [PubMed]

]. Counterpropagating beams in spun high-birefringence optical fibers can be used to excite stable kink solitons [21

21. S. Wabnitz, “Chiral polarization solitons in elliptically birefringent spun optical fibers,” Opt. Lett. 34, 908–910 (2009). [CrossRef] [PubMed]

]. The sharp wavefront of kink wave is crucial for high intensity short pulses exhibiting nonlinear responses or self-steepening effects in fibers [22

22. G. Agrawal, Nonlinear Fiber Optics (Academic Press, 1995).

, 23

23. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (Izdatel Nauka, 1988).

].

So far, researches on kink solitons in spatial domain are very rare, with the only exceptions reported in [24

24. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface lattice kink solitons,” Opt. Express 14, 12365–12372 (2006). [CrossRef] [PubMed]

] where surface kink solitons in defocusing Kerr media were predicted, and in [25

25. C. Huang, J. Zheng, S. Zhong, and L. Dong, “Interface kink solitons in defocusing saturable nonlinear media,” Opt. Commun. 284, 4225–4228 (2011). [CrossRef]

] where kink solitons in semi-infinite lattice modulated media with defocusing saturable nonlinearity were investigated. In Ref. [24

24. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface lattice kink solitons,” Opt. Express 14, 12365–12372 (2006). [CrossRef] [PubMed]

], Kartashov et al. found two different types of kink solitons, i.e., “in-phase kink solitons” and “out-of-phase kink solitons”, termed by the relative height of the pedestal and primary maxima located in the nearest-to-interface lattice channel. The former type is always unstable while the latter type is stable in a wide parameter window. Later, Ye and his coworkers proposed a defected lattice model for linking localized defect solitons and symmetric kink pairs [26

26. F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Bragg guiding of domainlike nonlinear modes and kink arrays in lower-index core structures,” Opt. Lett. 33, 1288–1290 (2008). [CrossRef] [PubMed]

].

In this article, we demonstrate the existence and stability properties of symmetric and antisymmetric solitons supported by a finite lattice imprinted into a defocusing saturable medium. Several families of solitons, such as out-of-phase and in-phase solitons with symmetric and antisymmetric profiles, are found in the first and higher finite band gaps of infinite lattices. Such solitons resemble a pair of kink solitons connected each other symmetrically or antisymmetrically. Although the lattice here is finite, the existence and stability of solitons still depend strongly on the band-gap structure of infinite lattice. Especially, we reveal that the instability of solitons can be suppressed significantly by increasing the lattice site number. Furthermore, we find two branches of in-phase solitons in finite lattices, which is in sharp contrast to the kink solitons in semi-infinite lattices where two branches of out-of-phase solitons exist only in shallow lattices [25

25. C. Huang, J. Zheng, S. Zhong, and L. Dong, “Interface kink solitons in defocusing saturable nonlinear media,” Opt. Commun. 284, 4225–4228 (2011). [CrossRef]

]. Rigorous linear stability analysis is performed on the various families of solitons and the results show that out-of-phase and the first branch of in-phase solitons in the band gap can propagate stably in a wide parameter window.

We consider a beam propagation along the z axis in a finite optical lattice embedded into a defocusing saturable medium. The evolution of laser beam is governed by the nonlinear Schrödinger equation for the normalized field amplitude q:
iqz=122qx2+q|q|21+s|q|2pR(x)q
(1)
where the transverse x and longitudinal z coordinates are normalized to the width and diffraction length of beam, respectively; s stands for the saturation parameter; p characterizes the lattice depth and the function R(x) describes the profile of linear refractive index modulation.

Stationary solutions of Eq. (1) can be searched in the form of q(x,z) = w(x)exp(ibz), where w(x) is a real function depicting the soliton profiles and b is a nonlinear propagation constant. Substituting the expression into Eq. (1), one obtains an ordinary differential equation. Note that the Neumann boundary conditions are applied for numerically solving the soliton profiles. Solitons are characterized by the propagation constant b, saturation parameter s, lattice distribution R, lattice depth p, lattice frequency Ω and lattice site number n. Without loss of generality, we vary b,s,n, p and fix lattice frequency Ω in numerical analysis.

We are interested in nonlinear waves composed by two pedestals at x → ±∞ and decaying oscillatory tails in lattice modulated region. According to Eq. (1), one can deduce that symmetric solitons drop off from two pedestals whose height is determined by w(x → ±∞) = [−b/(1+ sb)]1/2, which also gives out the existence condition of solitons, i. e., solitons can exist only when the relations b ≤ 0 and 1 + sb > 0 (or b > −1/s) are satisfied simultaneously. To quantitatively depict the energy flow of the wave fronts and oscillatory tails trapped in lattices, we define the concept of “renormalized energy flow” based on the profiles of solitons [24

24. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface lattice kink solitons,” Opt. Express 14, 12365–12372 (2006). [CrossRef] [PubMed]

]:
Ur=[w(x)|b1+bs|1/2H(x)]2dx
(3)
where H(x) = 0 in lattice modulated region and 1 elsewhere.

Firstly, we address the properties of symmetric nonlinear waves supported by a defocusing saturable medium with an imprinted finite lattice. Such waves are composed of two pedestals with the same height connected by fading tails residing in lattice modulated region. To realize the symmetric nonlinear waves, we take R(x) = cos2x) for |x| = /2Ω and R(x) = 0 otherwise as a linear refractive index modulation. Here, Ω is the modulation frequency; n is an odd positive integer and denotes the lattice site number.

Figure 1 shows some typical examples of stationary solutions of symmetric solitons. Out-of-phase solitons shown in Fig. 1(a)-1(c) drop from two symmetric pedestals and exhibit decaying tails in lattice modulated region. The pedestal height of solitons decreases with the propagation constant. The localization of solitons in lattice region depends on the position of propagation constant b inside the existence region blowbbupp and weakens with the decrease of b. Soliton will degenerate into chirped truncated Bloch wave [27

27. J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, “Truncated-Bloch-wave solitons in optical lattices,” Phys. Rev. A 79, 043610 (2009). [CrossRef]

] when its propagation constant enters into the second band of infinite lattice [Fig. 1(b) and 2(a)]. At fixed propagation constant, the pedestal height of solitons increases with the saturation parameter s [Fig. 1(c)]. Yet, the saturation parameter does not affect the localization of solitons in lattices evidently. One can also infer from the relation w(x → ±∞) = [−b/(1 + sb)]1/2 that the pedestal height of solitons approaches to infinity when propagation constant goes to its lower cutoff. For example, w(x → ±∞) = [−b/(1 + sb)]1/2 → ∞ when b → −2 for s = 0.5.

Fig. 1 Profiles of symmetric solitons. s = 0.5 for out-of-phase (a–c and f) and 0.2 for in-phase (d, e) solitons. The shaded stripes represent the regions R ≥ 0.5. b = −0.3 in (c, d) and −1.5 in (e), p = 16 in (f) and 4 in other panels. In all cases Ω = 2 and n = 13.
Fig. 2 (a) Band-gap structure of infinite lattice R(x) = cos2x). Bands are shown gray and gaps are shown white. Existence domains of out-of-phase solitons at s = 0.5 (b) and in-phase solitons at s = 0.2 (c). (d) Renormalized energy flow versus propagation constant for out-of-phase solitons in lattices with different site number n. Inset: Ur versus b for different saturation parameter s. p = 4 in (d) and Ω = 2,n = 13 in all panels.

In the outermost lattice channels, in-phase solitons contain two sharp peaks higher than the pedestal height. In contrast to the kink solitons in shallow semi-infinite lattices [25

25. C. Huang, J. Zheng, S. Zhong, and L. Dong, “Interface kink solitons in defocusing saturable nonlinear media,” Opt. Commun. 284, 4225–4228 (2011). [CrossRef]

], where two branches of out-of-phase kink solitons were found, we find two branches of in-phase solitons in finite lattices [Fig. 1(d) and 1(e)]. We term them as “in-phase I” solitons and “in-phase II” solitons respectively. In-phase solitons can also be seen as truncated Bloch modes when their propagation constant resides in the band of lattice spectrum. Out-of-phase and in-phase solitons in higher band gaps are also possible. Typical profiles of out-of-phase solitons in the second band gap are illustrated in Fig. 1(f). The peak number of oscillatory tails in the second band gap is twice of those in the first band gap.

To compare with the kink solitons in semi-infinite lattice modulated media, we plot the band-gap structure of the linearized version of Eq. (1) with R(x) = cos2x) in Fig. 2(a). The finite gaps expand with the growth of lattice depth. As is well known, localized solitons in infinite lattices always reside in the gaps while Bloch modes exist in the bands. Solitons can exist in both semi-infinite gap and finite gaps in periodically modulated focusing media. For defocusing media with imprinted optical lattices, solitons can only be found in finite gaps [19

19. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008). [CrossRef]

]. Thus, we focus our discussions on the dynamics of solitons in the first finite gap. Similar to the cases in infinite lattices, symmetric solitons cannot be found in the first band. However, solitons can exist in the second band of infinite lattices. The reason may be attributed to that the finite lattice here is partially periodic.

The existence domain of out-of-phase solitons is determined by the relation bupp = blow + 1/s when the lattice depth exceeds a critical value [Fig. 2(b)]. Deviation of this relation at small p is due to the fact that the upper cutoff of propagation constant is restricted by the upper edge of the first gap of lattice spectrum. Out-of-phase solitons stop to exist when bupp approaches to the upper edge of the second band. Nonlinear waves will degenerate into linear ones when bbupp. Namely, out-of-phase solitons are nonlinear modes bifurcating from the linear modes. Unlike out-of-phase solitons, the existence domain of in-phase solitons expands with the lattice depth firstly and shrinks later when the lattice depth exceeds a critical value [Fig. 2(c)]. Note that the hoofed existence domain does not satisfy the relation bupp = blow + 1/s.

The renormalized energy flow Ur of out-of-phase solitons is a monotonically decreasing function of the propagation constant b. It vanishes at the upper cutoffs and approaches to infinity at the lower cutoffs [Fig. 2(d)]. Meanwhile, Ur increases with the growth of the lattice site number n. The inset plot illustrates the dependence of the existence domain of out-of-phase solitons on the saturation parameter. Solitons with higher renormalized energy flow can be found in the vicinity of bupp in the high-saturation regime. Comparing with the kink solitons in defocusing cubic media with imprinted semi-infinite lattices [24

24. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface lattice kink solitons,” Opt. Express 14, 12365–12372 (2006). [CrossRef] [PubMed]

], this property can be utilized to realize “higher power” symmetric solitons in the lattice gap.

Now, we study the stability properties of symmetric solitons. Figure 3(a) shows the stability and instability domains of out-of-phase solitons on the bs plane. Out-of-phase solitons are almost completely stable in the first gap. There exists a narrow instability area near the lower edge of the first gap for small saturation parameters. Solitons shaped as chirped truncated Bloch waves in the second band are completely unstable. On the other hand, solitons are completely stable when the saturation parameter exceeds a critical value. This property illustrates that the saturation of the nonlinear media can be utilized to suppress the instability of solitons. To understand the influence of the lattice site number on the existence and stability of the symmetric waves, we exhaustively solve the stationary solutions of solitons and comprehensively conduct linear stability analysis on them. The results are summarized in Fig. 3(b). The instability domain of out-of-phase solitons in the first gap shrinks rapidly with the growth the lattice sites and vanishes when the lattice site number exceeds a critical value (ncr = 25) which constitutes one of our central results. That is to say, one can improve the stability of symmetric out-of-phase solitons by increasing the lattice site number. Yet, the growth of lattice site number cannot change the instability of solitons residing in the second band.

Fig. 3 Areas of stability and instability (shaded) on the (b,s) plane (a) and (b,n) plane (b) for out-of-phase solitons. Dotted line denotes the upper edge of the second band. Renor-malized energy flow versus propagation constant for two types of in-phase solitons (c) and in-phase I solitons at different s (d). Inset: Ur of in-phase I solitons near bupp at s = 0.2. n = 13 in (a, b, d), s = 0.2 in (b, c) and p = 4 in all panels.

Figure 3(c) displays the renormalized energy flow of in-phase I and II solitons versus propagation constant. The renormalized energy flow of the second branch of solitons is higher than that of the first branch. As can also be inferred from their profiles shown in Fig. 1(d) and 1(e). The upper and lower propagation constant cutoffs of in-phase I solitons are the same as those of in-phase II [also see Fig. 2(c)]. The dependence of renormalized energy flow of in-phase I solitons on the propagation constant for different saturation parameters is illustrated in Fig. 3(d) and the inset plot in it. The existence domain of solitons shrinks with the saturation parameter. An interesting feature of in-phase solitons is that there exist inflexions at bbupp. These inflexions indicate that in-phase solitons have threshold energy flow. In other words, such solitons are purely nonlinear modes and cannot be bifurcated from linear modes, which differs from out-of-phase solitons [Fig. 2(d)].

Examples of linear stability analysis results are presented in Fig. 4. We apply both Dirichlet and Neumann boundary conditions for solving the eigenvalue functions and derive the same results. The eigen-modes of perturbation corresponding to the unstable growth rates are localized. The instability of out-of-phase solitons can be suppressed by increasing the lattice site number [Fig. 4(a)]. While in-phase II solitons are always unstable, in-phase I solitons can be stable when they reside in the first band gap [Fig. 4(b)], which is different from the surface kink solitons in semi-infinite lattices, where in-phase solitons are completely unstable [24

24. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface lattice kink solitons,” Opt. Express 14, 12365–12372 (2006). [CrossRef] [PubMed]

]. Spectrum of the linearization operator for out-of-phase soliton at b = −1,n = 13, p = 4 is plotted in Fig. 4(c). To verify the linear stability analysis results, we extensively perform propagation simulations of solitons by the split-step Fourier method. Typical stable and unstable propagation examples of out-of-phase and in-phase solitons are presented in Fig. 4(d)-4(f). Note the soliton shown in Fig. 4(f) suffers weak oscillatory instability since the eigen-spectrum is complex and has small real parts [Re(δ)max = 0.052].

Fig. 4 Perturbation growth rate versus propagation constant for out-of-phase solitons at s = 0.5 (a) and two types of in-phase solitons at s = 0.2 (b). (c) Spectrum of the linearization operator for out-of-phase soliton at b = −1.0. (d, e) Stable propagations of out-of-phase soliton at b = −0.45 and in-phase I soliton at b = −0.25. (f) Unstable propagation of out-of-phase soliton at b = −1.0. White noises with σnoise2=0.01 were added into the initial inputs. p = 4 in (a–f) and n = 13 in (b–f).

Dark solitons are 1D nonlinear waves which can preserve their envelops during propagation. They possess a smooth transition region linking two antisymmetric pedestals. The amplitude of dark soliton at its center is zero [14

14. Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81–197 (1998). [CrossRef]

]. Here, we suggest that a finite lattice embedded into a defocusing saturable medium can support dark-like solitons with an oscillatory transition region. To realize such modes, one needs to select a finite lattice in the form of R(x) = 1 – cos(Ωx) for |x| ≤ /Ω and R(x) = 0 otherwise. In the following, discussions, we set Ω = 4, p = 2 in numerical analysis. Even n is needed for dark-like antisymmetric solitons.

The renormalized energy flow of out-of-phase dark-like solitons is also a monotonically decreasing function of propagation constant. Contrary to the intuition and the symmetric solitons, the growth of lattice sites leads to the decrease of renormalized energy flow of antisymmetric solitons [Fig. 5(b)]. We also find two types of in-phase dark-like solitons whose profiles are shown in Fig. 5(d). The threshold renormalized energy flow of in-phase I and II solitons is of the same value (Pth = 11.26) [inset in Fig. 5(b)]. The main features, such as the location of oscillatory peaks and localization of oscillatory tails, are similar to those of symmetric solitons. The amplitudes at the symmetric center of out-of-phase and in-phase dark-like solitons are zero which is different from those of symmetric waves [Fig. 5(c) and 5(d)].

Fig. 5 (a) Band-gap structure of periodic lattice R(x) = 1 – cos(Ωx). (b) Renormalized energy flow versus propagation constant for out-of-phase solitons. Inset: Ur versus b for two types of in-phase solitons. (c) Examples of out-of-phase solitons. (d) Profiles of two types of in-phase solitons at b = −0.3. The shaded stripes represent the regions R ≥ 1. s = 0.5 for out-of-phase solitons and 0.2 for in-phase solitons. p = 2,n = 12 in (b–d) and Ω = 4 in all cases.

Representative examples of linear stability analysis results of antisymmetric solitons are shown in Fig. 6. The stability of out-of-phase solitons can also be improved by increasing the lattice site number [Fig. 6(a)]. In fact, the instability will vanish provided that the lattice site number exceeds a critical value, just similar to the aforementioned symmetric solitons. In-phase I solitons in the first gap can be stable in a wide propagation constant window [Fig. 6(b)] and in-phase II solitons are always unstable. Stable propagation examples of out-of-phase and in-phase I solitons are plotted in Fig. 6(c) and 6(d) respectively.

Fig. 6 Perturbation growth rate versus propagation constant for out-of-phase solitons (a) and in-phase solitons (b). Stable propagation examples of out-of-phase soliton at b = −0.5 (c) and in-phase I soliton at b = −0.55 (d). s = 0.5 for out-of-phase solitons and 0.2 for in-phase solitons. In all the cases p = 2,n = 12,Ω = 4.

Finally, we briefly discuss the influence of lattice site number on the existence of solitons. The finite lattices with fewer sites (e.g. n < 5) do not exhibit periodicity. The corresponding band-gap structure of infinite lattice exerts no influence on the solitons. With the increase of lattice sites, periodicity becomes stronger and nonlinear modes are partly affected by the band-gap of infinite lattices. It is impossible to realize symmetric and antisymmetric solitons with nonvanishing amplitudes in infinite lattices. However, our results show that solitons strongly “feel” the restriction of infinite lattices when the lattice site number is larger (e.g. n = 12). Thus, we deem that finite lattices should support diverse types of localized solitons, such as fundamental, dipole, triple solitons in 1D nonlinear systems and multipole, vortex, vector solitons in 2D nonlinear systems.

In conclusion, we addressed the existence, stability and propagation dynamics of various families of symmetric and antisymmetric nonlinear waves supported by a finite lattice embedded into a defocusing saturable medium. While out-of-phase solitons bifurcate from the linear modes, in-phase solitons are purely nonlinear modes and have threshold renormalized energy flow. Such solitons contain two pedestals in bulk media and decaying tails in the lattice modulated region. The existence and stability properties of solitons depend strongly on the band-gap structure of infinite lattice when the lattice site number is not very small. Solitons residing in the bands shape as modulated truncated Bloch waves with nonvanishing amplitude in bulk media. The pedestal height and the renormalized energy flow of solitons can be enhanced evidently by increasing the saturation degree of nonlinear media. Symmetric out-of-phase solitons in the first band gap will be completely stable provided that the lattice site number or saturation parameter exceeds a critical value. Specially, we found two branches of in-phase solitons in finite lattices and one branch of them can propagate stably for certain parameters. Our findings can be easily generalized into asymmetric nonlinear waves when different nonlinearities or lattices beside the center of system are applied. This work may bridge the gap between solitons in finite and infinite lattices.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 11074221 and 10704067) and Natural Science Foundation of Zhejiang Province, China (Grant No. Y6100381).

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Y. Kominis and K. Hizanidis, “Lattice solitons in self-defocusing optical media: analytical solutions of the nonlinear Kronig-Penney model,” Opt. Lett. 31, 2888–2890 (2006). [CrossRef] [PubMed]

18.

J. Belmonte-Beitia and J. Cuevas, “Existence of dark solitons in a class of stationary nonlinear Schrödinger equations with periodically modulated nonlinearity and periodic asymptotics,” J. Math. Phys. 52, 032702 (2011). [CrossRef]

19.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008). [CrossRef]

20.

Y. S. Kivshar and B. A. Malomed, “Raman-induced optical shocks in nonlinear fibers,” Opt. Lett. 18, 485–487 (1993). [CrossRef] [PubMed]

21.

S. Wabnitz, “Chiral polarization solitons in elliptically birefringent spun optical fibers,” Opt. Lett. 34, 908–910 (2009). [CrossRef] [PubMed]

22.

G. Agrawal, Nonlinear Fiber Optics (Academic Press, 1995).

23.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (Izdatel Nauka, 1988).

24.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface lattice kink solitons,” Opt. Express 14, 12365–12372 (2006). [CrossRef] [PubMed]

25.

C. Huang, J. Zheng, S. Zhong, and L. Dong, “Interface kink solitons in defocusing saturable nonlinear media,” Opt. Commun. 284, 4225–4228 (2011). [CrossRef]

26.

F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Bragg guiding of domainlike nonlinear modes and kink arrays in lower-index core structures,” Opt. Lett. 33, 1288–1290 (2008). [CrossRef] [PubMed]

27.

J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, “Truncated-Bloch-wave solitons in optical lattices,” Phys. Rev. A 79, 043610 (2009). [CrossRef]

OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(190.6135) Nonlinear optics : Spatial solitons

ToC Category:
Nonlinear Optics

History
Original Manuscript: July 6, 2011
Manuscript Accepted: August 5, 2011
Published: August 17, 2011

Citation
Shunsheng Zhong, Changming Huang, Chunyan Li, and Liangwei Dong, "Symmetric and antisymmetric solitons in finite lattices," Opt. Express 19, 17179-17188 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-17179


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References

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  17. Y. Kominis and K. Hizanidis, “Lattice solitons in self-defocusing optical media: analytical solutions of the nonlinear Kronig-Penney model,” Opt. Lett. 31, 2888–2890 (2006). [CrossRef] [PubMed]
  18. J. Belmonte-Beitia and J. Cuevas, “Existence of dark solitons in a class of stationary nonlinear Schrödinger equations with periodically modulated nonlinearity and periodic asymptotics,” J. Math. Phys. 52, 032702 (2011). [CrossRef]
  19. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008). [CrossRef]
  20. Y. S. Kivshar and B. A. Malomed, “Raman-induced optical shocks in nonlinear fibers,” Opt. Lett. 18, 485–487 (1993). [CrossRef] [PubMed]
  21. S. Wabnitz, “Chiral polarization solitons in elliptically birefringent spun optical fibers,” Opt. Lett. 34, 908–910 (2009). [CrossRef] [PubMed]
  22. G. Agrawal, Nonlinear Fiber Optics (Academic Press, 1995).
  23. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (Izdatel Nauka, 1988).
  24. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface lattice kink solitons,” Opt. Express 14, 12365–12372 (2006). [CrossRef] [PubMed]
  25. C. Huang, J. Zheng, S. Zhong, and L. Dong, “Interface kink solitons in defocusing saturable nonlinear media,” Opt. Commun. 284, 4225–4228 (2011). [CrossRef]
  26. F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Bragg guiding of domainlike nonlinear modes and kink arrays in lower-index core structures,” Opt. Lett. 33, 1288–1290 (2008). [CrossRef] [PubMed]
  27. J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, “Truncated-Bloch-wave solitons in optical lattices,” Phys. Rev. A 79, 043610 (2009). [CrossRef]

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