Nonlinear surface waves have been investigated in the context of fluids [1], solids [2], plasmas [3] and nonlinear optics [4–6], 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 [7–11]. One important generalization of these ideas is the concept of surface gap solitons [12, 13]. Meanwhile, dark solitons in uniform or lattice modulated media with defocusing nonlinearity were investigated [14–18]. For a review of early works, see [19] and references therein.

Of particular interest is that optical pulses in the form of kink solitons may propagate stably in optical fibers [20, 21]. 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]. Counterpropagating beams in spun high-birefringence optical fibers can be used to excite stable kink solitons [21]. The sharp wavefront of kink wave is crucial for high intensity short pulses exhibiting nonlinear responses or self-steepening effects in fibers [22, 23].

So far, researches on kink solitons in spatial domain are very rare, with the only exceptions reported in [24] where surface kink solitons in defocusing Kerr media were predicted, and in [25] where kink solitons in semi-infinite lattice modulated media with defocusing saturable nonlinearity were investigated. In Ref. [24], 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].

Most of the previous studies about lattice solitons focused on the localized nonlinear modes in infinite or semi-infinite lattices. Advancement of laser written technology allows one to realize arbitrary linear refractive index modulation, including (non)periodic lattices, chirped lattices, supper lattices and finite lattices. A challenging open problem is to link nonlinear modes in infinite and finite lattices.

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]. 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:

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.

To elucidate the stability properties of solitons, we search for perturbed solutions of Eq. (1) in the form q(x,z) = [w(x) + u(x) + iv(x)]exp(ibz), here u,v ≪ w stand for the real and imaginary parts of perturbation, which may grow with a complex index δ during propagation. Substituting the perturbed solution into Eq. (1) and linearizing it around the stationary solution w yield a system of coupled equations for perturbations u and v:

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]:

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) = cos²(Ωx) for |x| = nπ/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 b_low ≤ b ≤ b_upp and weakens with the decrease of b. Soliton will degenerate into chirped truncated Bloch wave [27] 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.

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Fig. 2 (a) Band-gap structure of infinite lattice R(x) = cos²(Ωx). 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: U_r versus b for different saturation parameter s. p = 4 in (d) and Ω = 2,n = 13 in all panels.

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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], 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) = cos²(Ωx) 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]. 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 b_upp = b_low + 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 b_upp approaches to the upper edge of the second band. Nonlinear waves will degenerate into linear ones when b → b_upp. 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 b_upp = b_low + 1/s.

The renormalized energy flow U_r 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, U_r 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 b_upp in the high-saturation regime. Comparing with the kink solitons in defocusing cubic media with imprinted semi-infinite lattices [24], 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 b – s 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 (n_cr = 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: U_r of in-phase I solitons near b_upp at s = 0.2. n = 13 in (a, b, d), s = 0.2 in (b, c) and p = 4 in all panels.

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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 b → b_upp. 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]. 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 ${\sigma }_{\mathit{\text{noise}}}^2\hspace{0.17em}=\hspace{0.17em}0.01$ were added into the initial inputs. p = 4 in (a–f) and n = 13 in (b–f).

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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]. 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| ≤ nπ/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 (P_th = 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: U_r 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.

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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.

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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.