1. Introduction

The optical lattice is an ideal study tool for optical solitons and matter-wave solitons because lattice depth, spacing, and potential can be altered or switched off during the experiment. In optics, the optical lattice provides novel physics and light-routing applications [1,2] and interesting properties of solitons [3, 4]. Recently, optical superlattices have been applied to the studies of solitons in Bose-Einstein condensate [5, 6] and optical solitons in Bragg grating [7, 8]. By changing the relative depths of the superlattices’ wells, one can finely tune the effective dispersion of the matter waves and, therefore, effectively control both the peak density and the spatial width of the gap solitons [5]. The existence and stability of bright, dark, and gap matter-wave solitons in optical superlattices have been demonstrated [6]. Optical lattices can be generated by the methods of Refs. [9,10].

Defect solitons are the nonlinear defect modes that bifurcate out from linear defect modes [11]. Such solitons have been investigated in waveguide arrays with defocusing cubic nonlinearity [12]. In particular, the defect of a one-dimensional optical lattice features unique properties of solitons [13].

In this paper, we theoretically find that defect superlattice solitons (DSSs) exist at the defect site in one-dimensional optical superlattices with focusing saturable nonlinearity. Solitons with some unique properties exist in superlattices with defect. The stable domains of solitons with different defect depths are given. When the peak intensity of defect has a specific value, solitons in the semi-infinite gap can be equally divided into two parts at a lower power.

2. The model

Usually there are two ways to optically induce photonic lattices. One is created by a pair of beams or multibeams, and the probe beam is launched in the perpendicular direction. Another is by the amplitude mask method [14]; in this case, the lattice beam can input parallel to or perpendicular to the propagation direction of the probe beam. Here we use an ordinary polarized beam, which passes through an amplitude mask to generate superlattices with a single defect, to launch into a photorefractive crystal with focusing saturable nonlinearity [13]. The amplitude mask can control the distribution of optical intensity, which forms superlattices with lattice defect on the photorefractive crystal. The beam of superlattices with defect is assumed to be uniform along the direction of propagation. Meanwhile, we consider an extraordinary polarized probe beam, which is launched into the defect site. The probe beam is incoherent with the lattice beam and propagates collinearly with it. Therefore, the probe beam at the defect site in one-dimensional optical superlattices with focusing saturable nonlinear media is described by the nonlinear Schrödinger equation. The nondimensionalized model equation for the probe beam is [13,15]

Here, q is the slowly varying amplitude of the probe beam, z is the propagation distance (in units of 2k ₁ D ²/π ²), k ₁ = k ₀ n_e, n_e is the unperturbed refractive index, k ₀ = 2π/λ ₀ is the wave number (λ ₀ is the wavelength in a vacuum), D is the lattice spacing, x is the transverse distance (in units of D/π), E ₀ is the applied dc field [in units of π ²/(k ² ₀ n ⁴ _e D ² γ ₃₃)], and γ ₃₃ is the electro-optic coefficient of the crystal. I_L is the intensity profile of the optical lattice described by

Here, I₀ is the peak intensity without modulating the uniform photonic lattice (i.e. ε ₁=1). ε ₁ represents the modulation parameter of superlattice peak intensity. Therefore, the peak intensity of superlattices is determined by the I₀ and ε ₁. ε ₂ represents the modulation parameter of the peak intensity of defect. The constant c is introduced to make the peak intensity of the defect at ε ₂=0 approximate to that of superlattices at a given value of ε ₁. We take c = 0.85 when ε ₁=0.3. We consider the intensity of defect with a single oscillation of a sine function, and the defect locates at the center of superlattices and is modulated by ε ₂, as shown in Figs. 1(a)-1(c). In Fig. 1, different defect optical superlattice profiles with I₀=3 and ε ₁=0.3 are shown at (a) ε ₂=0.45, (b) ε ₂= -0.45, and (c) ε ₂=0. The superlattice potential given by Eq. (2) can be induced optically by launching a beam into the amplitude mask whose intensity distribution of transmission light is the same as the superlattice potential.

 

Fig. 1. Lattice intensity profile with I₀=3 and ε ₂=0.3: (a) ε ₂=0.45, (b) ε ₂= -0.45, and (c) ε ₂=0; (d) applied dc field parameter E₀ versus the propagation constant μ. Gray regions are Bloch bands.

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We take typical parameters as D = 30μm, ε ₁=0.3, λ=0.5μm, n_e=2.3, γ ₃₃=280pm/V, then x=1, z=1, and E₀=1 correspond to 9.55μm, 5.3 mm, and 8.86 V/mm, respectively. We take I₀=3 and E₀=6, which are typical in experimental conditions as shown in Ref. [16].

To show the existing conditions for DSSs, we search the Floquet-Bloch spectrum by substituting a solution q(x, z) = f(x) exp(ikx + iμz) to the linear version of Eq. (1) where μ is the real propagation constant, k is the Bloch wave number, and f(x) is the complex periodic function [here f(x) = f(x + π)]. The substitution of the light field in such form yields the eigenvalue problem [17]

We numerically solve Eq. (3) to obtain the Floquet-Bloch spectrum μ(E₀) of the (infinite) superlattice as shown in Fig. 1(d), which shows the bandgap structure in the uniform (defect-free) superlattice.

We search for the stationary soliton profiles in the form of q(x,z) = f(x)exp(iμz/z), where f(x) is the real function satisfying the equation

The power P of a soliton is defined as P = ∫^+∞ _-∞ f ²(x)dx. By numerically solving Eq. (4) using the shooting method, we get the soliton profiles in Section 3 as in Figs. 2(c)-2(e). To indicate the stability of DSSs, we search for the perturbed solution of Eq. (1) in the form q(x,z) = [f(x) + h(x,z) + ie(x,z)]exp(iμz), where h(x,z), and e(x,z) are the real and imaginary parts of perturbation that can grow with complex rate δ upon propagation. Omit the neglectable nonlinear terms in Eq. (1), the eigenmodes of coupled equations as follows:

These equations are solved numerically to get the perturbation growth rate Re(δ).

3. Numerical results

To further study the DSSs’ robustness, in all numerical simulations based on Eq. (3) we add a noise to the inputted DSSs by multiplying them with [1 + ρ(x)], where ρ(x) is a Gaussian random function with <ρ>=0 and <ρ ²> =δ ² (we choose that δ is equal to 10% of the input soliton amplitude).

 

Fig. 2. (a) Power versus propagation constant (blue regions are Bloch bands) in the semi-infinite gap for ε ₂=0.45; the solid curve is stable and the dashed curve is unstable. (b) Perturbation growth rate Re(δ). Stable DSSs at (c) A: μ=-2.35, (d) B: μ=-1.8, and unstable DSSs at (e) C: μ=-1.5. (f)-(h) DSSs propagate corresponding to (c)-(e), respectively.

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First, we choose ε ₂= 0.45 as a typical case for the positive defect. Figure 2(a) shows that the power of DSSs increase with the increase of propagation constant μ. Figures 2(c)-2(e) show the profiles of DSSs with different propagation constants μ=-2.35, -1.8, and -1.5, respectively. Figures 2(f)-2(h) show the solitons’ propagations corresponding to Figs. 2(c)-2(e), respectively. In the range of propagation constant -2.35≤μ≤-1.8, the DSSs can stably propagate, but the propagations of DSSs are unstable when the propagation constant is μ>-1.8 (corresponding to higher power). This phenomenon is similar to defect solitons in a regular lattice [13]. We find that in the semi-infinite gap, solitons with higher power have amplitude oscillations, and in this case the δ has an imaginary part. The amplitude oscillations of solitons are not persistent, and finally, the solitons gradually destruct upon propagation, as shown in Fig. 2(h). The stability of DSSs is analyzed by solving Eq. (5) to get the growth rate Re(δ) , as shown Fig. 2(b), which is in agreement with the above analysis. Therefore, at high power, DSSs are unstable in the positive defect, which is the same as defect solitons in regular photonic lattices [13], while at low power, DSSs can stably propagate.

When ε ₂= 0, DSSs exist in the semi-infinite gap and their stability is similar to the positive defect except in the case of the lower power. Figure 3(a) shows that the power of DSSs increases with the increase of the propagation constant μ. Figures 3(c)-3(e) show the profiles of DSSs with different propagation constants μ=-2.4, -1.9, and -2.55, respectively, and Figs. 3(f)-3(h) show their propagations. In the range of propagation constant -2.52≤μ≤-1.85, the DSSs can be stably propagated, but they are unstable with propagation constants μ>-1.85 (corresponding to higher power) and μ<-2.52 (corresponding to lower power). The power P increases with the increase of propagation constant μ when μ≥-2.52. At high power, the propagations of solitons are unstable, which is different from the usual Vakhitov-Kolokolov criterion [13], while at low power, solitons’ stability can be judged by the usual Vakhitov-Kolokolov criterion of instability based on the sign of the slope in the power curve. Apparently, in the range of propagation constant -2.52≤μ≤-1.85, the propagations of solitons are stable because of dP/dμ > 0, but unstable when μ<-2.52 (such as near point A) because of dP/dμ < 0. The stability of DSSs is analyzed by solving Eq. (5) to get the growth rate Re(δ), as shown in Fig. 3(b), which is in agreement with the above analytic result.

 

Fig. 3. (a) Power versus propagation constant (blue regions are Bloch bands) in the semi-infinite gap for ε ₂=0; solid curve is stable except for near point A; near point A and the dashed curve are unstable. (b) Perturbation growth rate Re(δ) Stable DSSs at (c) B: μ= -2.4, (d) C: μ= -1.9, and unstable DSSs at (e) A: μ= -2.55. (d)-(f) DSSs propagate corresponding to (f)-(h), respectively.

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When the peak intensity of defect decreases to a certain value in the negative defect, DSSs can stably exist in the first finite gap (between 1^st and 2^nd bands). As a typical case, we consider ε ₂= -0.45. Figure 4(a) shows that the power of DSSs increases with the increase of μ. Figures 4(b) and 4(c) show the profile of DSSs for μ= -3.31 and μ= -3.85, respectively. For the negative defect, most of DSSs exist in the first finite gap and can be stable in propagation, which is the same as that of uniform photonic lattices [13]. Note that for the negative defect, the existent bandgap of DSSs shifts gradually from the semi-infinite gap to the first finite gap with the increase of defect depth. For this shift, the critical value of the modulation parameter is ε ₂=-0.2; i.e., for ε ₂>-0.2, the DSSs exist in the semi-infinite gap, while for ε ₂<-0.2, DSSs exist in the first finite gap. For ε ₂=-0.2, DSSs exist not only in the semi-infinite gap but also in the first finite gap, as shown in Figs. 5(a) and (b). We give the DSSs’ stable/unstable domains according to the relation defect of the modulation parameter to the propagation constant in Fig. 5, where the stable domains are shown in gray.

 

Fig. 4. Power versus propagation constant (blue regions are Bloch bands) in the first finite gap for ε ₂=-0.45. Stable DSSs at (b) B: μ= -3.31, (c) A: μ= -3.85.

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Fig. 5. Stable (gray) and unstable (white) domains of DSSs in the semi-infinite gap (a) and the first finite gap (b), respectively.

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Finally, we find that the DSSs can be divided equally into two parts along the propagation direction by changing ε ₂ from zero to negative when ε ₁ is changed from high to low. As an example, we consider the cases of ε ₁=0.1 and ε ₂=-0.2. Figures 6(a) and 6(b) show the profiles of DSSs for μ=-2.67 and μ=-2.7, respectively, and Figs. 6(c) and 6(d) show that propagation splits. It is worthy of discussion as to whether the defect of superlattices can be used as the Y waveguide by this phenomenon. Table 1 shows the regions of parameters in which DSSs split upon propagation.

 

Fig. 6. For ε ₁=0.1 and ε ₂=-0.2, DSSs with different propagation constants (a) μ=-2.67 and (b) μ=-2.7. (c) and (d) DSSs propagate corresponding to (a) and (b), respectively.

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Table 1. Regions of DSSs dividing into two parts along propagation direction.

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4. Summary

We demonstrate that DSSs exist at the defect site in one-dimensional photonic superlattices with focusing saturable nonlinearity. For the positive defect, solitons exist in the semi-infinite gap, and solitons stably exist at low power but are unstable at high power. For the negative defect, most solitons exist in the first finite gap, and in the whole bandgap, solitons can stably propagate. The stable domains of solitons with different defect depths are given. We also find that the DSSs can be split upon propagation in a specific region. The defect of superlattices can be used as the Y waveguide in a special case. The combination of such superlattices with the surface models will be as useful to study as the surface gap solitons [18–20].