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We find the existence of two kinds of solitons at the interface of optical superlattices with both spatially modulated nonlinearity and linear refraction index. The first kind of solitons can either drift across the lattice, or deflect to the uniform nonlinear medium. The dynamics of such solitons mainly depends on their powers. The other kind of solitons can stably propagate along the surface, and can be controlled by additional Gaussian beams. In addition, we demonstrate the input-angle-dependent reflection, trapping, and refraction with nearly no losses by launching sech-shaped solitons.
©2011 Optical Society of America
1. Introduction
Recently, Surface solitons at the interface of periodic nonlinear medium have attracted lots of attention due to their fundamental physics and potential application. Such surface solitons were first predicted theoretically [1,2] and subsequently observed in experiments [3,4]. Since then, there is increasing interest in this field [5–18]. Apart from the fundamental solitons [1–6], the research contents have extended to surface dipole [7], kink [8], vector solitons [9–11], polychromatic solitons [12], and even surface light bullets [13]. In addition, the structure has ranged from simple periodic waveguide arrays [1,3,5,6] and photonic lattices [2,4], to binary waveguide arrays [14] and supperlattices [15], as well as the complex case with two dissimilar lattices [16]. Many types of material are also found to support the surface solitons, such as Kerr [1,3,5,6], quadratic [17], photorefractive [2,4], and even the media with nonlocality [7].
However, in all the abovementioned cases, the nonlinearity is spatially uniform inside the material. The solitons at the interface of lattice with both spatially modulated nonlinearity (SMN) and linearity have not been explored. On the other hand, little effort has been made to study the mobility (especially for long range and arbitrary direction) and the direct control of the solitons at interface, which are essential for all-optical applications.
In this work, we investigate the dynamics and all-optical control of the solitons at the interface of superlattices with both spatially modulated nonlinearity and linearity. We find that there exist two kinds of solitons, whose properties are distinct from each other. The first kind of solitons can drift across the lattice, trap near the surface, or deflect to the uniform medium, depending on their powers; while the other kind of solitons can stably propagate along the surface. Moreover, we demonstrate the steering of the solitons by simply adding Gaussian beams.
2. The model
The light propagation at the interface between uniform nonlinear medium and superlattice with SMN, is described by the following Nonlinear Schrödinger equation:
where X, Z stand for the transverse and longitudinal coordinates; A is the envelope of the light field, β0 = k0n0 = 2πn 0/λ is the propagation constant in terms of the optical wavelength λ; n0 and n2 are the linear and nonlinear refractive index. R(X) is the modulation profile for both the linear refractive index and nonlinearity, whose modulation depths are Δχ and σ, respectively. This modulated lattice can be created by Ti indiffusion in LiNbO3 crystals [19], or femtosecond laser pulses writing in fused silica [20,21]. Recently, similar model with both of the SMN and linear refraction modulation was also introduced into 2D surface solitons [22]. The SMN has been used for modifying the stability of lattice solitons [23,24] and steering the lattice solitons freely [25] (for the topic of nonlinear lattices, see also review [26]).With x = X/w0, z = Z/Ld, q = (k0|n2|Ld)1/2A, and Ld = β0w0 2 (w0 and Ld are scaling parameters related to the input beam width and the diffraction length), Eq. (1) can be normalized to the dimensionless form:
where R(x) = 0 for x<0, and for x≥0 describe the semi-infinite surperlattice profile [Fig. 1(b) ]. p is the modulation depth of the refractive index. In the general case of incommensurate periods d 1 and d2, lattice R(x) is quasiperiodic [27]. However, we focus on the double-period lattice (e.g. see Ref. [15]) with d 1 = 2d2. Varying the parameter ε between 0 and 1, one can change the shape and relative depth of the superlattice. Unless otherwise stated, we set d1 = 2, d2 = 1, and ε = 0.3 in this work.
Fig. 1 The lateral movable solitons: (a) Solitons power U versus propagation constant b with p = 2. The white area is the semi-infinite gap. The red dots correspond to the threshold value bd in (b). (b), (c) Low-cutoff bc, Uc, trap threshold bt, Ut, and deflection threshold bd, Ud, depend on σ at p = 2, the dash line in (b) indicates the lower edge of the semi-infinite gap. Insert: the modulation profile R(x) with ε = 0.3, d1 = 2, and d2 = 1. Profiles of solitons for (d) b = 5.5, (e) b = 6.4, (f) b = 9.2 and (j) b = 14 for σ = 0.35, p = 2; (k) b = 4.1 for σ = 0.8; (l) b = 8.9 for σ = 1.1. Panels below them are solitons propagation: (g) drifting across lattice; (h) trapping at the inside lattice; (i) trapping near the surface; (m), (n) deflection from the surface; (o) splitting.
We search for the solitons solutions in the form of q(x,z) = f(x)exp(ibz), where b is the propagation constant and f(x) is a real function with boundary conditions f( ± ∞) = 0. f(x) is numerically solved by the relaxation method. The normalized soliton power is . The propagation of the solitons solutions is then studied by direct simulation of Eq. (2) with initial q(x,0).
3. Numerical results
We find that in the system described by Eq. (2), there exist two kinds of solitons at the interface: one is the lateral movable solitons; the other is the stable surface solitons. Figure 1 shows the typical results of the lateral movable solitons, whose positions locate near the first peak of the lattice. It can be seen in Fig. 1(a) that these solitons exist only when the propagation constant b exceeds a low cut-off value bc (e.g. bc = 5.49 for σ = 0.35), which is different from the case of combined linear and nonlinear bulk lattice [28]. For different σ, the power U of the solitons increases with the growth of b [Fig. 1(a)]. The most unique property of such solitons is the lateral move from the surface. The dynamics can be grouped into three types, according to three b regions, which are divided by the critical b values, bc, bt, and bd, as shown in Fig. 1(b) (their corresponding powers are shown in Fig. 1(c)). Take σ = 0.35 for example. When b is above bc, the solitons drift across the lattice [Figs. 1(d), 1(g)]. As b increases but lower than bt, the solitons become to be captured inside the lattice after drifting [Figs. 1(e), 1(h)]. While b reaches to the region between bt and bd, the solitons will be trapped near the surface [Figs. 1(f), 1(i)]. Once b exceeds the threshold value bd, the solitons turn to deflect from the surface and propagate stably into the uniform nonlinear medium [Figs. 1(j), 1(m)] (this is also called solitons emission [29,30]). The deflection angles turn larger with the growth of b. As σ increases, the solitons profiles become boarder [Figs. 1(k), 1(n)], and the corresponding threshold bd is lower [Fig. 1(b)], which means the solitons deflection is easier to occur if σ is larger. When σ>0.9, the solitons profiles change from single-peak to double peaks [Fig. 1(l)], and the solitons undergo splitting upon propagation [Fig. 1(o)]. In our results, the solitons at the interface can spontaneously move laterally, while the lateral motion of their bulk counterparts in Ref. [28] is driven by adding a transverse momentum.
While tuning the superlattice parameter ε, we also find the similar phenomenon. However, the thresholds vary for different ε (not shown in figure). They will achieve minimum values when ε = 0.7, and have much higher values when ε→0 or ε→1, which means that the phenomenon mentioned above is more difficult to be observed in regular lattice (this is also the reasons why we choose supperlattice in this work). The locations of the border between the lattice and uniform medium also affect the results. Generally speaking, as the border location varies from x = 0 to x = 2 (only one period needs to be considered here), the positions of the solitons will move towards the nonlinear and linear lattice. Thus the solitons undergo different effect of the refractive index and nonlinearity of the lattice, and the power threshold of solitons deflection will vary for different locations of the border. We find that the power threshold decreases when the locations of the solitons approach to the lattice peak, while the power threshold increases when the solitons are away from the lattice peak.
Besides the lateral movable solitons shown above, the system also supports the stable surface solitons for the same parameters. The power U of such solitons increases with b except for a very narrow region near the cut-off [Fig. 2(a) ]. To study the stability of the surface solitons, we substitute the perturbed solutions in the form of q = [f(x) + u(x)exp(δz) + iv(x) exp(δz)]exp(ibz) into Eq. (2) and make linearization around f(x), then solve the resulting eigenvalue problem for the growth rate δ [2]. We find the stable condition Re(δ) = 0 occurs when b exceeds a threshold bs [Fig. 2(b)], which coincides with the condition dU/db>0. Direct numerical propagation tests also confirm the stability analysis. Hence the solitons are stable in almost the entire existent domain [Fig. 2(e)]. Figures 2(c) and 2(d) show an example of the stable surface solitons, whose parameters are identical to those of Figs. 1(k) and 1(n). Notice that the stable surface solitons are slightly away from the surface [Fig. 2(c)] and shift a little to the uniform medium as b increases. This means they reside much deeper into the uniform medium than the lateral movable solitons, which locate near the lattice peak. Thus they can be excited independently by a localized input [Fig. 2(f)].
Fig. 2 The stable surface solitons: (a) Solitons power U versus propagation constant b for p = 2, σ = 0.8, the white area is the semi-infinite gap. (b) Real part of perturbation growth rate δ vs b for (a). (c, d) Stable soliton for b = 4.1, p = 2 and σ = 0.8. (e) Low cut-off bc and stable threshold bs depend on σ at p = 2. Domain above bs is for stable propagation. The dash line indicates the lower edge of the semi-infinite gap. (f) Excitation of surface soliton by inputting a Gaussian beam with A = 2.86, w = 0.6, x0 = −0.07.
Interestingly, the stable surface solitons can be controlled by adding another Gaussian beam. For example, if we add a Gaussian beam in the form of with A = 1.1, x0 = 0.3 and w = 0.3 as a control beam, then the original stable soliton in Fig. 2(d) can be deflected [Fig. 3(a) ] at an angle θ = 23.1°. Moreover, varying the parameters of the Gaussian beam can switch the soliton to different angles. As shown in Fig. 3(b), the deflection angles θ increases with the growth of A, and the Gaussian beam can effectively switch the solitons to large θ when x0≈0.4 (near the peak of the lattice). Note that the beams’ amplitudes in Fig. 3(b) should exceed a threshold to drift the solitons away from the surface, e.g. A>0.75, (more than 26% that of the solitons’ amplitudes, and cannot be considered as small perturbation); otherwise the solitons still propagate along the surface [Fig. 3(c)]. The results will be useful for all-optical applications, such as all-optical switching and routing.
Fig. 3 (a) Turning the stable soliton corresponding to Fig. 2(c) to deflect by additional Gaussian beam with A = 1.1, x0 = 0.3 and w = 0.3. (b) Deflection angle θ controlled by the parameters of Gaussian beams: amplitude A and center x0, with the width w = 0.3. (c) The same with (a) except for A = 0.3.
Next, we study the direct excitation of the system by launching Gaussian beams along the z direction. To excite the solitons, we choose the input positions of the Gaussian beams to be close to those of the solitons solutions. Considering the practical conditions, we keep the widths unvaried and only tune the amplitudes. For example, by inputting Gaussian beams with w = 0.55, x0 = 0.4 and with different amplitudes: A = 3.1, A = 3.4 and A = 3.5, we obtain solitons drifting, trapping, and deflection [Fig. 4(a) ], respectively. Figure 4(b) shows the deflection angles θ depending on the amplitude A. The power thresholds of the beams coincide qualitatively with those of the solitons [Fig. 1(c)]. However, the mismatch between the beams’ profiles and the solitons’ profiles results in the deviation of power thresholds. Notice that the solitons’ widths shrink as b increases in Fig. 1, while the beams’ widths in Fig. 4(a) are unvaried for practical experiment conditions. If the widths of the beams are wider, it is easier to observe the solitons deflection. We find the phenomenon [Fig. 4(a)] occurs only when the input positions are near the first lattice peak, and the solitons excited belong to the lateral movable solitons; if the input positions are near the surface, the input beams will excite the stable surface solitons [Fig. 2(f)].
Fig. 4 (a) Superimposed results: solitons drifting, trapping and deflection by inputting Gaussian beams (x0 = 0.4, w = 0.55) with A = 3.1, A = 3.4, and A = 3.5. (b) Deflection angle θ versus A, the white region is the solitons deflection case. (c) Soliton refraction by launching a sech-shaped soliton from x0 = −1.4 with A = 4.2, v = 0.2, for σ = 0.2 and p = 2, (d) refractive angle versus incident angle. (e) Soliton reflection by launching a sech-shaped soliton with A = 4, v = 0.2 for σ = 0.4 and p = 2; (f), (g) Soliton trapping with v = 0.55 and v = 0.64; (h) the output channel positions x versus v, the grey region is for solitons reflection.
The system also behaves uniquely when launching sech-shaped solitons from the uniform medium, in the form of (solitons transmission across two dissimilar lattices with homogenous nonlinearity is reported recently in Ref. [31]). For small σ (e.g. σ = 0.2), the solitons will refract and propagate across the lattice with nearly no losses [Fig. 4(c)]. Figure 4(d) shows the relation between the incident and refractive angles. For moderate σ (e.g. σ = 0.4), the solitons dynamics is classified into different types according to v [Fig. 4(h)]. If v is small, the solitons do not have enough energy to penetrate into the lattice and will be reflected [Fig. 4(e)]. When v becomes moderate, the solitons penetrate into the lattice and are trapped at the minimum of the lattice [Figs. 4(f), 4(g)]. The x coordinate of the output channel increases dramatically with v [Fig. 4(h)], which means the trapping are very sensitive to the incident angle. If v is not too large, the solitons experience almost without energy losses. The low losses and angular sensitivity make the system promising for all-optical devices.
Finally, we estimate the power needed to launch and control the solitons in practical experimental conditions. The material considered here is chalcogenide glasses. This is a promising material for all-optical device, due to its subpicosecond response, and large nonlinear refractive index coefficient n2, e.g. n2 = 2.4 × 10−13cm2/W, which is greater than 1000 times that of fused silica [32,33]. It has been demonstrated that not only the linear refractive index [34], but also the nonlinear coefficient [33] can be tailored by laser process. The real power of the solitons is
where λ = 1.55μm, n0 = 2.45. WY and wy are the real and normalized transverse mode size along the Y direction, which are approximately 2μm and 0.1. With these particular parameters, the every normalized unit of power U corresponds to 0.8W. Then the power needed to form soliton in Fig. 2(c) is 4.7W, while the power of the control light in Fig. 3(a) is 0.4W. Since these can be easily achieved by semiconductor pulse laser, the all-optical devices based on surface solitons can be probably realized.4. Summary
In summarization, we investigated the solitons at the interface of optical superlattices with spatially modulated nonlinearity and linearity. We found two kinds of solitons featuring different properties. One can drift across the lattice, trap near the surface, or deflect to the uniform medium, depending on their powers. The other can stably propagate along the surface, and can be controlled by adding Gaussian beams. Almost lossless and input-angle-dependent reflection, trapping, and refraction dynamics of solitons were also demonstrated. We expect these results may find their applications in all-optical devices.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (NSFC) (Grant Nos. 10874250, 11074311, and 10804131).
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