1. Introduction

Optical solitons have played an important role in the field of high-bit data transmission systems and in fundamental studies of modern nonlinear science. A variety of research on optical solitons in fibers, bulk materials, film waveguides and arrays of waveguides has been conducted and led to the observation of spatial, temporal and spatiotemporal solitons [1–5]. The concept of nonlinear tunneling of solitons was firstly proposed by Newell in 1978 [6]. Subsequently, soliton tunneling was investigated, theoretically and experimentally, in hydrodynamics, BEC and optics [7–16]. In optics, the tunneling of solitons has been studied in the temporal domain through a longitudinal junction [9], and in the frequency domain across a forbidden normal-dispersion barrier [10, 11]. The process of nonlinear tunneling of optical solitons through a strong nonlinear organic thin film exhibits jumplike nonadiabatic evolution, which eventually leads to the soliton “fission reactions” [12]. Optical soliton tunneling through dispersion and nonlinear barrier (or well) has been studied in detail [13, 14]. More recently, spatial soliton tunneling and ejection through a potential barrier have been observed experimentally [15] and multisoliton ejection from an amplifying potential trap has been investigated theoretically [16]. These studies on soliton tunneling have opened up an exciting area in the applications.

During the past years, a great deal of attention has been devoted to light propagation in periodic nonlinear lattices because of potential applications in optical information processing [17–28]. Optical lattices with periodic modulation of the refractive index strongly affect the diffraction properties of light beams and, in combination with nonlinearity, lead to the localization of light in the form of spatial solitons. Spatial solitons in optical lattices exhibit a number of interesting propagation scenarios including oscillation, switching and routing [20–23]. Recently, optical lattices with longitudinal refractive index modulation have received significant attention [29–35]. Nonlinear lattices built of the properly designed segments support diffraction-managed solitons [29, 30]. Fading optical lattices can give rise to soliton steering and soliton fission [31]. The parametric amplification of spatial soliton steering can be achieved in harmonic nonlinear lattices with longitudinal modulation [32], and controllable soliton dragging occurs in dynamical lattices produced by three imbalanced interfering plane waves [33,34]. Variation of the lattice shape in longitudinal direction offers more opportunities for the applications in all-optical devices based on spatial solitons.

In the present paper, we focus on the spatial soliton tunneling through a steep potential barrier created by an exponential rising lattice followed by a decaying lattice. By performing extensively numerical simulations, we show that spatial solitons can exhibit different behaviors such as compression and splitting when passing through the barrier under proper barrier height and input soliton parameters. Based on the properties, we propose a lattice system for compressing soliton and splitting soliton into twin beams. The obtained results are useful in developing novel all-optical devices based on the soliton signals.

2. Model

The propagation of light beams in the focusing Kerr nonlinear medium with the linear refractive index modulation in both transverse x and longitudinal z directions is described by the nonlinear Schrödinger equation [31, 32]:

Here q(η,ξ)=(L _dif/L _nl)^1/2 A(η,ξ)I-^1/2 ₀, where A(η,ξ) is the slowly varying envelope, I ₀ is the input peak intensity, η=x/r ₀,r ₀ is the input beam width, ξ=z/L _dif, L _dif=n ₀ ω r ² ₀/c is the diffraction length, L _nl=2c/ωn ₂ I ₀ is the nonlinear length, ω is the carrying frequency, p=L _dif/L _ref, L _ref=c/δnω is the linear refraction length and δn is the refractive index modulation depth. R(η,ξ) describes the profile of the refractive index along the transverse and longitudinal axes. Here we consider an optical lattice with the profile:

where Ω_η denotes the transverse modulation frequency and δ is the rise (or decay) rate along the longitudinal direction. For a large δ>0, the lattice given by Eq. (2) can describe a steep lattice potential barrier at ξ=ξ _B, as shown in Fig. 1. Such kind of lattice with high longitudinal barrier can be realized by an exponential rising lattice followed by a decaying lattice, which can be technologically fabricated or induced optically in photorefractive crystals [36, 37]. The lattice parameters, such as the transverse frequency and rise (or decay) rate, can be experimentally tuned by changing intensities, intersection angles, carrying wavelength of lattice-forming plane waves and the crystal temperature [31,33,36,37]. The value of parameter p in Eq. (1) can be estimated according to the experimental parameters of Ref. [38]. For a beam width 5 µm and wavelength 532nm, a lattice period ~5µm, the parameter p=1 corresponds to a refractive index variation δn≈0.0002 and ξ=1 corresponds to a lattice length z≈0.7mm. Therefore, the modulation depth of the refractive index can be considered small for a short lattice (e.g.ξ=5) and is of the order of the nonlinear correction to the refractive index due to the Kerr effect.

 

Fig. 1. Profile of a steep lattice potential barrier described by Eq. (2) with Ω_η=4,δ=1 and ξ_B=5.

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It should be noted that Eq. (1) is known in physics and mathematics as the nonautonomous model with varying in space and time external potential [13, 39, 40]. Based on Eq. (1) with complex and nonuinform potentials, exact analytic solutions to the problem of optimal soliton amplification and soliton management have been investigated in detail [13, 39]. Recently, novel soliton solutions for the nonautonomous model [Eq. (1)] with linear and harmonic oscillator potentials under exact integrability condition have been found [40]. However, for the potential presented in Eq. (3), it is difficult to obtain the exact solution of Eq. (1). In order to study the tunneling of spatial solitons through a steep lattice barrier given by Eq.(2), we adopt the split-step Fourier method to perform direct evolution of Eq.(1) with the typical sechtype beam q(η,ξ=0)=Asec h[A(η-η ₀)], where A is the amplitude or inverse beam width and η ₀ the center position, respectively. In the subsequent analysis, we assume the parameters of the lattice δ=1.0 and ξ _B=5, respectively.

3. Numerical results and discussion

3.1 Soliton tunneling

The propagation dynamics of spatial solitons with A=1 and η ₀=0 in the lattice with longitudinal barrier given by Eq. (2) are illustrated in Fig. 2. In this case the input soliton width is comparable with the transverse modulation period $\frac{\pi }{{\Omega }_{\eta }}$ and the soliton center is located at the lattice high-index site. For lower potential barrier p=6, as shown in Fig. 2 (a) and 2(c), the tunneling occurs and the soliton recovers its original shape after it passes through the barrier, where the beam is modulated and presents a main peak and two side peaks. When the potential exceeds the value of p=6, the soliton starts to spread rapidly after passing through the barrier due to the imbalance between the diffraction and the nonlinear effect. The higher the potential barrier is, the more seriously the soliton broadens. Figure 2(b) and 2(d) present the evolution scenarios of the soliton across the barrier with higher potential p=36. Form it one can clearly see that the soliton spreads and radiates rapidly after passing through the barrier. When we increase the transverse frequency of the lattice to Ω_η=8, which corresponds to the case that input soliton width is much larger than the transverse modulation period of the lattice, the soliton can easily pass through the barrier with p=36 and then recover its initial shape, as shown in Fig. 3(a) and 3(c). This result clearly shows that for the same modulation depth, the modulation period of the refractive index strongly affects the diffraction properties of light beams. Naturally, for high enough potentials, for example p=150, the soliton is also difficult to tunnel through the barrier with large transverse frequency Ω_η=8 [see Fig. 3(b) and 3(d)].

 

Fig. 2. Propagation of spatial solitons with A=1and η ₀=0 across the barrier given by Eq. (2) with Ω_η=4, (a) p=6; (b) p=36; (c) and (d) corresponding to the cases of (a) and (b), respectively.

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Fig. 3. Propagation of spatial solitons with A=1 and η ₀=0 across the barrier given by Eq. (2) with Ω_η=8, (a) p=36; (b) p=150; (c) and (d) corresponding to the cases of (a) and (b), respectively.

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Fig. 4. Tunneling of spatial solitons with A=2 and η ₀=0 through the barrier given by Eq.(2) with (a) Ω_η=4, p=36; (b) Ω_η=8, p=150; (c) and (d) corresponding to the cases of (a) and (b), respectively.

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Next, we consider the cases of spatial solitons with A=2 and η ₀=0, as shown in Fig. 4. In contrast with Fig. 2(b), 2(d), Fig. 3(b) and 3(d), we find that the solitons with A=2 can easily overcome and tunnel through the lattice potential barriers which the solitons with A=1 can not pass through, respectively. The result means that for a certain potential, the larger the transverse frequency of the lattice is and the higher peak intensity the soliton possesses, the easier the soliton will tunnel through.

It should be noted in Fig. 2(c), 2(d), 3(c) and 3(d) that whether spatial solitons with A=1 and η ₀=0 can tunnel through the potential barrier or not, they exhibit odd number of peaks at ξ=ξ_B. Physically, these peaks are caused by the mutual interaction between the high lattice potentials and the solitons. The larger the transverse frequency of the lattice is, the more waveguides the input beam covers, and the more peaks the solitons exhibit at ξ=ξ_B. Also, the higher the lattice potential, the more serious the mutual interaction, hence, the deeper dips solitons appear. Interestingly, for solitons with A=2, the side peaks can be effectively suppressed and the energy is collected in the centric peak when they cross the barriers [see Fig. 4(c) and 4(d)]. This property may provide the possibility for realizing the soliton compression.

Furthermore, we investigate the tunneling of spatial solitons with initial shift ${\eta }_0=\frac{\pi }{2{\Omega }_{\eta }}$ through the longitudinal lattice barrier described by Eq. (2). In this case, the soliton is centered in between two lattice high-index sites. Similar to the case of η ₀=0, for lower potential barrier, the soliton with A=1 can successfully pass through the barrier and then revive without changing its shape, as shown in Fig. 5(a) and 5(d). For high enough potential barrier, the intensity of the soliton with A=1 is insufficient to overcome the potential barrier, hence it broadens rapidly, which is illustrated in Fig. 5(b) and 5(e). By increasing the soliton peak intensity we find that the soliton with A=2 successfully passes through the barrier and recovers its shape [see Fig. 5(c) and 5(f)]. However, it is worth noticing in Fig. 5(d), 5(e) and 5(f) that the solitons with ${\eta }_0=\frac{\pi }{2{\Omega }_{\eta }}$ exhibit different behaviors from the case of η ₀=0. These solitons experience the strong interaction with the potential barriers at ξ=ξ _B and appear even number of peaks owing to the property that the beams are symmetrically attracted towards the higher refractive index lattice sites. From Fig. 5(d), 5(e) and 5(f), we can clearly see that at ξ=ξ _B, higher potential results in more energy concentration in the left and right adjacent waveguides and the soliton with A=2 can be clearly divided into twin beams accompanied by compression. These results provide the possibility for spliting soliton and achieving twin beams with high intensity and narrow width.

 

Fig. 5. Propagation of spatial solitons with ${\eta }_0=\frac{\pi }{2{\Omega }_{\eta }}$ across the barrier given by Eq. (2) with Ω_η=4, (a) A=1, p=10; (b) A=1, p=50; (c) A=2, p=50; (d), (e) and (f) corresponding to the cases of (a), (b) and (c), respectively

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3.2 Soliton compression and splitting

As mentioned above, the tunneling properties of spatial solitons through the lattice with longitudinal barrier given by Eq. (2) imply that it is possible to compress and split the spatial solitons. In what follows, we demonstrate a scheme that spatial solitons can be effectively compressed and clearly split by a lattice system with the profile:

Here Ω_η,δ and ξ_B stand for the same physical meanings as denoted in Eq. (2), respectively. This lattice system (3) can be built by a segment of exponential increasing lattice followed by a segment of harmonic lattice. Also, we assume Ω_η=4,δ=1.0 and ξ_B=5. Figure 6(a) displays the evolution scenarios of spatial soltion with A=2,η ₀=0 in the lattice system (3).

 

Fig. 6. Compression of spatial solitons with A=2 and η ₀=0 through the lattice system given by Eq. (3) with Ω_η=4 (a) δ=1, p=30; (b) δ=0 (i.e. harmonic lattice), p=30. (c) and (d) corresponding to the beam shapes of (a) and (b) at different distances and different potentials, respectively.

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From it one can clearly see that the spatial soliton is effectively compressed and stably travels along the lattice. Figure 6(c) shows the compressed beam shapes corresponding to the input soliton shape for different potentials. It can be seen from it that in this lattice system (3), higher lattice potential results in more efficient compression for the same input soliton. However, in harmonic lattice without longitudinal barrier, it is difficult to achieve the compression as effective as that in Fig. 6(c) due to large radiative loss at the start stage, as shown in Fig. 6(b). When we further increase the potential to p=60, the compression ratio decreases instead of increasing due to much more radiative loss [see Fig. 6(d)]. The results presented here may have promising applications in soliton compression.

Also, we are interested in the results shown in Fig. 5 that even though spatial solitons with ${\eta }_0=\frac{\pi }{2{\Omega }_{\eta }}$ can not tunnel through the barrier, it is also split and compressed at ξ=ξ_B. To further explore this point, we study the transmission of the soliton with ${\eta }_0=\frac{\pi }{2{\Omega }_{\eta }}$ in the lattice system (3), as shown in Fig. 7(a) and 7(c). Form these two figures, one can see that the soliton is gradually attracted towards the neighboring waveguides and forms stable twin beams with much narrower width and higher peak intensity than the input soliton. This effect is difficult to achieve in harmonic lattice. For comparing, Figure 7(b) and 7(d) present the cases in the harmonic lattice. At p=50, the input soliton is strongly attracted to the neighboring waveguides and rapidly split into two beams, which is accompanied by too large radiative loss to form two stable beams in harmonic lattice [see Fig. 7(b)]. Numerical simulations show that in harmonic lattice, even if the potential is increased to p=100, spatial solitons can not be efficiently split without radiative loss [see Fig. 7(d)]. These results can be used to obtain twin beams with high peak intensity and narrow width by splitting the input soliton.

 

Fig. 7. Splitting of spatial solitons with A=2 and ${\eta }_0=\frac{\pi }{2{\Omega }_{\eta }}$ through the lattice system given by Eq. (3) with Ω_η=4 (a) δ=1, p=50 ; (b) δ=0 (i.e. harmonic lattice), p=50 ; (c) beam shapes at different distances corresponding to the case of (a);(d) the parameters are the same as in (b) except p=100.

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4. Conclusion

In conclusion, we have numerically investigated the nonlinear tunneling of spatial solitons through a focusing Kerr nonlinear optical lattice with harmonic transverse refractive index modulation and steep longitudinal potential barrier. The results have shown that the position of input beams apparently affects the tunneling behaviors of spatial solitons, which exhibit compression or splitting when passing through the barrier. Moreover, the transverse modulation frequency of lattice and the intensity of input beam strongly affect the ability of tunneling. Based on these properties, we have discussed the feasibility of compressing solitons and splitting solitons into stable twin beams through a lattice system consisting of an exponential increasing lattice followed by a harmonic lattice. The results presented here offer a new scheme for implementation of soliton compression and splitting and may have promising applications in future all-optical devices based on soliton signals.