1. Introduction

In 1998, Bender and Boettcher firstly introduced the concept of non-Hermitian parity-time (PT) symmetric complex potentials, whose real part must be an even function of position whereas the imaginary component is odd. There exists a critical threshold below which the eigenvalues are entirely real, which is referred to a phase transition point [1]. Since then, there has been a lot activity in this area especially in the field of optics and photonics [2–5 ]. Beam dynamics in PT-symmetric periodic lattices exhibit unique properties. PT-symmetric periodic potentials supporting defect modes [6,7], bright spatial solitons in 1D and 2D defocusing Kerr media with PT-symmetric potentials [8], binary PT-symmetric nonlinear lattices with balanced gain and loss [9] have been studied. Besides, solitons in PT symmetric mixed linear-nonlinear Optical lattices [10, 11] and PT-symmetric potentials with nonlocal nonlinearity were also investigated [12, 13]. In [14] it was investigated the stability of solitons in PT-symmetric nonlinear optical couplers. Also, it was shown recently that PT-symmetric coupled optical waveguides with gain and loss support localized oscillatory structures similar to breathers of a classical model [15].

Optical lattices with refractive index modulation have also intrigued the interest of researchers [16–21 ]. It was revealed that truncated periodic complex potentials with homogeneous losses can support stable surface solitons in both focusing and defocusing media [22]. Optical solitons in optical lattices with periodic modulation of the refractive index offer more opportunities for the applications in all-optical devices based on spatial solitons.

In this Letter, we investigate the propagation of spatial solitons in PT-symmetric optical lattices with a longitudinal barrier. We demonstrate the transmission drift of a spatial soliton when it propagates through a special lattice barrier with different parameters modulation. We explore the relations of the transmission deflection angle to various parameters of the potential barrier. The imaginary part coefficient of the PT symmetric potential barrier induces an anti-symmetric force upon soliton propagating and pushes it to a certain direction accordingly. Soliton dynamics in such complex potential barrier could be tuned through adjusting the parameters of the barrier, which may have the possibility for controllable deflection of spatial solitons.

2. The model

Light propagation in a focusing Kerr nonlinear medium with linear refractive index modulation in both transverse and longitudinal directions is described by the dimensionless nonlinear Schrodinger equation [6, 7]

The normalized transverse x and longitudinal z coordinates are scaled to the input beam width a and diffraction length L _diff = n ₀ k ₀ a ²/2, respectively, where k ₀ = 2π/λ ₀ and n ₀ is the background refractive index. $q(x,z)={\left(L_{\text{diff}}/L_{\mathrm{nl}}\right)}^{1/2}A(x,z)I_0^{-1/2}$ where A(x,z) is the slowly varying envelope, I ₀ is the input peak intensity, L _nl = 2/k ₀ n ₂ I ₀ is the nonlinear length, n ₂ is the Kerr nonlinear coefficient. p = L _diff /L _ref, L _ref = 1/δ nk ₀ is the linear refraction length and δn is the refraction index modulation depth. In this way, R(x,z) describes the complex refractive-index distribution along the transverse and longitudinal axes. We consider the following PT symmetric optical potential barrier similar to that in [23]:

 

Fig. 1 Schematic Profile of a PT symmetric lattice potential. barrier described by Eq. (2)

Download Full Size | PDF

3. Numerical results and analysis

In order to investigate the influence of the PT symmetric potential barrier on the propagation of spatial solitons, we adopt the split-step Fourier method to simulate Eq. (1) which has been used in our previous work [27]. We assume the profile of the input beam to be a fundermental soliton solution with an amplitude A,

For simplicity, we consider the A = 1 soliton, whose FWHM is about 1.76, with an initial energy of 2, incident normally into the PT symmetric potential barrier at the site x ₀ = 0. The slope rate, gain/loss coefficient and the potential depth are firstly set as δ = 1, ω ₀ = 1 and p = 6, respectively. In Figs. 2(a)–2(c), we demonstrate three cases with typical modulation frequencies Ω_x = 2.4, 2.8 and 4. The period of the PT symmetric optical potential barrier T = π/Ω_x corresponds to 1.31, 1.12 and 0.79, respectively. Our simulation results demonstrate that solitons are more likely to penetrate through the potential barrier with a smaller Ω_x, accompanied by a deflection towards the right side of the propagation direction. The reason for this is because we takes ω ₀ > 1 here, which makes the imaginary part of the lattice Im {V} an increasing function of x around the point x ₀. Otherwise, if a negative ω ₀ is used, solitons could be expected to bend to the opposite side. However, for larger Ω_x, no such deflection is observed (see Fig. 2(c)). We also drew the beam profile of the solitons at z = 0, z_b, and 2z_b shown in Figs. 2(d)–2(f). Interestingly, after passing through the barrier, the soliton not only endures a deflection, but also its FWHM is narrowed to match the lattice period, a certain amount of energy is also obtained from the barrier meanwhile.

 

Fig. 2 Propagation of spatial soliton through lattice barriers with typical transverse modulation frequencies (a) Ω_x = 2.4, (b) 2.8 and (c) 4; (d–f) Beam profiles of solitons at z = 0, z_b, and 2z_b accordingly. Other parameters of the barrier are set as ω ₀ = 1, p = 6 and δ = 1.

Download Full Size | PDF

To explore the inherent physical dynamics of solitons in such barrier, we define the bending angle φ as in Fig. 3(a) to show the deflection of the beam transmission, which satisfies tanφ = Δx/Δz. We set Δx and Δz to be the beam traverse and longitudinal displacement respect to its initial location as shown in Fig. 3(a). As a result, the dependence of bending angle on the modulation frequency Ω_x is depicted in Fig. 3(b) with blue line. It should be noted here that the bending angles are not the real angles in unscaled units. In a real spatial scheme, Δx and Δz should be multiplied by the input beam width a and diffraction length L _diff, respectively. The emergent beam energy can be integrated by a similar equation to Eq. (4), and its dependence on Ω_x is shown in the same figure in red line as well. Obviously, there exists a critical point Ω_cr ≈ 3 for such transversely drift motion. If the modulation frequency is larger than Ω_cr, the bending angle is close to zero, as demonstrated in Fig. 2(c). The reason for this apparent by comparing the beam profiles of solitons at z = 0, z_b, and 2z_b shown in Figs. 2(d)–2(f). The beam waist covers several lattice sites of V(x) at larger values of Ω_x. As a result, the variation of Im{V} is averaged and its influence on the dynamic motion of solitons becomes negligibly small. According to the red line in Fig. 3(b), the emergent beam energy decreases with the modulation frequency, a similar trend with the bending angle. Such a beam dynamics is valid in a modulation frequency ranging from 2.5 to 3.

 

Fig. 3 (a) Definition of bending angle; (b) Dependence of soliton bending angle (blue line) and emergent beam energy (red line) on modulation frequency.

Download Full Size | PDF

Intuitively, the gain/loss coefficient of the PT symmetric potential barrier ω ₀ could be the main cause of such beam deflection. Upon this purpose, we still consider the A = 1 fundamental soliton, which is incident on site at the point x ₀ = 0. The barrier length, the modulation frequency and the potential depth are fixed at Ω_x = 2 and p = 6, respectively. We swept ω ₀ from -1 to 1, and drew the dependence of bending angle on ω ₀ in Fig. 4(a). Apparently, the motion of negative ω ₀ demonstrates an anti-symmetric way due to the lattice property. Furthermore, the larger ω ₀ is, the bending angle becomes larger. It is well known that ω ₀ = 0.5 is a phase transition point of a PT symmetric lattice with a form of V(x) = [cos²(Ω_xx) + iω ₀ sin(2Ω_xx)]. When ω ₀ < 0.5, the eigenvalues are entirely real, which makes the band-gap structure of such PT symmetric lattice resembles that of a traditional lattice. While for ω ₀ > 0.5, the first two bands start to merge together and form an oval-like structure with a related complex spectrum [6]. Actually, it is clearly seen from the curve in Fig. 4(a) that for ω ₀ below such transition point, solitons experience only a small deflection and the bending angle increases very slowly. Once ω ₀ reached above 0.5, the soltion could feel an increasing force caused by the barrier, and bend to the right side of the propagation direction.

 

Fig. 4 (a) Dependence of soliton bending angle(blue line) and emergent beam energy (red line) on the gain/loss coefficients; (b) and (c) are solitons propagating through lattice barriers with ω ₀ = 1 and 0, respectively. other lattice parameters are Ω_x = 2 and p = 3, and δ = 1. (d) and (e) are beam profiles corresponding to the cases in (b) and (c).

Download Full Size | PDF

To verify the fact that the above-mentioned soliton transversely drift motion results from the spatially inhomogeneous loss, we investigate the soliton propagation with the optical lattice barrier without such PT symmetry, i.e., ω ₀ = 0 in Eq. 2. It is discovered that the soliton bending angle equals to zero and the energy remains to be 2 invariably whatever the modulation frequency Ω_x changes. Actually in this case, the imaginary part of V(x) becomes 0, and the system turns to non-dispersive. As the lattice potential is symmetric about the beam center, and solitons are incident normally into the lattice, no transverse forces could be expected. We drew such differences in Figs. 4(b)–4(e) for typical barriers both with and without gain/losses. Though the modulation frequency Ω_x = 2 is far below the critical modulation frequency Ω_cr, no transverse drift is observed in lattices without such PT symmetry, as shown in Figs. 4(d) and 4(e). In addition, the emergent beam energy in Fig. 4(b) is stronger than that of Fig. 4(c), which is proved to be a unique feature of PT symmetric lattice barriers. Physically, when ω ₀ > 0, the imaginary part of V(x) is positive on the right half side of the soliton, while negative on the other side. The soliton would obtain energy from the lattice on the right side while lose energy on the other side. Such an asymmetric effect generally increases upon propagation, and interacts severely with the lattice barrier at z = 5. As for ω ₀ < 0, the bending angle would be reflected back towards left side.

Another essential factor that causes a soliton self-bending is its incident position relative to optical lattices, which has been reported in our previous work [27]. Thus in the following study, we fixed Ω_x = 2.4, ω ₀ = 1, A = 1, p = 6, δ = 1, and let the incident position varies in a lattice period, ranging from −0.5T to 0.5T. Typical examples are shown in Fig. 5. It is interesting to find that the bending angle still has a similar variation trend with the beam energy, as shown in Fig. 5(a). Notably, there exists a range near 0.5T where the beam dynamics change dramatically, which is different from other cases. Typically we drew the case of x ₀ = 0.4T and 0.3T in Figs. 5(b) and 5(c) for comparison in the same plot. Although the soliton is incident at two points very closely, their propagation dynamics are totally different. This phenomenon reflects the instability of solitons in a PT symmetric lattice barrier. Physically, this result can be understood by the modulation instability of the beam itself. When the beam is incident near the position of 0.5T, where the local refractive index (RI) reaches its minimum, the beam tends to bend towards regions with higher RI. Upon this fact, the soliton dynamics is much more sensitive to its incident positions compared with other cases, especially at a position around x ₀ /T~ 0.3, where the real part of V(x) is asymmetric across the beam profile. As the splitting of beams occurs at the same time, the variation in beam power could be expected. In a word, the bending of solitons could be a combined effect of the RI modulation and the loss/gain anti-symmetric properties of the lattice barriers, which can be tuned sharply.

 

Fig. 5 Dependence of soliton bending angle (blue line) and emergent beam energy (red line) on the incident position; (b–c) are soliton propagation through the lattice barrier at x ₀ = 0.4T and 0.3T respectively, other parameters are Ω_x = 2.4, ω ₀ = 1, A = 1, p = 6, δ = 1; (d) and (e) are the beam profiles corresponds to the cases of (b) and (c), respectively.

Download Full Size | PDF

There might have some other factors that could influence the transmission deflection of solitons, for example the lattice modulation depth, which determines the relative refractive index variation in structures. Here we fix a set of parameters as Ω_x = 2, ω ₀ = 1, A = 1, x ₀ = 0 and δ = 1. The dependence of bending angle on the potential depth is shown in Fig. 6(a). When the potential depth p is below 1, the soliton propagation direction remains almost unchanged due to the fact that the lattice barrier is too weak to affect the soliton propagation. However, as the lattice modulation depth becomes larger, the bending angle of the solitons would be increased accordingly. Especially when p is larger than 2, the bending angle grows very sharply with the increase of the lattice modulation depth. In the Figs. 6(b)–6(e), we give two typical examples of the soliton propagation at p = 2 and 4, respectively. Apparently, in the latter case, the soliton not only bend to its right side, but also gained a certain amount of energy from the PT-symmetric lattice barrier, the corresponding beam energy reaches 3.78 in the latter case, which is almost twice of the initial soltion energy.

 

Fig. 6 Dependence of soliton bending angle (blue line) and emergent beam energy (red line) on the potential depth; (b–c) are soliton propagation through the lattice barrier at p = 2 and 4 respectively, other parameters are Ω_x = 2.4, ω ₀ = 1, A = 1, x ₀ = 0, δ = 1; (d) and (e) are the beam profiles corresponds to the cases of (b) and (c), respectively.

Download Full Size | PDF

Lastly, we investigate the influence of the barrier length, which is determined by the slope rate of the lattice δ. Optically, when the barrier length approaches zero, the resulted barrier is close to a uniform PT-symmetric lattice. At this time, the modulation of the PT-symmetric lattice is neglectable and solitons will not deflect when passing through the lattice. As the barrier length shrinks with the slope rate, we only consider the slope rate to be a range of 1 < δ < 5. Other parameters are set as x ₀ = 0, ω ₀ = 1,A = 1,Ω_x = 2.4, and p = 6. Figure 7(a) demonstrates a decay of bending angle with the barrier length. Actually, as the barrier length increases, the height of the barrier will damp much faster. The interaction time between the incident soliton and the barrier would be shortened. In Figs. 7(b)–7(e), we give two typical examples of the soliton propagation with δ = 2 and δ = 4. The difference in emergent beam energy and the bending angle could be clearly seen in Figs. 7(d) and 7(e), which can be tuned by the barrier length.

 

Fig. 7 Dependence of soliton bending angle (blue line) and emergent beam energy (red line) on slope rate δ of the barrier; (b–c) are soliton propagation through the lattice barrier at δ = 2 and δ = 4, respectively, other parameters are Ω_x = 2.4, ω ₀ = 1, p = 6, A = 1, x ₀ = 0.;(d) and (e) are the beam profiles corresponds to the cases of (b) and (c), respectively.

Download Full Size | PDF

4. conclusion

In conclusion, we have investigated the soliton dynamics in PT-symmetric optical lattices with a longitudinal barrier. The results proved that solitons can penetrate through the barrier and exhibit a transverse drift together with a gain of energy from the PT symmetric lattice barrier. Such a transverse drift motion of solitons arises from an external force resulting from the inhomogeneous gain and loss of the potential. The bending angle of the soliton propagation can be efficiently tuned by adjusting barrier parameters such as the modulation frequency, potential depth, gain/loss coefficient, incident position, slope rate and so on. These results suggest new possibilities for experimental and theoretical studies of the solitons dynamics in media with a PT symmetric barrier. It may be interesting to extend the analysis for similar two-dimensional settings.