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We study the solitons in parity-time symmetric potential in the medium with spatially modulated nonlocal nonlinearity. It is found that the coefficient of the spatially modulated nonlinearity and the degree of the uniform nonlocality can profoundly affect the stability of solitons. There exist stable solitons in low-power region, and unstable solitons in high-power region. In the unstable cases, the solitons exhibit jump from the original site to the next one, and they can continue the motion into the other lattices. The region of the stable soliton can be expanded by increasing the coefficient of the modulated nonlocality. Finally, critical amplitude of the imaginary part of the linear PT lattices is obtained, above which solitons are unstable and decay immediately.
©2012 Optical Society of America
1. Introduction
Bender and Boettcher in their pioneering work in 1998 found that even a non-Hermitian Hamiltonian can also show an entirely real eigenvalue spectrum, and it is called parity-time (PT) symmetry [1]. In 2008, Mussilimani studied the optical solitons in one-dimension (1D) and two-dimensional (2D) PT-symmetric periodic optical lattices, and they also have demonstrated that there is a critical threshold [2]. Above this threshold the PT symmetry will be broken, and the eingenvalue spectrum becomes partially complex [2]. Since that, the PT symmetry, the linear modes and the nonlinear modes are investigated widely in theory and experiments [2–10].
Recently, the stability analysis of gap solitons in 1D and 2D PT-symmetric optical lattices have been investigated [11]. It has found families of analytical solutions for symmetric and antisymmetric solitons in a dual-core system in Kerr medium with PT-balanced gain and loss [12]. The Bragg gap solitons in PT-symmetric potentials with competing nonlinearity have been also studied [13]. Moreover, solitons can also exist in PT nonlinear optical lattices [14]. The transformations among PT- symmetric systems by rearrangements of waveguide arrays with gain and losses has been recently reported, which shows that the transformations do not affect their pure real linear spectra [15]. In previous work, by using PT-symmetric potentials, we have revealed effect of the relative strength of superlattices on stability of soliton [16], also found existence of gray solitons [17], obtained stable 1D and 2D solitons in defocusing Kerr media [18], and investigated solitons dynamics in PT-symmetric mixed linear-nonlinear optical lattices [19, 20].
On the other way, solitons in nonlocal nonlinearity are also widely studied in recent years. Surface fundamental, dipole solitons in one-dimensional (1D) [21] and dipole, vortex solitons in two-dimensional nonlocal nonlinearity media also are revealed [22]. Some unique properties are discovered in this nonlinearity. For example, the nonlocality of the nonlinear response can affect the soliton mobility profoundly [23]. Recently, the defect solitons in PT-symmetric potentials with nonlocal nonlinearity are investigated [24].
At the same time, the spatially modulation of nonlinearity have attracted much attention. The interplay between linear and out-of-phase nonlinear refractive modulations can lead to new soliton properties, such as the modifications of transverse mobility and the soliton characteristic of stability [25]. Solitons can drift, rebound, penetration, and trapping at the interface between media with uniform and spatially modulated nonlinearities [26].
In this work, we investigate the spatial solitons in parity-time symmetric potentials with spatially modulation of nonlocal nonlinearity. It is found that the modulation coefficient, and the degree of uniform nonlocality will obviously affect the existence and the stability of the solitons. With the different modulation coefficients, the domains and the channels of instability solitons are different.
2. The model
The normalized 1D nonlinearity equation for PT symmetric linear optical lattices with spatial modulation of nonlocal nonlinearity about the light field q is [2, 21–23]
withwhere d is the degree of the uniform nonlocality. When d→0, Eq. (1) describes a local nonlinear response, and d→∞, it describes a strongly nonlinear response. We select the real part of the PT-symmetric potential is V(x) = 4cos(2x), while its imaginary part is W(x) = W0sin(2x). The last term of the Eq. (1) is the spatial modulated nonlocal nonlinearity, here, f(x) is the real modulated function, we choose the modulated function f(x) = kcos2(2x) (k is the coefficient of the modulation).In fact, the parity-time-symmetric potentials (optical lattices) can be realized through using the complex refractive index distribution, here represents the background refractive index. According to the PT condition, must be even while should be odd [2–4]. In experiment, the PT symmetry is observed [5]. On the other hand, we add periodic nonlinear modulation in Eq. (1). In experiments, the nonlinear optical lattices can be created by fs-laser writing in fused silica [27]. And it can be created in Bose-einstein condensates (BECs) by the Feshbach resonance (FR) modulation of the scattering length [28]. The solitons in nonlinear optical lattices have been studied in many systems [14, 26, 29–32].
We search for the stationary soliton solutions of Eq. (1) in the form of , here μ is the real propagation constant and is the complex function. By substituting into Eq. (1), we get the equation
The power of soliton is defined as, and the refractive index distribution induced by the nonlocality can be written as . We obtain the numerical solutions of Eq. (3) by the spectral renormalization method [33].The linear part of Eq. (3) is
The Bloch theorem indicates that the eigenfunctions of Eq. (4) are in the form of , here K is the Bloch wave number, and is a periodic function of with the same period as the lattices and . By Substituting the Bloch solution into Eq. (4), we can obtain the eigenvalue equationEquation (5) can be numerically solved by the plane wave expansion method. The PT lattices and the corresponding band structure are displayed in Figs. 1(a) and 1(b), respectively. When, the semi-infinite gap as , the first gap as , the second gap as, etc. The PT symmetry can be retained as 0≤W0≤4.
Fig. 1 (a) PT-symmetric optical lattices (OLs) with W0 = 2 (the bue line is the real part, and the red line is the imaginary part). (b)The band structure corresponding to the lattice profile shown in panel (a).
3. Numerical results and analysis of the generic outcomes
First, by setting and, we find that for various coefficients of the modulated nonlocality k, solitons can exist in the semi-infinite gap. These solitons are stable in the low-power region but unstable in the high-power region. Figure 2 shows the power diagram for k = −0.8, k = 0, and k = 1, which indicates that the region (the value of) of stable soliton is expanded with the growth of k. The nonlinearity modulation changes the nonlinearity effects, so that the stability domains of solitons are different.
Fig. 2 Power P versus propagation constant μ for various k with d = 1 and W0 = 2.The solid line represents the stable regions and the dash line represents the unstable regions.
Here, we give some typical examples of stable and unstable solitons evolutions for various values of k. We take μ = 1.8 as a stable case and μ = 2.5 as an unstable case for k = −0.8, respectively, shown in Figs. 3(a,c) and 3(b,d). Figure 3(d) shows that the soliton jump into the adjacent channels during propagation. For k = 0, Figs. 4(a,c) and 4(b,d) are the stable case and the unstable case, respectively, corresponding to μ = 2.0 and μ = 2.8. For μ = 2.8, the soliton is unstable. For k = 1, Figs. 5(a) , 5(c) and 5(b,d) are the stable case and the unstable case, respectively, corresponding to μ = 2.2 and μ = 3.0. From Figs. 4(b,d) and 5(b,d), the unstable solitons, similar to the Figs. 3(b,d) for k = −0.8, also exhibit jump into the adjacent channels during propagation. And it displays that the jump of unstable solitons is more obvious with the decrease of the k. The phenomenon of the unstable solitons shift to neighboring channel arises from the combination of the modulation nonlocal nonlinearity and the PT symmetric potentials. In addition, Figs. 6 (a, b, c) show that with the increase the modulation coefficient k, the amplitudes of the real part and imaginary part of the soliton profile will decrease.
Fig. 3 Soliton profiles (blue line is the real part, the red line is the imaginary part, and the green dash line is the refractive index profile) and Soliton evolution for μ = 1.8 (a, c), and μ = 2.5 (b, d). The other parameters are d = 1, W0 = 2, and k = −0.8.
Fig. 4 The soliton profiles for μ = 2.0 (a), and μ = 2.8 (b). (c) and (d) are the corresponding propagations. Here, k = 0. The other parameters are the same as those in Fig. 3.
Fig. 5 The soliton profiles for μ = 2.2 (a), and μ = 3.0 (b). (c), (d) are the corresponding propagations. Here, k = 1. The other parameters are the same as those in Fig. 3.
Fig. 6 When μ = 2.0. (a). (b), (c) are the soliton profiles for k = −0.8, k = 0, and k = 1, respectively.
Next, we increase the degree of uniform nonlocality to d = 3 to study soliton property (fixing other parameters). It is demonstrated that solitons exist also in the semi-infinite gap in Fig. 7(a) , and the stable region is narrower than the case of d = 1, W0 = 2, and k = 1 by comparison with Fig. 2. The stable case with μ = 1.9 is shown in Figs. 7 (b) and 7(d), and for μ = 2.5 soliton is unstable shown in Figs. 7(c) and 7(e).
Fig. 7 Power P versus propagation constant μ (a). The soliton profiles for μ = 1.9 (b), and μ = 2.5 (c). (d), (e) are the corresponding propagations. Here, d = 3. The other parameters are the same as those in Fig. 5.
Moreover, we find that with an increase of the W0, the stability domain will be shrunk. When d = 1, Figs. 8(a) to 8(c) are the stability domain (μ, k) for W0 = 1.2, W0 = 2, and W0 = 2.4, respectively. Figure 8(d) is the stability domain (μ, d) for W0 = 2 When k = 1. The stability domains are shrunk when d increases. And when W0 = 1.2 and W0 = 2.4, the stability domains are also shrunk with an increase of the nonlocality degree d.
Fig. 8 Stability domain (μ, k) for (a) W0 = 1.2, (b) W0 = 2, and (c) W0 = 2.4 when d = 1. (d) stability domain (μ, d) for W0 = 2 and k = 1. The gray regions are stable domains.
Finally, we increase the value of the W0. The PT symmetric OLs with W0 = 4.2 and the band structure are shown in Figs. 9(a) and 9(b), respectively. We numerically discover that when W0 increases to a critical value (W0 = 4.0), the PT symmetry is broken. In this case, soliton solution can also be found, for example for μ = 1.0, and μ = 2.0, which are displayed in Figs. 10(a) and 10(b), respectively, but these solitons are unstable and decay quickly. The critical value of W0 separating the regions of the stable and unstable solitons has also been found in previous works [2,4,18,19].
Fig. 9 (a) PT symmetric OLs with W0 = 4.2 (the blue line is the real part, and the red line is the imaginary part). (b) The band structure corresponding to the lattice profile shown in panel (a).
Fig. 10 The soliton profiles for μ = 1.0 (a) and μ = 2.0 (b), and. (c) and (d) are their corresponding propagations. The parameters are d = 1, W0 = 4.2, and k = 1.
4. Conclusions
We investigate the existence and the stability of solitons in a parity-time symmetric potential with the spatially modulation of nonlocal nonlinearity. It is found that solitons are stable in low-power region and unstable in high-power region. The region of the stable soliton can be controlled by changing the coefficient of the spatially modulated nonlocal nonlinearity and the degree of the uniform nonlocality. In particular, for unstable case, solitons can jump from original channel into adjacent channels. In addition, when the amplitude of the imaginary part of the linear PT lattices exceeds a critical value (phase transition point), solitons are unstable and decay quickly upon propagation.
Acknowledgments
We thank Dr. Xing Zhu for his helpful discussions. This work was supported by the National Natural Science Foundation of China (Grant No.11104086) and the Natural Science Foundation of Guangdong Province of China (Grant No.S2011040001908). The work of Yingji was supported by the National Natural Science Foundation of China (Grant No. 11174061) and the Guangdong Province Natural Science Foundation of China (Grant No. S2011010005471).
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