Two-dimensional solitons in parity-time-symmetric optical lattices with nonlocal defocusing nonlinearity

Hong Wang; Zhen Li; Xiaoping Ren; Yuanhang Weng

doi:10.1364/OE.24.023063

1. Introduction

Spatial soliton is a phenomenon occurred in the nonlinear optics, and it is a wave packet that could maintain its profile during propagation and has drawn a lot of attention due to its potential application in all-optical communication and optical switch [1]. It is obvious that the properties of the solitons can be significantly affected by a particular model of the nonlinear response of the medium. Nonlocality is associated with a variety of optical materials, such as nematic liquid crystals, thermo-optic materials, atomic vapors, and photorefractive crystals [2–4]. Nonlinear response at a certain point in nonlocal medium is induced by its surrounding regions rather than a single point, which is quite different from the Kerr (local) medium. This character can bring some new features to solitons, such as suppression of modulation instability, arrest of catastrophic collapse of multidimentional beams, stabilization of complex solitons [5–8] etc.

Parity-time symmetry is another research focus since Bender and Boettcher found in 1998 that non-Hermitian Hamiltonians can also have entirely real spectra [9]. To achieve PT symmetry, the potential of a system must meet condition $V (x, y) = V^{*} (- x, - y)$ , that is, the real part of this complex potential is an even function while the imaginary part is an odd one. Such systems are appealing from both fundamental and practical perspectives [10–12]. In optics, this system has drew much more attentions because of the ease of realizing complex Hamiltonians [13], and the experiments based on theoretical findings have become possible. In optical lattices, the imaginary part of PT symmetric potential can be achieved by adding gain and loss [14,15]. There are a lot of exotic properties accompanied with PT symmetry including double refraction, nonreciprocal behavior and PT symmetry breaking [14,16–19] etc. Very recently, a comprehensive review on nonlinear waves in PT-symmetric systems has been reported [20]. In addition, some overview works on the unique dynamics of spatial solitons in PT-symmetric optical lattices have also been done [21, 22]. The solitons in both one- and two-dimensional PT-symmetric mixed linear-nonlinear optical lattices have also been reported [23,24], and it has been found that the nonlinear lattices play an important role on the dynamics of solitons.

In theoretical work, solitons in the nonlocal optical lattice have been studied as well, however, most studies are focused on one-dimensional solitons [25–27], the nonlocal soltions’ stability and mobility in two dimensional is still an open issue. In this paper, we investigated the existence and stability of 2D solitons in PT symmetric optical lattices with the nonlocal nonlinearity, and found that PT symmetry can have great impact on the dynamics of solitons. And we also indicated how the degree of nonlocality affects the power and the stability of solitons.

2. Theoretical model

The beam propagation in the 2D PT-symmetric optical lattice with defocusing nonlocal nonlinearity can be described by the following normalized 2D nonlinear Schrodinger coupled equations:

\begin{array}{l} i \frac{\partial U}{\partial z} + \frac{\partial^{2} U}{\partial x^{2}} + \frac{\partial^{2} U}{\partial y^{2}} + V (x, y) U + σ n U = 0, \\ d \frac{\partial^{2} n}{\partial x^{2}} - n + | U |^{2} = 0, \end{array}

where

U

is the complex electric field envelop and

n

is nonlinear refractive index,

d

indicates the degree of nonlocality,

z

is the scaled propagation distance,

x, y

are the scaled transverse coordinates, and

σ = - 1

represents defocusing nonlinearity.

V (x, y)

represents the PT-symmetric optical lattice, We set potential as

V (x, y) = V_{0} (R + i W_{0} I)

, where

R (x, y) = \sin^{2} (x) + \sin^{2} (y)

is the real part of PT symmetric potential and

I (x, y) = \sin (2 x) + \sin (2 y)

is the imaginary part,

V_{0}

controls the depth of the PT-symmetric lattice and

W_{0}

indicates the strength of the imaginary part of the potential. We choose

V_{0} = - 10

and

W_{0} = 0.1

in our discussion.

Bloch bands will emerge in the periodic optical lattice, and solitons can exist in the band gaps. According to the Bloch theory, we can write the Bloch modes as $U = F (x, y) e^{i μ z} e^{i (k_{x} x + k_{y} y)}$ . Here, $μ$ is the propagation constant, $F (x, y)$ is a periodic function with the same period of the optical lattice, $k_{x}$ and $k_{y}$ are the Bloch wave numbers along to the $x, y$ axis, respectively. Substituting $U = F (x, y) e^{i μ z} e^{i (k_{x} x + k_{y} y)}$ into the linear version of Eq. (1), we can get the following eigenvalue equation:

μ F - (\frac{\partial^{2}}{\partial x^{2}} + 2 i k_{x} \frac{\partial}{\partial x} - k_{x}^{2}) F - (\frac{\partial^{2}}{\partial y^{2}} + 2 i k_{y} \frac{\partial}{\partial y} - k_{y}^{2}) F - V F = 0

By solving the above equation with the plane wave expansion method, we finally get the band structure of the optical lattice, as depicted in Fig. 1. According to our calculation, the first gap is $- 10.4 \leq μ \leq - 5.9$ , and the semi-infinite gap is $- 5.8 \leq μ$ .

Fig. 1 The band structure of the PT-symmetric optical lattice for $V_{0} = 10$ and $W_{0} = 0.1$ , zone 1 is the semi-infinite band gap and zone 2 is the first band gap.

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Soliton solutions of Eq. (1) are sought in the form $U (x, y, z) = u (x, y) e^{i μ z}$ . Here $u (x, y)$ is a complex periodic function and $μ$ is the propagation constant. Hence the Eq. (1) can be reduced as:

\begin{array}{l} - μ u + (\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}) u + V u + σ n u = 0 \\ d (\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}) u - n + | u |^{2} = 0 \end{array}

Equation (3) can be solved by a developed modified squared-operator method [28]. And the power of solitons is defined as $P = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} | u |^{2} d x d y$

There are three types nonlocality in our discussion: weak-nonlocal, general-nonlocal and strong nonlocal. Types of the nonlocality are classified by the relative width between light beam and nonlocal response. Weak-nonlocal means that the width of the nonlocal response is much less than the width of the light beam, and general-nonlocal means that the width of the nonlocal response is in the same magnitude with the light beam, and when the width of the nonlocal response is much greater than the light beam width, the strong-nonlocal condition is satisfied. The 2D nonlocal response described by $g (x, y) = (1 / 2 π \sqrt{d}) \exp (- \sqrt{x^{2} + y^{2}} / \sqrt{d})$ , the two extreme situations are: $d \to 0$ represents a local system, and $d \to \infty$ represents a highly strong nonlocal one. Figure 2 shows that three different types nonlocality [Figs. 2(b), 2(c), 2(d)] under different value of $d = 0.01, 0.5, 5$ . And the corresponding refractive index n are depicted in Fig. 3.

Fig. 2 The corresponding diagrams of soliton and nonlocal response profiles with degree of nonlocality (a) $d = 0.01$ (weak-nonlocal), (b) $d = 0.5$ (general-nonlocal), (c) $d = 5$ (strong-nonlocal). The blue lines represent soliton profiles and the red lines represent nonlocal response profiles.

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Fig. 3 (a), (b), (c) are the profiles of the corresponding refractive index of the different degree of nonlocality [Figs. 2(a), 2(b), 2(c)], respectively.

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Stability is an important characterization of the soliton dynamics. In order to investigate the stability, we add a small perturbation $(F, G)$ into the stationary solution [28]:

U (x, y, z) = e^{i μ z} [u (x, y) + F (x, y) e^{λ z} + G^{*} (x, y) e^{λ^{*} z}]

Where $| F |, | G | < < 1$ , $λ$ is the complex-valued instability growth rate, and the superscript $*$ is the complex conjugation. Substituting Eq. (4) into Eq. (1) and linearizing, we can obtain the following eigenmode equations:

\begin{array}{l} λ F = i [- μ F + (\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial x^{2}}) F + (V + i W) F + σ n F + σ u Δ n] \\ λ G = i [μ G - (\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial x^{2}}) G - (V - i W) G - σ n F - σ u^{*} Δ n] \end{array}

Where

\begin{matrix} n = \int_{- \infty}^{\infty} g (x - x^{'}, y - y^{'}) | q (x^{'}, y^{'}) |^{2} d x^{'} d y^{'} \\ Δ n = \int_{- \infty}^{\infty} g (x - x^{'}, y - y^{'}) [u (x^{'}, y^{'}) G (x^{'}, y^{'}) + u^{*} (x^{'}, y^{'}) F (x^{'}, y^{'})] d x^{'} d y^{'} \\ g (x, y) = (1 / 2 π \sqrt{d}) \exp (- \sqrt{x^{2} + y^{2}} / \sqrt{d}) \end{matrix}

Equation (5) can be solved numerically [28]. Soliton is linearly unstable if the real part of $λ$ is nonzero ( $Re (λ) > 0$ ), otherwise, it is linearly stable.

3. Numerical analysis

We studied the properties of solitons in PT symmetric optical lattice with different degree of defocusing nonlocality( $d = 0.01, d = 0.5, d = 5$ ) nonlinearity. In this case, solitons can be found in the first band gap only. The relationships between the solitons power and propagation constants for both PT-symmetric and conservative systems are illustrated in Figs. 4(a) and 4(b), respectively. As we can see, the power of solitons decreased with the increase of the propagation constant in both PT-symmetric and conservative systems. The stable and unstable regions of soliton are also indicated in Fig. 4 by red and other colorful lines, respectively. It is obvious that the solitons in the conservative system are more robust than the solitons in the PT-symmetric potentials. Furthermore, the degree of nonlocality has little effect on the stability of solitons in the conservative system.

Fig. 4 The power diagrams for $d = 0.01$ (yellow lines), $d = 0.5$ (green lines), $d = 5$ (blue lines) with (a) $W_{0} = 0.1$ ,the solid and dashed lines indicate stable and unstable regions, respectively. (b) $W_{0} = 0$ , red lines indicate the solitons are unstable and blue, green and yellow lines indicate the solitons are stable. The blue shaded regions are the Bloch bands.

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For weak nonlocality ( $d = 0.01$ ), solitons are stable at $- 6.2 < μ < - 6.1$ and $- 10.2 < μ < - 6.1$ for $W_{0} = 0.1$ and $W_{0} = 0$ , respectively. Figure 5 shows the propagating dynamics of $μ = - 6.1$ (first row) and $μ = 8.6$ (second row), corresponding points B and A in Fig. 4(a), respectively. At $μ = 6.1$ , soliton remains its original profile after propagating 400 units, as shown in Figs. 5(a) and 5(b). Figure 5(c) is the growth rate of this soliton, and we can see that the real part of the growth rate is zero, which indicates that the soliton is stable. At $μ = - 8.6$ , however, soliton diffracts significantly at $z = 400$ , see Figs. 5(d) and 5(e). This instability is also verified by the result of the growth rate, as shown in Fig. 5(f).

Fig. 5 (a) is the soliton profile for $μ = - 6.1$ at $z = 0$ , (b) is the soliton for $μ = - 6.1$ at $z = 400$ , (c) indicates the growth rate for $μ = - 6.1$ , and (d) is the soliton profile for $μ = - 8.6$ at $z = 0$ , (e) is the soliton for $μ = - 8.6$ at $z = 400$ , (f) indicates the growth rate for $μ = - 8.6$ .

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Next, we studied the properties of solitons in PT optical lattice with general nonlocality ( $d = 0.5$ ). The relationship between the solitons power and propagation constants is also illustrated in Fig. 4. The stable region for $d = 0.5$ with $W_{0} = 0.1$ is $- 6.9 < μ < - 6.3$ . Compared with $d = 0.01$ [Fig. 4(a)], this stable region is much wider. The stable region with $W_{0} = 0$ is $- 10.2 < μ < - 6.0$ . It is much wider than the stable region of solitons in PT-symmetric potentials. Figure 6 shows the evolution of solitons at $μ = - 6.4$ (first row) and $μ = - 8.4$ (second row),corresponding point D and C in Fig. 4(a). For $μ = - 6.4$ , soliton remains its original shape after propagating 400 units, see Figs. 6(a) and 6(b). Figure 6(c) shows the growth rate of that point. For $μ = - 8.4$ , unlike the situation in real lattice, soliton doesn’t diffract after propagating for 400 units distance, instead it splits and shifts. We can see that from Figs. 6(d) and 6(e), after traveling for 400 units, the soliton deviated from its initial position in terms of transverse coordinates. Figure 6(f) is the growth rate of that soliton, and it indicates soliton is unstable.

Fig. 6 (a) is the soliton profile for $μ = - 6.4$ at $z = 0$ , (b) is the soliton for $μ = - 6.4$ at $z = 400$ , (c) indicates the growth rate for $μ = - 6.4$ , and (d) is the soliton profile for $μ = - 8.4$ at $z = 0$ , (e) is the soliton for $μ = - 8.4$ at $z = 400$ , (f) indicates the growth rate for $μ = - 8.4$ .

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Then we studied the soliton’s properties when nonlocality $d = 5$ . The stable region of soliton in PT-symmetric lattice with $W_{0} = 0.1$ is $- 6.2 < μ < - 6.1$ . And the stable region of soliton in conservative system is $- 10.1 < μ < - 5.9$ . This indicate the solitons in the conservative system are more robust. Figure 7 shows solitons’ evolution at $μ = - 6.2$ (first row) and $μ = - 7.2$ (second row),corresponding point F and E in Fig. 4(a). Soliton at $μ = - 6.2$ is stable after propagating 400 units seen as Figs. 7(a) and 7(b). Figure 7(c) shows the real part of growth rate of this soliton is zero. Solitons at $μ = - 7.2$ will experience splitting and shifting. This phenomenon is shown in Figs. 7(d) and 7(e). And the result of the growth rate shows that soliton is unstable, see Fig. 7(f).

Fig. 7 (a) is the soliton profile for $μ = - 6. 2$ at $z = 0$ , (b) is the soliton for $μ = - 6.2$ at $z = 400$ , (c) indicates the growth rate for $μ = - 6.2$ , and (d) is the soliton profile for $μ = - 7.2$ at $z = 0$ , (e) is the soliton for $μ = - 7.2$ at $z = 400$ , (f) indicates the growth rate for $μ = - 7.2$ .

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We also investigate how the degree of nonlocality and how the different $W_{0}$ affect the solitons’ stable region, respectively, as shown in Fig. 8. Figure 8(a) shows how the degree of nonlocality affects the stability of solitons, we can see that the stable region decreases after an increase at first. Figure 8(b) shows that the stable region constantly shrinks with the increase of the strength of the imaginary part of PT symmetric lattice potential.

Fig. 8 (a) shows the stable region of solitons with the change of degree of nonlocality, and (b) shows the stable region of solitons with the change of $W_{0}$ , the shaded region represents stable region.

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Finally, we studied factors that affect soliton’s split and shift when $W_{0} = 0.1$ . We found that, the stronger the nonlocality is, the easier for the soliton to experience splitting and shifting during propagation. Figure 9 shows the states of soltion ( $μ = - 7$ ) at different propagation distance for different nonlocality. The first row is for $d = 1$ , we can see that at $z = 200$ , soliton maintained its profile; at $z = 300$ , the soliton tended to split; and at $z = 400$ , the soliton split and shifted. The second row is for $d = 5$ , we can see that at $z = 200$ , the soliton already tended to split; and at $z = 300$ and afterward, the soliton experienced splitting and shifting. The third row is for $d = 10$ , the soliton already experienced splitting and shifting at $z = 200$ , and this kind of split and shift strengthened afterwards. This shows the shift of soliton during the evolution can be influenced significantly by the nonlocality.

Fig. 9 The first column are the soliton profiles of $μ = - 7$ for $d = 1, 5, 10$ at $z = 200$ , respectively, the second column are the soliton profiles of $μ = - 7$ for $d = 1, 5, 10$ at $z = 300$ , respectively, the third column are the soliton profiles of $μ = - 7$ for $d = 1, 5, 10$ at $z = 400$ , respectively.

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4. Conclusions

We investigated the evolution of soliton in defocusing nonlocal PT symmetric optical lattice. It is found that solitons can only exist in the first band gap, and the stable region of solitons first increases and then decreases with the increase of the degree of nonlocality. And the stable region decreases with the increase of imaginary part of the lattice potential. The soliton have some bizarre behavior when propagating in the PT symmetric optical lattice with a relatively higher degree of nonlocality, such as soliton split and shift, which is easier to emerge with a relatively higher degree of nonlocality.

Funding

The Strategic Emerging Industry Special funds program of Guangdong Province (Nos. 2012A080302003, 2012A080304015), the Key technologies R&D Program of Guangdong Province(No. 2014B010119002), and the Applied technologies R&D Projects of Guangdong Province (No. 2015B010127013).

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Two-dimensional solitons in parity-time-symmetric optical lattices with nonlocal defocusing nonlinearity

Abstract

1. Introduction

2. Theoretical model

3. Numerical analysis

4. Conclusions

Funding

References and links

Cited By

Figures (9)

Equations (6)

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