Stability of in-phase quadruple and vortex solitons in the parity-time-symmetric periodic potentials

Xiaoping Ren; Hong Wang; Hongcheng Wang; Yingji He

doi:10.1364/OE.22.019774

1. Introduction

Vortices are fundamental objects in many branches of physics [1]. In optics, vortices are associated with the screw phase dislocations carried by diffracting optical beams [2]. When such optical vortices propagate in nonlinear media, the vortex cores become self-trapped and optical vortex solitons form [3]. In recent years, the investigations of optical waves carrying angular momentum and vortex soliton generating in optical lattices have been carried to reality because of their ability to create nonlinear waveguide arrays in 2D. The experimental observation of vortex solitons in nonlinear lattices has been reported [4]. In a periodic potentials optical lattice, the self-focusing or defocusing nonlinearity may balance the diffraction of the lattice, which can result in the stable vortex solitons. In particular, both on-site vortex solitons (vortices whose screw phase dislocation is located on a site) [5] and off-site vortex solitons (vortices whose screw phase dislocation is located between sites) [6] in an optical lattice with Kerr nonlinearity have been shown theoretically. The vortex solitons in photorefractive crystal are quickly observed in experiments [7, 8]. Recently, a theoretical work demonstrated that both on- and off-site vortex solitons can be stable with certain range of parameters in a photorefractive crystal [9].

On the other hand, light propagation in PT-symmetric potentials has attracted intensive investigations in recent years. In 1998, Bender and Boettcher showed that if the system is parity-time (PT)-symmetry, non-Hermitian Hamiltonians can have entirely real spectra [10]. In recent, the existence of stable one-dimensional (1D) and two-dimensional (2D) optical solitons in a PT-symmetric nonlinear [11] lattice and linear lattice [12, 13] have been investigated. In the context of PT-symmetry, the gap solitons in optical lattices have been investigated extensively [14–20]. Very recently, the composite vortices and surface modes in PT-symmetric photonic lattices have been reported [21]. However, the properties of vortex solitons in the PT-symmetric separable periodic optical potentials have not been studied.

Stability of vortex solitons in PT-symmetric periodic potentials is obviously an important issue. With the PT-symmetry, there are some characteristics which cannot be observed in the conservative media for vortex solitons in lattices. The study of vortex solitons in PT-symmetric periodic potentials lays the foundation for the future exploration in the interplay of PT-symmetry, nonlinearity and angular momentum. The evolution of the power along the propagation distance which is in accordance with the stability of solitons can serve as another condition to verify whether the solitons in PT-symmetric lattices are stable or not.

In this paper, we numerically investigate the stability of in-phase quadruple solitons and off-site vortex solitons with unit and double charge in the PT-symmetric periodic potentials based on defocusing Kerr nonlinearity. The interplay of angular momentum, diffraction of beams, nonlinearity, and PT-symmetry can lower the power of vortex solitons. The stable region of vortex solitons shrinks by increasing the topological charge. Furthermore, the influence of lattice depth and gain-loss component on the stability of vortex solitons has been analysed. For fixed lattice depth or fixed gain-loss component, both exists a critical value, below or above which all vortex solitons are unstable. The power evolution of solitons along the propagation distance has been studied and it is in accordance with the stability of solitons, and this is very different from the case in conservative media.

2. Theoretical model

The mathematical model for light propagation in the PT-symmetric periodic potentials with defocusing Kerr nonlinearity is the normalized nonlinear Schrödinger equation, which can be written as [22]

i U_{z} + U_{x x} + U_{y y} + V (x, y) U - {| U |}^{2} U = 0

where

U

is the slowly varying amplitude of the beam, z is the normalized longitudinal coordinate,

(x, y)

are the normalized distances along the transverse directions, and

V (x, y)

is the 2D PT-symmetric periodic potential. In particular, we take the 2D PT-symmetric periodic potential

V (x, y)

as [22]

V (x, y) = V_{0} {[\sin^{2} (x) + \sin^{2} (y)] + i W_{0} [\sin (2 x) + \sin (2 y)]},

Here $V_{0}$ is the parameter which controls the depth of the PT-symmetric optical lattice, and $W_{0}$ is the parameter corresponding to the amplitude of the imaginary part compared with the real part. The band structure of the PT-symmetric optical lattice can be calculated by the plane wave expansion method. The band structures corresponding to this separable PT-symmetric periodic potential for $V_{0} = 9$ with $W_{0} = 0.1$ and $W_{0} = 0.5$ are displayed in Figs. 1(a) and 1(b), respectively. Numerical analysis shows that the critical threshold for this potential is $W_{0}^{t h} = 0.5,$ which is identical to the case $V (x, y) = V_{0} {[\cos^{2} (x) + \cos^{2} (y)] + i W_{0} [\sin (2 x) + \sin (2 y)]}$ [22,23]. If $W_{0}$ is above 0.5, the PT-symmetric will be destroyed.

Fig. 1 The band structure of the PT-symmetric optical lattices associate with (a) $W_{0} = 0.1$ , (b) $W_{0} = 0.5$ , for all cases $V_{0} = 9$ .

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In this paper, we take $V_{0} = 9, W_{0} = 0.1$ . As shown in Fig. 1(a), the first gap is $8.331 \leq μ \leq 12.337$ , and the semi-infinite gap is $12.539 \leq μ < + \infty$ .

The lattice solitons in Eq. (1) are sought in the form of $U = u (x, y) \exp (i μ z)$ , Where $μ$ is the real propagation constant, $u (x, y)$ is a complex-value localized function. Thus, the function $u (x, y)$ satisfies the following equation

u_{x x} + u_{y y} + V (x, y) u - {| u |}^{2} u - μ u = 0

These soliton solutions are resolved numerically by using the modified squared-operator iteration method [24]. Define $P = {\int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} | u |}^{2} d x d y$ as the power of a soliton [22].

In order to investigate the linear stability of these solitons, we add the perturbations $p (x, y)$ and $q (x, y)$ to the solution, which is written as:

U (x, y, z) = \exp (i μ z) {u (x, y) + [p (x, y) - q (x, y)] \exp (λ z) + {[p (x, y) + q (x, y)]}^{*} \exp (λ^{*} z)}

where

| p | < < | u |, | q | < < | u |

. By substituting

U (x, y, z)

into Eq. (1) and linearizing, we can obtain the following coupled equations:

{\begin{matrix} λ p = - i [- i V_{0} W_{0} p + \frac{1}{2} u^{2} p - \frac{1}{2} {(u^{*})}^{2} p - μ q + q_{x x} + q_{y y} + V_{0} q - 2 {| u |}^{2} q + \frac{1}{2} u^{2} q + \frac{1}{2} {(u^{*})}^{2} q] \\ λ q = - i [- μ p + p_{x x} + p_{y y} + V_{0} p - 2 {| u |}^{2} p - \frac{1}{2} u^{2} p - \frac{1}{2} {(u^{*})}^{2} p - i V_{0} W_{0} q - \frac{1}{2} u^{2} q + \frac{1}{2} {(u^{*})}^{2} q] \end{matrix}

Equation (3) can be solved by the Fourier collocation method [25]. If the complex value $λ$ with $Re (λ) > 0$ , solitons are linearly unstable; Otherwise, they are linearly stable.

3. Numerical results

3.1 In-phase quadruple solitons

First, we investigate the in-phase quadruple solitons (i.e. $m = 0$ ). For this kind of solitons, there are four intensity peaks which are in phase with each other. The profile of real part is shown in Fig. 3(a). We find that it can exist in the first gap. Its power diagram versus the propagation constant $μ$ is displayed in Fig. 2 (red line). It can be stable in a wide region of first gap ( $11.330 \leq μ \leq 12.337$ ).

Fig. 2 Power P of in-phase quadruple solitons (red line) and vortex solitons with unit (blue line) and double charge (green line) versus propagation constant $μ$ for $V_{0} = 9, W_{0} = 0.1$ . The stable branches are plotted by solid curves and the unstable branches are plotted by dashed curves. The black shaded region is the first Bloch bands. The inset shows power curves near the Bloch band.

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In order to investigate the longitudinal evolution of this soliton, the robustness of the soliton propagation is numerically simulated by adding a random noise (5% of the soliton amplitude) to solitons. We take $μ = 11.96$ as a stable case, the evolutions of the soliton at $z = 0$ and $z = 500$ are displayed in Figs. 3(b) and 3(d). To demonstrate, solving Eq. (3) numerically, we can get the linear-stability spectrum of the soliton, which is shown in Fig. 3(c). It is shown that there is not any unstable eigenvalues in its spectrum, thus the soliton is stable.

Fig. 3 (a) The profile of the real part of the in-phase quadruple soliton. (b) and (d) are the profiles ( $| u |$ ) of the perturbed in-phase quadruple soliton for $μ = 11.96$ at $z = 0$ and $z = 500$ , respectively. (c) The corresponding linear-stability spectrum.

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In the high power regime, the soliton will become unstable. To demonstrate, we take $μ = 10.90$ as an example. During the propagation, the soliton will not maintain its original shape at $z = 300$ [Fig. 4(b)]. The corresponding linear-stability spectrum is shown in Fig. 4(c). It is shown that the soliton contains a quadruple of complex eigenvalues. It will suffer oscillatory (the unstable eigenvalues are complex) instability seriously.

Fig. 4 The profile of in-phase quadruple soliton for $μ = 10.90$ , (a) $z = 0$ (b) $z = 300$ . (c) Linear-stability spectrum.

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3.2 Vortex solitons

In this section, we first study the vortex solitons with unit charge (i.e. topological charge $m = \pm 1$ , where $m = 1$ indicates the direction of the angular momentum of vortex solitons is counterclockwise while $m = - 1$ corresponds to clockwise direction). We learn that the vortex solitons with topological charge $m = \pm 1$ have the same properties, so we discuss vortex solitons with topological charge $m = 1$ only hereafter. We find that one type of unit charge vortex solitons has four intensity peaks locating at four adjacent lattice sites and forming a compact square configuration. Its profile of real part is shown in Fig. 5(a). And Fig. 5(b) is its phase structure. Its power curve is shown in Fig. 2 (blue line).

Fig. 5 (a) The profile of real part of unit charge vortex soliton, (b) the corresponding phase structure.

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From Fig. 2 (blue line), we see that unit charge vortex solitons also exists in the first gap. It is obvious that this type of vortex soliton cannot bifurcate from the Bloch band and its power curve has a local extremum near the first Bloch band. It is different from the in-phase quadruple soliton, which can bifurcate from the Bloch band and whose power increases monotonically with the decrease of propagation constant. The unit charge vortex solitons can be stable in the lower power regime ( $12.21 \leq μ \leq 12.31$ ) on the side of the power minimum point away from the first Bloch band. Apparently, the stable region of in-phase quadruple soliton is larger than that of the vortex solitons. It is reason that the vortex solitons carry the angular momentum, and will suffer the azimuthal modulation instability during propagation. By comparing their power curves, we find that the in-phase quadruple soliton have the higher power than that of the vortex solitons at the same propagation constant. It is indicated that the interaction of angular momentum, the diffraction of beams, and the nonlinearity in the PT-symmetric periodic potentials can lower the power of vortex solitons.

In addition, we also study the double charge vortex solitons (i.e. $m = 2$ ). It is found that the vortex solitons with double charge also exist in the first gap and only can be stable in a very small region $12.315 \leq μ \leq 12.325$ . Its power curve is shown in Fig. 2 with green line. From Fig. 2, we learn that the power of vortex solitons decreases with the increase of its topological charge, and its stability region shrinks by increasing the topological charge of vortex solitons. This further explains that the interplay of angular momentum, the diffraction of beams, and nonlinearity in the PT-symmetric lattices can lower the power of vortex solitons and the angular momentum has a great effect on the stability of vortex solitons.

To determine the stability of unit charge vortex solitons, we also add a 5% random noise and simulate the propagation by numerical method. The evolution of a stable case ( $μ = 12.28$ ) at $z = 0$ and $z = 200$ are shown in Figs. 6(a) and 6(b). Figures 6(d) and 6(e) is corresponding to an unstable case ( $μ = 11.50$ ), where Figs. 6(d) and 6(e) are the evolutions of soliton at $z = 0$ and $z = 30$ , respectively. It is shown that the vortex soliton will be out of shape after a short distance when it is unstable. The linear-stability spectrums for stable and unstable cases are displayed in Figs. 6(e) and 6(f), respectively. From Figs. 6(c) and 6(f), we see that the linear-stability spectrum contains none unstable eigenvalues for stable case, and thus it can propagate stably. While for unstable case $μ = 11.50$ , the linear-stability spectrum has two couples of real eigenvalues.

Fig. 6 The evolution of unit charge vortex solitons, the first row is for a stable case ( $μ = 12.28$ ). The second row is for an unstable case ( $μ = 11.50$ ). The last row is the corresponding linear-stability spectrum.

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In order to further explain the problem, the vortex solitons’ power evolution versus normalized longitudinal coordinate $z$ is plotted in Figs. 7(a) and 7(b) for unit charge vortex solitons with stable case $μ = 12.28$ and unstable case $μ = 11.50$ , respectively.

Fig. 7 The evolution of unit charge vortex solitons’ power versus the propagation distance, (a) is the stable case $μ = 12.28$ . (b) is for the unstable case $μ = 11.50$ .

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In Fig. 7(a), when the solitons are stable, the variation of the power versus the propagation distance is so small that it can hardly observed. However, for unstable cases, as shown in Fig. 7(b), the oscillations of the power occur, which mainly due to its linear-stability spectrum having two couples of real eigenvalues and its symmetry is broken at $z = 30$ [Fig. 6(e)]. The relation between the power evolution and longitudinal coordinate is consistent with the results of their stability discussed periviously. This property is very different from the case in the conservative media. Indeed, in the conservative media, the power of vortex solitons will remain unchanged during propagation no matter it is stable or not. Take $V_{0} = 6, W_{0} = 0, μ = 7.40$ (unstable) as an example, the evolution of vortex soliton at $z = 0$ and $z = 130$ are shown in Figs. 8(a) and 8(b), respectively. Figures 8(c) and 8(d) are the corresponding power evolution along propagation distance and linear-stability spectrum. From Fig. 8, we learn that the vortex solitons is unstable and its linear-stability spectrum contains a quadruple of complex eigenvalues. However, its power has no change during propagation.

Fig. 8 The evolution of unit charge vortex soliton with $V_{0} = 6, W_{0} = 0, μ = 7.40$ , (a) and (b) are evolution of unit charge vortex soliton at $z = 0$ and $z = 130$ ,respectively. (c) is its power evolution along the propagation distance and (d) is the corresponding linear-stability spectrum.

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The stability region of vortex solitons shrinks with increasing the gain-loss component. In particular, numerically analysis shows when $V_{0} = 9, W_{0} = 0.2$ , the stable region is $11.89 \leq μ \leq 11.91$ . It is much smaller than the case $W_{0} = 0.1$ . In addition, for a fix lattice depth $V_{0} = 9$ , there exists a critical value $W_{0} = 0.3$ , above which all vortex solitons will be unstable. Thus the gain-loss component has a great effect on the stability of vortex solitons. The stability domains of vortex solitons on $W_{0}$ values are presented in Fig. 9. Obviously, the stability domain of vortex solitons shrinks with increasing the gain-loss component $W_{0}$ and vanishes when $W_{0} \leq 0.3$ .

Fig. 9 The stability domain of unit charge vortex solitons on $W_{0}$ values. The blue shade region is the stability domain of unit charge vortex solitons. Fixed $V_{0} = 9$ .

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Furthermore, when decreasing the lattice depth, the stable region of vortex solitons also shrinks. Indeed, at the shallow lattice depth $V_{0} = 7$ , the vortex soliton cannot be stable in its whole exist region with either $W_{0} = 0.2$ or $W_{0} = 0.1$ . This means there exist a critical lattice depth, below which the stable region of vortex soliton disappears. In addition, the imaginary part of the potential has an effect on the phase of vortex soliton regardless of the lattice depth, which is shown in Fig. 10. We see that the phase of vortex soliton is related to the gain-loss component of the lattice.

Fig. 10 The phase of unit charge vortex soliton, the first row is for $W_{0} = 0.1$ and the second for $W_{0} = 0.2$ . The first column is for $V_{0} = 9$ and the second for $V_{0} = 7$ .

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In order to elucidate the phenomenon clearly, we plot the real, imaginary part and phase of unit charge vortex solitons with $V_{0} = 9, W_{0} = 0.1$ and $W_{0} = 0.4$ in Fig. 11. Comparing with these two cases, we see that when $W_{0} = 0.4$ , there are many tails around every intensity peak of both its real and imaginary part, and its phase is fuzzy.

Fig. 11 The real, imaginary part and phase of unit charge vortex solitons, the first row is for $V_{0} = 9, W_{0} = 0.1$ and the second for $V_{0} = 9, W_{0} = 0.4$ .

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4. Conclusion

We investigate the stability of in-phase quadruple and off-site vortex solitons in the PT-symmetric optical lattice with defocusing Kerr nonlinearity numerically. It is shown that they exist in the first gap. By comparing their power and stability region, we find that the interplay of angular momentum, diffraction of beams, nonlinearity and PT-symmetry can lower the power of vortex solitons and the angular momentum has a great effect on the stability of vortex solitons. Especially for double charge vortex solitons, its stable region is very small. By studying their power evolution along the propagation distance, we show that their power evolution versus propagation distance is associated with their stability. This is different from the cases in the conservative media. We also reveal that decreasing the lattice depth or increasing the gain-loss component of the potential shrinks the stability region of vortex soliton; and for both there exist a critical value, below or above which all vortex solitons will become unstable. The gain-loss component also has an influence on the phase of vortex solitons.

Acknowledgments

This work was supported by the Strategic Emerging Industry Special funds of Guangdong Province (Nos. 2011A081301004, 2012A080302003, 2012A080304015), and the Fundamental Research Funds for the Central Universities (No. 2013ZP0017), and the Key Technologies R&D Program of Guangzhou City (No. 2011Y5-00006).

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Stability of in-phase quadruple and vortex solitons in the parity-time-symmetric periodic potentials

Abstract

1. Introduction

2. Theoretical model

3. Numerical results

3.1 In-phase quadruple solitons

3.2 Vortex solitons

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (11)

Equations (5)

Optics Express