## Abstract

We report the evolution of higher-order nonlinear states in a focusing cubic medium, where both the linear refractive index and the nonlinearity are spatially modulated by a complex optical lattice exhibiting a parity-time (𝒫𝒯) symmetry. We reveal that introduction of out-of-phase nonlinearity modulation makes possible the stabilization of higher-order solitons with number of poles up to 7, which are highly unstable in linear 𝒫𝒯 lattices. Under appropriate conditions, multipole-mode solitons with out-of-phase components in the neighboring lattice sites are completely stable provided that their power or propagation constant exceeds a critical value. Thus, our findings suggest an effective way for the realization of stable multipole-mode solitons in periodic potentials with gain-loss components.

© 2013 Optical Society of America

## 1. Introduction

Since the experimental realization of optical lattices [1], dynamics of lattice solitons have been intensively studied. For a review of early works, see e.g., [2] and references therein. Recent findings demonstrated that the transverse nonlinearity modulation of optical materials can substantially affect the existence conditions and stability properties of solitons [3]. Various types of one-dimensional solitons (in the form of odd, even, multipole, and vector solitons), two-dimensional solitons (in the form of fundamental, multipole, and vortex solitons), and three-dimensional solitons (in the form of light-bullets and optical tandems) were predicted in mixed linear-nonlinear lattices [4–6] or in purely nonlinear lattices [7].

Dynamics of nonlinear waves in media modulated by 𝒫𝒯 lattices has attracted considerable interest because 𝒫𝒯–symmetric non-Hermitian Hamiltonian can display entirely real spectra [8–16]. Solitons in periodic or nonperiodic potentials with gain-loss components were reported in a variety of forms, including, e.g., fundamental [8, 9], dipole [17, 18], multi-stable [19] and defect [20–23] solitons. Two-dimensional bright solitons in focusing/defocusing Kerr media [18, 24] were predicted. In addition to the above progress, solitons in nonlinear 𝒫𝒯 lattices [17, 25] were also investigated. Although solitons in *in-phase* mixed linear-nonlinear 𝒫𝒯 lattices were studied in [26], the discussions were limited to the fundamental solitons. Moreover, stable fundamental states can be found only at lower power.

Thus far, most studies of solitons in 𝒫𝒯 lattices have focused on fundamental states, with few exceptions reported in Refs. [17, 18, 23], where dipole modes were addressed. In linear 𝒫𝒯 lattices, the stability region of multipole-mode solitons shrinks rapidly with the increase of pole number. No stable solitons with pole number larger than 3 were found.

In bulk or lattice-modulated nonlinear media, to support stable multipole-mode solitons, one usually needs media with a saturable, competing or nonlocal nonlinearity [27, 28]. Considering the fact that, in purely real lattices, the competition between the out-of-phase linear and nonlinear lattices usually favor the stabilization of solitons in them [3], it is natural to ask: Whether one can use the out-of-phase mixed linear-nonlinear 𝒫𝒯 lattices to suppress the instability of higher-order solitons?

To answer this question, we investigate the existence, stability and propagation dynamics of multipole-mode solitons in focusing Kerr media modulated by out-of-phase mixed linear-nonlinear 𝒫𝒯 lattices. A variety of families of soliton solutions with a different number of poles are found. Linear stability analysis corroborated by direct propagation simulations reveals that suitable modulation of out-of-phase nonlinearity can remarkably suppress the instability of lattice solitons, either for fundamental states or for higher-order states. Fundamental solitons can propagate stably in their entire existence domain under appropriate conditions. Multipole-mode solitons with out-of-phase soliton components are completely stable provided that their power or propagation constant exceeds a critical value. Stable higher-order solitons with 7 poles were found. The excitation of multipole-mode solitons by a tilted input Gaussian beam and the splitting of solitons into their soliton constituents are also addressed. Our findings, thus, provide the first example of stable multipole-mode solitons in 𝒫𝒯-symmetric lattices.

## 2. Theoretical model

Beam propagation in the focusing Kerr-type nonlinear media with mixed linear-nonlinear lattices can be described by the dimensionless nonlinear Schrödinger equation [3]:

*q*is the complex field amplitude;

*x*and

*z*are the normalized transverse and longitudinal coordinates, respectively;

*p*and

*σ*denote the depths of modulation of refractive index and nonlinearity, respectively; the refractive index profile obeys:

*R*(

*x*) =

*V*(

*x*) +

*iW*(

*x*) = cos

^{2}(Ω

*x*/2)+

*iχ*sin(Ω

*x*), where Ω is the frequency of lattice and

*χ*is the relative magnitude of gain-loss component.

*R*(

*x*) satisfies the 𝒫𝒯 symmetry, i.e.,

*V*(−

*x*) =

*V*(

*x*) and

*W*(−

*x*) = −

*W*(

*x*). The linear modulation of refractive index with a 𝒫𝒯 symmetry has been realized in experiment by Rüter

*et al.*[11]. The out-of-phase nonlinearity modulation may be implemented by means of proper spatial modulation of nonlinear gain and losses. The optical system described by Eq. (1) may be realized in nonlinear waveguides by employing concatenated semiconductor optical amplifier and semiconductor-doped two-photon absorber sections [17]. Very recently, a non-Hamiltonian system with both linear and nonlinear potentials satisfying a 𝒫𝒯 symmetry was realized experimentally in the frame of electronics [29].

We search for stationary solutions of Eq. (1) in the form of *q*(*x,z*) = *u*(*x*)exp(*ibz*), which can be characterized by the complex soliton profile *u*(*x*) and the propagation constant *b*. After substituting *q*(*x,z*) into Eq. (1), one obtains the following nonlinear differential equation:

*b*, relative magnitude

*χ*, nonlinearity-modulation depth

*σ*, lattice depth

*p*, and modulation frequency Ω. Through the rest of this paper, we are going to use a specific configuration by setting

*p*≡ 4 and Ω ≡ 4. Note that the lattice period can obviously be rescaled. When the linear lattice is deep (for large

*p*), the nonlinearity modulation can be ignored and the system degenerates into the case reported in [8, 18], where stable fundamental and dipole solitons were found. On the other hand, if the linear lattice is very shallow (for small

*p*), the system is similar to the model proposed by Fatkhulla

*et al.*[17], in which the evolution of solitons in focusing cubic media modulated by a nonlinear lattice with a 𝒫𝒯 symmetry was discussed. However, the dynamics of higher-order solitons are very different since the sign before the nonlinearity modulation is opposite. The properties of solitons in weak linear lattice will be discussed later. The power of a soliton can be defined as $P\left(z\right)={P}_{r}\left(z\right)+{P}_{i}\left(z\right)={\int}_{-\infty}^{\infty}{\left|q\left(x,z\right)\right|}^{2}dx$[8,9]. The subscripts

*r*and

*i*here and below stand for the real and imaginary parts.

The stability of solitons in a system described by the nonlinear Schrödinger equation can be analyzed by considering small perturbations on the stationary solutions in the form: *q*(*x,z*) = {*u*(*x*) + [*v*(*x*) − *w*(*x*)]exp(*λz*) + [*v*(*x*) + *w*(*x*)]^{*} exp(*λ*^{*}*z*)}exp(*ibz*), where *v*, *w* ≪ 1 are the infinitesimal perturbations, and superscript ^{*} represents the complex conjugation. Substituting the perturbed solution into Eq. (1) and linearizing it around *u* yields an eigenvalue problem:

The coupled equations can be solved numerically by a finite-difference method. Stability of a soliton is determined by the spectrum of the above linearization operator. Solitons can

## 3. Multipole-mode solitons and their stabilities

Due to the periodicity of linear lattice, the existence of solitons is confined by the Floquet-Bloch spectrum of the complex potential. We show the Band-gap structures of the linear version of Eq. (1) in Fig. 1(a). There exists a critical value *χ* = 0.5 (phase transition point), above which the eigenvalue spectra become complex, and linear waves amplify exponentially during propagation. Thus, any solitons would also be unstable to perturbations. In the following discussions, we will focus on the properties of solitons in the semi-infinite gap of 𝒫𝒯 lattice with *χ* < 0.5. We first consider a special configuration with the relative magnitude of gain-loss component *χ* = 0.2 and nonlinearity-modulation depth *σ* = 0.2. The influence of *χ* and *σ* on the stability of solitons will be addressed later.

Before we discuss the evolution of higher-order solitons, it is instructive to understand the dynamics of fundamental states in out-of-phase mixed linear-nonlinear 𝒫𝒯 lattices. We summarize the properties of fundamental solitons in Fig. 1. At fixed *χ* and *σ*, power *P* increases monotonically with the growth of propagation constant *b*. There exists a lower cutoff of propagation constant(*b _{co}* = 2.24), below which no soliton solutions can be found [Fig. 1(b)]. The amplitude ratio between the imaginary and real parts is a nonmonotonic function of propagation constant. It increases at small

*b*and reaches a maximum, afterwards, it decreases with the growth of

*b*.

Two typical soliton profiles are shown in Figs. 1(c) and 1(d). As *b* grows, the peak of soliton increases and soliton becomes more localized. Transverse power-flow density due to the nontrivial phase structure of complex solitons may arise in complex potentials. Following Ref. [8], it can be defined as:
$S=\left(i/2\right)\left(\varphi {\varphi}_{x}^{*}-{\varphi}_{x}{\varphi}^{*}\right)$. At small *b*, the power-flow density is positive in the lattice sites and negative in the space between them. It becomes completely positive if *b* exceeds a critical value. Interestingly, a local minimum appears in the central lattice channel at large *b* [Fig. 1(e)]. This property has not been revealed in other systems.

Linear-stability analysis based on Eq. (3) is conducted on the stationary solutions. Fundamental solitons in the present scheme are stable in their entire existence domain, which is in sharp contrast to the stability of fundamental solitons in in-phase mixed linear-nonlinear 𝒫𝒯 lattices [26], where stable solitons exist only at a narrow region near the edge of semi-infinite gap, i.e., at lower soliton power. It indicates that the out-of-phase modulation of nonlinearity can dramatically improve the stability of solitons. To confirm the prediction of linear-stability analysis, we solve Eq. (1) numerically by the standard beam propagation method with the input condition *q*(*x,z* = 0) = *u*(*x*)[1 + *ρ*(*x*)], where *ρ*(*x*) is a random function with Gaussian distribution. An example of stable propagation is illustrated in Fig. 1(f).

Next, we examine the properties of dipole solitons in out-of-phase mixed linear-nonlinear 𝒫𝒯 lattices. Unlike fundamental solitons, there exists an inflexion point on the power curve of dipole solitons near the lower cutoff of *b* [Fig. 2(a)]. In comparison with Fig. 1(b), the amplitude ratio between the imaginary and real parts increases at small and moderate *b* and decreases rapidly when the nonlinearity is strong. It implies that the strong modulation of nonlinearity can suppress the imaginary part of solitons. In other words, the nonlinearity modulation effectively counteracts the role played by imaginary lattice, since the emergence of imaginary part of soliton is solely due to the existence of imaginary lattice. Two representative profiles of dipole solitons are shown in Figs. 2(b) and 2(c). The symmetric center of dipoles is shifted to *x*_{0} = *π*/4 and the symmetries of the real and imaginary parts exhibit as: *u _{r}*[−(

*x*+

*π*/4)] = −

*u*(

_{r}*x*+

*π*/4) and

*u*[−(

_{i}*x*+

*π*/4)] =

*u*(

_{i}*x*+

*π*/4). The two peaks of the modulus of dipole soliton are of the same height, due to the symmetries of the real and imaginary parts [Fig. 2(c)]. With the decrease of

*b*, dipole soliton becomes broader and extends to several lattice sites.

The inflexion point on the power curve shown in Fig. 2(a) implies that dipoles may suffer a Vakhitov-Kolokolov (V-K) instability [31]. We examine the real and imaginary parts of the unstable eigenvalue *λ* and find that all imaginary parts of *λ* equal zero when *b* is smaller than the critical value *b _{in}*. It confirms the prediction according to the V-K criterion. Oscillatory instability corresponding to

*λ*with nonzero real and imaginary parts may arise if

*b*>

*b*[Fig. 2(d)]. By comparing with fundamental solitons, the instability of dipoles may be attributed to the repulsive interaction between the two parts beside the symmetrical center. Dipole solitons are completely stable when the propagation constant exceeds a critical value. In other words, dipoles are stable at higher power, which is again in contrast to the stability of fundamental solitons in in-phase mixed linear-nonlinear 𝒫𝒯 lattices [26]. Thus, we get the central finding of this paper, i.e., out-of-phase mixed linear-nonlinear 𝒫𝒯 lattices can remarkably suppress the instability of higher-order solitons in them, especially for solitons at higher power.

_{in}Unstable and stable propagation examples of dipoles at *b* = 3.44 and 5.70 are displayed in Figs. 2(e) and 2(f), respectively. After a short distance, the unstable dipole soliton undergoes an asymmetric distortion, similar to the evolution of unstable solitons in linear 𝒫𝒯 lattices [18,21].

Now, we study the triple solitons in mixed linear-nonlinear lattices with a 𝒫𝒯 symmetry. The power of triple solitons is higher than that of fundamental/dipole ones [Fig. 3(a)]. As can be explained by borrowing the concept of “soliton trains” in the saturable nonlinear media modulated by a purely real lattice [27]. That is, a multipole-mode soliton can be viewed as a nonlinear superposition of several out-of-phase fundamental solitons. Thus, at the same *b*, the power of triple soliton approximates to three times of the power of fundamental soliton. From Fig. 3(a), one can also infer that the amplitude of imaginary part increases with the growth of the pole number of solitons.

The difference between the peaks of soliton modulus decreases with the growth of propagation constant [Figs. 3(b) and 3(c)]. The three peaks are of the same height at large *b* and increase simultaneously with *b*. The physical reason is that the out-of-phase nonlinearity modulation partly weakens the modulation of linear lattice, and thus the poles of high-power solitons experience a saturation effect, just similar to the profiles of solitons in competing or saturable nonlinear media.

For triple solitons, the precise structure of instability regions is rather complicated. There exist multiple narrow stability “windows” at lower *b* [Fig. 3(d)]. Similar to dipole solitons, triples can propagate stably provided that their power exceeds a critical value. We should note that the instability region expands slowly with the growth of the number of soliton poles. Direct propagation simulations on triple solitons verify the prediction of linear stability analysis, see e.g., Figs. 3(e) and 3(f).

While stable families of higher-order solitons can exist in mixed linear-nonlinear 𝒫𝒯 lattices (below the phase transition point), increasing the gain-loss component has an overall destabilizing effect on soliton propagation. Specifically, when the gain-loss component increases, the parameter range of stable solitons shrinks as new regions of instability appear. On the other hand, for fixed linear lattices, increasing the modulation of nonlinearity may lead to an instability of high-power solitons, since the strong nonlinear effect on the beams cannot be balanced by the linear potentials and diffraction.

To elucidate the influence of the relative magnitude of gain-loss component *χ* and the depth of nonlinearity modulation *σ* on the stability of multipole-mode solitons, we investigate the evolution of solitons in lattices with relatively large *χ* and *σ*, taking dipole and triple solitons as examples. At fixed *b* and *σ*, while the real part of dipole soliton remains unchanged, the peak of imaginary part increases with *χ* [Fig. 4(a)]. Yet, dipole solitons are still completely stable at higher power [Fig. 4(c)]. Thus, the growth of *χ* leads to the shrink of stability region. At fixed *χ* and *b*, the real part of triple soliton decreases and the imaginary part increases with the growth of *σ* [Fig. 4(b)]. If *σ* is large, an instability region of triple solitons appears at higher power [Fig. 4(d)], since too strong nonlinearity modulation destroys the stability of high-power solitons. The growth of the depth of nonlinearity modulation *σ* can be seen as an effective decrease of the depth of linear lattice *p*. Thus, one can infer from Fig. 4(d) that, for fixed *σ*, the decrease of *p* will lead to the shrinkage of the stability domain of higher-order solitons.

From the above discussions, one can expect that solitons with more poles can still be stable under appropriate conditions. To gain further insight into the multipole-mode solitons in mixed linear-nonlinear 𝒫𝒯 lattices, we illustrate the profiles of two solitons with 5– and 7– poles and their stable propagations in Figs. 5(a) and 5(b), respectively. Again, the same height of soliton peaks demonstrates that the competition between the out-of-phase linear and nonlinear potentials results in a saturable effect. The large amplitude of imaginary part does not introduce a new instability, which is impossible in linear 𝒫𝒯 lattices. We emphasis that, in 𝒫𝒯 lattices, stable solitons with pole number larger than 3 have never been reported. Such higher-order nonlinear modes with out-of-phase poles in the neighboring lattice sites can be seen as a generalization of “soliton trains” [27] into complex lattices.

Besides the aforementioned higher-order solitons, we find another type of nonlinear localized modes, which resemble the dual-core solitons reported in Ref. [32]. Such solitons can be seen as a nonlinear superposition of two in-phase or out-of-phase fundamental solitons whose central positions are separated by several lattice sites. We show the profiles of symmetric and antisymmetric dual-core solitons together with their stable propagations in Figs. 5(c) and 5(d), respectively. Both the symmetric and the antisymmetric solitons in mixed linear-nonlinear 𝒫𝒯 lattices are stable in a wide parameter window.

Finally, we reveal that multipole-mode solitons can be excited by a tilted input Gaussian beam in the form of *u*(*x*) = *A*exp[−(*x/d*)^{2}]exp(*iθx*), where *A* and *d* are the amplitude and width of the input beam, respectively; *θ* characterizes the tilt angle or the phase. Figure 6(a) displays an example of the excitation of stable fundamental soliton by a Gaussian beam. As expected, the excited stable soliton immediately transforms into a sech-type soliton if the 𝒫𝒯 lattice is removed at a certain propagation distance. Similar phenomenon occurs for the excited dipole soliton by a broader input Gaussian beam [Fig. 6(b)]. The asymmetric splitting of the excited dipole soliton in the absence of optical lattice manifest its initial inner phase structure.

Similar to the multicolor solitons in real lattices [33], multipole-mode solitons are tightly packed only by the optical lattices. Since the interaction between the neighboring constituents of soliton is always repulsive, removal of the lattice will cause fast splitting (or unpacking) of the soliton. The asymmetric propagation directions of the unpacked soliton components are again due to the original inner phase structure of the complex solitons [Figs. 6(c) and 6(d)]. Such splitting can be used to obtain a set of diverging solitons with controllable escape angles. The splitting shown in Figs. 6(c) and 6(d) is reversible, i.e., spatial solitons launched into a focusing Kerr medium with appropriately adjusted relative phases and incidence angles can be tightly packed into the mixed linear-nonlinear 𝒫𝒯 lattice and can propagate further as a compact multipole-mode soliton.

## 4. Summary

To conclude, we studied the stability properties of multipole-mode solitons in a focusing cubic medium modulated spatially by both the linear refractive index and the nonlinearity. The out-of-phase mixed linear and nonlinear complex periodic lattices satisfy a 𝒫𝒯 symmetry. Stable higher-order solitons with various numbers of poles were found. Under appropriate conditions, fundamental solitons can propagate stably in their entire existence domain. Multipole-mode solitons with out-of-phase soliton components in the neighboring lattice sites are completely stable provided that their power or propagation constant exceeds a critical value. We demonstrated that suitable modulation of out-of-phase nonlinearity can remarkably suppress the instability of higher-order solitons. In addition, we revealed that the excitation of multipole-mode solitons by a tilted input Gaussian beam and the splitting of solitons into their soliton constituents are possible in 𝒫𝒯 lattices. Our findings, thus, provide an effective way for the realization of stable multipole-mode solitons in the complex potentials with gain-loss components.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 11074221) and the Program for Innovative Research Team, Zhejiang Normal University, Jinhua, Zhejiang Province, P. R. China.

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