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Abstract
The symmetry breaking bifurcation of solitons in an optical waveguide with focusing saturable nonlinearity and parity-time (𝒫𝒯)-symmetric complex-valued external potentials is investigated. As the soliton power increases, it is found that the branches of asymmetric solitons split off from the base branches of 𝒫𝒯-symmetric fundamental soliton. The bifurcation diagrams, consisting essentially of the propagation constants of optical solitons, indicate that symmetric fundamental and multipole solitons, as well as asymmetric solitons can exist. The stabilities and the dynamics characteristics of solitons are comprehensively investigated. We find the different instability scenarios of the symmetric solitons, but the symmetry breaking bifurcation is caused only by the onset of instability of the symmetric fundamental solitons. This result is further confirmed by the numerical examples with the different saturable nonlinearity parameters. In particular, we find that the soliton power and the stability of soliton at the bifurcation points are significantly changed by varying the strength of the saturable nonlinearities. These results provide additional way to control symmetry breaking bifurcations in 𝒫𝒯-symmetric optical waveguide.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Parity-time (𝒫𝒯) symmetric systems originate from an observation in non-Hermitian quantum mechanics and they have attracted a growing interest in recent years. A simple one-dimensional 𝒫𝒯-symmetric system can be described by the Schrödinger equation with a complex potential, where the real part of the potential is an even function of the coordinates and the imaginary part is an odd function. Although 𝒫𝒯-symmetric systems are non-Hermitian, they possess all-real eigenvalue spectra, as demonstrated in seminal works by Bender and Boettcher [1–3]. The analysis was initially motivated by the development of quantum theories, then it soon spread out to the various 𝒫𝒯-symmetric physical systems including optics and photonics [4, 5], Bose-Einstein condensates [6], plasmonic waveguides and meta-materials in which losses appear due to the metallic absorption [7–11], and superconductivity [12], etc.
In optical waveguides, the propagation of light can often be described theoretically by Schrödinger-type equation under the paraxial approximation, where the refractive indices of optical waveguides can be described by real-valued potential functions and the imaginary parts of complex potentials stand for the effects of losses and gains in the optical waveguides. A judicious balance of gain and loss constitutes a 𝒫𝒯-symmetric optical waveguide, then the propagation of light can be governed by a Schrödinger equation with a 𝒫𝒯-symmetric Hamiltonian. 𝒫𝒯-symmetric optical system, constructed by two coupled optical waveguides with equal gain and loss, was firstly investigated theoretically and experimentally in the linear regime [4, 13–15]. Beyond the paraxial approximation, rigorous theory based on Maxwell’s equations was also presented [16]. When nonlinear effects are involved in the 𝒫𝒯-symmetric optical waveguides, the interplay between 𝒫𝒯-symmetry and nonlinear effect will give rise to novel phenomena, such as 𝒫𝒯-symmetric stationary nonlinear modes [17–27] and novel nonlinear dynamic properties [28–30]. Furthermore, theoretical studies were extended to the 𝒫𝒯-symmetric spatiotemporal localizations. Spatiotemporal solitons (light bullets) were found in 𝒫𝒯-symmetric potentials with cubic nonlinearity and inhomogeneous diffraction [31], as well as an inhomogeneous nonlinear medium with unbounded gain and loss distributions [32].
Recently, non-𝒫𝒯-symmetric solitons (asymmetric solitons) in 𝒫𝒯-symmetric potentials have aroused great attention. Different from the aforementioned solitons, they possess asymmetric profiles due to the symmetry breaking of solitons. It has been shown that this symmetry breaking cannot occur in generic 𝒫𝒯-symmetric potentials. But for a special class of 𝒫𝒯-symmetric potentials, a family of stable 𝒫𝒯-symmetry breaking solitons with real eigenvalue spectra have been found [33]. For this type of potentials, a precondition of the existence of non-𝒫𝒯-symmetric solitons is miraculously satisfied [34,35], where the model describing the symmetry breaking bifurcations of PT-symmetric solitons is mainly focused on cubic nonlinearity. In theoretical works, the model has been considered in 𝒫𝒯-symmetric optical waveguides with saturable nonlinearities [36, 37], however, only the properties of the symmetric solitons have been investigated. In previous papers, the symmetry breaking of PT-symmetric solitons was decided by the shapes or the parameters of the PT-symmetric potentials. The main aim of this paper is to discuss how the effect of the saturable nonlinearity can vary the symmetry breaking bifurcation, which could provide an additional way to transform the soliton from a symmetric to an asymmetric profile in 𝒫𝒯-symmetric system. In the following sections, we will carry out a detailed investigation of symmetric and asymmetric solitons in a 𝒫𝒯-symmetric potential with focusing saturable nonlinearity and study the key features of two types of solitons in one dimensional Schrödinger equation. Especially, we will show that the points of the symmetry breaking are to be significantly changed by adjusting the strength of the saturable nonlinearity and present the dependence of the soliton power at the point of symmetry breaking bifurcation on the modulation strength of the potential. This may provide another way to control the symmetry breaking bifurcations in 𝒫𝒯-symmetric optical waveguides by changing the strength of the saturable nonlinearity.
The paper is organized as follows. In Sect. 2, the governing model is introduced. In Sect. 3, we show that asymmetric solitons, as well as the fundamental and multipole symmetric solitons can exist in one dimensional Schrödinger equation with the focusing saturable nonlinearity. The bifurcation phenomena and the dependence between the propagation constants and the input power are discussed. Numerical results on the dependence of the soliton power at the point of symmetry breaking bifurcation on the modulation strength of the potential are also presented. In Sect. 4, the stability of symmetric and asymmetric solitons with different parameters of saturable nonlinearity are analyzed systematically, and the corresponding nonlinear evolution dynamics are studied by performing direct numerical simulations. Finally, we will drawn our conclusions in Sect. 5.
2. The governing model
In paraxial regime, optical wave propagation in a planar graded-index waveguide with focusing saturable nonlinearity is governed by the following one dimensional normalized nonlinear Schrödinger equation
where ψ (ζ, ξ) is the normalized optical field envelope function, U(ξ) ≡ V (ξ) + iW (ξ) is the normalized complex potential, σ = 1 represents the focusing nonlinearity, and S is the saturable parameter standing for the degree of saturable nonlinearity [38]. When the parameter of focusing nonlinearity σ = 0, Eq. (1) describes a linear 𝒫𝒯-symmetric system, where it has purely real eigenvalues below an exceptional point and a pair of complex conjugated eigenvalues above the point. Only for the purely real eigenvalues, the Hamiltonian commutes with 𝒫𝒯 operator. When the parameter of focusing nonlinearity σ = 1, we focus on the solutions with physical relevance, i.e., the solutions are stationary, and the corresponding eigenvalues are all real. To obtain optical solitary wave solutions, we assume the stationary solutions in the form of ψ(ζ, ξ) = ϕ(ξ) eiβζ, where ϕ (ξ) ≡ ϕR + iϕI is a complex function with the real part ϕR and imaginary part ϕI, and β is the propagation constant. Substitution into Eq. (1) yields In this paper, we consider a special class of 𝒫𝒯-symmetric potential where g (ξ) is a real and even function. We take g (ξ) in the form of where W0 represents the modulation strength of the potential in Eq. (3), ξ0 and χ0 are related to the separation and width of 𝒫𝒯-symmetric potential, respectively. Then the real and imaginary parts of the 𝒫𝒯-symmetric potential can be constructed as3. Solitons and symmetry breaking bifurcations
In this section, we aim to search for the symmetric and asymmetric solitons, and illustrate the characteristics of the bifurcations of symmetry breaking in the special class of 𝒫𝒯-symmetric potential in Eq. (3).
First, we consider the focusing nonlinearity with the saturable parameter S = 1. As an example, the real and even function g (ξ) is chosen in the form of a double-hump hyperbolic secant function, and the 𝒫𝒯-symmetric potential in Eqs. (5) and (6) is in the form of a Scarff-II potential with double-hump profiles. We consider the 𝒫𝒯-symmetric potential with parameters ξ0 = 2, χ0 = 1, and W0 = 1.5. The profiles of the real function g (ξ), real and imaginary parts of the 𝒫𝒯-symmetric potential are shown in Fig. 1(a).
Fig. 1 Diagram of bifurcation and the corresponding symmetric and asymmetric solitons. (a) Profiles of the real and even function g (ξ) (black dashed curve), real (blue solid curve) and imaginary (thin red solid curve) components of the potential. (b) The propagation constants versus the soliton power, where the symmetric fundamental solitons (blue solid curve labeled with SS1), symmetric dipole solitons for the excited state (black dash-dot curve labeled with SS2), symmetric tripole solitons for the excited state (thin gray solid curve labeled with SS3), as well as asymmetric solitons (red dotted curve labeled with AS) are shown. (c) and (d) Distributions of fundamental symmetric and asymmetric solitons with the soliton power P = 1.5, where the blue solid and dashed red curves represent the real part and imaginary part, respectively. Here, the parameters are ξ0 = 2, χ0 = 1, and W0 = 1.5.
The propagation constants and the corresponding solitons can be obtained by solving numerically the stationary Eq. (2) with the power-conserving squared-operator iteration method [42]. Numerical results indicate that there exist two families of solitons, i.e., 𝒫𝒯-symmetric solitons (symmetric solitons) and non-𝒫𝒯-symmetric solitons (asymmetric solitons). The diagrams of bifurcation, consisting essentially of the propagation constants of solitons, show that there only exist symmetric solitons at a low soliton power (). The solitons with symmetric profiles are found, including the symmetric fundamental solitons (denote with SS1), the symmetric dipole and tripole solitons for the excited states (denote with SS2 and SS3). With the increase of soliton power, a new branch splits off from the base branch of 𝒫𝒯-symmetric fundamental solitons at the point of P = 0.7. Beyond this point, the symmetry of the fundamental 𝒫𝒯-symmetric solitons are broken with the increasing of soliton power, but the symmetry of the multipole soliton is still preserved. The propagation constants of symmetric and asymmetric solitons (denote with AS) form a pitchfork bifurcations shown in Fig. 1(b), which is also called as symmetry-breaking bifurcations in the models with real potential [39–41]. We present the profiles of symmetric and asymmetric solitons with soliton power P = 1.5 in Fig. 1(c) and Fig. 1(d), respectively. It is a natural result for the symmetric fundamental soliton that the complex-valued function possesses symmetric real profile and antisymmetric imaginary profile, as shown in Fig. 1(c). For asymmetric soliton, the profiles of the real and imaginary parts are all asymmetric, as shown in Fig. 1(d).
Figure 2 shows the symmetric and asymmetric solitons with the soliton power varying from 0.1 to 3. Fig. 2(a) presents the profiles of the fundamental solitons. The profiles of the symmetric dipole and tripole solitons for the excited states are shown in Fig. 2(b) and Fig. 2(c), respectively. It is worth to note that the profiles of the symmetric fundamental solitons and the symmetric dipole solitons for the excited state are all double humps, but the profiles of the real and imaginary parts have an opposite symmetry. The profiles of asymmetric solitons are presented in Fig. 2(d), where the results show that asymmetric solitons can exist ranging from 0.7 to 3 for the parameter of soliton power. With the increasing of soliton power, the profiles of asymmetric solitons are gradually concentrated into one of the humps of the 𝒫𝒯-symmetric potential.
Fig. 2 Profiles of solitons for different soliton power. (a) Symmetric fundamental solitons, (b) symmetric dipole solitons for the excited state, (c) symmetric tripole solitons, (d) asymmetric solitons. Here, the parameters are the same as those in Fig. 1.
To further illustrate the effect of the focusing saturable nonlinearity on the symmetry breaking of solitons, the diagrams of bifurcation are depicted with the different saturable nonlinear parameters. In Fig. 3(a), the diagram of bifurcation is presented with a weak saturable nonlinearity S = 0.1. The fundamental solitons are symmetric when the soliton power is below 0.5 (shown with blue solid curve), and the asymmetric solitons begin to emerge when the soliton power exceeds 0.5 (shown with red dotted curve). The point of symmetry breaking bifurcation appears at P = 0.5 as shown by the black vertical dashed line. The profiles of solitons, corresponding to the blue solid curve for P ⩽ 0.5 and red dotted curve in Fig. 3(a), present a transformation of the fundamental solitons from symmetric to asymmetric distributions, as shown in Fig. 3(d). Diagram of bifurcation with moderate saturable nonlinearity (S = 0.5) is depicted in Fig. 3(b), where the point of symmetry breaking bifurcation appears at P = 0.6. For comparison, diagram of bifurcation with saturable nonlinear parameter S = 1 is also plotted in Fig. 3(c) with the point of symmetry breaking bifurcation P = 0.7. The corresponding solitons of a transformation from symmetric to asymmetric distributions are also shown in Fig. 3(e) and Fig. 3(f), respectively. These numerical results indicate that the soliton power at the bifurcation points can be changed by the different values of the saturable parameter. Therefore, the the symmetry breaking bifurcation may be controlled by the different strengths of the saturable nonlinearities instead of the shape and parameter of the 𝒫𝒯-symmetric potential.
Fig. 3 Diagrams of bifurcation and profiles of solitons for different saturable nonlinear parameters. (a), (b) and (c) are diagrams of bifurcations with saturable nonlinear parameters S = 0.1, S = 0.5, and S = 1, respectively, where the vertical dashed lines show the soliton powers at bifurcation points. (d), (e) and (f) depict that the profiles transform from the symmetric fundamental solitons into asymmetric solitons due to symmetry breaking. The vertical plane shows the soliton power at bifurcation points, where the saturable nonlinear parameters are S = 0.1, S = 0.5, and S = 1, respectively. Here, the other parameters are the same as those in Fig. 1.
It is also relevant to address the dependence of the symmetry breaking bifurcation on the modulation strength of the potential W0 with different saturable nonlinear parameters. In Fig. 4, the top row shows the symmetry breaking bifurcations with modulation strength of the potential W0 = 2, where Figs. 4(a), 4(b) and 4(c) present the diagrams of bifurcation with S = 0.1, 0.5 and 1, respectively. The bottom row presents the symmetry breaking bifurcations with modulation strength of the potential W0 = 2, where Figs. 4(d), 4(e) and 4(f) present the diagrams of bifurcation with S = 0.1, 0.5 and 1, respectively. The numerical results demonstrate that the soliton power at the point of symmetry breaking bifurcation increases with the decrease of the modulation strength of the potential W0, which indicates that the symmetry of soliton needs a much higher soliton power to break for the weak modulation strength of the potential.
Fig. 4 Diagrams of bifurcation depending on the different modulation strength of the potential W0 with different saturable nonlinear parameters. (a), (b) and (c) depict the diagrams of bifurcation with modulation strength of the potential W0 = 2, and the focusing saturable nonlinearity parameters S = 0.1, 0.5 and 1, respectively. The red dotted curves and blue solid curves represent the propagation constants of symmetric and asymmetric solitons. (d), (e) and (f) show the diagrams of bifurcation with modulation strength of the potential W0 = 1, the focusing saturable nonlinearity parameters S = 0.1, 0.5 and 1, respectively. Here, the other parameters of potential are the same as those in Fig. 1.
4. Linear stability analysis and dynamics
In order to examine the stability of these solitons, it is necessary to explore the perturbed solutions for a known solution ϕ (ξ) with the form
where ϕ (ξ) is the stationary solution with a real propagation constant β, u (ξ) and v (ξ) are small perturbations with |u|, |v| ≪ |ϕ|. Substituting Eq. (7) into Eq. (1) and keeping only the linear terms, we obtain the following linear eigenvalue problem Here, L11, L12, L21 and L22 are expressed as the following forms where δ is a complex eigenvalue of the linear problem for Eq. (8). The positive real part of complex eigenvalue can be used to measure the instability growth rate of the perturbation. If δ contains a real part, it is indicated that the soliton ϕ (ξ) is linearly unstable, otherwise, ϕ (ξ) is linearly stable.Figure 5 presents the dependence of the largest real part Max(δR) of the eigenvalues of linear stability analysis for the symmetric and asymmetric solitons on the soliton power P with different saturable nonlinearity parameters S. The largest real part of the eigenvalues of linear stability analysis for the symmetric fundamental solitons, the symmetric dipole and tripole solitons for the excited states, as well as the asymmetric solitons are plotted with blue solid curves, black short dot-dashed curves, gray dashed curves and red curves with circles for S = 0.1, S = 0.5 and S = 1 in Figs. 5(a), 5(b) and 5(c), respectively. For the saturable parameter S = 0.1, all of the symmetric solitons are linearly stable when the soliton power is lower than 0.5, then the value of Max(δR) begins to increase, which indicates that the symmetric fundamental solitons turn into a unstable region ranging from P = 0.5 to 3. It is in this region that the asymmetric solitons can coexist with the symmetric solitons, as shown in Fig. 5(a). The symmetric dipole solitons for the excited states are linearly stable in the region of P = 0.5 to 3, but the symmetric tripole solitons possess a weak unstable region ranging from P = 1.6 to 3. Asymmetric solitons become unstable when the soliton power increases upto P = 2.5. The results of the linear stability analysis for S = 0.5 and S = 1 are shown in Figs. 5(b) and 5(c), where the symmetric fundamental solitons turn into a unstable region at the points of P = 0.6 and P = 0.7, respectively. We also found that the stable regions for symmetric tripole solitons and asymmetric solitons shrink with the increasing of the saturable nonlinearity parameter.
Fig. 5 Eigenvalues of linear stability analysis for the different parameters of saturable nonlinearity. (a), (b) and (c) depict the largest real part of δ for the symmetric and asymmetric solitons with focusing saturable nonlinearity parameters S = 0.1, 0.5 and 1, respectively. Here, the parameters are the same as in Fig. 3.
By comparing Fig. 3 with Fig. 5, it is found the asymmetric solitons always exist in that regions where the symmetric fundamental solitons begin to turn into an unstable region. This indicates that the above mentioned a special class of 𝒫𝒯-symmetric potential is only one of the necessary conditions for the symmetry-breaking bifurcations, and the symmetry breaking has been induced by the instability of the symmetric fundamental solitons. Furthermore, the points of symmetry breaking bifurcation are changed by the different values of the saturable parameter in Fig. 3, but essentially, these bifurcation points are caused by the onset of instability of the symmetric fundamental solitons with different saturable parameters, as shown those in Fig. 5.
The results of linear stability analysis for solitons are further confirmed by employing direct numerical simulations of Eq. (1). Numerical results are summarized in Fig. 6 and Fig. 7 with the saturable parameter S = 1. The eigenvalue spectra of Eq. (8) for symmetric and asymmetric solitons are shown in the left column in Fig. 6 with the soliton power P = 1, where the eigenvalue spectra of linear stability analysis for the symmetric solitons are displayed in Figs. 6(a), 6(b) and 6(c). It is found that the real component (δR) and imaginary component (δI) of the eigenvalues of linear stability analysis appear in pairs for the symmetric solitons, so these eigenvalue spectra are quartet symmetry. The eigenvalues of linear stability analysis also appear in pairs for the asymmetric soliton, however, the real components of these eigenvalues are asymmetric, as depicted in Fig. 6(d). From the eigenvalue spectra of linear stability analysis, we can found that the symmetric fundamental soliton is strongly linearly unstable in Fig. 6(a). The corresponding evolution plot shows that the optical field intensities oscillate periodically between two humps of the 𝒫𝒯-symmetric potential after a short distance of stationary evolution, as shown in Fig. 6(e). The symmetric solitons for the excited states, as well as the asymmetric soliton, are weakly and linearly unstable. To confirm that, the initial profiles are perturbed by a 5% random noise. The results of the direct numerical simulations indicate that these solitons can propagate stably at the initial stage, then the amplitudes of soliton appear small oscillations due to the weak instability, and these small oscillations are continually amplified with the increasing of propagation distance which eventually leads to the broken-down profiles, as presented in Figs. 6(f), 6(g) and 6(h), respectively.
Fig. 6 The eigenvalue spectra of linear stability analysis for the solitons with saturable parameter S = 1 and the corresponding evolution plots. (a), (b) and (c) show the eigenvalue spectra of linear stability analysis for SS1, SS2 and SS3, (d) shows the eigenvalue spectrum of AS with the soliton power P = 1, respectively. (e), (f), (g), and (h) The corresponding evolution plots of the field intensity. Here the other parameters are the same as in Fig. 1.
Fig. 7 The eigenvalue spectra of linear stability analysis for the solitons with saturable parameter S = 1 and the corresponding evolution plots. (a), (b) and (c) show the eigenvalue spectra of linear stability analysis for SS1, SS2 and SS3, (d) shows the eigenvalue spectrum of AS with the soliton power P = 2.5. (e), (f), (g), and (h) The corresponding evolution plots of the field intensity. Here the other parameters are the same as those in Fig. 1.
The eigenvalue spectra and the dynamics evolutions of symmetric and asymmetric solitons are also presented in Fig. 7 with different soliton power. By comparing Fig. 7(a) with Fig. 6(a), we can see that the symmetric fundamental soliton have a larger instability growth rate for the soliton power P = 2.5. The evolution plot shows that the symmetric fundamental soliton has eventually broken down after oscillating periodically in Fig. 7(e). Similar to Fig. 6(b), the symmetric dipole soliton for the excited state is still weakly and linearly unstable for the soliton power P = 2.5. The eigenvalue spectrum of linear stability analysis and the corresponding evolution perturbed by a 5% random noise are shown in Figs. 7(b) and 7(f), respectively. The stability of the symmetric tripole soliton and the asymmetric soliton decrease when the soliton power P = 2.5, as shown in Figs. 7(c) and 7(d). The evolution plots present these unstable propagations, even though the initial profiles haven’t been perturbed by the random noise, as shown in Figs. 7(g) and 7(h).
Finally, we would like to note that, the control of symmetry breaking bifurcations can be realized experimentally by considering the waveguide fabrication with photorefractive crystals, where the PT-symmetric Scarff-II potential with double-hump profiles in Eqs. (5) and (6) can be realized by Ti in-diffusion, gain and loss can be realized with Fe-doped [15]. The strength of the saturable nonlinearity can be dynamically adjusted with an applied voltage in the biased photorefractive crystals [43,44]. For the controllable symmetry breaking bifurcations of solitons, it includes two steps: generating a symmetric or an asymmetric soliton and adjusting the point of symmetry breaking bifurcation. Then soliton can be transformed from a symmetric to an asymmetric profile by varying the strength of the saturable nonlinearity, and vice versa.
5. Conclusions
In conclusion, we have performed a systematic study for the symmetry breaking of solitons in 𝒫𝒯-symmetric optical waveguides with focusing saturable nonlinearity. Our analysis has shown that the branches of asymmetric soliton bifurcated from the base branches of 𝒫𝒯-symmetric fundamental soliton with the increasing of the soliton power, causing by the onset of instability of the symmetric fundamental solitons.
The stabilities of the symmetric and asymmetric soliton solutions have been comprehensively analyzed by employing linear stability analysis. By comparing numerical examples of the different saturable nonlinearity parameters, we found that the above-mentioned a special class of 𝒫𝒯-symmetric potentials is a necessary condition for the symmetry breaking bifurcations. Under such a condition, the symmetry breaking of the solitons have been essentially caused by the instabilities of the fundamental solitons. Especially, these numerical results of the linear stability analysis have also indicated that the soliton power at the bifurcation point can be changed by the different saturable nonlinearity parameters.
The different instability scenarios of solitons, including the eigenvalue spectra of linear stability analysis and the corresponding evolution plots, have been revealed by using direct numerical simulations. The results have shown that the stability of the symmetric and asymmetric solitons decrease in 𝒫𝒯-symmetric optical waveguides with the focusing saturable nonlinearity when the soliton power increases.
Thereby, a further study with these effects is of great interest for the controllable symmetry breaking bifurcations of solitons in 𝒫𝒯-symmetric optical waveguides with saturable nonlinearity. Extension of this symmetry breaking in saturable nonlinear media to higher spatial dimensions is also an important direction.
Funding
Doctoral Scientific Research Foundation of Taiyuan Normal University (Grant No. I170144); Zhejiang Provincial Natural Science Foundation of China (Grant No. LY17F050011); Open Fund of IPOC (BUPT); Doctoral Scientific Research Foundation of Taiyuan University of Science and Technology (Grant No. 20152043); National Natural Science Foundation of China (Grant No. 11747063).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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