Open Access
Add to BibSonomyAbstract
We address the existence and stability properties of optical solitons in a competing cubic-quintic medium with an imprinted complex lattice featuring a parity-time (𝒫𝒯) symmetry. Various families of solitons with even and odd geometrical symmetries are found in both the semi-infinite and the first finite gaps. Linear stability analysis corroborated by direct propagation simulations reveals that solitons with different symmetries and different number of humps can propagate stably at the same propagation constants, i.e., multi-stable solitons can exist in this scheme. Interestingly enough, in sharp contrast to the stability of solitons in a conventional (real) lattice, both even and odd solitons with the same propagation constant belonging to different branches can be stable in the first gap of 𝒫𝒯 lattice, which indicates that the imaginary part of lattice plays an important role for the stabilization of solitons.
© 2012 Optical Society of America
1. Introduction
In quantum mechanics, every physical observable is associated with a real spectrum and thus must be Hermitian. Intriguingly, the pioneering work done by Bender et al. verified that a class of non-Hermitian Hamiltonian can also display entirely real spectra provided they respect a 𝒫𝒯-symmetry [1]. 𝒫𝒯-symmetry demands that a Hamiltonian belonging to it shares a common set of eigenfunction with the P̂T̂ operator. The parity operator P̂, responsible for spatial reflection, is defined through the operations p̂ → −p̂, x̂ → −x̂, while the time reversal operator T̂ leads to p̂ → −p̂, x̂ → x̂ and to complex conjugation î → −î. Given the fact T̂Ĥ = p̂2/2 + V*(x) [1,2], a necessary condition for the Hamiltonian to be 𝒫𝒯-symmetric is that the potential function V(x) should satisfy the condition V(x) = V*(−x) [3–5].
In recent years, light propagation in 𝒫𝒯-symmetric potentials within the framework of nonlinear optics has been studied theoretically and observed experimentally [4–10]. These activities were due to the connection, bridged by the same Schrödinger equation, between the quantum mechanics and optical field. 𝒫𝒯 potentials can be realized by introducing a complex refractive index into a wave-guided system: n(x) = nr(x) + ini(x), where nr(x) = nr(−x) is the real index profile, and ni(x) = −ni(−x) stands for the imaginary one [10]. Thus far, various types of solitons in diverse optical potentials, accompanied by the gain or loss regions, have been reported. These works include, for example, solitons in purely nonlinear lattices [11–13], solitons in mixed linear-nonlinear lattices [14], defect nonlinear modes in linear lattices [15–18], 2D bright solitons in defocusing Kerr media [19], Bragg gap solitons in competing cubic-quintic media [20], and optical modes in a double-channel waveguide [21]. We should note that single-and double-humped solitons in cubic-quintic media with imprinted 𝒫𝒯 lattices were found to be completely unstable [20].
Very recently, Nixon et al. systematically studied the stability properties of solitons in 𝒫𝒯 lattices and found that both 1D and 2D solitons can propagate stably under appropriate conditions [22]. They revealed that, in the first gap, fundamental and dipole solitons with the same propagation constant can be stable. According to the discussions in Refs. [23, 24], such solitons can be termed as “bistable solitons”. Recalling the existence of multi-stable solitons in a real lattice built in a cubic-quintic medium, a natural question that arises is, can multi-stable solitons exist in 𝒫𝒯 potentials? If they can, what is their dynamics and how does the imaginary part of the complex lattice influence on them?
In the present paper, we investigate the existence domains, stability conditions and propagation dynamics of solitons with different geometrical symmetries in a cubic-quintic medium with an imprinted complex lattice satisfying the 𝒫𝒯-symmetry. Note that the cubic-quintic medium exhibits focusing nonlinearity at lower light intensity and defocusing nonlinearity at higher light intensity. This property allows one to research the dynamics of solitons in the semi-infinite and finite gaps. Various families of soliton solutions with a different number of humps, residing in the semi-infinite and the first gaps, are found. Numerical results illustrate that multi-stable solitons can exist either in the semi-infinite gap or in the first gap, which is in sharp contrast to the cases in purely real lattices [24].
2. Model
We consider light propagation along the z axis in a competing cubic-quintic medium with a transverse complex refractive-index modulation. Dynamics of the beam can be described by the nonlinear Schrödinger equation for the dimensionless complex field amplitude ψ:
Here, the longitudinal z and transverse x coordinates are scaled in the terms of diffraction length and beam width, respectively; V(x) = V0sin2(x) and W(x) = W0 sin(2x) are the real and imaginary periodic refractive-index modulations, respectively, satisfying the 𝒫𝒯-symmetry, i.e., V(−x) = V(x) and W(−x) = −W(x). V0 and W0 denote the amplitudes of the real lattice and imaginary lattice. Without loss of generality, through the rest of this paper, we are going to use a specific configuration by setting V0 = 3, W0 = 0.3 unless stated otherwise.To understand the basic properties of the guided modes, it is instructive to consider the Floquet-Bloch spectrum of the complex periodic potentials. Band-gap structures of the linear version of Eq. (1) with V0 = 3, W0 = 0.3 and 1.55 are shown in Fig. 1(b). There exists a critical value W0 = 1.5 (phase transition point), above which the eigenvalue spectra become complex, and all Bloch bands merge together simultaneously. Typical profiles of the real and imaginary refractive-index modulation are displayed in Fig. 1(a). Stationary solutions of Eq. (1) can be solved by assuming ψ(x, z) = ϕ(x)exp(−ibz), where ϕ(x) is a complex function and b is a real propagation constant (below the phase transition point). Substitution of the expression into Eq. (1) yields an ordinary differential equation describing the stationary solution:
which can be solved numerically by applying the boundary conditions: ϕr,i(x → ±∞) = 0.
Fig. 1 (a) Typical profile of 𝒫𝒯 lattice. Solid: real part; dashed: imaginary part. (b) Band-gap structure of lattices. Solid: V0 = 3, W0 = 0.3 [corresponding to (a)]; dashed: V0 = 3, W0 = 1.55. Inset: imaginary part of complex eigenvalues at W0 = 1.55.
To study the stability properties of stationary solutions, we consider a perturbed solution in the form of ψ(x, z) = {ϕ(x) + [v(x) − w(x)]exp(λz) + [v(x) + w(x)]* exp(λ*z)} exp(−ibz), where v, w ≪ 1 are the infinitesimal perturbations which may grow with a common complex rate λ upon propagation, and * is the complex conjugate operation. Substituting the perturbed solution into Eq. (1) and linearizing it around the stationary solution ϕ(x), one obtains a coupled set of linear eigenvalue equations:
where L̂ = d2/dx2/2 + b − V + 2|ϕ|2 − 3|ϕ|4. Obviously, if there exist nonzero real parts of λ, the stationary solution is unstable, otherwise it is stable.All the solitons can be categorized into two classes according to the geometrical symmetries of their amplitude profiles beside the symmetrical center, i.e., even solitons and odd solitons. This means of classification may contradict with the conventional classification method according to the symmetry of soliton poles. In the following discussions, we will illustrate the properties of such solitons with a different number of humps residing in the semi-infinite and first gaps.
3. Multi-stable solitons in the semi-infinite gap
First, we consider the properties of even solitons in the semi-infinite gap. Figure 2(a) displays the dependence of power [defined as ] of the first five families of solitons with a different number of humps on the propagation constant. Unlike the solitons in a lattice modulated Kerr medium, the existence domain of solitons in a cubic-quintic medium does not occupy the whole semi-infinite gap. There are some turning points on the successive power curve. These turning points divide the solitons between them into different families. The lowest branch of solitons with a single hump bifurcate from the linear Bloch wave at the lower edge of the first band. While the real part of soliton satisfies Re[ϕ(−x)] =Re[ϕ(x)], the symmetry of the imaginary part is Im[ϕ(−x)] =−Im[ϕ(x)] [Figs. 2(b)–2(f)]. It means that the symmetry of even solitons is consistent with that of 𝒫𝒯 lattice. The peak of the real part grows with the decrease of propagation constant. Meanwhile, the two main peaks of the imaginary part increase with a slower speed [Fig. 2(b)].
Fig. 2 (a) Power P of even solitons versus propagation constant b in the semi-infinite gap. The first five branches are plotted. Shaded region denotes the first band. (b–f) Profiles of solitons marked by dots in (a) at b = 0.881 (left) and b = 0.948 (right). Real parts are shown by solid lines and imaginary parts are shown by dashed ones. Amplitude profile of soliton marked by f 1 is shown in (f). To distinguish with Re(ϕf1), it is enhanced by 0.1.
When the propagation constant approaches to a lower threshold value (turning point, b1low = 0.878), the peak of soliton reaches a maximum. The single-humped soliton stops to exist if the propagation constant is below the threshold value. Reversely, the two lobes beside the main lobe grow slowly with the increase of propagation constant. In this process, the peak of the main lobe still increases until it reaches another maximum value larger than 1 [Fig. 2(c)]. The 2nd branch of solitons stop to exist when the propagation constant reaches an upper threshold value (b2upp = 0.949). Now, the defocusing nonlinearity becomes dominant for the main lobe of soliton which in turn prevents the further increase of soliton peak located in the central lattice site. Thus, the two lobes beside the central one grow rapidly with the decrease of propagation constant for solitons belonging to the 3rd branch. Meanwhile, the central peak decreases due to the defocusing quintic nonlinearity. The soliton features a complete three-humped structure [Fig. 2(d)]. With the further increase of power, solitons will exhibit a structure featuring three main lobes with the same peak values (slightly larger than 1) accompanying by two lower lobes beside them [Fig. 2(e)]. The outer two lobes begin to grow if the solitons switch to the 5th branch [Fig. 2(f)]. Similar to the 3rd branch, the central peaks of the 5th branch decrease with the decrease of propagation constant. This process may be continued until the soliton solution becomes completely periodic.
We plot an example of soliton amplitude in Fig. 2(f) [corresponding to f1 marked in Fig. 2(a)]. To avoid the superposition with the corresponding real part, it is enhanced by 0.1. Although the soliton includes imaginary part with odd symmetry, its amplitude profile keeps even symmetric.
To study the stability properties of even solitons in the semi-infinite gap, we conduct linear stability analysis, according to Eqs. (3), on the five branches of stationary solutions. Numerical results reveal that the 3rd and 5th branches of even solitons are unstable in their entire existence domains [Fig. 3(a)]. However, for the 1st, 2nd, and 4th branches of solitons, no eigenvalues with nonzero real parts are found. In other words, these three branches of solitons are completely stable in their whole existence domains, which constitutes the central idea of this paper. That is, solitons with different power or Hamiltonian can be stable at the same propagation constant.
Fig. 3 Instability growth rate of even solitons in the semi-infinite gap. (b) Unstable propagation of soliton at b = 0.948 marked by f1 in Fig. 2(a). (c, d) Power-flow densities of solitons at b = 0.948 marked by c1 and d1 in Fig. 2(a). Inset: the corresponding stable and unstable propagations. Scaled real part of lattice is shown. The evolutions of amplitude profiles are plotted in (b–d).
To verify the linear stability analysis results, we have checked the robustness of each nonlinear mode using a standard beam propagation method. Random noises are added into the initial inputs. Typical simulation examples, shown in Figs. 3(b)–3(d), are in good agreements with the prediction of linear stability analysis. The unstable soliton undergoes a quick expansion after a quasi-stable propagation [Figs. 3(b) and 3(d)].
In complex lattices, transverse power-flow density arises due to the nontrivial phase structure of complex nonlinear modes. Transverse power-flow density or Poynting vector across the complex solitons can be defined as: [8]. Two typical examples of power-flow densities associating with solitons marked by c1 and d1 in Fig. 2(a) are shown in Figs. 3(c) and 3(d), respectively. The direction of the flow from gain to loss regions varies across the lattice. It is negative in the lattice sites and positive in the space between lattice sites. Comparing the two curves, the slight higher amplitude of power-flow density shown in Fig. 3(d) may account for the occurrence of instability.
Now, we elucidate the properties of solitons with an odd symmetry in the semi-infinite gap of 𝒫𝒯 lattice. Unlike the odd solitons in [24] and even solitons mentioned above, the lowest branch of odd solitons are purely nonlinear modes having a threshold power, which means that such nonlinear modes cannot bifurcate from the linear modes [Fig. 4(a)]. The real parts of odd solitons exhibit an odd symmetry, whereas the imaginary parts feature an even symmetry [Figs. 4(b)–4(f)]. This is exactly opposite to the symmetry of even solitons. Profiles of odd solitons vary from two humps to four humps along the path b1 – b2 – d2 – d1 – e1 – e2 and from four humps to four humps along the path c1 – c2 – f2 – f1. Solitons belonging to the 2nd branch and the 4th branch converge together at the lower cutoff of propagation constant (left turning point) [Fig. 4(a)]. The 5th branch of solitons stop to exist when the propagation constant is smaller than 0.884.
Fig. 4 (a) Power P of odd solitons versus b in the semi-infinite gap. Shaded region denotes the first band. (b–f) Profiles of solitons marked by dots in (a) at b = 0.884 (left) and b = 0.949 (right). Real parts are shown by solid lines and imaginary parts are shown by dashed ones. Amplitude profile of soliton marked by f1 is shown in (f). To distinguish with Re(ϕf1), it is enhanced by 0.1.
Linear stability analysis results demonstrate that odd solitons in the semi-infinite gap are unstable in their entire existence domains, with the only exception that the lowest branch can be stable in a wide region [Figs. 5(a)]. As can be understood from the profiles of solitons. The humps of solitons beside the central channel of lattice are out-of-phase, then the interaction between them is repulsive, which may lead to the occurrence of instability, especially for solitons with higher power. Figure 5(b) shows the distribution of power-flow density of soliton marked by f1 in Fig. 4(a). Though the left part and right part of soliton are out-of-phase, the power-flow density is an even function of x. Two typical stable and unstable propagations of solitons belonging to 1st and 5th branches are illustrated in Figs. 5(c) and Fig. 5(d), respectively. The unstable soliton can survive for a long propagation distance since the instability growth rate is very small (max[Re(λ)] < 0.05).
Fig. 5 (a) Instability growth rate for odd solitons in the semi-infinite gap. (b) Power-flow density of soliton corresponding to (d). Scaled real part of lattice is shown. Stable (c) and unstable (d) propagations of solitons at b = 0.949 [marked by b1 and f1 in Fig. 4(a)]. The evolutions of amplitude profiles are plotted in (c, d).
4. Multi-stable solitons in the first gap
We next examine the properties of solitons with even and odd symmetries in the first finite gap. The dependence of power on the propagation constant for both classes of solitons are displayed in Figs. 6(a) and 6(b), respectively. In comparison with the solitons in the semi-infinite gap, the power curves of even and odd solitons in the first gap are not successive. They are all monotonously increasing functions of propagation constant, which implies that the defocusing nonlinearity plays a dominant role in the first gap. The existence domains of both classes of solitons are much broader than those in the semi-infinite gap. Solitons with a different number of humps can be found at the same region b ∈ [1.16, 2.38]. The power threshold values of different families of solitons indicate that they are not a continuum of linear modes in bands.
Fig. 6 Dependence of power P on b for even (a) and odd (b) solitons with a different number of humps in the first gap. Shaded regions denote the first and second bands. Solid curves correspond to the stable solitons.
We show several profiles of even and odd solitons at b = 1.40 in Fig. 7. Here, the peak values of the humps are obviously larger than 1. It agrees with the prediction according to the slopes of power curves, i.e., solitons in the first gap experience strong defocusing nonlinearity. By comparing the real parts of even and odd solitons in the semi-infinite gap, the minima of the real parts between the neighboring lattice channels are enhanced evidently, due to the dominant defocusing nonlinearity. One can infer from Figs. 2, 4, and 7 that the imaginary parts of solitons in the first gap are very different from those in the semi-infinite gap. Note that the multi-humped solitons resemble the truncated-Bloch-wave solitons in purely real lattices [25], again due to the dominant defocusing nonlinearity.
Fig. 7 Profiles of even solitons (a–c) and odd solitons (d–f) at b = 1.40 marked by dots in Fig. 6. Examples of amplitude profiles are shown in (c, f).
One of the most interesting findings in this paper is that, in the first gap, even and odd solitons with a different number of humps can be stable in a certain range of propagation constant [Fig. 6]. This is in sharp contrast to the cases in purely real lattices, where only the lowest branches of even and odd solitons can be stable under appropriate conditions [24]. The comparison reveals that the imaginary part of the complex lattice can stabilize the solitons in the first gap. As the number of humps increases, the stability areas of both even and odd solitons shrink. Stable solitons with the number of humps larger than 8 cannot be found. To corroborate the linear stability analysis results, we exhaustively simulate the propagations of solitons belonging to different families. Some representative propagation examples are displayed in [Fig. 8]. Obviously, solitons with the number of humps equaling to 1, 2...6 can propagate stably at the same propagation constant (b = 2.13) without any distortions.
Fig. 8 Stable propagations of multi-stable solitons marked by hollow circles in Fig. 6. (a–c) Even solitons. (d–f) Odd solitons. b = 2.13 in all panels. The evolutions of amplitude profiles are plotted.
5. Conclusions
To summarize, we investigated the dynamics of solitons supported by a complex optical lattice featuring a 𝒫𝒯 symmetry imprinted in a competing cubic-quintic nonlinear medium. Various families of solitons with even and odd symmetries were found in both the semi-infinite and the first finite gaps. In the semi-infinite gap, different branches of even and odd solitons exhibit successive power curves with turning points at the lower and upper cutoffs of propagation constant, due to the competing nonlinearity. In the first gap, such turning points no longer exist for both classes of solitons, since the quintic defocusing nonlinearity are dominant. Linear stability analysis corroborated by direct propagation simulations revealed that: (i) In the semi-infinite gap, the 1st, 2nd, and 4th branches of even solitons are stable in their entire existence domain, whereas the 3rd and 5th branches are completely unstable. Stable odd solitons are possible only when they belong to the 1st branch. (ii) In the first gap, both classes of solitons can propagate stably under appropriate conditions. The stability area shrinks with the increase of the number of soliton humps. Therefore, multi-stable solitons can exist in the following three cases: (i) Even solitons in the semi-infinite gap; (ii) Even solitons in the first gap; (iii) Odd solitons in the first gap. We concluded that the imaginary part of the complex 𝒫𝒯 lattice can be utilized to suppress the instability of solitons in the finite gaps.
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.
References and links
1. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998). [CrossRef]
2. C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002). [CrossRef]
3. C. M. Bender, G. V. Dunne, and P. N. Meisinger, “Complex periodic potentials with real band spectra,” Phys. Lett. A 252, 272–276 (1999). [CrossRef]
4. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007). [CrossRef] [PubMed]
5. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008). [CrossRef] [PubMed]
6. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 -symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009). [CrossRef] [PubMed]
7. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “𝒫𝒯 -symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010). [CrossRef]
8. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008). [CrossRef] [PubMed]
9. K. Makris, R. El-Ganainy, D. Christodoulides, and Z. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. of Theor. Phys. 50, 1019–1041 (2011). [CrossRef]
10. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010). [CrossRef]
11. F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in 𝒫𝒯 -symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011). [CrossRef]
12. D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, “Stability of solitons in PT-symmetric nonlinear potentials,” EPL-Europhys. Lett. 96, 64003 (2011). [CrossRef]
13. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Solitons in PT-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun. 285, 3320–3324 (2012). [CrossRef]
14. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in 𝒫𝒯 -symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85, 013831 (2012). [CrossRef]
15. K. Zhou, Z. Guo, J. Wang, and S. Liu, “Defect modes in defective parity-time symmetric periodic complex potentials,” Opt. Lett. 35, 2928–2930 (2010). [CrossRef] [PubMed]
16. H. Wang and J. Wang, “Defect solitons in parity-time periodic potentials,” Opt. Express 19, 4030–4035 (2011). [CrossRef] [PubMed]
17. Z. Lu and Z.-M. Zhang, “Defect solitons in parity-time symmetric superlattices,” Opt. Express 19, 11457–11462 (2011). [CrossRef] [PubMed]
18. S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85, 043826 (2012). [CrossRef]
19. Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with 𝒫𝒯 -symmetric potentials,” Phys. Rev. A 84, 053855 (2011). [CrossRef]
20. S. Liu, C. Ma, Y. Zhang, and K. Lu, “Bragg gap solitons in symmetric lattices with competing nonlinearity,” Opt. Commun. 285, 1934–1939 (2012). [CrossRef]
21. L. Chen, R. Li, N. Yang, D. Chen, and L. Li, “Optical modes in PT-symmetric double-channel waveguides,” Proc. Romanian Acad. A 13, 46–54 (2012).
22. S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in 𝒫𝒯 -symmetric optical lattices,” Phys. Rev. A 85, 023822 (2012). [CrossRef]
23. A. E. Kaplan, “Bistable solitons,” Phys. Rev. Lett. 55, 1291–1294 (1985). [CrossRef] [PubMed]
24. J. Wang, F. Ye, L. Dong, T. Cai, and Y.-P. Li, “Lattice solitons supported by competing cubic-quintic nonlinearity,” Phys. Lett. A 339, 74–82 (2005). [CrossRef]
25. J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, “Truncated-Bloch-wave solitons in optical lattices,” Phys. Rev. A 79, 043610 (2009). [CrossRef]
