Discrete solitons in saturable nonlinearity media with parity-time symmetric lattices

Optics Communications 331 (2014) 105–110

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Discrete solitons in saturable nonlinearity media with parity-time symmetric lattices Qianggui Song a, Zhiwei Shi a, Yang Li a, Huagang Li b,n a b

Faculty of Information Engineering, Guangdong University of Technology, Guangzhou 510006, PR China Department of Physics, Guangdong University of Education, Guangzhou 510303, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 21 April 2014 Received in revised form 23 May 2014 Accepted 2 June 2014 Available online 12 June 2014

We study the existence of discrete solitons in saturable nonlinearity media with parity-time symmetric lattices. We find that there exist two kinds of discrete solitons, they share the same parameters but different energy flows. The degree of saturable nonlinearity can influence the properties of such discrete solitons. Stable discrete solitons are found near the anti-continuum limit. & 2014 Published by Elsevier B.V.

Keywords: Parity-time Saturable nonlinearity Discrete solitons

1. Introduction The discrete nonlinear Schrödinger (DNLS) equation [1], namely a discrete self-trapping equation [2], has been widely studied over the last decade due to its applications in nonlinear optics [3] and Bose–Einstein condensates [4]. Discrete soliton, governed by DNLS equation, can be shaped by the interaction between lattice discrete diffraction (or dispersion) and material nonlinearity in the context of optical waveguide [3]. Such solitons were first predicted in 1988 by Christodoulides and Joseph [5] and have been successfully observed in one- and two-dimensional nonlinear waveguide arrays [6–8]. Since Bender and his co-workers found that non-Hermitian Hamiltonian can exhibit entirely real eigenvalue spectra provided they satisfy parity-time symmetry in 1998 [9], much attention has been paid to the concept. If the complex potential of a Hamiltonian has symmetric real and antisymmetric imaginary parts [10–12], we say that the Hamiltonian holds PT symmetry. PT symmetry also has applications in many areas, such as nonlinear optics [13], nuclear physics [14], condensed matter [15] and quantum field theory [16]. Recently, a PT symmetric coupled system has been reported in the field of nonlinear optics [17]. Such system, described by the PT symmetric DNLS equation, is composed of periodic coupled waveguides with one gain and another loss. In the past few years, the PT symmetric DNLS equations have attracted much attention. Many topics have been considered, such

n

Corresponding author. E-mail address: [email protected] (H. Li).

http://dx.doi.org/10.1016/j.optcom.2014.06.002 0030-4018/& 2014 Published by Elsevier B.V.

as stable discrete solitons in nonlinear binary arrays with PT symmetric waveguides [18], stationary solution in PT symmetric oligomers [19,20], a necklace of coupled optical waveguides with PT symmetry [21] and classify different solutions in PT symmetric DNLS equations with alternating coupling coefficients [22]. A systematic way of analyzing the nonlinear stationary states with the implicit function theorem is developed to discuss the PT symmetric DNLS equations [23]. PT symmetric dimers and quadrimers in the range of the gain and loss coefficients in which the zero equilibrium state is neutrally stable are studied in [24]. The existence and dynamics of the PT symmetric solitons with a finite number of sites are analyzed in [25]. However, the above researches are based on the Kerr type nonlinearity. In the present work, we address the PT symmetric DNLS equations with saturable nonlinearity and alternating coupling coefficients by analytical continuation method [22]. There exist two solutions under the same parameters. The properties of these solutions are affected by the degree of the saturable nonlinearity. Stability near and in the anti-continuum limit is also examined by linear stability analysis. To substantiate our linear stability analysis, we perform simulation of the propagation of PT symmetric discrete soliton. We find that there exist stable solutions.

2. Theoretical model We consider such a PT symmetric waveguide array illustrated in Fig. 1. The white waveguides represent gain, the gray ones represent loss. All these waveguides can play the roles of complex PT symmetric potentials [26]. This system is governed by two

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coupled PT symmetric DNLS equations written as 8 > ∂U n jU n j2 U n > > þ C 0 ðV n U n Þ þ C 1 ðV n  1 U n Þ   iγ U n ¼ 0; >i < ∂z 1 þ sjU n j2 > ∂V n jV n j2 V n > > > : i ∂z þ C 0 ðU n V n Þ þ C 1 ðU n þ 1 V n Þ  1 þsjV j2 þ iγ V n ¼ 0:

ð1Þ

n

Without loss of generality, we fix C 0 ¼ 1 and let C 1 ¼ ϵ such that Eq. (1) becomes 8 > ∂U n jU n j2 U n > > þ V n  U n þ ϵðV n  1  U n Þ   iγ U n ¼ 0; >i < ∂z 1 þsjU n j2 ð2Þ > ∂V n jV n j2 V n > > > : i ∂z þ U n  V n þ ϵðU n þ 1  V n Þ  1 þ sjV j2 þ iγ V n ¼ 0; n where Un and Vn are the evolutions of complex normalized amplitude in the waveguide n, z represents longitudinal coordinates, ϵ is the coupling coefficient which must be positive, γ stands for the gain–loss coefficient, s is the degree of saturable nonlinearity and should also be positive [27]. The anti-continuum

limit corresponds to ϵ ¼0 [28]. Such structure in the linear case [29] and the Kerr type nonlinearity [22] has been discussed. This model depicts a waveguide array with alternating coupling coefficient. The PT symmetry is unbroken if [18] j1  ϵjZ γ :

ð3Þ

To do analytical continuation, we should first find the analytical solution of Eq. (2) in the anti-continuum limit. We seek for stationary solutions in the form of ðU n ; V n Þ ¼ ðun ; vn Þe  iEz , where E ¼ 1 þ ϵ þ β, β is propagation constant, un and vn are complex functions and independent of z. Boundary conditions are un ; vn -0 as n-1. By substituting ðun ; vn Þe  iEz into Eq. (2) we obtain 8 > jun j2 un > > > < ðβ  iγ Þun þ vn þ ϵvn  1 ¼ 1 þsju j2 ; n ð4Þ > jvn j2 vn > ðβ þ iγ Þv þ u þ ϵu > : > n n nþ1 ¼ 2 : 1 þ sjvn j We suppose that the PT symmetric condition on the whole infinite lattice is un ¼ v  n ; vn ¼ u  n ; …:

ð5Þ

Considering that only one center dimer is excited which means only u0 ; v0 a 0 and ϵ ¼0, we obtain 8 > ju0 j2 u0 > > ðβ  iγ Þu0 þ v0 ¼ ; > < 1 þ sju0 j2 ð6Þ > jv0 j2 v0 > > β þ i γ Þv þ u ¼ : ð > 0 0 : 1 þ sjv0 j2 If PT symmetry holds i.e., u0 ¼ v 0 and v0 ¼ u 0 , we can rewrite the first equation of Eq. (6) as ðβ  iγ Þu0 þ u 0 ¼

ju0 j2 u0 : 1 þsju0 j2

ð7Þ

Eq. (7) has two solutions u07 ¼ A 7 eiα 7 ;

5

5

4

4

|un|,|vn|

|un|,|vn|

Fig. 1. Sketch of the PT symmetric array of waveguides with gain (white ones) and loss (gray ones).

3 2 1

3 2 1

0 −4

0

0 −4

4

30

60

15

30

0

5

0

4

n

P

P

n

0

ð8Þ

10

0

0

5

10

Fig. 2. Discrete solitons are illustrated in (a) with s ¼0.01 and (b) with s ¼0.05, the other parameters are ϵ ¼0.5, β ¼10 and γ ¼0.1. The filled circles correspond to jun j and empty circles correspond to jvn j. (c) and (d) are P versus ϵ of (a) and (b) respectively. The curves with red are plus solutions while blue ones correspond to minus solutions. All quantities are plotted in arbitrary dimensionless unit. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Q. Song et al. / Optics Communications 331 (2014) 105–110

pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi where A27 ¼ ðβ 7 1  γ 2 Þ=ð1 sðβ 7 1  γ 2 ÞÞ and  γ ¼ sin ð2α 7 Þ. v0 can be obtained by v0 ¼ u 0 . These solutions exist for γ A ½0; 1Þ if

qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1  γ2 o β o 8 1  γ2: s

ð9Þ

3. Numerical results and discussion In this section, we study the discrete solitons numerically. We define ðu0þ ; v0þ Þ as plus solution, while ðu0 ; v0 Þ corresponds to minus solution. These solutions bifurcate from the anti-continuum limit, which means that these solutions come from the point ϵ ¼0 (see Eq. (7)). Solutions with different s are shown in Fig. 2 with

1000

800

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0

P

P

8

0

40

0

80

0

30

8

0

0

0

40

10

20

β

f

f

β

−30

107

−8

80

0

β

10

20

β

500

400

400

300

300

z

z

500

200

200

100

100

0 −40

0

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40

n

0

40

n

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400

400

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300

z

z

500

200

200

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100

0 −40

0

n

40

0 −40

0

40

n

Fig. 3. f versus β, P versus β and propagations of Fig. 2(a) and (b). The curves with red are plus solutions while blue ones correspond to minus solutions. The points in (c) and (d) marked with green circles correspond to β ¼ 10. (a) and (b) are P versus β of Fig. 2(a) and (b) respectively. (e) and (f) are propagations with variant noise δnoise ¼ 0:05 of plus solutions, while (g) and (h) correspond to those of minus solutions respectively. All quantities are plotted in arbitrary dimensionless unit. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

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(a) at s¼0.01 and (b) at s¼0.05. Apparently we can see that two solitons with the same parameters show different shapes, the amplitude of the plus solutions is always larger than that of minus solutions. In Fig. 2(c) and (d), we illustrate P versus

ϵ, where

P ¼ ∑n ðjun j2 þjvn j2 Þ corresponds to power or energy flow. In addition, large degree of the saturable nonlinearity results large energy flow for both solutions. For plus solutions, powers decrease

saturable nonlinearity is close to the Kerr nonlinearity. Note that there exists a termination point where the localized solution vanishes due to the fact that the Jacobian operator becomes singular for plus solutions [22]. In Fig. 2(c) and (d), the termination points are 3.99 and 3.15 respectively. While for different s, all minus solutions show the same behaviors. Powers of all minus

increase but slowly (see Fig. 2(c)), since s is smaller enough,

ϵcr ¼ 9:00. To show the influence of the degree of saturable

5

5

4

4

3

3

|un|,|vn|

|un|,|vn|

with the increase of ϵ, if s 4 0:01. Otherwise powers will always

solutions decrease with the increase of ϵ and disappear at the edge qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 of the linear spectrum, i.e., ϵ ¼ ϵcr ¼ β þ γ 2  1 [22], here

2

1

1 0 −4

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0

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n

0

4

n 500

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z

z

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n

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z

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Fig. 4. Panels in the left (right) correspond to plus (minus) solutions. Discrete solitons are illustrated in (a) and (b), the red line and green line correspond to ϵ¼ 0 and ϵ ¼0.05, while the blue line is to ϵ ¼3. The other parameters are the same as those of Fig. 2(b). (c), (e) and (g) are propagations of (a) with ϵ¼ 0, ϵ¼ 0.05 and ϵ¼ 3 respectively while (d), (f) and (h) correspond to propagations of (b). All propagations are with variant noise δnoise ¼ 0:05. All quantities are plotted in arbitrary dimensionless unit. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Q. Song et al. / Optics Communications 331 (2014) 105–110

nonlinearity in more detail, we also visualize P versus β with different s in Fig. 3(a) and (b). The result shows that powers of all solutions increase very quickly with the increase of of P versus

β. The curves

β with larger degree of saturable are more steeper.

Comparing with the Kerr case in [22], the propagation constant β cannot take on any value and has a certain range affected by the degree of saturable nonlinearity s (see inequality (9)). Now we study the stability of one center dimer case in the anticontinuum limit. At the ϵ ¼ 0; n a 0, zero solution is stable for γ o 1 [22]. For ϵ ¼ 0 and n ¼ 0, we assume that w0 ¼ ðu0 ; v0 ÞT and perturb them as W 0 ¼ w0 þ ϕ0 eλz þ ϕ1 eλ z :

ð10Þ

By substituting Eq. (10) into Eq. (2) at eigenvalue problem

L

!

ϕ0 ϕ1

ϵ ¼0, we obtain the

!

ϕ0 ; ϕ1

¼ iλ

ð11Þ

where L¼

L0

L1

 L1

 L0

! ;

ð12Þ

0

2ju07 j2 þ sju07 j4  β þiγ B B ð1 þ sju 7 j2 Þ2 0 B L0 ¼ B B @ 1

1 1

C C C C; C 2jv07 j2 þ sjv07 j4 A  β  i γ ð1 þ sjv07 j2 Þ2

ð13Þ

0

B B ð1 þ sju 7 j2 Þ2 0 B L1 ¼ B B @0

0 ðv07 Þ2

ð1 þ sjv07 j2 Þ2

C C C C: C A

ð14Þ

If inequality (9) and f 4 0 hold, we say that these solitons are stable for γ A ½0; 1Þ [19]. We define f as stability parameter. We can use the above discussion to ensure the stability of one dimer soliton in or near the anti-continuum limit ð0 o ϵ{1Þ [22]. When we fix γ and β, s can affect f, namely can also affect the soliton stability. If

γ and s are fixed, Eq. (15) becomes a quadratic function

which can show the relationship between f and β. To confirm the results of the above linear stability discussion, we show simulation of the propagation dynamics of the PT symmetric discrete solitons perturbed by input noise with variance δnoise ¼ 0:05 that means W 0 ¼ w0 ð1 þ δnoise Þ, where w0 is the soliton solution. Propagations of plus solutions with s ¼0.01 and s¼0.05 are illustrated in Fig. 3(e) and (f), while those of minus solutions are illustrated in Fig. 3(g) and (h). We can see that plus (minus) solutions are stable (unstable) due to f 4 0 ðf o 0Þ (see Fig. 3(a) and (b)). And there occur energy transfer periodically between two center waveguides in minus solution (see Fig. 3 (g) and (h)), since coupling coefficient between two center waveguides is large enough and power is low enough, one of the center waveguides overlaps with another one [30,31]. Moreover, the result can support the idea that the anti-Vakhitov–Kolokolov criterion ðdP=dβ 4 0Þ is a necessary (but, generally, not sufficient) condition for the stability of solitons [32]. In Fig. 4, we depict discrete solitons with ϵ ¼0 (decouple state), ϵ ¼0.05 and ϵ ¼ 3, the

0.1

|un|,|vn|

|un|,|vn|

1

u0 and v0 can be obtained by Eq. (7). If s¼ 0, we can derive the Kerr state [22]. We can use the characteristic polynomial DðλÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi λ2 ðλ2 þ 4 1  γ 2 f Þ to determine the stability, where qffiffiffiffiffiffiffiffiffiffiffiffiffi f ¼ ð1  βsÞð2 1  γ 2 7 β Þ 8 sð1  γ 2 Þ: ð15Þ

2

1

0 −4

ðu07 Þ2

109

0

0.05

0 −4

4

n

0

4

n 500

400

400

300

300

z

z

500

200 100 0 −40

200 100

0

n

40

0 −40

0

40

n

Fig. 5. Solitons with s ¼ 0.05, ϵ ¼ 10  3 , γ ¼0.1 and β¼ 1. (a) is plus solution and (b) is minus solution. The filled circles correspond to jun j and empty circles correspond to jvn j. (c) and (d) are propagations of (a) and (b) with variant noise δnoise ¼ 0:05. All quantities are plotted in arbitrary dimensionless unit.

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other parameters are identical with those of Fig. 2(b). The stability of two solitons with ϵ ¼0 and ϵ ¼ 0.05 is the same as that of Fig. 2 (b). While soliton with ϵ ¼ 3 shows different properties. Both solutions are unstable because larger ϵ make energy transfer occur easily between two nearby waveguides. Note that the established stability results resemble those found for the case s¼ 0 in [22] where the plus modes with propagation constant equal to 10 are also typically more stable than the minus modes. All modes tend to lose stability with the increase of ϵ. However, the minus solutions are not always unstable. In Fig. 5, we illustrate solitons and their propagations with ϵ ¼ 10  3 and β ¼1, while the other parameters are the same as Fig. 2(b). The value of stability parameter of plus (minus) solution is 2.79 (0.99). In Fig. 5(c) and (d), we can find that even minus solution with small value of coupling coefficient can be stable. 4. Conclusion In conclusion, discrete solitons in saturable nonlinearity media with PT symmetric lattices are addressed. Two kinds of discrete solitons are found (plus and minus solutions), they have difference in energy flow. These modes vanish at different points when their coupling coefficients reach a certain value. The degree of saturable nonlinearity can affect the properties of both discrete solitons. In fact, discrete solitons with large degree of saturable nonlinearity have large energy flow. Furthermore, the propagation constant cannot take on any value and has a certain range affected by the degree of saturable nonlinearity. Also we study the stability of these discrete solitons and find that there exist stable discrete solitons near the anti-continuum limit, however, the stability lost with the increase of the coupling efficient due to the fact that the interaction between two nearby waveguides become larger and there occurs energy transfer easily. Plus solutions are more stable, whereas minus ones can be stable at small value of coupling coefficient. The degree of the saturable nonlinearity can affect the stability parameter, can also affect the soliton stability. When the stability parameter is more than zero, the solitons are stable; conversely, the ones are unstable. Acknowledgments This research is supported by the Postdoctoral Science Foundation of China (Grant no. 2013M531822) and the Natural Science Foundation of Guangdong Province of China (Grant no. S2012040007188).

References [1] D. Henning, G.P. Tsironis, Phys. Rep. 307 (1999) 333. [2] J.C. Eilbeck, P.S. Lomdahl, A.C. Scott, Phys. Rev. B 30 (1984) 4703. [3] F. Lederer, G.I. Stegeman, D.N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, Phys. Rep. 463 (2008) 1. [4] P.G. Keverkidis, D.J. Frantzeskakis, Mod. Phys. Lett. B 18 (2004) 173. [5] D.N. Christodoulides, R.I. Joseph, Opt. Lett. 13 (1988) 794. [6] H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, J.S. Aitchison, Phys. Rev. Lett. 81 (1998) 3383. [7] J.W. Fleischer, M. Segev, N.K. Efremidis, D.N. Christodoulides, Nature 422 (2003) 147. [8] J.W. Fleischer, T. Carmon, M. Segev, N.K. Efremidis, D.N. Christodoulides, Phys. Rev. Lett. 90 (2003) 023902. [9] C.M. Bender, S. Boettcher, Phys. Rev. Lett. 80 (1998) 5243. [10] C.M. Bender, S. Boettcher, P.N. Meisinger, J. Math. Phys. 40 (1999) 2201. [11] C.M. Bender, D.C. Brody, H.F. Jones, Am. J. Phys. 71 (2003) 1095. [12] A. Ruschhaupt, F. Delgado, J.G. Muga, J. Phys. A Math. Gen. 38 (2005) 171. [13] S.K. Moayedi, A. Rostami, Eur. Phys. J. B 36 (2003) 359. [14] D. Baye, G. Levai, J.M. Sparenberg, Nucl. Phys. A 599 (1996) 435. [15] N. Hatano, D.R. Nelson, Phys. Rev. Lett. 77 (1996) 570. [16] C. Bernard, V.M. Savage, Phys. Rev. D 64 (2001) 085010. [17] C.E. Rüter, K.G. Makris, R. El-Ganainy, D.N. Christodoulides, M. Segev, D. Kip, Nat. Phys. 6 (2010) 192. [18] S.V. Dmitriev, A.A. Suhkorukov, Y.S. Kivshar, Opt. Lett. 35 (2010) 2976. [19] K. Li, P.G. Keverkidis, Phys. Rev. E 83 (2011) 066608. [20] D.A. Zezyulin, V.V. Konotop, Phys. Rev. Lett. 108 (2012) 213906. [21] I.V. Barashenkov, L. Baker, N.V. Alexeeva, Phys. Rev. A. 87 (2013) 033819. [22] V.V. Konotop, D.E. Pelinovsky, D.A. Zezyulin, Europhys. Lett. 100 (2012) 56006. [23] P.G. Kevrekidis, D.E. Pelinovsky, D.Y. Tyugin, SIAM J. Appl. Dyn. Syst. 12 (2013) 1210. [24] P.G. Kevrekidis, D.E. Pelinovsky, D.Y. Tyugin, J. Phys. A Math. Theor. 46 (2013) 365201. [25] D.E. Pelinovsky, D.A. Zezyulin, V.V. Konotop, J. Phys. A Math. Theor. 47 (2014) 085204. [26] K.G. Makris, R. El-Ganainy, D.N. Christodoulides, Z.H. Musslimani, Phys. Rev. Lett. 100 (2008) 103904. [27] S. Gatz, J. Herrmann, J. Opt. Soc. Am. B 8 (1991) 2296. [28] R.S. MacKay, S. Aubry, Nonlinearity 7 (1994) 1623. [29] Z.H. Musslimani, K.G. Makris, R. El-Ganainy, D.N. Christodoulides, Phys. Rev. Lett. 100 (2008) 030402. [30] A. Yariv, IEEE J. Quantum Electron. 9 (1973) 919. [31] S. Somekh, E. Garmire, A. Yariv, H.L. Garvin, R.G. Hunsperger, Appl. Phys. Lett. 22 (1973) 46. [32] H. Sakaguchi, B.A. Malomed, Phys. Rev. A 81 (2010) 013624.