Defect solitons in saturable nonlinearity media with parity-time symmetric optical lattices

Physica B 429 (2013) 28–32

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Defect solitons in saturable nonlinearity media with parity-time symmetric optical lattices Sumei Hu a,b, Wei Hu b,n a b

Department of Physics, Guangdong University of Petrochemical Technology, Maoming 525000, PR China Laboratory of Nanophotonic Functional Materials and Devices, South China Normal University, Guangzhou 510631, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 8 May 2013 Received in revised form 19 July 2013 Accepted 24 July 2013 Available online 1 August 2013

We reported the existence and stability of defect solitons in saturable nonlinearity media with paritytime (PT) symmetric optical lattices. Families of fundamental and dipole solitons are found in the semiinfinite gap and the first gap. The power of solitons increases with the increasing of the propagation constant and saturation parameter. The existence areas of fundamental and dipole solitons shrink with the growth of saturation parameter. The instability of dipole solitons for positive and no defect induced by the imaginary part of PT symmetric potentials can be suppressed by the saturation nonlinearity, but for negative defect it cannot be suppressed by the saturation nonlinearity. & 2013 Elsevier B.V. All rights reserved.

Keywords: Soliton Parity-time symmetry Optical lattice Saturable nonlinearity Defect soliton

1. Introduction In 1998, Bender et al. found that a wide class of non-Hermitian Hamiltonians can actually possess entirely real spectra as long as they respect parity-time (PT) symmetry [1]. The concept was introduced to optics by many people [2,4–6]. A necessary condition for a Hamiltonian to be PT symmetry is that its complex potential satisfies V (x) ¼V n(  x) [7,8]. This implies that the real part of the potential must be an even function of position and that the imaginary part must be odd. In optics, PT-symmetric structures can be constructed by inclusion of gain or loss regions into waveguides, which make the complex refractive-index distribution obeying the condition nðxÞ ¼ nn ðxÞ [3–6]. Experimental realizations of such PT systems have been reported recently. Guo et al. have observed passive PT-symmetry breaking and phase transition that lead to a loss-induced optical transparency in specially designed pseudo-Hermitian guiding potentials [9]. Rüter et al. have observed the spontaneous PT symmetry breaking and power oscillations violating left-right symmetry in PT optical coupled linear system involving a complex index potential [10]. PT symmetries have been realized in the LRC circuits [11,12], and dual behavior of PT-symmetric scattering has been observed [13]. Recently, Regensburger et al. represent the application of PTsymmetry to a new generation of multifunctional optical devices

n

Corresponding author. Tel./fax: +86 20 39310083. E-mail address: [email protected] (W. Hu).

0921-4526/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2013.07.023

and networks experimentally [14]. Those theoretical and experimental results led to the proposal of a new class of PT-symmetric synthetic materials with intriguing and unexpected properties [10,14,15]. In optics, nonlinearities in the PT-symmetric systems have been considered by many authors [15–19], especially in the PT-symmetric optical lattices [5,6,15], and some new kinds of soliton were found and investigated [5,20–23]. Defect solitons in optical lattices with specially designed defect have attracted special attention due to their novel and unique characteristics in diverse areas of physics and have been applied extensively for steering of optical beams [24–27], all-optical switches [28,29], filtering [30] and routing of optical signals [31]. Defect solitons in local Kerr nonlinearity media with PT symmetric optical lattices have been studied, and stable solitons are found mainly in the semi-infinite gap [22]. Recently, we have studied defect solitons in nonlocal Kerr nonlinearity media with PT symmetric optical lattices and found that the nonlocal nonlinearity can expand stable ranges of solitons [23]. However, the optical properties with PT symmetric potentials in saturable nonlinearity media have not been studied. It is noteworthy that the nonlinearity in the photorefractive media, in which Rüter et al. have observed the non-reciprocal wave propagation [10], is saturable nonlinearity. The nonlinearity saturation suppresses the collapse of fundamental solitons in two and three dimensions [32,33], which opens the door for their experimental observation in multidimensional optical beams. The instability of higher-order (multihump) solitons is not suppressed by the nonlinearity saturation in general lattices [34–36]. However, in this paper, we find

S. Hu, W. Hu / Physica B 429 (2013) 28–32

that the instability of dipole solitons can be suppressed by the saturation nonlinearity in the PT symmetric optical lattices. The saturable nonlinearity can expand stable ranges of fundamental and dipole solitons, especially in the first gap. It is found that the stability of defect solitons depends on the defect, the degree of saturable nonlinearity, PT potential and the symmetry of solitons.

2. Theoretical model We consider the propagation of light beam in PT symmetric defective lattices embedded into a focusing saturable medium. The evolution of complex normalized amplitude U of the light field can be described by the following nonlinear Schrödinger equation: i

∂U 1 ∂2 U UjUj2 þ þ p½V ðxÞ þ iW ðxÞU þ ¼ 0; 2 ∂z 2 ∂x 1 þ sjUj2

ð1Þ

where the transverse x and longitudinal z coordinates are normalized to the width and diffraction; p is the depth of PT symmetric potentials; s stands for the degree of saturable nonlinearity; according to Ref. [37], s is depended on the light-induced maximum refractive-index change and s is positive. V(x) and W(x) are the real and imaginary parts of PT symmetric defective potentials, respectively, which are assumed in this paper as VðxÞ ¼ cos 2 ðxÞ½1 þ ϵ expðx8 =128Þ;

WðxÞ ¼ W 0 sin ð2xÞ;

ð2Þ

here ϵ represents the strength of the defect, and defect is expressed as a super-Gaussian profile [22]. ϵ ¼ 0 corresponds to uniform lattice, and the soliton in this lattice is a gap soliton, so we define this lattice as no defect. When ϵ 40, the center refractive index is greater than that of both sides, and the defect is defined as positive defect. When ϵ o 0, the center refractive index is lower than that of both sides, and the defect is defined as negative defect. The parameter W0 represents the strength of the imaginary part compared with the real part. The linearized normalization relation of Eq. (1) is given in Ref. [4]. The linearized version in Eq. (1) has a Bloch band structure when ϵ ¼ 0. The band diagram can be entirely real when the system is operated below the phase transition point (W0 ¼0.5) [5]. In this paper, only PT lattice with its Bloch spectrum below the phase transition point is considered, and without loss of generality, parameters W0 ¼ 0.15 and W0 ¼0.25 are adopted throughout the paper. Typical PT lattice profile and its Bloch band structure with W0 ¼0.15 are shown in Fig. 1(a) and (b), respectively. From Fig. 1(b), we can see that the region of semi-infinite gap for p¼ 4 and W0 ¼0.15 is b4 2:7, and the first and the second gaps locate in 0:83 o b o 2:63 and 0:55 o b o 0:15, respectively, where b is the propagation constant.

4 2 2

b

PT Lattices

4

0 0 -10

-5

0 x

5

10

-2 -1.0

-0.5

0.0 k

0.5

1.0

Fig. 1. For p ¼4 and W0 ¼ 0.15, (a) profile of the PT lattice (blue and red lines represent the real and imaginary parts, respectively) and (b) band structure of the lattice in (a). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

29

We search for stationary solutions to Eq. (1) in the form U ¼ f ðxÞ expðibzÞ, where f(x) is the complex function satisfies equations: bf ¼

1 ∂2 f f jf j2 þ p½V ðxÞ þ iW ðxÞf þ : 2 2 ∂x 1 þ sjf j2

ð3Þ

The solutions of defect solitons are gotten numerically from Eq. (3). Families of solitons are determined by the propagation constant b, saturation parameter s, lattice depth p, and the strength of the imaginary part of PT symmetric potential W0. Without loss of generality, we fixed lattice depth p ¼4 and varied b, s, W0 throughout the paper unless stated otherwise. To elucidate the stability of defect solitons, we search for the perturbed solution to Eq. (1) in the form Uðx; zÞ ¼ ½f ðxÞ þ uðx; zÞþ ivðx; zÞ expðibzÞ, where the real [uðx; zÞ] and imaginary [vðx; zÞ] parts of the perturbation can grow with a complex rate δ upon propagation, i.e. uðz; xÞ ¼ p1 ðxÞeδz and vðz; xÞ ¼ p2 ðxÞeδz , respectively. Linearization of Eq. (1) around the stationary solution f(x) yields the eigenvalue problem δv ¼

1 ∂2 u bu þ pðVuWvÞ 2 ∂x2 þ 

f½Reðf Þ2 ½Imðf Þ2 þ 2jf j2 gu þ 2 Reðf Þ Imðf Þv ð1 þ sjf j2 Þ

2sjf j2 Reðf Þ½Reðf Þu þ Imðf Þv ð1 þ sjf j2 Þ2

;

ð4Þ

1 ∂2 v þ bvpðVu þ WvÞ 2 ∂x2 f½Imðf Þ2 ½Reðf Þ2 þ 2jf j2 gv þ 2 Reðf Þ Imðf Þu  ð1 þ sjf jÞ

δu ¼ 

þ

2sjf j2 Imðf Þ½Reðf Þu þ Imðf Þv ð1 þ sjf j2 Þ2

:

ð5Þ

The above coupled equations can be solved numerically to find the maximum value of ReðδÞ. If ReðδÞ 4 0, solitons are unstable. Otherwise, they are stable.

3. Defect solitons In the saturable nonlinearity media with PT symmetric defective lattices, we find two types of defect solitons for positive, no, and negative defect cases, as shown in Figs. 2 and 3. The first type is nodeless fundamental solitons, which can exist stably in the semi-infinite gap. We consider that fundamental solitons are PT symmetric, because their real parts are even and imaginary parts are odd, that is similar to the PT symmetric potential. The other type of defect solitons, which exists in the first gap, is called dipole solitons in this paper, because they have two significant intensity peaks. The real parts of dipole solitons are odd and the imaginary parts are even, that is opposite to the PT symmetric potentials and fundamental solitons. In this paper we state that dipole solitons are PT antisymmetric. For positive, no, and negative defect, we take ϵ ¼ 0:5, ϵ ¼ 0 and ϵ ¼ 0:5, respectively. The field profiles of the defect solitons are shown in Fig. 2(a)–(d) for different defects, saturation parameter s and symmetric. Fig. 2(a) shows fundamental soliton in the semiinfinite gap, while dipole soliton in the first gap is shown in Fig. 2 (b) for positive defects. One can see that the poles of dipole solitons are located inside the central channel of the lattice. The properties of defect solitons with zero defects are similar to those with positive defects. Differing from the positive and zero defects, fundamental solitons can exist in both the semi-infinite gap and the first gap, as shown in Fig. 2(c) and (d).

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S. Hu, W. Hu / Physica B 429 (2013) 28–32

18 s=1

P

12 s=1

6

0.5 0

1

0.5 2

3

4

5

6

b 18

18

12

12

6 0

2

s=1 6 0.5

0.5 3

4

5

0

6

b

Fig. 2. The complex fields (solid blue: real part; dotted red: imaginary part) for soliton solutions at (a) b¼ 5, s ¼1; (b) b ¼2.3, s ¼1; (c) b ¼2.8, s ¼0.2; and (d) b ¼1.8, s ¼1, respectively. For all cases p ¼4. For (a) and (b) W0 ¼0.15, ϵ ¼ 0:5, and (c) and (d) W0 ¼ 0.6, ϵ ¼ 0:5. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

20

3

2

4

9 s=0.2

s=0.2

6 s=1

P

P

15 10

3

5 0

1

0.5

b

25

R þ1

The power of solitons is defined as P ¼ 1 jf ðxÞj2 dx. Fig. 3 shows the power of solitons for positive, no, and negative defect. For the positive defect, Fig. 3(a) and (b) shows the power versus propagation constant for different saturation parameter s with W0 ¼0.15 and W0 ¼0.25, respectively. The results show that the power of solitons for both fundamental and dipole solitons increases with the increasing of the propagation constant b and saturation parameter s, but it decreases with the increasing of W0, when comparing Fig. 3(a) with (b). Fundamental and dipole solitons vanish when the propagation constant is below the cutoff point, whose value does not depend on the saturation parameter. This feature is similar to the case of the traditional uniform lattices and PT symmetric lattices in the nonlocal media [23]. The power of defect solitons with no defect is similar to that with positive defects except that the cutoff points approach the edges of gaps, as shown in Fig. 3(c). It shows that the propagation constants of cutoff points decrease with decreasing of the defect peak. So for the negative defect, the cutoff points of fundamental solitons shift to the first gap. Fig. 3(d) shows the power versus propagation constant for different saturation parameter s with W0 ¼0.6 for fundamental solitons in the semi-infinite gap and dipole solitons in the first gap. When s 40:3, the fundamental solitons in the semi-infinite gap and dipole solitons in the first gap cannot exist, so we choose s ¼0.1 and s ¼0.2 in Fig. 3(d). Fig. 3(e) shows the power of the fundamental solitons in the first gap. We can see that there exists a cutoff point of propagation constant above which the fundamental solitons can exist in the first gap. We find that there exists a critical propagation constant bmax of fundamental and dipole solitons for a fixed saturation parameter s, above which the solitons vanish. Fig. 4(a), (b), and (c) shows the relation of the bmax and s for positive, no, and negative defect, respectively. One can see that the existence range of fundamental and dipole solitons is b o bmax for a fixed s, and bmax decreases with increasing of saturation degree s. The gray dense regions show the existence regions of solitons. We can see that the existence areas of fundamental and dipole solitons shrink with the growth of saturation parameter s. When the saturation degree s increases,

s=1

0.5 1

s=1

P

P

s=1

0.1

0.1 1

2

3

4

b

0.5 5

6

7

0

0.5 1.0 1.5 2.0 2.5

b

Fig. 3. The power versus propagation constant for fundamental solitons in the semi-infinite gap and dipole solitons in the first gap with (a) ϵ ¼ 0:5, W0 ¼ 0.15; (b) ϵ ¼ 0:5, W0 ¼ 0.25; (c) ϵ ¼ 0, W0 ¼ 0.15; (d) ϵ ¼ 0:5, W0 ¼ 0.15, and fundamental solitons in the first gap with (e) ϵ ¼ 0:5, W0 ¼0.15. For all cases p¼ 4.

the term sjUj2 b1, so the system approximatively becomes linear, and no solitons can exist in the system. Fig. 5(a), (b), and (c) shows the linear stability analysis of fundamental solitons in the semi-infinite gap and dipole solitons in the first gap for positive, zero, and negative defect, respectively. For positive defect, we find that fundamental solitons are stable in the whole regime where solitons exist in the semi-infinite gap, like its counterparts in nonlocal PT lattices. Dipole solitons are found to exist stably in the whole existence regime in the first gap with W0 ¼0.15 and W0 ¼0.25 for s¼ 1. For comparison, dipole solitons in Kerr media with PT symmetric defective lattices are stable only in a small region near the edge of the first gap [22]. As W0 increases, the stability region of defect soliton decreases. From Fig. 5(a), we can see that dipole solitons are unstable for the large propagation constant with low saturation parameter s, and the stable region increases with increasing of saturation parameter s. It shows that the saturation nonlinearity expands the stable ranges of dipole solitons. However, the saturation nonlinearity cannot expand the stable ranges of higher-order (multihump) solitons in general lattices [34–36]. So the instability of multihump solitons induced by the large imaginary part of PT symmetric potentials can be suppressed by the saturation nonlinearity. The instability of dipole solitons for no defect can also be suppressed by the saturation nonlinearity, as shown in Fig. 5(b). However, from Fig. 5(c), we can

S. Hu, W. Hu / Physica B 429 (2013) 28–32

12

31

12 ε=0.5

9

9

b

b

ε=0 6 3 0

6 3

0.5

1.0 s

1.5

0

2.0

0.5

1.0 s

1.5

2.0

9

6 b

ε=-0.5

3

0

0.5

1.0 s

1.5

2.0

Fig. 4. For p ¼ 4 and W0 ¼0.15, the existence areas of fundamental solitons in the semi-infinite gap and dipole solitons in the first gap versus saturation parameter s with (a) ϵ ¼ 0:5; (b) ϵ ¼ 0; and fundamental solitons in the semi-infinite gap and the first gap with (c) ϵ ¼ 0:5. (The blue regions are bands while the blank regions are gaps, and the gray dense regions are the existence areas of solitons.) (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

Fig. 6. Evolutions of fundamental solitons in the semi-infinite gap and dipole solitons in the first gap with (a) ϵ ¼ 0:5, W0 ¼ 0.25, b ¼5, s ¼1; (b) ϵ ¼ 0:5, W0 ¼0.15, b¼ 2.8, s ¼0.2; (c) ϵ ¼ 0:5, W0 ¼0.25, b¼ 2.3, s ¼0.1; (d) ϵ ¼ 0:5, W0 ¼0.25, b¼ 2.3, s ¼ 1; (e) ϵ ¼ 0, W0 ¼0.15, b¼ 1.7, s ¼0.1; and (f) ϵ ¼ 0, W0 ¼0.15, b¼ 1.7, s ¼ 1. For all cases p ¼ 4.

instability of multihump solitons induced by the large imaginary part of PT symmetric potentials mainly. The propagations of defect solitons are simulated in Fig. 6 based on Eq. (1), and 1% random-noise perturbation is added into the initial input to verify the results of linear stability analysis. We can see that fundamental solitons for positive defect can propagate stably and fundamental solitons at the edge of the gap for negative defect are unstable, which is shown in Fig. 6(a) and (b). For positive and no defect, dipole solitons are unstable for s¼0.1 but they are stable for s¼ 1, as shown in Fig. 6(c)–(f).

4. Summary Fig. 5. The unstable growth rate ReðδÞ for fundamental solitons in the semi-infinite gap and dipole solitons in the first gap with (a) W0 ¼0.25, ϵ ¼ 0:5; (b) W0 ¼ 0.15, ϵ ¼ 0; and (c) W0 ¼0.15, ϵ ¼ 0:5. For all cases p¼ 4.

see that the instability of dipole solitons for negative defect cannot be suppressed by the saturation nonlinearity. It is because that the instability of dipole solitons for negative defect is induced not only by the imaginary part of PT symmetric potentials but also by the negative defect [22], and the saturation nonlinearity can stress the

In conclusion, we have studied the existence and stability of defect solitons supported by PT symmetric optical lattices with saturation nonlinearity. It is found that the saturation nonlinearity expands stable regions of defect solitons, especially the dipole solitons in the first gap. The power of solitons increases with the increasing of the propagation constant and saturation parameter. The existence areas of fundamental and dipole solitons shrink with the growth of saturation parameter s. The instability of dipole

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S. Hu, W. Hu / Physica B 429 (2013) 28–32

solitons for positive and no defect induced by the imaginary part of PT symmetric potentials can be suppressed by the saturation nonlinearity, but for negative defect, it cannot be suppressed by the saturation nonlinearity. These properties of the defect solitons in PT symmetric lattices with saturation nonlinearity are obviously different from those in the general lattices.

Acknowledgments This research was supported by the National Natural Science Foundation of China (Grant nos. 10804033, 11174090 and 11174091). References [1] C.M. Bender, S. Boettcher, Phys. Rev. Lett. 80 (1998) 5243. [2] A. Ruschhaupt, F. Delgado, J.G. Muga, J. Phys. A 38 (2005) 171. [3] M. Kulishov, J.M. Laniel, N. Bálanger, J. Azańa, D.V. Plant, Opt. Express 13 (2005) 3068. [4] R. El-Ganainy, K.G. Makris, D.N. Christodoulides, H. Ziad, Opt. Lett. 32 (2007) 2632. [5] Z.H. Musslimani, K.G. Makris, R. El-Ganainy, D.N. Christodoulides, Phys. Rev. Lett. 100 (2008) 030402. [6] K.G. Makris, R. El-Ganainy, D.N. Christodoulides, Z.H. Musslimani, Phys. Rev. Lett. 100 (2008) 103904. [7] C.M. Bender, D.C. Brody, H.F. Jones, J. Math. Phys. 40 (1999) 2201. [8] C.M. Bender, D.C. Brody, H.F. Jones, Am. J. Phys. 71 (2003) 1095. [9] A. Guo, G.J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, D.N. Christodoulides, Phys. Rev. Lett. 103 (2009) 093902. [10] C.E. Ruter, K.G. Makris, R. El-Ganainy, D.N. Christodoulides, M. Segev, D. Kip, Nat. Phys. 6 (2010) 192.

[11] J. Schindler, A. Li, M.C. Zheng, F.M. Ellis, T. Kottos, Phys. Rev. A 84 (2011) 040101 (R). [12] H. Ramezani, J. Schindler, F.M. Ellis, U. Gunther, T. Kottos, Phys. Rev. A 85 (2012) 062122. [13] Z. Lin, J. Schindler, F.M. Ellis, T. Kottos, Phys. Rev. A 85 (2012) 050101(R). [14] A. Regensburger, C. Bersch, MA. Miri, G. Onishchukov, D.N. Christodoulides, U. Peschel, Nature 488 (2012) 167. [15] Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, D.N. Christodoulides, Phys. Rev. Lett. 106 (2011) 213901. [16] A.A. Sukhorukov, Z.Y. Xu, Y.S. Kivshar, Phys. Rev. A 82 (2010) 043818. [17] A.E. Miroshnichenko, B.A. Malomed, Y.S. Kivshar, Phys. Rev. A 84 (2011) 012123. [18] F.Kh. Abdullaev, Y.V. Kartashov, V.V. Konotop, D.A. Zezyulin, Phys. Rev. A 83 (2011) 041805(R). [19] D.A. Zezyulin, Y.V. Kartashov, V.V. Konotop, Europhys. Lett. 96 (2011) 64003. [20] X. Zhu, H. Wang, L.X. Zheng, H.G. Li, Y.J. He, Opt. Lett. 36 (2011) 2680. [21] K.Y. Zhou, Z.Y. Guo, J.C. Wang, S.T. Liu, Opt. Lett. 35 (2010) 2928. [22] W. Hang, J.D. Wang, Opt. Express 19 (2011) 4030. [23] S.M. Hu, X.K. Ma, D.Q. Lu, Y.Z. Zheng, W. Hu, Phys. Rev. A 85 (2012) 043826. [24] V.A. Brazhnyi, V.V. Konotop, V.M. Pérez-García, Phys. Rev. Lett. 96 (2006) 060403. [25] V.A. Brazhnyi, V.V. Konotop, V.M. Pérez-García, Phys. Rev. A 74 (2006) 023614. [26] A. Piccardi, G. Assanto, L. Lucchetti, F. Simoni, Appl. Phys. Lett. 93 (2008) 171104. [27] V. Kartashov Yaroslav, Torner Lluis, A. Vysloukh Victor, Opt. Lett. 29 (2004) 1102. [28] R. Rosberg Christian, L. Garanovich Ivan, A. Sukhorukov Andrey, N. Neshev Dragomir, Krolikowski Wieslaw, S. Kivshar Yuri, Opt. Lett. 31 (2006) 1150. [29] F. Ye, Y.V. Kartashov, V.A. Vysloukh, L. Torner, Phys. Rev. A 78 (2008) 013847. [30] A. Sukhorukov Andrey, S. Kivshar Yuri, Opt. Lett. 30 (2005) 1849. [31] D.K. Campbell, S. Flach, Y.S. Kivshar, Phys. Today 57 (2004) 43. [32] J.M. Soto-Crespo, E.M. Wright, N.N. Akhmediev, Phys. Rev. A 45 (1992) 3168. [33] M. Mitchell, M. Segev, D.N. Christodoulides, Phys. Rev. Lett. 80 (1998) 4657. [34] J.M. Rost, J.S. Briggs, J.M. Feagin, Phys. Rev. Lett. 66 (1991) 1642. [35] W.J. Firth, D.V. Skryabin, Phys. Rev. Lett. 79 (1997) 2450. [36] J.K. Yang, Phys. Rev. E 66 (2002) 026601. [37] S. Gatz, J. Herrmann, J. Opt. Soc. Am. B 8 (1991) 2296.