DYNAMICS OF SPATIAL SOLITONS IN PARITY-TIME-SYMMETRIC OPTICAL LATTICES: A SELECTION OF RECENT THEORETICAL RESULTS

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1 DYNAMICS OF SPATIAL SOLITONS IN PARITY-TIME-SYMMETRIC OPTICAL LATTICES: A SELECTION OF RECENT THEORETICAL RESULTS YING-JI HE 1, *, XING ZHU 2, DUMITRU MIHALACHE 3,# 1 Guangdong Polytechnic Normal University, School of Electronics and Information, Guangzhou, China 2 Guangdong University of Education, Department of Physics, Guangzhou , China 3 Horia Hulubei National Institute for Physics and Nuclear Engineering, P.O. Box MG-6, RO , Bucharest-Magurele, Romania * heyingji8@126.com; # Dumitru.Mihalache@nipne.ro Received December 17, 2015 We provide a brief overview of selected recent theoretical studies, which were performed in diverse relevant optical settings, on the key features and unique dynamics of spatial solitons in parity-time-symmetric optical lattices. Key words: localized optical structures, spatial optical solitons, parity-time-symmetric lattices. 1. INTRODUCTION During the past years a new level of understanding has been achieved about conditions for the existence, stability, excitation, and robustness of localized structures in optical and matter-wave media, see, for example, a series of representative works performed in this very broad area, by several research groups [1 23]. Studies of beam dynamics in parity-type-symmetric (PT-symmetric) periodic optical lattices (OLs) have attracted a lot of attention and some unique phenomena were put forward, such as double refraction, power oscillations, nonreciprocal diffraction patterns, spatial soliton formation, etc. Both onedimensional (1D) and two-dimensional (2D) PT-symmetric synthetic linear OLs can be generated in Kerr nonlinear media [24 34]. The intense experimental efforts during the past decade and the corresponding new results have inspired and triggered the theoretical investigations in the area of PT-symmetric optical structures. In the following we briefly mention a series of relevant results reported during the past few years in this fast growing field. Defect modes (both positive and negative defects) in PT-symmetric periodic complexvalued potentials have been studied [35] and spatial solitons in PT-symmetric complex-valued periodic OLs with the real part of the linear superlattice potential were investigated in Ref. [36]. Rom. Journ. Phys., Vol. 61, Nos. 3 4, P , Bucharest, 2016

2 596 Ying-Ji He, Xing Zhu, Dumitru Mihalache 2 Stable 1D and 2D bright spatial solitons in defocusing Kerr media with PT-symmetric potentials have been found, too [37]. Also, it has been found that gray solitons in PT-symmetric complex-valued external potentials can be stable in certain parts of their existence domains [38]. The analysis of stability properties of solitons in PT-symmetric lattices indicates that both 1D and 2D solitons can propagate stably under appropriate conditions [39]. Achilleos et al. [40] considered nonlinear analogs of PT-symmetric linear systems exhibiting defocusing optical nonlinearities. They studied both the ground state and odd excited states (dark- and vortex-solitons) of the system and they put forward the unique features of PT-symmetric optical structures exhibiting selfdefocusing nonlinearities. Driben and Malomed [41] investigated in detail the problem of stability of solitons in PT-symmetric nonlinear optical couplers and reported families of analytic solutions for both symmetric and antisymmetric solitons in dual-core systems with Kerr nonlinearity and PT-balanced gain and loss. Stabilization of solitons in PT-symmetric models with supersymmetry by periodic management in a system based on dual-core nonlinear waveguides with balanced gain and loss acting separately in the cores was investigated in Ref. [42]. Zezyulin and Konotop [43] studied in detail the characteristics of nonlinear modes in finite-dimensional PT-symmetric systems consisting of multi-waveguides of PT-symmetric lattices. The transformations among PT-symmetric systems by rearrangements of waveguide arrays with gain and loss do not affect their pure real linear spectra; however, the nonlinear features of such PT-symmetric systems undergo significant changes, see Ref. [43]. Chen et al. [44] reported the key features of optical modes in PT-symmetric double-channel waveguides. Barashenkov et al. [45] showed that PT-symmetric coupled optical waveguides with gain and loss support localized oscillatory structures similar to the breathers of the classical model. Alexeeva et al. [46] studied spatial and temporal solitons in the PT-symmetric coupler with gain in one waveguide and loss in the other one. It was shown in Ref. [46] that stability properties of both high- and low-frequency solitons are completely determined by a single combination of the soliton s amplitude and the gain-loss coefficient of the coupled waveguides. Bragg gap solitons in PT-symmetric lattices with competing optical nonlinearities of the cubic-quintic (CQ) type have been also investigated in Ref. [47]. Various families of solitons in a CQ medium with an imprinted OL with even and odd geometrical symmetries were found in both the semi-infinite gap and the first gap [48]. Lattice solitons in optical media described by the complex Ginzburg-Landau model with PT-symmetric periodic potentials were studied by He and Mihalache [49]. These solitons can exhibit either a transverse (lateral) drift or a persistent

3 3 Dynamics of spatial solitons in parity-time-symmetric optical lattices 597 swing around the input launching point due to gradient force arising from the spatially inhomogeneous loss [50]. These features are intimately related to the dissipative nature of the system under consideration because they do not arise in the conservative counterpart of the nonlinear dynamical model. Solitons in PT-symmetric external potentials with nonlocal nonlinearity were also investigated [51 55]. The degree of nonlocality can significantly affect the soliton power and the region of stability of PT-symmetric lattice solitons [51]. Defect solitons in PT-symmetric potentials with nonlocal nonlinearity were investigated by Hu et al. [52]. For positive or zero defects, fundamental and dipole solitons can exist stably in the semi-infinite gap and the first gap, respectively, see Ref. [52]. Yin et al. [53] studied the soliton features in PT-symmetric potentials with spatially modulated nonlocal nonlinearity and revealed that there exist stable solitons in the low-power region, and unstable ones in the high-power region. In the unstable cases, the solitons exhibit jump from the original site (channel) to the next one, and they can continue the motion into the other adjacent channels, see Ref. [53]. It should be mentioned that PT-symmetric nonlinear OLs can also support stable discrete solitons [56]. A series of relevant works in the area of PT-symmetric nonlinear optical lattices in various physical settings have been reported [57 59]. The existence of localized modes, including multipole solitons, supported by PT-symmetric nonlinear lattices was investigated [57]. Such PT-symmetric nonlinear OLs can be implemented by means of proper periodic modulation of nonlinear gain and losses, in specially engineered nonlinear optical waveguides, see also Refs. [58, 60, 61]. Solitons in mixed PT-symmetric linear-nonlinear lattices have been investigated, too [62, 63]. The combination of PT-symmetric linear and nonlinear lattices can stabilize lattice solitons and the parameters of the linear lattice periodic potential play a significant role in controlling the extent of the stability domains; see the overview paper [64]. Multipeaked solitons in 1D and 2D cases forming in different media with PT-symmetric optical lattices have been studied, too [65, 66]. Such multipeaked solitons can be easily made stable in defocusing nonlinear media but the stability is rather difficult to achieve in focusing media. Recently, several interesting and counterintuitive features were found in PTsymmetric optical arrangements, e.g., selective mode lasing in microring resonator systems [67, 68]. Moreover, unidirectional invisibility [69, 70] and defect states [71] with unconventional properties have been also demonstrated. PT-symmetric external potentials have also been introduced into the fast growing fields of plasmonics and optical metamaterials [72]. It has been put forward that operating close to the exceptional point of a PT-symmetric coupled microring arrangement

4 598 Ying-Ji He, Xing Zhu, Dumitru Mihalache 4 can significantly affect thermal nonlinearities and Raman lasing [73]. Nonreciprocal light propagation and diode behavior was observed in two coupled PT-symmetric whispering-gallery microcavities with a saturable nonlinearity, thus enabling new possibilities for on chip signal processing [74, 75]. In this paper, we present an outline of a few basic theoretical results on the rich dynamics of lattice solitons that can be supported by various types of PTsymmetric optical potentials. In Sec. 2 we consider lattice solitons in optical media described by the complex Ginzburg-Landau model with PT-symmetric periodic potentials. In Sec. 3 we briefly overview 2D multipeak gap solitons supported by PT-symmetric complex-valued periodic potentials. Then, mixed-gap vector solitons in PT-symmetric mixed linear-nonlinear lattices are discussed in Sec. 4. We then briefly overview in Sec. 5 recent studies of nonlocal multihump solitons in PT-symmetric periodic potentials. In Sec. 6 we overview a series of recent theoretical and experimental developments in the area of PT-symmetric photonic structures. Finally, Sec. 7 concludes this paper. 2. PT-SYMMETRIC LATTICE SOLITONS IN OPTICAL MEDIA DESCRIBED BY THE COMPLEX GINZBURG-LANDAU MODEL The existence, stability, and rich dynamics of dissipative lattice solitons in optical media described by the CQ complex Ginzburg-Landau (CGL) model with PT-symmetric external potentials have been investigated in detail in Ref. [49]. Generic spatial soliton propagation scenarios were put forward by changing (i) the linear loss coefficient in the CGL model, (ii) the amplitudes, and (iii) the periods of real and imaginary parts of the complex-valued PTsymmetric optical lattice potential. When the period of the real part of the PT-symmetric optical lattice potential is close to π, the spatial solitons are tightly bound and they can propagate straightly along the lattice. However, when the period of the real part of the PT-symmetric optical lattice potential is larger than π, the launched solitons are loosely bound and they can exhibit either a transverse (lateral) drift or a persistent swing around the input launching point due to gradient force arising from the spatially inhomogeneous loss [49]. The above-mentioned generic propagation scenarios of spatial lattice solitons can be effectively managed by properly changing the profile of the spatially inhomogeneous loss; see Ref. [49] for a detailed study of these issues.

5 5 Dynamics of spatial solitons in parity-time-symmetric optical lattices A GENERIC DYNAMICAL MODEL We consider spatial beam propagation in optical media described by the (1+1)-dimensional CGL model with PT-symmetric periodic potentials [49]: 2 4 iu (1/ 2) u u u u u in[ u] L( x) u, (1) z xx where u is the complex-valued optical field, z is the propagation distance and x is the transverse coordinate. Further, ν is the quintic self-defocusing coefficient, and the combination of the CQ nonlinear terms is N[u] = αu + ε u 2 u + μ u 4 u. Here α is the linear loss coefficient, μ is the quintic-loss parameter, and ε is the cubic-gain coefficient. The last term in Eq. (1) represents the effect on light wave of the PT -symmetric linear OLs, L(x) = R(x) + ii (x). As a typical example we consider here periodic potentials of the form R (x) = = A 1 cos 2 (x/t 1 ) and I (x) = A 2 sin(x/t 2 ), where A 1 and A 2 are amplitudes of real and imaginary parts of the PT-symmetric lattice potential, respectively, and πt 1 and 2πT 2 are the corresponding periods [49] NUMERICAL RESULTS We next fix the following set of parameters: ν = 0.2, μ = 1, ε = 1.6, A 1 = = 0.2, and A 2 = 0.2 [49]. The typical soliton propagation scenarios are shown in Figs. 1 and 2. In Fig. 1 we show the dependence of the linear loss coefficient α on the period T 2 and the unique soliton dynamics for the case of a tight binding lattice potential with a relatively small period T 1 = 1 of the real part of the PTsymmetric potential. We see in Fig. 1 the typical propagation scenarios: excess gain propagation, soliton drift, straight propagation, and soliton decay. We display in Fig. 2 the dependence of the linear loss coefficient α on the period T 1 and the rich soliton dynamics for the case of a large lattice period T 1 > 1 of the real part of the PT-symmetric OL potential and for T 2 = 0.5. We see in Fig. 2 the unique propagation scenarios for this set of parameters: excess gain propagation, soliton drift, soliton persistent swing, and soliton decay.

6 600 Ying-Ji He, Xing Zhu, Dumitru Mihalache 6 Fig. 1 (a) The dependence of the linear loss coefficient α on the period T 2 ; soliton excess gain propagation (region A), soliton drift to adjacent lattice (region B), stable straight propagation (region C), and soliton decay (for α > 0.54). (b) Excess gain propagation for α = 0.2 and T 2 = (c) Soliton drift for α = 0.25 and T 2 = (d) Soliton drift for α = 0.4 and T 2 = (e) Straight propagation for α = 0.5 and T 2 = (f) Soliton decay for α = 0.55 and T 2 = 0.55 (as per Ref. [49]).

7 7 Dynamics of spatial solitons in parity-time-symmetric optical lattices 601 Fig. 2 (a) The dependence of the linear loss coefficient α on the period T 1 ; soliton excess gain propagation (region A), soliton drift (region B), soliton persistent swing (region C), and soliton decay (for α > 0.54). (b) Soliton excess gain propagation for α = 0.2 and T 1 =4. (c) Soliton drift for α = 0.3 and T 1 = 4. (d) Soliton persistent swing for α = 0.4 and T 1 = 4. (e) Soliton persistent swing for α = 0.5 and T 1 = 4. (f) Soliton decay for α = 0.55 and T 1 = 4 (as per Ref. [49]).

8 602 Ying-Ji He, Xing Zhu, Dumitru Mihalache 8 3. TWO-DIMENSIONAL MULTIPEAK GAP SOLITONS SUPPORTED BY PARITY-TIME-SYMMETRIC PERIODIC POTENTIALS In Ref. [65] we reported on the existence and stability of the 2D multipeak gap solitons in a PT-symmetric periodic potential with defocusing Kerr nonlinearity. We investigated the multipeak solitons with all the peaks of the real parts locked in-phase. These solitons can be stable in the first gap. The optical system can support not only stable solitons with an even number of peaks, but also stable solitons with an odd number of peaks [65]. The normalized 2D nonlinear Schrödinger equation that describes beam propagation in a PT-symmetric potential with defocusing Kerr nonlinearity can be written as iu U U V x, y U U 2 ( ) 0. z xx yy (2) Here U is the complex-valued field amplitude, z is the normalized longitudinal coordinate and the 2D potential V(x, y) is PT-symmetric. We choose a PTsymmetric potential as V(x, y) = V 0 {[cos(2x) + cos(2y)] + iw 0 [sin(2x) + sin(2y)]}, where V 0 is the parameter that controls the depth of the optical lattice and W 0 is the parameter that stands for the amplitude of the imaginary part. We fix V 0 = 8 and W 0 = 0.1. The band structure is plotted in Fig. 3(a). The critical threshold of this system is W th 0 = 0.5. The power diagram for four-peak solitons is displayed in Fig. 3(b) (the blue line). In this case, solitons exist in the first gap, and can be stable in the moderate power region ( 6.35 μ 5.85). We take μ = 6.0 as a typical case of stable soliton. The real and imaginary parts of the stable four-peak soliton are shown in Figs. 3(c) and 3(d), respectively. The peaks of the real part are all inphase with each other, see Fig. 3(c). For the imaginary part, some peaks are out-ofphase with the other ones, as shown in Fig. 3(d). For the family of six-peak solitons, the power versus propagation constant is shown in Fig. 3(b) (the pink line). We see that the stable region ( 6.10 μ 5.94) of these solitons shrinks a lot. For μ = 6.0, the real and imaginary parts of the six-peak soliton are displayed in Figs. 4(a) and 4(b), respectively. The sixpeak solitons can stably propagate, as exhibited in Fig. 5(a c). The system can also support stable three-peak solitons in a relatively wide region of the parameter μ ( 7.02 μ 5.73). The power diagram for this family of solitons is shown in Fig. 3(b) (the green line). Figures 4(c) and 4(d) show the real and imaginary parts of the three-peak soliton for μ = 6.0, respectively. The three-peak solitons can also stably propagate, as shown in Fig. 5(d f).

9 9 Dynamics of spatial solitons in parity-time-symmetric optical lattices 603 Fig. 3 (a) The typical band structure. (b) The power versus the propagation constant for three-, four-, and six-peak solitons (red shaded regions are the Bloch bands, the solid lines represent the stable regions while the dashed lines represent the unstable regions). (c) and (d) The real and imaginary parts of the four-peak solitons for μ = 6.0 (as per Ref. [65]). Fig. 4 (a) and (b) The real and imaginary parts of the six-peak soliton. (c) and (d) The real and the imaginary parts of the three-peak soliton (as per Ref. [65]).

10 604 Ying-Ji He, Xing Zhu, Dumitru Mihalache 10 Fig. 5 (a) and (d) The linear stability spectra of the six- and three-peak solitons, respectively. The profiles of the perturbed six- and three-peak solitons at z = 0 (b) and (e) and at z = 500 (c) and (f), respectively (as per Ref. [65]). 4. MIXED-GAP VECTOR SOLITONS IN PT-SYMMETRIC MIXED LINEAR-NONLINEAR OPTICAL LATTICES Mixed-gap vector solitons in PT-symmetric mixed linear-nonlinear optical lattices have been investigated in Ref. [63]. The first component of the mixed-gap vector soliton is the fundamental mode, whereas the second component is the outof-phase dipole mode. The propagation constants of the two components are in the semi-infinite gap and the first finite gap, respectively. The imaginary part, the depth of the PT-symmetric nonlinear optical lattice, and the propagation constant of the first component of the vector soliton can change the soliton s existence and stability domains [63]. Also, the stability of vector solitons is affected by the imaginary part of the PT-symmetric linear optical lattice potential THE GENERIC MODEL The coupled normalized 1D nonlinear Schrödinger equations for describing two mutually incoherent light beams propagating in PT-symmetric mixed linearnonlinear periodic potentials are [57, 62, 76 77]:

11 11 Dynamics of spatial solitons in parity-time-symmetric optical lattices i U / z / x ( ) ( ) U U 1 2 U V ( x) iw ( x) U 1,2 1, ,2 V x iw x U ,2 Here, U 1 and U 2 are the complex field amplitudes of two components, and z and x are the normalized longitudinal and transverse coordinates, respectively. The real and imaginary parts of the PT-symmetric linear and nonlinear optical lattices are described by V 1 (x), W 1 (x), V 2 (x), and W 2 (x), respectively. The PT-symmetry condition requires that V 1 (x) = V 1 ( x), W 1 (x) = W 1 ( x), V 2 (x) = V 2 ( x), and W 2 (x) = W 2 ( x). The stationary vector soliton solutions of Eq. (3) are searched as U 1,2 = q 1,2 exp(iμ 1,2 z). Here, μ 1,2 are the real propagation constants of the two components U 1,2 and q 1,2 are complex-valued functions that satisfy the coupled equations 2 2 q1,2 / x V1 iw1 q1,2 2 2 (4) 1 V iw q q q q ,2 1,2 1,2 Equation (4) can be solved numerically by the modified squared-operator method [78]. The total and partial powers of the vector soliton are defined as P and P 1, 2, respectively. (3) 4.2. NUMERICAL RESULTS We choose V 1 = 6 cos(2x), W 1 = 2.1 sin(2x), V 2 = cos 2 (x), W 2 = sin(2x), and μ 1 = 5.0. The propagation constant of the single-peaked component is in the semiinfinite gap (μ 1 = 5.0), and the propagation constant of the out-of-phase dipole component (μ 2 ) belongs to the first finite gap. The existence domain of the vector solitons is 1.94 μ With the increase of the propagation constant of the out-of-phase dipole component (μ 2 ), the total soliton power will increase, as shown in Fig. 6(a). The power of the single-peaked component (P 1 ) decreases and the power of the out-ofphase dipole component (P 2 ) increases as the propagation constant μ 2 increases, see Fig. 6(b). The vector solitons can be stable in the low-power region but are unstable in the high-power region. The stable region is 1.94 μ Figure 7(a) shows the max [Re(δ)] versus the propagation constant μ 2. When μ 2 = 0.9 [point A in Fig. 6(a)], the profile of the first component of the vector soliton is shown in Fig. 7(b). Figure 7(c) shows the profile of the second component (the out-of-phase dipole). This vector soliton is stable; see Figs. 7(d) and 7(e). In the high-power region, the vector solitons shown in Figs. 7(f) and 7(g) are unstable. For μ 2 = 0 [point B in Fig. 6(a)], the soliton cannot propagate stably as seen from Fig. 7(a). The soliton instability is clearly shown in Figs. 7(h) and 7(i), respectively.

12 606 Ying-Ji He, Xing Zhu, Dumitru Mihalache 12 Fig. 6 (a) The power versus propagation constant μ 2. (b) The powers of fundamental component (P 1 ) and out-of-phase dipole component (P 2 ). The shaded regions are Bloch bands (as per Ref. [63]). Fig. 7 (a) max(re(δ) versus μ 2. (b), (c) Profiles of the first component (solid line is for the real part, while the dashed line is for the imaginary part) and the second component of the vector soliton for μ 2 = 0.9. (d), (e) Stable propagation of the two perturbed components for μ 2 = 0.9. (f), (g) Profiles of the two components for μ 2 = 0. (h), (i) Unstable propagations for μ 2 = 0 (as per Ref. [63]).

13 13 Dynamics of spatial solitons in parity-time-symmetric optical lattices NONLOCAL MULTIHUMP SOLITONS IN PT-SYMMETRIC PERIODIC POTENTIALS The existence and stability of nonlocal multihump gap solitons in 1D PTsymmetric periodic potentials have been investigated in detail in Ref. [54]. These spatial solitons exist in the first gap in the case of defocusing nonlocal nonlinearity and in the semi-infinite gap in the case of focusing nonlocal nonlinearity. The solitons can be stable for defocusing nonlinearity but are unstable for focusing nonlinearity. The degree of nonlocality affects the stability domains and the intensity distribution of these spatial multihump solitons THEORETICAL MODEL The beam propagation in PT-symmetric complex-valued periodic potentials with nonlocal nonlinearity can be written in the form of normalized 1D nonlinear coupled equations [79 81] 2 2 iu / z U / x V iw U nu 0, (5a) d n/ x n U 0. (5b) Here, U is the complex field amplitude, n is the nonlinear contribution to refractive index, d is the degree of nonlocality [for d = 0 the system (5) describes a local nonlinear response whereas for d it describes the case of strong nonlocality], and x and z are the normalized transverse and longitudinal coordinates, respectively. The normalized parameter 1 represents either the focusing or defocusing nonlinearity. Next we consider the real part of the PT-symmetric potential as V(x) = V 0 sin 2 (x), and the imaginary part as W(x) = V 0 W 0 sin(2x), where V 0 is the parameter that controls the depth of PT-symmetric optical lattice and W 0 is the relative amplitude of the imaginary part. In the following we choose V 0 = 10 and W 0 = 0.1. The critical threshold of this PT-symmetric optical lattice is W th 0 = 0.5. Above this threshold, the PT-symmetry will be broken. PT-symmetric potentials can be created by using complex refractive index distributions with gain or loss: n 2 (x) = n 2R (x) + in 2I (x), where n 2I represents the gain or loss component. According to the PT-symmetry condition, n 2R (x) = n 2R ( x) and n 2I (x) = n 2I ( x). We search for the stationary multihump soliton solutions of Eqs. (5) in the form: U = q(x) exp(iμz), where q(x) is a complex function and μ is the corresponding real propagation constant. Thus, q (x) obeys the coupled system of equations 2 2 q / x V iw q nq q 0, (6a)

14 608 Ying-Ji He, Xing Zhu, Dumitru Mihalache d n / x n q 0. (6b) In order to check the stability of solitons, they are perturbed as: z z iz U x, z q( x) F( x) e G ( x) e e, (7) where F, G «1 and the superscript * denotes the complex conjugation. Substituting (7) into Eqs. (5) and linearizing the corresponding equations, we get the eigenvalue equations: 2 2 F i F F / x V iw F nf qn, (8a) 2 2 G i G G / x ( V iw ) G ng q n. (8b) 2 * Here n g( x ) q( ) d, n g( x )[ q( ) G( ) q ( ) F( )] d, and 1/2 1/2 ( ) 1/ (2 )exp( / ). g d x d Equations (8a) and (8b) can be solved numerically. If the real part of δ is greater than zero (Re(δ) > 0), the soliton is linearly unstable. Otherwise, it is linearly stable NUMERICAL RESULTS In Fig. 8(a) we show the band structure for V 0 = 10 and W 0 = 0.1. The semiinfinite gap is in the region μ 2.91 and the first gap is in the domain 7.48 μ 3.0. First, we investigate the multihump solitons for defocusing nonlocal nonlinearities (σ = 1). In this case, the solitons can exist in the first gap. In Figs. 8(b) and 8(c), we plotted the power diagrams for the fundamental and multihump solitons when d = 0.5 and d = 3, respectively. In Figs. 8(d) and 8(g), we show the profiles of the three-hump solitons (solid lines are for the real parts and dashed lines are for the imaginary parts) for μ = 3.35, when d = 0.5 [point A in Fig. 8(b)] and d = 3 [point F in Fig. 1(c)], respectively. The shapes of nonlinear contribution to refractive index also display three-hump structures, which are shown in Figs. 8(e) and 8(h), respectively. In Figs. 8(f) and 8(i), we display the corresponding transverse power flows that result from the nontrivial phase structures of these solitons. The distributions of intensities I = q 2 of the corresponding solitons are displayed in Figs. 9(a) and 9(d), respectively. The corresponding stable propagation of the perturbed solitons (when 5% random noises were added to the input solitons) are shown in Figs. 9(c) and 9(f), respectively. The linear stability spectra are shown in Figs. 9(b) and 9(e), respectively. The spectra indicate that the three-hump solitons are stable.

15 15 Dynamics of spatial solitons in parity-time-symmetric optical lattices 609 Fig. 8 (a) The band structure. (b) and (c) The power diagrams (the shaded regions are the Bloch bands, the solid lines represent stable cases whereas the dashed lines represent unstable cases) for one-hump, three-hump, and seven-hump solitons when d = 0.5 and d = 3, respectively. (d), (e), and (f) The soliton profile (the solid line is for the real part whereas the dashed line is for the imaginary part), the refractive index shape, and the soliton transverse power for σ = 1, μ = 3.35, and d = 0.5, respectively. (g), (h), and (i) The soliton profile, the refractive index shape, and the soliton transverse power flow for σ = 1, μ = 3.35, and d = 3, respectively (as per Ref. [54]). Fig. 9 (a) and (d) The intensity distributions of the three-hump solitons for d = 0.5 and d = 3, respectively. (b) and (e) The corresponding linear stability spectra. (c) and (f) The stable propagation of the perturbed solitons. Here σ = 1 and μ = 3.35 (as per Ref. [54]).

16 610 Ying-Ji He, Xing Zhu, Dumitru Mihalache RECENT DEVELOPMENTS In this Section we overview some selected recent theoretical and experimental results in the area of PT-symmetric photonic structures. The concept of PT-symmetry has been introduced in photonics settings as a means to ensure stable energy flow in optical systems that simultaneously employ both gain and loss. Wimmer et al. [34] have experimentally demonstrated stable optical discrete solitons in PT-symmetric mesh lattices. Unlike other non-conservative nonlinear systems where dissipative solitons appear as fixed points in the parameter space of the governing equations, the discrete PT-symmetric solitons in optical lattices form a continuous parametric family of solutions [34]. Hassan et al. [82] have studied both theoretically and experimentally the problem of nonlinear reversal of the PT-symmetric symmetry breaking in a system of coupled semiconductor microring resonators. It was revealed that nonlinear processes such as nonlinear saturation effects are capable of reversing the order in which the symmetry breaking occurs [82]. Next we briefly mention a series of relevant theoretical developments in this area. Yang [83] investigated the necessity of PT-symmetry for soliton families in 1D complex-valued potentials and argued that the PT-symmetry of such complex potentials is a necessary condition for the existence of soliton families. The existence and stability of 2D fundamental, dipole-mode, vortex and multipole solitons in triangular photonic lattices with PT-symmetry were investigated by Wang et al. [84]. Vector soliton solutions in PT-symmetric coupled waveguides and the corresponding Newton s cradle dynamics were studied by Liu et al. [85]. The study of interactions of bright and dark solitons with localized PTsymmetric potentials has been reported by Karjanto et al. [86] and the existence and stability of defect solitons in nonlinear OLs with PT-symmetric Bessel potentials were investigated in Ref. [87]. Recent works deal with the soliton dynamics in PT-symmetric OLs with longitudinal potential barriers [88], the study of spatial solitons in both selffocusing and self-defocusing Kerr nonlinear media with generalized PT-symmetric Scarff-II potentials [89], the problem of interplay between PT-symmetry, supersymmetry, and nonlinearity [90], the study of solitons supported by 2D mixed linear-nonlinear complex OLs [91], the nonlinear tunneling of spatial solitons in PT-symmetric potentials [92], the study of asymmetric solitons in 2D PT-symmetric potentials [93], and the study of 2D linear modes and solitons in PT-symmetric Bessel complex-valued potentials [94]. The concept of PT-symmetry was recently extended in other interesting research directions [95 98]. Kartashov et al. [95] have introduced partially-pt-symmetric azimuthal potentials and have studied the corresponding nonlinear topological states. Also, recent studies deal with the optical properties of bulk, three-dimensional

17 17 Dynamics of spatial solitons in parity-time-symmetric optical lattices 611 PT-symmetric plasmonic metamaterials [96], the problem of guiding surface plasmon polaritons with PT-symmetry and the realization of waveguides and cloaks [97], and the observation of Bloch oscillations in PT-symmetric photonic lattices [98]. 7. CONCLUSIONS In this work, we reviewed some selected recent results concerning the rich spatial soliton dynamics in PT-symmetric periodic potentials within the context of optics and photonics for both 1D and 2D lattice geometries. We conclude with the hope that this overview on recent developments in the area of localized optical structures in PT-symmetric systems will inspire further studies. Acknowledgments. This work was supported by the National Natural Science Foundation of China (Grant No ) and the Guangdong Province Education Department Foundation of China (Grant No. 2014KZDXM059). The work of D.M. was supported by CNCS-UEFISCDI, Project No. PN-II-ID-PCE REFERENCES 1. P. Mandel and M. Tlidi, J. Opt. B 6, R60 (2004). 2. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, J. Opt. B 7, R53 (2005). 3. N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, Appl. Phys. B 81, 937 (2005). 4. F.W. Wise, A. Chong, and W.H. Renninger, Laser Photon. Rev. 2, 58 (2008). 5. F. Lederer et al., Phys. Rep. 463, 1 (2008). 6. G. P. Agrawal, J. Opt. Soc. Am. B 28, A1 (2011). 7. Y. V. Kartashov, B. A. Malomed, and L. Torner, Rev. Mod. Phys. 83, 247 (2011). 8. P. Grelu and N. Akhmediev, Nature Photon. 6, 84 (2012). 9. Z. Chen, M. Segev, and D. N. Christodoulides, Rep. Prog. Phys. 75, (2012). 10. D. Mihalache, Rom. J. Phys. 57, 352 (2012). 11. H. Leblond and D. Mihalache, Phys. Rep. 523, 61 (2013). 12. Rujiang Li, Pengfei Li, and Lu Li, Proc. Romanian Acad. A 14, 121 (2013). 13. Y. J. He, B. A. Malomed, and D. Mihalache, Phil. Trans. R. Soc. A 372, (2014). 14. M. Tlidi et al., Phil. Trans. R. Soc. A 372, (2014). 15. D. Mihalache, Rom. J. Phys. 59, 295 (2014). 16. D. J. Frantzeskakis, H. Leblond, and D. Mihalache, Rom. J. Phys. 59, 767 (2014). 17. B.A. Malomed, J. Opt. Soc. Am. B 31, 2460 (2014). 18. Ruo-Yun Cao et al., Rom. Rep. Phys. 67, 375 (2015). 19. D. Mihalache, Proc. Romanian Acad. A 16, 62 (2015). 20. V. S. Bagnato et al., Rom. Rep. Phys. 67, 5 (2015). 21. D. Mihalache, Rom. Rep. Phys. 67, 1383 (2015). 22. S. Amarande, Rom. Rep. Phys. 67, 1431 (2015). 23. S. T. Popescu, A. Petris, and V. I. Vlad, Proc. Romanian Acad. A 16, 437 (2015). 24. C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998). 25. Z. Ahmed, Phys. Lett. A 282, 343 (2001). 26. C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, (2002).

18 612 Ying-Ji He, Xing Zhu, Dumitru Mihalache C. M. Bender, D. C. Brody, and H. Jones, Am. J. Phys. 71, 1095 (2003). 28. C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, Phys. Rev. Lett. 98, (2007). 29. C. E. Rüter et al., Nat. Phys. 6, 192 (2010). 30. Z. H. Musslimani et al., Phys. Rev. Lett. 100, (2008). 31. K. G. Makris et al., Phys. Rev. Lett. 100, (2008). 32. K. G. Makris et al., Phys. Rev. A 81, (2010). 33. O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 103, (2009). 34. M. Wimmer et al., Nat. Commun. 6, 7782 (2015). 35. K. Zhou, Z. Guo, J. Wang, and S. Liu, Opt. Lett. 35, 2928 (2010). 36. X. Zhu, H. Wang, L.-X. Zheng, H. Li, and Y.-J. He, Opt. Lett. 36, 2680 (2011). 37. Z. Shi, X. Jiang, X. Zhu, and H. Li, Phys. Rev. A 84, (2011). 38. H. Li, Z. Shi, X. Jiang, and X. Zhu, Opt. Lett. 36, 3290 (2011). 39. S. Nixon, L. Ge, and J. Yang, Phys. Rev. A 85, (2012). 40. V. Achilleos et al., Phys. Rev. A 86, (2012). 41. R. Driben and B. A. Malomed, Opt. Lett. 36, 4323 (2011). 42. R. Driben and B. A. Malomed, Europhys. Lett. 96, (2011). 43. D. A. Zezyulin and V. V. Konotop, Phys. Rev. Lett. 108, (2012). 44. L. Chen, R. Li, N. Yang, D. Chen, and L. Li, Proc. Romanian Acad. A 13, 46 (2012). 45. I. V. Barashenkov et al., Phys. Rev. A 86, (2012). 46. N. V. Alexeeva et al., Phys. Rev. A 85, (2012). 47. S. Liu, C. Ma, Y. Zhang, and K. Lu, Opt. Commun. 285, 1934 (2012). 48. C. Li, H. Liu, and L. Dong, Opt. Express 20, (2012). 49. Y. He and D. Mihalache, Phys. Rev. A 87, (2013). 50. Y. He and D. Mihalache, J. Opt. Soc. Am. B 29, 2554 (2012). 51. H. Li, X. Jiang, X. Zhu, and Z. Shi, Phys. Rev. A 86, (2012). 52. S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, Phys. Rev. A 85, (2012). 53. C. Yin, Y. He, H. Li, and J. Xie, Opt. Express 20, (2012). 54. X. Zhu, H. Li, H. Wang, and Y. He, J. Opt. Soc. Am. B 30, 1987 (2013). 55. J. Xie, W. Chen, J. Lv, Z. Su, C. Yin, and Y. He, J. Opt. Soc. Am. B 30, 1216 (2013). 56. S. V. Dmitriev, A. A. Sukhorukov, and Y. S. Kivshar, Opt. Lett., 35, 2976 (2010). 57. F. K. Abdullaev et al., Phys. Rev. A 83, (2011). 58. A. E. Miroshnichenko, B. A. Malomed, and Y. S. Kivshar, Phys. Rev. A 84, (2011). 59. S. V. Suchkov et al., Phys. Rev. E 84, (2011). 60. D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, Europhys. Lett. 96, (2011). 61. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, Opt. Commun. 285, 3320 (2012). 62. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, Phys. Rev. A 85, (2012). 63. X. Zhu, P. Cao, L. Song, Y. He, and H. Li, J. Opt. Soc. Am. B 31, 2109 (2014). 64. Y. He and D. Mihalache, Rom. Rep. Phys. 64, 1243 (2012). 65. X. Zhu, H. Wang, H. Li, W. He, and Y. He, Opt. Lett. 38, 2723 (2013). 66. H. Wang, D. Ling, and Y. He, Chin. Phys. Lett. 32, (2015). 67. H. Hodaei et al., Science 346, 975 (2014). 68. L. Feng et al., Science 346, 972 (2014). 69. Z. Lin et al., Phys. Rev. Lett. 106, (2011). 70. L. Feng et al., Nat. Mater. 12, 108 (2013). 71. A. Regensburger et al., Phys. Rev. Lett. 110, (2013). 72. G. Castaldi, S. Savoia, V. Galdi, A. Alu, and N. Engheta, Phys. Rev. Lett. 110, (2013). 73. B. Peng et al. Science 346, 328 (2014). 74. B. Peng et al., Nat. Phys. 10, 394 (2014). 75. L. Chang et al., Nat. Photon. 8, 524 (2014). 76. Y. V. Kartashov, Opt. Lett. 38, 2600 (2013). 77. F. K. Abdullaev, V. V. Konotop, M. Salerno, and A. V. Yulin, Phys. Rev. E 82, (2010). 78. J. Yang and T. I. Lakoba, Stud. Appl. Math. 118, 153 (2007).

19 19 Dynamics of spatial solitons in parity-time-symmetric optical lattices Y. V. Kartashov, L. Torner, and V. A. Vysloukh, Opt. Lett. 31, 2595 (2006). 80. F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, Opt. Lett. 35, 628 (2010). 81. Y. He and B. A. Malomed, Phys. Rev. A 87, (2013). 82. A. U. Hassan et al., Phys. Rev. A 92, (2015). 83. J. Yang, Phys. Lett. A 378, 367 (2014). 84. H. Wang et al., Opt. Commun. 335, 146 (2015). 85. B. Liu, L. Li, and D. Mihalache, Rom. Rep. Phys. 67, 802 (2015). 86. N. Karjanto et al., Chaos 25, (2015). 87. H. Wang et al., Eur. Phys. J. D 69, 31 (2015). 88. K. Zhou et al., Opt. Express 23, (2015). 89. Z. Yan et al., Phys. Rev. E 92, (2015). 90. P. G. Kevrekidis et al., Phys. Rev. E 92, (2015). 91. X. Ren et al., Opt. Commun. 356, 230 (2015). 92. Yun-Jie Xu and Chao-Qing Dai, Opt. Commun. 318, 112 (2014). 93. Haibo Chen and Sumei Hu, Phys. Lett. A 380, 162 (2016). 94. Haibo Chen and Sumei Hu, Opt. Commun. 355, 50 (2015). 95. Y. V. Kartashov, V. V. Konotop, and L. Torner, Phys. Rev. Lett. 115, (2015). 96. H. Alaeian and J. A. Dionne, Phys. Rev. A 89, (2014). 97. Fan Yang and Zhong Lei Mei, Sci. Rep. 5, (2015). 98. M. Wimmer et al., Sci. Rep. 5, (2015).