dynamics of spatial solitons in parity time symmetric ... · bragg gap solitons in pt-symmetric...

DYNAMICS OF SPATIAL SOLITONS IN PARITY-TIME-SYMMETRIC

OPTICAL LATTICES: A SELECTION

OF RECENT THEORETICAL RESULTS

YING-JI HE1,*, XING ZHU2, DUMITRU MIHALACHE3,#

1Guangdong Polytechnic Normal University, School of Electronics and Information,

510665 Guangzhou, China 2Guangdong University of Education, Department of Physics, Guangzhou 510303, China

3 “Horia Hulubei” National Institute for Physics and Nuclear Engineering, P.O. Box MG-6,

RO-077125, Bucharest-Magurele, Romania

*E-mail: [email protected]; #Email: [email protected]

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 one-

dimensional (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 complex-

valued 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. 595–613, Bucharest, 2016

mailto:[email protected]

mailto:[email protected]

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 self-

defocusing 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 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 PT-

symmetric 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


can significantly affect thermal nonlinearities and Raman lasing [73]. Non-

reciprocal 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 PT-

symmetric 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 PT-

symmetric 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.


2.1. 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

(1/ 2) [ ] ( ) ,z xx

u u u u u u iN u L x ui (1)

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|2u + μ|u|

4u. 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) =

= A1 cos2 (x/T1) and I (x) = A2 sin(x/T2), where A1 and A2 are amplitudes of real and

imaginary parts of the PT-symmetric lattice potential, respectively, and πT1 and

2πT2 are the corresponding periods [49].

2.2. NUMERICAL RESULTS

We next fix the following set of parameters: ν = − 0.2, μ = −1, ε = 1.6, A1 =

= 0.2, and A2 = 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 T2 and the unique soliton dynamics for the case of a tight binding lattice

potential with a relatively small period T1 = 1 of the real part of the PT-

symmetric 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 T1 and the rich soliton dynamics for the case of a large lattice period

T1 > 1 of the real part of the PT-symmetric OL potential and for T2 = 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.


Fig. 1 – (a) The dependence of the linear loss coefficient α on the period T2; 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 T2 = 0.55. (c) Soliton

drift for α = 0.25 and T2 = 0.55. (d) Soliton drift for α = 0.4 and T2 = 0.55. (e) Straight propagation

for α = 0.5 and T2 = 0.55. (f) Soliton decay for α = 0.55 and T2 = 0.55 (as per Ref. [49]).


Fig. 2 – (a) The dependence of the linear loss coefficient α on the period T1; 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 T1 =4. (c) Soliton drift for α = 0.3

and T1 = 4. (d) Soliton persistent swing for α = 0.4 and T1 = 4. (e) Soliton persistent swing for α = 0.5

and T1 = 4. (f) Soliton decay for α = 0.55 and T1 = 4 (as per Ref. [49]).


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

2

( ) 0., | |z xx yy

iU U U V x y U U (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 PT-

symmetric potential as V(x, y) = V0{[cos(2x) + cos(2y)] + iW0[sin(2x) + sin(2y)]},

where V0 is the parameter that controls the depth of the optical lattice and W0 is the

parameter that stands for the amplitude of the imaginary part. We fix V0 = 8 and

W0 = 0.1.

The band structure is plotted in Fig. 3(a). The critical threshold of this system

is Wth

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 in-

phase with each other, see Fig. 3(c). For the imaginary part, some peaks are out-of-

phase 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 six-

peak 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).


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]).


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 out-

of-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.

4.1. THE GENERIC MODEL

The coupled normalized 1D nonlinear Schrödinger equations for describing

two mutually incoherent light beams propagating in PT-symmetric mixed linear-

nonlinear periodic potentials are [57, 62, 76–77]:


1 2

2 2

1,2 1,2 1 1 1,2

2 2

2 2 1,2

/ /

| | | | 0.

( ) ( )

1 ( ) ( )

i U z x

U U

U V x iW x U

V x iW x U

(3)

Here, U1 and U2 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 V1(x), W1(x), V2(x), and W2(x), respectively. The

PT-symmetry condition requires that V1(x) = V1(−x), W1(x) = −W1(−x), V2 (x) =

V2(−x), and W2(x) = −W2(−x). The stationary vector soliton solutions of Eq. (3) are

searched as U1,2 = q1,2 exp(iμ1,2 z). Here, μ1,2 are the real propagation constants of

the two components U1,2 and q1,2 are complex-valued functions that satisfy the

coupled equations

1 2

2 2

1,2 1 1 1,2

2 2

2 2 1,2 1,2 1,2

/

1 | | | | 0.

V W

V W

q x i q

i q q q q

(4)

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

P1,2, respectively.


We choose V1 = 6 cos(2x), W1 = 2.1 sin(2x), V2 = cos2(x), W2 = sin(2x), and

μ1 = 5.0. The propagation constant of the single-peaked component is in the semi-

infinite gap (μ1 = 5.0), and the propagation constant of the out-of-phase dipole

component (μ2) belongs to the first finite gap. The existence domain of the vector

solitons is −1.94 ≤ μ2 ≤ 0.26.

With the increase of the propagation constant of the out-of-phase dipole

component (μ2), the total soliton power will increase, as shown in Fig. 6(a). The

power of the single-peaked component (P1) decreases and the power of the out-of-

phase dipole component (P2) increases as the propagation constant μ2 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 ≤ μ2 ≤ −0.08.

Figure 7(a) shows the max [Re(δ)] versus the propagation constant μ2. When

μ2 = − 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 μ2 = 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.


Fig. 6 – (a) The power versus propagation constant μ2. (b) The powers of fundamental component

(P1) and out-of-phase dipole component (P2). The shaded regions are Bloch bands (as per Ref. [63]).

Fig. 7 – (a) max(Re(δ) versus μ2. (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

μ2 = −0.9. (d), (e) Stable propagation of the two perturbed components for μ2 = −0.9. (f), (g) Profiles

of the two components for μ2 = 0. (h), (i) Unstable propagations for μ2 = 0 (as per Ref. [63]).


5. NONLOCAL MULTIHUMP SOLITONS IN PT-SYMMETRIC PERIODIC POTENTIALS

The existence and stability of nonlocal multihump gap solitons in 1D PT-

symmetric 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.

5.1. 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/ / 0,i U z U V iW U nUx (5a)

2 2 2

0/ | | .d n Un x (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) = −V0 sin2(x), and the imaginary part as W(x) = −V0W0 sin(2x),

where V0 is the parameter that controls the depth of PT-symmetric optical lattice

and W0 is the relative amplitude of the imaginary part. In the following we choose

V0 = 10 and W0 = 0.1. The critical threshold of this PT-symmetric optical lattice is

Wth

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: n2(x) = n2R (x) + in2I (x), where n2I represents the gain or loss component.

According to the PT-symmetry condition, n2R(x) = n2R (−x) and n2I (x) = −n2I (−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/ 0,V iW nq qq x q (6a)


2 2 2

0/ | | .d nn x q (6b)

In order to check the stability of solitons, they are perturbed as:

, ( ) ( ) ( ) ,z z i zU 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 q n (8a)

2 2/ ( ) .G i G G x V iW G nG q n

(8b)

Here2

( ) ( ) ,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.


In Fig. 8(a) we show the band structure for V0 = 10 and W0 = 0.1. The semi-

infinite 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 = |q2| 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.


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]).


6. 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 PT-

symmetric 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 self-

focusing 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


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. 11174061) 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-2011-3-0083.

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