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
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
598 Ying-Ji He, Xing Zhu, Dumitru Mihalache 4
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.
5 Dynamics of spatial solitons in parity-time-symmetric optical lattices 599
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.
600 Ying-Ji He, Xing Zhu, Dumitru Mihalache 6
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]).
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 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]).
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
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).
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]).
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 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]:
11 Dynamics of spatial solitons in parity-time-symmetric optical lattices 605
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.
4.2. NUMERICAL RESULTS
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.
606 Ying-Ji He, Xing Zhu, Dumitru Mihalache 12
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]).
13 Dynamics of spatial solitons in parity-time-symmetric optical lattices 607
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)
608 Ying-Ji He, Xing Zhu, Dumitru Mihalache 14
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.
5.2. NUMERICAL RESULTS
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.
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]).
610 Ying-Ji He, Xing Zhu, Dumitru Mihalache 16
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
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. 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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