Effect of Dispersive Reflectivity on the Dynamics of Interacting Solitons
in Dual-Core Systems with Separated Bragg Grating and Nonlinearity
Tanvir Ahmed and Javid Atai
School of Electrical and Information Engineering, The University of Sydney, NSW 2006, Sydney, Australia
Keywords:
Gap Soliton, Fiber Bragg Grating, Dispersive Reflectivity.
Abstract:
The interactions of in-phase and π-out-of-phase quiescent Bragg grating solitons in a dual-core system where
one core has Kerr nonlinearity and other is linear and has a Bragg grating with dispersive reflectivity are
systematically investigated. The effect of dispersive reflectivity on the outcomes of the interactions is analyzed.
It is found that above a certain value of dispersive reflectivity solitons develop sidelobes. The presence of
sidelobes has a significant effect on the outcomes of the interactions. In the absence of sidelobes, in-phase
soliton-soliton interactions may result in several outcomes such as merger into a quiescent soliton, symmetric
or asymmetric separation of solitons or destruction of both solitons. However, the interaction of solitons in the
presence of sidelobes produces other outcomes such as repulsion of solitons or formaion of a temporary bound
state followed by separation of two solitons. π-out-of-phase solitons generally repel each other. However, in
the presence of sidelobes, interactions of π-out-of-phase solitons may lead to the formation of temporary
bound state and subsequent generation of two separating solitons.
1 INTRODUCTION
A key characteristic of fiber Bragg gratings (FBGs)
is that they exhibit strong effective dispersion due to
resonant reflection of light. This effective dispersion
can be six orders of magnitude larger than the chro-
matic dispersion of silica fiber (de Sterke and Sipe,
1994; Eggleton et al., 1997). The interplay of FBG-
induced dispersion and Kerr nonlinearity gives rise
to the formation of gap solitons (GSs). Over the
past few decades, gap solitons have been investigated
extensively both theoretically (Aceves and Wabnitz,
1989; Christodoulides and Joseph, 1989; Malomed
and Tasgal, 1994; Barashenkov et al., 1998; De Rossi
et al., 1998; Neill and Atai, 2006) and experimen-
tally (Eggleton et al., 1996; de Sterke C. M. and
Krug, 1997; Eggleton et al., 1999) due to their po-
tential applications in optical signal processing, opti-
cal buffer elements and logic gates (Krauss, 2008). A
major property of gap solitons is that they can pos-
sess any velocity from zero (quiescent) to the speed
of light in the medium. To date, gap solitons with
a velocity in excess of 23% of the speed of light in
the medium have been observed experimentally (Mok
et al., 2006).
Gap solitons have been studied in more complex
media and structures such as photonic crystals (Monat
et al., 2010; Skryabin, 2004; Neill and Atai, 2007),
quadratic nonlinearity (Mak et al., 1998b; Conti et al.,
1997; He and Drummond, 1997), waveguide ar-
rays (Mandelik et al., 2004; Tan et al., 2009; Dong
et al., 2011), dual core fibers (Atai and Malomed,
2000; Atai and Malomed, 2001; Mak et al., 1998a;
Tsofe and Malomed, 2007), and Bragg gratings (BGs)
with dispersive reflectivity (Atai and Malomed, 2005;
Neill et al., 2008). Dual-core nonlinear couplers with
dissimilar cores exhibit rich nonlinear dynamics and
switching characteristics (Atai and Chen, 1992; Atai
and Chen, 1993; Atai and Malomed, 1998). There-
fore, slow GSs in grating-assisted couplers can be ex-
ploited to build novel optical devices for signal pro-
cessing and switching.
In this paper, we investigate the effect of disper-
sive reflectivity on the interaction dynamics of gap
solitons in a dual-core fiber where one core has only
Kerr nonlinearity and the other core is linear and is
equipped with a BG with dispersive reflectivity.
2 THE MODEL
Propagation of light in a dual-core fiber where one
core has only Kerr nonlinearity and other one is linear
and contains a BG with dispersive reflectivity is de-
Ahmed T. and Atai J.
Effect of Dispersive Reflectivity on the Dynamics of Interacting Solitons in Dual-Core Systems with Separated Bragg Grating and Nonlinearity.
DOI: 10.5220/0006099000490053
In Proceedings of the 5th International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS 2017), pages 49-53
ISBN: 978-989-758-223-3
Copyright
c
 2017 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
49

-80 -40 0 40 80
x
t
2000
0
(a)
-80 -40 0 40 80
x
t
4000
0
(b)
-36
-18 0 18
36
x
t
4000
0
(c)
.
-64
-32 0 32
64
x
t
4000
0
(d)
Figure 1: Examples of the outcomes of soliton-soliton interaction for in-phase quiescent solitons without sidelobes for λ = 1.0,
c = 0.2 and △x= 12. (a) Destruction for ω = 1.570, m = 0.020 (b) Merger for ω = 1.610, m = 0.060 (c) Symmetric separation
for ω = 1.570, m = 0.160 (d) Asymmetric separation for ω = 1.590, m = 0.260. Only |u| component is shown here.
scribed by the following set of normalized equations:
iu
t
+ iu
x
+

|v|
2
+
1
2
|u|
2

u+ φ = 0,
iv
t
− iv
x
+

|u|
2
+
1
2
|v|
2

v+ ψ = 0,
iφ
t
+ icφ
x
+ u + λψ+ mψ
xx
= 0,
iψ
t
− icψ
x
+ v + λφ+ mφ
xx
= 0,
(1)
where u(x,t) and v(x, t) are the forward- and
backward-propagating waves in the nonlinear core,
and φ(x, t) and ψ(x,t) are their counterparts in the
linear core which is equipped with a Bragg grating.
The coefficient of linear coupling between the cores
is normalized to 1 and the FBG-induced linear cou-
pling coefficient between the forward- and backward-
propagating waves is represented by a real parameter
λ > 0. The group velocity in the nonlinear core is
set equal to 1 and c represents the relative group ve-
locity in the linear core. The real parameter m > 0
controls the strength of dispersive reflectivity. Sub-
stituting u, v, φ, ψ ∼ exp(ikx− iωt) into Eqs. (1) and
linearizing results in the following form of dispersion
relation:
ω
4
−

2+

λ− mk
2

2
+

1+ c
2

k
2

ω
2
+

λk− mk
3

2
+

ck
2
− 1

2
= 0 (2)
The dispersion relation results a genuine central
band gap along with upper and lower band gaps
(which are not genuine band gaps). No stationary
solutions are available in the central bang gap. The
soliton solutions are sought numerically in both up-
per and lower band gaps by relaxation algorithm in
the range 0 ≤ m ≤ 0.5.
PHOTOPTICS 2017 - 5th International Conference on Photonics, Optics and Laser Technology
50

3 INTERACTIONS OF GAP
SOLITONS
We have investigated the interactions between two
identical stable quiescent GSs by numerically solv-
ing Eqs. (1) using a symmetrized split-step Fourier
method for in-phase and π-out-of-phase solitons sub-
ject to the following initial conditions:
u(x, 0) = u

x−
△x
2
, 0

+ u

x+
△x
2
, 0

exp(i△θ),
v(x, 0) = v

x−
△x
2
, 0

+ v

x+
△x
2
, 0

exp(i△θ),
φ(x, 0) = φ

x−
△x
2
, 0

+ φ

x+
△x
2
, 0

exp(i△θ),
ψ(x, 0) = ψ

x−
△x
2
, 0

+ ψ

x+
△x
2
, 0

exp(i△θ),
(3)
where △x and △θ represent the initial separa-
tion and phase difference between the solitons respec-
tively, and u, v, φ and ψ belong to the stable regions.
As is shown in Fig. 1, for the in-phase solitons, the
interactions may result in several outcomes such as
merger into a single soliton, destruction of both soli-
tons, symmetric and asymmetric separation of soli-
tons for low to moderate values of dispersive reflec-
tivity. Fig. 2 summarizes the outcomes of in-phase
and π-out-of-phase soliton-soliton interactions in the
upper bandgap in the plane (m, ω) for λ = 1, c = 0.2
and initial separation of ∆x = 12. The simulations
demonstrate that the outcomes of interactions are sig-
nificantly influenced by strong dispersive reflectivity.
This is mainly due to the fact that beyond a certain
value of m (i.e. the region to the right of the dash-
dotted curve in Fig. 2) solitons develop sidelobes.
The presence of sidelobes drastically alters the in-
teraction dynamics. As is shown in Fig. 2(a), the
bound state formation and repulsion are generally ob-
served when sidelobes are present. Typical examples
of repulsion and formation a temporary bound state
followed by generation of two separating solitons are
shown in Fig 3(a) and 3(b), respectively.
In the case of π-out-of-phase solitons, in the ab-
sence of sidelobes solitons repel each other (see Fig.
2(b)). However, when solitons have sidelobes the in-
teractions may also result in the formation of a bound
state followed by two separating solitons. Figs. 3(c)
and 3(d) respectively show examples of repulsion and
formation of a bound state followed by generation of
two moving solitons for initially π-out-of-phase soli-
tons.
0
0.25 0.5
m
1.50
1.55
1.6
1.618
ω
D
A
M
R
B
U
S
(a)
0
0.25 0.5
m
1.50
1.55
1.6
1.618
ω
R
B
U
(b)
Figure 2: The outcomes of soliton-soliton interactions for
λ = 1.0, c = 0.2 and △x = 12 in the upper bandgap for
(a) in-phase and (b) π−out-of-phase solitons on the (m, ω)
plane. The labeled regions are merger (M), symmetric sep-
aration (S), asymmetric separation (A), destruction (D), re-
pulsion (R), and formation of bound state followed by sepa-
ration (B). In the region U, GSs are unstable. GSs have side-
lobes in the region to the right of the dashed-dotted curve.
4 CONCLUSIONS
The interaction dynamics of quiescent Bragg grating
solitons are investigated in a systematic way to test
the effect of dispersive reflectivity in a dual-core sys-
tem where one core contains Kerr nonlinearity and an-
other core is equipped with a Bragg grating and dis-
persive reflectivity. It is found that, for low to moder-
ate dispersive reflectivity (i.e. in the absence of side-
lobes), in-phase quiescent Bragg grating solitons at-
tract each other and generate different outcomes such
as merger into a quiescent soliton, separation of soli-
tons with equal and unequal velocities after initial at-
traction or destruction of both solitons. For high dis-
Effect of Dispersive Reflectivity on the Dynamics of Interacting Solitons in Dual-Core Systems with Separated Bragg Grating and
Nonlinearity
51

-50 -25
0
25 50
x
t
0
(a)
t
4000
0
t
-34 -17 0 17 34
x
t
6000
0
(b)
-80 -40 0 40 80
x
t
5000
0
(c)
-40 -20 0 20 40
x
t
15000
0
(d)
Figure 3: Examples of the outcomes of soliton-soliton interaction for λ = 1.0, c = 0.2 and △x = 12 for solitons with sidelobes.
(a) Repulsion for ∆θ = 0, ω = 1.590, m = 0.360, (b) Formation of temporary bound state followed by separation for ∆θ = 0,
ω = 1.590, m = 0.420, (c) Repulsion for ∆θ = π, ω = 1.601, m = 0.280 and (d) Formation of a bound state followed by
separating solitons for ∆θ = π, ω = 1.610, m = 0.380. Only |u| component is shown here.
persive reflectivity solitons have side-lobes in their
profile. The presence of sidelobes significantly af-
fects the interaction dynamics. In particular, in-phase
solitons with sidelobes may repel each other or form
of a temporary bound state followed by generation of
two separating solitons. In the case of π-out-of-phase
solitons, solitons without sidelobes always repel each
other. However, in the region where solitons have
sidelobes, the interactions may also give rise to the
formation of a bound state that subsequently evolves
in to two separating solitons.
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Effect of Dispersive Reflectivity on the Dynamics of Interacting Solitons in Dual-Core Systems with Separated Bragg Grating and
Nonlinearity
53