Dynamics of Interacting Bragg Grating Solitons in a Semilinear
Dual-core System with Cubic-quintic Nonlinearity
Md Jahirul Islam and Javid Atai
School of Electrical and Information Engineering, The University of Sydney, NSW 2006, Sydney, Australia
Keywords:
Bragg Grating Soliton, Dual Core System, Cubic-quintic Nonlinearity.
Abstract:
The interaction dynamics of in-phase Bragg grating gap solitons in a semilinear dual-core optical waveguide,
where one core has cubic-quintic nonlinearity and equipped with Bragg grating and the other is linear, are
investigated. The model supports two disjoint families of Bragg grating solitons (referred as Type 1 and
Type 2). It is found that the interactions of two stable in-phase (∆θ = 0) quiescent solitons result in several
outcomes. The possible interaction outcomes between two solitons may include symmetric or asymmetric
separation, merger into one quiescent or moving soliton, destruction of one or both solitons and the formation
of three solitons. It is found that the outcomes of the interactions are dependent upon the strength of quintic
nonlinearity (q), initial separation (∆x) of the solitons, coupling-coefficient (κ) between the cores and the group
velocity term (c) in the linear core.
1 INTRODUCTION
Fiber Bragg gratings (FBGs) are generated through
the periodic variation of the refractive index of the
core in an optical fiber. In recent years, FBGs based
devices have played an important role in optical sys-
tems due to their applications in high speed switching,
pulse compression, high-bit-rate optical communica-
tions, filtering, sensing and signal processing (Radic
et al., 1995; Christiansen et al., 2000; Loh et al., 1996;
Kashyap, 1999).
One of the key characteristics of FBGs is that the
cross coupling between the forward- and backward-
propagating waves gives rise to strong dispersion
(Desterke and Sipe, 1994). At high intensities, the
strong FBG-induced dispersion and the nonlinearity
may be balanced leading to the formation of Bragg
grating (BG) solitons. BG solitons have been investi-
gated extensively in Kerr nonlinear media both theo-
retically (Christadoulides and Joseph, 1989; Aceves
and Wabnitz, 1989; Mak et al., 2003; Neill et al.,
2007; Neill and Atai, 2006) and experimentally
(Eggleton et al., 1997; Taverner et al., 1998; Mok
et al., 2006). More recently, the existence andstability
of BG solitons have been investigated in other non-
linear systems such as quadratic nonlinearity (Conti
et al., 1997) and cubic-quintic nonlinearity (Atai and
Malomed, 2001; Atai, 2004; Dasanayaka and Atai,
2013b; Dasanayaka and Atai, 2013a). They have also
been studied in more sophisticated structures such as
dual-core fibers where Bragg grating exists in one
or both cores (Mak et al., 1998; Atai and Malomed,
2000).
Couplers with dissimilar cores have received
much attention due to their potential applications
in switching and signal processing (Bertolotti et al.,
1995; Atai and Chen, 1992; Atai and Chen, 1993;
Nistazakis et al., 2002). The presence of a Bragg grat-
ing in such couplers results in a system that supports
BG solitons whose stability properties are governed
by other parameters such as the strength of the cou-
pling coefficient and relative group velocity in the lin-
ear core (Atai and Malomed, 2000). As a result, such
systems exhibit a very rich dynamics.
In this paper, we investigate the interactions of BG
solitons in a dual-core system that is composed of a
linear core coupled to a nonlinear core with a Bragg
grating and cubic-quintic nonlinearity.
2 THE MODEL
Starting with the model put forward in (Atai and
Malomed, 2000) and employing the approach de-
scribed in (Atai and Malomed, 2001), the following
dimensionless coupled-mode equations for two lin-
early coupled cores can be derived, assuming BG is
present only in the nonlinear core and the other being
Islam, M. and Atai, J.
Dynamics of Interacting Bragg Grating Solitons in a Semilinear Dual-core System with Cubic-quintic Nonlinearity.
DOI: 10.5220/0005651502270230
In Proceedings of the 4th International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS 2016), pages 229-232
ISBN: 978-989-758-174-8
Copyright
c
 2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
229

-80 -40 0 40 80
1000
t
0
x
(a)
t
0
x
t
0
x
t
0
x
-80 -40 0 40 80
1000
t
0
x
(b)
-80 -40 0 40 80
500
t
0
x
(c)
500
t
0
x
-80 -40 0 40 80
500
t
0
x
(d)
Figure 1: Typical examples of the interactions of quiescent solitons. (a) Asymmetric separation for κ = 0.10, c = 0.1,
∆x = 10.0, q = 0.35, ω = 0.85; (b) symmetric separation for κ = 0.50, c = 0.20, ∆x = 12.0, q = 0.16, ω = 1.10; (c) merger
into a quiescent soliton for κ = 1.0, c = 0.0, ∆x = 8.0, q = 0.40, ω = 1.35 and (d) formation of three solitons (a quiescent and
two moving solitons) for κ = 0.1, c = 0.1, ∆x = 10.0, q = 0.26, ω = 0.60. In all figures only the u component is shown.
linear:
iu
t
+ iu
x
+

|v|
2
+
1
2
|u|
2

u−
q

1
4
|u|
4
+
3
2
|u|
2
|v|
2
+
3
4
|v|
4

u+ v+ κφ = 0,
iv
t
− iv
x
+

|u|
2
+
1
2
|v|
2

v−
q

1
4
|v|
4
+
3
2
|v|
2
|u|
2
+
3
4
|u|
4

v+ u+ κψ = 0,
iφ
t
+ icφ
x
+ κu = 0
,
iψ
t
−icψ
x
+ κv = 0,
(1)
where u and v are the forward- and backward-
propagating waves in the nonlinear core and φ and
ψ are their counterparts in the linear core, respec-
tively. q > 0 is a real parameter that controls the
strength of the quintic nonlinearity and κ is the co-
efficient of linear coupling between the cores. Also,
c represents the relative group velocity in the linear
core (group velocity in the nonlinear core has been
set to 1). It is worth noting that the cubic-quintic
nonlinearity has been observed in various organic ma-
terials and chalcogenide glass (Boudebs et al., 2003;
Zhan et al., 2002; Lawrence et al., 1994). Assuming
a typical value of ∆n = 5× 10
−4
and using the values
of nonlinear coefficients from these references, it is
PHOTOPTICS 2016 - 4th International Conference on Photonics, Optics and Laser Technology
230

found that q can range from 0.05 to 0.6. As a result,
in our simulations we have assumed that q varies in
the range 0 ≤ q ≤ 1.
Substituting u, v, φ, ψ ∼ exp(ikx−iωt) into the sys-
tem of Eqs. (1) and linearizing, a dispersion equation
for ω(k) can be obtained as (Islam and Atai, 2015):
ω
4
−

1+ 2κ
2
+ (1+ c
2
)k
2

ω
2
+ κ
4
+
(c
2
−2cκ
2
)k
2
+ c
2
k
4
= 0.
(2)
Analyzing this equation, for c = 0 it is easy to con-
clude that the spectrum contains two set of disjoint
bandgaps; one in upper half and the other in lower
half of the spectrum and the limits of the gaps are (Is-
lam and Atai, 2015):



−
1
2
+
q
1
4
+ κ
2
≤ ω ≤
1
2
+
q
1
4
+ κ
2
ω > 0,
−
1
2
−
q
1
4
+ κ
2
≤ ω ≤
1
2
−
q
1
4
+ κ
2
ω < 0.
(3)
For c 6= 0, the shapes of the branches of the dispersion
diagram change, and a central gap (which is a gen-
uine gap) is formed. In this case, the lower and upper
gaps overlap with one branch of continuous spectrum
and therefore they are not genuine bandgaps. It has
been found that soliton solutions exist in the upper
and lower gaps only (Islam and Atai, 2015).
3 INTERACTION OF SOLITONS
The model of Eqs. 1 is nonintegrable. Thus, the
interactions between solitons are more complex. To
simulate the interaction of solitons we have utilized
a symmetrized split-step Fourier method to numeri-
cally solve Eqs. 1 subject to the following initial con-
ditions:
u(x, 0) = u

x+
∆x
2
, 0

+ u

x−
∆x
2
, 0

e
i∆θ
,
v(x, 0) = v

x+
∆x
2
, 0

+ v

x−
∆x
2
, 0

e
i∆θ
,
φ(x, 0) = φ

x+
∆x
2
, 0

+ φ

x−
∆x
2
, 0

e
i∆θ
,
ψ(x, 0) = ψ

x+
∆x
2
, 0

+ ψ

x−
∆x
2
, 0

e
i∆θ
,
(4)
where ∆x and ∆θ are the initial separation and the
phase difference between the two quiescent solitons,
respectively and u(x, 0), v(x,0), φ(x, 0) and ψ(x,0)
belong to the stable regions. We have previously
found that Eqs. 1 support two disjoint families of soli-
tons, namely Type 1 and Type 2, and that only Type 1
solitons are stable (Islam and Atai, 2015). Therefore,
we have only considered the interactions of Type 1
solitons.
0 0.2 0.4
0.6
0.8 1
0.8
0.9
1
1.1
1.2
q
ω
S
M
A
Unstable (Type 2)
Unstable(Type 1)
T
M
Figure 2: Results of the interactions of in-phase (∆θ = 0)
quiescent solitons in (q, ω) plane for κ = 0.5, c = 0.2 and
∆x = 12.0. The labeled regions are asymmetric separation
(A), symmetric separation (S), merger into a quiescent soli-
ton (M) and the formation of three solitons (T) i.e. a quies-
cent soliton and two moving solitons.
We have conducted a systematic investigation of
the interactions of in-phase quiescent solitons. Typ-
ical examples of the interaction outcomes are dis-
played in Figure (1). The interactions may lead to the
destruction of one or both solitons, generation of two
asymmetrically (Figure 1(a)) or symmetrically (Fig-
ure 1(b)) separating solitons , merger of solitons into
a single quiescent soliton (Figure 1(c)) and the forma-
tion of three solitons (Figure 1(d)) [one quiescent soli-
ton and two moving solitons with equal velocities].
Figure 2 displays the soliton interaction outcomes
in the (q,ω) plane for κ = 0.50. In the region A where
the outcome of the interactions is generation of two
asymmetrically separating solitons (e.g. Figure 1(a)),
the solitons may undergo multiple collisions and then
separate with unequal velocities and magnitudes, fol-
lowed by subsequent radiation of energy. In case of
symmetric separation (i.e. region S), the two solitons
temporarily merge into a single one and then splits
into two moving solitons with equal velocities. An
interesting feature of the interactions is that region S
is principally observed in the upper bandgap. On the
other hand, the transformation of 2 → 3 solitons oc-
curs in both upper and lower gaps. It is also found
that the interaction regions are greatly affected by ∆x,
c, and κ. The interplay of these parameters and their
effect on the outcomes are currently under investiga-
tion.
Dynamics of Interacting Bragg Grating Solitons in a Semilinear Dual-core System with Cubic-quintic Nonlinearity
231

4 CONCLUSIONS
In this paper, we have investigated the interaction
dynamics of two in-phase quiescent BG solitons in
a semilinear dual core system, where one core has
cubic-quintic nonlinearity with BG and the other is
linear. By means of systematic numerical simula-
tions, the soliton interactions have been studied for
different soliton parameters. It is found that, the inter-
actions between stable Type 1 solitons may produce
diverse results such as destruction of one or both the
solitons, symmetric or asymmetric separation, merger
into one quiescent or moving soliton and even the
transformation of 2 → 3 solitons. It is also observed
that the interaction outcomes strongly dependent on
the strength of quintic nonlinearity (q), coupling co-
efficient (κ) between the cores and the the group ve-
locity term (c) in the linear core. The interactions
may result in the formation of three solitons (a quies-
cent soliton and two moving ones) in both the upper
and lower bandgaps.
REFERENCES
Aceves, A. B. and Wabnitz, S. (1989). Self induced trans-
parency solitons in nonlinear refractive periodic me-
dia. Phys. Lett. A, 141:37–42.
Atai, J. (2004). Interaction of Bragg grating solitons in a
cubic-quintic medium. J. Opt. B Quantum Semiclass.,
6:S177–S181.
Atai, J. and Chen, Y. (1992). Nonlinear couplers composed
of different nonlinear cores. J. Appl. Phys., 72:24–27.
Atai, J. and Chen, Y. (1993). Nonlinear mismatches be-
tween two cores of saturable nonlinear couplers. IEEE
J. Quant. Elec., 29:242–249.
Atai, J. and Malomed, B. A. (2000). Bragg-grating soli-
tons in a semilinear dual-core system. Phys. Rev. E,
62:8713–8718.
Atai, J. and Malomed, B. A. (2001). Families of Bragg-
grating solitons in a cubic-quintic medium. Phys. Lett.
A, 284:247–252.
Bertolotti, M., Monaco, M., and Sibilia, C. (1995). Role of
the asymmetry in a third-order nonlinear directional
coupler. Opt. Comm., 116:405–410.
Boudebs, G., Cherukulappurath, S., Leblond, H., Troles, J.,
Smerktala, F., and Sanchez, F. (2003). Experimental
and theoretical study of higher-order nonlinearities in
chalcogenide glasses. Opt. Commun., 219:427–433.
Christadoulides, D. N. and Joseph, R. I. (1989). Slow Bragg
solitons in nonlinear periodic structures. Phys. Rev.
Lett., 62:1746–1749.
Christiansen, P., Sorensen, M., and Scott, A., editors (2000).
Nonlinear Science and the Dawn of the 21st Century.
Springer, Verlag Berlin Heidelberg New York, 1st edi-
tion.
Conti, C., Trillo, S., and Assanto, G. (1997). Doubly res-
onant Bragg simultons via second-harmonic genera-
tion. Phys. Rev. Lett., 78:2341–2344.
Dasanayaka, S. and Atai, J. (2013a). Moving Bragg grat-
ing solitons in a cubic-quintic nonlinear medium with
dispersive reflectivity. Phys. Rev. E, 88:022921.
Dasanayaka, S. and Atai, J. (2013b). Stability and collisions
of moving Bragg grating solitons in a cubic-quintic
nonlinear medium. J. Opt. Soc. Am. B, 30:396–404.
Desterke, C. M. and Sipe, J. E. (1994). Gap solitons.
Progress in Optics, 33:203–260.
Eggleton, B. J., Desterke, C. M., and Slusher, R. E. (1997).
Nonlinear pulse propagation in Bragg gratings. J. Opt.
Soc. Am. B, 14:2980–2993.
Islam, M. J. and Atai, J. (2015). Bragg grating solitons in
semilinear dual-core system with cubic-quintic non-
linearity. In International Conference on Photonics,
Optics and Laser Technology, Berlin, Germany. IN-
STICC.
Kashyap, R. (1999). Fiber Bragg Gratings. Academic
Press, San Diego.
Lawrence, B. L., Cha, M., Torruellas, W. E., Stegeman,
G. I., Etemad, S., Baker, G., and Kajzar, F. (1994).
Measurement of the complex nonlinear refractive-
index of single-crystal p-toluene sulfonate at 1064-
nm. Appl. Phys. Lett., 64:2773–2775.
Loh, W., Laming, R., Robinson, N., Cavaciuti, A., C. Va-
ninetti, J. A., Zervis, M., and Cole, M. (1996). Dis-
persion compensation over distances in excess of 500
km for 10 gb/s systems using chirped fiber gratings.
IEEE Photon. Technol. Lett., 8:944–946.
Mak, W. C. K., Malomed, B. A., and Chu, P. L. (1998).
Solitary waves in coupled nonlinear waveguides with
Bragg gratings. J. Opt. Soc. Am. B, 15:1685–1692.
Mak, W. C. K., Malomed, B. A., and Chu, P. L. (2003).
Formation of a standing-light pulse through collision
of gap solitons. Phys. Rev. E, 68:02669.
Mok, J. T., Desterke, C. M., Litler, I. C. M., and Eggleton,
B. J. (2006). Dispersionless slow light using gap soli-
tons. Nat. Phys., 2:775–780.
Neill, D. R. and Atai, J. (2006). Collision dynamics of gap
solitons in kerr media. Phys. Lett. A, 353:416–421.
Neill, D. R., Atai, J., and Malomed, B. A. (2007). Gap soli-
tons in a hollow optical fiber in the normal dispersion
regime. Phys. Lett. A, 367:73–82.
Nistazakis, H. E., Frantzeskakis, D. J., Atai, J., Malomed,
B. A., Efremidis, N., and Hizanidis, K. (2002). Mul-
tichannel pulse dynamics in a stabilized ginzburg-
landau system. Phys. Rev. E, 65:036605.
Radic, S., George, N., and Agrawal, G. P. (1995). The-
ory of low-threshold optical switching in nonlinear
phase-shifted periodic structures. J. Opt. Soc. Am. B,
12:671–680.
Taverner, D., Broderick, N. G. R., Richardson, D. J., Lam-
ing, R. I., and Ibsen, M. (1998). Nonlinear self-
switching and multiple gap-soliton formation in a
fiber Bragg grating. Opt. Lett., 23:328–330.
Zhan, C., Zhang, D., Zhu, D., Wang, D., Li, Y., Li, D., Lu,
Z., Zhao, L., and Nie, Y. (2002). Third- and fifth-order
optical nonlinearities in a new stilbazolium derivative.
J. Opt. Soc. Am. B, 19:369–375.
PHOTOPTICS 2016 - 4th International Conference on Photonics, Optics and Laser Technology
232