Moving Gap Solitons in Semilinear Coupled Bragg Gratings with a

Phase Mismatch

Shuvashis Saha and Javid Atai

School of Electrical and Information Engineering, The University of Sydney, NSW 2006, Australia

Keywords:

Moving Gap Solitons, Bragg Gratings, Phase Mismatch, Dual-core System.

Abstract:

We consider the existence and stability of the moving gap solitons in a semilinear coupler where one core

has Kerr nonlinearity and the other is linear and both cores are equipped with a Bragg grating with a phase

mismatch between them. We analyze the effect of the phase mismatch and the soliton velocity on the existence

and stability of moving gap solitons. It is found that larger phase mismatch leads to the expansion of the

stability region of the moving gap solitons.

1 INTRODUCTION

Nonlinear photonic structures equipped with ﬁber

Bragg gratings (FBGs) have attracted much interest

due to their potential applications in the slow light

applications (Aceves and Wabnitz, 1989; de Sterke

and Sipe, 1994; Sukhorukov and Kivshar, 2006). One

of the most interesting features of FBGs is that there

is a band gap in their linear spectrum where no lin-

ear waves can propagate. Moreover, a strong effec-

tive dispersion is generated due to the cross-coupling

between forward and backward propagating waves

(de Sterke and Sipe, 1994). This strong effective dis-

persion can be counterbalanced by the nonlinearity of

the medium when the pulse intensity is sufﬁciently

high giving rise to solitary waves known as gap soli-

tons (GSs). A very important property of GSs is that

they can propagate with any velocities ranging from

zero to the speed of light in the medium (Aceves and

Wabnitz, 1989). Experimentally GSs were observed

in a ∼ 6cm−long FBGs (Eggleton et al., 1996). Thus

far, moving 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).

The properties of GSs have been studied theo-

retically for different photonic structures and non-

linear media such as cubic-quintic nonlinear media

(Dasanayaka and Atai, 2013; Islam and Atai, 2014),

photonic crystals (Skryabin, 2004; Atai et al., 2006;

Neill and Atai, 2007), nonuniform Bragg gratings

(Atai and Malomed, 2005; Ahmed and Atai, 2017),

waveguide arrays (Mandelik et al., 2004; Dong et al.,

2011) and dual-core ﬁbers with gratings in one or both

cores (Mak et al., 1998; Atai and Malomed, 2005;

Baratali and Atai, 2012).

Semilinear coupled systems exhibit superior

switching characteristics and support a wide range

of GSs (Chowdhury and Atai, 2017). It has been

demonstrated that very slow GSs can be generated in

a grating-assisted semilinear coupler (Atai and Mal-

omed, 2000; Shnaiderman et al., 2011). In the case

of dual-core systems made of coupled identical Bragg

gratings, both symmetric and asymmetric GSs exist in

the system (Mak et al., 2004). However, in the pres-

ence of a ﬁnite phase shift between the gratings, the

symmetric GSs are transformed into quasi-symmetric

ones (Tsofe and Malomed, 2007).

In this paper we analyze the existence and dynam-

ics the moving GSs in a semilinear coupled system

where both cores are equipped with a BG but there is

a phase mismatch between the gratings and one core

is linear while other one has Kerr type nonlinearity.

2 THE MODEL

The system model which was introduced in Ref.

(Saha and Atai, 2021) is given by the following cou-

pled partial differential equations:

iu

t

+ iu

x

+

|v|

2

+

1

2

|u|

2

u+ v+ κφ = 0

iv

t

− iv

x

+

|u|

2

+

1

2

|v|

2

v+ u+ κψ = 0

iφ

t

+ icφ

x

+ ψe

iθ/2

+ κu = 0

iψ

t

−icψ

x

+ φe

−iθ/2

+ κv = 0

(1)

38

Saha, S. and Atai, J.

Moving Gap Solitons in Semilinear Coupled Bragg Gratings with a Phase Mismatch.

DOI: 10.5220/0010819900003121

In Proceedings of the 10th International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS 2022), pages 38-41

ISBN: 978-989-758-554-8; ISSN: 2184-4364

Copyright

c

2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

-8 -4 0 4 8

k

-8

-4

0

4

8

Ω

δ = 0.10

δ = 0.50

δ = 0.70

(a)

-8 -4 0 4 8

k

-8

-4

0

4

8

Ω

δ = 0.10

δ = 0.50

δ = 0.70

(b)

Figure 1: Examples of the spectra generated by the disper-

sion relation for κ = 0.5, c = 1, and different values of soli-

ton velocity δ; (a) θ = 0 and (b) θ = 2π.

where u and v represent the forward and backward

waves in the nonlinear core and φ and ψ are their

counterparts in the linear core. κ accounts for the

linear coupling coefﬁcient between two cores. Rel-

ative group velocity in the nonlinear core is set to 1

and c denotes the relative group velocity mismatch.

The phase mismatch between two Bragg gratings is

denoted by θ.

To determine the moving GS solutions, Eqs.

(1)

need to be transformed to the moving frame using

the transformation

{X,T} = {x− δt,t}. δ accounts

for the normalized velocity of moving solitons (δ = 1

corresponds to velocity of light in the medium).

The transformation gives rise to the following set of

partial differential equations in the moving reference

frame:

-80 -40 0 40 80

x

(a)

0

t

1200

0

t

-80 -40 0 40 80

x

800

0

t

(b)

Figure 2: Examples of the evolution of moving gap soli-

tons for zero phase mismatch (θ = 0). (a) Stable soliton

corresponding to Ω = 0.20 and (b) Unstable soliton corre-

sponding to Ω = −0.45. The other parameters are κ = 0.5,

c = 1.0 and δ = 0.1. Only the u components are shown.

iu

T

+ i (1 − δ)u

X

+

|v|

2

+

1

2

|u|

2

u+ v+ κφ = 0

iv

T

− i (1 + δ)v

X

+

|u|

2

+

1

2

|v|

2

v+ u+ κψ = 0

iφ

T

+ i (c− δ) φ

X

+ ψe

iθ/2

+ κu = 0

iψ

T

− i (c+ δ) ψ

X

+ φe

−iθ/2

+ κv = 0

(2)

To determine the linear spectrum in which the

moving soliton solutions may reside, we substitute the

plane wavesolutions of {u,v, φ,ψ} ∼ exp(ikX − iΩT)

into the linearized form of Eqs.

(2), which gives the

following dispersion relation:

Ω

4

+4Ω

3

δk−Ω

2

c

2

k

2

+6Ω

2

δ

2

k

2

−Ω

2

k

2

−2Ω

2

+k

2

−2Ω

2

κ

2

−2Ωc

2

δk

3

+4Ωδ

3

k

3

−2Ωδk

3

−4Ωδkκ

2

Moving Gap Solitons in Semilinear Coupled Bragg Gratings with a Phase Mismatch

39

-80 -40 0 40 80

x

(a)

0

t

1200

0

t

-80 -40 0 40 80

x

600

0

t

(b)

Figure 3: Examples of the evolution of moving gap solitons

for θ = 2π. (a) Stable soliton corresponding to Ω = 0.20

and (b) Unstable soliton corresponding to Ω = −0.70. The

other parameters are κ = 0.5, c = 1.0 and δ = 0.1. Only the

u components are shown.

−4Ωδk−c

2

δ

2

k

4

+c

2

k

4

+c

2

k

2

−2ck

2

κ

2

+δ

4

k

4

+κ

4

−δ

2

k

4

− 2δ

2

k

2

κ

2

− 2δ

2

k

2

− 2cos

θ

2

κ

2

+ 1 = 0,

(3)

where k denotes the wave number and Ω is the fre-

quency detuning in the moving reference frame and it

is related to the frequency detuning (ω) in the station-

ary frame by Ω(k) = ω(k) −δk.

Fig. 1 shows the dis-

persion relation curves for different values of δ, when

c = 1, κ = 0.5 for θ = 0 and 2π. From straightforward

mathematical analysis of Eq.

(3), it is concluded that

when c = 1, the moving GS solutions exist only in

the central band gap. However, there exists a critical

value of soliton velocity, δ

cr

< 1 for which the central

band gap completely closes.

0

0.5

1

1.5

2

θ/π

-0.2

0

0.2

0.4

0.6

0.8

1

Ω

No Soliton Solution

Stable

Unstable

Figure 4: Stability diagram of the moving gap solitons in

the (θ,Ω) plane for κ = 0.5, c = 1.0 and δ = 0.10.

3 STABILITY OF MOVING

SOLITONS

To obtain the moving soliton solutions, Eqs. (2) were

solved numerically using a relaxation algorithm. We

then utilized the split-step Fourier method to investi-

gate the stability of moving GS solutions through di-

rect simulation of Eqs. (1). Figs. 2 and 3 show some

examples of the evolution of stable and unstable mov-

ing GSs for θ = 0 and 2π, respectively. Our analysis

shows that the model supports stable moving solitons

for values of θ in the range 0 ≤ θ ≤ 2π. A noteworthy

ﬁnding is that in the absence of the phase shift, i.e.

θ = 0, the unstable solitons may either decay com-

pletely (Fig. 2(b)) or shed some energy in the form

of radiation and evolve to another stable moving soli-

ton. However, as is shown in Fig. 3(b), when θ = 2π,

an unstable soliton loses energy upon propagationand

then splits and evolves into a quiescent and a moving

soliton.

Fig. 4 summarizes the result of stability diagram

on the (θ,Ω) plane when c = 1, κ = 0.50 and δ =

0.1. A notable ﬁnding is that, increasing θ leads to

the expansion of the stability region. The interplay of

θ and other parameters and their effect on the stability

of solitons is currently under investigation.

4 CONCLUSIONS

We have investigated the existence and stability of

moving GSs in a semilinear dual-core system where

one core has Kerr nonlinearity and the other is lin-

PHOTOPTICS 2022 - 10th International Conference on Photonics, Optics and Laser Technology

40

ear and both cores are equipped with BGs but there

is phase shift between them. We have focused on

the effect of phase mismatch and soliton velocity on

the existence and stability of moving GSs. A notable

ﬁnding is that the higher phase mismatch leads to the

expansion of the stable regions for the moving GSs.

Another ﬁnding is that for certain parameters unsta-

ble solitons may evolve into a quiescent and a moving

soliton. This outcome and its prevalence is currently

under investigation.

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.

Ahmed, T. and Atai, J. (2017). Bragg solitons in systems

with separated nonuniform bragg grating and nonlin-

earity. Phys. Rev. E, 96:032222.

Atai, J. and Malomed, B. A. (2000). Bragg-grating soli-

tons in a semilinear dual-core system. Phys. Rev. E,

62:8713.

Atai, J. and Malomed, B. A. (2005). Gap solitons in bragg

gratings with dispersive reﬂectivity. Phys. Lett. A,

342:404 – 412.

Atai, J., Malomed, B. A., and Merhasin, I. M. (2006). Sta-

bility and collisions of gap solitons in a model of a

hollow optical ﬁber. Opt. Commun., 265:342–348.

Baratali, B. H. and Atai, J. (2012). Gap solitons in dual-core

bragg gratings with dispersive reﬂectivity. J. Opt.,

14:065202.

Chowdhury, S. A. M. S. and Atai, J. (2017). Moving bragg

grating solitons in a semilinear dual-core system with

dispersive reﬂectivity. Sci. Rep., 7:1–12.

Dasanayaka, S. and Atai (2013). Stability and collisions of

moving bragg grating solitons in a cubic-quintic non-

linear medium. J. Opt. Soc. Am. B, 30:396–404.

de Sterke, C. M. and Sipe, J. E. (1994). Gap solitons. Prog.

Optics, 33:203–260.

Dong, R., R¨uter, C. E., Kip, D., Cuevas, J., Kevrekidis,

P. G., Song, D., and Xu, J. (2011). Dark-bright gap

solitons in coupled-mode one-dimensional saturable

waveguide arrays. Phys. Rev. A, 83:063816.

Eggleton, B. J., Slusher, R. E., Krug, P. A., and Sipe, J. E.

(1996). Bragg grating solitons. Phys. Rev. Lett.,

76:1627–1630.

Islam, M. J. and Atai, J. (2014). Stability of gap solitons

in dual-core bragg gratings with cubic-quintic nonlin-

earity. Laser Phys. Lett., 12:015401.

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. (2004).

Symmetric and asymmetric solitons in linearly cou-

pled bragg gratings. Phys. Rev. E, 69:066610.

Mandelik, D., Morandotti, R., Aitchison, J. S., and Silber-

berg, Y. (2004). Gap solitons in waveguide arrays.

Phys. Rev. Lett., 92:093904.

Mok, J. T., de Sterke, C. M., Littler, I. C. M., and Eggle-

ton, B. J. (2006). Dispersionless slow light using gap

solitons. Nature Phys., 2:775–780.

Neill, D. R. and Atai, J. (2007). Gap solitons in a hollow

optical ﬁber in the normal dispersion regime. Phys.

Lett. A, 367:73–82.

Saha, S. and Atai, J. (2021). Effect of phase mismatch be-

tween the bragg gratings on the stability of gap soli-

tons in semilinear dual-core system. In Proceedings of

the 9th International Conference on Photonics, Optics

and Laser Technology- PHOTOPTICS, pages 36–39.

Shnaiderman, R., Tasgal, R. S., and Band, Y. B. (2011).

Creating very slow optical gap solitons with a grating-

assisted coupler. Opt. Lett., 36:2438–2440.

Skryabin, D. V. (2004). Coupled core-surface solitons in

photonic crystal ﬁbers. Opt. Express, 12(20):4841–

4846.

Sukhorukov, A. A. and Kivshar, Y. S. (2006). Slow-light

optical bullets in arrays of nonlinear bragg-grating

waveguides. Phys. Rev. Lett., 97:233901.

Tsofe, Y. J. and Malomed, B. A. (2007). Quasisymmetric

and asymmetric gap solitons in linearly coupled bragg

gratings with a phase shift. Phys. Rev. E, 75:056603.

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