Dynamics of moving Bragg grating solitons in

cubic-quintic nonlinear media

Sahan Dasanayaka, Javid Atai

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

Abstract—Stability and collisions of moving Bragg gratingsolitons in a cubic-quintic medium are investigated. The effectof solitons’ initial velocity on the outcome of the collisions isanalyzed.

Keywords—Bragg grating; solitons; cubic-quintic nonlinearity

I. INTRODUCTION

A periodic variation of refractive index, such as in a fiber

Bragg grating, gives rise to a photonic bandgap inside which

low intensity light experiences strong reflection [1]. However,

material nonlinearity allows light pulses known as Bragg

grating solitons to propagate within the normally forbidden

frequency band. These solitons can have any velocity between

zero (quiescent) and the speed of light in the medium [2], [3].

The slowest experimentally observed Bragg grating soliton to

date has a velocity of 0.23c, where c is the speed of light in

the medium [4].

The generation of quiescent or very low velocity soli-

tons is challenging. One possibility is to introduce grating

nonuniformities such as grating defects [5]. Another approach

is through the collision of two counterpropagating solitons.

For example, it has been theoretically shown that in a Kerr

nonlinear medium collision of solitons with a velocity less

than 0.2c will result in the formation of quiescent solitons [6].

Bragg grating solitons have been predicted in more sophis-

ticated nonlinearities such as quadratic nonlinearity [7] and

cubic-quintic nonlinearity [8], [9], [10]. For instance, it has

been shown that, in a cubic-quintic nonlinear medium, there

exist two disjoint families of quiescent solitons. In this paper

we investigate the existence and stability of moving solitons

in cubic-quintic media. We also analyze the outcomes of the

collisions between solitons.

II. CUBIC-QUINTIC MODEL

Bragg gratings written in cubic-quintic media can be mod-

eled using the cubic-quintic nonlinear coupled-mode equations

[8]:

iut + iux +

[

1

2|u|2 + |v|2

]

u

−ν

[

1

4|u|4 +

3

2|u|2|v|2 +

3

4|v|4

]

u + v = 0,

(1)

ivt − ivx +

[

1

2|v|2 + |u|2

]

v

−ν

[

1

4|v|4 +

3

2|v|2|u|2 +

3

4|u|4

]

v + u = 0,

where u and v are the forward- and backward-propagating

waves, and ν > 0 is a parameter controlling the strength of

quintic nonlinearity. It is worth noting that cubic-quintic non-

linear response has been experimentally observed in chalco-

genide glasses [11] and some transparent organic materials

[12].

To find moving solitons in this system, the equations must

be written in the moving coordinates {X, T } = {x − ct, t},where c is the velocity of the soliton normalized such that

c = 1 is the speed of light in the medium. Substituting

{u (X, T ) , v (X, T )} = {U(X), V (X)} e−iω√

1−c2T into the

transformed system results in the following system of ordinary

differential equations:

ω√

1 − c2U + i (1 − c)UX +

[

1

2|U |2 + |V |2

]

U

−ν

[

1

4|U |4 +

3

2|U |2|V |2 +

3

4|V |4

]

U + V = 0,

(2)

ω√

1 − c2V − i (1 + c)VX +

[

1

2|V |2 + |U |2

]

V

−ν

[

1

4|V |4 +

3

2|V |2|U |2 +

3

4|U |4

]

V + U = 0.

The frequency parameter, ω, is the same as that used in the

Kerr analytical solutions [2]. For quiescent solitons (c = 0),Eqs. 2 can be solved analytically to find two disjoint families

of solitons filling the entire bandgap [8]. The two families,

known as Type 1 and Type 2 solitons, are separated on the

(ν, ω) plane by ν = 27/ (160 (1 − ω)). Type 1 solitons,

are a generalization of the solitons in Kerr media, while

Type 2 solitons occur for larger values of ν. In the case of

moving solitons (c 6= 0), there are no analytical solutions

available. Therefore, we have solved Eqs. 2 numerically using

a relaxation algorithm. Similar to the quiescent solitons, we

have found that above a certain value of ν there exist two

disjoint families of solitons. The border separating the soliton

families has numerically been determined to be:

ν =27

160 (1 − ω)

[

1√1 − c2

]

. (3)

362

TuR3 (Contributed Oral)2:15 PM – 2:30 PM

978-1-4577-0733-9/12/$26.00 ©2012 IEEE

III. STABILITY AND COLLISIONS OF SOLITONS

Stability of the numerically obtained moving solitons was

assessed by numerically solving Eqs. (1). The stability analysis

has revealed that stable solitons exist in both families and

that the stability regions are dependent upon the velocity of

solitons. In Fig.1 the unstable region has been identified with

the letter “U”.

Figure 1 displays the outcomes of the collisions between

in-phase counterpropagating stable solitons with velocity of

0.1 in the (ν, ω) plane. Firstly, all stable Type 2 solitons are

destroyed by collisions. In the case of Type 1 solitons, there are

several regions where collisions result in two solitons moving

in opposite directions with varying velocities. There are also

regions where the collisions result in the merger of solitons

into a quiescent one (Region M in Fig. 1) or the formation of

two moving solitons and a quiescent one (Region T in Fig. 1).

0 0.2 0.4 0.6 0.8 1

ν

-1

-0.5

0

0.5

1

ω

D

E

TA

S

D F

M

A

S

U

U

Figure 1. The outcomes of collisions between in-phase solitons for c = 0.1.The labeled regions are asymmetrically moving solitons (A), soliton destruc-tion (D), quasi-elastic collision (E), faster solitons (F), merger to form a singlequiescent soliton (M), slower solitons (S), formation of three solitons (T),unstable single solitons (U) and quiescent soliton formation (shaded). Thedashed curve indicates the boundary between Type 1 and Type 2 regions.

A noteworthy feature of Fig. 1 is that the merger of solitons

occurs only when quintic nonlinearity is weak. On the other

hand, above a certain value of ν, the collisions give rise to

the formation of three solitons (see Fig. 2(b)). In addition, our

simulations demonstrate that as the initial velocity of solitons

increases the region M shrinks and completely disappears for

c ≥ 0.3. However, the effect of initial velocity on the region

T is very small.

-80 -40 0 40 80

x

2000

0

t

0

t

(a)

-80 -40 0 40 80

x

1000

0

t

0

t

(b)

Figure 2. Examples of quiescent soliton formation via (a) the merger ofsolitons (ω = 0.65, ν = 0.05, c = 0.10) and (b) formation of three solitons(ω = 0.55, ν = 0.30, c = 0.10)

Another key finding is that when there is a small phase

difference between the solitons, the merger process no longer

occurs. It usually results in two solitons moving in opposite

directions with different velocities that are highly sensitive to

the amount of phase difference. However for the three soliton

process, a small phase difference only causes the central

soliton to attain a small velocity.

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