Interaction of Bragg grating solitons in a cubic–quintic medium

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2004 J. Opt. B: Quantum Semiclass. Opt. 6 S177

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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF OPTICS B: QUANTUM AND SEMICLASSICAL OPTICS

J. Opt. B: Quantum Semiclass. Opt. 6 (2004) S177–S181 PII: S1464-4266(04)74883-1

Interaction of Bragg grating solitons in acubic–quintic mediumJavid Atai

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

E-mail: atai@ee.usyd.edu.au

Received 8 January 2004, accepted for publication 12 March 2004Published 4 May 2004Online at stacks.iop.org/JOptB/6/S177DOI: 10.1088/1464-4266/6/5/003

AbstractWe investigate the interaction of Bragg grating solitons in a cubic–quinticnonlinear medium. In a previous work two different families of solitonshave been identified in such a medium, which are separated by a border1 − ω = 27/(160ν), where −1 < ω < 1 is the frequency and ν > 0 controlsthe strength of quintic nonlinearity. We find that two zero-velocity solitonsbelonging to the region 1 − ω < 27/(160ν) with initial phase difference ofzero (i.e.�φ = 0) attract each other and when�φ = π and π/2 the solitonsrepel each other. When �φ = 0, for certain parameter values, theinteraction results in a single zero velocity soliton. But in many cases twomoving solitons emerge as a result of interaction. The interaction of solitonsbelonging to the region 1 − ω > 27/(160ν) is similar to the other family inthat when �φ = 0 they attract, and they repel when �φ = π and π/2.However, in the case of �φ = 0 and π/2 the interaction always results inthe deformation and subsequent destruction of solitons.

Keywords: Bragg grating solitons, cubic–quintic nonlinearity, solitoninteraction

1. Introduction

It is well known that the periodic variation of linear dielectricconstant in an optical medium gives rise to a photonic bandstructure. Such a band structure induces a linear cross couplingbetween the counterpropagating waves which results in a largeeffective dispersion (see [1, 2]). An example of such a mediumis a fibre Bragg grating (FBG). It has been experimentallydemonstrated that the effective dispersion in an FBG can besix orders of magnitude greater than the inherent materialdispersion of the uniform optical medium [3]. Since solitonsare formed as a result of a balance between nonlinearity anddispersion, one may expect to observe solitons in FBGs if theintensity of light is sufficiently high. In particular, since theeffective dispersion in an FBG is very large, a soliton may beformed in a very short distance (∼6 cm) [4]. Such solitons aregenerally termed Bragg-grating (BG) solitons.

The existence and stability of Bragg solitons inperiodic structures with Kerr nonlinearity have been thesubject of much interest both theoretically [2, 5–8] and

experimentally [3, 4, 9, 10]. On the theoretical side, exactBG solitons were reported in [7] and [8]. It was pointed outin [8] that the exact BG solitons of nonlinear coupled modeequations form a two-parameter family of solutions. One ofthese parameters determines the soliton’s velocity, which canrange between zero and the speed of light in the medium, andthe other one embodies the detuning frequency, peak power andsoliton width. It has also been shown that approximately halfof these solitons are stable against oscillatory perturbations[11–13]. On the other hand, a main focus of experimentalefforts in this area has been the generation of zero-velocityBG solitons. The driving force behind these experimentalactivities has been the intriguing possibility of ‘full stoppageof light’ through dynamical equilibrium between forward-and backward-propagating waves. The recent discovery ofultraslow light in ultracold gases [14] has provided a significantimpetus for experimental observation of zero-velocity BGsolitons. To date, however, BG solitons with a velocity inexcess of 50% of the speed of light have been observed [4].

Due to their potential applications, the possibility of Bragggrating solitons in more sophisticated nonlinear systems have

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J Atai

also been considered [15–17]. For example, in [15] a dualcore system equipped with identical BG carrying cores wereconsidered. It was shown that a bifurcation at a critical valueof the soliton’s energy may give rise to an asymmetric soliton.It has also been shown that a family of BG solitons exists ina dual core system with nonidentical cores where one core islinear [16]. A special class of solitons has also been reportedin the case of a dual core system when one core is linear andcontains a Bragg grating, and the other is nonlinear [17].

BG solitons have been predicted in more sophisticatednon-Kerr nonlinearities such as quadratic nonlinearity [18, 19]and nonlinearity due to narrow layers of resonantly absorbingtwo-level atoms whose spacing is equal to the BG period [20].More recently, the existence and stability of zero velocitysolitons in a cubic–quintic medium has been considered [21].It has been shown that in a cubic–quintic medium there existtwo disjoint families of BG solitons, one of which may beregarded as the generalization of the usual BG solitons in Kerrmedia, while the other family is comprised of novel solitonswhere quintic nonlinearity is dominant.

In this paper we study the interactions and dynamics ofzero velocity BG solitons reported in [21]. In section 2 wepresent the model and briefly review the general characteristicsof the soliton solutions. In section 3 we discuss the results ofsimulations of interactions of two solitons placed initially atsome distance from each other. Section 4 concludes the paper.

2. The model and soliton solutions

We first summarize some of the results from [21] which arepertinent to this paper. The starting point is the followingsystem of normalized coupled equations for amplitudes of twocounterpropagating waves u(x, t) and v(x, t):

iut + iux + [ 12 |u|2 + |v|2]u

− ν[ 14 |u|4 + 3

2 |u|2|v|2 + 34 |v|4]u + v = 0 (1)

ivt − ivx + [ 12 |v|2 + |u|2]u

− ν[ 14 |v|4 + 3

2 |v|2|u|2 + 34 |u|4]v + u = 0 (2)

where ν > 0 is a parameter to control the strength of thequintic nonlinearity and the BG-induced coupling coefficientis normalized to unity. It should be pointed out that althoughthe analysis is performed in the temporal domain, the resultsare readily applicable to the spatial domain where the BG iscreated in a planar waveguide by a set of parallel scores. In thespatial domain, x and t are interpreted as the transverse andlongitudinal coordinates. In fact, the spatial realization of themodel may be more practical since cubic–quintic nonlinearityhas already been demonstrated in the p-toluene sulfonate (PTS)crystal [22].

In order to obtain zero-velocity solitons the followingansatz is substituted into equations (1) and (2):

u(x, t) = A(x) exp(iφ(x) − iωt),

v(x, t) = B(x) exp(iψ(x) − iωt),

where −1 < ω < 1. After some simple algebraicmanipulations it can be shown that A(x) = B(x) and ψ(x) =−φ(x) and that A(x) and φ(x) satisfy the following equations:

da

dx= 2a sin(2φ), a ≡ A2, (3)

dφ

dx= ω +

3

2a − 5

2νa2 + cos(2φ). (4)

From equations (3) and (4) the soliton solution can be obtainedin an implicit form,

x(a) =∫ a0

a

db

2b√

1 − (ω + 3

4 b − 56νb2

)2. (5)

This integral describes the soliton solution in the interval0 < x < ∞. It readily follows from (5) that a(x) attainsits maximum value a0 at x = 0 and that a0 must be a root ofthe expression under the square root in (5) i.e. ω + (3/4)a −(5ν/6)a2 = ±1. The physically relevant roots of this quadraticequation are given by:

(a0)1 = 3

5ν

[3

4−

√9

16− 10ν

3(1 − ω)

], (6)

(a0)2 = 3

5ν

[3

4+

√9

16+

10ν

3(1 + ω)

]. (7)

It can be deduced from (5) that root (6) exists in the region

1 − ω <27

160ν, (8)

while root (7) becomes irrelevant. On the other hand, root (7)determines the amplitude of the soliton in the region

1 − ω >27

160ν, (9)

where root (6) does not exist. Exactly at the border, 1 − ω =27

160ν , no soliton exists. Thus in this model there exist twodisjoint families of solitons which are separated by the border1 − ω = 27

160ν . Henceforth, the solitons corresponding to (6)and (7) will be referred to as type I and type II, respectively.

An important difference between these solutions is in theirphase structures and parities. In the case of type I solitons,Re(u(x)) and Re(v(x)) are odd, and Im(u(x)) and Im(v(x))are even functions of x . On the other hand, in type II solitons,the parities of the real and imaginary parts of the solutionsare opposite to those of type I solitons. Stability of type I andtype II solitons was investigated in [21] by means of systematicnumerical simulations. The result of the numerical stabilityanalysis is shown in figure 1.

3. Interaction of solitons

Due to the nonintegrability of the model, interactions betweenBG solitons may be quite complex. For example, it has beenshown that in a FBG with cubic nonlinearity (which is also anonintegrable system), under certain circumstances, collisionof two moving BG solitons may lead to various scenariossuch as generation of a zero velocity soliton or emergence ofmoving solitons whose velocities are greater than those of theoriginal ones [23]. It has also been shown that the outcome ofinteraction of two copropagating moving BG solitons in cubicnonlinearity is strongly dependent on the initial separation ofthe solitons [24]. In view of these results and the characteristics

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Interaction of Bragg grating solitons in a cubic–quintic medium

Stable (Type I)U

nsta

ble

(Typ

e I)

0.5−

1−

0.5

0

1

0 0.80.60.40.2 1ν

Stable (Type II)

Unstable (Type II)

Figure 1. Stability diagram for type I and type II solitons. Thedashed curve, whose equation is 1 − ω = 27/(160ν), separates thetwo families of solitons.

of the soliton solutions in cubic–quintic nonlinearity, oneanticipates a rich interaction dynamics in this system.

In order to simulate the interactions of BG solitons weutilize the split-step Fourier approach to numerically solveequations (1) and (2) subject to the following initial conditions:

u(x, 0) = u

(x − �x

2, 0

)+ u

(x +

�x

2, 0

)exp(i�φ),

v(x, 0) = v

(x − �x

2, 0

)+ v

(x +

�x

2, 0

)exp(i�φ),

where �x and �φ are the initial separation and phasedifference between the solitons, and u and v belong to thestable regions shown in figure 1.

-40 -20 0 20 40x

1000

t

0

(a)

-40 -20 0 20 40x

200

t

0

(b)

-40 -20 0 20 40x

200

t

0

(c)

Figure 2. Interaction of type I solitons with ν = 0.1, ω = 0.5 and�x = 6. (a)�φ = 0; (b) �φ = π/2; (c) �φ = π . In this figure, and allfigures that follow, only the u-component is shown.

3.1. Type I solitons

An example of interaction of type I solitons is displayedin figure 2. It is found that in this case the interaction ofsolitons has similar features to nonlinear Shrodinger solitonsin that when �φ = 0 the solitons attract each other, and for�φ = π/2 and π they repel each other. The most interestingoutcome is shown in figure 2(a) where the attraction resultsin a zero-velocity soliton. In fact, we find that for solitonswith ω = 0.5 and ν = 0.1 mutual attraction leads to a zero-velocity soliton provided that the initial separation is within therange 6 � �x � 12. Generation of zero-velocity solitons hasalso been reported in [23] for collisions of BG solitons in Kerrnonlinearity. However, a feature that has not been observedin [23] and is present here is that the outcome of interactionis also dependent on the initial separation between solitons.As is shown in figure 3, depending on the initial separation(within the range 6 � �x � 12) the solitons may mergeinto a single zero-velocity soliton (figure 3(a)) or may undergomultiple collisions leading to two moving solitons with unequalamplitudes and velocities (figure 3(b)).

A noteworthy feature of interaction in figure 3(b) is thebreaking of initial symmetry between the two solitons. Thiscollision-induced symmetry-breaking may be attributed to thefact that the ‘lump’ that is temporarily formed as a result ofattraction is subject to modulational instability. Therefore, anyasymmetry in the noise may give rise to symmetry breaking.

It is clear from figures 2 and 3 that the interactions arealways accompanied by some loss of energy in the form ofradiation. One may quantify this emission of radiation bycomparing the initial and final energies of solitons, i.e.

Ef

Ei=

∫ +w−w(|uf |2 + |vf |2) dx∫ +w−w(|ui|2 + |vi|2) dx

where w is the half-width of the computational window. Itshould be noted that absorbing boundary conditions are usedin order to attenuate the radiation reaching the edges of thecomputational window. In the case of figure 3(a) Ef/Ei is

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J Atai

-40 -20 0 20 40x

2000

t

0

(a)

-40 -20 0 20 40x

1000

t

0

(b)

Figure 3. Interaction of type I solitons with ν = 0.1, ω = 0.9 and �φ = 0. (a)�x = 8; (b)�x = 12.

-40 -20 0 20 40x

60t

0

(a)

-40 -20 0 20 40x

60t

0

(b)

-40 -20 0 20 40x

200t

0

(c)

Figure 4. Interaction of type II solitons with ν = 0.8, ω = −0.32 and �x = 6. (a) �φ = 0; (b)�φ = π/2; (c)�φ = π .

found to be 0.93, i.e. approximately 7% of the initial energyis lost to radiation during the interaction process. On theother hand, in figure 3(b) only 1% of the initial energy isemitted as radiation. In particular, when �φ = π/2, therepulsion of solitons results in symmetry breaking in termsof the velocities of eventual moving solitons. This symmetrybreaking originates from the fact that the symmetry of theinitial configuration is not compatible with equations (1)and (2) and is therefore broken upon propagation.

3.2. Type II solitons

We have also studied the interaction of type II solitons forvarious parameters. Figure 4 shows a typical example ofinteraction of solitons of this type. The results are quitestriking. In the case �φ = 0 the solitons attract one anotherand during this process one of them (the right soliton) getsdeformed which then leads to destabilization of the left solitonand subsequently both solitons are destroyed. When �φ =

π/2 (see figure 4(b)) the solitons repel one another but asthey start to move apart the right soliton is deformed whichleads to the emission of significant amount of radiation anddestabilization of the left soliton. The subsequent evolution ofsolitons and their eventual destruction is due to their continuousinteraction with radiation. When �φ = π the solitons repeleach other and two moving type II solitons are generated. Thedestruction of solitons when �φ = 0 and π/2 may indicatethat type II solitons (both quiescent and moving solitons) areless stable than their type I counterparts.

4. Conclusion

In this paper, we have investigated the interactions of the twofamilies of solitons in a cubic–quintic nonlinearity which werereported in [21]. It is found that in-phase type I solitons,which may be regarded as the generalization of the usual BGsolitons in Kerr media, attract each other, and when the initialphase difference is π/2 or π they repel. Depending on the

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Interaction of Bragg grating solitons in a cubic–quintic medium

parameters, particularly the initial separation, the interactionof in-phase type I solitons may either result in a single zerovelocity soliton or two moving solitons. In the case of type IIsolitons, which have no counterpart in periodic Kerr media,it is shown that when initially the solitons are in-phase, orthe phase difference is π/2, the interaction always results indeformation and subsequent destruction of solitons. However,when the initial phase difference is π the solitons repel oneanother and two moving type II solitons are generated. It istherefore concluded that the type II solitons are less stable thanthe type I solitons.

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