DISCRETE VECTOR SPATIAL SOLITONS IN A NONLINEAR WAVEGUIDE ARRAY.pdf

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Discrete vector spatial solitons in a nonlinear waveguide array

Mark J. Ablowitz and Ziad H. MusslimaniDepartment of Applied Mathematics, University of Colorado, Campus Box 526, Boulder, Colorado 80309-0526

Received 2 November 2001; published 21 May 2002

A vector discrete diffraction managed soliton system is introduced. The vector model describes propagation

of two polarization modes interacting in a nonlinear waveguide array with varying diffraction via the cross-

phase modulation coupling. In the limit of strong diffraction we derive averaged equations governing the slowdynamics of the beams amplitudes, and their stationary in the form of bright-bright vector bound state and

traveling wave solutions are found. Through an extensive series of direct numerical simulations, interactions

between diffraction-managed solitons for different values of velocities, diffraction, and cross-phase modulation

coefficient are studied. We compare each collision case with its classical counterpart constant diffraction and

find that in both the scalar and vector diffraction management cases, the interaction picture involves beam

shaping, fusion, fission, nearly elastic collisions, and, in some cases, multihump structures. The collision

scenario is found, in both the scalar and vector diffraction managed cases, to be rather different from the

classical case.

DOI: 10.1103/PhysRevE.65.056618 PACS numbers: 42.65.Wi

I. INTRODUCTION

Recently, there has been considerable interest in the studyof discrete spatial solitons in nonlinear media. Such solitonsare localized modes of nonlinear lattices that form when dis-crete diffraction is balanced by nonlinearity. Discrete solitonshave been demonstrated to exist in a wide range of physicalsystems; cf. Ref. 1. An array of coupled optical waveguidesis a setting that represents a convenient laboratory for experi-mental observations.

Discrete solitons in an optical waveguide array were theo-retically predicted in 2. Later on, many theoretical studiesof discrete solitons in a waveguide array reported switching,steering, and other collision properties of these solitons 3,4see also the review papers 5. In all of the above cases, the

localized modes are solutions of the well-known discretenonlinear Schrodinger DNLS equation that describes beampropagation in Kerr nonlinear media according to coupledmode theory. However, discrete bright and dark solitonshave also been found in quadratic media 6. In some cases,their properties differ from their Kerr counterparts 7.

It took almost a decade until self-trapping of light in dis-crete nonlinear waveguide array was experimentally ob-served 8,9. When a low intensity beam is injected into oneor a few waveguides, the propagating field spreads over theadjacent waveguides, hence experiencing discrete diffrac-tion. However, at sufficiently high power, the beam self-trapsto form a localized state a soliton in the center waveguides.

Subsequently, many interesting properties of nonlinear lat-tices and discrete solitons were reported. For example, theexperimental observation of linear and nonlinear Bloch os-cillations in AlGaAs waveguides 10, polymer waveguides11, and in an array of curved optical waveguides 12. Dis-crete systems have unique properties that are absent in con-tinuous media such as the possibility of producing anoma-lous diffraction 13. Hence, self-focusing and defocusingprocesses can be achieved in the same medium structureand wavelength. This also leads to the possibility of observ-ing discrete dark solitons in self-focusing Kerr media 14.

The recent experimental observations of discrete solitons

8 and diffraction management 13 have inspired furtherinterest in discrete solitons in nonlinear lattices. This in-

cludes the newly proposed model of discrete diffraction man-aged nonlinear Schrodinger equation 15 whose width andpeak amplitude vary periodically; optical spatial solitons innonlinear photonic crystals 16 and the possibility of creat-ing discrete solitons in Bose-Einstein condensation 17.Also, very recently it was shown that discrete solitons intwo-dimensional networks of nonlinear waveguides can beused to realize intellegent functional operations such asblocking, routing, logic functions, and time gating 18.

Here, we propose a vector discrete diffraction managedsoliton system. The vector model describes propagation oftwo polarization modes interacting in a waveguide array withKerr nonlinearity in the presence of varying diffraction. Thecoupling of the two fields is via the cross-phase modulation

coefficient. In the limit of strong diffraction we derive aver-aged equations governing the slow dynamics of the beamsamplitudes. Stationary in the form of bright-bright vectorbound state and traveling wave solutions are obtained.Through an extensive series of direct numerical simulations,we study interactions between periodic diffraction-managedsolitons for different values of velocities, diffraction, andcross-phase modulation coefficient. In both the scalar andvector cases, we find that the interaction picture involvesbeam shaping, fusion, fission, nearly elastic collision, and, insome cases, multihump structures. The collision results, forthe diffraction managed case, are often very different fromtheir classical constant diffraction counterpart.

The paper is organized as follows. In Sec. II we set up a

physical model that describes the propagation of two inter-acting optical fields in nonlinear waveguide array with vary-ing diffraction. In Sec. III we discuss first the scalar limit ofdiffraction managed solitons. Section IV includes a deriva-tion of the averaged equations that describe the slow dynam-ics of the beams amplitude, followed in Sec. V by a discus-sion of the stationary solutions of a vector bright-brightbound state. Analysis of traveling wave solutions is dis-cussed in Sec. VI. Different scenarios involving collisionsbetween scalar and vector periodic diffraction-managed dis-crete solitons are presented in Sec. VII. Finally, we concludein Sec. VIII.

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II. MODEL DESCRIPTION

We begin our analysis by considering an infinite array ofweakly coupled optical waveguides with equal separation d.The equation that governs the evolution of two interacting

electric fields En(1 ) and En

(2 ) , according to nonlinear coupledmode theory 2,15,20,21, is given by

dEn(j)

dz iCEn1

(j)En1

(j) ikw(j)En

(j)iEnjEn

(j) ,

j1,2, 1

where is a 22 matrix with j j and jl , jl are theself- and cross-phase modulation coefficients, respectively,which result from the nonlinear index change; En(En

(1 )2, En(2 )2)T, Cis a coupling constant, z is the propa-

gation distance, and kw(1,2) are the propagation constants of

the waveguides. When a cw mode of the form

En(1,2)z A 1,2expi kzznkxd , 2

is inserted into the linearized version of Eqs. 1, it yields

kzkw(1,2)2Ccos kxd,

kz2Cd2 cos kxd, 3

where in close analogy to the definition of dispersion, dis-

crete diffraction is given by kz. An important consequence

of Eqs. 3 is that kz can have a negative sign if /2 kxd, hence, a light beam can experience anomalousdiffraction. Experimentally, the sign and local value of the

diffraction can be controlled and manipulated by launchinglight at a particular angle with respect to the normal to thewaveguides or equivalently by tilting the waveguide array.To build a nonlinear model of diffraction management, weuse a cascade of different segments of the waveguide, eachpiece being tilted by an angle zero and , respectively. Theactual physical angle the waveguide tilt angle is relatedto the wave number kx by the relation 13 sinkx/kwherek2n0/0 (01.53 m is the central wavelength invacuum and we take n03.3 to be the linear refractive in-dex. In this way, we generate a waveguide array with alter-nating diffraction. Next, we define the dimensionless ampli-

tudes Un(j) (Un

(1 )Un,Un(2 )Vn) by

En(j)P

*Un

(j)e i(kw(j )2C)z, zz/z

*, 4

where P*max(Unmax

2 ,Vnmax2 ) is the characteristic power

andz*

is the nonlinear length scale. Substituting these quan-tities into Eqs. 1 yields the following dropping the primediffraction-managed vector discrete nonlinear Schrodingerequations:

idUn

dz

Dz/z w

2h2 n

2Un Un2Vn

2Un0,

idVn

dz

Dz/zw

2h 2 n

2VnUn2 Vn

2Vn0, 5

with

n2A nA n1A n12A n, 6

where 12/11 we take 1122 , 1221 and z*1/(11P

*). We assume that z

*Ccos(kxd)D(z/zw)/(2h

2)whereD (z/zw) is a piecewise constant periodic function thatmeasures the local value of diffraction. Here z w2L/z*

withLbeing the physical length of each waveguide segment seeFig. 1a for a schematic representation. Equations 5 de-scribe the dynamical evolution of coupled beams in a Kerrmedium with varying diffraction. When the effective non-linearity balances the average diffraction then brightvector discrete solitons can form. The dependence ofthe coupling constant C on the waveguide width (l ) andseparation d is given by for AlGaAs waveguide C(0.009 84/l )exp(0.22d) see Eq. 13.8-10 on pp. 523 ofRef. 19. Therefore, the coupling constant C that corre-sponds to the experimental data reported in Ref. 14 for

2.5 m waveguide separation and width is found to be C2.27 mm1. For typical power P

*300 W; typical non-

linear Kerr coefficient113.6 m1 W1 and typical wave-

guide length L100 m we find z*

1 mm and z w0.2,which suggests the use of asymptotic theory based on smallzw. Model 5 admits stationary soliton solution even whenzw is of order one. To this end, we consider here the case inwhich the diffraction takes the form

Dz/zw a1

zwz/z w, 7

wherea is the average diffraction taken to be positive and

(z/zw) is a periodic function see Fig. 1b.

III. SCALAR CASE

In this section we review the scalar case, i.e., for whichthe cross-phase modulation coefficient vanishes (0). Inthis case, Eq. 5 yields the following diffraction-managedDNLS DM-DNLS equation ( UnVnn) 15:

dn

dz i

Dz/zw

2h 2 n

2ni n2n. 8

FIG. 1. Schematic presentation of the waveguide array a and

of the diffraction map b.

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Since we are considering the case in which the diffractionchanges rapidly (zw1) and that Eq. 8 contains bothslowly and rapidly varying terms, we introduce new fast andslow scales z/zw and Zz , respectively, and expand nin powers ofzw,

nn(0 ),Zz wn

(1 ),ZOz w2 . 9

Substituting Eqs. 9 and 7 into Eq. 8 we find that theleading order in 1/z w and the order 1 equations are, respec-tively, given by

Ln(0 )0, L n

(1 )Fn, 10

where

LA niA n

2h 2 n

2A n, 11

and

Fni n

(0 )

Z a

2h2n

2n(0 )n

(0 )2n(0 ) .

To solve at order 1/zw, we introduce the discrete Fouriertransform

w q ,,Z n

w n,Zeiqnh, 12

with the inverse transform given by

w n,Zh

2/h

/h

w q ,,Ze iqnhdq. 13

The solution is therefore given in the Fourier representation

by ( 0 being the Fourier transform ofn(0 )

0 q ,,ZZ,q expi q C , 14

with (q)1cos(qh)/h2 and C()0()d. The

amplitude(Z,q) is an arbitrary function whose dynamicalevolution will be determined by a secularity condition asso-ciated with Eq. 10. In other words, the condition of theorthogonality ofFn to all eigenfunctions n of the adjointhomogeneous linear problem, which, when written in theFourier domain, takes the form

0

1

dF q,,Z q ,,Z0, 15

where F, are the Fourier transform ofFn, n, respec-

tively. Here,J 0, with J being the adjoint operator to

Li d/d()(q). Substituting Eq. 14 into F using

expi(q)C() and performing the integration in condi-tion 15 yields the following nonlinear evolution equation

for (Z,q):

idZ,q

dZ a q Z,q RZ,q ,

R dqK q ,q 1 ,q 2 q 1 q 2* q 1q 2q ,16

wheredqdq1

dq2

and the kernel K is defined by

K q,q 1 ,q2h 2

42

0

1

dexp iC q,q 1 ,q2 ,

4

h 2cos h q 1q 2

2

j1

2

sin h qjq 2

. 17Equation 16 governs the evolution in Fourier space of asingle optical beam in a coupled nonlinear waveguide arrayin the regime of strong diffraction. In the special case oftwo-step diffraction map shown in Fig. 1b, i.e., when twowaveguide segments are tilted by angle zero and alterna-

tively, we have ()1 for 0 /2 and 2 in theregion/21/2, whereis the fraction of the map withdiffraction 1. In this case, the kernel K takes the simpleform

Kh 2

42

sin s

s , 18

with s1(1)2/4. Importantly, these parameterscan be related to the experiments reported in Ref. 13. Toachieve a waveguide configuration with alternate diffraction,we use two values ofkxd0 and 2/3 (h1), which cor-respond to waveguide tilt angles0 and 3.43. In this case

we find, for 0.5, 120.681, a1.135 and s0.17. Different sets of parameters with a smaller angle are also realizable. Next, we look for a stationary solution forEq. 16 for the particular kernel given above in the form

(Z,q)s(q)exp(isZ). Inserting this ansatz into Eq. 16leads to

s q 1

a q sRs q Qs q , 19

which implies that the mode s(q) is a fixed point of thenonlinear functional Q. To numerically find the fixed point,we employ a modified Neumann iteration scheme 2224

and write Eq. 19 in the form

s(m1) q sLs(m)

sRs(m)

3/2

Qs(m) q , m0,

sLs(m) q 2dq , sR s(m) q Qs(m) q dq .The factorssL and sRare introduced to stabilize an otherwisedivergent Neumann iteration scheme. This method is used tofind stationary soliton solutions to the integral equation 16,

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which in turn provides an asymptotic description of thediffraction-managed DNLS Eq. 8. It should also be notedthat we can obtain periodic diffraction-compensated solitonsolutions directly from Eq. 8. The technique is an alterna-tive for finding periodic dispersion-managed solitons in com-munications problems 2729. The averaging proceduredoes not require that the map period, z w, be small. To imple-

ment the method, initially we start with a guess, say n(0 )

sech(nh ) with E0 n sech2(nh). Over one period

this initial ansatz will evolve ton(0 ) , which in general will

have a chirp 30. We then define an average:

n(0 )

1

2n

(0 )n

(0 )expin,

where n(0 ) n

(0 )exp(in), which has power E0. Then

n(1 )n

(0 )E0/E0 is the new guess and in general the mthiteration takes the form

n(m1)

E0/Emn(m) , Em

n

n(m)2. 20

In Fig. 2 the mode profiles associated with a stationary solu-tion are depicted for two typical parameter values. The pro-files are obtained by using both the integral equation ap-proach as well as the averaged method. The evolutions ofthese discrete diffraction managed solitons are illustrated inFigs. 3 and 4 for the same two set of parameter values as inFig. 2, respectively. We note that initially the beam has zerochirp 30. During propagation a chirp develops and the peak

amplitude of the beam begins to decrease and, as a result, thebeam becomes wider due to conservation of power. A fullrecovery of the solitons initial amplitude and width isachieved at the end of the map period. This breathing behav-ior is shown in Figs. 3 and 4 for both strongly and moder-ately confined beams, respectively.

To establish the relation between the two approaches andto highlight the periodic nature of these new solitons, wecalculate the nonlinear chirp by both the integral equation

approach and the averaging method. It is clear that for smallvalues of the map period, z w , the asymptotic analysis is ingood agreement with the averaging method, as shown in Fig.5. We also mention briefly that the method of analysis asso-ciated with Eq. 8 can be modified to account for situationswhere the average diffraction is small, i.e., a1 and thelocal diffraction is O(1). In such a situation we write aD a, DD a(z), z/, and nn. Then it isfound that n satisfies:

dn

d i

D/

2h2 n1 n12n in

2n,

21

where D(/)D a(1/)(/). The model 21 is validin parameter regimes, which applies to a physical situationwhere the average diffraction is small and local diffraction isof order one.

IV. VECTOR CASE: AVERAGED EQUATIONS

In this section we will discuss the full vector system 5 inthe limit of strong diffraction. Following Sec. III we againexpand the fields Un, Vn in powers ofzw,

UnUn(0 ),Zz wUn

(1 ),ZOz w2 ,

VnVn(0 ),Zz wVn(1 ),ZOz w2 . 22

Substituting Eqs. 22 and 7 into Eqs. 5 we find that atleading order O(1/z w) ,

L Un(0 )0, L Vn

(0 )0, 23

while the order 1 equations are

L Un(1 )Gn

(1 ) , L Vn(1 )Gn

(2 ) , 24

where the linear operator L is given in Eq. 11 and

FIG. 2. Mode profile in physical space obtained from Eq. 19

solid line and from the average method dashed line. Parameters

are: a s1, h1, s0.17, 1 20.681, a1.135, and

zw0.2; b s1, h0.5, s1, 124, a1, and zw0.2.

FIG. 3. Beam propagation over one period using Eq. 37 as

initial condition obtained by a direct numerical simulation of Eq.

8. Parameters used are the same as in Fig. 2a.

FIG. 4. Beam propagation over one period a and stationary

evolution b obtained by a direct numerical simulation of Eq. 8evaluated at end map period. Parameters used are the same as in

Fig. 2b.


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Gn(1 )i

Un(0 )

Z

a

2h 2n

2Un(0 ) Un

(0 )2Vn(0 )2Un

(0 ) ,

Gn(2 )i

Vn(0 )

Z

a

2h 2n

2Vn(0 )Un

(0 )2Vn(0 )2Vn

(0 ) .

To solve at order 1/zw, we make use of the discrete Fouriertransform 12. The solution to Eqs. 23 is thus given in theFourier representation by

U 0Z,q ei(q)C(), V0Z,q e

i(q)C(),25

where U 0 , V0 are the Fourier transforms of Un(0 ) and Vn

(0 ) .

The amplitudes and are arbitrary functions whose dy-namical evolution will be determined by a secularity condi-tion associated with Eqs. 24. In other words, the condition

of the orthogonality ofGn(1,2) to all eigenfunctions n of the

homogeneous adjoint linear problem that, when written inthe Fourier domain, takes the form

0

1

dG(1,2) q,,Z q,,Z0, 26

where G(1,2),are the Fourier transforms ofGn(1,2) and n,

respectively; satisfies L0 with L being the adjoint

operator to Li(/)()(q). Substituting Eq. 25

into G(1,2) and performing the integration in condition 26

withexpi(q)C() yields the following coupled nonlin-

ear evolution equations for the amplitudes and :

idZ,q

dZ a q Z,q A, ,

idZ,q

dZ a q Z,q B, , 27

with

A dqK q ,q 1 ,q 2 q1 q 2* q1q 2q q 1 q 2* q 1q 2q ,

B dqK q ,q 1 ,q 2 q1 q2* q1q 2q q 1 q 2* q 1q2q ,

where as before dqdq 1dq 2 and the kernel K is given byEq. 17. Equations 27 govern the averaged evolution inFourier space of coupled optical beams in a nonlinear wave-guide array in the regime of strong diffraction. The aboveresults are general and hold for any diffraction map. How-ever, the analysis simplifies significantly in the special caseof two-step diffraction map shown in Fig. 1b, in which casethe kernel is given by Eq. 18.

V. STATIONARY SOLUTIONS

We now look for a stationary solution for Eqs. 27 forthe particular kernel given in Eq. 18 in the form

Z,q seiZ, Z,q se

iZ. 28

Inserting this ansatz into Eq. 27 leads to

s q A/a q M,

s q B/a q N. 29

To numerically find the modes s(q) ands(q), we employthe modified Neumann iteration scheme mentioned in Sec.III and write Eqs. 29 in the form

s(m1)

sL(1 )

sR(1 )

Mm, s(m1)

sL(2 )

sR(2 )

Nm, 30

sL(1 )s(m)2dq, sL(2 )s(m)2dq,

sR(1 ) s(m)Mmdq, sR(2 ) s(m)Nmdq .

The factor 3/2 is chosen to make the right-hand side ofEqs. 30 of degree zero, which yields convergence of the

scheme 22,23. Clearly, when s(m)

(q)s(q), s(m)

(q)s(q) as m then sL

(1 )/sR(1 )1, sL

(2 )/sR(2 )1 and in

turn s(q), s(q) will be the solutions to Eqs. 29. The

factors sL(j) and sR

(j) , j1,2 are introduced to stabilize anotherwise divergent simple Neumann iteration scheme. Notethat when applying the continuous Fourier transform to findstationary solutions, the numerical scheme based on Eq. 30does not converge which indicates that, contrary to the vectorintegrable DNLS 25,26, continuous stationary solutionsmay not exist. Figures 6,7 show typical solutions to Eq. 29both in the Fourier domain Fig. 6 and in physical space

FIG. 5. Periodic evolution of the beam chirp versus maximum

peak power. Solid line represents the leading order approximation,

i.e., Eq. 14; dashed line represents numerical solution of the DM-

DNLS, and dotted line depicts the evolution of the leading order

solution under Eq. 8. Parameters used are the same as in Fig. 2b

with z w 0.2 a and 0.1 b.


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Fig. 7 for lattice spacing h0.5. The evolution of thesediscrete diffraction managed solitons is illustrated in Fig. 8for the same set of parameter values as in Fig. 6. We notethat initially the vector solitons have zero chirp. Duringpropagation a chirp develops and the peak amplitude of eachbeam begins to decrease and, as a result, the two beams

become wider due to conservation of power. A full recov-ery of the vector solitons initial amplitudes and widths isachieved at the end of the map period. This breathing behav-ior is shown in Fig. 8 for different values of cross-phasemodulation (). Finally, in Fig. 9 we show the stationaryevolution of these beams evaluated at each map period.

VI. TRAVELING SOLITONS

In this section we will discuss traveling wave solution toEqs.5 in the limit of strong diffraction management. Withsoliton collisions in mind we consider here the case wherethe two interacting beams are initially well separated, in

which case, the overlap termsVn2

Un and Un2

Vn are verysmall compared to the self-phase modulation term. There-fore, in what follows we find travelling wave solutions in thecase0, i.e., in the scalar diffraction managed model. Ourapproach can be generalized to the 0 case for which thetwo polarizations travel possibly at different speed. We lookfor a traveling wave solution in the form

Unz u,z ein, nhV

0

z

D , 31

with nnh0zD . Note that we allow the amplitude u

to depend explicitly on z. The reason is that when the dif-fraction is constant (DD0const) and the amplitude u isindependent ofz, then the problem reduces to finding travel-ing wave solution to the DNLS equation 3136. However,recent studies 24 of traveling solitons of the DNLS equa-tion reveal the important conclusion that a uniformly movingcontinuous wave solution is unlikely to exist. These resultsdiffer from those of Ref. 31 in which continuous travel-ing solitary waves were reported using Fourier series expan-sions with finite period L while assuming convergence asL . Even then, for moderately localized pulses, approxi-mate traveling waves are found to persist for long distanceswithout radiation. However, for strongly localized beams, the

wave front gets distorted during propagation and it deceler-ates. Nevertheless, the pulse travels almost uniformly overthe experimental distance 6 mm.

Mathematically, we analzse the problem as follows. Sub-stituting Eq. 31 into Eq. 5 leads to

FIG. 6. Mode profiles,ssolid line andsdashed line in the

Fourier space obtained from Eqs. 30. Parameters are s1, h

0.5, s1, 124, a1, andz w0.2. a 2/3 and b

1.

FIG. 7. Mode profiles, Un solid line and Vn dashed line in

physical space. Parameters are the same as in Fig. 6 with 2/3

a and 1 b. Circles denote grid points.

FIG. 8. Beam propagation over one period using Eq. 25 as

initial conditions obtained by a direct numerical simulation of Eqs.

5. Parameters used are the same as in Fig. 6 with 2/3 a, b

and 1 c, d.

FIG. 9. Stationary evolution obtained by a direct numerical

simulation of Eqs. 5 evaluated at end map period. Parameters used

are the same as in Fig. 6 with 2/3 a, b and 1 c, d.


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iu

zDz/z wi Vu Tu2h2 u u2u , 32

where D(z/zw) is the diffraction function given in Eq. 7and

Tueihuh e ihuh 2u.

Since in this case, Eq. 32 contains both slowly and rapidlyvarying terms, we proceed as before and define new fast andslow scales z/z w andZz, respectively, and expand u inpowers ofz w,

u,z u 0,,Zzwu 1,,ZOz w2 . 33

Substituting Eq. 33 into 32 we find that the leading order O(1/z w) and the order 1 equations are, respectively, givenby

J u 00, J u1U, 34

where

JA iA

iV

A

A

2h2TA ,

Uiu0

Zai Vu 0 Tu02h 2 u 0 u02u 0 .

To solve at order 1/z w, we again introduce the discreteFou-rier transformtreating as a discrete variable

u0 q ,,Z mhZ

u 0,,Zeiq,

u0,,Zh

2/h

/h

u0 q ,,Zeiq dq . 35

At this stage it is useful to make some further comments onthe Fourier transform. Since is a continuous variable itimplies that Eq. 32 is a continuous equation in both and z.Therefore it seems natural to solve Eqs. 34 using the con-tinuousFourier transform given by

u0 q,,Z

u 0,,Zeiq d,

u 0,,Z1

2

u0 q,,Zeiqdq . 36

But when applying the continuous Fourier transform, ournumerical scheme fails to converge 24. However, when weuse the discrete Fourier transform our method yields an ap-proximate but notexact solution. The solution at O (1/zw) isgiven in the Fourier representation by

u0 q,,ZUZ,q expiS q ,C , 37

with S(q ,)qV1cos(q)h/h2. As in Sec. III

the amplitude U(Z,q) is an arbitrary function whose dy-namical evolution is determined by a secularity conditionassociated with the order 1 equation. In other words, thecondition of the orthogonality ofU to all eigenfunctions of the adjoint linear problem, which, when written in theFourier domain, takes the form

01

dU q,,Z q ,,Z0, 38

whereU, are the Fourier transform ofU, respectively.

Here, J 0 with J being the adjoint operator to J

id/d()S(q,). Substituting Eq. 37 into U using

expiS(q,)C() and performing the integration in con-dition 38 yields the following nonlinear evolution equation

for U(Z,q):

idUZ,q

dZ aS q ,UZ,q PUZ,q 0,

P dqK q ,q 1 ,q 2U q 1U q2U* q1q 2q ,39

wheredqdq 1dq 2 and the kernel Kis defined by

K q,q1 ,q2h 2

42

0

1

dexp iC q ,q 1 ,q 2,

4

h 2cos h q 1q 22

2

j1

2

sin h qjq 2

.Restricting the discussion to the two-step diffraction mapsee Fig. 1 and Sec. III, we find Kh

2sin(s)/(42s).

To find a moving solitary wave solution to system 39, wewrite

UZ,q U TW q eiZ U e q U o q e

iZ, 40

with U e(q) and U e(q) being the even and odd parts of

U TW(q). This choice of ansatz is consistent with the problemof finding traveling solitary waves for the DNLS equationconstant diffraction where it was shown 24 that the modeprofile in physical space can be written as UF()iG()exp(nhz) with F,G being even and odd func-

tions, respectively. Substituting Eq. 40 into Eq. 39 wearrive at

U TW(m1) q sL

sR

3/2 P U TW(m) q

aS q,, 41

where sL and sR are the convergence factors introduced inSec. III. For a given set of parameters h,, and 0, themode shapes and soliton velocity are found by iterating Eq.

41 with an initial guess, e.g., U e(0 )(q)sech(q), U o

(0 )(q)sech(q)tanh(q), and VV

*0. The iterations are carried


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out until the condition sLsR is satisfied with 0being a prescribed tolerance. However, unlike the stationarycase, here, the soliton velocity is still to be determined. Forany choice of V

*0 if sLsR, we seek a different

value ofV*

at whichsLsR changes sign. Then, we use thebisection method to changeV

*in order to locate the correct

velocityVand modes U e, U o for each , , and h. A typi-

cal soliton mode is shown in Fig. 10 for velocity V

0.5and lattice spacing h0.5.

VII. COLLISIONS BETWEEN DISCRETE SOLITONS

Having found a moving discrete diffraction managed soli-tary wave we next study interactions between them Fig. 11.We have conducted a series of numerical simulations both inthe scalar for different soliton velocities and vector fordifferent values of the cross-phase modulation coefficientand different velocities. Below we discuss each case sepa-rately. To keep the paper self-contained, we consider firstinteractions between scalar and vector discrete solitarywaves in the case of constant diffraction, i.e., when 1 and

2 are both zero ( s0), which implies constant kernel Kh2/42.

A. Scalar interaction: Constant diffraction

Scalar discrete excitations propagating in a media withnormal diffraction are solutions of the DNLS equation seeEq. 8 with constant D(z/z w)1:

dWn

dz

i

2h2n

2WniWn2Wn. 42

To simulate the collision process between discrete solitonswe first need to obtain a traveling wave solution to the aboveequation. To do so, we look for traveling localized modes inthe form

Wnz wexpinhiz , 43

with nhVz whereVand are the soliton velocity and

wave number shift, respectively. Assuming w is complex,i.e.,w ()wR()iwI() then Eq. 42 takes the form

VwI1wR2wI wR2wI

2wRwR,

VwR1wI2wR wR2wI

2wIwI, 44

where prime denotes derivative with respect to and

1X1

h 2cosh EEX2X ,

2X

sinh

h 2 E

EX,

45

with EX()X(h). Note that system 44 is invariantunder the transformation: , wIwI, and VV, which will play a central role in collision processes. Tofind the mode shapes and soliton velocity, we proceed asbefore by taking the discrete Fourier transform of Eqs. 44,which yields the following iteration scheme:

wR(m1) q

2 q

1 q wI

(m) q sL(1 )/sR

(1 )3/2I1 ,

w

I

(m1)

q

2 q

1 q w

R

(m)

q

sL(2 )

/sR(2 )

3/2

I2 , 46

where wR(q) and wI(q)iwI(q) are the Fourier trans-forms ofwR() and wI(), respectively, and

I1h2

421 q wR*w

R*w

RwI*w

I*w

R ,

I2h 2

421 q wR*w

R*w

IwI*w

I*w

I. 47

FIG. 10. Traveling wave solution to Eq. 39 in real space. a

shows the even part and b the odd part. Parameters are s1,

1 24, a1, zw0.1, 1, h0.5, 0.5, and V

0.5. Circles denote grid points.

FIG. 11. Evolution of a moderately localized soliton in physical

domain obtained by a direct numerical simulation of Eq. 8 evalu-

ated at end map period. Parameters are s1, 124, a1, zw0.1, 1, h0.5, 0.5, and V0.5. Shown on the

vertical axis is the beam intensity Un2.


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The convergence factors sL(j) and sR

(j) , j1,2 are

sL(1 ) wR(m) q wR(m) q 2 q 1 q wI(m) q dq ,

sR(1 )

wR

(m) q I1dq, sR(2 )

wI

(m) q I2dq, 48

and sL(2 ) is obtained from sL

(1 ) by interchanging wR and wI.

Here, 1(q)(2/h2)1cos(hq)cos(h) and 2(q)

(2/h 2)sin(hq)sin(h)Vq. For a given set of parametersh , , and 0, the mode shapes and soliton velocity arefound by iterating system 46 together with the bisectiontechnique outlined in Sec. VI. Typical traveling solitonmodes are shown in Fig. 12. Having obtained moving soli-tary wave solutions, we investigated next the collision pro-cess. If the soliton speed is less than a critical velocity, VVc0.2, then the solitons are found to fuse after collisionaccompanied with radiation modes, see Figs. 13a, 13b.However, when the two modes move sufficiently fast (VVc) then they pass through each other. This collision re-sults in two nonstationary travelling waves moving in theopposite directions with modulated envelopes Fig. 13c.

B. Vector interaction: Constant diffraction

The vector discrete system with constant diffraction isdescribed by the following coupled discrete NLS equations

idUn

dz

1

2h2n

2Un Un2Vn

2Un0,

idVn

dz

1

2h 2n

2VnUn2 Vn

2Vn0, 49

where Un, Vn are the two interacting beams and is thecross-phase modulation coefficient. To study interaction be-tween vector solitons, we consider the case in which the twomodes are initially well separated and orthogonally polar-ized. In this situation, the two modes are completely decou-pled and each one of them satisfies a scalar DNLS equationfor which a traveling solitary wave was obtained in the pre-ceding section. In Fig. 14 we show a typical collision experi-ment for different values of cross-phase modulation coeffi-cient and different soliton velocities. In all cases the Unmode is always stationary while the other one moves withspeed V. When 2/3 and the second mode ( Vn) movesslowly (V0.1), we observe power exchange see Fig.14a and Fig. 15a, i.e., the outgoing beams have ampli-tudes different from the one before the collision. TheVnfieldmostly survived the collision whereas Un undergoes a fissionprocess and two solitons are born. Each one of them hasamplitude almost half the original one. The interaction pic-ture is more clean at higher speed ( V0.4) in which thetwo modes pass each other nearly unaffected except for aphase shift in the soliton position, Fig. 14b and Fig. 15b.Decreasing the value ofto 0.5 leads to a different picture.In this case whenV0.1 the the two beams fuse see Fig.14c and Fig. 15c and form a single fused traveling wave

FIG. 12. Mode shapes in physical space for 1 and 0.5.

Solid line corresponds to h0.5 and velocity V0.5, whereas

dashed line for h1 and V0.3. FIG. 13. Interaction between scalar classical discrete solitons

for different values of velocity obtained by a direct numerical simu-

lation of Eq. 42. Parameters are h0.5, and V0.1,

0.1 a; V0.25, 0.25 b; and V0.5, 0.5 c.

Shown on the vertical axis is the intensity Wn2


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whereas the two modes simply go through each other athigher velocity (V0.5) as shown in Fig. 14d and Fig.

15d. Notice that the collision process reported in Fig. 15dis nearly elastic. Unlike the symmetric discrete vector NLSequation, where the nonlinear terms are taken to be (Un

2

Vn2)(Un1Un1) and (Un

2 Vn

2)(Vn1Vn1) 25, here vector nonintegrable case, we did notfind a bouncing process. This could be attributed to the spe-cial properties of the symmetric vector NLS system.

C. Scalar interaction: Diffraction management

In this section we study interactions between scalar dis-crete diffraction managed solitons. In what follows, we high-

FIG. 14. Interaction between discrete classical vector solitons

for different values of cross-phase modulation coefficient and

different soliton velocity. Shown on the vertical axis is the intensity

of the two colliding beams, i.e., InUn2Vn

2. Before collision,

Un is the left beam, whereas Vn is the right beam. In all the parts,

the incoming Un mode is stationary, whereas Vn travels with a

speed V. Parameters used are 2/3, V0.1, 0.1 a;

2/3, V0.4, 0.4 b; 0.5, V0.1, 0.1 c; and

1, V0.5, 0.5 d.

FIG. 15. Snapshot showing the final state of the collision illus-

trated in Fig. 14. ad correspond to Figs. 14a14d, respec-

tively. Dashed solid line corresponds to Un (Vn).

FIG. 16. Interaction between scalar diffraction-managed solitons

for different values of velocity obtained by a direct numerical simu-

lation of Eq. 8 evaluated at end map period. Parameters are

1, h0.5, s1, 124, a1, and zw0.1. The ve-

locity Vand are, respectively, equal to 0.1, 0.1 a; 0.25, 0.25

b; and 0.5, 0.5 c. Shown on the vertical axis is the intensity

Un2.


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light the similarity and differences between classical sca-

lar discrete solitons and diffraction managed solitons. Figure16 illustrates different possible interactions for which the twocolliding beams are identical and move in opposite directionswith identical speed. For small relative velocity the two col-liding beams fuse and form a new state that is characterizedby a localized core and oscillating tail. When propagatingalong thez direction, the core starts to develop double-humppattern and becomes wider. As a result of an instability, thecore beam reshapes itself into a Gaussian-like form and be-comes narrower as more radiation is emitted see Fig. 17.This behavior repeats and represents a periodic diffractionmanaged analog of the fused classical case compare withFig. 13a. For higher velocities, the two beams cross eachothers trajectory. The outgoing traveling waves are charac-

terized by modulated envelopes. The higher the collision ve-locity the larger the modulation amplitude. This behavior issimilar to the one observed with classical soliton com-pare with Figs. 13b, 13c.

D. Vector interaction: Diffraction management

Next, we investigate what happens in the vector case,when the two beams interact via the cross-phase coupling.Figure 18 depicts different soliton collision scenarios forvarious values of but fixed soliton velocities. In otherwords, the mode Un is always stationary while Vn moveswith fixed velocity V0.1 and the only parameter wechange is . It turns out that interactions between discretediffraction-managed solitons differ fundamentally from theirclassical counterparts, as can be seen from Fig. 18 see,for example, Fig. 14. For small value of0.1 the interac-tion results in a facinating process. Immediately after thecollision, the two modes partially pass each other see Figs.19a, 19b and then, as a result of an attractive force, bothbeams turn over and come back to the initial state. Instead ofcontinuing to go away from each other the two polarizationmodes turn back again. This behavior persists for very longdistances and demonstrate a bound state recurrencelike phe-nomena.

Increasing the value of to 2/3 changes the interactionpicture as more radiation is emitted and the resulting colli-sion products exhibit multihump structure and constantlychange their relative position. When the interaction takesplace, the Un dashed line intensity has a double-humpedprofile while Vn solid line is single humped see Figs.20a, 20b. After propagating for some distance, then ex-actly the opposite happens. Now the intensity of the Vn po-larization is double-humped whereas Un has a single hump.When the cross- and self-phase modulation coefficients areidentical then the collision product is a bound state com-posed of single- and double-humped intensity profiles. Uponpropagation, this bound state hops to the right and left, backand forth in a periodic manner see Figs. 21a, 21b. Moni-toring the evolution of the Vn mode shows that its doublehump structure in intensity is preserved but exhibits aninternal dynamics, i.e., the distance between the first andsecond maximum changes with z. In all of the above cases,

FIG. 17. Snapshot showing the evolution stages of the collision

illustrated in Fig. 16a.

FIG. 18. Interaction between diffraction managed vector soli-

tons for different values of cross-phase modulation coefficient ( )

obtained by a direct numerical simulation of Eq. 5 evaluated at

each map period. Shown on the vertical axis is the intensity verti-cal axis of the two colliding beams InUn

2Vn

2. Before col-

lision, Un is the stationary beam on the left and Vn is the mode on

the right that travels with velocity V. Parameters are 1, h

0.5, s1, 1 24, a1, and z w0.1. The velocity V

0.1 and 0.1 with equal to 0.1 a, 2/3 b, 1 c, and 2 d.


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the resulting beam moves with a negative velocity. Interest-ingly, for 2, the collision process results in a fusedtravelinglike wave that moves in the positive direction withan additional small residual beam that propagates in thenegative direction Fig. 18d. This fused state evolved froman intricate process. After collision, the intensities of bothpolarizations develop multihump structures Fig. 22a,which decay as a result of instability and form two lumpstogether with a side wave Fig. 22b. This residual waveescapes away from the center and, as a result, both Un andVn relax into a fused state Figs. 22c, 22d with emis-sion of additional radiation.

VIII. CONCLUSIONS

In this paper we have investigated scalar and vector dis-crete diffraction managed systems. We have compared the

diffraction managed cases with their classical counter-parts. The proposed vector model describes propagation oftwo polarization modes interacting in a waveguide array withKerr nonlinearity in the presence of varying diffraction. Thecoupling of the two fields is described via the cross-phase

modulation coefficient. In the limit of strong diffraction man-agement we derive averaged equations governing the slow

dynamics of the beams amplitudes and stationary in theform of bright-bright vector bound state as well as ap-

proximate traveling wave solutions are obtained. Throughan extensive series of direct numerical simulations, we studyinteractions between periodic diffraction-managed solitonsfor different values of velocities, diffraction, and cross-phasemodulation coefficients. In both the scalar and vector cases,we find that the interaction picture involves beam shaping,fusion, fission, nearly elastic collision, and, in some cases,multihump structures.

ACKNOWLEDGMENTS

M.J.A. was partially supported by the Air Force Office ofScientific Research under Grant No. F49620-00-1-0031 andby the NSF under Grant No. DMS-0070792. Z.H.M. wouldlike to thank the Rothschild Foundation for financial support.


illustrated in Fig. 18a. Dashed solid lines depicts Un2 ( Vn

2).

a is the initial condition and bd shows interactions at different

increasing distances fromz0, respectively.


illustrated in Fig. 18b. Dashed solid lines depict Un2 (Vn

2).

FIG. 21. Snapshot showing the intensity profile of both modes,

Un2 dashed andVn

2 solid right after the collision a and the

last computational stage of the collision b. These snapshots corre-

spond to Fig. 18c.


illustrated in Fig. 18d. Dashed solid lines depict Un2 (Vn

2).


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