Switching of Discrete Solitons in Engineered Waveguide Arrays

0.33 MB·English

by  R. A. Vicencio

Switching of discrete optical solitons in engineered waveguide arrays
Rodrigo A. Vicencio1, Mario I. Molina1, and Yuri S. Kivshar2
1Departamento de F´ısica, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile
2
Nonlinear Physics Group and Centre for Ultra-high bandwidth Devices for Optical Systems (CUDOS),
Research School of Physical Sciences and Engineering,
Australian National University, Canberra ACT 0200, Australia
We demonstrate simple methods for controlling nonlinear switching of discrete solitons in arrays
4 of weakly coupled optical waveguides, for both cubic and quadratic nonlinearities. Based on the
0 effective discrete nonlinear equations describing the waveguide arrays in the tight-binding approx-
0 imation, we develop the concept of the array engineering by means of a step-like variation of the
2 waveguide coupling. We demonstrate the digitized switching of a narrow input beam for up to
eleven neighboring waveguides, in the case of the cubic nonlinearity, and up to ten waveguides,
n
in the case of quadratic nonlinearity. We discuss our predictions in terms of the physics of the
a
engineered Peierls-Nabarro (PN) potential experienced by strongly localized nonlinear modes in a
J
lattice, and calculate thePN potential for thequadraticnonlinear array for thefirst time. Wealso
1
confirmourconceptandmajor findingsforafull-scaled continuousmodelandrealistic parameters,
2
bymeans of thebeam propagation method.

]
s PACSnumbers: 42.82.Et,42.65.Sf,42.65.Tg
c
i
t
p I. INTRODUCTION properties of localized modes introduced by discreteness
o such as the soliton steering in and discreteness-induced
.
s trapping by the lattice (see, e.g., Ref. [15]). As a result,
Discrete nonlinear systems are known to support self-
c
a very little effort has been made so far to link these
i localized modes that exist due to an interplay between
s findingswithrealisticapplicationsofdiscretesolitonsfor
a coupling between the lattice sites and nonlinearity [1].
y
multi-port all-optical switching.
h Such spatially localized modes of discrete nonlinear lat-
p tices existing without defects are known as discrete soli- Indeed, one of the major problems for achieving con-
[ tons or intrinsic localized modes; they appear in many trollable multi-port all-optical switching of discrete soli-

diverseareasofphysics suchasbiophysics,nonlinearop- tons in waveguide arrays is the existence of an effective
3
v tics, and solid state physics [2, 3]. More recently, such periodic Peierls-Nabarro (PN) potential which appears
9 modes have been predicted in the studies of the Bose- due to the lattice discreteness. As a consequence of this
8 Einstein condensatesin opticallattices [4]and photonic- potential, a narrowlarge-amplitude discrete soliton does
0 crystal waveguides and circuits [5]. not propagate freely in the lattice and, instead, it be-
1 One of the most important applications of discrete comes trapped by the array. Several ideas to exploit
0
solitons is found in nonlinear optics where discrete the discreteness properties of the array for digitized all-
4
0 optical solitons were first suggested theoretically by opticalswitching have been suggested[16, 17]. However,
/ Christodoulides and Joseph [6] for an array of weakly the main result of those earlier studies is the observa-
s coupled optical waveguides. Because the use of discrete tionthatthediscretesolitonscanbe wellcontrolledonly
c
i solitons promises an efficient way to realize and control in the limit of broad beams whereas the soliton dynam-
s
y multi-port nonlinear switching in systems of many cou- ics in highly discrete arrays has been shown to be more
h pled waveguides,this field has been exploredextensively complicated and even chaotic [17].
p during last ten years in a number of theoretical papers In this paper, we explore in detail an effective way to
v: (see, e.g., Refs. [7, 8, 9], as an example). More im- control nonlinear switching of discrete solitons in arrays
i portantly, the discrete solitons have also been generated ofweaklycoupledopticalwaveguidesearliersuggestedin
X
experimentally in fabricated periodic waveguide struc- ourbrief letter [18]. First, using the discrete modelvalid
r tures (see, e.g., some original papers reporting on the in the tight-binding approximation, we estimate the PN
a
experimental observations [10, 11] and also the recent potential experienced by a strongly localized nonlinear
review papers [12, 13, 14]). mode that is kicked initially in a cubic nonlinear waveg-
The majorityof theoreticalstudies conducted so far is uide array. The result suggests a possible control mech-
devoted to the analysis of different types of stationary anism for the switching of strongly localized excitations
localized modes in discrete nonlinear models and their by means of a step-like variation of the waveguide cou-
stability. Consequently, experimental papers have re- pling. For particular types of the engineered arrays, we
ported on the observation of self-trapped states in the are able to demonstrate the digitized switching of a nar-
periodicsystemswithbrokentranslationalsymmetryand row input beam for up to eleven waveguides. Second,
some of their properties, in both focusing and defocus- we demonstrate the validity of the predictions made in
ing regimes [12, 13, 14]. However, only very few studies theframeworkofthediscretemodelbyperformingafull-
andexperimentaldemonstrationsaddressedmorespecific scaled continuous simulation using realistic parameters.
2
Last but not least, we extend the concept of control-
labledigitizedswitchingofdiscreteopticalsolitonstothe
case of quadratic nonlinear waveguide arrays, where the
experimental observation of discrete optical solitons has
beenreportedveryrecently[19]. Here,weobtain,forthe
firsttime toourknowledge,the PNpotentialforthe dis-
crete soliton and demonstrate numerically the digitized
switching for up to ten waveguides.
The paper is organized as following. In Sec. II we
study the arrays of cubic nonlinear waveguides. First,
we consider the system dynamics described by the dis-
crete nonlinear Schr¨odinger equation, and show how to
modulate the waveguide coupling in order to suppress FIG. 1: Exampleof a homogeneous waveguide array and the
the chaotic dynamics and achieve fully controllable dig- generationofadiscretesolitonbyexcitingasinglewaveguide.
itized switching. We also employ the beam propagation
methodandsimulatenumericallyamorerealisticcontin-
uousmodelofthewaveguidearrayswithrealisticparam- boring waveguides, and z is the propagation distance
eters,andconfirmthatourconceptcanbeveryusefulfor alongthewaveguide. Theparameterγ =ω0n2/(cAeff)is
optimizationofthesolitonswitchinginrealisticsettings. the effective waveguide nonlinearity associated with the
Next, in Sec. III we extend our analysis to the arrays of Kerr nonlinearity of the core material. Figure 1 shows a
weakly coupled quadratic nonlinear waveguides, where typicalexperimentalstructureofaquasi-one-dimensional
discrete quadratic solitons are composed of the coupled homogeneouswaveguidearrayandtheexcitationscheme
beams of the fundamental and second-harmonic fields. for generating a discrete optical soliton.
Finally, Sec. IV concludes the paper. Steering and trapping of discrete optical solitons have
been analyzed in the framework of the model (1) in a
number of theoretical studies. Being kicked by an ex-
II. CUBIC NONLINEAR WAVEGUIDES ternal force, the discrete soliton propagates through the
lattice for some distance, but then it gets trapped by
The most common theoretical approach to study the the lattice due to the discreteness effects. For a stronger
discreteopticalsolitonsinarraysofweaklycoupledopti- kick, the output soliton position fluctuates between two
calwaveguidesisbasedonthedecompositionoftheelec- (or more) neighboring waveguides making the switching
tric field of the periodic photonic structure into a sum uncontrollable [17].
of weakly coupled fundamental modes excited in each In order to show this feature, first we consider homo-
waveguide of the array; in solid-state physics this ap- geneous arrays and select an input profile in the form of
proach is known as the tight-binding approximation. Ac- a narrow sech-like beam localized on a few waveguides,
cordingtothisapproach,thewavedynamicsisdescribed
u (0)=A sech[A(n n )/√2] e−ik(n−nc), (2)
by an effective discrete nonlinear Schr¨odinger (DNLS) n − c
equation that possesses spatially localized stationary so-
for n n = 0, 1, and u (0) = 0, otherwise. For the
c n
lutions in the form of discrete localized modes. Many − ±
particular results presented below, we consider an array
propertiesofthediscreteopticalsolitonscanbeanalyzed
of 101 waveguidesand place the beam at the middle po-
in the framework of this approach and the DNLS equa-
sition, n = 50. The maximum normalized propagation
c
tion [6, 9].
distance used in our simulations is z = 45. Parame-
max
ter k in the ansatz (2) has the meaning of the transverse
steeringvelocityofthebeam,inanalogywiththecontin-
A. Discrete model
uousapproximation. Itdescribesthevalueofaneffective
kickofthebeaminthetransversaldirectionattheinput,
1. Homogeneous Arrays in order to achieve the beam motion and shift into one
of the neighboring(or other desired) waveguideoutputs.
A standard model of a weakly coupled array of cubic In our simulations, we control the numerical accuracy
nonlinear waveguides is described by the DNLS equa- bymonitoringthetwoconservedquantitiesofmodel(1),
tion [6] that we write in the normalized form [20], the soliton power
idduzn +V(un+1+un−1)+γ|un|2un =0, (1) P =Xn |un(z)|2, (3)
where u is the effective envelope of the electric field in
n and the system Hamiltonian,
the n-th waveguide, the normalized parameter V is pro-
portional to the propagation constant of a single waveg- H = V(u u∗ +u∗u )+(γ/2)u 4 . (4)
uideanditcharacterizesthecouplingbetweentheneigh- − n n+1 n n+1 | n|
Xn (cid:8) (cid:9)
3
A=1.414 A=1.416 A=1.418 “sees”a potential barrier (the PN barrier),whose height
depends on the effective discreteness of the system as
seen by the excitation [21]. Thus, for wide modes, the
barrier will be smaller that for narrow modes. A rough
estimate of this PN barrier can be obtained by equating
it to the difference in the values of the Hamiltonian, be-
tweenthemodecenteredatawaveguide(oddmode)and
the mode centered between two neighboring waveguides
(even mode) [21].
Inorderto evaluateachangeofthe PNbarrierforthe
50 57 60 50 5860 50 57 60 mode initially kicked by an external force, we introduce
an initial phase tilt that is proportional to the factor
FIG. 2: An example of erratic switching of a localized in- exp( ikn)inthediscretecase. Ourpurposeisnotonly
∼ −
put beam with a slight variation of the beam intensity in a provide an extension to the earlier results [21], but also
homogeneous array. study,forthefirsttimetoourknowledge,thevariationof
theeffectivePNpotentialforaninitiallykickedlocalized
mode.
The input condition (2) does not correspondto an ex- Odd modes. We consider a strongly localized mode
act stationary solution of the discrete equation (1) even (SLM) in the form of three excited cites,
for k = 0 and, as the input kick (k = 0) forces the lo-
calized wave move to the right (k <60) or left (k > 0), un(z)=un eiλ1z
its motion is accompanied by some radiation. The effec- u 0..,0,ǫ eik,1,ǫ e−ik,0,...0 eiλ1z, (5)
0 1 1
tive lattice discreteness can be attributed to an effective ≈ { }
periodic potential, the PN potential, which is dynamic whereu isthemodeamplitude,kistheparameterofthe
0
and changes in time. Due to both the strong radiation initial“kick”(aneffectivetransverseangle)appliedtothe
andthepresenceofthePNbarrierwhichshouldbeover- mode, λ is the longitudinal propagation constant, and
1
taken in order to move the beam transversally, the dis- ǫ is a small parameter, to be determined from Eq. (1).
1
cretesolitongetstrappedatoneofthewaveguidesinthe After substituting Eq. (5) into Eq. (1) and keeping only
array. In most of the cases, the shift of the beam posi- linear terms in ǫ , we obtain
1
tion to the neighboring waveguide is easy to achieve, as
showninmanystudies[17]. However,thesolitonswitch- λ1 =2ǫ1V cos(k)+γu20
ing becomes rather complicated and even chaotic. This
and ǫ =V cos(k)/λ , so that
is shown in Fig. 2 where, for a fixed value of the input 1 1
angle, a slight variation in the beam intensity results in
λ γu2, ǫ V cos(k)/γu2 1. (6)
a erratic switching of the beam. 1 ≈ 0 1 ≈ 0 ≪
Even modes. Inthis case,the SLMmode hasthe form
2. Modulated Arrays u˜n(z)=u˜n eiλ2z
u˜ 0..,0,ǫ eik,1,e−ik,ǫ e−2ik,0,...0 eiλ2z (7)
0 2 2
≈ { }
In this paper, we suggest to modulate the coupling in
where, as above, u˜ is the amplitude of the even mode,
the waveguide array in order to achieve a controllable 0
k is the initial angle or effective parameter of the initial
output and to engineer the switching results. What this
“kick”,λ isthelongitudinalpropagationconstantofthe
modulationofthe couplingsdoesistoaffectthePNbar- 2
evenmode, andǫ is asmallparameter. After substitut-
rier, providing us with a simple physical mechanism for 2
ing Eq. (7) into Eq. (1) and keeping only linear terms in
fine tuning and control of the beam self-trapping.
ǫ , we obtain
To justify the validity of our concept, we perform 2
a qualitative estimate of the PN barrier in the frame- λ =(1+ǫ ) V cos(k)+γu˜2
work of the applicability of the discrete model and per- 2 2 0
turbation theory. We study the case of strongly local- and ǫ =V cos(k)/λ , so that
2 2
izedmodes[21]propagatinginahomogeneouswaveguide
array with identical coupling between the neighboring V cos(k)
λ V cos(k)+γu˜2,ǫ 1. (8)
waveguides, described by Eq. (1). We consider a gen- 2 ≈ 0 2 ≈ V cos(k)+γu˜2 ≪
0
eral localized mode that we want to propagate through-
out the array. Due to discreteness, our system lacks the ¿FromEqs.(6)and(8)wecometotheconclusionthat,
translational invariance and, as a result, some of its en- in order to have strongly localized modes, the nonlinear
ergymustbesuppliedinordertoforcethemodemoving. contribution described by the term γu2( or γu˜2) must
0 0
Another way to look at this problem is to consider that, be much larger than the linear term described by the
because of the lattice discreteness, the localized mode term V cos(k). Now, for calculating the PN barrier, we
4
shouldrelatetheamplitudesofthemodesoftwodifferent
Vn
symmetries. One way is to think of both the modes as 10 er
differentstatesofasingleeffectivemodeshiftedbyahalf et 1.0106
m 1.0086
lattice site along the chain. This means that the power 8 ara
cPoordn=eternPitnonǫf1|ubanon(tzdh)mǫ|22o,idsweeasomcboutnasstienbrveeiddeqnutiacnatli,tsyi.ncTeothtehepofiwresrt Displacement 46 CouplingP0 3 1 6 9 12
P =u2+O(ǫ2), P =2u˜2+O(ǫ2) (9) Waveguide
odd 0 1 even 0 2
2
Thus, the relation P = P , implies u2 2u˜2. We
odd even 0 ≈ 0 k=-0.9
are now in position to compute H and H for a
odd even 0
strongly localized mode, using the above relation and
1.5 1.6 1.7 1.8 1.9
Eqs. (4), (6), and (8),
AmplitudeA
γ
H u4+O(ǫ2)
odd ≈ −2 0 1 FIG. 3: Digitized controlled switching of a discrete soliton
γ in a cubic nonlinear waveguide array by varying the beam
Heven ≈ −4 u40−2u20V cos(k)+O(ǫ1·ǫ2), (10) intensity, for a fixed angle (k = −0.9). Inset: Optimized
modulation of thecouplings Vn.
which implies that the PN barrier∆(3) for the nonlinear
cubic array is given by A=1.414 A=1.416 A=1.418
γ
∆(3) =H H u4+2u2V cos(k). (11)
odd− even ≈−4 0 0
In comparison with the previously obtained result for
the PNbarrier[21], Eq.(11) adds an extra,albeit small,
term that shows how the PN barrier is modified for the
mode initially kicked in the lattice. Indeed, besides the
firsttermdependentonthemodeamplitude,Eq.(11)in-
cludes a linear term proportional to the factor V cos(k),
50 6164 50 60 64 50 59 64
whose magnitude could be modified by a judicious ad-
justment of the waveguide couplings and/or the value of
the initial kick. FIG.4: SameasinFig.2,butintheengineeredwaveguidear-
raywiththecouplingmodulationshownintheinsetofFig.3.
Dependence of the PN barrier on the mode coupling
suggests that, if we wish to find a way to engineer the
valueofthePNbarrierinthelattice,weshouldstudythe
waveguide super-lattices. An example of one of such op-
propertiesofamodifiedmodeldescribedbytheevolution
timized structure, where we modulate the coupling pa-
equation
rameterV inastep-likemanner,isshownintheinsetof
n
du Fig. 3 which also shows the discrete position of the soli-
i n +Vn+1un+1+Vn−1un−1+γ un 2un =0, (12) ton at the output as a function of the amplitude of the
dz | |
input beam, at a fixed value of the steering parameter
where the coupling V between two neighboring guides k = 0.9. In a remarkable contrast with other studies
n
−
is assumed to vary either through the effective propaga- (see, e.g., Ref. [17]), the coupling modulation allows to
tion constant or by a change in the spacing between the achieveacontrollabledigitizedswitchingofdiscreteopti-
neighboring waveguides. To study the beam steering in cal solitons in the array with very little or no distortion.
this novelmodel,we useagainasaninitialconditionthe As is shown in Fig. 4, by decreasing the amplitude
sech-likeprofile(2),althoughthisisnotreallyfundamen- of the input pulse at a fixed value of the steering pa-
tal limitation, as argued below. rameter k (in our example fixed to be k = 0.9), it is
−
Wementionthatavariationofthewaveguidecoupling possible to achieve self-trapping of the discrete soliton
inthearrayconstitutesthestartingpointforourconcept by the lattice at some (short) distance from the input at
ofthewaveguidearrayengineering. Achangeofthecou- different waveguide positions. Due to the step-like mod-
plings breaks the symmetry between the beam motion to ulated coupling, we create a selection between the beam
the right and left at the moment of trapping, thus elimi- motion to the right and left at the moment of trapping
nating chaotic trapping observed in the case of homoge- thus suppressing or eliminating the chaotic trapping ob-
neous arrays. servedinhomogeneouswaveguidearrays. Inthisway,we
We have tested different types of modulation in the achievea controllable digitized nonlinear switchingwhere
array coupling and the corresponding structures of the thecontinuouschangeoftheamplitudeoftheinputbeam
5
the observation that for any system with local nonlin-
Vn
8 1.02 er earity a narrow initial profile will render the system into
et an effective linear one containing a small nonlinear clus-
m
ra ter (or even a single site); the bound state will therefore
nt 6 1.01 gPa strongly resemble that corresponding to a nonlinear im-
e n
Displacem 4 0 31 6 9 Coupli purity [22].
Waveguide B. Continuous model
2
A=1.55 In Sec. IIA we have obtained exceedingly interesting
results for the switching of discrete solitons, v`ia the use
0
of the discrete model and tight-binding approximation.
0.5 0.6 0.7 0.8 0.9 1 1.1
Effectivekick k In this section, in order to confirm our predictions, we
perform the corresponding numerical simulations of the
continuous evolution equations of the electric field in-
FIG.5: Digitizedcontrolledswitchingofacubicdiscretesoli-
side a waveguide array, by means of the Beam Propa-
ton byvaryingthebeam inputangle, for afixedbeam inten-
sity. Inset: Optimized modulation of the couplings Vn. gation Method (BPM) [23]. As we show below, these
results support our major findings concerning the digi-
tized switching of an optical beam in engineered nonlin-
resultsina quantized systematic displacement oftheout- ear waveguide arrays.
put beam by an integer number of waveguides. Conse- In the continuous model, the starting point is an op-
quently, for the parameters discussed above we observe tical beam propagating in a three-dimensional medium
almost undistorted switching up to eleven waveguides. with the refractive index that varies in space and is also
Incidentally,we notice here thatthe use ofa linearramp intensity-dependent accounting for the Kerr effect. For
potential(e.g.,intheformV =an)forthispurposedoes a nonlinear waveguide array, the beam is assumed to
n
not lead to an effective switching but, instead, it makes propagate along the z direction and to diffract or self-
the soliton switching even more chaotic due to the phe- focus along the transversal directions x and y. Assum-
nomenonofBlochoscillationswhichbecomerandomized ing that the beam envelope A(x,y,z) varies with z on a
in the nonlinear regime. scalemuchlongerthatthewavelengthλ,thebeamenve-
In Fig. 5 we show another example of the optimized lopeisfoundtoobey[24]thethree-dimensionalnonlinear
coupling modulation, this time as a function of the ef- Schr¨odinger (NLS) equation,
fective input ‘kick’, for a fixed beam intensity. In this
case, we can achieve completely controlled switching up 2iβ ∂A + ∂2A + ∂2A + 2β0k0n2 A2A=0 (14)
to nine waveguides. 0∂z (cid:18)∂x2 ∂y2(cid:19) S | |
Iftheinputbeamwastoexciteinitiallyfivewaveguides
where β = 2πn /λ is the beam propagation constant,
instead of three creating in this manner a wider excita- 0 0
n is the Kerr coefficient of the nonlinear guide, S is the
tion(i.e.,being closertothe continuumlimit), onecould 2
area of the mode, and A2 is the beam power.
expect a smaller amount of radiation emitted. However,
| |
The waveguide structure creates a periodic modula-
this would imply a longer distance before the beam gets
tion ofthe refractive index in only one of the transversal
trapped by one of the waveguides in the array due to
directions (say, x). Along the other transversal direc-
the effective PN potential. Also, this means that one
tion y we assume the beam confinement. Therefore, the
could, in principle, switch the soliton beam to any de-
real dimensionality of the system is two (i.e., the lon-
sired waveguide in the waveguide array, no matter how
gitudinal propagation and the transversal spreading or
far;itwouldbejustamatterofchoosinganinitialbeam
self-trapping). We write the electric fieldenvelopein the
wide enough, i.e., closer to the continuum (in addition
form A(x,y,z) = a(x,z)b(y) and use the effective index
to optimize the coupling in a step-wise manner), by re-
method[25]to formallytransformEq.(14)into aneffec-
moving the randomselection between the directions and
tive two-dimensional equation
suppressing the beam random switching.
Another observation is that the sech-like initial pro-
∂a 1 ∂2a k n
file is not really fundamental. We have verified that the i + + 0 2 a2a=0, (15)
∂z 2k n (x)∂x2 A | |
similar dynamics is observedfor other types of the input 0 eff eff
beam profiles, including a nonlinear impurity-like input where A = S ∞ b2dy/ ∞ b4dy, is the effective
of the form eff −∞| | −∞| |
area over which Rthe nonlineaRr interaction occurs. Pa-
1 A2 |n−nc|/2 rameter neff is the effective, space-varying linear index
un(0)=A(cid:18)1−+A2(cid:19) e−ik(n−nc). (13) of refraction for the one-dimensional problem. For our
problem, n (x) consists of a periodic array of parallel
eff
The reason for this universal behavior seems to rest on slabs with the indices n and n +∆n.
0 0
6
FIG.6: ResultsofnumericalBPMsimulationsofthecontinuousmodelfortheswitchingofa1.5µmGaussianbeampropagating
in an array of 41 AlGaAs waveguides as a function of the input beam intensity, for a fixed input beam angle (0.86o). Left:
2200 W, center: 2330 W,right: 2700 W.
WesolveEq.(15)numericallybythewell-knownbeam phase modulation of the fundamental as well as the sec-
propagation method (BPM), using typical experimen- ond harmonic waves [26]. This phase modulation ac-
tal parameter values. In our simulations, we use an companies the familiar amplitude modulation, being the
array of 41 waveguides, 10mm long, a Kerr coefficient basis of any frequency conversion, and it may produce
n = 1.5 10−17m2/W, assuming a Gaussian beam of the effects which resemble those known to occur in cu-
2
×
the height 3µm, 8µm wide, at λ = 1.5µm, and 0.86o bic nonlinear materials. Typical examples are all-optical
input angle. The effective one-dimensional medium con- switching phenomena in interferometric or coupler con-
sists of a periodic array of slabs, 4µm wide, with 6µm figurationsaswellastheformationofspatialandtempo-
center-to-center separation, n = 3.27, with a modula- ral solitons in planar waveguides (see, e.g., Ref. [26] and
0
tion of the refractive index ∆n=0.0014. the references therein).
Figure6 showsthe beamswitchingas the initial input Recently,itwasdemonstratedtheoretically[27,28,29]
beam power is varied: Switching to seven, five and three thatarraysofquadraticnonlinearwaveguidesrepresenta
guides wasobservedfor a powerof2200W, 2330W and convenientsystemtoverifyexperimentallymanytheoret-
2700 W, respectively. As expected from the estimates icalpredictionsforthedynamicsofnonlinearlaticeswith
of section IIA2, as power is decreased, so does the PN cubic nonlinearity. The first experimentalobservationof
barrier,whichallowsthebeamtogetself-trappedfarther discrete quadraticsolitons hasbeen reportedrecentlyby
away from the vicinity of the input guide. Stegemanandco-authors[19],whodemonstratedthefor-
The above constitute a strong confirmation of the ro- mationofdiscretequadraticsolitonsinperiodicallypoled
bustnessofthe switchingmechanismpredictedby asim- Lithium Niobate waveguide arrays, excited with funda-
ple theory, against more realistic effects. Fine tuning of mental wave pulses at a wavelength of 1572 nm. These
the switching by an appropriate modulation of the effec- experimental observations open many perspectives for
tive waveguide couplings will allow a fine control of the employing much larger nonlinearities provided by non-
positionofthedestinationwaveguide. However,thistask linearquadraticmaterials. Inthis section,weextendthe
is beyond the main scope of the present paper. concept of the controlled digitized soliton switching dis-
cussed above to the case of quadratic discrete solitons.
III. QUADRATIC NONLINEAR WAVEGUIDES
A. Discrete Model
Up to now we discussed the arrays of weakly coupled
waveguideswiththe cubic nonlinearity. However,during The standard discrete model for an array of weakly
last years a growing interest is observed in the study of coupledquadraticnonlinearwaveguideshastheform[9]:
nonlinear optical effects based on the so-calledquadratic
nonlinearities. In contrast to the conventional studies da
i n +V (a +a )+2γ b a∗ = 0
of quadratic nonlinearities where the main attention is dz a n+1 n−1 2 n n
centered primarily on parametric processes and the fre- db
quency conversion,morerecentworksarefocusedonthe i dzn +Vb(bn+1+bn−1)+βbn+γ2a2n = 0, (16)
7
where a and b representthe amplitudes for the funda- Even modes. Nowwesearchforapproximatesolutions
n n
mental (ω) and second harmonic (2ω) fields in the n-th of Eqs. (16) of the form
guide, V and V stand for the linear couplings between
a b a,˜ =a˜ ..,0,˜a eik,1,e−ik,a˜ e−2ik,0,... eiλ2z
thenearest-neighborwaveguides. Parameterγ describes n 0 1 1
2 { }
thenonlinearsecond-ordercoefficientproportionaltothe ˜b = ˜b ..,0,˜b e2ik,1,e−2ik,˜b e−4ik,0,... e2iλ2z (21)
n 0 1 1
{ }
second-order dielectric susceptibility, and β is the effec-
tive mismatch between the fields in the array. where a˜ and ˜b are the amplitudes of the coupled har-
0 0
As in the case of the cubic nonlinearity, the system monics, k is the initial beam angle or effective ‘kick’,
(16) possesses two conserved quantities: the total power, λ2 is the longitudinal propagation constant, a˜1 and ˜b1
are small parameters determined from the equations
of motion. After substituting Eq. (21) into Eq. (16)
P = a (z)2+2b (z)2 (17) and keeping only linear terms in a˜ and ˜b , we obtain:
n n 1 1
Xn (cid:0)| | | | (cid:1) λ2 = (1 + a˜1)Vacos(k) + 2γ2˜b0, a˜20 = (˜b0/λ2)(2λ2 −
β (1 +˜b )V cos(2k)), a˜ = (V /λ )cos(k) and ˜b =
1 b 1 a 2 1
and the system Hamiltonian, V −cos(2k)/(2λ β). From these relations, we find
b 2
−
λ V cos(k)+2γ ˜b , which implies
2 a 2 0
H = (V a∗a +V b∗b +(β/2)b 2+ ≈
−Xn a n n+1 b n n+1 | n| a˜20 ≈ 4˜b20−(˜b0/γ2)[β−2Vacos(k)+Vbcos(2k)],
+ γ2a2nb∗n+c.c.). (18) a˜ Vacos(k) 1,
1 ≈ V cos(k)+2γ ˜b ≪
a 2 0
However, unlike the case of the cubic nonlinear waveg-
V cos(2k)
uidearrayswhereitispossibletofindanalyticalsolutions ˜b b 1. (22)
in the continuum limit which can be used as input pro- 1 ≈ 2Vacos(k)+4γ2˜b0 β ≪
−
files for numerical simulations of discrete solitons, in the
FromEq.(20)andEq.(22),itiseasytoseethat,inor-
caseofthequadraticnonlinearitiesnoexactsolutionsare
dertoobtainSLM,thenonlineartermγ b (˜b )shouldbe
available. Thus, we shouldresortto the limit of strongly 2 0 0
be much larger than the linear coupling terms, V cos(k)
localizedmodes(SLMs)inordertocalculatethePNbar- a
and V cos(2k).
rierandusethe SLMprofileasaninput beamprofilefor b
From Eq. (17) we calculate the total power
the numerical computation of the soliton switching.
P a2+2b2+O(a2,b2), (23)
odd ≈ 0 0 1 1
B. Localized modes and the PN barrier
P 2a˜2+4˜b2+O(a˜2,˜b2), (24)
As in the case of the cubic nonlinearity, we calculate even ≈ 0 0 1 1
the PN barrier as a difference between the values of the and the Hamiltonian of each mode,
Hamiltonianfortheoddandevenstronglylocalizedtwo-
H 8a2a V cos(k) 8b2b V cos(2k)
component modes. odd ≈ − 0 1 a − 0 1 b
Odd modes. We search for approximate solutions of −βb20−2γ2a20b0+O(a21,b21), (25)
Eq. (16) of the form
H 4a˜2(1+2a˜ )V cos(k) 4γ a˜2˜b
a = a ....,0,a eik,1,a e−ik,0,... eiλ1z even ≈− 0 1 a − 2 0 0
n 0{ 1 1 } 4˜b2(1+2˜b )V cos(2k) 2β˜b2+O(a˜2,˜b2). (26)
bn = b0 ....,0,b1e2ik,1,b1e−2ik,0,... e2iλ1z, (19) − 0 1 b − 0 1 1
{ }
We follow the same reasoning as in the case of the cu-
where a0 and b0 are the amplitudes of two harmonics bicnonlinearwaveguidearraysandcalculatetheeffective
composing a localized mode, k is the initial beam angle PNbarrier. Suchcalculationslooksimplerfor the physi-
oreffective‘kick’,λ1 isthelongitudinalpropagationcon- callyimportantcaseofvanishingmismatch,β 0. With
stant,anda1 andb1 aresmallparametersthatshouldbe thatassumption,andimposingthatthepower≈contentof
determined from the equations. After substituting the both, odd and even modes, are equal, P = P , we
odd even
ansatz (19) into Eqs. (16) and keeping only linear terms obtain3b2 6˜b2+(˜b /γ )(2V cos(k) V cos(2k)),and
in a1 andb1, we obtain: λ1 =2a1Vacos(k)+2γ2b0, a20 = then 0 ≈ 0 0 2 a − b
(b /γ )[2λ β 2b V cos(2k)],a =(V /λ )cos(k),and
0 2 1 1 b 1 a 1
− − b ( 2V cos(k) V cos(2k) )
b1 =Vbcos(2k)/(2λ1 β). From these relations, we find ˜b 0 a − b . (27)
− 0
λ1 2γ2b0, which implies ≈ √2 − 12γ2
≈
In terms of b , the Hamiltonian of both the modes can
a2 4b2 (β/γ )b , 0
0 ≈ 0− 2 0 be approximated as
V cos(k) V cos(2k)
a b
a1 ≈ 2γ2b0 ≪1, b1 ≈ 4γ2b0−β ≪1. (20) Hodd ≈−8γ2b30+O(a21,b21) (28)
8
10 Vn ao= 0.912 ao= 0.955 ao= 0.965
r
e
8 met 0.315
a
r
a
P
ment 6 pling 0.3
Displace 4 Cou0 1 2Wave3guide4 5 6
2
20 26 29 20 25 29 20 24 29
k=-1
0
FIG. 8: Switching to 6, 5 and 4 sites of a discrete quadratic
0.8 0.9 1 1.1 1.2 1.3 1.4
Fundamentalharmonicamplitudeuo soliton (SLM) with a slight intensity variation of the funda-
mental mode.
FIG.7: Controlled digitizedswitchingofadiscretequadratic
soliton by a variation of the intensity of the fundamental
discretetwo-frequency(fundamental+secondharmonic)
mode, for a fixed input ‘kick’. Inset: Optimized modulation
solitoncanbe achievedforupto10waveguides,byvary-
of Vn =Va,n=Vb,n.
ing the intensity of the input fundamental mode for a
fixed parameter k = 1. The inset of Fig. 7 shows the
−
H 4√2γ b3 8V b2cos(k) coupling modulation required to achieve this type of en-
even ≈ − 2 0− a 0 gineered soliton switching, which is particularly simple
2V b2cos(2k)+O(a a˜ ,b ˜b ). (29)
− b 0 1· 1 1· 1 and consists of only a single change of about 5% in the
value of the coupling parameter. Figure 8 demonstrates
Finally,we calculate,inthis approximation,the PNbar-
the switching ofthe discrete mode,composedofthe fun-
rier of the strongly localized modes,
damental and second-harmonic fields, to six, five, and
∆(2) =H H four neighboring waveguides, as the intensity of the in-
odd even
− ≈ put fundamental mode is increased. In this respect, it
8Cγ b3+2b2( 4V cos(k)+V cos(2k) ), (30)
− 2 0 0 a b is interesting to point out that in all cases of the digi-
talswitchingboththefundamentalandsecond-harmonic
where C = (1 √2/2). The PN barrier of an array of
− fieldsactasastronglycoupledstate,andthatno‘lagging
nonlinear quadratic waveguides (30) has been obtained,
behind’ was observed of any of the modes with respect
to the best of our knowledge,for the first time. It shows
to the other.
someinterestingfeatures: Themainterm(30)iscubicin
We have performed other simulations with the
themodeamplitude,whileforthecubiccaseitwasquar-
quadraticnonlineararrayincludingthecasesV =0(de-
tic [see Eq.(11)]. Also wenotice thatthe firstcorrection b
coupled second-harmonic fields in the array) and V =
to the PN barrier (30) is linear in the couplings, and it b
αV (reduced coupling of the second-harmonic fields)
dependsonthesquareoftheSLMamplitude. Thisisex- a
with α < 1, etc. In all of those cases, we have observed
actly the same term as in the case of the nonlinearcubic
the digitized switching of the discrete solitons by engi-
array. Thisimpliesthatthefirst-ordercorrectionismore
neering the coupling in the array as discussed above.
important in the nonlinear quadratic array than that in
the nonlinear cubic array suggesting that the appropri-
ate engineering of the couplings and/or input ‘kick’ to
achieve digitized switching should be easier to achieve. IV. CONCLUSIONS
For the numericalsimulations,we use the initial input
in the form of an odd mode,
We have suggested and demonstrated numerically a
simple but yet effective method for controlling nonlinear
a (0) = a a|n−nc|e−i(n−nc)k,
n 0· 1 switching of discrete solitons in arrays of weakly cou-
b (0) = b b|n−nc|e−2i(n−nc)k, (31) pled optical waveguides. We have demonstrated how to
n 0· 1
achievethedigitizedswitchingofdiscreteopticalsolitons
for n nc = 0, 1, and an(0) = bn(0) = 0, otherwise. in weakly coupled arrays of cubic and quadratic nonlin-
− ±
In Eq. (31) we use a 4b2 (β/γ )b ,a V /2γ b ear waveguides described, in the framework of the tight-
0 ≈ 0− 2 0 1 ≈ a 2 0
and b1 Vb/(4γ2b0 βp). We consider an array of 41 binding approximation, by discrete models such as the
≈ −
waveguides with the initial input centered at the mid- DNLS equation with a step-like variation of the waveg-
dle, n = 20. For simplicity, we also assume the case uide coupling parameter. Our approach involves a weak
c
of complete phase-matching, i.e. β 0, and identical step-like modulation of the coupling strength (or, equiv-
≈
coupling for both the harmonic fields, V = V . Fig- alently, distance between the waveguides) in the arrays
a,n b,n
ure7showsthatthe controlleddigitizedswitchingofthe withthe periodlargerthanthe waveguidespacing. Such
9
kindofasuper-latticewaveguide structureallowstomod- Acknowledgements
ify the trapping properties of the array due to discrete-
ness as well as engineer the strength of the effective
Peierls-Nabarro potential arising due to the lattice dis-
creteness. In particular, we have demonstrated the digi-
tized switching of a narrow input beam for up to eleven RodrigoVicencio acknowledgesa supportfromaCon-
waveguides,in the case of the cubic nonlinear array,and icytdoctoralfellowship. MarioMolina andYuri Kivshar
up to ten waveguides, in the case of quadratic nonlinear acknowledgeasupportfromtheFondecytgrants1020139
array. Wehaveconfirmedourpredictionsforafull-scaled and 7020139. Yuri Kivshar thanks the Department of
continuous model and realistic parameters by employing Physics of the University of Chile for a warmhospitality
the beam propagationmethod. in Santiago.
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