. INTRODUCTIONn recent years, a lot of attention has been paid to photo-ic crystal fibers (PCFs) due to the interesting propertieshey present in contrast with the conventional optical fi-ers [1,2]. A PCF is a tubular structure of a substrate ma-erial with a periodic array of air holes running parallel tohe optical axis and presenting a central defect. Amonghe properties that make these structures very interest-ng are the endlessly monomode character; i.e., they sup-ort only the fundamental mode regardless the wave-ength for a very broad range of geometrical parameters,he flexibility of dispersion control that allows to bring theoint of minimum dispersion to a desired wavelength onlycting on the geometrical parameters, and the possibilityo enhance and precisely design the birefringence. Accord-ng to the type of the central defect there can be two dif-erent kinds of guiding mechanisms. When the core defects made of a hole of different size or shape (hollow fibers),uiding properties rely on Bragg scattering which leads tohe existence of bandgaps in the frequency spectrum. Onhe other hand, when the defect consists in a lack of somef the central holes, so that the fiber core is made of theubstrate material (solid core fibers), the guiding is ac-omplished by the conventional total internal reflectionechanism due to the fact that the air holes in the clad-
ing region make it to have a lower mean refractive indexhan the core.
The particular geometry of PCFs, constituting a systemith discrete symmetry, makes them present a modal
pectrum different from that of conventional fibers [3] notnly because of the specific symmetry properties but alsoecause discreteness induces limitations in the number ofodes [4]. The possibility of making PCFs of a material
resenting a nonlinear response to the optical field makeshose structures very appropriate for all-optical process-ng, since nonlinear effects are enhanced due to the strongonfinement of the field inside the fiber core. In this way,
ptical fibers are known to support different kinds of non-inear modes in the form of fundamental solitons [5] andortices [6]. Importantly, the structure of the PCF allowsn to stabilize nonlinear modes that are unstable in con-entional homogeneous media. In fact, the stabilization ofpatial optical solitons when the PCF substrate presentsKerr nonlinear response was numerically demonstrated
7,8], which are known to undergo spreading or collapsen homogeneous media.
As it concerns vortex solitons, i.e., optical fields pre-enting a phase dislocation point and an increasing phaserom 0 to 2�� around the dislocation, � being an integeramed vorticity or winding number, they may present inonlinear media either radial symmetry [6,9] or a multi-
obed structure in the form of clusters [10–13] made ofundamental solitons presenting the characteristic vortexhase structure. Vortex solitons and soliton clusters wereound to be stable in PCFs under a power threshold14,15]. In a similar context, two-dimensional solitons andortices were stabilized using a periodic square lattice16,17].
In this paper we carry out a study of the vortexlikeodes in a nonlinear Kerr-type PCF with two defects
lack of two consecutive or close air holes). The fundamen-al solutions for this kind of dual-core PCFs were alreadytudied [18] resulting in, along with the usual symmetricnd antisymmetric modes, a new asymmetric mode ap-earing over a power threshold bifurcating from the sym-etric one. This new state is the key for the all-optical
witching operations since it is stable whereas the sym-etric one turns unstable beyond the bifurcation pointaking possible the suppression of the coupling. This ef-
ect of destabilization of the original solution to form atable asymmetric state is known as spontaneous symme-ry breaking (SSB) and was already studied for two-omponent nonlinear systems with a superposed periodicattice [19]. On the other hand, it is worth mentioning
009 Optical Society of America
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2302 J. Opt. Soc. Am. B/Vol. 26, No. 12 /December 2009 J. R. Salgueiro and F. Santos
hat experimental results on applications using dual-coreCFs were already reported, as for instance velocity mea-uring [20] or nonlinear optical switching [21].
The solutions with a vortex in either one or two of theCF cores are, in principle, expected to be modulationallytable under a power threshold due to the PCF structurehough azimuthally unstable. Nevertheless, there is theossibility of modifying this kind of systems introducinglements to enhance the stability properties. In fact, vec-or systems with an additional incoherently coupled com-onent have been demonstrated to be suitable for stabi-izing vortices [22,23]. Also, particular interest hasecently been devoted to nonlocal nonlinear media, wherehe nonlinear response at a particular position does notxclusively depend on the field at that point but also onhe field in the surroundings. This kind of media supportstable vortices [24–26] and also other types of solitaryaves carrying angular momentum, such as azimuthallyodulated rotating singular optical beams or azimuthons
27], or multipole and spiraling solitons [28], as well aspiraling multivortices [29]. The possibility of using suchedia for the fabrication of PCFs or the generation of
uch nonlinear response by means of complementaryechniques such as filling up the PCF holes with nonlin-ar liquids makes the consideration of this basic and sim-ler system a necessary previous step for the study ofelds with phase structure in nonlinear PCF structures.In Section 2 we will present the model used to describe
he nonlinear fields with vorticity in the dual-core nonlin-ar PCF. In Section 3 we find, classify, and study the dif-erent families of solutions and their bifurcation schemes.inally, in Section 4, we analyze the stability of the differ-nt families of solutions.
. MODEL AND STATIONARY STATESe consider a PCF made of a material with refractive in-
ex ns and with the nonlinear Kerr response structuredith a triangular network of air holes (index na). The lack
f two of those consecutive or close holes constitutes twoefects forming a dual-core coupler (see Fig. 1). The effec-ive increase in the linear index inside the cores with re-pect to the surroundings makes the PCF device of thendex-guiding or solid-core type. We choose the opticalxis of the PCF in the direction Z (propagation direction),o that the triangular structure lies on the transversallane �X ,Y� and comes given by the function W�X ,Y�na+�V�X ,Y�, where �=ns−na is the index difference and
Λ
r
ig. 1. (Color online) Sketch of the central part of the PCF withriangular structure and two close defects (double solid core)howing the basic parameters.
�X ,Y� is a normalized function which takes the value=1 in the substrate and the value V=0 inside the airoles. Thus, the optical scalar field E�X ,Y ,Z� propagating
nside the PCF along the Z direction can be modeled byhe following nonlinear wave equation:
2ik�E
�Z+ ��
2 E + 2k2�W�X,Y� + �V�X,Y��E�2�E = 0, �1�
here ��2 =�2 /�X2+�2 /�Y2 is the Laplace operator, k
2� /� is the wavenumber related to the wavelength �,nd � is a parameter describing the nonlinear Kerr re-ponse of the substrate. To further simplify the model weill choose na=0 without loss of generality. In fact, choos-
ng a particular value for the base index na only has theffect of a shift in the modal spectrum. On the other hand,he parameters k, �, and � can be dropped from the equa-ion by a proper rescaling of the spatial variables and theeld. In fact, taking the new variables and field as xk�2�X, y=k�2�Y, z=k�Z, and U=�� /�E, the equation
akes the following canonical form:
i�U
�z+ ��
2 U + V�x,y��1 + �U�2�U = 0, �2�
here the Laplace operator is now referred to the newpatial variables x and y.
We are interested in solutions which remain stationarylong the propagation direction z, being of the form
U�x,y,z� = u�x,y�exp�i�z�, �3�
here � is the propagation constant. Also, we are inter-sted in solutions presenting phase dislocations (vortices)o that the phase performs a number of windings aroundhe dislocation. In principle, we seek stationary solutions�x ,y� with phase dislocations located at the center ofach defect core, so that the phase increases from 0 to 2��round any or both core centers, with � being the vorticityr winding number. According to this, the function u de-cribing the transversal amplitude of the stationary fieldhould be a complex function. In this work we will focusn the simplest case of first-order vortices, considering �1. Substituting Eq. (3) into Eq. (2) we come up with the
ollowing z-independent equation for the transversal fieldmplitude:
− �u + ��2 u + V�x,y��1 + �u�2�u = 0. �4�
ow considering the real and imaginary parts of u sepa-ately, u=u1+ iu2, and substituting into Eq. (4) we obtainsystem of two identical equations for u1 and u2,
− �ui + ��ui + V�1 + u12 + u2
2�ui = 0, i = 1,2. �5�
lthough both equations are identical, they are coupledy the nonlinear term and, consequently, the functions u1nd u2 representing real and imaginary parts of the fieldre in general different. The model is equivalent to thene describing a vector system where the two incoher-ntly coupled components present the same propagationonstant.
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J. R. Salgueiro and F. Santos Vol. 26, No. 12 /December 2009 /J. Opt. Soc. Am. B 2303
. FAMILIES OF NONLINEAR MODESe solved the system described by Eq. (5) numerically
30]. The nonlinearity makes solutions of Eq. (4) depen-ent on power, P=��u1
2+u22�dxdy, and there exist different
amilies, each one described by a curve in the plane �� ,P�,hich show different field amplitudes or phase configura-
ions. Different kinds of symmetric and asymmetric solu-ions containing vortices in either one or both cores wereound. There also exist first-order solutions without anyhase structure which can also be symmetric or asymmet-ic and present the shape of double or single dipoles. Inubsections 3.A–3.C we will describe the different typesf solutions classifying them in three main groups. First,e briefly examine the solutions without any vorticity.econd, we will pay attention to those solutions present-
ng a vortex centered at each of the cores. We will generi-ally name these states double vortices (DVs) and will ex-mine the different subfamilies according to themplitude distribution (shape) and phase structure. Fi-ally, we will study the high asymmetric solutions con-aining vortices that we name asymmetric vortices (AVs).hey are composed of a vortex in one of the cores and aeld distribution without any phase structure in the otherore.
The power curves correspondent to the families we aretudying are plotted in Fig. 2. The numerical calculationsere carried out considering a PCF with a pitch (distanceetween closest hole centers) of �=10 and hole radius r4. These particular values of the parameters, which lead
o holes filling a large fraction of the fiber section, are nec-ssary in order that the PCF supports the first excitedodes (as is the case of vortices). Otherwise, the strongonomode character of PCFs would not allow the exis-
ence of modes except for the fundamental.
40
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. First-Order Solutions without Vorticityf we focus on first-order solutions, the simplest station-ry states are real-valued functions—with one of the com-onents in Eq. (5) identically zero—in the form of doubleipoles (DDs) which can present the four different con-gurations shown in Figs. 3(a)–3(d). Each configurationorresponds to a family of solutions on the power diagramepresented in Fig. 2 as a dashed curve. There are actu-lly four close-together lines (one for each solution type)s is shown in Fig. 4 (dashed lines) where the low poweregion of the diagram is zoomed.
In the linear limit �P→0� the four DDs are the uniquerst-order solutions possible since the double-core struc-ure reduces the original discrete symmetry of the PCF,6v, into C2v, and consequently they must remain invari-nt upon rotations of � radians or reflections with respecto the Cartesian coordinate axes. Besides, the states illus-rated in Figs. 3(a) and 3(b) will be named bounding �b�nd antibounding �a�, respectively, regarding the molecu-ar orbital theory, since they are characterized, respec-ively, by a high and low field density in the space be-ween both cores. The other two families of solutionsFigs. 3(c) and 3(d)] will be named parallel �p� and crossedx�, respectively, due to the fact that both dipoles presentame or opposite sign distribution. So, these four types ofolutions will be denoted b, a, p, and x, respectively.
In a nonlinear regime, as a consequence of a SSB, therelso exist asymmetric real-valued states in the form ofDs characterized by a different power on each of the
ores, which can take the shape of single dipoles at highnough power. We name them asymmetric dipoles, andhey can present one of the two shapes shown in Figs. 3(e)nd 3(f) and bifurcate from the symmetric DDs at points5 and O6 (see Fig. 4).
. Symmetric Vortex Solutionsn the linear limit it is not possible to find vortexlike fieldswith dislocations in one or two of the cores). In fact, ashown above, the reduction in symmetry induced by theual-core structure makes the four DD solutions to beondegenerate and consequently they cannot be com-ined to form a stationary state. When nonlinearity isresent, however, it is possible to get stationary vortex so-utions if real and imaginary parts are formed by DDsontributing with a different power; i.e., they are asym-etric with respect to the real and imaginary compo-
ents. Examples of such stationary states with the shape
0.7 0.8 0.9 1 1.1propagation constant, β
0
20
A
AVAD
ig. 2. Power versus propagation constant for the differentamilies of vortexlike solutions (continuous curves), the double-ipole solutions (dashed curves), and the asymmetric dipolesdashed-dotted curves). Big capital letters label points corre-ponding to the different examples shown in other figures below.he italic legends label the branches correspondent to the differ-nt types of solutions: double dipoles (DDs), asymmetric dipolesADs), DVs, and AVs. Beware that curves may actually constitute
bunch of close-together curves (not distinguished due to thecale of the figure) describing families of solutions with similarthough different) power. Inset: detail of the DV branch at higherower, where double-quadrupole solutions originate.
ig. 3. (Color online) (a)–(d) Four DD states of a dual-core PCFor �=0.8: (a) bounding type �b�, (b) antibounding �a�, (c) parallelp�, (d) crossed �x�. (e),(f) Two asymmetric dipoles (AD1 and AD2)hat bifurcate from the DDs at points O6 and O7 (Fig. 4) and takehe shape of single dipoles at high enough power.
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2304 J. Opt. Soc. Am. B/Vol. 26, No. 12 /December 2009 J. R. Salgueiro and F. Santos
f double-doughnut vortices are shown in Fig. 5. Therere four types of DV solutions according to the type ofDs forming their real and imaginary parts; to be precise
here are bounding and antiboundinglike solutions—weenote them as b and a—which have a bounding or anti-ounding DD, respectively, as one of the components, andach of those types can host two vortices of the same orhe opposite vorticity, regarded as positive �+� and nega-ive (�), respectively. Consequently, we denote the fourossible DV solutions as b+, b−, a+, and a−. The ampli-ude and phase diagrams of Fig. 5 account for these fourypes of DVs.
The four different DV solutions are nondegenerated, sohat the curve on the power diagram describing the DVamily (Fig. 2) is actually formed by four close curves,ach one correspondent to one of the types of DVs de-cribed above. They exist over a particular power thresh-ld and bifurcate from the DDs at points O1–O4 as shownn the zoomed diagram of Fig. 4. The bounding-positiveb+� and bounding-negative �b−� DV families bifurcate
0.755 0.76 0.765propagation constant, β
0.0
0.5
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0.53
0.54
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O6
O1
AD1
AD2AV
O7
x p
a-
O3O2O4
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b
O5
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ig. 4. Detail of the power curves for the low power regime.ashed lines are the four DD families (b, a, p, and x stand for theounding, antibounding, parallel, and crossed families, respec-ively). Points labeled O1–O4 are those from where the four DVamilies bifurcate [the four types bounding positive �b+�, bound-ng negative �b−�, antibounding positive �a+�, and antiboundingegative �a−� are indicated]. O5 and O6 are the bifurcation points
or the two asymmetric DD families (labeled as AD1 and AD2 andlotted as dashed-dotted lines). O7 is the bifurcation point for theV. Inset: zoom of the region close to point O5 to show thatranches AD1 and AD2 are noncoincident.
ig. 5. (Color online) (a),(b) Intensity-level plots of two DVtates for �=0.85 (point B in Fig. 2), one of the bounding type (a)nd another of the antibounding type (b). (c)–(f) Phase patternsor each of the DV states correspondent to that point. Images (c)nd (e) show the phase of the same vorticity states �+� and im-ges (d) and (f) the phase of those with opposite vorticity (�), sohat images (c)–(f) correspond to the states b+, b−, a+, and a−,espectively.
rom the crossed and parallel DDs, respectively, whereasoth antibounding solutions (a+ and a−) bifurcate fromhe antibounding DD. Close to the bifurcation points,ower carried by real and imaginary components is in-reasingly different up to the point that one of them van-shes at the bifurcation point turning the state into theD mode. An example of a DVs state of the b+ type close
o the bifurcation point is shown in Fig. 6 (image A).At higher powers we found DV solutions in the form ofultipoles particularly with the shape of double tripoles
nd double quadrupoles (see examples in Fig. 6; cases Cnd E, respectively). The double-tripole solutions arise atoderate power and they are described in the power dia-
ram (Fig. 2) by branches that originated from the mainV curves. The rise of solutions with a tripole shape is re-
ated to the symmetry of the PCF network. In fact, aingle tripole (ST) presents the symmetry described byhe group C3 which is a subgroup of the C6v symmetryroup characteristic of the PCF. Nevertheless, the pres-nce of the second core further limits the symmetry to C2vnd consequently only two different double tripoles canxist; those shown in Fig. 6 (images labeled C1 and C2).gain, the four combinations b+, b−, a+, and a− are pos-ible.
It is important to notice that the point where theouble-tripole solutions start is not an actual bifurcationoint. Instead, the branch of the doughnutlike vorticespens, so that both extremes join to one different double-ripole branch [see Figs. 2 and 7(a)]. As power increases,he lobes of the double-tripole solutions become increas-ngly narrower and their maximum amplitude increas-ngly larger, becoming independent of the PCF structurend so both curves merge. The transition between dough-utlike and tripole solutions is gradual and the absence ofproper bifurcation point is related to the fact that both
ouble-tripole solutions are nondegenerated. In fact, theuppression of a bifurcation point due to an asymmetryhat lifts a degeneration is already known in the contextf asymmetric couplers [31]. We would like to remark athis point that this phenomenon is described in [19] as aseudobifurcation in the context of a two-component sys-em in a 2D square lattice. In that system, however, thesymmetry is not introduced by two defects in theattice—the use of the coupled mode theory would not al-ow us to model the effect of the two side-by-side defectsnyway. Instead, the asymmetry is introduced by a phaseismatch between the two lattices affecting each compo-
ig. 6. (Color online) Some examples of DV states plotted asntensity-level images and correspondent to different branches inhe power diagram. Two double-doughnut states (A, D)—onelose to the bifurcation point (A)—two double tripoles (C1, C2),nd two double quadrupoles (E1, E2) are shown. Labels corre-pond to points (A, C, D, and E) in Fig. 2.
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J. R. Salgueiro and F. Santos Vol. 26, No. 12 /December 2009 /J. Opt. Soc. Am. B 2305
ent, but the result is similar: the suppression of the bi-urcation point.
For even higher powers, double quadrupoles exist andgain they are described by branches joining to the mainV curves [see Fig. 2 (inset) and Fig. 7(b)]. In this case
he original symmetry of a single-core quadrupole �C4v�oes not belong to the group of the PCF symmetry, al-hough they have in common the symmetry of the sub-roup C2v. As in the case of the double tripoles, theouble-core structure of the PCF imposes a C2v symmetrynd the solution types shown in Fig. 6 (E1 and E2) con-titute the unique possibilities.
. Asymmetric Solutions with Vorticitye have also found asymmetric solutions in the form of
ingle-core vortices (SVs) and combinations of a vortex inne of the cores and a fundamental field in the othervortex-fundamental (VF)]. In Fig. 8 we present some ex-mples to illustrate different kinds of configurations. Thexistence of symmetric DVs as well as SVs suggests thexistence of asymmetric DVs that could be generatedrom the DVs via a SSB. Nevertheless, we were unable tond such solutions. A possible explanation for the lack of
ig. 7. (a) Detail of the power diagram at the junction whereouble-tripole branches (labeled DT) originate from the DV ones.ifferent curves correspondent to families of different phase
tructure are shown: bounding positive �b+�, bounding negativeb−�, antibounding positive �a+�, and antibounding negativea−�. For the low branch lines corresponding to the families a+nd a− are very close and not resolved at the scale of the plot. (b)ame for the junction point where double-quadrupole (DQ) solu-ions originate. Again curves corresponding to a+ and a− are toolose to be resolved.
ig. 8. (Color online) Some examples of single vortices and VFtates with different shapes (shown as intensity plots): two typesf STs (F1, F2), single doughnut (G), two types of TF states (H1,2), and doughnut-fundamental state (I). Labels correspond tooints in Fig. 2.
olutions with such a shape is the reduced symmetry ofhe dual-core system which makes the four DDs nonde-enerated. Consequently, as commented above, the gen-ration of a stationary vortex requires a specific power ra-io between real and imaginary parts. Additionally, ansymmetric state also requires a specific ratio betweenower carried by each core and both conditions cannot beulfilled simultaneously to form an asymmetric DV sta-ionary solution. It is possible, however, to have a stateith a vortex in one core and a field without phase struc-
ure in the other one.At low power (see Fig. 2) there is a single curve of AVs
escribing the solutions of a vortex with a doughnuthape [of the form similar to the one shown in Fig. 8 (G)].his curve bifurcates from one of the asymmetric DDs atoint O7 (see Fig. 4). Close to the bifurcation point, pow-rs of real and imaginary parts become increasingly dif-erent up to the point (bifurcation) where one of themanishes and the field takes the shape of an AD.
For higher power the asymmetric tripole solutions areound. In this case the junction region is much more com-licated since several branches corresponding to severalultipolar solutions appear (Fig. 8). On one hand, there
re STs of two different shapes (images F1 and F2) whichre nondegenerated due to the presence of the secondore. On the other hand, there are also nondegeneratedombinations of a tripole and a fundamental field [tripole-undamental (TF)] also with two different shapes (images1 and H2). These TF solutions, along with the VF with aoughnut vortex (image I), can also present two possiblehase distributions that we name bounding �b� or anti-ounding �a� depending, respectively, on whether the fun-amental field has the same or opposite sign as the clos-st lobe of the dipole forming the vortex. This means thathe corresponding curves on the power diagram are actu-lly formed by two close lines. In Fig. 9 we show the re-ion of the junction point as a zoomed diagram represent-ng all the lines describing the different family solutions
0.91 0.92 0.93propagation constant, β
18
19
20
21
22
pow
er
a
VF
TF
ST
dipoles
SV
b
b (2)
a (1)b (1)
a (2)
AV
ig. 9. Detail of the power diagram at the junction where dif-erent branches of ST and TF solutions originate from thesymmetric-vortex family. The shape of the different solutions isndicated with labels: single (doughnut) vortex (SV), ST, TF, anddoughnut) VF. Additionally, the type of solution according to thehase structure is also indicated with labels: bounding �b� andntibounding �a�. The numbers in brackets indicate the type ofolution according to symmetry criteria (cases F and H inig. 8).
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2306 J. Opt. Soc. Am. B/Vol. 26, No. 12 /December 2009 J. R. Salgueiro and F. Santos
nd how they originate in a nontrivial way at the junc-ion. Again, the breakage of symmetry induced by theual-core structure prevents the existence of actual bifur-ation points.
For higher powers, in a similar fashion as in the case ofhe DVs, the upper branches correspondent to the single-oughnut and doughnut-fundamental solutions gohrough new junction zones where single quadrupolesnd combinations of quadrupoles and fundamental fieldsrise. Some examples are presented in Fig. 10. Again, theymmetry of the structure allows only two kinds of singleuadrupoles (images J1 and J2) and two quadrupole-undamental states (images L1 and L2). The latter onesan be of two types, bounding or antibounding.
. STABILITYe have checked the stability of the states of the different
amilies by simulating their propagation using a standardeam propagation method. The main conclusion is the ex-stence of two different scenarios. States characterized by
power over a threshold undergo collapse after a certainropagation distance. The higher the power the shorterhe distance they evolve before collapsing as is shown in
ig. 10. (Color online) Examples of single vortices and VF statesorrespondent to families of the doughnut and quadrupole types.he two types of single quadrupoles (J1, J2) originate from theV branch in Fig. 9 at higher power. The two types ofuadrupole-fundamental states (L1, L2) originate from the VFranch at higher power. The doughnut-shaped single-vortex (K)nd VF (M) are also examples related to a higher power.
0 50 100 150 200 2500
5
10
max
imum
inte
nsity
0 100 200 300 400 5000
0.5
1
1.5
max
imum
inte
xity
0 100 200 300 400 500propagation distance, z
0
0.05
0.1
max
imum
inte
xity
(a) (b)
(c)
(f)
(g)
(h)
(d)
(e)
ig. 11. Maximum intensity versus propagation distance for dif-erent double vortices (continuous curves) and single vorticesdashed curves). DV curves correspond to P=34.4 (a), P=30.1 (b),=27.9 (c), and P=5.8 (d). Single-vortex simulations are for=27.3 (e), P=15.8 (f), P=15.1 (g), and P=2.76 (h).
ig. 11 cases (a) and (b) and (e) and (f) for DV and single-ortex states, respectively. Remarkably, this property isependent on power density and not on the family underonsideration. In fact, families with power distribution onoth cores—DV and VF—present a collapse thresholdbout P�30 while those families with power on a singleore (single vortices) present a threshold around P15.75. In both cases the propagation constant at the
hreshold point is about ��0.88. The power threshold ispproximately at the point where the tripole solutionsriginate and consequently they undergo collapse uponropagation.On the other hand, below the power threshold, theodes do not develop collapse and the PCF structure pre-
ents them to spread out as in a bulk Kerr medium. Nev-rtheless, after a distance they develop the azimuthal in-tability so that the vortices break into fundamentalolitons that remain spinning inside the PCF core. Theistance for this breakup to occur depends on power, sohat the lower the power the longest distance they survives shown in Fig. 11 cases (c) and (d) and (g) and (h). Forow powers (close to the linear limit) the distance beforehe breakup can be quite long.
Since the stationary states for a reasonable nonlinearegime are unstable, since they collapse or at least de-elop the azimuthal instability, an interesting furthertep is to seek ways to enhance the stability properties ofhe medium, as considering vectorial systems of incoher-ntly coupled components or using media with nonlocalonlinearities. Anyway, the study of the present system,ue to the simplicity of the model, constitutes a necessaryrevious step for the study of vortices in dual-core PCFs.
. CONCLUSIONSn this work we have studied and classified the differenttationary nonlinear vortex-type families of solutions in aCF with two consecutive defects constituting a dual-coreonlinear coupler. We found solutions in the form of DVsith shapes of double doughnut, double tripole, andouble quadrupole. Additionally, asymmetric solutions inhe form of single vortices (located in one of the cores) andombination of vortices and fundamental states, also withoughnut, tripole, and quadrupole shapes, were also cal-ulated. The power diagram with the different bifurca-ions and junction points was obtained and the differentolution families were classified. Finally, we presented atability study determining different instability scenarios.
CKNOWLEDGMENTShe authors thank Yuri Kivshar from The Australian Na-
ional University for the very useful discussions and sug-estions. This work was supported by the Ministerio deiencia e Innovación of Spain through the Acción Comple-entaria Internacional grant PCI2006-A7-0561, projectAT2008-06870, and the Ramón y Cajal contract granted
o J. R. Salgueiro, and also by Xunta de Galicia, Spainhrough project PGIDIT06PXIB239155PR.
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1
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1
1
2
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J. R. Salgueiro and F. Santos Vol. 26, No. 12 /December 2009 /J. Opt. Soc. Am. B 2307
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