Vortex creation without stirring in coupled ring resonators with gain and loss
Aleksandr Ramaniuk, Nguyen Viet Hung, Michael Giersig, Krzysztof Kempa, Vladimir V. Konotop, Marek Trippenbach
VVortex creation without stirring in coupled ring resonators with gain and loss
Aleksandr Ramaniuk, Nguyen Viet Hung, Michael Giersig,
3, 4
Krzysztof Kempa, Vladimir V. Konotop, and Marek Trippenbach Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL-02-093 Warszawa, Poland Advanced Institute for Science and Technology, Hanoi University of Science and Technology, 100803 Hanoi, Vietnam Department of Physics, Freie Universit¨at Berlin, Arnimalle 14, D-14195 Berlin, Germany International Academy of Optoelectronics at Zhaoqing,South China Normal University, 526238 Guangdong, P. R. China Boston College, Department of Physics, Chestnut Hill, MA 02467 USA Centro de F´ısica Te´orica e Computacional and Departamento de F´ısica, Faculdade de Ciˆencias,Universidade de Lisboa, Campo Grande, Edif´ıcio C8, Lisboa 1749-016, Portugal (Dated: October 3, 2018)We present study of the dynamics of two ring waveguide structure with space dependent coupling,linear gain and nonlinear absorption - the system that can be implemented in polariton condensates,optical waveguides, and nanocavities. We show that by turning on and off local coupling betweenrings one can selectively generate permanent vortex in one of the rings. We find that due to themodulation instability it is also possible to observe several complex nonlinear phenomena, includingspontaneous symmetry breaking, stable inhomogeneous states with interesting structure of currentsflowing between rings, generation of stable symmetric and asymmetric circular flows with variousvorticities, etc. The latter can be created in pairs (for relatively narrow coupling length) or as singlevortex in one of the channels, that is later alternating between channels. a r X i v : . [ n li n . PS ] M a y I. INTRODUCTION
Coupled microrings (microdiscs, or more generally microcavities) are standard basic elements in diverse physicalapplications. In optics they are used for nonreciprocal devices [1], switches [2], loss control of lasing [3], and ringlasers [4, 5], to mention a few. Recently the attention was also turned to more sophisticated devices which includeseveral coupled microcavities and which are referred to as photonic molecules [6]. Coupled non-Hermitian microcavitiesare also used for study of chiral modes in exciton-polariton condensates [7], as well as for modeling coupled circulartraps for Bose-Einstein condensates (BEC), where gain corresponds to adding atoms while nonlinear losses occursdue to inelastic two-body interactions. They can also be realized in nanoplasmonic systems [8] and slow-light opticalmicrocavities [9].Like in case of any coupled subsystems, the characteristics of coupling between microrings (or to an externalelement, for instance to a bus fiber) may strongly affect the stationary regimes as well as the dynamics supported bythe system. The coupling can be modified in various ways. It depends on the geometry (i.e. on the mutual locationsof the rings), on the wave-guiding characteristics of the rings which determine the field decay outside the cavities, onthe medium between the cavities (it can be homogeneous or gradient; active, absorbing, or conservative), etc. Thus,it is of natural interest to understand how the characteristics of coupling affect the field distribution and dynamicsinside the ring cavities. This is the question which is addressed in the present work. The main emphasis is made onthe interplay between the size of the coupling domain and other spatial scales of the system (i.e. on the ring lengthsand on the characteristic scales of the excited modes).Before we move on to specific model (formulated in Sec. II) whose stationary solutions are investigated in Sec. III,the dynamical regimes in Sec. IV and Sec. V, and vortices in Sec. VI, we would like to advertise our effort to designup a nanostructure of coupled rings to explore experimentally this and similar settings and its dynamics. In Sec. VIIwe discuss specific experimental proposal based on the structures that were already manufactured.
II. THE MODEL
In the present study we consider a model described by two coupled nonlinear Schr¨odinger equations with gain andnonlinear loss (depending on applications they also can be termed Gross-Pitaevski or Ginzburg-Landau equations),which we write down in scaled dimensionless units i∂ t ψ = − ∂ x ψ + iγψ + (1 − i Γ) | ψ | ψ + J ( x ) ψ ,i∂ t ψ = − ∂ x ψ + iγψ + (1 − i Γ) | ψ | ψ + J ( x ) ψ . (1)Here ψ and ψ are the fields in the first and second waveguides, γ is the linear gain and Γ is the nonlinear loss, bothconsidered constants along the waveguides, and J ( x ) is position depending coupling.Extended discussion of model (1) and of its applications can be found in a previous publication [10] where therings were homogeneously coupled, i.e. where it was assumed that J ( x ) is constant. We also mentioned that (1)with constant coupling is analogous to the model introduced earlier in [11] where it was considered on the wholeaxis subject to the zero boundary conditions. In this paper we focus on expanding the study of the model throughintroducing coupling modulation J ( x ). We consider rings assuming, without loss of generality, that x ∈ [ − π, π ].This implies periodic boundary conditions for both channels: ψ i ( x, t ) = ψ i ( x + 2 π, t ), and the coupling function J ( x )is extended only in the certain region of the rings. In particular, for numerical simulations we shall consider localGaussian coupling in the following form J ( x ) = J √ πw exp (cid:18) − x w (cid:19) (2)where w is the width of the coupling, while J characterizes the coupling strength. Our results are not sensitive tothe particular shape of the wavefunction, as we have checked using supergaussian functions with very high power.An important remark about the used terminology is in order. For all applications mentioned in the Introduction,the meaning of the variable x is an angle defining a point on the circumference. The functions ψ , are rather envelopes of the field distributions than the total fields (see e.g. [12] for optical resonators and [13, 14] for BEC applications).Thus, solutions for ψ , having nonzero topological charge (see [10]) may correspond to the total field distributionshaving phase singularities in the centers of the rings. In other words, such solutions describe vortices . Taking thisinto account the respective solutions are referred below as vortices. System (1) is simple, but possesses surprisinglyrich and diverse set of stable states (some of them nonstationary). For the limit of very wide coupling ( w (cid:29) π ) weexpect the same results dynamics as for the constant coupling (it is described in [10]). On the other side, very narrowcoupling ( w (cid:28) π ) allows to approximate coupling with delta function. FIG. 1. Absolute values of antisymmetric stationary states after propagation time T = 100 in the coupled double-ring system(1) obtained for the initial conditions (4) with γ = 3 and Γ = 1. Left panel: Antisymmetric states calculated for differentcoupling strengths and fixed coupling width ( w = 1). Black line represents homogeneous state for the respective uncoupledsystem. Right panel: Antisymmetric states calculated for different coupling widths and fixed normalized coupling ( J = 1).. Most of the results found in the present study are numerical (using propagation techniques). In this context thereis one important issue that we need to address before we present the outcome of our investigations. As discussed in[10], for uncoupled case ( J = 0) one can find stable background solutions in the form ψ , ( t ) = (cid:114) γ Γ e − i γ Γ t . (3)Once the rings become coupled, modulation instability occurs mostly due to the interplay between gain and nonlinearabsorption. In the case of constant coupling in Ref. [10] two distinct classes of solutions have been found analytically:symmetric, characterized by ψ = ψ , and anti-symmetric with ψ = − ψ . The anti-symmetric solutions are alwaysstable, and symmetric are usually unstable. Therefore we decided to perform numerical studies using symmetric stateas initial condition. This led us to plethora of new states and attractors [10].We found that even if the coupling is not uniform, dynamics starting from anti-symmetric states lead to anti-symmetric stationary stable solution (see the examples of such attractors in Fig. 1). On the other hand, starting froma symmetric state, in some regions of parameters, does not necessarily lead to the antisymmetric stationary states,but can traverse to the more interesting attractors, like limit cycles or even chaotic states. Hence below we focus onthe dynamics starting with the symmetric initial state with small perturbation imposed in the form of ψ , ( x, t = 0) = (cid:114) γ Γ (1 + β sin( kx )) , (4)where the perturbation β was typically of order of 10 − . In our simulations we took the value of the gain parameter γ larger than loss Γ and we checked that all results are qualitatively the same, regardless of the particular valuesof this parameters. The results also do not depend on particular values of the amplitude of the perturbation β orperturbation wavenumber k . In the case of stronger loss (Γ > γ ) results seem to be different and they are not includedhere. In this article we assume γ = 3 , Γ = 1 for all later considerations and propagate from initial state defined with(4), unless stated otherwise. All simulations were performed via Split-Step Fourier method [15].
III. STATIONARY SOLUTIONS
When the coupling between the rings is weak ( J ≤ J , as presented in Fig. 1.In this figure we plot the modulus of the wavefunction for unitary width ( w = 1) and various coupling constant J (Fig. 1 left panel). In the right panel of Fig. 1 we fix J = 1 and change w , going towards narrow distributions, toshow what one can expect when the coupling has a form of Dirac delta function. We also show background level,plotting it as a black horizontal reference line, in left panel of Fig. 1.In order to interpret the results of Fig. 1, we first notice that for antisymmetric solutions ψ = − ψ the couplingplays the role of the linear potential well, − J ( x ), thus leading to the equation i∂ t ψ = − ∂ x ψ − J ( x ) ψ + iγψ + (1 − i Γ) | ψ | ψ (5)Let us now rewrite this model in the hydrodynamic form, introducing the amplitude distribution ρ ( x ) as well as thephase gradient v ( x ) = ∂ x arg[ ψ ( x )], through the relation ψ ( x ) = ρ ( x ) exp (cid:0) i (cid:82) v ( x ) dx (cid:1) . For the stationary solutionwe obtain from (5):2 v ( x ) ρ ( x ) dρ ( x ) dx + Γ ρ ( x ) + dv ( x ) dx − γ = 0 (6)1 ρ ( x ) d ρ ( x ) dx − ρ ( x ) − v ( x ) + J ( x ) + µ = 0 (7)Due to the parity symmetry of the problem ψ ( x ) = ψ ( − x ) and taking into account the continuity of the solution(i.e. of the functions ρ ( x ) and v ( x ), and of their derivatives) we have the relations. ρ x (0) = ρ x ( ± π ) = 0 and v (0) = v ( ± π ) = 0 (8)Now we observe that the maximal density ρ max = ρ (0), is achieved at x = 0, i.e. in the point of the minimum of thepotential energy landscape − J ( x ). Thus, at x = 0 we obtain from (6) that ρ max = ( γ − v x (0)) / Γ, and thus v x (0) < v ( x ) > x ∈ ( − π,
0) and v ( x ) < x ∈ (0 , π ) (9)Similarly one can analyze the point x = ± π , located at the opposite side of the ring diameter. Denoting ρ ( ± π ) = ρ (cid:48) we have ρ (cid:48) = ( γ − v x ( ± π )) / Γ. Now v x ( ± π ) ≥ ρ (cid:48) ≤ ρ = (cid:112) γ/ Γ. This is what we observe in both panels ofFig. 1. In particular, in the left panel of Fig. 1 we observe decrease of ρ (cid:48) with increase of J /w .An interesting feature in the density distributions shown in both panels of Fig. 1 is the appearance of non-monotonicdependence of ρ ( x ) in the intervals x ∈ ( − π,
0) and x ∈ (0 , π ). In particular, the minimal density is achieved in twosymmetric points x = ± x m of the ring, different from x = ± π . As it follows from (6), at these points the absolutevalue of the velocity gradient v x ( ± x m ) is maximal. Since the solution separating monotonic and non-monotonicdensities (in the intervals ( − π,
0) and (0 , π )) is characterized by ρ xx ( ± π ) = 0 (at this solution the curvature of ρ ( x )changes its sign), it is not difficult to make an estimate of the parameters for that solution in the case of sufficientlynarrow coupling. Indeed in that case J ( π ) and v x ( π ) become negligibly small and one obtains from (6) and (7) µ ≈ ρ ( π ) ≈ γ/ Γ.In following subsections we present results of our studies of the dynamics of our system with increasing couplingstrength. We shall distinguish two separate regimes of very narrow ( w (cid:28)
1) and extended ( w (cid:38)
1) couplings. As weshall see in these limiting cases the dynamics can be of a quite different character.
IV. NARROW COUPLING DYNAMICS
In this paragraph we present results obtained in simulations with very narrow coupling function J ( x ), where weused Gaussian (2) of the width equal to w = 0 .
01. Our results are summarized in Figs. 2, 3 and 4. They are alsodescribed in the concise form below.Depending on the value of J we can distinguish several different patterns characteristic for the long (asymptotic)regular behavior. It can be stationary or periodic. In all simulations, the system reaches respective attractor stateafter a transient period, which depends on initial conditions and perturbation, and typically doesn’t exceed t trans ≈ J (cid:46)
1) asymptotically long time dynamics lead to stable anti-symmetric state shown with theblack curve in Fig. 2. This state exhibits non-trivial phase structure, which is formed as a result of non-Hermiticityof our model.For slightly larger coupling ( J (cid:39) .
5) when we propagate initial symmetric distribution for as long as it takes toreach steady state, we observe symmetry breaking (between channels) and our system approaches a stable asymmetric state, represented by blue ( | ψ | ) and red ( | ψ | ) curves in Fig. 2. We observe that local minima of the densities ρ , are achieved at x = π for the first ring and for intermediate points ± x m where | x m | < π for the second ring. Tomake a plot of the phase we excluded an x − independent component of the phase linearly growing in time. Note thathydrodynamic formulation allows us to discuss energy flow in the system. An interesting observation is that in thecenter of the coupling, i.e. in the vicinity x = 0, the energy flows in the two rings have opposite directions. Indeed FIG. 2. Absolute value (left) and the phase (right) of stationary solutions observed in case of narrow coupling w = 0 .
01. Blackcurve represents antisymmetric solution ψ ( x ) = − ψ ( x ) ( J = 1), blue and red curves show absolute value of both channelsin asymmetric state ( J = 1 . φ ( x = ± π ) = 0. Relative phase of the secondchannel in antisymmetric case is not shown, as φ ,antisym = φ ,antisym + π . Note that φ asym − φ asym equals zero at x = ± π . these flows are determined by v j and in the vicinity x = 0 in the first ring with higher field density the current isdirected outwards from the center ( xv > xv < J (cid:38)
2) our system tends to the limit cycle dynamics and we observe oscillationssymmetrical with respect to the center of the coupling region. It is an interesting class of solutions, and analogousdynamics was found in the broad coupling regime (see the next paragraph). It seems to be universal and the generalpicture of the dynamics in this regime does not depend on the width of the coupling. Due to this universality wedecided to discuss this class of solutions only for the broad coupling in the next paragraph.
V. BROAD COUPLING DYNAMICS
In the opposite limit, when the range of the coupling is comparable to the length of the ring (but not uniform yet)we also observe few classes of characteristic steady state dynamics and we classify them according to the (increasing)value of coupling strength J . We present results for w = 1 and show asymptotic (in time) dynamics. In this case,since the most interesting steady state exhibits rather complex oscillations, corresponding to the limit cycle, we chooseto present contour plots (see figure 3) and dynamical snapshots at various times in figures 4 and 6. Note that wealways start from perturbed symmetric state given in Eq. (4), but our predictions are not sensitive to this particularchoice.As it is expected, increase of the coupling strength leads to the increase of the oscillation frequency. An interestingobservation, however, is that the transient period, i.e. time necessary for establishing oscillations, also increases with J . The oscillatory dynamics is symmetric with respect to both rings, with alternating field characteristics (amplitudeand relative phase) exchanging after each half-period.In the case of small coupling strength ( J (cid:46) . J (cid:39) . ψ is shown (wavefunction in the second channel is just shifted by halfperiod), next we present the evolution of the norm in each channel ( N i = (cid:82) | ψ i | dx ) as function of time and finally,we present more details, the evolution of the norm of each of the wavefunctions and the average within the full periodof oscillations.These results are complemented by the full view of the wavefunction during its half-period oscillations in theasymptotic regime in figure 4. The first and the third rows show the modulus of the wavefunctions (blue and redcurves correspond to the first and second channels correspondingly), and second and fourth rows show the phase FIG. 3. Top row: Contour plots of absolute values and phases of the propagated wavefunction ψ in the stationary (asymptotic)regime for two different coupling stengths J and the fixed width w = 1. Phase oscillation term is eliminated so that φ ( x = π ) = 0. Central row: Norms in both channels ( N and N ) during propagation from initial perturbed symmetric homogeneousstates, defined in Eq. (4). Bottom row: Norms of both channels and average ( N + N ) / J = 4 and panels on the right show asymmetric oscillations for J = 5. structure. We can trace the dynamics in which one of the channels develops two symmetric dips (see Fig. 4 (a1)and (b1)), that develop slowly, reach bottom (c1) and eventually flatten to make exchange with its partner from thesecond channel (see frames (e1)-(h1)). In parallel we show the phase structure (frames (a2) though (h2)), and perhapsthe most prominent feature is very steep phase profile in frames (c2)-(d2). It happens exactly at the time when thetwo dip structures in one of the moduli (blue curve in (c1)) reach zero at the minimum. Notice that the solution stayssymmetric all the time. This will no longer be true when we go to the higher coupling regime. FIG. 4. Snapshots of wavefunction propagation, representing half period of symmetric oscillations with coupling strength J = 4, w = 1, γ = 3 and Γ = 1. All frames are presented in pairs, where top frames (a1-h1) show absolute values of bothwavefunctions (blue and red curves) and rescaled coupling potential (green curve), while bottom frames (a2-h2) show relativephases in both rings. Phases are plotted so that point x = π for blue curve is fixed at 0, to eliminate phase oscillations term.FIG. 5. Left panel: Frequency of the oscillations of the limit cycle solutions for three different values of γ . Symmetricoscillations, see Fig. 4 are marked as continuous lines and asymmetric (see Fig. 6) with dashed lines. The other parameters: w = 1, Γ = 1. Right panel: The norm in both channels at J = 4 .
5, where the blue line on left panel has discontinuity.We observe that after initial propagation (not shown) system develops into symmetrically oscillating state (shown in the timeinterval from t = 0 to t ≈
10) and after several (typically 3 or 4, depending on initial perturbation) periods of oscillation systemfinally evolves into asymmetrical oscillations.
We investigated the frequency of oscillations of the periodic solutions. Results are presented in figure 5. It is acollective plot containing the results not only for our central example of γ = 3, but also two different values of thisparameter (notice that we keep the value of Γ = 1). At small value of the coupling frequency grows rapidly, then asudden jump occurs and further growth is linear. This jump is associated with yet another bifurcation (symmetrybreaking), and as a matter of fact solutions marked with dashed lines no longer belong to the class described above, asit exhibits asymmetric oscillations. We will describe it in more details in the next paragraph. In our leading exampleof γ = 3 this phase transition occurs at coupling J ≈ .
5. In the transition region around this value of the couplingwe observe that dynamics leads first to the symmetric oscillations and after the transient period of several symmetricoscillations it follows to the asymmetric oscillations, which are asymptotically stable. This transient behaviour canbe identified in the right panel of figure 5, where we plot norm of the wavefunctions.For values of the coupling above J ≈ . γ = 3, see fig.5) phasestructure becomes asymmetric with respect to the center of the coupling function, see Eq. (2). This leads to periodic, FIG. 6. Snapshots of wavefunction propagation, representing full period of asymmetric oscillations with coupling strength J = 5 in the broad coupling regime, w = 1, γ = 3 and Γ = 1. All frames are presented in pairs; top frames (a1-l1) showabsolute values of both wavefunctions (blue and red curves) and rescaled coupling potential (green curve), bottom frames(a2-l2) show relative phases in both rings. Phase oscillations term was eliminated as in figure 4. limited in time, appearance of vortex in one of the channels, which then (on the regular basis) reappears in theopposite channel, with inverted topological charge. The whole dynamics has again a form of regular oscillations andwe show various phases of the evolution of the wavefunctions in figures 3 and 6. In Fig. 3 we show the time evolutionof the modulus of the wavefunction and its phase on the contour plot. Note that in this regime there is single dipthat appears once when a vortex is created (vortex in our case, as we mentioned above is equivalent to excitation)and again when vortex disappears, only to show up a bit later in the opposite ring. One can follow this process evencloser looking at figure 6. In panel (a1) we see the dip which is just about to reach zero at around x = −
1. The phasearound this point is very steep (panel (b)), and vortex is created (there is a phase across the ring equal to 2 π ). Thenthe wavefunction flattens, until second dip starts to develop at around x = 1. Once second dip reaches zero, vortexdisappears. Then the whole process repeats itself in the second ring in reversed order (panels (g)-(l)). This type ofbehavior was not observed in the case of narrow coupling, when we only have a symmetric structure. It seems thatthere is some distance between the edges of the coupling function necessary for this spacial symmetry to be broken. VI. ON VORTEX CREATION WITHOUT STIRRING.
Our system is non-Hermitian and as such it does not conserve topological charge; vortices can be created duringthe dynamics, even if the system is rotationally invariant. It happens due to the modulational instability, and as wementioned above in this 1D system vortex is equivalent to higher momentum excitations. Previously [10], for somevalues of the coupling constant we demonstrated that the system can, starting from perturbed symmetric state, arrive
FIG. 7. Snapshots of wavefunction propagation, representing relaxation from point of vortex generation (point b from Figure 6)after turning off the coupling, for γ = 3 and Γ = 1. All frames are presented in pairs; top frames (a1-d1) show absolute valuesof both wavefunctions (blue and red curves), bottom frames (a2-d2) show relative phases in both rings. Phase oscillations termwas eliminated as in figure 4. at the stationary states defined asΨ ( x, t ) = − Ψ ( x, t ) = (cid:114) γ Γ e i [ κx − ( γ Γ + κ − c ) t ] . (10)Here κ is an integer number expressing the value of the topological charge. This antisymmetric state is stable againstinitial perturbations. There is also symmetric solution, where Ψ ( x, t ) = Ψ ( x, t ), but they are usually unstable andwe never observed that in the asymptotic limit of our simulations.So far we could create vortices with equal topological charges in both channels. But here, when we define inhomo-geneous local coupling, additionally varying in time (it is enough to switch it on for some time and then switch it off)there is even more exciting possibility of creating vortex only in one ring, accompanied by constant solution in theother. The idea is based on the results obtained for the broad coupling function, described in Sec. V. Imagine thatwe start from perturbed symmetric state (as we did in all the cases described in this manuscript) and we allow thesystem to evolve towards the regime of asymmetric oscillations. When the system is already in this regime, we willabruptly turn off the coupling between rings. Dynamics corresponding to this scenario is illustrated in figure 7. Herewe present series of snapshots of both wavefunctions and their phases. In Fig. 6 in panels (a1) and (b1) the coupling isstill on, and one of the wavefunctions is at the stage when vortex is created and dip in the amplitude is fully developed(reaches the bottom). In the series of panels (columns) in figure 7 the coupling between rings is off and we observeslow, independent relaxation in each of them separately. In this circumstances one of the wavefunctions preserves itsvortex structure and tends to the solution with smooth modulus, defined in Eq. (10) with J = 0, and the oppositering develops wavefunction defined by Eq. (3). Turning off coupling in any other moment, while non-zero topologicalcharge is present in the system, leads to even faster relaxation into described state. In this scenario we showed howto selectively create vortex in one ring. In the remaining part of the manuscript we present experimental proposal,where such dynamics may be investigated, and propose similar coupled ring systems that we intend to investigate inthe near future. VII. EXPERIMENTAL PROPOSAL: PLASMONIC AND METAMATERIAL EFFECTS IN ARRAYS OFNANORINGS
The effects discussed above can be studied via electromagnetic response of arrays of metallic nanorings. Theseform photonic/plasmonic crystals with enhanced electromagnetic response [8]. Such structures can be made by avariety of techniques, including the electron beam lithography (EBL), as well as the Shadow Nanosphere Lithography(SNL) [16, 17]. This last technology provides an inexpensive route to complex periodic nanostructures, includingnanorings. Figures 8 (a) and (b) show scanning electron microscopy (SEM) images of arrays of metallic nanorings(circular and c-shaped, respectively) deposited on a substrate using SNL. Such arrays can be used as a basis forcorresponding arrays of coupled nanorings, which will have a very pronounced electromagnetic response dependingon the physics of the inter-ring coupling and non-linear, intra-ring losses. To obtain the coupled nanoring arrays, wemake two copies of an array (e.g. circular rings), as shown schematically in figure 8 (c), left panel. Each copy isan array, deposited on a transparent but lossy substrate (blue), and with the rings (orange) coated with a dielectricfilm (yellow). The two copies are sandwiched together, but depending on their relative offset, various configuration0
FIG. 8. (a) and (b) SEM images of the circular and c-shaped nanorings produced by SNL. (c) Schematic of the assembly ofthe coupled nanoring arrays (left panel), and the resulting examples of possible offset-dependent ring alignment configurations. overlaps can be achieved. For example, one can perfectly align the rings (not shown) and therefore realize a uniformcoupling ( J ( x ) = constant ). However, by shifting the two arrays relative to each other, one can achieve localizedcoupling as discussed above (configuration 1), or multiple localized coupling (configuration 2). In configuration 2 onehas a more complicated situation, with some rings coupled only to single rings in the other array, and some to multiplerings (in both arrays). In addition, even in the configuration 1, one can achieve a pair of localized coupling regionsby making horizontal shift adjustments. The local coupling can be also realized by tilting rings with respect to eachother, since the strength of the coupling is proportional to the distance. One can also use inhomogeneous filling ofthe inter-ring space.Experimentally realizable, more complex, configurations could be of interest, and we will study the correspondingmodels elsewhere.The physics described in previous sections relies on sufficiently large non-linear absorption losses. These canbe controlled by choosing highly non-linear materials for the substrates (or substrate coatings), such as organicsemiconductors, e.g. acetoacetanilide [18]. Another way to control the intra-ring nonlinear absorption would be toemploy lossy metals for the body of the ring, or use the c-shaped rings, as shown in figure 8 (b), with a nonlinearmaterial coating in the opening. Such processing is possible [19]. The inter-ring coupling can be easily controlled viathe dielectric coating (yellow colored layer in the left panel of figure 8 (c)). Also, if the substrates could be madeconducting, the bias across the pair could control the inter-ring coupling as well, relying on the non-linearity of thecurrent-voltage characteristics of these metal-insulator-metal (MIM) structures.1 VIII. CONCLUSIONS
In this work we continued our investigation of simple ring-shaped nonlinear waveguides in the presence of thelinear gain and nonlinear dissipation, adding local coupling between two channels. We have identified stationary anddynamic solutions both in case of narrow and broad coupling. These solutions represent different types of symmetrybreaking, corresponding to bifurcations of fixed points (stationary solutions) and of limiting cycles (symmetric andasymmetric oscillatory solutions). Dynamic solutions, connected with vortex generation, allow us both to controlchannel populations and generate vortex states in the system. We propose experimental realization of our model innanoplasmonic system.
ACKNOWLEDGMENTS
The work was supported by the Polish National Science Centre 2016/22/M/ST2/00261 (A. Ramaniuk and M.Trippenbach.) N.V.H. was supported by Vietnam National Foundation for Science and Technology Development(NAFOSTED) under grant number 103.01-2017.55. M.G. acknowledges the funding by the Guangdong Innovativeand Entrepreneurial Team Program titled ”Plasmonic Nanomaterials and Quantum Dots for Light Management inOptoelectronic Devices” (No.2016ZT06C517).The authors declare no conflict of interest. [1] Peng, B., ¨Ozdemir, S¸. K.; Lei, F.; Monifi, F.; Gianfreda, M.; Lu, G.; Long, L. G.; Fan, S.; Nori, F.; Bender, C. M.; YangL. Paritytime-symmetric whispering-gallery microcavities.
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