Discrete Light Bullets in Passively Mode-Locked Semiconductor Lasers
Thomas G. Seidel, Auro M. Perego, Julien Javaloyes, Svetlana V. Gurevich
DDiscrete Light Bullets in Passively Mode-Locked Semiconductor Lasers
Thomas G. Seidel, Auro M. Perego, Julien Javaloyes, and Svetlana V. Gurevich
1, 4, 3, a) Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Str. 9, D-48149 Münster,Germany Aston Institute of Photonic Technologies, Aston University, Birmingham B4 7ET, United Kingdom Departament de Física, Universitat de les Illes Balears, & Institute of Applied Computing and Community Code (IAC-3),C/ Valldemossa km 7.5, 07122 Mallorca, Spain Center for Nonlinear Science (CeNoS), University of Münster, Corrensstrasse 2, D-48149 Münster,Germany
In this paper, we analyze the formation and dynamical properties of discrete light bullets (dLBs) in an array of passivelymode-locked lasers coupled via evanescent fields in a ring geometry. Using a generic model based upon a system ofnearest-neighbor coupled Haus master equations we show numerically the existence of dLBs for different couplingstrengths. In order to reduce the complexity of the analysis, we approximate the full problem by a reduced set of discreteequations governing the dynamics of the transverse profile of the dLBs. This effective theory allows us to perform adetailed bifurcation analysis via path-continuation methods. In particular, we show the existence of multistable branchesof discrete localized states (dLSs), corresponding to different number of active elements in the array. These branchesare either independent of each other or are organized into a snaking bifurcation diagram where the width of the dLSgrows via a process of successive increase and decrease of the gain. Mechanisms are revealed by which the snakingbranches can be created and destroyed as a second parameter, e.g., the linewidth enhancement factor or the couplingstrength are varied. For increasing couplings, the existence of moving bright and dark dLSs is also demonstrated.
I. INTRODUCTION
Discrete localized states in nonlinear lattices appear inmany areas of research such as biological molecular chainsor energy transfer in protein α -helices , conducting polymerchains , solid-state systems , Bose-Einstein condensate oroptical wave-guides just to mention a few. In nonlinearoptical systems these states are often referred to as discretesolitons (dSs) and they have been a subject of intense inves-tigation in recent years both theoretically and experimentall,see e.g., Ref. for a review. In particular, one- and two-dimensional dSs were predicted theoretically and observedexperimentally in arrays of weakly coupled nonlinear cavitieswith Kerr, saturable cubic, and quadratic nonlinearities .The properties of dSs usually differ from those of continu-ous systems. In particular, the lack of translational symmetryin discrete systems usually causes the trapping of dSs by theso-called Peierls-Nabarro potential so that they remain atrest unless the coupling between the array elements exceedssome critical value. In the limit of strong coupling, the mo-bility properties of dSs were discussed in Ref. , whereas inRef. the uniformly moving as well as chaotic oscillatory dSswere observed in an array of coupled Kerr-nonlinear cavities.The chimera-like localized states consisting of spatiotempo-ral chaos embedded in a homogeneous background were re-cently studied in a discrete model for an array of coupled-waveguide resonators subject to optical injection. These inter-mittent spatiotemporal chaotic states are shown to coexist withstationary dSs corresponding to different numbers of activewaveguides. The multistability and snaking behaviour of dSswere also reported in the model for optical cavities with focus-ing saturable nonlinearity . The interaction properties of a) Electronic mail: [email protected] short pulse trains in an array of nearest-neighbor coupled pas-sively mode-locked lasers were recently addressed in Ref. .It was shown that this array can produce a periodic train ofclusters consisting of two or more closely packed pulses withthe possibility to change the interval between them via thevariation of the coupling parameter.Passive mode locking (PML) is a well-known method forachieving short optical pulses . For proper parameters, thecombination of a laser amplifier providing gain and a non-linear loss element, usually a saturable absorber, leads tothe emission of temporal pulses much shorter than the cavityround-trip. However, if operated in the so-called long-cavityregime, the PML pulses become individually addressable tem-poral localized states coexisting with the off solution . Inthis regime, the round-trip time is much longer than the semi-conductor gain recovery time, which is the slowest variable.This temporal confinement regime was found to be compati-ble with an additional spatial localization mechanism, leadingto the formation of stable three-dimensional light bullets, i.e.localized pulses of light that are simultaneously confined inthe transverse and propagation directions . Light bulletshave attracted a lot of attention in the last two decades; In par-ticular they should be addressable, i.e., one can envision thatthey would circulate independently within an optical cavity aselementary bits of information.In this paper we study the formation and dynamical proper-ties of discrete light bullets (dLBs) in an array of PML laserscoupled via evanescent fields in a ring, see Fig. 1. Here theblue and red parts correspond to the gain and absorber sec-tions of the individual PML laser whereas the arrows indi-cate the next neighbor coupling with the coupling strength c . We perform the analysis in this paper in two steps: First,using an ensemble of nearest-neighbor coupled Haus masterequations we show the existence of dLBs for a wide rangeof coupling strengths. To understand the localization mecha-nism in details, we approximate the solution of the full sys- a r X i v : . [ n li n . PS ] J a n FIG. 1. Schematic representation of a ring array of coupled mode-locked lasers. Blue and red parts of each PML laser correspond tothe gain and absorber sections, respectively. The arrows indicate thecoupling via evanescent fields with the coupling strength c . tem by the product of a slowly evolving discrete transverseprofile and of a short temporal pulse propagating inside thecavity. This allows us to obtain a reduced discrete model gov-erning the dynamics of the transverse profile. This effectivemodel termed the discrete Rosanov equation allows for a de-tailed multi-parameter bifurcation study. It also enables us toidentify the different mechanisms of instabilities of transversedLBs. II. MODEL
We describe the PML laser array in Fig. 1 using nearest-neighbor coupled Haus master equations for the evolu-tion of the field profile E j = E j ( z , σ ) , j = , . . . N , over theslow time scale σ that corresponds to the number of round-trips in the cavity ∂ σ E j = (cid:26) √ κ (cid:20) + − i α G j − − i β Q j (cid:21) − + γ ∂ z (cid:27) E j + i c (cid:0) E j − + E j + (cid:1) , (1)whereas z is a fast time-like variable representing the evolu-tion of the field within the round-trip. The carrier dynamicsfor G j = G j ( z ) and Q j = Q j ( z ) reads ∂ z G j = Γ G − G j (cid:16) Γ + (cid:12)(cid:12) E j (cid:12)(cid:12) (cid:17) , (2) ∂ z Q j = Q − Q j (cid:16) + s (cid:12)(cid:12) E j (cid:12)(cid:12) (cid:17) . (3)Here, κ is the fraction of the power remaining in the cavityafter each round-trip, γ is the bandwidth of the spectral filter, α and β are the linewidth enhancement factors of the gainand absorber sections, respectively and c denotes the nearestneighbor coupling constant. Further, G is the pumping rate, Γ is the gain recovery rate, Q is the value of the unsaturatedlosses, and s the ratio of the saturation energy of the gain andof the saturable absorber sections.For proper parameters, Eqs. (1)-(3) sustain the existence ofstable dLBs as depicted in Fig. 2. Here, the intensity profile j a) 101520 b)0.5 0.0 0.5z101520 j c) 0.5 0.0 0.5z101520 d)
012 012012 012
FIG. 2. Exemplary solutions of Eqs. (1)-(3) showing the intensityprofile of a stable dLB existing in an array of N =
30 elementsfor different couplings strength c : a) c=0.05, b) c=0.10, c) c=0.15,d) c=0.20. Other parameters are: ( γ , κ , α , β , G , Γ , Q , s , L z , N z ) =( , . , . , . , . , . , . , , , ) ,where L z is the lengthof the cavity and N z are the number of grid points. If not otherwisestated, all the data represented in the figures are dimensionless. of a stable dLB in the array of N =
30 elements is shown forfour different values of the coupling strength c .To understand the formation mechanism of dLBs in detail,we start the analysis with the dynamics of the transverse pro-file of a dLB, that in the following we refer to as a discretelocalized state (dLS). To this aim we follow to derivean approximate model governing the shape of the transverseprofile. We assume that each field E j = E j ( z , σ ) is representedas a product of a short normalized temporal localized pulse p ( z ) upon which the dLB is built and a slowly evolving ampli-tude of the transverse field A j ( σ ) , i.e., E j ( z , σ ) = p ( z ) A j ( σ ) .Note that this is a strong approximation as we assume the tem-poral pulse p ( z ) to be identical in width and timing for all thearray elements. Separating the temporal evolution into the fastand slow parts corresponding to the pulse emission and thesubsequent gain recovery allows us to find the discrete equa-tion governing the dynamics of A j = A j ( σ ) , j = , . . . N , as ∂ σ A j = i c ( A j + − A j + A j − ) + F ( | A j | ) A j . (4)Defining h ( p j ) = ( − e − p j ) / p j with p j = | A j | the nonlinearfunction F reads F ( p j ) = √ κ (cid:20) + − i α G h ( p j ) − − i β Q h ( sp j ) (cid:21) . (5)Note that the continuous counterpart of the discrete equa-tion (4), obtained taking the limit c → ∞ with a nonlinearfunction F corresponding to a static saturated nonlinearity,i.e. h = / ( + | A | ) is a so-called Rosanov equation that is known in the context of static transverse autosolitonsin bistable interferometer. III. RESULTS
A single solution of Eqs. (4, 5) can be found in the form A j ( σ ) = a j e − i ωσ , (6) P b)c)d)a) H H H | A | d) 0.00.51.0 | A | c)20 25 30 35j 0.00.51.0 | A | b)SNH H FIG. 3. Bifurcation diagram of Eqs. (4, 5) showing one- (red),two- (green) and three-sites (blue) dLSs. The parameters used are ( Q , α , β , c , s , κ ) = ( . , . , . , . , , . ) and N =
51 array el-ements. Thick lines describe stable solution branches, while thinlines stand for unstable ones. Green squares denote Andronov-Hopf(AH) bifurcation points, whereas black dots stand for Saddle-Node(SN) bifurcations. The inset gives a zoomed view on the area aroundthe folds. The panels b), c) and d) show exemplary intensity profilesmarked in a). where a j is a complex amplitude of each array element and ω represents the carrier frequency of the solution. SubstitutingEq. (6) into Eqs. (4, 5) we are left searching for unknowns a j and ω of the following algebraic equations set i c ( a j + − a j + a j − ) + i ω a j + F ( | a j | ) a j = . (7)We followed the solutions of Eqs. (7) in parameter space, byusing pseudo-arclength continuation within the AUTO-07Pframework . Here, the spectral parameter ω becomes anadditional free parameter that is automatically adapted dur-ing the continuation. We define G th = √ κ − + Q as thethreshold gain value above which the off solution a j = j = , . . . N becomes unstable. The primary continuationparameter could be e.g., the gain normalized to the threshold g = G / G th , the linewidth enhancement factor α or the cou-pling strength c . Multistability of dLSs
One can start at, e.g., a numericallygiven solution, continue it in parameter space, and obtain adLS solution branch. The result for an array of N =
51 ele-ments is presented in Fig. 3, where in the panel a) the power P = ∑ i | a i | of three different dLSs is depicted as a functionof the normalized gain g for the fixed small coupling strength c . One can see that dLSs only occur in discrete widths cor-responding to different numbers of lasing lasers in the array,see Fig. 3(b,d), where the exemplary profiles of one-, two- andthree-sites dLSs are shown. Further, the system is multistable,and we find separate branches for solution profiles containingdifferent number of lasing nodes. Each of the branches bifur-cates from the threshold g =
1, possesses a fold at some fixedvalue (marked as a black circle in Fig. 3), and goes to higherintensities. The stability properties of different dLSs branchesare similar: The one-site-dLS (red line) is stable between thesaddle-node bifurcation point (SN) and the Andronov-Hopf(AH) bifurcation point H (marked as a green square) close tothe threshold. However, for two- and three-sites dLSs, otherAH bifurcations occur around the SN point (e.g., H , and H , j0.30.00.3 R e () , I m () a) jc) je)20 25 30j0.30.00.3 R e () , I m () b) 20 25 30jd) 20 25 30jf) FIG. 4. Real and imaginary parts of the critical eigenfunctions ψ ofthe double Hopf bifurcations H (a,b), H (c,d) and H (e,f) ( cf. Fig.3). The red points correspond to the Re ( ψ ) , whereas the blue pointsto Im ( ψ ) , respectively. see inset in Fig. 3(a) that can limit the stability of the dLSs forlow values of the gain. For the higher bias and intensities, thestability is again limited by the AH bifurcations (cf. H and H points) close to g = H − H points actually corresponds to a double AHbifurcation. Here, the imaginary parts of the two eigenvaluesare the same which means that both eigenmodes exhibit thesame frequency. The real and imaginary parts of the corre-sponding eigenmodes for the H − H bifurcation points areshown in Fig. 4. One can see that there is always one even(upper row) and one odd (lower row) eigenmode. Snaking bifurcation of dLSs
Interestingly, the bifurcationstructure of the branches becomes different if the linewidthenhancement factor α is varied. In particular, reducing α re-veals a snaking structure in the bifurcation diagram (see Fig.5). Here, the stable parts of the branches for odd, i.e., one-, three-, five-, etc. sites dLS (thick lines) are connected viaSN bifurcations by unstable connections (thin lines). The sta-bility on the left side is limited by a SN bifuraction for theone-site (SN , black dot) or AH bifurcations (H , H , greensquare, for three or more lasing dLSs). For the increasingvalue of the control parameter g , the dLSs become unstable inSN bifurcations (see e.g., SN and SN points). Furthermore,the branches with an odd dLSs are not connected to the oneswith an even dLSs, see Fig. 5(b), because with increasing g the neighboring nodes of the array are excited symmetricallysuch that switching from an even number of lasing lasers toan odd number is not possible. Note that the snaking bifurca-tion of dLSs was also reported in , where a discrete modelfor optical cavities with focusing saturable nonlinearity wasstudied. Bifurcation analysis of the snaking
Now we want to un-derstand the transition between the independent branches fordifferent dLSs as presented in Fig. 3 to the snaking structureas shown in Fig. 5. To this aim, we analyze in details thebehavior of the branches of dLSs corresponding to differentvalues of α . For simplicity here we focus on the transition be-tween the solution profiles with one- and three-sites dLSs and P SN H H SN SN a) 0.6 0.7 0.8g0123 b) FIG. 5. a) Bifurcation diagram in the ( g , P ) plane for α = . g = .
67 for the one-, three- and five-site dLSs, respectively.b) Bifurcation diagram for α = .
6, where the branches for bothodd (red) and even (blue) number of lasing nodes is shown (seethe Supplementary Material for the video showing the profile evo-lution along the branches). Other parameters are: ( Q , β , c , s , κ ) =( . , . , . , , . ) and N=51. Fig. 6 shows the resulting bifurcation diagrams in the ( g , P ) plane obtained for different α . Figure 6(a,b) indicates that forsmall values of α the branches for one-site (red) and three-sites (blue) dLSs are not connected to each other. However,the stability on the branches here is different to those shownin Fig. 3: While a one-site dLS is stable between a SN andan AH bifurcation points, for a three-sites dLS the situationis different. In particular, the three-sites dLS gains the sta-bility in a AH bifurcation after the SN point, and looses thestability in a pitchfork bifurcation (marked as a magenta tri-angle in Fig. 6(a-c). At the pitchfork bifurcation point twobranches (red) corresponding to left- and right-site- asymmet-rical dLSs emerge (see Fig. 7). Because both dLSs shownin Fig. 7(b,c) correspond to the same power P , the branchescoincide for the norm chosen. One can see that the range ofstability for these solutions is very small (thick red line) asthe branch looses its stability quickly in an AH bifurcationmarked with H in Fig. 7(a). For increasing values of α thebifurcation structure becomes more complicated and two ad-ditional SN bifurcations appear on both red and blue branchesas shown in Fig. 6(c). These bifurcation pairs separate fur-ther from each other with α ,cf. Fig. 6(d), until two of the SNbifurcations, corresponding to the rightmost fold of the blue,three-sites dLS branch and the leftmost of the red one, cor-responding to the one-site dLS solutions, merge. This leadsto the reconnection of the lower part of the one-site dLS andthe upper part of the three-sites dLS branches and a snakingstructure emerges, see the red branch in Fig. 6(e). As thistakes place, the residual (upper part of the red branch) con-nects to the lower part of the blue one and an unstable branch,shown in green in Fig. 6(e) appears. However, at this point thepart of the branch corresponding to three-sites dLS is unstableand can be stabilized by an AH bifurcation if α is increased,see Fig. 6(f). Note that further branches corresponding tolarger odd number of lasing elements are created in a simi-lar way. However, further increases in α lead to the break-up of the snaking behavior via the same mechanism the branchwas created. In particular, for increasing α a part of unstableresidual (green) branch goes close to the main snaking branchand hits it at some fixed α . This leads to connection of thered and green branches so that two folds appear as shownin Fig. 6(g). Note that during this reconnection, the appear-ing branch of the one-site dLS (red) becomes separated fromthe still snaking branch of three- and five-sites dLSs (blue).For even larger α the SN bifurcations annihilate and disap-pear (Fig. 6(h) and finally separate, independent branches, forone-site and three-sites dLSs are formed Fig. 6(i). Note thatthe five-sites dLSs branch separates from the three-sites dLSsbranch via the same scenario. For more details see the Sup-plementary Material section where the video showing the cre-ation and destruction of the snaking structure is shown. Influence of the coupling strength
Next, we are interestedin the influence of the coupling strength c on the dynam-ics of dLSs. To this aim we reconstructed the branches ofthe dLSs for different values of c and for the fixed values of ( α , β ) = ( . , . ) . In this case the branches of all dLSs areindependent, see Fig. 3. We start with the branch of the one-site dLS and small coupling strength and look to its evolutionin c . The results are presented in Fig. 8(a), where the branchesfor fourteen different values of c are collected in a three-dimensional ( c , g , P ) bifurcation diagram. One can observethat with increasing coupling strength c the branch reveals asimilar transition to a snaking as in the case of changing α ,cf. Fig. 6. In particular, one can see, inspecting Fig. 8(a) thatwith increasing c the stability region of the one-site dLS de-creases as one of the AH bifurcation points (green square),limiting the stability region is moving towards smaller val-ues of g with increasing c . Then, similar to the case of vary-ing α , two SN points appear on the branch and the AH pointdisappears. One of the appearing folds is then connected tothe fold of the three-site dLS branch and the snaking bifurca-tion structure emerges, see Fig. 8(b-d), where three branchesfor three values of c illustrating this transition are presented.Here, stable branches for solution profiles corresponding toone-, three-, five-, etc. sites dLSs are interconnected by un-stable branches. One can see that with increasing of c newAH bifurcation points appear leading to the decrease of thestability region of small-sites dLSs. This is an expected resultas with increasing c the systems tends to the continuous limitwhere these dLSs do not exist. Moving dLSs
Finally, we consider the case of even largercoupling strengths. As was mentioned in the Introduction sec-tion, the discreteness breaks the translational symmetry thatusually causes the trapping of dLSs so that they remain at restunless e.g., the coupling strength between the nodes exceedssome critical value. An example of a drifting dLS that we re-fer to as a bright dLS in the following, obtained by a directnumerical integration of Eqs. (4)-(5) is shown in Fig. 9(a,b).A space-time plot is presented in Fig. 9(a) where the time evo-lution of the position of each element j in the array is shown,whereas the color corresponds to the intensity. One can seethat after a dLS is formed in the array it becomes unstable,accelerates slowly and drifts to the right. After some time theacceleration phase ends and the dLS moves with a constant P a) b) c)0.00.51.01.5 P d) e) f)0.6 0.7 0.8 0.9 1.0g0.00.51.01.5 P g) 0.6 0.7 0.8 0.9 1.0gh) 0.6 0.7 0.8 0.9 1.0gi) FIG. 6. Bifurcation diagrams in ( g , P ) plane for different values of α = ( , . , . , . , . , . , . , . , . ) for the panels a)-i),respectively. The creation/destruction cycle of the snaking structure is shown. Red and blue branches correspond to one- and three-sitesdLS solutions. Thick and thin lines stand for stable and unstable branch parts, respectively. The green branch in e) and f) appears by thereconnection of the one- and three-sites dLS branches and is unstable. See the Supplementary Material section for the visualization of thetransition between independent branches and snaking behavior. Other parameters are ( Q , β , c , s , κ , N ) = ( . , . , . , , . , ) . P a) | A | b) 25 35j | A | c)stableH FIG. 7. a) Branch of a three-sites dLS in ( g , P ) plane for α = . H areplotted in b) and c). Both branches have the same norm P and coin-cide in the bifurcation diagram in a) (see the Supplementary Materialfor a video showing the evolution of the solution profiles along thebranch). Other parameters as in Fig. 6. velocity. An exemplary solution profile is plotted in Fig. 9(b).Note that for the coupling strength used, the dLSs correspond-ing to smaller number of nodes are unstable. One can demon-strate that the drift velocity of the dLS is determined by thecoupling c and the inset in Fig. 9(b) clearly shows that thevelocity increases linearly with c .Interestingly, besides bright dLSs as shown in Fig. 9,the system exhibits also dark , or grey , moving dLSs, seeFig. 9(c,d), which are formed for the same value of c but g0.5 0.6 0.7 0.8 0.9 1.0 1.1 c P b) P c)0.56 0.60 0.64 g P d) FIG. 8. a) Evolution of the branch of the one-site dLS in the ( g , P ) plane with the coupling strength c . The panels b),c) and d) show thebranches close to the snaking transition which are marked blue in a)for c = ( . , . , . ) , respectively. Similarly to Fig. 6 onecan observe snaking branches which occur above the critical cou-pling c (cid:39) . ( Q , α , β , s , κ = . , . , . , , . ) and N=51. slightly larger gain values. These dLSs are characterized byone non-lasing node, whereas all other nodes have non-zerointensity. Note that dark moving dLSs were also found in ar-rays of coupled quadratic nonlinear resonators driven by aninclined holding beam or in coupled in optical cavities withfocusing saturable nonlinearity . The time evolution of thedark moving dLS is shown in Fig. 9(c). One can see that inthe initial phase of the time evolution - because of the strong t i m e a)
35 40 45 50 j | A | b)0 10 20 30 40 50j0246810 t i m e c) j | A | d) v C O M FIG. 9. Time evolution of a bright a) and dark c) moving dLS cal-culated by direct numerical integration of (4),(5) for c = . G = .
33 and c) G = .
36. Panels b) and d) represent the exemplaryprofiles for both cases at the last time step of the numerical simula-tion. The inset in b) displays the center of mass velocity v COM as afunction of the coupling c , showing the linear dependence. See theSupplementary Material for more details of the time evolution. Otherparameters are ( Q , α , β , s , κ , N ) = ( . , . , . , , . , ) . coupling c and large gain value - more and more laser nodesbecome unstable until all but one have a non-zero intensity,cf. Fig. 9(d). This state remains stationary until the dark dLSspontaneously starts to move with a constant velocity. Themotion is facilitated by switching between odd and even num-ber of nodes with zero intensity (see the Supplementary Mate-rial section for a video of the moving dark dLS). Furthermore,one sees that the intensity profile is asymmetric and exhibitsoscillatory tail on the right side, cf. Fig. 9(d). This can poten-tially lead to the formation of bound states between two darkdLSs in arrays with larger number of elements. Notice thatgenerally the formation mechanisms of the bright and darkmoving dLSs are different: The linear stability analysis re-veals that for bright dLSs several AH bifurcations trigger thetranslation while for dark dLS real eigenvalues appear unsta-ble in the spectrum making motion possible. Multistability of dLB in the discrete Haus model
The re-sults of the discrete Rosanov model (4),(5) indicate the mul-tistability between different dLSs corresponding to the trans-verse profile of a dLB. To prove the possible co-existence ofdifferent dLBs we go back to the original coupled Haus equa-tions (1)-(3) and conduct direct numerical simulations for thecase of small coupling c , cf. Fig. 3. We show the resultingbranches of one- three- and five-sites dLBs in Fig. 10. Onecan clearly see that also in the coupled Haus model (1)-(3)the multistability occurs and dLBs corresponding to differentnumber of odd lasing lasers can form, see Fig. 10(b,d). Notethat this scenario also occurs for an even number of nodes.However, and at variance with the results of Fig. 2, thesmall values of the coupling using in Fig. 10 make it so that theindividual lasing nodes are separated by a certain offset alongthe z-axis. This can correspond to the effective repulsive in- P a)b)c)d) 101520 j b)101520 j c)0.7 0.0 0.7z101520 j d) 012012012 FIG. 10. a) Branches of one- (red), three- (green) andfive-sites (blue) dLB found by the direct numerical integra-tion of the coupled Haus Eqs. (1)-(3) with N=30 array el-ements. b),c),d): Exemplary profiles of dLBs at g=0.672(black cross in a)). Parameters are ( γ , κ , α , β , T , Γ , Q , s , c , N z ) =( , . , . , . , , . , . , , . , ) . teraction along the fast time axis between individual elements.The latter is induced by the gain dynamics . Recently itwas shown that even in the case when the pulses in an indi-vidual PML system exhibit strong repulsion, the formation ofbound pulse trains can be achieved between the elements ofan array of mode-locked lasers coupled via evanescent fields.This way the pulses interact not only within one system butalso with those in the neighboring nodes, leading to a differentbalance between attraction and repulsion. Since the coupledHaus Eqs. (1)-(3) can be seen as an effective master equationfor the delay differential equation model used in in the longdelay limit , the observed dLBs can be interpreted as thefully localized analogues of the periodic train of pulse clustersconsisting of two or more closely packed pulses in the arrayas found in . IV. CONCLUSION
We studied the formation and the dynamical properties ofdLBs in an array of PML lasers coupled via evanescent fieldsin a ring geometry. Using nearest-neighbor coupled Hausmaster equations, we demonstrated the existence of dLBs forthe wide range of coupling strength. To understand the for-mation mechanisms in details, the dynamics of dLBs was ap-proximated by a simplified discrete model governing the dy-namics of the transverse profile of the dLB, that we calleda dLS. This effective discrete Rosanov equation has allowedfor a detailed bifurcation analysis. In particular, for smallcoupling strengths, our results revealed the multistability be-tween branches corresponding to different kind of dLSs witha varying number of active elements. These branches beingindependent from each other for one parameter set can be-come connected in the snaking bifurcation structure if one ad-ditional parameter, e.g. the linewidth enhancement factor isvaried. The reconnection procedure is very involved and sev-eral intricate discrete states, including stationary unsymmet-rical dLSs were disclosed. Furthermore, it was demonstratedthat the snaking behavior between different dLSs branches canalso be achieved by changing the coupling strength. More-over, further increasing of the coupling strength was shown tolead to the formation of the moving bright and dark dLSs. Fi-nally, the multistability of several dLBs was demonstrated inthe original coupled Haus model. In contrast to the transversemultistable dynamics, where all the temporal pulses were sup-posed to synchronized for all the array elements, the elementsof the resulting dLBs are not in phase because of the repul-sive underlying gain dynamics. These dLBs can be seen asa localized version of the periodic train of clusters consist-ing of closely packed localized pulses reported recently inRef. . There, one could change the interval between individ-ual pulses via the variation of the coupling phase parameter,which is missing in the coupled Haus model (1)-(3) as we as-sumed the coupling to be evanescent. This interesting issue isout of the scope of this paper and will be discussed elsewhere. ACKNOWLEDGMENTS
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