Bound Pulse Trains in Arrays of Coupled Spatially Extended Dynamical Systems

Abstract

We study the dynamics of an array of nearest-neighbor coupled spatially distributed systems each generating a periodic sequence of short pulses. We demonstrate that unlike a solitary system generating a train of equidistant pulses, an array of such systems can produce a sequence of clusters of closely packed pulses, with the distance between individual pulses depending on the coupling phase. This regime associated with the formation of locally coupled pulse trains bounded due to a balance of attraction and repulsion between them is different from the pulse bound states reported earlier in different laser, plasma, chemical, and biological systems. We propose a simplified analytical description of the observed phenomenon, which is in a good agreement with the results of direct numerical simulations of a model system describing an array of coupled mode-locked lasers.Nonlinear temporal pulses and spatial dissipative localized structures appear in various optical, plasma, hydrodynamic, chemical, and biological systems [1][2][3][4][5][6][7][8][9][10][11][12][13]. Being well-separated from each other these structures can interact locally via exponentially decaying tails and, as a result of this interaction, they can form bound states, known also as "dissipative soliton molecules" [14], characterized by fixed distances and phase differences between individual structures. Such bound states can emerge due to the oscillatory character of the interaction force which is related to the presence of oscillating tails. Another scenario occurs in the case of monotonic repulsive interaction when either the pulse tails decay monotonically, or a strong nonlocal repulsive interaction between the pulses is present. In this case the pulses tend to distribute equidistantly in time or space leading to periodic pulse trains [15][16][17][18] which, in contrast to closely packed bound states, exhibit large distances between the consequent pulses.In this Letter we show that even in the case when the pulses in an individual system exhibit strong repulsion, the formation of bound pulse trains can be achieved by arranging several systems in an array with nearestneighbor coupling. As a result, the pulses interact not only within one system, but also with those in the neighboring ones leading to a different balance of attraction and repulsion. More specifically, we demonstrate that this array can produce a periodic train of clusters consisting of two or more closely packed pulses with the possibility to change the interval between the pulses via the variation of coupling phase parameter. We show that the observed pulse train states coexist with the regimes which are amplitude synchronized and possess fixed phase shifts between the pulses emitted by neighboring array elements. In contrast to the pulse bound state regimes predicted and observed experimentally previously [14,[19][20][21][22][23][24][25][26][27][28][29][30][31][32], this regime cannot exist in a solitary pulse-generating system. We illustrate this general result by considering a part...

Discrete light bullets in passively mode-locked semiconductor lasers

“…Since the coupled Haus equations limit, [25][26][27] the observed dLBs can be interpreted as the fully localized analogues of the periodic train of pulse clusters consisting of two or more closely packed pulses in the array as found in Ref. 22.…”

“…These dLBs can be seen as a localized version of the periodic train of clusters consisting of closely packed localized pulses reported recently in Ref. 22. There, one could change the interval between Chaos ARTICLE scitation.org/journal/cha individual pulses via the variation of the coupling phase parameter, which is missing in coupled Haus model ( 1)-( 3) as we assumed the coupling to be evanescent.…”

“…The latter is induced by the gain dynamics. 25,39,40 Recently, it was shown 22 that even in the case when the pulses in an individual PML system exhibit strong repulsion, the formation of bound pulse trains can be achieved between the elements of an array of mode-locked lasers coupled via evanescent fields. This way the pulses interact not only within one system but also with those in the neighboring nodes, leading to a different balance between attraction and repulsion.…”

“…The interaction properties of short pulse trains in an array of nearest-neighbor coupled passively mode-locked lasers were recently addressed in Ref. 22. It was shown that this array can produce a periodic train of clusters consisting of two or more closely packed pulses with the possibility to change the interval between them via the variation of the coupling parameter.…”

Hopf bifurcation of relative periodic solutions in a system of five passively mode-locked lasers

“…This delay differential system has been extensively applied to analyze instabilities [38,41] and hysteresis [21,43] in mode-locked lasers, optically injected lasers [2,45], hybrid mode locking [3], noise reduction [25], resonance to delayed feedback [1], and Fourier domain mode locking [46]. The same system has been used to model four identical lasers coupled in D 4 -symmetric fashion [44]. An increasing interest in small systems of symmetrically coupled lasers and large synchronized laser arrays motivates the analysis of other symmetric configurations of coupled lasers [26,33,42,49].…”