Emulating the local Kuramoto model with an injection-locked photonic crystal laser array

Abstract

The Kuramoto model is a mathematical model for describing the collective synchronization phenomena of coupled oscillators. We theoretically demonstrate that an array of coupled photonic crystal lasers emulates the Kuramoto model with non-delayed nearest-neighbor coupling (the local Kuramoto model). Our novel strategy employs indirect coupling between lasers via additional cold cavities. By installing cold cavities between laser cavities, we avoid the strong coupling of lasers and realize ideal mutual injection-locking with effective non-delayed dissipative coupling. First, after discussing the limit cycle interpretation of laser oscillation, we demonstrate the synchronization of two indirectly coupled lasers by numerically simulating coupled-mode equations. Second, by performing a phase reduction analysis, we show that laser dynamics in the proposed device can be mapped to the local Kuramoto model. Finally, we briefly demonstrate that a chain of indirectly coupled photonic crystal lasers actually emulates the one-dimensional local Kuramoto chain. We also argue that our proposed structure, which consists of periodically aligned cold cavities and laser cavities, will best be realized by using state-of-the-art buried multiple quantum well photonic crystals.

Synchronization of Two Indirectly Coupled Singly Resonant Optical Parametric Oscillators

“…The Kuramoto-like model Eq. ( 20) is a case of the coupled phase model [20,23,24] in which phase differences between oscillators are coupled through a harmonic odd function. The coupling term G c sin(θ 1 − θ 2 ) contribution in the first relation of Eq.…”

“…Eq. ( 20) have solved analytically to give the critical coupling constant as [20,23,24]. Henceforward we have G c = (4k 2 c )/Γ = ∆/2.…”

“…As a result, we couple two SROs of slightly detuned frequencies, indirectly by incorporating a cold cavity between them. This coupling technique avoids any time delay and strong coupling effects such as mode competition and unstable perturbation of the SROs [20]. We demonstrate that the system is modeled by two Stuart-Landau oscillators coupled to each other through the cold cavity (treated as a harmonic oscillator) and synchronize their frequencies to a common value at a coupling strength threshold.…”

A Phase Model of the Bio-Inspired NbOx Local Active Memristor under Weak Coupling Conditions

“…In the past 40 years, the research has shifted to using the oscillator phase model to study the dynamic characteristics of the network, which will reduce the number of differential equations to , saving a lot of simulation time. Among them, the most representative is the Kuramoto network model [ 41 , 42 , 43 , 44 , 45 , 46 ]. The advantage of this model is that the equation is simple, and it is easy to predict the phase evolution of the system.…”

Effects of correlated and uncorrelated quenched disorder on nearest-neighbor coupled lasers

“…Introduction -Many different physical systems, both quantum and classical, are well described by many-body interacting oscillators: Transverse field spin models describe spins effectively rotating around the local magnetic field, which can synchronize to reach finite magnetization even in the presence of a spatially varying magnetic field [1][2][3][4][5]. The synchronization of classical phase oscillators has been theoretically studied for decades through the Kuramoto model [6], and is manifested in many different systems such as arrays of Josephson junctions [7,8], phase locking of coupled laser arrays [9,10], and even human networks [11]. In all of these, disorder in the system is one of the main obstacles to synchronization, acting against the interaction between the individual members of the ensemble.…”