Abstract
We report here for the first time (to our knowledge), a new and universal mechanism by which a two-element laser array is locked to external optical injection and admits stably injection-locked states within a nontrivial trapezoidal region. The rate equations for the system are studied both analytically and numerically. We derive a simple mathematical expression for the locking conditions, which reveals that two parallel saddle-node bifurcation branches, not reported for conventional single lasers subject to optical injection, delimit the injection locking range and its width. Important parameters are the linewidth enhancement factor, the laser separation, and the frequency offset between the two laterally-coupled lasers; the influence of these parameters on locking conditions is explored comprehensively. Our analytic approximations are validated numerically by using a path continuation technique as well as direct numerical integration of the rate equations. More importantly, our results are not restricted by waveguiding structures and uncover a generic locking behavior in the lateral arrays in the presence of injection.
Highlights
Coupled nonlinear oscillators/systems have received considerable attention due to their rich dynamics including stable continuous wave operation, oscillatory states and chaos, as well as collective dynamical behavior, e.g. synchronization of periodic and even chaotic oscillations; see, e.g.1–3, and references therein
The stable locking conditions are well understood: the stable region is bounded by a saddle-node (SN) and a Hopf bifurcation line, and outside of this region the system exhibits a wealth of dynamical behaviour which can include pulsations, chaos, periodic oscillations and multi-stability[34,40]
Following[12] with a straightforward modification to account for the optical injection we restrict ourselves to the case where a solitary laser supports a single transverse mode and extend the basic coupled-mode equations to include an externally injected field kinjEinje−iΔωt, where Δω = ωinj − ω, with ωinj as the injected angular frequency and ω as the free-running angular frequency of the total electric field of the system in the absence of injection, Einj as the injected field, and kinj as a coupling rate for the injected signal
Summary
Coupled nonlinear oscillators/systems have received considerable attention due to their rich dynamics including stable continuous wave (cw) operation, oscillatory states and chaos, as well as collective dynamical behavior, e.g. synchronization of periodic and even chaotic oscillations; see, e.g.1–3, and references therein. Researchers are interested in this basic array because practically it can be readily fabricated on a single chip, and it can be accurately modelled by a set of ordinary differential equations, usually called the coupled laser model This basic set of simple rate equations makes extending the investigation to larger 1-D or even 2-dimensional arrays more tractable and, when coupled with a detailed bifurcation analysis, regions of stability and dynamics and their nature can be revealed. Aside from its technological applications, from the physics viewpoint the investigation of such a configuration has fundamental interest and can clarify the complex dynamical behaviour as mentioned above (see[9] and references therein) These devices turn out to be sources for uncovering novel physical phenomena such as gain tuning and parity-time symmetry breaking[10], turbulent chimeras[11], as well as a periodicity of behavior with laser separation[12]. We consider the influence of the four waveguide systems introduced in[12] and some key parameters, including linewidth enhancement factor, laser separation and frequency offset (frequency difference between the two waveguide lasers), on the locking range and width
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