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Register a new userLambert function methods for laser dynamics with time-delayed feedback
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Time-delayed differential equations arise frequently in the study of nonlinear dynamics of lasers with optical feedback. Traditionally, one has resorted to numerical methods because the analytical solution of such equations are intractable. In this manuscript, we show that under some conditions, the rate equations model that is used to model semiconductor lasers with feedback can be analytically solved by using the Lambert W function. In particular, we discuss the conditions under which the coupled rate equations for the intra-cavity electric field and excess carrier inversion can be reduced to a single equation for the field, and how this single rate equation can be cast in a form that is amenable to the use of the Lambert W function. We conclude the manuscript with a similar discussion for two lasers coupled via time-delayed feedbacks.
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We show that time-delayed feedback methods, which have successfully been used to control unstable periodic ortbits, provide a tool to stabilize unstable steady states. We present an analytical investigation of the feedback scheme using the Lambert function and discuss effects of both a low-pass filter included in the control loop and non-zero latency times associated with the generation and injection of the feedback signal.
The influence of time delay in systems of two coupled excitable neurons is studied in the framework of the FitzHugh-Nagumo model. Time-delay can occur in the coupling between neurons or in a self-feedback loop. The stochastic synchronization of instantaneously coupled neurons under the influence of white noise can be deliberately controlled by local time-delayed feedback. By appropriate choice of the delay time synchronization can be either enhanced or suppressed. In delay-coupled neurons, antiphase oscillations can be induced for sufficiently large delay and coupling strength. The additional application of time-delayed self-feedback leads to complex scenarios of synchronized in-phase or antiphase oscillations, bursting patterns, or amplitude death.
A proposal for an all-optical memory based on a bistability of single-mode states in a dual-mode diode laser with time-delayed optical feedback is presented. The system is modeled using a multimode extension of the Lang-Kobayashi equations with injected optical pulses. We uncover the bifurcation structure by deriving analytical expressions for the boundaries of the bistable region and demonstrate how the delay time in the external cavity determines an optimal pulse duration for efficient switching of the memory element. We also show the relevant role played by gain saturation and by the dual-mode solutions of the Lang-Kobayashi equations for the existence of the bistable regions. Our results demonstrate that feedback induced bistability can lead to significant performance improvements when compared to memory elements based on the injection locking bistability in dual-mode devices.
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