We study the behaviour of a multi-transverse-mode vertical-cavity surface-emitting laser subject to optical
feedback in which the optical modes are coupled through the external round-trip. Starting from a delayed
partial differential equation description of the spatial optical mode profiles and the carrier diffusion, we first
use eigenfunction expansion techniques to resolve the spatial dependence. The resulting system of delay differential
equations is then amenable to a full nonlinear bifurcation analysis by means of numerical continuation
techniques. As illustration, we present bifurcation diagrams of a two-mode VCSEL in the plane of feedback
strength versus feedback phase. In this way, we identify a number of changes in the structure and bifurcations
of the VCSEL's dynamics. In particular, we find coexisting stable steady state solutions, which bifurcate to
stable in-phase and anti-phase periodic solutions with vastly differing frequencies. We show how these periodic
solutions give rise to quasiperiodic and chaotic laser dynamics.
We present a detailed analysis of the external filter mode (EFM) structure of a semiconductor laser subject to filtered optical feedback (FOF). These EFMs form the 'backbone' of the dynamics of the system. Specifically, from the governing delay differential equations, we find analytic, transcendental expressions for both the solution curves, which define the frequency and amplitude of the EFMs, and their envelopes. We use numerical continuation to find and follow solutions of these equations. This approach allows us to show how the structure depends on the key parameters of filter width, filter detuning, and feedback phase. In other words, we identify the external influence of the filter on an otherwise fixed laser.
We study the behaviour of a semiconductor laser subject to phase-conjugate feedback when the interaction time of the phase-conjugating mirror changes. With continuation techniques we present two-parameter bifurcation diagrams in the plane of feedback strength versus pump current, which change qualitatively as the interaction time of the mirror is increased. This reveals that for small interaction times the assumption of instantaneous feedback is justified. On the other hand, increasingly larger interaction times lead to considerable changes in the locking region. By investigating how curves of Hopf bifurcations change with the interaction time, we show how more complicated, chaotic dynamics become suppressed. One-parameter bifurcation diagrams as a function of the pump current, obtained by simulation, complement the continuation analysis.
The rate equations describing a laser with phase conjugate feedback are analyzed in the case of non-zero detuning. For low feedback rates and detuning, the stability diagram of the steady state is similar to the laser subject to injection. A stable steady state may loose its stability through a Hopf bifurcation exhibiting a frequency close to the relaxation oscillation frequency of the solitary laser. We also construct time-periodic pulsating intensity solutions exhibiting frequencies close to an integer multiple of the external cavity frequency. These solutions have been found numerically for the zero detuning case and play an important role in the bifurcation diagram.
We present a theoretical study into the dynamics and bifurcations of a semiconductor laser subject to delayed optical feedback, as modelled by the Lang-Kobayashi equations. For the case of a short external cavity, of the order of a few centimeters, there is a limited number of external cavity modes (ECMs), which makes it possible to apply advanced techniques from dynamical systems, such as the continuation of ECMs and their bifurcations, and the computation of unstable manifolds. From the physical point of view, a short cavity is characterized by the fact that the delay time in the external cavity is of the same order of magnitude as the period of the relaxation oscillation of the laser. In this regime the optical feedback phase is known to play an important role. We provide a detailed overview of how the dynamics depends on the feedback phase, which is in good agreement with recent experimental measurements.
A semiconductor laser subject to phase-conjugate optical feedback can be described by rate equations, which are mathematically delay differential equations (DDEs) with an infinite-dimensional phase space. We employ new numerical continuation techniques for DDEs to study the exact nature of the locking region in the parameter plane given by the feedback strength and the pump current. This reveals interesting dynamics, including heteroclinic bifurcations, near the locking region, leading to different scenarios of possible transitions into and out of locking. We show how several special points act as organizing centers for the dynamics.
A semiconductor laser subject to phase-conjugate optical feedback can be described by rate equations, which are mathematically delay differential equations (DDEs) with an infinite dimensional phase space. This is why, from the theoretical point of view, this system was only studied by numerical simulation up to now. We employ new numerical techniques for DDEs, namely the continuation of periodic orbits and the computation of unstable manifolds, to study bifurcations and routes to chaos in the system. Specifically we compute 1D unstable manifolds of a saddle-type periodic orbit as intersection curves in a suitable Poincare section. We are able to explain in detail a transition to chaos as the feedback strength is increased, namely the break-up of a torus and a sudden transition to chaos via a boundary crisis. This allows us to make statements on properties of the ensuing chaotic attractor, such as its dimensionality. Information of this sort is important for applications of chaotic laser signals, for example, in communication schemes.
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