We present a comprehensive study of the emission dynamics of semiconductor lasers induced by dela... more We present a comprehensive study of the emission dynamics of semiconductor lasers induced by delayed optical feedback from a short external cavity. Our analysis includes experiments, numerical modelling and bifurcation analysis by means of computing unstable manifolds. This provides a unique overview and a detailed insight into the dynamics of this technologically important system and into the mechanisms leading to delayed feedback instabilities. By varying the external cavity phase we find a cyclic scenario leading from stable intensity emission via periodic behavior to regular and irregular pulse packages, and finally back to stable emission. We reveal the underlying interplay of localized dynamics and global bifurcations.
This work shows how the computation of 1D unstable manifolds of saddle external cavity modes (ECM... more This work shows how the computation of 1D unstable manifolds of saddle external cavity modes (ECMs) can be used to provide insight into global features and bifurcations in the transition to pulse packages (PPs) in a semiconductor laser with optical feedback. This study of 1D unstable manifolds of saddle ECMs agrees well with experimental measurements reported previously and allows the identification of global bifurcations in the transition to PPs. Computing these manifolds is a new tool for the study of systems of delay differential equations (DDEs) in general, and lasers with delays in particular.
We present an analysis of the external cavity mode (ECM) structure of a vertical-cavity surface-e... more We present an analysis of the external cavity mode (ECM) structure of a vertical-cavity surface-emitting laser subject to optical feedback. We consider a model in which two transverse optical modes are excited. Furthermore, we allow weak coupling of the modes via the feedback term. (In addition to the coupling through the inversion.) We use numerical continuation techniques to find and follow solutions of the governing partial, delay differential equations. This approach allows us to show how the ECM structure depends on the key parameters of feedback strength, feedback phase and the amount of coupling via the feedback term.
We present a detailed study of a route to chaos via quasiperiodicity on a torus in a semiconducto... more We present a detailed study of a route to chaos via quasiperiodicity on a torus in a semiconductor laser with phase-conjugate feedback. Highlighting the use of new tools that go far beyond mere simulation, we compute bifurcation diagrams and unstable manifolds of saddle periodic orbits. In this way, we show how a torus breaks up with a final sudden onset of chaos in a crisis bifurcation. We also identify regions of bistability between periodic solutions and other attractors in the system.
Part I Coastal Morphodynamics Long-lived Transients in Transitional Pipe Flow Dynamics of Pattern... more Part I Coastal Morphodynamics Long-lived Transients in Transitional Pipe Flow Dynamics of Patterns with Delayed Feedback Optical Delay Dynamics and its Applications Symbolic Dynamics in Genetic Oscillation Patterns Translocation Dynamics and Randomness Part II Entropy Production, the Breaking of Detailed Balance, and the Arrow of Time Monodromy and the Complexity of Quantum Systems Dynamics in Materials Science Synchronization on the Circle Conclusion from the Editors Discussions Results of the Conference, Future Perspectives
This report demonstrates how to compute 1D unstable manifolds in delay differential equations (DD... more This report demonstrates how to compute 1D unstable manifolds in delay differential equations (DDEs) with discrete, fixed delays. Specifically, using the Matlab continuation package DDE-BIFTOOL to compute the necessary starting data, we first show how to compute unstable manifolds of saddle steady states using time integration. Secondly, we use a recently developed algorithm to compute the unstable manifold of a saddle periodic orbit in a DDE. As illustration we consider two DDE models describing semiconductor lasers subject to phase-conjugate feedback and conventional optical feedback.
We present a detailed study of a route to chaos via quasiperiodicity involving tori in the system... more We present a detailed study of a route to chaos via quasiperiodicity involving tori in the system. More concretely, we show how chaos arises when a torus breaks up. This is made possible by computing what are called unstable manifolds of saddle periodic orbits, for which we developed a new numerical method.
We present a detailed analysis of the external filter mode (EFM) structure of a semiconductor las... more 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.
Magnetic resonance techniques have been applied to study the stability of the complexes formed be... more Magnetic resonance techniques have been applied to study the stability of the complexes formed between Mn(II) ions and NADP in aqueous solutions at a pH of 7.5 and 20 °C. The electron paramagnetic resonance (epr) data indicate that at low Mn(II) ion concentrations ([Mn(II)] 
We present an analytical study of the mode structure of a semiconductor laser subject to filtered... more We present an analytical study of the mode structure of a semiconductor laser subject to filtered optical feedback (FOF). First, we derive expressions for the solution sets that define the structure of the modes. We then employ continuation techniques to investigate how this structure changes as filter width, filter detuning and feedback phase are varied. This leads to a comprehensive picture of the FOF laser's mode structure. In particular, we find an equation for the boundary of the parameter region in which one finds two separate components of modes.
We study the behaviour of a multi-transverse-mode vertical-cavity surface-emitting laser subject ... more 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 inphase and anti-phase periodic solutions with vastly differing frequencies. We show how these periodic solutions give rise to quasiperiodic and chaotic laser dynamics.
Physical review. E, Statistical, nonlinear, and soft matter physics, 2002
We investigate dynamics and bifurcations of a single-mode semiconductor laser subject to phase-co... more We investigate dynamics and bifurcations of a single-mode semiconductor laser subject to phase-conjugate feedback near the locking region. The system is described by rate equations which are a three-dimensional system with a delay. With tools that go much beyond mere simulation, we find and follow steady states regardless of their stability and compute unstable manifolds of saddle points. Furthermore, we identify heteroclinic bifurcations, which turn out to be responsible for bistability and excitability at the locking boundaries.
The rate equations describing a laser with phase conjugate feedback are analyzed in the case of n... more 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 comprehensive study of the emission dynamics of semiconductor lasers induced by dela... more We present a comprehensive study of the emission dynamics of semiconductor lasers induced by delayed optical feedback from a short external cavity. Our analysis includes experiments, numerical modelling and bifurcation analysis by means of computing unstable manifolds. This provides a unique overview and a detailed insight into the dynamics of this technologically important system and into the mechanisms leading to delayed feedback instabilities. By varying the external cavity phase we find a cyclic scenario leading from stable intensity emission via periodic behavior to regular and irregular pulse packages, and finally back to stable emission. We reveal the underlying interplay of localized dynamics and global bifurcations.
This work shows how the computation of 1D unstable manifolds of saddle external cavity modes (ECM... more This work shows how the computation of 1D unstable manifolds of saddle external cavity modes (ECMs) can be used to provide insight into global features and bifurcations in the transition to pulse packages (PPs) in a semiconductor laser with optical feedback. This study of 1D unstable manifolds of saddle ECMs agrees well with experimental measurements reported previously and allows the identification of global bifurcations in the transition to PPs. Computing these manifolds is a new tool for the study of systems of delay differential equations (DDEs) in general, and lasers with delays in particular.
We present an analysis of the external cavity mode (ECM) structure of a vertical-cavity surface-e... more We present an analysis of the external cavity mode (ECM) structure of a vertical-cavity surface-emitting laser subject to optical feedback. We consider a model in which two transverse optical modes are excited. Furthermore, we allow weak coupling of the modes via the feedback term. (In addition to the coupling through the inversion.) We use numerical continuation techniques to find and follow solutions of the governing partial, delay differential equations. This approach allows us to show how the ECM structure depends on the key parameters of feedback strength, feedback phase and the amount of coupling via the feedback term.
We present a detailed study of a route to chaos via quasiperiodicity on a torus in a semiconducto... more We present a detailed study of a route to chaos via quasiperiodicity on a torus in a semiconductor laser with phase-conjugate feedback. Highlighting the use of new tools that go far beyond mere simulation, we compute bifurcation diagrams and unstable manifolds of saddle periodic orbits. In this way, we show how a torus breaks up with a final sudden onset of chaos in a crisis bifurcation. We also identify regions of bistability between periodic solutions and other attractors in the system.
Part I Coastal Morphodynamics Long-lived Transients in Transitional Pipe Flow Dynamics of Pattern... more Part I Coastal Morphodynamics Long-lived Transients in Transitional Pipe Flow Dynamics of Patterns with Delayed Feedback Optical Delay Dynamics and its Applications Symbolic Dynamics in Genetic Oscillation Patterns Translocation Dynamics and Randomness Part II Entropy Production, the Breaking of Detailed Balance, and the Arrow of Time Monodromy and the Complexity of Quantum Systems Dynamics in Materials Science Synchronization on the Circle Conclusion from the Editors Discussions Results of the Conference, Future Perspectives
This report demonstrates how to compute 1D unstable manifolds in delay differential equations (DD... more This report demonstrates how to compute 1D unstable manifolds in delay differential equations (DDEs) with discrete, fixed delays. Specifically, using the Matlab continuation package DDE-BIFTOOL to compute the necessary starting data, we first show how to compute unstable manifolds of saddle steady states using time integration. Secondly, we use a recently developed algorithm to compute the unstable manifold of a saddle periodic orbit in a DDE. As illustration we consider two DDE models describing semiconductor lasers subject to phase-conjugate feedback and conventional optical feedback.
We present a detailed study of a route to chaos via quasiperiodicity involving tori in the system... more We present a detailed study of a route to chaos via quasiperiodicity involving tori in the system. More concretely, we show how chaos arises when a torus breaks up. This is made possible by computing what are called unstable manifolds of saddle periodic orbits, for which we developed a new numerical method.
We present a detailed analysis of the external filter mode (EFM) structure of a semiconductor las... more 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.
Magnetic resonance techniques have been applied to study the stability of the complexes formed be... more Magnetic resonance techniques have been applied to study the stability of the complexes formed between Mn(II) ions and NADP in aqueous solutions at a pH of 7.5 and 20 °C. The electron paramagnetic resonance (epr) data indicate that at low Mn(II) ion concentrations ([Mn(II)] 
We present an analytical study of the mode structure of a semiconductor laser subject to filtered... more We present an analytical study of the mode structure of a semiconductor laser subject to filtered optical feedback (FOF). First, we derive expressions for the solution sets that define the structure of the modes. We then employ continuation techniques to investigate how this structure changes as filter width, filter detuning and feedback phase are varied. This leads to a comprehensive picture of the FOF laser's mode structure. In particular, we find an equation for the boundary of the parameter region in which one finds two separate components of modes.
We study the behaviour of a multi-transverse-mode vertical-cavity surface-emitting laser subject ... more 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 inphase and anti-phase periodic solutions with vastly differing frequencies. We show how these periodic solutions give rise to quasiperiodic and chaotic laser dynamics.
Physical review. E, Statistical, nonlinear, and soft matter physics, 2002
We investigate dynamics and bifurcations of a single-mode semiconductor laser subject to phase-co... more We investigate dynamics and bifurcations of a single-mode semiconductor laser subject to phase-conjugate feedback near the locking region. The system is described by rate equations which are a three-dimensional system with a delay. With tools that go much beyond mere simulation, we find and follow steady states regardless of their stability and compute unstable manifolds of saddle points. Furthermore, we identify heteroclinic bifurcations, which turn out to be responsible for bistability and excitability at the locking boundaries.
The rate equations describing a laser with phase conjugate feedback are analyzed in the case of n... more 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.
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Papers by Kirk Green