We present an analysis of a semiconductor laser subject to filtered optical feedback from two filtering elements (2FOF).
The motivation for this study comes from applications where two filters are used to control and stabilise the laser output.
Compared to a laser with a single filtered optical feedback loop, the introduction of the second filter significantly influences
the structure of the basic continuous-wave solutions, which are also known as external filtered modes (EFMs). We compute
and represent the EFMs of the underlying delay differential equation model as surfaces in the space of frequency <i>ω<sub>s</sub></i> and
inversion level <i>N<sub>s</sub></i> of the laser, and feedback phase difference <i>dC<sub>p</sub></i>. The quantity <i>dC<sub>p</sub></i> is a key parameter since it is
associated with interference between the two filter fields and, hence, controls the effective feedback strength. We further
show how the EFM surface in (<i>ω<sub>s</sub></i>, <i>dC<sub>p</sub></i>, <i>N<sub>s</sub></i>)-space changes upon variation of other filter parameters, in particular, the two
delay times. Overall, the investigation of the EFM-surface provides a geometric approach to the multi-parameter analysis
of the 2FOF laser, which allows for comprehensive insight into the solution structure and dynamics of the system.
We investigate the stability of an array of three laterally coupled semiconductor lasers. This study of the
simplest system with an underlying structure that is also found in larger arrays constitutes a first step towards
understanding the stability properties of large arrays. We use a composite-cavity model, where the individual
lasers are described by the transverse modes of the entire composite-cavity system. Specifically, we analyze
the stable locking region, where the laser array exhibits continuous wave emission for different detunings and
coupling strengths between the individual lasers. We find that the optical fields in the outer lasers are out of
phase with the middle laser.
This paper investigates sensitivity of semiconductor lasers to external optical signals. Bifurcation analysis of ordinary rate
equations, describing noise-free lasers with pure coherent external signal, reveals that adjusting the extend and type of
externally-induced bifurcations and chaos to desired state is possible by tailoring of the laser active medium and resonator
configurations. Extending the analysis to stochastic rate equations, which describe lasers with spontaneous emission noise
and noisy external signal, reveals a dramatic impact of phase-fluctuations (incoherence) in the external signal on induced
bifurcations and chaos. A nonlinear optics approach is proposed where the strong sensitivity of laser instabilities to the
intensity and coherence of external signal are used to detect ultra low levels of laser radiation.
We study a semiconductor laser subject to filtered optical feedback from two separate filters. This work is motivated by an
application where two fiber gratings are used to stabilize the output of a laser source. Specifically, we consider the structure
of the external filtered modes (EFMs), which are the basic cw-states of the system. The system is modelled by a set of
four delay differential equations with two delays that are due to the travel times of the light in each of the external cavities.
Here, each filter is approximated by a Lorentzian and we assume that there is no interaction between the two filters.
We derive a transcendental equation for the EFMs as a function of the widths, detunings and the feedback strengths
of the two filters. With continuation techniques we investigate how the number of EFMs changes with parameters. In
particular, we consider the equation for its envelope. This allows us to determine regions in the plane of the two detunings
that correspond to one, two or three EFM components - disjoint closed curves that are traced out by the EFMs as a
function of the feedback phase.
We consider a semiconductor laser device, where the active region consists of parallel stripes in the longitudinal
direction. In the composite cavity model, the stripes are coupled via the transversal modes of the entire
compound laser device. By calculating the spatial mode profiles we accurately account for the frequency
detuning between the modes as well as for the gain and coupling of the individual modes, which are determined
by spatial overlap integrals of the mode profiles. In particular, we show the nonlinear dependence of these
quantities on the geometry of the laser device. The temporal dynamics of the composite cavity modes are
described by corresponding rate equations. Bifurcation analysis of these rate equations, which are coupled to
the spatial mode equations, unravels the dynamics of a twin-stripe laser. We identify different locking regions
as well as regions with possibly chaotic dynamics.
The paper is concerned with a theoretical study of synchronization
between two end-to-end coupled lasers. Lasers are treated within the framework of a special multimode theory, valid for arbitrary coupling between lasers, where the laser field is decomposed in terms of the composite-cavity modes of the entire coupled-laser system. Bifurcation continuation techniques are used to systematically investigate the resulting equations under class-A, and further under class-B approximation. We discovered that the mechanism leading to laser synchronization changes from strong composite-cavity mode competition in class-A regime to frequency locking of composite-cavity modes in class-B regime.
We find that, although inversion noise has only a marginal effect on the linewidth of a semiconductor laser in CW operation, in the presence of dynamics it may play a key role in determining the final dynamical state. It is therefore essential to include both field noise and carrier noise of realistic strength when analysing semiconductor laser dynamics. Next we investigate the influence of quantum noise, both field and carrier noise, on the highly complex nonlinear dynamics that arise in a single-mode semiconductor laser subject to filtered optical feedback. Our numerical study based on stochastic rate equations shows that for a wide range of filter widths the noise may lead to qualitatively different dynamics than predicted by a deterministic analysis. In particular, we find that certain attractors that are predicted in the absence of noise may no longer be available when the effects of noise are correctly incorporated, while others show remarkable robustness instead. In general, the results confirm that carrier noise in the laser can influence the dynamics quite substantially. Finally, we present numerical results of noise-induced pulsations in a semiconductor laser with optical injection. We show that, close to the locking edge, patterns of single, two and three pulses can be excited and we suggest that experimental study of this multi-pulse excitability be based on pulse timing statistics.
We show that a single-mode semiconductor laser subject to
optical injection, and described by rate equations, can produce
excitable multipulses, where the laser emits a certain number of
pulses after being triggered from its steady state by a single
This phenomenon occurs in experimentally accessible regions in
parameter space that are bounded by curves of n-homoclinic
bifurcations, connecting a saddle to itself only at the n-th
return to a neighborhood of the saddle. These regions are organised in what we call 'homoclinic teeth' that grow in size and shape with the linewidth enhancement factor.
In this paper unprecedented agreement is reported between a theoretical two-dimensional bifurcation diagram and the corresponding experimental stability map of an optically injected semiconductor laser over a large range of relevant injection parameter values. The bifurcation diagram encompasses both local and global bifurcations mapping out regions of regular, chaotic and multistable behavior in considerable detail.
We use advanced techniques from bifurcation theory to examine the dynamics of single-mode semiconductor lasers with optical injection, modeled by three-dimensional rate equations. Key bifurcations, namely saddle-node, Hopf, period-doubling, saddle-node of limit cycle and torus bifurcations, are followed over a wide range of injection strengths and detunings for different fixed values of the linewidth enhancement factor (alpha) . Combining the stability diagram in parameter space with phase portraits provides a global and detailed view of complex dynamics of injected semi-conductor lasers. In particular, we concentrate here on different routes to phase locking, which can be surprisingly complicated. Our analysis reveals many regions of chaotic behavior and multistability in good agreement with experimental studies.
Chaotic dynamics have been found in a single mode semiconductor laser subject to optical injection experimentally or by numerical simulation. In this paper we study this laser system by means of rate equations, which mathematically are a three-dimensional vector field. To study different routes to chaos we start from the knowledge of bifurcation curves in the plane of injection strength and detuning in Ref.  of this issue. Our main tool is combining the continuation of bifurcation curves with computing the respective phase space objects. In this way, we obtain detailed knowledge of regions in parameter space of different types of chaos, and what transitions can be found at the boundaries of such regions. This gives new insight into chaotic output found in experiments. Furthermore, it allows relatively easy access to chaotic dynamics for applications such as chaotic data encryption schemes.