It is shown that dynamics of delay-coupled quantum cascade lasers can be described in good approximation by the simpler set of rate equations for conventional interband diode lasers. A comparison of the steady-state solutions for locking shows excellent agreement with results obtained by time integrations of the full rate equations. The delay introduces an effective coupling phase equal to the product of operation frequency and delay time and the detuning width of the locking range is shown to periodically change when the delay time changes over an interval equal to half of the operation oscillation period.
Rate equations are derived for two lasers with a linear coupling element. A strategy and scheme is indicated for iterative self-consistent numerical solution of the steady-state equations. The presence of the linewidth-enhancement parameter is explicitly taken into account. Locking in stable operation for a large range of coupling phases is numerically demonstrated and locking ranges are given. Numerical and analytical results are given for the output powers and operation frequency as functions of the pump strengths of the individual lasers and their frequency detuning.
Theoretical analysis of the modulation performance of this wave-guide device shows great potential when combined with a single-mode laser on a monolithic optical chip. On the basis of the reversed-bias electro-optic effect, modulation speeds surmounting 25 Gbit/s with 10 dB extinction ratio are feasible.
Semiconductor lasers often make use of external optical feedback, for various reasons, like single-mode selection,
stabilization or linewidth narrowing. A draw-back of such method is that delayed feedback can easily lead to sustained
relaxation oscillations (RO). The occurrence of the RO is sensitive to, among other things, the applied settings of the
phase of the feedback light. It is known from early studies that the onset of ROs already occurs with weak feedback.
However, the onset of ROs is shifted to higher feedback strengths under resonance conditions, i.e. when the product of
RO-frequency and external delay time equals an integer. This finding was based upon certain numerical and analytical
considerations, but no simple explanation was given. From a theoretical analysis based on the Lang and Kobayashi
equations, the existence of RO-free bias-current intervals of substantial width will be demonstrated for realistic pumping
values, irrespective of the feedback phase. It will be shown that for conventional semiconductor lasers with weak optical
feedback under RO-resonance condition the laser with feedback behaves as if no feedback is present. Therefore, under
these conditions the RO is damped, hence will be suppressed. This result is valid in the regime of sufficiently weak
feedback, such that the RO frequency is not deviating significantly from its value in the solitary laser and is supported by
calculations of the RIN spectrum.
A set of rate equations is derived describing the deterministic multi-mode dynamics of a semiconductor laser. Mutual interactions among the lasing modes, induced by high-frequency modulations of the carrier distribution, are described by carrier-inversion moments and lead to special spectral content of each spatial mode. The diffusion of carriers is shown to play an important role in determining the spectral properties of the field. The Bogatov effect of asymmetric gain suppression in semiconductor lasers will be derived. We will explicitly discuss the nontrivial relationship between the modes of the nonlinear cavity and the optical spectrum of the laser output and illustrate this for a two and three-mode laser.
Vertical Cavity Surface Emitting Lasers (VCSELs) often present switching between two orthogonal polarization states when varying parameters like e.g. current or temperature. Around such a switching point, the system randomly jumps between these two polarization states (mode hopping), driven by noise. In this contribution, we present experimental and numerical results showing the effect of coloured noise, externally added to the current, on the switching characteristics of a VCSEL.
A reflective semiconductor optical amplifier is modeled using a rate equation for the inversion and an expression for the outgoing field in terms of the incoming field and the instantaneous inversion, for both the cases of linear and nonlinear gain, while assuming modulation speeds well below the round-trip frequency. Simulations are performed on the basis of these equations. Results for both the static and dynamic performance are presented and discussed.
In the filtered optical feedback (FOF) scheme a part of the emission of the laser is spectrally filtered, for example by a Fabry-Perot filter, and than fed back into the laser. If a semiconductor laser is subject to such delayed FOF qualitative different types of oscillations are possible: the well known relaxation oscillations and, more remarkably, frequency oscillations. We explain how the continuous wave operation of the FOF laser - the external filtered modes - lose their stability and the different types of oscillations arise due to the presence of the filter. This study is restricted to the case of a narrow filter. This means that there are only a few external filtered modes within the width of the filter, so that the influence of the feedback phase can be studied explicitly.
We investigate the continuous wave solutions of a system of two mutually delay coupled semiconductor lasers. These continuous wave solutions, which we refer to as compound laser modes (CLMs), are locked solutions of the coupled laser system where both lasers lase at a common frequency. We model the system by a set of delay differential rate equations, where we assume that, apart from a possible detuning in their free running optical frequencies, the lasers are identical. We show how the structure and the stability of the CLMs depend on the main parameters, namely, the feedback phase, the feedback rate, the pump parameter, and the detuning. We identify two mechanisms for creating CLMs. First, CLMs emerge from the off-state of the coupled laser system in Hopf bifurcations. Second, CLMs are created in pairs in saddle-node bifurcations. For the special case of zero detuning we also find pitchfork bifurcations that organize the CLM structure. We show in which parameter regions CLMs exist, where they are stable, and which bifurcation curves form the boundary of the stable locking region.
Semiconductor lasers subject to filtered feedback are examined in the limit of a narrow filter. Under particular conditions, the laser dynamical equations reduce to the equations of a laser subject to an externally injected signal. The conditions for steady state locking and Hopf bifurcation can be determined analytically allowing a deeper understanding of the role of the laser parameters on the perturbed semiconductor laser dynamics. The success of our analysis motivates the study of other feedback cases such as the laser subject to slow phase-conjugated feedback.
A method for the investigation of the dynamics of two semiconductor lasers, grown side-by-side on the same wafer to enhance the lateral optical coupling, is presented. Using steady state analysis, parameter regimes of relevant dynamics are identified. This is completed by a spectral analysis, were two routes to chaos are implicated. Finally, we confirm the calculations by showing an avoided crossing type of behavior for the coupling strength.
Laterally coupled laser-diode pairs, also called "twin stripe lasers", are coupled by means of the evanescent em-field in between two dielectric laser cavities. The coupling strength between the two lasers is determined by the decay length of this evanescent em-field.
This length depends on the indices of refraction of the material
which separates the two cavities compared to the cavity material. However the index of refraction of the cavity material also depends on the inversion density in the lasing material. Therefore the coupling coefficient in the rate equation for the twin stripe device can be written as κ<sub>0</sub>[1 + ½(α<sub>c</sub><sub>(1)</sub> ρ<sub>1</sub> + α<sub>c</sub><sub>(2)</sub> ρ<sub>2</sub>)], where ρ<sub>1</sub> and ρ<sub>2</sub> are the inversion densities in the two stripes. We give an expression for α<sub>c</sub<sub>>(<i>i</i>)</sub> in terms of the properties of the device.
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.
We theoretically investigate the dynamical properties of a system
of two semiconductor lasers that are mutually coupled via
their optical fields. An intrinsic feature of the coupling is its
time delay which generically arises from the finite propagation
time of the light form one laser to the other. In our system the
coupling time is in the sub-ns range, which is of the
same order of magnitude as the period of laser's internal relaxation
oscillations. We model this system with Lang-Kobayashi-type rate equations where we account for the mutual coupling of the two lasers by a delay term. The resulting set of nonlinear delay differential equations is analyzed by using recently developed numerical continuation. We consider the case of two nearly identical lasers with symmetrical coupling conditions but different frequencies, and present an analysis of the coupled laser modes (CLMs) of the system.
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.
A multimode model is necessary to describe the behavior observed in
a twin stripe diode laser. We will use a single-stripe version of the
device to calibrate the parameters enabling the model to be used in
the description and analysis of the twin-stripe lasers.
A novel multi-longitudinal-mode rate-equations description of the Fabry-Perot type semiconductor laser is presented. The model includes gain dynamics among the longitudinal modes due to e.g. spatial hole burning.
Due to the large energy splitting of the single-electron levels in a small quantum dot, only one single electron level and one single hole level can be made resonant with the levels in the conduction band
and valence band. This results in a closed system with nine distinct levels, which are split by the Coulomb interactions. We show that flat and tall cylindrically symmetric dots have level schemes
with different selection rules. In both cases entangled photon pairs can be efficiently produced.
We explain how a semiconductor optical amplifier in a Sagnac-interferometric arrangement can be used for switching of 200 fs optical pulses. The switching principles are based on gain and index saturation dynamics on a sub-picosecond timescale. We present a model that accounts for bi-directional propagation of ultrashort optical pulses through the amplifier as well as free-carrier absorption and two-photon absorption. We have also carried out pump and probe experiments to measure the ultrafast refractive index dynamics of a multi-quantum well InGaAsP-InGaAs semiconductor optical amplifier that is operated in the gain regime. The pump and probe pulses are cross-linearly polarized. We observe a phase shift of 200 degrees if the amplifier is pumped with 120 mA of current, but find that the phase shift vanishes if the injection current is increased to 160 mA. Our results indicate a contribution of two-photon absorption to the nonlinear phase shift that opposes the phase shift introduced by the gain. Finally, we observe that the phase shift comes up and disappears within a picosecond.
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.
Telecommunication systems using 200 fs optical pulses for ultrahigh bit-rate optical transmission require new concepts to be developed for ultrafast all-optical switching. On the basis of numerical experiments, we discuss how a semiconductor optical amplifier in a Sagnac interferometric arrangement can be used for switching these short pulses. The switching principle is based on gain and index saturation dynamics on a sub-picosecond timescale. The model accounts for bi-directional propagation of ultrashort optical pulses through the amplifier as well as free-carrier absorption and two-photon absorption.
Through simulations based on the rate equations for diode lasers with filtered optical feedback, we show that in the Coherence Collapse regime a large variety of dynamics is predicted such as periodic and quasiperiodic oscillations and chaos. The control of the transition through these dynamical regimes is achieved through the filter parameters : the filter's spectral width and its central frequency.
Our report focuses on the strong kink found in the P-I output of an asymmetric twin stripe semiconductor laser. A multi longitudinal mode model is used to describe the system. The model allows for asymmetric coupling of the two lasers and also accounts for multi longitudinal effects within and in-between the lasers.
A passively Q-switched laser produces regular pulses above threshold. Here we show that this type of laser, described by the Yamada model, exhibits excitability below threshold. There is a stable equilibrium, the off-solution, from which the system can be triggered by a sufficiently large, but small perturbation, to produce a single pulse after which it settles back to the off-solution. In order to study possible applications, such as pulse reshaping and clock recovery, we consider the reaction of the laser to a triggering input pulse. Furthermore, we demonstrate that when the laser is subjected to injected optical noise below threshold it reacts by producing a train of pulses, whose coherence is maximal and, equivalently, the normalized jitter is minimal for a particular noise level. This shows that the Yamada model with optically injected noise constitutes an example of all-optical coherence resonance.
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.
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.
For a diode laser subjected to feedback from a phase- conjugating mirror we present (1) the first exact stability- analysis and (2) various spectra. The stability properties are intermediate between those of the injection laser and the laser with conventional optical feedback. The role of a finite response-time is to drastically enhance the steady- state stability. For moderate feedback the frequency noise is suppressed by several orders of magnitude, and the main relaxation frequency of the laser shows a crossover from the usual relaxation oscillation frequency to a new frequency determined by the amount of feedback. This may be of technological importance since it improves the modulation bandwidth.
This paper presents the dynamics of a semiconductor laser subject to instantaneous phase-conjugate feedback. Recently, the behavior of such a laser have been explored by means of bifurcation diagrams. However, the exact nature of the involved dynamics and bifurcations remained unclear. Here we present a detailed study of the changes of the dynamics as the feedback strength is varied.
We study the wave propagation through a semiconductor optical amplifier. It is investigated how the anisotropy caused by the quantum wells and the waveguiding together determine the polarization sensitivity of the amplification. We also analyze how the tensile strain in the quantum well can be optimized such that the resulting amplifier operates independent of the state of polarization of the incident field. The promising properties of non-(001) grown quantum wells are discussed.
We show that there exists phase-locked motion on a torus in the full single-mode rate equations of a diode laser subject to weak optical injection. The stability of the torus depends on the linewidth-enhancement factor (alpha) . The case of unstable torus motion leads to a bistability, which gives rise to hysteresis when the detuning of the input signal is varied for suitable injection strength. The torus becomes unlocked in a heteroclinic bifurcation. Our results confirm the validity of a previous analysis of an averaged phase/amplitude equation by means of bifurcation theory. We present the relevant dynamics by several panels, which include plots of the time series of the power, phase plots, and field and RIN spectra.
There are two main reasons which explain why in a semiconductor laser amplifier the amplification can depend on the state of polarization: (1) Waveguiding can give rise to an amplification that differs for TE polarization and for TM polarization. This can happen even if the confinement to the active layer is comparable for the two polarization states. (2) The interaction process between light and matter in a quantum well is, in general, anisotropic. This is because the electrons are confined in only one direction. So the response to an electromagnetic field will have a tensor character, rather than a scalar. We analyze these two causes and show how they can be balanced, so that the desired polarization amplification can be achieved.
Most of the previous treatments of semiconductor lasers subject to optical feedback from a phase-conjugate mirror (PCM) have assumed the PCM responds instantaneously. Furthermore, the mechanism responsible for phase conjugation does not usually enter into the analysis. In this paper are derived the time-dependent reflectivity from a PCM created through non-degenerate four-wave mixing. The resulting laser dynamics are compared to the case of the ideal PCM, as a function of PCM mirror interaction depth, distance to the PCM, and laser current. The time-responsive PCM tends to suppress otherwise chaotic output and produces power pulses whose frequency is tunable by varying laser current or PCM reflectivity.
This presentation consists of three parts: (1) A new mode solver for structures with high gain and/or losses is discussed. By using 'scattering matrices,' great numerical robustness is obtained. (2) In planar waveguides, the amplification for TE modes can be much larger than that for TM modes, even though the confinement to the active layer is comparable. The connection between confinement factors and gain is elucidated. We show that an often used approximation for the TM gain is not valid for practical configurations. An improved approximation is given. (3) We describe how a waveguide with an anisotropic active layer can be modeled using a scattering approach. This model should then tell how much anisotropy is needed to get a gain that is polarization-independent.
A fully quantum electrodynamical model is presented for the spontaneous emission of light from a quantum well system in a vertical cavity device. The model solves the combined Maxwell and Schrodinger equations to describe the time evolution of both the carrier populations in the quantum well and the radiation field. Structures that present major challenges to other models can be handled by our model without any difficulty. Our model is validated against more standard approaches to the spontaneous emission problem for a simple case.
We have analyzed the behavior of a diode laser with low-amplitude optical injection. The noisy injection field has phase fluctuations only, with a linewidth larger than both the solitary linewidth and the locking range. Simulations show that the laser output spectrum can be much narrower than the injection spectrum, and may have a dip at the solitary laser frequency. A model describing these features is based on linear superposition of the stationary responses to monochromatic injection. The mechanism of line narrowing is that the response to injection ouside the locking range lies mainly inside.
The Hopf bifurcation points for single mode solutions of the Lang and Kobayashi equations are determined using asymptotic methods. The approximation is based on a small parameter (epsilon) which is defined as the ratio of the photon and carrier lifetimes. The remaining parameters are scaled with respect to (epsilon) . The critical feedback rate for a Hopf bifurcation is studied in terms of the pump parameter and is either an 0((epsilon) ) or an 0((epsilon) <SUP>1/2</SUP>) quantity. At a fixed value of the pump parameter, we obtain an expression of the Hopf bifurcation point in terms of the effective feedback strength and the feedback phase. In addition, we investigate the Hopf bifurcation point near and at the lasing threshold.
In this paper, the various physical mechanisms in low-frequency intensity fluctuations are identified, that occur when a diode laser is subject to moderate optical feedback while operating close to its solitary threshold. The maximum gain mode, which surprisingly often stable, serves as a seemingly unreachable goal for the system. In attempting to reach this mode, the system forms mode-locked pulses. In between pulses the mode-locking is frustrated and inevitably the system passes too close to one of the many saddle points, that will take the system back to the low power solitary laser state.
A theoretical analysis of the optically injected single-mode diode laser outside the locking regime is presented. After a short overview of our model and its description of the locking regime, we concentrate on the nonlinear interaction between diode and injection signal that occurs outside this locking regime. When the injection is sufficiently weak this process can be approximated by four-wave mixing (FWM). We will present an extensive analytical treatment of the FWM behavior and find good agreement with the results of a recent experiment.
A theoretical study is presented of multi-wave mixing dynamics in a single-mode semiconductor laser with monochromatic weak external injection. Three relevant regimes are overviewed, i.e., corresponding to locking, four-wave mixing, and multi-wave mixing. Moreover, much emphasis is put on four-wave mixing. For this regime, several analytical expressions are presented, some of which are new. A detailed theoretical explanation is given for the peculiar relaxation oscillation resonance behavior that was recently observed in experiments.
Space charge build-up in the well is shown to be the cause of the inductive effects in double- barrier diodes. A new impedance model for the diode is presented, built on a static model of coherent tunneling in a selfconsistent electron potential. The corresponding equivalent circuit is made up of two capacitances--related to the charge accumulations in the emitter and in the well--, and two conductances--one for each barrier. The numerical results of this circuit model are in qualitative agreement with experimental data. The success of the earlier quantum inductance model of Brown et al. is explained in terms of the presented model, without the need of introducing such a quantum inductance.
In nearly every application of semiconductor lasers some externally reflected light is coupled back into the laser with a certain time delay. The noise and coherence properties are very sensitive to optical feedback. Coupling to external cavity modes induces frequency shifts linenarrowing and broadening competition among different external cavity modes or dynamical instabilities. Strongly dispersive gain makes instabilities efficiently generate phase noise leading to coherence collapse. We present an overview of these phenomena and their theoretical descriptions.