class-B Laser Research Articles

Contagious photons: Laser dynamics and the susceptible-infected-recovered model.

We explore a complete analogy between the classic susceptible-infected-recovered epidemiological model with natural birth and death rates, and class-B laser equations. As a result, recently derived asymptotic formulas in the former context can be used to describe the switch-on intensity pulse of a laser suddenly brought well above the lasing threshold, as in active Q-switching. Conversely, the well-studied laser relaxation oscillations find a companion behavior in epidemiology, emphasizing nontrivial timescales. Finally, we discuss the possible correspondence between multistrain outbreaks and multimode lasing.

Transient nonlinear dynamics in an electrically modulated quantum cascade laser with optical injection

Based on rate equations, we mainly simulate the transient instability characteristics of an 8 µm quantum cascade laser (QCL) subject to optical injection with alternating current (AC) electrical modulation. Simulation results show that the period-one oscillation of the optical injection-locked QCL is broken by applying an AC current to the direct current (DC) bias. Combining an external optical injection and induced period current modulation can cause period-one oscillation dropouts and can display chaotic states outside the stable locking region, owing to the cooperative interplay between the AC frequency and the periodic oscillation frequency caused by the optical injection. To give a clear physical picture of the chaos under different line-width enhancement factors (LEFs), we use temporal series, Poincaré bifurcation diagrams, Fourier spectra, phase portraits, and first return maps to carefully analyze. These analytical methods are effective for the dynamical behaviors of QCLs with low LEFs, which show that the chaos of QCLs strongly rely on external modulation compared with class-B laser systems. This work paves a new way for realizing chaotic signal generation and has an important application in secure communication in the mid-infrared and terahertz frequency band.

Routes to chaos in a class-B laser coupled to a neutral resonator

We study the dynamics of a class-B semiconductor microring laser coupled to a neutral microring resonator for common fabrication parameters. For zero detuning between the resonators, we identify five dynamical regions controlled by the normalized pump rate of the laser and coupling strength between the resonators. These regions show stable lasing with phase locking for either sufficiently small or large coupling strength parameter values. Multistability and chaos arise in between these stable regions where at least one fixed point is attractive. The transition from stable to unstable lasing regions in the parameter space phase diagram is a crossover.

The transition from conservative to dissipative flows in class-B laser model with fold-Hopf bifurcation and coexisting attractors

Dynamical behaviors of a class-B laser system with dissipative strength are analyzed for a model in which the polarization is adiabatically eliminated. The results show that the injected signal has an important effect on the dynamical behaviors of the system. When the injected signal is zero, the dissipative term of the class-B laser system is balanced with external interference, and the quasi-periodic flows with conservative phase volume appear. And when the injected signal is not zero, the stable state in the system is broken, and the attractors (period, quasi-period, and chaos) with contractive phase volume are generated. The numerical simulation finds that the system has not only one attractor, but also coexisting phenomena (period and period, period and quasi-period) in special cases. When the injected signal passes the critical value, the class-B laser system has a fold-Hopf bifurcation and exists torus “blow-up” phenomenon, which will be proved by theoretical analysis and numerical simulation.

Delay-differential-equation model for mode-locked lasers based on nonlinear optical and amplifying loop mirrors

Delay differential equation model of a nonlinear optical--nonlinear amplifying loop mirror mode-locked laser is developed that takes into account the finite relaxation rate of the gain medium and asymmetric beam splitting at the entrance of the nonlinear mirror loop. Asymptotic linear stability analysis of the continuous wave solutions performed in the limit of large delay indicates that in a class-B laser flip instability is preceded by the modulational instability and therefore cannot give rise to stable square wave patterns. Numerically it is shown that the model can demonstrate large windows of regular fundamental and harmonic mode-locked regimes with single and multiple pulses per cavity round trip time separated by domains of irregular pulsing.

Low- and high-β lasers in the class-A limit: photon statistics, linewidth, and the laser-phase transition analogy

Nanocavity lasers are commonly characterized by the spontaneous coupling coefficient β that represents the fraction of photons emitted into the lasing mode. While β is conventionally discussed in semiconductor lasers where the photon lifetime is much shorter than the carrier lifetime (class-B lasers), little is known about β in atomic lasers where the photon lifetime is much longer than the other lifetimes and only the photon degree of freedom exists (class-A lasers). We investigate the impact of the spontaneous coupling coefficient β on lasing properties in the class-A limit by extending the well-known Scully–Lamb master equation. We demonstrate that in the class-A limit all the photon statistics are uniquely characterized by β and that the laser phase transition-like analogy becomes transparent. In fact, β perfectly represents the “system size” in phase transition. Finally, we investigate the laser-phase transition analogy from the standpoint of a quantum dissipative system. Calculating a Liouvillian gap, we clarify the relation between β and the continuous phase symmetry breaking.

Lasing in para-Fermi class-B microring resonator arrays

We demonstrate lasing in arrays of microring resonators with underlying para-Fermi symmetry. The properties of the algebra allow the analytic prediction of propagation constants and normal modes of an array of passive resonators that show the optical analogue of a zero-energy mode that is its own chiral pair. The rest of the normal modes in the linear model form chiral pairs that are evenly distributed around the pseudo zero-energy mode. We use this information to construct a class-B laser model where even and odd sites are driven with different strength in a pattern following para-Fermi algebra chirality to demonstrate lasing with enhancement or suppression of the zero-energy mode. Zero-energy mode lasing is stable to initial conditions for a large set of driving ratios. Lasing with zero-energy mode suppression is unstable and more susceptible to initial field configurations.

Deterministic Optical Rogue Waves in Nd:YVO4 Lasers Induced by Near-Degenerate Transverse Modes

We identify the role of transverse modes in causing temporal instability in an all-solid-state laser system and thus propose a new method to trigger deterministic optical rogue waves. Experimental investigations show that the near-degenerate transverse modes with mode frequency difference close to the relaxation oscillation frequency is responsible for the rogue waves. By tilting the reflective mirror, the transverse modes beating frequency decreases and gets close to relaxation oscillation frequency, thus the system exhibits self-pulsing behavior and rarely-arising giant pulses. The statistical distribution of the pulse intensity deviates from the exponential statistics to the L-shaped distributions with long tails, indicating the appearance of optical rogue waves. The conclusion can be extended to other Class-B lasers where the population inversion cannot be eliminated adiabatically.

Lasing thresholds and photon statistics in high-βburied multiple quantum well photonic crystal nanocavity lasers

We investigate the lasing thresholds and photon statistics of buried multiple quantum well (MQW) photonic crystal (PhC) lasers with a high spontaneous emission coupling coefficient $\ensuremath{\beta}$ and a large carrier transparency number ${N}_{0}$. Using measured input-output curves and delay-dependent second-order photon correlations ${g}^{(2)}(\ensuremath{\tau})$, we estimate that our population inversion threshold is near the kink threshold of the input-output curve, which indicates that the buried MQW PhC lasers operate close to the ``lasers without inversion'' (intensity jump without inversion) regime proposed by Yamamoto and Bj\ork. With respect to static photon statistics ${g}^{(2)}(0)$, the super-Poissonian photon statistics ${g}^{(2)}(0)g1$ are observed even one order of magnitude above the kink threshold. The super-Poissonian photon statistics are specific to the class-B lasers where the photon decay rate ${\ensuremath{\gamma}}_{c}$ is much larger than that of carriers ${\ensuremath{\gamma}}_{\ensuremath{\parallel}}$ and the relaxation oscillation occurs. We also propose that the pump power needed to maximize the damping time of the relaxation oscillation could be the noise threshold at which the photon statistics change from thermal to Poissonian.

Localized patterns in star networks of class-B lasers with optoelectronic feedback

We analyze how a star network topology shapes the dynamics of coupled CO\textsubscript{2} lasers with an intracavity electro-optic modulator that exhibit bistability. Such a network supports spreading and stationary activation patterns. In particular, we observe an activation spreading where the activated periphery turns on the center element, an activated center which drifts the periphery into the active region and an activation of the whole system from the passive into the active region. Pinned activation, namely activation localized only in the center or the peripheral elements is also found. Similar dynamical behavior has been observed recently in complex networks of coupled bistable chemical reactions. The current work aims at revealing those phenomena in laser arrays, giving emphasis on the essential role of the coupling structure in fashioning the overall dynamics.

Extreme events in systems with discontinuous boundaries

We describe a class of extreme events emerging in systems with discontinuous boundaries that exhibit stick-slip dynamics. This kind of systems is capable of generating extreme events when the system trajectory sticks to a sliding set and slides for a relatively large distance along the discontinuous boundary to one of the system's subspaces. We present two examples of such systems, a microelectromechanical cantilever and a driven class-B laser, where this type of extreme events appear. We show that their forecasting is possible by monitoring the sliding trajectory along the discontinuous boundary. In both cases, the probability distribution of recurrence times exhibits a power-law behavior, typical for extreme events. The results of this study can be of interest for engineering applications, for example, to predict extremely large-amplitude oscillations in cantilevers, systems with friction, lasers, and other systems with discontinuous boundaries.

Theoretical and experimental analysis of rare earth whispering gallery mode laser relative intensity noise

The relative intensity noise (RIN) of a solid state whispering-gallery-mode class-B laser is studied both theoretically and experimentally under different pumping regimes. In particular, we show that harmonics of the spiking frequency are observed in the RIN spectrum. A rate equation model including Langevin forces and the nonlinear coupling between inverted ion and photon number fluctuations has been developed to reproduce the experimental results and to extract relevant physical parameters from the fitting of the RIN spectrum.

Spatially inhomogeneous pattern formation due to parametric modulation in large broad-area lasers

The article discusses the formation of the optical patterns by modulation of the pump parameter, which may give rise to the Faraday instability in the wide-aperture lasers. The dynamics of class-B lasers is considered in the area of stability of the steady on-axis laser generation. Conditions for the appearance of spatio-temporal structures including stripes and oscillating hexagons are obtained.

Suppression of spatio-temporal instabilities in broad-area class-B lasers

In this paper we investigated the spatio-temporal instabilities of stationary lasing in broad-area class-B lasers. Conditions for the onset of filamentary instability and its spatio-temporal characteristics were obtained analytically. The lasing stabilization capabilities through varying laser parameters were considered. We have also shown that effective suppression of unstable field components may be achieved by external optical injection into the cavity.

Stabilization of flat-mirror vertical-external-cavity surface-emitting lasers by spatiotemporal modulation of the pump profile

We propose and demonstrate theoretically that vertical-external-cavity surface-emitting lasers (VECSELs) with external flat mirrors can be stabilized by applying a periodic spatiotemporal modulation of the pump current. Such pump modulation is shown to suppress the pattern-forming instabilities (modulation instabilities), which eventually results in stable beam emission. A modified Floquet linear stability analysis is used to characterize the dynamics of the modulated system and to evaluate its stabilization performance. Stability maps identify the regions in parameter space for complete and partial stabilization of VECSELs operating in different regimes depending on the external-cavity length. In particular, the stabilization method is shown to operate most efficiently in Class-A laser limit (for relatively long VECSEL resonators), while it becomes ineffective in Class-B laser limit (for relatively short resonators). The stabilization effect is further confirmed through direct integration of the dynamical equations.

Bifurcation phenomena and control analysis in class-B laser system with delayed feedback

In this paper, we study dynamics in delayed class-B laser system, with particular attention focused on Hopf and double Hopf bifurcations. Firstly, we identify the critical values for stability switches, Hopf and double Hopf bifurcations and derive the normal forms near the Hopf and double Hopf bifurcations critical points. By analyzing local dynamics near bifurcation critical points, we show how the delayed feedback control parameters effect the dynamical behaviors of the system. Furthermore, detailed numerical analysis using MATLAB extends the local bifurcation analysis to a global picture, and stable windows are observed as we change control parameter. Namely, even for parameter values not chosen in the neighborhood of the Hopf bifurcation critical points, two families of stable periodic solutions, which are resulted from Hopf bifurcation, exist in a large region of delay, and they merge into a family of stable and globally existed periodic solutions. Finally, by choosing proper control parameters, numerical simulations, including stable equilibrium, stable periodic solutions and stable quasiperiodic solutions are presented to demonstrate the theoretical results. Therefore, in accordance with above theoretical analysis, reasonable lasers with proper control parameters can be designed in order to achieve various applications.

Formation mechanism of bifurcation in mode-locked class-B laser

The random oscillations of many longitudinal modes are inevitable in both class −A and −B lasers due to their broadened atomic bandwidths. The destructive superposition of electric field components that are incoherently oscillating at the different longitudinal modes can be converted into a constructive one by using the mode-locking technique. Here, the Maxwell—Bloch equations of motion are solved for a three-mode class-B laser under the mode-locking conditions. The results indicate that the cavity oscillating modes are shifted by changing the laser pumping rate. On the other hand, the frequency components of cavity electric field simultaneously form the various bifurcations. These bifurcations satisfy the well-known mode-locking conditions as well. The atomic population inversion forms only one bifurcation, which is responsible for shaping the cavity electric field bifurcations.

Non-resonant double Hopf bifurcation of a class-B laser system

We investigate the double Hopf bifurcation at positive equilibrium point of a class-B laser system. Firstly, we give the critical values of Hopf and double Hopf bifurcations. Secondly, we implement the normal form method and the center manifold theory for a class-B laser system, and derive the universal unfolding of the double Hopf bifurcation. Thirdly, some dynamic behaviors, such as periodic and quasi-periodic solutions, are observed using numerical simulation.

Noise fluxes produced by the cavity Langevin force in above-threshold single-mode class-B lasers

It has been customary to ignore the key role of the cavity Langevin force on the laser noise fluxes. Here, the noise fluxes of the laser pump and spontaneous and stimulated emission (laser) radiation are calculated for class-B lasers by considering the cavity Langevin force alone. They have a symmetrical double-peak profile around the cavity resonance frequency, which is in complete agreement with the experimental results of the heterodyne gain profile of class-B laser amplifiers. This agreement indicates the lasers undertake the same mechanism in amplifying an input signal as superimposing noise fluxes on their variables. Meanwhile, the noise flux profiles of the cavity Langevin force are consistent with those produced by two other Langevin forces, associated with atomic population inversion and the atomic dipole moment. Finally, the noise fluxes generated by the cavity Langevin force are confirmed by demonstrating the flux conservation law.

Theory of gain and mode locking in free-running class-B lasers with simultaneous oscillation of three longitudinal modes

The operation of a three-mode laser with a homogeneous broad bandwidth is modeled as three different harmonic oscillators coupled to each other by the usual Maxwell–Bloch equations. The main aim is to extend and compare the laser variables when it moves from a single to a three-mode state. A multi-component model for the cavity electric field and the atomic population inversion is introduced to describe the simultaneous oscillation of the laser central and adjacent modes. The phase relations and the frequency shifts of these components in mode-locking conditions are determined. We also investigate the effect of the laser pumping rate and the mode frequency separation on the amplification of modes. Finally, the amplification gains of all three modes are calculated and their agreement with the energy conservation law is demonstrated.