Set Of Nonlinear Rate Equations Research Articles

Relative Intensity Noise Reduction by Optimizing Fiber Grating Fabry–Perot Laser Parameters

A set of nonlinear rate equations that can describe an external cavity laser with any arbitrary external optical feedback (OFB) level are derived. A comprehensive study on the relative intensity noise (RIN) characteristics of a fiber grating Fabry-Perot is performed numerically. In this paper, fiber Bragg grating (FBG) is used as a wavelength lasing selective element to control the external OFB level, thereby control the RIN. In addition to the external OFB level, the effect of other external cavity parameters such as temperature, injection current, cavity volume, gain compression factor, and FBG parameters on RIN characteristics is investigated. The temperature dependence (TD) of RIN is calculated according to TD of laser parameters instead of well-known Parkove relationship. Results show that by optimization, the peak value of the RIN can be reduced down to around -150 dB/Hz. The optimum and the shortest external cavity length that provides the minimum RIN is found to be around 3.1 cm. In addition, by optimization, the relaxation oscillation frequency of RIN spectra is shifted toward around 5.6 GHz.

Emergent Properties of Interacting Populations of Spiking Neurons

Dynamic neuronal networks are a key paradigm of increasing importance in brain research, concerned with the functional analysis of biological neuronal networks and, at the same time, with the synthesis of artificial brain-like systems. In this context, neuronal network models serve as mathematical tools to understand the function of brains, but they might as well develop into future tools for enhancing certain functions of our nervous system. Here, we present and discuss our recent achievements in developing multiplicative point processes into a viable mathematical framework for spiking network modeling. The perspective is that the dynamic behavior of these neuronal networks is faithfully reflected by a set of non-linear rate equations, describing all interactions on the population level. These equations are similar in structure to Lotka-Volterra equations, well known by their use in modeling predator-prey relations in population biology, but abundant applications to economic theory have also been described. We present a number of biologically relevant examples for spiking network function, which can be studied with the help of the aforementioned correspondence between spike trains and specific systems of non-linear coupled ordinary differential equations. We claim that, enabled by the use of multiplicative point processes, we can make essential contributions to a more thorough understanding of the dynamical properties of interacting neuronal populations.

Fast Numerical Calculation for a Set of Nonlinear Rate Equations: Application to Rapid Ionizaton Phase in Plasma

To make a useful contribution to the development of the compact X-ray laser, a modified quasi-steady-state approximation is applied to the fast numerical calculation of rate equations for the rapid ionization phase of a laser-produced plasma, to which the usual quasi-steady-state model by Bates et al. is not applicable. The set of nonlinear rate equations for the excited level populations coupled with the electron density and temperature for the plasma are numerically calculated with sufficient accuracy for their temporal evolution. The calculation also shows that a suitable combination is possible between the power and pulse width of a heating laser so that a high-density plasma can be obtained as the X-ray laser medium. The mathematical meaning of our approximation is briefly mentioned and its applicability is discussed.

Dynamics of a high-Q vertical-cavity organic laser

We investigate the dynamics of the organic laser guest-host composite of tris-(8-hydroxy quinoline) aluminium and 4-(dicyanomethylene)-2-methyl-6-(p-dimethylaminostyryl)-4H-pyran embedded in a high-Q (Q≈4500) double distributed Bragg reflector microcavity using subpicosecond up-conversion techniques. Lasing is observed at a threshold of 0.4nJ∕20μJcm−2 with a linewidth of 0.05 nm (resolution limit). We observe a strongly nonlinear intensity-dependent delay of the emitted radiation burst. All experimental results are successfully modeled by a set of nonlinear rate equations, emphasizing the importance of a feedback mechanism for lasing.

Low frequency fluctuation with two external cavity reflectors

The dynamics of a semiconductor laser with two optical feedbacks is studied in this paper. A new set of nonlinear rate equations that can describe external cavity semiconductor lasers with any amount of two optical feedbacks is proposed. It is found that when the laser is biased above the threshold and subjected to one feedback, the other feedback can induce low-frequency fluctuations.

Phase transition of a photogenerated electron gas in semiconductors

We study a nonequilibrium phase-transition-like behavior of a photogenerated electron gas in semiconductors. The kinetics of the electron gas is given by a set of nonlinear rate equations. For low temperatures we show that they have three steady-state solutions when the photoexcitation energy is in a certain interval that depends on the electron-electron interaction. Two of them are stable and the other is unstable. By defining a generalized free potential we obtain the Maxwell construction that determines the order parameter, namely, the difference of the electron population in the bottom of the conduction band of the two steady-stable states. Hence, this phase transition is a nonequilibrium first-order phase transition.

Inter-Exciton Exchange Interaction and Nonlinear Luminescence Polarization Dynamics in III–V Semiconductor Quantum Wells

The effects of the inter-exciton exchange interaction in the time evolution of the exciton luminescence intensity and polarization in III–V semiconductor quantum wells are investigated theoretically. The luminescence dynamics in the resonant and high density excitation regime is simulated with a set of nonlinear rate equations for the occupation numbers of a finite set of exciton spin states, and studied as function of the polarization and intensity of the exciting pulse. The spin flip transitions are driven by both intra- and inter-exciton exchange interaction. Two contributions to the latter are included in the simulation, namely the spin mixing in the symmetrized biexciton state and the spin dependent free exciton–exciton scattering. It is found that the fast decay observed experimentally at sufficient high excitation intensities, in both luminescence intensity and polarization, is reproduced with reasonable values for the relaxation times only when both contributions to the inter-exciton exchange interaction are turned on.

Relative intensity noise for laser diodes with arbitrary amounts of optical feedback

A unified study of the noise characteristics of semiconductor lasers with optical feedback and short external cavity length is presented. A new set of nonlinear rate equations that can describe a laser diode with any amount of optical feedback is proposed. The relative intensity noise (RIN) is calculated by using a numerical solution of these equations. This paper concentrates mainly on the moderate and strong feedback regimes. The spectral phenomena observed during the transition from the weak feedback to the coherence collapse regime and then to the strong feedback regime are studied and explained. The effect of the variation of some of the laser diode parameters on the RIN characteristics is also investigated.

Nonlinear waves in a rotating plasma cylinder

Previous work on exact cylindrical surface waves in a nonneutral cold-electron plasma bounded by a dielectric is extended to include plasma rotation. A set of nonlinear rate equations describing the temporal behavior of the system is derived by first determining an appropriate spatial wave structure. Physically relevant periodic solutions are obtained. >

Excess electron motion in a molecular chain and nonlinear rate equations

The nonlinear quantum dynamics of the simultaneous transfer of two electrons in a molecular chain are studied. The discussion concentrates on the interplay between the mutual Coulomb interaction and the dephasing of the electronic wave function. A nonexponential redistribution of the electronic occupation probability has been obtained by the numerical solution of appropriate equations of motion of the electronic density matrix. Such a behavior can be explained in deriving a set of nonlinear rate equations which take into account the shift of the electronic site energies due to the mutual Coulomb interaction. This approximate description agrees well with the exact one for dephasing times equal and smaller than the coherent transfer time.

Model for hot-electron phenomena: Theory and general results.

We propose a model for the description of the hot-electron phenomenon in semiconductors. Based on this model we are able to reproduce accurately the main characteristics observed in experiments of electric-field transport, optical absorption, steady-state photoluminescence, and relaxation processes. Our theory contains neither free, nor adjustable parameters, it is very fast computationally, and it incorporates the main collision mechanisms including screening and phonon-heating effects. Our description is based on a set of nonlinear rate equations in which the interactions are represented by coupling coefficients or effective frequencies. We calculate these coefficients from the characteristic constants and the band structure of the material.

Mean-field theory of multilayer physisorption. III. Desorption kinetics

Starting from a set of nonlinear rate equations with phonon-mediated transition probabilities calculated from the self-consistent solutions of temperature-dependent mean-field theory, we calculate temperature- and coverage-dependent desorption times for the $^{3}\mathrm{He}$-graphite, $^{4}\mathrm{He}$-graphite, $^{3}\mathrm{He}$-Ar(100), and $^{3}\mathrm{He}$-Constantan systems up to a coverage of two monolayers. We find (a) a significant drop in the heat of adsorption and in the prefactor, (b) a prononunced compensation effect, and (c) a change over from first-order desorption at low coverage to zero-order evaporation for $\ensuremath{\bigominus}>1.5$.

Physisorption Kinetics from Mean-Field Theory: Compensation Effect near Monolayer Coverage

Based on a set of nonlinear rate equations with phonon-mediated transition rates calculated from mean field theory, the coverage-dependent isothermal desorption time for $^{3}\mathrm{He}$ on graphite up to a coverage of 1.5 adlayers is determined. Prefactors in the Arrhenius parametrization drop by 2 orders of magnitude as the heat of adsorption decreases for increasing coverage, establishing a compensation effect in physisorption kinetics.

Small prefactors and compensation effect in physisorption kinetics

Starting from a set of nonlinear rate equations with phonon-mediated transition probabilities calculated from the self-consistent solutions of temperature-dependent mean field theory, we calculate temperature and coverage dependent desorption times for 3He desorbing from constantan (a Cu-Ni alloy) up to a coverage of θ = 1.5 layers. We find (a) a pronounced compensation effect; (b) a drop in the prefactor v from 10 10 s -1 at θ = 0 to 10 7 s -1 at θ = 1.5; (c) a changeover from first order desorption at low coverage to zero order evaporation for θ ≳ 1.2.

Small prefactors and compensation effect in physisorption kinetics

Starting from a set of nonlinear rate equations with phonon-mediated transition probabilities calculated from the self-consistent solutions of temperature-dependent mean field theory, we calculate temperature and coverage dependent desorption times for 3He desorbing from constantan (a Cu-Ni alloy) up to a coverage of θ = 1.5 layers. We find (a) a pronounced compensation effect; (b) a drop in the prefactor ν from 10 10 s −1 at θ = 0 to 10 7 s −1 at θ = 1.5; (c) a changeover from first order desorption at low coverage to zero order evaporation for θ ≳ l.2.

Kinetics of order–disorder transformations in alloys. II

As a continuation of previously published work, isothermal ordering and disordering processes (of B2 type structure) in bcc binary alloys from initial states both near and far from equilibrium are investigated theoretically. The vacancy mechanism for atomic migration is assumed, interactions among nearest neighbors are taken into account, and the pair approximation of the path probability method of irreversible, cooperative phenomena is utilized. This leads to a set of nonlinear rate equations derived in a systematic, statistical fashion which describes the dynamical behavior of the system. The rate equations can be linearized near equilibrium and are shown to obey in this form the formalism of irreversible thermodynamics of Onsager. The driving forces are identified as the partial derivatives of the free energy function of the system with respect to state variables. Several relaxation times are obtained which characterize near equilibrium relaxation, and their meanings are discussed. Nonlinear coupling of relaxation modes occurs in relaxation processes from states far from equilibrium. Especially, a distinct three step relaxation of internal energy and a two step relaxation of the long range order are predicted when the alloy is quenched from states having zero or small long range order. The behavior is strikingly similar to the three step relaxation process of the internal energy observed in Ni3Fe near the order–disorder transition point. These relaxation effects are related to the geometry of the free energy surface.

Turbulent heating in a moderate energy electron beam plasma experiment

The excitation and dissipation of electron beam driven electrostatic turbulence has been investigated experimentally and theoretically. Optimization of the beam to wave power transfer efficiency is shown to depend on the details of the linear dispersion of plasma waves on the bounded plasma column. In addition, the time variation of the cw operated beam plasma system has been modeled by a coupled set of nonlinear rate equations for the intensity of unstable field fluctuations, the electron temperature, and the plasma density. Agreement with the experiment is excellent allowing identification of the important dynamics. In particular, the dominant dissipation mechanism is shown to be stochastic heating of (nonresonant) thermal electrons.

On the theory of selection of coupled macromolecular systems

In this paper we examine a set of nonlinear rate equations (devised by M. Eigen (1971) ) which describe the process of selection in a collection of self-reproducing, macromolecular information carriers. We construct exact solutions to the equations for the case of constant rate parameters and constant error distributions. The solutions allow the direct assessment of the effect of mutations on the “selective value” parameters discussed by Eigen as well as the distribution of the molecular species selected in steady state. In addition we show that the selection process may be characterized by an extremal principle.

Dynamics of the CO2 Atmospheric Pressure Laser with Transverse Pulse Excitation

The dynamical processes responsible for laser emission in the pulsed pumping of a transversely excited atmospheric (TEA) CO2 laser are investigated. An explanation for the formation of the giant pulse is proposed on the basis of a gain-switching mechanism in which it is assumed that with short strong-current pulses a high population inversion can be achieved prior to the onset of laser action. The kinetics of the mechanism are described by means of a set of nonlinear rate equations idealized to a four-energy-state system. With suitable initial conditions on the populations, the transient solution of these equations for the mixtures CO2–He and CO–N2–He appears to be consistent with the major features of experimental observation.