Rate Equation Formalism Research Articles

Optical injection locking and optical-fiber data transmission by directly modulated wavelength tunable laser transmitters

Enhancement of the small- and large-signal modulation performance of wavelength tunable laser diode (TLD) transmitters under strong optical injection locking (OIL) is investigated numerically in back-to-back and optical-fiber transmission schemes. Our model is based on the spatiotemporal description of laser dynamics as due to the composite cavity design of TLDs, the usual rate equation formalism is not directly applicable. We demonstrate that TLD transmission strongly depends on wavelength tuning, which was investigated over a 21-nm range between 1529 and 1550 nm emission wavelengths. The best performance for both free-running (FR) and OIL TLDs is achieved at shorter wavelengths, 1529 nm for our device. Although in both cases this is due to larger differential material gain at shorter wavelengths, the underlying physics of the effect is completely different. For an FR TLD, it is the resonance oscillation frequency (ROF) that defines the best modulation speed, while for an OIL TLD, the achievable modulation speed depends on the cavity mode shift due to optical injection. Both the ROF and the cavity mode shift increase when the differential gain increases. However, the ROF is the device’s fixed parameter, while the cavity mode shift is defined by the OIL conditions, and thus, it can be optimized. The superior performance of the optical fiber digital data transmission with the OIL TLD is demonstrated at around 20-Gb/s modulation speed for standard fibers. This result is attributed to an enhanced modulation response and suppressed frequency chirping of the OIL TLD, and it is important for practical utilization of TLD transmitters.

Overview of kinetic Monte Carlo methods used to simulate microstructural evolution of materials under irradiation

Kinetic Monte Carlo (KMC) methods are commonly used to simulate the microstructure evolution of metals under irradiation due to their ability to generate the random walks underlying defect-mediated diffusion processes at the atomic scale. However, the range of applicability of KMC methods is severely limited by the kinetic trapping of the simulated trajectories within low energy basins presenting small intra-basin barriers. This results in dramatically reducing the efficiency of the classical KMC algorithm. Kinetic trapping can be alleviated by implementing non-local jumps relying on the theory of absorbing Markov chains. A factorisation of an auxiliary absorbing transition matrix then allows to generate escaping paths and first-passage times out of trapping basins. Although, the speed-up can be of several orders of magnitudes, this is sometimes not enough for very long-term prediction. We must then turn to homogenised rate-equation formulation of the problem. Usually solved deterministically, the corresponding large ordinary differential equation system often suffers from the curse of dimensionality. Dedicated Monte Carlo schemes can simulate the coarse-grained rate equations based on a chemical master equation. Finally, we show the relevance of relaxing the rigid-lattice assumption in the calculation of the free energy barriers and attempt frequencies to capture elastic effects that are important for certain systems, such as high entropy alloys. The activation-relaxation technique can be used for this purpose in kinetic Monte Carlo studies of slow diffusion processes.

Rate-equation approach for multi-level quantum systems

Strong driving of quantum systems opens opportunities for both controlling and characterizing their states. For theoretical studying of these systems properties we use the rate-equation formalism. The advantage of such approach is its relative simplicity. We used the formalism for description of a two-level system with further expanding it on a case of a multi-level system. Obtained theoretical results have good agreement with experiments. The presented approach can also be considered as one more way to explore properties of quantum systems and underlying physical processes such as, for instance, Landau–Zener–Stückelberg–Majorana transitions and interference.

Electron-boson-interaction induced particle-hole symmetry breaking of conductance into subgap states in superconductors

Particle-hole symmetry (PHS) of conductance into subgap states in superconductors is a fundamental consequence of a noninteracting mean-field theory of superconductivity. The breaking of this PHS has been attributed to a noninteracting mechanism, i.e., quasiparticle poisoning (QP), a process detrimental to the coherence of superconductor-based qubits.Here, we show that the ubiquitous electron-boson interactions in superconductors can also break the PHS of subgap conductances. We study the effect of such couplings on the PHS of subgap conductances in superconductors using both the rate equation and Keldysh formalism, which have different regimes of validity. In both regimes, we found that such couplings give rise to a particle-hole $asymmetry$ in subgap conductances which increases with increasing coupling strength, increasing subgap-state particle-hole content imbalance and decreasing temperature. Our proposed mechanism is general and applies even for experiments where the subgap-conductance PHS breaking cannot be attributed to QP.

Upconversion luminescence in transparent oxyfluoride glass ceramics containing hexagonal NaErF4

Transparent oxyfluoride glass ceramics containing hexagonal NaErF4 nanocrystals were synthesized by a heat treatment of a precursor glass obtained by the melt quenching technique. Combined X-ray diffraction (XRD) and transmission electron microscopy (TEM) analysis revealed the formation of single phase nanocrystals in the glass ceramics. The enhancement of intensity and spectral change of upconversion luminescence (UCL) confirmed the presence of Er3+ ions in the crystalline phase of the glass ceramics. Dominant energy transfer processes were identified using the rate equation formalism. Time-resolved site-selective spectroscopy studies at low temperatures were employed to elucidate the local structure of Er3+ ions in the glass ceramics and microcrystalline NaErF4. The origin of multi-site formation in hexagonal NaErF4 lattice is discussed.

Intensity noise and bandwidth analysis of nanolasers via optical injection.

A measurement method that can be used to extract the relative intensity noise of a nanolaser is introduced and analyzed. The method is based on optical injection of emission from a nanolaser, serving as a master oscillator, transferring its intensity fluctuations to a low-noise semiconductor laser serving as a slave oscillator. Using the stochastic rate equation formalism, we demonstrate that the total relative intensity noise of the system is a weighted superposition of the relative intensity noise of individual lasers. We further discuss the analytical relations that can be used to extract the relative intensity noise spectrum of a nanolaser. Finally, we use mutual correlation as a mathematical tool to quantify the degree of resemblance between the injected and extracted intensity fluctuations, theoretically confirming that the spectra are at least 97% correlated within the 3-dB bandwidth when an injection strength is chosen properly.

Reduction Rate of 1-Phenyl Phospholane 1-Oxide Enhanced by Silanol Byproducts: Comprehensive DFT Study and Kinetic Modeling Linked to Reagent Design.

Important stoichiometric transformations like Wittig and Appel reactions have been implemented in a catalytic fashion in the past decade. The phosphine oxide generated in situ can be reintroduced as phosphine into the catalytic cycle using mild and selective silane reagents (redox-driven catalysis). While the field of experimental investigation has been fully expanding in the past decade, theoretical studies are still sparse. In this present work, density functional theory (DFT) has been used to characterize the free energy surfaces of the reduction of 1-phenyl phospholane 1-oxide with four different silanes. Found stationary points have been studied in-depth to highlight mechanistic peculiarities, like the effect of substituents at the silicon center and the parallel and competitive reactivity between the precursor silanes and their semioxidized byproducts. Calculated thermodynamic parameters in combination with "real" values for concentrations have been used in the formulation of rate equations for simple bimolecular and monomolecular steps of the mechanism. The deterministic integration concentrations versus time of such rate equations led to a realistic description of the systems under study and paved the way to strategic and rational design of new silanes with increased reactivity.

Importance of coherence in models of mid-infrared quantum cascade laser gain spectra

We present a three-level model based on a density matrix to examine the influence of coherence and dephasing on the gain spectrum of mid-infrared quantum cascade lasers. The model is used to examine a quantum cascade active region with multiple optical transitions. We show how coherence can explain the origin of additional peaks in the gain spectrum. We also analyze the spectra calculated using the three-level model with a rate equation formalism to demonstrate the importance of considering interface roughness and limitations of the rate equation formalism. Specifically, we present how interface roughness influences the broadening and oscillator strength that are recovered using a rate equation analysis. The results of this work are important when considering the design of active regions with multiple optical transitions and could lead to devices with improved performance.

Stochastic Model of Energy-Transfer Processes Among Rare-Earth Ions. Example of Al2O3:Tm3+

Energy-transfer processes strongly affect the performance of lanthanide-doped photonic devices. In this work, we introduce a simple stochastic model of energy-transfer processes and successfully apply it to the example of cross-relaxation (CR) and energy-transfer upconversion (ETU) in amorphous Al2O3:Tm3+ waveguides on silicon intended for lasers operating at ∼2 μm. The stochastic model is based on the rate-equation formalism and considers two spectroscopically distinct ion classes, namely single ions and ions with neighbors (pairs and clusters), with the corresponding ion fractions being dependent on the doping concentration. We prove that a more accurate description of the luminescence properties of amorphous Al2O3:Tm3+ is obtained when accounting for the presence of these distinct ion classes. Based on the developed model, we derive microscopic CR and ETU parameters of CCR = 5.83 × 10–38 cm6 s–1, CETU1 = 0.93 × 10–40 cm6 s–1, and CETU2 = 7.81 × 10–40 cm6 s–1, and determine the laser quantum efficiency ...

Optimal pump pulse shapes for optical parametric oscillators

In this paper we theoretically investigate the most efficient pump temporal shape for pulsed operation of optical parametric oscillators (OPOs). The theoretical analysis consists of solving the optimal control problem using the rate equation formalism. The OPO optimal buildup behavior is derived for any OPO configuration, and we show that the optimal pump pulse shape is a double rectangle. The first rectangle is optimized for the buildup process, while the second one is optimized for the steady state, and both depend differently on the OPO mirror reflectivities as well as the available pump energy and maximum peak power. This double rectangle is quantitatively described and is consistent with previous results reported in the literature. Also, numerical simulations are used to validate our analytical model on a doubly resonant OPO and show consistent results. The numerical simulations also show that an optimal top-hat temporal profile could provide performances very close to the ones achieved with the optimal double rectangle.

Pump duration optimization for optical parametric oscillators

In this paper we theoretically and experimentally investigate the optimal pulse settings to pump optical parametric oscillators (OPOs) in the transient regime. The theoretical analysis is carried out with an extended OPO rate-equation formalism that yields simple expressions when applied to rectangular pump pulses while fully taking into account pump depletion and back-conversion effects. The OPO buildup time is investigated in detail, leading to a very simple analytical formula. The study then focuses on engineering rules for the choice of the optimal rectangular-pump-pulse power and duration giving the best conversion efficiency for given input pump energy and OPO features. Making the most of a master oscillator-fiber power amplifier’s pulse-tailoring opportunities, experimental investigations covering pulsed operation ranging from nanosecond to microsecond pump duration are carried out in the case of a doubly-resonant OPO. Very good agreement with the theoretical analysis is obtained.

Electron transport in nanoscale junctions with local anharmonic modes.

We study electron transport in nanojunctions in which an electron on a quantum dot or a molecule is interacting with an N-state local impurity, a harmonic ("Holstein") mode, or a two-state system ("spin"). These two models, the Anderson-Holstein model and the spin-fermion model, can be conveniently transformed by a shift transformation into a form suitable for a perturbative expansion in the tunneling matrix element. We explore the current-voltage characteristics of the two models in the limit of high temperature and weak electron-metal coupling using a kinetic rate equation formalism, considering both the case of an equilibrated impurity, and the unequilibrated case. Specifically, we show that the analog of the Franck-Condon blockade physics is missing in the spin-fermion model. We complement this study by considering the low-temperature quantum adiabatic limit of the dissipative spin-fermion model, with fast tunneling electrons and a slow impurity. While a mean-field analysis of the Anderson-Holstein model suggests that nonlinear functionalities, bistability and hysteresis may develop, such effects are missing in the spin-fermion model at the mean-field level.

Temperature dependence of 3P0 Pr3+ fluorescence dynamics in Y4Al2O9 crystals

Temperature-dependent emission spectra and fluorescence dynamics profiles have been investigated in Pr3+:Y4Al2O9 crystals in order to better understand the processes responsible for quenching of the praseodymium 3P0 emissions. The cross-relaxation transfer rates were experimentally determined as a function of temperature. Using the rate equations formalism, the dynamics of the observed emissions were modeled. Basing on comparison between the measured and calculated decays, the energy transfer rates between Pr3+ ions were evaluated. The role of the backward process in explanation of the complicated character of 3P0 decays and its temperature dependence, especially its unexpectedly slow decaying component, were established.

An indirectly pumped terahertz quantum cascade laser with low injection coupling strength operating above 150 K

We designed and demonstrated a terahertz quantum cascade laser based on indirect pump injection to the upper lasing state and phonon scattering extraction from the lower lasing state. By employing a rate equation formalism and a genetic algorithm, an optimized active region design with four-well GaAs/Al0.25Ga0.75As cascade module was obtained and epitaxially grown. A figure of merit which is defined as the ratio of modal gain versus injection current was maximized at 150 K. A fabricated device with a Au metal-metal waveguide and a top n+ GaAs contact layer lased at 2.4 THz up to 128.5 K, while another one without the top n+ GaAs lased up to 152.5 K (1.3ℏω/kB). The experimental results have been analyzed with rate equation and nonequilibrium Green's function models. A high population inversion is achieved at high temperature using a small oscillator strength of 0.28, while its combination with the low injection coupling strength of 0.85 meV results in a low current. The carefully engineered wavefunctions enhance the quantum efficiency of the device and therefore improve the output optical power even with an unusually low injection coupling strength.

Z-scan study of nonlinear absorption in novel lanthanide bis-phthalocyanines

We report our results on the nonlinear absorption of novel sandwich lanthanide(III) double-decker complexes BuPc2Ln (BuPc=2,3,9,10,16,17,23,24-octabutyl-phthalocyaninate; Ln=Eu, Dy, Er, Lu) studied in tetrahydrofuran (THF) solution using open aperture z-scan technique with 350ps laser pulses at 532nm. All the compounds exhibit reverse saturable (RSA) absorption. With approaching to the focus a weak saturation of absorption was observed, while at the focus a strong decrease in transmittance took place. Employing the rate equations formalism we demonstrate RSA being mainly contributed by the triplet state absorption.

Temporal dynamics of IR-to-visible up-conversion in LiNbO_3:Er^3+/Yb^3+: a path to phosphors with tunable chromaticity

In this work, a study of the temporal dynamical behavior of IR-visible up-conversion in the LiNbO3: Er3+/Yb3+ system under modulated IR excitation at 980 nm is presented. It is shown that modulation characteristics, including the relative green-to-red emission ratio (GRR), can be quantitatively explained by using the rate equations formalism. The relevant spectroscopic magnitudes are identified and this provides a general framework to explore the properties of other Er3+/Yb3+ doped up-converting phosphors.

The effects of energy transfer on the Er3+ 1.54 μm luminescence in nanostructured Y2O3 thin films with heterogeneously distributed Yb3+ and Er3+ codopants

We report the effects of heterogeneous Yb3+ and Er3+ codoping in Y2O3 thin films on the 1535 nm luminescence. Yb3+:Er3+:Y2O3 thin films were deposited using sequential radical enhanced atomic layer deposition. The Yb3+ energy transfer was investigated for indirect and direct excitation of the Yb 2F7/2 state using 488 nm and 976 nm sources, respectively, and the trends were described in terms of Forster and Dexter's resonant energy transfer theory and a macroscopic rate equation formalism. The addition of 11 at. % Yb resulted in an increase in the effective Er3+ photoluminescence (PL) yield at 1535 nm by a factor of 14 and 42 under 488 nm and 976 nm excitations, respectively. As the Er2O3 local thickness was increased to greater than 1.1 Å, PL quenching occurred due to strong local Er3+ ↔ Er3+ excitation migration leading to impurity quenching centers. In contrast, an increase in the local Yb2O3 thickness generally resulted in an increase in the effective Er3+ PL yield, except when the Er2O3 and Yb2O3 layers were separated by more than 2.3 Å or were adjacent, where weak Yb3+ ↔ Er3+ coupling or strong Yb3+ ↔ Yb3+ interlayer migration occurred, respectively. Finally, it is suggested that enhanced luminescence at steady state was observed under 488 nm excitation as a result of Er3+ → Yb3+ energy back transfer coupled with strong Yb3+ ↔ Yb3+ energy migration.

A phonon scattering assisted injection and extraction based terahertz quantum cascade laser

A lasing scheme for terahertz quantum cascade lasers, based on consecutive phonon-photon-phonon emissions per module, is proposed and experimentally demonstrated. The charge transport of the proposed structure is modeled using a rate equation formalism. An optimization code based on a genetic algorithm was developed to find a four-well design in the GaAs/Al0.25Ga0.75As material system that maximizes the product of population inversion and oscillator strength at 150 K. The fabricated devices using Au double-metal waveguides show lasing at 3.2 THz up to 138 K. The electrical characteristics display no sign of differential resistance drop at lasing threshold, which, in conjunction with the low optical power of the device, suggest—thanks to the rate equation model—a slow depopulation rate of the lower lasing state, a hypothesis confirmed by non-equilibrium Green’s function calculations.

Picosecond pump-probe of carrier relaxation time in deep levels of HgCdTe

The non-equilibrium carrier dynamics in HgCdTe has been studied with picosecond pump-probe measurements.The delay-time dependent differential transmission shows a negative minimum after the absorption saturation maximum,which can be attributed to an additional absorption process because of the deep levels.A rate equation formalism has been employed to model the carrier dynamics.The experimental differential transmission spectrum has been fitted very well with the calculations when two deep level relaxation time constants are introduced,implying two types of the deep levels.The variation of the relaxation time with the pump intensity has also been studied.

Kinetic analyses and mathematical modeling of primary photochemical and photoelectrochemical processes in plant photosystems

In this paper the model and simulation of primary photochemical and photo-electrochemical reactions in dark-adapted intact plant leaves is presented. A descriptive algorithm has been derived from analyses of variable chlorophyll a fluorescence and P700 oxidation kinetics upon excitation with multi-turnover pulses (MTFs) of variable intensity and duration. These analyses have led to definition and formulation of rate equations that describe the sequence of primary linear electron transfer (LET) steps in photosystem II (PSII) and of cyclic electron transport (CET) in PSI. The model considers heterogeneity in PSII reaction centers (RCs) associated with the S-states of the OEC and incorporates in a dark-adapted state the presence of a 15–35% fraction of QB-nonreducing RCs that probably is identical with the S0 fraction. The fluorescence induction algorithm (FIA) in the 10μs–1s excitation time range considers a photochemical O-J-D, a photo-electrochemical J-I and an I-P phase reflecting the response of the variable fluorescence to the electric trans-thylakoid potential generated by the proton pump fuelled by CET in PSI. The photochemical phase incorporates the kinetics associated with the double reduction of the acceptor pair of pheophytin (Phe) and plastoquinone QA [PheQA] in QB nonreducing RCs and the associated doubling of the variable fluorescence, in agreement with the three-state trapping model (TSTM) of PS II. The decline in fluorescence emission during the so called SMT in the 1–100s excitation time range, known as the Kautsky curve, is shown to be associated with a substantial decrease of CET-powered proton efflux from the stroma into the chloroplast lumen through the ATPsynthase of the photosynthetic machinery.