Photon Rate Equations Research Articles

The Investigation of Carrier Mobility Effect on the Performance Characteristics of the InGaN-Based Vertical Cavity Surface Emitting Laser (VCSEL) by Solving the Rate Equations

Background: Among the parameters that play an important role in describing the performance of many devices is carrier mobility which is a criterion for the easy movement in semiconductor crystals. Objective: The effect of carrier mobility on the performance characteristics of InGaN quantum well vertical-cavity surface-emitting laser was analytically investigated. Methods: By solving the Poisson’s equation, current density equation, charge concentration continuity equation and carrier and photon rate equations, the variation of current density and carrier density with respect to the position and time and the effects of carrier mobility and temperature on these parameters were investigated. Furthermore, the effect of mobility on the variation of output power versus the injection current and on the time variation of photon and carrier density and the output power was investigated. Results: By increasing the carrier mobility, the threshold current is reduced and the output power is increased. In studying the effect of temperature on the desired parameters, the variation of carrier density with respect to time and position was affected by the temperature change. This phenomenon is due to the dependence of these parameters on the diffusion coefficients and consequently on the mobility of the carriers and the dependence of mobility on temperature. Conclusions: The output power increased, and the time delay in accruing the laser decreased. Consequently, the carrier recombination increased, further resulting in a rapid laser operation.

Modeling power and linewidth of quantum dot superluminescent light emitting diode

A model for quantum dot (QD) superluminescent light emitting diode (SLED) is presented to study an optical power output and linewidth over a wide range of injection currents. The analysis is based on the photon and carrier rate equations including the effects of homogeneous and inhomogeneous broadening, carrier escape process, and high-current heating. The model is validated using experimental data available from the literature. The results show non-monotonic variations of the output optical power and linewidth of the SLED with the injection current density. It is seen that there exists an optimum injection current density for which the power–linewidth product becomes maximum for a given device length of the QD SLED.

Drift-Diffusion Simulation of High-Speed Optoelectronic Devices

In this paper, we address the problem of research and development of the advanced optoelectronic devices designed for on-chip optical interconnections in integrated circuits. The development of the models, techniques, and applied software for the numerical simulation of carrier transport and accumulation in high-speed AIIIBV (A and B refer to group III and V semiconductors, respectively) optoelectronic devices is the purpose of the paper. We propose the model based on the standard drift-diffusion equations, rate equation for photons in an injection laser, and complex analytical models of carrier mobility, generation, and recombination. To solve the basic equations of the model, we developed the explicit and implicit techniques of drift-diffusion numerical simulation and applied software. These aids are suitable for the stationary and time-domain simulation of injection lasers and photodetectors with various electrophysical, constructive, and technological parameters at different control actions. We applied the model for the simulation of the lasers with functionally integrated amplitude and frequency modulators and uni-travelling-carrier photodetectors. According to the results of non-stationary simulation, it is reasonable to optimize the parameters of the lasers-modulators and develop new construction methods aimed at the improvement of photodetectors’ response time.

Modelling and analysis of polarization noise in vertical cavity surface emitting LASERs

Previous researches have provided mathematical models of vertical cavity surface emitting LASER’s (VCSEL’s) that evaluated the effect of spontaneous noise on the Laser’s performance but completely neglected the effect of polarization noise. Also the results presented were in terms of RIN variation with respect to frequency. In this paper,a model is proposed to analyse the effect of spontaneous and polarization noise on VCSEL’s output power. Spontaneous noise is included by augmenting feedback in photon rate equations. Polarization noise is incorporated in the dynamics with the help of self-stabilizing and gain recovering coefficients. Characteristics of VCSEL’s output power are studied under continuous wave (CW) and sinusoidal modulation. VCSEL dynamics are simulated for both the cases with and without noise , firstly without noise and then with the effect of noise. The output light’s power curves depict the effect of spontaneous and polarization noise in VCSEL. For high frequency modulation, the outputs have fluctuations throughout the simulation time but noise amplifies these fluctuations, affecting the performance of LASER. For CW modulation, the output light shows exponential behaviour with respect to bias.

Quantum Efficiency of Quantum Dot Lasers

The quantum efficiency relates the calculated to the measured external threshold current of a laser. This quantity is often estimated from the length dependence of the external differential efficiency above threshold, assuming the carrier density is pinned. Often it is also assumed that the internal current varies linearly with the external current; it is shown here this is not the case due to the effect of stimulated emission on the current-voltage relation of the active region. Furthermore, it has been observed that spontaneous emission from inhomogeneous quantum dots does not pin above threshold which questions the determination of their optical loss. This nonpinning is reproduced by a model in which the laser photon rate equation is coupled to rate equations for the occupation of dot states mediated by a thermal phonon distribution. The threshold current from this model agrees with a Fermi–Dirac calculation but the external efficiency above threshold is lower and its length dependence gives a smaller value of mode loss than the input value. The reasons for this behavior are analyzed and it is concluded that a Fermi–Dirac calculation does not represent the light–current characteristics in quantum dot lasers at room temperature.

Self-consistent electro-opto-thermal model of quantum cascade lasers with coupled electron and phonon interactions far from equilibrium

A self-consistent model of quantum cascade lasers (QCLs) is presented here for the study of the QCL’s behavior in the far from equilibrium conditions. The approach is developed by employing a number of physics-based models such as the carrier and photon rate equations, the energy balance equation, the heat transfer equation and a simplified rate equation for the creation and annihilation of nonequilibrium optical phonons. The temperature dependency of the relevant physical effects such as stimulated gain cross section, longitudinal optical (LO) phonons and hot-phonon generation rates are included in the model. Using the presented model, the static and transient device characteristics are calculated and analyzed for a wide range of heat sink temperatures. Besides the output characteristics, this model also provides a way to study the hot-phonon dynamics in the device, and to explore the electron temperature and thermal roll-over in the QCLs.

Energy efficiency analysis of ArF excimer laser system

The reliable functioning and continual optimizing of ArF excimer laser system is of importance when it comes to productization into the market from a laboratory test machine. The analysis of dynamic characteristics of the system is vital to understanding its operating mechanism and optimizing the design theoretically. In this article, one-dimensional fluid model is used to analyze the excimer laser discharge mechanism, and the content ratio of fluorine gas, argon gas, and neon gas, which constitute a gas mixture, is studied in a simulated ArF excimer laser system. Particles are treated as a fluid, which significantly reduces the computing cost in fluid model, and therefore is suitable for high-pressure situation. Four equations are included in one-dimensional fluid model, i.e., Boltzmann equation that describes electron energy distribution, ion continue equation that illustrates ion number density, Poisson's equation that shows the distribution of electric field, and photon rate equation that demonstrates laser outputting process. By combining these four equations, high pressure plasma discharge process and particles stimulated radiation process are studied, and calculation continues from one time step to another until the end of discharging process. The result of the calculation presents energy transfer process from three aspects:energy deposition efficiency, ArF* formation, and laser outputting. In the energy deposition process, the energy deposition efficiency is sensitive to the change of fluorine gas ratio while the variation of the content ratio of other two gases has a less influence on this process. In addition, there exists an optimal fluorine gas ratio that causes the highest energy deposition efficiency. In the ArF* formation process, the reaction between excited argon ions and fluorine gas is the main channel that generates ArF*. The proper increasing of fluorine gas ratio helps form ArF*. In the laser outputting process, photon loss is mainly because of the reaction between fluorine negative ions and photons. Therefore superfluous fluorine gas in the mixture leads to less photons, which eventually results in low energy efficiency of laser. By summarizing the three aspects of energy transfer process, the fluorine gas ratio in the gas mixture plays a significant role in determining the energy efficiency of ArF excimer laser system. This theory is verified by experiments, showing that the deviation of the optimized fluorine gas ratio severely reduces energy efficiency. This conclusion can guide us in optimizing the design and steady reliable function of ArF excimer laser system.

Quantum Mechanical Treatment of Optical Amplifiers Based on Population Inversion

Optical amplification in a population-inverted medium is quantum-mechanically described, using the Heisenberg picture. Unlike conventional approaches that use a photon rate equation or a phenomenological field equation, the time evolution of the field operator, the photon number operator, and their squares are directly derived from first principles. The results give a logical theoretical base for the traditional phenomenological expressions describing quantum-mechanical characteristics of optical amplifiers.

Genetic algorithm applied to the optimization of quantum cascade lasers with second harmonic generation

A computational model for the optimization of the second order optical nonlinearities in GaInAs/AlInAs quantum cascade laser structures is presented. The set of structure parameters that lead to improved device performance was obtained through the implementation of the Genetic Algorithm. In the following step, the linear and second harmonic generation power were calculated by self-consistently solving the system of rate equations for carriers and photons. This rate equation system included both stimulated and simultaneous double photon absorption processes that occur between the levels relevant for second harmonic generation, and material-dependent effective mass, as well as band nonparabolicity, were taken into account. The developed method is general, in the sense that it can be applied to any higher order effect, which requires the photon density equation to be included. Specifically, we have addressed the optimization of the active region of a double quantum well In0.53Ga0.47As/Al0.48In0.52As structure and presented its output characteristics.

Optimizing optical nonlinearities in GaInAs/AlInAs quantum cascade lasers

Regardless of the huge advances made in the design and fabrication of mid-infrared and terahertz quantum cascade lasers, success in accessing the ~3-4 mm region of the electromagnetic spectrum has remained limited. This fact has brought about the need to exploit resonant intersubband transitions as powerful nonlinear oscillators, consequently enabling the occurrence of large nonlinear optical susceptibilities as a means of reaching desired wavelengths. In this work, we present a computational model developed for the optimization of second-order optical nonlinearities in In0.53Ga0.47As/Al0.48In0.52As quantum cascade laser structures based on the implementation of the Genetic algorithm. The carrier transport and the power output of the structure were calculated by self-consistent solutions to the system of rate equations for carriers and photons. Both stimulated and simultaneous double-photon absorption processes occurring between the second harmonic generation-relevant levels are incorporated into rate equations and the material-dependent effective mass and band non-parabolicity are taken into account, as well. The developed method is quite general and can be applied to any higher order effect which requires the inclusion of the photon density equation.

Fast periodic modulations in the photon correlation of single-mode vertical-cavity surface-emitting lasers

We report on the observation of periodic modulations in the second-order photon correlation of single-mode vertical-cavity surface-emitting lasers (VCSELs) in steady-state operations. We realized the fast photon correlation measurement of VCSELs by using a streak camera operated in a photon-counting mode. Fast modulation signals were observed in second-order photon correlations at and around the threshold.The calculations using photon rate equations that include a gain medium dynamics shows that the oscillating signals originate from relaxation oscillations.

Analytical theory of a small signal modulation response of a transistor laser with dots-in-well in the base

We have developed the theory for the threshold base current, light power output and small signal modulation response of a transistor laser, the base of which contains a quantum well (QW) with a layer of quantum dots (QDs) inserted within it. Our theory involves the solution of the continuity equation for an arbitrary location of the QW in the base, and the solution of coupled rate equations for carriers and photons in the QD. The excitonic recombination model is assumed to be valid in the QD. The estimated threshold base current for InAs QDs embedded in the InGaAs QW is found to be smaller than the value for the InGaAs QW in the GaAs base. The values of the modulation bandwidths for different values of the base current however do not reach the high values obtained for the QW system.

Simulation of quantum cascade lasers

We report a simulation of quantum cascade lasers based on the integration of a number of optoelectronic models on both microscopic and macroscopic scales. On the microscopic scale, quantum mechanical computation was performed to find the quantization states and a rate equation approach was used to compute the optical gain. On the macroscopic scale, we solved the drift-diffusion equations with modification of current density to account for long-range carrier transport, including quantum tunneling, mini-band tunneling, and hot carrier transport. Multiple lateral optical modes were computed by solving a scalar wave equation as an eigenvalue problem. Finally, multiple lateral mode laser cavity photon rate equations were solved with the drift-diffusion equations in a self-consistent manner to predict the lasing characteristics of a quantum cascade laser. The simulation compared the integrated models with experimental data from a number of AlInGaAs/InP systems with variable quantum wells and at different temperatures. Reasonable agreements with experiments have been obtained for both electrical and lasing characteristics.

Modeling of Self-Sustained Pulsation Based on Two-Section Distributed Feedback Lasers with Shift-Layer

We propose a numerical model to estimate the self-sustained pulsation frequency (SSP frequency) for a two-section distributed feedback laser (TS-DFB). A modulation transfer function is derived from the rate equations for carriers and photons. The SSP frequency can be obtained from the singularity condition of the transfer function. A useful but simple systematic design procedure is proposed for investigating the effects of various structural parameters on the SSP condition. The device parameters varied in the analysis include carrier density, section length ratio, grating coupling coefficient, and the refractive index change caused by adding a shift-layer. The device structure used for the SSP experiments and analysis is a TS-DFB laser with a shift-layer. This type of lasers can have stable lasing mode with large side-mode suppression ratio such that the inherent mode instability in a conventional index-guide DFB laser can be eliminated in this structure. This can simplify the operation of the SSP laser. The analysis model can also be applied to other laser structures. The results of numerical analysis match well with the experimental data.

Novel DFB laser structure suitable for stable single longitudinal mode operation

This paper shows that a suitable design of a single phase-shift (1PS) distributed coupling coefficient (DCC) distributed feedback (DFB) laser structure can strongly improve both the normalized mode selectivity S and the flatness of the electrical field profile of DFB lasers—usually quantified by a flatness parameter F —in order to ensure the required criteria for single longitudinal mode operation ( S ≥ 0.25 and F ≤ 0.05 ) through an extended range of biasing current injection ( I). It is shown that a symmetric longitudinal laser cavity should be used in order to accomplish the requirements imposed by the modern optical communication systems. Photon and carrier rate equations have been used to evaluate the performance of the proposed laser structure in the above-threshold regime. The variations of the selectivity, the flatness, the lasing-wavelength, the emitted power ( P) and the side-mode suppression ratio (SMSR) versus current have been assessed. For I=5 I th , where I th is the laser threshold current, substantial improvements in S (60% higher), in F (40% better), in P (45% higher) and in the SMSR (about 4 dB higher) are achieved in the proposed 1PS-DCC-DFB laser when compared to similar single phase-shifted DFB structures referred elsewhere. The improvements are even better when compared to the standard QWS-DFB laser.

Optimisation of a corrugation-pitch-modulated DFB laser structure with inhomogeneous coupling coefficient for stable single longitudinal mode operation

This paper shows that a suitable design of a corrugation-pitch-modulated (CPM) distributed-coupling-coefficient (DCC) distributed feedback (DFB) laser structure can strongly improve the mode selectivity $(\mathfrak{G})$ and the flatness $(\mathfrak{F})$ of DFB laser structures in order to ensure the required criteria for single longitudinal mode operation ( $\mathfrak{G}$ ≥ 0.25 and $\mathfrak{F}$ ≤ 0.05), through an extended range of current injection. It is shown that a symmetric structure should be used in order to accomplish the requirements imposed by the modern optical communication systems. Photon and carrier rate equations have been used in order to evaluate the performance of the proposed laser structure in the above-threshold regime. The variations of $\mathfrak{G}$ , $\mathfrak{F}$ , the lasing wavelength, the emitted power ( P ) and the side-mode-suppression-ratio (SMSR) with the current injection ( I ) have been evaluated. For I = 5 I th , where I th is the laser threshold current, substantial improvements in $\mathfrak{G}$ ( 3.8 times better), in $\mathfrak{F}$ ( 2 times better), in P (15% higher) and in the SMSR (about 3.4 dB higher) are achieved in the proposed CPM-DCC-DFB laser when compared to an optimised CPM-DCC-DFB laser referred elsewhere. The improvements are even better when compared to the standard QWS-DFB laser.

Gain-Switched Pulse Response of Quantum-Well Lasers

An analytical model is established for the gain-switched pulse response of quantum-well lasers (QWLs). The electron and photon rate equations are coupled through the dynamic variations of the laser gain represented by the stimulated lifetime in terms of the Fermi energies. The gain switching response is obtained analytically using approximate quiescent boundary conditions and a perturbation technique which enables solution of the differential equations. A theoretical sech <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> ) pulse shape for the photon response is obtained analytically for the first time. The condition for obtaining identical repetitive pulses is discussed and found to be determined by both the electrical direct current (dc) bias and alternating current (ac) pulse amplitude and width. The analytical solution is shown to be excellent by comparison to a numerical solution of the standard equation. To establish the utility of the model, pulse data on vertical cavity surface emitting lasers (VCSELs) were taken and found to be well predicted within the error of the measurement.

Experimental and numerical study on AlGaInAs/AlGaAs distributed feedback lasers with GaInP gratings

Ridge-waveguide AlGaInAs/AlGaAs distributed feedback lasers with lattice-matched GaInP gratings were fabricated and their light-current characteristics, spectrum and far-field characteristics were measured. On the basis of our experimental results we analyze the effect of the electron stopper layer on light-current performance using the commercial laser simulation software PICS3D. The simulator is based on the self-consistent solution of drift diffusion equations, the Schrodinger equation, and the photon rate equation. The simulation results suggest that, with the use of a 80 nm-width p-doped Al0.6GaAs electron stopper layer, the slope efficiency can be increased and the threshold current can be reduced by more than 10 mA.

Coupled optoelectronic modeling and simulation of nanowire lasers

A self-consistent, coupled optoelectronic simulation approach for studying microcavity nanowire lasers is presented. To focus the discussion, we use GaN nanowires to illustrate the methodology. The comprehensive model solves, simultaneously and self-consistently, the carrier transport equations and the photon rate equations. The basic physical model takes into account both bulk and surface dark carrier recombination processes. It includes stimulated emission, the anisotropic optical gain typical of the wurtzite GaN structure, the modified spontaneous emission, and its coupling into the lasing modes as a consequence of microcavity effects. The model further incorporates band-gap shrinkage effects due to band renormalization; it includes the complex dispersion and reflectivity relations of the guided modes, and the effects of multiple lateral and longitudinal lasing and nonlasing optical modes. We provide a detailed discussion of the influence of other important effects on the operation of optically pumped GaN nanowire lasers. Finally, we want to point out that the method is sufficiently general that it can be used to study any class of nanowire lasers.

Influence of photon recycling on semiconductor luminescence refrigeration

Luminescence refrigeration in semiconductors is studied using a model that includes the rate equations for carriers and photons as well as the influence of spectral dependent photon recycling. Expressions are derived for cooling efficiency, cooling power density, and the minimum external quantum efficiency required for cooling. These results show that net cooling is accessible and that photon recycling significantly contributes to luminescence refrigeration when the luminescence extraction is small.