Single-mode Equations Research Articles

Fixed-flux Rayleigh–Bénard convection in doubly periodic domains: generation of large-scale shear

This work studies two-dimensional fixed-flux Rayleigh–Bénard convection with periodic boundary conditions in both horizontal and vertical directions and analyses its dynamics using numerical continuation, secondary instability analysis and direct numerical simulation. The fixed-flux constraint leads to time-independent elevator modes with a well-defined amplitude. Secondary instability of these modes leads to tilted elevator modes accompanied by horizontal shear flow. For $Pr=1$ , where $Pr$ is the Prandtl number, a subsequent subcritical Hopf bifurcation leads to hysteresis behaviour between this state and a time-dependent direction-reversing state, followed by a global bifurcation leading to modulated travelling waves without flow reversal. Single-mode equations reproduce this moderate Rayleigh number behaviour well. At high Rayleigh numbers, chaotic behaviour dominated by modulated travelling waves appears. These transitions are characteristic of high wavenumber elevator modes since the vertical wavenumber of the secondary instability is linearly proportional to the horizontal wavenumber of the elevator mode. At a low $Pr$ , relaxation oscillations between the conduction state and the elevator mode appear, followed by quasi-periodic and chaotic behaviour as the Rayleigh number increases. In the high $Pr$ regime, the large-scale shear weakens, and the flow shows bursting behaviour that can lead to significantly increased heat transport or even intermittent stable stratification.

Uniform spike trains in optically injected quantum cascade oscillators.

It has been experimentally and theoretically analyzed that noise-induced excitability in quantum well and quantum dot semiconductor laser systems usually produces sharp spike patterns of non-uniform amplitude. In this paper, we experimentally record that a quantum cascade oscillator injected externally with a monochromatic laser beam exhibits a series of highly uniform spike trains, which occur in the proximity of the saddle-node bifurcation. Theoretical analysis based on a properly designed single-mode rate equation model endowed with quantum noise reveals that this high uniformity has its primary origin in the ultrashort carrier lifetime of the quantum cascade laser gain medium that is typically close to 1 ps.

Novel Optical bi-directional solutions to the new dual-mode derivative nonlinear Schrödinger equation

In recent times, a novel category of nonlinear physical models known as dual-mode nonlinear equations has emerged. These equations include various real-valued dual-mode equations linked to widely-known single-mode equations like KdV, mKdV, Schrödinger and Burger’s. Extensive research has been conducted to establish and investigate these equations. This study presents a novel dual-mode derivative nonlinear Schrödinger equation that incorporates new parameters for dissipative effects, nonlinearity, and interaction phase velocity. Various methods such as the tanh-coth scheme, extended exponential method, Kudryashov-scheme and the sine-cosine function methods are employed to investigate the solutions of the model. The obtained solutions are illustrated through graphical 2D and 3D and to demonstrate their dynamics and shapes. Furthermore, the interaction of the dual-waves is correlated with changes in the phase-velocity parameter. This model describes propagation of two simultaneously directional waves instead of as in standard Schrödinger equation. For the propagation of solitons in nonlinear optics, the solutions found in this study have important significance. All the resulting solutions can help to comprehend the underlying mechanisms for numerous nonlinear phenomena in diverse domains, including nonlinear optics, plasma physics, Bose–Einstein condensates and others.

Dynamic performance of terahertz quantum cascade laser with double optical feedbacks: Role of the external cavity

The dynamic performance of a terahertz (THz) quantum cascade laser (QCL) with double optical feedbacks (DOFs) is carefully studied within the framework of single-mode rate equations. The results have shown that the system displays three types of nonlinear dynamic phenomena under the control of different feedback strengths, which are envelope oscillations, spiking oscillations, and high-frequency small-amplitude oscillations, respectively. The fundamental frequency of oscillation is related to the ratio of the round-trip times from two external cavities and the ratio of their feedback strengths. By solving the steady-state equations, we find that the high-frequency, small-amplitude oscillations arise because a pair of modes appear at almost the same point. The output performance of THz QCL system is also studied with square wave signal modulation. The results show that the fundamental frequency of oscillation at the output is only related to the frequency of the square wave signal. And low frequency fluctuations (LFFs) dynamics can be excited by combination of DOFs and adjusting the duty cycle of the square wave signal. This work provides a novel insight into high data-bit THz communication based on QCLs.

Single-Mode Solutions for Convection and Double-Diffusive Convection in Porous Media

This work employs single-mode equations to study convection and double-diffusive convection in a porous medium where the Darcy law provides large-scale damping. We first consider thermal convection with salinity as a passive scalar. The single-mode solutions resembling steady convection rolls reproduce the qualitative behavior of root-mean-square and mean temperature profiles of time-dependent states at high Rayleigh numbers from direct numerical simulations (DNS). We also show that the single-mode solutions are consistent with the heat-exchanger model that describes well the mean temperature gradient in the interior. The Nusselt number predicted from the single-mode solutions exhibits a scaling law with Rayleigh number close to that followed by exact 2D steady convection rolls, although large aspect ratio DNS results indicate a faster increase. However, the single-mode solutions at a high wavenumber predict Nusselt numbers close to the DNS results in narrow domains. We also employ the single-mode equations to analyze the influence of active salinity, introducing a salinity contribution to the buoyancy, but with a smaller diffusivity than the temperature. The single-mode solutions are able to capture the stabilizing effect of an imposed salinity gradient and describe the standing and traveling wave behaviors observed in DNS. The Sherwood numbers obtained from single-mode solutions show a scaling law with the Lewis number that is close to the DNS computations with passive or active salinity. This work demonstrates that single-mode solutions can be successfully applied to this system whenever periodic or no-flux boundary conditions apply in the horizontal.

Staircase solutions and stability in vertically confined salt-finger convection

Bifurcation analysis of confined salt-finger convection using single-mode equations obtained from a severely truncated Fourier expansion in the horizontal is performed. Strongly nonlinear staircase-like solutions having, respectively, one (S1), two (S2) and three (S3) regions of mixed salinity in the vertical direction are computed using numerical continuation, and their stability properties are determined. Near onset, the one-layer S1 solution is stable and corresponds to maximum salinity transport among the three solutions. The S2 and S3 solutions are unstable but exert an influence on the statistics observed in direct numerical simulations (DNS) in larger two-dimensional (2-D) domains. Secondary bifurcations of S1 lead either to tilted-finger (TF1) or to travelling wave (TW1) solutions, both accompanied by the spontaneous generation of large-scale shear, a process favoured for lower density ratios and Prandtl numbers ($Pr$). These states at low$Pr$are associated, respectively, with two-layer and three-layer staircase-like salinity profiles in the mean. States breaking reflection symmetry in the midplane are also computed. In two dimensions and for low$Pr$, the DNS results favour direction-reversing tilted fingers resembling the pulsating wave state observed in other systems. Two-layer and three-layer mean salinity profiles corresponding to reversing tilted fingers and TW1 are observed in 2-D DNS averaged over time. The single-mode solutions close to the high wavenumber onset are in an excellent agreement with 2-D DNS in small horizontal domains and compare well with 3-D DNS.

Optimizing Equivalent Circuit Model Parameters of DFB Lasers With RSM Model and NSGA-II Algorithm

The parameters group of the distributed feedback (DFB) laser equivalent circuit model based on the single-mode rate equations is the key to precisely presenting DFB response characteristics, so a novel optimization solution based on the response surface methodology (RSM) is proposed to rapidly select the optimized parameters by the multi-objective algorithm. The RSM model is designed to match the DFB laser characteristics related to the direct-current and small-signal frequency response, and non-dominated sorting genetic algorithm-II (NSGA-II) attributes to elevating the RSM model optimizing to screen out an optimal set of parameters by Pareto sorting. To further verify the accuracy of the model, the resonant frequency ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f_r</i> ) and the threshold current ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I_th</i> ) are considered the objective optimization variables to set the target values as 18 GHz and 11.5 mA. The single-objective and multi-objective optimization are analyzed and compared to each other, and the optimized results have shown good agreement with predicted values, such as lower <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I_th</i> in the multi-objective optimization while close <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f_r</i> in both cases. It has been demonstrated that optimization makes it possible not only to exploit the potential of existing DFB lasers but also to provide guidance for the inverse design of laser.

Continuous current-injected waveforms shaping for suppressing relaxation oscillations of direct modulation based on equivalent circuit model.

An improved technique of continuous shaping current-injected waveforms based on the single-mode rate equations is proposed to suppress relaxation oscillations (ROs) from direct modulation of distributed feedback laser (DFB). The signal expression of shaping current is deduced theoretically from the dependence of DFB desired output waveforms in detail, and the specific parameters derivation of the different polynomial degree is also discussed necessarily. Furthermore, a polynomial p-function with inverse operation is adopted to construct the Fourier series corresponding to injection current waveform signal. The equivalent circuit model with DFB phenomenological description is injected into shaping current signal to verity the proposed validity by evaluating the static and dynamic characteristics. The simulation results of the optimized shaping signal show the good agreement with the desired output pulse including rising and falling edge and suppress the ROs amplitude dramatically at the two jump edges.

Relative Intensity Noise and Linewidth for Hybrid-Cavity Semiconductor Lasers

Low relative intensity noise (RIN) was demonstrated experimentally for hybrid square-rectangular lasers (HSRLs). In this paper, RIN of the HSRL is numerically simulated and the laser linewidth is investigated theoretically and experimentally. The non-zero delayed self-heterodyne method is utilized to measure the linewidth, and the laser linewidth of 2.9 MHz is obtained experimentally. A two-section single-mode rate equation model including Langevin noise sources is established to study the noise characteristics theoretically. The time-varying mode photon density and frequency are simulated, then the intensity and frequency noise (FN) spectra are calculated. With the rising of bias currents, RIN decreases and approaches the standard quantum limit. Large mode <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> -factor, high bias current, and passive square microcavity are effective methods to reduce FN and linewidth of the HSRLs. By inducing a passive square microcavity, we can expect a low linewidth of 37 kHz for the hybrid-cavity laser with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> = 1.17 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> .

Time delay concealment and unpredictability enhancement of nanolasers under external cavity regulation

As an important optical element of the optical integration in the future, nanolasers has been a research hotspot in recent years, and the corresponding structural engineering and output characteristics have been widely investigated. However, the nonlinear dynamical performances of nanolasers are rarely reported. Only some preliminary analyses of the dynamic behavior under the optical feedback, optical injection and mutual injection can be found. Some researches pointed out the future prospect of nanolasers, however, some chaos-based applications have not been explored. Therefore, we numerically investigate chaos dynamics in a nanolaser subjected to optical feedback and in another nanolaser subjected to chaotic injection from the former structure by using single mode rate equation, which includes the Purcell cavity-enhanced spontaneous emission factor &lt;i&gt;F&lt;/i&gt; and spontaneous emission coupling factor &lt;i&gt;β&lt;/i&gt;. The &lt;i&gt;F&lt;/i&gt; denotes the ratio of the spontaneous emission rate into the cavity mode to the total spontaneous emission rate in the bulk medium in the absence of a cavity and &lt;i&gt;β&lt;/i&gt; represents the fraction of spontaneous emitted photons which are coupled into cavity mode. Specifically, chaos time delay signature (TDS) and unpredictability are evaluated by the peak size of autocorrelation function (ACF) and permutation entropy (PE) respectively. Such kinds of calculations have the advantage of fast operation speed and anti-noise robustness. The results show that the increasing of bias current and the decreasing of gain saturation factor &lt;i&gt;ε&lt;/i&gt;, &lt;i&gt;F&lt;/i&gt; and &lt;i&gt;β&lt;/i&gt; are beneficial to improving the unpredictability and suppressing TDS because the weak damping of the relaxation oscillation leads to strong oscillation. Large linewidth enhancement factor &lt;i&gt;α&lt;/i&gt; will increase the number of laser oscillating modes, sideband modes, the spectral components, and enhance the dispersion effect, which will also weaken the information about outer cavity and improve the complexity of chaos. In addition, the above-mentioned chaos properties can be enhanced by injecting the chaos output from a nanolaser subjected to optical feedback into another (slave) nanolaser, which is due to the nonlinear interaction between the driving chaotic signal and the internal electric field of the slave nanolaser. Finally, two-dimensional maps depicting high unpredictability and TDS concealment in the parameter space of the frequency detuning and the injection strength are obtained. It can be found that unpredictability degree can be enhanced by choosing high detuning frequency and intermediate injection strength in the non-injection locking area. The numerical results pave the way for generating the high-quality chaotic sources on a chip or the photonic integrated circuits based on novel semiconductor nanolaser and its related applications.

Study on Fast Response of Semiconductor Laser Driven by Impulse Current

A simulation model based on single-mode rate equations of semiconductor laser and its typical parameters shows that, the photon stimulated emission rate at the early turn-on time of the semiconductor laser is increased with the increase of the injection current, and the corresponding rise time is reduced with the increase of injection current. However, for an output power is fixed semiconductor laser, increasing the injection current to reduce the rise time of the transmitted pulse leads to increase in temperature of semiconductor laser. We show that adding an impulse current in the front of the normal luminous pulse current can reduce the rise time of the transmitted pulse and ensure the stable output of the semiconductor laser. This method is verified by simulation and an experiment using PLTB450B semiconductor laser. The simulation and test results show that for an output power is fixed semiconductor laser, the rise time of the transmitted pulse can be reduced significantly by adding impulse current.

Transfer function replacement of phenomenological single-mode equations in semiconductor microcavity modeling.

Semiconductor microcavities are frequently studied in the context of semiconductor lasers and in application-oriented fundamental research on topics such as linear and nonlinear polariton systems, polariton lasers, polariton pattern formation, and polaritonic Bose-Einstein condensates. A commonly used approach to describe theoretical properties includes a phenomenological single-mode equation that complements the equation for the nonlinear optical response (interband polarization) of the semiconductor. Here, we show how to replace the single-mode equation by a fully predictive transfer function method that, in contrast to the single-mode equation, accounts for propagation, retardation, and pulse-filtering effects of the incident light field traversing the distributed Bragg reflector (DBR) mirrors, without substantially increasing the numerical complexity of the solution. As examples, we use cavities containing GaAs quantum wells and transition-metal dichalcogenides (TMDs).

Tunable Frequency Comb Generation Using Quantum Cascade Lasers Subject to Optical Injection

We dissect the enabling capabilities of the tunable period-one (P1) limit cycles of optically injected quantum cascade laser (QCL) oscillators for the generation of optical frequency combs. As such, we theoretically investigate the P1 dynamics of a QCL using a single-mode rate equation model. We find that such a P1 limit cycle occupies a rather large and wide region of the optical frequency detuning and injection level ratio map. We have not recorded evidence of chaos in this injected laser system, in marked contrast with quantum well and quantum dot cases. Contrary to interband semiconductor lasers, the QCL's oscillation frequency is generally smaller than the detuning frequency, and is reduced with increasing injection strength, due to the strong injection pulling effect. When the optical injection is operated in the vicinity of the Hopf bifurcation , the P1 oscillation produces dense optical frequency combs, owing to both the frequency pulling effect and the four-wave mixing effect. The comb spacing is continuously tunable from subgigahertz up to a few gigahertz, via fine control of either the detuning frequency and/or the injection ratio. This novel approach of the frequency comb generation is of prime importance for high-resolution detection of narrow absorption lines of gas molecules.

Design of ZnO Nanowire Laser Single Mode and Study Its Properties

Designing of single-mode ZnO nanowire laser has been achieved. Studying its properties has also been considered. Analysis of single-mode rate equations indicates that the laser has threshold current of 53 mA, the output power of 30 mW at bias current of 70 mA and the slope efficiency around 1.77 mW/mA for the output power from both mirrors. The critical diameter for nanowire has been calculated and it is found to be 128 nm. The number of nanowire rods through designed area of dimensions (21µm×21µm) has also been calculated. The cavity of nanowire laser is (F-P). The results are compared with some experimental work of ZnO nanowire lasers and good agreement is found.

Step-Pulse Modulation of Gain-Switched Semiconductor Pulsed Laser

To improve the peak power and extinction ratio and produce ultra-short pulses, a novel approach is presented in this paper offers a highly effective modulated method for a gain-switched semiconductor laser by using step-pulse signal modulation. For the purpose of single pulse output, then the effects on the output from the gain-switched semiconductor laser are studied by simulating single mode rate equation when changing the amplitude and width of the modulated signal. The results show that the proposed method can effectively accelerate the accumulation speed of the population inversion and we can acquire the output pulse with higher peak power and shorter width. Compared with the traditional rectangular wave modulation, this method is advantageous to obtain a high gain switching effect by increasing the second modulation current and reduce the pulse width to saturation at the best working point. It can be incorporated as a practical and cost-effective approach for many fields which need high extinction ratio short pulse, such as the optical time domain reflectometry.

Dark and singular optical solutions with dual-mode nonlinear Schrödinger's equation and Kerr-law nonlinearity

Recently, a new family of nonlinear physical models under the name of dual-mode nonlinear equations has been arisen. Several real-valued dual-mode equations associated with well-known single-mode equations such as KdV, mKdV, and Burgers’ have been established and studied. In this study, we provide the first attempt to establish a complex-valued dual-mode equation that corresponds to the single-mode nonlinear Shrödinger equation. Accordingly, necessary conditions on the nonlinearity and dispersion parameters for the existence of two right-left (R-L) moving wave solutions are explored. Finally, graphical analysis is provided to validate the findings of this work.

Dynamics of two identical mutually delay-coupled semiconductor lasers in photonic integrated circuits

We theoretically investigate a system of two mutually delay-coupled semiconductor lasers, in a face to face configuration for integration in a photonic integrated circuit. This system is described by single-mode rate equations, which are a system of delay differential equations with one fixed delay. Several bifurcation scenarios involving multistabilities are presented, followed by a comprehensive frequency analysis of the symmetric and symmetry-broken, one-color and two-color states.

Purcell Effect in the Stimulated and Spontaneous Emission Rates of Nanoscale Semiconductor Lasers

Nanoscale semiconductor lasers have been developed recently using either metal, metallo-dielectric or photonic crystal nanocavities. While the technology of nanolasers is steadily being deployed, their expected performance for on-chip optical interconnects is still largely unknown due to a limited understanding of some of their key features. Specifically, as the cavity size is reduced with respect to the emission wavelength, the stimulated and the spontaneous emission rates are modified, which is known as the Purcell effect in the context of cavity quantum electrodynamics. This effect is expected to have a major impact in the 'threshold-less' behavior of nanolasers and in their modulation speed, but its role is poorly understood in practical laser structures, characterized by significant homogeneous and inhomogeneous broadening and by a complex spatial distribution of the active material and cavity field. In this work, we investigate the role of Purcell effect in the stimulated and spontaneous emission rates of semiconductor lasers taking into account the carriers' spatial distribution in the volume of the active region over a wide range of cavity dimensions and emitter/cavity linewidths, enabling the detailed modeling of the static and dynamic characteristics of either micro- or nano-scale lasers using single-mode rate-equations analysis. The ultimate limits of scaling down these nanoscale light sources in terms of Purcell enhancement and modulation speed are also discussed showing that the ultrafast modulation properties predicted in nanolasers are a direct consequence of the enhancement of the stimulated emission rate via reduction of the mode volume.

Small-Signal Analysis of Ultra-High-Speed Multi-Mode VCSELs

In this paper, we show that the dynamic performance of multi-mode vertical-cavity surface-emitting lasers (VCSELs) can be modeled by single-mode rate equations developed for edge emitters as long as the lasing modes share a common carrier reservoir. However, this assumption does not hold for ultra-high performing VCSEL devices. Due to the high photon densities inside these optimized VCSELs, the common carrier reservoir splits up as a result of the spatial hole burning effect. This is caused by the high intensity of the multiple transverse modes. In this case, a small-signal modulation response with a different shape is expected. We derive an easy-to-apply fitting function, which allows the extraction of consistently expanded figures of merit. This novel function works for all VCSELs, particularly, including devices with carrier reservoir splitting. Furthermore, we use this new model to perform a detailed analysis of our latest VCSEL generation with a modulation bandwidth of up to 32.7 GHz.

Multimode modeling of digital modulation in nearly single-mode semiconductor lasers

We present multimode modeling of the digitalmodulation characteristics of nearly single-mode semiconductor lasers. The model takes into account the mechanisms of spectral suppression of modal gain; namely, symmetric and asymmetric gain suppressions. The digital modulation performance of the laser is quantitatively examined in terms of the turn-on jitter, on-off ratio and a Q-factor. We study the effect of the modulation parameters on the modulation characteristics of the laser. We also clarify the effect of the asymmetric gain suppression on the modulated signals of the oscillating modes. In addition, we examine the validity of modeling the digital modulation of the laser via a single-mode rate equation model. We show that the modulation characteristics improve with an increase in the bias and/or modulation current or with a decrease in the bit rate due to reduction in the bit-pattern effect. The asymmetric gain suppression does not affect the characteristics of the total laser signal, however it enhances the modulation of the nearest modes on the long-wavelength side of the gain spectrum.