Bifurcation and locking in an multi-quantum-well laser subjected to external injection

Bifurcation and dynamic stability as well as locking in an external light injection multi-quantum-well (MQW) laser are studied. Different dynamic regimes in locking diagram are analyzed with the injection level and frequency detuning. Bifurcate behavior is numerically simulated via the external injection light intensity, frequency detuning, current, linewidth enhance factor, photon loss rate and carrier loss rate, respectively. And the route to chaos from bifurcation, period-doubling and quasi-period are described by numerical analysis. A perturbation equation of four-dimension model and the bifurcation condition are demonstrated. Dynamic stability of the laser is theoretically and numerically analyzed. The bifurcate expression is theoretically given while the maximum locking frequency domain is given. The variational characteristic of the oscillation frequency in the self-pulse regimes versus the injection and detuning are numerically analyzed to find that the relaxation frequency is reduced with adding the detuning in the smaller detuning or with adding the injection in the smaller injection while the relaxation frequency will be increased with adding the detuning in the larger detuning or with adding the injection in the larger injection. We find also that the chaotic spectra are broadened with adding the detuning or narrowed with adding the injection or the current.     

Periodic and chaotic oscillations in mutual-coupled mid-infrared quantum cascade lasers

Dynamic states in mutual-coupled mid-infrared quantum cascade lasers (QCLs) were numerically investigated in the parameter space of injection strength and detuning frequency based on the Lang—Kobayashi equations model. Three types of period-one states were found, with different periods of injection time delay τ_inj, 2τ_inj, and reciprocal of the detuning frequency. Besides, square-wave, quasi-period, pulse-burst and chaotic oscillations were also observed. It is concluded that external-cavity periodic dynamics and optical modes beating are the mainly periodic dynamics. The interaction of the two periodic dynamics and the high-frequency dynamics stimulated by strong injection induces the dynamic states evolution. This work helps to understand the dynamic behaviors in QCLs and shows a new way to mid-infrared wide-band chaotic laser.     

Dynamical tracking of unstable periodic orbits

Tracking unstable periodic orbits and its stabilization by large periodic modulation of a control parameter are studied numerically in the Hénon map and laser equations. Some important scaling relations linking the tracking range to the modulation amplitude and frequency are deduced. The results obtained with both models are compared. Experimental realization of dynamical tracking is demonstrated in a loss-driven CO₂ laser where cavity detuning or losses are periodically modulated.     

Effect of Material Response Time on Optical Bistability in Nonlinear Distributed Feedback Gratings

The dynamic behaviors of nonlinear periodic waveguides with a finite material response time have been investigated by numerically solving both the coupled-mode equations and Debye relaxation equation. When the response time of the nonlinearity is more than the transit time through the grating, a modulational instability is sufficiently suppressed and one can obtain optical bistability. Unlike a nonlinear Fabry-Perot resonator or ring resonators, overshoot in the transmission never takes place in the case of zero detuning. However, a relaxation (or damped) oscillation always takes place in the presence of detuning. It is also found that the switch-on time depends on the material response time and the switch-off time is almost independent of it.     

光纤激光器自组织相干阵列的理论分析

 分析了光纤激光器自相干阵列最简单的情况——两个光纤激光器之间的相位锁定过程。建立了该过程的数学模型,并对它的时间特性和输出光束的模式进行了理论和数值分析,讨论了运行频率差和耦合系数对锁相过程的影响。对于运行频率相等的两个光纤激光器,当两个光纤激光器初始相位差不为0时,相位锁定后的激光器阵列输出模式为异相模;对于运行频率不等的情形,给出了锁相运行对两个光纤激光器运行频率差的要求,并分析了降低这种要求的几种方法。     

三能级原子系统中单光子频率失谐对光减速的影响

在Λ型三能级Rb原子介质中，观察到了由电磁感应透明（EIT）效应导致的光减速现象并测 量了光延迟对单光子频率失谐量的依赖关系. 结果表明，由于多普勒展宽效应的存在，在单 光子频率失谐±600MHz的范围内，光减速效应较为显著. 在考虑多谱勒频移的情况下，数值 计算了光延迟随单光子频率失谐量的变化曲线，实验结果与理论曲线很好地符合. 这一研 究结果为利用单光子频率失谐控制光的群速度提供了理论与实验参考. 关键词： 光减速 电磁感应透明 多普勒展宽     

Experimental and numerical evidence for the existence of wide and deep phononic gaps induced by inertial amplification in two-dimensional solid structures

The aim of this paper is to show that a two-dimensional periodic solid structure with embedded inertial amplification mechanisms can possess a wide and deep phononic gap at low frequencies. The width and depth of the inertial amplification induced phononic gaps (stop bands) are determined both analytically using a distributed parameter model and numerically using one-dimensional (1D) and two-dimensional (2D) finite element models. The inertial amplification mechanisms are optimized to yield wide and deep gaps at low frequencies. These optimized mechanisms are used to form one- and two-dimensional periodic structures. Frequency responses of these periodic structures are obtained numerically using 1D and 2D finite element models. A deeper gap is generated with the two-dimensional periodic structure when compared with the one-dimensional periodic structure that has the same number of unit cells along the excitation direction. Then, the two-dimensional periodic structure is manufactured and its frequency response is determined via experimental modal analysis. The experimental and numerical frequency response results match quite well, which validate that the two-dimensional periodic solid structure has a wide and deep phononic gap.     

光纤激光器自组织相干阵列的理论分析

分析了光纤激光器自相干阵列最简单的情况——两个光纤激光器之间的相位锁定过程。建立了该过程的数学模型,并对它的时间特性和输出光束的模式进行了理论和数值分析,讨论了运行频率差和耦合系数对锁相过程的影响。对于运行频率相等的两个光纤激光器,当两个光纤激光器初始相位差不为0时,相位锁定后的激光器阵列输出模式为异相模;对于运行频率不等的情形,给出了锁相运行对两个光纤激光器运行频率差的要求,并分析了降低这种要求的几种方法。     

Numerical study of the spiking instability caused by modulation frequency detuning in an actively mode-locked solid-state laser

We have studied both analytically and numerically the transient active mode locking in the case of a modulation frequency detuning. Analytical results for pulse width and position, extending previous works, are obtained within the frame of the stationary theory by assuming that a short pulse restores its shape after each round trip. Numerical evaluation leads to relaxation oscillations and change of the short-pulse shape. The results of both approaches agree with each other only for small detunings, when laser is operated in the steady state. For larger detunings the laser dynamics change dramatically, and the three operation modes observed earlier in experiments were demonstrated numerically: sequences of damped relaxation oscillations, deep periodic spiking and chaotic spiking. The periodic spiking can be used to increase the intensity of mode-locked pulses.     

主副垂直腔面发射激光器动力学系统混沌输出的时延特征及带宽分析

基于垂直腔面发射激光器(VCSEL)的自旋反转模型, 数值研究了由一个光反馈VCSEL (定义为主VCSEL, M-VCSEL)输出的混沌光单向注入到另一个VCSEL (定义为副VCSLE, S-VCSEL)所构成的主副VCSELs系统的混沌动力学特性, 分析了注入强度、M-VCSEL与S-VCSEL之间的频率失谐以及M-VCSEL所受到的光反馈强度对系统混沌输出时延特征(包括强度时延特征(I-TDS) 和相位时延特征(P-TDS))以及输出带宽(BW)的影响. 结果显示: 通过调节注入强度和频率失谐, 该系统混沌输出的两个偏振分量(X-PC和Y-PC)的P-TDS和I-TDS可以同时得到抑制; 进一步分析注入强度和频率失谐对混沌BW的影响, 发现在较大负频率失谐区域, 系统可输出BW超过30 GHz 的X-PC和Y-PC混沌信号; 结合系统混沌输出信号的TDS与BW在注入强度和频率失谐参量空间下的演化特性, 可确定宽带宽、低时延特征混沌信号输出的参量空间区域. 此外, 通过合理调节M-VCSEL 所受到的光反馈强度, 可以显著优化系统的混沌输出信号质量.     

Nonlinear resonance and synchronization in the ring of unidirectionally coupled Toda oscillators

In the ring of unidirectionally coupled Toda oscillators the nonlinear resonance and the synchronization are investigated. It is shown how the nonlinear resonance affects the structure of the main synchronization region. As a result of nonlinear resonance we observe the coexistence of two stable limit cycles near the resonant frequency, which leads to coexistence of periodic and quasi-periodic regimes within the synchronization region.     

Stabilization of periodic and quasiperiodic motion in chaotic systems through changes in the system variables

This work deals with the suppression of chaos in dissipative systems that exhibit a transition from the coexistence of several periodic oscillations to deterministic chaos. The application of changes in the system variables is able to yield the prechaotic behaviour, that can be either quasiperiodic (two inconmensurate frequencies) or periodic (frequency locking), in the same way as for the original system. The performance of the method is shown by application to the two-dimensional Burgers map.     

Bistable phase locking in rocked lasers

We investigate analytically and numerically the dynamics of single mode lasers with periodic ac injection (rocked lasers). Such lasers show phase bistability as the phase of the light emitted by such lasers can lock to either of two values shifted by π. Locking regimes for different lasers are studied showing that the system response is strongly modified in class B lasers due to the influence of relaxation oscillations.     

Nonparaxial effects on the propagation and scattering of a polarized optical pulse

Propagation characteristics of a polarized optical solitary pulse are analyzed by taking into account the effect of nonparaxiality and mutual interaction. To start with, a pair of generalized nonlinear Schrodinger equations is deduced through an operator approach. Stationary solutions of such a system are then analyzed numerically through a boundary value problem in two stages, with and without the nonparaxial effect. In the second stage, the propagating form of the corresponding spatial soliton is studied by an extended split step algorithm ETDRK. The initial profile is considered to be both a one- and two-soliton solution, to visualize the event of scattering and fusion. From this data, we have computed the intensity, root mean square spectral width, and chirp of a single soliton as it propagates. In the case of the two-soliton solution, we observe that for source parameter values, the fusion is more favored than scattering. It is observed that nonparaxiality and the interaction between A(x) and A(y) tends to destroy the periodic behaviors of these parameters. Lastly, we have investigated the modulational instability of the system as function of frequency detuning and nonparaxiality. The form of the gain is discussed as a function of nonparaxiality.     

Spatial symmetry breaking and coexistence of attractors in a nonlinear ring cavity

We consider a passive optical system consisting of a ring cavity and a homogeneously broadened two-level medium. We find that a spatial modulation of the input beam imposes processes of competition between the external modulation frequency and the internal space frequencies which emerge from modulation instabilities. This leads to symmetry breaking phenomena when both the amplitude and the wavelength of the modulation are increased and results in the coexistence of periodic attractors. We numerically analyze the corresponding bifurcations and find that they can be explained by a generalization of the concept of cooperative frequency locking although the bifurcating attractors are periodic and are not related to linear resonator modes.     

Exact periodic solutions and chaos in the semiclassical Jaynes-Cummings model

Exact periodic solutions for the semiclassical Jaynes-Cummings model without the assumption of the rotating-wave approximation are studied. They are valid for some specific values of the coupling constant, detuning parameter and the energy of the system. The behaviour of the system in the vicinity of this special solution is studied numerically. By computation of the Fourier spectrum, it is shown that the behaviour of the system changes from a non-chaotic to a chaotic one when the energy of the system increases.     

The response of two-degree-of-freedom systems with quadratic non-linearities to a combination parametric resonance

The response of two-degree-of-freedom systems with quadratic non-linearities to a combination parametric resonance in the presence of two-to-one internal resonances is investigated. The method of multiple scales is used to construct a first order uniform expansion yielding four first order non-linear ordinary differential equations governing the modulation of the amplitudes and the phases of the two modes. Steady state responses and their stability are computed for selected values of the system parameters. The effects of detuning the internal resonance, detuning the parametric resonance, the phase and magnitude of the second mode parametric excitation, and the initial conditions are investigated. The first order perturbation solution predicts qualitatively the trivial and non-trivial stable steady state solutions and illustrates both the quenching and saturation phenomena. In addition to the steady state solutions, other periodic solutions are predicted by the perturbation amplitude and phase modulation equations. These equations predict a transition from constant steady state non-trivial responses to limit cycle responses (Hopf bifurcation). Some limit cycles are also shown to experience period doubling bifurcations. The perturbation solutions are verified by numerically integrating the governing differential equations.     

Grazing instabilities and post-bifurcation behavior in an impacting string

A theoretical and experimental investigation of the nonlinear dynamic response of a periodically excited string subject to a knife-edge amplitude restraint is presented. The amplitude restraint creates an impact condition as the amplitude of the response grows. The focus of this work is on the influence of a grazing instability; this zero-velocity impact event leads to complicated, post-bifurcation behavior ranging from multifrequency, periodic motion to chaos. In addition to looking at the response numerically, parameter combinations leading to an incidence of grazing are clearly identified in the excitation force excitation frequency parameter space using a multiple scales perturbation analysis. Modeling issues, numerical difficulties, and experimental limitations are also discussed.