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图 1是半导体环形激光器交叉光反馈示意图,CW为顺时针方向,CCW为逆时针方向。如图 1所示,左边的两个反馈环路将光反馈回了激光器,其中CW方向的光反馈到CCW方向,而CCW方向的光反馈到CW方向,此时的光反馈结构为交叉光反馈。结合参考文献[13]中提出的SRL模型,考虑顺时针传播的模式ECW,逆时针传播的模式ECCW和载流子数N,并引入交叉反馈项,半导体环形激光器光反馈下的速率方程表示为:
$ \begin{aligned} \frac{\mathrm{d} E_{\mathrm{CW}}}{\mathrm{d} t} &=\kappa(1+\mathrm{i} \alpha)\left(g_{\mathrm{CW}} N-1\right) E_{\mathrm{CW}}-\left(k_{\mathrm{d}}+\right.\\ &\left.\mathrm{i} k_{\mathrm{e}}\right) E_{\mathrm{CCW}} \mathrm{c}^{\mathrm{i} \varphi}+\eta_{\mathrm{CCW}} E_{\mathrm{CCW}}\left(t-T_{\mathrm{CCW}}\right) \mathrm{e}^{-\mathrm{i} \omega T_{\mathrm{CCW}}} \end{aligned} $
(1) $ \begin{array}{c} \frac{{{\rm{d}}{E_{{\rm{CCW}}}}}}{{{\rm{d}}t}} = \kappa (1 + {\rm{i}}\alpha )\left( {{g_{{\rm{CCW}}}}N - 1} \right){E_{{\rm{CCW}}}} - \\ \left( {{{\rm{k}}_d} + i{{\rm{k}}_c}} \right){{\rm{E}}_{CW}}{e^{i\varphi }} + {\eta _{CW}}{{\rm{E}}_{CW}}\left( {{\rm{t}} - {{\rm{T}}_{CW}}} \right){e^{ - i\omega {{\rm{T}}_{CCW}}}} \end{array} $
(2) $ \frac{\mathrm{d} N}{\mathrm{d} t}=\gamma\left(\mu-N-g_{\mathrm{CW}} N\left|E_{\mathrm{CW}}\right|^{2}-g_{\mathrm{CCW}} N\left|E_{\mathrm{CCW}}\right|^{2}\right) $
(3) $ g_{\mathrm{CW}}=1-s\left|E_{\mathrm{CW}}\right|^{2}-m\left|E_{\mathrm{CCW}}\right|^{2} $
(4) $ g_{\mathrm{CCW}}=1-s\left|E_{\mathrm{CCW}}\right|^{2}-m\left|E_{\mathrm{CW}}\right|^{2} $
(5) 式中, κ为电场衰减率,γ为载流子衰减率, (kd+ikc)为反向散射系数,kd和kc为耗散系数和保守系数, ηCW与ηCCW为两个方向的反馈系数,TCW与TCCW代表两个方向的反馈延迟时间,ECW(t-TCW)与ECCW(t-TCCW)为两个方向反馈回激光器的电场复振幅, α为线宽增强因子,φ为耦合相位,ω为激光器自由运行角频率,ωTCW与ωTCCW为时间延迟引起的两个方向的相位差, gCW与gCCW为两个方向的增益系数,s和m分别代表增益自饱和互饱和系数, μ为归一化的偏置电流,μ=1为阈值电流。根据参考文献[13],本文中仿真所使用的参量取值为:κ=100ns-1,α=3.5,γ=0.2ns-1,s=0.005,m=0.01,kd=0.033ns-1,kc=0.44ns-1,μ=2.4。
半导体激光器在外腔光反馈下,存在由外腔长度确定的与外腔模式周期性有关的时延特征,而抑制时延特征是混沌安全通信中的关键。通常研究混沌信号时延特征的方法是计算时间序列在各个时刻的自相关值,自相关函数的数学定义式为:
$ \begin{array}{c} C(\Delta t) = \\ \frac{{\left\langle {[x(t) - \langle x(t)\rangle ]\left[ {{x_{\rm{s}}}(t) - \left\langle {{x_{\rm{s}}}(t)} \right\rangle } \right]} \right\rangle }}{{\sqrt {\left. {[x(t) - {{\langle x(t)]}^2}\rangle \left\langle {{{\left[ {{x_{\rm{s}}}(t) - \left\langle {{x_{\rm{s}}}(t)} \right\rangle } \right]}^2}} \right\rangle } \right\rangle } }} \end{array} $
(6) 式中,x(t)为任一时间序列,Δt为时间延迟,〈·〉表示时间平均。xs(t)=x(t+Δt)为时间移动Δt后时间序列的值,C(Δt)是时间延迟为Δt时的自相关系数的值,自相关系数无量纲,绝对值小于等于1。
基于半导体环形激光器的光反馈动力学研究
Study on optical feedback dynamics based on semiconductor ring lasers
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摘要: 为了研究半导体环形激光器在外部交叉光反馈情况下的非线性动力学特性, 基于环形激光器的速率方程, 采用数值仿真的方法, 得到了各种反馈参量下的时间序列及功率谱, 并研究了反馈系数对混沌带宽及时间延迟特性的影响。结果表明, 在各种不同外部反馈参量的情况下, 环形激光器展现了单周期、多周期、低频反相波动以及混沌态等丰富的动力学态; 在对称光反馈时, 可得到最大为3.7GHz的混沌带宽; 在时间不对称光反馈时, 产生的混沌信号的时延特征被很好地抑制。该研究结果可为环形激光器的实际应用提供一定的理论参考。Abstract: In order to study nonlinear dynamic characteristics of a semiconductor ring laser (SRL) under external cross-optical feedback, the time series and power spectra under various feedback parameters were obtained by numerical simulation based on the SRL rate equation, and the effect of feedback coefficient on chaotic bandwidth and time delay characteristics was studied. The results show that SRL exhibits a variety of dynamic states such as single period, multi-period, low-frequency anti-phase fluctuation and chaotic state under various external feedback parameters. The maximum chaotic bandwidth of 3.7GHz can be obtained with symmetric optical feedback, and the time delay characteristics of chaotic signals generated by time-asymmetric optical feedback are well suppressed. The results of this study can provide some theoretical reference for the practical application of SRLs.
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Key words:
- nonlinear optics /
- dynamics /
- semiconductor ring laser /
- optical feedback /
- bandwidth
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