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图 1是利用半导体激光器产生窄线宽光子微波信号的示意图。其中, 可调激光器(tunable laser,TL)的输出光经过分束器(beam splitter,BS)BS1后注入到SL中,使SL进入P1振荡。SL输出的光经过分束器BS2分成两部分,一部分光经FBG反馈后再回到SL来压缩线宽,一部分光输出后被探测。透镜L1和L2分别为准直和汇聚透镜。基于SL的Lang-Kobayashi模型并引入FBG光反馈后,描述基于SL的窄线宽光子微波信号产生的速率方程为[21]:
$ \begin{gathered} \frac{\mathrm{d} a}{\mathrm{~d} t}=\frac{1-\mathrm{i} b}{2}\left[\frac{\gamma \rho n}{\sigma J}-\mathit{\Gamma }\left(|a|^{2}-1\right)\right] a+ \\ \xi \gamma a_{0} \exp (-\mathrm{i} 2 \pi \Delta \nu t)+\eta \gamma \exp (\mathrm{i} \theta)[r(t) \times \\ \quad \exp (-\mathrm{i} \Delta \mathit{\Omega } t)] \cdot a(t-\tau)+F \end{gathered} $
(1) $ \begin{gathered} \frac{\mathrm{d} n}{\mathrm{~d} t}=-\left(\sigma+\rho|a|^{2}\right) n- \\ \sigma J\left(1-\frac{\mathit{\Gamma }}{\gamma}|a|^{2}\right)\left(|a|^{2}-1\right) \end{gathered} $
(2) $ \begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;r(\mathit{\Omega }) = \\ \frac{{\kappa \sinh \left( {\sqrt {{\kappa ^2} - {\delta ^2}} L} \right)}}{{ - \delta \sinh \left( {\sqrt {{\kappa ^2} - {\delta ^2}} L} \right) + {\rm{i}}\sqrt {{\kappa ^2} - {\delta ^2}} \cosh \left( {\sqrt {{\kappa ^2} - {\delta ^2}} L} \right)}} \end{array} $
(3) 式中,a为电场复振幅,n为载流子数,a0为激光器自由运行时的电场强度,η为滤波反馈强度,τ为外腔反馈时间,Ω表示角频率,ΔΩ为激光器频率与FBG中心频率的角频率失谐,θ为相位变化,ξ为注入强度,Δν为注入频率失谐,γ为腔衰减速率,σ为自发载流子弛豫速率,ρ代表微分载流子弛豫速率,Γ代表非线性载流子弛豫速率,b是线宽增强因子,J为归一化的电流,κ为FBG的耦合系数,L为FBG腔长,r(t)为FBG在时域中的响应,r(Ω)为FBG在频域中的响应,δ=NΩ/c表示反向传播模式与FBG布喇格频率的相位失配,N代表光纤折射率,c表示光速。FBG的带宽可由κc/(πN)得出,F为噪声项,“*”表示卷积运算。参量的物理含义和取值请见表 1。
Table 1. Simulation parameters of the laser and FBG
parameters value cavity decay rate γ 5.36×1011s-1 spontaneous carrier relaxation rate σ 5.96×109s-1 differential carrier relaxation rate ρ 7.53×109s-1 nonlinear carrier relaxation rate Γ 1.91×1010s-1 linewidth enhancement factor b 3.2 normalized bias current J 1.222 coupling coefficient of FBG κ 200m-1 refractive index of optical fiber N 1.45 length of FBG L 20mm velocity of light c 3×108m/s feedback time τ 2ns 值得一提的是,在上述速率方程推导的过程中,TL和SL使用的复电场ETL(t)和ESL(t)分别为:
$ E_{\mathrm{TL}}(t)=\sqrt{S_{\mathrm{TL}}} \exp \left[-\mathrm{i} \varphi_{\mathrm{TL}}(t)\right] $
(4) $ E_{\mathrm{SL}}(t)=\sqrt{S_{\mathrm{SL}}} \exp \left[-\mathrm{i} \varphi_{\mathrm{SL}}(t)\right] $
(5) 式中,STL和SSL分别表示TL和SL输出的光子数,φTL(t)和φSL(t)分别表示TL和SL的相位, φTL(t)和φSL(t)是与时间相关的函数,由于TL工作在稳定态,所以假设它的相位φTL(t)为常数0。尽管在TL的注入下,SL的相位φSL(t)随时间波动,但是它的波动较小,因此也假设为一常数。在这样的假设下两束激光的相位差在仿真过程中是保持恒定的[22]。
为了量化微波线宽,文中使用常用的微波频率随微波功率分布的标准方差进行计算,具体的表达式为:
$ \Delta f=\left[\left\langle\nu^{2}\right\rangle-\langle\nu\rangle^{2}\right]^{1 / 2} $
(6) $ \left\{\begin{array}{l} \langle\nu\rangle=\frac{\int_{-\infty}^{\infty} \nu P(\nu) \mathrm{d} \nu}{\int_{-\infty}^{\infty} P(\nu) \mathrm{d} \nu} \\ \left\langle\nu^{2}\right\rangle=\frac{\int_{-\infty}^{\infty} \nu^{2} P(\nu) \mathrm{d} \nu}{\int_{-\infty}^{\infty} P(\nu) \mathrm{d} \nu} \end{array}\right. $
(7) 式中,Δf表示光子微波的线宽,ν和P(ν)表示功率谱中的频率和相应的功率。
基于半导体激光器窄线宽光子微波信号获取
Generation of narrow linewidth photonic microwave signal using semiconductor laser with optical feedback
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摘要: 为了研究光注入半导体激光器(SL)产生的光子微波信号的性能, 基于SL的速率方程和光纤布喇格光栅(FBG)滤波理论, 采用数值仿真的方法进行了理论分析, 得到了各种注入参量下的光谱、功率谱和线宽, 并讨论了反馈参量对微波线宽的影响, 考虑到光注入下产生的微波线宽较宽, 进一步引入FBG光反馈窄化了微波信号的线宽。结果表明, 当SL仅在光注入作用下时, 通过改变注入参量, 可实现微波频率连续可调谐和微波强度最大化; 微波线宽随着反馈强度的增加逐渐变窄, 通过适当调节反馈参量可将微波线宽压缩到10kHz以下。该研究结果可为半导体激光器在光生微波中的应用提供一定的理论参考。Abstract: In order to study the performance of the photonic microwave signal generated by a semiconductor laser (SL) under optical injection, based on the rate equation of SL and the fiber Bragg grating (FBG) filter theory, the optical spectra, power spectra, and linewidth under different injection parameters were obtained by numerical simulation and theoretical analysis, and the effect of the feedback parameters on the microwave linewidth was also studied. Considering that the generated microwave signal has wide linewidth, a FBG optical feedback was further introduced to narrow the microwave linewidth. The results show that, for the SL subject to optical injection only, the microwave frequency can be continuously tuned and the microwave intensity can be maximized by changing the injection parameters; the microwave linewidth gradually decreases with the increase of feedback strength. Through properly adjusting the feedback parameters, the microwave linewidth can be compressed below 10kHz. The results can provide a theoretical reference for the application of semiconductor laser in the photonic microwave signals generation.
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Key words:
- nonlinear optics /
- photonic microwave /
- optical injection /
- fiber Bragg grating /
- period one
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Table 1. Simulation parameters of the laser and FBG
parameters value cavity decay rate γ 5.36×1011s-1 spontaneous carrier relaxation rate σ 5.96×109s-1 differential carrier relaxation rate ρ 7.53×109s-1 nonlinear carrier relaxation rate Γ 1.91×1010s-1 linewidth enhancement factor b 3.2 normalized bias current J 1.222 coupling coefficient of FBG κ 200m-1 refractive index of optical fiber N 1.45 length of FBG L 20mm velocity of light c 3×108m/s feedback time τ 2ns -
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