Harmonic frequency combs, in which the lasing modes are separated by a period of tens of free spectral ranges from each other, have been recently discovered in quantum cascade lasers (QCLs). There is an ongoing debate how the harmonic combs can be formed and stable under continuous pumping. Here we reproduce the harmonic state of lasing in QCLs by space-time-domain numerical simulations and show that the corresponding optical wave has a frequency-modulated nature, with the fundamental beat note in the electron population being suppressed. To understand the physics behind the formation and stability of the harmonic state, we develop a frequency-domain linearized analytic theory which analyzes the stability of single mode lasing with respect to the harmonic state formation and the instabilities of the resulting harmonic state. Our analysis shows how the instability of a single lasing mode can result in the growth of harmonic side modes. The coupling between the two side modes leads to enhancement of the gain for frequency-modulated waves as compared to the gain for amplitude-modulated waves. The dependence of the sideband gain spectrum on group velocity dispersion is also studied. While the instability of a single lasing mode with respect to the growth of side bands separated by many free spectral ranges does not prove that the harmonic state can be self-starting, we analyze the instability of three strong harmonic modes and demonstrate that the harmonic state can be stable and self-supported.

中文翻译：

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量子级联激光器中的谐波频率梳：时域和频域理论

最近在量子级联激光器（QCL）中发现了谐波频率梳，其中激光模式彼此间隔数十个自由光谱范围。正在进行的辩论是如何在连续泵浦下形成和稳定谐波梳。在这里，我们通过空时域数值模拟来再现QCL中激光的谐波状态，并表明相应的光波具有调频特性，电子群中的基本拍音被抑制。为了了解谐波状态形成和稳定性背后的物理原理，我们开发了一种频域线性分析理论，该理论分析了单模激光相对于谐波状态形成的稳定性以及所得谐波状态的不稳定性。我们的分析表明，单个激光模式的不稳定性如何导致谐波侧模的增长。与振幅调制波的增益相比，两侧模式之间的耦合导致频率调制波的增益提高。还研究了边带增益谱对群速度色散的依赖性。尽管单个激光模式相对于由许多自由光谱范围分开的边带的增长所产生的不稳定性并不能证明谐波状态可以自启动，但我们分析了三种强谐波模式的不稳定性并证明了谐波状态可以稳定并自给自足。与振幅调制波的增益相比，两侧模式之间的耦合导致频率调制波的增益提高。还研究了边带增益谱对群速度色散的依赖性。尽管单个激光模式相对于由许多自由光谱范围分开的边带的增长所产生的不稳定性并不能证明谐波状态可以自启动，但我们分析了三种强谐波模式的不稳定性并证明了谐波状态可以稳定并自给自足。与振幅调制波的增益相比，两侧模式之间的耦合导致频率调制波的增益提高。还研究了边带增益谱对群速度色散的依赖性。尽管单个激光模式相对于由许多自由光谱范围分开的边带的增长所产生的不稳定性并不能证明谐波状态可以自启动，但我们分析了三种强谐波模式的不稳定性并证明了谐波状态可以稳定并自给自足。