  Frequency combs show various applications in molecular fingerprinting,
imaging, communications, and so on. In the terahertz frequency range,
semiconductor-based quantum cascade lasers (QCLs) are ideal platforms for
realizing the frequency comb operation. Although self-started frequency comb
operation can be obtained in free-running terahertz QCLs due to the four-wave
mixing locking effects, resonant/off-resonant microwave injection, phase
locking, and femtosecond laser based locking techniques have been widely used
to broaden and stabilize terahertz QCL combs. These active locking methods
indeed show significant effects on the frequency stabilization of terahertz QCL
combs, but they simultaneously have drawbacks, such as introducing large phase
noise and requiring complex optical coupling and/or electrical circuits. Here,
we demonstrate Farey tree locking of terahertz QCL frequency combs under
microwave injection. The frequency competition between the Farey fraction
frequency and the cavity round-trip frequency results in the frequency locking
of terahertz QCL combs, and the Farey fraction frequencies can be accurately
anticipated based on the downward trend of the Farey tree hierarchy.
Furthermore, dual-comb experimental results show that the phase noise of the
dual-comb spectral lines is significantly reduced by employing the Farey tree
locking method. These results pave the way to deploying compact and low phase
noise terahertz frequency comb sources.
