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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., Comment: 22 page, 7 figures

Spectral/angular emissivity $e$ and absorptivity ${\alpha}$ of an object are widely believed to be identical by Kirchhoff's law of thermal radiation in reciprocal systems, but this introduces an intrinsic and inevitable energy loss for energy conversion and harvesting devices. So far, experimental evidences of breaking this well-known balance are still absent, and previous theoretical proposals are restricted to narrow single-band nonreciprocal radiation. Here we observe for the first time, to our knowledge, the violation of Kirchhoff's law using ultrathin ($<{\lambda}/40$, ${\lambda}$ is the working wavelength) magnetized InAs semiconductor films at epsilon-near-zero (ENZ) frequencies. Large difference of $|{\alpha}-e|>0.6$ has been experimentally demonstrated under a moderate external magnetic field. Moreover, based on magnetized ENZ building blocks supporting asymmetrically radiative Berreman and surface ENZ modes, we show versatile shaping of nonreciprocal thermal radiation: single-band, dual-band, and broadband nonreciprocal emission spectra at different wavebands. Our findings of breaking Kirchhoff's law will advance the conventional understanding of emission and absorption processes of natural objects, and lay a solid foundation for more comprehensive studies in designing various nonreciprocal thermal emitters. The reported recipe of diversely shaping nonreciprocal emission will also breed new possibilities in renovating next-generation nonreciprocal energy devices in the areas of solar cells, thermophotovoltaic, radiative cooling, etc., Comment: 21 pages, 4 figures

Broadband dual-comb spectroscopy has attracted increasing interests due to its unique advantages in high spectral resolution, fast detection, and so on. Although the dual-comb technique is relatively mature in the infrared wavelengths, it is, currently, not commercially capable of practical applications in the terahertz regime due to the lack of high performance broadband terahertz dual-comb sources. In the terahertz frequency range, the electrically pumped quantum cascade laser (QCL) is a suitable candidate for the dual-comb operation. However, free running terahertz QCL dual-comb sources normally show limited optical bandwidths ($\sim$100-200 GHz). Although the resonant microwave injection locking has been widely used to broaden the emission spectra of terahertz QCLs by modulating the laser drive current at the cavity round-trip frequency, it is hard to be employed to broaden the dual-comb bandwidths due to the large phase noise induced by the resonant injection and non-ideal microwave circuits. Therefore, it is challenging to obtain broadband terahertz dual-comb sources that can fully exploits the laser gain bandwidth. Here, we employ an off-resonant microwave injection to significantly broaden the dual-comb bandwidth of a terahertz QCL dual-comb source emitting around 4.2 THz. The measured optical dual-comb bandwidth is broadened from 147 GHz in free running to $>$450 GHz under the off-resonant injection. The broadened dual-comb bandwidth is experimentally proved by the transmission measurements of a filter and a GaAs etalon. By performing a simple numerical analysis based on a rate equation model, we explain that the broadband dual-comb operation under the off-resonant microwave injection could be resulted from a wider lasing bandwidth and a higher degree of phase matching., Comment: 22 pages, 6 figures

Dual-comb sources with equally spaced and low phase noise frequency lines are of great importance for high resolution spectroscopy and metrology. In the terahertz frequency range, electrically pumped semiconductor quantum cascade lasers (QCLs) are suitable candidates for frequency comb and dual-comb operation. For a single laser frequency comb, the repetition rate can be locked using a microwave injection locking and the carrier frequency can be locked to a highly stable source. However, for the locking of two laser combs, four frequencies (two repetition rates and two carrier offset frequencies) should be simultaneously locked; If one only refers to the dual-comb signal, two relative frequencies, i.e., the offset frequency and repetition frequency of one laser against those of the other laser, should be locked. Although the locking techniques that have been successfully used for a single laser comb can be, in principle, applied to a dual-comb laser source, the complete locking considerably complicates the implementation of such a system. Here, we propose a method to stabilize a terahertz QCL dual-comb source by phase locking one of the dual-comb lines to a radio frequency (RF) synthesizer. This technique forces one of the lasers to follow the tone of the other one (keeping the sum of the carrier offset frequency difference and repetition frequency difference between the two laser combs as a constant) by exploiting a laser self-detection that avoids the use of an external detector. Through the demonstration of this locking technique, we demonstrate that the dual-comb can generate periodic pulses over a 2 us time scale, showing that the terahertz QCL comb without a control of the repetition rate can produce pulsed-type waveforms., Comment: 21 pages, 5 figures

Due to its fast and high resolution characteristics, dual-comb spectroscopy has attracted an increasing amount of interest since its first demonstration. In the terahertz frequency range where abundant absorption lines (finger prints) of molecules are located, multiheterodyne spectroscopy that employs the dual-comb technique shows an advantage in real-time spectral detection over the traditional Fourier transform infrared or time domain spectroscopies. Here, we demonstrate compact terahertz dual-comb spectroscopy based on quantum cascade lasers (QCLs). In our experiment, two free-running QCLs generate approximately 120 GHz wide combs centered at 4.2 THz, with slightly different repetition frequencies. We observe that $\sim$490 nW terahertz power coupling of one laser into the other suffices for laser-self-detecting the dual-comb spectrum that is registered by a microwave spectrum analyzer. Furthermore, we demonstrate practical terahertz transmission dual-comb spectroscopy with our device, by implementing a short air path at room temperature. Spectra are shown of semiconductor samples and of moist air, the latter allowing rapid monitoring of the relative humidity. Our devices should be readily extendable to perform imaging, microscopy and near-field microscopy in the terahertz regime., Comment: 7 pages, 4 figures