## 2 Experimental Setup and Results  
The QCL under study was grown on an InP substrate by solid-source molecular beam epitaxy [33]. The active region of the laser consists of 30 stages of InGaAs/InAlAs quantum wells. The laser has a ridge width of 8.5 \(\upmu\)m and a cavity length of 2.0 mm. The first-order buried distributed feedback grating is fabricated on the upper InGaAs layer for achieving the single-mode emission. The front facet of the laser has an anti-reflection coating with a reflectivity of 1%, and the rear facet has a high-reflection coating with a reflectivity of 95%. Figure 1a shows the experimental setup for the generation of P1 oscillations. The QCL is powered by a low-noise current source (Newport, LDC-3736), and the operation temperature is maintained at 20 \({}^{\circ}\)C by using a thermo-electric cooler. The emitted light is collimated by an aspherical lens with a focal length of 4.0 mm. The optical feedback is provided by a reflector with a reflectivity of 47.5% that is placed about 61 cm away from the QCL. The tilt-angle of the reflector is finely adjusted in the yaw direction for the excitation of P1 oscillations. The optical spectrum is measured by a high-resolution (0.08/cm) Fourier transform infrared spectrometer (FTIR, Bruker Vertex 80). The optical signal is detected by an HgCdTe photodetector with a nominal bandwidth of 560 MHz. The temporal waveform is recorded on a digital oscilloscope (OSC, Keysight DSOS604A, 6 GHz bandwidth), and the electrical spectrum is measured on an electrical spectrum analyzer (ESA, Keysight N9040B, 50 GHz bandwidth). As shown in Figure 1b, the free-running QCL exhibits a lasing threshold of 245 mA and emits a single mode around 2210/cm or 4.53 \(\upmu\)m (see inset). The optical linewidth is extracted from the analysis of the measured frequency noise spectrum [16]. Figure 1c shows that the optical linewidth is around 16 MHz. The slight increase in the linewidth with the pump current can be attributed to the non-uniform carrier distribution induced by the spatial hole-burning effect [34; 35]. On the other hand, the small fluctuations of the linewidth at 270 mA and 290 MA are due to the fact that the operation temperature is slightly tuned at each pump current to meet the frequency requirement of the frequency discriminator [16; 36]. When the reflector in Figure 1a is well aligned with the optical path (tilt angle is zero), the QCL remains stable and produces continuous-wave output. When the reflector is tilted in the yaw direction, however, the QCL loses its stability and produces periodic oscillations at small tilt angles, quasi-periodic oscillations at moderate angles, and low-frequency oscillations at large angles. For details of the oscillation dynamics, refer to [22; 23]. The generation of the above nonlinear dynamic is attributed to the non-degeneracy of odd-order round-trip reflections with the even-order specimens, which arises from the tilted optical feedback [37; 38]. This work focuses on the P1 dynamic, which is a type of periodic oscillation with only one period in the temporal waveform. Figure 2a shows that the P1 oscillation of the QCL exhibits a sinusoidal waveform, which is similar to the waveform of lasers with direct sinusoidal current modulation. The electrical spectrum in Figure 2b shows that the modulation frequency of the laser is 246.5 MHz. This modulation frequency equals the exact external cavity frequency, which is the inverse of the round-trip propagation time from the QCL to the reflector. Therefore, the P1 oscillation frequency is solely determined by the external cavity frequency due to the absence of relaxation oscillation resonance in QCLs [39]. This differs from common near-infrared laser diodes, where the P1 frequency is governed by the interaction of the relaxation oscillation frequency and the external cavity frequency [1]. Figure 1: (**a**) Experimental setup for the modulation using P1 dynamics. FTIR, Fourier transform infrared spectrometer; PD, photodetector; ESA, electrical spectrum analyzer; OSC, oscilloscope.