## 3 Simulations and Discussion  
In order to show the optical sidebands induced by the P1 oscillation, we simulate the optical spectrum using the rate equation approach, which takes into account the effect of tilted optical feedback [22; 23; 37]. Rate equations for the photon number \(S(t)\) and the phase of the electric field \(\phi(t)\) are given by  
\[\frac{dS}{dt}=\bigg{(}mG_{0}\Delta N-\frac{1}{\tau_{p}}\bigg{)}S+m\beta\frac{ N_{3}}{\tau_{sp}}+2k_{c}f_{n}\sqrt{r_{ext}S(t)}\sum_{n=1}^{\infty}\Big{(} \sqrt{Rr_{ext}}\Big{)}^{n-1}\sqrt{S(t-n\tau_{ext})}\cos(\Delta\phi_{n}) \tag{1}\]  
\[\frac{d\phi}{dt}=\frac{\alpha_{H}}{2}\bigg{(}mG_{0}\Delta N-\frac{1}{\tau_{p}} \bigg{)}-\frac{2k_{c}f_{n}}{\sqrt{r_{ext}S(t)}}\sum_{n=1}^{\infty}\Big{(} \sqrt{Rr_{ext}}\Big{)}^{n-1}\sqrt{S(t-n\tau_{ext})}\sin(\Delta\phi_{n}) \tag{2}\]  
Figure 4: Modulation depth of the temporal waveform for pump currents of 280, 290, and 300 mA. Figure 5: Electrical spectra of P1 oscillations for various single-trip feedback lengths. The pump current is 290 mA. with  
\[f_{n}=\left\{ \begin{array}{ccc}\frac{\sin(\theta\pi)}{\theta\pi}&\text{for}&n=1,3,5\ldots\\ 1&\text{for}&n=2,4,6\ldots\end{array} \right. \tag{3}\]  
where \(m\) is the number of cascading gain stages, \(G_{0}\) is the gain coefficient, \(\Delta N\) is the population inversion, \(\tau_{p}\) is the photon lifetime, \(\tau_{sp}\) is the spontaneous emission lifetime, \(\beta\) is the spontaneous emission factor, \(N_{3}\) is the carrier number in the upper lasing level, \(\alpha_{H}\) is the linewidth broadening factor (LBF), and \(R\) is the facet reflectivity. The feedback ratio is \(r_{ext}\), \(\tau_{ext}\) is the feedback delay time, and \(k_{c}\) is the feedback coupling coefficient to the laser cavity. The phase difference is \(\Delta\phi_{n}=n\phi_{0}+\phi(t)-\phi(t-n\tau_{ext})\), with \(\phi_{0}\) being the initial feedback phase, and \(n\) being the order of roundtrip reflection. The tilted angle of the reflection mirror is \(\theta\), which is normalized by the tilt angle at which the first diffractive coupling minimum occurs. Details of the rate equation model and the parameter values will be described elsewhere. Figure 6a simulates the optical spectrum of the QCL with P1 oscillation, where the LBF is 2.0 and the single-trip feedback length is 60 cm. It is shown that the laser exhibits multiple sidebands with a spacing of 250 MHz, which is determined by the P1 frequency or the external cavity frequency. The optical sidebands are asymmetric with respect to the main lasing mode due to the non-zero LBF. Because of this phase-amplitude coupling effect, the P1 oscillations introduce not only amplitude modulation (AM) but also frequency modulation (FM) [45]. Taking advantage of this feature, the P1 oscillations of near-infrared laser diodes have been used for the AM-to-FM conversion of photonic microwaves [46; 47]. The LBF of the QCL under study increases almost linearly with the pump current, from 1.4 at 250 mA to 2.5 at 310 mA (see [23] for details). In comparison with the AM measurement in Figure 2, experimental characterization of the FM is much more challenging, usually requiring a frequency discriminator to convert the FM into AM [48]. Therefore, we confirm the existence of FM using rate Equations (1) and (2) as well. Based on the temporal waveform in Figure 6b, the FM index is deduced to be 17%, and the AM index (or the modulation depth) is 1.9%. Therefore, the P1 oscillation-induced modulation is dominated by the FM. This behavior is also similar to that of QCLs with direct current modulation, where the ratio of the AM index to the FM index is around 0.07 [49]. Finally, we summarize the potential application of P1 oscillations in the modulation spectroscopy technique. The WMS usually requires a modulation frequency in the range of kHz to MHz [25; 32], as well as a very large modulation depth [50; 51; 52].