This work theoretically investigates the frequency noise (FN) characteristics of quantum cascade lasers subject to the optical injection through a set of coupled rate equations with Langevin noise sources. It is shown that the low-frequency FN is completely suppressed by the optical injection, and the suppression bandwidth increases with the increasing injection ratio. The optimal FN peak suppression ratio at an injection ratio of 10 dB reaches 2.9 dB. In addition, it is found that the optical injection at positive frequency detunings close to the locking boundary invokes an additional peak in the FN spectrum, which can be higher than the carrier noise-induced one of free-running lasers. This peak amplitude strongly depends on the value of the linewidth broadening factor. Unlike injection-locked interband lasers, the FN peak does not necessarily exhibit a resonance.
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The optical injection technique synchronizes one laser of high spectral purity (master laser) with another laser (slave laser) through an optical isolator. Semiconductor lasers subject to the optical injection produces a large variety of nonlinear dynamics including stable injection locking, periodic and aperiodic pulsations, and chaotic oscillations . The stable locking regime is bounded by the Hopf bifurcation and the saddle-node bifurcation, where the phase of the slave laser is synchronized to that of the master laser . Inside the injection locking regime, the optical injection can considerably improve dynamical performances of semiconductor lasers, including enhancement of the modulation bandwidth, reduction of the frequency chirp, as well as suppression of the relative intensity noise (RIN) and the frequency noise (FN, or phase noise) . Owing to the above benefits, the injection locking technique attracts increasing interests for quantum cascade lasers (QCLs) as well. In recent years, researchers have theoretically investigated the injection locking regime [4,5], the modulation bandwidth enhancement properties , and the RIN suppression characteristics  of QCLs subject to the optical injection. In experiments, Taubman et al. demonstrated a locking range of 1.0 GHz at an injection ratio of −16 dB . Juretzka et al. proved that the RIN of a QCL was reduced up to 10 dB by the optical injection . On the other hand, most work employs the optical injection to narrow the spectral linewidth of QCLs. Bielsa et al. locked a 9.2 μm QCL to a single-mode CO2 laser, and reduced the laser linewidth from several MHz down to the kilohertz range . An alternative popular method is locking mid-infrared or terahertz QCLs to near-infrared optical frequency combs through the difference-frequency generation process, where the linewidth of QCLs is governed by the frequency comb [11–13].
In our recent work, it was demonstrated theoretically that the FN spectrum of free-running QCLs exhibited a broad peak due to the carrier noise induced carrier variation . In this work, we theoretically investigate the FN characteristics of a QCL subject to the optical injection. It is found that the optical injection significantly suppresses the FN of QCLs in the low frequency range, and the suppression bandwidth increases with the injection strength. At an injection ratio of 10 dB, the optimal FN peak suppression ratio reaches 2.9 dB, which is achieved at detuning frequency of −3.3 GHz. In addition, the optical injection at frequency detuning close to the positive locking boundary invokes a peak in the FN spectrum, and the peak amplitude is strongly dependent on the linewidth broadening factor.
2. Rate equation model of injection-locked QCLs
The electronic structure model of the QCL takes into account the upper laser level, the lower laser level, and the bottom level . Electrons are firstly injected from the injector region into the upper laser level in the gain region. Then, electrons scatter into the lower laser level with a time τ32, and into the bottom level with a time τ31 through longitudinal-optical phonon emissions . Subsequently, electrons in the lower laser level deplete into the bottom level with a time τ21, and finally escape the gain region with a tunneling out time τout into the next stage. The stimulated emission occurs upon the population inversion between the upper and the lower laser levels. Correspondingly, the rate equations describing the carrier numbers in the upper level (N3), in the lower level (N2), and in the bottom level (N1) are given by16], the photon number (S) and the phase (ϕ) of the injection-locked QCL are derived as17,18].
The rate equation model includes the carrier noise and the spontaneous emission noise through the Langevin approach . The time averages of all the carrier (F3,2,1), photon (FS), and phase (Fϕ) noise sources are zero due to the random nature , and the correlation relations of the noise sources are:21,22]. In the following sections, the steady-state solutions and the optical spectra are calculated through the numerical approach, while the FN spectra are obtained through the semi-analytical approach.
3. Results and discussion
Table 1 lists the QCL parameters used for the simulations in this work, unless stated otherwise [23–27]. The free-running QCL exhibits a lasing threshold current of Ith = 223 mA, and the laser is pumped at 1.5 × Ith in all the simulations of this work. Based on the analysis of the integrated time series of the photon number, Fig. 1 illustrates the stable injection-locking boundaries, which are formed by the Hopf bifurcation and the saddle-node bifurcation [2,4]. The same as interband semiconductor lasers, the two bifurcations strongly depend on the value of LBF. QCLs usually show near-zero LBFs, and the reported values range from 0 up to 3.0 [28, 29]. The non-zero LBF in QCLs has been found to originate from the non-parabolicity of the band structure, the many-body effect, the resonant tunneling transport, as well as the counter-rotating wave contribution [30, 31]. The bifurcation diagram is symmetric for a zero LBF, while a non-zero LBF leads to an asymmetric diagram. The stable locking regime broadens with the increasing LBF—at an injection ratio of 0 dB, the locking range increases from 6.4 GHz for αH = 0 to 15.3 GHz for αH = 2.0. In the following sections, we discuss the injection strength effect and the frequency detuning effect on the FN spectrum of QCLs within the stable locking regime, respectively.
3.1 Injection strength effect on the FN
Within the stable locking regime, the steady-state solutions of the injection-locked QCL are derived asFigure 2 shows the injection strength effect on the photon number at zero frequency detuning. As suggested in Eq. (8), the photon of the injection-locked QCL at zero detuning increases with the injection ratio while decreases with the LBF .
In rating Eqs. (1)-(5), all the noise sources perturb the laser system slightly away from the steady-state condition. Based on the standard small-signal analysis in the Appendix, the FN spectrum of the injection-locked QCL is calculated by :
Figure 3(a) shows that the FN of the free-running QCL is almost constant at 930 Hz2/Hz for low frequencies up to 1.0 GHz. Interestingly, the FN exhibits a broad peak in the frequency range of 10 GHz to 1.0 THz, which is due to the carrier noise perturbation based on our previous analysis . Beyond 10 THz, the FN becomes almost flat at 746 Hz2/Hz, which is determined solely by the spontaneous emission noise. The difference between the low- and high-frequency FN is governed by a factor of () . When the QCL is subject to the optical injection, the FN in the whole frequency range is reduced, partly owing to the increase of the photon number, the decrease of the carrier number in the upper laser level, as well as the reduction of the threshold gain . The low-frequency FN undergoes the largest reduction, down to almost zero as expressed in Eq. (12). This is because the low-frequency phase of the slave laser is strongly locked to that of the master laser without any noise. In case the rate equations take into account the optical noise of the master laser, which has no correlation with that of the slave laser, the low-frequency FN will be the same as that of the master laser [34–36]. Therefore, like injection-locked interband lasers, the spectral linewidth of the injection-locked QCL is only governed by the FN of the master laser, and no longer follows the Schawlow-Townes limit of the free-running laser . It is worthwhile to mention that the relative intensity noise of injection-locked interband lasers is dominated by the master laser as well, and the injection-locked QCL is expected to exhibit a similar behavior [37,38].
In order to quantitatively characterize this low frequency range, we simply define the suppression bandwidth, within which the FN is lower than −3 dB of the FN in the free-running QCL (arrow in Fig. 3(a)). Thus, the FN below the suppression bandwidth is primarily determined by the master laser, while that above the bandwidth is mainly governed by the slave laser. Figure 3(b) presents that the FN suppression bandwidth (circles) increases with the injection strength from 4.0 MHz at Rinj = −60 dB up to 40 GHz at Rinj = 10 dB. However, a more rigorous definition of the FN suppression bandwidth for QCLs is required in future work. For interband lasers, the FN suppression bandwidth is found to be half-width of the stable locking bandwidth at αH = 0 [34, 35]. In order to characterize the FN suppression effect, the suppression ratio of the FN peak is defined as the peak amplitude of the free-running laser divided by that of the injection-locked laser. Obviously, the FN peak suppression ratio (squares) in Fig. 3(b) increases with the injection strength and reaches 2.7 dB at Rinj = 10 dB.
3.2 Frequency detuning effect on the FN
Figure 4 shows the frequency detuning effect on the photon number of the QCL at an injection ratio of 0 dB. For αH = 0, the maximum photon number occurs at zero frequency detuning, and the curve is symmetric for both negative and positive detunings as indicated in Eq. (8). On the other hand, the maximum photon number for non-zero LBFs is achieved at a negative detuning, where the phase difference ϕ between the master laser and the slave laser is found to be zero. For any LBF, the photon number of the injection-locked laser converges to the free running one (dashed line) at the detuning of (3.2 GHz), where the population inversion meets in Eq. (8).
Figure 5 shows the frequency detuning effect on the FN at Rinj = 0 dB for different values of LBF. For αH = 0 in Fig. 5(a), the FN of the free-running laser (dashed line) is constant over the whole frequency range. The minimum FN is achieved at zero frequency detuning while both the negative and the positive detunings raise the FN symmetrically. In addition, both frequency detunings lead to the appearance of a peak in the FN spectrum, and the peak amplitude becomes higher than the free-running case when the detuning is operated close to the locking boundaries ( ± 3.2 GHz). For non-zero LBFs in Figs. 5(b) and 5(c), the minimum FN is also achieved at a phase difference of zero, where the photon number S reaches maximum (see Fig. 4) and the carrier number in the upper laser level N3 reaches minimum (see Eq. (13)) . In addition, the FN peak increases with increasing frequency detuning. At some positive detunings (4.0 GHz in Fig. 5(b) and 5.0, 7.0, 9.0 GHz in Fig. 5(c)), the optical injection invokes another peak in the FN spectrum in addition to the carrier noise induced one of the free-running laser. The frequency of the peak arising from the optical injection is smaller, while the amplitude of the peak is higher than the latter one. Especially for αH = 2.0 in Fig. 5(c), the FN near the positive locking boundary (9.0 GHz) exhibits a giant peak, which is five times higher than the free-running case. The appearance of the giant peak near the positive locking boundary is similar to that of injection-locked interband lasers [18,39]. The peak in interband lasers is attributed to the underdamped resonance arising from the transient interference between the injection-locked field and the shifted cavity-resonance field . However, it is found that the injection-locked QCL only exhibits a resonance for detunings at 0, 5.0, 7.0, and 9.0 GHz in Fig. 5(c), while the other cases in Fig. 5 do not have any resonance. Thus, the physical mechanism for the peak appearance in QCLs is different to that in interband lasers, which requires further investigation in future work. In addition, the FN peak frequency decreases with increasing detuning frequency at the positive detuning side, which is opposite in interband lasers. The peak frequency eventually merges with the Hopf frequency at the positive locking boundary.
Figure 6(a) illustrates the suppression ratio diagram of the FN peak formed by the injection ratio and the detuning frequency at αH = 0.5. It is shown that the FN peak of the injection-locked QCL is mostly lower than that of the free-running laser, except at very positive detuning frequencies. The detuning frequency for achieving the maximum suppression ratio (dashed line) reduces with the injection ratio, from −1.4 GHz at Rinj = 0 dB down to −3.3 GHz at Rinj = 10 dB. Figure 6(b) shows that the detuning frequency for obtaining the maximum suppression bandwidth (dashed line) also decreases with the injection ratio, from −0.9 GHz at Rinj = 0 dB down to −3.3 GHz at Rinj = 10 dB.
Figure 7 simulates the optical spectra of the injection-locked QCL at Rinj = 0 dB, through integrating rate Eqs. (1)-(5) using the Euler-Maruyama method . The time integration step is 0.1 ps, the integration length is 1.0 μs, and each integration costs about nine-hour computational time. The spectral linewidth of the injection-locked QCL is almost zero for any LBF, because the linewidth is determined only by the low-frequency (<100 MHz) FN, which is zero in Fig. 5 . For αΗ = 2.0, the optical spectrum of the injection-locked QCL exhibits two side peaks (arrows), which arise from the giant FN peak in Fig. 5(c). The frequency separation between the side peaks and the central lasing peak is about 6.0 GHz, corresponding to the FN peak frequency.
In conclusion, the injection strength and the frequency detuning effects on the FN characteristics of the QCL have been numerically investigated through the rate equations. It is shown that the stable locking regime is broadened by a large LBF. The FN decreases with the increasing injection ratio, and the low-frequency FN reduces to zero when the master laser has no optical noise. The optical injection at positive detuning close to the locking boundary induces a peak in the FN spectrum, and the peak amplitude strongly depends on the LBF. In contrast to interband lasers, the injection-induced FN peak in QCLs does not necessarily have a resonance. In addition, the giant FN peak leads to the appearance of side peaks in the optical spectrum of the injection-locked QCL. Future work will take into account the optical noise of the master laser, and will investigate the optical feedback effect on the FN of QCLs.
Appendix, : Small-signal analysis of the rate equations
In the small-signal analysis of rate Eqs. (1)-(5), all the Langevin noise sources drive the laser system away from its steady-state solutions. The small-signal responses of the carriers, the photon, and the phase are written asEq. (14) into rate Eqs. (1)-(5), and neglecting harmonic frequency terms, we obtain the linearized differential rate equations in the form of matrix asEq. (11) based on the semi-analytical approach.
Shanghai Pujiang Program (17PJ1406500).
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