Single-mode instability in standing-wave lasers: The quantum cascade laser as a self-pumped parametric oscillator

We report the observation of a clear single-mode instability threshold in continuous-wave Fabry-Perot quantum cascade lasers (QCLs). The instability is characterized by the appearance of sidebands separated by tens of free spectral ranges (FSR) from the first lasing mode, at a pump current not much higher than the lasing threshold. As the current is increased, higher-order sidebands appear that preserve the initial spacing, and the spectra are suggestive of harmonically phase-locked waveforms. We present a theory of the instability that applies to all homogeneously broadened standing-wave lasers. The low instability threshold and the large sideband spacing can be explained by the combination of an unclamped, incoherent Lorentzian gain due to the population grating, and a coherent parametric gain caused by temporal population pulsations that changes the spectral gain line shape. The parametric term suppresses the gain of sidebands whose separation is much smaller than the reciprocal gain recovery time, while enhancing the gain of more distant sidebands. The large gain recovery frequency of the QCL compared to the FSR is essential to observe this parametric effect, which is responsible for the multiple-FSR sideband separation. We predict that by tuning the strength of the incoherent gain contribution, for example by engineering the modal overlap factors and the carrier diffusion, both amplitude-modulated (AM) or frequency-modulated emission can be achieved from QCLs. We provide initial evidence of an AM waveform emitted by a QCL with highly asymmetric facet reflectivities, thereby opening a promising route to ultrashort pulse generation in the mid-infrared. Together, the experiments and theory clarify a deep connection between parametric oscillation in optically pumped microresonators and the single-mode instability of lasers, tying together literature from the last 60 years.

Figure 1

(a) The emission spectrum at the instability threshold comprises a primary mode and two weak sidebands. (b) The temporal behavior of the field $E\left(t\right)$ depends on the relative phases of the three modes ${\varphi }_{-},{\varphi }_0$, and ${\varphi }_{+}$, and shown are the AM and FM configurations. (c) The AM and FM fields can be understood in terms of the constructive and destructive addition of two beat note phasors, where each phasor represents a contribution to the intensity modulation at the difference frequency $\delta \omega$ resulting from the superposition of each sideband with the primary mode. (d) In a standing-wave cavity, the intensity of each mode varies with position, and the spatial modes corresponding to different frequencies do not perfectly overlap.

Figure 2

Total power output of each QCL (from both facets) vs current, color coded to indicate the range over which the laser operates in a single mode (the region following threshold), harmonic state (the middle region), or dense state (the highest-power region). Inset: the intracavity power normalized to the saturation intensity (calculated from the measured output power and the best estimates for $\kappa ,T_1,T_2$, and the facet reflectivities) is plotted vs $J/J_{\mathrm{th}}$.

Figure 3

Spectra of the three uncoated QCLs (a) LL-9.8, (b) TL-4.6, and (c) DS-3.8 as the current is incremented, starting from below threshold.

Figure 4

Spectra of the three uncoated QCLs (a) LL-9.8, (b) TL-4.6, and (c) DS-3.8 as the current is decremented, starting from the current reached at the end of the upward current ramp shown in Fig. 3.

Figure 5

The IV curve of DS-3.8 exhibits a hysteresis as the current is increased (red, upper curve) and decreased (blue, lower curve). The hysteresis is correlated with the transition from the noisy harmonic state to the dense state on the ramp up, and from the dense state to the noisy harmonic state on the ramp down.

Figure 6

Spectra of TL-4.6:HR/AR as the current is (a) incremented, starting from below threshold, and (b) decremented, starting from the current reached at the end of the upward current ramp in (a).

Figure 7

The magnitude, phase, and real part of ${\chi }^{\left(3\right)}$ are plotted vs $\delta \omega T_2$ for three different ratios $T_1/T_2=40,10,5$. The parametric gain seen by the sidebands is determined by $\mathrm{Re}\left[{\chi }^{\left(3\right)}\right]$. Low-frequency PPs lead to a parametric suppression of the gain. For the gain to be parametrically enhanced, $\delta \omega$ must be large enough that the inversion can no longer follow the intensity in antiphase; in other words, the phase of ${\chi }^{\left(3\right)}$ must be between $-\pi /2$ and $\pi /2$.

Figure 8

Overview of the three different types of instabilities. (a) The incoherent instability relies only on the unclamped gain due to the PG, and occurs when parametric effects can be neglected. (b) The FM instability occurs when the gain due to a strong PG, despite the parametric suppression of the gain of small-detuned sidebands, allows the less-suppressed FM sidebands to reach threshold. (c) The AM instability occurs when the gain due to a weak PG, together with the parametric enhancement of the gain of large-detuned sidebands, allows the more strongly enhanced AM sidebands to reach threshold. In (b) and (c), the value $T_1/T_2=20$ was used. The FP modes (green, bottom of each panel) are not associated with the ordinate, and simply provide a sense of the mode spacing.

Figure 9

The GVD is extracted from the subthreshold spectrum for (a) LL-9.8 at 973 mA, (b) TL-4.6 at 424 mA, and (c) DS-3.8 at 354 mA. The subthreshold spectra are plotted for reference on an arbitrary linear scale.

Figure 10

The beat note of LL-9.8 at 1434 mA, the current at which the dense state appeared. The resolution bandwidth of the spectrum analyzer is 430 Hz.

Figure 11

The sideband gain ${\overline{g}}_{\text{AM}}/\overline{\ell }$ of a traveling-wave laser, given in Eq. (G23), is plotted at various pump strengths, for three different values of $T_1/T_2$: 1, 10, and 100. The largest value of $p$ in each plot is equal to the instability threshold given in Eq. (G26).

Figure 12

Numerical solutions of the instability threshold obtained by setting the gain $\overline{g}$ in Eq. (G38) equal to the loss $\overline{\ell }$, yielding both (a) the sideband separation $\delta {\omega }_{\text{sb}}{T}_2$ and (b) the pumping $p_{\text{sb}}$. The experimentally measured values of $\delta {\omega }_{\mathrm{sb}}{T}_2$ are compared to the theory to infer $T_1/T_2$, which also gives the theoretical prediction for the instability threshold $p_{\text{sb}}$.