Observing the Intrinsic Linewidth of a Quantum-Cascade Laser: Beyond the Schawlow-Townes Limit

A comprehensive investigation of the frequency-noise spectral density of a free-running midinfrared quantum-cascade laser is presented for the first time. It provides direct evidence of the leveling of this noise down to a white-noise plateau, corresponding to an intrinsic linewidth of a few hundred hertz. The experiment is in agreement with the most recent theory on the fundamental mechanism of line broadening in quantum-cascade lasers, which provides a new insight into the Schawlow-Townes formula and predicts a narrowing beyond the limit set by the radiative lifetime of the upper level.

Figure 1

Three-level model of one stage of the QCL active region. The lasing transition is between levels 3 and 2, with a radiative lifetime ${\tau }_r$, while ${\tau }_3=(1/{\tau }_{31}+1/{\tau }_{32}{)}^{-1}$ is the lifetime of the nonradiative relaxation processes of the upper level. Its total lifetime is ${\tau }_t=(1/{\tau }_r+1/{\tau }_3{)}^{-1}$. $P=I/e$ is the pump rate corresponding to the current $I$, and $\eta$ is the injection efficiency from the previous stage.

Figure 2

Principle of the frequency-noise PSD measurement. The black line is the molecular absorption profile, the light gray arrow is the laser frequency ${\nu }_0$, and the small dark gray arrows represent the noise component at frequency $f_n$. The strong-absorbance regime allows one to maximize both the slope (i.e., the gain) and the dynamic range of the measurement. Inset: Spectral response of the frequency-to-amplitude converter. The flat response region corresponds to the region where the absorption profile can be approximated as linear. The 3-dB cutoff lies between 40 and 50 MHz.

Figure 3

Frequency-noise PSD of the QCL (gray) acquired between 10 Hz and 100 MHz, compared with the expected contribution arising from the driving-current noise (straight black line). The dashed lines show the three main trends: the $1/f$ and $1/f^2$ trends fit well the experimental data up to a few MHz, while at higher frequencies the flattening to a white-noise level $N_w$ is evident.

Figure 4

The experimental data set (black dots) is compared with the prediction of the theoretical model by Yamanishi et al. [Eq. (2)]. The dark gray curve corresponds to the set of parameters considered as the most realistic for our device: ${\tau }_{21}=0.25\mathrm{ps}$, ${\tau }_{31}=2.0\mathrm{ps}$, ${\tau }_{32}=3.4\mathrm{ps}$, ${\tau }_t=1.3\mathrm{ps}$ (see Ref. 11). The gray area in the plot covers the uncertainty space, as inferred by the parameters’ variability for similar devices reported in the literature: $0.15<{\tau }_{21}<0.35\mathrm{ps}$, $1.8<{\tau }_{31}<4.5\mathrm{ps}$, $2.5<{\tau }_{32}<4.3\mathrm{ps}$, $7<{\tau }_r<8\mathrm{ns}$, $0.6<\eta <0.8$, and $1.2\times {10}^{11}<\gamma <1.4\times {10}^{11}{\mathrm{s}}^{-1}$. Inset: Comparison with the model calculated for ${\beta }_{\mathrm{eff}}=\beta$ and with the conventional ST formula evaluation described in the text.