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The recently discovered ability of the quantum cascade laser to produce a harmonic frequency comb has attracted new interest in these devices for both applications and fundamental laser physics. In this review we present an extensive experimental phenomenology of the harmonic state, including its appearance in mid-infrared and terahertz quantum cascade lasers, studies of its destabilization induced by delayed optical feedback, and the assessment of its frequency comb nature. A theoretical model explaining its origin as due to the mutual interaction of population gratings and population pulsations inside the laser cavity will be described. We explore different approaches to control the spacing of the harmonic state, such as optical injection seeding and variation of the device temperature. Prospective applications of the harmonic state include microwave and terahertz generation, picosecond pulse generation in the mid-infrared, and broadband spectroscopy.

We investigate the nonlinear optical response of a commercial extended-wavelength In$_{0.81}$Ga$_{0.19}$As photodetector. Degenerate two-photon absorption in the mid-infrared range is observed at room temperature using a quantum cascade laser emitting at $\lambda=4.5~\mu$m as the excitation source. From the measured two-photon photocurrent signal we extract a two-photon absorption coefficient $\beta^{(2)} = 0.6 \pm 0.2$ cm/MW, in agreement with the theoretical value obtained from the $E_g^{-3}$ scaling law. Considering the wide spectral range covered by extended-wavelength In$_x$Ga$_{1-x}$As alloys, this result holds promise for new applications based on two-photon absorption for this family of materials at wavelengths between 1.8 and 5.6 $\mu$m.