The self-mixing (SM) effect describes the interference of the intracavity electromagnetic field of a laser with its emitted radiation partially reinjected into the laser cavity [1]. Although such optical feedback (OF) can be detrimental to laser operation [2], the SM effect can also be exploited for metrological applications [3] through a technique known as laser-feedback interferometry (LFI) [4]. This homodyne technique, in which the laser acts simultaneously as source, mixer, and shot-noise limited detector, allows the retrieval of information about the external cavity, composed of the target and the external medium, by monitoring the response of the laser to OF. Thanks to the universal character of the SM phenomenon, its functionality has been demonstrated from the visible to the microwave range using laser systems as diverse as gas lasers [3], solid-state lasers [5], and semiconductor lasers [6]. This has led to the development of a large number of sensing applications [7] ranging from displacement measurement [8] to material analysis [9], laser emission spectrometry [10], and coherent imaging [11,12]. Among all kinds of semiconductor lasers, quantum cascade lasers (QCLs) allow further simplification of the SM scheme, thanks to their intrinsic voltage sensitivity to OF. This allows the SM modulation to be measured directly and with high sensitivity [13] via the voltage variation across the active region with no need for an external photodetector [14]. Moreover, QCLs exhibit peculiar ultra-stability to OF [15] due to their small linewidth enhancement factor [16] ($ 0 \lt |\alpha | \lt 1 $) and long photon-to-carrier lifetime ratio. These unique and versatile characteristics offer an opportunity for the development of high-performance LFI schemes operating at mid-infrared and terahertz (THz) frequencies for coherent imaging [17] and displacement sensing [18,19]. In fact, despite the relatively long wavelength, nanometer target displacements as low as $ \lambda /100 $ have been measured by adding a fast-moving etalon in the external cavity [20] and 60–70 nm in-plane spatial resolution with both amplitude and phase contrast through THz scattering-type scanning near-field optical microscopy [21].

In this Letter, we report a significant improvement of nanometer displacement sensing by showing the detection of deeply subwavelength vibrations ($ \lt \lambda /6000 $) of a suspended mechanical resonator by employing a 3.34 THz QCL operating in continuous-wave (CW). The experimental results have been validated by solving the Lang–Kobayashi (LK) model [22] in the steady-state regime after calibration of the absolute membrane position. Furthermore, the measured oscillation amplitudes are shown to be in agreement with independent measurements performed using a laser Doppler vibrometer (LDV) operating in the visible region.

We chose as an external element a $ {{\rm Si}_3}{{\rm N}_4} $ trampoline membrane, which is an optimum candidate for interferometric readout and optomechanical applications [23]. Such membranes have already been used to observe optomechanical features in an infrared laser diode SM configuration [24]. A scanning electron microscope (SEM) image of the fabricated sample is shown in Fig. 1(a). The structure is composed of a 300 nm thick, 300 µm wide suspended central pad held at each corner by 10 µm wide tethers anchored at the vertices of a 1 mm side square window. The details of the sample fabrication can be found elsewhere [24]. The membrane pad dimensions were chosen in order to increase the level of OF for SM measurements by creating a reflective surface of size comparable to the focused THz beam spot size ($ \sim 250\,\,{\rm \unicode{x00B5}m} $). To further increase the reflectivity, a 10 nm thick gold layer was thermally evaporated over the entire membrane surface to produce an almost totally reflective target. The sample was directly glued on the top surface of a piezoelectric ceramic actuator with the fundamental out-of-plane mechanical resonance at $ \sim 100\,\,{\rm kHz} $. The system composed of the piezo-actuator and membrane sample was mounted inside a vacuum chamber allowing control of the environmental pressure. An initial membrane characterization was performed using a commercial high-spatial-resolution LDV (Polytech UHF 120).

 

Fig. 1. (a) SEM image of the $ {{\rm Si}_3}{{\rm N}_4} $ membrane. (b) Membrane mechanical spectrum measured at atmospheric pressure at the membrane center using the LDV. Inset: zoom of the fundamental resonance and corresponding fit (dashed line) using Eq. (2). (c) $ {\omega _{\rm m}}/2\pi $, $ Q $, and $ {A^{{\rm max}}} $ as a function of $ P $ in logarithmic scale. The dashed curve is a quadratic fit for the data down to $ P = 1\,\,{\rm mbar} $, while the dashed line indicates the saturation effect observed below this pressure.

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From the spectrum shown in Fig. 1(b), obtained at atmospheric pressure by applying a flat spectrum voltage excitation to the piezo-actuator, the fundamental mechanical mode (highlighted in the inset) is observed to be well separated from the higher-order modes. Moreover, with the membrane oscillating orthogonally to the chip-plane, this particular mode ensures that the full THz wavefront reflected from the membrane experiences the same external optical path. This makes the fundamental vibrational mode particularly suitable for coherent measurements of membrane vibrations by LFI [24] and, as such, subsequent measurements here focus on this mode. A LDV characterization of the membrane fundamental mode as a function of pressure was performed. The data were analyzed by modelling the membrane as a one-dimensional driven harmonic oscillator. Defining $ A(t) $ and $ a(t) $ as the positions of the membrane trampoline and of the tether clamps with respect to their common rest positions $ A = a = 0 $, respectively, the membrane equation of motion can be written as

By fitting the displacement spectra using Eq. (2), we obtain for each investigated pressure the values of the resonance frequency $ {\omega _{\rm m}}/2\pi $, the quality factor $ Q = {\omega _{\rm m}}/\gamma $, and the oscillation amplitude at the resonance frequency $ {A^{{\rm max}}} $. As the environmental pressure decreases, these three quantities monotonically increase up to a saturation level [see Fig. 1(c)]. The resonance frequency and quality factor change from $ {\omega _{\rm m}}/2\pi \sim 67.9\,\,{\rm kHz} $ and $ Q \sim 40 \pm 2 $, respectively, at atmospheric pressure to $ \sim 73.4\,\,{\rm kHz} $ and $ \sim 200 \pm 10 $ at pressure $ P \sim 7.3 \times {10^{ - 3}}\,\,{\rm mbar} $.

A schematic diagram of the experimental apparatus for the THz LFI measurements is presented in Fig. 2(a). We used a THz QCL consisting of a 14 µm thick GaAs-AlGaAs active region with growth details reported in the literature [25]. The device was fabricated as a single-metal ridge with longitudinal and transversal dimensions of 1.8 mm and 150 µm, respectively. The laser was driven in the CW mode with a current equal to 510 mA and was maintained at a temperature of 25 K using a continuous-flow cryostat. Under these conditions, the QCL provided stable single-mode emission at 3.34 THz and a large SM response [13,25]. The THz beam passing through the polythene window of the cryostat was collimated and focused (using two $ f/2 $ off-axis parabolic reflectors with a diameter of 50.8 mm) onto the trampoline membrane which constituted the mirror of the external cavity. The vacuum chamber containing the membrane was fixed to a motorized stage allowing the external optical cavity length $ L $ between the target and the QCL emission facet to be micrometrically changed. The LFI apparatus was used in two configurations. In the first configuration [conf. 1 in Fig. 2(a)], the membrane motion was not excited, and the membrane was moved towards the laser facet in 2 µm long steps parallel to the beam propagation direction. The QCL SM voltage $ {V_{{\rm SM}}} $ was measured by a lock-in amplifier synchronized to the optical modulation frequency ($ {\nu _{{\rm mod}}} = 212\,\,{\rm Hz} $) imposed to the THz beam by a mechanical chopper placed in front of the vacuum chamber. In the second configuration [conf. 2 in Fig. 2(a)], the membrane equilibrium position instead was kept fixed, but the membrane vibrated at the frequency of the piezo driving excitation which, in turn, was used as the reference of the lock-in amplifier.

 

Fig. 2. (a) Sketch of the two configurations of the SM apparatus. (b) Calculated $ \Delta N $ (blue curve), and measured $ {V_{{\rm SM}}} $ (red points) as a function of $ \Delta L $ using configuration 1.

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Fig. 3. (a) Measured $ {V_{{\rm SM}}} $ as a function of the drive frequency for different applied voltages at $ P = 0.5 $ mbar. Inset: computed (red dashed line) and experimental (red crosses) $ {V_{{\rm SM}}} $ spectrum with 1 $ {V_{{\rm RMS}}} $ applied to the piezo-actuator. (b) Measured (red squares) and calculated (dashed line) $ V_{{\rm SM}}^{{\rm max}} $ as a function of $ {A^{{\rm max}}} $. Inset: LDV-measured $ {A^{{\rm max}}} $.

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To model the SM voltage signal as a function of the target position, we used the steady-state solutions of the LK equations describing the laser under OF [22]. This was done by evaluating the laser emission frequency under OF, $ {\omega _{\rm F}} $, by numerically solving the so-called excess phase equation:

SM measurements of the membrane oscillation amplitude were then performed using configuration 2. $ {V_{{\rm SM}}} $ measured at an environmental pressure of $ P = 0.5\,\,{\rm mbar} $ is reported as a function of the drive frequency and for several RMS drive voltages in Fig. 3(a). Using a Lorentzian spectral line-shape as the fitting function for the data (valid in the experimentally verified high-$ Q $ limit), the vibration spectra were observed to have the same resonance frequency $ {\omega _{\rm m}}/2\pi \sim 73.43 \pm 0.01\,\,{\rm kHz} $ and $ Q \sim 198 \pm 2 $, independent of the bias applied to the piezo-actuator. These quantities agree with those obtained previously with the LDV. The $ {V_{{\rm SM}}} $ signal measured in response to the membrane vibrations can be modelled with Eqs. (3) and (4) for the steady-state solution of the LK model. In fact, with the limit $ 2\pi /{\tau _{{\rm ext}}} \gg {\omega _{\rm m}} $ satisfied, the membrane position at each instant in time can be considered fixed, contributing as a static external optical path for the LK equations. The time-dependent variation of carrier number $ \Delta N(t) $ corresponding to a certain membrane oscillation amplitude can thus be numerically obtained by inserting into Eqs. (3)–(4) the following expression for $ L $: