Membrane-Mediated Conversion of Near-Infrared Amplitude Modulation into the Self-Mixing Signal of a Terahertz Quantum Cascade Laser

Abstract

A platform for converting near-infrared (NIR) laser power modulation into the self-mixing (SM) signal of a quantum cascade laser (QCL) operating at terahertz (THz) frequencies is introduced. This approach is based on laser feedback interferometry (LFI) with a THz QCL using a metal-coated silicon nitride trampoline membrane resonator as both the external QCL laser cavity and the mechanical coupling element of the two-laser hybrid system. We show that the membrane response can be controlled with high precision and stability both in its dynamic (i.e., piezo-electrically actuated) and static state via photothermally induced NIR laser excitation. The responsivity to nanometric external cavity variations and robustness to optical feedback of the QCL LFI apparatus allows a highly sensitive and reliable transfer of the NIR power modulation into the QCL SM voltage, with a bandwidth limited by the thermal response time of the membrane resonator. Interestingly, a dual information conversion is possible thanks to the accurate thermal tuning of the membrane resonance frequency shift and displacement. Overall, the proposed apparatus can be exploited for the precise opto-mechanical control of QCL operation with advanced applications in LFI imaging and spectroscopy and in coherent optical communication.

1. Introduction

Laser feedback interferometry (LFI) [1] is a homodyne technique where the response of a laser to optical feedback (OF), namely, the self-mixing (SM) signal [2], is detected to perform ultrasensitive metrology, such as obtaining the distance and motion of reflective targets acting as external mirrors or the permittivity of the external medium. The universal nature of the SM effect enabled LFI applications throughout laser systems as diverse as solid-state [3], gas [4], and semiconductor [5,6,7] laser sources. Thanks to their intrinsic stability to OF [8] and the highly sensitive detection of the SM signal in their voltage modulation without the need of an external photodetector, quantum cascade lasers (QCLs) emerged as the most efficient and compact tools [8] for LFI at mid-infrared (IR) and terahertz (THz) frequencies. A QCL can act as a source, mixer, and shot-noise-limited detector, allowing a broad range of applications [9], including the following: characterization of internal laser properties [10,11], laser emission spectroscopy [12], displacement sensing [13,14], trace gas detection [15,16], material analysis [17,18], and far-field [19,20] and near-field coherent imaging [21,22,23].

In general, LFI is primarily used to characterize external cavities and monitor their temporal evolution through the SM signal. In this study, we target the reverse approach: exploiting OF to precisely modulate the SM voltage of a THz QCL. By controlling the feedback dynamics through the photothermally induced motion (driven by a pumping laser) of a trampoline resonator (acting as the external mirror for the QCL cavity), we aim to demonstrate a robust platform for signal transfer from the near-infrared (NIR) domain to the THz frequency range. In this configuration, the power modulation of the NIR laser commands the evolution of the SM signal of the THz QCL. The link between these two frequency regimes intends to show the broadest coverage of the optical spectrum for the proposed hybrid laser scheme, spanning from near-telecom wavelengths (in principle extendable to optical frequencies) to the ultimate band limit for semiconductor lasers. Thanks to the peculiar properties of THz light, the exploitation of LFI with THz QCLs enables the possible implementation of advanced LFI-based imaging and spectroscopy (in fields like biomedicine/biology and security screening), where the mechanical resonator could act as a tuning element of the THz QCL operation, providing adaptive control over the sensing or imaging process.

For the resonator, we choose a thin, high-stress stoichiometric, metal-coated silicon nitride (${\mathrm{Si}}_3{\mathrm{N}}_4$) trampoline membrane. The metal coating on the membrane front side serves as both the reflective surface for the THz beam and the absorption layer for NIR laser radiation. This resonator platform has been originally introduced as ideal for optomechanics [24], such as for quantum experiments at room temperature [25,26], and it has been later exploited for sensing experiments [27], where a compound cavity laser diode was used to measure the thermal noise of the membrane acting as the external vibrating mirror. More recent examples show the potential of the trampoline mechanical resonator design for the high-performance sensing of position [28], acceleration [29], and electron spin resonance [30]; and for light detection in ultra-broadband graphene-based [31] NEMS or electrode-integrated ${\mathrm{Si}}_3{\mathrm{N}}_4$ membrane MEMS operating in the NIR [32] and sub-THz band [33,34]. In the framework of LFI, reflective ${\mathrm{Si}}_3{\mathrm{N}}_4$ trampoline membranes were introduced to demonstrate the ability of measuring oscillation amplitudes with deeply subwavelength precision within the SM voltage of a THz QCL [35], while, in another work, the SM signal of an NIR laser was exploited to detect photothermally induced resonant frequency variations in a membrane used as a micromechanical bolometer [36].

In the present experiment, we implement these previous SM configurations by performing THz QCL LFI from the metal-coated facet of a ${\mathrm{Si}}_3{\mathrm{N}}_4$ trampoline membrane that is collinearly excited from the rear side by a NIR diode laser. This second laser controls the membrane movement, which, in turn, determines the OF phase evolution of the THz QCL and, therefore, its SM voltage. The absorption of NIR light by the metal layer generates heat, inducing thermomechanical stress that displaces the membrane equilibrium position and shifts its resonance frequency. The recording of the THz QCL SM signal allows the measurement of the displacement of the resonator at both its static and dynamic (in the presence of piezo-electric excitation) equilibrium under continuous wave (CW) NIR illumination. By sweeping the piezo-excitation frequency and using it as the reference for the lock-in amplified detection of the SM signal, we characterize the membrane resonant motion. This includes measuring the maximum trampoline oscillation amplitude, quality factor and resonance frequency, therefore, the photothermally-induced resonance red-shift with respect to the non-illuminated state. The measured SM signal is found to be in agreement with the numerical solutions of the Lang–Kobayashi equations (governing the laser-population inversion under OF) in both the static and actuated states of the membrane. These results agree with finite element (FEM) simulations of the resonator thermomechanical dynamics under illumination. The reproducibility of the membrane mechanical response (both in amplitude and frequency) after several cycles of NIR-induced thermal deformations guarantees the consistent control of the THz QCL SM signal over time via the custom and tailored modulation of the NIR beam power. Interestingly, we show the possibility of independently encoding external laser modulations within the THz SM signal via both photothermally induced membrane frequencies and displacement shifts. As a demonstration of the potential of the proposed platform, we show the feasibility of successfully transferring a signal encoded in the NIR laser power to the SM voltage of the THz QCL, with a bandwidth limited by the thermal response of the mechanical resonator.

2. Experimental Apparatus

The experimental setup, comprising both optics and electronics for laser powering and signal readout, is depicted in Figure 1a. The optical scheme is composed of two branches, a NIR and a THz QCL beam path, impinging on the same suspended mechanical resonator as the final reflective movable target. As shown in the optical image reported in Figure 1b, the resonator consists of a $300\mathrm{nm}$ thick ${\mathrm{Si}}_3{\mathrm{N}}_4$ trampoline membrane with a $300\mathsf{\mu }\mathrm{m}\times 300\mathsf{\mu }\mathrm{m}$ central plate and $490\mathsf{\mu }\mathrm{m}\times 10\mathsf{\mu }\mathrm{m}$ tethers clamped to a $1\mathrm{mm}\times 1\mathrm{mm}$ Si supporting frame. On the membrane surface, a $3/50\mathrm{nm}\mathrm{Cr}/\mathrm{Au}$ thick disc layer with a radius of $130\mathsf{\mu }\mathrm{m}$ acts as a nearly perfect reflective mirror for the THz beam and as an absorption layer for NIR excitation. All membrane fabrication details can be found in [33]. The resonator is fixed on a pass-through-holed piezo-electric actuator and installed inside a room temperature vacuum cell equipped with two $1\mathrm{mm}$ thick cyclic olefin copolymer (COC) windows to allow transparency and the access of both NIR and THz beams relative to the membrane. The $1064\mathrm{nm}$ NIR light—CW emitted by a laser diode that operates as the “master” laser (ML) source commanding the resonator deformation and consequent displacement—is aligned and focused on the rear side of the membrane by two flat mirrors and a 10×, 0.28 NA, 34 mm focal length objective lens. The THz beam is generated by a single metal $1.8\mathrm{mm}$ long, $150\mathsf{\mu }\mathrm{m}$ wide ridge QCL with a $10\mathsf{\mu }\mathrm{m}$ thick GaAs-AlGaAs active region based on longitudinal–optical phonon-assisted miniband transitions [37]. This device was already calibrated and demonstrated to be suitable for imaging and LFI applications [38,39]. The CW-powered THz QCL is fixed at a 40 mA higher current with respect to the lasing threshold (∼810 mA) and a temperature of $30\mathrm{K}$ (using a continuous-flow helium-gas-cooled cryostat) to achieve stable single-mode emission at $\sim 3.48\mathrm{THz}$ (${\lambda }_0\sim 86\mathsf{\mu }\mathrm{m}$) with an output power of ∼450 μW. After passing through the cryostat COC window, the THz beam is collimated and focused on the center of the membrane metal side using two off-axis gold-coated parabolic mirrors with a $50.8\mathrm{mm}$ diameter and a unitary focal ratio (f-number $=1$), resulting in a less than $100\mathsf{\mu }\mathrm{m}$ laser spot size. The vacuum chamber is mounted on a three-axis motorized stage achieving a sub-micrometer resolution of the membrane positioning with respect to the THz source. While the the x- and y-axis stages facilitate the sample alignment on the orthogonal plane with respect to the laser beam, the z-axis stage allows the membrane to be scanned along the optical path to vary the external laser cavity length (L, defined as the distance between the membrane and the QCL emission facet) of the THz LFI apparatus. The atmosphere inside the vacuum cell is constantly kept at a pressure of $P\sim 8.5\times {10}^{-5}\mathrm{mbar}$ throughout all performed experiments described in the following sections.

3. LFI Without NIR Membrane Excitation

Two distinct LFI measurements of the THz QCL SM signal are performed, and we defined them as stationary SM (SSM) and alternate SM (ASM) detection. In the SSM configuration, the SM signal is measured by directly recording the lock-in amplified QCL voltage ($V_{\mathrm{SM}}$) synchronized to the QCL laser power modulation frequency ($f_{\mathrm{ref}}=211\mathrm{Hz}$) imposed by a mechanical chopper placed in front of the QCL emission facet (as shown in Figure 1a). An example of the measured $V_{\mathrm{SM}}$ obtained by sweeping the membrane rest position (with a $2\mathsf{\mu }\mathrm{m}$ step size) toward the QCL emission facet is shown in Figure 2a. $V_{\mathrm{SM}}$ is plotted as a function of the external cavity variation, $\Delta L=L_0-L$, showing the characteristic ${\lambda }_0/2\sim 43\mathsf{\mu }\mathrm{m}$ fringe periodicity of the SM response. The SM signal can be computed by solving the Lang–Kobayashi equations, which describe the laser operation under OF [40]. The OF-affected laser frequency (${\omega }_{\mathrm{F}}$) can be numerically estimated using the following equation:

$${\omega }_0-{\omega }_{\mathrm{F}}={\displaystyle \frac{\kappa }{{\tau }_{\mathrm{c}}}}\sqrt{1+{\alpha }^2}sin({\omega }_{\mathrm{F}}{\tau }_{\mathrm{ext}}+arctan\left(\alpha \right)),$$

(1)

where ${\omega }_0=3.48\mathrm{THz}$ is the emission frequency without OF; ${\tau }_{\mathrm{c}}$ and ${\tau }_{\mathrm{ext}}=2L/c$ are the laser cavity and external cavity round-trip times, respectively; $\alpha$ is the Henry’s linewidth enhancement factor; and $\kappa$ is the OF coupling rate. The variation in the laser inversion population due to OF is given by:

$$\Delta N=-{\displaystyle \frac{2\kappa }{G{\tau }_{\mathrm{c}}}}cos\left({\omega }_{\mathrm{F}}{\tau }_{\mathrm{ext}}\right),$$

(2)

where G is the modal gain factor. Starting from the initial external cavity optical length, $L_0=34\mathrm{cm}$, $\Delta N$ is calculated from Equation (2) as a function of $\Delta L$ after computing the solution for ${\omega }_{\mathrm{m}}$ from Equation (1) at each L. The numerical results for $\Delta N$ are reported via the solid curve in Figure 2a. A good agreement with the measured $V_{\mathrm{SM}}$ is found using the following best-fitting parameters: ${\tau }_{\mathrm{c}}=43.8\mathrm{ps}$, $\kappa =1.69\times {10}^{-2}$. G and $\alpha$ are fixed at $1.42\times {10}^4{\mathrm{s}}^{-1}$ and 0.5, respectively, according to the THz QCL LFI literature [9,11,41]. In agreement with the experimental quasi-sinusoidal shape of $V_{\mathrm{SM}}$, the fit confirms that the laser is in the weak OF coupling regime with feedback coefficient $C=\kappa {\tau }_{\mathrm{ext}}/{\tau }_{\mathrm{c}}\sqrt{1+{\alpha }^2}\sim 0.98<1$, where only a single stable solution for ${\omega }_{\mathrm{F}}$ is found.

In the ASM measurements, the membrane position is kept fixed, and a voltage modulation is applied to the piezo-electric actuator. This modulation serves two purposes: it excites the membrane motion and it acts as the reference signal for the lock-in amplified detection of the QCL $V_{\mathrm{SM}}$. Unlike the SSM configuration, no chopper-induced power modulation is required in this case. Figure 2b shows the $V_{\mathrm{SM}}$ amplitude obtained from the lock-in amplifier by sweeping the piezo-electric actuator driving frequency (${\omega }_{\mathrm{PZT}}$) in the $[90.05;90.45]\mathrm{kHz}$ range. The reported curves present a rising signal (from blue to red colors) by increasing the piezo-driving voltage amplitudes ($V_{\mathrm{PZT}}$) from 10 to $280\mathrm{mV}$. The $|V_{\mathrm{SM}}|$ spectra can be analyzed using the model of a damped mechanical oscillator, where the mechanical susceptibility is defined as follows:

$$\chi \left(\omega \right)={\displaystyle \frac{{\omega }_{\mathrm{m}}^2}{({\omega }_{\mathrm{m}}^2-{\omega }^2)-i\omega {\gamma }_{\mathrm{m}}}},$$

(3)

with ${\omega }_{\mathrm{m}}$ and ${\gamma }_{\mathrm{m}}/2\pi$ representing the membrane resonance frequency and damping coefficient, respectively. The oscillation amplitude of the membrane trampoline in Fourier space can be thus described as $A\left(\omega \right)=\chi \left(\omega \right)a_0$, where $a_0$ is the maximum oscillation amplitude. By using Equation (3), ${\omega }_{\mathrm{m}}$ is found to be peaked at $\sim 90.25\mathrm{kHz}$ with a quality factor of $Q\sim 1400$. The observed resonance corresponds to the excitation of the membrane fundamental mode where the membrane trampoline oscillates as a whole orthogonally to the chip plane thanks to the tethers elongation, as confirmed via FEM simulations (reported in Appendix A) and consistent with a previous work with similar resonator designs [35]. The latter experiment demonstrated the independence of ${\omega }_{\mathrm{m}}$ from the THz impinging power by detecting no resonance variation frequency between independent THz LFI and laser Doppler vibrometry measurements. Therefore, a photothermal shift of the resonance frequency due to THz radiation absorption can be neglected, and the measured mechanical response can be considered that of the bare resonator. A linear mechanical response is observed up to $V_{\mathrm{PZT}}=140\mathrm{mV}$ while Duffing-like features start to arise beyond this voltage amplitude. $V_{\mathrm{PZT}}=100\mathrm{mV}$ is therefore maintained throughout all following experiments to guarantee coherence between measurements and avoid any frequency shifts from a piezo-voltage-induced mechanical non-linearity. In order to maximize the signal shown in Figure 2b, the membrane positions giving the maximum $|V_{\mathrm{SM}}|$ values are identified. This is obtained by sweeping $\Delta L$ at fixed $V_{\mathrm{PZ}}$ and ${\omega }_{\mathrm{PZT}}$ values, and monitoring $V_{\mathrm{SM}}$. The recorded signal is reported in Figure 2c. These data are in agreement with the derivative of the numerical solution for $\Delta N$ ($\Delta N^{\prime }=\mathrm{d}N\left(L\right)/\mathrm{d}L$), whose peak and dip positions indicate $\Delta L$ values where the membrane vibration amplitudes (A) produce the maximum $|V_{\mathrm{SM}}|$ response in the ASM configuration, i.e., the highest LFI sensitivity to vibrations. Specifically, the maximized ASM signal reported in Figure 2b is achieved by fixing the membrane at $\Delta L\sim 42.5\mathsf{\mu }\mathrm{m}$, as highlighted by the dashed vertical line of Figure 2c. Interestingly, by obtaining the derivative of the static $V_{\mathrm{SM}}$ in Figure 2a, a maximum slope coefficient of $\sim 200\mathsf{\mu }\mathrm{V}/\mathsf{\mu }\mathrm{m}$ is obtained, which reveals that oscillation amplitudes as low as a few nanometers can in principle be detected. Using an independent laser Doppler calibration to match the oscillation amplitude to the piezo-electric voltage ($A/V_{\mathrm{PZT}}\sim 1\mathrm{nm}/\mathrm{mV}$), we can estimate that our THz QCL LFI is capable of detecting a minimum of $10\mathrm{nm}$ oscillation amplitude.

4. LFI with NIR Membrane Excitation

In this section, we report the THz QCL LFI characterization of the membrane mechanical response under the NIR illumination provided by the ML diode. The wavelength of NIR radiation falls inside the band gap of the THz QCL active region avoiding possible alterations in the QCL carrier density due to interband transitions induced by collinear NIR radiation, therefore avoiding the need for a NIR optical filter in the THz beam path.

First, with the membrane fixed at $L=L_0+42.5\mathsf{\mu }\mathrm{m}$ and actuated by applying $V_{\mathrm{PZT}}=100\mathrm{mV}$, THz QCL $|V_{\mathrm{SM}}|$ spectra are acquired in the ASM configuration for different NIR ML powers ($P_{\mathrm{IR}}$, in the $[0;8.25]\mathrm{mW}$ range) focused on the rear side of the trampoline membrane. The corresponding results are shown in Figure 3a. Two distinct effects associated with the NIR power illumination can be observed. As $P_{\mathrm{IR}}$ is increased, a resonance red-shift and a reduction in $|V_{\mathrm{SM}}|$ peak amplitude can be clearly observed. Both effects originate from the NIR-induced photothermal heating of the trampoline membrane. The metal bilayer on the trampoline absorbs a fraction of the NIR ML light, leading to a relaxation in tension on the membrane compared to room-temperature conditions, thereby resulting in a red-shift relative to the resonance frequency. While (at 1 $\mathsf{\mu }\mathrm{m}$ wavelength) the $50\mathrm{nm}$ gold layer mostly contributes to the membrane reflection (with an estimated absorption of $<1\%$ [42]), the $3\mathrm{nm}$ Cr layer dominates the membrane optical absorption with an amplitude that can increase by several percentages relative to the incoming NIR power [43] due to multiple reflection interference in the thin ${\mathrm{Si}}_4{\mathrm{N}}_3$/Cr/Au-layer stack. From FEM thermo–mechanical simulations, where thermal material properties are fixed as reported in

Table A1. Global parameters for materials and external environment properties exploited in each module of the FEM simulation model.

Module	Description	Symbol	Values
Geometry	Au disc radius	R	100 μm
	Au thickness	$h_{\mathrm{Au}}$	50 nm
	${\mathrm{Si}}_3{\mathrm{N}}_4$ thickness	$h_{\mathrm{SiN}}$	300 nm
	Au density	${\rho }_{\mathrm{Au}}$	19,300 kg/m3
	${\mathrm{Si}}_3{\mathrm{N}}_4$ density	${\rho }_{\mathrm{SiN}}$	3100 kg/m3
Solid Mechanics	Au Young’s modulus	$E_{\mathrm{Au}}$	90 GPa
	${\mathrm{Si}}_3{\mathrm{N}}_4$ Young’s modulus	$E_{\mathrm{SiN}}$	260 GPa
	Au Poisson ratio	${\nu }_{\mathrm{Au}}$	0.42
	${\mathrm{Si}}_3{\mathrm{N}}_4$ Poisson ratio	${\nu }_{\mathrm{SiN}}$	0.23
	Au initial stress	${\sigma }_{\mathrm{Au}}$	$0.1$ GPa
	${\mathrm{Si}}_3{\mathrm{N}}_4$ initial stress	${\sigma }_{\mathrm{SiN}}$	1039 GPa
Heat Transfer	Au thermal conductivity	$k_{\mathrm{Au}}$	300 W/(m*K)
	${\mathrm{Si}}_3{\mathrm{N}}_4$ thermal conductivity	$k_{\mathrm{SiN}}$	30 W/(m*K)
	Au specific heat	$c_{\mathrm{Au}}$	125 J/(kg*K)
	${\mathrm{Si}}_3{\mathrm{N}}_4$ specific heat	$c_{\mathrm{SiN}}$	700 J/(kg*K)
	Surface thermal emissivity	${ϵ}_{\mathrm{th}}$	0.9
	IR excitation power	$P_{\mathrm{IR}}$	1–10 mW
	Reference temperature	$T_{\mathrm{ref}}$	293.15 K
	IR metal absorption	${\beta }_{\mathrm{IR}}$	5%
Events	Event start time	$t_0$	50 ms
	IR pulse ON/OFF state duration (short)	$t_{\mathrm{s}}$	250 ms
	IR pulse ON/OFF state duration (long)	$t_{\mathrm{l}}$	660 ms
	IR metal absorption	${\beta }_{\mathrm{IR}}$	5%
Multiphysics	Au thermal expansion	${\delta }_{\mathrm{Au}}$	$14.2\times {10}^{-6}$ 1/K
	${\mathrm{Si}}_3{\mathrm{N}}_4$ thermal expansion	${\delta }_{\mathrm{SiN}}$	$3\times {10}^{-6}$ 1/K

of Appendix A, we phenomenologically observe that an effective absorption of the metal bilayer of $5\%$ correctly reproduces the membrane thermo–mechanical response, as described in the following sections. The frequency shift as a function of $P_{\mathrm{IR}}$ is reported in Figure 3b, showing a linear trend with $\Delta {\omega }_{\mathrm{m}}/\Delta P_{\mathrm{IR}}\sim 0.46\mathrm{kHz}/\mathrm{mW}$. The simulated photothermal frequency shift (obtained as described in Appendix A) is also reported in the same graph. A good agreement is found up to $P_{\mathrm{IR}}=6\mathrm{mW}$, while a deviation is observable at a higher power level that is ascribed to a geometric non-linearity arising in the simulations, which is not yet observable in the experiment throughout the investigated power range of irradiation.

The second effect of NIR CW illumination is a decrease in the $|V_{\mathrm{SM}}|$ peak amplitude when an increase in NIR ML power occurs, which is highlighted by the inset of Figure 4. The $|V_{\mathrm{SM}}|$ peak values are obtained via a Lorentzian fit (using Equation (3)) of the spectra of Figure 3a. Starting from $P_{\mathrm{IR}}>2\mathrm{mW}$, the data show a linear dispersion with a slope coefficient of $-2.24\mathsf{\mu }\mathrm{V}/\mathrm{mW}$. As the CW NIR excitation power is not expected to affect the membrane oscillation amplitude (as confirmed by [36]), this effect can be explained by a shift in the membrane dynamical equilibrium position along the cavity axis. FEM simulations show that this membrane deformation points toward the metal-coated side, which in turn reduces the external optical path, in both the static and actuated states of the resonator, as shown in Figure A2 of Appendix A. The direction of the displacement in the membrane equilibrium position is determined by the z-axis geometric asymmetry induced by the metallization. Accordingly to the fixed absolute membrane position $L=L_0+42.5\mathsf{\mu }\mathrm{m}$, this membrane displacement direction implies a consequent quasi-linear reduction in $|V_{\mathrm{SM}}|$ (as can be observed in Figure 2c), which is indeed confirmed by the $|V_{\mathrm{SM}}|$ spectra as a function of NIR power. In order to experimentally quantify the relative displacement of the membrane equilibrium position (${\Delta }^{\prime }L=L-\Delta L$) as a function of $P_{\mathrm{IR}}$ in the linear response regime, we convert the measured Lorentzian amplitudes, recorded in volts, into cavity displacements using the calculated average derivative of the $V_{\mathrm{SM}}$ of Figure 2c between its minimum and maximum values (only considering the linear fitting of the positive slopes). This leads to an average $|V_{\mathrm{SM}}|$ slope coefficient of $-1.36\pm 0.05\mathsf{\mu }\mathrm{V}/\mathsf{\mu }\mathrm{m}$. The volts-to-power ratio and volts-to-micrometer coefficients establishes the link between the photothermally induced membrane relative displacement ${\Delta }^{\prime }L$ and $P_{\mathrm{IR}}$. Figure 4 reports the calculated ${\Delta }^{\prime }L$ as a function of $P_{\mathrm{IR}}$ power with a quantified slope coefficient of $\sim 1.65\mathsf{\mu }\mathrm{m}/\mathrm{mW}$. This change in the membrane dynamical equilibrium position is also found in agreement with the increment of the external cavity length needed to recover the same $|V_{\mathrm{SM}}|$ spectrum amplitude at the original reference membrane equilibrium position without NIR excitation.

In the subsequent study, we investigate the reproducibility of the performed measurements by showing the repeatability of the observed effects in the mechanical response of the membrane under NIR illumination, such as its corresponding resonance frequency and position shift. Figure 5 displays the spectra recorded for five different powers in order to compare the response after repeated photothermal cycles. Each plot reports the spectra after two representative ON/OFF excitation cycles. For the investigated range of powers, the spectra in the excited states are observed to always recover the same resonance frequency and signal/oscillation amplitude. A little response variation in the not-excited state is observed for $P>2\mathrm{mW}$, but with almost the same small blue-shift ($\sim 20\mathrm{Hz}$) and amplitude decrease ($\sim 1\mathsf{\mu }\mathrm{V}$), which we ascribe to a systematic hysteresis of the membrane equilibrium position after thermal relaxation. Overall, the observed stability of the membrane response to NIR excitation demonstrates the consistency of the acquired $|V_{\mathrm{SM}}|$ signals over time, therefore guaranteeing a reliable transfer of the NIR-induced membrane response in terms of frequency and position shifts in the THz QCL SM signal.

The possibility of converting a specific ML NIR power modulation into the THz QCL SM signal is then investigated. In particular, a SOS Morse signal is encoded in the ML NIR-emitted power by directly driving its current, and it is transferred into THz QCL voltage through the membrane motion, solely driven by the NIR photothermal excitation (i.e., with no piezo-electric actuation applied). The ML driving signal, along with the detected QCL $V_{\mathrm{SM}}$, using as reference the chopper modulation frequency, is reported in Figure 6a. The duration of the short and long ON/OFF states of the SOS signal is 125 and $300\mathrm{ms}$, respectively, while the emitted power in the ON state is fixed at $8\mathrm{mW}$. A close alignment is observed between the detected and reference signals, with a calculated $V_{\mathrm{SM}}$ rise/fall time of approximately $300\mathrm{ms}$. FEM simulations suggest that the response time of the signal transfer in the setup is intrinsically limited by the thermal response time of the membrane resonator. The simulated thermal response times are approximately ∼$150\mathrm{ms}$ and $250\mathrm{ms}$ depending on whether geometrical non-linearity is excluded or included from the model, respectively. Moreover, the performed simulations (reported in Figure A3 of the Appendix A) show that the thermal response time of the membrane does not depend on the NIR power (or membrane heating rate) for the investigated range of power and corresponding range of temperature reached by the membrane. The fastest achievable SOS signal transfer (with a total duration of ∼$5.8\mathrm{s}$), preserving the same ON/OFF time ratio of the experimental one, is reported in Figure 6b in terms of both simulated temperature and deformation amplitudes reached by the central point of the trampoline membrane metal disc. A temperature increase per unit power of ∼$16\mathrm{K}/\mathrm{mW}$, corresponding to a maximum oscillation amplitude of ∼$14\mathrm{nm}/\mathrm{mW}$, is computed and reported in more detail in Appendix A. The significant discrepancy in the photothermally induced displacement of the membrane equilibrium position between the static and dynamic/actuated states can be ascribed to the different initial strain conditions of the two states. Notably, analogous signal transfer can be also performed in the ASM configuration by applying a fixed piezo-electric actuation amplitude at a frequency matching that of its photothermally excited state, which is already known for the specific impinging NIR power; by demodulating $V_{\mathrm{SM}}$ at the piezo-driving frequency, the signal can indeed be detected only during the ON states of the NIR ML. Alternatively, both membrane resonance red-shifts and the relative displacement can be detected by the real-time tracking of both phase and amplitude variations in the ASM-$V_{\mathrm{SM}}$, analogously with what was carried out in reference [36] for the recording of the solely photothermally induced frequency variation.

5. Conclusions

A platform to achieve the conversion of a NIR (∼$1\mathsf{\mu }\mathrm{m}$) laser power modulation into the self-mixing signal of a QCL operating at THz frequencies (∼$86\mathsf{\mu }\mathrm{m}$) is introduced. This is realized by exploiting the NIR laser-driven motion of a mechanical resonator (a metal-coated ${\mathrm{Si}}_3{\mathrm{N}}_4$ trampoline membrane) used as the external cavity mirror of a THz QCL LFI scheme.

Fine control over the resonator mechanical response via NIR photothermal heating is demonstrated. In particular, the resonance red-shift and the relative membrane displacement can be tuned with CW NIR power with a high sensitivity of ∼$460\mathrm{kHz}/\mathrm{W}$ and ∼$1.6\mathsf{\mu }\mathrm{m}/\mathrm{mW}$ (in the linear range of $V_{\mathrm{SM}}$), respectively. A controlled deformation is also shown via solely photothermal excitation with a responsivity of $14\mathrm{nm}/\mathrm{mW}$, in agreement with FEM simulations. The shown stability of the membrane mechanical response to the cycles of CW NIR irradiation guarantees the reliable control of the membrane motion over time. This allows the exploitation of the NIR laser source as a master laser to infer tailored modulations in the SM signal of the THz QCL acting as a slave laser source. As a demonstration of the potential of the developed master–slave configuration, we show the successful transfer of a specific signal encoded in the power amplitude of the NIR ML to the THz QCL SM voltage. NIR ML excitation power as low as $100\mathrm{s}\mathsf{\mu }\mathrm{W}$ (corresponding to a static membrane displacement of a few nanometers) can in principle be used to perform the proposed signal conversion thanks to the high membrane NIR photothermal responsivity and the high sensitivity of the THz QCL LFI apparatus. The bandwidth of NIR power modulations relative to SM signal conversion is shown to be limited by the photothermal response time of the membrane, calculated to be within the 150–300 ms range. The response time of the signal conversion can be thus decreased by lowering the heat capacity and increasing the thermal conductance of the mechanical resonator. This can be engineered by reducing the ${\mathrm{Si}}_3{\mathrm{N}}_4$ mass portion of the resonator (i.e., by decreasing the overall ${\mathrm{Si}}_3{\mathrm{N}}_4$ thickness and the size of the central trampoline pad, keeping the rest of the geometry fixed), which provides the highest contribution to heat capacities, and by making metal paths on the tethers in order to enhance the overall thermal conductance. Interestingly, as the NIR power controls both membrane frequency and position shifts in the excited state of the resonator, the ML modulation can provide dual information-encoding processes in the phase and amplitude of the detected SM voltage.

In summary, the presented mechanical LFI configuration represents a suitable platform for information transfer between NIR and THz laser frequencies, coupled via a finely controllable thermo–mechanical element. In general, by providing detailed insights on the QCL LFI detection of the photothermally driven dynamics of a ${\mathrm{Si}}_3{\mathrm{N}}_4$-suspended membrane prototype, the present study can be beneficial for opto-mechanical experiments in which high precision control over the resonator frequency and position is crucial, such as for coherent optical communication within heterogeneous/hybrid laser systems. Moreover, as the resonator dynamics can be directly inferred from the voltage and power amplitude of the QCL, the proposed platform represents a versatile proof-of-concept scheme of a master–slave configuration in realizing the external laser-driven mechanically-mediated control of QCL operations, particularly suitable for advanced QCL LFI-based imaging and spectroscopic applications.