1. Introduction

Self-mixing (SM) interferometry within terahertz (THz) quantum-cascade lasers (QCLs) has garnered significant attention in the past decade for various sensing applications. In SM interferometry, the output beam of a laser is intentionally reinjected back into the laser cavity, which coherently perturbs both the amplitude and phase of the intra-cavity field. The reinjected beam contains information regarding its journey, which it then imparts onto the laser’s measurable properties such as its power output and terminal voltage. The latter property is particularly useful for sensing, since the response is quite fast, and is fundamentally limited by the relevant laser electron and photon lifetimes (typically tens of picoseconds at most), and the RC-time constants associated with the laser packaging. The sensing is coherent and can be very sensitive, which is particularly useful considering the lack of convenient room-temperature fast detectors in the THz. In this way, sensing applications become achievable using THz QC-lasers without the need for a separate detector. Some of the first set of demonstrations were in THz imaging, where SM was used to perform long distance imaging, real-time imaging, and depth-resolved sensing [1–4]. SM has also been used to demonstrate material characterization, gas spectroscopy, displacement sensing, velocimetry, and near-field microscopy [2,5–11]. Beyond sensing, SM has been used for laser self-characterization [12–14]. This is because the laser response to optical feedback is dependent on key laser parameters such as the linewidth enhancement factor (LEF), laser linewidth, and the emission spectrum. The LEF in particular has been shown to play a key role in optical frequency comb generation in THz QCLs [15,16]. Self-mixing is an important tool towards studying the effect that optical feedback can have on the laser itself. Depending on the feedback conditions, and the type of semiconductor laser, optical feedback is known to cause the laser to mode hop, multi-mode, stabilize, destabilize, or even drive the laser into chaos [17–22]. Due to extremely fast gain recovery times, and near-zero LEFs, QCLs have been shown to be particularly stable under optical feedback compared to semiconductor diode lasers [23]. Nonetheless, optical feedback has been shown to have a significant effect on frequency comb operation and stability in THz QC-lasers [21,24,25]. The intentional or accidental presence of feedback must always be considered, since there are no readily available optical isolators in the 1–10 THz range. To date, almost all demonstrations of SM interferometry in THz QCLs has been performed via surface-plasmon waveguide edge-emitting lasers, which were favored over metal-metal waveguide lasers because of their larger waveguide mode, which has eased optical coupling upon retroreflection [26].

In this work, self-mixing in a THz QC vertical-external-cavity surface-emitting laser (QC-VECSEL) is studied for the first time. The QC-VECSEL architecture is based on an amplifying metasurface composed of a periodic sub-wavelength array of microstrip patch antennas loaded with QC gain material [27]. The metasurface, which typically has areas on the order of 1 mm$^2$, acts as an amplifying mirror in an external Fabry-Pérot cavity with a highly reflective output coupler. This architecture exhibits excellent beam quality, scalable output powers, and the capability for broadband single-mode tuning [28,29]. Prior to this investigation, it was not immediately obvious how the QC-VECSEL would differ, if at all, from a ridge-waveguide QCL in the context of optical feedback. For example, the VECSEL requires output coupler reflectances typically greater than 90%, which suggests the device may be less sensitive to feedback than an edge-emitting laser with facet reflectance ~30%. At the same time, the large emitting-aperture and the excellent beam-quality favors more efficient coupling. We have already observed the influence of optical feedback in previous experiments on QC-VECSELs, primarily as an unwanted feature. For instance, there was evidence of optical feedback modulating injection locking bandwidths in both optical injection-locking and RF injection-locking schemes, as well as triggering multi-mode oscillation from a nominally single-mode QC-VECSEL [24,30]. In this work, self-mixing is studied under a variety of experimental conditions, and phenomena unique to the QC-VECSEL architecture are investigated and discussed. Specifically, we observe significant evidence of feedback associated with multiple round-trips within the external cavity. A concise analytical model for self-mixing in the QC-VECSEL is established, which is shown to be in good agreement with experimental data.

2. Theoretical model for multi-bounce self-mixing in THz QC-VECSELs

A theoretical model for self-mixing that includes the effect of the metasurface resonance can be established by adapting the three-mirror model [31]. While a Lang-Kobayashi-based reduced rate equation model can provide a complete time-dependent picture, the three-mirror model can arrive at analytical steady-state solutions that are in good agreement with experiment [14,21,32]. This is especially true in THz quantum-cascade lasers because these devices have been shown to remain stable across a wide range of feedback levels [23].

Consider the three-mirror Fabry-Pérot cavity shown in Fig. 1. The reflection coefficients of each mirror are given by $\Gamma _{\mathrm {MS}}$, $\Gamma _{\mathrm {OC}}$, and $\Gamma _{\mathrm {ext}}$ respectively. If light transmitted through the output coupler undergoes $n$ round-trips (RTs) in the external feedback cavity, we can write an effective reflection coefficient at the output coupler facet that subsumes the effect of the external cavity:

 

Fig. 1. Schematic of the three-mirror model including the QC-VECSEL. $\Gamma _\text {MS}$, $\Gamma _\text {OC}$, and $\Gamma _\text {ext}$ are the reflection coefficients of the metasurface, output coupler, and external target respectively. $\tau _\text {int}$ and $\tau _\text {ext}$ are the internal and external cavity round-trip times. $L_\text {c}$ is the VECSEL cavity length, and $L_\text {ext}$ is the external feedback cavity length.

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Fig. 2. The metasurface reflectance (left) and reflection phase (right) for the device used in this work. The solid lines are from a full-wave FEM simulation. The dashed lines are based on an analytical Lorentz oscillator model. Reflectance is shown for various levels of applied material gain to compensate for losses.

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Plugging everything back into Eq. (5), we arrive at an intuitive form of the phase equation:

The second term is the phase contribution from the feedback-induced frequency-modulation (FM) of the laser. The third term is the phase contribution from the feedback-induced amplitude-modulation (AM) of the laser. Note that this term is a consequence of a finite LEF which results in an amplitude-phase coupling inside the oscillator and is an integral part of self-mixing phenomena. However, the factor $\omega _{\text {r}}/{\omega _0}$ in the AM term is not a factor that appears in conventional ridge lasers, but appears in the VECSEL analysis due to the single-mode detunability away from the metasurface peak reflectance via the Fabry-Pérot VECSEL cavity mode. This detuning factor contributes to an effective linewidth enhancement factor of the laser which we define as $\alpha _{{\rm eff}}: = (\omega _{\rm r}/\omega _0)\alpha$. This dependence is not unique to the VECSEL, and is a result of the frequency dependence of the LEF due to the odd-symmetric phase profile [35,36]. In fact, the material LEF is itself also frequency dependent, and so the full dependence on $\alpha _{\text {eff}}$ with detuning is not known.

Defining $\phi _\text {FB} : = \omega \tau _\text {ext}$ and $\phi _0 : = \omega _0 \tau _\text {ext}$, Eq. (8) can be rearranged into the more practical and familiar form of the generalized Adler phase equation,

For a single RT, the parameter $C_i$ reduces to the familiar Acket parameter, $C$, which subsumes the various feedback variables and can succinctly describe the regimes of operation of a self-mixing system [37,38]. Therefore, the parameter $C_i$ is an extension of the Acket parameter that describes the feedback level associated with the $i$^th RT of the external cavity. While Eq. (11) describes the perturbation of the emission frequency due to feedback, the SM signal can be obtained from the terminal voltage $V_{\text {SM}}$ of the laser at constant current injection due to the modulation of the threshold gain. If we assume the total gain is proportional to the carrier concentration relative to threshold, then

Some illustrative examples of the relation provided in Eq. (11) are shown in Fig. 3. In particular, we demonstrate the relationship between $\phi _\text {FB}$ and $\phi _0$ for the case of $n = 2$ and varying $C_2$ for a given $C_1$. Fig. 3 illustrates the case for $C_1 = 0$, and Fig. 3 shows the case for $C_1 = 1$. If we only consider reflections from the second RT, the self-mixing dynamics resemble the conventional case, but with two main distinctions. One, the periodicity is cut by half, resulting in a period of $\pi$. In an optical system such as that of Fig. 1, this will manifest as a $\lambda /4$ spatial periodicity of the self-mixing fringes as the external target is translated. Two, the phase relation steps into the regime of multistability sooner than the case of a single RT, resulting in a bistable regime for $0.5 < C_2 < 2.3$. The presence of multiple RTs in the self-mixing analysis precludes the simple delineation of feedback regimes based on a single feedback parameter [39–42]. Thus, for the purposes of this work, we classify the weak feedback regime as the set of $C_i$ $n$-tuples that maintain bijectivity of the phase relation in Eq. (11). The case for $n = 2$ is illustrated in Fig. 3. As $\phi _\text {FB}$ becomes multi-valued, the system enters the moderate feedback regime, typically characterized by mode-hopping instabilities, multimoding, and hysteresis [15,17,20].

 

Fig. 3. (a) Feedback phase relation for $C_1 = 0$, and varying $C_2$ from 0 to 4. (b) Feedback phase relation for $C_1 = 1$, and varying $C_2$ from 0 to 4. (c) Mapping of the weak feedback regime in the $C_1$-$C_2$ plane. Outside this regime, the phase relation becomes multi-valued and enters the moderate feedback regime.

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A few representative self-mixing voltage waveforms according to Eq. (13) are calculated in Fig. 4 for the case of $n = 2$. Fig. 4 shows the case for $\alpha = 0$, and Fig. 4 shows the case for $\alpha = 0.5$. When both $C_1$ and $C_2$ are present, the self-mixing waveform features fringes that don’t necessarily peak at the same value. This is indicative of the multi-bounce self-mixing effect and is experimentally demonstrated in Section 4. As the feedback strength gets larger, the waveform develops discontinuities associated with the path-dependence of $\phi _\text {FB}$ with respect to $\phi _0$. Additionally, comparing Fig. 4 and 4 illustrates the effect of the LEF on the SM signal morphology; a non-zero $\alpha$ breaks the symmetry and produces a tilt to the signal.

 

Fig. 4. (a) Calculated SM voltage signals for $C_1 = 0$ and $C_1 = 0.2$ as $C_2$ is varied and $\alpha _\text {eff} = 0$. (b) Same as (a) but for $\alpha _\text {eff} = 0.5$.

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3. Methods

The QC-VECSEL used in this work is based on an elliptical patch metasurface that couples to surface-incident radiation. The electric field profile of a unit cell is illustrated in Fig. 5. The patches are composed of 5-°m-thick QC gain material based on a 4-well phonon-depopulation design grown by IQE plc and was featured in Ref. [43]. The epitaxial growth is 80 repetitions of the layer sequence given, in Å, by 106/20/106/37/88/40/172/51 (GaAs/Al_0.15Ga_0.45As), where the middle 88 Å of the underlined layer is nominally doped at $5\times 10^{16}$ cm^-3. A microscope image of the entire metasurface and a corresponding SEM image of a single patch are shown in Fig. 5. The period of the patch array is 50 °m, and the major and minor axes of the ellipse are 21.4 °m and 14.6 °m respectively. Only patches within a central circular area with a diameter of 0.75 mm are biased, so as to preferentially pump the fundamental Gaussian mode of the VECSEL cavity. The output coupler is based on a metallic mesh evaporated on a quartz substrate to give a reflectance of $|\Gamma _{\text {OC}}|^2=0.95$. The output coupler is fixed and clamped onto the metasurface die copper submount with a 1 mm thick copper spacer, which eliminates any instabilities or thermal drift associated with a kinematic mount. The laser has a single mode emission peak at 2.80 THz and peak output power of 0.5 mW with threshold current of 385 mA and max current of 460 mA (Fig. 5). The laser is driven in continuous-wave mode by a low noise (3 nA/$\sqrt {\text {Hz}}$) current source from Wavelength Electronics (QCL OEM1000+) and is cooled with liquid nitrogen to 77 K inside a cryostat with a TPX window.

 

Fig. 5. (a) Simulated electric field profile of the metasurface unit cell. The period of the metasurface is 50 $\mu$m in both $x$ and $y$ dimensions. The dimensions of the ellipse are given by $a = 21.4$ $\mu$m and $b = 14.6$ $\mu$m. (b) Optical microscope image of the full metasurface, and an SEM image of a single patch. The patches are selectively biased in a circular bias area in the center of the metasurface. (c) Power-current-voltage measurement of the device under test in continuous-wave operation at 77 K. The inset shows the emission spectrum obtained from an FTIR. (d) Schematic of the experimental setup. Crossed wire-grid polarizers are used whenever additional attenuation is desired.

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Light leaving the laser is collimated by a 4” focal length f/2 off-axis parabolic (OAP) mirror and reflects off a flat mirror mounted on a 25 mm translation stage, allowing for stepping the external cavity length. Moving the translation stage along a fixed rail mount allows for cavity extension from 10 to 80 cm while maintaining the alignment between the two mirrors. The laser emission frequency being near the edge of an atmospheric transmission window necessitates the use of a nitrogen purge enclosure in which relative humidity is reduced to $\sim$1%. Dual wire-grid polarizers (Microtech G30-S) in the collimated beam path are used whenever additional attenuation is desired.

As the translation stage reduces the external cavity length, the SM signal can be recorded as the ac component of the QCL terminal voltage. This signal can be simply acquired using a benchtop multimeter, as was done for the results in Section 4.1. We also employed the conventional method of lock-in detection with an optical chopper (modulated at 200 Hz), which was used for data presented in Section 4.2. However, as will be discussed in Section 4.3, these measurements were accompanied by unexpectedly large noise at higher feedback levels. In response we employed an alternate method, detailed in Section 4.4, involving lock-in differential resistance detection made using small-signal modulation of the current source with a 100 kHz sinusoidal signal.

4. Experimental results

4.1 Multi-bounce self-mixing voltage signal

Without any intentional effort to produce the phenomenon, the experimentally observed SM voltage signal consistently demonstrated feedback coupling from light that underwent two RTs in the external cavity. In fact, when the external cavity length was made sufficiently short ($L_\text {ext} < 40\; \text {cm}$), we observed SM signals that were dominated by the second RT ($C_2 \gg C_1$). An example of such a signal is shown in Fig. 6 for a cavity length of $17.8\;\text {cm}$. Fitting the data to Eq. (11) and Eq. (13) with appropriate normalization, we obtain $C_2 = 0.56 \pm 0.09$ and $\alpha _\text {eff} = 0.8 \pm 0.3$. The value of $\alpha _\text {eff}$ is consistent with previously reported SM measurements of the LEF in THz QCLs, ranging from $-0.2$ to $+1$ [8,12].

 

Fig. 6. (a) Normalized SM voltage signal. The green curve is a fit to the collected data based on Eq. (11) and Eq. (13). The extracted parameters are $C_1 \approx 0$, $C_2 = 0.56 \pm 0.09$, and $\alpha _\text {eff} = 0.8 \pm 0.3$. (b) Coupling strength, $\kappa$, versus output coupler reflectance. The green curve corresponds to a surface-plasmon waveguide laser. The orange curve corresponds to the QC-VECSEL. The labeled points show exactly the point of operation.

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Based on the fit parameters, we can quantify both the total field attenuation, $\epsilon$, and the coupling strength using Eq. (12). The attenuation factor includes loss from the cryostat window, diffraction, and the mismatch of the spatial mode of the reinjected field. The internal RT time can be calculated from the known laser cavity length ($L_\text {c}$ = 1.0 mm) and the simulated metasurface group delay. We obtain an attenuation coefficient of $\epsilon \approx 0.072$, which results in a coupling strength of $3.6 \times 10^{-3}$. A recent work by Keeley et al. reported an attenuation coefficient of $\epsilon = 0.0033$ when using a SP waveguide emitting at 3.4 THz [44]. Assuming a facet reflectivity of 33%, the obtained coupling strength is $3.8 \times 10^{-3}$ — almost identical to that obtained with the VECSEL [45]. At first, the large output coupler reflectances required for the VECSEL might suggest the architecture is not well-suited for SM interferometry. However, the high beam-quality does well to compensate by reducing the amount of reinjected power lost to diffraction and modal-mismatch. Fig. 6 illustrates this by comparing the $\kappa$ vs $R_\text {OC}$ curves for the respective attenuation coefficients for the SP waveguide and the VECSEL. In fact, while the maximum achievable $\kappa$ for the SP waveguide is fixed, the VECSEL has some freedom in moving along this curve. VECSELs with OC reflectance as low as 82% have been demonstrated, which would correspond to $\kappa = 1.3 \times 10^{-2}$ [46]. Conversely, the reflectance can be made to be larger to reduce the laser’s sensitivity to optical feedback.

4.2 Asymmetric cavity test

The observation of multiple RTs in the external cavity as shown in Section 4.1 warrants further investigation of the influence of optical alignment. The external target is deliberately misaligned by an angle $\theta$, and a SM signal is collected over a 500 $\mu$m translational scan. Fig. 7 illustrates (to scale) a ray trace up to the second RT for a target misalignment angle of $\theta = 2^\circ$. It is evident that in such an asymmetric cavity, the feedback ray from the first RT is displaced further from the initial source compared to the feedback from the second RT. This will cause the coupling strength from the second RT to be more robust to target misalignment. This effect remains true in general for all odd-numbered versus even-numbered RTs, but higher-order contributions were observed to be much weaker and are not considered in this work [47,48].

 

Fig. 7. (a) Ray tracing of the optical setup for a misaligned external target. The red trace corresponds to the first round-trip beam, and the blue trace corresponds to the second round-trip beam. The dotted portions of the path represent rays traveling away from the laser. The solid portions are rays traveling towards the laser. (b) Measured self-mixing voltage signals. The bottom signal is obtained when the mirror is rotated 2 degrees toward the laser. (c) Energy in the SM signal as a function of relative mirror angle for two different cavity lengths. The vertical dashed lines represent the angles that correspond to reflections incident at the physical edges of the OAP. (d) Spatial Fourier transform of the SM signal when $L_\text {30}$ = 30 cm. The arrows indicate the spatial frequencies that correspond to the first and second round-trip.

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Figure 7 shows two SM signals obtained at an external cavity length of 30 cm. The top plot shows the case for when the target is close to perfectly aligned, and the contribution from the first and second RTs are comparable. The bottom plot was obtained by misaligning the mirror $2^\circ$ off normal, which results in a SM signal that is dominated by the second RT as indicated by the near-$\lambda$/4 periodicity. Although it was left out in the analysis performed in Section 2. for simplicity, in general there can be additional phase offsets between each round-trip due to varied propagation lengths in an imperfectly aligned system. Data was collected across a range of misalignment angles from $-10^\circ$ to $10^\circ$, and the total energy in the SM signal is plotted for two external cavity lengths, $L_\text {ext} = 30$ cm and $L_\text {ext} = 53$ cm (Fig. 7). The vertical dashed lines correspond to the mirror angles that would result in the initial ray walking off the boundaries of the OAP (which has inherent asymmetry). In contrast to SM interferometry with edge-emitting lasers, the SM signal strength from the VECSEL appears to be significantly tolerant to target misalignment. This is due to two reasons. First, the output coupler serves as a large flat mirror that the initial ray can reflect off of to undergo secondary RTs. Second, the high beam-quality reduces the diffraction loss in the external cavity that would otherwise significantly attenuate the power from the second RT. The contribution from the second RT can be seen in Fig. 7, where the spatial Fourier transform is plotted against the relative misalignment angles for the case of $L_\text {ext} = 30$ cm. The spatial frequency labeled $i = 2$ corresponds to the component that is periodic with $\lambda$/4, and remains relatively independent of the angle until it approaches the physical edges of the OAP. One interesting observation is the spectral broadening of the higher order RT components. We speculate that this is caused by, in part, the higher sensitivity of higher order bounces in generating regions of multistability for lower feedback levels $C_i$ as suggested by the model in Fig. 3.

4.3 Observation of self-mixing enhanced voltage instability

For higher signal-to-noise ratios, and faster signal acquisition, it is customary to implement an optical chopper for lock-in detection in a THz SM interferometric system [2,12]. However, in our experiments, we observed moments of enhanced noise at the modulation frequency regimes accessible by the optical chopper (< 1 kHz). To investigate this further, we monitored the terminal voltage of the device in the frequency domain via an oscilloscope. Without the presence of optical feedback, a particular tone at ~920 Hz was always visible, as shown in Fig. 8. This tone is caused by the mechanical resonance of the quartz output coupler, and can vary depending on the thermo-mechanical state of the device. However, when optical feedback is introduced, this tone is amplified, and additional harmonics are generated due to the nonlinearity introduced by self-mixing at large feedback levels (Fig. 8).

 

Fig. 8. (a) AC-coupled averaged voltage spectrum of the device biased at 410 mA without optical feedback. The inset shows a resonance at ~920 Hz. The spectrum is collected with a span of 40 KHz and a resolution bandwidth of 300 mHz. (b) Same conditions as (a), but with the presence of feedback. The harmonic tones are separated by 920 Hz. (c) Strength of the 920 Hz tone versus target mirror position. The orange dashed line shows the no feedback case as reference.

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Additionally, we find that across all bias points and external cavity lengths, the presence of feedback always amplified the voltage instability. Fig. 8 plots the peak of the 920 Hz tone as the target mirror is swept across a half wavelength. Without feedback, the strength of the tone is measured to be -96 dBV with a 300 mHz resolution bandwidth. With feedback, the peak ranges from -85 dBV to -60 dBV, once again with a $\lambda$/4 periodicity. These results can be predicted based on the model established in Section 2. The mechanical vibration of the output coupler can be represented as small perturbations to $L_\text {c}$. The corresponding perturbations to the SM voltage can be written as

The first term in Eq. (15) corresponds to the changes in threshold gain due to perturbations in the emission frequency without feedback. This can come from the frequency dependence of the metasurface resonance, output coupler reflectance, and the material gain lineshape. The frequency instability due to $L_\text {c}$ for various feedback conditions is given by Eq. (16) and can be experimentally observed [49]. The expression suggests that the frequency stability can always be made better or worse depending on the feedback phase. However, in the case of the voltage stability, the perturbations to the threshold gain and the emission frequency are coupled, resulting in a more complicated dynamic.

4.4 Weak and moderate feedback regimes

Due to the phenomenon observed in Section 4.3, it is important that any SM interferometric system using the QC-VECSEL is configured to measure its signals spectrally far from the mechanical noise regime. This is especially true for higher levels of feedback. One way to achieve this is via small-signal modulation of the current source. Because the intrinsic frequency of the laser is a function of injected current, modulating the current modulates the intrinsic phase, $\phi _0 = \omega _0 \tau _\text {ext}$. This, in turn, produces a modified SM voltage signal given by

While $V_\text {SM}$ is a signal produced by feedback induced changes in the threshold gain, $V'_\text {SM}$ is the signal produced by feedback induced changes in the differential resistance associated with photon-assisted transport. We modulate the current source by 0.80 mA peak-to-peak at 100 kHz (limited by the operating range of the lock-in amplifier). This current modulation corresponds to a frequency modulation of ~13.5 MHz/mA [49] of the laser mode. At an external cavity length of 61 cm, this corresponds to a phase modulation of $\Delta \phi _0$ = 0.09$\pi$.

While this alternate measurement scheme was motivated by operating away from mechanical noise, it comes with two additional benefits. One, modulation at higher frequencies allows faster lock-in detection and faster data acquisition rates. Two, signals corresponding to the moderate feedback regime become easily identifiable due to infinite discontinuities in differential resistance at the jump points (see Fig. 3). Fig. 9 shows an SM signal collected in the weak feedback regime, and one collected in the moderate feedback regime. The feedback level was controlled via crossed polarizers, allowing for additional RT field attenuation, $\epsilon _a$, all the way down to the noise floor. Fig. 9 plots the maximum slope of the SM signal as the feedback level is gradually increased. The curvature follows from Eq. (17), but eventually saturates due to the discrete step size in $z$. Nevertheless, since our step size is much smaller than the periodicity ($\Delta z$ = 1 $\mu$m), it allows us to easily identify excursions into the moderate feedback regime. We find that despite the large reflectance of the output coupler, the VECSEL is susceptible to moderate levels of feedback for cavity lengths up to 81 cm. Additionally, the relative feedback contributions from the first and second RT can vary significantly with attenuation and cavity length.

 

Fig. 9. (a) Measured self-mixing signals according to Eq. (17) in the weak feedback regime and the moderate feedback regime. (b) Maximum differential voltage as a function of field attenuation. The dashed curve show a trajectory according to Eq. (17). The blue shaded region corresponds to the moderate feedback regime. (c) Color plots of the SM signal for various levels of additional field attenuation via crossed-polarizers. The horizontal dashed lines indicate the approximate border between weak and moderate feedback. The dashed oval indicates a region where the feedback contribution from the second RT is much smaller compared to the first. The three color plots show data obtained for three different external cavity lengths: $L_\text {ext} = 41$ cm, $L_\text {ext} = 61$ cm, and $L_\text {ext} = 81$ cm.

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The color plots in Fig. 9 illustrate the obtained $V'_\text {SM}$ signal for various levels of added attenuation. The three color plots, from left to right, show the case for $L_\text {ext} = 41$ cm, $L_\text {ext} = 61$ cm, and $L_\text {ext} = 81$ cm respectively. The horizontal dashed line indicate the border between weak and moderate feedback regimes. When the feedback level is high, and the cavity length is sufficiently short, the SM signals appear to be dominated by the second RT term. However, as further attenuation is added, $C_1$ becomes much larger than $C_2$, and the signal begins to recover $\lambda$/2 periodicity because the second RT term is attenuated by $\epsilon _a^2$. Finally, as the external cavity length is further extended, the diffracted loss experienced by the second RT beam becomes large enough that its effects become extinguished. We also note that the device was biased near threshold, but these trends were similar across different bias points of the laser. These color plots exemplify the distinct behavior of a QC-VECSEL SM interferometer under various feedback conditions. Many of the well-developed analytical and analog processing techniques only work when higher-order RT reflections can be ignored [2,12,40,50–53]. Therefore, extra care must be taken in the interpretation and processing of SM signals acquired from these devices.

5. Conclusion

We report the first investigation of self-mixing in THz QC-VECSELs: new phenomenology is observed, and a steady-state theoretical model is presented to describe it. Specifically, multi-bounce self-mixing is found to be prevalent at a large range of feedback conditions — something that has not been reported in conventional ridge waveguide THz QCLs. This can be understood as a consequence of the VECSEL’s excellent beam quality, large millimeter-scale emitting aperture, and the fact that the output coupler is a large flat mirror which allows multiple round trips to survive in the feedback cavity. These effects are well described by a modified Lang-Kobayashi steady-state model which includes multi-bounce feedback, which in turn allowed extraction of an effective value for the linewidth enhancement factor of the gain-loaded metasurface.

These studies suggest that this external cavity architecture comes with particular challenges associated with its sensitivity to feedback, and its effect on laser instabilities. Nonetheless, the QC-VECSEL has the potential to be a powerful SM sensor. The combination of excellent beam pattern with the ability to choose the output coupler reflectance, allows one to deliberately reach high feedback coupling strengths, and obtain the concomitant high sensitivity. Being able to modify the feedback sensitivity with choice of OC is a unique feature of the VECSEL and is a potential avenue for future study. Furthermore, QC-VECSELs have been shown to exhibit broadband single-mode tunability (~19%) by using ultrashort cavities with piezoelectrically tunable lengths [28,29]. Such a laser is potentially equipped to perform broadband SM interferometry for hyperspectral or 3D depth resolved imaging, solid-state or multi-species gas-phase spectroscopy, and measurement of complex refractive indices in the THz band. The device may also serve as a versatile platform for self-characterization and studying feedback effects across the gain bandwidth.

Appendix: Parameters used in this work

Table 1. Parameter symbols and their descriptions

View Table

Funding

National Science Foundation (2041165); National Aeronautics and Space Administration (80NSSC22K1894).

Acknowledgments

The authors are grateful to Christopher Curwen from JPL for assistance with the active material as well as valuable discussions on QC-VECSEL feedback. We also acknowledge Professor Frédéric Grillot from Institut Polytechnique de Paris for his counsel and valuable insights. Finally, special thanks to Eilam Morag from UCLA for helpful discussions in the theoretical modeling.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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