Laser feedback interferometry in multi-mode terahertz quantum cascade lasers

: The typical modal characteristics arising during laser feedback interferometry (LFI) in multi-mode terahertz (THz) quantum cascade lasers (QCLs) are investigated in this work. To this end, a set of multi-mode reduced rate equations with gain saturation for a general Fabry-Pérot multi-mode THz QCL under optical feedback is developed. Depending on gain bandwidth of the laser and optical feedback level, three diﬀerent operating regimes are identiﬁed, namely a single-mode regime, a multi-mode regime, and a tuneable-mode regime. When the laser operates in the single-mode and multi-mode regimes, the self-mixing signal amplitude (peak to peak value of the self-mixing fringes) is proportional to the feedback coupling rate at each mode frequency. However, this rule no longer holds when the laser enters into the tuneable-mode regime, in which the feedback level becomes suﬃciently strong (the boundary value of the feedback level depends on the gain bandwidth). The mapping of the identiﬁed feedback regimes of the multi-mode THz QCL in the space of the gain bandwidth and feedback level is investigated. In addition, the dependence of the aforementioned mapping of these three regimes on the linewidth enhancement factor of the laser is also explored, which provides a systematic picture of the potential of LFI in multi-mode THz QCLs for spectroscopic sensing applications.

Laser feedback interferometry (LFI) utilizing THz QCLs has been investigated comprehensively over the past decade and has established itself as one of the most promising technologies for sensing and imaging applications in this part of the electromagnetic spectrum [14][15][16][17][18][19][20][21][22]. In a typical LFI system, the laser emission from the THz QCL reflects back into the laser cavity from an external target, mixes with the intra-cavity electric field, and generates a measurable self-mixing (SM) signal in both the emission power and terminal voltage of the laser [20]. As a coherent sensing technique, LFI technology has the capacity to provide both amplitude and phase information about the external target or the external cavity without the need for external detectors, which is very attractive at THz frequencies for which existing detectors are either not sufficiently sensitive or are not fast enough, such as pyroelectric detectors or Golay cells, respectively [22]. In addition, the SM signal in THz QCLs is typically two orders of magnitude larger than that in laser diodes (LDs) [23] with high detection sensitivity [24]. Consequently, THz QCL-based LFI technology has been demonstrated for a wide variety of applications, including high-resolution THz imaging [23,25], three-dimensional image reconstruction [26,27], biomedical imaging [28,29], and materials analysis [30][31][32].
Most of the reported THz QCL-based LFI systems have employed a single-mode THz QCL. However, a wide range of biomedical tissues and chemicals have a characteristic spectrum which often contains multiple "spectral fingerprints" in the THz region [33,34]. Frequency tuneability in coupled-cavity THz QCLs has been demonstrated theoretically and experimentally [35,36], and applications of multi-spectral terahertz sensing have also been proposed [37], where a set of SM signals at multiple THz frequencies could be obtained through tuning the driving current applied on the passive cavity of the laser. More conveniently than LFI in a coupled-cavity THz QCL, LFI in a single-cavity multi-mode THz QCL has been demonstrated to measure the emission spectrum of the device itself [38], which offers a simple alternative to Fourier transform infrared (FTIR) spectroscopy approaches typically employed for spectral characterization of midand far-infrared lasers. With an external cavity extension of 200 mm, a frequency resolution of 750 MHz has been achieved in that work, which in principle could be further improved to less than 15 MHz with further cavity extension up to 10 m. By using the same principle, gas spectroscopy has been demonstrated in a multi-mode THz QCL [39]. In that work, spectral features of methanol at two longitudinal modes at 3.362 THz and 3.428 THz were measured simultaneously. Although multi-spectral sensing with LFI in a multi-mode THz QCL has been experimentally demonstrated, the optical feedback effects in a multi-mode THz QCL are not yet fully understood.
All current theoretical studies of THz laser-feedback interferometers are based on the assumption of single-mode QCLs. A model for a pulsed THz QCL under optical feedback was proposed and the interplay between electro-optical, thermal, and feedback phenomena was investigated to understand the laser behavior [40]. In [41], it was concluded that the high value of the photon to carrier lifetime ratio and the negligible linewidth enhancement factor determines the ultrastable of QCLs against optical feedback. In addition, the multi-mode instability threshold of the stationary emission of a QCL with optical feedback was studied [42] where it should be noted the multi-mode regimes in that work refers to external cavity modes rather than laser cavity longitudinal modes. The laser behavior in a multi-longitudinal-mode (indicated as multi-mode in the following sections) THz QCL under optical feedback are very different from that of the single-mode case, due to mode competition of the optical gain in the active region of the laser created by stimulated emission through transitions of the injected carriers. Indeed, it is much more complicated than single-mode case when optical feedback occurs between reflected multi-mode and intra-cavity modes in a multi-mode THz QCL, which results in mode cooperation or mode suppression, depending on optical feedback conditions.
It should be pointed out that mode instabilities, including spatial hole burning (SHB) [43] and Risken-Nummedal-Graham-Haken (RNGH) [44] have been demonstrated dominant in multi-longitudinal-mode THz QCLs due to fast gain recovery. They could lead to a proliferation of modes with random phases and destroy coherence. The influence of SHB in mode-locked QCLs were emphasized since it can reduce pulse duration and deteriorate pulse quality [45,46]. However, SHB was not clearly observed experimentally in a multi-mode THz QCL under feedback when it was used for multi-spectral applications as mentioned above [38,39], to a large extent due to the laser is driven very close to its threshold current to maximize self-mixing signal. For the multi-spectral sensing applications, the steady-state Lang-Kobayashi model [47] has been used to analyze the self-mixing signal properties in multi-mode THz QCLs with optical feedback [19,38] where it was assumed that different laser modes operated independently in the Fabry-Pérot (FP) resonator of the THz QCL.
In this work, we explore the characteristics of a FP multi-mode THz QCL under optical feedback and investigate the boundary feedback conditions within which the multi-mode THz QCL could be used for multi-spectral sensing applications. Multi-mode reduced rate equations (RREs) model have been successfully used for the prediction of mode switching dynamics in coupled-cavity THz QCLs with and without optical feedback [35,37,48]. Here we improved the multi-mode RREs model by introducing gain saturation at each of the longitudinal modes. Depending on the optical feedback level and gain spectral bandwidth, three typical feedback regimes are identified, which are the single-mode regime, the multi-mode regime, and the tuneable-mode regime. In single-mode and multi-mode regimes, the SM signal amplitude at each mode is proportional to the feedback coupling coefficient at that particular frequency. Since the feedback coupling coefficients are determined by the coupling factor, reflection coefficients, and phase shift induced by the target, the LFI technology could be used for measurements of the absorption coefficient and complex refractive index of the media in the external cavity or of the target itself. More importantly, the dependence of the absorption coefficient and complex refractive index within a range of THz frequencies can be extracted directly from the multi-mode SM amplitude by extending the external cavity length when the laser operates in the multi-mode regime. However, unlike the single-mode and multi-mode cases, the amplitude of the SM signal is no longer proportional to the feedback coupling coefficient when the feedback is sufficiently strong that the laser enters into tuneable-mode regime. This suggests that a multi-mode laser operating in the tuneable-mode regime cannot be used for spectroscopic sensing directly via the external cavity extension method. However, it could be used as a tuneable laser for spectroscopic sensing when combined with current sweeping for SM signal generation [30].
In addition to the above, a second-order amplitude modulation on the SM signal at the optical path length of the laser cavity is investigated for the laser operating in all these three regimes. This phenomenon could be used to extract the actual lasing cavity length or the effective refractive index of the laser if one of these two is known. The mapping of the feedback regimes of the multi-mode THz QCL in the space of the gain spectral bandwidth and feedback level is investigated as well. The influence of the linewidth enhancement factor on the feedback regime mapping is also explored, which indicates that a multi-mode THz QCL with a larger linewidth enhancement factor has larger multi-mode regime that could be used for spectroscopic sensing.
The remainder of this article is organized as follows. Section 2 describes the multi-mode RRE model in the presence of optical feedback for a multi-mode THz QCL and the analytical expression of the SM signal derived from single-mode RREs. This is followed by an exploration of SM signal characteristics in different feedback regimes in multi-mode THz QCLs in Sec. 3.1 and Sec. 3.2. The feedback regime mappings in the space of feedback level and gain bandwidth, and its dependence on the linewidth enhancement factor are presented in Sec. 3.3. Finally, Sec. 4 presents our conclusions.

Theoretical model of LFI in multi-mode THz QCLs
The exemplar device selected for simulation is based on a GaAs/AlGaAs bound-to-continuum active region structure as described in Ref. [49]. The wafer was grown on a semi-insulating GaAs substrate by molecular beam epitaxy and processed into a surface plasmon ridge waveguide using optical lithography and wet chemical etching at University of Leeds, UK . A standard FP type THz QCL without external feedback normally lases on multiple longitudinal modes with the mode spacing determined by c/(2n in L in ), where n in and L in are the refractive index and laser cavity length of the laser, respectively. We consider a laser feedback interferometer in such a single-cavity multi-mode THz QCL as shown schematically in Fig. 1. We considered seven central longitudinal modes from Mode 1 (M1) to Mode 7 (M7) which have the material gain larger than the total loss in the laser cavity at each of the mode frequencies to demonstrate emission mode structure changes due to optical feedback condition varying. The multi-mode THz beam emitted from the laser transmits through the external cavity and reflected from the external target with the reflection coefficient R(ν) and phase change θ R (ν) respectively, which are generally frequency (ν) dependent. The electric field sustained within the laser cavity mixes with the reinjected field from the target at each mode frequency and causes measurable changes in the total emission power and terminal voltage. The variations in the emission power and terminal voltage contain information of the reflection coefficient and phase change of the target, as well as the external cavity optical path length and re-injection loss. These variations are also influenced by some of the laser intrinsic parameters, such as linewidth enhancement factor, photon and electron lifetimes, number of the active region stages, and electron injection efficiencies, as shown below. The absorption spectrum of the target (or the media in the external cavity) can be extracted from the variations in the emission power or terminal voltage of the multi-mode THz QCL which are induced by optical feedback from an extending external cavity length. Nevertheless, this only occurs when the optical feedback condition is appropriate under which the modes are co-existing and the SM signal characteristics are not changed due to mode competition. Fig. 1. Laser feedback interferometry model for a multi-mode THz QCL. The laser cavity has refractive index n in and cavity length L in with corresponding internal cavity round trip time τ in . Seven longitudinal modes from Mode 1 (M1) to Mode 7 (M7) which have the material gain larger than the total loss in the laser cavity are considered in this work. This multi-mode emission is transmitted through the facet on the right-hand side of the laser cavity and traverses the external cavity of refractive index n ext and length L ext with corresponding external cavity round trip time τ ext before reflecting back toward the QCL from the target. The power reflectivity and phase change introduced by the target are R(ν) and θ R (ν) respectively, which are generally frequency dependent. A portion of the reflected light, dictated by the re-injection coupling factor ε, re-enters the laser and mixes with the field inside the laser cavity, generating a set of SM signals at multi-mode frequencies through extending external cavity length with a moving target, which can be used to extract R(ν) and θ R (ν).
In order to fully explore the characteristics of the SM signal at each mode in multi-mode THz QCLs, we used the following set of multi-mode RREs, which is a updated version of the multi-mode RREs [35,37,48] by involving gain saturation in gain factor at each mode: where N 3 (t) and N 2 (t) are the carrier populations in the upper and lower laser levels (ULL/LLL) of the active cavity, respectively, while S m (t) and φ m (t) represent the photon population and the phase of the electric field in mode m. Equations (3) and (4) each represent N equations for the photon population and the phase for each of the longitudinal eigen-mode. The electron transport parameters (η 3 , η 2 , τ 3 , τ 2 , τ 32 ) are obtained from full self-self-consistent energy-balance Schrödinger-Poisson scattering transport calculations for the given driving current I described in Refs. [40,[50][51][52]. The driving current of the multi-mode laser is a constant at 0.75 A during external cavity extension process so that the dependence of the input parameters on the voltage and temperature can be ignored. The initial eigen-mode frequencies of the multi-mode THz QCL that are determined by the optical path in the laser cavity are updated at each time step by solving the set of phase rate Eq. (4) which in turn are used in the set of electron population and photon population rate Eqs. (1), (2), and (3). The entire set of the RREs is solved numerically using the fourthorder Runge-Kutta method. Once the equations are solved, the emission output power per mode can be calculated by P m (t) = η 0,m ω m S m (t)/τ p,m , where η 0,m = α m,m /2/(α a + α m,m ) is power output coupling coefficient for mode m [53]. However, the dependence of the feedback coupling coefficient κ on mode frequencies has been neglected to investigate the influence of the gain spectrum on the emission mode structure. The photon lifetime τ p,m is calculated from τ p,m = (v g,m (α a + α m,m )) −1 , where α m,m = ln(r 1,m r 2,m ) −1 /L in is the mirror loss of the laser cavity at mode m, and r 1,m = r 2,m = (n in (ν m ) − 1)/(n in (ν m ) + 1) is the reflectivities of the laser facets. α a is the waveguide loss (the dependence on frequency has been neglected) and v g,m = c/n in (ν m ) is the group velocity for mode m.
In order to gain a better understanding of the characteristics of the SM signal, we determined the steady state solutions of the carrier populations and the emission power that all contain an SM signal from the single-mode RREs when m = 1 in Eqs. (1)-(4). Defining the rate at which reinjected light is coupled into the laser cavity as κ = κ/τ in , and neglecting spontaneous emission terms and gain compression effect, the steady state solutions are given by where , and all the other symbols have their usual meanings. It should be noted that here G 0 is the gain factor of the single-mode, which is chosen at the peak gain frequency.
When 2 κτ p cos(φ FB ) 1, the photon population can be further derived: where S 2s = (G 0 Aτ 3 τ 2 ) −1 , S 1s = (η 3 IMτ p )q −1 − (η 3 + η 2 )IMτ p (qAτ 3 ) −1 , and φ FB is the steady state phase under optical feedback. The term φ FB in turn satisfies the excess phase equation [20]: Accordingly, Eqs. (5) and (6) can be rewritten as: The steady-state photon population as shown in Eq. (8) contains three terms: DC component S 1s − S 2s , first-order AC component S 1s 2 κτ p cos(φ FB ) and second-order AC component S 2s (2 κτ p ) 2 cos 2 (φ FB ). It is observed from Eq. (8) that, although physically the photon population is defined as the square of the electric field, namely, S(t) = |E(t)| 2 [20], mathematically the dominant term of S s is the first-order AC component term S 1s 2 κτ p cos(φ FB ), which is proportional to the electric field due to the definition of the feedback coupling rate κ. The SM amplitude, which is defined as the peak-to-peak values of the SM fringes, is mainly determined by S 1s 2 κτ p . Furthermore, the feedback coupling rate is proportional to the square root of the target reflectivity √ R and the re-injection coupling factor ε. Therefore, when the target is probed by a multi-mode THz beam at a group of THz frequencies, the SM signal at each frequency, which contains the reflectivity and the phase change at each particular frequency, can serve as a "fingerprint" to extract the refractive index at multiple THz frequencies of the target. This has clear application in material identification. It is also noted that the first-order AC component of S s is not influenced by the gain factor. However, the larger the gain factor, the larger the DC component of the S s , but the weaker the second-order AC component term of S s . Nevertheless, the SM amplitude contained in carrier populations are influenced by the gain factor, as shown in first terms of Eqs. (10) and (11). The larger the gain factor, the smaller the SM amplitude on both N 2s and N 3s .
It is worth pointing out that the SM amplitude is influenced by the gain factor at each mode frequency in multi-mode THz QCLs due to mode competition of the optical gain. As shown in Sec. 3.1, the mode with highest gain factor has highest SM amplitude. Therefore, a flat gain spectrum for multi-mode THz QCLs would be preferred for spectroscopic sensing applications. Multi-stack THz QCLs provide just such a broadened gain spectral bandwidth for such applications [10][11][12]. The simultaneous influence of gain bandwidth and optical feedback level on the SM signal characteristics in multi-mode THz QCLs is investigated in Sec. 3.
The gain saturation at high photon populations and the associated gain compression coefficient ε g is taken into account in the model to avoid emission energy being overly concentrated on the central mode in the gain spectrum. The gain compression coefficient ε g for photon populations in a multi-mode THz QCL can be estimated by: where I s,m is the saturation intensity for mode m [54] and V is the active region volume. The meaning of other symbols, together with typical values used in the simulations are summarized in Table 1. It is noted that the key parameters which influence the modal characteristics or the side mode suppression ratio (SMSR) of the multi-mode emission include the spontaneous lifetime τ sp , the linewidth enhancement factor α, and active region refractive index dispersion n in (ν m ). By choosing reasonable values for the parameters above for THz QCLs as shown in Table 1, and when the multi-mode RREs are solved forward in time domain to their steady state, the values of the gain compression coefficient had to be chosen to be two orders of magnitude larger than the a Varies as G m = G 0 /(1 + ((ν p − ν r,m )/(∆ν/2)) 2 )/(1 + (ε g S m (t))). estimated value from (12) to allow multi-mode emission. This may be partially explained by the fact that we neglect the symmetric cross gain saturation from neighboring modes [55,56] and the spatial hole burning effect [57] in the model.

Typical modal characteristics in multi-mode THz QCLs
The emission power from a multi-mode THz QCL under optical feedback contains a set of SM signals at multi-mode THz frequencies. The set of SM signals is created by extending the length of the external cavity from an initial value for L ext of 0.5 m, as illustrated in Fig. 1. The shape of the SM temporal waveform at each mode and consequently the emission spectrum from the multi-mode THz QCL is determined by both the gain spectrum of the laser and the level of optical feedback. The emission power at each mode was calculated by solving the set of RREs with fourth-order Runge-Kutta method at each external cavity length for 2 µs to reach temporal steady state. In order to investigate changes of the SM signal waveform at each mode, the external cavity length was extended by an amount ∆L ext of 250 µm with a step size of 2 µm in this section. It should be noted that we used three modifications as follows to speed up (in silico) the dynamical process as in order to reach the temporal steady state with reduced computational effort: 1) we solved the rate equation for the mode frequency with optical feedback ν m instead of the mode phase φ m (t) to avoid its accumulation with the time, considering that dφ m (t)/dt = 2π(ν m − ν r,m ); 2) the optical feedback terms in the rate equations of photon population and the frequency at each mode were replaced by their steady state versions as (2κ/τ in )S m (t) cos (2πν m τ ext ) and −(κ/τ in ) sin (2πν m τ ext ), respectively [20]; 3) the photon and carrier populations were normalized (divided) by 10 6 so that they are on the same order of magnitude as the mode frequency. Depending on the gain spectrum bandwidth and optical feedback level, a multi-mode THz QCL with an extending external cavity can exhibit characteristics of either a single-mode laser, a multi-mode laser, or a tuneable laser. Narrow gain spectrum bandwidth usually leads to large SMSR in the multi-mode emission power, and we defined the modal structure to be single-mode when SMSR of the THz emission power between the strongest two modes are larger than 30 dB. As shown in Fig. 2(a), when the gain bandwidth (FWHM) ∆ν is 360 GHz with the re-injection coupling factor ε of 0.001(C=0.0228), the SMSR between Mode 4 and 5 is around 50 dB, indicating single-mode emission. The emission power at each of the modes as a function of varying ∆L ext is shown on both a logarithmic and linear scale in Fig. 2(a) and Fig. 2(b), respectively, for the purpose of illustrating the typical modal structure and visualization of the SM signal amplitude at each mode, respectively. The logarithmic scale is good to show the definition of the modal structure but fails to clearly display the vast difference in the SM signal amplitude for each of the modes due to nonlinear amplification of the values larger and smaller than 1. Around 4.5 SM fringes generated due to the external cavity extension ∆L ext of 250 µm can be observed from the insets of Fig. 2(b) on Mode 4, 5, and 3 with decreasing SM amplitude from 3 µW, 0.0258 µW, to 0.00214 µW as a result of frequency dependent gain factor. The DC component of emission power at Mode 4, 5, and 3 are 6.59 mW, 0.157 µW, and 0.0528 µW, respectively. In this scenario, most of the gain contributes to the central mode frequency (Mode 4). Although the SM signal is carried on each of the longitudinal modes, the DC component of the side modes are too weak to be used for spectroscopic sensing applications.
The SM signal amplitude at each mode increases with increasing feedback level as observed from Eq. (1). When the SMSR reduces to 30 dB and smaller, the laser emission exhibits multi-mode modal structure. To show a typical multi-mode SM signal waveform with the coexistance of effective emission power at more than two modes, the simulation result with the gain bandwidth ∆ν of 800 GHz and the re-injection coupling factor ε of 0.001 (C = 0.0228) is given in Fig. 3, with the SMSR among the two strongest modes only 3.45 dB. The SM amplitude at Mode 4, 5, and 3 are 0.108 mW, 0.0780 mW, and 0.0287 mW, respectively at the DC component of emission power of 4.51 mW, 1.98 mW, and 0.111 mW. In this regime, since there is more than one mode that has relatively large DC emission power and AC component of the SM signal, it can be used for spectroscopic sensing to measure the absorption "fingerprints" at multiple THz frequencies [38,39]. The asymmetry in the modal emission power of Mode 3 and 5 is due to refractive index dispersion, which also leads to higher output coupling coefficient for Mode 5 compared to Mode 3. Since the amplitude of SM signal depends on both the feedback coupling coefficient κ and the gain factor G m at each mode frequency ν m , the emission spectrum obtained by performing a Fourier transform of the total emission power, couples the gain spectrum of the laser and is influenced by the feedback level. By choosing a proper reference target with known reflection coefficient, the multi-mode regime can be used to measure the reflection coefficient spectrum for an unknown target [39]. When the influence of the gain factor on the SM amplitude at each mode is removed by using the reference target, SM signal amplitude will be proportional to the reflection coefficient of the target under study in the multi-mode regime.
When the optical feedback level is further increased, the multi-mode THz QCL enters into the tuneable mode region, where the mode competition is so strong that only part of the set of LFI-induced fringes for each mode remains; most of the SM fringes are suppressed by other modes, which leads to mode switching. As shown in Fig. 4, when the re-injection coupling factor ε increases to 0.1 (C = 2.283) with a gain bandwidth ∆ν of 800 GHz, mode switching among Mode 4(3), 5, 6, 7 and 2 occurs cyclically when pulling back the target every half wavelength. The relationships between the gain spectrum and the position of the eigen-mode frequencies on the threshold gain spectrum (which is determined by the feedback parameters), determines the next dominant mode with varying ∆L ext , which hops to the mode with the largest net gain. In this regime, the SM signal amplitude is on the order of mW, which is much larger than that for singleand multi-mode cases, and is not proportional to the feedback coupling coefficient κ anymore.

Laser cavity modulations on the SM signal in multi-mode THz QCLs
In addition to the periodic modulation on the emission power with the SM fringes which occurs on the order of a half wavelength on each mode frequencies discussed in Sec. 3.1, it is found that there is additional modulation phenomenon which occurs on the order of optical path in the intra-laser cavity in the emission power of each mode, as shown in Fig. 5. This longer amplitude modulation occurring periodically with the external cavity extensions of around n in (ν m )L in , which are 10.8342 mm, 10.8364 mm, 10.8387 mm, 10.8410 mm, 10.8432 mm, 10.8455 mm, 10.8477 mm, respectively for each of the modes. It is also worth pointing out that this laser cavity modulation on the emission power is a universal phenomenon no matter whether the laser is operating in single-mode regime, multi-mode regime, or tuneable-mode regime, which are demonstrated in Fig. 5 (a), (b), and (c), respectively. This phenomenon is induced by light wave propagating and reflected by the laser facets in intra-laser cavity and could be used to measure the actual lasing cavity length in a single-mode or multi-mode THz QCL if the refractive index of the active region material is known or vice versa. This phenomenon was also observed in multi-mode LDs [58].

Mapping of the three regimes from LFI in multi-mode THz QCLs
Considering a Lorenzian gain spectrum centered at Mode 4 (the central mode), we simulated the multi-mode SM waveforms with external cavity extendsion with varying gain spectral bandwidth and feedback coupling coefficient ε, and calculate the SMSR between the central mode (Mode 4) and the other modes accordingly. The regime is defined as tuneable mode region when any of the SMSRs between the central mode and other modes is negative, which means there is mode switching during the target pull back process. Otherwise, it is defined as single-mode regime or multi-mode regime if the minimum of the SMSR between the central mode and other modes is larger or smaller than 30 dB, respectively. We indicate the single-mode regime, multi-mode regime, and tuneable-mode regime as Regime 1, 2, 3, respectively in the feedback regime maps in Fig. 6. We choose 100 × 20 equispaced points for the feedback coupling coefficient ε varying from 0 to 0.2 and the gain bandwidth varying from 100 GHz to 1100 GHz in the mapping. For each point in the map, we solve the RREs for 2 µs at each external target position and external cavity extension of n in L in of 10.85mm with a step of 2 µm. For the purpose of multi-spectral sensing applications, the interesting feedback condition is in the range of weak and moderate feedback levels defined for single-mode LDs with the corresponding feedback parameter C varying from 0 to 4.566 for a THz QCL if the initial external cavity length was used to calculate C (same applys for the values of C mentioned below) [20]. When the gain bandwidth is too narrow (< 150 GHz with the intrinsic laser parameters chosen in Table 1), the laser exhibits high stability to feedback perturbations so that it is always in single-mode regime with no influence by optical feedback due to limited optical gain, as shown in Fig. 6. However, if the laser has a moderate gain bandwidth between 150 GHz to around 400 GHz, although it is still single-mode in the absence of optical feedback (the line when ε = 0 in Fig. 6), it transits to multi-mode with growing feedback level. For example, the laser with gain bandwidth of 300 GHz enters into multi-mode regime when ε = 0.04 (C = 0.913). The phenomenon that mode structures of SM waveform changes from single-mode to multi-mode with increasing optical feedback level was observed experimentally in LDs and THz QCLs as well [38,59], which leads to subperiodicity in the shape of the SM waveform. As mentioned in Sec. 3.1, there are two routes to enter into the multi-mode regime: increasing feedback level for a fixed gain bandwidth or increasing the gain bandwidth for a fixed feedback level, which can be clearly observed from the regime mappings in Fig. 6. The SMSR is decreasing with the growing feedback level until it is below zero that the laser enters into tuneable mode regime when ε is larger than 0.064 (C = 1.461), as shown in Fig. 7. The value of ε when the laser transits from multi-mode regime (Regime 2) to tuneable-mode regime (Regime 3) is decreasing with increasing gain bandwith. When the gain bandwidth is larger than 400 GHz, the laser is multi-mode without optical feedback, and the SMSR is decreasing with increasing feedback level until the laser enters into tuneable mode regime. For example, when gain bandwidth is 500 GHz, the laser becomes tunealbe-mode when ε is around 0.042 (C = 0.959), and it reduces to 0.012 (C = 0.274) when the gain bandwidth increases to 1000 GHz. Therefore, although broad gain bandwidth has less gain spectrum influence on spectroscopic sensing spectrum extraction, the multi-mode regime that can be used for this application is smaller. The tiny region of Regime 2 (multi-mode regime) in the Regime 3 (tuneable-mode regime) in Fig. 6 is due to changes of SMSR from negative to positive values smaller than 1 dB, which is the definition of the boundary condition between multi-mode and tuneable-mode regime. Although the single-mode THz QCL with intermediate optical feedback has been demonstrated for gas sensing recently [60], multi-mode THz QCL with intermediate optical feedback is hardly used for sensing applications due to mode competition.  The regime mapping with a increased linewidth enhancement factor of 0.5 in the space of gain bandwidth and feedback coupling coefficient ε (0 ≤ C ≤ 5.080) is shown in Fig. 8 with the SMSR mapping in the same space shown in Fig. 9. It is observed that the multi-mode regime (Regime 2) is broadened with larger linewidth enhancement factor due to that the SMSR is increasing with increasing linewidth enhancement factor for the fixed gain bandwidth and feedback level, which gives more tolerence to optical feedback before the transition to tuneable-mode regime and makes the multi-mode regime larger. This suggests that FP THz QCLs with larger linewidth enhancement factor have more tolerance to optical feedback level that can be used for spectroscopic sensing due to larger multi-mode regime.

Conclusion
In conclusion, we explored the general characteristics of a FP multi-mode THz QCL under optical feedback through developing an original theoretical model that involves multi-mode RREs and gain saturation. The emission modal structure which contains a set of SM signals at each of the mode frequencies is determined by both gain spectrum of the laser and the optical feedback levels.
By varying optical feedback level and gain bandwidth, we identified three typical operation regimes, which includes a single-mode, a multimode, and a tuneable-mode regime, depending on the gain bandwidth and optical feedback levels. We discovered that only the multi-mode regime with a set of SM signals whose amplitudes are proportional to the reflection coefficient of the target at each of the THz frequencies could be used for spectroscopic sensing. The feedback regime map of the investigated modal structures in the space of gain spectral bandwidth and feedback level is also explored. The results provide design criteria for multi-mode THz QCLs when they are used for spectroscopic sensing in terms of the operating feedback level at varying gain bandwidth. The regime map with larger linewidth enhancement factor suggests that the multi-mode regime extended into higher feedback levels and would give more tolerance to optical feedback when used for spectroscopic sensing at the same gain bandwidth.