Terahertz Quantum cascade lasers (THz QCLs) are compact and coherent THz light sources that generate optical transitions between conduction subbands in semiconductor multiple-quantum-well structures [1]. Together with their unipolar nature and wide coherent sensing range, QCLs can meet the increasing needs of applications in materials imaging, THz communication technology, atmospheric science, spectroscopy, and frequency metrology [2].

Self-mixing interference (SMI) (also known as laser feedback interference) is a sensing method that uses measurements of the change in the operating parameters of the laser under optical feedback. In contrast to traditional sensing systems, which employ the laser as a source and an optical interferometer to split and recombine the beam, SMI is based on the interaction of the in-cavity field with the back-scattered field from a target, which induces a modulation in amplitude of the optical emission frequency, power, and terminal voltage [3]. The use of SMI in THz QCLs has been studied in-depth, and it could be a promising solution for THz sensing of displacement, vibration, and velocity, and 2D/3D THz imaging [4]. Under optical feedback, QCLs can maintain a more stable working state than diode lasers because of the absence of relaxation oscillations. This is attributed both to high photon-to-carrier lifetime ratios and a negligible linewidth enhancement factor (α < 1) [5, 6].

Typical SMI prefers a stable, single-mode laser with low linewidth, which makes the data processing in a concise way. However, most Fabry–Pérot (FP) QCLs work with a broad emission linewidth in a multi-mode regime. Common candidates are distributed feedback quantum cascade lasers (DFB-QCLs) incorporating a first-order Bragg DFB grating into a standard QCL waveguide, which can provide more strictly single-mode emission with a high side mode suppression ratio (SMSR) [7–9]. Moreover, based on a second- or fourth-order Bragg gratings, some researches achieved high power surface emitting THz QCLs [10]. Rencently, a new model was designed to predict resonant mode characteristics of THz QCLs with a first, second, and third-order DFB-QCLs [11]. And wavelength beam-combining of four terahertz THz DFB-QCLs is demonstrated using low-cost THz components [12]. Base on the self-mixing technique, an experiment has been made to measure the linewidth enhancement factor α [13]. A newest experiment has made an extensive study of the linewidth enhancement factor α of a DFB-QCL, and it used the SMI technique to obtain α factors for current biases up to more than 100% of the threshold current [14]. Typically, the DFB gratings are either purely index coupled, purely gain coupled, or complex coupled, according to adjustments in the etching process based on theory and experience. From this point of view, some early studies investigated the properties of SMI in index-coupled diode lasers using the coupled wave theory, and the results show that the cosine-like self-mixing signal is similar to that from a single-mode FP laser [15, 16]. Furthermore, some studies also have shown that SMI sensors based on gain-coupled DFB lasers exhibit high accuracy [17]. However, a DFB laser with a grating etched into its upper cladding layer has an increased waveguide loss through the top contact layer, and this may in turn decrease its performance [18]. Recently, as a result of their performance, complex-coupled DFB-QCLs have received considerable attention in terms of both theory and experiments; in contrast to FP-QCLs and index-coupled DFB-QCLs, they have a reduced threshold current density and an increased power output via the introduction of built-in longitudinal modulation of the optical gain [19, 20].

A complex-coupled DFB THz QCL has a DFB grating etched into the top surface of the active region across the upper cladding layer; therefore, these devices have mixed index and gain coupling, and they also have a complex coupling factor [18]. Complex-coupled and gain-coupled DFB lasers have shown significant improvements in some studies in terms of reduced spatial hole burning and enhancement of modulation bandwith [18, 21]. In addition, experimental and theoretical works have also reported that complex-coupled and gain-coupled DFB lasers with a very large gain coupling show the potential for lower feedback sensitivity when compared with other DFB lasers [22, 23].

As noted above, complex-coupled and gain-coupled DFB lasers are more appropriate for use in SMI systems, especially in the case of weak optical feedback. However, in experiments, it is difficult to fabricate pure gain-coupled gratings because variations in gain cause variations in carrier density, which in turn cause variations in refractive index. In experiment, the complex coupling of DFB-QCLs has been attainable by chemical wet etching of the top contact layer to a certain depth, with the DFB grating etched close to the active region [24]. Recent studies on complex-coupled DFB-QCLs have reported that they exhibit excellent performance, with high-power continuous-wave, room temperature operation, and single-mode emitting with a high SMSR [25, 26]. However, there have so far been few in-depth studies of the dynamics of complex-coupled DFB-QCLs. In particular, the differences between complex-coupled and index-coupled DFB-QCLs in terms of their response rules for optical feedback are in need of detailed investigation.

The Lang–Kobayashi (L–K) equations are generally employed to study the dynamical behavior of SMI in FP-QCLs. However, the L–K equations cannot be simply applied to DFB-QCLs under optical feedback, despite the fact that several studies have used modified L–K equations to this end [17]. Here, we use the coupled wave theory to describe the mode-coupling phenomenon and calculate the emission power of complex-coupled THz DFB-QCLs under SMI using the multi-mode rate equation method. This model was implemented in our early study on SMI in index-coupled DFB-QCLs with a purely real index-coupling factor [27].

The remainder of this article is organized as follows. In Section 2, the coupled wave theory and the multi-mode rate equation method for the simulation of SMI in DFB-QCLs are presented. In Section 3, the basic output characteristics of SMI in DFB-QCLs of pure index coupling, complex coupling, and pure gain coupling are discussed. Finally, Section 4 presents our conclusions.

The coupled wave theory is an important method for simulating the longitudinal modes distributed in a DFB structure. Figure 1 shows a schematic drawing of a complex-coupled DFB-QCL with a first-order grating, which mainly radiates in the vicinity of Bragg wavevector β ₀ = 2π/(Λn _eff), where Λ is the grating period, and n _eff is the effective refractive index of the medium. Generally, the first-order grating for a QCL is pure real index-coupled grating or complex-coupled one with a small imaginary part that depends on the etching depth. At least in theory, there are also gratings with a pure imaginary coupling factor, i.e., gain-coupled gratings, although they are not common for QCLs.

In accordance with the coupled wave theory, starting from the scalar wave equation and using small-perturbation assumptions, the coupled wave equations are given as [22]: ∂ F ( z ) ∂ z = i ( k − β 0 ) F ( z ) + i κ F B B ( z ) , (1) − ∂ B ( z ) ∂ z = i ( k − β 0 ) B ( z ) + i κ B F F ( z ) , (2) where F(z) is the forward-running envelop wave and B(z) is the backward-running envelop wave. These two counter-running waves grow from the presence of gain, and they feed energy into each other due to Bragg scattering. The parameter k is the wave vector inside the medium of the laser and κ is the coupling factor, which measures the coupling strength between F(z) and B(z). For a complex-coupled DFB-Laser, κ is considered as a complex number satisfying κ F B = κ B F ∗ = κ index + i κ gain (3) with κ _index and κ _gain being real numbers measuring the coupling strength of the index and gain coupling of the grating, respectively. The laser modes in the cavity match the boundary conditions F ( 0 ) = r 1 B ( 0 ) (4) B ( L ) = r 2 F ( L ) , (5) where L is the length of the laser cavity, and r ₁ and r ₂ are the reflection coefficients of the laser facets. From Eqs 1–5 we get ( γ + r 1 ) ( γ + r 2 ) ( 1 + r 1 γ ) ( 1 + r 2 γ ) e − 2 i ( k − β 0 ) 2 − κ 2 L = 1 , (6) where we adopt the expression of κ = κ _index + iκ _gain from Eq. 3, and γ = κ / ( ( k − β 0 ) 2 − κ 2 + k − β 0 ) . From Eq. 6, we can solve the complex wave vectors k indicating the modes existing in the laser cavity. In accordance with the expression k = k ₀ n _eff + ig _th, we can calculate the emitting wave vector k ₀ in free space, the effective refractive index n _eff, and the threshold gain gth for each mode.

As illustrated in Figure 1, when a target with reflection coefficient r ₃ reflects part of the light back into the laser cavity, on the consideration of weak optical feedback (r ₂ r ₃ ≪ 1), we introduce an equivalent reflection coefficient to the emitting facet of the laser, describing the effect of SMI as [28] r 2 ′ = r 2 + ( 1 − r 2 2 ) r 3 e − 2 i k 0 L ex , (7) where L _ex is the distance from the laser emission facet to the target. In the case of strong optical feedback level, we suggest to refer to the expression of the equivalent reflection coefficient in [29]. From Eqs 6, 7, we can solve the wave vector k ₀ and the corresponding threshold gain gth with the influence of self-mixing feedback, and simultaneously solve the corresponding envelop waves F(z) and B(z). We can then obtain the self-mixing frequency signal with the formula Δ ν = ν l − ν l 0 (8) where ν _l is the mode frequency with optical feedback and ν _l0 is the solitary mode frequency.

In addition to the frequency signal, we use the multi-mode rate equation method to calculate the self-mixing power signal [30, 31]: d S l d t = Z g l S l − Γ S l τ l + Z β N 3 τ 32 + N 3 τ 31 , (9) d N 3 d t = I in q − N 3 τ 32 − N 3 τ 31 − ∑ l g l S l , (10) d N 2 d t = N 3 τ 32 − N 2 τ 21 + ∑ l g l S l , (11) d N 1 d t = N 3 τ 31 + N 2 τ 21 − N 1 τ out , (12) where: S _l is photon number of mode l; N ₃, N ₂, and N ₁ are the carrier numbers in the upper radiative, lower radiative, and collector levels, respectively; Z is the number of gain stages in the QCL; g _l is the mode gain; Γ is the confinement factor; τ _l is the photon lifetime; β is the spontaneous emission factor; τ _ij is the scattering lifetime between levels i and j; I _in is the injected current into level 3; q is the electron charge; and τ _out is the lifetime from level 1 into the subsequent miniband. The mode gain g _l = G (N ₃–N ₂), where G is the differential gain coefficient. In this study, we use the assumption of no carrier losses between the subsequent stages with N ₁/τ _out = I _in/q. In order to gain a better understanding of the mode competition derived from the DFB grating structure with optical feedback, we assume that the value of g _l is the same for the total modes in the laser cavity. The photon lifetime is written as 1 τ l = 1 τ w + 1 τ m l , (13) where: τ _w is the waveguide loss, which is the same for the total modes; and τ m l is the lifetime of a mode within the laser cavity due to mirror loss [30] τ m l ( L ex ) = ∫ z 0 z 2 < U ( z ) > d z + ∫ z 0 z 1 + L ex < U ( z ) > d z 2 ϵ 0 μ 0 n 0 2 | B ( z 0 ) | 2 ( 1 − r 1 ) 2 + | F ( z 2 ) | 2 ( 1 − r 2 ) 2 , (14) in which the energy density < U ( z ) > = 2 ϵ 0 n ( z ) 2 | F ( z ) | 2 + | B ( z ) | 2 , (15) where ϵ ₀, μ ₀, and n ₀ are the dielectric constant, permeability, and refractive index of a vacuum, and n(z) is the refractive index at coordinate z.From Eqs 9–12, the optical power P _l is obtained as [32] P l = h ν l S l ( 1 − R ) c / ( L ⋅ n e ff ) , (16) where hν _l is the photon energy and R is the reflectivity of the output facet. We can then obtain the self-mixing power signal by the formula Δ P = P l − P l 0 (17) where P _l and P _l0 are the optical power with and without optical feedback, respectively. It should be also noted that we assume the linewidth enhancement factor α = 0 for THz QCLs [5].

We assume the facet reflection coefficients to be r ₁ = r ₂ = 0.5 [33] and r ₃ = 0.001 [3], and the initial distance L _ex = 47 cm from z ₁ to z ₂, which is a typical precondition in the application of SMI. In addition, the reflection coefficient of the emitting facet can be changed by the optimized reflectivity facet coatings [9]. Based on the definition of the feedback parameter C = ( 1 − r 2 2 ) r 3 L ex / ( r 2 L n eff ) in the L-K equations, the above parameters correspond to C = 0.07 which is within the weak feedback regime. The feedback parameter C is a typical parameter being chose in the theory and experiment studies of the self-mixing interference [34, 35]. Because of L _ex/L being nearly unchanged if the target moves several wavelengths (ΔL _ex ≪ L _ex and L), we use the reflection coefficient r ₃ to describe the optical feedback strength in our investigation. Unless stated otherwise, the parameters used in the calculations are those presented in Table 1.

This study explored the output characteristics of self-mixing interference in terahertz distributed feedback quantum cascade lasers in the index-, complex-to gain-coupling regimes. Keeping the modulus of the coupling factor fixed while varying its argument from 0 to π/2, we found that extreme points occur at π/9 in the self-mixing frequency and power signals of DFB-QCLs. We also showed that the self-mixing frequency and power signals change with DFB coupling factor before and after a mode hoping phenomenon occurs. In the case of index-coupling dominated DFB-QCLs with a fixed modulus, the amplitudes of the self-mixing frequency and power signals increase while increasing argument. For gain-coupling dominated DFB-QCLs, when the argument of coupling factor is increased, the amplitude of the self-mixing power signal increases; however, the varying range of the self-mixing frequency signal decreases. With a fixed coupling factor argument, for index-coupling dominated DFB-QCLs, the varying ranges of the self-mixing frequency signals decrease with the increasing modulus. For coupling dominated DFB-QCLs, increasing the modulus of coupling factor decreases the varying ranges of the self-mixing frequency signal; however, the amplitude of the self-mixing power signals increase increases with increasing modulus. These findings will be helpful in investigating the nonlinear dynamics of complex-coupled self-mixing interference in THz DFB-QCLs, and this may be valuable for the application to THz DFB-QCLs in self-mixing sensing systems.