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Abstract
The phase-locking, noise, and modulation properties of two face-to-face optically mutual-injected terahertz (THz) quantum cascade lasers (QCLs) are analyzed theoretically. In the phase-locking range, the two THz QCLs are in stationary states working at the same frequency. Outside the phase-locking range, the amplitude and the instantaneous frequency of the optical field oscillate with time, and the power spectrum shows a series of discrete peaks. For strong mutual injection, the optical field of the THz QCL array also exhibits oscillatory behavior. Coherent collapse or chaotic behavior is not observed within the range of the parameters used in this simulation. The spontaneous emission noise of phase-locked THz QCLs is higher than that of THz QCLs at free-running operation, and mutual injection may introduce additional modulation peaks in the noise spectrum. The modulation response of the mutual-injected THz QCLs to an individual modulation is investigated. It is found that the modulation bandwidth and the phase difference are significantly dependent on the modulation parameters. These results are helpful for further understanding the nonlinear dynamic behaviors of THz QCLs under optical injection and provide theoretical support for the development of THz QCL phase-locked arrays.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In the last two decades, quantum cascade lasers (QCLs) have become important sources in the mid-infrared and terahertz spectral range [1, 2]. QCL is a new type of unipolar laser based on intersubband transition and resonant tunneling in a repeated stack of semiconductor multiple quantum well heterostructures. Compact size, widely tunable frequency range, and high coherence make THz QCLs useful in imaging [3, 4], material analysis [4, 5], gas monitoring [6], and THz-communication [7].
In applications, high-performance THz QCLs are highly desired. High power THz QCLs have been achieved through design optimization and fabrication improvements [8–10]. The output power of pulsed devices can reach Watt level [8], and the continuous wave operation can reach hundreds of milliwatts [9]. However, these devices usually work with multi-modes, which may limit their application in some areas where spectral purity is highly required. The power of single-mode THz QCLs is still low, and the beam quality of facet-emitting THz QCLs is bad. In order to improve the single-mode output power and beam quality of THz QCLs, many efforts have been made, such as using graded photonic heterostructures [10], master-oscillator power-amplifier [11], tapered waveguide [12], and symmetric wafer bonded active regions [13].
It is an important method by using optically mutual-injected phase-locked laser array to improve the output power and beam quality of single-mode devices. The dynamical behavior of phase-locked lasers has long been concerned due to the requirement of phase locking between independent laser sources, and the diode laser arrays have been extensively studied [14, 15]. The symmetry-breaking [16, 17], stability condition [18, 19], and chaos synchronization [20, 21] in laser arrays have been investigated in detail. It has been found that the phase-locking properties of mutual-injected lasers are strongly dependent on the delay time, the coupling strength and the detuning frequency [22–24].
For THz-QCLs, great efforts have been made in developing Y-coupled devices [25], global antenna mutual coupling [26], phase-locked surface-emitting arrays [27], external cavity coherent quantum cascade laser array [28], and ring resonator arrays [29], et al. However, compared with the developed mutual-injected diode laser arrays, the studies on mutual-injected THz QCLs are still limited. The optical mutual injection could be regarded as the basic mechanism for face-to-face injected, lateral coupled or even some other kinds of phase-locked QCL arrays. Ultra-short carrier lifetime and small linewidth enhancement factor may make THz QCL arrays different from traditional semiconductor lasers. The lack of understanding for the basic dynamics of optically mutual-injected QCLs is an obstacle for developing corresponding devices. In this work, we will investigate the dynamics of two face-to-face mutual-injected THz QCLs to reveal the basic phase-locking property of THz QCL arrays.
Noise is a key factor affecting the performance of THz QCLs in potential applications [30–32]. Noises in THz QCLs come from spontaneous emission, non-radiative losses of carriers, interactions between photons and carriers, fluctuations of carriers, and so on. The spontaneous emission is an intrinsic noise source for lasers. The spontaneous emission noise of free-running and master-slave injection locking mid-infrared QCL has been investigated [33–35]. For mutual-injected THz QCLs, an important problem is how mutual injection affects the noise level. So we will characterize the spontaneous emission noise of mutual-injected THz QCLs.
The frequency response and phase characteristics of THz QCLs under modulation are closely related to applications in high-speed data transmission. Although mid-infrared wireless communication has been implemented by employing the mid-infrared QCLs as transmitter [36], it is still a long-standing challenge to realize terahertz wireless transmission. At present, the research on modulation properties of THz QCLs has attracted much attention [37–39]. For the mutual-injected THz QCL array, an interesting problem is whether the whole arrays can respond to an individual modulation. We will discuss this problem and fundamental modulation properties for THz QCL arrays.
The paper is organized as follows: a theoretical model of optically mutual-injected THz QCLs based on the rate equations is described in Section 2. In Section 3, we firstly study the phase locking boundary and basic working properties of optically mutual-injected THz QCLs within and out of the phase locking range. Then we investigate the spontaneous emission noise of the mutual-injected THz QCLs under different injection current and coupling strength. Finally, we perform a small signal analysis of the rate equations to discuss the frequency response and phase difference of mutual-injected THz QCLs, followed with a conclusion in Section 4.
2. Theoretical model of optically mutual-injected terahertz quantum cascade lasers
The basic scheme of two face-to-face optically mutual-injected THz QCLs system is shown in the inset of Fig. 1. We assume that the two QCLs are single-mode ones and have identical parameters but differ in free-running frequencies. The rate equations describing the dynamics of the system are written as:
where εA(B) is the complex electric field of THz QCL A (B). NA(B) is the carrier number of one single period of active region in THz QCL A (B). ΩA0(B0) is the angular frequency of the free-running THz QCL A(B). ΔΩ = ΩB0 − ΩA0 is the detuning frequency. Z is the number of stages of the THz QCLs. g is the differential gain coefficient of the active region. NT is the carrier number of one single active region at transparency and τp is the lifetime of photon in the cavity. κc and Cp represent the coupling coefficient and coupling phase of the mutual injection, respectively. where R is the reflectivity of the emitting facet of the laser, ζ is the fraction of the injected light coupled into the lasing mode. In general, Cp does not qualitatively influence the results [40]. So in the latter calculation Cp is taken as 0.1π for simplification. It is worthwhile to note that we ignore the multiple longitudinal-mode effects caused by the cavity length L in our simulations. Then the cavity length only affects the round-trip time τl = 2nef f L/c, where nef f is the effective refractive index of the active region, and c is the speed of light in vacuum. The carrier decay rate is expressed as γe = 1/τe where τe is the lifetime of carrier. α is the linewidth enhancement factor. α is taken as 0.1 in simulations, which is a typical value for QCLs[41, 42].
Fig. 1 The phase-locking range of face-to-face optically mutual-injected THz QCLs. The shadow region is the phase-locked range of the mutual-injected THz QCLs. Inset: the schematic of two THz QCLs with mutual injection. The red dashed line indicates the limitation of κ due to stability requirements.
The delay time is expressed as τext = d/c, where d is the distance between the light-emitting facets of laser A and B. The value of d significantly affects the dynamics of laser array. It has been shown that for a very long d, coupled lasers exhibit periodic modulation [40] and low-frequency fluctuation [43]. In our simulation, d is fixed at 0.1m, which is small enough to avoid the complicated dynamical behaviors. This value is typical for an experimental setup of mutual-injected lasers on separated cryostats. For monolithic QCL arrays, the value of d may be further smaller. But even a smaller d will not change the dynamics of mutual-injected QCLs qualitatively.
Iin, A and Iin,B are the corresponding injection currents. η denotes the current injection efficiency into the lasing levels. q is the electronic charge. In our model, the processes of carrier’s injection are not discussed in detail. The phenomenon that the injection efficiency η decreases with the increase of injection current in realistic QCL is not considered here. so η is kept asa constant in the simulation. In general, the injection current can be expressed as (1 + a)Ith , where a is the pump parameter [33]. So in the later simulations, Ith is used as the unit of injection current directly.
For the convenience of numerical simulation, the use of dimensionless parameters is necessary. Denoting the instantaneous frequencies of the two THz QCLs by ΩA and ΩB, the slowly-varying complex electric fields can be expressed as EA(t) = |εA| exp[−i(ΩA − ΩA0)t] and EB(t) = |εB | exp[−i(ΩB − ΩB0)t]. We introduce the dimensionless time s and normalize the time to the photon lifetime by s = t/τp. Then the variables and parameters in Eqs. (1)–(4) are nondimensionalized in accordance with the following rules:
where is the normalization factor of optical field. Nc = γ/Zg is the normalization factor of carrier number where γ = 1/τp is the decay rate of phonon. Nc0 = NT + γ/Zg is a constant shift to ensure n = 0 at the laser threshold, i.e., nA and nB are the excess carrier number of THz QCLs A and B, respectively. The dimensionless rate equations for EA, EB and the excess carrier number nA, nB can be written as:Where T ≡ τe/τp = γ/γe is defined as the ratio of carrier lifetime and photon lifetime. κ = κc(τp/τl) is defined as the coupling strength. It should be noted that κ is controllable because ζ and R can be changed by light path and experimental design. κ is quite similar to the feedback parameter C in Lang-Kobayashi equation [44] of self-mixing [45]. The coupling strength κ may be also calibrated in the similar manner of the calibration of feedback parameter C in self-mixing experiments [46]. So in this paper, κ is directly used as a simulation parameter for simplicity. θ ≡ τext /τp is the dimensionless external cavity one-trip time. pA(B) = (Zg/γ)(ηIin, A(B)/qγe − NT)− 1 is the pump rate of THz QCL A (B). ωA0,ωB0, and Δω are the dimensionless parameters of ΩA0, ΩB0 and ΔΩ respectively. The meaning of all parameters and typical values for the simulation of THz QCLs are listed in Table 1.
Table 1. Parameters values used in the simulations.
3. Results and discussion
3.1. Stability of optically mutual-injected terahertz quantum cascade lasers
In order to study the stability properties of the system, we first determine the phase-locking range by analyzing the stationary solutions of Eqs. (7)–(10). When the QCLs are phase-locked, their fields and excess carrier numbers are assumed to have the form as follows:
where ωL is the locked frequency of the coupled laser system, and ϕL is the locked phase of the field EAL (s) relative to EBL (s). FAL(BL) and nAL(BL) are the steady-state solutions of the corresponding amplitude of electric field and excess carrier number of THz QCL A(B). Iin, A = Iin,B = Iin and pA = pB = p are assumed in order to simplify the analysis. By inserting Eqs. (11)–(12) into Eqs. (7)–(10), we have:Although those coupled equations cannot be solved analytically, they provide some useful information about the condition of the phase-locking.
We further assume that the amplitude of the electric field satisfies such a relationship that FAL /FBL ≈ 1. This approximation is reasonable when the two phase-locked THz QCLs are identical except their free-running frequencies. Based on the Eqs. (15) and (16), the detuning frequency can be expressed as
According to the boundedness of the trigonometric function, the boundary of the phase-locked detuning frequency can be obtained by:
The phase-locking range is shown as the shaded area in Fig. 1. With the increase of the coupling strength κ, the model allows larger frequency detuning for phase locking.
The working properties of mutual-injected THz QCLs depend on the coupling strength κ and detuning frequency ΔΩ/2π. In order to reveal the dynamic properties of optically mutual-injected THz QCLs, we calculate the time evolution of |EA|, |EB |, instantaneous frequency, and power spectral density from Eqs. (7)–(10) for different detuning frequency ΔΩ/2π and coupling strength κ. As shown in Figs. 2(a)–2(c), the equations are solved numerically with ΔΩ/2π = 0.5GHz and κ = 9.87 × 10−3. The coupling strength is moderate and the frequency detuning is within the locking range. In this case for κ = 9.87 × 10−3, the maximum allowable locking range is 0.547 GHz based on Eq. (18). So the two THz QCLs with ΔΩ/2π = 0.5GHz are phase-locked. From the time evolution of optical fields in Fig. 2(a), it can be confirmed that the lasers are rapidly stabilized after a short period of interaction. The sharp peak at ΔΩ/2π = 0 in the power spectral density curve in Fig. 2(b) indicates that the DC component of the output power is dominant. The instantaneous frequencies of the two THz QCLs are equal and stable.
Fig. 2 Time evolution of |EA|, |EB | (first column), the corresponding power spectral density (second column), and instantaneous frequency (third column) with different coupling strength κ and detuning frequency ΔΩ/2π. The coupling strength in the first three rows are set as κ = 9.87 × 10−3, which is the case of moderate coupling. (a)–(c) within the phase-locking regime, ΔΩ/2π = 0.5GHz, (d)–(f) out of the pahse-locking regime, ΔΩ/2π = 5GHz, (g)–(i) out of the pahse-locking regime, ΔΩ/2π = 0.55GHz. The fourth row is the case of strong coupling with κ = 0.247. (j)–(l) out of the pahse-locking regime, ΔΩ/2π = 0.2GHz. The effective injection current is 1.5Ith in all simulations.
In Figs. 2(d)–2(f), we keep the coupling strength κ unchanged and increase the detuning frequency ΔΩ/2π to 5.0GHz which is far greater than the locking boundary. It is found that the electric fields oscillate with time. Since |EA|≈|EB |, the power spectral density of the two THz QCLs is almost equal as shown in Fig. 2(d). In addition, the power spectral density has sidebands at n · ΔΩ/2π(n = 1, 2, 3…). It indicates that the coupling of THz QCLs is equivalent to mutual frequency modulation of frequency ΔΩ/2π if the detuning frequency is large enough. In this case, the instantaneous frequencies of the two lasers exhibit periodic oscillations near their free-running frequencies.
Then, as shown in Figs. 2(g)–2(i), we still keep κ unchanged and set the detuning frequency ΔΩ/2π to 0.55GHz which is a little greater than the locking boundary determined by Eq. (18). The optical field cannot reach a stable state, but remains a bounded oscillation even after a long time evolution. In the frequency domain, the power spectral density of both THz QCLs is split into many separated lines. These peaks do not correspond to detuning frequencies n · ΔΩ/2π(n = 1, 2, 3…). The frequencies of the two lasers are still not locked. The instantaneous frequencies of the two THz QCLs show periodic oscillations.
Figures 2(j)–2(l) show the strong coupling case with κ = 0.247 and ΔΩ/2π = 0.2GHz. The lasers are in the phase-locking range according to Eq. (18), but unlike the case in Figs. 2(a)–2(c), the two lasers cannot achieve their steady states with such a large coupling strength. In time domain, the amplitude of optical field still has a very small oscillation with time, as shown in Fig. 2(j). Moreover, although the instantaneous frequencies of the two lasers are very close, they also behave as small oscillations and cannot be locked. Compared with Fig. 2(h), the power spectral density of the non-DC component is much lower, and the sidebands do not correspond to n · ΔΩ/2π(n = 1, 2, 3…). Conventional semiconductor lasers exhibit coherent collapse or chaotic behavior under strong light injection. Similar behavior is not found in the simulation of mutual-injected THz QCL array. By comparison, the stability of the THz QCL array under strong injection is better than that of diode laser arrays.
3.2. Spontaneous emission noise of optically mutual-injected terahertz quantum cascade lasers
After discussing the phase-locking property of mutual-injected THz QCLs, we continue to analyze the noise property of the model in the phase-locking range. An intrinsic source of noise in THz QCLs is the fluctuations in the complex field EA, EB caused by the spontaneous emission of photons. This kind of noise is very important for dynamics and can not be ignored in general. The equations of electric fields with noise are expressed as:
Where XA(B) denotes the spontaneous emission noise of THz QCL A which follows the Langevin force formulation. Time evolutions of the real and imaginary parts of XA(B) are independent random processes. XA and XB have zero mean < XA >=< XB >= 0 and the following correlations:
where β = 10−6 is spontaneous emission factor representing the fraction of spontaneous emission processes contribute to the lasing mode. nA0 = nB0 = gNT /γ is the dimensionless carrier number at threshold. Considering the symmetry of the detuning frequency in the model, we only calculate the relative intensity noise (RIN) of THz QCL A. PA is the normalized photon number so that PA = |EA|2. The RIN spectrum can be calculated by:The RINs are obtained by averaging over 20 independent calculations in order to avoid fluctuation caused by randomness and limited time evolution.
In order to reveal the effect of mutual injection on the spontaneous emission noise of devices, we compare the RIN between the mutual-injected array and the free-running laser. The RIN of a free-running QCL can be solved analytically by a semiclassical noise model [33, 34]. The RIN of the single free-running THz QCL A is calculated and shown in Fig. 3. The RIN spectrum is flat at low frequencies and exhibits white noise, which is consistent with the results in [33]. For κ = 9.87 × 10−3 and ΔΩ/2π = 0.2GHz, the RIN of the optically mutual-injected THz QCLs is also given in Fig. 3. The RIN spectrum of phase-locked mutual-injected THz QCLs is still frequency-independent in low-frequency region. There is no characteristic peak corresponding to relaxation oscillations on the RIN spectrum. The overdamping of relaxation oscillations is attributed to the ultra-short electron lifetime in THz QCLs. Compared with the free-running case, the mutual injection scheme makes the noise level of the laser increase by 57dB/Hz.
Fig. 3 The comparison of RINs for mutual-injected THz QCLs and the free-running one. Black line: moderate coupling within the pahse-locking regime with κ = 9.87 × 10−3 and ΔΩ/2π = 0.2GHz; Red line: moderate coupling out of the phase-locking regime with κ = 9.87 × 10−3 and ΔΩ/2π = 20GHz; Blue line: the RIN of a free-running THz QCL. And in all cases, Iin = 1.5Ith.
According to the numerical calculation in Section 3.1, the two THz QCLs cannot be locked steadily when the detuning frequency is too large, such as κ = 9.87 × 10−3 and ΔΩ/2π = 20GHz. The feedback term in Eq. (19) can be regarded as a modulation signal of 20GHz mixed in the noise XA(s). Then the coupled lasers are approximately free-running with the noise containing modulated signals. We calculate the RIN of the THz QCL A in this case, and compare it to the RIN with κ = 9.87 × 10−3 and ΔΩ/2π = 0.2GHz where the laser can be locked steadily. As shown in Fig. 3, in these two cases, the RINs are approximately equal below several GHz and show the characteristics of white noise. There are peaks at 20, 40, 60GHz in the case κ = 9.87 × 10−3 , ΔΩ/2π = 20GHz resulted from the frequency modulation.
The RINs of THz QCL A with coupling coefficients κ = 9.87 × 10−4 and κ = 9.87 × 10−2 are shown in Fig. 4(a). The RINs increase with coupling coefficient. Although increasing the coupling strength κ makes it easier for the array to achieve phase locking, the spontaneous emission noise increases.
Fig. 4 RINs of two mutual-injected THz QCLs with different coupling strengths and injection currents. (a) κ = 9.87 × 10−4 , 9.87 × 10−2, with the effective injection current and detuning frequency set as Iin = 1.5Ith and ΔΩ/2π = 0.01GHz, respectively. (b) Iin = 1.5Ith, 2Ith, with the coupling strength and detuning frequency set as κ = 9.87 × 10−3, ΔΩ/2π = 0.2GHz, respectively.
It is known that the spontaneous emission noises of THz QCLs are sensitive to current changes. Fig. 4(b) shows the RIN of THz QCL A with injection currents 1.5Ith and 2Ith with κ = 9.87 × 10−3 , ΔΩ/2π = 0.2GHz. It is found that an appropriate increase of current can reduce the spontaneous emission noise of phase-locked THz QCLs.
3.3. Modulation response characteristics of optically mutual-injected terahertz quantum cascade lasers
The modulation response of THz QCLs is an important aspect of laser performance. Under the condition of high-frequency modulation, ensuring high output signal amplitude is essential to high-speed data transmission. Theoretical and experimental researches on the modulation responses of free-running and injection-locked mid-infrared QCLs have been discussed in [47, 50]. The optically mutual-injected THz QCLs model is one of the most typical delay-coupled systems, and its frequency response characteristics are of great concern. In order to investigate the modulation response of mutual-injected THz QCLs, a small-signal analysis is carried out to Eqs. (7)–(10). ΔFA, FB, ΔϕA, ΔϕB, ΔnA and ΔnB are small variances around the steady state values, i.e.,
the rate equations for small signal modulation can be derived by substituting Eq. (24) into Eqs. (7)–(10).After taking the Laplace transform and rewriting them into a matrix form we can get:
where the matrix elements are expressed as with ψAL = ωL θ + Cp + ϕL and ψBL = ωL θ + Cp – ϕL . Considering the symmetry of our model, we define the output signal as the electric field amplitude of THz QCL A. When only adding the modulation signal ΔJA to the laser A, the transfer function is written asThen the frequency response is expressed as |HAA(iω)| and the phase difference between the output signal and the modulation signal is ∠HAA(iω). Due to the injection coupling of the two THz QCLs, there is frequency response in THz QCL A even when the modulation signal is only added to THz QCL B, the corresponding transfer function is
And the corresponding modulation response and phase difference are |HAB(iω)| and ∠HAB(iω), respectively.
The transfer function method is widely used to describe the frequency response of linear time-invariant systems. In our case, because the matrix Q contains delay term e−λθ , the transfer function contains e−λθ to e−3λθ terms which make the system complex. Fortunately, we do not particularly need the analytic zeros and poles of the transfer function. We are more concerned about the common nature of this kind of coupled system under modulation. So the numerical method is applied to calculate the modulation response of mutual-injected THz QCLs. The frequency responses of the lasers under different coupling strength κ and injection current Iin are calculated. The obtained Bode diagrams are presented in Fig. 5.
Fig. 5 Within the phase-locking regime, the normalized frequency response |HAA| and |HAB | with different coupling strengths and injection currents. The frequency detunning ΔΩ/2π is set as 0.01GHz. (a) Iin = 1.5Ith, different coupling strength with κ = 9.87×10−4, 9.87 × 10−2. (b) κ = 9.87 × 10−3, different injection current Iin = 1.5Ith,2.0Ith . The −3 dB modulation renponse is indicated by the blue dashed line.
The mutual-modulation frequency response |HAB(iω)| is always less than the self-modulation frequency response |HAA(iω)|. When the modulation is added to THz QCL A, we notice that ΔJA(s) linearly affects ΔnA(s) in Eq. (29) through pA = (Zg/γ)(ηIin, A/qγe − NT)− 1. So ΔFA(s) is linearly dependent on ΔJA(s) through the term 1/2ΔnA(s)FAL in Eq. (25). According to the symmetry of the equations, the relationship between ΔFB(s) and ΔJB(s) is similar. However, when the modulation signal is added to THz QCL B, ΔJB(s) changes ΔFA(s) through the term κΔFB(s − θ)cos(ωL θ + Cp + ϕL) in Eq. (25). Because κ is very small, it causes 1/2ΔnA(s)FAL >κΔFB(s − θ)cos(ωL θ + Cp + ϕL) in general. This is the main reason for |HAB(iω)| < |HAA(iω)|.
The 3dB bandwidth f3dB is defined as the frequency at which the modulation response |H(2πif3dB)| = 1/2|H(0)|. It is found that increasing the coupling strength and injection current can increase the modulation bandwidth of both modulation method. The modulation bandwidth of conventional semiconductor laser is restricted by the relaxation oscillation resulting from the interaction of the carriers and the photons. The normalized frequency response of the mutual-injected THz QCL array does not contain a characteristic peak corresponding to relaxation oscillation due to the ultra-short carrier lifetime of THz QCLs. Then for phase-locking arrays of THz QCLs, the absence of relaxation oscillations makes them very promising in high-frequency applications.
Since the model is a delayed coupling system, it is necessary to study the phase difference between input and output signals. ∠HAA(iω) and ∠HAB(iω) are shown in Figs. 6(a) and 6(b) respectively. As shown in Fig. 6, in both cases the low-frequency limit of phase difference is π, which comes from the negative coefficient −qγγe/Zg in Eqs. (33) and (34). It means that the phase of ΔFA is completely synchronized with ΔJA and ΔJB by low-frequency modulation. However, with the increase of modulation frequency, phase asynchrony is generated between response and modulation, and the phase difference decreases with the increase of the modulation frequency. It is found that ∠HAB(iω) drops faster under the same delay time parameter. The results show that when d = 0.05m and the modulation frequency is 20 GHz, ∠HAA decreases by less than 30 degrees, but ∠HAB has fallen by more than 360 degrees.
Fig. 6 The phase difference between input and output signals of the mutual-injected THz-QCLs under frequency modulation with d = 0.05m, 0.1m, 0.15m. (a) self-modulation ∠HAA(iω), (b) mutual-modulation ∠HAB(iω). The coupling strength, detuning frequency, and effective injection current are set as κ = 9.87 × 10−3, ΔΩ/2π = 0.01GHz, and 1.5Ith , respectively.
Figure 6 also shows that ∠HAA(iω) is independent of delay time, since 1/2ΔnA(s)FAL in Eq. (25) does not contain the delay time θ. On the contrary, ∠HAB(iω) shows the dependence of delay time, and drops faster as the delay time increases. The reason behind this is that the instantaneous value of κΔFB(s − θ)cos(ωL θ + Cp + ϕL) in Eq. (25) is related to the history of ΔFB, a phase difference is inevitable. It is found that ∠HAA(iω) gradually tends to 180 degrees as the modulation frequency increases, which is similar to the properties of free-running and master-slave injection arrays. In contrast, ∠HAB(iω) still depends on the modulation frequency.
4. Conclusion
In conclusion, the theoretical investigation of two face-to-face optically mutual-injected THz QCLs is carried out. It is shown that coupling coefficient κ and linewidth enhancement factor α determine the maximum allowable frequency detuning of phase-locking operation. According to the working parameters, the mutual-injected array can operate in the mode of stable phase locking or periodic oscillation. Particularly when the detuning frequency is far away from the locking range boundary, the effect of mutual injection on each laser is equivalent to a modulation of the detuning frequency. By studying the spontaneous emission noise in the model, it is found that the mutual-injected THz QCLs show a higher white noise level than free-running operation within several GHz frequency range. And a series of peaks corresponding to n · ΔΩ/2π(n = 1, 2, 3…) appear on the noise spectrum when the array is working far away from the locking region. By the method of small signal analysis, we get the modulation response property of the phase-locked THz QCLs. Increasing the injection current and coupling strength can increase the modulation bandwidth of the whole array. The modulation and response of the two THz QCLs at low-frequency limit are completely synchronous. With the increase of modulation frequency, phase difference is generated within the array. These results may help to further understand the nonlinear dynamic behavior and noise characteristics of THz QCLs under light injection, and provide theoretical support for the development of THz QCL phase-locked array.
Funding
National Natural Science Foundation of China (NSAF) Joint Fund (U1730246); National Natural Science Foundation of China (NSFC-RS) (F040302); Foundation of Director of IAPCM (ZYSZ1518-16).
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