1. Introduction

Terahertz quantum-cascade lasers (THz QCLs) have been attracting much attention as compact semiconductor sources of terahertz radiation for a variety of applications [1]. In addressing such application requirements, high frequency-stability of THz sources is obviously demanded. In this context, knowledge on the intrinsic line-broadening due to quantum noise such as spontaneous emissions and thermal photons is crucially important since it ultimately limits their achievable spectral resolution and coherence length. Following the demonstration of a narrow intrinsic linewidth 260 Hz in a mid-infrared (4.3 μm) DFB QCL at room temperature [2], an even narrower intrinsic linewidth 90 Hz in a 2.5 THz QCL at a low temperature has been recently demonstrated by Vitiello et al. [3]. The obtained linewidth has been evaluated by the available linewidth formula derived by taking account of spontaneous emissions, noisy stimulated emissions due to thermal photons existing inside the cavity, and incoming fluctuations of thermal photons [4]. However, two different models for describing the intrinsic linewidth (LW) were proposed so far. Haug and Haken [5] and Liu et al [6] revealed LW formulas that do not count the stimulated emission term due to thermal photons but only incoming fluctuations of thermal photons as they solved standard van der Pol equations and Langevin equations, respectively, with an external noisy driving term. In reference [6], a memory effect for spontaneous emission events, i.e., a kind of non-Markovian process is counted but it is numerically so small, at most, a few percent correction in real QCLs. On the other hand, Herzog and Bergou [7] explored an atomic system in which excited two-level atoms injected into a cavity interact with photon field during time interval much shorter than photon decay time. Equivalently, perfect population-inversion (n_sp = 1, see the definition for n_sp in the next section) and short dephasing time for polarization dipole like in QCLs were assumed. Their assumption of the cavity-damping time longer than all other relevant time scales ensures that the time dependence of the polarization and population-inversion of the gain medium can be adiabatically eliminated and the Scully-Lamb model of the laser is applicable. Consequently, they obtained a LW formula by solving the standard master equation for density operator that represents the coupling of a cavity field to the environment, modeled by a reservoir in thermal equilibrium (existing thermal photons inside the cavity). The LW formula in reference [7] includes the stimulated emission term due to thermal photons that supports the model adopted in reference [4]. Furthermore, in the thermal limit, i.e., photon energy much lower than thermal energy ħω_opt<<k_BT, both of the LW formulas in references [4] and [7] approaches the Shimoda-Takahashi-Townes LW for a maser oscillator [8, 9].

Nevertheless, there has been a perplexity on the validities of the models as in reference [10]. In this context, the debate on model validities has to be finished promptly by confirming the validity of each model. In a theoretical point-of-view, the model including stimulated emission noises [4, 7] seems to have an advantage since thermal photons are primarily generated in the active region inside the cavity of a THz QCL as explicitly described in the next section. A more convincing way to confirm the model validities, however, is to compare the theoretical results with available experimental one [3]. In all the theoretical works [4–7], the thermal photon population N_bb is treated as a given parameter by assuming that the whole systems are thermally equilibrated. In contrast to that, in a THz QCL operating in CW mode, the thermal reservoirs in the cavity and ambient are not necessarily thermally equilibrated. The latter fact demands further modifications of LW formulas to reveal the real situation relevant to thermal photons.

The main aim of this article is to exhibit the modified LW formula derived by taking account of fluctuation-dissipation balance for thermal photons in a THz QCL. The obtained LW based on the formula is compared with the experimental result [3] to confirm the validity of the model. The modified formula is useful in future experiments. The inclusion of noisy stimulated emissions due to thermal photons is justified by intuitive consideration for a population-inverted two-level system in Appendix A. The key physical parameters, i.e., internal power and population-inversion parameter n_sp in the LW-estimation are precisely estimated, indicating consistency with separate experimental results for the 2.5 THz QCL [3] in Appendix B.

2. Linewidth formulas

We start with the intrinsic LW formula (for the full width at half maximum; FWHM) developed by assuming that the thermal reservoirs in the laser cavity and ambient are thermally equilibrated relevant to thermal photons [4],

As already pointed out in the introduction, the thermal reservoirs in the laser cavity and ambient system are not always equilibrated in a THz laser. In order to reveal the real situation, the following points have to be figured out: separate counting of generation rates of thermal photons from reservoirs with different temperatures and of thermal photon population in the laser cavity. Physical processes relevant to quantum fluctuations and dissipations of thermal photons, referring to Figs. 1 and 2 are summarized as follows.

 

Fig. 1 Band diagram of the active region in a THz BTC-QCL [3].

Download Full Size | PDF

 

Fig. 2 Illustration for fluctuation and dissipation flows of thermal photons in the THz QCL.

Download Full Size | PDF

For the stationary state, the total fluctuation and dissipation (average) rates for thermal photons N_bb,cavity in the laser cavity are balanced,

Now, we have four quantum noise-source terms for laser frequency fluctuations inside the cavity under the condition of the gain-loss balance above threshold, Μ(β/τ_r)N₃/n_sp = γ _, (with the notations, M is the number of cascade-modules, β the coupling efficiency of spontaneous emissions into a lasing mode, and τ_r the spontaneous-emission life-time); (1) spontaneous emissions, γn_sp, (2) stimulated emissions due to thermal photons γn_spN_bb,cavity, (3) incoming fluctuations from the electron systems, γ _activeN_bb,0(T_e) + γ _cladN_bb,0(T_L), and (4) incoming fluctuations from the ambient systems γ_mfN_bb,a-f(T_a-f) + γ _mbN_bb,a-b(T_a-b). The fluctuating photon generation events with random phase angles at random timings contribute to the frequency fluctuations through interferences between the intense coherent laser field (N_photon>>1) and fluctuating fields. These fluctuating processes additively contribute to the frequency noise (power spectral density) with weightings given by their average rates since they are random stochastic ones, being uncorrelated to each other. On the other hand, the dissipations as well as photon absorptions due to electronic transitions from the lower to upper laser-states do not play considerable roles in laser frequency fluctuations for N_photon>>1, as discussed extensively in reference [4]. Consequently, according to the treatment developed in Appendix B of reference [4], the intrinsic LW (FWHM) is represented by

3. Comparison with experimental results and discussion

For the assignment of LWs, we use numerical values for the internal power and population-inversion parameter at the bias current I₀−I_th = 90 mA, that are justified in Appendix B; P_int = (ħω_photon)γΝ_photon~15.06 mW and n_sp~1.22 for the present 2.5 THz BTC-QCL with a Fabry-Pérot cavity emitting single-mode photons in CW mode [3]. By using numerical values for physical parameters in Eq. (4); α_c~0.35, ħω_photon~10.4 meV, N_bb,a−f(T_a−f) = N_bb,a−b(T_a-b) = N_bb(T_H = 47.5K)~0.0855, N_bb,0(T_e = 90K)~0.354, N_bb,0(T_L = 76K)~0.257, γ ~7.3 × 10¹⁰ 1/s, α_m = α_mf + α_mb~4.55 cm⁻¹, as in reference [3], and α_active~1.9 cm⁻¹ and α_clad~2.6 cm⁻¹ [14], in addition to P_int ~15.06 mW and n_sp~1.22, the intrinsic LW of the 2.5 THz BTC-QCL [3] is estimated to be δf ~86.6 Hz. The estimated intrinsic LW (FWHM) value ~86.6 Hz is in very good agreement with the experimental (mean) value, 29π = 91.1 Hz with errors, ± 30 Hz [3]. The excellent agreement supports the model [4, 7] that includes the influence of noisy stimulated emissions due to thermal photons existing inside the cavity. On the other hand, the broader LW ~105 Hz, obtained by plugging N_bb(T_e = 90K)~0.354 (>N_bb,cavity~0.191) in Eq. (1) as in reference [3] deviates from the experimental value ~91.1 Hz. This indicates the significance of the consideration based on fluctuation-dissipation balance for thermal photons in LW estimation of the THz QCL. Also, complete ignorance of the thermal photon terms N_bb leads to δf~64.2 Hz. In other words, the thermal photons push up the LW by ~22.4 Hz (~35% enhancement for δf~64.2 Hz) in the 2.5 THz QCL at low temperatures, T_e~90 K, T_L~76 K and T_H~47.5 K. The LW enhancement, ~22.4 Hz is primarily brought about by the electron temperature, T_e~90K in the miniband of the active region and by T_L~76 K in the doped contact layer in the THz QCL. Furthermore, when the stimulated emission term n_spN_bb,cavity is ignored, i.e., Eq. (5) is used, one obtains, δf~74.3 Hz, substantially smaller than ~91.1 Hz. If the thermal photon-stimulated emission term n_spN_bb is ignored in Eq. (1) with N_bb(T_e = 90K)~0.354, the LW is obtained to be δf ~82.9 Hz, relatively close to ~91.1 Hz. However, this should be viewed as an occasional coincidence caused by relying on the inadequate model and wrong assignment of thermal photon population.

Now, we discuss the influences of parameter values on the intrinsic LW, focusing on the α-parameter α_c, optical losses, and electronic and lattice temperatures. The α-parameter was identified to range from α_c~0.2 to ~0.5 for a similar BTC-QCL [15]. The uses of α_c = 0.2-0.35(mean value)-0.5 in Eq. (4) lead to only small changes in the LW, δf = 80.2-86.6-96.4 Hz within the experimental errors, ± 30 Hz [3], that are smaller than the LW enhancement due to thermal photons, ~22.4 Hz. The optical losses would affect more seriously the LW value. However, in reference [3], the errors involved in the mirror-loss estimation (α_m = α_mf + α_mb~4.55 cm⁻¹) could be only a few percents coming from the facet reflectivity. The waveguide loss α_w = α_active + α_clad ~4.5 cm⁻¹ was actually measured by changing the length of the laser cavity [14]. The electronic and lattice temperatures were measured on the same BTC-QCL [12, 16], taking account of the operating conditions. The errors involved in the temperature estimations are so small, only less than ± 5 K for T_e~90 K, namely only less than ± 10% errors in thermal photon population that lead to very small errors, only ± 2 Hz (0.1 × 22.4 Hz) in the LW. Similarly, for the fixed waveguide loss α_w = α_active + α_clad ~4.5 cm⁻¹, errors (probably 20%) in the estimation of the loss-ratio, α_active/α_clad = (α_w-α_clad)/α_clad may not seriously affect the factor γ_activeN_bb,0(T_e) + γ _cladN_bb,0(T_L) (small errors, ~6% in the factor) in the LW since the population values, N_bb,0(T_e = 90K)~0.354 and N_bb,0(T_L = 76K)~0.257 are close to each other. Of course, specific device and operating conditions affect largely the optical losses and the electronic and lattice temperatures. In particular, hypothetically increasing electronic and lattice temperatures, for instance, up to ~150 K result in substantially larger thermal photon populations, N_bb,0(T_e = 150K) = N_bb,0(T_L = 150K)~0.809 for ħω_photon~10.4 meV and, consequently, lead to, for fixed values of other parameters, much larger LW broadening of about 52 Hz (80% broadening with respect to δf~64.2 Hz due to only spontaneous emissions) induced by the larger thermal photon populations. Obviously, the intentional temperature changes affect most significantly and directly the LW broadening, keeping the spontaneous emission term constant.

We, also, have to take care about measurement accuracy of the intrinsic LW. Any effect on the LW related with optical feedback [17] can be excluded in the experiment [3] since their experimental configuration (the mirrors, polarizer and gas-cell) was carefully engineered to avoid the optical feedback and in particular, the gas cell window was properly tilted for this purpose [14]. Furthermore, the LW ~91.1 Hz was extracted from the totally flat frequency noise power spectral density (FNPSD) ~29 Hz²/Hz in the high frequency range, 8-60 MHz, much higher than cut-off frequencies, typically ~100 kHz for any types of temperature fluctuations. Hence, there is no room for doubt in the intrinsic LW, ~91.1 Hz [3]. On the other hand, other results [17] published after the original submission of this article would be inappropriate for the present purpose, namely the confirmation of model-validities on thermal photons by the following reasons. First, the operating temperature of the QCL was T_H~20 K that is substantially lower than T_H~47.5 K in reference [3] and, in turn, the thermal photon population in the cavity of the QCL would be too small (N_bb,cavity<0.1) to discuss the model-validities. Second, as pointed out in reference [17], an unrealistically large α-parameter, α_c~1.5, despite the similarity of the active region structures between the two cases [3, 17], is required to interpret the LW, ~235 Hz for a detected output power per facet of 2 mW (corresponding P_int~12 mW), as an intrinsic one. The required α-parameter, α_c~1.5 is substantially larger than α_c~0.2-0.5 experimentally identified for a similar THz BTC-QCL [15]. The LW was obtained with a white noise plateau ~75 Hz²/Hz in the FNPSD in the limited frequency range, 20-100 kHz which is much (three-orders of magnitude) lower than 8-60 MHz in reference [3]. These facts imply that there might be problems in measurement accuracy for assignment of the intrinsic frequency noise. For instance, a similar plateau in the same limited frequency range is known to exist, that is caused by temperature fluctuations induced by electron density fluctuations in bipolar lasers (not by the intrinsic noise) as shown by curve D in the insertion of Fig. 1 in reference [18]. In QCLs, noise levels due to electron density fluctuations would be quite low, compared with those in bipolar lasers, because of fast recovery (~10 ps due to nonradiative relaxation) of density fluctuations to the steady state value in QCLs [4]. Nevertheless, one could not exclude the possibility of noise plateau due to temperature fluctuations induced by electron density fluctuations in the frequency range since the FNPSD levels due to 1/f²-noise are already very low, ~10² Hz²/Hz around 10 kHz in these THz QCLs [3, 17], five-orders of magnitude lower than that in a mid-infrared DFB QCL at higher temperatures [18]. In order to assign convincingly the intrinsic LW, white noise measurements in such higher frequency range (>1MHz) are indispensable.

Obviously, improved measurement techniques and/or special device configurations might lead to deeper understanding of physics, relevant to thermal photons behind the intrinsicLW. Possible experiments are, here, suggested to make knowledge about the influence of thermal photons more abundant and deeper. The first trial is to intentionally elevate the ambient temperature, for instance, T_a-b of the thermal photon source by a heater outside the cavity for fixed N_bb,0(T_e) and N_bb,0(T_L). The second trial is to use a different type of QCLs based on resonant phonon extraction that show a substantial increase in electron-temperature, ~100 K higher than the lattice temperature (for instance, T_L~150 K), in the upper laser-subband [19]. It is interesting to clarify experimentally how the temperature rise in the upper laser-subband, the higher lattice temperature and a larger population-inversion parameter n_sp~3 [10] affect the LW broadening. Equation (4) is quite useful in evaluation of experimental results obtained in these experiments as it is represented by different reservoir temperatures. The LW formula would be, if necessary, further modified to include the substantial electron-temperature rise in the upper laser-subband. The third trial is to engineer THz thermal photons and to suppress LW-broadening by thermal photons. We consider a 2.5 THz QCL, for instance, with a higher T_e (~T_L) of 150 K, and a lower waveguide loss, α_active + α_clad~2 cm⁻¹, a low mirror loss α_mb~0 cm⁻¹ at a HR-coated rear mirror and a higher mirror loss α_mf ~15 cm⁻¹ at a LR-coated front mirror. In such a THz QCL with the designed cavity, (γ _active + γ _clad)/γ <<1, and γ _mf/γ~1 and γ _mb~0, the thermal photon term in Eq. (4) is kept to be small, N_bb,cavity~0.17 for N_bb,0(T_e = T_L = 150K)~0.809 and N_bb,a−f(T_a−f = 47.5 K)~0.0855, which is even smaller than that, ~0.191 for the examined THz QCL with the lower T_e = 90 K and T_L = 76 K. The low ambient temperature T_a−f~47.5 K in the front side would be obtainable by cooling a hollow waveguide coupled to the front mirror [20] even if the heat-sink temperature is higher, T_H~100 K. This is quite similar to substantial reduction of dark currents due to incoming thermal photons by cooling of a low loss optical fiber coupled to the photo-active plane in a near-infrared photo-multiplier tube [21]. Such a trial to reduce the influence of thermal photons on the intrinsic LW might be more important as the laser frequency is lower.

4. Conclusions

The modified intrinsic linewidth formula derived by taking account of fluctuation-dissipation balance for thermal photons in a THz QCL has been exhibited. The linewidth formula based on the model [4] that counts explicitly the influence of noisy stimulated emissions due to thermal photons existing inside the laser cavity has interpreted experimental results ~90 Hz reported recently with a 2.5 THz BTC-QCL [3] by using reliable parameter values. The line-broadening induced by thermal photons is estimated to be ~22.4 Hz, i.e., 34% broadening with respect to the linewidth by only spontaneous emissions. Possible experiments have been suggested for systematic understanding of the influence of thermal photons on the intrinsic linewidth. As a bench mark in engineering of THz thermal photons inside laser cavities, the modified linewidth formula would be quite useful in evaluation of experimental results obtained with the suggested experiments.

Appendix A: Roles of spontaneous emissions, stimulated emissions due to thermal photons, and incoming thermal photon fluctuations in frequency fluctuations

In a two-level system consisting of upper and lower laser-states, 3 and 2 shown in Fig. 3 , the time evolution of total (average) photon population, N_photon(coherent)+N_bb,cavity(fluctuating), at a lasing mode is represented by

(6)$$\begin{array}{l}\text{d}\left(N_{\text{photon}}+N_{\text{bb,cavity}}\right)/\text{d}t\hfill \\ =-[\gamma -M(\beta /{\tau }_{\text{r}})(N_{\text{3}}-N_2)]N_{\text{photon}}+M\beta N_3/{\tau }_{\text{r}}+M(\beta /{\tau }_{\text{r}})(N_3-N_2)N_{\text{bb,cavity}}+\gamma N_{\text{bb,eff}}-\gamma N_{\text{bb,cavity}}\hfill \\ =-[\gamma -M(\beta /{\tau }_{\text{r}})(N_3-N_2)]\left(N_{\text{photon}}+N_{\text{bb,cavity}}\right)+M\beta N_3/{\tau }_{\text{r}}+\gamma N_{\text{bb,eff}}.\hfill \end{array}$$

In the relation, the fluctuation term γN_bb,eff for thermal photons at the lasing mode is defined as γN_bb,eff = γ _activeN_bb,0(T_e) + γ _cladN_bb,0(T_L) + γ _mfN_bb,a-f(T_a-f) + γ _mbN_bb,a-b(T_a-b). With a hypothetical upper-level population N₃₀ = N₂exp(−ħω_photon/k_BT₀) in which N_bb,cavity = 1/[exp(ħω_photon/k_BT₀)−1], one obtains fluctuation-dissipation balance on average,

(7)$$\beta N_{30}/{\tau }_r+\beta N_{30}{N}_{\text{bb,cavity}}/{\tau }_r=\beta N_2{N}_{\text{bb, cavity}}/{\tau }_r.$$

Consequently, Eq. (6) is rewritten, eliminating the absorption term −MβN₂N_bb,cavity/τ_r,

(8)$$\begin{array}{c}\text{d}\left(N_{\text{photon}}+N_{\text{bb,cavity}}\right)/\text{d}t=-[\gamma -M(\beta /{\tau }_{\text{r}})(N_3-N_2)]N_{\text{photon}}\\ +M\beta (N_3-N_{30})(1+N_{\text{bb,cavity}})/{\tau }_{\text{r}}+\gamma N_{\text{bb, eff}}-\gamma N_{\text{bb,cavity}}.\end{array}$$

If the two-level system is in thermal equilibrium, i.e., N₃ = N₃₀ and N_photon = 0, one obtains dN_bb,cavity/dt = γ N_{bb, eff}−γN_bb,cavity = 0.

 

Fig. 3 Population-inverted two-level system consisting of upper and lower laser-states subjected to quantum fluctuations.

Download Full Size | PDF

The second term on the right-hand side of Eq. (8) is the sum of excess (nonequilibrium) spontaneous and thermal photon-stimulated emission rates that acts as a noisy driving force for lasing photons N_photon. This alters the steady-state stimulated-emission rate from M(β/τ_r)(N₃−N₂)=γ to M(β/τ_r)(N₃−N₂)=γ −M(β/τ_r)(N₃−N₃₀)(1+N_bb,cavity)/N_photon required for maintaining of lasing. The photon population N_photon in the denominator of the correction factor indicates decreasing frequency fluctuations with increasing photon population (increasing internal power). Noisy photons with random phase angles generated at random timings by spontaneous emissions and stimulated emissions due to thermal photons, M(β/τ_r)(N₃−N₃₀)(1+N_bb,cavity), are funneled into lasing photons, and, in turn, the average thermal photon population N_bb,cavity is unchanged, dN_bb,cavity/dt=0 as it is independent of the pump level. Obviously, both of fluctuating spontaneous emission events with average rate MβN₃/τ_r (not net rate M(β/τ_r)(N₃−N₃₀)) and stimulated emission events with M(β/τ_r)N₃N_bb,cavity in addition to incoming fluctuations of thermal photons γN_bb,eff are to contribute to the frequency fluctuations of laser fields N_photon+N_bb,cavity. The dissipations for thermal photons −γN_bb,cavity as well as photon absorptions −M(β/τ_r)N₂N_bb,cavity are not influential in both of frequency and intensity fluctuations for N_photon≥10⁷>>1 which is satisfied even near above threshold (I₀/I_th−1)≥0.01 in a QCL, as clarified in reference [4]. In references [5, 6], fluctuating stimulated emissions M(β/τ_r)N₃N_bb,cavity are not counted since an external noise source term is used in their analyses.

Appendix B: Rate equation analysis for a THz bound-to-continuum quantum-cascade laser

For the active region of a module in a THz BTC-QCL, we restrict the rate equation analysis to a three-level system that consists of the following states: level 1 merged to states with electron population N₁ in a lower miniband M_l with sheet doping N_inj and the two states, levels 3 and, 2 of the lasing transition with electron populations N₃ and N₂, as depicted in Fig. 4 . A fraction of current flow, η_intI₀/e (η_int~0.15 in [3]) pumps the upper-laser level 3 and the remaining flow (1−η_int)I₀/e is assumed to pass through the higher miniband M_h and quickly relax down, by LO-phonon emissions, to the lower miniband M_l [3], as shown by the dashed arrows in Fig. 4. The latter assumption is supported by the fact that the photoluminescence (PL) due to electronic transitions from higher miniband states was observed to be, despite the substantial leakage flow (1−η_int)~0.85, much weaker (one order-of-magnitude) than that due to transitions from lower miniband states, as shown in Figs. 3 and 4(b) of reference [12]. This can be represented by the lower miniband pumping in the analysis, as shown by the bold arrow labeled (1−η_int)I₀/e in Fig. 4.

 

Fig. 4 Three-level model in a module of the THz BTC-QCL used in the rate equation analysis.

Download Full Size | PDF

For the steady state case, the rate equations for the electron populations and the photon population N_photon in a single lasing mode read as follows:

(9)$${\eta }_{\text{int}}\left(I_0/e\right)=N_3/{\tau }_{31}-N_1/{\tau }_{13}+N_3/{\tau }_{32}-N_2/{\tau }_{23}+(\beta /{\tau }_r)(N_3-N_2)N_{\text{photon}},$$

(10)$$(1-{\eta }_{\text{int}})\left(I_0/e\right)+N_3/{\tau }_{31}+N_2/{\tau }_{21}=\left(I_0/e\right)+N_1/{\tau }_{13}+N_1/{\tau }_{12}.$$

Equation (10) is rewritten, N₃/τ₃₁ + N₂/τ₂₁ = η_int(I₀/e) + N₁/τ₁₃ + N₁/τ₁₂, indicating that the current flow (1−η_int)(I₀/e) does not affect the electron population N₁ just like a simple leakage flow as depicted in Fig. 4. The particle conservation (global charge neutrality in each module) is represented by

(11)$$N_{\text{inj}}=N_1+N_2+N_3.$$

In order to avoid inessential complication for the present discussion, we assume that the nonradiative relaxation N₃/τ₃₂ and relevant back scattering N₂/τ₂₃ are negligible, compared with N₃/τ₃₁ and N₁/τ₁₃, respectively [3] because of E₃₂~10.4 meV<<LO-phonon energy~36 meV. With this assumption, below threshold I₀<I_th, i.e., N_photon=0, the electron populations at the upper- and lower-laser states are given by

(12)$$N_3=\frac{{\eta }_{\text{int}}\left(I_0/e\right){\tau }_{31}[1+({\tau }_{21}/{\tau }_{12})]+({\tau }_{31}/{\tau }_{13})N_{\text{inj}}}{1+({\tau }_{21}/{\tau }_{12})+({\tau }_{31}/{\tau }_{13})},$$

and

(13)$$N_2=({\tau }_{21}/{\tau }_{12})N_1=N_{\text{2therm}}.$$

The ratio of thermal back-filling population N_2therm to the injector one N₁ is represented explicitly by physical parameters as follows. As the PL measurements in reference [12] show that the population distribution in the lower miniband M_l is represented by a Boltzmann function, exp[−(E−E_F)/k_BT_e], the population N₁ is given by counting out electron populations in the subbands from the bottom to the second top, the 6-th one of the lower miniband consisting of 7-subbands,

(14)$$\begin{array}{c}{N}_1=\left(k_{\text{B}}{T}_{\text{e}}\right)({\rho }_{\text{2D}}S)\text{exp}\left[E_{\text{F}}/k_{\text{B}}{T}_{\text{e}}\right]{\displaystyle {\sum }_{i=1}^{i=6}\text{exp}[-(i-1)\Delta E/k_{\text{B}}{T}_{\text{e}}]}\\ =\frac{\left(k_{\text{B}}{T}_{\text{e}}\right)({\rho }_{\text{2D}}S)\text{exp}\left[E_{\text{F}}/k_{\text{B}}{T}_{\text{e}}\right]\{1-\text{exp}[-6\Delta E/k_{\text{B}}{T}_{\text{e}}]\}}{1-\text{exp}[-\Delta E/k_{\text{B}}{T}_{\text{e}}]},\end{array}$$

in which ΔE = E_M/6 is the average gap between adjacent subbands ans ρ_2D is the two-dimensional density-of-state function. On the other hand, the electron population at the lower-laser subband, i.e., the top one in the miniband is given by

(15)$$N_2=N_{\text{2therm}}=\left(k_{\text{B}}{T}_{\text{e}}\right)({\rho }_{\text{2D}}S)\exp \left[E_{\text{F}}/k_{\text{B}}{T}_{\text{e}}\right]\cdot \exp [-E_{\text{M}}/k_{\text{B}}{T}_{\text{e}}].$$

Hence, the back-filling factor for level 2 is given by

(16)$$\begin{array}{l}{f}_{\text{bf2}}={\tau }_{21}/{\tau }_{12}=N_{\text{2therm}}/N_1\hfill \\ =\frac{\exp [-E_{\text{M}}/k_{\text{B}}{T}_{\text{e}}]\cdot \{1-\exp [-\Delta E/k_{\text{B}}{T}_{\text{e}}]\}}{1-\text{exp}[-E_{\text{M}}/k_{\text{B}}{T}_{\text{e}}]}, \hfill \end{array}$$

which is smaller than exp[−E_M/k_BT_e]. Similarly, the back-filling factor for level 3 is given by

(17)$$f_{\text{bf3}}={\tau }_{31}/{\tau }_{13}=N_{\text{3therm}}/N_1=\frac{\exp [-(E_{\text{M}}+E_{32})/k_{\text{B}}{T}_{\text{e}}]\cdot \{1-\text{exp}[-\Delta E/k_{\text{B}}{T}_{\text{e}}]\}}{1-\exp [-E_{\text{M}}/k_{\text{B}}{T}_{\text{e}}]}.$$

The numerical values for the back-filling factors are given by using E_M~15 meV, E₃₂~10.4 meV and T_e~90 K as in reference [3]; f_bf2~0.0466 and f_bf3~0.0122.

At threshold, with the gain-loss balance relation, M(β/τ_r)(N₃−N₂)=γ, one obtains the threshold current, electron population at level 3 and population-inversion parameter,

(18)$$I_{\text{th}}=\left(\frac{e}{{\eta }_{\text{int}}}\right)\left\{\left(\frac{\gamma {\tau }_{\text{r}}}{M\beta {\tau }_{31}}\right)\left(\frac{1+f_{\text{bf2}}+f_{\text{bf3}}}{1+2f_{\text{bf2}}}\right)+\left(\frac{N_{\text{inj}}(f_{\text{bf2}}-f_{\text{bf3}})}{{\tau }_{31}(1+2f_{\text{bf2}})}\right)\right\},$$

(19)$$N_{\text{3th}}=\left(\frac{\gamma {\tau }_{\text{r}}}{M\beta }\right)\left(\frac{1+f_{\text{bf2}}}{1+2f_{\text{bf2}}}\right)+\left(\frac{N_{\text{inj}}{f}_{\text{bf2}}}{1+2f_{\text{bf2}}}\right),$$

(20)$$n_{\text{spth}}=\frac{N_{\text{3th}}}{{(N_3-N_2)}_{\text{th}}}=\left(\frac{1+f_{\text{bf2}}}{1+2f_{\text{bf2}}}\right)+\left(\frac{M\beta }{\gamma {\tau }_{\text{r}}}\right)\left(\frac{N_{\text{inj}}{f}_{\text{bf2}}}{1+2f_{\text{bf2}}}\right).$$

Above threshold I₀>I_th, relations for the photon flux γΝ_photon, electron population at level 3 and population-inversion parameter are given as follows,

(21)$$\gamma N_{\text{photon}}=\frac{M[{\eta }_{\text{int}}(I_0-I_{\text{th}})/e][1+2f_{\text{bf}2}]}{(1+2f_{\text{bf}2})+({\tau }_{21}/{\tau }_{31})(1+2f_{\text{bf3}})},$$

(22)$$N_3-N_{3th}=\frac{[{\eta }_{\text{int}}(I_0-I_{\text{th}})/e]{\tau }_{21}}{\left(1+2f_{\text{bf2}}\right)+({\tau }_{21}/{\tau }_{31})\left(1+2f_{\text{bf3}}\right)}$$

(23)$$\Delta n_{\text{sp}}=n_{\text{sp}}-n_{\text{spth}}=\frac{(M\beta /{\tau }_{\text{r}}\gamma )[{\eta }_{\text{int}}(I_0-I_{\text{th}})/e]{\tau }_{21}}{\left(1+2f_{\text{bf2}}\right)+({\tau }_{21}/{\tau }_{31})\left(1+2f_{\text{bf3}}\right)},$$

The key parameter β /τ_r is given explicitly by using Eq. (A15) in reference [4],

(24)$$\beta /{\tau }_{\text{r}}=\frac{4\pi e^2{f}_0{Z}_{32}{}^2{\Gamma }_{\text{conf}}}{{\epsilon }_0{n}_{\text{eff}}{}^2{V}_{\text{c}}(\hslash {\Gamma }_{\text{FWHM}})}~2.12\times {10}^{-4}{\text{μs}}^{-1},$$

where f₀ is the frequency of lasing photons ~2.5 THz, Z₃₂ the dipole moment relevant to laser transitions ~10 nm, Γ_conf the confinement factor ~0.7, n_eff the effective refractive index of the surface plasmon waveguide ~3.65, V_c the cavity volume given by a product of the total thickness of the active region and the area, ~15 μm × 150 μm × 2.5 mm, ħΓ_FWHM the emission linewidth (measured)~2.5 meV [3]. It is noted that the cumbersome waveguide factor existing in the denominator and numerator of the expressions for β and 1/τ_r (Eqs. (A14) and (A11) of reference [4]) is cancelled in the form for β /τ_r (Eq. (24)). Equation (24) is linked to the standard form for the gain coefficient as in Eq. (A16a) in reference [4].

For assignments of the threshold current, the internal power and the population-inversion parameter, the physical parameters, f_bf2~0.0466, f_bf3~0.0122, M=110, β /τ_r~2.12×10⁻⁴ 1/μs, η_int~0.15, N_inj/S ~5×10⁹ 1/cm² [22], and γ~7.3×10¹⁰ 1/s are precisely determined. The ratio of the relaxation times, τ₂₁/τ₃₁ is given to be small, τ₂₁/τ₃₁~0.0273 by τ₂₁~0.3 ps due to fast electron-electron scatterings in the miniband [3, 23] and τ₃₁~11 ps. With an assumed relaxation time of τ₃₁~11 ps together with all the other parameter values in Eq. (18), the threshold current of the present 2.5 THz BTC-QCL [3] is estimated to be I_th~351 mA (~93.7 A/cm²) which is in very good agreement with experimental threshold current 375 mA (~100 A/cm²) [3]. The agreement obviously justifies the use of τ₃₁~11 ps. The relaxation time τ₃₁~11 ps in the present 2.5 THz BTC-QCL with an energy separation between the upper laser-state and bottom state of the miniband E₃₂+E_M~25.4 meV(=ΔE_{2.5 THz}) (<LO-phonon energy ~36 meV) [3] is reasonably long, compared to τ₃₁~4.3 ps identified at a lattice temperature ~80 K in a similar 3.5 THz BTC-QCL [24] with a larger energy separation E₃₂+E_M~33.4 meV(=ΔE_{3.5 THz}) (~LO-phonon energy) [25]. In fact, the assumed life-time ratio 11/4.3~2.56 is very close to a factor exp[(ΔE_{3.5 THz} −ΔE_{2.5 THz})/k_BT_e]~2.80 based on thermally activated LO-phonon process. The maximum current reasonably higher than the threshold current is obtained to be I_max~eN_inj(1−f_bf3)/2η_intτ₃₁(1+f_bf2/2)~900 mA~2.4I_th given by the condition N₃~N₁ in which the laser output shows roll-over. The simultaneous agreements in threshold and maximum currents justify the numerical values, N_inj/S ~5×10⁹ 1/cm² and τ₃₁~11 ps.

The internal power at the bias current I₀−I_th=90 mA is obtained, with these reliable parameters in Eq. (21), P_int=(ħω_photon)γΝ_photon~15.06 mW, that leads to the output power per facet, P_out=[α_m/2(α_m+α_active+α_clad)]P_int~3.8 mW. By considering a collection efficiency of the optical setup and optical beam divergence, the estimated output power P_out~3.8 mW is regarded as being reasonably higher (~34%) than a detected power of ~2.5 mW in the 2.5 THz BTC-QCL with a Fabry-Pérot cavity [3]. This may also justify the use of η_int~0.15 that was determined by a separate manner (laser-induced hot-electron cooling) [23]. The population-inversion parameter at threshold is, with Eq. (20), estimated to be n_spth~1.213, being close to 1 with recourse to the small back-filling factor f_bf2~0.0466. Similarly, the deviation in the population-inversion parameter for the bias current I₀−I_th=90 mA is, with Eq. (23), obtained to be very small Δn_sp=(n_sp−n_spth)~0.0072 for τ₂₁~0.3 ps identified in reference [23] and ~0.0144 even for an assumed longer τ₂₁~0.6 ps. Hence, the population-inversion parameter is assigned convincingly to be n_sp~1.22 at the bias current I₀−I_th=90 mA. In the present case, the population-inversion parameter, n_spth is insensitive to temperature variation, n_spth=1.068−1.268 for T_e=50 K−110 K confirmed by using Eq. (20). The temperature-insensitive strong population-inversion (N₃>>N₂), n_spth~1 is consistent with the experimental evidence in reference [24]; both the inverse ~eτ_r/η_intMIτ₃₁(T) of luminescence intensity for a fixed current I and threshold current I_th~en_spthτ_rγ /η_intMβτ₃₁(T) have very similar temperature dependences over the temperature range from 50 K to 110 K, demonstrating that the BTC design has a performance limited by the upper state life-time τ₃₁(T). To summarize, the key parameters, P_int~15.06 mW and n_sp~1.22 are estimated precisely, indicating consistency with the separate experimental evidences for threshold current, maximum current, detected output power, and temperature dependences of the inverse of luminescence intensity and threshold current.

Acknowledgments

The author expresses his thanks to Miriam S. Vitiello, CNR-Instituto Nazionale di Ottica and LENS, Italy for her discussion and for providing information on optical parameters of the 2.5 THz bound-to-continuum quantum-cascade laser. He acknowledges Tooru Hirohata, Tadataka Edamura, and Kazuue Fujita, Hamamatsu Photonics KK for discussion on suppression of thermal photons in a cooled fiber, expertise comments on Si-doping, and band-structure computations for THz bound-to-continuum quantum-cascade lasers, respectively. The work has been, in part, supported by the Japan Society for the Promotion of Science (JSPS) through the “Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program).”

References and links

1. M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007). [CrossRef]  

2. S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express 19(19), 17996–18003 (2011). [CrossRef]   [PubMed]  

3. M. S. Vitiello, L. Consolino, S. Bartalini, A. Taschin, A. Tredicucci, M. Inguscio, and P. De Natale, “Quantum-limited frequency fluctuations in a terahertz laser,” Nat. Photonics 6(8), 525–528 (2012) (and “Supplementary information” for the reference). [CrossRef]  

4. M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44(1), 12–29 (2008). [CrossRef]  

5. H. Haug and H. Haken, “Theory of noise in semiconductor laser emission,” Z. Phys. 204(3), 262–275 (1967). [CrossRef]  

6. T. Liu, K. E. Lee, and Q. J. Wang, “Effects of resonant tunneling and dynamics of coherent interaction on intrinsic linewidth of quantum cascade lasers,” Opt. Express 20(15), 17145–17159 (2012). [CrossRef]  

7. U. Herzog and A. Bergou, “Quantum-limited linewidth of a good-cavity laser: an analytical theory from near to above threshold,” Phys. Rev. A 62(6), 063814 (2000). [CrossRef]  

8. K. Shimoda, H. Takahashi, and C. H. Townes, “Fluctuations in amplification of quanta with application to maser amplifiers,” J. Phys. Soc. Jpn. 12(6), 686–700 (1957). [CrossRef]  

9. For an explicit form of LW of a maser oscillator, see Eq. (10).62) in Chap. 10, “Fundamentals of noise processes” by Yoshihisa Yamamoto in quantum information lecture series (2010), http://first-quantum.net/e/forStudents/lecture/index.html. It is noted that the correction factor 1/2 due to stabilization of amplitude fluctuations above threshold is not included in the equation and n_sp = 1 is also assumed.

10. C. Jirauschek, “Monte Carlo study of intrinsic linewidths in terahertz quantum cascade lasers,” Opt. Express 18(25), 25922–25927 (2010). [CrossRef]   [PubMed]  

11. Equation (3).14) in reference [7] is applicable to THz-QCLs even near above thresholds since amplitude fluctuations are stabilized by very fast nonradiative relaxation of upper laser-state electrons as pointed out in Appendix B in reference [4] and also in Appendix A of this article.

12. M. S. Vitiello, G. Scamarcio, V. Spagnolo, T. Losco, R. P. Green, A. Tredicucci, H. E. Beere, and D. A. Ritchie, “Electron-lattice coupling in bound-to-continuum THz quantum-cascade lasers,” Appl. Phys. Lett. 88(24), 241109 (2006). [CrossRef]  

13. C. H. Henry, “Line broadening of semiconductor lasers,” in Coherence, amplification, and quantum effects in semiconductor lasers, Y. Yamamoto, Ed. (Wiley, 1991) pp. 5–76.

14. M. S. Vitiello, private communication on optical and electronic parameters.

15. R. P. Green, J. Xu, L. Mahler, A. Tredicucci, F. Beltran, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92(7), 071106 (2008). [CrossRef]  

16. M. S. Vitiello, G. Scamarcio, V. Spagnolo, C. Worrall, H. E. Beere, D. A. Ritchie, C. Sirtori, J. Alto, and S. Barbieri, “Subband electronic temperatures and electron-lattice energy relaxation in terahertz quantum cascade lasers with different conduction band offsets,” Appl. Phys. Lett. 89(13), 131114 (2006). [CrossRef]  

17. M. Ravaro, S. Barbieri, G. Santarelli, V. Jagtap, C. Manquest, C. Sirtori, S. P. Khanna, and E. H. Linfield, “Measurement of the intrinsic linewidth of terahertz quantum cascade lasers using a near-infrared frequency comb,” Opt. Express 20(23), 25654–25661 (2012). [CrossRef]   [PubMed]  

18. S. Borri, S. Bartalini, P. C. Pastor, I. Galli, G. Giusfredi, D. Mazzotti, M. Yamanishi, and P. De Natale, “Frequency-noise dynamics of mid-infrared quantum cascade lasers,” IEEE J. Quantum Electron. 47(7), 984–988 (2011). [CrossRef]  

19. M. S. Vitiello, G. Scamarcio, V. Spagnolo, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Measurement of subband electronic temperatures and population inversion in THz quantum-cascade lasers,” Appl. Phys. Lett. 86(11), 111115 (2005). [CrossRef]  

20. M. S. Vitiello, J.-H. Xu, M. Kumar, F. Beltram, A. Tredicucci, O. Mitrofanov, H. E. Beere, and D. A. Ritchie, “High efficiency coupling of Terahertz micro-ring quantum cascade lasers to the low-loss optical modes of hollow metallic waveguides,” Opt. Express 19(2), 1122–1130 (2011). [CrossRef]   [PubMed]  

21. The substantial reduction of dark currents by suppression of incoming thermal photons in a cooled low loss fiber coupled to a near infrared photomultiplier tube was in fact confirmed experimentally; T. Hirohata, Y. Negi, and M. Niigaki, Japan patent application No. 2009–047909 (2009) (will be published elsewhere).

22. The Si doping level in injectors of the THz BTC-QCL is designed to be N_inj/S~10¹⁰ 1/cm² [3]. However, in the analysis, the lower doping density 5 × 10⁹ 1/cm² is used by taking account of compensation by residual defect states. In fact, the assumed (effective) doping density 5 × 10⁹ 1/cm² leads to good agreements in threshold and maximum currents between theory and experiments.

23. M. S. Vitiello, G. Scamarcio, J. Faist, G. Scalari, C. Walther, H. E. Beere, and D. A. Ritchie, “Probing quantum efficiency by laser-induced hot-electron cooling,” Appl. Phys. Lett. 94(2), 021115 (2009). [CrossRef]  

24. G. Scalari, L. Ajili, J. Faist, H. Beere, E. Linfield, D. Ritchie, and G. Davies, “Far-infrared (λ~87 μm) bound-to-continuum quantum-cascade lasers operating up to 90 K,” Appl. Phys. Lett. 82(19), 3165–3167 (2003). [CrossRef]  

25. The band-structure computations for the 2.5 THz and 3.5 THz BTC-QCLs were performed by solving Schrödinger/Poisson equations; K. Fujita, private communication.