## Abstract

Intrinsic linewidth formula modified by taking account of fluctuation-dissipation balance for thermal photons in a THz quantum-cascade laser (QCL) is exhibited. The linewidth formula based on the model that counts explicitly the influence of noisy stimulated emissions due to thermal photons existing inside the laser cavity interprets experimental results on intrinsic linewidth, ~91.1 Hz reported recently with a 2.5 THz bound-to-continuum QCL. The line-broadening induced by thermal photons is estimated to be ~22.4 Hz, i.e., 34% broadening. The modified linewidth formula is utilized as a bench mark in engineering of THz thermal photons inside laser cavities.

©2012 Optical Society of America

## 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*_{B}*T*, 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],

*α*is the LW enhancement factor,

_{c}*ħω*

_{photon}the lasing photon energy,

*γ*the photon decay rate in the cavity,

*N*

_{bb}the thermal photon population,

*N*

_{photon}the photon population at a lasing mode, and

*P*

_{int}the internal laser power. The population-inversion parameter (or spontaneous emission factor)

*n*

_{sp}is given by the electron populations

*N*

_{3}and

*N*

_{2}at the upper and lower laser-states,

*n*

_{sp}=

*N*

_{3}/(

*N*

_{3}−

*N*

_{2})

_{th}. If the population-inversion parameter is assumed to be unity, i.e., perfect population-inversion

*n*

_{sp}= 1, one obtains,

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.

- (a) Quantum fluctuation sources in the laser cavity. We focus on a THz bound-to-continuum (BTC) QCL used in the LW measurements [3]. In the BTC-QCL, electron populations in the lower miniband M
_{1}and upper laser-subband 3 shown in Fig. 1 are well-thermalized separately, sharing a common electron temperature*T*_{e}~90 K, slightly higher than the lattice temperature,*T*_{L}~76 K at the lasing condition with the heat sink temperature*T*_{H}~47.5 K. These temperatures were identified experimentally by micro-probe photoluminescence measurements [12]. Obviously, the miniband electrons tend to generate thermal photons characterized by the electron temperature,*T*_{e}~90 K. In other words, the electron system in the miniband acts as a (quasi)-thermal reservoir for thermal photon generation. The average generation rate of thermal photons at a lasing mode in the cavity is given by*γ*_{active}N_{bb,0}(*T*_{e}) where N_{bb,0}(*T*_{e}) = 1/[exp(*ħω*_{photon}*/k*_{B}*T*_{e})−1]. Similarly, the free carrier system in the doped contact layer in the THz QCL with a surface plasmon waveguide acts as another reservoir for thermal photon generation and their generation rate is given by*γ*_{clad}N_{bb,0}(*T*_{L}) where N_{bb,0}(*T*_{L}) = 1/[exp(*ħω*_{photon}*/k*_{B}*T*_{L})−1]. The decay constants,*γ*_{active}=*c*_{0}*α*_{active}/*n*_{eff}due to a modal loss in the active layers and*γ*_{clad}=*c*_{0}*α*_{clad}/*n*_{eff}due to another modal loss in the doped contact layer are applicable to the generation rates because of the reciprocity in fluctuation-dissipation processes in general. The resultant average generation rate of thermal photons from the electron systems is given by*γ*_{active}N_{bb,0}(*T*_{e}) +*γ*_{clad}N_{bb,0}(*T*_{L}). Electronic transitions inside the upper laser-subband would not be directly relevant to considered photons with the*z*-polarization perpendicular to quantum-well planes.However, the electron system in the upper laser-subband 3 does not equilibrate thermally with that in the lower miniband

*M*_{1}, resulting in strong population-inversion*N*_{3}>*N*_{2}. Hence, electronic transitions from the upper laser-subband 3 to lower laser-subband 2 generate additional photons due to spontaneous emissions and also noisy stimulated emissions by thermal photons*N*_{cavity}at a lasing mode. The thermal photon population*N*_{cavity}inside the cavity is in general, different to N_{bb,0}(*T*_{e}) and N_{bb,0}(*T*_{L}), as will be shown below. These emitted photons are no longer viewed as thermal photons but are funneled into the single-mode laser-field, as clarified explicitly in Appendix A. The photon generation processes should be viewed as fluctuating noise sources for frequency fluctuations of the laser-field. In this view, as in references [4, 7], the inclusion of fluctuating stimulated emissions due to thermal photons as well as spontaneous emissions into phase (frequency) fluctuations is physically quite natural and reasonable. - (b) Incoming fluctuations from ambient systems. The average incoming rate of thermal photons into the cavity from ambient systems is given by
*γ*_{mf}N_{bb,a-f}(*T*_{a-f}) +*γ*_{mb}N_{bb,a-b}(*T*_{a-b}). In the relation,*γ*_{mf}and*γ*_{mb}are the decay rates at front- and rear-mirrors and are given more explicitly,*γ*_{mf}=*c*_{0}*α*_{mf}/*n*_{eff}=*c*_{0}ln(1/*R*_{f})/2*Ln*_{eff}, and*γ*_{mb}=*c*_{0}*α*_{mb}/*n*_{eff}=*c*_{0}ln(1/*R*_{b})/2*Ln*_{eff}, and*T*_{a-f}and*T*_{a-b}are the reservoir temperatures in front- and rear-side ambient systems.

- (c) Dissipation into electron systems. The average decay rate of thermal photons
*N*_{bb,cavity}in the cavity into the electron systems in the miniband*M*_{1}and the doped contact layer is given by (*γ*_{active}+*γ*_{clad})*N*_{bb,cavity}. - (d) Dissipation into ambient systems. The average decay rate of thermal photons into the ambient systems is given by (
*γ*_{mf}+*γ*_{mb})*N*_{bb,cavity}.

For the stationary state, the total fluctuation and dissipation (average) rates for thermal photons *N*_{bb,cavity} in the laser cavity are balanced,

*γ = γ*

_{active}+

*γ*

_{clad}+

*γ*

_{mf}+

*γ*

_{mb}is the total decay constant for photons inside the cavity. The thermal photon population

*N*

_{bb,cavity}inside the cavity takes a value different from N

_{bb,0}(

*T*

_{e}) and N

_{bb,0}(

*T*

_{L}), depending on

*γ*

_{mf}N

_{bb,a-f}(

*T*

_{a-b}) and

*γ*

_{mb}N

_{bb,a-b}(

*T*

_{a-b}). The thermal photon population

*N*

_{bb,cavity}in the laser cavity is represented explicitly by optical losses and system temperatures.

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})*N*_{3}/*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*_{sp}*N*_{bb,cavity}, (3) incoming fluctuations from the electron systems, *γ* _{active}N_{bb,0}(*T*_{e}) + *γ* _{clad}N_{bb,0}(*T*_{L}), and (4) incoming fluctuations from the ambient systems *γ*_{mf}N_{bb,a-f}(*T*_{a-f}) + *γ* _{mb}N_{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

*n*

_{sp}

*N*

_{bb,cavity}is ignored (as in references [5,6]), the LW is given by

_{bb,0}(

*T*

_{e}) = N

_{bb,,0}(

*T*

_{L}) = N

_{bb,a-f}(

*T*

_{a-f}) = N

_{bb,a-b}(

*T*

_{a-b}) =

*N*

_{bb}or

*γ*

_{mf}=

*γ*

_{mb}= 0 and N

_{bb,0}(

*T*

_{e}) = N

_{bb,0}(

*T*

_{L}) =

*N*

_{bb}, Eq. (4) becomes Eq. (1). In other words, in the previous work, we assumed equilibration in the whole system or equilibration inside the cavity (zero mirror loss). These LW formulas can be easily rewritten in the forms of current ratio

*I*

_{0}/

*I*

_{th}as in references [3, 4] but we hold the present forms represented by optical parameters which show clear correspondences with the Schawlow-Townes formula [13].

## 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 *I*_{0}−*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 × 10

^{10}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*

_{sp}

*N*

_{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*

_{sp}

*N*

_{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*_{active}N_{bb,0}(*T*_{e}) + *γ* _{clad}N_{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 Hz^{2}/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 Hz^{2}/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/*f ^{2}*-noise are already very low, ~10

^{2}Hz

^{2}/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

*γN*

_{bb,eff}for thermal photons at the lasing mode is defined as

*γN*

_{bb,eff}=

*γ*

_{active}

*N*

_{bb,0}(

*T*

_{e}) +

*γ*

_{clad}

*N*

_{bb,0}(

*T*

_{L}) +

*γ*

_{mf}

*N*

_{bb,a-f}(

*T*

_{a-f}) +

*γ*

_{mb}

*N*

_{bb,a-b}(

*T*

_{a-b}). With a hypothetical upper-level population

*N*

_{30}=

*N*

_{2}exp(−

*ħω*

_{photon}/

*k*

_{B}

*T*

_{0}) in which

*N*

_{bb,cavity}= 1/[exp(

*ħω*

_{photon}

*/k*

_{B}

*T*

_{0})−1], one obtains fluctuation-dissipation balance on average,

*MβN*

_{2}

*N*

_{bb,cavity}

*/τ*

_{r},

*N*

_{3}=

*N*

_{30}and

*N*

_{photon}= 0, one obtains d

*N*

_{bb,cavity}/d

*t*=

*γ N*

_{bb, eff}−

*γN*

_{bb,cavity}= 0.

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})(*N*_{3}−*N*_{2})=*γ* to *M*(*β/τ*_{r})(*N*_{3}−*N*_{2})=*γ* −*M*(*β/τ*_{r})(*N*_{3}−*N*_{30})(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})(*N*_{3}−*N*_{30})(1+*N*_{bb,cavity}), are funneled into lasing photons, and, in turn, the average thermal photon population *N*_{bb,cavity} is unchanged, d*N*_{bb,cavity}/d*t*=0 as it is independent of the pump level. Obviously, both of fluctuating spontaneous emission events with average rate *MβN*_{3}*/τ*_{r} (not net rate *M*(*β/τ*_{r})(*N*_{3}−*N*_{30})) and stimulated emission events with *M*(*β/τ*_{r})*N*_{3}*N*_{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})*N*_{2}*N*_{bb,cavity} are not influential in both of frequency and intensity fluctuations for *N*_{photon}≥10^{7}>>1 which is satisfied even near above threshold (*I*_{0}/*I*_{th}−1)≥0.01 in a QCL, as clarified in reference [4]. In references [5, 6], fluctuating stimulated emissions *M*(*β/τ*_{r})*N*_{3}*N*_{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 *N*_{1} 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 *N*_{3} and *N*_{2}, as depicted in Fig. 4
. A fraction of current flow, *η*_{int}*I*_{0}/*e* (*η*_{int}~0.15 in [3]) pumps the upper-laser level 3 and the remaining flow (1−*η*_{int})*I*_{0}/*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})*I*_{0}/*e* in Fig. 4.

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:

*N*

_{3}/

*τ*

_{31}+

*N*

_{2}/

*τ*

_{21}=

*η*

_{int}(

*I*

_{0}/

*e*) +

*N*

_{1}/

*τ*

_{13}+

*N*

_{1}/

*τ*

_{12}, indicating that the current flow (1−

*η*

_{int})(

*I*

_{0}/

*e*) does not affect the electron population

*N*

_{1}just like a simple leakage flow as depicted in Fig. 4. The particle conservation (global charge neutrality in each module) is represented by

In order to avoid inessential complication for the present discussion, we assume that the nonradiative relaxation *N*_{3}/*τ*_{32} and relevant back scattering *N*_{2}/*τ*_{23} are negligible, compared with *N*_{3}/*τ*_{31} and *N*_{1}/*τ*_{13}, respectively [3] because of *E*_{32}~10.4 meV<<LO-phonon energy~36 meV. With this assumption, below threshold *I*_{0}<*I*_{th}, i.e., *N*_{photon}=0, the electron populations at the upper- and lower-laser states are given by

The ratio of thermal back-filling population *N*_{2therm} to the injector one *N*_{1} 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*_{B}*T*_{e}], the population *N*_{1} 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,

*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

*E*

_{M}/

*k*

_{B}

*T*

_{e}]. Similarly, the back-filling factor for level 3 is given by

*E*

_{M}~15 meV,

*E*

_{32}~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})(*N*_{3}−*N*_{2})=*γ*, one obtains the threshold current, electron population at level 3 and population-inversion parameter,

Above threshold *I*_{0}>*I*_{th}, relations for the photon flux *γΝ*_{photon}, electron population at level 3 and population-inversion parameter are given as follows,

The key parameter *β* /*τ*_{r} is given explicitly by using Eq. (A15) in reference [4],

*f*

_{0}is the frequency of lasing photons ~2.5 THz,

*Z*

_{32}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* ~5×10^{9} 1/cm^{2} [22], and *γ*~7.3×10^{10} 1/s are precisely determined. The ratio of the relaxation times, *τ*_{21}/*τ*_{31} is given to be small, *τ*_{21}/*τ*_{31}~0.0273 by *τ*_{21}~0.3 ps due to fast electron-electron scatterings in the miniband [3, 23] and *τ*_{31}~11 ps. With an assumed relaxation time of *τ*_{31}~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/cm^{2}) which is in very good agreement with experimental threshold current 375 mA (~100 A/cm^{2}) [3]. The agreement obviously justifies the use of *τ*_{31}~11 ps. The relaxation time *τ*_{31}~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 *E*_{32}+*E*_{M}~25.4 meV(=*ΔE*_{2.5 THz}) (<LO-phonon energy ~36 meV) [3] is reasonably long, compared to *τ*_{31}~4.3 ps identified at a lattice temperature ~80 K in a similar 3.5 THz BTC-QCL [24] with a larger energy separation *E*_{32}+*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*_{B}*T*_{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})/2*η*_{int}*τ*_{31}(1+*f*_{bf2}/2)~900 mA~2.4*I*_{th} given by the condition *N*_{3}~*N*_{1} in which the laser output shows roll-over. The simultaneous agreements in threshold and maximum currents justify the numerical values, *N*_{inj}/*S* ~5×10^{9} 1/cm^{2} and *τ*_{31}~11 ps.

The internal power at the bias current *I*_{0}−*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 *I*_{0}−*I*_{th}=90 mA is, with Eq. (23), obtained to be very small *Δn*_{sp}=(*n*_{sp}−*n*_{spth})~0.0072 for *τ*_{21}~0.3 ps identified in reference [23] and ~0.0144 even for an assumed longer *τ*_{21}~0.6 ps. Hence, the population-inversion parameter is assigned convincingly to be *n*_{sp}~1.22 at the bias current *I*_{0}−*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 (*N*_{3}>>*N*_{2}), *n*_{spth}~1 is consistent with the experimental evidence in reference [24]; both the inverse ~*eτ*_{r}/*η*_{int}*MIτ*_{31}(*T*) of luminescence intensity for a fixed current *I* and threshold current *I*_{th}~*en*_{spth}*τ*_{r}*γ* /*η*_{int}*Mβτ*_{31}(*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 *τ*_{31}(*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).”

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**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.

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**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~10^{10} 1/cm^{2} [3]. However, in the analysis, the lower doping density 5 × 10^{9} 1/cm^{2} is used by taking account of compensation by residual defect states. In fact, the assumed (effective) doping density 5 × 10^{9} 1/cm^{2} 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.