A theoretical model for calculation of the intrinsic linewidth of QCLs is built on the basis of the quantum Langevin approach. It differs from the traditional rate equation model in that the resonant tunneling and the dynamics of coherent interaction can be considered. Results show that the coupling strength and the dephasing rate associated with resonant tunneling strongly affect the linewidth of THz QCLs in the incoherent resonant-tunneling transport regime but only induce little influence in the coherent regime. The dynamics of coherent interaction and resonant-tunneling transport show insignificant effects on the linewidth calculation of mid-infrared QCLs due to strong coupling in resonant tunneling. We also demonstrate that by properly designing the active regions of QCLs, one can reduce the intrinsic linewidth according to our model.
©2012 Optical Society of America
Quantum cascade lasers (QCLs) have been a subject of great interest since their invention in 1994 . Since then, the emission wavelength can be engineered across the mid-infrared (mid-IR) (~3-24 μm) and Terahertz (THz) (~1.2-5 THz, or ~60-250 μm) regions. A thorough understanding of the linewidth of QCLs is increasingly important, as it is related to many practical applications e.g. trace-gas absorption spectroscopy and optical free-space data communication [2–7]. Recently, an ultra-narrow intrinsic linewidth of ~510 Hz has been experimentally observed in a distributed feedback mid-IR QCL at I0/Ith = 1.54 (I0 is the operation current and Ith is the laser threshold current) . The experimental results are fitted by theoretical calculations derived from the rate equation model. Although the rate equation model can explain the reason behind the narrow intrinsic linewidth well , it cannot adequately describe the electron transport characteristics of QCLs, and therefore cannot accurately predict the actual linewidth value.
In the rate equation model, localization of wavefunctions due to the dephasing scattering is overlooked which can lead to unrealistic results and limit the usefulness of the model. As reported in Refs 10–12, the effects of resonant tunneling and dynamics of coherent interaction (influenced by dephasing) are important when describing the transport of QCLs. The coherent (or incoherent) injection coupling plays an important role in the electrical injection process to populate the upper radiative level, which affects the electron populations associated with stimulated emission processes and hence the ratio of spontaneous emission coupled into the lasing mode to the net stimulated emission. As a result, the resonant tunneling and the dynamics of coherent interaction may finally affect the intrinsic linewidth of QCLs. In addition, the rate equation model uses the operating pumping current as an input parameter, thus knowledge on linewidth reduction through changing the active regions of QCLs, e.g. the coupling strength and the doping level, is restricted before obtaining experimental data.
The rate equation model also assumes that the polarization follows the other dynamic variables adiabatically, and therefore the dynamics of coherent interaction associated with laser transitions are neglected. But the polarization dynamics may help us interpret some physical phenomena such as electron memory effect induced intrinsic linewidth reduction [13,14]. Therefore, in some circumstances a careful consideration of coherent interaction dynamics associated with the laser transition is essential.
Since the resonant-tunneling transport and dynamics of coherent interaction cannot be properly included in the classical rate equation model, one has to refer to the more fundamental quantum mechanical model. In this paper, we report a new model for calculation of intrinsic linewidth of QCLs based on the quantum Langevin equations. It differs from the results derived from the classical rate equation model in that the resonant-tunneling effects and dynamics of coherent interaction are considered. The results show that the coupling strength and the dephasing rate associated with resonant tunneling strongly affect the linewidth of QCLs in the incoherent resonant-tunneling transport regime but only induce little influence in the coherent regime. Finally, the intrinsic linewidth reduction through optimization of the doping density and lifetime of different energy levels is presented.
2. Quantum Langevin equations
We consider a three-level QCLs system, as shown in Fig. 1 . denotes the interaction of the resonant tunneling, is the electron-light interaction. The electrons in level 1′ are injected into the upper laser level 3 by resonant-tunneling transport. Then, they relax from level 3 to level 2 by emitting photons. The Hamiltonian of this three-level QCLs system in the rotating-wave approximation can be obtained from Ref. 15 by adding the resonant-tunneling term (the fourth item on the right side of the equation):
The operators corresponding to the observables of our interest are the single-mode intracavity slowly varying laser field operator, electron polarization operator and electron population operator in level j. By including the noise operators and , the quantum Langevin equations which rule the dynamics of the laser are derived according to the Hershberger’s equation 
In the above equations, we neglect the polarization driven by higher quantum coherence effects. This assumption is, in general, valid for mid-IR designs due to a large value of and also reasonably valid for THz QCLs due to thicker injector barriers .
3. Laser intrinsic linewidths
3.1. Equivalent c-number Langevin equations
To solve the present problem, we have to convert the above operator equations into c-number equations . For this we have to choose certain particular ordering for field and electron density operators, because the c-number variables commute with each other while the operators do not. Here we choose the normal ordering of field and electron density operators to be,,,,,,,, that is, the stochastic c-number variables corresponding to the operators,,,, are replaced with their classical counterparts ,,,,. Then the c-number Langevin equations of Eqs. (3) can be written as
The diffusion coefficients of c-number Langevin forces may be different from the corresponding diffusion coefficients of the operator Langevin forces defined by Eqs. (A5). The diffusion coefficients are determined in such a way that the second moments calculated from the c-number equations agree with those calculated from the operator equations as explained in Appendix B.
3.2. Steady-state solution for above-threshold operation
The steady-state solutions for the mean values of the field and electron number variables above threshold are obtained by setting the time derivatives to be zero and dropping the noise terms in Eqs. (6). Neglecting the correlations between the electron populations and the photons, one then finds for the steady-state mean photon numbers, population inversion and population of the upper and lower laser levels
On the other hand, the steady-state polarization can be expressed in terms of the mean value of the field as
From Eq. (9a), we find that if is real, is purely imaginary. Therefore, we can choose the mean value of the phase of the laser field to be zero for simplification.
The current at and above threshold can be derived as, respectively
3.3. Dynamics of fluctuations around steady state
To determine the frequency noise spectrum and linewidth we need first of all to investigate the small fluctuations of the field and electron number variables around the steady state. Neglecting terms of the second and higher order in the fluctuations, we set
These equations can now be solved exactly by taking the Fourier transform of,
3.4. Noise spectra
To calculate noise spectra, we use the semiclassical expression of the laser field 
Then the dynamical equations of phase fluctuations are obtained as
By taking Fourier transform of Eq. (17) on both sides, one gets
According to Eq. (14a), the Fourier amplitudes of the field fluctuations can be expressed as
The autocorrelation function of the phase fluctuations can be expressed as
According to the Wiener-Khinchine theorem , the relations between autocorrelation function and power spectral density satisfy
The frequency noise spectral density Sf, therefore, can be derived as
Once we know the power spectrum density of the frequency noise, the linewidth can then be obtained by the following derivation. The autocorrelation function of the lasing field can be explicitly derived for single-mode QCLs when the phase fluctuations are considered as a Gaussian distribution .18]. The overall linewidth should be multiplied by 1 + α2 if this factor is considered. This parameter is not included in our model since the α-value is negligible in QCLs, which has been confirmed by experiments .
In general, if the spectral density of the frequency fluctuation is known, the mean squared value of the phase change can be obtained as 
The linewidth is then obtained from the following spectral density by taking the Fourier transform of Eq. (26)
4. Results and discussions
As shown in Eq. (28), fluctuations of laser field can be attributed to three sources, i.e. thermal photons (2κnth), vacuum fluctuations (κ), and spontaneous emission processes (2gg*/γ23). These sources induce linewidth broadening in lasers, which cannot be overcome due to fundamental quantum limitations. It needs to be mentioned that the vacuum fluctuations as one of noise sources are not included in rate equation model. Although the fact that vacuum fluctuations cannot be detected directly, the interaction of this vacuum field and the laser field can result in a modulation of the photon flux, which causes noise in the cavity . As a result, our model based on quantum Langevin equations provides more information about the fundamental physical origins of the laser linewidth.
In contrast to the traditional average squared phase fluctuation expression (such as Eq. (B10) in Ref. 9), there is an additional term (the second line in Eq. (28)). This term can finally determine the intrinsic linewidth, which depends on the time of measurement. If the time of measurement is shorter than the characteristic time τ = (γ23 + κ/2)−1, i.e. using ultrafast photodetectors, the phase fluctuation caused by spontaneous emission will disappear. This is attributed to suppression of spontaneous emission quantum noise caused by the electron memory effect associated with the transient behavior of the polarization. Therefore, for short-time measurements, the laser linewidth will reduce due to the disappearance of the spontaneous emission contribution. This property is not shown in the rate equation model. Since the rate equation model cannot include the dynamics of coherent interaction, the linewidth of the laser can only be based on the long-time measurement. Owing to the limit of bandwidth of the photodetector, we will not consider the cases of short-time measurements, and only investigate the laser linewidth of a long-time measurement in the following discussion.
For a long-time measurement, the exponential term becomes zero, and the phase fluctuation increases linearly with increasing time. The linewidth of the laser can then be expressed as
According to above Eq. (30) and Eq. (B12) in Ref. 9, we see that the linewidth deduced from our quantum mechanical Langevin model is different from the expression based on the rate equation model. The explicit difference between these two formulas is caused by the coefficient (1 + κ/2γ23)2. In the rate equation model, is assumed, and the polarization of the active medium is adiabatically eliminated. Hence, the information on dynamics of coherent interaction is missed. The inclusion of the coherent interaction can induce a smaller linewidth by a factor of (1 + κ/2γ23)2 if the coherent time is comparable to the cavity loss according to Eq. (30). The intrinsic difference between these two models is from the resonant-tunneling effects. Electrons in the ground injector level are injected into the upper laser level in QCLs through resonant tunneling. Since coherence plays an important role in the resonant-tunneling mechanism, which can significantly influence the electron transport [10–12], it is necessary to include this factor in laser dynamics simulations of QCLs. Comparing these two formulas, the rate equation model overestimates the influence of thermal photon, which lead to a higher intrinsic linewidth calculation for THz QCLs during high temperature operation.
Figure 2 shows the comparisons of the laser linewidths derived from our quantum mechanical model and the classical rate equation model at different operation currents in both THz (Fig. 2(a)) and Mid-IR (Fig. 2(b)) QCLs. Comparing the two models, the apparent difference for THz QCLs and relatively smaller difference for Mid-IR QCLs are caused by including the resonant-tunneling transport effect in our model. These differences increase as the operation current increases. Because the effects of resonant tunneling increases when the injector level and upper laser level tend to be aligned as the increasing operation current. It is noted that the effects of dynamics of coherent interaction associated with laser transitions on the linewidth of QCLs are not significant according to our calculation with the present parameters.
The role of coherence in resonant-tunneling transport in QCLs has been investigated [10–12]. It has been suggested that the inclusion of coherent transport and dephasing in calculations is essential especially when transport is dominated by transitions between weakly coupled states. We take THz QCLs structures [10,12] based on resonant-tunneling injection to investigate how coupling strength and dephasing rate associated with resonant tunneling influence the intrinsic linewidth. Figure 3 shows the effects of coupling strength on the linewidth at resonance. As shown in Fig. 3(a), the linewidth decreases as the injection coupling strength increases. When the injection coupling strength is below 3 meV, the linewidth diminishes rapidly. But once the coupling strength exceeds 3 meV, there only exists slight changes. This can be attributed to the coherence of electrons transport across the injector barrier, and this coherence can be determined by the factor ( coherent) and ( incoherent) . The small injection coupling strength corresponds to the regime of incoherent resonant-tunneling transport. But when the coupling becomes strong, the resonant-tunneling transport tends to be coherent. As a result, more excited electrons are injected into upper laser level, and then stimulated to the lower laser level by emitting photons, current density and photon number increases, as shown in Fig. 3(b). Hence, the noise associated with spontaneous emission is strongly suppressed, leading to a major reduction in the linewidth. Therefore, THz QCLs will have a much larger linewidth for structures with largely incoherent resonant-tunneling transport, but a smaller linewidth with coherent resonant-tunneling transport. Since the injector coupling strength of THz QCLs based on resonant-tunneling injection is typically 1~2 meV, the resonant tunneling can strongly influence the linewidth of THz QCLs.
Since the resonant-tunneling transport can be strongly influenced by the scattering, the linewidth has a strong dependence on the damping of the coherent interaction. Figure 4(a) shows the effects of dephasing rate associated with resonant-tunneling transport on the linewidth. When the dephasing rate γ31′ increases, only a small fraction of the electrons tunnels through the injector barrier into the upper laser level. As a result, the current density and photon number decreases (Fig. (4b)) and the noise associated with spontaneous emission becomes strong. Therefore, the linewidth increases as the dephasing rate increases.
As demonstrated in the above analysis, the characteristics of resonant-tunneling transport can strongly influence the linewidth of THz QCLs in the incoherent resonant-tunneling transport regime, but only have small effect on the linewidth in the coherent resonant-tunneling transport regime. For the mid-IR QCLs, the injector barrier is usually designed to be thinner, and the electrons are injected into the upper laser level by coherent resonant-tunneling processes. We can deduce that the resonant-tunneling transport can exhibit negligible effect on the linewidth of mid-IR QCLs. Figures 5 and 6 show the effects of coupling strength and dephasing rate associated with resonant tunneling on the linewidth. Owing to the much larger coupling strength of mid-IR QCLs, the electron can be injected into the upper laser level by coherent resonant tunneling. As a result, further increasing the coupling strength and dephasing rate γ31′ can only induce few changes of photon numbers, and hence only introduce little influence on the linewidth of mid-IR QCLs.
In the rate equation model, the operation current is taken as an input parameter. However, many other parameters e.g. coupling strength, dephasing rate and doping level can influence the current. Therefore quantum design with the optimization of the active region of a QCL to reduce the intrinsic linewidth is limited by using rate equation model, but it can be easily made according to our quantum mechanical Langevin model. To take the effect of doping on the linewidth as an example, we can optimize the doping density to achieve a reduced linewidth. Increasing doping density results in an increase in free-carrier absorption and waveguide loss. According to Ref. 22, the loss can be assumed to be roughly proportional to the doping density. Doping not only affects the cavity loss (thus the intrinsic linewidth) but also the electron population distribution and photon number according to our model (we neglect the effects of doping density on relaxation rates in our chosen ranges of doping). For the rate equation model, doping affects the linewidth only by the cavity loss. Figure 7(a) shows the self-induced linewidth variation by doping, the linewidth variation due to the change of doping induced cavity loss, and the overall effects from these two factors. The linewidth is enhanced as the doping density increases. Doping density does not cause the change of photon number but induce the increasing of current density in this calculation, as shown in Fig. 7(b). This shows that more electrons populate the upper laser level and take part in the spontaneous emission, then induce the linewidth broadening. The simulation tells us that we can optimize the doping density to reduce the linewidth of QCLs. It is noted that the optimization of the injector barrier is complex, since the thickness of barrier not only influences the coupling strength, but also it affects the lifetimes of the upper and lower laser level. Figure 8 shows the effects of lifetimes of the upper and laser levels on the linewidth. The linewidth increases and decreases as increasing the relaxation rate of upper laser and lower laser level, respectively. This is due to the variation of ratio of current to threshold current and photon number caused by the change of their relaxation rates, as shown in Fig. (9) . Thus, a careful design of active region should be made to reduce the intrinsic linewidth of QCLs.
In conclusion, we have developed a new model for the calculation of intrinsic linewidth of QCLs based on quantum Langevin equations. It includes the effects of resonant-tunneling transport and dynamics of coherent interaction. We conclude that the linewidth of the laser field can be attributed to three sources, e.g. thermal photons, vacuum fluctuations and spontaneous emission processes. The results show that for the short time measurement, the effects of electron memory can lead to suppression of spontaneous emission quantum noise; and the intrinsic linewidth of QCLs are reduced with the consideration of dynamics of coherent interaction. We also demonstrate that the coupling strength and dephasing rate have significant effects on the linewidth of THz QCLs in the incoherent resonant-tunneling regime, but small effects on that of mid-IR QCLs due to their strong coherent resonant-tunneling. The linewidth decreases with the increase of the injection coupling strength and reduction of the dephasing rate associated with resonant tunneling. According to our model, a reduced intrinsic linewidth can be easily designed through optimization of the injector barrier and doping density of the active region of QCLs.
Appendix A: operator diffusion coefficients
For the dynamical variation , we have the following quantum Langevin equations
For example, the diffusion coefficient corresponding to the fluctuation force,Eq. (A3)
Applying the same procedure to other Langevin forces, one can obtain
Appendix B: c-number diffusion coefficients
For c-number corresponding to the operator , we have the following c-number quantum Langevin equations
The diffusion coefficients can be calculated using the generalized Einstein relations
If is normally ordered, its expectation value is equal to the expectation value of the corresponding c-number product . Therefore, we have
If is not normally ordered, we can use the commentator relationship of operator to bring it into the chosen order.
Then the c-number second moments have
We would like to acknowledge financial support from Nanyang Technological University (NTU) (M58040017), Defense Research and Technology Office, Singapore (TL-9009105606-01), and Ministry of Education, Singapore (MOE2011-T2-2-147). Support from the CNRS International-NTU-Thales Research Alliance (CINTRA) Laboratory, UMI 3288, Singapore 637553, is also acknowledged.
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