## Abstract

Based on a coupled simulation of carrier transport and optical cavity field, the intrinsic linewidth in resonant phonon terahertz quantum cascade lasers is self-consistently analyzed. For high power structures, values on the order of Hz are obtained. Thermal photons are found to play a considerable role at elevated temperatures. A linewidth enhancement factor of 0.5 is calculated for the investigated designs.

© 2010 OSA

## 1. Introduction

The terahertz (THz) quantum cascade laser (QCL) is a versatile source of coherent THz radiation, offering the advantages of a solid state device, like robustness and compactness. Featuring high achievable output powers of several 100mW [1, 2] and potentially narrow linewidths below 1kHz [3], the THz QCL is a promising tool for various applications, such as telecommunications or high resolution THz spectroscopy. In this context, the intrinsic linewidth due to quantum noise plays a fundamental role, ultimately limiting the achievable spectral resolution and coherence length. Recently, the intrinsic linewidth has been measured for a 4.33 *μ*m QCL, yielding ∼ 500Hz [4]. To date, such comprehensive data are still lacking for THz structures. In fact, the measured linewidths are dominated by extrinsic noise and exceed the Schawlow-Townes limit by far [5–8]. These noise sources can be suppressed by locking the THz QCL to an external reference, and frequency stabilization to the Hz level has recently been demonstrated [9,10]. Theoretically, intrinsic linewidths in the sub-kHz regime have been predicted for THz QCLs [3]. However, these results are based on a rate equation approach containing intrinsic laser parameters like the upper laser level lifetime and the linewidth enhancement factor, which had to be estimated from available experimental data.

Recently we have developed a method to perform self-consistent coupled simulations of the carrier transport and the optical dynamics in QCLs [11]. This approach is based on a well established Monte Carlo carrier transport simulation supplemented by evolution equations for the optical cavity field, and only relies on well known material parameters and the device specifications. Here we extend this approach to also account for spontaneous emission and black-body radiation, allowing us to self-consistently investigate optical fluctuations due to these effects. We study the intrinsic linewidth and related properties like the linewidth enhancement factor for various resonant phonon THz QCL designs. For high power structures, especially low intrinsic linewidths in the Hz range are obtained.

## 2. Method

The carrier transport in the QCL structure is modeled using a semiclassical ensemble Monte Carlo (EMC) approach. The subband wave functions and energies are computed with a Schrödinger-Poisson solver [12], coupled to the EMC simulation tool. All relevant scattering mechanisms are included [13, 14]. Self-consistent results for the carrier transport and the laser field are obtained by coupling the EMC simulation to an optical evolution equation [11]. For studying optical fluctuations and the intrinsic linewidth, we have to consider thermal photons and spontaneous photon emission events in addition to the stimulated transitions. The evolution of the photon number *N* in the laser mode is then given by

*a*=

*a*

_{w}+

*a*

_{m}describes the total cavity loss (comprising waveguide loss

*a*

_{w}and outcoupling at the end mirrors

*a*

_{m}), and Γ is the overlap factor. Furthermore,

*n*

_{0},

*g*and

*n*

_{sp}are the refractive index, net power gain coefficient and spontaneous emission factor of the gain medium at the (angular) lasing frequency

*ω*= 2

*πf*=

*ω*

_{L}, and

*c*is the vacuum speed of light. The term (Γ

*g*–

*a*)

*N*describes amplification in the gain medium and cavity damping, Γ

*gn*

_{sp}contains spontaneous emission processes in the gain medium, and

*an*

_{th}describes the injection of thermal photons from the cavity into the lasing mode. Here,

*n*

_{th}= [exp (

*ħω/*(

*k*

_{B}

*T*)) – 1]

^{−1}is the photon occupation number in thermal equilibrium at temperature

*T*, with

*k*

_{B}and

*ħ*denoting the Boltzmann and reduced Planck constant, respectively.

The optical evolution equation Eq. (1) is coupled to the carrier transport via the quantities *g* and *n*_{sp}, which are self-consistently computed in the EMC simulation. There, the inter- and intrasubband scattering events are stochastically evaluated for a large ensemble of discrete particles. Each carrier *n* is at a given time described by its quantum state |*i _{n}*,

**k**

*〉, i.e., subband*

_{n}*i*and in-plane wave vector

_{n}**k**

*. The net gain*

_{n}*g*=

*g*

^{+}–

*g*

^{−}is then evaluated by summing over all available final states |

*j*,

**k**

*〉 and performing an ensemble average over all carriers*

_{n}*n*[11], with

*g*

^{+}and

*g*

^{−}are the material gain and loss in the gain medium due to stimulated emission and absorption, respectively.

*Z*

_{0}denotes the impedance of free space, and

*V*is the simulated volume. Moreover,

*E*

_{inj}=

*E*–

_{in}*E*=

_{j}*ħω*

_{inj}corresponds to the energy difference between the initial and final state,

*d*

_{inj}denotes the optical transition matrix element,

*γ*

_{inj}, and

*f*

_{jkn}is the occupation probability of state |

*j*,

**k**

*〉. The spontaneous emission rate into an optical mode (*

_{n}*c*/

*n*

_{0}) Γ

*gn*

_{sp}equals the stimulated emission rate per one photon in this mode (

*c*/

*n*

_{0}) Γ

*g*

^{+}[3, 15]. Thus we obtain

*n*

_{sp}=

*g*

^{+}(

*ω*

_{L})

*/g*(

*ω*

_{L}), allowing us to extract

*n*

_{sp}directly from the EMC simulation via Eq. (2).

The back-action of the optical cavity field on the carrier transport is considered by adding scattering rates for photon emission and absorption to EMC. The optical intensity in a gain medium with volume *V*_{g} is obtained from the photon number *N* in Eq. (1) by

*N*= 1 in Eq. (4). Thus, the spontaneous transition rate for an electron in a state |

*i*,

**k**〉 to a state |

*j*,

**k**〉 follows from the stimulated electron-photon scattering rate given in [11] as

*E*> 0, and ${r}_{i\to j}^{\text{sp}}=0$ otherwise. To account for the total spontaneous emission, we sum in Eq. (5) over all cavity modes, characterized by their frequencies

_{ij}*ω*. However, for the investigated QCLs, ${r}_{i\to j}^{\text{sp}}$ is far too low to contribute significantly to the carrier transport.

_{m}## 3. Intrinsic linewidth analysis

The intrinsic linewidth is given by the modified Schawlow-Townes formula [3, 15]

*|*

_{t}N_{sp}= Γ

*gn*

_{sp}

*c/n*

_{0}(see Eq. (1)) and substituting the gain-loss balance relation

*γ*≈ Γ

*gc/n*

_{0}, where

*γ*=

*ac/n*

_{0}is the photon decay rate. The excess spontaneous emission factor

*K*is a correction factor for non-orthogonal cavity modes, and the linewidth enhancement factor

*α*quantifies the strength of amplitude-phase coupling in the gain medium. As pointed out in [3], blackbody radiation, i.e., thermal photons, play a role in the THz regime, where the thermal photon number

*n*

_{th}reaches unity around or even below room temperature. The inclusion of thermal photons leads to a further modified linewidth formula, which is for

*K*= 1 given by [16] It should be mentioned that in [3], a somewhat different result is obtained,

#### 3.1. Simulation results for *3THz* QCL

In the following, simulation results are presented for a 3THz resonant phonon design [17]. The gain medium cross section and cavity length are *A* = 10*μ*m × 23*μ*m and *L* = 1.22mm, respectively; furthermore, Γ = 0.93, *a*_{w} = 18.7cm^{−1}, and *a*_{m} = 1.3cm^{−1} [17]. The bias is set to 10.9kV/cm, where the simulation yields maximum output power.

The excess spontaneous emission factor can be written as *K* = *K*_{P}*K*_{H}, where *K*_{P} and *K*_{H} account for the non-orthogonality of transverse and longitudinal modes due to gain guiding and mirror outcoupling, respectively [15]. *K*_{P} has been determined using a mode solver based on the effective-index method and the Drude model parameters given in [18], yielding *K*_{P} = 1 in very good approximation. *K*_{H} can be obtained from the facet reflectivities *R*_{1,2} as [15]

*K*

_{H}= 1.0022 for the investigated design (

*R*

_{1}=

*R*

_{2}= 0.85 [17]). Thus we can set

*K*≈ 1.

Next, the linewidth enhancement factor *α* is computed at *ω* = *ω*_{L} using [16, 19]

*χ*(

*ω*) = −i

*n*

_{0}

*cg*

_{c}(

*ω*)/

*ω*is the contribution of the intersubband transitions to the gain medium susceptibility $\chi +{n}_{0}^{2}-1$. The complex gain

*g*

_{c}is evaluated based on Eq. (2), replacing the lineshape function ℒ by the full complex expression in Eq. (3). Simulations are performed at slightly varied cavity losses, yielding

*g*

_{c}for different photon numbers

*N*. Various lattice temperatures

*T*

_{L}between 10 and 170K are considered. In Fig. 1(a),

*χ*is shown for different values of

*N*at

*T*

_{L}= 100K. Eq. (10) is evaluated at the position of the peak gain, i.e., the frequency

*ω*

_{L}where ℑ{

*χ*} reaches its minimum. For all simulated temperatures, we obtain

*α*≈ 0.5, which agrees with the only experimental value available for THz QCLs [20].

In the following, the temperature dependence of the intrinsic linewidth *δ _{f}* is investigated for the 3THz design, using

*K*= 1 and

*α*= 0.5. In Fig. 1(b),

*δ*is shown, as obtained without considering thermal photons (Eq. (6)) and with thermal photons included (Eqs. (7), (8)). At elevated temperatures, the laser operation is impaired, and the photon number

_{f}*N*decreases strongly from 1.52 × 10

^{9}(10K) to 3.08 × 10

^{8}(170K), while

*n*

_{sp}only changes slightly from 3.94 to 3.55, and

*γ*and

*α*are constant in good approximation. According to Eq. (6), this results in an increase of

*δ*from 40.5Hz to 180.6Hz (solid curve). Thermal photons cause additional broadening of 18% at 170K according to Eq. (7) (dashed curve), and even 80% according to Eq. (8) (dotted curve). At this point, it would be desirable to clarify which of the two models Eqs. (7), (8) is valid for THz QCLs; however, this goes beyond the scope of the present paper.

_{f}As discussed in [3], no comprehensive experimental results are available for the intrinsic linewidth of THz QCLs. However, recently measurements have been performed for a 4.33 *μ*m design, yielding *δ _{f}* ≈ 500Hz [4]. This value is consistent with the rate equation approach given in [3]. Thus we compare our results to that model, yielding
${\delta}_{f}={(4\pi )}^{-1}\gamma \beta {\tau}_{\text{t}}{\tau}_{\text{r}}^{-1}{({I}_{0}/{I}_{\text{th}}-1)}^{-1}$ for THz QCLs at moderate electron temperatures up to

*T*

_{e}≈ 100K [3]. Here,

*K*= 1 and

*α*= 0 is assumed, and thermal photons are neglected. Estimating an above threshold current

*I*

_{0}

*/I*

_{th}= 1.2, a photon decay rate

*γ*= 2 × 10

^{11}s

^{−1}, a coupling efficiency

*β*= 10

^{−2}and upper laser level lifetimes

*τ*

_{r}= 10

*μ*s and

*τ*

_{t}= 10ps due to spontaneous emission and non-lasing transitions, respectively,

*δ*= 800Hz is obtained for a 3THz QCL [3]. We compare this value to our EMC simulation at

_{f}*T*

_{L}= 10K (corresponding to

*T*

_{e}≈ 100K), yielding a much smaller value

*δ*= 32.4Hz for our investigated 3THz design. However, using parameter values directly extracted from this simulation (

_{f}*I*

_{0}/

*I*

_{th}= 2.5,

*τ*

_{t}= 1.83ps, $\beta {\tau}_{\text{r}}^{-1}=1950{\text{s}}^{-1}$,

*γ*=

*ac/n*

_{0}= 1.58 ×10

^{11}s

^{−1}), above formula yields

*δ*= 30Hz in excellent agreement with our numerical result.

_{f}#### 3.2. Simulation results for high power QCLs

As evident from Eq. (6), the intrinsic linewidth *δ _{f}* can be reduced by increasing the photon number

*N*, i.e., the generated optical power. This can be achieved by designing the waveguide and active region correspondingly [1, 2]. For example,

*N*can be directly enhanced by increasing the waveguide width

*w*, without affecting

*γ*and

*n*

_{sp}much. Besides, QCLs work best at low temperatures, also eliminating the influence of thermal photons (

*n*

_{th}≈ 0). In the following,

*δ*is investigated for two high power QCLs at

_{f}*T*

_{L}= 10K, providing a lower estimate for the obtainable intrinsic linewidth of current THz QCL designs. Both structures are based on the same active region design, but use different types of waveguides with comparable widths

*w*≈ 100

*μ*m, lengths

*L*≈ 2mm, and active region thicknesses

*d*≈ 10

*μ*m [1, 2]. The cavity parameters have been determined at 4.5THz using a mode solver based on the effective-index method, along with the Drude model parameters and facet reflectivities

*R*given in [18]. Assuming a coated rear facet, the extracted values for the surface plasmon (SP) waveguide [1] are

*R*= 0.32,

*a*

_{m}= 2.8cm

^{−1},

*a*

_{w}= 3.7cm

^{−1}, Γ = 0.27, and for the metal-metal (MM) waveguide [2]

*R*= 0.7,

*a*

_{m}= 0.9cm

^{−1},

*a*

_{w}= 11.5cm

^{−1}, Γ = 1.0. Furthermore, we obtain

*K*

_{P}= 1.0 for both structures, and Eq. (9) yields

*K*

_{H}= 1.11 and

*K*

_{H}= 1.01 for the SP- and MM-based QCL, respectively. We simulate

*δ*and the linewidth enhancement factor

_{f}*α*as described in Section 3.1. We obtain again

*α*≈ 0.5 for both devices, furthermore

*δ*= 3.0Hz for the MM structure and even

_{f}*δ*= 0.64Hz for the SP structure, as compared to

_{f}*δ*= 40.5Hz for the 3THz MM QCL (see Fig. 1(b)). The ∼ 7 times higher gain medium volume of the high power QCLs results in high photon numbers

_{f}*N*= 3.13 × 10

^{10}and 2.07 × 10

^{10}for the SP- and MM-based design, as compared to

*N*= 1.52 × 10

^{9}for the 3THz QCL. Furthermore, the low photon decay rate

*γ*= 5.1×10

^{10}s

^{−1}of the SP-based device contributes directly to its extremely narrow linewidth (see Eqs. (6)–(8)), and also indirectly through the increased photon number. We note that there are uncertainties associated with determining

*a*

_{w}[17], affecting the exact value of

*δ*. However, our results clearly indicate that intrinsic linewidths on the order of Hz or even below can be obtained for high power THz QCLs.

_{f}## 4. Conclusion

Based on coupled carrier transport and optical simulations, a method is developed to self-consistently investigate optical fluctuations due to spontaneous emission and thermal photons in QCLs. The intrinsic linewidth in resonant phonon THz designs is analyzed, yielding values on the order of Hz or even below for high power structures. At elevated temperatures, thermal photons are found to contribute significantly to linewidth broadening. For the investigated designs, a linewidth enhancement factor of around *α* ≈ 0.5 is extracted.

## Acknowledgments

We acknowledge support from P. Lugli at the TUM. This work was funded by the Emmy Noether program of the German Research Foundation under Grant No. DFG, JI115/1-1.

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