Monte Carlo study of intrinsic linewidths in terahertz quantum cascade lasers

Christian Jirauschek

doi:10.1364/OE.18.025922

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 μm 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

(n_{0} / c) \partial_{t} N = (Γ g - a) N + Γ g n_{sp} + {a n}_{th},

where 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, n₀, 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 = 2πf = ω_L, and c is the vacuum speed of light. The term (Γg – a)N describes amplification in the gain medium and cavity damping, Γgn_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_BT)) – 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_n〉, i.e., subband i_n and in-plane wave vector k_n. The net gain g = g⁺ – g⁻ is then evaluated by summing over all available final states |j, k_n〉 and performing an ensemble average over all carriers n [11], with

g^{\pm} (ω) = \frac{π Z_{0} ω}{V n_{0} ħ} \sum_{\underset{\pm E_{i_{n} j} > 0}{n, j}} (1 - f_{j k_{n}}) | d_{i_{n} j} |^{2} ℒ_{i_{n} j} (ω) .

Here, g⁺ and g⁻ are the material gain and loss in the gain medium due to stimulated emission and absorption, respectively. Z₀ denotes the impedance of free space, and V is the simulated volume. Moreover, E_{i_nj} = E_{i_n} – E_j = ħω_{i_nj} corresponds to the energy difference between the initial and final state, d_{i_nj} denotes the optical transition matrix element,

ℒ_{i_{n} j} (ω) = ℜ {\frac{1}{π} \frac{1}{ω - | ω_{i_{n} j} | + i γ_{i_{n} j}}} = \frac{1}{π} \frac{γ_{i_{n} j}}{γ_{i_{n} j}^{2} + {(ω - | ω_{i_{n} j} |)}^{2}}

is the Lorentzian lineshape function with linewidth γ_{i_nj}, and f_{jk_n} is the occupation probability of state |j, k_n〉. The spontaneous emission rate into an optical mode (c/n₀) Γgn_sp equals the stimulated emission rate per one photon in this mode (c/n₀) Γg⁺ [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

I_{N} (ω) = (Γ N / V_{g}) ħ ω c / n_{0} .

As mentioned above, the spontaneous emission rate equals the stimulated emission rate for a single photon, corresponding to 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

r_{i \to j}^{sp} = \frac{π Z_{0}}{n_{0} ħ^{2}} | d_{i j} |^{2} (1 - f_{j k}) \sum_{m} I_{N = 1} (ω_{m}) ℒ_{i j} (ω_{m})

for E_ij > 0, and

r_{i \to j}^{sp} = 0

otherwise. To account for the total spontaneous emission, we sum in Eq. (5) over all cavity modes, characterized by their frequencies ω_m. However, for the investigated QCLs,

r_{i \to j}^{sp}

is far too low to contribute significantly to the carrier transport.

3. Intrinsic linewidth analysis

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

δ_{f} = {(4 π)}^{- 1} K (1 + α^{2}) \partial_{t} N |_{sp} N^{- 1} = {(4 π)}^{- 1} K (1 + α^{2}) γ n_{sp} N^{- 1},

valid for single-mode lasing. The second expression is obtained from the first one by inserting the spontaneous emission rate into the lasing mode ∂_tN|_sp = Γgn_spc/n₀ (see Eq. (1)) and substituting the gain-loss balance relation γ ≈ Γgc/n₀, where γ = ac/n₀ 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]

δ_{f} = {(4 π)}^{- 1} (1 + α^{2}) γ (n_{sp} + n_{th}) N^{- 1} .

It should be mentioned that in [3], a somewhat different result is obtained,

δ_{f} = {(4 π)}^{- 1} (1 + α^{2}) γ [n_{sp} + (1 + n_{sp}) n_{th}] N^{- 1} .

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 = 10μm × 23μm 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_PK_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} = {[\frac{(\sqrt{R_{1}} + \sqrt{R_{2}}) (1 - \sqrt{R_{1} R_{2}})}{\sqrt{R_{1} R_{2}} ln (R_{1} R_{2})}]}^{2},

giving K_H = 1.0022 for the investigated design (R₁ = R₂ = 0.85 [17]). Thus we can set K ≈ 1.

Next, the linewidth enhancement factor α is computed at ω = ω_L using [16, 19]

α = \frac{\partial_{N} ℜ {χ (ω_{L})}}{\partial_{N} ℑ {χ (ω_{L})}} = - \frac{\partial_{N} ℑ {g_{c} (ω_{L})}}{\partial_{N} ℜ {g_{c} (ω_{L})}} .

Here, χ(ω) = −in₀cg_c (ω)/ω is the contribution of the intersubband transitions to the gain medium susceptibility

χ + 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].

Fig. 1 (a) Susceptibility χ vs frequency f = ω/(2π) for internal powers of 145, 160 and 175mW, corresponding to N = 1.01 × 10⁹, 1.11 × 10⁹, and 1.21 × 10⁹, respectively. (b) Intrinsic linewidth as a function of temperature without and with thermal photons included.

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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_f 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 N decreases strongly from 1.52 × 10⁹ (10K) to 3.08 × 10⁸ (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_f 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.

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 μm 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 $δ_{f} = {(4 π)}^{- 1} γ β τ_{t} τ_{r}^{- 1} {(I_{0} / I_{th} - 1)}^{- 1}$ 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 I₀/I_th = 1.2, a photon decay rate γ = 2 × 10¹¹ s⁻¹, a coupling efficiency β = 10⁻² and upper laser level lifetimes τ_r = 10μs and τ_t = 10ps due to spontaneous emission and non-lasing transitions, respectively, δ_f = 800Hz is obtained for a 3THz QCL [3]. We compare this value to our EMC simulation at T_L = 10K (corresponding to T_e ≈ 100K), yielding a much smaller value δ_f = 32.4Hz for our investigated 3THz design. However, using parameter values directly extracted from this simulation (I₀/I_th = 2.5, τ_t = 1.83ps, $β τ_{r}^{- 1} = 1950 s^{- 1}$ , γ = ac/n₀ = 1.58 ×10¹¹ s⁻¹), above formula yields δ_f = 30Hz in excellent agreement with our numerical result.

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_f is investigated for two high power QCLs at 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 ≈ 100μm, lengths L ≈ 2mm, and active region thicknesses d ≈ 10μm [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_f and the linewidth enhancement factor α as described in Section 3.1. We obtain again α ≈ 0.5 for both devices, furthermore δ_f = 3.0Hz for the MM structure and even δ_f = 0.64Hz for the SP structure, as compared to δ_f = 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 N = 3.13 × 10¹⁰ and 2.07 × 10¹⁰ for the SP- and MM-based design, as compared to N = 1.52 × 10⁹ for the 3THz QCL. Furthermore, the low photon decay rate γ = 5.1×10¹⁰ 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_f. However, our results clearly indicate that intrinsic linewidths on the order of Hz or even below can be obtained for high power THz QCLs.

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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Monte Carlo study of intrinsic linewidths in terahertz quantum cascade lasers

Abstract

1. Introduction

2. Method

3. Intrinsic linewidth analysis

3.1. Simulation results for 3THz QCL

3.2. Simulation results for high power QCLs

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (1)

Equations (10)

Optics Express