1. Introduction

Quantum cascade lasers (QCLs) are unipolar light sources emitting coherent radiation in the mid-infrared to THz spectral range [1]. They show very remarkable and special features like large tuning-ranges [2–5] or optical frequency-comb operation regimes [6–9]. Another very peculiar characteristic of QCLs is their capability for fast modulation experiments up to GHz-frequencies because of the ultrashort (subpicosecond) lifetime of the upper state combined with the very short stimulated emission time. The same active region dynamics is also responsible for the absence of any resonance-oscillation peak in electrical modulation experiments [10], in strong contrast to near-IR interband diode lasers [11].

GHz-modulation of mid-IR QCLs is of particular interest for different applications, e.g chirped laser dispersion spectroscopy (CLaDS) [12, 13]. It is a baseline-free trace gas detection scheme, which is immune to source intensity-fluctuations. Moreover an intrinsic linear response to the sample concentration is obtained, for laser sources operated in the single sideband regime (SSB). In this specific case one sideband is suppressed by more than 30 dB. To scan broad spectral lines in the mid-infrared range high modulation frequencies are needed which is also beneficial for other applications like high bit-rate optical free-space communication [14, 15] and active device mode-locking [16, 17].

Previous studies either analyzed the modulation response of mid-IR QCLs only at liquid-nitrogen temperatures: gain-switching experiments with picosecond pulses [18], direct modulation measurements of the optical response up to 10 GHz [10] or electrical response measurements of microwave-cavity optimized double-metal waveguide devices up to 14 GHz [19]. Room-temperature experiments were also performed, but limited to low modulation frequencies such as the heterodyne-beating experiment of distributed-feedback (DFB-) QCLs by Aellen et al. [20] (∼1 GHz), or the direct modulation measurements up to 1.7 GHz by Hangauer et al. [21].

In this study we overcome previous limitations and present the electrical and optical modulation-response of Peltier-cooled mid-IR DFB-QCLs in continuous-wave operation up to modulation frequencies of 26.5 GHz, i.e. the cut-off frequency of the setup. The devices rely on a previously presented special waveguide-design for low electrical dissipation operation [22], leading to reduced capacitance values which are needed for high modulation bandwidths.

The device’ electrical and optical response under different driving conditions is measured, analyzed and modeled. In addition the observation of a particular resonance peak in one sideband of the optical response of the DFB-QCL upon modulating at a frequency close to the round-trip frequency of the laser-cavity, will be discussed and is shown to be consistent with the resonant excitation of a sidemode below threshold. Design-guidelines for DFB-QCLs on how this resonance can be avoided, or exploited to lead to the SSB regime, are given.

2. Device design and electrical characterization

The devices used in this study are based on an In_0.62Ga_0.38As/In_0.4Al_0.6As 3-quantum well active region (AR) design grown at 515°C on InP substrate by MBE [23]. After fabrication of a buried-heterostructure (BH) configuration [22] including a 1^st order DFB-grating in the top InGaAs-layer for single-mode emission around 4.5 μm (cp. [22]), the topside of the devices is processed into a three terminal coplanar waveguide (CPW) configuration [24] for optimized RF-current injection. Figure 1(a) shows the schematized front-view of such a CPW-BH-QCL. Two ground planes run in parallel on each side of the 40 μm wide central contact at a fixed distance of w = 15 μm. For such a configuration the QCLs are etched on both sides of the top-contact down to the substrate (∼10 μm) and individual Au-contacts are deposited followed by a 4–5 μm thick electroplated Au-layer for better heat extraction and improved device robustness. Figure 1(b) shows the top-view image of a fabricated sample including all 3 terminals.

 

Fig. 1 (a) Front-view sketch of a typical CPW-BH-QCL. (b) Top-view of a processed device. (c) Three-terminal current-injection probe (z-probe [26]) for RF-frequencies up to 40 GHz.

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The advantages of the CPW-configuration compared to other RF-injection schemes like strip-or microstrip-lines [25], are the planar and all topside geometry and the resulting simple circuit integration. A commercially available high-frequency three terminal contacting probe (z-probe, Fig. 1(c), f_max = 40 GHz [26]) is used for current-injection from the circuit to the device. Consequently no wire-bonding is needed and additional inductive contributions possibly limiting the modulation bandwidth at high frequencies are avoided.

The low-capacitance BH devices [22] fabricated for this study have 3–5 μm wide waveguides and typical cleaved lengths are 1–2 mm. To lower the threshold, they include a high-reflectivity (HR−) back facet coating.

Two particular devices are investigated in detail, their dimensions are: 3 μm × 2 mm (+HR-coating) and 4 μm × 1 mm (+HR-coating).

The whole electrical driving circuit is shown in Fig. 2. An up to 26.5-GHz bias-T combines the output of a 43.5-GHz RF-generator (Rohde & Schwarz SMF 100A) and a low-noise DC laser-diode current-source (Wavelength Electronics, QCL2000). The combined signal is injected via a 50-Ω line (R_L) into the QCL device, modeled by two parallel RC-circuits in parallel. One represents the AR, where also the optical transition takes place (’AR’, red), and the other one the InP:Fe insulation layer of the BH-waveguide (’InP’, blue). To avoid strong attenuation of the RF signal, special high-frequency cables are used (Huber+Suhner Sucoflex 100).

 

Fig. 2 Experimental setup for RF-current-injection including a 26.5-GHz bias-T which combines DC- (current-source) and RF-current and injects it through a 50 Ω line (R_L) into the DFB-QCL (Z_QCL). In case of the electrical (rectification) measurement, which is described in the main text and in [27], a lock-in amplifier is added as shown.

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The response function of a QCL to a RF-modulation can be seen as the product of its electrical and optical response. The small signal optical transfer function h(ω) is here defined as the ratio of the modulation amplitude of the photon flux S̃ to the injection current density J̃, normalized by their steady-state ratio $S^0/J^0:\frac{\tilde{S}}{\tilde{J}}=h(\omega )\frac{S^0}{J^0}$ [28]. Following a rate equation approach, assuming vanishing lower lasing state lifetime and negligible thermal backfilling, the absolute value of h(ω) is given by: $\left|h(\omega )\right|=\frac{1}{\sqrt{1+{\omega }^4{\tau }_p^2{\tau }_{\mathit{st}}^2+{\omega }^2{\tau }_{\mathit{st}}{\tau }_p\left(\frac{{\tau }_p}{{\tau }_{\mathit{st}}}+2\frac{{\tau }_p}{{\tau }_{\mathit{up}}}+\frac{{\tau }_p{\tau }_{\mathit{st}}}{{\tau }_{\mathit{up}}^2}-2\right)}}$ with ω = 2πν (ν: modulation frequency) and the upper-state, photon-cavity and stimulated lifetime τ_up, τ_p and τ_st. It is their interplay that reflects the shape of the response-curve.

The applied current-density J to the QCL is subject to the electrical cut-off function c(ν) of the circuit which is typically given by a RC-type cut-off function: $c(\nu )=\frac{1}{1+{\left(2\pi \nu {\tau }_{\mathit{RC}}\right)}^2}$ time-constant of the circuit including R_AR,InP and C_AR,InP from Fig. 2).

To characterize the electrical device-properties, the fundamental modulation limit of the AR is distinguished first. Assuming no parasitic effects, i.e. the AR itself without considering the process geometry through R_InP and C_InP in Fig. 2, a RC-type of roll-off behavior of the AR yields a fundamental 3-dB cut-off limit defined through the normalized differential resistance $\mathrm{\Delta }F/\mathrm{\Delta }J:{\left(\mathit{RC}\right)}_{\mathit{AR}}=\frac{{\epsilon }_0{\epsilon }_{\mathit{AR}}\mathrm{\Delta }F}{\mathrm{\Delta }J}\to {\text{f}}_{3\mathit{dB}}=\frac{1}{2\pi {\left(\mathit{RC}\right)}_{\mathit{AR}}}=53\hspace{0.17em}\text{GHz}$ [28]. In this expression ε₀ is the vacuum-permittivity, ε_AR the active region dielectric constant, ΔF the range of applied electric field and DeltaJ the current density range above threshold.

Using the values corresponding to the devices used in this study, regardless of the 3-dB cutoff of h(ω), a fundamental limit will be hit at 53 GHz for this AR-design due to pure electrical considerations.

Including the BH geometry shown in Fig. 1a, we estimate a total capacitance $C_{\mathit{tot}}=C_{\mathit{AR}}+C_{\mathit{InP}:\mathit{Fe}}=\frac{{\epsilon }_0{\epsilon }_s\cdot A}{d}$ (ε_s: sheet-permittivity (i.e. AR or InP:Fe), A: pumped area (QCL-AR or InP:Fe), d: thickness of the layer (AR or InP:Fe)) of C_tot1 = 3.2 pF (3 μm × 2 mm) and C_tot2 = 1.6 pF (4 μm × 1 mm), for the two different devices.

Such a simplified model that assumes a constant active region capacitance implies that the charges are completely fixed in the active region. In fact, a change in the applied bias is expected to redistribute the electrons across the various energy states. This redistribution of the electrons with respect to the ionized impurities will change the partial screening of the applied field inside the AR. This effect is especially strong if the dopants are set back from the ground state of the injector region. In a full computation, the electron distribution would take into account the change with the applied field of the populations of all energy levels, including the effects of electron injection and the photon field. In this article, we used a model in which the populations are computed assuming a thermal distribution inside each active region. We estimate that this captures correctly between 80–90% of the electron population, depending on the applied bias. As an example, Fig. 3(a) shows the potential drop along one AR-period calculated with a self-consistent Schrödinger Poisson (SP) solver at an external field of 40 kV/cm. The approximate location of negative and positive charges are indicated as well as the dipole L_S. As described in more details in Appendix A, we used the results of the self-consistent SP-solver to compute the change in capacitance as a function of applied field.

 

Fig. 3 (a) Electrical potential along one AR-period at an applied external field of 40 kV/cm calculated by the self-consistent Schrödinger Poisson solver. Approximate location of negative and positive charges are indicated as well as the dipole L_S. (b) Capacitance versus DC-current for the calculated AR-model (red) and the fitted rectification model (black). Threshold-current I_th and the rollover current I_rollover of the structure are indicated.

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To the value extracted from this self-consistent model, the parasitic capacitance from the InP:Fe is added. A bias-field dependent modified capacitance C_self as shown by the red curve in Fig. 3(b) is obtained, which displays the calculated capacitance as function of applied DC-current to the QCL. The maximum capacitance for a typical 3 μm × 2 mm device, obtained at the lowest field of 47 kV/cm (at 42 mA), is C_self,42mA ∼ 4 pF and it decreases linearly to C_self,632mA = 3.5 pF at 94 kV/cm (at 632 mA). Finally the bias-field dependent 3-dB cut-off frequencies can be calculated using these capacitance values together with the corresponding device differential resistance at the two operational points, i.e. R_42mA = 50.4 Ω and R_632mA = 9.2 Ω, extracted from the devices’ current-voltage characteristics. The results are: f_3dB,42mA = 0.8 GHz and f_3dB,632mA = 4.9 GHz.

The same model can be used to distinguish design guidelines for optimizing the QCL-AR in terms of high RF-bandwidth operation, i.e. low capacitance values. As shown in Eq. (12), increasing the number of periods and the period lengths as well as keeping the doping to a moderate level, reduces C_self. It is also beneficial to minimize L_S or at least its dependence on voltage. But the RF-bandwidth f_3dB is also a function of the resistance of the device: $f_{3\mathit{dB}}=\frac{1}{2\pi \mathit{RC}}$, which will in general increase with an increased N_P as well as a reduction in the doping n_S and counter-balances the positive effects of the lowered capacitance.

Because of it enables to measure devices with a poor impedance matching, a microwave rectification technique was used to measure the electrical response characteristics, as already reported in [19,27,29]. In summary: a small signal analysis of the modulated bias-voltage V(I(t)) yields a DC rectification-voltage V_rect which can be measured on a lock-in amplifier [27] (see Fig. 2). The derivation of V_rect for our experimental setup is carried-out in Appendix B. Its final expression which is used to model the experimental results later on is given by:

The parameters are defined as: |V″| is the absolute value of the second derivative of the voltage-current characteristic of the QCL, P_RF stands for the RF-power of the RF-generator, R_L is the line-resistance of 50 Ω, R,C_QCL = R_AR + R_InP, C_AR + C_InP are: total device resistance/capacitance including AR and InP:Fe) and ω is the angular modulation-frequency.

To measure the DC-rectification signal with a lock-in amplifier, a 50-kHz amplitude-modulation (AM) signal is added to the RF-current. The frequency of this additional modulation is high enough to avoid significant heating-effects. To attenuate electrical resonances coming from the circuit as much as possible, the lock-in amplifier is placed in the DC-arm of the setup (see Fig. 2) with the inductance of the bias-T acting as a filter for high frequencies (GHz-range). Since still a significant rectification signal could be measured on the lock-in amplifier, the attenuation from the high-frequency optimized bias-T at low frequencies, is minor. Also the resistance of the DC-source at 50 kHz plays a minor role in the experimental results.

Figure 4(a) shows the normalized electrical rectification curves for a typical CPW-DFB-QCL (3 μm × 2 mm +HR-coating) measured at widespread driving-currents ranging from below (42 mA to 200 mA) to above (400 mA to 632 mA) and at lasing threshold I_th = 320 mA, for a fixed RF-injection level of 18 dBm. At 42 mA a roll-off of −18.6 db/decade is extracted which is in good agreement with the expected −20db/decade roll-off of a typical second order system (RC-circuit). Further increase in modulation bandwidth with increasing DC-current is observed. This trend follows the above calculations of the capacitance C_self. It is also in good agreement with literature [19], where similar results were obtained above lasing-threshold. Our measurements show, that the same trend is also observed below J_th.

 

Fig. 4 (a) Normalized electrical rectification data of a typical 3 μm × 2 mm device (+HR-coating) at different bias-currents between I_DC = 42 mA (below lasing threshold I_th = 320 mA) and 632 mA (= I_max). The roll-off is −18.6 dB/decade at I_DC = 42 mA. (b) Normalized electrical rectification signals at 42 mA and 632 mA (dots) fit by the V_rect-model (dashed lines) with very good agreement between simulation and experiment.

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In Fig. 4(b) the direct comparison of the 42 mA and the 632 mA measurement (dots) is shown. When fitting the measurements with the theoretical curves (dashed lines) from the rectification-model, the corresponding capacitance values can be extracted (see Eq. (1)). Very good agreement with the experimental data is obtained. The used parameters for the two fits in Fig. 4(b) as defined in Eq. (1) are listed in Table 1.

Table 1. Parameters for the rectification measurement fits in Fig. 4(b) for the 3 μm × 2 mm at 42 mA and 632 mA.

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Even though the lock-in measures the rectification signal in the DC-arm of the electrical circuit, electrical resonances are observed in the experimental data in Fig. 4(a) and (b), comparable to what was observed by Calvar et al. [19]. They correspond to waves on the centimeter-scale and are coming from reflections in the electrical circuit and are not inherent to the device. The extracted capacitance-values from the different driving-currents in Fig. 4(a) can be found in Fig. 3(b) as function of the applied DC-current. They are compared to the calculated values from the AR-model (SP-solver). The overestimated capacitance values of the AR-model (red trace) compared to the extracted values from the actual measurements with the rectification model (black trace), are attributed to the limitations of our model. It takes only into account the thermal population of electrons in the AR but no effects of electrical transport in the QC structure or of the optical field. The latter show a continuous decrease of capacitance with increasing drive-current. Above I_th the values of the V_rect-model are behaving similarly but with a stronger decrease. Below lasing threshold the capacitance values from the rectification-model show no consistent behavior and more detailed investigations are needed, which are beyond the scope of this paper. From Fig. 4(a) a maximum 3-dB cut-off value of f_3dB,632mA = 6.6 GHz at I_DC = 632 mA is obtained, by using the extracted corresponding resistance value as differential resistance from the measured voltage-current characteristics. The 3-dB cut-off value is in reasonable agreement with the predicted 4.9 GHz from the AR-model.

3. Optical characterization

In order to analyze the optical response function of the CPW-DFB-QCLs, two different experiments are performed as shown in Fig. 5: i) a FTIR-measurement (Bruker Vertex80, maximum resolution: 0.075 cm⁻¹ = 2.25 GHz) on a liquid-nitrogen cooled mercury-cadmium-telluride (MCT) photodetector, see Fig. 5(a), and ii) a heterodyne beating experiment as shown in Fig. 5(b). Both rely on the inherent single-mode emission of the devices. While the first one directly measures the spectral evolution of the RF-generated sidebands, the latter combines the test laser (RF-modulated, DFB-QCL that is investigated) with a reference laser (DC only) using a beam-splitter and analyzes their beating signal on a fast room-temperature MCT-detector (Vigo SA, Peml 3) by a spectrum analyzer (SA) (Agilent E4402B, 38 dB pre-amplifier). For this purpose, the beating-frequency, i.e. the spectral difference between reference laser emission and the investigated sideband/carrier signal of the test laser, has to lie within the resolution-bandwidth of the detection system (MCT, SA, pre-amplifier), which is 3 GHz in this experiment. Consequently, by changing the Peltier-temperature and the injected DC-current, the reference laser is spectrally tuned closer than 3 GHz to the RF-sidebands/carrier of the test laser and their beating signal can be observed on the SA. In contrast to the FTIR-measurement, no absolute spectral position is measured with this technique for which a referencing of the frequency would be needed. A single peak is obtained instead on the SA, representing the spectral mismatch between reference laser and the respective sideband/carrier of the test laser. This measurement is performed for both sidebands as well as the carrier signal at each RF-modulation frequency and further analyzed later on.

 

Fig. 5 Experimental setup for: (a) FTIR- and (b) heterodyne-experiment.

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To avoid optical feedback into the QCLs, the detector is slightly tilted. The laser beams are overlapped along the entire beam-path of 60 cm after the beamsplitter, for a maximum beating-signal. The modulation-measurements are performed for two device-geometries: 4 μm × 1 mm +HR-coating and the 3 μm × 2 mm +HR-coating device from the electrical measurements. The reference device for the heterodyne experiment is a 4 μm × 1.5 mm +HR-coating device from the same fabrication process. It is driven at variable DC-currents and temperatures to tune its spectral emission close to the test laser. The DC-current of the latter is fixed and only the RF-modulation frequency ν is varied. To avoid strong fluctuations of the beating signal, both devices are temperature-stabilized (typically between 245K and 250K) and driven by individual, low-noise QCL-drivers (Wavelength Electronics, test laser: QCL2000, reference laser: QCL1000). All heterodyne data is obtained at a fixed RF injection level of 18 dBm which is the maximum value for a stable beating-signal. Higher values lead to strong fluctuations in the signal. The reference laser is driven with a CW optical output-power of up to 12 mW (measured directly in front of the device). The measurements of the FTIR-spectra are performed at varying injection levels between 17 dBm and 24 dBm, always keeping the highest RF-power that can be supplied by the RF source.

Those two types of experiments, i.e. the FTIR- and the heterodyne beating measurement, have been performed to compare their deviation when measuring the same quantities but also since they complement each other. While the FTIR-experiment allows the direct spectral observation of the sideband-evolution under RF-modulation conditions without additional referencing, it is limited by the maximum resolution of the FTIR (2.25 GHz). Even though the heterodyne experiment demands for a higher degree of alignment of the setup as well as the spectral emission of the reference laser with respect to the test laser, the minimum RF-modulation frequency is given by the experimental linewidth of the reference laser and the sideband/carrier of the test laser and the minimum observable limit of the SA, which typically results in significantly lower values than for the FTIR-measurement. In addition, smaller signals can in principle be measured with a well temperature-stabilized beating-experiment, since the small intensities of the measured sidebands (especially at high RF-frequencies), are mixed with the strong intensity signal of the reference laser beam. This allows e.g. the direct measurement of the linewidth enhancement factor α as it was shown by Aellen et al. [20].

From both experiments, the relative sideband-ratio with respect to the carrier-signal is extracted by normalizing to the carrier-peaks for each individual measurement condition and shown in Fig. 6(c) and (d) as function of the RF-frequency for the 4 μm × 1 mm +HR-coating and the 3 μm × 2 mm +HR-coating device, respectively. The green open data-points correspond to the FTIR-measurements and the blue filled points to the heterodyne-experiment. Diamonds stand for the −1^st and squares for the +1^st order of the sidebands. Details on the theory of the sideband-generation including their derivation and relation to intensity and frequency modulation (IM and FM) of the devices can be found in [21].

 

Fig. 6 (a) Electroluminescence of the 4 μm × 1 mm device with a 2.14 cm⁻¹ wide stopband on the high-energy side of the DFB-mode. The next lower-energy mode is spaced by ∼40 GHz. (b) Electroluminescence of the 3 μm × 2 mm device (stopband-width: 1.87 cm⁻¹, high-energy side of lasing DFB-mode). The next lower-energy mode is spaced by about 20 GHz. (c) Comparison of the measured relative sideband-intensity (FTIR: green (open symbols), Heterodyne: blue (filled symbols); −1^st order: diamonds, +1^st order: squares), i.e. normalized to the respective carrier-signal, with the simulated curve of the −1^st order sideband in red for the 4 μm × 1 mm device. (d) Comparison of the measured relative sideband-intensity, with the simulated curve of the −1^st order sideband (red) for the 3 μm × 2 mm device. A strong resonance in the −1^st order is observed around 15 GHz.

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In Fig. 6(c), which shows the measurement at a DC-current of about 430 mA (= 1.7 × I_th) injected to the RF-device, a continuous reduction of the sideband intensity for both types of measurements and both orders of sidebands with increasing ν is observed. This reduction runs in parallel for all shown data, with higher signal-levels for the FTIR-data as well as higher observed maximum modulation frequencies. The intensity-offset is attributed to the fact, that the measurements are not performed at exactly the same driving conditions of the devices. And also, mainly thermal, fluctuations were relatively pronounced in the heterodyne experiment even when averaging 5 pulses and extracting the peak-value. Since the SA data was obtained on a dBm scale, those fluctuations are a mayor contribution to the offset observed in Fig. 6(c) and (d) and also to the variations of the signal within one measurement series.

Thermal fluctuations are also responsible for the ’variations’ in the FTIR-data at different modulation frequencies of the device. They lead to small variations in the measured relative intensities as observed. To overcome such thermally induced limitations, the devices will be put in a closed box in the future, where the temperature of the devices and the environment is stabilized more accurately as well as the temperature of the cooling water cycle.

Better thermal stabilization of the devices, their housing and also e.g. of the cooling water temperature will help to reduce those variations in the future.

For the 2 mm long device in Fig. 6(d) again a similar intensity-reduction is found in the +1^st order with increasing ν for both experiments. The −1^st order remains flat up to about 8 to 10 GHz. The main difference compared to the 1 mm device is a resonance peak around 15 GHz which follows in the −1^st order. It shows an offset between the heterodyne and the FTIR data of about 1.5 GHz, which is attributed to the different DC-driving currents in both experiments (I_DC,FTIR = 632 mA ≙ 2 × I_th, I_DC,testlaser = 506 mA ≙ 1.6 × I_th), because the experimentally reachable beating-range in the heterodyne experiment is limited. Such a difference leads to a shift of the peak position ν_peak, which could be verified by additional FTIR- and heterodyne-measurements at different drive-currents.

In general, the observed decay of the relative sideband-signals is for some driving conditions more pronounced than expected. This indicates that further thermal but also electrical (current injection, electrical circuit) optimizations of the devices and the setup will lead to a more extended (flat) response curve of the devices at GHz modulation-frequencies.

To understand the origin of the resonance-peak in Fig. 6(d), which in QCLs is not the result of damping oscillations in the modulation response [10], the sub-threshold spectra of the two devices are investigated. Figure 6(a) and (b) show those spectra for the 1 mm and the 2 mm long device, respectively. The lasing DFB-mode is indicated and the DFB-stopband is on the high-energy side for both devices. Its measured spectral width is Δν_S ∼ 64 GHz (1 mm device) and 56 GHz (2 mm device), respectively. On the low-energy side, the next mode is spaced by about Δν ∼ 40 GHz (Fig. 6(a)) and Δν ∼ 20 GHz (Fig. 6(b)). This leads to the assumption that the peak is a result from the coupling of the lasing DFB-mode to its neighboring (non-lasing) FP-mode, when modulating at frequency close to the roundtrip frequency of the cavity. The absence of a peak for the 1 mm device (f_RT = 40 GHz) and the observation of the peak on the opposite side of the stopband only, support this assumption.

To support the hypothesis, the response-function of the devices is theoretically modeled. Starting from the rate equation response-function h(ω) of the lasers, we extend it by a Maxwell-Bloch formalism [30, 31] to include the mode-coupling mechanism. This approach combines the population-densities evolution of upper and lower lasing state [30], with the RF current-modulation and a modal decomposition of the cavity field in a dispersed Fabry-Pérot-cavity [31]. We simplify this system and substitute the cavity-field decomposition by only two coupled modes, with the modal detuning, i.e. spacing between those two modes, as parameter.

The model computes the amplitude of the mode excited by the coupling through the obtained modulation term m(ν) (ν = ω/(2π)), which results in the following expression:

The total response function F_resp,tot(ν) for a DFB-QCL as function of the modulation frequency ν is finally expressed by including the previously defined rate equation response function h(ν) and cut-off behavior of the electrical circuit c(ν):

Table 2. Parameters for simulating F_resp,tot(ν) (3 μm × 2 mm and the 4 μm × 1 mm device).

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Their values are either design-/ typical device-parameters for the given geometry (α_m, α_wg, n, τ_p, R_L, τ_up and τ_stim, κ_n,n+1) or they are extracted from measurements of the device (ν₀, R, C_Vrect, τ_RC) or a reasonable estimation is performed to obtain them ( $g_0^n$, δ′, δ). This is validated by comparison of the simulation results with the experimental data (see Fig. 6(c) and (d)). Good agreement between the simulated curve of the −1^st order sideband in red (cp. with Eq. (3)) and the experimental data (green and blue points) is found in the case without resonance in Fig. 6(c). The reduction of sideband intensity is reproduced very well and the main discrepancy is a slight offset in the absolute values.

The parameters from modeling the non-resonance data are used as input for simulating the −1^st order data with resonance-peak (Fig. 6(d), red trace) and can be found together with the parameters for the resonance-device in Table 2 in Appendix C. The slight intensity-reduction of the −1^st order signal at modulation-frequencies below resonance is reproduced well, but with a stronger pronounced difference in the absolute values. The location, width and height of the peak are in good agreement between model and −1^st order sideband of the FTIR-measurement as well as the very strong decrease after the resonance.

Concerning the shape and position of the peak the main parameters are the detuning of the non-lasing FP-mode from lasing-threshold δ′ together with the photon-cavity lifetime τ_p. They define the ”sharpness”, i.e. bandwidth and height, of the resonance. The modal detuning ν₀, distinguished by the free spectral range of the cavity, sets the ”frequency-position” of the peak.

In the last part the observation of the SSB regime [21] is analyzed. It is characterized by only one of the two first-order sidebands being present in the emission of the QCL, while the second one is fully suppressed (SMSR >30 dB). This regime is observed for the first time in mid-IR QCLs. In Fig. 7 the FTIR-spectra for this regime at a DC-current of 632 mA and different modulation frequencies between 15 GHz and 21 GHz are shown for the 3 μm × 2 mm device. For better visibility, the carrier-frequency is offset to zero for all spectra.

 

Fig. 7 FTIR-spectra of the 3 μm × 2 mm device for RF-modulation frequencies between 15 GHz and 21 GHz and a DC-current of 632 mA. The SSB-regime can clearly be observed where no +1^st order sideband is observed for this particular device and driving-conditions.

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The −1^st order sidebands are clearly observed (vanished at 21 GHz for this driving conditions in the FTIR-signal, see also Fig. 6(d), green diamonds), while the +1^st order has vanished already at 15 GHz. This regime is enabled in such devices due to the resulting enhancement of the signal of one sideband only.

4. Conclusion

In conclusion the electrical and optical characterization of 4.5-μm emitting CPW-DFB-QCLs for modulation frequencies up to 26.5 GHz (electrical) and 23.5 GHz (optical) was shown. The electrical measurements in a microwave-rectification scheme reveal 3-dB cut-off values of up to 6.6 GHz and in general an increase of the cut-off bandwidth with increasing DC-current. The optical measurements by FTIR-spectroscopy and in a heterodyne-beating experiment show a resonance peak in the sideband on one side, which is the result of a mode-coupling between the lasing DFB- and its neighboring sub-threshold FP-mode. This could be shown by modeling the experimental data with a simplified two-mode Maxwell-Bloch formalism including the modal-detuning as a parameter. The result of the mode-coupling is the first experimental observation of the SSB-regime in mid-IR QCLs, due to the enhanced modulation-frequencies of one sideband only.

Appendices

A. Calculation of the capacitance of a QCL-AR

To calculate the capacitance of a QCL-AR under applied bias, we have to add the corrections due to the self-consistent field E_SC to the applied field F_ext in the active region of the QCL. The resulting effect of those individual field contributions is summarized schematically in Figure 8.

 

Fig. 8 (a)–(d) One QCL active-region period (AR- and injector-part) for different conditions of self-consistent (E_SC) and external field (F_ext): (a) E_sc = F_ext = 0, (b) E_sc ≠0, F_ext = 0, (c) E_sc = 0, F_ext ≠ 0, (d) E_sc = F_ext ≠ 0. (e) First active-region period after the ohmic contact: (top) band diagram, (middle) net charge distribution and (bottom) electrical potential, as function of geometrical position.

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To minimize the impurity broadening of the optical transition and of the injection resonance, the dopants are usually placed in the center of the injector region. For this reason, at zero applied bias, the electrons will mainly reside in the active region, and create a self-consistent dipole field with the ionized donors, as shown in Figure 8(e). As the bias is increased, the electrons are progressively transferred into the injector and the self-consistent field is reduced.

The used Schrödinger-Poisson (SP) solver assumes that each period of the active region is electrically neutral, and for this reason for any applied average field F_ext the local electric field E_edge should be identical at the beginning and the end of each period. The capacitance is then extracted from the computation of E_edge as a function of bias using the following argument:

As shown schematically in Figure 8(e), we consider the transition between the doped cladding and the first period of the AR. We can consider the cladding as being field-free, and therefore there will be an accumulation of charge ρedge of opposite signs at each side of the AR to satisfy Poissons equation at the transition:

(4)$${\rho }_{\mathit{edge}}={\epsilon }_0{\epsilon }_{\mathit{AR}}\cdot \left|E_{\mathit{edge}}\right|.$$

The change of the total charge Aρ_edge as a function of applied bias V_tot is the capacitance of the active region:

(5)$$C_{\mathit{self}}=\frac{\delta {\rho }_{\mathit{edge}}}{\delta V_{\mathit{tot}}}=\frac{{\epsilon }_0{\epsilon }_s\cdot A\cdot \delta E_{\mathit{edge}}}{\delta V_{\mathit{tot}}}$$

where A is the area of the active region.

The total voltage V_tot across the AR can be expressed by the average electrical Field E_avg across one entire period (i.e. the slope of the self-consistent potential along one period, because E = −∇ϕ) times the total length (= thickness d) of the quantum-structure: $V_{\mathit{tot}}=E_{\mathit{avg}}\cdot \underset{=d}{\underbrace{N_p\cdot L_p}}\Rightarrow \delta V_{\mathit{tot}}=\delta E_{\mathit{avg}}\cdot N_p\cdot L_p$ (N_p: number of periods, L_p: period-length):

(6)$$C_{\mathit{self}}=\frac{{\epsilon }_0{\epsilon }_s\cdot A}{N_p\cdot L_p}\cdot \frac{\delta E_{\mathit{edge}}}{\delta E_{\mathit{avg}}}=C_{\mathit{tot}}\cdot \frac{\delta E_{\mathit{edge}}}{\delta E_{\mathit{avg}}}$$

Equation 6 shows that C_self is a function of the ”intrinsic” capacitance $C_{\mathit{tot}}=\frac{{\epsilon }_0{\epsilon }_s\cdot A}{N_p\cdot L_p}$, multiplied by the ratio of the change of the edge-field and average field (along one entire period). Both values can be extracted from the change in self-consistent potential which is obtained from the SP-solver.

From the Schrödinger-Poisson solving finally a ratio δE_edge/δE_avg as function of external applied field showing a linear decrease (cp. Fig. 3(b), red curve) of −2.2 × 10⁻³ (kV/cm)⁻¹ is obtained for the used active region. The resulting capacitance-values are shown in Fig. 3(b).

The QCL-AR can also be evaluated in terms of design-guidelines for optimized RF-bandwidth, i.e. low capacitance. In this respect relevant parameters are the period-length L_P, the number of periods N_P and the ”length of the dipole”’ L_S, i.e. the effective distance between the center of mass of the negative and the positive charges.

To evaluate the AR and obtain a formula depending on L_P, N_P and L_S the following calculation is performed. For definition of the different parameters see Fig. 8(e).

(7)$$E_S-E_{\mathit{edge}}=\frac{-n_S}{{\epsilon }_0\cdot {\epsilon }_S}\cdot q$$

(8)$$\to E_S=E_{\mathit{edge}}-q{n}_S/({\epsilon }_0\cdot {\epsilon }_S)$$

The corresponding voltage drop V for a given electrical field E over a distance L is given by: V = N_P · E · L. To attribute for the cascading structure of a QCL, N_P has to be included:

(9)$$\Rightarrow \left(E_{\mathit{edge}}\underset{L_{\mathit{edge}}}{\underbrace{(L_P-L_S)}}+\underset{E_S}{\underbrace{\left(E_{\mathit{edge}}-\frac{q{n}_S}{{\epsilon }_0\cdot {\epsilon }_S}\right)}}\cdot L_S\right)=\frac{V_{\mathit{tot}}}{N_P}$$

(10)$$\frac{q{\rho }_{\mathit{edge}}}{{\epsilon }_0\cdot {\epsilon }_S}=E_{\mathit{edge}}=\frac{V_{\mathit{tot}}}{N_P\cdot L_P}+\frac{q{n}_S}{{\epsilon }_0\cdot {\epsilon }_S}\cdot \frac{L_S}{L_P}$$

The capacitance C_mod (per device area A) is then finally calculated to be:

(11)$$\frac{C_{\mathit{self}}}{A}=\frac{\partial q{\rho }_{\mathit{edge}}}{\partial V_{\mathit{tot}}}=\frac{\partial E_{\mathit{edge}}}{\partial V_{\mathit{tot}}}{\epsilon }_0\cdot {\epsilon }_S$$

(12)$$=\frac{{\epsilon }_0\cdot {\epsilon }_S}{L_P}\left(\frac{1}{N_P}+\frac{q{n}_S}{{\epsilon }_0\cdot {\epsilon }_S}\cdot \frac{\partial L_S}{\partial V_{\mathit{tot}}}\right)$$

Consequently it is beneficial to increase the number of periods N_P and the period-length L_P as well as use a moderate doping to lower n_S in order to reduce the capacitance of a QCL-AR. According to this criterion, it would also be beneficial to minimize L_S or at least its dependence on voltage.

But the RF-bandwidth f_3dB is also a function of the resistance of the device: $f_{3\mathit{dB}}=\frac{1}{2\pi \mathit{RC}}$, which will in general increase with an increased N_P as well as a reduction in the doping n_S and counter-balance the positive effects on the capacitance.

B. Rectification voltage of a buried-heterostructure QCL

To calculate the rectification voltage of a typical BH-QCL a small signal analysis of the total biasing voltage V(I(t)) is applied to the QCL as function of current I(t) around DC-current I₀:

(13)$$V(I(t))=V(I_0)+{\frac{\partial V}{\partial I}|}_{I_0}\left(I(t)-I_0\right)+\frac{1}{2}{\frac{{\partial }^{2}V}{\partial I^2}|}_{I_0}{\left(I(t)-I_0\right)}^2$$

(14)$$\text{with}:\hspace{0.17em}\hspace{0.17em}I(t)=I_0+\mathrm{\Delta }I\cdot \mathit{sin}(2\pi \cdot \nu t)$$

(15)$$\to V(I(t))=V_0+{V^{\prime }|}_{I_0}\mathrm{\Delta }I\cdot \mathit{sin}(2\pi \cdot \nu t)+\frac{1}{2}{V}_{I_0}^{″}{\left(\mathrm{\Delta }I\cdot \mathit{sin}(2\pi \cdot \nu t)\right)}^2$$

with the the DC bias-voltage V(I₀) = V₀, the modulation-current amplitude ΔI and the modulation-frequency ν.

The voltage V(I(t)) is analyzed by a lock-in amplifier and consequently the difference of the averaged values of applied DC-voltage V₀ and RF-voltage V(I(t)) is measured:

(16)$$V_{\mathit{lock}–\mathit{in}}={\overline{V}}_0-\overline{V\left(I(t)\right)}$$

(17)$$V_{\mathit{lock}–\mathit{in}}=V_0-V_0-\underset{=0}{\underbrace{\overline{{V^{\prime }|}_{I_0}\mathrm{\Delta }I\cdot \mathit{sin}(2\pi \cdot \nu t)}}}-\underset{=\frac{1}{2}{V^{″}|}_{I_0}\mathrm{\Delta }{I}^2\cdot \frac{1}{2}}{\underbrace{\overline{{V^{″}|}_{I_0}{\left(\mathrm{\Delta }I\cdot \mathit{sin}(2\pi \cdot \nu t)\right)}^2}}}$$

(18)$$\Rightarrow V_{\mathit{lock}–\mathit{in}}=V_{\mathit{rect}}=-\frac{1}{4}{V^{″}|}_{I_0}\underset{\mathrm{\Delta }I=\sqrt{2}{I}_{\mathit{eff}}=\sqrt{2}{I}_{\mathit{RF},\mathit{QCL}}}{\underbrace{\mathrm{\Delta }{I}^2}}$$

(19)$$=-\frac{1}{2}{V^{″}|}_{I_0}{I}_{\mathit{RF},\mathit{QCL}}^2=\frac{1}{2}{V^{″}|}_{I_0}{I}_{\mathit{RF},\mathit{QCL}}^2$$

V_rect corresponds to the measured rectification-voltage on the lock-in amplifier. It is proportional to the second derivative of the VI-curve V″ evaluated at the current I₀ and the square of the effective current applied by the RF-source to the QCL-device I_RF,QCL. In the last step of Eq. (19) V″ is substituted by its absolute value knowing, that the second derivative of a concave function is always negative (see 2^nd derivative in Fig. 9). It should be explicitly mentioned, that V″ becomes very small for high currents applied to the QCL but is not zero. Consequently applying a current-modulation to a QCL, results in a DC-response which can be evaluated along the entire bias-current range of the device.

 

Fig. 9 VI- (left-hand scale, black) and 2^nd derivative of the VI-curve (right-hand scale, red) of a typical BH-QCL (plot in the investigated range between 42 mA and 632 mA).

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It is further important to note that both, V″ and I_RF,QCL, still depend on the angular modulation-frequency ω = 2πν.

To derive the RF-current I_RF at the device, the electrical circuit which is shown in Fig. 2 is analyzed. Fig. 10(b) shows the same circuit with all the components and their labeling.

 

Fig. 10 (a) Thévenin equivalent circuit for a perfect voltage source. (b) Equivalent electrical driving circuit including all components and their labeling.

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The RF-source is designed to have a certain power into a 50 Ω load resistance. Figure 10(a) shows the modeling of the RF-source as a perfect voltage source followed by a 50 Ω resistor in series (Thévenin equivalent circuit) and by the 50 Ω output load. This load is represented by R_L in Fig. 2 and Fig. 10(b).

The electrical output-power P_RF of the RF-source can be written in the following way:

(20)$$P_{\mathit{RF}}=\underset{=50\mathrm{\Omega }}{\underbrace{R}}{I}_{\mathit{RF},\mathit{tot}}^2\Rightarrow V_{\mathit{RF},\mathit{tot}}=2\cdot \sqrt{R\cdot P_{\mathit{RF}}}$$

(21)$$\text{with}:\hspace{0.17em}\hspace{0.17em}{I}_{\mathit{RF},\mathit{tot}}^2=\frac{V_{\mathit{RF},\mathit{tot}}^2}{R_{\mathit{tot}}^2}\hspace{0.17em}\hspace{0.17em}\text{and}:\hspace{0.17em}\hspace{0.17em}{R}_{\mathit{tot}}=2\cdot R$$

with the (fictitious) RF-voltage V_RF,tot.

The total current which is supplied by the RF-source I_RF,tot is given by the RF-voltage V_RF,tot which is biasing the load R_L (= 50 Ω) and the impedance of the QCL Z_Q:

(22)$$I_{\mathit{RF},\mathit{tot}}=\frac{V_{\mathit{RF},\mathit{tot}}}{R_L+Z_Q}\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}{V}_{\mathit{RF},\mathit{QCL}}=Z_Q\cdot I_{\mathit{RF},\mathit{tot}}$$

(23)$$\Rightarrow V_{\mathit{RF},\mathit{QCL}}=V_{\mathit{RF},\mathit{tot}}-R_L\cdot I_{\mathit{RF},\mathit{tot}}=\frac{V_{\mathit{RF},\mathit{tot}}\cdot Z_Q}{R_L+Z_Q}$$

In the next step we assume that the source of the non-linear response of the QCL is the current through the resistive part of the device only and the capacitance has no contribution (capacitor is a linear device). Consequently the current through the device I_RF,QCL as function of the total resistance of the QCL R_QCL (1/R_QCL = 1/R_AR + 1/R_InP:Fe) is given by:

(24)$$I_{\mathit{RF},\mathit{QCL}}=\frac{V_{\mathit{RF},\mathit{QCL}}}{R_{\mathit{QCL}}}=V_{\mathit{RF},\mathit{tot}}\cdot \frac{Z_0}{R_{\mathit{QCL}}\cdot \underset{=Z_{\mathit{tot}}}{\underbrace{(R_L+Z_Q)}}}$$

After some algebra to calculate |Z_tot|² and inserting V_RF,tot from Eq. (20) (with R = R_L), I_RF,QCL from Eq. (24) and $Z_Q=\frac{1}{1/R_{\mathit{QCL}}+i\omega C_{\mathit{QCL}}}$ (C_QCL: total capacitance of the QCL) the rectification voltage V_rect from Eq. (19) is finally calculated to be:

(25)$$\begin{array}{r}\hfill V_{\mathit{rect}}=2\cdot \left|V^{″}\right|\cdot P_{\mathit{RF}}\cdot \frac{R_L}{R_{\mathit{QCL}}}\cdot \frac{1}{{\left(R_L+R_{\mathit{QCL}}\right)}^2}\cdot \left(\frac{1+{\left(\omega R_{\mathit{QCL}}{C}_{\mathit{QCL}}\right)}^2}{1+{\left(\omega \overline{R}{C}_{\mathit{QCL}}\right)}^2}\right)\cdot \\ \hfill \frac{R_{\mathit{QCL}}^2}{1+{\left(\omega R_{\mathit{QCL}}{C}_{\mathit{QCL}}\right)}^2}\end{array}$$

with $\overline{R}=\frac{R_L\cdot R_{\mathit{QCL}}}{R_L+R_{\mathit{QCL}}}$.

It consists of two frequency-dependent terms. The latter corresponds to a typical RF roll-off at −20 dB/decade. The first term goes to unity for high ω. Depending on the resistance and capacitance of the device R_QCL and C_QCL the onset of the −20 dB/decade roll-off is shifted towards higher ω.

C. Derivation of the resonance peak according to Maxwell-Bloch formalism

In this part the theoretical response function, including resonance peak, of a DFB-QCL, is derived which is used to model the measured experimental results. Starting from the rate equation response function h(ω) and extending it by the additional contributions, namely the cut-off behavior of the electrical current-injection circuit c(ω) and the mode coupling term between lasing DFB- and below threshold FP-mode, m(ω), the total response response function F_resp,tot is obtained.

The function h(ω), which is calculated from a rate-equation approach including a small signal analysis of the time-dependent quantities: photon flux, pump current density and the population densities, yields the following transfer function [28]:

(26)$$\left|h(\omega )\right|=\frac{1}{\sqrt{1+{\omega }^4{\tau }_p^2{\tau }_{\mathit{st}}^2+{\omega }^2{\tau }_{\mathit{st}}{\tau }_p\left(\frac{{\tau }_p}{{\tau }_{\mathit{st}}}+2\frac{{\tau }_p}{{\tau }_{\mathit{up}}}+\frac{{\tau }_p{\tau }_{\mathit{st}}}{{\tau }_{\mathit{up}}^2}-2\right)}}$$

with ω = 2πν, and the upper state, photon cavity and stimulated lifetimes τ_up, τ_p and τ_st.

To include the effect of DFB- to FP-mode coupling, the model is extended by a Maxwell-Bloch formalism, which was first introduced to QCLs by Khurgin et al. [30] and extended later on by Villares and Faist [31].

According to Eq. (64) in [31], the amplitudes A₀ and A₁ of both modes can be written by neglecting the four-wave mixing term as:

(27)$$2{\tau }_p{\dot{A}}_0-i2{\tau }_p\underset{\text{cavity}\hspace{0.17em}\text{dispersion}}{\underbrace{\left(\frac{{\omega }_0^2-{\omega }_{0c}^2}{2{\omega }_0}\right)}}{A}_0=\underset{\text{net}\hspace{0.17em}\text{gain}}{\underbrace{\left(g_0{\tilde{G}}_0-1\right)}}{A}_0+2g_0{\tilde{G}}_0\underset{\text{modulation}\hspace{0.17em}\text{term}}{\underbrace{\left[A_1{M}_{+}{\delta }^{*}{\kappa }_{0,1}+A_{-1}{M}_{-}\delta {\kappa }_{0,1}^{\prime }\right]}}$$

where three main contributions are identified. The dispersion introduced by the cavity, the contribution of the gain of the active medium (net gain) and the modulation term, which comes from (amplitude-)modulation of the injected current to the device.

The parameters in the equation are: A₀, A₁: field of the (non-lasing) FP-, (lasing) DFB-mode, the photon-cavity lifetime τ_p ( ${\tau }_p=\frac{n_{\mathit{eff}}}{c\cdot ({\alpha }_m+{\alpha }_{wg})}$; ; c: speed of light, α_m,wg: mirror-/waveguide-losses), ω₀ and ω_0c are the dispersion-less and including cavity-dispersion (cavity-)resonances, respectively, g₀G̃₀ is the normalized gain with respect to lasing-threshold, $M_{\pm }=\frac{{\gamma }_{22}}{{\gamma }_{22}\pm i\omega }$ contains the scattering rate from the upper lasing-state (modulation bandwidth) γ₂₂ and the mode-spacing ω from the mode-expansion, δ is the modulation depth (strength of the RF-signal) and the coefficient κ_0,±1 is a value for the relative coupling of the modes. A detailed description and analysis of all parameters from Eq. (27) can be found in [31].

Without loss of generality a few simplifications can be made to this equation: First of all A₋₁ is set to zero (two-mode case). Ȧ₀ is also set to zero to analyze the steady-state solution. Moreover g₀G̃₀ is re-written as the gain normalized to the lasing-threshold, neglecting its imaginary part, which is assumed to be small $g_0{\tilde{G}}_0=g_0^n$, which finally yields:

(28)$$-2i{\tau }_p\left(\frac{{\omega }_0^2-{\omega }_{0c}^2}{2{\omega }_0}\right)A_0=\left(g_0^n-1\right)A_0+2g_0^n{M}_{+}{\delta }^{*}{\kappa }_{0,1}{A}_1$$

(29)$$\Rightarrow {\left|A_0\right|}^2=\frac{4{\left(g_0^n\right)}^2{\delta }^{*2}{\kappa }_{0,1}^2}{{\left(1-g_0^n\right)}^2-4{\tau }_p^2{\left(\frac{{\omega }_0^2-{\omega }_{0c}^2}{2{\omega }_0}\right)}^2}{\left|A_1\right|}^2$$

The net gain is written as: ${\delta }^{\prime }=1-g_0^n$. Instead of the angular frequencies ω_i, the frequencies ν (modal detuning) and ν₀ (mode-spacing) are used, which leads to the final expression for the response function of the mode coupling:

(30)$$\begin{array}{lll}m(\nu )={\left|A_0\right|}^2\hfill & =\hfill & \underset{\mathit{modulation}}{\underbrace{4{\left(g_0^n\right)}^2{\delta }^{*2}{\kappa }_{0,1}^2}}\cdot \underset{\text{roll}-\text{off}\hspace{0.17em}\text{lasing}\hspace{0.17em}\text{mode}}{\underbrace{{\left|A_1\right|}^2}}\hfill \\ \hfill & \cdot \hfill & \frac{1}{\underset{\text{resonance}}{\underbrace{{\delta }^{\prime 2}{\left(1+\frac{16{\tau }_p^2{\pi }^2{\left(\frac{{\nu }^2-{\nu }_0^2}{\nu }\right)}^2}{{\delta }^{\prime 2}}\right)}^2}}}\cdot \frac{1}{\underset{=\left|M_{+}\right|}{\underbrace{1+4{\pi }^2{\nu }^2{\tau }_{\mathit{up}}^2}}}\hfill \end{array}$$

Four contributions to the intensity of the (non-lasing) FP-mode |A₀|² are identified. First there is the modulation term, the second term gives the roll-off behavior of the lasing mode, which is assumed to be RC-like ( $=\frac{1}{1+4{\pi }^2{\nu }^2{\tau }_{\mathit{RC}}^2}$, characteristic lifetime τ_RC = R · C). Third there is an ”Lorentzian-shaped” peak-function, which enhances the photon-cavity lifetime τ_p by the gain (i.e. detuning of the non-lasing mode from threshold) by $\frac{1}{{\delta }^{\prime 2}}$ at a bandwidth of $1/\left(\frac{16{\pi }^2{\tau }_p^2}{{\delta }^{\prime 2}}\right)$.The fourth and last term gives the roll-off behavior of the upper-state, defined by its lifetime τ_up and which is also of a RC-type (see above for definition of M₊).

The total response-function which also includes the circuit cut-off behavior c(ν) is finally given by:

(31)$$F_{\mathit{resp},\mathit{tot}}(\nu )=c(\nu )\cdot \left(\left|h(\nu )\right|+m(\nu )\right)$$

where the equation is written as function of the modulation frequency ν.

Table 2 lists all the parameters used for modeling the experimental data in Fig. 6(c) and (d). They are either design-/ typical device-parameters for the given geometry (α_m, α_wg, n, τ_p, R_L, τ_up and τ_stim, κ_n,n+1) or they are extracted from measurements of the device (ν₀, R, C_Vrect, τ_RC) or a reasonable estimation is performed to obtain them ( $g_0^n$, δ′, δ), which is validated by comparison of the simulation results with the experimental data (see Fig. 6(c) and (d)).

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