1. Introduction

Quantum Cascade Laser (QCL), invented in the mid-1990s [1], is now not only a high-performance device, which has found applications in many fields of human activity, but also a nice system for studying the quantum coherence of electronic states, induced by both dc and ac electromagnetic (e-m) fields. Theoretically, it was proven that when both these fields are present in the QCL structure, the coherent coupling of the injector and upper states mixes them in such a way that an effective interaction of the ac field with both these states is possible even when the direct injector-to-lower state transition is forbidden due to the zero dipole element, $\mu _{il}=0$. This is similar to what may appear in the resonant $P$ scheme of a three-level amplification medium, where a microwave field is scattered into the anti-Stokes optical component [2]. In QCL, the pump is the dc field, which is converted into the laser field, which is the Raman anti-Stokes component [3,4]. Analyses predict that the amplification in the $P$ scheme is possible even in the absence of population inversion in the laser doublet. This may happen provided the upper level is less populated than the injector level, which makes the anti-Stokes component fulfill the amplification condition [2].

Accounting for these concepts requires advanced methods of theoretical analysis, which, due to their conceptual and numerical complexity, can be applied to small, schematic-rather-than-realistic structures. On the other hand, the performance of real devices is equally described by all quantum and classical phenomena occurring in the whole QCL structure, which often reaches a micrometer size. For such a case, the use of simplified models is a must. Modeling techniques for QCLs have been reviewed in [5]. Starting from the simplest, the carrier transport models which account for light-matter interaction include (i) rate equations (RE) [6–11], (ii) Monte Carlo [12,13], (iii) density matrix (DM) [11,14–20], and (iv) non-equilibrium Green’s function (NEGF) [21–25] methods. When analyses go beyond the small-signal response, most of them (exceptions include [17,19–22]) use a simplified model of optical gain, in which it is proportional to the population inversion between laser states. In this paper, improved (but still simple) models of the optical gain in QCL are proposed which, to some extent, take into account the Raman anti-Stokes component of the laser field. These models preserve the load-saving, scattering-like approach to the light-matter interaction and so are highly demanded in multi-scale modeling of light-emitting devices. It is also shown that under certain conditions the small-signal formulation of the optical gain gives its realistic estimate also for larger field strengths. This observation validates the use of the NEGF-based approaches, in which the interaction with the optical field is included through electron-photon selfenergy (SE). Such an approach to light-matter interaction generates only little increase in numerical load as the electron-photon SE enters the formalism like other (scattering) SEs [26–29]. This is in contrast to the exact treatment of time-dependent potentials, which within NEGF requires solving of the full set of NEGF equations for several higher harmonics of fundamental frequency [21,22]; in SE approach, the NEGF formalism is solved only for the steady-state. This improvement makes the method a highly valuable and efficient tool for device evaluation [25] and optimization [24,30]. The outcomes from the large-scale computer models that use this method are supposed to present the best physics-based approximations of electronic transport because the NEGF method does not need any phenomenological parameters: the pure dephasing times, necessary (and usually arbitrarily assumed) in DM methods, in the NEGF method inherently arise as a result of the interplay between coherent and incoherent processes included in the formalism.

The appropriate analyses are provided in Sections 2 and 3, where various gain models in QCL structures are considered. Section 4 contains the summary of the findings and the corresponding conclusions.

2. Gain models

2.1 Rate equation

The simplest model of QCL is a three-level system illustrated in Fig. 1 [6].

 

Fig. 1. Three-state DM model of QCL. Carriers out/in-scattering from/to the states $|i \rangle / |j \rangle$ are controlled by the scattering times, $\tau _{ij}$, and the stimulated emission time, $\tau _{st}$.

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This description includes injector states $|1\rangle$ and $|1'\rangle$, lower state $|2\rangle$ and upper state $|3\rangle$. The injector and upper states are coupled incoherently, so in the RE model, the coupling term $\Omega$ has the meaning of a particle current injected to the upper state. The light-matter interaction is represented by the stimulated emission time $\tau _{st}$, which controls the radiative channel of the upper-to-lower transition. This channel opens itself once the population inversion, i.e., the difference $\Delta n_{32}= n_{3}-n_{2}$ between the populations $n_{i}$ in the states $|i \rangle = |3 \rangle$ and $|i \rangle = |2 \rangle$, exceeds its threshold value $\Delta n^{th}_{32}=g_{th} / g_{c}$; $g_{th}$ is the threshold gain, $g_{c}$ is the gain cross-section. The stimulated emission time $\tau _{st}$, which describes the rate of this transition, is inversely proportional to the photon flux $\Phi$ (per unit active region width), the length $d$ of one QCL module, and the gain cross-section, $\tau _{st}=1 / (g_{c} d \Phi )$. Then, in the RE model, the gain is given by the product of the gain cross-section and the population inversion

2.2 Density matrix with dc coherence

The formulation of Eq. (1) for the optical gain in the RE model is fully maintained in a more advanced DM model of the QCL, which includes dc coherences of electronic states involved in the pumping process [15,31–35]. A three-state version of such a model is depicted in Fig. 1. Now the injector state $|1 \rangle$ with the energy $E_{1}$ and the upper laser state $|3 \rangle$ with the energy $E_{3}$ are coupled by the external dc field [36]. The strength of this coupling is given by the coupling energy $\hbar \Omega$, which can be derived from isolated wave functions by the tight-binding approach. Details specific to a QCL can be found, e.g., in [5,31].

2.3 Density matrix with dc and ac coherences

One step further is to consider the model, which accounts for the coherences of electronic states induced by the laser (ac) field. They couple only the states that are separated by the energy close to the ac field frequency, $|E_{i}-E_{j}| / \hbar \approx \omega$. For the three-state model of Fig. 1, this condition selects the pairs: $|3 \rangle$-$|2\rangle$ and $|1 \rangle$-$|2\rangle$. Within the rotating wave approximation (RWA), the density matrix that includes the ac coherences of these states takes the form [35,37]

Note that, due to the coherent coupling of the states $|1 \rangle$ and $|2 \rangle$, the coherences $\rho _{12}, \rho _{21}$ are allowed to vary at the driving frequency, although the dipole momentum $\mu _{12}$ is set to 0. The time evolution of the system interacting with the e-m field can be described by the Lindblad equation

The solution for the steady state can be found using the unitary transformation [39]

This leads to the set of eight linear equations, which can be solved for $\rho$. Optical gain is determined by the imaginary part of the linear susceptibility $\chi$, which is related to the density matrix element $\rho _{23}$ (recall that $\mu _{12}=0$) [35,37]

 

Fig. 2. Gain spectra (symbols) and occupations $n_{i}= \rho _{ii}$ (solid lines) of the QCL states calculated for the three-state model interacting with the monochromatic laser field with the magnitude characterized by the Rabi frequency $\Omega _{32} \rightarrow 0$ (left) and $\Omega _{32}$ $\cong 7.6$ Trad/s ($F\cong 100$ kV/cm, right). Decomposition of the gain into linear and nonlinear parts is made using Eq. (9). In the left panel, the analytical gain curves calculated with Eq. (10) are also shown (dashed lines). Calculations were performed for the model parameters: $\hbar \Omega =2$ meV, $\tau _{32}=1$ ps, $\tau _{31}=4$ ps, $\tau _{21}= \tau ^{pure}_{iu}= \tau ^{pure}_{ul}= \tau ^{pure}_{il}=0.15$ ps, $E_{32}=0.24$ eV (reasonable for m-IR QCL) and $\delta =0$.

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Once Eq. (5) is transformed to the super-operator representation, one can easily derive the relations

From Eq. (8), the imaginary part of $\mathrm {\mathbf {\rho }}$ can be calculated

Analytical solution of Eq. (9) can be found in [3,42,43].

3. Discussion

3.1 Gain components

It is straightforward with Eq. (9) to decompose the gain into the linear part (proportional to the population inversion, $\Delta n_{32}=\rho _{33} - \rho _{22}$) and the remaining nonlinear part. In Fig. 2, such decomposition is illustrated for both field intensities. For the vanishing field, the calculations were compared with the analytical result provided in Eq. (22) of [3]

The agreement is excellent what confirms the correctness of Eqs. (8) and (9). Equation (10) provides justification for the used nomenclature: the component proportional to the population inversion between laser levels scales linearly with $\Omega _{32}$ and gives the main contribution to the linear susceptibility of the medium $\chi ^{(1)}(\omega )$ and thus the linear gain. The component, proportional to $\Delta n_{13}=\rho _{11}-\rho _{33}$, can be identified as connected with the stimulated coherent transfer of carriers directly from the injector state $|1\rangle$ to the lower laser state $|2\rangle$ [3,4]. This process can occur only when the states $|1 \rangle$ and $|3 \rangle$ are coherently coupled and it requires triple mixing of the fields: $\Omega ^{2}\Omega _{32}$. Then, even for the vanishing fields where the populations are field independent, it is a third-order process, which contributes to the third-order susceptibility $\chi ^{(3)}(\omega \!:\!0,\omega,0)$ and constitutes the nonlinear contribution to the gain [3].

One can learn from left Fig. 2 that the nonlinear component non-negligibly contributes even to the small-signal response of the medium. For the increased field intensity (right Fig. 2), the gain experiences suppression and power broadening. The linear component decreases whereas the nonlinear component increases. In general, these changes should be correlated to the changes in states occupation. It can be deduced from Fig. 3 that the optical field affects these correlations in different ways. For the linear component, they are almost field independent; the contributions to gain calculated with either Eq. (9) or Eq. (10) are almost identical. This means that for the linear contribution, the gain-inversion relation is hardly affected by the strength of the optical field. For the nonlinear part, the situation is quite different; the increase of the optical field makes the population-to-gain conversion less efficient. This can be inferred from the plots calculated with either Eq. (9) or Eq. (10), and considered as the gain compression. The respective gain compression function can be calculated as the ratio of these two curves. The result of such calculations presented in Fig. 4 shows that this function can be well fitted with the Lorentzian function, $f(\Omega _{32})=1/(1+(\epsilon \Omega _{32})^{2})$, where $\epsilon$ is the compression coefficient. A similar functional form of the gain-field dependence is known to describe the gain saturation in a two-level homogeneously broadened laser resulting from the changes of the carrier density [44]. Here, its meaning is different; it describes a suppression of the gain in addition to changes caused by the changes in the population inversion. This effect is known as the gain compression. In this context, the Lorentzian form of the compression function was proposed by Channin [45]. Due to this fundamental difference, the FWHM-related parameter is termed as the gain compression parameter in contrast to the saturation field intensity used to characterize the process of gain saturation.

 

Fig. 3. Gain at $\hbar \omega$ $\equiv E_{32}=0.24$ eV and its components calculated with Eq. (9) (symbols) or with Eq. (10) (black dashed lines) or with Eq. (14) (red solid line) as the function of the optical Rabi frequency $\Omega _{32}$ (optical field intensity).

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Fig. 4. Gain compression function for the nonlinear component. Symbols refer to the numerical calculations. Line is the plot of the function $f(\Omega _{32})=1/(1+(\varepsilon \Omega _{32})^{2})$ with $\varepsilon =(8.8 \, \mathrm {Trad} / \mathrm {s})^{-1}$.

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3.2 Improved RE model

With the RE method, the common practice to account for nonlinear gain components is to complement Eq. (1) by a compression factor $f$ (read [46] for a review)

A very recent example that employs such modeling can be found in [47], where the RE model which uses Eq. (11) was employed to correlate the linewidth enhancement factor with the gain compression coefficient in QCL. The analysis in sec. 3.1 suggests that for the systems described by the three-state model, a better formulation for the inclusion of gain nonlinearity is

3.3 Improved dc-DM model

The gain model proposed in Eq. (12) requires additional parameters (namely, $a$ and $\varepsilon$). This drawback can be easily waived by using the static (dc) coherence $\rho _{13}$, rather than the population difference $\Delta n_{13}$. More specifically, when the relation

In Fig. 5, the results of the calculations performed with such an improved dc-DM model (for the set of parameters used to produce Fig. 3) are shown. Although not identical, the plots are very similar: the nonlinear component contributes to the gain by 10-20% and increases with the increasing optical flux. At large field intensities, the linear term drops below zero due to the lack of population inversion on the laser levels, but lasing is still possible (as $g>0$) due to the inclusion of the nonlinear gain component.

 

Fig. 5. Gain resolved for linear and nonlinear components versus square root of the stimulated emission rate (proportional to the optical Rabi frequency) calculated with DM model with dc coherences only. The interaction with optical radiation was included through Eq. (16). Gain and its components were calculated with Eq. (15).

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3.4 NEGF model

One more observation about Fig. 3 is that once the populations and dc coherences are reliably estimated, the formulation of Eq. (14) - derived for the small optical signal - well reproduces the true response of the medium for arbitrary large signal (calculated with Eq. (9)). This is of some importance as the small-signal formulations can be obtained with the perturbation methods that exploit steady-state solutions instead of explicit solving of the system for the time-dependent potentials. The first implementation of this concept was by Kumar and Hu [35], who calculated gain spectra of the THz QCL using a four-level ac-DM model for the vanishing ac field whereas the actual (high) field intensity was accounted for in the model through the scattering process controlled by the appropriate stimulated emission time (see sec. 2.2). The observation we made justifies this approach.

In this section, a more elaborated model that employs NEGF formalism is proposed. Within this formalism, the small-signal formulation of the gain in QCLs was developed in [48]. The reliable estimation of occupations and dc coherences, which accounts for the interaction with the laser field, can be obtained by incorporating into NEGF formalism the electron-photon selfenergies and solving the NEGF equations for the steady-state. Such an approach generates only little increase in numerical load as electron-photon selfenergies enter the formalism like other (scattering) selfenergies [26–29]. This is in contrast to the exact treatment of time-dependent potentials, which within NEGF approach requires solving of the full set of NEGF equations for several higher harmonics of fundamental frequency [21,22].

The exemplary calculations made with the NEGF model were thought to reproduce the results presented in Fig. 3. These results were obtained for the three-state model, in which all parameters, except laser field intensity, i.e., the site energies, coupling strength and detuning $\delta$, were kept unchanged. In the NEGF model, this requirement translates to unchanged potential, Fermi levels (in the leads) and external bias. Under such conditions, the total charge in the device may not be conserved. Therefore, the plots in Fig. 6 do not exactly reproduce those in Fig. 3. Nevertheless, they clearly demonstrate that gain properties derived from the time-dependent analysis are preserved. Namely, (i) positive gain is observed also in the absence of the population inversion between the laser levels, (ii) gain disappears once the population difference between the injection and the upper level $\Delta n_{13}$ becomes negative. More details of this model and calculations are provided in the Appendix.

 

Fig. 6. Populations of the injector ($n_{1}$) and the laser levels ($n_{2}$, $n_{3}$) (left axis) and the total gain (right axis) versus the photon flux of the stimulated emission calculated for the exemplary m-IR QCL device with the NEGF method, in which the interaction with the optical field was incorporated through the electron-photon selfenergy.

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4. Summary and conclusions

For optoelectronic devices, the light-matter interaction is crucial. Its rigorous treatment requires the use of time-dependent potentials and appropriate methods of problem solution. For realistic modeling of nowadays structures, it can lead to extraordinary numerical loads. Therefore, any approach that reduces this load and simultaneously keeps the essential physics of the phenomena is highly demanded. The paper adds some novelty in this field: the improved gain models defined by Eqs. (12), (15) and (16) enable accounting for its nonlinear components while preserving the load-saving, scattering-like approach to the light-matter interaction. Such simple models are still being used to solve nontrivial problems of device physics like, e.g., gain recovery dynamics [9,10], photon-driven transport [11,15,49], linewidth enhancement [47], and dumping of Rabi oscillations [30,31].

It was also conjectured that as long as the populations and dc coherences account for the interaction with the optical field, the small-signal formulation of the gain gives its realistic estimate also for larger field strengths. This observation validates the use of the NEGF-based approaches, in which the interaction with the optical field is included through the electron-photon selfenergies, whereas the small-signal formulation of the gain can be used to monitor the saturation process, estimate the clamping flux, and eventually the light-current characteristic. These estimates are supposed to present the best physics-based approximation because the NEGF method does not need any phenomenological parameters. The pure dephasing times, necessary (and usually arbitrarily assumed) in DM methods, in the NEGF method inherently arise as a result of the interplay between coherent and incoherent processes included in the formalism. Other advantages of the NEGF method over few-level DM models, in which only the dominant pathways (both coherent (terms $\Omega _{ij}$) and incoherent (terms $\tau _{ij}$)) are kept, are that it accounts for several tens of basis states and all inter-state dc coherences. It is quite important in the context of the nonlinear gain component as the injector-upper state coherence is not the only one, which can produce this component [3]. All these make the model of sec. 3.4 a valuable and efficient tool for the evaluation and the optimization of the QCL devices emitting in either m-IR or THz range.

Appendix: NEGF model and calculations

For simplicity, the device was modeled with the one-dimensional (1D) one-band effective-mass Hamiltonian

(17)$$H= \frac{-\hbar^2}{2}\frac{1}{m}\frac{d^{2}}{dz^{2}}+V(z)+\frac{\hbar^2k^2_{||}}{2m},$$

from which the spatial and energy dispersion of the effective mass was removed. The calculations were performed in real space for the exemplary structure, which emits the radiation of $\cong 5\;\mathrm{\mu}$m wavelength. The band diagram of the simulated device is shown in Fig. 7.

 

Fig. 7. Band diagram of a QCL device connected to the leads. Important levels are schematically depicted: blue - injector, red - upper, pink - lower.

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For proper operation, the structure needs to be connected to the leads, which extend to $\pm$ infinity. In the NEGF formalism, this extension can be mimicked by the so-called contact selfenergies [50–52]. For one-band 1D calculations, these selfenergies are the complex numbers, which add to the left/rightmost diagonal element of a discretized Hamiltonian ($\mathbf {H}$).

The NEGF formalism enables the inclusion of scattering processes. Similarly to the leads, they enter the formalism through appropriate selfenergies. In these calculations, only the electron-photon selfenergy was included. The equations of the NEGF formalism, discretized on the nonuniform real space grid, read [53,54]

(18)$$\begin{aligned} (E\mathbf{I}-\mathbf{H}-\mathbf{\Sigma}^{\mathbf{R}}_{\mathbf{lead}}-\mathbf{\Sigma}^{\mathbf{R}}_{\mathbf{e-ph}})\mathbf{G}^{\mathbf{R}}=\pmb{\lambda}, \\ \mathbf{G}^{<}=\mathbf{G}^{\mathbf{R}} \pmb{\lambda}^{\mathbf{-1}}(\mathbf{\Sigma}^{\mathbf{<}}_{\mathbf{lead}}+\mathbf{\Sigma}^{<}_{\mathbf{e-ph}})\mathbf{G}^{\mathbf{R} \mathbf{\dagger}}, \end{aligned}$$

where $\dagger$ is the Hermitian operator, I is the unity matrix, $\pmb {\lambda }$ is a diagonal matrix with the elements described by the local grid spacing, $\mathbf {\lambda }_{ii} \equiv \mathbf {\lambda }_{i}=2/(z_{i+1}-z_{i-1})$, and $\mathbf {\Sigma }$s and Gs are selfenergy and Green’s function (GF) matrices. As already mentioned, the only nonzero elements of the $\mathbf {\Sigma }_{\mathbf {lead}}$ matrix are its first and last diagonal elements. In contrast, $\mathbf {\Sigma }_{\mathbf {e-ph}}$ is a full matrix. In QCLs, the radiative intersubband transitions are stimulated by the $z$-polarized light propagating along the $y$-axis. The e-m field can be described in terms of the vector potential $\textbf{A}=[0\;0\;A_{z}]$, which can be related to the photon flux $\mathrm {\Phi }$ through the time-averaged Poynting vector $S$: for the monochromatic field, $S=\mathrm {\Phi } E_{\gamma }=2n_{r}c \epsilon _{0} E^{2}_{\gamma } \hbar ^{-2}|A_{z}|^{2}$, where $n_r$ is the material refractive index, $c$ is the speed of light, and $E_{\gamma }=h\nu$ is the photon energy. The respective electron-photon selfenergies are given by [23]

(19)$$\begin{aligned} \mathbf{\Sigma}^{\mathbf{<,R}}_{\mathbf{e-ph}}=e^{2}\hbar^{{-}2}|A_{z}|^{2}\mathbf{W} (\mathbf{G}^{\mathbf{<,R}}(E+E_{\gamma}) + \\ +\mathbf{G}^{\mathbf{<,R}}(E-E_{\gamma})) \pmb{\lambda}^{{-}1} \mathbf{W}, \end{aligned}$$

where $\mathbf {W}=\mathbf {HZ}-\mathbf {ZH}$ is the commutator, and $\mathbf {Z}$ is a diagonal matrix with elements $Z_{ii}=z_{i}$.

It stems from Eqs. (17) and (18) that all Gs have the dimension of $\mathrm {energy}^{-1}\mathrm {length}^{-1}$ and are functions of two parameters, i.e., total energy $E$ and in-plane momentum modulus $k_{||}$, $\mathbf {G}=\mathbf {G}(E,k_{||})$. Then, densities of states (DOSs) and electrons (DOEs), which are related to GFs, are the position-energy-momentum-resolved quantities. Specifically, the 1D DOE can be calculated from the diagonal elements of $\mathbf {G}^{<}$ matrix [50,55]

(20)$$n(z_{i},E,k_{||})={-}\frac{i}{2\pi} G^{<}_{ii}.$$

As the aim is to demonstrate the issues discussed in Section 3, the calculations were limited to the vanishing $k_{||}=0$. Results are shown in Fig. 8.

 

Fig. 8. Density of electrons calculated with the NEGF method for the device shown in Fig. 7 (color map). Wave functions (module-squared) of major electronics states and device band structure are also shown (lines). Rectangles show the areas of the integration used in the estimation of the populations $n_{1}, \, n_{2}, \, n_{3}$.

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Populations $n_{1}, \, n_{2}, \, n_{3}$ were obtained by integrating DOE over the specified $E$-$z$ windows. All parameters used in the simulations are gathered in Table 1.

Table 1. Model and physical parameters.

View Table

Optical gain can be calculated with the perturbation method [48]. First, the complex conductivity can be obtained as $\sigma (\omega )=\delta J(\omega ) / F(\omega )$, where $\delta J$ is the current perturbation in the response to the external radiation field $F$. For the system defined by Eq. (18), the current perturbation is given by

(21)$$\begin{array}{c} \delta J(\omega)=\frac{e}{\hbar V}\int\frac{dE}{2\pi}\mathrm{Tr}\{(\delta\mathbf{U}(\omega)\mathbf{Z} \\ -\mathbf{Z}\delta\mathbf{U}(\omega))\boldsymbol{\lambda}^{{-}1}\mathbf{G^<}(E)+\boldsymbol{\lambda}^{{-}1}\mathbf{W}\delta \mathbf{G^<}(\omega,E)\}, \end{array}$$

where $\delta \mathbf {U}$ is the perturbing potential, which in the Coulomb gauge can be calculated as $\delta \mathbf {U}=-eF(\omega )\mathbf {W}/\hbar \omega$. The perturbed lesser Green’s function $\delta \mathbf {G^<}$ in the first approximation is given by

(22)$$\begin{array}{c} \delta \mathbf{G^<}(\omega,E)=\mathbf{G^R}(E+\hbar\omega)\boldsymbol{\lambda}^{{-}1}\delta\mathbf{U}(\omega)\mathbf{G^<}(E)\\ +\mathbf{G^<}(E+\hbar\omega)\boldsymbol{\lambda}^{{-}1}\delta\mathbf{U}(\omega)\mathbf{G^A}(E). \end{array}$$

Eventually, the absorption coefficient can be obtained from the real part of the complex conductivity, i.e., $\alpha = \Re \left \{ \sigma (\omega ) \right \}/c\epsilon _{0} n_{r}$.

Funding

Narodowe Centrum Nauki (OPUS-19 UMO-2020/37/B/ST7/01830).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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