# Analytic Characterization of the Dynamic Regimes of Quantum-Dot Lasers

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## Abstract

**:**

## 1. Introduction

## 2. Minimal Quantum-Dot Laser Model

## 3. Numerical Results

**Figure 1.**Relaxation oscillation (RO) frequency ${\omega}_{\mathrm{RO}}=\left|\mathrm{Im}\phantom{\rule{0.166667em}{0ex}}\lambda \right|$ (solid red) and damping rate ${\Gamma}_{\mathrm{RO}}=-\mathrm{Re}\phantom{\rule{0.166667em}{0ex}}\lambda $ (solid blue) determined from Equation (7) for the (

**a**) shallow and (

**b**) deep quantum-dot (QD) laser as a function of the effective electron scattering rate R, for constant photon number S. The dashed black line denotes purely real eigenvalues which are not related to ROs. The circles denote the RO parameters extracted numerically from the full microscopically based balance equation (MBBE) model [19]. The green, yellow, and red shaded areas denote the constant-reservoir, overdamped, and synchronized regimes, respectively. ${T}_{\mathrm{sp}}=2\phantom{\rule{0.166667em}{0ex}}$ns, $g=230\phantom{\rule{0.166667em}{0ex}}$ns${}^{-1}$, $\kappa =50\phantom{\rule{0.166667em}{0ex}}$ns${}^{-1}$. (a) ${T}_{1}=0.17\phantom{\rule{0.166667em}{0ex}}$ns, $S=0.068$, $d=0.035$; (b) ${T}_{1}=0.2\phantom{\rule{0.166667em}{0ex}}$ns, $S=0.009$, $d=0.25$.

**Figure 2.**RO frequency ${\omega}_{\mathrm{RO}}$ (red) and damping rate ${\Gamma}_{\mathrm{RO}}$ (blue) determined from Equation (7) for different QD-quantum-well (QW) carrier distribution coefficients: $d=0.035$ (solid line) describing small energy separation, $d=0.2$ (dashed line), and $d=0.5$ (dotted line) corresponding to a large energetic distance. Other parameters as in Figure 1a.

## 4. Analytic Approximation

#### 4.1. Slow Scattering – Constant-Reservoir Regime

**Figure 3.**Comparison between the exact eigenvalue solutions of Equation (7) (solid lines) and the analytic approximations of Equation (11) (dashed), and Equation (19) (dotted) for the

**(a)**shallow and

**(b)**deep QD laser in dependence of the effective electron scattering rate R (keeping S constant), cf. Figure 1.

#### 4.2. Fast Scattering – Synchronized Regime

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

#### A. Analytic first-order corrections in the synchronized regime

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Lingnau, B.; Lüdge, K.
Analytic Characterization of the Dynamic Regimes of Quantum-Dot Lasers. *Photonics* **2015**, *2*, 402-413.
https://doi.org/10.3390/photonics2020402

**AMA Style**

Lingnau B, Lüdge K.
Analytic Characterization of the Dynamic Regimes of Quantum-Dot Lasers. *Photonics*. 2015; 2(2):402-413.
https://doi.org/10.3390/photonics2020402

**Chicago/Turabian Style**

Lingnau, Benjamin, and Kathy Lüdge.
2015. "Analytic Characterization of the Dynamic Regimes of Quantum-Dot Lasers" *Photonics* 2, no. 2: 402-413.
https://doi.org/10.3390/photonics2020402