# Alternation of Defects and Phase Turbulence Induces Extreme Events in an Extended Microcavity Laser

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## Abstract

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## 1. Introduction

## 2. Extreme Events in a Microcavity Laser

## 3. Theoretical Description of a One-Dimensional Spatially Extended Laser

## 4. Characterization of Spatiotemporal Dynamics of an Extended Laser with a Saturable Absorber: Alternation of Defects and Phase Turbulence

## 5. Alternation of Defects and Phase Turbulence Induces Extreme Events

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Experimental set up. (

**a**) Schematic representation of an extended planar vertical cavity surface emitting laser with an integrated saturable absorber medium (VCSEL-SA). (

**b**) Right panels account for the top-view camera snapshots of the one-dimensional line VCSEL-SA surface below (upper image, with the mask visible) and above laser threshold (lower image).

**Figure 2.**Typical temporal evolution of the experimentally recorded intensity and semi-log graph of the associated probability density distribution of the intensity height H for different normalized pump values (adapted from [12]): (

**a**) $P/{P}_{th}=1.02$; (

**b**) $P/{P}_{th}=1.17$; (

**c**) $P/{P}_{th}=1.20$; and (

**d**) $P/{P}_{th}=1.25$. Normal and extreme events are shown in orange and green, respectively ($AI>2$).

**Figure 3.**Numerical characterization of the emergence of extreme events in an extended, planar vertical-cavity surface-emitting laser with an integrated saturable absorber medium obtained from Equation (1). Graph of the proportion of extreme events ${p}_{\mathrm{EE}}$ (×) (

**a**) and excess kurtosis ${\gamma}_{2}$ (∗) (

**b**) as a function of pump parameter $\mu =P/{P}_{th}$ considering the height H of the laser intensity. Graph of the largest Lyapunov exponent $max\left({\lambda}_{i}\right)$ (squares) (

**c**) and Kaplan–Yorke dimension ${D}_{\mathrm{KY}}$ (

**d**) as a function of pump parameter $\mu $. Graph of the proportion of extreme events ${p}_{\mathrm{EE}}$ (

**e**) and excess kurtosis ${\gamma}_{2}$ (

**f**) as a function of pump parameter $\mu $ considering the local intensity spatiotemporal maxima (adapted from [10]).

**Figure 4.**Alternation of defects and phase turbulence in the laser with saturable absorber model expressed by Equation (1). Spatiotemporal evolution of the electric field intensity, together with the spatiotemporal positions of phase singularities of the electric field envelope $E(x,t)$ and of the extreme events (blue and red dots, respectively; temporal location of respective events are highlighted by dash signs) in the spatiotemporal complex regime with $\alpha =2$, $\beta =0$, ${\gamma}_{g}=0.005$, ${\gamma}_{q}=0.005$, $\gamma =0.5$, $s=10$, and the following $\mu $ values: (

**a**) $\mu =2.1$; (

**b**) $\mu =2.3$; (

**c**) $\mu =2.4$; (

**d**) $\mu =2.6$.

**Figure 5.**Turbulence dynamics of the one-dimensional microcavity laser with a saturable absorber medium. Spatiotemporal diagram and the average spectrum ${\overline{\phi}}_{k}$ of the phase of the electric field envelope of Equation (1) by $\alpha =2$, $\beta =0$, ${\gamma}_{g}=0.005$, ${\gamma}_{q}=0.005$, $\gamma =0.5$, $s=10$, and the following $\mu $ values: (

**a**) $\mu =2.2$; (

**b**) $\mu =3.4$; (

**c**) $\mu =4.0$. (

**d**) The average spectrum ${\overline{\phi}}_{k}$ of the phase of the electric field envelope for different pumping parameters.

**Figure 6.**Complex dynamics exhibited by the laser with saturable absorber model computed with Equation (1) in a large time window. Spatiotemporal progression of the electric field magnitude and spatiotemporal positions of the defects of the electric field envelope $E(x,t)$ and of the extreme events (blue and red dots, respectively; temporal location of respective events are highlighted by dash signs). Parameters are identical to those in Figure 4, with pumping: (

**a**) $\mu =2.1$; (

**b**) $\mu =2.4$; (

**c**) $\mu =3.2$; and (

**d**) $\mu =3.6$.

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**MDPI and ACS Style**

Barbay, S.; Coulibaly, S.; Clerc, M.G.
Alternation of Defects and Phase Turbulence Induces Extreme Events in an Extended Microcavity Laser. *Entropy* **2018**, *20*, 789.
https://doi.org/10.3390/e20100789

**AMA Style**

Barbay S, Coulibaly S, Clerc MG.
Alternation of Defects and Phase Turbulence Induces Extreme Events in an Extended Microcavity Laser. *Entropy*. 2018; 20(10):789.
https://doi.org/10.3390/e20100789

**Chicago/Turabian Style**

Barbay, Sylvain, Saliya Coulibaly, and Marcel G. Clerc.
2018. "Alternation of Defects and Phase Turbulence Induces Extreme Events in an Extended Microcavity Laser" *Entropy* 20, no. 10: 789.
https://doi.org/10.3390/e20100789