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

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 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

- Nicolis, G.; Prigogine, I. Self-Organization in Nonequilibrium Systems; Wiley: New York, NY, USA, 1977. [Google Scholar]
- Cross, M.C.; Hohenberg, P.C. Pattern formation outside of equilibrium. Rev. Mod. Phys.
**1993**, 65, 851–1112. [Google Scholar] [CrossRef][Green Version] - Pismen, L.M. Patterns and Interfaces in Dissipative Dynamics; Springer: Berlin, Germany, 2006. [Google Scholar]
- Cross, M.; Greenside, H. Pattern Formation and Dynamics in Nonequilibrium Systems; Cambridge University Press: New York, NY, USA, 2009. [Google Scholar]
- Hoyle, R.B. Pattern Formation: An Introduction to Methods; Cambridge University Press: New York, NY, USA, 2006. [Google Scholar]
- Newell, A.C. Envelope equations. Lect. Appl. Math.
**1974**, 15, 157–163. [Google Scholar] - Newell, A.C.; Passot, T.; Lega, J. Order parameter equations for patterns. Annu. Rev. Fluid Mech.
**1993**, 25, 399–453. [Google Scholar] [CrossRef] - Coullet, P.; Iooss, G. Instabilities of one-dimensional cellular patterns. Phys. Rev. Lett.
**1990**, 64, 866–869. [Google Scholar] [CrossRef] [PubMed] - Clerc, M.G.; Verschueren, N. Quasiperiodicity route to spatiotemporal chaos in one-dimensional pattern-forming systems. Phys. Rev. E
**2013**, 88, 052916. [Google Scholar] [CrossRef] [PubMed] - Coulibaly, S.; Clerc, M.G.; Selmi, F.; Barbay, S. Extreme events following bifurcation to spatiotemporal chaos in a spatially extended microcavity laser. Phys. Rev. A
**2017**, 95, 023816. [Google Scholar] [CrossRef] - Panajotov, K.; Clerc, M.G.; Tlidi, M. Spatiotemporal chaos and two-dimensional dissipative rogue waves in Lugiato-Lefever model. Eur. Phys. J. D
**2017**, 71, 176. [Google Scholar] [CrossRef] - Selmi, F.; Coulibaly, S.; Loghmari, Z.; Sagnes, I.; Beaudoin, G.; Clerc, M.G.; Barbay, S. Spatiotemporal chaos induces extreme events in an extended microcavity laser. Phys. Rev. Lett.
**2016**, 116, 013901. [Google Scholar] [CrossRef] [PubMed] - Aranson, I.S.; Kramer, L. The world of the complex Ginzburg–Landau equation. Rev. Mod. Phys.
**2002**, 74, 99–143. [Google Scholar] [CrossRef] - Chate, H. Spatiotemporal intermittency regimes of the one-dimensional complex Ginzburg–Landau equation. Nonlinearity
**1994**, 7, 185–204. [Google Scholar] [CrossRef] - Kuramoto, Y. Chemical Oscillations, Waves, and Turbulence; Springer: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Bensimon, D.; Shraiman, B.I.; Croquette, V. Nonadiabatic effects in convection. Phys. Rev. A
**1988**, 38, 5461–5464. [Google Scholar] [CrossRef] - Clerc, M.G.; Falcon, C.; Tirapegui, E. Additive noise induces front propagation. Phys. Rev. Lett.
**2005**, 94, 148302. [Google Scholar] [CrossRef] [PubMed] - Clerc, M.G.; Falcon, C. Localized patterns and hole solutions in one-dimensional extended systems. Physica A
**2005**, 356, 48–53. [Google Scholar] [CrossRef][Green Version] - Barbay, S.; Hachair, X.; Elsass, T.; Sagnes, I.; Kuszelewicz, R. Homoclinic Snaking in a Semiconductor-Based Optical System. Phys. Rev. Lett.
**2008**, 101, 253902. [Google Scholar] [CrossRef] [PubMed] - Solli, D.R.; Ropers, C.; Koonath, P.; Jalali, B. Optical rogue waves. Nature
**2007**, 450, 1054–1057. [Google Scholar] [CrossRef] [PubMed] - Akhmediev, N.; Kibler, B.; Baronio, F.; Belić, M.; Zhong, W.-P.; Zhang, Y.; Chang, W.; Soto-Crespo, J.-M.; Vouzas, P.; Grelu, P.; et al. Roadmap on optical rogue waves and extreme events. J. Opt.
**2016**, 18, 063001. [Google Scholar] [CrossRef][Green Version] - Lecaplain, C.; Grelu, P.H.; Soto-Crespo, J.M.; Akhmediev, N. Dissipative Rogue Waves Generated by Chaotic Pulse Bunching in a Mode-Locked Laser. Phys. Rev. Lett.
**2012**, 108, 233901. [Google Scholar] [CrossRef] [PubMed] - Bonatto, C.; Feyereisen, M.; Barland, S.; Giudici, M.; Masoller, C.; Leite, J.R.R.; Tredicce, J.R. Deterministic Optical Rogue Waves. Phys. Rev. Lett.
**2011**, 107, 053901. [Google Scholar] [CrossRef] [PubMed] - Bonazzola, C.R.; Hnilo, A.A.; Kovalsky, M.G.; Tredicce, J.R. Features of the extreme events observed in an all-solid-state laser with a saturable absorber. Phys. Rev. A
**2015**, 92, 053816. [Google Scholar] [CrossRef] - Elsass, T.; Gauthron, K.; Beaudoin, G.; Sagnes, I.; Kuszelewicz, R.; Barbay, S. Control of cavity solitons and dynamical states in a monolithic vertical cavity laser with saturable absorber. Eur. Phys. J. D
**2010**, 59, 91–96. [Google Scholar] [CrossRef] - Barbay, S.; Ménesguen, Y.; Sagnes, I.; Kuszelewicz, R. Cavity optimization of optically pumped broad-area microcavity lasers. Appl. Phys. Lett.
**2005**, 86, 151119. [Google Scholar] [CrossRef] - Dubbeldam, J.L.; Krauskopf, B. Self-pulsations of lasers with saturable absorber: Dynamics and bifurcations. Opt. Commun.
**1999**, 159, 325–338. [Google Scholar] [CrossRef] - Kharif, C.; Pelinovsky, E.; Slunyaev, A. Rogue Waves in the Ocean; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Pomeau, Y.; Manneville, P. Intermittent transition to turbulence in dissipative dynamical systems. Commun. Math. Phys.
**1980**, 74, 189–197. [Google Scholar] [CrossRef] - Bache, M.; Prati, F.; Tissoni, G.; Kheradmand, R.; Lugiato, L.A.; Protsenko, I.; Brambilla, M. Cavity soliton laser based on VCSEL with saturable absorber. Appl. Phys. B
**2005**, 81, 913–920. [Google Scholar] [CrossRef] - Elsass, T.; Gauthron, K.; Beaudoin, G.; Sagnes, I.; Kuszelewicz, R.; Barbay, S. Fast manipulation of laser localized structures in a monolithic vertical cavity with saturable absorber. Appl. Phys. B
**2010**, 98, 327–331. [Google Scholar] [CrossRef] - Chow, W.; Koch, S.; Sargent, M. Semiconductor-Laser Physics; Springer: Berlin, Germany, 1994. [Google Scholar]
- Nicolis, G. Introduction to Nonlinear Science; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Ott, E. Chaos in Dynamical Systems; Cambridge University Press: Cambridge, UK, 2002. [Google Scholar]
- Liu, Z.; Ouali, M.; Coulibaly, S.; Clerc, M.G.; Taki, M.; Tlidi, M. Characterization of spatiotemporal chaos in a Kerr optical frequency comb and in all fiber cavities. Opt. Lett.
**2017**, 42, 1063–1066. [Google Scholar] [CrossRef] [PubMed] - Manneville, P. Liapunov exponents for the Kuramoto-Sivashinsky model. In Macroscopic Modelling of Turbulent Flows; Springer: Berlin/Heidelberg, Germany, 1985; pp. 319–326. [Google Scholar]
- Metayer, C.; Serres, A.; Rosero, E.J.; Barbosa, W.A.S.; de Aguiar, F.M.; Rios Leite, J.R.; Tredicce, J.R. Extreme events in chaotic lasers with modulated parameter. Opt. Express
**2014**, 22, 19850–19859. [Google Scholar] [CrossRef] [PubMed] - Granese, N.M.; Lacapmesure, A.; Agüero, M.B.; Kovalsky, M.G.; Hnilo, A.A.; Tredicce, J.R. Extreme events and crises observed in an all-solid-state laser with modulation of losses. Opt. Lett.
**2016**, 41, 3010–3012. [Google Scholar] [CrossRef] [PubMed]

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

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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