Advances in Condensed Matter Physics

Volume 2019, Article ID 7632852, 6 pages

https://doi.org/10.1155/2019/7632852

## Extreme Events in Lasers with Modulation of the Field Polarization

^{1}Departamento de Fisica, Universidad de Buenos Aires, Intendente Guiraldes 2160, CABA, Argentina^{2}Universite de la Nouvelle Caledonie, ISEA, BP R4, 98851 Noumea Cedex, France

Correspondence should be addressed to Jorge R. Tredicce; rf.srnc.nlni@eccidert.egroj

Received 25 September 2018; Revised 13 December 2018; Accepted 6 January 2019; Published 4 February 2019

Academic Editor: Jan A. Jung

Copyright © 2019 Alexis Gomel et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We develop a theoretical model for a unidirectional ring laser consisting of an isotropic active medium inside a cavity containing a birefringent Kerr cell. We analyze the dynamical behavior of the system as we modulate the voltage applied to the Kerr cell. We discuss the bifurcation diagram and we study the regions of control parameter space where it becomes possible to observe and predict extreme events.

#### 1. Introduction

Lasers have been used as test benches for nonlinear dynamics in many different configurations, some of them requiring a complicated set up or involving a very large number of degrees of freedom. Lasers with optical feedback, laser with saturable absorbers, and lasers with large Fresnel number are typical examples appearing in recent literature. In particular, lasers with a modulated parameter are able to display a large variety of dynamical regimes [1–4]. Periodic behavior, period doubling transition to chaos [2–4], intermittency, crisis of chaotic attractors [5–7], and optical rogue waves [8–10] are among possible observed phenomena. Modulation of cavity losses [3], cavity length [4], and pump rate [11] have been reported as mechanisms generating chaotic behavior in Class B lasers.

Today there is an increasing interest on the study of the so called optical rogue waves. Optical rogue waves are high intensity pulses much larger than average and therefore rare events [8–10, 12–17]. Several optical systems have been reported [9, 13, 14, 18–20] as showing such type of pulses more frequently than what would be expected for a normal distribution probability of the light intensity. However the analysis of the physical mechanism at the origin of extreme events remains difficult in those experiments and models.

Here we analyze a theoretical model of a laser in which the modulation is applied to the relative phase between the two components of the linear polarizations of the field. Modulation of such parameter is usually achieved by introducing inside the cavity a birefringent material whose extraordinary refractive index is changed through a sinusoidal voltage. If we assume that the active medium and the cavity are isotropic, the laser may operate in principle at any polarization of the field. We identify the existence of generalized multistability, period doubling transition to chaos, and three types of crises of strange attractors. However the main objective of this work is to show the appearance of optical rogue waves and to identify the physical mechanism at their origin. Special interest is put also in establishing our ability to predict them [21–25]. We establish clearly the relevance of the existence of generalized multistability and the role played by an external crisis of the chaotic attractor in order to generate optical rogue waves. We construct also bifurcation diagrams taking the amplitude of the modulation as the main control parameter. All other parameter values like gain, losses, and dissipation are compatible with Class B lasers [26].

#### 2. Theoretical Model

The theoretical model is based on a single mode, Class B unidirectional ring laser with an electro optic modulator (EOM) placed inside the cavity. After applying the rotating wave and slowly varying amplitude approximations and without taking into account diffraction, the set of Maxwell-Bloch equations describing the interaction between a single mode electromagnetic field and a two-level atom arewhere and are the vectorial form of the electromagnetic field and the atomic polarization respectively; is the atomic population inversion, is the cavity loss rate, is the detuning between the atomic frequency and the cavity resonance, is the population inversion given by the pump, and is the relaxation rate of the population inversion. It is worthwhile noting that time and all relaxation rates in (1) are normalized to the atomic linewidth . A birefringent material (EOM) placed inside the cavity changes the relative phase of one component of the linear polarization with respect to the other without changing their amplitudes. Writing the electromagnetic field at the input of the EOM asit becomes at the output:where and are the components of the electric field in the direction of the axis corresponding to the ordinary and extraordinary indexes of refraction of the EOM crystal, respectively, and for the EOM is given bywhere is the voltage making a variation on the birefringence equivalent to a phase shift of , is the medium birefringency at , and are the extraordinary and ordinary indexes of refraction, respectively, is the length of the EOM crystal, and is the wavelength. If the voltage applied to the EOM changes sinusoidally with frequency , (4) produces a periodic modulation of . The relative phase of the y-component with respect to the x-component of the field is then changed bywhere is the modulation amplitude normalized to the free spectral range of the cavity. Thus implies a change in frequency corresponding to the intermode spacing. Introducing this effect into the boundary conditions for the field, we obtain, after some algebra, the following set of equations: It is worthwhile noting that an adiabatic elimination of the atomic polarization is not easy to perform even if it decays much faster than the other variables because the phase is modulated and therefore the frequency of the electromagnetic field and the atomic polarization depends on time. We notice also that the modulation appears only in the equation of the y-component of the field polarization because we choose x and y as the axes corresponding to the ordinary and extraordinary indexes of refraction of the electrooptic device.

#### 3. Results

Figure 1 shows a typical bifurcation diagram corresponding to the dynamical system described above. We plot the maxima of the intensity, I, as a function of the amplitude of the modulation, m, for a fixed modulation frequency and detuning.