To be updated until april 30th

Thorsten Ackemann

## Complex solitons in vertical-cavity surface-emitting lasers with frequency-selective feedback

Thorsten Ackemann
thorsten.ackemann@strath.ac.uk
University of Strathclyde, SUPA and Department of Physics, 107 Rottenrow East, Glasgow G4 0NG, Scotland, UK
Broad-area vertical-cavity surface-emitting lasers (VCSELs) with feedback from an external cavity emerged as a highly controllable and versatile tool to investigate spatio-temporal self-organization. The external cavity provides natural means for control and to induce time-dependent behavior, which – via space-time analogy in systems with delay- can lead to quasi-3D spatial dynamics, in particular spatio-temporal solitons and potentially light bullets. The high circular symmetry of VCSELs also allows for the investigation of vectorial effects in light-matter coupling. I will present investigations on the spontaneous appearance of vector vortex beams in a VCSEL with frequency-selective feedback with a radial, spiral and hyperbolic polarization structure and their interpretation as high order vectorial solitons. The role of the feedback loop in controlling birefringence will be elucidated. Potential extensions to spin-orbit coupling of light, topological states and the inclusion of carrier spin are discussed as well as mode-locking these states.

Shalva Amiranashvili

## Controlling light by light

Shalva Amiranashvili
shalva.amiranashvili@wias-berlin.de
Weierstrass Institute, Mohrenstr. 39, 10117 Berlin, Germany
We discuss propagation of ultrashort solitary pulses in nonlinear single-mode optical fibers. Each pulse creates a moving perturbation of the refractive index; the perturbation is capable to scatter co-propagating pulses. An ultrashort optical soliton serves, under suitable conditions, as an impenetrable mirror for the group-velocity matched small-amplitude waves. Reflection of such waves by a quickly moving mirror in dispersive media is a rich source of the intriguing phenomena including analogue event horizons and radiation at negative frequencies. On the other hand, energy exchange between the scattered pump waves and the soliton provides an effective way to manipulate the soliton, e.g., to fix its frequency or to compress it to a large extent.

Pierre Azam

## Quantum Fluids of light in atomic vapors

Pierre Azam
pierre.azam@inphyni.cnrs.fr
Université Côte d'Azur, INPHYNI, CNRS, 1361 route des lucioles, 06560 Valbonne, France

Since its discovery in 1995, Bose-Einstein Condensation (BEC) is a powerful object for quantum experiments. Its coherence offers a lot of possibilities for measuring quantum phenomena. Even though BEC is well studied with ultracold atoms cloud, an analogy for classical waves propagating in a non-linear medium can be established and condensation of classical waves has been predicted. Our experiment is based on the use of an atomic vapor as a non linear medium. By heating a Rubidium cell, we create a nonlinear medium with adjustable non linearity. By modifying the properties of the incident laser beam (shape, size, frequency, etc) we are able to study a wide range of phenomena. After the observation of precondensation of classical waves in this system, we turned to a study of shock wave creation in this system. We will present first results on this investigation, including numerical and experimental comparisons.

## Towards the generation of light-bullets in semiconductor lasers

niversité Côte d'Azur, INPHYNI, CNRS, 1361 route des lucioles, 06560 Valbonne, France

Localized structures (LS) are nonlinear solutions of dissipative systems characterized by a correlation range much shorter than the size of the system. Since they are individually addressable, LS can be used as fundamental bits for information processing in optical resonators. While spatial LS are confined peaks of light appearing in the transverse section of broad-area resonators, temporal LSs are short pulses travelling back and forth in the longitudinal direction of the cavity. Spatial and temporal LS have been observed independently in semiconductor lasers systems based on a gain medium coupled to a saturable absorber.

In this work we present preliminary results for the generation of spatio-temporal localized structures, also called “Light bullets”, in semiconductor lasers. In this case, light is stored in the three spatial dimensions, leading to information processing with disruptive performances in terms of bit rate, resilience and agility. Despite the effort made in nonlinear optics, only fading LB have been observed so far experimentally. Our approach consists of chasing “dissipative” LB, which will be robust and suitable to applications. Accordingly, once LB will be obtained and characterized, their application to information processing will be addressed by targeting a three-dimensional electro/optical buffer. The results shown were obtained using a vertical external cavity surface emitting laser, composed by a gain mirror and a semiconductor saturable absorber mirror (SESAM). These components have been properly engineered for matching the parameters requirements for implementing light bullets, which require a cavity roundtrip time much larger than the carrier relaxation time, a large Fresnel number and a bistable response of the system. We show that self-imaging condition between the gain section and the SESAM enables the first two conditions, while bistability can be obtained by designing the modulation depth of the SESAM.

Neil Broderick

## Soliton Explosions and Optical Rogue Waves

Neil Broderick
n.broderick@auckland.ac.nz
Dodd-Walls Centre for Photonic and Quantum Technologies, Department of Physics, University of Auckland
We will present our recent results regarding the observation of soliton explosions in an all-normal dispersion fibre laser. Using a real time dispersive Fourier transform we were able to make single shot measurements of the spectrum showing how the explosion happens in frequency space and how this translates to the temporal behaviour of the pulse. Simulations of the generalised nonlinear Schrodinger equation agree well with the experimental results and highlight the regions between stability and chaos in such systems. Further investigations highlight the presence of optical rogue waves and chimera states in such a laser which will be discussed in the presentation.

Stephane Coen

## Asymmetric balance in symmetry breaking

Stephane Coen
s.coen@auckland.ac.nz
The University of Auckland, Physics Department, Auckland, New Zealand, and Dodd Walls Centre
Spontaneous symmetry breaking is central to our understanding of physics and explains many natural phenomena, from cosmic scales to sub-atomic particles. Its use for applications requires devices with a high level of symmetry, but engineered systems are always imperfect. Surprisingly, the impact of such imperfections has barely been studied, and restricted to a single asymmetry. Here, we experimentally study spontaneous symmetry breaking in presence of two controllable asymmetries. We remarkably find that the characteristic features of spontaneous symmetry breaking, while dramatically destroyed by one asymmetry, can be entirely restored when a second asymmetry is introduced. In essence, asymmetries are found to balance each other. Our study illustrates aspects of the universal unfolding of the pitchfork bifurcation, and provides new insights into a key fundamental process. It also has practical implications, showing that asymmetry can be exploited as a new degree of freedom. In particular, it would enable sensors based on symmetry breaking or exceptional points to reach divergent sensitivity even in presence of imperfections. Our experimental implementation built around an optical fibre ring additionally constitutes the first observation of the polarization symmetry breaking of passive driven nonlinear resonators.

Germán J. De Valcárcel

## Laser modelocking beyond Haus: the coherent master equation

Germán_J. De_Valcárcel
german.valcarcel@uv.es
Universitat de València, Departament d'Òptica, Dr. Moliner 50, 46100 Burjassot, Spain
Haus master equation (HME) is the standard theoretical approach to laser modelocking which has proven successful in many different situations. However HME is unable to account for light-matter coherent effects which are at the basis of certain types of laser modelocking and frequency comb generation in, e.g., quantum-dot and quantum-cascade lasers. Here we present a new theoretical framework for master equation laser modelocking modeling, which consistently incorporates coherent effects: the coherent master equation (CME). As a proof of its coherent nature, CME captures the Risken-Nummedal-Graham-Haken self-modelocking instability. We apply the new approach to an amplitude-modulated modelocked laser, whose CME yields predictions that differ from HME which are most prominent when the gain recovery time is comparable or shorter than the cavity roundtrip time. These divergent predictions include the existence of non-Gaussian pulses and the asymmetric effect of the modulation frequency, which are verified experimentally in an actively modelocked semiconductor laser with a long external cavity.

Pascal DelHaye

## Nonlinear Interaction and Symmetry Breaking of Light in Optical Microresonators

Pascal Del'Haye
pascal.delhaye@npl.co.uk
National Physical Laboratory (NPL), Hampton Road, Teddington, TW11 0LW, UK
Ultra-high-Q microresonators can confine extremely large amounts of optical energy in tiny mode volumes. This talk will focus on recent realizations of nonlinear interaction of counterpropagating light in these resonators. Particularly, above a certain threshold power, light of a given frequency can only circulate in one direction. Experimental and theoretical results show spontaneous symmetry breaking that follows from the interaction of the counterpropagating light. The resulting nonreciprocity of the light propagation in the microresonators can be used for novel applications including integrated photonic isolators and circulators.

Eugenio Delre

## Giant broadband refraction and nonlinear optics in ferroelectric super-crystals

Eugenio Delre
eugenio.delre@uniroma1.it
Dipartimento di Fisica, Università di Roma La Sapienza, 00185 Rome, Italy
We review recent progress in the study of linear and nonlinear propagation in ferroelectric super-crystals. The three-dimensional super-lattice of spontaneous polarization vortices leads to giant broadband refraction across the entire visible spectrum. Absence of diffraction, chromatic dispersion, and propagation normal to the crystal facets is compatible with giant values of index of refraction larger that 25. The huge values of optical susceptibility greatly enhance second-harmonic generation efficiency with broadband spectral and angular acceptance.

Cornelia Denz

## Spatio-temporal molding of light in caustic networks

Cornelia Denz
denz@uni-muenster.de
Institute of Applied Physics, University of Muenster, Corrensstr. 2-4, 48149 Münster, Germany

Caustic light revolutionized optics in the last decade in the areas of structured light and random waves. On the one hand, tailored caustic beams serve as fabricating light for (nonlinear) material processing, transfer complex momentum flows for advanced micro-manipulation, and enable novel high-resolution imaging methods. On the other hand, the random focusing of light rays forms networks of caustics that appear as high-intensity ramifications in many optical systems. This linear focusing, caused by strong wavefront aberrations and denoted as branched flow, yields waves with extreme amplitudes – so called rogue waves, originally studied in oceanography. Optics has proven to be a vast testbed to investigate different linear and nonlinear mechanisms for the formation of rogue waves as spatio-temporal wave phenomena. Though there are indications that the two different mechanisms described above, branched flows and nonlinear modulation instabilities, contribute to the formation of rogue waves, the influence of their mutual interplay on the rogue wave statistic is still an open question.

In our contribution, we exploit a nonlinear photorefractive material as an optical platform to investigate these different mechanisms for rogue wave formation simultaneously in a single system. We show that free-space branched flows of light caused by wavefront distortions in form of correlated Gaussian random fields (GRFs) focus to caustic networks with controllable extension and sharpness, which in turn determine the probability for the occurrence of optical rogue waves. This focusing can be enhanced by propagating GRFs in a nonlinear refractive index structure with focusing nonlinearity. Beyond propagating in homogeneous media, we fabricate two-dimensional tailored photonic disorder in such a photorefractive crystal and investigate the mutual interplay of linear focusing by GRFs and scattering. We find optimal conditions for enhanced focusing of waves with extreme intensities by controlling the size and strength of the disordered photonic refractive index structure.

Thus, in our contribution, we will link different mechanisms for rogue wave formation that are commonly studied separately and discuss their interplay. Our work demonstrates that different focusing mechanisms can enhance or depress the formation of rogue waves, thereby introducing an optical platform that allows exploring rogue waves far beyond the optical realization, and allows new insights into general spatio-temporal wave dynamics.

Lorenzo Dominici

## Nonlinear polariton fluids

Lorenzo Dominici
lorenzo.dominici@gmail.com
CNR NANOTEC, Institute of Nanotechnology

Polaritons are very interesting quasiparticles, that are generated in semiconductors as a hybrid mixture of light and the material’s optical excitation. They inherit a strong nonlinearity from the exciton component while keeping a high coherence as well as a nonparabolic dispersion from the photon counterpart. These features can activate, among other effects, Bose-Einstein condensation [1], nonlinear quantum fluid dynamics [2] and even quantum correlations [3]. In this talk we will show variegated nonlinear spatiotemporal reshaping phenomena in microcavity polaritons, where the whole fluid can be described by a collective wavefunction characterised by bistability regions, solitons and quantum vortices. We will also discuss the fundamental repulsive nonlinearity of exciton-polaritons, which can trigger the formation of two-dimensional X-waves [4], or ignite expanding shock waves and sustain stable dark soliton rings [5]. In particular, we will describe a novel effect of retarded nonlinearity inversion, that results in the dynamical formation of a bright soliton [5]. The simultaneous presence of the central density singularity and the radially-expanding cloud recall the exotic structures that are also seen in condensed matter bosonic supernovas. Finally, we will show how we can seed and track quantum vortices in the polariton fluid on the picosecond timescale. These quantum vortices are characterized by a central phase singularity surrounded by an azimuthally-winding cloud. The observations highlight a rich nonlinear phenomenology, such as the vortex spiralling, splitting, and the ordered branching into newly generated secondary couples [6]. These events remind of the particle pair generation effect. Remarkably, we also observe that vortices placed in close proximity experience attractive-repulsive scenarios. Such nonlinear vortex pair-interactions can be described by a tuneable effective potential [7], reminiscent of Lennard-Jones potential existing between molecules.

[1] Kasprzak et al., Nature 443, 409 (2006)

[2] Lerario et al., Nat. Phys. 13, 837 (2017)

[3] Delteil et al., Nat. Materials 18, 219 (2019)

[4] Gianfrate et al., Light Sci. Appl. 7, e17119 (2018)

[5] Dominici et al., Nat. Commun. 6, 8993 (2015)

[6] Dominici et al., Sci. Adv. 1, e1500807 (2015)

[7] Dominici et al., Nat. Commun. 9, 1467 (2018)

John M. Dudley

## Real-time measurement of instabilities in optical fibres and optical fibre lasers

John_M. Dudley
john.dudley@univ-fcomte.fr
FEMTO-ST Institute, UMR 6174 CNRS-Université Bourgogne Franche-Comté, Besançon, France
There have been many recent dramatic advances in the real-time measurement of ultrafast non-repetitive optical signals based on the use of the dispersive Fourier transform in the frequency domain, or time-lens approaches and related techniques in the time domain. In the context of propagation in nonlinear optical fibres, these real-time methods were initially used to study modulation instability, supercontinuum generation and rogue-wave phenomena, but were rapidly applied to study instabilities in modelocked lasers. In this presentation, we will review our recent work in this area, including results studying single-pass instabilities in optical fibre, as well as recent work studying complex pulse evolution behavior observed during the generation of dissipative soliton structures in a fiber laser. These results provide a unique picture of the internal evolution of dissipative solitons in a laser system, and we anticipate further applications in understanding the underlying laser dynamics and optimizing laser performance and stability.

Daniele Faccio

## Analogue gravity in rotating spacetimes

Daniele Faccio
daniele.faccio@glasgow.ac.uk
University of Glasgow

Superradiant gain is the process in which waves are amplified via their interaction with a rotating body, examples including evaporation of a spinning black hole and electromagnetic emission from a rotating metal sphere. We will first discuss the case of photon fluids, i.e. room temperature superfluids generated by a laser beam propagating in a nonlinear defocusing material. Prior work has already demonstrated the superfluid nature of the 2D beam profile in this setting and we have recently studied that by injecting a vortex pump beam, it is possible to generate a rotating spacetime metric and experimentally identify the horizon and ergosphere. Numerical studies based on the Nonlinear Schrodinger equation now illustrate the conditions under which experiments are expected to observe superradiance by analyzing the optical currents in the system. Finally, we will examine a different scenario, more akin to the sutation examined in 1971 by Zel’dovich, i.e. a rotating cylinder. We elucidate theoretically how superradiance may be realized in the field of acoustics, and predict the possibility of non-reciprocally amplifying or absorbing acoustic beams carrying orbital angular momentum by propagating them through an absorbing medium that is rotating. We discuss a possible geometry for realizing the superradiant amplification process using existing technology.

Goery Genty

## Predicting Extreme Events in Modulation Instability Using Machine Learning

Goery Genty
goery.genty@tuni.fi
Tampere University, Photonics Laboratory, FI-33104 Tampere, Finland

The study of instabilities that drive extreme events is central to nonlinear science. Perhaps, the most canonical form of nonlinear instabilities is modulation instability (MI) describing the exponential growth of a weak perturbation on top of a continuous background. In optical fibres, when driven initially by small-amplitude noise, MI has been shown to lead to the emergence of localized temporal breathers with random statistics. It has also been suggested that these dynamics may be associated with the emergence of extreme events or rogue waves [1,2]. However, direct measurement in the time-domain of the breather properties is extremely challenging, requiring complex time-lens systems that typically suffer from drastic experimental constraints [3,4]. Real-time spectral measurement techniques such as the dispersive Fourier transform (DFT) on the other hand are commonly used to measure ultrafast instabilities [5]. Although relatively simple to implement, the DFT only provides spectral information. Here, we show how machine learning can overcome this restriction to study time-domain properties of optical fibre modulation instability based only on spectral intensity measurements. Specifically, we demonstrate that it is possible to train a supervised neural network to correlate the spectral and temporal properties of modulation instability using numerical simulations, and then apply the trained neural network to the analysis of high dynamic range experimental MI spectra and yield the temporal probability distribution for the highest peaks in the instability field [6].

[1] D.R. Solli, C. Ropers, P. Koonath and B. Jalali, "Optical rogue waves", Nature 450, 1054-057 (2007).

[2] J.M. Dudley, F. Dias, M. Erkintalo, and G. Genty, "Instabilities, breathers and rogue waves in optics," Nat. Photonics 8, 755–764 (2014).

[3] K. Goda and B. Jalali, "Dispersive Fourier transformation for fast continuous single-shot measurements", Nat. Photon. 7, 102-112 (2013).

[4] M. Närhi, et al. "Real-time measurements of spontaneous breathers and rogue wave events in optical fibre modulation instability," Nat. Commun. 7, 13675 (2016).

[5] P. Suret et al., "Single-shot observation of optical rogue waves in integrable turbulence using time microscopy," Nat. Commun. 7, 13136 (2016).

[6] M. Narhi et al., ''Machine learning analysis of extreme events in optical fibre modulation instability,'' Nat. Commun. 9, 4923 (2018)

Giovanni Giacomelli

## The LANER: optical networks as complex lasers

Giovanni Giacomelli
giovanni.giacomelli@isc.cnr.it
Istituto dei Sistemi Complessi -CNR, via Madonna del Piano 10, 50019 Sesto F.no, Firenze (Italy)
We present the main features of a recently introduced system capable of laser action: the complex active optical network, or lasing network (LANER). The system is experimentally realized with optical fibers linked each other with couplers and with one or more coherently amplifying sections. A linear theoretical description shows how the LANER can be considered as a generalization of the laser with the physical network acting as a complicated cavity, and can be represented by directed graphs disclosing the analogies with the problem of quantum chaos on graphs. Experiments in simple configurations are reported, with evidence of lasing action and its characterization. Examples of spectra of the detected emitted intensity are obtained in different cases, in a phenomenological agreement with the numerical findings of the theory.

Mathieu Isoard

## Short-distance propagation of nonlinear optical pulses

Mathieu Isoard
mathieu.isoard@u-psud.fr
LPTMS, UMR 8626, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
We theoretically describe the quasi one-dimensional transverse spreading of a light pulse propagating in a defocusing nonlinear optical material in the presence of a uniform background light intensity. For short propagation distances the pulse can be described within a nondispersive approximation by means of Riemann's approach. We are also able to calculate the wave-breaking time, at which nonlinear nondispersive spreading leads to a gradient catastrophe. The theoretical results are in excellent agreement with numerical simulations. Experimental and theoretical studies have demonstrated the occurence of wave breaking even in absence of background. Our results exhibit this feature and the corresponding theoretical wave-breaking time agrees very well with simulations.

Raphael Jauberteau

## 2D spatiotemporal extreme event in quadratic nonlinear crystal

Raphael Jauberteau
raphael.jauberteau@etu.unilim.fr
Dipartimento di Ingegneria dell’Informazione, Università di Brescia, 25123 Brescia, Italy / Université de Limoges, Institut Xlim, UMR CNRS 7252, 87060 Limoges, France

Solitonic waves are nonlinear self-sustained waves observable in a large number of conditions and various fields of physics, from electronics to optics via fluidics. Quadratic quasi-solitons have been early predicted by Karamzin et al. [1] and later observed by Torruellas et al. [2]. These types of self-guided beams have been seen, after modulation instability, in 2D spatial structures [3]. More recently, it has been shown that Peregrine solitons, and Akhmediev Breathers, could be obtained in quadratic materials [4].

In this paper we show spontaneous 2D quadratic extreme events, generated and controlled with non-collinear beams. We launched a large collimated beam (R = 200 µm, 30 ps) in a 8X8X30 mm KTP crystal cut for type II second harmonic generation. Beams first experienced a strong self-focusing leading to a stable 2D confined propagation. Because of the spatial walk-off due to the nonlinear crystal anisotropy, the trapped beams come with spatial reorientation, controlable by the initial polarization state. Additional self-confined events can appear in the transverse output pattern by increasing the input peak power. Such nonlinear spatial reshaping of the initial beam can also provide a way to control the apparition of 2D nonlinear periodic structures, a situation that reminds the Akhmediev Breathers solution, only valid in 1D.

These effects could be used to implement all-optical logic functions with ultrafast switching, but also to mimic the effect of a nonlinear saturable absorber able to realize ultrafast temporal pulse reshaping. The self-trapping process acts like a spatial self-cleaning process, which changes a set of initial non-collinear beams into a single one.

[1] Yu. N. Karamzin et al., Sov. Phys. JETP 41,414 (1976).

[2] W. E. Torruellas et al., Phys. Rev. Lett. 74, 5036 (1995).

[3] M. Delqué, et al., Optics Comm. 284, 1401–1404 (2011).

[4] F. Baronio et al., Opt. Lett., 42, 1756-1759 (2017).

Julien Javaloyes

## Third Order Dispersion in Time-Delayed Systems: Applications to the Passive Mode-locking of VECSELs

Julien Javaloyes
julien.javaloyes@uib.es
Dept. de Física & IAC3 - Universitat de les Illes Balears. Cra. de Valldemossa, km 7.5 E-07122 Palma, SPAIN.

Time-Delayed dynamical systems (DDSs) materialize in situations where distant, point-wise, nonlinear nodes exchange information that propagates at a finite speed. They describe a large number of phenomena in nature and they exhibit a wealth of dynamical regimes such as localized structures, fronts and chimera states. A fertile perspective lies in their interpretation as spatially extended diffusive systems which holds in the limit of long delays. However, DDSs are considered devoid of dispersive effects, which are known to play a leading role in pattern formation and wave dynamics. In particular, second order dispersion in nonlinear extended media governs the Benjamin-Feir (modulational) instability and also controls the appearance of cavity solitons in injected Kerr fibers. Third order dispersion is the lowest order non-trivial parity symmetry breaking effect, which leads to convective instabilities and drifts.

In this contribution, we review our recent results regarding how second and third order dispersion may appear naturally in DDSs by using a more general class of Delayed Systems, the so-called Delay Algebraic Delay Differential Equations. This class of DDS appears for instance in the modeling of Vertical External-Cavity Surface-Emitting Lasers (VECSELs) and we illustrate our general result studying the effect of third order dispersion onto the optical pulses found in the output of a passively mode-locked VECSEL and link our results with the Gires-Tournois interferometer. We show that third order dispersion leads to the creation of satellites on one edge of the pulse which induces a new form of pulse instability. Our results are in good agreement with the experiment. Finally, we connect these results with the possibility of obtaining Light bullets, that is to say, pulses of light that are simultaneously confined in the transverse and the propagation directions, in mode-locked VECSELs.

Dmitry Krizhanovskii

## Nonlinear polariton phenomena in semiconductor microcavities and slab waveguides

Dmitry Krizhanovskii
d.krizhanovskii@sheffield.ac.uk
Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, United Kingdom

When light propagates through an optically active semiconductor material hybridisation of the optical and electronic excitations (photons and excitons) may occur. This leads to the formation of novel quasi-particles, so-called polaritons. The exciton component in the polariton wavefunction leads to giant repulsive interactions between the two colliding quasi-particles (giant Kerr-like nonlinearity), which enable control of light by light at ultrafast speeds. This is potentially useful for applications in all-optical signal processing. The strong polariton nonlinearity also results in many-body phenomena ranging from superfluid-like behaviour of light to Bose-Einstein condensation and ultra-low power soliton physics which develop on short time- and length-scale at very weak excitation powers. In my talk I am going to review several nonlinear polariton phenomena including backward Cherenkov radiation by polariton solitions, spin domain formation, vortex-vortex generation, polygon pattern formation and spatio-temporal continuum generation [1-5] .

References:

1. “Spatiotemporal continuum generation in polariton waveguides” PM Walker et al., DN Krizhanovskii Light: Science & Applications 8 (1), 6 (2019)

2. “Spin domains in one-dimensional conservative polariton solitons” M Sich et al., DN Krizhanovskii ACS Photonics 5 (12), 5095-5102 (2019)

3. “Backward Cherenkov Radiation Emitted by Polariton Solitons in a Microcavity” D. V. Skryabin, Y. Kartashov, O. Egorov, D. Krizhanovskii, M. Sich, J. Chana, L. E. Tapia-Rodriguez, M. S. Skolnick, P. M. Walker, E. Clarke, and B. Royall. Nature Comm. 8, 1554 (2017)

4. “Transition from propagating polariton solitons to a standing wave condensate induced by interactions” M Sich, JK Chana, et al., D N Krizhanovskii Phys. Rev. Letters 120 (16), 167402 (2018)

5. “Ultra-low-power hybrid light–matter solitons”, P. M. Walker, L. Tinkler, D. V. Skryabin et al., and D. N. Krizhanovskii, Nature Comm. 6, 8317 (Oct 2015)

Cristina Masoller

## Statistical properties of the speckle pattern at the output of a multimode optical fiber

Cristina Masoller
cristina.masoller@upc.edu
Departament de Física, Universitat Politècnica de Catalunya. Rambla St. Nebridi 22, Terrassa 08222, Barcelona, Spain
Speckle patterns are intensity patterns produced by coherent waves interfering with each other. They typically occur due to reflections of coherent laser light in rough surfaces or in media with scattering particles on the scale of the wavelength. Speckle is often undesired in imaging because of the grainy image produced. On the other hand, the spatial correlations present in the speckle pattern contain information that can be used to reconstruct the object that generates the speckle. Low-cost vibration sensors have been demonstrated, which measure the frequency of the vibrations by monitoring the speckle pattern that changes in time. As speckle patterns are wavelength-dependent, after calibration they can also be exploited for implementing low-cost high precision wavemeter. In this contribution we study experimentally how the statistical properties of the speckle pattern at the output of a multimode fiber, generated by using as light source a diode laser in the visible range, depend on the exposure time of the CCD camera and on the degree of coherence of the light, which is controlled by varying the laser pump current from below to above the threshold. Using the standard speckle contrast measure (the mean intensity of the pattern normalized to the standard deviation), we determine under which conditions the speckle pattern can be either minimized or maximized.

Guy Millot

## Parametric interactions in multimode fibers

Guy Millot
Guy.Millot@u-bourgogne.fr
Université Bourgogne Franche-Comté, ICB, UMR CNRS 6303, 9 Avenue A. Savary, 21078 Dijon, France

Over the last few years, it has been demonstrated that multimode fibers (MMFs) offer novel opportunities to explore the nonlinear coupling between the temporal and spatial effects. In particular, the process of periodic self-imaging (SI) of light occurring inside graded-index (GRIN) MMFs has been found to play a major role in the nonlinear propagation of optical pulses with normal dispersion. In this talk, we focus on the spectral evolutions of an input narrowband multimode beam induced by the SI effect. First, we show that when a large number of modes is initially excited in a highly multimode fiber, SI leads to an original phenomenon of geometric parametric instability characterized by the generation of an intense frequency comb spanning from the near-ultraviolet to the near infrared. On the other hand, for powerful pulses, all parametric sidebands are characterized by a bell-shape beam similar to that emerging from a single-mode fiber. By limiting the nonlinear interactions to the lowest order fiber modes only, we study the influence of a superimposed seed centered on the first-order parametric Stokes sideband, on the efficiency of the multiple sideband generation processes. We show that the injected seed can stimulate the generation of new spectral sidebands in the visible and near-infrared regions of the spectrum. The second part of the talk is dedicated to intermodal four-wave-mixing and modulational instability that occur in a few-mode GRIN fiber. We show that far-detuned (from 200 up to 450 THz) frequency conversion is obtained via intermodal four-wave-mixing with an important role played by a secondary pump in the subsequent supercontinuum generation. Moreover, we observe a strong power dependence of intermodal modulational instability. Finally, we introduce the concept of spectral control of parametric sidebands in GRIN MMFs by tailoring their linear refractive index profile with a Gaussian dip into the refractive index profile.

Arnaud Mussot

## Symmetry breaking of the non nonlinear stage of modulation instability : a complete experimental characterization in optical fibers

Arnaud Mussot
arnaud.mussot@univ-lille.fr
Université de Lille, PHLAM/IRCICA
We report an original method enabling a non invasive characterization in phase and intensity of the longitudinal evolution of the main spectral components involved in the Fermi Pasta Ulam recurence process. We will show that it allows to evidence the symmetry breaking of the process. Future prospects and recent results will be presented.

Alioune Niang

## Combination of Kerr Beam Self-Cleaning and Supercontinuum Generation in Tapered Ytterbium-doped Multimode Fiber with Parabolic Core Refractive Index and Doping Profile

Alioune Niang
alioune.niang@unibs.it
Dipartimento di Ingegneria dell’Informazione, Università degli Studi di Brescia, via Branze 38, 25123, Brescia, Italy
The non-linear multimode optical fibers are opened a new window to study the spectral, spatial and temporal degrees of freedom of light beams that has been received a great fundamental and applicative interest during the last decades. In this article, we demonstrate spatial beam self-cleaning and supercontinuum generation in a new type of multimode fiber amplifier, consisting of a Ytterbium-doped (Yb-doped) multimode fiber taper with parabolic index refractive and doping profile, and a length of 9.5 m with a core diameter exponentially decreasing along its length from 120 to 40 microns. The beam self-cleaning, a bell-shaped output beam profile, has been achieved in passive configuration with an input beam peak power threshold of 20 kW and further increasing the input power causes no significant frequency conversion that can be attributed to the first Raman Stokes sideband. In active configuration, the gain leads to combine self-cleaning with supercontinuum generation, spanning from the visible to the mid-infrared (520-2600 nm), which is due to the geometric parametric instability and the Raman stokes sideband. In both configurations, the self-beam cleaning in the tapered fiber can be ascribed to the accelerated self-imaging. Finally, we studied the evolution of self-cleaning and super continuum generation as a function of taper length in active configuration to analyze the spatial and spectral beam dynamics resulting accelerated self- imaging. We observed that the speckled output spatial distribution in the first meters evolved into a dual lobe, LP11 mode, and finally into the fundamental mode (LP01). The results obtained confirm the combination of accelerated self-imaging with landscape dissipative in the tapered Yb-doped multimode fiber with graded index profile that leads to control spectral and spatial light beams in the active mode.

Gian-Luca Oppo

## Rotating spatio-temporal structures and rotating cavity solitons in scalar and vectorial Kerr resonators

Gian-Luca Oppo
g.l.oppo@strath.ac.uk
University of Strathclyde, Department of Physics, Glasgow, G4 0NG, Scotland, E.U.

We consider generalisations of the Lugiato-Lefever models for transverse Kerr cavities with one or two field components and pumped by beams carrying optical angular momentum (OAM). These studies complete early investigations that focused on optical parametric oscillators, semiconductor heterostructures and photorefractive materials, respectively [1]. In particular we find analytical expressions that fully describe two-dimensional rotating Turing structures and rotating cavity solitons in single field (scalar) Kerr resonators. Rotating localised states on a transverse ring can be considered as slow light pulses with fully controllable speed and structure for use in optical quantum memories and delay lines.

The inclusion of a second field component in the light-matter interaction inside the cavity offers further degrees of control in the shape, rotation and polarization of the nonlinear structures. Numerical simulations of coupled circularly polarized beams with inputs of equal, opposite and different OAM, result in fully-structured optical beams made of periodic or localised nonlinear structures and a multitude of shapes, phases, polarization, singularities and dynamics. Applications of these rotating structures to particle manipulation, optical beam shaping and photonic devices will also be discussed.

[1] G.-L. Oppo et al., Phys. Rev. E 63, 066209 (2001); R. Kheradmand et al., Opt. Express 11, 3612 (2003); V. Caullet et al., Phys. Rev. Lett. 108, 263903 (2012)

Nicolas Pavloff

## Non Linear Diffraction

Nicolas Pavloff
nicolas.pavloff@u-psud.fr
LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
In a nonlinear medium, the phase and amplitude of a coherent beam are affected by nonlinearity. This modifies the laws of geometrical optics and may also lead to a gradient catastrophe resulting in complex optical patterns. I will discuss two such configurations resulting in the formation of a dispersive shock in a local nonlinear medium and of an Airy beam in a highly nonlocal one.

Auro Michele Perego

## Dissipation induced modulation instabilities: gain-through-losses in nonlinear optics

Auro_Michele Perego
a.perego1@aston.ac.uk
Aston University, Aston Institute of Photonics Technologies, B4 7ET Aston Triangle, Birmingham, UK
We present results on a dissipative modulation instability caused by the presence of asymmetric spectral losses for signal and idler waves. Such instability can occur without satisfying standard phase-matching conditions and has applications in a variety of nonlinear optical systems especially in the design of a novel class of fiber optics parametric amplifiers, optical parametric oscillators and optical frequency combs sources.

Antonio Picozzi

## Disorder-induced acceleration of wave condensation in multimode fibers

Antonio Picozzi
Antonio.Picozzi@u-bourgogne.fr
Université Bourgogne Franche-Comté, ICB, 9 Av. A. Savary, 21078 Dijon, France

Recent studies on wave turbulence revealed that a purely classical system of random waves can exhibit a process of condensation that originates from the divergence of the Rayleigh-Jeans (RJ) equilibrium distribution, in analogy with the quantum Bose-Einstein condensation (see references in [1]). However, the observation of optical wave condensation in a conservative (cavity-less) configuration is hindered by the prohibitive large propagation lengths required to achieve the RJ thermalization.

A phenomenon of spatial beam self-cleaning has been recently discovered in multimode optical fibers (MMFs), whose underlying mechanism still remains debated [2]. Light propagation in MMFs is affected by a structural disorder of the material. We formulate a wave turbulence kinetic description of the random waves accounting for the impact of the disorder. The theory unexpectedly reveals a dramatic acceleration of thermalization and condensation by several orders of magnitudes, which can probably explain the effect of spatial beam self-cleaning as a macroscopic population of the fundamental mode of the MMF [1]. The theory also explains why spatial beam self-cleaning has not been observed in step-index MMFs.

Our experiments in MMFs evidence the transition to light condensation: By decreasing the kinetic energy ('temperature') below a critical value, we observe a transition from the incoherent thermal RJ distribution to wave condensation [1]. These observations are corroborated by the experimental evidence that beam self-cleaning is characterized by a turbulence cascade of kinetic energy toward the higher-order modes of the MMF [3].

[1] A. Fusaro et al., Dramatic acceleration of wave condensation mediated by disorder in multimode fibers, PRL 122, 123902 (2019)

[2] K. Krupa et al., Spatial beam self-cleaning in multimode fibres, Nature Phot. 11, 237 (2017)

[3] E. V. Podivilov et al., Hydrodynamic 2D turbulence and spatial beam condensation in multimode optical fibers, PRL 122, 103902 (2019)

Alexander Pimenov

## Temporal solitons in a delayed model of a semiconductor laser

Alexander Pimenov
pimenov@wias-berlin.de
Weierstrass Institute, Mohrenstr. 39, 10117 Berlin
In the last few years temporal localised structures such as dissipative solitons observed in optical ring cavities received significant experimental and theoretical attention. Under some simplifying assumptions these solitons can be studied phenomenologically in the standard PDE frameworks of Lugiato-Lefever equation (LLE) and Haus master equation. On the other hand, delayed differential equation (DDE) models of semiconductor lasers proved to be very useful in qualitative analysis of various dynamical regimes for very wide realistic parameter ranges, and they can adequately represent different experimental set-ups. Recently, we demonstrated how the effect of chromatic dispersion arising due to dispersive element in the cavity such as a fiber loop can be modelled using a time-delay system, and derived a condition for modulational instability in the anomalous dispersion regime. This result allows us to make a theoretical connection between DDE models and LLE, and discuss the conditions under which solitons can be observed in semiconductor lasers.

Carlos Quintero-Quiroz

## Can the state space of spatially extended systems and of time delayed systems be reconstructed from the time series of a scalar variable?

Carlos Quintero-Quiroz
carlos.alberto.quintero@upc.edu
Universitat Politècnica de Catalonia
The space-time representation of high-dimensional dynamical systems that have a well defined characteristic time scale has proven to be very useful to deepen the understanding of such systems and to uncover hidden features in their output signals. By using the space-time representation many analogies between one-dimensional spatially extended systems (1D SESs) and time delayed systems(TDSs) have been found, including similar pattern formation and propagation of localized structures.An open question is whether such analogies are limited to the space-time representation, or it is also possible to recover similar evolution in a low-dimensional pseudo-space. To address this issue, we analyze a 1D SES (a bistable reaction-diffusion system), a scalar TDS (a bistable system with delayed feedback), and a non-scalar TDS (a model of two delay-coupled lasers). In these three examples, we show that we can reconstruct the dynamics in a three-dimensional phase space, where the evolution is governed by the same polynomial potential. We also discuss the limitations of the analogy between1D SESs and TDSs.

Stephane Randoux

## Single-shot observations of modulation instability in optical fibres : full complex field acquisition and space-time evolution

Stephane Randoux
stephane.randoux@univ-lille.fr
Laboratoire de Physique des Lasers, Atomes et Molecules (PhLAM), Université de Lille, Cité Scientifique, 59655 Villeneuve d'Ascq
Light propagating in optical fibers might undergo a modulation instability which leads to the break-up of a continuous wave field. In this presentation, we review recent experiments where both the intensity and phase of the field are recorded at the output of a fiber thanks to an improved temporal imaging system, showing in details the formation of ultra-fast nonlinear structures. We also show how the spatio-temporal evolution of the intensity can be revealed using a recirculating fiber loop.

Amy Roche

## Nozaki-Bekki Holes in a Long Laser

Amy Roche
amy.roche@mycit.ie
Université Côte d'Azur, INPHYNI, CNRS, 1361 route des lucioles, 06560 Valbonne, France
Long cavity fiber-based swept source lasers are promising devices with a wide range of potential applications ranging from communications to life sciences. For example, Fourier Domain Mode-Locked (FDML) lasers, which are commonly used for Optical Coherence Tomography (OCT) imaging applications, are long cavity lasers incorporating an intra-cavity resonator which is driven in resonance with the cavity round trip time. The coherence properties of such swept sources are of major importance as they define the image quality. The purpose of this work is to analyze the mechanism that deteriorates the coherence of long lasers. In our experiment, the laser included a 100nm wide semiconductor optical amplifier at 1310nm and a fiber cavity that could vary from 20m to 20km. The laser emission wavelength was controlled using a fiber based intra-cavity filter with a bandwidth of 10GHz. Near the lasing threshold and/or for fast carrier decay rate, we observed the appearance of periodic power dropouts with stable Nozaki-Bekki holes (NBH) that circulate in the laser cavity. As a function of the injection current, the laser could operate in various regimes including bi-stability between NBH and stable (cw) operation, unstable NBH or chaotic operation. Such behavior indicates that the interplay between the injection current and carrier decay rate can lead to highly coherent emission of a long cavity laser.

Jordi Tiana-Alsina

## Neuron-like dynamics of semiconductor lasers with optical feedback

Jordi Tiana-Alsina
jordi.tiana@upc.edu
Universitat Politècnica de Catalunya, Nonlinear Dynamics (UPC), Nonlinear Optics and Lasers Research group (DONLL), Rambla St. Nebridi 22, 08222 Terrassa, Barcelona, Spain
Neuromorphic photonics is a new paradigm for ultra-fast neuro-inspired optical computing that can revolutionize information processing and artificial intelligence systems. To implement practical photonic neural networks is crucial to identify low-cost energy-efficient laser systems that can mimic neuronal activity. Here we study experimentally the spiking dynamics of a semiconductor laser with optical feedback under periodic modulation of the pump current, and compare with the dynamics of a neuron that is simulated with the stochastic FitzHugh-Nagumo model, with an applied periodic signal whose waveform is the same as that used to modulate the laser current. Sinusoidal and pulse-down waveforms are tested. We find that the laser response and the neuronal response to the periodic forcing, quantified in terms of the variation of the spike rate with the amplitude and with the period of the forcing signal, is qualitatively similar. We also compare the laser and neuron dynamics using symbolic time series analysis. The characterization of the statistical properties of the relative timing of the spikes in terms of ordinal patterns unveils similarities, and also some differences. Our results indicate that semiconductor lasers with optical feedback can be used as low-cost, energy-efficient photonic neurons, the building blocks of all-optical signal processing systems; however, the external cavity needed for optical feedback limits the laser integration in photonic integrated circuits

Stefano Trillo

## Competing mechanisms of nonlinear modulation instability

Stefano Trillo
trlsfn@unife.it
Department of Engineering, University of Ferrara, Via Saragat 1, 44122 Ferrara, Italy
The nonlinear stage of modulation instability (MI) is extremely rich. For periodic perturbations multiple recurrences occurs according to a complex homoclinic structure that represents the continuation of MI in the depleted stage. When the perturbation becomes localized the MI recurrences break down and different scenarios are possible. A quite universal scenario is the development of an auto-modulation, i.e. a strongly oscillating structure within a characteristic wedge-shaped region that smoothly connect to the background. However, for sufficiently generic perturbations the auto-modulation can be accompanied by the emission of breather pairs. In this talk we discuss how to predict the parameters of such breathers in terms of simple formulas.

Sergei Turitsyn

## Spatio-temporal dynamics in fibre lasers

Sergei Turitsyn
s.k.turitsyn@aston.ac.uk
Aston Institute of Photonic Technologies, Aston University, B4 7ET, Birmingham, UK
Understanding of the properties of nonlinear photonic systems is important both for the fundamental science and because of their relevance to numerous applications of light technology. Nonlinearity is an essential component in the design of numerous photonic devices, but it is often shunned by engineers in view of its practical intractability and greatly increased difficulty of comprehension of system behavior. The understanding and mastering of nonlinear effects can translate into improving performance of the existing devices and enabling a new generation of engineering concepts. However, many measurement techniques and signal processing methods have been developed and optimised for linear systems. Understanding of nonlinear dynamics would greatly benefit from new measurement approaches. I will review our recent works on the nonlinear science of fibre lasers, including spatio-temporal dynamics and new approaches for theoretical and experimental analysis of such systems.

Stelios Tzortzakis

## Molded nonlinear light wave packets and applications

Stelios Tzortzakis
stzortz@iesl.forth.gr
Texas A&M University at Qatar, P.O. Box 23874, Doha, Qatar

The nonlinear propagation of ultrashort laser pulses in the form of solitons, filaments and light bullets is an exciting research field [1]. Beyond the basic studies on the complex spatio-temporal phenomena involved, the field is driven significantly by its numerous applications, like for example in materials engineering, remote spectroscopy, but also for their use as powerful secondary sources across the electromagnetic spectrum [2]. Here we discuss our recent advances in molding the shape, temporal and spectral properties of filaments [3] and some corresponding applications enabled through these advances. We demonstrate how it becomes possible, for the first time after 20 years of research, to achieve localized and controlled modification of the index of refraction in the bulk of silicon [4]. This advance opens the way for laser processing in the exciting field of silicon photonics. We also discuss our recent advances in developing intense THz secondary sources using tailored laser filaments. We demonstrate that one may obtain powerful THz radiation using unconventional media, like liquids, where the medium presents strong linear absorption [5]. The mechanism responsible for this counterintuitive result is a phase locked second harmonic component in the filament that results in strong transient electron currents that radiate intense THz fields. Finally, we will also be discussing the way in achieving extreme THz electric and magnetic fields, in excess of GV/cm and kilo-Tesla strengths respectively, using intense two-color mid-infrared filaments [6,7].

[1] P. Panagiotopoulos et al., Nat. Commun. 4, 2622 (2013)

[2] K. Liu et al., Optica 3, 605-608 (2016)

[3] A. D. Koulouklidis et al., Phys. Rev. Lett. 119, 223901 (2017)

[4] M. Chanal et al., Nat. Commun. 8, 773 (2017)

[5] I. Dey et al., Nat. Commun. 8, 1184 (2017)

[6] V. Fedorov and S. Tzortzakis, Phys. Rev. A 97, 063842 (2018)

[7] V. Y. Fedorov, and S. Tzortzakis, Opt. Express 26, 31150-31159 (2018)

Alexis Verschelde

## Multisection semiconductor laser for optical coherence tomography

Alexis Verschelde
alexs.verschelde@inphyni.cnrs.fr
Université Côte d'Azur, INPHYNI, CNRS, 1361 route des lucioles, 06560 Valbonne, France

Optical coherence tomography (OCT) is a non-invasive three-dimensional imaging technique of scattering media used in applications such as medical diagnostics and industrial testing in manufacturing lines. Swept Source-OCT (SS-OCT) requires a laser whose wavelength can be rapidly and continuously swept over a broad spectral range. Nowadays, most swept source lasers (SSL) technologies rely on mechanical filters whose sweeping speed is limited to 100 kHz. Multisection semiconductor lasers are electrically tunable lasers that offer the possibility to reach sweeping speeds up to the MHz regime. The technology is based on semiconductor slot mirrors having comb reflectivity spectra. The spacing of the comb spectral lines is imposed by the periodicity of the slots. The electrical injection of these mirror sections allows to shift the reflectivity spectra by the variation of the refractive index of the medium. By ensuring that the period of the slots are different between the front and back mirrors, two incommensurate comb reflection spectra can be formed. The Vernier effect occurs due to the interference of the two offset combs when independent electrical tuning of the two mirror sections is realised. This Vernier effect is responsible for wide and fast frequency sweeps. However such SS lasers based on the Vernier effect display mode hops during the laser operation that induce a loss of coherence.

In this work, we analyse the spectral features of semiconductor multisection slot lasers when the mirror sections are electrically tuned. Based on our cartographies of the laser emission wavelength as a function of the mirrors currents, we intend to provide an electrical path for a rapid and quasi-continuous wavelength sweep over a broad bandwidth. This work paves the way for further explorations of the opto-electronic control of the multisection lasers coherence during a full wavelength sweep.

## Nonlinear wave phenomena in delay differential models of multimode lasers

Weierstrass Institute, Mohrenstrasse 39, D-10117 Berlin, Germany
Multimode lasers are widely used in medical, industrial, and technological applications. In particular, mode-locked semiconductor lasers are low cost, compact, and efficient sources of short optical pulses with high repetition rates suitable for application in telecommunication networks. A conventional technique to the theoretical studies of these lasers is based on numerical integration of a system of partial differential equations for the electric field envelope and carrier density. Here we use an alternative approach to describe multimode lasers, based on the use of delay differential equations (DDEs). We investigate DDE models of different multimode laser devices, - nonlinear mirror mode-locked lasers generating short optical pulses, frequency swept lasers with a long dispersive fiber delay line, and broad area external cavity semiconductor lasers. In addition to numerical simulations of these models we perform an analytical linear stability analysis that reveals modulational, Turing-type, and flip instabilities of CW regimes. We demonstrate the existence of bistability, chaotic regimes, square waves, as well as temporal and spatio-temporal (light bullets) localised structures of light and discuss their properties and interaction.

Stefan Wabnitz

## Spatiotemporal multimode light waves

Stefan Wabnitz
stefan.wabnitz@uniroma1.it
Dipartimento di Ingegneria dell'Informazione, Elettronica e Telecomunicazioni, Sapienza Università di Roma, Via Eudossiana 18, 00184 Rome, Italy
Nonlinear propagation of optical pulses in multimode fibers is subject to complex spatio-temporal phenomena. We outline different strategies for the control and optimization of nonlinear mode coupling. The first approach involves transverse wavefront shaping of the input beams, which permits to launch an optimized mode combination, that results in the generation of a stable nonlinear mode alphabet at the fiber output. The second approach involves the longitudinal variation of the core diameter of multimode active and passive tapers, which leads to tailored supercontinuum generation with high spatial beam quality.

Michael Woodley

## Spontaneous Symmetry Breaking, Instability, and Chaos in Ring Resonators

Michael Woodley
michael.woodley@npl.co.uk
National Physical Laboratory, Hampton Road, Teddington, Middlesex, TW11 0LW, UK; Heriot-Watt University, Edinburgh Campus, Edinburgh, EH14 4AS, UK.

When a ring resonator is pumped with laser light of sufficient intensity, then the refractive index -- and so the resonant frequency -- of the resonator can be modulated by the intensity of the light within it -- a phenomenon known as the Kerr nonlinearity. If the resonator is pumped with two laser beams, then this effect can give rise to spontaneous symmetry breaking in the two optical modes within the resonator. We present analytical, numerical, and experimental evidence for a rich range of exotic behaviours exhibited by this symmetry-broken light, including oscillations (implying periodic energy exchange between the modes), period-doubling, and chaos. These optical modes are described by the following coupled system of ordinary differential equations:

$$\dot{e}_{1,2}=\tilde{e}_{1,2} -[1+i(A|e_{1,2}|^{2}+B|e_{2,1}|^{2}-\Delta_{1,2})]e_{1,2},$$

where $\tilde{e}_{1,2}$ and $e_{1,2}$ are the input and coupled electric field amplitudes for each beam, respectively, and $\Delta_{1,2}$ are the frequency detunings of the laser beams, with respect to the non-Kerr-shifted cavity resonance frequency. The coefficients $A$ and $B$ denote the strengths of self- and cross-phase modulation, respectively -- i.e., the extent to which the modes interact with themselves and with each other. The physics of this dynamical system is not only of fundamental interest, but is also important for the construction of integrated all-optical circuitry and devices, such as isolators, circulators, logic gates, advanced sensors, oscillators, and scramblers.