# Spatiotemporal phenomena in nonlinear optics contributions

*To be updated until april 30th*

## Controlling light by light

## Quantum Fluids of light in atomic vapors

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

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.

## Soliton Explosions and Optical Rogue Waves

## Asymmetric balance in symmetry breaking

## Laser modelocking beyond Haus: the coherent master equation

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

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

## Spatio-temporal molding of light in caustic networks

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.

## Nonlinear polariton fluids

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)

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

## Analogue gravity in rotating spacetimes

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.

## Predicting Extreme Events in Modulation Instability Using Machine Learning

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)

## The LANER: optical networks as complex lasers

## Short-distance propagation of nonlinear optical pulses

## 2D spatiotemporal extreme event in quadratic nonlinear crystal

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).

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

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.

## Nonlinear polariton phenomena in semiconductor microcavities and slab waveguides

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:

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

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

“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)

“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)

“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)

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

## Parametric interactions in multimode fibers

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.

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

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

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

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)

## Non Linear Diffraction

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

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

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)

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

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

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

## Nozaki-Bekki Holes in a Long Laser

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

## Competing mechanisms of nonlinear modulation instability

## Spatio-temporal dynamics in fibre lasers

## Molded nonlinear light wave packets and applications

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)

## Multisection semiconductor laser for optical coherence tomography

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

## Spatiotemporal multimode light waves

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

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.