Temporal localized states and square-waves in semiconductor micro-resonators with strong time-delayed feedback.

In this paper, we study the dynamics of a vertically emitting micro-cavity operated in the Gires-Tournois regime that contains a semiconductor quantum-well and that is subjected to strong time-delayed optical feedback and detuned optical injection. Using a first principle time-delay model for the optical response, we disclose sets of multistable dark and bright temporal localized states coexisting on their respective bistable homogeneous backgrounds. In the case of anti-resonant optical feedback, we identify square-waves with a period of twice the round-trip in the external cavity. Finally, we perform a multiple time scale analysis in the good cavity limit. The resulting normal form is in good agreement with the original time-delayed model.

Square waves and Bykov T-points in a delay algebraic model for the Kerr-Gires-Tournois interferometer.

We study theoretically the mechanisms of square wave formation of a vertically emitting micro-cavity operated in the Gires-Tournois regime that contains a Kerr medium and that is subjected to strong time-delayed optical feedback and detuned optical injection. We show that in the limit of large delay, square wave solutions of the time-delayed system can be treated as relative homoclinic solutions of an equation with an advanced argument. Based on this, we use concepts of classical homoclinic bifurcation theory to study different types of square wave solutions. In particular, we unveil the mechanisms behind the collapsed snaking scenario of square waves and explain the formation of complex-shaped multistable square wave solutions through a Bykov T-point. Finally, we relate the position of the T-point to the position of the Maxwell point in the original time-delayed system.

A normal form for frequency combs and localized states in Kerr-Gires-Tournois interferometers.

We elucidate the mechanisms that underlay the formation of temporal localized states and frequency combs in vertical external-cavity Kerr-Gires-Tournois interferometers. We reduce our first-principles model based upon delay algebraic equations to a minimal pattern formation scenario. It consists in a real cubic Ginzburg-Landau equation modified by high-order effects such as third-order dispersion and nonlinear drift, which are responsible for generating localized states via the locking of domain walls connecting the high and low intensity levels of the injected micro-cavity. We interpret the effective parameters of the normal form in relation with the configuration of the optical setup. Comparing the two models, we observe an excellent agreement close to the onset of bistability.

Square-wave generation in vertical external-cavity Kerr-Gires-Tournois interferometers.

We study theoretically the mechanisms of square-wave (SW) formation in vertical external-cavity Kerr-Gires-Tournois interferometers in the presence of anti-resonant injection. We provide simple analytical approximations for their plateau intensities and for the conditions of their emergence. We demonstrate that SWs may appear via a homoclinic snaking scenario, leading to the formation of complex-shaped multistable SW solutions. The resulting SWs can host localized structures and robust bound states.

Wetting dynamics under periodic switching on different scales: characterization and mechanisms.

The development of substrates with a switchable wettability is proceeding and the limit of switching frequencies and contact angle differences between substrate states has developed in the past years. In this paper we investigate the behavior of a droplet on a homogeneous substrate, which is switched between two wettabilities for a large range of switching frequencies. Here, we are particularly interested in the dependence of the wetting behavior on the switching frequency. We show that results obtained on the particle level via molecular dynamics simulations and on the continuum level via the thin-film model are consistent. Predictions of simple models as the molecular theory of wetting (MKT) and analytical calculations based on the MKT also show good agreement and offer deeper insights into the underlying mechanisms.

Influence of time-delayed feedback on the dynamics of temporal localized structures in passively mode-locked semiconductor lasers.

In this paper, we analyze the effect of optical feedback on the dynamics of a passively mode-locked ring laser operating in the regime of temporal localized structures. This laser system is modeled by a set of delay differential equations, which include delay terms associated with the laser cavity and the feedback loop. Using a combination of direct numerical simulations and path-continuation techniques, we show that the feedback loop creates echoes of the main pulse whose position and size strongly depend on the feedback parameters. We demonstrate that in the long-cavity regime, these echoes can successively replace the main pulses, which defines their lifetime. This pulse instability mechanism originates from a global bifurcation of the saddle-node infinite-period type. In addition, we show that, under the influence of noise, the stable pulses exhibit forms of a behavior characteristic of excitable systems. Furthermore, for the harmonic solutions consisting of multiple equispaced pulses per round-trip, we show that if the location of the pulses coincides with the echo of another, the range of stability of these solutions is increased. Finally, it is shown that around these resonances, branches of different solutions are connected by period-doubling bifurcations.

Wiggling instabilities of temporal localized states in passively mode-locked vertical external-cavity surface-emitting lasers.

We analyze the emergence of wiggling temporal localized states in a passively mode-locked vertical external-cavity surface-emitting laser composed by a gain chip and a resonant saturable absorber mirror. We show that the wiggling instability stems from the interplay between the third-order dispersion induced by the micro-cavities and their frequency mismatch. The latter is identified as an experimentally crucial parameter defining the range of existence of stable emission. We reveal the homoclinic scenario underlying the wiggling phenomenon, and we show how it allows us to control the oscillation parameters.

How carrier memory enters the Haus master equation of mode-locking.

We present a generalization of the Haus master equation in which a dynamical boundary condition allows to describe complex pulse trains, such as the Q-switched and harmonic transitions of passive mode-locking, as well as the weak interactions between localized states. As an example, we investigate the role of group velocity dispersion on the stability boundaries of the Q-switched regime and compare our results with that of a time-delayed system.

Bifurcations of front motion in passive and active Allen-Cahn-type equations.

The well-known cubic Allen-Cahn (AC) equation is a simple gradient dynamics (or variational) model for a nonconserved order parameter field. After revising main literature results for the occurrence of different types of moving fronts, we employ path continuation to determine their bifurcation diagram in dependence of the external field strength or chemical potential. We then employ the same methodology to systematically analyze fronts for more involved AC-type models. In particular, we consider a cubic-quintic variational AC model and two different nonvariational generalizations. We determine and compare the bifurcation diagrams of front solutions in the four considered models.

Bound states of light bullets in passively mode-locked semiconductor lasers.

In this paper, we analyze the dynamics and formation mechanisms of bound states (BSs) of light bullets in the output of a laser coupled to a distant saturable absorber. First, we approximate the full three-dimensional set of Haus master equations by a reduced equation governing the dynamics of the transverse profile. This effective theory allows us to perform a detailed multiparameter bifurcation study and to identify the different mechanisms of instability of BSs. In addition, our analysis reveals a non-intuitive dependence of the stability region as a function of the linewidth enhancement factors and the field diffusion. Our results are confirmed by direct numerical simulations of the full system.

Phase-field-crystal description of active crystallites: Elastic and inelastic collisions.

The active Phase-Field-Crystal (aPFC) model combines elements of the Toner-Tu theory for self-propelled particles and the classical Phase-Field-Crystal (PFC) model that describes the transition between liquid and crystalline phases. In the liquid-crystal coexistence region of the PFC model, crystalline clusters exist in the form of localized states that coexist with a homogeneous background. At sufficiently strong activity (related to self-propulsion strength), they start to travel. We employ numerical path continuation and direct time simulations to first investigate the existence regions of different types of localized states in one spatial dimension. The results are summarized in morphological phase diagrams in the parameter plane spanned by activity and mean density. Then we focus on the interaction of traveling localized states, studying their collision behavior. As a result, we distinguish "elastic" and "inelastic" collisions. In the former, localized states recover their properties after a collision, while in the latter, they may completely or partially annihilate, forming resting bound states or various traveling states.

Discrete light bullets in passively mode-locked semiconductor lasers.

In this paper, we analyze the formation and dynamical properties of discrete light bullets in an array of passively mode-locked lasers coupled via evanescent fields in a ring geometry. Using a generic model based upon a system of nearest-neighbor coupled Haus master equations, we show numerically the existence of discrete light bullets for different coupling strengths. In order to reduce the complexity of the analysis, we approximate the full problem by a reduced set of discrete equations governing the dynamics of the transverse profile of the discrete light bullets. This effective theory allows us to perform a detailed bifurcation analysis via path-continuation methods. In particular, we show the existence of multistable branches of discrete localized states, corresponding to different number of active elements in the array. These branches are either independent of each other or are organized into a snaking bifurcation diagram where the width of the discrete localized states grows via a process of successive increase and decrease of the gain. Mechanisms are revealed by which the snaking branches can be created and destroyed as a second parameter, e.g., the linewidth enhancement factor or the coupling strength is varied. For increasing couplings, the existence of moving bright and dark discrete localized states is also demonstrated.

Sliding Drops: Ensemble Statistics from Single Drop Bifurcations.

Ensembles of interacting drops that slide down an inclined plate show a dramatically different coarsening behavior as compared to drops on a horizontal plate: As drops of different size slide at different velocities, frequent collisions result in fast coalescence. However, above a certain size individual sliding drops are unstable and break up into smaller drops. Therefore, the long-time dynamics of a large drop ensemble is governed by a balance of merging and splitting. We employ a long-wave film height evolution equation and determine the dynamics of the drop size distribution towards a stationary state from direct numerical simulations on large domains. The main features of the distribution are then related to the bifurcation diagram of individual drops obtained by numerical path continuation. The gained knowledge allows us to develop a Smoluchowski-type statistical model for the ensemble dynamics that well compares to full direct simulations.

Nudged elastic band calculation of the binding potential for liquids at interfaces.

The wetting behavior of a liquid on solid substrates is governed by the nature of the effective interaction between the liquid-gas and the solid-liquid interfaces, which is described by the binding or wetting potential g(h) which is an excess free energy per unit area that depends on the liquid film height h. Given a microscopic theory for the liquid, to determine g(h), one must calculate the free energy for liquid films of any given value of h, i.e., one needs to create and analyze out-of-equilibrium states, since at equilibrium there is a unique value of h, specified by the temperature and chemical potential of the surrounding gas. Here we introduce a Nudged Elastic Band (NEB) approach to calculate g(h) and illustrate the method by applying it in conjunction with a microscopic lattice density functional theory for the liquid. We also show that the NEB results are identical to those obtained with an established method based on using a fictitious additional potential to stabilize the non-equilibrium states. The advantages of the NEB approach are discussed.

Instabilities of Layers of Deposited Molecules on Chemically Stripe Patterned Substrates: Ridges versus Drops.

A mesoscopic continuum model is employed to analyze the transport mechanisms and structure formation during the redistribution stage of deposition experiments where organic molecules are deposited on a solid substrate with periodic stripe-like wettability patterns. Transversally invariant ridges located on the more wettable stripes are identified as very important transient states and their linear stability is analyzed accompanied by direct numerical simulations of the fully nonlinear evolution equation for two-dimensional substrates. It is found that there exist two different instability modes that lead to different nonlinear evolutions that result (i) at large ridge volume in the formation of bulges that spill from the more wettable stripes onto the less wettable bare substrate and (ii) at small ridge volume in the formation of small droplets located on the more wettable stripes. In addition, the influence of different transport mechanisms during redistribution is investigated focusing on the cases of convective transport with no-slip at the substrate, transport via diffusion in the film bulk and via diffusion at the film surface. In particular, it is shown that the transport process does neither influence the linear stability thresholds nor the sequence of morphologies observed in the time simulation, but only the ratio of the time scales of the different process phases.

Stationary broken parity states in active matter models.

We demonstrate that several nonvariational continuum models commonly used to describe active matter as well as other active systems exhibit nongeneric behavior: each model supports asymmetric but stationary localized states even in the absence of pinning at heterogeneities. Moreover, such states only begin to drift following a drift-transcritical bifurcation as the activity increases. Asymmetric stationary states should only exist in variational systems, i.e., in models with gradient structure. In other words, such states are expected in passive systems, but not in active systems where the gradient structure of the model is broken by activity. We identify a "spurious" gradient dynamics structure of these models that is responsible for this nongeneric behavior, and determine the types of additional terms that render the models generic, i.e., with asymmetric states that appear via drift-pitchfork bifurcations and are generically moving. We provide detailed illustrations of our results using numerical continuation of resting and steadily drifting states in both generic and nongeneric cases.

Structure formation by dynamic self-assembly.

This review summarizes the work conducted in the last decade on the fabrication of mesostructured patterns, which have lateral dimensions within the nano- and microscales, over a wafer-scaled size by means of dynamic self-assembly using Langmuir-Blodgett (LB) transfer or dip-coating. First, strategies to form mesostructures from a homogeneous Langmuir monolayer with controlled shape, size, and patterns alignment will be presented, followed by a detailed theoretical explanation of the pattern formation. In addition, the patterning of nanocrystals and other chemicals with LB transfer or other dynamic processes, such as dip-coating, will be summarized.

Two-dimensional localized states in an active phase-field-crystal model.

The active phase-field-crystal (active PFC) model provides a simple microscopic mean field description of crystallization in active systems. It combines the PFC model (or conserved Swift-Hohenberg equation) of colloidal crystallization and aspects of the Toner-Tu theory for self-propelled particles. We employ the active PFC model to study the occurrence of localized and periodic active crystals in two spatial dimensions. Due to the activity, crystalline states can undergo a drift instability and start to travel while keeping their spatial structure. Based on linear stability analyses, time simulations, and numerical continuation of the fully nonlinear states, we present a detailed analysis of the bifurcation structure of resting and traveling states. We explore, for instance, how the slanted homoclinic snaking of steady localized states found for the passive PFC model is modified by activity. Morphological phase diagrams showing the regions of existence of various solution types are presented merging the results from all the analysis tools employed. We also study how activity influences the crystal structure with transitions from hexagons to rhombic and stripe patterns. This in-depth analysis of a simple PFC model for active crystals and swarm formation provides a clear general understanding of the observed multistability and associated hysteresis effects, and identifies thresholds for qualitative changes in behavior.

Pattern formation in monolayer transfer systems with substrate-mediated condensation.

The formation of regular stripe patterns during transfer of surfactant monolayers onto solid substrates is investigated. Two coupled differential equations describing the surfactant density and the height profile of the water subphase are derived within the lubrication approximation. If the transfer is carried out in the vicinity of a first order phase transition of the surfactant, the interaction with the substrate plays a key role. This effect is included in the surfactant free-energy functional via a height-dependent external field. Using transfer velocity as a control parameter, a bifurcation from a homogeneous transfer to regular stripe patterns arranged parallel to the contact line is investigated in one and two dimensions. Moreover, in the two-dimensional case, a secondary bifurcation to perpendicular stripes is observed in a certain control parameter range.

Effects of time-periodic forcing in a Cahn-Hilliard model for Langmuir-Blodgett transfer.

The influence of a temporal forcing on the pattern formation in Langmuir-Blodgett transfer is studied employing a generalized Cahn-Hilliard model. The occurring frequency-locking effects allow for controlling the pattern formation process. In the case of one-dimensional (i.e., stripe) patterns one finds various synchronization phenomena such as entrainment between the distance of deposited stripes and the forcing frequency. In two dimensions, the temporal forcing gives rise to the formation of intricate complex patterns such as vertical stripes, oblique stripes, and lattice structures. Remarkably, it is possible to influence the system in the spatial direction perpendicular to the forcing direction leading to synchronization in two spatial dimensions.