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We study the collective dynamics of swarmalators subjected to periodic (sinusoidal) forcing. Although previous research focused on the simplified case of motion in a one-dimensional (1D) periodic domain, we extend this analysis to the more realistic scenario of motion in two and three spatial dimensions with periodic boundary conditions. In doing so, we identify analogues of the 1D states and characterize their dynamics and stability boundaries analytically. Additionally, we investigate the forced swarmalators model with power-law interaction kernels, finding that the analytically tractable model with periodic boundary conditions can reproduce the observed dynamic behaviors of this more complex model., Comment: 11 pages, 8 figures

We study a variant of the one-dimensional swarmalator model where the units' interactions have a controllable length scale or range. We tune the model from the long-range regime, which is well studied, into the short-range regime, which is understudied, and find diverse collective states: sync dots, where the swarmalators arrange themselves into k>1 delta points of perfect synchrony, q-waves, where the swarmalators form spatiotemporal waves with winding number q>1, and an active state where unsteady oscillations are found. We present the phase diagram and derive most of the threshold boundaries analytically. These states may be observable in real-world swarmalator systems with low-range coupling such as biological microswimmers or active colloids., Comment: 9 pages, 9 figures, 2 tables

The global stability of oscillator networks has attracted much recent attention. Ordinarily, the oscillators in such studies are motionless; their spatial degrees of freedom are either ignored (e.g. mean field models) or inactive (e.g geometrically embedded networks like lattices). Yet many real-world oscillators are mobile, moving around in space as they synchronize in time. Here we prove a global synchronization theorem for such swarmalators for a simple model where the units' movements are confined to a 1d ring. This can be thought of as a generalization from oscillators connected on random networks to oscillators connected on temporal networks, where the edges are determined by the oscillators' movements.

We study the 1d swarmalator model in the continuum limit. We examine the stability of its collective states which have compact support: synchrony, where the swarmalators lie in two sync dots (zero dimensional support), and the phase wave, where the swarmalators line up in a ring with uniformly spaced positions and phases (one dimensional support). The compact support imposes analytic difficulties that occur in many other swarmalator models and thus is blocking progress in the field. We show how to overcome this difficulty, deriving the two states' stability spectra exactly.

We study a simple model of swarmalators subject to periodic forcing and confined to move around a one-dimensional ring. This is a toy model for physical systems with a mix of sync, swarming, and forcing such as colloidal micromotors. We find several emergent macrostates and characterize the phase boundaries between them analytically. The most novel state is a swarmalator chimera, where the population splits into two sync dots, which enclose a `train' of swarmalators that run around a peanut-shaped loop., Comment: 9 pages, 7 figures

Swarmalators are oscillators that swarm through space as they synchronize in time. Introduced a few years ago to model many systems which mix synchrony with self-assembly, they remain poorly understood theoretically. Here we obtain the first analytic results on swarmalators moving in two-dimensional (2D) plane by enforcing periodic boundary conditions; this simpler topology allows expressions for order parameters, stabilities, and bifurcations to be derived exactly. We suggest some future directions for swarmalator research and point out some connections to the Kuramoto model and the Vicsek model from active matter; these are intended as a call-to-arms for the sync community and other researchers looking for new problems and puzzles to work on.

We present a case study of swarmalators (mobile oscillators) which move on a 1D ring and are subject to pinning. Previous work considered the special case where the pinning in space and the pinning in the phase dimension were correlated. Here we study the general case where the space and phase pinning are uncorrelated, both being chosen uniformly at random. This induces several new effects, such as pinned async, mixed states, and a first order phase transition. These phenomena may be found in real world swarmalators such as systems of vinegar eels, Janus matchsticks, electrorotated Quincke rollers or Japanese tree frogs., Comment: 9 pages, 6 figures

Similar to sperm, where individuals self-organize in space while also striving for coherence in their tail swinging, several natural and engineered systems exhibit the emergence of swarming and synchronization. The arising and interplay of these phenomena have been captured by collectives of hypothetical particles named swarmalators, each possessing a position and a phase whose dynamics are affected reciprocally and also by the space-phase states of their neighbors. In this work, we introduce a solvable model of swarmalators able to move in two-dimensional spaces. We show that several static and active collective states can emerge and derive necessary conditions for each to show up as the model parameters are varied. These conditions elucidate, in some cases, the displaying of multistability among states. Notably, in the active regime, individuals behave chaotically, maintaining spatial correlation under certain conditions, and breaking it under others on what we interpret as a dimensionality transition., Comment: 18 pages, 6 figures

We study a population of swarmalators, mobile variants of phase oscillators, which run on a ring and have both attractive and repulsive interactions. This one-dimensional (1D) swarmalator model produces several of collective states: the standard sync and async states as well as a splaylike "polarized" state and several unsteady states such as active bands or swirling. The model's simplicity allows us to describe some of the states analytically. The model can be considered as a toy model for real-world swarmalators such as vinegar eels and sperm which swarm in quasi-1D geometries.

We present a simple model of driven matter in a 1D medium with pinning impurities, applicable to magnetic domains walls, confined colloids, and other systems. We find rich dynamics, including hysteresis, reentrance, quasiperiodicity, and two distinct routes to chaos. In contrast to other minimal models of driven matter, the model is solvable: we derive the full phase diagram for small $N$, and for large $N$, derive expressions for order parameters and several bifurcation curves. The model is also realistic. Its collective states match those seen in the experiments of magnetic domain walls, and its force-velocity curve imitates those of superconductor vortices.

We investigate a population of swarmalators, a mobile version of phase oscillators that both sync in time and swarm through space. We focus on a XY-type model of identical swarmalators running on a one-dimensional ring and subject to thermal noise. We uncover four distinct collective states, some of which capture the behavior of real-world swarmalators such as vinegar eels and sperm. Among these, the most intriguing is the `mixed state', which blends two of the other states. We present a comprehensive phase diagram from the Fourier mode analysis with a high accuracy, which is in excellent agreement with numerical simulation results. Our model serves as a tractable toy model for thermal systems that both self-synchronize and self-assemble interdependently., Comment: 10 pages, 4 figures

We study the emergent behaviors of a population of swarming coupled oscillators, dubbed 'swarmalators'. Previous work considered the simplest, idealized case: identical swarmalators with global coupling. Here we expand this work by adding more realistic features: local coupling, non-identical natural frequencies, and chirality. This more realistic model generates a variety of new behaviors including lattices of vortices, beating clusters, and interacting phase waves. Similar behaviors are found across natural and artificial micro-scale collective systems, including social slime mold, spermatozoa vortex arrays, and Quincke rollers. Our results indicate a wide range of future use cases, both to aid characterization and understanding of natural swarms, and to design complex interactions in collective systems from soft and active matter to micro-robotics.

We study a population of swarmalators (swarming/mobile oscillators) which run on a ring and are subject to random pinning. The pinning represents the tendency of particles to stick to defects in the underlying medium which competes with the tendency to sync / swarm. The result is rich collective behavior. A highlight is low dimensional chaos which in systems of ordinary, Kuramoto-type oscillators is uncommon. Some of the states (the phase wave and split phase wave) resemble those seen in systems of Janus matchsticks or Japanese tree frogs. The others (such as the sync and unsteady states) may be observable in systems of vinegar eels, electrorotated Quincke rollers, or other swarmalators moving in disordered environments., Comment: Accepted for publication in Physical Review E (2023)

We investigate the effects of delayed interactions in a population of ``swarmalators", generalizations of phase oscillators that both synchronize in time and swarm through space. We discover two steady collective states: a state in which swarmalators are essentially motionless in a disk arranged in a pseudo-crystalline order, and a boiling state in which the swarmalators again form a disk, but now the swarmalators near the boundary perform boiling-like convective motions. These states are reminiscent of the beating clusters seen in photoactivated colloids and the living crystals of starfish embryos.

We study a simple model of identical swarmalators, generalizations of phases oscillators that swarm through space. We confine the movements to a one-dimensional (1D) ring and consider distributed (non-identical) couplings; the combination of these two effects captures an aspect of the more realistic 2D swarmalator model \cite{o2017oscillators}. We find new collective states as well as generalizations of previously reported ones which we describe analytically. These states imitate the behavior of vinegar eels, catalytic microswimmers, and other swarmalators which move on quasi-1D rings.

We study the collective behavior of swarmalators, generalizations of phase oscillators that both sync and swarm, confined to move on a 1D ring. This simple model captures some of the essence of movement in 2D or 3D but has the benefit of being solvable: most of the collective states and their bifurcations can be specified exactly. The model also captures the behavior of real-world swarmalators which swarm in quasi-1D rings such as bordertaxic sperm and vinegar eels.

In 2020 and 2021, the spread of COVID-19 was globally addressed by imposing restrictions on the distance of individual travel. Recent literature has uncovered a clear pattern in human mobility that underlies the complexity of urban mobility: $r \cdot f$, the product of distance traveled $r$ and frequency of return $f$ per user to a given location, is invariant across space. This paper asks whether the invariant $r\cdot f$ also serves as a driver for epidemic spread, so that the risk associated with human movement can be modeled by a unifying variable $r\cdot f$. We use two large-scale datasets of individual human mobility to show that there is in fact a simple relation between $r$ and $f$ and both speed and spatial dispersion of disease spread. This discovery could assist in modeling spread of disease and inform travel policies in future epidemics -- based not only on travel distance $r$ but also on frequency of return $f$., Comment: 9 pages, 4 figures

Seasonal influenza presents an ongoing challenge to public health. The rapid evolution of the flu virus necessitates annual vaccination campaigns, but the decision to get vaccinated or not in a given year is largely voluntary, at least in the United States, and many people decide against it. In early attempts to model these yearly flu vaccine decisions, it was often assumed that individuals behave rationally, and do so with perfect information -- assumptions that allowed the techniques of classical economics and game theory to be applied. However, the usual assumptions are contradicted by the emerging empirical evidence about human decision-making behavior in this context. We develop a simple model of coupled disease spread and vaccination dynamics that instead incorporates experimental observations from social psychology to model annual vaccine decision-making more realistically. We investigate population-level effects of these new decision-making assumptions, with the goal of understanding whether the population can self-organize into a state of herd immunity, and if so, under what conditions. Our model agrees with established results while also revealing more subtle population-level behavior, including biennial oscillations about the herd immunity threshold., Comment: 20 pages, 6 figures

We analyse the darkweb and find its structure is unusual. For example, $ \sim 87 \%$ of darkweb sites \emph{never} link to another site. To call the darkweb a "web" is thus a misnomer -- it's better described as a set of largely isolated dark silos. As we show through a detailed comparison to the World Wide Web (www), this siloed structure is highly dissimilar to other social networks and indicates the social behavior of darkweb users is much different to that of www users. We show a generalized preferential attachment model can partially explain the strange topology of the darkweb, but an understanding of the anomalous behavior of its users remains out of reach. Our results are relevant to network scientists, social scientists, and other researchers interested in the social interactions of large numbers of agents., Comment: arXiv admin note: substantial text overlap with arXiv:1704.07525

Human mobility patterns are surprisingly structured. In spite of many hard to model factors, such as climate, culture, and socioeconomic opportunities, aggregate migration rates obey a universal, parameter-free, `radiation' model. Recent work has further shown that the detailed spectral decomposition of these flows -- defined as the number of individuals that visit a given location with frequency $f$ from a distance $r$ away -- also obeys simple rules, namely, scaling as a universal inverse square law in the combination, $rf$. However, this surprising regularity, derived on general grounds, has not been explained through microscopic mechanisms of individual behavior. Here we confirm this by analyzing large-scale cell phone datasets from three distinct regions and show that a direct consequence of this scaling law is that the average `travel energy' spent by visitors to a given location is constant across space, a finding reminiscent of the well-known travel budget hypothesis of human movement. The attractivity of different locations, which we define by the total number of visits to that location, also admits non-trivial, spatially-clustered structure. The observed pattern is consistent with the well-known central place theory in urban geography, as well as with the notion of Weber optimality in spatial economy, hinting to a collective human capacity of optimizing recurrent movements. We close by proposing a simple, microscopic human mobility model which simultaneously captures all our empirical findings. Our results have relevance for transportation, urban planning, geography, and other disciplines in which a deeper understanding of aggregate human mobility is key.

Taxis spend a significant amount of time idle, searching for passengers. The routes vacant taxis should follow in order to minimize their idle times are hard to calculate; they depend on complex quantities like passenger demand, traffic conditions, and inter-taxi competition. Here we explore if reinforcement learning (RL) can be used for this purpose. Using real-world data to characterize passenger demand, we show RL-taxis indeed learn to how to reduce their idle time in many environments. In particular, a single RL-taxi operating in a population of regular taxis learns to out-perform its rivals by a significant margin.

Data on vehicular mobility patterns have proved useful in many contexts. Yet generative models which accurately reproduce these mobility patterns are scarce. Here, we explore if recurrent neural networks can cure this scarcity. By training networks on taxi from NYC and Shanghai, and personal cars from Michigan, we show most aspects of the mobility patterns can be reproduced. In particular, the spatial distributions of the street segments usage is well captured by the recurrent neural networks, which other models struggle to do.

The arXiv has collected 1.5 million pre-print articles over 28 years, hosting literature from scientific fields including Physics, Mathematics, and Computer Science. Each pre-print features text, figures, authors, citations, categories, and other metadata. These rich, multi-modal features, combined with the natural graph structure---created by citation, affiliation, and co-authorship---makes the arXiv an exciting candidate for benchmarking next-generation models. Here we take the first necessary steps toward this goal, by providing a pipeline which standardizes and simplifies access to the arXiv's publicly available data. We use this pipeline to extract and analyze a 6.7 million edge citation graph, with an 11 billion word corpus of full-text research articles. We present some baseline classification results, and motivate application of more exciting generative graph models., Comment: 7 pages, 3 tables, 2 figures, ICLR 2019 workshop RLGM submission

From fireflies to heart cells, many systems in Nature show the remarkable ability to spontaneously fall into synchrony. By imitating Nature's success at self-synchronizing, scientists have designed cost-effective methods to achieve synchrony in the lab, with applications ranging from wireless sensor networks to radio transmission. A similar story has occurred in the study of swarms, where inspiration from the behavior flocks of birds and schools of fish has led to 'low-footprint' algorithms for multi-robot systems. Here, we continue this 'bio-inspired' tradition, by speculating on the technological benefit of fusing swarming with synchronization. The subject of recent theoretical work, minimal models of so-called 'swarmalator' systems exhibit rich spatiotemporal patterns, hinting at utility in 'bottom-up' robotic swarms. We review the theoretical work on swarmalators, identify possible realizations in Nature, and discuss their potential applications in technology.

We study a new random search process: the \textit{taxi-drive}. The motivation for this process comes from urban sensing, in which sensors are mounted on moving vehicles such as taxis, allowing urban environments to be opportunistically monitored. Inspired by the movements of real taxis, the taxi-drive is composed of both random and regular parts; passengers are brought to randomly chosen locations via deterministic (i.e. shortest paths) routes. We show through a numerical study that this hybrid motion endows the taxi-drive with advantageous spreading properties. In particular, on certain graph topologies it offers reduced cover times compared to persistent random walks.

Sensors can measure air quality, traffic congestion, and other aspects of urban environments. The fine-grained diagnostic information they provide could help urban managers to monitor a city's health. Recently, a `drive-by' paradigm has been proposed in which sensors are deployed on third-party vehicles, enabling wide coverage at low cost} Research on drive-by sensing has mostly focused on sensor engineering, but a key question remains unexplored: How many vehicles would be required to adequately scan a city? Here, we address this question by analyzing the sensing power of a taxi fleet. Taxis, being numerous in cities and typically equipped with some sensing technology (e.g. GPS), are natural hosts for the sensors. Our strategy is to view drive-by sensing as a spreading process, in which the area of sensed terrain expands as sensor-equipped taxis diffuse through a city's streets. In tandem with a simple model for the movements of the taxis, this analogy lets us analytically determine the fraction of a city's street network sensed by a fleet of taxis during a day. Our results agree with taxi data obtained from nine major cities, and reveal that a remarkably small number of taxis can scan a large number of streets. This finding appears to be universal, indicating its applicability to cities beyond those analyzed here. Moreover, because taxi motions combine randomness and regularity (passengers' destinations being random, but the routes to them being deterministic), the spreading properties of taxi fleets are unusual; in stark contrast to random walks, the stationary densities of our taxi model obey Zipf's law, consistent with the empirical taxi data. Our results have direct utility for town councilors, smart-city designers, and other urban decision makers.

Cerebral autoregulation refers to the brain's ability to maintain cerebral blood flow at an approximately constant level, despite changes in arterial blood pressure. The performance of this mechanism is often assessed using a ten-scale index called the ARI (autoregulation index). Here, $0$ denotes the absence of, while $9$ denotes the strongest, autoregulation. Current methods to calculate the ARI do not typically provide error estimates. Here, we show how this can be done using a bayesian approach. We use Markov-chain Monte Carlo methods to produce a probability distribution for the ARI, which gives a natural way to estimate error.

Synchronization is a universal phenomenon, seen in systems as diverse as superconducting Josephson junctions and discharging pacemaker cells. Here the elements have rhythmic state variables whose mutual influence promotes temporal order. A parallel form of order is seen in swarming systems, such as schools of fish or flocks of birds. Now the degrees of freedom are the individuals' positions, which get redistributed through interactions to form spatial structures. Systems capable of both swarming and synchronizing, dubbed swarmalators, have recently been proposed [O'Keeffe, Kevin P., and Steven H. Strogatz. "Swarmalators: Oscillators that sync and swarm." arXiv preprint arXiv:1701.05670 (2017)] and analyzed in the continuum limit. Here we extend this work by studying finite populations of swarmalators, whose phase similarity affects both their spatial attraction and repulsion. We find ring states, and compute criteria for their existence and stability. Larger populations can form annular distributions, whose density and inner and outer radii we calculate explicitly. These states may be observable in groups of Japanese tree frogs, magnetic colloids, and other systems with an interplay between swarming and synchronization.

We report the operation of an optical parametric amplifier (OPA) capable of producing gigawatt peak-power laser pulses with tunable wavelength in either the visible or near-infrared spectrum. The OPA has two distinct operation modes (i) generation of >350 uJ, sub 100 fs pulses, tunable between 1250 - 1550 nm; (ii) generation of >190 uJ, sub 150 fs pulses tunable between 490 - 530 nm. We have recorded high-order harmonic spectra over a wide range of driving wavelengths. This flexible source of femtosecond pulses presents a useful tool for exploring the wavelength-dependence of strong-field phenomena, in both the multi-photon and tunnel ionization regimes., Comment: 14 pages, 9 figures, This paper was published in Proceedings of SPIE 10088, Nonlinear Frequency Generation and Conversion: Materials and Devices XVI, doi 10.1117/12.2250775

Synchronization occurs in many natural and technological systems, from cardiac pacemaker cells to coupled lasers. In the synchronized state, the individual cells or lasers coordinate the timing of their oscillations, but they do not move through space. A complementary form of self-organization occurs among swarming insects, flocking birds, or schooling fish; now the individuals move through space, but without conspicuously altering their internal states. Here we explore systems in which both synchronization and swarming occur together. Specifically, we consider oscillators whose phase dynamics and spatial dynamics are coupled. We call them swarmalators, to highlight their dual character. A case study of a generalized Kuramoto model predicts five collective states as possible long-term modes of organization. These states may be observable in groups of sperm, Japanese tree frogs, colloidal suspensions of magnetic particles, and other biological and physical systems in which self-assembly and synchronization interact.

High harmonic generation (HHG) is an established means of producing coherent, short wavelength, ultrafast pulses from a compact set-up. Table-top high-harmonic sources are increasingly being used to image physical and biological systems using emerging techniques such as coherent diffraction imaging and ptychography. These novel imaging methods require coherent illumination, and it is therefore important to both characterize the spatial coherence of high-harmonic beams and understand the processes which limit this property. Here we investigate the near- and far-field spatial properties of high-harmonic radiation generated in a gas cell. The variation with harmonic order of the intensity profile, wavefront curvature, and complex coherence factor is measured in the far-field by the SCIMITAR technique. Using the Gaussian-Schell model, the properties of the harmonic beam in the plane of generation are deduced. Our results show that the order-dependence of the harmonic spatial coherence is consistent with partial coherence induced by both variation of the intensity-dependent dipole phase as well as finite spatial coherence of the driving radiation. These findings are used to suggest ways in which the coherence of harmonic beams could be increased further, which would have direct benefits to imaging with high-harmonic radiation., Comment: 12 pages, 3 figures, journal article

We consider a mean-field model of coupled phase oscillators with quenched disorder in the natural frequencies and coupling strengths. A fraction $p$ of oscillators are positively coupled, attracting all others, while the remaining fraction $1-p$ are negatively coupled, repelling all others. The frequencies and couplings are deterministically chosen in a manner which correlates them, thereby correlating the two types of disorder in the model. We first explore the effect of this correlation on the system's phase coherence. We find that there is a a critical width $\gamma_c$ in the frequency distribution below which the system spontaneously synchronizes. Moreover, this $\gamma_c$ is independent of $p$. Hence, our model and the traditional Kuramoto model (recovered when $p=1$) have the same critical width $\gamma_c$. We next explore the critical behavior of the system by examining the finite-size scaling and the dynamic fluctuation of the traditional order parameter. We find that the model belongs to the same universality class as the Kuramoto model with deterministically (not randomly) chosen natural frequencies for the case of $p<1$., Comment: 7 pages, 6 figures

We analyze a variant of a model proposed by Kuramoto, Shinomoto, and Sakaguchi for a large population of coupled oscillatory and excitable elements. Using the Ott-Antonsen ansatz, we reduce the behavior of the population to a two-dimensional dynamical system with three parameters. We present the stability diagram and calculate several of its bifurcation curves analytically, for both excitatory and inhibitory coupling. Our main result is that when the coupling function is broad, the system can display bistability between steady states of constant high and low activity, whereas when the coupling function is narrow and inhibitory, one of the states in the bistable regime can show persistent pulsations in activity.

We consider a mean-field model of coupled phase oscillators with quenched disorder in the coupling strengths and natural frequencies. When these two kinds of disorder are uncorrelated (and when the positive and negative couplings are equal in number and strength), it is known that phase coherence cannot occur and synchronization is absent. Here we explore the effects of correlating the disorder. Specifically, we assume that any given oscillator either attracts or repels all the others, and that the sign of the interaction is deterministically correlated with the given oscillator's natural frequency. For symmetrically correlated disorder with zero mean, we find that the system spontaneously synchronizes, once the width of the frequency distribution falls below a critical value. For asymmetrically correlated disorder, the model displays coherent traveling waves: the complex order parameter becomes nonzero and rotates with constant frequency different from the system's mean natural frequency. Thus, in both cases, correlated disorder can trigger phase coherence., Comment: 6 pages, 6 figures

We study the coupling of a Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) equation to a separate, advection-only transport process. We find that the front dynamics can be described by an FKPP-like equation only at sufficiently fast diffusion or large coupling strength. For such parameter regimes, we find a mapping to an effective FKPP equation. We also find that FKPP equation is fragile with respect to the coupling to an advection-only mechanism, discover conditions when the front width diverges, and when front speed is insensitive to the coupling. At zero diffusion in this mean-field description, the downwind front speed goes to a finite value as the coupling goes to zero.

We consider the transient behavior of globally coupled systems of identical pulse coupled oscillators. Synchrony develops through an aggregation phenomenon, with clusters of synchronized oscillators forming and growing larger in time. Previous work derived expressions for these time dependent clusters, when each oscillator obeyed a linear charging curve. We generalize these results to cases where the charging curves have nonlinearities

We consider models of identical pulse-coupled oscillators with global interactions. Previous work showed that under certain conditions such systems always end up in sync, but did not quantify how small clusters of synchronized oscillators progressively coalesce into larger ones. Using tools from the study of aggregation phenomena, we obtain exact results for the time-dependent distribution of cluster sizes as the system evolves from disorder to synchrony., Comment: Supplemental Materials available on request

We present a technique for measuring the transverse spatial properties of an optical wavefront. Intensity and phase profiles are recovered by analysis of a series of interference patterns produced by the combination of a scanning X-shaped slit and a static horizontal slit; the spatial coherence may be found from the same data. We demonstrate the technique by characterizing high harmonic radiation generated in a gas cell, however the method could be extended to a wide variety of light sources., Comment: 4 pages, 3 figures, 1 table

A scheme for quasi-phase-matching high-harmonic generation is proposed in which polarization beating within a hollow core birefringent waveguide modulates the generation of harmonics. The evolution of the polarization of a laser pulse propagating in a birefringent waveguide is calculated and is shown to periodically modulate the harmonic generation process. The optimum conditions for achieving quasi-phase-matching using this scheme are explored and the growth of the harmonic intensity as a function of experimental parameters is investigated.

The first scheme for quasi-phase-matching high harmonic generation of circularly polarized radiation is proposed: optical rotation quasi-phase-matching (ORQPM). In ORQPM propagation of the driving radiation in a system exhibiting circular birefringence causes its plane of polarization to rotate; by appropriately matching the period of rotation to the coherence length it is possible to avoid destructive interference of the generated radiation. It is shown that ORQPM is approximately 5 times more efficient than conventional QPM, and half as efficient as true phase-matching.

The generalization of quasi-phase-matching using polarization beating and of multimode quasi-phase-matching (MMQPM) for the generation of high-order harmonics is explored, and a method for achieving polarization beating is proposed. If two (and in principle more) modes of a waveguide are excited, modulation of the intensity, phase, and/or polarization of the guided radiation will be achieved. By appropriately matching the period of this modulation to the coherence length, quasi-phase-matching of high-order-harmonic radiation generated by the guided wave can occur. We show that it is possible to achieve efficiencies with multimode quasi-phase-matching greater than the ideal square wave modulation. We present a Fourier treatment of QPM and use this to show that phase modulation, rather than amplitude modulation, plays the dominant role in the case of MMQPM. The experimental parameters and optimal conditions for this scheme are explored.