Sensitivity to initial conditions at bifurcations in one-dimensional nonlinear maps: rigorous nonextensive solutions

Using the Feigenbaum renormalization group (RG) transformation we work out exactly the dynamics and the sensitivity to initial conditions for unimodal maps of nonlinearity $\zeta >1$ at both their pitchfork and tangent bifurcations. These functions have the form of $q$-exponentials as proposed in Tsallis' generalization of statistical mechanics. We determine the $q$-indices that characterize these universality classes and perform for the first time the calculation of the $q$-generalized Lyapunov coefficient $\lambda_{q}$. The pitchfork and the left-hand side of the tangent bifurcations display weak insensitivity to initial conditions, while the right-hand side of the tangent bifurcations presents a `super-strong' (faster than exponential) sensitivity to initial conditions. We corroborate our analytical results with {\em a priori} numerical calculations.Comment: latex, 4 figures. Updated references and some general presentation improvements. To appear published in Europhysics Letter

Frontiers of beam diagnostics in plasma accelerators: measuring the ultra-fast and ultra-cold

Advanced diagnostics are essential tools in the development of plasma-based accelerators. The accurate measurement of the quality of beams at the exit of the plasma channel is crucial to optimize the parameters of the plasma accelerator. 6D electron beam diagnostics will be reviewed with emphasis on emittance measurement, which is particularly complex due to large energy spread and divergence of the emerging beams, and on femtosecond bunch length measurements

Acceleration with Self-Injection for an All-Optical Radiation Source at LNF

We discuss a new compact gamma-ray source aiming at high spectral density, up to two orders of magnitude higher than currently available bremsstrahlung sources, and conceptually similar to Compton Sources based on conventional linear accelerators. This new source exploits electron bunches from laser-driven electron acceleration in the so-called self-injection scheme and uses a counter-propagating laser pulse to obtain X and gamma-ray emission via Thomson/Compton scattering. The proposed experimental configuration inherently provides a unique test-bed for studies of fundamental open issues of electrodynamics. In view of this, a preliminary discussion of recent results on self-injection with the FLAME laser is also given.Comment: 8 pages, 10 figures, 44 references - Channeling 2012 conferenc

Time-resolved characterization of ultrafast electrons in intense laser and metallic-dielectric target interaction

High-intensity ultrashort laser pulses interacting with thin solid targets are able to produce energetic ion beams by means of extremely large accelerating fields set by the energetic ejected electrons. The characterization of such electrons is thus important in view of a complete understanding of the acceleration process. Here, we present a complete temporal-resolved characterization of the fastest escaping hot electron component for different target materials and thicknesses, using temporal diagnostics based on electro-optical sampling with 100 fs temporal resolution. Experimental evidence of scaling laws for ultrafast electron beam parameters have been retrieved with respect to the impinging laser energy (0.4-4 J range) and to the target material, and an empirical law determining the beam parameters as a function of the target thickness is presented

Plasma ramps caused by outflow in gas-filled capillaries

Plasma confinement inside capillaries has been developed in the past years for plasma-based acceleration to ensure a stable and repeatable plasma density distribution during the interaction with either particles or laser beams. In particular, gas-filled capillaries allow a stable and almost predictable plasma distribution along the interaction with the particles. However, the plasma ejected through the ends of the capillary interacts with the beam before the inner plasma, affecting the quality of the beam. In this article we report the measurements on the evolution of the plasma flow at the two ends of a 1 cm long, 1 mm diameter capillary filled with hydrogen. In particular, we measured the longitudinal density distribution and the expansion velocity of the plasma outside the capillary. This study will allow a better understanding of the beam-plasma interaction for future plasma-based experiments.Comment: 5 pages, 6 figures, EAAC 201

Longitudinal phase-space manipulation with beam-driven plasma wakefields

The development of compact accelerator facilities providing high-brightness beams is one of the most challenging tasks in field of next-generation compact and cost affordable particle accelerators, to be used in many fields for industrial, medical and research applications. The ability to shape the beam longitudinal phase-space, in particular, plays a key role to achieve high-peak brightness. Here we present a new approach that allows to tune the longitudinal phase-space of a high-brightness beam by means of a plasma wakefields. The electron beam passing through the plasma drives large wakefields that are used to manipulate the time-energy correlation of particles along the beam itself. We experimentally demonstrate that such solution is highly tunable by simply adjusting the density of the plasma and can be used to imprint or remove any correlation onto the beam. This is a fundamental requirement when dealing with largely time-energy correlated beams coming from future plasma accelerators

A recent appreciation of the singular dynamics at the edge of chaos

We study the dynamics of iterates at the transition to chaos in the logistic map and find that it is constituted by an infinite family of Mori's $q$-phase transitions. Starting from Feigenbaum's $\sigma$ function for the diameters ratio, we determine the atypical weak sensitivity to initial conditions $\xi _{t}$ associated to each $q$-phase transition and find that it obeys the form suggested by the Tsallis statistics. The specific values of the variable $q$ at which the $q$-phase transitions take place are identified with the specific values for the Tsallis entropic index $q$ in the corresponding $\xi_{t}$. We describe too the bifurcation gap induced by external noise and show that its properties exhibit the characteristic elements of glassy dynamics close to vitrification in supercooled liquids, e.g. two-step relaxation, aging and a relationship between relaxation time and entropy.Comment: Proceedings of: Verhulst 200 on Chaos, Brussels 16-18 September 2004, Springer Verlag, in pres

Overview of Plasma Lens Experiments and Recent Results at SPARC_LAB

Beam injection and extraction from a plasma module is still one of the crucial aspects to solve in order to produce high quality electron beams with a plasma accelerator. Proper matching conditions require to focus the incoming high brightness beam down to few microns size and to capture a high divergent beam at the exit without loss of beam quality. Plasma-based lenses have proven to provide focusing gradients of the order of kT/m with radially symmetric focusing thus promising compact and affordable alternative to permanent magnets in the design of transport lines. In this paper an overview of recent experiments and future perspectives of plasma lenses is reported

Universal renormalization-group dynamics at the onset of chaos in logistic maps and nonextensive statistical mechanics

We uncover the dynamics at the chaos threshold $\mu_{\infty}$ of the logistic map and find it consists of trajectories made of intertwined power laws that reproduce the entire period-doubling cascade that occurs for $\mu <\mu_{\infty}$. We corroborate this structure analytically via the Feigenbaum renormalization group (RG) transformation and find that the sensitivity to initial conditions has precisely the form of a $q$-exponential, of which we determine the $q$-index and the $q$-generalized Lyapunov coefficient $\lambda _{q}$. Our results are an unequivocal validation of the applicability of the non-extensive generalization of Boltzmann-Gibbs (BG) statistical mechanics to critical points of nonlinear maps.Comment: Revtex, 3 figures. Updated references and some general presentation improvements. To appear published as a Rapid communication of PR