There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Femtosecond filamentation in transparent media

Particle accelerators are used in a wide variety of fields, ranging from medicine and biology to high-energy physics. The accelerating fields in conventional accelerators are limited to a few tens of MeV m(-1), owing to material breakdown at the walls of the structure. Thus, the production of energetic particle beams currently requires large-scale accelerators and expensive infrastructures. Laser-plasma accelerators have been proposed as a next generation of compact accelerators because of the huge electric fields they can sustain (>100 GeV m(-1)). However, it has been difficult to use them efficiently for applications because they have produced poor-quality particle beams with large energy spreads, owing to a randomization of electrons in phase space. Here we demonstrate that this randomization can be suppressed and that the quality of the electron beams can be dramatically enhanced. Within a length of 3 mm, the laser drives a plasma bubble that traps and accelerates plasma electrons. The resulting electron beam is extremely collimated and quasi-monoenergetic, with a high charge of 0.5 nC at 170 MeV.

A laser-plasma accelerator producing monoenergetic electron beams

Laser-driven accelerators, in which particles are accelerated by the electric field of a plasma wave (the wakefield) driven by an intense laser, have demonstrated accelerating electric fields of hundreds of GV m-1 (refs 1–3) These fields are thousands of times greater than those achievable in conventional radio-frequency accelerators, spurring interest in laser accelerators4,5 as compact next-generation sources of energetic electrons and radiation To date, however, acceleration distances have been severely limited by the lack of a controllable method for extending the propagation distance of the focused laser pulse The ensuing short acceleration distance results in low-energy beams with 100 per cent electron energy spread1,2,3, which limits potential applications Here we demonstrate a laser accelerator that produces electron beams with an energy spread of a few per cent, low emittance and increased energy (more than 109 electrons above 80 MeV) Our technique involves the use of a preformed plasma density channel to guide a relativistically intense laser, resulting in a longer propagation distance The results open the way for compact and tunable high-brightness sources of electrons and radiation

https://escholarship.org/content/qt9c55r46s/qt9c55r46s.pdf?t=lnras5

High-quality electron beams from a laser wakefield accelerator using plasma-channel guiding

An overview is given of the physics issues relevant to the plasma wakefield accelerator, the plasma beat-wave accelerator, the laser wakefield accelerator, including the self-modulated regime, and wakefield accelerators driven by multiple electron or laser pulses. Basic properties of linear and nonlinear plasma waves are discussed, as well as the trapping and acceleration of electrons in the plasma wave. Formulas are presented for the accelerating field and the energy gain in the various accelerator configurations. The propagation of the drive electron or laser beams is discussed, including limitations imposed by key instabilities and methods for optically guiding laser pulses. Recent experimental results are summarized.

Overview of plasma-based accelerator concepts

A nonlinear theory of intense laser-plasma interactions is developed and used to describe relativistic optical guiding, coherent harmonic radiation production, and nonlinear plasma wakefield generation. Relativistic optical guiding is found to be ineffective in preventing the leading portion (\ensuremath{\le} a plasma wavelength) of a laser pulse from diffracting. Coherent harmonic generation is found to be most efficient for short laser pulses. Optical guiding and harmonic generation may be enhanced by the presence of large amplitude plasma wakefields. These phenomena may be important in laser-driven plasma accelerators, x-ray sources, and fusion schemes.

Nonlinear theory of intense laser-plasma interactions.

Several features of intense, short-pulse (/spl lsim/1 ps) laser propagation in gases undergoing ionization and in plasmas are reviewed, discussed, and analyzed. The wave equations for laser pulse propagation in a gas undergoing ionization and in a plasma are derived. The source-dependent expansion method is discussed, which is a general method for solving the paraxial wave equation with nonlinear source terms. In gases, the propagation of high-power (near the critical power) laser pulses is considered including the effects of diffraction, nonlinear self-focusing, ionization, and plasma generation. Self-guided solutions and the stability of these solutions are discussed. In plasmas, optical guiding by relativistic effects, ponderomotive effects, and preformed density channels is considered. The self consistent plasma response is discussed, including plasma wave effects and instabilities such as self-modulation. Recent experiments on the guiding of laser pulses in gases and in plasmas are briefly summarized.

Self-focusing and guiding of short laser pulses in ionizing gases and plasmas

An injector and accelerator is analyzed that uses three collinear laser pulses in a plasma: an intense pump pulse, which generates a large wake field $(\ensuremath{\ge}20\mathrm{GV}/\mathrm{m})$, and two counterpropagating injection pulses. When the injection pulses collide, a slow phase velocity beat wave is generated that injects electrons into the fast wake field for acceleration. Particle tracking simulations in 1D with injection pulse intensities near ${10}^{17}\mathrm{W}/\mathrm{cm}{}^{2}$ indicate the production of relativistic electrons with bunch durations as short as 3 fs, energy spreads as small as 0.3%, and densities as high as ${10}^{18}\mathrm{cm}{}^{\ensuremath{-}3}$.

/pdf/electron-injection-into-plasma-wakefields-by-colliding-laser-1oj8n5bqy9.pdf

Electron Injection into Plasma Wakefields by Colliding Laser Pulses

An electron acceleration method is investigated which employs a short (τL ∼2πωω−1p ∼1 ps), high‐power (P≥1015 W), single frequency laser pulse to generate large amplitude (E≥1 GeV/m) plasma waves (wakefields). At sufficiently high laser powers [P≥17(ω/ωp )2 GW], relativistic optical guiding may be used to prevent the pulse from diffracting within the plasma.

Antonio Ting

Papers

Overview of plasma-based accelerator concepts

Nonlinear theory of intense laser-plasma interactions.

Self-focusing and guiding of short laser pulses in ionizing gases and plasmas

Electron Injection into Plasma Wakefields by Colliding Laser Pulses

Laser wakefield acceleration and relativistic optical guiding