Plasma density shaping for attosecond electron bunch generation

High energy attosecond electron bunches from the laser-plasma wakefield accelerator (LWFA) are potentially useful sources of ultra-short duration X-rays pulses, which can be used for ultrafast imaging of electron motion in biological and physical systems. Electron injection in the LWFA depends on the plasma density and gradient, and the laser intensity. Recent research has shown that injection of attosecond electron bunches is possible using a short plasma density ramp. For controlled injection it is necessary to keep both the laser intensity and background plasma density constant, but set to just below the threshold for injection. This ensures that injection is only triggered by an imposed density perturbation; the peak density should also not exceed the threshold for injection. A density gradient that only persists over a short range can lead to the injection of femtosecond duration bunches, which are then Lorentz contracted to attoseconds on injection. We consider an example of a sin^2 shaped modulation where the gradient varies until the downward slope exceeds the threshold for injection and then reduces subsequently to prevent any further injection. The persistence above the threshold determines the injected bunch length, which can be varied. We consider several designs of plasma media including density perturbations formed by shaped Laval nozzles and present an experimental and theoretical study of the modulated media suitable for producing attosecond-duration electron bunches.