Laser driven ion channel x-ray laser.

An intense short pulse laser, undergoing relativistic self-focusing in a plasma, pushes the low energy electrons radially outward to create an ion channel, while accelerates, directly by itself or by the wakefield driven plasma wave, the energetic ones axially to multi-MeV energies. When the energy spread of energetic electrons is contained, they form a beam that executes betatron oscillations in the channel and drives x-ray radiation unstable via betatron resonance, $\omega $-kv$_{z}=\omega _{b}$, producing coherent x-rays of frequency $\omega \mbox{ }=\mbox{ 2}\gamma _\mbox{0}^\mbox{2} \mbox{ }\omega _\mbox{b} \mbox{ ,}$ with optimum growth rate $\Gamma _\mbox{m} \mbox{ }\approx \mbox{ }\omega _\mbox{p} \mbox{ }\left( {\mbox{n}_{\mbox{ob}} \mbox{/ n}_\mbox{o} } \right)^{\mbox{1/3}}\mbox{ / 2 }\gamma _\mbox{0}^{\mbox{5/6}} \mbox{ ,}$ where $\mbox{k }=\mbox{ }\omega \mbox{/ c, }\omega _\mbox{b} \mbox{ }=\mbox{ }\omega _\mbox{p} \mbox{ / }\left( {\mbox{2 }\gamma _\mbox{0} } \right)^{\mbox{1/2}}\mbox{ }$ is the betatron frequency, $\omega _{p}$ is the plasma frequency, $\gamma _{0}$ is the Lorentz factor of the beam, v$_{z}$ is its axial velocity, and n$_{0b}$/ n$_{0}$ is the ratio of beam density to plasma density. The fractional energy spread in excess of 2 $\Gamma _{m}$/ 3 $\omega _{b}$ reduces the growth rate.